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A uniform model for KR crystals III: Nonsymmetric Macdonald polynomials]
A uniform model for Kirillov–Reshetikhin crystals III:
Nonsymmetric Macdonald polynomials at $t=0$ and
Demazure characters
2010 Mathematics Subject Classification.
Primary: 17B37; Secondary: 17B67, 81R50, 81R10.
C. Lenart]Cristian Lenart
[Cristian Lenart]
Department of Mathematics and Statistics,
State University of New York at Albany,
Albany, NY 12222, U.S.A.
S. Naito]Satoshi Naito
[Satoshi Naito]
Department of Mathematics, Tokyo Institute of Technology,
2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan
D. Sagaki]Daisuke Sagaki
[Daisuke Sagaki]
Institute of Mathematics, University of Tsukuba,
Tsukuba, Ibaraki 305-8571, Japan
A. Schilling]Anne Schilling
[Anne Schilling]
Department of Mathematics, University of California, One Shields Avenue,
Davis, CA 95616-8633, U.S.A.
M. Shimozono]Mark Shimozono
[Mark Shimozono]
Department of Mathematics, MC 0151, 460 McBryde Hall, Virginia Tech,
225 Stanger St., Blacksburg, VA 24061 USA
We establish the equality of the specialization $\Mac{w\lambda}$ of
the nonsymmetric Macdonald polynomial $\Mact{w\lambda}$ at $t=0$ with the graded character
$\gch U_{w}^{+}(\lambda)$ of a certain Demazure-type submodule
$U_{w}^{+}(\lambda)$ of a tensor product of “single-column” Kirillov–Reshetikhin
modules for an untwisted affine Lie algebra, where $\lambda$ is a dominant integral weight and
$w$ is a (finite) Weyl group element; this generalizes our previous result,
that is, the equality between the specialization $P_{\lambda}(x\,;\,q,\,0)$ of the
symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$
at $t=0$ and the graded character of a tensor product
of single-column Kirillov–Reshetikhin modules. We also give two combinatorial formulas
for the mentioned specialization of nonsymmetric Macdonald polynomials:
one in terms of quantum Lakshmibai–Seshadri paths and the other in terms of the quantum alcove model.
§ INTRODUCTION.
In our previous paper <cit.>, we proved that
the specialization $P_{\lambda}(x\,;\,q,\,0)$ of the
symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$ at $t=0$
is identical to the graded character of a certain tensor product
of Kirillov–Reshetikhin (KR for short) modules of one-column type
for an untwisted affine Lie algebra $\Fg_{\af}$,
where $\lambda$ is a dominant integral weight
for the finite-dimensional simple Lie algebra $\Fg \subset \Fg_{\af}$.
The purpose of this paper is to generalize this result to
the specialization $\Mac{w\lambda}$ of
the nonsymmetric Macdonald polynomial
$\Mact{w\lambda}$ at $t=0$, where $w$ is an element
of the (finite) Weyl group $W$ of $\Fg$;
note that if $w$ is the longest element $\lng$ of $W$, then
Let us explain our result more precisely. Let $\Fg$ be
a finite-dimensional simple Lie algebra (over $\BC$),
with $X$ its integral weight lattice,
and $\Fg_{\af}$ the associated untwisted affine Lie algebra.
We denote by $\bigl\{\alpha_{i}\bigr\}_{i \in I}$ and
$\bigl\{\alpha_{i}^{\vee}\bigr\}_{i \in I}$
the simple roots and simple coroots of $\Fg$, respectively,
and by $\vpi_{i}$, $i \in I$, the fundamental weights for $\Fg$.
For a dominant integral weight $\lambda=\sum_{i \in I} m_{i} \vpi_{i} \in X$
with $m_{i} \in \BZ_{\ge 0}$, let $\QLS(\lambda)$ denote
the crystal of quantum Lakshmibai-Seshadri (QLS for short) paths
of shape $\lambda$; for details, see Definition <ref> below.
Then we know from <cit.> that
the crystal $\QLS(\lambda)$ provides a realization of the crystal basis of
the tensor product $\bigotimes_{i \in I} W(\vpi_{i})^{\otimes m_{i}}$
of the level-zero fundamental representations $W(\vpi_{i})$, $i \in I$,
of the quantum affine algebra $U_{q}'(\Fg_{\af})$ associated to $\Fg_{\af}$.
The main result of <cit.> states that the specialization
$P_{\lambda}(x\,;\,q,\,0)$ of the symmetric Macdonald polynomial
at $t=0$ is identical to the graded character of the crystal $\QLS(\lambda)$,
where the grading on $\QLS(\lambda)$ is given by the degree function,
or equivalently, by the (global) energy function.
Let $W=\langle r_{i} \mid i \in I \rangle$ denote the (finite) Weyl group of $\Fg$,
and set $W_{J}:=\langle r_{i} \mid i \in J \rangle \subset W$, where
$J:=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0 \bigr\}$.
Also, let $W^{J}$ denote the set of minimal(-length) coset representatives
for the cosets in $W/W_{J}$; for $w \in W$,
we denote by $\mcr{w}=\mcr{w}^{J} \in W^J$ the minimal coset representative
for the coset $w W_J$ in $W/W_J$. Now, for $w \in W^{J}$, we set
\begin{equation*}
\QLS_{w}(\lambda) :=
\bigl\{ \eta \in \QLS(\lambda) \mid
\iota(\eta) \le w
\bigr\},
\end{equation*}
where for a QLS path
$\eta = (x_{1},\,\dots,\,x_{s} \,;\, \sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in \QLS(\lambda)$, we define the initial direction $\iota(\eta)$ of $\eta$
to be $x_{1} \in W^{J}$; here the symbol $\le$ is used to
denote the Bruhat order on $W$. Furthermore, we define the graded character
$\gch \QLS_{w}(\lambda)$ of $\QLS_{w}(\lambda) \subset \QLS(\lambda)$ by
\begin{equation*}
\gch \QLS_{w}(\lambda) := \sum_{\eta \in \QLS_{w}(\lambda)}
q^{-\Deg(\eta)} e^{\wt (\eta)},
\end{equation*}
where $\wt:\QLS(\lambda) \rightarrow X$ and
$\Deg:\QLS(\lambda) \rightarrow \BZ_{\le 0}$ denote
the weight function and the degree function on $\QLS(\lambda)$, respectively;
for the definitions, see (<ref>) and (<ref>) below.
Now, the main result of this paper is as follows.
For each $w \in W^{J}$, the equality
\begin{equation*}
\gch \QLS_{w}(\lambda) = \Mac{w\lambda}
\end{equation*}
holds, where $\Mac{w\lambda}$ denotes the specialization of
the nonsymmetric Macdonald polynomial $\Mact{w\lambda}$ at $t=0$.
We should mention that this result generalizes
since it holds that $\QLS_{\mcr{\lng}}(\lambda) = \QLS(\lambda)$ and
where $\lng \in W$ denotes the longest element. On the other hand,
in Theorems <ref> and <ref>, we express
$\Mac{w\lambda}$ in terms of the so-called quantum alcove model <cit.>.
In the following, we explain the representation-theoretic meaning of
Theorem <ref>; see <ref> for details.
Let $V(\lambda)$ denote the extremal weight module of extremal weight $\lambda$
over the quantum affine algebra $U_{q}(\Fg_{\af})$ associated to $\Fg_{\af}$, and set
$V_{w}^{+}(\lambda):=U_{q}^{+}(\Fg_{\af})S_{w}^{\norm}v_{\lambda} \subset V(\lambda)$
for $w \in W$, which is the Demazure submodule generated by the extremal weight vector
$S_{w}^{\norm}v_{\lambda} \in V(\lambda)$ of weight $w\lambda$
over the positive part $U_{q}^{+}(\Fg_{\af})$ of $U_{q}(\Fg_{\af})$;
note that $V_{w}^{+}(\lambda) \subset V_{\lng}^{+}(\lambda)$ for all $w \in W$.
For $w \in W$, we define $U_{w}^{+}(\lambda)$
to be the image of $V_{w}^{+}(\lambda)$ under the canonical projection
$V_{\lng}^{+}(\lambda) \twoheadrightarrow
for the definition of $Z_{\lng}^{+}(\lambda)$, see <ref>.
Then, $U_{\lng}^{+}(\lambda)$ is isomorphic, as a $U_{q}(\Fg)$-module, to
the tensor product $\bigotimes_{i \in I} W(\vpi_{i})^{\otimes m_{i}}$
of level-zero fundamental representations $W(\vpi_{i})$, $i \in I$;
note that this is not an isomorphism of $U_{q}^{+}(\Fg_{\af})$-modules.
Because the module $V_{w}^{+}(\lambda)$ is generated
by the extremal weight vector $S_{w}^{\norm}v_{\lambda} \in V(\lambda)$
over $U_{q}^{+}(\Fg_{\af})$, it follows that the module
$U_{w}^{+}(\lambda) \subset U_{\lng}^{+}(\lambda)$ is also generated by
the image of $S_{w}^{\norm}v_{\lambda}$ over $U_{q}^{+}(\Fg_{\af})$.
Thus, in a sense,
we can think of $U_{w}^{+}(\lambda) \subset U_{\lng}^{+}(\lambda)$
as a Demazure-type submodule of $U_{\lng}^{+}(\lambda)$,
which is isomorphic as a $U_{q}(\Fg)$-module to
$\bigotimes_{i \in I} W(\vpi_{i})^{\otimes m_{i}}$.
Also, if we define the graded character
$\gch U_{w}^{+}(\lambda)$ of $U_{w}^{+}(\lambda)$ by
\begin{equation*}
\gch U_{w}^{+}(\lambda) :=
\sum_{\gamma \in Q,\,k \in \BZ}
\dim U_{w}^{+}(\lambda)_{\lambda-\gamma+k\delta}\,x^{\lambda-\gamma}q^{k},
\end{equation*}
where $Q:=\bigoplus_{i \in I}\BZ \alpha_{i}$ is the root lattice for $\Fg$,
$\delta$ denotes the null root of $\Fg_{\af}$, and $q:=x^{\delta}$, then we have
(see Theorem <ref>)
\begin{equation*}
\gch U_{w}^{+}(\lambda) =
\gch \QLS_{w}(\lambda) \stackrel{\text{Theorem~\ref{thm:Mac0}}}{=} \Mac{w\lambda}.
\end{equation*}
In <ref>, we give a bijective proof of Theorem <ref>
by making use of the Orr-Shimozono formula for the specialization at $t=0$ of
nonsymmetric Macdonald polynomials <cit.>.
The outline of our proof is as follows.
In <ref>, we briefly review
the Orr-Shimozono formula (see Theorem <ref>),
which expresses the specialization $\Mac{\mu}$ of
the nonsymmetric Macdonald polynomial $\Mact{\mu}$ at $t=0$
in terms of the set $\QBX$ of quantum alcove paths
from $e$ to $m_{\mu}$ for an integral weight $\mu$,
where $m_{\mu}$ denotes the element of the (extended) affine
Weyl group that is of minimal length in the coset $t_{\mu} W$,
with $t_{\mu}$ the translation by $\mu$.
Next, for a dominant integral weight $\lambda \in X$,
we show in Lemma <ref> that there exists a canonical bijection
between the particular set $\QBM_{\lex}$ and the set $\CA(-\lamm)$;
here, $\QBM_{\lex}$ is defined by using a specific reduced expression
for $m_{\lamm} = t_{\lamm}$ corresponding to
a lexicographic $(-\lamm)$-chain of roots.
Also, we give an explicit bijection
$\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$
in such a way that the diagram below
is commutative (see Proposition <ref>).
Furthermore, in Lemma <ref>
combined with Proposition <ref>,
we show that there exists a natural embedding
$\QBw \hookrightarrow \QBM_{\lex}$ for an arbitrary $w \in W^{J}$.
\begin{equation*}
\begin{diagram}
\node{\QBw \quad}
\arrow{e,b}{\begin{subarray}{c}
\text{Embedding} \\
\text{(Lemma~\ref{lem:embed}} \\
\text{and Proposition~\ref{prop:Theta})}
\end{subarray}
\node{\quad \QBM_{\lex} \quad}
\arrow{e,b}{\begin{subarray}{c}
\text{Bijection} \\
\text{(Lemma~\ref{lem:A-QB})}
\end{subarray}
\arrow{se,b}{\begin{subarray}{c}
\text{Bijection $\Xi$} \\
\text{(Proposition~\ref{prop:Xi})}
\end{subarray}
\node{\CA(-\lamm)}
\arrow{s,r}{\begin{subarray}{c}
\text{Bijection $\Pi$} \\
\text{(\cite[\S8.1]{LNSSS2})}
\end{subarray}
} \\
\node{} \node{}
\node{\QLS(\lambda)}
\end{diagram}
\end{equation*}
Finally, in Proposition <ref> and Lemma <ref>,
we show that the image of $\QBw$ under the composite of the maps
$\QBw \hookrightarrow \QBM_{\lex} \stackrel{\Xi}{\longrightarrow} \QLS(\lambda)$
is identical to $\QLS_{w}(\lambda)$; we also show in
Proposition <ref>, Lemma <ref>, and Proposition <ref> that
both of the embedding $\QBw \hookrightarrow \QBM_{\lex}$ and
the bijection $\Xi:\QBM \rightarrow \QLS(\lambda)$
preserve “weights” and “degrees”.
This implies that the graded character of $\QLS_{w}(\lambda)$ is
identical to that of $\QBw$. Because we know from the Orr-Shimozono formula
that the graded character of $\QBw$ is identical to $\Mac{w\lambda}$,
we conclude from the above that the graded character of
$\QLS_{w}(\lambda)$ is identical to $\Mac{w\lambda}$.
In Appendix <ref>, using the crystal structure on the set $\QLS(\lambda)$,
we obtain a recursive formula (see Proposition <ref>)
for the graded characters $\gch \QLS_{w}(\lambda)$, $w \in W^{J}$,
which is described in terms of Demazure operators.
Here we note that in view of Theorem <ref> above, this recursive formula
is equivalent to the one (see Proposition <ref>)
for nonsymmetric Macdonald polynomials $\Mac{w\lambda}$, $w \in W^{J}$,
specialized at $t=0$; in Appendix <ref>,
we provide a sketch of how to derive this recursive formula for
$\Mac{w\lambda}$ by using the polynomial representation of
the double affine Hecke algebra.
§.§ Acknowledgments
C.L. was partially supported by the NSF grant DMS–1362627.
S.N. was partially supported by Grant-in-Aid for Scientific Research (C), No. 24540010, Japan.
D.S. was partially supported by Grant-in-Aid for Young Scientists (B), No. 23740003, Japan.
A.S. was partially supported by NSF grants OCI–1147247 and DMS–1500050.
M.S. was partially supported by the NSF grant DMS–1200804.
§ PROOF OF THEOREM <REF>.
§.§ Setting.
Let $\Fg$ be a finite-dimensional simple Lie algebra (over $\BC$).
We denote by $\bigl\{\alpha_{i}\bigr\}_{i \in I}$ and
$\bigl\{\alpha_{i}^{\vee}\bigr\}_{i \in I}$
the simple roots and simple coroots of $\Fg$, respectively,
and by $\vpi_{i}$, $i \in I$, the fundamental weights for $\Fg$;
we set
\begin{equation*}
Q:=\bigoplus_{i \in I} \BZ\alpha_{i}, \qquad
Q^{\vee}:=\bigoplus_{i \in I} \BZ \alpha_{i}^{\vee},
\quad \text{and} \quad
X:=\bigoplus_{i \in I} \BZ \vpi_{i}.
\end{equation*}
Let $\Phi^{+}$ (resp., $\Phi^{\vee+}$) denote
the set of positive roots (resp., coroots), and
$\Phi^{-}$ (resp., $\Phi^{\vee-}$)
the set of negative roots (resp., coroots).
We set $\rho:=(1/2) \sum_{\alpha \in \Phi^{+}}\alpha$.
Let $W=\langle r_{i} \mid i \in I \rangle$ be
the (finite) Weyl group of $\Fg$,
with length function $\ell:W \rightarrow \BZ_{\ge 0}$;
we denote by $\lng \in W$ the longest element,
and by $e \in W$ the identity element. Also, let us
denote by $\omega:I \rightarrow I$ the Dynkin diagram automorphism
given by: $w_{\circ}\alpha_{i} = -\alpha_{\omega(i)}$ for $i \in I$.
For a subset $J \subset I$, we set
\begin{align*}
& \Phi_{J}^{+}:=\Phi^{+} \cap \biggl(\bigoplus_{i \in J} \BZ\alpha_{i}\biggr), & &
\rho_{J}:=\frac{1}{2}\sum_{\alpha \in \Phi_{J}^{+}} \alpha, \\[1mm]
& \Phi_{J}^{\vee+}:=\Phi^{\vee+} \cap \biggl(\bigoplus_{i \in J} \BZ\alpha_{i}^{\vee}\biggr), & &
W_{J}:=\langle r_{i} \mid i \in J \rangle \subset W;
\end{align*}
let $\lngJ$ denote the longest element of $W_{J}$.
Also, let $W^J$ denote the set of minimal(-length) coset
representatives for the cosets in $W / W_J$; recall that
\begin{equation} \label{eq:mcrs}
W^{J}=\bigl\{ w \in W \mid
\text{$w\alpha \in \Phi^{+}$ for all $\alpha \in \Phi_{J}^{+}$} \bigr\},
\end{equation}
\begin{equation}
\ell(wz) = \ell(w) + \ell(z) \qquad
\text{for all $w \in W^{J}$ and $z \in W_{J}$}.
\end{equation}
For $w \in W$, we denote by $\mcr{w}=\mcr{w}^{J} \in W^J$
the minimal coset representative for the coset $w W_J$ in $W/W_J$.
We use the symbol $\le$ for the Bruhat order on the Weyl group $W$.
§.§ Quantum Lakshmibai-Seshadri paths.
In this subsection, we recall the definition of quantum Lakshmibai-Seshadri paths
from <cit.>.
Let $J$ be a subset of $I$.
The (parabolic) quantum Bruhat graph $\QB(W^{J})$ is
the $(\Phi^{+} \setminus \Phi_{J}^{+})$-labeled,
directed graph with vertex set $W^J$ and
$(\Phi^{+} \setminus \Phi_{J}^{+})$-labeled, directed edges
of the following form:
$w \edge{\beta} \mcr{wr_{\beta}}$
for $w \in W^{J}$ and $\beta \in \Phi^{+} \setminus \Phi_{J}^{+}$,
where either
(i) $\ell(\mcr{wr_{\beta}})=\ell(w)+1$, or
(ii) $\ell(\mcr{wr_{\beta}})=\ell(w)-2\pair{\beta^{\vee}}{\rho-\rho_{J}}+1$;
if (i) holds (resp., (ii) holds), then the edge is called a Bruhat edge
(resp., a quantum edge). If $J$ is the empty set $\emptyset$, then
we simply write $\QB(W^{J}) = \QB(W^{\emptyset})$ as $\QB(W)$.
(1) We have $\pair{\beta^{\vee}}{\rho-\rho_{J}} > 0$
for all $\beta \in \Phi^{+} \setminus \Phi_{J}^{+}$.
Indeed, since $\pair{\alpha_{i}^{\vee}}{\alpha} \le 0$
for all $i \in I \setminus J$ and $\alpha \in \Phi_{J}^{+}$,
we see that $\pair{\alpha_{i}^{\vee}}{\rho_{J}} \le 0$ for all
$i \in I \setminus J$, and hence
$\pair{\alpha_{i}^{\vee}}{\rho-\rho_{J}} > 0$ for all $i \in I \setminus J$.
Also, we have $\pair{\alpha_{i}^{\vee}}{\rho-\rho_{J}} = 1 -1 = 0$ for all $i \in J$.
Therefore, $\pair{\beta^{\vee}}{\rho-\rho_{J}} > 0$
for all $\beta \in \Phi^{+} \setminus \Phi_{J}^{+}$.
As a consequence, if $w \edge{\beta} \mcr{wr_{\beta}}$ is a quantum edge,
then $\ell(\mcr{wr_{\beta}}) < \ell(w)$.
(2) If $w \edge{\beta} \mcr{wr_{\beta}}$ is a Bruhat edge,
then $wr_{\beta} \in W^{J}$, and hence $\mcr{wr_{\beta}} = wr_{\beta}$
(see <cit.>).
(3) Let $x,\,y \in W^{J}$ be such that $x \le y$ in the Bruhat order on $W$. If
\begin{equation} \label{eq:dpA}
x=x_{0} \edge{\beta_{1}} x_{1} \edge{\beta_{2}} \cdots
\edge{\beta_{k}} x_{k}=y
\end{equation}
is a shortest directed path from $x$ to $y$
in $\QB(W^{J})$, then all of its edges are Bruhat edges. Indeed,
by Definition <ref> (for Bruhat edges) and part (1) of this remark
(for quantum edges), we have
\begin{equation} \label{eq:length}
\ell(y) - \ell(x) = \sum_{q=1}^{k}
(\ub{\ell(x_{q})-\ell(x_{q-1})}{=1 \text{ or } < 0})
\le \sum_{q=1}^{k} 1 = k;
\end{equation}
note that the equality holds if and only if
$\ell(x_{q})-\ell(x_{q-1})=1$ for all $1 \le q \le k$,
or equivalently, all the edges are Bruhat edges.
Since $x \le y$ by the assumption,
we deduce from the chain property (see <cit.>)
that there exists a directed path from $x$ to $y$ in $\QB(W^{J})$
all of whose edges are Bruhat edges; the length of this directed path
is equal to $\ell(y)-\ell(x)$. Therefore, we obtain $k \le \ell(y)-\ell(x)$
since the directed path (<ref>) is a shortest one.
Combining this inequality and (<ref>), we obtain
$k=\ell(y)-\ell(x)$, and hence all the edges
in the shortest directed path (<ref>) are Bruhat edges.
Now, we fix a dominant integral weight $\lambda \in X$ for $\Fg$, and set
\begin{equation*}
J=J_{\lambda}:=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0\bigr\} \subset I.
\end{equation*}
As above, we simply write $\mcr{w}^{J}=\mcr{w}^{J_{\lambda}} \in W^{J}$
for $w \in W$ as $\mcr{w}$, unless stated otherwise explicitly.
For a given rational number $\sigma$, we define $\QB_{\sigma\lambda}(W^{J})$ to be the subgraph
of the parabolic quantum Bruhat graph $\QB(W^{J})$ with the same vertex set but
having only the edges:
\begin{equation*}
w \edge{\beta} \mcr{wr_{\beta}} \quad \text{with} \quad
\pair{\beta^{\vee}}{\sigma\lambda}=\sigma \pair{\beta^{\vee}}{\lambda} \in \BZ.
\end{equation*}
A quantum Lakshmibai-Seshadri (QLS for short) path of shape $\lambda$ is a pair
\begin{equation} \label{eq:QLS}
\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\end{equation}
of a sequence $x_{1},\,x_{2},\,\dots,\,x_{s}$ of elements
in $W^{J}$ with $x_{u} \ne x_{u+1}$ for $1 \le u \le s-1$ and
a sequence $0 = \sigma_{0} < \sigma_{1} < \cdots < \sigma_{s} = 1$
of rational numbers satisfying the condition that there exists
a directed path from $x_{u+1}$ to $x_{u}$ in $\QB_{\sigma_{u}\lambda}(W^{J})$
for each $1 \le u \le s-1$; we denote this $x_u \blarrl{\sigma_u\lambda} x_{u+1}$.
Let $\QLS(\lambda)$ denote the set of all QLS paths of shape $\lambda$.
We identify $\eta \in \QLS(\lambda)$ of the form (<ref>)
with the following piecewise-linear, continuous map
$\eta:[0,1] \rightarrow \BR \otimes_{\BZ} X$:
\begin{equation} \label{eq:path}
\eta(t)=\sum_{p=1}^{u-1}
(t-\sigma_{u-1})x_{u}\lambda \quad
\text{for $\sigma_{u-1} \le t \le \sigma_{u}$, $1 \le u \le s$}.
\end{equation}
In <cit.>, we proved that
$\QLS(\lambda)$ is identical (as a set of piecewise-linear,
continuous maps from $[0,1]$ to $\BR \otimes_{\BZ} X$)
to the set $\BB(\lambda)_{\cl}$ of “projected” Lakshmibai-Seshadri
paths of shape $\lambda$; for the definition
of $\BB(\lambda)_{\cl}$, see <cit.>.
Let $\eta = (x_{1},\,\dots,\,x_{s}\,;\,\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in \QLS(\lambda)$. We define the weight $\wt(\eta)$ of $\eta \in \QLS(\lambda)$ by
\begin{equation} \label{eq:wt}
\wt (\eta) := \eta(1) =\sum_{u=1}^{s}(\sigma_{u}-\sigma_{u-1})x_{u}\lambda;
\end{equation}
we can show in exactly the same way as <cit.> that
$\wt(\eta) \in X$.
Also, we define the degree $\Deg(\eta)$ as follows
(see <cit.>). First, let $x,\,y \in W^{J}$, and let
\begin{equation*}
x=y_{0} \edge{\beta_{1}} y_{1} \edge{\beta_{2}} \cdots
\edge{\beta_{k}} y_{k}=y
\end{equation*}
be a shortest directed path from $x$ to $y$ in $\QB(W^{J})$.
Then we set
\begin{equation}\label{defwtlam}
\wt_{\lambda}(x \Rightarrow y):=
\sum_{
\begin{subarray}{c}
1 \le p \le k \\[1.5mm]
\text{$y_{p-1} \stackrel{\beta_{p}}{\rightarrow} y_{p}$ is a quantum edge}
\end{subarray}
} \pair{\beta_{p}^{\vee}}{\lambda} \in \BZ_{\ge 0};
\end{equation}
we see from <cit.> that
this value does not depend on the choice of a shortest directed path
from $x$ to $y$ in $\QB(W^{J})$. For $\eta =
\in \QLS(\lambda)$, we define
\begin{equation} \label{eq:Deg}
\Deg(\eta):=-\sum_{u=1}^{s-1} (1-\sigma_{u})
\wt_{\lambda}(x_{u+1} \Rightarrow x_{u}) \in \BZ_{\le 0}.
\end{equation}
For $\eta = (x_{1},\,\dots,\,x_{s}\,;\,\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in \QLS(\lambda)$, we set $\iota(\eta):=x_{1} \in W^{J}$, and
call it the initial direction of $\eta$.
Now, for each $w \in W^{J}$, we set
\begin{equation}
\QLS_{w}(\lambda):=\bigl\{ \eta \in \QLS(\lambda) \mid
\iota(\eta) \le w \bigr\},
\end{equation}
and define the graded character
$\gch \QLS_{w}(\lambda)$ of $\QLS_{w}(\lambda) \subset \QLS(\lambda)$ by
\begin{equation*}
\gch \QLS_{w}(\lambda) := \sum_{\eta \in \QLS_{w}(\lambda)}
q^{-\Deg(\eta)} e^{\wt (\eta)}.
\end{equation*}
We will prove that for each $w \in W^{J}$, the equality
\begin{equation}
\gch \QLS_{w}(\lambda) = \Mac{w\lambda}
\end{equation}
holds, where $\Mac{w\lambda}$ denotes the specialization of
the nonsymmetric Macdonald polynomial $\Mact{w\lambda}$ at $t=0$.
§.§ Orr-Shimozono formula.
In this subsection, we review a formula (<cit.>) for
the specialization at $t=0$ of nonsymmetric Macdonald polynomials.
Let $\ti{\Fg}$ denote the dual Lie algebra of $\Fg$, and
let $\bigl\{\ti{\alpha}_{i}\bigr\}_{i \in I}$ and
$\bigl\{\ti{\alpha}_{i}^{\vee}\bigr\}_{i \in I}$ be
the simple roots and the simple coroots of $\ti{\Fg}$, respectively.
We denote by $\ti{W}$ the Weyl group of $\ti{\Fg}$;
note that $W \cong \ti{W}$. As is well-known,
for $w \in W \cong \ti{W}$ and $i \in I$,
\begin{equation} \label{eq:iden2}
w \ti{\alpha}_{i} = \sum_{j \in I} c_{j}\ti{\alpha}_{j}
\quad \text{if and only if} \quad
w \alpha_{i}^{\vee} =
\sum_{j \in I} c_{j}\alpha_{j}^{\vee}.
\end{equation}
Hence we identify $w\ti{\alpha}_{i}$ with $w\alpha_{i}^{\vee}$
for $w \in W \cong \ti{W}$ and $i \in I$:
\begin{equation} \label{eq:iden}
w \ti{\alpha}_{i}
\quad \stackrel{\text{identify}}{\longleftrightarrow} \quad
w \alpha_{i}^{\vee}.
\end{equation}
Let $\ti{\Phi}^{+}$ denote the set of positive roots of $\ti{\Fg}$,
which we identify with the set $\Phi^{\vee+}$ of
positive coroots of $\Fg$ by (<ref>).
Now, let $\ti{\Fg}_{\af}$ denote
the untwisted affine Lie algebra associated to $\ti{\Fg}$.
Let $\bigl\{\ti{\alpha}_{i}\bigr\}_{i \in I_{\af}}$
be the simple roots of $\ti{\Fg}_{\af}$,
where $I_{\af}=I \sqcup \{0\}$,
and $\ti{\delta}$ the null root of $\ti{\Fg}_{\af}$.
We denote by $\ti{\Phi}_{\af}^{+}$ (resp., $\ti{\Phi}_{\af}^{-}$)
the set of positive (resp., negative) real roots of $\ti{\Fg}_{\af}$;
note that
\begin{equation*}
\ti{\Phi}_{\af}^{+}=
\bigl(\underbrace{\BZ_{\ge 0}\ti{\delta}+\ti{\Phi}^{+}}_{%
\begin{subarray}{c}
\text{identified with} \\[1mm]
\BZ_{\ge 0}\ti{\delta}+\Phi^{\vee+}
\end{subarray}
} \bigr) \sqcup
\bigl(\underbrace{\BZ_{> 0}\ti{\delta}-\ti{\Phi}^{+}}_{%
\begin{subarray}{c}
\text{identified with} \\[1mm]
\BZ_{> 0}\ti{\delta}-\Phi^{\vee+}
\end{subarray}
} \bigr).
\end{equation*}
Denote by $\ti{W}_{\af}$ the Weyl group of $\ti{\Fg}_{\af}$;
note that $\ti{W}_{\af} \cong Q \rtimes \ti{W} \cong Q \rtimes W$.
Also, we denote by $\ti{W}_{\ext}:=X \rtimes \ti{W} \cong
X \rtimes W$ the extended affine Weyl group of $\ti{\Fg}_{\af}$,
and by $t_{\mu} \in \ti{W}_{\ext}$ the translation by $\mu \in X$.
For $x \in \ti{W}_{\ext}$, define $\wt(x) \in X$ and $\dir(x) \in W$ by:
\begin{equation*}
x = t_{\wt(x)}\dir(x).
\end{equation*}
For an integral weight $\mu \in X$ for $\Fg$, we set
\begin{equation*}
m_{\mu}:=t_{\mu}v(\mu)^{-1} \in X \rtimes W \cong \ti{W}_{\ext},
\end{equation*}
where $v(\mu)$ denotes the shortest element in $W$ such that $v(\mu)\mu$ is
an antidominant integral weight (see <cit.>).
The following lemma will be used later.
Let $\lambda \in X$ be a dominant integral weight, and let $w \in W^{J}$,
where $J=J_{\lambda}=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0\bigr\}$.
Then, $v(w\lambda)= \mcr{\lng}w^{-1}$, and hence
\begin{equation*}
m_{w\lambda}=t_{w\lambda} (\mcr{\lng}w^{-1})^{-1} =
w (\mcr{\lng})^{-1} t_{\lng\lambda}.
\end{equation*}
In particular,
\begin{equation*}
\begin{cases}
v(\lamm)=v(\mcr{\lng}\lambda)=e, \qquad m_{\lamm}=t_{\lamm}, \\[1.5mm]
v(\lambda) = \mcr{\lng}, \qquad m_{\lambda} =
(\mcr{\lng})^{-1} t_{\lng\lambda},
\end{cases}
\end{equation*}
and $m_{w\lambda}=wm_{\lambda}$.
It is obvious that $(\mcr{\lng}w^{-1})w\lambda
= \lng\lambda$ is antidominant. Hence it suffices to show that
$\ell(x) \ge \ell( \mcr{\lng}w^{-1} )$
for all $x \in W$ such that $xw\lambda=\lng\lambda$.
If $xw\lambda=\lng\lambda$, then $\lng xw \in W_{J}$, and hence
$x=\lng zw^{-1}$ for some $z \in W_{J}$; note that
$\ell( zw^{-1} ) = \ell(wz^{-1}) = \ell(w) + \ell(z^{-1})$ since
$w \in W^{J}$ and $z \in W_{J}$. Therefore,
\begin{equation*}
\ell(x) = \ell(\lng) - \ell(zw^{-1}) =
\ell(\lng) - \ell(w) - \ell(z^{-1}).
\end{equation*}
Here we remark that $\mcr{\lng} = \lng \lngJ$,
where $\lngJ \in W_{J}$ is the longest element.
Hence it follows from the computation above
(with $z$ replaced by $\lngJ$) that
\begin{equation*}
\ell(\mcr{\lng}w^{-1}) =
\ell(\lng \lngJ w^{-1}) =
\ell(\lng) - \ell(w) - \ell(\lngJ^{-1}).
\end{equation*}
Since $\ell(z^{-1}) \le \ell(\lngJ^{-1})$, we obtain
$\ell(x) \ge \ell(\mcr{\lng}w^{-1})$, as desired.
We fix an arbitrary $\mu \in X$, and apply the argument in <cit.>
to the case that $u=e$ (the identity element) and $w=m_{\mu}$;
we generally follow the notation thereof. Let
\begin{equation} \label{eq:redw}
m_{\mu} = \pi
\underbrace{ r_{i_1} r_{i_2} \cdots r_{i_{\ell}} }_{\in \ti{W}_{\af}}
\end{equation}
be a reduced expression for $m_{\mu}$,
where $\pi$ is an (affine) Dynkin diagram
automorphism of $\ti{\Fg}_{\af}$, and set
\begin{equation} \label{eq:betak}
\beta_{k}^{\OS}
:= r_{i_{\ell}} \cdots r_{i_{k+1}}\ti{\alpha}_{i_{k}}
\quad \text{for $1 \le k \le \ell$},
\end{equation}
which is a positive real root of $\ti{\Fg}_{\af}$
contained in $\BZ_{> 0} \ti{\delta} - \ti{\Phi}^{+}$
(see <cit.>). Then we can write $\beta_{k}^{\OS}$ as:
\begin{equation} \label{eq:ak}
\beta_{k}^{\OS}=a_{k}\ti{\delta}+\ol{\beta_{k}^{\OS}}
\quad \text{for $a_{k} \in \BZ_{> 0}$ and
$\ol{\beta_{k}^{\OS}} \in \ti{\Phi}^{-}$, \quad $1 \le k \le \ell$};
\end{equation}
we think of $\ol{\beta_{k}^{\OS}}$ as an element of $\Phi^{\vee-}$
under the identification (<ref>)
of $\ti{\Phi}^{+}$ and $\Phi^{\vee+}$, and set
$\gamma_{k}^{\OS}:=-(\ol{\beta_{k}^{\OS}})^{\vee} \in \Phi^{+}$.
Let $A=\bigl\{j_{1} < j_{2} < \cdots < j_{r}\bigr\}$
be a subset of $\bigl\{1,\,2,\,\dots,\,\ell\bigr\}$.
Following <cit.>
(recall that $u=e$ and $w = m_{\mu}$),
we set
\begin{equation*}
z_{0}:=m_{\mu}, \qquad
z_{k}:=z_{k-1}r_{\beta_{j_{k}}^{\OS}} \quad
\text{for $1 \le k \le r$};
\end{equation*}
or equivalently, $z_{0} = m_{\mu}$, and
$z_{k}$ is obtained from the reduced expression (<ref>)
by removing the $j_{1}$-th reflection,
the $j_{2}$-th reflection, $\dots$,
and the $j_{k}$-th reflection.
We express these data as:
\begin{equation} \label{eq:pJ}
z_{0} \edge{\beta_{j_1}^{\OS}}
z_{1} \edge{\beta_{j_2}^{\OS}} \cdots
\edge{\beta_{j_r}^{\OS}} z_{r}
\Bigr).
\end{equation}
Keep the notation and setting above.
We say that $p_{A}$ is an element of $\QBX$ if
\begin{equation*}
\dir(z_{0}) \edge{ \gamma_{j_{1}}^{\OS} }
\dir(z_{1}) \edge{ \gamma_{j_{2}}^{\OS} } \cdots
\edge{ \gamma_{j_{r}}^{\OS} } \dir(z_{r})
\end{equation*}
is a directed path in
the quantum Bruhat graph $\QB(W)=\QB(W^{\emptyset})$ for $W$.
For an element $p_{A} \in \QBX$,
we set (see <cit.>)
\begin{equation} \label{eq:A-}
\bigl\{j_{k} \in A \mid
\text{$\dir(z_{k-1}) \edge{\gamma_{j_k}^{\OS}} \dir(z_{k})$
is a quantum edge}
\bigr\} \subset A,
\end{equation}
and then set (see <cit.>)
\begin{equation} \label{eq:qwt}
\qwt (p_{A}) : = \sum_{j \in A^{-}} \beta_{j}^{\OS},
\end{equation}
which is contained in
$\BZ_{> 0}\ti{\delta} - \ti{Q}^{+}$ if $A^{-} \ne \emptyset$,
where $\ti{Q}^{+}:=\sum_{i \in I} \BZ_{\ge 0} \ti{\alpha}_{i}$.
Furthermore, in view of equation (<ref>), we set
(in the notation of <cit.>)
\begin{equation} \label{eq:degr}
\degr(\qwt (p_{A})):=\sum_{j \in A^{-}} a_{j} \in \BZ_{\ge 0}.
\end{equation}
Also, if $p_{A} \in \QBX$ is of the form (<ref>), then
we set
\begin{equation} \label{eq:wtp}
\enp{A}:=z_{r} \in \ti{W}_{\ext} = X \rtimes W \quad \text{and} \quad
\wt(p_{A}): = \wt(\enp{A}).
\end{equation}
Keep the notation and setting above. We have
\begin{equation*}
\Mac{\mu} = \sum_{p \in \QBX} e^{\wt(p)}q^{\degr(\qwt(p))}.
\end{equation*}
§.§ Bijective correspondence between $\QBM$ and $\CA(-\lamm)$.
First, we recall the quantum alcove model from <cit.>
(see also <cit.>).
We set
$H_{\alpha,\,n}:=\bigl\{\zeta \in \Fh^{\ast}_{\BR} \mid
\pair{\alpha^{\vee}}{\zeta}=n \bigr\}$ for $\alpha \in \Phi$ and $n \in \BZ$,
where $\Fh^{\ast}_{\BR}:=\BR \otimes_{\BZ} X=
\bigoplus_{i \in I} \BR\alpha_{i}$.
An alcove is, by definition, a connected component
(with respect to the usual topology on $\Fh^{\ast}_{\BR}$) of
\begin{equation*}
\Fh^{\ast}_{\BR} \setminus
\bigcup_{\alpha \in \Phi^{+},\,n \in \BZ} H_{\alpha,\,n}.
\end{equation*}
We say that two alcoves are adjacent if they are distinct and
have a common wall. For adjacent alcoves $A$ and $B$,
we write $A \edge{\alpha} B$, with $\alpha \in \Phi$,
if their common wall is contained in the hyperplane
$H_{\alpha,\,n}$ for some $n \in \BZ$, and if $\alpha$ points
in the direction from $A$ to $B$. An alcove path is a sequence of
alcoves $(A_{0},\,A_{1},\,\dots,\,A_{s})$ such that $A_{u-1}$ and $A_{u}$
are adjacent for each $u=1,\,2,\,\dots,\,s$. We say that
$(A_{0},\,A_{1},\,\dots,\,A_{s})$ is reduced if it has minimal length
among all alcove paths from $A_{0}$ to $A_{s}$.
Recall that $\ti{W}_{\ext} \cong X \rtimes W$
acts (as affine transformations) on $\Fh^{\ast}_{\BR}$ by
\begin{equation*}
(t_{\xi}w) \cdot \zeta = w\zeta + \xi \qquad
\text{for $\xi \in X$, $w \in W$, and $\zeta \in \Fh^{\ast}_{\BR}$}.
\end{equation*}
For $\beta = \alpha^{\vee} + n\ti{\delta} \in \ti{\Phi}_{\af}^{+}$ with
$\alpha \in \Phi^{+}$ and $n \in \BZ_{\ge 0}$
(here we identify $\ti{\Phi}^{+}$ with $\Phi^{\vee+}$ under (<ref>)),
we have
$r_{\alpha^{\vee}+n\ti{\delta}} \cdot \zeta =
(t_{-n\alpha}r_{\alpha^{\vee}}) \cdot \zeta =
r_{\alpha^{\vee}}\zeta-n\alpha = r_{\alpha}\zeta-n\alpha$
for $\zeta \in \Fh^{\ast}_{\BR}$.
Hence $r_{\alpha^{\vee}+n\ti{\delta}} \in \ti{W}_{\ext}$
acts on $\Fh^{\ast}_{\BR}$ as
the affine reflection with respect to the hyperplane
$H_{\alpha,\,-n} = H_{-\alpha,\,n}$.
Now, let $\lambda \in X$ be a dominant integral weight;
note that $\lng\lambda \in X$ is antidominant,
where $\lng \in W$ denotes the longest element. We set
\begin{equation*}
\zeta \in \Fh^{\ast}_{\BR} \mid
\text{$0 < \pair{\alpha^{\vee}}{\zeta} < 1$ for all $\alpha \in \Phi^{+}$}
\bigr\},
\end{equation*}
and $A_{\lamm}:=A_{\circ}+\lamm$.
The sequence of roots $(\gamma_{1},\,\gamma_{2},\,\dots,\,\gamma_{\ell})$ is
called a $(-\lng\lambda)$-chain of roots if
\begin{equation*}
A_{\circ}=A_{0} \edge{-\gamma_{1}} A_{1}
\edge{-\gamma_{2}} \cdots \edge{-\gamma_{\ell}} A_{\ell}=A_{\lamm}
\end{equation*}
is a reduced alcove path.
Here we note that $m_{\lamm}=t_{\lamm}$ by Lemma <ref>.
It follows from <cit.> that
there exists a bijection:
\begin{equation} \label{eq:1to1}
\bigl\{ \text{reduced expressions for $m_{\lamm}=t_{\lamm}$} \bigr\}
\quad \stackrel{\text{1:1}}{\longleftrightarrow} \quad
\bigl\{ \text{$(-\lamm)$-chains of roots} \bigr\}.
\end{equation}
More precisely, let $m_{\lamm}=t_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$
be a reduced expression for $m_{\lamm}=t_{\lamm} \in \ti{W}_{\ext}$. We set
$A_{k}:=(\pi r_{i_1}r_{i_2} \cdots r_{i_k}) \cdot A_{\circ}$
for $0 \le k \le \ell$, and
\begin{equation} \label{eq:betaL0}
\beta_{k}^{\Le}:=\pi r_{i_1} \cdots r_{i_{k-1}}(\ti{\alpha}_{i_k}) =
r_{\p{1}} \cdots r_{\p{k-1}}(\ti{\alpha}_{\p{k}})
\quad
\text{for $1 \le k \le \ell$};
\end{equation}
note that $\beta_{k}^{\Le}$ is a positive real root of
$\ti{\Fg}_{\af}$ contained in $\BZ_{\ge 0}\ti{\delta}+\ti{\Phi}^{+}$.
In fact, by <cit.>, we have
\begin{align}
\bigl\{ \beta_{k}^{\Le} \mid 1 \le k \le \ell \bigr\}
& = \ti{\Phi}_{\af}^{+} \cap m_{\lamm}\ti{\Phi}_{\af}^{-}
= \ti{\Phi}_{\af}^{+} \cap t_{\lamm}\ti{\Phi}_{\af}^{-} \nonumber \\
& = \bigl\{ b \ti{\delta} + \beta^{\vee} \mid
\beta \in \Phi^{+},\,0 \le b < -\pair{\beta^{\vee}}{\lamm} \bigr\} \label{eq:inv}
\end{align}
under the identification (<ref>)
of $\ti{\Phi}^{+}$ and $\Phi^{\vee+}$.
Therefore, we can write $\beta_{k}^{\Le}$ in the form
\begin{equation} \label{eq:bk}
\beta_{k}^{\Le} = b_{k}\ti{\delta}+\ol{\beta_{k}^{\Le}}, \
\text{
with $b_{k} \in \BZ_{\ge 0}$ and
$\ol{\beta_{k}^{\Le}} \in -\lng\bigl(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}\bigr)$},
\end{equation}
for each $1 \le k \le \ell$. If we set $\gamma_{k}^{\Le}:=
(\ol{\beta_{k}^{\Le}})^{\vee} \in -\lng\bigl(\Phi^{+} \setminus \Phi^{+}_{J}\bigr)$, then
\begin{equation} \label{eq:chain1}
A_{\circ}=A_{0} \edge{-\gamma_{1}^{\Le}} A_{1}
\edge{-\gamma_{2}^{\Le}} \cdots \edge{-\gamma_{\ell}^{\Le}} A_{\ell}=A_{\lamm}
\end{equation}
is a $(-\lamm)$-chain of roots.
Let $1 \le k \le \ell$. We see from Remark <ref> that
the action of $r_{\beta_{k}^{\Le}} \in \ti{W}_{\af}$ on $\Fh^{\ast}_{\BR}$ is
the affine reflection with respect to
the hyperplane $H_{\gamma_{k}^{\Le},\,-b_{k}}$.
Also, we know that
\begin{equation} \label{eq:bk2}
0 \le b_{k}=\# \bigl\{ 1 \le p < k \mid \gamma_{p}^{\Le} = \gamma_{k}^{\Le} \bigr\} <
\pair{\ol{\beta_{k}^{\Le}}}{-\lamm};
\end{equation}
the sequence $(b_{1},\,\dots,\,b_{\ell})$ is
called the height sequence
for the $(-\lamm)$-chain (<ref>).
Keep the notation and setting above.
If we define $\beta_{k}^{\OS}$, $1 \le k \le \ell$,
by (<ref>) for the reduced expression
$m_{\lamm}=t_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$, then we have
$\beta_{k}^{\Le}=-t_{\lamm}(\beta_{k}^{\OS})$ for all $1 \le k \le \ell$.
In particular, $\ol{\beta_{k}^{\Le}} = - \ol{\beta_{k}^{\OS}}$
(see (<ref>) and (<ref>)),
and hence $\gamma_{k}^{\Le}=\gamma_{k}^{\OS}=:\gamma_{k}$.
Also, we have $b_{k}=\pair{\gamma_{k}^{\vee}}{-\lamm}-a_{k}$.
Now, let
\begin{equation} \label{eq:chain}
A_{\circ}=A_{0} \edge{-\gamma_{1}} A_{1}
\edge{-\gamma_{2}} \cdots \edge{-\gamma_{\ell}} A_{\ell}=A_{\lamm}
\end{equation}
be a $(-\lamm)$-chain of roots.
Let $\CA(-\lamm)$ denote the set of all subsets
$A=\bigl\{j_{1} < \cdots < j_{r}\bigr\}$ of
$\bigl\{1,\,2,\,\dots,\,\ell\bigr\}$ such that
\begin{equation}\label{deffinal}
e \edge{\gamma_{j_1}} r_{\gamma_{j_1}}
\edge{\gamma_{j_2}} r_{\gamma_{j_1}}r_{\gamma_{j_2}}
\edge{\gamma_{j_3}} \cdots \edge{\gamma_{j_r}}
r_{\gamma_{j_1}}r_{\gamma_{j_2}} \cdots r_{\gamma_{j_r}}=:\phi(A)
\end{equation}
is a directed path in the quantum Bruhat graph $\QB(W)$ for $W$.
The subsets $A$ are called admissible subsets, and
$\phi(A)$ is called the final direction of $A$.
For $A=\bigl\{j_{1} < \cdots < j_{r}\bigr\} \in \CA(-\lamm)$,
we define $\wt(A) \in X$, $\Ht(A) \in \BZ_{\ge 0}$
(see <cit.>),
and $\cHt(A) \in \BZ_{\ge 0}$ as follows:
\begin{equation} \label{eq:wtA}
\begin{split}
\wt(A)
& :=-r_{\beta_{j_1}^{\Le}}r_{\beta_{j_2}^{\Le}} \cdots r_{\beta_{j_r}^{\Le}} \cdot (\lamm) \\
& =-r_{\gamma_{j_1}^{\Le},\,-b_{j_1}}r_{\gamma_{j_2}^{\Le},\,-b_{j_2}} \cdots
r_{\gamma_{j_r}^{\Le},\,-b_{j_r}} \cdot (\lamm),
\end{split}
\end{equation}
\begin{equation} \label{eq:height}
\Ht(A):= \sum_{j \in A_{-}} \Bigl(
\pair{(\gamma_{j}^{\Le})^{\vee} }{ -\lamm } - b_{j} \Bigr),
\end{equation}
\begin{equation}\label{defcoheight}
\cHt(A):= \sum_{j \in A_{-}} b_{j},
\end{equation}
\begin{equation} \label{eq:a-}
\bigl\{ j_{k} \in A \mid
\text{
$r_{\gamma_{j_1}} \cdots r_{\gamma_{j_{k-1}}} \edge{ \gamma_{j_k}^{\Le} }
r_{\gamma_{j_1}} \cdots r_{\gamma_{j_{k-1}}}r_{\gamma_{j_{k}}}$
is a quantum edge} \bigr\}.
\end{equation}
Let $m_{\lamm}=t_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$ be
the reduced expression for $m_{\lamm}=t_{\lamm}$ corresponding
to the $(-\lamm)$-chain of roots (<ref>) under
the correspondence (<ref>).
We define $\QBM$ by using this reduced expression
for $m_{\lamm}=t_{\lamm}$. Note that
\begin{equation} \label{eq:gamma}
\gamma_{k} = \gamma_{k}^{\Le} = \gamma_{k}^{\OS} \qquad \text{for $1 \le k \le \ell$}.
\end{equation}
Keep the notation and setting above.
\begin{equation*}
A \in \CA(-\lamm) \quad \text{\rm if and only if} \quad p_{A} \in \QBM.
\end{equation*}
Hence we have a bijection from $\CA(-\lamm)$ onto
$\QBM$ that maps $A \in \CA(-\lamm)$ to $p_{A} \in \QBM$.
Moreover, we have
\begin{equation} \label{eq:HtWt}
\wt(A) = - \wt(p_{A}) \quad \text{\rm and} \quad
\Ht(A)=\deg(\qwt(p_{A})) \quad \text{\rm for all $A \in \CA(-\lamm)$}.
\end{equation}
Let $A=\bigl\{j_{1} < \cdots < j_{r}\bigr\}$. Then, we have
\begin{align*}
& p_{A} \in \QBM \iff
\underbrace{\dir(z_0)}_{=e}
\edge{\gamma_{j_1}^{\OS}}
\dir(z_1) \edge{\gamma_{j_2}^{\OS}}
\cdots \edge{\gamma_{j_r}^{\OS}}
\dir(z_r) \quad \text{in $\QB(W)$} \\[3mm]
& \quad \iff
e \edge{\gamma_{j_1}} r_{\gamma_{j_1}}
\edge{\gamma_{j_2}} \cdots \edge{\gamma_{j_r}}
r_{\gamma_{j_1}}r_{\gamma_{j_2}} \cdots r_{\gamma_{j_r}}
\quad \text{in $\QB(W)$ by \eqref{eq:gamma}} \\
& \quad \iff A \in \CA(-\lamm).
\end{align*}
Next, we prove that $\Ht(A)=\deg(\qwt(p_{A}))$ for all $A \in \CA(-\lamm)$.
Let $A=\bigl\{j_{1} < \cdots < j_{r}\bigr\} \in \CA(-\lamm)$;
we see from the argument above that the set $A^{-}$ in (<ref>) is
identical to the set $A_{-}$ in (<ref>). Then, we see that
\begin{align*}
\Ht(A) & = \sum_{j \in A_{-}} \Bigl(
\pair{(\gamma_{j}^{\Le})^{\vee} }{ -\lamm } - b_{j} \Bigr)
\quad \text{by definition \eqref{eq:height}} \\[3mm]
& = \sum_{j \in A^{-}} \Bigl(
\underbrace{\pair{\gamma_{j}^{\vee} }{ -\lamm } - b_{j}}_{=a_{j}} \Bigr)
\quad \text{by Remark~\ref{rem:betaLPOS}} \\[3mm]
& = \sum_{j \in A^{-}} a_{j} = \deg(\qwt(p_{A}))
\quad \text{by \eqref{eq:degr}}.
\end{align*}
Finally, we show that $\wt(A) = - \wt(p_{A})$
for all $A \in \CA(-\lamm)$;
we proceed by induction on
the cardinality of $A \in \CA(-\lamm)$.
First, observe that this equality is obvious if $A=\emptyset$.
Now, let us take $A=\bigl\{j_1 < \cdots < j_{r-1} < j_{r} \bigr\} \in \CA(-\lamm)$,
and set $A':=\bigl\{j_1 < \cdots < j_{r-1}\bigr\}$,
which is also an element of $\CA(-\lamm)$.
By direct computation, together with definition (<ref>),
we can show that
\begin{equation} \label{eq:wt1}
\wt (A) = \wt (A')-
\bigl( \pair{ \gamma_{j_{r}}^{\vee} }{-\lamm}-b_{j_{r}} \bigr)
r_{\gamma_{j_1}} \cdots r_{\gamma_{j_{r-1}}} (\gamma_{j_{r}});
\end{equation}
or, we may refer the reader to the proof of <cit.>.
Also, we have
\begin{equation*}
z_{r} = z_{r-1} r_{\beta_{j_{r}}^{\OS}} =
z_{r-1}r_{ a_{j_r}\ti{\delta}+\ol{\beta_{j_r}^{\OS}} } =
z_{r-1} \bigl( t_{ -a_{j_r}(\ol{\beta_{j_r}^{\OS}})^{\vee} }
r_{ \ol{\beta_{j_r}^{\OS}} } \bigr) =
z_{r-1} t_{ a_{j_r}\gamma_{j_{r}} }r_{ \gamma_{j_{r}} }.
\end{equation*}
Therefore, if we write $z_{r}=t_{\wt (z_{r})}\dir(z_{r})$ and
$z_{r-1}=t_{\wt (z_{r-1})}\dir(z_{r-1})$, then we deduce that
\begin{align*}
& = t_{\wt(z_{r-1})}\dir(z_{r-1}) t_{ a_{j_r}\gamma_{j_{r}} }r_{ \gamma_{j_{r}} }
= t_{\wt(z_{r-1})}t_{ a_{j_r}\dir(z_{r-1})\gamma_{j_{r}} }
\dir(z_{r-1}) r_{ \gamma_{j_{r}} } \\
& = t_{\wt(z_{r-1}) + a_{j_r}\dir(z_{r-1})\gamma_{j_{r}} }
\bigl(\dir(z_{r-1}) r_{ \gamma_{j_{r}} }\bigr),
\end{align*}
and hence
\begin{equation*}
\wt(p_{A})=\wt(z_{r})=\wt(z_{r-1}) + a_{j_r}\dir(z_{r-1})\gamma_{j_{r}}.
\end{equation*}
Here, since $a_{j_{r}}=\pair{\gamma_{j_{r}}^{\vee}}{-\lamm}-b_{j_{r}}$ by
Remark <ref>, we obtain
\begin{align*}
\wt(p_{A}) & = \wt(z_{r-1}) +
\bigl(\pair{\gamma_{j_{r}}^{\vee}}{-\lamm}-b_{j_{r}}\bigr)
\dir(z_{r-1})\gamma_{j_{r}} \\
& = \wt(p_{A'})+ \bigl(\pair{\gamma_{j_{r}}^{\vee}}{-\lamm}-b_{j_{r}}\bigr)
\dir(z_{r-1})\gamma_{j_{r}};
\end{align*}
note that $\dir(z_{r-1})=r_{\gamma_{j_{1}}} \cdots r_{\gamma_{j_{r-1}}}$ since
$\dir(z_{0})=\dir(m_{\lamm})=e$. Hence it follows that
\begin{align*}
\wt (p_{A}) & =
\wt(p_{A'})+ \bigl(\pair{\gamma_{j_{r}}^{\vee}}{-\lamm}-b_{j_{r}}\bigr)
r_{\gamma_{j_{1}}} \cdots r_{\gamma_{j_{r-1}}}(\gamma_{j_{r}}) \\
& = -\wt (A') + \bigl(\pair{\gamma_{j_{r}}^{\vee}}{-\lamm}-b_{j_{r}}\bigr)
r_{\gamma_{j_{1}}} \cdots r_{\gamma_{j_{r-1}}}(\gamma_{j_{r}}) \\
& \hspace*{50mm} \text{by our induction hypothesis} \\
& = -\wt (A) \quad \text{by \eqref{eq:wt1}},
\end{align*}
as desired. This completes the proof of the lemma.
§.§ Lexicographic (lex) $(-\lamm)$-chains of roots.
We keep the notation and setting of the previous subsection;
we fix a dominant integral weight $\lambda \in X$, and set
$J=J_{\lambda}=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0\bigr\}$.
For $w \in W$, we simply write $\mcr{w}^{J}=\mcr{w}^{J_{\lambda}} \in W^{J}$ as
$\mcr{w}$, unless stated otherwise explicitly.
In <cit.> (see also <cit.>),
we introduced a specific $(-\lamm)$-chain of roots, called
a lexicographic (lex for short) $(-\lamm)$-chain of roots.
We will frequently make use of the following property
of a lex $(-\lamm)$-chain of roots: If $m_{\lamm}=\pi
r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$ is the reduced expression for
$m_{\lamm}$ corresponding to a lex $(-\lamm)$-chain of roots
(recall from (<ref>) the one-to-one correspondence between
the reduced expressions for $m_{\lamm}=t_{\lamm}$ and
the $(-\lamm)$-chains of roots), then we have
\begin{equation} \label{eq:bkle}
0 \le \frac{b_{1}}{ \bpair{ \ol{\beta_{1}^{\Le}} }{-\lamm} } \le
\frac{b_{2}}{ \bpair{ \ol{\beta_{2}^{\Le}} }{-\lamm} } \le \cdots
\le \frac{b_{\ell}}{ \bpair{ \ol{\beta_{\ell}^{\Le}} }{-\lamm}} < 1,
\end{equation}
where $\beta_{k}^{\Le} = b_{k}\ti{\delta} + \ol{\beta_{k}^{\Le}}$
for $1 \le k \le \ell$ is given as in (<ref>) and (<ref>)
(see also Remark <ref>).
We know from Lemma <ref> that
$m_{\lamm} = t_{\lamm} = \mcr{\lng} (t_{\lambda} \mcr{\lng}^{-1})$ and
$m_{\lambda} = t_{\lambda} \mcr{\lng}^{-1}$. It follows that
$m_{\lamm} = \mcr{\lng}m_{\lambda}$.
Also, since $\ell(t_{\lambda})= \ell(m_{\lambda} \mcr{\lng} ) =
\ell(m_{\lambda}) + \ell(\mcr{\lng})$ by <cit.>,
we have
\begin{equation} \label{eq:elltmu}
\ell(m_{\lamm}) = \ell(t_{\lamm})= \ell(t_{\lambda}) = \ell(m_{\lambda})+\ell(\mcr{\lng});
\end{equation}
note that $\ell(t_{\lamm})=\ell(t_{\lambda})$ by <cit.>.
This implies that the concatenation of a reduced expression for $\mcr{\lng}$ with
a reduced expression for $m_{\lambda}$ is
a reduced expression for $m_{\lamm}=t_{\lamm}$.
We set $M:=\ell(\mcr{\lng})$.
Let $m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$ be
the reduced expression for $m_{\lamm}$ corresponding to a lex $(-\lamm)$-chain of roots under
the correspondence (<ref>). Then,
\begin{equation*}
\mcr{\lng} = r_{\p{1}} \cdots r_{\p{M}} \quad \text{\rm and} \quad
m_{\lambda} = \pi r_{i_{M+1}} \cdots r_{i_{\ell}}.
\end{equation*}
\begin{equation*}
m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}=
( \underbrace{r_{\p{1}} \cdots r_{\p{M}}}_{=\mcr{\lng} \ \text{\rm (reduced)}} )
( \underbrace{\pi r_{i_{M+1}} \cdots r_{i_{\ell}}}_{=m_{\lambda} \ \text{\rm (reduced)}}).
\end{equation*}
We make use of (<ref>).
Let $K$ be the maximal index such that
$b_{K} / \pair{ \ol{\beta_{K}^{\Le}} }{-\lamm} = 0$.
Then we see that
\begin{equation} \label{eq:lex1a}
\bigl\{ \beta_{k}^{\Le} \mid 1 \le k \le \ell \bigr\} \cap
\Bigl(-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})\Bigr) =
\bigl\{ \beta_{k}^{\Le} \mid 1 \le k \le K \bigr\}.
\end{equation}
Also, we see from (<ref>) that
$-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}) \subset
\bigl\{\beta_{k}^{\Le} \mid 1 \le k \le \ell \bigr\}$. Hence
the left-hand side of (<ref>) is identical to
$-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})$.
From these, by noting that
$\#(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}) =
\ell(\lng)-\ell(\lngJ)=\ell(\mcr{\lng}) = M$
(recall that $\lngJ$ is the longest element of $W_{J}$),
we conclude that $K=M$, and hence that
$\bigl\{ \beta_{k}^{\Le} \mid 1 \le k \le M \bigr\} =
-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})$.
In addition, since
\begin{equation} \label{eq:btLe1}
\beta_{k}^{\Le}
= \pi r_{i_{1}} \cdots r_{i_{k-1}}(\ti{\alpha}_{i_{k}})
= r_{\p{1}} \cdots r_{\p{k-1}}(\ti{\alpha}_{\p{k}}) \in
-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})
\end{equation}
for all $1 \le k \le M$, we see easily that $\p{1},\,\dots,\,\p{M} \in I$.
We will show that $v:=r_{\p{1}} \cdots r_{\p{M}} \in W$
is identical to $\mcr{\lng}$. By the argument above, we have
\begin{equation*}
\bigl\{\alpha \in \Phi^{\vee +} \mid
v^{-1}\alpha \in \Phi^{\vee-}\bigr\} =
\bigl\{ \beta_{k}^{\Le} \mid 1 \le k \le M \bigr\} =
-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}).
\end{equation*}
From this, we see that
\begin{equation} \label{eq:lexA}
-v^{-1}\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}) \subset \Phi^{\vee-},
\quad \text{so that} \quad
v^{-1}\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}) \subset \Phi^{\vee+}.
\end{equation}
Hence it follows that $\bigl\{\alpha \in \Phi^{\vee +} \mid
v^{-1}\lng\alpha \in \Phi^{\vee-}\bigr\} \subset \Phi^{\vee+}_{J}$.
Since $v=r_{\p{1}} \cdots r_{\p{M}}$ is a reduced expression,
we have $\ell(v)=M$, and hence
\begin{equation*}
\# \bigl\{\alpha \in \Phi^{\vee +} \mid
v^{-1}\lng\alpha \in \Phi^{\vee-}\bigr\} = \ell(v^{-1}\lng)=N-M.
\end{equation*}
Also, we have $\# \Phi^{\vee+}_{J} = \ell(\lngJ) = \ell(\lng)-\ell(\mcr{\lng})=N-M$.
Therefore, we deduce that
\begin{equation} \label{eq:lexB}
\bigl\{\alpha \in \Phi^{\vee +} \mid
v^{-1}\lng\alpha \in \Phi^{\vee-}\bigr\} = \Phi^{\vee+}_{J}.
\end{equation}
Since $\lngJ(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}) \subset
\Phi^{\vee+} \setminus \Phi^{\vee+}_{J}$ and
$\lngJ(\Phi^{\vee+}_{J}) \subset \Phi^{\vee-}_{J}$, we have
\begin{align*}
& v^{-1}\lng \lngJ (\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})
\subset v^{-1}\lng (\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})
\subset \Phi^{\vee+} \qquad \text{by \eqref{eq:lexA}}, \\
& v^{-1}\lng \lngJ (\Phi^{\vee+}_{J})
\subset v^{-1}\lng (\Phi^{\vee-}_{J})
\subset \Phi^{\vee+}_{J} \qquad \text{by \eqref{eq:lexB}}.
\end{align*}
From these, we obtain $v^{-1}\lng \lngJ (\Phi^{\vee+}) \subset \Phi^{\vee+}$,
which implies that $v^{-1}\lng \lngJ = e$, and hence that
$v=\lng\lngJ = \mcr{\lng}$, as desired.
Finally, because $\ell(m_{\lambda})=\ell(m_{\lamm})-\ell(\mcr{\lng})=\ell-M$
and $m_{\lamm}=\mcr{\lng}m_{\lambda}$, it follows that
$m_{\lambda} = \pi r_{i_{M+1}} \cdots r_{i_{\ell}}$ is
a reduced expression for $m_{\lambda}$.
This proves the lemma.
Fix a lex $(-\lamm)$-chain of roots.
We construct $\QBM$ from the reduced expression
$m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$
corresponding to the lex $(-\lamm)$-chain of roots under (<ref>),
which we denote by $\QBM_{\lex}$; recall from
(<ref>), (<ref>), and Remark <ref>
that for $1 \le k \le \ell$,
\begin{equation*}
\begin{cases}
\beta_{k}^{\OS}
= r_{i_{\ell}} \cdots r_{i_{k+1}}\ti{\alpha}_{i_{k}}
= a_{k}\ti{\delta} + \ol{\beta_{k}^{\OS}}, &
\text{with $a_{k} \in \BZ_{> 0}$ and $\ol{\beta_{k}^{\OS}} \in \ti{\Phi}^{-}$}, \\[1.5mm]
\gamma_{k}=\gamma_{k}^{\OS}=-(\ol{\beta_{k}^{\OS}})^{\vee} \in \Phi^{+}, \\[1.5mm]
\end{cases}
\end{equation*}
We see from (<ref>) that
\begin{equation} \label{eq:gam}
\gamma_{k}=\gamma_{k}^{\Le} = (\beta_{k}^{\Le})^{\vee} =
r_{\p{1}} \cdots r_{\p{k-1}}(\alpha_{\p{k}}) \qquad
\text{for $1 \le k \le M=\ell(\mcr{\lng})$},
\end{equation}
and hence
\begin{equation} \label{eq:ga2}
\bigl\{\gamma_{1},\,\dots,\,\gamma_{M}\bigr\} =
\bigl\{(\beta_{1}^{\Le})^{\vee},\,\dots,\,(\beta_{M}^{\Le})^{\vee}\bigr\} =
-\lng(\Phi^{+} \setminus \Phi^{+}_{J}) = \Phi^{+} \setminus \Phi^{+}_{\omega(J)},
\end{equation}
where $\omega:I \rightarrow I$ is the Dynkin diagram automorphism given by:
$\lng \alpha_{i} = - \alpha_{\omega(i)}$ for $i \in I$. Also, it follows from
the equality “$K=M$” (shown in the proof of Lemma <ref>),
together with (<ref>), that
\begin{equation} \label{eq:bkle2}
0 = \frac{b_{1}}{ \pair{ \gamma_{1}^{\vee} }{-\lamm} } = \cdots =
\frac{b_{M}}{ \pair{ \gamma_{M}^{\vee} }{-\lamm} } <
\frac{b_{M+1}}{ \pair{ \gamma_{M+1}^{\vee} }{-\lamm} } \le \cdots \le
\frac{b_{\ell}}{ \pair{ \gamma_{\ell}^{\vee} }{-\lamm}} < 1.
\end{equation}
Now, let $\mcr{\lng}=r_{p_1}r_{p_2} \cdots r_{p_M}$ be
an (arbitrary) reduced expression for $\mcr{\lng}$,
and set $i_{k}':=\pi^{-1}(p_{k})$ for $1 \le k \le M$.
We see from Lemma <ref> that
\begin{equation} \label{eq:red-mlamm}
( \underbrace{r_{p_{1}} \cdots r_{p_{M}}}_{=\mcr{\lng}} )
( \underbrace{\pi r_{i_{M+1}} \cdots r_{i_{\ell}}}_{=m_{\lambda}})=
\pi r_{i_1'} \cdots r_{i_M'}r_{i_{M+1}} \cdots r_{i_{\ell}}
\end{equation}
is a reduced expression for $m_{\lamm}$, which we denote by $R$.
We construct $\QBM$ from this reduced expression $R$ of $m_{\lamm}$,
and denote it by $\QBM_{R}$. Then,
\begin{align*}
\beta_{k}^{\OS, R} & :=
\begin{cases}
\underbrace{r_{i_{\ell}} \cdots r_{i_{M+1}}}_{=m_{\lambda}^{-1}\pi}
r_{i_{M}'} \cdots r_{i_{k+1}'}\ti{\alpha}_{i_{k}'} =
m_{\lambda}^{-1}r_{p_{M}} \cdots r_{p_{k+1}}\ti{\alpha}_{p_{k}} &
\text{\rm for $1 \le k \le M$}, \\[5mm]
r_{i_{\ell}} \cdots r_{i_{k+1}}\ti{\alpha}_{i_{k}} = \beta_{k}^{\OS} &
\text{\rm for $M+1 \le k \le \ell$},
\end{cases} \\[5mm]
& = a_{k}^{R}\ti{\delta} + \ol{\beta_{k}^{\OS, R}} \quad
\text{for some $a_{k}^{R} \in \BZ_{> 0}$ and $\ol{\beta_{k}^{\OS, R}} \in \ti{\Phi}^{-}$},
\end{align*}
\begin{equation*}
\gamma_{k}^{\OS, R} : =
- (\ol{\beta_{k}^{\OS, R}})^{\vee} \in \Phi^{+}.
\end{equation*}
Also, for the reduced expression $R$ of $m_{\lamm}$ in (<ref>),
we define $\beta_{k}^{\Le, R}$, $1 \le k \le \ell$, as in (<ref>),
and write it as:
$\beta_{k}^{\Le, R} = b_{k}^{R}\ti{\delta} + \ol{\beta_{k}^{\Le, R}}$,
with $b_{k}^{R} \in \BZ_{\ge 0}$ and $\ol{\beta_{k}^{\Le, R}} \in
-\lng(\Phi^{\vee+} \setminus \Phi^{\vee+}_{J})$ (see (<ref>)).
Then we set $\gamma_{k}^{\Le, R} : = (\ol{\beta_{k}^{\Le, R}})^{\vee}$
for $1 \le k \le \ell$. By Remark <ref>, we have
\begin{equation*}
\gamma_{k}^{\Le, R} = \gamma_{k}^{\OS, R}=:\gamma_{k}^{R}
\quad \text{and} \quad
b_{k}^{R}=\pair{(\gamma_{k}^{R})^{\vee}}{-\lng \lambda} - a_{k}^{R}
\quad \text{for $1 \le k \le \ell$}.
\end{equation*}
Notice that $\beta_{k}^{\Le, R} = r_{p_{1}} \cdots r_{p_{k-1}}(\ti{\alpha}_{p_{k}})$
for $1 \le k \le M$. Since $p_{1},\,\dots,\,p_{M} \in I$, we see that
$\beta_{k}^{\Le, R} \in \Phi^{\vee+}$ for all $1 \le k \le M$,
which implies that $b_{k}^{R} = 0$ and
\begin{equation} \label{eq:gamR}
\gamma_{k}^{R} =
\gamma_{k}^{\Le, R} =
(\ol{\beta_{k}^{\Le, R}})^{\vee} =
(\beta_{k}^{\Le, R})^{\vee}=
r_{p_1} \cdots r_{p_{k-1}}(\alpha_{p_{k}})
\end{equation}
for all $1 \le k \le M$.
Keep the notation and setting above. We have
\begin{align}
& \bigl\{\beta^{\OS,R}_{k} \mid 1 \le k \le M\bigr\} = \bigl\{\beta^{\OS}_{k} \mid 1 \le k \le M\bigr\},
\label{eq:R1} \\
& \beta^{\OS,R}_{k} = \beta^{\OS}_{k} \qquad \text{\rm for all $M+1 \le k \le \ell$}.
\label{eq:R2}
\end{align}
\begin{align}
& \bigl\{\gamma^{R}_{k} \mid 1 \le k \le M\bigr\}
= \bigl\{\gamma_{k} \mid 1 \le k \le M\bigr\}
= \Phi^{+} \setminus \Phi^{+}_{\omega(J)}, \label{eq:R3} \\
& \gamma^{R}_{k} = \gamma_{k} \quad \text{\rm for all $M+1 \le k \le \ell$}, \label{eq:R4} \\[1.5mm]
& b_{k}^{R}=
\begin{cases}
0 & \text{\rm for $1 \le k \le M$}, \\[1.5mm]
b_{k} > 0 & \text{\rm for $M+1 \le k \le \ell$}.
\end{cases} \label{eq:R5}
\end{align}
It is obvious from the definitions that
$\beta^{\OS,R}_{k} = \beta^{\OS}_{k}$ for all $M+1 \le k \le \ell$.
We see from these equalities and (<ref>) that
\begin{equation*}
\gamma_{k}^{R} = \gamma_{k} \quad \text{and} \quad
b_{k}^{R}=b_{k} > 0 \quad \text{for all $M+1 \le k \le \ell$}.
\end{equation*}
Also, we have shown that $b_{k}^{R} = 0$ for all $1 \le k \le M$
(see the comment preceding this lemma).
It remains to show (<ref>) and (<ref>).
Since $\mcr{\lng}=r_{p_1} \cdots r_{p_M} = r_{\p{1}} \cdots r_{\p{M}}$
are reduced expressions, it follows that
\begin{align*}
\bigl\{r_{p_{M}} \cdots r_{p_{k+1}}(\ti{\alpha}_{p_{k}}) \mid 1 \le k \le M\bigr\}
& = \bigl\{ \ti{\alpha} \in \Phi^{\vee+} \mid \mcr{\lng}\ti{\alpha} \in -\Phi^{\vee+} \bigr\} \\
& = \bigl\{r_{\p{M}} \cdots r_{\p{k+1}}(\ti{\alpha}_{\p{k}}) \mid 1 \le k \le M\bigr\};
\end{align*}
notice that
\begin{equation} \label{eq:invw0}
\bigl\{ \ti{\alpha} \in \Phi^{\vee+} \mid \mcr{\lng}\ti{\alpha} \in -\Phi^{\vee+} \bigr\} =
\Phi^{\vee+} \setminus \Phi_{J}^{\vee+}.
\end{equation}
Indeed, we see from (<ref>) that
$\bigl\{ \ti{\alpha} \in \Phi^{\vee+} \mid \mcr{\lng}\ti{\alpha} \in -\Phi^{\vee+} \bigr\}
\subset \Phi^{\vee+} \setminus \Phi^{\vee+}_{J}$. Conversely,
if $\ti{\alpha} \in \Phi^{\vee+} \setminus \Phi^{\vee+}_{J}$, then
$\lngJ \ti{\alpha} \in \Phi^{\vee+}$, and hence
$\mcr{\lng} \ti{\alpha} = \lng \lngJ \ti{\alpha} \in -\Phi^{\vee+}$, as desired.
Therefore, we deduce from the definitions that
\begin{align*}
\bigl\{\beta^{\OS,R}_{k} \mid 1 \le k \le M\bigr\}
& = m_{\lambda}^{-1}(\Phi^{\vee +} \setminus \Phi^{\vee+}_{J})
= \bigl\{\beta^{\OS}_{k} \mid 1 \le k \le M\bigr\},
\end{align*}
and hence that
\begin{equation*}
\bigl\{\gamma^{R}_{k} \mid 1 \le k \le M\bigr\}
= \bigl\{\gamma_{k} \mid 1 \le k \le M\bigr\}
\stackrel{\eqref{eq:ga2}}{=} \Phi^{+} \setminus \Phi_{\omega(J)}^{+}.
\end{equation*}
This proves the lemma.
We set $\ell(\lng):=N$; since $\lng = \mcr{\lng} \lngJ$, it follows that
$\ell(\lngJ) = N-M$. Fix a reduced expression
$\lngJ = r_{t_{M+1}} r_{t_{M+2}} \cdots r_{t_{N}}$ for $\lngJ$. Then,
\begin{equation*}
\lng =
\ub{r_{\p{1}} \cdots r_{\p{M}}}{=\mcr{\lng}}
\ub{r_{t_{M+1}} r_{t_{M+2}} \cdots r_{t_{N}}}{=\lngJ} =
\ub{r_{p_1} \cdots r_{p_M}}{=\mcr{\lng}}
\ub{r_{t_{M+1}} r_{t_{M+2}} \cdots r_{t_{N}}}{=\lngJ}
\end{equation*}
are reduced expressions for $\lng$. Now we set
\begin{equation} \label{eq:gamM}
\xi_{k}=\xi_{k}^{R}:=\mcr{\lng}r_{t_{M+1}} \cdots r_{t_{k-1}}\alpha_{t_{k}} \in \Phi^{+}
\qquad \text{for $M+1 \le k \le N$}.
\end{equation}
Then, by (<ref>) and (<ref>),
both of the sets
$\bigl\{\gamma_{k} \mid 1 \le k \le M \bigr\} \cup
\bigl\{\xi_{k} \mid M+1 \le k \le N \bigr\}$ and
$\bigl\{\gamma_{k}^{R} \mid 1 \le k \le M \bigr\} \cup
\bigl\{\xi_{k}^{R} \mid M+1 \le k \le N \bigr\}$
are identical to $\Phi^{+}$.
Hence it follows from (<ref>) that
\begin{equation} \label{eq:gamM2}
\bigl\{\xi^{R}_{k} \mid M+1 \le k \le N\bigr\}
= \bigl\{\xi_{k} \mid M+1 \le k \le N\bigr\}
= \Phi^{+}_{\omega(J)}.
\end{equation}
If we define total orders $\prec$ and $\prec_{R}$ on $\Phi^{+}$ by:
\begin{align}
& \ub{\gamma_{1} \prec \cdots \prec \gamma_{M}}
{\in \Phi^{+} \setminus \Phi_{\omega(J)}^{+}} \prec
\ub{\xi_{M+1} \prec \cdots \prec \xi_{N}}{\in \Phi_{\omega(J)}^{+}},
\label{eq:prec} \\[3mm]
& \ub{\gamma_{1}^{R} \prec_{R} \cdots \prec_{R} \gamma_{M}^{R}}
{\in \Phi^{+} \setminus \Phi_{\omega(J)}^{+}} \prec
\ub{\xi_{M+1}^{R} \prec_{R} \cdots \prec_{R} \xi_{N}^{R}}{\in \Phi_{\omega(J)}^{+}},
\label{eq:precR}
\end{align}
respectively, then these total orders are reflection orders
(see, for example, <cit.>).
Let $A=\bigl\{j_{1},\,j_{2},\,\dots,\,j_{r}\bigr\} \subset \bigl\{1,\,2,\,\dots,\,\ell\bigr\}$
be such that
\begin{equation*}
\begin{split}
& p_{A}=\Bigl(
m_{\lamm} = t_{\lamm} = z_{0} \edge{\beta_{j_1}^{\OS}} \cdots
\edge{\beta_{j_r}^{\OS}} z_{r}
\Bigr) \in \QBM_{\lex}, \\[3mm]
& \text{(resp.,\ }
m_{\lamm} = t_{\lamm} = z_{0}^{R} \edge{\beta_{j_1}^{\OS,R}} \cdots
\edge{\beta_{j_r}^{\OS,R}} z_{r}^{R}
\Bigr) \in \QBM_{R} \text{ )};
\end{split}
\end{equation*}
we set $j_{0}:=0$ by convention.
By the definition (see Definition <ref>),
we have a directed path
\begin{equation*}
\begin{split}
e=\dir(z_{0}) \edge{ \gamma_{j_{1}} } \cdots
\edge{ \gamma_{j_{r}} } \dir(z_{r}) \\[3mm]
\text{(resp.,\ }
e=\dir(z_{0}^{R}) \edge{ \gamma_{j_{1}}^{R} } \cdots
\edge{ \gamma_{j_{r}}^{R} } \dir(z_{r}^{R})
\text{)}
\end{split}
\end{equation*}
in the quantum Bruhat graph $\QB(W)$. Let us take
$0 \le s \le r$ such that $j_{s} \le M$ and $j_{s+1} \ge M+1$,
and set
\begin{equation} \label{eq:tii}
\begin{split}
& \ti{\iota}(p_{A}):=\dir(z_{s}) =r _{\gamma_{j_1}} \cdots r _{\gamma_{j_s}} \in W \\
& \text{(resp., \ }
\ti{\iota}(p_{A}):=\dir(z_{s}^{R}) =r _{\gamma_{j_1}^{R}} \cdots r _{\gamma_{j_s}^{R}} \in W\text{)}.
\end{split}
\end{equation}
Because $\gamma_{j_1} \prec \gamma_{j_2} \prec \cdots \prec \gamma_{j_s}$
with respect to the reflection order $\prec$ on $\Phi^{+}$ (see (<ref>)),
we deduce from <cit.> that
$e= \dir(z_{0}) \edge{ \gamma_{j_{1}} } \cdots \edge{ \gamma_{j_{s}} } \dir(z_{s})=\ti{\iota}(p_{A})$
is a shortest directed path from $e$ to $\ti{\iota}(p_{A})$
in the quantum Bruhat graph $\QB(W)$. Therefore, all the edges in this directed path
are Bruhat edges by Remark <ref> (3).
We show by induction on $u$ that
$\dir(z_{u}) \in W^{\omega(J)}$ for all $0 \le u \le s$.
If $u=0$, then it is obvious that $\dir(z_{0}) = e \in W^{\omega(J)}$.
Assume that $0 < u \le s$. Since $\dir(z_{u-1}) \in W^{\omega(J)}$
by our induction hypothesis, and
since $\dir(z_{u-1}) \edge{ \gamma_{j_{u}} }
\dir(z_{u}) = \dir(z_{u-1})r_{\gamma_{j_u}}$ is a Bruhat edge in $\QB(W)$,
we see by <cit.> that
$\dir(z_{u}) \in W^{\omega(J)}$ or
$\dir(z_{u}) = \dir(z_{u-1})r_{i}$ for some $i \in \omega(J)$.
Suppose that $\dir(z_{u}) = \dir(z_{u-1})r_{i}$ for some $i \in \omega(J)$.
Since $\dir(z_{u-1})r_{\gamma_{j_u}} = \dir(z_{u}) = \dir(z_{u-1})r_{i}$,
we have $r_{\gamma_{j_u}} = r_{i}$, and hence $\gamma_{j_u} =
\alpha_{i} \in \Phi_{\omega(J)}^{+}$, which contradicts the fact that
$\gamma_{j_u} \in \Phi^{+} \setminus \Phi_{\omega(J)}^{+}$.
Thus we obtain $\dir(z_{u}) \in W^{\omega(J)}$, as desired.
In particular, $\ti{\iota}(p_{A}) = \dir (z_{s}) \in W^{\omega(J)}$.
The same argument works also for the reduced expression $R$.
Here we define a map $\Theta_{R}^{\lex} : \QBM_{R} \rightarrow \QBM_{\lex}$.
Let $A=\bigl\{j_{1},\,j_{2},\,\dots,\,j_{r}\bigr\} \subset \bigl\{1,\,2,\,\dots,\,\ell\bigr\}$
be such that
\begin{equation} \label{eq:pA}
m_{\lamm} = t_{\lamm} =
\edge{\beta_{j_1}^{\OS,R}} \cdots
\edge{\beta_{j_r}^{\OS,R}}
\Bigr) \in \QBM_{R},
\end{equation}
that is,
\begin{equation} \label{eq:Theta0}
e=\dir(z_{0}^{R}) \edge{ \gamma_{j_{1}}^{R} }
\dir(z_{1}^{R}) \edge{ \gamma_{j_{2}}^{R} } \cdots
\edge{ \gamma_{j_{r}}^{R} } \dir(z_{r}^{R})
\end{equation}
is a directed path in the quantum Bruhat graph $\QB(W)$.
If we take $0 \le s \le r$ such that
$j_{s} \le M$ and $j_{s+1} \ge M+1$, then we have a shortest directed path
\begin{equation*}
e=\dir(z_{0}^{R}) \edge{ \gamma_{j_{1}}^{R} }
\dir(z_{1}^{R}) \edge{ \gamma_{j_{2}}^{R} } \cdots
\edge{ \gamma_{j_{s}}^{R} } \dir(z_{s}^{R}) = \ti{\iota}(p_{A}) \in W^{\omega(J)}
\end{equation*}
in the quantum Bruhat graph $\QB(W)$;
note that $\gamma_{j_{1}}^{R} \prec_{R} \cdots \prec_{R} \gamma_{j_{s}}^{R}$
with respect to the reflection order $\prec_{R}$ on $\Phi^{+}$ (see (<ref>)).
We know from <cit.> that
there exists a unique shortest directed path
\begin{equation*}
e = x_{0} \edge{ \gamma_{q_1} } \cdots \edge{ \gamma_{q_u} } x_{u}
\edge{ \xi_{q_{u+1}} } \cdots \edge{ \xi_{q_{s}} }x_{s} = \ti{\iota}(p_{A})
\end{equation*}
from $e$ to $\ti{\iota}(p_{A})$ in $\QB(W)$ such that
$1 \le q_1 < \cdots < q_u \le M < q_{u+1} \le \cdots \le q_{s} \le N = \ell(\lng)$
(see (<ref>) and (<ref>)) for some $0 \le u \le s$; note that
all the edges in this directed path are Bruhat edges
by Remark <ref> (3).
We claim that $u=s$. Indeed, suppose for a contradiction that $u < s$;
in this case, $\xi_{q_{s}} \in \Phi_{\omega(J)}^{+}$ by (<ref>),
and hence $r_{\xi_{q_s}} \in W_{\omega(J)}$.
We write $x_{s-1} = \mcr{x_{s-1}}^{\omega(J)} z$ for some $z \in W_{\omega(J)}$;
note that $\ell(x_{s-1}) = \ell(\mcr{x_{s-1}}^{\omega(J)}) + \ell(z)$.
We see that
\begin{equation*}
\ti{\iota}(p_{A}) = x_{s} = x_{s-1} r_{\xi_{q_s}} =
\ub{\mcr{x_{s-1}}^{\omega(J)}}{\in W^{\omega(J)}}
\ub{zr_{\xi_{q_s}}}{\in W_{\omega(J)}},
\end{equation*}
and hence $\ell(x_{s}) = \ell(\mcr{x_{s-1}}^{\omega(J)}) + \ell(zr_{\xi_{q_s}})$.
Because $x_{s-1} \edge{ \xi_{q_s} } x_{s}$ is a Bruhat edge in $\QB(W)$ as seen above, we have
$\ell(x_{s})=\ell(x_{s-1})+1$. Combining these equalities, we obtain
\begin{equation*}
\ell(\mcr{x_{s-1}}^{\omega(J)}) + \ell(zr_{\xi_{q_s}}) =
\ell(x_{s}) = \ell(x_{s-1})+1 =
\ell(\mcr{x_{s-1}}^{\omega(J)}) + \ell(z) + 1,
\end{equation*}
and hence $\ell(zr_{\xi_{q_s}}) = \ell(z) + 1 \ge 1$.
Hence it follows that $zr_{\xi_{q_s}} \ne e$,
which implies that $\ti{\iota}(p_{A}) = x_{s} =
\mcr{x_{s-1}}^{\omega(J)} zr_{\xi_{q_s}} \notin W^{\omega(J)}$.
However, this contradicts the fact that $\ti{\iota}(p_{A}) \in W^{\omega(J)}$
(see Remark <ref>). Thus, we obtain $u=s$, and hence a directed path
\begin{equation} \label{eq:Theta1}
e = x_{0} \edge{ \gamma_{q_1} } \cdots \edge{ \gamma_{q_s} } x_{s}
= \ti{\iota}(p_{A})
\end{equation}
such that $1 \le q_1 < \cdots < q_s \le M$.
Now, we set $B:=\bigl\{q_{1},\,\dots,\,q_{s},\,j_{s+1},\,\dots,\,j_{r}\bigr\}$,
and consider
\begin{equation} \label{eq:pB}
m_{\lamm} = t_{\lamm} = z_{0}
\edge{\beta_{q_1}^{\OS}} \cdots \edge{\beta_{q_s}^{\OS}}
z_{s} \edge{\beta_{j_{s+1}}^{\OS}} \cdots
\edge{\beta_{j_r}^{\OS}} z_{r}
\Bigr).
\end{equation}
Since $M+1 \le j_{s+1} < \cdots < j_{r} \le \ell$,
we see from (<ref>) that $\gamma_{j_{u}} = \gamma_{j_{u}}^{R}$
for all $s+1 \le u \le r$. Therefore, by replacing the first $s$ edges in
(<ref>) with (<ref>), we obtain a directed path
\begin{equation*}
e=\dir(z_{0}) \edge{ \gamma_{q_{1}} } \cdots
\edge{ \gamma_{q_{s}} } \dir(z_{s}) = \ti{\iota}(p_{A})
\edge{ \gamma_{j_{s+1}} } \cdots \edge{ \gamma_{j_{r}} } \dir(z_{r})
\end{equation*}
in the quantum Bruhat graph $\QB(W)$.
Hence we conclude that $p_{B} \in \QBM_{\lex}$;
we set $\Theta_{R}^{\lex}(p_{A}):=p_{B}$.
The map $\Theta_{R}^{\lex} : \QBM_{R} \rightarrow \QBM_{\lex}$ is bijective.
Moreover, for every $p \in \QBM_{R}$,
\begin{equation*}
\wt (\Theta_{R}^{\lex}(p)) = \wt (p), \qquad
\qwt (\Theta_{R}^{\lex}(p)) = \qwt (p), \qquad
\ti{\iota}(\Theta_{R}^{\lex}(p)) = \ti{\iota}(p).
\end{equation*}
Let $A=\bigl\{j_{1},\,j_{2},\,\dots,\,j_{r}\bigr\}$ and
$B = \bigl\{q_{1},\,\dots,\,q_{s},\,j_{s+1},\,\dots,\,j_{r}\bigr\}$
be as in the definition above of the map $\Theta_{R}^{\lex}$.
Recall that
\begin{equation*}
\begin{split}
& p_{A} =\Bigl(
m_{\lamm}=z_{0}^{R} \edge{\beta_{j_1}^{\OS,R}} \cdots
\edge{\beta_{j_s}^{\OS,R}} z_{s}^{R} \edge{\beta_{j_{s+1}}^{\OS,R}}
\cdots \edge{\beta_{j_r}^{\OS,R}} z_{r}^{R} = \enp{A}
\Bigr), \\[3mm]
e=\dir(z_{0}^{R}) \edge{ \gamma_{j_{1}}^{R} } \cdots
\edge{ \gamma_{j_{s}}^{R} } \dir(z_{s}^{R}) = \ti{\iota}(p_{A})
\edge{ \gamma_{j_{s+1}}^{R} } \cdots \edge{ \gamma_{j_{r}}^{R} } \dir(z_{r}^{R}),
\end{split}
\end{equation*}
and that
\begin{equation*}
\begin{split}
& p_{B} =\Bigl(
m_{\lamm}=z_{0} \edge{\beta_{q_1}^{\OS}} \cdots
\edge{\beta_{q_s}^{\OS}} z_{s} \edge{\beta_{j_{s+1}}^{\OS}} \cdots
\edge{\beta_{j_r}^{\OS}} z_{r}^{R} = \enp{B}
\Bigr), \\[3mm]
e=\dir(z_{0}) \edge{ \gamma_{q_{1}} } \cdots
\edge{ \gamma_{q_{s}} } \dir(z_{s}) = \ti{\iota}(p_{A})
\edge{ \gamma_{j_{s+1}} } \cdots \edge{ \gamma_{j_{r}} } \dir(z_{r}).
\end{split}
\end{equation*}
Since $q_{s} \le M$ and $j_{s+1} \ge M+1$, it follows that
Next, we prove that
\begin{equation} \label{eq:Theta}
\wt (p_{B}) = \wt (p_{A}) \quad \text{and} \quad
\qwt (p_{B}) = \qwt (p_{A}).
\end{equation}
Recall from (<ref>) and (<ref>) that
\begin{equation*}
\qwt (p_{B}) = \sum_{j \in B^{-}} \beta_{j}^{\OS}
\quad \text{and} \quad
\qwt (p_{A}) = \sum_{j \in A^{-}} \beta_{j}^{\OS,R}.
\end{equation*}
We know from Remark <ref> that
all the edges in
$e=\dir(z_{0}^{R}) \edge{ \gamma_{j_{1}}^{R} } \cdots
\edge{ \gamma_{j_{s}}^{R} } \dir(z_{s}^{R}) = \ti{\iota}(p_{A})$ and
$e=\dir(z_{0}) \edge{ \gamma_{q_{1}} } \cdots
\edge{ \gamma_{q_{s}} } \dir(z_{s}) = \ti{\iota}(p_{A})$
are Bruhat edges, which implies that
$A^{-},\,B^{-} \subset \bigl\{j_{s+1},\,\dots,\,j_{r}\bigr\}$.
Since $M+1 \le j_{s+1} < \cdots < j_{r} \le \ell$,
we see from (<ref>) that
$\gamma_{j_{u}}^{R} = \gamma_{j_{u}}$ for all $s+1 \le u \le r$.
Therefore, the directed paths
$\dir(z_{s}^{R}) = \ti{\iota}(p_{A})
\edge{ \gamma_{j_{s+1}}^{R} } \cdots \edge{ \gamma_{j_{r}}^{R} } \dir(z_{r}^{R})$
and $\dir(z_{s}) = \ti{\iota}(p_{A})
\edge{ \gamma_{j_{s+1}} } \cdots \edge{ \gamma_{j_{r}} } \dir(z_{r})$ are identical,
which implies that $A^{-}=B^{-}$. Since $\beta_{j_{u}}^{\OS, R} = \beta_{j_{u}}^{\OS}$
for all $s+1 \le u \le r$ by (<ref>), we obtain $\qwt (p_{B}) = \qwt (p_{A})$.
Finally, we prove that $\wt (p_{B}) = \wt (p_{A})$; it suffices to show that
$\enp{B}=\enp{A}$ (see (<ref>)). Since $b_{k}^{R}=b_{k}=0$
for all $1 \le k \le M$ by (<ref>) and (<ref>), we see that
\begin{equation*}
\beta_{k}^{\OS,R}=
\pair{(\gamma_{k}^{R})^{\vee}}{-\lng \lambda} \ti{\delta} -
(\gamma_{k}^{R})^{\vee}, \qquad
\beta_{k}^{\OS}=
\pair{\gamma_{k}^{\vee}}{-\lng \lambda} \ti{\delta} -
\gamma_{k}^{\vee}
\end{equation*}
for $1 \le k \le M$, which implies that
\begin{equation*}
= \bigl(t_{ \pair{(\gamma_{k}^{R})^{\vee}}{-\lng \lambda} \gamma_{k}^{R} }\bigr)
r_{\gamma_{k}^{R}}, \qquad
= \bigl(t_{ \pair{\gamma_{k}^{\vee}}{-\lng \lambda} \gamma_{k}}\bigr)
\end{equation*}
Using these equalities, together with $z_{0}=z_{0}^{R}=m_{\lamm} = t_{\lamm}$,
we can show by induction on $0 \le u \le s$ that
\begin{equation*}
z_{u}^{R}= t_{ \dir (z_{u}^{R}) \lng \lambda} \dir (z_{u}^{R}), \qquad
z_{u}=t_{ \dir (z_{u}) \lng \lambda} \dir (z_{u}).
\end{equation*}
Since $\dir (z_{s}^{R}) = \ti{\iota}(p_{A}) = \dir (z_{s})$, we deduce that
\begin{equation*}
t_{ \dir (z_{s}^{R}) \lng \lambda} \dir (z_{s}^{R}) =
t_{ \dir (z_{s}) \lng \lambda} \dir (z_{s}) = z_{s}.
\end{equation*}
Since $\beta_{j_{u}}^{\OS, R} = \beta_{j_{u}}^{\OS}$
for all $s+1 \le u \le r$ as seen above, we obtain
\begin{equation*}
\enp{A} = z_{s}^{R}r_{\beta_{j_{s+1}}^{\OS,R}} \cdots r_{\beta_{j_{r}}^{\OS,R}}
= z_{s}r_{\beta_{j_{s+1}}^{\OS}} \cdots r_{\beta_{j_{r}}^{\OS}} = \enp{B}.
\end{equation*}
This proves (<ref>).
If we define a map $\Theta_{\lex}^{R} : \QBM_{\lex} \rightarrow \QBM_{R}$
in exactly the same manner as for the map $\Theta_{R}^{\lex}:\QBM_{R} \rightarrow \QBM_{\lex}$,
then from the uniqueness of a directed path in $\QB(W)$
whose labels are increasing in a reflection order (see <cit.>),
we deduce that both of the composites
$\Theta_{\lex}^{R} \circ \Theta_{R}^{\lex}$ and
$\Theta_{R}^{\lex} \circ \Theta_{\lex}^{R}$ are the identity maps.
This proves the bijectivity of the map $\Theta_{R}^{\lex}$,
and hence completes the proof of the proposition.
§.§ Embedding of $\QBw$ into $\QBM_{\lex}$.
We keep the notation and setting of the previous subsection.
Recall that $m_{\lamm} = t_{\lamm}= \pi r_{i_{1}}r_{i_{2}} \cdots r_{i_{\ell}}$
is the reduced expression for $m_{\lamm} = t_{\lamm}$ corresponding to
the (fixed) lex $(-\lamm)$-chain of roots; we know from Lemma <ref> that
\begin{equation} \label{eq:mlamm2}
m_{\lamm}=t_{\lamm}=\pi r_{i_{1}}r_{i_{2}} \cdots r_{i_{\ell}} =
\bigl( \ub{ r_{\p{1}} \cdots r_{\p{M}} }{=\mcr{\lng}} \bigr)
\bigl( \ub{ \pi r_{i_{M+1}} \cdots r_{i_{\ell}} }{=m_{\lambda}} \bigr),
\end{equation}
where $M=\ell(\mcr{\lng})$.
Let $w \in W^{J}$, and set $L:=\ell(w) \le M$.
We can take a reduced expression
$\mcr{\lng} = r_{p_1}r_{p_2} \cdots r_{p_M}$ for $\mcr{\lng}$
such that $w=r_{p_{M-L+1}} \cdots r_{p_{M}}$;
\begin{equation} \label{eq:re}
\mcr{\lng} = \ub{r_{p_1} \cdots r_{p_{M-L}}}{=\mcr{\lng}w^{-1}}
\ub{r_{p_{M-L+1}} \cdots r_{p_{M}}}{=w}.
\end{equation}
Indeed, recall that $\lng = \mcr{\lng} \lngJ$,
with $\ell(\lng) = \ell(\mcr{\lng}) + \ell(\lngJ)$.
Since $w \in W^{J}$, we have
$\ell(w \lngJ) = \ell(w) + \ell(\lngJ)$. Hence it follows that
\begin{align*}
\ell(\mcr{\lng}) + \ell(\lngJ)
& = \ell(\lng) = \ell(\lng (w\lngJ)^{-1}) + \ell(w \lngJ) \\
& = \ell(\mcr{\lng}w^{-1}) + \ell(w) + \ell(\lngJ),
\end{align*}
so that $\ell(\mcr{\lng}) = \ell(\mcr{\lng}w^{-1}) + \ell(w)$,
which implies that $\ell(\mcr{\lng}w^{-1}) = M-L$.
Therefore, if $\mcr{\lng}w^{-1} = r_{p_1} \cdots r_{p_{M-L}}$ is
a reduced expression for $\mcr{\lng}w^{-1}$, and
$w= r_{p_{M-L+1}} \cdots r_{p_{M}}$ is a reduced expression for $w$,
then $\mcr{\lng} = r_{p_1} \cdots r_{p_{M-L}}
r_{p_{M-L+1}} \cdots r_{p_{M}}$ is a reduced expression for $\mcr{\lng}$.
Now, we set $i_{k}':=\pi^{-1}(p_{k})$ for $1 \le k \le M$;
we see that
\begin{equation} \label{eq:mlamm3}
m_{\lamm} =
\bigl( \underbrace{ r_{p_1}r_{p_2} \cdots r_{p_M} }_{=\mcr{\lng}} \bigr)
\bigl( \underbrace{ \pi r_{i_{M+1}} \cdots r_{i_{\ell}} }_{=m_{\lambda}} \bigr)
= \pi r_{i_{1}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}}
\end{equation}
is a reduced expression for $m_{\lamm}$. As in <ref>,
we construct $\QBM$ from this reduced expression $R$ for $m_{\lamm}$, and
denote it by $\QBM_{R}$; recall from Proposition <ref> the bijection
$\Theta_{R}^{\lex}:\QBM_{R} \rightarrow \QBM_{\lex}$.
We set $A_{0}:=\bigl\{1,\,2,\,\dots,\,M-L\bigr\} \subset
\bigl\{1,\,2,\,\dots,\,\ell\bigr\}$, and consider $p_{A_0}$.
Using Lemma <ref>, we see by direct computation that
\begin{align*}
& z_{0}^{R}=m_{\lamm}=
\pi r_{i_{1}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}} =t_{\lamm}, \\
& z_{1}^{R}=
\pi r_{i_{2}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}} = r_{p_1} t_{\lamm}, \\
& z_{2}^{R}=
\pi r_{i_{3}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}} = r_{p_2} r_{p_1} t_{\lamm}, \\
& \qquad \vdots\\
& z_{M-L-1}^{R}=
\pi r_{i_{M-L}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}} =
r_{p_{M-L-1}} \cdots r_{p_2} r_{p_1} t_{\lamm}, \\
& z_{M-L}^{R}=
\pi r_{i_{M-L+1}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}} =
\underbrace{r_{p_{M-L}} \cdots r_{p_2} r_{p_1}}_{=w\mcr{\lng}^{-1}} t_{\lamm} = m_{w\lambda}.
\end{align*}
From these, we deduce that $\dir(z_{K}^{R})= r_{p_{K}} \cdots r_{p_{2}}r_{p_1}$
for $0 \le K \le M-L$, and that
\begin{equation} \label{eq:dp5}
e \edge{\gamma_{1}^{R}} r_{p_1}
\edge{\gamma_{2}^{R}}
\cdots
\edge{\gamma_{M-L-1}^{R}} r_{p_{M-L-1}} \cdots r_{p_{2}}r_{p_{1}}
\edge{\gamma_{M-L}^{R}} w\mcr{\lng}^{-1}
\end{equation}
is a directed path from $e$ to $w\mcr{\lng}^{-1}$
in the quantum Bruhat graph $\QB(W)$;
since $\ell(\dir(z_{K})) = \ell(\dir(z_{K-1})) + 1$
for $1 \le K \le M-L$, all the edges in this directed path
are Bruhat edges. Hence we obtain
\begin{equation} \label{eq:pA0}
\Bigl(
m_{\lamm} =
z_{0}^{R} \edge{\beta_{1}^{\OS,R}}
z_{1}^{R} \edge{\beta_{2}^{\OS,R}} \cdots
\edge{\beta_{M-L}^{\OS,R}} z_{M-L}^{R}=m_{w\lambda}
\Bigr) \in \QBM_{R}.
\end{equation}
Since $m_{w\lambda}=w \mcr{\lng}^{-1} t_{\lng \lambda} = w m_{\lambda}$
by Lemma <ref>, we have
\begin{equation} \label{eq:mw}
m_{w\lambda} =
\bigl( \underbrace{ r_{p_{M-L+1}} \cdots r_{p_M} }_{=w} \bigr)
\bigl( \underbrace{ \pi r_{i_{M+1}} \cdots r_{i_{\ell}} }_{=m_{\lambda}} \bigr)
= \pi r_{i_{M-L+1}'} \cdots r_{i_{M}'} r_{i_{M+1}} \cdots r_{i_{\ell}};
\end{equation}
since (<ref>) is a reduced expression (for $m_{\lamm}$),
we see that (<ref>) is also a reduced expression (for $m_{w\lambda}$).
Let us construct $\QBw$ from this reduced expression.
Namely, for a subset $B=\bigl\{j_{1} < j_{2} < \cdots < j_{r}\bigr\}
\subset \bigl\{M-L+1,\,M-L+2,\,\dots,\,\ell\bigr\}$, we define
\begin{equation*}
y_{0}^{R} = m_{w\lambda}, \qquad
y_{k}^{R} = y_{0}r_{\beta_{j_1}^{\OS,R}} \cdots r_{\beta_{j_{k}}^{\OS,R}} \quad
\text{for $1 \le k\le r$},
\end{equation*}
where $\beta_{k}^{\OS,R}$, $M-L+1 \le k \le \ell$, are
those used in the definition of $\QBM_{R}$, and set
\begin{equation*}
p_{B}:= \Bigl(
m_{w\lambda}=y_{0}^{R} \edge{\beta_{j_1}^{\OS,R}} y_{1}^{R}
\edge{\beta_{j_2}^{\OS,R}} y_{2}^{R} \edge{\beta_{j_3}^{\OS,R}}
\cdots
\edge{\beta_{j_r}^{\OS,R}} y_{r}^{R}\Bigr).
\end{equation*}
Then, $p_{B} \in \QBw$ if
\begin{equation*}
w \mcr{\lng}^{-1}=\dir(y_{0}^{R}) \edge{ \gamma_{j_1}^{R} }
\dir(y_{1}^{R}) \edge{ \gamma_{j_2}^{R} }
\cdots \edge{ \gamma_{j_r}^{R} } \dir(y_{r}^{R})
\end{equation*}
is a directed path in the quantum Bruhat graph $\QB(W)$.
Since $\enp{A_{0}}=m_{w\lambda}$,
we can “concatenate” $p_{A_{0}}$ with an arbitrary $p_{B} \in \QBw$,
which is just $p_{A_{0} \sqcup B}$;
we see easily that $p_{A_{0} \sqcup B} \in \QBM_{R}$.
There exists an embedding
$\QBw \hookrightarrow \QBM_{R}$, which maps
$p_{B} \in \QBw$ to $p_{A_{0} \sqcup B} \in \QBM_{R}$.
$\wt(p_{A_{0} \sqcup B}) = \wt (p_{B})$, and
$\qwt(p_{A_{0} \sqcup B}) = \qwt (p_{B})$ (and hence
$\degr(\qwt(p_{A_{0} \sqcup B})) =
\degr(\qwt (p_{B}))$).
The injectivity of the map is obvious.
Since $\enp{A_{0} \sqcup B} = \enp{B}$ by the definition,
we have $\wt(p_{A_{0} \sqcup B}) = \wt (p_{B})$.
Because all the edges in the directed path (<ref>)
are Bruhat edges, we see from the definition (<ref>)
that $(A_{0} \sqcup B)^{-}= B^{-}$. Hence we obtain
$\qwt(p_{A_{0} \sqcup B}) = \qwt (p_{B})$ by the definition
(<ref>) of $\qwt$. This proves the lemma.
We set
\begin{equation*}
\QBM_{R, w}:=
\bigl\{ p_{A} \in \QBM_{R} \mid
\bigl\{1,\,2,\,\dots,\,M-L\bigr\} \subset A
\bigr\}.
\end{equation*}
We see from the argument above that
$\QBM_{R, w}$ is identical to the image of
the embedding $\QBw \hookrightarrow \QBM_{R}$ of Lemma <ref>.
Let $p_{A} \in \QBM_{R}$. Then, $p_{A} \in \QBM_{R, w}$ if and only if
$\ti{\iota}(p_{A}) \ge w \mcr{\lng}^{-1}$ with respect to
the Bruhat order $\ge$ on $W$.
First, we prove the “only if” part.
Since $p_{A} \in \QBM_{R, w}$, it follows that
$A$ is of the form: $A=\bigl\{1,\,2,\,\dots,\,M-L,\,j_{1},\,\dots,\,j_{r}\bigr\}$
for $M-L+1 \le j_{1} < \cdots < j_{r} \le \ell$;
we set $j_{0}=0$ by convention. Take $0 \le s \le r$ such that
$j_{s} \le M$ and $j_{s+1} \ge M+1$. Then, by (<ref>) and
the definition of $\ti{\iota}(p_{A})$, we have a directed path
\begin{equation*}
e \edge{\gamma_{1}^{R}} r_{p_1}
\edge{\gamma_{2}^{R}} \cdots \edge{\gamma_{M-L}^{R}} w\mcr{\lng}^{-1}
\edge{ \gamma_{j_{1}}^{R} } \cdots \edge{ \gamma_{j_{s}}^{R} } \ti{\iota}(p_{A})
\end{equation*}
in the quantum Bruhat graph $\QB(W)$. Since all the edges in this directed path
are Bruhat edges (see Remark <ref>), we obtain
$\ti{\iota}(p_{A}) \ge w\mcr{\lng}^{-1}$, as desired.
Next, we prove the “if” part.
Assume that $\ti{\iota}(p_{A}) \ge w \mcr{\lng}^{-1}$,
with $A=\bigl\{j_{1},\,\dots,\,j_{r}\bigr\} \subset
\bigl\{1,\,2,\,\dots,\,\ell\bigr\}$. If we take $0 \le s \le \ell$ such that
$j_{s} \le M$ and $j_{s+1} \ge M+1$, then we have a shortest directed path
\begin{equation} \label{eq:dp7}
e=\dir(z_{0}^{R}) \edge{\gamma_{j_1}^{R}}
\dir(z_{1}^{R}) \edge{\gamma_{j_2}^{R}} \cdots \edge{\gamma_{j_s}^{R}}
\dir(z_{s}^{R}) = \ti{\iota}(p_{A})
\end{equation}
in the quantum Bruhat graph $\QB(W)$ all of whose edges are Bruhat edges
(see Remark <ref>); note that $s=\ell(\ti{\iota}(p_{A}))$.
Here, because $\ti{\iota}(p_{A}) \ge w \mcr{\lng}^{-1}$
with respect to the Bruhat order on $W$, we deduce
by the chain property of the Bruhat order (see <cit.>) that
there exists a directed path $w \mcr{\lng}^{-1} = x_{0} \edge{ \xi_1 } \cdots
\edge{ \xi_{s-M+L} } x_{s-M+L} = \ti{\iota}(p_{A})$
of length $\ell(\ti{\iota}(p_{A})) - \ell(w \mcr{\lng}^{-1}) = s-(M-L)$
from $w \mcr{\lng}^{-1}$ to $\ti{\iota}(p_{A})$
in the quantum Bruhat graph $\QB(W)$ all of whose edges are Bruhat edges.
Concatenating this directed path with the directed path (<ref>),
we get the directed path
\begin{equation} \label{eq:dp6}
e \edge{\gamma_{1}^{R}} r_{p_1}
\edge{\gamma_{2}^{R}} \cdots \edge{\gamma_{M-L}^{R}} w\mcr{\lng}^{-1}
= x_{0} \edge{ \xi_1 } \cdots \edge{ \xi_{s-M+L} } x_{s-M+L} = \ti{\iota}(p_{A}).
\end{equation}
Since the length of this directed path is equal to
$s=\ell(\ti{\iota}(p_{A})) - \ell(e)$,
this directed path is also a shortest directed path from
$e$ to $\ti{\iota}(p_{A})$ in the quantum Bruhat graph $\QB(W)$.
Because the labels in the directed path (<ref>)
are strictly increasing with respect to the reflection
order $\prec_{R}$ (see (<ref>)),
that is, $\gamma_{j_1}^{R} \prec_{R} \cdots \prec_{R} \gamma_{j_s}^{R}$,
it follows from <cit.> that
the directed path (<ref>) is lexicographically minimal among
all shortest directed paths from $e$ to $\ti{\iota}(p_{A})$;
in particular, the directed path (<ref>) is less than
or equal to the directed path (<ref>),
which implies that $j_{1}=1,\,j_{2}=2,\,\dots,\,j_{M-L}=M-L$.
Thus, we obtain $\bigl\{1,\,2,\,\dots,\,M-L\bigr\} \subset A$,
and hence $p_{A} \in \QBM_{R, w}$. This completes the proof of the lemma.
From Lemma <ref> (together with the comment preceding it),
Lemma <ref>, and Proposition <ref>,
we obtain the following proposition.
The image of $\QBw$ under the composite
\begin{equation*}
\QBw \stackrel{\text{\rm Lemma~\ref{lem:embed}}}{\hookrightarrow}
\QBM_{R} \stackrel{\Theta_{R}^{\lex}}{\longrightarrow} \QBM_{\lex}
\end{equation*}
is identical to
\begin{equation} \label{eq:QBMlexw}
\QBM_{\lex, w}:=\bigl\{ p \in \QBM_{\lex} \mid \ti{\iota}(p) \ge w \mcr{\lng}^{-1} \bigr\}.
\end{equation}
Hence we have
\begin{equation} \label{eq:prf1}
\sum_{p \in \QBM_{\lex,w}} e^{\wt(p)} q^{\degr(\qwt(p))} =
\sum_{p \in \QBw} e^{\wt(p)} q^{\degr(\qwt(p))} = \Mac{w\lambda}.
\end{equation}
§.§ Bijection between $\QBM_{\lex}$ and $\QLS(\lambda)$.
As in the previous subsection, we fix a lex $(-\lamm)$-chain of roots
\begin{equation} \label{eq:lex1}
A_{\circ}=A_{0} \edge{-\gamma_{1}} A_{1}
\edge{-\gamma_{2}} \cdots \edge{-\gamma_{\ell}} A_{\ell}=A_{\lamm},
\end{equation}
and let $m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$ be
the corresponding reduced expression for $m_{\lamm}$ under (<ref>).
We construct $\CA(-\lamm)$ from this reduced expression
$m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$,
which we denote by $\CA(-\lamm)_{\lex}$;
recall from Remark <ref> and (<ref>) that
\gamma_{k}^{\Le}=(\ol{\beta_{k}^{\Le}})^{\vee} \in -\lng(\Phi^{+} \setminus \Phi^{+}_{J})$
for all $1 \le k \le \ell$. We set (see Remark <ref>)
\begin{equation} \label{eq:dk}
d_{k}:=\frac{b_{k}}{ \bpair{ \ol{\beta_{k}^{\Le}} }{-\lamm} }
= 1 - \frac{a_{k}}{ \bpair{ \ol{\beta_{k}^{\OS}} }{\lamm} }
\qquad \text{for $1 \le k \le \ell$}.
\end{equation}
Because $m_{\lamm}=\pi r_{i_1}r_{i_2} \cdots r_{i_{\ell}}$ is the reduced
expression corresponding to the lex $(-\lamm)$-chain of roots,
it follows from (<ref>) that
\begin{equation*}
0 \le d_{1} \le d_{2} \le \cdots \le d_{\ell} < 1.
\end{equation*}
Let $1 \le k < p \le \ell$ be such that $d_{k}=d_{p}$.
Then we know from <cit.> that
$\gamma_{k} \prec \gamma_{p}$
in the reflection order $\prec$ (see (<ref>)).
In the following,
we define a map $\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$
(resp., $\Pi:\CA(-\lamm)_{\lex} \rightarrow \QLS(\lambda)$; see <cit.>).
Let $A=\bigl\{j_{1} < \cdots < j_{r}\bigr\} \subset
\bigl\{1,\,\dots,\,\ell\bigr\}$ be such that
\begin{equation*}
m_{\lamm} = z_{0} \edge{\beta_{j_1}^{\OS}}
z_{1} \edge{\beta_{j_2}^{\OS}} \cdots \edge{\beta_{j_r}^{\OS}} z_{r}
\Bigr)
\end{equation*}
is an element of $\QBM_{\lex}$ (resp., $A \in \CA(-\lamm)_{\lex}$);
if we set $x_{k}:=r_{\gamma_{j_1}} \cdots r_{\gamma_{j_k}} = \dir(z_{k}) \in W$
for $0 \le k \le r$, then
\begin{equation*}
e=x_{0} \edge{\gamma_{j_1}} x_{1} \edge{\gamma_{j_2}} \cdots
\edge{\gamma_{j_r}} x_{r}
\end{equation*}
is a directed path in the quantum Bruhat graph $\QB(W)$.
Take $0=u_{0} \le u_{1} < u_{2} < \cdots < u_{s-1} < u_{s}=r$ (with $s \ge 1$)
in such a way that
\begin{equation} \label{eq:ap}
\begin{split}
& \underbrace{0 = d_{j_1} = \cdots =d_{j_{u_1}}}_{=:\sigma_{0}} <
\underbrace{d_{j_{u_1+1}}= \cdots = d_{j_{u_2}}}_{=:\sigma_{1}} < \\[1.5mm]
& \hspace{20mm}
\underbrace{d_{j_{u_2+1}}= \cdots = d_{j_{u_3}}}_{=:\sigma_{2}} <
\cdots \cdots \cdots
< \underbrace{d_{j_{u_{s-1}+1}}= \cdots =d_{j_{r}}}_{=:\sigma_{s-1}} < 1=:\sigma_{s};
\end{split}
\end{equation}
note that $u_{1} = 0$ if $d_{j_1} > 0$.
We set $w_{p}':=x_{u_p}$ for $1 \le p \le s-1$, and
$w_{s}':=x_{r}$. For each $1 \le p \le s-1$, we have
a directed path
\begin{equation*}
w_{p}'=x_{u_p} \edge{\gamma_{j_{u_p+1}}}
x_{u_p+1} \edge{\gamma_{j_{u_p+2}}} \cdots
\edge{\gamma_{j_{u_{p+1}}}} x_{u_{p+1}}=w_{p+1}'
\end{equation*}
in the quantum Bruhat graph $\QB(W)$.
We claim that this directed path is a shortest directed path
from $w_{p}'$ to $w_{p+1}'$. Indeed,
since $d_{j_{u_p+1}}= \cdots = d_{j_{u_{p+1}}}$ by (<ref>),
it follows from Remark <ref> that
$\gamma_{j_{u_p+1}} \prec \cdots \prec \gamma_{j_{u_{p+1}}}$
in the reflection order $\prec$ (see (<ref>)).
Therefore, we deduce from <cit.> that
the directed path above is a shortest directed path
from $w_{p}'$ to $w_{p+1}'$, as desired. Hence it follows that
\begin{equation} \label{eq:achain}
\begin{split}
& w_{p}:=w_{p}'\lng=x_{u_p}\lng
\edger{-\lng\gamma_{j_{u_p+1}}} x_{u_p+1}\lng
\edger{-\lng\gamma_{j_{u_p+2}}} \\[1.5mm]
& \hspace{50mm} \cdots \cdots
\edger{-\lng\gamma_{j_{u_{p+1}}}}
\end{split}
\end{equation}
is also a shortest directed path in the quantum Bruhat graph $\QB(W)$,
where $-\lng\gamma_{j_{u}} \in \Phi^{+} \setminus \Phi^{+}_{J}$
for all $u_{p}+1 \le u \le u_{p+1}$
since $\gamma_{j_{u}} \in -\lng\bigl(\Phi^{+} \setminus \Phi^{+}_{J}\bigr)$
as mentioned at the beginning of this subsection.
Moreover, for $u_p+1 \le u \le u_{p+1}$, we have
\begin{equation*}
\sigma_{p}\pair{ -\lng\gamma_{j_{u}}^{\vee} }{\lambda} =
d_{j_{u}}\pair{ \gamma_{j_{u}}^{\vee} }{-\lamm} =
\frac{b_{j_u}}{ \bpair{ \ol{\beta_{j_u}^{\Le}} }{-\lamm} }
\times \bpair{ \ol{ \beta_{j_u}^{\Le} } }{-\lamm} = b_{j_u} \in \BZ.
\end{equation*}
Hence the directed path (<ref>) is
a directed path in $\QB_{\sigma_p\lambda}(W)$.
We deduce from <cit.> that
there exists a directed path from $\mcr{w_{p+1}} = \mcr{w_{p+1}}^{J}$ to
$\mcr{w_{p}} = \mcr{w_{p}}^{J}$ in $\QB_{\sigma_{p}\lambda}(W^{J})$.
Therefore, we conclude that
\begin{equation*}
\eta:=
(\mcr{w_{1}},\,\mcr{w_{2}},\,\dots,\,\mcr{w_{s}} \,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in \QLS(\lambda);
\end{equation*}
we set $\Xi(p_{A}):=\eta$.
Keep the setting above. Because
$0 = d_{j_1} = \cdots =d_{j_{u_1}} < d_{j_{u_1 + 1}}$ by the definition of $u_{1}$,
we see from (<ref>) that $j_{u_{1}} \le M$ and $j_{u_1 +1} \ge M+1$,
where $\ell(\mcr{\lng})=M$. Therefore, by definition (<ref>),
$\ti{\iota}(p_{A})$ is just $\dir (z_{u_1}) = x_{u_{1}} = w_{1}'$.
Hence we obtain
\begin{equation} \label{eq:iota}
\iota(\Xi(p_{A})) = \iota(\eta) = \mcr{w_{1}} = \mcr{w_{1}'\lng} =
\mcr{\ti{\iota}(p_{A})\lng}.
\end{equation}
The map $\Pi:\CA(-\lamm)_{\lex} \rightarrow \QLS(\lambda)$ is bijective.
Moreover, for every $A \in \CA(-\lamm)_{\lex}$,
\begin{equation} \label{eq:Pi-wtht}
\wt(\Pi(A)) = - \wt(A) \qquad \text{\rm and} \qquad
\Deg (\Pi(A)) = - \Ht(A).
\end{equation}
The map $\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$ is bijective.
Moreover, for every $p_{A} \in \QBM_{\lex}$,
\begin{equation*}
\wt(\Xi(p_{A}))= \wt(p_{A}) \qquad \text{\rm and} \qquad
\Deg (\Xi(p_{A})) = - \deg(\qwt(p_{A})).
\end{equation*}
From the constructions, we see that
the map $\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$ above
is identical to the composite of the bijection
$\QBM_{\lex} \stackrel{\sim}{\rightarrow} \CA(-\lamm)_{\lex}$ of Lemma <ref> and
the bijection $\Pi:\CA(-\lamm)_{\lex} \stackrel{\sim}{\rightarrow} \QLS(\lambda)$
in Proposition <ref>. Hence the map
$\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$ is also bijective.
\begin{equation*}
\begin{diagram}
\node{\quad \QBM_{\lex} \quad}
\arrow{e,b}{\begin{subarray}{c}
\text{Bijection} \\
\text{in Lemma~\ref{lem:A-QB}}
\end{subarray}
\arrow{se,b}{\Xi}
\node{\CA(-\lamm)_{\lex}}
\arrow{s,r}{\Pi} \\
\node{}
\node{\QLS(\lambda)}
\end{diagram}
\end{equation*}
We know from (<ref>) that
$\wt (A) = - \wt (p_{A})$ and $\Ht (A) = \deg (\qwt(p_{A}))$
for all $A \in \CA(-\lamm)$.
Combining this equality and (<ref>),
we obtain the equalities
$\wt(\Xi(p_{A}))= \wt(p_{A})$ and
$\Deg(\Xi(p_{A}))= - \degr(\qwt(p_{A}))$
for all $p_{A} \in \QBM_{\lex}$, as desired.
The image of $\QBM_{\lex,w}$ (see Proposition <ref>)
under the bijection $\Xi:\QBM_{\lex} \rightarrow \QLS(\lambda)$
of Proposition <ref> is identical to $\QLS_{w}(\lambda)$.
Let $p \in \QBM_{\lex}$. Then,
\begin{align*}
p \in \QBM_{\lex,w}
& \stackrel{\eqref{eq:QBMlexw}}{\iff} \ti{\iota}(p) \ge w \mcr{\lng}^{-1}
\iff \ti{\iota}(p) \lng \le w \mcr{\lng}^{-1} \lng.
\end{align*}
Since $\ti{\iota}(p) \in W^{\omega(J)}$ (see Remark <ref>),
it follows by (<ref>) that
\begin{equation*}
\ti{\iota}(p) \lng \lngJ (\Phi^{+}_{J}) =
\ti{\iota}(p) \lng (- \Phi^{+}_{J}) =
\ti{\iota}(p) (\Phi^{+}_{\omega(J)}) \subset \Phi^{+}.
\end{equation*}
From this, we deduce that
$\ti{\iota}(p) \lng \lngJ \in W^{J}$ again by (<ref>),
which implies that
$\mcr{\ti{\iota}(p) \lng} \lngJ =
\mcr{\ti{\iota}(p) \lng \lngJ} \lngJ =
(\ti{\iota}(p) \lng \lngJ) \lngJ =
\ti{\iota}(p) \lng$.
\begin{equation*}
\ti{\iota}(p) \lng \le w \mcr{\lng}^{-1} \lng
\iff \mcr{\ti{\iota}(p) \lng} \lngJ \le w \lngJ.
\end{equation*}
Here we have
\begin{equation*}
\mcr{\ti{\iota}(p) \lng} \lngJ \le w \lngJ \quad \iff \quad
\mcr{\ti{\iota}(p) \lng} \le w.
\end{equation*}
Indeed, the “only if” part ($\Rightarrow$) follows immediately from
<cit.>. Let us show the “if” part ($\Leftarrow$).
Fix reduced expressions for $\lngJ \in W_{J}$ and $w \in W^{J}$, respectively,
and then take a reduced expression of $\mcr{\ti{\iota}(p) \lng} \in W^{J}$
that is a “subword” of the fixed reduced expression of $w$
(see <cit.>). By <cit.>,
the concatenation of this reduced expression for $\mcr{\ti{\iota}(p) \lng}$
(resp., $w \in W^{J}$) with a reduced expression for $\lngJ$ is
a reduced expression for $\mcr{\ti{\iota}(p) \lng} \lngJ$
(resp., $w \lngJ$); observe that the obtained reduced expression for
$\mcr{\ti{\iota}(p) \lng} \lngJ$ is a subword of
the obtained reduced expression for $w\lngJ$.
Therefore, by <cit.>, we see that
$\mcr{\ti{\iota}(p) \lng} \lngJ \le w \lngJ$, as desired.
Finally, we have
\begin{align*}
\mcr{\ti{\iota}(p) \lng} \le w
& \iff \iota(\Xi(p)) \le w \qquad \text{by \eqref{eq:iota}} \\
& \iff \Xi(p) \in \QLS_{w}(\lambda).
\end{align*}
This proves the lemma.
We compute:
\begin{align*}
\sum_{\eta \in \QLS_{w}(\lambda)}
& = \sum_{p \in \QBM_{\lex,w}}
e^{\wt(p)}q^{\degr(\qwt(p))} \\
& \hspace{30mm}
\text{by Lemma~\ref{lem:Xi} and Proposition~\ref{prop:Xi}} \\[3mm]
& =\Mac{w\lambda} \quad \text{by \eqref{eq:prf1}}.
\end{align*}
This completes the proof of Theorem <ref>.
§.§ The formula in terms of the quantum alcove model.
We start with some review from <cit.>.
Recall the Dynkin diagram automorphism $\omega:I \rightarrow I$
induced by $\lng \alpha_{j}=-\alpha_{\omega(j)}$ for $j \in I$.
Note that $\omega$ acts as $-\lng$
on the integral weight lattice $X$. There exists a group automorphism,
denoted also by $\omega$, of the Weyl group $W$ such that
$\omega(r_{j})=r_{\omega(j)}$ for all $j \in I$.
Now, fix $\lambda \in X$ be a dominant integral weight
with $J=\bigl\{ i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda} = 0 \bigr\}$,
and let
\begin{equation} \label{eq:se1}
\eta=(x_{1},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in \QLS(\lambda),
\end{equation}
with $x_{1},\,\dots,\,x_{s} \in W^{J}$ and
rational numbers $0=\sigma_{0} < \cdots < \sigma_{s}=1$.
Then we define
\begin{equation} \label{eq:dual}
\eta^{\ast}:=
\end{equation}
We also define $\omega(\eta)$ by
\begin{equation} \label{eq:se2}
\omega(\eta):=
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s}).
\end{equation}
Both maps, $*$ and $\omega$, are
bijections between $\QLS(\lambda)$ and $\QLS(-w_{\circ}\lambda)$,
and they change the weight of a path by a negative sign and $\omega$, respectively.
Finally, we set $S(\eta):=\omega(\eta^{\ast})=(\omega(\eta))^{\ast}$,
which turns out to be the Lusztig involution on $\QLS(\lambda)$.
Replacing $\lambda$ by $-\lamm$
in <ref> and <ref>,
let us consider a lex $\lambda$-chain of roots, and
the quantum alcove model $\CA(\lambda)_{\lex}$ associated to it.
Recall the map $\Pi$ (in Proposition <ref>
with $\lambda$ replaced by $-\lamm$) and
the corresponding commutative diagram:
\begin{equation}
\begin{diagram}
\node{\CA(\lambda)_{\lex} } \arrow{e,t}{\Pi} \arrow{se,b}{\Pi^{\ast}}
\node{\QLS(-w_{\circ}\lambda)} \arrow{s,r}{\ast} \\
\node{} \node{\QLS(\lambda).}
\end{diagram}
\end{equation}
We need an analogue of <cit.> for the coheight statistic,
which was defined in (<ref>). This is stated as follows,
and is proved in a completely similar way, based on the results in <cit.>.
Consider an admissible subset $A \in \CA(\lambda)_{\lex}$, and
the corresponding QLS path $\Pi(A) \in \QLS(-\lamm)$.
Write $\Pi(A)$ as follows (cf. Definition <ref>):
\begin{equation} \label{qlsp}
\blarrl{-\sigma_1w_{\circ}\lambda} x_2
\blarrl{-\sigma_2w_{\circ}\lambda} \ldots
\blarrl{-\sigma_{s-1}w_{\circ}\lambda} x_s,
\end{equation}
with $x_i \in W^{\omega(J)}$ and
$0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s} = 1$. Then, we have
\begin{equation}
\cHt(A)=\sum_{i=1}^{s-1} \sigma_i\wt_{-w_\circ\lambda}(x_{i+1}\Rightarrow x_{i}),
\end{equation}
where $\wt_{-w_\circ\lambda}(x_{i+1}\Rightarrow x_{i})$
was defined in (<ref>).
We will now express the nonsymmetric Macdonald polynomial
in terms of the quantum alcove model.
Recall that the final direction $\phi(A)$ of
an admissible subset $A$ was defined in (<ref>).
We have
\begin{equation} \label{e3}
\sum_{
\begin{subarray}{c}
A \in \CA(\lambda) \\
\mcr{\phi(A)}^{J} \le w
\end{subarray}
} q^{\cHt(A)} x^{\wt(A)}.
\end{equation}
We derive this formula directly from Theorem <ref>, based on the map $\Pi^\ast$,
which is known to be a weight-preserving bijection, by <cit.>.
Using the very explicit description of the map $\Pi^\ast$ in <cit.>,
we can see that it switches initial and final directions, i.e.,
for $A \in \CA(\lambda)$ we have
\begin{equation*}
\iota(\Pi^\ast(A))= \mcr{\phi(A)}^{J}.
\end{equation*}
Finally, by using the notation (<ref>) for $\Pi(A)$,
we deduce:
\begin{align*}
\cHt(A)
& = \sum_{u=1}^{s-1} \sigma_{u} \wt_{-\lamm}(x_{u+1}\Rightarrow x_{u})
= - \Deg(S(\Pi(A))) \\[1.5mm]
& = - \Deg(\omega(\Pi^\ast(A)))
= - \Deg(\Pi^\ast(A)).
\end{align*}
Here the first equality is based on Theorem <ref>,
the second one on <cit.>,
the third one on the above definition of the Lusztig involution $S$, and
the last one on <cit.>.
In <cit.>, we realized an appropriate tensor product of
Kirillov–Reshetikhin crystals $\BB$ in terms of $\QLS(\lambda)$.
Based on this, we expressed the so-called “right” energy function on $\BB$ as
$\Deg(\eta)$ for $\eta\in\QLS(\lambda)$. In these terms,
$\Deg(S(\eta))$ expresses the corresponding “left” energy function,
see <cit.>. We also realized $\BB$ in terms of
the quantum alcove model, and in this setup the two energy functions are
expressed by the height and coheight statistics.
When $\Gamma$ is an (arbitrary) $\lambda$-chain of roots,
we denote by $\CA(\lambda)_{\Gamma}$ the quantum alcove model
associated to $\Gamma$.
In <cit.>, we defined certain combinatorial moves
(called quantum Yang-Baxter moves) in the quantum alcove model,
namely on the collection of $\CA(\lambda)_{\Gamma}$,
where $\Gamma$ is any $\lambda$-chain (of roots).
We showed that these define an affine crystal isomorphism
between $\CA(\lambda)_{\Gamma}$ and $\CA(\lambda)_{\Gamma'}$
for any two $\lambda$-chains $\Gamma$ and $\Gamma'$.
We also showed that the moves preserve the weight,
the height and coheight, as well as the final direction of
(the path in $\QB(W)$ associated with) an admissible subset.
Based on these facts, we can generalize Theorem <ref>.
Theorem <ref> still holds if we replace the admissible subsets
$\CA(\lambda)_{\lex}$ for a lex $\lambda$-chain with
the ones for an arbitrary $\lambda$-chain $\Gamma$,
namely $\CA(\lambda)_{\Gamma}$.
The formulas in Theorems <ref> and <ref>
(in fact, the latter can be replaced with the mentioned generalization) specialize,
upon setting $q=0$, to the formulas for Demazure characters in terms of LS paths
<cit.> and the alcove model <cit.>.
§ GRADED CHARACTERS OF QUOTIENTS OF DEMAZURE MODULES.
§.§ Additional setting.
The untwisted affine Lie algebra $\Fg_{\af}$ is written as:
$\Fg_{\af}=\Fg \otimes \BC[t,\,t^{-1}] \oplus \BC c \oplus \BC D$,
where $c=\sum_{j \in I_{\af}} a^{\vee}_{j} \alpha_{j}^{\vee}$
is the canonical central element, and
$D$ is the scaling element (or the degree operator); note that
the Cartan subalgebra $\Fh_{\af}$ of $\Fg_{\af}$ is
$\Fh \oplus \BC c \oplus \BC D$,
where $\Fh$ is the Cartan subalgebra of $\Fg$.
Let us denote by $\bigl\{\alpha_{i}\bigr\}_{i \in I_{\af}}$ and
$\bigl\{\alpha_{i}^{\vee}\bigr\}_{i \in I_{\af}}$
the simple roots and simple coroots of $\Fg_{\af}$, respectively, and
by $\Lambda_{j} \in \Fh_{\af}^{\ast}$, $j \in I_{\af}$,
the fundamental weights for $\Fg_{\af}$;
note that $\pair{D}{\alpha_{j}}=\delta_{j,0}$ and
$\pair{D}{\Lambda_{j}}=0$ for $j \in I_{\af}$.
We take a weight lattice $X_{\af}$ for $\Fg_{\af}$ as follows:
\begin{equation} \label{eq:lattices}
X_{\af} =
\left(\bigoplus_{j \in I_{\af}} \BZ \Lambda_{j}\right) \oplus
\BZ \delta \subset \Fh_{\af}^{\ast},
\end{equation}
where $\delta \in \Fh_{\af}^{\ast}$ denotes the null root of $\Fg_{\af}$.
We think of a weight $\mu \in \Fh^{\ast}$ for $\Fg$
as a weight ($\in \Fh_{\af}^{\ast}$) for $\Fg_{\af}$ by:
$\pair{c}{\mu}=\pair{D}{\mu}=0$. Then, for each $i \in I$,
the fundamental weight $\vpi_{i}$ for $\Fg$ is identical to
$\Lambda_{i}-a_{i}^{\vee}\Lambda_{0} \in \Fh_{\af}^{\ast}$;
we call the weights $\vpi_{i}=\Lambda_{i}-a_{i}^{\vee}\Lambda_{0}
\in \Fh_{\af}^{\ast}$, $i \in I$, the level-zero fundamental weights.
The (affine) Weyl group $W_{\af}$ of $\Fg_{\af}$ is the subgroup
$\langle r_{j} \mid j \in I_{\af} \rangle \subset \GL(\Fh_{\af}^{\ast})$
generated by the simple reflections $r_{j}$
associated to $\alpha_{j}$ for $j \in I_{\af}$,
with length function $\ell:W_{\af} \rightarrow \BZ_{\ge 0}$ and
unit element $e \in W_{\af}$; recall that $W_{\af} \cong W \ltimes Q^{\vee}$.
We denote by $\rr$ the set of real roots,
and by $\prr \subset \rr$ the set of positive real roots.
Let $x \in W_{\af} \cong W \ltimes Q^{\vee}$, and
write it as $x = w t_{\xi}$ for $w \in W$ and $\xi \in Q^{\vee}$.
Then we define the semi-infinite length $\sell(x)$ of $x$ by
$\sell (x) := \ell (w) + 2 \pair{\xi}{\rho}$.
Now, let $J$ be a subset of $I$. Following <cit.>
(see also <cit.>), we define
\begin{align}
&:= \biggl(\bigoplus_{i \in J} \BZ \alpha_{i} + \BZ\delta \biggr) \cap \prr, \\
\label{eq:stabilizer}
&:= \bigl\{ x \in W_{\af} \mid
\text{$x\beta \in \prr$ for all $\beta \in (\Phi_J)_{\af}^+$} \bigr\}.
\end{align}
(1) The (parabolic) semi-infinite Bruhat graph $\SB$ is
the $\prr$-labeled, directed graph with vertex set $(W^J)_{\af}$
and $\prr$-labeled, directed edges of the following form:
$x \edge{\beta} r_{\beta} x$ for $x \in (W^J)_{\af}$ and $\beta \in \prr$,
where $r_{\beta } x \in (W^J)_{\af}$ and
$\sell (r_{\beta} x) = \sell (x) + 1$.
The semi-infinite Bruhat order is a partial order
$\sile$ on $(W^J)_{\af}$ defined as follows:
for $x,\,y \in (W^J)_{\af}$, we write $x \sile y$
if there exists a directed path from $x$ to $y$ in $\SB$;
also, we write $x \sil y$ if $x \sile y$ and $x \ne y$.
Finally, let $U_{q}(\Fg_{\af})$ denote the quantum affine algebra associated to $\Fg_{\af}$
with integral weight lattice $X_{\af}$, and $E_{j},\,F_{j},\,j \in I_{\af}$,
the Chevalley generators of $U_{q}(\Fg_{\af})$.
Also, let $U_{q}^{+}(\Fg_{\af})$ denote
the subalgebra of $U_{q}(\Fg_{\af})$ generated by $E_{j}$, $j \in I_{\af}$.
§.§ Extremal weight modules and Demazure modules.
For an arbitrary integral weight $\lambda \in X_{\af}$ of $\Fg_{\af}$,
let $V(\lambda)$ denote the extremal weight module of
extremal weight $\lambda$ over $U_{q}(\Fg_{\af})$, which is
an integrable $U_{q}(\Fg_{\af})$-module generated
by a single element $v_{\lambda}$ with
the defining relation that $v_{\lambda}$ is
an “extremal weight vector” of weight $\lambda$
(for details, see <cit.> and <cit.>).
We know from <cit.> that $V(\lambda)$ has
a crystal basis $(\CL(\lambda),\,\CB(\lambda))$ with corresponding global basis
$\bigl\{G(b) \mid b \in \CB(\lambda)\bigr\}$;
we denote by $u_{\lambda}$ the element of $\CB(\lambda)$
such that $G(u_{\lambda})=v_{\lambda} \in V(\lambda)$.
Now, let $\lambda$ be a dominant integral weight for $\Fg$, and set
$J=J_{\lambda}=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0\bigr\}$;
note that $\lambda$ is regarded as an element of $X_{\af}$ by
For each $x \in W_{\af}$, we set
\begin{equation} \label{eq:dem}
\subset V(\lambda),
\end{equation}
where $S^{\norm}_{x}$ denotes the action of the (affine) Weyl group $W_{\af}$ on
the set of extremal weight vectors (see <cit.>).
We know from <cit.> (see also <cit.>) that
there exists a subset $\CB_{x}^{+}(\lambda)$ of
the crystal basis $\CB(\lambda)$ such that
$\bigl\{G(b) \mid b \in \CB_{x}^{+}(\lambda)\bigr\}$ is
the global basis of $V_{x}^{+}(\lambda)$.
§.§ Quotients of Demazure modules and their graded characters.
We fix a dominant integral weight $\lambda$ for $\Fg$.
As in <cit.>, we set
\begin{equation*}
Z_{\lng}^{+}(\lambda) :=
\sum_{
\begin{subarray}{c}
\bc \in \ol{\Par(\lambda)} \\[1.5mm]
\bc \ne (\emptyset)_{i \in I}
\end{subarray}
} U_{q}^{+}(\Fg_{\af}) S_{\bc}S_{\lng}^{\norm}v_{\lambda};
\end{equation*}
notice that our notation differs slightly from that in <cit.>.
Here, $\ol{\Par(\lambda)}$ denotes a certain set of multi-partitions
indexed by $I$ (see <cit.>), and
$S_{\bc} \in U_{q}^{+}(\Fg_{\af})$ denotes the PBW-type basis element
of weight $|\bc|\delta$ corresponding to
the “purely imaginary part” (see <cit.>),
where $|\bc|$ is the sum of all parts
in the multi-partition $\bc$.
Notice that $Z_{\lng}^{+}(\lambda) \subset V_{\lng}^{+}(\lambda)=
U_{q}^{+}(\Fg_{\af})S_{\lng}^{\norm}v_{\lambda}$ since
$S_{\bc} \in U_{q}^{+}(\Fg_{\af})$ for all $\bc \in \ol{\Par(\lambda)}$.
Now, let $w \in W$; in what follows, we may assume that
$w \in W^{J} \subset (W^{J})_{\af}$ since
$V_{w}^{+}(\lambda) = V_{\mcr{w}}^{+}(\lambda)$
for $w \in W$ by <cit.>.
Then, noting that
\subset V_{\mcr{\lng}}^{+}(\lambda) = V_{\lng}^{+}(\lambda)$
by <cit.> since $\mcr{w} \sile \mcr{\lng}$,
we define $U_{w}^{+}(\lambda)$ to be the image of
$V_{w}^{+}(\lambda)$ under the canonical projection
$V_{\lng}^{+}(\lambda) \twoheadrightarrow
We write the weight space decomposition of $U_{w}^{+}(\lambda)$
with respect to $\Fh_{\af}$ as:
\begin{equation*}
U_{w}^{+}(\lambda) = \bigoplus_{\gamma \in Q,\,k \in \BZ}
\end{equation*}
and define the graded character
$\gch U_{w}^{+}(\lambda)$ of $U_{w}^{+}(\lambda)$ to be
\begin{equation*}
\gch U_{w}^{+}(\lambda) := \sum_{\gamma \in Q,\,k \in \BZ}
\dim U_{w}^{+}(\lambda)_{\lambda-\gamma+k\delta}\,x^{\lambda-\gamma}q^{k},
\qquad \text{where $q=x^{\delta}$}.
\end{equation*}
The following is the main result of this section.
Keep the notation and setting above. We have
\begin{equation*}
\gch U_{w}^{+}(\lambda) = \Mac{w\lambda}.
\end{equation*}
§.§ Semi-infinite Lakshmibai-Seshadri paths.
We keep the notation and setting of <ref>;
recall that $\lambda = \sum_{i \in I} m_{i} \vpi_{i}$ is
a dominant integral weight for $\Fg$, and
$J=J_{\lambda}= \bigl\{ i \in I \mid
\pair{\alpha_i^{\vee}}{\lambda}=0 \bigr\} \subset I$.
For a rational number $0 < \tau < 1$,
define $\SBb{\tau}$ to be the subgraph of $\SB$
with the same vertex set but having only the edges of the form:
$x \edge{\beta} y$ with
$\tau \pair{\beta^{\vee}}{x\lambda} \in \BZ$.
A semi-infinite Lakshmibai-Seshadri path (SiLS path for short) of
shape $\lambda $ is, by definition, a pair $(\bm{y}\,;\,\bm{\tau})$ of
a (strictly) decreasing sequence $\bm{y} : y_1 \sig \cdots \sig y_s$
of elements in $(W^J)_{\af}$ and an increasing sequence
$\bm{\tau} : 0 = \tau_0 < \tau_1 < \cdots < \tau_s =1$
of rational numbers satisfying the condition that
there exists a directed path from $y_{u+1}$ to $y_u$ in
$\SBb{\tau_u}$ for each $u = 1,\,2,\,\dots,\,s-1$.
We denote by $\sLS$ the set of all SiLS paths of shape $\lambda$.
In <cit.>, we defined root operators
$e_{j}$ and $f_{j}$, $j \in I_{\af}$, on $\sLS$,
and proved that the set $\sLS$, equipped with these root operators,
is a crystal with weights in $X_{\af}$.
Keep the notation and setting above.
There exists an isomorphism of crystals
between the crystal basis $\CB(\lambda)$ of the extremal weight module
$V(\lambda)$ of extremal weight $\lambda$ and the crystal $\sLS$ of
SiLS paths of shape $\lambda$.
For $\pi=(y_1,\,\dots,\,y_s\,;\,\tau_{0},\,\tau_{1},\,\dots,\,\tau_{s}) \in \sLS$,
we define the piecewise-linear, continuous map
$\ol{\pi}: [0,1] \rightarrow \BR \otimes_{\BZ} X_{\af}$ by
\begin{equation} \label{eq:path2}
\ol{\pi}(t)=\sum_{p=1}^{u-1}
(t-\tau_{u-1})y_{u}\lambda \quad
\text{for $\tau_{u-1} \le t \le \tau_{u}$, $1 \le u \le s$}.
\end{equation}
Then we know from <cit.> that
$\ol{\pi}$ is a Lakshmibai-Seshadri (LS for short) path of shape $\lambda$;
for the definition of LS paths of shape $\lambda$,
see <cit.> and <cit.>.
We denote by $\BB(\lambda)$ the set of all LS paths of shape $\lambda$.
In fact, the map $\ol{\phantom{\pi}} :
\sLS \rightarrow \BB(\lambda)$, $\pi \mapsto \ol{\pi}$, is
a surjective, strict crystal morphism.
Define a surjective map
$\cl : (W^{J})_{\af} \twoheadrightarrow W^{J}$ by
\begin{equation*}
\cl (x) := w \quad \text{if $x = wzt_{\xi}$
for $w \in W^{J}$, $z \in W_{J}$, and $\xi \in Q^{\vee}$.}
\end{equation*}
Then, for $\pi = (y_{1},\,\dots,\,y_{s}\,;\,\tau_{0},\,\tau_{1},\,\dots,\,\tau_{s})
\in \sLS$, we set (see <cit.>)
\begin{equation*}
\cl(\pi): =
\end{equation*}
for each $1 \le p < q \le s$ such that $\cl(y_{p})= \cdots = \cl(y_{q})$,
we drop $\cl(y_{p}),\,\dots,\,\cl(y_{q-1})$ and $\tau_{p},\,\dots,\,\tau_{q-1}$
from this expression of $\cl(\pi)$.
Let $\BB_{0}^{\si}(\lambda)$ denote the connected component of $\sLS$
containing $\pi_{e}:=(e\,;\,0,\,1)$. We know from
<cit.> that for each $\eta \in \QLS(\lambda)=\BB(\lambda)_{\cl}$
(see Remark <ref>), there exists a unique element
$\pi_{\eta}=(y_1,\,\dots,\,y_s\,;\,\bm{\tau}) \in \BB_{0}^{\si}(\lambda)$ such that
$\iota(\pi_{\eta}):=y_1 \in W^{J}$ and $\cl(\pi_{\eta})=\eta$.
We claim that
\begin{equation} \label{eq:deg}
\wt(\pi_{\eta}) = \lambda-\beta - \Deg(\eta) \delta, \qquad
\text{where $\beta \in Q^{+}:=\sum_{j \in I} \BZ_{\ge 0} \alpha_{j}$}.
\end{equation}
Indeed, since $\wt(\pi_{\eta}) = \wt(\ol{\pi_{\eta}})$ by their definitions,
it suffices to show that $\ol{\pi_{\eta}} \in \BB(\lambda)$ satisfies
the following conditions (see <cit.> and
(a) $\cl(\ol{\pi_{\eta}}(t)) = \eta(t)$ for all $t \in [0,1]$,
where $\cl:\BR \otimes_{\BZ} X_{\af} \twoheadrightarrow
(\BR \otimes_{\BZ} X_{\af})/\BR\delta$ denotes the canonical projection;
(b) $\ol{\pi_{\eta}}$ is contained
in the connected component $\BB_{0}(\lambda)$ of
$\BB(\lambda)$ containing $\pi_{\lambda}$, where
$\pi_{\lambda}(t):=t\lambda$ for $t \in [0,1]$;
(c) $\iota(\ol{\pi_{\eta}})=y_{1}\lambda \in \lambda-Q^{+}$.
If $x \in W_{\af}$ is of the form
$x = wzt_{\xi}$ with $w \in W^{J}$, $z \in W_{J}$, and $\xi \in Q^{\vee}$,
then $x\lambda = w\lambda - \pair{\xi}{\lambda}\delta$
(recall that $\pair{c}{\lambda}=0$), and hence
$x\lambda \equiv w\lambda$ modulo $\BR\delta$.
Therefore, assertion (a) is obvious from the definitions of
$\ol{\phantom{\pi}} : \sLS \rightarrow \BB(\lambda)$ and the maps $\cl$.
Also, since $\pi_{\eta} \in \BB^{\si}_{0}(\lambda)$,
there exists a monomial $Y$
in root operators such that $\pi_{\eta}=Y\pi_{e}$.
Because $\ol{\phantom{\pi}} : \sLS \rightarrow \BB(\lambda)$
commutes with the action of root operators,
we have $\ol{\pi_{\eta}}=\ol{Y\pi_{e}}=Y\ol{\pi_{e}}=Y\pi_{\lambda}$.
Hence we obtain $\ol{\pi_{\eta}} \in \BB_{0}(\lambda)$.
Finally, since $\iota(\pi_{\eta})=y_{1} \in W^{J}$ and
$\lambda$ is a dominant integral weight for $\Fg$,
it follows that $\iota(\ol{\pi_{\eta}}) = y_{1}\lambda$
is contained in $\lambda-Q^{+}$. This proves (<ref>).
§.§ Proof of Theorem <ref>.
We know from <cit.> that
there exists a subset $\CB(Z_{\lng}^{+}(\lambda))$ of $\CB(\lambda)$
such that $\bigl\{G(b) \mid b \in \CB(Z_{\lng}^{+}(\lambda))\bigr\}$ is
the global basis of $Z_{\lng}^{+}(\lambda)$. Also, recall that
$\bigl\{G(b) \mid b \in \CB_{w}^{+}(\lambda)\bigr\}$ is
the global basis of $V_{w}^{+}(\lambda)$. Therefore,
\begin{equation*}
\bigl\{G(b) \ \mathrm{mod} \ Z_{\lng}^{+}(\lambda) \mid
b \in \CB(U_{w}^{+}(\lambda)):=
\CB_{w}^{+}(\lambda) \setminus \CB(Z_{\lng}^{+}(\lambda)) \bigr\}
\end{equation*}
is the global basis of $U_{w}^{+}(\lambda)$,
which is the image of $V_{w}^{+}(\lambda)$ under
the canonical projection
$V_{\lng}^{+}(\lambda) \twoheadrightarrow
We know from <cit.> that
there exists an isomorphism $\Psi_{\lambda}^{\vee}:
\CB(\lambda) \stackrel{\sim}{\rightarrow} \BB^{\si}(\lambda)$
of crystals, which maps $\CB(U_{w}^{+}(\lambda)) \subset \CB(\lambda)$ to
\begin{equation*}
\bigl\{\pi_{\eta} \mid
\text{\rm $\eta \in \QLS(\lambda)$ such that $w \ge \iota(\pi_{\eta})$} \bigr\}
\subset \sLS.
\end{equation*}
Since $\iota(\pi_{\eta}) \in W^{J}$, we see that $\iota(\pi_{\eta}) = \iota(\eta)$.
Therefore, the subset above is identical to the set
$\bigl\{\pi_{\eta} \mid \eta \in \QLS_{w}(\lambda) \bigr\}$.
Hence we compute:
\begin{align*}
\gch U_{w}^{+}(\lambda)
& = \left(\sum_{b \in \CB(U_{w}^{+}(\lambda))} x^{\wt (b)}\right)\Biggm|_{x^{\delta}=q}
= \left(\sum_{\eta \in \QLS_{w}(\lambda)} x^{\wt(\pi_{\eta})}\right)\Biggm|_{x^{\delta}=q} \\[3mm]
& = \left(\sum_{\eta \in \QLS_{w}(\lambda)} x^{\wt(\eta)-\Deg(\eta)\delta}\right)\Biggm|_{x^{\delta}=q}
\qquad \text{by \eqref{eq:deg}} \\[3mm]
& = \sum_{\eta \in \QLS_{w}(\lambda)} q^{-\Deg(\eta)} x^{\wt(\eta)}
= \Mac{w\lambda} \qquad \text{by Theorem~\ref{thm:Mac0}}.
\end{align*}
This completes the proof of Theorem <ref>.
§ APPENDIX.
§ RECURSIVE FORMULAS IN TERMS OF DEMAZURE OPERATORS.
We use the notation of <ref> and <ref>.
Fix a dominant integral weight $\lambda \in X$, and
set $J=J^{\lambda}=\bigl\{i \in I \mid \pair{\alpha_{i}^{\vee}}{\lambda}=0\bigr\}$.
For each $i \in I$, we define a $\BZ[q]$-linear operator $D_{i}$
(called a Demazure operator) on $\bigl(\BZ[q]\bigr)[e^{\xi}\,;\,\xi \in X]$ by:
\begin{align}
D_{i}(e^{\xi}) & :=
\frac{ e^{\xi+\rho} - e^{r_{i}(\xi+\rho)} }
{ 1-e^{-\alpha_{i}} } e^{-\rho} \nonumber \\[3mm]
& =
\begin{cases}
e^{\xi}\bigl(1+e^{-\alpha_{i}}+ \cdots +e^{-n\alpha_{i}}\bigr)
& \text{if $n=\pair{\alpha_{i}^{\vee}}{\xi} \ge 0$}, \\[1.5mm]
0 & \text{if $n=\pair{\alpha_{i}^{\vee}}{\xi}=-1$}, \\[1.5mm]
-e^{\xi}\bigl(e^{\alpha_{i}}+ \cdots +e^{(-n-1)\alpha_{i}}\bigr)
& \text{if $n=\pair{\alpha_{i}^{\vee}}{\xi} \le -2$}.
\end{cases} \label{eq:Demazure}
\end{align}
In this appendix, we give a recursive formula
for $\gch \QLS_{w}(\lambda)$ (Proposition <ref>)
and one for $\Mac{w\lambda}$ (Proposition <ref>),
both of which are in terms of Demazure operators.
§.§ Recursive formula for $\gch \QLS_{w}(\lambda)$.
Let $w \in W^{J}$ and $i \in I$ be such that $w > r_{i}w$;
note that $r_{i}w \in W^{J}$ by <cit.>.
Then we have
\begin{equation*}
\gch \QLS_{w}(\lambda) = D_{i} \gch \QLS_{r_{i}w}(\lambda).
\end{equation*}
Let $U_{q}'(\Fg_{\af})$ denote the quantum affine algebra
without the degree operator associated to $\Fg_{\af}$.
We know that the set $\QLS(\lambda)=\BB(\lambda)_{\cl}$ (see Remark <ref>),
equipped with root operators $e_{j}$, $f_{j}$, $j \in I_{\af}$,
is a $U_{q}'(\Fg_{\af})$-crystal; for the definition of root operators,
see <cit.> and <cit.>.
We prove Proposition <ref>
by using this $U_{q}'(\Fg_{\af})$-crystal structure on
(cf. <cit.>).
Let $w \in W^{J}$ and $i \in I$ be such that $w > r_{i}w$.
We have
\begin{equation*}
\QLS_{w}(\lambda) =
\bigcup_{p \ge 0} f_{i}^{p}\QLS_{r_{i}w}(\lambda) \setminus \{\bzero\}.
\end{equation*}
First we prove the inclusion $\subset$.
Let $\eta \in \QLS_{w}(\lambda)$, and set $\eta':=e_{i}^{\max}\eta$.
It suffices to show that $\eta' \in \QLS_{r_{i}w}(\lambda)$;
for simplicity of notation, we set $x:=\iota(\eta) \in W^{J}$.
If $\iota(\eta')=\iota(\eta)=x$, then it follows from
the definition of the root operator $e_{i}$ that
\begin{equation*}
\pair{\alpha_{i}^{\vee}}{x\lambda} =
\pair{\alpha_{i}^{\vee}}{\iota(\eta)\lambda} =
\pair{\alpha_{i}^{\vee}}{\iota(\eta')\lambda} \ge 0,
\end{equation*}
since $e_{i}\eta' = \bzero$.
Because $\eta \in \QLS_{w}(\lambda)$ by the assumption,
we have $x=\iota(\eta) \le w$.
Also, from the assumption that $w > r_{i}w$ and $w \in W^{J}$,
it follows that $r_{i}w \in W^{J}$ and $\pair{\alpha_{i}^{\vee}}{w\lambda} < 0$
by <cit.>.
Therefore, we deduce from <cit.>
applied to $x \le w$ that
$x \le r_{i}w$, and hence $\iota(\eta') = \iota(\eta) = x \le r_{i}w$.
Thus we obtain $\eta' \in \QLS_{r_{i}w}(\lambda)$,
as desired.
If $\iota(\eta') \ne \iota(\eta)$, then
it follows from the definition of
the root operator $e_{i}$ that
$\iota(\eta')=r_{i}\iota(\eta)=r_{i}x$ and
\begin{equation*}
\pair{\alpha_{i}^{\vee}}{x\lambda}
= - \pair{\alpha_{i}^{\vee}}{r_{i}x\lambda}
= - \pair{\alpha_{i}^{\vee}}{r_{i}\iota(\eta)\lambda}
= - \pair{\alpha_{i}^{\vee}}{r_{i}\iota(\eta')\lambda} < 0.
\end{equation*}
Since $x=\iota(\eta) \le w$ by the assumption
and $\pair{\alpha_{i}^{\vee}}{w\lambda} < 0$ as seen above,
we deduce from <cit.> applied to $x \le w$ that
$r_{i}x \le r_{i}w$, and hence $\iota(\eta') = r_{i}x \le r_{i}w$.
Therefore, we obtain $\eta' \in \QLS_{r_{i}w}(\lambda)$,
as desired. This proves the inclusion $\subset$.
Next we prove the opposite inclusion $\supset$.
Let $\eta' \in \QLS_{r_{i}w}(\lambda)$, and
assume that $\eta:=f_{i}^{p}\eta' \ne \bzero$
for some $p \ge 0$. If $\iota(\eta)=\iota(\eta')$,
\begin{equation*}
\iota(\eta)=\iota(\eta') \le r_{i}w < w,
\end{equation*}
and hence $\eta \in \QLS_{w}(\lambda)$.
Assume now that $\iota(\eta) \ne r_{i}\iota(\eta')$, and hence
for simplicity of notation,
we set $x':=\iota(\eta') \in W^{J}$.
Then we see from the definition of the root operator $f_{i}$ that
$\pair{\alpha_{i}^{\vee}}{x'\lambda} = \pair{\alpha_{i}^{\vee}}{\iota(\eta')\lambda} > 0$.
Also, we have $\pair{\alpha_{i}^{\vee}}{r_{i}w\lambda} =
- \pair{\alpha_{i}^{\vee}}{w\lambda} > 0$ as seen above and
$x' = \iota(\eta') \le r_{i}w$ by the assumption.
It follows from <cit.> applied to $x' \le r_{i}w$
that $r_{i}x' \le w$, and hence $\iota(\eta) = r_{i}x' \le w$.
Therefore, we obtain $\eta \in \QLS_{w}(\lambda)$, as desired.
This proves the opposite inclusion $\supset$.
This completes the proof of the lemma.
For $i \in I$ and $\eta \in \QLS(\lambda)$,
let $S_{i}(\eta)$ denote the $\alpha_{i}$-string through $\eta$, that is,
\begin{equation*}
S_{i}(\eta) : =
\bigl\{ e_{i}^{p}\eta,\,f_{i}^{q}\eta \mid p,\,q \ge 0 \bigr\}
\setminus \{\bzero\}.
\end{equation*}
Let $\eta \in \QLS(\lambda)$, and $i \in I$.
For $z \in W^{J}$,
\begin{equation*}
\QLS_{z}(\lambda) \cap S_{i}(\eta)=
\emptyset,\
\bigl\{e_{i}^{\max}\eta\bigr\},\ \text{\rm or}\
\end{equation*}
For simplicity of notation, we set $\eta':=e_{i}^{\max}\eta$.
We will prove that if
$\QLS_{z}(\lambda) \cap S_{i}(\eta)$ is neither $\emptyset$ nor
$\bigl\{e_{i}^{\max}\eta\bigr\}$, then
$\QLS_{z}(\lambda) \cap S_{i}(\eta) = S_{i}(\eta)$,
or equivalently, $S_{i}(\eta) \subset \QLS_{z}(\lambda)$.
By our assumption,
$\QLS_{z}(\lambda) \cap S_{i}(\eta)$ contains
an element $\eta''$ that is not $\eta'$.
We can write the element $\eta''$ as $\eta''= f_{i}^{p}\eta'$
for some $p \ge 1$.
Here, from the definition of the root operator $f_{i}$,
we can deduce that
\begin{equation*}
\iota(f_{i}\eta') = \iota(f_{i}^{2}\eta') = \cdots =
\iota(f_{i}^{p}\eta') = \cdots = \iota(f_{i}^{\max}\eta').
\end{equation*}
Since $\iota(f_{i}^{p}\eta') = \iota(\eta'') \le z$ by the assumption that
$\eta'' \in \QLS_{z}(\lambda)$,
we see that the elements $f_{i}\eta',\,f_{i}^{2}\eta',\,\dots,\,
f_{i}^{\max}\eta'$ are all contained in $\QLS_{z}(\lambda)$. Namely,
\begin{equation} \label{eq:lemdem1}
S_{i}(\eta) \setminus \bigl\{\eta'\bigr\} \subset
\QLS_{z}(\lambda).
\end{equation}
Hence it remains to show that $\eta' \in \QLS_{z}(\lambda)$.
If $\iota(\eta')=\iota(\eta'')$, then we have
$\iota(\eta') \le z$ since $\iota(\eta'') \le z$ by
the assumption that $\eta'' \in \QLS_{z}(\lambda)$.
This implies that $\eta' \in \QLS_{z}(\lambda)$.
Assume now that $\iota(\eta'') \ne r_{i}\iota(\eta')$, and hence
that $\iota(\eta'')=r_{i}\iota(\eta')$.
Then, by the definition of the root operator $f_{i}$,
we see that $\pair{\alpha_{i}^{\vee}}{\iota(\eta')\lambda} > 0$.
Therefore, we deduce that $\pair{\alpha_{i}^{\vee}}{\iota(\eta'')\lambda} =
\pair{\alpha_{i}^{\vee}}{r_{i}\iota(\eta')\lambda} < 0$, and hence
that $\iota(\eta')=r_{i}\iota(\eta'') < \iota(\eta'') \le z$.
Thus we obtain $\eta' \in \QLS_{z}(\lambda)$.
Combining this with (<ref>), we conclude that
$S_{i}(\eta) \subset \QLS_{z}(\lambda)$, as desired.
This completes the proof of the lemma.
First, we show that for each $\eta \in \QLS(\lambda)$,
\begin{equation} \label{eq:string}
\QLS_{w}(\lambda) \cap S_{i}(\eta)=
\emptyset \text{ or } S_{i}(\eta).
\end{equation}
Now, assume that
$\QLS_{w}(\lambda) \cap S_{i}(\eta) \ne \emptyset$.
Then, we see from Lemma <ref> that
$\QLS_{w}(\lambda) \cap S_{i}(\eta) =
\bigl\{e_{i}^{\max}\eta\bigr\}$ or $S_{i}(\eta)$;
in both cases, we have $e_{i}^{\max}\eta \in \QLS_{w}(\lambda)$.
Here we recall from the proof of Lemma <ref>
that if $\psi \in \QLS_{w}(\lambda)$,
then $e_{i}^{\max}\psi \in \QLS_{r_{i}w}(\lambda)$.
Hence it follows that
is contained in $\QLS_{r_{i}w}(\lambda)$.
Therefore, we see from Lemma <ref> that
$f_{i}^{p}e_{i}^{\max}\eta \in \QLS_{w}(\lambda)$
for all $p \ge 0$ unless
$f_{i}^{p}e_{i}^{\max}\eta = \bzero$.
From this, we conclude that $S_{i}(\eta)
\subset \QLS_{w}(\lambda)$, as desired.
From (<ref>), we deduce that
$\QLS_{w}(\lambda)$ decomposes into a disjoint union of $\alpha_{i}$-strings:
\begin{equation*}
\QLS_{w}(\lambda) = S^{(1)} \sqcup S^{(2)} \sqcup \cdots \sqcup S^{(n)},
\quad
\text{where $S^{(m)}$ is an $\alpha_{i}$-string for each $1 \le m \le n$}.
\end{equation*}
Since $i \in I$, the degree function $\Deg$ is
constant on $S^{(m)}$ for each $1 \le m \le n$ (see <cit.>);
we set $d_{m}:=\Deg|_{S^{(m)}}$
for $1 \le m \le n$. Then we have
\begin{equation*}
\gch \QLS_{w}(\lambda) =
\sum_{m=1}^{n} q^{-d_{m}} \sum_{\eta \in S^{(m)}} e^{\wt(\eta)}.
\end{equation*}
Next, let us consider the intersection
$\QLS_{r_{i}w}(\lambda) \cap S^{(m)}$
for each $1 \le m \le n$. Recall that
if $\psi \in \QLS_{w}(\lambda)$, then
$e_{i}^{\max}\psi \in \QLS_{r_{i}w}(\lambda)$.
Since $S^{(m)} \subset \QLS_{w}(\lambda)$,
it follows from the above that
$\QLS_{r_{i}w}(\lambda)$ contains
a unique element $\eta_{m} \in S^{(m)}$
such that $e_{i}\eta_{m} = \bzero$;
in particular, $\QLS_{r_{i}w}(\lambda) \cap S^{(m)} \ne \emptyset$.
Therefore, from Lemma <ref>, we deduce that
\begin{equation*}
\QLS_{r_{i}w}(\lambda) \cap S^{(m)} =
\bigl\{\eta_{m}\bigr\} \text{ or } S^{(m)}
\quad \text{for each $1 \le m \le n$};
\end{equation*}
here we assume that
\begin{equation*}
\QLS_{r_{i}w}(\lambda) \cap S^{(m)} =
\begin{cases}
\bigl\{\eta_{m}\bigr\} & \text{for $1 \le m \le p$}, \\[1.5mm]
S^{(m)} & \text{for $p+1 \le m \le n$},
\end{cases}
\end{equation*}
for some $0 \le p \le n$ for simplicity of notation.
Then, we have
\begin{equation*}
\gch \QLS_{r_{i}w}(\lambda) =
\sum_{m=1}^{p} q^{-d_{m}} e^{\wt(\eta_{m})}+
\sum_{m=p+1}^{n} q^{-d_{m}} \sum_{\eta \in S^{(m)}} e^{\wt(\eta)}.
\end{equation*}
Combining all the above, we compute:
\begin{align*}
D_{i} \gch \QLS_{r_{i}w}(\lambda) & =
\sum_{m=1}^{p} q^{-d_{m}} D_{i} e^{\wt(\eta_{m})} +
\sum_{m=p+1}^{n} q^{-d_{m}} D_{i}
\Biggl(
\underbrace{\sum_{\eta \in S^{(m)}} e^{\wt(\eta)}}_{%
\begin{subarray}{c}
=D_{i}e^{\wt(\eta_{m})} \\[1mm]
\text{by \eqref{eq:Demazure}}
\end{subarray}
\Biggr) \\[1mm]
& =
\sum_{m=1}^{p} q^{-d_{m}} D_{i} e^{\wt(\eta_{m})} +
\sum_{m=p+1}^{n} q^{-d_{m}}
\underbrace{D_{i}D_{i}e^{\wt(\eta_{m})}}_{ =D_{i}e^{\wt(\eta_{m})} } \\[3mm]
& = \sum_{m=1}^{n} q^{-d_{m}} D_{i} e^{\wt(\eta_{m})}
= \sum_{m=1}^{n} q^{-d_{m}} \sum_{\eta \in S^{(m)}} e^{\wt(\eta)}
\quad \text{by \eqref{eq:Demazure}} \\[3mm]
& = \gch \QLS_{w}(\lambda).
\end{align*}
This completes the proof of the proposition.
§.§ Recursive formula for $\Mac{w\lambda}$.
In view of Theorem <ref>,
Proposition <ref> is equivalent to
the following proposition.
Let $w \in W^{J}$ and $i \in I$ be such that $w > r_{i}w$;
note that $r_{i}w \in W^{J}$ by <cit.>.
Then we have
\begin{equation*}
\Mac{w\lambda} = D_{i} \Mac{r_{i}w\lambda}.
\end{equation*}
We can also show this proposition
by using the polynomial representation of
the double affine Hecke algebra as follows.
Note that $r_{i}w \in W^{J}$ and $\pair{\alpha_{i}^{\vee}}{r_{i}w\lambda} > 0$
by <cit.>. We set $\mu:=r_{i}w\lambda$.
Since $\pair{\alpha_{i}^{\vee}}{\mu} =
\pair{\alpha_{i}^{\vee}}{r_{i}w\lambda} > 0$ as seen above,
it follows from <cit.> (in the notation thereof) that
\begin{equation}
\left(t^{2}r_{i}+(t^{2}-1)\frac{1-r_{i}}{ 1-e^{\alpha_{i}} }-
(t^{2}-1)\frac{1}{1-Y^{-\alpha_{i}}}\right) \cdot \Mact{\mu}
= \Mact{r_{i}\mu}. \label{eq:demM1}
\end{equation}
Also, we know from <cit.>
(in the notation thereof) that
\begin{equation*}
Y^{-\alpha_{i}}\Mact{\mu} =
q^{ \pair{\alpha_{i}^{\vee}}{\mu} } t^{ -2 \pair{v(\mu)\alpha_{i}^{\vee}}{\rho} }
\Mact{\mu}.
\end{equation*}
Since $\pair{\alpha_{i}^{\vee}}{\mu} > 0$ as seen above, it follows that
$\pair{v(\mu)\alpha_{i}^{\vee}}{v(\mu)\mu} = \pair{\alpha_{i}^{\vee}}{\mu} > 0$.
Since $v(\mu)\mu$ is antidominant by the definition of $v(\mu)$,
we see that $v(\mu)\alpha_{i}^{\vee}$ is a negative coroot, and hence
$-2 \pair{v(\mu)\alpha_{i}^{\vee}}{\rho} > 0$.
Therefore, by taking the limit $t \to 0$, we deduce from
(<ref>) that
\begin{equation*}
\left( \frac{r_{i}-1}{1-e^{\alpha_{i}}} + 1 \right) \Mac{\mu} = \Mac{r_{i}\mu}.
\end{equation*}
We see by direct computation that
\begin{equation*}
\frac{r_{i}-1}{1-e^{\alpha_{i}}} + 1 = D_{i}.
\end{equation*}
Thus, we obtain
$D_{i} \Mac{\mu} = \Mac{r_{i}\mu}$, and hence
$\Mac{w\lambda} = D_{i} \Mac{r_{i}w\lambda}$, as desired.
This proves Proposition <ref>.
If we employ the special case $w=e$ of Theorem <ref> as
the start of the induction and use Proposition <ref> and
Proposition <ref> (proved as above) in the induction step,
then we can give an inductive proof of Theorem <ref>.
J. Beck and H. Nakajima,
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1511.00387
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We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in $\Real^4$ with entropy less than or equal to that of $\mathbb{S}^2\times \Real$, the round cylinder in $\Real^4$, are diffeomorphic to $\mathbb{S}^3$.
§ INTRODUCTION
If $\Sigma$ is a hypersurface, that is, a smooth properly embedded codimension-one submanifold of $\Real^{n+1}$, then the
Gaussian surface area of $\Sigma$ is
\begin{equation}
F[\Sigma]=\int_{\Sigma}\Phi\, d\mathcal{H}^{n}=(4\pi)^{-\frac{n}{2}}\int_{\Sigma}e^{-\frac{|\xX|^2}{4}} d\mathcal{H}^n,
\end{equation}
where $\mathcal{H}^n$ is $n$-dimensional Hausdorff measure. Following Colding-Minicozzi <cit.>, define the entropy of $\Sigma$ to be
\begin{equation*}
\lambda[\Sigma]=\sup_{(\yY,\rho)\in\Real^{n+1}\times\Real^+}F[\rho\Sigma+\yY].
\end{equation*}
That is, the entropy of $\Sigma$ is the supremum of the Gaussian surface area over all translations and dilations of $\Sigma$. Observe that the entropy of a hyperplane is one. In <cit.>, we show that, for $2\leq n\leq 6$, the entropy of a closed (i.e. compact and without boundary) hypersurface in $\Real^{n+1}$ is uniquely (modulo translations and dilations) minimized by $\mathbb{S}^n$, the unit sphere centered at the origin. This verifies a conjecture of Colding-Ilmanen-Minicozzi-White <cit.> (cf. <cit.>). We further show, in <cit.>, that surfaces in $\Real^3$ of small entropy are topologically rigid. That is, if $\Sigma$ is a closed surface in $\Real^3$ and $\lambda[\Sigma]\leq\lambda[\mathbb{S}^1\times\Real]$, then $\Sigma$ is diffeomorphic to $\mathbb{S}^2$.
In this article, we use a weak mean curvature flow (see <cit.> and <cit.>) to obtain new topological rigidity for closed hypersurfaces in $\Real^4$ of small entropy. This generalizes a result of Colding-Ilmanen-Minicozzi-White <cit.> for closed self-shrinkers to arbitrary closed hypersurfaces and contrasts with the methods of both <cit.> and <cit.>, which both use only the classical mean curvature flow.
If $\Sigma\subset\Real^{4}$ is a closed hypersurface with $\lambda[\Sigma]\leq\lambda[\mathbb{S}^2\times\Real]$, then $\Sigma$ is diffeomorphic to $\mathbb{S}^3$.
One of the key ingredients in the proof of Theorem <ref> is a refinement of <cit.> about the topology of asymptotically conical self-shrinkers of small entropy. Recall, a hypersurface $\Sigma$ is said to be asymptotically conical if it is smoothly asymptotic to a regular cone; i.e., $\lim_{\rho\to 0} \rho\Sigma= \mathcal{C} (\Sigma)$ in $C^{\infty}_{loc}(\Real^{n+1}\setminus\set{\OO})$ for $\mathcal{C} (\Sigma)$ a regular cone. A self-shrinker, $\Sigma$, is a hypersurface that satisfies
\begin{equation}\label{SSEqn}
\mathbf{H}_\Sigma+\frac{\xX^\perp}{2}=\mathbf{0},
\end{equation}
where $\mathbf{H}_\Sigma=-H_{\Sigma} \nN_\Sigma=\Delta_\Sigma \xX$ is the mean curvature vector of $\Sigma$ and $\xX^\perp$ is the normal component of the position vector. Let us denote the set of self-shrinkers in $\Real^{n+1}$ by $\mathcal{S}_n$ and the set of asymptotically conical self-shrinkers by $\mathcal{ACS}_n$. Self-shrinkers generate solutions to the mean curvature flow that move self-similarly by scaling. That is, if $\Sigma\in\mathcal{S}_n$, then
\begin{equation*}
\set{\Sigma_t}_{t\in(-\infty,0)}=\set{\sqrt{-t}\, \Sigma}_{t\in(-\infty,0)}
\end{equation*}
moves by mean curvature. Important examples are the maximally symmetric self-shrinking cylinders with $k$-dimensional spine,
\begin{equation*}
\mathbb{S}^{n-k}_{*}\times\Real^k=\set{(\xX,\yY)\in\Real^{n-k+1}\times\Real^k=\Real^{n+1}: |\xX|^2=2(n-k)},
\end{equation*}
where $0\leq k\leq n$. As $\mathbb{S}^{n-k}_{*}\times\Real^k$ are self-shrinkers, their Gaussian surface area and entropy agree (cf. <cit.>). That is,
\begin{equation*}
\lambda_n=\lambda [\mathbb{S}^n]=F[\mathbb{S}_*^n]=F[\mathbb{S}_*^n\times\Real^l]=\lambda[\mathbb{S}^n\times\Real^l].
\end{equation*}
Hence, a computation of Stone <cit.>, gives that
\begin{equation*}
\end{equation*}
Let $\Sigma\in \mathcal{ACS}_n$ for $n\geq 2$. If $\lambda[\Sigma]\leq\lambda_{n-1}$, then $\Sigma$ is contractible and $\mathcal{L} (\Sigma)$, the link of the asymptotic cone $\mathcal{C} (\Sigma)$, is a homology $(n-1)$-sphere.
We always consider homology with integer coefficients.
For $n=3$, the classification of surfaces and Alexander's theorem <cit.> gives
Let $\Sigma\in \mathcal{ACS}_3$. If $\lambda[\Sigma]\leq\lambda_{2}$, then $\Sigma$ is diffeomorphic to $\Real^3$.
To prove Theorem <ref> we first establish a topological decomposition, i.e., Theorem <ref>, constructed from the weak mean curvature flow associated to $\Sigma$. Together with Corollary <ref> this allows one to perform a surgery procedure which immediately gives the result. Both these steps require $n=3$. For $n\geq 4$, one can use Theorem <ref> and this surgery procedure to show a (strictly weaker) extension of Theorem <ref> valid in any dimension where the two hypotheses below are satisfied. These hypotheses ensure the existence of topological decomposition. Specifically, they ensure that if the entropy of an initial hypersurface is small enough, then tangent flows at all singularities are modeled by self-shrinkers that are either closed or asymptotically conical.
In order to state these hypotheses, first let $\mathcal{S}_n^*$ denote the set of non-flat elements of $\mathcal{S}_n$ and, for any $\Lambda>0$, let
\mathcal{S}_n(\Lambda)=\set{\Sigma\in \mathcal{S}_n: \lambda[\Sigma]<\Lambda} \mbox{ and } \mathcal{S}_n^*(\Lambda)=\mathcal{S}^*_n \cap \mathcal{S}_n(\Lambda).
Next, let $\mathcal{RMC}_n$ denote the space of regular minimal cones in $\Real^{n+1}$, that is $\mathcal{C}\in \mathcal{RMC}_n$ if and only if it is a proper subset of $\Real^{n+1}$ and $\mathcal{C}\backslash\set{\OO}$ is a hypersurface in $\Real^{n+1}\backslash\set{\OO}$ that is invariant under dilation about $\OO$ and with vanishing mean curvature. Let $\mathcal{RMC}_n^*$ denote the set of non-flat elements of $\mathcal{RMC}_n$ – i.e., cones whose second fundamental forms do not identically vanish. For any $\Lambda>0$, let
\mathcal{RMC}_n(\Lambda)=\set{\mathcal{C}\in \mathcal{RMC}_n: \lambda[\mathcal{C}]< \Lambda} \mbox{ and } \mathcal{RMC}_n^*(\Lambda)=\mathcal{RMC}^*_n \cap \mathcal{RMC}_n(\Lambda).
Let us now fix a dimension $n\geq 3$ and a value $\Lambda>1$.
The first hypothesis is
\begin{equation} \label{Assump1}
\mbox{For all $3\leq k\leq n$, }\mathcal{RMC}_{k}^*(\Lambda)=\emptyset \tag{$\star_{n,\Lambda}$}.
\end{equation}
Observe that all regular minimal cones in $\mathbb{R}^2$ consist of unions of rays and so $\mathcal{RMC}^*_1=\emptyset$. Likewise, as great circles are the only geodesics in $\mathbb{S}^2$, $\mathcal{RMC}_2^*=\emptyset$. The second hypothesis is
\begin{equation} \label{Assump2}
\mathcal{S}_{n-1}^*(\Lambda) =\emptyset. \tag{$\sstar_{n,\Lambda}$}
\end{equation}
Obviously this holds only if $\Lambda\leq\lambda_{n-1}$.
We then show the following conditional result:
Fix $n\geq 4$ and $\Lambda\in (\lambda_{n}, \lambda_{n-1}]$. If (<ref>) and (<ref>) both hold and $\Sigma$ is a closed hypersurface in $\Real^{n+1}$ with $\lambda[\Sigma]\leq\Lambda$, then $\Sigma$ is a homology $n$-sphere.
If (<ref>) and (<ref>) hold for $\Lambda\leq\lambda_{n}$, then it follows from Huisken's monotonicity formula and the results of <cit.> and <cit.> that there does not exist a closed hypersurface $\Sigma$ so that $\lambda[\Sigma]\leq\Lambda$ unless $\Lambda=\lambda_n$ and $\Sigma$ is a round sphere. Thus, we require $\Lambda>\lambda_n$ in order to make Theorem <ref> non-trivial.
For general $n$ and $\Lambda\in (\lambda_n, \lambda_{n-1}]$, neither the validity of (<ref>) nor that of (<ref>) is known. However, both can be established for $n=3$ and $\Lambda=\lambda_2$. First, as part of their proof of the Willmore conjecture, Marques-Neves gave a lower bound on the density of non-trivial regular minimal cones in $\Real^4$. In particular, it follows from <cit.> that if $\mathcal{C}\in\mathcal{RMC}_3^*$, then $\lambda[\mathcal{C}] >\lambda_2$ and so $\rstar{3,\lambda_2}$ holds. Furthermore, it follows from <cit.> that $\mathcal{S}_2^*(\lambda_2)=\emptyset$ and so $\rsstar{3,\lambda_2}$ holds.
For $n\geq 4$, some partial results suggest that (<ref>) and (<ref>) hold for $\Lambda=\lambda_{n-1}$. For instance, Ilmanen-White <cit.>, have shown that if $\mathcal{C}\in \mathcal{RMC}_n^*$ and is area-minimizing and topologically non-trivial, then $\lambda[\mathcal{C}]\geq \lambda_{n-1}$. Additionally, <cit.> says that the self-shrinking sphere has the lowest entropy among all compact self-shrinkers and <cit.> posits that $\rsstar{n,\lambda_{n-1}}$ holds for $n\leq 7$.
It is important to note that there exist many topologically trivial elements of $\mathcal{RMC}_n^*$. Indeed, the work of Hsiang <cit.> and Hsiang-Sterling <cit.>, shows that there exist topologically trivial elements of $\mathcal{RMC}_n^*$ for $n=5,7$ and for all even $n\geq 4$.
The paper is organized as follows. In Section <ref>, we introduce notation and recall some basic facts about the mean curvature flow. In Section <ref>, we show regularity of self-shrinking measures of low entropy. In Section <ref>, we study the structure of the singular set for weak mean curvature flows of small entropy. Importantly, we give a topological decomposition, Theorem <ref>, of the regular part of the flow which is the basis of the surgery procedure. In Section <ref>, we prove Theorem <ref> and Corollary <ref>. Finally, in Section <ref>, we carry out the surgery procedure and prove Theorems <ref> and <ref>.
§ NOTATION AND BACKGROUND
In this section, we fix notation for the rest of the paper and recall some background on mean curvature flow. Experts should feel free to consult this section only as needed.
§.§ Singular hypersurfaces
We will use results from <cit.> on weak mean curvature flows. For this reason, we follow the notation of <cit.> as closely as possible.
Denote by
* $\M(\Real^{n+1})=\set{\mu: \mu\mbox{ is a Radon measure on $\Real^{n+1}$}}$ (see <cit.>);
* $\IM_k(\Real^{n+1})=\set{\mu: \mu\mbox{ is an integer $k$-rectifiable Radon measure on $\Real^{n+1}$}}$ (see <cit.>);
* $\IV_k(\Real^{n+1})=\set{V: V\mbox{ is an integer rectifiable $k$-varifold on $\Real^{n+1}$}}$ (see <cit.> or <cit.>).
The space $\M(\Real^{n+1})$ is given the weak* topology. That is,
\mu_i\to\mu\iff\int f\, d\mu_i\to\int f \, d\mu\mbox{ for all $f\in C^0_c(\Real^{n+1})$}.
And the topology on $\IM_k(\Real^{n+1})$ is the subspace topology induced by the natural inclusion into $\M(\Real^{n+1})$. For the details of the topologies considered on $ \IV_k(\Real^{n+1})$, we refer to <cit.> or <cit.>. There are natural bijective maps
V: \IM_k(\Real^{n+1})\to \IV_k(\Real^{n+1}) \mbox{ and } \mu:\IV_k(\Real^{n+1})\to\IM_k(\Real^{n+1}).
The second map is continuous, but the first is not. Henceforth, write $V(\mu)=V_\mu$ and $\mu(V)=\mu_V$.
If $\Sigma\subset\Real^{n+1}$ is a $k$-dimensional smooth properly embedded submanifold, we denote by $\mu_\Sigma=\mathcal{H}^k\lfloor\Sigma\in\IM_k(\Real^{n+1})$. Given $(\yY,\rho)\in\Real^{n+1}\times\Real^+$ and $\mu\in\IM_k(\Real^{n+1})$, we define the rescaled measure $\mu^{\yY,\rho}\in\IM_k(\Real^{n+1})$ by
\mu^{\yY,\rho}(\Omega)=\rho^{k}\mu\left(\rho^{-1}\Omega+\yY\right).
This is defined so that if $\Sigma$ is a $k$-dimensional smooth properly embedded submanifold, then
\mu^{\yY,\rho}_\Sigma=\mu_{\rho (\Sigma-\yY)}.
One of the defining properties of $\mu\in \IM_k(\Real^{n+1})$ is that for $\mu$-a.e. $\xX\in\Real^{n+1}$, there is an integer
$\theta_\mu(\xX)$ so that
\lim_{\rho\to \infty}\mu^{\xX,\rho}=\theta_\mu(\xX)\mu_{P},
where $P$ is a $k$-dimensional plane through the origin. When such $P$ exists, we denote it by $T_{\xX} \mu$ the approximate tangent plane at $\xX$. The value $\theta_\mu(\xX)$ is the multiplicity of $\mu$ at $\xX$ and by definition, $\theta_\mu(\xX)\in\mathbb{N}$ for $\mu$-a.e. $\xX$. Notice that if $\mu=\mu_{\Sigma}$, then $T_{\xX}\mu=T_{\xX}\Sigma$ and $\theta_\mu(\xX)=1$. Given a $\mu\in \IM_n(\Real^{n+1})$, set
\reg(\spt(\mu))=\set{\xX\in\spt(\mu): \exists\rho>0 \mbox{ s.t. $B_\rho(\xX)\cap\spt(\mu)$ is a hypersurface}},
and $\sing(\spt(\mu))=\spt(\mu)\setminus\reg(\spt(\mu))$. Here $B_\rho(\xX)$ is the open ball in $\Real^{n+1}$ centered at $\xX$ with radius $\rho$. Likewise,
\reg(\mu)=\set{\xX\in \reg(\spt(\mu)): \theta_\mu(\xX)=1} \mbox{ and } \sing(\mu)=\spt(\mu)\setminus\reg(\mu).
For $\mu\in\IM_n(\Real^{n+1})$, we extend the definitions of $F$ and $\lambda$ in the obvious manner, namely,
F[\mu]=F[V_\mu]=\int \Phi\, d\mu \mbox{ and } \lambda[\mu]=\lambda[V_\mu]=\sup_{(\yY,\rho)\in\Real^{n+1}\times\Real^+} F[\mu^{\yY,\rho}].
§.§ Gaussian densities and tangent flows
Historically, the first weak mean curvature flow was the measure-theoretic flow introduced by Brakke <cit.>. This flow is called a Brakke flow. Brakke's original definition considered the flow of varifolds. We use the (slightly stronger) notion introduced by Ilmanen <cit.>. For our purposes, the Brakke flow has two important roles. The first is the fact that Huisken's monotonicity formula <cit.> holds also for Brakke flows; see <cit.>. The second is the powerful regularity theory of Brakke <cit.> for such flows. In particular, we will often refer to White's version of Brakke's local regularity theorem <cit.>. We emphasize that White's argument is valid only for a special class of Brakke flows, but that all Brakke flows considered in this paper are within this class.
A consequence of Huisken's monotonicity formula is that if a Brakke flow $\mathcal{K}=\set{\mu_t}_{t\geq t_0}$ has bounded area ratios, then $\mathcal{K}$ has a well-defined Gaussian density at every point $(\yY,s)\in\Real^{n+1}\times (t_0,\infty)$ given by
\Theta_{(\yY,s)}(\mathcal{K})=\lim_{t\to s^-}\int\Phi_{(\yY,s)}(\xX,t)\, d\mu_t(\xX),
\Phi_{(\yY,s)} (\xX,t)=(4\pi)^{-\frac{n}{2}} e^{\frac{|\xX-\yY|^2}{4(t-s)}}.
Furthermore, the Gaussian density is upper semi-continuous.
Combining the compactness of Brakke flows (cf. <cit.>) with the monotonicity formula, one establishes the existence of tangent flows. For a Brakke flow $\mathcal{K}=\set{\mu_t}_{t\geq t_0}$ and a point $(\yY,s)\in\Real^{n+1}\times(t_0,\infty)$, define a new Brakke flow
\mathcal{K}^{(\yY,s),\rho}=\set{\mu_t^{(\yY,s),\rho}}_{t\geq\rho^2(t_0-s)},
\mu_t^{(\yY,s),\rho}=\mu_{s+\rho^{-2}t}^{\yY,\rho}.
Let $\mathcal{K}=\set{\mu_t}_{t\geq t_0}$ be an integral Brakke flow with bounded area ratios. A non-trivial Brakke flow $\mathcal{T}=\set{\nu_t}_{t\in\Real}$ is a tangent flow to $\mathcal{K}$ at $(\yY,s)\in\Real^{n+1}\times(t_0,\infty)$, if there is a sequence $\rho_i\to\infty$ so that $\mathcal{K}^{(\yY,s),\rho_i}\to\mathcal{T}$. Denote by $\mathrm{Tan}_{(\yY,s)}\mathcal{K}$ the set of tangent flows to $\mathcal{K}$ at $(\yY,s)$.
The monotonicity formula implies that any tangent flow is backwardly self-similar.
Given an integral Brakke flow $\mathcal{K}=\set{\mu_t}_{t\geq t_0}$ with bounded area ratios, a point $(\yY,s)\in\Real^{n+1}\times({t_0},\infty)$ with $\Theta_{(\yY,s)}(\mathcal{K})\geq 1$, and a sequence $\rho_i\to\infty$, there exists a subsequence $\rho_{i_j}$ and a $\mathcal{T}\in\mathrm{Tan}_{(\yY,s)}\mathcal{K}$ so that $\mathcal{K}^{(\yY,s),\rho_{i_j}}\to\mathcal{T}$.
Furthermore, $\mathcal{T}=\set{\nu_t}_{t\in\Real}$ is backwardly self-similar with respect to parabolic rescaling about $(\OO,0)$. That is, for all $t<0$ and $\rho>0$,
\nu_t=\nu_t^{(\OO,0),\rho}.
Moreover, $V_{\nu_{-1}}$ is a stationary point of the $F$ functional and
\Theta_{(\yY,s)}(\mathcal{K})=F[\nu_{-1}].
§.§ Level set flows and boundary motions
We will also need a set-theoretic weak mean curvature flow called the level-set flow. This flow was first studied in the context of numerical analysis by Osher-Sethian <cit.>. The mathematical theory was developed by Evans-Spruck <cit.> and Chen-Giga-Goto <cit.>. For our purposes, it has the important advantages of being uniquely defined and satisfying a maximum principle.
A technical feature of the level-set flow is that the level sets ${L}(\Gamma_0)=\set{\Gamma_t}_{t\geq 0}$ may develop non-empty interiors for positive times. This phenomena is called fattening and is unavoidable for certain initial sets $\Gamma_0$ and is closely related to non-uniqueness phenomena of weak solutions of the flow. We say $L(\Gamma_0)$ is non-fattening, if each $\Gamma_t$ has no interior.
It is relatively straightforward to see that the non-fattening condition is generic; see for instance <cit.>.
In <cit.>, Ilmanen synthesized both notions of weak flow. In particular, he showed that for a large class of initial sets, there is a canonical way to associate a Brakke flow to the level-set flow, and observed that this allows, among other things, for the application of Brakke's partial regularity theorem. For our purposes, it is important that the Brakke flow constructed does not vanish gratuitously. A similar synthesis may be found in <cit.>. The result we need is the following:
If $\Sigma_0$ is a closed hypersurface in $\Real^{n+1}$ and the level-set flow ${L}(\Sigma_0)$ is non-fattening, then there is a set $E\subset \Real^{n+1}\times \Real$ and a Brakke flow $\mathcal{K}=\set{\mu_t}_{t\geq 0}$ so that:
* $E=\set{(\xX,t): u(\xX,t)>0}$, where $u$ solves the level set flow equation with initial data $u_0$ that satisfies $E_0=\set{\xX: u_0(\xX)>0}$ and $\partial E_0=\set{\xX: u_0(\xX)=0}=\Sigma_0$;
* each $E_t=\set{\xX: (\xX,t)\in E}$ is of finite perimeter and $\mu_t=\mathcal{H}^n\lfloor\partial^\ast E_t$, where $\partial^* E_t$ is the reduced boundary of $E_t$.
§ REGULARITY OF SELF-SHRINKING MEASURES OF SMALL ENTROPY
We establish some regularity properties of self-shrinking measures of small entropy when $n\geq 3$. We restrict to $n\geq 3$ in order to avoid certain technical complications coming from the fact that $\lambda_1>\frac{3}{2}$.
§.§ Self-shrinking measures
We will need a singular analog of $\mathcal{S}_n$. To that end, we define the set of self-shrinking measures on $\Real^{n+1}$ by
\begin{equation*}
\mathcal{SM}_n=\set{\mu\in\IM_n(\Real^{n+1}): V_\mu\mbox{ is stationary for the ${F}$ functional}, \spt(\mu)\neq\emptyset}.
\end{equation*}
Clearly, if $\Sigma\in \mathcal{S}_n$, then $\mu_\Sigma\in\mathcal{SM}_n$. There are many examples of singular self-shrinkers. For instance, any element of $\mathcal{C}\in\mathcal{RMC}_n$ satisfies $\mu_{\mathcal{C}}=\mathcal{H}^n\lfloor\mathcal{C}\in \mathcal{SM}_n$.
For $\mu\in\mathcal{SM}_n$, we define the associated Brakke flow $\mathcal{K}=\set{\mu_t}_{t\in\Real}$ by
\begin{equation*}
\mu_t= \left \{ \begin{array}{cc}
0 & t\geq 0 \\
\mu^{\OO, \sqrt{-t}} & t<0.
\end{array} \right.
\end{equation*}
One can verify that this is a Brakke flow.
Given $\Lambda>0$, set
\begin{equation*}
\mathcal{SM}_n(\Lambda)=\set{\mu\in\mathcal{SM}_n: \lambda[\mu]<\Lambda} \mbox{ and } \mathcal{SM}_n[\Lambda]=\set{\mu\in\mathcal{SM}_n: \lambda[\mu]\leq \Lambda}.
\end{equation*}
§.§ Regularity and asymptotic properties of self-shrinking measures of small entropy
A $\mu\in\IM_n(\Real^{n+1})$ is a cone, if $\mu^{\OO,\rho}=\mu$. Likewise, $\mu\in\IM_n(\Real^{n+1})$ splits off a line, if, up to an ambient rotation of $\Real^{n+1}$, $\mu=\hat{\mu}\times\mu_\Real$ for $\hat{\mu}\in\IM_{n-1}(\Real^{n})$. Observe that if $\mu\in\mathcal{SM}_n$ is a cone, then $V_\mu$ is stationary (for area). Similarly, if $\mu\in\mathcal{SM}_n$ splits off a line, then $\hat{\mu}\in\mathcal{SM}_{n-1}$ and $\lambda[\mu]=\lambda[\hat{\mu}]$.
Standard dimension reduction arguments give the following:
Fix $n\geq 3$ and $\Lambda\leq 3/2$ and suppose that (<ref>) holds. If $\mu\in\mathcal{SM}_n(\Lambda)$ is a cone, then $\mu=\mu_P$ for some hyperplane $P$.
We will prove this by showing that if (<ref>) holds, then for all $3\leq m\leq n$, if $\mu\in\mathcal{SM}_m(\Lambda)$ is a cone, then $\mu=\mu_{P}$ for a hyperplane $P$ in $\mathbb{R}^{m+1}$.
We proceed by induction on $m$. When $m=3$, note that $\Lambda\leq\frac{3}{2}$ and so we have that $\mu=\mu_{\mathcal{C}}$ for some $\mathcal{C}\in \mathcal{RMC}_3$ by <cit.>. Hence, by the assumption that $\mathcal{RMC}_3^*(\Lambda)=\emptyset$, we must have that $\mathcal{C}$ is a hyperplane through the origin.
To complete the induction argument, we observe that it suffices to show that if $\mu\in \mathcal{SM}_m(\Lambda)$ is a cone, then $\mu=\mu_{\mathcal{C}}$ for some $\mathcal{C}\in \mathcal{RMC}_m(\Lambda)$. Indeed, such a $\mathcal{C}$ must be a hyperplane because (<ref>) holds and so, by definition, $\mathcal{RMC}^*_m(\Lambda)=\emptyset$ for $3\leq m \leq n$.
To complete the proof, we argue by contradiction. Suppose that $\spt(\mu)$ is not a regular cone. Then there is a point $\yY\in\sing(\mu)\setminus\set{\OO}.$ As $V_\mu$ is stationary, and $\mu\in\IM_m$ with $\lambda[\mu]<\Lambda$, we may apply Allard's integral compactness theorem (see <cit.>) to conclude that there exists a sequence $\rho_i\to\infty$ so that $\mu^{\yY,\rho_i}\to\nu$ and $V_\nu$ is a stationary integral varifold. Moreover, it follows from the monotonicity formula <cit.> that $\nu$ is a cone; see also <cit.>.
As $\mu$ is a cone, $\nu$ splits off a line. That is, $\nu=\hat{\nu}\times\mu_\Real$, where $\hat{\nu}\in\mathcal{IM}_{m-1}$ and $V_{\hat{\nu}}$ is a stationary cone and so $\hat{\nu}\in \mathcal{SM}_{m-1}$. Moreover, by the lower semi-continuity of entropy,
\lambda[\hat{\nu}]=\lambda[\hat{\nu}\times\mu_\Real]\leq\lambda[\mu]<\Lambda.
Thus, it follows from the induction hypotheses that $\hat{\nu}=\mu_{\hat{P}}$, where $\hat{P}$ is a hyperplane in $\Real^m$ and so $V_\nu$ is a multiplicity-one hyperplane. Hence, by Allard's regularity theorem (see <cit.>), $\yY\in\reg(\mu)$, giving a contradiction. Therefore, $\mu=\mu_{\mathcal{C}}$ for a $\mathcal{C}\in\mathcal{RMC}_m(\Lambda)$.
As a consequence, we obtain regularity for elements of $\mathcal{SM}_n(\Lambda)$ under the hypothesis that (<ref>) holds.
Fix $n\geq 3$ and $\Lambda\leq 3/2$ and suppose that (<ref>) holds. If $\mu\in \mathcal{SM}_n(\Lambda)$, then $\mu=\mu_\Sigma$ for some $\Sigma\in \mathcal{S}_n(\Lambda)$.
Observe that for $\mu\in\mathcal{SM}_n(\Lambda)$, the mean curvature of $V_\mu$ is locally bounded by (<ref>). Following the same reasoning in the proof of Lemma <ref>, given $\yY\in\sing (\mu)$, there exists a sequence $\rho_i\to\infty$ so that $\mu^{\yY,\rho_i}\to\nu$ and $V_\nu$ is a stationary cone and so $\nu\in \mathcal{SM}_{n}$. By the lower semi-continuity of entropy, $\lambda[\nu]\leq\lambda[\mu]<\Lambda$. Hence, together with Lemma <ref>, it follows that $\sing(\mu)=\emptyset$. That is, $\spt(\mu)$ is a smooth submanifold of $\Real^{n+1}$ that, moreover, satisfies (<ref>). Finally, the entropy bound on $\mu$ implies that $\mu(B_R)\leq C R^n$ for some $C>0$ and so, by <cit.>, $\spt(\mu)$ is proper. That is, $\mu=\mu_\Sigma$ for some $\Sigma\in \mathcal{S}_n$.
If, in addition, (<ref>) holds:
Fix $n\geq 3$ and $\Lambda\leq\Lambda_{n-1}$ and suppose that both (<ref>) and (<ref>) hold. If $\mu\in\mathcal{SM}_n(\Lambda)$, then $\mu=\mu_\Sigma$ for some $\Sigma\in\mathcal{S}_n(\Lambda)$, and either $\Sigma$ is diffeomorphic to $\mathbb{S}^n$ or $\Sigma\in\mathcal{ACS}_n$.
First observe that, by Proposition <ref>, $\mu=\mu_\Sigma$ for some $\Sigma\in \mathcal{S}_n(\Lambda)$. If $\Sigma$ is closed, then it follows from <cit.> that $\Sigma$ is diffeomorphic to $\mathbb{S}^n$. On the other hand, if $\Sigma$ is not closed, then it is non-compact.
Let $\mathcal{K}=\set{\mu_t}_{t\in\Real}$ be the Brakke flow associated to $\mu$. Note that $\mu_t=\mu_{\sqrt{-t}\, \Sigma}$ for $t<0$. Let $\mathcal{X}=\set{\yY: \yY\neq\OO, \Theta_{(\yY,0)}(\mathcal{K})\geq 1}\subset \Real^{n+1}\setminus\set{\OO}$. As $\Sigma$ is non-compact, $\mathcal{X}$ is non-empty. Indeed, pick any sequence of points $\yY_i\in\Sigma$ with $|\yY_i|\to\infty$. The points $\hat{\yY}_i=|\yY_i|^{-1}\yY_i\in |\yY_i|^{-1}\Sigma$. Hence, $\Theta_{(\hat{\yY}_i, -|\yY_i|^{-2})}(\mathcal{K})\geq 1$. As the $\hat{\yY}_i$ are in a compact subset, up to passing to a subsequence and relabeling, $\hat{\yY}_i\to\hat{\yY}$, and so the upper semi-continuity of Gaussian density implies that $\Theta_{(\hat{\yY},0)}(\mathcal{K})\geq 1$.
We next show that $\mathcal{X}$ is a regular cone. The fact that $\mathcal{X}$ is a cone readily follows from the fact that $\mathcal{K}$ is invariant under parabolic scalings. To see that $\sing(\mathcal{X})\subset \set{\OO}$, we note that, by <cit.>, for any $\yY\in\mathcal{X}$ and $\mathcal{T}\in\mathrm{Tan}_{(\yY,0)}\mathcal{K}$, $\mathcal{T}=\set{\nu_t}_{t\in\Real}$ splits off a line. That is, up to an ambient rotation, $\nu_{t}=\hat{\nu}_t\times\mu_\Real$ with $\set{\hat{\nu}_t}_{t\in\Real}$ the Brakke flow associated to $\hat{\nu}_{-1}\in\mathcal{SM}_{n-1}(\Lambda)$. Here we use the lower semi-continuity of entropy. Note that $\Lambda\leq\lambda_{n-1}<3/2$. Thus, by Proposition <ref> and the hypothesis that $\rstar{n,\Lambda}$ holds, $\hat{\nu}_{-1}=\mu_\Gamma$ for $\Gamma\in\mathcal{S}_{n-1}(\Lambda)$. Hence, as we assume that (<ref>) holds, $\Gamma$ is a hyperplane through the origin. Therefore, it follows from Brakke's regularity theorem that, for $t<0$ close to $0$, $\spt(\mu_t)$ has uniformly bounded curvature near $\yY$ and so $\sqrt{-t}\, \Sigma\to\mathcal{X}$ in $C^\infty_{loc}\left(\Real^{n+1}\backslash \set{\OO}\right)$, concluding the proof.
As a consequence, we establish the following compactness theorem for asymptotically conical self-shrinkers of small entropy.
Fix $n\geq 3$, $\Lambda\leq\Lambda_{n-1}$, and $\epsilon_0>0$. If both (<ref>) and (<ref>) hold, then the set
\mathcal{ACS}_n[\Lambda-\epsilon_0]=\set{\Sigma: \Sigma \in \mathcal{ACS}_n \mbox{ and } \lambda[\Sigma]\leq \Lambda-\epsilon_0}
is compact in the $C^\infty_{loc}(\Real^{n+1})$ topology.
Consider a sequence $\Sigma_i\in \mathcal{ACS}_n[\Lambda-\epsilon_0]$ and let $\mu_i=\mu_{\Sigma_i}\in\mathcal{SM}_n[\Lambda-\epsilon_0]$. By the integral compactness theorem for $F$-stationary varifolds, up to passing to a subsequence, $\mu_i\to \mu$ in the sense of Radon measures. Moreover, by the lower semi-continuity of the entropy, $\mu\in \mathcal{SM}_n[\Lambda-\epsilon_0]$. Hence, by Proposition <ref>, $\mu=\mu_\Sigma$ for $\Sigma\in \mathcal{S}_n[\Lambda-\epsilon_0]$ and so, by Allard's regularity theorem, $\Sigma_i\to \Sigma$ in $C^\infty_{loc}(\Real^{n+1})$. Finally, as each $\Sigma_i$ is non-compact and connected, so is $\Sigma$ and so, by Proposition <ref>, $\Sigma\in \mathcal{ACS}_n[\Lambda-\epsilon_0]$, proving the claim.
Recall that $\mathcal{C}(\Sigma)$ denotes the asymptotic cone of any $\Sigma\in \mathcal{ACS}_n$. Denote the link of the asymptotic cone by $\mathcal{L}(\Sigma)=\mathcal{C}(\Sigma)\cap \mathbb{S}^n$.
Fix $n\geq 3$, $\Lambda \leq \lambda_{n-1}$, and $\epsilon_0>0$. If both (<ref>) and (<ref>) hold, then the set
\mathcal{L}_n[\Lambda-\epsilon_0]=\set{\mathcal{L}(\Sigma): \Sigma\in \mathcal{ACS}_n[\Lambda-\epsilon_0]}
is compact in the $C^\infty(\mathbb{S}^n)$ topology.
Consider a sequence $L_i\in \mathcal{L}_n[\Lambda-\epsilon_0]$ and let $\Sigma_i\in \mathcal{ACS}_n[\Lambda-\epsilon_0]$ be chosen so that $\mathcal{L}(\Sigma_i)=L_i$ (observe that the $\Sigma_i$ are uniquely determined by <cit.>). By Corollary <ref>, up to passing to a subsequence, $\Sigma_i\to \Sigma\in \mathcal{ACS}_n[\Lambda-\epsilon_0]$. We claim that $L_i\to L=\mathcal{L}(\Sigma)$ in $C^\infty(\mathbb{S}^n)$.
To see this, let $\mu_i=\mu_{\Sigma_i}$ and $\mu=\mu_\Sigma$ be the corresponding elements of $\mathcal{SM}_n[\Lambda-\epsilon_0]$ and let $\mathcal{K}_i$ and $\mathcal{K}$ be the associated Brakke flows. Clearly, $\mu_i\to \mu$ in the sense of measures. Hence, by construction, the $\mathcal{K}_i$ converge in the sense of Brakke flows to ${\mathcal{K}}$. Since
$$\mathcal{C}(\Sigma)=\set{\xX\in \Real^{n+1}: \Theta_{(\xX,0)}(\mathcal{K}) \geq 1}$$
and likewise for $\mathcal{C}(\Sigma_i)$, we have by Brakke's regularity theorem that $\mathcal{C}(\Sigma_i)\to \mathcal{C}(\Sigma)$ in $C^\infty_{loc}(\Real^{n+1}\backslash \set{\OO})$, that is $\mathcal{L}(\Sigma_i)\to \mathcal{L}(\Sigma)$ in $C^\infty(\mathbb{S}^n)$ as claimed.
Let $B_R$ denote the open ball in $\mathbb{R}^{n+1}$ centered at the origin with radius $R$. Combining Corollary <ref> and Proposition <ref> gives that
Fix $n\geq 3$, $\Lambda \leq \lambda_{n-1}$, and $\epsilon_0>0$. Suppose that (<ref>) and (<ref>) hold. There is an $R_0=R_0(n, \Lambda, \epsilon_0)$ and $C_0=C_0(n, \Lambda, \epsilon_0)$ so that if $\Sigma \in \mathcal{ACS}_n[\Lambda-\epsilon_0]$, then
* $\Sigma\setminus\bar{B}_{R_0}$ is given by the normal graph of a smooth function $u$ over $\mathcal{C}(\Sigma)\setminus\Omega$, where $\Omega$ is a compact set, satisfying that for $p\in\mathcal{C}(\Sigma)\setminus\Omega$,
\abs{\xX(p)} \abs{u(p)}+ \abs{\xX(p)}^2 \abs{\nabla_{\mathcal{C}(\Sigma)} u(p)}+\abs{\xX(p)}^3 \abs{\nabla_{\mathcal{C}(\Sigma)}^2 u(p)}\leq C_0;
* given $\delta>0$, there is a $\kappa\in (0,1)$ and $\mathcal{R}>1$ depending only on $n,\Lambda, \epsilon_0$ and $\delta$ so that if $p\in\Sigma\setminus B_{\mathcal{R}}$ and $r=\kappa |\xX (p)|$, then $\Sigma\cap B_r(p)$ can be written as a connected graph of a function $v$ over a subset of $T_p\Sigma$ with $|Dv|\leq\delta$.
As such, for any $R\geq R_0$, $\Sigma\backslash B_R$ is diffeomorphic to $\mathcal{L}(\Sigma)\times [0, \infty)$.
For any sequence $\Sigma_i\in\mathcal{ACS}_n[\Lambda-\epsilon_0]$, by Corollary <ref> and Proposition <ref>, up to passing to a subsequence, $\Sigma_i\to\Sigma$ in $C^\infty_{loc} (\Real^{n+1})$ for some $\Sigma\in\mathcal{ACS}_n[\Lambda-\epsilon_0]$, and $\mathcal{L}(\Sigma_i)\to\mathcal{L}(\Sigma)$ in $C^\infty (\mathbb{S}^n)$. Let $\mathcal{K}_i$ and $\mathcal{K}$ be the associated Brakke flows to $\Sigma_i$ and $\Sigma$, respectively. As $\Sigma\in\mathcal{ACS}_n$, $\mathcal{K}\lfloor (B_2\setminus \bar{B}_1)\times [-1,0]$ is a smooth mean curvature flow. Furthermore, since $\mathcal{K}_i\to\mathcal{K}$, it follows from Brakke's local regularity theorem that $\Sigma_i$ have uniform curvature decay, more precisely, there exist $R, C>0$ so that for all $i$ and $p\in\Sigma_i\setminus B_R$,
\sum_{k=0}^2 \abs{\xX(p)}^{k+1}\abs{\nabla^k_{\Sigma_i} A_{\Sigma_i}(p)}\leq C,
where $A_{\Sigma_i}$ is the second fundamental form of $\Sigma_i$. As the $\mathcal{C}(\Sigma_i)\to\mathcal{C}(\Sigma)$, by <cit.> and <cit.>, there exist $R^\prime, C^\prime>0$ so that Items (1) and (2) in the statement hold for all $\Sigma_i$. This establishes the corollary by the arbitrariness of the $\Sigma_i$.
Finally, we need the fact that closed self-shrinkers of small entropy have an upper bound on their extrinsic diameter.
Fix $n\geq 3$, $\Lambda\leq\lambda_{n-1}$, and $\epsilon_0>0$. Suppose that both (<ref>) and (<ref>) hold. Then there is a $R_D=R_D(n,\Lambda, \epsilon_0)$ so that if $\Sigma\in \mathcal{S}_n[\Lambda-\epsilon_0]$ is closed, then $\Sigma\subset \bar{B}_{R_D}$.
We argue by contradiction. If this was not true, then there would be a sequence of $\Sigma_i\in \mathcal{S}_n[\Lambda-\epsilon_0]$ with the property that there are points $p_i\in \Sigma_i$ with $|p_i|\to \infty$. In particular, for each $R>\sqrt{2n}$, there is an $i_0=i_0(R)$ so that if $i>i_0(R)$, then $\Sigma_i\cap \partial B_R\neq \emptyset$. Indeed, if this was not the case, then the mean curvature flows $\set{\sqrt{-t}\, \Sigma}_{t\in [-1,0)}$ and $\set{\partial B_{\sqrt{R^2-2n(t+1)}}}_{t\in [-1,0)}$ would violate the avoidance principle.
Now, let $\mu_i=\mu_{\Sigma_i}\in \mathcal{SM}_n[\Lambda-\epsilon_0]$. By the integral compactness theorem for $F$-stationary varifolds, up to passing to a subsequence the $\mu_i$ converge to a $\mu\in \mathcal{SM}_n[\Lambda-\epsilon_0]$. By Proposition <ref>, $\mu=\mu_\Sigma$ for some $\Sigma\in \mathcal{S}_n [\Lambda-\epsilon_0]$. Furthermore, up to passing to a further subspace, $\Sigma_i\to \Sigma$ in $C^\infty_{loc}(\Real^{n+1})$. It follows that $\Sigma \cap \partial B_R\neq \emptyset$ for all $R>\sqrt{2n}$. In other words, $\Sigma$ is non-compact and so, by Proposition <ref>, $\Sigma\in \mathcal{ACS}_n$. However, this implies that $\Sigma$ is non-collapsed (cf. <cit.>), while the $\Sigma_i$ are collapsed by <cit.>. This contradicts <cit.> and completes the proof.
§ SINGULARITIES OF FLOWS WITH SMALL ENTROPY
Given a Brakke flow $\mathcal{K}=\set{\mu_t}_{t\in I}$ and a point $(\xX_0,t_0)\in \sing(\mathcal{K})$ with $t_0\in \mathring{I}$, a tangent flow $\mathcal{T}\in\mathrm{Tan}_{(\xX_0,t_0)}\mathcal{K}$ is of compact type if $\mathcal{T}=\set{\nu_t}_{t\in (-\infty,\infty)}$ and $\spt(\nu_{-1})$ is compact. Otherwise, the tangent flow is of non-compact type. If every element of $\mathrm{Tan}_{(\xX_0, t_0)}\mathcal{K}$ is of compact type, then $(\xX_0,t_0)$ is a compact singularity. Likewise, if every element of $\mathrm{Tan}_{(\xX_0, t_0)}\mathcal{K}$ is of non-compact type, then $(\xX_0,t_0)$ is a non-compact singularity.
For the remainder of this section, we fix a dimension $n\geq 3$ and constants $\Lambda\in (\lambda_n, \lambda_{n-1}]$[The reader may refer to Remark <ref> for the reason that we restrict to $\Lambda>\lambda_n$.] and $\epsilon_0>0$, and suppose that both (<ref>) and (<ref>) hold. We further assume that $\Sigma_0\subset \Real^{n+1}$ is a closed connected hypersurface with $\lambda[\Sigma_0]\leq\Lambda-\epsilon_0$ and with the property that the level set flow $L(\Sigma_0)$ is non-fattening and that $(E,\mathcal{K})$ is the pair given by Theorem <ref>.
Let $(\xX_0,t_0)\in \sing(\mathcal{K})$ and $\mathcal{T}\in \mathrm{Tan}_{(\xX_0,t_0)}\mathcal{K}$. If $\mathcal{T}=\set{\nu_t}_{t\in (-\infty,\infty)}$ is of non-compact type, then $\nu_{-1}=\mu_\Sigma$ for some $\Sigma\in\mathcal{ACS}_n$. Moreover, there is a constant $R_1=R_1(n, \Lambda, \epsilon_0)$ so that for all $R\geq R_1$,
$$\mathcal{T} \lfloor \left(B_{16R}\setminus\bar{B}_{R}\right)\times (-1,1)$$
is a smooth mean curvature flow. Moreover, for all $\rho\in (R,16R)$ and $t\in (-1,1)$, $\partial B_\rho$ meets $\spt(\nu_t)$ transversally and $\partial B_\rho\cap \spt(\nu_t)$ is connected.
First, invoking Theorem <ref> and the monotonicity formula, $\mathcal{T}$ is backwardly self-similar with respect to parabolic scalings about $(\mathbf{0},0)$ and $\nu_{-1}\in\mathcal{SM}_n [\Lambda-\epsilon_0]$. Furthermore, by Proposition <ref>, we have $\nu_{-1}=\mu_\Sigma$ for some $\Sigma\in\mathcal{ACS}_n[\Lambda-\epsilon_0]$. Finally, by Corollary <ref>, the pseudo-locality property of mean curvature flow <cit.>[The proof of <cit.> uses the local regularity theorem of White, which is also applicable to the Brakke flows in Theorem <ref> and their tangent flows – see <cit.>.] and Brakke's local regularity theorem, there is an $R_1>0$ depending only on $n,\Lambda,\epsilon_0$ so that for $R>R_1$,
$$\mathcal{T} \lfloor \left(B_{16R}\setminus \bar{B}_R\right)\times (-1,1)$$
is a smooth mean curvature flow. Indeed, for all $t\in (-1,1)$, $\spt(\nu_t)\cap \left(B_{16R}\setminus \bar{B}_R\right)$ is the graph of a function over a subset of $\mathcal{C}(\Sigma)$ the asymptotic cone of $\Sigma$ with small $C^2$ norm. As such, for all $\rho\in (R,16R)$ and $t\in (-1,1)$, $\partial B_\rho$ meets $\spt(\nu_t)$ transversally. As $\lambda[\Sigma]\leq \lambda[\Sigma_0]<\lambda_{n-1}$ it follows from <cit.> that $\mathcal{L}(\Sigma)$, the link of $\mathcal{C}(\Sigma)$, is connected and, hence, so is $\partial B_\rho \cap \spt (\nu_t)$.
Next we observe that singularities are either compact or non-compact.
Each $(\xX_0,t_0)\in\sing(\mathcal{K})$ is either a compact or a non-compact singularity.
Suppose that $(\xX_0,t_0)$ is not a non-compact singularity. Then there is a $\mathcal{T}=\{\nu_t\}_{t\in\Real}\in\mathrm{Tan}_{(\xX_0,t_0)}\mathcal{K}$ of compact type. By the monotonicity formula and Theorem <ref>, $\nu_{-1}\in\mathcal{SM}_n[\Lambda-\epsilon_0]$.
It follows from Proposition <ref> that $\nu_{-1}=\mu_\Sigma$ for some $\Sigma\in\mathcal{S}_n[\Lambda-\epsilon_0]$ and $\Sigma$ is closed. Hence, by <cit.>, $\mathcal{T}$ is the only element of $\mathrm{Tan}_{(\xX_0,t_0)}\mathcal{K}$ and so $(\xX_0,t_0)$ is a compact singularity, proving the claim.
We further prove that
Given $(\xX_0, t_0)\in \sing(\mathcal{K})$, there exist $\rho_0=\rho_0(\xX_0,t_0, \mathcal{K})>0$ and $\alpha=\alpha(n,\Lambda,\epsilon_0)>1$ so that:
* If $(\xX_0,t_0)$ is a compact singularity and $\rho<\rho_0$, then
$$\mathcal{K}\lfloor \left(B_{2\alpha \rho}(\xX_0)\times (t_0-4\alpha^2\rho^2, t_0+4\alpha^2\rho^2) \backslash \set{(\xX_0,t_0)} \right) $$
is a smooth mean curvature flow. Furthermore, for all $R\in (\frac{1}{2}\alpha \rho, 2\alpha \rho)$ and $t\in (t_0 -\rho^2, t_0+\rho^2)$, $\spt(\mu_t)\cap \partial B_{R}(\xX_0)=\emptyset$.
* If $(\xX_0,t_0)$ is a non-compact singularity and $\rho<\rho_0$, then
$$\mathcal{K}\lfloor \left(B_{2\alpha \rho}(\xX_0)\times (t_0-4\alpha^2\rho^2, t_0] \backslash \set{(\xX_0,t_0)} \right) $$
$$\mathcal{K}\lfloor \left(B_{2\alpha\rho}(\xX_0)\backslash \bar{B}_{\frac{1}{2}\alpha\rho}(\xX_0)\right)\times (t_0-\rho^2, t_0+\rho^2)$$
are both smooth mean curvature flows. Furthermore, for all $R\in (\frac{1}{2}\alpha \rho, 2\alpha \rho)$ and $t\in (t_0 -\rho^2, t_0+\rho^2)$, $\partial B_{R}(\xX_0)$ meets $\spt(\mu_t)$ transversally and the intersection is connected.
Finally, for all $t\in (t_0-\rho^2, t_0)$, $\spt(\mu_t)\cap \bar{B}_{\alpha \rho}(\xX_0)$ is diffeomorphic (possibly as a manifold with boundary) to $\Gamma\cap \bar{B}_{\alpha}$, where $\Gamma\in \mathcal{S}_n^*[\Lambda-\epsilon_0]$ and, if $\Gamma\in \mathcal{ACS}_n$, then $\Gamma\backslash B_\alpha$ is diffeomorphic to $\mathcal{L}(\Gamma)\times[0,\infty)$.
Set $\alpha=4\max\set{R_1, R_D,1}$ where $R_1$ is given by Proposition <ref> and $R_D$ is given by Proposition <ref>. Without loss of generality, we may assume that $(\xX_0,t_0)=(\mathbf{0},0)$.
We establish the regularity near (but not at) $(\OO,0)$ by contradiction. To that end, suppose that there was a sequence of points $(\xX_i,t_i)\in \sing(\mathcal{K})\backslash \set{(\OO, 0)}$ such that $(\xX_i,t_i)\to (\mathbf{0},0)$. If $(\mathbf{0},0)$ is a non-compact singularity, we further assume $t_i\leq 0$. Let $r_i^2=|\xX_i|^2+|t_i|$. Then, up to passing to a subsequence, it follows from Theorem <ref> that $\mathcal{K}^{(\mathbf{0},0),r_i}\to\mathcal{T}$ in the sense of Brakke flows and $\mathcal{T}=\{\nu_t\}_{t\in\Real}\in\mathrm{Tan}_{(\mathbf{0},0)}\mathcal{K}$. Let $\tilde{\xX}_i=r^{-1}_i \xX_i$ and $\tilde{t}_i=r^{-2}_i t_i$. Then $|\tilde{\xX}_i|^2+|\tilde{t}_i|=1$, that is, $(\tilde{\xX}_i,\tilde{t}_i)$ lies on the unit parabolic sphere in space-time. Thus, up to passing to a subsequence, $(\tilde{\xX}_i,\tilde{t}_i)\to (\tilde{\xX}_0,\tilde{t}_0)$, where $|\tilde{\xX}_0|^2+|\tilde{t}_0|=1$. Moreover, the upper semi-continuity of Gaussian density implies that $\Theta_{(\tilde{\xX}_0,\tilde{t}_0)}(\mathcal{T})\geq 1$.
As $\nu_{-1}\in \mathcal{SM}_n [\Lambda-\epsilon_0]$, Proposition <ref> implies that $\sing(\nu_t)=\emptyset$ for $t<0$. That is, $(\tilde{\xX}_0,\tilde{t}_0)$ is a regular point of $\mathcal{T}$ if $\tilde{t}_0<0$.
If $(\mathbf{0},0)$ is a non-compact singularity, then $\mathcal{T}$ is of non-compact type and $\tilde{t}_0\leq 0$. Hence, either $(\tilde{\xX}_0,\tilde{t}_0)$ is a regular point or $\tilde{t}_0=0$ and $|\tilde{\xX}_0|=1$. However in the later case, Proposition <ref> applied to $ \mathcal{T}^{(\mathbf{0}, 0), \alpha}\in\mathrm{Tan}_{(\mathbf{0},0)}\mathcal{K}$ implies that $(\tilde{\xX}_0,\tilde{t}_0)$ is also a regular point of $\mathcal{T}$.
If $(\mathbf{0},0)$ is a compact singularity, then $\mathcal{T}$ is of compact type and $\nu_{-1}=\mu_\Gamma$ for some $\Gamma\in\mathcal{S}_n(\Lambda)$ by Proposition <ref>. This implies that $\mathcal{T}$ is extinct at time $0$ and $\sing(\mathcal{T})=\{(\mathbf{0},0)\}$, again implying that $\tilde{t}_0\leq 0$ and $(\tilde{\xX}_0,\tilde{t}_0)$ is a regular point of $\mathcal{T}$.
Hence, it follows from Brakke's local regularity theorem that for all $i$ sufficiently large, $(\tilde{\xX}_i,\tilde{t}_i)\notin\sing(\mathcal{K}^{(\mathbf{0},0),r_i})$, or equivalently, $(\xX_i,t_i)\notin\sing(\mathcal{K})$. This is the desired contradiction. Therefore, for $\rho_0'>0$ sufficiently small, if $\rho<\rho_0'$ and $(\mathbf{0},0)$ is a non-compact singularity, then
\mathcal{K} \lfloor \left(B_{2\alpha\rho}\times (-4\alpha^2\rho^2,0]\setminus\{(\mathbf{0},0)\}\right)
is a smooth mean curvature flow, while, if $\rho<\rho_0'$ and $(\mathbf{0},0)$ is a compact singularity, then
\mathcal{K} \lfloor \left(B_{2\alpha\rho}\times (-4\alpha^2 \rho^2,4\alpha^2 \rho^2)\setminus\{(\mathbf{0},0)\}\right)
is a smooth mean curvature flow.
We continue arguing by contradiction and again consider a sequence, $\rho_i$, of positive numbers with $\rho_i\to 0$ and $\rho_i<\rho_0'$. Up to passing to a subsequence, $\mathcal{K}^{(\mathbf{0},0),\rho_i}$ converges, in the sense of Brakke flows, to some $\mathcal{T}=\{\nu_t\}_{t\in\Real}\in\mathrm{Tan}_{(\mathbf{0},0)}\mathcal{K}$. If $(\OO,0)$ is a compact singularity, then, as $\alpha\geq 4 R_D$, $\partial B_R \cap \spt(\nu_{t})=\emptyset$ for $R\geq\frac{1}{2}\alpha$ and $t\in (-1,1)$ by Proposition <ref> and the avoidance principle. Hence, the nature of the convergence implies that, for $\rho_i$ sufficiently large, $\partial B_{R}\cap \spt(\mu_t)=\emptyset$ for $t\in (-\rho^2_i,\rho^2_i)$ and $R\in (\frac{1}{2} \alpha \rho_i, 2\alpha \rho_i)$.
If $(\OO, 0)$ is a non-compact singularity, then Proposition <ref>, implies that
\mathcal{T} \lfloor \left(B_{4\alpha}\setminus \bar{B}_{\frac{1}{4}\alpha}\right)\times (-1,1)
is a smooth mean curvature flow and for all $R\in (\frac{1}{4}\alpha,4\alpha)$ and $t\in (-1,1)$, $\partial B_R$ meets $\spt(\nu_t)$ transversally and as a connected set. Thus, by Brakke's local regularity theorem, for all $i$ sufficiently large,
\mathcal{K}^{(\mathbf{0},0),\rho_i} \lfloor \left(B_{2\alpha}\setminus\bar{B}_{\frac{1}{2} \alpha}\right)\times (-1,1)
is a smooth mean curvature flow, and hence so is
\mathcal{K} \lfloor \left(B_{2\alpha \rho_i}\setminus\bar{B}_{\frac{1}{2}\alpha \rho_i}\right) \times (-\rho_i^2,\rho_i^2).
Moreover, for all $R\in (\frac{1}{2}\alpha \rho_i,2\alpha\rho_i )$ and $t\in (-\rho_i^2,\rho_i^2)$, $\partial B_R$ meets $\mu_t$ transversally and as a connected set.
Hence, as the sequence $\rho_i$ was arbitrary, there is a $\rho_0''<\rho_0'$ so that Items (1) and (2) hold for $\rho<\rho_0''$.
To complete the proof, we observe that again arguing by contradiction, there is a $\rho_0<\rho_0''$
so that if $\rho<\rho_0$, $B_{2\alpha }\cap \rho^{-1} \spt(\mu_{-\rho^2})$ is a normal graph over a domain $\Omega$ in $\Gamma$ with small $C^2$ norm for some $\Gamma\in \mathcal{S}_n [\Lambda-\epsilon_0]$. In particular, by Corollary <ref>, $\partial\Omega$ is a small normal graph over $\partial B_{\alpha} \cap \Gamma$, so $\bar{B}_{\alpha\rho }\cap \spt(\mu_{-\rho^2})$ is diffeomorphic to $\bar{B}_\alpha \cap \Gamma$. Furthermore, the choice of $\alpha$ ensures that if $\Gamma\in \mathcal{ACS}_n$, then $\Gamma\backslash B_{\alpha}$ is diffeomorphic to $\mathcal{L}(\Sigma)\times [0, \infty)$.
It remains only to show that $\bar{B}_{\alpha\rho }\cap \spt(\mu_{t})$ is diffeomorphic to $\bar{B}_\alpha \cap \Gamma$ for $t\in (-\rho^2, 0)$. This follows from the fact that, as already established, the flow is smooth in $\bar{B}_{2\alpha \rho} \times [-2\rho^2, 0)$ and, for all $t\in [-\rho^2,0)$, either $\partial B_{\alpha\rho} \cap \spt(\mu_t)=\emptyset$ (if the singularity is compact) or the intersection is transverse (if the singularity is non-compact). As such, the flow provides a diffeomorphism between $\bar{B}_{\alpha\rho }\cap \spt(\mu_{t})$ and $\bar{B}_{\alpha\rho }\cap \spt(\mu_{-\rho^2})$ – see Appendix A.
We obtain a direct consequence of Theorem <ref>.
For each $t_0>0$, $\sing_{t_0}(\mathcal{K})=\set{\xX: (\xX,t_0)\in \sing(\mathcal{K})}$ is finite.
Given a manifold $M$ we say a subset $U\subset M$ is a smooth domain if $U$ is open and $\partial U$ is a smooth submanifold.
There is an $N=N(\Sigma_0)\in\mathbb{N}$ and a sequence of closed connected hypersurfaces $\Sigma^1, \ldots, \Sigma^N$ so that:
* $\Sigma^1=\Sigma_0$;
* $\Sigma^N$ is diffeomorphic to $\mathbb{S}^n$;
* For each $i$ with $1\leq i \leq N-1$, there is an $m=m(i)\in \mathbb{N}$ and open connected pairwise disjoint smooth domains $U_1^i, \ldots, U_{m(i)}^i \subset \Sigma^i$ and $V_1^i, \ldots, V_{m(i)}^i \subset \Sigma^{i+1}$ so that:
* There are orientation preserving diffeomorphisms
$$\hat{\Phi}^{i}:\Sigma^{i+1}\backslash \cup_{j=1}^{m(i)} V_j^{i}\to \Sigma^{i}\backslash \cup_{j=1}^{m(i)} U_j^i;$$
* Each $\bar{U}_j^i$ is diffeomorphic to $\bar{B}_{R_j^i}\cap \Gamma_j^i$ where $\Gamma_j^i\in \mathcal{ACS}_n^*(\Lambda)$ and $\Gamma_j^i\backslash B_{R_j^i}$ is diffeomorphic to $\mathcal{L}(\Gamma_j^i)\times [0,\infty)$.
Let us denote the set of compact singularities of $\mathcal{K}$ by $\sing^C(\mathcal{K})$ and the set of non-compact singularities by $\sing^{NC}(\mathcal{K})$. By Lemma <ref>, $\sing(\mathcal{K})=\sing^{NC}(\mathcal{K})\cup \sing^C(\mathcal{K})$. We note that if $X\in \sing^{NC}(\mathcal{K})$, then, by Proposition <ref>, every element of $\mathrm{Tan}_X \mathcal{K}$ is the flow of an element of $\mathcal{ACS}_n$ and so the tangent flows are non-collapsed at time $0$ in the sense of <cit.>. Hence, by <cit.>, $\sing^C(\mathcal{K})\neq \emptyset$. In fact, if we define the extinction time of $\mathcal{K}$ to be
T(\mathcal{K})=\sup\set{t: \spt(\mu_t)\neq \emptyset},
\emptyset \neq \set{\xX\in \Real^{n+1}: \Theta_{(\xX_0,T(\mathcal{K}))} (\mathcal{K})\geq 1}=\set{\xX\in \Real^{n+1}: (\xX, T(\mathcal{K}))\in \sing{}^C (\mathcal{K})}.
It follows from Theorem <ref> that $\sing^C(\mathcal{K})$ consists of at most a finite number of points.
Observe that if $\sing(\mathcal{K})$ consists of exactly one point $X_0$, then we can take $N=1$. Indeed, by the above discussion, this singularity must be compact and hence, by Proposition <ref>, there is a $\Gamma\in \mathcal{S}_n(\Lambda)$ diffeomorphic to $\mathbb{S}^n$ so that one of the tangent flows at $X_0$ is the flow associated to $\mu_{\Gamma}$. In this case we may write $\mathcal{K}=\set{\mu_{\Sigma_t}}_{t\in [0, T(\mathcal{K}))}$ where $\set{\Sigma_t}_{t\in [0, T(\mathcal{K}))}$ is a smooth mean curvature flow. By Brakke's regularity theorem, there is a $t$ near $T(\mathcal{K})$ so that $\Sigma_t$ is a small normal graph over $\Gamma$ and hence $\Sigma^1=\Sigma_0$ is diffeomorphic to $\Gamma$, verifying the claim.
Now let $\mathrm{ST}(\mathcal{K})=\set{t\in \Real: (\xX,t)\in \sing(\mathcal{K})}$ be the set of singular times. Notice that by Corollary <ref> there are at most a finite number of singular points associated to each singular time. We observe that as $\Sigma^1=\Sigma_0$ is smooth, there is a $\delta>0$ so that $\mathrm{ST}(\mathcal{K})\subset [\delta, T(\mathcal{K})]$. Furthermore, as $\sing(\mathcal{K})$ is a closed set, so is $\mathrm{ST}(\mathcal{K})$.
For each $t\in \mathrm{ST}(\mathcal{K})$, let
\rho(t)=\min\set{\rho_0(\xX, t, \mathcal{K}): \xX\in \sing{}_t(\mathcal{K})}>0,
where $\rho_0(\xX, t, \mathcal{K})$ is the constant given by Theorem <ref>. This minimum is positive as $\sing_{t}(\mathcal{K})$ is a finite set. Observe that by Theorem <ref>,
\begin{equation} \label{DisjointEqn}
B_{\alpha\rho(t)}(\xX)\cap B_{\alpha\rho(t)}(\xX^\prime)=\emptyset
\end{equation}
when $\xX,\xX^\prime$ are distinct elements of $\sing_t(\mathcal{K})$ and $\alpha=\alpha(n, \Lambda, \epsilon_0)$ is given by Theorem <ref>. Next, choose $\tau(t)\in (0,\rho^2(t))$ so that
\mathcal{K} \lfloor \left(\Real^{n+1}\setminus\bigcup_{\xX\in\sing_t(\mathcal{K})} \bar{B}_{\alpha\rho(t)}(\xX)\right)\times \left(t-\tau(t),t+\tau(t)\right)
is a smooth mean curvature flow. Such a $\tau$ exists as $\sing(\mathcal{K})$ is a closed set.
As $\mathrm{ST}(\mathcal{K})$ is a closed subset of $[0,T(\mathcal{K})]$, it is a compact set and so the open cover
\set{(t-\tau(t), t+\tau(t)): t\in \mathrm{ST}(\mathcal{K})}
of $\mathrm{ST}(\mathcal{K})$ has a finite subcover. That is, there are a finite number of times $t_1, \ldots, t_{N^\prime}\in \mathrm{ST}(\mathcal{K})$, labeled so that $t_i<t_{i+1}$ and chosen so that
\mathrm{ST}(\mathcal{K})\subset \bigcup_{i=1}^{N^\prime} (t_i-\tau(t_i),t_i+\tau(t_i)).
Furthermore, we can assume that for each $i$:
* For all $j>i$, $t_i-\tau(t_i)< t_j- \tau(t_j)$,
* For all $j<i$, $t_i+\tau(t_i)>t_j+\tau(t_j)$, and
* For all $j<i<j'$, $t_j+\tau(t_j)<t_{j'}-\tau(t_{j'})$.
As otherwise, we could delete $(t_i-\tau(t_i),t_i+\tau(t_i))$ and still have an open cover.
Note that, by the definition of $\tau(t)$, one must have $t_{N^\prime}=T(\mathcal{K})$.
By Theorem <ref> we may choose a sequence of points $s^\pm_1, \ldots, s_{N^\prime}^\pm$ with $t_i\in (s_i^-, s_i^+)$, $|s_i^\pm-t_i|<\tau(t_i)$, $s_i^+\leq s_{i+1}^-$ and so that
\left([0, s_1^-]\cup \bigcup_{i=1}^{N^\prime-1}[s_i^+, s_{i+1}^-]\right)\cap \mathrm{ST}(\mathcal{K})=\emptyset.
More concretely, first take $s_1^-\in (t_1-\tau(t_1), t_1)$ with $s_1^->0$ and $s_{N^\prime}^+=t_{N^\prime}+\frac{1}{2}\tau(t_{N^\prime})$.
For $1\leq i\leq N^\prime-1$, let
$$\tilde{s}_i^+=\sup \left( \mathrm{ST}(\mathcal{K})\cap (t_i-\tau(t_i), t_i+\tau(t_i))\right)$$
and for $2\leq i \leq N^\prime$, let
$$\tilde{s}_i^-=\inf \left( \mathrm{ST}(\mathcal{K})\cap (t_i-\tau(t_i), t_i+\tau(t_i))\right).$$
The definition of $\tau(t_i)$ and Theorem <ref> imply that $\tilde{s}_i^-=t_i$.
As the set of singular times is closed and $t_i\in \mathrm{ST}(\mathcal{K})$, $\tilde{s}_i^+\in \mathrm{ST}(\mathcal{K})$ and $t_i \leq \tilde{s}^+_i$.
We treat two cases. In the first case we suppose that $t_{i+1}-\tau(t_{i+1})<t_i+\tau(t_i) $. As $\tilde{s}_{i+1}^-=t_{i+1}$, there are then no singular times in the interval $(t_{i+1}-\tau(t_{i+1}),t_i+\tau(t_i))$ and so we may take $s_i^+=s_{i+1}^-$ to be the same point in this interval. In the second case, we suppose that $t_i+\tau(t_i)\leq t_{i+1}-\tau(t_{i+1})$ and observe that $\tilde{s}_i^+\leq t_i+\tau(t_i)\leq t_{i+1}-\tau(t_{i+1})$. In fact, $\tilde{s}_i^+<t_i+\tau(t_i)$ as otherwise in order to cover $\mathrm{ST}(\mathcal{K})$ assumption (3) from above would not hold.
Pick $s_i^+$ as some point in $(\tilde{s}_i^+, t_i+\tau(t_i))$ and $s_{i+1}^-$ as some point in $( t_{i+1}-\tau(t_{i+1}), t_i)$. The lack of singular times in $[0,s_0^-]$ and in each $[s_i^+, s_{i+1}^-]$ follows by our choices and assumptions (1) and (3) above.
For $1\leq i \leq N^\prime$ set $\Sigma^i_\pm = \spt(\mu_{s_i^\pm})$. By the choice of $s_i^\pm$, each $\Sigma^i_\pm$ is a closed hypersurface and, as there are no singular times between $s_i^+$ and $s_{i+1}^-$, we have for $1\leq i \leq N^\prime-1$ diffeomorphisms $\Phi^i: \Sigma_+^i \to \Sigma_-^{i+1}$ coming from the flow and, for the same reason, a diffeomorphism $\Phi^0: \Sigma^1\to \Sigma^1_-$. Observe that, a priori, the $\Sigma^i_\pm$ need not consist of one component (indeed, $\Sigma^{N^\prime}_+$ is empty). By Corollary <ref>, $\sing_{t_i}(\mathcal{K})$ is finite for each $1\leq i \leq N^\prime$ and we write
\set{\xX_i^1, \ldots, \xX_i^{M(i)}}=\sing{}_{t_i}(\mathcal{K})
i.e., the $(\xX_i^j,t_i)$ are the singular points of the flow at time $t_i$. Up to relabeling, there is an $0\leq m(i)\leq M(i)$ so that for $1\leq j \leq m(i)$, $(\xX_i^j, t_i)\in \sing^{NC}(\mathcal{K})$ while for $m(i)<j\leq M(i)$, $(\xX_i^j,t_i)\in \sing^C(\mathcal{K})$. Set $R^i=\alpha\rho(t_i)$ and, for each $\xX_i^j$, let $U^{i}_{j,\pm}\subset \Sigma^i_\pm$ be the sets $ B_{R^i}(\xX_i^j)\cap \Sigma^i_\pm$. By (<ref>) for fixed $j$, these are pairwise disjoint sets and, by Theorem <ref>, these intersections are transverse and so the $\sigma^{i}_{j,\pm}=\partial U^i_{j,\pm}$ are submanifolds of $\Sigma^i_\pm$. Hence, the $U^i_{j,\pm}$ are smooth pairwise disjoint domains.
Furthermore, by Theorem <ref> and fact that $\tau(t)<\rho(t)$, each $\bar{U}^i_{j,-}$ is diffeomorphic to $\bar{B}_{\alpha}\cap \Gamma_{j}^i$ for some $\Gamma_{j}^i \in \mathcal{S}_n$. In particular, for $j>m(i)$ we have that $\bar{U}^i_{j,-}$ is a closed connected hypersurface, while for $1\leq j\leq m(i)$, $\partial \bar{U}^i_{j,-}$ is non-empty and connected. Hence, for $j>m(i)$, $\bar{U}^i_{j,+}=\emptyset$, while for $1\leq j\leq m(i)$, $\partial \bar{U}^i_{j,+}$ is non-empty and connected.
Furthermore, Theorem <ref> implies that there are diffeomorphisms (see Appendix A)
$$\Psi^i: \Sigma_-^i\backslash \bigcup_{j=1}^{M(i)} U_{j,-}^i \to \Sigma_+^i \backslash \bigcup_{j=1}^{M(i)} U_{j,+}^i.$$
As $\Sigma^1$ is connected and $\Phi^0(\Sigma^1)=\Sigma^1_-$, $\Sigma^1_-$ is also connected. As each $\sigma^1_{j,-}$ is connected, we obtain that $\hat{\Sigma}^1_-=\Sigma^1_-\backslash \bigcup_{j=1}^{M(1)} U_{j,-}^1$ is connected. Let $\tilde{\Sigma}^1_+$ be the connected component of $\Sigma^1_+$ that contains $\Psi^1(\hat{\Sigma}^1_{-})$. Inductively, let $\tilde{\Sigma}^{i+1}_-=\Phi^i(\tilde{\Sigma}^i_+)$ and $\hat{\Sigma}^{i+1}_-=\tilde{\Sigma}^{i+1}_-\backslash \bigcup_{j=1}^{M(i+1)} U_{j,-}^{i+1}$ and define $\tilde{\Sigma}^{i+1}_+$ to be the connected component of $\Sigma^{i+1}_+$ that contains $\Psi^{i+1}(\hat{\Sigma}^{i+1}_-)$. Here we adopt the convention that if $\hat{\Sigma}^{i+1}_-=\emptyset$, then $\tilde{\Sigma}^{i+1}_+=\emptyset$. It follows inductively that each $\tilde{\Sigma}^i_{\pm}$ is connected.
Let $\tilde{\Phi}^i: \tilde{\Sigma}^i_+\to \tilde{\Sigma}^{i+1}_-$ be the diffeomorphisms given by restricting the $\Phi^i$. To be consistent we also set $\tilde{\Sigma}^1_-=\Sigma^1_-$ and $\tilde{\Phi}^0=\Phi^0$.
Finally let
N=\max\set{1\leq i\leq N^\prime: \tilde{\Sigma}^k_-\neq\emptyset\mbox{ for all $1\leq k\leq i$}}.
If $N<N^\prime$, then, by constructions, $\hat{\Sigma}^N_-=\emptyset$ and $\tilde{\Sigma}^N_- = U^N_{j,-}$ for some $j>m(N)$. If $N=N^\prime$, then $t_N=T(\mathcal{K})$ at which all singularities are compact. Thus it follows from <cit.> that $\tilde{\Sigma}^N_-$ is diffeomorphic to $\mathbb{S}^n$. The theorem now follows by taking $\Sigma^i=\tilde{\Sigma}^i_-$ for $2\leq i \leq N$ and $\hat{\Phi}^i$ are the diffeomorphisms given by $(\tilde{\Phi}^i\circ\Psi^i)^{-1}$.
§ A SHARPENING OF <CIT.>
In order to prove Theorem <ref>, we begin with an elementary lemma.
If $\xX_1, \ldots, \xX_{m+1}\in \Real^{n+1}$ is a sequence of points so that
\begin{equation} \label{StupidHyp}
|\xX_i-\xX_{i+1}|\leq \hat{K} (1+|\xX_i|)^{-1}
\end{equation}
for $1\leq i \leq m$ and some $\hat{K}\geq 0$, then
\begin{equation} \label{StupidClaim}
|\xX_1-\xX_{m+1}|\leq K(m) (1+|\xX_1|)^{-1}
\end{equation}
where $K(m)=(\hat{K}+1)^m-1$.
We proceed by induction on $m$. The lemma is obviously true when $m=1$. Suppose (<ref>) holds for $m=m'$. Using this induction hypothesis with (<ref>) implies that
|\xX_1-\xX_{m'+2}|\leq |\xX_1-\xX_{m'+1}|+|\xX_{m'+1}-\xX_{m'+2}|\leq K(m') (1+|\xX_1|)^{-1}+\hat{K} (1+|\xX_{m'+1}|)^{-1}.
Furthermore, by the induction hypothesis and triangle inequality
$$ |\xX_{1}|\leq K(m') (1+|\xX_{1}|)^{-1}+ |\xX_{m'+1}|.$$
As $K(m')\geq 0$ and $(1+|\xX_{1}|)^{-1}\leq 1$, this implies that
1+ |\xX_{1}| \leq 1+K(m')+|\xX_{m'+1}|\leq (1+K(m')) (1+|\xX_{m'+1}|).
That is,
$$ (1+|\xX_{m'+1}|)^{-1}\leq (1+K(m')) (1+|\xX_{1}|)^{-1}.
|\xX_1-\xX_{m'+2}|\leq (K(m')+\hat{K}(1+K(m'))) (1+|\xX_1|)^{-1}
and, by the induction hypothesis, $K(m')=(\hat{K}+1)^{m'}-1$ and so setting
verifies that (<ref>) holds for $m=m'+1$ and finishes the proof.
We next observe that the proof of the main result of <cit.> actually allows us to make the following more refined conclusion.
Fix $n\geq 2$, if $\Sigma\in \mathcal{ACS}_n[\lambda_{n-1}]$, then there is a homeomorphic involution $\phi:\mathbb{S}^{n}\to \mathbb{S}^{n}$ which fixes $\mathcal{L} (\Sigma)$, the link of the asymptotic cone, $\mathcal{C} (\Sigma)$, of $\Sigma$, and swaps the two components of $\mathbb{S}^n\backslash \mathcal{L}(\Sigma)$.
By <cit.>, the link $\mathcal{L}(\Sigma)$ is connected and separates $\mathbb{S}^n$ into two components $\Omega_+$ and $\Omega_-$. In particular, $\mathcal{L}(\Sigma)=\partial \bar{\Omega}_+=\partial \bar{\Omega}_-$. In order to construct $\phi$, it is enough to show the existence of a homeomorphism ${\psi}: \bar{\Omega}_+\to \bar{\Omega}_-$ so that ${\psi}|_{\mathcal{L}(\Sigma)} : \mathcal{L}(\Sigma)\to \mathcal{L}(\Sigma)$ is the identity map.
Indeed, if such a ${\psi}$ exists, one defines $\phi$ by
\phi(p)=\left\{ \begin{array}{ll} {\psi}(p) & p\in \bar{\Omega}_+ \\ {\psi}^{-1}(p) & p \in \Omega_- \end{array}\right.
To explain the construction of ${\psi}$ let us first summarize the main objects used in the proof of <cit.>. First, recall that it is shown there that associated to $\Sigma$ are two smooth mean curvature flows $\set{\Gamma_t^{\pm }}_{t\in[-1,0]}$ with $\Gamma_{-1}^{+}$ the normal exponential graph over $\Sigma$ of a small positive multiple of the lowest eigenfunction of the self-shrinker stability operator of $\Sigma$ (normalized to be positive) and $\Gamma^-_{-1}$ to be a small negative multiple of this function. In particular, by choosing the multiple small enough, one can ensure both that
$\Gamma^+_{-1}$ is the exponential normal graph of some function on $\Gamma^-_{-1}$ and that $\Gamma^-_{-1}$ is the exponential normal graph of some function on $\Gamma^+_{-1}$.
Furthermore, up to relabeling, each $\Gamma^{\pm}=\Gamma^{\pm}_0$ is diffeomorphic to $\Omega^\pm$ the components of $\mathbb{S}^n\backslash \mathcal{L}(\Sigma)$. Moreover, these diffeomorphisms, which we denote by $\Pi^\pm$, are given by restricting the map
\begin{equation*}
\Pi(p)=\frac{\xX(p)}{|\xX(p)|}
\end{equation*}
to $\Gamma^\pm$.
We next use the flow $\set{\Gamma^\pm_{t}}_{t\in[-1,0]}$ to construct a natural diffeomorphism ${\Psi}: \Gamma^+\to \Gamma^-$ which has the property that there is a constant $K>0$ so that
\begin{equation}
\label{DistortionEst}\left|\xX(p) -\xX({\Psi}(p))\right|\leq \frac{K}{1+|\xX(p)|}.
\end{equation}
We do so iteratively. Specifically, by <cit.> there is a constant $\tilde{C}_0>0$ so that
\begin{equation}
\label{GammaCurvEst}
\sup_{t\in [-1,0]} \sup_{\Gamma_t^\pm} \left( |A_{\Gamma^\pm_t}|+|\nabla_{\Gamma^\pm_t} A_{\Gamma^\pm_t}|\right)<\tilde{C}_0.
\end{equation}
This, together with <cit.>, implies that there is a $\rho>0$ so that for each $t\in [-1,0]$,
$\mathcal{T}_{\rho}(\Gamma_t^\pm)$ is a regular tubular neighborhood of $\Gamma_t^\pm$. It follows from this and (<ref>) that there is a $\delta>0$ so that if $t_1, t_2\in [-1,0]$ and $|t_1-t_2|<\delta$, then $\Gamma^\pm_{t_1}$ is a normal exponential graph over $\Gamma^\pm_{t_2}$ and vice versa. As such, for all $t_1, t_2\in [-1,0]$ with $|t_1-t_2|<\delta$, there is a diffeomorphism
$${\Psi}^\pm_{t_2,t_1}: \Gamma_{t_1}^\pm \to \Gamma_{t_2}^\pm$$
defined by nearest point projection from $\Gamma_{t_1}^\pm$ to $\Gamma^{\pm}_{t_2}$.
Pick $M\in \mathbb{N}$ so $M\delta>1$ and choose $0=s_0>s_1>\ldots >s_M=-1$ so that $|s_i-s_{i+1}|<\delta$ and define a diffeomorphism $\Psi^-: \Gamma^-_{-1}\to \Gamma^-$ by
\Psi^-= \Psi^-_{s_0, s_1}\circ \Psi^-_{s_1,s_2} \circ \cdots \circ \Psi^-_{s_{M-1}, s_{M}}.
Likewise, define a diffeomorphism $\Psi^+:\Gamma^+ \to \Gamma^+_{-1}$ by
\Psi^+=\Psi^+_{s_{M}, s_{M-1}}\circ \Psi^+_{s_{M-1}, s_{M-2}}\circ \cdots \circ \Psi^+_{s_1,s_0}
and let $\Psi^{+,-}:\Gamma^+_{-1}\to \Gamma^-_{-1}$ be given by nearest point projection. By construction, this is also a diffeomorphism and so the map
\Psi=\Psi^-\circ \Psi^{+,-}\circ \Psi^+
is a diffeomorphism $\Psi:\Gamma^+\to \Gamma^-$.
By construction, if $t_1,t_2\in [-1,0]$ and $|t_1-t_2|<\delta$, then for all $p\in \Gamma_{t_1}^\pm$,
\begin{equation}
\label{SillyEst}
|\xX(p)-\xX(\Psi_{t_2, t_1}^\pm(p) )|<\rho.
\end{equation}
Furthermore, <cit.> implies that for $t\in [-1,0]$ each $\Gamma_t^\pm $ is smoothly asymptotic to $\mathcal{C}(\Sigma)$. In particular, there is a $R>0$ and functions $u_t^\pm$ on $\mathcal{C}(\Sigma)\backslash B_R$ whose normal exponential graph over $\mathcal{C}(\Sigma)$ sits inside of $\Gamma^\pm_t$ and contains $\Gamma^\pm_t\backslash B_{2R}.$
Moreover, by <cit.> and <cit.>
there is a constant $K^\prime>0$ so that for $p\in \mathcal{C}(\Sigma)\backslash B_R$,
|u_t^\pm(p)|\leq K^\prime (1+|\xX(p)|)^{-1}.
Hence, for any $t_1,t_2\in [-1,0]$, if $p \in \Gamma^\pm_{t_1}\backslash B_{2R}$, then there is a point $p'\in \mathcal{C}(\Sigma)\backslash B_R$ so that
\begin{equation}
\label{DecayEst}
|\xX(p)-\xX(p')|\leq K^\prime (1+ |\xX(p')|)^{-1}
\end{equation}
and also a point $p''\in \Gamma^\pm_{t_2}$ so that
\begin{equation}
|\xX(p')-\xX(p'')|\leq K^\prime (1+ |\xX(p')|)^{-1}.
\end{equation}
Hence, if $|t_1-t_2|<\delta$, then as $\Psi^\pm_{t_2,t_1}$ is given by nearest point projection,
\begin{align*}
|\xX(p)-\xX(\Psi^\pm_{t_2,t_1}(p))| &\leq |\xX(p)-\xX(p'')|\\
&\leq |\xX(p)-\xX(p')|+|\xX(p')-\xX(p'')|\\
&\leq 2 K^\prime (1+|\xX(p')|)^{-1}.
\end{align*}
As $K^\prime>0$ and $1+|\xX(p')|\geq 1$, (<ref>) implies that
(1+|\xX(p')|)^{-1}\leq (1+K^\prime)(1+|\xX(p)|)^{-1},
and so
|\xX(p)-\xX(\Psi^\pm_{t_2,t_1}(p))|\leq 2 K^\prime (1+K^\prime)(1+|\xX(p)|)^{-1}.
Combining this with (<ref>)
one obtains that for all $p \in \Gamma^\pm_{t_1}$,
|\xX(p)-\xX(\Psi^\pm_{t_2,t_1}(p))|\leq \hat{K}(1+|\xX(p)|)^{-1}
where $\hat{K}=2K^\prime (1+K^\prime)+\rho(1+2R)$.
By the same arguments, for all $p\in \Gamma^{+}_{-1}$,
|\xX(p)-\xX(\Psi^{+,-}(p))|\leq \hat{K}(1+|\xX(p)|)^{-1}.
Hence, it follows from Lemma <ref>, that
|\xX(p)-\xX(\Psi(p))|\leq K(1+|\xX(p)|)^{-1}
where $K=(1+\hat{K})^{2M+2}-1$.
To complete the proof set
\psi(p)=\left\{\begin{array}{cc} \Pi^-(\Psi((\Pi^+)^{-1}(p))) & p \in \Omega_+ \\ p & p\in \partial \Omega_+. \end{array} \right.
We claim that $\psi$ is a homeomorphism.
First, note that, by <cit.>, there is an $R>1$ and $\tilde{C}_1>1$ so that if $p\in \Gamma^\pm \backslash B_{R}$, then
\tilde{C}_1^{-1} |\xX(p)|^{2\mu} < \dist_{\Real^{n+1}}(p, \mathcal{C}(\Sigma)) < \tilde{C}_1 |\xX(p)|^{-1}
where $\mu<-1$. Hence,
\begin{equation}\label{TwoSidedEst}
C^{-1} |\xX(p)|^{2\mu-1}<\dist_{\mathbb{S}^n}(\Pi^\pm (p), \mathcal{L}(\Sigma)) < C |\xX(p)|^{-2}
\end{equation}
where $C\geq\tilde{C}_1$.
Hence, for $q\in \Omega^+$, with $\dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))$ sufficiently small, if we set $q'=(\Pi^+)^{-1}(q)\in \Gamma^+$, then
|\xX(q')|\geq C^{\frac{1}{2\mu-1}} \dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))^{\frac{1}{2\mu-1}}.
By (<ref>),
\begin{align*}
| |\xX(\Psi(q'))|-|\xX(q')|| &\leq
&\leq K C^{-\frac{1}{2\mu-1}} \dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))^{-\frac{1}{2\mu-1}}.
\end{align*}
Hence, for $\dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))$ sufficiently small,
\dist_{\mathbb{S}^n}(q, \psi(q))\leq 4K C^{-\frac{1}{2\mu-1}} \dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))^{-\frac{1}{2\mu-1}} |\xX(q')|^{-1} .
Using (<ref>), again gives
\dist_{\mathbb{S}^n}(q, \psi(q))\leq 4K C^{-\frac{2}{2\mu-1}} \dist_{\mathbb{S}^n}(q, \mathcal{L}(\Sigma))^{-\frac{2}{2\mu-1}}.
As $\mu<-1$, for any $q_0\in \mathcal{L}(\Sigma)$, the right hand side goes to $0$ as $q\to q_0$. By the triangle inequality
\dist_{\mathbb{S}^n}(q_0, \psi(q))\leq \dist_{\mathbb{S}^n}(q, \psi(q))+\dist_{\mathbb{S}^n}(q, q_0)
and so the right hand side goes to $0$ as $q\to q_0$. Hence, $\psi$ is continuous. Finally, as $\bar{\Omega}_+$ is compact and $\bar{\Omega}_-$ is Hausdorff, $\psi$ is a closed map and hence, as $\psi$ is a bijection, it is a homeomorphism.
Theorem <ref> is a standard topological consequence of Proposition <ref>.
(of Theorem <ref>)
Observe that as $\mathcal{L}(\Sigma)$ is connected, by <cit.>, there are exactly two components of $\mathbb{S}^n\backslash \mathcal{L}(\Sigma)$, which we denote by $U^\pm$. Let $\phi:\mathbb{S}^n\to \mathbb{S}^n$ be the homeomorphism given by Proposition <ref> so $\phi(U^-)=U^+$. Pick a regular tubular neighborhood $T\subset \mathbb{S}^n$ of $\mathcal{L}(\Sigma)$. We let $V^\pm =U^\pm \cup T$ and observe that $\bar{U}^\pm$, the closure of $U^\pm$, is a retract of $V^\pm$ and that $\mathcal{L}(\Sigma)$ is a retraction of $T=V^-\cap V^+$.
As $\bar{U}^\pm$ is a retraction of $V^\pm$ and $\mathcal{L}(\Sigma)$ is a retraction of $T$, the natural inclusion maps induce isomorphisms between the reduced homology groups $\tilde{H}_k(\bar{U}^\pm)$ and $\tilde{H}_k(V^\pm)$ and between $\tilde{H}_k(\mathcal{L}(\Sigma))$ and $\tilde{H}_k(T)$. As such, there is a natural map $\Phi: \tilde{H}_k(V^-)\to \tilde{H}_k(V^+)$ defined by the following diagram,
\begin{equation*}
\begin{tikzcd}
{} & \tilde{H}_k(T) \arrow{r}{j^-_*} \arrow[pos=0.3]{dr}{j^+_*} & \tilde{H}_k(V^-) \arrow{d}{\Phi}\\
\tilde{H}_k(\mathcal{L}(\Sigma)) \arrow[leftrightarrow]{ur}{\simeq}\arrow{r}{i^-_*} \arrow{dr}{i^+_*} & \tilde{H}_k(\bar{U}^-) \arrow[leftrightarrow, crossing over, pos=0.3]{ur}{\simeq} \arrow{d}{\phi_*} & \tilde{H}_k(V^+) \\
{} & \tilde{H}_k(\bar{U}^+) \arrow[leftrightarrow]{ur}{\simeq}&
\end{tikzcd}
\end{equation*}
where $i^\pm: \mathcal{L}(\Sigma)\to \bar{U}^\pm$ and $j^\pm:T\to V^\pm$ denote the natural inclusion maps and we used that $\phi\circ i^-=i^+$. As $\phi$ is a homeomorphism, both $\phi_*$ and $\Phi$ are isomorphisms. This implies that the map
\begin{equation*}
J=(j^-_*, -j^+_*):\tilde{H}_k(T) \to \tilde{H}_k(V^-)\oplus \tilde{H}_k(V^+)
\end{equation*}
is surjective if and only if $\tilde{H}_k(V^-)=\tilde{H}_k(V^+)=\set{0}$. Indeed, if the map is surjective, then for any element $\alpha\in \tilde{H}_k(V^-)$ there is an element $\beta \in \tilde{H}_k(T)$ so that $J(\beta)=(\alpha,0)$. That is, $j_*^-(\beta)=\alpha$ and $j_*^+(\beta)=0$. Hence, $0=j_*^+(\beta)=\Phi(j_*^-(\beta))=\Phi(\alpha)$. In other words, as $\Phi$ is an isomorphism, $\alpha\in \ker(\Phi)=\set{0}$ and so $\tilde{H}_k(V^-)=\set{0}$. The proof that $\tilde{H}_k(V^+)=\set{0}$ is the same. The converse is immediate.
We next recall several standard facts about the reduced homology of manifolds and of manifolds with boundary. First of all, as $\mathcal{L}(\Sigma)$ is a connected, oriented $(n-1)$-dimensional manifold, $\tilde{H}_k(\mathcal{L}(\Sigma))=\tilde{H}_k(T)=\set{0}$ for $k=0$ and $k\geq n$ and $\tilde{H}_{n-1}(\mathcal{L}(\Sigma))=\tilde{H}_{n-1}(T)=\mathbb{Z}$. Likewise, as the $\bar{U}^\pm$ are connected, oriented $n$-manifolds with boundary, $\tilde{H}_k(\bar{U}^\pm)=\tilde{H}_k(V^\pm)=0$ for $k=0$ and $k\geq n$.
In order to compute the remaining reduced homology groups, we use the Mayer-Vietoris long exact sequence for the reduced homology of $(V^-,V^+,\mathbb{S}^n)$. This gives the following exact sequences for $k\geq 0$
\begin{equation}\label{ExactSeq}
\begin{tikzcd}
\tilde{H}_{k+1}(\mathbb{S}^n)\arrow{r} & \tilde{H}_k(T) \arrow{r}{J} & \tilde{H}_k(V^-)\oplus \tilde{H}_k(V^+)
\arrow{r} & \tilde{H}_k(\mathbb{S}^n).
\end{tikzcd}
\end{equation}
As $\tilde{H}_k(\mathbb{S}^n)=\mathbb{Z}$ for $k=n$ and is otherwise $\set{0}$, (<ref>) implies that $J$ is surjective for $0\leq k\leq n-1$. Hence, for these $k$, $\tilde{H}_k(\bar{U}^\pm)=\tilde{H}_k(V^\pm)=\set{0}$ and so the $U^\pm$ are homology $n$-balls as claimed. As such, (<ref>) further implies that $\tilde{H}_k(\mathcal{L}(\Sigma))=\tilde{H}_k(T)=\set{0}$ for $0\leq k \leq n-2$ completing the verification that $\mathcal{L}(\Sigma)$ is a homology $(n-1)$-sphere.
To conclude the proof, it is enough, by the Hurewicz theorem, to show that $\pi_1(U^\pm)=\pi_1(\bar{U}^\pm)=\set{1}$.
To that end first observe that the maps
$F^\pm: \mathbb{S}^n\to \bar{U}^\pm$ defined by
F^\pm(p)=\left\{ \begin{array}{ll} p & p\in \bar{U}^\pm \\ {\phi}(p) & p \in U^\mp \end{array} \right.
are continuous. Now suppose $\gamma$ is a closed loop in $\bar{U}^\pm$. As $\pi_1(\mathbb{S}^n)=\set{1}$, there is a homotopy $H: \mathbb{S}^1\times[0,1]\to \mathbb{S}^n$ taking $\gamma$ to a point. Clearly, $F^\pm\circ H: \mathbb{S}^1\times[0,1]\to \bar{U}^\pm$ is also a homotopy taking $\gamma$ to a point. That is, $\pi_1(\bar{U}^\pm)=\set{1}$.
(of Corollary <ref>)
By Theorem <ref>, $\mathcal{L}(\Sigma)$ is a homology $2$-sphere. By the classification of surfaces this means that $\mathcal{L}(\Sigma)$ is diffeomorphic to $\mathbb{S}^2$ and so Alexander's Theorem <cit.> implies that both components of $\mathbb{S}^3\backslash \mathcal{L}(\Sigma)$ are diffeomorphic to $\Real^3$, proving the claim.
§ SURGERY PROCEDURE
We prove Theorem <ref> using Corollary <ref> and Theorem <ref>.
(of Theorem <ref>)
We first observe that $\rstar{3,\lambda_2}$ holds by <cit.> and that $\rsstar{3,\lambda_2}$ holds by <cit.>. If $\Sigma$ is (after a translation and dilation) a self-shrinker, then, by <cit.>, $\Sigma$ is diffeomorphic to $\mathbb{S}^3$, proving the theorem. Otherwise, flow $\Sigma$ for a small amount of time by the mean curvature flow (using short time existence of for smooth closed initial hypersurfaces) to obtain a hypersurface, $\Sigma^\prime$, diffeomorphic to $\Sigma$ and, by Huisken's monotonicity formula, with $\lambda[\Sigma']<\lambda[\Sigma]$. On the one hand, if the level set flow of $\Sigma'$ is non-fattening, then we set $\Sigma_0=\Sigma'$. On the other hand, if the level set flow of $\Sigma'$ is fattening, then we can take $\Sigma_0$ to be a small normal graph over $\Sigma'$ so that $\lambda[\Sigma_0]<\lambda[\Sigma]$ and, because the non-fattening condition is generic, the level set flow of $\Sigma_0$ is non-fattening.
Hence, the hypotheses of Section <ref> hold and we may apply Theorem <ref> unconditionally to obtain a family of hypersurfaces $\Sigma^1, \ldots, \Sigma^N$ in $\Real^4$. As $\Sigma^N$ is diffeomorphic to $\mathbb{S}^3$, if $N=1$, then there is nothing more to show and so we may assume that $N>1$. We will now show that $\Sigma^{N-1}$ is diffeomorphic to $\Sigma^N$ and hence to $\mathbb{S}^3$.
Let us denote by $V= \cup_{j=1}^{m(N-1)} V_j^{N-1}$ and by $\hat{\Sigma}^N=\Sigma^N\backslash V$ and let $U= \cup_{j=1}^{m(N-1)} U_j^{N-1}$ and $\hat{\Sigma}^{N-1}=\Sigma^{N-1}\backslash U$ so $\hat{\Phi}^{N-1}: \hat{\Sigma}^N \to \hat{\Sigma}^{N-1}$ is the orientation preserving diffeomorpism given by Theorem <ref>. By Corollary <ref>, each component of $\bar{U}$ is diffeomorphic to a closed three-ball $\bar{B}^3$. Hence, each component of $\partial \hat{\Sigma}^{N-1}$ and $\partial \hat{\Sigma}^{N}$ is diffeomorphic to $\mathbb{S}^2$. That is, for $1\leq j \leq m(N-1)$, $\partial V_j^{N-1}$ is diffeomorphic to $\mathbb{S}^2$ and so, as $\Sigma^N$ is diffeomorphic to the three-sphere, Alexander's theorem <cit.> implies that each $\bar{V}_j^{N-1}$ is diffeomorphic to the closed three-ball. Hence, there are orientation preserving diffeomorphisms $\Psi_j^{N-1}: \bar{V}_j^{N-1}\to \bar{U}_j^{N-1}$.
Denote by $\hat{\phi}^{N-1}_j: \partial V_j^{N-1} \to \partial U_j^{N-1}$ the diffeomorphism given by restricting $\hat{\Phi}^{N-1}$ and, likewise, let $\psi^{N-1}_j: \partial V_j^{N-1}\to \partial U_j^{N-1}$ denote the diffeomorphisms given by restricting $\Psi^{N-1}_j$. Observe, that the orientation of $\hat{\Sigma}^{N}$ and the orientation on $\bar{V}$ induce opposite orientations on $\partial \bar{V}$. Likewise, the orientation of $\hat{\Sigma}^{N-1}$ and that of $\bar{U}$ induce opposite orientations on $\partial \bar{U}$. By construction, the $\hat{\phi}^{N-1}_j$ preserve the orientations induced from $\hat{\Sigma}^N$ and $\hat{\Sigma}^{N-1}$. Hence, as the orientations induced by $\bar{V}_j^{N-1}$ and $\bar{U}_j^{N-1}$ are opposite to those induced by $\hat{\Sigma}^N$ and $\hat{\Sigma}^{N-1}$, the $\hat{\phi}^{N-1}_j$ also preserve these orientations. The same is true of the $\psi^{N-1}_j$. As such, $\xi_j^{N-1}=(\psi_{j}^{N-1})^{-1}\circ \hat{\phi}_j^{N-1}\in \mathrm{Diff}_+(\partial V_j^{N-1})$, where $\mathrm{Diff}_+(M)$ is the space of orientation preserving self-diffeomorphisms of an oriented manifold $M$ (here we may use the orientation on $\partial V_j^{N-1}$ induced by either $\bar{V}$ or $\hat{\Sigma}^N$). By <cit.> – see also <cit.> and <cit.> – the space $\mathrm{Diff}_+(\mathbb{S}^2)$ is path-connected and so any element of $\mathrm{Diff}_+(\mathbb{S}^2)$ extends to an element of $\mathrm{Diff}_+(\bar{B}^3)$. That is, there are diffeomorphism $\Xi_j^{N-1}\in \mathrm{Diff}_+( \bar{V}_j^{N-1})$ that restrict to $\xi_{j}^{N-1}$ on $\partial V_j^{N-1}$. Thus, the maps $\hat{\Psi}_j^{N-1}=\Psi_j^{N-1}\circ \Xi_{j}^{N-1}: \bar{V}_j^{N-1} \to \bar{U}_j^{N-1}$ are diffeomorphisms that agree with $\hat{\Phi}^{N-1}$ on the common boundary.
Define $\Phi^{N-1}:\Sigma^N \to \Sigma^{N-1}$ by
\Phi^{N-1}(p)=\left\{ \begin{array}{cc} \hat{\Phi}^{N-1}(p) & p \in \hat{\Sigma}^N \\ \hat{\Psi}_j^{N-1}(p) & p\in V_j^{N-1}. \end{array} \right.
By construction, this map is a homeomorphism. However, it is a standard procedure to construct a diffeomorphism between $\Sigma^N$ and $\Sigma^{N-1}$ by smoothing this map out (see for instance <cit.>). Hence, $\Sigma^{N-1}$ is diffeomorphic to $\mathbb{S}^3$ and iterating this argument shows that $\Sigma=\Sigma^1$ is diffeomorphic to $\mathbb{S}^3$ as claimed.
Theorem <ref> follows from Theorem <ref>, Theorem <ref> and the Mayer-Vietoris long exact sequence for reduced homology. For completeness, we include a proof of the following standard topological fact which we will need to use.
Let $M$ be a closed $n$-dimensional manifold and $\Sigma \subset M$ a closed hypersurface. If $M$ is a homology $n$-sphere and $\Sigma$ is a homology $(n-1)$-sphere, then each component of $M\backslash \Sigma$ is a homology $n$-ball.
Our hypotheses ensure that both $M$ and $\Sigma$ are connected and oriented. Hence, $\Sigma$ is two-sided and there is an open $U^+\subset M$ so that $\Sigma =\partial U^+$. Let $U^-=M\backslash \bar{U}^+$. To prove the lemma we will need to compute the Mayer-Vietoris long exact sequence for $(\bar{U}^-, \bar{U}^+, M)$. Strictly speaking, we should “thicken" $\bar{U}^+$ and $\bar{U}^-$ up with a regular tubular neighborhood of $\Sigma=\partial \bar{U}^\pm$ as in the proof of Theorem <ref>, but we leave the details of this to the reader.
The Mayer-Vietoris long exact sequence and the fact that $M$ is a homology $n$-sphere and $\Sigma$ is a homology $(n-1)$-sphere gives the sequences
\begin{equation*}
\begin{tikzcd}
\tilde{H}_{k+1}(M)\arrow{r}{\partial} \arrow[leftrightarrow]{d}{=} & \tilde{H}_k(\Sigma) \arrow{r} \arrow[leftrightarrow]{d}{=}& \tilde{H}_k(\bar{U}^-)\oplus \tilde{H}_k(\bar{U}^+)
\arrow{r} \arrow[leftrightarrow]{d}{=} & \tilde{H}_k(M) \arrow[leftrightarrow]{d}{=}\\
\tilde{H}_{k+1}(\mathbb{S}^n) \arrow{r}{\partial} & \tilde{H}_k(\mathbb{S}^{n-1}) \arrow{r} & \tilde{H}_k(\bar{U}^-)\oplus \tilde{H}_k(\bar{U}^+)
\arrow{r} & \tilde{H}_k(\mathbb{S}^{n}).
\end{tikzcd}
\end{equation*}
For $0\leq k\leq n-2$ and $k\geq n+1$ this immediately gives that $\tilde{H}_k(\bar{U}_\pm)=\set{0}$. When $k={n-1}$, the map $\partial$ is necessarily generated by $[M]\mapsto [\Sigma]$ where $[M] $ is the fundamental class of $M$ and $[\Sigma]$ is the fundamental class of $\Sigma$. In particular, this map is an isomorphism and so we conclude that $\tilde{H}_{n-1}(\bar{U}^\pm)=\set{0}$. For the same reason, $\tilde{H}_n(\bar{U}^\pm)=\set{0}$, which verifies the claim.
(of Theorem <ref>)
Arguing as in the first paragraph of the proof of Theorem <ref>, we obtain $\Sigma^1,\ldots, \Sigma^N$ the hypersurfaces given by Theorem <ref>. As $\Sigma^N$ is diffeomorphic to $\mathbb{S}^n$, it is a homology $n$-sphere. In particular, if $N=1$, then there is nothing further to show. As such, we may assume that $N>1$.
Let us show that $\Sigma^{N-1}$ is a homology $n$-sphere. First, set $V= \cup_{j=1}^{m(N-1)} V_j^{N-1}$ and $\hat{\Sigma}^N=\Sigma^N\backslash V$ and let $U= \cup_{j=1}^{m(N-1)} U_j^{N-1}$ and $\hat{\Sigma}^{N-1}=\Sigma^{N-1}\backslash U$. Next observe that, as $\partial U_j^{N-1}=\mathcal{L}(\Gamma_j^{N-1})$ for some $\Gamma_j^{N-1}\in \mathcal{ACS}_n^*(\Lambda)$, Theorem <ref> implies that each component of $\partial \hat{\Sigma}^{N-1}$ is a homology $(n-1)$-sphere. Hence, as $\partial U=\partial \hat{\Sigma}^{N-1}$ is diffeomorphic to $\partial \hat{\Sigma}^N=\partial V$, we see that each component of $\partial V=\partial \hat{\Sigma}^N$ is a homology $(n-1)$-sphere and so Lemma <ref> implies that each component of $\bar{V}$ is a homology $n$-ball.
We may now use the Mayer-Vietoris long exact sequence to compute that $\tilde{H}_k(\hat{\Sigma}^N)=\set{0}$ for $k\neq n-1$ and $\tilde{H}_{n-1}(\hat{\Sigma}^N)=\mathbb{Z}^{m(N-1)-1}$. To see this, consider the Mayer-Vietoris long exact sequence of $(\bar{V}, \hat{\Sigma}^N, \Sigma^N)$. This long exact sequence and the fact that $\bar{V}$ is the union of homology $n$-balls gives, for $k>0$, the exact sequences
\begin{equation*}
\begin{tikzcd}
\tilde{H}_{k+1}(\Sigma^N)\arrow{r}{\partial} \arrow[leftrightarrow]{d}{=} & \tilde{H}_k(\partial V) \arrow{r} \arrow[leftrightarrow]{d}{=}& \tilde{H}_k(\bar{V})\oplus \tilde{H}_k(\hat{\Sigma}^N)
\arrow{r}\arrow[leftrightarrow]{d}{=} & \tilde{H}_k(\Sigma^N)\arrow[leftrightarrow]{d}{=}\\
\tilde{H}_{k+1}(\mathbb{S}^n)\arrow{r}{\partial} & \bigoplus\limits_{j=1}^{m(N-1)}\tilde{H}_k(\mathbb{S}^{n-1}) \arrow{r} & \tilde{H}_k(\hat{\Sigma}^N)
\arrow{r} & \tilde{H}_k(\mathbb{S}^n).
\end{tikzcd}
\end{equation*}
Hence, for $1\leq k\leq n-2$ and $k\geq n+1$, $\tilde{H}_k(\hat{\Sigma}^N)=\set{0}$. When $k=n-1$, the map $\partial$ is generated by $ [\Sigma^N]\mapsto([\partial V_{1}^{N-1}], \ldots, [\partial V_{m(N-1)}^{N-1}])$ where $[\Sigma^N]$ is the fundamental class of $\Sigma^N$ and $[\partial V_{j}^{N-1}]$ is the fundamental class of $\partial V_{j}^{N-1}$. It follows that $ \tilde{H}_{n-1}(\hat{\Sigma}^N)=\mathbb{Z}^{m(N-1)-1}$ and, as this map is injective, that $\tilde{H}_{n}(\hat{\Sigma}^N)=\set{0}$.
Finally, as $\hat{\Sigma}^N$ is connected, $\tilde{H}_0(\hat{\Sigma}^N)=\set{0}$, which completes the computation.
By Theorem <ref>, $\hat{\Sigma}^N$ is diffeomorphic to $\hat{\Sigma}^{N-1}$ and so $\tilde{H}_k(\hat{\Sigma}^{N-1})=0$ for $k\neq n-1$ and $\tilde{H}_{n-1}(\hat{\Sigma}^{N-1})=\mathbb{Z}^{m(N-1)-1}$. Furthermore, Theorem <ref> implies that each component of $\bar{U}$ is contractible. Hence, applying the Mayer-Vietoris long exact sequence to $(\hat{\Sigma}^{N-1}, \bar{U}, \Sigma^{N-1})$ gives, for $k>0$,
\begin{equation*}
\begin{tikzcd}[column sep=12pt]
\tilde{H}_k(\partial \bar{U}) \arrow{r} \arrow[leftrightarrow]{d}{=} & \tilde{H}_k(\bar{U})\oplus \tilde{H}_k(\hat{\Sigma}^{N-1})
\arrow{r} \arrow[leftrightarrow]{d}{=}& \tilde{H}_k(\Sigma^{N-1}) \arrow{r}\arrow[leftrightarrow]{d}{=} &\tilde{H}_{k-1}(\partial \bar{U})\arrow[leftrightarrow]{d}{=}\\
\bigoplus\limits_{j=1}^{m(N-1)}\tilde{H}_k(\mathbb{S}^{n-1}) \arrow{r} & \tilde{H}_k(\hat{\Sigma}^{N-1}) \arrow{r} & \tilde{H}_k(\Sigma^{N-1})\arrow{r} & \bigoplus\limits_{j=1}^{m(N-1)} \tilde{H}_{k-1} (\mathbb{S}^{n-1}).
\end{tikzcd}
\end{equation*}
In particular, for $1\leq k\leq n-2$ and $k\geq n+1$, we obtain that $\tilde{H}_k(\Sigma^{N-1})=\set{0}$. The Mayer-Vietoris long exact sequence further gives the exact sequences
\begin{equation*}
\begin{tikzcd}
\tilde{H}_{n-1}(\partial \bar{U}) \arrow{r}{\delta} \arrow[leftrightarrow]{d}{=}& \tilde{H}_{n-1}(\bar{U} )\oplus \tilde{H}_{n-1}(\hat{\Sigma}^{N-1}) \arrow{r} \arrow[leftrightarrow]{d}{=} & \tilde{H}_{n-1}(\Sigma^{N-1}) \arrow{r} \arrow[leftrightarrow]{d}{=} &\tilde{H}_{n-2}(\partial \bar{U}) \arrow[leftrightarrow]{d}{=}\\
\mathbb{Z}^{m(N-1)} \arrow{r}{\delta} & \mathbb{Z}^{m(N-1)-1}
\arrow{r} & \tilde{H}_{n-1}(\Sigma^{N-1}) \arrow{r} &\set{0}.
\end{tikzcd}
\end{equation*}
Here $\delta$ is given by $(l_1, \ldots, l_{m(N-1)})\mapsto (l_1-l_{m(N-1)}, \ldots, l_{m(N-1)-1}-l_{m(N-1)})$. As $\delta$ is surjective, it follows that $\tilde{H}_{n-1}(\Sigma^{N-1})=\set{0}$.
Finally, as $\Sigma^{N-1}$ is an oriented, connected $n$-dimensional manifold $\tilde{H}_n(\Sigma^{N-1})=\mathbb{Z}$ and $\tilde{H}_0(\Sigma^{N-1})=\set{0}$. Hence, $\Sigma^{N-1}$ is a homology $n$-sphere.
As our argument only used that ${\Sigma}^N$ was a homology $n$-sphere, we may repeat it to see that each of the ${\Sigma}^i$ is a homology $n$-sphere and so conclude that $\Sigma$ is one as well.
Fix an open subset $U\subset \Real^{n+1}$. A hypersurface in $U$, $\Sigma$, is a proper, codimension-one submanifold of $U$. A smooth mean curvature flow in $U$, $S$, is a collection of hypersurfaces in $U$, $\set{\Sigma_t}_{t\in I}$, $I$ an interval, so that:
* For all $t_0\in I$ and $p_0\in \Sigma_{t_0}$ there is a $r_0=r_0(p_0,t_0)$ and an interval $I_0=I_0(p_0,t_0)$ with $(p_0,t_0)\in B_{r_0}^{n+1}(p_0)\times I_0\subset U\times I$;
* There is a smooth map $F: B_{1}^n\times I_0\to \Real^{n+1}$
so that $F_t(p)=F(p,t): B_1^n\to \Real^{n+1}$ is a parameterization of $B_{r_0}^{n+1}(p_0)\cap \Sigma_t$; and
* $\left(\frac{\partial}{\partial t} F (p,t)\right)^\perp= \mathbf{H}_{\Sigma_t}(F(p,t)).$
It is convenient to consider the space-time track of $S$ (also denoted by $S$):
\begin{equation}
{S}=\set{(\xX(p),t)\in\mathbb{R}^{n+1}\times \mathbb{R}: p\in\Sigma_t}\subset U\times I.
\end{equation}
This is a smooth submanifold of space-time and is transverse to each constant time hyperplane $\Real^{n+1}\times \set{t_0}$. Along the space-time track ${S}$, let $\frac{d}{dt}$ be the smooth vector field
\begin{equation}
\left.\frac{d}{dt}\right|_{(p,t)}=\frac{\partial}{\partial t}+\mathbf{H}_{\Sigma_t}(p).
\end{equation}
It is not hard to see that this vector field is tangent to ${S}$ and the position vector satisfies
\begin{equation} \label{MCFEqn}
\frac{d}{dt} \xX(p,t)=\mathbf{H}_{\Sigma_t}(p).
\end{equation}
It is a standard fact that if each $\Sigma_t$ in $S$ is closed, i.e. is compact and without boundary, then there is a smooth map
F:M\times I \to \Real^{n+1}
so that each $F_t=F(\cdot,t): M\to \Real^{n+1}$ is a parameterization of $\Sigma_t$ a closed $n$-dimensional manifold $M$. As a consequence, each $\Sigma_t$ is diffeomorphic to $M$.
We will need the following generalization of this last fact to manifolds with boundary.
Fix $R\in (0,\infty]$ and let $\set{\bar{B}_{2r_1}(\xX_1), \ldots, \bar{B}_{2r_m}(\xX_m)}$ be a collection of pairwise disjoint balls in $B_{R}\subset \Real^{n+1}$ and let $U=B_{2R}\backslash \bigcup_{i=1}^m \bar{B}_{r_i}(\xX_i)$.
If $\set{\Sigma_{t}}_{t\in (-\tau,\tau)}$ is a smooth mean curvature flow in $U$ with the property that
* Each $\hat{\Sigma}_t=\Sigma_t\cap \left(\bar{B}_R\backslash
\bigcup_{i=1}^m B_{2 r_i}(\xX_i)\right)$ is compact,
* For each $1\leq i \leq m$, $\partial B_{2 r_i} (\xX_i)$ intersects $\Sigma_t$ transversally and non-trivially for all $t\in (-\tau,\tau)$,
* If $R<\infty$, then $\partial B_R$ intersects $\Sigma_t$ transversally and non-trivially for all $t\in (-\tau,\tau)$,
then, for any $t_1,t_2\in (-\tau,\tau)$, $\hat{\Sigma}_{t_1}$ and $\hat{\Sigma}_{t_2}$ are diffeomorphic as compact manifolds with boundary.
For simplicity, we consider only $R=\infty$, $m=1$, $\xX_1=\OO$ and $r_1=\frac{1}{2}$. It is straightforward to extend the argument to the general case.
Let $S$ be the space-time track of the flow, so $S$ is a smooth hypersurface in $(\Real^{n+1}\backslash \bar{B}_{1/2})\times(-\tau,\tau)$. As each $\Sigma_t$ intersects $\partial B_1$ transversally, it is clear that $S$ meets $\partial B_1 \times (-\tau,\tau)$ transversally. In particular, $\tilde{S}=S\backslash \left({B}_1 \times (-\tau,\tau)\right)$ is a smooth hypersurface with boundary. Let $\tilde{B}=\partial \tilde{S}=\set{(p,t):p\in \partial B_1\cap \Sigma_t, t\in (-\tau,\tau)}$.
Without loss of generality we may assume that the given $t_1,t_2$ satisfy $t_1<t_2$. Let $\hat{S}= \tilde{S}\cap \left(\Real^{n+1}\times [t_1,t_2]\right)$ and $\hat{B}=\tilde{B}\cap \left(\Real^{n+1}\times [t_1,t_2]\right)$. Observe that $\hat{S}$ is a compact manifold with corners and $\hat{B}$ is one of its boundary strata. The other two boundary strata are $\hat{\Sigma}_{t_1}\times \set{t_1}$ and $\hat{\Sigma}_{t_2}\times \set{t_2}$.
As $\partial B_1$ meets each $\Sigma_t$ transversally and $\hat{B}$ is compact, there is an $\epsilon>0$ so that, for $(p,t)\in \hat{B}$, $|\xX^\top(p,t)|\geq 2\epsilon$, where $\xX^\top$ is the tangential component of the position vector. By continuity there is a $\frac{1}{2}>\delta>0$ so that, for any $t\in [t_1,t_2]$ and $p\in\left(\bar{B}_{1+\delta} \backslash B_{1-\delta}\right)\cap \Sigma_t$, $|\xX^\top(p,t)|\geq \epsilon$.
Now let $\eta \in C^\infty_0(\Real^{n+1})$ be a smooth function with $0\leq \eta\leq 1$, $\eta=1$ on $\partial B_1$ and $\spt(\eta)\subset\bar{B}_{1+\delta} \backslash B_{1-\delta}$.
For $(p,t)\in \hat{S}$ consider the vector
\mathbf{V}(p,t)= - \eta(\xX(p,t)) \frac{ (\xX(p,t)\cdot \mathbf{H}_{\Sigma_t}(p))}{|\xX^\top(p,t)|^2} \xX^\top(p,t)
and observe this gives a smooth vector field on $S$ that restricts to a smooth compactly supported vector field on each $\Sigma_t$.
Let $\mathbf{W}=\frac{d}{dt}+\mathbf{V}$ which is a smooth vector field on ${S}$.
We claim that $\mathbf{W}$ is tangent to $\hat{B}$ and transverse to $\hat{\Sigma}_{t_1}\times\set{t_1}\cup \hat{\Sigma}_{t_2}\times \set{t_2}$. As $\mathbf{V}$ is tangent to $\Sigma_t\times\set{t}$, the transversality of $\mathbf{W}$ follows from the transversality of $\frac{d}{dt}$. This transversality follows immediately from the definition of $\frac{d}{dt}$. To see the tangency note that, by construction, $\hat{B}=\set{(p,t)\in \hat{S}: |\xX(p,t)|^2=1}$. For $(p,t)\in \hat{B}$, one computes
\begin{align*}
\mathbf{W}\cdot |\xX(p,t)|^2& = 2 \xX(p,t) \cdot \nabla_\mathbf{W} \xX(p,t)\\
% &= 2\xX(p,t) \cdot \frac{d}{dt} \xX(p,t)+2 \xX(p,t) \cdot \mathbf{V}\cdot \xX(p,t)\\
&=2\xX(p,t) \cdot \mathbf{H}_{\Sigma_t}(p)-2\eta(\xX(p,t)) \frac{ (\xX(p,t)\cdot \mathbf{H}_{\Sigma_t}(p))}{|\xX^\top(p,t)|^2} \xX(p,t)\cdot \xX^\top(p,t) \\
\end{align*}
where the last equality used that $(p,t)\in \hat{B}$ so $\eta(\xX(p,t))=1$. This verifies the claim.
To conclude the proof observe that, as $\hat{S}$ is compact and $\mathbf{W}$ is tangent to $\hat{B}$ and transverse to $\hat{\Sigma}_{t_1}\times\set{t_1}\cup \hat{\Sigma}_{t_2}\times\set{t_2}$, standard ODE theory gives that for any $P_0=(p_0,t_0)\in \hat{S}$ the initial value problem
\left\{ \begin{array}{c} \dot{\gamma}(s)=\mathbf{W}(\gamma(s)) \\ \gamma_{P_0}(0)=P_0 \end{array}
\right.
has a unique smooth solution $\gamma_{P_0}:[t_1-t_0,t_2-t_0]\to \hat{S}$ which depends smoothly on $P_0$. These solutions satisfy $t(\gamma_{P_0}(s))=s+t_0$ and so there is a diffeomorphism $\phi:\Sigma_{t_1}\to \Sigma_{t_2}$ given by $(\phi(p),t_2)=\gamma_{(p,t_1)}(t_2-t_1)$.
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1511.00251
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In the author's work <cit.>, it has been shown that solutions of Maxwell-Klein-Gordon equations in $\mathbb{R}^{3+1}$ possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointwise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad-Sterbenz <cit.>, in which smallness was assumed both for the scalar field and the Maxwell field.
§ INTRODUCTION
In this paper, we study the pointwise decay of solutions to the Maxwell-Klein-Gordon equations on $\mathbb{R}^{3+1}$ with large Cauchy data. To define the equations, let $A=A_\mu dx^\mu$ be a $1$-form. The covariant derivative associated to this 1-form is
\begin{equation*}
D_\mu =\pa_\mu+\sqrt{-1}A_\mu,
\end{equation*}
which can be viewed as a $U(1)$ connection on the complex line bundle over $\mathbb{R}^{3+1}$ with the standard flat metric $m_{\mu\nu}$. Then the curvature $2$-form $F$ associated to this connection is given by
\begin{equation*}
F_{\mu\nu}=-\sqrt{-1}[D_{\mu}, D_{\nu}]=\pa_\mu A_\nu-\pa_\nu A_\mu=(dA)_{\mu\nu}.
\end{equation*}
This is a closed $2$-form, that is, $F$ satisfies the Bianchi identity
\begin{equation}
\label{bianchi}
\pa_\ga F_{\mu\nu}+\pa_\mu F_{\nu\ga}+\pa_\nu F_{\ga\mu}=0.
\end{equation}
The Maxwell-Klein-Gordon equations (MKG) is a system for the connection field $A$ and the complex scalar field $\phi$:
\begin{equation}
\label{EQMKG}\tag{MKG}
\begin{cases}
\pa^\nu F_{\mu\nu}=\Im(\phi \cdot\overline{D_\mu\phi})=J_\mu;\\
D^\mu D_\mu\phi=\Box_A\phi=0.
\end{cases}
\end{equation}
These are Euler-Lagrange equations of the functional
\[
L[A, \phi]=\iint_{\mathbb{R}^{3+1}}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\phi\overline{D^{\mu}\phi}dxdt.
\]
A basic feature of this system is that it is gauge invariant under the following gauge transformation:
\[
\phi\mapsto e^{i\chi}\phi; \quad A\mapsto A-d\chi.
\]
More precisely, if $(A, \phi)$ solves (<ref>), then $(A-d\chi, e^{i\chi}\phi)$ is also a solution for any potential function $\chi$. Note that $U(1)$ is abelian. The Maxwell field $F$ is invariant under the above gauge transformation and (<ref>) is said to be an abelian gauge theory. For the more general theory when $U(1)$ is replaced by a compact Lie group, the corresponding equations are referred to as Yang-Mills-Higgs equations.
In this paper, we consider the Cauchy problem to (<ref>). The initial data set $(E, H, \phi_0, \phi_1)$ consists of the initial electric field $E$, the magnetic field $H$, together with initial data $(\phi_0, \phi_1)$ for the scalar field. In terms of the solution $(F, \phi)$, on the initial hypersurface, these are:
\begin{equation*}
% \label{IDset}
F_{0i}=E_i,\quad \leftidx{^*}F_{0i}=H_i,\quad \phi(0, x)=\phi_0,\quad D_t\phi(0, x)=\phi_1,
\end{equation*}
where $\leftidx{^*}F$ is the Hodge dual of the 2-form $F$. In local coordinates $(t, x)$,
\[
(H_1, H_2, H_3)=(F_{23}, F_{31}, F_{12}).
\]
The data set is said to be admissible if it satisfies the compatibility condition
\begin{equation}
\label{eq:comp:cond}
div(E)=\Im(\phi_0\cdot \overline{\phi_1})=\left.J_0\right|_{t=0},\quad div (H)=0,
\end{equation}
where the divergence is taken on the initial hypersurface $\mathbb{R}^3$. For solutions of (<ref>), the energy
\[
E[F, \phi](t):=\int_{\mathbb{R}^3}|E|^2+|H|^2+|D\phi|^2dx
\]
is conserved. Another important conserved quantity is the total charge
\begin{equation}
\label{defcharge}
q_0=\frac{1}{4\pi}\int_{\mathbb{R}^3}\Im(\phi\cdot \overline{D_t\phi})dx=\frac{1}{4\pi}\int_{\mathbb{R}^3}div (E)dx,
\end{equation}
which can be defined at any fixed time $t$. The existence of nonzero charge plays a crucial role in the asymptotic behavior of solutions of (<ref>). It makes the analysis more complicated and subtle. This is obvious from the above definition as the electric field $E_i=F_{0i}$ has a tail $q_0r^{-3}x_i$ at any fixed time $t$.
The Cauchy problem to (<ref>) has been studied extensively. One of the most remarkable results is due to Eardley-Moncrief in <cit.>, <cit.>, in which it was shown that there is always a global solution to the general Yang-Mills-Higgs equations for sufficiently smooth initial data.
This has later been improved to data merely bounded in the energy space for MKG by Klainerman-Machedon in <cit.> and for the non-abelian case of Yang-Mills equations in e.g. <cit.>, <cit.>,
<cit.>. Since then there has been an extensive literature on generalizations and extensions of this classical result, aiming at improving the regularity of the initial data in order to construct a global solution, see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and references therein. A common feature of all these works is to
construct a local solution with rough data. Then the global well-posedness follows by establishing a priori bound for some appropriate norms of the solution. For example, a local solution was constructed in <cit.> while in <cit.>, they showed that the $L^\infty$ norm of the solution never blows up even though it may grow in time $t$. As a consequence the solution can be extended to all time; however the decay property of the solution is unknown. In view of this, although the solution of (<ref>) exists
globally with rough initial data, very little is known about the decay properties.
Asymptotic behavior and decay estimates are well understood for linear fields (see e.g.<cit.>) and nonlinear fields with sufficiently small initial data (see e.g.<cit.>, <cit.>). These mentioned results rely on the conformal symmetry of the system, either by conformally compactifying the Minkowski space or by using the conformal killing vector field $(t^2+r^2)\pa_t+2tr\pa_r$ as multiplier. Nevertheless the use of the conformal symmetry requires strong decay of the initial data and thus in general does not allow the presence of nonzero charge except when the initial data are essentially compactly supported. For the case with nonzero charge, the first related work regarding the asymptotic properties was due to W. Shu in <cit.>. However, that work only considered the case when the solution is trivial outside
a fixed forward light cone. Details for general case were not carried out. A complete proof towards this program was later contributed by Lindblad-Sterbenz in <cit.>, also see a more recent work <cit.>.
The presence of nonzero charge has a long range effect on the asymptotic behavior of the solutions, at least in a neighbourhood of the spatial infinity. This can be seen from the conservation law
of the total charge as the electric field $E$ decays at most $ r^{-2}$ as $r\rightarrow\infty$ at any fixed time. This weak decay rate makes the analysis more complicated even for small initial data. To deal with this difficulty, Lindblad-Sterbenz constructed a global chargeless field and made use of the fractional Morawetz estimates obtained by using the vector fields $u^p\pa_u+v^p\pa_v$ as multipliers. The latter work <cit.> relied on the observation that the angular derivative of the Maxwell field has zero charge. The Maxwell field then can be estimated by using Poincaré inequality.
The asymptotic behavior of solutions of MKG with general large data remains unknown until recently in <cit.> quantitative decay estimates have been obtained for solutions with data bounded in some weighted energy space. Pointwise decay requires the energy estimates for the derivatives of the solution. However, commuting the equations with derivatives generates nonlinear terms. The aim of this paper is to identify a class of large data for MKG equations such that we can derive the pointwise decay of the solutions.
We define some necessary notations in order to state our main result.
We use the standard polar local coordinate system $(t, r,
\om)$ of Minkowski space as well as the null coordinates $u=\frac{t-r}{2}$, $v=\frac{t+r}{2}$. Let $\nabla$ denote the derivative on $\mathbb{R}^3$ and $\Om$ be the set of angular momentum vector fields $\Om_{ij}=x_i\pa_j-x_j\pa_i$.
Without loss of generality we only prove estimates in the future, i.e., $t\geq 0$.
Next we introduce a null frame $\{L, \Lb, e_1, e_2\}$, where
\[
L=\pa_v=\pa_t+\pa_r,\quad \Lb=\pa_u=\pa_t-\pa_r
\]
and $\{e_1, e_2\}$ is an orthonormal basis of the sphere with
constant radius $r$. We use $\D$ to denote the covariant derivative associated to the connection field $A$ on the sphere with radius $r$. For any 2-form $F$, denote the null decomposition under the above null frame by
\begin{equation}
\label{eq:curNull}
\a_i=F_{Le_i},\quad\underline{\a}_i=F_{\Lb e_i},\quad \rho=\f12 F_{\Lb L}, \quad \si=F_{e_1 e_2},\quad i\in{1, 2}.
\end{equation}
We assume that the initial data set $(E, H, \phi_0, \phi_1)$ is admissible. Let $q_0$ be the charge defined in (<ref>) which is
uniquely determined by the initial data of the scalar field $(\phi_0, \phi_1)$. We assume that the data for the scalar field is small but the data for the Maxwell field is large. However the data can not be assigned freely. They satisfy the compatibility condition (<ref>). To measure the size of the initial data for the scalar field and the Maxwell field, let $(E^{df}, E^{cf})$ be the Hodge decomposition of the electronic field $E$ with $E^{df}$ the divergence free part and $E^{cf}$ the curl free part. Then the compatibility condition (<ref>) on $E$ is equivalent to
\[
\div E^{cf}=\Im(\phi_0\cdot \overline{\phi_1}).
\]
This implies that $E^{cf}$ can be uniquely determined by $(\phi_0, \phi_1)$ (with suitable decay assumption on $E$). Therefore for the initial data set $(E, H, \phi_0, \phi_1)$ for (<ref>) we can freely assign $\phi_0$, $\phi_1$ and $E^{df}$, $H$ as long as $\div H=0$, $\div E^{df}=0$. The total charge $q_0$ is a constant determined by $(\phi_0, \phi_1)$.
We now define the norms of the initial data. For some positive constant $0<\ga_0<1$, we define the second order weighted Sobolev norm respectively for the initial data of the Maxwell field $(E, H)$ and the initial data of the scalar field $(\phi_0, \phi)$:
\begin{align*}
\mathcal{M}:&=\sum\limits_{l\leq 2}(1+r)^{1+\ga_0}(|\Om^l E^{df}|^2+|\Om^l H|^2+|\nabla^l E^{df}|^2+|\nabla^l H|^2)dx,\\
\mathcal{E}:&=\sum\limits_{l\leq 2}(1+r)^{1+\ga_0}(|\nabla\Om^l\phi_0|^2+|\Om^l \phi_1|^2+|\nabla^{l+1} \phi_0|^2+|\nabla^{l} \phi_1|^2+|\phi_0|^2)dx.
\end{align*}
We remark here that the definition for $\mathcal{E}$ is not gauge invariant. The gauge invariant norm depends on the connection field $A$ which up to to a gauge transformation can be determined by the initial data of the Maxwell field $(E^{df}, H)$. However in our setting $\mathcal{M}$ can be arbitrarily large while $\mathcal{E}$ is assumed to be small depending on $\mathcal{M}$. It it hence much clear to use a norm for the scalar field so that it does not depend on $\mathcal{M}$. However we will show later (see Lemma <ref> in Section <ref>) that the gauge invariant norm is equivalent to the above Sobolev norm up to a constant depending on $\mathcal{M}$.
We now can state our main theorem:
Consider the Cauchy problem to (<ref>) with admissible initial data set $(E, H, \phi_0, \phi_1)$. Then there exists a positive constant $\ep_0$, depending on $\mathcal{M}$, $\ga_0$, such that for all $\mathcal{E}<\ep_0$, the solution $(F, \phi)$ of (<ref>) satisfies the following decay estimates:
\begin{align*}
|D_{\Lb}(r\phi)|^2(u, v, \om)\leq C\mathcal{E}u_+^{-1-\ga_0},&\quad |r\ab|^2(u, v, \om)\leq C u_+^{-1-\ga_0};\\
r^p(|D_L(r\phi)|^2+|\D(r\phi)|^2)(u, v, \om)\leq C\mathcal{E} u_+^{p-1-\ga_0},&\quad 0\leq p\leq 1+\ga_0;\\
r^p(|r\a|^2+|r\si|^2)(u, v, \om)\leq C u_+^{p-1-\ga_0},&\quad 0\leq p\leq 1+\ga_0;\\
r^{p+2}|\rho-q_0 r^{-2}\chi_{\{t+R\leq r\}}|^2(u, v, \om)\leq C u_+^{p-1-\ga_0},&\quad 0\leq p< 1,\\
r^p|\phi|^2(u, v,\om)\leq C\mathcal{E} u_+^{p-2-\ga_0},&\quad 1\leq p\leq 2\\
|D\phi|^2(t, x)+|\phi|^2(t, x)\leq C\mathcal{E}(1+t)_+^{-1-\ga_0},&\quad |F|^2(t, x)\leq C(1+t)_+^{-1-\ga_0} ,\quad \forall |x|\leq R
\end{align*}
for all $(u, v, \om)\in\mathbb{R}^{3+1}\cap\{|x|\geq R\}$ and for some constant $C$ depending on $\mathcal{M}$, $\ga_0$, $p$. Here $q_0$ is the total charge and $\chi_{\{t+2\leq r\}}$ is the characteristic function on the exterior region $\{t+2\leq r\}$ and $u_+=2+|u|$.
We make several remarks:
The second order derivatives of the initial data is the minimum regularity we need to derive the above pointwise decay of the solution. Similar decay estimates hold for the higher order derivatives of the solution if higher order weighted Sobolev norms of the initial data are known.
The restriction on $\ga_0$, that is $0<\ga_0<1$, is merely for the sake of brevity. If $\ga_0\geq 1$, then the decay property of the solutions propagates in the exterior region ($t+2\leq r$). In other words, we have the same decay estimates as in the theorem for $\tau\leq 0$. However in the interior region where $\tau>0$, the maximal decay rate is $\tau_+^{-2}$ (corresponding to $\ga_0=1$), that is, the decay rate in the interior region for $\ga_0\geq 1$ in general can not be better than that of $\ga_0=1$.
Since we assume the scalar field is small, the charge is also small by definition. Combined with the techniques in <cit.>, our approach can be adapted to the case with large charge. There are two ways of generalizations. The first one is to relax the assumption on the scalar field as in <cit.> so that the charge can be large. Secondly we can consider the following unphysical equations:
\begin{equation*}
\begin{cases}
\pa^\nu F_{\mu\nu}=\la\Im(\phi \cdot\overline{D_\mu\phi});\\
\Box_A\phi=0
\end{cases}
\end{equation*}
for some constant $\la$.
Compared to the previous result of Lindblad-Sterbenz <cit.>, we have made the following improvements: first of all, we obtain pointwise decay estimates for solutions of (<ref>) for a class of large initial data. We only require smallness on the scalar field. In particular our initial data for (<ref>) can be arbitrarily large. Combining the method in <cit.>, we can even make the data on the scalar field large in the energy space. Secondly we have lower regularity on the initial data. In <cit.>, it was assumed that the derivative of the initial data decays one order better, that is, $\nabla^{I}(E^{df}, H)$, $D^{I}(D\phi_0, \phi_1)$ belong to the weighted Sobolev space with weights $(1+r)^{1+\ga_0+2|I|}$, while in this paper we only assume that the angular derivatives of the data obey this improved decay (see the definition of $\mathcal{M}$, $\mathcal{E}$). For the other derivatives, the weights is merely $(1+r)^{1+\ga_0}$. This makes the analysis more delicate. Moreover, as the solution decays weaker initially, our decay rate is weaker than that in <cit.> (only decay rate in $\tau$, the decay in $r$ is the same). However if we assume the same decay of the initial data as in <cit.>, then we are able to obtain the same decay for the solution.
We use a new approach developed in <cit.> to study the asymptotic behavior of solutions of (<ref>). This new method was originally introduced by Dafermos-Rodnianski in <cit.> for the study of decay of linear waves on black hole spacetimes. This novel method starts by proving the energy flux decay of the solutions of linear equations through the forward light cone $\Si_{\tau}$ (see definitions in Section <ref>). The pointwise decay then follows by commuting the equation with $\pa_t$ and the angular momentum $\Om$. In the abstract framework set by Dafermos-Rodnianski in <cit.>, the energy flux decay relies on three kinds of basic ingredients and estimates: a uniform energy bound, an integrated local energy decay estimate and a hierarchy of $r$-weighted energy estimates in a neighbourhood of the null infinity, which can be obtained by using the vector fields $\pa_t$, $f(r)\pa_r$, $r^p(\pa_t+\pa_r)$ as multipliers respectively. Combining these three estimates, a pigeon hole argument then leads to the energy flux decay.
As the initial data for the scalar field is small, we can use perturbation method to prove the pointwise decay of the solution. With a suitable bootstrap assumption on the nonlinearity $J=\Im(\phi\cdot \overline{D\phi})$, we first can use the new method to prove energy decay estimates for the Maxwell field up to the second order derivatives. Once we have these decay estimates for the Maxwell field, we then can show the energy decay as well as pointwise decay for the scalar field, which can then be used to improve the bootstrap assumption. The smallness of the scalar field is used here to close the bootstrap assumptions.
The existence of nonzero chargel has a long range effect on the asymptotic behavior of the solution in the exterior region $\{t+R\leq r\}$, which has been discussed in <cit.> when the charge is large. To deal with this difficulty, we define the chargeless 2-form
\[
\bar F=F-\chi_{\{t+R\leq r\}}q_0 r^{-2}dt\wedge dr.
\]
We first carry out estimates for $\bar F$ on the exterior region $\{t+R\leq r\}$, which in particular controls the energy flux through $\{t+R=r\}$ (the intersection of the interior region and the exterior region). We then can use the new method to obtain estimates for the Maxwell field $F$ in the interior region. The Maxwell equation commutes with the Lie derivatives of $F$ (see Lemma <ref>). It is not hard to obtain energy decay estimates for the derivatives of the Maxwell field under suitable bootstrap assumptions on the nonlinearity $J$ by using the new approach.
The main difficulty lies in showing the energy decay estimates for the scalar field due to fact that the covariant derivative $D$ is not commutable. The interaction terms of the Maxwell field and the scalar field arise from the commutator. To control those interaction terms, previous results (<cit.>, <cit.>) rely on the smallness of the Maxwell field and those terms could be absorbed. The key observation that the Maxwell field is allowed to be large in this paper is that the robust new method makes use of the decay in $\tau$ (equivalent to $1+|t-r|$ up to a constant) and those terms could be controlled by using Gronwall's inequality without smallness assumption on the Maxwell field. Traditionally, the Gronwall's inequality is used with respect to the foliation $t=constant$. Therefore strong decay in $t$ is necessary. As the new method foliates the spacetime by using the null hypersurfaces $S_{\tau}$, it enables us to make use of the weaker decay in $\tau$ in order to apply Gronwall's inequality.
The paper is organized as follows: we define additional notations and derive the transport equations for the curvature components in Section <ref>; we then review the energy identities respectively for the scalar field and the Maxwell field in Section <ref>; in Section <ref>, we use the new method to obtain energy decay estimates first for the Maxwell field and then for the scalar field; finally in the last section, we improved the bootstrap assumption and conclude our main theorem.
The author would like to thank Pin Yu for helpful discussions.
§ PRELIMINARIES AND NOTATIONS
We define some additional notations used in the sequel. Recall the null frame $\{L, \Lb, e_1, e_2\}$ defined in the introduction. At any fixed point $(t, x)$, we may choose $e_1$, $e_2$ such that
\begin{equation*}
[L, e_i]=-\frac{1}{r}e_i,\quad [\Lb, e_i]=\frac{1}{r}e_i,\quad
\left.[e_1, e_2]\right|_{(t, x)}=0,\quad i\in\{1, 2\}.
\end{equation*}
This helps to compute those geometric quantities which are
independent of the choice of the local coordinates. We then can compute the covariant derivatives for the null frame at any fixed point:
\begin{equation}
\label{eq:nullderiv}
\begin{split}
&\nabla_L L=0,\quad \nabla_L \Lb=0, \quad \nabla_L e_i=0,\quad \nabla_{\Lb}\Lb=0, \quad \nabla_{\Lb}e_i=0,\\
&\nabla_{e_i}L=r^{-1}e_i,\quad \nabla_{e_i}\Lb=-r^{-1}e_i, \quad \nabla_{e_1}e_2=\nabla_{e_2}e_1=0,\quad \nabla_{e_i}e_i=-r^{-1}\pa_r.
\end{split}
\end{equation}
We use $\pa$ to abbreviate $(\pa_t,
\pa_{1}, \pa_{2}, \pa_{3}) =(\pa_t, \nabla)$ and $\nabb$ to denote the covariant derivative on the sphere with radius $r$.
Now we define the foliation of the spacetime. Let $H_u$ be the outgoing null hypersurface $\{t-r=2u\}$ and $\underline{H}_{\underline{u}}$ as the incoming null hypersurface $\{t+r=2 \underline{u}\}$. Let $R>1$ be a fixed constant. We now use this fixed constant $R$
to define the foliation of the future of the initial hypersurface $t=0$. Let $\tau^*=\frac{\tau-R}{2}$. In the exterior region where $t+R\leq r$, we use the foliation
\[
\Si_{\tau}:=H_{\tau^*}\cap \{t\geq 0\},\quad \tau\leq 0.
\]
In the interior region where $t+R\geq r$, let $\Si_{\tau}$ be the foliation defined as follows:
\begin{align*}
\Si_{\tau}:=\{t=\tau, \quad |x|\leq R\}\cup (H_{\tau^*}\cap \{|x|\geq R\}).
\end{align*}
In particular, the future spacetime $t\geq 0$ is foliated by $\Si_{\tau}$, $\tau\in \mathbb{R}$ where $\tau\leq 0$ foliates the exterior region and $\tau\geq 0$ gives the foliation in the interior region.
Note that the boundary of the region bounded by $\Si_{\tau_1}$ and $\Si_{\tau_2}$ is part of the future null infinity where the decay behavior of the solution is unknown. To make the energy estimates rigorous, we consider the finite truncated region. For any $v_0\geq \frac{\tau+R}{2}$, denote the truncated $\Si_{\tau}$ as
\[
\Si_{\tau}^{v_0}=\Si_{\tau}\cap\{v\leq v_0\}.
\]
Unless we specify it, in the following the outgoing null hypersurface $H_u$ stands for $H_{u}\cap \{t\geq 0\}$ in the exterior region and
$H_{u}\cap \{|x|\geq R\}$ in the interior region. On the initial hypersurface $t=0$, we denote the annulus with radii $r_1<r_2$ as
\[
B_{r_1}^{ r_2}=\{r_1\leq |x|\leq r_2\}, \quad B_{r}=B_{r}^{\infty}.
\]
Next we define the domains.
In the exterior region, for $\tau_2\leq \tau_1\leq 0$, denote $\mathcal{D}_{\tau_1}^{ \tau_2}$ to be the Cauchy development of the annulus $\{R-\tau_1\leq |x|\leq R- \tau_2\}$ or more precisely
\[
\mathcal{D}_{\tau_1}^{ \tau_2}=\{(t, x)|||x|+\tau_1^*+\tau_2^*|+t\leq \tau_1^*-\tau_2^*\}.
\]
The boundary of this domain consists of the spacelike initial surface $B_{R-\tau_1}^{R-\tau_2}$ and the truncated outgoing and incoming null hypersurfaces which we denote as:
\begin{align*}
H_{\tau_1^*}^{\tau_2^*}=H_{\tau_1^*}\cap \mathcal{D}_{\tau_1}^{ \tau_2},\quad\underline{H}_{-\tau_2^*}^{\tau_1^*}=\underline{H}_{-\tau_2^*}\cap \mathcal{D}_{\tau_1^*}^{ \tau_2^*}.
\end{align*}
In the interior region, for any $\tau_2\geq \tau_1\geq 0$, we denote $\mathcal{D}_{\tau_1}^{\tau_2}$ to be the region bounded by $\Si_{\tau_1}$ and $\Si_{\tau_2}$:
\begin{align*}
\mathcal{D}_{\tau_1}^{\tau_2}=\{(t, x)|(t, x)\in \Si_\tau, \tau_1\leq \tau\leq \tau_2\}.
\end{align*}
We use
$\bar{\mathcal{D}}_{\tau_1}^{\tau_2}=\mathcal{D}_{\tau_1}^{\tau_2}\cap\{|x|\geq R\}$ to denote the region outside the cylinder.l
The incoming null boundary of this finite region is denoted by $\underline{H}_{v}^{\tau_1, \tau_2}=\underline{H}_v\cap \{\tau_1^*\leq u\leq \tau_2^*\}$.
Finally, for $\tau\in\mathbb{R}$, let $\mathcal{D}_{\tau}=\mathcal{D}_{\tau}^{+\infty}$ when $\tau\geq 0$ and $\mathcal{D}_{\tau}=\mathcal{D}_{\tau}^{-\infty}$ when $\tau< 0$.
We use $E[\phi](\Si)$ to denote the energy flux of the complex scalar field $\phi$ and $E[F](\Si)$ the energy flux of the 2-form $F$
through the hypersurface $\Si$ in Minkowski space. The derivative on the scalar field is with respect to the covariant derivative $D$. For our interested hypersurfaces, we can comlpute
\begin{align*}
E[\phi](\mathbb{R}^3)&=\int_{\mathbb{R}^3}|D\phi|^2dx,\quad E[F](\mathbb{R}^3)=\int_{\mathbb{R}^3}\rho^2+|\si|^2+\frac{1}{2}(|\a|^2+|\underline{\a}|^2)dx,\\
E[\phi](H_u)&=\int_{H_u}(|D_L\phi|^2+|\D\phi|^2)r^2dvd\om,\quad E[F](H_u)=\int_{H_u}(\rho^2+\si^2+|\a|^2) r^2dvd\om,\\
E[\phi](\underline{H}_{\underline{u}})&=\int_{\underline{H}_{\underline{u}}}(|D_{\Lb}\phi|^2+|\D\phi|^2)r^2dvd\om,\quad E[F](\Hb_{\ub})=\int_{S_\tau}(\rho^2+\si^2+|\ab|^2) r^2dvd\om.
\end{align*}
Here $\rho$, $\si$, $\a$, $\ab$ are the null components of the 2-form $F$ defined in line (<ref>).
Next we define weighted Sobolev norm either on domains or on surfaces. For any function $f$ (scalar or vector valued or tensors) we denote the spacetime integral on $\mathcal{D}$ in Minkowski space
\[
I^{p}_{q}[f](\mathcal{D}):=\int_{\mathcal{D}}u_+^{q}r_+^{p}|f|^2,\quad r_+=1+r,\quad u_+=1+|u|.
\]
for any reall numbers $p$, $q$.
Here $\mathcal{D}$ can be the domain or hypersurface in the Minkowski space. For example, when $\mathcal{D}$ is $H_{u}$, then
\[
I^{p}_{q}[f](H_u):=\int_{H_u} r_+^{p}u_+^{q}|f|^2 r^2 dvd\om.
\]
To define the norms of the derivatives of the solution, we need vector fields used as commutators which, in this paper,
are the killing vector field $\pa_t$ together with the angular momentum $\Om$ with components $\Om_{ij}=x_i\pa_j-x_j\pa_i$. We denote the set
\[
Z=\{\pa_t, \Om_{ij}\}.
\]
For the scalar field, it is nature to take the
covariant derivative $D_Z$ associated to the connection $A$ for any vector field $Z$. This covariant derivative has already been defined for the purpose of defining the equations in the beginning of the introduction. For the Maxwell field $F$ which is a 2-form, we define the Lie derivative
\begin{align*}
(\mathcal{L}_Z F)_{\mu\nu}=Z(F_{\mu\nu})-F(\mathcal{L}_Z \pa_{\mu}, \pa_\nu)-F(\pa_\mu, \mathcal{L}_{Z}\pa_\nu), \quad (\mathcal{L}_Z J)_\mu= Z(J_\mu)-J(\mathcal{L}_Z \pa_\mu)
\end{align*}
respectively for any two form $F$ and any one form $J$. Here $\mathcal{L}_Z X=[Z, X]$.
If the vector field $Z$ is killing, that is, $\pa^\mu Z^\nu+\pa^\nu Z^\mu=0$,
then we can show that
\begin{align*}
\pa^\mu(\mathcal{L}_Z F)_{\mu\nu}&=Z(\pa^\mu F_{\mu\nu})+\pa^\mu Z^\a\pa_\a F_{\mu\nu}+\pa_\mu Z^\a\pa^\mu F_{\a\nu}+\pa_\nu Z^\a\pa^\mu F_{\mu\a}\\
&=Z(\pa^\mu F_{\mu\nu})+\pa_\nu Z^\a\pa^\mu F_{\mu\a}=(\mathcal{L}_Z \delta F)_\nu.
\end{align*}
Here $\delta F_{\nu}=\pa^\mu F_{\mu\nu}$ is a one form which is the divergence of the $2$-form $F$. We use $\mathcal{L}_Z^k$ or $D^k_Z$ to denote the $k$-th derivatives, that is ,
\[
\mathcal{L}_Z^k=\mathcal{L}_{Z^{1}}\mathcal{L}_{Z^{2}}\ldots\mathcal{L}_{Z^k}.
\]
Similarly for $D_Z^k$. The vector fields $Z^j$ is $\pa_t$ or the angular momentum $\Om_{ij}$.
Based on these calculations, we have the following commutator lemma:
For any killing vector field $Z$, we have
\begin{align*}
[\Box_A, D_Z]\phi&=2i Z^\nu F_{\mu \nu}D^\mu \phi+i \pa^\mu(Z^\nu F_{\mu\nu})\phi,\\
\pa^\mu(\mathcal{L}_Z G)_{\mu\nu}&=(\mathcal{L}_Z \delta G)_\nu
\end{align*}
for any complex scalar field $\phi$ and any two form $G$.
For the energy estimates of the solutions of (<ref>), the initial energies $\mathcal{M}$, $\mathcal{E}$ defined in the introduction can not be used directly as $\mathcal{E}$ is not gauge invariant. Note that the vector fields used as commutators are $Z=\{\pa_t, \Om\}$, for any two form $F$ satisfying the Bianchi identity and any scalar field $\phi$, for the given connection field $A$, we define the following weighted $k$-th order initial energies:
\begin{equation}
\label{eq:def4E0kFphi}
E^k_0[F]:=\sum\limits_{l\leq k}\int_{\mathbb{R}^3}r_+^{1+\ga_0}|\mathcal{L}_{Z}^l F|^2(0, x) dx,\quad E^k_0[\phi]:=\sum\limits_{l\leq k, j\leq 3}\int_{\mathbb{R}^3}r_+^{1+\ga_0}|D_{Z}^l D_j\phi|^2(0, x) dx.
\end{equation}
Here $D_j$ denotes the spatial covariant derivative and $0<\ga_0<1$ is the constant in the main Theorem. We remark here that $F$ may not be the full Maxwell field of the solution of (<ref>). In application, it can be the chargeless part of the full solution. However, the connection field $A$ is associated to the full Maxwell field. In fact the full Maxwell field does not belong to this weighted Sobolev space due to the existence of nonzero charge.
We end this section by writing the Maxwell equation under the null frame $\{L, \Lb, e_1, e_2\}$. In other words, we derive the transport equations for the curvature components. Let $F_{\mu\nu}$ be the two-form verifying the Bianchi identity. Denote $J=\delta F$, that is, $J_{\mu}=\pa^\nu F_{\mu \nu}$. Then we have
Under the null frame $\{L, \Lb, e_1, e_2\}$, the MKG equations are the following transport equations for the curvature components:
\begin{align}
\label{eq:eq4rhoCu}
&\Lb(r^2\rho)-\divs(r^2\ab)=r^2 J_{\Lb}, \quad L(r^2\rho)+\divs(r^2\a)=r^2 J_L,\\
\label{eq:eq4ab}
&\nabla_{L}(r{\ab}_i)-r\nabb_{e_i}\rho-r\nabb_{e_j}F_{e_ie_j}=rJ_{e_i},\quad i=1, 2,\\
\label{eq:eq4si}
&\Lb(r^2\si)=r^2 (e_2\ab_1-e_1\ab_2),\quad L(r^2\si)=r^2 (e_2\a_1-e_1\a_2),\\
\label{eq:eq4a}
&\nabla_{\Lb}(r{\a}_i)+r\nabb_{e_i}\rho-r\nabb_{e_j}F_{e_ie_j}=rJ_{e_i},\quad i=1, 2.
\end{align}
Here $\divs$ is the divergence operator on the sphere with radius $r$.
From the Maxwell equation $J_L=(\delta F)(L)$. Use the formula
\[
(\nabla_{X}F)(Y, Z)=XF(Y, Z)-F(\nabla_{X}Y, Z)-F(Y, \nabla_{X}Z)
\]
for all vector fields $X$, $Y$, $Z$. By using (<ref>), we then can compute
\begin{align*}
-(\delta F)(\Lb)&=-\f12(\nabla_{L}F)(\Lb, \Lb)-\f12 (\nabla_{\Lb}F)(L, \Lb)+(\nabla_{e_i}F)(e_i, \Lb)\\
&=\Lb\rho-e_i\ab_i-F(-2r^{-1}\pa_r, \Lb)-F(e_i, -r^{-1}e_i)\\
\end{align*}
Multiple both sides by $r^2$. We then get the first equation of (<ref>). The second equation follows similarly.
For (<ref>) and (<ref>), we need to use the Bianchi identity (<ref>) which is equivalent to
\[
(\nabla_X F)(Y, Z)+(\nabla_Y F)(Z, X)+(\nabla_Z F)(X, Y)=0
\]
for all vector fields $X$, $Y$, $ Z$. Let's only prove (<ref>). We can show that
\begin{align*}
-(\delta F)(e_i)&=-\f12(\nabla_{L}F)(\Lb, e_i)-\f12 (\nabla_{\Lb}F)(L, e_i)+(\nabla_{e_j}F)(e_j, e_i)\\
&=-\f12 L\ab_i+\f12 (\nabla_{L}F)(e_i, \Lb)+\f12(\nabla_{e_i}F)(\Lb, L)+e_j F_{e_je_i}-F(-2r^{-1}\pa_r, e_i)-F(e_i, -r^{-1}\pa_r)\\
&=-L\ab_i+e_i\rho-\f12 F(-r^{-1}e_i, L)-\f12 F(\Lb, r^{-1}e_i)+e_j F_{e_je_i}+r^{-1}F(\pa_r, e_i)\\
&=-L\ab_i+e_i\rho+e_j F_{e_je_i}-r^{-1}\ab_i.
\end{align*}
This leads to (<ref>).
The first transport equation (<ref>) for $\si$ follows from the Bianchi identity:
\begin{align*}
0&=(\nabla_{\Lb} F)(e_1, e_2)+(\nabla_{e_1}F)(e_2, \Lb)+(\nabla_{e_2}F)(\Lb, e_1)\\
&=\Lb\si-e_1\ab_2-F(e_2, -r^{-1}e_1)+e_2\ab_1-F(-r^{-1}e_2, e_1)\\
\end{align*}
The dual one follows if we replace $\Lb$ with $L$.
§ ENERGY METHOD
In this section, we review the energy method for solutions of the covariant linear wave equations and Maxwell equations by using the new method developed in <cit.>.
Denote $d\vol$ the
volume form in the Minkowski space. In the local coordinate system $(t, x)$, we have $
$. Here we have chosen $t$ to be the time orientation.
§.§ Eergy identity for the scalar field
For any complex scalar field $\phi$, we define the associated energy momentum tensor
\begin{equation*}
\begin{split}
T[\phi]_{\mu\nu}=\Re\left(\overline{D_\mu\phi}D_\nu\phi\right)-\f12 m_{\mu\nu}\overline{D^\ga\phi}D_\ga\phi.
\end{split}
\end{equation*}
Here $m_{\mu\nu}$ is the flat metric of Minkowski spacetime and the covariant derivative $D$ is defined with respect to the given connection field $A$. For any vector field $X$, we have the following identity
\[
\pa^\mu(T[\phi]_{\mu\nu}X^\nu) = \Re(\Box_A \phi X^\nu \overline{D_\nu\phi})+X^\nu F_{\nu\ga}J^\ga[\phi]+T[\phi]^{\mu\nu}\pi^X_{\mu\nu},
\]
where $\pi_{\mu\nu}^X=\f12 \mathcal{L}_X m_{\mu\nu}$ is the deformation tensor of the vector field $X$ in Minkowski space, $\Box_A$ is the covariant wave operator associated to the connection $A$, $F=dA$ is the exterior derivative
of the one-form $A$ which gives us a two-form and $J^\ga[\phi]=Im(\phi\cdot \overline{D^\ga \phi})$.
For any function $\chi$, we have
\begin{align*}
\f12\pa^{\mu}\left(\chi \pa_\mu|\phi|^2-\pa_\mu\chi|\phi|^2\right)= \chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2+\chi \Re(\Box_A\phi\cdot \overline\phi).
\end{align*}
We now define the vector field $\tilde{J}^X[\phi]$ with components
\begin{equation}
\label{eq:mcurent}
\tilde{J}^X_\mu[\phi]=T[\phi]_{\mu\nu}X^\nu -
\f12\pa_{\mu}\chi \cdot|\phi|^2 + \f12 \chi\pa_{\mu}|\phi|^2+Y_\mu
\end{equation}
for some vector field $Y$ which may also depend on the complex scalar field $\phi$. We then have the equality
\[
\pa^\mu \tilde{J}^X_\mu[\phi] =\Re(\Box_A \phi(\overline{D_X\phi}+\chi\overline \phi))+div(Y)+X^\nu F_{\nu\a}J^\a[\phi]+T[\phi]^{\mu\nu}\pi^X_{\mu\nu}+
\chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2.
\]
Here the operator $\Box$ is the wave operator in Minkowski space. Now for any region $\mathcal{D}$ in $\mathbb{R}^{3+1}$, using Stokes' formula, we derive the following energy identity:
\begin{align}
\notag &\iint_{\mathcal{D}}\Re(\Box_A \phi(\overline{D_X\phi}+\chi\overline \phi))+div(Y)+X^\nu F_{\nu\ga}J^\ga+T[\phi]^{\mu\nu}\pi^X_{\mu\nu}+
\chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2d\vol\\
&=\iint_{\mathcal{D}}\pa^\mu \tilde{J}^X_\mu[\phi]d\vol=\int_{\pa \mathcal{D}}i_{\tilde{J}^X[\phi]}d\vol,
\label{energyeq}
\end{align}
where $\pa\mathcal{D}$ denotes the boundary of the domain $\mathcal{D}$ and $i_Z d\vol$ denotes the contraction of the volume form $d\vol$
with the vector field $Z$ which gives the surface measure of the
boundary. For example, for any basis $\{e_1, e_2,
\ldots, e_n\}$, we have
$$i_{e_1}( de_1\wedge de_2\wedge\ldots
de_k)=de_2\wedge de_3\wedge\ldots\wedge de_k.
Throughout this paper, the domain $\mathcal{D}$ will be regular regions bounded by the $t$-constant slices, the outgoing null hypersurfaces $u=constant$, the incoming null
hypersurfaces $v=constant$ or the surface with constant $r$. We now compute $i_{\tilde{J}^{X}[\phi]}d\vol$ respectively on these hypersurfaces.
$t=constant$ slice, the surface measure is a function times $dx$. Recall the volume form
\[
\]
Here note that $dx$ is a $3$-form. We thus can show that
\begin{equation}
\label{eq:curlessR}
\begin{split}
D_X\phi)-\f12 X^0\overline{D^\ga\phi}D_\ga\phi-\f12 \pa^t\chi
\end{split}
\end{equation}
On the surface with constant $r$, the surface measure is $r^2dtd\om$. Therefore we have
\begin{equation}
\label{eq:currcon}
D_X\phi)-\f12 X^{r}\overline{D^\ga\phi}D_\ga\phi-\f12
\pa^{r}\chi|\phi|^2+\f12\chi \pa^{r}|\phi|^2+Y^{r})r^2dt d\om.
\end{equation}
On the outgoing null hypersurface $H_u$, we can write the volume form
\[
d\vol=dxdt=r^2drdt d\om=2r^2dvdud\om=-2dudvd\om.
\]
Here $d\om$ is the standard surface measure on the unite sphere.
Notice that $\Lb=\pa_u$. We can compute
\begin{equation}
\label{eq:curStau}
D_X\phi)-\f12 X^{\Lb}\overline{D^\ga\phi}D_\ga\phi-\f12
\pa^{\Lb}\chi|\phi|^2+\f12\chi \pa^{\Lb}|\phi|^2+Y^{\Lb}l)r^2dvd\om.
\end{equation}
Similarly, on the $v$-constant incoming null hypersurfaces $\Hb_{\ub}$, we
\begin{equation}
\label{eq:curnullinfy}
D_X\phi)-\f12 X^{L}\overline{D^\ga\phi}D_\ga\phi-\f12
\pa^{L}\chi|\phi|^2+\f12\chi \pa^{L}|\phi|^2+Y^{L})r^2dud\om.
\end{equation}
We remark here that the above formulae hold for any vector fields $X$, $Y$ and any function $\chi$.
§.§ Energy identities for the Maxwell field
Let $F$ be any 2-form satisfying the Bianchi identity (<ref>). The associated the energy momentum tensor is
\[
\]
For any vector field $X$, we have the divergence formula
\[
\pa^\mu{T[F]_{\mu\nu}X^\nu}=\pa^\mu F_{\mu\ga}F_{\nu}^{\;\ga}X^\nu+T[F]^{\mu\nu}\pi^X_{\mu\nu},
\]
where as defined previously, $\pi_{\mu\nu}^X=\f12 \mathcal{L}_X m_{\mu\nu}$ is the deformation tensor of the vector field $X$ in Minkowski space.
Define the vector field $J^X[F]$ as follows:
\[
\]
Then for any domain $\mathcal{D}$ in $\mathbb{R}^{3+1}$, we have the following energy identity for the Maxwell field $F$:
\begin{align}
\iint_{\mathcal{D}}\pa^\mu F_{\mu\a}F_{\nu}^{\;\a}X^\nu+T[F]^{\mu\nu}\pi^X_{\mu\nu}d\vol=\iint_{\mathcal{D}}\pa^\mu J^X_\mu[F]d\vol=\int_{\pa \mathcal{D}}i_{J^X[F]}d\vol.
\label{energyeqcur}
\end{align}
For the terms on the boundary, similar to (<ref>) to (<ref>), we can compute
\begin{equation}
\label{eq:curF}
\begin{split}
i_{J^{X}[F]}d\vol&=-(F^{0\mu}F_{\nu\mu}X^\nu-\frac{1}{4}X^0 F_{\mu\nu}F^{\mu\nu})dx;\\
i_{J^{X}[F]}d\vol&=(F^{r\mu}F_{\nu\mu}X^\nu-\frac{1}{4}X^r F_{\mu\nu}F^{\mu\nu})r^2dtd\om;\\
i_{J^{X}[F]}d\vol&=-2(F^{\Lb \mu}F_{\nu\mu}X^\nu-\frac{1}{4}X^{\Lb}F_{\mu\nu}F^{\mu\nu})r^2dvd\om;\\
i_{J^{X}[F]}d\vol&=2( F^{L \mu}F_{\nu\mu}X^\nu-\frac{1}{4}X^{L}F_{\mu\nu}F^{\mu\nu})r^2dud\om
\end{split}
\end{equation}
respectively on the $t=constant$ slice, surface with constant $r$, the outgoing null hypersurface $H_u$ and the incoming null hypersurface $\Hb_{\ub}$.
§.§ The integrated local energy estimates using the multiplier $f(r)\pa_r$
For the full solution $(\phi, F)$ of the Maxwell Klein-Gordon equations, including the case with large charge, the integrated local energy estimates together with the $r$-weighted energy estimates
in the next subsection have been studied in the author's work <cit.>. To obtain estimates for higher order derivatives of the solutions, we need to commute the equations with derivatives and
nonlinear terms hence arise. Furthermore, in our setting, the data for the Maxwell field are large while the data for the complex scalar field are small. We thus need to obtain estimates separately for the
Maxwell field and the scalar field.
We first consider the integrated local energy estimates for the scalar field. In the energy identity (<ref>) for the scalar field,
we choose the vector fields
$$X=f(r)\pa_r,\quad Y=0$$
for some function $f(r)$. We then can compute
\begin{align*}
&T[\phi]^{\mu\nu}\pi^X_{\mu\nu}+ \chi \overline{D_\mu\phi}D^\mu\phi-\f12 \Box\chi |\phi|^2\\
=&(r^{-1}f-\chi+\f12 f')|D_t\phi|^2+(\chi+\f12 f'-r^{-1}f)|D_r\phi|^2+(\chi-\f12 f')|\D\phi|^2-\f12 \Box\chi |\phi|^2.
\end{align*}
The idea is to choose the functions $f$, $\chi$ so that the coefficients are positive. Let $\ep$ be a small positive constant, depending only on $\ga_0$. Construct the functions $f$ and $\chi $ as follows:
$$f=2\ep^{-1}-\frac{2\ep^{-1}}{(1+r)^{\ep}},\quad \chi=r^{-1}f.$$
We can compute
\begin{align*}
\chi-r^{-1}f + \f12 f'=r^{-1}f + \f12 f' - \chi=\frac{1}{(1+r)^{1+\ep}},\quad-\f12 \Box \chi=\frac{1+\ep}{r(1+r)^{2+\ep}}.
\end{align*}
When $r>1$, we have the following improved estimate for
$\chi-\f12 f'$
\begin{equation}\label{improvnabb}\chi- \f12
f'\geq \frac{2\ep^{-1}}{r} - \frac{1+2\ep^{-1}}{r(1+r)^\ep}\geq \frac{1}{r},
\quad r\geq 1.
\end{equation}
This improved estimate will be used to show the improved integrated local
energy estimate for the covariant angular derivative of the
scalar field $\phi$.
From the above calculation, we see that for this particular choice of vector field $X$ and the function $\chi$, the last three terms in the first line of equation (<ref>) have positive signs. We
treat the first two terms as nonlinear terms. To get an integrated local energy estimate for the scalar field $\phi$, it the suffices to control the boundary terms arising from the Stokes' formula (<ref>). This requires a version of Hardy's inequality. Before stating the lemma, we make a convention
that the notation $A\les_{\Gamma} B$ means that there exists a constant $C$, depending only on the constants $R$, $\ga_0$, $\ep$ and the set $\Gamma$ such that $A\leq CB$. For the particular case when $\Gamma$ is empty, we omit the index $\Gamma$.
Assume $0\leq\ga<1$ and the complex scalar field $\phi$ vanishes at null infinity, that is,
\[
\lim_{v\rightarrow\infty}\phi(v, u, \om)=0
\]
for all $u$, $\om$. Then we have
\begin{equation}
\label{eq:Hardyga}
\int_{H_u}r^{\ga}|\phi|^2dvd\om \les \int_{\om}r^{1+\ga}|\phi|^2(u, v(u), \om)d\om +\int_{H_u} r^{\ga}|D_L(r\phi)|^2dvd\om
\end{equation}
for all $u\in\mathbb{R}$. Here $v(u)=-u$ when $u\leq -\frac{R}{2}$ that is in the exterior region and $v(u)=2R+u$ when $u>-\frac{R}{2}$ that is in the interior region. In particular, we have
\begin{equation}
\label{eq:Hardy}
\int_{H_u}|\phi|^2dvd\om\les E[\phi](H_u),\quad \int_{\Si_{\tau}}|\phi|^2dv'd\om \les E[\phi](\Si_{\tau}).
\end{equation}
Here $v'=v$ when $r\geq R$ and $v'=r$ otherwise.
It suffices to notice that the connection $D$ is compatible with the inner product $<,>$ on the complex plane. Then the proof when $\ga=0$ goes the same as the case when the connection field $A$ is trivial,
see e.g. Lemma 2 of <cit.> or Proposition 11.2 of <cit.>. Another quick way to reduce the proof of the Lemma to the case with trivial connection field $A$ is to choose a particular gauge such that the scalar field
$\phi$ is real. We can do this is due to the fact that all the norms we considered in this paper are gauge invariant.
For general $\ga$, based on the above argument, the proof goes similar to the proof of the standard Hardy's inequality. Let $\psi=r\phi$. Note that $\ga <1$. We can show that
\begin{align*}
\int_{v_0}^{\infty}\int_{\om }r^{\ga-2}|\psi|^2dvd\om &=\frac{1}{\ga-1}\int_{v_0}^{\infty}\int_{\om}|\psi|^2d\om d r^{\ga -1}\\
&=\frac{1}{\ga-1}\left. r^{\ga -1}\int_{\om }|\psi|^2d\om \right|_{v_0}^{\infty}+\frac{2}{1-\ga}\int_{v_0}^{\infty}\int_{\om }r^{\ga -1}D_L\psi \cdot \psi dv d\om \\
&\leq \frac{1}{1-\ga} \int_{\om }r^{1+\ga}|\phi|^2(u, v_0, \om ) d\om +\frac{1}{2}\int_{v_0}^{\infty}\int_{\om}r^{\ga-2}|\psi|^2dvd\om \\
&\quad \quad +\frac{8}{(1-\ga)^2}\int_{v_0}^{\infty}\int_{\om}r^{\ga}|D_L\psi|^2dvd\om.
\end{align*}
Estimate (<ref>) then follows by absorbing the second term and taking $v_0=v(u)$.
We then can derive the following integrated local energy estimate for the scalar field $\phi$.
Assume the complex scalar field $\phi$ vanishes at null infinity and the spatial infinity initially. Then in the interior region $\{r\leq R+t\}$, we have the following energy estimates:
\begin{equation}
\label{eq:ILE:Sca:in}
\begin{split}
& I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})+E[\phi](\Si_{\tau_2})+E[\phi](\Hb^{\tau_1, \tau_2}_v)+\iint_{\mathcal{D}_{\tau_1}^ {\tau_2}}\frac{|\D\phi|^2}{1+r} dxdt\\
&\les E[\phi](\Si_{\tau_1})+I^{1+\ep}_0[\Box_A\phi](\mathcal D_{\tau_1}^{\tau_2})+
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|F_{L\nu}J^\nu[\phi]|+|F_{\Lb\nu}J^\nu[\phi]|dxdt
\end{split}
\end{equation}
for all $0\leq \tau_1<\tau_2$, $v\geq\frac{\tau_2+R}{2}$, where we denote $\bar D\phi=(D\phi, r_+^{-1}\phi)$ and $F=dA$. Similarly, in the exterior region $\{r>t+R\}$, we have
\begin{equation}
\label{eq:ILE:Sca:ex}
\begin{split}
& I^{-1-\ep}_0[\bar D\phi](\mathcal D_{\tau_1}^{\tau_2})+E[\phi](H_{\tau_1^*}^{-\tau_2^*})+E[\phi](\Hb_{-\tau_2^*}^{\tau_1^*})\\
&\les E[\phi](B_{R-\tau_1})+I^{1+\ep}_0[\Box_A\phi](\mathcal D_{\tau_1}^{\tau_2})+
\iint_{\mathcal{D}_{\tau_1}^{ \tau_2}}|F_{L\nu}J^\nu[\phi]|+|F_{\Lb\nu}J^\nu(\phi)| dxdt
\end{split}
\end{equation}
for all $\tau_2\leq \tau_1\leq 0$. Here the notations have been defined in Section <ref> and $J^\mu[\phi]=\Im(\phi\cdot \overline{D^\mu \phi})$.
For all $v_0\geq \frac{\tau_2+R}{2}$, take the region $\mathcal{D}$ to be $\mathcal{D}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}$ which is bounded by the surfaces $\Si_{\tau_1}$, $\Si_{\tau_2}$
, $\Hb_{v_0}^{\tau_1, \tau_2}$ and the functions $f$, $\chi$ as above and the vector field $Y=0$ in the energy identity (<ref>).
The boundary terms can be controled by the energy flux according to Hardy's inequality of Lemma <ref>. For more details regarding this bound, we refer to e.g. Proposition 1 in <cit.>.
Therefore the above calculations lead to the following integrated local energy estimate:
\begin{align*}
&\iint_{\mathcal{D}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}}\frac{|D\phi|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2}{1+r}+\frac{|\phi|^2}{r(1+r)^{2+\ep}}dxdt\\
&\les E[\phi](\Si_{\tau_1}^{ v_0})+E[\phi](\Si_{\tau_2}^{v_0})+E[\phi](\Hb^{\tau_1, \tau_2}_{v_0})
+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|\Box_A \phi (\overline{D_X\phi}+\chi\overline{\phi})|+|F_{r\nu}J^\nu[\phi]|dxdt
\end{align*}
Next, we take the vector fields $X=\pa_t$, $Y=0$ and the function $\chi=0$ in the energy identity (<ref>) for the scalar field. Consider the region
$\mathcal{D}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}$. We retrieve the classical energy estimate
\begin{equation*}
E[\phi](\Si_{\tau_2}^{v_0})+E[\phi](\Hb_{v_0}^{\tau_1, \tau_2})=E[\phi](\Si_{\tau_1}^{v_0})-2\iint_{\mathcal{D}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}}\Re\left(\Box_A\phi \overline{D_t\phi}\right)+F_{0\nu}J^\nu[\phi] dxdt.
\end{equation*}
Combined with the previous integrated local energy estimate and letting $v_0\rightarrow \infty$, we derive that
\begin{align*}
I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})\les E[\phi](\Si_{\tau_1})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|\Box_A\phi \overline{\bar D\phi}|+|F_{L\nu}J^\nu[\phi]|+|F_{\Lb\nu}J^\nu[\phi]| dxdt.
\end{align*}
We apply Cauchy-Schwarz's inequality to the integral of $\Box_A\phi \overline{\bar D\phi}$:
\[
2|\Box_A\phi \overline{\bar D\phi}|\leq \ep_1 (1+r)^{-1-\ep}|\bar D\phi|^2+\ep_1^{-1}(1+r)^{1+\ep}|\Box_A\phi|^2,\quad \forall \ep_1>0.
\]
Choose $\ep_1$ to be sufficiently small depending only on $\ep$, $\ga_0$, $R$ so that the integral of the first term can be absorbed. We thus can derive the integrated local
energy estimate for the scalar field. Then in the above classical energy estimate, we can use Cauchy-Schwarz's inequality again to bound $\Re\left(\Box_A\phi \overline{D_t\phi}\right)$ which gives
control of the energy flux $E[\phi](H_{\tau_2^*})$. This energy estimate together with the previous integrated local energy estimate imply the energy estimate (<ref>) of the Proposition in the interior region.
The improved estimate for the angular covariant derivative is due to the improve estimate (<ref>).
The proof for the estimate (<ref>) in the exterior region is similar. The only point we need to emphasize is that we use the fact that the
$\phi$ goes to zero as $r\rightarrow \infty$ on the initial
hypersurface. We thus can use the Hardy's inequality to control the integral of $\frac{|\phi|^2}{(1+r)^{2}}$. This is also the reason that we have $E[\phi](B_{R-\tau_1})$
instead of $E[\phi](B_{R-\tau_1}^{R-\tau_2})$ on the right hand side of (<ref>).
In our setting, $F$ is the Maxwell field which is no longer small. In particular this means that the integral of $|F_{L\nu}J^\nu[\phi]|$ on the right hand side of (<ref>), (<ref>) could not be absorbed. The key to control those terms is to use the $r$-weighted energy estimates in the next section.
Let $F$ be any 2-form satisfying the Bianchi identity (<ref>). Denote $J=\delta F$ or $J_{\mu}=\pa^\nu F_{\nu\mu}$ be the divergence of $F$. This notation $J$ can be viewed as inhomogeneous term of the linear Maxwell equation. In (<ref>), this $J$ is identity to $J[\phi]$ which is quadratic in the scalar field $\phi$. Under the null frame $\{L, \Lb, e_1, e_2\}$, ldenote $\J=(J_{e_1}, J_{e_2})$. We derive an analogue of Proposition <ref>.
Then in the interior region $\{r\leq t+R\}$, we have the integrated local energy estimates
\begin{equation}
\label{eq:ILE:cur:in}
\begin{split}
&I^{-1-\ep}_0[F](\mathcal{D}_{\tau_1}^{\tau_2})+ \int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}\frac{\rho^2+\si^2}{1+r} dxd\tau+E[F](\Hb_{v_0}^{\tau_1,\tau_2})+E[F](\Si_{\tau_2})\\
\les & E[F](\Si_{\tau_1})+I^{1+\ep}_0[|J_{L}|+|\J|](\mathcal{D}_{\tau_1}^{\tau_2})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt.
\end{split}
\end{equation}
for all $0\leq \tau_1<\tau_2$, $v_0\geq \frac{\tau_2+R}{2}$. In the exterior region $\{r\leq R+t\}$ for all $\tau_2\leq \tau_1\leq 0$, we have
\begin{equation}
\label{eq:ILE:cur:ex}
\begin{split}
&\les E[F](B_{R-\tau_1}^{R-\tau_2})+I^{1+\ep}_0[|J_{L}|+|\J|](\mathcal{D}_{\tau_1}^{\tau_2})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt.
\end{split}
\end{equation}
The idea to prove this proposition is the same as that of the previous proposition for the scalar field. However, the calculations is different for the Maxwell field $F$.
In the energy identity (<ref>) for the Maxwell field, we take the vector field
\[
\]
Denote $\om_i=r^{-1}x_i$. We then can compute
\begin{align*}
\end{align*}
where the Latin indices $\mu$, $\nu$ run from $0$ to $3$ and the Greek indices $i$, $j$ run from 1 to 3. Using the null decomposition of the 2-form under the null frame
$\{L, \Lb, e_1, e_2\}$ defined in line (<ref>), we can show that
\begin{align*}
F_{\mu\nu}F^{\mu\nu}&=-2 \rho^2-2\a\cdot \underline{\a}+2\si^2,\\
F_{0\nu}F^{0\nu}&=-\frac{1}{4}(4\rho^2+2\a\cdot \underline{\a}+|\a|^2+|\underline\a|^2),\\
\end{align*}
Therefore we have
\begin{equation}
\label{TFXr}
T[F]^{\mu\nu}\pi_{\mu\nu}^X=(r^{-1}f-\f12 f')(\rho^2+\si^2)+\frac{1}{4}f'(|\a|^2+|\underline\a|^2).
\end{equation}
The calculations before line (<ref>) imply that the coefficients $r^{-1}f-\f12 f'$, $f'$ have positive signs.
To obtain the similar integrated local energy estimates for the Maxwell field $F$, we need to control the boundary terms arising from the Stokes' formula (<ref>).
Using the formulae (<ref>), we can compute that
\begin{align*}
2| i_{J^{X}[F]}d\vol|&= f\left||\a|^2-|\underline{\a}|^2\right|dx\leq |F|^2 dx=2f i_{J^{\pa_t}[F]}d\vol,\\
2|i_{J^{X}[\phi]}d\vol|&=f \left|-\rho^2+|\a|^2-\si^2\right|r^2dvd\om\leq f(\rho^2+|\a|^2+\si^2)r^2dvd\om=2fi_{J^{\pa_t}[\phi]}d\vol,\\
2|i_{J^{X}[\phi]}d\vol|&=f\left|-\rho^2+|\underline{\a}|^2-\si^2\right|r^2dud\om\leq f(\rho^2+|\underline{\a}|^2+\si^2)r^2dud\om=2fi_{J^{\pa_t}[\phi]}d\vol
\end{align*}
respectively on the $t=constant$ slice, the outgoing null hypersurface and the incoming null hypersurface for all positive function $f$. This in particular implies that
the boundary terms corresponding to the vector field $f\pa_r$ can be bounded by the energy flux for all positive bounded function $f$. Therefore for the particular choice of vector field $X$,
the energy identity (<ref>) on the domain $\mathcal{D}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}$ for all $0\leq \tau_1<\tau_2$, $v_0\geq \frac{\tau_2+R}{2}$ leads to
\begin{align*}
\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}^{v_0}}\frac{|F|^2}{(1+r)^{1+\ep}}+\frac{\rho^2+\si^2}{1+r} dxd\tau\les & E[F](\Si_{\tau_1}^{v_0})+E[F](\Si_{\tau_2}^{v_0})+E[F](\Hb_{v_0}^{\tau_1, \tau_2})\\
&+\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}|J^{\ga}||F_{L\ga}-F_{\Lb \ga})|dxd\tau.
\end{align*}
Here notice that we have the improved estimate (<ref>) for the coefficient of $\rho^2+\si^2$. If we take the vector field $X=\pa_t$ on the same domain, we then can derive the classical energy identity
\begin{align*}
\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}^{v_0}}J^{\ga}(F_{L \ga}+F_{\Lb\ga})dxd\tau=E[F](\Si_{\tau_1}^{v_0})-E[F](\Hb_{v_0}^{\tau_1,\tau_2})-E[F](\Si_{\tau_2}^{v_0}).
\end{align*}
Let $v_0\rightarrow \infty$ and apply Cauchy-Schwarz to the inhomogeneous term $J^{\mu} (|F_{L\mu}|+|F_{\Lb \mu}|)$ for $\mu=\Lb$, $e_1$, $e_2$:
\[
|J^{\Lb}| |F_{L\Lb}|+|J^{e_i}| (|F_{L e_i}|+|F_{\Lb e_i}|)\les \ep_1^{-1}(|J_{L}|+|\J|)r_+^{1+\ep}+\ep_1 |F|^2r_+^{-1-\ep},\quad \ep_1>0.
\]
The integral of second term could be absorbed for sufficiently small $\ep_1$. For the component when $\mu=L$, we estimate:
\[
|J^{L}| |F_{L\Lb}|\les |J_{\Lb}||\rho|.
\]
Then the above energy identity together with the integrated local energy estimates imply the integrated local energy estimate (<ref>) in the interior region. The energy estimate
(<ref>) in the exterior region follows in a same way.
§.§ The $r$-weighted energy estimates using the multiplier $r^pL$
In this section, we establish the robust $r$-weighted energy estimates both for the scalar field and the Maxwell field. This estimate for solutions of linear wave equation in Minkowski space
was first introduced by Dafermos-Rodnianski in <cit.>. We study the $r$-weighted energy estimate either in the exterior region $\{r\geq R+t\}$ for the domain $\mathcal{D}_{\tau_1}^{\tau_2}$ for $\tau_2\leq \tau_1\leq 0$ or in the interior region for
domain $\bar{\mathcal{D}}_{\tau_1}^{\tau_2}$ for $0\leq \tau_1<\tau_2$ which is bounded by the outgoing null hypersurfaces $H_{\tau_1^*}$, $H_{\tau_2^*}$ and the cylinder $\{r=R\}$.
We have the following $r$-weighted energy estimates for the complex scalar field.
Assume that the complex scalar field $\phi$ vanishes at null infinity. Then in the interior region, for all $0\leq \tau_1\leq \tau_2$, $v_0\geq \frac{\tau_2+R}{2}$, we have the $r$-weighted energy estimate:
\begin{equation}
\label{eq:pWE:sca:in}
\begin{split}
&\int_{H_{\tau_2^*}}r^p|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{p-1}(p|D_L\psi|^2+(2-p)|\D\psi|^2)dvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}r^p|\D\psi|^2dud\om\\
\les &\int_{H_{\tau_1^*}}r^p|D_L\psi|^2dvd\om+I^{\max\{1+\ep, p\}}_{\min\{1+\ep, p\}}[\Box_A\phi](\mathcal{D}_{\tau_1}^{\tau_2})+E[\phi](\Si_{\tau_1})+I^{1+\ep}_0[\Box_A\phi](\mathcal{D}_{\tau_1}^{\tau_2})\\
\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}r^{p}|F_{L\mu}J^\mu[\phi]|dxdt.
\end{split}
\end{equation}
for all $0\leq p\leq 2$. Similarly in the exterior region, for all $\tau_2\leq \tau_1\leq 0$, we have
\begin{equation}
\label{eq:pWE:sca:ex}
\begin{split}
&\int_{H_{\tau_1^*}^{-\tau_2^*}}r^p|D_L\psi|^2dvd\om+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p-1}(p|D_L\psi|^2+(2-p)|\D\psi|^2)dvd\om du+\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^p|\D\psi|^2dud\om\\
\les & \int_{B_{R-\tau_1}^{R-\tau_2}}r^p(|D_L\psi|^2+|\D\psi|^2)drd\om+I^{\max\{p, 1+\ep\}}_{\min\{1+\ep, p\}}[\Box_A \phi](\mathcal{D}_{\tau_1}^{\tau_2})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p}|F_{L\mu}J^\mu[\phi]| dxdt.
\end{split}
\end{equation}
Apply the energy identity (<ref>) to the region $\bar{\mathcal{D}}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}$ which
bounded by $H_{\tau_1^*}$, $H_{\tau_2^*}$, $\{r=R\}$ and $\Hb_{v_0}^{\tau_1, \tau_2}$ with the vector fields $X$, $Y$ and the function $\chi$ as follows:
\[
X=r^{p}L, \quad Y=\frac{p}{2}r^{p-2}|\phi|^2L,\quad \chi=r^{p-1}.
\]
Denote $\psi=r\phi$ as the weighted scalar field. Note that we have the equality
\[
r^2|D_L\phi|^2=|D_L\psi|^2-L(r|\phi|^2),\quad r^2|\D\phi|^2=|\D\psi|^2,\quad r^2|D_{\Lb}\phi|^2=|D_{\Lb}\psi|^2+\Lb(r|\phi|^2).
\]
We then can compute
\begin{align*}
& div(Y)+T[\phi]^{\mu\nu}\pi_{\mu\nu}^X+\chi \overline{D^{\mu}\phi}D_\mu\phi-\f12\Box\chi |\phi|^2\\
&=\frac{p}{2}r^{-2}L(r^{p}|\phi|^2)+\f12r^{p-1}\left(p|D_L\phi|^2+(2-p)|\D\phi|^2\right)-\f12 p(p-1)r^{p-3}|\phi|^2\\
&=\f12 r^{p-3}\left(p|D_L\psi|^2+(2-p)|\D\psi|^2\right).
\end{align*}
We next compute the boundary terms according to the formula (<ref>). We have
\begin{align*}
\int_{H_{\tau^*}}i_{\tilde{J}^X[\phi]}d\vol&=\int_{H_{\tau^*}}r^{p}|D_L\psi|^2-\f12 L(r^{p+1}\phi) \quad dvd\om,\\
\int_{\Hb_{v_0}^{\tau_1, \tau_2}}i_{\tilde{J}^X[\phi]}d\vol&=-\int_{\Hb_{v_0}^{\tau_1, \tau_2}}r^{p}|\D\psi|^2+\f12\Lb(r^{p+1}|\phi|^2)\quad dud\om,\\
\int_{\{r=R\}\cap\{ \tau_1\leq t\leq \tau_2\} }i_{\tilde{J}^X[\phi]}d\vol&=\int_{\tau_1}^{\tau_2}\int_{\om}\f12 r^p(|D_L\psi|^2-|\D\psi|^2)-\f12\pa_t(r^{p+1}|\phi|^2) \quad d\om dt.
\end{align*}
Now notice that there is a cancellation for the boundary terms:
\begin{align*}
&-\int_{H_{\tau_1^*}}L(r^{p+1}|\phi|^2)dvd\om-\int_{\Hb_{v_0}^{\tau_1, \tau_2}}\Lb(r^{p+1}|\phi|^2)dud\om\\
&+\int_{H_{\tau_2^*}}L(r^{p+1}|\phi|^2)dvd\om+\int_{\tau_1}^{\tau_2}\int_{\om}\pa_t(r^{p+1}|\phi|^2)d\om dt=0.
\end{align*}
Therefore in the interior region for the domain $\bar{\mathcal{D}}_{\tau_1}^{\tau_2}\cap\{v\leq v_0\}$, the above calculations lead to the following $r$-weighted energy identity:
\begin{equation}
\label{eq:pWE:sca:in:id}
\begin{split}
&\int_{H_{\tau_2^*}^{v_0}}r^p|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^{p-1}(p|D_L\psi|^2+(2-p)|\D\psi|^2)dvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}r^p|\D\psi|^2dud\om\\
=&\int_{H_{\tau_1^*}^{v_0}}r^p|D_L\psi|^2dvd\om-\f12\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|D_L\psi|^2-|\D\psi|^2) d\om dt\\
&-\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^{p-1}\Re\left(\Box_A\phi \overline{D_L\psi}\right)+r^{p}F_{L\mu}J^\mu[\phi] dxdt.
\end{split}
\end{equation}
Similarly, in the exterior region $\{r\leq R+t\}$ for the domain $\mathcal{D}_{\tau_1}^{\tau_2}$ for all $\tau_2\leq \tau_1\leq 0$, we have
\begin{equation}
\label{eq:pWE:sca:ex:id}
\begin{split}
&\int_{H_{\tau_1^*}^{-\tau_2^*}}r^p|D_L\psi|^2dvd\om+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p-1}(p|D_L\psi|^2+(2-p)|\D\psi|^2)dvd\om du+\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^p|\D\psi|^2dud\om\\
=&\f12\int_{B_{R-\tau_1}^{R-\tau_2}}r^p(|D_L\psi|^2+|\D\psi|^2)drd\om-\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p-1}\Re\left(\Box_A\phi \overline{D_L\psi}\right)+r^{p}F_{L\mu}J^\mu[\phi] dxdt.
\end{split}
\end{equation}
For the inhomogeneous term, when $p\geq 1+\ep$, we apply Cauchy-Schwarz inequality directly
\[
2r^{p+1}|\Box_A\phi \cdot \overline{D_L\psi}|\les r^{p} u_+^{-1-\ep}|D_L\psi|^2+r^{p+2}u_+^{1+\ep}|\Box_A\phi|^2.
\]
The integral of the first term in the above inequality can be controlled by using Gronwall's inequality both in (<ref>) and (<ref>). In particular this shows that estimate (<ref>)
follows from (<ref>).
When $p<1+\ep$, we note that
\[
2p-1-\ep<p\cdot \frac{p}{1+\ep}+(p-1)(1-\frac{p}{1+\ep}).
\]
Then we can estimate the inhomogeneous term as follows:
\begin{align*}
2r^{p+1}|\Box_A\phi \cdot \overline{D_L\psi}|&\leq \ep_1 r^{2p- 1-\ep} u_+^{-p}|D_L\psi|^2+\ep_1^{-1}r^{1+\ep+2}u_+^{ p}|\Box_A\phi|^2\\
&\leq \ep_1 (r^p u_+^{-1-\ep})^{\frac{p}{1+\ep}}(r^{p-1})^{ 1-\frac{p}{1+\ep}}|D_L\psi|^2+\ep_1^{-1}r^{1+\ep+2}u_+^{ p}|\Box_A\phi|^2\\
&\leq \ep_1 r^p u_+^{-1-\ep}|D_L\psi|^2+\ep_1 r^{p-1}|D_L\psi|^2+\ep_1^{-1}r^{1+\ep+2}u_+^{ p}|\Box_A\phi|^2
\end{align*}
for all $\ep_1>0$. The integral of the first term lcan be controlled by using Gronwall's inequality. The integral of
the second term can be absorbed for sufficiently small $\ep_1$. Then estimate (<ref>) follows.
For the $r$-weighted energy estimate (<ref>) in the interior region, we need to control the boundary term on $\{r=R\}$. It suffices to estimate it for $p=0$ in (<ref>) by making use of the energy estimates (<ref>). From Hardy's inequality in Lemma <ref>,l we note that
\[
\int_{H_{\tau}}|D_L\psi|^2d\om dv \les E[\phi](\Si_\tau).
\]
By using the itegrated local energy estimate (<ref>), we therefore can show that
\begin{align*}
&\left|\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|D_L\psi|^2-|\D\psi|^2) d\om dt\right|\\
\les & R^p \int_{H_{\tau_2^*}^{v_0}}|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^{-1}|\D\psi|^2 dvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}|\D\psi|^2dud\om\\
&\qquad +\int_{H_{\tau_1^*}^{v_0}}|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^{-1}|\Re\left(\Box_A\phi \overline{D_L\psi}\right)|+|F_{L\mu}J^\mu[\phi]| dxdt\\
\les & E[\phi](\Si_{\tau_1})+I^{1+\ep}_0[\Box_A\phi](\mathcal{D}_{\tau_1}^{\tau_2})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|F_{\Lb\mu}J^\mu[\phi]|+|F_{L\mu}J^\mu[\phi]| dxdt.
\end{align*}
The inhomogeneous term can be bounded by using Cauchy-Schwarz inequality and the integrated local energy estimates. Once we have bound for the boundary terms on $\{r=R\}$, the $r$-weighted energy estimate (<ref>) follows from
the identity (<ref>) and Gronwall's inequality.
Next we establish the $r$-weighted energy estimate for the Maxwell field.
Let $F$ be any 2-form satisfying the Bianchi identity (<ref>). Then in the interior region, for all $0\leq \tau_1\leq \tau_2$, $v_0\geq \frac{\tau_2+R}{2}$, we have the $r$-weighted energy estimate:
\begin{align}
\notag
&\int_{H_{\tau_2^*}}r^{p+2}|\a|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{p+1}(p|\a|^2+(2-p)(\rho^2+\si^2))dvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}r^{p+2}(\rho^2+\si^2)dud\om\\
\label{eq:pWE:cur:in}
\les &\int_{H_{\tau_1^*}^{v_0}}r^{p+2}|\a|^2dvd\om+I^{\max\{p, 1+\ep\}}_{\min\{1+\ep, p\}}[\J](\mathcal{D}_{\tau_1}^{\tau_2})+(2-p)^{-1}I^{p+1}_0[J_{L}](\mathcal{D}_{\tau_1}^{\tau_2})\\
\notag
\end{align}
for all $0\leq p\leq 2$. Similarly in the exterior region, for all $\tau_2\leq \tau_1\leq 0$, $0\leq p\leq 2$, we have
\begin{equation}
\label{eq:pWE:cur:ex}
\begin{split}
&\int_{H_{\tau_1^*}^{-\tau_2^*}}r^p|\a|^2r^2dvd\om+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p+1}(p|\a|^2+(2-p)(\rho^2+\si^2))dvd\om du+\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^p(\rho^2+\si^2)r^2dud\om\\
\les &\int_{B_{R-\tau_1}^{R-\tau_2}}r^p |F|^2 dx+I^{\max\{p, 1+\ep\}}_{\min\{p,1+\ep\}}[\J ](\mathcal{D}_{\tau_1}^{\tau_2})+(2-p)^{-1}I^{p+1}_0[J_{L}](\mathcal{D}_{\tau_1}^{\tau_2}).
\end{split}
\end{equation}
Take the vector field
\[
X=r^p L=f\pa_t+f\pa_r
\]
in the energy identity (<ref>) for the Maxwell field. Using the computations before line (<ref>), we have
\begin{align*}
&=(r^{-1}f-\f12 f')(\rho^2+\si^2)+\frac{1}{4}f'(|\a|^2+|\underline\a|^2)+\frac{f'}{4}(|\a|^2-|\underline{\a}|^2)\\
&=\f12 r^{p-1}\left((2-p)(\rho^2+\si^2)+ p|\a|^2\right).
\end{align*}
For the boundary terms corresponding to the vector field $X=r^p L$, we have
\begin{align*}
i_{J^{X}[F]}d\vol=\f12 r^p (|\a|^2+\rho^2+\si^2)dx,\quad & i_{J^{X}[F]}d\vol=\f12 r^p (|\a|^2-\rho^2-\si^2)r^2dtd\om,\\
i_{J^{X}[\phi]}d\vol=r^p|\a|^2 r^2dvd\om,\quad & i_{J^{X}[\phi]}d\vol=-r^p (\rho^2+\si^2)r^2dud\om
\end{align*}
respectively on the $t=constant$ slice, $r=constant$ surface, the outgoing null hypersurface and the incoming null hypersurface. Therefore for all $0\leq \tau_1\leq \tau_2$, $v_0\geq \frac{\tau_2+R}{2}$ if we take the region $\mathcal{D}$
bounded by $H_{\tau_1^*}$, $H_{\tau_2^*}$, $\{r=R\}$, $\Hb_{v_0}^{\tau_1, \tau_2}$, we get the $r$-weighted energy identity in the interior region $\{r\leq R+t\}$:
\begin{align}
\notag
&\int_{H_{\tau_2^*}^{v_0}}r^p|\a|^2r^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^{p-1}(p|\a|^2+(2-p)(\rho^2+\si^2))r^2dvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}r^p(\rho^2+\si^2)r^2dud\om\\
\label{eq:pWE:cur:in:id}
=&\int_{H_{\tau_1^*}^{v_0}}r^p|\a|^2r^2dvd\om-\f12\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|\a|^2-\rho^2-\si^2) r^2d\om dt-\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}r^p \pa^\mu F_{\mu\nu}F_{L}^{\;\nu} dxdt.
\end{align}
Similarly, in the exterior region $\{r\geq R+t\}$, consider the region $\mathcal{D}_{\tau_1}^{\tau_2}$ for $\tau_2\leq \tau_1\leq 0$. We have the following identity:
\begin{align}
\notag
&\int_{H_{\tau_1^*}^{-\tau_2^*}}r^p|\a|^2r^2dvd\om+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p-1}(p|\a|^2+(2-p)(\rho^2+\si^2))dvd\om du+\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^p(\rho^2+\si^2)r^2dud\om\\
\label{eq:pWE:cur:ex:id}
=&\f12\int_{B_{R-\tau_1}^{R-\tau_2}}r^p(|\a|^2+\rho^2+\si^2)dx-\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p} \pa^\mu F_{\mu\nu}F_{L}^{\;\nu} dxdt.
\end{align}
To obtain (<ref>), we note that under the null frame $\{L, \Lb, e_1, e_2\}$
\[
F_{L}^{\;L}=-\f12 F_{L\Lb}=\rho,\quad F_{L}^{\;\Lb}=0, \quad F_{L}^{\;e_j}=\a_j,\quad j=1, 2.
\]
We can use the same method to treat the term $r^p|\pa^\mu F_{\mu e_j}F_{L}^{\;e_j}|$ as that for $\Box_A\phi \cdot \overline{D_L\psi}$ in the previous Proposition <ref> (simply replace $\Box_A\phi$ with $\pa^\mu F_{\mu e_j}$ and
$\overline{D_L}\psi$ with $r \a_j$).
For the term involving $\rho$, we estimate
\[
r^{p+2} |J_{L}\cdot \rho|\leq \f12 (2-p)r^{p+1}|\rho|^2+\frac{2}{2-p}r^{p+3}|J_{L}|^2.
\]
The integral of the first term could be absorbed. Then the $r$-weighted energy estimate (<ref>) follows from the above $r$-weighted energy identity (<ref>).
We can treat the inhomogeneous term the same way for the $r$-weighted energy estimate in the interior region from the $r$-weighted energy identity (<ref>). Like the case for the scalar field, the boundary term on $\{r=R\}$ can be bounded by taking
$p=0$ in (<ref>) and then by making use of the integrated local energy estimate (<ref>):
\begin{align*}
|\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|\a|^2-\rho^2-\si^2) r^2d\om dt|&\les \int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}(\rho^2+\si^2)rdvd\om d\tau+\int_{\Hb_{v_0}^{\tau_1, \tau_2}}(\rho^2+\si^2)r^2dud\om\\
&+R^p\int_{H_{\tau_2^*}^{v_0}}|\a|^2r^2dvd\om+ \int_{H_{\tau_1^*}^{v_0}}|\a|^2r^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}^{v_0}}|\pa^\mu F_{\mu\nu}F_{L}^{\;\nu} |dxdt\\
& \les E[F](\Si_{\tau_1})+I^{1+\ep}_0[|J_{L}|+|\J|](\mathcal{D}_{\tau_1}^{\tau_2})+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt.
\end{align*}
This together with the above argument by using Gronwall's inequality imply the $r$-weighted energy estimate for the Maxwell field in the interior region.
§ DECAY ESTIMATES FOR THE LINEAR SOLUTIONS
In this section we derive energy flux decay both for the linear Maxwell field and linear complex scalar field under appropriate assumptions.
We use bootstrap argument to construct global solutions of the nonlinear (<ref>). The first step is to study the decay properties of the linear solutions. Recall that $F=dA$ with $A$ the connection used to define the covariant derivative
$D$. Our strategy is that we make assumptions on $J_{\mu}=\pa^\nu F_{\mu\nu}$ to obtain estimates for the linear solution $F$. We then use these estimates to derive estimates for the solutions of the linear
covariant wave equation $\Box_A\phi=0$. As in the equation (<ref>) the nonlinearity $J$ is quadratic in $\phi$, by making use of the smallness of the scalar field we then can close our bootstrap assumption on $J$. The difficulty is that
the Maxwell field $F$ is no longer small and the existence of nonzero charge.
Assume that $F=dA$ has charge $q_0$ and splits into the charge part and chargeless part
\[
F=\chi_{\{r>t+R\}}q_0 r^{-2}dt\wedge dr+\bar F.
\]
Let $\J=(J_{e_1}, J_{e_2})$ be the angular component of $J$. Denote
\begin{equation}
\label{eq:def4NkF}
\begin{split}
m_k&=\sum\limits_{l\leq k}I^{1+\ga_0}_{1+\ep}[\mathcal{L}_Z^l \J](\{r\geq R\})+I^{2+\ga_0}_0[\mathcal{L}_Z^l J_L](\{ r\geq R\})+I^{1+\ep}_{1+\ga_0}[|\mathcal{L}_Z^l \J|+|\mathcal{L}_Z^l J_L|](\{t\geq 0\})\\
& +I^{1-\ep}_{1+\ga_0+2\ep}[\mathcal{L}_Z^l J_{\Lb}](\{t\geq 0\})+I_{1+\ga_0}^0[\nabla \mathcal{L}_Z^{l-1}J](\{r\leq 2R\})+|q_0|\sup\limits_{\tau\leq 0}\tau_+^{1+\ga_0}\iint_{\mathcal{D}_{\tau}^{-\infty}}|J_{\Lb}|r^{-2}dxdt,\\
M_k &=m_k+ \sum\limits_{l\leq k}E^k_0[\bar F]+1+|q_0|,
\end{split}
\end{equation}
where we recall from line (<ref>) in Section <ref> that $E_0^k[F]$ denotes the weighted Sobolev norm of the Maxwell field with weights $r_+^{1+\ga_0}$ on the initial hypersurface $t=0$. The integral of $|J_{\Lb}|r^{-2}$ is used to control the interaction of the nonzero charge with the nonlinearity $J$ in the exterior region.
To derive the energy decay for the Maxwell field, we assume that $M_k$ is finite. Note that $m_k$ can be viewed as the assumption on the inhomogeneous term $J$ which depends on the scalar field $\phi$ in the equation (<ref>). According
to our assumption, it is small. $E^k_0[\bar F]$ denotes the size of the initial data for the chargeless part of the Maxwell field. Hence it is large in our setting. The charge $q_0$ is a constant depending on the initial data of the
scalar field.
§.§ Energy decay for the Maxwell field
We derive energy flux decay for the Maxwell field $F$ under the assumption that $M_k$ is finite.
In the interior region for all $0\leq \tau_1\leq \tau_2$, $v_0\geq\frac{\tau_2+R}{2}$, we have the following energy flux decay for the Maxwell field:
\begin{equation}
\label{eq:Endecay:cur:in}
I^{-1-\ep}_0[F](\mathcal{D}_{\tau_1}^{\tau_2})+ \int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}\frac{\rho^2+\si^2}{1+r} dxd\tau+E[F](\Hb_{v_0}^{\tau_1,\tau_2})+E[F](\Si_{\tau_1})\les (\tau_1)_+^{-1-\ga_0}M_0.
\end{equation}
In the exterior region $\{r\leq R+t\}$ for all $\tau_2\leq \tau_1\leq 0$, $0\leq p\leq 1+\ga_0$, we have
\begin{align}
\label{eq:Endecay:cur:ex}
&I^{-1-\ep}_0[\bar F](\mathcal{D}_{\tau_1}^{\tau_2})+E[\bar F](\Hb_{-\tau_2^*}^{\tau_1^*})+E[\bar F](H_{\tau_1^*})+(\tau_1)_+^{-p}\int_{H_{\tau_1^*}}r^{p+2}|\a|^2dvd\om
\les (\tau_1)_+^{-1-\ga_0}M_0.
\end{align}
Let's first consider the estimates in the exterior region. By the definition of $M_0$, we derive that
\begin{align*}
\int_{B_{R-\tau_1}^{R-\tau_2}}r^p |\bar F|^2 dx+&I^{p}_{1+\ep}[\J](\mathcal{D}_{\tau_1}^{\tau_2})+I^{p+1}_0[J_{L}](\mathcal{D}_{\tau_1}^{\tau_2})\les (\tau_1)_+^{p-1-\ga_0}M_0,\quad 0\leq p\leq 1+\ga_0.
\end{align*}
Here note that in the exterior region $r\geq \frac{1}{2}u_+$. Then the $r$-weighted energy estimate (<ref>) implies that
\begin{equation*}
\begin{split}
\int_{H_{\tau_1^*}^{-\tau_2^*}}r^{p+2}|\a|^2dvd\om+\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{p+1}(|\a|^2+\bar\rho^2+\si^2)dvd\om du
\les (\tau_1)_+^{p-1-\ga_0}M_0.
\end{split}
\end{equation*}
This estimate can be used to bound the integral of $|J_{\Lb}||\rho|$ on the right hand side of (<ref>). Recall that $\rho=q_0 r^{-2}+\bar \rho$ when $r\geq R+t$. We then can show thatl
\begin{align*}
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt\les \iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|q_0||J_{\Lb}|r^{-2}+|\bar\rho|^2 r^{\ep-1}u_+^{-\ep}+
|J_{\Lb}|^2r^{1-\ep}u_+^{\ep}dxdt\les M_0(\tau_1)_+^{-1-\ga_0}.
\end{align*}
The decay estimate (<ref>) then follows from the energy estimate (<ref>) as
\[
E[\bar F](B_{R-\tau_1}^{R-\tau_2})+I^{1+\ep}_0[|\J|+|J_{L}|](\mathcal{D}_{\tau_1}^{\tau_2})\les (\tau_1)_+^{-1-\ga_0}M_0.
\]
For the decay estimates in the interior region, we use the pigeon hole argument in <cit.>. First by interpolation, we derive from the definition of $M_0$ that
\begin{align*}
I^{\max\{p, 1+\ep\}}_{\min\{p, 1+\ep\}}[\J](\mathcal{D}_{\tau_1}^{\tau_2})+I^{p+1}_0[J_{L}](\mathcal{D}_{\tau_1}^{\tau_2})\les (\tau_1)_+^{p-1-\ga_0}M_0
\end{align*}
for all $\ep\leq p\leq 1+\ga_0$. To bound $|J_{\Lb}||\rho|$, we use Cauchy-Schwaz:
\[
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt\les \iint_{\mathcal{D}_{\tau_1}^{\tau_2}}\ep_1|\rho|^2 r_+^{\ep-1}+\ep_1^{-1}r_+^{1-\ep}|J_{\Lb}|^2 dxdt,\quad \forall \ep_1>0.
\]
Here note that in the interior region $\rho=\bar \rho$. For $\ep\leq p\leq 1+\ga_0$ and sufficiently small $\ep_1$ the first term could be absorbed from the $r$-weighted energy estimates (<ref>) and the second term is bounded above by $M_0(\tau_1)_+^{-1-\ga_0}$ by the definition of $M_2$.
To apply the pigeon hole argument, we need to control the weighted energy flux through the initial hypersurface $\Si_0$ of the interior region. Note that $H_{-\frac{R}{2}}=H_{0^*}$. The weighted energy flux bound through $H_{0^*}$ follows from the decay estimate (<ref>) in the exterior region:
\begin{align*}
E[F](H_{0^*})+\int_{H_{0^*}}r^{3+\ga_0}|\a|^2dvd\om\les M_0.
\end{align*}
Here we note that on the boundary $H_{0^*}$ the charge part has bounded energy. Hence
take $p=1+\ga_0$, $\tau_1=0$ in the $r$-weighted energy estimate (<ref>). We derive that
\begin{equation*}
\int_{H_{\tau_2^*}}r^{3+\ga_0}|\a|^2dvd\om+\int_{0}^{\tau_2}\int_{H_{\tau^*}}r^{\ga_0+2}(|\a|^2+\si^2+\rho^2)dvd\om\les M_0,\quad \forall \tau_2\geq 0.
\end{equation*}
We conclude that there exists a dyadic sequence $\{\tau_{n}\}$, $n\geq 3$ such that
\begin{equation*}
\int_{H_{\tau_n^*}}r^{\ga_0+2}|\a|^2dvd\om \les (\tau_n)_+^{-1}M_0,\quad \la^{-1}\tau_n\leq \tau_{n+1}\leq \la \tau_n.
\end{equation*}
for some constant $\la$ depending only on $\ga_0$, $\ep$, $R$. Interpolation implies that
\[
\int_{H_{\tau_n^*}}r^{1+2}|\a|^2dvd\om \les (\tau_n)_+^{-\ga_0}M_0.
\]
To bound $|J_{\Lb}||\rho|$ on the right hand side of the energy estimate (<ref>), we interpolate $|\rho|$ between the integrated local energy estimate and the above $r$-weighted energy estimate:
\begin{align*}
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|J_{\Lb}||\rho|dxdt &\les \iint_{\mathcal{D}_{\tau_1}^{\tau_2}}\ep_1|\rho|^2 (r_+^{-\ep-1}+r_+^{\ga_0}\tau_+^{-1-\ga_0})+\ep_1^{-1}\tau_+^{2\ep}r_+^{1-\ep}|J_{\Lb}|^2 dxdt\\
&\les \ep_1 I^{-1-\ep}_0[F](\mathcal{D}_{\tau_1}^{\tau_2})+\ep_1^{-1}M_0(\tau_1)_+^{-1-\ga_0},\quad \forall 1>\ep_1>0.
\end{align*}
Here we have used the bound
\[
r_+^{\ep-1}\tau_+^{-2\ep}\leq r_+^{-1-\ep}+\tau_+^{-1-\ga_0}r_+^{\ga_0}.
\]
Take $\ep_1$ to be sufficiently small. From the energy estimate (<ref>), we then obtain
\begin{align*}
I^{-1-\ep}_0[F](\mathcal{D}_{\tau_1}^{\tau_2})+E[F](\Si_{\tau_2})\les E[F](\Si_{\tau_1})+(\tau_1)_+^{-1-\ga_0}M_0
\end{align*}
for all $0\leq \tau_1\leq \tau_2$, $0<\ep_1<1$. In particular, we have
\[
\int_{\tau_n}^{\tau_2}\int_{\{r\leq R\}\cap \{t=\tau\}}|F|^2 dx d\tau\les E[F](\Si_{\tau_1})+(\tau_1)_+^{-1-\ga_0}M_0.
\]
Then combine this integrated local energy estimate with the $r$-weighted energy estimate (<ref>) with $p=1$. For all $\tau_n\leq \tau_2$, we derive that
\begin{align*}
\int_{\tau_n}^{\tau_2}E[F](\Si_{\tau})d\tau&\les \int_{\tau_n}^{\tau_2}\int_{\{r\leq R\}\cap \{t=\tau\}}|F|^2 dx d\tau+\int_{\tau_n}^{\tau_2}\int_{H_{\tau^*}}
&\les \int_{H_{\tau_n^*}}r^{1+2}|\a|^2dvd\om+E[F](\Si_{\tau_n})+(\tau_n)_+^{-\ga_0}M_0\\
&\les E[F](\Si_{\tau_n})+(\tau_n)_+^{-\ga_0}M_0.
\end{align*}
On the other hand, for all $\tau\leq \tau_2$, we have
\[
E[F](\Si_{\tau_2})\leq E[F](\tau)+(\tau)_+^{-1-\ga_0}M_0\les E[F](\Si_0)+M_0\les M_0.
\]
Then from the previous estimate, we can show that
\begin{align*}
(\tau_2-\tau_n)E[F](\Si_{\tau_2})\les E[F](\Si_{\tau_n})+(\tau_n)_+^{-\ga_0}M_0\les M_0.
\end{align*}
The above estimate holds for all $\tau_2\geq \tau_n$. In particular, we obtain the coarse bound
\[
E[F](\Si_{\tau})\les \tau_+^{-1}M_0,\quad \forall \tau\geq 0.
\]
Based on this coarse bound, we can take $\tau_2=\tau_{n+1}$ in the previous estimate. We then can show that
\[
(\tau_{n+1}-\tau_n)E[F](\Si_{\tau_{n+1}})\les (\tau_n)^{-\ga_0}M_0.
\]
As $\{\tau_n\}$ is dyadic, we conclude that
\[
E[F](\Si_{\tau_n})\les (\tau_n)^{-1-\ga_0}M_0,\quad \forall n\geq 3.
\]
Then using the energy estimate, we can show that for $\tau\in[\tau_n, \tau_{n+1}]$ we have
\[
E[F](\Si_{\tau})\les E[F](\tau_{n})+(\tau_n)_+^{-1-\ga_0}M_0\les (\tau_n)_+^{-1-\ga_0}M_0\les \tau_+^{-1-\ga_0}M_0.
\]
Having this energy flux decay, the integrated local energy decay (<ref>) follows from the integrated local energy estimate (<ref>).
Since the Lie derivative $\mathcal{L}_{Z}$ commutes with the linear Maxwell equation from the commutator Lemma <ref>, as a corollary of the above energy decay proposition, we also have the
energy decay estimates for the higher order derivatives of the Maxwell field.
We have the following energy flux decay for the $k$-th derivative of the Maxwell field:
\begin{equation}
\label{eq:Endecay:cur:hide}
E[\mathcal{L}_Z^k \bar F](\Si_{\tau})\les (\tau)_+^{-1-\ga_0}M_k,\quad \forall \tau\in\mathbb{R}.
\end{equation}
This decay estimate then leads to the integrated local energy and $r$-weighted energy estimates for the Maxwell field.
By using the finite speed of propagation, the estimates in the above proposition and corollary in the exterior region depend only on the data and $J$
in the exterior region $\{t+R\leq r\}$ instead of the whole spacetime. Therefore the quantity $M_k$ can be replaced by the corresponding one defined in the exterior region. However, the estimates in the interior region rely on the data in the whole space.
§.§ Pointwise bound for the Maxwell field
The energy decay estimates derived in the previous section are sufficient to obtain pointwise bound for the Maxwell field $F$ after commuting the equation with $Z=\{\pa_t, \Om\}$ for sufficiently many times, e.g., in <cit.>, four derivatives were used to show the pointwise bound for the solution. The aim of this section is to derive the pointwise bound for the Maxwell field $F$ merely assuming $M_2$ is finite, that is, we commute the equation with $Z$ for only twice. The difficulty is that we are lack of Klainerman-Sobolev embedding which leads to the decay of the solution directly, see e.g. <cit.>. Our idea is that in the inner region $r\leq R$ we rely on elliptic estimates. In the outer region $r\geq R$, write the solutions as functions of $(u, v, \om)$. The angular momentum $\Om$ can be viewed as derivative on $\om$. The pointwise bound then follows by using a trace theorem on the null hypersurfaces and a Sobolev embedding on the sphere. Since we do not commute the equation with $L$ nor $\Lb$, those necessary energy estimates heavily rely on the null equations given in Lemma <ref>.
Let's first consider the pointwise bound for the Maxwell field in the inner region $\{r\leq R\}$. To derive the pointwise bound, we use the vector fields $\pa_t$ and the angular momentum $\Om$ as commutators. Note that the angular momentum vanishes at $r=0$. In particular we are not able to get the robust estimates for the solution in the bounded region
$r\leq R$ merely from the angular momentum. We thus rely on the killing vector field $\pa_t$ andl elliptic estimates. The following proposition gives the estimates for the Maxwell field $F$ on the bounded region $\{r\leq R\}$.
For all $0\leq \tau$, $0\leq \tau_1<\tau_2$, we have
\begin{align}
\label{eq:Est4F:in:R}
\int_{\tau_1}^{\tau_2}\sup_{|x|\leq R}|F|^2(\tau, x)d\tau&\les \int_{\tau_1}^{\tau_2}\int_{r\leq R}|\pa^2 F|^2 dxdt \les M_2(\tau_1)_+^{-1-\ga_0},\\
\label{eq:Est4F:in:R:p}
|F|^2(\tau, x)&\les M_2 \tau_+^{-1-\ga_0},\quad \forall |x|\leq R.
\end{align}
Estimate (<ref>) gives the pointwise bound for $F$ in the inner region $\{r\leq R\}$ but it is weaker than the integral version (<ref>) in the sense of decay rate. It is this integrated decay estimate that allows us to control the nonlinearities in the inner region. In other words, it is not necessary to show the improved decay of the solution in the inner region by using our approach, see e.g. <cit.>. However this does not mean that our method is not able to obtain the improved decay in the inner region. The improved decay can be derived by commuting the equation with the vector field $L$. For details about this, we refer to <cit.>.
We use elliptic estimates to prove this proposition. At fixed time $t$, let $E$, $H$ be the electric and magnetic part of the Maxwell field $F$. Let $B_r$ be the ball with radius $r$, that is, $B_r=\{t, |x|\leq r\}$. The Maxwell equation can be written as follows:
\begin{align*}
div(E)=J_0,\quad \pa_t H +cur(E)=0,\\
div(H)=0,\quad \pa_t E-cur(H)=\bar J,
\end{align*}
where $\bar J=(J_1, J_2, J_3)$ is the spatial part of $J$. Therefore by using elliptic theory we derive that
\begin{align*}
\sum\limits_{k\leq 1}\|\pa_t^k F\|_{H^1_x(B_{ \frac{3R}{2}})}^2\leq \sum\limits_{k\leq 1}\|\pa_t^k H\|_{H^1_x(B_{ \frac{3R}{2}})}^2+\|\pa_t^k E\|_{H^1_x(B_{ \frac{3R}{2}})}^2\les \sum\limits_{k\leq 1}
\|\pa_t^k J\|_{L^2_x(B_{ 2R})}^2+\|\pa_t^{k+1} F\|_{L^2_x(B_{ 2R})}^2.
\end{align*}
Make use of the above estimates with $k=1$. Differentiate the linear Maxwell equation with the spatial derivative $\nabla$. Using elliptic estimates again, we then obtain
\begin{align*}
\|\pa F\|_{H^1_x(B_{ R})}^2\les \|\pa J\|_{L^2_x(B_{ 2R})}^2+\|\pa^2_t F\|_{L^2_x(B_{ 2R})}^2.
\end{align*}
Here we omitted the lower order terms. Integrate the above inequality from time $\tau_1$ to $\tau_2$. We derive
\begin{align*}
\int_{\tau_1}^{\tau_2}\int_{r\leq R}|\pa^2 F|^2 dxdt &\les \int_{\tau_1}^{\tau_2}\int_{r\leq 2R}|\pa^2_t F|^2+|\pa J|^2 dxdt\\
&\les I_0^{-1-\ep}[\pa_t^2 F](\mathcal{D}_{\tau_1^+}^{\tau_2})+I_0^{-1-\ep}[\pa_t J](\mathcal{D}_{\tau_1^+}^{\tau_2})+I_0^0[\nabla J](\mathcal{D}_{\tau_1}^{\tau_2}\cap\{r\leq 2R\})\\
&\les M_2 (\tau_1)_+^{-1-\ga_0}.
\end{align*}
Here $\tau_1^+=\max\{\tau_1-R, 0\}$. The estimate (<ref>) then follows by using Sobolev embedding.
For the pointwise bound (<ref>), first we note that
\begin{align*}
\int_{r\leq 2R}|\nabla J|^2 dx\les \sum\limits_{k\leq 1}\int_{\tau}^{\tau+1}|\nabla \mathcal{L}_Z^kJ|^2dxdt\les M_2\tau_+^{-1-\ga_0}.
\end{align*}
Consider energy estimate on the region $\mathcal{D}_1$ bounded by $\Si_{\tau^+}$, $\tau^+=\max\{\tau-R, 0\}$ and $t=\tau$, $\tau\geq 0$. From the energy estimate (<ref>), we conclude that
\[
\int_{r\leq 2R}|\mathcal{L}_Z^2 F|^2 dx=E[\mathcal{L}_Z^2 F](r\leq 2 R)\les E[\mathcal{L}_Z^2 F](\Si_{\tau^+})+I^{1+\ep}_0[\mathcal{L}_Z^2J](\mathcal{D}_1)\les M_2\tau_+^{-1-\ga_0}.
\]
Thus the pointwise bound (<ref>) holds.
To show the decay of the solution via the energy flux through the null hypersurface, we rely on the following trace theorem.
Let $f(r, \om)$ be a smooth function defined on $[a, b]\times \mathbb{S}^2$. Then
\begin{equation}
\label{eq:trace}
\left(\int_{\om}|f|^4(r_0, \om)d\om\right)^{\f12}\leq C\int_{a}^{b}\int_{\om}|f|^2+|\pa_rf|^2+|\pa_{\om}f|^2d\om dr,\quad \forall r_0\in[a, b]
\end{equation}
for some constant $C$ independent of $r_0$.
The condition implies that $f\in H^1_{r, \om}$.
By using trace theorem,
\[
\|f(r_0, \cdot)\|_{H^{\f12}_{\om}}\leq C\|f\|_{H^1_{r, \om}},\quad \forall r_0\in[a, b].
\]
The lemma then follows by using Sobolev embedding on the sphere.
Using this lemma, we are now able to show the pointwise bound for the Maxwell field when l$\{r\geq R\}$.
Let $\bar{\mathcal{D}}_{\tau_1}=\mathcal{D}_{\tau_1}\cap\{r\geq R\}$. Then we have
\begin{align}
\label{eq:supab:I}
\|r \mathcal{L}_{Z}^k\ab\|_{L_u^2L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau_1})}^2&\les M_2(\tau_1)_+^{-1-\ga_0+2\ep},\quad k=0, 1,\\
\label{eq:supab:p}
|r\ab|^2(\tau, v, \om)&\les
% \sum\limits_{k\leq 1}\|r\mathcal{L}_Z^k\ab(\tau, v, \om)\|_{L_{\om}^4}^2 &\les
\label{eq:suparhosi:p}
r^p(|r\a|^2+|r\si|^2)(\tau, v, \om)&\les
%\sum\limits_{k\leq 1}r^p\|r\mathcal{L}_Z^k\a(\tau, v, \om)\|_{L_{\om}^4}^2 &\les
M_2\tau_+^{p-1-\ga_0},\quad 0\leq p\leq 1+\ga_0,\\
\label{eq:supbarrho:p}
r^{p}|r\bar\rho|^2(\tau, v, \om)&\les M_2\tau_+^{p-1-\ga_0},\quad 0\leq p\leq 1-\ep,\\
\label{eq:suprhosi:I}
\|r\mathcal{L}_Z^k\si\|_{L_v^2L_u^\infty L_\om^2(\bar{\mathcal{D}}_{\tau})}^2 &\les M_2\tau_+^{-1-\ga_0+\ep},\quad k\leq 1.
\end{align}
In terms of decay rate, the integral version (<ref>) is stronger than the pointwise bound (<ref>). We are not able to improve the $u$ decay of the Maxwell field is due to the weak decay rate of the initial data as initially the pointwise bound (<ref>) is the best we can get. However the integral version improves one order of decay in $u$. This is the key point that we are able to construct the global solution with the weak decay rate of the initial data.
For the integral estimate (<ref>), we rely on the transport equation (<ref>) for $\ab$. For the case in the exterior region, one can choose the
initial hypersurface $\{t=0\}$. In the interior region, for all $0\leq \tau_1\leq \tau_2$, we can choose the incoming null hypersurface $\Hb_{\frac{\tau_2+R}{2}}^{\tau_1, \tau_2}$. Let's only consider the case in the interior region. From the equation (<ref>) for $\ab$ under the null frame, for $k=0$ or $1$, we can show that
\begin{align*}
\|r \mathcal{L}_{Z}^k\ab\|_{L_u^2L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau_1}^{\tau_2})}^2&\les E[\mathcal{L}_Z^k\ab](\Hb_{\frac{\tau_2+R}{2}}^{\tau_1, \tau_2})+I^{-1-\ep}_0[\mathcal{L}_Z^k\ab](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})
& \les M_2(\tau_1)_+^{-1-\ga_0}+\|r^{\frac{1+\ep}{2}}(|\mathcal{L}_Z^{k+1}\rho|+|\mathcal{L}_Z^{k+1}\si|)\|_{L_u^2L_v^2L_{\om}^2(\bar{\mathcal{D}}_{\tau_1}^{\tau_2})}^2
+I^{1+\ep}_0[\mathcal{L}_{Z}^k \J](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})\\
&\les M_2(\tau_1)_+^{-1-\ga_0+2\ep}.
\end{align*}
Here we used interpolation to bound $\rho$ and $\si$. Indeed, the integrated local energy estimate implies that
\[
\iint r^{-\ep+1}(|\mathcal{L}_Z^{k+1}\rho|^2+|\mathcal{L}_Z^{k+1}\si|^2)dudvd\om\les M_2 (\tau_1)_+^{-1-\ga_0}.
\]
On the other hand the $r$-weighted energy estimate shows that
\[
\iint r^{2+\ga_0}(|\mathcal{L}_Z^{k+1}\rho|^2+|\mathcal{L}_Z^{k+1}\si|^2)dudvd\om\les M_2.
\]
Interpolation then implies the estimate for $\rho$ and $\si$. Thus estimate (<ref>) holds.
For the pointwise bound (<ref>) for $\ab$, we rely on the energy flux on the incoming null hypersurface together with Lemma <ref>. Consider the point $(\tau, v, \om)$. In the exterior region when $\tau<0$, let $\Hb_{\tau}$ be the incoming null hypersurface $\Hb_{v}^{\tau^*, -v}$, which is the incoming null hypersurface extending to the initial hypersurface $t=0$. In the interior region when $\tau\geq 0$, let $\Hb_{\tau}$ be $\Hb_{v}^{\tau, 2v-R}$ which is the incoming null hypersurface truncated by $r=R$. From the energy estimate (<ref>), (<ref>), we conclude that
\[
\int_{\Hb_{\tau}}|r\mathcal{L}_Z^k\ab|^2dud\om\les E[\mathcal{L}_Z^k F](\Hb_{\tau})\les M_2\tau_+^{-1-\ga_0},\quad \forall k\leq 2.
\]
As $Z$ consists of $\pa_t$ and the angular momentum $\Om$, to apply Lemma <ref>, we need the energy flux of the tangential derivative $\Lb(\ab)$. We make use of the structure equation (<ref>) which implies that
\begin{align*}
\int_{\Hb_{\tau}}|\Lb\mathcal{L}_Z^k(r\ab)|^2dud\om&\les \int_{\Hb_{\tau}}|L\mathcal{L}_Z^k(r\ab)|^2+|\mathcal{L}_{\pa_t}\mathcal{L}_Z^k(r\ab)|^2dud\om\\
&\les \int_{\Hb_{\tau}}|\mathcal{L}_Z^{k+1}\rho|^2+|\mathcal{L}_Z^{k+1}\si|^2+|\mathcal{L}_Z^k(r\J)|^2+|\mathcal{L}_Z^{k+1}(r\ab)|^2dud\om\\
&\les E[\mathcal{L}_Z^{k+1}F](\Hb_{\tau})+I^{0}_{0}[\mathcal{L}_Z^{k+1}\J](\mathcal{D}_{\tau})\\
&\les M_2 \tau_+^{-1-\ga_0},\quad k\leq 1.
\end{align*}
Here note that $\Om=(r e_1, re_2)$. Then by Lemma <ref>, for all $v$ and fixed $\tau$,
\[
\left(\int_{\om}|r\mathcal{L}_Z^k\ab|^4(\tau, v, \om)d\om\right)^{\f12}\les M_2\tau_+^{-1-\ga_0},\quad k\leq 1.
\]
Estimate (<ref>) then follows by using Sobolev embedding on the sphere.
For the pointwise bound (<ref>), (<ref>) for $\a$, $\si$, $\bar\rho$, the proof for $\a$ is slightly different to that of $\si$ and $\bar\rho$. However the idea is the same. Let's consider $\a$ first. Consider $H_{\tau^*}$, $\tau\in \mathbb{R}$. The $r$-weighted energy estimates (<ref>), (<ref>) imply that
\[
\int_{H_{\tau^*}}r^{p}|r\mathcal{L}_Z^k\a|^2dvd\om\les M_2\tau_+^{p-1-\ga_0},\quad \forall 0\leq p\leq 1+\ga_0,\quad k\leq 2.
\]
To apply Lemma <ref>, we need the energy flux of the tangential derivative $L(r\a)$. Similar to the case of $\a$, we make use of the equation (<ref>) and $\pa_t$ derivative:
\begin{align*}
\int_{H_{\tau^*}}r^{p}|L(r\mathcal{L}_Z^k\a)|^2dvd\om &\les \int_{H_{\tau^*}}r^p(|\Lb(r\mathcal{L}_Z^k\a)|^2+|r\pa_t\mathcal{L}_Z^k\a|^2)dvd\om\\
&\les \int_{H_{\tau^*}}r^p(|\mathcal{L}_Z^{k+1}\rho|^2+|\mathcal{L}_Z^{k+1}\si|^2+|\mathcal{L}_Z^k(r\J)|^2+|\mathcal{L}_Z^{k+1}(r\a)|^2)dvd\om\\
&\les M_2\tau_+^{p-1-\ga_0}+\int_{H_{\tau^*}}r^2(|\mathcal{L}_Z^{k+1}\rho|^2+|\mathcal{L}_Z^{k+1}\si|^2)+r^p|\mathcal{L}_Z^k(r\J)|^2dvd\om\\
&\les M_2\tau_+^{p-1-\ga_0}+E[\mathcal{L}_Z^k F](H_{\tau^*})+I^{p}_0[\mathcal{L}_Z^{k+1}\J](\mathcal{D}_{\tau})\\
&\les M_2\tau_+^{p-1-\ga_0}
\end{align*}
for $k\leq 1$. Estimate for $\a$ then follows by Lemma <ref> together with Sobolev embedding on the sphere.
For $\bar \rho$, $\si$, we make use of the $r$-weighted energy estimates (<ref>), (<ref>) through the incoming null hypersurface $\Hb_{\tau}$ defined as above. First we have
\[
\int_{\Hb_{\tau}}r^{p-2}(|\mathcal{L}_Z^k(r^2\bar\rho)|^2+|\mathcal{L}_Z^k(r^2\si)|^2)dud\om\les M_2\tau_+^{p-1-\ga_0},\quad k\leq 2.
\]
To derive the tangential derivative $\Lb(r^2\bar\rho)$, $\Lb(r^2\si)$, we use the equations (<ref>), (<ref>). We can show that
\begin{align*}
\int_{\Hb_{\tau}}r^{p-2}(|\Lb(r^2\mathcal{L}_Z^k\bar\rho)|^2dud\om &\les \int_{\Hb_{\tau}}r^{p-2}(|r\mathcal{L}_Z^{k+1}\ab|^2+|r^2\mathcal{L}_Z^k J_{\Lb}|^2dud\om\\
&\les E[\mathcal{L}_Z^{k+1}F](\Hb_{\tau})+I^{p}_{0}[\mathcal{L}_Z^{k+1} J_{\Lb}](\mathcal{D}_{\tau})\\
&\les M_2\tau_+^{p-1-\ga_0},\quad k=0, \quad 1
\end{align*}
for all $0\leq p\leq 1-\ep$. We are not able to extend $p$ to the full range of $[0, 1+\ga_0]$ is due to the assumption on $J_{\Lb}$. The equation (<ref>) for $\si$ does not involve $J_{\Lb}$. We hence have the full range $0\leq p\leq 1+\ga_0$ for $\si$.
Lemma <ref> and Sobolev embedding on the sphere then lead to the pointwise bound for $\bar\rho$ and $\si$. We thus have shown (<ref>), (<ref>).
Finally for the integrated decay estimates (<ref>), we show it by integrating along the incoming null hypersurface. In the interior region case we integrate from from $r=R$ while in the exterior region we integrate from the initial hypersurface $t=0$. Let's only prove (<ref>) for the interior region case. In particular take $\bar{\mathcal{D}}_{\tau}$ to be $\bar{\mathcal{D}}_{\tau_1}^{\tau_2}$ for $0\leq \tau_1< \tau_2$. First by using the decay estimate (<ref>) for $F$ when $r\leq R$, we can show that on the boundary $r=R$
\begin{align*}
\int_{\tau_1}^{\tau_2}|\mathcal{L}_Z^k F|^2(\tau, R,\om)d\om d\tau\les \int_{\tau_1}^{\tau_2}\int_{r\leq R}|\pa \mathcal{L}_Z^k F|^2dxd\tau\les M_2( \tau_1)_+^{-1-\ga_0}.
\end{align*}
Then from the transport equation (<ref>), (<ref>), we can show that
\begin{align*}
\|r\mathcal{L}_Z^k\si\|_{L_v^2L_u^\infty L_\om^2(\bar{\mathcal{D}}_{\tau_1}^{\tau_2})}^2 %+\|r^{\frac{p}{2}+1}\bar\rho\|_{L_v^2L_u^\infty L_\om^2(\bar{\mathcal{D}}_{\tau_1}^{\tau_2})}^2\\
&\les \int_{\tau_1}^{\tau_2}|\mathcal{L}_Z^k F|^2(\tau, R, \om)d\om d\tau+\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}r|\mathcal{L}_Z^k\si|^2 +|(\mathcal{L}_Z^k\si)\cdot \Lb(r^2\mathcal{L}_Z^k\si)|dudvd\om\\
&\les M_2(\tau_1)_+^{-1-\ga_0}++\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}r^{1+\ep}|\mathcal{L}_Z^k\si|^2+r^{1-\ep}|\mathcal{L}_Z^{k+1}\ab|^2dudvd\om\\
&\les M_2(\tau_1)_+^{-1-\ga_0}+M_2(\tau_1)_+^{-1-\ga_0+\ep}\les M_2(\tau_1)_+^{-1-\ga_0+\ep}.
\end{align*}
Here we have used the $r$-weighted energy estimates for $\si$ with $p=\ep$ and the integrated local energy estimates to bound $\ab$. This proves (<ref>).
§.§ Energy decay for the scalar field
In this section, we study the energy decay for the complex scalar field $\phi$ satisfying the linear covariant wave equation $\Box_A \phi=0$. When the connection field $A$ is trivial, the energy decay has
been well studied by using the new approach, see e.g. <cit.>. For general connection field $A$, presumably not small, new difficulty arises as there are interaction terms between the curvature $dA$ and the
scalar field. In the previous subsection, we derived the energy flux decay for the Maxwell field $F=dA$ with appropriate bound on $J$. The purpose of this section is to derive energy flux decay for the complex scalar field.
In addition to the assumption that $M_k$ is finite, for the complex scalar field, we assume the inhomogeneous term $\Box_A\phi$ and the initial data are bounded in the following norm:
\begin{equation}
\label{eq:def4Nkphi}
\begin{split}
\mathcal{E}_k[\phi]=\sum\limits_{l\leq k}E^k_0[\phi]&+I^{1+\ga_0}_{1+\ep}[D_Z^l\Box_A\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[D_Z^l\Box_A \phi](\{t\geq 0\}).
\end{split}
\end{equation}
For solutions of (<ref>), $\mathcal{E}_{k}[\phi]$ denotes the weighted Sobolev norm of the initial data for the complex scalar field.
As the estimates in the interior region requires the information on the boundary $\Si_{0}$ of which $H_{0^*}$ is the boundary of the exterior region. Thus we need to obtain the energy decay estimates,
at least the boundedness of the energy flux in the exterior region. The main difficulty in the presence of nontrivial connection field is to control the interaction term $(dA)_{\cdot \nu}J^\nu[\phi]$ under very
weak estimates on the curvature $dA$. In the integrated local energy estimate (<ref>) for the scalar field, it is not possible to control or absorb those terms as there is no smallness assumption on $dA$.
The idea is to make use of the null structure of $J^\nu[\phi]$ together with the $r$-weighted energy estimate (<ref>). More precisely, we first control those terms in the $r$-weighted energy estimate through Gronwall's
inequality. Then we estimate those terms in the integrated local energy estimates. Once we have control on those interaction terms, the decay of the energy flux follows from the standard argument of the new approach, similar to that
of the energy decay for the Maxwell field in the previous section.
We need a lemma to control the scalar field $\phi$ by using the $r$-weighted energy.
Assume $\phi$ vanishes at null infinity. In the exterior region on $H_{u}$, we have
\begin{equation}
\label{eq:Est4phipWE:ex}
\int_{\om}|r\phi|^2(u, v, \om)d\om\les \int_{\om}|r\phi|^2(u, -u,\om)d\om+\b^{-1}u_+^{-\b}\int_{-u}^{v}\int_{\om}r^{1+\b}|D_L(r\phi)|^2 dvd\om,\quad \forall \b>0.
\end{equation}
In the interior region on $\Si_{\tau}$, for $1\leq p\leq 2$, we have
\begin{equation}
\label{eq:Est4phipWE:in:p}
\int_{\om}r^p|\phi|^2d\om\les \left(E[\phi](\Si_{\tau})\right)^{\delta_p}\left(I^{1+\ga_0}_0[r^{-1}D_L\psi](H_{\tau^*})\right)^{1-\delta_p},\quad \delta_p=\frac{2+\ga_0-p}{1+\ga_0}.
\end{equation}
Moreover on $\Si_{\tau}$, $\tau\in\mathbb{R}$, we have
\begin{equation}
\label{eq:Est4phiE:small}
r\int_{\om}|\phi|^2d\om\les \ep_1^{-1}\int_{\Si}|\phi|^2d\tilde{v}d\om+\ep_1 E[\phi](\Si_{\tau})
\end{equation}
for all $1\geq \ep_1> 0$. Here $(\tilde{v}, \om)=(v, \om)$ when $r\geq R$ and $(r,\om)$ when $r\leq R$.
Estimate (<ref>) follows from the inequality
\[
|r\phi|(u, v,\om)\leq |r\phi|(u, -u, \om)+\int_{-u}^{v}|D_L(r\phi)|dv
\]
followed by the Cauchy-Schwarz's inequality.
In the interior region, for $\psi=r\phi$, the problem is that we can not integrate from the initial hypersurface or the boundary $H_{0^*}$ or the null infinity as the behaviour of $r\phi$ at null infinity is unknown (generically not zero).
However the scalar field $\phi$ vanishes at null infinity. We thus can bound $r|\phi|^2$ by the energy flux through $\Si$ with $\Si=\Si_{\tau}$ or $H_u$. More
precisely, on $\Si$ we can show that
\begin{align*}
r\int_{\om}|\phi|^2d\om&\les \int_{\Si}|\phi|^2d\tilde{v}d\om +\int_{\Si}r|D_{\tilde{v}}\phi||\phi|d\tilde{v}d\om\\
&\les \ep_1\int_{\Si}|D_{\tilde{v}}\phi|^2 r^2 d\tilde{v}d\om +(\ep_1^{-1}+1)\int_{\Si}|\phi|^2d\tilde{v}d\om\\
&\les \ep_1 E[\phi](\Si)+\ep_1^{-1}\int_{\Si}|\phi|^2d\tilde{v}d\om.
\end{align*}
This gives estimate (<ref>). In particular for $\ep_1=1$, from Hardy's inequality (<ref>), we conclude that estimate (<ref>) holds for $p=1$. To prove it for all $1\leq p\leq 2$, it suffices to show the estimate with $p=2$.
Consider the sphere with radius $r=\frac{\tau^*+v}{2}$ on
$H_{\tau^*}\subset \Si_{\tau}$. Choose the sphere with radius $r_1=\frac{\tau^*+v_1}{2}$ such that
\[
\]
If $r\leq r_1$, then (<ref>) with $p=2$ follows from (<ref>) with $p=1$. Otherwise we have $r_1< r$.
\begin{align*}
\int_{\om}|r\phi|^2(\tau^*, v,\om)d\om&\les \int_{\om}|r\phi|^2(\tau^*, v_1, \om )+r_1^{-\ga_0}\int_{H_{\tau^*}}r^{1+\ga_0}|D_L(r\phi)|^2dvd\om\\
&\les r_1 E[\phi](\Si_{\tau})+r_1^{-\ga_0}I^{1+\ga_0}_0[r^{-1}D_L\psi](H_{\tau^*})\\
&\les \left(E[\phi](\Si_{\tau})\right)^{\frac{\ga_0}{1+\ga_0}}\left(I^{1+\ga_0}_0[r^{-1}D_L\psi](H_{\tau^*})\right)^{\frac{1}{1+\ga_0}}.
\end{align*}
Here we recall the notation $I$ defined in Section <ref>.
The following lemma is quite simple but it turns out to be very useful.
Suppose $f(\tau)$ is smooth. Then for any $\b\neq 0$, we have the identity
\begin{equation*}
\int_{\tau_1}^{\tau_2}s^\b
\end{equation*}
§.§.§ Energy decay in the exterior region
In the exterior region, as $r\geq \frac{1}{3}u_+$, it suffices to consider the $r$-weighted energy estimate for the largest $p=1+\ga_0$. First we can show that
In the exterior region for all $\tau_2\leq \tau_1\leq 0$, we have
\begin{equation}
\label{eq:Est4FJ:ex}
\begin{split}
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{1+\ga_0}| F_{L\mu}J^{\mu}[\phi] |dxdt\les & M_2 E_0^0[\phi]+M_2 \int_{u}u_+^{-1-\ep}\int_{v}r^{1+\ga_0}|D_L\psi|^2dvd\om du\\
&+ |q_0|\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}r^{\ga_0}(|D_L(r\phi)|^2 +|\D(r\phi)|^2)dvdud\om.
\end{split}
\end{equation}
As $F=dA$ has different decay properties for different components, we estimate the integral according to the index $\mu$. Note that $ r^2 J[\phi]=J[r\phi]$.
For $\mu=\Lb$, we have
\begin{equation}
\label{eq:FLLbJ}
|F_{L\Lb}J^{\Lb}[\phi]|\les r^{-2}|q_0| |D_L\psi||\psi|+|\bar \rho||D_L\psi||\psi|,\quad \psi=r\phi.
\end{equation}
The first term on the right hand side will be absorbed with the smallness assumption on the charge $q_0$ (as the data for the scalar field is small). In deed, by using Lemma <ref>, we can show that
\begin{align*}
2\iint r^{\ga_0-1}|D_L\psi||\psi|dudvd\om & \leq \iint r^{\ga_0} |D_L\psi|^2dvdud\om +\iint r^{\ga_0}|\phi|^2dvdud\om \\
&\les \iint r^{\ga_0} |D_L\psi|^2dvdud\om+\int_{u}\int_{\om }(r^{1+\ga_0}|\phi|^2)(u, -u, \om )d\om du\\
&\les \iint r^{\ga_0} |D_L\psi|^2dvdud\om+E_0^0[\phi].
\end{align*}
For the second term on the right hand side of (<ref>), the idea is that we use Cauchy-Schwarz inequality and make use of the $r$-weighted energy estimate. First we can estimate that
\begin{align*}
2r^{1+\ga_0}|\bar \rho||D_L\psi||\psi| \leq r^{1+\ga_0}|D_L\psi|^2 u_+^{-1-\ep}+u_+^{1+\ep}r^2|\bar\rho|^2r^{1+\ga_0}|\phi|^2.
\end{align*}
The first term will be controlled through Gromwall's inequality. For the second term, we can first use Sobolev embedding on the unit sphere to bound $\bar \rho$ and then apply Lemma <ref>:
\begin{align*}
&\iint u_+^{1+\ep}r^2|\bar\rho|^2r^{1+\ga_0}|\phi|^2 dudvd\om \\
&\les \int_{u}u_+^{1+\ep}\int_{v} \sum\limits_{j\leq 2}r^2\int_{\om}|\mathcal{L}_{\Om}^j\bar \rho|^2 d\om \cdot \int_{\om}r^{1+\ga_0}|\phi|^2d\om dvd u\\
&\les \int_{u} u_+^{1+\ep -1} E^2[\bar F](H_{u}) (u_+^{\ga_0}\int_{\om}|r\phi|^2(u, -u, \om)d\om +\int_{v}\int_{\om}r^{1+\ga_0}|D_L\psi|^2 dvd\om) du\\
&\les M_2\int_{|x|\geq R} r_+^{1+\ga_0-\ep-2}|\phi|^2(0, x)dx+M_2\int_{u}u_+^{-1-\ep}\int_{v}r^{1+\ga_0}|D_L\psi|^2dvd\om du.
\end{align*}
The first term is bounded by the weighted Sobolev norm of the initial data. The second term can be controlled by using Gronwall's inequality. Thus estimate (<ref>) holds for the case of $\mu=L$.
For $\mu=e_1$ or $e_2$, first we can bound
\begin{align*}
r^{1+\ga_0}|F_{L e_j}||J^{e_j}[\psi]|\leq \ep_1 r^{\ga_0}|\nabb \psi|^2+\ep_1^{-1} r^{3+\ga_0}|\a|^2 r|\phi|^2, \quad \forall \ep_1>0.
\end{align*}
We choose sufficiently small $\ep_1$ so that the integral of the first term can be absorbed. For the second term, we first use Sobolev embedding on the unit sphere to bound $\a$ and then
Lemma <ref> to control $\phi$:
\begin{align*}
&\iint r^{3+\ep}|\a|^2 r^{1+\ga_0-\ep}|\phi|^2 dudvd\om\\
& \les \int_{u} u_+^{\ga_0-\ep-1}\int_{v} r^{3+\ep}\sum\limits_{j\leq 2}\int_{\om }|\mathcal{L}_{\Om}^j \a|^2d\om \cdot \left(\int_{\om}|r\phi|^2(u, -u, \om )d\om +u_+^{-\ga_0}
\int_{v}\int_{\om}r^{1+\ga_0}|D_L\psi|^2dvd\om \right)du\\
&\les M_2\int_{|x|\geq R}r_+^{1+\ga_0-\ep -2}|\phi|^2(0, x)dx+M_2\int_{u}u_+^{-1-\ep}\int_{v}r^{1+\ga_0}|D_L\psi|^2dvd\om du\\
&\les M_2E_0^0[\phi]+M_2\int_{u}u_+^{-1-\ep}\int_{v}r^{1+\ga_0}|D_L\psi|^2dvd\om du.
\end{align*}
As the data for the scalar field is small, the charge is also small. In particular we can choose $\ep_1=|q_0|$ (if $q_0=0$, let $\ep_1$ small depending only on $\ep$, $\ga_0$ land $R$).
Therefore estimate (<ref>) holds for the case when $\mu=e_1$ or $e_2$. We thus finished the proof for the Proposition.
As a corollary, we show the $r$-weighted energy flux decay of the scalar field in the exterior region.
Assume that the charge $q_0$ is sufficiently small, depending only on $\ep$, $R$, $\ga_0$. Then in the exterior region, we have the energy flux decay
\begin{equation}
\label{eq:pWEdecay:sca:ex}
\begin{split}
&\int_{H_{\tau_1^*}}r^p|D_L\psi|^2dvd\om+\iint_{\mathcal{D}_{\tau_1}}r^{p-1}(p|D_L\psi|^2+|\D\psi|^2)dvd\om du+\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^p|\D\psi|^2dud\om \\
&\les_{M_2} \mathcal{E}_0[\phi] (\tau_1)_+^{p-1-\ga_0},\quad \forall 0\leq p\leq 1+\ga_0,\quad \forall \tau_2\leq \tau_1\leq 0.
\end{split}
\end{equation}
It suffices to prove the corollary for $p=1+\ga_0$. For sufficiently small $q_0$ depending only on $\ep$, $\ga_0$ and $R$, from the $r$-weighted energy estimate (<ref>) and the estimate (<ref>) for the error term,
the integral of $r^{\ga_0}|D_L(r\phi)|^2$ can be absorbed. Then estimate (<ref>) follows from Gronwall's inequality.
Next we make use of the $r$-weighted energy decay to show the energy flux decay and the integrated energy decay for the scalar field in the exterior region. In the sequel we assume the charge $q_0$ is sufficiently small so that estimate (<ref>) of Corollary <ref> holds in the exterior region $\tau_2\leq \tau_1\leq 0$. From the integrated energy estimate (<ref>), it suffices to bound
the interaction term of the gauge field and the scalar field.
In the exterior region for all $\tau_2\leq \tau_1\leq 0$, we have
\begin{equation}
\label{eq:Est4bFJ:ex}
\begin{split}
&\iint_{\mathcal{D}_{\tau_1}^{ \tau_2}}|F_{L\nu}J^\nu[\phi]|+|F_{\Lb\nu}J^\nu(\phi)| dxdt\\
&\les \ep_1 I^{-1-\ep}_0[D\phi](\mathcal{D}_{\tau_1}^{\tau_2})+C_{M_2, \ep_1}\left(\mathcal{E}_0[\phi](\tau_1)_+^{-1-\ga_0}+(\tau_1)_+^{\ep}\int_{-\tau_1^*}^{-\tau_2^*}v^{-1-\ep}E[\phi](\Hb_{v}^{-\tau_1^*})dv\right)
\end{split}
\end{equation}
for all $\ep_1>0$ and some constant $C_{M_2, \ep}$ depending on $M_2$ and $\ep_1$.
The integral of $(dA)_{L\nu}J^\nu[\phi]$ has been controlled in the previous Proposition <ref> as Corollary <ref> implies that the right hand
side of (<ref>) can be bounded by a constant depending on $M_2$, $\ep$, $\ga_0$ and $R$. Since in the exterior region $r\geq \frac{1}{3} u_+$, we easily obtain the desired bound:
\begin{equation*}
\iint_{\mathcal{D}_{\tau_1}^{ \tau_2}}|F_{L\nu}J^\nu[\phi]|dxdt\les_{M_2}(\tau_1)_+^{-1-\ga_0}\mathcal{E}_0[\phi].
\end{equation*}
It remains to estimate the integral of $F_{\Lb \nu}J^{\nu}[\phi]$. The $r$-weighted energy decay gives control for the "good" derivative of the scalar field. The problem is that we do not have any control for the "bad" derivative $D_{\Lb}\phi$. What worse is the existence of nonzero charge. Although the
charge can be small, we are not able to absort the charge part $q_0 r^{-2}J_{\Lb}[\phi]$ in the integrated local energy estimate (<ref>) as there is a small $\ep$ loss of decay in $I^{-1-\ep}_0[\bar D\phi]$ on the
left hand side. The idea to treat this term is to make use of the energy flux on the incoming null hypersurface $\Hb_{-u_2}^{u_1}$ and then apply Gronwall's inequality. Let's first consider the easier terms in the integral of
$F_{\Lb \nu}J^{\nu}[\phi]$. For $\nu=e_1$ or $e_2$, we have
\[
|F_{\Lb \nu}J^{\nu}[\phi]|\les |\ab||\D\phi||\phi|.
\]
Note that from estimate (<ref>) of Lemma <ref> and Corollary <ref>, we obtain
\[
\int_{\om}|r\phi|^2(u, v, \om)d\om \les_{M_2}u_+\int_{\om}(-2u)|\phi(0, -2u, \om)|^2 d\om +\mathcal{E}_0[\phi] u_+^{-\ga_0}.
\]
Here we parametrize $\phi$ in $(t, r, \om )$ coordinates. We then use Sobolev embedding on the initial hypersurface $t=0$ to derive the decay of $\phi$:
\begin{equation}
\label{eq:SPHDecay:sca:ex}
\int_{\om}|r\phi|^2(u, v, \om)d\om \les_{M_2}\mathcal{E}_0[\phi] u_+^{-\ga_0}.
\end{equation}
From the $r$-weighted energy estimate (<ref>), we have the weighted angular derivative of the scalar field on the incoming null hypersurface
\[
\int_{\Hb_{-\tau_2^*}^{\tau_1^*}}r^{1+\ga_0}|\D(r\phi)|^2dud\om\les_{M_2}\mathcal{E}_0[\phi].
\]
In the exterior region, note that $r\geq\frac{v}{2}$.
Therefore we can show that
\begin{align*}
&\iint_{\mathcal{D}_{\tau_1}^{ \tau_2}}|F_{\Lb e_j}||J^{e_j}[\phi]|dxdt \\
&\les \int_{-\tau_1^*}^{-\tau_2^*}\int_{-v}^{\tau_1^*}\int_{\om}r^2|\ab||\D\phi||\phi|d\om dudv\\
&\les \int_{-\tau_1^*}^{-\tau_2^*}\int_{-v}^{\tau_1^*}r^{-\frac{3+\ga_0}{2}}\left(r^2\sum\limits_{j\leq 2}\int_{\om}|\mathcal{L}_{\Om}^j \ab|^2d\om\right)^{\f12}\left(r^{3+\ga_0}\int_{\om}|\D\phi|^2d\om\cdot \int_{\om }|r\phi|^2d\om\right)^{\f12} dudv\\
&\les_{M_2}\mathcal{E}_0[\phi](\tau_1)_+^{-\frac{\ga_0}{2}-\frac{1+\ga_0}{2}-\frac{1}{2}}\les_{M_2}\mathcal{E}_0[\phi] (\tau_1)_+^{-1-\ga_0}.
\end{align*}
When $\nu=L$, first we have
\[
|F_{\Lb L}||J^{L}[\phi]|\les |q_0|r^{-2}|D_{\Lb }\phi||\phi|+|\bar \rho||D_{\Lb}\phi||\phi|.
\]
The second term is easy to bound. We may use Cauchy-Schwarz to absorb $D_{\Lb}\phi$. In deed,
\[
2|\bar \rho||D_{\Lb}\phi||\phi|\leq \ep_1 |D_{\Lb}\phi|^2 r^{-1-\ep}+\ep_1^{-1}|\bar \rho|^2|\phi|^2 r^{1+\ep},\quad \forall \ep_1>0.
\]
For sufficiently small $\ep_1$, the integral of the first term on the right hand side can be absorbed from the integrated energy estimate (<ref>). For the second term, we make use of estimate (<ref>) to show
\begin{align*}
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|\bar \rho|^2r^{3+\ep}|\phi|^2 dvdud\om &\les \int_{\tau_1^*}^{\tau_2^*}\int_{-u}^{-\tau_2^*}\sum\limits_{j\leq 2}\int_{\om}r^2|\mathcal{L}_{\Om}^j \bar\rho|^2d\om\cdot r^{1+\ep}\int_{\om}|\phi|^2d\om dvdu\\
&\les_{M_2} \int_{\tau_1^*}^{\tau_2^*}E^2[\bar F](H_u^{-\tau_2^*})u_+^{-1+\ep-\ga_0}\mathcal{E}_0[\phi] du\\
& \les_{M_2} \mathcal{E}_0[\phi]\int_{u_1^*}^{u_2^*}u_+^{-2-\ga_0}du\les_{M_2}\mathcal{E}_0[\phi] (\tau_1)_+^{-1-\ga_0}.
\end{align*}
Finally we need to bound the charge part, namely the integral of $|q_0|r^{-2}|D_{\Lb }\phi||\phi|$. As we have explained previously, this term can not be absorbed even though the charge $q_0$ is small due to the loss of decay
in the integrated local energy $I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})$ in (<ref>). The idea is to make use of the energy flux in the incoming null hypersurface $\Hb_{-\tau_2^*}^{\tau_1^*}$ and then apply Gronwall's inequality.
From estimate (<ref>) and noting that $r\geq \frac{1}{2}v$ in the exterior region, we can show that
\begin{align*}
\iint_{\mathcal{D}_{\tau_1}^{ \tau_2}}r^{-2}|D_{\Lb}\phi||\phi|dxdt &\les \int_{-\tau_1^*}^{-\tau_2^*}\int_{-v}^{\tau_1}\int_{\om}|D_{\Lb}\phi||\phi|d\om dudv\\
&\les \int_{-\tau_1^*}^{-\tau_2^*}\int_{-v}^{\tau_1}r^{-2}\left(r^{2}\int_{\om}|D_{\Lb}\phi|^2d\om\cdot \int_{\om }|r\phi|^2d\om\right)^{\f12} dudv\\
&\les_{M_2} \mathcal{E}_0[\phi]^{\f12} \int_{-\tau_1^*}^{-\tau_2^*}v^{-\frac{3+\ga_0-\ep}{2}}\int_{-v}^{\tau_1^*}r^{-\frac{1-\ga_0+\ep}{2}}\left(r^{2}\int_{\om}|D_{\Lb}\phi|^2d\om\right)^{\f12} u_+^{-\frac{\ga_0}{2}} dudv\\
&\les_{M_2} \mathcal{E}_0[\phi]^{\f12}\int_{-\tau_1^*}^{-\tau_2^*}v^{-\frac{3+\ga_0-\ep}{2}}(E[\phi](\Hb_{v}^{-\tau_1^*}))^{\f12}(\tau_1)_+^{-\frac{\ep}{2}}dv\\
\end{align*}
Combining all the previous estimates, we then have shown (<ref>).
As a corollary we then can show the energy flux decay as well as the integrated local energy decay of the scalar field in the exterior region.
For all $\tau_2<\tau_1\leq 0$, we have:
\begin{equation}
\label{eq:ILEdecay:sca:ex}
\begin{split}
& I^{-1-\ep}_0[\bar D\phi](\mathcal D_{\tau_1}^{\tau_2})+E[\phi](H_{\tau_1^*}^{-\tau_2^*})+E[\phi](\Hb_{-\tau_2^*}^{\tau_1^*})\les_{M_2} (\tau_1)_+^{-1-\ga_0}\mathcal{E}_0[\phi].
\end{split}
\end{equation}
First choose $\ep_1$ in the estimate (<ref>) to be sufficiently small, depending only on $\ep$, $\ga_0$ and $R$, so that after combing
estimate (<ref>) and the integrated energy estimate (<ref>), the term $\ep_1 I^{-1-\ep}_0[D\phi](\mathcal{D}_{\tau_1}^{\tau_2})$ on the right hand side of (<ref>) can be absorbed by
$I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})$ on the left hand side of (<ref>). Then notice that we have the uniform bound:
\[
(\tau_1)_+^{\ep}\int_{-\tau_1^*}^{-\tau_2^*}v^{-1-\ep}dv\les 1,\quad \forall \tau_2<\tau_1\leq 0.
\]
By using Gronwall's inequality (fix $\tau_1\leq 0$ and take $\tau_2\leq \tau_1$ as variable), we then obtain (<ref>).
§.§.§ Energy decay in the interior region
Once we have the energy flux and the $r$-weighted energy decay estimates for the scalar field in the exterior region, we in particular have the energy flux bound for the scalar field on the boundary $H_{-\frac{R}{2}}$. This is necessary to
consider the energy flux decay in the interior region. Compared to the case in the exterior region, the charge is not a problem as the charge only effects the decay property of the Maxwell field in the exterior region. However, new difficulties
arise in the interior region. First of all there is no lower bound for $\frac{r}{\tau_+}$. That means we may need estimates for general $p$ for the $r$-weighted energy estimate instead of simply the largest $p$. Secondly as we
have explained before that we are not able to absorb the interaction term between the gauge field $A$ and the scalar field due to the fact that $dA$ is no longer small in our setting. Thus we need to rely on the $r$-weighted
energy estimates and make use of the null structure of $J[\phi]$. In the exterior region, the idea is first to derive the
$r$-weighted energy decay and then to obtain the integrated local energy and energy flux decay. In the interior region, we see from the $r$-weighted
energy estimates (<ref>) that the term $|F_{\Lb \mu}J^{\mu}[\phi]|$ also appears on the right hand side. This suggests that we have to consider the $r$-weighted energy estimate and the integrated local energy estimates
As the boundedness of the energy flux on the boundary $H_{-\frac{R}{2}}=H_{0^*}$ requires the smallness of the assumption that the charge $|q_0|$ is small. In the sequel when we
consider estimates in the interior, we always assume this without repeating it. Let's first estimate the interaction terms of $dA$ and the scalar field in the $r$-weighted energy estimate (<ref>).
In the interior region, for all $0\leq \tau_1\leq \tau_2$ and $1\leq p\leq 1+\ga_0$, we have
\begin{align}
\notag
\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}&r^p|F_{L\mu}J^{\mu}[\phi]|^2 dxdt\les \ep_1\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}r^{p-1}|\D(r\phi)|^2dvd\om d\tau+
\label{eq:Est4FJ:sca:in}
&+M_2\ep_1^{-1}\left(\delta_p\int_{\tau_1}^{\tau_2}E[\phi](\Si_{\tau})\tau_+^{\delta_p^{-1}-1-\ep}d\tau+(1-\delta_p)I^{1+\ga_0}_{-1-\ep}[r^{-1}D_L\psi](\bar {\mathcal{D}}_{\tau_1}^{\tau_2})\right)
\end{align}
for all $\ep_1>0$. Here $\delta_p=\frac{2+\ga_0-p}{1+\ga_0}$ is given in Lemma <ref> in line (<ref>).
Denote $\psi=r\phi$ and $F=dA$. First we have
\begin{align*}
2r^p|F_{L\mu}J^{\mu}[\phi]|r^2\leq r^p|D_L\psi|^2\tau_+^{-1-\ep}+r^p |\rho|^2 |\psi|^2 \tau_+^{1+\ep}+\ep_1 r^{p-1}|\D\psi|^2+\ep_1^{-1}r^{p+3}|\a|^2|\phi|^2
\end{align*}
for all $\ep_1>0$. The first term can be absorbed by using Gronwall's inequality. The third term will be absorbed for sufficiently small $\ep_1$ depending only on $\ep$, $\ga_0$ and $R$. For the second term,
we use the energy flux of $\rho$ on $H_{\tau^*}$ to bound $\rho$ and estimate (<ref>) of Lemma <ref> to bound $\phi$. For the last term, we use the $r$-weighted energy estimate to bound $\a$.
Then similar to the proof of Proposition <ref> we can show that
\begin{align*}
&\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}\tau_+^{1+\ep}r^p |\rho|^2|\psi|^2+r^{p+3}|\a|^2|\phi|^2dvd\om d\tau\\
&\les \int_{\tau_1}^{\tau_2}\int_{2R+\tau^*}^{\infty}\sum\limits_{j\leq 2}\int_{\om}\tau_+^{1+\ep} r^2|\mathcal{L}_{\Om}^j \rho|^2+r^3|\mathcal{L}_{\Om}^j\a|^2d\om\cdot
\int_{\om} r^{p}|\phi|^2d\om dvdu\\
&\les M_2\int_{\tau_1}^{\tau_2}\tau_+^{-\ep} \left(E[\phi](\Si_{\tau})\right)^{\delta}\left(I^{1+\ga_0}_0[r^{-1}D_L\psi](H_{\tau^*})\right)^{1-\delta} d\tau\\
&\les M_2\left(\delta\int_{\tau_1}^{\tau_2}E[\phi](\Si_{\tau})\tau_+^{\delta^{-1}-1-\ep}d\tau+(1-\delta)I^{1+\ga_0}_{-1-\ep}[r^{-1}D_L\psi](\bar {\mathcal{D}}_{\tau_1}^{\tau_2})\right).
\end{align*}
The proposition then follows.
Next we estimate the interaction terms in the energy estimate (<ref>). We show the following:
We have
\begin{equation}
\label{eq:Est4FJ:sca:in:0}
\begin{split}
&\les \ep_1 I^{-1-\ep}_0[D\phi](\mathcal{D}_{\tau_1}^{\tau_2}) +\ep_1^{-1} \int_{\tau_1}^{\tau_2} g(\tau)E[\phi](\Si_\tau)d\tau+I_{-2-\ga_0}^{1+\ga_0}[r^{-1}D_L(r\phi)]
\end{split}
\end{equation}
for all $0<\ep_1<1$. Here
\[
g(\tau):=\sum\limits_{j\leq 2}I^{-1-\ep}_{1+2\ep}[\mathcal{L}_{\Om}^j F](\Si_{\tau})+\int_{H_{\tau^*}}r^{2+\ep}(|\mathcal{L}_{\Om}^j\a|^2+|\mathcal{L}_{\Om}^j\rho|^2)dvd\om+\sup\limits_{|x|\leq R}|F|^2(\tau, x).
\]
For the integral on $\{r\geq R\}$, we use Sobolev embedding on the unit sphere to bound the curvature and the proof is quite similar to that of
the previous proposition. On the finite region $r\leq R$, we make use of the $L^2_t L_x^\infty$ norm of the curvature given in Proposition <ref>. For the case when $r\geq R$, first we have
\begin{align*}
|F_{L\nu}J^{\nu}[\phi]|+|F_{\Lb\nu}J^{\nu}[\phi]|&\les (|\rho|+|\a|)|D\phi||\phi|+|\ab||\D\phi||\phi|\\
&\les \ep_1 r_+^{-1-\ep}|D\phi|^2+\ep_1^{-1}(|\rho|^2+|\a|^2)r_+^{1+\ep}|\phi|^2+|\ab||\D\phi||\phi|.
\end{align*}
The first term can be absorbed in the energy estimate (<ref>) for sufficiently small $\ep_1$.
For the second term, we can use estimate (<ref>) to bound $\phi$ by the energy flux through $H_{\tau^*}$ and the $r$-weighted energy to control the curvature terms. The last term is the most difficult term to control. The reason is that we do not have powerful estimates for $\ab$. The only estimates we have are the integrated local energy estimate and the energy flux through the incoming null hypersurface. Unlike the case in the
exterior region where we can make use of the energy flux through the incoming null hypersurface for $\ab$, this method fails in the interior region. The main reason is that the energy flux $E[F](\Hb_{v}^{\tau_1, \tau_2})$ decays in $\tau_1$ instead of $v$. A possible way to solve this issue is to assume pointwise bound for $\ab$. However the problem is that the pointwise decay for $\ab$ is too weak (due to the assumption on the initial data. We have explained this in the introduction) to be useful. We thus can only rely on the integrated local energy estimate for $\ab$. As there is a $r^{\ep}$ ldecay loss in the integrated local energy estimate for $\ab$, we are not able to bound $\phi$ simply by using the energy flux through $H_{\Si_{\tau^*}}$ but instead we need to make use of the $r$-weighted energy estimate. This means that we can not obtain uniform energy bound from the energy estimate (<ref>). We need to combine it with the $r$-weighted energy estimate.
For the integral of $|\ab||\D\phi||\phi|$, from estimate (<ref>) with $p=1+\ep$, we can show that
\begin{align*}
&\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}|\ab||\D\phi||\phi|r^2d\om dv d\tau\\
&\les \int_{\tau_1}^{\tau_2}\int_{2R+\tau^*}^\infty \left(\sum\limits_{j\leq 2}\int_{\om}r^{1-\ep}|\mathcal{L}_{\Om}^j\ab|^2d\om\right)^\f12\left(\int_{\om}r^2|\D\phi|^2d\om\cdot \int_{\om}r^{1+\ep}|\phi|^2d\om \right)^\f12 dv d\tau\\
&\les \sum\limits_{j\leq 2}\int_{\tau_1}^{\tau_2}\left(I^{-1-\ep}_0[\mathcal{L}_{\Om}^j \ab](\Si_{\tau})E[\phi](\Si_\tau)\right)^\f12 \left(E[\phi](\Si_\tau)\right)^{\f12\delta}\left(I_0^{1+\ga_0}[r^{-1}D_L(r\phi)](H_{\tau^*})\right)^{\f12-\f12\delta}d\tau\\
&\les \sum\limits_{j\leq 2}\int_{\tau_1}^{\tau_2} I^{-1-\ep}_{1+2\ep}[\mathcal{L}_{\Om}^j \ab](\Si_{\tau})E[\phi](\Si_\tau)d\tau+\int_{\tau_1}^{\tau_2}\tau_+^{-1-\ep}E[\phi](\Si_\tau)d\tau+I_{-2-\ga_0}^{1+\ga_0}[r^{-1}D_L(r\phi)]
\end{align*}
Here $\delta=\frac{1+\ga_0-\ep}{1+\ga_0}$ and in the last step we have used Jensen's inequality as well as the relation
\[
\f12+\ep-\f12 \delta(1+\ep)-(2+\ga_0)(\f12-\f12\delta)=\f12\frac{\ep}{1+\ga_0}>0.
\]
In the above estimate the first two terms will be estimated by using Gronwall's inequality. We keep the last term which involves the $r$-weighted energy estimates.
For the integral of $(|\rho|^2+|\a|^2)r_+^{1+\ep}|\phi|^2$, we use estimate (<ref>) to bound $\phi$. We have
\begin{align*}
&\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}(|\rho|^2+|\a|^2)r_+^{1+\ep}|\phi|^2r^2d\om dv d\tau\\
&\les \int_{\tau_1}^{\tau_2}\int_{2R+\tau^*}^\infty \sum\limits_{j\leq 2}\int_{\om}r^{2+\ep}(|\mathcal{L}_{\Om}^j\a|^2+|\mathcal{L}_{\Om}^j\rho|^2)d\om\cdot
\int_{\om}r|\phi|^2d\om dv d\tau\\
&\les \sum\limits_{j\leq 2}\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{2+\ep}(|\mathcal{L}_{\Om}^j\a|^2+|\mathcal{L}_{\Om}^j\rho|^2)dvd\om \cdot E[\phi](\Si_{\tau})d\tau.
\end{align*}
This term will be controlled in the energy estimate (<ref>) by using Gronwall's inequality.
For the integral on the region $r\leq R$, we can show that
\begin{align*}
\int_{\tau_1}^{\tau_2}\int_{r\leq R}|F_{L\nu}J^{\nu}[\phi]|+|F_{\Lb\nu}J^{\nu}[\phi]|dx d\tau &\les \ep_1\int_{\tau_1}^{\tau_2}\int_{r\leq R}|D\phi|^2 dxd \tau+\ep_1^{-1}\int_{\tau_1}^{\tau_2}\int_{r\leq R}|F|^2|\phi|^2dxd\tau\\
&\les \ep_1\int_{\tau_1}^{\tau_2}\int_{r\leq R}\frac{|D\phi|^2}{r_+^{1+\ep}} dxd \tau+\ep_1^{-1}\int_{\tau_1}^{\tau_2}\sup\limits_{|x|\leq R}|F|^2 \cdot E[\phi](\Si_{\tau}) d\tau
\end{align*}
for all $\ep_1>0$. The first term will be absorbed for small $\ep_1$. The second term
can be controlled by using Gronwall's inequality. Combining all these estimates above, we thus have shown estimate (<ref>) of the Proposition.l
As a corollary the energy estimate (<ref>) lcan be reduced to the following:
In the interior region, we have the following integrated local energy estimate
\begin{equation}
\label{eq:ILE:sca:in:sim0}
\begin{split}
%+E[\phi](\Hb^{\tau_1, \tau_2}_v)+\iint_{\mathcal{D}_{\tau_1}^ {\tau_2}}\frac{|\D\phi|^2}{1+r} dxdt
&I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})+E[\phi](\Si_{\tau_2})+\int_{\tau_1}^{\tau_2}\tau_+^{-1-\ep}E[\phi](\Si_{\tau})d\tau+
\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}|F_{L\nu}J^{\nu}[\phi]|+|F_{\Lb\nu}J^{\nu}[\phi]|dxdt\\
&\les_{M_2} E[\phi](\Si_{\tau_1})+(\tau_1)_+^{-1-\ga_0}\mathcal{E}_0[\phi]+I_{-2-\ga_0}^{1+\ga_0}[r^{-1}D_L(r\phi)]
\end{split}
\end{equation}
First choose $\ep_1$ sufficiently small in the estimate (<ref>) so that combining the energy estimate (<ref>) with (<ref>) the integrated local energy
$I^{-1-\ep}_0[D\phi](\mathcal{D}_{\tau_1}^{\tau_2})$ could be absorbed. By our notation, the smallness of $\ep_1$ depends only on $\ep$, $\ga_0$ and $R$. Then for the second term on the right hand side of (<ref>), to
apply Gronwall's inequality, we show that $g(\tau)$ (defined after line (<ref>)) is integrable. From the integrated local energy estimates (<ref>) and the $r$-weighted energy estimates (<ref>) for the Maxwell field, we conclude from the previous section that
\begin{align*}
I^{-1-\ep}_0[\mathcal{L}_{Z}^k F](\mathcal{D}_{\tau_1}^{\tau_2})&\les M_k (\tau_1)_+^{-1-\ga_0},\quad \int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{2+\ep}(|\mathcal{L}_{Z}^k \a|^2+|\mathcal{L}_Z^k \rho|^2)dvd\om d\tau\les M_k(\tau_1)_+^{-\ga_0+\ep}.
\end{align*}
Therefore by using Lemma <ref> and Proposition <ref>, we can show that
\begin{align*}
\int_{\tau_1}^{\tau_2}g(\tau)d\tau&\les M_2(\tau_1)_+^{-\ga_0+\ep}+\sum\limits_{j\leq 2}I^{-1-\ep}_{1+2\ep}[\mathcal{L}_{\Om}^j F](\mathcal{D}_{\tau_1}^{\tau_2})\\
&\les M_2(\tau_1)_+^{-\ga_0+\ep}+\sum\limits_{j\leq 2}\int_{\tau_1}^{\tau_2}\tau_+^{2\ep}I^{-1-\ep}_{0}[\mathcal{L}_{\Om}^j F](\mathcal{D}_{\tau}^{\tau_2})d\tau+(\tau_1)_+^{1+2\ep}I^{-1-\ep}_{0}[\mathcal{L}_{\Om}^j F](\mathcal{D}_{\tau_1}^{\tau_2})\\
&\les M_2(\tau_1)_+^{-\ga_0+\ep}+M_2\int_{\tau_1}^{\tau_2}\tau_+^{-1-\ga_0+2\ep}d\tau+M_2(\tau_1)_+^{-\ga_0+2\ep}\les M_2(\tau_1)_+^{-\ga_0+2\ep}.
\end{align*}
By using this uniform bound, the second term on the right hand side of (<ref>) can be absorbed by using Gronwall's inequality. The corollary then follows.
We now can use Proposition <ref> and the above corollary to obtain the necessary $r$-weighted energy estimates. To derive energy decay estimates, we at least need the $r$-weighted energy estimates with $p=1$ and $p=1+\ga_0$
(some $p$ bigger than one. However, the decay rate depends on this largest $p$). In any case, we first choose $\ep_1$ in estimate (<ref>) sufficiently small so that combining it with the $r$-weighted energy estimate
(<ref>), the first term on the right hand side of (<ref>) can be absorbed (note that $\ga_0<1$). The second term on the right hand side of (<ref>) can be controlled by using
Gronwall's inequality. Let's first combine the $r$-weighted energy estimate (<ref>) for $p=1$ with the integrated local energy estimate (<ref>) to derive the bound for the integral of
the energy flux.
In the interior region for all $0\leq \tau_1<\tau_2$ we have
\begin{equation}
\label{eq:pWE:sca:in:1}
\begin{split}
\int_{\tau_1}^{\tau_2}E[\phi](\Si_\tau)d\tau\les_{M_2} &\int_{H_{\tau_1^*}}r|D_L\psi|^2dvd\om+E[\phi](\Si_{\tau_1})+(\tau_1)_+^{-\ga_0}\mathcal{E}_0[\phi]
\end{split}
\end{equation}
In $(t, r, \om )$ coordinate, using Sobolev embedding, we have
\[
\int_{\om}|\phi|^2(\tau, R, \om)d\om\les \int_{r\leq R}|\phi|^2+|D\phi|^2dx.
\]
Then we can show that
\begin{align*}
\int_{\tau_1}^{\tau_2}E[\phi](\Si_\tau)d\tau&\les \int_{\tau_1}^{\tau_2}\int_{r\leq R}|D\phi|^2dx d\tau +\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}|D_L(r\phi)|^2+|\D(r\phi)|^2dvd\om d\tau\\
&\qquad+\int_{\tau_1}^{\tau_2}\int_{\om}|\phi|^2(\tau, R, \om )d\om\\
&\les I^{-1-\ep}_0[\bar D\phi](\mathcal{D}_{\tau_1}^{\tau_2})+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}|D_L(r\phi)|^2+|\D(r\phi)|^2dvd\om d\tau.
\end{align*}
Therefore take $p=1$ in the $r$-weighted energy esimate (<ref>). From the above argument, we obtain the following bound for the integral of the enrgy flux:
\begin{equation*}
\begin{split}
&\int_{\tau_1}^{\tau_2}E[\phi](\Si_\tau)d\tau\les \int_{H_{\tau_1^*}}r|D_L\psi|^2dvd\om+M_2\int_{\tau_1}^{\tau_2}E[\phi](\Si_{\tau})\tau_+^{-\ep}d\tau\\
\end{split}
\end{equation*}
for some constant $C_{M_2}$ depending on $M_2$, $\ep$, $\ga_0$ and $R$. For the second term, we further can bound:
\[
\tau_+^{-\ep}=(\ep_1^{-\frac{1}{\ep}}\tau_+^{-1-\ep})^{\frac{\ep}{1+\ep}}\cdot (\ep_1)^{\frac{1}{1+\ep}}\leq \frac{\ep}{1+\ep} \ep_1^{-\frac{1}{\ep}}\tau_+^{-1-\ep}+\frac{\ep_1}{1+\ep},\quad \forall \ep_1>0.
\]
Choose $\ep_1$ sufficiently small so that the second term can be absorbed. Then the first term can be bounded by using Corollary <ref>. Therefore the previous estimates amounts to
estimate (<ref>).
As a corollary, we show that
We have
\begin{equation}
\label{eq:pWE:sca:in:1:Wga}
\begin{split}
\int_{\tau_1}^{\tau_2}\tau_+^{\ga_0-\ep}E[\phi](\Si_\tau)d\tau\les_{M_2}& \int_{H_{\tau_1^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+(\tau_1)_+^{1+\ga_0-\ep}E[\phi](\Si_{\tau_1})\\
\end{split}
\end{equation}
By using estimate (<ref>) of Lemma <ref>, we have the bound
\[
\int_{H_{\tau^*}}|D_L(r\phi)|^2dvd\om \leq \int_{H_{\tau^*}}|D_L\phi|^2r^2dvd\om +\lim\limits_{r\rightarrow\infty}\int_{\om}r|\phi|^2d\om\les E[\phi](\Si_{\tau}).
\]
For all $\ep_1>0$, we have the following inequality:
\[
\tau_+^{\ga_0-1-\ep}r=(\ep_1^{-\ga_0}r^{1+\ga_0}\tau_+^{-1-\ep})^{\frac{1}{1+\ga_0}}(\ep_1\tau_+^{\ga_0-\ep})^{\frac{\ga_0}{1+\ga_0}}\leq
\frac{\ep_1^{-\ga_0}r^{1+\ga_0}\tau_+^{-1-\ep}}{1+\ga_0}+\frac{\ga_0 \ep_1\tau_+^{\ga_0-\ep}}{1+\ga_0}.
\]
In particular the above inequality holds for $r=1$.
Moreover we also have
\[
(\tau_1)_+^{\ga_0-\ep}r=(r^{1+\ga_0})^{\frac{1}{1+\ga_0}}\left((\tau_1)_+^{1+\ga_0-\frac{1+\ga_0}{\ga_0}\ep}\right)^{\frac{\ga_0}{1+\ga_0}}\leq r^{1+\ga_0}+(\tau_1)_+^{1+\ga_0-\ep}.l
\]
Therefore from estimate (<ref>), we can show that
\begin{align*}
&\les \ep_1^{-\ga_0}\int_{\tau_1}^{\tau_2}\tau_+^{-1-\ep}\int_{H_{\tau^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om d\tau+\ep_1\int_{\tau_1}^{\tau_2}\tau_+^{\ga_0-\ep}E[\phi](\Si_\tau)d\tau\\
\end{align*}
On the right hand side of the above estimate, the first term can be grouped with the last term. The second term will be absorbed for small $\ep_1$. The third term can be bounded by using estimate (<ref>). Therefore
by using Lemma <ref> and Proposition <ref>, we can show that
\begin{align*}
\int_{\tau_1}^{\tau_2}\tau_+^{\ga_0-\ep}E[\phi](\Si_\tau)d\tau\les_{M_2}& \ep_1\int_{\tau_1}^{\tau_2}\tau_+^{\ga_0-\ep}E[\phi](\Si_\tau)d\tau+
\ep_1^{-\ga_0} \int_{H_{\tau_1^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\ep_1^{-\ga_0}\mathcal{E}_0[\phi]\\
\end{align*}
Let $\ep_1$ be sufficiently small, depending on $M_2$, $\ep$, $\ga_0$ and $R$. We obtain estimate (<ref>).
Estimate (<ref>) will be used to derive the $r$-weighted energy estimate with $p=1+\ga_0$.
We have
\begin{equation}
\label{eq:pWE:sca:in:1ga}
\begin{split}
&\int_{H_{\tau_2^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{\ga_0}(|D_L\psi|^2+|\D\psi|^2)dvd\om d\tau\\
&\les_{M_2} \int_{H_{\tau_1^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\mathcal{E}_0[\phi]+(\tau_1)_+^{1+\ga_0-\ep}E[\phi](\Si_{\tau_1}).
\end{split}
\end{equation}
By taking $\ep_1$ in estimate (<ref>) to be sufficiently small and combining it with the $r$-weighted energy estimate (<ref>) for $p=1+\ga_0$, from corollary <ref>, we obtain
\begin{align*}
&\int_{H_{\tau_2^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{\ga_0}(|D_L\psi|^2+|\D\psi|^2)dvd\om d\tau\\
&\les \int_{H_{\tau_1^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\mathcal{E}_0[\phi]
+M_2\left(\int_{\tau_1}^{\tau_2}E[\phi](\Si_{\tau})\tau_+^{\ga_0-\ep}d\tau+I^{1+\ga_0}_{-1-\ep}[r^{-1}D_L\psi](\bar {\mathcal{D}}_{\tau_1}^{\tau_2})\right).\\
\end{align*}
for some constant $C_{M_2}$ depending on $M_2$. Estimate (<ref>) then follows from estimate (<ref>) together with Gronwall's inequality.
If we take $\tau_1$ on the right hand side of (<ref>) to be $0$, from the energy estimate (<ref>) and the $r$-weighted energy estimate (<ref>) in the exterior region, we conclude that the right hand side of (<ref>) is bounded with $\tau_1=0$. In particular the $r$-weighted energy estimate in the interior region is bounded as we expected.
Let $\psi=r\phi$. For all $0\leq \tau_1\leq \tau_2$, we have
\begin{equation}
\label{eq:pWEdecay:sca:in:1ga}
\begin{split}
\int_{H_{\tau_2^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om+\int_{\tau_1}^{\tau_2}\int_{H_{\tau^*}}r^{\ga_0}(|D_L\psi|^2+|\D\psi|^2)dvd\om d\tau
%+ (\tau_1)_+^{\ep}I_{-1-\ep}^{1+\ga_0}[r^{-1}D_L\psi](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})
\les_{M_2} \mathcal{E}_0[\phi].
\end{split}
\end{equation}
From the $r$-weighted energy decay estimate (<ref>) for $p=1+\ga_0$, $\tau_1=0$ in the exterior region, let $u_2\rightarrow \infty$. We obtain
\[
\int_{H_{0^*}}r^{1+\ga_0}|D_L\psi|^2dvd\om=\int_{H_{-\frac{R}{2}}}r^{1+\ga_0}|D_L\psi|^2\les_{M_2}\mathcal{E}_0[\phi].
\]
From the energy estimate (<ref>) in the exterior region, we conclude that
\[
E[\phi](\Si_{0})=E[\phi](\{t=0, r\leq R\})+E[\phi](H_{-\frac{R}{2}})\les \mathcal{E}_0[\phi].
\]
Then estimate (<ref>) follows from (<ref>) by taking $\tau_1=0$ on the right hand side.
This uniform bound for the $r$-weighted energy estimate in the interior region is crucial for the energy flux decay. It in particular implies that the terms involving the $r$-weighted energy flux on the right hand side of the energy estimate (<ref>) and the integral of the energy flux estimate (<ref>) have the right decay to show the energy flux decay.
In the interior region, we have the energy flux decay:
\begin{equation}
\label{eq:Enerdecay:sca:in}
E[\phi](\Si_\tau)\les_{M_2}\mathcal{E}_0[\phi]\tau_+^{-1-\ga_0},\quad \forall \tau\geq 0.
\end{equation}
Estimate (<ref>) implies that
\[
I_{-2-\ga_0}^{1+\ga_0}[r^{-1}D_L\psi](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})\les_{M_2}(\tau_1)_+^{-1-\ga_0}\mathcal{E}_0[\phi],\quad \forall 0\leq \tau_1\leq \tau_2.
\]
Then using the pigeon hole argument similar to the proof of Proposition <ref> for the energy flux decay of
the Maxwell field in the interior, the energy decay estimate (<ref>) for the scala field follows from the energy estimate (<ref>), the integral of the energy flux estimate (<ref>) and the $r$-weighted energy estimate (<ref>). For a detailed proof for this, we refer to Proposition 2 of <cit.>.
§.§.§ Energy decay estimates for the first order derivative of the scalar field
In this section, we derive the energy flux decay estimates for the derivative of the scalar field. The difficulty is that the covariant wave operator $\Box_A$ does not commute with $D_{Z}$. Commutators are quadratic in the Maxwell field and
the scalar field. In our setting, the Maxwell field is large. In particular those terms can not be absorbed. The idea is to exploit the null structure of the commutators and to use Gronwall's inequality adatped to our foliation $\Si_{\tau}$.
In the following, we always use $\psi$ to denote the weighted scalar field $r\phi$, that is, $\psi=r\phi$. The first order derivative of $\phi$ is shorted as $\phi_1$, similarly, for $\phi_2$.
More precisely, we denote $\phi_1=D_Z\phi$, $\phi_2=D_{Z}^2\phi$. Same notation for the weighted scalar field $\psi$, e.g., $\psi_1=r D_Z\phi$. For any function $f$, under the null coordinates $(u, v, \om)$, we denote
\[
\|f\|_{L_v^2 L_u^{\infty}L_{\om}^2(\mathcal{D})}^2:=\int_{v}\sup\limits_{u}\int_{\om}|f|^2d\om dv,
\]
where $(u, v,\om)$ are the null coordinates on the region $\mathcal{D}$. Similarly we have the notation of $\|f\|_{L_u^2 L_v^\infty L_{\om}^2(\mathcal{D})}$. We can also define $L_u^pL_v^q L_{\om}^r$ norms for general $p$, $q$, $r$.
To apply Corollary <ref> for the exterior region and Proposition <ref> for the interior region, it suffices to control the commutator terms.
However, we are not able to bound the commutator terms directly by using the zero's order energy estimates. One has to make use of the energy flux of the first order derivative of the solution and then apply Gronwall's inequality.
However for the energy estimate for the first order derivative of the solution, the key is to understand the commutator $[\Box_A, D_Z]$ with $Z=\pa_t$ or the angular momentum. The cases of $\pa_t$ and the angular momentum are
quite different. The main reason is that the angular momentum contains weights in $r$ while $\pa_t$ does not. For the case when $Z=\pa_t$, it is easy to bound $[\Box_A, D_{\pa_t}]\phi$. The only place we need to be careful is the charge part. For the case of $Z=\Om$, the problem is that the commutator $[\Box_A, D_{\Om}]$ will produce a term of the form $Z^\nu F_{\mu\nu}D^{\mu}\phi$ which can not be written as a linear term of $D_Z\phi$. The estimate for the commutator terms heavily rely on the null structure. We first show the following lemma for the commutator terms.
When $|x|\geq R$, we have
\begin{equation}
\label{eq:Est4commu:1}
|[\Box_A, D_Z]\phi|\les |\a||D_{\Lb}\psi|+(|\ab|+r^{-1}|\rho|)|D_{L}\psi|+|F||\D\phi|+(|J|+r|\J|+|\si|+r^{-1}|\rho|)|\phi|.
\end{equation}
When $r\leq R$, we have
\begin{equation}
\label{eq:Est4commu:1:in}
|[\Box_A, D_Z]\phi|\les |F||\bar{D} \phi|+|J||\phi| .
\end{equation}
In this paper, all the quantity involving $Z$ should be interpreted as the sum of the quantity for all possible vector fields $Z$ unless we specify it.
Let $\psi=r\phi$. First from Lemma <ref>, we can write
\begin{equation}
\label{eq:com1:nullst}
[\Box_A, D_{Z}]\phi=2i r^{-1}Z^\nu F_{\mu\nu}D^{\mu}\psi+i \pa^\mu F_{\mu\nu}Z^\nu\phi+i\phi\left(-2 Z^\nu F_{\mu\nu}r^{-1}\pa^{\mu}r+\pa^{\mu}Z^\nu F_{\mu\nu}\right).
\end{equation}
We write the commutator terms as above is to exploit the null structure. The first term is the main term. Since we will rely on the $r$-weighted energy estimates, this suggests to write the main term of the commutator in terms of the weighted solution $r\phi$. The second term is easy as $\pa^\mu F_{\mu\nu}$ is nonlinear term of $\phi$ by the Maxwell equation. Let's first estimate the third term. When $Z=\Om$, note that $r^{-1}\Om$ is linear combination of $e_1$, $e_2$. We then can show that
\begin{align*}
|r^{-1}Z^\nu F_{\mu\nu}D^{\mu}(r\phi)|\les |\a||D_{\Lb}(r\phi)|+|\ab||D_{L}(r\phi)|.
\end{align*}
This is the null structure we need: the "bad" component $\ab$ of the curvature does not interact with the "bad" component $D_{\Lb}(r\phi)$ of the scalar field. Similarly, when $Z=\pa_t$, the "bad" term $r^{-1}\ab D_{\Lb}(r\phi)$ does not appear. More precisely, we have
\begin{align*}
|r^{-1}Z^\nu F_{\mu\nu}D^{\mu}(r\phi)|\les r^{-1}(|\a|+|\ab|)|\D(r\phi)|+r^{-1}|\rho||D_r(r\phi)|.
\end{align*}
For the second term on the right hand side of (<ref>), we note that $\pa^\mu F_{\mu\nu}$ is nonlinear term of $\phi$. We have
\begin{equation*}
|\pa^\mu F_{\mu\nu}Z^\nu\phi|\les (|J|+r|\J|)|\phi|.
\end{equation*}
For the third term on the right hand side of (<ref>), we show that
\begin{equation*}
|i\phi\left(-2 Z^\nu F_{\mu\nu}r^{-1}\pa^{\mu}r+\pa^{\mu} Z^\nu F_{\mu\nu}\right)|\les (|\si|+r^{-1}|\rho|)|\phi|.
\end{equation*}
The case when $Z=\pa_t$ is trivial. To check the above inequality for the case when $Z=\Om$, it suffices to prove it for the component $\Om_{jk}=x_j \pa_k-x_k\pa_j$. Then we can show that
\begin{align*}
-2\Om^\nu F_{\mu\nu}r^{-1}\pa^{\mu}r+\pa^{\mu}\Om^\nu F_{\mu\nu}&=2F_{jk}-2F(\pa_r, \Om_{ij})\\
&=2F(\om_{j}\pa_r+\pa_j-\om_j\pa_r, \om_k\pa_r+\pa_k-\om_k\pa_r)-2F(\pa_r, \Om_{jk})\\
&=2F(\pa_j-\om_j\pa_r, \pa_k-\om_k\pa_r).
\end{align*}
Here recall that $\om_j=r^{-1}x_j$. Since for all $j=1$, $2$, $3$, $\pa_j-\om_j\pa_r$ is orthogonal to $L$, $\Lb$, we conclude that $\pa_j-\om_j\pa_r$ is a linear combination of $e_1$ and $e_2$. The desired estimate then follows as the norm of the vector fields $\pa_j-\om_j\pa_r$ is less than 1.
We begin a series of propositions in order to estimate the weighted spacetime norm of the commutators. The estimates in the bounded region $r\leq R$ are easy to obtain as the weights are finite.
We now concentrate on the region $r\geq R$. Let $\bar{\mathcal{D}}_{\tau}=\mathcal{D}_{\tau}\cap\{|x|\geq R\}$. We first consider
For all $\ep_1>0$, we have
\begin{align}
\label{eq:supDLbphi}
\|D_{\Lb}(r\phi)\|_{L_u^2 L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}& \les_{M_2}\mathcal{E}_0[\phi]\ep_1^{-1} \tau_+^{-1-\ga_0}+\ep_1 I^{1+\ep}_0[r^{-1}D_LD_{\Lb}\psi](\bar{\mathcal{D}}_{\tau}).
\end{align}
The idea is to bound $\sup|D_{\Lb}(r\phi)|$ by the $L^2$ norm of $D_{L}D_{\Lb}(r\phi)$. In the exterior region when $\bar{\mathcal{D}}_{\tau}=\mathcal{D}_{\tau}^{-\infty}$, we can integrate from the initial hypersurface $t=0$.
In the interior region, choose the incoming null hypersurface $\Hb_{\frac{\tau_2+R}{2}}^{\tau_1, \tau_2}$ as the starting surface. Denote $\psi=r\phi$. We show estimate (<ref>) for the interior region case, that is, when
$0\leq \tau_1<\tau_2$. On the outgoing null hypersurface $H_{\tau^*}$, for all $0\leq \tau_1\leq \tau\leq \tau_2$, we have
\begin{align*}
\sup\limits_{v\geq\frac{\tau+R}{2}}\int_{\om}|D_{\Lb}(r\phi)|^2(\tau^*, v, \om)d\om & \les \int_{\om}|D_{\Lb}(r\phi)|^2(\tau^*, \frac{\tau_2+R}{2}, \om)d\om+\int_{H_{\tau^*}}|D_L D_{\Lb}(r\phi)|\cdot |D_{\Lb}(r\phi)|dvd\om.
\end{align*}
Integrate the above estimate from $\tau_1$ to $\tau_2$ and apply Cauchy-Schwarz's inequality to the last term. From the integrated local energy estimate
(<ref>) and the energy decay estimate (<ref>), we then derive
\begin{align*}
\int_{\tau_1}^{\tau_2}\sup\limits_{v\geq \frac{R+\tau}{2}}\int_{\om}|D_{\Lb}(r\phi)|^2 d\om d\tau\les_{M_2}\mathcal{E}_0[\phi]\ep_1^{-1} (\tau_1)_+^{-1-\ga_0}+\ep_1 I^{1+\ep}_0[r^{-1}D_LD_{\Lb}\psi](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})
\end{align*}
for all $\ep_1>0$. The case in the exterior region follows in a similar way.
We also need the analogous estimates for $D_{L}(r\phi)$.
For all $\ep_1>0$ and $0\leq p\leq 1+\ga_0$, we have
\begin{align}
\label{eq:supDlphi}
\|r^{\frac{p}{2}}D_{L}(r\phi)\|_{L_v^2 L_u^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}^2 &\les_{M_2}\ep_1^{-1} \mathcal{E}_0[\phi](\tau)_+^{-1-\ga_0}+\ep_1 I^{p_1}_{p_2}[r^{-1}D_{\Lb}D_{L}\psi](\bar{\mathcal{D}}_{\tau}).
\end{align}
Here $p_1=\max\{1+\ep, p\}$, $p_2=\min\{1+\f12\ep, p\}$.
Similar to the proof of the previous proposition, we choose the starting surface for $D_{L}(r\phi)$ to be $H_{\tau_1^*}$ in the interior region and the initial hypersurface $t=0$ in the exterior region.
We only prove the proposition for the exterior region case. On $\Hb_{v}^{\tau^*}$, $v\geq -\tau^*$, we can show that
\begin{align*}
r^p\int_{\om}|D_{L}\psi|^2d\om\les \int_{\om}(r^p|D_L\psi|^2)(-v, v, \om)d\om+\int_{\Hb_v^{\tau^*}}r^{p-1}|D_L\psi|^2+r^p|D_L\psi||D_{\Lb}D_L\psi|dud\om.
\end{align*}
The integral of the first term can bounded by the assumption on the data. We control the second term by using the $r$-weighted energy estimate. We bound the last term as follows:
\begin{align*}
r^p|D_L\psi||D_{\Lb}D_L\psi|\les \ep_1 r^{p_1} u_+^{p_2}|D_{\Lb}D_L\psi|^2+\ep_1^{-1}r^{2p-p_1}u_+^{-p_2}|D_L\psi|^2,\quad \forall \ep_1>0.
\end{align*}
When $2p\geq p_1$, we can use the $r$-weighted energy estimate (<ref>) to bound the weighted integral of $|D_L\psi|$. Otherwise one can use interpolation and the integrated local energy decay
estimate (<ref>). For any case, from the energy decay estimates (<ref>), (<ref>), (<ref>), (<ref>) for $\phi$, one can always show that
\begin{align*}
\iint_{\mathcal{D}_{\tau}}r^{2p-p_1}u_+^{-p_2}|D_L\psi|^2 dudvd\om\les_{M_2}\mathcal{E}_0[\phi]\tau_+^{p-1-\ga_0}.
\end{align*}
Another way to understand the above estimate is to use interpolation. It suffices to show the above estimate with $p=0$ and $p=1+\ga_0$. The former case follows by using the integrated local energy estimates for $\phi$ while the later situation relies on the $r$-weighted energy estimate.
Estimate (<ref>) for the exterior region case then follows. The interior region case holds in a similar way.
As we only commute the equation with $\pa_t$ or the angular momentum $\Om$, to estimate the weighted spacetime integral of $D_LD_{\Lb}(r\phi)$ in terms of $D_Z\phi$, we use the equation of $\phi$ under the null frame.
Under the null frame, we can write the covariant wave operator $\Box_A$ as follows:
\begin{equation}
\label{eq:EQ4sca:null}
r\Box_A \phi=r D^\mu D_\mu\phi=-D_L D_{\Lb}(r\phi)+\D^2(r\phi)-i\rho \cdot r\phi=-D_{\Lb} D_{L}(r\phi)+\D^2(r\phi)+i\rho \cdot r\phi
\end{equation}
for any complex scalar field $\phi$. Here $\D^2=\D^{e_1}\D_{e_1}+\D^{e_2}\D_{e_2}$ and $\rho=\f12 (dA)_{\Lb L}$.
The lemma follows by direct computation.
This lemma leads to the following estimates for $D_L D_{\Lb}(r\phi)$ and $D_{\Lb}D_{L}(r\phi)$.
For all $1+\ep\leq p\leq 1+\ga_0$, we have
\begin{align}
\label{eq:Est4DLDLbphi}
\end{align}
Here $\phi_1=D_Z\phi$ and $\psi_1=D_Z(r\phi)$.
Let's only consider estimate for $D_LD_{\Lb}(r\phi)$ in the interior region. The proof easily implies the estimates for $D_{\Lb}D_L(r\phi)$. The case in the exterior region is easier as in that region $r\geq\frac{1}{3}u_+$ and
it suffices to show the estimate for $p=1+\ga_0$ which is similar to the proof for the interior region case. Take $\bar{\mathcal{D}}_{\tau}$ to be $\bar{\mathcal{D}}_{\tau_1}^{\tau_2}$ for $0\leq \tau_1=\tau<\tau_2$. From the equation (<ref>) for $\phi$ under the null frame, we derive
\[
r^p|D_L D_{\Lb}(r\phi)|^2 \les r^p|\Box_A \phi|^2 r^2+r^p|r\phi \rho|^2+r^p|r^{-1}\D D_{\Om}\psi|^2.
\]
Here we note that $|\D^2\psi|^2\les |r^{-1}\D D_{\Om}\psi|$. The integral of the first term on the right hand side can be bounded by $\mathcal{E}_0[\phi]$. For the second term, we control $\phi$ by using Lemma <ref>.
The last term is favorable as it is a form of $\D D_Z\psi$. We will absorb those terms with the help of the small constant $\ep_1$ from Propositions <ref>, <ref>. According to our notation in this section, let $\psi_1=D_{\Om}\psi$. For all $1+\ep\leq p\leq 1+\ga_0$, we have
\[
\tau_+^{2+\ga_0-p-2\ep}r^{p-2}\les r^{\ga_0}+\tau_+^{1+\ga_0-\ep}r^{-1-\ep}, \quad r\geq R.
\]
Since the energy flux for $\phi$ decays from Proposition <ref>, by using Lemma <ref>, we conclude that
\[
\int_{\om}r^p|\phi|^2d\om\les_{M_2}\mathcal{E}_0[\phi](\tau_1)_+^{p-2-\ga_0}.
\]
Therefore for all $1+\ep\leq p\leq 1+\ga_0$ we can show that
\begin{align*}
&\iint_{\bar{\mathcal{D}}_{\tau_1}^{\tau_2}}\tau_+^{2+\ga_0-p-2\ep}r^{p}|D_L D_{\Lb}(r\phi)|^2 dvdud\om\\
&\les I^{p}_{2+\ga_0-p- 2\ep}[\Box_A\phi](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})+I^{-1-\ep}_{1+\ga_0-\ep}[D\phi_1](\bar{\mathcal{D}}_{\tau_1}^{\tau_2})
&\qquad+\int_{\tau_1}^{\tau_2}\tau_+^{2+\ga_0-p-2\ep}\int_{\frac{\tau+R}{2}}^{\infty} \int_{\om}r^{p}|\phi|^2d\om\cdot \sum\limits_{j\leq 2}\int_{\om}r^2|\mathcal{L}_{\Om}^2 \bar \rho|^2d\om dvdu\\
\end{align*}
This finished the proof.
Next we estimate the weighted spacetime norm of $|\a||D_{\Lb}(r\phi)|$.
For all $1+\ep\leq p\leq 1+\ga_0$, $\ep_1>0$, we have
\begin{align}
\label{eq:com1:aDLbphi}
\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{2+\ga_0+\ep-p}r^p|\a|^2 |D_{\Lb}(r\phi)|^2 dxdt &\les_{M_2}\mathcal{E}_0[\phi] \ep_1^{-1} \tau_+^{-\ga_0+\ep}+\ep_1 I^{1+\ep}_{1+\ep}[r^{-1}D_LD_{\Lb}\psi](\bar{\mathcal{D}}_{\tau} ).
\end{align}
Make use of Proposition <ref>. For all $1+\ep\leq p\leq 1+\ga_0$, we can show that
\begin{align*}
\iint_{\bar{\mathcal{D}}_{\tau}} r^p|\a|^2 |D_{\Lb}(r\phi)|^2 dxdt &\les \|D_{\Lb}\psi\|_{L_u^2 L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}^2\cdot
\|r^{\frac{p}{2}+1}\a\|_{L_u^{\infty}L_v^2 L_{\om}^{\infty}(\bar{\mathcal{D}}_{\tau})}^2\\
&\les \|D_{\Lb}\psi\|_{L_u^2 L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}^2\cdot
\sum\limits_{j\leq 2}\|r^{\frac{p}{2}+1}\mathcal{L}_{\Om}^j\a\|_{L_u^{\infty}L_v^2 L_{\om}^{2}(\bar{\mathcal{D}}_{\tau})}^2\\
&\les_{M_2}\mathcal{E}_0[\phi] \ep_1^{-1} \tau_+^{p-2-2\ga_0}+\ep_1 \tau_+^{p-1-\ga_0}I^{1+\ep}_0[r^{-1}D_LD_{\Lb}\psi](\bar{\mathcal{D}}_{\tau} )
\end{align*}
for all $\ep_1>0$. As the above estimate holds for all $\tau\in\mathbb{R}$, from Lemma <ref>, we conclude that
\begin{align*}
\iint_{\bar{\mathcal{D}}_{\tau}} u_+^{2+\ga_0+\ep-p}r^p|\a|^2 |D_{\Lb}(r\phi)|^2 dxdt
&\les_{M_2}\mathcal{E}_0[\phi] \ep_1^{-1} \tau_+^{-\ga_0+\ep}+\ep_1 I^{1+\ep}_{1+\ep}[r^{-1}D_LD_{\Lb}\psi](\bar{\mathcal{D}}_{\tau} ).
\end{align*}
This finished the proof for estimate (<ref>).
Next we estimate the weighted spacetime integral of $(|\ab|+r^{-1}|\rho|)|D_L(r\phi)|$. One possible way to bound this term, in particular $\ab$, is to make use of the energy flux through the incoming null hypersurface.
It turns out that we loss a little bit of decay in $u$ and we are not able to close the bootstrap argument later. An alternative way is to use $\sup\limits_{v}\int_{\om}|\ab|^2d\om$ which has to exploit the equation for $F$.
For $\tau\in\mathbb{R}$, let
\begin{equation}
\label{eq:defofgu}
g(\tau)=\sum\limits_{k\leq 1}\|\mathcal{L}_{\Om}^k(r\ab)\|_{L_v^\infty L_{\om}^2(H_{\tau^*})}^2+\sum\limits_{k\leq 2}\int_{\Si_{\tau}}\frac{|\mathcal{L}_{Z}^k \bar{F}|^2}{r_+^{1+\ep}}r^{2}d\tilde{v}d\om.
\end{equation}
Here $(\tilde{v}, \om)$ are coordinates of $\Si_{\tau}$. This notation should not be confused with the one in Proposition <ref> in the previous section and this function will only be used in this section.
We can not show that $g(\tau)$ decays in $\tau$. However as a corollary we can show that
The function $g(\tau)$ is integrable in $\tau$:
\begin{equation}
\label{eq:L2gu}
\int_{\tau_1}^{\tau_2}\tau_+^{1+\ep}g(\tau)d\tau\les M_2\tau_+^{-\ga_0+3\ep},\quad \int_{\tilde{\tau}\leq \tau}\tilde{\tau}_+^{1+\ep}g(\tilde{\tau})d\tilde{\tau}\les M_2\tau_+^{-\ga_0+3\ep}
\end{equation}
for all $0\leq \tau_1\leq \tau_2$, $\tau\leq 0$.
By using Lemma <ref>, the corollary follows from estimate (<ref>) and the integrated local energy estimates (<ref>), (<ref>) for the curvature.
We now can estimate the weighted spacetime integral of $|\ab||D_{L}\psi|$.
For all $1+\ep\leq p\leq 1+\ga_0$, $\ep_1>0$, we have
\begin{align}
\notag
&\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{2+\ga_0+\ep-p}|\ab|^2 |D_{L}(r\phi)|^2 r^{p} dxdt \\
\label{eq:com1:abDLphi}
&\les_{M_2}\ep_1 \iint_{\bar{\mathcal{D}}_{\tau}} \tilde{\tau}_+^{2+\ga_0+\ep-p}g(\tilde{\tau}) r^{p}|D_L\psi_1|^2 dvd\om d\tilde{\tau}+\ep_1^{-1}\mathcal{E}_0[\phi]\tau_+^{-\ga_0+3\ep}.
\end{align}
We first use Sobolev embedding on the unit sphere to bound that
\[
\||r\ab||D_L\psi|\|_{L_{\om}^2}^2\les (\|r\ab\|_{L_{\om}^2}^2+\|r\mathcal{L}_{\Om}\ab\|_{L_{\om}^2}^2)\cdot(\ep_{1}^{-1}\|D_L\psi\|_{L_{\om}^2}^2+\ep_1\|D_{\Om}D_{L}\psi\|_{L_{\om}^2}^2),\quad \ep_1>0
\]
The proof for this estimate for all connection $A$ follows from the case when $A$ is trivial as the norm is gauge invariant. We in particular can choose a gauge so that the function is real.
Then make use of estimate (<ref>) of Proposition <ref>. We therefore can show that
\begin{align*}
&\les\|r^{\frac{p}{2}}\sum\limits_{k\leq 1}\|\mathcal{L}_{\Om}^k(r\ab)\|_{L_\om^2}\cdot u_+^{\frac{1}{2}(2+\ga_0+\ep-p)}(\ep_1^{\f12}\|D_{\Om}D_L\psi\|_{L_{\om}^2}+\ep_1^{-\f12}\|D_L\psi\|_{L_{\om}^2})\|_{L_u^2L_v^2}^2\\
&\les \| \tilde{\tau}_+^{2+\ga_0+\ep-p}g(\tilde{\tau})(\ep_1\|r^{\frac{p}{2}}D_LD_{\Om}\psi\|_{L_v^2L_{\om}^2}^2+\ep_1\|r^{\frac{p}{2}}r\a \psi\|_{L_v^2L_{\om}^2}^2+\ep_1^{-1}\|r^{\frac{p}{2}}D_L\psi\|_{L_v^2L_{\om}^2}^2)\|_{L_u^1}\\
&\les \ep_1\int_{\tilde{\tau}} \tilde{\tau}_+^{2+\ga_0+\ep-p}g(\tilde{\tau})\int_{H_{\tilde{\tau}^*}}r^{p}|D_L\psi_1|^2 dvd\om d\tilde{\tau}+\ep_1^{-1}\|\tilde{u}_+^{1+\ep}g(\tilde{u})\|_{L_u^1}\|u_+^{\frac{1+\ga_0-p}{2}}r^{\frac{p}{2}}D_L\psi\|_{L_u^\infty L_v^2 L_\om^{2}}^2 \\
&\qquad+\ep_1\|\tilde{u}_+^{1+\ep-\ga_0}g(\tilde{u})\|_{L_u^1}\|u_+^{\frac{1+\ga_0-p}{2}}r^{\frac{p}{2}}r\a\|_{L_u^\infty L_v^2 L_\om^{\infty}}^2\|u_+^{\frac{\ga_0}{2}}\psi\|_{L_u^\infty L_v^\infty L_{\om}^2}^2\\
&\les_{M_2}\ep_1 \iint_{\bar{\mathcal{D}}_{\tau}} \tilde{\tau}_+^{2+\ga_0+\ep-p}g(\tilde{\tau}) r^{p}|D_L\psi_1|^2 dvd\om du+\ep_1^{-1}\mathcal{E}_0[\phi]\tau_+^{-\ga_0+3\ep}.
\end{align*}
Here we have used the $r$-weighted energy estimates (<ref>), (<ref>) and estimate (<ref>) to bound $\phi$.
For $|r^{-1}\rho||D_{L}\psi|$, we have extra decay in $r$ which allows us to use Proposition <ref>.
For all $\ep_1>0$, we have
\begin{align}
\label{eq:com1:rhophi}
\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{1+\ga_0}|r^{-1}\rho|^2 |D_{L}(r\phi)|^2 r^{1+\ga_0} dxdt \les_{M_2}\ep_1 I^{1+\ep}_{\ep}[r^{-1}D_{\Lb}D_L\psi](\mathcal{D}_{\tau_1}^{\tau_2})+\ep_1^{-1}\mathcal{E}_0[\phi].
\end{align}
The idea is that we bound $\rho$ by using the energy flux through the incoming null hypersurface and $D_{L}\psi$ by using Proposition <ref>. In the exterior region, we need to specially consider
the effect of the nonzero charge. Other than that, the proof is the same for the interior region case. We thus take $\bar{\mathcal{D}}_{\tau}$ to be $\mathcal{D}_{\tau}$ with $\tau\leq 0$. For all $1+\ep\leq p\leq 1+\ga_0$, we then can show that
\begin{align*}
\iint_{\mathcal{D}_{\tau}}|r^{-1}\rho|^2 |D_{L}\psi|^2 r^{p} dxdt&\les \iint_{\mathcal{D}_{\tau}}|\bar{\rho}|^2 |D_{L}\psi|^2 r^{p} dudvd\om +\iint_{\mathcal{D}_{\tau}}|q_0|^2|D_{L}\psi|^2 r^{p-4} dudvd\om\\
&\les_{M_2} \|D_L\psi\|_{L_v^2 L_u^\infty L_{\om}^2(\mathcal{D}_{\tau})}^2\|r\bar\rho\|_{L_v ^\infty L_u^2 L_{\om}^\infty(\mathcal{D}_{\tau})}^2+\mathcal{E}_0[\phi]\tau_+^{-1-2\ga_0}\\
&\les_{M_2} \ep_1\tau_+^{-1-\ga_0} I^{1+\ep}_0[r^{-1}D_{\Lb}D_L\psi](\mathcal{D}_{\tau})+\ep_1^{-1}\mathcal{E}_0[\phi]\tau_+^{-1-2\ga_0}.
\end{align*}
The above estimate also holds for the interior region case when $\bar{\mathcal{D}}_{\tau}=\mathcal{D}_{\tau_1}^{\tau_2}$ for all $0\leq \tau_1<\tau_2$. As the estimates holds for all $\tau$, from Lemma <ref>, we then can show that (take the interior region for example)
\begin{align*}
&\iint_{\mathcal{D}_{\tau_1}^{\tau_2}}\tau_+^{1+\ga_0}|r^{-1}\rho|^2 |D_{L}\psi|^2 r^{p} dxdt\\
&\les_{M_2} \ep_1 I^{1+\ep}_0[r^{-1}D_{\Lb}D_L\psi](\mathcal{D}_{\tau_1}^{\tau_2})+\ep_1\int_{\tau_1}^{\tau_2}\tau_+^{-1}I^{1+\ep}_0[r^{-1}D_{\Lb}D_L\psi]
&\les_{M_2}\ep_1 I^{1+\ep}_{\ep}[r^{-1}D_{\Lb}D_L\psi](\mathcal{D}_{\tau_1}^{\tau_2})+\ep_1^{-1}\mathcal{E}_0[\phi].
\end{align*}
Here we note that $\ln\tau_+\les \tau_+^{\ep}$.
Next we estimate $r^{-1}|F||\D(r\phi)|$.
For all $\ep_1>0$, we have
\begin{align}
\label{eq:com1:FDZphi}
\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{1+\ga_0}|r^{-1}F|^2|\D(r\phi)|^2r^{1+\ga_0} dxdt &\les_{M_2}\ep_1^{-1}\mathcal{E}_0[\phi]+\ep_1\int_{\tilde{\tau}}\tilde{\tau}_+^{1+\ga_0}g(\tilde{\tau})E[D_Z\phi](H_{\tilde{\tau}^*})d\tilde{\tau}.
\end{align}
The idea is that we use the energy flux through the outgoing null hypersurface to bound $\D(r\phi)=D_{\Om}\phi$ and the integrated local energy to control $F$. We only show the estimate in the exterior region.
Take $\bar{\mathcal{D}_{\tau}}$ to be $\mathcal{D}_{\tau}$ for any $\tau\leq 0$. In the exterior region we have the relation $r\geq \frac{1}{3}u_+$. Therefore from estimate (<ref>) and recalling the
definition (<ref>) of $g(u)$, we can show that
\begin{align*}
&\iint_{\mathcal{D}_{\tau}}u_+^{1+\ga_0}|F|^2|\D(r\phi)|^2r^{1+\ga_0} dudvd\om\\
&\les \int_{u}u_+^{1+\ga_0}\int_{v}\sum\limits_{k\leq 2}r^{1-\ep}\int_{\om}|\mathcal{L}_{\Om}^k\bar{F}|^2+|q_0r^{-2}|^2 d\om \cdot \int_{\om}r|D_{\Om}\phi|^2d\om dudv\\
&\les |q_0|^2 \iint_{\mathcal{D}_{\tau}}\frac{|\D\phi|^2}{r^{1+\ep}}dxdt+\int_{\tilde{\tau}}(\tilde{\tau})_+^{1+\ga_0}g(\tilde{\tau})(\ep_1^{-1}\int_{H_{\tilde{\tau}^*}}|\D\phi|^2 r^2dvd\om+\ep_1E[D_{Z}\phi](H_{\tilde{\tau}^*}))d\tilde{\tau}\\
\end{align*}
Here we assumed that $\ga_0<1$ and $\ep$ is sufficiently small. For the case $\ga_0=1$, the above estimate also holds but in a different form where we have to rely on the $r$-weighted energy estimate. For the sake of simplicity, we would not discuss in details when $\ga_0\geq 1$.
Finally we estimate the weighted spacetime norm of $(|J|+|r\J|+|\si|+|r^{-1}\rho|)|\phi|$. We show that
For all $1+\ep\leq p\leq 1+\ga_0$, we have
\begin{align}
\label{eq:com1:Jphi}
\iint_{\bar{\mathcal{D}}_{\tau}}(|J|^2+|r\J|^2+|\si|^2+|r^{-1}\rho|^2)|\phi|^2r^{p}u_+^{2+\ga_0+\ep-p} dxdt &\les_{M_2}\mathcal{E}_0[\phi]\tau_+^{-\ga_0+\ep}.
\end{align}
Let's first consider $(|J|^2+|\si|^2+|r^{-1}\rho|^2)|\phi|^2$. The idea is that we bound $\phi$ by the energy flux. Note that the nonzero charge only affects the estimates
in the exterior region where $r\geq \frac{1}{3}u_+$. From the embedding (<ref>) and the energy decay estimates (<ref>), (<ref>), we can show that
\begin{align*}
&\iint_{\bar{\mathcal{D}}_{\tau}}(|J|^2+|\si|^2+|r^{-1}\rho|^2)|\phi|^2r^{p+2}u_+^{2+\ga_0+\ep-p} dudvd\om\\
&\les \int_{u}u_+^{2+\ga_0+\ep-p}\int_{v}\sum\limits_{k\leq 2}r^{p+1}\int_{\om}|\mathcal{L}_{\Om}^k J|^2+|\mathcal{L}_{\Om}^k\si|^2+
|r^{-1}\mathcal{L}_{\Om}^k\bar \rho|^2+|q_0r^{-3}|^2 d\om \cdot \int_{\om}r|\phi|^2d\om dvdu\\
&\les_{M_2} \mathcal{E}_0[\phi]\int_{u}u_+^{1+\ep-p}\int_{v}\sum\limits_{k\leq 2}r^{p+1}\int_{\om}|\mathcal{L}_{\Om}^k J|^2+|\mathcal{L}_{\Om}^k\si|^2+
|r^{-1}\mathcal{L}_{\Om}^k\bar \rho|^2+|q_0r^{-3}|^2 d\om dvdu\\
\end{align*}
Here we used the $r$-weighted energy estimates (<ref>), (<ref>) to bound the curvature components and the definition for $M_2$ to control $J$.
For $|r\J|^2|\phi|^2$, the only difference is that we need to put more $r$ weights on $\phi$. By using the embedding inequality (<ref>) and the energy decay estimates (<ref>), (<ref>),
we conclude that
\[
\int_{\om}r^{1+p-\ga_0}|\phi|^2d\om \les_{M_2}\tau_+^{p-1-2\ga_0}.
\]
Therefore we have
\begin{align*}
\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{2+\ga_0+\ep-p}|r\J|^2|\phi|^2r^{p+2}dudvd\om&\les \int_{u}u_+^{2+\ga_0+\ep-p}\int_{v}\sum\limits_{k\leq 2}r^{3+\ga_0}\int_{\om}
|\mathcal{L}_{\Om}^k \J|^2d\om\cdot \int_{\om}r^{1+p-\ga_0}|\phi|^2d\om dudv\\
&\les_{M_2} \mathcal{E}_{0}[\phi]\iint_{\bar{\mathcal{D}}_{\tau}}u_+^{1-\ga_0+\ep}r^{3+\ga_0}\int_{\om}
|\mathcal{L}_{\Om}^k \J|^2d\om dvdu\\
\end{align*}
Estimate (<ref>) then follows.
Now we are remained to consider the spacetime norm on the bounded region $r\leq R$.
On the bounded region $r\leq R$, for all $0\leq \tau_1<\tau_2$ we have
\begin{equation}
\label{eq:com1:R}
\int_{\tau_1}^{\tau_2}\tau_+^{1+\ga_0}\int_{r\leq R}|[\Box_A, D_Z]\phi|^2dxdt\les_{M_2}\mathcal{E}_0[\phi](\tau_1)_+^{-1-\ga_0}.
\end{equation}
First we conclude from the energy estimate (<ref>) that the energy flux of the scalar field decays:
\[
E[\phi](\Si_{\tau})\les_{M_2}\tau_+^{-1-\ga_0},\quad \forall \tau\geq 0.
\]
From the commutator estimates (<ref>), we have
\[
|[\Box_A, D_Z]\phi|\les |F||\bar D\phi|+|J||\phi|.
\]
For the first term, we make use of estimate (<ref>):
\begin{align*}
\int_{\tau_1}^{\tau_2}\tau_+^{1+\ga_0}\int_{r\leq R}|F|^2|\bar D\phi|dxdt &\les \int_{\tau_1}^{\tau_2}\sup\limits_{|x|\leq R}|F|^2(\tau, x) E[\phi](\Si_{\tau})\tau_+^{1+\ga_0}d\tau\\
&\les_{M_2}\mathcal{E}_0[\phi]\int_{\tau_1}^{\tau_2}\sup\limits_{|x|\leq R}|F|^2(\tau, x)d\tau\\
\end{align*}
For $|J||\phi|$, we use Sobolev embedding on the ball $B_R$ with radius $R$ at fixed time $\tau$:
\begin{align*}
\int_{\tau_1}^{\tau_2}\tau_+^{1+\ga_0}\int_{r\leq R}|J|^2|\phi|^2dxdt &\les \int_{\tau_1}^{\tau_2}\tau_+^{1+\ga_0}\|J\|_{H^1_x(B_R)}^2\cdot \|\phi\|_{H^1_x(B_R)}^2d\tau\\
&\les_{M_2}\int_{\tau_1}^{\tau_2}\mathcal{E}_0[\phi]\int_{r\leq R}|\nabla J|^2+|J|^2 dx d\tau\\
\end{align*}
Thus estimate (<ref>) holds.
Now from Lemma <ref>, combine estimates (<ref>), (<ref>)-(<ref>). We can bound the first order commutator.
For all positive constant $\ep_1<1$, we have
\begin{equation}
\label{eq:Est4:com1}
\begin{split}
&I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi](\{t\geq 0\})+ I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi](\{t\geq 0\}\\
&\les_{M_2}\ep_1 I^{-1-\ep}_{1+\ga_0-\ep}[D\phi_1](\{t\geq 0\})+\ep_1 I^{\ga_0}_{0}[\D\psi_1](\{t\geq 0\}\cap\{r\geq R\})+\mathcal{E}_0[\phi] \ep_1^{-1}\\
&+\ep_1\int_{\mathbb{R}}\tau_+^{1+\ga_0}g(\tau)E[D_Z\phi](\Si_{\tau})d\tau+\ep_1 \int_{\mathbb{R}}\int_{H_{\tau^*}} \tau_+^{2+\ga_0+\ep-p}g(\tau) r^{p}|D_L\psi_1|^2 dvd\om d\tau
\end{split}
\end{equation}
From Lemma <ref>, estimate (<ref>) is a consequence of estimates (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>). The term $I^{1+\ep}_{1+\ep}[r^{-1}D_{\Lb}D_L\psi](D)$ can further be controlled by using Proposition <ref> with $p=1+\ep$.
Now we are able to derive the energy decay estimates for the first order derivative of the scalar field. Based on the result for the decay estimates for $\phi$ in the previous subsection, it suffices to bound $\mathcal{E}_0[D_Z\phi]$.
We have the following bound:
\begin{equation}
\label{eq:bd4EDZphi}
\mathcal{E}_0[D_Z\phi]\les_{M_2}\mathcal{E}_1[\phi].
\end{equation}
First by definition,
\begin{align*}
\mathcal{E}_0[D_Z\phi]&\les \mathcal{E}_1[\phi]+I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi](\{t\geq 0\})+ I^{1+\ep}_{1+\ga_0}[[\Box_A, D_Z]\phi](\{t\geq 0\}.
\end{align*}
Then from previous estimate (<ref>), the above inequality leads to
\begin{align*}
\mathcal{E}_0[D_Z\phi]&\les_{M_2} \ep_1 I^{-1-\ep}_{1+\ga_0-\ep}[DD_Z\phi](\{t\geq 0\})+\ep_1 I^{\ga_0}_{0}[\D\psi_1](\{t\geq 0\}\cap\{r\geq R\})+\mathcal{E}_1[\phi] \ep_1^{-1}\\
&+\ep_1\int_{\mathbb{R}}\tau_+^{1+\ga_0}g(\tau)E[D_Z\phi](\Si_{\tau})d\tau+\ep_1 \int_{\mathbb{R}}\int_{H_{\tau^*}} \tau_+^{2+\ga_0+\ep-p}g(\tau) r^{p}|D_L\psi_1|^2 dvd\om d\tau
\end{align*}
for all $0<\ep_1<1$. By our notation, the implicit constant is independent of $\ep_1$ and $\phi_1=D_Z\phi$.
Now from the integrated local energy estimates (<ref>), (<ref>) combined with Lemma <ref>, we can show that
\begin{align*}
I^{-1-\ep}_{1+\ga_0-\ep}[DD_Z\phi](\{t\geq 0\})\les_{M_2} \mathcal{E}_0[D_Z\phi].
\end{align*}
By the energy decay estimate (<ref>), (<ref>), we have the energy decay for $D_Z\phi$:
\[
E[D_Z\phi](\Si_{\tau})\les_{M_2}\mathcal{E}_0[D_Z\phi]\tau_+^{-1-\ga_0},\quad \forall \tau\in \mathbb{R}.
\]
Moreover, the $r$-weighted energy estimates (<ref>), (<ref>) imply that
\begin{align*}
\tau_+^{1+\ga_0-p}\int_{H_{\tau^*}}r^p|D_{L}(rD_Z\phi)|^2dvd\om+\int_{\mathbb{R}}\int_{H_{\tau^*}}r^{\ga_0}|\D(rD_{Z}\phi)|^2 dvd\om d\tau\les_{M_2}\mathcal{E}_0[D_Z\phi].
\end{align*}
Recall the definition for $g(\tau)$ in line (<ref>). By Lemma <ref>, we then can demonstrate that
\begin{align*}
&\int_{\mathbb{R}}\tau_+^{1+\ga_0}g(\tau)E[D_Z\phi](\Si_{\tau})d\tau\les_{M_2} \mathcal{E}_0[D_Z\phi] \int_{\mathbb{R}}g(\tau)d\tau\les_{M_2}\mathcal{E}_0[D_Z\phi],\\
&\int_{\mathbb{R}}\int_{H_{\tau^*}} \tau_+^{2+\ga_0+\ep-p}g(\tau) r^{p}|D_L\psi_1|^2 dvd\om d\tau\les_{M_2}\mathcal{E}_0[D_Z\phi] \int_{\mathbb{R}}\tau_+^{1+\ep}g(\tau)d\tau\les_{M_2} \mathcal{E}_0[D_Z\phi].
\end{align*}
We therefore derive that
\[
\mathcal{E}_0[D_Z\phi]\les_{M_2} \ep_1 \mathcal{E}_0[D_Z\phi]+\ep_1^{-1}\mathcal{E}_1[\phi],\quad \forall 0<\ep_1<1.
\]
Take $\ep_1$ to be sufficiently small, depending only on $M_2$, $\ga_0$, $R$ and $\ep$. We then obtain estimate (<ref>).
The argument then implies all the desired energy decay estimates for the first order derivative of the scalar field in terms of $\mathcal{E}_1[\phi]$. Moreover estimate (<ref>) can be improved to be the following:
For all positive constant $\ep_1<1$, we have
\begin{equation}
\label{eq:Est4:com1:b}
\begin{split}
I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi](\{t\geq 0\})+ I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi](\{t\geq 0\}\les_{M_2}\ep_1 \mathcal{E}_1[\phi]+\mathcal{E}_0[\phi] \ep_1^{-1}.
\end{split}
\end{equation}
§.§.§ Energy decay estimates for the second order derivatives of the scalar field
In this subsection, we establish the energy decay estimates for the second order derivative of the scalar field. Note that the definition of $M_2$ records the size and regularity of the connection field $A$ which is
independent of the scalar field. In particular Proposition <ref> and Corollary <ref> apply to $\phi_1=D_Z\phi$:
\begin{align*}
I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi_1](\{t\geq 0\})+& I^{1+\ga_0}_{1+\ep}[[\Box_A, D_Z]\phi_1](\{t\geq 0\}\les_{M_2}\ep_1 \mathcal{E}_1[\phi_1]+\mathcal{E}_1[\phi] \ep_1^{-1}
\end{align*}
for all $0<\ep_1<1$. Here by Proposition <ref>, $\mathcal{E}_0[\phi_1]\les_{M_2} \mathcal{E}_1[\phi]$. To derive the energy decay estimates for the second order derivative of the solution, it suffices to bound $\mathcal{E}_1[\phi_1]$.
As $\phi_1=D_Z\phi$, by definition
\begin{align*}
\mathcal{E}_1[\phi_1]&=\mathcal{E}_0[\phi_1]+E^1_0[\phi_1]+I^{1+\ga_0}_{1+\ep}[D_Z\Box_A\phi_1](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[D_Z\Box_A \phi_1](\{t\geq0\})\\
&\les \mathcal{E}_2[\phi]+I^{1+\ga_0}_{1+\ep}[D_Z[\Box_A,D_Z]\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[D_Z[\Box_A,D_Z] \phi](\{t\geq0\})\\
&\les \mathcal{E}_2[\phi]+I^{1+\ga_0}_{1+\ep}[[D_Z, [\Box_A,D_Z]]\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[[D_Z,[\Box_A,D_Z]] \phi](\{t\geq0\})\\
&\qquad+I^{1+\ga_0}_{1+\ep}[[\Box_A,D_Z]D_Z\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[[\Box_A,D_Z] D_Z\phi](\{t\geq0\})\\
&\les_{M_2} \mathcal{E}_2[\phi]+I^{1+\ga_0}_{1+\ep}[[D_Z, [\Box_A,D_Z]]\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[[D_Z,[\Box_A,D_Z]] \phi](\{t\geq0\})\\
&\qquad+\ep_1 \mathcal{E}_1[\phi_1]+\mathcal{E}_1[\phi] \ep_1^{-1}
\end{align*}
for $0<\ep_1<1$.
Let $\ep_1$ to be sufficiently small. We then conclude that
\begin{align*}
\mathcal{E}_1[\phi_1]&\les_{M_2} \mathcal{E}_2[\phi]+I^{1+\ga_0}_{1+\ep}[[D_Z, [\Box_A,D_Z]]\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[[D_Z,[\Box_A,D_Z]] \phi](\{t\geq0\}).
\end{align*}
Therefore to bound $\mathcal{E}_1[\phi_1]$, it is reduced to control the second order commutator $[D_Z, [\Box_A,D_Z]]\phi$.
First we have the following analogue of Lemma <ref>:
For all $X, \, Y\in Z$, when $r\geq R$, we have
\begin{equation}
\label{eq:Est4commu:2}
\begin{split}
|[D_X,[\Box_A, D_Y]]\phi|\les & |[\Box_{\mathcal{L}_Z A}, D_Z]\phi|+(|F|^2+|r\a||r\ab|+|r\si|^2+|r\rho|(|\ab|+|\a|))|\phi|.
\end{split}
\end{equation}
When $r\leq R$, we have
\begin{equation}
\label{eq:Est4commu:2:in}
|[D_X, [\Box_A, D_Y]]\phi|\les |[\Box_{\mathcal{L}_Z A}, D_Z]\phi|+|[\Box_{A}, D_Z]\phi|+|F|^2|\phi|.
\end{equation}
Here we note that $\mathcal{L}_Z F=\mathcal{L}_Z d A=d\mathcal{L}_Z A$.
First from Lemma <ref>, we can write
\begin{equation*}
[\Box_A, D_{X}]\phi=2i X^\nu F_{\mu\nu}D^{\mu}\phi+i \pa^\mu( F_{\mu\nu}X^\nu)\phi.
\end{equation*}
We need to compute the double commutator $[D_Y, [\Box_A, D_X]]\phi$ for $X$, $Y\in Z$. We can compute that
\begin{align*}
[D_Y, [\Box_A, D_{X}]]\phi&=\mathcal{L}_Y\left(2i X^\nu F_{\mu\nu}D^{\mu}+i \pa^\mu( F_{\mu\nu}X^\nu)\right)\phi\\
&=2i (\mathcal{L}_Y F)(D\phi, X)+2i F([D_Y, D\phi], X)+2iF(D\phi, [Y, X])+iY(\pa^\mu( F_{\mu\nu}X^\nu))\phi.
\end{align*}
\begin{align*}
[D_Y, D\phi]=D^{\mu}\phi [Y, \pa_{\mu}]+[D_Y, D^{\mu}]\phi \pa_{\mu}=i F_{Y\pa_{\mu}}\phi\pa^{\mu}-(\pa^{\mu}Y^{\nu}+\pa^{\nu}Y^{\mu})D_{\mu}\phi\pa_{\nu}
\end{align*}
As $X$, $Y\in Z$, $Z=\{\pa_t, \Om_{ij}\}$, we conclude that $X$, $Y$ are killing:
\[
\pa^\mu X^\nu+\pa^\nu X^\mu=0,\quad \pa^\mu Y^\nu+\pa^\nu Y^\mu=0.
\]
This implies that the following term can be simplified:
\begin{align*}
Y(\pa^\mu( F_{\mu\nu}X^\nu))&=[Y, \pa^\mu]F(\pa_{\mu}, X)+\pa^{\mu} (\mathcal{L}_Y F)(\pa_{\mu}, X)+\pa^{\mu} F(\mathcal{L}_Y \pa_{\mu}, X)+\pa^{\mu} F( \pa_{\mu}, \mathcal{L}_Y X)\\
&=\pa^{\mu} (\mathcal{L}_Y F)(\pa_{\mu}, X)+\pa^{\mu} F( \pa_{\mu}, [Y,X])
\end{align*}
Therefore we can write the double commutator as follows:
\begin{align*}
[D_Y, [\Box_A, D_{X}]]\phi&=2i (\mathcal{L}_Y F)(D\phi, X)+i\pa^{\mu} (\mathcal{L}_Y F)(\pa_{\mu}, X)\\
&+2iF(D\phi, [Y, X])+i\pa^{\mu} F( \pa_{\mu}, [Y,X])\phi-2 F^{\mu}_{\ X}F_{Y\mu}\phi.
\end{align*}
Note that $[X, Y]\in Z$ for $X$, $Y\in Z=\{\pa_t, \Om_{ij}\}$. We thus can write $2iF(D\phi, [Y, X])+i\pa^{\mu} F( \pa_{\mu}, [Y,X])\phi$ as $[\Box_A, D_{[Y, X]}]\phi$ and can be bounded by
using Lemma <ref>. The term $2i (\mathcal{L}_Y F)(D\phi, X)+i\pa^{\mu} (\mathcal{L}_Y F)(\pa_{\mu}, X)$ has the same form with $[\Box_A, D_{X}]\phi$ if we replace $F$ with $\mathcal{L}_Y F$. In particular the bound follows from Lemma <ref>. Therefore to show this lemma, it remains to control $F^{\mu}_{\ X}F_{Y\mu}\phi$ for $X$, $Y\in Z$. This term has crucial null structure we need to exploit when $r\geq R$. The main difficulty is that the angular momentum $\Om$ contains weights in $r$. If both $X, \, Y\in \Om$, then
\[
|F^{\mu}_{\ X}F_{Y\mu}|\les |r\a||r\ab|+|r\si|^2.
\]
If $X=Y=\pa_t$, then
\[
|F^{\mu}_{\ X}F_{Y\mu}|\les |F|^2.
\]
If one and only one of $X,\, Y$ is $\pa_t$, then the null structure is as follows:
\begin{align*}
|F^{\mu}_{\ X}F_{Y\mu}|&\les r|F^{\mu}_{\ L}F_{e_i\mu}|+r|F^{\mu}_{\ \Lb}F_{e_i\mu}|\les r(|\rho|+|\si|)(|\ab|+|\a|).
\end{align*}
We see that the "bad" term $r|\ab|^2$ does not appear on the right hand side. Hence
\[
|F^{\mu}_{\ X}F_{Y\mu}|\les |F|^2+|r\a||r\ab|+|r\si|^2+|r\rho|^2,\quad \forall X,\, Y\in Z.
\]
Therefore estimate (<ref>) holds. On the bounded region $\{r\leq R\}$, null structure is not necessary and estimate (<ref>) follows trivially.
The above lemma shows that the double commutator $[D_Z, [\Box_A, D_Z]]\phi$ consists of quadratic part $[\Box_{\mathcal{L}_Z^k A}, D_Z]\phi$ which can be bounded similar to $[\Box_A, D_Z]\phi$ as
we can put one more derivative $D_Z$ on the scalar field $\phi$ when we do Sobolev embedding. It thus suffices to control those cubic terms in (<ref>).
We have
\begin{equation}
\label{eq:Est4:com2:quad}
\begin{split}
&\sum\limits_{k\leq 1}I_{1+\ga_0}^{1+\ep}[[\Box_{\mathcal{L}_Z^k A}, D_Z]\phi](\{t\geq 0\})+ I^{1+\ga_0}_{1+\ep}[[\Box_{\mathcal{L}_Z^k A}, D_Z]\phi](\{t\geq 0\}\\
&\les_{M_2}\ep_1 I^{-1-\ep}_{1+\ga_0-\ep}[D\phi_2](\{t\geq 0\})+\ep_1 I^{\ga_0}_{0}[\D\psi_2](\{t\geq 0\}\cap\{r\geq R\})+\mathcal{E}_1[\phi] \ep_1^{-1}\\
&+\ep_1\int_{\mathbb{R}}\tau_+^{1+\ga_0}g(\tau)E[\phi_2](\Si_{\tau})d\tau+\ep_1 \int_{\mathbb{R}}\int_{H_{\tau^*}} \tau_+^{2+\ga_0+\ep-p}g(\tau) r^{p}|D_L\psi_2|^2 dvd\om d\tau
\end{split}
\end{equation}
for all positive constant $\ep_1$. Here $\phi_2=D_Z^2\phi$, $\psi_2=rD_Z^2\phi$. The function $g(\tau)$ is defined in line (<ref>).
From Corollary <ref> and the decay estimates for the first order derivative of the scalar field, it suffices to consider estimate (<ref>) with $k=1$. The difference between estimate (<ref>) and estimate (<ref>) is that $F$ is replaced with $\mathcal{L}_Z F$ in (<ref>). However we are allowed to put one more derivative on the scalar field ($\phi_1=D_Z\phi$ is replaced with $D_Z^2\phi$). Note that for the proof of estimate (<ref>) the higher order derivative comes in when we use Sobolev embedding on the sphere to bound $\|F\cdot D\phi\|_{L_{\om}^2}$:
\[
\|F\cdot D\phi\|_{L_{\om}^2}\les \sum\limits_{k\leq 2}\|\mathcal{L}_Z^k F\|_{L_\om^2}\cdot \|D\phi\|_{L_{\om}^2} \textnormal{ or }
\sum\limits_{k\leq 1}\|\mathcal{L}_Z^k F\|_{L_\om^2}\cdot \|DD_{Z}^k\phi\|_{L_{\om}^2}.
\]
For estimate (<ref>), the corresponding term $\mathcal{L}_Z F\cdot D\phi$ can be bounded as follows:
\[
\|\mathcal{L}_Z F\cdot D\phi\|_{L_{\om}^2}\les \sum\limits_{k\leq 1}\|\mathcal{L}_Z^k \mathcal{L}_Z F\|_{L_\om^2}\cdot \|D D_Z^k \phi\|_{L_{\om}^2} \textnormal{ or }
\|\mathcal{L}_Z F\|_{L_\om^2}\cdot \sum\limits_{k\leq 2}\|DD_{Z}^k\phi\|_{L_{\om}^2}.
\]
This is how we can transfer one derivative on $F$ to the scalar field $\phi$. In particular estimate (<ref>) holds.
From Lemma <ref>, to bound the double commutator, it suffices to control the cubic terms in (<ref>), (<ref>). We rely on the pointwise bound for the Maxwell field summarized in Propositions <ref>, <ref>.
For all $1+\ep\leq p\leq 1+\ga_0$, we have
\begin{equation}
\label{eq:Est4:com2:cubic}
\begin{split}
&I^p_{2+\ga_0+\ep-p}[(|F|^2+|r\a||r\ab|+|r\si|^2+|r\rho|(|\ab|+|\a|))|\phi|](\{t\geq 0, \, r\geq R\})\\
&+I^p_{2+\ga_0+\ep-p}[|F|^2|\phi|](\{t\geq 0, \, r\leq R\})\les_{M_2}\mathcal{E}_1[\phi].
\end{split}
\end{equation}
On the finite region $r\leq R$, the weights $r^p$ have an upper bound. The Maxwell field $F$ can be bounded by using the pointwise estimate (<ref>). We then can estimate the scalar field by using the integrated local energy estimates. In deed for all $0\leq \tau_1<\tau_2$, we can show that
\begin{align*}
\int_{\tau_1}^{\tau_2}\int_{r\leq R}\tau_+^{1+\ga_0}|F|^4|\phi|^2dxd\tau\les\int_{\tau_1}^{\tau_2}\tau_+^{1+\ga_0}\sup|F|^4|\phi|^2dx d\tau\les_{M_2}(\tau_1)_+^{-2-2\ga_0} \mathcal{E}_1[\phi].
\end{align*}
For the cubic terms on the region $r\geq R$, let's first consider $|r\a||r\ab||\phi|$. We use the $r$-weighted energy estimates (<ref>), (<ref>) for the Maxwell field to control $\a$ and the
integrated decay estimate (<ref>) of Proposition <ref> to bound $\ab$. The reason that we can not use the pointwise bound (<ref>) is the weak decay rate. The scalar field $\phi$ can be bounded by using Lemma <ref>. In deed, for $1+\ep\leq p\leq 1+\ga_0$, we can show that
\begin{align*}
&I^{p}_{2+\ga_0+\ep-p}[|r\a||r\ab|| \phi|](\{t\geq 0\}\cap\{r\geq R\})\\
&\les\iint u_+^{2+\ga_0+\ep-p}r^{p+2}|r\a|^2|r\ab|^2|\phi|^2dudvd\om\\
& \les\sum\limits_{k\leq 1} \int_{u}\int_{v}u_+^{2+\ga_0+\ep-p}r^{1+\ga_0}\int_{\om}|r\mathcal{L}_Z^k \a|^2d\om \cdot \int_{\om}|r\mathcal{L}_Z^k \ab|^2d\om \cdot
\int_{\om}r^{p+1-\ga_0}|\mathcal{L}_Z^k \phi|^2d\om dvdu\\
&\les_{M_2}\mathcal{E}_1[\phi]\sum\limits_{k\leq 1} \int_{u}\int_{v}u_+^{1+\ep-\ga_0}r^{1+\ga_0}\int_{\om}|r\mathcal{L}_Z^k \a|^2d\om \cdot \int_{\om}|r\mathcal{L}_Z^k \ab|^2d\om dvdu\\
&\les_{M_2}\mathcal{E}_1[\phi]\sum\limits_{k\leq 1} \int_{u}u_+^{1+\ep-\ga_0}\int_{v}r^{1+\ga_0}\int_{\om}|r\mathcal{L}_Z^k \a|^2d\om dv \cdot\sup\limits_{v} \int_{\om}|r\mathcal{L}_Z^k \ab|^2d\om \, du\\
\end{align*}
Here recall the definition of $g(\tau)$ in line (<ref>) and the last step follows from Corollary <ref>.
For $|F|^2|\phi|$, we use the pointwise estimates (<ref>), (<ref>) of Proposition <ref> to bound the Maxwell field $F$. The scalar field $\phi$ can be bounded by using Lemma <ref> as
above. In the exterior region where the the Maxwell field contains the charge part $q_0 r^{-2}dt\wedge dr$, we have the relation $r_+\geq \frac{1}{2}u_+$. We can show that
\begin{align*}
&I^{p}_{2+\ga_0+\ep-p}[|F|^2 \cdot \phi](\{t\geq 0\}\cap\{r\geq R\})\\
&\les\iint u_+^{2+\ga_0+\ep-p}r^{p+2}|\bar F|^4|\phi|^2dudvd\om+|q_0|^2\iint_{t+R\leq r} u_+^{2+\ga_0+\ep-p}r^{p+2-8}|\phi|^2dudvd\om\\
& \les_{M_2}\int_{u}\int_{v}u_+^{2+\ga_0+\ep-p-2-2\ga_0}r^{p+2-4-1}\int_{\om}r|\phi|^2d\om dvdu+\mathcal{E}_0[\phi]\\
&\les_{M_2}\mathcal{E}_1[\phi] \int_{u}\int_{v}u_+^{\ep-p-1-2\ga_0}r^{p-3}dvdu\les_{M_2}\mathcal{E}_1[\phi].
\end{align*}
For $|r\si|^2|\phi|$, same reason as the case $|r\a||r\ab||\phi|$, we are not allowed to use the pointwise bound (<ref>) to control $\si$ due to the strong $r$ weights here. Instead we use the $r$-weighted energy estimate for $\si$ on the incoming null hypersurface together with the integrated decay estimate (<ref>).
Then we can show that
\begin{align*}
\iint_{\bar{\mathcal{D}}_{\tau_1}}r^p|r\si|^4|\phi|^2dxdt
& \les\sum\limits_{k\leq 1} \int_{u}\int_{v}r^{1+\ga_0}\int_{\om}|r\mathcal{L}_Z^k \si|^2d\om \cdot \int_{\om}|r\mathcal{L}_Z^k \si|^2d\om \cdot
\int_{\om}r^{p+1-\ga_0}|\mathcal{L}_Z^k \phi|^2d\om dvdu\\
&\les_{M_2}\mathcal{E}_1[\phi](\tau_1)_+^{-1+p-2\ga_0}\sum\limits_{k\leq 1} \int_{v}\int_{u}r^{1+\ga_0}\int_{\om}|r\mathcal{L}_Z^k \si|^2d\om du\cdot \sup\limits_{u}\int_{\om}|r\mathcal{L}_Z^k \si|^2d\om dv\\
&\les_{M_2}\mathcal{E}_1[\phi](\tau_1)_+^{-1+p-2\ga_0}\sum\limits_{k\leq 1} \|r\mathcal{L}_Z^k \si\|_{L_v^2L_u^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau_1})}^2\\
\end{align*}
This holds for all $\tau_1\in \mathbb{R}$. Since
\[
2+3\ga_0-\ep-p>2+\ga_0+\ep-p,\quad 0\leq p\leq 1+\ga_0,
\]
from Lemma <ref>, we obtain
\[
I^{p}_{2+\ga_0+\ep-p}[|r\si|^2 \phi](\{t\geq 0\}\cap\{r\geq R\})\les_{M_2}\mathcal{E}_1[\phi].
\]
Finally for $|r\rho|(|\ab|+|\a|)|\phi|$, we need to take into consideration of the charge effect in the exterior region. Except this charge, the proof for the interior region case is the same. Let's merely estimate this cubic term in the exterior region. In particular take $\bar{\mathcal{D}}_{\tau_1}$ to be $\mathcal{D}_{\tau_1}$ for some $\tau_1<0$. By using the $r$-weighted energy estimate for $\bar \rho$ and the pointwise bound (<ref>), (<ref>) for $F$, for $0\leq p\leq 1+\ga_0$ we then can show that
\begin{align*}
&\iint_{\mathcal{D}_{\tau_1}} r^{p+2}|r\rho|^2(|\a|^2+|\ab|^2)|\phi|^2dudvd\om\\
&\les \iint_{\mathcal{D}_{\tau_1}} |q_0|r^{p}(|\a|^2+|\ab|^2)|\phi|^2dudvd\om+\iint_{\mathcal{D}_{\tau_1}} r^{p+2}|r\bar\rho|^2(|\a|^2+|\ab|^2)|\phi|^2dudvd\om\\
& \les_{M_2}\mathcal{E}_1[\phi](\tau_1)_+^{-1-2\ga_0}+\sum\limits_{k\leq 1} \int_{u}\int_{v}r^{p-1}\int_{\om}|r\mathcal{L}_Z^k \bar{\rho}|^2 d\om \cdot \sup (|r\ab|^2+|r\a|^2) \cdot
\int_{\om}r|\mathcal{L}_Z^k \phi|^2d\om dvdu\\
&\les_{M_2}\mathcal{E}_1[\phi](\tau_1)_+^{-1-2\ga_0}+\mathcal{E}_1[\phi](\tau_1)_+^{-2-2\ga_0}\sum\limits_{k\leq 1} \iint_{\mathcal{D}_{\tau_1}}r^{p-1}|r\mathcal{L}_Z^k \bar{\rho}|^2 du dv d\om \\
\end{align*}
Here the last term is bounded by using the $r$-weighted energy estimates for $\bar \rho$. As $\tau_1$ is arbitrary, from Lemma <ref>, we derive that
\[
I^{p}_{2+\ga_0+\ep-p}[r\rho\cdot (|\ab|+|\a| )\cdot \phi](\{t\geq 0\}\cap\{r\geq R\})\les_{M_2}\mathcal{E}_1[\phi], \quad 1+\ep\leq p\leq 1+\ga_0.
\]
To summarize, we have shown (<ref>).
The above two propositions <ref>, <ref> together with Lemma <ref> lead to the desired estimates for the double commutator $[D_X, [\Box_A, D_Y]]$ for $X, \, Y\in Z$.
Then by the argument at the beginning of this section, we have control of $\mathcal{E}_0[D_X D_Y \phi]$. By using the same argument of Proposition <ref>, we then can bound $\mathcal{E}_0[D_X D_Y \phi]$ by $\mathcal{E}_2[\phi]$ which then implies the decay of the second order derivative of the scalar field.
For any $X, \, Y\in Z$, we have the following bound:
\begin{equation}
\label{eq:bd4DZZphi}
\mathcal{E}_0[D_XD_Y\phi]\les_{M_2}\mathcal{E}_2[\phi].
\end{equation}
From the argument at the beginning of this section (before Lemma <ref>), we derive that
\begin{align*}
\mathcal{E}_0[D_X D_Y\phi]&\les_{M_2} \mathcal{E}_2[\phi]+I^{1+\ga_0}_{1+\ep}[[D_X, [\Box_A,D_Y]]\phi](\{t\geq 0\})+I^{1+\ep}_{1+\ga_0}[[D_X,[\Box_A,D_Y]] \phi](\{t\geq0\}).
\end{align*}
Then by Lemma <ref> and Proposition <ref>, for all $0<\ep_1<1$, $X$, $Y\in Z$, we conclude that
\begin{align*}
&\mathcal{E}_0[D_X D_Y\phi]\les_{M_2} \ep_1 I^{-1-\ep}_{1+\ga_0-\ep}[D\phi_2](\{t\geq 0\})+\ep_1 I^{\ga_0}_{0}[\D\psi_2](\{t\geq 0\}\cap\{r\geq R\})\\
&\quad+\mathcal{E}_1[\phi] \ep_1^{-1}+\ep_1\int_{\mathbb{R}}\tau_+^{1+\ga_0}g(\tau)E[\phi_2](\Si_{\tau})d\tau+\ep_1 \int_{\mathbb{R}}\int_{H_{\tau^*}} \tau_+^{2+\ga_0+\ep-p}g(\tau) r^{p}|D_L\psi_2|^2 dvd\om d\tau,
\end{align*}
where $\phi_2=D_XD_Y\phi$, $\psi_2=r\phi_2$. The proposition follows by the same argument of Proposition <ref>.
§.§ Pointwise bound for the scalar field
Once we have the bound (<ref>), from Proposition <ref> and Corollary <ref>, we obtain the energy flux decay estimates as well as the $r$-weighted energy estimates for
the second order derivatives of the scalar field. In another word, simply assuming $M_2$ is finite (see definition of $M_2$ in line (<ref>)), we then can derive the energy decay estimates for
the second order derivatives of the scalar field. For the MKG equations, $J=\delta F$ is quadratic in $\phi$. To construct global solutions, we need to bound these nonlinear terms.
In this section, we show the pointwise bound for the scalar field with the assumption that $M_2$ is finite.
We start with an analogue of Proposition <ref> regarding the pointwise bound of the scalar field in the finite region $r\leq R$. Similar to the pointwise bound of the Maxwell field, we use elliptic estimates. However as the connection field $A$ is general, we are not able to apply the elliptic estimates for the flat case directly. We therefore establish an elliptic lemma for the operator $\Delta_A=\sum\limits_{i=1}^3 D_iD_i$ first. Let $B_{R_1}$ be the ball with radius $R_1$ in $\mathbb{R}^3$. Define
\[
\|\phi\|_{H^k(B_{R_1})}=\sum\limits_{1\leq j_{l}\leq 3}\|D_{j_1}D_{j_2}\ldots D_{j_{k}}\phi\|_{L^2(B_{R_1})}+\|\phi\|_{H^{k-1}(B_{R_1})},\quad k\geq 1.
\]
Then we have
We have the following elliptic estimates:
\begin{equation}
\label{eq:elliptic:A}
\|\phi\|_{H^2(B_{R_1})}\les_{M_2, R_1, R_2}\|\Delta_A\phi\|_{L^2(B_{R_2})}+(1+\|F\|_{L^{\infty}(B_{R_2})}+\|J\|_{H^1(B_{R_2})}) \|\phi\|_{H^1(B_{R_2})}
\end{equation}
for all $R_1<R_2$. Here the constant $M_2$ is defined in line (<ref>) and $J=\delta(dA)$ or $J_{j}=\pa^{i}(dA)_{ij}$.
The proof is similar to the case when the connection field $A$ is trivial. For the case when the scalar field $\phi$ is compactly supported in some ball $B_{R_1}$, using integration by parts, we can show that
\begin{align*}
\int_{B_{R_1}}D_i D_j\phi\cdot \overline{D_iD_j\phi}dx&=-\int_{B_{R_1}} D_iD_iD_j\phi \cdot \overline{D_j\phi}dx\\
&=-\int_{B_{R_1}} D_jD_iD_i\phi \cdot \overline{D_j\phi}dx-\int_{B_{R_1}} [D_iD_i, D_j]\phi \cdot \overline{D_j\phi}dx\\
&=\int_{B_{R_1}}|\Delta_A\phi|^2dx-\int_{B_{R_1}} \sqrt{-1}(2F_{ij}D_i\phi+\pa_{i}F_{ij}\phi)\cdot \overline{D_j\phi}dx.
\end{align*}
Estimate (<ref>) then follows.
For general complex function $\phi$, we can choose a real cut-off function $\chi$ which is supported on the ball $B_{R_2}$ and equal to $1$ on the smaller ball $B_{R_1}$. By direct computation, we can show that:
\begin{align*}
\|\Delta_A(\chi\phi)\|_{L^2(B_{R_2})}&=\|\chi \Delta_A\phi+2\pa_i\chi \cdot D_i\phi+\Delta \chi \cdot \phi\|_{L^2(B_{R_2})}\\
&\les \|\Delta_A \phi\|_{L^2(B_{R_2})}+\|\phi\|_{H^1(B_{R_2})}.
\end{align*}
The lemma then follows from the above argument for the compactly supported case.
We assume $\Box_A\phi$ verifies the following extra bound
\[
\int_{\tau_1}^{\tau_2}\int_{r\leq 2R}|D\Box_A\phi|^2+|D_Z D\Box_A\phi|^2dxd\tau\leq C \mathcal{E}_2[\phi](\tau_1)_+^{-1-
\ga_0},\quad 0\leq \tau_1<\tau_2
\]
for some constant $C$ depending only on $R$.
The above elliptic estimate adapted to the connection field $A$ implies the following pointwise bound for the scalar field $\phi$ on the compact region $r\leq R$.
For all $0\leq \tau$, $0\leq \tau_1<\tau_2$, we have
\begin{align}
\label{eq:Est4phi:in:R}
\int_{\tau_1}^{\tau_2}\sup_{|x|\leq R}(|D\phi|^2+|\phi|^2)(\tau, x)d\tau&\les \int_{\tau_1}^{\tau_2}\int_{r\leq R}|D^2 D\phi|^2+|\phi|^2 dxdt \les_{ M_2}\mathcal{E}_2[\phi](\tau_1)_+^{-1-\ga_0},\\
\label{eq:Est4phi:in:R:p}
|D\phi|^2(\tau, x)+|\phi|^2(\tau, x)&\les_{ M_2}\mathcal{E}_2[\phi] \tau_+^{-1-\ga_0},\quad \forall |x|\leq R.
\end{align}
At the fixed time $\tau\geq 0$, consider the elliptic equation for the scalar field $\phi_k=D_Z^k\phi$:
\[
\Delta_A \phi_k=D_tD_t\phi_k+D_k\Box_A\phi+[\Box_A, D_Z^k]\phi.
\]
Proposition <ref> and <ref> indicate that the Maxwell field $F$ is bounded. The definition of $M_2$ shows that
\[
\|J\|_{H^1(B_{2R})}^2\les \int_{\tau}^{\tau+1}|\nabla J|^2+|\pa_t \nabla J|^2+|J|^2 +|\pa_t J|^2dx dt\les M_2.
\]
Here $B_{R_1}$ denotes the ball with radius $R_1$ at time $\tau$. Then by the previous Lemma <ref>, we conclude that
\begin{align*}
\|\phi_k\|_{H^2(B_{\frac{3}{2}R})}^2\les_{M_2} \|D_tD_t\phi_k\|_{L^2(B_{2R})}^2+\|D_Z^k\Box_A\phi\|_{L^2(B_{2R})}^2+\|[\Box_A, D_Z^k]\phi\|_{L^2(B_{2R})}^2+\|\phi_k\|_{H^1(B_{2R})}^2.
\end{align*}
This gives the $H^2$ estimates for $D_t\phi$, $\phi$. To obtain estimates for $D_j\phi$, commute the equation with $D_j$:
\[
\Delta_A D_j\phi=D_jD_tD_t\phi+D_j\Box_A\phi+[\Delta_A, D_j]\phi=D_jD_tD_t\phi+D_j\Box_A\phi+\sqrt{-1}(2F_{ij}D_i\phi+\pa_{i}F_{ij}\phi).
\]
Then by using Lemma <ref> again, we obtain
\begin{align*}
\|D_j\phi\|_{H^2(B_{R})}^2&\les_{M_2} \|D_j\phi\|_{H^1(B_{1.5R})}^2+\|\Delta_A D_j\phi\|_{L^2(B_{1.5R})}^2\\
&\les_{M_2} \|\phi\|_{H^2(B_{1.5R})}^2+\|D_j D_t^2\phi\|_{L^2(B_{1.5R})}^2+\| D_j\Box_A\phi\|_{L^2(B_{1.5R})}^2.
\end{align*}
Here we have used the fact $|F|^2\les M_2$, $\|J\|_{H^1(B_{2R})}^2\les M_2$. Then for the pointwise bound (<ref>), we need to show the energy flux decay through $B_{2R}$ at time $\tau$. This can be fulfilled by considering the energy estimate obtained by using the vector field $\pa_t$ as multiplier on the region bounded by $t=\tau$ and $\Si_{\tau-R}$ (recall that $\Si_{\tau}=H_{\tau^*}$ for negative $\tau<0$). Corollary <ref> together with Proposition <ref>, <ref> imply that
\begin{align*}
E[D_Z^k \phi](B_{2R})\les E[D_Z^k\phi](\Si_{\tau-R})+(\tau-R)_+^{-1-\ga_0}\mathcal{E}_0[D_Z^k\phi]\les_{M_2}\mathcal{E}_k[\phi]\tau_+^{-1-\ga_0},\quad k\leq 2.
\end{align*}
For the flux of the inhomogeneous term $D\Box_A\phi$ and the commutator term $[D_Z, l\Box_A]\phi$, we can make use of the integrated local energy estimates. More precisely combine the above $H^2$ estimates for $\phi_k=D_Z^k\phi$, $k=0$, $1$ and $D_j\phi$. We can show that
\begin{align*}
&\|D_j\phi\|_{H^2(B_{R})}^2+\sum\limits_{k\leq 1}\|\phi_k\|_{H^2(B_{R})}^2\\
&\les_{M_2} \sum\limits_{l\leq 2}E[D_Z^l\phi](B_{2R})+\| D\Box_A\phi\|_{L^2(B_{2R})}^2+\| [\Box_A, D_Z]\phi\|_{L^2(B_{2R})}^2\\
&\les_{M_2} \mathcal{E}_2[\phi]\tau_+^{-1-\ga_0}+\int_{\tau}^{\tau+1}\int_{r\leq 2R}|D\Box_A\phi|^2+|D_tD\Box_A\phi|^2dxd\tau+I^0_0[D_Z [\Box_A, D_Z]\phi](D_{\tau-R}^{\tau})\\
\end{align*}
Here we have used the following bound:
\[
I^{1+\ep}_{1+\ga_0}[D_Z^k[\Box_A, D_Z]\phi](\{t\geq 0\})\les_{M_2} \mathcal{E}_{k+1}[\phi],\quad k=0, 1.
\]
which is a consequence of the proof in the previous section (see the argument in the beginning of Section <ref>). Then Sobolev embedding implies the pointwise bound (<ref>) for $\phi$.
For the integrated decay estimates (<ref>), we integrate the $H^2$ norm of $D_j\phi$ from time $\tau_1$ to $\tau_2$:
\begin{align*}
\int_{\tau_1}^{\tau_2}\|D\phi\|_{H^2(B_{R})}^2 d\tau&\les_{M_2}\int_{\tau_1}^{\tau_2}\sum\limits_{l\leq 2}\|D_Z^l\phi\|_{L^2(B_{2R})}^2+\| D\Box_A\phi\|_{L^2(B_{2R})}^2+\| [\Box_A, D_Z]\phi\|_{L^2(B_{2R})}^2 d\tau\\
&\les_{M_2}\sum\limits_{l\leq 2}I^{-1-\ep}_0[D_Z^l\phi](D_{\tau_1-R}^{\tau_2})+\int_{\tau_1}^{\tau_2}\int_{r\leq 2R}|D\Box_A\phi|^2dxd\tau+I^{0}_0[[D_Z, \Box_A]\phi](D_{\tau_1-R}^{\tau_2})\\
\end{align*}
Here we have used the integrated local energy estimates for the second order derivative of the scalar field. Then Sobolev embedding implies the integrated decay estimate (<ref>).
For the Sobolev embedding adapted to the connection $A$, it suffices to establish the $L^p$ embedding in terms of the $H^1$ norm. As the norm is gauge invariant we can choose a particular gauge so that the function is real. For real function $f$ we have the trivial bound $\|D_A f\|_{L^2}\geq \|\pa f\|_{L^2}$. This explains the Sobolev embedding we have used in this paper adapted to the general connection field $A$.
Next we consider the pointwise bound for the scalar field outside the cylinder $r\leq R$. The decay estimate for $\phi$ easily follows from Lemma <ref> as we have energy decay estimates for second order derivatives of $\phi$. However this does not apply to the derivative of $\phi$ due to the limited regularity (only two derivatives). Like the Maxwell field in Proposition <ref>, we rely on Lemma <ref>.
We have the following pointwise bound:
\begin{align}
\label{eq:supDLbphi:I}
\|D_{\Lb}(rD_Z^k\phi)\|_{L_u^2 L_v^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}& \les_{M_2}\mathcal{E}_{k+1}[\phi] (\tau_1)_+^{-1-\ga_0+3\ep},\quad k=0, 1,\\
\label{eq:supDLphi:I}
\|r^{\frac{p}{2}}D_{L}(rD_Z^k\phi)\|_{L_v^2 L_u^\infty L_{\om}^2(\bar{\mathcal{D}}_{\tau})}^2 &\les_{M_2} \mathcal{E}_{k+1}[\phi](\tau_1)_+^{p+4\ep-1-\ga_0},\quad 0\leq p\leq 1+\ga_0-4\ep, \quad k=0, 1,\\
\label{eq:supDLpsi:p}
r^p(|D_L(r\phi)|^2+|\D(r\phi)|^2)(\tau, v, \om)&\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{p-1-\ga_0},\quad 0\leq p\leq 1+\ga_0,\\
\label{eq:supDLbpsi:p}
|D_{\Lb}(r\phi)|^2(\tau, v, \om)&\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{-1-\ga_0},\\
\label{eq:supphi:p}
r^p|\phi|^2(\tau, v,\om)&\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{p-2-\ga_0},\quad 1\leq p\leq 2.
\end{align}
If we have one more derivative (assume $M_3$), then we have better estimate for $\D(r\phi)$ as we can write it as $D_{Z}\phi$.
Estimate (<ref>) follows from (<ref>), (<ref>) together with the $r$-weighted energy and integrated local energy estimates for the scalar field $D_Z^k\phi$, $k\leq 2$. Estimate (<ref>) is a consequence of (<ref>), (<ref>).
For the pointwise bound for the scalar field, let $\phi_k=D_Z^k\phi$, $\psi_k=r\phi_k$, $k\leq 2$. First the $r$-weighted energy estimates (<ref>), (<ref>) imply that
\[
\int_{H_{\tau^*}}r^p|D_L\psi_k|^2dvd\om\les_{M_2}\mathcal{E}_k[\phi]\tau_+^{p-1-\ga_0},\quad k\leq 2, \quad 0\leq p\leq 1+\ga_0.
\]
From the $r$-weighted energy estimate for $F$ and Lemma <ref>, we can bound the commutator:
\begin{align*}
\int_{H_{\tau^*}}r^p|[D_Z^2, D_L]\psi|^2dvd\om&\les \int_{H_{\tau^*}}r^p(|F_{ZL}D_Z\psi|^2+|\mathcal{L}_Z F_{ZL}\psi|^2)dvd\om\\
&\les \sum\limits_{l\leq 1}\int_{H_{\tau^*}}r^p(|\mathcal{L}_Z^l\rho \psi_{1-l}|^2+|r\mathcal{L}_Z^l\a||\psi_{l-1}|)dvd\om\\
\end{align*}
Here the charge part only appears when $\tau<0$. The previous two estimates lead to:
\begin{align*}
\int_{H_{\tau^*}}r^p|D_Z^kD_L D_Z^l\psi|^2dvd\om\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{p-1-\ga_0},\quad k+l\leq 2, \quad 0\leq p\leq 1+\ga_0.
\end{align*}
To apply Lemma <ref>, we need the energy flux for $D_LD_L\psi$. From the null equation (<ref>) for the scalar field, on the outgoing null hypersurface $H_{\tau^*}$, for $k=0,\, 1$, we can show that
\begin{align*}
\int_{H_{\tau^*}}r^p|D_{\Lb}D_{L}\psi_k|^2dvd\om&\les\int_{H_{\tau^*}}r^p(|\rho\cdot r\phi_k|^2+|r^{-1}\D D_{\Om}\psi_k|^2+|r\Box_A\phi_k|^2)dvd\om\\
\end{align*}
Here the first term $\rho\cdot r\psi_k$ has been bounded in the above commutator estimate for $[D_Z^2, D_L]\psi $. The second term $|r^{-1}\D D_{\om}\psi_k|^2$ can be bounded by the energy flux of $\mathcal{L}_Z^2 F$ through $H_{\tau^*}$ as $p\leq 2$. The bound for $\Box_A\phi_k$ follows from the argument in the previous section <ref> where we have shown that $\mathcal{E}_1[\phi_k]\les_{M_2}\mathcal{E}_{2}[\phi_{k-1}]$, $k=0$, $1$. Now commute $D_L$ with $\psi_k=D_Z^k\psi$. First we can show that
\[
|D_L[D_L, D_Z]\psi|\les |L F_{LZ}||\psi|+|F_{ZL}||D_L\psi|\les (|\Lb(r\a)|+|r\mathcal{L}_Z\a|+|L\rho|)|\psi|+(|\rho|+|r\a|)|D_L\psi|.
\]
On the right hand side of the above inequality, the second term is easy to bound as we can control the Maxwell field $\rho$, $r\a$ by the $L^\infty$ norm shown in Proposition <ref> and the scalar field $\psi$ by the $r$-weighted energy estimates. For the first term, we have to use the null structure equations of Lemma <ref> to control $L(r\a)$, $L\rho$. We can show that
\begin{align*}
\int_{H_{\tau^*}}r^p|D_L[D_L, D_Z]\psi|^2dvd\om &\les_{M_2}\int_{H_{\tau^*}}r^p |D_Z^2 D_L\psi|^2+r^p(|\Lb(r\a)|^2+|r\mathcal{L}_Z\a|^2+|L\rho|^2)\mathcal{E}_2[\phi]dvd\om\\
&\les_{M_2} \mathcal{E}_{2}[\phi](\tau_+^{p-1-\ga_0}+\int_{H_{\tau^*}}r^p(|\mathcal{L}_{\Om}(\rho, \si, \a)|^2+|rJ|^2+|\rho|^2)dvd\om)\\
&\les_{M_2} \mathcal{E}_{2}[\phi]\tau_+^{p-1-\ga_0}.
\end{align*}
Here we can bound $\rho$, $\a$, $\si$ by the energy flux as $p\leq 2$. For the inhomogeneous term $J$ we can use one more derivative $\mathcal{L}_{\pa_t}$. In particular we have shown that
\begin{align*}
&\sum\limits_{k\leq 1}\int_{H_{\tau^*}}r^p(|D_{L}D_Z^k D_{L}\psi|^2+|D_{\Om}D_Z^k D_L\psi|^2+|D_Z^kD_L\psi|^2)dvd\om\\
&\les \sum\limits_{l\leq 2}\int_{H_{\tau^*}}r^p(|D_Z^l D_L\psi|^2+|D_{L}[D_Z, D_{L}]\psi|^2+|D_{\Lb}D_LD_Z \psi|^2+|D_{\pa_t}D_LD_Z \psi|^2)dvd\om\\
\end{align*}
Then by using Lemma <ref> and Sobolev embedding we derive the pointwise estimate for $D_{L}\psi$ (see Remark <ref> for the Sobolev embedding adapted to the connection $A$). This proves the first part of (<ref>).
For $D_{\Lb}\psi$ and $\D(r\phi)$, we make use of the energy flux through the incoming null hypersurface $\Hb_{\tau}$ which is defined as $\Hb_{v}^{\tau^*, -v}$ when $\tau<0$ or $\Hb_{v}^{\tau, 2v-R}$ when $\tau\geq 0$. From the energy estimates (<ref>), (<ref>), (<ref>), (<ref>), we obtain the energy flux decay
\begin{align*}
\int_{\Hb_{\tau}}|D_{\Lb}D_Z^k\psi|^2+|\D D_Z^k\psi|^2 +\tau_+^{-p} r^p|D_{\Om} D_Z^k\phi|^2+r^2|D_{\Lb}D_Z^k\phi|^2dud\om\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{-1-\ga_0}
\end{align*}
for $k\leq 2,\quad 0\leq p\leq 1+\ga_0$. For $\D(r\phi)=D_{\Om}\phi$, the above estimates together with Lemma <ref> indicate that
\[
r^{p}|D_{\Om}\phi|^2\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{p-1-\ga_0},\quad 0\leq p\leq 1+\ga_0.
\]
Thus the second part of (<ref>) holds.
For $D_{\Lb}\psi$, we need to pass the $D_{\Lb}$ derivative to $\psi$. We can compute the commutator:
\begin{align*}
|[D_Z^2, D_{\Lb}]\psi|\les (|r\mathcal{L}_Z\ab|+|\mathcal{L}_Z\rho|)|\psi|+(|r\ab|+|\rho|)|D_Z\psi|.
\end{align*}
We can bound $\psi$ by using Lemma <ref> and $\rho$, $\ab$ by using the energy flux through $\Hb_{\tau}$. Then the previous energy estimates imply that
\begin{align}
\label{eq:DZDLbDZpsi}
\int_{\Hb_{\tau}}|D_Z^k D_{\Lb}D_Z^l\psi|^2+|r^{-1}D_Z^{k+1}\psi|^2 dud\om \les_{M_2}\mathcal{E}_2[\phi]\tau_+^{-1-\ga_0},\quad k+l\leq 2.
\end{align}
To apply Lemma <ref>, we also need the energy flux of $D_{\Lb}D_{\Lb}\psi$. We use the null equation (<ref>) to show that
\begin{align*}
\int_{\Hb_{\tau}}|D_L D_{\Lb}\psi_k|^2dud\om\les_{M_2}\mathcal{E}_2[\phi]\tau_+^{-1-\ga_0},\quad k\leq 1.
\end{align*}
The proof of this estimate is similar to that through the outgoing null hypersurface we have done above. To pass the $D_{\Lb}$ derivative to $\psi$, we commute $D_{\Lb}$ with $\psi_1=D_Z\psi$:
\[
|D_{\Lb}[D_{\Lb}, D_Z]\psi|\les |D_{\Lb}\psi|(|r\ab|+|\rho|)+|\psi|(|\Lb\rho|+|L(r\ab)|+|\pa_t(r\ab)|).
\]
Again we can bound $D_{\Lb}\psi$ by using the energy flux and $r\ab$, $\rho$ by the $L^\infty $ norm. For the second term, $\psi$ can be bounded by using Lemma <ref> and the curvature components $\Lb\rho$, $L(r\ab)$ are controlled by using the null structure equations (<ref>), (<ref>). More precisely, we can show that
\begin{align*}
\sum\limits_{k\leq 1}\int_{\Hb_{\tau}}|D_{\Lb}D_Z^k D_{\Lb}\psi|^2dud\om&l\les\int_{\Hb_{\tau}}|D_{\Lb} [D_Z, D_{\Lb}]\psi|^2+|D_{L} D_{\Lb}D_Z^k\psi|^2+|D_{\pa_t}D_{\Lb}D_Z^k\psi|^2dud\om\\
\end{align*}
This estimate and (<ref>) combined with Lemma <ref> imply the pointwise bound (<ref>) for $D_{\Lb}\psi$.
The pointwise bound (<ref>) for $\phi$ follows from Lemma <ref>:
\[
\int_{\om}r^p|D_Z^k\phi|^2(\tau, v, \om)d\om\les_{M_2}\mathcal{E}_k[\phi]\tau_+^{p-2-\ga_0}, \quad k\leq 2, \quad 1\leq p\leq 2
\]
together with Sobolev embedding on the sphere.
§ BOOTSTRAP ARGUMENT
We use bootstrap argument to prove the ourl theorem. In the exterior region, we decompose the full Maxwell field $F$ into the chargeless part and the charge part:
\begin{equation*}
F=\bar F+q_0\chi_{\{r\geq t+R\}} r^{-2}dt\wedge dr.
\end{equation*}
We make the following bootstrap assumptions on the nonlinearity $J_{\mu}=\pa^\nu F_{\nu\mu}=\Im(\phi\cdot\overline{D_\mu\phi})$:
\begin{equation}
\label{eq:btstrap:assum}
m_2\leq 2\mathcal{E}.
\end{equation}
Here recall the definition of $m_2$ in (<ref>). Since the nonlinearity $J$ is quadratic in $\phi$, $m_2$ has size $\mathcal{E}^2$. By assuming small $\mathcal{E}$, we then can improve the above bootstrap assumption and hence conclude our main theorem. The smallness of $\mathcal{E}$ depends on $\mathcal{M}$. Without loss of generality, we assume $\mathcal{E}\leq 1$, $\mathcal{M}>1$.
In the definition (<ref>) for $M_2$, the main contribution is $E_0^2[\bar F]$ with $\bar F$ the chargeless part of the Maxwell field. As the scalar field $\phi$ solves the linear equation $\Box_A\phi=0$, we derive from the definition (<ref>) for $\mathcal{E}_2[\phi]$ that $\mathcal{E}_2[\phi]=E_0^2[\phi]$. The definition for $E_0^k[\bar F]$, $E_0^k[\phi]$ has been given in (<ref>). To proceed, we need to bound $E_0^2[\bar F]$, $\mathcal{E}_2[\phi]$ in terms of $\mathcal{M}$ and $\mathcal{E}$ which is shown in the following lemma.
We can bound $E_0^2[\bar F]$, $E_0^2[\phi]$ as follows:
\begin{equation*}
E_0^2[\bar F]\les \mathcal{M},\quad E_0^2[\phi]\les_{\mathcal{M}} \mathcal{E}.
\end{equation*}
To define the norm $E^k_0[\phi]$, we need to know the connection field $A$ at least on the initial hypersurface $t=0$. As the norm $E_0^k[\phi]$ is gauge invariant, we may choose a particular gauge. Let $\bar A=(A_1, A_2, A_3)(0, x)$, $A_0=A_0(0, x)$. We want to choose a particular connection field $( A_0, \bar A)$ on the initial hypersurface to define the gauge invariant norm $E_0^k[\phi]$ so that we are able to prove this Lemma.
It is convenient to choose Coulomb gauge to make use of the divergence free part $E^{df}$ and the curl free part $E^{cf}$ of $E$. More precisely on the initial hypersurface $t=0$, we choose $( A_0, \bar A)$ so that $\div(\bar A)=0$.
Then the compatibility condition (<ref>) is equivalent to
\[
\Delta A_0=-\Im(\phi_0\cdot \overline{\phi_1})=-J_0(0),\quad \nabla \times \bar A=H.
\]
Define the weighed Sobolev space
\[
W^{p}_{s, \delta}:=\{f|\sum\limits_{|\b|\leq s}\|(1+|x|)^{\delta+|\b|}|\pa^\b f\|_{L^p}<\infty\}.
\]
For the special case $p=2$, let $H_{s, \delta}=W^2_{s, \delta}$. Denote $\tilde{Z}=\{\Om, \pa_j\}$, $\delta=\frac{1+\ga_0}{2}$. By the definition of $\mathcal{M}$:
\[
\|\mathcal{L}_{\tilde{Z}}^k H\|_{H_{0, \delta}}\les \mathcal{M}^{\f12},\quad k\leq 2.
\]
Then from Theorem 0 of <cit.> or Theorem 5.1 of <cit.>, we conclude that
\begin{equation*}
\|\mathcal{L}_{\tilde{Z}}^k\bar A\|_{H_{1, \delta-1}}\les \mathcal{M}^{\f12}, \quad k\leq 2.
\end{equation*}
This is the desired estimate for the gauge field $\bar A$. With this connection field $\bar A$, we then can define the covariant derivative $\tilde{D}=\nabla+\sqrt{-1}\bar A$ in the spatial direction. Therefore
\begin{align*}
\|D\phi(0, \cdot)\|_{H_{0, \delta}}=\|\tilde{D}\phi_0\|_{H_{0, \delta}}+\|\phi_1\|_{H_{0, \delta}}\les \mathcal{E}^{\f12}+\|\bar A\|_{W^3_{0, \delta}}\| \phi_0\|_{W^6_{0, \delta}}\les\mathcal{E}^{\f12}(1+\mathcal{M}^{\f12})\les\mathcal{E}^{\f12}\mathcal{M}^{\f12}.
\end{align*}
By the same argument and commute the equations with $D_{\tilde Z}$, we obtain same estimates for $D_{\tilde{Z}}\phi$:
\begin{equation*}
\|DD_{\tilde{Z}}^k\phi(0, \cdot)\|_{H_{0, \delta}}\les \mathcal{E}^{\f12}\mathcal{M}^{\f12},\quad k\leq 2.
\end{equation*}
To define the covariant derivative $D_0$, we need estimates for $A_0$. The difficulty is the nonzero charge. Take a cut off function $\chi(x)=\chi(|x|))$ such that $\chi=1$ when $|x|\geq R$ and vanishes for $|x|\leq \frac{R}{2}$. Denote the chargeless part of $A_0$ and $J_0$ as follows:
\[
\bar A_0=A_0+\chi q_0r^{-1}, \quad \bar J_0(0):=J_0-\Delta (\chi q_0r^{-1}).
\]
By the definition of the charge $q_0$, we then have
\[
\Delta \bar A_0=-\bar J_0(0),\quad \int_{\mathbb{R}^3}\bar J_0(0)dx=0.
\]
Recall that $J_0(0)=\Im(\phi_0\cdot \overline{\phi_1})$. By using Sobolev embedding, we can bound
\begin{align*}
\|\bar J_0(0)\|_{W^{\frac{3}{2}}_{0, 2\delta}}\les |q_0|+\|\phi_1\|_{W^{2}_{0, \delta}}\|\phi_0\|_{W^6_{0, \delta}}\les |q_0|+\|\phi_1\|_{W^{2}_{0, \delta}}\|\phi_0\|_{W^2_{1, \delta}}\les \mathcal{E}.
\end{align*}
Then from Theorem 0 of <cit.>, we conclude that
\begin{equation*}
\|\bar A_0\|_{W^{\frac{3}{2}}_{2, 2\delta-2}}\les \mathcal{E}.
\end{equation*}
Here the condition that $\bar A_0$ is chargeless guarantees $\bar A_0$ to belong to the above weighted Sobolev space. Then by using Gagliardo-Nirenberg interpolation inequality, we derive that
\begin{align*}
\|\nabla\bar A_0\|_{H_{0, 2\delta-\frac{1}{2}}}\les \|\nabla \bar A_0\|_{W^{\frac{3}{2}}_{0, 2\delta-1}}^\f12\cdot\|\nabla\nabla \bar A_0\|_{W^{\frac{3}{2}}_{0, 2\delta}}^\f12\les \mathcal{E} .
\end{align*}
By definition $E=\pa_t \bar A-\nabla A_0$. By our gauge choice, $\pa_t \bar A$ is divergence free, $\nabla A_0$ is curl free. In particular we derive that $E^{df}=\pa_t \bar A$, $E^{cf}=-\nabla A_0$. Take the chargeless part. We obtain that $\bar E^{cf}=\nabla \bar A_0$ when $|x|\geq R$. Therefore we can bound the weighted Sobolev norm of the chargeless part of the Maxwell field $\bar F$ on the initial hypersurface as follows:
\begin{align*}
\|\bar F\|_{H_{0, \delta}}&\leq \|F\chi_{\{|x|\leq R\}}\|_{H_{0, \delta}}+\|(\bar E, H)\chi_{\{|x|\geq R\}}\|_{H_{0, \delta}}\\
&\les \|F\chi_{\{|x|\leq R\}}\|_{H_{0, \delta}}+\|(E^{df}, H)\chi_l{\{|x|\geq R\}}\|_{H_{0, \delta}}+\|\bar E^{cf}\chi_{\{|x|\geq R\}}\|_{H_{0, \delta}}\\
&\les \mathcal{M}^{\f12}+\|\nabla\bar A_0\|_{H_{0, \delta}}\les \mathcal{M}^{\f12}.
\end{align*}
Similarly we have the same estimates for $\mathcal{L}_{\tilde Z}^k \bar F$, $k\leq 2$, that is
\begin{equation*}
\|\mathcal{L}_{\tilde Z}^k\bar F\|_{H_{0, \delta}}\les \mathcal{M}^{\f12},\quad k\leq 2.
\end{equation*}
To derive estimates for $D_{Z}^k\phi$, $\mathcal{L}_{Z}^k \bar F$ on the initial hypersurface, we use the equations:
\[
\pa_t E-\nabla\times H=\Im(\phi\cdot \tilde{D}\phi),\quad \pa_t H+\nabla\times E=0,\quad D_t\phi_1=\tilde{D}\tilde{D}\phi
\]
to replace the time derivatives with the spatial derivatives. The inhomogeneous term $\Im(\phi\cdot \tilde{E}\phi)$ or the commutators $[D_t, \tilde{D}]$ could be controlled by using Sobolev embedding together with Hölder's inequality. The lemma then follows.
The above lemma then leads to the following:
Let $(\phi, F)$ be the solution of (<ref>). Under the bootstrap assumption (<ref>), we have
\begin{equation*}
M_2\les \mathcal{M},\quad \mathcal{E}_2[\phi]\les_{\mathcal{M}} \mathcal{E}.
\end{equation*}
The corollary follows from the definition of $M_2$, $\mathcal{E}_2[\phi]$ in line (<ref>), (<ref>) together lemma <ref>.
From now on, we allow the implicit constant $\les$ also depends on $\mathcal{M}$, that is, $B\les K$ means that $B\leq C K$ for some constant $C$ depending on $\ga_0$, $R$, $\ep$ and $\mathcal{M}$. The rest of this section is devoted to improve the bootstrap assumption.
To improve the bootstrap assumption, we need to estimate $m_2$ defined in line (<ref>). On the finite region $r\leq R$, the null structure of $J$ is not necessary as the weights of $r$ is bounded above. When $r\geq R$, the null structure of $J$ plays a crucial role. Note that $J_{L}$, $\J=(J_{e_1}, J_{e_2})$ are easy to control as they already contain "good" terms $\D\phi$ or $D_L(r\phi)$. The difficulty is to exploit the null structure of the component $J_{\Lb}$ which is not a standard null form defined in <cit.>, <cit.>. The null structure of the system is that $J_{\Lb}$ does not interact with the "bad" component $\ab$ of the Maxwell field.
For nonnegative integers $k$, denote $\phi_k=D_Z^k\phi$, $\psi_k=r\phi_k$, $F_k=\mathcal{L}_Z^k$ in this section. First we expand the second order derivative of $J=\Im(\phi\cdot \overline{D\phi})$.
Let $X$ be $L$, $\Lb$, $e_1$, $e_2$. Then we have
\begin{equation*}
\begin{split}
|\mathcal{L}_Z^2 J|+|\nabla\mathcal{L}_Z J|&\les |D\phi_1||D\phi|+|\phi_1||D^2\phi|+|\phi||D^2\phi_{1}|+|\pa F||\phi|^2+|F||D\phi||\phi|,\quad |x|\leq R;\\
r^2|\mathcal{L}_Z^2 J_X|&\les \sum\limits_{k\leq 2} |\psi_k||D_X\psi_{2-k}|+\sum\limits_{l_1+l_2+l_3\leq 1}|\mathcal{L}_Z^{l_1} F_{ZX}||\psi_{l_2}||\psi_{l_3}|,\quad |x|>R.
\end{split}
\end{equation*}
By the definition of the Lie derivative $\mathcal{L}_Z$, we can compute
\begin{align*}
\mathcal{L}_Z J_X= Z(J_X)-J_{\mathcal{L}_Z X}&=\Im(D_Z\phi \cdot \overline{D_X\phi}+\phi\cdot\overline{D_Z D_X\phi}-\phi\cdot \overline{D_{[Z, X]\phi}})\\
&=\Im(\phi_1 \cdot \overline{D_X\phi}+\phi\cdot\overline{D_X\phi_1}+\phi\cdot \overline{([D_Z, D_X]-D_{[Z, X]})\phi})\\
\end{align*}
Here we note that $[D_Z, D_X]-D_{[Z, X]}=\sqrt{-1}F_{ZX}$ for any vector fields $Z$, $X$ and we omitted the summation sign for $l=0,\, 1$. Take one more derivative $\nabla$. The estimate on the region $\{r\leq R\}$ then follows.
Similarly the second order derivative expands as follows:
\begin{align*}
\mathcal{L}_Y\mathcal{L}_Z J_X&= Y \mathcal{L}_Z J_X- \mathcal{L}_Z J_{[Y, X]}\\
&= Y\Im(\phi_l\cdot\overline{D_X}\phi_{1-l})-Y (F_{ZX}|\phi|^2)-\Im(\phi_l\cdot\overline{D_{[Y,X]}}\phi_{1-l})+F_{Z[Y, X]}|\phi|^2\\
&= \Im(\phi_k\cdot\overline{D_X}\phi_{2-k})-(\mathcal{L}_Y F_{ZX}+F_{[Y, Z]X})|\phi|^2+\Im(\sqrt{-1}\phi_l\cdot\overline{F_{YX}\phi_{1-l}})-F_{ZX}Y|\phi|^2
\end{align*}
for any $Y\in Z$. Here we omitted the summation sigh for $k=0, 1, 2$ and $l=0, 1$. Note that
\[
\Im(\phi\cdot \overline{D_X\phi})=r^{-2}\Im(r\phi\cdot \overline{D_X (r\phi)}),\quad [Y, Z]=0 \textnormal { or }\in Z.
\]
The estimate on the region $r\geq R$ then follows. We thus finished the proof of the lemma.
Next we use the above bound for $J$ to improve the bootstrap assumption.
We have
\begin{equation}
\label{eq:btstrap:imp}
m_2\leq C \mathcal{E}^2
\end{equation}
for some constant $C$ depending on $\mathcal{M}$, $\ep$, $R$ and $\ga_0$.
Since $M_2\les \mathcal{M}$, all the estimates in the previous section hold. In particular we have the energy flux and the $r$-weighted energy decay estimates for the scalar field and the chargeless part of the Maxwell field up to second order derivatives. Moreover the pointwise estimates in Proposition <ref>, <ref>, <ref>, <ref> hold.
Let's first consider the estimate of $|J_{\Lb}|r^{-2}$ on the exterior region. We have the simple bound that $|J_{\Lb}|\leq |D_{\Lb}\phi||\phi|$. We can control $D_{\Lb}\phi$ by using the energy flux through the incoming null hypersurface $\Hb_{v}$ and $\phi$ by the $L^\infty$ norm. In particular for any $\tau<0$ we can show that
\begin{align*}
\iint_{\mathcal{D}_{\tau}^{-\infty}}|J_{\Lb}|r^{-2}dxdt&\les\int_{-\tau^*}^{\infty}\left(\int_{\Hb_{v}}|D_{\Lb}\phi|^2r^2 dud\om\right)^{\f12}\cdot \left(\int_{\Hb_{v}}|\phi|^2r^{-2}dud\om\right)^{\f12}dv \\
&\les \mathcal{E}\int_{-\tau^*}^{\infty}\tau_+^{-\frac{1+\ga_0}{2}}\left(\int_{\Hb_{v}}r^{-4}\tau_+^{-\ga_0}dud\om\right)^{\f12}dv\\
&\les\mathcal{E}\int_{-\tau^*}^{\infty}\tau_+^{-\frac{1+2\ga_0}{2}}r^{-\frac{3}{2}}dv\les \mathcal{E}\tau_+^{-1-\ga_0}.
\end{align*}
We remark here that we can not use the integrated local energy to bound the above term due to the exact total decay rate of $|J_{\Lb}|r^{-2}$. As the charge $|q_0|\les \mathcal{E}$, we therefore obtain
\[
|q_0|\sup\limits_{\tau\leq 0}\tau_+^{1+\ga_0}\iint_{\mathcal{D}_{\tau}^{-\infty}}|J_{\Lb}|r^{-2}dxdt\les \mathcal{E}^2,\quad \forall \tau\leq 0.
\]
Next we consider the estimates on the compact region $r\leq 2R$. As $|\phi_1|=|D_Z\phi|\les |D\phi|$ when $|x|\leq R$, we can bound $\phi_1$, $\phi$, $D\phi$ and $F$ by the $L^\infty$ norm obtained in (<ref>), (<ref>). Then $D^2\phi_1$, $\pa F$ can be controlled by using the integral decay estimates (<ref>), (<ref>) on $r\leq R$. To derive estimates for $D^2\phi_k$ or $\pa F$ on the region $\{R\leq r\leq 2R\}$, we use the equation (<ref>). From Lemma <ref> and Lemma <ref>, we can show that
\begin{align*}
|D^2\phi_1|+\mathcal{E}|\pa F|&\les |D\phi_2|+|D_{L}D_{\Lb}\psi_1|+|F||\phi_1|+\mathcal{E}(|\mathcal{L}_Z F|+|L(r^2\rho, r^2\si, r\ab)|+|L(r\ab)|)\\
&\les |D\phi_2|+|\Box_A\phi_1|+|F||\phi_1|+\mathcal{E}(|\mathcal{L}_Z F|+|J|).
\end{align*}
Here we omitted the easier lower order terms. On the region $\{R\leq R\leq 2R\}$, $Z$ only miss one derivative which could be recovered from the equation. From Lemma <ref>, we can show that
\begin{align*}
&I^0_{1+\ga_0+2\ep}[|\mathcal{L}_Z^2 J|+|\nabla \mathcal{L}_Z J|](\{r\leq 2R\})\\
&\les \mathcal{E}\int_{0}^{\infty}\tau_+^{2\ep}\int_{r\leq 2R}|D^2\phi_1|^2+\mathcal{E}|\pa F|^2+|D\phi|^2 dxd\tau\\
&\les \mathcal{E}^2+\mathcal{E}\int_{0}^{\infty}\tau_+^{2\ep}\int_{R\leq r\leq 2R}|D\phi_2|^2+|\Box_A\phi_1|^2+|F|^2|\phi_1|^2+\mathcal{E}(|\mathcal{L}_Z F|^2+|J|^2)dxd\tau\\
&\les \mathcal{E}^2+\mathcal{E}I^{0}_{2\ep}[\Box_A\phi_1](\{r\geq R\})+\mathcal{E}^2I^{0}_{2\ep}[J](\{r\geq R\})\les \mathcal{E}^2.
\end{align*}
Here the implicit constant also depends on $\mathcal{M}$ and we only consider the highest order terms. The second last step follows as the integral from time $\tau_1$ to $\tau_2$ decays in $\tau_1$. Hence the spacetime integral is bounded by using Lemma <ref>. The bound for $\Box_A\phi_1$ follows from Proposition <ref> and the spacetime norm for $J$ is controlled by the bootstrap assumption.
Next, we consider the case when $|x|\geq R$ where the null structure of $J$ plays a crucial role.
For $|\mathcal{L}_Z^2 J_{\Lb}|$, Lemma <ref> implies that
\[
r^2|\mathcal{L}_Z^2 J_{\Lb}|\les |\psi_k||D_{\Lb}\psi_{2-k}|+(|r\mathcal{L}_Z^{l_1}\ab|+|\mathcal{L}_Z^{l_1}\rho|)|\psi_{l_2}||\psi_{1-l_1-l_2}|.
\]
Here the indices $k$, $2-k$, $1-l_1-l_2$, $l_1$, $l_2$ are nonnegative integers and we only consider the highest order term as the lower order terms are easier and could be bounded in a similar way. On the right hand side of the above inequality, after using Sobolev embedding on the sphere, we can bound $|\psi|$ by using Lemma <ref> and $D_{\Lb}\psi$, $|\ab|$, $\rho$ by using the integrated local energy estimates. In deed we can show that
\begin{align*}
&I^{1-\ep}_{1+\ga_0+2\ep}[\mathcal{L}_Z^2 J_{\Lb}](\{r\geq R\})=\int_{\tau}\int_{H_{\tau^*}}r_+^{-\ep-1}\tau_+^{1+\ga_0+2\ep}|r^2\mathcal{L}_Z^2J_{\Lb}| ^2dud\om d\tau\\ &\les\int_{\tau}\int_{v}r_+^{-1-\ep}\tau_+^{1+\ga_0+2\ep} \int_{\om}|\psi_2|^2d\om\cdot \int_{\om}|D_{\Lb}\psi_{2}|^2d\om +\int_{\om}|r\mathcal{L}_Z^{2}\ab|^2+|\mathcal{L}_Z^{2}\bar\rho|^2d\om(\int_{\om}|\psi_{2}|^2d\om)^2 dv d\tau\\
&\qquad+\int_{\tau\leq 0}\int_{v}r^{-1-\ep}|q_0|^2r^{-4} \tau_+^{1+\ga_0+2\ep}(\int_{\om}|\psi_{2}|^2d\om)^2 dvd\tau\\
&\les \mathcal{E}\int_{\tau}\tau_+^{1+2\ep}\int_{H_{\tau^*}}\frac{|\bar D\phi_2|^2}{r_+^{1+\ep}}dxd\tau+\mathcal{E}^2\int_{\tau}\tau_+^{1+2\ep}\int_{H_{\tau^*}}\frac{|\mathcal{L}_Z^2 \bar F|^2}{r_+^{1+\ep}}dxd\tau
+\mathcal{E}^2|q_0|^2\int_{\tau\leq 0}\int_{v}r_+^{-4+\ep+\ga_0}dvd\tau\\
&\les \mathcal{E} I^{-1-\ep}_{1+2\ep}[\bar D \phi_2](\{t\geq 0\})+\mathcal{E}^2 I^{-1-\ep}_{1+2\ep}[\mathcal{L}_Z^2\bar F](\{t\geq 0\})+|q_0|^2\mathcal{E}^2\les \mathcal{E}^2.
\end{align*}
Here after using Sobolev embedding on the sphere, we dropped the lower order terms like $\psi_1$, $\psi$. In the above estimate, we have used the decay estimates $\int_{\om}|\psi_k|^2d\om\les \mathcal{E}\tau_+^{-\ga_0}$ by Lemma <ref>. The last step follows from the integrated local energy decay (see e.g. estimate (<ref>)) and Lemma <ref>. We also note that in the exterior region $r_+\geq \frac{1}{2}\tau_+$.
For $J_{L}$, Lemma <ref> indicates that
\[
r^2|\mathcal{L}_Z^2 J_{L}|\les |\psi_k||D_L\psi_{2-k}|+(|r\mathcal{L}_Z^{l_1}\a|+|\mathcal{L}_Z^{l_1}\rho|)|\psi_l{l_2}||\psi_{1-l_1-l_2}|.
\]
Similarly, after using Sobolev embedding, we control $\psi_k$ by using Lemma <ref>. Then for $D_{L}\psi_k$, $|\mathcal{L}_Z^k\a|$ we can apply the $r$-weighted energy estimates. For $\rho$, we split it into the charge part $q_0r^{-2}$ and the chargeless part which can be bounded by using the energy flux decay estimates. More precisely for $\ep\leq p\leq 1+\ga_0$ we can show that
\begin{align*}
&I^{1+p}_{1+\ga_0+\ep-p}[\mathcal{L}_Z^2 J_{L}](\{r\geq R\})=\int_{\tau}\int_{H_{\tau^*}}r_+^{p-1}\tau_+^{1+\ga_0+\ep-p}|r^2\mathcal{L}_Z^2J_{L}| ^2dvd\om d\tau\\ &\les\mathcal{E}\int_{\tau}\int_{H_{\tau^*}}r_+^{p}\tau_+^{\ep-p} |D_{L}\psi_{2}|^2d\om dvd\tau +\mathcal{E}^2\int_{\tau}\int_{H_{\tau^*}}r_+^{p}\tau_+^{\ep-p-\ga_0}(|r\mathcal{L}_Z^{2}\a|^2+|\mathcal{L}_Z^{2}\bar\rho|^2)d\om dv d\tau\\
&\qquad+\mathcal{E}^2\int_{\tau\leq 0}\int_{v}r^{p-1}|q_0|^2r^{-4} \tau_+^{1-\ga_0+\ep-p} dvd\tau\\
&\les \mathcal{E}^2\int_{\tau}\tau_+^{\ep-p-1-\ga_0+p}+\tau_+^{\ep-p-\ga_0-1-\ga_0+p}+\tau_+^{\ep-p-1-\ga_0}d\tau+\mathcal{E}^2|q_0|^2\les \mathcal{E}^2.
\end{align*}
Here for $\psi_k$, we have used the estimate $r^{-1}\int_{\om}|\psi_k|^2d\om \les \mathcal{E}\tau_+^{-1-\ga_0}$.
Next for $\J$, Lemma <ref> shows that
\[
r^2|\mathcal{L}_Z^2 \J|\les |\psi_k||\D\psi_{2-k}|+(|r\mathcal{L}_Z^{l_1} \si|+|\mathcal{L}_Z^{l_1} \a|+|\mathcal{L}_Z^{l_1} \ab|)|\psi_{l_2}||\psi_{1-l_1-l_2}|.
\]
Like the previous estimates for $J_{\Lb}$, $J_{L}$, for all $\ep\leq p\leq \ga_0$ we can show that
\begin{align*}
&I^{1+p}_{1+\ga_0+\ep-p}[\mathcal{L}_Z^2 \J_{L}](\{r\geq R\})=\int_{\tau}\int_{H_{\tau^*}}r_+^{p-1}\tau_+^{1+\ga_0+\ep-p}|r^2\mathcal{L}_Z^2 \J|^2dvd\om d\tau\\ &\les\mathcal{E}\int_{\tau}\int_{H_{\tau^*}}r_+^{p}\tau_+^{\ep-p} |\D\psi_{2}|^2d\om dvd\tau +\mathcal{E}^2\int_{\tau}\int_{H_{\tau^*}}r_+^{p}\tau_+^{\ep-p-\ga_0}(|r\mathcal{L}_Z^{l_1} \si|^2+|\mathcal{L}_Z^{l_1} (\ab,\a)|^2)d\om dv d\tau\\
&\les\mathcal{E}\int_{\tau}\int_{H_{\tau^*}}r_+^{\ga_0} (|\D\psi_{2}|^2+\mathcal{E}|r\mathcal{L}_Z^{l_1} (\si,\a)|^2)d\om dvd\tau +\mathcal{E}^2\int_{\tau}\int_{H_{\tau^*}}r_+^{1-\ep}|\mathcal{L}_Z^{l_1} \ab|^2d\om dv d\tau\\
&\les \mathcal{E}^2.
\end{align*}
Here $l_1\leq 1$. The last term is bounded by using the integrated local energy estimates. This relies on the assumption that $\ga_0\leq 1-\ep<1$. For $\ga_0\geq 1$, we then can use the improved integrated local energy estimate for the angular derivatives of $\phi$ or $\si$ or we can move the $r$ weights to $\phi_k$.
Combining the above estimates, we have (<ref>).
By choosing $\mathcal{E}$ sufficiently small depending only on $\mathcal{M}$, $\ep$, $R$ and $\ga_0$, we then can improve the bootstrap assumption (<ref>). To prove theorem <ref>, we can choose $R=2$. Then for sufficiently small $\mathcal{E}$, we can bound $m_2$ and $M_2$. The pointwise estimates in the main Theorem <ref> follow from Propositions <ref>, <ref>, <ref>, <ref>.
DPMMS, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, UK CB3 0WA
Email address: [email protected]
|
1511.00194
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Bridy et al.]Andrew Bridy, Patrick Ingram, Rafe Jones, Jamie Juul, Alon Levy, Michelle Manes, Simon Rubinstein-Salzedo, and Joseph H. Silverman
University of Rochester, Rochester, New York
Colorado State University, Fort Collins, Colorado
[email protected][Corresponding author.]
Carleton College, Northfield, Minnesota
Amherst College, Amherst, Massachusetts
KTH Royal Institute of Technology, Stockholm, Sweden
University of Hawaii, Honolulu, Hawaii
Stanford University, Stanford, CA
Brown University, Providence, Rhode Island
The authors would like to thank AIM and the organizers of the March
2014 AIM workshop on Postcritically finite maps in complex and
arithmetic dynamics at which this research was started.
Bridy's research partially supported by NSF Grant #EMSW21-RTG.
Ingram's research partially supported by Simons Collaboration Grant #283120.
Juul's research partially supported by DMS-1200749.
Levy's research partially supported by the Göran Gustafsson Foundation.
Manes's research partially supported by NSF DMS #1102858.
Silverman's research partially supported by Simons Collaboration Grant #241309.
Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$ generated by the preimages of $\alpha$ under all iterates of $\varphi$. In particular when $\varphi$ is post-critically finite, i.e. there exists a non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$ and $\varphi : W \to X$ is étale, we prove that $K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire <cit.> in the case $X = \mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes <cit.> in the case $X = \mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a local argument.
§ INTRODUCTION
For a number field $K$ and a finite set $S$ of places of $K$, the arithmetic fundamental group $\Gal(K_S/K)$ is the Galois group of the maximal algebraic extension of $K$ unramified outside of $S$. These groups are objects of great number-theoretic interest, and well-known quotients of them arise through algebraic geometry, for example in the form of linear representations attached to abelian varieties. Recently another approach to constructing such quotients has arisen, one that relies on arithmetic dynamical systems. It was first observed in <cit.> that adjoining iterated pre-images of $\alpha \in K$ under a polynomial $f \in K[x]$ gave rise to infinite extensions unramified outside a finite set of primes, provided that $f$ is post-critically finite, i.e., the forward orbit of each critical point of $f$ is finite. In <cit.> and <cit.> this approach was extended to the case where $f$ is a rational function. In this paper, we extend this result even further, to the case where $f$ is allowed to be any finite endomorphism of a smooth, irreducible variety, and $K$ is allowed to be the fraction field of any Dedekind domain. We conjecture, however, that this finite ramification phenomenon for arithmetic dynamical systems is limited to those that are post-critically finite, and we prove this in the case where $f$ is a rational function.
We begin by extending the
definition of post-critical finiteness of a rational function to the case of endomorphisms of more general varieties.
Let $X$ be an irreducible, smooth variety, and let $\varphi:X\to X$ be a finite morphism. We say that $\varphi$ is post-critically finite (PCF) if there is a non-empty Zariski-open $W\subseteq X$ such that $\varphi^{-1}(W)\subseteq W$, and such that $\varphi:W\to X$ is étale.
Note that a PCF map must be separable, since an inseparable morphism is nowhere étale. In the case $X=\PP^N$, over the complex numbers, it is easy to see that Definition <ref> is equivalent to the usual one, namely that if $C_\varphi\subseteq \PP^N_\CC$ is the critical locus of $\varphi$, then the sequence $\varphi^n(C_\varphi)$ is supported on a finite collection of irreducible subvarieties of $\PP^N_\CC$ as $n\to\infty$. Indeed, this latter condition holds if and only if the union
\begin{equation} \label{union}
\bigcup_{n > 0} \varphi^n(C_\varphi)\subseteq\PP^N_\CC
\end{equation}
is an algebraic subvariety of $\PP^N_\CC$, and thus the condition in Definition <ref> is satisfied, where we take $W$ to be the complement of the union in (<ref>).
Although the definition of PCF maps is purely geometric, PCF maps have special arithmetic properties besides the finite ramification of preimage fields explored in this article. See <cit.> for some recent examples.
In this paper, we consider the primes that ramify in extensions obtained by taking backward images of $K$-points of $X$, where we typically take $K$ to be a global field.
Let $K$ be any field, and let $\varphi: X \to X$ be defined over $K$. If $\alpha \in X(K)$ and $n$ is a positive integer, we set $K_{n, \varphi}(\alpha) = K(\varphi^{-n}(\alpha))$, and $K_{\infty, \varphi}(\alpha) = \bigcup_{n > 0}K(\varphi^{-n}(\alpha))$. If $K$ is a global field, we set $S_{n, \varphi}(\alpha)$ to be the set of primes of $K$ that ramify in $K_{n, \varphi}(\alpha)$ for $n \in \N \cup \{\infty\}$. When $\varphi$ is clear from context, we suppress it and write $K_n(\alpha)$ and $S_n(\alpha)$.
Our first theorem says that if a map is PCF, then the field generated by the full iterated backward orbit of an algebraic point is ramified at only finitely many primes.
Let $X$ be a smooth, irreducible, projective variety defined over the field of fractions $K$ of a Dedekind domain $R$, and suppose that $\varphi: X \to X$ is a PCF morphism defined over $K$. Then there exists a non-empty Zariski-open $W\subseteq X$ such that the following holds: for any $\alpha \in W(K)$, $S_\infty(\alpha)$ is a finite set.
An equivalent way of stating Theorem <ref> is that the ramification of the associated so-called arboreal representation (see <cit.>) has finite support.
Theorem <ref> is in the same spirit as Beckmann's theorem <cit.> on ramified primes in specializations of branched covers of $\PP^1$. It is also interesting to consider the dependence of $S_\infty(\alpha)$ on the point $\alpha$. The existence of primes that ramify independently of the choice of $\alpha$ is related to ramification in the field of moduli of $\phi$. See <cit.>, <cit.>, and <cit.> for further discussions. For certain maps it is possible to show that there are primes (for example primes of bad reduction) that lie in $\bigcap_{\alpha\in W(K)}S_\infty(\alpha)$. It is not true in general that $\bigcup_{\alpha\in W(K)}S_\infty(\alpha)$ is a finite set (see Examples <ref> and <ref>).
The subvariety $W\subseteq X$ in Theorem <ref> can, at worst, be taken to be the $W$ from Definition <ref> (i.e., the complement of the postcritical set), as shown in the proof of Theorem <ref>. In the case $X=\PP^1$, we show in Theorem <ref> that we may in fact take $W = X$, thus reproducing in this case the existing results in the literature (<cit.>). Levy and Tucker <cit.> have announced a proof of a related conjecture about the behavior of PCF morphisms under restriction that would imply that Theorem <ref> is true with $W = X$.
We conjecture that the finite ramification of preimage fields given by Theorem <ref> in fact characterizes PCF morphisms:
Let $X$ be a smooth, irreducible, projective variety defined over the field of fractions $K$ of a Dedekind domain $R$. Suppose that $\varphi: X \to X$ is a non-PCF morphism defined over $K$, $\alpha \in X(K)$, and $\bigcup_{n > 0}\varphi^{-n}(\alpha)$ is an infinite set. Then $S_\infty(\alpha)$ is an infinite set.
If $\bigcup\varphi^{-n}(\alpha)$ is not Zariski-dense then we say that $\alpha$ is an exceptional point for $\varphi$. For $X = \PP^1$ this means that $\bigcup\varphi^{-n}(\alpha)$ is a finite set, which occurs only when, after coordinate change, $\alpha = \infty$ and $\varphi$ is a polynomial, or when $\alpha \in \{0, \infty\}$ and $\varphi(z) = z^d$. If $\dim X > 1$ and $\alpha$ is exceptional for $\varphi$ but $\bigcup_{n > 0}\varphi^{-n}(\alpha)$ is infinite, then the Zariski-closure of $\bigcup_{n > 0}\varphi^{-n}(\alpha)$ is a positive-dimension subvariety $Y$ that is fixed under $\varphi$, and $\alpha$ is not exceptional under $\varphi|_Y$.
Conjecture <ref> is known only in the case where $K$ is a function field over a number field $k$, $\varphi$ is a univariate polynomial defined over $k$, and $\alpha$ is transcendental over $k$ <cit.>. We prove the following:
Conjecture <ref> holds when $K$ is a number field and $X = \PP^1$.
We also raise a second conjecture here: in Theorem <ref>, we get an infinite field extension of $K$ that is ramified over finitely many primes of $K$. It is a longstanding problem to study extensions of number fields with restricted tame ramification, and it would be interesting to have a dynamical source of such extensions. We conjecture, however, that this is not possible:
In the situation of Theorem <ref>, there always exists at least one prime of $K$ in $S_\infty(\alpha)$ where the ramification is wild.
§ PROOF OF THEOREM <REF>
Before giving the proof of Theorem <ref>, we state and prove an appropriate version of the Chevalley-Weil Theorem.
Let $\varphi: \mathscr{X} \to \mathscr{Y}$ be a finite étale map of schemes defined over $A$, and let $P\in \mathscr{Y}(A)$. Then there is a finite, unramified extension $B/A$ of rings and a point $Q\in\mathscr{X}(B)$ such that $f(Q)=P$.
First, let $\mathscr{Z}=\mathscr{X}\times_{\mathscr{Y}}\Spec(A)$ be the fiber product relative to the maps $\varphi:\mathscr{X}\to\mathscr{Y}$ and $P:\Spec(A)\to\mathscr{Y}$. Since $P$ and $\varphi$ are finite, so is the map $\mathscr{Z}\to\Spec(A)$, and hence $\mathscr{Z}=\Spec(B)$ for some finite $A$-module $B$ <cit.>. By definition, the fiber product gives a map $\Spec(B)\to \scX$, i.e., a point $Q\in \scX(B)$ with $P=\varphi(Q)$. Here, we abuse notation and use $P$ to refer to the canonical map $\Spec(B)\to\mathscr{Y}$ induced by composing the canonical map $\Spec(B)\to\Spec(A)$ with $P:\Spec(A)\to\mathscr{Y}$. Finally, the fact that $B/A$ is unramified follows from the fiber product of étale morphisms being étale.
Let $R$ be a Dedekind domain, with fraction field $K$, let $X/K$ be an irreducible, smooth, projective variety, let $\varphi:X\to X$ be a PCF morphism defined over $K$, let $W\subseteq X$ be a set as in Definition <ref>, and let $\alpha\in W(K)$. We now describe a finite set of primes which will contain all primes ramifying in any of the extensions $K_n/K$. Specifically, let $S\subseteq \Spec(R)$ be the set of places $\mfp$ for which $X$ has bad reduction modulo $\mfp$, $\varphi$ has bad reduction modulo $\mfp$, the ramification subscheme of $\varphi$ has a fibral component modulo $\mfp$, or $\alpha$ collides with the complement of $W$ modulo $\mfp$. In other words, if $R_S$ is the localization of $R$ at $S$, then $X$ is the generic fiber of a scheme $\scX\to\Spec(R_S)$, $W$ is the generic fiber of a scheme $\scW\to\Spec(R_S)$, $\alpha$ extends to a morphism $\widetilde{\alpha}:\Spec(R_S)\to \scW$, and $\varphi$ extends to an étale morphism $\varphi:\scW\to\scX$ over $R_S$. Now, note that if $\scU=\varphi^{-n}(\scW)$, then since the inclusion map $\scU\to\scW$ is étale, so is the map $\varphi^n:\scU\to\scW$.
Theorem <ref> is the application of the Chevalley-Weil Theorem to the point $\widetilde{\alpha}\in \scW(R_S)$ and the map $\varphi^n:\scU\to\scW$.
The ramification subscheme of $\varphi$ mod $\p$ has a fibral component if and only if the reduction of $\varphi$ mod $\p$ is inseparable.
If $X = \PP^1$ then Theorem <ref> holds with $W = X$.
The proof of Theorem <ref> explicitly constructs $W$ to be $X$ minus the postcritical set of $\varphi$, which when $X = \PP^1$ and $\varphi$ is PCF is a finite set of points. The critical locus $C_\varphi$ consists of a finite set of points, and there exists some constant $m$ such that if $\beta \in C_\varphi$ then $\alpha = \varphi^i(\beta)$ for some $i \leq m$ or $\alpha$ is not in the forward orbit of $\beta$.
Suppose first that $\alpha$ is not a periodic point of $\varphi$. Then $\varphi^{-m}(\alpha)$ consists of a finite set of points, none of which is of the form $\varphi^i(\beta)$ for any $\beta \in C_\varphi$. Thus, $K_\infty(\alpha)/K_m(\alpha)$ is finitely ramified. $K_m(\alpha)/K$ is finitely ramified since it is a finite extension, and therefore $K_\infty(\alpha)/K$ is finitely ramified, as required.
If $\alpha$ is periodic, then we first observe that $K_n(\alpha) \subseteq K_m(\alpha)$ if $n < m$, and therefore
$$K_{\infty, \varphi}(\alpha) = \bigcup_{i = 1}^\infty K_{i, \varphi}(\alpha) = \bigcup_{i = 1}^\infty K_{ni, \varphi}(\alpha) = K_{\infty, \varphi^n}(\alpha)$$
for every $n \geq 1$. Since $K_{\infty, \varphi}(\alpha) = K_{\infty, \varphi^n}(\alpha)$, we may freely replace $\varphi$ with an iterate, and in particular we may assume that $\alpha$ is fixed. But now $K_n(\alpha)$ is the compositum of $K_{n-1}(\beta)$ over all $\beta \in \varphi^{-1}(\alpha) \setminus \{\alpha\}$. Since $\beta$ is not periodic, $K_\infty(\beta)$ is finitely ramified, and this implies $K_\infty(\alpha)$ is finitely ramified.
§ EXAMPLES
The proof of Theorem <ref> is sufficiently explicit that a set of primes outside of which the preimage extensions are unramified can often be presented with relative ease.
Dupont <cit.> produces an example of a quadratic map $\varphi:\PP^2\to\PP^2$ which is post-critically finite, namely
\[\varphi[x:y:z]=\left[(x-y+z)^2:(x+y-z)^2:(-x+y+z)^2\right].\]
One checks that the post-critical locus of this morphism is precisely that defined by
\[xyz(x-y)(y-z)(z-x)=0.\]
Thus, in the proof of Theorem <ref>, we may take $W\subseteq\PP^2$ to be the complement of this divisor.
Given a number field $K$, and a point $P\in W(K)$, let $S$ be any set of places of $K$ containing all places above 2, above the coordinates of $P$, and above their pairwise differences. Then if $\mathcal{O}_{K, S}$ is the set of $S$-integers of $K$, the morphism $\varphi:W\to W$ is the generic fiber of an étale map $\varphi:\mathcal{W}\to \mathcal{W}$ of $\mathcal{O}_{K, S}$-schemes, and $P$ extends to a point $P\in \mathcal{W}(\mathcal{O}_{K, S})$. It follows from the proof of Theorem <ref> that the extensions $K(\varphi^{-n}(P))$ are all unramified above primes outside of $S$.
Incidentally, we note that, given a set $S$, there are only finitely many points $P\in W(K)$ for which this set of places suffices, which in this case follows from the finiteness of solutions to $S$-unit equations. More generally, if $W\subseteq\PP^2$ is an affine open set containing infinitely many $S$-integral points, this places substantial restrictions on the complement $\PP^2\setminus W$ (in our case, the postcritical set); see, e.g., work of Corvaja and Zannier <cit.>.
Observe that the periodic postcritical components are $x = y$, $y = z$, and $z = x$, which $\varphi$ permutes in a $3$-cycle. Let us look at how $\varphi^3$ acts on these components. Since $\varphi$ has a $\Z/3\Z$-automorphism by the coordinate change $x \mapsto y \mapsto z \mapsto x$, it suffices to look at the action of $\varphi^3$ on just one of these components. Let us look at the component $x = y$, and let us dehomogenize by setting $z = 1$. It is a straight forward computation that
$$\varphi^3(x) = \frac{(4x^2 - 4x - 1)^4}{(16x^4 - 32x^3 + 40x^2 - 24x + 1)^2}$$
There are $14$ critical points, counted with multiplicity: the two roots of $4x^2 - 4x - 1$ are quadruple zeros and triple critical points, the roots of $16x^4 - 32x^3 + 40x^2 - 24x + 1$ are double poles and simple critical points, $\infty$ is a simple critical point (the top two terms of both numerator and denominator are the same), and after factoring those out we obtain $1/2$ as a triple critical point. $0, 1/2, \infty$ all map to the fixed point $1$. More specifically, the critical values are $0, 1, \infty$, which are precisely the intersections of the other postcritical components with $x = y$. As expected based on the in-progress work in <cit.>, the restriction of $\varphi^3$ to $x = y$ is PCF. If we pull back a point on this line, then again we obtain finite ramification.
As another example, consider the generalized Tchebyshev map $\varphi:\PP^2\to\PP^2$ given by
\[\varphi[x:y:z]=[x^2-2yz:y^2-2xz:z^2],\]
whose postcritical set consists of the quintic curve
\[z(x^2y^2-4x^3z-4y^3z+18xyz^2-27z^4)=0.\]
If $W$ is the complement of this curve, and $P\in W(K)$, then there is a finite set $S$ of primes such that over $\mathcal{O}_{K, S}$, $\varphi$ extends to an étale map $\varphi:\mathcal{W}\to\mathcal{W}$, and $P$ extends to a point $\mathcal{W}(\mathcal{O}_{K, S})$. The extensions $K(\varphi^{-n}(P))$ are again unramified above primes outside of $S$. It follows again from the result of Corvaja and Zannier <cit.> that a given set $S$ will work for only finitely many points.
In this example, there is in fact a simpler way of accessing this fact (noting that computing the set of primes modulo which a given point intersects the above quintic is perhaps not trivial). The Tchebyshev example above fits into a commutative diagram
\[\begin{CD}
\mathbb{G}_\mathrm{m}^2 @>{Q\mapsto Q^2}>> \mathbb{G}_\mathrm{m}^2\\
@V{\pi}VV @VV{\pi}V\\
\PP^2 @>>{\varphi}> \PP^2
\end{CD}\]
for $\pi(x, y)=\left(x+y+\frac{1}{xy}, \frac{1}{x}+\frac{1}{y}+xy\right)$, which is ramified exactly where $x=y$, $x=y^{-2}$, or $y=x^{-2}$. Thus if $P\in \PP^2(K)$ satisfies $P=\pi(Q)$, and $S$ is a set of places of $K$ for which $Q$ is an $S$-unit, then the extension $K(\varphi^{-n}(P))$, contained in $K(Q^{1/2^n})$, will be unramified outside of $S$.
§ A CHARACTERIZATION OF PCF MORPHISMS ON $\MATHBB{P}^1$
Let us now turn to the proof of Theorem <ref>; throughout this section, unless otherwise noted, we assume that $K$ is a number field. The key ingredient of the proof of Theorem <ref> is that a given $\alpha \in X(K)$ will only collide with a proper closed algebraic subset of $X$ modulo finitely many primes. If $f: X \to X$ is postcritically infinite, then we expect $\alpha$ to collide with the postcritical set modulo infinitely many primes. However, such collision modulo a prime alone is insufficient to guarantee ramification of pre-image fields at that prime.
Let $p$ be a prime number, $f(z) = z(z-p)$, and $\alpha = 0$. The unique finite critical point is $p/2$, with critical value $-p^2/2$. Although $-p^2/2 \equiv 0 \mod p$, we have $\Q(f^{-1}(0)) = \Q$, and so there is no ramification in $\Q(f^{-1}(0))$.
It is not hard to prove that, if $X = \PP^1$, then for every infinite postcritical orbit and every $\alpha$ that does not lie on it, there exist infinitely many primes modulo which $\alpha$ does lie in the orbit. Indeed, by results of Silverman <cit.>, this is true of every infinite forward orbit. Our proof of Theorem <ref> requires the following stronger version of this:
Let $\varphi(z): \PP^1_K \to \PP^1_K$, let $e \geq 2$ be an integer, suppose that $0$ is not a postcritical point under $\varphi$, suppose that $a\in\mathbb{P}^1(K)$ is not preperiodic, and let $S$ be any finite set of places of $K$. Then there exist $\mathfrak{p}\not\in S$ and $n\geq 0$ such that $v_{\mathfrak{p}}(\varphi^n(a))$ is positive and not divisible by $e$.
Before beginning the proof, we note that it is enough to establish the claim for some iterate of $\varphi$. If $0$ is not post-critical for $\varphi$, then it is not post-critical for $\varphi^k$ either, for any $k\geq 1$. Since every iterate of $\varphi^k$ is also an iterate of $\varphi$, the result for $\varphi$ now follows from that for $\varphi^k$. We may replace $\varphi$ with an iterate, then, to assume that $\deg(\varphi)\geq 5$.
Note also that the numerator of $\varphi$ has no repeated roots, since such a root $\beta$ would satisfy $\varphi(\beta)=\varphi'(\beta)=0$, which contradicts our hypothesis that $0$ is not post-critical for $\varphi$. Since $\infty$ is also not mapped by $\varphi$ to 0 with multiplicity, we may assume that the numerator of $\varphi$ (written in lowest terms as a quotient of two polynomials) has at least 4 distinct simple roots.
Now fix $\varphi$, $e$, and $a$ as in the statement and suppose, toward a contradiction, that there is some finite set $S$ of places of $K$ such that for all $\mathfrak{p}\not\in S$ and all $n\geq 0$, if $v_{\mathfrak{p}}(\varphi^n(a))>0$ then $e\mid v_{\mathfrak{p}}(\varphi^n(a))$. We will enlarge our set $S$ below, which clearly does not disrupt this property.
Write $\varphi^n(a_0)=\alpha_n/\beta_n$ with $\alpha_n, \beta_n\in \mathcal{O}_K$ chosen so that $\alpha_n\mathcal{O}_K+\beta_n\mathcal{O}_K$ all divide some fixed ideal $I$ (we can do this by the finiteness of the class group). Write $\varphi(x/y)=F(x, y)/G(x, y)$ with integral coefficients, and enlarge the set $S$ of places enough that $\operatorname{Res}(F, G)\in \mathcal{O}_{K, S}^\times$ and such that $\mathcal{O}_{K, S}$ is a PID in which $I$ generates the trivial ideal.
For a given $n$, write $\varphi^n(a_0)\mathcal{O}_{K, S}=\mathfrak{a}/\mathfrak{b}$, where $\mathfrak{a}$ and $\mathfrak{b}$ are coprime ideals in $\mathcal{O}_{K, S}$. Note that, by our construction of $S$, we have
\[F(\alpha_{n-1}, \beta_{n-1})\mathcal{O}_{K, S}=\mathfrak{a}.\]
We have assumed that $e\mid v_{\mathfrak{p}}(\varphi^n(a_0))$ for every $n\geq 0$ and every $\mathfrak{p}\not\in S$ with $v_{\mathfrak{p}}(\varphi^n(a_0))>0$, and so (since $\mathcal{O}_{K, S}$ is a PID) we have
\[F(\alpha_{n-1}, \beta_{n-1})=sy^e,\]
for some $S$-unit $s$. Choosing a finite set $\msS$ of coset representatives of $\mathcal{O}_{K, S}^\times/(\mathcal{O}_{K, S}^\times)^e$, we in fact have that for each $n\geq 0$,
\begin{equation}\label{eq:dgeq}
s^{-1}F(\alpha_{n-1}, \beta_{n-1})=y^e,
\end{equation}
for some $y\in K$ and some $s\in \msS$. Since $\msS$ is finite, there is one particular $s\in \msS$ such that (<ref>) has a solution with $y\in K$ for all $n$ in some infinite set $Z$. Since $\alpha_n \mathcal{O}_K+\beta_n \mathcal{O}_K$ all divide some fixed ideal $I$, and since we have that $F(x, 1)$ has four distinct simple roots, we may apply a result of Darmon and Granville <cit.> to conclude that there are only finitely many distinct values $\varphi^n(a_0)=\alpha_n/\beta_n$ with $n\in Z$. But $Z$ is infinite, and this would mean that $a_0$ is preperiodic for $\varphi$. The lemma follows from this contradiction.
From Lemma <ref>, we now prove Theorem <ref>.
We first make some reductions of the theorem. Note that the compositum of a finite extension of $K$ with a finitely-ramified extension will again be finitely ramified, so it suffices to prove the theorem after adjoining some algebraic values to $K$. In particular, we will assume that $\varphi$ has a critical point $\zeta\in\mathbb{P}^1(K)$ whose orbit is infinite, and we denote by $e\geq 2$ the local degree of $\varphi$ at this point.
We write $K_n(\alpha)=K(\varphi^{-n}(\alpha))$. If $\varphi^k(\beta)=\alpha$, then $K_n(\beta)\subseteq K_{n+k}(\alpha)$, so it suffices to prove the theorem after replacing $\alpha$ with any of its preimages. Since $\alpha$ is not exceptional, it has infinitely many preimages, only finitely many of which can be postcritical. Without loss of generality, then, we will suppose that $\alpha$ is not post-critical.
Finally, making a change of coordinates if necessary, we assume for convenience that $\alpha=0$ and that $\varphi(\infty)=\infty$.
When we write
\[\varphi([x:y])=\frac{F(x, y)}{G(x, y)},\]
with $F$ and $G$ coprime homogeneous forms, this last assumption gives that $G(x, y)$ is divisible by $y$.
Toward a contradiction, suppose that there exists a finite set $S$ of places of $K$ such that the fields $K_n(0)/K$ are all unramified outside of $S$. We will replace $S$ with a larger finite set such that the coefficients of $F$ and $G$ are $S$-integral; $\operatorname{Res}(F, G)$ is an $S$-unit; $\zeta$ and $\varphi(\zeta)$ are $S$-integral; and the constant terms of $G(x+\zeta, 1)$ and $F(1, x)$, and the quantity $\varphi^{(e)}(\zeta)/e!$, are $S$-units, where $\varphi^{(e)}$ denotes the $e$th derivative of $\varphi$ evaluated at $\zeta$ (which is necessarily non-zero). Note that it follows from this that $\varphi(z+\zeta)$ admits a power series expansion with $S$-integral coefficients of the form
\[\varphi(z+\zeta)=\varphi(\zeta)+\frac{\varphi^{(e)}(\zeta)}{e!}z^e+O\left(z^{e+1}\right),\]
with the coefficient of $z^e$ an $S$-unit.
Also note that if we set
\[F_{n+1}(x, y)=F(F_n(x, y), G_n(x, y))\]
\[G_{n+1}(x, y)=G(F_n(x, y), G_n(x, y))\]
then $\operatorname{Res}(F_n, G_n)$ is always an $S$-unit, and $F_n$ and $G_n$ always have $S$-integral coefficients.
By Lemma 12 there exists a prime $\mathfrak{p}\not\in S$ such that $v_{\mathfrak{p}}(\varphi^n(\zeta))$ is positive and not divisible by $e$. We have
\[0<v_{\mathfrak{p}}(\varphi^n(\zeta))=v_{\mathfrak{p}}(F_{n-1}(\varphi(\zeta),1))-v_{\mathfrak{p}}(G_{n-1}(\varphi(\zeta),1)).\]
But $F_{n-1}(\varphi(\zeta), 1)$ and $G_{n-1}(\varphi(\zeta), 1)$ are $S$-integers with no common factor, and so in fact
\[v_{\mathfrak{p}}(\varphi^n(\zeta))=v_{\mathfrak{p}}(F_{n-1}(\varphi(\zeta),1)).\]
Let $\mathfrak{P}$ be a prime of $K_{n}(0)$ dividing $\mathfrak{p}$. Note that, since $\mathfrak{p}$ does not ramify, we have $v_{\mathfrak{P}}(x)=v_{\mathfrak{p}}(x)$ for all $x\in K$.
If we write $F_n(x, 1)=s_nx^{d^n}+\cdots$ for each $n$, then a simple induction using the fact that $y$ divides $G(x, y)$ shows that $s_n=s_1^{1+d+d^2+\cdots+d^{n-1}}$. In particular, this value is an $S$-unit for all $n$ (given our assumption that it is for $n=1$).
Factoring in $K_{n-1}(0)$, we have
\[F_{n-1}(x, y)=s_{n-1}(x-\delta_1 y)\cdots (x-\delta_{d^{n-1}} y),\]
where $\delta_1, ..., \delta_{d^{n-1}}$ are necessarily $S$-integral.
If $e\mid v_{\mathfrak{P}}(\varphi(\zeta)-\delta_i)$ for each $i$, then the same would be true for
\[v_{\mathfrak{P}}(F_{n-1}(\varphi(\zeta), 1))=\sum_{i=1}^rv_{\mathfrak{P}}(\varphi(\zeta)-\delta_i),\]
which we know not to be the case. So we have some $i$ with $e\nmid v_{\mathfrak{P}}(\varphi(\zeta)-\delta_i)>0$.
Now consider the function $\psi(z)=\varphi(z+\zeta)-\delta_i$.
The roots of $\psi(z)$ are all $K_n(0)$-rational, since each is of the form $\gamma-\zeta$, with $\gamma$ an $n$th preimage of $0$ (more specifically, a first preimage of $\delta_i$).
By hypothesis, this function admits a power series expansion which converges on the $\mathfrak{P}$-adic open unit disk. On the other hand, we know that this power series has the form
\[\psi(z)=\left(\varphi(\zeta)-\delta_i\right)+\frac{\varphi^{(e)}(\zeta)}{e!}z^e+O(z^{e+1}),\]
where the remaining terms all have $\mathfrak{P}$-integral coefficients. The Newton Polygon for $\psi$, then, has an initial segment of length $e$ and slope $-v_{\mathfrak{P}}(\varphi(\zeta)-\delta_i)/e$, followed by an infinite segment of slope 0. Consequently, one of the roots of $\psi$ has valuation $v_{\mathfrak{P}}(\varphi(\zeta)-\delta_i)/e$ (indeed, $e$ of them do), but such roots are $K_n(0)$-rational, and thus have integral $\mathfrak{P}$-valuation. We conclude that $v_{\mathfrak{P}}(\varphi(\zeta)-\delta_i)$ is divisible by $e$. This is a contradiction.
In higher dimension, if $\alpha$ is exceptional, then its backward image is contained in a positive-codimension closed subvariety $Y \subsetneq X$. In line with Conjecture <ref>, we conjecture that whenever the restriction of $\varphi$ to $Y$ is postcritically infinite, we again get ramification at infinitely many primes. The reason for the restriction to non-exceptional points in the statement of Conjecture <ref> is that even when $\varphi$ is postcritically infinite, its restriction to $Y$ may be PCF.
If Lemma <ref> is true for a higher-dimensional variety $X$, it is not too difficult to generalize Theorem <ref>. The proof of Theorem <ref> can be reinterpreted in Newton polygon language, which readily generalizes to higher dimension; for a survey, see <cit.>.
The other step in the proof requires showing that a given $\alpha$ collides with one postcritical component at a time; this is why $S$ contains the set of primes where two critical points of $\varphi$ collide. To generalize this to higher dimension, first, we note that at all but finitely many primes — indeed, generically, for all primes — no two components of the critical locus, a finite union of irreducible hypersurfaces, will have the same reduction.
Now, we need to deal with the case in which $\alpha$ collides with the intersection of $\varphi^{n}(C_1)$ and $\varphi^{n}(C_2)$ where $C_1$ and $C_2$ are two distinct critical components, mapping to their images with local degrees $e_1$ and $e_2$. In that case, $\alpha$ would have a preimage of multiplicity $e_1 e_2$, which is clearly divisible by $e_1$. More generally, the proper intersection of critical components of mapping degrees $e_1, \ldots, e_k$ maps to its preimage with degree $e_1\ldots e_k$; improper intersection will only happen modulo finitely many primes, which we can exclude. We can thus make $e_1$ the $e$ we use in the proof of Theorem <ref>.
The difficulty is in generalizing Lemma <ref>. To obtain the same argument, we first must assume that the Bombieri-Lang conjecture is true. Although the conjecture merely says that general-type varieties have non-Zariski dense sets of rational points, rather than finite sets, in the particular example of $sy^e = f(\varphi(C))$, where $C$ is a hypersurface, $\varphi$ is its image, and $f(\varphi(C))$ is a normalized defining polynomial for $\varphi(C)$, Bombieri-Lang does in fact imply finiteness.
The problem is that we cannot place all the defining polynomials for the components of the postcritical locus into one equation. The components will have growing degrees, so they will come from Chow varieties of growing dimension. The converse, i.e. colliding a specified hypersurface with a Zariski-dense orbit of points, follows trivially from Bombieri-Lang, but for our purposes, we need to show that a specified non-postcritical $\alpha$ collides with the postcritical locus, to degrees not divisible by certain mapping degrees, modulo infinitely many primes. We cannot make any use of stronger conjectures, such as a uniform bound on the number of rational points of general-type varieties: such conjectures only apply for families with fixed discrete parameters, just as the uniform bound conjecture for rational points of general-type curves lets the number of points depend on the genus.
|
1511.00173
|
Department of Physics, Ben-Gurion University
of the Negev,
Beer-Sheva 84105, Israel
Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
Department of Chemistry,
Ben-Gurion University
of the Negev, Beer-Sheva 84105, Israel
We examine the role of interactions for a Bose gas trapped in a double-well potential (“Bose-Josephson junction”) when external noise is applied and the system is initially delocalized with equal probability amplitudes in both sites. The noise may have two kinds of effects: loss of atoms from the trap, and random shifts in the relative phase or number difference between the two wells. The effects of phase noise are mitigated by atom-atom interactions and tunneling, such that the dephasing rate may be reduced to half its single-atom value. Decoherence due to number noise (which induces fluctuations in the relative atom number between the wells) is considerably enhanced by the interactions. A similar scenario is predicted for the case of atom loss, even if the loss rates from the two sites are equal. In fact, interactions convert the increased uncertainty in atom number (difference) into (relative) phase diffusion and reduce the coherence across the junction. We examine the parameters relevant for these effects using a simple model of the trapping potential based on an atom chip device. These results provide a framework for mapping the dynamical range of barriers engineered for specific applications and sets the stage for more complex circuits (“atomtronics”).
§ INTRODUCTION
The development of circuits for neutral atoms (coined “atomtronics”) has received tremendous impetus from recent advances in the control and manipulation of ultracold atoms using magnetic and optical fields <cit.>.
The idea of atomtronics is inspired by the analogy between ultracold atoms confined in optical or magnetic
potentials and solid-state systems based on electrons in various forms of conductors, semiconductors or superconductors.
For example, ultracold atoms in optical lattices exhibit a Mott insulator to superfluid transition, or display spin-orbit
coupling as in solid-state systems.
Another example is a Bose-Einstein condensate (BEC) of neutral atoms in a double-well potential, which is
analogous to a Josephson junction of coupled superconductors. On the other hand, the quantum
properties of ultracold atoms as coherent matter waves enable
systems that are equivalent to
optical circuits, which are based on waveguides and beam-splitters for interferometric precision measurements
in fundamental science and technological applications.
A promising platform for accurately manipulating matter waves in a way that would enable integrated
circuits for neutral atoms is an atom chip <cit.>.
Such a device facilitates precise control over magnetic or optical potentials on the micrometer scale. This length
scale, which is on the order of the de-Broglie wavelength of ultracold atoms under typical conditions, permits
control of important dynamical parameters, such as the tunneling rate through a potential barrier.
For a network built of static magnetic fields, such control over the dynamics requires loading the atoms into
potentials just a few $\mu$m from the surface of the chip <cit.>.
The ability to load such potentials, while maintaining spatial coherence, was recently shown to be possible <cit.>.
This achievement is facilitated by the weak coupling of neutral atoms to the environment <cit.>.
Yet, in view of the fact that spatial coherence is one of the most vulnerable properties of quantum systems made
of massive particles, it is quite surprising that a BEC of thousands of atoms preserves spatial coherence for
a relatively long time in the very close proximity of a few micrometers from a conducting surface at room temperature.
Here we examine the interplay between coupling to external noise and the internal parameters – tunneling rate
and atom-atom interactions – of a BEC in a double-well potential (a “Bose-Josephson junction”).
Such a system, consisting of a potential barrier between two potential wells, provides one of the fundamental
building blocks of atomic circuits and comprises one of the basic models for studying a
simple system of many interacting particles occupying only two modes.
This study unravels some general many-body effects,
and at the same time enables insights into the limits of
the practical use of circuits of a trapped BEC near an atom chip surface.
Macroscopic one-particle coherence is the hallmark of Bose-Einstein condensation.
The Penrose-Onsager criterion states that as the condensate forms, one of the eigenvalues of the reduced
one-particle density matrix becomes dominant, resulting in a pure state in which all atoms occupy the
same quantum “orbital”.
Once a condensate is prepared however, its one-particle coherence can be lost via entanglement with an external environment (“decoherence”) or by internal entanglement between condensate atoms due to interactions (“phase diffusion”). The interplay between these two processes, namely non-Hamiltonian decoherence and the Hamiltonian dynamics of interacting particles, is often rich and intricate. The combined effect is rarely additive and depends strongly on the details of the coupling mechanisms. For example, decoherence may be used to protect one-particle coherence by suppressing interaction-induced squeezing in a quantum-Zeno-like effect <cit.>.
It may also induce stochastic resonances which enhance the system's response to external driving <cit.>.
Reversing roles, one may ask how interactions affect the dephasing or dissipation of a BEC due to its coupling to the environment. In this work we consider a BEC in a double well using a two-mode Bose-Hubbard approximation, and investigate how the loss of one-particle coherence due to the external noise is affected by interparticle interactions. We find that many-body dynamics may indeed either enhance or suppress decoherence, depending on the nature of the applied noise.
In light of these fundamental effects, this work also attempts
to construct a framework for combining our theoretical model with practical experimental parameters for realistic magnetic potentials
and magnetic noise on an atom chip. Finally, we present some experimental results concerning
atom loss at distances of a few micrometers from an atom chip, which may enable some concrete conclusions
regarding the issue of coherence in atomic circuits using similar platforms.
This paper is structured as follows:
in Sec. <ref>, we describe the basic constituents
of the system we are about to study.
In Sec. <ref> we review the theoretical model and
fundamental properties of a BEC in a double well and in Sec. <ref> we derive its coupling to magnetic noise.
Section <ref> then combines these effects to present the main results of this work:
how decoherence in an atomic Josephson junction can be suppressed or enhanced by atom-atom interactions.
The range of validity and accessible range of parameters of our
model are discussed in Sec. <ref>.
This discussion is supplemented by experimental measurements of magnetic noise in atomic traps (Sec. <ref>).
Finally, our discussion in Sec. <ref> includes examples of practical and fundamental implications of the
predicted effects.
§ DESCRIPTION OF THE SYSTEM
We consider a Bose-Einstein condensate (BEC) of atoms with mass $m$ in a double-well potential.
The potential is modeled by a cylindrically symmetric harmonic
transverse part $V_{\perp}=\frac{1}{2}m\omega_{\perp}^2[y^2+(z-z_0)^2]$
centered at a distance $z_0$ from the surface of an atom chip,
and a longitudinal part $V_{\parallel}$ representing a barrier of height $V_0$ between two wells with minima
at $x=\pm d/2$. This potential, which is symmetric under $x\to -x$ reflections, may be modeled as:
V_∥(x)={ V_0cos^2(πx/d) |x|≤d/2
1/2mω_x^2(|x|-d/2)^2 |x|>d/2
. ,
where the longitudinal frequency $\omega_x$ characterizing the curvature of the potential far from the barrier
is typically smaller than the transverse frequency $\omega_{\perp}$.
Such a system is usually referred to as a Josephson junction; it exhibits Josephson oscillations with frequency $\omega_J$
when the number of atoms in the two wells deviates slightly from equilibrium.
Some of the most important properties of the condensate in the potential can be derived from the assumption that a “macroscopic" number of atoms occupy a single mode whose wave function $\phi_0({\bf r})$
satisfies the Gross-Pitaevskii (GP) equation, whose static form is
where $\mu$ is the chemical potential, which is the energy of a single atom in the effective (mean-field) potential $V_{\rm eff}=V({\bf r})+gN|\phi_0({\bf r})|^2$.
Here $N$ is the total number of atoms
and $g=4\pi\hbar^2 a_s/m$ is the collisional interaction strength, with $a_s$ being the $s$-wave scattering length.
As long as the barrier height is not too high and the temperature is low enough, the atoms predominantly occupy the
condensate mode $\phi_0({\bf r})$ and the system is coherent, namely, the phase
between the two sites is well defined. However, when the barrier height grows and tunneling is suppressed,
more atoms occupy other spatial modes and the one-particle coherence drops.
In order to understand this effect, we use a set of spatial modes defined by higher-energy solutions of
the GP equation
where $\phi_0({\bf r})$ is the condensate mode with $E_0=0$ and the other modes ($j>0$) represent excited
single-atom states in the mean-field potential <cit.>. These modes form a complete set which may serve as a basis for any calculation.
Furthermore, this specific choice is useful because it can describe the ground state of the system for all barrier
When the barrier is low (or does not exist) the condensate approximation holds for the ground state.
However, the nature of the ground state changes when the barrier becomes higher than the longitudinal ground-state
\begin{eqnarray}
\mu_{\parallel} &\equiv& \int d^3 r\ \phi_0({\bf r})\hat{H}_{\parallel}^{\rm eff}\phi_0({\bf r})\, ; \nonumber \\
\hat{H}_{\parallel}^{\rm eff} &\equiv & -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+
\label{eq:def-H-parallel}
\end{eqnarray}
As demonstrated in Fig. <ref>, when $\mu_{\parallel}\lesssim V_0$
the energy $E_1$ of the (anti-symmetric) first excited mode $\phi_1$ becomes very small
compared to the other excited modes and the configuration space can be described by
the two modes $\phi_0$ and $\phi_1$, or alternatively by their superpositions $\phi_L$ and $\phi_R$
localized predominantly in the left- and right-hand wells, respectively.
In this regime, collisional interactions
play a major role in determining the ground state configuration beyond the spatial effects accounted for
by the mean-field potential, as will be described in Sec. <ref>.
Mean-field energy levels of the lowest energy spatial modes for $N=200$ $^{87}$Rb atoms
as a function of barrier height $V_0$
in a double-well potential with transverse frequency $\omega_{\perp}=2\pi\times 500\,{\rm Hz}$ and a longitudinal
potential [Eq. (<ref>)] with $d=5\,\mu$m and $\omega_x=2\pi\times 200\,{\rm Hz}$.
The lowest solid curve represents the longitudinal energy $\mu_{\parallel}$
of the condensate
[Eq. (<ref>)],
and the other energies are higher than
this energy by $E_j$ for $j=1,2,3$
[Eq. (<ref>)]. When the barrier height grows, the energy splitting between the
lowest energy pair decreases and the anti-symmetric mode $\phi_1$ becomes
significantly occupied even at very low temperatures or even at absolute zero due to mixing between the
two lowest levels caused by atom-atom interactions.
The inset shows the shape of the potential for a barrier height
$V_0\approx 470\,{\rm Hz}$
where $\mu_{\parallel}\approx 425\,{\rm Hz}$
(dashed line) is slightly lower than the barrier.
The symmetric condensate wavefunction $\phi_0=(\phi_R+\phi_L)/\sqrt{2}$ and the antisymmetric wavefunction
$\phi_1=(\phi_R-\phi_L)/\sqrt{2}$ have approximately the same shape in the two wells, and
the energy splitting is $E_1\approx 1\,{\rm Hz}$.
In this work we focus on the interplay between the intrinsic parameters of the system and
the coupling to the environment. The latter may appear in different shapes and forms. Basically, the strongest coupling is between the magnetic moment of the atom
and magnetic field noise originating from current fluctuations in the atom chip device that creates the trapping potentials from current-carrying wires. We distinguish
between macroscopic current fluctuations generated by external
drivers (“technical noise”)
and Johnson noise, i.e., microscopic fluctuations of thermal origin
in the metallic layers of the chip itself, which are typically at room temperature or higher. Johnson noise has a short correlation length <cit.> and can
therefore cause direct loss of coherence over short length scales.
When the magnetic field fluctuates perpendicular to the
quantization axis
(along which the atomic spin is typically aligned), it may induce transitions between Zeeman sub-levels and
cause atoms to leave the trap, as discussed in more detail in Sec. <ref>.
This loss mechanism may also cause dephasing, as we discuss in Sec. <ref>.
Although technical noise has a long correlation length, it may also contribute to dephasing through the latter mechanism.
However, as we discuss in Sec. <ref>, technical noise may also lead to direct dephasing
if the corresponding current fluctuations
induce an asymmetric deformation in the trapping potential
as a result of summing
magnetic field vectors from nearby microwires and more distant sources.
§ THE TWO-MODE MODEL
Our framework for the analysis of decoherence of a Bose gas in a double well is the two-site Bose-Hubbard model,
which is based on the assumption that all atoms occupy one of two spatial modes,
as described in the previous section. Before we analyze decoherence in the system we review
this model and its main predictions relevant to this work. The validity of the model for typical
scenarios presented in this paper is further discussed in Sec. <ref>.
§.§ Interaction and tunneling Hamiltonian
Consider $N$ bosonic atoms in a double well such that two spatial modes may be occupied,
$\phi_L({\bf r})$ in the left well and $\phi_R({\bf r})$ in the right well. The dynamics is governed by the
two-site Bose-Hubbard Hamiltonian <cit.>
\begin{eqnarray} \hat{H} &=&
\frac{\epsilon}{2}(\hat{n}_L - \hat{n}_R)
-\frac{J}{2}(\hat{a}_L^{\dag}\hat{a}_R + \hat{a}_R^{\dag}\hat{a}_L) \nonumber \\
&& +
\frac{U}{2}
\sum_{j = L,R} \hat{n}_{j} (\hat{n}_{j} - 1)
\label{eq:Ham}
\end{eqnarray}
where $\epsilon$ is the energy imbalance (per particle)
between the two wells, $J$ is the tunneling matrix element and $U$ is the on-site interaction
energy per atom pair.
Here $\hat{a}_j$ ($j = L,R$)
are the bosonic annihilation operators of atoms in the two modes $\phi_j$ and $\hat{n}_j=\hat{a}_j^{\dag}\hat{a}_j$
are the corresponding number operators.
Since the Hamiltonian [Eq. (<ref>)] conserves the total number of particles in the
two wells, we can write it in terms of
the pseudo-spin operators $\hat{S}_{1} = \frac{1}{2}(\hat{a}_L^{\dag}\hat{a}_R+\hat{a}_R^{\dag}\hat{a}_L)$,
$\hat{S}_{2} = -\frac{i}{2}(\hat{a}_L^{\dag}\hat{a}_R-\hat{a}_R^{\dag}\hat{a}_L)$ and $\hat{S}_{3} =
\frac{1}{2}(\hat{n}_L - \hat{n}_R)\equiv \hat{n}$,
whereupon the Hamiltonian takes the form
Ĥ=ϵŜ_3 - JŜ_1 + UŜ_3^2. Note that the total spin $\hat{S}_{1}^2 + \hat{S}_{2}^2 + \hat{S}_{3}^2
= s(s+1)$ is fixed by the total atom number $2s = N = \hat{n}_{L}
+ \hat{n}_{R}$, as it commutes with
the Hamiltonian in Eqs. (<ref>)-(<ref>).
The Hilbert space of the two-mode model is spanned by the $\hat{S}_{3}$ eigenstates $\left | s, n\right\rangle$, corresponding to the number states basis $|n_L,n_R\rangle$ with
$n_{L,R} = N/2 \pm n=s\pm n$. It will prove advantageous to make use of dimensionless characteristic parameters which determine both stationary and dynamic properties; these are
u ≡NU/J
and ε≡ϵ/J
§.§ Equivalence to the top and pendulum Hamiltonians
The spin Hamiltonian in Eq. (<ref>) is equivalent to a quantized top, whose spherical phase space is described by the conjugate non-canonical coordinates $({\hat\theta},{\hat\varphi})$ defined through
\begin{eqnarray}
\hat S_{3} &=&
\frac{N}{2} % [s(s{+}1)]^{1/2}
\, \cos\hat{\theta},
\label{eq:def-S3}\\
\hat S_{1} &=&
\frac{N}{2} % [s(s{+}1)]^{1/2}
\, \sin\hat{\theta}\cos\hat{\varphi},
\label{eq:def-S1}
\end{eqnarray}
in analogy to the definitions on the Bloch sphere.
Equation (<ref>) is thus transformed into the top Hamiltonian <cit.>,
Ĥ(θ̂,φ̂) = NJ/2[
- sinθ̂cosφ̂
+ u /2 cos^2θ̂
] .
If the populations in the two wells remain nearly equal during the dynamics, i.e., $\langle \hat S_3 \rangle \ll N/2$,
then the spherical phase space reduces to the “equatorial
region” around $\theta = \pi/2$,
where it may be described by the cylindrical coordinates
$\hat{n}$ and its canonically conjugate angle $\hat{\varphi}$, satisfying
$[\hat n, \, \hat\varphi] = i$.
The Hamiltonian [Eq. (<ref>)] is then approximated by the
Josephson pendulum Hamiltonian <cit.>
Ĥ_J(n̂,φ̂) = U (n̂-n_ϵ)^2 - 1/2JN cosφ̂ ,
where $n_\epsilon = -\epsilon/2U$.
§.§ Classical phase space and interaction regimes
The classical states of the double-well system are SU(2) spin coherent states $|\theta,\varphi\rangle$ <cit.>.
Dynamics restricted to such states satisfies the GP mean-field approximation,
where all the atoms are in a single mode – a superposition of $\phi_L$ and $\phi_R$ –
and ${\cal O}(1/N)$ fluctuations are neglected.
The operators in Eqs. (<ref>), (<ref>), and (<ref>) are hence replaced by corresponding real numbers. Thus, the GP approximation may be viewed as the classical limit of the quantum many-body Hamiltonian, with an effective Planck constant
$\hbar\rightarrow 2/N$.
The qualitative features of the classical phase-space structure change drastically with the interaction strength <cit.>.
If $|{\varepsilon}|<\varepsilon_c\equiv
\left(u^{2/3}-1\right)^{3/2}$, then for a strong enough interaction (${u>1}$)
two types of dynamics appear, which are described in the spherical $(\theta,\varphi)$ phase space by three
regions on the Bloch sphere, divided by a separatrix, as shown in Fig. <ref>.
Motion within the region close to the equator (small population imbalance) is dominated by linear dynamics
and is characterized by Josephson oscillations of population between the wells. Conversely, motion between
the separatrix and the poles is dominated by nonlinear dynamics and is characterized by
self-trapped phase oscillations with non-vanishing time-averaged population imbalance.
In what follows, we focus on the case of zero energy imbalance ($\epsilon=0$). Accordingly we distinguish between three regimes depending on the strength of the interaction <cit.>:
\begin{eqnarray}
\mbox{Rabi regime:} &~& u<1~, \\
\mbox{Josephson regime:} &~& 1<u<N^2~,
\label{ineq:Josephson}
\\
\mbox{Fock regime:} &~& u>N^2~.
\end{eqnarray}
In the Rabi regime the separatrix disappears and the entire phase space is dominated by nearly linear population oscillations between the wells.
At the opposite extreme, in the Fock regime the nearly-linear region has an area less than $1/N$, and therefore it cannot accommodate quantum states.
Thus in the Fock regime, motion throughout the classical phase space corresponds to nonlinear phase oscillations with suppressed tunneling between the wells (self trapping).
In the absence of tunneling between the wells spatial coherence cannot be sustained even if external noise does not exist.
We will therefore focus on the dynamics within the Josephson oscillations phase-space region, which exists in the Rabi and Josephson interaction regimes.
Schematic illustration of iso-energy contours (solid lines) in the Josephson regime for a symmetric double-well potential ($\epsilon = 0$).
The spherical coordinates $\theta, \varphi$ map the Bloch sphere
of pseudospin operators $\hat{S}_\alpha$ ($\alpha = 1,2,3$)
[see Eqs. (<ref>)-(<ref>)]: $\varphi$ is the
relative phase between the wells, and the left-right number
difference is proportional to $\cos\theta$.
The equilibrium distribution (circles or ellipses) are centered
on the $S_1$-axis.
The aspect ratio between the principal axes of the elliptical Josephson oscillation trajectories is $\xi^2$ [Eq. (<ref>)].
The dashed circle denotes an equal energy contour in the absence of interaction.
The small filled circle denotes the Gaussian-like phase-space distribution corresponding to the coherent
ground state (solid angle ${\cal O}(4\pi/N)$). The effect of stochastic rotations about the $S_{3}$ ($\delta \varphi$ kicks) and $S_{2}$ ($\delta n$ kicks)
axes is illustrated by arrows. The combined action of kicking and nonlinear rotation is to spread the phase-space
distribution over the inner and outer ellipse, respectively.
§.§ Semiclassical dynamics
Beyond strictly classical evolution, much of the full quantum dynamics is captured by a semiclassical approach. This amounts to classical Liouville propagation of an ensemble of points corresponding to the initial Wigner distribution in phase space <cit.>. Thus, while classical GP theory assumes the propagation of minimal Gaussians centered at $(\theta,\varphi)$ with fixed uncertainties (coherent
states), the semiclassical truncated-Wigner-like method permits the deformation of the phase-space distribution, thus accounting for its squeezing, folding, spreading, and dephasing. The only parts of the dynamics missed by such semiclassical evolution are true interference effects which only become significant for long time scales when different sections of the phase-space distribution overlap.
§.§ Ground state and excitation basis
As an alternative point of view that complements the semiclassical approach we also use a fully quantum
treatment of the two-mode Bose-Hubbbard model, which takes a simple analytic form when the
deviation from the minimum energy eigenstates of the Hamiltonian is small. For simplicity,
we will assume zero energy imbalance ($\epsilon=0$) and derive the structure of the ground state and lowest
energy excitations.
In the Fock regime ($u>N^2$) the ground state and low-lying excitations are number states. However, throughout the Rabi and Josephson regime (i.e., for $u<N^2$) the ground state and low-lying excitations are nearly coherent and are characterized by a small phase uncertainty $\langle \hat{\varphi}^2\rangle\ll 1$.
We may therefore approximate the low-energy regime of the pendulum by a harmonic oscillator, i.e., (ignoring the constant $-1$) $-\cos\hat{\varphi}\to \hat{\varphi}^2/2$, converting the Josephson Hamiltonian [Eq. (<ref>)] to an oscillator in the canonical variables $\hat{n} \equiv \hat{S}_3$ and $\hat{\varphi}$,
Ĥ_J≈1/2M(ξn̂)^2+M/2ω_J^2(φ̂/ξ)^2, where $\xi$ is the squeezing factor, given by
$\omega_J$ is the Josephson frequency
ω_J=√(J(J+NU))=ξ^2 J, and $M\equiv N/2\omega_J$. The ground state of this Hamiltonian is Gaussian in $\xi \hat{n}$ and $\hat{\varphi}/\xi$,
and the excitation energies in this approximation are integer multiples of $\hbar\omega_J$.
More explicitly, we may define a bosonic operator
b̂≡ √( N ) / 2ξ φ̂
+ i ξ/ √( N ) n̂
such that
from $[\hat{\varphi}, \hat{n}]=i$ we obtain $[\hat{b},\hat{b}^{\dag}]=1$. The Josephson Hamiltonian can then be written
and the pseudo-spin operators of Eq. (<ref>) are identified as
\begin{eqnarray}
\hat{S}_{1} &=& \frac{N}{2} \sin\hat{\theta}\cos\hat{\varphi}
\nonumber \\
&\approx & \frac{N}{2}-\frac{1}{4}\left[\xi^2(\hat{b}+\hat{b}^{\dag})^2-\xi^{-2}(\hat{b}-\hat{b}^{\dag})^2-2\right]
\label{eq:Sxbb} \\
\hat{S}_{2} &=& \frac{N}{2} \sin\hat{\theta}\sin\hat{\varphi}\approx \frac{1}{2}\sqrt{N}\xi (\hat{b}+\hat{b}^{\dag})
\label{eq:Sybb} \\
\hat{S}_{3} &=& \hat{n}\approx \frac{\sqrt{N}}{2i\xi}(\hat{b}-\hat{b}^{\dag})
\label{eq:Szbb}
\end{eqnarray}
Here we have used a second-order expansion in $\hat{n}/N$ and
$\hat{\varphi}$ such that
$\sum_\alpha\hat{S}_{\alpha}^2 =\frac{N}{2}\left(\frac{N}{2}+1\right)$.
§.§ Coherence
We are interested in the one-particle spatial coherence, namely the visibility of a fringe pattern
formed by averaging many events where the atoms are released from the double-well trap.
If we neglect experimental imperfections related to the release process or imaging, then the
coherence is the relative magnitude of the interference term of the momentum distribution
of the atoms in the trap <cit.>. If the two spatial modes are confined primarily to the left and right sites
of the potential, then the coherence is given by
g^(1)_LR=|⟨â_L^†â_R⟩|/√(n_L n_R),
where $n_j\equiv \langle \hat{n}_j\rangle$ is the average number of atoms in the two modes.
For a symmetric double well ($\epsilon=0$) with equal populations of the two modes ($\langle \hat{S}_3 \rangle = 0$), the
definition of Eq. (<ref>) coincides with the normalized length $S/s
= 2 S / N$ of the
Bloch vector $\mathbf{S}\equiv (\langle{\hat S_{1}}\rangle, \langle{\hat S_{2}}\rangle, \langle{\hat S_{3}}\rangle)$.
Let us note that $s$ is the maximal allowed Bloch vector length (e.g., after preparation of a coherent state), whereas $S$ is the actual Bloch vector length, i.e., after time evolution under dephasing or if the prepared state was squeezed.
For the ground state preparations with $\langle\varphi\rangle=0$ considered below,
symmetry implies that the Bloch vector remains aligned along the $S_{1}$ axis, so that
$g^{(1)}_{LR}=S/s=\langle \hat{S}_{1}\rangle/s$
throughout the time evolution.
Semiclassically, for any Gaussian phase-space distribution, the one-particle coherence is related to the variances $\Delta_a,\Delta_b$ of the distribution along its principal axes $a,b$
as <cit.>
where $\Delta^2=\Delta_a^2+\Delta_b^2$. A coherent state with isotropic $\Delta_a^2=\Delta_b^2=1/N$ thus has $S=s$, i.e., $g^{(1)}_{LR}=1$.
Otherwise, for a squeezed Gaussian distribution with
$\Delta_a^2=\xi^2/N$, $\Delta_b^2=\xi^{-2}/N$, the coherence
drops to $g^{(1)}_{LR}=\exp[-\left[(\xi^2+\xi^{-2}-2)/2N\right]] < 1$.
For example, the ground state of the Josephson Hamiltonian is described by a Gaussian phase-space
distribution which is squeezed along the principal axes
with the squeezing factor
$\xi $ in Eq. (<ref>), corresponding to a coherent state ($\xi=1$) only
in the non-interacting case ($u=1$).
More generally, time evolution such as the decoherence process that will be described in Sec. <ref> may lead to
a Gaussian distribution which has spread out by a factor $D$
so that $\Delta_a^2=D\xi^2/N$, $\Delta_b^2=D\xi^{-2}/N$ gives an even smaller value
By using the approximation of Eq. (<ref>) for $\hat{S}_{1}$ in terms of the bosonic operator $\hat{b}$, one can see that
the coherence of the ground state is given by $g^{(1)}_{LR}\approx 1-(\xi^2+\xi^{-2}-2)/2N$, in agreement with the squeezed Gaussian semiclassical value. Thus, the ground state is coherent for the non-interacting case ($\xi=1$) and its coherence is reduced due to interactions. It is evident that in order to see significant reduction in the ground state coherence, $\xi^2=\sqrt{u+1}$ should be comparable to $N$. Thus, the coherence of the ground state drops only in the transition from the Josephson to the Fock regimes ($u > N^2$) <cit.>. This crossover in the finite size system is the
simplest version of the superfluid to Mott insulator
transition <cit.>.
The coherence drops further if excited states are populated.
This happens in thermal equilibrium at low temperatures, where the coherence factor is $g^{(1)}_{LR} \approx 1 -
[(\xi^2+\xi^{-2})(2n_T+1)-2]/2N$ with
$n_T\equiv \langle \hat{b}^{\dag}\hat{b}\rangle_T\approx [\exp(\hbar\omega_J/k_BT)-1]^{-1}$ being the thermal occupation of the excited levels. Reduction of
coherence also occurs due to external noise, as we shall see below.
§ DEPHASING AND LOSS DUE TO MAGNETIC NOISE
The interaction of the magnetic field fluctuations with an atom is given by the Zeeman Hamiltonian
V̂_Z(r,t) = -μ⃗·B(r,t), where ${\vec \mu}$ is the magnetic moment of the atom. If the magnetic field is not too strong then the
atom stays in a specific hyperfine state ($F$) and its magnetic moment is proportional to the angular momentum
${\bf F}$ through ${\vec \mu} = \mu_F {\bf F} = g_F\mu_B{\bf F}$,
$g_F$ being the Landé factor and $\mu_B$ the Bohr magneton.
For an ensemble of atoms with translational degrees of freedom, we adopt the language of second
quantization and
write the interaction Hamiltonian as
V̂_Z( t ) = -μ_F∑_q ∑_m,m'∫d^3r
where the three operators
∑_mm' F̂_q^mm'≡∑_mm'
are the components of the magnetization density.
Here, $q$ labels the components of the magnetic field and the
angular momentum operator; the indices $m,m'$ label the Zeeman
states of the hyperfine level
($-F\leq m,m' \leq F$). The operators
$\hat{\psi}_{m}({\bf r})$ are field operators for atoms in the internal sublevel $|m\rangle$, satisfying
bosonic commutation relations $[\hat{\psi}_m({\bf r}),\hat{\psi}_{m'}^{\dag}({\bf r}')]
=\delta_{m,m'}\delta({\bf r}-{\bf r}')$.
In magnetic traps the Zeeman Hamiltonian [Eq. (<ref>)], with ${\bf B}$ as the trapping magnetic field,
determines the trapping potential [e.g., Eq. (<ref>)] for atoms in a Zeeman sublevel whose magnetic moment
is aligned parallel to the magnetic field. We define the quantization axis as the local direction parallel to the trapping
magnetic field ($\hat{p}$ direction).
We will now examine the effect of magnetic field fluctuations, such that
${\bf B}$ in Eq. (<ref>) will represent changes in the magnetic field relative to an average trapping field.
are responsible either for fluctuations of the trapping potential (for parallel magnetic field components
$q=p$ or slowly varying fields in any direction) or for transitions between different Zeeman states (transverse components $q=\pm$
at the transition frequency for transitions with positive or negative angular momentum changes).
In this work we make two simplifying assumptions:
(a) only one spin component is trapped ($m=F$) and
once an atom is in another internal state it immediately escapes
from the trap;
(b) the atoms in the double-well
trap occupy one of two spatial states $\phi_L$ and $\phi_R$ corresponding to the two (left and right) wells.
For the interaction driven by the fluctuating
field $B_{p}$ parallel to the trapping magnetic field,
we only need the diagonal matrix element $F^{mm}_p = F$ ($m = F$).
Inserting the expansion of the field operator
$\hat\psi_F$ over the two (spatial rather than internal)
modes $\phi_{L,R}$,
we can therefore replace Eq. (<ref>) with
F̂_p^F,F →F∑_i,j=L,R
ϕ_i^*(r)ϕ_j(r)â_i^†â_j. Note that in the presence of $B_p$ the Hamiltonian [Eq. (<ref>)] conserves the total number of trapped atoms. The
diagonal terms $i = j$ correspond to fluctuating energy shifts
that potentially dephase coherent superpositions of left and
right sites.
Magnetic fields perpendicular to the $\hat{p}$-axis drive spin flip
betwen the trapped level $m=F$
and the untrapped level $m=F-1$
(matrix element $F^{F-1,F}_- = \sqrt{F}$).
We can replace
Eq. (<ref>) with
where $\zeta_k({\bf r})$ are the modes of the untrapped state
and $\hat{c}_k^{\dag}$ are the corresponding creation operators.
Note that slowly varying magnetic fields perpendicular to the average local quantization axis $\hat{p}$ would not
cause transitions to untrapped states because the atomic spin direction would adiabatically follow the local direction of the
magnetic field vector, such that the net effect of these slow fluctuations would be a change of the potential
in a manner similar to magnetic fluctuations parallel to $\hat{p}$.
We will now examine the master equation for the dynamics of the two processes: number-conserving processes
dominated by energy/phase fluctuations, and the loss processes.
§.§ Dephasing
The number-conserving stochastic evolution due to a fluctuating
magnetic field aligned with the atomic magnetic moment
is derived from the Hamiltonian [Eq. (<ref>)] keeping only
the component $B_{p}$ and the atomic operator given
in Eq. (<ref>).
The corresponding master equation for the density matrix
$\rho$ is of the form
$\dot{\rho}=i[\hat{H},\rho]+{\cal L}_N\rho$, where
${\cal L}_N$ is the number-conserving operator
L_Nρ=-∑_ijkl γ_N^ijkl/2[â_i^†â_jâ_k^†â_lρ+ρâ_i^†â_jâ_k^†â_l
and involves transition rates given by
\begin{eqnarray} \gamma_{N}^{ijkl} &=& \frac{\mu_F^2 F^2}{\hbar^2}\int d^3{r}\int d^3{r}'\,
\label{eq:dephasing-rate-1}
\\
&& \times \phi_i^*({\bf r})\phi_j({\bf r})
\phi_k^*({\bf r}')\phi_l({\bf r}')
{\cal B}_{pp}({\bf r},{\bf r}',\omega_{ijkl})
\,;
\nonumber
\\[6pt]
{\cal B}_{pp'}({\bf r},{\bf r}',\omega) &=&
\int d\tau e^{i\omega\tau} \langle B_{p}({\bf r},t)B_{p'}({\bf r}',t+\tau)\rangle
\,.
\end{eqnarray}
Here ${\cal B}_{pp}({\bf r},{\bf r}',\omega)$ is the two-point
correlation spectrum of the $p$-component of the magnetic field <cit.>. The frequencies $\omega_{ijkl}\equiv
\omega_i+\omega_k-\omega_j-\omega_l$ are the transition frequencies, which may be taken to
be zero in our case since
we assume that the two modes $\phi_L$ and $\phi_R$ are degenerate. Here we also neglect energy shifts (magnetic
Casimir-Polder interaction) which are typically quite
small <cit.>.
If the spatial variation of the magnetic fields is small across
the trap volume of both sites, we may take
${\cal B}_{pp}({\bf r},{\bf r}',\omega_{ijkl})$
in Eq. (<ref>) outside the double integral.
The orthogonality relations among the
modes $\phi_j$ imply
that only the terms with $i=j$ and $k=l$ survive.
We then obtain $\gamma^{ijkl}_{N} = \delta_{ij}\delta_{kl}
\gamma_{N}$ with
\begin{equation}
\gamma_{N} =
{\cal B}_{pp}({\bf r}_t,{\bf r}_t,0),
\label{eq:dephasing-rate}
\end{equation}
where ${\bf r}_t$ is a typical position in the trapping region.
The dissipative term in the
master equation becomes
L_Nρ=-γ_N/2∑_i,j=L,R[n_in_jρ+ρn_i n_j-2n_iρn_j].
If the total number of atoms $n_L+n_R=N$ is fixed, this term vanishes completely.
If ${\cal B}_{pp}({\bf r},{\bf r}',0)$ varies over the trap region, then $\gamma_N^{ijkl}$ may be non-zero for any set of indices.
However, if the modes $\phi_L({\bf r})$ and $\phi_R({\bf r})$ are well separated, having only a small overlap
in the region of the barrier, then the terms with $i=j$ and $k=l$ are much larger.
These terms may be written in the form
\begin{eqnarray}
\gamma_{N}^{iijj} &=& \frac{\mu_F^2F^2}{\hbar^2}{\cal B}_{pp}({\bf r}_t,{\bf r}_t,0)\alpha_{ij}
\nonumber
\\[6pt]
\alpha_{ij} &=& \int d^3{r}\int d^3{r}' {\cal A}({\bf r},{\bf r}')|\phi_i({\bf r})|^2|\phi_j({\bf r}')|^2\,,
\label{eq:alpha-ij}
\end{eqnarray}
where ${\bf r}_t$ represents a typical location in which ${\cal B}_{pp}$ may be maximal, and
the dimensionless function ${\cal A}({\bf r},{\bf r}')$ represents the spatial shape of ${\cal B}_{pp}$ relative to its
value at ${\bf r}_t$.
If all elements of the (symmetric) matrix $\alpha_{ij}$ were equal,
they would give no contribution to the master equation,
similar to the argument in Eq. (<ref>).
By subtracting $\alpha_{LR}$ from all elements,
we obtain the following dissipative terms in the master
\begin{eqnarray} {\cal L}_N\rho &=& -\frac{\gamma_N}{2}
\sum_{j = L, R} (\alpha_{jj} -\alpha_{LR})[n_j^2\rho+\rho n_j^2-2n_j\rho n_j] \nonumber \\[6pt]
&=& -\gamma_{p}(\hat{S}_{3}^2\rho+\rho\hat{S}_{3}^2-2\hat{S}_{3}\rho\hat{S}_{3}),
\label{eq:masterS3}
\end{eqnarray}
and we have used $\hat{n}_{L,R}=N/2\pm \hat{S}_3$ with
the total number $N$ being conserved.
As expected, magnetic fluctuations along the quantization ($\hat{p}$)
axis apply phase noise between the two wells. This can be
represented by random rotations about the $S_3$-axis of the Bloch
sphere, and the dissipative term [Eq. (<ref>)] generates
the corresponding phase diffusion in the density operator.
Let us now consider two typical types of noise. The first is Johnson noise from thermal current fluctuations
near a conducting surface.
Let us focus for simplicity on a double-well potential whose sites
are equally close to the surface. The correlation function
${\cal B}_{pp}({\bf r},{\bf r}')$ then depends only on the
distance $|{\bf r} - {\bf r}'|$ and decays to zero on a length
scale $\lambda_c$ comparable to the height of the trap above the surface <cit.>. Consider for example the simple
model ${\cal B}_{pp}({\bf r},{\bf r}')
\propto \exp(-|{\bf r}-{\bf r}'|/\lambda_c)$.
When the correlation length $\lambda_c$ is smaller than the
distance $d$ between the two sites, the off-diagonal terms
in the matrix $\alpha_{ij}$ [Eq. (<ref>)]
decay like $\alpha_{LR}\sim e^{-d/\lambda_c}$, while the diagonal
terms scale like
$\alpha_{jj}\sim V_c/V_{\phi_j}$, which is the fraction
of the mode volume ($V_{\phi_j}$) lying within a radius
$\lambda_c$ around the trap center. For very short correlation
lengths, $V_c\sim \lambda_c^3$.
It follows that when the correlation length is smaller than the
distance $d$, the off-diagonal terms $\alpha_{LR}$ become small
and a dephasing process takes place. This conclusion also
applies when the correlation function
${\cal B}_{pp}({\bf r},{\bf r}')$ is not exponential,
but Lorentzian, as computed in Ref. <cit.>.
The second example is noise with a long correlation length that changes the effective magnetic potential, as
happens typically with so-called “technical noise". Assume that one (or more) of the parameters
in the potential of Eq. (<ref>), such as $\omega_x$, $V_0$ or $d$ is fluctuating, or that another fluctuating
term such as $\delta V(t)=f(t)x$ is added to the potential.
If any of these variables (denoted by a generic name $v$)
fluctuates as $\delta v(t)$, where $\langle \delta v\rangle=0$ and $\int d\tau \langle \delta v(t)\delta v(t+\tau)\rangle=\eta>0$,
then the magnetic field correlation function has the form
${\cal B}_{pp}({\bf r},{\bf r}',0)\propto (\partial V({\bf r})/\partial v)(\partial V({\bf r}')/\partial v)\eta$
so that
${\cal A}({\bf r},{\bf r}')$ in Eq. (<ref>) can be factorized into a product of a function of ${\bf r}$ and the same function at ${\bf r}'$.
This implies that $\alpha_{ij}$ can be also factorized as $\alpha_{ij}=\beta_i \beta_j$,
where $\beta_j\propto \int d^3 r\ (\partial V({\bf r})/\partial v)|\phi_j({\bf r})|^2$.
We then get a dephasing rate $\gamma_p\propto (\beta_L-\beta_R)^2$.
It follows that for noise of technical origin, dephasing is expected
whenever the potential changes due to the magnetic fluctuations are asymmetric. For example,
if the magnetic field fluctuations create a linear slope,
$\delta V({\bf r}) = f(t) x$, then
with $\langle f(t)f(t')\rangle=\eta\delta(t-t')$ we obtain $\gamma_p=\frac{1}{2}(d/\hbar)^2\eta$, where $d$ is the
distance between the centers of the two wells. This is indeed what we expect for white phase noise $\delta\varphi(t)\sim f(t)d/\hbar$
between the two sites.
If the two wells are not fully separated, then we should expect other terms in the master equation, which
will typically reduce the dephasing rate. However, for magnetic
fields with short correlation lengths, we should also
expect the appearance of cross terms $\hat{a}_L^{\dag}\hat{a}_R$
in the master equation. These are driven by spatial gradients
of the magnetic field (see Refs. <cit.>
for a more detailed discussion).
On the Bloch sphere, cross terms correspond to random rotations around the $S_1$-axis (tunneling rate fluctuations) and around the $S_2$-axis (number noise).
For completeness, we include such terms in the following discussion.
Although they are expected to be small,
they may be amplified by atomic collisions,
as we shall see below.
§.§ Loss
The magnetic fields transverse to the quantization axis
lead to a loss interaction
V̂_loss =
∑_k ĉ_k^†
+ h.c., where
g_kj(t)=-μ_F√(F)∫d^3r B_-(r,t)ζ_k^*(r)ϕ_j(r)
, and $B_-$ is a complex-valued circular component
that lowers the angular momentum by one unit
[$B_- = (B_x + {\rm i} B_y)/\sqrt{2}$
if the quantization axis is along $z$].
The relevant loss term in the master equation for the atoms
in the two wells is then given by
L_lossρ=-∑_i,j=L,Rγ^ij_loss/2[â_i^†â_jρ+ρâ_i^†â_j-2â_iρâ_j^†], where the loss rates are given by
\begin{eqnarray} \gamma^{ij}_{\rm loss} &=& \frac{1}{\hbar^2}\sum_k \int d\tau e^{i\omega_k\tau}\langle g_{ki}(t) g^*_{kj}(t+\tau)\rangle
\nonumber \\[6pt]
&=&\frac{\mu_F^2 F}{\hbar^2}
\sum_k \int d^3{r}\int d^3{r}' \\
&&\times\ \zeta_k({\bf r})\phi_i^*({\bf r})
\phi_j({\bf r}')\zeta_k^*({\bf r}')
{\cal B}_{-+}({\bf r},{\bf r}',\omega_k)
\nonumber
\\[6pt]
{\cal B}_{-+}({\bf r},{\bf r}',\omega)
&=& \int d\tau e^{i\omega\tau} \langle B_-({\bf r},t)B_+({\bf r}',t+\tau)\rangle\,,
\end{eqnarray}
${\cal B}_{-+}({\bf r},{\bf r}',\omega)$
is the spectral density of transverse magnetic field fluctuations,
$B_+ = B_-^*$,
and $\omega_k$ contains the Zeeman and kinetic energies of the
non-trapped level.
If we assume that the magnetic spectrum is flat over the range
of energies $\omega_k$, then we can use the completeness relation
of the non-trapped states
$\sum_k \zeta^*_k({\bf r})\zeta_k({\bf r}')=
\delta({\bf r}-{\bf r}')$ to obtain
γ^ij_loss = μ_F^2 F/ħ^2
where $\omega_Z$ is the average transition energy to the untrapped
Zeeman level.
If the magnetic noise spectrum depends weakly on the position
${\bf r}$ over the trap region, the orthogonality relations
between the modes
$\phi_L$ and $\phi_R$ imply that
the off-diagonal loss rates vanish and we are left with
γ^ij_loss = δ_ijμ_F^2 F/ħ^2
where ${\bf r}_t$ represents the region occupied by the atoms.
Compared to the dephasing rate $\gamma_{p}$
[Eq. (<ref>)] of the previous subsection,
loss is harder to suppress: it occurs whenever the spectrum of the magnetic fluctuations has significant
components at the transition frequency $\omega_Z$ and does not depend on the correlation length of the field.
However, for Johnson noise at trap-surface distances of the same order as the distance between the wells,
the two rates happen to be similar
since the magnetic noise spectrum
is typically quite flat in frequency
and not strongly anisotropic <cit.>.
Let us also note that several suggestions exist on how to suppress the overall Johnson noise (e.g., Ref. <cit.>), and in addition, how to suppress specific components of the field, such as those contributing to $\gamma_p$ <cit.>. For experimental measurements of loss rates, see Sec. <ref>.
§ COMBINED DYNAMICS
In order to investigate the combined dynamics of tunneling and interactions in the presence of noise we make
some analytical estimations of the expected decoherence rate and also solve the master equation numerically
for specific configurations. We assume that the atomic gas
is initially in the lowest energy
eigenstate for a specific number of atoms $N$. Usually $N=50$
in our numerical simulations, and both sites are symmetric
with $N/2$ occupation on average. We then
turn on the noise source and follow the coherence of the Josephson junction as a function of time.
§.§ Number conserving noise (no loss)
For visualizing the combined effects of the nonlinear Hamiltonian
dynamics of Sec. <ref> and the magnetic noise,
a semiclassical picture involving the distribution over the
spherical phase space (Bloch sphere) is illuminating. For
example, the master equation generated by the dissipative
term of Eq. (<ref>) can be represented by adding
a term $f_3(t){\hat S}_3$ to the Hamiltonian where $f(t)$ is
an erratic driving amplitude (a random process with zero average
and short correlation time <cit.>).
This may be viewed as a realization of a Markovian stochastic process such that upon averaging,
⟨f_3(t)f_3(t')⟩= 2γ_3δ(t-t') .
On the Bloch
sphere, such a driving term corresponds to a random sequence
of rotations around the “north-south” (or $S_3$-) axis.
The phase-space distribution in Fig. <ref> then diffuses
in the $\varphi$-direction so that the relative phase between
the left and right wells gets randomized (phase noise).
The parameter $\gamma_3$
is the corresponding phase diffusion rate.
This is consistent with the fact that the additional Hamiltonian
corresponds to a random bias of the double-well potential,
\begin{equation}
f_3(t){\hat S}_3 = \frac{ f_3(t) }{ 2 } (\hat{n}_L - \hat{n}_R)\,.
\label{eq:interp-S3-drive}
\end{equation}
Similarly, we also consider random rotations around
other axes of the Bloch sphere, for example
\begin{equation}
f_2(t) \hat{S}_2 = - i \frac{ f_2(t) }{ 2 }
(\hat{a}_L^\dag \hat{a}_R - \hat{a}_R^\dag \hat{a}_L)\,.
\label{eq:S2-drive}
\end{equation}
This corresponds to inter-site hopping with a
random amplitude and is generated by fluctuating
inhomogeneous magnetic fields (Sec. <ref>).
On the Bloch sphere of Fig. <ref>,
we then have, near $\varphi=0$, random kicks that spread the phase-space
distribution along the “north-south” (or $\theta$) direction.
They change the population difference which is proportional (for $|\theta-\pi/2|\ll 1$)
to $\cos\theta \approx \pi/2 - \theta$, and we denote $\gamma_2$ as
the corresponding diffusion rate (number noise).
Note that the effect of rotations around the $\hat{S}_1$-axis [involving a $+$ sign on the right-hand side of Eq. (<ref>)] on phase-space points near $\varphi=0,\theta=\pi/2$ is more intricate. It amounts to random fluctuations of the tunneling rate $J$ in the Hamiltonian [Eq. (<ref>)] and will be discussed briefly at the end of this subsection.
The actual dynamics must
also consider the other parts of the Bose-Hubbard
Hamiltonian [Eq. (<ref>)].
Consider first the interaction-free case $U = 0$ with the squeezing
parameter $\xi=1$. Classical trajectories in the vicinity of the ground state are perfect circles
on the Bloch sphere
centered on the $S_{1}$-axis.
When the Rabi-Josephson rotation is faster than the diffusion/kicking rate, both types of noise above produce isotropic 2D diffusive spreading of the phase-space distribution, with a diffusion rate $\gamma_\alpha$ ($\alpha = 2,3$). Thus, at large times $t$, one obtains a circular Gaussian phase-space distribution whose radial variance grows linearly as $\Delta^2=2\gamma_\alpha t$. Substitution into Eq. (<ref>) gives the expected exponential decay $S/s\propto \exp{[-\gamma_\alpha t]}$.
This regime flattens out when the state has spread over the
entire Bloch sphere, $\Delta \sim \pi$.
Now, when interactions are present, it is evident that the two types of noise will give different results.
The Josephson trajectories are squeezed, with a $1/\xi^2$ ratio between their principal axes (Fig. <ref>).
The combined action of phase noise ($\alpha = 3$) and Josephson rotation gives an elliptical distribution whose variances are $\Delta_2^2 = 2\gamma_{3} t$ and $\Delta_{3}^2=2\gamma_{3} t/\xi^4$ (inner blue ellipse in Fig. <ref>). In contrast, for number noise ($\alpha = 2$), the distribution at time $t$ has variances $\Delta_2^2 = 2\xi^4 \gamma_{2} t$ and $\Delta_{3}^2=2\gamma_{2} t$ (outer blue ellipse in Fig. <ref>).
Therefore, using Eq. (<ref>) the one-particle coherence decays
S/s ∝exp(-Γ_3,2 t) ,
with effective decay rates
Γ_3,2 = γ_3,2/2(1+ξ^∓4).
Thus, for phase (number) noise the decoherence rate is suppressed (enhanced) in the presence of interactions with respect to the interaction-free case, respectively.
The best suppression factor one can expect in the former case is a factor of two,
i.e., $\Gamma_{3} = \gamma_{3}/2$.
This limiting factor of $2$ in the suppression can be understood in the following way: in Fig. <ref> the interactions can restrict the vertical spreading but not the horizontal spreading. This evolution then leads to a distribution with a relatively small $\Delta n$ but a large $\Delta \varphi$. In other words, observing the dashed circle and inner ellipse in Fig. <ref>, no matter how much interactions shrink the ellipse, its $\Delta\varphi$ width will be the same as it was without interactions. The coherence of a distribution tightly squeezed along the equatorial line with this $\Delta\varphi$ is precisely half that of the dashed circle.
It should be noted that our model does not take into account
processes that relax the system towards its ground state and
actually create phase coherence.
In the derivation of Eq. (<ref>), it was assumed that the Josephson oscillations spread the semiclassical ensemble throughout the pertinent classical orbit.
However, it may happen that diffusion due to the
noise leads to an ensemble with an oscillating width. This is particularly noticeable at short times and if the noise is switched on faster than the Josephson period.
The combination of anisotropic diffusion and motion along squeezed elliptical
classical trajectories then leads to a “breathing” ensemble and a coherence that oscillates
with half the Josephson period about the decaying coherence value of Eq. (<ref>), as we show below.
For a more quantitative analysis, we use a master equation with a dissipative term of the structure of Eq. (<ref>) to derive rate
equations for the relevant bilinear observables of the Josephson junction: the occupation $b^\dag b$ and the “anomalous occupation” $bb$.
Let us begin with number noise where the $\hat{S}_{3}$
operator is replaced by $\hat{S}_{2}$, according to the
random rotation picture of the preceding paragraph.
We obtain the dynamical equations
\begin{eqnarray} \frac{d}{dt}\langle \hat{b}^{\dag}\hat{b}\rangle &=& \gamma_{2}\frac{N\xi^2}{4} \\
\frac{d}{dt} \langle\hat{b}^2\rangle &=& -2i\omega_J\langle\hat{b}^2\rangle-\gamma_{2}\frac{N\xi^2}{4}~, \end{eqnarray}
where $\gamma_2$ is the diffusion rate related to number noise.
This has the simple solution,
\begin{eqnarray} \langle \hat{b}^{\dag}\hat{b}\rangle_t &=& \langle \hat{b}^{\dag}\hat{b}\rangle_0
+\gamma_{2}\frac{N}{4}\xi^2 t \\
\langle \hat{b}^2\rangle_t &=& \langle \hat{b}^2\rangle_0 e^{-2i\omega_J t}
+i\frac{\gamma_{2} N\xi^2}{8\omega_J}(1-e^{-2i\omega_J t})\,. \end{eqnarray}
Substitution into Eq. (<ref>) gives the coherence evolution for an initial state where $\langle \hat{b}^2\rangle_0 = 0$
(this applies to the ground state and in thermal equilibrium),
⟨Ŝ_1⟩_t= ⟨Ŝ_1⟩_0
-γ_2 ξ^2/4[(ξ^2+ξ^-2)t-sin2ω_J t/2ω_J (ξ^2-ξ^-2)],implying that
cos^2ω_J t+ξ^2 sin^2ω_J t].
Thus, the decay rate oscillates between $\gamma_{2}$ and $\gamma_{2}\xi^4$ with the average rate
given by $\gamma_{2} (1+\xi^4)/2$, as in Eq. (<ref>).
Similarly we obtain for phase noise (random kicks around
the $S_{3}$-axis),
cos^2ω_J t+ξ^-2 sin^2ω_J t], implying oscillations of the decay rate between $\gamma_{3}$ and $\gamma_{3}\xi^{-4}$ with an average rate of $\gamma_{3}(1+\xi^{-4})/2$.
The predictions of Eqs. (<ref>), (<ref>), and (<ref>) are confirmed by exact numerical solutions of the master equation with the dissipative term having the form of Eq. (<ref>).
As we showed in Sec. <ref>, magnetic noise in a double-well trap leads mainly to phase noise represented
by an erratic field proportional to $\hat{S}_{3}$. However, in the spirit of the general discussion above and for
completeness of the discussion, we have also
calculated the evolution of the coherence for $\hat{S}_{2}$ and $\hat{S}_{1}$
as driving forces: random rotations around the
$\hat{S}_{2}$-axis that generate number
noise, and around the $\hat{S}_{1}$-axis that generate tunneling-rate fluctuations.
In Fig. <ref> we present the evolution of coherence for given parameters (see caption) and different rotation axes.
The decay rates $\Gamma_{\alpha}(t)\equiv -\partial/\partial t[\log \langle \hat{S}_{1}\rangle_t]$
for the noise rotating around $\hat{S}_{\alpha}$ are presented in Fig. <ref>b ($\gamma_{\alpha}=\gamma$ for $\alpha = 1, 2, 3$).
The decoherence rate
for rotations around $\hat{S}_{1}$ (cyan curve) oscillates between $\Gamma_{1}=0$ and $\Gamma_{1}=\gamma$ and then starts
to increase gradually.
The decoherence rate $\Gamma_{3}$ (phase noise, blue curve) oscillates between $\gamma$ and $\gamma/\xi^4$
and then stabilizes with the average value $\Gamma_{3}\approx \gamma(1+\xi^{-4})/2$ of Eq. (<ref>).
The fastest decay of coherence is found for number noise (red curve, $\alpha = 2$) because random rotations around this axis significantly displace the squeezed ensemble from its equilibrium position.
The rate
first oscillates between $\gamma$ and $\gamma\xi^4$ and then stabilizes with the average value
It gradually decreases to smaller values when the coherence is already small.
The decay of the decoherence rate oscillations may be interpreted within the semiclassical approach as
following from the dispersion of Josephson periods for classical trajectories with different perimeters.
In the linearized excitation approach, this corresponds to
unequally spaced energy levels due to the deviation from the harmonic approximation.
Note that similar oscillations have been observed in Ref. <cit.>
for an oscillator and a certain scenario of spatial decoherence.
Decoherence process due to number-conserving noise in a double well. Time is scaled to the inverse of the tunneling rate $J$,
and the on-site interaction energy in the Hamiltonian [Eqs. (<ref>) and (<ref>)] is $U=0.25\,J$ with $N=50$ particles. The Josephson frequency is
$\omega_J=3.67\,J$, corresponding to a squeezing factor $\xi=1.92$.
(a) coherence as a function of time for random $S_{1}$, $S_{2}$ and $S_{3}$ rotations
with corresponding stochastic rates $\gamma_{\alpha}=\gamma=0.01\,J$.
For reference, we also plot the single particle
decoherence where $g^{(1)}_{LR}\propto e^{-\gamma t}$ (dashed line).
The inset shows the short time behavior. (b) instantaneous coherence decay rate for the 3 cases,
showing oscillations with approximately the Josephson frequency. The dashed line again shows the single-particle case for reference.
The decay of the coherence in the case of tunneling rate fluctuations
(random rotations around $\hat{S}_{1}$), which leave the system
invariant in the absence of interactions (circular distribution
in Fig. <ref>), can also be explained by the squeezing.
The rotation of an elliptical ground state changes it
and excites higher energy
states (or classical trajectories). After some time these excitations become similar to number excitations and
the decay rate increases.
§.§ Decoherence induced by loss
Loss of atoms from traps due to noise-induced transitions to untrapped internal atomic states is a very common
process in atom chip traps. Here we examine the possibility that a loss process leads not only to the reduction of the
total number of atoms in the trap, but also to decoherence of the remaining BEC. Let us begin by noting that while a loss process
does not usually heat up a BEC by transitions into higher-energy trap levels, it does increase the uncertainty in the number
of remaining atoms. If we imagine two BECs in two
separate traps which initially have a fixed number of atoms, then independent loss from both traps will lead to
uncertainty in the relative numbers in the two traps.
Due to interactions, this makes the chemical potential uncertain
and leads to dephasing if the phase between the two BECs is initially well defined. Does this process of independent loss from the two
traps, which is described by Eq. (<ref>), indeed lead to decoherence as
we have shown above for number noise?
Consider a trapped Bose gas of $N$ atoms in a double well. Atom loss from the left (right) well is described by the
application of the annihilation operator $\hat{a}_L$ ($\hat{a}_R$) to the system state. In the resulting state with $N-1$
atoms, the expectation value of a pseudo-spin operator is
⟨â_i^†Ŝ_αâ_i⟩_N/⟨n̂_i⟩_Nfor $i = L,R$.
Specifically we have
≈⟨Ŝ_3⟩_N∓1/2(1-4⟨n̂^2⟩_N/N), where the $+$ and $-$ refer to $L$ and $R$, respectively.
We have dropped terms of order $\langle \hat n\rangle/N$, assuming nearly equilibrated populations in the wells.
If the initial state is the ground state of non-interacting atoms,
it is also a eigenstate (coherent state) of
$\hat{S}_{1}$, and we have $\langle \hat{n}^2\rangle_N=N/4$.
The application of $\hat{a}_L$ or $\hat{a}_R$ does not change the coherence
nor the number difference: the state remains a coherent state of $N-1$ atoms.
However, if the $N$-atom state is the squeezed ground state of an interacting system,
then $\langle \hat{n}^2\rangle_N=N/4\xi^2$ and each atom lost from either well changes the
imbalance between the two wells by $\pm (1-1/\xi^2)$. It then follows that the loss process from an interacting
system is equivalent to random kicks in the number direction,
similar to the number noise considered above.
Then the decoherence rate would be
Γ_dec∼γ_loss1-1/ξ^2/2N[c(t)+s(t)ξ^4] ,
where $c(t)$ and $s(t)$ are functions that we would expect to behave like $\cos^2\omega_J t$ and $\sin^2\omega_Jt$
for short times and then stabilize with their average value $1/2$.
In Fig. <ref> we show the time evolution of the coherence due to a loss process described by a dissipative
term as in Eq. (<ref>), which is driven by the annihilation operators $\hat{a}_L$ and $\hat{a}_R$
from both wells. For the parameters considered in Fig. <ref>, the decoherence rate is much smaller than the
loss rate, but it grows with the strength of the interaction $u$. The results of the numerical
simulation show that the initial rate of decoherence is close to zero [$c(0)=0$ in Eq. (<ref>)],
rather than to the lower value $\gamma_{\rm loss}
(1 - 1/\xi^2)/2N$.
However, the value of $s(t)$ is close to the expected value $\sin^2\omega_J t$.
Decoherence induced by loss. A loss rate $\gamma_{\rm loss}=0.08\ J$ is applied, such
that after $t=10/J$ only $\sim 45$% of the atoms are left in the trap. During this time the coherence
drops due to interactions. We have used $N=50$ atoms in the numerical simulation, showing two curves, for
onsite interaction strengths $U = 0.2\,J$ [$u=10$] and $U = J$ [$u=50$]. (a) coherence as a function of time. (b) instantaneous rate of decoherence,
showing oscillations with half the Josephson period [$\omega_J = 3.32\,J$ and $7.14\,J$ for the two curves].
For relatively strong interactions,
the decoherence rate may become
larger than the loss rate itself. This occurs when $\xi^4/2N
\sim u / 2 N > 1$, and may even happen in the Josephson
regime [see Eq. (<ref>)].
§ REALIZABILITY AND VALIDITY
Model validity range: the parameters of the Bose-Hubbard model for
a trapping potential having the form of Eq. (<ref>) with parameters as in Fig. <ref>
(transverse frequency $\omega_{\perp}=2\pi\times 500$ Hz) and
the number of atoms $N=100,200,300,400,500,600$ [bottom to top in (a)]. (a) The longitudinal chemical potential $\mu_{\parallel}$
as a function of the barrier height $V_0$. We define
the validity range for each value of $N$ as the range where
$V_0>\mu_{\parallel}$ (to the right of the dashed line representing $\mu_{\parallel}=V_0$).
This validity criterion is presented in (b)-(d) by simply not drawing the curves where they do not adhere to this criterion.
(b) The tunneling matrix element $J$, (c) the Josephson frequency $\omega_J$, and
(d) the squeezing parameter $\xi^2$. The range where $\xi^2>N$ [dashed portions of the curves in (b)-(d)] is the Fock regime, where the coherence of the ground state drops to zero.
In this work we are interested in the
Josephson regime of Eq. (<ref>), where $\xi^2<N$.
When $\xi^2>2\sqrt{N}$, loss-induced decoherence is faster than the loss rate, as
happens for most of the curves in (d).
The two-site Bose-Hubbard model, which was used in the previous sections, is fully valid (namely, applicable over the
whole phase space shown in Fig. <ref>) only if the atom-atom interaction energy is much smaller than
the energy of excited modes in each well, such that higher-energy spatial modes are not
excited by the interaction.
For a single-well frequency $\omega$ this criterion implies
$gn\ll \hbar\omega$, where $n$ is the peak atomic density.
This requirement is equivalent to the healing length
$l_c=\hbar/\sqrt{mgn}$ being much longer than the trap width $L\sim \sqrt{\hbar/m\omega}$.
However, dynamics similar to that predicted by the two-mode model, such as Josephson oscillations and phase oscillations
of self-trapped populations, appear in mean-field (GP) calculations even if the interactions are much stronger,
as long as the atomic density changes during the evolution are small.
Since the GP approximation represents the classical limit of the two-site model, this suggests that the model
is valid for such dynamics, in which the populations in the two sites do not deviate much from their equilibrium
values and the spatial density can be derived from the two stationary modes $\phi_L$ and $\phi_R$.
Here we are interested in slow dynamics during which the population does not vary considerably, since the main change
is in the one-particle coherence.
With the aid of the mean-field approach described in Sec. <ref>,
we therefore require that the dynamics involves only low-lying excitations of the two-mode system, whose energy is lower
than the energy splitting of higher spatial modes $E_j$ ($j\geq 2)$
[see Eq. (<ref>)], such that only the pair $\phi_0$ and $\phi_1$ is populated.
In addition, the Hamiltonian [Eq. (<ref>)] neglects interaction terms involving different modes, such as
a term proportional to $\hat{n}_L\hat{n}_R$ <cit.>,
which may be significant if the two main modes $\phi_L$ and $\phi_R$ are not well separated in space.
We therefore define a simple validity criterion: that the barrier height $V_0$ is larger than the
longitudinal energy $\mu_{\parallel}$ of the condensate mode $\phi_0$.
With this condition we can show that the Josephson frequency $\omega_J$,
which defines the excitation energy in the two-mode system, is much smaller than the higher-mode excitation energy.
This ensures that, as long as low-energy states of the two-mode system are involved, higher spatial modes are
not excited and the system may still be described by the two modes.
Same as Fig. <ref> with transverse frequency $\omega_{\perp}=2\pi\times 100
$ Hz. Here $\xi^2<2\sqrt{N}$ for all curves, implying weak loss-induced decoherence over the parameter
In Figs. <ref> and <ref> we show the parameters of the Bose-Hubbard and
Josephson Hamiltonians in their validity range,
as calculated for $^{87}$Rb atoms in a potential having the form of Eq. (<ref>)
with $d=5\,\mu$m and $\omega_x=2\pi\times 200$ Hz, as in Fig. <ref>.
The strength of the interaction is determined both by the total number of atoms $N$
and by the transverse frequency $\omega_{\perp}/2\pi$, which is equal to 500 Hz in Fig. <ref>
and 100 Hz in Fig. <ref>. In all cases the Josephson frequency is not larger than $30$ Hz, which is smaller than
the excitation energy of higher modes, and thus ensures the validity of the model. However, for the
strongly interacting BEC in Fig. <ref> the squeezing factor $\xi^2$ may become larger than $N$, implying
that for such high barrier heights and low tunneling rates the system is in the Fock regime ($u>N^2$, represented
by the dashed curves). In this regime our model is not valid, since the coherence of the initial state
(the ground state) is already very low and the dynamics of coherence loss in general is dominated by phase diffusion.
Two differences between Fig. <ref> and Fig. <ref> are most noticeable. First,
the chemical potential of the BEC in tight confinement is much larger than that of the BEC in weaker
confinement, implying that the barrier height for which the two-mode model is valid is much lower for the latter.
Second, the amount of squeezing is correspondingly much larger for the tightly confined BEC in Fig. <ref>. It follows that for a given rate $\gamma_{\rm loss}$
of atoms from the trap, we would expect a high decoherence rate for the tightly confined BEC,
since $\gamma_{\rm dec}/\gamma_{\rm loss}\sim \xi^4/4N>1$ for most of the parameter range
in Fig. <ref>. Conversely, we should expect very low decoherence rates relative to the loss rate
in the case of weak confinement as in Fig. <ref>.
In the next section we present some experimental results that give realistic loss rates near
an atom chip. This will enable us to discuss expected decoherence rates for atomic
Josephson junctions in similar scenarios.
§ MEASUREMENTS OF ATOM LOSS NEAR AN ATOM CHIP
We investigate the expected dephasing due to the nearby surface by measuring the lifetime of atomic clouds held at varying atom-surface distances $z_0$.
The cloud temperature is about $\rm2\,\muK$; distances are measured by reflection imaging <cit.>.
Our data, adjusted for the lifetime due to vacuum of about $\rm30\,s$, are shown in Fig. <ref>.
For comparison with theory, let us for the moment ignore cascading effects ( transitions
$|2,2\rangle \longleftrightarrow |2,1\rangle \longrightarrow |2,0\rangle$
amongst Zeeman sub-levels that can re-populate the initial state),
and the finite lateral extent of the current-carrying wire ( we initially assume a layer of infinite extent).
We assume that the measured lifetimes $\tau_{\rm meas}$ are due to Johnson and technical noise, and we write <cit.>:
\begin{eqnarray}
\tau_{\rm meas}^{-1} &=&
\tau_{\rm Johnson}^{-1}+\tau_{\rm tech}^{-1}
\label{eq:measured}
\\[6pt]
\frac{ 1 }{ \tau_{\rm Johnson} } &=&
\frac{ c_1 }{ z_0^2 }
= \left(\frac{3}{8}\right)^2 \frac{\bar{n}_{\rm th}+1}{\tau_0} \left(\frac{c}{\omega}\right)^3 \frac{2h}{\delta^2 z_0^2}
\label{eq:Johnson}
\\[6pt]
\frac{ 1 }{ \tau_{\rm tech} } &=&
\frac{ c_2 }{ z_0^2 }
= \left(\frac{\mu_0\mu_F}{2\pi\hbar}\right)^2
\frac{ {\mathcal I}^2 }{ z_0^2 }\ ,
\label{eq:tech}
\end{eqnarray}
where $\bar{n}_{\rm th} = k_BT/\hbar\omega \approx 2\times10^7$,
and $\tau_0$ is
the free-space lifetime at the Larmor
frequency $\omega$ <cit.>.
Our experimental parameters are: gold wire with thickness
$h=\rm0.5\,\mum$; $\omega/2\pi=500\rm\,kHz$; and $T=\rm400\,K$ due
to Joule heating of the atom chip wire.
The skin depth $\delta \approx \rm130\,\mum$ is calculated with
a temperature-dependent resistivity $\rho( T ) = \varrho T$
for gold <cit.>. With these substitutions, the Johnson
lifetime is actually independent of $\omega$ and $T$. The
scaling with distance is valid in the intermediate regime
$h \ll z_0 \ll \delta$ <cit.>. The lifetime due
to technical noise involves the current noise spectrum ${\cal I}$
(in ${\rm A} / \sqrt{\rm Hz}$) <cit.>;
it scales with the same power of the distance $z_0$.
The experimental data of Fig. <ref> are fitted
on a logarithmic scale and conform to the exponent
$\tau_{\rm meas} \propto z_0^2$ very well. The fit yields
$c_1 + c_2 \approx 65\,{\rm \mu m^2/s}$.
From Eq. (<ref>), we estimate the coefficient $c_1$
for Johnson noise, finding $c_1 \approx 8.5\,\rm \mum^2/s$.
This yields in turn $c_2\approx56\,\rm \mum^2/s$ and
to lifetimes for Johnson and technical noise
of $\rm3\,s$ and $\rm0.4\,s$, respectively at $z_0=\rm5\,\mum$.
Re-introducing cascading and geometrical effects <cit.> doubles the estimated lifetime to $\rm6\,s$ for the Johnson noise component in these atom chip measurements
(thick green lines in Fig. <ref>, see caption for
details). The power law $\sim z_0^2$ for the lifetime
becomes invalid only for distances $z_0 < 2\,\rm\mum$, becoming comparable to the gold wire thickness.
Atom chip noise: lifetimes measured for a range of atom-surface distances. The black data points are for thermal $^{87}$Rb atoms, while the blue datum is for a BEC. For comparison, we show data for conditions similar to those of our measurements and over a similar range of distances, but using no current in the surface closest to the atoms and therefore exhibiting much weaker technical noise <cit.>). The solid red line is the best-fit line for an assumed quadratic dependence using our measurements for thermal atoms. The green curves are calculated for spin flips due to Johnson noise only. The solid green curve also accounts for cascading and
the lateral wire geometry <cit.>; the dashed
green curve is valid also at distances comparable to the gold
layer thickness but does not include cascading and the wire geometry;
the dotted green line is the simple power law of
Eq. (<ref>).
We conclude that in our experiment, technical noise is the dominant
cause of loss, by at least a factor of $6$.
We note that both the geometrical and cascading factors lengthen the calculated lifetime due to Johnson noise, so the conclusion that technical noise dominates our losses is reinforced. This may also explain
the difference in lifetime compared to Ref. <cit.>.
From Eq. (<ref>) we obtain a current noise spectral density of ${\mathcal I}=0.9\rm\,nA/\sqrt{Hz}$. This is much
larger than shot noise, by at least one order of magnitude,
strengthening the hypothesis of a technical origin for the
current noise.
§ DISCUSSION
In this work we have investigated the effect of noise on the decoherence of a BEC, initially in its ground state in a double-well
potential. Specifically, we have discussed three dephasing mechanisms:
(a) direct dephasing from short correlation length (Johnson) noise or asymmetric (technical or Johnson) noise. This can be represented by random rotations about the $S_3$-axis of the Bloch sphere. The phase-space distribution in Fig. <ref>
then diffuses in the $\varphi$-direction so that the relative phase between the left and right wells gets randomized (phase noise);
(b) dephasing due to population difference fluctuations (number noise).
On the Bloch sphere, these correspond to random rotations around the $S_2$-axis.
They may originate from an overlap between the spatial modes of atoms in the two sites and
fluctuating magnetic fields with a short correlation length.
While Fig. <ref> shows that rotation about the $S_2$-axis gives rise to an enhanced decoherence rate,
the physical source of these rotations is weak and its contribution may be assumed to remain small even when enhanced;
(c) dephasing due to losses induced by both Johnson and technical noise.
This decoherence process involves an enhancement effect similar to that of number noise, which may make it dominant
for strong atom-atom interactions.
Of the three dephasing mechanisms examined, we find that two are dominant: direct dephasing (a) and loss-induced decoherence (c). Direct dephasing can be suppressed by atom-atom interactions (up to a factor of 2). However, interactions play quite a different role in the context of loss-induced decoherence. While loss induced by noise has no effect on the coherence of the remaining atoms if the initial state is a coherent state of non-interacting atoms,
it may produce considerable decoherence if atom-atom interactions are dominant. Such enhanced decoherence would appear for any kind of noise which tends to change (randomize) the relative number of particles in the two wells.
It is worthwhile to note a general result following from our derivation. Although we have shown that symmetric noise with a long correlation length does not lead to direct dephasing
of the type (a) above if the density matrix of the system is diagonal in the basis of states with a well-defined total number $N$,
a master equation with a stochastic term as in Eq. (<ref>) may still lead to the decay of off-diagonal density matrix elements.
If one would adopt a symmetry-breaking approach to BEC <cit.>, where
the density matrix involves superpositions between different number states, these would decay under this
interaction. This scenario would provide a typical example of a dynamically emerging “superselection rule” stating that superpositions of different atom numbers are forbidden.
Our results are well established, as long as the two-mode model is valid, since their main
characteristics are derived independently
by three different approaches: semiclassical phase-space methods, analytical calculations using a
linear excitation approximation, and exact numerical calculations for relatively small atom numbers.
However, we note two limitations of our model. First, it does not take into account possible effects of heating or cooling of the Bose gas
due to transitions between the two main modes and higher-energy spatial modes in the trap.
Such transitions may be driven by components of the external noise which have correlation lengths on the order
of the single trap width, or by external magnetic fields which cause deformations of the potential with higher-order
spatial dependence. Such processes may affect the dynamics of decoherence in the double-well trap
but are beyond the scope of this paper.
Second, our model is valid for a BEC with a macroscopic number of
atoms, while dephasing in a system of a few atoms, which may be relevant to atomic circuits, would have to be
treated separately.
Finally, the results of this work allow an estimation of the accessible range of parameters for an atomic Josephson junction
permitting operation over a reasonable duration of time without significant decoherence. As we have shown in
Sec. <ref>, typical values of the Johnson noise at a distance of 5 $\mu$m from the surface
cause losses at a rate of less than
0.5 s$^{-1}$. At this distance, the rate of dephasing due to Johnson noise is expected to be on the same order
as the loss rate and we therefore expect that such dephasing will enable coherent operation for a time scale of a few seconds.
This time scale could even be doubled in the presence of squeezing due to interactions,
as predicted by our theory in Sec. <ref>. The main source of noise in our experiments is found to be technical
noise, which is not expected to directly cause dephasing due to its long correlation length, provided it does not contain strong
asymmetric components. Technical noise, however, induces loss, and decoherence due to loss is expected to be significant if the BEC is strongly interacting due to tight transverse confinement as in Fig. <ref>. In this case we expect that the squeezing factor is so large that the coherence time is shorter than the trapping lifetime. However, in the case of weak confinement, as in Fig. <ref>,
we expect that the decoherence rate due to loss is smaller by a factor of $\xi^4/4N\lesssim 4/N$, which is much
smaller than the loss rate.
To conclude, at distances of a few $\rm\mu m$ (for which accurately controllable tunneling barriers may be formed) and within the framework of a total spatial decoherence rate equal or smaller than the loss rate, we find that tunneling rates of about $0.1-10$ Hz, or Josephson oscillation frequencies $\omega_J/2\pi$ of about $2-25$ Hz, may be obtained (Fig. <ref>), depending on the barrier height, degree of transverse confinement, and number of atoms. This provides a wide dynamic range in the operation of a tunneling barrier for atomtronics.
We thank David Groswasser for support with the experimental system. This work is funded in part by the Israeli Science Foundation (1381/13), the European Commission “MatterWave” consortium (FP7-ICT-601180), and the German DFG through the DIP program (FO 703/2-1). We also acknowledge support from the PBC program for outstanding postdoctoral researchers of the Israeli Council for Higher Education. MK acknowledges support from the Ministry of Immigrant Absorption (Israel) and AV acknowledges support from the Israel Science Foundation (346/11).
|
1511.00374
|
The performance of scintillator counters with embedded wavelength-shifting
fibers has been measured in the Fermilab Meson Test Beam Facility using
$120~\mathrm{GeV}$ protons. The counters were extruded with a titanium dioxide
surface coating and two channels for fibers at the Fermilab NICADD facility.
Each fiber end is read out by a $2\times2~\mathrm{mm^2}$ silicon
photomultiplier. The signals were amplified and digitized by a custom-made
front-end electronics board. Combinations of $5\times2~\mathrm{cm^2}$ and
$6\times2~\mathrm{cm^2}$ extrusion profiles with $1.4$ and
$1.8~\mathrm{mm}$ diameter fibers were tested. The design is intended for the
cosmic-ray veto detector for the Mu2e experiment at Fermilab. The light yield as
a function of the transverse and longitudinal position of the beam will
be given.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The Mu2e experiment at Fermilab intends to make the most sensitive measurement
of the neutrinoless, coherent conversion of muons into electrons in the field of
a nucleus tdr<cit.>:
\[
\mu+N \to e+N
\]
This process is an example of charged lepton flavor violation, and its detection
would be unambiguous evidence for physics beyond the Standard Model.
The layout of the Mu2e experiment <cit.>.
The layout of the Mu2e experiment is shown in
Figure <ref> <cit.>. The high-intensity pulsed
proton beam impacting the production target produces muons among other
particles. The muons propagate through the transport solenoid and roughly half
are captured in the stopping target. If a muon is converted into an electron
via the conversion process, a $105~\mathrm{MeV}$ electron would be ejected
from the stopping target, which should be detected by the tracker and calorimeter.
One major background source for the Mu2e experiment is cosmic ray-induced muons
faking signal electrons by interacting with the detector materials <cit.>.
Simulations show such background events occur at a rate of approximately one per
day. For Mu2e to be able to reach its sensitivity goal, this background must
be suppressed to $\sim$0.1 event over the entire 3-year lifetime of the
experiment <cit.>. In order to combat this background, Mu2e relies on a
cosmic ray veto (CRV) system.
A schematic of the cosmic ray veto (CRV) detector. It covers the detector
solenoid and part of the transport solenoid shown in
Figure <ref>, but there is no coverage underneath the detector.
Figure <ref> shows a schematic view of the Mu2e CRV
detector <cit.>.
The CRV covers the majority of the detector solenoid and about half of the
transport solenoid shown in Figure <ref> <cit.>. Simulations
have shown that the vetoing efficiency of the CRV has to be at least $99.99\%$
and must withstand an intense radiation environment. Each side of the CRV
consists of four staggered layers of scintillator counters
(Figure <ref>). Each scintillator counter contains two embedded
wavelength shifting fibers.
A schematic drawing of the cross-sectional view of the CRV counters.
One rectangular box corresponds to a scintillator counter with a thickness of
$20~\mathrm{mm}$, a width of 50 or $60~\mathrm{mm}$, and a length of up to
$6,600~\mathrm{mm}$. Each counter contains two holes for the wavelength-shifting
fibers which propagate the light signals to the silicon photomultipliers on both
ends. Two scintillator counters are glued together to form one di-counter.
The scintillator counters were tested at Fermilab in a $120~\mathrm{GeV}$ proton
beam fermi<cit.>. The goal of the test was to measure the lateral and longitudinal response
of the scintillator counter at different beam angles of incidence with 50 and
60-mm wide counters containing 1.4 and 1.8-mm diameter fibers.
A small fraction of the counters in the detector are read out only from one end yet
all the counters tested have readouts on both ends.
§ TEST BEAM SETUP
Figure <ref> shows a photograph of the test beam setup. The
scintillator counters were mounted on the test stand with the coordinate system
shown in Figure <ref>. The 120 GeV protons are minimum ionizing, so
they deposit a similar energy as minimum-ionizing muons. When a
charged particle traverses the counter and deposits energy in the
scintillator, light is produced which is captured by the
wavelength-shifting fibers shown in Figure <ref>.
The light is retransmitted to the ends of the counter where it is detected and
amplified by Silicon Photo-Multipliers (SiPMs) shown in
Figure <ref>. The SiPMs are mounted on small carrier boards that sit
in wells which allow them to be pushed up against the fibers. A counter
motherboard (CMB) as shown in Figure <ref> with spring-loaded pins makes
electrical contact with the SiPM carrier boards. Signals from up to 16 counter
motherboards are sent to a front-end board by HDMI cables where they are
amplified and digitized.
A photograph of the test beam setup. The coordinate system is indicated
in red showing the direction of the $x$- and $y$-axes together with the origin.
The yellow arrow indicates the proton beam direction.
Top: Two scintillator counters are glued together to
form one di-counter. The extrusions are coated with titanium dioxide so
light cannot travel from one counter to the other. Two extrusion sizes were
tested: $20~\mathrm{mm}\times50\mathrm{mm}\times3000~\mathrm{mm}$ and
$20~\mathrm{mm}\times60\mathrm{mm}\times1820~\mathrm{mm}$. Each extrusion is
equipped with two wavelength-shifting fibers as indicated by the red circles. The
fibers (Kuraray Y11 non-s type, 175 ppm) are not glued in their channels. Two
fiber diameters were tested: $1.4~\mathrm{mm}$ and $1.8~\mathrm{mm}$.
Bottom: Fibers are glued in place to the fiber guide bar and polished using a
diamond flycutter.
Photograph of a Silicon Photo-Multiplier (SiPM) used to read out the fibers. We use the
Hamamatsu S13360-2050VE SiPM with a photosensitive area of
$2~\mathrm{mm}\times2~\mathrm{mm}$ and a pixel pitch of $50~\mathrm{\mu m}$.
Each SiPM contains approximately 1,600 pixels.
The counter motherboard (CMB) has two LED
flashers, a thermometer and spring-loaded pins that push the SiPM carrier boards
(not shown) against the fibers. One CMB is mounted on each end of a di-counter,
so there is a total of two CMBs per di-counter.
The counters are mounted on a rotatable support fixture on a table capable of
translational motion, which allows us to place the proton beam at different positions on the
counter and to vary incident angles. The change in incident angle changes the
path length of the protons through the scintillator and therefore
changes the light yield. A coincidence of beam scintillator counters read out
by photomultiplier tubes is used to provide a trigger. Upon receipt of a trigger
signal, the SiPMs were read out for $1.6~\mathrm{\mu s}$ with their signals
digitized every $12.6~\mathrm{ns}$. Four multiwire proportional chambers (MWPC),
two upstream and two downstream of the CRV counters, provide tracking
information <cit.>, which allow us to reconstruct the proton
trajectory to within $0.25~\mathrm{mm}$ at the counters.
§ DATA ANALYSIS
Figure <ref> shows a typical event with all four channels on an
extrusion. The pulses before the signal window are dark current peaks, which are
used for calibration.
A typical event is shown here. Each TDC corresponds to about
$12.6~\mathrm{ns}$. Nominal pedestal has been subtracted. In the figure four channels of a single scintillation
counter are shown: channels 2 and 3 of FEB1 are on the $+\hat x$ end; channels 2
and 3 of FEB2 are on the $-\hat x$ end. The peak in the signal region is caused by a
proton, and the small signal in the pre-signal region is a dark current pulse.
§.§ Calibration
Since the SiPMs produce resolvable single pixel peaks, a calibration can be
carried out to convert ADC units to the number of pixels. The distribution of
the dark current peaks allows such a conversion. As shown in
Figure <ref>, by fitting a function to the distribution, the
positions of the peaks can be extracted and the average ADC per pixel fired can
be calculated. The number of pixels fired is proportional to the number of
incident photons with crosstalk. To find the photoelectron yield the crosstalk
needs to be determined as discussed later.
The dark current (noise) spectrum of a single SiPM. The peaks
correspond to 0, 1, 2, 3 …pixels that fire in the SiPM.
§.§ Transverse and Longitudinal Response
After calibration the number of pixels fired in the SiPM for each
event can be calculated. Then the transverse
and longitudinal response of the extrusion can be studied.
Transverse response of a 100-mm wide di-counter. The
illustration shows the number of pixels fired (sum of the two SiPMs on the same
end of the extrusion) as the beam scanned through the width of the di-counter at
$x=1,400~\mathrm{mm}$. Features of the counter, including the gap between the
counters and the fiber holes are visible.
Figure <ref> shows the transverse response with the beam
incident at $x=1,400~\mathrm{mm}$. The total number of pixels fired in the
two channels on the $-\hat x$ end of the extrusion is plotted. From the figure it can
be observed that the number of fired pixels is relatively constant across the
width of the counter. The edges of the counter are well defined.
The signal size does not fall below 50% until the beam incident position
reaches about $0.5~\mathrm{mm}$ from the edges. The gap between the two
extrusions is clearly evident. It has an effective width of about
$1~\mathrm{mm}$. Dips of 20% to 25% in the signal are also present at the
locations where the fiber holes are. Such dips are expected and arise due to
the reduced path length in the scintillator when the proton passes through one
of the fiber holes. The difference between the total number of fired pixels in
the two counters is very likely due to the different crosstalk levels between the
SiPMs, which will be discussed in the following section.
Longitudinal response of two counters with different size fibers.
Figure <ref> shows the longitudinal response of the
extrusions with the beam centered between two fibers. Due to the limitation of
the test facility, a full-length scan was impossible; instead, half the length
of the extrusion was scanned ($0\sim1,400~\mathrm{mm}$). In
Figure <ref> points greater than $1,400~\mathrm{mm}$ come from
SiPM at the far ($+\hat x$) end. Each data point corresponds to the average
number of fired pixels of a single SiPM. The nice aggreement between the data
taken from different channels indicates a consistency among the SiPMs.
Simulations show the response of the counter along its length is consistent with
the fiber attenuation length. Figure <ref> also shows that the
light output of the 1.4-mm and 1.8-mm diameter fibers has a difference of about
15%.[Recent bench-test measurements using sources and cosmic ray muons
show a larger increase of almost 40%, which is expected. Misalignment at the
the $1.8~\mathrm{mm}$ fiber and the $2~\mathrm{mm}$ SiPMs is presumed to be
the reason for the lower increase observed here.]
Longitudinal response of two counters with different widths, both
with 1.4-mm diameter fibers.
Figure <ref> shows the small performance
difference between the 50-mm and 60-mm wide extrusions.
Close-up view of the longitudinal response at small x values. The
outputs of both SiPMs at the $-\hat x$ end have been summed.
Figure <ref> provides a
close-up view of the longitudinal response at small x values.A fall-off is
present in the longitudinal response towards the ends, which is
thought to be caused by photons being absorbed by the dark-colored fiber guide
bars. Detailed Monte Carlo studies are underway.
§.§ Photoelectron Yield and Cross Talk
We have been careful to only report the pixel yield of the devices. To obtain
the photoelectron yield from the pixel yield, the effect of cross talk needs to
be considered. By crosstalk we mean pixels that induce neighboring pixels to
fire, which inflates the pixel count. Hamamatsu, the SiPM provider, reports a
40% cross talk at their operating bias, where their definition is the ratio of
the 2-pixel peak to the 1-pixel peak in the noise response. The investigation of
the cross talk at the biases used in the test beam is in progress using several
different techniques. The results are not yet mature enough to be discussed
here. However, it should be pointed out that when using a preliminary crosstalk
correction, for example, Figure <ref> shows a much more uniform
response across the two counters. We find using these preliminary results that
photoelectron yields are roughly a factor of two less than pixel yield. On a
side note, the SiPM model tested is no longer being produced. A different device
with much lower crosstalk will be tested in a test-beam run early in 2016.
§ CONCLUSIONS
The Mu2e cosmic ray veto (CRV) system is an essential component of the Mu2e
experiment. The test beam measurements were carried out to understand the
performance of the scintillator counters used in the Mu2e CRV system to ensure
that they satisfy the design requirements.
The test results demonstrate a generally uniform response of the counter along
its width with a sharp drop-off $0.5~\mathrm{mm}$ away from the edges. Between
the two counters of a di-counter an effective gap of $1~\mathrm{mm}$
width is present. Fiber holes in the counters have a small effect on
the signal level when a particle passes through the hole. The response of the
counter along its length is consistent with the fiber attenuation length.
L. Bartoszek et al, “Mu2e Technical Design Report (TDR)”, arXiv:1501.05241.
“Fermilab Test Beam Facility”, $<$ftbf.fnal.gov/beam-overview/$>$.
“Presentation on Fenker Chambers”,
|
1511.00539
|
Gevery regularity with weight for Euler equation ]
Gevery regularity with weight for incompressible Euler equation in the half plane
F. Cheng W.-X. Li C.-J. Xu]
Feng Cheng Wei-Xi Li Chao-Jiang Xu
Feng Cheng, School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
Wei-Xi Li, School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, 430072 Wuhan, China
Chao-Jiang Xu,
School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
[2010]35M33, 35Q31, 76N10
In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted $L^2$- estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.
§ INTRODUCTION
In this paper we study the Gevrey propagation of solutions to incompressible Euler equation. Gevrey class is a stronger concept than the $C^\infty$-smoothness. In fact it is an intermediate space between analytic space and $C^\infty$ space.
There have been extensive mathematical investigations (cf.<cit.>, <cit.>, <cit.>, <cit.> <cit.>, <cit.>, <cit.>, <cit.>, <cit.> for instance and the references therein) on Euler equation in different kind of frames, such as Sobolev space, analytic space and Gevrey space. In this work we will consider the problems of Gevrey regularity with weight, and this is motivated by the study of the vertical boundary layer problem introduced in <cit.> which remains still open up to now. The related and preliminary work for the well-posedness of vertical boundary problem is to establish the Gevrey regularty with weight, and this is the main result of the present paper. In the future work we hope to investigate the vertical boundary layer problem, basing on the weighted Gevrey regularity of Euler equations.
Without loss of generality, we consider the incompressible Euler equation in half plane $\RR^2_+$, where $\RR^2_+= \{(x,y);x\in\RR, y\in\RR^+ \}$, and our results can be generalized to 3-D Euler equation. The velocity $(u(t,x,y),v(t,x,y))$ and the pressure $p(t,x,y)$ satisfy the following equation:
\begin{equation}\label{1.1}
\partial_t u + u\dd_x u+v\dd_y u+ \dd_x p = 0\quad \ \mbox{in}\ \RR^2_+\times(0,\infty),
\end{equation}
\begin{equation}\label{1.2}
\dd_t v+u\dd_x v+v\dd_y v+\dd_y p=0 \quad \ \mbox{in}\ \RR^2_+\times(0,\infty),
\end{equation}
\begin{equation}\label{1.3}
\partial_x u+\partial_y v=0,\quad \ \mbox{in}\ \RR^2_+\times(0,\infty),
\end{equation}
\begin{equation}\label{1.4}
v\big|_{y=0} =0, \quad \ \mbox{in}\ \RR\times(0,\infty),
\end{equation}
with initial data
\begin{equation}\label{1.5}
u|_{t=0}=u_0, \quad v|_{t=0}=v_0,\quad\ \mbox{on}\ \RR^2_+\times\{t=0\}.
\end{equation}
Here the initial data $(u_0,v_0)$ satisfy the compatibility condition:
\[
\dd_x u_0+\dd_y v_0=0;\quad v_0|_{y=0}=0.
\]
Before stating our main result we first introduce the (global) weighted Gevrey space.
Let $\ell_x,\ell_y\geq 0$ be real constants that independent of $x,y$, we say that $f\in G^s_{\tau,\ell_x,\ell_y}(\RR^2_+)$ if
\begin{equation*}
\sup_{\abs\alpha\geq 0} \frac{\tau^\alpha}{\abs\alpha!^s} \norm{\comii{x}^{\ell_x}\comii{y}^{\ell_y} \dd^\alpha f}_{L^2(\RR^2_+)}<\infty,
\end{equation*}
where and throughout the paper we use the notation $\comii{\cdot} =(1+|\cdot|^2)^{\frac{1}{2}}$.
In this work we present the persistence of weighted Gevrey class regularity of the solution, i.e., we prove that if the initial datum $(u_0,v_0)$ is in some weighted Gevrey space and satisfy the compatible condition, then the global solution belongs to the same space. With only minor changes, these results can also extend to 3-D Euler equation, and the global solution here will be replaced by a local solution.
Suppose the initial data $u_0\in G^s_{\tau_0,0,\ell_y}, v_0\in G^s_{\tau_0,\ell_x,0}$ for some $s\geq 1,\tau_0>0$ and $0\leq\ell_x,\ell_y\leq 1$. Then the Euler equation (<ref>)-(<ref>) admits a solution $u,v,p$ such that
\begin{eqnarray*}
u\in L^\infty([0,+\infty);~G^s_{\tau,0,\ell_y}), \quad {\rm and} ~v\in L^\infty([0,+\infty);~G^s_{\tau,\ell_x,0}),
\end{eqnarray*}
and p satisfies
\[
\dd_x p\in L^\infty([0,+\infty);\ G^s_{\tau,0,\ell_y}) \quad \dd_y p\in L^\infty([0,+\infty);\ G^s_{\tau,\ell_x,0}),
\]
where $\tau>0$ depends on the initial radius $\tau_0.$
We remark that the existence of smooth solutions to (<ref>)-(<ref>) is well developed (cf.<cit.>, <cit.>, <cit.>, <cit.>, <cit.> for instance), and in two-dimensional case smooth initial data can yield global solutions, while in the three-dimensional case the solution may be local in general condition. The appearance of the weight function increases the difficulty of estimating the pressure term, and for this part it is different from <cit.>. We also point out that in the whole space $\mathbb R^2$ or two dimensional torus $\mathbb T^2,$ the classical approach to analyticity or Gevrey regularity is that it makes crucial use of Fourier transformation, which can't apply to our case. Instead we will use the basic $L^2$ estimate (c.f. <cit.> for instance).
The paper is organized as follows. In section 2, we introduce the notation used to define the weighted Sobolev norms, and we prove the persistence of the weighted Sobolev regularity. In section 3, we state the priori estimate to prove the main theorem. Section 4 and 5 are consist of the proofs of these lemmas.
§ NOTATIONS AND PRELIMINARIES
In the following context, we use the conventional symbols for the standard Sobolev spaces \({ H}^m(\RR^2_+) \) with \(m\in \mathbb{N}\), and let $\norm{\cdot}_{H^m}$ be its norm. For the case \(m=0\), it was usually written as ${ L}^2(\RR^2_+)$. Denote $\left\|\cdot\right\|_{L^2}$ and $\left<\cdot,\cdot\right>$ be the norm and inner product in ${ L}^2(\RR^2_+)$. We usually write a vector function in bold type as ${\bf u}$ and a scalar function in it's conventional way as $u$. For a vector function ${\bf u}=(u,v)$ we denote
\[
\left\|{\bf u} \right\|_{H^m}=\sqrt{\norm{u}^2_{H^m}+\norm{v}^2_{H^m}}.
\]
And when we say that ${\bf u}\in { H}^m$, we mean that $u,v\in{ H}^m$.
With the notations above, we introduce the weighted Sobolev spaces ${ H}^m_{\ell_x}(\RR^2_+)$ and ${ H}^m_{\ell_y}(\RR^2_+)$, where $\ell_x,\ell_y$ are real constants. Let
{H}^m_{\ell_x}(\RR^2_+)=\left\{v\in{ H}^m(\RR^2_+);\quad \comii{x}^{\ell_x}\dd^\alpha v\in L^2, 1\leq\abs{\alpha}\leq m \right\},
and it's norm is defined by
\[
\norm{v}_{H^m_{\ell_x}}=\sqrt{\norm{v}_{L^2}^2+\sum_{1\leq\abs{\alpha}\leq m}\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}^2 }.
\]
Similarly, let
{ H}^m_{\ell_y}(\RR^2_+)=\left\{u\in{ H}^m(\RR^2_+);\quad \comii{y}^{\ell_y}\dd^\alpha u\in L^2, 1\leq\abs{\alpha}\leq m \right\},
equipped with the norm
\[
\norm{u}_{H^m_{\ell_y}}=\sqrt{\norm{u}_{L^2}^2+\sum_{1\leq\abs{\alpha}\leq m}\norm{\comii{y}^{\ell_y}\dd^\alpha u}_{L^2}^2 }.
\]
We then define space ${ H}^m_{\ell_x,\ell_y}$ of vector functions by
\[
{ H}^m_{\ell_x,\ell_y}=\left\{{\bf u}=(u,v)\in { H}^m_{\ell_x,\ell_y}: u\in { H}^m_{\ell_y}, v\in { H}^m_{\ell_x} \right\},
\]
which is equipped with the norm
\[
\norm{{\bf u}}_{H^m_{\ell_x,\ell_y}}=\sqrt{\norm{u}_{H^m_{\ell_y}}^2+\norm{v}_{H^m_{\ell_x}}^2 }.
\]
It's well known that the corresponding Cauchy problem to (1.1)-(1.5) is globally well posed in $H^k$ if $k>2$ with dimension $d=2$, see e.g.[<cit.>, Chapter 17, Section 2] and
\begin{equation*}
\norm{\bf u(t)}_{{ H}^m}\leq \norm{{\bf u}_0}_{{ H}^m}\exp\left(C_0\int_0^t \norm{\nabla {\bf u(s)}}_{{ L}^\infty}ds \right)
\end{equation*}
Where $C_0$ is a constant depending on ${\bf u}_0$.
Now we will show that if the initial data ${\bf u}_0\in{ H}^m_{\ell_x,\ell_y}$ for $0\leq \ell_x,\ell_y\leq1$, the solution is also in $H^m_{\ell_x,\ell_y}$. This is the first step for the Gevery regularity.
For fixed $m\geq3$ and $0\leq\ell_x,\ell_y\leq1$, let the initial data ${\bf u}_0\in {H}^m_{\ell_x,\ell_y}(\RR^2_+)$ and suppose the compatibility condition is fulfilled.
Then the ${ H}^m$-solution ${\bf u}$ to the Euler equation (<ref>)-(<ref>) is also in weighted Sobolev space:
{\bf u}(t)\in C\left([0,\infty);\ { H}^m_{\ell_x,\ell_y}(\RR^2_+)\right).
\begin{equation}\label{estq}
\begin{aligned}
\norm{{\bf u}(t)}_{H^m_{\ell_x,\ell_y}} &\leq \norm{{\bf u}_0}_{H^m_{\ell_x,\ell_y}}\exp\left[C_0 \int_0^t \left(\norm{{\bf u}(s)}_{{ L}^\infty}+\norm{\comii{y}^{\ell_y}\nabla u(s)}_{{ L}^\infty}\right.\right.\\
&+\left.\left.\norm{\comii{x}^{\ell_x}\nabla v(s)}_{{ L}^\infty} \right)ds\right],
\end{aligned}
\end{equation}
where $C_0$ is a constant depending on m.
It suffices to show (<ref>) holds. We begin with proving a priori estimate. First we have
\begin{equation}\label{2.2}
{1\over2}\frac{d}{dt} \left(\norm{u(t)}_{L^2}^2+\norm{v(t)}_{L^2}^2 \right)=0.
\end{equation}
Now let $\alpha\in \mathbb{N}_0^2$ be the multi-index such that $1\leq |\alpha|\leq m$. We apply $\partial^\alpha$ on both sides of (<ref>) and take ${ L}^2$ inner product with $\comii{y}^{2\ell_y}\dd^\alpha u$
\begin{equation}\label{2.3}
{1\over2}\frac{d}{dt} \norm{\comii{y}^{\ell_y}\dd^\alpha u }_{L^2}^2+ \left< \left<y\right>^{\ell_y}\partial^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right>+\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right>=0.
\end{equation}
And similarly for (<ref>) taking ${ L}^2$ inner product with $\comii{x}^{2\ell_x}\dd^\alpha v$ ,
\begin{equation}\label{2.4}
{1\over2}\frac{d}{dt}\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}^2+\left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right>+
\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>=0.
\end{equation}
Taking sum over $1\leq\abs{\alpha}\leq m$ in (<ref>) and (<ref>), and combining (<ref>), we have
\begin{equation}\label{2.5}
\begin{aligned}
&{\frac{1}{2}}\frac{d}{dt}\norm{{\bf u}(t)}_{H^m_{\ell_x,\ell_y}}^2+\sum_{1\leq\abs{\alpha}\leq m} \left[ \left< \left<y\right>^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right> +\right.\\
\left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right> \right] + \sum_{1\leq\abs{\alpha}\leq m} \left[\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right> \right.\\
+\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>\right]=0.
\end{aligned}
\end{equation}
It remains to estimate ${\rm I}_1$ and ${\rm I}_2$, with $I_j$ defined by
\begin{eqnarray*}
{\rm I}_1&=&\sum_{1\leq\abs{\alpha}\leq m} \left[ \left< \left<y\right>^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right> +
\left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right> \right],\\
{\rm I}_2&=&\sum_{1\leq\abs{\alpha}\leq m} \left[\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right>
+\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>\right].
\end{eqnarray*}
The estimate on ${\rm I}_1$:
Using Hölder inequality and divergence-free condition, we have
\begin{equation*}
\begin{aligned}
\abs{{\rm I}}_1 &\leq \norm{{\bf u}}_{H^m_{\ell_x,\ell_y}}\sum_{1\leq\abs{\alpha}\leq m}\left[\norm{\comii{y}^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u)-{\bf u}\cdot\nabla(\comii{y}^{\ell_y}\dd^\alpha u)}_{L^2}\right.\\
&\left.\quad+\norm{\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v)-{\bf u}\cdot\nabla(\comii{x}^{\ell_x}\dd^\alpha v)}_{L^2} \right].
\end{aligned}
\end{equation*}
Note the fact that $|\dd^\beta\comii{y}^{\ell_y}|,|\dd^\beta\comii{x}^{\ell_x}|\leq C_m$ for $1\leq\beta\leq\alpha$ and $C_m$ depending on m, then the weight function can be put in the bracket. And with the application of <cit.> we have
\begin{equation}\label{2.6}
\abs{{\rm I}_1}\leq C\left(\norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty} \right)\norm{\bf u}_{H^m_{\ell_x,\ell_y}}^2.
\end{equation}
The estimate on ${\rm I_2}$:
In order to estimate ${\rm I}_2$, we need to use Lemma 5.1 and Lemma 5.3 in Section 5. Observe $p$ satisfies the following Neumann problem.
\begin{equation*}
\left\{
\begin{aligned}
& -\Delta p=2(\dd_y u)\dd_x v-2(\dd_x u)\dd_y v\quad \mbox{in}\ \RR^2_+\times\{0,\infty \},\\
& \dd_y p|_{y=0}=0 \quad \mbox{on}\ \RR\times\{0,\infty \}.
\end{aligned}
\right.
\end{equation*}
We proceed to estimate ${\rm I}_2$ through two cases.
(a). If $\abs{\alpha}=1$, then we use Lemma 5.1 and classical argument of ${\rm H}^2$-regularity result of the above Neumann problem, to get
\begin{equation*}
\begin{aligned}
&\sum_{\abs{\alpha}=1}\left( \norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}\right)\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_y u\dd_x v}_{L^2}+C\norm{\comii{y}^{\ell_y}\dd_x u\dd_y v}_{L^2}+C\norm{\comii{x}^{\ell_x}\dd_y u\dd_x v}_{L^2}\\
&+C\norm{\comii{x}^{\ell_x}\dd_x u\dd_y v}_{L^2}+C\norm{\dd_x p}_{L^2}+C\norm{\dd_y p}_{L^2}\\
&\leq C\norm{\bf u}_{{\rm H}^m_{\ell_x,\ell_y}}\norm{\nabla{\bf u}}_{L^\infty},
\end{aligned}
\end{equation*}
where we used the Hodge decomposition of ${ L}^2(\RR^2_+)$ to estimate $\norm{\nabla p}_{L^2}$ and C is a constant.
(b). If $2\leq\abs{\alpha}=k\leq m$, then we use Lemma 5.3 and similar arguments as <cit.>; this gives
\begin{equation*}
\begin{aligned}
&\sum_{2\leq\abs{\alpha}\leq m}\left( \norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}\right)\\
&\leq C\sum_{k=2}^m\sum_{\abs{\beta}=k-1}\left(\norm{\comii{y}^{\ell_y}\dd^\beta(\dd_y u\dd_x v)}_{L^2}+\norm{\comii{y}^{\ell_y}\dd^\beta(\dd_x u\dd_y v)}_{L^2} \right.\\
&\left.+\norm{\comii{x}^{\ell_x}\dd^\beta(\dd_y u\dd_x v)}_{L^2}
+\norm{\comii{x}^{\ell_x}\dd^\beta(\dd_x u\dd_y v)}_{L^2}\right)
+C\sum_{k=2}^{m}\left(\norm{\dd_x^{k-2}(\dd_y u\dd_x v)}_{L^2} \right.\\
&\left.+\norm{\dd_x^{k-2}(\dd_x u\dd_y v)}_{L^2}\right)\\
&\leq C\norm{\bf u}_{{\rm H}^m_{\ell_x,\ell_y}}\left(\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\right).
\end{aligned}
\end{equation*}
Thus we combine the above two cases to conclude that
\begin{equation}\label{2.7}
\abs{{\rm I}_2}\leq C\norm{\bf u}_{{ H}^m_{\ell_x,\ell_y}}^2\left( \norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\right),
\end{equation}
where $C$ is a constant depending only on $m$.
And then by (<ref>), (<ref>) and (<ref>), we have
\begin{equation*}
\frac{d}{dt}\norm{{\bf u}(t)}_{{ H}^m_{\ell_x,\ell_y}}\leq C\left(\norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\right)\norm{{\bf u}(t)}_{H^m_{\ell_x,\ell_y}}
\end{equation*}
Then with Grownwall inequality we obtain (<ref>).
Now consider $u\in H^m$. Repeating the above arguments with $\comii y^{\ell_y}$ and $\comii x^{\ell_x}$ replaced, respectively, by
\begin{eqnarray*}
\frac{\comii y^{\ell_y} }{\comii {\eps y}^{\ell_y}}, \quad \frac{\comii x^{\ell_x} }{\comii {\eps x}^{\ell_x}}
\end{eqnarray*}
where $0<\eps<1$, then we can also deduce (<ref>) by letting $\eps\rightarrow 0$. We complete the proof of the proposition.
§ WEIGHTED GEVREY REGULARITY
We inherit the notations that used in <cit.> for $X_\tau$ and $Y_\tau$. That is to say for a multi-index $\alpha=(\alpha_1,\alpha_2)$ in $\NN^2$, and a vector function ${\bf u}=(u,v)$, define the Sobolev and semi-norms as follows:
\[
\abs{\bf u}_{m,\ell_x,\ell_y}=\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd^\alpha {u}}_{L^2}+\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}\right),
\]
\[
\abs{\bf u}_{m,\ell_x,\ell_y,\infty}=\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd^\alpha {u}}_{L^\infty}+\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^\infty}\right),
\]
where $\abs{\bf u}_m=\abs{\bf u}_{m,0,0}$ and $\abs{\bf u}_{m,\infty}=\abs{\bf u}_{m,0,0,\infty}.$
For $s\geq1$ and $\tau>0$, define a new weighted Gevrey spaces, which is equivalent to that in Definition <ref>, by
\[
X_{\tau,\ell_x,\ell_y}=\left\{{\bf u}\in C^\infty :\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}<\infty \right\},
\]
\[
\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}=\sum_{m=3}^\infty \abs{\bf u}_{m,\ell_x,\ell_y}\frac{\tau^{m-3}}{(m-3)!^s}.
\]
And let
\[
Y_{\tau,\ell_x,\ell_y}=\left\{{\bf u}\in C^\infty :\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}<\infty \right\},
\]
\[
\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}=\sum_{m=3}^\infty \abs{\bf u}_{m,\ell_x,\ell_y}\frac{(m-3)\tau^{m-4}}{(m-3)!^s}.
\]
We will denote $X_{\tau}=X_{\tau,0,0}$ and $Y_\tau=Y_{\tau,0,0}$.
In order to show the main result, Theorem <ref>, it suffices to show the following
Let the initial data ${{\bf u}_0}=(u_0,v_0)$ satisfy
\[{\bf u}_0\in X_{\tau_0,\ell_x,\ell_y}\]
for some $s\geq 1,\tau_0>0$ and $0\leq\ell_x,\ell_y\leq 1$. Then the Euler system (<ref>)-(<ref>) admits a solution
\begin{eqnarray*}
{\bf u}(t)\in C([0,\infty);~X_{\tau(t),\ell_x,\ell_y}),
\end{eqnarray*}
where $\tau(t)$ depends on the initial radius $\tau_0$.
We will prove Theorem 3.1 using the method of <cit.>, with main difference from the estimate on pressure. By Proposition <ref> we see $\norm{\bf u}_{H^m_{\ell_x,\ell_y}}<+\infty$ for each $m.$ With notations above we have
\begin{equation}\label{3.1}
\frac{d}{dt}\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}} =\dot{\tau}\norm{{\bf u}(t)}_{Y_{\tau(t),\ell_x,\ell_y}}+\sum_{m=3}^\infty \frac{d}{dt}|{\bf u}(t)|_{m,\ell_x,\ell_y} \frac{\tau(t)^{m-3}}{(m-3)!^s}.
\end{equation}
Recalling from (<ref>) and (<ref>) and using Hölder inequality, we obtain
\begin{equation*}
\begin{aligned}
\frac{d}{dt}\abs{{\bf u}(t)}_{m,\ell_x,\ell_y} &\leq \sum_{\abs{\alpha}=m}\sum_{\beta\leq\alpha,\beta\neq0}{\alpha\choose\beta}\left(\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} \right.\\
&\left.+\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \right)
+\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2} \right.\\
&\left.+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \right)+\norm{\bf u}_{L^\infty}\abs{\bf u}_m.
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Ccal&=\sum_{m=3}^\infty \sum_{\abs{\alpha}=m} \sum_{\beta\leq\alpha,\beta\neq0}{\alpha\choose\beta}\left(\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2}\right.\\
&\left.+\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \right)\frac{\tau^{m-3}}{(m-3)!^s}
\end{aligned}
\end{equation*}
\[
\Pcal=\sum_{m=3}^\infty\sum_{\abs{\alpha}=m} \left(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \right)\frac{\tau^{m-3}}{(m-3)!^s}.
\]
Combined with (<ref>), we have
\begin{equation}\label{3.2}
\frac{d}{dt}\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}}\leq \dot\tau\norm{{\bf u}(t)}_{Y_{\tau(t),\ell_x,\ell_y}}+\mathcal{C}+\mathcal{P}+C\tau\norm{\bf u}_{L^\infty}\norm{\bf u}_{Y_\tau}.
\end{equation}
We give the following Lemma to estimate $\Ccal$, the proof is postponed to Section 4.
There exists a sufficiently large constant $C>0$ such that
\[
\Ccal\leq C\left(\Ccal_1+\Ccal_2\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \right),
\]
\begin{equation*}
\begin{aligned}
\Ccal_1 &=\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+\tau \abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{u}_{3,\ell_x,\ell_y}\\
&+\tau^2\abs{\bf u}_3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}
\end{aligned}
\end{equation*}
\[
\Ccal_2=\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}+\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+\tau^2\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}+\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}.
\]
The following lemmas shall be used to estimate $\Pcal$. The proof is postponed to Section 5 below.
There exists a sufficiently large constant $C>0$ such that
$$\Pcal_{\ell_1,\ell_2}\leq C\left(\Pcal_{1}+\Pcal_{2}\norm{{\bf u}}_{Y_{\tau,\ell_x,\ell_y}}\right), $$
\begin{equation*}
\begin{aligned}
\Pcal_{1} &=\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}\\
&+\tau\left(\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{u}_{2,\ell_x,\ell_y}\right) \\
&+\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Pcal_{2} &=\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}+\tau^2\left(\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right)+\tau^3\left(\abs{\bf u}_{3,\ell_x,\ell_y,\infty} \right.\\
&\left.+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\right)
+\left(\tau^2+\tau^{5/2}+\tau^3 \right)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}+\tau^4\abs{\bf u}_{3,\ell_x,\ell_y,\infty}.
\end{aligned}
\end{equation*}
Let $m\geq6$ be fixed. With Sobolev embedding theorem and the lemmas above and (<ref>), we have
\begin{equation}\label{3.3}
\begin{aligned}
\frac{d}{dt}\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\tau_y}} &\leq \dot{\tau}(t) \norm{{\bf u}(t)}_{Y_{\tau(t),\ell_x,\ell_y}}+C(1+\tau(t)^3) \norm{\bf u}_{H^m_{\ell_x,\ell_y}}^2\\
&\quad+C \tau(t)\left(\norm{\bf u}_{L^\infty} +\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right) \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \\
&\quad+ C (\tau(t)^2+\tau(t)^4)\norm{\bf u}_{{ H}^m_{\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \\
&\quad+ C (\tau(t)^2+\tau(t)^3)\norm{\bf u}_{X_{\tau(t),\ell_x,\ell_y}} \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}},
\end{aligned}
\end{equation}
where the constant $C$ is independent of $u,v$. If $\tau(t)$ decreases fast enough such that
\begin{equation}\label{3.4}
\begin{aligned}
\dot{\tau}(t)+C\tau(t)\left(\norm{\bf u}_{L^\infty}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right)&+C(\tau(t)^2+\tau(t)^4)\norm{\bf u}_{ H^m_{\ell_x,\ell_y}}\\
&+C(\tau(t)^2+\tau(t)^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\leq 0.
\end{aligned}
\end{equation}
Then (<ref>) implies
\begin{equation*}
\frac{d}{ dt}\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}}\leq C(1+\tau(0)^3)\norm{\bf u}^2_{H^m_{\ell_x,\ell_y}}.
\end{equation*}
\begin{equation*}
\begin{aligned}
\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}}\leq \norm{{\bf u}_0}_{X_{\tau_0,\ell_x,\ell_y}}+&C_{\tau(0)}\int_0^t \norm{{\bf u}(s)}^2_{ H^m_{\ell_x,\ell_y}}
\end{aligned}
\end{equation*}
for all $0<t<\infty$, where $C_{\tau(0)}=C\left(1+\tau(0)^3\right)$. As $\tau$ is chosen to be a decrease function, a sufficient condition for (<ref>) to hold is that
\begin{equation}\label{3.5}
\begin{aligned}
\dot\tau(t) &+C\left(\norm{\bf u}_{L^\infty}+\abs{{\bf u}(t)}_{1,\ell_x,\ell_y,\infty}\right)\tau(t)\\
&+C\tau(t)^2\left[C_{\tau(0)}^\prime \norm{{\bf u}(t)}_{{ H}^m_{\ell_x,\ell_y}}
+C_{\tau(0)}^{\prime\prime} M(t) \right]\leq 0,
\end{aligned}
\end{equation}
where $C_{\tau(0)}^\prime=\left(1+\tau(0)^2\right), C_{\tau(0)}^{\prime\prime}=1+\tau(0)$ and
\[
M(t)=\norm{{\bf u}_0}_{X_{\tau_x,\ell_y}}+C_{\tau(0)}\int_0^t \norm{{\bf u}(s)}^2_{H^m_{\ell_x,\ell_y}}ds.
\]
\begin{equation*}
G(t)=\exp\left[C\int_0^t \left(\norm{\bf u(s)}_{L^\infty}+\abs{{\bf u}(s)}_{1,\ell_x,\ell_y,\infty}\right) ds \right].
\end{equation*}
By Proposition <ref> we can choose the constant $C>0$ is taken largely enough such that
\[\norm{{\bf u}(t)}^2_{{ H}^m_{\ell_x,\ell_y}}\leq\norm{{\bf u}_0}^2_{{ H}^m_{\ell_x,\ell_y}} G(t).\]
It then follows that (<ref>) is satisfied if we let
\begin{equation*}
\tau(t)=G(t)^{-1}\frac{1}{\frac{1}{\tau(0)}+C\int_0^t \left[C_{\tau(0)}^\prime \norm{{\bf u}(s)}_{{ H}^m_{\ell_x,\ell_y}}+C_{\tau(0)}^{\prime\prime}M(s)\right]G(s)^{-1} ds }.
\end{equation*}
The proof of Theorem <ref> is complete.
§ THE COMMUTATOR ESTIMATE
In this section we will prove Lemma <ref>, the method here is similar with <cit.> except for the parts involving the weight function.
We first write the sum as
\begin{equation*}
\Ccal=\sum_{m=3}^\infty\sum_{j=1}^m \Ccal_{m,j},
\end{equation*}
where we denote
\begin{equation*}
\begin{aligned}
\Ccal_{m,j} &=\frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\sum_{\abs{\beta}=j,\beta\leq\alpha}{\alpha\choose\beta}\left(\norm{\comii{y}^{\ell_y}\dd^\beta {\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} \right.\\
&\left.+\norm{\comii{x}^{\ell_x}\dd^\beta {\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \right).
\end{aligned}
\end{equation*}
Then we split the right side of the above inequality into seven terms according to the values of m and j, and prove the following estimates.
For small $j$, we have
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,1}\leq C\abs{\bf u}_{1,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty} \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,2}\leq C\abs{\bf u}_{2,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+C\tau\abs{ \bf u}_{2,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^2\abs{\bf u}_{2,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
For intermediate $j$, we have
\begin{equation*}
\sum_{m=6}^\infty\sum_{j=3}^{[{m\over2}]}\Ccal_{m,j}\leq C\tau^2 \norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation}\label{4.1}
\begin{aligned}
\sum_{m=7}^\infty \sum_{j=[m/2]+1}^{m-3}\Ccal_{m,j} &\leq C(\tau^2+\tau^{5/2}+\tau^3) \norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{aligned}.
\end{equation}
For higher j, we have
\begin{equation*}
\sum_{m=5}^\infty \Ccal_{m,m-2}\leq C\tau^2\abs{\bf u}_{3}\abs{\bf u }_{3,\ell_x,\ell_y,\infty}+C\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau},
\end{equation*}
\begin{equation*}
\sum_{m=4}^\infty \Ccal_{m,m-1}\leq C\tau\abs{\bf u}_{3}\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,m}\leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_3+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau}.
\end{equation*}
The proof of the above estimates is similar as in <cit.> and we just point out the difference due to the weight function. The main difference may be caused by the weight function is the estimation of (<ref>).
Note that for $[m/2]+1\leq j\leq m-3$, with Hölder inequality and [Proposition 3.8, Chapter 13,Section 3,<cit.>], one have
\begin{equation}\label{4.2}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} &\leq C\norm{\dd^\beta{\bf u}}_{L^2}\norm{\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u}_{L^2}^{1/2}\\
&\times\norm{D^2\left(\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta} u\right)}_{L^2}^{1/2},
\end{aligned}
\end{equation}
where we used the notation
\[
D^k u=\{\dd^\alpha u: \abs{\alpha}=k\},\quad\ \ \norm{D^k u}_{L^2}=\sum_{\abs{\alpha}=k}\norm{\dd^\alpha u}_{L^2}.
\]
Note that by Leibniz formula
\begin{equation}\label{4.3}
\begin{aligned}
\norm{D^2\left(\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u \right)}_{L^2} &\leq C\left(\norm{\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u}_{L^2}+\norm{\comii{y}^{\ell_y}D^1\nabla\dd^{\alpha-\beta}u}_{L^2}\right.\\
&\left.+\norm{\comii{y}^{\ell_y}D^2\nabla\dd^{\alpha-\beta}u}_{L^2} \right)
\end{aligned}
\end{equation}
where $C$ is a constant. And here we used the fact that, observing $0\leq\ell_y\leq1$,
\[
|\dd_y^2\comii{y}^{\ell_y}|\leq C,\quad\ \ |\dd_y\comii{y}^{\ell_y}|\leq1, \quad 1\leq\comii{y}^{\ell_y},
\]
for some constant $C$. And similar arguments also applied to $\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2}$.
With (<ref>) and (<ref>) , we have
\begin{equation*}
\begin{aligned}
\sum_{m=7}^\infty\sum_{j=[m/2]+1}^{m-3}\Ccal_{m,j}
&\leq C\sum_{m=7}^\infty \sum_{j=[m/2]+1}^{m-3}\abs{\bf u}_j \abs{\bf u}_{m-j+1,\ell_x,\ell_y}^{1/2}\left(\abs{\bf u}_{m-j+1,\ell_x,\ell_y}^{1/2}\right.\\
&\left.+\abs{\bf u}_{m-j+2,\ell_x,\ell_y}^{1/2}+\abs{\bf u}_{m-j+3,\ell_x,\ell_y}^{1/2} \right){m\choose j}\frac{\tau^{m-3}}{(m-3)!^s}
\end{aligned}
\end{equation*}
And the estimation of the right side of the above inequality is similar as in <cit.>. So we omit the details here. The proof of Lemma 4.3 is complete.
§ THE PRESSURE ESTIMATE
It can be deduced from the Euler system (<ref>)-(<ref>) that the pressure term $p$ satisfies
\begin{equation}\label{5.1}
-\Delta p=h(u,v)\quad\text{in}\ \RR^2_+,
\end{equation}
where $h(u,v)=2(\dd_y u)\dd_x v-2(\dd_x u)\dd_y v$. Take the values of (<ref>) on $\dd \RR^2_+$ and use (<ref>) to obtain
\begin{equation}\label{5.2}
\dd_y p|_{y=0}=0 \quad \text{on}\ \dd\RR^2_+ .
\end{equation}
In order to estimate $\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}$ and $\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}$,
we first consider the following Neumann problem, and here we hope to obtain a weighted $H^2$-regularity result.
Suppose $\phi$ is the smooth solution of the following equation with Neumann boundary condition, and $\psi\in {\rm C}^\infty$
\begin{equation}\label{5.3}
\left\{
\begin{aligned}
& -\Delta \phi=\psi\quad \mbox{in}\quad \RR^2_+,\\
& \dd_y \phi\big|_{y=0}=0 \quad \mbox{on}\ \dd\RR^2_+.
\end{aligned}
\right.
\end{equation}
Then there exist a constant $C$ such that for $\forall \alpha\in\NN^2_0$ with $\abs{\alpha}=2$
\begin{equation}\label{5.4}
\norm{\comii{y}^{\ell_y}\dd^\alpha\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y}\psi}_{L^2}+C\norm{\dd_x \phi}_{L^2},
\end{equation}
\begin{equation}\label{5.5}
\norm{\comii{x}^{\ell_x}\dd^\alpha\phi}_{L^2}\leq C\norm{\comii{x}^{\ell_x}\psi}_{L^2}+C\norm{\dd_y \phi}_{L^2},
\end{equation}
\begin{equation}\label{5.6}
\norm{\comii{y}^{\ell_y}\dd_y\dd^\alpha\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y}\dd_y\psi}_{L^2}+C\norm{\psi}_{L^2},
\end{equation}
\begin{equation}\label{5.7}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha\phi}_{L^2}\leq C\norm{\comii{x}^{\ell_x}\dd_y\psi}_{L^2}+C\norm{\psi}_{L^2}.
\end{equation}
The proof is similar with the classical $\rm{H^2}$- regularity arguments. Due to the symmetry it suffices to prove (<ref>) and (<ref>), since (<ref>) and (<ref>) can be proved similarly.
The method is to use integration by parts. We first multiply the first equation of (<ref>) by $\comii{y}^{2\ell_y}\dd_{xx}\phi$ and integrate over $\RR^2_+$, to obtain
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}^2+\int_{\RR^2_+}\comii{y}^{2\ell_y}\dd_{xx}\phi\dd_{yy}\phi dxdy=-\left<\comii{y}^{\ell_y}\dd_{xx}\phi ,\comii{y}^{\ell_y}\psi\right>.
Integrating by parts with the second term, we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y} \dd_{xx}p}_{L^2}^2+\norm{\comii{y}^{\ell_y} \dd_{xy}p}_{L^2}^2 &=-\left<\comii{y}^{\ell_y}\dd_{xx}\phi ,\comii{y}^{\ell_y}\psi\right>\\
\end{aligned}
\end{equation*}
Using Cauchy-Schwarz inequality and noticing that $|\dd_y\comii{y}^{\ell_y}|\leq1$ for $0\leq \ell_y\leq1$, we can obtain, for any $0<\eps,\eps'<1$,
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}^2+\norm{\comii{y}^{\ell_y} \dd_{xy}\phi}_{L^2}^2 &\leq C_\epsilon\norm{\comii{y}^{\ell_y} \psi}_{L^2}^2+\epsilon\norm{\comii{y}^{\ell_y}\dd_{xx}\phi}_{L^2}^2\\
&+\epsilon^\prime \norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}^2+C_{\epsilon^\prime}\norm{\dd_{x}\phi}_{L^2}^2,
\end{aligned}
\end{equation*}
and thus
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}+\norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y} \psi}_{L^2}+C\norm{\dd_{x}\phi}_{L^2}
for some constant $C>0$. Now if we multiply $\comii{y}^{2\ell_y}\dd_{yy}\phi$ on both sides of ((<ref>)) and do the procedure as above, we can obtain
\norm{\comii{y}^{\ell_y} \dd_{yy}p}_{L^2}+\norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y}\psi}_{L^2}+C\norm{\dd_{x}\phi}_{L^2}.
Then we have proven (<ref>) and (<ref>). To prove (<ref>) and (<ref>), we first apply $\dd_y$ on equation (<ref>) to get
\begin{equation}\label{5.8}
\left\{
\begin{aligned}
& -\Delta \dd_y\phi=\dd_y \psi\quad \text{in}\ \RR^2_+,\\
& \dd_y\phi\big|_{y=0}=0 \quad \mbox{on}\ \dd\RR^2_+.
\end{aligned}
\right.
\end{equation}
And with this Dirichlet equation for $\dd_y\phi$, we can also proceed like before. In this time we multiply $\comii{y}^{2\ell_y}\dd_{xxy}\phi$ and $\comii{y}^{2\ell_y}\dd_{yyy}\phi$ on both sides of (<ref>) and calculate as above, then with the use of the classical $H^2$ regularity result (<ref>) and ((<ref>)) can be obtained.
The terms of order one on right side of (<ref>)-(<ref>) are created by differentiating on the weight functions $\comii{x}^{\ell_x}$ and $\comii{y}^{\ell_y}$ when integrating by parts. And this is the main reason why we need the constants $\ell_x,\ell_y$ to be in the interval $[0,1]$.
For higher order regularity estimates, we need the following lemma.
Suppose $g$ is a smooth solution of
\begin{equation}\label{5.9}
\left\{
\begin{aligned}
&-\Delta g=f \quad in\ \RR^2_+,\\
&\dd_y g\big|_{y=0}=0 \quad \mbox{on}\ \dd\RR^2_+,
\end{aligned}
\right.
\end{equation}
with $f\in C^\infty$. Then there exist a universal constant $C>0$ such that the following estiamtes
\begin{equation}\label{5.10}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2}\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}
\end{equation}
\begin{equation}\label{5.11}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2}\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}
\end{equation}
hold for any $m\geq3$ and any multi-index $\alpha\in\NN^2_0$ such that $\abs{\alpha}=m$ .
In (<ref>) and (<ref>) we have summation over the set
\{\beta\in \NN_0^2:\ \abs{\beta}=m-1, \exists~ l\in\NN_0 \ \text{such that}\ \beta^\prime-\alpha^\prime=2l+1\}
\{\beta\in \NN_0^2:\ \abs{\beta}=m-1, \exists~ l\in\NN_0 \ \text{such that}\ \beta^\prime-\alpha^\prime=2l\},
and similar conventions are used throughout this section.
First by (<ref>), we use the following induction equality from <cit.>:
\begin{equation}\label{5.12}
\dd_y^{2k+2}g=(-1)^{k+1}\dd_x^{2k+2}g-\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j}f,
\end{equation}
and applying $\dd_y$ on the above equation gives
\begin{equation}\label{5.13}
\dd_y^{2k+3}g=(-1)^{k+1}\dd_x^{2k+2}\dd_y g-\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}f.
\end{equation}
Then for given $\abs{\alpha}=m$, we discuss the situations as the value of $\alpha_2$ varies.
Case 1. If $\alpha_2=0$ then $\comii{y}^{\ell_y}\dd_x\dd^\alpha g=\comii{y}^{\ell_y}\dd_x^{m+1}g$ and
$$\comii{x}^{\ell_x}\dd_y\dd^\alpha g=\comii{x}^{\ell_x}\dd_y\dd_x^m g$$
Letting $\phi=\dd_x^{m-1}g$ and applying Lemma 5.1, we obtain
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^m g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2}&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_y\dd_x^{m-1} g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
In such case, Lemma <ref> is proved.
Case 2. If $\alpha_2=1$ then $\comii{y}^{\ell_y}\dd_x\dd^\alpha g=\comii{y}^{\ell_y}\dd_x^{m}\dd_y g$ and
$$\comii{x}^{\ell_x}\dd_y\dd^\alpha g=\comii{x}^{\ell_x}\dd_y^2\dd_x^{m-1} g$$
Letting $\phi=\dd_x^{m-1}g$, we can obtain the same result by Lemma <ref> as above.
Case 3. If $\alpha_2=2k+2\geq2$, then by the induction (<ref>) we have
\begin{equation}\label{5.14}
\begin{aligned}
\comii{y}^{\ell_y}\dd_x\dd^\alpha g&=\comii{y}^{\ell_y}\dd_y^{2k+2}\dd_x^{\alpha_1+1}g\\
&=\comii{y}^{\ell_y}(-1)^{k+1}\dd_x^{2k+2}\dd_x^{\alpha_1+1}g-\comii{y}^{\ell_y}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j}\dd_x^{\alpha_1+1}f.
\end{aligned}\end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1+1}g$, we apply Lemma <ref> to obtain
\begin{equation}\label{5.15}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x^{2k+2}\dd_x^{\alpha_1+1}g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^m g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation}
Substituting (<ref>) into (<ref>), we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{y}^{\ell_y}\dd_x^{2k-2j}\dd_y^{2j}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
And similarly from the induction (<ref>) equality
\begin{equation}\label{5.16}
\begin{aligned}
\comii{x}^{\ell_x}\dd_y\dd^\alpha g &=\comii{x}^{\ell_x}\dd_y^{2k+3}\dd_x^{\alpha_1}g\\ &=\comii{x}^{\ell_x}(-1)^{k+1}\dd_x^{2k+2}\dd_y \dd_x^{\alpha_1}g-\comii{x}^{\ell_x}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1}f.
\end{aligned}\end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1}g$, we apply Lemma <ref> to get
\begin{equation}\label{5.17}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_y\dd_x^{\alpha_1}g}_{L^2}&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k}\dd_y\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_y^2\dd_x^{2k}\dd_x^{\alpha_1} g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k}\dd_y\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation}
Substituting (<ref>) into (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{x}^{\ell_x}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
Thus in such case the lemma is also proved.
Case 4. If $\alpha_2=2k+3\geq3$, then by the induction we have
\begin{equation}\label{5.18}
\begin{aligned}
\comii{y}^{\ell_y}\dd_x\dd^\alpha g&=\comii{y}^{\ell_y}\dd_y^{2k+3}\dd_x^{\alpha_1+1}g\\
&-\comii{y}^{\ell_y}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1+1}f.
\end{aligned} \end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1+1}$, then applying Lemma <ref>, we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x^{2k+2}\dd_y\dd_x^{\alpha_1+1}g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_y\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x\dd_y\dd_x^{2k+\alpha_1+1} g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_y\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
Thus substituting the above estimate into (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{y}^{\ell_y}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
On the other hand, observe
\begin{equation}\label{5.19}
\begin{aligned}
\comii{x}^{\ell_x}\dd_y\dd^\alpha g &=\comii{x}^{\ell_x}\dd_y^{2(k+1)+2}\dd^{\alpha_1}g\\
&=\comii{x}^{\ell_x}(-1)^{k+2}\dd_x^{2k+4} \dd_x^{\alpha_1}g-\comii{x}^{\ell_x}\sum_{j=0}^{k+1} (-1)^{k+1-j}\dd_x^{2k+2-2j}\dd_y^{2j}\dd_x^{\alpha_1}f.
\end{aligned} \end{equation}
Then, letting $\phi=\dd_x^{2k+2}\dd_x^{\alpha_1}g$ and applying Lemma <ref>, we obtain
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_x^{2k+4}\dd_x^{\alpha_1}g}_{L^2} &\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_y \dd_x^{2k+2+\alpha_1}g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2},
\end{aligned}
\end{equation*}
which along with (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^{k+1} \norm{\comii{x}^{\ell_x}\dd_x^{2k+2-2j}\dd_y^{2j}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
So in this case Lemma <ref> is also proved. Thus for all $\alpha$ such that $\abs{\alpha}=m$ we have proved Lemma <ref>.
Now we come to the proof of Lemma <ref>.
Apply Lemma <ref> with equation (<ref>)-(<ref>) we have
\begin{equation*}
\begin{aligned}
\Pcal &=\sum_{m=3}^\infty \frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \right)\\
&\leq C\sum_{m=3}^\infty \frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\left(\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\right.\\
&\left.+\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}+C\norm{\dd_x^{m-2}h}_{L^2} \right)\\
&\leq C\sum_{m=3}^\infty \frac{m\tau^{m-3}}{(m-3)!^s}\left(\sum_{\abs{\beta}=m-1}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\right.\\
&\left.+\sum_{\abs{\beta}=m-1}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}
+\norm{\dd_x^{m-2}h}_{L^2} \right),
\end{aligned}
\end{equation*}
where we denote $h=2(\dd_x v)\dd_y u-2(\dd_x u)\dd_y v$. If we exchange the order of the summation, we can obtain, by direct verification,
\[
\sum_{\abs{\alpha}=m}\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\leq Cm\sum_{\abs{\beta}=m-1}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}
\]
\[
\sum_{\abs{\alpha}=m}\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\leq Cm\sum_{\abs{\beta}=m-1}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}.
\]
And direct computation also gives
\[
\sum_{\abs{\alpha}=m}\norm{\dd_x^{m-2}h}_{L^2}\leq m\norm{\dd_x^{m-2}h}_{L^2}.
\]
Recall $h=-2(\dd_x u)\dd_y v+2(\dd_y u)\dd_x v$, then we have, for arbitrary $\beta\in\NN^2_0$
\[
\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\leq C\sum_{\gamma\leq\beta}{\beta\choose\gamma}\norm{\comii{y}^{\ell_y}|\dd^\gamma\nabla{ u}||\dd^{\beta-\gamma}{\nabla v} |}_{L^2},
\]
\[
\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\leq C\sum_{\gamma\leq\beta}{\beta\choose\gamma}\norm{\comii{x}^{\ell_x}|\dd^\gamma\nabla{ u}||\dd^{\beta-\gamma}{\nabla v} |}_{L^2}.
\]
So with these inequalities, we have
\begin{equation*}
\begin{aligned}
\Pcal_{w}&=C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\beta}=m-1}\left(\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}+\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\right)\\
&\leq C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\stackrel{ \abs{\beta}=m-1} {0\leq\gamma\leq\beta} } {\beta\choose\gamma}\left(\norm{\comii{y}^{\ell_y}|\dd^\gamma\nabla u||\dd^{\beta-\gamma}\nabla v|}_{L^2}\right.\\
&\left.+\norm{\comii{x}^{\ell_x}|\dd^\gamma\nabla u||\dd^{\beta-\gamma}\nabla v|}_{L^2}\right)
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Pcal_x&=C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\norm{\dd_x^{m-2} h}_{L^2}\\
&\leq C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{0\leq j\leq m-2}{m-2\choose j}\norm{|\dd_x^j\nabla u|| \dd_x^{m-2-j}\nabla v|}_{L^2}.
\end{aligned}
\end{equation*}
Then we have
\[
\Pcal\leq \Pcal_w+\Pcal_x.
\]
The rest part is to estimate $\Pcal_w$ and $\Pcal_x$.
We first estimate $\Pcal_w$. To do so we split the summation into
\begin{equation}\label{5.20}
\Pcal_{w}\leq C\sum_{m=3}^\infty\sum_{j=0}^{m-1}\Pcal_{w,m,j},
\end{equation}
\begin{equation*}
\begin{aligned}
\Pcal_{w,m,j}&=\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\beta}=m-1}\sum_{\abs{\gamma}=j}{m-1\choose j}\left(\norm{\comii{y}^{\ell_y}|\dd^\gamma\nabla u||\dd^{\beta-\gamma}\nabla v|}_{L^2}\right.\\
&\left.+\norm{\comii{x}^{\ell_x}|\dd^\gamma\nabla u||\dd^{\beta-\gamma}\nabla v|}_{L^2} \right)
\end{aligned}
\end{equation*}
we split the right side of (<ref>) into seven terms according to the values of $m$ and $j$. For lower $j$, we have
\begin{equation*}
\sum_{m=3}^\infty \Pcal_{w,m,0} \leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}},
\end{equation*}
\begin{equation*}
\begin{aligned}
\sum_{m=3}^\infty \Pcal_{w,m,1} &\leq C\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+C\tau\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}\\
&+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{aligned}
\end{equation*}
\begin{equation*}
\sum_{m=5}^\infty \Pcal_{w,m,2} \leq C\tau^2\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
For intermediate $j$, we have
\begin{equation*}
\sum_{m=8}^\infty \sum_{j=3}^{[m/2]-1}\Pcal_{w,m,j} \leq C(\tau^2+\tau^{5/2}+\tau^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}},
\end{equation*}
\begin{equation*}
\begin{aligned}
\sum_{m=6}^\infty \sum_{j=[m/2]}^{m-3}\Pcal_{w,m,j}\leq C(\tau^2+\tau^{5/2}+\tau^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{aligned}
\end{equation*}
For higher $j$, we have
\begin{equation*}
\sum_{m=4}^\infty \Pcal_{w,m,m-2}\leq C\tau \abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Pcal_{w,m,m-1}\leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
In these estimations we used the fact that for vector function ${\bf u}=(u,v)$, the norm of $u$ or $v$ can be bounded by the norm of ${\bf u}$, for example
\[
\norm{\comii{y}^{\ell_y}\nabla\dd^\gamma u}_{L^\infty}\leq \abs{\bf u}_{\abs{\gamma}+1,\ell_x,\ell_y,\infty}.
\]
With this consideration the estimations can be proved similarly by the method of <cit.> and the arguments of the commutator estimates, and we omit the details.
To estimate $\Pcal_{x}$, we proceed as above, and write
\begin{equation*}
\Pcal_x\leq C\sum_{m=3}^\infty\sum_{j=0}^{m-2} \Pcal_{x,m,j},
\end{equation*}
\begin{equation*}
\Pcal_{x,m,j}=\frac{m\tau^{m-3}}{(m-3)!^s}{{m-2}\choose j} \norm{|\dd_x^j\nabla u||\dd_x^{m-j-2}\nabla v|}_{L^2}.
\end{equation*}
For lower $j$, we have
\begin{align*}
& \sum_{m=3}^\infty \Pcal_{x,m,0}\leq C\abs{\bf u}_{1,\infty}\abs{\bf u}_2+C\tau\abs{\bf u}_{1,\infty}\abs{ \bf u}_3+C\tau^2\abs{\bf u}_{1,\infty}\norm{\bf u}_{Y_\tau},\\
& \sum_{m=4}^\infty \Pcal_{x,m,1}\leq C\tau\abs{\bf u}_{2,\infty}\abs{\bf u}_2+C\tau^2\abs{\bf u}_{2,\infty}\abs{\bf u}_3+C\tau^3\abs{\bf u}_{2,\infty}\norm{\bf u}_{Y_\tau},\\
& \sum_{m=6}^\infty \Pcal_{x,m,2}\leq C\tau^3\abs{\bf u}_{3,\infty}\abs{\bf u}_3+C\tau^4\abs{\bf u}_{3,\infty}\norm{\bf u}_{Y_\tau}.
\end{align*}
For mediate $j$, we have
\begin{align*}
& \sum_{m=8}^\infty\sum_{j=3}^{[m/2]-1} \Pcal_{x,m,j}\leq C\tau^3\norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_\tau}\\
& \sum_{m=6}^\infty\sum_{j=[m/2]}^{m-3} \Pcal_{x,m,j}\leq C\tau^3\norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_\tau}.
\end{align*}
Finally for higher $j$, we have
\begin{align*}
\sum_{m=5}^\infty \Pcal_{x,m,m-2}\leq C\tau^2\abs{\bf u}_{1,\infty}\norm{\bf u}_{Y_\tau}.
\end{align*}
These estimations can be proved similarly as <cit.> with the fact that $\norm{\bf u}_{X_\tau}\leq\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}$, $\norm{\bf u}_{Y_\tau}\leq\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}$ and $\norm{\bf u}_{H^m}\leq\norm{\bf u}_{H^m_{\ell_x,\ell_y}}$. With all these estimations, we can complete the proof Lemma <ref>.
Acknowledgments. W. Li would like to appreciate the support from NSF of China (No. 11422106), and C.-J. Xu was partially supported by “the Fundamental Research Funds
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1511.00539
|
Gevrey regularity with weight for Euler equation ]
Gevrey regularity with weight for incompressible Euler equation in the half plane
F. Cheng W.-X. Li C.-J. Xu]
Feng Cheng Wei-Xi Li Chao-Jiang Xu
Feng Cheng, School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
Wei-Xi Li, School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, 430072 Wuhan, China
Chao-Jiang Xu,
School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
[2010]35M33, 35Q31, 76N10
In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted $L^2$- estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.
§ INTRODUCTION
In this paper we study the Gevrey propagation of solutions to incompressible Euler equation. Gevrey class is a stronger concept than the $C^\infty$-smoothness. In fact it is an intermediate space between analytic space and $C^\infty$ space.
There have been extensive mathematical investigations (cf.<cit.>, <cit.>, <cit.>, <cit.> <cit.>, <cit.>, <cit.>, <cit.>, <cit.> for instance and the references therein) on Euler equation in different kind of frames, such as Sobolev space, analytic space and Gevrey space. In this work we will consider the problems of Gevrey regularity with weight, and this is motivated by the study of the vertical boundary layer problem. From a physical point of view, as well as from a mathematical point of view, when the direction of rotation is perpendicular to
the boundaries, the boundary equation is well developed and called Ekman layers. Up to now the Ekman layers (horizontal layer) are well understood, (c.f. <cit.> for instance and the references therein).
When the direction of rotation is parallel to
the boundaries, the situation is, however, different from the perpendicular case above: the vertical layers are very different and much more complicated, from a physical, analytical and mathematical point of view, and many open questions in all these directions remain open. We refer to <cit.> for detailed discussion on the vertical layers. Recently we try to investigate the well-posedness problem for vertical layer, and the related and preliminary step is to establish the Gevrey regularty with weight for the outer flow which is described by Euler equation. This is the main result of the present paper.
Without loss of generality, we consider the incompressible Euler equation in half plane $\RR^2_+$, where $\RR^2_+= \{(x,y);x\in\RR, y\in\RR^+ \}$, and our results can be generalized to 3-D Euler equation. The velocity $(u(t,x,y),v(t,x,y))$ and the pressure $p(t,x,y)$ satisfy the following equation:
\begin{align}
\dd_t u + u\dd_x u+v\dd_y u+ \dd_x p &=0\quad \ \mbox{in}\ (0,\infty)\times\RR^2_+\label{1.1}\\ \dd_t v + u\dd_x v+v\dd_y v+\dd_y p &=0\quad \ \mbox{in}\ (0,\infty)\times\RR^2_+\label{1.2}\\
\partial_x u+\partial_y v &=0\quad \ \mbox{in}\ (0,\infty)\times\RR^2_+\label{1.3}
\end{align}
The boundary condition
\begin{equation}\label{1.4}
v\big|_{y=0} =0 \quad \ \text{in}\ (0,\infty)\times\RR;\quad u,v\to0,\ \text{as}\ \sqrt{x^2+y^2}\to\infty
\end{equation}
With initial data
\begin{equation}\label{1.5}
u|_{t=0}=u_0, \quad v|_{t=0}=v_0,\quad\ \mbox{in}\ \RR^2_+.
\end{equation}
Here the initial data $(u_0,v_0)$ satisfy the compatibility condition:
\[
\dd_x u_0+\dd_y v_0=0,\quad v_0|_{y=0}=0,\quad u_0,v_0\to0,\quad\text{as}\quad \sqrt{x^2+y^2}\to\infty.
\]
Before stating our main result we first introduce the (global) weighted Gevrey space.
Let $\ell_x,\ell_y\geq 0$ be real constants that independent of $x,y$, we say that $f\in G^s_{\tau,\ell_x,\ell_y}(\RR^2_+)$ if
\begin{equation*}
\sup_{\abs\alpha\geq 0} \frac{\tau^\alpha}{\abs\alpha!^s} \norm{\comii{x}^{\ell_x}\comii{y}^{\ell_y} \dd^\alpha f}_{L^2(\RR^2_+)}<\infty,
\end{equation*}
where and throughout the paper we use the notation $\comii{\cdot} =(1+|\cdot|^2)^{\frac{1}{2}}$.
In this work we present the persistence of weighted Gevrey class regularity of the solution, i.e., we prove that if the initial datum $(u_0,v_0)$ is in some weighted Gevrey space and satisfy the compatible condition, then the global solution belongs to the same space. With only minor changes, these results can also be extended to 3-D Euler equation, and the global solution here will be replaced by a local solution.
Suppose the initial data $u_0\in G^s_{\tau_0,0,\ell_y}, v_0\in G^s_{\tau_0,\ell_x,0}$ for some $s\geq 1,\tau_0>0$ and $0\leq\ell_x,\ell_y\leq 1$. Then the Euler equation (<ref>)-(<ref>) admits a solution $u,v,p$ such that
\begin{eqnarray*}
u(t,\cdot)\in L^\infty\left([0,+\infty);~G^s_{\tau(t),0,\ell_y}\right), \quad v(t,\cdot)\in L^\infty\left([0,+\infty);~G^s_{\tau(t),\ell_x,0}\right)
\end{eqnarray*}
and p satisfies
\[
\dd_x p(t,\cdot)\in L^\infty\left([0,+\infty);\ G^s_{\tau(t),0,\ell_y}\right), \quad \dd_y p(t,\cdot)\in L^\infty\left([0,+\infty);\ G^s_{\tau(t),\ell_x,0}\right)
\]
where $\tau(t)$ is a decreasing function of $t$ with initial value $\tau_0$.
If the initial data $(u_0,v_0)$ were posed both horizontal and vertical weights, namely $u_0\in G^s_{\tau_0,\ell_x,\ell_y},v_0\in G^s_{\tau_0,\ell_x,\ell_y}$, the result of Theorem <ref> is also valid.
We remark that the existence of smooth solutions to (<ref>)-(<ref>) is well developed (c.f.<cit.>, <cit.>, <cit.>, <cit.>, <cit.> for instance), and in two-dimensional case smooth initial data can yield global solutions, while in the three-dimensional case the solution may be local in general condition. The appearance of the weight function increases the difficulty of estimating the pressure term, and for this part it is different from <cit.>. We also point out that in the whole space $\mathbb R^2$ or two dimensional torus $\mathbb T^2,$ the classical approach to analyticity or Gevrey regularity is that it makes crucial use of Fourier transformation, which can't apply to our case. Instead we will use the basic $L^2$ estimate (c.f. <cit.> for instance).
The paper is organized as follows. In section <ref>, we introduce the notation used to define the weighted Sobolev norms, and we prove the persistence of the weighted Sobolev regularity. In section <ref>, we state the priori estimate to prove the main theorem. Section <ref> and <ref> are consist of the proofs of these lemmas.
§ NOTATIONS AND PRELIMINARIES
In the following context, we use the conventional symbols for the standard Sobolev spaces \({ H}^m(\RR^2_+) \) with \(m\in \mathbb{N}\), and let $\norm{\cdot}_{H^m}$ be its norm. For the case \(m=0\), it was usually written as ${ L}^2(\RR^2_+)$. Denote $\left\|\cdot\right\|_{L^2}$ and $\left<\cdot,\cdot\right>$ be the norm and inner product in ${ L}^2(\RR^2_+)$. We usually write a vector function in bold type as ${\bf u}$ and a scalar function in it's conventional way as $u$. For a vector function ${\bf u}=(u,v)$ we denote
\[
\left\|{\bf u} \right\|_{H^m}=\sqrt{\norm{u}^2_{H^m}+\norm{v}^2_{H^m}}.
\]
And when we say that ${\bf u}\in { H}^m$, we mean that $u,v\in{ H}^m$.
With the notations above, we introduce the weighted Sobolev spaces ${ H}^m_{\ell_x}(\RR^2_+)$ and ${ H}^m_{\ell_y}(\RR^2_+)$, where $\ell_x,\ell_y$ are real constants. Let
{H}^m_{\ell_x}(\RR^2_+)=\left\{v\in{ H}^m(\RR^2_+);\quad \comii{x}^{\ell_x}\dd^\alpha v\in L^2, 1\leq\abs{\alpha}\leq m \right\},
and it's norm is defined by
\[
\norm{v}_{H^m_{\ell_x}}=\sqrt{\norm{v}_{L^2}^2+\sum_{1\leq\abs{\alpha}\leq m}\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}^2 }.
\]
Similarly, let
{ H}^m_{\ell_y}(\RR^2_+)=\left\{u\in{ H}^m(\RR^2_+);\quad \comii{y}^{\ell_y}\dd^\alpha u\in L^2, 1\leq\abs{\alpha}\leq m \right\},
equipped with the norm
\[
\norm{u}_{H^m_{\ell_y}}=\sqrt{\norm{u}_{L^2}^2+\sum_{1\leq\abs{\alpha}\leq m}\norm{\comii{y}^{\ell_y}\dd^\alpha u}_{L^2}^2 }.
\]
We then define space $H^m_{\ell_x,\ell_y}(\RR^2_+)$ of vector functions by
\[
H^m_{\ell_x,\ell_y}(\RR^2_+)=\left\{{\bf u}=(u,v)\in H^m_{\ell_x,\ell_y}(\RR^2_+): u\in H^m_{\ell_y}(\RR^2_+), v\in H^m_{\ell_x}(\RR^2_+)\right\}
\]
which is equipped with the norm
\[
\norm{{\bf u}}_{H^m_{\ell_x,\ell_y}}=\sqrt{\norm{u}_{H^m_{\ell_y}}^2+\norm{v}_{H^m_{\ell_x}}^2 }.
\]
It's well known that the corresponding Cauchy problem to (<ref>)-(<ref>) is globally well posed in $H^m$ if $m>2$ with dimension $d=2$, see e.g.[<cit.>, Chapter 17, Section 2] and
\begin{equation*}
\norm{\bf u(t)}_{{ H}^m}\leq \norm{{\bf u}_0}_{{ H}^m}\exp\left(C_0\int_0^t \norm{\nabla {\bf u(s)}}_{{ L}^\infty}ds \right)
\end{equation*}
Where $C_0$ is a constant depending on ${\bf u}_0$.
Now we will show that if the initial data ${\bf u}_0=(u_0,v_0)\in H^m_{\ell_x,\ell_y}$ for $0\leq \ell_x,\ell_y\leq1$, the solution is also in $H^m_{\ell_x,\ell_y}$. This is the first step for the weighted Gevery regularity.
For fixed $m\geq3$ and $0\leq\ell_x,\ell_y\leq1$, let the initial data ${\bf u}_0\in {H}^m_{\ell_x,\ell_y}(\RR^2_+)$ and suppose the compatibility condition is fulfilled.
Then the ${ H}^m$-solution ${\bf u}$ to the Euler equation (<ref>)-(<ref>) is also in weighted Sobolev space:
{\bf u}(t,\cdot)\in L^\infty\left([0,\infty);\ { H}^m_{\ell_x,\ell_y}(\RR^2_+)\right).
\begin{equation}\label{estq}
\begin{aligned}
\norm{{\bf u}(t,\cdot)}_{H^m_{\ell_x,\ell_y}} \leq \norm{{\bf u}_0}_{H^m_{\ell_x,\ell_y}}\exp &\bigg[C_0 \int_0^t \bigg(\norm{{\bf u}(s,\cdot)}_{{ L}^\infty}+\norm{\comii{y}^{\ell_y}\nabla u(s,\cdot)}_{{ L}^\infty}\\
&+\norm{\comii{x}^{\ell_x}\nabla v(s,\cdot)}_{{ L}^\infty} \bigg)ds\bigg],
\end{aligned}
\end{equation}
where $C_0$ is a constant depending on m.
It suffices to show (<ref>) holds. When no ambiguity arises, we suppress the time dependence of $u$ and $v$ on $t$. We begin with proving a priori estimate. First we have
\begin{equation}\label{2.2}
{1\over2}\frac{d}{dt} \left(\norm{u(t,\cdot)}_{L^2}^2+\norm{v(t,\cdot)}_{L^2}^2 \right)=0.
\end{equation}
Now let $\alpha\in \mathbb{N}_0^2$ be the multi-index such that $1\leq |\alpha|\leq m$. We apply $\partial^\alpha$ on both sides of (<ref>) and take $L^2-$ inner product with $\comii{y}^{2\ell_y}\dd^\alpha u$
\begin{equation}\label{2.3}
{1\over2}\frac{d}{dt} \norm{\comii{y}^{\ell_y}\dd^\alpha u }_{L^2}^2+ \left< \left<y\right>^{\ell_y}\partial^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right>+\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right>=0.
\end{equation}
And similarly taking $L^2-$ inner product with $\comii{x}^{2\ell_x}\dd^\alpha v$ on both sides of (<ref>),
\begin{equation}\label{2.4}
{1\over2}\frac{d}{dt}\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}^2+\left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right>+
\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>=0.
\end{equation}
Taking sum over $1\leq\abs{\alpha}\leq m$ in (<ref>) and (<ref>), and combining (<ref>), we have
\begin{equation}\label{2.5}
\begin{aligned}
&\quad{\frac{1}{2}}\frac{d}{dt}\norm{{\bf u}(t,\cdot)}_{H^m_{\ell_x,\ell_y}}^2\\
&+\sum_{1\leq\abs{\alpha}\leq m} \left[ \left< \left<y\right>^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right> + \left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right> \right] \\
&+\sum_{1\leq\abs{\alpha}\leq m} \left[\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right>
+\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>\right]=0.
\end{aligned}
\end{equation}
It remains to estimate ${\rm I}_1$ and ${\rm I}_2$, with $I_j$ defined by
\begin{eqnarray*}
{\rm I}_1&=&\sum_{1\leq\abs{\alpha}\leq m} \left[ \left< \left<y\right>^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u),
\left<y\right>^{\ell_y}\partial^\alpha u\right> +
\left<\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v),\comii{x}^{\ell_x}\dd^\alpha v\right> \right],\\
{\rm I}_2&=&\sum_{1\leq\abs{\alpha}\leq m} \left[\left<\left<y\right>^{\ell_y}\partial^\alpha\partial_x p ,\left<y\right>^{\ell_y}\partial^\alpha u \right>
+\left<\comii{x}^{\ell_x}\dd^\alpha\dd_y p,\comii{x}^{\ell_x}\dd^\alpha v\right>\right].
\end{eqnarray*}
The estimate on ${\rm I}_1$:
Using Hölder inequality and divergence-free condition, we have
\begin{equation*}
\begin{aligned}
\abs{{\rm I}}_1 &\leq \norm{{\bf u}}_{H^m_{\ell_x,\ell_y}}\sum_{1\leq\abs{\alpha}\leq m}\left[\norm{\comii{y}^{\ell_y}\dd^\alpha({\bf u}\cdot\nabla u)-{\bf u}\cdot\nabla(\comii{y}^{\ell_y}\dd^\alpha u)}_{L^2}\right.\\
&\left.\quad+\norm{\comii{x}^{\ell_x}\dd^\alpha({\bf u}\cdot\nabla v)-{\bf u}\cdot\nabla(\comii{x}^{\ell_x}\dd^\alpha v)}_{L^2} \right].
\end{aligned}
\end{equation*}
Note the fact that $\left|\dd^\beta\comii{y}^{\ell_y}\right|,\left|\dd^\beta\comii{x}^{\ell_x}\right|\leq C_m$ for $1\leq\beta\leq\alpha$ and $C_m$ depending on m, then the weight function can be put into the bracket. And with the application of <cit.> we have
\begin{equation}\label{2.6}
\abs{{\rm I}_1}\leq C\left(\norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty} \right)\norm{\bf u}_{H^m_{\ell_x,\ell_y}}^2.
\end{equation}
Where $C$ is a constant depending on $m$. In the following $C$ denotes a generic positive constant depending on $m$.
The estimate on ${\rm I_2}$:
In order to estimate ${\rm I}_2$, we need to use Lemma <ref> and Lemma <ref> in Section <ref>. Observe $p$ satisfies the following Neumann problem.
\begin{equation*}
\left\{
\begin{aligned}
\Delta p &=2(\dd_x u)\dd_y v-2(\dd_y u)\dd_x v \quad \text{in}\ \RR^2_+\times\{0,\infty \},\\
\dd_y p|_{y=0}&=0 \quad \mbox{on}\ \RR\times\{0,\infty \}.
\end{aligned}
\right.
\end{equation*}
We proceed to estimate ${\rm I}_2$ through two cases.
(a). If $\abs{\alpha}=1$, then we use Lemma <ref> and classical argument of $H^2$-regularity result of the above Neumann problem, to get
\begin{equation*}
\begin{aligned}
&\quad\sum_{\abs{\alpha}=1}\left( \norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}\right)\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_y u\dd_x v}_{L^2}+C\norm{\comii{y}^{\ell_y}\dd_x u\dd_y v}_{L^2}+C\norm{\comii{x}^{\ell_x}\dd_y u\dd_x v}_{L^2}\\
&\quad+C\norm{\comii{x}^{\ell_x}\dd_x u\dd_y v}_{L^2}+C\norm{\dd_x p}_{L^2}+C\norm{\dd_y p}_{L^2}\\
&\leq C\norm{\bf u}_{{\rm H}^m_{\ell_x,\ell_y}}\norm{\nabla{\bf u}}_{L^\infty},
\end{aligned}
\end{equation*}
where we used the Hodge decomposition of ${ L}^2(\RR^2_+)$ to estimate $\norm{\nabla p}_{L^2}$ and C is a constant.
(b). If $2\leq\abs{\alpha}=k\leq m$, then we use Lemma <ref> and similar arguments as <cit.>; this gives
\begin{equation*}
\begin{aligned}
&\quad\sum_{2\leq\abs{\alpha}\leq m}\bigg( \norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}\bigg)\\
&\leq C\sum_{k=2}^m\sum_{\abs{\beta}=k-1}\bigg(\norm{\comii{y}^{\ell_y}\dd^\beta(\dd_y u\dd_x v)}_{L^2}+\norm{\comii{y}^{\ell_y}\dd^\beta(\dd_x u\dd_y v)}_{L^2} \\
&\quad+\norm{\comii{x}^{\ell_x}\dd^\beta(\dd_y u\dd_x v)}_{L^2}
+\norm{\comii{x}^{\ell_x}\dd^\beta(\dd_x u\dd_y v)}_{L^2}\bigg)\\
&\quad+C\sum_{k=2}^{m}\bigg(\norm{\dd_x^{k-2}(\dd_y u\dd_x v)}_{L^2}+\norm{\dd_x^{k-2}(\dd_x u\dd_y v)}_{L^2}\bigg)\\
&\leq C\norm{\bf u}_{{\rm H}^m_{\ell_x,\ell_y}}\bigg(\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\bigg).
\end{aligned}
\end{equation*}
Thus we combine the above two cases to conclude that
\begin{equation}\label{2.7}
\abs{{\rm I}_2}\leq C\norm{\bf u}_{{ H}^m_{\ell_x,\ell_y}}^2\left( \norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\right),
\end{equation}
where $C$ is a constant depending only on $m$.
And then by (<ref>), (<ref>) and (<ref>), we have
\begin{equation*}
\frac{d}{dt}\norm{{\bf u}(t)}_{{ H}^m_{\ell_x,\ell_y}}\leq C\left(\norm{\bf u}_{L^\infty}+\norm{\comii{y}^{\ell_y}\nabla u}_{L^\infty}+\norm{\comii{x}^{\ell_x}\nabla v}_{L^\infty}\right)\norm{{\bf u}(t)}_{H^m_{\ell_x,\ell_y}}
\end{equation*}
Then with Grownwall inequality we obtain (<ref>).
Now we consider ${\bf u}\in H^m$. Repeating the above arguments with $\comii y^{\ell_y}$ and $\comii x^{\ell_x}$ replaced, respectively, by
\begin{eqnarray*}
\frac{\comii y^{\ell_y} }{\comii {\eps y}^{\ell_y}}, \quad \frac{\comii x^{\ell_x} }{\comii {\eps x}^{\ell_x}}
\end{eqnarray*}
where $0<\eps<1$, then we can also deduce (<ref>) by letting $\eps\rightarrow 0$. We then complete the proof of the proposition.
§ WEIGHTED GEVREY REGULARITY
We inherit the notations that used in <cit.> for $X_\tau$ and $Y_\tau$. That is to say for a multi-index $\alpha=(\alpha_1,\alpha_2)$ in $\NN^2$, and a vector function ${\bf u}=(u,v)$, define the Sobolev and semi-norms as follows:
\[
\abs{\bf u}_{m,\ell_x,\ell_y}=\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd^\alpha {u}}_{L^2}+\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^2}\right),
\]
\[
\abs{\bf u}_{m,\ell_x,\ell_y,\infty}=\sum_{\abs{\alpha}=m}\left(\norm{\comii{y}^{\ell_y}\dd^\alpha {u}}_{L^\infty}+\norm{\comii{x}^{\ell_x}\dd^\alpha v}_{L^\infty}\right),
\]
where $\abs{\bf u}_m=\abs{\bf u}_{m,0,0}$ and $\abs{\bf u}_{m,\infty}=\abs{\bf u}_{m,0,0,\infty}.$
For $s\geq1$ and $\tau>0$, define a new weighted Gevrey spaces, which is equivalent to that in Definition <ref>, by
\[
X_{\tau,\ell_x,\ell_y}=\left\{{\bf u}\in C^\infty :\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}<\infty \right\},
\]
\[
\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}=\sum_{m=3}^\infty \abs{\bf u}_{m,\ell_x,\ell_y}\frac{\tau^{m-3}}{(m-3)!^s}.
\]
And let
\[
Y_{\tau,\ell_x,\ell_y}=\left\{{\bf u}\in C^\infty :\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}<\infty \right\},
\]
\[
\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}=\sum_{m=3}^\infty \abs{\bf u}_{m,\ell_x,\ell_y}\frac{(m-3)\tau^{m-4}}{(m-3)!^s}.
\]
We will denote $X_{\tau}=X_{\tau,0,0}$ and $Y_\tau=Y_{\tau,0,0}$.
In order to show the main result, Theorem <ref>, it suffices to show the following
Let the initial data ${{\bf u}_0}=(u_0,v_0)$ satisfy
\[{\bf u}_0\in X_{\tau_0,\ell_x,\ell_y}\]
for some $s\geq 1,\tau_0>0$ and $0\leq\ell_x,\ell_y\leq 1$. Then the Euler system (<ref>)-(<ref>) admits a solution
\begin{eqnarray*}
{\bf u}(t,\cdot)\in L^\infty\left([0,\infty);~X_{\tau(t),\ell_x,\ell_y}\right),
\end{eqnarray*}
where $\tau(t)$ is a decreasing function depending on the initial radius $\tau_0$ and the $H^m_{\ell_x,\ell_y}-$ solution ${\bf u}$ for $m\geq 6$.
We will prove Theorem 3.1 using the method of <cit.>, with main difference from the estimate on pressure. By Proposition <ref> we see $\norm{\bf u}_{H^m_{\ell_x,\ell_y}}<+\infty$ for each $m>2$. When no ambiguity arises, we suppress the time dependence of $\tau$ and $u,v$ on $t$. With notations above we have
\begin{equation}\label{3.1}
\frac{d}{dt}\norm{{\bf u}(t,\cdot)}_{X_{\tau(t),\ell_x,\ell_y}} =\tau^\prime(t)\norm{{\bf u}(t,\cdot)}_{Y_{\tau(t),\ell_x,\ell_y}}+\sum_{m=3}^\infty \frac{d}{dt}|{\bf u}(t,\cdot)|_{m,\ell_x,\ell_y} \frac{\tau(t)^{m-3}}{(m-3)!^s}.
\end{equation}
Recalling from (<ref>) and (<ref>) and using Hölder inequality, we obtain
\begin{equation*}
\begin{aligned}
\frac{d}{dt}\abs{{\bf u}(t,\cdot)}_{m,\ell_x,\ell_y} &\leq \sum_{\abs{\alpha}=m}\sum_{\beta\leq\alpha,\beta\neq0}{\alpha\choose\beta}
\bigg(\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} \\
&\quad+\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \bigg)
+\sum_{\abs{\alpha}=m}\bigg(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2} \\
&\quad+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \bigg)+\norm{\bf u}_{L^\infty}\abs{\bf u}_m.
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Ccal&=\sum_{m=3}^\infty \sum_{\abs{\alpha}=m} \sum_{\beta\leq\alpha,\beta\neq0}{\alpha\choose\beta}\bigg(\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2}\\
&\quad+\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \bigg)\frac{\tau(t)^{m-3}}{(m-3)!^s}
\end{aligned}
\end{equation*}
\[
\Pcal=\sum_{m=3}^\infty\sum_{\abs{\alpha}=m} \left(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \right)\frac{\tau(t)^{m-3}}{(m-3)!^s}.
\]
Combined with (<ref>), we have
\begin{equation}\label{3.2}
\frac{d}{dt}\norm{{\bf u}(t,\cdot)}_{X_{\tau(t),\ell_x,\ell_y}}\leq \tau^\prime(t)\norm{{\bf u}(t,\cdot)}_{Y_{\tau(t),\ell_x,\ell_y}}+\mathcal{C}+\mathcal{P}+C\tau(t)\norm{{\bf u}(t,\cdot)}_{L^\infty}\norm{{\bf u}(t,\cdot)}_{Y_\tau}.
\end{equation}
We give the following Lemma to estimate $\Ccal$, the proof is postponed to Section <ref>.
There exists a sufficiently large constant $C>0$ such that
\[
\Ccal\leq C\left(\Ccal_1+\Ccal_2\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \right),
\]
\begin{equation*}
\begin{aligned}
\Ccal_1 &=\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+\tau \abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{u}_{3,\ell_x,\ell_y}\\
&\quad+\tau^2\abs{\bf u}_3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}
\end{aligned}
\end{equation*}
\[
\Ccal_2=\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}+\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+\tau^2\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}+\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}.
\]
The following lemmas shall be used to estimate $\Pcal$. The proof is postponed to Section <ref> below.
There exists a sufficiently large constant $C>0$ such that
$$\Pcal\leq C\left(\Pcal_{1}+\Pcal_{2}\norm{{\bf u}}_{Y_{\tau,\ell_x,\ell_y}}\right) $$
\begin{equation*}
\begin{aligned}
\Pcal_{1} &=\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}\\
&\quad+\tau\left(\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{u}_{2,\ell_x,\ell_y}\right) \\
&\quad+\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Pcal_{2} &=\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}+\tau^2\left(\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right)+\tau^3\left(\abs{\bf u}_{3,\ell_x,\ell_y,\infty} \right.\\
&\left.\quad+\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\right)
+\left(\tau^2+\tau^{5/2}+\tau^3 \right)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}+\tau^4\abs{\bf u}_{3,\ell_x,\ell_y,\infty}
\end{aligned}
\end{equation*}
Let $m\geq6$ be fixed. With Sobolev embedding theorem and the lemmas above and (<ref>), we have
\begin{equation}\label{3.3}
\begin{aligned}
\frac{d}{dt}\norm{{\bf u}(t,\cdot)}_{X_{\tau(t),\ell_x,\tau_y}} &\leq {\tau}^\prime(t) \norm{{\bf u}(t,\cdot)}_{Y_{\tau(t),\ell_x,\ell_y}}+C(1+\tau(t)^3) \norm{{\bf u}(t,\cdot)}_{H^m_{\ell_x,\ell_y}}^2\\
&\quad+C \tau(t)\left(\norm{\bf u}_{L^\infty} +\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right) \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \\
&\quad+ C (\tau(t)^2+\tau(t)^4)\norm{\bf u}_{{ H}^m_{\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}} \\
&\quad+ C (\tau(t)^2+\tau(t)^3)\norm{\bf u}_{X_{\tau(t),\ell_x,\ell_y}} \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{aligned}
\end{equation}
where the constant $C$ is independent of $u,v$. If $\tau(t)$ decreases fast enough such that
\begin{equation}\label{3.4}
\begin{aligned}
{\tau}^\prime(t)+C\tau(t)\left(\norm{\bf u}_{L^\infty}+\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\right)&+C(\tau(t)^2+\tau(t)^4)\norm{\bf u}_{ H^m_{\ell_x,\ell_y}}\\
&+C(\tau(t)^2+\tau(t)^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\leq 0
\end{aligned}
\end{equation}
Then (<ref>) implies
\begin{equation*}
\frac{d}{ dt}\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}}\leq C(1+\tau(0)^3)\norm{\bf u}^2_{H^m_{\ell_x,\ell_y}}
\end{equation*}
\begin{equation}\label{priori}
\begin{aligned}
\norm{{\bf u}(t)}_{X_{\tau(t),\ell_x,\ell_y}}\leq \norm{{\bf u}_0}_{X_{\tau_0,\ell_x,\ell_y}}+&C_{\tau(0)}\int_0^t \norm{{\bf u}(s,\cdot)}^2_{ H^m_{\ell_x,\ell_y}}ds
\end{aligned}
\end{equation}
for all $0<t<\infty$, where $C_{\tau(0)}=C\left(1+\tau(0)^3\right)$. As $\tau(t)$ is chosen to be a decrease function, a sufficient condition for (<ref>) to hold is that
\begin{equation}\label{3.5}
\begin{aligned}
\tau^\prime(t) &+C\left(\norm{\bf u}_{L^\infty}+\abs{{\bf u}(t)}_{1,\ell_x,\ell_y,\infty}\right)\tau(t)\\
&+C\tau(t)^2\left[C_{\tau(0)}^\prime \norm{{\bf u}(t)}_{{ H}^m_{\ell_x,\ell_y}}
+C_{\tau(0)}^{\prime\prime} M(t) \right]\leq 0,
\end{aligned}
\end{equation}
where $C_{\tau(0)}^\prime=\left(1+\tau(0)^2\right), C_{\tau(0)}^{\prime\prime}=1+\tau(0)$. Set
\[
M(t)=\norm{{\bf u}_0}_{X_{\tau_x,\ell_y}}+C_{\tau(0)}\int_0^t \norm{{\bf u}(s)}^2_{H^m_{\ell_x,\ell_y}}ds.
\]
and denote
\begin{equation*}
G(t)=\exp\left[C\int_0^t \left(\norm{\bf u(s)}_{L^\infty}+\abs{{\bf u}(s)}_{1,\ell_x,\ell_y,\infty}\right) ds \right].
\end{equation*}
By Proposition <ref> we can choose the constant $C>0$ is taken largely enough such that
\[\norm{{\bf u}(t)}^2_{{ H}^m_{\ell_x,\ell_y}}\leq\norm{{\bf u}_0}^2_{{ H}^m_{\ell_x,\ell_y}} G(t).\]
It then follows that (<ref>) is satisfied if we let
\begin{equation*}
\tau(t)=G(t)^{-1}\frac{1}{\frac{1}{\tau(0)}+C\int_0^t \left[C_{\tau(0)}^\prime \norm{{\bf u}(s)}_{{ H}^m_{\ell_x,\ell_y}}+C_{\tau(0)}^{\prime\prime}M(s)\right]G(s)^{-1} ds }.
\end{equation*}
With this decreasing function $\tau(t)$, we can conclude the a priori estimates that are used to prove of Theorem <ref>.
§ THE COMMUTATOR ESTIMATE
In this section we will prove Lemma <ref>, the method here is similar with <cit.> except for the parts involving the weight function.
We first write the sum as
\begin{equation*}
\Ccal=\sum_{m=3}^\infty\sum_{j=1}^m \Ccal_{m,j},
\end{equation*}
where we denote
\begin{equation*}
\begin{aligned}
\Ccal_{m,j} &=\frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\sum_{\abs{\beta}=j,\beta\leq\alpha}{\alpha\choose\beta}
\bigg(\norm{\comii{y}^{\ell_y}\dd^\beta {\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} \\
&\quad+\norm{\comii{x}^{\ell_x}\dd^\beta {\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2} \bigg).
\end{aligned}
\end{equation*}
Then we split the right side of the above inequality into seven terms according to the values of m and j, and prove the following estimates.
For small $j$, we have
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,1}\leq C\abs{\bf u}_{1,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty} \norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,2}\leq C\abs{\bf u}_{2,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+C\tau\abs{ \bf u}_{2,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^2\abs{\bf u}_{2,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
For intermediate $j$, we have
\begin{equation*}
\sum_{m=6}^\infty\sum_{j=3}^{[{m\over2}]}\Ccal_{m,j}\leq C\tau^2 \norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation}\label{4.1}
\begin{aligned}
\sum_{m=7}^\infty \sum_{j=[m/2]+1}^{m-3}\Ccal_{m,j} &\leq C(\tau^2+\tau^{5/2}+\tau^3) \norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{aligned}.
\end{equation}
For higher j, we have
\begin{equation*}
\sum_{m=5}^\infty \Ccal_{m,m-2}\leq C\tau^2\abs{\bf u}_{3}\abs{\bf u }_{3,\ell_x,\ell_y,\infty}+C\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau},
\end{equation*}
\begin{equation*}
\sum_{m=4}^\infty \Ccal_{m,m-1}\leq C\tau\abs{\bf u}_{3}\abs{\bf u}_{2,\ell_x,\ell_y,\infty}+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Ccal_{m,m}\leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_3+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_\tau}.
\end{equation*}
The proof of the above estimates is similar as in <cit.> and we just point out the difference due to the weight function. The main difference may be caused by the weight function is the estimation of (<ref>).
Note that for $[m/2]+1\leq j\leq m-3$, with Hölder inequality and [Proposition 3.8, Chapter 13,Section 3,<cit.>], one have
\begin{equation}\label{4.2}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}u}_{L^2} &\leq C\norm{\dd^\beta{\bf u}}_{L^2}\norm{\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u}_{L^2}^{1/2}\\
&\quad\times\norm{D^2\left(\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta} u\right)}_{L^2}^{1/2},
\end{aligned}
\end{equation}
where we used the notation
\[
D^k u=\{\dd^\alpha u: \abs{\alpha}=k\},\quad\ \ \norm{D^k u}_{L^2}=\sum_{\abs{\alpha}=k}\norm{\dd^\alpha u}_{L^2}.
\]
Note that by Leibniz formula
\begin{equation}\label{4.3}
\begin{aligned}
\norm{D^2\left(\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u \right)}_{L^2} &\leq C\bigg(\norm{\comii{y}^{\ell_y}\nabla\dd^{\alpha-\beta}u}_{L^2}
&\quad+\norm{\comii{y}^{\ell_y}D^2\nabla\dd^{\alpha-\beta}u}_{L^2} \bigg)
\end{aligned}
\end{equation}
where $C$ is a constant. And here we used the fact that, observing $0\leq\ell_y\leq1$,
\[
\left|\dd_y^2\comii{y}^{\ell_y}\right|\leq C,\quad\ \ \left|\dd_y\comii{y}^{\ell_y}\right|\leq1, \quad 1\leq\comii{y}^{\ell_y},
\]
for some constant $C$. And similar arguments also applied to $\norm{\comii{x}^{\ell_x}\dd^\beta{\bf u}\cdot\nabla\dd^{\alpha-\beta}v}_{L^2}$.
With (<ref>) and (<ref>) , we have
\begin{equation*}
\begin{aligned}
\sum_{m=7}^\infty\sum_{j=[m/2]+1}^{m-3}\Ccal_{m,j}
&\leq C\sum_{m=7}^\infty \sum_{j=[m/2]+1}^{m-3}\abs{\bf u}_j \abs{\bf u}_{m-j+1,\ell_x,\ell_y}^{1/2}\bigg(\abs{\bf u}_{m-j+1,\ell_x,\ell_y}^{1/2}\\
&\quad+\abs{\bf u}_{m-j+2,\ell_x,\ell_y}^{1/2}+\abs{\bf u}_{m-j+3,\ell_x,\ell_y}^{1/2} \bigg){m\choose j}\frac{\tau^{m-3}}{(m-3)!^s}
\end{aligned}
\end{equation*}
And the estimation of the right side of the above inequality is similar as in <cit.>. So we omit the details here. The proof of Lemma 4.3 is complete.
§ THE PRESSURE ESTIMATE
It can be deduced from the Euler system (<ref>)-(<ref>) that the pressure term $p$ satisfies
\begin{equation}\label{5.1}
\Delta p=h\quad\text{in}\ \RR^2_+,
\end{equation}
where $h=2(\dd_x u)\dd_y v-2(\dd_y u)\dd_x v$. Taking the values of (<ref>) on $\dd \RR^2_+$ and using (<ref>), we have the following Neumann boundary condition
\begin{equation}\label{5.2}
\dd_y p|_{y=0}=0 \quad \text{on}\ \dd\RR^2_+ .
\end{equation}
In order to estimate $\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}$ and $\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2}$,
we first consider the following Neumann problem, and here we hope to obtain a weighted $H^2$-regularity result.
Suppose $\phi$ is the smooth solution of the following equation with Neumann boundary condition, and $\psi\in {\rm C}^\infty$
\begin{equation}\label{5.3}
\left\{
\begin{aligned}
\Delta \phi &=\psi\quad \mbox{in}\quad \RR^2_+,\\
\dd_y \phi\big|_{y=0} &=0 \quad \mbox{on}\ \dd\RR^2_+.
\end{aligned}
\right.
\end{equation}
Then there exist a constant $C$ such that for $\forall \alpha\in\NN^2_0$ with $\abs{\alpha}=2$
\begin{align}
\norm{\comii{y}^{\ell_y}\dd^\alpha\phi}_{L^2} &\leq C\norm{\comii{y}^{\ell_y}\psi}_{L^2}+C\norm{\dd_x \phi}_{L^2},\label{5.4}\\
\norm{\comii{x}^{\ell_x}\dd^\alpha\phi}_{L^2} &\leq C\norm{\comii{x}^{\ell_x}\psi}_{L^2}+C\norm{\dd_y \phi}_{L^2},\label{5.5}\\
\norm{\comii{y}^{\ell_y}\dd_y\dd^\alpha\phi}_{L^2} &\leq C\norm{\comii{y}^{\ell_y}\dd_y\psi}_{L^2}+C\norm{\psi}_{L^2},\label{5.6}\\
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha\phi}_{L^2} &\leq C\norm{\comii{x}^{\ell_x}\dd_y\psi}_{L^2}+C\norm{\psi}_{L^2}.\label{5.7}
\end{align}
The proof is similar with the classical $H^2$- regularity arguments. Due to the symmetry it suffices to prove (<ref>) and (<ref>), since (<ref>) and (<ref>) can be proved similarly.
The method is to use integration by parts. We first multiply the first equation of (<ref>) by $\comii{y}^{2\ell_y}\dd_{xx}\phi$ and integrate over $\RR^2_+$,
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}^2+\int_{\RR^2_+}\comii{y}^{2\ell_y}\dd_{xx}\phi\dd_{yy}\phi dxdy=\left<\comii{y}^{\ell_y}\dd_{xx}\phi ,\comii{y}^{\ell_y}\psi\right>.
Integrating by parts with the second term, we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}^2+\norm{\comii{y}^{\ell_y} \dd_{xy}\phi}_{L^2}^2 &=\left<\comii{y}^{\ell_y}\dd_{xx}\phi ,\comii{y}^{\ell_y}\psi\right>\\
\end{aligned}
\end{equation*}
Using Cauchy-Schwarz inequality and noticing that $\left|\dd_y\comii{y}^{\ell_y}\right|\leq1$ for $0\leq \ell_y\leq1$, we can obtain, for some $0<\eps,\eps'<1$,
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}^2+\norm{\comii{y}^{\ell_y} \dd_{xy}\phi}_{L^2}^2 &\leq C_\epsilon\norm{\comii{y}^{\ell_y} \psi}_{L^2}^2+\epsilon\norm{\comii{y}^{\ell_y}\dd_{xx}\phi}_{L^2}^2\\
&\quad+\epsilon^\prime \norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}^2+C_{\epsilon^\prime}\norm{\dd_{x}\phi}_{L^2}^2,
\end{aligned}
\end{equation*}
and thus
\norm{\comii{y}^{\ell_y} \dd_{xx}\phi}_{L^2}+\norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y} \psi}_{L^2}+C\norm{\dd_{x}\phi}_{L^2}
for some constant $C>0$. Now if we multiply $\comii{y}^{2\ell_y}\dd_{yy}\phi$ on both sides of (<ref>) and do the procedure as above, we can obtain
\norm{\comii{y}^{\ell_y} \dd_{yy}\phi}_{L^2}+\norm{\comii{y}^{\ell_y}\dd_{xy}\phi}_{L^2}\leq C\norm{\comii{y}^{\ell_y}\psi}_{L^2}+C\norm{\dd_{x}\phi}_{L^2}.
Then we have proven (<ref>). To prove (<ref>), we first apply $\dd_y$ on equation (<ref>) to get
\begin{equation}\label{5.8}
\left\{
\begin{aligned}
\Delta \dd_y\phi &=\dd_y \psi\quad \text{in}\ \RR^2_+,\\
\dd_y\phi\big|_{y=0} &=0 \quad \mbox{on}\ \dd\RR^2_+.
\end{aligned}
\right.
\end{equation}
Denote $\Phi\triangleq\dd_y \phi$, then we have
\begin{equation}\label{Phi}
\left\{
\begin{aligned}
\Delta \Phi &=\dd_y \psi\quad \text{in}\ \RR^2_+,\\
\Phi\big|_{y=0} &=0 \quad \mbox{on}\ \dd\RR^2_+.
\end{aligned}
\right.
\end{equation}
And with this Dirichlet boundary problem for $\Phi$, we multiply (<ref>) by $\comii{y}^{2\ell_y}\dd_{xx}\Phi$ and integrate over $\RR^2_+$.
\begin{equation}\label{Phi1}
\norm{\comii{y}^{\ell_y}\dd_{xx}\Phi}_{L^2}^2+\int_{\RR^2_+}\dd_{yy}\Phi \comii{y}^{2\ell_y}\dd_{xx}\Phi dxdy =\int_{\RR^2_+}\dd_y \psi \comii{y}^{2\ell_y}\dd_{xx}\Phi dxdy
\end{equation}
Since $\Phi$ vanish at infinity and $\Phi\big|_{y=0}$, then $\dd_y\Phi \comii{y}^{2\ell_y}\dd_{xx}\Phi\bigg|_{y=0}=0$. We can integrate by parts to obtain
\begin{align*}
\int_{\RR^2_+}\dd_{yy}\Phi \comii{y}^{2\ell_y}\dd_{xx}\Phi dxdy &=\int_{\RR^2_+} \dd_{xy}\Phi \dd_y\left(\comii{y}^{2\ell_y} \dd_x\Phi \right)dxdy\\
&=\norm{\comii{y}^{\ell_y}\dd_{xy}\Phi}_{L^2}^2+2\int_{\RR^2_+}\comii{y}^{\ell_y}\dd_{xy}\Phi \bigg( \dd_y\comii{y}^{\ell_y}\dd_x\Phi\bigg)dxdy
\end{align*}
Substituting the above equality into (<ref>) and using Hölder inequality on the right hand side, we then obtain
\norm{\comii{y}^{\ell_y}\dd_{xx}\Phi}_{L^2}^2+\norm{\comii{y}^{\ell_y}\dd_{xy}\Phi}_{L^2}^2\leq C\norm{\comii{y}^{\ell_y}\dd_{y}\psi}_{L^2}^2+C^\prime\norm{\dd_{x}\Phi}_{L^2}^2
If we multiply (<ref>) by $\comii{y}^{2\ell_y}\dd_{yy}\Phi$, then we can proceed as above to obtain
\norm{\comii{y}^{\ell_y}\dd_{yy}\Phi}_{L^2}^2+\norm{\comii{y}^{\ell_y}\dd_{xy}\Phi}_{L^2}^2\leq C\norm{\comii{y}^{\ell_y}\dd_{y}\psi}_{L^2}^2+C^\prime\norm{\dd_{x}\Phi}_{L^2}^2
With the use of the classical $H^2$ regularity result, we have
\norm{\dd_x\Phi}_{L^2}=\norm{\dd_{xy}\phi}_{L^2}\leq C\norm{\psi}_{L^2}
Combining the above three equalities, we can prove (<ref>). (<ref>) and (<ref>) can be proved similarly.
The terms of order one on right side of (<ref>)-(<ref>) are created by differentiating on the weight functions $\comii{x}^{\ell_x}$ and $\comii{y}^{\ell_y}$ when integrating by parts. And this is the main reason why we need the constants $\ell_x,\ell_y$ to be in the interval $[0,1]$.
For higher order regularity estimates, we need the following lemma.
Suppose $g$ is a smooth solution of
\begin{equation}\label{5.9}
\left\{
\begin{aligned}
\Delta g &=f \quad in\ \RR^2_+,\\
\dd_y g\big|_{y=0} &=0 \quad \mbox{on}\ \dd\RR^2_+,
\end{aligned}
\right.
\end{equation}
with $f\in C^\infty$. Then there exist a universal constant $C>0$ such that the following estiamtes
\begin{equation}\label{5.10}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2}\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}
\end{equation}
\begin{equation}\label{5.11}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2}\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}
\end{equation}
hold for any $m\geq3$ and any multi-index $\alpha\in\NN^2_0$ such that $\abs{\alpha}=m$ .
In (<ref>) and (<ref>) we have summation over the set
\left\{\beta\in \NN_0^2:\ \abs{\beta}=m-1, \exists~ l\in\NN_0 \ \text{such that}\ \beta^\prime-\alpha^\prime=2l+1\right\}
\left\{\beta\in \NN_0^2:\ \abs{\beta}=m-1, \exists~ l\in\NN_0 \ \text{such that}\ \beta^\prime-\alpha^\prime=2l\right\}
and similar conventions are used throughout this section.
First by (<ref>), we use the following induction equality from <cit.>:
\begin{equation}\label{5.12}
\dd_y^{2k+2}g=(-1)^{k+1}\dd_x^{2k+2}g+\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j}f,
\end{equation}
and applying $\dd_y$ on the above equation gives
\begin{equation}\label{5.13}
\dd_y^{2k+3}g=(-1)^{k+1}\dd_x^{2k+2}\dd_y g+\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}f.
\end{equation}
Then for given $\alpha\in\NN_0^2$ with $\abs{\alpha}=m$, we discuss the situations as the value of $\alpha_2$ varies.
Case 1. If $\alpha_2=0$ then $\comii{y}^{\ell_y}\dd_x\dd^\alpha g=\comii{y}^{\ell_y}\dd_x^{m+1}g$ and
$$\comii{x}^{\ell_x}\dd_y\dd^\alpha g=\comii{x}^{\ell_x}\dd_y\dd_x^m g$$
Letting $\phi=\dd_x^{m-1}g$ and applying Lemma 5.1, we obtain
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^m g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2}&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_y\dd_x^{m-1} g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{m-1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
In such case, Lemma <ref> is proved.
Case 2. If $\alpha_2=1$ then $\comii{y}^{\ell_y}\dd_x\dd^\alpha g=\comii{y}^{\ell_y}\dd_x^{m}\dd_y g$ and
$$\comii{x}^{\ell_x}\dd_y\dd^\alpha g=\comii{x}^{\ell_x}\dd_y^2\dd_x^{m-1} g$$
Letting $\phi=\dd_x^{m-1}g$, we can obtain the same result by Lemma <ref> as above.
Case 3. If $\alpha_2=2k+2\geq2$, then by the induction (<ref>) we have
\begin{equation}\label{5.14}
\begin{aligned}
\comii{y}^{\ell_y}\dd_x\dd^\alpha g&=\comii{y}^{\ell_y}\dd_y^{2k+2}\dd_x^{\alpha_1+1}g\\
&=\comii{y}^{\ell_y}(-1)^{k+1}\dd_x^{2k+2}\dd_x^{\alpha_1+1}g+\comii{y}^{\ell_y}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j}\dd_x^{\alpha_1+1}f.
\end{aligned}\end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1+1}g$, we apply Lemma <ref> to obtain
\begin{equation}\label{5.15}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x^{2k+2}\dd_x^{\alpha_1+1}g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^m g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation}
Substituting (<ref>) into (<ref>), we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{y}^{\ell_y}\dd_x^{2k-2j}\dd_y^{2j}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
And similarly from the induction (<ref>) equality
\begin{equation}\label{5.16}
\begin{aligned}
\comii{x}^{\ell_x}\dd_y\dd^\alpha g &=\comii{x}^{\ell_x}\dd_y^{2k+3}\dd_x^{\alpha_1}g\\ &=\comii{x}^{\ell_x}(-1)^{k+1}\dd_x^{2k+2}\dd_y \dd_x^{\alpha_1}g+\comii{x}^{\ell_x}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1}f.
\end{aligned}\end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1}g$, we apply Lemma <ref> to get
\begin{equation}\label{5.17}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_y\dd_x^{\alpha_1}g}_{L^2}&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k}\dd_y\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_y^2\dd_x^{2k}\dd_x^{\alpha_1} g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k}\dd_y\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation}
Substituting (<ref>) into (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{x}^{\ell_x}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
Thus in such case the lemma is also proved.
Case 4. If $\alpha_2=2k+3\geq3$, then by the induction we have
\begin{equation}\label{5.18}
\begin{aligned}
\comii{y}^{\ell_y}\dd_x\dd^\alpha g&=\comii{y}^{\ell_y}\dd_y^{2k+3}\dd_x^{\alpha_1+1}g\\
&\quad+\comii{y}^{\ell_y}\sum_{j=0}^k (-1)^{k-j}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1+1}f.
\end{aligned} \end{equation}
Letting $\phi=\dd_x^{2k}\dd_x^{\alpha_1+1}$, then applying Lemma <ref>, we have
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x^{2k+2}\dd_y\dd_x^{\alpha_1+1}g}_{L^2}&\leq C\norm{\comii{y}^{\ell_y}\dd_y\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x\dd_y\dd_x^{2k+\alpha_1+1} g}_{L^2}\\
&\leq C\norm{\comii{y}^{\ell_y}\dd_y\dd_x^{2k}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
Thus substituting the above estimate into (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^k \norm{\comii{y}^{\ell_y}\dd_x^{2k-2j}\dd_y^{2j+1}\dd_x^{\alpha_1+1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
On the other hand, observe
\begin{equation}\label{5.19}
\begin{aligned}
\comii{x}^{\ell_x}\dd_y\dd^\alpha g &=\comii{x}^{\ell_x}\dd_y^{2(k+1)+2}\dd^{\alpha_1}g\\
&=\comii{x}^{\ell_x}(-1)^{k+2}\dd_x^{2k+4} \dd_x^{\alpha_1}g+\comii{x}^{\ell_x}\sum_{j=0}^{k+1} (-1)^{k+1-j}\dd_x^{2k+2-2j}\dd_y^{2j}\dd_x^{\alpha_1}f.
\end{aligned} \end{equation}
Then, letting $\phi=\dd_x^{2k+2}\dd_x^{\alpha_1}g$ and applying Lemma <ref>, we obtain
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_x^{2k+4}\dd_x^{\alpha_1}g}_{L^2} &\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_y \dd_x^{2k+2+\alpha_1}g}_{L^2}\\
&\leq C\norm{\comii{x}^{\ell_x}\dd_x^{2k+2}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2},
\end{aligned}
\end{equation*}
which along with (<ref>) yields
\begin{equation*}
\begin{aligned}
\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha g}_{L^2} &\leq C\sum_{j=0}^{k+1} \norm{\comii{x}^{\ell_x}\dd_x^{2k+2-2j}\dd_y^{2j}\dd_x^{\alpha_1}f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}\\
&\leq C\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{x}^{\ell_x}\dd^\beta f}_{L^2}+C\norm{\dd_x^{m-2}f}_{L^2}.
\end{aligned}
\end{equation*}
So in this case Lemma <ref> is also proved. Thus for all $\alpha$ such that $\abs{\alpha}=m$ we have proved Lemma <ref>.
Now we come to the proof of Lemma <ref>.
Apply Lemma <ref> with equation (<ref>)-(<ref>) we have
\begin{equation*}
\begin{aligned}
\Pcal &=\sum_{m=3}^\infty \frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\bigg(\norm{\comii{y}^{\ell_y}\dd_x\dd^\alpha p}_{L^2}+\norm{\comii{x}^{\ell_x}\dd_y\dd^\alpha p}_{L^2} \bigg)\\
&\leq C\sum_{m=3}^\infty \frac{\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\alpha}=m}\bigg(\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\\
&\quad+\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}+\norm{\dd_x^{m-2}h}_{L^2} \bigg)\\
&\leq C\sum_{m=3}^\infty \frac{m\tau^{m-3}}{(m-3)!^s}\bigg(\sum_{\abs{\beta}=m-1}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\\
&\quad+\sum_{\abs{\beta}=m-1}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}
+\norm{\dd_x^{m-2}h}_{L^2} \bigg),
\end{aligned}
\end{equation*}
If we exchange the order of the summation, we can obtain, by direct verification,
\[
\sum_{\abs{\alpha}=m}\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l+1}}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\leq Cm\sum_{\abs{\beta}=m-1}\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}
\]
\[
\sum_{\abs{\alpha}=m}\sum_{\substack{l\in\NN_0,\abs{\beta}=m-1\\ \beta^\prime-\alpha^\prime=2l}}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\leq Cm\sum_{\abs{\beta}=m-1}\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}.
\]
And direct computation also gives
\[
\sum_{\abs{\alpha}=m}\norm{\dd_x^{m-2}h}_{L^2}\leq m\norm{\dd_x^{m-2}h}_{L^2}.
\]
Since $h=2(\dd_x u)\dd_y v-2(\dd_y u)\dd_x v$, then we have, for arbitrary $\beta\in\NN^2_0$
\[
\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}\leq C\sum_{\gamma\leq\beta}{\beta\choose\gamma}\norm{\comii{y}^{\ell_y}\left|\dd^\gamma\nabla{ u}\right|\left|\dd^{\beta-\gamma}{\nabla v} \right|}_{L^2},
\]
\[
\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\leq C\sum_{\gamma\leq\beta}{\beta\choose\gamma}\norm{\comii{x}^{\ell_x}\left|\dd^\gamma\nabla{ u}\right|\left|\dd^{\beta-\gamma}{\nabla v} \right|}_{L^2}.
\]
So with these inequalities, we have
\begin{equation*}
\begin{aligned}
\Pcal_{w}&=C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\beta}=m-1}
\bigg(\norm{\comii{y}^{\ell_y}\dd^\beta h}_{L^2}+\norm{\comii{x}^{\ell_x}\dd^\beta h}_{L^2}\bigg)\\
&\leq C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\stackrel{ \abs{\beta}=m-1} {0\leq\gamma\leq\beta} } {\beta\choose\gamma}\bigg(\norm{\comii{y}^{\ell_y}\left|\dd^\gamma\nabla u\right|\left|\dd^{\beta-\gamma}\nabla v\right|}_{L^2}\\
&\quad+\norm{\comii{x}^{\ell_x}\left|\dd^\gamma\nabla u\right|\left|\dd^{\beta-\gamma}\nabla v\right|}_{L^2}\bigg)
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Pcal_x&=C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\norm{\dd_x^{m-2} h}_{L^2}\\
&\leq C\sum_{m=3}^{\infty}\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{0\leq j\leq m-2}{m-2\choose j}\norm{\left|\dd_x^j\nabla u\right|\left| \dd_x^{m-2-j}\nabla v\right|}_{L^2}.
\end{aligned}
\end{equation*}
Then we have
\[
\Pcal\leq \Pcal_w+\Pcal_x.
\]
The rest part is to estimate $\Pcal_w$ and $\Pcal_x$.
We first estimate $\Pcal_w$. To do so we split the summation into
\begin{equation}\label{5.20}
\Pcal_{w}\leq C\sum_{m=3}^\infty\sum_{j=0}^{m-1}\Pcal_{w,m,j},
\end{equation}
\begin{equation*}
\begin{aligned}
\Pcal_{w,m,j}&=\frac{m\tau^{m-3}}{(m-3)!^s}\sum_{\abs{\beta}=m-1}\sum_{\abs{\gamma}=j}{m-1\choose j}\bigg(\norm{\comii{y}^{\ell_y}\left|\dd^\gamma\nabla u\right|\left|\dd^{\beta-\gamma}\nabla v\right|}_{L^2}\\
&\quad+\norm{\comii{x}^{\ell_x}\left|\dd^\gamma\nabla u\right|\left|\dd^{\beta-\gamma}\nabla v\right|}_{L^2} \bigg)
\end{aligned}
\end{equation*}
we split the right side of (<ref>) into seven terms according to the values of $m$ and $j$. For lower $j$, we have
\begin{equation*}
\sum_{m=3}^\infty \Pcal_{w,m,0} \leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}},
\end{equation*}
\begin{equation*}
\begin{aligned}
\sum_{m=3}^\infty \Pcal_{w,m,1} &\leq C\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{2,\ell_x,\ell_y}+C\tau\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}\\
&\quad+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{aligned}
\end{equation*}
\begin{equation*}
\sum_{m=5}^\infty \Pcal_{w,m,2} \leq C\tau^2\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^3\abs{\bf u}_{3,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
For intermediate $j$, we have
\begin{equation*}
\sum_{m=8}^\infty \sum_{j=3}^{[m/2]-1}\Pcal_{w,m,j} \leq C(\tau^2+\tau^{5/2}+\tau^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}},
\end{equation*}
\begin{equation*}
\begin{aligned}
\sum_{m=6}^\infty \sum_{j=[m/2]}^{m-3}\Pcal_{w,m,j}\leq C(\tau^2+\tau^{5/2}+\tau^3)\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{aligned}
\end{equation*}
For higher $j$, we have
\begin{equation*}
\sum_{m=4}^\infty \Pcal_{w,m,m-2}\leq C\tau \abs{\bf u}_{2,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau^2\abs{\bf u}_{2,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}
\end{equation*}
\begin{equation*}
\sum_{m=3}^\infty \Pcal_{w,m,m-1}\leq C\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\abs{\bf u}_{3,\ell_x,\ell_y}+C\tau\abs{\bf u}_{1,\ell_x,\ell_y,\infty}\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}.
\end{equation*}
In these estimations we used the fact that for vector function ${\bf u}=(u,v)$, the norm of $u$ or $v$ can be bounded by the norm of ${\bf u}$, for example
\[
\norm{\comii{y}^{\ell_y}\nabla\dd^\gamma u}_{L^\infty}\leq \abs{\bf u}_{\abs{\gamma}+1,\ell_x,\ell_y,\infty}.
\]
With this consideration the estimations can be proved similarly by the method of <cit.> and the arguments of the commutator estimates, and we omit the details.
To estimate $\Pcal_{x}$, we proceed as above, and write
\begin{equation*}
\Pcal_x\leq C\sum_{m=3}^\infty\sum_{j=0}^{m-2} \Pcal_{x,m,j},
\end{equation*}
\begin{equation*}
\Pcal_{x,m,j}=\frac{m\tau^{m-3}}{(m-3)!^s}{{m-2}\choose j} \norm{\left|\dd_x^j\nabla u\right|\left|\dd_x^{m-j-2}\nabla v\right|}_{L^2}.
\end{equation*}
For lower $j$, we have
\begin{align*}
& \sum_{m=3}^\infty \Pcal_{x,m,0}\leq C\abs{\bf u}_{1,\infty}\abs{\bf u}_2+C\tau\abs{\bf u}_{1,\infty}\abs{ \bf u}_3+C\tau^2\abs{\bf u}_{1,\infty}\norm{\bf u}_{Y_\tau},\\
& \sum_{m=4}^\infty \Pcal_{x,m,1}\leq C\tau\abs{\bf u}_{2,\infty}\abs{\bf u}_2+C\tau^2\abs{\bf u}_{2,\infty}\abs{\bf u}_3+C\tau^3\abs{\bf u}_{2,\infty}\norm{\bf u}_{Y_\tau},\\
& \sum_{m=6}^\infty \Pcal_{x,m,2}\leq C\tau^3\abs{\bf u}_{3,\infty}\abs{\bf u}_3+C\tau^4\abs{\bf u}_{3,\infty}\norm{\bf u}_{Y_\tau}.
\end{align*}
For mediate $j$, we have
\begin{align*}
& \sum_{m=8}^\infty\sum_{j=3}^{[m/2]-1} \Pcal_{x,m,j}\leq C\tau^3\norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_\tau}\\
& \sum_{m=6}^\infty\sum_{j=[m/2]}^{m-3} \Pcal_{x,m,j}\leq C\tau^3\norm{\bf u}_{X_\tau}\norm{\bf u}_{Y_\tau}.
\end{align*}
Finally for higher $j$, we have
\begin{align*}
\sum_{m=5}^\infty \Pcal_{x,m,m-2}\leq C\tau^2\abs{\bf u}_{1,\infty}\norm{\bf u}_{Y_\tau}.
\end{align*}
These estimations can be proved similarly as <cit.> with the fact that $\norm{\bf u}_{X_\tau}\leq\norm{\bf u}_{X_{\tau,\ell_x,\ell_y}}$, $\norm{\bf u}_{Y_\tau}\leq\norm{\bf u}_{Y_{\tau,\ell_x,\ell_y}}$ and $\norm{\bf u}_{H^m}\leq\norm{\bf u}_{H^m_{\ell_x,\ell_y}}$. With all these estimations, we can complete the proof Lemma <ref>.
Acknowledgments. W. Li would like to appreciate the support from NSF of China (No. 11422106), and C.-J. Xu was partially supported by “the Fundamental Research Funds
for the Central Universities” and the NSF of China (No. 11171261).
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1511.00352
|
Leslie Pack Kaelbling
Many methods have been proposed for detecting emerging events in text streams using topic modeling. However, these methods have shortcomings that make them unsuitable for rapid detection of locally emerging events on massive text streams. We describe Spatially Compact Semantic Scan (SCSS) that has been developed specifically to overcome the shortcomings of current methods in detecting new spatially compact events in text streams. SCSS employs alternating optimization between using semantic scan (<cit.>) to estimate contrastive foreground topics in documents, and discovering spatial neighborhoods (<cit.>) with high occurrence of documents containing the foreground topics. We evaluate our method on Emergency Department chief complaints dataset (ED dataset) to verify the effectiveness of our method in detecting real-world disease outbreaks from free-text ED chief complaint data.
Event Detection, Latent Dirichlet Allocation, Topic Modeling
§ INTRODUCTION
Text streams are ubiquitous in data processing and knowledge discovery workflows. Their analysis and summarization is difficult because of their unstructured nature, the sparsity of the canonical bag-of-words representation, the massive scale of web-scale text streams like Twitter and Yelp Reviews, and the noise present due to word variations from mispellings, dialects, and slang.
Topic modeling is a mixed-membership model used to summarize a corpus of text documents from a set of latent topics, where each topic is a sparse distribution on words. However, traditional topic modeling methods like Latent Dirichlet Allocation (LDA) are too slow for analyzing web-scale text streams, and also assume that there is no concept drift in the topics being learned over time. Variations like Online LDA (<cit.>), Dynamic Topic Models (<cit.>), Topics over Time (<cit.>), and Non-parametric Topics over Time (<cit.>) relax the assumption that there is no concept drift in the learned topics with time, but make strong assumptions about the evolution of topics with time.
In this paper, we propose Spatially Compact Semantic Scan (SCSS) which was developed to overcome these shortcomings in the scalable detection of spatially localized emerging topics in text streams.
In the Background section (<ref>), we introduce important terminology used in the rest of the paper, describe Latent Dirichlet Allocation (LDA), and collapsed Gibbs sampling used in LDA inference. In the Related Work section (<ref>), we present a literature survey and compare SCSS to related previous work on event detection in text streams. In the Methodology section (<ref>), we motivate and describe Spatially Compact Semantic Scan (SCSS). In the Results section (<ref>), we present results comparing SCSS to state-of-the-art methods described in the Related Work section. We discuss the results and possible avenues for future work in the Discussions section (<ref>), and conclude the report in the Conclusions section (<ref>).
§ BACKGROUND
§.§ Terminology
Here, we introduce some terminology that we will encounter often through this report:-
* Latent Dirichlet Allocation (<cit.>): A Bayesian mixed-membership topic model which treats each text document as a mixture of various topics i.e. multinomial distributions over words. Described in section (<ref>).
* Semantic Scan (SS): A enhancement of the LDA topic model to detect emerging topics in text corpora. Described in detail in section (<ref>).
* Spatially Compact Semantic Scan (SCSS): A further improvement over Semantic Scan to detect emerging topics from spatio-temporal text corpora such that the emerging topic occurs in documents are spatially located close to each other. Described in detail in section (<ref>).
* Background Documents: To detect emerging topics, we assume that a portion of our corpus is composed of documents where the emerging topic does not occur. These documents are referred to as background documents, and the portion of the corpus is referred to as background corpus.
* Foreground Documents: The documents which may contain the emerging topics are called foreground documents, and the appropriate portion of the corpus is called foreground corpus. The division of the entire corpus into foreground and background corpora is designated by the user of the method. The user needs to specify the dividing timestamp such that the documents that were collected before the timestamp are designated background documents and the documents that were collected after the timestamp are designated foreground documents. For the ED dataset, we chose all documents from 2003 as the background documents and all documents from 2004 as the foreground documents.
* Background Topics: Also called old topics or static topics. These are topics that are considered to have generated the background documents through the LDA generative process.
* Foreground Topics: Also called new topics or emerging topics. These topics alongwith the background topics are considered to have generated the foreground documents through the LDA generative process.
§.§ Latent Dirichlet Allocation (LDA)
Here, we briefly describe the Latent Dirichlet Allocation (LDA) topic model (<cit.>) since it forms the foundation of SCSS. LDA assumes that documents are generated as a mixture of topics, where topics themselves are distributions over words. The model also has an intuitive polyhedral interpretation: the documents reside on a low-dimensional topic simplex embedded in the high-dimensional word simplex. Parameter inference on the LDA topic model therefore aims at dimensionality reduction using a small number of latent topics so as to best explain the observed documents using these latent topics. While this assumption is not necessarily true in practice,it is a very useful assumption that helps in denoising and discovering frequent distinctive word co-occurrences from text corpora.
Plate Diagram of the LDA Topic Model
We refer the reader to figure (<ref>) which describes the generative model for LDA. There are $T$ topics given by $\phi = \{\phi_i: i=1,..,T\}$, where each $\phi_i$ is a distribution over the $N_u$ unique words in the dictionary. Since $\phi_i$ is the parameter of a multinomial distribution over words, it is naturally assumed to be generated from a Dirichlet prior with parameter $\beta$. There are $N$ documents in total. The multinomial distribution over the $T$ topics in the $i^{th}$ document is parameterized by $\theta_i$, which is generated from a Dirichlet distribution with parameter $\alpha$. To generate the $j^{th}$ word of the $i^{th}$ document, we first sample the topic at that position $z_{ij}$ from $\theta_i$, and then use the sampled topic to sample the actual word $w_{ij}$ from a multinomial distribution with parameter $\phi_{z_{ij}}$.
The plate diagram shows plates for the $T$ topics, $N$ documents, and $N_d$ words within each documents, indicating that these templates need to be repeated the corresponding number of times to obtain the full probabilistic graphical model over which inference of the unknown parameters will be performed. The observed words $w_{ij}$ in the $j^{th}$ position of the $i^{th}$ document are indicated by darkened circles in the plate diagram, while the unobserved variables to be inferred are indicated by empty circles. Exact inference in LDA is intractable in general, and the posterior over the unobserved parameters is obtained using (collapsed) Gibbs sampling or variational inference. Recent research has found the LDA model to be identifiable under assumptions of separability i.e. each topic has an anchor word that only occurs in that topic with positive probability, and hence only appears in documents which have that topic in their generating mixture of topics (<cit.>). Further work has found that the model is identifiable under weaker assumptions (<cit.>).
§.§ Collapsed Gibbs Sampling for LDA Inference
The most common sampling-based approach to LDA inference is collapsed Gibbs sampling (<cit.>). In collapsed Gibbs sampling, we maintain the following variables during the inference:-
* $\bf{Z}$ $ = \{z_{ij}\}$: topic assignments in all documents indexed by $i$ and word positions within each document indexed by $j$.
* $n_{ik}$: the number of times topic $k$ is assigned to words in document $i$. Therefore, $n_{ik} = \sum_{j} [z_{ij} == k]$. Here, $[condition]$ is the indicator function that is 1 when the $condition$ is true and 0 when the $condition$ is false.
* $n_{kw}$: the number of times topic $k$ is assigned to word $w$ in the entire corpus $C$. Therefore, $n_{kw} = \sum_{i,j} [z_{ij} == k \text{ and } w_{ij} == w]$ i.e. we count all word positions in the corpus where the actual word is $w$ and the topic assignment is $k$.
* $n_k$: the number of times topic $k$ is assigned to any word in the entire corpus $C$. Therefore, $n_k = \sum_w n_{kw}$.
We note that $n_{ik}$, $n_{kw}$, and $n_k$ can be calculated given $\bf{Z}$ as explained for each one of them. However, these statistics are stored because having access to them at each step of the Gibbs sampling makes the process much faster.
Let ${\bf Z}_{-ij}$ denote a particular instance of topic assignments, excluding the assignment at the $j^{th}$ position of the $i^{th}$ document. Let $n^{(-ij)}_{ik}$, $n^{(-ij)}_{kw}$, and $n^{(-ij)}_k$ be the $n_{ik}$, $n_{kw}$, and $n_k$ aggregate statistics calculated also without considering the topic assignment at the $j^{th}$ position of the $i^{th}$ document. These can be easily calculated from $n_{ik}$, $n_{kw}$, and $n_k$ by subtracting the contribution to these counts resulting from the topic assignment $z_{ij}$ at $j^{th}$ position of the $i^{th}$ document.
Collapsed Gibbs sampling proceeds through all words in the corpus sequentially. At each word position $w_{ij}$ in document $i$, it calculates the multinomial topic probability $P(z_{ij})$ conditioned on the observed corpus $D$ and all other topic assignments in the corpus ${\bf Z}_{-ij}$. The calculation is governed by the following formula:
\begin{equation}
P(z_{ij} = k \mid {\bf Z}_{-ij}, {\bf D}) \propto (n^{(-ij)}_{ik}+\alpha_k) \left(
\frac{n^{(-ij)}_{kw} + \beta_w}{ n^{(-ij)}_{k} + \sum_w \beta_w} \right)
\label{eq:collapsed_gibbs_lda}
\end{equation}
Gibbs sampling then samples a topic from the topic multinomial distribution $P(z_{ij} = k \mid {\bf Z}_{-ij}, {\bf D})$ and updates the statistics $n_{ik}$, $n_{kw}$, and $n_k$ by adding the count contribution from the newly sampled topic $z_{ij}$ at word position $w_{ij}$. The algorithm then moves to the next word position and repeats the topic multinomial calculation, topic sampling, and statistics updates. This process of sequentially going through the words of the corpus and sampling $z_{ij}$ is known to eventually converge to sampling from the stationary distribution of the LDA topic model after an intial burn-in period typical of MCMC sampling methods.
§ RELATED WORK
In this section, we briefly describe some related papers and discuss some of their shortcomings in detecting spatially compact emerging topics in text streams.
§.§ Efficient topic model inference on streaming document collections
The Gibbs2 and Gibbs3 sampling-based inference methods described in (<cit.>) are very similar to the Semantic Scan setting we describe below, and perhaps the closest work in literature to which we can compare our method. Gibbs2, Gibbs3, and Semantic Scan begin by learning topic assignments for words in the background documents and then begin inference on the foreground documents. All three methods also hold the topic assignments for words in background documents fixed, while performing sampling for topic assignments in the foreground documents. However, a key distinction is that our method allows additional new topics to be assigned to words in foreground documents, while Gibbs2 and Gibbs3 do not. We will see that allowing new topics to be learned entirely from foreground documents leads to precise topics that characterize emerging events in the text stream well. In fact, setting the number of new topics in Semantic Scan to 0 gives us Gibbs3. This is because the background topics are not allowed to change once they have been learned in both Semantic Scan and Gibbs3. Thus, Semantic Scan generalizes Gibbs3 for the purpose of emerging event detection.
Schematic diagram showing the distinction between Gibbs1, Gibbs2, and Gibbs3. Note that SCSS is similar to Gibbs2 where we learn new topics from the foreground documents in a batch fashion. From (<cit.>)
§.§ Labeled LDA
Labeled LDA (<cit.>) is another paper which closely resembles our experimental setup. It is a supervised method in which different partitions of the corpus are constrained to contain different sets of topics. This is helpful in a multi-labeled text corpus where each text document can possibly be assigned multiple labels by human labelers to explicitly indicate the topics it contains. Each label is then associated with a topic and a document is assumed to contain only the topics corresponding to its labels during the Gibbs sampling. In our setup, we can associate the background documents with a subset of topics assigned to the foreground documents and the corrrespondence of our method with labeled LDA really becomes clear. However, this simplistic solution ignores the fact that the number of foreground documents available for emerging event detection may be orders of magnitude smaller than the number of background documents collected over years or even decades. Applying labeled LDA naively would mean performing Gibbs sampling on the entire corpus of background and foreground documents every time we receive a batch of new documents. In the SCSS method, we need to perform Gibbs sampling only on the foreground documents which can be much more computationally efficient than labeled LDA.
§.§ Topics over time
Topics over Time: A Non-Markov Continuous-Time Model of Topical Trends
(<cit.>) presents a graphical model which relaxes the assumption of Markovian evolution of the natural parameters of the topic model. Instead, each topic is associated with a continuous beta distribution over timestamps normalized to the interval $[0,1]$. The topics remain static over time, however, the occurrence of topics in the corpus varies with time. However, the assumption that the number of topics is constant over time and that only the topic parameters evolve smoothly with time is still present in this model.
Non-paramteric topics over time (<cit.>) is a variation of the algorithm that allows the number of topics to be determined from the corpus. However, the topics are still constrained to evolve smoothly over time.
§.§ Online LDA
Online LDA (<cit.>) employs online variational Bayes inference to determine the posterior distribution over the latent variables of the topic model. The algorithm is based on online stochastic optimization and is shown to provide equivalently good topics in lesser time compared to batch variational Bayes algorithm. The algorithm requires a learning rate $k \in (0.5,1]$ for convergence. This parameters specifies the rate at which the old parameters are forgotten. Thus, there is an assumption of parameter smoothness which can delay the detection of suddenly emerging topics in a text stream.
§.§ Online NMF
Latent Factor Detection and Tracking with Online Non-Negative Matrix Factorization (<cit.>) suggests using currently learnt topics (as a proxy to past documents) along with the new documents to learn the new set of topics. This approach is similar to a variant of semantic scan where the background topics are used in the initialization step of the MCMC procedure to find foreground topics, but are not held fixed through the inference of foreground topics. The drawback is that the foreground topics found using this method might include the background topics in their span but may not help us precisely find the emerging foreground topics, since the detected topics might be a mixture of both the old and the new latent factors.
§.§ Kernel Topic Models
Kernel Topic Models (<cit.>) is a topic model that can incorporate spatial, temporal, hierarchical, and social metadata about text documents in the topic model by assuming that a document is represented by real-valued features that are generated by real-valued functions sampled from a Gaussian process prior, and passing these features through a softmax function to obtain the document-topic proportions that lie on the probability simplex. It is a adaptation of Gaussian process latent variable model (GPLVM) (<cit.>) where the document-topic proportions are obtained in a manner similar to GPLVM but the topics are sampled from a Dirichlet prior as in LDA. Kernel Topic Models make no distinction between background and foreground documents and might be unable to detect spatially localized emerging events in a small number of foreground documents compared to a large background corpus. Like many other models, Kernel Topic Models also cannot be applied to event detection since it does not have the ability to detect when no event is emerging in the foreground documents without significant modification or additions to the algorithm.
§.§ Adaptive Topic Models
(<cit.>) propose an online version of LDA for topic detection and tracking. However, the method makes a strong assumption about the evolution of topics: the parameter of the Dirichlet prior generating a topic is a linear combination of the topic vector from the previous $\delta$ iterations of the algorithm. The smoothness and strict form imposed on the evolution of topics will not allow the method to detect rapidly emerging topics or subtle spatially localized topics hidden in the stream. In addition, the assumption is that the number of topics is constant over time and only the topic parameters evolve smoothly with time. There is no reason to believe that this is true, since the addition of a new topic does not mean that an old topic has disappeared from the corpus.
§.§ Dynamic Topic Models
Dynamic Topic Models (<cit.>) extends the LDA model by allowing the natural multinomial parameters of LDA to evolve over consecutive time slices. This is the standard Markovian assumption of state space models. The model is best illustrated using the plate diagram shown in figure (<ref>) which shows how DTM extends the LDA topic model shown in figure (<ref>) and clearly illustrates the Markovian evolution of parameters in the topic model.
Graphical representation of a dynamic topic model (for three time slices). From (<cit.>)
§.§ A Latent Variable Model for Geographic Lexical Variation
(<cit.>) propose a hierarchical LDA model consisting of a set of pure topics which suffer variations with region to form regional topics that finally generate the documents. The model assumes a fixed number of regions, and that pure topics exist in the form of regional variants in every region. A region is modeled using a bivariate Gaussian distribution which assumes that each region has a center where its regional topics are concentrated and the effect of the regional topics decays away from this center. While consideriing a bivariate Gaussian distribution for modeling the location of documents in SCSS is possible, it implies that the effect of a topic decays away from the epicenter. This might not be true in the case of steady state of a disease outbreak where all documents in the affected spatial neighborhood may be equally likely to contain the emerging topic.
§.§ Discovering Geographical Topics in the Twitter Stream
(<cit.>) propse another topic model for modeling text documents annotated with geospatial coordinates. As in (<cit.>), (<cit.>) also model the locations of documents as drawn from a bivariate Gaussian distribution. However, the goal is to model geographically localized topics where each topic is dominant in one region, rather than discover regional variants of pure topics as is the case with (<cit.>). Both (<cit.>) and (<cit.>) deal only with the spatial aspect of topic models and do not address detection of an emerging spatially localized topic.
§.§ Topic Posterior Contraction Analysis
Recent work (<cit.>) suggests a theoretical justification as to why typical topic models do not work well on short documents like tweets. This justifies the additional novel contributions we need to incorporate in topic modeling to improve the outcome of topic modeling on a spatio-temporal corpus of short text documents.
§ METHODOLOGY
Notation used in this paper
Symbol Explanation
$N_b$ number of background documents
$N_f$ number of foreground documents
$N = N_b + N_f$ total number of documents
$D_b$ corpus consisting of background documents
$D_f$ corpus consisting of foreground documents
$D$ composite corpus consisting of documents from $D_b$ and $D_f$
$D_{bi}$ $i^{th}$ background document in $D_b$
$D_{fi}$ $i^{th}$ foreground document in $D_f$
$D_i$ $i^{th}$ document in $D$
$w_{bi}$ words in $i^{th}$ background document in $D_b$
$w_{fi}$ words in $i^{th}$ foreground document in $D_f$
$w_i$ words in $i^{th}$ document in $D$
$T_b$ number of background topics
$T_f$ number of foreground topics
$T = T_b + T_f$ total number of topics
$N_d$ number of words in a single document
$N_u$ number of unique words in the dictionary
$\eta$ hyperparameter for distribution of $\gamma$
$\gamma$ severity hyperparameter for $\delta$
$s_c$ center of spatial region
$n$ size of spatial region
$S_{cn}$ set of nodes in spatial region
$p$ sparsity parameter
$S$ a subset of $S_{cn}$
$\delta$ variable capturing if doc has new topic
$\alpha_b$ dirichlet hyperparameter for mixture of old topics
$\alpha$ dirichlet hyperparameter for mixture of all topics
$\phi_b$ background topics from $1..T_b$
$\phi_f$ foreground topics from $1..T_f$
$\phi$ all topics from $1..(T_b+T_f)$
$\beta_b$ dirichlet hyperparameter for generating background topics
$\beta_f$ dirichlet hyperparameter for generating foreground topics
$\theta$ multinomial parameter for document-specific topic mixture
$z$ sampled topic per word position
$w$ sampled word at each position
(a) Semantic Scan First Phase (b) Semantic Scan Second Phase
Plate Diagram of Spatially Compact Semantic Scan
Plate Diagram of Spatially Compact Semantic Scan.
In our method, we aim to capture spatial coherence of the emerging topic i.e. we want to detect documents which contain the emerging topic and are spatially close to each other. This spatial proximity can be interpreted in a broad sense. In the simplest case, it can mean that documents that contain the new topic are actually generated at geospatial locations close to each other. In a more sophisticated case, we can consider that documents which contain the new topic are closer in other ways. For example, tweets generated by users that follow each other on Twitter can be considered to be close to each other in the heterogeneous network structure of the Twitter social graph. This could be useful in detecting topics that spread virally on a social network. For example, a stock market crash or undue market volatality could be a source of discussion and tweets among traders in New York, London, and Hong Kong who follow each other. Although these cities are geographically distributed, we believe that market-related tweets emerging from these cities will have similarity on other measures derived from the underlying network structure.
§.§ Semantic Scan
Semantic Scan (SS) <cit.> is a method which learns temporally emerging events from a text stream. The method learns a set of background topics from background documents which do not contain any emerging event of interest. When a new set of documents comes in, semantic scan learns a new set of topics for these documents while contrasting them with already learnt background topics so as not to relearn old topics. We describe SS in detail here, because our method Spatially Compact Semantic Scan (SCSS) builds on SS to incorporate spatial cohesion of documents containing the emerging topic as well.
The original semantic scan learns the set of emerging topics without taking spatial information into account, then performs a spatial scan to identify emerging spatial cluster of documents assigned to these topics. In contrast, SCSS coherently integrates a spatial model of the affected locations with the emerging topic model of semantic scan, thus enabling it to learnmore precisely focused, spatially localized topics which improve overall detection performance.
The Semantic Scan (SS) model for detecting emerging topics is illustrated using the plate diagram in figure (<ref>). The notation used in the model is explained in table (<ref>). The Bayesian generative model can be outlined as follows:-
* Generate background documents comprising $C_b$
* Choose a $T_b$-dimensional Dirichlet hyperparameter $\alpha_b$ for generating document-specific topic distributions, where $T_b$ is the number of background topics.
* Choose a $W$-dimensional Dirichlet hyperparameter $\beta_b$ for generating background topics, where $W$ is the number of words in the vocabulary.
* Sample $T_b$ background topics from $Dir(\beta_b)$ together denoted by $\phi_b$; each topic $\phi_{bi}$ is a $W$-dimensional multinomial distribution over words.
* For each of the $i=1,..,D_b$ background documents:
* Sample a $T_b$-dimensional multinomial distribution $\theta_i$ over background topics from $Dir(\alpha_b)$.
* For each of the $j=1,..,N_i$ words in the document:
* Sample a topic $z_{ij}$ for the word position $j$ from $Mult(\theta_i)$. $z_{ij}$ is one of the background topics $\phi_b$.
* Sample a word $w_{ij}$ for the word position $j$ from $Mult(\phi_{bz_{ij}})$.
* Generate foreground documents comprising $C_f$
* Choose a $T$-dimensional Dirichlet hyperparameter $\alpha$ for generating document-specific topic distributions, where $T=T_b+T_f$ is the total number of topics including background and foreground topics.
* Choose a $W$-dimensional Dirichlet hyperparameter $\beta_f$ for generating foreground topics.
* Sample $T_f$ foreground topics from $Dir(\beta_f)$ together denoted by $\phi_f$; each topic $\phi_{fi}$ is a $W$-dimensional multinomial distribution over words.
* Denote the set of all topics as $\phi$ which includes topics from $\phi_b$ indexed from $1,..,T_b$ and topics from $\phi_f$ indexed from $T_b+1,..,T_b+T_f$
* For each of the $i=1,..,D_f$ foreground documents:
* Sample a $T$-dimensional multinomial distribution $\theta_i$ over all topics from $Dir(\alpha)$.
* For each of the $j=1,..,N_i$ words in the document
* Sample a topic $z_{ij}$ for the word position $j$ from $Mult(\theta_i)$. $z_{ij}$ is one of the topics in $\phi$.
* Sample a word $w_{ij}$ for the word position $j$ from $Mult(\phi_{z_{ij}})$.
§.§ Inference for Semantic Scan
The inference procedure consists of two phases as shown in figure (<ref>). In the first phase, we learn a set of background LDA topics $\phi_{bi},~i=1,..,T_b$ using collapsed Gibbs sampling on a set of background documents. In the second phase, we keep the first $T_b$ topics fixed and learn new topics $\phi_{bi},~i=T_b+1,..,T_b+T_f$ while allowing the document-topic distributions $\theta_i$ for the foreground documents $C_f$ to change during the Gibbs sampling procedure. For details of the LDA collapsed Gibbs sampling procedure, we refer the reader to section (<ref>) and (<cit.>). If the number of background topics $T_b$ is sufficient and we have learnt the background topics well, then fixing the background topics during the detection of the new topic propels the new topic to capture emerging trends in the text stream. This is because the span of fixed background topics explains words in the documents that are produced by the background data-generating process and are irrelevant to the emerging topic.
The plate diagram in figure (<ref>.b) shows the model on which inference is performed in the second phase. The model shows that the words of the documents as well as the $T_b$ topics learned from the first phase are observed variables in the second phase, and we perform collapsed Gibbs sampling inference to learn the emerging topics $\phi_f$.
After the topic modeling step, SS (<cit.>) assigns each document to one of the topics using an EM-like approach, and performs circular expectation-based poission spatial scan in order to detect a circular neighborhood of zipcodes that are affected by the outbreak.
§.§ Spatially Compact Semantic Scan
In order to ensure that the emerging topic is also spatially regularized to occur in spatially nearby documents, we place a hierarchical prior over the spatial regions whose documents can be affected by the emergence of the new topic. The proposed topic model (which incorporates SS as a building block) is illustrated using the plate diagram in figure (<ref>). The notation used in the plate diagram is described in table (<ref>). The document generation process is as follows: We first select a subset of zipcodes where the documents will contain a new topic. To do this, we select a center of the spatial region $s_c$ and a neighborhood size $n$. The set of all zipcodes that are the $n$ nearest neighbors of $s_c$ form a circular neighborhood $S_{cn}$. To construct an arbitrarily shaped spatial region $S$ from $S_{cn}$, we choose a sparsity parameter $p$ and sample the zipcodes from $S_{cn}$ with probability $p$. This gives us a set of zipcodes $S$. Documents from zipcodes in $S$ may contain the foreground topics. The $i^{th}$ zipcode is associated with severity $\gamma_i$ sampled from a Beta distribution parameterized by $\eta$, which indicates the proposrtion of documents at that zipcode that will contain foreground topics.
For $j^{th}$ document located at the $i^{th}$ zipcode, we sample a document-specific $\delta_j$ as the output of a Bernoulli experiment indicating if the document should contain the new topic using the severity of the emerging topic indicated by $\gamma_i$. Documents outside $S$ are generated using the old topics and its distribution characteristics. For documents outside $S$, we set their $\delta$ to 0.
If $\delta_j$ is 1, we sample the distribution over topics $\theta_j$ using the hyperparameter $\alpha$ which indicates a distribution over all topics including the new ones. If $\delta_j$ is 0, we use $\alpha_b$ to sample the distribution over old topics only since the document is not supposed to contain the new topic. It is possible to constrain the new hyperparameter $\alpha$ for topic distributions using the old hyperparameter $\alpha_b$. However, in our case, both $\alpha_b$ and $\alpha$ are uniform symmetric priors.
Once we have sampled the multinomial parameters $\theta_j$, we can sample a topic for each word position in the document. This sampled topic can then be used to index into the set of topic vectors to get the parameters of a multinomial distribution over words. Using the topic chosen for the word position, we now can sample a word from the topic. This completes the generation process for the foreground documents in the text corpus with emerging spatially localized topics. The entire Bayesian generative model can be outlined as follows:
* Generate background documents comprising $C_b$
* Choose a $T_b$-dimensional Dirichlet hyperparameter $\alpha_b$ for generating document-specific topic distributions.
* Choose a $W$-dimensional Dirichlet hyperparameter $\beta_b$ for generating background topics.
* Sample $T_b$ background topics from $Dir(\beta_b)$ together denoted by $\phi_b$; each topic $\phi_{bi}$ is a $W$-dimensional multinomial distribution over words.
* For each of the $i=1,..,D_b$ background documents:
* Sample a timestamp unformly from the set of possible background timestamps $K_b$.
* Sample a zipcode uniformly from the set of zipcodes considered to generate the documents $Z$.
* Sample a $T_b$-dimensional multinomial distribution $\theta_i$ over background topics from $Dir(\alpha_b)$.
* For each of the $j=1,..,N$ words in the document:
* Sample a topic $z_{ij}$ for the word position $j$ from $Mult(\theta_i)$. $z_{ij}$ is one of the background topics $\phi_b$.
* Sample a word $w_{ij}$ for the word position $j$ from $Mult(\phi_{bz_{ij}})$.
* Generate foreground documents comprising $C_f$
* Choose a zipcode $s_c$ from possible zipcodes $Z$ and a neighborhood size $n$. The neighborhood of $n$ data-generating locations from $Z$ around $s_c$ is called $S_{cn}$.
* Choose a sparsity parameter $p \in (0,1]$. Choose locations from $S_{cn}$ with probability $p$ to form a subset $S$ of locations that will produce documents affected by the background and foreground topics. All other locations will produce documents generated from the background topics only.
* Choose a Bernoulli severity parameter $\gamma_i \in (0,1]$ for the $i^{th}$ zipcode in $S$ from $Beta(\eta)$.
* Choose a $T$-dimensional Dirichlet hyperparameter $\alpha$ for generating document-specific topic distributions.
* Choose a $W$-dimensional Dirichlet hyperparameter $\beta_f$ for generating foreground topics.
* Sample $T_f$ foreground topics from $Dir(\beta_f)$ together denoted by $\phi_f$; each topic $\phi_{fi}$ is a $W$-dimensional multinomial distribution over words.
* Denote the set of all topics as $\phi$ which includes topics from $\phi_b$ indexed from $1,..,T_b$ and topics from $\phi_f$ indexed from $T_b+1,..,T_b+T_f$
* For each of the $i=1,..,D_f$ foreground documents:
* Sample a timestamp uniformly from the set of possible foreground timestamps $K_f$.
* Sample a zipcode $l_i$ uniformly from the set of zipcodes $Z$ considered to generate the documents.
* If $l_i \notin S$ or ($l_i \in S$ and $Bern(\gamma_{l_i}) == 0$),
* Set new topic indicator $\delta_i=0$
* Sample a $T_b$-dimensional multinomial distribution $\theta_i$ over background topics from $Dir(\alpha_b)$.
* For each of the $j=1,..,N_i$ words in the document:
* Sample a topic $z_{ij}$ for the word position $j$ from $Mult(\theta_i)$. $z_{ij}$ is one of the background topics $\phi_b$.
* Sample a word $w_{ij}$ for the word position $j$ from $Mult(\phi_{bz_{ij}})$.
* If $l_i \in S$ and $Bern(\gamma_{l_i}) == 1$,
* Set new topic indicator $\delta_i=1$
* Sample a $T$-dimensional multinomial distribution $\theta_i$ over all topics from $Dir(\alpha)$.
* For each of the $j=1,..,N_i$ words in the document:
* Sample a topic $z_{ij}$ for the word position $j$ from $Mult(\theta_i)$. $z_{ij}$ is one of the topics in $\phi_i$, which could be either a background or a foreground topic.
* Sample a word $w_{ij}$ for the word position $j$ from $Mult(\phi_{z_{ij}})$.
§.§ Inference for Spatially Compact Semantic Scan
We note that the variables marked in a hatched texture in the plate diagram of figure (<ref>) are observed, while the other variables are to be inferred. The inference proceeds through MCMC sampling whose stationary distribution gives us the posterior distribution over the unobserved variables.
The inference of posterior distribution over variables $s_c$, $n$, $S_{cn}$, $p$, and $S$ (denoted collectively by $\mathcal{S}$) is done using the Generalized Fast Subset Sums framework (<cit.>) which allows for efficient inference of these variables given the likelihood ratio $LR_i$ of each document.
This results in an alternating MCMC where the inference over $\mathcal{S}$ happens conditioned on the values over the remaining variables $\Omega-\mathcal{S}$, and the sampling of the variables $\Omega-\mathcal{S}$ proceeds conditioned on the inference for $\mathcal{S}$. We iterate between these two conditional inference steps until convergence.
§.§.§ Inference over $\mathcal{S}$
The likelihood ratio of $i^{th}$ foreground document $C_{fi}$ is given as
\begin{equation}
LR_i = \frac{\mathcal{L}(C_{fi} \mid \theta_i,~\phi)}{\mathcal{L}(C_{fi} \mid \theta_{bi},~\phi_b)}
\end{equation}
Since we do not know the exact parameters $\theta_i$, $\phi$, $\theta_{bi}$, and $\phi_b$, we use MAP estimates of these parameters from the collapsed Gibbs sampling phase of the inference.
We calculate the likelihood of a document with words $\mathbf{w}$ as follows:
\begin{align}
\mathcal{L}(\mathbf{w} \mid \hat{\theta},~\hat{\phi}) & = \sum_{\mathbf{z}} Pr(\mathbf{w},~\mathbf{z} \mid \hat{\theta},~\hat{\phi}) \\
& = \prod_{i=1}^N \left\{ \sum_{z_i} Pr(w_i, z_i \mid \hat{\phi}_{z_i}, \hat{\theta}) \right\} \\
& = \prod_{i=1}^N \left\{ \sum_{z_i} Mult(w_i \mid \hat{\phi}_{z_i}) \cdot Mult(z_i \mid \hat{\theta}) \right\}
\end{align}
Here, the last step of interchanging sum of products with product of sums can be carried out because we are marginalizing over all possible values of each topic assignment variable $z_i$. This considerably speeds up the computation of the likelihood terms from exponential to linear time.
Once we have calculated the likelihood ratio for each document $LR_i$, we can calculate the posterior probability of the event $E_{cn}$ that emerging topics are localized in a given neighborhood $S_{cn}$ centered at $s_c$ and consisting of $n$ locations as follows (<cit.>):-
\begin{align}
Pr(E_{cn} \mid C_f) & \propto \sum_{S \subseteq S_{cn}} Pr(S \mid C_f) \\
& \propto \sum_{S \subseteq S_{cn}} Pr(S) \cdot \prod_{s_i \in S} LR_i \\
& \propto \sum_{S \subseteq S_{cn}} p^{|S|} \cdot (1-p)^{n-|S|} \cdot \prod_{s_i \in S} LR_i \\
& \propto (1-p)^n \sum_{S \subseteq S_{cn}} \left(\frac{p}{1-p}\right)^{|S|} \cdot \prod_{s_i \in S} LR_i \\
& \propto (1-p)^n \sum_{S \subseteq S_{cn}} \prod_{s_i \in S} \left(\frac{p}{1-p}\right) \cdot LR_i
\end{align}
Since we are summing over $2^n$ subsets of $S_{cn}$, we can reduce the time complexity from exponential to linear by writing sum of $2^n$ products as the product of $n$ sums (<cit.>).
\begin{align}
\therefore Pr(E_{cn} \mid C_f) & \propto (1-p)^n \prod_{s_i \in S_{cn}} \left\{ 1 + \left(\frac{p}{1-p}\right) \cdot LR_i \right\} \\
& \propto \prod_{s_i \in S_{cn}} \left\{ 1-p + p \cdot LR_i \right\}
\end{align}
Thus, the posterior probability of a neighborhood $S_{cn}$ showing an outbreak of foreground topics in its documents is proportional to the product of smoothed likelihood ratios $(1-p + p \cdot LR_i)$ for all documents in the neighborhood. Finally, we calculate the normalizer by marginalizing over all $S_{cn}$.
\begin{align}
Pr(E_{cn} \mid C_f) & = \frac{Pr(E_{cn} \mid C_f)}{\sum_{\forall~s_c,n} Pr(E_{cn} \mid C_f)}
\end{align}
Probability of an event $E_j$ that foreground topics occur in document at location $s_j$ can be calculated in a similar fashion by considering only those neighborhoods $S_{cn}$ where $s_j$ is included and summing over the $2^{n-1}$ subsets $S \subseteq S_{cn}$ such that $s_j \in S$. The average likelihood ratio in neighborhood $S_{cn}$ such that location $s_j$ is always included in any chosen subset $S$ of locations is given as $p \cdot LR_j \cdot \prod_{s_i \in S_{cn}-s_j} \left\{ 1-p + p \cdot LR_i \right\}$. The total posterior probability $Pr(E_j \mid C_f)$ of document in location $s_j$ containing the foreground topics can be calculated by marginalizing over all $s_c$ and $n$ such that the resultant neighborhood $S_{cn}$ contains location $s_j$.
§.§.§ Inference over $\Omega - \mathcal{S}$
This phase is very similar to the second phase of Semantic Scan inference described earlier with changes to incorporate only those foreground documents that we believe actually contain the emerging topics and are therefore composed of not just the background topics. Given the total posterior probability $Pr(E_i \mid C_f)$ of document in location $s_i$ containing the foreground topics calculated from the previous phase of inference, we generate a binary value $\delta_i$ by sampling from a Bernoulli distribution $Bern(Pr(E_i \mid C_f))$. If $\delta_i=0$, we do not believe that the document has foreground topics and therfore do not include it as a part of the foreground documents on which Gibbs sampling happens in the second phase of Semantic Scan. If $\delta_i=1$, we believe that the document has foreground topics as indicated by our spatial inference based on document-specific likelihood ratios and therfore we include the document as a part of the foreground documents on which Gibbs sampling happens in the second phase of Semantic Scan. Once we have decided the documents on which Gibbs sampling is to be done by assigning values to $\delta_i$ , we proceed with collapsed Gibbs sampling according to the procedure described in section (<ref>) to obtain foreground topics $\phi_f$ and document-specific distributions over topics $\theta_i$ that are used again in inference over $\mathcal{S}$ in our alternating inference mechanism.
(a) Spatial Precision (a) Document Precision
(b) Spatial Recall (b) Document Recall
(c) Spatial Overlap (c) Document Overlap
§ RESULTS
In this section, we describe the ED dataset and how we use it to compare Spatially Compact Semantic Scan (SCSS) with competing approaches such as Semantic Scan (SS), Topics over Time (ToT), Spatial Topics over Time (Spatial ToT), Online LDA (OLDA), and Naive Bayes (NB).
§.§ Emergency Department Chief Complaints Dataset
The ED dataset consists of text complaints noted by the staff of emergency departments of Allegheny County hospitals. The dataset includes complaints from 2003 to 2005. Each complaint is associated with the date on which it was recorded, the zipcode of the hospital where the complaint was recorded, and the ICD9 code to which it was assigned. The external manual classification of diseases using the ICD9 codes has been used to create semi-synthetic disease outbreaks as described below. This external piece of information associated with a text complaint is not assumed to be known by the detection methods and is used for evaluating the methods only. In practice, in many cases, the ICD9 code is unknown or incorrect until its final assignment for billing purposes after the patient's visit. In order to get geospatial coordinates for a complaint, we map its associated zipcode to the centroid latitude and longitude coordinates for the zipcode area. Thus, we have a dataset where each document is a short text complaint followed by the date on which it was recorded, the geospatial coordinates of the zipcode in which it was recorded, and its ICD9 code.
(a) Fraction of Outbreaks Detected (b) Days to Detection
Detection Power
§.§ Experimental Setup
We perform leave-one-out (LOO) validation of SCSS alongwith the baseline methods. We treat documents from 2003 as the background documents and documents from 2004 as the foreground documents. We pick the 10 most frequent ICD9 codes in the dataset. For each of these ICD9 codes, we remove all complaints from the background and foreground data corresponding to that ICD9 code. We then create outbreaks corresponding to the held-out ICD9 code in the foreground documents belonging to 2004. The outbreak is created by sampling $s_c$ and $n$, calculating the neighborhood $S_{cn}$, sampling sparsity parameter $p$, and sampling $S$ from $S_{cn}$. To generate a datapoint in an outbreak, we sample the text document uniformly from the held-out ICD9 text complaints, and the location uniformly from the zipcodes in $S$. For each of the 10 held-out ICD9 codes, we create 10 outbreaks each, resulting in a total of 100 outbreaks over which we run SCSS and each of the baselines. Each outbreak is 30-days long, and the number of cases generated for the $d^{th}$ day is $3*d$. While running any of the methods, we assume a 3-day moving window for outbreak detection. For methods like SCSS and Spatial ToT, it is not necessary to perform a spatial scan as the last step since these methods also output a detected spatial region. For other methods including the original semantic scan, we perform the assignment of documents to topics and circular spatial scan as outlined in the Semantic Scan paper (<cit.>).
§.§ Competing Approaches
We have chosen the following related work against which to compare SCSS:
* SS-Emerging: This is the version of Semantic Scan (SS) (<cit.>) that we described in section (<ref>). It allows for both background and foreground topics and holds background topics fixed while learning foregroudn emerging topics on incoming batch of documents. 25 background and 25 foreground topics are used in the evaluation.
* SS-Dynamic: This version of SS (<cit.>) does not allow for any background topics. The topic learning only focuses on foreground topics on an incoming batch of documents. 25 foreground topics are used in the evaluation; there are no background topics in this model.
* SS-Static: This version of SS (<cit.>) does not allow any foreground topics. Once static topics are learned at the beginning, SS-Static only performs document assignment and sptial scan steps for an incoming batch of documents. 25 background topics are used in the evaluation; there are no foreground topics in this model.
* Topics over Time (ToT): ToT (<cit.>) is an LDA variant that incorporates temporal aspect of a document set by modeling the timestamps assigned to a topic as being sampled from a $Beta$ distribution specific to the topic. 50 total topics are used in the evaluation of this this model.
* Spatial Topics over Time (Spatial ToT): In order to perform an apples-to-apples comparison to SCSS, we modified ToT (<cit.>) by additionally modeling the spatial coordinates assigned to a topic as being sampled from a 2D spatial $Gaussian$ distribution specific to the topic. 50 total topics are used in the evaluation of this this model.
* Online LDA: We compare SCSS to two versions of Online LDA (<cit.>) - one with $\kappa=0.55$, and one with $\kappa=0.95$. We refer to these two variants as "OLDA:0.55" and "OLDA:0.95" respectively. $\kappa$ is a hyperparameter of Online LDA algorithm that controls how quickly topics can adapt to changes in the topics of the text stream. 50 total topics are used in the evaluation of this this model.
* Naive Bayes: Finally, we compare SCSS to NB by considering background and foreground documents as belonging to two different classes, using the NB prediction as a document's assignment, and performing spatial scan on the obtained document assignments.
§.§ ED Dataset Results
We consider the following metrics to compare SCSS to our chosen baselines:
* Spatial Precision: $\left(\frac{tp}{tp+fp}\right)$ The fraction of zipcodes that are actually a part of the outbreak, out of the zipcodes that were detected to be a part of the outbreak.
* Spatial Recall: $\left(\frac{tp}{tp+fn}\right)$ The fraction of zipcodes that were detected to be a part of the outbreak, out of the zipcodes that are actually a part of the outbreak.
* Spatial Overlap: $\left(\frac{tp}{tp+fp+fn}\right)$ The fraction of zipcodes that are actually a part of the outbreak as well as detected to be a part of the outbreak, out of all zipcodes that were either actually a part of the outbreak or detected to be a part of the outbreak.
* Document Precision: $\left(\frac{tp}{tp+fp}\right)$ The fraction of foreground documents that actually contain the foreground topics out of the foreground documents that were detected to contain the foreground topics.
* Document Recall: $\left(\frac{tp}{tp+fn}\right)$ The fraction of foreground documents that were detected to contain the foreground topics out of the foreground documents that actually included the foreground topics during their generation.
* Document Overlap: $\left(\frac{tp}{tp+fp+fn}\right)$ The fraction of documents that were injected with foreground topics and detected to contain the foreground topics out of all documents that were either injected with foreground topics or detected to contain the foreground topics.
* Percentage of Outbreaks Detected: The percentage of outbreaks detected versus the number of false positives per year. This is a monotonically non-decreasing graph where a higher value represents a better outcome.
* Days to Detection: The number of days of data required to detect an outbreak versus the number of false positives per year. This is a monotonically non-increasing graph, where a lower value represents a better outcome.
The graphs for spatial precision, recall, and overlap can be found in figure (<ref>). The three metrics are plotted against the outbreak day on the X-axis which ranges from 1 to 25. We observe that SCSS has spatial precision, recall, and overlap almost double that of any of the baselines. All three metrics improve steadily as the duration and intensity of the outbreak increases.
The graphs for document precision, recall, and overlap can be found in figure (<ref>). The three metrics are again plotted against the duration of the outbreak on the X-axis measured in number of days. We notice that SCSS has significantly better document precision and overlap compared to the baselines. For document recall, SCSS and Naive Bayes have similar performance, and Naive Bayes exceeds the SCSS performance at several points of the graph. However, this just indicates that Naive Bayes is classifying a lot of documents as a part of the outbreak. Considered together with document precision and overlap, we still conclude that SCSS performs significantly better than the baselines we have compared to.
The graphs for the fraction of outbreaks detected and the number of days of data required to detect an outbreak can be found in figure (<ref>). Both metrics are plotted against the number of false positives per year on the X-axis. As expected, we see that the fraction of outbreaks detected increases as we allow more false positives per year. Similarly, the number of days of data required to detect an outbreak decreases as we allow more false positives per year. We note that SCSS performance on these metrics is comparable to that of SS-Emerging, and is better than SS-Emerging for low false positive rates. SCSS performance is not significantly better than the baselines on these two metrics. However, coupled with performance on precision, recall, and overlap metrics, SCSS beats the baselines that we have compared to. Many of these baselines like ToT, Spatial ToT (which performs better than ToT), and Online LDA are state-of-the-art methods in literature for (spatio-)temporal event detection in text streams.
§ FUTURE WORK
We envision the following possible investigations and refinements to SCSS:
* Combining spatio-temporal text data streams with other spatio-temporal data such as heat indices, medication sales, etc. to improve the detection power of SCSS.
* Testing the robustness of SCSS using other datasets and outbreak simulations.
* Scaling up the method so that it can be run on massive datasets such as Yelp reviews or Twitter streams.
* Ability to detect multiple spatial clusters. Prior work (<cit.>) incrementally detect a cluster and removes it to reveal other clusters. Incorporating this feature and testing its accuracy and efficiency merit further investigation.
§ CONCLUSIONS
We have proposed a topic model for finding spatially compact and temporally emerging topics in real-world text corpora. We have evaluated the model on real-world ED data from disease outbreak detection and presented our results in section (<ref>) to demonstrate the efficacy of our method.
One of the promising future directions we are considering is finding subtle emerging topics by mining the residuals of the new documents i.e. the component of the document vectors not explained by the currently learnt topics. A newly emerging topic will tend to create clusters in the residual space, which can then be mined for topics in high density regions using an algorithm like DBSCAN (<cit.>), a spatial data-structure like R-Tree (<cit.>), or using linear algebraic scan statistics that search for high density cones in the residual space.
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1511.00036
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Laboratoire de Physique, École Normale Supérieure de Lyon,
46 allée d'Italie, 69007 Lyon, France
Laboratoire de Physique, École Normale Supérieure de Lyon,
46 allée d'Italie, 69007 Lyon, France
Laboratoire de Physique des Solides, CNRS, Université Paris-Sud,
Université Paris-Saclay, 91405 Orsay Cedex, France
Laboratoire de Physique, École Normale Supérieure de Lyon,
46 allée d'Italie, 69007 Lyon, France
A continuous deformation of a Hamiltonian possessing at low energy two
Dirac points of opposite chiralities can lead to a gap opening by
merging of the two Dirac points. In two dimensions, the critical
Hamiltonian possesses a semi-Dirac spectrum: linear in one
direction but quadratic in the other.
We study the transport properties across such a transition, from a
Dirac semi-metal through a semi-Dirac phase towards a gapped
phase. Using both a Boltzmann approach and a diagrammatic Kubo
approach, we describe the conductivity tensor within the diffusive
regime. In particular, we show that both the anisotropy of the Fermi
surface and the Dirac nature of the eigenstates combine to give rise
to anisotropic transport times, manifesting themselves through an
unusual matrix self-energy.
§ INTRODUCTION
The discovery of graphene has triggered a lot of work on the exotic transport properties of Dirac-like particles in solids <cit.>.
Indeed, the graphene electronic spectrum is made of two sub-bands
which touch at two inequivalent points in reciprocal space.
Near the touching points, named Dirac points, the spectrum has a
linear shape and the electron dynamics is well described by a 2D
Dirac equation for massless particles.
Due to the structure of the honeycomb lattice, the wave functions have two components corresponding to the two inequivalent sites of the lattice, and the Hamiltonian is a $2 \times 2$ matrix.
To describe the low energy properties, the original Hamiltonian is replaced by two copies of a 2D Dirac equation
H= ±c . σ⃗
where the velocity $c \simeq 10^5~$m.s$^{-1}$.
This linearization is possible because the energy of the saddle point separating the two Dirac cones (valleys) is very large ($\simeq 3$ eV) compared to the Fermi energy and temperature scales.
Other realizations of Dirac-like physics in two dimensions have been proposed in the
organic conductor (BEDT-TTF)$_2$I$_3$ under
pressure<cit.>, and has been observed in
artificially assembled nanostructures<cit.> and ultracold
Besides these two dimensional realizations, the existence and properties of semi-metallic phases in three dimensions have recently been studied
To go beyond and in order to account for a structure which consists in two Dirac points separated by a saddle point, one needs an appropriate low energy $2 \times 2$ Hamiltonian. Moreover, such a description is mandatory in situations where, by varying band parameters, the Dirac points can be moved in reciprocal space.
Since these Dirac points are characterized by opposite topological charges, they can even annihilate each other <cit.>
. This merging is therefore a topological transition. It has been
shown that, at the transition, the electronic dispersion is quite
unusual since it is quadratic in one direction and linear in the other direction (the direction of merging). This "semi-Dirac" <cit.> spectrum has new properties intermediate between a Schrödinger and a Dirac spectrum. The vicinity of the topological transition can be described by the following Hamiltonian in two dimensions<cit.>:
H= (Δ+ p_x^2 2 m ) σ_x + c_y p_y σ_y . It has been coined "Universal Hamiltonian" since the merging scenario of two Dirac points related by time reversal symmetry is uniquely described by this Hamiltonian<cit.>.
The parameter $\Delta$ drives the transition ($\Delta=0$) between a semi-metallic phase ($\Delta<0$) with two Dirac points and a gapped phase ($\Delta>0$), see Figs. <ref>,<ref>. The evolution of several thermodynamic quantities like the specific heat and the Landau level spectrum has been studied in details <cit.>.
This work addresses the transport properties for an electronic spectrum undergoing a topological merging transition as
depicted in this figure, and commented in more details in Fig. <ref>.
In this paper we address the evolution of the conductivity tensor across the merging transition (Figs. <ref>,<ref>).
A first objective of this work is to characterize the transport properties as a possible signature of the evolution of the underlying band structure.
On a more fundamental perspective, an additional interest of this problem
stands from two important ingredients in the description of diffusive transport.
First, at low energy the electronic wave functions have a spinorial structure which leads to effective anisotropic scattering matrix elements (similar to the case of a scalar problem with anisotropic scattering due to a disorder potential with finite range).
This leads to a transport scattering time $\tau^{\tr}$ different from the elastic scattering time $\tau_e$ , as in graphene for point-like impurities where $\tau^{\tr}= 2 \tau_e$.
Second, the anisotropy of the dispersion relation leads to an additional complexity: the scattering times become themselves anisotropic and depend on the direction of the applied electric field. We show that within the Green's function formalism this anisotropy manifests itself into a rather unusual matrix
structure of the self-energy. A comparison between a Boltzmann approach and a perturbative Green's function formalism allows for a detailed understanding of this
The outline of the paper is the following. In the next section, we
recall the model, i.e. the Universal Hamiltonian with coupling
to impurities described by a point-like white noise potential. We
define a directional density of states and derive the angular dependence of the elastic scattering time. In section <ref>, we use the Boltzmann equation to calculate the conductivity tensor.
As a result of the two important ingredients mentioned above, the conductivity along a direction $\alpha$ is not simply proportional to the angular averaged squared velocity $\langle v_\alpha^2(\theta) \rangle$ because :
(i) the elastic scattering time has also an angular dependence due to the angular anisotropy of the spectrum, so that one should consider the average $\langle v_\alpha^2(\theta) \tau_e(\theta)\rangle$; (ii) since the matrix elements of the interaction get an angular dependence, it will lead to transport times different of the elastic time. These transport times depend on the direction $\alpha$ and, to obtain the conductivity, we will have to consider the average $\langle v_\alpha^2(\theta) \tau_\alpha^{\tr}(\theta)\rangle$. These results obtained from Boltzmann equation are confirmed by a diagrammatic calculation presented in section <ref>. We discuss our results in the last section.
§ THE MODEL
§.§ Hamiltonian and Fermi surface parametrization
Typical energy spectrum of the model (<ref>) for various $\Delta $ but a fixed
energy $\epsilon >0$.
Dirac phase with
(a) $\Delta < - \epsilon$,
(S) $\Delta = - \epsilon $ (saddle-point) ,
(b) $ - \epsilon < \Delta <0$.
Critical semi-Dirac metal
(M) $\Delta = 0$.
Gapped phase
(c) $\Delta > 0$.
We consider the model described by the Hamiltonian
\begin{equation}
\label{eq:HModel}
H=H^{0}+ V,
\end{equation}
where the disorder potential $V$ is defined and discussed in section <ref> and the Hamiltonian for the pure system is
defined as
\begin{equation}
\label{eq:PureModel}
H^{0} = \left[ \Delta + \frac{p_x^2}{2m} \right] \sigma_x + c_y p_y \, \sigma_y \ .
\end{equation}
In the present and the following sections (<ref> and <ref>)
we start by discussing a few properties of the Hamiltonian $H^{0}$ without disorder.
For $\Delta >0$ this Hamiltonian describes a gapped phase. When $\Delta < 0$, it describes
two Dirac cones with opposite chiralities, hereafter named a Dirac phase.
Note that these Dirac cones are in general anisotropic with respectives velocities in the $x$ and $y$ directions $c_x = \sqrt{2 | \Delta | /m}$ and
The energy spectrum is given by
\begin{equation}
\epsilon^{2} = \left( \frac{p_x^2}{2m} + \Delta \right)^{2} + \left( c_y p_y \right)^{2} \ .
\label{eq:EnergySpectrum}
\end{equation}
We will consider only the case of positive energies $\epsilon >0$, as the situation $\epsilon <0$ can be deduced from particle-hole symmetry.
Fig. <ref> presents the different regimes discussed in this paper.
Parametrization of the constant energy contours of Eq. (<ref>) is done by taking advantage of the $p_{x}$ parity.
For each half plane $p_{x} \lessgtr 0$ we use the parametrization
\begin{equation}
\frac{p_x^2}{2m} + \Delta = \epsilon \cos \theta
\ ; \
c_y p_y = \epsilon \sin \theta
\ ; \
\eta_{p}=\textrm{sign}(p_{x})=\pm
\label{eq:param}
\end{equation}
where $\theta \in [-\theta_{0},\theta_{0}]$ is a coordinate along the constant energy contour.
Its range depends on the topology of the constant energy contour,
and thus on the energy $\epsilon$, see Fig. <ref>. Specifying the discussion to the Fermi surface associated with the Fermi energy $\epsilon_{F}$,
we can distinguish two cases :
(i) Low energy metal with two disconnected Fermi surfaces when $\Delta <0$ and $\epsilon_F < - \Delta$. In this case $\theta_0 = \pi$.
This corresponds to the energy spectrum $(a)$ of Fig. <ref>.
(ii) High energy metal with a single connected Fermi surface for $\epsilon_F > | \Delta | $. In this case $\cos \theta_0 = \Delta/\epsilon_F $.
For $\Delta <0$,
$\theta_0$ varies from $\pi$ for $\epsilon_F = -\Delta $ to $\pi /2$ for $\epsilon_F \gg -\Delta$.
For $\Delta >0$, $\theta_0$ varies from $\pi/2$ for $\epsilon_F \gg \Delta$ to $0$
for $\epsilon_{F} \to \Delta$ .
This corresponds to the energy spectra $(b),M,(c)$ of Fig. <ref>.
Left : Constant energy contours $\epsilon(k_{x},k_{y})$ for different energies and $\Delta <0 $
corresponding to the situations (a), (b) and the saddle-point S defined on Fig. <ref>. The arrow field describes the phase $\theta$ parametrizing in a unambiguous way each half $k_x>0$ and $k_x<0$ of
the energy contour according to Eq. (<ref>). It also describes the
relative phase between the two components of the eigenstate (<ref>) of momentum $\vec{k}$ and energy $\epsilon$. Right : Same quantities at the merging point $\Delta=0$ (M) and for $\epsilon(k_{x},k_{y}) > \Delta >0 \ (c)$.
The eigenstates of positive energy corresponds to wave functions
conveniently expressed with the parametrization of the constant energy contour
\begin{equation}
\psi_{\vec{k}}(\vec{r}) = \frac{1}{\sqrt{2}L} \left( \begin{array}{c} 1 \\ e^{i \theta_{\vec{k}}} \end{array} \right) e^{i\vec{k}.\vec{r}} \ ,
\label{eq:eigenstate}
\end{equation}
where $\theta_{\vec{k}}$ is defined by inversion of Eq. (<ref>), and $\vec{p} = \hbar \vec{k}$.
From now on, we will set $L=1$.
The group velocity varies along the constant energy contour according to :
\begin{align}
v_{x} (\eta_p,\theta) & = \eta_p \sqrt{\frac{2\epsilon}{m}} \cos \theta \sqrt{\cos \theta - \delta } \ ,
\\
v_{y} (\eta_p,\theta) & = c_y \sin \theta \ ,
\end{align}
where throughout this paper we use the reduced parameter $\delta = \Delta /\epsilon$.
The evolution of the velocity along constant energy contours is shown on Fig. <ref>.
Velocity $\vec v(\vec k)$ along constant energy contours (here $\Delta <0$).
§.§ Density of States
We define a directional density of states along the constant energy contour parametrized
by $\theta$ from the equality
\begin{equation}
\int \frac{dp_x dp_y}{(2 \pi \hbar)^2}=
\int \frac{dk_x dk_y}{(2 \pi)^2} = \int \rho(\epsilon,\theta) ~d\epsilon d\theta \ ,
\end{equation}
\begin{equation}
\rho(\epsilon,\theta) =
\frac{\sqrt{2 m \epsilon}}{(2 \pi \hbar)^2 c_y} ~
\frac{1}{2 \sqrt{\cos \theta - \delta}} \ .
\label{eq:dos12}
\end{equation}
The density of states in then obtained by the integral
\begin{equation}
\rho(\epsilon) = 2 \int_{-\theta_{0}}^{\theta_{0}} d\theta ~\rho(\epsilon,\theta) \,
\end{equation}
where the extra factor $2$ accounts for the sign of $p_x$.
The integral gives
\begin{equation}
\rho(\epsilon) = \frac{\sqrt{2 m \epsilon}}{(2 \pi \hbar)^2 c_y} ~ {I}_1(\delta) \ ,
\label{eq:defRho}
\end{equation}
with the function
\begin{equation}
{I}_1 (\delta)= \int_{-\theta_0}^{\theta_0} {d \theta \over \sqrt{\cos \theta - \delta} } \ ,
\label{I1}
\end{equation}
where $\theta_0 = \Arccos (\delta)$ when $|\delta|<1$ and $\theta_0 = \pi$ otherwise.
From Eqs. (<ref>,<ref>), we can rewrite $\rho(\ep,\theta)$ as
\begin{equation}
\label{eq:DirectionalDensityRewriting}
\rho(\ep,\theta) = {\rho(\ep) \over 2 \, I_1(\delta) \sqrt{\cos \theta - \delta}} \ .
\end{equation}
§.§ Disorder Potential and Elastic Scattering Time
The disorder part $V$ of the Hamiltonian accounts for the inhomogeneities in the system.
This random potential $V(\r)$ is assumed to describe a gaussian
point-like uncorrelated disorder, characterized by two cumulants
\begin{equation}
\overline{V(\r)} = 0,
\quad
\overline{V(\vec{r})V(\vec{r'}) } = \gamDes \, \delta(\vec{r}-\vec{r'}) \ .
\end{equation}
where the overline denotes a statistical average over realizations of the random potential.
The presence of this random potential induces a finite lifetime for the eigenstates of momentum $\k$
of the pure model (\ref{eq:PureModel}), {called elastic scattering time, and
obtained from the Fermi golden rule~:
\begin{align}
\frac{\hbar}{\tau_{e}(\vec{k})} &=
2\pi \int \frac{d^2 \vec{k}'}{(2\pi)^2}
\delta (\epsilon_{\vec{k}} - \epsilon_{\vec{k}'} )
\overline{|\mathcal{A}(\vec{k},\vec{k}')|^2} \ ,
\end{align}
where the scattering amplitude is defined by
\begin{equation}
\mathcal{A}(\vec{k},\vec{k}') = \langle \psi_{\vec{k}} |V| \psi_{\vec{k}'}\rangle \ .
\end{equation}
For uncorrelated point-like disorder, the angular dependence of this scattering amplitude originates from the eigenstates overlap and one has
\begin{equation}
\label{eq:scattamplitude}
\overline{|\mathcal{A}(\vec{k},\vec{k}') |^2 } =
\frac{\gamDes}{2} \left( 1 + \cos (\theta_{\vec{k}}-\theta_{\vec{k}'}) \right) \ .
\end{equation}
%\frac{1}{\tau_{e}(\vec{k})} = \frac{2 \pi}{\hbar} \oint_{\mathcal{C}_{\epsilon}} d\theta \rho(\epsilon, \theta)
%|\langle \psi_k |V| \psi_k'\rangle |^2 ~ .
Defining $τ_e(ϵ,θ)=τ_e(k⃗)$ where $ϵ, θ$ and $k⃗$ are related through
Eq.~(\ref{eq:param}), we can express the elastic scattering time as an integral
\begin{equation}
\label{eq:tauepstheta}
\frac{\hbar}{\tau_{e}(\epsilon,\theta)}
2 \pi \gamDes \int_{-\theta_0}^{\theta_0} d\theta'~\rho(\epsilon,\theta')
\left[ 1 + \cos (\theta-\theta') \right] .
\end{equation}
Introducing the bare scattering time
\begin{equation}
\tau_{e}^{0} (\epsilon) = \frac{ \hbar }{ \pi \gamDes \rho({\epsilon })} \ ,
\label{eq:ElasticTimeMean}
\end{equation}
we can rewrite (\ref{eq:tauepstheta}) in the form
\begin{equation}
\tau_{e}(\epsilon,\theta)
=\frac{ \tau_{e}^{0} (\epsilon) }{ 1 + r(\delta ) \cos \theta } \ ,
\label{eq:ElasticTimeParam}
\end{equation}
where the density of states $\rho(\epsilon)$ is given by (\ref{eq:defRho}).
The denominator of this expression exactly accounts for the anisotropy of the scattering time. As a convenient parametrization of this property, we }
have introduced the anisotropy function $r(δ)$ which will be used throughout this
paper :
\begin{equation}
r(\delta) = {J}_1 \left(\delta \right) / {I}_1\left(\delta \right),
\label{eq:defR}
\end{equation}
with the function $I_1(δ)$ defined in (\ref{I1})
\begin{equation}
{J}_1 (\delta)= \int_{-\theta_0}^{\theta_0} {d \theta \cos \theta \over \sqrt{\cos \theta - \delta} } \ ,
\label{J1}
\end{equation}
where $θ_0 = (δ)$ when $|δ|<1$ and $θ_0 = π$ otherwise.
\begin{figure} [!h]
\centering
\includegraphics[width=8cm]{r.pdf}
\caption{Function $r(\delta)$ parametrizing the angular dependence of the elastic scattering time $\tau_e$ plotted as a
function of $\delta = \Delta / \epsilon$.
It has the limits
$r(\delta \rightarrow - \infty) \simeq 1/(4\delta)$, $r(-1)=-1$, $r(0)= 2 \Gamma(3/4)^4 /\pi^2 \simeq 0.456947$, $r(1)=1$.
In this figure, as in following figures, we systematically reserve the colors~: blue for the Dirac phase ($\delta < 0$), black for the gapped phase ($\Delta > 0$) and red for the semi-Dirac point.
\label{fig:r}
\end{figure}
The function $r(δ)$
is plotted in Fig.~\ref{fig:r}. Deep in the Dirac phase ($Δ≪0$), at low energy ($≪|Δ|$), one has
\begin{equation}
r(\delta) \rightarrow -{ 1 \over 4 |\delta|}=-{ \epsilon \over 4 |\Delta|} \ll 1 ,
\end{equation}
so that the anisotropy can be neglected in Eq.~(\ref{eq:ElasticTimeParam}) and we recover a scattering time independent of the direction of
propagation as standard for Dirac fermions.
%\begin{figure*} [!h]
%\caption{Energy as a function of $p_{x}$ for $p_{y}=0$ and $\Delta=0, \Delta =\pm 0.5$. Shape of constant energy curves for the same parameters. }
\section{Diffusive regime from the Boltzmann Equation}
\label{sect:Diffusive}
We now consider the transport properties of the model (\ref{eq:HModel}) at a fixed energy $ϵ$ large enough so that the condition $k l_e ≫1$ is fulfilled, $l_e$ being a typical elastic mean free path. Therefore we will not consider the close vicinity of a Dirac point and the associated physics of minimal conductivity
\cite{Katsnelson:2006b,Twordzylo:2006}.
For a system of typical size much larger than this mean free path $l_e$, this corresponds to the regime of classical diffusion.
We describe this regime first with a standard Boltzmann equation, before turning to a complementary but equivalent diagrammatic approach based on
Kubo formula for the conductivity. The use of these two approaches will reveal the physics hidden between the technical specificities of the diffusive
transport for the model we consider.
\subsection{Boltzmann equation}
We start from the Boltzmann equation \cite{Abrikosov:88,Ziman:79} expressing the evolution of the distribution function $f(k⃗,r⃗)$ :
\begin{equation}
\frac{d f}{dt} + \frac{d \vec{r}}{dt} ~ \nabla_{\vec{r}} f+ \frac{d \vec{k}}{dt} ~ \nabla_{\vec{k}} f =
I[f] \ ,
\label{eq:Boltzmann0}
\end{equation}
where $I[f]$ is the collision integral defined below.
The position $r⃗$ and momentum $k⃗$ parametrizing the distribution function $f(k⃗,r⃗)$ are classical variables, whose time evolutions
entering Eq.(\ref{eq:Boltzmann0}) are described by the semi-classical equations \cite{Xiao:2010,Son:2013}
\begin{align}
\frac{d \vec{r}}{dt} &= \vec{v}(\vec{k}) +
\frac{d \vec{k}}{dt} \times \vec{F}_{\vec{k}} \\
\hbar \frac{d \vec{k}}{dt} &= -e \vec{E} - e \frac{d \vec{r}}{dt} \times \vec{B}
\label{eq:semiclassic2}
\end{align}
with the group velocity
$ v⃗(k⃗) = ħ^-1∂ϵ(k⃗) / ∂k⃗$,
$B⃗$ is a local magnetic field, and
$F⃗_k⃗ = i ∇_k⃗×⟨ψ_k⃗ |∇_k⃗ ψ_k⃗ ⟩$
is the Berry curvature.
In the present case, we consider the response of the distribution function
$f(k⃗,r⃗)$ due to a uniform weak electric field $E⃗$ : we can neglect the gradient $ ∇_r⃗ f$
in Eq.~(\ref{eq:Boltzmann0}) and drop the spatial dependence of $f$.
Due to the absence of magnetic field, we deduce from Eqs.~(\ref{eq:Boltzmann0},\ref{eq:semiclassic2}) that
a stationary out-of-equilibrium distribution $f(k⃗)$ satisfies the simpler equation
\begin{equation}
- { e \over \hbar } \vec{E} .\nabla_{\vec{k}} f = I[f] \ .
\label{eq:Boltzmann}
\end{equation}
where $f$ is now as function of $k⃗$ and the collision integral is expressed as
\begin{multline}
I[f] =
2\pi \int \frac{d^2 \vec{k}'}{(2\pi)^2} \\
\delta (\epsilon_{\vec{k}} - \epsilon_{\vec{k}'} )
\overline{ |\mathcal{A}(\vec{k},\vec{k}') |^2 }
\left( f(\vec{k}') - f(\vec{k}) \right) .
\label{eq:Idef}
\end{multline}
By assuming the perturbation to be weak, we can expand the stationary out-of-equilibrium distribution $f(k⃗)$
around the equilibrium Fermi distribution $f^0(k⃗) = n_F(ϵ_k⃗)$ following
the ansatz\cite{Abrikosov:88,Ziman:79}
\begin{equation}
f(\vec{k}) = f^0(\vec{k}) + e \frac{\partial n_F }{\partial \epsilon } ~ \vec{\Lambda}(\vec{k}).\vec{E} \ ,
\label{eq:BoltzAnsatz}
\end{equation}
where the vector $Λ⃗$ has the dimension of a length, and its components correspond to transport lengths in the different spatial directions.
They are related to transport times through the definition
$Λ_α(k⃗) = v_α(k⃗) τ^tr_α (k⃗) $.
Eq.~(\ref{eq:BoltzAnsatz}) can be rewritten as a shift of energies by the field~:
$f(k⃗) = n_F(ϵ_k⃗ + e Λ⃗(k⃗).E⃗ )$. In the case of an isotropic Fermi surface, we do not expect this
shift to depend on the direction of application of the field $E⃗$ : in that case a unique transport time $τ^tr$ is necessary to
describe the stationary distribution\cite{Ziman:79}.
Here, for an anisotropic Fermi surface such as (\ref{eq:EnergySpectrum}), we generically expect the response of the distribution
function to
{depend on the direction of the electric field $\vec{E}$ \cite{Sondheimer:1962,Sorbello:1974,Sorbello:1975}.
For an electric field applied in the $x$ or $y$ direction, this leads}
to the definition of different anisotropic transport times $τ^tr_x,τ^tr_y$.
From Eqs. (\ref{eq:Boltzmann},\ref{eq:Idef},\ref{eq:BoltzAnsatz}), one obtains
\begin{widetext}
\begin{equation}
\vec{v}(\vec{k}) =
\left. \frac{1}{\hbar}\frac{\partial \epsilon}{\partial \vec{k}}\right|_{\epsilon=\epsilon_{\vec{k}}}
2\pi \int \frac{d^2 \vec{k}'}{(2\pi)^2}
\delta (\epsilon_{\vec{k}} - \epsilon_{\vec{k}'} )
\overline{ |\mathcal{A}(\vec{k},\vec{k}') |^2 }
\left( \vec{\Lambda}(\vec{k}) - \vec{\Lambda}(\vec{k}') \right) \ ,
\end{equation}
By using the parametrization (\ref{eq:param}) on the contour of constant energy $\ep$, each component $\alpha$ of the velocity
obeys the equation (to lighten notation, we omit the energy $\ep$ in the argument of the quantities in the next expressions)~:
% and assuming that
% $\Lambda_{\alpha}(\vec{k})$ possesses the same symmetries as $v_{\alpha}(\vec{k})$
\begin{equation}
v_\alpha(\eta_p,\theta) = {\Lambda_\alpha(\eta_p,\theta) \over \tau_e(\theta)}
- \frac{ \pi \gamDes}{\hbar} \sum_{\eta'_p=\pm} \int_{-\theta_0}^{\theta_0} d\theta' ~ \rho( \theta')
\left[1+\cos(\theta - \theta') \right]
\Lambda_\alpha(\eta'_p,\theta') .
\end{equation}
The transport times $\tau^{\textrm{tr}}_{\alpha} (\epsilon, \theta)$ are defined as
\begin{equation}
\Lambda_{\alpha}(\ep, \eta_p,\theta) = v_{\alpha}(\ep, \eta_p,\theta) \tau^{\textrm{tr}}_{\alpha} (\ep, \theta) \ .
\end{equation}
We now assume the following ansatz, namely that the transport times and the elastic scattering time have the same angular dependence:
\begin{equation}
\tau^{\textrm{tr}}_{ \alpha} (\ep, \theta) = \lambda_\alpha(\ep) \, \tau_e (\ep, \theta) \ ,
\end{equation}
so that the parameters $\lambda_\alpha(\ep)$ are obtained from the self-consistent equation (at fixed energy $\ep$)
\begin{equation}
v_\alpha (\eta_p,\theta)= \lambda_\alpha v_\alpha (\eta_p,\theta)
% \\
- \frac{ \pi \gamDes}{\hbar}\lambda_\alpha \sum_{\eta'_p=\pm} \int_{-\theta_0}^{\theta_0} d\theta' \rho( \theta')
\left[1+\cos(\theta - \theta') \right]
v_\alpha(\eta'_p,\theta') \tau_e(\theta')
\label{eq:Lambda}
\end{equation}
where $v_\alpha( \eta_p,\theta)$ is defined in Eq.~(\ref{eq:vitesse}).
Then from Eq.~{ (\ref{eq:DirectionalDensityRewriting})} and (\ref{eq:ElasticTimeParam}), we finally get
\begin{equation}
v_\alpha (\eta_p,\theta)= \lambda_\alpha v_\alpha (\eta_p,\theta)
% \\
- {\lambda_\alpha \over 2 I_1(\delta) }\sum_{\eta'_p=\pm} \int_{-\theta_0}^{\theta_0} d\theta'
{1+\cos(\theta - \theta') \over 1 +r(\delta) \cos \theta'}
{v_\alpha(\eta'_p, \theta') \over \sqrt{\cos \theta' - \delta}} \ .
\label{eq:lambda}
\end{equation}
\end{widetext}
We now consider the two directions $α= x,y$ separately.
{\it Along the $x$ direction}, since the velocity is an odd function of $k_x$,
the sum over $η_p$ in Eq.~(\ref{eq:lambda}) vanishes: we obtain $λ_x()=1$, {\it i.e.} the transport time is
equal to the scattering time\footnote{This is not true in graphene where $\lambda_x=2$. Note that this
peculiar result ($\lambda_x=1$) is due the fact that the matrix
elements of the disorder potential are supposed here to have no
momentum dependence. Assuming an opposite limit where the disorder
would not couple valleys, then in the Dirac limit $0 < \ep \ll -
\Delta$, one would recover $\lambda_x=1$ and $\tau_x^{\textrm{tr}}=
2 \tau_e$.}
{$ \tau^{\textrm{tr}}_{x} (\ep, \theta) = \tau_e (\ep, \theta) $}.
{\it Along the $y$ direction}, where $v_y(θ)= c_y sinθ$ independent of $η_p$,
Eq.~(\ref{eq:lambda}) possesses a self-consistent solution, and we obtain
\begin{equation}
\lambda_y(\delta) = \frac{1}{1 - \mathcal{I}_2( \delta )/ {I}_1( \delta )}
\label{eq:lambday}
\end{equation}
\begin{equation}
\mathcal{I}_2(\delta) =
\int_{-\theta_{0}}^{\theta_{0}} d \theta ~ \frac{\sin^2 \theta}{\sqrt{\cos \theta - \delta}~ (1 +r(\delta) \cos \theta)} ~ .
\label{eq:calI2}
\end{equation}
The function $I_1(δ)$ is defined in (\ref{I1}). Note that the expression (\ref{eq:lambday}) of the renormalization factor of
the transport time $λ_y(δ)$
reflects the iterative structure
of the vertex correction to the bare conductivity that will be obtained within
a diagrammatic treatment in section \ref{sec:DiagConductivity} (see Eqs.~\ref{eq:RenormJy},\ref{eq:sigma_yy_diag}). The dependence $λ_y(δ)$ is plotted in Fig.~\ref{fig:lambday}.
\begin{figure} [!h]
\centering
\includegraphics[width=8cm]{lambday.pdf}
\caption{
Dependence on $\delta = \Delta /\epsilon$ of the
renormalization factor of the transport time $\tau^{\textrm{tr}}_{ y}$ with respect to the elastic scattering time :
$\lambda_y(\delta)= \tau^{\textrm{tr}}_{ y}({\k}) / \tau_{e}({\k})$.
\label{fig:lambday}
\end{figure}
Having obtained the transport times along the $x$ and $y$ directions, we now turn to the calculation of the conductivities.
\subsection{Conductivity}
\label{sect:conductivity}
We can express the current density $j⃗$ occurring in response to the application of the electric field $E⃗$ as
\begin{equation}
\vec{j} =
\int \frac{d^2 \vec{k}}{(2\pi)^2}
\left[f(\vec{k}) - n_F (\epsilon_{\vec{k}})\right] (-e \vec{v}(\vec{k})) \ .
\end{equation}
By using $∂n_F / ∂ϵ≃- δ(ϵ-ϵ_F)$
and the ansatz (\ref{eq:BoltzAnsatz}) for the distribution function $f(k⃗)$ we obtain
\begin{multline}
\vec{j} =
e^2 \\ \sum_{\eta_p=\pm}
\int_{-\theta_0}^{\theta_0} d\theta ~ \rho(\epsilon_{F},\theta)
\vec{v}(\epsilon_{F},\eta_p, \theta) \left[ \vec{\Lambda}(\epsilon_{F},\eta_p,\theta).\vec{E} \right] \ .
\end{multline}
The symmetries of this equation imply that off-diagonal terms of the conductivity tensor vanish
($σ_α, β≠α = 0$) while the diagonal terms can be written as
\begin{equation}
\sigma_{\alpha \alpha} =
2 e^2
\int_{-\theta_0}^{\theta_0} d\theta ~ \rho(\epsilon_{F},\theta)
v_{\alpha}(\epsilon_{F},\theta) \Lambda_{\alpha}(\epsilon_{F},\theta) \ .
\end{equation}
where the factor $2$ originates from the two possible signs of $η_p=±$.
We end up with the Einstein relation
\begin{equation}
\sigma_{\alpha\alpha} = e^{2} \rho(\epsilon_{F}) D_{\alpha} ,
\end{equation}
with the diffusion coefficients
\begin{subequations}
\begin{align}
D_{\alpha } &=
2 \lambda_\alpha(\ep_F)
\int_{-\theta_0}^{\theta_0} d\theta ~ { \rho(\epsilon_{F},\theta) \over \rho(\epsilon_{F})}
v^2_{\alpha}(\theta) \tau_e(\theta) \\
\left< v^2_{\alpha}(\epsilon_{F},\theta) ~ \tau^{\textrm{tr}}_{\alpha}(\epsilon_{F},\theta) \right>_{\theta}
\\
\lambda_\alpha(\ep_F)
\left< v^2_{\alpha}(\epsilon_{F},\theta) \tau_e(\epsilon_{F},\theta) \right>_{\theta}
\end{align}
\end{subequations}
where we have defined the average along the constant energy contour
$\left< \cdots \right>_{\theta} = 2
\int_{-\theta_0}^{\theta_0} d\theta ~ \cdots \rho(\epsilon_{F},\theta) / \rho(\epsilon_{F})
This corresponds to the result announced in the introduction~: the diffusion coefficients $D_\alpha$ are obtained by an average over the Fermi surface
of $v^2_{\alpha}~ \tau^{\textrm{tr}}_{\alpha}$ instead of $v^2_{\alpha}~ \tau_e$. With our solution of the Boltzmann equation, this
difference is accounted for by a renormalization factor $\lambda_\alpha(\ep_F)$ of the diffusion coefficients, which does not depend
on the direction along the Fermi surface but \emph{on the direction $\alpha$ of application of the electric field}.
We now specify explicitly the conductivities along the two directions $x$ and $y$.
{\it Along the $x$ direction}, there is no renormalization of the transport time ($λ_x=1$, $τ_x^tr=τ_e$) and the conductivity $σ_xx$ reads
\begin{align}
\sigma_{xx} & =
2 e^2 \int_{-\theta_{0}}^{\theta_{0}} d\theta ~
\rho(\epsilon_{F},\theta) v_x^2(\theta) \tau_e(\theta) \nonumber \\
& =
\frac{ e^2 \hbar}{ \pi \gamDes} ~
\frac{2 \epsilon }{m } ~
\frac{ \mathcal{I}_3(\Delta/\epsilon)}{ {I}_1(\Delta/\epsilon)} \ ,
\label{eq:sigma_xx_boltzmann}
\end{align}
where we define
\begin{equation}
\mathcal{I}_3(\delta)=
\int_{-\theta_{0}}^{\theta_{0}} d \theta ~ \frac{\cos^2 \theta \sqrt{\cos \theta -\delta }}{1 + r(\delta) \cos \theta} \ ,
\end{equation}
and the function $I_1(δ)$ is given in (\ref{I1}).
For the conductivity {\it along the $y$ direction}, the renormalization of the transport time is given by (\ref{eq:lambday}) and we obtain
\begin{align}
\sigma_{yy} & =
2 e^2 \lambda_y(\Delta /\epsilon)
\int_{-\theta_{0}}^{\theta_{0}} d\theta ~
\rho(\epsilon_{F},\theta) v_y^2(\theta) \tau_e(\theta) \nonumber \\
& =
\frac{ e^2 \hbar}{ \pi \gamDes} ~ c_y^2 ~
\frac{ \mathcal{I}_2(\Delta/\epsilon)}{I_1(\Delta/\epsilon) - \mathcal{I}_2(\Delta/\epsilon) } \ ,
\label{eq:sigma_yy_boltzmann}
\end{align}
where the functions $I_1(δ)$ and $ℐ_2(δ)$ are respectively given by Eqs. (\ref{I1}) and (\ref{eq:calI2}).
Eqs.~(\ref{eq:sigma_xx_boltzmann}, \ref{eq:sigma_yy_boltzmann}) constitute the main results of this work.
We discuss them in section \ref{sect:discussion}.
In the next section, we use a diagrammatic approach
which proposes a complementary description of the anisotropy of transport and allows to
confirm the ansatz made to solve the Boltzman equation and recover exactly the results of Eqs.~(\ref{eq:sigma_xx_boltzmann}, \ref{eq:sigma_yy_boltzmann})
\section{Diagrammatic Approach}
\label{sec:diagrammatic}
An alternative approach to describe the diffusive transport of
electron consists in a perturbative expansion in disorder of the
conductivity tensor using a diagrammatic technique\cite{Akkermans}.
Beyond confirming the ansatz made to solve the Boltzmann equation
described above, this method allows for an instructive alternative
treatment of the different transport anisotropies.
In the diagrammatic approach, the transport coefficients of the model
are obtained from the Kubo formula. A perturbative expansion is then
used to express the transport coefficients using the average single particle
Green's function.
In this formalism, the
anisotropy of scattering and transport times are cast into a unusual
matrix form for the self-energy operator $\Sigma$.
Beyond the present model, such a technique allows to
describe anisotropy of diffusion of Dirac fermion models due {\it
e.g.} to the warping of the Fermi surface in topological insulators
\cite{Adroguer:12} or anisotropic impurity scattering, the study of
which goes beyond the scope of the present paper.
Nevertheless our work provides a physical understanding of the technicalities naturally occurring in these other problems.
In the next
subsections, we first discuss the self-energy and the single
particle Green's function. We then turn to the calculation of the
\subsection{Green's functions and self-energy}
The retarded and advanced Green's functions are defined by~:
\begin{multline}
G^{R/A}(\vec{k},\vec{k}', \epsilon_{F}) = \\
\left[
\left( (\epsilon_{F} \mp i 0) \mathbf{I} - H^0(\vec{k}) \right)\delta(\vec{k}-\vec{k}') - V(\vec{k},\vec{k}') \mathbf{I}
\right]^{-1}
\end{multline}
In the case of the model without disorder defined by Eq.~(\ref{eq:PureModel}), the Green's function is expressed as a $2 \times 2$ matrix~:
\begin{align}
G^0(\vec{k},\epsilon) &= \left( \epsilon \, \mathbf{I} - H^0(\vec{k}) \right)^{-1}
\nonumber \\
\frac{\epsilon \, \mathbf{I} + \left( \frac{\hbar^2 k_x^2}{2m}+\Delta \right) \sigma_x + c_y~ \hbar k_y \sigma_y }
{\epsilon^2 -\left( \frac{\hbar^2 k_x^2}{2m}+\Delta \right)^2- c_y^2 \hbar^2 k_y^2} \ ,
\end{align}
where $\mathbf{I}$ is the identity matrix.
Disorder is perturbatively incorporated in the averaged Green's
function through a self-energy matrix $\Sigma(\vec{k},\epsilon)$ such that
\begin{equation}
\overline{G}^{R/A}(\vec{k},\epsilon)=\left[(\epsilon \mp i 0) \, \mathbf{I} -H^0(\vec{k}) \mp i ~ \textrm{Im}~ \Sigma (\vec{k},\epsilon)\right]^{-1}.
\end{equation}
The real part of the self-energy has been neglected. The elastic
scattering rates will be defined below from the imaginary part of the
To lowest order in the disorder strength $\gamDes$, this self-energy, solution of a Dyson equation, reads
\begin{equation}
\Sigma (\vec{k} {, \epsilon}) = \int \frac{d\vec{k}'}{(2\pi)^2}
~\overline{ V(\vec{k}')V(-\vec{k}') } ~ G^0(\vec{k}-\vec{k}' {, \epsilon}) \ .
\end{equation}
Its imaginary part is then obtained as
\begin{align}
- \textrm{Im}~ \Sigma { (\epsilon)} &=
\pi \gamDes
\int_{-\theta_{0}}^{\theta_{0}}
d\theta ~ \rho(\epsilon,\theta) \left[ \mathbf{I} - \cos \theta ~ \sigma_x \right] \\
&= \frac{\hbar}{2 \tau_{e}^{0}{ (\epsilon)}} \left[ \mathbf{I} + r\left( \delta \right) \sigma_x \right] \ .
\label{eq:SigmaMatrix}
\end{align}
The densities of states $\rho(\epsilon,\theta)$ and $\rho(\epsilon)$, the bare scattering time $\tau_{e}^{0} (\epsilon)$
and the anisotropy factor $r(\delta)$ have been defined in Eqs.~(\ref{eq:dos12},\ref{eq:defRho},\ref{eq:ElasticTimeMean},\ref{eq:defR}).
It is worth noting that this self-energy acquires an unusual matrix structure in pseudo-spin space: this manifests within the diagrammatic
approach the anisotropy of the scattering time $\tau_e(\epsilon,\theta)$, which was described in Eq.~(\ref{eq:ElasticTimeParam}) previously.
Indeed, in the Green function formalism, the direction of propagation of eigenstates of the Hamiltonian (\ref{eq:PureModel}) is encoded into
their spinor structure (the relative phase between their components, see Eq.~(\ref{eq:eigenstate})).
Hence the scattering time in the corresponding direction will be obtained as the matrix element of the above self-energy in the associated spinor eigenstate.
%G^{R/A}(\vec{k},\epsilon) &= \left( \epsilon ~ \mathbf{I} - H^0(\vec{k} - \pm \textrm{Im}~ \Sigma ) \right)^{-1}
%\nonumber \\
%\frac{\epsilon ~ \mathbf{I} + \left( \frac{\hbar^2 k_x^2}{2m}+\Delta \right) \sigma^x + c_y~ \hbar k_y \sigma^y }
%{(\epsilon \pm i \frac{\hbar}{2 \tau_e^0})^2 -\left( \frac{\hbar^2 k_x^2}{2m}+\Delta \mp i \frac{\hbar r(\Delta/\epsilon)}{2 \tau_e^0} \right)^2- c_y^2 \hbar^2 k_y^2} .
%%---------------------------------- FIGURE ---------------------------------------------%%
\begin{figure}
\begin{center}
\begin{tikzpicture}
line around/.style={decoration={pre length=#1,post length=#1}},scale=1]
\draw[->,>=latex,thick] (0.,0) -- (1.5,0);
\node at (0.7,0.5) {$\left[ \overline{G}^{R} \right]_{ab}(\vec{k})$};
\node at (0.,-0.2) {$a$} ;
\node at (1.5,-0.2) {$b$} ;
\draw[<-,>=latex,dashed,thick] (2.5,0) -- (4.,0);
\node at (3.2,0.5) {$\left[ \overline{G}^{A} \right]_{cd}(\vec{k})$};
\node at (2.5,-0.2) {$d$} ;
\node at (4.,-0.2) {$c$} ;
\draw[thick,dotted] (5,0.) -- (7,0.);
\draw[fill=black!50] (6,0.) circle (0.15);
\node at (5.8,0.5) {$\overline{ V^{2}}(\vec{k})=\gamDes $};
\end{tikzpicture}
\end{center}
\caption{\label{fig:convention}
Conventions for the diagrammatic representation of perturbation theory of transport.
\end{figure}
%%---------------------------------- FIGURE ---------------------------------------------%%
%%---------------------------------- FIGURE ---------------------------------------------%%
\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
[decoration=snake,line around/.style={decoration={pre length=#1,post length=#1}},scale=1]
\begin{scope}
\draw[fill=black!50,fill opacity=0.5, thick]
(0.5,0) arc (160:90:1.5)
-- ++(0,-2)
arc (270:200:1.5);
\end{scope}
\draw[-,decorate, thick] (0.5,0) -- ++(-0.5,0);
\draw[-,decorate, thick] (3.3,0) -- ++(+0.5,0);
\begin{scope}
\draw[->, >=latex, thick] (1.9,1) arc (90:20:1.5);
\draw[<-, >=latex,dashed, thick] (1.9,-1) arc (270:340:1.5);
\end{scope}
\node at (1.3,0) {$J_\alpha(\vec{k})$};
\node at (2.9,1.2) {$\overline{G}^{R}(\vec{k})$};
\node at (2.9,-1.2) {$\overline{G}^{A}(\vec{k})$};
\node at (3.7,-.3) {$j_\alpha(\vec{k})$};
\end{tikzpicture}
\end{center}
\caption{\label{fig:classical conduct} Diagrammatic representation of the classical conductivity with the
conventions of Fig.~\ref{fig:convention}. The renormalized current operator
is defined in Fig.~\ref{fig:renormCurrent}.
\end{figure}
%%---------------------------------- FIGURE ---------------------------------------------%%
\subsection{Conductivity}
\label{sec:DiagConductivity}
\subsubsection{Kubo formula}
The longitudinal conductivity can be deduced from the Kubo formula ($\alpha = x,y$) :
\begin{multline}
\sigma_{\alpha \alpha} = \\
\frac{\hbar}{2 \pi L^2} \mathrm{Tr} \left[
G^{R} (\vec{k},\vec{k}', \epsilon_{F})
j_{\alpha} (\vec{k}')
G^{A} (\vec{k}',\vec{k} , \epsilon_{F})
\right],
\label{eq:Kubo}
\end{multline}
where $\mathrm{Tr}$ corresponds to a trace over the pseudo-spin and momentum quantum numbers :
$\mathrm{Tr}=\mathrm{tr} ~\sum_{\vec{k}} \simeq L^2 \mathrm{tr}~ \int d\vec{k}/(2\pi)^2 $ and
$\mathrm{tr}$ is a trace over the pseudo-spin indices only.
For clarity, throughout this section on transport coefficients, we will omit the dependence on the Fermi energy $\epsilon_{F}$ of various quantities.
The current density operators are also operators acting on both spin and momentum spaces. They are
deduced from the Hamiltonian (\ref{eq:PureModel}) as:
\begin{equation}
j_x (\vec{k}) = -\frac{e}{m} \hbar k_x ~\sigma_x
\quad ; \quad
j_y (\vec{k})= -e c_y ~\sigma_y .
\end{equation}
Note that $j_x$ is linear in momentum while $j_y$ depends only on spin quantum numbers.
Perturbation in the disorder amplitude of the conductivity ~(\ref{eq:Kubo}) is obtained by expanding the Green's function in the disorder potential $V$ before
averaging over the gaussian distribution.
In the classical diffusive limit, the dominant terms which determine the averaged classical conductivity are
represented diagrammatically on Fig.~\ref{fig:classical conduct} and lead to
\begin{equation}
\overline{\sigma}_{\alpha \alpha} =
\frac{\hbar}{2 \pi L^2} \mathrm{Tr} \left[
\overline{G}^{R}
\overline{G}^{A}
\right]
\label{eq:ConductRenorm}
\end{equation}
where $J_\alpha$ is the renormalized current density operator. The discrepancy between $J_\alpha$ and the bare
current operator $j_\alpha$ accounts for the appearance of transport time
$\tau^{\textrm{tr}}_{\alpha}$ in the Boltzmann
approach\cite{McCann:2006} due to the anisotropy of scattering.
This renormalized current operator is easier to define diagrammatically, as shown on Fig.~\ref{fig:renormCurrent}.
%%---------------------------------- FIGURE ---------------------------------------------%%
\begin{figure*}[!ht]
\begin{center}
\begin{tikzpicture}
[decoration=snake,line around/.style={decoration={pre length=#1,post length=#1}},scale=1]
\begin{scope}
\draw[-,decorate, thick] (5.5,0) -- ++(-0.5,0);
\draw[fill=black!50,fill opacity=0.5, thick]
(5.5,0) arc (160:90:1.5)
-- ++(0,-2)
arc (270:200:1.5);
\end{scope}
\node at (6.3,0) {$J_\alpha(\vec{k})$};
\node[scale=1] at (7.5,0.) {$=$};
\draw[-,decorate, thick] (8.5,0) -- ++(-0.5,0);
\draw[-,thick] (8.5,0) arc (160:150:1.5);
\draw[-,thick] (8.5,0) arc (-160:-150:1.5);
\node[scale=1] at (9.,0.) {$+$};
\draw[-,decorate, thick] (10.,0) -- ++(-0.5,0);
\draw[-,->,>=latex,thick] (10,0) arc (160:90:1.5);
\draw[-,<-,>=latex,dashed,thick] (10,0) arc (200:270:1.5);
\draw[-,dotted,thick] (11.4,1)--++(0,-2);
\draw[fill=black!50] (11.4,0.) circle (0.15);
\node[scale=1] at (12.,0.) {$+$};
\draw[-,decorate, thick] (13.,0) -- ++(-0.5,0);
\draw[-,->,>=latex,thick] (13,0) arc (160:90:1.5);
\draw[-,<-,>=latex,dashed,thick] (13,0) arc (200:270:1.5);
\draw[-,->,>=latex,thick] (14.4,1)-- (15,1);
\draw[-,<-,>=latex,dashed,thick] (14.4,-1)-- (15,-1);
\draw[-,dotted,thick] (14.4,1)--++(0,-2);
\draw[-,dotted,thick] (15.,1)--++(0,-2);
\draw[fill=black!50] (14.4,0.) circle (0.15);
\draw[fill=black!50] (15.,0.) circle (0.15);
\node[scale=1] at (15.5,0.) {$+$};
\draw[-,decorate, thick] (16.5,0) -- ++(-0.5,0);
\draw[-,->,>=latex,thick] (16.5,0) arc (160:90:1.5);
\draw[-,<-,>=latex,dashed,thick] (16.5,0) arc (200:270:1.5);
\draw[-,->,>=latex,thick] (17.9,1)-- (18.5,1);
\draw[-,->,>=latex,thick] (18.5,1)-- (19.1,1);
\draw[-,<-,>=latex,dashed,thick] (17.9,-1)-- (18.5,-1);
\draw[-,<-,>=latex,dashed,thick] (18.5,-1)-- (19.1,-1);
\draw[-,dotted,thick] (17.9,1)--++(0,-2);
\draw[-,dotted,thick] (18.5,1)--++(0,-2);
\draw[-,dotted,thick] (19.1,1)--++(0,-2);
\draw[fill=black!50] (17.9,0.) circle (0.15);
\draw[fill=black!50] (18.5,0.) circle (0.15);
\draw[fill=black!50] (19.1,0.) circle (0.15);
\node[scale=1] at (20.,0.) {$+ \cdots$};
\end{tikzpicture}
\begin{tikzpicture}
[decoration=snake,line around/.style={decoration={pre length=#1,post length=#1}},scale=1]
\begin{scope}
\draw[-,decorate, thick] (5.5,0) -- ++(-0.5,0);
\draw[fill=black!50,fill opacity=0.5, thick]
(5.5,0) arc (160:90:1.5)
-- ++(0,-2)
arc (270:200:1.5);
\end{scope}
\node at (6.3,0) {$J_\alpha(\vec{k})$};
\node at (7.1,1) {$a$};
\node at (7.1,-1) {$b$};
\node[scale=1] at (7.5,0.) {$=$};
\draw[-,decorate, thick] (8.5,0) -- ++(-0.5,0);
\draw[-,thick] (8.5,0) arc (160:150:1.5);
\draw[-,thick] (8.5,0) arc (-160:-150:1.5);
\node at (8.5,-1) {$j_\alpha(\vec{k})$};
\node at (8.7,0.4) {$a$};
\node at (8.7,-0.4) {$b$};
\node[scale=1] at (9.,0.) {$+$};
\draw[-,decorate, thick] (10.,0) -- ++(-0.5,0);
\draw[fill=black!50,fill opacity=0.5, thick]
(10,0) arc (160:90:1.5)
-- ++(0,-2)
arc (270:200:1.5);
\draw[-,->,>=latex,thick] (11.4,1)-- (12.1,1);
\draw[-,<-,>=latex,dashed,thick] (11.4,-1)-- (12.1,-1);
\draw[-,dotted,thick] (12.1,1)--++(0,-2);
\draw[fill=black!50] (12.1,0.) circle (0.15);
\node at (10.8,0) {$J_\alpha(\vec{k}')$};
\node at (11.7,1.4) {$\overline{G}^R(\vec{k}')$};
\node at (11.7,-1.5) {$\overline{G}^A(\vec{k}')$};
\node[scale=1] at (12.7,0.) {$\gamDes$};
\node at (12.3,1) {$a$};
\node at (12.3,-1) {$b$};
\node at (11.6,0.85) {$c$};
\node at (11.6,-0.8) {$d$};
\end{tikzpicture}
\end{center}
\caption{\label{fig:renormCurrent}
Schematic representation of renormalized current operator $[J_\alpha]_{ab}(\vec{k})$ as the infinite sum of vertex corrections to the
bare current operator (top), and corresponding recursive equation satisfied by $J_\alpha$ (bottom).
\end{figure*}
%%---------------------------------- FIGURE -----------------------------------f----------%%
\subsubsection{Conductivity along $x$}
In this direction, the current operator is linear in $k_{x}$, while the averaged Green's functions
$\overline{G}^{R} (\vec{k}),\overline{G}^{A} (\vec{k})$ are even functions of $k_x$. Hence all the terms in the expression of
the renormalized current $J_x$ with at least a Green's function vanish by $k_x \rightarrow -k_x$ symmetry, and
\begin{equation}
J_x(\vec{k}) = j_x(\vec{k}) =-\frac{e}{m} \hbar k_x ~\sigma_x \ .
\label{eq:RenormJx}
\end{equation}
There is no renormalization of the current operator, in agreement with the result $\tau^{\textrm{tr}}_x = \tau_e$ from
the Boltzmann equation approach.
In the $x$ direction, the expression (\ref{eq:ConductRenorm}) reduces to
\begin{multline}
\overline{\sigma}_{xx} =
\left( \frac{\hbar e}{m} \right)^2
\frac{\hbar}{2 \pi L^2}
\mathrm{Tr} \left[
%\int \frac{d\vec{k}}{(2 \pi)^2} ~
\sigma_x \overline{G}^{R} (\vec{k})
\sigma_x \overline{G}^{A} (\vec{k})
\right] .%;
\label{eq:SxxTemp}
\end{multline}
%where $ \mathrm{tr}$ runs over the spin indices only.
Using $L^{-2}\sum_{\vec{k}} \simeq \int \rho(\epsilon,\theta) d\epsilon d\theta $
the parametrization defined in Eq. (\ref{eq:param}) of the contours of constant energy $\epsilon$ we perform the
integration over energy to obtain
\begin{multline*}
\overline{\sigma}_{xx} =
\frac{e^2 \tau^{0}_e \epsilon_F}{m }
\int_{-\theta_0}^{+\theta_0} d\theta ~
\frac{ \rho(\epsilon , \theta)~ (\cos \theta - \delta)}{1+r(\delta) \cos \theta}
\\
\times \mathrm{tr} \biggl[
\sigma_x
\left[ \mathbf{I} + \cos \theta \sigma_x + \sin \theta \sigma_y \right]
\sigma_x
\left[ \mathbf{I} + \cos \theta \sigma_x + \sin \theta \sigma_y \right]
\biggr].
% \label{eq:SxxTemp2}
\end{multline*}
Performing the spin trace first, we obtain
\begin{multline}
\overline{\sigma}_{xx} =
4 \frac{e^2 \tau^{0}_e \epsilon_F}{m }
\int_{-\theta_0}^{+\theta_0} d\theta ~
\frac{ \rho(\epsilon , \theta)~ (\cos \theta - \delta)}{1+r(\delta) \cos \theta}
\cos^2 \theta \ .
\label{eq:SxxTemp3}
\end{multline}
By using eq.~(\ref{eq:DirectionalDensityRewriting}) for the directional density of states
we recover exactly the integral expression for the result (\ref{eq:sigma_xx_boltzmann}) of Boltzmann approach:
\begin{align}
\overline{\sigma}_{xx} =
\frac{ e^2 \hbar}{ \pi \gamDes} ~
\frac{2 \epsilon }{m } ~
\frac{ \mathcal{I}_3(\Delta/\epsilon)}{ {I}_1(\Delta/\epsilon)} \ .
\label{eq:sigma_xx_diag}
\end{align}
\subsubsection{Renormalized current operator along $y$}
In the $y$ direction, the current operator is renormalized : the bare current operator $j_y$ is independent of the momentum
$\vec{k}$ and the symmetry argument used for the $x$ direction does not hold anymore. This renormalized current operator satisfies a Bethe-Salpeter equation represented in Fig.~\ref{fig:renormCurrent}:
\begin{equation}
J_y = j_y+ J_y \Pi \gamDes
%\gamDes \left( \mathbf{I} \otimes \mathbf{I} - \gamDes P_0 \right)^{-1},
\label{eq:Bethe2}
\end{equation}
where tensor product in spin space are assumed and
\begin{equation}
\Pi(\epsilon, \Delta) = \int \frac{d\vec{k}}{(2 \pi)^2}
\overline{G}^{R} (\vec{k},\epsilon) \otimes \overline{G}^{A} (\vec{k},\epsilon )^T .
\end{equation}
Due to the spinorial structure of the wave functions, this propagator is here an operator acting as the tensor product of two spin $\frac12$ spaces. The notation
$\cdots^T$ corresponds to a transposition of spin matrices.
Using the parametrization defined in Eq. (\ref{eq:param}) of the contours of constant energy $\epsilon$ we perform the
integration over energy to obtain for $\Pi(\epsilon, \Delta) \equiv \Pi( \delta = \Delta / \epsilon)$:
\begin{multline}
\Pi(\delta) =
\frac{\pi \tau_e^0}{\hbar}
\int_{-\theta_0}^{+\theta_0} d\theta ~
\frac{ \rho(\epsilon , \theta)}{1+r(\delta) \cos \theta}
\\
\times \left[ \mathbf{I} + \cos \theta \sigma_x + \sin \theta \sigma_y \right] \otimes
\left[ \mathbf{I} + \cos \theta \sigma_x - \sin \theta \sigma_y \right] .
\label{eq:P0_3}
\end{multline}
The expression (\ref{eq:DirectionalDensityRewriting}) for the directional density of states allows to rewrite it as
\begin{multline}
\Pi(\delta) =
\frac{1}{2 \gamDes I_1(\delta)} \biggl[
\mathcal{I}_1(\delta) \mathbf{I} \otimes \mathbf{I}
+ (\mathcal{I}_1(\delta) - \mathcal{I}_2(\delta))\sigma_x \otimes \sigma_x
\\
- \mathcal{I}_2(\delta) \sigma_y \otimes \sigma_y
+\mathcal{J}_1(\delta)( \mathbf{I} \otimes \sigma_x + \sigma_x \otimes \mathbf{I}) \biggr] \ ,
\label{eq:P_D}
\end{multline}
where we introduced the functions:
\begin{align}
\mathcal{I}_1(\delta) &= \int_{-\theta_{0}}^{\theta_{0}} d \theta ~ \frac{1}{\sqrt{\cos \theta- \delta}\, (1+r(\delta) \cos \theta)} \ , \\
%\mathcal{I}_6(x) = \int_{-\theta_{0}}^{\theta_{0}} d \theta ~ \frac{\cos^2 \theta}{\sqrt{\cos \theta-x}(1+ r(x) \cos \theta)} \\
\mathcal{J}_1(\delta) &= \int_{-\theta_{0}}^{\theta_{0}} d \theta ~ \frac{\cos \theta}{\sqrt{\cos \theta- \delta}\, (1+ r(\delta) \cos \theta)}\ ,
\end{align}
whereas $I_1$ and $\mathcal{I}_2$ are defined in Eqs.~(\ref{I1},\ref{eq:calI2}).
The inversion of the Bethe-Salpeter equation (\ref{eq:Bethe2}) is done in the appendix \ref{sec:appCurrent} and we find
\begin{equation}
J_y = j_y \left(\mathbf{I} \otimes \mathbf{I} - \gamDes \Pi(\delta) \right)^{-1}
= \left( 1 - \frac{\mathcal{I}_2(\delta) }{ I_1(\delta)} \right)^{-1} j_y \ .
\label{eq:RenormJy}
\end{equation}
\subsubsection{Conductivity along $y$}
Following the formula (\ref{eq:ConductRenorm}), the average conductivity along $y$ is expressed
\begin{equation}
\overline{\sigma}_{yy} =
\frac{\hbar}{2 \pi } \mathrm{tr} \left[
J_{y}.\Pi(\delta) . j_y \right] \ .
\label{eq:ConductRenormyy}
\end{equation}
From the eq.~(\ref{eq:Bethe2}), we express $J_y.\Pi(\delta)=\gamDes^{-1}(J_y-j_y)$ to obtain
from (\ref{eq:ConductRenormyy}):
\begin{equation}
\overline{\sigma}_{yy} =
\frac{\hbar}{2 \pi \gamDes} \mathrm{tr} \left[(J_y-j_y) j_y \right] \ .
\end{equation}
The expression for the renormalized current operator (\ref{eq:RenormJy}) leads to the final result
\begin{equation}
\overline{\sigma}_{yy} =
\frac{e^2 \hbar}{ \pi \gamDes} ~ c_y^2 ~
\frac{\mathcal{I}_2(\delta) }{ I_1(\delta) - \mathcal{I}_2(\delta)} \ ,
\label{eq:sigma_yy_diag}
\end{equation}
which is precisely the result (\ref{eq:sigma_yy_boltzmann}) obtained within the Boltzmann equation approach.
This concludes the derivation of the conductivity tensor within the diagrammatic approach. In doing so, we have identified the encoding of the anisotropic scattering
rates through the matrix self-energy (\ref{eq:SigmaMatrix}), while the corresponding transport times are hidden into the renormalization of vertex operators
(\ref{eq:RenormJx},\ref{eq:RenormJy}). Comparison with the Boltzmann approach allows to unveil the physical meaning of these technical structures, which we believe to be applicable to other situations of anisotropic transport of Dirac-like states.
\section{Results and Discussion}
\label{sect:discussion}
We now turn to a discussion of our results for various situations corresponding to energy spectra represented in Fig.~\ref{fig:EnergySpectrum}.
%Discussion from the expressions
%\sigma_{xx} & =
% \frac{e^2 \hbar }{\pi \gamma} ~
% c_x^2 ~
% \left(
% \frac{\epsilon}{\Delta}~
% \frac{ \mathcal{I}_3(\Delta/\epsilon)}{ \mathcal{I}_1(\Delta/\epsilon)} \right) ,
% \\
% \sigma_{yy} & =
% \frac{e^2 \hbar }{\pi \gamma} ~
% c_y^2 ~
% \frac{ \mathcal{I}_2(\Delta/\epsilon)}{ \mathcal{I}_1(\Delta/\epsilon) - \mathcal{I}_2(\Delta/\epsilon) }
%where $c_x$ was defined after eq. ??? (model) and $\epsilon/|\Delta|$ is varied : we study the characteristics of different %phases.
%Interpretation : dependance of $\sigma_{yy}$ on energy follows qualitatively the
%energy dependence of $\lambda_y = \tau^{\textrm{tr}}/\tau_e$. Indeed, $c_y$ is independent of $\epsilon$
%and $\rho(\epsilon) \tau_e(\epsilon)$ is approximately a constant of $\epsilon$.
%Energy dependence follows from Einstein relation.
%In the $x$ direction : $\sigma_{xx}$ can be interpreted by the same Einstein relation but without any renormalization of
%transport time with respect to elastic scattering time. The only energy dependance arises from the group velocity:
%$v_x^2 \simeq \epsilon$.
\subsection{$\Delta =0$~: Semi-Dirac {spectrum} }
Focusing first on the merging point ($Δ=0$), we find that the conductivities are expressed, from Eqs. (\ref{eq:sigma_xx_boltzmann},\ref{eq:sigma_yy_boltzmann},\ref{eq:sigma_xx_diag},\ref{eq:sigma_yy_diag}) as~:
\begin{subequations}
\begin{align}
\sigma_{xx} (\epsilon) &= \frac{e^2 \hbar }{\pi \gamDes}\, \frac{2 \epsilon}{m} \, \frac{ \mathcal{I}_3 (0)}{I_1(0)} \simeq 0.197 \, \frac{e^2 \hbar }{\pi \gamDes}\, \frac{2 \epsilon}{m} \\
\sigma_{yy} (\epsilon) &= \frac{e^2 \hbar }{\pi \gamDes}\, c_y^2 \, \frac{ \mathcal{I}_2 (0)}{I_1(0)- \mathcal{I}_2(0)} \simeq 1.491 \frac{e^2 \hbar }{\pi \gamDes}\, c_y^2 \ .
\end{align}
\label{eq:sigma_SemiDirac}
\end{subequations}
This case, which corresponds to an hybrid dispersion relation, linear in one direction and quadratic in the other direction, has been previously studied in
Ref.~\onlinecite{Banerjee:12}. However these authors have neglected
both the spinorial structure of the wave function and the angular
dependence of the elastic scattering time caused by the anisotropic
whose importance is emphasized in the present paper.
Using the numerical values of the integrals given in appendix B, we find $λ_y(0) ≃2.4915$ and\footnote{Note the correspondance between our notations
$I_k(x), (k=1,2,3)$ and those $I_k$ of
Ref.~\onlinecite{Banerjee:12} : $I_k(0)= 4 I_k$.}~:
\begin{subequations}
\begin{align}
\sigma_{xx} &= {\mathcal{I}_3(0) \over I_3(0)} ~ \sigma_{xx}^B \simeq 0.781 ~ \sigma_{xx}^B \\
\sigma_{yy} &= \lambda_y(0) {\mathcal{I}_2(0) \over I_2(0)} ~ \sigma_{yy}^B \simeq 2.237 ~ \sigma_{yy}^B \ ,
\end{align}
\end{subequations}
where $σ_xx^B$ and $σ_yy^B$ are the values obtained in Ref.~[\onlinecite{Banerjee:12}].
The energy dependence of the conductivities (\ref{eq:sigma_SemiDirac})
arises from the energy dependence of the average squared velocities. It is therefore independent of the energy along the $y$ direction, while it
increases linearly with energy along the $x$ direction.
\subsection{$\Delta >0$~: gapped {spectrum}}
When $Δ>0$, the energy spectrum exhibits a gap and we study here the conductivity {above this gap} at energies $ϵ> Δ$.
Along the $x$ direction, the renormalization factor $λ_x=1$ so that $τ_x^tr(θ) = τ_e (θ)$. The energy dependence arises mainly from the energy dependence of the average squared velocity.
Therefore we expect a roughly linear dependence in energy\footnote{More precisely, for $\ep \ll \Delta$ we have~:
\sigma_{xx} (\epsilon)
\approx 0.197 \, e^2 \hbar /(\pi \gamDes) c_x^2 \, (\epsilon/\Delta - 0.76)
\begin{eqnarray}
\sigma_{xx} (\epsilon) &=& \frac{e^2 \hbar }{\pi \gamDes} \frac{2 \epsilon}{m} \frac{ \mathcal{I}_3 (\Delta/\epsilon)}{I_1(\Delta/\epsilon)} = \frac{e^2 \hbar }{\pi \gamDes} c_x^2 \, \frac{ \epsilon}{\Delta} \frac{ \mathcal{I}_3 (\Delta/\epsilon)}{I_1(\Delta/\epsilon)} \\
0.2 \, \frac{e^2 \hbar }{\pi \gamDes} c_x^2 \, \frac{\epsilon - \Delta}{\Delta}
\label{eq:sigmaxx-highE-Delta-pos}
\end{eqnarray}
$c_x = \sqrt{2 \Delta / m}$ is the velocity along $x$ of the massive Dirac equation describing the spectrum for small momenta.
The dependence in energy of the conductivity $σ_yy$ is mainly due to the energy dependence of the renormalization factor $λ_y$ between transport time and relaxation time~:
\begin{eqnarray}
\sigma_{yy} (\epsilon) &=& \frac{e^2 \hbar }{\pi \gamDes} c_y^2 \frac{ \mathcal{I}_2 (\Delta/\epsilon)}{I_1(\Delta/\epsilon)- \mathcal{I}_2(\Delta/\epsilon)} \\
&\approx& \frac{e^2 \hbar }{\pi \gamDes} c_y^2 \frac{ \mathcal{I}_2(0)}{I_1(0)}\, \lambda_y({\Delta/\epsilon}) \ .
\label{eq:sigmayy-highE-Delta-pos}
\end{eqnarray}
The energy dependence of the conductivities $σ_xx$ and $σ_yy$ is plotted in Fig.~\ref{fig:sigmas_e_delta_positif}.
\begin{figure} [!h]
\centering
\includegraphics[width=8cm]{sigma_e_delta_positifG.pdf}
\caption{(Conductivities $\sigma_{\alpha\alpha}$ in
units of $e^2 \hbar c_\alpha^2 / \gamDes$ for $\alpha = x,y$) as functions of $\epsilon /\Delta $
in the gapped phase ($\Delta > 0$). The energy dependence of $\sigma_{xx}$ arises from the energy dependence of the velocity along $x$, while for $\sigma_{yy}$ it comes mainly from the energy dependence of the renormalization factor $\lambda_y$.
\label{fig:sigmas_e_delta_positif}
\end{figure}
\subsection{$\Delta <0$ : Dirac {spectrum} }
\begin{figure} [!h]
\centering
\includegraphics[width=8cm]{sigma_e_delta_negatifG.pdf}
\caption{Conductivities $\sigma_{\alpha\alpha}$ in
units of $e^2 \hbar c_\alpha^2 / \gamDes$ for $\alpha = x,y$) as functions of $\epsilon /|\Delta |$
in the Dirac phase
($\Delta <0$). The conductivity vanishes at the saddle point ($\epsilon=|\Delta|$). The vicinity of the saddle point should be treated with a self-consistent Born approximation (see text).
\label{fig:sigmas_e_delta_negatif}
\end{figure}
In this phase, we have two regimes separated by the saddle point energy $|Δ|$.
At high energy above the saddle point, $ϵ≫|Δ|$, the energy dependence of the conductivities are still given by Eqs.
(\ref{eq:sigmaxx-highE-Delta-pos}, \ref{eq:sigmayy-highE-Delta-pos}).
In the low energy limit $ϵ≪|Δ|$, expanding these expressions using Eq. (\ref{limits}), we recover the conductivities associated to a conic
dispersion of characteristic velocities $c_x$ and $c_y$~:
\begin{eqnarray}
\sigma_{xx} (\epsilon \to 0) &=& \frac{e^2 \hbar }{\pi \gamDes} \, c_x^2\\
\sigma_{yy} (\epsilon \to 0) &=& 2 \frac{e^2 \hbar }{\pi \gamDes} \, c_y^2 \ .
\end{eqnarray}
Note however the factor $2$ between the two expressions. This is due to the fact that $τ^_y= 2 τ_e$ like in graphene
while $τ^_x= τ_e$. It is instructive to compare with this limit with the case of graphene, where it is known that
$τ^= 2 τ_e$ is all directions\cite{McCann:2006}, and where Einstein relation
$σ_αα= e^2 (c_α^2 τ^ / 2) ρ(_F)$ together with the Fermi golden rule
$τ^= 2 τ_e= 2 ħ/(πρ(ϵ_F) )$ leads to
\begin{equation}
\sigma(\textrm{graphene}) = \frac{e^2 \hbar }{\pi \gamDes} \, v_F^2 \ . \\
\end{equation}
Using the fact that $c_x^2 = c_y^2 = v_F^2/2$, we find the same result for $σ_yy$ but the conductivity is twice smaller
along the $x$ direction.
The difference by a factor 2 between $σ_xx$ and $σ_yy$
results from intervalley scattering taking place along the $x$ direction (see Ref.~\onlinecite{McCann:2006} or a related discussion of diffusion within
graphene with different intervalley and intravalley disorder rates).
It is important to note that our calculations predict vanishing
conductivities at the saddle point $ϵ= |Δ|$. That is the result of the logarithmic divergence of the density
of states producing a vanishing elastic scattering time in
Eq.~(\ref{eq:ElasticTimeParam}). However, in such a limit, $ k_F l_e
→0$, so our approximations are no longer valid. To describe
correctly the behavior of the scattering time in the vicinity of the
saddle point, it is necessary to go beyond second order perturbation
theory, using for instance the self-consistent Born
approximation\cite{carpentier2013}, in order to obtain a finite
density of states and a non-zero elastic scattering
time. Qualitatively, we expect that the zero of the conductivity will
be replaced by a minimum for $ϵ≃|Δ|$.
\subsection{Evolution of conductivites across the transition}
\label{sec:transition}
We are now in position to discuss the evolution of the conductivity at fixed energy $_F$, as
a function of the parameter $Δ$ as we cross the merging
transition. Such evolution, derived for eq.~(\ref{eq:sigma_xx_boltzmann},\ref{eq:sigma_yy_boltzmann}) is represented on
Fig.~\ref{fig:sigma-delta}, where we have plotted $σ_αα$ in
units of $e^2 ħc̃_α^2 / $ for $α= x,y$ and
$c̃_x = √(2 ϵ_F / m)$ and $c̃_y=c_y$. Below the saddle point for
$Δ< -ϵ_F$, $σ_yy$ is nearly constant, while
$σ_xx$ decreases almost linearly with $Δ$.
At the saddle point $Δ= -ϵ_F$ where the topology of the Fermi surface changes,
a dip in both $σ_xx$ and $σ_yy$ is visible, down to minimal values not quantitatively captured by the present approach.
Past the saddle point, while $σ_xx$ remains linearly decreasing
with $Δ$, albeit more slowly, $σ_yy$ is first increasing,
presenting a maximum for $Δ/ϵ_F ≃-0.39$ and then
decreases to zero. No signature of the underlying transition at $Δ=0$ is manifest in the transport at high Fermi energy $ϵ_F$.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{conductivitiesG.pdf}
\caption{Conductivities (in
units of $e^2 \hbar \tilde{c}_\alpha^2 / \gamDes$ for $\alpha = x,y$) as a function of the parameter $\Delta$ for
a fixed chemical potential $\epsilon$. The conductivity
$\sigma_{xx}$ decreases monotonically with $\Delta$,
in almost linear fashion with a change of slope at the saddle
point S. The conductivity $\sigma_{yy}$ is almost constant below
the saddle point. Above the saddle point, the behavior of $\sigma_{yy}$ becomes non-monotonous with
$\Delta$. At the metal-insulator transition, both $\sigma_{xx}$ and
$\sigma_{yy}$ vanish linearly. Symbols (a), (b), (c), M and S refer to the regions presented in Fig. \ref{fig:EnergySpectrum}. }
\label{fig:sigma-delta}
\end{figure}
\section{Conclusion}
We have studied the behavior of the conductivity in both the Dirac phase, the critical semi-Dirac and above the gapped phase.
Using the complementary Boltzmann and diagrammatic techniques we have identified the different nature of anisotropy of the elastic scattering times
and transport times.
Indeed the transport is inherently anisotropic due both to the spinorial structure of the eigenstates and the anisotropy of the dispersion relation.
The approaches developed in this paper can be generalized to study the diffusive transport in other semi-metallic phases, including the various three dimensional
species recently identified.
\begin{acknowledgments}
We thank the hospitality of the Institut Henri Poincar{\'e} where part of this work was completed.
This work was supported by the French Agence Nationale de la Recherche (ANR) under grants SemiTopo (ANR-12- BS04-0007), IsoTop (ANR-10-BLAN-0419).
\end{acknowledgments}
\appendix
\section{Current vertex renormalization}
\label{sec:appCurrent}
\subsection{Inverse of tensor product}
We use the notation $M^{ab}_{cd} = A_{ab} \otimes B_{cd} $ for the coefficients of a tensor product $M=A\otimes B$.
The inverse (for the outer product) $N=A^{-1}\otimes B^{-1}$ of $M$ satisfies the relation
$ M^{ab}_{cd} N^{be}_{df} = \delta_{ae} \delta_{cf} $.
In the obtention of the diffuson propagator, we need to invert a tensor product of the form
\begin{multline}
M =
a~ \mathbf{I} \otimes \mathbf{I}
+ b~\sigma_x \otimes \sigma_x
+ c~ \sigma_y \otimes \sigma_y
\\
+d~( \mathbf{I} \otimes \sigma_x + \sigma_x \otimes \mathbf{I}) \ .
\label{eq:KR_1}
\end{multline}
Its inverse $M^{-1}$ can be parametrized as
\begin{multline}
M^{-1} = \Delta^{-1} \biggl[
A~ \mathbf{I} \otimes \mathbf{I}
+ B~\sigma_x \otimes \sigma_x
+ C~ \sigma_y \otimes \sigma_y
\\
+D~( \mathbf{I} \otimes \sigma_x + \sigma_x \otimes \mathbf{I})
+ E~ \sigma_z \otimes \sigma_z \biggr] \ ,
\label{eq:P_2}
\end{multline}
\begin{subequations}
\begin{align}
A &=a^3 + 2 b d^2 - a (b^2 + c^2 + 2 d^2) \\
B &= - b (a^2 - b^2 + c^2) + 2 (a - b) d^2 \\
C &= c ~(-a^2 - b^2 + c^2 + 2 d^2) \\
D &= - d \left[ (a - b)^2 - c^2 \right] \\
E &= 2 c ~(-a b + d^2) \\
\Delta &= \left[ (a - b)^2 - c^2 \right] \left[ (a + b)^2 - c^2 - 4 d^2 \right] \ .
\end{align}
\end{subequations}
Let us now focus on the following contraction
%\left( \sigma_y\right)_{ab} \left[ M^{-1} \right]^{bc}_{ad} \left( \sigma_y \right)_{cd}
%&= 2~ \frac{A - B - C - E}{\Delta} \\
%&= \frac{2}{a - b - c}.
\begin{align}
\left[ M^{-1} \right]^{ab}_{cd} \left( \sigma_y \right)_{bd}
&= \frac{A - B - C - E}{\Delta} \left( \sigma_y \right)_{ac} \\
&= \frac{1}{a - b - c} \left( \sigma_y \right)_{ac} \ .
\label{eq:KronContraction2}
\end{align}
irrespective of $d$ and hence of $\Delta$. In particular eq.~(\ref{eq:KronContraction2}) is valid even if $\Delta$ vanishes.
\subsection{Current renormalization}
Let us now use the above parametrization (\ref{eq:KR_1}) for the tensor
$M=\mathbf{I} \otimes \mathbf{I} - \gamDes \Pi(\delta)$.
%\Pi(\delta) =
% \frac{1}{2 \gamDes I_1(\delta)} \biggl[
% \mathcal{I}_1(\delta) \mathbf{I} \otimes \mathbf{I}
% + (\mathcal{I}_1(\delta) - \mathcal{I}_2(\delta))\sigma_x \otimes \sigma_x
% - \mathcal{I}_2(\delta) \sigma_y \otimes \sigma_y
%+\mathcal{J}_1(\delta)( \mathbf{I} \otimes \sigma_x + \sigma_x \otimes \mathbf{I}) \biggr] \ ,
%to obtain the contractions:
%\mathrm{Tr} \left( \sigma_y \Pi \sigma_y \right) &=
%\left( \sigma_y\right)_{ab} \left[ \Pi \right]^{bc}_{ad} \left( \sigma_y \right)_{cd} =
% \frac{ 2 \mathcal{I}_2(\delta) }{\gamDes I_1(\delta)}
%\subsection{Diffuson structure factor and contribution to the conductivity $\sigma_{yy}$.}
%Let us now use the above analysis to identify the structure factor for the Diffuson, defined in (\ref{eq:Bethe}) : we want to invert
%$M = 1 - \gamDes \Pi(\delta)$.
From the parametrization of eq.~(\ref{eq:P_D}), we obtain the following identification of coefficients
\begin{subequations}
\begin{align}
a &= 1 - \frac{\mathcal{I}_1(\delta) }{2 I_1(\delta)} \ , &
b &= \frac{\mathcal{I}_2(\delta) - \mathcal{I}_1(\delta) }{2 I_1(\delta)} \ , \\
c &= \frac{\mathcal{I}_2(\delta) }{2 I_1(\delta)} \ , &
d &= - \frac{\mathcal{J}_1(\delta) }{2 I_1(\delta)} \ .
\end{align}
\label{eq:CoefKron}
\end{subequations}
Then the equation (\ref{eq:KronContraction2}) provides the expression for the renormalized current operator:
\begin{equation}
J_y = j_y \left(\mathbf{I} \otimes \mathbf{I} - \gamDes \Pi(\delta) \right)^{-1}
= \left( 1 - \frac{\mathcal{I}_2(\delta) }{ I_1(\delta)} \right)^{-1} j_y \ .
\end{equation}
A word of caution is necessary at this stage : $ \left( 1 - \gamDes \Pi(\vec{q}) \right)^{-1}$
is the structure factor which encodes the propagation of the diffuson modes\cite{Akkermans}.
In the symplectic class which we consider, there is one such mode which is diffusive : $1 - \gamDes \Pi(\vec{q})$
possesses a vanishing eigenvalue $\propto (Dq^2)$. Hence in the limit $q\to 0$ that we consider, $ 1 - \gamDes \Pi$ is no longer invertible. In principle, we should have kept a finite momentum $q$ during the calculation, and
taken the limit $q\to 0$ only in the result.
However, the vertex renormalization that we consider in this section
is not sensitive to this long-wavelength physics : it corresponds to a renormalization of the elastic scattering time into a transport time, which occurs
on short distances. This is manifest in the independence of the result (\ref{eq:KronContraction2}) on the determinant $\Delta$ of the matrix $M$: this is a classical contribution, which depends on these massive diffuson modes, while the diffusive long
distance modes enter the quantum correction not discussed in this paper.
\section{Special Functions}
In this appendix, we discuss a few useful results of the various integrals entering the expressions of transport coefficients in the paper and arising as
integrals along the constant energy contours of the model.
Let us first consider the integrals~:
\begin{equation*}
{I}_1 (\delta)= \int_{-\theta_0}^{\theta_0} {d \theta \over \sqrt{\cos \theta - \delta} }
\quad , \quad
{J}_1 (\delta)= \int_{-\theta_0}^{\theta_0} {d \theta \cos \theta
\over \sqrt{\cos \theta - \delta} },
\end{equation*}
with $cosθ_0=δ$ if $|δ|<1$ and $θ_0=π$
Defining $X= √(2 / (1 - δ))$, we find\footnote{We use the definition of the elliptic integrals from Gradshteyn and Ryzhik \cite{Gradstein}. They differ from those used in Mathematica :
$K_{Grad.}(x)= K_{Math.}(x^2)$, $E_{Grad.}(x)= E_{Math.}(x^2)$ and $\Pi_{Grad.}(\phi,n,x)=\Pi_{Math.}(n,\phi,x^2)$.}:
\begin{eqnarray}
{I}_1 (\delta)&=& {4 \over \sqrt{1 - \delta}} K(X) \qquad \delta<-1 \\
{I}_1 (\delta)&=&2 \sqrt{2} K(1/X) \qquad -1 < \delta < 1
\end{eqnarray}
\begin{multline}
{J}_1 (\delta)= \\
{4 \over \sqrt{1 - \delta}}[(1-\delta) E(X)+ \delta K(X)] \textrm{ for } \delta<-1 \ ,
\end{multline}
\begin{multline}
{J}_1 (\delta)= \\
2 \sqrt{2}[2 E(1/X) - K(1/X)] \textrm{ for } -1 < \delta < 1 \ .
\end{multline}
The anisotropy factor $r(δ)= J_1(δ)/I_1(δ)$ reads~:
\begin{align}
r(\delta)&= (1 - \delta) E(X)/K(X) + \delta && \textrm{ for } \delta <-1 \\
r(\delta)&= 2 E(1/X) / K(1/X) - 1 && \textrm{ for } -1 < \delta < 1 \ .
\end{align}
The functions $I_1(δ$ and $J_1(δ)$ are plotted in Fig.~\ref{fig:I1J1}, and the dependence on $δ$
of $r(δ)$ is shown on Fig.~\ref{fig:r}.
\begin{figure} [!h]
\centering
\includegraphics[width=4cm]{I1.pdf}
\includegraphics[width=4cm]{J1.pdf}
\caption{ Functions ${I}_1(\delta)$ and ${J}_1(\delta)$ }
\label{fig:I1J1}
\end{figure}
\begin{figure} [!h]
\centering
\includegraphics[width=4cm]{calI1.pdf}
\includegraphics[width=4cm]{calJ1.pdf}
\includegraphics[width=4cm]{calI2.pdf}
\includegraphics[width=4cm]{calI3.pdf}
\caption{ {Functions ${\cal I}_1(\delta)$, ${\cal J}_1(\delta)$, ${\cal I}_2(\delta)$ and ${\cal I}_3(\delta)$ }}
\label{fig:calIJ}
\end{figure}
Let us now consider the four integrals
\begin{align}
\mathcal{I}_1(r,\delta) & =\int_{-\theta_0}^{\theta_0} \frac{d\theta}{(1+r \cos \theta) \sqrt{\cos \theta-\delta}}
\label{eq:calligraphic-I1-integral}
\\
\mathcal{J}_1(r,\delta) & =\int_{-\theta_0}^{\theta_0} \frac{d\theta \cos \theta}{(1+r \cos \theta) \sqrt{\cos \theta-\delta}}
\label{eq:calligraphic-J1-integral}
\\
\mathcal{I}_2(r,\delta) &=\int_{-\theta_0}^{\theta_0} \frac{d\theta \sin^2 \theta}{(1+r \cos \theta) \sqrt{\cos \theta-\delta}} \label{eq:calligraphic-I2-integral}
\\
\mathcal{I}_3(r,\delta)&=\int_{-\theta_0}^{\theta_0}\frac{d\theta \cos^2 \theta \sqrt{\cos \theta-\delta} }{(1+r \cos \theta) }
\label{eq:calligraphic-I3-integral}
\end{align}
We first focus on first the integral $ℐ_1(r,δ)$. For $δ<-1$, it can be rewritten as:
\begin{align}
\mathcal{I}_1(r,\delta)=&\frac{4}{(1+r) \sqrt{1-\delta}} \nonumber \\
\int_{0}^{\frac \pi 2}
\frac{d\theta}{\left( 1-\frac{2r}{1+r} \sin^2 \theta\right)
\sqrt{1-\frac{2}{1-\delta}\sin^2 \theta}} \\
=& \frac{4}{(1+r) \sqrt{1-\delta}} ~\Pi\left(\frac \pi 2, \frac{2r}{1+r}, \sqrt{\frac 2
{1-\delta}}\right)\ ,
\end{align}
where $Π$ is an elliptic integral of the third kind.
For $|δ|<1$, we use the change of variable $sinθ/2 =
sinθ_0/2 sinφ$ to rewrite
this function as:
\begin{align}
\mathcal{I}_1(r,\delta)=&\frac{2 \sqrt{2}}{1+r} \nonumber \\
\int_0^{\frac \pi 2} &
\frac{d\varphi}{\left(1-\frac{2r}{1+r} \sin^2 \frac{\theta_0} 2
\sin^2 \varphi\right) \sqrt{1-\sin^2 \frac{\theta_0} 2
\sin^2 \varphi}} \nonumber \\
=& \frac{2 \sqrt{2}}{1+r} ~\Pi\left(\frac \pi 2,
\frac{2r}{1+r}\sin^2 \frac{\theta_0} 2,
\sin \frac{\theta_0} 2 \right)\ .
\end{align}
Moreover the integrals $𝒥_1(r,δ)$, $ℐ_2(r,δ)$ and
$ℐ_3(r,δ)$ can be expressed in terms of
$I_1(δ)$, $J_1(δ)$ and $ℐ_1(r,δ)$:
\begin{align}
\mathcal{J}_1(r,\delta) &= \frac 1 r \left[ I_1(\delta) - \mathcal{I}_1(r,\delta)\right]\ , \\
\mathcal{I}_2(r,\delta) &= \frac 1 {r^2} I_1(\delta) - \frac 1 r J_1(\delta)+ \left( 1-\frac 1 {r^2}\right)
\mathcal{I}_1(r,\delta)\ , \\
\mathcal{I}_3(r,\delta) &= \frac 1 r \left(\frac 1 3 + \frac \delta r
+ \frac 1 {r^2}\right) I_1(\delta) - \frac 1 r \left(\frac{\delta}{3}
+\frac 1 r\right) J_1(\delta)
\nonumber \\
& - \frac 1 {r^2} \left(\delta + \frac 1 {r}\right) \mathcal{I}_1(r,\delta) \ .
\end{align}
Finally, the integrals used in the text are
\begin{align*}
\mathcal{I}_1( \delta)&=\mathcal{I}_1[r(\delta),\delta]
\mathcal{J}_1( \delta)=\mathcal{J}_1[r(\delta),\delta] \\
\mathcal{I}_2( \delta)&=\mathcal{I}_2[r(\delta),\delta]
&&\mathcal{I}_3( \delta)=\mathcal{I}_3[r(\delta),\delta] .
\end{align*}
The following special values are of particular interest for the expressions in the text :
\begin{align*}
I_1(0) & = 2 \sqrt{2} K\left(\frac{1}{\sqrt{2}}\right) \simeq 5.2441 \\
J_1(0) & = \pi \sqrt{2} / K\left(\frac{1}{\sqrt{2}}\right) \simeq 2.3963 \\
r(0) & = J_1(0) / I_1(0) \simeq 0.457 \\
I_2(0) & = {4 \over 3} \sqrt{2} K(1/\sqrt{2}) \simeq 3.4961 \\
\mathcal{I}_2(0) & \simeq 3.1393 \\
I_3(0) & \simeq 1.4377 \\
\mathcal{I}_3(0) & \simeq 1.0322 \ .
\end{align*}
as well as the limits when $\delta \rightarrow - \infty$~:
\begin{align}
\frac{ \mathcal{I}_2 (\delta)}{I_1(\delta)- \mathcal{I}_2(\delta)} &\to 1
\frac{ \mathcal{I}_3 (\delta)}{I_1(\delta)} \to - \frac{\delta}{2} \ .
\label{limits}
\end{align}
%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
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{Hosur}}\ and\ \bibinfo {author} {\bibfnamefont {Xiaoliang}\ \bibnamefont
{Qi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Recent developments
in transport phenomena in weyl semimetals},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Comptes Rendus Physique}\ }\textbf {\bibinfo
{volume} {14}},\ \bibinfo {pages} {857} (\bibinfo {year} {2013})}\BibitemShut
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{Hasegawa}, \citenamefont {Konno}, \citenamefont {Nakano},\ and\
\citenamefont {Kohmoto}}]{Hasegawa:06}%
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\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Nakano}}, \ and\ \bibinfo
{author} {\bibfnamefont {M.}~\bibnamefont {Kohmoto}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Zero modes of tight-binding electrons on the
honeycomb lattice},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {74}},\ \bibinfo
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\citenamefont {Guinea},\ and\ \citenamefont {Sols}}]{Guinea:08}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Wunsch}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Guinea}}, \
and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Sols}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Dirac-point engineering and topological
phase transitions in honeycomb optical lattices},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {New. J. Phys.}\ }\textbf {\bibinfo {volume}
{10}},\ \bibinfo {pages} {103027} (\bibinfo {year} {2008})}\BibitemShut
\bibitem [{\citenamefont {Pereira}\ \emph {et~al.}(2009)\citenamefont
{Pereira}, \citenamefont {Neto},\ and\ \citenamefont {Peres}}]{Pereira:09}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.~M.}\ \bibnamefont
{Pereira}}, \bibinfo {author} {\bibfnamefont {A.~H.~Castro}\ \bibnamefont
{Neto}}, \ and\ \bibinfo {author} {\bibfnamefont {N.~M.~R.}\ \bibnamefont
{Peres}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Tight-binding
approach to uniaxial strain in graphene},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
{80}},\ \bibinfo {pages} {045401} (\bibinfo {year} {2009})}\BibitemShut
\bibitem [{\citenamefont {Montambaux}\ \emph
{et~al.}(2009{\natexlab{a}})\citenamefont {Montambaux}, \citenamefont
{{Pi\'echon}}, \citenamefont {Fuchs},\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Montambaux}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
{{Pi\'echon}}}, \bibinfo {author} {\bibfnamefont {J.-N.}\ \bibnamefont
{Fuchs}}, \ and\ \bibinfo {author} {\bibfnamefont {M.O.}\ \bibnamefont
{Goerbig}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Merging of
dirac points in a two-dimensional crystal},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
{80}},\ \bibinfo {pages} {153412} (\bibinfo {year}
{2009}{\natexlab{a}})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Montambaux}\ \emph
{et~al.}(2009{\natexlab{b}})\citenamefont {Montambaux}, \citenamefont
{{Pi\'echon}}, \citenamefont {Fuchs},\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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{{Pi\'echon}}}, \bibinfo {author} {\bibfnamefont {J.-N.}\ \bibnamefont
{Fuchs}}, \ and\ \bibinfo {author} {\bibfnamefont {M.O.}\ \bibnamefont
{Goerbig}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A universal
hamiltonian for the motion and the merging of dirac cones in a
two-dimensional crystal},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Eur. Phys. J. B}\ }\textbf {\bibinfo {volume} {72}},\ \bibinfo
{pages} {509} (\bibinfo {year} {2009}{\natexlab{b}})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Banerjee}\ \emph {et~al.}(2009)\citenamefont
{Banerjee}, \citenamefont {Singh}, \citenamefont {Pardo},\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Banerjee}}, \bibinfo {author} {\bibfnamefont {R.~R.~P.}\ \bibnamefont
{Singh}}, \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Pardo}}, \ and\
\bibinfo {author} {\bibfnamefont {W.~E.}\ \bibnamefont {Pickett}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Tight-binding modeling and
low-energy behavior of the semi-dirac point},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {103}},\ \bibinfo {pages} {016402} (\bibinfo {year}
{2009})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Katsnelson}(2006)}]{Katsnelson:2006b}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Katsnelson}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Zitterbewegung, chirality, and minimal conductivity in graphene},}\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Eur. Phys. J. B}\
}\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {157--160} (\bibinfo
{year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Tworzydlo}\ \emph {et~al.}(2006)\citenamefont
{Tworzydlo}, \citenamefont {Trauzettel}, \citenamefont {Titov}, \citenamefont
{Rycerz},\ and\ \citenamefont {Beenakker}}]{Twordzylo:2006}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Tworzydlo}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Trauzettel}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Titov}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Rycerz}}, \ and\ \bibinfo
{author} {\bibfnamefont {C.W.J}\ \bibnamefont {Beenakker}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Sub-poissonian shot noise in
graphene},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. Lett.}\ }\textbf {\bibinfo {volume} {96}},\ \bibinfo {pages} {246802}
(\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Abrikosov}(1988)}]{Abrikosov:88}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.A.}\ \bibnamefont
{Abrikosov}},\ }\href@noop {} {\emph {\bibinfo {title} {Fundamentals of the
Theory of Metals}}}\ (\bibinfo {publisher} {North Holland},\ \bibinfo
{address} {Amsterdam, Netherlands},\ \bibinfo {year} {1988})\BibitemShut
\bibitem [{\citenamefont {Ziman}(1972)}]{Ziman:79}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont
{Ziman}},\ }\href@noop {} {\emph {\bibinfo {title} {Principles of the Theory
of Solids}}}\ (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo
{address} {Cambridge},\ \bibinfo {year} {1972})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Xiao}\ \emph {et~al.}(2010)\citenamefont {Xiao},
\citenamefont {Chang},\ and\ \citenamefont {Niu}}]{Xiao:2010}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Di}~\bibnamefont
{Xiao}}, \bibinfo {author} {\bibfnamefont {Ming-Che}\ \bibnamefont {Chang}},
\ and\ \bibinfo {author} {\bibfnamefont {Qian}\ \bibnamefont {Niu}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Berry phase effects on
electronic properties},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo
{pages} {1959} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Son}\ and\ \citenamefont {Spivak}(2013)}]{Son:2013}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~T.}\ \bibnamefont
{Son}}\ and\ \bibinfo {author} {\bibfnamefont {B.~Z.}\ \bibnamefont
{Spivak}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Chiral anomaly
and classical negative magnetoresistance of weyl metals},}\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo
{volume} {88}},\ \bibinfo {pages} {104412} (\bibinfo {year}
{2013})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sondheimer}(1962)}]{Sondheimer:1962}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.H.}\ \bibnamefont
{Sondheimer}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The
boltzmann equation for anisotropic metals},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Proc. R. Soc. London, Ser. A}\ }\textbf
{\bibinfo {volume} {268}},\ \bibinfo {pages} {100} (\bibinfo {year}
{1962})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sorbello}(1974)}]{Sorbello:1974}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.S.}\ \bibnamefont
{Sorbello}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On the
anisotropic relaxation time},}\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {J. Phys. F}\ }\textbf {\bibinfo {volume} {4}},\
\bibinfo {pages} {503} (\bibinfo {year} {1974})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sorbello}(1975)}]{Sorbello:1975}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.S.}\ \bibnamefont
{Sorbello}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Effects of
anisotropic scattering on electronic transport properties},}\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Phys. Cond. Matter}\ }\textbf
{\bibinfo {volume} {19}},\ \bibinfo {pages} {303} (\bibinfo {year}
{1975})}\BibitemShut {NoStop}%
\bibitem [{Note1()}]{Note1}%
\BibitemOpen
\bibinfo {note} {This is not true in graphene where $\lambda _x=2$. Note that
this peculiar result ($\lambda _x=1$) is due the fact that the matrix
elements of the disorder potential are supposed here to have no momentum
dependence. Assuming an opposite limit where the disorder would not couple
valleys, then in the Dirac limit $0 < \epsilon \ll - \Delta $, one would
recover $\lambda _x=1$ and $\tau _x^{\protect \textrm {tr}}= 2 \tau
_e$.}\BibitemShut {Stop}%
\bibitem [{\citenamefont {Akkermans}\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Akkermans}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Montambaux}},\ }\href@noop {} {\emph {\bibinfo {title} {Mesoscopic Physics
of Electrons and Photons}}}\ (\bibinfo {publisher} {Cambridge Univ. Press},\
\bibinfo {year} {2007})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Adroguer}\ \emph {et~al.}(2012)\citenamefont
{Adroguer}, \citenamefont {Carpentier}, \citenamefont {Cayssol},\ and\
\citenamefont {Orignac}}]{Adroguer:12}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Adroguer}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Carpentier}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Cayssol}},
\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Orignac}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Diffusion at the surface of
topological insulators},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {New J. Phys.}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo
{pages} {103027} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {McCann}\ \emph {et~al.}(2006)\citenamefont {McCann},
\citenamefont {Kechedzhi}, \citenamefont {Fal'ko}, \citenamefont {Suzuura},
\citenamefont {Ando},\ and\ \citenamefont {Altshuler}}]{McCann:2006}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{McCann}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Kechedzhi}},
\bibinfo {author} {\bibfnamefont {V.~I.}\ \bibnamefont {Fal'ko}}, \bibinfo
{author} {\bibfnamefont {H.}~\bibnamefont {Suzuura}}, \bibinfo {author}
{\bibfnamefont {T.}~\bibnamefont {Ando}}, \ and\ \bibinfo {author}
{\bibfnamefont {B.~L.}\ \bibnamefont {Altshuler}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Weak-localization magnetoresistance and valley
symmetry in graphene},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo
{pages} {146805} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Banerjee}\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Banerjee}}\ and\ \bibinfo {author} {\bibfnamefont {W.~E.}\ \bibnamefont
{Pickett}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Phenomenology
of a semi-dirac semi-weyl semimetal},}\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {86}},\
\bibinfo {pages} {075124} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
\bibitem [{Note2()}]{Note2}%
\BibitemOpen
\bibinfo {note} {Note the correspondance between our notations $I_k(x),
(k=1,2,3)$ and those $I_k$ of Ref.~\protect \rev@citealpnum {Banerjee:12} :
$I_k(0)= 4 I_k$.}\BibitemShut {Stop}%
\bibitem [{Note3()}]{Note3}%
\BibitemOpen
\bibinfo {note} {More precisely, for $\epsilon \ll \Delta $ we have~: $
\sigma _{xx} (\epsilon ) \approx 0.197 \protect \tmspace +\thinmuskip
{.1667em} e^2 \hbar /(\pi \gamma ) c_x^2 \protect \tmspace +\thinmuskip
{.1667em} (\epsilon /\Delta - 0.76) $}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Carpentier}\ \emph {et~al.}(2013)\citenamefont
{Carpentier}, \citenamefont {Fedorenko},\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Carpentier}}, \bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
{Fedorenko}}, \ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Orignac}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Effect of
disorder on {2D} topological merging transition from a dirac semi-metal to a
normal insulator},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Europhys. Lett.}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo
{pages} {67010} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
\bibitem [{Note4()}]{Note4}%
\BibitemOpen
\bibinfo {note} {We use the definition of the elliptic integrals from
Gradshteyn and Ryzhik \cite {Gradstein}. They differ from those used in
Mathematica : $K_{Grad.}(x)= K_{Math.}(x^2)$, $E_{Grad.}(x)= E_{Math.}(x^2)$
and $\Pi _{Grad.}(\phi ,n,x)=\Pi _{Math.}(n,\phi ,x^2)$.}\BibitemShut {Stop}%
\bibitem [{\citenamefont {Gradshteyn}\ and\ \citenamefont
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Gradshteyn}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Ryzhik}},\ }\href@noop {} {\emph {\bibinfo {title} {Tables of integrals
series and products}}},\ \bibinfo {edition} {5th}\ ed.\ (\bibinfo
{publisher} {Academic Press},\ \bibinfo {address} {New-York},\ \bibinfo
{year} {1994})\BibitemShut {NoStop}%
\end{thebibliography}%
\end{document}
|
1511.00596
|
In this paper we obtain a result about the global existence of weak solutions for the
$d$-dimensional Bussinesq system, with viscosity dependent on
temperature. The initial temperature is just supposed to be bounded, while the initial velocity
belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity
close enough to a positive constant, and the $L^\infty$ norm of their difference plus the Besov norm of
the horizontal component of the initial velocity is supposed to be exponentially small with respect to
the vertical component of the initial velocity. On Preliminaries and
in the appendix we consider some $L^p L^q$ regularity Theorems for the heat kernel, which play
an important role in the main proof of this article.
§ INTRODUCTION
The general Boussinesq system turns out from a first approximation of a coupling system related to the Navier-Stokes and the thermodynamic equations. In such approximation, if we consider the structural coefficients to be constant, as for example the viscosity, we obtain a system between two parabolic equations with linear second order operators. Nevertheless, several fluids cannot be modeled in this way, for instance if we want to study the plasma evolution. Hence it should be necessary to consider a class of quasilinear parabolic systems coming from the general Boussinesq one.
This paper is devoted to the global existence of solutions for the Cauchy problem related to one of these models, namely:
\begin{equation}\label{Navier_Stokes_system}
\begin{cases}
\;\partial_t\theta + \Div\, (\theta u)=0 & \RR_+ \times\RR^d,\\
\;\partial_t u + u\cdot \nabla u -\Div\, (\nu(\theta)\MM) +\nabla\Pi=0 & \RR_+ \times\RR^d,\\
\;\Div\, u = 0 & \RR_+ \times\RR^d,\\
\;(u,\,\theta)_{|t=0} = (\bar{u},\,\bar{\theta}) & \;\;\quad \quad\RR^d,\\
\end{cases}
\end{equation}
where $\MM$ is defined by $\nabla u + \tr\nabla u$.
Here $\theta$, $u=(u^1,\dots,u^d)$ and $\Pi$ stand for the temperature, velocity field and pressure of the fluid respectively, depending on the time variable $t\in \RR_+=[0,+\infty)$ and on the space variables $x\in \RR^d$. We denote by $u^h:=(u^1,\dots,u^{d-1})$ the horizontal coordinates of the velocity field, while $u^d$ is the vertical coordinate. Furthermore $\nu(\cdot)$ stands for the viscosity coefficient, which is a smooth positive function on $\RR_+$.
Such system is useful as a model to describe many geophysical phenomena, like, for example, a composed obtained by mixing several incompressible immiscible fluids. Indeed the temperature fulfills a transport equation, while the velocity flow verifies a Navier-Stokes type equation which describes the fluids evolution. We consider here the case where the viscosity depends on the temperature, which allows to characterize the immiscibility hypotheses.
§.§.§ Some Developments in the Boussinesq System
The general Boussinesq system, derived in <cit.>, assumes the following form:
\begin{equation}\label{general_Boussinesq}
\begin{cases}
\;\partial_t\theta + \Div\, (\theta u)-\Delta \varphi (\theta)+ |D|^s\theta =0
& \RR_+ \times\RR^d,\\
\;\partial_t u + u\cdot \nabla u -\Div\, (\nu(\theta)\MM) +\nabla\Pi=F(\theta)
& \RR_+ \times\RR^d,\\
\;\Div\, u = 0 & \RR_+ \times\RR^d,\\
\;(u,\,\theta)_{|t=0} = (\bar{u},\,\bar{\theta}) & \;\;\quad \quad\RR^d,\\
\end{cases}
\end{equation}
An exhaustive mathematical justification of such system as a model of stratified fluids (as atmosphere or oceans) is given by Danchin and He in <cit.>. We present here a short (and of course incomplete) overview concerning some some well-posedness results.
Provided by some technical hypotheses, in <cit.> Díaz and Galiano establish the global existence of weak solution for system <ref> when $s=0$. Moreover they achieve the uniqueness of such solutions in a two dimensional domain, assuming the viscosity $\nu$ to be constant.
In <cit.> Hmidi and Keraani study system (<ref>) in a two dimensional setting, when the parameter $s$ is null, $\varphi(\theta)=\theta$ and $F(\theta)$ stands for a Buoyancy force, more precisely they considered $F(\theta)=\theta e_2$, with $e_2$ the classical element of the canonical basis of $\RR^2$. They prove the global existence of weak solutions when both the initial data belong to $L^2(\RR^2)$. Furthermore, they establish the uniqueness of such solutions under an extra regularity on the initial data, namely $H^r(\RR^2)$, for $r>0$.
In <cit.> Wang and Zhang consider system <ref> with Buoyancy force and constant viscosity, when the temperature $\theta$ satisfies
\begin{equation*}
\partial_t\theta + \Div\, (\theta u)-\Div( k\nabla \theta) =0,
\end{equation*}
where $k$ stands for the thermal diffusivity, which also depends on the temperature. They prove existence and uniqueness of global solutions when the initial data belong to $H^r(\RR^2)$, for $r>0$.
In <cit.> Chae considered system (<ref>) in two dimension, with constant viscosity and when $\varphi(\theta)$ is equal to $\theta$ or $0$. In this case the author establish the existence of smooth solutions.
System (<ref>) has also given interest in the Euler equation framework, when the viscosity $\nu$ is supposed to be null. In this direction, Hmidi, Keerani and Rousset <cit.> develop the existence and uniqueness of a solutions when $s=1$, provided that the initial velocity belongs to $\BB_{\infty,1}^{1}\cap \dot{W}^{1,p}_x$ while the initial temperature lives in $\BB_{\infty,1}^{0}\cap L^p_x$.
In <cit.> Abidi and Hmidi perform an existence and uniqueness result for system (<ref>) in two dimension, when $\varphi\equiv 0$, $s=0$ and the force $F(\theta)=\theta e_2$. Here, the initial velocity is supposed in $L^2\cap \BB_{\infty,1}^{-1}$ and the temperature belongs to $\BB_{2,1}^0$.
In <cit.> Paicu and Danchin consider the case of constant viscosity. Given a force $F(\theta)=\theta e_2$, imposing $s=2$ and $\phi=\theta$, the authors perform a global existence result for system (<ref>), on the condition that the initial data are of Yudovich's type, namely the initial temperature is in $L^2_x\cap \BB_{p,1}^{-1}$, the initial velocity is in $L^2_x$ and the initial vorticity $\partial_1 \bar{u}_{2}-\partial_2 \bar{u}_{1}$ is bounded and belongs to some Lebesgue space $L^r_x$ with $r\geq 2$.
We mention that a no constant viscosity has also been treated in the study of the inhomogeneous incompressible Navier Stokes equation with variable viscosity
\begin{equation}\label{System2_intro}
\begin{cases}
\;\partial_t \rho + \Div\, (\rho u)=0 & \RR_+ \times\RR^d,\\
\;\partial_t (\rho u) + \Div\{\rho u\otimes u\} -\Div\, (\eta(\rho)\MM) +\nabla\Pi=f
& \RR_+ \times\RR^d,\\
\;\Div\, u = 0 & \RR_+ \times\RR^d,\\
\;(u,\,\rho)_{|t=0} = (\bar{u},\,\bar{\rho}) & \;\;\quad \quad\RR^d.\\
\end{cases}
\end{equation}
In <cit.> Abidi and Paicu analyze the global well-posedness of (<ref>) in certain critical Besov spaces provided that the initial velocity is small enough and the initial density is strictly close to a positive constant.
In <cit.> Abidi and Zhang establish the existence and uniqueness of global solutions for system (<ref>), on the condition that the initial velocity belongs to $H^{-2\delta}\cap H^1$, for some $\delta\in (0,1/2)$, the initial density lives in $L^{2}_x\cap \dot{W}^{1,r}_x$, with $r\in (2,2/(1-2\delta)\,)$, and $\bar{\rho}-1$ belongs to $L^2_x$.
We finally mention that in <cit.> Huang and Paicu investigate the time decay behavior of weak solutions for (<ref>) in a two dimensional setting.
In this paper we are going to study the global existence of solutions for the system (<ref>) concerning standard and natural conditions on the initial data: the initial temperature is only assumed to be bounded and the initial velocity field is supposed to belong to certain critical homogeneous Besov space. More precisely we consider
\begin{equation}\label{initial_data}
\bar{\theta}\in L^\infty_x\quad\text{and}\quad \bar{u} \in \BB_{p,r}^{\frac{d}{p}-1}\quad\text{with}\quad r\in (1,\infty)\quad\text{and}\quad p\in (1,d).
\end{equation}
As the classical Navier-Stokes equation, system (<ref>) has also a scaling property, more precisely if $(\theta, u, \Pi)$ is a solution then, for
all $\lambda>0$,
\begin{equation*}
(\theta(\lambda^2 t, \lambda\,x), \lambda\, u(\lambda^2 t, \lambda\,x), \lambda^2\, \Pi(\lambda^2 t, \lambda\,x))
\end{equation*}
is also solution of (<ref>), with initial data $ (\bar{\theta}(\lambda\,x), \lambda \,\bar{u}(\lambda\,x))$. Hence it is natural to consider the initial data in a Banach space with a norm which is invariant under the previous scaling, as for instance $L^\infty_x\times \BB_{p,r}^{d/p-1}$. Let us remark that this initial data type allows $\theta$ to include discontinuities along an interface, an important physical case as a model that describes a mixture of fluids with different temperatures.
From here on we suppose the viscosity $\nu$ to be a bounded smooth function, close enough to a positive constant $\mu$, which we assume to be $1$ for the sake of simplicity. Then, we assume the following small condition for the initial data to be fulfilled:
\begin{equation}\label{smallness_condition}
\eta :=
\big(
\| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\big)
\exp\Big\{ c_r \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}^{4r} \Big\}
\leq c_0.
\end{equation}
where $c_0$ and $c_r$ are two suitable positive constants. This sort of initial condition is not new in literature, for instance it appears in <cit.>, where Huang, Paicu and Zhang study of an incompressible inhomogeneous fluid in the whole space with viscosity dependent on the density, and moreover in <cit.>, where Danchin and Zhang examine the same fluid typology, in the half-space setting.
Before enunciating our main results, let us recall the meaning of weak solution for system (<ref>):
We call $(\theta ,\,u,\,\Pi)$ a global weak solution of
(<ref>) if
(i) for any test function $\varphi\in \DD(\RR_+\times \RR^d)$, the following identities are
well-defined and fulfilled:
\begin{equation*}
\int_{\RR_+} \int_{\RR^d}
\{\theta \left( \partial_t \varphi + u \cdot \nabla \varphi \right)\}(t,x)\dd x\,\dd t +
\int_{\RR^d}
\bar{\theta}(x)\varphi(0,x)\dd x = 0\text{,}
\end{equation*}
\begin{equation*}
\int_{\RR_+} \int_{\RR^d}
\{u\cdot\nabla \varphi\}(t,x) \dd x\,\dd t = 0\text{,}
\end{equation*}
(ii) for any vector valued function
$\Phi=(\Phi_1,\dots,\Phi_d)\in \DD(\RR_+\times \RR^d)^d$
the following equality is well-defined and satisfied:
\begin{equation*}
\int_{\RR_+} \int_{\RR^d}\{
u\cdot \partial_t\Phi - (u\cdot \nabla u)\cdot \Phi
- \nu (\theta) \MM \cdot \nabla \Phi
+ \Pi\, \Div\, \Phi\}(t,x) \dd x\, \dd t
+ \int_{\RR^d}
\bar{u}(x)\cdot \Phi(0,x) \dd x = 0\text{,}
\end{equation*}
§.§.§ The smooth case
Some regularizing effects for the heat kernel, like the well-known $L^pL^q$-Maximal Regularity Theorem (see Theorem <ref>), play an key role in our proof as well as an useful homogeneous Besov Spaces characterization (see Theorem <ref> and Corollary <ref>). Indeed, we can reformulate the momentum equation of (<ref>) in the following integral form:
\begin{equation}
u(t) = e^{t\Delta}\bar{u} + \int_0^t e^{(t-s)\Delta} \big\{ -u\cdot \nabla u+\nabla\Pi\big\}(s)\dd s+
\int_0^t \Div\, e^{(t-s)\Delta}\big\{ \big((\nu(\theta)-1\big)\MM\big\}(s)\dd s.
\end{equation}
Thus, it is reasonable to assume the velocity $u$ having the same regularity of the heat kernel convoluted with the initial datum $\bar{u}$. The Maximal Regularity Theorem suggests us to look for a solution in a $L^{\bar{r}}_t L^q_x$ setting. Now, in the simpler case where $u$ just solves the heat equation with initial datum $\bar{u}$, having $\nabla u$ in some $L^{\bar{r}}_t L^q_x$ is equivalent to $\bar{u}\in \BB_{q,\bar{r}}^{d/q-1}$ on the condition $N/q-1=1-2/\bar{r}$. From the immersion $\BB_{p,r}^{d/p-1}\hookrightarrow \BB_{q,\bar{r}}^{d/q-1}$, for every $\bar{q}\geq p$ and $\bar{r}\geq r$, we deduce that this strategy requires $p\leq dr/(2r-1)$. Then, according to the above heuristics, our first result reads as follows:
Let $r\in (1,\infty)$ and $p\in (1,dr/(2r-1))$. Suppose that the initial data $(\bar{\theta},\,\bar{u})$ belongs to $L^infty_x\times \BB_{p,r}^{d/p-1}$.
There exist two positive constants $c_0$, $c_r$ such that, if the smallness condition (<ref>) is fulfilled, then
there exists a global weak solution $(\theta,\,u,\,\Pi)$ of (<ref>), in the sense of definition <ref> such that
\begin{equation*}
u \in L^{2r}_tL^{\frac{dr}{r-1}}_x,\quad\quad
\nabla u \in L^{2r}_tL^{\frac{dr}{2r-1}}_x\cap
\Pi\in L^{r}_t L^{\frac{dr}{2(r-1)}}_x.
\end{equation*}
Furthermore, the following inequalities are satisfied:
\begin{align*}
& \| \nabla u^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\|\nabla u^h\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} +
\| u^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_1\eta,\\
& \|\nabla u^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\|\nabla u^d\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} +
\| u^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + C_3,\\
& \|\Pi\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x}
\leq
\| \theta \|_{L^\infty_{t,x}}\leq \| \bar{\theta} \|_{L^\infty_x}.
\end{align*}
for some positive constants $C_1$, $C_2$, $C_3$ and $C_4$.
§.§.§ The general case
As we have already pointed out, the choice of a $L^{\bar{r}}_t L^{q}_x$ functional setting requires the condition $p< dr/(2r-1)$. The remaining case
$dr/(2r-1)\leq p<d$ can be handled by the addiction of a weight in time. Indeed, in the simpler case where $u$ just solves the heat equation with initial datum $\bar{u}$, having $u$ in some $\BB_{p_3,\bar{r}}^{d/p_3-1}$ for some $p_3\in (dr/(r-1),\infty)$ is equivalent to $t^{1/2( 1-d/p_3)-1/\bar{r})}u\in L^{\bar{r}}_t L^{p_3}_x$. In the same line having $\nabla \bar{u}$ in a suitable Besov space $\BB_{p_2,\bar{r}}^{d/p_2-1}$ is equivalent to have $t^{1/2( 2-d/p_3)-1/\bar{r})}u$ in $L^{\bar{r}}_t L^{p_2}_x$.
Hence, reformulating the smallness condition (<ref>) by
\begin{equation}\label{smallness_condition_general_case}
\eta :=
\big(
\| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\big)
\exp\Big\{ c_r \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}^{2r} \Big\}
\leq c_0,
\end{equation}
with similar heuristics proposed in the first case, our second results reads as follows:
Let $p,\,r$ be two real numbers in $ (2d/3,\,d)$ and $(1,\infty)$ respectively, such that
\begin{equation}\label{Main_Thm_r_restrictions}
\frac{2}{3}\frac{d}{p}-\frac{d}{6p}<\frac{1}{2}-\frac{1}{2r},\quad
\frac{1}{r}<\frac{1}{3}\big(\frac{d}{p}-1\big), \quad
\frac{1}{r}<\frac{4}{3}-\frac{d}{p}.
\end{equation}
Let us define $p_2:= 3pd/(2p+d)$ and $p_3:=3p^*/2 = 3pd/(2d-2p)$, so that $1/p=1/p_2 +1/p_3$ and
\begin{equation*}
\alpha:=\frac{1}{2}\big(3-\frac{d}{p_1} \big) -\frac{1}{r},\quad
\beta:=\frac{1}{2}\big(2-\frac{d}{p_2} \big) -\frac{1}{2r},\quad
\gamma_1:=\frac{1}{2}\big(1-\frac{d}{p_3} \big)-\frac{1}{2r},\quad
\gamma_2:=\frac{1}{2}\big(1-\frac{d}{p_3} \big).
\end{equation*}
There exist two positive constants $c_0$ and $c_r$ such that, if the smallness condition (<ref>) is fulfilled, then
there exists a global weak solution $(\theta,u, \Pi)$ of (<ref>),
in the sense of definition <ref> such that
\begin{equation*}
t^{\gamma_1} u \in L^{2r}_tL^{p_3}_x,\quad
t^{\gamma_2} u \in L^{\infty}_tL^{p_3}_x\quad
t^{\beta}\nabla u \in L^{2r}_tL^{p_2}_x\quad
t^{\alpha}\Pi\in L^{r}_t L^{p^*}_x.
\end{equation*}
Furthermore, the following
inequalities are satisfied:
\begin{equation}\label{Main_Theorem2_inequalities}
\begin{aligned}
& \|t^{\alpha}\nabla u^h\|_{L^{2r}_t L^{p^*}_x}+
\|t^{\beta}\nabla u^h\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^h \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^h \|_{L^{\infty}_t L^{p_3}_x}
\leq C_1\eta,\\&
\|t^{\alpha}\nabla u^d\|_{L^{2r}_t L^{p^*}_x}+
\|t^{\beta}\nabla u^d\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^d \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^d \|_{L^{\infty}_t L^{p_3}_x}
\leq C_2\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}} + C_3\\&
\|t^\alpha\Pi\|_{L^{r}_t L^{p^*}_x}
\leq
\| \theta \|_{L^\infty_{t,x}}\leq \| \bar{\theta} \|_{L^\infty_x}.
\end{aligned}
\end{equation}
for some positive constants $C_1$, $C_2$ and $C_3$.
We remark that the conditions on $p$ and $r$ in Theorem <ref> are not restrictive. Indeed, we can always embed $\BB_{p,r}^{d/p-1}$ into $\BB_{q,r}^{d/q-1}$ with
$q\geq p$ which satisfies $q\in (2d/3, d)$ (see Theorem <ref>). Moreover $\BB_{p,r}^{d/p-1}$ is embedded in $\BB_{p,\tilde{r}}^{d/p-1}$, with $\tilde{r}\geq r$,
then there is no lost of generality assuming the inequalities (<ref>).
Let us briefly describe the organization of this paper. In the second section we recall some technical Lemmas concerning the regularizing effects for the heat kernel, as the Maximal regularity Theorem, which will play an important role in the main proofs. We also mention some results regarding the characterization of the homogeneous Besov Spaces. In the third section we prove the existence of solutions for (<ref>), with stronger conditions on the initial data with respect to the ones of Theorem <ref>. In the fourth section we regularize our initial data by the dyadic partition, and, using the results of the third section with a compactness argument, we conclude the proof of Theorem <ref>. In the fifth and sixth sections we perform to the proof of Theorem <ref>, proceeding with a similar structure of the third and fourth sections.
In order to obtain the uniqueness about the solution of (<ref>), the more suitable strategy is to reformulate our system by Lagrangian coordinates, following for example <cit.>, <cit.> and <cit.>. The existence of such coordinates may be achieved supposing the velocity field with Lipschitz space condition, more precisely claiming $u$ belongs to $L^1_{loc}(\RR_+;{Lip}_x)$, or equivalently
$\nabla u \in L^{1}_{loc}(\RR_+;L^{\infty}_x)$. If we want to obtain this condition without controlling two derivatives of $u$ (in the same line of the existence part) and then without using Sobolev embedding, we need to bound terms like
\begin{equation}\label{problem_uniqueness_introduction1}
\int_0^t \Delta e^{(t-s)\Delta}\big\{ \big((\nu(\theta)-1\big)\nabla u\big\}(s)\dd s
\end{equation}
in some $L^{s}(0,T;L^\infty_x)$ space, with $s>1$. Unfortunately this is not allowed by the Maximal Regularity Theorem <ref> for the heat kernel, because of the critical exponents of this spaces. Then, we need to impose an extra regularity for the initial temperature, as $\nabla \bar{\theta}\in L^{l_1}_x$, for an opportune $l_1$, in order to obtain $\nabla \theta$ in $L^1_{loc}(\RR_+;{L^{l_1}_x})$ and then to split (<ref>) into
\begin{equation}
\int_0^t \Div\, e^{(t-s)\Delta} \big\{\nu'(\theta)\nabla \theta\cdot \nabla u\big\} (s)\dd s +
\int_0^t \Div\, e^{(t-s)\Delta} \big\{\big( \nu(\theta)-1\big)\nabla^2 u\big\} (s)\dd s.
\end{equation}
Hence we need to control the norm of $\nabla^2 u$ in some $L^{r_1}(0,T;L^{l_2}_x)$, with $r_1>1$ and also $l_2>d$ in order to fulfill the Morrey Theorem's hypotheses. It is necessary to do that starting from the approximate systems of the third section, however the only way to control two derivatives of the approximates solutions with some inequalities independent by the indexes $n\in \NN$ and $\varepsilon>0$ (present in the extra term of the perturbed transport equation) is to impose
$\nabla\bar{\theta}\in L^{l_1}_x$ with $l_1>d$. We conjecture that this is not the optimal condition for the initial data in order to obtain the uniqueness, indeed, inspired by <cit.>, we claim that, supposing $\nabla \bar{\theta}\in L^d_x$ and $\bar{u}\in$ $\dot{B}_{p,1}^{-1+\frac{d}{p}}$, it is possible to prove the uniqueness with the velocity field into the space
\begin{equation*}
L^\infty_t \dot{B}_{p,1}^{-1+\frac{d}{p}}\cap L^1_t \dot{B}_{p,1}^{1+\frac{d}{p}}.
\end{equation*}
However this needs to change the structure of the existence part, more precisely to change the functional space where we are looking for a solution. Since in our Theorem we suppose only the initial temperature to be bounded, then we have decided to devote this paper only to the existence part of a global weak solution for system (<ref>).
§ PRELIMINARIES
The purpose of this section is to present some lemmas concerning the regularizing effects for the heat kernel, which will be useful for the next sections. At first step let us recall the well-known Hardy-Littlewood-Sobolev inequality, whose proof is available in <cit.>, Theorem $1.7$.
Let $f$ belongs to $L^p_x$, with $1< p <\infty $, $\alpha\in ]0,d[$ and suppose $r\in ]0,\infty[$ satisfies
$1/p+\alpha/d= 1+1/r$. Then $|\cdot |^{-\alpha}* f$ belongs to $ L^r_x$ and there exists a positive constant $C$ such that
\left\|
|\cdot |^{-\alpha}* f
\right\| _{L^r_x}
\leq
\left\|
\right\|_{L^p_x}
From this Theorem we can infer the following corollary.
Let $f$ belongs to $L^p_x$, with $1< p <d $ and let $(\sqrt{-\Delta})^{-1}$ be the Riesz potential,
defined by $(\sqrt{-\Delta})^{-1}f(\xi) := \mathcal{F}^{-1}( \hat{f}(\xi)/|\xi|)$. Then $(\sqrt{-\Delta})^{-1}f$ belongs to $L^{dp/(d-p)}_x$
and there exists a positive constant $C$ such that $\|(\sqrt{-\Delta})^{-1}f\|_{L^{pd/(d-p)}_x}\leq C\|f \|_{L^p_x}$.
From the equality $ (\sqrt{-\Delta})^{-1}f(x)=c(|\cdot |^{-d+1}* f)(x)$, for almost every $x\in\RR^d$ and for an appropriate constante $c$, the theorem is a direct consequence of
Theorem <ref>, considering $\alpha =d-1$.
One of the key ingredient used in the proof of Theorem (<ref>) is the maximal regularity Theorem for the heat kernel. We recall here the statement (see <cit.>, theorem 7.3).
Let $T\in ]0,\infty]$, $1<p,q<\infty$ and $f\in L^p(0,T;L^q_x)$. Let the operator $A$ be
defined by
\begin{equation*}
Af(t,\cdot):=\int_0^t \Delta e^{(t-s)\Delta}f(s,\cdot)\dd s\text{.}
\end{equation*}
Then $A$ is a bounded operator from $L^p(0,T;L^q_x)$ to $L^p(0,T;L^q_x)$.
If instead of $\Delta$ on the definition of the operator $A$ we consider $\nabla$ (the operator $B$ of Lemmas <ref> and <ref> ) or even without derivatives (the operator $C$ of Lemma <ref>), then we can obtain similar results with respect to the maximal regularity Theorem, using a direct computation. We present here the proofs. At first step let us recall two useful identities:
Let us denote by $K$ the heat kernel, defined by $K(t,x)=e^{-|x|^2/(4t)}/(2\pi t)^{d/2}$, then
$\|K(t,\cdot)\|_{L^q_x}=\|K(1,\cdot)\|_{L^q_x}/t^{d/(2q')}$, for all $1\leq q <\infty$. Moreover considering the gradient of the heat kernel,
$\Omega(t,x):=\nabla K(t,x)=-xK(t,x)/(2t)$, we have $\|\Omega(t,\cdot)\|_{L^q_x}=\|\Omega(1,\cdot)\|_{L^q_x}/|t|^{d/(2q')+1/2}$.
Let us denote by $R:= \tr (R_1,\dots,R_d)$, where $R_j$ is the Riesz transform over $\RR^d$, defined by
\begin{equation*}
R_jf:= \mathcal{F}^{-1}\left(-i \frac{\xi_j}{|\xi|}\hat{f}\right)\text{.}
\end{equation*}
we recall that $R_j$ is a bounded operator from $L^q_x$ to itself, for every $1<q<\infty$ (for more details we refer to <cit.>).
Let $T\in\, ]0,\infty]$ and $f\in L^r(0,T;L^p_x)$, with $1<p<d$ and $1<r<\infty$.
Let the operator $B$ be defined by
\begin{equation*}
\mathcal{B}f(t,\cdot)\doteq \int_0^t \nabla e^{(t-s)\Delta}f(s,\cdot)\dd s\text{,}
\end{equation*}
Then $\mathcal{B}$ is a bounded operator from $L^r(0,T;L^p_x)$ to $L^r(0,T;L^{\frac{dp}{d-p}}_x)$.
From corollary <ref> we have that, for almost every $s\in (0,T)$,
\begin{equation*}
\big(\sqrt{-\Delta}\big)^{-1}f(s) \in L^{\frac{dp}{d-p}}_x.
\end{equation*}
Then, reformulating $\mathcal{B}$ by
\begin{equation*}
\mathcal{B}f(t,\cdot)= -\int_0^t \Delta e^{(t-s)\Delta}
R\big(\sqrt{-\Delta}\big)^{-1}f(s,\cdot)\dd s\text{,}
\end{equation*}
we deduce, by theorem <ref>, that
$\mathcal{B}f\in L^r( 0,T ;L^{\frac{dp}{d-p}}_x)$ and
\begin{equation*}
\left\|\,\mathcal{B}f\,\right\|_{L^r( 0,T ;L^{\frac{dp}{d-p}}_x)}
\leq C_1 \big\|\,R(\sqrt{-\Delta})^{-1}f\,\big\|_{L^r( 0,T ;L^{\frac{dp}{d-p}}_x)}
\leq C_2 \left\|\,f\,\right\|_{L^r( 0,T ;L^p_x)}
\text{,}
\end{equation*}
for opportune positive constant $C_1$ and $C_2$.
Let $T\in\, ]0,\infty]$ and $f\in L^r(0,T;L^{p}_x)$, with $1<r<\infty$ and $p\in [1,\frac{dr}{r-1}]$.
Let the operator $\mathcal{B}$ be defined as in Lemma <ref>.
Then, we have that $\mathcal{B}$ is a bounded operator from
$L^r(0,T;L^{p}_x)$ with values to $L^{2r}(0,T;L^{q}_x)$, where $1/q:=1/p-(r-1)/(dr)$.
Observe that, for every $t\in \RR_+$,
\begin{equation*}
\big\|\int_0^t \nabla e^{(t-s)\Delta}f(s)\dd s\,\big\|_{L^q_x}
\leq \int_0^t \|\,\Omega(t-s,\cdot)*f(s,\cdot)\,\|_{L^q_x}\dd s
\leq \int_0^t \|\,\Omega(t-s)\|_{L^{\tilde{q}}_x}\|\,f(s)\,\|_{L^{p}_x}\dd s,
\end{equation*}
with $1/\tilde{q}+1/p=1/q+1$ or equivalently $\tilde{q}'=dr/(r-1)$. Recalling Remark <ref>, we obtain
\begin{equation*}
\big\|\int_0^t \nabla e^{(t-s)\Delta}f(s)\dd s\,\big\|_{L^q_x}
\leq
\int_0^t \frac{\quad\|\,f(s)\,\|_{L^{p}_x}}{\quad|t-s|^{\frac{2r-1}{2r}}}\dd s
\leq
\int_{\RR} \frac{\quad\|\,f(s)\,\|_{L^{p}_x}}
{\quad|t-s|^{\frac{2r-1}{2r}}}1_{(0,T)}(s)\dd s.
\end{equation*}
Since by Theorem <ref>
\begin{equation*}
|\cdot|^{-\frac{2r-1}{2r}}*\|f(\cdot)1_{(0,T)}(\cdot)\|_{L^p_x}\in L^{2r}_t,
\end{equation*}
then there exists $\tilde{C}>0$ such that
\begin{equation*}
\|\,\mathcal{B}f\,\|_{L^{2r}(0,T;L^{q}_x)}
\leq C
\big\|\;|\cdot|^{-\frac{2r-1}{2r}}*\|f(\cdot)1_{(0,T)}(\cdot)\|_{L^p_x}\big\|_{L^{2r}_t}
\leq
\tilde{C} \|\,f\,\|_{L^{r}(0,T;L^{p}_x)}
\end{equation*}
Let $T\in\, ]0,\infty]$, $r\in (1,\infty)$ and $p\in (1,\frac{dr}{2r-1})$.
Let the operator $\mathcal{C}$ be defined by
\begin{equation*}
\mathcal{C}f(t,\cdot)\doteq \int_0^t e^{(t-s)\Delta}f(s,\cdot)\dd s\text{,}
\end{equation*}
Then, $\mathcal{C}$ is a bounded operator from
$L^r(0,T;L^{p}_x)$ with values to $L^{2r}(0,T;L^{q}_x)$, where $1/q:=1/p-(2r-1)/dr$.
For every $t\in \RR_+$, notice that
\begin{equation*}
\big\|\int_0^t e^{(t-s)\Delta}f(s)\dd s\,\big\|_{L^q_x}
\leq \int_0^t \|\,K(t-s,\cdot)*f(s,\cdot)\,\|_{L^q_x}\dd s
\leq \int_0^t \|\,K(t-s)\|_{L^{\tilde{q}}_x}\|\,f(s)\,\|_{L^{p}_x}\dd s,
\end{equation*}
with $1/\tilde{q}+1/p=1/q+1$, that is $\tilde{q}'=dr/(2r-1)$. Recalling Remark <ref>, we get
\begin{equation*}
\big\|\int_0^t e^{(t-s)\Delta}f(s)\dd s\,\big\|_{L^q_x}
\leq
\int_0^t \frac{\quad\|\,f(s)\,\|_{L^{p}_x}}{\quad|t-s|^{\frac{2r-1}{2r}}}\dd s
\leq
\int_{\RR} \frac{\quad\|\,f(s)\,\|_{L^{p}_x}}
{\quad|t-s|^{\frac{2r-1}{2r}}}1_{(0,T)}(s)\dd s.
\end{equation*}
Since by Theorem <ref>
\begin{equation*}
|\cdot|^{-\frac{2r-1}{2r}}*\|f(\cdot)1_{(0,T)}(\cdot)\|_{L^p_x}\in L^{2r}_t,
\end{equation*}
then there exists $\tilde{C}>0$ such that
\begin{equation*}
\|\, \mathcal{C}f \,\|_{L^{2r}(0,T;L^{q}_x)}
\leq
\big\|\;|\cdot|^{-\frac{2r-1}{2r}}*\|f(\cdot)1_{(0,T)}(\cdot)\|_{L^p_x}\big\|_{L^{2r}_t}
\leq
\tilde{C} \|\,f\,\|_{L^{r}(0,T;L^{p}_x)}.
\end{equation*}
For the definition and the main properties of homogeneous Besov Spaces we refer to <cit.>. However let us briefly recall two results which characterize such spaces in relation to the heat kernel.
Let $s$ be a negative real number and $(p,r)\in [1,\infty]^2$. $u$ belongs to $\dot{B}_{p,r}^s(\RR^d)$
if and only if $e^{t\Delta}u$ belongs to $L^p_x$ for almost every $t\in \RR_+$ and
\begin{equation*}
t^{-\frac{s}{2}}\left\|e^{t\Delta}u\right\|_{L^p_x}\in L^r\Big(\RR_+;\frac{\dd t}{t}\Big).
\end{equation*}
Moreover, there exists a positive constant $C$ such that
\begin{equation*}
\frac{1}{C}\left\| u\right\|_{\dot{B}_{p,r}^s(\RR^d)} \leq
\left\|\left\|t^{-\frac{s}{2}}e^{t\Delta}u\right\|_{L^p_x}
\right\|_{L^r(\RR_+;\frac{\dd t}{t})}\leq
C \left\| u\right\|_{\dot{B}_{p,r}^s(\RR^d)}\text{.}
\end{equation*}
Then, imposing the index $s$ equal to $-\frac{2}{r}$, the following Corollary is satisfied:
Let $p\in [1,\infty]$ and $r\in [1,\infty)$. $u$ belongs to $\dot{B}_{p,r}^{-\frac{2}{r}}(\RR^d)$
if and only if $e^{t\Delta}u\in L^r_t L^p_x$.
Moreover, there exists a positive constant $C$ such that
\begin{equation*}
\frac{1}{C}\left\| u\right\|_{\dot{B}_{p,r}^{-\frac{1}{2r}}(\RR^d)} \leq
\left\|e^{t\Delta}u
\right\|_{L^r_tL^p_x} \leq
C \left\| u\right\|_{\dot{B}_{p,r}^{-\frac{1}{2r}}(\RR^d)}\text{.}
\end{equation*}
At last, let us state the following Theorem concerning embedding features of Besov spaces, which proof is in
<cit.> Proposition $2.20$.
Let $1\leq p_1\leq p_2 \leq \infty$ and $1\leq r_1\leq r_2\leq \infty$. Then for any real
number $s$, the space $\dot{B}_{p_1,r_1}^{s}(\RR^d)$ is continuously embedded in
$\dot{B}_{p_2,r_2}^{s-d\big(\frac{1}{p_1}-\frac{1}{p_2} \big) }(\RR^d)$.
§ EXISTENCE OF SOLUTIONS FOR SMOOTH INITIAL DATES
In this section, by Proposition <ref> and Theorem <ref>, we prove the existence of weak solutions for system (<ref>), assuming more regularity for the initial data. The proofs proceed in the same line of <cit.> and <cit.>, however the novelty is to consider also an extra-term $-\ee \Delta$, with $\ee>0$, in the transport equation.
This perturbation is motivated by the necessity to control the norm of the gradient of the approximate temperatures, even without a space-Lipschitz condition on the approximate velocity field. Obviously this control depends on $\ee$. Hence we consider the following approximation of (<ref>).
\begin{equation}\label{Navier_Stokes_system_eps}
\begin{cases}
\;\partial_t\theta + \Div\, (\theta u) - \ee \Delta u=0 & \RR_+ \times\RR^d,\\
\;\partial_t u + u\cdot \nabla u -\Div\, (\nu(\theta)\MM) +\nabla\Pi=0 & \RR_+ \times\RR^d,\\
\;\Div\, u = 0 & \RR_+ \times\RR^d,\\
\;(u,\,\theta)_{|t=0} = (\bar{u},\,\bar{\theta}) & \;\;\quad \quad\RR^d,\\
\end{cases}
\end{equation}
Since $\Div\,u=0$, we observe that the momentum equation of system (<ref>) can be reformulated as follows
\begin{equation*}
\begin{cases}
\partial_t u^h -\Delta u^h +\nabla^h \Pi = - u^d\,\partial_d u^h - u^h\cdot\nabla u^h + \Div \big\{(\nu (\theta)-1)\MM^h \big\} & \RR_+ \times\RR^d,\\
\partial_t u^d -\Delta u^d +\partial_d \Pi = - \nabla^h u^d \cdot u^h + u^d\Div^h u^h + \Div \big\{(\nu (\theta)-1)\MM^d \big\} & \RR_+ \times\RR^d,\\
\end{cases}
\end{equation*}
where $\MM^h:= \nabla u^h + \tr \nabla^h u\quad\text{and}\quad \MM^d:= \partial^d u + \nabla u^d$.
Firstly, let us prove the existence of weak solutions for system (<ref>).
Let $1<r<\infty$ and $p\in (1,dr/(2r-1))$. Suppose that $\bar{\theta}$ belongs to $L^\infty_x $ and $\bar{u}$ belongs to
$\BB_{p,r}^{d/p-1}\cap \BB_{p,r}^{d/p-1+\ee} $ with $\ee<\min\{1/(2r), 1-1/r, 2(d/p -2 + 1/r)\}$.
If (<ref>) holds, then there exists a global weak solution $(\theta, u, \Pi)$
of (<ref>) such that
\begin{equation*}
u \in L^{2r}_tL^{\frac{dr}{r-1}}_x, \quad\quad
\nabla u \in L^{2r}_tL^{\frac{dr}{2r-1}}_x
\cap L^{r}_tL^\frac{dr}{2(r-1)}_x,
\quad\text{and}\quad
\Pi\in L^{r}_t L^{\frac{dr}{2(r-1)}}_x.
\end{equation*}
Furthermore, the following inequalities are satisfied:
\begin{equation}\label{inequalities_statement_prop_smooth_dates}
\begin{aligned}
& \| \nabla u^h \|_{L^{r}_t L^{ \frac{dr}{2(r-1)}}_x}+
\| \nabla u^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_1\eta\text{,}\\
& \| \nabla u^d \|_{L^{r}_t L^{ \frac{dr}{2(r-1)}}_x}+
\|\nabla u^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\| u^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + C_3\\
&\|\Pi\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x}
\leq
\|\theta\|_{L^\infty_{t,x}}\leq \|\bar{\theta}\|_{L^{\infty}_x}.
\end{aligned}
\end{equation}
for some suitable positive constants $C_1$, $C_2$, $C_3$ and $C_4$ which are independent by $n$ and $\ee$.
First, recalling remark <ref>, we approximate system (<ref>) by a sequence of linear systems:
we impose $(\theta_0,u_0,\Pi_0)=(0,0,0)$ and we consider
\begin{equation}\label{Transport_equation_navier_stokes_approximate}
\begin{cases}
\partial_t\theta_{n+1} -\ee\Delta \theta_{n+1}+ \Div (\theta_{n+1} u_n)=0 & \RR_+ \times\RR^d,\\
\theta_{n|t=0} = \bar{\theta} & \quad\,\;\, \quad\RR^d,\\
\end{cases}
\end{equation}
\begin{equation}\label{Navier_Stokes_system_approximate}
\begin{cases}
\partial_t u_{n+1} -\Delta u_{n+1} +\nabla\Pi_{n+1}=g_{n+1}+\Div \big\{ (\nu (\theta_{n+1})-1)\MM_n \big\} & \RR_+ \times\RR^d,\\
\Div\, u_{n+1} = 0 & \RR_+ \times\RR^d,\\
u_{n+1|t=0} = \bar{u} &\;\;\quad \quad \RR^d,\\
\end{cases}
\end{equation}
for all $n\in \NN$, where $g_{n+1}$ is a $d$-dimensional vector field, defined by
\begin{equation}\label{def_gn}
\left(
\begin{matrix}
u_{n}^d\,\partial_d u^h_{n+1} +u_n^h\cdot\nabla u_n^h\\
\nabla^h u_n^d \cdot u_{n+1}^h - u_n^d\Div^h u_{n+1}^h
\end{matrix}
\right) =:
\left(
\begin{matrix}
\end{matrix}
\right).
\end{equation}
Moreover we denote by $\MM_n^h:= \nabla u_n^h + \tr \nabla^h u_n$ and by $\MM^d:= \partial_d u_n + \nabla u^d_n$. For all $n\in \NN$, the global existence of a
weak solution $(\theta_{n+1}, u_{n+1}, \Pi_{n+1})$ of (<ref>) and (<ref>) is proved by induction,
using Theorem <ref>. Thanks to such results, we have that
$u_{n+1}$ belongs to $L^{2r}_t L^{dr/(r-1)}_x$, $\nabla u_{n+1}$ belongs to $L^{2r}_t L^{dr/(2r-1)}_x\cap L^{r}_tL^{dr/(2r-2)}_x$,
$\theta_{n+1}$ to $L^{\infty}_{t,x}$ and $\Pi_{n+1}$ to $L^{r}_t L_x^{dr/(2r-2)}$.
Step 1: Estimates not dependent on $\ee$. First, the Maximal Principle for parabolic equation implies, $\| \theta_n\|_{L^\infty_{t,x}}\leq \|\bar{\theta}\|_{L^\infty_x}$,
for any positive integer $n$. We want to prove that
\begin{equation}\label{horizantal_inequality_apprx_system}
\begin{alignedat}{2}
& \| \nabla u_n^h \|_{L^{r}_t L^{ \frac{dr}{2(r-1)}}_x}+
\| \nabla u_n^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_n^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
&&\leq C_1\eta \text{,}\\
& \|\nabla u_n^d\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x}+
\|\nabla u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\| u_n^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
&&\leq C_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + C_3,
\end{alignedat}
\end{equation}
for any $n\in\NN$ and for some suitable positive constants $C_1$, $C_2$ and $C_3$. First we will show by induction that, if $\eta$ is small enough then
\begin{equation}\label{indcution_one}
\begin{aligned}
& \| \nabla u_n^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_n^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq \bar{C}_1\tilde{\eta}\text{,}\\
& \|\nabla u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\| u_n^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq \bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + \bar{C}_3
\end{aligned}
\end{equation}
for all $n\in \NN$ and for some appropriate positive constant $\bar{C}_1$, $\bar{C}_2$, $\bar{C}_3$, where $\tilde{\eta}\leq \eta$ is defined by
\begin{equation}\label{def_tildeeta}
\tilde{\eta}:=\big(
\|\nu-1\|_\infty + \| \bar{u}^h \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\big)
\exp
\big\{
\frac{c_r}{2}\|\bar{u}^d\|^{4r}_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\big\}.
\end{equation}
Let $\lambda$ be a positive real number, and let $u_{n+1,\lambda}$, $\nabla u_{n+1, \lambda}$ and $\Pi_{n+1, \lambda}$ be defined by
\begin{equation}\label{def_ulambda}
(u_{n+1, \lambda},\,\nabla u_{n+1, \lambda},\,\Pi_{n+1, \lambda} )(t):= h_{n,\lambda }(0,t)(u_{n+1},\,\nabla u_{n+1},\,\Pi_{n+1} )(t),
\end{equation}
where, for all $0 \leq s<t<\infty$,
\begin{equation}\label{def_h}
h_{n, \lambda}(s,t):=
\exp
\big\{
-\lambda \int_s^t \|u_n^d(\tau)\|^{2r}_{L_x^{\frac{dr}{r-1}}}\dd \tau\,
-\lambda \int_s^t \|\nabla u_n^d(\tau)\|^{2r}_{L_x^{\frac{dr}{2r-1}}}\dd \tau\,
\big\}.
\end{equation}
Writing $u_{n+1}$ by the Mild formulation, we get
\begin{equation}\label{def_u_n+1}
\begin{aligned}
u_{n+1}(t) =
\underbrace{
e^{t \Delta}\bar{u}}_{u_{L}}+
\underbrace{
\int_{0}^t e^{(t-s)\Delta}\PP g_{n+1}(s)\dd s}_{F^1_{n+1}(t)}+
\underbrace{
\int_{0}^t \nabla e^{(t-s)\Delta} R\cdot R\cdot \{(\nu (\theta_{n+1})-1)\MM_n\}(s)\dd s}_{F^2_{n+1}(t)} + \\ +
\underbrace{
\int_{0}^t \Div\, e^{(t-s)\Delta} \{(\nu (\theta_{n+1})-1)\MM_n\}(s)\dd s}_{F^3_{n+1}(t)},
\end{aligned}
\end{equation}
where $R:=\nabla/\sqrt{-\Delta}$ is the Riesz transform ($R\cdot := \Div/\sqrt{-\Delta}$) and $\PP := I+R\,R\cdot$ is the Leray projection operator, which are
bounded operators from $L^q_x$ to $L^q_x$ for any $q\in(1,\infty)$. Thus
\begin{equation}\label{def_un+1lambda}
\underbrace{
h_{n,\lambda}(0,t)u_L(t) }_{u_{L,\lambda}(t)}+
\underbrace{
\int_{0}^th_{n,\lambda}(s,t) e^{(t-s)\Delta}\PP g_{n+1,\lambda}(s)\dd s }_{F^{1}_{n+1,\lambda}(t)} +
\underbrace{
h_{n,\lambda}(0,t)F_2(t) }_{F^{2}_{n+1, \lambda}(t)}
\underbrace{
h_{n,\lambda}(0,t)F_3(t) }_{F^{3}_{n+1,\lambda}(t)},
\end{equation}
where $g_{n+1,\lambda}(t)=g_{n+1}(t)h_{n,\lambda}(0,t)$. First, we want to estimate $\nabla u_{n+1,\lambda}^h$ in $L^{2r}_t L^{ dr/(2r-1)}_x$ and $ u_{n+1,\lambda}^h$ in
$L^{2r}_t L^{ dr/(r-1)}_x$. We begin observing that, by Corollary <ref> and Theorem <ref>,
\begin{equation}\label{estitmate_1a}
\begin{aligned}
\| u_{L,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{r-1 }}_x} +
\| \nabla u_{L,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{2r-1 }}_x}
\leq
\| u_L^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}+
\| \nabla u_L^h \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\leq
\end{aligned}
\end{equation}
for a suitable positive constant $C$. Furhtermore, by the definition of $g_{n+1}$ and by Lemma <ref>, Lemma <ref>, Lemma <ref> and
Lemma <ref>, we obtain
\begin{equation}\label{estitmate_1b}
\begin{aligned}
\| F^{1,h}_{n+1,\lambda} \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x } &+
\|\nabla F^{1,h}_{n+1,\lambda} \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}\leq
\frac{1}{\,\lambda^{\frac{1}{4r}}} \|\, u_n^d\,\|^{\frac{1}{2}}_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\|\,\nabla u_{n+1,\lambda}^h\,\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+ \\&+
\|\,u_n^h\,\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\|\,\nabla u_n^h\,\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\frac{1}{\,\lambda^{\frac{1}{4r}}}
\|\,\nabla u_n^d\,\|^{\frac{1}{2}}_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\|\,u_{n+1,\lambda}^h\,\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\big\}.
\end{aligned}
\end{equation}
Furthermore, By Corollary <ref> and Theorem <ref> we also obtain
\begin{equation}\label{estitmate_1c}
\begin{aligned}
\| F^{2,h}_{n+1,\lambda} + F^{3,h}_{n+1,\lambda} &\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\|
\int_{0}^t \Delta e^{(t-s)\Delta}\PP R\cdot (\sqrt{-\Delta})^{-1}\{(\nu (\theta_{n+1})-1)
\MM_{n}\}(s)\dd s\,
\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\\
(\sqrt{-\Delta})^{-1}(\nu (\theta_{n+1})-1)
\MM_{n}
\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\leq
C\|\,\nu-1\,\|_{\infty} \|\,\nabla u_{n}\,\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}.
\end{aligned}
\end{equation}
Similarly, recalling Theorem <ref>, we deduce that
\begin{equation}\label{estitmate_1d}
\begin{aligned}
\| \nabla F^{2,h}_{n+1,\lambda}+ \nabla F^{3,h}_{n+1,\lambda} \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}
\|\,
\int_0^t \Delta e^{(t-s)\Delta}R \,\PP\, R\cdot
\{(\nu(\theta_{n+1})-1)\MM_{n}\}(s)\dd s\,
\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}\\
\|\{(\nu(\theta_{n+1})-1)\MM_{n}\}
\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}\leq
C\|\nu -1\|_\infty\|\,\nabla u_{n}\,\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}.
\end{aligned}
\end{equation}
Summarizing (<ref>), (<ref>), (<ref>) and (<ref>), we deduce that there exists a positive constant $C$ such that,
for all $n\in \NN$
\begin{equation}\label{estimate_1}
\begin{aligned}
\|\nabla u_{n+1,\lambda}^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_{n+1,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq
C\big\{ \|\,\bar{u}^h\,\|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} +
\frac{1}{\,\lambda^{\frac{1}{4r}}}
\big(
\|\, u_n^d\,\|^{\frac{1}{2}}_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\|\,\nabla u_{n+1,\lambda}^h\,\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +\\ +
\|\,\nabla u_n^d\,\|^{\frac{1}{2}}_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\|\,u_{n+1,\lambda}^h\,\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\big)
+\|\,u_n^h\,\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\|\,\nabla u_n^h\,\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
+\|\nu -1\|_\infty\|\,\nabla u_{n}\,\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}\big\}.
\end{aligned}
\end{equation}
Recalling the induction hypotheses (<ref>), we fix a positive $\lambda$ such that
\begin{equation}\label{inequality_lambda1}
C \frac{1}{\,\lambda^{\frac{1}{4r}}}
\Big(\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}} + \bar{C}_3\Big)^{\frac{1}{2}}=
\frac{1}{4}\quad
\text{ namely } \lambda = (4C)^{4r}(\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} +
\bar{C}_3)^{2r}\,),
\end{equation}
so that we can absorb all the terms on the right-hands side with index $n+1$ by the left-hand side, obtaining
\begin{equation}\label{first_estimate}
\begin{aligned}
\| \nabla u_{n+1,\lambda}^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}&+
\| u_{n+1,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq \\
\| \bar{u}^h \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+
\bar{C}_1^2\tilde{\eta}^2 +
\|\nu-1\|_{L^\infty_x}( \bar{C}_1\tilde{\eta} +\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + \bar{C}_3 )
\end{aligned}
\end{equation}
thanks to the induction hypotheses (<ref>).
Now we reformulate (<ref>) without the index $\lambda$ on the left-hand side:
\begin{align*}
\| \nabla u_{n+1}^h& \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x} +
\| u_{n+1}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq
\sup_{t\in\RR_+}h_{n,\lambda}(0,t)^{-1}
\big(\,
\| \nabla u_{n+1,\lambda}^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_{n+1,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\,
\big)\\
\exp
\big\{ \lambda
\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}+\bar{C}_3
\big\}
\big(\,
\| \nabla u_{n+1,\lambda}^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_{n+1,\lambda}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\big),
\end{align*}
thanks to the second inequality of (<ref>). Hence, recalling (<ref>) and (<ref>), we obtain the following inequality
\begin{align*}
\| \nabla u_{n+1}^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x} +
\| u_{n+1}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq
\big\{
\bar{C}_2^{4r}\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}^{4r}+\bar{C}_3^{4r}
\big\}{\scriptstyle \times} \\ {\scriptstyle \times}
\| \bar{u}^h \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+
\bar{C}_1^2\tilde{\eta}^2 +
\|\nu-1\|_{L^\infty_x}( \bar{C}_1\tilde{\eta} +\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + \bar{C}_3 )
\end{align*}
Assuming that $c_r$ of (<ref>) fulfills $c_r\geq 1$ and $c_r/4\geq 2^{4r-1}(4C)^{4r} \bar{C}_2^{4r}$, we get that the right-hand side of the previous
inequality is bounded by
\begin{align*}
2C \exp
\big\{
C^{4r}_3 +
\frac{c_r}{4}\| \bar{u}^d \|^{4r} _{\dot{B}_{p,r}^{\frac{d}{p}-1}}
\big\}
\| \bar{u}^h \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+
\bar{C}_1^2\tilde{\eta}^2 +
\|\nu-1\|_{L^\infty_x}( \bar{C}_1\tilde{\eta} +\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} + \bar{C}_3 )
\leq
2C \exp
\left\{
\right\}
( 1+(\bar{C}_1^2+\bar{C}_1)\tilde{\eta} + \bar{C}_2 + \bar{C}_3 )\tilde{\eta},
\end{align*}
where we have used
$ \|\nu-1\|_{\infty}\|\bar{u}^d\|_{\dot{B}_{p,r}^{d/p-1}}
\leq
\|\nu-1\|_{\infty}
\exp
\{
\|\bar{u}^d\|_{\dot{B}_{p,r}^{d/p-1}}^{4r}/(4r)
\}
Imposing $\bar{C_1}$ big enough and $\eta$ small enough in order to have
\begin{equation*}
\exp
\big\{
\bar{C}^{4r} _3
\big\}2C(1+\bar{C}_2+\bar{C}_3) < \frac{\bar{C}_1}{2}
\quad\text{and}\quad
\exp
\big\{
\bar{C}^{4r} _3
\big\}
(\bar{C}_1+1)\tilde{\eta}\leq \frac{1}{2},
\end{equation*}
we finally obtain that the first equation of (<ref>) is true for any $n\in\NN$. Now we deal with the second equation of (<ref>) and we still proceed
by induction. Recalling (<ref>)and proceeding in a similarly way as done in the previous estimates, the following inequality is satisfied:
\begin{align*}
\|\nabla u_{n+1}^d &\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\| u_{n+1}^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\leq
C\big\{ \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{1}{p}}} +
\|g_{n+1}\|_{L^{r}_tL^{\frac{dr}{3r-2}}_x}+
\|\nu-1\|_{\infty}\|\nabla u_n \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}
\big\},
\end{align*}
for a suitable positive constant $C$. Hence, by the definition (<ref>) of $g_{n+1}$, we deduce that
\begin{align*}
&\|\nabla u_{n+1}^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+\| u_{n+1}^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\leq
\|\bar{u}^d\|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\|u_n^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\|\nabla u_n^h\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+
\|u_{n+1}^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\\&{\scriptstyle \times}
\|\nabla u_{n}^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
+\|u_{n}^d\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\|\nabla u_{n+1}^h\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+
\|\nu-1\|_{\infty}
\big(
\|\nabla u_n^h \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x} +
\|\nabla u_n^d \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}
\big)
\big\},
\end{align*}
so that, thanks to the induction hypotheses and the previous estimates, we bound the right hand-side by
\begin{equation*}
C + \bar{C}_1\bar{C}_2\tilde{\eta} + \|\nu-1\|_{\infty}\bar{C}_2\,
)\|\bar{u}^d\|_{\dot{B}_{p,r}^{\frac{d}{p}-1}} +
\bar{C}_1\bar{C}_3 + \bar{C}_1^2\tilde{\eta} +
\|\nu-1\|_{\infty}
\bar{C}_1 +\bar{C}_2
\end{equation*}
Finally, imposing $C<\bar{C}_2$ and $\eta$ small enough in order to fulfill $C + (\,\bar{C}_1\bar{C}_2\,+\,\bar{C}_2\,) \eta \leq \bar{C}_2$ and moreover
$( \bar{C}_1\bar{C}_3 + \bar{C}_1^2\eta + \eta( \bar{C}_1 +\bar{C}_2 \big) )\eta \leq \bar{C}_3$, then the second inequality of (<ref>) is
satisfied for any $n\in\NN$. Now, let us prove by induction that there exist three positive constants $\tilde{C}_1$, $\tilde{C}_2$ and $\tilde{C}_3$, such that
\begin{equation}\label{Finalestimate2r-2}
\|\nabla u_{n+1}^h\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq \tilde{C}_1 \eta\quad
\text{and}\quad
\|\nabla u_{n+1}^d\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq
\tilde{C}_2\|\bar{u}^d\|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+\tilde{C}_3,
\end{equation}
for any positive integer $n$. Recalling the mild formulation (<ref>) of $u_{n+1}$, Lemma (<ref>),
Corollary (<ref>) and Theorem (<ref>), it turns out that
\begin{equation}\label{uLh2r-1}
\|\nabla u_L^h \|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} +
\|\nabla F^{1,h}_{n+1} \|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq
\|\bar{u}^h\|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+
\|g_{n+1}\|_{L^{r}_t L^{\frac{dr}{3r-2}}_x}
\big),
\end{equation}
while Theorem <ref> implies
\begin{equation*}\label{F22r-1}
\| \nabla F^{2,h}_{n+1} + \nabla F^{3,h}_{n+1} \|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq \|\nu-1\|_{\infty}\|\nabla u_n\|_{L^r_tL^{\frac{dr}{2(r-1)}}_x}.
\end{equation*}
By the definition of $g_{n+1}$ (<ref>), its $L^r_t L^{dr/(3r-2)}_x$-norm is bounded by
\begin{equation*}
\|u_n^d \|_{L^{2r}_tL^{\frac{dr}{r-1}}_x} \|\nabla u_{n+1}^h \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}+
\|u_n^h \|_{L^{2r}_tL^{\frac{dr}{r-1}}_x} \|\nabla u_{n}^h \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}+
\|u_{n+1}^h \|_{L^{2r}_tL^{\frac{dr}{r-1}}_x} \|\nabla u_{n}^d \|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}
\end{equation*}
Hence, thanks to the uniform estimates given by (<ref>), we obtain
\begin{equation}\label{estimate_g}
\|g_{n+1}\|_{L^{r}_t L^{\frac{dr}{3r-2}}_x}
\leq
\big( \bar{C}_1\tilde{\eta}+\bar{C}_3 \big)\bar{C}_1\tilde{\eta}+ \bar{C}_1 \bar{C}_2
\|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}\tilde{\eta}\leq
\big(
\bar{C}_1\tilde{\eta}+\bar{C}_3+\bar{C}_2
\big)\bar{C}_1 \eta,
\end{equation}
Furthermore, by the induction hypotheses (<ref>), we remark that
\begin{equation}\label{recall_2}
\|\nu-1\|_{\infty}\|\nabla u_n\|_{L^r_t L^{\frac{dr}{2(r-1)}}_x} \leq
\|\nu-1\|_{\infty}\tilde{C}_1 \eta
\|\nu-1\|_{\infty}\tilde{C}_3.
\end{equation}
Those, summarizing (<ref>), (<ref>), (<ref>) and (<ref>), we finally obtain
\begin{equation}\label{induction_2r-1a}
\|\nabla u_{n+1}^h\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq
\|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}+
\big(
\bar{C}_1\tilde{\eta}+\bar{C}_3+ \bar{C}_2
\big)\bar{C}_1 \eta+
\|\nu-1\|_{\infty}\tilde{C}_1 \eta+
\tilde{C}_2\tilde{\eta}+
\|\nu-1\|_{\infty}\tilde{C}_3
\big\},
\end{equation}
hence, imposing $\tilde{C}_1> C\big(1 + \bar{C}_1\bar{C}_3+\bar{C}_1 \bar{C}_2 +\tilde{C}_2+\tilde{C}_3\big)$
and assuming $\eta$ small enough, we get that the first inequality of (<ref>) is true for any positive integer $n$.
Now, proceeding as to prove (<ref>), we get
\begin{equation*}
\|\nabla u_{n+1}^d\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq
\|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\big(
\bar{C}_1\tilde{\eta}+\bar{C}_3+\bar{C}_2
\big)\bar{C}_1 \eta
\|\nu-1\|_{\infty}\tilde{C}_1 \eta
\|\nu-1\|_{\infty}\tilde{C}_3.
\big\}
\end{equation*}
Hence, imposing $\tilde{C}_2> C$, $\tilde{C}_3>0$ such that
$C\{( \bar{C}_1\tilde{\eta}+\bar{C}_3+\bar{C}_2)\bar{C}_1 \eta+\|\nu-1\|_{\infty}\tilde{C}_1 \eta +\tilde{C}_2\tilde{\eta}\}< \tilde{C}_3$ and assuming
$\eta$ small enough, we finally establish that also the second inequality of (<ref>) is true for any $n\in\NN$. To conclude this first step,
denoting $C_1:=\bar{C}_1+\tilde{C}_1$, $ C_2:= \bar{C}_2 + \tilde{C}_2$, $C_3:=\bar{C}_3 + \tilde{C}_3$ and summarizing
(<ref>) and (<ref>), we finally obtain (<ref>). To conclude this first step we observe that
$\Pi_{n+1}$ is determined by
\begin{equation}\label{def_Pi_n}
\Pi_{n+1} :=
-\left(-\Delta\right)^{-\frac{1}{2}}R\cdot g_{n+1}
-R\cdot R\cdot \{(\nu(\theta_{n+1})-1)\nabla u_n \},
\end{equation}
so that, thanks to Corollary <ref> and (<ref>), we deduce that
\begin{equation}\label{estimate_Pin+1}
\| \Pi_{n+1} \|_{L^r_t L^{\frac{Nr}{2(r-1)}}}
\leq
\| g_{n+1} \|_{L^r_t L^{\frac{Nr}{3r-2)}}_x}+
\|\nu -1\|_{L^\infty_x}\| \nabla u_n \|_{L^r_t L^{\frac{Nr}{2(r-1)}}_x} )
\leq C_4 \eta,
\end{equation}
for any $n\in\NN$ and for a suitable positive constant $C_4$.
Step 2: $\ee$-Dependent Estimates. As second step, we are going to establish some $\ee$-dependent estimates which are useful for the third step, where we will
prove that $(\theta^n,\, u^n,\, \Pi^n)$ is a Cauchy sequence in a suitable space. Defining $\bar{r}:= 2r/(2-\ee r)>r$, then
we still have $ p<d\bar{r}/(2\bar{r}-1)=2dr/((4+\ee)r-2)$, since $\ee$ is bounded by $2(d/p -2+ 1/r)$. Since
$\BB_{p,r}^{d/p-1}\hookrightarrow \BB_{p,\bar{r}}^{d/p-1}$, then there exists a positive constant $C$ such that
\begin{equation*}
\bar{\eta}:= \big(
\| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,\bar{r}}^{\frac{d}{p}-1}}
\big)
\exp\Big\{ c_r \|\bar{u}^d\|_{\dot{B}_{p,\bar{r}}^{\frac{d}{p}-1}}^{4r} \Big\}
\leq C\eta.
\end{equation*}
Hence, arguing exactly as to prove (<ref>) with $\bar{r}$ instead of $r$, we get also
\begin{equation}\label{inequality_apprx2}
\begin{alignedat}{2}
& \| \nabla u_n^h \|_{L^{2\bar{r}}_t L^{ \frac{d\bar{r}}{2\bar{r}-1}}_x}+
\| u_n^h \|_{L^{2\bar{r}}_t L^{\frac{d\bar{r}}{\bar{r}-1}}_x}
&&\leq C_1\bar{\eta} \text{,}\\
& \|\nabla u_n^d\|_{L^{2\bar{r}}_t L^{\frac{dr}{2\bar{r}-1}}_x} +
\| u_n^d \|_{L^{2\bar{r}}_t L^{\frac{d\bar{r}}{\bar{r}-1}}_x}
&&\leq C_2\| \bar{u}^d \|_{\dot{B}_{p,\bar{r}}^{\frac{d}{p}-1}} + C_3.
\end{alignedat}
\end{equation}
First, we want to demonstrate by induction that there exists a positive constant $\bar{C}_5$ such that
\begin{equation}\label{induction_ee}
\| u_n \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla u_n \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\leq
\bar{C}_5 \| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\ee }}
\end{equation}
Let us remark that such spaces are well defined, since $\bar{r}(1-\ee)-1>0$ (from $\ee<1-1/r<1-1/\bar{r}$).
Recalling the mild formulation of $u_{n+1}$ (<ref>), Corollary <ref> and Theorem <ref>
\begin{equation*}
\| u_L \|_{ L^{2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla u_L \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\leq
C \| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\ee } },
\end{equation*}
for a suitable positive constant $C$. Moreover, thanks to Lemma <ref> and Lemma <ref>, we get
\begin{equation*}
\| F_{n+1}^1 \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla F_{n+1}^1 \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\leq
\| g_{n+1} \|_{L^{\bar{r}}_t L^{\frac{d\bar{r}}{(3-\ee)\bar{r}-2}}_x}.
\end{equation*}
From the definition of $g_{n+1}$ (<ref>) and the estimates (<ref>), we get
\begin{equation*}
\begin{aligned}
\| g_{n+1} \|_{L^{\bar{r}}_t L^{\frac{d\bar{r}}{(3-\ee)\bar{r}-2}}_x}
\leq
\big(\,
\| u_n^d \|_{ L^{2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} & +
\| u_{n+1}^d \|_{ L^{2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} + \\&+
\| \nabla u_n^d \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x} +
\| \nabla u_{n+1}^d \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\big),
\end{aligned}
\end{equation*}
so that, by the induction hypotheses (<ref>), we have the following bound
\begin{equation*}
\| g_{n+1} \|_{L^{\bar{r}}_t L^{\frac{d\bar{r}}{(3-\ee)\bar{r}-2}}_x}
\leq
\big(\,
\| u_{n+1} \|_{ L^{2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla u_{n+1} \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\big)
\| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\ee} }.
\end{equation*}
Moreover, thanks to Lemma <ref> and Theorem <ref>, we get
\begin{equation*}
\begin{aligned}
\| F_{n+1}^2 +F_{n+1}^3 \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla F_{n+1}^2 + \nabla F_{n+1}^3 \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\leq
\| \nu - 1 \|_{L^\infty_x}
\| \nabla u^n \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}.
\end{aligned}
\end{equation*}
Summarizing the previous estimates and absorbing the terms with indexes $n+1$ on the right side by the left-hand side, we get that there exists a positive constant $C$
such that
\begin{align*}
\| u_{n+1} \|_{ L^{2\bar{r} }_t L^{ \frac{d\bar{r}}{\bar{r}(1-\ee)-1} }_x} +
\| \nabla u_{n+1} \|_{ L^{ 2\bar{r} }_t L^{ \frac{d\bar{r}}{(2-\ee)\bar{r}-1} }_x}
\leq
(C(1+C_1\bar{\eta})+\bar{C}_5C_1\bar{\eta})\| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1+\ee} },
\end{align*}
thus (<ref>) is true for any positive integer $n$, assuming $\bar{C}_5>2C$ and $\bar{\eta}$ small enough. Now recalling that $\bar{r}=2r/(2-\ee r)$,
(<ref>) can be reformulated by
\begin{equation}\label{estimate_nee}
\| u_n \|_{ L^{ \frac{4r}{2-\ee r} }_t L^{ \frac{2dr}{(2-\ee)r-2} }_x} +
\| \nabla u_n \|_{ L^{ \frac{4r}{2-\ee r} }_t L^{ \frac{2dr}{(4-\ee)r-2} }_x}
\leq
\bar{C}_5\| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\ee }}\leq
C_5\| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\ee }},
\end{equation}
for a suitable positive constant $C_5$.
Now we want to prove the existence of a positive constant $C_6$ such that
\begin{equation}\label{induction_ee/2}
\| u_n \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{r-1} }_x} +
\| u_n \|_{ L^{2r }_t L^{ \frac{2dr}{(2-\ee)r-2} }_x} +
\| \nabla u_n \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x}+
\| \nabla u_n \|_{ L^{ 2r }_t L^{ \frac{2dr}{(4-\ee)r-2} }_x} \leq
C_6 \| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1+\frac{\ee}{2}} }.
\end{equation}
Let us remark that such spaces are well defined, since $2-\ee r>0$ (from $\ee< 2/r$) and $(2-\ee)r-2>0$ (from $\ee/2<\ee<1-1/r$).
Proceeding exactly as for proving (<ref>), with $r$ instead of $\bar{r}$ and $\ee/2$ instead of $\ee$, we get
\begin{equation}\label{estimate_ee/2_part1}
\| u_n \|_{ L^{ 2r }_t L^{ \frac{2dr}{(2-\ee)r-2} }_x} +
\| \nabla u_n \|_{ L^{ 2r }_t L^{ \frac{2dr}{(4-\ee)r-2} }_x}
\leq
\bar{C}_6 \| \bar{u} \|_{ \BB_{p,\bar{r}}^{\frac{d}{p}-1+\frac{\ee}{2} }},
\end{equation}
for a suitable positive constant $\bar{C}_6$. Furthermore, recalling the mild formulation of $u_{n+1}$ (<ref>), Corollary
<ref> and Theorem <ref> implies
\begin{equation*}
\| u_L \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{r-1} }_x} +
\| \nabla u_L \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x} +
\leq
C \| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1+\frac{\ee}{2}} },
\end{equation*}
for a suitable positive constant $C$. Thanks to Lemma <ref> and Lemma <ref>, we obtain
\begin{equation*}
\begin{aligned}
\| F_{n+1}^1 \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{r-1} }_x} +
\| \nabla F_{n+1}^1 \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x} +
\leq
C\| g_{n+1} \|_{L^{\frac{2r}{2-\ee r}}_t L^{\frac{dr}{3r-2}}_x}.
\end{aligned}
\end{equation*}
From the definition of $g_{n+1}$ (<ref>) and the estimates (<ref>), we get that
\begin{equation*}
\begin{aligned}
\| g_{n+1} \|_{L^{\frac{r}{1-\ee r}}_t L^{\frac{dr}{3r-2}}_x} \leq
\big(\,
\| u_n^d \|_{L^{\frac{4r}{2-\ee r}}_t L^{\frac{dr}{r-1}}_x} & +
\| u_{n+1}^d \|_{L^{\frac{4r}{2-\ee r}}_t L^{\frac{dr}{r-1}}_x} + \\&+
\| \nabla u_n^d \|_{L^{\frac{4r}{2-\ee r}}_t L^{ \frac{dr}{2r-1}}_x}+
\| \nabla u_{n+1}^d \|_{L^{\frac{4r}{2-\ee r}}_t L^{ \frac{dr}{2r-1}}_x}
\big),
\end{aligned}
\end{equation*}
so that, by the induction hypotheses of (<ref>), we have the following bound
\begin{equation*}
\| g_{n+1} \|_{L^{\frac{r}{1-\ee r}}_t L^{\frac{dr}{3r-2}}_x} \leq
\big(\,
\| u_{n+1} \|_{L^{\frac{4r}{2-\ee r}}_t L^{\frac{dr}{r-1}}_x } +
\| \nabla u_{n+1} \|_{L^{\frac{4r}{2-\ee r}}_t L^{ \frac{dr}{2r-1}}_x }
\big)
\| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1+\frac{\ee}{2}} }.
\end{equation*}
Finally, thanks to Lemma <ref> and <ref>, we get
\begin{equation*}
\begin{aligned}
\| F_{n+1}^2 + F_{n+1}^3 \|_{ L^{\frac{4r}{2-\ee r} }_t L^{\frac{dr}{r-1} }_x} +
\| \nabla F_{n+1}^2 + \nabla F_{n+1}^3 \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x}
\leq
\| \nu - 1 \|_{L^\infty}
\| \nabla u^n \|_{ L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x}.
\end{aligned}
\end{equation*}
Summarizing the previous estimates and absorbing the terms with indexes $n+1$ on the right side by the left-hand side, we get that there exists a positive constant
$C$ such that
\begin{equation}\label{estimate_ee/2_2part}
\| u_{n+1} \|_{L^{\frac{4r}{2-\ee r} }_t L^{\frac{dr}{r-1} }_x}+
\| \nabla u_{n+1} \|_{L^{\frac{4r}{2-\ee r} }_t L^{ \frac{dr}{2r-1} }_x}
\leq
(C(1+C_1\bar{\eta})+C_6C_1\bar{\eta})\| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1+\frac{\ee}{2}} }.
\end{equation}
Thus, recalling (<ref>) and (<ref>), we get that (<ref>) is true for any $n\in\NN$, with $C_6>\bar{C}_6+2C$
and $\eta$ small enough.
Step 3. Convergence of the Series.
We denote by $ \delta u_n:= u_{n+1}-u_{n}$ by $\delta \nu_n:= \nu(\theta_{n+1})-\nu(\theta_{n})$ and by $\delta \theta_n:= \theta_{n+1}-\theta_{n}$, for every
positive integer $n$. Moreover, fixing $\lambda>0$, we define
\begin{equation*}
\begin{aligned}
\delta U_{n,\lambda}(T):=
\|\delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
&+\|\delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}\\&
+\|\nabla \delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
+\|\nabla \delta u_{n,\lambda}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)},
\end{aligned}
\end{equation*}
where, recalling (<ref>), $\delta u_{n,\lambda}(t):=\delta u_n(t)h_{n,\lambda}(0,t)$. We want to prove that the series $\sum_{n\in \NN}\delta U_{n}(T)$ is finite.
Denoting by $\delta g_{n}:=g_{n+1}-g_{n}$, $\delta \MM_n:=\MM_{n+1}-\MM_n$, then, thanks to the equality (<ref>), we can formulate
$\delta u_{n,\lambda }=
f_{n,1}+f_{n,2}+f_{n,3}$, where
\begin{equation}\label{formulation_delta_u_n}
\begin{aligned}
f_{n,1}&:= h_{n,\lambda}(0,t)\int_{0}^t e^{(t-s)\Delta}\PP\delta g_{n}(s)\dd s ,\\
f_{n,2}&:=h_{n,\lambda}(0,t)\int_{0}^t\big[ \nabla e^{(t-s)\Delta} R\cdot R\cdot \{( \nu(\theta_n)-1) \delta \MM_{n-1}\}+
\Div\, e^{(t-s)\Delta} \{( \nu(\theta_n)-1) \delta \MM_{n-1}\}\big](s)\dd s,\\
f_{n,3}&:=h_{n,\lambda}(0,t)\big(\int_{0}^t \nabla e^{(t-s)\Delta} R\cdot R\cdot \{\delta \nu_{n}\MM_n\}(s)\dd s
+h_{n,\lambda}(0,t)\int_{0}^t \Div\, e^{(t-s)\Delta} \{\delta \nu_{n}\MM_n\}(s)\dd s\big).
\end{aligned}
\end{equation}
At first step let us estimate
\begin{equation*}
\|f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)},
\end{equation*}
Observing that
\begin{equation*}
\delta g_{n}=
\left(\,
\begin{matrix}
u_n^d \partial_d \delta u_n^h + \delta u_n^d \partial_d u_n^h+
\delta u_{n-1}^h\cdot \nabla u_n^h + u_{n-1}^h \cdot \nabla \delta u_{n-1}^h\\
\\
\nabla^h u_n^d\cdot\delta u_n^h+\nabla^h\delta u_n^d\cdot u_n^h-
u_n^d \Div^h\delta u_n^h-\delta u_{n-1}^d \Div^h u_n^h\\
\end{matrix}\,
\right),
\end{equation*}
then, by Lemma <ref> and Lemma <ref>, we obtain
\begin{align*}
\|f_{n,1}&\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}\leq\\
\frac{1}{\lambda^{\frac{1}{4r}}}
\Big(
\| u_n^d\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}^{\frac{1}{2}}
\| \partial_d \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\|\nabla^h u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}^\frac{1}{2}
\|\delta u_{n,\lambda}^h\|_{L^{2r}(0,T; L^{\frac{dr}{r-1}}_x)}+\\
\|u_n^d\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}^\frac{1}{2}
\|\nabla^h \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}\,
\Big)
\|\delta u_{n,\lambda}^d \|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
\| \partial_d u_n^h\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}+\\
\|\delta u_{n-1,\lambda}^h \|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
\|\nabla u_n^h \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\|u_{n-1}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\| \nabla \delta u_{n-1,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+\\
\|\nabla^h\delta u_{n,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\|u_n^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\|\delta u_{n-1,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
\|\nabla^h u_n^h\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\bigg\}.
\end{align*}
which yields, by (<ref>) and (<ref>)
\begin{align*}
\|f_{n,1}&\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\leq
\frac{1}{4}
\| \nabla \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
C\bar{C}_1\tilde{\eta} \|\delta u_{n,\lambda}^d \|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}+\\
\Big(
\|\delta u_{n-1,\lambda}^h \|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
\| \nabla \delta u_{n-1}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\Big)
\frac{1}{4}\|\delta u_{n,\lambda}^h\|_{L^{2r}(0,T; L^{\frac{dr}{r-1}}_x)}
C\bar{C}_1\tilde{\eta}\|\nabla^h\delta u_{n,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+
\frac{1}{4}
\|\nabla^h \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\|\delta u_{n-1,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}.
\end{align*}
Assuming $\eta$ small enough, the previous inequality yields
\begin{equation}\label{estimate_f_1}
\begin{aligned}
\|f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}&+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}\leq
\frac{1}{4}
\big\{
\|\delta u_{n,\lambda}\|_{L^{2r}(0,T; L^{\frac{dr}{r-1}}_x)}+\\
\|\delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}+
\|\nabla \delta u_{n,\lambda}\|_{L^{2r}(0,T; L^{\frac{dr}{2r-1}}_x)}+
\|\nabla \delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
\big\}
\end{aligned}
\end{equation}
Now, let us estimate $f_{n,1}$ and $\nabla f_{n,1}$ in $L^{2r}(0,T; L^{2dr/((2-\ee)r-2)}_x)$ and $L^{2r}(0,T; L^{2dr/((4-\ee)r-2)}_x)$ respectively.
Thanks to Lemma <ref> and <ref>, the following inequality is satisfied:
\begin{align*}
\|&f_{n,1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\leq\\
\frac{1}{\lambda^{\frac{1}{4r}}}
\Big(
\| u_n^d\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}^{\frac{1}{2}}
\| \partial_d \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\|\nabla^h u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}^\frac{1}{2}
\|\delta u_{n,\lambda}^h\|_{L^{2r}(0,T; L^{\frac{2dr}{(2-\ee)r-2}}_x)}+\\
\|u_n^d\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}^\frac{1}{2}
\|\nabla^h \delta u_{n,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\,
\Big)
\|\delta u_{n,\lambda}^d \|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}
\| \partial_d u_n^h\|_{L^{2r}_tL^{\frac{dr}{2r-1}}_x}+\\
\|\delta u_{n-1,\lambda}^h \|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}
\|\nabla u_n^h \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\|u_{n-1}^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\| \nabla \delta u_{n-1,\lambda}^h\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}+\\
\|\nabla^h\delta u_{n,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\|u_n^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\|\delta u_{n-1,\lambda}^d\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}
\|\nabla^h u_n^h\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}
\bigg\}.
\end{align*}
Hence, (<ref>), (<ref>) and the smallness condition on $\eta$ imply that
\begin{equation}\label{estimate2_f_1}
\begin{aligned}
\|f_{n,1}&\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\leq
\frac{1}{4}
\big\{
\|\delta u_{n,\lambda}\|_{L^{2r}(0,T; L^{\frac{2dr}{(2-\ee)r-2}}_x)}+\\
\|\delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}+
\|\nabla \delta u_{n,\lambda}\|_{L^{2r}(0,T; L^{\frac{2dr}{(4-\ee)r-2}}_x)}+
\|\nabla \delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\big\}
\end{aligned}.
\end{equation}
Thus, summarizing (<ref>) and (<ref>), we obtain
\begin{equation}\label{estimates_f_1}
\begin{aligned}
\|f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}\\&
+\|\nabla f_{n,1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\leq
\frac{1}{4}\delta U_{n,\lambda}(T)+\frac{1}{4}\delta U_{n-1,\lambda}(T).
\end{aligned}
\end{equation}
Now, we want to estimate $f_{n,2}$ in $L^{2r}(0,T;L^{dr/(r-1)}_x)\cap L^{2r}(0,T;L^{2dr/((2-\ee)r-2)}_x)$ and moreover $\nabla f_{n,2}$ in
$L^{2r}(0,T;L^{dr/(2r-1)}_x)\cap L^{2r}(0,T;L^{2dr/((4-\ee)r-2)}_x)$. From Lemma <ref> and Theorem <ref> we obtain
\begin{align*}
\|f_{n,2}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
+\|\nabla f_{n,2}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
+\|\nabla f_{n,2}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\leq \\
\Big(
\|\nabla \delta u_{n-1}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+
\|\nabla \delta u_{n-1}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\Big)\\
&\leq \tilde{C}_r\eta
\Big(
\|\nabla \delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+
\|\nabla \delta u_{n-1,\lambda}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\Big),
\end{align*}
hence, we deduce that
\begin{equation}\label{estimates_f_2}
\begin{aligned}
\|f_{n,2}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
&+\|\nabla f_{n,2}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+\\&
+\|\nabla f_{n,2}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\leq
\bar{C}_r\eta \delta U_{n-1,\lambda}(T).
\end{aligned}
\end{equation}
Now we deal with $f_{n,3}$ and $\nabla f_{n,3}$. At first, since $v\in C^{\infty}(\RR)$ and
$\|\theta_{n}\|_{L^\infty_{t,x}}\leq \|\bar{\theta}\|_{L^{\infty}_x}$, then there exists
$\tilde{c}>0$ (dependent on $\|\bar{\theta}\|_{L^{\infty}_x)}$) such that
$\|\delta \nu_n(t)\|_{L^{^\infty}_x}\leq \tilde{c}\|\delta \theta_n(t)\|_{L^{^\infty}_x}$, for almost every $t\in \RR_+$.
Moreover, by Lemma <ref> and Theorem <ref>, we have
\begin{align*}
\|&f_{n,3}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
+\|\nabla f_{n,3}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}+\\&
+\|\nabla f_{n,2}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)}\leq
\| \delta \nu_n \MM_n \|_{L^{2r}(0,T; L^{ \frac{dr}{2r-1} }_x)}+
\| \delta \nu_n \MM_n \|_{L^{2r}(0,T; L^{\frac{2dr}{(4-\ee)r-2}}_x)}
\big\}.
\end{align*}
Thus, recalling (<ref>) and (<ref>), we finally obtain
\begin{equation}\label{estimate_f_n3_partA}
\begin{aligned}
\|f_{n,3}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
+\|\nabla f_{n,3}\|_{L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)}
+\|\nabla f_{n,3}\|_{L^{2r}(0,T;L^{\frac{2dr}{(4-\ee)r-2}}_x)} \\
\leq
2C\tilde{c}\| \delta \theta_n\|_{L^{\frac{4}{\ee}}(0,T;L^{\infty}_x)}
\big\{
\| \nabla u_n \|_{L^{\frac{4r}{2-\ee r}}_t L^{ \frac{dr}{2r-1} }_x}+
\| \nabla u_n \|_{L^{\frac{4r}{2-\ee r}}_t L^{ \frac{2dr}{(4-\ee)r-2} }_x}
\big\} \leq \hat{C}_1(\bar{u})\| \delta \theta_n\|_{L^{\frac{4}{\ee}}(0,T;L^{\infty}_x)},
\end{aligned}
\end{equation}
where $\hat{C}_1(\bar{u}):=2C \tilde{c} (C_5 \| \bar{u} \|_{\BB_{p,r}^{d/p-1+\ee}} + C_6\| \bar{u} \|_{\BB_{p,r}^{d/p-1+\ee/2}} )$. Now, let us
observe that $\delta \theta_n$ is the weak solution of
\begin{equation*}
\begin{cases}
\partial_t \delta \theta_n -\ee \Delta \delta \theta_n =
-\Div (\,\delta \theta_n u_n \,) -\Div (\,\delta u_{n-1} \theta_n \,)
& \RR_+ \times\RR^d,\\
\delta \theta_{n\,|t=0} =0 & \;\;\,\quad \quad \RR^d,\\
\end{cases}
\end{equation*}
which implies
\begin{equation}\label{delta_theta_n}
\delta \theta_n(t)= -\int_0^t \Div\, e^{\ee(t-s)\Delta}\delta \theta_n(s)u_n(s)\dd s
-\int_0^t \Div\, e^{\ee(t-s)\Delta}\delta u_{n-1}(s)\theta_n(s)\dd s.
\end{equation}
By Remark <ref> we deduce then
\begin{equation*}
\| \delta \theta_n(t)\|_{L^\infty_x} \leq
\int_0^t
\frac{\|\delta \theta_n(s)u_n(s)\|_{L^{\frac{2dr}{(2-\ee )r -2}}_x}}{|\ee(t-s)|^{1-\frac{1}{2r}-\frac{\ee}{4}}}
\dd s
\int_0^t
\frac{\|\delta u_{n-1}(s)\theta_n(s)\|_{L^{\frac{2dr}{(2-\ee )r -2}}_x}}{|\ee(t-s)|^{1-\frac{1}{2r}-\frac{\ee}{4}}}
\dd s,
\end{equation*}
hence, defining $\alpha := (1-1/(2r) -\ee/4 )(2r)'<1$, $\| \delta \theta_n(t)\|_{L^\infty_x}^{2r}$ is bounded by
\begin{align*}
\int_0^t
\frac{1}{|\ee(t-s)|^\alpha}\dd s
\bigg)^{2r-1}
\int_0^t
\|\delta \theta_n(s)\|_{L^\infty_x}^{2r} \| u_n(s)\|_{L^{q*}_x}^{2r}
\dd s+
\int_0^t
\|\bar{\theta}\|_{L^\infty_x}^{2r} \|\delta u_{n-1}(s)\|_{L^{q*}_x}^{2r}
\dd s
\big\}.
\end{align*}
Then, using the Gronwall inequality, we have
\begin{align*}
\| \delta \theta_n(t)\|_{L^\infty_x}^{2r} \leq
\big( 2\frac{(1-\alpha)t^{1-\alpha}}{\ee^\alpha} \big)^{2r-1}\|\bar{\theta}\|_{L^\infty_x}^{2r}
\int_0^t
\|\delta u_{n-1}(s)\|_{L^{\frac{2dr}{(2-\ee)r-2}}_x}^{2r}
\dd s
\exp
\big\{
\int_0^t
\| u_n(s)\|_{L^{\frac{2dr}{(2-\ee)r-2}}_x}^{2r}\dd s
\big\},
\end{align*}
which yields
\|\delta \theta_n(t)\|_{L^\infty_x}\leq \chi (t)\delta U_{n-1}(t),
where $\chi$ is an increasing function defined by
\begin{equation*}
\chi(t):=\big( 2\frac{(1-\alpha)t^{1-\alpha}}{\ee^\alpha} \big)^{1-\frac{1}{2r}}
\exp
\Big\{
\frac{1}{2r}
\Big\}.
\end{equation*}
Hence, Recalling (<ref>), we deduce that
\begin{equation}\label{estimates_f_3}
\begin{aligned}
\|f_{n,3}\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)} +
\|f_{n,3}\|_{L^{2r}(0,T;L^{\frac{2dr}{(2-\ee)r-2}}_x)}&+
\|\nabla f_{n,3}\|_{L^{2r}(0,T ;L_x^{ \frac{2dr}{(4-\ee)r-2}})}\\
&+\|\nabla f_{n,3}\|_{L^{2r}(0,T ;L_x^{ \frac{dr}{2r-1} })}
\leq \hat{C}_1(\bar{u})\chi(T)\|\delta U_{n-1} \|_{L^{\frac{4}{\ee}}(0,T)}.
\end{aligned}
\end{equation}
Summarizing (<ref>), (<ref>) and (<ref>) we finally deduce that
\begin{equation}\label{est_deltaUn}
\delta U_{n,\lambda}(T) \leq \Big(\frac{1}{3}+\frac{4}{3}\tilde{C}_r \eta \Big)\delta U_{n-1,\lambda}(T)
+ \frac{4}{3}\hat{C}_1(\bar{u})\chi(T)\|\delta U_{n-1} \|_{L^{\frac{4}{\ee}}(0,T)},
\end{equation}
Supposing $\eta$ small enough, we can assume $ \mu:=(1/3+4\tilde{C}_r\eta/3 )<1$. Thus, fixing $T>0$ and denoting by $C_T$ the
constant $4\bar{C}_1(\bar{u})\chi(T) \exp\{ \lambda ( \bar{C}_2\|\bar{u}^d\|_{\BB_{p,r}^{\frac{d}{p}-1}}+\bar{C}_3)\}/3$,
then we have
\begin{equation*}
\delta U_{n,\lambda}(t) \leq \mu\,\delta U_{n-1,\lambda}(t)
+ C_T\|\delta U_{n-1,\lambda} \|_{L^{\frac{4}{\ee}}(0,t)},
\end{equation*}
for all $t\in [0,T]$, where we have used that $\chi$ is an increasing function.
Now, let us prove by induction that there exists $C=C(T)>0$ and $K=K(T)>0$ such that
\begin{equation}\label{induction_delta_U}
\delta U_{n,\lambda}(t)\leq C\mu^{\frac{n}{2}}\exp\big\{ K\frac{t}{\sqrt{\mu}}\big\},
\end{equation}
for all $t\in [0,T]$ and for all $n\in\NN$. The base case is trivial, since it is sufficient to find $C=C(T)>0$ such that
\delta U_{0,\lambda}(t)\leq C
$, for all $t\in [0,T]$.
\delta U_{0,\lambda}(t)\leq C\exp\{ Kt/\mu\}
$, for all $K>0$. Passing to the induction,
\begin{align*}
\delta U_{n+1,\lambda}(t)
&\leq\mu\delta U_{n-1,\lambda}(t)+ C_T\|\delta U_{n-1,\lambda} \|_{L^{\frac{4}{\ee}}(0,t)}
\leq \sqrt{\mu}C \mu^{\frac{n+1}{2}}+
\Big(
\int_0^t \exp\big\{\frac{4}{\ee} K\frac{s}{\sqrt{\bar{\eta}}} \big\}\dd s
\Big)^\frac{\ee}{4}\\
&\leq \big(
\sqrt{\mu} + \big( \frac{\ee}{4K}\big)^{\frac{4}{\ee}} \mu^{\frac{\ee}{8}-\frac{1}{2}} C_T
\big)C\mu^{\frac{n+1}{2}}\exp\big\{ K\frac{t}{\sqrt{\bar{\eta}}}\big\}.
\end{align*}
Chosen $K>0$ big enough, we finally obtain that (<ref>) is true for any positive integer $n$. Hence, the series
$\sum_{n\in\NN} \delta u_{n,\lambda}(T)$ is convergent, for any $T\in\RR_+$. This yields that
\begin{equation*}
\sum_{n\in\NN} \delta U_n(T)\leq
\exp\Big\{ \lambda \big( \bar{C}_2\|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}+\bar{C}_3 \big)^{2r}\Big\}
\sum_{n\in\NN} \delta U_{n,\lambda}(T)<\infty,
\end{equation*}
so that $(u_n)_\NN$ and $(\nabla u_n)_\NN$ are Cauchy sequences in $L^{2r}(0,T;L^{dr/(r-1)}_x)$ and
$L^{2r}(0,T;L^{\frac{dr}{2r-1}}_x)$ respectively. Furthermore, $(\theta_n)_\NN$ is a Cauchy sequence in $L^{\infty}(\, (0,T)\times \RR^d)$,
since $\|\delta \theta_n\|_{L^{\infty}(\, (0,T)\times \RR^d)}$ is bounded by $\chi(T)\delta U_{n-1}(T)$.
Recalling also the definition of $\delta g_n$ (<ref>), we get
\begin{equation*}
\sum_{n\in\NN}\|\delta g_n \|_{L^{r}(0,T;L^{dr/(3r-2)}_x)}<\infty,
\end{equation*}
for all $T>0$. Thus $(g_n)_\NN$ is a Cauchy sequence in $ L^{r}(0,T;L^{dr/(3r-2)}_x)$ and
$ (\,{(\sqrt{-\Delta})^{-1}}g_n)_\NN$ is a Cauchy sequence in $L^{r}(0,T;L^{dr/(2r-2)}_x)$, thanks to
Corollary <ref>. Recalling the Mild formulation (<ref>), by Lemma <ref> and Theorem
<ref>, there exist $C>0$ such that
\begin{align*}
\| \nabla \delta u_n \|_{L^{r}(0,T;L^{\frac{dr}{2(r-1)}}_x)}\leq C
\Big\{
\|\delta g_n\|_{L^{r}(0,T;L^{\frac{dr}{3r-2}}_x)}
\|\nabla \delta u_{n-1} \|_{L^{r}(0,T;L^{\frac{dr}{2(r-1)}}_x)}+\\
\|\delta \nu_n\|_{L^{\infty}(\,(0,T)\times\RR^d )}\|\nabla u_n \|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x)}
\Big\}
\end{align*}
for all $n\in\NN$. Hence the series
\sum_{n\in\NN}\|\nabla\delta u_n \|_{L^{r}(0,T;L^{dr/(2r-2)}_x)}
is finite, which implies that $(\nabla u_n)_\NN$ is a Cauchy sequence in
Finally $(\Pi_n)_\NN$ is a Cauchy sequence in $L^{r}(0,T;L^{dr/(2r-2)}_x)$, by (<ref>) and this concludes the proof of the Proposition.
Now, let us prove that system (<ref>) admits a weak solution, adding some regularity to the initial data.
Let $1<r<\infty$ and $p\in (1,dr/(2r-1))$. Suppose that $\bar{\theta}$ belongs to $L^\infty_x\cap L^2_x$ and $\bar{u}$ belongs to
$\BB_{p,r}^{d/p-1}\cap \BB_{p,r}^{d/p-1+\ee} $ with $\ee<\min\{1/(2r), 1-1/r, 2(d/p -2 + 1/r)\}$.
If (<ref>) holds, then there exists a global weak solution $(\theta, u, \Pi)$
of (<ref>) which satisfies the properties of Theorem <ref>.
By Proposition <ref>, there exist $u_\ee$ in $L^{2r}_t L^{dr/(r-1)}_x$ with $\nabla u_\ee$ in
$L^{2r}_t L^{dr/(2r-1)}_x\cap L^{r}_t L^{dr/(2r-2)}_x
$, and also $\theta_\ee \in L^{\infty}(\RR_+\times \RR^d),\quad \Pi_\ee\in L^{r}_t L^{\frac{dr}{2(r-1)}}_x$,
such that $(\theta_\ee,\,u_\ee,\,\Pi_\ee)$ is weak solution of (<ref>). Moreover, thanks to
(<ref>), we have the following weakly convergences:
\begin{equation*}
\begin{array}{lll}
u_{\ee_n} \rightharpoonup u \quad w-L^{2r}_t L^{\frac{dr}{r-1} }_x, &
\nabla u_{\ee_n} \rightharpoonup \nabla u \quad w-L^{2r}_t L^{\frac{dr}{2r-1} }_x, &
\nabla u_{\ee_n} \rightharpoonup \nabla u \quad w-L^{2r}_t L^{\frac{dr}{2(r-1)}}_x, \\
\theta_{\ee_n} \overset{*}{ \rightharpoonup} \theta \quad w*-L^{\infty}_{t,x}, &
\Pi_{\ee_n} \rightharpoonup \Pi \quad w-L^{r}_t L^{\frac{dr}{2(r-1)}}_x,
\end{array}
\end{equation*}
for a positive decreasing sequence $(\ee_n)_\NN$ which is convergent to $0$. We want to prove that $(\theta, u, \Pi)$ is a weak solution of
First let us observe that $\{ u_\ee\,|\,\ee>0\}$ is a compact set on
$C([0,T]; \dot{W}_x^{-1,dr/(2r-2)})$, for all $T>0$. Indeed, recalling the momentum equation of
(<ref>), $\partial_t {(\sqrt{-\Delta})^{-1}}u_\ee $ is uniformly bounded in $L^r(0,T;L^{dr/(2r-2)}_x)$. This yields that
$\{(\sqrt{-\Delta})^{-1}u_\ee\,|\,\ee>0\}$ is an equicontinuous and bounded family on
$C([0,T], L_x^{dr/(2r-2)})$. Hence we can assume that $(\sqrt{-\Delta})^{-1} u_{\ee_n}$ strongly converges to
$(\sqrt{-\Delta})^{-1} u$ in $L^{\infty}(0,T;L^{dr/(2r-2)}_x)$, namely $ u_{\ee_n}$ strongly converges to $u$ in $L^{\infty}(0,T; \dot{W}_x^{-1,dr/(2r-2)} )$.
We recall that $(\nabla u_{\ee_n})_\NN$ is a bounded sequence on $L^{r}_t L^{dr/(2r-2)}_x$, so that $(u_{\ee_n})_\NN$ is a bounded sequence on
$L^{r}_t \dot{W}^{1,dr/(2r-2)}_x$. Thus, passing through the following real interpolation
\begin{equation*}
\Big[ \dot{W}_x^{-1,\frac{dr}{2(r-1)}}, \dot{W}_x^{+1,\frac{dr}{2(r-1)}}\Big]_{\frac{1}{2r},1}=
\dot{B}_{\frac{dr}{2(r-1)},1}^{1-\frac{1}{r}},
\end{equation*}
(see <cit.>, Theorem $6.3.1$), and since
$ \dot{B}_{dr/(2r-2),1}^{1-\frac{1}{r}}\hookrightarrow L^{dr/(r-1)}_x $
(see <cit.>, Theorem $2.39$), we deduce that,
\begin{align*}
\|u_{\ee_n}-u\|_{L^{2r}(0,T;L^{\frac{dr}{r-1}}_x)}
C \Big\|
\|u_{\ee_n}-u\|_{\dot{W}_x^{-1,\frac{dr}{2(r-1)}}}^{1-\frac{1}{2r}}
\|u_{\ee_n}-u\|_{\dot{W}_x^{1,\frac{dr}{2(r-1)}}}^{\frac{1}{2r}}
\Big\|_{L^{2r}(0,T)}\\
C \|u_{\ee_n}-u\|_{L^{\infty}(0,T;\dot{W}^{-1,\frac{dr}{2(r-1)}}_x)}^{1-\frac{1}{2r}}
\|u_{\ee_n}-u\|_{L^{1}(0,T;\dot{W}^{1,\frac{dr}{2(r-1)}}_x)}^{\frac{1}{2r}},
\end{align*}
for all $T>0$. This implies that $u_{\ee_n}$ strongly converges to $u$ in $L^{2r}_{loc}(\RR_+;L^{\frac{dr}{r-1}}_x)$, for all $T>0$, and moreover that
$u_{\ee_n}\theta_{\ee_n}$ and $u_{\ee_n}\cdot \nabla u_{\ee_n}$ converge to $u\, \theta$ and $u\cdot \nabla u$, respectively, in the distributional sense.
We deduce that $\theta$ is a weak solution of
\begin{equation}\label{prop_smooth_data_transport_eqution}
\partial_t\theta + \Div (\theta u)=0\quad\text{in}\quad \RR_+ \times\RR^d,\quad\quad
\theta_{|t=0} = \bar{\theta} \quad\text{in }\quad\RR^d.
\end{equation}
Now, we claim that $\theta_{\ee_n}\rightarrow\theta$ almost everywhere on $\RR_+\times \RR^d$, up to a subsequence. Multiplying the first equation of
(<ref>) by $\theta/2$ and integrating in $[0,t)\times \RR^d$ we get
\begin{equation*}
\| \theta_{\ee_n}(t) \|_{L^2_x}^2 +
\ee_n \int_0^t \| \nabla \theta_\ee (s) \|_{L^2_x}^2 \dd s =
\| \bar{\theta} \|_{L^2_x},
\end{equation*}
which yields $\| \theta_{\ee_n} \|_{L^2((0,T)\times \RR^d)}\leq T^{1/2}\| \bar{\theta} \|_{L^2_x}$ for any $T>0$. Moreover, multiplying
(<ref>) by $\theta$ and integrating in $[0,t)\times \RR^d$, we achieve $\| \theta (t) \|_{L^2_x} = \| \bar{\theta} \|_{L^2_x}$
for any $t\in (0,T)$, hence
\begin{equation*}
\limsup_{n\rightarrow \infty} \| \theta_{\ee_n} \|_{L^\infty(0,T;L^2_x)} \leq
T^\frac{1}{2}\| \bar{\theta} \|_{L^2_x} =
\| \theta \|_{L^2(0,T;L^2_x)}.
\end{equation*}
Thus we can extract a subsequence (which we still call it $\theta_{\ee_n}$) such that $\theta_{\ee_n}$ strongly converges to $\theta$ in $L^2_{loc}(\RR_+\times\RR^d)$.
We deduce that $\theta_{\ee_n}$ converges almost everywhere to $\theta$, up to a subsequence, and $\nu(\theta_{\ee_n})$ strongly converges to
$\nu(\theta)$ in $L^m_{loc}(\RR_+\times \RR^d)$, for every $1\leq m<\infty$, thanks to the Dominated Convergence Theorem.
Then $\nu(\theta_{\ee_n})\MM_{\ee_n}$ converges to $\nu(\theta)\MM$ in the distributional sense.
Summarizing all the previous considerations we finally conclude that $(\theta,\,u,\,\Pi)$ is a weak solution of
(<ref>) and it satisfies the inequalities given by (<ref>).
§ PROOF OF THEOREM <REF>
In this section we present the proof of Theorem (<ref>). Because of the low regularity of the initial temperature, by the dyadic partition we approximate our initial data and by Theorem <ref> we construct a sequence of approximate solutions. A step one, still using the mentioned Theorem, we observe that such solutions fulfill inequalities which are dependent only on the initial data. Therefore, using a compactness argument, we establish that the approximate solutions converge, up to a subsequence, and that the limit is the solution we are looking for.
Recalling the Besov embedding $L^\infty_x\hookrightarrow \dot{B}_{\infty,\infty}^0$, we define
\begin{equation*}
\bar{\theta}_n :=\chi_n \sum_{|j|\leq n}\dot{\Delta}_j\bar{\theta}\quad\text{and}\quad
\bar{u}_n:=\sum_{|j|\leq n}\dot{\Delta}_j\bar{u}, \quad\text{for every}\quad n\in\NN,
\end{equation*}
where $\chi_n\leq 1$ is a cut-off function which has support on the ball $B(0,n)\subset \RR^d$. Thus $\bar{\theta}_n\in L^\infty_x\cap L^2_x$ and $\bar{u}_n \in \dot{B}_{p,r}^{d/p} \cap \dot{B}_{p,r}^{d/p-1+\ee}$, with $\ee<\min\{1/(2r), 1-1/r, 2(d/p -2 + 1/r)\}$.
Then, by Theorem <ref>, there exists $(\theta_n, u_n, \Pi_n)$ weak solution of
\begin{equation*}
\begin{cases}
\partial_t\theta_n + \Div (\theta_n u_n)=0 & \RR_+ \times\RR^d,\\
\partial_t u_n + u_n\cdot \nabla u_n -\Div (\nu(\theta_n)\nabla u_n) +\nabla\Pi_n=0 & \RR_+ \times\RR^d,\\
\Div\, u_n = 0 & \RR_+ \times\RR^d,\\
(\theta_n,\,u_n)_{t=0} = (\bar{\theta}_n,\,\bar{u}_n) & \;\;\quad \quad\RR^d,
\end{cases}
\end{equation*}
such that $\theta_n\in L^{\infty}(\RR_+\times\RR^d)$, $u_n \in L^{2r}_tL^{dr/(r-1)}_x$, $\nabla u_n \in L^{2r}_tL^{dr/(2r-1)}_x\cap L^{r}_tL^{dr/(2r-2)}_x $
and moreover $\Pi_n\in L^{r}_t L^{\frac{dr}{2(r-1)}}_x$.
Furthermore the following inequalities are satisfied:
\begin{align*}
& \| \nabla u_n^h \|_{L^{2r}_t L^{ \frac{dr}{2(r-1)}}_x}+
\| \nabla u_n^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\| u_n^h\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_1\eta\text{,}\\
& \| \nabla u_n^d \|_{L^{2r}_t L^{ \frac{dr}{2(r-1)}}_x}+
\|\nabla u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\| u_n^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}} + C_3,\\
& \|\Pi_n\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq C_4\eta,\quad
\|\theta_n\|_{L^\infty(\RR_+\times\RR^d)}\leq C\|\bar{\theta}\|_{L^{\infty}_x}
\end{align*}
for all $n\in\NN$ and for some positive constants $C_1$, $C_2$, $C_3$, $C_4$, $C_5$ and $C$.
Then there exists a subsequence (which we still denote by $( \,(\theta_n, u_n, \Pi_n)\,)_\NN$ ) and $(\theta,\, u,\,\Pi)$ in the same space of
$(\theta_n, u_n, \Pi_n)$, such that
\begin{equation*}
\begin{array}{lll}
u_{n} \rightharpoonup u \quad w-L^{2r}_t L^{\frac{dr}{r-1}}_x,
&\nabla u_{n} \rightharpoonup \nabla u \quad w-L^{2r}_t L^{\frac{dr}{2r-1}}_x,
&\nabla u_{n} \rightharpoonup \nabla u \quad w-L^{2r}_t L^{\frac{dr}{2(r-1)}}_x,\\
\theta_{n} \overset{*}{\rightharpoonup} \theta \quad w^*-L^{\infty}_{t,x},
&\Pi_{n} \rightharpoonup \Pi \quad w-L^{r}_t L^{\frac{dr}{2(r-1)}}_x.&
\end{array}
\end{equation*}
Moreover, proceeding as in Theorem <ref>, $u_n$ strongly converges to $u$ in $L^{2r}_{loc, t} L^{dr/(r-1)}_x$, so that $\theta$ is weak
solution of
\begin{equation}\label{transport_equation}
\partial_t\theta + \Div (\theta u)=0 \quad\text{in}\quad \RR_+ \times\RR^d \quad\text{and}\quad
\theta_{|t=0} = \bar{\theta}\quad\text{in} \quad\RR^d.
\end{equation}
Now, we claim that $\theta^2_n \overset{*}{\rightharpoonup} \theta^2$ in $L^\infty (\RR_+ \times\RR^d)$. Observing that $\|\theta^2\|_{L^\infty(\RR_+ \times\RR^d)}\leq
C^2\|\bar{\theta}\|^2_{L^\infty_x}$, there exists $\omega\in L^\infty_{t,x}$ such that $\theta^2_n \overset{*}{\rightharpoonup} \omega$
in $L^\infty _{t,x}$, up to a subsequence.
Now, let us remark that $\theta_n^2$ is weak solution of
\begin{equation*}
\partial_t\theta_n^2 + \Div (\theta_n^2 u_n)=0 \quad\text{in}\quad \RR_+ \times\RR^d\quad\text{and}\quad
\theta^2_{n|t=0} = \bar{\theta}^2 \quad\text{in}\quad \RR^d,
\end{equation*}
then, passing through the limit as $n$ goes to $\infty$, we deduce that $\omega$ is weak solution of
\begin{equation*}
\partial_t \omega + \Div (\omega u)=0 \quad\text{in}\quad \RR_+ \times\RR^d\quad\text{and}\quad
\omega_{|t=0} = \bar{\theta}^2 \quad\text{in}\quad \RR^d.
\end{equation*}
Moreover, multiplying (<ref>) by $\theta$, we get
\begin{equation*}
\partial_t\theta^2 + \Div (\theta^2 u)=0 \quad\text{in}\quad \RR_+ \times\RR^d\quad\text{and}\quad
\theta^2_{|t=0} = \bar{\theta}^2 \quad\text{in}\quad \RR^d,
\end{equation*}
which yields $\omega=\theta^2$, from the uniqueness of the transport equation.
Summarizing the previous considerations, we deduce that $\theta_n \rightarrow \theta$ $s-L^2_{loc}(\RR_+ \times\RR^d)$, so that $\theta_n$ converges to $\theta$ almost everywhere in $\RR_+ \times\RR^d$ up to a subsequence, thus $\nu(\theta_n)$ converges to $\nu(\theta)$ almost everywhere in $\RR_+ \times\RR^d$. We conclude that and $\nu(\theta_n)$ strongly converges to $\nu(\theta_n)$ in $L^m_{loc}(\RR_+ \times\RR^d)$, for every $m\in [1,\infty)$, thanks to the Dominated Convergence Theorem. Therefore, passing through the limit as $n$ goes to $\infty$, we deduce that
\begin{equation*}
\Div (\nu(\theta_n)\nabla u_n) \rightarrow \Div (\nu(\theta)\nabla u),
\end{equation*}
in the distributional sense, which allows to conclude that $(\theta, u, \Pi)$ is a weak solution of (<ref>).
If we replace the two first equations of system (<ref>) by
\begin{equation*}
\partial_t\theta + \Div\, (\theta u)+a\theta=0 \quad \text{in}\quad \RR_+ \times\RR^d\quad \text{and}\quad
\partial_t u + u\cdot \nabla u -\Div\, (\nu(\theta)\MM) +\nabla\Pi=a \theta e_d \quad \text{in}\quad \RR_+ \times\RR^d,
\end{equation*}
where $e_d=\,^t(0,\dots,1)\in\RR^d$ and $a$ is a positive real constant, then we can adapt our strategy in order to establish the existence of weak solutions for
such new system. In the case of the original system, a term as $\theta e_d$ can be assumed only to be bounded both in time and space, hence it does not provide a time
integrability, which is necessary in order to achieve the existence result. However, adding the damping term $a \theta$ to the classical transport equation, and supposing
$\bar{\theta}$ to belongs to $L^{2d/(3r-2)}_x$, then
\begin{equation*}
\|\theta(t)\|_{L^{\frac{dr}{3r-2}}_x}\leq
\|\bar{\theta}\|_{L^{\frac{dr}{3r-2}}_x}
\exp\big\{-a\,t\big\},
\end{equation*}
for every $t\in\RR_+$. Thus $\theta$ belongs to $L^r_tL^{dr/(3r-2)}_x$ and we can proceed as in the previous proofs, obtaining a global weak solution
$(\theta,\,u,\, \Pi)$ which belongs to the space defined by Theorem $\ref{Main_Theorem}$. Moreover, increasing $\eta$ by
\begin{equation*}
\eta_2:=\Big(
\| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
\Big)
\exp\Big\{ c_r \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}^{4r} \Big\},
\end{equation*}
the solution $(\theta,\,u,\, \Pi)$ fulfills
\begin{align*}
& \| \nabla u^h \|_{L^{2r}_t L^{ \frac{dr}{2r-1}}_x}+
\|\nabla u^h\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} +
\| u^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_1\eta_2\text{,}\\
& \|\nabla u^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x} +
\|\nabla u^d\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x}
\| u^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
\leq C_2\Big(\| \bar{u}^d \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}
+a\|\bar{\theta}\|_{L^{\frac{dr}{3r-2}}}\Big) + C_3,\\
& \|\Pi\|_{L^{r}_t L^{\frac{dr}{2(r-1)}}_x} \leq
\end{align*}
for some positive constants $C_1$, $C_2$, $C_3$ and $C_4$.
§ THE GENERAL CASE: SMOOTH INITIAL DATA
As preliminary, before starting the proof of the main Theorem, we enunciate three fundamental Lemma concerning the regularizing effects of the
heat kernel, which will be useful. We recall that $\Bb$ and $\Cc$ are defined by
\begin{equation*}
\Bb f(t):= \int_0^t \nabla e^{(t-s)\Delta} f(s)\dd s,\quad
\Cc f(t):= \int_0^t e^{(t-s)\Delta} f(s)\dd s.
\end{equation*}
Let us assume that $p$, $p_3$, $r$, $\alpha$, $\gamma_1$, $\gamma_2$ fulfill the hypotheses of Theorem <ref> and let $\ee$
be a non-negative constant bounded by $\min\{ 1/r, 1-1/r, d/p-1\}$. If $t^\alpha f(t)$ belongs to $L^{2r/(1-\ee r)}(0,T;L^p_x)$ then
$t^{\gamma_1}\Cc f(t)$ belongs to $L^{2r/(1-\ee r)}(0,T;L^{p_3}_x)$ and there exists a positive constant $C$ such that
\begin{equation*}
\|t^{\gamma_1} \Cc f(t)\|_{L^{\frac{2r}{1-\ee r}}(0,T; L^{p_3}_x)}
\leq C
\|t^\alpha f(t)\|_{L^{\frac{2r}{1-\ee r}}(0,T; L^{p }_x)}.
\end{equation*}
Moreover, if $\ee$ is null then $t^{\gamma_2}\Cc f(t)$ belongs to $L^{\infty}(0,T;L^{p_3}_x)$ and
\begin{equation*}
\|t^{\gamma_2} \Cc f(t)\|_{L^{\infty }(0,T; L^{p_3}_x)}
\leq C
\|t^\alpha f(t)\|_{L^{\frac{2r}{1-\ee r} }(0,T; L^{p }_x)}.
\end{equation*}
Let us assume that $p$, $p_2$, $r$, $\alpha$, $\beta$ fulfill the hypotheses of Theorem <ref> and let $\ee$ be a non-negative
constant bounded by $\min\{ 1/r, 1-1/r, d/p-1\}$. If $t^\alpha f(t)$ belongs to $L^{2r/(1-\ee r)}(0,T;L^p_x)$ then $t^\beta\Bb f(t)$
belongs to $L^{2r/(1-\ee r)}(0,T;L^{p_2}_x)$ and there exists a positive constant $C$ such that
\begin{equation*}
\|t^\beta \Bb f(t)\|_{L^{\frac{2r}{1-\ee r}}(0,T; L^{p_2}_x)}
\leq C
\|t^\alpha f(t)\|_{L^{\frac{2r}{1-\ee r}}(0,T; L^{p }_x)}.
\end{equation*}
Let us assume that $p$, $p_2$, $r$, $\alpha$, $\beta$, $\gamma_1$, $\gamma_2$ fulfill the hypotheses of Theorem <ref> and
let $\ee$ be a non-negative constant bounded by $\min\{ 1/r, 1-1/r, d/p-1\}$.
If $t^{\beta}f$ belongs to $L^{2r/(1-\ee r)}(0,T; L^{p_2}_x)$ then $t^{\gamma_1}\Bb f(t)$ belongs to $L^{2r/(1-\ee r)}(0,T;L^{p_3}_x)$
\begin{equation}\label{Lemma6_inequality}
\|t^{\gamma_1 }\Bb f \|_{L^{\frac{2r}{1-\ee r} }(0,T;L^{p_3}_x)}
\leq C
\|t^{\beta } f \|_{L^{\frac{2r}{1-\ee r} }(0,T;L^{p_2}_x)}.
\end{equation}
Furthermore, if $\ee=0$ then there exists a positive $C$ such that
\begin{equation}\label{Lemma6_inequality2}
\|t^{\gamma_2 }\Bb f \|_{L^{\infty }(0,T;L^{p_3}_x)}
\leq C
\|t^{\beta } f \|_{L^{\frac{2r}{1-\ee r} }(0,T;L^{p_2}_x)}.
\end{equation}
The proofs of these lemmas are a direct consequence of Remark <ref>. We perform the one of Lemma <ref>, while the others can
be achieved thanks to a similar procedure.
We begin controlling the $L^{2r/(1-\ee r)}(0,T; L^{p_3}_x)$-norm. First Remark (<ref>) yields
\begin{equation*}
\|t^{\gamma_1 }\Bb f(t) \|_{L^{p_3}_x}
\leq C
\int_{0}^t
\frac{t^{\gamma_1}}{| t-s |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}}\|f(s) \|_{L^{p_2}_x}
\dd s= C
\int_{0}^1
\frac{t^{\gamma_1-\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}-\beta}}
{| 1-\tau |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}\tau^{\beta}}
F(t \tau)\dd \tau,
\end{equation*}
where $F(s):= s^{\beta}\|f(s) \|_{L^{p_2}_x} $. Now, since
$\gamma_1-d(1/p_2-1/p_3)/2+1/2-\beta$ is null, we have
\begin{align*}
\|t^{\gamma_1 }\Bb f \|_{L^{\frac{2r}{1-\ee r} }(0,T;L^{p_3}_x)}&\leq C
\int_{0}^1
\frac{1}{| 1-\tau |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}\tau^{\beta}}
\|F(t \tau)\|_{L^{\frac{2r}{1-\ee r} }_t(0,T;L^{p_2}_x)}\dd \tau\\&\leq C
\int_{0}^1
\frac{1}{| 1-\tau |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}\tau^{\beta+\frac{1}{2r}-\frac{\ee}{2}}}
\dd \tau
\|F\|_{L^{\frac{2r}{1-\ee r} }(0,T;L^{p_2}_x)},
\end{align*}
thanks to the Minkowski inequality. Thus (<ref>) is true, since $\beta+1/(2r)-\ee/2<1$ and moreover
$d(1/p_2-1/p_3)/2+1/2=2/3-d/(6p)+1/2<1-1/(2r)<1$. Finally, observing that
\begin{align*}
\|t^{\gamma_2 }\Bb f(t) \|_{L^{p_3}_x}
&\leq C
\int_{0}^t
\frac{t^{\gamma_2}}
{| t-s |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}}\|f(s) \|_{L^{p_2}_x}
\dd s\\
&\leq C
\Big(
\int_{0}^t
\Big|
\frac{t^{\gamma_2}}
{| t-s |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}s^{\beta}}
\Big|^{(2r)'}
\dd s
\Big)^{1-\frac{1}{2r}}
\|F\|_{L^{2r }(0,T;L^{p_2}_x)}
\end{align*}
we obtain
\begin{equation*}
\|t^{\gamma_2 }\Bb f(t) \|_{L^{p_3}_x}
\leq C
\Big(
\int_{0}^1
\Big|
\frac{1}{| 1-\tau |^{\frac{d}{2}\big( \frac{1}{p_2}-\frac{1}{p_3} \big)+\frac{1}{2}}\tau^{\beta}}
\Big|^{(2r)'}
\dd \tau
\Big)^{1-\frac{1}{2r}}
\|F\|_{L^{2r }(0,T;L^{p_2}_x)}
\end{equation*}
by the change of variable $s=t\tau$, since $(2r)'\{\gamma_2-d(1/p_2-1/p_3)/2-1/2-\beta\}+1$ is null. Hence
(<ref>) turns out from $\{d( 1/{p_2}-1/{p_3})/2+1/2\}(2r)'<1$ and $\beta(2r)'<1$.
We present the statement of a modified version of the Maximal Regularity Theorem, whose proof can be found in <cit.>.
Let $T\in ]0,\infty]$, $1<\bar{r},q<\infty$ and $\alpha\in (0,1-1/\bar{r})$. Let the operator
$\Aa$ be defined as in Theorem <ref>. Suppose that
$t^\alpha f(t)$
belongs to $L^{\bar{r}}(0,T;L^q_x)$. Then $t^\alpha \Aa f(t)$ belongs to $L^{\bar{r}}(0,T;L^q_x)$ and there
exists $C>0$ such that
\begin{equation*}
\|t^\alpha \Aa f(t)\|_{L^{\bar{r}}(0,T;L^q_x)}\leq C \| t^\alpha f(t)\|_{L^{\bar{r}}(0,T;L^q_x)}.
\end{equation*}
As last part of this preliminaries, we have the following corollary, which will be useful in order to control the pressure $\Pi$.
Let $p\in (1,d)$, $\bar{r}\in(1,\infty)$ and $\alpha\in (0,1-1/\bar{r})$. If $t^{\alpha}f$ belongs to $L^{\bar{r}}(0,T;L^p_x)$ then $t^{\alpha}\Bb f$ belongs to
$L^{\bar{r}}(0,T; L^{p^*}_x$ and there exists a positive constant $C$ (not dependent by $f$) such that
\begin{equation*}
\|t^{\alpha} \Bb f(t)\|_{L^{2r}(0,T; L^{p^*}_x)}
\leq C
\|t^{\alpha} f(t)\|_{L^{2r}(0,T; L^{p }_x)}.
\end{equation*}
It is sufficient to observe that $\Bb f(t)$ reads as follows:
\begin{equation*}
\Bb f(t) =
-(\sqrt{-\Delta})^{-1} R \int_0^t \Delta e^{(t-s)\Delta }f(s) \dd s =
-(\sqrt{-\Delta})^{-1} R \Aa f(t).
\end{equation*}
Recalling that $R$ is a bounded operator from $L^q_x$ to itself for any $q\in (1,\infty)$ and $(\sqrt{-\Delta})^{-1}$ from $L^p_x$ into
$L^{p^*}_x$, the lemma is a direct consequence of Theorem <ref>.
Let $p,\,r,\,p_2,\,p_3$ be as in Theorem <ref>.
Suppose that $\bar{\theta}$ belongs to $L^\infty_x$ and $\bar{u}$ belongs to $\BB_{p,r}^{d/p-1}$. If the smallness condition (<ref>) holds,
then there exists a global weak solution $(\theta,\,u,\,\Pi)$ of (<ref>) such that it belongs to the functional framework defined by
Theorem <ref> and moreover it
\begin{equation}\label{inequalities_statement_prop_smooth_dates_general_data}
\begin{aligned}
& \|t^{\beta} \nabla u^h\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\alpha} \nabla u^h\|_{L^{2r}_t L^{p^*}_x}+
\|t^{\gamma_1} u^h \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^h \|_{L^{\infty}_t L^{p_3}_x}
\leq C_1\eta,\\&
\|t^{\beta} \nabla u^d\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\alpha} \nabla u^d\|_{L^{2r}_t L^{p^*}_x}+
\|t^{\gamma_1} u^d \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^d \|_{L^{\infty}_t L^{p_3}_x}
\leq C_2\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}} + C_3\\&
\|t^\alpha\Pi\|_{L^{2r}_t L^{p^*}_x}
\leq
\| \theta \|_{L^\infty_{t,x}}\leq \| \bar{\theta} \|_{L^\infty_x}.
\end{aligned}
\end{equation}
for some positive constants $C_1$, $C_2$ and $C_3$.
We proceed as in the proof of Proposition <ref>, considering the sequence of solutions for systems
(<ref>) and (<ref>). We claim that such solutions belong to
the same space defined in Theorem <ref> and moreover that:
\begin{equation}
\begin{aligned}
& \| t^{\beta } \nabla u^{h}_n \|_{L^{2r }_t L^{p_2}_x }+
\| t^{\gamma_1 } u^{h}_n \|_{L^{2r }_t L^{p_3}_x }+
\| t^{\gamma_2 } u^{h}_n \|_{L^{\infty }_t L^{p_3}_x }
\leq C_1\eta,\\&
\| t^{\beta } \nabla u^{d}_n \|_{L^{2r }_t L^{p_2}_x }+
\| t^{\gamma_1 } u^{d}_n \|_{L^{2r }_t L^{p_3}_x }+
\| t^{\gamma_2 } u^{d}_n \|_{L^{\infty }_t L^{p_3}_x }
\leq C_2\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}} + C_3,
\end{aligned}
\end{equation}
for some suitable positive constants $C_1$, $C_2$ and $C_3$, and for any positive integer $n$.
Step 1: Estimates. First, the maximal principle for parabolic equation implies that
$\|\theta_n\|_{L^\infty_{t,x}}$ is bounded by $\|\bar{\theta}\|_{L^\infty_x}$. Now, we want to prove by
induction that
\begin{equation}\label{prop_smooth-general_induction}
\begin{aligned}
& \|t^{\beta}\nabla u^{h}_{n}\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^{h}_{n} \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{h}_{n} \|_{L^{\infty}_t L^{p_3}_x}
\leq \frac{C_1}{2}\tilde{\eta}
\leq \frac{C_1}{2}\eta,\\&
\|t^{\beta}\nabla u^{d}_{n}\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^{d}_{n} \|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{d}_{n} \|_{L^{\infty}_t L^{p_3}_x}
\leq \frac{C_2}{2}\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}} + \frac{C_3}{2},
\end{aligned}
\end{equation}
for some positive constant $C_1$, $C_2$ and $C_3$, where $\tilde{\eta}$ is defined by
\begin{equation*}
\tilde{\eta}:= (\|\bar{u}^h\|_{\BB_{p,r}^{\frac{d}{p}-1}}+\|\bar{\theta}\|_{L^\infty_x}+\| \nu -1 \|_{\infty})
\exp\big\{\frac{c_r}{2}\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}^{2r}\big\}<\eta.
\end{equation*}
We begin with the horizontal component $u_n^{h}$.
Let $\lambda$ be a positive real number, and let $u_{n+1,\lambda}$, $\nabla u_{n+1, \lambda}$ and $\Pi_{n+1, \lambda}$ be defined by
\begin{equation}\label{def_ulambda2}
(u_{n+1, \lambda},\,\nabla u_{n+1, \lambda},\,\Pi_{n+1, \lambda} )(t):= h_{n,\lambda }(0,t)(u_{n+1},\,\nabla u_{n+1},\,\Pi_{n+1} )(t),
\end{equation}
where, for all $0 \leq s<t<\infty$,
\begin{equation}\label{def_h2}
h_{n, \lambda}(s,t):=
\exp
\big\{
-\lambda \int_s^t t^{2r\gamma_1}\|u_n^d(\tau)\|^{2r}_{L_x^{p_3}}\dd \tau\,
-\lambda \int_s^t t^{2r\beta }\|\nabla u_n^d(\tau)\|^{2r}_{L_x^{p_2}}\dd \tau\,
\big\}.
\end{equation}
We decompose $u_{n+1,\lambda}$ as in (<ref>),
$u_{n+1,\lambda}=u_L + F^{1}_{n+1,\lambda}+F^{2}_{n+1,\lambda}+F^{3}_{n+1,\lambda} $, the first estimate is given by Theorem
<ref> and Theorem <ref>:
\begin{equation}\label{Prop_smooth_data_est1}
\begin{aligned}
\| t^{\beta } \nabla u^{h}_{L,\lambda} \|_{L^{2r }_t L^{p_2}_x}+
\| t^{\gamma_1} u^{h}_{L,\lambda} \|_{L^{2r }_t L^{p_3}_x}&+
\| t^{\gamma_2} u^{h}_{L,\lambda} \|_{L^{\infty}_t L^{p_3}_x}
\leq
\| t^{\beta } \nabla u^{h}_{L,\lambda} \|_{L^{2r }_t L^{p_2}_x}+\\&+
\| t^{\gamma_1} u^{h}_{L,\lambda} \|_{L^{\infty}_t L^{p_3}_x}+
\| t^{\gamma_2} u^{h}_{L,\lambda} \|_{L^{\infty}_t L^{p_3}_x}
\leq
C\| \bar{u}^h \|_{\BB_{p,r}^{\frac{d}{p}-1}},
\end{aligned}
\end{equation}
for a positive constant $C$. Moreover, recalling the definition (<ref>) of $g_{n+1}$, we get
\begin{equation}\label{Prop_smooth_data_est2}
\begin{aligned}
\| t^{\beta } \nabla F^{1,h}_{n+1,\lambda} \|_{L^{2r }_t L^{p_2}_x} +
\| t^{\gamma_1} F^{1,h}_{n+1,\lambda} &\|_{L^{2r }_t L^{p_3}_x} +
\| t^{\gamma_2} F^{1,h}_{n+1,\lambda} \|_{L^{\infty}_t L^{p_3}_x}\leq
\frac{1}{ \lambda^{\frac{1}{2r}} }
\| t^{\beta } \nabla u_{n+1,\lambda}^h \|_{L^{2r }_t L^{p_2}_x} +\\&+
\| t^{\gamma_2} u_{n }^h \|_{L^{\infty}_t L^{p_3}_x}
\| t^{\beta } \nabla u_{n }^h \|_{L^{2r }_t L^{p_2}_x} +
\frac{1}{\,\lambda^{\frac{1}{2r}}}
\| t^{\gamma_1} u_{n+1,\lambda}^h \|_{L^{2r }_t L^{p_3}_x}
\big\}.
\end{aligned}
\end{equation}
thanks to Lemma <ref>, Lemma <ref>, Lemma <ref> and Lemma <ref>. Moreover,
\begin{equation}\label{Prop_smooth_data_est3}
\begin{aligned}
\| t^{\gamma_1} F^{2,h}_{n+1,\lambda} \|_{L^{2r }_t L^{p_3}_x} +
\| t^{\gamma_2} F^{2,h}_{n+1,\lambda} \|_{L^{\infty}_t L^{p_3}_x} +
\| t^{\gamma_1} F^{3,h}_{n+1,\lambda} \|_{L^{2r }_t L^{p_3}_x} +
\| t^{\gamma_2} F^{3,h}_{n+1,\lambda} \|_{L^{\infty}_t L^{p_3}_x}\leq \\
\leq
\| t^{\beta }(\nu(\theta_{n+1})-1) \MM_n \|_{L^{2r }_t L^{p_2}_x}\leq
\| \nu -1 \|_{\infty}
\| t^{\beta } \nabla u_n \|_{L^{2r }_t L^{p_2}_x}
\end{aligned}
\end{equation}
by Lemma <ref> and Lemma. Finally, Theorem <ref>
\begin{equation}\label{Prop_smooth_data_est4}
\begin{aligned}
\| t^{\beta} \nabla F^{2,h}_{n+1,\lambda} \|_{L^{2r }_t L^{p_2}_x} +
\| t^{\beta} \nabla F^{3,h}_{n+1,\lambda} \|_{L^{2r }_t L^{p_2}_x}\leq
\| \nu -1 \|_{\infty}
\| t^{\beta } \nabla u_n \|_{L^{2r }_t L^{p_2}_x}.
\end{aligned}
\end{equation}
Summarizing (<ref>), (<ref>), (<ref>) and
(<ref>), we deduce that
\begin{equation}\label{prop_smooth-general_case_horiz_est1}
\begin{aligned}
\|t^{\beta}\nabla u^{h}_{n+1,\,\lambda}\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^{h}_{n+1,\,\lambda}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{h}_{n+1,\,\lambda} \|_{L^{\infty}_t L^{p_3}_x}
&\leq \\
\leq
\Big\{
\| \bar{u}^h \|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\frac{1}{ \lambda^{\frac{1}{2r}} }
\| t^{\beta } \nabla u_{n+1,\lambda}^h \|_{L^{2r }_t L^{p_2}_x} +
\| t^{\gamma_2} u_{n }^h \|_{L^{\infty}_t L^{p_3}_x}&
\| t^{\beta } \nabla u_{n }^h \|_{L^{2r }_t L^{p_2}_x} +\\+
\frac{1}{\,\lambda^{\frac{1}{2r}}}
\| t^{\gamma_1} u_{n+1,\lambda}^h \|_{L^{2r }_t L^{p_3}_x} &+
\| \nu -1 \|_{\infty}
\| t^{\beta } \nabla u_n \|_{L^{2r }_t L^{p_2}_x}
\Big\}
\end{aligned}
\end{equation}
for a suitable positive constant $C$. Setting $\lambda := (2C)^{2r} $, we can absorb the terms with index $n+1$ on the right-hand side
by the the left-hand side, hence there exists a positive constant $\tilde{C}$ such that
\begin{align*}
\|t^{\beta}\nabla u^{h}_{n+1,\,\lambda}\|_{L^{2r}_t L^{p_2}_x}+&
\|t^{\gamma_1} u^{h}_{n+1,\,\lambda}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{h}_{n+1,\,\lambda} \|_{L^{\infty}_t L^{p_3}_x}
\leq \\
\tilde{C}
\Big\{
\| \bar{u}^h \|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\frac{C_1^2}{4}\tilde{\eta}^2+
\| \nu -1 \|_{\infty}
(\frac{C_1}{2}\tilde{\eta} +
\frac{C_2}{2}
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}
\Big\}.
\end{align*}
Then we deduce that
\begin{align*}
&\|t^{\beta}\nabla u^{h}_{n+1}\|_{L^{2r}_t L^{p_2}_x}+
\|t^{\gamma_1} u^{h}_{n+1}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{h}_{n+1} \|_{L^{\infty}_t L^{p_3}_x}
\leq \\
\tilde{C}
\sup_{t\in (0,\infty)}h_{n,\lambda}(0,t)^{-1}
\Big\{
\| \bar{u}^h \|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\frac{C_1^2}{4}\tilde{\eta}^2+
\| \nu -1 \|_{\infty}
(\bar{C}_1\tilde{\eta} +
\frac{C_2}{2}
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}
\Big\}\\
\tilde{C}
\exp\big\{
\frac{C_2}{2}
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}
\big\}
\Big\{
1+ (\frac{C_1^2}{4} +\frac{C_1}{2})\tilde{\eta} + \frac{C_2}{2} +\frac{C_3}{2}
\Big\}\tilde{\eta}.
\end{align*}
Imposing $C_1$ big enough and $\tilde{\eta}$ small enough in order to have
\begin{equation*}
\tilde{C}
\exp\big\{
\frac{C_2}{2}
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}
\big\}
\Big\{
1+ (\frac{C_1^2}{4} +\frac{C_1}{2})\tilde{\eta} + \frac{C_2}{2} +\frac{C_3}{2}
\Big\}\leq \frac{C_1}{2} \tilde{\eta},
\end{equation*}
we finally deduce that the first inequality of (<ref>) is true for any positive integer $n$.
Now, let us handle the vertical component $u_n^d$. Proceeding as in the proof of (<ref>),
we obtain that the following inequality is satisfied:
\begin{align*}
\|t^{\beta}\nabla u^{d}_{n+1}\|_{L^{2r}_t L^{p_2}_x}&+
\|t^{\gamma_1} u^{d}_{n+1}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{d}_{n+1} \|_{L^{\infty}_t L^{p_3}_x}
\leq \\
\Big\{
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\| t^{\alpha } g_{n+1} \|_{L^{2r }_t L^{p}_x}+
\| \nu -1 \|_{\infty}
\| t^{\beta } \nabla u_n \|_{L^{2r }_t L^{p_2}_x}
\Big\},
\end{align*}
for a suitable positive constant $C$, where $g_{n+1}$ is defined by (<ref>). Recalling that
$\alpha=\beta+\gamma_1$ and $1/p=1/p_2+1/p_3$ we get
\begin{align*}
\|t^{\beta}\nabla u^{d}_{n+1}\|_{L^{2r}_t L^{p_2}_x}&+
\|t^{\gamma_1} u^{d}_{n+1}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{d}_{n+1} \|_{L^{\infty}_t L^{p_3}_x}
\leq
\Big\{
\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}+\\+
\| t^{\gamma_2 } u_n^h &\|_{L^{\infty }_t L^{p_3}_x}
\| t^{\beta }\nabla u_n^h \|_{L^{2r }_t L^{p_2}_x}+
\| t^{\gamma_2 } u_{n+1}^h \|_{L^{\infty }_t L^{p_3}_x}
\| t^{\beta }\nabla u_n^d \|_{L^{2r }_t L^{p_2}_x}+ \\&+
\| t^{\gamma_2 } u_n^h \|_{L^{\infty }_t L^{p_3}_x}
\| t^{\beta }\nabla u_{n+1}^d \|_{L^{2r }_t L^{p_2}_x}+
\| \nu -1 \|_{\infty}
\| t^{\beta } \nabla u_n \|_{L^{2r }_t L^{p_2}_x}
\Big\},
\end{align*}
which yields that
\begin{align*}
\|t^{\beta}\nabla u^{d}_{n+1}\|_{L^{2r}_t L^{p_2}_x}&+
\|t^{\gamma_1} u^{d}_{n+1}\|_{L^{2r}_t L^{p_3}_x}+
\|t^{\gamma_2} u^{d}_{n+1} \|_{L^{\infty}_t L^{p_3}_x}
\leq\\
)\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}+
\frac{C_1C_3}{4} + \frac{C_1^2}{4}\tilde{\eta} +
\|\nu-1\|_{\infty}
\frac{C_1}{2} +\frac{C_2}{2}
\end{align*}
Hence the second inequality of (<ref>) is true for any positive integer $n$ if we assume $\bar{C}_2$ big
enough and $\eta$ small enough in order to have
\begin{equation*}
C(1+\frac{C_1C_2}{4} \tilde{\eta})<\frac{C_2}{2}
\quad
\text{and}
\quad
C(\frac{C_1C_3}{2} +\frac{C_1^2}{4}\eta +\eta(\frac{C_1}{2}+\frac{C_2}{2}))\eta\leq\frac{C_3}{2}.
\end{equation*}
Proceeding again by induction, we claim that
\begin{equation}\label{ind_talphanabla}
\|t^{\alpha}\nabla u^h_n\|_{L^{2r}_t L^{p^*}_x}\leq \frac{C_1}{2}\eta
\quad\text{and}\quad
\|t^{\alpha}\nabla u^d_n\|_{L^{2r}_t L^{p^*}_x}\leq \frac{C_2}{2}\| \bar{u}^d \|_{\BB_{p,r}^{\frac{d}{p}-1}}+\frac{C_3}{2},
\end{equation}
for any positive integer $n$. First, we remark that $\nabla u_L$ can be rewritten as $\nabla u_L = -(\sqrt{-\Delta})^{-1}R\Delta u_L $. Hence,
recalling that $(\sqrt{-\Delta})^{-1}$ is a bounded operator from $L^p_x$ into $L^{p^*}_x$ and $R$ is a bounded operator from $L^q_x$ into itself,
for any $q\in (1,\infty)$, there exists a positive constant $C$ such that
\begin{equation}
\|t^{\alpha}\nabla u_L\|_{L^{2r}_t L^{p^*}_x}\leq C
\|t^{\alpha}\Delta u_L\|_{L^{2r}_t L^{p}_x}\leq
\|\bar{u}\|_{\BB_{p,r}^{\frac{d}{p}-1}},
\end{equation}
thanks to Theorem <ref>. Moreover Theorem <ref> and
Corollary <ref> imply
\begin{equation*}
\|t^{\alpha}(\nabla F^{2}_{n+1}+\nabla F^{3}_{n+1})\|_{L^{2r}_t L^{p^*}_x}
\leq
\|t^{\alpha}\nabla u_{n}\|_{L^{2r}_t L^{p^*}_x},\,
\|t^{\alpha}\nabla F^{1}_{n+1}\|_{L^{2r}_t L^{p^*}_x}\leq C \|t^{\alpha} g_{n+1} \|_{L^{2r}_t L^{p}_x}
\leq C\eta.
\end{equation*}
Assuming $\eta$ small enough we get that (<ref>) is true for any $n\in\NN$.
Finally, recalling that $\Pi_{n+1}$ is determined by
\begin{equation*}
\Pi_{n+1}=(-\Delta)^{-1}R\cdot g_{n+1}-R\cdot R\cdot\{(\nu(\theta_{n+1})-1)\nabla u_n\},
\end{equation*}
we get
\begin{align*}
\|t^{\alpha} \Pi_{n+1} \|_{L^{2r}_tL^{p^* }_x}
\|t^{\alpha} g_{n+1} \|_{L^{2r}_tL^{p }_x}+
\|\nu-1 \|_{\infty}
\|t^{\beta} \nabla u_n \|_{L^{2r}_tL^{p_2 }_x}
\big\}
\leq
\end{align*}
for a suitable positive constant $C_4$ and for any positive integer $n$.
Step 2: $\ee$-Dependent Estimates. As second step, we establish some $\ee$-dependent estimates which will be useful in order
to show that $(\theta_n,\,u_n,\,\Pi_n)_\NN$ is a Cauchy sequence in a suitable space. First, we claim that
\begin{equation}\label{ee_estimates2_induction}
\|t^{ \gamma_1 } u_{n,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\|t^{ \beta } \nabla u_{n,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq \bar{C}_4
\| \bar{u} \|_{ \BB_{p,r}^{ \frac{d}{p}-1+\ee } },
\end{equation}
where $u_{n,\lambda}(t)=u_n(t)h(0,t)$, with $h$ is defined by (<ref>).
Recalling the characterization of the homogenous Besov spaces given by Theorem <ref> and the
embedding of Theorem <ref>, we get
\begin{equation}\label{ee_estimates2_1}
\|t^{ \gamma_1 } u_L \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\|t^{ \beta } \nabla u_L \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq C
\| \bar{u} \|_{ \BB_{p,r}^{ \frac{d}{p}-1+\ee } },
\end{equation}
for a suitable $C>0$. Furthermore, Lemma <ref> and Lemma <ref> yields
\begin{align*}
\|&t^{ \gamma_1 } F^1_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\| t^{ \beta } \nabla F^1_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq \bar{C}
\Big\{
\frac{1}{\lambda^{\frac{1}{2r}}}
\|t^{ \beta } \nabla u^h_{ n+1 ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +
\|t^{ \gamma_2 } u^h_{ n } \|_{ L^{ \infty }_t L^{ p_3 }_x }
{ \scriptstyle \times}\\ &{ \scriptstyle \times}
\|t^{ \beta } \nabla u^h_{ n ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +
\|t^{ \gamma_2 } u^h_{ n+1 } \|_{ L^{ \infty }_t L^{ p_3 }_x }
\|t^{ \beta } \nabla u^d_{ n ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +
\|t^{ \gamma_2 } u^h_{ n } \|_{ L^{ \infty }_t L^{ p_3 }_x }
\|t^{ \beta } \nabla u^d_{ n+1 ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\Big\},
\end{align*}
for a positive constant $\bar{C}$. Imposing $\lambda:= (2\bar{C})^{2r}$, we deduce that
\begin{equation}\label{ee_estimates2_2}
\begin{aligned}
\|&t^{ \gamma_1 } F^1_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\| t^{ \beta } \nabla F^1_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq
\frac{1}{2}
\|t^{ \beta } \nabla u^h_{ n+1 ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +\\&+
\bar{C}C_1\eta
\|t^{ \beta } \nabla u^h_{ n ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +
\bar{C}C_1\eta
\|t^{ \beta } \nabla u^d_{ n ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } +
\bar{C}C_1\eta
\|t^{ \beta } \nabla u^d_{ n+1 ,\lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }.
\end{aligned}
\end{equation}
Moreover, Theorem <ref> and Lemma <ref> imply
\begin{equation}\label{ee_estimates2_3}
\begin{aligned}
\| t^{ \gamma_1 } ( F^2_{n+1,\lambda} &+ F^3_{n+1,\lambda} ) \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\| t^{ \beta } \nabla ( F^2_{n+1,\lambda} + F^3_{n+1,\lambda} ) \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq \\ \leq
\| t^{ \gamma_1 } &( F^2_{n+1 } + F^3_{n+1 } ) \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\| t^{ \beta } \nabla ( F^2_{n+1 } + F^3_{n+1 } ) \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x } \\
&\leq C
\| \nu - 1 \|_{\infty}
\|t^{ \beta } \nabla u_{ n } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq
\tilde{C}\eta
\|t^{ \beta } \nabla u_{ n, \lambda } \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\end{aligned}
\end{equation}
assuming $C_r$ in the definition of $\eta$ big enough. Summarizing (<ref>), (<ref>) and
(<ref>), there exists a positive constant $C$ such that
\begin{align*}
\|t^{ \gamma_1 } u_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\|t^{ \beta } \nabla u_{n+1,\lambda} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq
C \bar{C}_4 \eta \| \bar{u} \|_{ \BB_{p,r}^{\frac{d}{p}-1}+\ee },
\end{align*}
so that (<ref>) is true for any positive integer $n$. Finally, multiplying both the left and right-hand sides
of (<ref>) by $sup_{t\in\RR} h^{-1}(0,t)$, we get
\begin{equation}\label{ee_estimates2_conclusion}
\|t^{ \gamma_1 } u_{n} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_3 }_x } +
\|t^{ \beta } \nabla u_{n} \|_{ L^{ \frac{2r}{1-\ee r} }_t L^{ p_2 }_x }
\leq C_5
\| \bar{u} \|_{ \BB_{p,r}^{ \frac{d}{p}-1+\ee } }
\exp
\big\{ C_6
\| \bar{u}^d \|_{ \BB_{p,r}^{ \frac{d}{p}-1 } }^{2r}
\big\},
\end{equation}
for two suitable positive constant $C_5$ and $C_6$.
Step 3. Convergence of the Series.
We proceed as in the third step of Theorem <ref>, denoting $\delta u_n:= u_{n+1}-u_{n}$,
$\delta \nu_n:= \nu(\theta_{n+1})-\nu(\theta_{n})$ and $\delta \theta_n:= \theta_{n+1}-\theta_{n}$. We define
\begin{align*}
\delta U_{n,\lambda}(T):=
\| t^{ \gamma_1 } \delta u_{n,\lambda} \|_{L^{ 2r }(0,T;L^{p_3}_x)}
+\| t^{ \gamma_2 } \delta u_{n,\lambda} \|_{L^{ \infty }(0,T;L^{p_3}_x)}
+\| t^{ \beta } \nabla \delta u_{n,\lambda} \|_{L^{ 2r }(0,T;L^{p_2}_x)},
\end{align*}
where $\delta u_{n,\lambda}(t):=\delta u_n(t)h_{n,\lambda}(0,t)$. We claim that the series $\sum_{n\in \NN}\delta U_{n}(T)$ is
convergent. First, we split $\delta u_n$ into $\delta u_{n,\lambda }=f_{n,1}+f_{n,2}+f_{n,3}$, where $f_{n,i}$ is defined by
(<ref>), for $i=1,2,3$. We begin estimating $f_{n,1}$. Lemma <ref> and Lemma <ref> yield that
\begin{align*}
\| &t^{ \gamma_1 } \delta f_{n,1} \|_{L^{ 2r }(0,T;L^{p_3}_x)}+
\| t^{ \gamma_2 } \delta f_{n,1} \|_{L^{ \infty }(0,T;L^{p_3}_x)}+
\| t^{ \beta } \nabla \delta f_{n,1} \|_{L^{ 2r }(0,T;L^{p_2}_x)}
\leq\\
&\leq C
\Big\{
\frac{1}{\lambda^{\frac{1}{2r}}}
\big(
\| t^{\beta } \partial_d \delta u_{n,\lambda}^h \|_{L^{2r }(0,T; L^{p_2}_x) }+
\| t^{\gamma_1 } \delta u_{n,\lambda}^h \|_{L^{2r }(0,T; L^{p_3}_x) }+
\| t^{\beta } \nabla^h \delta u_{n,\lambda}^h \|_{L^{2r }(0,T; L^{p_2}_x) }
\big)+\\&\quad+
\| t^{\gamma_2 } \delta u_{n,\lambda}^d \|_{L^{\infty }(0,T; L^{p_3}_x) }
\| t^{\beta } \partial_d u_n^h \|_{L^{2r }_t L^{p_2}_x }+
\| t^{\gamma_2 } \delta u_{n-1,\lambda}^h \|_{L^{\infty }(0,T; L^{p_3}_x) }
\| t^{\beta } \nabla u_n^h \|_{L^{2r }_t L^{p_2}_x }+\\&\quad+
\| t^{\gamma_2 } u_{n-1}^h \|_{L^{2r }_t L^{p_3}_x }
\| t^{\beta } \nabla \delta u_{n-1,\lambda}^h \|_{L^{2r }(0,T; L^{p_2}_x) }+
\| t^{\beta } \nabla^h \delta u_{n,\lambda}^d \|_{L^{2r }(0,T; L^{p_2}_x) }
\| t^{\gamma_2 } u_n^h \|_{L^{\infty }_t L^{p_3}_x }+\\&\quad+
\| t^{\gamma_2 } \delta u_{n-1,\lambda}^d \|_{L^{\infty }(0,T; L^{p_3}_x) }
\| t^{\beta } \nabla^h u_n^h \|_{L^{2r }_t L^{p_2}_x }
\Big\}.
\end{align*}
which yields,
\begin{equation}\label{estimates_f_1_general_case}
\begin{aligned}
\| t^{ \gamma_1 } \delta f_{n,1} \|_{L^{ 2r }(0,T;L^{p_3}_x)}&+
\| t^{ \gamma_2 } \delta f_{n,1} \|_{L^{ \infty }(0,T;L^{p_3}_x)} +\\&+
\| t^{ \beta } \nabla \delta f_{n,1} \|_{L^{ 2r }(0,T;L^{p_2}_x)}
\leq
\frac{1}{4}
\big(
\delta U_{n,\lambda}(T)+\delta U_{n-1,\lambda}(T)
\big),
\end{aligned}
\end{equation}
assuming $\eta$ small enough. Now, we carry out the estimate of $f_{n,2}$. Lemma <ref> and Theorem
<ref> imply
\begin{align*}
\| t^{ \gamma_1 } \delta f_{n,2} \|_{L^{ 2r }(0,T;L^{p_3}_x)}&+
\| t^{ \gamma_2 } \delta f_{n,2} \|_{L^{ \infty }(0,T;L^{p_3}_x)} +
\| t^{ \beta } \nabla \delta f_{n,2} \|_{L^{ 2r }(0,T;L^{p_2}_x)}
\leq \\ &\leq C
\|\nu-1\|_{\infty}
\| t^{ \beta } \nabla \delta u_{n-1} \|_{L^{2r }(0,T;L^{p_2}_x)}
\leq \tilde{C}_r\eta
\| t^{ \beta } \nabla \delta u_{n-1,\lambda} \|_{L^{2r }(0,T;L^{p_2}_x)},
\end{align*}
hence, we deduce that
\begin{equation}\label{estimates_f_2_general_case}
\| t^{ \gamma_1 } \delta f_{n,2} \|_{L^{ 2r }(0,T;L^{p_3}_x)} +
\| t^{ \gamma_2 } \delta f_{n,2} \|_{L^{ \infty }(0,T;L^{p_3}_x)} +
\| t^{ \beta } \nabla \delta f_{n,2} \|_{L^{ 2r }(0,T;L^{p_2}_x)}\leq
\bar{C}_r\eta \delta U_{n-1,\lambda}(T).
\end{equation}
Now we deal with $f_{n,3}$. Thanks to Lemma <ref> and Theorem <ref>, we have
\begin{equation}
\begin{aligned}\label{estimates_f3_general_case_partA}
\|&t^{ \gamma_1 } \delta f_{n,3} \|_{L^{ 2r }(0,T; L^{p_3 }_x)}+
\| t^{ \gamma_2 } \delta f_{n,3} \|_{L^{ \infty }(0,T; L^{p_3 }_x)} +
\| t^{ \beta } \nabla \delta f_{n,3} \|_{L^{ 2r }(0,T; L^{p_2 }_x)}
\leq \\& \leq
\| t^{\beta } \delta \nu_n \MM_n \|_{L^{2r }(0,T; L^{ p_2 }_x)}
\leq C
\| \delta \nu_n \|_{L^{\frac{2}{\ee} }(0,T; L^{\infty }_x)}
\| t^{\beta } \nabla u_n \|_{L^{2r }(0,T; L^{ p_2 }_x)}
\leq \hat{C}_1(\bar{u})
\| \delta \theta_n \|_{L^{\frac{2}{\ee} }(0,T; L^{\infty }_x)}
\end{aligned}
\end{equation}
where $\hat{C}_1(\bar{u})$ is a positive constant which depends on $\| \bar{u} \|_{\BB_{p,r}^{d/p-1+\ee}}$. Now, recalling that
$\delta \theta_n$ is determined by (<ref>), we get
\begin{equation*}
\| \delta \theta_n(t)\|_{L^\infty_x} \leq
\int_0^t
\frac{s^{\gamma_1}\|\delta \theta_n(s)u_n(s) \|_{L^{p_3}_x}}{s^{\gamma_1}|\ee(t-s)|^{\frac{d}{2}\frac{1}{p_3}+\frac{1}{2}}}
\dd s
\int_0^t
\frac{s^{\gamma_1}\|\delta u_{n-1}(s)\theta_n(s) \|_{L^{p_3}_x}}{s^{\gamma_1}|\ee(t-s)|^{\frac{d}{2}\frac{1}{p_3}+\frac{1}{2}}}
\dd s,
\end{equation*}
hence, defining $\alpha := (d/(2p_3)+1/2)(2r)'<1$, $\| \delta \theta_n(t)\|_{L^\infty_x}^{2r}$ is bounded by
\begin{align*}
\Big(
\int_0^t
\frac{1}{s^{\gamma_1(2r)'}|\ee(t-s)|^\alpha}\dd s
\Big)^{2r-1}
\Big\{
\int_0^t &
\|\delta \theta_n(s)\|_{L^\infty_x}^{2r} s^{2r\gamma_1 }\| u_n (s) \|_{L^{p_3}_x}^{2r}
\dd s+\\&+
\int_0^t
\| \bar{\theta}\|_{L^\infty_x}^{2r} s^{2r\gamma_1 }\| \delta u_{n-1} (s) \|_{L^{p_3}_x}^{2r}
\dd s
\Big\}.
\end{align*}
Then, using the Gronwall inequality, we have
\begin{align*}
\| \delta \theta_n(t)\|_{L^\infty_x}^{2r}
\leq \hat{C}_2(t)
\|\bar{\theta}\|_{L^\infty_x}^{2r}
\int_0^t
s^{2r\gamma_1} \|\delta u_{n-1}(s)\|_{L^{p_3}_x}^{2r}
\dd s
\exp
\Big\{
\int_0^t
s^{2r \gamma_1}\| u_n(s)\|_{L^{p_3}_x}^{2r}
\dd s
\Big\},
\end{align*}
which yields
\|\delta \theta_n(t)\|_{L^\infty_x}\leq \chi (t)\delta U_{n-1}(t),
where $\chi$ is an increasing function.
Hence, Recalling (<ref>), we deduce that
\begin{equation*}
\| t^{ \gamma_1 } \delta f_{n,3} \|_{L^{ 2r }(0,T; L^{p_3 }_x)}+
\| t^{ \gamma_2 } \delta f_{n,3} \|_{L^{ \infty }(0,T; L^{p_3 }_x)} +
\| t^{ \beta } \nabla \delta f_{n,3} \|_{L^{ 2r }(0,T; L^{p_2 }_x)}
\leq \hat{C}_1(\bar{u})\chi(T)\|\delta U_{n-1} \|_{L^{\frac{4}{\ee}}(0,T)}.
\end{equation*}
Summarizing the last inequality with (<ref>) and (<ref>),
we finally deduce that
\begin{equation*}
\delta U_{n,\lambda}(T) \leq \Big(\frac{1}{3}+\frac{4}{3}\tilde{C}_r \eta \Big)\delta U_{n-1,\lambda}(T)
+ \frac{4}{3}\hat{C}_1(\bar{u})\chi(T)\|\delta U_{n-1} \|_{L^{\frac{2}{\ee}}(0,T)},
\end{equation*}
which is equivalent to to (<ref>). Thus we can conclude proceeding as in the last part of Theorem
Now, we want to prove that system (<ref>) admits a weak solution, adding some regularity to the initial data.
Let us assume that the hypotheses of Theorem <ref> are fulfilled. Suppose that $\bar{\theta}$ belongs to
$L^2_x \cap L^\infty_x $ and $\bar{u}$ belongs to $\BB_{p,r}^{d/p-1}\cap \BB_{p,r}^{d/p-1+\ee} $ with $\ee<\min\{1/(2r), 1-1/(2r), d/p-1\}$.
If the smallness condition (<ref>) holds then there exists a global weak solution $(\theta, u, \Pi)$
of (<ref>) which satisfies the properties of Theorem <ref>.
By Proposition <ref>, there exists $(\theta_\ee,\,u_\ee,\,\Pi_\ee)$, solution of
(<ref>), such that $t^{\gamma_1}u_\ee$ belongs to $L^{2r}_t L^{p_3}_x$, $t^{\gamma_2} u_\ee$ belongs to $L^{\infty}_t L^{p_3}_x$, $t^{\beta}\nabla u_\ee$
lives in $L^{2r}_t L^{p_2}_x$, $t^{\alpha}\nabla u_\ee$ in $L^{2r}_t L^{p^*}_x$, $\theta_\ee$ in $L^{\infty}_{t,x}$ and $t^\alpha \Pi_\ee$ in $L^{2r}_t L^{p^*}_x$.
Then, thanks to inequalities (<ref>), there exists $(\theta,\,u,\,\Pi)$ in the same space of $(\theta_\ee,\,u_\ee,\,\Pi_\ee)$, such
\begin{equation*}
\begin{array}{lll}
t^{\gamma_1} u_{\ee_n} \rightharpoonup t^{\gamma_1} u \quad w-L^{2r }_t L^{p_3 }_x, &
t^{\gamma_2} u_{\ee_n} \rightharpoonup t^{\gamma_2} u \quad w-L^{\infty }_t L^{p_3 }_x, &
t^{\beta }\nabla u_{\ee_n} \rightharpoonup t^{\beta }\nabla u \quad w-L^{2r }_t L^{p_2 }_x, \\
t^{\alpha }\nabla u_{\ee_n} \rightharpoonup t^{\alpha }\nabla u \quad w-L^{2r }_t L^{p^* }_x, &
\theta_{\ee_n} \overset{*}{ \rightharpoonup} \theta \quad w*-L^{\infty}_{t,x}, &
t^{\alpha } \Pi_{\ee_n} \rightharpoonup t^{\alpha }\Pi \quad w-L^{2r}_t L^{p^* }_x,
\end{array}
\end{equation*}
for a positive decreasing sequence $(\ee_n)_\NN$ convergent to $0$. We claim that $(\theta, u, \Pi)$ is weak solution of (<ref>).
First, we show that $ u_{\ee_n}$ strongly converges to $u$ in $L^{\tau_3}(0,T;L^{p_3}_x)$, up to
a subsequence, with a suitable $\tau_3>1$. We proceed establishing that $\{ u_\ee-u_L\,|\,\ee>0\}$ is a compact set in $C([0,T]; \dot{W}_x^{-1,p^*})$, for all $T>0$. Applying
$(\sqrt{-\Delta})^{-1}$ to the momentum equation of (<ref>), we observe that $t^{\alpha}\partial_t {(\sqrt{-\Delta})^{-1}}u_\ee $ is uniformly bounded in
$L^{2r}(0,T;L^{p^*}_x)$. Hence, observing that $\alpha (2r)'<1$, we get
\begin{equation*}
\|\partial_t (\sqrt{-\Delta})^{-1} u_\ee \|_{L^{1 }(0,T; L^{p^*}_x) } \leq
\frac{T^{1-\alpha(2r)'}}{1-\alpha(2r)'}
\| t^{ \alpha }\partial_t (\sqrt{-\Delta})^{-1} u_\ee \|_{L^{ 2r }(0,T L^{p^*}_x) }
\end{equation*}
Thus $\{(\sqrt{-\Delta})^{-1}(u_\ee-u_L)\,|\,\ee>0\}$ is an equicontinuous and bounded family of $C([0,T], L_x^{p^*})$, namely it is a compact family.
Then we can extract a subsequence (which we still denote by $u_{\ee_n}$) such that $(\sqrt{-\Delta})^{-1}(u_{\ee_n}-u_L)$ strongly converges to
$(\sqrt{-\Delta})^{-1}(u-u_L)$ in $L^{\infty}(0,T;L^{p^*}_x)$, that is $ u_{\ee_n}-u_L$ strongly converges to $u-u_L$ in $L^{\infty}(0,T; \dot{W}_x^{-1,p^*} )$.
Now, passing through the following real interpolation
\begin{equation*}
\Big[ \dot{W}_x^{-1,p^*}, \dot{W}_x^{1,p^*}\Big]_{\mu,1}=
\dot{B}_{p^*,1}^{\frac{d}{p^*}-\frac{d}{p_3}}\hookrightarrow L^{p_3}_x,
\end{equation*}
with $\mu:= (d/p^* -d/p_3)+1/2<1$ (see <cit.>, Theorem $6.3.1$ and <cit.>, Theorem $2.39$), we deduce that
\begin{align*}
\| u_{\ee_n}-u \|_{L^{\tau}(0,T;L^{p_3}_x)}
C \Big\|
\|u_{\ee_n}-u\|_{\dot{W}_x^{-1,p^* }}^{1-\mu}
\|u_{\ee_n}-u\|_{\dot{W}_x^{1,p^*}}^{\mu}
\Big\|_{L^{\tau}(0,T)}\\
C \|u_{\ee_n}-u\|_{L^{\infty}(0,T;\dot{W}^{-1,p^*}_x)}^{1-\mu}
\| t^{-\alpha} \|_{L^{\frac{2r\tau}{2r-\tau}}(0,T)}^\mu
\|t^{\alpha}\nabla (u_{\ee_n}-u)\|_{L^{2r}(0,T;L^{p^*}_x)}^{\mu},
\end{align*}
for all $T>0$, where we have considered $\tau\in (1, 2r/(1+ 2\alpha r))$ so that $\alpha 2r \tau /(2r-\tau)<1$. Moreover, we choose $\tau$ such that there exist
$\tau_2$ in $(1, 2r/(1+2\beta r ))$ and $\tau_3$ in $(1, 2r/(1+2\gamma_1 r) )$ which fulfill $1/\tau_3 + 1/\tau_2 = 1/\tau_1$. Let us remark that the norms
\begin{align*}
\| u_{\ee_n} \|_{L^{\tau_3}(0,T; L^{p_3}_x)} &\leq \| t^{\gamma_1}\|_{L^{\frac{2r\tau_3}{2r-\tau_3}}(0,T)}\|t^{\gamma_1} u_{\ee_n} \|_{L^{2r}_t L^{p_3}_x}<\infty,\\
\|\nabla u_{\ee_n} \|_{L^{\tau_2}(0,T; L^{p_2}_x)} &\leq \| t^{\beta}\|_{L^{\frac{2r\tau_2}{2r-\tau_2}}(0,T)}\|t^{\beta} u_{\ee_n} \|_{L^{2r}_t L^{p_2}_x}<\infty,
\end{align*}
that is they are uniformly bounded in $n$. Now, we consider $\tau<\sigma <\tau_3$ strictly closed to $\tau_3$ so that it still fulfills $1/\sigma + 1/\tau_2 >1$. Then the following
interpolation inequality
\begin{equation*}
\| u_{\ee_n} - u \|_{L^\sigma (0,T;L^{p_3}_x)} \leq
\| u_{\ee_n} - u \|_{L^\tau (0,T;L^{p_3}_x)}^{\frac{\tau_3 -\sigma}{\tau - \tau_3}}
\| u_{\ee_n} - u \|_{L^{\tau_3} (0,T;L^{p_3}_x)}^{\frac{\sigma -\tau_3 }{\tau - \tau_3}},
\end{equation*}
which converges to $0$ as $n$ goes to $\infty$, so that $u_{\ee_n}$ strongly converges to $u$ in $L^{\sigma}_{loc}(\RR_+;L^{p_3}_x)$. This yields that
$u_{\ee_n}\theta_{\ee_n}$ and $u_{\ee_n}\cdot \nabla u_{\ee_n}$ converge to $u\, \theta$ and $u\cdot \nabla u$, respectively, in the distributional sense.
We deduce that $\theta$ is weak solution of
\begin{equation*}
\partial_t\theta + \Div (\theta u)=0\quad\text{in}\quad \RR_+ \times\RR^d,\quad\quad
\theta_{|t=0} = \bar{\theta} \quad\text{in }\quad\RR^d.
\end{equation*}
Arguing as in theorem <ref>, $\theta_{\ee_n}$ converges almost everywhere to $\theta$, up to a subsequence, so that
$\nu(\theta_{\ee_n})$ strongly converges to $\nu(\theta)$ in $L^m_{loc}(\RR_+\times \RR^d)$, for every $1\leq m<\infty$, thanks to the Dominated Convergence Theorem.
Then $\nu(\theta_{\ee_n})\MM_{\ee_n}$ converges to $\nu(\theta)\MM$ in the distributional sense.
Summarizing all the previous considerations we finally conclude that $(\theta,\,u,\,\Pi)$ is a weak solution of
(<ref>) and it satisfies (<ref>).
§ PROOF OF THEOREM <REF>
In this section we present the proof of Theorem (<ref>). We proceed similarly as in the proof of Theorem <ref>, approximating our initial data by
\begin{equation*}
\bar{\theta}_n := \chi_n\sum_{|j|\leq n}\dot{\Delta}_j\bar{\theta}\quad\text{and}\quad
\bar{u}_n:=\sum_{|j|\leq n}\dot{\Delta}_j\bar{u}, \quad\text{for every}\quad n\in\NN,
\end{equation*}
where $\chi_n\leq 1$ is a cut-off function which has support on the ball $B(0,n)\subset \RR^d$, so that $\bar{\theta}_n\in L^\infty_x\cap L^2_x$ and $\bar{u} \in \dot{B}_{p,r}^{d/p} \cap \dot{B}_{p,r}^{d/p-1+\ee}$, with $\ee<\min\{1/(2r), 1-1/r, 2(d/p -2 + 1/r)\}$.
Then, by Theorem <ref>, there exists $(\theta_n, u_n, \Pi_n)$ weak solution of
\begin{equation*}
\begin{cases}
\partial_t\theta_n + \Div (\theta_n u_n)=0 & \RR_+ \times\RR^d,\\
\partial_t u_n + u_n\cdot \nabla u_n -\Div (\nu(\theta_n)\nabla u_n) +\nabla\Pi_n=0 & \RR_+ \times\RR^d,\\
\Div\, u_n = 0 & \RR_+ \times\RR^d,\\
(\theta_n,\,u_n)_{t=0} = (\bar{\theta}_n,\,\bar{u}_n) & \;\;\quad \quad\RR^d,
\end{cases}
\end{equation*}
which belongs to the functional space defined in Theorem <ref> and it fulfills the inequalities (<ref>), uniformly in $n\in\NN$. Then there exists a subsequence (which we still denote by $(\theta_n, u_n, \Pi_n)_\NN$) and an element $(\theta,\,u,\,\Pi)$ in the same space of $(\theta_n, u_n, \Pi_n)$, such that
\begin{equation*}
\begin{array}{lll}
t^{\gamma_1} u_{n} \rightharpoonup t^{\gamma_1} u \quad w-L^{2r }_t L^{p_3 }_x, &
t^{\gamma_2} u_{n} \rightharpoonup t^{\gamma_2} u \quad w-L^{\infty }_t L^{p_3 }_x, &
t^{\beta }\nabla u_{n} \rightharpoonup t^{\beta }\nabla u \quad w-L^{2r }_t L^{p_2 }_x, \\
t^{\alpha }\nabla u_{n} \rightharpoonup t^{\alpha }\nabla u \quad w-L^{2r }_t L^{p^* }_x, &
\theta_{\ee_n} \overset{*}{\rightharpoonup} \theta \quad w*-L^{\infty}_{t,x}, &
t^{\alpha } \Pi_{n} \rightharpoonup t^{\alpha }\Pi \quad w-L^{2r}_t L^{p^* }_x.
\end{array}
\end{equation*}
In order to complete the proof, we claim that $(\theta,\,u,\,\Pi)$ is weak solution of (<ref>).
We first rewrite $u_n= t^{-\gamma_1} t^{\gamma_1}u_n$, $\nabla u= t^{-\beta} t^{\beta}\nabla u$ and $\Pi_n= t^{-\alpha} t^{\alpha}\Pi_n$, so that the Hölder inequality guarantees that
$u_n$, $\nabla u_n$ and $\Pi_n$ are uniformly bounded in $L^{\tau_3}(0,T; L^{p_3}_x)$, $L^{\tau_2}(0,T; L^{p_2}_x)$ and $L^{\tau_1}(0,T; L^{p^*}_x)$ respectively, with $T\in (0,\infty)$ and
\begin{equation*}
\tau_1 \in \big( 1, \frac{2r}{1+2\alpha r} \big), \quad \tau_2 \in \big( 1, \frac{2r}{1+2\beta r} \big),\quad \tau_3 \in \big( 1, \frac{2r}{1+2\gamma_1 r} \big),
\quad\text{such that}\quad
\frac{1}{\tau_1} = \frac{1}{\tau_2} + \frac{1}{\tau_3}.
\end{equation*}
The same properties are preserved by $(\theta,\,u,\,\Pi)$. Moreover, arguing as in Theorem <ref>, $u_n$ strongly converges to $u$ in $L^{\sigma}_{loc}(\RR_+;L^{p_3}_x)$, with $\sigma\in (\tau_1,\tau_3)$ strictly closed to $\tau_3$ so that $1/\sigma + 1/\tau_2 >1$. This yields that $u_n\cdot \nabla u_n$ and $u_n\theta_n$ converge to
$u\cdot \nabla u$ and $u\,\theta$ respectively, in the distributional sense. Moreover, proceeding as in theorem <ref>, $\theta_{n}$ converges almost everywhere to $\theta$, up to a subsequence, so that $\nu(\theta_{n})$ strongly converges to $\nu(\theta)$ in $L^m_{loc}(\RR_+\times \RR^d)$, for every $1\leq m<\infty$, thanks to the Dominated Convergence Theorem. Then $\nu(\theta_{n})\MM_{n}$ converges to $\nu(\theta)\MM$ in the distributional sense and this allows us to conclude that $(\theta,\,u,\,\Pi)$ is weak solution of (<ref>). Finally, passing through the limit as $n$ goes to $\infty$, $(\theta,\,u,\,\Pi)$ still fulfills inequalities (<ref>) and this concludes the proof of the Theorem.
§ INEQUALITIES
In this section we improve Lemma <ref> and Lemma <ref> for a particular choice of the function $f$ and also with a perturbation of the operators, which is dependent on a parameter $\lambda>0$. This Lemmas are useful for the Theorem of section $3$, more precisely during the proof of the inequalities, since, for an opportune choice of $\lambda$, they permit to “absorb” some uncontrolled terms. Here the statements and the proofs.
Let $ 1< r < \infty$ and $q_1,\, q_2\in (1,\infty]$ such that $1/q=1/q_1+1/q_2\in ((2r-1)/dr, 1)$. Let $v \in L^{2r}_t L_x^{q_1}$ and
for all $\lambda>0 $ let $h=h_{\lambda} $ be defined by
\begin{equation*}
h(s,t):= \exp\big\{ -\lambda\int_s^t
\|v\|_{L_x^{q_1}}^{2r}\big\}\text{,}
\end{equation*}
for all $0\leq s\leq t<\infty $ and consider $\mathcal{C}_\lambda$, the operator defined by
\begin{equation*}
\mathcal{C}_\lambda (f)(t) :=
\int_0^t h(s,t)e^{(t-s)\Delta} f(s)\dd s.
\end{equation*}
Then there exists a positive constant $C_r$, such that
\begin{equation*}
\|\mathcal{C}_\lambda(v\omega) \|_{L^{2r}_t L_x^{q_3}}
\leq C_r\frac{1}{\,\lambda^{\frac{1}{4r}}}\|v \|_{L^{2r}_t L_x^{q_1}}^\frac{1}{2}
\|\omega \|_{L^{2r}_t L_x^{q_2}},
\end{equation*}
where $q_3$ is defined by $1/q_3= 1/q-(2r-1)/dr$.
Notice that
\begin{align*}
\|\int_0^t h(s,t) K(t-s)*v\omega (s)\dd s \,\|_{L^{q_3}_x}
\int_0^th(s,t) \|K(t-s)*v\omega (s)\|_{L^{q_3}_x}\dd s\\
\int_0^th(s,t)\|K(t-s)\|_{L^{\tilde{q}}_x}\|v \omega (s)\|_{L^q_x}\dd s,
\end{align*}
where $ 1/\tilde{q}'=1-1/\tilde{q}=1/q-1/q_3=(2r-1)/(dr)$.
By Remark <ref> and Holder inequality, we obtain
\begin{equation}\label{inequatliy_A1}
\begin{aligned}
\|\mathcal{C}(v \omega)(t)\|_{L^{q}_x}
\int_0^th(s,t)\|v (s)\|_{L^{p_1}}^\frac{1}{2}
\frac{1}{\quad| t-s |^{\frac{2r-1}{2r}}}
\|v (s)\|_{L^{p_1}}^\frac{1}{2}
\|\omega (s)\|_{L^{p_2}_x}\dd s\\
\bigg(
\int_0^th(s,t)^{4r}\| v(s)\|^{2r}_{L^{p_1}_x}\dd s
\bigg)^\frac{1}{4r}
\bigg(\,
\int_{\RR_+}
\frac{ (\,
\|v (s)\|_{L^{q_1}}^{\frac{1}{2}}
\|\omega (s)\|_{L^{q_2}_x} )^{\frac{4r}{4r-1}}
{\quad| t-s |^{\frac{2r-1}{2r}\frac{4r}{4r-1}}}
\dd s
\bigg)^{1-\frac{1}{4r}}.
\end{aligned}
\end{equation}
\begin{equation*}
\Big(\,
\|v (\cdot)\|_{L^{q_1}}^{\frac{1}{2}}
\|\omega (\cdot)\|_{L^{q_2}_x}
\Big)^{\frac{4r}{4r-1}}
\in
\end{equation*}
by Hardy-Littlewood-Sobolev inequality,
\begin{equation*}
|\cdot|^{-\frac{4r-2}{4r-1}}*g \in L_t^\frac{4r-1}{2},
\end{equation*}
and then
\begin{equation*}
\left(
\right)^{1-\frac{1}{4r}}\in L_t^{2r}.
\end{equation*}
Moreover there exists $C>$ such that
\begin{align*}
\|
\|_{L_t^{2r}}&=
\|
\|_{L_t^{\frac{4r-1}{2}}}^{1-\frac{1}{4r}}
\leq
\bigg(
\int_{\RR_+}
\|v (t)\|_{L^{q_1}_x}^{\frac{1}{2}}
\|\omega (t)\|_{L^{q_2}_x}
\,)^{\frac{4}{3}r}\dd t
\bigg)^{\frac{3}{4r}}\\
C\|\,\|v\|_{L^{q_1}_x}^\frac{1}{2} \|_{L^{4r}_t}
\|\omega \|_{L^{2r}_t L^{q_2}_x}\leq
C\|v\|_{L_t^{2r} L^{q_1}_x}^\frac{1}{2}
\|\omega \|_{L^{2r}_t L^{q_2}_x}.
\end{align*}
Observing that
\begin{equation*}
\Big(
\int_0^th(s,t)^{4r}\| v(s)\|^{2r}_{L^{q_1}_x}\dd s
\Big)^\frac{1}{4r}
\leq
\Big(\frac{1}{4r\lambda}\Big)^{\frac{1}{4r}},
\end{equation*}
the Lemma is proved.
Let $ 1< r < \infty$ , $q_1\in [1,\frac{dr}{r-1}]$ and $v \in L^{2r}_t L_x^{q_1}$.
For all $\lambda>0 $ let $h=h_{\lambda} $ be defined as in Lemma <ref> and let $\Bb_\lambda$ the operator defined by
\begin{equation*}
\mathcal{B}_\lambda(f)(t) :=
\int_0^t h(s,t)\nabla e^{(t-s)\Delta}f(s)\dd s.
\end{equation*}
For all $q_2\in [q_1',\infty]$, there exists a positive constant $C_r$, such that
\begin{equation*}
\|\Bb_\lambda(v\omega) \|_{L^{2r}_t L_x^{q}}
\leq C_r\frac{1}{\,\lambda^{\frac{1}{4r}}}
\|v \|_{L^{2r}_t L_x^{p_1}}^\frac{1}{2}
\|\omega \|_{L^{2r}_t L_x^{p_2}},
\end{equation*}
where $q$ is defined by $1/q:=1/q_1+1/q_2-(r-1)/dr$.
Let $r\in (1,\infty)$, $p_1\in (d/2,d)$, $p_3> dr/(r-1)$ and $p_2$ be given by $1/p_1+1/p_2=1/p_3$. Let
$t^\gamma_1 v \in L^{2r}_t L^{L^{p_3}}_x$ and $t^{\beta} \omega\in L^{2r}_t L^{p_2}_x$. Defining
\begin{equation*}
h_{\lambda}(s,t) :=
\exp
\Big\{
- \lambda\int_s^t \tau^{2r\gamma_1}\|v (\tau)\|_{L^{p_3}_x}^{2r}\dd\tau
- \lambda\int_s^t \tau^{2r\beta}\|\omega (\tau)\|_{L^{p_2}_x}^{2r}\dd\tau
\,\Big\},
\end{equation*}
where $\lambda$ is a positive constant, there exists a positive constant $C_r$ such that
\begin{align}
\|t^{\beta_1}\Bb(v\omega)_\lambda (t)\|_{L^{2r}_t L^{p_2}_x}
\frac{C_r}{\lambda^{\frac{1}{2r}}}\|t^{\beta}\omega_\lambda \|_{L^{2r}_t L^{p_2}_x},\label{Lemma_A.3_est1}\\
\|t^{\beta_1}\Bb(v\omega)_\lambda (t)\|_{L^{2r}_t L^{p_2}_x}
\frac{C_r}{\lambda^{\frac{1}{2r}}}\|t^{\gamma_1}v_\lambda \|_{L^{2r}_t L^{p_3}_x}.\label{Lemma_A.3_est2}
\end{align}
Remark <ref> yields that there exists a positive constant $C$ such that
\begin{equation}\label{Lemma_A.3_first_inequality}
\begin{aligned}
t^{\beta}\|&\Bb(v\omega)_\lambda (t)\|_{L^{p_2}_x}
\leq
\int_0^t \frac{t^{\beta_1}}{|t-s |^{\frac{d}{2p_3}+\frac{1}{2}}s^{\alpha_2}}
s^{\beta_1}\|\omega_\lambda(s)\|_{L^{p_2}_x}\dd s\\
\Big(
\int_0^t h_\lambda (s,t)^{2r}s^{2r\gamma_1}\|v(s)\|_{L^{p_3}_x}\dd s
\Big)^{\frac{1}{2r}}
\Big(
\int_0^t\Big| \frac{t^{\beta_1}}{|t-s |^{\frac{d}{2p_3}+\frac{1}{2}}s^{\alpha_2}}
F(s)\Big|^{(2r)'}\dd s
\Big)^{\frac{1}{(2r)'}}.
\end{aligned}
\end{equation}
Hence, raising to the power of $(2r)'$ both the left-hand and the right-hand sides, we get
\begin{equation*}
\begin{aligned}
t^{(2r)'\beta_1}\|\Bb(v\omega)_\lambda (t)\|_{L^{p_2}_x}^{(2r)'}
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}
\int_0^t\Big| \frac{t^{\beta_1}}{|t-s |^{\frac{d}{2p_3}+\frac{1}{2}}s^{\alpha_2}}
s^{\beta_1}\|\omega(s)\|_{L^{p_2}_x}\Big|^{(2r)'}\dd s\\
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}\int_0^1
\Big|
\frac{t^{\beta_1-\alpha_2 - \frac{N}{2p_3}-\frac{1}{2}}}
{ |1-\tau |^{\frac{d}{2p_3}+\frac{1}{2}}\tau^{\alpha_2}}
\end{aligned}
\end{equation*}
where $F(s) = s^{\beta}\|\omega_\lambda(s)\|_{L^{p_2}_x}$. Observing that
$ \beta-\alpha_2 - N/(2p_3)-1/2 = 1/(2r)-1 = -1/(2r)'$,
we get
\begin{equation}\label{estimate_lambda1}
t^{(2r)'\beta_1}\|\Bb(v\omega)_\lambda (t)\|_{L^{p_2}_x}^{(2r)'}\lesssim
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}
\int_0^1
\Big|
\frac{1}{ |1-\tau |^{\frac{d}{2p_3}+\frac{1}{2}}\tau^{\alpha_2}}
\end{equation}
Hence, applying the $L^{(2r)/(2r)'}_t$-norm to both the left and right-hand sides,
\begin{equation*}
\begin{aligned}
\| t^{\beta_1}\Bb(v\omega)_\lambda (t)\|_{L^{2r}_tL^{p_2}_x}^{(2r)'}&\lesssim
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}
\int_0^1
\Big|
\frac{1}{ |1-\tau |^{\frac{d}{2p_3}+\frac{1}{2}}\tau^{\alpha_2}}\Big|^{(2r)'}
\Big(\int_0^\infty F(t\tau)^{2r}\,\dd\tau\Big)^{\frac{1}{2r-1}}\dd t\\&\lesssim
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}
\int_0^1
\Big| \frac{1}{ |1-\tau |^{\frac{d}{2p_3}+\frac{1}{2}}\tau^{\alpha_1}}\Big|^{(2r)'}
\dd \tau\|t^{\beta}\omega_\lambda \|_{L^{2r}_t L^{p_2}_x}^{(2r)'},
\end{aligned}
\end{equation*}
thanks to Minkowski inequality. Since $\alpha_1(2r)'<1$ and $(d/(2p_3)+1/2)(2r)'<1$ we finally obtain (<ref>). Now, defining
$F(t):=s^{\gamma_1}\|v_\lambda(s)\|_{L^{p_3}_x}$, we also have
\begin{equation*}
t^{\beta}\|\Bb(v\omega)_\lambda (t)\|_{L^{p_2}_x}
\leq
\Big(
\int_0^t h_\lambda (s,t)^{2r}s^{2r\beta}\|\omega(s)\|_{L^{p_2}_x}\dd s
\Big)^{\frac{1}{2r}}
\Big(
\int_0^t\Big| \frac{t^{\beta}}{|t-s |^{\frac{d}{2p_3}+\frac{1}{2}}s^{\alpha_2}}
F(s)\Big|^{(2r)'}\dd s
\Big)^{\frac{1}{(2r)'}},
\end{equation*}
which is equivalent to (<ref>). Thus, arguing as for proving (<ref>), we also obtain (<ref>).
Let $r\in (2,\infty)$, $p_1\in (dr/(2r-2),N)$ and $p_3\geq Nr/(r-2)$ such that $1/p_1+1/p_2=1/p_3$. Let
$h_\lambda$, $v$ and $\omega$ be defined as in the previous Lemma. Then there exists $C_r>0$ such that
\begin{equation}\label{Lemma_A.4_est}
\begin{aligned}
\|t^{\gamma_1}\Cc(v\omega)_\lambda(t)\|_{L^{2r}_t L^{p_3}_x} +
\|t^{\gamma_2}\Cc(v\omega)_\lambda(t)\|_{L^{\infty}_t L^{p_3}_x}\leq
\frac{C_r}{\lambda^{\frac{1}{2r}}}\|t^{\beta_1}\omega_\lambda \|_{L^{2r}_t L^{p_2}_x},\\
\|t^{\gamma_1}\Cc(v\omega)_\lambda(t)\|_{L^{2r}_t L^{p_3}_x} +
\|t^{\gamma_2}\Cc(v\omega)_\lambda(t)\|_{L^{\infty}_t L^{p_3}_x}\leq
\frac{C_r}{\lambda^{\frac{1}{2r}}}\|t^{\beta_1}v_\lambda \|_{L^{2r}_t L^{p_3}_x}.
\end{aligned}
\end{equation}
We control the $L^{2r}_t L^{p_3}_x$ norm arguing as in previous proof. Indeed we have
\begin{equation*}
t^{(2r)'\gamma_1}\|\Cc(v\omega)_\lambda (t)\|_{L^{p_3}_x}^{(2r)'}
\leq
\frac{1}{\lambda^{\frac{(2r)'}{2r}}}
\int_0^1
\Big|
\frac{1}{ |1-\tau |^{\frac{d}{2p_2}}\tau^{\alpha_2}}
\end{equation*}
where $F(s)= s^\beta \|\omega_\lambda\|_{L^{p_2}_x}$ or $F(s)= s^{\gamma_1} \|v_\lambda\|_{L^{p_3}_x}$
instead of (<ref>). Let us take in consideration the $L^\infty_t L^{p_3}_x$ norm. With
a direct computation we get
\begin{align*}
\|t^{\gamma_2}\Cc (t)\|_{L^{p_3}_x}
\leq
\Big(\int_0^t \Big| \frac{t^{\gamma_2}}{|t-s|^{\frac{N}{2p_2}}s^{\alpha_2}}\Big|^{r'}\dd
\Big(\int_0^th(s,t)^r s^{r\gamma_1}\|v(s)\|_{L^{p_3}_x}^rs^{r\beta_1}\|\omega(s)\|_{L^{p_2}_x}^r
\dd s\Big)^{\frac{1}{r}}\\
\leq
C \Big(\int_0^1 \Big| \frac{t^{\gamma_2-\alpha_2-\frac{N}{2p_2}}}
{|1-\tau|^{\frac{N}{2p_2}}\tau^{\alpha_2}}\Big|^{r'}t\, \dd \tau\Big)^{\frac{1}{r'}}
\Big(\int_0^th(s,t)^r s^{r\gamma_1}\|v(s)\|_{L^{p_3}_x}^rs^{r\beta_1}\|\omega(s)\|_{L^{p_2}_x}^r
\dd s\Big)^{\frac{1}{r}}
\end{align*}
Thus, observing that $\gamma_2-\alpha_2-d/(2p_2)+1/r'=0$, $dr'/(2p_2)<1$ and $\alpha_2r'<1$, we conclude that
\begin{align*}
\|t^{\gamma_2}\Cc (t)\|_{L^{p_3}_x}&\leq
\bar{C}_r
\Big(
\int_0^t h_\lambda (s,t)^{2r}s^{2r\gamma_1}\|v(s)\|_{L^{p_3}_x}\dd s
\Big)^{\frac{1}{2r}}
\|t^{\beta}\omega_\lambda \|_{L^{2r}_t L^{p_2}_x} \quad\text{and}\\
\|t^{\gamma_2}\Cc (t)\|_{L^{p_3}_x}&\leq
\bar{C}_r
\Big(
\int_0^t h_\lambda (s,t)^{2r}s^{2r\beta}\|\omega(s)\|_{L^{p_2}_x}\dd s
\Big)^{\frac{1}{2r}}
\|t^{\gamma_1}\omega_\lambda \|_{L^{2r}_t L^{p_3}_x},
\end{align*}
for a suitable positive constant $\bar{C}_r$, which finally yields (<ref>).
Let $r\in(1,\infty)$, $p\in (1,dr/(2r-1))$ and $\bar{u}\in\dot{B}_{p,r}^{d/p-1}$. Le us suppose that
\begin{equation*}
f_1\in (L^{r}_t L^{\frac{dr}{3r-2}}_x)^d\cap (L^{r}_t L^{\check{p}}_x)^d,\quad
f_2\in (L^{2r}_tL^{\frac{dr}{2r-1}}_x)^{d\times d}\cap(L^{r}_tL^{\frac{dr}{2(r-1)}}_x)^{d\times d},
\end{equation*}
Let $v$ belongs to $L^{2r}_t L^{dr/(r-1)}_x$ with $\nabla v\in L^{2r}_t L^{\frac{dr}{2r-1}}_x$. Then system
\begin{equation} \label{Stokes_system_with_linear_perturbation}
\begin{cases}
\partial_t u^h + v\,\partial_d u^h -\Delta u^h +\nabla^h\Pi
=f_1^h+\Div f_2^h & \RR_+ \times\RR^d,\\
\partial_t u^d + \nabla^h v \cdot u^h - v\,\Div^h u^h -\Delta u^d +\partial_d\Pi
=f_1^d+\Div f_2^d & \RR_+ \times\RR^d,\\
\Div u = 0 & \RR_+ \times\RR^d,\\
u_{|t=0} = \bar{u} &\;\;\quad \quad \RR^d,\\
\end{cases}
\end{equation}
admits a weak solution $(u,\Pi)$, such that $u$ belongs to $L^{2r}_tL^{\frac{dr}{r-1}}_x$ with $\nabla u$ in $L^{2r}_tL^{\frac{dr}{2r-1}}_x$ and
$\Pi$ in $L^{r}_tL^{\frac{dr}{2(r-1)}}_x$.
For all $u$ in $L^{2r}_t L^{dr/(r-1)}_x)^d $ with $ \nabla u \in L^{2r}_t L^{dr/(2r-1)}_x$, let $g(u)$ be defined by
\begin{equation}\label{def_g_apx}
g(u):= (- v\,\partial_d u^h ,
-\nabla^h v\cdot u^h +v\,\Div ^h u^h )\in L^{r}_t L^{\frac{dr}{3r-2}}_x.
\end{equation}
Then, the momentum equations of (<ref>) reads as follows:
\begin{equation}\label{Stokes_system_with_linear_perturbation_g}
\partial_t u -\Delta u +\nabla \Pi =g(u)+f_1+\Div f_2 \quad\text{in}\quad \RR_+ \times\RR^d,
\end{equation}
We want to prove the existence of a weak solution for this system, using the Fixed-Point Theorem. We define the functional space $Y_r$ by
\begin{equation*}
Y_r:= \Big\{ u\in L^{2r}_t L^{\frac{dr}{r-1}}_x
\quad \text{such that}\quad
\nabla u\in L^{2r}_t L^{\frac{dr}{2r-1}}_x \Big\},
\end{equation*}
then, fixing a positive constant $\lambda$, we consider the norm $\|\cdot \|_\lambda$ on $Y_r$, defined by
\begin{equation*}
\|u\|_\lambda :=
\| u(t)\,h_\lambda (0,t) \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}
+\| \nabla u(t) \,h_\lambda (0,t) \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x},
\end{equation*}
where, for all $0\leq s\leq t\leq \infty$,
\begin{equation}\label{definition_h}
\begin{aligned}
h_\lambda (s,t):= \exp
\bigg\{
&-\lambda\Big( \int_s^t \| v(\tau)\|_{L^{\frac{dr}{r-1}}_x}^{2r}
+ \int_s^t \| \nabla v(\tau)\|_{L^{\frac{dr}{2r-1}}_x}^{2r}
+ \int_s^t \| \nabla v(\tau)\|_{L^{q}_x}^{2r}\Big)
\Big\}\leq 1.
\end{aligned}
\end{equation}
Let $\Psi$ be the operator from $Y_r$ to itself, such that, for all $\omega\in Y_r$, $\Psi(\omega)$ is the velocity of the weak solution of
\begin{equation*}
\begin{cases}
\partial_t u -\Delta u +\nabla \Pi
=g(\omega)+f_1+\Div f_2 & \RR_+ \times\RR^d,\\
\Div u = 0 & \RR_+ \times\RR^d,\\
u_{|t=0} = \bar{u} &\;\;\quad \quad \RR^d.\\
\end{cases}
\end{equation*}
Let us prove that, for a good choice of $\lambda$, $\Psi$ is a contraction on $Y_r$.
First of all, for all $\omega_1,\,\omega_2\in Y_r$, the difference $\delta\Psi:=\Psi(\omega_1)-\Psi(\omega_2)$ is the velocity field of the weak solution of
\begin{equation*}
\begin{cases}
\partial_t \delta\Psi -\Delta \delta\Psi +\nabla \Pi
=g(\delta \omega) & \RR_+ \times\RR^d,\\
\Div\, \delta\Psi = 0 & \RR_+ \times\RR^d,\\
\delta \Psi_{|t=0} = 0 &\;\;\quad \quad \RR^d,\\
\end{cases}
\end{equation*}
where $\delta \omega := \omega_1-\omega_2$. Since the Mild formulation yields
\begin{equation*}
\delta\Psi(t)= \int_{0}^t e^{(t-s)\Delta}\PP g(\delta \omega)(s)\dd s,
\end{equation*}
then, by the definition (<ref>) of $g$, Lemma <ref> and Lemma <ref> the following inequality is fulfilled:
\begin{equation*}
\|\delta\Psi\|_\lambda\leq
\frac{C}{\lambda^{4r}}\Big\{
\| v \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}^\frac{1}{2}
\| \delta\nabla\omega (t)h(0,t)\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+
\| \nabla v\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}^{\frac{1}{2}}
\| \delta\omega(t)h(0,t)\|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\Big\}.
\end{equation*}
Imposing $\lambda>0$ big enough we finally obtain $\|\delta\Psi\|_\lambda\leq \|\delta\omega\|_\lambda/2$, namely $\Psi$ is a contraction on $Y_r$.
Then, by the Fixed-Point Theorem, there exists a function $u$ in $Y_r$ such that, $u$ is the velocity field of the weak solution $(u,\Pi)$ of
(<ref>). Let us remark that $\nabla u$ belongs also to $L^{r}_tL^{dr/(2r-2)}_x$. Indeed $\nabla u$ is formulated by
\begin{equation*}
\begin{aligned}
\nabla u(t):= e^{t \Delta}\nabla \bar{u}\,+&\int_{0}^t \nabla e^{(t-s)\Delta}\PP
\left(f_1(s)+g(u)(s)\right)\dd s\,+\\
&-\int_{0}^t \Delta e^{(t-s)\Delta}R R R\cdot R\cdot f_2(s)\dd s-
\int_{0}^t \Delta e^{(t-s)\Delta} R\cdot R\cdot f_2(s)\dd s,
\end{aligned}
\end{equation*}
then the result holds thanks to Corollary <ref>, Lemma <ref> and Theorem <ref>.
Finally, recalling that $\Pi$ is determined by
\begin{equation*}
\Pi :=
-\left(-\Delta\right)^{-\frac{1}{2}}R\cdot \left(f_1 + g(u)\right)
-R\cdot R\cdot f_2,
\end{equation*}
we deduce that $\Pi$ belongs to $L^{r}_t L^{dr/(2r-2)}_x$, by Corollary <ref>.
If we add a small extra regularity on $\bar{u}$ in Theorem <ref> assuming $\bar{u}$ in
$\BB_{p,r}^{d/p-1+\ee}$, with $\ee<\min\{ 1/(2r), 1-1/r, 2(d/p-2+1/r)\}$, the weak solution $(u,\,\Pi)$ fulfills also
\begin{equation*}
u\in L^{2r}_t L^{\frac{2dr}{(2-\ee)r-2}}_x\cap L^{\frac{4r}{2-\ee r}}_t L^\frac{dr}{r-1}_x \quad\text{with}\quad
\nabla u\in L^{2r}_t L^{\frac{2dr}{(4-\ee)r-2}}_x\cap L^{\frac{4r}{2-\ee r}}_t L^\frac{dr}{2r-1}_x.
\end{equation*}
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1511.00521
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In randomized experiments with noncompliance, tests may focus on compliers rather than on the overall sample. Rubin (1998) put forth such a method, and argued that testing for the complier average causal effect and averaging permutation based $p$-values over the posterior distribution of the compliance status could increase power, as compared to general intent-to-treat tests. The general scheme is to repeatedly do a two-step process of imputing missing compliance statuses and conducting a permutation test with the completed data. In this paper, we explore this idea further, comparing the use of discrepancy measures, which depend on unknown but imputed parameters, to classical test statistics and exploring different approaches for imputing the unknown compliance statuses. We also examine consequences of model misspecification in the imputation step, and discuss to what extent this additional modeling undercuts the permutation test's model independence. We find that, especially for discrepancy measures, modeling choices can impact both power and validity. In particular, imputing missing compliance statuses assuming the null can radically reduce power, but not doing so can jeopardize validity. Fortunately, covariates predictive of compliance status can mitigate these results. Finally, we compare this overall approach to Bayesian model-based tests, that is tests that are directly derived from posterior credible intervals, under both correct and incorrect model specification. We find that adding the permutation step in an otherwise Bayesian approach improves robustness to model specification without substantial loss of power.
§ INTRODUCTION
Noncompliance refers to the situation when the actual treatment received does not perfectly correspond to treatment assigned in a randomized experiment.
With noncompliance, a simple intent-to-treat analysis, which compares outcomes based on the assignment, can fail to estimate the effect of the treatment itself.
The usual approach in these circumstances is to use instrumental variables (IV)<cit.>, which can be thought of as a special case of principal stratification <cit.>, where the units are partitioned into subpopulations defined by the compliance behavior, and where the focus is then on the effect of the treatment among compliers, i.e., those who would always comply to the treatment assigned irrespective of the arm.
This subpopulation effect is often referred to as the complier average causal effect (CACE).
In an IV analysis, identification of such effect hinges on the absence of defiers (monotonicity assumption) and the lack of any effect of the assignment for noncompliers (the exclusion restriction assumption).
Under these assumptions, a zero intent-to-treat effect (ITT) is a necessary and sufficient condition for CACE being zero as well.
Therefore, a hypothesis test for ITT will be a valid test to assess a zero treatment effect for compliers.
Nevertheless, ITT tests ignore all compliance information.
This is where <cit.> attempted to obtain more power by focusing on testing procedures that incorporate compliance information.
This was motivated by the observation that, taking the presence of noncompliance into account, rather than just estimating an overall average effect, would exploit the information of the data to a larger extent.
On the other hand, such a test seems challenging to construct because the compliance status is typically not known for all units.
Rubin's proposal was to average $p$-values over the posterior predictive distribution of the vector of compliance statuses.
Under an MCMC perspective this translates into including an imputation or data augmentation step, where each unit's compliance status is imputed from its posterior predictive distribution and then a $p$-value for the complier effect is generated from the complete data within each MCMC iteration.
This Bayesian approach to $p$-values has its roots in the posterior predictive model-checking method <cit.>, a popular model-checking tool that can account for nuisance parameters by comparing the observed data to synthetic data drawn from the posterior predictive distribution of a hypothesized model marginalized over the nuisance parameters.
<cit.> and <cit.> extended posterior predictive checks by replacing classical test statistics with discrepancy variables, that is, variables that depend on the nuisance parameters, and this extension can also be applied to testing.
Finally, <cit.> applied the idea of Bayesian posterior $p$-values to Fisher randomization tests <cit.> for experimental designs in the presence of noncompliance.
Fisher randomization tests (FRTs), or permutation tests, are nonparametric tests that can be used to test a null hypothesis on the outcome distribution of a randomized experiment.
The reference distribution of a test statistic is derived by computing its value for each possible permutation of the assignment vector under the known assignment mechanism.
In this paper, we explore this general idea of posterior predictive Fisher randomization tests more in depth, and conduct extensive simulation studies to show how these tests play out in practice in randomized experiments with noncompliance.
The combination of Fisher randomization tests with posterior predictive $p$-values lead to a sequence of both a permutation and an imputation step.
At each iteration,
a test statistic, in its classical view, is computed from the data that would have been observed under the permuted assignment vector and the imputed compliance status vector.
<cit.> proposed the use of any estimator of the complier average casual effect as classical tests statistic.
Here, we first examine the replacement of such test statistics with discrepancy measures.
In noncompliance settings, a discrepancy measure for obtaining posterior predictive FRTs would be an estimate of CACE conditional on the imputed compliance statuses.
These measures seem promising because they can directly estimate CACE from the complete data.
We investigate the benefits and disadvantages of using discrepancy measures for testing as compared to test statistics.
We also closely examine the imputation step.
Different methods are possible, such as imputing under the null or under the alternative.
Imposing the null seems a natural choice from a testing approach, and is also in line with posterior predictive checks, where nuisance parameters are typically drawn from the assumed model.
Imposing the null also protects the validity of the overall testing procedure.
However, this approach turns out to have some potential costs in terms of power.
We explore this, and discuss how to mitigate these costs.
The imputation step without the permutation step is nothing more than what would typically be used for the direct estimation of the posterior distribution of CACE.
Credible intervals of this posterior distribution could themselves lead to nominal $p$-values.
These model-based $p$-values are less computationally demanding than the posterior predictive $p$-values obtained by a Fisher randomization test within the imputation step.
We compare these two approaches, and see if the addition of the Fisher randomization test has anything extra to offer.
In particular, we compare the performance of these two testing methods under both correct and incorrect model specification.
Simulation studies suggest several “best practices” as well as help elucidate the reasons of why each approach can work.
Finally, we discuss how predictive covariates are of particular importance for the performance of discrepancy-based FRTs.
Predictive covariates help alleviate model misspecification concerns which can arise in the imputation step, and help address many of the concerns found in our investigations.
The paper is organized as follows.
In Section <ref>, we give a brief review of posterior predictive checks.
We then introduce Fisher randomization tests in Section <ref> and illustrate the use of posterior predictive $p$-values to deal with the unknown compliance status in noncompliance settings.
In Section <ref>, we describe the simulation studies we constructed to compare the use of discrepancy measures versus tests statistics, assess the impact of different modeling assumptions, and determine the potential benefits of predictive covariates.
Simulation results are shown in Section <ref>.
In Section <ref> we compare the validity of FRTs in combination with posterior predictive $p$-values to the corresponding Bayesian model-based tests.
We discuss common patterns and what they suggest for practice, in Section <ref>.
§ POSTERIOR PREDICTIVE CHECKS
Classical $p$-values were extended to the Bayesian framework by <cit.> and <cit.>.
The Bayesian framework is particularly appealing for investigating the compatibility of a posited model with observed data when the model has unknown nuisance parameters (composite null model).
While classical method would typically plug-in a point estimate of the parameter and rely on known reference distributions of pivotal quantities or on asymptotic results, Bayesian tests average over the posterior distribution of the unknown parameters and use the posterior predictive distribution to simulate the reference distribution for any test statistic.
Suppose we have a realization $Y^{obs}$ of a random variable $Y$ and we posit a parametric null model,
$H_0: \vY \sim f(\vY\mid \vtheta), \vtheta \in \Theta_0$. The essence of model assessment lies in comparing the observed data with hypothetical replicates that could be observed under the assumed model.
The classical approach amounts to measuring the discrepancy between the observed value of a test statistic, $T(\vYobs)$, and its reference (i.e., sampling) distribution derived under the posited model.
A Bayesian model checking approach uses the posterior predictive distribution under the null hypothesis, $p(\vY|\vYobs, H_0)$, to derive the posterior distribution of the test statistic. Assuming a test statistic $T(\vYobs)$ where larger values contraindicate the null, the posterior predictive $p$-value based on the test statistic, $p_{B_T}$, is then
\begin{equation}
\begin{aligned}
\label{eq:pBT}
p_{B_T}=&Pr_{\vY}\left\{T(\vY)\geq T(\vYobs) \mid \vYobs, H_0\right\} \\
=&E_{\vtheta}\left\{Pr_{\vY}\left\{T(\vY)\geq T(\vYobs) \mid \vYobs, H_0, \theta \right\} \mid \vYobs, H_0 \right\} .
%=& \int_{\vtheta \in \Theta_0} \int_{\vY \in A_T}f(\vY\mid \vtheta)\pi(\theta\mid \vYobs)d\vY d\vtheta
\end{aligned}
\end{equation}
The presence of unknown parameters has been taken into account by averaging over their posterior distribution under the null hypothesis.
A Monte-Carlo simulation-based approach would draw $K$ values of the parameters, {$\vtheta^k;\,\,\, k=1,\ldots,K\}$, from their posterior distribution $\pi(\vtheta|\vYobs, H_0)$, simulate replications of the data under the conditional distribution $f(\vY\mid \vtheta^k)$ and compare the new values of the test statistic with the observed value.
This approach follows from an equivalent expression of equation (<ref>):
\begin{equation}
p_{B_T}= \int_{\vtheta \in \Theta_0} \int_{\vY}\mathbf{1}\big[T(\vY)\geq T(\vYobs)\big]f(\vY\mid \vtheta)\pi(\theta\mid \vYobs, H_0)d\vY d\vtheta
\end{equation}
where $\mathbf{1}[\cdot]$ is the indicator function.
Meng (1994) and later Gelman, Meng and Stern (1996) proposed to replace classical test statistics, $T(\vY)$, with parameter-dependent statistics, $D(\vY, \vtheta)$, referred to as discrepancy variables.
These cannot translate to the classical framework because the parameter values are unknown.
In the Bayesian framework, however, they can be used as the posterior gives us predictions for these parameters and so both the “observed” discrepancy variables as well as its reference distribution can be calculated.
In particular, the posterior predictive $p$-value based on a discrepancy variable, $p_{D_T}$, is
\begin{equation}
\begin{aligned}
\label{eq:pBD}
p_{B_D}=&E_{\vtheta}\left\{Pr_{\vY}\left\{D(\vY, \vtheta)\geq D(\vYobs, \vtheta) \mid \vYobs, H_0, \vtheta \right\} \mid \vYobs, H_0 \right\}\\
=& \int_{\vtheta \in \Theta_0} \int_{\vY}\mathbf{1}\big[ D(\vY, \vtheta)\geq D(\vYobs, \vtheta)\big]f(\vY\mid \vtheta)\pi(\theta\mid \vYobs, H_0)d\vY d\vtheta .
\end{aligned}
\end{equation}
Note how both $D(\vY, \vtheta)$ and $D(\vYobs, \vtheta)$ are random under the posterior distribution.
This approach has two advantages. First, a discrepancy variable requires smaller computational effort than a test statistic, given that the latter often involves an additional estimation of the parameters.
Second, classical test statistics are typically computed by plugging-in an estimate of the nuisance parameter, thus they assess the `discrepancy' between the data and the best fit of the model. Conversely, the use of a parameter-dependent statistic directly checks the `discrepancy' between the data and the overall model.
<cit.> and <cit.> derived several results on the frequency evaluation of discrepancy $p$-values under the null. If $D(\vY, \vtheta)$ is a pivotal quantity with known distribution $\mathcal{D}_0$ under the null, then the distribution of $p$-values under the null would still be uniform and their expression would simplify to $p_{B_D}=Pr\left\{\mathcal{D}_0 \geq D(\vYobs, E[\vtheta\mid \vYobs, H_0]) \right\}$.
On the other hand, in the more common situation where the discrepancy is not a pivotal quantity, averaging over the parameters on which the discrepancy depends leads to a distribution of $p$-values that is no longer uniform. Meng investigates the behavior of such $p$-values under the prior predictive distribution conditional on the null, i.e., $p(\vY\mid H_0)$. His main result is that, under this distribution, discrepancy $p$-values are centered around $\frac{1}{2}$, i.e., $E_{\vY}\{p_{B_D} \mid H_0\}=1/2$ and that $Pr_{\vY}\left\{p_{B_D} \leq \alpha \mid H_0 \right\} \leq 2\alpha$.
This means that there are cases in which $p$-values are conservative and other cases in which they are anti-conservative, but there is a bound for the Type I error of twice the nominal level.
His further discussion suggests, however, that in practice we would expect error rates to rarely be this high.
As the posterior $p$-values are stochastically less variable than $U[0,1]$, we expect the tails to be lighter, and the error rates to often be conservative for low values of $\alpha$.
Extending this work, <cit.> show that discrepancy-based $p$-values can be seriously conservative even when the discrepancy has asymptotic mean 0 for all values of the nuisance parameters, whereas posterior predictive $p$-values based on test statistics are conservative whenever the asymptotic mean of the test statistic depends on the parameters.
Arguably, the conservativeness of a test is not a bad thing per se, because it means that it would not reject a true hypothesis more often than indicated by the nominal level.
Indeed, such tests are considered valid <cit.>, as the Type I error would be less or equal to $\alpha$.
According to <cit.>, the typical conservativeness when using discrepancies, noted by <cit.> and <cit.>, arises from the `extra' information carried by the imputations of $\vtheta$.
This information can be traced to both modeling and structural assumptions used to define the posterior distributions used for the imputation; a fundamental role is played by the the fact that imputations are performed under the null hypothesis.
This argument can be connected to the one for potential conservatism of multiple imputation in <cit.>, where these informative imputations are called `superefficient'.
§ FISHER RANDOMIZATION TESTS USING POSTERIOR PREDICTIVE $P$-VALUES
<cit.> proposed a distribution-free technique to test a sharp null hypothesis of zero treatment effect at the unit level for randomized experiments.
<cit.> then showed how these Fisher's randomization tests (FRTs) can be formally viewed as a posterior predictive check.
The Bayesian justification is that they are based on the posterior predictive distribution of the test statistic induced by the random assignment $\vZ$.
Although Fisher never used the potential outcomes framework—a method of articulating causal effects originally proposed by
Neyman in the context of randomized experiments <cit.> and then formalized and extended to observational studies by Rubin <cit.>—FRTs can be phrased in terms of potential outcomes.
Under the potential outcome framework, the potential outcomes, denoted $Y_i(1)$ and $Y_i(0)$, represent the
outcomes for individual $i$ had he received the treatment ($Z_i=1$) or control ($Z_i=0$) respectively. Let $Z^{obs}_i$ be the treatment that was actually assigned to unit $i$.
The “fundamental problem of causal inference” <cit.> is that, for each individual, we can observe only one of these potential outcomes, i.e., $Y_i^{obs}=Y_i(1)Z^{obs}_i +Y_i(0)(1-Z^{obs}_i)$, because each unit will receive either treatment or control.
As first formalized in <cit.>, all causal effects are inherently a comparison of potential outcomes. Thus, Fisher's sharp null hypothesis $H_0$ of no treatment effect can be formalized as $Y_i(0)=Y_i(1) \, \, \forall i$ and Fisher's $p$-values can be stated as $Pr_{\vZ}\left\{T(\vY(\vZ),\vZ)\geq T(\vYobs, \vZobs) \mid \vZobs, \vYobs, H_0 \right\}$. Fisher's hypothesis is said to be sharp because it allows one to perfectly impute $Y_i(1-Z_i)$ for any value of $Z_i$ (i.e., the missing potential outcome).
<cit.> then formalized the idea of Fisher randomization-based tests using posterior predictive $p$-values (FRT-PP) in the context of noncompliance. Let $D_i$ be the actual treatment received by unit $i$; with noncompliance $D_i$ may differ from $Z_i$. The compliance type for each unit is then defined by the joint values of the treatment receipt if assigned to control, $D_i(0)$, or to treatment $D_i(1)$. Because we can never observe both $D_i(0)$ and $D_i(1)$, however, the compliance status is not generally known for all units.
These are the unknown variables we will average over to obtain posterior predictive $p$-values.
<cit.> took this approach for the case of one-sided noncompliance, where $D_i(0)=0$ for all units.
With one-sided noncompliance, we have two compliance types: `compliers', for whom $D_i(1)=1$ and `never-takers', for whom $D_i(1)=0$. In this case compliance statuses are unknown only for those units in the control arm.
We also discuss one-sided noncompliance.
Let $C_i$ denote the compliance status indicator, being $0$ for a never-taker or $1$ for a complier. The complier average causal effect (CACE) can then be written as
\begin{equation}
CACE:=E[Y_i(1)-Y_i(0) | C_i=1]
\end{equation}
Assuming the exclusion restriction for non-compliers, i.e., $Y_i(0)\!=Y_i(1) \,\, \forall i\!\!: C_i=0$,
the null hypothesis we wish to test here is a null effect for compliers, i.e., $H_0: Y_i(0)=Y_i(1) \,\, \forall i\!\!: C_i=1$. It is worth noting that, in one-sided noncompliance settings under the exclusion restriction, CACE can be expressed as the ratio between the intent-to-treat effect, i.e., $ITT=E[Y_i(1)-Y_i(0)]$, and the probability of being a complier, i.e, $\pi_c=Pr(C_i=1)$. Therefore, in principle a rejection of a zero ITT would necessarily mean a rejection of a zero CACE.
The goal of <cit.> was to show that in some circumstances tests that take noncompliance into account can be more powerful than the ones based on the ITT only.
Rubin proposed the use of a test statistic $T$, which depends on the observed data $O(\vZ)$ for each assignment vector $\vZ$, with $O(\vZ)=[\vY(\vZ), \vD(\vZ), \vZ]$, and not on the imputed compliance statuses.
Test statistics can be any estimator of the complier average causal effect (CACE): posterior mean, posterior median or posterior mode as well as MLE or IV estimates.
Regardless of the choice, the Bayesian procedure averages $p$-values over the posterior predictive distribution of the unknown compliance statuses, $\prob(\vC\mid \vYobs, \vDobs, \vZobs, H_0)$, which will in turn depend on other unknown parameters:
\begin{equation}
\label{eq:pBT_c}
p_{B_T}=E_{\vtheta}\left\{E_{\vC}\left\{p_{B_T}(\vC) \mid \vYobs, \vDobs, \vZobs, \vtheta \right\} \mid \vYobs, \vDobs, \vZobs, H_0 \right\} ,
\end{equation}
\begin{equation*}
\begin{aligned}
p_{B_T}(\vC)&=Pr_{\vZ}\left\{T(\vY(\vZ), \vD(\vZ), \vZ)\geq T(\vYobs, \vDobs, \vZobs) \mid \vYobs, \vDobs, \vZobs, \vC, H_0 \right\}\\
&=Pr_{\vZ}\left\{T(\vYobs, \vC\vZ, \vZ)\geq T(\vYobs, \vDobs, \vZobs) \mid \vC \right\} .
\end{aligned}
\end{equation*}
The last expression follows from two observations: (1) under the null hypothesis $Y_i(Z_i)$ is equal to the observed outcome; and (2) $D_i(Z_i)=C_i Z_i$ thanks to the definition of the compliance status $C_i$ in one-sided noncompliance settings.
We can also use the equivalent expressions of
\begin{equation}
\begin{aligned}
p_{B_T}= \int_{\vtheta \in \Theta} \int_{\vC}\int_{\vZ} &\bigg[\int_{\vY(\vZ)}\int_{D(\vZ)}\mathbf{1}\big[ T\big(\vY(\vZ), \vD(\vZ), \vZ\big)\geq T\big(\vYobs, \vDobs, \vZobs\big)\big]\times\\
&\quad \prob(\vY(\vZ), \vD(\vZ)\mid \vYobs,\vDobs, \vZobs, \vC,H_0)\,d\vY(\vZ)\,d\vD(\vZ)\bigg] \\
&\prob(\vZ)\prob(\vC \mid \vYobs, \vDobs, \vZobs, \vtheta)\pi(\theta\mid \vYobs, \vDobs, \vZobs, H_0)\,d\vZ \,d\vC\, d\vtheta ,
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
p_{B_T}= \int_{\vtheta \in \Theta} \int_{\vC}\int_{\vZ} &\mathbf{1}\big[ T\big(\vYobs, \vC\vZ, \vZ\big)\geq T\big(\vYobs, \vDobs, \vZobs\big)\big]\\
&\prob(\vZ)\prob(\vC \mid \vYobs, \vDobs, \vZobs, \vtheta)\pi(\theta\mid \vYobs, \vDobs, \vZobs, H_0)\,d\vZ \,d\vC\, d\vtheta .
\end{aligned}
\end{equation}
An MCMC approach proceeds, at each iteration $k$, by first drawing the parameters from their posterior distribution and then imputing the missing compliance statuses in the control arm using the posterior predictive distribution conditional on these draws.
Then, for the permutation step, the assignment vector is permuted and the test statistic computed based on the outcome and treatment that would be observed under the new assignment vector, if the imputed compliance statuses were true and the individual treatment effect were null, i.e., $T^k=T\big(\vYobs, \vC\vZ, \vZ\big)$.
The $p$-value $p_{B_T}$ is then the proportion of iterations where the test statistic $T^k$ is greater than or equal to the observed statistic $T^{obs}=T\big(\vYobs, \vDobs, \vZobs \big)$.
Following <cit.> and <cit.>, we explore replacing parameter-independent test statistics with parameter-dependent discrepancy variables.
Rubin (1998) already mentioned the possibility of using discrepancies, such as difference-in-means estimate of the effect among compliers, i.e. $\overline{Y}_{1,c}-\overline{Y}_{0,c} $, but he only used test statistics dependent on $O(\vZ)$ in his examples.
Compliance-dependent discrepancy $p$-values can be written as:
\begin{equation}
\label{eq:pBD_c}
p_{B_D}=E_{\vtheta}\left\{E_{\vC}\left\{p_{B_D}(\vC) \mid \vYobs, \vDobs, \vZobs, \vtheta \right\} \mid \vYobs, \vDobs, \vZobs, H_0 \right\}
\end{equation}
\begin{equation*}
\begin{aligned}
p_{B_D}(\vC)&=Pr_{\vZ}\left\{D(\vY(\vZ), \vC, \vZ)\geq D(\vYobs, \vC, \vZobs) \mid \vYobs, \vDobs, \vZobs, \vC, H_0 \right\}\\
&=Pr_{\vZ}\left\{D(\vYobs, \vC, \vZ)\geq D(\vYobs, \vC, \vZobs) \mid \vC \right\} .
\end{aligned}
\end{equation*}
In the MCMC, at each iteration the imputed compliance statuses will affect both values of discrepancy variable, the one for the permuted assignment vector under the null hypothesis, $D^k=D(\vYobs, \vC, \vZ)$, and the one for the observed values, $D^{k,obs}=D(\vYobs, \vC, \vZobs)$. $p_{B_D}$ is then the proportion of iterations where $D^k\geq D^{k,obs}$.
Our purpose here is to compare the frequentist performance of FRT-PP based on test statistics $T(\vY(\vZ), \vD(\vZ), \vZ)$ or on imputed compliance-dependent discrepancy variables $D(\vY(\vZ), \vC, \vZ)$. In particular, we will use the typical IV estimator of CACE <cit.> as our test statistic:
\begin{equation}
\label{eq:T}
T(\vY(\vZ), \vD(\vZ), \vZ)=\frac{\widehat{ITT}_Y}{\hat{\pi}_c}=\frac{\overline{\vY}_{1}-\overline{\vY}_{0}}{\overline{\vD}_{1}-\overline{\vD}_{0}} ,
\end{equation}
where $\widehat{ITT}_Y$ is the method of moments estimator of the effect of the assignment on the outcome, $\hat{\pi}_c$ is the method of moments estimator of the proportion of compliers,
\[\overline{\vY}_{1}=\frac{\sum_{i}Y_i(Z_i)Z_i}{\sum_{i}Z_i} \qquad\overline{\vY}_{0}=\frac{\sum_{i}Y_i(Z_i)(1-Z_i)}{\sum_{i}(1-Z_i)} \]
\[\overline{\vD}_{1}=\frac{\sum_{i}D_i(Z_i)Z_i}{\sum_{i}Z_i} \qquad\overline{\vD}_{0}=\frac{\sum_{i}D_i(Z_i)(1-Z_i)}{\sum_{i}(1-Z_i)} . \]
Similarly, the discrepancy variable will be the method of moment estimator of the complier average causal effect if compliance status were known:
\begin{equation}
\label{eq:D}
D(\vY(\vZ), \vC, \vZ)=\overline{\vY}_{c1}-\overline{\vY}_{c0}
\end{equation}
\[\overline{\vY}_{c1}=\frac{\sum_{i}Y_i(Z_i)C_iZ_i}{\sum_{i}C_iZ_i} \qquad\overline{\vY}_{c0}=\frac{\sum_{i}Y_i(Z_i)C_i(1-Z_i)}{\sum_{i}C_i(1-Z_i)} \]
Based on our previous discussion on the typical conservativeness of discrepancy-based $p$-values, we expect the use of these $p$-values for problems of noncompliance to lead to conservative tests. Simulations under the null, not shown here, confirm our hypotheses. In fact, in the hypothetical situation where compliance statuses were known for all units, the distribution of $p$-values would still look uniform. On the other hand, when compliance statuses were imputed for units assigned to control, the distribution of $p$-values seemed to be concentrated around 0.5. We tested this result in both the hypothetical case where
compliance statuses are imputed from the correct model with known parameters $\prob(\vC\mid \vYobs, \vDobs, \vZobs; \vtheta^{\star})$ and the more realistic situation where the parameters are not known.
In the latter case, the Bayesian procedure that follows from the definition of $p$-values in (<ref>) and (<ref>) uses the posterior predictive distribution of the unknown compliance statuses, which in turn depends on the posterior distribution of the parameters:
$\prob(\vC\mid \vYobs, \vDobs, \vZobs, H_0)= \int_{\theta} \prob(\vC\mid \vYobs, \vDobs, \vZobs, \vtheta) \pi(\vtheta\mid \vYobs, \vDobs, \vZobs, H_0) d\vtheta$.
Imputation was still conducted under the correct model.
The conservativeness that was seen in this case is presumably due to the fact that the imputed replications of compliers would carry information on the null hypothesis, given that the parameters were estimated under the correct imputation model (consisting of a correct model specification, the exclusion restriction assumption, and the null hypothesis). In fact, the discrepancy measure makes use of the information provided by both the data and the model that is stored in the imputed compliance statuses.
Most of the past literature focuses on the comparison of $p$-values based on discrepancies and classical statistics under the null hypothesis. There seems to be less work concerning power.
We should bear in mind that type I error and power are usually traded off against each other:
a test that is less likely to reject a correct hypothesis is usually also less likely to reject an incorrect one.
The purpose of this paper is to shed light on this trade-off for discrepancy-based $p$-values in noncompliance settings.
Does a reduction in the type I error to below $\alpha$ levels signal a loss of power? When is this trade-off more of an issue?
Power simulations in the hypothetical case with known parameters and correct (alternative) model specification showed that discrepancy measures can give a large increase in power.
However, in the more realistic situation with unknown parameters, the literature on posterior predictive $p$-values suggests to estimate the parameters under the null hypothesis.
Conditioning on the null-hypothesis is motivated by the need for obtaining a valid test under the null.
However, under a true alternative, imputation of compliance statuses would then be conducted under the wrong model which could result in a loss of power to detect a non-zero CACE.
The intuition is that there might be scenarios where the distributions of outcomes for compliers and never-takers under active and control treatment assignment are such that in the control arm units imputed as compliers from the posterior distribution conditional on the null hypothesis end up being the ones with outcomes close to the compliers in the treatment group, resulting in small values of the observed complier average causal effect, i.e., $D(\vYobs, \vDobs, \vZobs)$.
This potential problem motivated us to investigate an imputation of the compliance statuses from the posterior distribution $\prob(\vC\mid \vYobs, \vDobs, \vZobs)$ without imposition of the null as an alternative.
This is akin to a “plug-in” style approach in classical testing.
However, due to the same trade-off mentioned earlier, this relaxation could lead to an increase in type I error, possibly giving an invalid test.
We will also investigate how the use of observed covariates, when available, can improve the imputation and thus the performance of $p$-values under either approach.
Given these tensions, we conducted a simulation study in order to assess whether these trade-offs indeed occur, and in order to provide clear recommendations on what is best procedure.
In particular, we wish to compare the overall accuracy of discrepancy-based and statistic-based FRT-PP under different compliance imputation models.
§ SIMULATION STUDY
Our simulation study is designed to assess the performance of different types of randomization-based $p$-values in testing the sharp null hypothesis of no treatment effect for compliers, i.e., $H_0: Y_i(0)=Y_i(1) \,\,\forall i: C_i=1$, under the exclusion restriction assumption, i.e., $Y_i(0)=Y_i(1) \forall i: C_i=0$.
Specifically, we compared $p$-values from (<ref>) using the test statistic in (<ref>) and $p$-values from (<ref>) based on the discrepancy variable in (<ref>).
We also used four different methods for imputing the compliance status in the control arm:
|
1511.00172
|
We present results on the broadband nature of the power spectrum $S(\omega)$, $\omega\in(0,2\pi)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps $f:[0,1]\to[0,1]$ with $f(x)\approx x^{1+\gamma}$ for $x\approx 0$, where $\gamma\in(0,1)$.
Such maps have summable decay of correlations when $\gamma\in(0,\frac12)$, and $S(\omega)$ extends to a continuous function on $[0,2\pi]$ by the classical Wiener-Khintchine Theorem. We show that $S(\omega)$ is typically bounded away from zero for Hölder observables.
Moreover, in the nonsummable case $\gamma\in[\frac12,1)$, we show that $S(\omega)$ is defined almost everywhere with a continuous extension $\tilde S(\omega)$ defined on $(0,2\pi)$, and $\tilde S(\omega)$ is typically nonvanishing.
§ INTRODUCTION
Let $f:X\to X$ be a measure preserving transformation of a probability space
$(X,\mu)$ and let $v:X\to\R$ be an $L^2$ observable.
The power spectrum $S:[0,2\pi]\to\R$ is given by
\[
S(\omega)=\lim_{n\to\infty}\frac1n\int_X\Bigl|\sum_{j=0}^{n-1} e^{ij\omega}v\circ f^j\Bigr|^2\,d\mu.
\]
By the Wiener-Khintchine Theorem <cit.>,
$S(\omega)=\sum_{k=-\infty}^\infty e^{ik\omega}\rho(k)$ where
$\rho(k)=\int_X v\circ f^k\,v\,d\mu-\Bigl(\int_X v\,d\mu)^2$ is the autocorrelation function of $v$.
In particular, the power spectrum is analytic if and only if the autocorrelations decay exponentially.
More generally, the power spectrum is well-defined and continuous provided the autocorrelations are summable.
The power spectrum is often used by experimentalists to distinguish periodic and quasiperiodic dynamics (discrete power spectrum with peaks at the harmonics and subharmonics) and chaotic dynamics (broadband power spectra).
See for example <cit.>.
In the atmospheric and oceanic sciences and in climate science, power spectra have been widely used to detect variability in particular frequency bands (see, for example, <cit.>).
Spectral analysis was successful in detecting dominant time scales in teleconnection patterns, revealing intraseasonal variability in time series of the global atmospheric angular momentum <cit.>, interannual variability in the El Niño/Southern Oscillation system <cit.>, and the Atlantic Multidecadal Variability <cit.>.
On millenial temporal scales the power spectrum was instrumental in unraveling dominant cycles in paleoclimatic records <cit.>.
Despite this widespread applicability across these disparate temporal scales, there are surprisingly few rigorous results on the nature of power spectra of complex systems.
The (quasi)periodic case with its peaks at discrete frequencies is well understood.
The nature of power spectra for chaotic systems was first treated in <cit.> in the case of uniformly hyperbolic (Axiom A) systems.
In our previous paper <cit.>, we considered in more detail the broadband nature of power spectra for chaotic dynamical systems and showed that for certain classes of dynamical systems $f$ and observables $v$, the power spectrum is bounded away from zero.
The main results in <cit.> are for nonuniformly expanding/hyperbolic dynamical systems with exponential decay of correlations. These results are summarised below in Subsection <ref>. The current paper is concerned with systems possessing subexponential — even nonsummable — decay of correlations.
A prototypical example is the class of Pomeau-Manneville intermittent maps <cit.>, specifically the class
considered in <cit.>. These are maps $f:[0,1]\to[0,1]$ given by
\begin{align} \label{eq-LSV}
f(x)=\begin{cases} x(1+2^\gamma x^\gamma), & x\in[0,\frac12) \\
2x-1, & x\in[\frac12,1]
\end{cases},
\end{align}
where $\gamma>0$ is a parameter. For $\gamma\in(0,1)$, there is a unique absolutely continuous invariant probability measure $\mu$, and autocorrelations decay at the rate $O(1/n^\beta)$ for Hölder observables
where $\beta=\gamma^{-1}-1$.
In particular, if $\gamma\in(0,\frac12)$, then the
autocorrelation function is summable and the Wiener-Khintchine Theorem assures that the power spectrum is well-defined and continuous.
We show that the power spectrum is typically bounded away from zero.
Moreover, we show that the same result holds for all $\gamma\in(0,1)$ provided the Hölder
exponent[Recall that if $(X,d)$ is a metric space and $\eta\in(0,1]$, then $v:X\to\R$ is $C^\eta$
(Hölder with exponent $\eta$) if $|v|_\eta=\sup_{x\neq y}|v(x)-v(y)|/d(x,y)^\eta<\infty$.]
of $v$ is sufficiently large. More precisely, we prove:
Suppose that $f:[0,1]\to[0,1]$ is of the type (<ref>) where $\gamma\in(0,1)$.
Suppose that $v:[0,1]\to\R$ is $C^\eta$ where $\eta>0$.
If $\gamma\in(0,\frac12)$,
then the power spectrum is continuous on $(0,2\pi)$ with a continuous extension to $[0,2\pi]$, and is
typically[Throughout this paper “typically” means lying outside a closed subspace of infinite codimension within the Banach space of Hölder observables of a given exponent. Thus Theorem <ref> fails only for infinitely degenerate observables.]
bounded away from zero.
If $\gamma\in[\frac12,1)$, and $\eta>(3\gamma-1)/2$, then the power spectrum is defined almost everywhere with a continuous extension to $(0,2\pi)$. Typically this extension is nonvanishing.
When $\gamma\in(0,\frac12)$,
the result is a special case of a more general result, Theorem <ref>, stated below.
The only part of the result that is new for $\gamma\in(0,\frac12)$ is that the power spectrum is typically bounded below.
The case $\gamma\in[\frac12,1)$ depends more strongly on the details of the system, and seems to be a new result in its entirety.
§.§ The results in <cit.> for systems with exponential decay
The simplest case considered in <cit.> is when $f$ is either a (noninvertible) uniformly expanding map
or a uniformly hyperbolic (Axiom A) diffeomorphism and
$\Lambda$ is a locally maximal transitive subset of $X$.
Suppose first that $\Lambda$ is mixing.
Then $f:\Lambda\to\Lambda$ has exponential decay of correlations for Hölder observables; in particular, the autocorrelation function of $v$ decays exponentially provided $v$ is Hölder.
In this situation, we showed <cit.> that the power spectrum
is bounded away from zero for typical Hölder observables.
Still in the uniformly expanding/hyperbolic setting, it was shown in <cit.> that the mixing condition is unnecessary.
There is an integer $q\ge1$ such that (i) $\Lambda$ is a disjoint union
$\Lambda=\Lambda_1\cup\cdots\cup \Lambda_q$, (ii) $f(\Lambda_i)= \Lambda_{i+1}$ (computing subscripts $\bmod\, q$), and (iii) $f^q:\Lambda_i\to\Lambda_i$ is mixing for each $i$.
Moreover, $f^q:\Lambda_i\to\Lambda_i$ has exponential decay of correlations for Hölder observables. By <cit.>, the power spectrum is analytic with removable singularities at $2\pi j/q$, $j=0,\dots,q$, and typically bounded away from zero.
Large classes of nonuniformly expanding/hyperbolic systems can be treated in the same manner, namely those modelled by the tower construction of Young <cit.>.
These systems include the logistic family, Hénon-like attractors, and planar periodic dispersing billiards. In such cases, the power spectrum is again analytic (up to removable singularities) and typically bounded away from zero.
§.§ Systems with subexponential but summable decay
The current paper is concerned with the case when exponential decay of correlations fails. Young <cit.> considers nonuniformly expanding maps with subexponential decay of correlations.
The precise definition of nonuniformly expanding map is given in Section <ref>.
Although the power spectrum is no longer analytic, we may still discuss its boundedness properties.
If the decay is summable, then the power spectrum is continuous and extends to a continuous (and hence bounded) function on $[0,2\pi]$.
We prove:
Let $f$ be a nonuniformly expanding map with
polynomial decay of correlations at a rate $O(1/n^\beta)$ where $\beta>1$.
Then the power spectrum is bounded away from zero for typical Hölder observables.
This result was claimed in <cit.> but the proof sketched there is incomplete (the iterates of $L^k\hat v$ are summable as claimed, but only in $L^p$ spaces, so the step that involves evaluation at periodic data is problematic).
A full proof is given in this paper.
Moreover, we consider also the case $\beta\in(0,1]$.
The remainder of this paper is organised as follows.
In Section <ref>, we give the definition for nonuniformly expanding map, and state Theorem <ref> which implies Theorem <ref>.
Also, we show how Theorem <ref> follows from Theorem <ref>.
In Section <ref>, we prove Theorem <ref>.
§ NONUNIFORMLY EXPANDING MAPS
Let $(X,d)$ be a locally compact separable bounded metric space with
Borel probability measure $m_0$ and let $f:X\to X$ be a nonsingular
transformation for which $m_0$ is ergodic.
Let $Y\subset X$ be a measurable subset with $m_0(Y)>0$, and
let $\alpha$ be an at most countable measurable partition
of $Y$. We suppose that there is an $L^1$
return time function $r:Y\to\Z^+$, constant on each $a\in\alpha$ with
value $r(a)\ge1$, and constants $\lambda>1$, $\eta\in(0,1)$, $C\ge1$,
such that for each $a\in\alpha$,
(1) $F=f^{r(a)}:a\to Y$ is a measure-theoretic bijection.
(2) $d(Fx,Fy)\ge \lambda d(x,y)$ for all $x,y\in a$.
(3) $d(f^\ell x,f^\ell y)\le Cd(Fx,Fy)$ for all $x,y\in a$,
$0\le \ell <r(a)$.
(4) $g_a=\frac{d(m_0|{a}\circ F^{-1})}{dm_0|_Y}$
satisfies $|\log g_a(x)-\log g_a(y)|\le Cd(x,y)^\eta$ for all
Such a dynamical system $f:X\to X$ is called nonuniformly expanding.
The induced map $F=f^r:Y\to Y$ is uniformly expanding and
there is a unique $F$-invariant probability measure $\mu_Y$ on $Y$ equivalent
to $m_0|_Y$ with density bounded above and below. Moreover $\mu_Y$ is mixing.
This leads to a unique $f$-invariant probability measure $\mu$ on $X$ equivalent
to $m_0$ (see for example <cit.>).
We assume throughout that
$\gcd\{r(a)-r(b):a,b\in\alpha\}=1$. In particular, $\mu$ is mixing.
The assumption is trivially satisfied for the maps (<ref>) since $\{r(a):a\in\alpha\}=\Z^+$. Such a restriction is not completely avoidable, since
the power spectrum has a finite number of (removable) singularities <cit.> when $F$ is not mixing.
The results in <cit.> apply directly when
$\mu_Y(y\in Y:r(y)>n)$ decays exponentially.
In this paper, we show that the same results hold when $r\in L^{2+}(Y)$, thus proving
Theorem <ref>.[Throughout, we write $\phi\in L^{p+}(Y)$ as shorthand for $\phi\in L^{p+\eps}(Y)$ for some $\eps>0$.]
In addition, we obtain results in the case $r\in L^{1+}(Y)$.
Given $v\in L^\infty(X)$ and
$\omega\in[0,2\pi]$, we define
the induced observable $V_\omega:Y\to\C$,
\begin{align*}
V_\omega(y)=\sum_{\ell=0}^{r(y)-1}e^{i\ell\omega}v(f^\ell y).
\end{align*}
For all $\omega_0,\omega\in[0,2\pi]$, $a\in\alpha$,
\begin{align*}
|1_aV_\omega|_\infty\le |v|_\infty r(a), \quad
2|v|_\infty r(a)^2|\omega-\omega_0|.
\end{align*}
The first estimate is immediate. Also, for $y\in a$,
\[
\sum_{\ell=0}^{r(a)-1}|e^{i\ell\omega}-e^{i\ell\omega_0}||v|_\infty
\le 2\sum_{\ell=0}^{r(a)-1}|\ell(\omega-\omega_0)||v|_\infty
\le 2|v|_\infty r(a)^2|\omega-\omega_0|,
\]
as required.
Let $p\ge1$. If $r\in L^p(Y)$ then $V_\omega\in L^p(Y)$ for all $\omega\in[0,2\pi]$.
$\omega\mapsto V_\omega$ is a continuous
map from $[0,2\pi]$ to $L^p(Y)$.
We have
$|V_\omega|_p^p\le \sum_{a\in\alpha} \mu_Y(a)|1_aV_\omega|^p_\infty
\le \sum_{a\in\alpha} \mu_Y(a) |v|^p_\infty r(a)^p=|v|_\infty^p|r|_p^p<\infty$,
so $V_\omega\in L^p(Y)$.
Next we prove continuity. Let $\omega_0\in[0,2\pi]$. Then
\begin{align*}
& \le \sum_{a\in\alpha}\mu_Y(a)|1_aV_\omega-1_aV_{\omega_0}|_\infty^p \\
& \le \sum_{a\,:\,r(a)\le R}\mu_Y(a)2^p|v|_\infty^p R^{2p}|\omega-\omega_0|^p
+ \sum_{a\,:\,r(a)>R}\mu_Y(a)2^p|v|_\infty^pr(a)^p \\
& \le 2^p|v|_\infty^p\Bigl(R^{2p}|\omega-\omega_0|^p + \int_{\{r(a)>R\}}r^p\,d\mu_Y\Bigr).
\end{align*}
Fix $R$ large so that the second term is as small as desired. For this fixed $R$, the first term converges to zero as $\omega\to\omega_0$, proving continuity at $\omega_0$.
Define $V^*_\omega:Y\to\C$,
\begin{align*}
V^*_\omega(y)=\max_{0\le j\le r(y)-1}\Bigl|\sum_{\ell=0}^je^{i\ell\omega}v(f^\ell y)\Bigr|.
\end{align*}
Again $|1_aV^*_\omega|_\infty\le |v|_\infty r(a)$, so
if $r\in L^p(Y)$, then $V^*_\omega\in L^p(Y)$ for all $\omega\in[0,2\pi]$.
We now state our main result; this is proved in Section <ref>.
Let $f:X\to X$ be a nonuniformly expanding map and let
$v:X\to\R$ be a Hölder observable.
(a) Suppose that $r\in L^{2+}(Y)$.
the limit $S(\omega)=\lim_{n\to\infty}n^{-1}\int_X|e^{ij\omega}v\circ f^j|^2\,d\mu$ exists for all $\omega\in (0,2\pi)$ and extends to a continuous
function on $[0,2\pi]$.
Typically, $\inf_{\omega\in (0,2\pi)} S(\omega)>0$.
(b) Suppose that $r\in L^a(Y)$ and that
$V^*_\omega\in L^{bp}(Y)$ for all $\omega\in (0,2\pi)$, where
$a\in(1,\infty]$, $1/a+1/b=1$ and $p>2$.
Suppose further that $\omega\mapsto V_\omega$ is a continuous map from $(0,2\pi)$ to $L^2(Y)$.
Then the limit $S(\omega)=\lim_{n\to\infty}n^{-1}\int_X|e^{ij\omega}v\circ f^j|^2\,d\mu$ exists for almost every $\omega\in (0,2\pi)$ and extends to a continuous
function $\tilde S(\omega)$ on $(0,2\pi)$.
Typically, $\tilde S(\omega)$ is nonvanishing on $(0,2\pi)$.
(i) Young <cit.> considers the case where $\mu_Y(r>n)=O(1/n^{\beta+1})$ for some $\beta>0$ and deduces decay of correlations at rate $O(1/n^\beta)$. The case $\beta>1$ is the setting of Theorem <ref>.
It is easily seen that $r\in L^{p+}(Y)$ if and only if $\mu(r>n)=O(1/n^{p+})$, so Theorem <ref> is a restatement of Theorem <ref>(a).
(ii) If $r\in L^2(Y)$, then summable decay of correlations follows from <cit.> and hence the Wiener-Khintchine Theorem guarantees that the power spectrum extends to a continuous function on $[0,2\pi]$. However, our proof that the spectrum is typically bounded below requires that $r\in L^{2+}(Y)$.
(iii) If $r\not\in L^2(Y)$, then we expect (but have been unable to prove) that $S(\omega)\to\infty$ as $\omega\to0$ and $\omega\to 2\pi$. It would then follow that typically the power spectrum is bounded below also in Theorem <ref>(b).
(iv) The proof of Theorem <ref>(b) shows that
the limit $S(\omega)$ exists and is continuous on the set of irrational angles $\omega$. A different argument, which we have not included, shows that the limit exists also for rational angles $\omega\in(0,2\pi)$ and we conjecture that the resulting function $\omega\to S(\omega)$ is continuous on $(0,2\pi)$.
§.§ Application to intermittent maps
For the intermittent maps (<ref>), it is convenient to take $Y=[\frac12,1]$ and to let $r:Y\to\Z^+$ be the first return time
$r(y)=\inf\{n\ge1:f^ny\in Y\}$. It is standard that $f$ is a nonuniformly expanding map and that
$r\in L^p(Y)$ for any $p<\frac{1}{\gamma}$. Hence if $\gamma\in(0,\frac12)$, then Theorem <ref> applies directly.
This completes the proof of Theorem <ref> for $\gamma\in(0,\frac12)$.
For $\gamma\in[\frac12,1)$, we still have that $r\in L^{1+}(Y)$.
To apply Theorem <ref>, we require the next result.
Suppose that $\gamma\in(0,1)$ and $v\in C^\eta$ where $0<\eta<\gamma$.
Let $p< 1/(\gamma-\eta)$.
Then $V^*_\omega\in L^p(Y)$ for all $\omega\in(0,2\pi)$ and $\omega\mapsto V_\omega$ is a continuous map from $(0,2\pi)$ to $L^p(Y)$.
This is analogous to the situation in <cit.> (see also <cit.>).
Writing $v=(v-v(0))+v(0)$, we may consider the cases $v(0)=0$ and $v\equiv v(0)$ separately.
For $v(0)=0$, we show that $V^*_\omega\in L^p(Y)$ for all $\omega\in[0,2\pi]$
and that
$\omega\mapsto V_\omega$ is a continuous map from $[0,2\pi]$ to $L^p(Y)$.
For $v\equiv v(0)$, we show that $V^*_\omega\in L^\infty(Y)$ for all
$\omega\in(0,2\pi)$ and that
$\omega\mapsto V_\omega$ is a continuous map from $(0,2\pi)$ to $L^q(Y)$ for
all $q<\infty$.
First, suppose that $v(0)=0$, so $|v(y)|\le |v|_\eta|y|^\eta$.
It is well known that
$|f^\ell y|\ll (r(y)-\ell )^{-1/\gamma}$ for $\ell =1,\dots,r(y)$.
(See for example <cit.>.)
Hence for $y\in Y$, $\omega\in[0,2\pi]$,
\begin{align*}
|V^*_\omega(y)| & \le
\sum_{\ell=0}^{r(y)-1}|v(f^\ell y)| \le
\sum_{\ell=0}^{r(y)-1}|v|_{\eta}|f^\ell y|^\eta\ll
\sum_{\ell=1}^{r(y)-1}(r(y)-\ell)^{-\eta/\gamma}\ll
\end{align*}
If $p<1/(\gamma-\eta)$, then $p(1-\eta/\gamma)<1/\gamma$ and
\begin{align*}
\int_Y|V^*_\omega(y)|^p\,d\mu_Y
\ll \int_Y r^{p(1-\eta/\gamma)}\,d\mu_Y
\end{align*}
Next, we recall the estimate $|e^{ix}-1|\le 2\min\{1,|x|\}\le 2|x|^\eps$, which holds for all $x\in\R$, $\eps\in[0,1]$.
For all $y\in Y$, $\omega_0,\omega\in[0,2\pi]$,
\begin{align*}
|V_\omega(y)-V_{\omega_0}(y)| & \le
\max_{0\le \ell<r(y)} |e^{i\ell\omega}-e^{i\ell\omega_0}|
\sum_{\ell=0}^{r(y)-1} |v|_\eta|v(f^\ell y)|^\eta
\\ & \ll r(y)^\eps|\omega-\omega_0|^\eps \sum_{\ell=0}^{r(y)-1}(r(y)-\ell)^{-\eta/\gamma}
\ll r(y)^{1-\eta/\gamma+\eps}|\omega-\omega_0|^\eps,
\end{align*}
so for $\eps$ sufficiently small, $|V_\omega-V_{\omega_0}|_p\ll |\omega-\omega_0|^\eps$.
It remains to consider the case $v\equiv v(0)$. We have
\[
V^*_\omega(y)=|v(0)|\max_{0\le \ell<r(y)}|(1-e^{i\omega \ell})/(1-e^{i\omega})|\le
\]
for all $y\in Y$, $\omega\in(0,2\pi)$.
Moreover, for $\omega_0,\omega\in(0,2\pi)$, regarding $\omega_0$ as fixed,
$|V_\omega(y)-V_{\omega_0}(y)|\le g_1(\omega,y)+g_2(\omega)$
\begin{align*}
& g_1(\omega,y) = |v(0)||1-e^{i\omega_0}|^{-1} |e^{i\omega r(y)}-e^{i\omega_0 r(y)}|, \\
& g_2(\omega) =2|v(0)||(1-e^{i\omega})^{-1}-(1-e^{i\omega_0})^{-1}|.
\end{align*}
Clearly, $g_2(\omega)\to0$ as $\omega\to\omega_0$.
Also, taking $\eps=1/q$, we have that $g_1(\omega,y)\ll r(y)^{1/q}|\omega-\omega_0|^{1/q}$, so
$\int_Y|g_1(\omega,y)|^q\,d\mu_Y\ll |r|_1|\omega-\omega_0|$.
It follows that $g_1(\omega,\cdot)\to0$ in $L^q(Y)$, and hence similarly for
$|V_\omega-V_{\omega_0}|$, as $\omega\to\omega_0$.
This completes the proof.
Theorem <ref>
As already mentioned, the case $\gamma\in(0,\frac12)$ follows directly from Theorem <ref>(a).
When $\gamma\in[\frac12,1)$, we require that $\eta>(3\gamma-1)/2$. Without loss, $\eta\in((3\gamma-1)/2,\gamma)$.
Note that $1/(\gamma-\eta)>2/(1-\gamma)>2$.
By Proposition <ref>,
$\omega\mapsto V_\omega$ is a continuous map from $(0,2\pi)$ to $L^2(Y)$.
Let $a=1/(\gamma+\eps)$, $b=1/(1-\gamma-\eps)$ where $\eps\in(0,1-\gamma)$.
Then $r\in L^a(Y)$.
Since $1/(\gamma-\eta)>2/(1-\gamma)$, it follows from Proposition <ref> that $V^*_\omega\in L^{2/(1-\gamma-\delta)}(Y)$ for $\delta>0$ sufficiently small. Hence we can choose $p>2$, $\eps>0$ such that
$V^*_\omega\in L^{bp}(Y)$ for all $\omega\in(0,2\pi)$.
Now apply Theorem <ref>(b).
§ PROOF OF THEOREM <REF>
In Subsection <ref>, we prove a version of our main results
for the induced system $F=f^r:Y\to Y$. This result is lifted to the original system $f:Y\to Y$
in Subsection <ref>.
In Subsection <ref>, we complete the proof of Theorem <ref>.
§.§ The induced system
Let $f:X\to\ X$ be a nonuniformly expanding map as in Section <ref> with induced map $F=f^r:Y\to Y$ and partition $\alpha$.
Define $r_n=\sum_{j=0}^{n-1}r\circ F^j$. The induced power spectrum
$S^Y:(0,2\pi)\to\R$ is given by
\begin{align*}
S^Y(\omega) & =\lim_{n\to\infty}n^{-1}\int_Y\Bigl|\sum_{j=0}^{n-1}e^{i\omega r_j}V_{\omega}\circ F^j\Bigl|^2\,d\mu_Y.
\end{align*}
We can now state the main result in this subsection.
Suppose that $r\in L^1(Y)$ and
that $\omega\mapsto V_\omega$ is continuous as a function from $(0,2\pi)$ to $L^2(Y)$.
The pointwise limit
$S^Y:(0,2\pi)\to[0,\infty)$ exists and is continuous.
(b) Typically, $\{\omega\in(0,2\pi):S^Y(\omega)=0\}=\emptyset$.
In the remainder of this subsection, we prove Lemma <ref>.
If $a_0,\dots,a_{n-1}\in\alpha$, we define the $n$-cylinder
Fix $\theta\in(0,1)$ and define the symbolic
metric $d_\theta(x,y)=\theta^{s(x,y)}$ where the separation time
$s(x,y)$ is the least integer $n\ge0$ such that $x$ and $y$ lie in distinct $n$-cylinders.
For convenience we rescale the metric $d$ on $X$ so that $\diam(Y)\le1$.
Let $\eta\in(0,1]$ and fix $\theta\in[\lambda^{-\eta},1)$. Then
$d(x,y)^\eta\le d_\theta(x,y)$ for all $x,y\in Y$.
Let $n=s(x,y)$. By condition (2),
\[
1\ge \diam Y\ge d(F^nx,F^ny)\ge \lambda^nd(x,y)\ge (\theta^{1/\eta})^{-n}d(x,y).
\]
Hence $d(x,y)^\eta\le \theta^n=d_\theta(x,y)$.
An observable $\phi:Y\to\R$ is Lipschitz if $\|\phi\|_\theta=|\phi|_\infty+|\phi|_\theta<\infty$ where
$|\phi|_\theta=\sup_{x\neq y}|\phi(x)-\phi(y)|/d_\theta(x,y)$.
The set $F_\theta(Y)$ of Lipschitz observables is a Banach space.
More generally, we say that $\phi:Y\to\R$ is locally Lipschitz, and write
$\phi\in F_\theta^{\rm loc}(Y)$, if $\phi|_a\in F_\theta(a)$ for each $a\in\alpha$. Accordingly, we define
$D_\theta\phi(a)=\sup_{x,y\in a:\,x\neq y}|\phi(x)-\phi(y)|/d_\theta(x,y)$.
Let $v:X\to\R$ be a $C^\eta$ function, $\eta\in(0,1]$.
Set $\theta=\lambda^{-\eta}$. Then
$V_\omega\in F_\theta^{\rm loc}(Y)$ for all $\omega\in[0,2\pi]$, and
there is a constant $C\ge1$ such that
\begin{align*}
D_\theta V_\omega(a)\le C|v|_\eta r(a),
\end{align*}
for all $\omega\in[0,2\pi]$, $a\in\alpha$.
Let $y,y'\in a$. Then $r(y)=r(y')=r(a)$. By condition (3) and Proposition <ref>,
\begin{align*}
& |V_\omega(y)-V_\omega(y')|
\le \sum_{\ell=0}^{r(a)-1}|v(f^\ell y)-v(f^\ell y')|
\le |v|_\eta\sum_{\ell=0}^{r(a)-1}d(f^\ell y,f^\ell y')^\eta
\\ & \qquad \le C^\eta |v|_\eta r(a)d(Fy,Fy')^\eta
\le C^\eta |v|_\eta r(a)d_\theta(Fy,Fy')
= C^\eta \theta^{-1}|v|_\eta r(a)d_\theta(y,y'),
\end{align*}
yielding the required estimate for $D_\theta V_\omega(a)$.
The transfer operator $P:L^1(Y)\to L^1(Y)$ corresponding to $F$ is given by
$\int_Y P\phi\,\psi\,d\mu_Y=\int_Y \phi\,\psi\circ F\,d\mu_Y$ for all $\psi\in L^\infty$.
It can be shown that $(P\phi)(y)=\sum_{a\in\alpha}g(y_a)\phi(y_a)$
where $y_a$ denotes the unique preimage of $y$ in $a$ under $F$ and $\log g$ is the potential.
Moreover, there exists a constant $C_1$ such that
\begin{align} \label{eq-GM}
g(y)\le C_1\mu_Y(a), \quad\text{and}\quad |g(y)-g(y')|\le C_1\mu_Y(a)d_\theta(y,y'),
\end{align}
for all $y,y'\in a$, $a\in\alpha$.
For $\omega\in[0,2\pi]$, we define the twisted transfer operator
$P_\omega:L^1(Y)\to L^1(Y)$ given by
$P_\omega v=P(e^{-i\omega r}v)$.
Let $J\subset(0,2\pi)$ be a closed subset.
Viewing $P_\omega$ as an operator on $F_\theta(Y)$,
there exists $C\ge1$ and $\tau\in(0,1)$ such that
$\|P_\omega^n\|\le C\tau^n$ for all $\omega\in J$, $n\ge1$.
This result is a combination of standard and elementary observations.
By <cit.>,
$\omega\mapsto P_\omega$ is a continuous map from
$[0,2\pi]$ to $F_\theta(Y)$. Hence it suffices to show that the spectral radius of $P_\omega$ is less than $1$ for $\omega\in(0,2\pi)$.
It is easily checked that the spectral radius of $P_\omega$ is at most $1$, and that the essential spectral radius is at most $\theta$ (see for example <cit.>).
Hence it remains to rule out eigenvalues on the unit circle.
Suppose for contradiction that $P_\omega v=e^{i\psi}v$ for some eigenfunction $v\in F_\theta(Y)$ and some
A calculation using the fact that $v\mapsto e^{i\omega r}v\circ F$ is the $L^2$ adjoint of $P_\omega$
(see for example <cit.>) shows that
$e^{i\omega r}v\circ F=e^{-i\psi}v$. By ergodicity of $F$, $|v|$ is constant and hence
$v$ is nonvanishing.
Since $F|_a:a\to Y$ is onto for each $a$, there exists $y_a\in a$ with $Fy_a=y_a$.
Evaluating at $y_a$ and using the fact that $v(y_a)\neq0$, we obtain that
$e^{i\omega r(a)}=e^{-i\psi}$ for each $a\in\alpha$.
Hence $\omega (r(a)-r(b))=0\bmod 2\pi$ for all $a,b\in\alpha$.
Since $\omega\in(0,2\pi)$, it is immediate that $\omega=2\pi p/q$ where $p,q$ are integers with $\gcd(p,q)=1$ and $1\le p<q$.
But then $\frac{p}{q}(r(a)-r(b)=0\bmod\Z$ and so $q$ divides $r(a)-r(b)$ for all $a,b\in\alpha$. This contradicts the assumption that
Suppose that $r\in L^1(Y)$. Then
(a) $P_\omega V_\omega\in F_\theta(Y)$ for all $\omega\in[0,2\pi]$
and $\sup_{\omega\in [0,2\pi]}\|P_\omega V_\omega\|_\theta<\infty$.
(b) $\sum_{n=1}^\infty \sup_{\omega\in J}
\bigl|\int_Y P_\omega^nV_\omega\,\bar V_\omega\,d\mu_Y\bigr|<\infty$
for any closed subset $J\subset(0,2\pi)$.
(a) Write
$(P_\omega \phi)(y)=\sum_{a\in\alpha}g(y_a)e^{-i\omega r(a)}\phi(y_a)$.
We use Propositions <ref> and <ref> and the estimates (<ref>). First,
\[
|P_\omega V_\omega|_\infty\le \sum_{a\in\alpha} |1_ag|_\infty|1_aV_\omega|_\infty\le C_1 \sum_{a\in\alpha}\mu_Y(a)|v|_\infty r(a)=C_1|v|_\infty|r|_1.
\]
Also, $P_\omega V_\omega(y)-P_\omega V_\omega(y')=I_1+I_2+I_2$ where
\[
I_1 = \sum_{a\in\alpha} (g(y_a)-g(y_a'))e^{-i\omega r(a)}V_\omega(y_a), \quad
I_2 = \sum_{a\in\alpha} g(y_a')e^{-i\omega r(a)}(V_\omega(y_a)-V_\omega(y_a')).
\]
We have
\[
|I_1|\le C_1 \sum_{a\in\alpha} \mu_Y(a)d_\theta(y_a,y'_a)|v|_\infty r(a)=\theta C_1|v|_\infty|r|_1d_\theta(y,y'),
\]
\begin{align*}
|I_2| & \le C_1 \sum_{a\in\alpha} \mu_Y(a)D_\theta V_\omega(a) d_\theta(y_a,y_a')
\\ & \le \theta C_1C \sum_{a\in\alpha} \mu_Y(a)|v|_\eta r(a) d_\theta(y,y')
=\theta C_1C |v|_\eta |r|_1 d_\theta(y,y').
\end{align*}
We deduce that $|P_\omega V_\omega|_\theta\ll (|v|_\infty+|v|_\eta)|r|_1$,
completing the proof of part (a).
For $n\ge1$,
\begin{align*}
& \Bigl|\int_Y P_\omega^nV_\omega\,\bar V_\omega\,d\mu_Y\Bigr|
\le
|P_\omega^nV_\omega|_\infty \,|V_\omega|_1
\le \|P_\omega^{n-1}\| \|P_\omega V_\omega\|_\theta \,|V_\omega|_1.
\end{align*}
The result follows from Corollary <ref>, Proposition <ref> and part (a),
Lemma <ref>(a)
Let $\omega\in (0,2\pi)$.
For $0\le j<k$,
\begin{align*}
\int_Y e^{-i\omega(r_k-r_j)}V_{\omega}\circ F^j\,\bar V_{\omega} & \circ F^k\,d\mu_Y
= \int_Y e^{-i\omega r_{k-j}\circ F^j}V_{\omega}\circ F^j\,\bar V_{\omega}\circ F^k\,d\mu_Y \\
& =\int_Y e^{-i\omega r_{k-j}}V_{\omega}\,\bar V_{\omega}\circ F^{k-j}\,d\mu_Y
=\int_Y P_\omega^{k-j}V_{\omega}\,\bar V_{\omega}\,d\mu_Y.
\end{align*}
\begin{align*}
& \int_Y|\sum_{j=0}^{n-1}e^{i\omega r_j}V_{\omega}\circ F^j|^2\,d\mu_Y =
\sum_{j,k=0}^{n-1}\int_Y e^{i\omega(r_j-r_k)}V_{\omega}\circ F^j\,\bar V_{\omega}\circ F^k\,d\mu_Y \\
& \quad\qquad = \sum_{j=0}^{n-1}\int_Y|V_{\omega}\circ F^j|^2\,d\mu_Y
+ 2\sum_{0\le j<k<n}\Re\int_Y e^{-i\omega(r_k-r_j)}V_{\omega}\circ F^j\,\bar V_{\omega}\circ F^k\,d\mu_Y \\
& \quad\qquad = n\int_Y|V_{\omega}|^2\,d\mu_Y
+ 2\sum_{0\le j<k<n}\Re\int_Y P_\omega^{k-j}V_{\omega}\,\bar V_{\omega}\,d\mu_Y \\
& \quad\qquad = n\int_Y |V_{\omega}|^2\,d\mu_Y
+ 2\sum_{m=1}^{n-1}(n-m)\Re\int_Y P_\omega^mV_{\omega}\,\bar V_{\omega}\,\,d\mu_Y.
\end{align*}
\begin{align*}
+ 2\lim_{n\to\infty}\sum_{m=1}^{n-1}\bigl(1-\frac{m}{n}\bigr)\Re\int_Y P_\omega^mV_{\omega}\,\bar V_{\omega}\,d\mu_Y.
\end{align*}
By Proposition <ref>(b), this converges uniformly on compact subsets of $(0,2\pi)$
to the sum
\begin{align} \label{eq-unif} \nonumber
S^Y(\omega) & =\int_Y|V_{\omega}|^2\,d\mu_Y+2\sum_{n=1}^\infty\Re\int_Y P_\omega^nV_{\omega}\,\bar V_{\omega}\,d\mu_Y \\
& =\int_Y|V_{\omega}|^2\,d\mu_Y+2\sum_{n=1}^\infty\Re\int_Y e^{-i\omega r_n}V_{\omega}\,\bar V_{\omega}\circ F^n\,d\mu_Y.
\end{align}
Fix $n\ge1$ and let $I_\omega=e^{-i\omega r_n} V_{\omega}\,\bar V_{\omega}\circ F^n$.
Note that $|I_\omega|=|V_{\omega}|\,|V_{\omega}|\circ F^n$ and
$\int_Y |I_\omega|\,d\mu_Y\le |V_{\omega}|_2|V_{\omega}\circ F^n|_2
= |V_{\omega}|_2^2<\infty$ for each $\omega$.
We claim that $\omega\mapsto \int_Y I_\omega \,d\mu_Y$ is continuous on $(0,2\pi)$.
It then follows from uniform convergence of the series (<ref>) that
$S^Y:(0,2\pi)\to[0,\infty)$ is continuous.
To prove the claim, fix $n\ge1$ and $\omega_*\in (0,2\pi)$.
Let $\omega_k$ be a sequence in $(0,2\pi)$ converging to $\omega_*$.
We show that $\int_Y I_{\omega_k}\,d\mu_Y\to \int_Y I_{\omega_*}\,d\mu_Y$ as $k\to\infty$.
Certainly $I_{\omega_k}\to I_{\omega_*}$ pointwise.
\begin{align*}
\int_Y |I_{\omega_k}|\,d\mu_Y -\int_Y |I_{\omega_*}|\,d\mu_Y
& = \int_Y(|V_{\omega_k}|-|V_{\omega_*}|)\,|V_{\omega_k}|\circ F^n\,d\mu_Y
\\ & \qquad \qquad + \int_Y |V_{\omega_*}|\,(|V_{\omega_k}|-|V_{\omega_*}|)\circ F^n\,d\mu_Y,
\end{align*}
and so
\begin{align*}
\Bigl|\int_Y |I_{\omega_k}|\,d\mu_Y -\int_Y |I_{\omega_*}|\,d\mu_Y \Bigr|
& \le \bigl||V_{\omega_k}|-|V_{\omega_*}|\bigr|_2 |V_{\omega_k}|_2
+|V_{\omega_*}|_2\bigl||V_{\omega_k}|-|V_{\omega_*}|\bigr|_2 \\
& \le |V_{\omega_k}-V_{\omega_*}|_2(|V_{\omega_k}|_2+|V_{\omega_*}|_2) \\
& \le |V_{\omega_k}-V_{\omega_*}|_2(|V_{\omega_k}-V_{\omega_*}|_2+2|V_{\omega_*}|_2).
\end{align*}
It follows that $\int_Y|I_{\omega_k}|\,d\mu_Y\to
\int_Y|I_{\omega_*}|\,d\mu_Y$ as $k\to\infty$.
By the dominated convergence theorem,
$\int_Y I_{\omega_k}\,d\mu_Y\to \int_Y I_{\omega_*}\,d\mu_Y$ completing the proof of the claim.
Lemma <ref>(b)
Let $\omega\in(0,2\pi)$ and define
$\chi_\omega =\sum_{j=1}^\infty P_\omega^jV_\omega$ and
$\tilde V_\omega = V_\omega+\chi_\omega- e^{i\omega r}\chi_\omega\circ F$.
Now $\|P_\omega^jV_\omega\|_\theta\le \|P_\omega^{j-1}\| \|P_\omega V_\omega\|_\theta$, so it follows from
Propositions <ref> and <ref>(a) that
$\chi_\omega$ is absolutely summable in $F_\theta(Y)$. In particular
$\tilde V_\omega\in L^2(Y)$.
A calculation shows that $\tilde V_\omega\in\ker P_\omega$ and so
$|\sum_{j=1}^{n-1}e^{i\omega r_j}\tilde V_\omega\circ F^j|_2^2=n|\tilde V_\omega|_2^2$. Moreover, $\sum_{j=1}^{n-1}e^{i\omega r_j}\tilde V_\omega\circ F^j
= \sum_{j=1}^{n-1}e^{i\omega r_j}V_\omega\circ F^j +\chi_\omega-e^{i\omega r_n}\chi_\omega\circ F_\omega^n$ and so
\[
\Bigl|\,\bigl|\sum_{j=1}^{n-1}e^{i\omega r_j}V_\omega\circ F^j\bigr|_2-
\bigl|\sum_{j=1}^{n-1}e^{i\omega r_j}\tilde V_\omega\circ F^j\bigr|_2\,\Bigr|\le 2|\chi_\omega|_2.
\]
Hence $S^Y(\omega)=|\tilde V_\omega|_2^2$ for all $\omega\in (0,2\pi)$.
In particular, for a fixed $\omega\in (0,2\pi)$ we have that if $S^Y(\omega)=0$, then equivalently $\tilde V_\omega=0$ and so
\begin{align} \label{eq-zero}
V_\omega=e^{i\omega r}\chi_\omega\circ F-\chi_\omega.
\end{align}
It remains to exclude the possibility that (<ref>)
holds for some $\omega$.
Following <cit.>,
let $y_0\in Y$ be a periodic point of period $p$ and let $y_n$ be a sequence with
$Fy_n=y_{n-1}$ and such that $d_\theta(y_{np},y_0)\le \theta^{np}$.
This ensures in particular that $r_p(y_{jp})=r_0(y_0)$ for all $j$.
Set $A_\omega=\sum_{j=0}^{p-1}e^{i\omega r_j}V_\omega\circ F^j$
and define
$g(\omega)=\sum_{j=1}^\infty e^{-ij\omega r_p(y_0)}(A_\omega(y_{jp})-A_\omega(y_0))$.
Note that for each fixed $y_n$, the function $\omega\mapsto A_\omega(y_n)$
is a finite trigonometric polynomial and hence is analytic on $[0,2\pi]$.
We claim that $g:[0,2\pi]\to\C$ is analytic.
Suppose that $d_\theta(y,y')\ge p$. Then
\begin{align*}
& \le \sum_{j=0}^{p-1}\sum_{\ell=0}^{r(F^jy)-1}|v|_\theta d_\theta(F^jy,F^jy')
= |v|_\theta\sum_{j=0}^{p-1} r(F^jy)\theta^{-j}d_\theta(y,y')
\\ & \le |v|_\theta \,\theta^{-p}(1-\theta)^{-1}r_p(y)d_\theta(y,y'),
\end{align*}
\begin{align*}
& \le |v|_\theta \,\theta^{-p}(1-\theta)^{-1}r_p(y_0)\theta^j,
\end{align*}
proving the claim.
If (<ref>) holds, then
$A_\omega=e^{i\omega r_p}\chi_\omega\circ F^p-\chi_\omega$ and
\begin{align*}
& \sum_{j=1}^n e^{-ij\omega r_p(y_0)}A_\omega(y_{jp})=
\chi_\omega(y_0)-e^{-in\omega r_p(y_0)}\chi_\omega(y_{np}), \\
& \sum_{j=1}^n e^{-ij\omega r_p(y_0)}A_\omega(y_0) =
\chi_\omega(y_0)-e^{-in\omega r_p(y_0)}\chi_\omega(y_0).
\end{align*}
\[
= \lim_{n\to\infty} e^{-in\omega r_p(y_0)}(\chi_\omega(y_0)-\chi_\omega(y_{np}))=0.
\]
Still keeping $\omega$ fixed, we can perturb the value of $v$ at $y_1$ (say), independently of any other values of $v$ involved in the computation of $g$, so that $g(\omega)\neq0$. Hence typically (<ref>) does not hold (and so $S^Y(\omega)$ is nonzero) for any fixed value of $\omega$.
Considering two such analytic functions $g_1$ and $g_2$ (for two distinct periodic points)
we can perturb so that $g_1$ and $g_2$ have no common zeros on $[0,2\pi]$ and hence $S^Y$
is nonvanishing on $(0,2\pi)$. By considering infinitely many periodic points, and hence infinitely many functions of the form $g$, we obtain infinitely many independent obstructions to the existence of an $\omega\in(0,2\pi)$ such that $S^Y(\omega)=0$.
§.§ Relation between $S^Y$ and $S$
In this subsection, we relate the power spectrum $S^Y$ of the induced map $F:Y\to Y$ with the power spectrum $S$ of the underlying nonuniformly expanding map $f:X\to X$.
Let $\bar r= \int_Y r\,d\mu_Y$.
We say that $\omega$ is an irrational angle if
Let $\omega$ be an irrational angle.
Suppose either that $r\in L^{2+}(Y)$, or that $r\in L^a(Y)$ and
that $V^*_\omega\in L^{bp}(Y)$ for all $\omega\in (0,2\pi)$, where
$a\in(1,\infty]$, $1/a+1/b=1$ and $p>2$.
Then $S(\omega)=S^Y(\omega)/\bar r$.
From now on, we work with a fixed irrational angle $\omega\in(0,2\pi)\setminus\pi\Q$.
We suppose throughout that $v:X\to\R$ is Hölder.
Let $d\varphi$ denote Haar measure on $S^1$,
Consider the circle extensions
\begin{align*}
& f_\omega:X\times S^1\to X\times S^1, \qquad
f_\omega(x,\varphi)=(fx,\varphi+\omega), \\
& F_\omega:Y\times S^1\to Y\times S^1, \qquad
F_\omega(y,\varphi)=(Fy,\varphi+\omega r(y)),
\end{align*}
with invariant probability measures $\nu=\mu\times d\varphi$ and
$\nu_Y=\mu_Y\times d\varphi$ respectively.
Recall that $F=f^r:Y\to Y$ is the induced map obtained from $f:X\to X$ with return time $r:Y\to\Z^+$.
Extend $r$ to a return time on $Y\times S^1$ by setting $r(y,\varphi)=r(y)$. Then $F_\omega=f_\omega^r$ is the induced map obtained from $f_\omega$.
Let $v:X\to \R$ be an observable.
We associate to $v$ the observable $u:X\times S^1\to\C$ given by
This leads to the induced observable
$U_\omega:Y\times S^1\to\C$ given by $U_\omega(q)=\sum_{\ell=0}^{r(q)-1}u(f_\omega^\ell q)$.
Note that
\[
U_\omega(y,\varphi)=\sum_{\ell=0}^{r(y)-1}u(f^\ell y,\varphi+\ell\omega)
=e^{i\varphi}\sum_{\ell=0}^{r(y)-1}e^{i\ell\omega}v(f^\ell y)
\]
and similarly that
\[
\sum_{j=0}^{n-1}U_\omega\circ F_\omega^j(y,\varphi)=e^{i\varphi}\sum_{j=0}^{n-1}e^{i\omega r_j(y)}V_\omega\circ F^j(y).
\]
Let $\omega$ be an irrational angle.
Suppose that $r\in L^{1+}(Y)$ and that $V^*_\omega\in L^2(Y)$. Then
\[
n^{-1}\Bigl|\sum_{j=0}^{n-1}u\circ f_\omega^j|^2\to_d Z_\omega \quad\text{on $(X\times S^1,\nu)$},
\]
where $Z_\omega$ is a random variable with $\E Z_\omega=S^Y(\omega)/\bar r$.
Since $\omega\in [0,2\pi]\setminus\pi\Q$, the circle extension $f_\omega:X\times S^1\to X\times S^1$ is ergodic, and we are in a position to apply <cit.>. Note that $G,\phi,V,V^*,f_h$ in <cit.> correspond to
$S^1,u,V_\omega,V^*_\omega,f_\omega$ here.
By <cit.>, we obtain a functional central limit theorem (weak invariance principle) as follows.
=n^{-1/2}\sum_{j=0}^{[nt]-1}u\circ f_\omega^j$ for $t=0,1/n,\dots,1$ and linearly interpolate to obtain $W_{n,\omega}\in C([0,1],\R^2)$.
Then $W_{n,\omega}\to_w W_\omega$ in $C([0,1],\R^2)$
on $(X\times S^1,\nu),$ where
$W_\omega$ is a two-dimensional Brownian motion with some covariance matrix $\Sigma_\omega$.
Moreover, it follows from the proof of <cit.>
(see the statements of <cit.>)
that $\Sigma_\omega=\hat\Sigma_\omega/\bar r$ where
\[
\hat\Sigma_\omega =
\lim_{n\to\infty}n^{-1}\int_{Y\times S^1}
\Bigl(\sum_{j=0}^{n-1}U_\omega\circ F_\omega^j\Bigr)
\otimes \Bigl(\sum_{j=0}^{n-1}U_\omega\circ F_\omega^j\Bigr)
\,d\nu_Y.
\footnote{For $a,b\in\C\cong\R^2$, we define
$a\otimes b=ab^T=\left(\begin{array}{cc} \Re a\Re b & \Re a\Im b \\ \Im a\Re b & \Im a \Im b \end{array}\right)$.}
\]
Consider the functional $\chi:C([0,1],\R^2)\to\R$, $\chi(g)=|g(1)|^2$.
By the continuous mapping theorem,
$\chi(W_{n,\omega})\to_d \chi(W_\omega)$, so
\[
n^{-1}|\sum_{j=0}^{n-1}u\circ f_\omega^j|^2\to_d Z_\omega,
\quad\text{where $Z_\omega=|W_\omega(1)|^2$}.
\]
In particular,
\begin{align*}
\E Z_\omega=\E|W_\omega(1)|^2=\Sigma_\omega^{11}+\Sigma_\omega^{22}
=(\hat\Sigma_\omega^{11}+\hat\Sigma_\omega^{22})/\bar r.
\end{align*}
\begin{align*}
S^Y(\omega) & =\lim_{n\to\infty}n^{-1}\int_Y|\sum_{j=0}^{n-1}e^{i\omega r_j}V_\omega\circ F^j|^2\,d\mu_Y \\ & =
\lim_{n\to\infty}n^{-1}\int_{Y\times S^1}|\sum_{j=0}^{n-1}U_\omega\circ F_\omega^j|^2\,d\nu_Y
\end{align*}
completing the proof.
Suppose that either
(a) $r\in L^{2+}(Y)$ and choose $p>2$ such that $r\in L^{\frac{p}{2}+1+}(Y)$, or
(b) $r\in L^a(Y)$ and
$V^*_\omega\in L^{bp}(Y)$ for all $\omega\in (0,2\pi)$, where
$a\in(1,\infty]$, $1/a+1/b=1$, and $p>2$.
Then there is a constant $C\ge1$ such that
$|\sum_{j=0}^{n-1}u\circ f_\omega^j|_p\le Cn^{1/2}$.
The method in both cases is to obtain a martingale coboundary decomposition <cit.> and then to apply Burkholder's inequality <cit.> to the martingale part. In case (a), the decomposition is done on $X\times S^1$ following <cit.>. In case (b), we pass to a tower extension and reduce to the induced system on $Y\times S^1$.
Case (a):
By Markov's inequality, the assumption $r\in L^{\frac{p}{2}+1+}(Y)$ guarantees that $\mu(r>n)=O(n^{-(\beta+1)})$ for some $\beta>p/2$.
Let $L:L^1(X)\to L^1(X)$ and $L_\omega:L^1(X\times S^1)\to L^1(X\times S^1)$
be the transfer operators corresponding to $f$ and $f_\omega$ respectively.
Regard $u$ as fixed, and
let $u'\in L^\infty(X\times S^1)$ be of the form $u'(y,\varphi)=e^{i\varphi}v'(x)$ where $v'\in L^\infty(X)$.
By <cit.>,
$\int_{X\times S^1} L_\omega^n u\,\bar u'\,d\nu=
\int_{X\times S^1} u\,\bar u'\circ f_\omega^n\,d\nu=O(n^{-\beta}|u'|_\infty)$.
Equivalently $\int_X L^nv \,\bar v'\,d\mu=O(n^{-\beta}|v'|_\infty)$.
By duality
$\int_{X\times S^1}|L_\omega^nu| \,d\nu=
\int_X|L^nv| \,d\mu=O(n^{-\beta})$.
Now we proceed as in the proof of <cit.>.
$|L_\omega^nu|_1 =O(n^{-\beta})$ and $|L_\omega^nu|_\infty =O(1)$,
it follows by interpolation that
$|L_\omega^nu|_q$ is summable for $q<\beta$, in particular for $q=p/2$.
$\chi_\omega=\sum_{n=1}^\infty L_\omega^nu\in L^{p/2}(X\times S^1)$
and we obtain $u=\tilde u_\omega+\chi_\omega\circ f_\omega-\chi_\omega$
where $\tilde u_\omega\in\ker L_\omega$.
Continuing as in <cit.> (in particular, see <cit.>) we obtain the desired result.
Case (b):
Define the Young tower <cit.>
\begin{align*}
\Delta=\{(y,\varphi,\ell)\in Y\times S^1\times\Z:0\le\ell<r(y)\},
\end{align*}
and the tower map $\hat f_\omega:\Delta\to\Delta$,
\[
\hat f_\omega(y,\varphi,\ell)=\begin{cases} (y,\varphi+\omega,\ell+1), & \ell\le r(y)-2
\\ (Fy,\varphi+\omega,0), & \ell=r(y)-1 \end{cases}
\]
with invariant probability measure
$\nu_\Delta=(\nu\times{\rm counting})/\bar r$.
The projection $\pi:\Delta\to X\times S^1$ given by $\pi(y,\varphi,\ell)=(f^\ell y,\varphi)$ is a measure-preserving semiconjugacy between $\hat f_\omega$ and $f_\omega$, with
Let $\hat u=u\circ \pi$.
Then $\int_{X\times S^1} |\sum_{j=0}^{n-1}u\circ f_\omega^j|^p\,d\nu=\int_{\Delta} |\sum_{j=0}^{n-1}\hat u\circ\hat f_\omega^j|^p\,d\nu_\Delta$.
Next, let $N_n:\Delta\to\{0,1,\dots,n\}$ be the number of laps by time $n$,
\[
N_n(y,\varphi,\ell)=\#\{j\in\{1,\dots,n\}:\hat f_\omega^j(y,\varphi,\ell)\in Y\times S^1\times\{0\}\}.
\]
\[
e^{i\ell\omega}\sum_{j=0}^{n-1}\hat u\circ \hat f_\omega^j(y,\varphi,\ell)=
\sum_{k=0}^{N_n(y,\varphi,\ell)-1}U_\omega\circ F_\omega^k(y,\varphi)+H_\omega\circ \hat f_\omega^n(y,\varphi,\ell)-H_\omega(y,\varphi,\ell)
\]
where $H_\omega(y,\varphi,\ell)=e^{i\varphi}\sum_{\ell'=0}^{\ell-1} e^{i\ell'\omega}v(f^{\ell'}y)$.
Note that $|H_\omega(y,\varphi,\ell)|\le V^*_\omega(y)$.
\begin{align} \label{eq-H} \nonumber
\int_\Delta |H_\omega\circ \hat f_\omega^n|^p\,d\mu_\Delta & =
\int_\Delta |H_\omega|^p\,d\mu_\Delta=(1/\bar r)\int_Y\sum_{\ell=0}^{r(y)-1}|H_\omega(y,\varphi,\ell)|^p\,d\nu_Y(y,\varphi)
\\ & \le \int_Y r \ |V^*_\omega|^p\,d\mu_Y
\le |r|_a |{V^*_\omega}^p|_b = |r|_a |V^*_\omega|_{bp}^p.
\end{align}
Next, by Hölder's inequality,
\begin{align*}
\int_{\Delta} &
\Bigl|\sum_{k=0}^{N_n-1}U_\omega\circ F_\omega^k\Bigr|^p\,d\nu_\Delta
\le \int_{\Delta}
\max_{j\le n}\Bigl|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k\Bigr|^p\,d\nu_\Delta
\\
& =(1/\bar r) \int_{Y\times S^1} r \
\max_{j\le n}\Bigl|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k\Bigr|^p\,d\nu_Y
\le
|r|_{L^a(Y)} \ \Biggl|\max_{j\le n}\Bigl|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k\Bigr|\Biggr|_{L^{bp}(Y\times S^1)}^p.
\end{align*}
Let $Q_\omega:L^1(Y\times S^1)
\to L^1(Y\times S^1)$ denote the transfer operator corresponding
to $F_\omega:Y\times S^1\to Y\times S^1$.
For an observable $U:Y\times S^1\to\C$ of the form
$U(y,\varphi)=e^{i\varphi}V(y)$, we have
$(Q_\omega U)(y,\varphi)=e^{i\varphi}(P_\omega V)(y)$.
By Propositions <ref> and <ref>(a),
\[
\chi_\omega=\sum_{n=1}^\infty Q_\omega^nU_\omega
=e^{i\varphi}\sum_{n=1}^\infty P_\omega^nV_\omega\in L^\infty(Y\times S^1).
\]
Hence we can write
\[
U_\omega= m_\omega+\chi_\omega\circ F_\omega-\chi_\omega,
\]
where $m_\omega\in\ker Q_\omega$.
Note also by the hypothesis on $V^*_\omega$ that
$U_\omega\in L^{bp}(Y\times S^1)$ and hence
$m_\omega\in L^{bp}(Y\times S^1)$ where $bp>2$.
By Burkholder's inequality <cit.>,
\[
\Bigl|\max_{j\le n}\bigl|\sum_{k=0}^{j-1}m_\omega\circ F_\omega^k\bigr|\Bigr|_{L^{bp}(Y\times S^1)}\ll |m_\omega|_{L^{bp}(Y\times S^1)} \ n^{1/2},
\]
and so
$\bigl|\max_{j\le n}|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k|\bigr|_{L^{bp}(Y\times S^1)}\ll n^{1/2}$. Hence we have shown that
\begin{align} \label{eq-maxU}
\Bigl|\sum_{k=0}^{N_n-1}U_\omega\circ F_\omega^k\Bigr|_{L^p(\Delta)}\le |r|_a^{1/p} \
\Biggl|\max_{j\le n}\Bigl|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k\Bigr|\Biggr|_{L^{bp}(Y\times S^1)}\ll n^{1/2}.
\end{align}
By the triangle inequality, it follows from (<ref>) and (<ref>) that
\[
\Bigl|\sum_{j=0}^{n-1}u\circ f_\omega^j\Bigr|_{L^p(X\times S^1)}=\Bigl|\sum_{j=0}^{n-1}\hat u\circ\hat f_\omega^j\Bigr|_{L^p(\Delta)}
\ll n^{1/2},
\]
as required.
Lemma <ref>
The bound on moments in Proposition <ref> together with
the distributional limit law in Proposition <ref> implies convergence of lower moments,
(see for example <cit.>),
and so
\[
\lim_{n\to\infty}n^{-1}\Bigl|\sum_{j=0}^{n-1} u\circ f_\omega^j\Bigr|^2_{L^2(X\times S^1)}=
S^Y(\omega)/\bar r.
\]
Now $\sum_{j=0}^{n-1} u(f_\omega^j(x,\varphi))=e^{i\varphi}
\sum_{j=0}^{n-1} e^{ij\omega}v(f^j(x))$ and so
$|\sum_{j=0}^{n-1} u\circ f_\omega^j|_{L^2(X\times S^1)}=
|\sum_{j=0}^{n-1} e^{ij\omega}v\circ f^j|_{L^2(X)}$.
\[
S(\omega)=\lim_{n\to\infty}n^{-1}\Bigl|\sum_{j=0}^{n-1} e^{ij\omega}v\circ f^j\Bigr|^2_{L^2(X)}=
S^Y(\omega)/\bar r,
\]
as required.
§.§ Completion of the proof
First assume the hypotheses of Theorem <ref>(b).
By Lemma <ref>, $S^Y$ exists and is continuous on $(0,2\pi)$,
and typically $S^Y$ is nonvanishing on $(0,2\pi)$.
By Lemma <ref>, $S$ coincides with $S^Y/\bar r$ almost everywhere on
Theorem <ref>(b) follows immediately.
Under the hypotheses of Theorem <ref>(a) we have the same properties, but in addition
$S(\omega)$ is continuous on $(0,2\pi)$ by the Wiener-Khintchine Theorem, so it follows that
$S(\omega)=S^Y(\omega)/\bar r$ for all $\omega\in(0,2\pi)$.
Hence $S$ is typically nonvanishing on $(0,2\pi)$.
Moreover, by Wiener-Khintchine, $S$ extends to a continuous function
$S_0(\omega)=\sum_{k=-\infty}^\infty e^{ik\omega}\rho(k)$ on $[0,2\pi]$.
By the Green-Kubo formula, $S_0(0)=S_0(2\pi)$ coincides with the variance
$\sigma^2=\lim_{n\to\infty}n^{-1}\int_X|\sum_{j=0}^{n-1}v_0\circ f^j|^2\,d\mu$
where $v_0=v-\int_X v\,d\mu$.
Typically $\sigma^2>0$, see for example <cit.>.
Hence typically $S_0$ is bounded away from zero on $[0,2\pi]$ and so
$S$ is bounded away from zero on $(0,2\pi)$.
This completes the proof of Theorem <ref>(a).
This research was supported in part by an International Research Collaboration Award at the University of Sydney. GAG acknowledges funding from the Australian Research Council. The research of IM was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977).
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1511.00542
|
This paper deals with vector linear index codes for multiple unicast index coding problems where there is a source with $K$ messages and there are $K$ receivers each wanting a unique message and having symmetric (with respect to the receiver index) two-sided antidotes (side information). Optimal scalar linear index codes for several such instances of this class of problems for one-sided antidotes(not necessarily adjacent) have been reported in <cit.>. These codes can be viewed as special cases of the symmetric unicast index coding problems discussed in <cit.> with one sided adjacent antidotes. In this paper, starting from a given multiple unicast index coding problem with with $K$ messages and one-sided adjacent antidotes for which a scalar linear index code $\mathfrak{C}$ is known, we give a construction procedure which constructs a sequence (indexed by $m$) of multiple unicast index problems with two-sided adjacent antidotes (for the same source) for all of which a vector linear code $\mathfrak{C}^{(m)}$ is obtained from $\mathfrak{C}.$ Also, it is shown that if $\mathfrak{C}$ is optimal then $\mathfrak{C}^{(m)}$ is also optimal for all $m.$ We illustrate our construction for some of the optimal scalar linear codes of <cit.> though the construction is applicable for all the codes of <cit.>.[The authors are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India. Email:[email protected].]
§ INTRODUCTION AND BACKGROUND
The problem of index coding with side information was introduced by Birk and Kol <cit.> and Bar-Yossef et al. <cit.> studied the class of index coding problems in which each receiver demands only one single message and the number of receivers equals number of messages. Ong and Ho <cit.> classify the binary index coding problem depending on the demands and the side information possessed by the receivers. An index coding problem is unicast if the demand sets of the receivers are disjoint. If the problem is unicast and if the size of each demand set is one, then it is said to be single unicast. Any unicast index problem can be equivalently reduced to an single unicast problem discussed in <cit.>. For this canonical unicast index coding problem, it was shown that the length of the optimal linear index code is equal to the minrank of the side information graph of the index coding problem but finding the minrank is NP hard.
Maleki et al. <cit.> found the capacity of symmetric multiple unicast index problem with neighboring antidotes (side information). In a symmetric multiple unicast index coding problem with equal number of $K$ messages and source-destination pairs, each destination has a total of $U+D=A<K$ antidotes, corresponding to the $U$ messages before (“up" from) and $D$ messages after (“down" from) its desired message. In this setting, the $k^{th}$ receiver $R_{k}$ demands the message $x_{k}$ having the antidotes
\begin{equation}
\label{antidote}
{\cal K}_k= \{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1}, x_{k+2},\dots,x_{k+D}\}.
\end{equation}
The symmetric capacity $C$ of this index coding problem setting is shown to be as follows:
$U,D \in$ $\mathbb{Z},$
$0 \leq U \leq D$,
\begin{array}{ll}
{1,\qquad\quad\ A=K-1}\\
{\frac{U+1}{K-A+2U}},A\leq K-2\qquad $per message.$
\end{array}
\right.$
The above expression for capacity per message can be equivalently expressed as:
\begin{equation}
\label{capacity}
\begin{array}{ll}
{1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \mbox{if} ~~ U+D=K-1}\\
{\frac{min(U,D)+1}{K+min(U,D)-max(U,D)}} ~~~ \mbox{if} ~~U+D\leq K-2.
\end{array}
\right.
\end{equation}
In the setting of <cit.> with one sided antidote cases, i.e., the cases where $U$ or $D$ is zero, without loss of generality, we can assume that $max(U,D)= D$ and $min(U,D)=0$ (all the results hold when $max(U,D)=U$). In this setting, the $k^{th}$ receiver $R_{k}$ demands the message $x_{k}$ having the antidotes,
\begin{equation}
\label{antidote1}
{\cal K}_k =\{x_{k+1}, x_{k+2},\dots,x_{k+D}\},
\end{equation}
for which (<ref>) reduces to
\begin{equation}
\label{capacity1}
\begin{array}{ll}
{1 ~~~~~~~~~~~~ \mbox{if} ~~ D=K-1}\\
{\frac{1}{K-D}} ~~~~~~~ \mbox{if} ~~D\leq K-2
\end{array}
\right.
\end{equation}
symbols per message.
§.§ Contributions
In the capacity expression given in (<ref>) if $\small{U+1}$ divides $\small{K-D+U}$, capacity can be achieved by using scalar linear codes. In the scalar linear coding one packs $K$ messages in $\frac{K-D+U}{U+1}$ dimensions (code symbols). If $\small{U+1}$ does not divide $\small{K-D+U}$, vector linear coding can only achieve capacity. In the vector linear code, one needs to pack $K(U+1)$ message symbols corresponding to $K$ users in $K-D+U$ dimensions.
In <cit.> Maleki $et\ al.$ proved that vector linear coding exists for any arbitrary $U$ and $D$ over a sufficiently large field size by imposing conditions on encoding and decoding matrices $U_{m,k}$ and $V_{m}$. In Section II we prove that vector linear solution exists for a two-sided symmetric antidote problem with $U$ antidotes above and $D$ antidotes below if a scalar linear solution exists for one-sided antidote problem with same number of messages and number of one-sided antidotes $\Delta =\vert D-U \vert$.
We give a construction procedure which constructs a sequence of multiple unicast index problems with two-sided antidotes starting from a given multiple unicast index coding problem of one-sided antidote. It is shown that if there is an optimal scalar linear index code for the starting problem then this code can be used to construct an optimal vector linear index code for all the extended problems. In <cit.> the authors proposed optimal scalar linear index codes for ten classes of one sided (not necessarily adjacent) symmetric multiple unicast index coding problems with optimal scalar linear index codes. The antidotes assumed in this work is a proper subset of (<ref>) for eight classes. These optimal codes continue to be optimal if the antidotes are taken to be adjacent as given in (<ref>). We illustrate our construction to some of the ten cases of the symmetric multiple unicast problems studied in <cit.> and demonstrate the new classes of symmetric multicast problems created for all of which an optimal vector linear code is exhibited.
In <cit.>, we proposed a lifting construction which constructs a sequence of multiple unicast index problems with one-sided antidotes with a scalar linear index code starting from a given multiple unicast index coding problem with a known scalar linear index code. The construction in this paper is different in the following two respects:
* The construction in <cit.> starts from a problem with $K$ messages and gives a sequence of problems with $mK$ number of messages for $m=2,3, \dots,$ i.e., the number of messages goes on increasing as the index $m$ moves. Whereas in this work the number of messages remains $K$ and only the size of the antidote sets increase.
* In <cit.> new scalar linear index codes are obtained starting from a scalar linear index code for problems of different source sizes whereas in this paper new vector linear index codes are obtained starting from a scalar linear index code for the problem with the same source size.
* The lifting construction in <cit.> results in index coding problems with one sided antidotes where as the construction in this paper results in index coding problems with two sided antidotes.
Throughout the paper WLOG we consider the case $D \geq U$. All the codes discussed in this paper also applicable for $U \geq D$. The decoding procedure in this paper is considered for the binary field. However the index codes considered in this paper works for any finite field.
§ EXTENSION OF SCALAR LINEAR CODE INTO VECTOR LINEAR CODES
Let the messages symbols be $\{x_1, x_2, \dots x_K \}.$ For every $x_k,$ when we deal with vector linear index codes the different messages symbols corresponding to $x_k$ are denoted by $x_{k,1}, x_{k,2}, x_{k,3}, \dots$ etc.
If $U+1$ do not divide $K-D+U$, scalar linear codes can not achieve capacity. In this case capacity can be achieved by using vector linear coding. In the vector linear code, we require to pack $K(U+1)$ message symbols corresponding to $K$ users in $K-D+U=K-\Delta$ dimensions.
We prove that we can pack $K(U+1)$ message symbols into $K$ symbols $\{y_{1},y_{2},\cdots,y_{K}\}$ and then convert these $K$ symbols into $K-\Delta$ code symbols by using one side adjacent antidote scalar linear code. Define the symbol $y_{k}$ for $k=1,2,\cdots,K$ as
\begin{equation}
\end{equation}
The symbol $y_{k}$ comprises of $(U+1)$ message symbols and each of the $K(U+1)$ message symbols appear exactly once in one of the $y_{k}$. In the symbol $y_{k}$, there exists $(U+1)$ message symbols and these $(U+1)$ message symbols are required by the receivers $R_{j}$, for $ j =k,k-1,\cdots, k-U.$
Vector coding message alignment.
For a multiple unicast index coding problem with $K$ messages $\{y_1,y_2,\cdots,y_K\}$ and the same number of receivers with the receiver $R_k$ wanting the message $y_k$ and having a symmetric antidote pattern ${\cal K}_k$ given by
\begin{equation}
\label{antidoteext}
{\cal K}_k =\{y_{k+1}, y_{k+2},\dots,y_{k+D}\},
\end{equation}
let $\mathfrak{C} = \{ t_1,t_2, \cdots, t_l \}$ be a scalar linear code of length $l.$ Define $\overline{x_k}=\{x_{k,1},x_{k,2},\cdots,x_{k,U+1}\}$ as $k^{th}$ vector message symbol for $k=1,2,\cdots,K$. For an arbitrary positive integer $U$ consider the index coding problem with $K$ number of messages $\{\overline{x_1},\overline{x_2},\cdots,\overline{x_{K}}\}$ and the number of receivers being $K,$ and receiver $R_k$ ($k=1,2, \cdots ,K$) having antidote pattern given by
\begin{equation}
\label{antidoteextension}
{\cal K}_k =\{\overline{x_{k-U}},\dots,\overline{x_{k-2}},\overline{x_{k-1}}\}~\cup~\{\overline{x_{k+1}}, \overline{x_{k+2}},\dots,\overline{x_{k+D+U}}\}.
\end{equation}
For this index coding problem the code $\mathfrak{C}^{(U+1)}$ is obtained by replacing every message symbol $y_k$ in the code symbols of $\mathfrak{C}$ with $\sum_{i=1}^{U+1} x_{k+1-i,i}$ for $1 \leq k \leq K,$ i.e., by making the substitution $y_k= \sum_{i=1}^{U+1} x_{k+1-i,i}.$
The given scalar linear code $\mathfrak{C} = \{ t_1,t_2, \cdots, t_l \}$ of length $l$ for a multiple unicast index coding problem with $K$ messages $\{y_1,y_2,\cdots,y_K\}$ and the same number of receivers with the receiver $R_k$ wanting the message $y_k$ and having a symmetric antidote pattern ${\cal K}_k$ given in (<ref>) enables the decoding of $K$ messages $\{y_1,y_2,\cdots,y_K\}$. That is, with linear decoding there exist combination of code symbols $\{ t_1,t_2, \cdots, t_l \}$ to get the sum of the form given in (<ref>)
\begin{equation}
\label{sum}
S_{k}=y_{k}+y_{k+a_{k,1}}+y_{k+a_{k,2}}+\cdots+y_{k+a_{k,d}} \ \vert \ k=1,2,\cdots,K, \\
\end{equation}
for $1 \leq a_{k,1}<a_{k,2}< \cdots <a_{k,d} \leq D,$ such that $y_{k+a_{k,1}}, y_{k+a_{k,2}},\cdots,y_{k+a_{k,d}}$ are in antidotes of receiver $R_{k}$. Since $a_{k,1},a_{k,2},\cdots,a_{k,d}$ are less than or equal to $D$ for $k=1,2,\cdots,K$ from the sum $y_{k}+y_{k+a_{k,1}}+y_{k+a_{k,2}}+\cdots+y_{k+a_{k,d}}$, receiver $R_{k}$ can decode its wanted message $y_{k}$.
Given an arbitrary positive integer $U$ consider the index coding problem with $K$ number of vector message symbols $\{\overline{x_1},\overline{x_2},\cdots,\overline{x_{K}}\}$ and the number of receivers being $K,$ and the receiver $R_k$ ($k=1,2, \cdots ,K$) having antidote pattern given by (<ref>). The code $\mathfrak{C}^{(U+1)}$ is the index code obtained by making the substitution $y_k= \sum_{i=1}^{U+1} x_{k+1-i,i}$ in the available code $\mathfrak{C}$. We prove that the receiver $R_{k}$ can decode its wanted message set $\overline{x_k}$ ($U+1$ message symbols $x_{k,1},x_{k,2},\cdots,x_{k,U+1}$) by using the code $\mathfrak{C}^{(U+1)}$. For the code $\mathfrak{C}^{(U+1)}$, the sum in (<ref>) can be written by
\begin{equation}
\label{sumextension}
S_{k}^{(U+1)}=\sum\limits_{i=1}^{U+1} x_{k+1-i,i}+\sum\limits_{i=1}^{U+1} x_{k+a_{k,1}+1-i,i}+\cdots+\sum\limits_{i=1}^{U+1} x_{k+a_{k,d}+1-i,i}
\end{equation}
We describe the entire decoding process in the following three steps. The Fig.1 will be helpful to trace these steps.
Step 1. Decoding of $(U+1)^{th}$ message symbol by receiver $R_{k}$
The sum $S_{k+U}^{(U+1)}$ similar to (<ref>) can be written as
$S_{k+U}^{(U+1)}=x_{k,U+1}+\sum_{i=1}^{U} x_{k+U+1-i,i}+\sum_{i=1}^{U+1} x_{k+U+a_{k,1}+1-i,i}+\cdots+\sum_{i=1}^{U+1} x_{k+U+a_{k,d}+1-i,i}.$
In $S_{k+U}^{(U+1)}$, only one message symbol is present which is required by $R_{k}.$ We have $1 \leq a_{k,1}<a_{k,2}< \cdots <a_{k,d} \leq D$ and thus all other message symbols present in $S_{k+U}^{(U+1)}$ are in antidotes of receiver $R_{k}$ according to antidote pattern as in (<ref>). (Note that if $S_{k+U}^{U+1}$ comprise $\geq 2$ message symbols which belong to $\overline{x_{k}}$ and $R_{k}$ requires at least two of them, then messages interfere and $R_{k}$ can not decode any of them). Thus by using $S_{k+U}^{(U+1)}$, receiver $R_{k}$ can decode its $(U+1)^{th}$ message symbol $x_{k,U+1}$. The code $\mathfrak{C}$ enables the decoding of $\{y_{1},y_{2},\cdots,y_{K}\}$ from the sums $S_{1},S_{2},\cdots,S_{K}$, which implies that code $\mathfrak{C}^{(U+1)}$ enables the the decoding of $\{x_{1,U+1},x_{2,U+1},\cdots,x_{k-U,U+1},\cdots,x_{K,U+1}\}.$
Step 2. Decoding of $U^{th}$ message symbol by receiver $R_{k}$
Receiver $R_{k}$ uses sum $S_{k+U-1}^{(U+1)}$ to decode its $U^{th}$ message symbol $x_{k,U}$. The sum $S_{k+U-1}^{(U+1)}$ similar to (<ref>) can be written as
$S_{k+U-1}^{(U+1)}=x_{k,U}+a(x_{k,U+1})+\sum_{i=1}^{U} x_{k+U-i,i}+\sum_{i=1}^{U+1} x_{k+U+a_{k,1}-i,i}+\cdots+\sum_{i=1}^{U+1} x_{k+U+a_{k,d}-i,i},$
where $a=1$ if $a_{k,1}=1$, else $a=0$. That is, depending on the value of $a_{k,1}$, sum $S_{k+U-1}^{(U+1)}$ comprises of either one message symbol or two message symbols belongs to vector message symbol $\overline{x_{k}}$.
If $a_{k,1}>1$, $S_{k+U-1}^{(U+1)}$ comprises only one message symbol ($x_{k,U}$) which belongs to $\overline{x_{k,U}}$. All other messages present in $S_{k+U-1}^{(U+1)}$ are antidotes to $R_{k}$. Thus receiver $R_{k}$ can decode message symbol $x_{k,U}$.
If $a_{k,1}=1$, $S_{k+U-1}^{(U+1)}$ comprises two message symbols ($x_{k,U+1},x_{k,U}$) belongs to vector message symbol $\overline{x_{k}}.$ Receiver $R_{k}$ has already decoded the message symbol $x_{k,U+1}$ and all other messages present in $S_{k+U-1}^{(U+1)}$ are antidotes to $R_{k}$. Thus receiver $R_{k}$ can decode message symbol $x_{k,U}$.
Step l. Decoding of $(U+2-l)^{th} (1 \leq l \leq U+1)$ message symbol by receiver $R_{k}$
Receiver $R_{k}$ uses sum $S_{k+U+1-l}^{(U+1)}$ to decode its $(U+2-l)^{th}$ message symbol $x_{k,U+2-l}$. The sum $S_{k+U+1-l}^{(U+1)}$ similar to (<ref>) can be written as
\begin{equation}
\begin{aligned}
S_{k+U+1-l}^{(U+1)}=x_{k,U+2-l}&+1_{A_{1}}(l)x_{k,U+2-l+a_{k,1}} \\
&+1_{A_{2}}(l)x_{k,U+2-l+a_{k,2}}+ \\
\cdots &+ 1_{A_{d}}(l)x_{k,U+2-l+a_{k,d}}\\
&+\sum\limits_{i=1, i\neq U+2-l}^{U+1} x_{k+U+2-l-i,i}\\&+
\sum\limits_{i=1,i \neq U+2-l+a_{k,1}}^{U+1}x_{k+U+2-l+a_{k,1}-i,i}+
\cdots \\&+\sum\limits_{i=1, i \neq U+2-l+a_{k,d}}^{U+1} x_{k+U+2-l+a_{k,d}-i,i}.
\end{aligned}
\end{equation}
Where $1_{A}$ is the indicator function such that $1_{A}(x)=1$ if $x \in A$, else it is zero. $A_{j}=\{a_{k,j}+1,a_{k,j}+2,\cdots,U+1\}$ if $U \geq a_{k,j}$, else $A_{j}=\{\Phi\}$ for $j=1,2,\cdots,d$. In the above sum $x_{k,U+2-l}$ is the required message.
The quantity $1_{A_{j}}(l)x_{k,U+2-l+a_{k,j}}$ for $j=1,2,\cdots,d$ is the interference to the receiver $R_{k}$ from its wanted vector message symbol $\overline{x_{k}}$. Receiver $R_{k}$ already knows the $l-1$ message symbols $\{x_{k,U+1}, x_{k,U},\cdots,x_{k,U+3-l}\}$ and thus the interference from the wanted message symbol $\overline{x_{k}}$ in the sum $S_{k+U+1-l}^{(U+1)}$ can be canceled.
The message symbols in the terms $\tiny{\sum_{i=1, i\neq U+2-l+a_{k,j}}^{U+1} x_{k+U+2-l+a_{k,j}-i,i}}$ for $j=1,2,\cdots,d$ is in antidotes for receiver $R_{k}$. Thus receiver $R_{k}$ can decode $x_{k,U+2-l}$ from $S_{k+U+1-l}^{(U+1)}$.
This procedure continued until all receiver $R_{k}$ decode its $U+1$ wanted messages $\{x_{k,1},x_{k,2},\cdots,x_{k,U+1}\}$. Thus receiver $R_{k}$ decodes its wanted vector message symbol $\overline{x_{k}}$. This completes the proof for decoding.
In the construction of Theorem <ref>, if the given code $\mathfrak{C}$ has optimal length $l=K-D,$ then all extended vector linear codes of given code are of optimal length and hence capacity achieving.
The number of code symbols in the extended code is equal to the number of code symbols in $\mathfrak{C}$ which is equal to $K-\Delta$ = $K-D+U$. By using $K-D+U$ code symbols, every receiver gets $U+1$ of its wanted messages. The capacity achieved by this code is $\frac{U+1}{K-D+U}$ per message, which is equal to the capacity mentioned in (<ref>). It follows that the extended vector linear codes are of optimal length.
The following examples illustrates Theorem <ref> and Theorem <ref>
$K=20, \ U=0, \ D=4.$
We have one-sided antidote scalar code $\mathfrak{C}$=$\{y_{1}+y_{5}, ~y_{2}+y_{6}, ~y_{3}+y_{7}, ~y_{4}+y_{8}, y_{5}+y_{9}, ~y_{6}+y_{10}, ~y_{7}+y_{11}, ~y_{8}+y_{12}, ~y_{9}+y_{13}, ~y_{10}+y_{14}, ~y_{11}+y_{15}, ~y_{12}+y_{16}, ~y_{13}+y_{17}, ~y_{14}+y_{18}, ~y_{15}+y_{19}, ~y_{16}+y_{20}\}$.
$Case~ I$: $K=20,\ U=1,\ D=5.$
$\Delta=4$, Capacity=$\frac{2}{16}$. Define $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,20.$ We have
$y_{1}=x_{1,1}+x_{20,2}, ~~ y_{11}=x_{11,1}+x_{10,2}$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~y_{12}=x_{12,1}+x_{11,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{13}=x_{13,1}+x_{12,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{14}=x_{14,1}+x_{13,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{15}=x_{15,1}+x_{14,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{19}=x_{19,1}+x_{18,2}$,
$y_{10}=x_{10,1}+x_{9,2}, ~~y_{20}=x_{20,1}+x_{19,2}$.
We get the code $\mathfrak{C^{(2)}}$ given by
$Case~ II$: $K=20,\ U=2,\ D=6.$
$\Delta=4$, Capacity=$\frac{3}{16}$. Defining $y_{i}=x_{i,1}+x_{i-1,\ 2}+x_{i-2,\ 3}$ for $i=1,2,\dots,20$ we get
$y_{1}=x_{1,1}+x_{20,2}+x_{19,3}, ~~ y_{11}=x_{11,1}+x_{10,2}+x_{9,3}$,
$y_{2}=x_{2,1}+x_{1,2}+x_{20,3}, ~~~y_{12}=x_{12,1}+x_{11,2}+x_{10,3}$,
$y_{3}=x_{3,1}+x_{2,2}+x_{1,3}, ~~~~y_{13}=x_{13,1}+x_{12,2}+x_{11,3}$,
$y_{4}=x_{4,1}+x_{3,2}+x_{2,3}, ~~~~y_{14}=x_{14,1}+x_{13,2}+x_{12,3}$,
$y_{5}=x_{5,1}+x_{4,2}+x_{3,3}, ~~~~y_{15}=x_{15,1}+x_{14,2}+x_{13,3}$,
$y_{6}=x_{6,1}+x_{5,2}+x_{4,3}, ~~~~y_{16}=x_{16,1}+x_{15,2}+x_{14,3}$,
$y_{7}=x_{7,1}+x_{6,2}+x_{5,3}, ~~~~y_{17}=x_{17,1}+x_{16,2}+x_{15,3}$,
$y_{8}=x_{8,1}+x_{7,2}+x_{6,3}, ~~~~y_{18}=x_{18,1}+x_{17,2}+x_{16,3}$,
$y_{9}=x_{9,1}+x_{8,2}+x_{7,3}, ~~~~y_{19}=x_{19,1}+x_{18,2}+x_{17,3}$,
$y_{10}=x_{10,1}+x_{9,2}+x_{8,3}, ~~y_{20}=x_{20,1}+x_{19,2}+x_{18,3}$.
For this case we get the extended code $ \mathfrak{C^{(3)}}$ to be
$ \mathfrak{C^{(3)}}=\{x_{1,1}+x_{20,2}+x_{19,3}+x_{5,1}+x_{4,2}+x_{3,3},
~~ {x_{2,1}}+x_{1,2}+x_{20,3}+x_{6,1}+x_{5,2}+x_{4,3},
~~ {x_{3,1}}+x_{2,2}+x_{1,3}+x_{7,1}+x_{6,2}+x_{5,3},
~~ {x_{4,1}}+x_{3,2}+x_{2,3}+x_{8,1}+x_{7,2}+x_{6,3},
~~ {x_{5,1}}+x_{4,2}+x_{3,3}+x_{9,1}+x_{8,2}+x_{7,3},
~~ {x_{6,1}}+x_{5,2}+x_{4,3}+x_{10,1}+x_{9,2}+x_{8,3},
~~ {x_{7,1}}+x_{6,2}+x_{5,3}+x_{11,1}+x_{10,2}+x_{9,3},
~~ {x_{8,1}}+x_{7,2}+x_{6,3}+x_{12,1}+x_{11,2}+x_{10,3},
~~ {x_{9,1}}+x_{8,2}+x_{7,3}+x_{13,1}+x_{12,2}+x_{11,3},
~~ {x_{10,1}}+x_{9,2}+x_{8,3}+x_{14,1}+x_{13,2}+x_{12,3},
~~ {x_{11,1}}+x_{10,2}+x_{9,3}+x_{15,1}+x_{14,2}+x_{13,3},
~~ {x_{12,1}}+x_{11,2}+x_{10,3}+x_{16,1}+x_{15,2}+x_{14,3},
~~ {x_{13,1}}+x_{12,2}+x_{11,3}+x_{17,1}+x_{16,2}+x_{15,3},
~~ {x_{14,1}}+x_{13,2}+x_{12,3}+x_{18,1}+x_{17,2}+x_{16,3},
~~ {x_{15,1}}+x_{14,2}+x_{13,3}+x_{19,1}+x_{18,2}+x_{17,3},
~~ {x_{16,1}}+x_{15,2}+x_{14,3}+x_{20,1}+x_{19,2}+x_{18,3}\}$.
$Case ~ III$: $K=20,\ U=3,\ D=7.$
$\Delta=4$, Capacity=$\frac{3}{16}$. With defining $y_{i}=x_{i,1}+x_{i-1,\ 2}+x_{i-2,\ 3}+x_{i-3,\ 4}$ for $i=1,2,\dots,20,$ we end with
$y_{1}=x_{1,1}+x_{20,2}+x_{19,3}+x_{18,4}, ~~ y_{11}=x_{11,1}+x_{10,2}+x_{9,3}+x_{8,4}$,
$y_{2}=x_{2,1}+x_{1,2}+x_{20,3}+x_{19,4}, ~~~y_{12}=x_{12,1}+x_{11,2}+x_{10,3}+x_{9,4}$,
$y_{3}=x_{3,1}+x_{2,2}+x_{1,3}+x_{20,4}, ~~~~y_{13}=x_{13,1}+x_{12,2}+x_{11,3}+x_{10,4}$,
$y_{4}=x_{4,1}+x_{3,2}+x_{2,3}+x_{1,4}, ~~~~y_{14}=x_{14,1}+x_{13,2}+x_{12,3}+x_{11,4}$,
$y_{5}=x_{5,1}+x_{4,2}+x_{3,3}+x_{2,4}, ~~~~y_{15}=x_{15,1}+x_{14,2}+x_{13,3}+x_{12,4}$,
$y_{6}=x_{6,1}+x_{5,2}+x_{4,3}+x_{3,4}, ~~~~y_{16}=x_{16,1}+x_{15,2}+x_{14,3}+x_{13,4}$,
$y_{7}=x_{7,1}+x_{6,2}+x_{5,3}+x_{4,4}, ~~~~y_{17}=x_{17,1}+x_{16,2}+x_{15,3}+x_{14,4}$,
$y_{8}=x_{8,1}+x_{7,2}+x_{6,3}+x_{5,4}, ~~~~y_{18}=x_{18,1}+x_{17,2}+x_{16,3}+x_{15,4}$,
$y_{9}=x_{9,1}+x_{8,2}+x_{7,3}+x_{6,4}, ~~~~y_{19}=x_{19,1}+x_{18,2}+x_{17,3}+x_{16,4}$,
$y_{10}=x_{10,1}+x_{9,2}+x_{8,3}+x_{7,4}, ~~y_{20}=x_{20,1}+x_{19,2}+x_{18,3}+x_{17,4}$.
The resulting extended vector linear code is
$ \mathfrak{C^{(4)}}=\{x_{1,1}+x_{20,2}+x_{19,3}+x_{18,4}+x_{5,1}+x_{4,2}+x_{3,3}+x_{2,4},
~~ {x_{2,1}}+x_{1,2}+x_{20,3}+x_{19,4}+x_{6,1}+x_{5,2}+x_{4,3}+x_{3,4},
~~ {x_{3,1}}+x_{2,2}+x_{1,3}+x_{20,4}+x_{7,1}+x_{6,2}+x_{5,3}+x_{4,4},
~~ {x_{4,1}}+x_{3,2}+x_{2,3}+x_{1,4}+x_{8,1}+x_{7,2}+x_{6,3}+x_{5,4},
~~ {x_{5,1}}+x_{4,2}+x_{3,3}+x_{2,4}+x_{9,1}+x_{8,2}+x_{7,3}+x_{6,4},
~~ {x_{6,1}}+x_{5,2}+x_{4,3}+x_{3,4}+x_{10,1}+x_{9,2}+x_{8,3}+x_{7,4},
~~ {x_{7,1}}+x_{6,2}+x_{5,3}+x_{4,4}+x_{11,1}+x_{10,2}+x_{9,3}+x_{8,4},
~~ {x_{8,1}}+x_{7,2}+x_{6,3}+x_{5,4}+x_{12,1}+x_{11,2}+x_{10,3}+x_{9,4},
~~ {x_{9,1}}+x_{8,2}+x_{7,3}+x_{6,4}+x_{13,1}+x_{12,2}+x_{11,3}+x_{10,4},
~~ {x_{10,1}}+x_{9,2}+x_{8,3}+x_{7,4}+x_{14,1}+x_{13,2}+x_{12,3}+x_{11,4},
~~ {x_{11,1}}+x_{10,2}+x_{9,3}+x_{8,4}+x_{15,1}+x_{14,2}+x_{13,3}+x_{12,4},
~~ {x_{12,1}}+x_{11,2}+x_{10,3}+x_{9,4}+x_{16,1}+x_{15,2}+x_{14,3}+x_{13,4},
~~ {x_{13,1}}+x_{12,2}+x_{11,3}+x_{10,4}+x_{17,1}+x_{16,2}+x_{15,3}+x_{18,4},
~~ {x_{14,1}}+x_{13,2}+x_{12,3}+x_{11,4}+x_{18,1}+x_{17,2}+x_{16,3}+x_{15,4},
~~ {x_{15,1}}+x_{14,2}+x_{13,3}+x_{12,4}+x_{19,1}+x_{18,2}+x_{17,3}+x_{16,4},
~~ {x_{16,1}}+x_{15,2}+x_{14,3}+x_{13,4}+x_{20,1}+x_{19,2}+x_{18,3}+x_{17,4}\}$.
Consider a multiple unicast index coding problem with $K$ messages and the same number of receivers with the receiver $R_k$ wanting the message $x_k$ and having a symmetric antidote pattern ${\cal K}_k$ given in (<ref>). For this index coding problem vector linear solution exists if scalar linear solution exists for one-sided antidote problem as mentioned in (<ref>) with same number of messages and number of one-sided antidotes $\Delta =\vert D-U \vert$. The vector linear index code is optimal if the scalar linear code is optimal.
Proof follows from Theorem <ref> and Theorem <ref>.
Proposed codes in <cit.> can be extended for all $U$ and $D$ with $\Delta$ = $max(U,D)-min(U,D)$ for the following problem instances:
* $\Delta$ divides $K$
* $K-\Delta$ divides $K$
* $\frac{K}{2}-\Delta$ divides $\Delta$
* $\Delta - \frac{K}{2}$ divides $\frac{K}{2}$
* $\Delta$ divides $K-\lambda$ and $\lambda$ divides $\Delta$ where $\lambda$ is an integer
* $K-\Delta$ divides $K-\lambda$ and $\lambda$ divides $K-\Delta$
* $\Delta+\lambda$ divides $K$ and $\lambda$ divides $\Delta$
* $K-\Delta+\lambda$ divides $K$ and $\lambda$ divides $K-\Delta$
* $\Delta$ divides $K+\lambda$ and $\lambda$ divides $\Delta$
* $K-\Delta$ divides $K+\lambda$ and $\lambda$ divides $K-\Delta$
In <cit.> codes for symmetric instances of one side antidote problem for a given $K$ and $D$ satisfying the above mentioned conditions with $\Delta$ replaced with $D$ were proposed. The proposed codes in <cit.> can also be used for the one-sided adjacent antidotes as given in (<ref>) and these codes achieve the capacity in (<ref>). Then the proof follows from Theorem <ref>.
Corollary 1. If $D$ divides $K$ then the proposed optimal length scalar linear code is
$\mathfrak{C}=\{x_{i+(j-1)D}+x_{i+jD}|~i=1,2,\dots,D,~ j=1,2,\dots, \frac{K}{D}-1\}$
$K=20,\ U=2,\ D=6.$
$K=20, \Delta=4$, capacity=$\frac{3}{16}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}+x_{i-2,\ 3}$ for $i=1,2,\dots,20.$
$y_{1}=x_{1,1}+x_{20,2}+x_{19,3}, ~~ y_{11}=x_{11,1}+x_{10,2}+x_{9,3}$,
$y_{2}=x_{2,1}+x_{1,2}+x_{20,3}, ~~~y_{12}=x_{12,1}+x_{11,2}+x_{10,3}$,
$y_{3}=x_{3,1}+x_{2,2}+x_{1,3}, ~~~~y_{13}=x_{13,1}+x_{12,2}+x_{11,3}$,
$y_{4}=x_{4,1}+x_{3,2}+x_{2,3}, ~~~~y_{14}=x_{14,1}+x_{13,2}+x_{12,3}$,
$y_{5}=x_{5,1}+x_{4,2}+x_{3,3}, ~~~~y_{15}=x_{15,1}+x_{14,2}+x_{13,3}$,
$y_{6}=x_{6,1}+x_{5,2}+x_{4,3}, ~~~~y_{16}=x_{16,1}+x_{15,2}+x_{14,3}$,
$y_{7}=x_{7,1}+x_{6,2}+x_{5,3}, ~~~~y_{17}=x_{17,1}+x_{16,2}+x_{15,3}$,
$y_{8}=x_{8,1}+x_{7,2}+x_{6,3}, ~~~~y_{18}=x_{18,1}+x_{17,2}+x_{16,3}$,
$y_{9}=x_{9,1}+x_{8,2}+x_{7,3}, ~~~~y_{19}=x_{19,1}+x_{18,2}+x_{17,3}$,
$y_{10}=x_{10,1}+x_{9,2}+x_{8,3}, ~~y_{20}=x_{20,1}+x_{19,2}+x_{18,3}$.
The proposed code is $\mathfrak{C}=\{y_{1}+y_{5}, ~y_{2}+y_{6}, ~y_{3}+y_{7}, ~y_{4}+y_{8}, ~y_{5}+y_{9}, ~y_{6}+y_{10}, ~y_{7}+y_{11}, ~y_{8}+y_{12}, ~y_{9}+y_{13}, ~y_{10}+y_{14}, ~y_{11}+y_{15}, ~y_{12}+y_{16}, ~y_{13}+y_{17}, ~y_{14}+y_{18}, ~y_{15}+y_{19}, ~y_{16}+y_{20}\}$.
$ ~ \mathfrak{C^{(3)}}=\{x_{1,1}+x_{20,2}+x_{19,3}+x_{5,1}+x_{4,2}+x_{3,3},~~{x_{2,1}}+x_{1,2}+x_{20,3}+x_{6,1}+x_{5,2}+x_{4,3},~~ {x_{3,1}}+x_{2,2}+x_{1,3}+x_{7,1}+x_{6,2}+x_{5,3},~~ {x_{4,1}}+x_{3,2}+x_{2,3}+x_{8,1}+x_{7,2}+x_{6,3},~~ {x_{5,1}}+x_{4,2}+x_{3,3}+x_{9,1}+x_{8,2}+x_{7,3},~~ {x_{6,1}}+x_{5,2}+x_{4,3}+x_{10,1}+x_{9,2}+x_{8,3},~~ {x_{7,1}}+x_{6,2}+x_{5,3}+x_{11,1}+x_{10,2}+x_{9,3},~~ {x_{8,1}}+x_{7,2}+x_{6,3}+x_{12,1}+x_{11,2}+x_{10,3},~~ {x_{9,1}}+x_{8,2}+x_{7,3}+x_{13,1}+x_{12,2}+x_{11,3},~~ {x_{10,1}}+x_{9,2}+x_{8,3}+x_{14,1}+x_{13,2}+x_{12,3},~~ {x_{11,1}}+x_{10,2}+x_{9,3}+x_{15,1}+x_{14,2}+x_{13,3},~~ {x_{12,1}}+x_{11,2}+x_{10,3}+x_{16,1}+x_{15,2}+x_{14,3},~~ {x_{13,1}}+x_{12,2}+x_{11,3}+x_{17,1}+x_{16,2}+x_{15,3},~~ {x_{14,1}}+x_{13,2}+x_{12,3}+x_{18,1}+x_{17,2}+x_{16,3},~~ {x_{15,1}}+x_{14,2}+x_{13,3}+x_{19,1}+x_{18,2}+x_{17,3},~~ {x_{16,1}}+x_{15,2}+x_{14,3}+x_{20,1}+x_{19,2}+x_{18,3}\}$.
Corollary 2. If divides $K$, then the proposed code is
$${\mathfrak{C}}=\{x_{i}+x_{i+m}+\dots+x_{i+(n-1)m}\ |\ {i = {1,2, \dots, m}\}}$$
where $K-D=m$ and .
$K=20,\ U=1,\ D=16$.
$K=20,\ \Delta=15$, capacity=$\frac{2}{5}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,20.$
$y_{1}=x_{1,1}+x_{20,2}, ~~ y_{11}=x_{11,1}+x_{10,2}$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~y_{12}=x_{12,1}+x_{11,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{13}=x_{13,1}+x_{12,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{14}=x_{14,1}+x_{13,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{15}=x_{15,1}+x_{14,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{19}=x_{19,1}+x_{18,2}$,
$y_{10}=x_{10,1}+x_{9,2}, ~~y_{20}=x_{20,1}+x_{19,2}$.
The proposed code is,
$\mathfrak{C}=\{y_{1}+y_{6}+y_{11}+y_{16}, \\
~~~~~~~~ y_{2}+y_{7}+y_{12}+y_{17}, \\
~~~~~~~~ y_{3}+y_{8}+y_{13}+y_{18}, \\
~~~~~~~~ y_{4}+y_{9}+y_{14}+y_{19},\\
~~~~~~~~ y_{5}+y_{10}+y_{15}+y_{20}\}.$
$ ~ \mathfrak{C^{(2)}}=\{{x_{1,1}}+x_{20,2}+x_{6,1}+x_{5,2}+x_{11,1}+x_{10,2}+x_{16,1}+x_{15,2},\
~~ {x_{2,1}}+x_{1,2}+x_{7,1}+x_{6,2}+x_{12,1}+x_{11,2}+x_{17,1}+x_{16,2},\
~~ {x_{3,1}}+x_{2,2}+x_{8,1}+x_{7,2}+x_{13,1}+x_{12,2}+x_{18,1}+x_{17,2},\
~~ {x_{4,1}}+x_{3,2}+x_{9,1}+x_{8,2}+x_{14,1}+x_{13,2}+x_{19,1}+x_{18,2},\
~~ {x_{5,1}}+x_{4,2}+x_{10,1}+x_{9,2}+x_{15,1}+x_{14,2}+x_{20,1}+x_{19,2}\}.$
Corollary 3. For the case $\frac{K}{2}-D$ divides $D$, the proposed scalar linear code is,
$\mathfrak{C}=\{x_{i}+x_{i+m}+\dots+x_{i+pm}, \\
~~~~~~~~ x_{i+m}+x_{i+2m}+\dots+x_{i+(p+1)m}, \\
~~~~~~~~~~~~~~~~~~~~ \vdots \\
~~~~~~~~ x_{i+m(p+1)}+x_{i+m(p+2)}+\dots+x_{i+(n-1)m}
| i=1,2, \dots ,m\},$
where $m=\frac{K}{2}-D$, $n=\frac{K}{\frac{K}{2}-D}$ and $\frac{D}{\frac{K}{2}-D}=p$.
$K=20,\ U=1,\ D=9$.
$K=20,\ \Delta=8$, capacity=$\frac{2}{12}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,20.$
$y_{1}=x_{1,1}+x_{20,2}, ~~~y_{2}=x_{2,1}+x_{1,2}, ~~~~y_{3}=x_{3,1}+x_{2,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{6}=x_{6,1}+x_{5,2},$
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{9}=x_{9,1}+x_{8,2}, $
$y_{10}=x_{10,1}+x_{9,2}, ~~~y_{11}=x_{11,1}+x_{10,2},~y_{12}=x_{12,1}+x_{11,2},$
$y_{13}=x_{13,1}+x_{12,2}, ~~y_{14}=x_{14,1}+x_{13,2}, ~y_{15}=x_{15,1}+x_{14,2}$,
$y_{16}=x_{16,1}+x_{15,2}, ~~y_{17}=x_{17,1}+x_{16,2}, ~y_{18}=x_{18,1}+x_{17,2}$,
$y_{19}=x_{19,1}+x_{18,2}, ~y_{20}=x_{20,1}+x_{19,2}$.
The proposed code is $\mathfrak{C}=$
$\{y_{1}+y_{3}+y_{5}+y_{7}+y_{9}, ~~ y_{2}+y_{4}+y_{6}+y_{8}+y_{10}, \\
y_{3}+y_{5}+y_{7}+y_{9}+y_{11}, ~~ y_{4}+y_{6}+y_{8}+y_{10}+y_{12},\\
y_{5}+y_{7}+y_{9}+y_{11}+y_{13}, ~~ y_{6}+y_{8}+y_{10}+y_{12}+y_{14},\\
y_{7}+y_{9}+y_{11}+y_{13}+y_{15}, ~~ y_{8}+y_{10}+y_{12}+y_{14}+y_{16},\\
y_{9}+y_{11}+y_{13}+y_{15}+y_{17}, ~~ y_{10}+y_{12}+y_{14}+y_{16}+y_{18},\\
y_{11}+y_{13}+y_{15}+y_{17}+y_{19}, ~~ y_{12}+y_{14}+y_{16}+y_{18}+y_{20}\}$.
$ ~ \mathfrak{C^{(2)}}=\{~~{x_{1,1}}+x_{20,2}+x_{3,1}+x_{2,2}+x_{5,1}+x_{4,2}+x_{7,1}+x_{6,2}+x_{9,1}+x_{8,2},\
~~ {x_{2,1}}+x_{1,2}+x_{4,1}+x_{3,2}+x_{6,1}+x_{5,2}+x_{8,1}+x_{7,2}+x_{10,1}+x_{9,2},\
~~ {x_{3,1}}+x_{2,2}+x_{5,1}+x_{4,2}+x_{7,1}+x_{6,2}+x_{9,1}+x_{8,2}+x_{11,1}+x_{10,2},\
~~ {x_{4,1}}+x_{3,2}+x_{6,1}+x_{5,2}+x_{8,1}+x_{7,2}+x_{10,1}+x_{9,2}+x_{12,1}+x_{11,2},\
~~ {x_{5,1}}+x_{4,2}+x_{7,1}+x_{6,2}+x_{9,1}+x_{8,2}+x_{11,1}+x_{10,2}+x_{13,1}+x_{12,2},\
~~ {x_{6,1}}+x_{5,2}+x_{8,1}+x_{7,2}+x_{10,1}+x_{9,2}+x_{12,1}+x_{11,2}+x_{14,1}+x_{13,2},\
~~ {x_{7,1}}+x_{6,2}+x_{9,1}+x_{8,2}+x_{11,1}+x_{10,2}+x_{13,1}+x_{12,2}+x_{15,1}+x_{14,2},\
~~ {x_{8,1}}+x_{7,2}+x_{10,1}+x_{9,2}+x_{12,1}+x_{11,2}+x_{14,1}+x_{13,2}+x_{16,1}+x_{15,2}, \
~~ {x_{9,1}}+x_{8,2}+x_{11,1}+x_{10,2}+x_{13,1}+x_{12,2}+x_{15,1}+x_{14,2}+x_{17,1}+x_{16,2},\
~~ {x_{10,1}}+x_{9,2}+x_{12,1}+x_{11,2}+x_{14,1}+x_{13,2}+x_{16,1}+x_{15,2}+x_{18,1}+x_{17,2},\
~~ {x_{11,1}}+x_{10,2}+x_{13,1}+x_{12,2}+x_{15,1}+x_{14,2}+x_{17,1}+x_{16,2}+x_{19,1}+x_{18,2},\
~~ {x_{12,1}}+x_{11,2}+x_{14,1}+x_{13,2}+x_{16,1}+x_{15,2}+x_{18,1}+x_{17,2}+x_{20,1}+x_{19,2}\}.$
Corollary 4. If $D$ divides $K-\lambda$ and $\lambda$ divides $D$, then the scalar linear code is given by $\mathfrak{C}=\{{x_{i+(j-1)D}+x_{i+jD}}|\ {i = {1,2,\dots,D}},\ {j = {1,2,\dots,n-1}}\}\\ \cup \{{x_{K-\lambda+r}+x_{K-\lambda+r-\lambda}+\dots+x_{K-\lambda+r-t\lambda}}|\ {r = {1,2,\dots,\lambda}},\\ \ t = 1,2,\dots,\frac{D}{\lambda}$} for $\frac{K-\lambda}{D}>1$ and $\frac{K-\lambda}{D}=n$.
$K=21,\ U=2,\ D=6$
$K=21,\Delta=4, \lambda=1$, capacity=$\frac{3}{17}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}+x_{i-2,\ 3}$ for $i=1,2,\dots,21.$
$y_{1}=x_{1,1}+x_{21,2}+x_{20,3}, ~~ y_{11}=x_{11,1}+x_{10,2}+x_{9,3}$,
$y_{2}=x_{2,1}+x_{1,2}+x_{21,3}, ~~~y_{12}=x_{12,1}+x_{11,2}+x_{10,3}$,
$y_{3}=x_{3,1}+x_{2,2}+x_{1,3}, ~~~~y_{13}=x_{13,1}+x_{12,2}+x_{11,3}$,
$y_{4}=x_{4,1}+x_{3,2}+x_{2,3}, ~~~~y_{14}=x_{14,1}+x_{13,2}+x_{12,3}$,
$y_{5}=x_{5,1}+x_{4,2}+x_{3,3}, ~~~~y_{15}=x_{15,1}+x_{14,2}+x_{13,3}$,
$y_{6}=x_{6,1}+x_{5,2}+x_{4,3}, ~~~~y_{16}=x_{16,1}+x_{15,2}+x_{14,3}$,
$y_{7}=x_{7,1}+x_{6,2}+x_{5,3}, ~~~~y_{17}=x_{17,1}+x_{16,2}+x_{15,3}$,
$y_{8}=x_{8,1}+x_{7,2}+x_{6,3}, ~~~~y_{18}=x_{18,1}+x_{17,2}+x_{16,3}$,
$y_{9}=x_{9,1}+x_{8,2}+x_{7,3}, ~~~~y_{19}=x_{19,1}+x_{18,2}+x_{17,3}$,
$y_{10}=x_{10,1}+x_{9,2}+x_{8,3}, ~~y_{20}=x_{20,1}+x_{19,2}+x_{18,3}$,
and $y_{21}=x_{21,1}+x_{20,2}+x_{19,3}.$
The proposed code is $\mathfrak{C}=\{y_{1}+y_{5}, ~y_{2}+y_{6}, ~y_{3}+y_{7}, ~y_{4}+y_{8}, ~y_{5}+y_{9}, ~y_{6}+y_{10}, ~y_{7}+y_{11}, ~y_{8}+y_{12}, ~y_{9}+y_{13}, ~y_{10}+y_{14}, ~y_{11}+y_{15}, ~y_{12}+y_{16}, ~y_{13}+y_{17}, ~y_{14}+y_{18}, ~y_{15}+y_{19}, ~y_{16}+y_{20}, ~y_{17}+y_{18}+y_{19}+y_{20}+y_{21}\}.$
$ ~ \mathfrak{C^{(3)}}=\{{x_{1,1}}+x_{21,2}+x_{20,3}+x_{5,1}+x_{4,2}+x_{3,3},\
~~ {x_{2,1}}+x_{1,2}+x_{21,3}+x_{6,1}+x_{5,2}+x_{4,3},\
~~ {x_{3,1}}+x_{2,2}+x_{1,3}+x_{7,1}+x_{6,2}+x_{5,3},\
~~ {x_{4,1}}+x_{3,2}+x_{2,3}+x_{8,1}+x_{7,2}+x_{6,3},\
~~ {x_{5,1}}+x_{4,2}+x_{3,3}+x_{9,1}+x_{8,2}+x_{7,3},\
~~ {x_{6,1}}+x_{5,2}+x_{4,3}+x_{10,1}+x_{9,2}+x_{8,3},\
~~ {x_{7,1}}+x_{6,2}+x_{5,3}+x_{11,1}+x_{10,2}+x_{9,3},\
~~ {x_{8,1}}+x_{7,2}+x_{6,3}+x_{12,1}+x_{11,2}+x_{10,3},\
~~ {x_{9,1}}+x_{8,2}+x_{7,3}+x_{13,1}+x_{12,2}+x_{11,3},\
~~ {x_{10,1}}+x_{9,2}+x_{8,3}+x_{14,1}+x_{13,2}+x_{12,3},\
~~ {x_{11,1}}+x_{10,2}+x_{9,3}+x_{15,1}+x_{14,2}+x_{13,3},\
~~ {x_{12,1}}+x_{11,2}+x_{10,3}+x_{16,1}+x_{15,2}+x_{14,3},\
~~ {x_{13,1}}+x_{12,2}+x_{11,3}+x_{17,1}+x_{16,2}+x_{15,3},\
~~ {x_{14,1}}+x_{13,2}+x_{12,3}+x_{18,1}+x_{17,2}+x_{16,3},\
~~ {x_{15,1}}+x_{14,2}+x_{13,3}+x_{19,1}+x_{18,2}+x_{17,3},\
~~ {x_{16,1}}+x_{15,2}+x_{14,3}+x_{20,1}+x_{19,2}+x_{18,3},\
~~ {x_{17,1}}+x_{16,2}+x_{15,3}+x_{18,1}+x_{17,2}+x_{16,3}+x_{19,1}+x_{18,2}+x_{17,3}+x_{20,1}+x_{19,2}+x_{18,3}+x_{21,1}+x_{20,2}+x_{19,3}\}.$
Corollary 5. If $K-D$ divides $K-\lambda,$ $\lambda$ divides $(K-D)$, then the proposed scalar linear code is $\mathfrak{C}=\{x_{i}+x_{i+m}+\dots+x_{i+(q-1)m}+x_{qm+1+(i-1) mod \lambda)}
|\ i = {1,2,\dots,m}\},$ where $K-D=m$, and $\frac{K-\lambda}{K-D}=q$.
$K=21,\ U=1,\ D=17$.
$K=21,\ \Delta=16,$ capacity=$\frac{2}{5}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,21.$
$y_{1}=x_{1,1}+x_{21,2}, ~~ y_{11}=x_{11,1}+x_{10,2}$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~y_{12}=x_{12,1}+x_{11,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{13}=x_{13,1}+x_{12,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{14}=x_{14,1}+x_{13,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{15}=x_{15,1}+x_{14,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{19}=x_{19,1}+x_{18,2}$,
$y_{10}=x_{10,1}+x_{9,2}, ~~y_{20}=x_{20,1}+x_{19,2}$,
and $y_{21}=x_{21,1}+x_{20,2}.$
The proposed code is $\mathfrak{C}=\{y_{1}+y_{6}+y_{11}+y_{16}+y_{21}, \
y_{2}+y_{7}+y_{12}+y_{17}+y_{21}, \
y_{3}+y_{8}+y_{13}+y_{18}+y_{21}, \
$ ~ \mathfrak{C^{(2)}}=\{{x_{1,1}}+x_{21,2}+x_{6,1}+x_{5,2}+x_{11,1}+x_{10,2}+x_{16,1}+x_{15,2}+x_{21,1}+x_{20,2},\
~~ {x_{2,1}}+x_{1,2}+x_{7,1}+x_{6,2}+x_{12,1}+x_{11,2}+x_{17,1}+x_{16,2}+x_{21,1}+x_{20,2},\
~~ {x_{3,1}}+x_{2,2}+x_{8,1}+x_{7,2}+x_{13,1}+x_{12,2}+x_{18,1}+x_{17,2}+x_{21,1}+x_{20,2},\
~~ {x_{4,1}}+x_{3,2}+x_{9,1}+x_{8,2}+x_{14,1}+x_{13,2}+x_{19,1}+x_{18,2}+x_{21,1}+x_{20,2},\
~~ {x_{5,1}}+x_{4,2}+x_{10,1}+x_{9,2}+x_{15,1}+x_{14,2}+x_{20,1}+x_{19,2}+x_{21,1}+x_{20,2}\}.$
Corollary 6. For the case $D+\lambda$ divides $K,$ $\lambda$ divides $D$, the scalar linear code is
$\mathfrak{C}=\{x_{i+j\lambda}+x_{i+(j+1)\lambda}+\dots+x_{i+(j+p)\lambda}|\ {i =1,2,\dots,\lambda},\\ j= 1,2,\dots,\frac{K-D-\lambda}{\lambda}$}
where $\frac{D}{\lambda}=p$ and $\frac{K}{D+\lambda}=n.$
$K=18,\ U=1,\ D=6.$
$K=18,\ \Delta=5, \lambda=1$, capacity=$\frac{2}{13}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,18.$
$y_{1}=x_{1,1}+x_{18,2}, ~~ y_{10}=x_{10,1}+x_{9,2},$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~y_{11}=x_{11,1}+x_{10,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{12}=x_{12,1}+x_{11,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{13}=x_{13,1}+x_{12,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{14}=x_{14,1}+x_{13,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{15}=x_{15,1}+x_{14,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
The proposed code is $\mathfrak{C}=\{y_{1}+y_{2}+y_{3}+y_{4}+y_{5}+y_{6},~ y_{2}+y_{3}+y_{4}+y_{5}+y_{6}+y_{7}, \
y_{3}+y_{4}+y_{5}+y_{6}+y_{7}+y_{8}, ~ y_{4}+y_{5}+y_{6}+y_{7}+y_{8}+y_{9}, \
y_{5}+y_{6}+y_{7}+y_{8}+y_{9}+y_{10}, ~ y_{6}+y_{7}+y_{8}+y_{9}+y_{10}+y_{11}, \
y_{7}+y_{8}+y_{9}+y_{10}+y_{11}+y_{12}, ~ y_{8}+y_{9}+y_{10}+y_{11}+y_{12}+y_{13}, \
y_{9}+y_{10}+y_{11}+y_{12}+y_{13}+y_{14}, ~ y_{10}+y_{11}+y_{12}+y_{13}+y_{14}+y_{15}, \
y_{11}+y_{12}+y_{13}+y_{14}+y_{15}+y_{16}, ~ y_{12}+y_{13}+y_{14}+y_{15}+y_{16}+y_{17}, \
$ ~ \mathfrak{C^{(2)}}=\{{x_{1,1}}+x_{18,2}+x_{2,1}+x_{1,2}+x_{3,1}+x_{2,2}+x_{4,1}+x_{3,2}+x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2},\
~~ {x_{2,1}}+x_{1,2}+x_{3,1}+x_{2,2}+x_{4,1}+x_{3,2}+x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2},\
~~ {x_{3,1}}+x_{2,2}+x_{4,1}+x_{3,2}+x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}+x_{8,1}+x_{7,2},\
~~ {x_{4,1}}+x_{3,2}+x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}+x_{8,1}+x_{7,2}+x_{9,1}+x_{8,2},\
~~ {x_{5,1}}+x_{4,2}+x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}+x_{8,1}+x_{7,2}+x_{9,1}+x_{8,2}+x_{10,1}+x_{9,2},\
~~ {x_{6,1}}+x_{5,2}+x_{7,1}+x_{6,2}+x_{8,1}+x_{7,2}+x_{9,1}+x_{8,2}+x_{10,1}+x_{9,2}+x_{11,1}+x_{10,2},\
~~ {x_{7,1}}+x_{6,2}+x_{8,1}+x_{7,2}+x_{9,1}+x_{8,2}+x_{10,1}+x_{9,2}+x_{11,1}+x_{10,2}+x_{12,1}+x_{11,2},\
~~ {x_{8,1}}+x_{7,2}+x_{9,1}+x_{8,2}+x_{10,1}+x_{9,2}+x_{11,1}+x_{10,2}+x_{12,1}+x_{11,2}+x_{13,1}+x_{12,2},\
~~ {x_{9,1}}+x_{8,2}+x_{10,1}+x_{9,2}+x_{11,1}+x_{10,2}+x_{12,1}+x_{11,2}+x_{13,1}+x_{12,2}+x_{14,1}+x_{13,2},\
~~ {x_{10,1}}+x_{9,2}+x_{11,1}+x_{10,2}+x_{12,1}+x_{11,2}+x_{13,1}+x_{12,2}+x_{14,1}+x_{13,2}+x_{15,1}+x_{14,2},\
~~ {x_{11,1}}+x_{10,2}+x_{12,1}+x_{11,2}+x_{13,1}+x_{12,2}+x_{14,1}+x_{13,2}+x_{15,1}+x_{14,2}+x_{16,1}+x_{15,2},\
~~ {x_{12,1}}+x_{11,2}+x_{13,1}+x_{12,2}+x_{14,1}+x_{13,2}+x_{15,1}+x_{14,2}+x_{16,1}+x_{15,2}+x_{17,1}+x_{16,2},\
~~ {x_{13,1}}+x_{12,2}+x_{14,1}+x_{13,2}+x_{15,1}+x_{14,2}+x_{16,1}+x_{15,2}+x_{17,1}+x_{16,2}+x_{18,1}+x_{17,2}\}.$
Corollary 7. If divides $K$ and $\lambda$ divides , then the scalar linear code is given by $\mathfrak{C}=\{x_{i}+x_{i+\lambda}+x_{i+\lambda+(K-D)}+x_{i+2\lambda+(K-D)}+x_{i+2\lambda+2(K-D)}+x_{i+3\lambda+2(K-D)}+\dots+x_{i+(p-1)\lambda+(p-1)(K-D)}+x_{i+p\lambda+(p-1)(K-D)}|\ $} where $\frac{K}{K-D+\lambda}=p$ and $\frac{K-D}{\lambda}=m$.
$K=24,\ U=1,\ D=20.$
$K=24,\ \Delta=19, \lambda=1$, capacity=$\frac{2}{5}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,24.$
$y_{1}=x_{1,1}+x_{24,2}, ~~ ~y_{13}=x_{13,1}+x_{12,2}$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~~y_{14}=x_{14,1}+x_{13,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{15}=x_{15,1}+x_{14,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{19}=x_{19,1}+x_{18,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{20}=x_{20,1}+x_{19,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{21}=x_{21,1}+x_{20,2}$,
$y_{10}=x_{10,1}+x_{9,2}, ~~y_{22}=x_{22,1}+x_{21,2}$,
$y_{11}=x_{11,1}+x_{10,2}, ~y_{23}=x_{23,1}+x_{22,2}$,
$y_{12}=x_{12,1}+x_{11,2}, ~y_{24}=x_{24,1}+x_{23,2}$.
The proposed code is $\mathfrak{C}=\{y_{1}+y_{2}+ y_{7}+y_{8}+y_{13}+y_{14}+y_{19}+y_{20},\ y_{2}+y_{3}+ y_{8}+y_{9}+y_{14}+y_{15}+y_{20}+y_{21},\ y_{3}+y_{4}+ y_{9}+y_{10}+y_{15}+y_{16}+y_{21}+y_{22},\ y_{4}+y_{5}+ y_{10}+y_{11}+y_{16}+y_{17}+y_{22}+y_{23},\ y_{5}+y_{6}+ y_{11}+y_{12}+y_{17}+y_{18}+y_{23}+y_{24}\}$.
$ \mathfrak{C^{(2)}}=\{{x_{1,1}}+x_{24,2}+x_{2,1}+x_{1,2}+x_{7,1}+x_{6,2}+x_{8,1}+x_{7,2}+x_{13,1}+x_{12,2}+x_{14,1}+x_{13,2}+x_{19,1}+x_{18,2}+x_{20,1}+x_{19,2},\
~~ {x_{2,1}}+x_{1,2}+x_{3,1}+x_{2,2}+x_{8,1}+x_{7,2}+x_{9,1}+x_{8,2}+x_{14,1}+x_{13,2}+x_{15,1}+x_{14,2}+x_{20,1}+x_{19,2}+x_{21,1}+x_{20,2},\
~~ {x_{3,1}}+x_{2,2}+x_{4,1}+x_{3,2}+x_{9,1}+x_{8,2}+x_{10,1}+x_{9,2}+x_{15,1}+x_{14,2}+x_{16,1}+x_{15,2}+x_{21,1}+x_{20,2}+x_{22,1}+x_{21,2},\
~~ {x_{4,1}}+x_{3,2}+x_{5,1}+x_{4,2}+x_{10,1}+x_{9,2}+x_{11,1}+x_{10,2}+x_{16,1}+x_{15,2}+x_{17,1}+x_{16,2}+x_{22,1}+x_{21,2}+x_{23,1}+x_{22,2},\
~~ {x_{5,1}}+x_{4,2}+x_{6,1}+x_{5,2}+x_{11,1}+x_{10,2}+x_{12,1}+x_{11,2}+x_{17,1}+x_{16,2}+x_{18,1}+x_{17,2}+x_{23,1}+x_{22,2}+x_{24,1}+x_{23,2}\}.$
Corollary 8. If $D$ divides $K+\lambda$ and $\lambda$ divides $D$, then the scalar linear code is given by $\mathfrak{C}=\{{x_{i+(j-1)D}+x_{i+jD}}|\ i = \{1,2,\dots,D\},\ j = \{1,2,\dots,n-2\}\}\\ \cup \{x_{K-2D+1+\lambda+i'}+x_{K-D+1+i'}+x_{K-\lambda+1+i' mod \lambda}|\ i' = \{0,1,2,\dots,p-1\}\}$ where $\frac{K+\lambda}{D}=n(>2)$, $p=K$ mod $D$ = $D-\lambda$.
$K=19, U=2,\ D=7.$
$K=19,\ \Delta=5,\ \lambda=1$, capacity=$\frac{3}{14}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}+x_{i-2,\ 3}$ for $i=1,2,\dots,19.$
$y_{1}=x_{1,1}+x_{19,2}+x_{18,3}, ~~ y_{11}=x_{11,1}+x_{10,2}+x_{9,3}$,
$y_{2}=x_{2,1}+x_{1,2}+x_{20,3}, ~~~y_{12}=x_{12,1}+x_{11,2}+x_{10,3}$,
$y_{3}=x_{3,1}+x_{2,2}+x_{1,3}, ~~~~y_{13}=x_{13,1}+x_{12,2}+x_{11,3}$,
$y_{4}=x_{4,1}+x_{3,2}+x_{2,3}, ~~~~y_{14}=x_{14,1}+x_{13,2}+x_{12,3}$,
$y_{5}=x_{5,1}+x_{4,2}+x_{3,3}, ~~~~y_{15}=x_{15,1}+x_{14,2}+x_{13,3}$,
$y_{6}=x_{6,1}+x_{5,2}+x_{4,3}, ~~~~y_{16}=x_{16,1}+x_{15,2}+x_{14,3}$,
$y_{7}=x_{7,1}+x_{6,2}+x_{5,3}, ~~~~y_{17}=x_{17,1}+x_{16,2}+x_{15,3}$,
$y_{8}=x_{8,1}+x_{7,2}+x_{6,3}, ~~~~y_{18}=x_{18,1}+x_{17,2}+x_{16,3}$,
$y_{9}=x_{9,1}+x_{8,2}+x_{7,3}, ~~~~y_{19}=x_{19,1}+x_{18,2}+x_{17,3}$,
The proposed code is $\mathfrak{C}=\{y_{1}+y_{6},\ y_{6}+y_{11},\ y_{11}+y_{15}+y_{19},\ y_{2}+y_{7},\ y_{7}+y_{12},\ y_{12}+y_{16}+y_{19},\ y_{3}+y_{8},\ y_{8}+y_{13},\ y_{13}+y_{17}+y_{19},\ y_{4}+y_{9},\ y_{9}+y_{14},\ y_{14}+y_{18}+y_{19}, \ y_{5}+y_{10},\ y_{10}+y_{15}\}.$
$ ~ \mathfrak{C^{(3)}}=\{{x_{1,1}}+x_{19,2}+x_{18,3}+x_{6,1}+x_{5,2}+x_{4,3},\
~~ {x_{2,1}}+x_{1,2}+x_{20,3}+x_{7,1}+x_{6,2}+x_{5,3},\
~~ {x_{3,1}}+x_{2,2}+x_{1,3}+x_{8,1}+x_{7,2}+x_{6,3},\
~~ {x_{4,1}}+x_{3,2}+x_{2,3}+x_{9,1}+x_{8,2}+x_{7,3},\
~~ {x_{5,1}}+x_{4,2}+x_{3,3}+x_{10,1}+x_{9,2}+x_{8,3},\
~~ {x_{6,1}}+x_{5,2}+x_{4,3}+x_{11,1}+x_{10,2}+x_{9,3},\
~~ {x_{7,1}}+x_{6,2}+x_{5,3}+x_{12,1}+x_{11,2}+x_{10,3},\
~~ {x_{8,1}}+x_{7,2}+x_{6,3}+x_{13,1}+x_{12,2}+x_{11,3},\
~~ {x_{9,1}}+x_{8,2}+x_{7,3}+x_{14,1}+x_{13,2}+x_{12,3},\
~~ {x_{10,1}}+x_{9,2}+x_{8,3}+x_{15,1}+x_{14,2}+x_{13,3},\
~~ {x_{11,1}}+x_{10,2}+x_{9,3}+x_{15,1}+x_{14,2}+x_{13,3}+x_{19,1}+x_{18,2}+x_{17,3},\
~~ {x_{12,1}}+x_{11,2}+x_{10,3}+x_{16,1}+x_{15,2}+x_{14,3}+x_{19,1}+x_{18,2}+x_{17,3},\
~~ {x_{13,1}}+x_{12,2}+x_{11,3}+x_{17,1}+x_{16,2}+x_{15,3}+x_{19,1}+x_{18,2}+x_{17,3},\
~~ {x_{14,1}}+x_{13,2}+x_{12,3}+x_{18,1}+x_{17,2}+x_{16,3}+x_{19,1}+x_{18,2}+x_{17,3}\}.$
Corollary 9. If $K-D$ divides $K+\lambda$ and $\lambda$ divides $K-D$, then the scalar linear code
$\mathfrak{C}$=$\{x_{k}+x_{k+m}+x_{k+2m}+ \dots+x_{k+(q-1)m}+x_{k+(q-1)m+\lambda}+x_{k+(q-1)m+2\lambda}+ \dots + x_{k+(q-1)m+(s-2)\lambda}| k =1,2,\dots,\lambda\}\cup \{x_{k}+x_{k+m}+x_{k+2m}\dots+x_{k+(q-2)m}+x_{k+(q-1)m-\lambda}|\ k = \lambda+1,\lambda+2,\dots,p\}\cup \{x_{k}+x_{k+m}+x_{k+2m}+ \dots+x_{k+(q-2)m}+x_{k+(q-2)m+\lambda}+x_{k+(q-2)m+2\lambda}+\dots + x_{k+(q-2)m+(s-1)\lambda}|\ k =p+1,p+2,\dots,m\}$
where $K-D=m$, $K-D-\lambda=p$, $\frac{K+\lambda}{K-D}=q$ and .
$K=28,\ U=1,\ D=19.$
$K=28,\ \Delta=18,\lambda=2$, capacity=$\frac{2}{10}$.
Let $y_{i}=x_{i,1}+x_{i-1,\ 2}$ for $i=1,2,\dots,28.$
$y_{1}=x_{1,1}+x_{28,2}, ~~ ~y_{15}=x_{15,1}+x_{14,2}$,
$y_{2}=x_{2,1}+x_{1,2}, ~~~~y_{16}=x_{16,1}+x_{15,2}$,
$y_{3}=x_{3,1}+x_{2,2}, ~~~~y_{17}=x_{17,1}+x_{16,2}$,
$y_{4}=x_{4,1}+x_{3,2}, ~~~~y_{18}=x_{18,1}+x_{17,2}$,
$y_{5}=x_{5,1}+x_{4,2}, ~~~~y_{19}=x_{19,1}+x_{18,2}$,
$y_{6}=x_{6,1}+x_{5,2}, ~~~~y_{20}=x_{20,1}+x_{19,2}$,
$y_{7}=x_{7,1}+x_{6,2}, ~~~~y_{21}=x_{21,1}+x_{20,2}$,
$y_{8}=x_{8,1}+x_{7,2}, ~~~~y_{22}=x_{22,1}+x_{21,2}$,
$y_{9}=x_{9,1}+x_{8,2}, ~~~~y_{23}=x_{23,1}+x_{22,2}$,
$y_{10}=x_{10,1}+x_{9,2}, ~~y_{24}=x_{24,1}+x_{23,2}$,
$y_{11}=x_{11,1}+x_{10,2}, ~y_{25}=x_{25,1}+x_{24,2}$,
$y_{12}=x_{12,1}+x_{11,2}, ~y_{26}=x_{26,1}+x_{25,2}$,
$y_{13}=x_{13,1}+x_{12,2}, ~y_{27}=x_{27,1}+x_{26,2}$,
$y_{14}=x_{14,1}+x_{13,2}, ~y_{28}=x_{28,1}+x_{27,2}$.
The proposed code is $\mathfrak{C}=\{y_{1}+y_{11}+y_{21}+y_{23}+y_{25}+y_{27},\ y_{2}+y_{12}+y_{22}+y_{24}+y_{26}+y_{28},\ y_{3}+y_{13}+y_{21},\ y_{4}+y_{14}+y_{22},\ y_{5}+y_{15}+y_{23},\ y_{6}+y_{16}+y_{24},\ y_{7}+y_{17}+y_{25},\ y_{8}+y_{18}+y_{26},\ y_{9}+y_{19}+y_{21}+y_{23}+y_{25}+y_{27},\ y_{10}+y_{20}+y_{22}+y_{24}+y_{26}+y_{28}\}.$
$ \mathfrak{C^{(2)}}=\{{x_{1,1}}+x_{28,2}+x_{11,1}+x_{10,2}+x_{21,1}+x_{20,2}+x_{23,1}+x_{22,2}+x_{25,1}+x_{24,2}+x_{27,1}+x_{26,2},\
~~ {x_{2,1}}+x_{1,2}+x_{12,1}+x_{11,2}+x_{22,1}+x_{21,2}+x_{24,1}+x_{23,2}+x_{26,1}+x_{25,2}+x_{28,1}+x_{27,2},\
~~ {x_{3,1}}+x_{2,2}+x_{13,1}+x_{12,2}+x_{21,1}+x_{20,2},\
~~ {x_{4,1}}+x_{3,2}+x_{14,1}+x_{13,2}+x_{22,1}+x_{21,2},\
~~ {x_{5,1}}+x_{4,2}+x_{15,1}+x_{14,2}+x_{23,1}+x_{22,2},\
~~ {x_{6,1}}+x_{5,2}+x_{16,1}+x_{15,2}+x_{24,1}+x_{23,2},\
~~ {x_{7,1}}+x_{6,2}+x_{17,1}+x_{16,2}+x_{25,1}+x_{24,2},\
~~ {x_{8,1}}+x_{7,2}+x_{18,1}+x_{17,2}+x_{26,1}+x_{25,2},\
~~ {x_{9,1}}+x_{8,2}+x_{19,1}+x_{18,2}+x_{21,1}+x_{20,2}+x_{23,1}+x_{22,2}+x_{25,1}+x_{24,2}+x_{27,1}+x_{26,2},\
~~ {x_{10,1}}+x_{9,2}+x_{20,1}+x_{19,2}+x_{22,1}+x_{21,2}+x_{24,1}+x_{23,2}+x_{26,1}+x_{25,2}+x_{28,1}+x_{27,2}\}.$
§ DISCUSSION
In this paper a construction is given for vector linear index codes of multiple unicast index problems from scalar linear codes which results in a sequence of index coding problems with same number of messages and receivers and two sided antidote patterns. Moreover, it is shown that if the problem with which the construction begins has an optimal linear index code then it induces an optimal linear index code for the vector index coding problem. This construction has been used on few classes of index coding problems given in <cit.> for which optimal linear index codes are known and new classes codes have been obtained starting from these classes of codes.
Another interesting direction of further research is to study the suitability of the new classes of codes presented in this paper for application to noise broadcasting problem. Recently, it has been observed that in a noisy index coding problem it is desirable for the purpose of reducing the probability of error that the receivers use as small a number of transmissions from the source as possible and linear index codes with this property have been reported in <cit.>, <cit.>. While the report <cit.> considers fading broadcast channels, in <cit.> AWGN channels are considered and it is reported that linear index codes with minimum length (capacity achieving codes or optimal length codes) help to facilitate to achieve more reduction in probability of error compared to non-minimum length codes for receivers with large amount of side-information. These aspects remain to be investigated for the new classes of sequences of vector codes presented in this paper.
H. Maleki, V. Cadambe, and S. Jafar, “Index coding – an interference alignment perspective", in IEEE Trans. Inf. Theory,, vol. 60, no.9, pp.5402-5432, Sep. 2014.
Y. Birk and T. Kol, “Informed-source coding-on-demand (ISCOD) over broadcast channels", in Proc. IEEE Conf. Comput. Commun., San Francisco, CA, 1998, pp. 1257-1264.
Z. Bar-Yossef, Z. Birk, T. S. Jayram and T. Kol, “Index coding with side information", in Proc. 47th Annu. IEEE Symp. Found. Comput. Sci., Oct. 2006, pp. 197-206.
M. Tahmasbi, A. Shahrasbi and A. Gohari, “Critical graphs in index coding,” in Proc. IEEE Int. Symp. Inf. Theory, Honolulu, Jul. 2014, pp. 281-285.
L Ong and C K Ho, “Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information”, in Proc. IEEE ICC, 2012, pp. 2213-2218.
Roop Kumar Bhattaram, Mahesh Babu Vaddi and B. Sundar Rajan, “A Lifting construction for Scalar Linear Index Codes,” arXiv:1510.08592 [cs.IT] 29, Oct' 2015.
Anoop Thomas, Kavitha Radhakumar, Attada Chandramouli and B. Sundar Rajan,
“Optimal Index Coding with Min-Max Probability of Error over Fading,”
PIMRC 2015, Hong Kong, August 2015.
Anjana A. Mahesh and B. Sundar Rajan, “Index Coded PSK Modulation,” arXiv preprint arXiv:1509.05874, October 2015.
Kavitha Radhakumar and B. Sundar Rajan,
“On the number of optimal index codes,”
ISIT 2015, Hong Kong, June 2015.
Mahesh Babu Vaddi, Roop Kumar Bhattaram and B. Sundar Rajan, “Optimal Scalar Linear Index Codes for Some Symmetric Multiple Unicast Problems,” arXiv:1510.05435v1 [cs.IT] 19, Oct' 2015.
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1511.00473
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In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalised) grid classes that have two monotone cells in the same row.
§ INTRODUCTION
In recent years, the emerging theory of grid classes has led to some of the major structural and enumerative developments in the study of permutation patterns. Particular highlights include the characterisation of all possible “small” growth rates <cit.> and the subsequent result that all classes with these growth rates have rational generating functions <cit.>.
To support results such as these, the study of grid classes themselves has gained importance. Restricting one's attention to monotone grid classes, it is known that the structure of the matrix defining a grid class determines both its growth rate <cit.>, and whether it is well-partially-ordered <cit.>.
One remaining open question about monotone grid classes concerns their bases, that is, the sets of minimal forbidden permutations of the classes. Backed up by some computational evidence, it is widely believed that all monotone grid classes are finitely based, but this is only known to be true for certain families, most notably those whose row-column graphs[The row-column graph of a $\{0,\pm1\}$-matrix $\M$ is the bipartite graph whose biadjacency matrix has $ij$-th entry equal to $|\M_{ij}|$.] are forests <cit.>. To date, the only other instances of monotone grid classes that are known to have a finite basis are two $2\times 2$ grid classes. The first concerns the class of skew-merged permutations, $\Av(2143,3412)$, in <cit.>, while the second is in Waton's PhD thesis <cit.>. Inspired by Waton's approach, we show that a certain family of (non-monotone) $2\times 2$ grid classes are all finitely based, from which we can conclude the following result.
Every $2\times 2$ monotone grid class is finitely based.
The rest of this section covers a number of prerequisite definitions. In Section <ref> we introduce a more general construction than grid classes, based on juxtapositions, that are known to be finitely based, and use these to characterise the grid classes they contain. In Section <ref> we consider three separate cases that will enable us to prove our more general result (Theorem <ref>), and thence Theorem <ref>.
Writing permutations in one-line notation, we say that the permutation $\sigma$ is contained in a permutation $\pi$, denoted $\sigma\leq\pi$, if there is a subsequence of the entries of $\pi$ that have the same relative ordering as the entries of $\sigma$. A specific instance of a set of entries of $\pi$ witnessing this containment is called a copy of $\sigma$ in $\pi$. Containment forms a partial order on the set of all permutations, and sets of permutations which are closed downwards in this order are called permutation classes. Specifically, if $\C$ is a permutation class, $\pi\in\C$ and $\sigma\leq\pi$, then we must have $\sigma\in\C$. For convenience later, we regard the empty permutation as belonging to every permutation class.
While permutation classes can be defined in a number of ways (for example, the set of all permutations that can be sorted by a stack forms a permutation class), a convenient characterisation can be given in terms of the unique set of minimal forbidden permutations that do not lie in the class. We call the set $B$ the basis of a class $\C$ if
\[
\C = \{\pi: \beta\not\leq\pi\text{ for all }\beta\in B\},
\]
and $B$ is minimal with this property, and we write $\C=\Av(B)$. By its minimality, the set $B$ must form an antichain under $\leq$, but since infinite antichains are know to exist in the containment partial order, $B$ need not be finite. When the basis of $\C$ is finite, we say that $\C$ is finitely based.
We frequently make use of a graphical perspective, in which we represent a permutation $\pi$ by plotting the points $(i,\pi(i))$ ($i=1,\dots,|\pi|$) in the plane. Indeed, we do not distinguish between the permutation $\pi$ written in one-line notation, and the graphical representation of $\pi$.
For $m,n\geq 1$, let $\M$ be an $m\times n$ matrix whose entries are permutation classes (including possibly the empty class). The grid class of the matrix $\M$, denoted $\Grid(\M)$, is the permutation class consisting of all permutations $\pi$ for which (in the graphical perspective) there exist $m-1$ horizontal and $n-1$ vertical lines which divide the entries of $\pi$ into $mn$ rectangles, so that the (possibly zero) entries of $\pi$ in each rectangle form a copy of a permutation from the class in the corresponding entry of $\M$. When the entries of $\M$ are all either $\Av(12)$, $\Av(21)$ or $\emptyset$, then $\Grid(\M)$ is a monotone grid class.
We are mostly concerned with $2\times 2$ matrices in this paper, and in this case it will prove convenient to refer to these grid classes more succinctly. If $\M = \begin{pmatrix}\A&\B\\\C&\D\end{pmatrix}$ is a matrix consisting of permutation classes, then we write
\[
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(2,2);\tclass(1,2){$\A$};\tclass(2,2){$\B$};\tclass(1,1){$\C$};\tclass(2,1){$\D$};}
\]
to mean $\Grid(\M)$. Additionally, when (say) $\A = \Av(21)$, then we may refer to the cell [scale=0.5,baseline=0.2cm-0.5ex](1,1);(1,1)$\A$; using [scale=0.5,baseline=0.2cm-0.5ex](1,1);(1,1);, reflecting the fact that all points in this cell are increasing. Similarly, we may write [scale=0.5,baseline=0.2cm-0.5ex](1,1);(1,1); when $\A=\Av(12)$. Finally, where the entries of the $2\times 2$ matrix $\M$ are either arbitrary or clear from the context, we may also simply refer to $\Grid(\M)$ as .
We are ready to state our general theorem, from which Theorem <ref> will follow.
Let $\C$ and $\D$ be finitely based permutation classes. Then the three grid classes
are all finitely based.
Our approach makes use of an existing result, which although not originally presented in this way, can be cast in terms of grid classes. For permutation classes $\C$ and $\D$, the (horizontal) juxtaposition of $\C$ and $\D$ is the $1\times 2$ grid class [scale=0.5,baseline=0.2cm-0.5ex](2,1);(1,1)$\C$;(2,1)$\D$;. Similarly, the vertical juxtaposition of $\C$ and $\D$ is the $2\times 1$ grid class [scale=0.5,baseline=0.5cm-0.5ex](1,2);(1,2)$\C$;(1,1)$\D$;.
Whenever $\C$ and $\D$ are finitely based, so are the horizontal and vertical juxtapositions of $\C$ and $\D$.
For clarity, we occasionally write $\hjuxta \C\D$ for the horizontal juxtaposition [scale=0.5,baseline=0.2cm-0.5ex](2,1);(1,1)$\C$;(2,1)$\D$; (we do not need the corresponding vertical juxtaposition notation).
§ JUXTAPOSITIONS AND RELATIVE BASES
In this section, we give a characterisation of $2\times 2$ grid classes of the form
\[\E = \tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(2,2);\tclass(1,2){$\A$};\tclass(2,2){$\B$};\tclass(1,1){$\C$};\tclass(2,1){$\D$};}\]
where $\A$, $\B$, $\C$ and $\D$ are four fixed (but arbitrary) permutation classes.
We begin by considering the following related class, formed by the horizontal juxtaposition of two vertical juxtapositions:
\[\F = \hjuxta{\tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(1,2);\tclass(1,2){$\A$};\tclass(1,1){$\C$};}}{\tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(1,2);\tclass(1,2){$\B$};\tclass(1,1){$\D$};}}.\]
Note that if $\A,\B,\C$ and $\D$ are finitely based, then by repeated application of Lemma <ref> so too is $\F$.
Clearly, $\E\subseteq \F$. We are interested in the basis of $\E$, which we can separate into two parts: those basis elements of $\E$ that lie within $\F$, and those basis elements of $\E$ that are not in $\F$. By minimality and since $\E\subseteq\F$, this latter set must also be basis elements of $\F$. The set of basis elements of $\E$ that are contained in $\F$ we call the relative basis of $\E$ in $\F$, and we have the following observation.
Let $\C$ and $\D$ be two permutation classes such that $\D$ finitely based, and $\C\subseteq\D$. Then $\C$ is finitely based if and only if the relative basis of $\C$ in $\D$ is finite.
Consider any permutation $\pi$ in the set $\F\setminus \E$. Since $\pi$ lies in the juxtaposition class $\F$, we can write $\pi=\pi_1\pi_2$ with
\[\pi_1\in
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(1,2);\tclass(1,2){$\A$};\tclass(1,1){$\C$};} \text{ and }\pi_2\in
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{\tgrid(1,2);\tclass(1,2){$\B$};\tclass(1,1){$\D$};}.\]
We refer to the division line $v$ that separates $\pi_1$ from $\pi_2$ as a v-line. Additionally, any horizontal division line in $\pi_1$ that demonstrates $\pi_1$ as a member of the vertical juxtaposition is called a left h-line of $\pi$, and similarly any valid horizontal division line in $\pi_2$ is called a right h-line. Thus, we can recognise $\pi\in\F$ by means of a division triple, $(v,r,\ell)$, where $v$ is the v-line, $r$ the right h-line, and $\ell$ the left h-line.
The condition that $\pi\in\F\setminus\E$ can now be described as follows: for every division triple $(v,r,\ell)$ that recognises $\pi\in\F$, the right h-line $r$ and the left h-line $\ell$ cannot be at the same height. We use the symbol to denote the set of permutations in $\F$ which have a division triple $(v,r,\ell)$ where $\ell$ is no higher than $r$, and to denote those permutations which have a division where $\ell$ is no lower than $r$. Note that and are both in fact permutation classes, and also that $\F = \squintA \cup \squintB$.
Our main result of this section now follows. It shows in particular that $\pi\in\F\setminus\E$ cannot simultaneously lie in and , and hence the relative basis of $\E$ in $\F$ can be divided into two disjoint parts: those that lie in and those that lie in .
Any $2\times 2$ grid class $\E=\twobytwo$ is equal to the intersection of the corresponding classes $\squintA$ and $\squintB$. That is,
\[\E = \twobytwo = \squintA \cap \squintB\,.\]
First, it is clear that $\twobytwo\subseteq \squintA \cap \squintB$, so suppose that we have a permutation $\pi$ in $\squintA \cap \squintB$.
(0,0) rectangle (10,10);
[dashed] (4,0) – (4,10);
[dashed] (0,7) – (4,7);
[dashed] (4,6) – (10,6);
(6,0) – (6,10);
(0,3) – (6,3);
(6,5) – (10,5);
[gray, ->] (.5,7.5) – (.5,7);
[gray, ->] (9.5,5.5) – (9.5,5);
[fill=none,draw=none] (r) at (10,5) [label=right:$r$] ;
[fill=none,draw=none] (r) at (10,6) [label=right:$r'$] ;
[fill=none,draw=none] (r) at (0,3) [label=left:$\ell$] ;
[fill=none,draw=none] (r) at (0,7) [label=left:$\ell'$] ;
[fill=none,draw=none] (r) at (6,10) [label=above:$v$] ;
[fill=none,draw=none] (r) at (4,10) [label=above:$v'$] ;
The relationship between the division $(v,r,\ell)$ and $(v',r',\ell')$ in the proof of Lemma <ref>. The small arrows indicate that the corresponding division lines have been chosen to be extremal in the direction specified by the arrows.
Consider $\pi$ first as a member of . There exists at least one division triple $(v,r,\ell)$ which recognises this, and we choose any valid v-line $v$, together with the lowest right h-line $r$ and the highest left h-line $\ell$. Note in particular that for any right h-line that is lower than $r$, there must exist a basis element in the top right cell. If $\ell$ and $r$ coincide, then we have $\pi\in \twobytwo$ and we are done, so we may assume that $\ell$ is strictly lower than $r$.
Next, consider $\pi$ as an element of . We pick a division $(v',r',\ell')$ by first choosing any v-line $v'$ which either coincides with $v$ or lies further to the left (the case where $v'$ is to the right of $v$ will follow upon rotating the picture by $180^\circ$). Next choose any valid $r'$,
noting that $r'$ must be at least as high as $r$ to avoid introducing a basis element into the top right cell. Finally, choose $\ell'$ to be as low as possible, subject to the division triple $(v',r',\ell')$ remaining a valid division for membership of (see Figure <ref>). We claim that $\ell'$ is at the same height as $r'$.
Suppose, for a contradiction, that $\ell'$ lies strictly above $r'$, and let $\ell''$ be the left h-line that has the same height as $r'$. Since the division triple $(v',r',\ell'')$ does not witness $\pi\in \squintB$ (but $(v',r',\ell')$ does), there must exist some basis element in the top left region defined by $(v',r',\ell'')$. However, this region is contained in the top left region defined by $(v,r,\ell)$, so this is impossible.
Thus $\ell'$ has the same height as $r'$, and $(v',r',\ell')$ is a division triple that recognises $\pi\in\squintB$, and hence $\pi\in\twobytwo$.
§ MAIN RESULTS
We are ready to start proving our three main results.
For finitely based classes $\C$ and $\D$, the class
\[\E=
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{
\tgrid(2,2);
\tup(1,1);
\tup(2,1);
\tclass(1,2){$\C$};
\tclass(2,2){$\D$};
is finitely based.
First, let $B$ denote the relative basis of $\E$ inside the juxtaposition
\[\F=
\hjuxta{\tikz[scale=0.5,baseline=0.5cm-0.5ex]{
\tgrid(1,2);
\tup(1,1);
\tclass(1,2){$\C$};
\tgrid(1,2);
\tup(1,1);
\tclass(1,2){$\D$};
Since $\F$ is finitely based, by Observation <ref> it suffices to show that $B$ is finite. By Lemma <ref> and the comments preceding it, any $\pi\in B$ lies in exactly one of or . Consider first the case where $\pi\in \squintA$. We will identify a bounded number of points in $\pi$ that demonstrate $\pi\not\in\E$.
We begin by identifying two division triples, $(v_L,r_L,\ell_L)$ and $(v_R,r_R,\ell_R)$: $v_L$ is the leftmost v-line recognising $\pi\in\squintA$, and $v_R$ is the rightmost such v-line. Subject to these choices, we pick $\ell_L$ and $\ell_R$ to be as high as possible, and $r_L$ and $r_R$ as low as possible.
We now prove the following claim: if $(v,r,\ell)$ is any other division triple recognising $\pi\in\squintA$ where the left h-line $\ell$ is chosen as high as possible, then $\ell$ is at the same height as either $\ell_L$ or $\ell_R$.
If $\ell_L$ and $\ell_R$ are at the same height, the claim follows immediately, so we can assume that $\ell_L$ is strictly higher than $\ell_R$. The situation is as depicted in Figure <ref>: we identify four points, $a$, $b$, $c$ and $d$, which are distinct (except possibly $b=c$) and which form the copies of 21 that define $\ell_L$ and $\ell_R$. Note that $a$ and $c$ lie immediately above $\ell_L$ and $\ell_R$, and, except that the relative positions of $a$ and $c$ can be interchanged providing $b\neq c$, the points must be arranged in the way shown in Figure <ref> in order that $\pi\in\squintA$. For the same reason, all other points of $\pi$ that lie in the marked rectangular regions 1, 2, 3 and 4 (defined by the bounding dotted and dashed lines) in Figure <ref> must lie on the diagonal segments indicated.
[gray, dashed] (3,0) – (3,10);
[gray, dashed] (0,7) – (3,7);
[gray, dashed] (3,9) – (10,9);
[gray, dotted] (7,0) – (7,10);
[gray, dotted] (0,3) – (7,3);
[gray, dotted] (7,5) – (10,5);
[gray, ->] (3.5,2) – (3,2);
[gray, ->] (6.5,2) – (7,2);
[gray, ->] (.2,6.5) – (.2,7);
[gray, ->] (.2,2.5) – (.2,3);
[gray, ->] (9.8,9.5) – (9.8,9);
[gray, ->] (9.8,5.5) – (9.8,5);
at (.5,7.3) [label=45:$a$] ;
(b) at (2.5, 6.3) [label=left:$b$] ;
(c) at (1,3.3) [label=left:$c$] ;
(d) at (5,2.5) [label=right:$d$] ;
(b) – (c);
(d) – (4.45,1.4);
(0,0) – (.7,1.4);
(5.3,3.5) – (6.5,5.9);
at (10,9) [invis_nd,label=right:$r_L$] ;
at (10,5) [invis_nd,label=right:$r_R$] ;
at (0,7) [invis_nd,label=left:$\ell_L$] ;
at (0,3) [invis_nd,label=left:$\ell_R$] ;
at (3,10) [invis_nd,label=above:$v_L$] ;
at (7,10) [invis_nd,label=above:$v_R$] ;
[fill=none,inner sep=1pt] at (6,8) $1$;
[fill=none,inner sep=1pt] at (0,5) $2$;
[fill=none,inner sep=1pt] at (2,0) $3$;
[fill=none,inner sep=1pt] at (6,0) $4$;
The relationships between the division triples $(v_L,r_L,\ell_L)$ and $(v_R,r_R,\ell_R)$, the points defining $\ell_L$ and $\ell_R$, and the restrictions on the placement of points in the four rectangular regions 1—4.
Consider any division triple $(v,r,\ell)$ recognising $\pi\in\squintA$ where $\ell$ is chosen as high as possible. If $v$ lies further left than all points in the region labelled 4 in Figure <ref>, then we can choose $\ell$ at the same height as $\ell_L$. On the other hand, if any point from region 4 lies to the left of $v$, then $c$ must lie above $\ell$, and thus $\ell$ is at the same height as $\ell_R$. This completes the claim.
We can now identify the following bounded collection of points of $\pi$: (i) a basis element of [scale=0.4,baseline=0.4cm-0.5ex]
which defines $v_R$, (ii) a basis element of
to define $v_L$, and (iii) at most 4 points $a,b,c$ and $d$ defining the two left h-lines $\ell_R$ and $\ell_L$.
It remains to identify a bounded number of points to ensure that any division triple $(v,r,\ell)$ recognising $\pi\in\squintA$ has $\ell$ strictly lower than $r$. For this, it suffices to consider only the extremal triples $(v,r,\ell)$ where $\ell$ is as high as possible, and $r$ is as low as possible. We identify the extremal triple $(v_X,r_X,\ell_X)$ where the v-line $v_X$ is chosen to lie immediately to the left of all points in region 4 of Figure <ref>. By the earlier claim, $\ell_X$ has the same height as $\ell_L$. The lowest right h-line $r_X$ must lie strictly above $\ell_X$, and is defined by a basis element of $\D$ to the right of $v_X$, with one point lying immediately below $r_X$. Observe that for any extremal triple $(v,r,\ell)$ where $v$ lies to the left of $v_X$, we have that $\ell$ is at the same height as $\ell_X$, and $r$ can be no lower than $r_X$. In particular, since $\pi$ as a basis element is minimally not in $\E$, if $r$ is higher than $r_X$ then it is because of points in $\pi$ that we have already identified.
Similarly, the position of the line $r_R$ is fixed by a basis element of $\D$ to the right of $v_R$. For any extremal triple $(v,r,\ell)$ where $v$ is further right than $v_X$, we know that $\ell$ is at the same height as $\ell_R$, and $r$ can be no lower than $r_R$ (because of the basis element of $\D$). Thus, again by the minimality of $\pi$, if $r$ is strictly higher than $r_R$ it is because of points that we have already identified.
From this, we conclude that if $\pi\in\squintA$ is a basis element of $\E$ relative to $\F$ then the number of points in $\pi$ is bounded, as $\pi$ comprises the points identified in (i), (ii) and (iii) above, and by at most two basis elements of $\D$.
[gray, dashed] (3,0) – (3,10);
[gray, dashed] (0,3) – (3,3);
[gray, dashed] (3,3) – (10,3);
[gray, dotted] (7,0) – (7,10);
[gray, dotted] (0,7) – (7,7);
[gray, dotted] (7,5) – (10,5);
[gray, ->] (3.5,9) – (3,9);
[gray, ->] (6.5,9) – (7,9);
[gray, ->] (.2,7.5) – (.2,7);
[gray, ->] (7.2,2.5) – (7.2,3);
[gray, ->] (7.2,4.5) – (7.2,5);
(a) at (4.3,3.3) ;
(b) at (5.5,6.7) ;
(c) at (9.5,4.7) ;
(d) at (8,5.3) ;
(e) at (8.3,2.7) ;
(a) – (b);
(c) – (8.7,3.3);
(4,1.7) – (3.3,1);
(0,0) – (2.7,.7);
(e) – (7.3,2);
at (10,3) [invis_nd,label=right:$r_L$] ;
at (10,5) [invis_nd,label=right:$r_R$] ;
at (0,7) [invis_nd,label=left:$\ell_R$] ;
at (0,3) [invis_nd,label=left:$\ell_L$] ;
at (3,10) [invis_nd,label=above:$v_L$] ;
at (7,10) [invis_nd,label=above:$v_R$] ;
[invis_nd] at (1.5,5) $\varnothing$;
[gray, dashed] (3,0) – (3,10);
[gray, dashed] (0,5) – (3,5);
[gray, dashed] (3,4) – (10,4);
[gray, dotted] (7,0) – (7,10);
[gray, dotted] (0,7) – (7,7);
[gray, dotted] (7,4) – (10,4);
[gray, ->] (3.5,2) – (3,2);
[gray, ->] (6.5,2) – (7,2);
[gray, ->] (.2,7.5) – (.2,7);
[gray, ->] (.2,5.5) – (.2,5);
[gray, ->] (7.2,3.5) – (7.2,4);
(a) at (2.5,4.7) ;
(b) at (5.5,6.7) ;
(c) at (9.5,3.7) ;
(d) at (8,4.3) ;
(a) – (0,0);
(b) – (3.5,5.3);
(c) – (7.3,0);
at (10,4) [invis_nd,label=right:$r_L=r_R$] ;
at (0,5) [invis_nd,label=left:$\ell_L$] ;
at (0,7) [invis_nd,label=left:$\ell_R$] ;
at (3,10) [invis_nd,label=above:$v_L$] ;
at (7,10) [invis_nd,label=above:$v_R$] ;
[invis_nd] at (5,1.5) $\varnothing$;
[invis_nd] at (1.5,6) $\varnothing$;
The relationships between the division triples $(v_L,r_L,\ell_L)$ and $(v_R,r_R,\ell_R)$ when $\pi\in\squintB$. On the left, if $r_L$ and $r_R$ are at different heights, then $\ell_L$ is at the same height as $r_L$. On the right, if $r_L$ and $r_R$ are at the same height, then the points defining $\ell_L$ guarantee $\pi\not\in\E$ for every triple $(v,r,\ell)$ recognising $\pi\in\squintB$.
The argument for a basis element $\pi$ that lies in is similar, and we omit some of the details. The process begins by identifying the leftmost and rightmost v-lines $v_L$ and $v_R$, and the corresponding highest right h-lines $r_L$ and $r_R$. The left hand picture in Figure <ref> illustrates that $r_L$ and $r_R$ cannot have different heights (else $\pi\in\twobytwo$ ). In the right hand picture of Figure <ref>, the points forming a basis element of $\C$ that defines the line $\ell_L$ ensures that in any extremal triple $(v,r,\ell)$, $r$ is lower than $\ell$. Thus $\pi$ consists of (i) a basis element of [scale=0.4,baseline=0.4cm-0.5ex]
which defines $v_R$, (ii) a basis element of
to define $v_L$, (iii) a copy of $21$ to define $r_R$, and (iv) a basis element of $\C$ to define $\ell_L$.
A similar approach, of bounding the number of possible left and right h-lines, can be applied for the other two cases, so we only sketch the proofs.
For finitely based classes $\C$ and $\D$, the class
\[\E=
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{
\tgrid(2,2);
\tup(1,1);
\tdown(2,1);
\tclass(1,2){$\C$};
\tclass(2,2){$\D$};
is finitely based.
We need only consider relative basis elements of $\E$ that lie in , as the argument for is symmetric. Thus, consider a basis element $\pi\in\squintA$ of $\E$.
[gray, dashed] (3,0) – (3,10);
[gray, dashed] (0,3) – (3,3);
[gray, dashed] (3,7) – (10,7);
[gray, dotted] (7,0) – (7,10);
[gray, dotted] (0,3) – (7,3);
[gray, dotted] (7,4) – (10,4);
[gray, ->] (3.5,9) – (3,9);
[gray, ->] (6.5,9) – (7,9);
[gray, ->] (.5,2.5) – (.5,3);
[gray, ->] (9.5,4.5) – (9.5,4);
[gray, ->] (9.5,7.5) – (9.5,7);
(a) at (8.5,3.7) [label=-135:$c$] ;
(b) at (4.5,6.7) ;
(c) at (2,2.5) [label=-45:$b$] ;
(d) at (1,3.3) [label=90:$a$] ;
(a) – (10,0);
(b) – (6.7,5.7);
(8.3,4.3) – (7.3,5.3);
(0,0) – (c);
at (10,7) [invis_nd,label=right:$r_L$] ;
at (10,4) [invis_nd,label=right:$r_R$] ;
at (0,3) [invis_nd,label=left:$\ell_R=\ell_L$] ;
at (3,10) [invis_nd,label=above:$v_L$] ;
at (7,10) [invis_nd,label=above:$v_R$] ;
[invis_nd] at (5,1.5) $\varnothing$;
The left h-line $\ell_R$ is defined by the points $a$ and $b$ which form a copy of 21. Both $a$ and $b$ must lie to the left of $v_L$, so this also defines $\ell_L$.
Define the division triples $(v_R,r_R,\ell_R)$ and $(v_L,r_L,\ell_L)$ recognising $\pi\in\squintA$ by choosing $v_R$ to be the rightmost v-line, and $v_L$ the leftmost, and then selecting $r_L$ and $r_R$ as low as possible, and $\ell_L$ and $\ell_R$ as high as possible.
We claim that $\ell_R$ and $\ell_L$ have the same height. In Figure <ref>, the point $c$ which defines the line $r_R$, forces the region below $\ell_R$ and between $v_L$ and $v_R$ to be empty. Consequently, the pair of points $a$ and $b$ (which forms a copy of 21 and hence defines the height of $\ell_R$) must lie to the left of $v_L$. This means that $a$ and $b$ also define the highest position of every left h-line $\ell$ in a division triple $(v,r,\ell)$ recognising $\pi\in\squintA$.
The proof concludes by noting that we can demonstrate $\pi\not\in\E$ by the following points: (i) a basis element of [scale=0.4,baseline=0.4cm-0.5ex]
which defines $v_R$, (ii) a basis element of
to define $v_L$, (iii) a copy of $21$ to define $\ell_R$, and (iv) a basis element of $\D$ to define $r_R$.
For finitely based classes $\C$ and $\D$, the class
\[\E=
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{
\tgrid(2,2);
\tdown(1,1);
\tup(2,1);
\tclass(1,2){$\C$};
\tclass(2,2){$\D$};
is finitely based.
As before, by symmetry it suffices to consider a relative basis element $\pi\in\squintA$ of $\E$. Define the division triples $(v_R,r_R,\ell_R)$ and $(v_L,r_L,\ell_L)$ as in earlier proofs.
We claim that in any division triple $(v,r,\ell)$ recognising $\pi\in\squintA$ where $\ell$ is as high as possible, $\ell$ has the same height as either $\ell_L$ or $\ell_R$. The situation is illustrated in Figure <ref>: if $v$ lies to the right of the point $a$ then $\ell$ can be no higher than $\ell_R$. On the other hand, if $v$ lies to the left of $a$, then the only available copy of 12 has $b$ as the `2', so $\ell$ has the same height as $\ell_L$.
[gray, dashed] (3,0) – (3,10);
[gray, dashed] (0,6) – (3,6);
[gray, dashed] (3,8) – (10,8);
[gray, dotted] (7,0) – (7,10);
[gray, dotted] (0,3) – (7,3);
[gray, dotted] (7,5) – (10,5);
[gray, ->] (3.5,9) – (3,9);
[gray, ->] (6.5,9) – (7,9);
[gray, ->] (2.5,2.5) – (2.5,3);
[gray, ->] (2.5,5.5) – (2.5,6);
[gray, ->] (7.5,5.5) – (7.5,5);
[gray, ->] (7.5,8.5) – (7.5,8);
(a) at (8,4.7) [label=-45:$c$] ;
(b) at (9.5,7.7) ;
(c) at (1.5,2.5) ;
(d) at (5.7,3.3) [label=135:$a$] ;
(e) at (5,1) ;
at (2.5,6.3) [label=135:$b$] ;
(a) – (7.3,4.3);
(b) – (8.5,5.3);
(2.5,1.5) – (c);
(6.7,4) – (d);
(1.3,3.3) – (0.5,5.7);
at (10,8) [invis_nd,label=right:$r_L$] ;
at (10,5) [invis_nd,label=right:$r_R$] ;
at (0,6) [invis_nd,label=left:$\ell_L$] ;
at (0,3) [invis_nd,label=left:$\ell_R$] ;
at (3,10) [invis_nd,label=above:$v_L$] ;
at (7,10) [invis_nd,label=above:$v_R$] ;
The left h-line $\ell_R$ is defined by the points $a$ and $b$ which form a copy of 12. Since $a$ lies to the left of $v_L$, the left h-line $\ell_L$ can be no higher than $\ell_R$.
With these two left h-lines defined, we need only identify two copies of basis elements of $\D$ to define corresponding lowest right h-lines in each case. Thus, $\pi\not\in\E$ is identified by the following points: (i) a basis element of [scale=0.4,baseline=0.4cm-0.5ex]
to defines $v_R$, (ii) a basis element of
to define $v_L$, (iii) at most two copies of $21$ to define $\ell_R$ and $\ell_L$, and (iv) at most two basis elements of $\D$ to define $r_R$ and $r_L$.
First, the only $2\times 2$ monotone grid classes whose row-column graphs are not forests (and hence finitely based by <cit.>) are those where all four cells are non-empty.
Any such $2\times 2$ monotone grid class can be described as a grid class in one of the three forms covered by Lemmas <ref>, <ref> and <ref>, upon taking the classes $\C$ and $\D$ to be $\Av(12)$ or $\Av(21)$, and possibly appealing to symmetry.
§ CONCLUDING REMARKS
Non-monotone 2 $\times$ 2 grids One obvious question arising from this work is how far one might be able to extend Theorem <ref> within the context of $2\times 2$ grids: in particular, can one replace the two monotone classes in the lower row by something more general? Any approach to this question would need to bear in mind that there do exist $2\times 2$ grid classes which are not finitely based, even though each entry of the matrix is finitely based. The primary example of this, given both in Murphy's PhD thesis <cit.> and in <cit.>, is
\[
\tikz[scale=0.5,baseline=0.5cm-0.5ex]{
\tgrid(2,2);
\tclass(1,1){$\C$};
\tclass(2,2){$\C$};
\tclass(1,2){$\varnothing$};
\tclass(2,1){$\varnothing$};
where $\C=\Av(321654)$. (Note this example is more normally written as a direct sum, $\C\oplus\C$.) This example can likely be adapted to produce other instances where the grid class is not finitely based, even though its individual entries are.
Larger grids There are a number of difficulties encountered when one tries to extend our results here to larger grids. Even in the “next” case of $2\times 3$ grids, there seems to be no obvious analogue to Lemma <ref> to enable us to consider relative bases inside some larger class. The primary issue is that our proof relied on the fact that the heights of all possible left-h-lines (or, analogously, right-h-lines) form a contiguous set of values, but this need no longer be the case.
Acknowledgements We are grateful to Mike Atkinson for several fruitful discussions about this problem, from which most of the ideas for this note emerged.
|
1511.00548
|
We prove that, for a finitely generated group hyperbolic relative to virtually
abelian subgroups, the generalised word problem for a parabolic subgroup is
the language of a real-time Turing machine. Then, for a hyperbolic group, we
show that the generalised word problem for a quasiconvex subgroup is a real-time
language under either of two additional hypotheses on the subgroup.
By extending the Muller-Schupp theorem we show that the generalised word
problem for a finitely generated subgroup of a finitely generated virtually
free group is context-free. Conversely, we prove that a hyperbolic group must
be virtually free if it has a torsion-free quasiconvex subgroup
of infinite index with context-free generalised word problem.
2010 Mathematics Subject Classification: 20F10, 20F67, 68Q45
Key words: generalised word problem, relatively hyperbolic group, context-free language, real-time Turing machine
§ INTRODUCTION
Let $G=\langle X \rangle$ with $|X|<\infty$ be a group and $H\leq G$.
The word problem $\WP(G,X)$ and generalised word problem
$\GWP(G,H,X)$ are defined to be the preimages $\phi^{-1}(\{1_G\})$ and
$\phi^{-1}(H)$ respectively, where $\phi$ is the natural map from the set of
words over $X$ to $G$.
We are interested in the relationship between the algebraic properties
of $G$ (and $H$) and the formal language classes containing $\WP(G,X)$ and
$\GWP(G,H,X)$. These questions have already been well studied for the word
problem, but relatively little for the generalised word problem. Since, as is
well known for $\WP(G,X)$, the question of membership of
$\WP(G,X)$ and $\GWP(G,H,X)$ in a formal language family $\F$ is typically
independent of the choice of
the finite generating set $X$,
we shall usually use the simpler notations $\WP(G)$ and $\GWP(G,H)$. It will
be convenient to assume throughout the paper that all generating sets $X$ of
groups $G$ are closed under inversion; i.e. $x \in X \Rightarrow x^{-1} \in X$.
It is elementary to prove that $\WP(G)$ is regular if and only if $G$ is
finite and, more generally, that $\GWP(G,H)$ is regular if and only if $|G:H|$
is finite. It is well-known that $\WP(G)$ is context-free if and only if $G$
is virtually free <cit.>, and it is shown in
<cit.> that $\WP(G)$ is a real-time language
(that is, the language of a real-time Turing machine) for several
interesting classes of groups, including hyperbolic and geometrically finite
hyperbolic groups. The solvability of $\GWP(G,H)$ has been established for numerous classes of groups, most recently for all (compact and connected) $3$-manifold groups <cit.>.
In this paper, we study the conditions under which $\GWP(G,H)$ is a real-time or
a context-free language for subgroups $H$ of hyperbolic and relatively
hyperbolic groups. There are several definitions of relatively hyperbolic groups, and the one
that we are using here is that of <cit.>; so in particular the
Bounded Coset Penetration Property holds. Our first result is the following.
Suppose that the finitely generated group $G$ is hyperbolic relative to a set
$\{H_i: i \in I\}$ of virtually abelian (parabolic) subgroups of $G$,
and that $H$ is a selected parabolic subgroup.
Then $\GWP(G,H)$ is a real-time language.
The proof uses a combination of results for relatively hyperbolic groups
that were developed by Antolín and Ciobanu in <cit.> and the
extended Dehn algorithms that were introduced by Goodman and
Shapiro in <cit.>. By choosing $H=H_0$ to be the trivial
subgroup of $G$, we obtain a generalisation of the result proved
(also using extended Dehn algorithms) in <cit.>
that $\WP(G)$ is a real-time language when $G$ is a geometrically finite
hyperbolic group.
Since Goodman and Shapiro's techniques, such as the $N$-tight Cannon's algorithm
and related results (Theorem 37 in <cit.>), only apply to virtually abelian groups and not, for example, to all virtually nilpotent groups,
our proof cannot be extended to parabolics beyond those that are virtually abelian.
Furthermore, even with a different approach to the proof there must be some
limitations on the choice of parabolics, since examples exist
of relatively hyperbolic groups with
generalised word problems that are not real-time.
If one considers, for example, the free product $G=H \ast K$ of two groups $H$
and $K$, where $H$ is hyperbolic and $K$ has unsolvable word problem,
then $G$ is hyperbolic relative to $K$, but it is easy to see that the subgroup membership problem for $K$ in $G$ is unsolvable, so it cannot be real-time.
Recall that a subgroup $H \le G$ is called quasiconvex in $G$ if
geodesic words over $X$ that represent elements of $H$
lie within a bounded distance of $H$ in the Cayley graph $\Cay(G,X)$.
And $H$ is called almost malnormal in $G$ if
$|H \cap H^g|$ is finite for all $g \in G \setminus H$.
We conjecture that, for a hyperbolic group $G$, the set $\GWP(G,H)$ is a real-time
language for any quasiconvex subgroup $H$ of $G$, but we are currently only
able to prove this under either of two
additional hypotheses:
Let $G$ be a hyperbolic group, and
$H$ a quasiconvex subgroup of $G$. Suppose that either
(i) $H$ is almost malnormal in $G$; or
(ii) $|C_G(h):C_H(h)|$ is finite for all $1 \ne h \in H$.
Then $\GWP(G,H)$ is a real-time language.
Our proofs of the result in the two cases are quite distinct, and so we write
them separately. Under assumption (i), $G$ is
hyperbolic relative to $\{H\}$ <cit.>, and
the proof is similar to that of Theorem <ref> (although $H$ is not
usually virtually abelian, so we cannot apply that result directly). Under
assumption (ii), we make use of some results of Foord <cit.> about the
Schreier graph of $G$ with respect to $H$. Note that, since
centralisers of elements of infinite order in hyperbolic groups are
virtually cyclic, $|C_G(h):C_H(h)|$ is always finite for such elements
$h$ and so, in particular, assumption (ii) holds whenever $G$ is torsion-free.
Our next result, which concerns context-free generalised word problems, is
straightforward to prove and may be known already, but it does not appear
to be in the literature.
Let $G$ be finitely generated and virtually free, and let $H$ be a finitely
generated subgroup of $G$. Then $\GWP(G,H)$ is deterministic context-free.
The conclusion of Theorem <ref> may or may not hold if we drop the
condition that $H$ is finitely generated. Suppose that $G$ is the free group
on two generators $a,b$. For $H_1=[G,G]$, the set
$\GWP(G,H_1)$ consists of all words whose exponent sums in both $a$ and $b$
are zero, and is not context-free, but for the subgroup $H_2$ of words whose
exponent sum in $a$ is zero, the set $\GWP(G,H_2)$ is context-free.
We would like to know to what extent Theorem <ref> is best possible
when $H$ is finitely generated.
We observe that, where $H_G := \cap_{g \in G} H^g$ is the core of $H$ in
$G$, the set $\GWP(G,H,X)$ is the same set of words as $\GWP(G/H_G,H/H_G,X)$.
We know of no examples for which $H$ is finitely generated with trivial core
and $\GWP(G,H)$ is context-free, but $G$ is not virtually free.
The following result is an attempt at a converse to Theorem <ref>.
Let $G$ be a hyperbolic group, and let $H$ be a quasiconvex subgroup of
infinite index in $G$ such that $|C_G(h):C_H(h)|$ is finite for all
$1 \ne h \in H$. If $\GWP(G,H)$ is context-free then $G$ is virtually free.
Note that we make the same assumption on centralisers of elements $h \in H$
as in Theorem <ref> (ii), and again we conjecture that this
is not necessary for the conclusion of the theorem.
However the necessity of quasiconvexity is demonstrated by a
construction found in <cit.>, as follows.
Let $Q$ be any finitely presented group, and
choose $\lambda >0$. We can define a finitely presented group $G$ and a normal
$2$-generated subgroup $H$ of $G$, such that $G/H \cong Q$, and $G$ satisfies
the small cancellation condition $C'(\lambda)$; in particular we can choose
$Q$ with insoluble word problem, and choose $\lambda \leq 1/6$ to ensure
that $G$ is hyperbolic, and in that case $\GWP(G,H)$ is not even recursive.
The closure properties of context-free languages
<cit.> and of real-time languages <cit.>
ensure that all of the above results are independent of the choice of the finite
generating set $X$ of $G$, so we are free to choose $X$ to suit our own
purposes in the proofs. (More generally, for membership of $\WP(G,X)$ or
$\GWP(G,H,X)$ in a formal language class $\mathcal{F}$ to be independent of
the choice of $X$, we need $\mathcal{F}$ to be closed under inverse
homomorphism. This property holds for all of the most familiar formal
language classes, including regular, context-free, deterministic
context-free, real-time, context-sensitive, deterministic context-sensitive,
and recursive languages.)
This article is structured as follows.
In Section <ref>, we summarise the basic properties of relatively
hyperbolic groups that we shall need, and we
recall some of their properties that are proved in <cit.>.
In Section <ref>, we introduce the concept of
extended Dehn algorithms for solving the word and generalised
word problems in groups and we recall some results pertaining to relatively
hyperbolic groups that are proved in <cit.>.
Sections <ref>, <ref>, <ref>,
<ref> and <ref> contain the proofs of Theorems
<ref>, <ref> (i), <ref> (ii),
<ref> and <ref>, respectively.
Finally, in Section <ref>, we sketch a proof of a result of
Foord <cit.> that we shall need, since the original
source might not be readily available to readers.
§ RELATIVELY HYPERBOLIC GROUPS
We follow <cit.> for notation and the definition of relatively hyperbolic
groups. This definition is equivalent to what Farb calls
“strong relative hyperbolicity” in <cit.>.
Suppose that $G$ is a group, $X$ a finite generating set,
and $\{H _i : i \in I\}$ a
collection of subgroups of $G$, which we call parabolic subgroups. We
define $X_i := X \cap H_i$ to be the set of generators in $X$ that lie within
the subgroup $H_i$, and $X_I := \cup_{i \in I} X_i$. Then we define $\H$ and
$X$ as the sets
\[ \H := \bigcup_{i \in I} (H_i \setminus \{1\}),\quad
\widehat{X} = X \cup \H. \]
Much of our argument involves the comparison of lengths of various words
that represent an element $g ∈G$ written over different sets, namely
$X$, $X$, a set $Z ⊇X$ that is introduced during the extended
Dehn algorithm (which, when $Z ⊃X$, is not actually a generating set
for $G$, since only some of the words over $Z$ correspond to elements of $G$),
and certain subsets of these sets.
So we shall consider the Cayley graphs $$ and $$ for $G$
over the generating sets $X$ and $X$,
and view words over $X$ and $X$ also as paths in $$ and
$$. We denote by $d_$ the graph distance in $$ and
by $d_$ the graph distance in $$. For a word
$w$ written over a set $Y$ (that is, an element of $Y^*$), we write $|w|$ to
denote the length of $w$
and, for a group element $g$, we write $|g|_Y$ to denote the length of a
shortest word over $Y$ that represents the group element $g$ (assuming that
such a word exists). We call $|g|_Y$ the {\em $Y$-length} of $g$, and a
shortest word over $Y$ that represents $g$ a {\em $Y$-geodesic} for $g$.
Most words will be written over either $X$ or $X$ and, in order to
make a clear distinction between those two types of words we will normally use
Roman letters as names
for words over $X$ and paths in $$, and Greek letters as names for words
over $X$ and paths in $$, with the exception that we will
write $w$ for the word over $X$ that is derived from a word $w$
over $X$ by a process called {\em compression}, which will be described
later in this section.
We refer to \cite{Osin06} for a precise definition of relative hyperbolicity of
$G$ with respect to ${H_i: i ∈I}$. Under that definition, relative
hyperbolicity is known to be equivalent to the fact that the Cayley graph
$$ is $δ$-hyperbolic
for some $δ$ together with the Bounded Coset Penetration Property, stated
below as Property \ref{prop:BCP}.
From now on we shall assume that $G$ is relatively hyperbolic in this sense.
It is proved in \cite{Osin06} that, under our assumption that $G$ is finitely
generated, the subgroups $H_i$ are finitely generated, the set $I$
is finite, and any two distinct parabolic subgroups have finite intersection.
%DFH: added assumption H_i=<X_i>
We assume that the generating set $X$ is chosen such that
$H_i = ⟨X_i ⟩$ for all $i ∈I$.
We need some terminology relating to paths in $$.
\begin{enumerate}
\item
We call a subpath of a path $\pi$ an {\em $H_i$-component}, or
simply a {\em component}, of $\pi$ if it is written as a word over $H_i$ for
some $i \in I$, and is not contained in any longer such subpath of $\pi$.
\item
Two components of (possibly distinct) paths
are said to be {\em connected} if both are
$H_i$-components for some $i\in I$, and both start within the same left coset
of $H_i$. % $(\sigma_-)H_i=(\rho_-)H_i$.
\item
A path is said to {\em backtrack} if it has a pair of
connected components.
A path is said to {\em vertex-backtrack} if it has a subpath of length
greater than 1 that is labelled by a word representing an element of some $H_i$.
Note that if a path does not vertex-backtrack, then it does not backtrack and
all of its components are edges.
\item
For $\kappa \ge 0$ we say that two paths
are {\em $\kappa$-similar} if the $X$-distances between their two initial
vertices and between their two terminal vertices are both at most $\kappa$.
\end{enumerate}
\begin{property}
[Bounded Coset Penetration Property {\cite[Theorem 3.23]{Osin06}}]
\label{prop:BCP}
For any $\lambda \geq 1, c \geq 0, \kappa \geq 0$, there exists a constant
$\epsilon = \epsilon (\lambda, c, \kappa)$ such that, for any two
$\kappa$-similar $(\lambda, c)$-quasi-geodesic
paths $\pi$ and $\pi'$ in $\HatCay$ that do not backtrack,
the following conditions hold:
\begin{enumerate}
\item[{\rm(1)}] The sets of vertices of $\pi$ and $\pi'$ are contained in the
closed $\epsilon$-neighbourhoods (with respect to the metric $d_\Cay$) of
each other.
\item[{\rm(2)}] For any $H_i$-component $\sigma$ of $\pi$ for which the $X$-distance between its endpoints is greater than $ \epsilon$,
some $H_i$-component $\sigma'$ of $\pi'$ is connected to $\sigma$.
\item[{\rm(3)}] Whenever $\sigma$ and $\sigma'$ are connected $H_i$-components of $\pi$ and $\pi'$ respectively,
the paths $\sigma$ and $\sigma'$ are $\epsilon$-similar.
\end{enumerate}
\end{property}
We shall need to use two more properties of relatively hyperbolic groups
that are proved in \cite{AntolinCiobanu}.
\begin{property}[{\cite[Theorem 5.2]{AntolinCiobanu}}] \label{prop:AC1}
Let $Y$ be a finite generating set for $G$. Then for some $\lambda \geq 1, c
\geq 0$ there exists a finite set $\Psi$ of non-geodesic words over
$Y\cup \H$ such that:
\begin{quote}
every 2-local geodesic word over $Y\cup \H$ not containing any element of
$\Psi$ as a subword labels a $(\lambda,c)$-quasi-geodesic path in
$\HatCay$ without vertex-backtracking.
\end{quote}
\end{property}
Now suppose that $v$ is any word in $X^*$.
Then following \cite[Construction 4.1]{AntolinCiobanu} we define $\widehat{v}$
to be the word over $\widehat{X}$
that is obtained from $v$ by replacing (working from the left) each
subword $u$ that is maximal as a subword over some $X_i$ ($i \in I$) by
the element $h_u$ of $\H$ that the subword represents.
We call these $X_i$-subwords $u$ of $v$ its {\em parabolic segments},
and use the term {\em compression} for the process that converts $v$ to
$\widehat{v}$. A word $v$ is said to have {\em no parabolic shortenings} if each
of its parabolic segments is an $X_i$-geodesic.
In order to avoid confusion we comment that a parabolic segment (which is a
subword over some $X_i$ of a word over $X$) is not quite the same as a component
(which is a maximal subpath/subword over some $H_i$ of a path/word over
$\widehat{X}$); but clearly the two concepts are close.
%DFH: replaced "second" by "other required"
The other required property is proved in \cite[Lemma 5.3]{AntolinCiobanu};
the precise description of $\Phi$ is taken from the proof of that lemma, rather
than from its statement.
\begin{property}\label{prop:AC2}
Let $Y$ be a finite generating set for $G$. Then for some
$\lambda \geq 1$,
$c\geq 0$, some finite subset
$\H'$ of $\H$, and any finite generating set $X$ of $G$
with $$Y\cup \H'\subseteq X\subseteq Y\cup \H,$$ there is a finite subset
$\Phi$ of non-geodesic words over $X$ such that:
\begin{quote}
if a word $w \in X^*$ has no parabolic shortenings and no subwords in $\Phi$,
then the word $\widehat{w} \in \widehat{X}^*$ is a 2-local geodesic and labels a
$(\lambda, c)$-quasi-geodesic path in $\HatCay$ without vertex-backtracking.
Furthermore, for every $i\in I$ and $h\in H_i$, we have $|h|_{X}=|h|_{X_i}$.
\end{quote}
In fact $\Phi = \Phi_1 \cup \Phi_2$, where $\Phi_1$ is the set of non-geodesic
words in $X^*$ of length 2, and $\Phi_2$ is the set of all words $u \in X^*$
with no parabolic shortening and for which $\widehat{u} \in \Psi$,
where $\Psi$ (together with $\lambda$ and $c$) is given by
\end{property}
\section{Extended Dehn algorithms}
\label{sec:eda}
Our proofs of Theorem~\ref{thm:relhyp} and both parts of
Theorem~\ref{thm:qcsubhyp} depend on the construction of an
{\em extended Dehn algorithm} (\eda) \cite{GoodmanShapiro} for $G$ with
respect to $H$. In each case, we then need to show that the \eda satisfies a
particular condition that allows us to apply Proposition~\ref{prop:realtime}
in order to verify both that the algorithm solves $\GWP(G,H)$ and that it
can be programmed on a real-time Turing machine.
Proposition~\ref{prop:realtime} is derived from \cite[Theorem 4.1]{HoltRees},
which was used to prove the solubility of the word problem in
real-time for various groups with \edas to solve that problem. We restate that
%DFH: clarify->clarity
result as a proposition in this paper for greater clarity of exposition.
Our definition of an extended Dehn algorithm (which is defined with respect to
a specific finite generating set $X$ of $G$) is modelled on the
definition of \cite{GoodmanShapiro} (where it is called a {\em Cannon's
algorithm}), with the difference that
we are using our algorithm to solve a generalised word problem $\GWP(G,H,X)$
rather than a word problem $\WP(G,X)$. Elsewhere in the literature
\cite{HoltRees} the same concept is called a {\em generalised Dehn algorithm};
our decision to introduce a new name is based on both our recognition that
there are many various different algorithms attributed to (more than one)
Cannon, and our desire to avoid overuse of the term `generalised'.
%DFH added "Noetherian" in brackets to ensure that R(w) exists and deal
%with referee's quibble. Also replaced "rewrite system" with "rewriting
%system", which is more standard.
For a (Noetherian) rewriting system $R$ with alphabet $Z$ and $w \in Z^*$, we
write $R(w)$ for the reduction of the word $w$ using the rules of $R$. In
general, $R(w)$ may depend on the order in which the rules are applied, and
we shall specify that order shortly. A word $w$ is called {\em ($R$-)reduced}
if $R(w)=w$; that is, if $w$ does not contain the left hand side
of any rule as a subword.
We define an \eda for a finitely generated group $G= \langle X \rangle$
with respect to a subgroup $H \le G$ to be
a finite rewriting system $S$ consisting of rules $u \rightarrow v$, where
\begin{mylist}
\item[(i)] $u,v \in Z^*$ for some
finite alphabet $Z \supseteq X \cup \{H\}$;
\item[(ii)] $|u|>|v|$; and
\item[(iii)] either $u,v \in (Z \setminus \{H\})^*$
or $u=Hu_1,v=Hv_1$, with $u_1,v_1 \in (Z \setminus \{H\})^*$.
\end{mylist}
We say that the \eda $S$ solves the generalised word problem $\GWP(G,H,X)$ if,
for every word $w$ over $X$, we have $S(Hw) = H$ if and only if $w$ represents
an element of $H$. If $H=\{1\}$ (in which case we may assume only that
$Z \supseteq X$), then we call $S$ an \eda for $G$.
As observed earlier, for $S(w)$ to be well-defined, we need to specify the
order in which the reduction rules are applied to words $w$.
In the terminology of \cite[Section 1.2]{GoodmanShapiro}, $S$ with the order
we specify below is an
{\it incremental rewriting algorithm} and, since we shall only apply
it to words of the form $Hw$ with $w \in X^*$, the rules $Hu_1 \to Hv_1$
are effectively {\em anchored} rules.
We assume that no two distinct rules have the same left hand sides.
Then we require that when a word $Hw$ contains several left hand sides of
$S$, the rule which is applied is one that ends closest to the start of $Hw$;
if there are several such rules, the one with the longest left hand side is
In our applications, the rules of the form $u \to v$ with
$u,v \in (Z \setminus \{H\})^*$ will form an \eda $R$
for $G$, of a type that is considered in
\cite{GoodmanShapiro}, and which solves the word problem $\WP(G,X)$;
i.e. $R(w)$ is the empty word if and only if $w=_G 1$.
The properties of the \eda $R$ that we shall use are
described in more detail in Proposition \ref{prop:P(D,E)} below.
Many of the technical results of \cite{GoodmanShapiro} apply without
modification to \edas that solve a $\GWP$ rather than $\WP$.
Note that the set $Z$ may properly contain $X \cup \{H\}$, and so contain
symbols that do not correspond to either elements or subsets of $G$.
But it is a consequence of \cite[Proposition 3]{GoodmanShapiro} that a word
in $(Z \setminus \{H\})^*$ that arises from applying these rules to a word
$w \in X^*$ unambiguously corresponds to the element of $G$ represented
by $w$, and so we may interpret such words as elements in $G$.
%DFH: added sentence, SR corrected `Hz \to Z' to `Hz \to H'
We shall see shortly that our rewrite rules $Hu \to Hv$ will all be of the
form $Hz \to H$ for words $z \in (Z \setminus \{H\})^*$ that represent
elements of $H$, and so words derived by applying rules of the \eda to $Hw$
with $w \in X^*$ unambiguously represent the coset of $H$ in $G$ defined by
Following \cite[Corollary 27]{GoodmanShapiro}, for a positive integer $D$, we
say that an \eda $S$ that solves the word problem for a group
$K=\langle Y \rangle$ is {\em $D$-geodesic} if a word $w$ that is $S$-reduced
and represents an element $g$ of $Y$-length at most $D$ must
in fact be written over $Y$, and be a $Y$-geodesic for $g$.
In this section so far, we have not made any assumptions on $G,H,X$, beyond
the finiteness of the generating set $X$.
Suppose now that the parabolic subgroups $H_i$ are all virtually abelian,
and that $R$ is an \eda for $G$ that solves $\WP(G,X)$.
For integers $D \ge E \ge 0$, we say that $R$ satisfies $\P(D,E)$ if the
following conditions hold.
\begin{mylist}
\item[(1)] For each $i$, $X_i := X \cap H_i$ generates $H_i$, and
the alphabet $Z$ of $R$ has the form $Z = \cup_{i \in I} Z_i \cup X$
with $X_i \subseteq Z_i$.
\item[(2)] For each rule $u \to v$ of $R$, we either have $u,v \in X^*$,
or $u,v \in Z_i^*$ for a unique $i \in I$.
\item[(3)] For each $i \in I$, the rules $u \to v$ with $u,v \in Z_i^*$ form
a $D$-geodesic \eda $R_i$ that solves $\WP(H_i,X_i)$.
\item[(4)] All $R$-reduced words $w \in X^*$ that have length at most $E$
are $X$-geodesic.
\end{mylist}
The following result is proved under slightly more general conditions
on the parabolic subgroups in \cite{GoodmanShapiro}, and is stated here in the
form in which we need it:
\begin{proposition}[{\cite[Theorem 37]{GoodmanShapiro}}]
\label{prop:P(D,E)}
Suppose that $G = \langle Y \rangle$ is hyperbolic relative to the virtually
abelian parabolic subgroups $H_i$.
Then, there is a finite generating set $X$ of $G$, consisting of the
generators in $Y$ together with some additional elements from the $H_i$
(that include all non-trivial elements from the intersections $H_i \cap H_j$
with $i \ne j$) with the following property: for all sufficiently large
integers $D,E$ with $D \ge E \ge 0$, there is an $\eda$ for $G$ that solves
$\WP(G,X)$ and satisfies $\P(D,E)$.
\end{proposition}
We need to extend our definition of the property $\P(D,E)$ to our wider
definition of an \eda for a group with respect to a subgroup.
For $G$ satisfying the above hypotheses, and given any subgroup $H \le G$, we
shall say that an \eda $S$ for $G$ with respect to $H$, with alphabet
$Z \cup\{H\}$, satisfies $\P(D,E)$ if the rules in $S$ of the form $u \to v$
with $u,v \in Z^*$ form an \eda $R$ satisfying $\P(D,E)$.
In the proofs of each of Theorems
\ref{thm:relhyp}, \ref{thm:qcsubhyp} (i), \ref{thm:qcsubhyp} (ii),
we shall apply the following result, which is essentially (part of)
\cite[Theorem 4.1]{HoltRees}.
\begin{proposition}
\label{prop:realtime}
Let $G$ be a group, finitely generated over $X$, $H$ a subgroup of $G$, and
let $S$ be an extended Dehn algorithm for $G$ with respect to $H$.
Suppose that there exists a constant $k$ such that, for any word $w$ over $X$,
we have
%DFH: g \in Hw -> g \in G,\, g \in Hw
\[ |w_1| \leq k \min \{ |g|_X : g\in G,\, g \in Hw \}, \]
where $w_1$ is the word over $Z$ defined by $Hw_1=S(Hw)$.
Then $S$ solves the generalised word problem and can be programmed on
a real-time Turing machine.
\end{proposition}
\begin{proof}
Since, for any $w \in \GWP(G,H)$ the minimal length representative of $Hw$
is the identity element, it is
immediate from the inequality that $S$ solves $\GWP(G,H)$.
That an \eda satisfying that condition can be programmed in real-time is
then an immediate consequence of \cite[Theorem 4.1]{HoltRees}; in fact as stated
that theorem applies only to \edas to solve the word problem, but it is clear
from the proof that it applies also to $\GWP(G,H)$.
\end{proof}
\section{The proof of Theorem \ref{thm:relhyp}}
\label{sec:main_thm}
Suppose that $G = \langle X \rangle$ satisfies the hypotheses of
Theorem \ref{thm:relhyp}, and that $H=H_0$ is the selected parabolic subgroup,
generated by $X_0 \subset X$.
We start with some adjustments to $X$ that are necessary to ensure that it
satisfies the conditions we need for our arguments.
These adjustments all consist of appending generators that lie in one of
the parabolic subgroups.
Firstly, we extend $X$ to contain the finite subset $\H'$ of $\H$ defined
in Property \ref{prop:AC2}.
Secondly, we adjoin to $X$ the elements of the $H_i$ that are required by
Proposition \ref{prop:P(D,E)}.
(As stated in Proposition \ref{prop:P(D,E)}, these include all elements in
the finite intersections $H_i \cap H_j$ (for $i \ne j$) of pairs of parabolic
Associated with this choice of $X$, Properties \ref{prop:AC1} and
\ref{prop:AC2} specify sets $\Psi$ and $\Phi$
of non-geodesic words over $\widehat{X}$ and $X$ respectively, and
associated parameters $\lambda,c$.
We then define $\epsilon=\epsilon(\lambda,c+1,0)$
to be the constant in the conclusion of Property~\ref{prop:BCP}.
We now apply Proposition~\ref{prop:P(D,E)} to find an \eda $R$ for
$G$ that solves $WP(G,X)$ and that satisfies $\P(D,E)$ for parameters $D,E$
with $D \ge E$, where $E >\max\{ \epsilon, 2\}$,
and $E$ is also greater than the length of any word in $\Phi$,
and greater than the $X$-length of any component of any word in
$\Psi$. (The reasons for these conditions will become clear during the proof.)
We can use all properties of $R$ that are proved in
\cite[Section 5]{GoodmanShapiro} and we observe in particular that,
by \cite[Lemma 48]{GoodmanShapiro}, for a word $w \in X^*$, if
$R(w) = u_1vu_2$ where $v$ is a maximal $Z_i$-subword for some $i \in I$,
then there exists $v' \in X_i^*$ with $R_i(v') = v$,
where $R_i$ is the associated \eda for $H_i$. So $v$ unambiguously represents
the element $v' \in H_i$.
We create an \eda $S$ for $G$ with respect to $H$ by adding to $R$ all
rules of the form $Hz \rightarrow H$ with $z \in Z_0$.
Since none of these new rules actually applies to words over $X$,
it is clear that the \eda $S$ also satisfies $\P(D,E)$.
We shall verify that $S$ solves $\GWP(G,H,X)$, and that it can be programmed
on a real-time Turing machine.
Now suppose that $w \in X^*$ and that $S(Hw) = Hw_1$.
In order to verify that our \eda $S$ solves $\GWP(G,H,X)$
and can be programmed on a real-time Turing machine, it is sufficient by
Proposition~\ref{prop:realtime} to establish the existence of a
constant $k$ that is independent of the choice of $w$, such that
\[ |w_1| \leq k \min \{ |g|_X : g \in Hw \}.\quad (\dagger) \]
So the aim of the rest of the proof is
to prove the inequality $(\dagger)$.
%DFH: added sentence - I hope this enough!
Observe that a maximal subword $p_i$ of $w_1$ that is written over $Z_i^*$
for some $i$ may contain symbols from $Z_i \setminus X_i$, but any such
symbols must have arisen from application of the rules in the \eda $R_i$ to
words over $X_i^*$, and so $p_i$ unambiguously represents an element of $H_i$.
Following \cite{GoodmanShapiro}, we decompose $w_1$ as a concatenation
\[ w_1=v_0p_1v_1\cdots p_mv_m, \quad(*) \]
where $p_1$ is defined to be the first subword of $w_1$ (working from the left)
that is written over $Z_{i_1}$ for some $i_1 \in I$,
has maximal length as such a subword, and represents an
element of $H_{i_1}$ of $X_{i_1}$-length greater than $E$.
(Note that the words $v_i$ are denoted by $g_i$ in \cite{GoodmanShapiro}.)
The subwords $p_i$ for $i>1$ are defined correspondingly with respect to the
suffix remaining after removing the prefix $v_0p_1v_1\cdots p_{i-1}$ from $w_1$.
Then any (maximal) subword of any $v_j$ that is written over any $Z_i$
represents an element of $H_i$ of $X_i$-length at
most $E \leq D$, and so $\P(D,E)\,(3)$ ensures that the subword is
written over $X_i$ and is an $X_i$-geodesic. Hence
$v_j$ is a word over $X$ with no parabolic shortenings.
Then (since $E>2$) property $\P(D,E)\,(4)$ ensures that $v_j$
is also a 2-local geodesic. Also, since $\Phi$ is a set of non-geodesic
words over $X$ of length at most $E$,
$v_j$ cannot contain any subword in $\Phi$. It follows by
Property~\ref{prop:AC2} that each $\widehat{v_j}$ is a 2-local geodesic.
Now, for each $j$, choose $q_j$ to be a geodesic word over $X_{i_j}$ that
represents the same element $h_j$ of $H_{i_j}$ as $p_j$
(so, by Property \ref{prop:AC2}, $q_j$ is also an $X$-geodesic).
Define \[ w_2 = v_0q_1v_1\cdots q_mv_m.\]
We already observed that each $v_j$ is written over $X$, and hence so is $w_2$.
Now according to \cite[Lemma 23]{GoodmanShapiro},
the $X$-lengths of non-identity elements of $H_i$ are bounded below by an
exponential function on the lengths of words over $Z_i$ that are their
reductions by the \eda for $H_i$.
So there is certainly a
%DFH: added "positive"
positive constant $k_1$ %(independent of $w$)
such that $|q_j| \ge k_1 |p_{i_j}|$ for all $j$,
and hence $|w_2| \ge k_1|w_1|$.
Hence it is sufficient to prove the inequality $(\dagger)$ above for the word
$w_2$ rather than $w_1$,
that is, for some $k'$, show that
\[ |w_2| \leq k' \min \{ |g|_X : g \in Hw \}.\quad (\dagger\!\dagger) \]
With this in mind, our next step is to construct a word $\widetilde{w_2}$
over $\widehat{X}$, representing the same element of $G$ as $w_2$,
and for which we can use Property~\ref{prop:AC1}.
We define
\[\widetilde{w_2} = \widehat{v_0}h_1\widehat{v_1}\cdots h_m\widehat{v_m}.\]
Note that we would have $\widetilde{w_2}=\widehat{w_2}$ if the subwords $q_j$
were parabolic segments of $w_2$, but this might not be true if the
first generator in some $q_j$ were in more than one parabolic subgroup,
and then we could not be sure that $\widehat{w_2}$ would satisfy the required
conditions. We call the process of conversion of $w_2$ to $\widetilde{w_2}$
{\em modified compression} and we call the subwords $q_j$ of $w_2$
together with the parabolic segments of the subwords $v_j$ the
{\em modified parabolic segments} of $w_2$. Observe that these
modified parabolic segments are all $X$-geodesics.
We want to apply Property~\ref{prop:AC1} to $\widetilde{w_2}$, so we must show
first that $\widetilde{w_2}$ is a 2-local geodesic over $\widehat{X}$.
If not, then $\widetilde{w_2}$ has a non-geodesic subword
$\zeta$ of length 2, equal in $G$ to an $\widehat{X}$-geodesic word $\eta$ of
length at most $1$. We saw earlier that the subwords $\widehat{v_j}$ are
2-local geodesics, so $\zeta$ must contain some $h_j$; that is, $\zeta=yh_j$ or
$\zeta = h_jy$, where $y \in \widehat{X}$.
Then the definition of the $p_j$ as maximal $Z_{i_j}$-subwords of $w_1$
ensures that $y \not \in H_{i_j}$, so $|\eta| = 1$.
So, since
$\lambda,c+1 \ge 1$, $\zeta$ and $\eta$ are both $(\lambda,c+1)$-quasigeodesics.
But now, since $h_j$ is a component in $\zeta$ of $X$-length greater
than $E>\epsilon = \epsilon(\lambda,c+1,0)$,
we can apply Property~\ref{prop:BCP} to the paths in $\HatCay$ labelled
by $\zeta$ and $\eta$, and deduce that $\eta$ contains a component connected
to the component $h_j$, and so $\eta$ represents an
element of $H_{i_j}$; hence $\zeta \in H_{i_j}$, and we have a contradiction.
(Alternatively, we could apply \cite[Lemma 4.2]{AntolinCiobanu} to deduce
that $|\zeta|=2$, a contradiction.)
To verify the second requirement of Property~\ref{prop:AC1}, we need to check
that $\widetilde{w_2}$ contains no subword in $\Psi$.
So suppose that $\xi$ is such a subword in $\Psi$.
Then $\xi$ cannot contain any of the generators $h_j$, since $h_j$ would
then be a component in $\xi$ of $X$-length greater than $E$, by the conditions
imposed on the decomposition $(*)$ of $w_1$; but this contradicts the
choice of $E$ earlier in this proof
to be greater than the $X$-length of any component of any word in $\Psi$.
So $\xi$ must be a subword of some $\widehat{v_j}$; but in that case,
$\xi=\widehat{u}$ for some subword $u$ of $v_j$.
We saw earlier that $v_j$ has no parabolic shortenings, and hence
neither does $u$.
So from the definition of $\Phi_2$ we have $u \in \Phi_2 \subseteq \Phi$.
But we also observed earlier that $v_j$ has no subword in $\Phi$, so
we have a contradiction.
It now follows using Property \ref{prop:AC1} that $\widetilde{w_2}$ is a
$(\lambda,c)$-quasigeodesic over $\widehat{X}$ without vertex-backtracking.
Let $w_3$ be a geodesic over $X$ that represents
an element of $Hw_2 = Hw$; since it is geodesic, $w_3$ cannot contain any
subwords in $\Phi$, and it cannot have any parabolic shortenings. So we can
apply Property~\ref{prop:AC2} to deduce that $\widehat{w_3}$ is a 2-local
geodesic over $\widehat{X}$ and a $(\lambda,c)$-quasigeodesic without
Now $\widetilde{w_2} =_G h\widehat{w_3}$ for some $h \in H$
and so, since $\widetilde{w_2}$ is a $(\lambda,c)$-quasigeodesic,
\[ |\widetilde{w_2}| \leq \lambda |h\widehat{w_3}|+c. \]
We note that $\widetilde{w_2}$ and $h\widehat{w_3}$ are both
$(\lambda,c+1)$-quasigeodesics over $\widehat{X}$ without vertex-backtracking,
and the initial and terminal vertices of the paths in $\HatCay$ that
they label coincide. So we have Properties \ref{prop:BCP}\,(2) and (3)
concerning the components of the two paths.
(This is why we chose $\epsilon =\epsilon(\lambda,c+1,0)$ at the beginning
of the proof.)
We choose a geodesic word $w_h$ over $X_0$ that represents $h$,
and consider the words $w_2$ and $w_hw_3$.
We want to compare the lengths of $w_2$ and $w_hw_3$, and we do this by
examining the processes of (modified) compression of $w_2$ and $w_hw_3$ to
$\widetilde{w_2}$ and $\widehat{w_hw_3}=h\widehat{w_3}$.
The word $w_2$ can be decomposed as a concatenation of
disjoint subwords that are its {\em long} modified parabolic segments, its
{\em short} modified parabolic segments and its maximal subwords over $X
\setminus X_I$; we define a modified parabolic segment to be {\em long} if its
length is greater than $4 \epsilon$ and {\em short} otherwise.
Now modified compression reduces the total length of subwords of the second
type, which are replaced by single elements of $\widehat{X}$, by a
factor of at most $4\epsilon$, while the subwords of the third type are
So the total length of the subwords of $w_2$ of the second and third
types is bounded by
\[ 4\epsilon | \widetilde{w_2}| \leq 4 \epsilon(\lambda
|h\widehat{w_3}| +c) \leq 4 \epsilon (\lambda | w_hw_3| + c),\]
where the second equality follows from the fact that $h\widehat{w_3}
= \widehat{w_hw_3}$.
Now let $u$ be a long modified parabolic segment of $w_2$.
Then Property \ref{prop:BCP} applied to $\widetilde{w_2}$ and $h\widehat{w_3}$
ensures that there is a corresponding parabolic segment $u'$ of
$w_hw_3$ such that $\widehat{u}$ and $\widehat{u'}$ are connected components
of length $1$ of $\widetilde{w_2}$ and $h\widehat{w_3}$
Then since $\widehat{u}$ and $\widehat{u'}$ must be $\epsilon$-similar, it
follows that the initial and terminal points of $u'$ must be
within $X$-distance $\epsilon$ of the initial and terminal points
(respectively) of $u$.
Then $4\epsilon < |u| \leq 2\epsilon + |u'|$, and so $|u| \leq 2|u'|$
(see Fig. \ref{fig:BCP}).
\begin{figure}
\begin{center}
\setlength{\unitlength}{1.0pt}%
\begin{picture}(200,120)(-79,-60)%
\put(-140,60){$\widetilde{w_2}$ (quasi-geodesic)}%
\put(-145,0){\circle*{5}}%
\put(-147,-13){$\IdHatCay$}%
\put(-140,-60){$h\widehat{w_3}$ (quasi-geodesic)}%
\put(5,58){component $\hat{u}$, $|u|_X > 4\epsilon$}%
\put(30,-55){$\widehat{u'}$}%
\put(-145,0){\line(2,1){100}}
\put(-45,50){\line(1,0){145}}
\put(100,50){\line(3,-1){60}}
\put(160,30){\line(1,-1){30}}
\put(-145,0){\line(3,-2){60}}
\put(-85,-40){\line(1,0){195}}
\put(110,-40){\line(2,1){80}}
\multiput(0,-40)(3.32,19.95){5}{\line(1,6){1.7}}
\multiput(70,-40)(6.65,19.95){5}{\line(1,3){3.4}}
\put(15,3){$\le \varepsilon$}
\put(95,3){$\le \varepsilon$}
\put(15,50){\line(1,0){85}}
\put(0,-40){\line(1,0){70}}
\end{picture}%
\end{center}%
\caption{The paths $\widetilde{w_2}$ and $h\widehat{w_3}$ in
$\HatCay$: bounded coset penetration}
\label{fig:BCP}
\end{figure}
Since $\widetilde{w_2}$ does not backtrack, distinct components of
$\widetilde{w_2}$ must correspond to distinct components of $h\widehat{w_3}$,
and we deduce that the total length of the long
parabolic segments in $w_2$ is bounded above by $2|w_hw_3|$. So
\[ |w_2| \leq (4 \epsilon \lambda +2) | w_hw_3| + 4\epsilon c.\]
Now we consider $w_h$, which is a parabolic segment of $w_hw_3$.
If $|w_h| > \epsilon$ then Property \ref{prop:BCP} applied to (the paths
labelled by) $h\widehat{w_3}$ and $\widetilde{w_2}$ ensures the existence of
a corresponding parabolic segment $u_2$ in $w_2$, whose initial vertex is within
$X$-distance $\epsilon$ of the basepoint $\IdHatCay$ of the Cayley graph
$\HatCay$, and whose terminal vertex must be in $H$. Let $w_2$
factorise as a concatenation of subwords $u_1u_2u_3$.
Since $u_1u_2$ is a prefix of $w_2$ that represents an element of $H$,
the fact that $\widetilde{w_2}$ does not vertex-backtrack ensures that its
subword $\widehat{u_1}\widehat{u_2}$ must have length at most 1, and hence
$u_1$ is empty. But now the prefix $u_2$ of $w_2$ is written over the
generators of $H$, and $w_2$ cannot have a non-trivial such prefix, since
$w_1$ (from which it was derived) was reduced by the \eda $S$, and we have
a contradiction.
So now $|w_h| \leq \epsilon$, and we can deduce from the inequality
above that
\[ |w_2| \leq A |w_3| + B \] for some constants $A,B$.
Provided that $|w_3| \neq 0$, it follows that
\[ |w_2| \leq (A+B)|w_3|. \]
But if $|w_3|=0$, then $w_2$ must represent an element of $H$ and so, since
$\widetilde{w_2}$ has already been proved not to vertex-backtrack,
$\widetilde{w_2}$ must be a word written over $H$ of length at most 1. It
follows that $w_2$ is a word over $X_0$, and so, just as above, we deduce that
$|w_2|=0$, and so the same inequality holds. This
completes our verification of the condition of $(\dagger\!\dagger)$ (and hence $(\dagger)$),
and the theorem is proved.
\section{The proof of Theorem \ref{thm:qcsubhyp} (i)}
\label{sec:qcthmi}
Let $H$ be a quasiconvex and almost malnormal subgroup of a hyperbolic group
$G = \langle X \rangle$. It is observed in
\cite[Section 1, Example (III)]{Osin06} that $G$ is hyperbolic relative to
$\{ H \}$ so we can apply the results of Section \ref{sec:relhypgps}.
The proof of Theorem \ref{thm:qcsubhyp} (i) is very similar to that
of Theorem \ref{thm:relhyp} (although we are no longer assuming that $H$ is
virtually abelian) but is more straightforward, so we shall only
summarise it here. In particular, we have $Z=X$ so the complications arising
from the elements of $Z \setminus X$ that do not necessarily represent group
elements do not arise.
We start by extending $X$ as before to include the finite subset $\H'$ of
$\H$ defined in Property \ref{prop:AC2}. Since we are not applying
Proposition \ref{prop:P(D,E)} in this proof, the other adjustment to $X$
is not necessary. We define $\Psi, \Phi, \lambda, c, \epsilon, E$ as
before and put $D=E$.
The standard Dehn algorithm for solving $\WP(G,X)$
consists of all rules $u \to v$ with $u,v \in X^*$ such that $u =_G v$ and
$4\delta \ge |u| > |v|$, where $\delta$ is the `thinness' constant of
$G$ with respect to $X$ (i.e. all geodesic triangles in $\Cay$ are
$\delta$-thin); see \cite[Theorem 2.12]{AL}.
Words $w$ that are reduced by this algorithm are $4\delta$-local geodesics,
and it is proved in \cite[Proposition 2.1]{Holt} that, if $w$ represents
the group element $g$, then $|w| \le 2|g|$.
We define our Dehn algorithm $R$ for $\WP(G,X)$ to consist of all rules $u \to v$ as above,
with $k \ge |u| > |v|$, where $k = \max(2D,4 \delta)$.
Then $R$-reduced words have the property that subwords representing
group elements of $X$-length at most $D$ are $X$-geodesics.
Since $D=E$, it is also true that $R$-reduced words of length at most $E$ are
geodesic, so $R$ has the required property $\P(D,E)$.
As in the previous proof, we define $S$ to be the \eda for
$\GWP(G,H,X)$ consisting of $R$ together with rules $Hx \to H$ for all
$x \in X_0 := X \cap H$.
As before, we suppose that $S$ reduces the input word $Hw$ to $Hw_1$ and
define the decomposition $(*)$ of $w_1$ with $p_i$ being maximal $X_0$-subwords
of $w_1$ that represent group elements of $X$-length greater than $E$.
Again we let $q_i$ be geodesic words over $X_0$ (and hence also over $X$)
with $q_i =_G p_i$. By \cite[Proposition 2.1]{Holt}, we have
$|q_i| \ge k_1|p_i|$ with $k_1 = 1/2$, so again we have
$|w_2| \ge k_1|w_1|$. The remainder of the proof is identical to that
of Theorem \ref{thm:relhyp}.
\section{Proof of Theorem \ref{thm:qcsubhyp} (ii)}\label{sec:qcthmii}
As we did for Part (i) of this theorem, we prove Theorem \ref{thm:qcsubhyp}
(ii) by constructing an \eda over the alphabet $X \cup \{ H \}$,
where $G = \langle X \rangle$. As in the two earlier proofs, we verify that
the conditions of Proposition~\ref{prop:realtime} hold, to complete the proof.
For a finite (inverse-closed) generating set $X$ of an arbitrary group $G$, we
define an {\em $X$-graph} to be a graph with directed edges labelled by
elements of $X$, in which, for each vertex $p$ and each $x \in X$, there is a
single edge labelled $x$ with source $p$ and, if this edge has target $q$,
then there is an edge labelled $x^{-1}$ from $q$ to $p$. So the {\em Cayley
graph} $\Cay(G,X)$ and, for a subgroup $H \le G$, the {\em Schreier graph}
$\Sch(G,H,X)$ of $G$ with respect to $H$ are examples of $X$-graphs.
We shall denote the base points of the Cayley and Schreier graphs by $\IdC$ and
$\IdS$ respectively.
Following \cite[Chapter 4]{Foord}, for $k \in \N$, we define the condition
$\GIB(k)$ (which stands for \emph{group isomorphic balls}) for $\Sch:=\Sch(G,H,X)$ as follows.
\begin{quote}
$\GIB(k)$: there exists $K \in \N$
such that, for any vertex $p$ of $\Sch$ with $d(\IdS, p) \ge K$,
the closed $k$-ball $B_k(p)$ of $\Sch$ is $X$-graph isomorphic to the $k$-ball
$B_k(\IdC)$ of $\Cay(G,X)$.
\end{quote}
We say that $\Sch$ satisfies
$\GIB(\infty)$ if it satisfies $\GIB(k)$ for all $k \ge 0$.
The following result is proved in \cite[Theorem 4.3.1.1]{Foord}.
Since its proof may not be readily available, we shall sketch it in
Section \ref{sec:foord}.
\begin{proposition}\label{prop:foord}
Let $H$ be a quasiconvex subgroup of the hyperbolic group $G$. Then
$\Sch(G,H,X)$ satisfies $\GIB(\infty)$ if and only if,
for all $1 \ne h \in H$, the index
$|C_G(h):C_H(h)|$ is finite.
\end{proposition}
Suppose that $G,H$ satisfy the hypotheses of Theorem~\ref{thm:qcsubhyp} (ii).
So $\Sch := \Sch(G,H,X)$ satisfies $\GIB(\infty)$ and, by \cite[Theorem
4.1.3.3]{Foord} or \cite{Kapovich}, $\Sch$ is $\delta$-hyperbolic for some
$\delta>0$ (that is, geodesic triangles in $\Sch$ are $\delta$-thin).
Let $k$ be an integer with $k \geq 4\delta$.
Let $K$ be an integer that satisfies the condition in the definition of
$\GIB(k)$, and let $R=2K$. We can assume that $K \ge \max(k,2)$.
We define our \eda to consist of all rules of the following two forms:
\begin{eqnarray*}
Hv_1 \rightarrow Hv_2,&& |v_2|<|v_1| \leq R,\quad(1) \\
u_1 \rightarrow u_2,&& |u_2| < |u_1|\leq k,\quad(2)
\end{eqnarray*}
where $v_1,v_2,u_1,u_2 \in X^*, v_1v_2^{-1} \in H, u_1=_G u_2$.
In order to apply Proposition~\ref{prop:realtime}
we need to verify that,
whenever $w \in X^*$ and $Hw$ is reduced according to the above \eda,
the length of the shortest string $v$ over $X$ with $Hv=Hw$ is bounded below
by a linear function of $|w|$.
We shall use \cite[Proposition 2.1]{Holt}: if $u$ (of length $>1$) is a
$k$-local geodesic in a $\delta$-hyperbolic graph, with $k\geq 4\delta$, then
the distance between the endpoints of $u$ is at least $|u|/2 + 1$.
So suppose that $Hw$ is reduced according to the \eda.
If $w$ has length at most $R$, then since $Hw$ is reduced by rules of type (1),
$Hw$ is geodesic in $\Sch$, and
the inequality in Proposition~\ref{prop:realtime} holds with $k=1$.
So suppose that $|w|>R$, and let $w_1$ be the prefix of length $R$ of
$w$. We aim to show that every vertex of $\Sch$ that comes after $w_1$
on the path from $\IdS$ labelled $w$ lies outside of $B_K(\IdS)$.
Choose $w_2$ so that $w_1w_2$ is maximal as a prefix of $w$ subject to all
vertices of $w_2$ lying outside of $B_K(\IdS)$.
Then, since $w_1$ is geodesic of length $R=2K$, we have $|w_2| \geq K-1$,
and so $|w_1w_2| \geq 3K-1$. Since $w_1$ is geodesic in $\Sch$, and
that part of the
path labelled $w_1w_2$ that lies outside of the $K$-ball is a $k$-local
geodesic in $\Sch$ (because, by $\GIB(k)$, it is isometric to the corresponding
word in the Cayley graph, and
our inclusion of the rules of type (2) in the \eda ensures that
the reduced words over $X$ of length $\leq k$ are geodesics),
we see that the whole of the path labelled $w_1w_2$ is a $k$-local geodesic
in $\Sch$.
So we can apply \cite[Proposition 2.1]{Holt} to deduce from the
$\delta$-hyperbolicity of $\Sch$ that
\[ d_{\Sch}(\IdS,Hw_1w_2) \geq |w_1w_2|/2 + 1\geq (3K+1)/2>K+1.\]
It follows that, if $w_1w_2$ were not already equal to $w$, then
it would be extendible to a longer prefix of $w$ subject to all
vertices of $w_2$ lying outside of $B_K(\IdS)$.
So $w_1w_2=w$, and the above inequality gives us the linear lower bound
$d_{\Sch}(\IdS,Hw) \ge |w|/2+1 \ge |w|/2$ on $d_{\Sch}(\IdS,Hw)$.
So the inequality in Proposition~\ref{prop:realtime} holds with $k=2$
and hence, by the preceding paragraph, it holds with $k=2$ for all
words $w$ such that $Hw$ is reduced according to the \eda.
The result now follows from Proposition~\ref{prop:realtime}, and this
completes the proof of Theorem~\ref{thm:qcsubhyp} (ii).
\section{Proof of Theorem \ref{thm:subvf}} \label{sec:subvf}
Let $F$ be a free subgroup of $G = \langle X \rangle$ with $|G:F|$ finite,
and let $Y$ be the inverse closure of a free generating set for $F$;
that is the union of a free generating set with its inverses.
Let $K=F \cap H$.
The subgroup $K$ has finite index in $H$,
and so must (like $H$) be finitely generated.
It easy to see that the elements in any right transversal of $K$ in $H$ lie
in different cosets of $F$ in $G$, so we can extend a right transversal
$T'=\{t_1,\ldots,t_m\}$ of $K$ in $H$ to a right transversal
$T = \{t_1,\ldots,t_n\}$ of $F$ in $G$.
Then any word $w \in X^*$ can be expressed (in $G$) as a word in $U^*T$,
where $U$ is the set $\{ u(i,x) : 1 \le i \le n,\,x \in X \}$ of Schreier
generators for $F$ in $G$ defined by the equations $t_i x= u(i,x)t_j$.
(Note that $X$ inverse-closed implies that $U$ is inverse-closed.)
Then, by substituting the reduced word in $Y^*$ for each $u(i,x)$, the word
$w$ can be written as a word $vt$ in $Y^*T$
(where $v$ is not necessarily freely reduced).
The first step to recognise whether $w \in H$ is to rewrite it to the form
$vt$, as above, using a transducer. Then $w \in H$ if and only if $t \in
T'$ and $v \in K$. It remains for us to describe the
operation of a deterministic pushdown automaton (\pda) $N$ to recognise those
words $v$ in $Y^*$ that lie in the subgroup $K$ of the free group $F$.
Note that this machine operates simultaneously, rather than sequentially, with
the transducer, and it follows from the fact that context-free languages are
closed under inverse {\gsm}s \cite[Example 11.1, Theorem 11.2]{HopcroftUllman}
that the combination of the two machines is a \pda.
By \cite{AnS75} or \cite[Proposition 4.1]{GerstenShort91b}, any finitely
generated subgroup $K$ of a free group is {\em $L$-rational}, where $L$ is the
set of freely reduced words over a free generating set; that is, the set
$K \cap L$ is a regular language.
In our case, we choose $L$ to ber the freely reduced words over $Y$.
We shall build our \pda $N$ out of a finite state automaton (\fsa) $M$ for which
$L(M) \cap L = K \cap L$ (where $L(M)$ is the set of words accepted by $M$).
The construction is (in effect) described in the proof of
\cite[Theorem 2.2]{GerstenShort91b} that $K$ is $L$-quasiconvex.
The $L$-quasiconvexity condition is equivalent to the property that
all prefixes of freely reduced words that represent elements of $K$ lie within a
bounded distance of $K$ in the Schreier graph $\Sch := \Sch(F,K,Y)$.
Equivalently, a freely reduced word $v$ over $Y$ represents an element
of $K$ precisely if it labels a loop in $\Sch$ from $\IdS$ to $\IdS$
that does not leave a particular bounded neighbourhood $B=B_d(\IdS)$ of $\IdS$.
Suppose that $g_1=1$ and that $K=Kg_1,Kg_2,\ldots,Kg_r$ are the right cosets
corresponding to the bounded neighbourhood $B$ of $\IdS$ within $\Sch$ that is
identified above.
The \fsa $M$ is defined as follows.
\begin{mylist}
\item[(i)] The states of $M$ are denoted by $\sigma_1,\ldots,\sigma_r,
\hat{\sigma}$.
\item[(ii)] The states $\sigma_1,\ldots,\sigma_r$ correspond to the
right cosets $Kg_i$ of $K$ in $F$, where each $g_i$ is in the
finite subset $B$ identified above; indeed we may identify $\sigma_i$ with
the coset $Kg_i$, and then use the name $B$ both for the set
$\{Kg_1,\ldots,Kg_r\}$ of cosets and for the set
$\{\sigma_1,\sigma_2,\ldots,\sigma_r\}$ of states. The state $\sigma_1$ (which
corresponds to the subgroup $K$) is the start state
and the single accepting state.
\item[(iii)] For $1\le i,j \le r$ and $y \in Y$, there is a transition
$\sigma_i^y = \sigma_j$ if and only if $Kg_iy=Kg_j$. It follows from this that
$\sigma_i^y = \sigma_j$ if and only if $\sigma_j^{y^{-1}} = \sigma_i$.
\item[(iv)] $\hat{\sigma}$ is a failure
state, and is the target of all transitions that are not
defined in (iii), including those from $\hat{\sigma}$.
\end{mylist}
We see that, as a word is read by $M$, the automaton keeps track of
the coset of $\Sch$ that contains $Kw$, where $w$ is the prefix that has been
read so far, so long as that coset is within the finite neighbourhood $B$ of
and in addition so are all cosets $Kw'$ for which $w'$ is a prefix of $w$.
The $L$-quasiconvexity of $K$ ensures that
a word in $L$ is accepted by $M$ if and only
if it represents an element of $K$.
In fact any word over $Y$ that is accepted by $M$ must represent an element of
However words over $Y$ that are not freely-reduced (that is, not in $L$)
and do not stay inside of $B$ will be rejected
by $M$, even when they represent elements of $K$.
In order to construct a machine that accepts all words $v$ over
$Y$ within $K$, and not simply those that are also freely-reduced,
we need to combine the operation of the \fsa $M$ above with a stack,
which we use to compute the free reduction.
We construct our
\pda $N$ to have the same state set $B \cup \{\hat{\sigma}\}$ as
$M$, again with $\sigma_1=Kg_1=K$ as the start state and sole accepting state.
The transitions from the states $\sigma_i=Kg_i$ are as in $M$. We need however
to describe the operation of the stack, and transitions from the state
$\hat{\sigma}$, which is non-accepting, but no longer a failure state.
The stack alphabet is the set $Y \cup (Y \times B)$. The second component of
an element of $Y \times B$ is used to record the state $M$ is in immediately
before it enters the state $\hat{\sigma}$.
In addition, we use the stack to store the free reduction of
the prefix of $v$ that has been read so far.
The operations of the \pda $N$ that correspond to the various transitions of
$M$ are described in the following table. The absence of an entry in the
`push' column indicates that nothing is pushed.
\begin{center}
\begin{tabular}{|l| lllll|}
\hline
Transition & Input & Input & Pop & Push & Output \\
of $M$ & state & symbol & & & state \\
\hline
$\sigma_i^y = \sigma_j$ & $\sigma_i$ & $y$ & $y^{-1}$ & & $\sigma_j$ \\
& $\sigma_i$ & $y$ & $y' \neq y^{-1}$ & $y'y$ & $\sigma_j$ \\
\hline
$\sigma_i^y = \hat{\sigma}$ & $\sigma_i$ & $y$ & $y'$ &
$y'(y,\sigma_i)$ & $\hat{\sigma}$ \\
\hline
$\hat{\sigma}^y = \hat{\sigma}$ & $\hat{\sigma}$ &
$y$ & $(y^{-1},\sigma_i)$ & & $\sigma_i$ \\
& $\hat{\sigma}$ & $y$ & $(y',\sigma_i),\quad y' \neq y^{-1}$ &
$(y',\sigma_i)y$ & $\hat{\sigma}$ \\
& $\hat{\sigma}$ & $y$ & $y^{-1}$ & & $\hat{\sigma}$ \\
& $\hat{\sigma}$ & $y$ & $y' \neq y^{-1}$ & $y'y$ & $\hat{\sigma}$ \\
\hline
\end{tabular}
\end{center}
Note that we have not specified that $y'\neq y^{-1}$ in line 3, but in fact the
condition $y'=y^{-1}$ does not arise in this situation.
Since there can be a symbol in $Y \times B$ on the stack only when $N$ is
in state $\hat{\sigma}$, it is not possible to pop such a symbol when
$N$ is in state $\sigma_i$, so there are no such entries in the table.
The fact that $N$ recognises $\GWP(F,K,Y)$ follows from the fact that $N$
accepts $w$ if and only if $M$ accepts $\overline{w}$, where $\overline{w}$ is
the free reduction (in $L$) of $w$. We prove this by induction on the number
$k$ of reductions of the form $w_1yy^{-1}w_2 \rightarrow w_1w_2$ with $y \in
Y$ that we need to apply to reduce $w$ to $\overline{w}$.
The case $k=0$ of our induction follows from the fact that $L(M)\cap L=K\cap L$,
combined with the observation that, if $w$ is freely reduced, then
$w$ leads to the same state of $M$ as it does of $N$. For in that case
the only possible transitions as we read $w$ are of the types
described in lines 2,3,5,7 of the table.
For $k>0$ it is enough to prove the statement
\begin{quote}
$(*)$: if $wy \in L$, then the configuration of $N$ after reading $wyy^{-1}$
is identical to the configuration after reading $w$.
\end{quote}
It follows from $(*)$ that a word $w_1yy^{-1}w_2$ in which $yy^{-1}$
is the leftmost cancelling pair is accepted by $N$ if and only if $w_1w_2$ is accepted by
$N$, and hence we have the inductive step we need.
We can check the statement $(*)$ with reference to the table. There
are up to seven possibilities for the type of transition of $N$ as the final
symbol $y$ of $wy$ is read.
For the first two of these,
$N$ is in state $\sigma_i$ after reading $w$,
and moves to a state $\sigma_j$.
Since $wy$ is in $L$, the top stack symbol after reading $w$ is not $y^{-1}$.
Hence the transition must be of the type described in line 2 of the table,
and not as in line 1, that is, $y' \neq y^{-1}$ is popped, and then $y'y$ is
Recalling that $\sigma_i^y = \sigma_j$ in $M$ if and only if
$\sigma_j^{y^{-1}}=\sigma_i$, we see that the next transition of $N$,
from $wy$ on $y^{-1}$, is of the type described in line 1. Then the symbol
$y$ is popped, the symbol $y'$ is again on the top of the stack,
and $N$ returns to the state $\sigma_i$
We consider similarly the remaining five possibilities for the transition from
$w$ on $y'$, and the subsequent transitions on $y'^{-1}$, and verify $(*)$ for
each of those configurations.
This completes the proof of Theorem~\ref{thm:subvf}.
\section{Proof of Theorem \ref{thm:subvf_conv}} \label{sec:subvf_conv}
Our proof of Theorem \ref{thm:subvf_conv} has the same structure as the proof
in \cite{MullerSchupp83} that groups with $\WP(G)$ context-free are virtually
free, and it would be helpful for the reader to be familiar with that proof.
We shall prove that $G$ has more than one end. Assuming that to be true,
we use Stalling's theorem \cite{Stallings71} to conclude that $G$ has
a decomposition as an amalgamated free product $G = G_1 *_K G_2$, or
as an HNN-extension $G = G_1*_{K,t}$, over a finite subgroup $K$.
Since $G_1$ (and $G_2$) are easily seen to be quasiconvex
subgroups of $G$, it is not hard to show that the hypotheses of the
theorem are inherited by the subgroup $H \cap G_1$ of $G_1$ (and
$H \cap G_2$ of $G_2$), and so they too have more than
one end, and we can apply the Dunwoody accessibility result to
conclude that $G$ is virtually free.
So we just need to prove that $G$ has more than one end.
Fix a finite inverse-closed generating set $X$ of $G$. Then, as in
\cite{MullerSchupp83}, we consider a context-free grammar in Chomsky normal
form with no useless variables that derives $\GWP(G,H,X)$.
More precisely, we suppose that each rule has the form $\start \to \nullstring$
(where $\start$ is the start symbol), $z \rightarrow z'z''$ or
$z \rightarrow a$, where $z,z',z''$ are variables, and $a$ is terminal, and we
assume that $\start$ does not occur on the right hand side of any
derivation. When a word $w'$ can be derived from a word $w$ by application of
a single grammatical rule we write $w \Rightarrow w'$, and when a
sequence of such rules is needed we write $w \Rightarrow ^* w'$.
Let $z_1,\ldots,z_n$ be the variables of the grammar other than $\start$
and, for each $z_i$ let $u_i$ be a shortest word in $X^*$ with
$z_i \Rightarrow^* u_i$. Let $L$ be the maximum length of the words $u_i$.
Let $w \in \GWP(G,H,X)$ with $|w|>3$, and fix a derivation of $w$ in the
grammar. We shall define a planar $X$-graph $\Delta$ with an associated
$X$-graph homomorphism $\phi:\Delta \to \Sch:= \Sch(G,H,X)$. We start with a simple
plane polygon with a base point, and edges labelled by the letters of $w$, and
with $\phi$ mapping the base-point of $\Delta$ to $\IdS$. Note that $\phi$ is
not necessarily injective.
If $z_i$ occurs in the chosen derivation of $w$, then we have $w = vv_iv'$
with $z_i \Rightarrow^* v_i$; two such words $v_i$ and $v_j$ are either
disjoint as subwords of $w$ or related by containment.
Since $z_i \Rightarrow^* u_i$, we also have $vu_iv' \in \GWP(G,H,X)$. So we
can draw a chord labelled $u_i$ in the interior of $\Delta$ between the two
ends of the subpath labelled $v_i$, and $\phi$ extends to this extension of
If we do this for each such $z_i$ for which $1 < |v_i| < |w|-1$ then, as in
\cite[Theorem 1]{MullerSchupp83}, we get a `diagonal triangulation' of $\Delta$,
in which the sides are either boundary edges of $\Delta$ or internal
chords of length at most $L$.
(But note that, for the first derivation $\start \to z_1z_2$, say,
if $|v_1|>1$ and $|v_2|>1$ then, to avoid an internal bigon, we
omit the chord labelled $u_2$.)
Suppose, for a contradiction, that $G$ has just one end; that is, for any
$R$, the complement in $\Cay(G,X)$ of any ball of radius $R$ is connected.
Then, for any $R$,
we can find a word $w_1w_2w_3$ over $X$ with $w_1w_2w_3=_G 1$ which, starting
at $\IdC$, labels a simple closed path in
$\Cay(G,X)$, where $|w_1|=|w_3|=R$, $w_3w_1$ is geodesic, and no vertex in
the path labelled $w_2$ is at distance less than $R$ from $\IdC$.
Choose such a path with $R=3L+1$. Choose $k'$ such that the whole path
lies in the ball $B_{k'}(\IdC)$ of $\Cay(G,X)$, and let $k = k'+L$. Then,
since by Proposition \ref{prop:foord} $\Sch(G,H,X)$ satisfies $\GIB(k)$, there
exists $K$
such that, for any vertex $p$ of $\Sch(G,H,X)$ with
$d(\IdS,p) \ge K$, the ball $B_k(p)$ of $\Sch(G,H,X)$ is $X$-graph isomorphic
to the ball $B_k(\IdC)$ of $\Cay(G,X)$.
Choose such a vertex $p$, and consider the path labelled $w_1w_2w_3$ of
$\Sch(G,H,X)$ that is based at $p$, as in Fig. \ref{fig:triangulation}.
Choose a vertex $q$ on the path labelled $w_1w_2w_3$ with $d(\IdS,q)$ minimal,
and let $w_4$ be the label of a geodesic path in $\Sch(G,H,X)$ from
$\IdS$ to $q$. Then, for some cyclic permutation $w'$ of $w_1w_2w_3$, we have
a closed path in $\Sch(G,H,X)$ based at $\IdS$ and labelled $w_4 w' w_4^{-1}$.
We apply the above triangulation process to a planar $X$-graph $\Delta$
for $w_4 w' w_4^{-1}$.
Since $q$ is the closest vertex to $\IdS$ on the loop labelled by $w'$, and the
path from $\IdS$ to $q$ in the Cayley graph labelled by $w_4$ is geodesic, every vertex on that path is
as close to $q$ as to any other vertex of $w'$, and so any vertex of $w_4$ that can be connected by the image of a chord of $\Delta'$ to a vertex of $w'$ must
be within distance at most $L$ of $q$. Let $r$ be the first such vertex on $w_4$
(as we move from $\IdS$ to $q$), and let $w_5$ be the suffix of $w_4$ that labels
the path along $w_4$ from $r$ to $q$. Then $|w_5|\leq L$.
So we can derive from our triangulation of $\Delta$ a triangulation
of a planar diagram $\Delta'$ for the word $w_5w'w_5^{-1}$, and there is
an associated $X$-graph homomorphism $\phi'$ that maps this to the corresponding
subpath in $\Sch(G,H,X)$. (Note that the images of $w_5$ and $w_5^{-1}$
under $\phi'$ are equal, but that $\phi'$ is injective when restricted to $w'$.)
By our choice of $k = k' + L$, the image of $\phi'$ lies entirely within
$B_k(p)$, which is $X$-graph isomorphic to $B_k(\IdC)$.
So the distances in $\Sch(G,H,X)$ between vertices in this image are the
same as in any path with the same label in $\Cay(G,X)$.
\begin{figure}
\begin{center}%
\setlength{\unitlength}{1.0pt}%
\begin{picture}(100,210)(-25,40)%
\put(15,45){\circle*{5}}%
\put(20,45){$\IdS$}%
\put(15,45){\line(1,3){37}}%
\put(33.5,100.5){\vector(1,3){0}}%
\put(25,105){\vector(1,3){17}}%
\put(17.5,97.5){$w_4$}%
\put(22,94){\line(-1,-3){15}}%
\put(52,156){\circle*{5}}%
\put(55,145){$q$}%
\put(46,138){\vector(1,3){0}}%
\put(50,133){$w_5$}%
\put(42,126){\circle*{5}}%
\put(45,115){$r$}%
\put(61,163){\vector(-3,1){31}}%
\put(67,151){\vector(-3,1){0}}%
\put(64,159){$w_3$}%
\put(77.5,157.5){\line(3,-1){14}}%
\put(22,166){\circle*{5}}%
\put(25,155){$p$}%
\put(-14,178){\vector(-3,1){0}}%
\put(-13,182){$w_1$}%
\put(97,141){\line(-3,1){141}}%
\qbezier(97,141)(62,320)(-44,188)%
\put(66,226){\vector(1,-1){0}}%
\put(61,233){$w_2$}%
\end{picture}%
\caption{Triangulation in $\Sch(G,H,X)$}
\label{fig:triangulation}
\end{center}%
\end{figure}
As in \cite{MullerSchupp83}, we colour, using three colours, the vertices of
the boundary paths of $\Delta'$ that are labelled
$w_1,w_2,w_3$ (where vertices on two of these subwords get both associated
colours), and we colour the vertices on $w_5$ and $w_5^{-1}$ with
the same colour (or colours) as $q$. As in \cite[Lemma 5]{MullerSchupp83},
we conclude that there is a triangle in the triangulation
whose vertices use all three colours between them. One (or even two) of these
vertices could be on the subpath labelled $w_5$, and two 2-coloured vertices
in the triangle might coincide, but, since any vertex on $w_5$ is within
distance $L$ of $q$, replacing vertices on $w_5$ by $q$ as necessary, we end
up with a triangle of three (not necessarily distinct) vertices
$p_1, p_2, p_3$ with $p_i$ on $w_i$, and
with $d(p_i,p_j) \le 2L$ for each $i,j$. At least one of $p_1,p_3$
must be within distance $L$ of $p$. But then $d(p,p_2) \leq 3L$, contradicting
our assumption that $w_2$ is outside $B_{3L}(p)$.
This completes the proof of Theorem~\ref{thm:subvf_conv}.
\section{Sketch of proof of Proposition \ref{prop:foord}} \label{sec:foord}
Suppose first that $|C_G(h):C_H(h)|$ is infinite for some $1 \ne h \in H$,
and let $w$ be a word representing $h$.
Then, for any $K>0$, there exists a word $v \in C_G(h)$ labelling a
path in $\Sch:=\Sch(G,H,X)$ from $\IdS$ to a vertex $p$ with
$d(\IdS,p)>K$, and there is a loop labelled $w$ based at $p$ in
$\Sch$, but no such loop based at $\IdC$ in $\Cay:= \Cay(G,X)$. So $\GIB(|w|)$
fails in $\Sch$.
Suppose conversely that $\GIB(k)$ fails in $\Sch$ for some $k$.
Then there are vertices $p$ of $\Sch$ at arbitrarily large distance from
%DFH: added more detail in following
$\IdS$ such that the ball $B_k(p)$ in $\Sigma$ is not $X$-graph isomorphic to
the ball $B_k(\IdC)$ in $\Cay$.
So the natural labelled graph morphism $B_k(\IdC) \to B_k(p)$ with
$\IdC \mapsto p$ is not injective, and hence two distinct vertices
of $B_k(\IdC)$ map to the same vertex of $B_k(p)$.
So, for any such vertex $p$, there is at least one labelled loop based at $p$,
within $B_k(p)$, such that the corresponding labelled path based at
in $B_k(\IdC)$ in $B_k(\IdC)$ is not a loop in $\Cay$.
Since the number of words that can label loops in a ball of radius $k$ in
$\Sch$ is finite, some word $w$ with $w \ne_G 1$ must label loops based at $p$
for infinitely many vertices $p$ of $\Sigma$,
and we can choose $w$ to be geodesic over $X$.
Now for any integer $N$, there is a word $v$ of length greater that $N$,
labelling a geodesic in $\Sch$ from $\IdS$ to a vertex $p$, from which there
is a loop in $\Sch$ labelled by $w$.
For such a word $v$, we have $hvw = v$ for some $h \in H$. Let $u$ be a
geodesic word labelling $h$. Then we have a geodesic quadrilateral
with vertices $A=\IdC,B,C,D$ in $\Cay(G,X)$ with sides
$AB$, $BC$, $CD$, $AD$ labelled $u$, $v$, $w$, $v$, respectively,
as shown in Fig. \ref{fig:ABCD}.
\begin{figure}
\begin{center}
\setlength{\unitlength}{1.0pt}%
\begin{picture}(150,180)(-70,5)%
\put(50,10){\circle*{5}}%
\put(57,8){$A=1_\Cay$}%
\put(-50,10){\circle*{5}}%
\put(-67,8){$B$}%
\qbezier(50,10)(10,15)(12,50)%
\qbezier(-50,10)(-10,15)(-12,50)%
\qbezier(-50,10)(-15,10)(-5,30)%
\qbezier(50,10)(15,10)(5,30)%
\qbezier(-5,30)(0,35)(5,30)%
\put(-2.5,32.2){\vector(-1,0){0}}%
\put(-3,36){$u$}%
\put(-12,50){\line(0,1){85}}%
\put(-12,87){\vector(0,1){0}}%
\put(-25,82){$v$}%
\put(12,50){\line(0,1){85}}%
\put(12,87){\vector(0,1){0}}%
\put(17,82){$v$}%
\put(-40,160){\circle*{5}}%
\put(-57,158){$C$}%
\put(40,160){\circle*{5}}%
\put(47,158){$D$}%
\qbezier(40,160)(12,155)(12,135)%
\qbezier(-40,160)(-12,155)(-12,135)%
\qbezier(40,160)(5,155)(5,135)%
\qbezier(-40,160)(-5,155)(-5,135)%
\qbezier(-5,135)(0,125)(5,135)%
\put(2,130.2){\vector(1,0){0}}%
\put(-5,121){$w$}%
\end{picture}%
\end{center}%
\caption{The geodesic quadrilateral $ABCD$}
\label{fig:ABCD}
\end{figure}
By the hyperbolicity of $G$, each vertex of $AB$ lies within a distance
$2\delta$ of some vertex on $BC$, $CD$ or $DA$, where $\delta$ is the
constant of hyperbolicity. Furthermore, since $H$ is quasiconvex in $G$,
there is a constant $\lambda$, such that each vertex of $AB$ is within a
distance $\lambda$ of a vertex of $\Cay$ representing an element of $H$.
Since each vertex of $w$ lies at distance at least $|v|-k$ from any vertex
in $H$, by choosing $|v| > k+2\delta+\lambda$ we can ensure that none of the
vertices of $AB$ is $2\delta$-close to any vertex of $CD$. So the vertices
of $AB$ must all be $2\delta$-close to vertices in $BC$ or $DA$.
But, since $v$ labels a geodesic path from $\IdS$ in $\Sch$, at most
%DFH: replace \gamma by \lambda three times
$2\delta + \lambda$ vertices on $BC$ or on $DA$ can be within $2\delta+\lambda$
of a vertex in $H$. So each vertex of $AB$ is at distance at most $2\delta$
from one of at most $4\delta + 2\lambda$ vertices and, since the total number of
vertices in $\Cay$ with that property is bounded, we see that
$|AB| = |u|$ is bounded by some expression in $|X|$, $\delta$ and $\lambda$.
By hyperbolicity of $G$, the two paths $BC$ and $AD$ labelled $v$
must synchronously $L$-fellow travel for some $L$ (which depends on the upper
bounds on $|w|$ and $|u|$).
Let $m>0$. Then, by choosing $v$ sufficiently long, we can ensure that
some word $u'$ appears as a word-difference between $BC$ and $AD$ at least $m$
that is, $v$ has consecutive subwords $v_0,\ldots,v_m,v'$, such that
$v = v_0v_1v_2 \cdots v_m v'$, and $hv_0v_1v_2 \cdots v_iu' =_G v_0v_1v_2
\cdots v_i$ for each $i$ with $0 \le i \le m$.
The case $i=0$ gives $u' = v_0^{-1}h^{-1}v_0$, and
it follows from this that
$g_i := v_0(v_1v_2 \cdots v_i)v_0^{-1} \in C_G(h)$ for $1 \le i \le m$.
Also, since $v$ labels a geodesic in $\Sch$, the elements
$v_0v_1, v_0v_1v_2, \ldots v_0v_1v_2 \cdots v_m$ lie in distinct cosets of
$H$ and hence so do the $g_i$. Since we can choose $m$ arbitrarily
large, this contradicts the finiteness of $|C_G(h):C_H(h)|$.
\section*{Acknowledgements}
The first author was supported by the Swiss National Science
Foundation grant Professorship FN PP00P2-144681/1, and would like to thank the
mathematics departments of the Universities of Newcastle and Warwick for their
support and hospitality.
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of Dehn's algorithm. {\em Internat. J. Algebra Comput.}, 18 (2008) 1137--1177.
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{\em Internat. J. Algebra Comput.} 10 (2000) 221--227.
\bibitem{HoltRees} D.F.\ Holt and S.\ Rees, Solving the word problem in real
time, {\em J. London Math Soc.} 63 (2001) 623--639.
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Addison-Wesley, 1979.
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\verb+http://arxiv.org/abs/math/0201045v2+.
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D.E.\ Muller and P.E.\ Schupp. Groups, the theory of ends, and context-free
languages. {\em J. Comp.\ System Sci.}, 26:295--310, 1983.
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algebraic properties and algorithmic problems. Mem. Amer. Math. Soc. 179
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\end{thebibliography}
\textsc{Laura Ciobanu,
Mathematical and Computer Sciences,
Colin McLaurin Building,
Heriot-Watt University,
Edinburgh EH14 4AS, UK}
\emph{E-mail address}{:\;\;}\texttt{[email protected]}
\bigskip
\textsc{Derek Holt,
Mathematics Institute,
Zeeman Building,
University of Warwick,
Coventry CV4 7AL, UK
\emph{E-mail address}{:\;\;}\texttt{[email protected]}
\bigskip
\textsc{Sarah Rees,
School of Mathematics and Statistics,
University of Newcastle,
Newcastle NE1 7RU, UK
\emph{E-mail address}{:\;\;}\texttt{[email protected]}
\end{document}
|
1511.00318
|
The moduli spaces of
stable sheaves on projective schemes
gluing data of Kapranov's NC structures,
which we call quasi NC structures.
formal completion of the quasi NC structure
at a closed point
coincides with the pro-representable hull
of the non-commutative deformation functor of the
corresponding sheaf.
In this paper,
we show the existence of
non-commutative dg-resolutions
of the above quasi NC structures,
and call them
quasi NCDG structures.
When there
are no higher obstruction spaces,
the quasi NCDG structures
define the notion of
NC virtual structure sheaves,
the non-commutative analogue of virtual
structure sheaves.
We show that
the NC virtual structure sheaves are described in terms of
usual virtual structure sheaves together with
Schur complexes of the perfect obstruction theories.
§ INTRODUCTION
The purpose of this paper is to introduce the notion of
non-commutative virtual structure sheaves on the
moduli spaces of stable sheaves on projective schemes
without higher obstruction spaces.
The motivation introducing this concept is
to construct non-commutative analogue
of the enumerative invariants
of sheaves,
e.g. Donaldson-Thomas (DT) invariants <cit.>,
involving non-commutative
deformations of sheaves.
In this introduction, we first recall some background
of commutative virtual structure sheaves
via smooth commutative dg-schemes, and
explain quasi
NC structures on the
moduli spaces of stable
sheaves obtained in <cit.>.
We then state
the existence of
smooth non-commutative dg-enhancements
on the moduli spaces of stable sheaves,
called quasi NCDG structures,
which govern the commutative
dg-enhancements and the quasi NC structures.
The construction of quasi NCDG structures leads to
the definition of NC virtual structure sheaves.
§.§ Commutative virtual structure sheaves
Let $X$
be a projective scheme,
and $M_{\alpha}$ the moduli space of
stable sheaves on $X$ with Hilbert polynomial $\alpha$.
In general, the moduli space
may not have the expected dimension
at $[E] \in M_{\alpha}$:
\begin{align}\label{exp.dim}
\mathrm{exp.dim}_{[E]}M_{\alpha}
\cneq
\dim \Ext^1(E, E)-\dim \Ext^2(E, E).
\end{align}
Here $\Ext^1(E, E)$ is the tangent space of $M_{\alpha}$
at $[E] \in M_{\alpha}$, and $\Ext^2(E, E)$ is the obstruction space.
As long as
there are no higher obstruction spaces, i.e.
\begin{align}\label{vanish:higher}
\Ext^{\ge 3}(E, E)=0
\
\mbox{ for any }
[E] \in M_{\alpha}
\end{align}
expected dimension (<ref>)
locally constant on $M_{\alpha}$, and
in this case
the virtual fundamental class
$[M_{\alpha}]^{\rm{vir}} \in A_{\ast}(M_{\alpha})$
with the expected dimension
(<ref>) can be constructed
using the notion of perfect obstruction theory <cit.>.
The integration of the virtual fundamental class
yields interesting enumerative invariants of sheaves, such as
DT invariants <cit.>.
From the construction of the virtual class, it admits a
natural K-theoretic enhancement (cf. <cit.>)
\begin{align}\label{intro:K}
\oO_{M_{\alpha}}^{\rm{vir}} \in K_0(M_{\alpha})
\end{align}
called the
virtual structure sheaf of $M_{\alpha}$.
It recovers the virtual fundamental class
by applying the cycle map to (<ref>).
It was also suggested by Kontsevich <cit.>
that $M_{\alpha}$ may be obtained as
the zero-th truncation
of a smooth commutative dg-scheme
$(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$,
i.e. $\oO_{M_{\alpha}}=\hH_0(\oO_{N_{\alpha}, \bullet})$,
and the virtual structure sheaf (<ref>)
may be described as
\begin{align}\label{intro:Kid}
\oO_{M_{\alpha}}^{\rm{vir}}
=\sum_{i\in \mathbb{Z}} (-1)^i [\hH_i(\oO_{N_{\alpha}, \bullet})].
\end{align}
The dg-scheme
$(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$ was
constructed by
To$\ddot{\textrm{e}}$n-Vaquié <cit.>
Behrend-Fontanine-Hwang-Rose <cit.>.
Also the identity (<ref>)
was established by Fontanine-Kapranov <cit.>.
§.§ Quasi NC structures on $M_{\alpha}$
In the previous paper <cit.>, the moduli space
$M_{\alpha}$ turned out to admit
a certain non-commutative structure,
giving an enhancement of $M_{\alpha}$
from the commutative dg-enhancement
$(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$.
Such a non-commutative structure was formulated
in terms of Kapranov's NC schemes <cit.>, which are
ringed spaces whose structure sheaves are possibly non-commutative,
but formal in the non-commutative direction.
We refer to <cit.>, <cit.> for the recent developments
on Kapranov's NC schemes.
The above non-commutative structure can be
naturally observed from the formal deformation theory.
For $[E] \in M_{\alpha}$,
the formal deformation theory of $E$ is governed by
the dg-algebra $\dR \Hom(E, E)$, which
is quasi-isomorphic to
a minimal $A_{\infty}$-algebra
\begin{align}\label{intro:A}
(\Ext^{\ast}(E, E), \{m_n\}_{n\ge 2}).
\end{align}
The formal solution of the Mauer-Cartan
equation of the $A_{\infty}$-algebra (<ref>)
the not necessary commutative algebra
\begin{align}\label{intro:R}
R_E^{\n} \cneq \frac{\widehat{T}(\Ext^1(E, E)^{\vee})}
{\left(\sum_{n\ge 2}m_n^{\vee} \right)}.
\end{align}
Here $m_n^{\vee}$ is the
dual of the $A_{\infty}$-product
\begin{align*}
m_n \colon
\Ext^1(E, E)^{\otimes n} \to
\Ext^2(E, E).
\end{align*}
algebra (<ref>)
is an enhancement of
the commutative algebra $\widehat{\oO}_{M_{\alpha}, [E]}$
in the sense that
\begin{align}\label{intro:RM}
(R_E^{\n})^{ab} \cong \widehat{\oO}_{M_{\alpha}, [E]}.
\end{align}
Indeed, the algebra $R_E^{\n}$ is a
pro-representable hull of the non-commutative deformation functor
of $E$ developed in <cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>.
The main result of <cit.> was
to construct a kind of globalization of the isomorphism (<ref>),
as follows:
There exists an affine open cover
$\{V_i\}_{i\in \mathbb{I}}$
of $M_{\alpha}$,
ringed spaces $V_i^{\n}$ and isomorphisms
\begin{align}\label{intro:qnc}
V_i^{\n} = (V_i, \oO_{V_i}^{\n}), \
\phi_{ij} \colon V_j^{\n}|_{V_{ij}} \stackrel{\cong}{\to}
\end{align}
where $V_i^{\n}$ is Kapranov's
affine NC scheme <cit.>,
such that $\phi_{ij}^{ab}=\id$
$\widehat{\oO}_{V_i, [E]}^{\n} \cong R_E^{\n}$
for any $[E] \in V_i$.
The data (<ref>)
was called a quasi NC structure of $M_{\alpha}$
in <cit.>.
§.§ Quasi NCDG structures
Now we have obtained
two kinds of enhancement of $M_{\alpha}$:
a commutative dg structure
and a quasi NC structure.
It is a natural
question to construct a
further enhancement which
these two structures.
We introduce the notion of
quasi NCDG structures
on commutative dg-schemes, and answer
this question.
Roughly speaking,
a quasi NCDG structure on a commutative
dg-scheme $(N, \oO_{N, \bullet})$
is an affine open cover
$\{U_i\}_{i\in \mathbb{I}}$ of $N$
together with
sheaves of non-commutative dg-algebras
$\oO_{U_i, \bullet}^{\n}$ on $U_i$ satisfying the
* We have $(\oO_{U_i, \bullet}^{\n})^{ab}=\oO_{N, \bullet}|_{U_i}$.
* We have the isomorphisms
$\phi_{ij, \bullet} \colon
(U_{ij}, \oO_{U_j, \bullet}^{\n}|_{U_{ij}})
\stackrel{\cong}{\to}(U_{ij}, \oO_{U_i, \bullet}^{\n}|_{U_{ij}})$
satisfying $\phi_{ij, \bullet}^{ab}=\id$.
We will show the following result:
(Theorem <ref>)
There is a smooth quasi NCDG structure on
the dg-moduli space
$(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$
whose zero-th
truncation gives a quasi NC structure (<ref>).
The quasi NCDG structure in Theorem <ref>
fits into the upper half of the
picture in Figure <ref>.
Relations of DG structures, quasi NC structures and quasi NCDG
\begin{align}\notag
\xymatrix{
\fbox{$\begin{array}{c}
\mbox{Quasi NC structure} \\
\{(V_i, \oO_{V_i}^{\n})\}_{i\in \mathbb{I}}
\end{array}$}
\ar[rr]^-{\rm{abelization}} & &
\ovalbox{$\begin{array}{c}
\mbox{Moduli space of stable sheaves} \\
\end{array}$} \\
\doublebox{$\begin{array}{c}\mbox{Quasi NCDG structure} \\
\{(U_i, \oO_{U_i, \bullet}^{\n})\}_{i\in \mathbb{I}}
\end{array}$}
\ar[u]^{\rm{truncation}}
\ar[d]_{\rm{cohomology}}
\ar[rr]^-{\rm{abelization}} & &
\ovalbox{$\begin{array}{c}
\mbox{Commutative DG structure} \\
(N_{\alpha}, \oO_{N_{\alpha}, \bullet})
\end{array}$} \ar[u]^{\rm{truncation}}
\ar[d]_{\rm{cohomology}} \\
\doublebox{$\begin{array}{c}
\mbox{NC virtual structure sheaves} \\
(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d} \in K_0(M_{\alpha})
\end{array}$
} & &
\ovalbox{$\begin{array}{c}
\mbox{Virtual structure sheaf} \\
\oO_{M_{\alpha}}^{\rm{vir}} \in K_0(M_{\alpha})
\end{array}$}
\end{align}
§.§ NC virtual structure sheaves
The quasi NCDG structure in Theorem <ref>
is interpreted as a smooth
dg-resolution of the quasi NC structure (<ref>).
For simplicity, let us assume that
the quasi NC structure
$\oO_{U_i, \bullet}^{\n}$
in Theorem <ref> glue to give
a global sheaf of non-commutative dg-algebras
$\oO_{N_{\alpha}, \bullet}^{\n}$
on $N_{\alpha}$, i.e.
the isomorphisms $\phi_{ij, \bullet}$ satisfy the
cocycle condition.
As an analogy of the identity (<ref>),
one may try to define the NC virtual structure sheaf
\begin{align}\label{intro:sum}
\oO_{M_{\alpha}}^{\rm{ncvir}} =
\sum_{i\in \mathbb{Z}}(-1)^i [\hH_i(\oO_{N_{\alpha}, \bullet}^{\n})].
\end{align}
The issue of the above construction is that
the sum (<ref>)
may be an infinite sum, so
does not make sense, even if the
condition (<ref>) is satisfied.
Instead of (<ref>),
if the condition (<ref>) is satisfied,
the following sum turns out to be finite
for each $d\in \mathbb{Z}_{\ge 0}$:
\begin{align}\label{intro:dvir}
(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d}
\cneq \sum_{i\in \mathbb{Z}}
(-1)^i [\hH_i((\oO_{N_{\alpha}, \bullet}^{\n})^{\le d})].
\end{align}
$(\oO_{N_{\alpha}, \bullet}^{\n})^{\le d}$
is the quotient of $\oO_{N_{\alpha}, \bullet}^{\n}$
by its $d$-th step
NC filtration
(cf. Subsection <ref>).
The quotient $(\oO_{N_{\alpha}, \bullet}^{\n})^{\le d}$
is interpreted as a $d$-smooth dg-resolution of the
quasi NC structure
$\{(V_i, (\oO_{V_i}^{\n})^{\le d})\}_{i\in \mathbb{I}}$
on $M_{\alpha}$,
hence (<ref>) is regarded as a $d$-smooth
thickening of (<ref>).
Moreover the sum (<ref>)
also makes sense even if
the quasi NCDG structure in Theorem <ref>
does not satisfy the cocycle condition (cf. Definition <ref>).
We call the sum (<ref>)
as d-th NC virtual structure sheaf of
the quasi NC structure (<ref>).
§.§ Descriptions via perfect obstruction theories
We will prove that
the $d$-th NC virtual structure sheaf (<ref>)
is described in terms of
the usual virtual structure sheaf (<ref>)
together with
the perfect obstruction theory $\eE_{\bullet}
\to \tau_{\ge -1} \dL_{M_{\alpha}}$
induced by the cotangent complex of
the commutative dg-scheme $(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$.
We have the following result:
(Theorem <ref>)
We have the following formula
\begin{align}\label{intro:ncvir:formula}
(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d}=\oO_{M_{\alpha}}^{\rm{vir}}
\otimes_{\oO_{M_{\alpha}}}
(\eE_{\bullet})^{\le d}_{\bullet}].
\end{align}
Here $L_{\oO_M}(\eE_{\bullet})$ is the
sheaf of super Lie algebras
in $T_{\oO_M}(\eE_{\bullet})$ generated by
$\eE_{\bullet}$, and
$S_{\oO_M}(-)$ is the super symmetric product
over $\oO_M$.
We refer to Subsection <ref>
for details of the notation of the RHS of (<ref>).
The formula (<ref>)
implies that (<ref>)
is described using the perfect obstruction theory,
without referring to quasi NCDG structures.
Also it is described by
Schur complexes ${\bf S}_{\lambda}(\eE_{\bullet})$
for partitions
For example
in the $d=2$ case, the RHS of (<ref>)
is written as
(cf. Corollary <ref>)
\begin{align*}
\oO_{M_{\alpha}}^{\rm{vir}} \otimes_{\oO_{M_{\alpha}}} \left(1+
{\bf S}_{(1, 1)}(\eE_{\bullet}) + {\bf S}_{(2, 1)}(\eE_{\bullet})
+{\bf S}_{(2, 2)}(\eE_{\bullet}) + {\bf S}_{(1, 1, 1, 1)}(\eE_{\bullet})
\right).
\end{align*}
In particular, the formula (<ref>)
implies that
$(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d}=\oO_{M_{\alpha}}^{\rm{vir}}$ if
the expected dimension of $M_{\alpha}$ is zero.
This implies that,
if $X$ is a Calabi-Yau 3-fold,
the integrations of NC virtual structure sheaves
coincide with the usual (commutative)
DT invariants.
On the other hand, if we consider
moduli spaces of stable sheaves
on algebraic surfaces or Fano 3-folds
so that they have the positive expected dimensions,
the resulting NC virtual structure sheaves are in general
different from the commutative virtual structure sheaves.
In such cases, integrations of their Chern characters may yield
interesting enumerative invariants.
In the next paper <cit.>,
we will pursue another approach
in constructing interesting enumerative
invariants of sheaves involving non-commutative
structures on the moduli spaces of stable sheaves.
We can consider motives of
Hilbert schemes of points on
a quasi NC structure (<ref>),
and construct
certain enumerative invariants
by integrating the Behrend functions on them.
If $X$ is a Calabi-Yau 3-fold,
using wall-crossing argument, we
relate these invariants
with generalized DT invariants
counting semistable sheaves on $X$ <cit.>, <cit.>
whose definition involves motivic Hall algebras.
This would give an intrinsic understanding
of the dimension formula <cit.>
of Donovan-Wemyss's non-commutative widths <cit.>
for floppable rational curves,
whose detail will be included in <cit.>.
§.§ Plan of the paper
In Section <ref>, we introduce the notion of
quasi NCDG structures on commutative dg-schemes, and
use it to define the NC virtual structure sheaves.
In Section <ref>, we
describe the NC virtual structure
sheaves via perfect obstruction theory, and
prove Theorem <ref>.
In Section <ref>,
we construct quasi NCDG structures on
the moduli spaces of representations of a certain quiver.
In Section <ref>, using the result of Section <ref>,
we prove Theorem <ref>.
§.§ Acknowledgement
The author would like to thank
Tomoyuki Abe, Will Donovan, Zheng Hua
and Michael Wemyss
for the discussions related to this paper.
This work is supported by World Premier
International Research Center Initiative
(WPI initiative), MEXT, Japan,
and Grant-in Aid
for Scientific Research grant (No. 26287002)
from the Ministry of Education, Culture,
Sports, Science and Technology, Japan.
§.§ Notation and convention
In this paper, an algebra always means an associative,
not necessary commutative,
The tensor product $\otimes$ is over $\mathbb{C}$ if
there is no subscript.
Also all the varieties or schemes
are defined over $\mathbb{C}$.
§ NC VIRTUAL STRUCTURE SHEAVES
In this section,
we recall
virtual structure sheaves associated to
commutative dg-schemes,
and introduce its NC version.
We prepare the following convention on the
super symmetric product.
For a graded vector space $W_{\bullet}$,
let $S_n$ acts on $W_{\bullet}^{\otimes n}$
in the super sense, i.e.
the action of the permutation $(i, j) \in S_n$
\begin{align*}
x_1 \otimes \cdots \otimes x_i \otimes \cdots \otimes
x_j &\otimes \cdots \otimes x_n \\
(-1)^{\deg x_i \deg x_j}
x_1 \otimes \cdots \otimes x_j \otimes \cdots \otimes
x_i\otimes \cdots \otimes x_n
\end{align*}
for homogeneous elements $x_1, \cdots, x_n \in W_{\bullet}$.
The super symmetric product of
$W_{\bullet}$ is defined by
\begin{align*}
S(W_{\bullet}) \cneq \bigoplus_{n\ge 0}
(W_{\bullet}^{\otimes n})^{S_n}.
\end{align*}
The above super
symmetric product is obviously
for graded vector bundles $\qQ_{\bullet}$ on a scheme $N$,
and we obtain
the sheaf of super commutative graded algebras
$S_{\oO_N}(\qQ_{\bullet})$ on $N$.
Here a graded algebra $A_{\bullet}$ is called
super graded commutative if
$a_e \cdot a_{e'}=(-1)^{e e'} a_{e'} \cdot a_e$
for $a_e \in A_e$, $a_{e'} \in A_{e'}$.
§.§ Commutative dg-schemes
Let $(N, \oO_{N, \bullet})$
be a
smooth commutative dg-scheme, i.e.
$N$ is a smooth scheme and $\oO_{N, \bullet}$ is
a sheaf of super commutative dg-algebras of the form[It may be
more natural to use the upper index $\oO_N^{\bullet}$
to denote the grading
of the sheaf of dg-algebras.
In this paper, we use the lower index
$\oO_{N, \bullet}$ as we will use several
other upper gradings.]
\begin{align*}
\oO_{N, \bullet}=S_{\oO_N}(\qQ_{-1} \oplus \cdots \oplus \qQ_{-k})
\end{align*}
for vector bundles $\qQ_{i}$ on $N$
located in degree $i$.
The zero-th cohomology of $\oO_{N, \bullet}$
is written as $\oO_N/J$ for the ideal
sheaf $J \subset \oO_N$, hence
determines a closed subscheme
$M \subset N$.
We write
\begin{align*}
\tau_0(N, \oO_{N, \bullet}) \cneq M
\end{align*}
and call it the zero-th truncation of $(N, \oO_{N, \bullet})$.
Let $E_{\bullet}$ be a finitely generated dg-$\oO_{N, \bullet}$-module.
We will use two kinds of restrictions of it to $M$
\begin{align}\label{restrict}
E_{\bullet}|_{M} \cneq E_{\bullet} \otimes_{\oO_N} \oO_M, \
\overline{E}_{\bullet}|_{M} \cneq E_{\bullet} \otimes_{\oO_{N, \bullet}} \oO_M. \end{align}
By setting $
{\bf O}_{\bullet} \cneq \oO_{N, \bullet}|_{M}$,
the two restrictions (<ref>)
are related by
\begin{align*}
\overline{E}_{\bullet}|_{M}=E_{\bullet}|_M/({\bf O}_{\le -1} E_{\bullet}|_{M}).
\end{align*}
Let $\Omega_{N, \bullet}$ be the cotangent complex of
$(N, \oO_{N, \bullet})$.
The complex $\overline{\Omega}_{N, \bullet}|_{M}$ is described as
\begin{align*}
\overline{\Omega}_{N, \bullet}|_{M}
=\left(0 \to \qQ_{-k}|_{M} \to
\cdots \to \qQ_{-2}|_{M} \stackrel{d_{-1}}{\to}
\qQ_{-1}|_{M} \stackrel{d_0}{\to} \Omega_N|_{M} \to 0 \right).
\end{align*}
Let $T_{N, \bullet} \cneq
\hH om_{\oO_{N, \bullet}}(\Omega_{N, \bullet}, \oO_{N, \bullet})$
be the tangent complex of $(N, \oO_{N, \bullet})$.
A smooth commutative dg-scheme $(N, \oO_{N, \bullet})$ is
called a $[0, 1]$-manifold if the cohomologies of the complex
$\overline{T}_{N, \bullet}|_{M}$
are concentrated on $[0, 1]$.
Suppose that $(N, \oO_{N, \bullet})$ is a $[0, 1]$-manifold.
Then $\overline{T}_{N, \bullet}|_{M}$
is quasi-isomorphic to the complex
\begin{align}\label{T:qis}
0 \to T_{N}|_{M} \stackrel{(d_0)^{\vee}}{\to} K \to 0
\end{align}
where $K$ is the kernel of
$(d_{-1}|_{M})^{\vee} \colon
(\qQ_{-1}|_{M})^{\vee} \to (\qQ_{-2}|_{M})^{\vee}$,
which is a locally free sheaf on $N$.
We define $\eE_{\bullet}$ to be
the dual of the complex (<ref>)
\begin{align}\label{E:qis}
\eE_{\bullet} \cneq (0 \to
K^{\vee} \stackrel{\overline{d}_0}
\to \Omega_{N}|_{M} \to 0)
\end{align}
where $\Omega_{N}|_{M}$
is located in degree zero
and $\overline{d}_0$
is induced by $d_0$.
We have the quasi-isomorphism
$\overline{\Omega}_{N, \bullet}|_{M}
\stackrel{\sim}{\to} \eE_{\bullet}$, and
by <cit.>,
we have the morphism of complexes
\begin{align}\label{perfect}
\eE_{\bullet} \to (0 \to J/J^2 \to \Omega_{N}|_{M} \to 0)
\end{align}
giving a perfect obstruction theory on $M$
in the sense of Behrend-Fantechi <cit.>.
§.§ Commutative virtual structure sheaves
For a $[0, 1]$-manifold
$(N, \oO_{N, \bullet})$, let us recall the
virtual fundamental class
and its K-theoretic enhancement associated
to the perfect obstruction theory (<ref>).
Let $C_{M/N}$ be the normal cone of $M$ in $N$
defined by
\begin{align*}
C_{M/N} \cneq \Spec_{\oO_M} \bigoplus_{k\ge 0} J^k /J^{k+1}.
\end{align*}
By <cit.>,
we have the closed embedding
$C_{M/N} \subset K$, hence obtain the following diagram
\begin{align*}
\xymatrix{
C_{M/N} \ar@{^{(}->}[rr] \ar[rd] & & K \ar@<0.5ex>[ld] \\
& M \ar@<0.5ex>[ur]^-j. &
\end{align*}
Here $j$ is the zero section.
The virtual fundamental class associated to (<ref>) is defined by
\begin{align}\label{def:Ktheory}
[M]^{\rm{vir}} \cneq j^{!}[C_{M/N}] \in A_{\bullet}(M).
\end{align}
By the above definition, the virtual
fundamental class has a K-theoretic enhancement,
called the virtual
structure sheaf
\begin{align}\label{Kth:vir}
\oO_M^{\rm{vir}} \cneq [\dL j^{\ast} \oO_{C_{M/N}}] \in K_0(M).
\end{align}
Note that the virtual
structure sheaf
recovers the virtual fundamental class
\begin{align*}
\cl(\oO_{M}^{\rm{vir}})=[M]^{\rm{vir}} \in A_{\bullet}(M).
\end{align*}
Here $\cl$ is the cycle map.
On the other hand,
note that each cohomology sheaf
$\hH_i(\oO_{N, \bullet})$ is a coherent $\oO_M$-module,
which vanishes for $i\ll 0$ by <cit.>.
Hence for a finitely generated
dg-$\oO_{N, \bullet}$-module
each cohomology $\hH_i(E_{\bullet})$ is a coherent
$\oO_M$-module, and vanishes for $\lvert i \rvert \gg 0$.
the following definition makes sense:
\begin{align*}
\cneq \sum_{i \in \mathbb{Z}}(-1)^i [\hH_i(E_{\bullet})] \in K_0(M).
\end{align*}
The above K-theory classes from
the dg-schemes
are related to the virtual structure sheaf
as follows:
Let $E_{\bullet}$ be a finitely generated locally free
dg-$\oO_{N, \bullet}$-module.
Then we have the equality in $K_0(M)$:
\begin{align*}
\oO_M^{\rm{vir}} \otimes_{\oO_M} [\overline{E}_{\bullet}|_{M}].
\end{align*}
In particular, we have the identity
$\oO_M^{\rm{vir}}=[\hspace{-0.5mm}[\oO_{N, \bullet}]\hspace{-0.5mm}]$
in $K_0(M)$.
Note that
for a (not necessary differential)
graded $\oO_{N, \bullet}$-module
we can similarly
define the
$\overline{F}_{\bullet}|_{M}=F_{\bullet} \otimes_{\oO_{N, \bullet}} \oO_M$.
\begin{align}\label{K:grade}
\in K_0(\oO_{N, \bullet})
\end{align}
be its class in the K-group of
finitely generated graded $\oO_{N, \bullet}$-modules.
We have the following corollary of
Theorem <ref>:
In the situation of Theorem <ref>,
let $F_{\bullet}$ be a
locally free graded $\oO_{N, \bullet}$-module.
Suppose that
in $K_0(\oO_{N, \bullet})$.
Then we have the identity
\begin{align*}
\oO_M^{\rm{vir}} \otimes_{\oO_M} [\overline{F}_{\bullet}|_{M}].
\end{align*}
The corollary follows from
Theorem <ref>
$[\overline{E}_{\bullet}|_{M}] \in K_0(M)$
is independent of the differential on $E_{\bullet}$.
§.§ Graded NC filtrations
We introduce the non-commutative version of
some notions recalled in the previous subsections.
Let $R$ be an
algebra which is not necessary commutative,
and $W_{\bullet}$ a finite dimensional
graded vector space
with $W_i=0$ for $i\ge 0$.
Below, we call a grading induced
by the grading on $W_{\bullet}$
as $|_{\bullet}$-grading.
We set
the $|_{\bullet}$-graded algebra $\Lambda_{\bullet}$ to be
\begin{align}\label{Lambda:bull}
\Lambda_{\bullet} \cneq R \ast T(W_{\bullet}).
\end{align}
Here $T(W_{\bullet})$ is the tensor algebra
\begin{align*}
T(W_{\bullet}) \cneq
\bigoplus_{n\ge 0} W_{\bullet}^{\otimes n}
\end{align*}
$\ast$ is the free product as $\mathbb{C}$-algebras.
Note that
\begin{align*}
\Lambda_{>0}=0, \
\Lambda_0=\Lambda, \
\Lambda_{-1}=R \otimes W_{-1} \otimes R
\end{align*}
and so on.
We regard $\Lambda_{\bullet}$ as a
$|_{\bullet}$-graded super Lie algebra
by setting
\begin{align*}
[x, y] \cneq xy-(-1)^{ab} yx, \
x \in \Lambda_a, \ y \in \Lambda_b.
\end{align*}
The subspace
$\Lambda_{\bullet, k}^{\rm{Lie}} \subset \Lambda_{\bullet}$
is defined to be spanned by the elements of the form
\begin{align*}
[x_1, [x_2, \cdots, [x_{k-1}, x_k]\cdots ]]
\end{align*}
for $x_i \in \Lambda_{\bullet}$, $1\le i\le k$.
The $|_{\bullet}$-graded NC filtration of $\Lambda_{\bullet}$ is the decreasing
\begin{align}\label{NCfilt}
\Lambda_{\bullet}=F^0\Lambda_{\bullet} \supset F^1 \Lambda_{\bullet}
\supset \cdots
\supset F^{d} \Lambda_{\bullet} \supset \cdots
\end{align}
where $F^d \Lambda_{\bullet}$ is the two-sided $|_{\bullet}$-graded
ideal of $\Lambda_{\bullet}$
defined by
\begin{align*}
F^d \Lambda_{\bullet} \cneq
\sum_{m\ge 0}\sum_{i_1+\cdots+i_m=m+d}\Lambda_{\bullet}
\cdot
\Lambda_{\bullet, i_1}^{\rm{Lie}} \cdot \Lambda_{\bullet} \cdot
\cdots \cdot \Lambda_{\bullet, i_m}^{\rm{Lie}} \cdot \Lambda_{\bullet}.
\end{align*}
Note that
$\Lambda_{\bullet}/F^1 \Lambda_{\bullet}$ is the abelization
$\Lambda^{ab}_{\bullet}$ of $\Lambda_{\bullet}$,
which is a super commutative $|_{\bullet}$-graded algebra written as
\begin{align}\label{abeliz}
\Lambda^{ab}_{\bullet}=R^{ab} \otimes S(W_{\bullet}).
\end{align}
We set $\Lambda^{\le d}_{\bullet} \cneq \Lambda_{\bullet}/F^{d+1}
\Lambda_{\bullet}$,
and $N^{\le d}_{\bullet} \cneq \Lambda^{\le d}_{\bullet}
\otimes_{\Lambda_{\bullet}}N_{\bullet}$
for a
graded left $\Lambda_{\bullet}$-module $N_{\bullet}$.
By the definition of the filtration (<ref>),
the subquotient
\begin{align}\label{grFlam}
\gr_{F}(\Lambda_{\bullet}) \cneq
\bigoplus_{d\ge 0}
F^d \Lambda_{\bullet}/F^{d+1} \Lambda_{\bullet}
\end{align}
is a bi-graded algebra:
it is a direct sum of
\begin{align*}
\gr_F(\Lambda_{\bullet})_{e}^{d}
\cneq \left(F^d \Lambda_{\bullet}/F^{d+1} \Lambda_{\bullet}\right)_{e}
\end{align*}
$e$ is the $|_{\bullet}$-grading, and
we call
the degree $d$ as $|^{\bullet}$-grading.
§.§ Quasi NC structures
In the notation of the previous subsection,
suppose that
$W_{\bullet}=0$ so that
$\Lambda_{\bullet}=R$ holds.
We recall some notions on NC algebras following <cit.>.
An algebra $R$
is called NC nilpotent of degree $d$
(resp. NC nilpotent)
if $F^{d+1}R=0$
(resp. $F^n R=0$ for $n\gg 0$).
(ii) An algebra $R$ is called NC complete
if the following natural map is an isomorphism
\begin{align*}
R \to R_{[\hspace{-0.5mm}[ab]\hspace{-0.5mm}]}
\cneq \lim_{\longleftarrow} R^{\le d}.
\end{align*}
(iii) An NC nilpotent algebra $R$ is
called of finite type if $R^{ab}$ is a
finitely generated $\mathbb{C}$-algebra and
each $\gr_F^d(R)$ is a finitely
generated $R^{ab}$-module.
Let $R$ be an NC complete algebra.
For any multiplicative set
$S \subset R^{ab}$ without zero divisor,
its pull-back
by the natural surjection $R^{\le d} \twoheadrightarrow
determines the multiplicative set
in $R^{\le d}$, which
satisfies the Ore localization condition (cf. <cit.>).
In particular,
one can define the localization
$S^{-1}R^{\le d}$
of $R^{\le d}$ by $S$.
Therefore, similarly to the case of usual affine
schemes, the NC nilpotent algebra
$R^{\le d}$ determines the sheaf of
algebras $\widetilde{R}^{\le d}$
on $\Spec R^{ab}$
(cf. <cit.>).
the topological basis
of $\Spec R^{ab}$
is given by
\begin{align*}
U_f \cneq \{\mathfrak{p} \in \Spec R^{ab} :
f \notin \mathfrak{p}\}
\end{align*}
$\widetilde{R}^{\le d}$ is the sheafication of the
$U_f \mapsto (f)^{-1} R^{\le d}$,
where $(f)$ is the multiplicative
set $\{f^{n} : n\ge 0\}$ in $R^{ab}$.
Similarly, for any left $R^{\le d}$-module
$P$, the sheaf $\widetilde{P}$ is
defined to be the sheafication of the presheaf
\begin{align*}
U_f \mapsto (f)^{-1} R^{\le d} \otimes_{R^{\le d}}
\end{align*}
The ringed space
\begin{align*}
\Spf R \cneq
(\Spec R^{ab}, \widetilde{R}), \
\widetilde{R} \cneq
\lim_{\longleftarrow}\widetilde{R}^{\le d}
\end{align*}
is called
an affine NC scheme, or
an affine NC structure
on the affine scheme $\Spec R^{ab}$.
The sheaf $\widetilde{R}$ is determined by the localization
a multiplicative set $S \subset R^{ab}$,
given by
\begin{align*}
S^{-1} R \cneq \lim_{\longleftarrow}
S^{-1}R^{\le d}.
\end{align*}
A ringed space is called an NC scheme if it is
locally isomorphic to affine NC schemes.
For a scheme $M$, an NC structure
is an NC scheme $(M, \oO_M^{\n})$ with
In <cit.>, a weaker
notion of the NC structures was considered:
Let $M$ be a commutative scheme.
A quasi NC structure on $M$
consists of an affine
open cover $\{V_i\}_{i\in \mathbb{I}}$
of $M$,
affine NC structures
$V_i^{\n}=(V_{i}, \oO_{V_i}^{\rm{nc}})$
on $V_i$
for each $i\in \mathbb{I}$,
and isomorphisms of NC schemes
\begin{align}\notag
\phi_{ij} \colon
(V_{ij}, \oO_{V_j}^{\rm{nc}}|_{V_{ij}})
\stackrel{\cong}{\to}
(V_{ij}, \oO_{V_i}^{\rm{nc}}|_{V_{ij}})
\end{align}
satisfying $\phi_{ij}^{ab}=\id$.
Let $\nN_{d}$ be the category of NC nilpotent algebras
of degree $d$, and $\nN$ the category of NC nilpotent algebras.
An exact sequence
\begin{align}\label{central}
0 \to J \to R_1 \to R_2 \to 0
\end{align}
in $\nN$
is called a central extension if $J^2=0$ and
$J$ lies in the center of $R_1$.
An NC nilpotent algebra $R$
of degree $d$ is called $d$-smooth if
it is of finite type and the functor
\begin{align*}
h_{R} \cneq \Hom(R, -) \colon \nN \to \sS et
\end{align*}
is formally $d$-smooth, i.e.
for any central extension (<ref>) in $\nN_d$,
the map $h_{R}(R_1) \to h_{R}(R_2)$ is surjective.
(ii) An NC complete algebra $R$ is called smooth if
$R^{\le d}$ is $d$-smooth for any $d\ge 0$.
A quasi NC structure in Definition <ref> is called smooth
if each $U_i^{\n}$ is written as
$U_i^{\n}=\Spf R_i$ for
a smooth algebra $R_i$.
If $M$ admits a smooth quasi NC structure, then $M$ must be smooth.
Conversely, any smooth variety admits a smooth quasi
NC structure by <cit.>.
§.§ Quasi NCDG structures
$R$ be an NC complete algebra and
a graded algebra
given by (<ref>).
Suppose that there is a degree one differential
\begin{align*}
Q \colon \Lambda_{\bullet} \to \Lambda_{\bullet +1}
\end{align*}
giving a dg-algebra structure on $\Lambda_{\bullet}$.
By the Leibniz rule,
the differential $Q$ preserves the
filtration (<ref>),
hence we have the induced
$(\Lambda_{\bullet}^{\le d}, Q^{\le d})$.
Note that each $|_{\bullet}$-degree term
of $\Lambda_{\bullet}^{\le d}$ is a left
$R^{\le d}$-module,
and $Q^{\le d}$ is a left
$R^{\le d}$-module homomorphism.
we have the associated
sheaf of dg-algebras
$\widetilde{\Lambda}_{\bullet}^{\le d}$ on
$\Spec R^{ab}$,
which is a complex of
quasi-coherent left $\widetilde{R}^{\le d}$-modules,
and the dg-ringed space
\begin{align*}
\Spf \Lambda_{\bullet}^{\le d} \cneq
(\Spec R^{ab}, \widetilde{\Lambda}_{\bullet}^{\le d}).
\end{align*}
We see that the above dg-ringed space is
also locally written
of the above form.
For any multiplicative set
$S \subset R^{ab}$ without zero divisor, we have
the canonical isomorphism
\begin{align}\label{id:d}
S^{-1}R^{\le d} \otimes_{R^{\le d}} \Lambda_{\bullet}^{\le d}
\stackrel{\cong}{\to}
\left(S^{-1}R \ast T(W_{\bullet}) \right)^{\le d}.
\end{align}
Note that there exists a canonical morphism from the LHS
to the RHS of (<ref>)
by the universality of the localization.
We prove the isomorphism (<ref>) by the induction of $d$.
For $d=0$, the claim
is obvious since both sides coincide with
$S^{-1} R^{ab} \otimes S(W_{\bullet})$.
Suppose that the isomorphism (<ref>) holds for
$d\ge 0$.
Since $S^{-1}R^{\le d+1}$ is a flat
right $R^{\le d+1}$-module, we have the exact sequence
\begin{align}\notag
0 \to S^{-1} R^{ab} \otimes_{R^{ab}}
\gr_F(\Lambda_{\bullet})^d \to
S^{-1} R^{\le d+1} &\otimes_{R^{\le d+1}}
\Lambda_{\bullet}^{\le d+1} \\
\label{exact:lam}
S^{-1}R^{\le d}\otimes_{R^{\le d}}
\Lambda_{\bullet}^{\le d} \to 0.
\end{align}
On the other hand, we have the
exact sequence
\begin{align}\notag
0 \to \gr_F(S^{-1}R \ast T(W_{\bullet}))^{d}
\to& \left( S^{-1}R \ast T(W_{\bullet}) \right)^{\le d+1} \\
\label{exact:lam2}
&\qquad \qquad
\to \left( S^{-1}R \ast T(W_{\bullet}) \right)^{\le d} \to 0.
\end{align}
By the assumption of the induction,
(<ref>), (<ref>), and the five lemma,
it is enough to show the isomorphism
\begin{align*}
S^{-1} R^{ab} \otimes_{R^{ab}}
\gr_F(\Lambda_{\bullet})^d
\stackrel{\cong}{\to}\gr_F(S^{-1}R \ast T(W_{\bullet}))^{d}.
\end{align*}
In Subsection <ref>, we will
see that
the subquotients of the NC
described by
Poisson envelopes.
Using Lemma <ref>
in Subsection <ref>,
it is enough to show the isomorphism
\begin{align*}
S^{-1} R^{ab} \otimes_{R^{ab}}
P(R^{ab} \otimes S(W_{\bullet}))^{d}
\stackrel{\cong}{\to}
P(S^{-1}R^{ab} \otimes S(W_{\bullet}))^{d}.
\end{align*}
The above isomorphism follows since
taking the Poisson envelope commutes with
the localization (cf. <cit.>).
We define the following dg-ringed
\begin{align}\label{affine:ncdg}
\Spf \Lambda_{\bullet} \cneq
(\Spec R^{ab},
\widetilde{\Lambda}_{\bullet}), \
\widetilde{\Lambda}_{\bullet} \cneq
\lim_{\longleftarrow}
\widetilde{\Lambda}_{\bullet}^{\le d}.
\end{align}
The sheaf $\widetilde{\Lambda}_{\bullet}$
is a sheaf of dg-algebras on $\Spec R^{ab}$,
which is a complex of
left $\widetilde{R}$-modules.
Note that
\begin{align}\label{af:com:dg}
\Spec \Lambda_{\bullet}^{ab} \cneq
(\Spec R^{ab},
\end{align}
is an affine
commutative dg-scheme.
the dg-ringed space (<ref>)
as an affine NCDG scheme
an affine
NCDG structure on
the commutative dg-scheme (<ref>).
We call it smooth if the ungraded
algebra $R$ is smooth.
In this case, (<ref>) is a smooth affine
commutative dg-scheme.
The zero-th truncation of (<ref>)
is defined by
\begin{align}\notag
\tau_0(\Spf \Lambda_{\bullet}) \cneq
(\Spec \hH_0(\Lambda_{\bullet}^{ab}),
\lim_{\longleftarrow}\hH_0(\widetilde{\Lambda}^{\le d}_{\bullet})).
\end{align}
Since $\hH_0(\Lambda_{\bullet}^{\le d})=\hH_0(\Lambda_{\bullet})^{\le d}$,
we have
$\tau_0(\Spf \Lambda_{\bullet})=\Spf \hH_0(\Lambda_{\bullet})$,
which is an affine
NC structure on $\Spec \hH_0(\Lambda_{\bullet}^{ab})$.
Let $R=\mathbb{C}[x]$ and
set $W_{\bullet}=W_{-1}=\mathbb{C} \cdot y$.
Then $\Lambda_{\bullet}=\mathbb{C}\langle x, y \rangle$
where $x$ is degree zero and $y$ is degree $-1$.
For $n\ge 1$, let $Q$ be the differential given by
\begin{align}\label{ex:Cxy}
Q \colon \Lambda_{\bullet} \to \Lambda_{\bullet+1}, \
Q(x)=0, \ Q(y)=x^n.
\end{align}
We have the associated affine NCDG scheme
\begin{align}\label{ex:ncdg}
\Spf \Lambda_{\bullet}=
(\Spec \mathbb{C}[x], \widetilde{\mathbb{C}\langle x, y \rangle})
\end{align}
which is an affine NCDG structure on
the commutative dg-scheme
\begin{align}\label{ex:cdg}
\Spec \Lambda_{\bullet}^{ab}=
(\Spec \mathbb{C}[x], \widetilde{\mathbb{C}[x]} \oplus
\widetilde{\mathbb{C}[x]}y).
\end{align}
The zero-th truncation of (<ref>) is
$\Spec \mathbb{C}[x]/(x^n)$.
We define the following NCDG analogue of NC schemes.
A dg-ringed space $(N, \oO_{N, \bullet}^{\n})$
is called an NCDG scheme if it is locally
isomorphic to affine NCDG schemes.
By Lemma <ref>,
for any open subset $U \subset \Spec R^{ab}$,
the restriction $(U, \widetilde{\Lambda}_{\bullet}|_{U})$
is an NCDG scheme.
We can also define the NCDG analogue of Definition <ref>:
Let $(N, \oO_{N, \bullet})$ be a commutative dg-scheme.
A quasi NCDG structure on $(N, \oO_{N, \bullet})$
consists of an affine
open cover $\{U_i\}_{i\in \mathbb{I}}$
of $N$,
affine NCDG structures
$(U_{i}, \oO_{U_i, \bullet}^{\rm{nc}})$
on $(U_i, \oO_{N, \bullet}|_{U_i})$
for each $i\in \mathbb{I}$,
and isomorphisms
of NCDG schemes
\begin{align}\label{phiij}
\phi_{ij, \bullet} \colon
(U_{ij}, \oO_{U_j, \bullet}^{\rm{nc}}|_{U_{ij}})
\stackrel{\cong}{\to}
(U_{ij}, \oO_{U_i, \bullet}^{\rm{nc}}|_{U_{ij}}).
\end{align}
satisfying $(\phi_{ij, \bullet})^{ab}=\id$.
A quasi NCDG structure in Definition <ref>
is called smooth if each $(U_{i}, \oO_{U_i, \bullet}^{\rm{nc}})$
is smooth.
If $(N, \oO_{N, \bullet})$ admits a smooth quasi NCDG structure, then
it must be smooth.
If the isomorphisms (<ref>)
satisfy the cocycle condition,
the sheaves of dg-algebras
$\oO_{U_i, \bullet}^{\n}$ glue to give
the sheaf of dg-algebras $\oO_{N, \bullet}^{\n}$ on $N$.
In this case, a pair $(N, \oO_{N, \bullet}^{\n})$
is an NCDG scheme, and
called a NCDG structure on $(N, \oO_{N, \bullet})$.
Let $M \subset N$ be the closed subscheme
given by the zero-th truncation of $(N, \oO_{N, \bullet})$,
and set $V_i=M \cap U_i$
for the quasi NCDG structure
in Definition <ref>.
By the definition,
we have the induced quasi
NC structure
\begin{align}\label{trunc}
\{V_i^{\n}=\tau_0(U_i, \oO_{U_i, \bullet}^{\n})\}_{i\in \mathbb{I}}, \
\hH_0(\phi_{ij, \bullet}) \colon
V_j^{\n}|_{V_{ij}} \stackrel{\cong}{\to}
\end{align}
on $M$
in the sense of Definition <ref>.
We call the quasi NC structure (<ref>) as
the zero-th truncation of the quasi NCDG structure
in Definition <ref>.
§.§ NC virtual structure sheaves
Let $(N, \oO_{N, \bullet})$ be a smooth commutative
which admits
a smooth quasi NCDG structure in Definition <ref>.
For simplicity, suppose that
the quasi NCDG structure glues to give an NCDG structure
$(N, \oO_{N, \bullet}^{\n})$ on $(N, \oO_{N, \bullet})$.
its zero-th truncation is an NC
structure $M^{\n}=(M, \oO_M^{\n})$ on $M$. As an analogy of
the identity $\oO_M^{\rm{vir}}
=[\hspace{-0.5mm}[\oO_{N, \bullet}]\hspace{-0.5mm}]$
in Theorem <ref>,
one would like to define the `NC virtual structure sheaf'
of $M^{\n}$
to be
\begin{align}\label{ncvir:1}
\oO_{M}^{\rm{ncvir}} =
\sum_{i\in \mathbb{Z}} (-1)^i [\hH_i(\oO_{N, \bullet}^{\n})].
\end{align}
Note that each $\hH_i(\oO_{N, \bullet}^{\n})$
is a quasi coherent left $\oO_M^{\n}$-module.
Hence if they are coherent and vanish for $\lvert i \rvert \gg 0$,
then the sum (<ref>)
makes sense as an element
of $K_0(M^{\n})$, where
$K_0(M^{\n})$ is the Grothendieck group of
the category of coherent left $\oO_M^{\n}$-modules.
However in general, the
sum (<ref>) is an infinite sum
even if
$(N, \oO_{N, \bullet})$ is a $[0, 1]$-manifold.
For example, the cohomologies of the
complex (<ref>)
are not bounded while the commutative dg-scheme (<ref>)
is a
$[0, 1]$-manifold.
Instead of
$(N, \oO_{N, \bullet}^{\n})$, consider
the NCDG scheme $(N, (\oO_{N, \bullet}^{\n})^{\le d})$
for $d \in \mathbb{Z}_{\ge 0}$. It is
regarded as a $d$-smooth dg-resolution of
$d$-th order NC thickening $(M^{\n})^{\le d}=(M, (\oO_M^{\n})^{\le d})$
of $M$.
Moreover we have the following:
If $(N, \oO_{N, \bullet})$ is a
$[0, 1]$-manifold, then
$(\oO_{N, \bullet}^{\n})^{\le d}$ is
quasi-isomorphic to a
bounded complex.
The subquotient
of the NC
filtration of $(\oO_{N, \bullet}^{\n})^{\le d}$
is the direct sum of
$\gr_F(\oO_{N, \bullet}^{\n})^j_{\bullet}$
for $0\le j\le d$.
It is easy to see that $\gr_F(\oO_{N, \bullet}^{\n})^j_{\bullet}$ is a
finitely generated dg $\oO_{N, \bullet}$-module, hence
bounded if $(N, \oO_{N, \bullet})$ is a $[0, 1]$-manifold.
Therefore the lemma holds.
By the above lemma, the sum
\begin{align}\label{ncvir:2}
(\oO_{M}^{\rm{ncvir}})^{\le d} =
\sum_{i\in \mathbb{Z}} (-1)^i [\hH_i((\oO_{N, \bullet}^{\n})^{\le d})]
\end{align}
makes sense in $K_0((M^{\n})^{\le d})$,
which is identified as an element of $K_0(M)$
since any left $(\oO_M^{\n})^{\le d}$-module
has a finite filtration whose subquotients
are $\oO_M$-modules.
We call (<ref>) as
$d$-th NC virtual structure sheaf
of the $d$-th NC thickening $(M^{\n})^{\le d}$
of $M$.
In general,
a quasi NCDG structure in Definition <ref>
may not glue to
give an NCDG structure.
In such a case, the
zero-th truncation (<ref>)
only gives a quasi NC
structure $M^{\n}$ on $M$.
We generalize the
above notion of $d$-th NC virtual structure
sheaves to the quasi NC structure $M^{\n}$.
Even if the isomorphisms (<ref>)
do not satisfy the cocycle condition,
we have the following lemma:
The isomorphisms (<ref>)
induce the isomorphisms
\begin{align*}
\gr_F(\phi_{ij, \bullet})^d_{\bullet} \colon
\gr_F(\oO_{U_j, \bullet}^{\rm{nc}}|_{U_{ij}})^d_{\bullet}
\stackrel{\cong}{\to}
\gr_F(\oO_{U_j, \bullet}^{\rm{nc}}|_{U_{ij}})^d_{\bullet}
\end{align*}
of dg-$\oO_{N, \bullet}|_{U_{ij}}$-modules
satisfying the cocycle condition.
Let $\Lambda_{\bullet}$ be a graded algebra and
$\phi_{\bullet} \colon \Lambda_{\bullet} \to \Lambda_{\bullet}$
an isomorphism
of graded algebras
satisfying $\phi^{ab}_{\bullet}=\id$.
Then the
induced isomorphism
\colon \gr_F(\Lambda_{\bullet}) \to \gr_F(\Lambda_{\bullet})$
is the identity
by <cit.>.
The lemma obviously follows from this fact.
By the above lemma,
the sheaves
$\gr_F(\oO_{U_i, \bullet}^{\n})^d_{\bullet}$ glue to
give the global
dg-$\oO_{N, \bullet}$-module on $N$
\begin{align}\label{gr:global}
\gr_F(\oO_{N, \bullet}^{\n})^d_{\bullet} \in
\rm{dg} \
\oO_{N, \bullet} \mbox{-}\modu.
\end{align}
Since (<ref>) is finitely generated,
we have the following element
\begin{align*}
[\hspace{-0.5mm}[\gr_F(\oO_{N, \bullet}^{\n})^d_{\bullet}]\hspace{-0.5mm}]
\in K_0(M).
\end{align*}
the following definition makes sense:
$(N, \oO_{N, \bullet})$
be a $[0, 1]$-manifold and $M \subset N$ its zero-th truncation.
Suppose that it admits a quasi NCDG structure,
let $M^{\n}$ be its zero-th truncation (<ref>).
The $d$-th NC virtual structure sheaf of $M^{\n}$ is defined
to be
\begin{align}\label{ncvir:3}
(\oO_{M}^{\rm{ncvir}})^{\le d} \cneq
\sum_{j=0}^{d}[\hspace{-0.5mm}[\gr_F(\oO_{N, \bullet}^{\n})^j_{\bullet}]
\hspace{-0.5mm}] \in K_0(M).
\end{align}
If a quasi NCDG structure gives
the NCDG structure, then the class (<ref>)
coincides with (<ref>) by taking the NC filtration of
$(\oO_{N, \bullet}^{\n})^{\le d}$.
By Theorem <ref>,
for $d=0$ we have the identity
$(\oO_{M}^{\rm{ncvir}})^{\le 0}=\oO_M^{\rm{vir}}$,
where $\oO_M^{\rm{vir}}$ is the commutative
virtual structure sheaf given in (<ref>).
In the situation of Example <ref>,
$M=\Spec \mathbb{C}[x]/(x^n)$.
By definition, the
$d$-th NC virtual structure sheaf of
$M$ is
\begin{align}\label{ex:ncvir}
(\oO_M^{\rm{ncvir}})^{\le d}=\sum_{j=0}^{d}
\left[ \hspace{-0.5mm} \left[
\gr_F \left( \mathbb{C}\langle x, y \rangle \right)^j \right] \hspace{-0.5mm}
\right].
\end{align}
By the identification
$K_0(M)=\mathbb{Z}$, we have
$\oO_M^{\rm{vir}}=[\hspace{-0.5mm}[\mathbb{C}[x, y]]\hspace{-0.5mm}]=n$ and
\begin{align*}
(\oO_M^{\rm{ncvir}})^{\le 1}&
=[\hspace{-0.5mm}[\mathbb{C}[x, y]\oplus \mathbb{C}[x, y][x, y]
\oplus \mathbb{C}[x, y][y, y]]\hspace{-0.5mm}] \\
\end{align*}
In general, one can show that
(<ref>) coincides with $n$ for all $d\ge 1$
(cf. Corollary <ref>).
§ DESCRIPTION OF NC VIRTUAL STRUCTURE SHEAVES
In this section, we give
an explicit description of the NC virtual structure sheaves
in terms of the perfect obstruction theory (<ref>),
and prove Theorem <ref>.
§.§ Graded Poisson envelope
Let $\Lambda_{\bullet}$ be a graded algebra (<ref>),
and take the NC filtration (<ref>).
For $x \in (F^d \Lambda_{\bullet})_{e}$,
$x' \in (F^{d'} \Lambda_{\bullet})_{e'}$,
it is easy to see that
\begin{align*}
x \cdot x' \in
(F^{d+d'} \Lambda_{\bullet})_{e+e'}, \
[x, x'] \in (F^{d+d'+1} \Lambda_{\bullet})_{e+e'}.
\end{align*}
Here $e, e'$ are $|_{\bullet}$-gradings.
Therefore the bracket $[-, -]$ induces the pairing
\begin{align*}
\{-, -\} \colon
\gr_F(\Lambda_{\bullet})_{e}^d
\times \gr_F(\Lambda_{\bullet})_{e'}^{d'} \to
\gr_F(\Lambda_{\bullet})_{e+e'}^{d+d'+1}
\end{align*}
which is super anti-symmetric
with respect to the
In general,
we introduce the following definition:
A $\lvert_{\bullet}$-graded Poisson algebra
is a
\begin{align}\label{triple}
(P_{\bullet}, \cdot, \{-, -\})
\end{align}
where $(P_{\bullet}, \cdot)$
is a super commutative graded algebra,
$\{-, -\}$ is a grade preserving,
super anti-symmetric
\begin{align}\label{pairing}
\{-, -\} \colon P_{\bullet} \times P_{\bullet} \to
\end{align}
satisfying the
super Jacobi identity
and $\{x, -\}$ is a super derivation for any $x \in P_{\bullet}$.
(ii) A $|_{\bullet}^{\bullet}$-graded Poisson algebra is a
$\lvert_{\bullet}$-graded Poisson algebra
$(P_{\bullet}, \cdot, \{-, -\})$
endowed with another grading
(called $\lvert^{\bullet}$-grading)
such that
the multiplication $\cdot$
preserves the $\lvert^{\bullet}$-degree, and
the paring (<ref>)
sends $P_{e}^{d} \times P_{e'}^{d'}$ to
A $\lvert_{\bullet}$-graded Poisson algebra
$P_i=0$ for $i\neq 0$
is nothing but a
Poisson algebra.
It is easy to see that the algebra (<ref>)
is a $|_{\bullet}^{\bullet}$-graded Poisson algebra.
Also a $|_{\bullet}^{\bullet}$-graded
Poisson algebra $(P_{\bullet}^{\bullet}, \cdot, \{-, -\})$
is interpreted as a
$|_{\bullet}$-graded Poisson algebra by forgetting
the $\lvert^{\bullet}$-degree.
We consider the functor
\begin{align}\label{Pois:fun}
(\rvert_{\bullet}\mbox{-graded Poisson algebras})
\to (\mbox{super commutative graded algebras})
\end{align}
defined by
forgetting (<ref>), i.e. it
the triple (<ref>)
to the super commutative graded algebra
$(P_{\bullet}, \cdot)$.
The functor (<ref>)
extends the
forgetting functor from the
category of Poisson algebras to
the category of commutative algebras.
It is well-known that the latter functor
has a left adjoint, called Poisson envelope.
We see that the graded version of the
similar construction
gives the left adjoint of (<ref>).
For a graded vector space $W_{\bullet}$,
\begin{align}\label{def:LW}
L(W_{\bullet}) \subset
\end{align}
be the $|_{\bullet}$-graded
super Lie algebra generated by $W_{\bullet}$.
It is a direct sum of $L_{e}^{d}(W_{\bullet})$, where
$e$ is the $|_{\bullet}$-grading, and
$d$ is the grading
determined by
\begin{align}\label{upper:grade}
L^0_{\bullet}(W_{\bullet})=W_{\bullet}, \
L^{d+1}_{\bullet}(W_{\bullet})=[W_{\bullet}, L^d_{\bullet}(W_{\bullet})].
\end{align}
For example, $L^1_{e}(W_{\bullet})$
is spanned by
\begin{align*}
[x_0, x_1]=x_0 \otimes x_1-(-1)^{e_0 e_1} x_1 \otimes x_0, \
x_i \in W_{e_i}, \ e_0+e_1=e.
\end{align*}
The free Poisson algebra generated by $W_{\bullet}$ is defined by
\begin{align}\label{free:Poiss}
\mathrm{Poiss}(W_{\bullet}) \cneq SL(W_{\bullet})
\end{align}
where $S(-)$ is the super symmetric product with
respect to the $|_{\bullet}$-grading.
Note that (<ref>)
is tri-graded:
it is the
direct sum of $S^n L(W_{\bullet})_{e}^{d}$
spanned by
elements of the form
\begin{align*}
\prod_{i=1}^{n}
[x_0^{(i)}, [x_1^{(i)}, \cdots, [x_{d_i-1}^{(i)}, x_{d_i}^{(i)}]\cdots ]]
\end{align*}
for $x_{j}^{(i)} \in W_{e_{ij}}$
with $1\le i\le n$, $0\le j\le d_i$ satisfying
\begin{align*}
\sum_{i=1}^{n}d_i=d, \quad
\sum_{i=1}^{n} \sum_{j=1}^{d_i} e_{ij}=e.
\end{align*}
Here $e$ is $|_{\bullet}$-degree,
$d$ is $|^{\bullet}$-degree, and
$n$ is called $^{\bullet}|$-degree.
By the $|_{\bullet}$-graded Leibniz rule,
the bracket $[-, -]$
on $L(W_{\bullet})$ extends
to the $|_{\bullet}$-graded bracket $\{-, -\}$
Then the algebra (<ref>)
is a $|_{\bullet}^{\bullet}$-graded Poisson algebra
with respect to the gradings $|^{\bullet}$
and $|_{\bullet}$.
Let $A_{\bullet}$ be a
super commutative graded algebra.
By regarding it as a graded vector space,
we obtain the $|_{\bullet}^{\bullet}$-graded Poisson algebra
Let $I_{A_{\bullet}} \subset S(A_{\bullet})$ be the
ideal given by the exact sequence
\begin{align*}
0 \to I_{A_{\bullet}} \to S(A_{\bullet}) \stackrel{\eta}{\to}
A_{\bullet} \to 0.
\end{align*}
Here $\eta$ is given by
the multiplication in $A_{\bullet}$.
The ideal $I_{A_{\bullet}}$ is generated by
elements of the form $a \cdot b -ab$ for
$a, b \in A_{\bullet}$.
We then define the ideal
\begin{align}\label{ideal:IA}
\langle \hspace{-0.5mm} \langle
\rangle \hspace{-0.5mm}
\rangle \subset \mathrm{Poiss}(A_{\bullet})
\end{align}
to be generated by elements of the form
\begin{align*}
\{x_1, \{x_2, \cdots, \{x_{k-1}, y\} \cdots \} \}, \
x_1, \cdots, x_{k-1} \in A_{\bullet}, \ y \in I_{A_{\bullet}}.
\end{align*}
The $|_{\bullet}^{\bullet}$-graded
Poisson envelope of $A_{\bullet}$ is defined by
\begin{align}\label{bi-Poi}
\cneq \mathrm{Poiss}(A_{\bullet})/\langle \hspace{-0.5mm}
\langle
\rangle \hspace{-0.5mm} \rangle.
\end{align}
The ideal (<ref>)
is homogeneous with respect to the
$|_{\bullet}$ and $|^{\bullet}$ grading
(but not for $^{\bullet}|$-grading),
so the algebra (<ref>)
is $|_{\bullet}^{\bullet}$-graded:
it is the direct sum of $P(A_{\bullet})_{e}^d$, where $d$ is
and $e$ is $|_{\bullet}$-degree.
By the definition of (<ref>),
the Poisson structure on $\mathrm{Poiss}(A_{\bullet})$
descends to the $|_{\bullet}^{\bullet}$-graded Poisson structure on
By forgetting the $|^{\bullet}$-grading on
(<ref>), we
obtain the $|_{\bullet}$-graded
Poisson algebra $P(A_{\bullet})_{\bullet}$.
The functor
$A_{\bullet} \mapsto P(A_{\bullet})_{\bullet}$
is the left adjoint of (<ref>).
The proof is straightforward and left to the reader.
Let $W_{\bullet}$ be a finite dimensional graded vector space
\begin{align*}
\end{align*}
Note that $A_{\bullet}$ is a super commutative
graded algebra.
In this case, we have
the canonical
isomorphism of
Poisson algebras
\begin{align}\label{isom:PG}
P(A_{\bullet})_{\bullet} \stackrel{\cong}{\to}
\gr_F
\end{align}
The above isomorphism is proved
in <cit.>
when $W_{\bullet}$ consists of degree zero part,
and the same argument works in the general case.
§.§ Description of graded Poisson envelopes
Let $A_{\bullet}$ be a super commutative graded algebra
given by
\begin{align}\label{Lam:ab}
A_{\bullet} =R \otimes S(W_{\bullet})
\end{align}
a smooth commutative algebra $R$ and
a finite dimensional graded vector space $W_{\bullet}$.
Let $\Omega_{A_{\bullet}}$ be the graded module of
differential forms on $A_{\bullet}$.
Similarly to (<ref>), let
\begin{align}\label{def:LA}
L_{A_{\bullet}}(\Omega_{A_{\bullet}}) \subset
\bigoplus_{n\ge 0}
\overset{n}{\overbrace{\Omega_{A_{\bullet}} \otimes_{A_{\bullet}}
\cdots \otimes_{A_{\bullet}}
\Omega_{A_{\bullet}}}}
\end{align}
be the super $A_{\bullet}$-Lie subalgebra generated by
It has $|_{\bullet}$-grading induced by the grading on $W_{\bullet}$,
and also $|^{\bullet}$-grading
similarly to (<ref>).
Let $L_{A_{\bullet}}^{+}(\Omega_{A_{\bullet}})$
be the positive degree part
of (<ref>)
with respect to the $|^{\bullet}$-grading.
If $R$ is local, we have an isomorphism of $|_{\bullet}^{\bullet}$-graded
Poisson algebras
\begin{align*}
P(A_{\bullet})_{\bullet}^{\bullet} \cong
\end{align*}
The case of $W_{\bullet}=0$ is
proved in <cit.>.
The case of $W_{\bullet} \neq 0$ is similarly
proved without any modification. Indeed
let ${\bf m} \subset R$ be
the maximal ideal of $R$,
and set $V={\bf m}/{\bf m}^2$.
We have the identification
\begin{align}\label{id:omega}
\Omega_{A_{\bullet}}=A_{\bullet} \otimes (V \oplus W_{\bullet}).
\end{align}
By the identification (<ref>),
we have
\begin{align*}
=A_{\bullet} \otimes S L^{+}(V \oplus W_{\bullet}).
\end{align*}
The above identification
gives a substitute of the third line
of the proof of <cit.>, and
the rest of the proof is the same.
Next we consider the case that $R$ is not necessary local.
\begin{align}\label{pi}
\pi \colon S L (A_{\bullet})_{\bullet}^{\bullet} \to
\end{align}
be the natural projection. The map (<ref>)
vanishes on $I_A$, hence it
factors through the map
\begin{align*}
\overline{\pi} \colon A_{\bullet} \otimes
S L^{+}(A_{\bullet})_{\bullet}^{\bullet}
\twoheadrightarrow P(A_{\bullet})_{\bullet}^{\bullet}.
\end{align*}
The above map
both of $|_{\bullet}$ and $|^{\bullet}$ degrees.
For $n\ge 0$, let $I^n$ be the ideal of $P(A_{\bullet})_{\bullet}^{\bullet}$
\begin{align}\label{ideal:I}
I^n \cneq \overline{\pi} \left( A_{\bullet} \otimes \bigoplus_{p\ge n}
S^p L^{+}(A_{\bullet})_{\bullet}^{\bullet} \right).
\end{align}
The ideal $I^n$ is homogeneous in both of
$|_{\bullet}$ and $|^{\bullet}$ degrees.
We define
\begin{align}\label{def:G}
\cneq \bigoplus_{n\ge 0}
\end{align}
Note that (<ref>) is a tri-graded algebra:
it is a direct sum of
\begin{align}\label{Ged}
G^{n} P(A_{\bullet})_{e}^d \cneq
\left( I^n/I^{n+1} \right)_{e}^{d}
\end{align}
where $e$ is $|_{\bullet}$-degree and
$d$ is $|^{\bullet}$-degree.
The degree $n$ is called $^{\bullet}|$-degree.
Note that (<ref>) vanishes for $n>d$ so
the filtration
\begin{align}\label{stabilize}
P(A_{\bullet})_{\bullet}^{d} = (I^0)^{d}_{\bullet} \supset
(I^1)^d_{\bullet} \supset
\cdots \supset (I^n)^d_{\bullet} \supset \cdots
\end{align}
stabilizes for $n>d$.
There is a natural isomorphism of
tri-graded algebras
\begin{align}\label{tri:gra}
\stackrel{\cong}{\to}
\end{align}
If $W_{\bullet}$ consists of degree zero part, then
$|_{\bullet}$-degrees of both
sides of (<ref>)
consists of degree zero, and
the result is proved in <cit.>.
The case of general $|_{\bullet}$-graded $W_{\bullet}$ is
similarly proved without any modification.
§.§ Graded NC filtration via graded Poisson envelope
Let $R$ be a smooth
(not necessary commutative) algebra, and
$W_{\bullet}$ a finite dimensional graded vector space.
the graded algebra
$\Lambda_{\bullet}$ given by
(<ref>), and set
\begin{align*}
A_{\bullet} \cneq (\Lambda_{\bullet})^{ab}.
\end{align*}
By (<ref>),
the algebra $A_{\bullet}$ is a super commutative graded algebra
of the form (<ref>).
We have the following lemma:
We have the canonical
isomorphism of $|_{\bullet}^{\bullet}$-graded Poisson algebras
\begin{align*}
P(A_{\bullet})^{\bullet}_{\bullet} \stackrel{\cong}{\to}
\gr_F(\Lambda_{\bullet})_{\bullet}^{\bullet}.
\end{align*}
If $W_{\bullet}=0$, the result is
the consequence of <cit.>,
and almost the same proof is applied
for $W_{\bullet} \neq 0$.
Note that $\gr_F(\Lambda_{\bullet})^0=A_{\bullet}$.
By the universality of the $|_{\bullet}$-graded Poisson envelope, we
have the canonical morphism of $|_{\bullet}$-graded
Poisson algebras
$\phi \colon P(A_{\bullet})_{\bullet} \to
\gr_F(\Lambda_{\bullet})_{\bullet}$,
which also preserves $|^{\bullet}$-grading.
Since both of
$P(A_{\bullet})^{\bullet}_{\bullet}$ and
are $|_{\bullet}$-graded $A_{\bullet}$-modules,
we can interpret them as $R^{ab}$-modules
by the algebra
\begin{align*}
R^{ab} \to A_{\bullet}, \
\lambda \mapsto \lambda \otimes 1.
\end{align*}
It is enough to show that,
for any closed point $x \in \Spec R^{ab}$,
$\phi$ induces
the isomorphism
\begin{align}\label{induced}
P(A_{\bullet})^{\bullet}_{\bullet} \otimes_{R^{ab}}
R^{ab}/{\bf m}_x \stackrel{\cong}{\to}
\gr_F(\Lambda_{\bullet})^{\bullet}_{\bullet}
\otimes_{R^{ab}}
R^{ab}/{\bf m}_x.
\end{align}
Here ${\bf m}_x \subset R^{ab}$ is the maximal ideal
which defines $x$.
Let $A_{x, \bullet} \cneq
\Lambda_{x}^{ab} \otimes S(W_{\bullet})$.
Then we have
\begin{align*}
P(A_{\bullet})^{\bullet}_{\bullet} \otimes_{R^{ab}}
R^{ab}/{\bf m}_x=
P(A_{x, \bullet})^{\bullet}_{\bullet} \otimes_{R^{ab}}
R^{ab}/{\bf m}_x.
\end{align*}
By Lemma <ref>, the RHS is computed as
\begin{align}\label{isom:P}
P(A_{x, \bullet})^{\bullet}_{\bullet} \otimes_{R^{ab}}
R^{ab}/{\bf m}_x=
S(W_{\bullet})_{\bullet} \otimes
S L^{+}(V \oplus W_{\bullet})_{\bullet}^{\bullet}.
\end{align}
Here $V={\bf m}_x/{\bf m}_x^2$.
Applying the same argument for
$S(V) \otimes S(W_{\bullet})=S(V\oplus W_{\bullet})$,
we obtain
\begin{align}\label{isom:P2}
P(S(V \oplus W_{\bullet}))^{\bullet}_{\bullet}
\otimes_{S(V)} S(V)/{\bf m}_0
=S(W_{\bullet})_{\bullet} \otimes
S L^{+}(V \oplus W_{\bullet})_{\bullet}^{\bullet}.
\end{align}
Here ${\bf m}_0$ is the maximal ideal of $S(V)$
corresponding to the origin.
On the other hand, by <cit.>, we have
the isomorphism for $j>d$
\begin{align*}
\gr_F(\Lambda_{\bullet})^{d}_{\bullet}
\otimes_{R^{ab}}
R^{ab}/{\bf m}_x
\stackrel{\cong}{\to}
\gr_F(\Lambda_{\bullet}/{\bf m}_{x}^j)^{d}_{\bullet}
\otimes_{R^{ab}}
R^{ab}/{\bf m}_x.
\end{align*}
Since we have
\begin{align*}
\Lambda_{\bullet}/{\bf m}_x^j \cong
T(V \oplus W_{\bullet})/{\bf m}_0^j
\end{align*}
we have the identification
\begin{align}\label{isom:g}
\gr_F(\Lambda_{\bullet})_{\bullet}^{\bullet}
\otimes_{R^{ab}}
R^{ab}/{\bf m}_x =
\gr_F(T(V \oplus W_{\bullet}))^{\bullet}_{\bullet}
\otimes_{S(V)}
S(V)/{\bf m}_0.
\end{align}
By (<ref>), (<ref>),
(<ref>) and
the isomorphism (<ref>),
$\phi$ induces the isomorphism (<ref>).
§.§ NC virtual structure sheaves via perfect obstruction theory
We now return to the situation of Definition <ref>.
Similarly to (<ref>), (<ref>), for a
graded vector bundle $\pP_{\bullet} \to M$ on a scheme $M$,
we set
\begin{align}\label{LOE}
L_{\oO_M}(\pP_{\bullet}) \subset T_{\oO_M}(\pP_{\bullet})
\end{align}
to be
the sheaf of super $\oO_M$-Lie
algebras generated by $\pP_{\bullet}$, i.e.
each fiber of $L_{\oO_M}(\pP_{\bullet})$ at $x \in M$
is the super
Lie algebra $L(\pP_{\bullet}|_{x})$.
Note that the grading on $\pP_{\bullet}$ induces the $|_{\bullet}$-grading on
Similarly to (<ref>), we also have the $|^{\bullet}$-grading on
and denote by $L_{\oO_M}^{+}(\pP_{\bullet})$ its positive degree part
with respect to the $|^{\bullet}$-grading.
The following is the main result in this section:
In the situation of Definition <ref>, we have the
following formula in $K_0(M)$:
\begin{align}\label{ncvir:formula}
(\oO_{M}^{\rm{ncvir}})^{\le d}=\oO_M^{\rm{vir}} \otimes_{\oO_M}
[S_{\oO_M}L_{\oO_M}^{+}(\eE_{\bullet})^{\le d}_{\bullet}].
\end{align}
is the two term complex (<ref>), and
$(-)^{\le d}$ is the degree $\le d$ part
with respect to the $|^{\bullet}$-grading.
For a commutative dg-scheme $(N, \oO_{N, \bullet})$,
the construction of the graded Poisson envelope
yields the sheaf of graded $\oO_{N, \bullet}$-module
$P(\oO_{N, \bullet})^{d}_{\bullet}$
for each $d \in \mathbb{Z}_{\ge 0}$.
By Lemma <ref>, we have the
isomorphism of
graded $\oO_{N, \bullet}$-modules
\begin{align}\label{pf:1}
P(\oO_{N, \bullet})^{\le d}_{\bullet}
\stackrel{\cong}{\to}
\gr_F(\oO_{N, \bullet}^{\n})_{\bullet}^{\le d}.
\end{align}
Also the construction of the ideals (<ref>)
the filtration
of graded $\oO_{N, \bullet}$-modules
\begin{align}\notag
P(\oO_{N, \bullet})^{\le d}_{\bullet}
=(\iI^0)_{\bullet}^{\le d} \supset (\iI^1)_{\bullet}^{\le d}
\supset \cdots \supset (\iI^n)^{\le d}_{\bullet} \supset \cdots
\end{align}
which stabilizes due to
the stabilization of (<ref>).
Hence using the notation (<ref>) and (<ref>),
we have the identity
in $K_0(\oO_{N, \bullet})$
\begin{align}\label{pf:2}
[P(\oO_{N, \bullet})^{\le d}_{\bullet}]
=[G^{\bullet}P(\oO_{N, \bullet})^{\le d}_{\bullet}].
\end{align}
Then by Lemma <ref>,
we have the isomorphism
of graded $\oO_{N, \bullet}$-modules
\begin{align}\label{pf:3}
S_{\oO_{N, \bullet}}^{\bullet}L_{\oO_{N, \bullet}}^{+}
(\Omega_{N, \bullet})_{\bullet}^{\le d} \stackrel{\cong}{\to}
G^{\bullet}P(\oO_{N, \bullet})^{\le d}_{\bullet}.
\end{align}
By (<ref>), (<ref>), (<ref>)
and Corollary <ref>,
we obtain the identity in $K_0(M)$:
\begin{align}\notag
(\oO_M^{\rm{ncvir}})^{\le d}=\oO_M^{\rm{vir}}
\otimes_{\oO_M} [S_{\oO_M} L_{\oO_M}^{+}
(\overline{\Omega}_{N, \bullet}|_{M})^{\le d}_{\bullet}].
\end{align}
We are left to show the identity
\begin{align}\label{id:S}
[S_{\oO_M} L_{\oO_M}^{+}
(\overline{\Omega}_{N, \bullet}|_{M})^{\le d}_{\bullet}]
=[S_{\oO_M} L_{\oO_M}^{+}
(\eE_{\bullet})^{\le d}_{\bullet}].
\end{align}
For a partition of $n$
\begin{align}\label{parti}
\lambda=(\lambda_1, \lambda_2, \cdots, \lambda_k), \
\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k
\end{align}
$V_{\lambda}$ be the corresponding
irreducible representation of $S_n$.
Let ${\bf S}_{\lambda}$ be
the Schur functor defined on the category of
complexes of vector bundles on $M$ to itself
(cf. <cit.>):
\begin{align}\label{Schur}
{\bf S}_{\lambda} \colon
\pP_{\bullet} \mapsto
(V_{\lambda} \otimes_{\oO_M} \pP_{\bullet}^{\otimes n})^{S_n}.
\end{align}
Since the functor
$\pP_{\bullet} \mapsto
S_{\oO_M} L_{\oO_M}^{+}(\pP_{\bullet})_{\bullet}^{\le d}$
is a polynomial functor
on the category of graded vector bundles on $M$
in the sense of <cit.>,
it is described as
\begin{align}\label{polynomial}
\bigoplus_{n\ge 0}
\bigoplus_{\lvert \lambda \rvert=n} \uU_{\bullet, \lambda}
\otimes_{\oO_M} {\bf S}_{\lambda}(\pP_{\bullet})
\end{align}
for graded vector bundles
$\uU_{\bullet, \lambda}$
Here for a partition (<ref>),
we set
$\lvert \lambda \rvert=\lambda_1+ \lambda_2+ \cdots +\lambda_k$.
Therefore the identity (<ref>)
follows from
Lemma <ref>
We have used the following lemma, which is
probably well-known, but
include the proof because of the lack of a reference.
Let $\pP_{\bullet}, \qQ_{\bullet}$ be bounded
complexes of vector bundles on a scheme $M$
and $s \colon \pP_{\bullet} \to \qQ_{\bullet}$ a
Then we have the identity
$[{\bf S}_{\lambda}(\pP_{\bullet})]=[{\bf S}_{\lambda}(\qQ_{\bullet})]$
in $K_0(M)$.
By the representation theory of $S_n$ (cf. <cit.>),
we have the decomposition
of complexes
\begin{align*}
(\pP_{\bullet})^{\otimes n} =
\bigoplus_{\lvert \lambda \rvert=n}
{\bf S}_{\lambda}(\pP_{\bullet}) \otimes V_{\lambda}.
\end{align*}
if $\pP_{\bullet}$ is acyclic,
then ${\bf S}_{\lambda}(\pP_{\bullet})$ is
also acyclic.
Let $\Cone(s)_{\bullet}$ be the cone of $s$, which is
acyclic as $s$ is
By the above argument, the complex
${\bf S}_{\lambda}(\Cone(s)_{\bullet})$ is
also acyclic, hence it is zero in $K_0(M)$.
On the other hand, as
$\Cone(s)_{\bullet}$ is
$\qQ_{\bullet} \oplus \pP_{\bullet}[1]$ as a
graded vector bundle,
${\bf S}_{\lambda}(\Cone(s)_{\bullet})$ is isomorphic to
${\bf S}_{\lambda}(\qQ_{\bullet} \oplus \pP_{\bullet}[1])$
as graded vector bundles.
We have the decomposition
as graded vector bundles (cf. <cit.>)
\begin{align}\label{rule:1}
{\bf S}_{\nu}(\qQ_{\bullet} \oplus \pP_{\bullet}[1])
=\bigoplus_{\lvert \lambda \rvert + \lvert \mu \rvert=\lvert \nu \rvert}
\left({\bf S}_{\lambda}(\qQ_{\bullet})
\otimes_{\oO_M} {\bf S}_{\mu}
(\pP_{\bullet}[1])\right)^{\oplus N_{\lambda, \mu}^{\nu}}.
\end{align}
Here $N_{\lambda, \mu}^{\nu} \in \mathbb{Z}_{\ge 0}$
is determined by the Littlewood-Richardson rule.
Hence we obtain the identity in $K_0(M)$:
\begin{align}\label{rule:2}
\sum_{\lvert \lambda \rvert + \lvert \mu \rvert=\lvert \nu \rvert}
N_{\lambda, \mu}^{\nu}[{\bf S}_{\lambda}(\qQ_{\bullet})] \otimes_{\oO_M}
[{\bf S}_{\mu}(\pP_{\bullet}[1])]=0.
\end{align}
Applying the above identity to $\id \colon \pP_{\bullet} \to \pP_{\bullet}$, we obtain
\begin{align}\label{rule:3}
\sum_{\lvert \lambda \rvert + \lvert \mu \rvert=\lvert \nu \rvert}
N_{\lambda, \mu}^{\nu}[{\bf S}_{\lambda}(\pP_{\bullet})] \otimes_{\oO_M}
[{\bf S}_{\mu}(\pP_{\bullet}[1])]=0.
\end{align}
Noting that $N_{\lambda, \emptyset}^{\lambda}=1$,
the identities (<ref>), (<ref>) together with
the induction on $\lvert \lambda \rvert$
shows that
$[{\bf S}_{\lambda}(\pP_{\bullet})]=[{\bf S}_{\lambda}(\qQ_{\bullet})]$.
Using Theorem <ref>, we
can compute NC virtual structure sheaves
in terms of Schur complexes
${\bf S}_{\lambda}(\eE_{\bullet})$,
given by (<ref>).
Note that we have
\begin{align*}
{\bf S}_{\lambda}
(\eE_{\bullet})=\bigwedge^d \eE_{\bullet}, \
\lambda=\overset{d}{\overbrace{(1, 1, \cdots, 1)}}.
\end{align*}
For $d=1$, we have the following formula:
\begin{align*}
(\oO_M^{\rm{ncvir}})^{\le 1}=
\oO_M^{\rm{vir}} \otimes_{\oO_M}
\left(1+ {\bf S}_{(1, 1)}(\eE_{\bullet}) \right).
\end{align*}
(ii) For $d=2$, we have the following formula:
\begin{align*}
(\oO_M^{\rm{ncvir}})^{\le 2}=
\oO_M^{\rm{vir}} \otimes_{\oO_M} \left(1+
{\bf S}_{(1, 1)}(\eE_{\bullet}) + {\bf S}_{(2, 1)}(\eE_{\bullet})
+{\bf S}_{(2, 2)}(\eE_{\bullet}) + {\bf S}_{(1, 1, 1, 1)}(\eE_{\bullet})
\right).
\end{align*}
The formula (<ref>)
\begin{align*}
&(\oO_M^{\rm{ncvir}})^{\le 1}=
\oO_M^{\rm{vir}} \otimes_{\oO_M} \left(1+
L_{\oO_M}(\eE_{\bullet})^1 \right) \\
&(\oO_M^{\rm{ncvir}})^{\le 2}=
\oO_M^{\rm{vir}} \otimes_{\oO_M} \left(1+
+S^2_{\oO_M}L_{\oO_M}(\eE_{\bullet})^1 \right).
\end{align*}
The formula for $d=1$ then follows from
\begin{align}\label{comp:L1}
L_{\oO_M}(\eE_{\bullet})^{1}=[\eE_{\bullet}, \eE_{\bullet}]=\bigwedge^2 \eE_{\bullet}={\bf S}_{(1, 1)}(\eE_{\bullet}).
\end{align}
For $d=2$,
we have the exact sequences of complexes
\begin{align*}
0 \to \bigwedge^3 \eE_{\bullet} \to \left(\bigwedge^2 \eE_{\bullet} \right)
\otimes
\eE_{\bullet} \to L_{\oO_M}(\eE_{\bullet})^2 \to 0 \\
0 \to {\bf S}_{(2, 1)}(\eE_{\bullet}) \to
\left(\bigwedge^2 \eE_{\bullet} \right)\otimes
\eE_{\bullet} \to \bigwedge^3 \eE_{\bullet} \to 0
\end{align*}
showing that
$[L_{\oO_M}(\eE_{\bullet})^2]=[{\bf S}_{(2, 1)}(\eE_{\bullet})]$.
Here the former sequence easily follows from
$L_{\oO_M}(\eE_{\bullet})^2=[\eE_{\bullet}, [\eE_{\bullet}, \eE_{\bullet}]]$
and the latter sequence follows from <cit.>.
Also by <cit.>,
we have the identity
\begin{align*}
[S^2_{\oO_M} {\bf S}_{(1, 1)}(\eE_{\bullet})]
=[{\bf S}_{(2, 2)}(\eE_{\bullet})]+[{\bf S}_{(1, 1, 1, 1)}(\eE_{\bullet})].
\end{align*}
By combining these identities, we obtain the desired formula for $d=2$.
Suppose that
$M^{\rm{vir}}$ has virtual dimension zero.
Then we have the identity
\begin{align}\label{id:vir0}
(\oO_{M}^{\rm{ncvir}})^{\le d}=\oO_M^{\rm{vir}}.
\end{align}
The assumption implies that the complex
given by (<ref>) is of rank zero,
hence any Schur complex ${\bf S}_{\lambda}(\eE_{\bullet})$
is of rank zero.
$\oO_M^{\rm{vir}}$ is
written as $[Q]-[Q']$ for zero dimensional sheaves
$Q, Q'$,
we obtain the desired
identity (<ref>)
by Theorem <ref>.
§.§ NC virtual structure sheaves associated to perfect obstruction theory
The result of Theorem <ref>
indicates that one may define the NC virtual structure sheaf
from the perfect obstruction theory,
without using a quasi NCDG structure.
Let $M$ be a
scheme and
\begin{align}\label{perfect2}
\phi \colon
\eE_{\bullet} \to \tau_{\ge -1}\dL_M
\end{align}
a perfect obstruction theory in the sense of <cit.>, i.e.
$\eE_{\bullet}$ is a two term complex of vector bundles on $M$,
$\tau_{\ge -1}\dL_M$ is the truncated cotangent complex
of $M$,
and $\phi$ is the
morphism in the derived
category such that
$\hH_0(\phi)$ is an isomorphism
and $\hH_{-1}(\phi)$ is surjective.
Similarly to (<ref>), one can define the
virtual structure sheaf $\oO_M^{\rm{vir}} \in K_0(M)$
using data (<ref>) as pointed out in <cit.>.
The result of Theorem <ref> naturally leads to the following
For a perfect obstruction theory (<ref>) on a scheme $M$,
the $d$-th NC virtual structure sheaf is defined by
\begin{align}\label{ncvir:formula22}
(\oO_{M}^{\rm{ncvir}})^{\le d}=\oO_M^{\rm{vir}} \otimes_{\oO_M}
[S_{\oO_M}L_{\oO_M}^{+}(\eE_{\bullet})^{\le d}_{\bullet}].
\end{align}
Suppose that $M$ is non-singular,
hence $\dL_M=\Omega_M$, and
the perfect obstruction theory (<ref>) is given by
the identity
$\eE_{\bullet}=\Omega_M \to \Omega_M$.
Then we have $\oO_M^{\rm{vir}}=\oO_M$ and
\begin{align}\label{smooth:K}
(\oO_{M}^{\rm{ncvir}})^{\le d}=
[S_{\oO_M}L_{\oO_M}^{+}(\Omega_M)^{\le d}].
\end{align}
If $M$ admits a $d$-smooth NC thickening
$(M, \oO_M^{\le d})$, then the RHS of (<ref>)
coincides with the K-theory class of $\oO_M^{\le d}$.
Suppose that
$M^{\rm{vir}}$ has virtual dimension zero, or
equivalently $\eE_{\bullet}$ is rank zero.
Similarly to Corollary <ref>,
we have the identity
\begin{align*}
(\oO_{M}^{\rm{ncvir}})^{\le d}=\oO_M^{\rm{vir}}.
\end{align*}
In particular
if (<ref>)
is a symmetric perfect obstruction theory (cf. <cit.>),
$\eE_{\bullet}$ is rank zero and
$(\oO_{M}^{\rm{ncvir}})^{\le d}$
coincides with the commutative virtual structure sheaf.
The definition of (<ref>)
also makes sense in the equivariant situation, and
gives non-trivial examples of
NC virtual structure sheaves
with virtual dimension zero.
Let $T=(\mathbb{C}^{\ast})^3$
acts on $\mathbb{C}^3$
by weight $(1, 1, 1)$.
By regrading $\mathbb{C}^3$ as
the moduli space of skyscraper sheaves
$\oO_x$ for $x \in \mathbb{C}^3$, we have
the $T$-equivariant
perfect obstruction theory
\begin{align*}
\Omega_{\mathbb{C}^3} \oplus \bigwedge^2 \Omega_{\mathbb{C}^3}[1]
\to \Omega_{\mathbb{C}^3}.
\end{align*}
Let $(t_1, t_2, t_3)$ be the $T$-equivariant parameters.
By localization, we obtain
the identity in $K_{T}(\mathbb{C}^3)$
\begin{align*}
\oO_{\mathbb{C}^3}^{\rm{vir}}=
\frac{(1-t_1^{-1}t_2^{-1})(1-t_1^{-1}t_3^{-1})(1-t_2^{-1}t_3^{-1})}{(1-t_1^{-1})(1-t_2^{-1})(1-t_3^{-1})}.
\end{align*}
By Corollary <ref> (i),
we have the identities in $K_{T}(\mathbb{C}^3)$
\begin{align*}
&(\oO_{\mathbb{C}^3}^{\rm{ncvir}})^{\le 1} \\
\otimes_{\oO_{\mathbb{C}^3}}
\left(1+\bigwedge^2 \Omega_{\mathbb{C}^3} - \Omega_{\mathbb{C}^3}
\otimes_{\oO_{\mathbb{C}^3}} T_{\mathbb{C}^3} +
S_{\oO_{\mathbb{C}^3}}^2(T_{\mathbb{C}^3}) \right) \\
\cdot
\left(-2+t_1^{-1}t_2^{-1}+t_1^{-1}t_3^{-1}+t_2^{-1}t_3^{-1}
-t_1^{-1}t_2 \right. \\
&\left. \hspace{15mm}
-t_2^{-1}t_3-t_1^{-1}t_3 -t_1 t_2^{-1} -t_2 t_3^{-1} -t_1 t_3^{-1}
+t_1^2 +t_2^2 +t_3^2+t_1 t_2+t_1 t_3 +t_2 t_3 \right).
\end{align*}
§ CONSTRUCTIONS OF QUASI NCDG STRUCTURES
In the previous section, we introduced the notion of
NC virtual structure sheaves (cf. Definition <ref>)
of a quasi NC structure,
using the notion of a quasi NCDG structure.
Although NC virtual structure sheaves
turned out
to be described
using the perfect obstruction theory (cf. Theorem <ref>),
still the validity
of Definition <ref>
relies on the existence
of a quasi NCDG structure.
In this section, we
show that the moduli spaces
of graded modules over graded
algebras admit
quasi NCDG structures.
The results in this section will be
in the next section
to show a similar result in a geometric context.
§.§ Graded algebras and quivers
$A$ be a graded algebra
\begin{align}\label{galg}
A=\bigoplus_{i\ge 0}A_i
\end{align}
such that $A_0=\mathbb{C}$ and each $A_i$ is finite dimensional.
We denote
\begin{align*}
\mathfrak{m} \cneq A_{>0} \subset A
\end{align*}
the maximal ideal of $A$.
Let $A \modu_{\gr}$
be the category of finitely generated
graded left $A$-modules.
For $M \in A \modu_{\rm{gr}}$, we denote by $M_i$ the
degree $i$-part of $M$, and write $\lvert a \rvert=i$
for non-zero $a \in M_i$.
For $q>p>0$, we define
\begin{align}\label{def:Apq}
A \modu_{[p, q]} \subset A \modu_{\rm{gr}}
\end{align}
to be
the subcategory of $M \in A \modu_{\rm{gr}}$
with $M_i=0$ for $i\notin [p, q]$.
The category (<ref>) is
also interpreted as
the category of representations of
some quiver, defined as follows:
For $q>p>0$,
the quiver $Q_{[p, q]}$
is defined as follows:
the set of vertices
\begin{align*}
\{p, p+1, \cdots, q\}.
\end{align*}
The number of arrows
in $Q_{[p, q]}$ from $i$ to $j$ is given by
The set of arrows in $Q_{[p, q]}$ is denoted by $Q_1$.
Below we fix bases of $\mathfrak{m}_k$
for each $k\in \mathbb{Z}_{\ge 1}$,
and identify the set of arrows from $i$ to $j$ with
the set of
basis elements of $\mathfrak{m}_{j-i}$.
Let $\mathbb{C}[Q_{[p, q]}]$
be the path algebra of $Q_{[p, q]}$.
The multiplication
\begin{align}
\vartheta \colon
\mathfrak{m}_{j-i} \otimes \mathfrak{m}_{k-j}
\to \mathfrak{m}_{k-i}
\end{align}
in $A$
defines the relation in $Q_{[p, q]}$,
by defining the two sided
\begin{align*}
I \subset \mathbb{C}[Q_{[p, q]}]
\end{align*}
to be generated by elements of the form
\begin{align*}
\vartheta(\alpha \otimes \beta)-\alpha \cdot \beta, \
\alpha \in \mathfrak{m}_{j-i}, \ \beta \in \mathfrak{m}_{k-j}.
\end{align*}
Here we have regarded $\alpha$, $\beta$ as
formal linear combinations of paths from $i$ to
$j$, $j$ to $k$, respectively
and $\alpha \cdot \beta$ is the multiplication in
$\mathbb{C}[Q_{[p, q]}]$.
We define $\mathrm{Rep}(Q_{[p, q]})$
be the category of representations of $Q_{[p, q]}$,
its objects consist of
\begin{align}\label{collect}
(\{W_k\}_{k=p}^{q}, \{\phi_a\}_{a \in Q_1}),
\ \phi_a \colon W_{t(a)} \to W_{h(a)}
\end{align}
where $W_k$ is a finite dimensional vector space
$\phi_a$ is a linear map
for each $a \in Q_1$.
Here $t(a)$ is the tail of $a$, and $h(a)$
is the head of $a$.
For a collection (<ref>),
its dimension vector is defined by
\begin{align*}
\dim W \cneq (\dim W_p, \dim W_{p+1}, \cdots, \dim W_q) \in
\mathbb{Z}_{\ge 0}^{q-p+1}.
\end{align*}
Given a collection (<ref>),
there is the natural map
\begin{align}\label{nmap}
\mathbb{C}[Q_{[p, q]}] \to \End (W_{\bullet}), \
W_{\bullet} \cneq \bigoplus_{k=p}^{q} W_k
\end{align}
sending $a\in Q_1$ to $\phi_a$.
The subcategory
of $(Q_{[p, q]}, I)$-representations
\begin{align*}
\mathrm{Rep}(Q_{[p, q]}, I) \subset \mathrm{Rep}(Q_{[p, q]})
\end{align*}
is defined
to be the category of collections (<ref>)
such that the map (<ref>) is zero on $I$.
By the construction, sending a collection (<ref>)
to $W_{\bullet}$ gives the equivalence
\begin{align}\label{equiv}
\mathrm{Rep}(Q_{[p, q]}, I) \stackrel{\sim}{\to}
A \modu_{[p, q]} .
\end{align}
§.§ Constructions of commutative dg-schemes
We are going construct to quasi NCDG structures on
the moduli spaces of
representations of $Q_{[p, q]}$.
Before that, following <cit.>,
we recall the constructions of
smooth commutative dg-structures on smooth
schemes using the notion of
curved DGLA.
In this subsection, we assume that
$N$ is a commutative smooth scheme.
A bundle of
curved DGLA over $N$
is a graded
vector bundle $\lL_{\bullet}$ on $N$,
endowed with the following data:
\begin{align*}
\mu \in \Gamma(\lL_2), \ \delta \colon \lL_{\bullet} \to \lL_{\bullet}, \
[-, -] \colon \wedge^2 \lL_{\bullet} \to \lL_{\bullet}
\end{align*}
where $\delta$ is an $\oO_N$-module homomorphism
of degree one (called twisted differential),
$[-, -]$ is an $\oO_N$-linear super alternating bracket of degree zero,
which subject to the following axioms:
* $\delta(\mu)=0$ as an element of $\Gamma(\lL_3)$.
* $\delta \circ \delta=[\mu, -]$.
* $\delta$ is a super derivation with respect to the
bracket $[-, -]$.
* The bracket $[-, -]$ satisfies the super Jacobi identity.
Given a curved DGLA $\lL_{\bullet}$ over $N$,
we associate a sheaf of super
commutative dg-algebras
whose underlying $\oO_N$-algebra is
\begin{align*}
\oO_{N, \bullet} \cneq
\end{align*}
By the Leibniz rule, the differential
on $\oO_{N, \bullet}$
is determined by
its restriction to
\begin{align*}
q=q_0+q_1+q_2 \colon \lL_{\bullet}[1]^{\vee} \to
\oO_N \oplus \lL_{\bullet}[1]^{\vee} \oplus
S_{\oO_N}^2 \left(\lL_{\bullet}[1]^{\vee}\right)
\end{align*}
where $q_0$ is given by $\mu$, $q_1$ is given by $\delta$ and $q_2$ is given by $[-, -]$. The
axiom of the curved DGLA shows that $q^2=0$, hence
we obtain the sheaf of super commutative dg-algebras
$(\oO_{N, \bullet}, q)$ on $N$.
We construct a curved DGLA on $N$
using a graded vector bundle
\begin{align*}
\vV_{\bullet} \to N
\end{align*}
together with the graded algebra (<ref>).
Note that $\eE nd_{\oO_N}(\vV_{\bullet})$ is a
graded vector bundle on $N$
degree $i$ piece consists of morphisms
$\vV_{\bullet} \to \vV_{\bullet}$ sending $\vV_j$ to $\vV_{j+i}$.
We define
\begin{align}\notag
\lL_{n} \cneq \Hom_{\rm{gr}}(\mathfrak{m}^{\otimes n},
\eE nd_{\oO_N}(\vV_{\bullet})), \ \lL_{\bullet}
\cneq \bigoplus_{n>0} \lL_n.
\end{align}
Here for graded vector spaces
$W_1, W_2$, we denote by
$\Hom_{\rm{gr}}(W_1, W_2)$ the
space of linear maps $W_1 \to W_2$ preserving
the degrees.
For example,
$\lL_1$ is written as
\begin{align*}
\lL_1=\bigoplus_{i\ge 1, j\in \mathbb{Z}}
\hH om_{\oO_N}(\mathfrak{m}_i \otimes \vV_j , \vV_{j+i}).
\end{align*}
Note that
$\lL_{\bullet}$ is a
graded vector bundle on $N$.
We see that $\lL_{\bullet}$ is a sheaf of
dg-algebras on $N$.
The differential
$d \colon \lL_n \to \lL_{n+1}$ is given by
\begin{align*}
df (a_1 \otimes \cdots \otimes a_{n+1})
=\sum_{i=1}^{n} (-1)^{n-i} f(a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes
\cdots \otimes a_{n+1}).
\end{align*}
The composition
$\circ \colon \lL_m \times \lL_n \to \lL_{m+n}$
is given by
\begin{align*}
f \circ f'(a_1 \otimes \cdots \otimes a_{m+n})
=(-1)^{mn} f(a_1 \otimes \cdots \otimes a_m) \circ
f'(a_{m+1} \otimes \cdots \otimes a_{m+n}).
\end{align*}
It is easy to check that the triple
\begin{align}\label{triple2}
(\lL_{\bullet}, d, \circ)
\end{align}
determines the sheaf of dg-algebras on $N$.
Now suppose that $e$ is a section of $\lL_1 \to N$,
i.e. $e$ is a degree preserving linear map
\begin{align*}
e \colon
\mathfrak{m} \to \End_{\oO_N}(\vV_{\bullet}).
\end{align*}
We will construct a curved DGLA associated to the above data, with the
underlying graded vector bundle is
\begin{align*}
\lL_{\ge 2} \cneq \bigoplus_{n\ge 2} \lL_n.
\end{align*}
The element $\mu \in \Gamma(\lL_2)$ is defined by
\begin{align*}
\mu \cneq de +e \circ e, \ \mu(a_1 \otimes a_2)=e(a_1 a_2) -e(a_1)
\circ e(a_2).
\end{align*}
The bracket is
\begin{align*}
[f, f']= f \circ f' -(-1)^{mn} f' \circ f
\end{align*}
for $f \in \lL_n$, $f' \in \lL_m$. The twisted
differential is defined by
\begin{align*}
\delta \cneq d +[e, -] \colon \lL_{\ge 2} \to \lL_{\ge 2}.
\end{align*}
It is easy to see
the triple $(\lL_{\ge 2}, \delta, [-, -])$ is
a curved DGLA, hence determines the
commutative dg-scheme
\begin{align}\label{construct:dg}
(N, S_{\oO_N}(\lL_{\ge 2}[1]^{\vee}))
=\left(N, S_{\oO_N}\left(\bigoplus_{n\ge 2}
\Hom_{\rm{gr}}\left(\mathfrak{m}^{\otimes n}, \eE nd_{\oO_N}(\vV_{\bullet})
\right)[1]^{\vee} \right) \right).
\end{align}
The zero-th truncation of the above dg-scheme
is the scheme theoretic zero locus of the section
§.§ (DG) moduli spaces of graded modules
Let us fix $q>p>0$ and
non-negative integers
\begin{align*}
\gamma=(\gamma_p, \gamma_{p+1}, \cdots, \gamma_q)
\in \mathbb{Z}^{q-p+1}_{\ge 0}.
\end{align*}
Let $W_{\bullet}$ be a finite dimensional graded vector space
written as
\begin{align*}
W_{\bullet}=\bigoplus_{k=p}^{q} W_k, \quad \dim W_k=\gamma_k.
\end{align*}
Then $W_{\bullet}$ is a graded vector bundle on a point,
hence the
construction of the previous subsection
yields the dg-algebra
\begin{align*}
L \cneq
\bigoplus_{n> 0}L_n, \
L_n \cneq \Hom_{\rm{gr}}(\mathfrak{m}^{\otimes n}, \End(W_{\bullet})).
\end{align*}
We define the following scheme
theoretic Mauer-Cartan locus
\begin{align}\label{def:Mau}
MC(L) \cneq \{ x \in L^1 : dx + x \circ x=0\}.
\end{align}
Note that an element
\begin{align*}
x \in L^1=\Hom_{\rm{gr}}(\mathfrak{m}, \End(W_{\bullet}))
\end{align*}
to a representation of $Q_{[p, q]}$,
and it
is contained in $MC(L)$
if and only if
it corresponds to an object in the
subcategory $\mathrm{Rep}(Q_{[p, q]}, I) \subset \mathrm{Rep}(Q_{[p, q]})$.
We next
consider the stability condition on
$\mathrm{Rep}(Q_{[p, q]})$.
An object $W \in \mathrm{Rep}(Q_{[p, q]})$
is called (semi)stable if for any
object $0\neq W' \subsetneq W$
in $\mathrm{Rep}(Q_{[p, q]})$, we have the inequality
\begin{align*}
\dim W_p \cdot \dim W'_{q}> (\ge ) \dim W_q \cdot \dim W'_{p}.
\end{align*}
We have the Cartesian square
\begin{align}\label{dia:MC}
\xymatrix{
MC(L)^{s} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & (L^1)^{s} \ar@{^{(}->}[d] \\
MC(L) \ar@{^{(}->}[r] & L^1.
\end{align}
Here $MC(L)^s$, $(L^1)^s$ correspond to
stable objects in $\mathrm{Rep}(Q_{[p, q]}, I)$,
$\mathrm{Rep}(Q_{[p, q]})$ respectively.
The vertical inclusions in (<ref>)
are open immersions and
the horizontal inclusions are closed embeddings.
Let $G$ be the group of
degree preserving linear isomorphisms
$W_{\bullet} \to W_{\bullet}$, i.e.
\begin{align*}
G \cneq \prod_{k=p}^{q} \GL(W_k).
\end{align*}
Then $L$ admits the action of $G$
\begin{align*}
(g\cdot f)(a_1 \otimes \cdots \otimes a_n)=
g \circ f(a_1 \otimes \cdots \otimes a_n) \circ g^{-1}
\end{align*}
where $f\in L_n$ and $g \in G$.
The dg-algebra structure on
$L$ is $G$-equivariant, hence
the diagram (<ref>) is also $G$-equivariant.
Since the automorphisms
of stable representations
are $\mathbb{C}^{\ast}$, the
group of the $G$-action on $(L^1)^s$
the diagonal subgroup
$\mathbb{C}^{\ast} \subset G$, hence
the $G$-action on $(L^1)^s$ descends to
the free action of the
quotient group
The free quotients
\begin{align}\label{scheme:s}
M_{\gamma} \cneq MC^{s}(L)/\overline{G},\ N_{\gamma} \cneq
\end{align}
are indeed obtained as GIT quotients (cf. <cit.>), hence
they are quasi projective schemes.
By <cit.>,
the scheme $N_{\gamma}$
is the coarse moduli space of
stable $Q_{[p, q]}$-representations,
and $M_{\gamma}$ is the closed subscheme of $N_{\gamma}$
corresponding to stable $(Q_{[p, q]}, I)$-representations.
Note that $N_{\gamma}$ is non-singular, since
it is a free quotient of a smooth variety.
Now suppose that $\gamma$ is a primitive dimension vector,
\begin{align*}
\mathrm{g.c.d.}\{ \gamma_i : p\le i\le q\} =1.
\end{align*}
Then by <cit.>, $N_{\gamma}$ admits
a universal representation
\begin{align}\label{univ:rep}
\vV=\left( \{\vV_i\}_{i=p}^{q}, \{\phi_a\}_{a \in Q_1} \right), \
\ \phi_a \colon \vV_{t(a)} \to \vV_{h(a)}
\end{align}
i.e. each $\vV_i$ is a vector
bundle on $N_{\gamma}$,
$\phi_a$ is a morphism of vector bundles
such that for any $x \in N_{\gamma}$, the restriction
$\vV|_{x}$ is the representation of $Q_{[p, q]}$ corresponding to $x$.
Note that
\begin{align*}
\vV_{\bullet}=\bigoplus_{i=p}^{q}\vV_i \to N_{\gamma}
\end{align*}
is a graded vector bundle, and the
collection of
morphisms $\phi_a$ corresponds to the
graded preserving linear map
\begin{align}\label{map:e}
e \colon \mathfrak{m} \to \End_{\oO_{N_{\gamma}}}(\vV_{\bullet}).
\end{align}
the construction of the dg-scheme (<ref>)
yields the commutative dg-structure on $N_{\gamma}$
\begin{align}\label{com:dga}
(N_{\gamma}, \oO_{N_{\gamma}, \bullet}), \
\oO_{N_{\gamma}, \bullet}=S_{\oO_{N_{\gamma}}} \left(\bigoplus_{n\ge 2}
\hH om_{\rm{gr}}
(\mathfrak{m}^{\otimes n} \otimes \vV_{\bullet}, \vV_{\bullet})[1]^{\vee}
\right).
\end{align}
By (<ref>) and
the construction of $M_{\gamma}$, the zero-th truncation of
$(N_{\gamma}, \oO_{N_{\gamma}, \bullet})$
coincides with the
closed subscheme $M_{\gamma} \subset N_{\gamma}$.
§.§ Quasi NC structures on $M_{\gamma}$
As before, we assume that $\gamma$ is a primitive dimension
of $Q_{[p, q]}$,
so that there exists a universal
$Q_{[p, q]}$-representation (<ref>).
Let $U \subset N_{\gamma}$ be an affine open subset
such that each $\vV_k|_{U}$ is a trivial
vector bundle
\begin{align*}
\vV_k=\oO_U \otimes W_k, \ p\le k\le q
\end{align*}
where $W_k$ is a vector space with dimension $\gamma_k$.
Since $U$ is a smooth affine scheme,
there is an NC smooth thickening (cf. <cit.>)
\begin{align*}
U^{\rm{nc}}=(U, \oO_{U}^{\rm{nc}})
\end{align*}
on $U$, which is unique up to non-canonical isomorphisms.
We set
\begin{align*}
\vV_{U, k}^{\rm{nc}} \cneq \oO_U^{\rm{nc}} \otimes W_k, \
p\le k\le q
\end{align*}
and regard them as left $\oO_{U}^{\rm{nc}}$-modules.
Since $\oO_{U}^{\rm{nc}} \twoheadrightarrow \oO_U$ is surjective,
each morphism $\phi_a$ lifts to a
left $\oO_{U}^{\rm{nc}}$-module homomorphism
\begin{align}\label{phi:nc}
\phi_a^{\rm{nc}} \colon \vV_{t(a)}^{\rm{nc}}
\to \vV_{h(a)}^{\rm{nc}}, \ a\in Q_1.
\end{align}
Then the data
\begin{align}\label{lift}
\vV^{\n}_U \cneq (\{\vV_{U, k}^{\rm{nc}}\}_{k=p}^{q}, \{\phi_a^{\rm{nc}}
\}_{a\in Q_1} )
\end{align}
is a flat family
of representations of $Q_{[p, q]}$ over the
NC scheme $U^{\rm{nc}}$.
Here we refer to <cit.> for the definition of
flat representations of quivers over NC schemes.
As before, we set
\begin{align*}
\vV_{U, \bullet}^{\n} \cneq \bigoplus_{k=p}^{q}
\vV_{U, k}^{\n}.
\end{align*}
two sided ideal $\jJ_{U, I} \subset \oO_U^{\n}$
is defined
by the image of
\begin{align}\label{ideal:J}
\vV_{U, \bullet}^{\n} \otimes I \otimes (\vV_{U, \bullet}^{\n})^{\vee}
\to \vV_{U, \bullet}^{\n} \otimes \End_{\oO_{U}^{\n}}(\vV_{U, \bullet}^{\n})
\otimes
(\vV_{U, \bullet}^{\n})^{\vee} \to
\oO_{U}^{\n}.
\end{align}
Here the first map of (<ref>)
is induced by
\begin{align*}
I \subset
\mathbb{C}[Q_{[p, q]}] \to \End_{\oO_{U}^{\n}}(\vV_{U, \bullet}^{\n})
\end{align*}
sending $a \in Q_1$ to $\phi_a^{\n}$,
and the second map
of (<ref>)
given by $x \otimes g \otimes f \mapsto f \circ g(x)$.
We set
\begin{align}\label{nc:V}
V\cneq M_{\gamma} \cap U, \
\oO_V^{\n} \cneq \oO_U^{\n}/\overline{\jJ}_{U, I}, \
V^{\n} \cneq (V, \oO_V^{\n}).
\end{align}
Here $\overline{\jJ}_{U, I}$ is the topological
closure of $\jJ_{U, I}$ with respect to the NC filtration
of $\oO_U^{\n}$.
Then $V^{\n}$ is an NC structure on $V$.
Note that giving
a collection of
morphisms (<ref>)
is equivalent to giving an element
\begin{align}\label{hate}
\widehat{e}
\in \Hom_{\rm{gr}}(\mathfrak{m}, \End_{\oO_{U}^{\n}}(\vV_{U, \bullet}^{\n}))
\end{align}
such that $\widehat{e}^{ab}=e|_{U}$, where
$e$ is the universal map (<ref>).
Similarly to the construction (<ref>),
the direct sum
\begin{align*}
\bigoplus_{n> 0}
\Hom_{\rm{gr}}(\mathfrak{m}^{\otimes n}, \End_{\oO_{U}^{\n}}(\vV_{U, \bullet}^{\n}))
\end{align*}
is a dg-algebra.
We set
\begin{align*}
\widehat{\mu} \cneq d\widehat{e} +\widehat{e} \circ \widehat{e} \in
\Hom_{\rm{gr}}(\mathfrak{m}^{\otimes 2}, \End_{\oO_{U}^{\n}}
(\vV_{U, \bullet}^{\n})).
\end{align*}
Then we have the natural morphism
of $\oO_U^{\n}$ bi-module
\begin{align}\label{map:mu}
\left(\vV_{U, \bullet}^{\n} \otimes \mathfrak{m}^{\otimes 2}
\otimes (\vV_{U, \bullet}^{\n})^{\vee} \right)_{0}
\to \oO_U^{\n}.
\end{align}
Here $(-)_{0}$ means the degree zero part,
and the map (<ref>) is
given by
\begin{align*}
a_1 \otimes a_2 \otimes f \mapsto
f \circ \widehat{\mu}(a_1 \otimes a_2)(x).
\end{align*}
From the construction, it is easy
to see that the image of (<ref>)
coincides with $\jJ_{U, I} \subset \oO_U^{\n}$.
We can give a moduli theoretic
interpretation of the NC thickening $V^{\n}$
of $V$.
Let $\nN$ be the category of NC nilpotent algebras and
\begin{align*}
h_{\gamma}|_{V} \colon \nN \to \sS et
\end{align*}
the functor
sending $R$
to the isomorphism classes of triples $(f, \wW, \psi)$:
* $f$ is a morphism of schemes
$f \colon \Spec R^{ab} \to V$.
* $\wW$ is a flat representation of $(Q_{[p, q]}, I)$
over $\Spf R$.
* $\psi$ is an isomorphism
$\psi \colon \wW^{ab} \stackrel{\cong}{\to} f^{\ast}\vV$
as $(Q_{[p, q]}, I)$-representations over $\Spec R^{ab}$.
An isomorphism
$(f, \wW, \psi) \to (f', \wW', \psi')$
exists if $f=f'$, and
there is an isomorphism $\wW \to \wW'$
as representations of $(Q_{[p, q]}, I)$ over $\Spf R$
commuting $\psi$, $\psi'$.
The natural transformation
\begin{align}\label{nat:trans}
h_{V^{\n}} \cneq \Hom(\Spf(-), V^{\n}) \to h_{\gamma}|_{V}
\end{align}
sending $g \colon \Spf R \to V^{\rm{nc}}$ to
$(g^{ab}, g^{\ast}\vV_U^{\n}, \id)$
is an NC hull of $h_{\gamma}|_{V}$, i.e.
(<ref>) is an isomorphism
on the category of commutative algebras,
for any
central extension (<ref>)
in $\nN$,
we have the surjection:
\begin{align*}
h_{V^{\n}}(R_1) \twoheadrightarrow h_{\gamma}|_{V}(R_1)
\times_{h_{\gamma}|_{V}(R_2)}
\end{align*}
Let $\{U_i\}_{i \in \mathbb{I}}$ be an affine open
cover of $N_{\gamma}$ such that
each $\vV_k|_{U_i}$ is trivial, and
set $V_i \cneq M_{\gamma} \cap U_i$.
Applying the construction (<ref>), we obtain
affine NC structures on each $V_i$
\begin{align*}
V_i^{\n}=(V_i, \oO_{V_i}^{\n}), \ i \in \mathbb{I}.
\end{align*}
Using Proposition <ref>, we proved the following in <cit.>:
There exist isomorphisms
\begin{align*}
\phi_{ij} \colon
V_j^{\n}|_{V_{ij}} \stackrel{\cong}{\to}
V_i^{\n}|_{V_{ij}}, \ g_{ij} \colon
\phi_{ij}^{\ast}\vV_{U_i}^{\n}|_{V_{ij}} \stackrel{\cong}{\to}
\vV_{U_j}^{\n}|_{V_{ij}}
\end{align*}
where $\phi_{ij}$ are isomorphisms of NC schemes
giving a quasi NC structure
on $M_{\gamma}$,
and $g_{ij}$ are isomorphisms
of representations of $(Q_{[p, q]}, I)$
over $V_j^{\n}|_{V_{ij}}$.
The constructions of this subsection and
the previous subsection are summarized by the following diagram:
\begin{align}\notag
\xymatrix{
\fbox{\mbox{Quasi NC structure} $\{V_i^{\n}\}_{i\in \mathbb{I}}$} \ar[rr]^-{\rm{abelization}} & &
\ovalbox{Classical moduli space $M_{\gamma}$} \\
\doublebox{? Quasi NCDG structure ?} \ar[u]^{\rm{truncation}}
\ar[rr]^-{\rm{abelization}} & &
\ovalbox{\mbox{DG moduli space} $(N_{\gamma}, \oO_{N_{\gamma}, \bullet})$}
\ar[u]^{\rm{truncation}}
\end{align}
Below, we
are going to construct a quasi NCDG structure
which fits into the above diagram.
§.§ Constructions of non-commutative dg-algebras
Let $A$ be
a graded algebra (<ref>),
and $R$
an another associative (not necessary commutative)
Let $P$ be a graded free right $R$-module, and
set $P^{\vee} \cneq \Hom_{R}(P, R)$
is a graded free left $R$-module.
We set
\begin{align*}
\mathfrak{\mathfrak{P}} \cneq \bigoplus_{n\ge 2} \left(P^{\vee} \otimes
\mathfrak{m}^{\otimes n}
\otimes P\right)_{0}
\end{align*}
which is a free graded $R$ bi-module.
the grading of $\mathfrak{P}$ on
$(P^{\vee} \otimes \mathfrak{m}^{\otimes n}
\otimes P)_{0}$
is set to be $1-n$.
We define the graded algebra $\mathfrak{A}$ to be
the tensor algebra of $\mathfrak{P}$ over $R$
\begin{align*}
\mathfrak{A} \cneq
\bigoplus_{m\ge 0}
\mathfrak{P}^{\otimes_{R} m}, \
\mathfrak{P}^{\otimes_{R} m} \cneq
\overset{m}{\overbrace{\mathfrak{P} \otimes_{R} \mathfrak{P} \otimes_{R}
\cdots \otimes_{R} \mathfrak{P}}}.
\end{align*}
The grading on $\mathfrak{A}$ is induced by that of $\mathfrak{P}$,
the degree zero part of $\mathfrak{A}$ is
$\mathfrak{P}^{\otimes_{R} 0} \cneq R$.
The algebra structure on $\mathfrak{A}$ is
given by
\begin{align*}
(b_1 \otimes \cdots \otimes b_m)
\cdot (b_{m+1} \otimes \cdots \otimes b_n)
=b_1 \otimes \cdots \otimes
b_{m} \otimes b_{m+1} \otimes \cdots \otimes b_n.
\end{align*}
\begin{align}\label{egrade}
\widehat{e} \colon {\mathfrak{m}} \to \End_{R}(P)
\end{align}
be a grade preserving linear map.
For $a \in \mathfrak{m}$,
$x \in P$ and
$f \in P^{\vee}$, we set
\begin{align}\label{convent}
ax \cneq \widehat{e}(a)(x) \in P, \
fa \cneq f \circ \widehat{e}(a) \in P^{\vee}.
\end{align}
We also set the linear map
\begin{align*}
\widehat{\mu} \colon \mathfrak{m}^{\otimes 2} \to \End_{R}(P)
\end{align*}
as follows:
\begin{align*}
\widehat{\mu}(a_1 \otimes a_2)=\widehat{e}(a_1 a_2)-\widehat{e}(a_1)
\circ \widehat{e}(a_2).
\end{align*}
We define the degree one ${R}$ bi-module map
\begin{align}\label{map:Q}
Q=Q_0+Q_1+Q_2 \colon
\mathfrak{P} \to {R} \oplus \mathfrak{P} \oplus (\mathfrak{P}\otimes_{R} \mathfrak{P})
\end{align}
in the following way.
The map $Q_0$ is defined by
\begin{align}
\label{def:q0}
Q_0 \colon \left(P^{\vee} \otimes \mathfrak{m}^{\otimes 2} \otimes P \right)_0
&\to {R} \\
\notag
f \otimes a_1 \otimes a_2 \otimes x &\mapsto f \circ
\widehat{\mu}(a_1, a_2)(x).
\end{align}
The map $Q_1$ is defined by
\begin{align*}
Q_1 \colon \left(P^{\vee} \otimes \mathfrak{m}^{\otimes n} \otimes P\right)_{0}
& \to
\left(P^{\vee} \otimes \mathfrak{m}^{\otimes n-1} \otimes P\right)_{0} \\
f \otimes a_1 \otimes \cdots \otimes a_n \otimes x
(-1)^{n+1}fa_1 \otimes a_2 \otimes \cdots \otimes a_{n} \otimes x \\
&\quad +\sum_{j=1}^{n-1} (-1)^{n+1-j} f \otimes a_1
\otimes \cdots \otimes a_j a_{j+1}
\otimes \cdots \otimes a_{n} \otimes x \\
&\hspace{40mm} -f \otimes a_1 \otimes \cdots \otimes a_{n-1} \otimes a_n x.
\end{align*}
Here we have used the convention in (<ref>).
Finally the map $Q_2$ is defined by
\begin{align*}
Q_2 \colon \left(P^{\vee} \otimes
\mathfrak{m}^{\otimes n} \otimes P\right)_{0} &\to
\bigoplus_{k=2}^{n-2}
\left(P^{\vee} \otimes \mathfrak{m}^{\otimes k} \otimes P \right)_0
\otimes_{R} \left( P^{\vee}
\otimes \mathfrak{m}^{\otimes n-k} \otimes P \right)_0
\\
f \otimes a_1 \otimes \cdots \otimes a_n \otimes x
&\mapsto \sum_{k=2}^{n-2} (-1)^{n(k-2)+1} f
\otimes a_1 \otimes \cdots \otimes a_k \otimes \widehat{\id}_P \\
&\hspace{40mm} \otimes a_{k+1} \otimes \cdots \otimes a_{n} \otimes x.
\end{align*}
is defined as follows:
we decompose $\id_P \in \Hom_{R}(P, P)=P \otimes_{R} P^{\vee}$
\begin{align*}
\id_P=\sum_{i} u_i \otimes _{R} v_i
\end{align*}
homogeneous elements $u_i \in P$, $v_i \in P^{\vee}$ with
$\lvert u_i \rvert + \lvert v_i \rvert =0$, and
\begin{align*}
\widehat{\id}_P \cneq
\sum_{\lvert f \rvert + \lvert a_1 \rvert+\cdots +
\lvert a_k \rvert+ \lvert u_i \rvert=0}
u_i \otimes_{R} v_i.
\end{align*}
By the Leibniz rule, the map (<ref>) extends to the
degree one ${R}$ bi-module map
\begin{align}\label{map:QB}
Q \colon \mathfrak{A} \to \mathfrak{A}.
\end{align}
We have the following proposition:
The map $Q$ in (<ref>)
satisfies $Q^2=0$.
Hence $(\mathfrak{A}, Q)$ is a non-commutative
differential graded algebra.
is straightforward to check $Q^2=0$,
and we leave the details to the readers.
The first few terms of the
complex $(\mathfrak{A}, Q)$ is
\begin{align*}
\cdots \to (P^{\vee} \otimes
\mathfrak{m}^{\otimes 3} \otimes P)_{0}
&\oplus \left(P^{\vee} \otimes
\mathfrak{m}^{\otimes 2}
\otimes P \right)^{\otimes_{R} 2}_0 \\
&\to (P^{\vee} \otimes
\mathfrak{m}^{\otimes 2} \otimes P)_{0}
\stackrel{Q_0}{\to} {R} \to 0.
\end{align*}
In particular, we have
\begin{align}\label{isom:h0}
\hH_0(\mathfrak{A}, Q)={R}/J
\end{align}
where $J$ is the two sided ideal
given by the image of $Q_0$ in (<ref>).
§.§ Abelization of $\mathfrak{A}$
We describe the abelization of the
non-commutative dg-algebra $\mathfrak{A}$.
We set
\begin{align*}
\mathfrak{P}^{ab} &\cneq \bigoplus_{n\ge 2} \left(\mathfrak{m}^{\otimes n} \otimes
\End_{{R}^{ab}}(P^{ab}) \right)_0 \\
& =\bigoplus_{n\ge 2}
\Hom_{\rm{gr}}(\mathfrak{m}^{\otimes n}, \End_{{R}^{ab}}(P^{ab}))^{\vee}.
\end{align*}
which is a graded free ${R}^{ab}$-module.
The grading on $(\mathfrak{m}^{\otimes n} \otimes \End_{{R}^{ab}}(P^{ab}))_0$
is $1-n$, and $\ast^{\vee}$ is the dual of $\ast$ over ${R}^{ab}$.
As a graded algebra, we have
\begin{align}\label{Bab}
\mathfrak{A}^{ab}=S_{{R}^{ab}}(\mathfrak{P}^{ab}).
\end{align}
We write $P$ as
$P=W \otimes{R}$
for a graded vector space $W$, and set
\begin{align*}
\overline{W} \cneq
\bigoplus_{n\ge 2} \left(W^{\vee} \otimes \mathfrak{m}^{\otimes n}
\otimes W\right)_0.
\end{align*}
Then we have
$\mathfrak{P}={R} \otimes \overline{W} \otimes {R}$, and
\begin{align*}
\mathfrak{A}={R} \ast T(\overline{W}).
\end{align*}
On the other hand, we have
$\mathfrak{P}^{ab}={R}^{ab} \otimes \overline{W}$, hence
\begin{align*}
S_{{R}^{ab}}(\mathfrak{P}^{ab})={R}^{ab} \otimes
\end{align*}
Therefore we obtain (<ref>).
By the Leibniz rule, the derivation (<ref>)
induces a degree one ${R}^{ab}$-linear derivation
\begin{align*}
q \cneq Q^{ab} \colon \mathfrak{A}^{ab} \to \mathfrak{A}^{ab}.
\end{align*}
From the description of $Q$,
it is easy to describe $q$
under the identity (<ref>).
By Lemma <ref>, the
map $q$ is determined by
its restriction to $\mathfrak{P}^{ab}$
\begin{align*}
q_0+q_1+q_2 \colon
\mathfrak{P}^{ab} \to {R}^{ab} \oplus \mathfrak{P}^{ab} \oplus S^2_{{R}^{ab}}(\mathfrak{P}^{ab}).
\end{align*}
Let $e$, $\mu$ be the compositions of $\widehat{e}$, $\widehat{\mu}$,
the natural map $\End_{R}(P) \to \End_{{R}^{ab}}(P^{ab})$:
\begin{align}\label{data:eu}
e \colon \mathfrak{m} \to \End_{{R}^{ab}}(P^{ab}), \
\mu \colon \mathfrak{m}^{\otimes 2} \to \End_{{R}^{ab}}(P^{ab}).
\end{align}
The map $q_0$ is described as
\begin{align*}
q_0 \colon \left(\mathfrak{m}^{\otimes 2} \otimes \End_{{R}^{ab}}(P^{ab})\right)_0
&\to {R}^{ab} \\
a_1 \otimes a_2 \otimes g &\mapsto
\tr(g \circ \mu(a_1, a_2)).
\end{align*}
The map $q_1$ is described as
\begin{align*}
q_1 \colon \left(\mathfrak{m}^{\otimes n} \otimes \End_{{R}^{ab}}(P^{ab})
\right)_{0}
&\to \left(\mathfrak{m}^{\otimes n-1} \otimes \End_{{R}^{ab}}(P^{ab})
\right)_{0} \\
a_1 \otimes \cdots \otimes a_n \otimes g &\mapsto
(-1)^{n+1}a_2 \otimes \cdots \otimes a_n \otimes (g \circ e(a_1)) \\
+\sum_{j=1}^{n-1} (-1)^{n+1-j} a_1 \otimes \cdots \otimes a_j a_{j+1} \otimes
\cdots \otimes a_n \otimes g \\
-a_1 \otimes \cdots \otimes a_{n-1} \otimes (e(a_n) \circ g).
\end{align*}
The map $q_2$ is described as
\begin{align*}
&q_2 \colon
\left(\mathfrak{m}^{\otimes n} \otimes \End_{{R}^{ab}}(P^{ab})\right)_0 \\
&\hspace{20mm} \to
\bigoplus_{k=2}^{n-2}
\left(\mathfrak{m}^{\otimes k} \otimes
\End_{{R}^{ab}}(P^{ab}) \right)_0 \otimes_{{R}^{ab}}
\left( \End_{{R}^{ab}}(P^{ab}) \otimes
\mathfrak{m}^{\otimes n-k} \right)_0 \\
&a_1 \otimes \cdots \otimes a_n \otimes g \\
& \hspace{20mm} \mapsto
\sum_{k=2}^{n-2}
a_1 \otimes \cdots \otimes a_k \otimes
\widehat{\circ}^{\vee} g \otimes
a_{k+1} \otimes \cdots \otimes a_n.
\end{align*}
$\circ^{\vee}$ is the dual of
the composition map
\begin{align*}
\circ^{\vee} \colon
\End_{{R}^{ab}}(P^{ab}) \to \End_{{R}^{ab}}(P^{ab}) \otimes_{{R}^{ab}}
\End_{{R}^{ab}}(P^{ab})
\end{align*}
and writing $\circ^{\vee} g$ as
the sum of $u_i \otimes_{{R}^{ab}} v_i$
for homogeneous elements
$u_i, v_i \in \End_{{R}^{ab}}(P^{ab})$,
we set
\begin{align*}
\widehat{\circ}^{\vee} g
=\sum_{\lvert a_1 \rvert + \cdots + \lvert a_k \rvert + \lvert u_i \rvert=0}
u_i \otimes_{{R}^{ab}} v_i.
\end{align*}
On the other hand, note that
\begin{align*}
\widetilde{P}^{ab} \to \Spec {R}^{ab}
\end{align*}
is a graded vector bundle
on $\Spec R^{ab}$.
The data of $e$ in (<ref>)
together with
the construction of (<ref>)
yield the affine commutative dg-scheme
\begin{align}\label{dg:affine}
\left(\Spec R^{ab}, S_{\widetilde{R}^{ab}}\left( \bigoplus_{n\ge 2}
\Hom_{\rm{gr}}
(\mathfrak{m}^{\otimes n}, \eE nd_{\widetilde{R}^{ab}}
(\widetilde{P}^{ab}))[1]^{\vee} \right) \right).
\end{align}
By Lemma <ref>
together with the above description of $q=Q^{ab}$,
the global section
of the dg-structure sheaf of (<ref>)
coincides with
$\mathfrak{A}^{ab}$ as a dg-algebra.
§.§ Quasi NCDG structures on $N_{\gamma}$
Now we return to the situation of Subsection <ref>.
As in Subsection <ref>, we
take an
affine open subset
$U \subset N_{\gamma}$
such that each $\vV_k|_{U}$
is trivial $\vV_k|_{U}=\oO_{U} \otimes W_k$.
We take an NC smooth thickening $U^{\n}$ of $U$,
and a lift
$\vV_{U, \bullet}^{\n}$ of
to a flat representation of $Q_{[p, q]}$
over $U^{\n}$, as in (<ref>).
We apply the construction
in Subsection <ref>
by setting
\begin{align*}
R=\Gamma(\oO_U^{\n}), \
P=\Gamma(\vV_{U, \bullet}^{\n})^{\vee}
\end{align*}
where $\ast^{\vee}$ is the dual of $\ast$
over $R$.
Note that $P$ is a graded free right $R$-module.
Using (<ref>) instead of (<ref>),
the construction in Proposition <ref>
yields the non-commutative dg-algebra
structure on
\begin{align}\label{dga:lambda}
\Lambda_{U, \bullet}^{\n} &\cneq
\bigoplus_{m\ge 0}
\left(\bigoplus_{n\ge 2}
\Gamma(\vV_{U, \bullet}^{\n}) \otimes
\mathfrak{m}^{\otimes n}
\otimes\Gamma(\vV_{U, \bullet}^{\n})^{\vee}
\right)^{\otimes_{\oO_U^{\n}}m}_0.
\end{align}
By the proof of Lemma <ref>,
we have
\begin{align}\label{U:ncdg}
\Lambda_{U, \bullet}^{\n}=
\Gamma(\oO_U^{\n}) \ast T(\overline{W})
\end{align}
$\overline{W}$ is the finite dimensional graded
vector space given by
\begin{align*}
\overline{W}=\bigoplus_{n\ge 2}
\left(\bigoplus_{p\le j, k \le q}W_j \otimes \mathfrak{m}^{\otimes n}
\otimes W_k^{\vee} \right)_0.
\end{align*}
Here $W_j$ is a vector space with dimension $\gamma_j$,
located in degree $j$.
We define the following
affine NCDG scheme
\begin{align}\label{SpfRUn}
(U, \oO_{U, \bullet}^{\n}) \cneq \Spf \Lambda_{U, \bullet}^{\n}.
\end{align}
Note that (<ref>) is smooth
as $U^{\n}$ is a smooth NC thickening of $U$.
By the identity (<ref>), we have
\begin{align}\label{id:V}
\tau_0(U, \oO_{U, \bullet}^{\n})=
(V, \oO_V^{\n})
\end{align}
where $\oO_V^{\n}$
is given in (<ref>),
as it is given by the NC completion
of the cokernel of (<ref>).
Also the argument in the previous subsection
shows that
\begin{align}\label{Uab}
(\oO_{U, \bullet}^{\n})^{ab} =
\oO_{N_{\gamma}, \bullet}|_{U}
\end{align}
where $\oO_{N_{\gamma}, \bullet}$ is the
sheaf of commutative dg-algebras on $N_{\gamma}$
given in (<ref>).
Hence (<ref>) is
an affine NCDG structure on $(U, \oO_{N_{\gamma}, \bullet}|_{U})$.
Let $\{U_i\}_{i\in \mathbb{I}}$ be an affine open
cover of $N_{\gamma}$,
such that each $\vV_k|_{U_i}$ is trivial.
Applying the above construction, we obtain
affine NCDG schemes
\begin{align*}
(U_i, \oO_{U_i, \bullet}^{\n}), \ i \in \mathbb{I}.
\end{align*}
On the other hand, we have isomorphisms
of NC schemes and $Q_{[p, q]}$-representations
over $U_j^{\n}|_{U_{ij}}$
(cf. <cit.>)
\begin{align}\label{isom:rep}
\phi_{ij} \colon U_j^{\n}|_{U_{ij}} \stackrel{\cong}{\to}
U_i^{\n}|_{U_{ij}}, \
g_{ij} \colon \phi_{ij}^{\ast}\vV_{U_i}^{\n}|_{U_{ij}} \stackrel{\cong}{\to}
\vV_{U_j}^{\n}|_{U_{ij}}
\end{align}
such that $\phi_{ij}^{ab}=\id$
is the
gluing isomorphism
of the universal object $\vV$
in (<ref>).
Since the dg-algebra (<ref>)
is determined by the
algebra $\Gamma(\oO_U^{\n})$ together
with the $Q_{[p, q]}$-representation
$\vV_{U}^{\n}$ over $U^{\n}$,
the isomorphisms (<ref>)
induce the isomorphisms
of NCDG schemes
\begin{align*}
\phi_{ij, \bullet} \colon
(U_{ij}, \oO_{U_j, \bullet}^{\n}|_{U_{ij}}) \stackrel{\cong}{\to}
(U_{ij}, \oO_{U_i, \bullet}^{\n}|_{U_{ij}})
\end{align*}
giving a quasi NCDG structure on $(N_{\gamma}, \oO_{N_{\gamma}, \bullet})$.
Also under (<ref>),
the isomorphisms $\hH_0(\phi_{ij, \bullet})$
give a quasi NC structure on $M_{\gamma}$
considered in Theorem <ref>.
As a summary, we have obtained the following:
There exists a smooth quasi NCDG
structure on the smooth
commutative dg-moduli space
$(N_{\gamma}, \oO_{N_{\gamma}, \bullet})$
gives a quasi NC structure on $M_{\gamma}$
in Theorem <ref>.
§ QUASI NCDG STRUCTURES ON THE MODULI SPACES OF STABLE SHEAVES
In this section, we
show that quasi NC structures on the moduli spaces of
stable sheaves
on projective schemes constructed in <cit.>
obtained as the zero-th truncations of
smooth quasi NCDG structures on
smooth commutative dg-moduli spaces of stable sheaves.
Throughout this section,
we assume that
$(X, \oO_X(1))$ is a connected polarized
projective scheme over $\mathbb{C}$.
§.§ Moduli spaces of stable sheaves
For $F \in \Coh(X)$,
let $\alpha(F, t)$ be its Hilbert polynomial
\begin{align*}
\alpha(F, t) \cneq \chi(F \otimes \oO_X(t))
\end{align*}
and $\overline{\alpha}(F, t) \cneq \alpha(F, t)/c$
its reduced Hilbert polynomial, where
$c$ is the leading coefficient of $\alpha(F, t)$.
Recall the (semi)stability on $X$:
coherent sheaf $F$ on $X$
is called (semi)stable if
it is a pure sheaf,
and for any
subsheaf $0 \subsetneq F' \subsetneq F$, we have
\begin{align}\label{def:stab}
\overline{\alpha}(F', k) <(\le) \overline{\alpha}(F, k), \
k\gg 0.
\end{align}
Let us take a polynomial $\alpha \in \mathbb{Q}[t]$,
which is a Hilbert polynomial of some coherent sheaf on $X$.
\begin{align}\label{mfunct}
\mM_{\alpha} \colon
\sS ch/\mathbb{C} \to \sS et
\end{align}
be the functor
defined by
\begin{align*}
\mM_{\alpha}(T) \cneq \left\{ \fF \in \Coh(X \times T) :
\begin{array}{c}
\fF \mbox{ is } T \mbox{-flat, }
\fF_t \mbox{ for any }
t\in T \mbox{ is }\\
\mbox{ stable with Hilbert polynomial }
\alpha
\end{array} \right\}/(\mbox{equiv}).
\end{align*}
$\fF$ and $\fF'$
are equivalent if there is an line bundle $\lL$ on $T$
such that $\fF \cong \fF' \otimes p_T^{\ast}\lL$,
where $p_T \colon X \times T \to T$ is the projection.
The moduli functor (<ref>) is not always
representable by a scheme, but
if we assume that
\begin{align}\label{primitive}
\mathrm{g. c. d.}\{\alpha(m) : m\in \mathbb{Z}\}=1
\end{align}
then (<ref>)
is represented by a projective scheme $M_{\alpha}$
(cf. <cit.>), i.e.
there is an isomorphism of functors
\begin{align}\label{funct:isom}
\Hom(-, M_{\alpha}) \stackrel{\cong}{\to}
\mM_{\alpha}.
\end{align}
We call $\alpha$ satisfying the condition
(<ref>) as primitive.
Below, we always assume
that $\alpha$ is primitive.
Note that the isomorphism (<ref>) is
induced by a universal family
\begin{align*}
\uU \in \Coh(X \times M_{\alpha}).
\end{align*}
§.§ DG moduli spaces of stable sheaves
Note that $X=\mathrm{Proj}(A)$ for
the graded algebra
\begin{align*}
A=\bigoplus_{i\ge 0}H^0(X, \oO_X(i)).
\end{align*}
We use the
quiver with
relation $(Q_{[p, q]}, I)$
constructed in Definition <ref>
from the above graded algebra $A$.
For $q > p >0$, we set
\begin{align}\label{GammaU}
\Gamma_{[p, q]}(\uU)
\cneq \bigoplus_{i=p}^{q}
p_{M\ast}(\uU \otimes p_X^{\ast}\oO_X(i)).
\end{align}
$p_M$, $p_X$ are the projections from
$X \times M_{\alpha}$ to $M_{\alpha}$, $X$
If we take $q \gg p \gg 0$, then
(<ref>) is a flat representation of
$(Q_{[p, q]}, I)$
over $M_{\alpha}$
with the primitive dimension vector
\begin{align}\label{gam:prim}
\gamma=(\alpha(p), \alpha(p+1), \cdots, \alpha(q)).
\end{align}
The object (<ref>)
defines the morphism of schemes
\begin{align}\label{Upsilon}
\Upsilon \colon
M_{\alpha} \to M_{\gamma}
\end{align}
which is an open immersion by <cit.>.
Let $M_{[p, q]} \subset M_{\gamma}$ be the image of $\Upsilon$.
Since both sides of (<ref>) are projective, the image
$M_{[p, q]}$
is a union of connected components of $M_{\gamma}$.
We have the isomorphism of schemes
\begin{align}\label{Upsilon:isom}
\Upsilon \colon M_{\alpha} \stackrel{\cong}{\to}
M_{[p, q]}.
\end{align}
In particular, the object (<ref>)
is a universal $(Q_{[p, q]}, I)$-representation
restricted to $M_{[p, q]}$.
By replacing $\uU$ by $\uU \otimes p_M^{\ast}\lL$
for some line bundle $\lL$ on $M_{\alpha}$,
we may assume that
the universal family $\vV_{\bullet}$ given in (<ref>)
restricted to $M_{[p, q]}$
coincides with (<ref>).
Recall that $M_{\gamma}$ is obtained as the
zero-th truncation of the
smooth commutative dg-scheme
$(N_{\gamma}, \oO_{N_{\gamma}, \bullet})$
given by (<ref>).
By taking a suitable open
subset $N_{\alpha} \subset N_{\gamma}$
containing $M_{[p, q]}$,
the following result was proved in <cit.>:
There is an open subset
$N_{\alpha} \subset N_{\gamma}$ such that the
smooth commutative dg-scheme
\begin{align}\label{com:dga2}
(N_{\alpha}, \oO_{N_{\alpha}, \bullet} \cneq
\oO_{N_{\gamma}, \bullet}|_{N_{\alpha}})
\end{align}
satisfies the following:
* The zero-th truncation of (<ref>)
is isomorphic to $M_{\alpha}$.
* For any $[E] \in M_{\alpha}$, the tangent
complex of (<ref>) at $[E]$ is
\begin{align}\label{tan:com}
\mathrm{Cone}(\mathbb{C} \to \dR \Hom(E, E))[1].
\end{align}
§.§ Existence of a quasi NCDG structure
One can also extend the isomorphism (<ref>) to their
NC thickenings.
\begin{align*}
h_{\alpha} \colon \nN \to \sS et
\end{align*}
be the functor sending $R\in \nN$
to the isomorphism classes of
triples $(f, \fF, \psi)$:
* $f$ is a morphism of schemes
$f \colon \Spec R^{ab} \to M_{\alpha}$.
* $\fF$ is an object of $\Coh(X_{R})$
which is flat over $R$, where $X_{R}\cneq X \times \Spf R$.
* $\psi$ is an isomorphism $\psi \colon \fF^{ab}
\stackrel{\cong}{\to} f^{\ast}\uU$.
An isomorphism $(f, \fF, \psi) \to (f', \fF', \psi')$
exists if $f=f'$,
and there is an isomorphism $\fF \to \fF'$ in $\Coh(X_{R})$
commuting $\psi$, $\psi'$.
The isomorphism (<ref>)
extends to the isomorphism of functors
\begin{align*}
\Gamma_{[p, q]} \colon
h_{\alpha} \stackrel{\cong}{\to} h_{\gamma}|_{M_{[p, q]}}.
\end{align*}
By combining Theorem <ref>, Theorem <ref> and
Proposition <ref>,
we obtain the following:
There is a smooth quasi NCDG structure
$\{(U_i, \oO_{U_i, \bullet}^{\n})\}_{i\in \mathbb{I}}$
on the smooth commutative dg-moduli space
$(N_{\alpha}, \oO_{N_{\alpha}, \bullet})$
such that
the zero-th truncations
\begin{align}\label{qnc:V}
\{(V_i, \oO_{V_i}^{\n})\}_{i\in \mathbb{I}} \cneq
\{\tau_0(U_i, \oO_{U_i, \bullet}^{\n})\}_{i \in \mathbb{I}}
\end{align}
is a quasi NC structure on $M_{\alpha}$
which fit into NC hulls $h_{V_i^{\n}} \to h_{\alpha}|_{V_i}$.
The quasi NC
structure in (<ref>)
is the one constructed in <cit.>.
By <cit.>,
it satisfies the following
For $[E] \in V_i$, let
$\widehat{\oO}_{V_i, [E]}^{\n}$
be the completion of $\oO_{V_i}^{\n}$ at $[E]$.
Then it coincides with the
pro-representable hull of the
NC deformation functor of $E$
developed in <cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>.
This implies that we have an isomorphism of algebras
\begin{align*}
\widehat{\oO}_{V_i, [E]} \cong
\end{align*}
where $R_E^{\n}$ is the algebra (<ref>)
constructed by the $A_{\infty}$-structure
§.§ An example
We take $X=\mathbb{P}^2$ and
$\alpha$ to be the constant function $1$.
Note that a stable sheaf on $X$ has Hilbert polynomial $1$
if and only if it is a skyscraper sheaf $\oO_x$
for $x\in \mathbb{P}^2$.
Then the moduli space $M_{\alpha}$ is isomorphic to
$\mathbb{P}^2$ itself.
On the other hand, by Beilinson's theorem <cit.>,
we have the derived equivalence
\begin{align*}
\dR \Hom(\eE, -) \colon D^b(\Coh(\mathbb{P}^2)) \stackrel{\sim}{\to}
D^b (\modu A)
\end{align*}
where $\eE$ and $A$ are given by
\begin{align*}
\eE=\oO_{\mathbb{P}^2} \oplus \oO_{\mathbb{P}^2}(-1) \oplus
\oO_{\mathbb{P}^2}(-2), \ A=\End(\eE).
\end{align*}
By the above equivalence, one can take $p=0$ and $q=2$ in the
argument of the previous subsection.
The quiver $Q_{[0, 2]}$
is described as
\begin{align}\label{fig:Q}
\xymatrix{ \stackrel{0}{\bullet}
\ar@<2ex>[r]_{x_1}
\ar@<0ex>[r]_{x_2}
\ar@<-2ex>[r]_{x_3}
\ar@/^/@<3ex>[rr]_{z_{11}}
\ar@/^/@<5ex>[rr]_{z_{22}}
\ar@/^/@<7ex>[rr]_{z_{33}}
\ar@/_/@<-3ex>[rr]^{z_{12}}
\ar@/_/@<-5ex>[rr]^{z_{13}}
\ar@/_/@<-7ex>[rr]^{z_{23}}
& \stackrel{1}{\bullet}
\ar@<2ex>[r]_{y_1}
\ar@<0ex>[r]_{y_2}
\ar@<-2ex>[r]_{y_3}
& \stackrel{2}{\bullet}
\end{align}
with relations
given by
\begin{align*}
z_{ij}=y_{j} x_i=y_i x_j, \ 1\le i \le j \le 3.
\end{align*}
The dimension vector
(<ref>) is $\gamma=(1, 1, 1)$.
The moduli space $N_{\gamma}$ of representations of
$Q_{[0, 2]}$ without relation
is the quotient of the stable locus of
$\mathbb{C}^3 \times \mathbb{C}^3 \times \mathbb{C}^6$
by $(\mathbb{C}^{\ast})^{\times 2}$.
It contains an open subset $U \subset N_{\gamma}$ which
parametrizes representations of (<ref>)
with $x_3=y_3=1$ and
$x_1, x_2, y_1, y_2, z_{ij} \in \mathbb{C}$, i.e.
\begin{align*}
U=\Spec \mathbb{C}[x_1, x_2, y_1, y_2, z_{ij} : 1\le i \le j \le 3].
\end{align*}
An NC smooth thickening of $U$ is given by
\begin{align*}
U^{\n}=\Spf R, \ R=\mathbb{C}\langle
x_1, x_2, y_1, y_2, z_{ij} : 1\le i \le j \le 3
\rangle_{[\hspace{-0.5mm}[ab]\hspace{-0.5mm}]}.
\end{align*}
Let $\vV$ be the universal representation
of $Q_{[0, 2]}$ on $N_{\gamma}$, which is a rank three
vector bundle.
A lift of $\vV|_{U}$ to $U^{\n}$ is given by the
following representation
\begin{align}\notag
\xymatrix{ R
\ar@<2ex>[r]_{\cdot x_1}
\ar@<0ex>[r]_{\cdot x_2}
\ar@<-2ex>[r]_{1}
\ar@/^/@<3ex>[rr]_{\cdot z_{11}}
\ar@/^/@<5ex>[rr]_{\cdot z_{22}}
\ar@/^/@<7ex>[rr]_{\cdot z_{33}}
\ar@/_/@<-3ex>[rr]^{\cdot z_{12}}
\ar@/_/@<-5ex>[rr]^{\cdot z_{13}}
\ar@/_/@<-7ex>[rr]^{\cdot z_{23}}
& R
\ar@<2ex>[r]_{\cdot y_1}
\ar@<0ex>[r]_{\cdot y_2}
\ar@<-2ex>[r]_{1}
& R.
\end{align}
The algebra (<ref>) is then given by
\begin{align*}
\Lambda_{U, \bullet}^{\n}=R \ast T(\mathfrak{m}_1^{\otimes 2})
\end{align*}
where $\mathfrak{m}_1=H^0(\mathbb{P}^2, \oO_{\mathbb{P}^2}(1))$
and $\mathfrak{m}_1^{\otimes 2}$
is located in degree $-1$.
It is written as
a suitable NC completion of
\begin{align*}
\mathbb{C}\langle x_1, x_2, y_1, y_2, z_{ij}, w_{kl} :
1\le i \le j \le 3, 1\le k, l \le 3 \rangle
\end{align*}
where $\deg x_i=\deg y_i=\deg z_{ij}=0$ and $\deg w_{kl}=-1$ and the
differential is given by
\begin{align*}
Q \colon
w_{kl} \mapsto z_{kl}-y_k x_l
\end{align*}
where we set $z_{kl}=z_{lk}$ if $k>l$ and
$\Spf \Lambda_{U, \bullet}^{\n}$ is
a smooth affine NCDG structure on
its abelization
$(U, \oO_U \otimes S(\mathfrak{m}_1^{\otimes 2}))$.
The zero-th
of $\Spf \Lambda_{U, \bullet}^{\n}$
gives $\mathbb{C}^2 \subset \mathbb{P}^2=M_{\alpha\equiv 1}$.
§.§ NC virtual structure sheaves on moduli spaces of stable sheaves
Now we assume that
the smooth
commutative dg-scheme (<ref>) is a $[0, 1]$-manifold,
which means that
the tangent complex of (<ref>) has
amplitude in $[0, 1]$.
By (<ref>),
this is equivalent to the condition
\begin{align}\label{high:ob}
\Ext^{i}(E, E)=0, \ i\ge 3
\end{align}
for any $[E] \in M_{\alpha}$.
Applying Definition <ref>
to the quasi NCDG structure in Theorem <ref>,
we obtain the
$d$-th NC virtual structure sheaf
\begin{align*}
(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d} \in K_0(M_{\alpha}).
\end{align*}
By Theorem <ref>
and (<ref>),
we have the following:
The $d$-th NC virtual structure sheaf
associated to the quasi NCDG structure in
Theorem <ref> is written as
\begin{align}\label{ncvir:formula2}
(\oO_{M_{\alpha}}^{\rm{ncvir}})^{\le d}=\oO_{M_{\alpha}}^{\rm{vir}}
\otimes_{\oO_{M_{\alpha}}}
(\eE_{\bullet})^{\le d}_{\bullet}].
\end{align}
Here $\eE_{\bullet} \to \tau_{\ge -1} \dL_{M_{\alpha}}$
is a perfect obstruction theory on $M_{\alpha}$
such that
for any $[E] \in M_{\alpha}$ we have
\begin{align*}
\hH_0(\eE_{\bullet}^{\vee}|_{[E]})=\Ext^1(E, E), \
\hH_1(\eE_{\bullet}^{\vee}|_{[E]})=\Ext^2(E, E).
\end{align*}
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Kavli Institute for the Physics and
Mathematics of the Universe, University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan.
E-mail address: [email protected]
|
1511.00365
|
Department of Physics, Tohoku University, Sendai 980-8578, Japan
CREST-JST, Sendai 980-8578, Japan
Photoinduced carrier dynamics in a correlated electron system on a coupled two-leg ladder lattice are studied.
The two-leg ladder Hubbard model is analyzed by utilizing the exact diagonalization method based on the Lanczos algorithm in finite size clusters.
In order to reveal the transient carrier dynamics after photoirradiation, we calculate the low-energy components of the hole kinetic energy, the pair-field correlation function, the optical conductivity spectra and others.
It is shown that the photoinduced metallic-like state appears in a half filled Mott insulating state, while the low-energy carrier motion is suppressed by photoirradiation in hole doped metallic states.
These photoinduced changes in electron dynamics are associated with changes in the carrier-pair coherence,
and are not attributed to a naive thermalization but to a ladder-lattice effect.
Based on the numerical results, optical controls of hole pairs by using the double-pulse pumping are demonstrated.
Implications to the recent optical pump-probe experiments are presented.
78.47.J-, 78.20.Bh, 74.72.-h
§ INTRODUCTION
Optical manipulations of macroscopic phenomena in solids are widely accepted as greatly attracted subjects in recent condensed matter physics. <cit.>
Ultrafast controls of magnetism <cit.>, ferroelectricity <cit.>
and electrical conductivity <cit.>, as well as superconductivity <cit.>,
have tried to be realized in a wide variety of materials by utilizing the recently developed laser pulse technology.
Strongly correlated electron systems, such as transition-metal oxides, organic molecular solids, and rare-earth compounds,
are the plausible candidate materials, where a number of degrees of freedom are entangled with each other under strong electron-electron and electron-lattice interactions.
Observations of transient changes in macroscopic quantities and comprehensions of the microscopic mechanisms are required as problems of great urgency from the fundamental and application viewpoints.
Low dimensional correlated electron systems are attractive targets for the ultrafast optical controls of electronic states.
Quasi one- and two-dimensional cuprate oxides <cit.>,
halogen-bridged complexes <cit.>
and BEDT-TTF molecular solids <cit.> are the examples.
This is attributed to a rich variety of equilibrium states and the simple lattice structures.
From the theoretical viewpoints, transient electron dynamics have been examined numerically and analytically
in the one- and two-dimensional
Hubbard <cit.> and $t-J$ <cit.> models which are accepted as minimal models for the low-dimensional correlated systems.
Correlated electrons on a ladder lattice are recognized to be in between the one- and two-dimensional systems.
The superconductivity observed in the two-leg ladder cuprates, Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$, suggests
unique electronic structures on a ladder lattice.
Insulating states associated with a charge order at $x=0$ is changed into a metallic state by substitution of Sr by Ca,
and superconductivity appears under high pressure. <cit.>
Short-range singlet spin correlation and spin excitation gap have attracted much attentions from the viewpoint of the superconductivity caused by doping of holes into the ladders. <cit.>
A number of theoretical examinations for the equilibrium electric and magnetic structures, as well as doublon–holon dynamics have been performed so far, in the two-leg ladder Hubbard and $t-J$ models. <cit.>
Recently, the photoinduced transient states were examined in the insulating ($x=0$) and metallic ($x=10$) two-leg ladder cuprates by utilizing the ultrafast optical pump-probe experiments. <cit.>
The transient optical measurements provide a hint to reveal unique spin and charge dynamics as well as their roles on the superconductivity in the two-leg ladder cuprates.
In this paper, the photoinduced carrier dynamics in a correlated electron system on a two-leg ladder lattice are studied.
The two-leg Hubbard model is analyzed by utilizing the exact-diagonalization method base on the Lanczos algorithm in finite size clusters.
In order to reveal the transient dynamics, we calculate the low-energy components of the kinetic energy, the pair-field correlation function, the optical conductivity spectra and others as functions of time.
It is found that the low energy carrier dynamics in metallic states are distinguish from those in the insulating states;
the photoinduced metallic-like state appears at half filled Mott insulating state, while the low-energy carrier motion is suppressed by photoirradiation in hole doped metallic states.
These photoinduced changes in the dynamics are associated with the changes in pair coherence of carriers.
Based on the calculated results, the optical controls of hole pairs by utilizing the double-pulse pumping are demonstrated.
Implications to the recent optical pump-probe experiments are presented.
In Sect. <ref>, the theoretical model and the numerical calculation method are introduced.
In Sect. <ref>, numerical results for the photoinduced carrier dynamics are presented.
In Sect. <ref>, numerical demonstrations of the photo-control carrier dynamics by utilizing the double-pulse pumping are shown.
Section <ref> is devoted to discussion and summary.
§ MODEL AND METHOD
The Hubbard model on a two-leg ladder lattice is defined by
\begin{align}
{\cal H}=
&- \sum_{\langle ij \rangle \alpha \sigma} \left ( t c_{i \alpha \sigma}^\dagger c_{j \alpha \sigma} + H. c. \right) \nonumber \\
&- \sum_{i \sigma} \left ( t' c_{i 1 \sigma}^\dagger c_{i 2 \sigma} + H.c. \right)
+U \sum_{i \alpha} n_{i \alpha \uparrow} n_{i \alpha \downarrow} ,
\label{eq:ham}
\end{align}
where $c_{i \alpha \sigma}^\dagger$ and $c_{i \alpha \sigma}$ are the creation and annihilation operators for an electron at the $i$-th rung and the right $(\alpha=1)$ or left $(\alpha=2)$ leg with spin $\sigma(=\uparrow, \downarrow)$, and
$n_{i \alpha \sigma}=c_{i \alpha \sigma}^\dagger c_{i \alpha \sigma}$ is the number operator.
The first and second terms represent the electron hoppings along the leg and rung in the two-leg ladder lattice, respectively,
and the third term describes the on-site Coulomb interaction.
A symbol $\langle ij \rangle$ represents the nearest-neighbor $ij$ pair along a leg.
We introduce an anisotropy of the hopping integral as $r_t = t'/t$ which is nearly equal to one in the ladder cuprates.
The photoinduced dynamics are examined by introducing the pump pulse photons in the first term in Eq. (<ref>) as
the Peierls phase given by
\begin{align}
t \rightarrow t e^{i A(\tau)} ,
\label{eq:peierls}
\end{align}
where a lattice constant, light velocity, and elementary charge are taken to be units,
and the light polarization is parallel to the leg direction.
The vector potential of the pump photon pulse at time $\tau$ is taken to be a Gaussian form as
\begin{align}
A(\tau)=A_p e^{-\tau/(2\gamma_p^2)} \cos \omega_{p} \tau ,
\label{eq:vp}
\end{align}
with amplitude $A_p$, frequency $\omega_{p}$ and pulse width $\gamma_p$.
A center of the pump photon pulse is located at $\tau=0$.
Electronic states before and after the photon pumping are calculated by using the exact-diagonalization method based on the Lanczos algorithm.
We adopt finite-size clusters of $N(=2 \times N/2)$ sites up to $N=14$, where the periodic- and open-boundary conditions are imposed along the leg.
The time evolutions of the wave function under the time-dependent Hamiltonian are calculated as <cit.>
\begin{align}
| \Psi(\tau+\delta \tau) \rangle & =e^{-i{\cal H}(\tau) \delta \tau } |\Psi(\tau) \rangle
\nonumber \\
& = \sum_i^M e^{-i { \varepsilon}_i \delta \tau} |\phi_i \rangle \langle \phi_i | \Psi(\tau)\rangle ,
\label{eq:wave}
\end{align}
where $|\phi_i \rangle$ and ${ \varepsilon}_i$, respectively, are the eigen-state and the eigen-energy in the order-$M$ Krylov subspace in the Lanczos process, $\delta \tau$ is a time step, and ${\cal H}(\tau)$ is the time-dependent Hamiltonian where the pump pulse is introduced as Eq. (<ref>).
We chose, in most of the numerical calculations, $M=15$ and $\delta \tau=0.01/t$, which are sufficient to obtain the results with high enough accuracy.
All energy and time parameters in the numerical calculations, respectively, are given as units of $t$ and $1/t$, which are about 0.5eV and 8fs in the ladder cuprates.
In most of the calculations, we chose $U/t=6$ and $\gamma_{p} t = 5$.
Hole density measured from the half filling are denoted by $x_h(\equiv 1-N_e/N)$ with the number of electrons $N_e$.
§ RESULTS
(Color online)
Optical conductivity spectra before photon pumping.
Dashed, solid and dotted curves represent the spectra at $x_h=0$ (half filling), $x_h=1/6$ and $1/3$, respectively.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter values are chosen to be $U/t=6$, and $t'/t=1$.
Bold arrows represent energies of the pump photons adopted in the numerical calculations for the
real time photoinduced dynamics.
In this section, the transient electronic states after the photon pumping are presented.
The ground states before photon pumping are studied by calculating the spin-gap energy, the holon-binding energy, the optical conductivity spectra and the one-particle excitation spectra and others.
Most of the results are consistent with the previous calculations in the two-leg ladder Hubbard models. <cit.>
In Fig. <ref>, the optical conductivity spectra before pumping at half filling $(x_h=0)$ and away from the half filling are shown.
The optical gap at half filling is collapsed by hole doping.
A sharp low-energy peak around $\omega/t=0.5$ which corresponds to the Drude component appears.
Finite but small energy of the “Drude" peak is due to the finite size effects in the open boundary condition,
and is confirmed to decrease with increasing the system size.
The pump photon energies ($\omega_p$) are tuned at the optical gap energy in the the half filling case, and
at the energy of the remnant gap in the hole doped metallic state as shown by bold arrows in Fig. <ref>.
We define the absorbed photon density as $n_{p}(\tau) =(E(\tau)-E_{0})/(N \omega_{p})$
where $E(\tau)$ is the total energy at time $\tau$, and $E_{0}=E(\tau \ll 0)$.
Numerical values of the pump photon amplitudes are chosen to satisfy the condition
$n_{p}(\tau \gg \gamma_{p}) \sim 0.05$.
§.§ Photoinduced change in carrier dynamics
(Color online)
(a)-(c) Increments of several terms of energy for several hole densities.
The Coulomb-interaction energy $(E_U)$, the kinetic energy $(E_t)$, and the kinetic energy for the low-energy hole motions $(E_t^{(h)})$ are plotted.
The Coulomb interaction parameter is chosen to be $U/t=6$.
(d) Time dependences of the Coulomb interaction energy at $x_h=1/6$ for several values of $U$.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter value is chosen to be $t'/t=1$.
Shaded areas represent the time interval when the pump pulse is introduced.
First, we show the photoinduced changes in the carrier dynamics.
In order to measure the low-energy dynamics of carriers, the projected kinetic energy expectations are introduced as <cit.>
\begin{align}
E_{t}^{(h)}=-t\sum_{\langle ij \rangle \alpha \sigma} \langle P_{ij \alpha {\bar \sigma}}^{(h)} c_{i \alpha \sigma}^\dagger c_{j \alpha \sigma} P_{ij \alpha {\bar \sigma}}^{(h)}+H.c. \rangle ,
\label{eq:kine}
\end{align}
\begin{align}
E_{t}^{(d)}=-t\sum_{\langle ij \rangle \alpha \sigma}
\langle P_{ij \alpha {\bar \sigma}}^{(d)} c_{i \alpha \sigma}^\dagger c_{j \alpha \sigma}
P_{ij \alpha {\bar \sigma}}^{(d)} +H.c. \rangle ,
\label{eq:kine1}
\end{align}
where ${\bar \sigma}=\uparrow (\downarrow)$ for $\sigma=\downarrow (\uparrow)$.
We define the projection operators as
P_{ij \alpha \sigma}^{(h)}=(1-n_{i \alpha \sigma})(1-n_{j \alpha \sigma})
P_{ij \alpha \sigma}^{(d)}=n_{i \alpha \sigma}n_{j \alpha \sigma}
which project onto the states where both the $i$ and $j$ sites are unoccupied and occupied by electrons with spin $\sigma$, respectively.
The projected kinetic energies $E_t^{(h)}$ and $E_t^{(d)}$, respectively, measure the kinetic energies of holes and doublons along the leg, where the number of the double occupancies are not changed,
for examples,
$|(\uparrow )_i \rangle \rightarrow |(\uparrow)_j \rangle$ and
$|(\uparrow \downarrow )_i (\downarrow)_j \rangle \rightarrow |(\downarrow)_i (\uparrow \downarrow )_j\rangle$.
In Fig. <ref>, several components of the energies are plotted as functions of time.
The Coulomb interaction energy ($E_U$) and the kinetic energy ($E_t$), respectively, are defined as the expectation values of the third and first terms in Eq. (<ref>).
We define the energy differences as $\Delta E_U=E_U(\tau)-E_U(\tau \ll 0)$ and others.
As shown in Fig. <ref>(a), $E_U$ increases by photon pumping in all values of $x_h$, which implies that the number of the doublely occupied sites increases.
Time dependences of $E_U$ after photon pumping strongly depend on $x_h$: large $E_U$ is maintained at half filling, while it gradually decreases with time in hole doped cases.
The reductions of $E_U$ imply recombinations of the photo-generated doublons and holons, and the excess Coulomb-interaction energy due to the photon pumping is transfered into the kinetic energy.
At vicinity of the half filling, there are no channels through which the large excess Coulomb-interaction energies are released. This is shown clearly in Fig. <ref>(d) where $E_U$ at $x_h=1/6$ is plotted for several values of $U$.
Reduction of $E_U$ is observed in the cases of small $U$.
The life times of the photo-generated doublons and holons are prolonged by increasing $U$, as suggested to be $\sim e^{\alpha U}$ with a positive constant $\alpha$ in Ref. strohmaier.
The kinetic energies, $E_t$ and $E_t^{(h)}$, respectively, are presented in Figs. <ref>(b) and (c).
We confirm that results of $E_t^{(h)}+E_t^{(d)}$ (not shown) show similar behaviors to $E_{t}^{(h)}$.
At half filling, both $E_t$ and $E_t^{(h)}$ decrease, implying increment of the carrier motions by photon pumping.
Opposite changes are seen in the hole doped cases where both $E_t$ and $E_t^{(h)}$ increase. That is, the carrier motions are suppressed by photon pumping.
(Color online)
(a)-(c) Kinetic energies of the low-energy hole motions for several values of $t'/t$.
(d)-(f) Low energy weights of the optical conductivity spectra for several values of $t'/t$.
Hole densities are chosen to be $x_h=0$ in (a) and (d), $1/6$ in (b) and (e), and $1/3$ in (c) and (f).
Finite size cluster of $N=12$ sites with the open boundary conditions is adopted.
Parameter value is chosen to be $U/t=6$.
Shaded areas represent the time interval when the pump pulse is introduced.
Let us focus on the photoinduced dynamics of the low-energy carrier motion.
In Figs. <ref>(a)-(c), the transient kinetic energies for the low-energy hole motion, $E_t^{(h)}$, are presented for several values of $r_t=t'/t$.
At half filling (Fig. <ref>(a)), reductions of $E_t^{(h)}$ by the photon pumping occur commonly in all values of $t'/t$. Remarkable $t'/t$ dependences are shown in the hole doped cases of $x_h=1/6$ [see Fig. <ref>(b)].
Monotonic reductions of $|E_t^{(h)}|$ after photon pumping
are observed around $r_t=1$.
In the cases of the weakly coupled two chains corresponding to small $r_t$,
$|E_t^{(h)}|$ increases by photon pumping as similar to the results at half filling.
The reductions of $|E_t^{(h)}|$ are more pronounced in $x_h=1/3$ as shown in Fig. <ref>(c).
Therefore, the two-leg ladder lattice plays an essential role for the suppression of the low-energy carrier dynamics by photon pumping.
Qualitatively similar behaviors of $E_t^{(h)}$ are confirmed in the calculations, in which
the cluster sizes are $N=5 \times 2$ to $7 \times 2$ with the periodic and open boundary conditions, and $U/t$ is chosen to be 6 and 8.
(Color online)
Optical conductivity spectra for several times at (a) $x_h=0$ and (b) $1/6$ .
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter values are chosen to be $U/t=6$ and $t'/t=1$.
(Color online)
(a) Time dependences of the low-energy weights of the optical conductivity spectra $(D)$ for several hole densities.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
(b) Changes in the low-energy weights of the optical conductivity spectra for several cluster sizes and hole densities in the hole-doped cases.
We chose $(N, x_h)=(10, 1/5)$, $(12, 1/6)$, $(12, 1/3)$, $(14, 1/7)$, and $(14, 2/7)$, respectively.
Parameter values are chosen to be $U/t=6$, $t'/t=1$ and $\omega_c/t=2$.
Shaded areas represent the time interval when the pump pulse is introduced.
Carrier dynamics after photon pumping are also examined by calculating the transient excitation spectra.
The transient optical responses are simulated by using the formula based on the linear response theory where the wave functions at time $\tau$ are used.
This has been utilized widely to examine the transient electronic structures in correlated electron systems, <cit.> and its validity was checked in Ref. ohara.
The regular part of the optical conductivity spectra is given by
\begin{align}
\sigma_{reg}(\omega)=-\frac{1}{N \omega} {\rm Im}\chi(\omega),
\end{align}
\begin{align}
\chi(\omega)=- \sum_{m n}
\biggl ( & \frac{\langle \Psi (\tau ) | \phi_m \rangle \langle \phi_m | j | \phi_n \rangle \langle \phi_n |j | \Psi(\tau) \rangle}{\omega-\varepsilon_m+\varepsilon_n+i \eta}
\nonumber \\
&\frac{\langle \Psi(\tau) |j| \phi_n \rangle \langle \phi_n | j | \phi_m \rangle \langle \phi_m | \Psi(\tau) \rangle}{\omega-\varepsilon_n+\varepsilon_m+i \eta} \biggr ) ,
\label{eq:chi}
\end{align}
where we introduce the current operator along the leg defined by
$j=i\sum_{i \alpha \sigma}c_{i \alpha \sigma}^\dagger c_{i+1 \alpha \sigma}+H.c.$, and a small positive constant $\eta$.
The calculated optical conductivity spectra at $x_h=0$ and $1/6$
are shown in Figs. <ref>(a) and (b), respectively.
At half filling, new peaks appear inside of the optical gap by photon pumping and grow with time.
This is consistent with the results of $E_t^{(h)}$ shown in Fig. <ref>(b).
This change in the optical conductivity spectra implies a transition from a Mott insulator to a metallic state.
(Color online)
(a) One particle DOS at $r_t=1$ and (b) those at $r_t=0.2$ before $(\tau t=-20)$ and after $(\tau t =20)$ pumping.
Dashed and solid lines represent the total DOS $[\rho^<(\omega)+\rho^>(\omega)]$ before and after pumping, respectively, and black and red shaded areas correspond to the electron parts of DOS $[\rho^<(\omega)]$ before and after pumping, respectively.
Dotted lines denote the Fermi level before pumping.
(c) and (d) One particle excitation spectra $A^>({\bm k},\omega)$ (solid lines) and $A^<({\bm k}, \omega)$ (dotted lines) before pumping,
and (e) and (f) those after pumping in the case of $r_t=1$.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter value is chosen to be $U/t=6$.
On the other hand, in the hole-doped case, a sharp low-energy peak observed before pumping is diminished just after the photon pumping, and its intensity decreases with time.
Weak change is seen in high-energy excitation spectra, that are identified as remnants of the Mott gap excitations.
We note that negative values around $\omega/t=3$ in Fig. <ref>(a) is attributable to be due to the optical emission from the photoexcited states.
In order to examine photoinduced changes in the low-energy spectral weight, we define the integrated spectral weight defined by <cit.>
\begin{align}
D=-\frac{\pi E_{t}}{N}-2 \int_{\omega_c}^{\infty} d \omega \sigma_{reg}(\omega) ,
\end{align}
in the calculations with the open boundary condition, where $\omega_c$ is a cut off energy.
In Fig. <ref>(a), the low-energy integrated spectral weights are shown for several hole densities.
We chose $\omega_c/t=2$, and do not observe any qualitative differences from the results with $1 \lesssim \omega_c/t \lesssim 3$.
Increase of $D$ from zero at half filling implies an appearance of the photo-generated metallic-like carrier motions.
Opposite behaviors are observed in $x_h=1/6$ and $1/3$; $D$ is redued after photon pumping.
Low energy intensities continuously decrease at least up to $\tau=50/t$.
These characteristics in the hole doped cases are widely observed in several size clusters and hole densities as shown in Fig. <ref>(b).
We confirmed that the reduction of $D$ is monotonically increased with the pump pulse amplitude $A_p$,
and is qualitatively insensitive to the parameter values of the photon energy between $4.9 \lesssim \omega_p/t \lesssim 7.5$ in $(N, x_h)=(12,1/6)$.
The low-energy spectral weights for several $x_h$ and $r_t=t'/t$ are summarized in Figs. <ref>(d)-(f). The results are qualitatively similar to the kinetic energies for the low-energy hole motion
shown in Figs. <ref>(a)-(c).
Photoinduced changes in the electronic structures are directly observed by calculating the one-particle excitation spectra. We calculate the transient electronic density of states (DOS)
where the wave function at time $\tau$ is used.
This is given by
\begin{align}
\rho^{\gtrless}(\omega)=\frac{1}{N}\sum_{\bm k} A^{\gtrless}({\bm k}, \omega) ,
\end{align}
where $\rho^> (\omega)$ and $\rho^< (\omega)$, respectively, are the electron and hole parts, and are
given by the one-particle excitation spectra defined by
\begin{align}
A^{<}({\bm k}, \omega)&=-\frac{1}{\pi} {\rm Im}
\sum_{m n }\sum_{\alpha \sigma} \nonumber \\
& \times
\frac{\langle \Psi (\tau ) | \phi_m \rangle
\langle \phi_m |c_{{\bm k} \alpha \sigma} | \phi_n \rangle
\langle \phi_n |c_{{\bm k} \alpha \sigma}^\dagger| \Psi(\tau) \rangle}
{\omega-\varepsilon_n+\varepsilon_m+i \eta} ,
\label{eq:chi1}
\end{align}
\begin{align}
A^{>}({\bm k}, \omega)&=-\frac{1}{\pi} {\rm Im}
\sum_{m n }\sum_{\alpha \sigma} \nonumber \\
& \times
\frac{\langle \Psi(\tau) |c_{{\bm k} \alpha \sigma}^\dagger| \phi_n \rangle
\langle \phi_n | c_{{\bm k} \alpha \sigma} | \phi_m \rangle
\langle \phi_m | \Psi(\tau) \rangle}
{\omega-\varepsilon_m+\varepsilon_n+i \eta} .
\label{eq:chi2}
\end{align}
Calculated DOS and one-particle excitation spectra are shown in Fig. <ref>.
We take that the $x$ and $y$ axes are parallel to the leg and rung directions, respectively.
In the isotropic ladder lattice (Fig. <ref>(a)), a sharp peak around the Fermi level (FL) before pumping is attributed to the two quasi-particle bands originating from the bonding $(k_y=0)$ and anti-bonding $(k_y=\pi)$ bands, <cit.> which cut the FL around
$k_x=0.6\pi$ and $0.3\pi$, respectively [see Figs. <ref>(c) and (d)].
After pumping, as shown in Figs. <ref>(a), (e) and (f), sharp quasi-particle peaks are smeared out, and are merged into the incoherent parts.
Remarkable changes in high energy structure are not seen, although electronic and hole parts distribute to high and low energy regions, respectively.
On the other hand, at $r_t=t'/t=0.2$, i.e. the case of the weakly coupled two chains, sharp peak around FL before pumping remains and shifts toward the low energy after photon pumping.
In the numerical calculation shown in Fig. <ref>(b),
the pump photon amplitude is chosen to a value in which $n_{p}$ is taken to be about $0.08$ in order to see the characteristics clearly.
The differences between the results in $r_t=1$ and $0.2$ shown above are consistent with
the $r_t$ dependence of the low-energy kinetic energies as well as the low-energy weight of the optical conductivity spectra shown in Fig. <ref>.
(Color online)
Time dependences of the pair-field correlation functions for several distances $|i-j|$.
Hole densities are (a) $x_h=0$, (b) $1/7$ and (c) $1/6$.
Finite size clusters of $N=12$ and $14$ sites with the open boundary condition are adopted.
Parameter values are chosen to be $U/t=6$ and $t'/t=1$.
Results at $x_h=1/6$ and $r_t=0.2$ are shown in (d).
Shaded area represents the time intervals when the pump pulses are introduced.
Finally, we show the paring properties of charge carriers and the photon pumping effect. <cit.>
We introduce the pair-field correlation function between sites $i$ and $j$ defined by
\begin{align}
P(|i-j|)=\langle \Psi(\tau) |\left ( \Delta_j^\dagger \Delta_i + H.c. \right ) |\Psi(\tau) \rangle ,
\end{align}
where the $d_{x^2-y^2}$-wave pair-field operator is given by
\begin{align}
\Delta_i=c_{i 1\downarrow}c_{i 1 \uparrow}-c_{i 2 \uparrow}c_{i 1 \downarrow} .
\end{align}
This function measures the correlation between the carrier pairs created at sites $j$ and annihilated at $i$.
As shown by the previous calculations, <cit.> the pair correlations in the equilibrium states at zero temperature are
damped within few sites at half filling, and are long ranged in the doped cases.
At half filling [see Fig. <ref>(a)], while the long-range correlation is reduced a little after pumping,
the short-range correlations are changed to be oscillating with large amplitude.
Away from the half filling [see Figs <ref>(b) and (c)],
the correlations for all distances monotonically decrease by photon pumping,
while the oscillatory behaviors observed at $x_h=0$ are weak.
The reductions in $P(|i-j|)$ are not pronounced in the case of small $r_t=t'/t$ as shown in Figs. <ref>(d).
These results are summarized that i) coherent oscillations of the carrier pairs are induced by the photon pumping at half filling,
and ii) the pair correlation becomes a short ranged by the photon pumping at hole doped cases.
Based on the results, the optical control of the carrier pair coherence will be demonstrated in Sect. <ref>.
§.§ Double pulse pumping
In this section, we demonstrate the electronic state changes by the double pulse pumping, where the two photon pulses with a time interval are introduced in the two-leg ladder Hubbard model.
The vector potential for the double-pulse pumping is given by
\begin{align}
A_{p}(\tau)&=A_1 e^{-\tau/(2\gamma_1^2)} \cos \omega_{1} \tau
\nonumber \\
A_2 e^{-(\tau-\tau_d) /(2\gamma_2^2)} \cos \omega_{2} \tau ,
\label{eq:vp2}
\end{align}
where $\tau_d$ is the time interval between the first and second pulses,
$A_{1(2)}$, $\omega_{1(2)}$, and $\gamma_{1(2)}$ are amplitude, frequency and damping factor
for the first (second) pulse, respectively.
In the following calculations, we take
$\gamma_1 = \gamma_2 = 1/t$.
Photon energies in both the first and second pulses are chosen to be the optical gap energy before the first pumping as $\omega_1 = \omega_2 = 4t$.
(Color online)
Optical conductivity spectra in the double pulse pumping at half filling ($x_h=0$).
Dotted, solid and dashed lines represent the spectra before pumping $(\tau t=-20)$, between the first and second pulse pumpings $(\tau t=12)$, and after the second pulse pumping $(\tau t= 28)$, respectively.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter values are chosen to be $U/t=6$, $t'/t=1$, $A_1=A_2=0.6$, $\omega_1=\omega_2=4$, and $\tau_d=15$. Bold arrow represents the energy of the first and second photon pulses.
The optical conductivity spectra at half filling before the first pulse, between the first and second pulses, and after the 2nd pulse are shown in Fig. <ref>.
By introduction the first pump pulse, the low-energy spectral weight appears inside the optical gap. After the second pump pulse, reduction of the low-energy spectral weight occurs.
In brief, this is a sequential change of the electronic structures as (a Mott insulator) $\rightarrow$ (a metallic state) $\rightarrow$ (a suppression of the metallic state).
In this sense, the photo-doped carriers by the first pump pulse play similar roles with the chemically doped carriers.
(Color online)
The calculated results in the double pulse pumping at half filling.
(a) Low energy weights of the optical conductivity spectra, (b) the absorbed photon density and (c) the Coulomb interaction energy.
The first pulse amplitudes are varied as $A_1=0.2-0.6$ and the second pulse amplitude is fixed to be $A_2=0.6$.
(d) Low energy weights of the optical conductivity spectra for several values of $\tau_d$.
The first and second pulse amplitudes are fixed to be $A_1= A_2=0.6$.
Finite size cluster of $N=12$ sites with the open boundary condition is adopted.
Parameter values are chosen to be $U/t=6$, and $t'/t=1$.
Shaded areas represent the time intervals when the two pump pulses are introduced.
We calculate the low-energy weights of the optical conductivity spectra, $D$, as functions of time
and show the results in Fig. <ref>(a), in which $A_1$ are varied and $A_2$ is fixed.
The spectral weight induced by the first pulse increases with increasing $A_1$.
On the other hand, the second pulse brings about non-monotonic changes in $D$; when $A_1$ is small (large), the second pulse increases (decreases) $D$.
The absorbed photon density and the Coulomb interaction energy are shown in Figs. <ref>(b) and (c), respectively.
Not only the first pulse pumping but also the second pulse pumping induces increases both of $n_p$ and $E_U$, in contrast to the results of $D$ shown in Fig. <ref>(a).
This implies that even in the case for the large $A_1$,
the second pumping realizes a higher energy excited state than the state before the second pumping, but suppresses the low energy carrier motion.
In Fig. <ref>(d), we show the low energy spectral weight
in the case of the large $A_1$ where the time interval between the first and second pulses is varied.
The characteristic reduction of $D$ after the second pumping is commonly observed in all cases of $\tau_d$.
Thus, the observed phenomena are not due to the interference effects between the first and second pulse excitations.
(Color online)
(a) Time dependences of the pair-field correlation functions in the double-pulse pumping at half filling. Upper, middle and lower panels show $P(|i-j|)$ for $|i-j|=1$, $3$, and $5$, respectively.
(b) Time averaged pair-field correlation functions in the single-pulse pumping at half filling ($x_h=0$) and at hole doped case $(x_h=1/6)$.
Dashed and solid lines represent ${\bar P}(|i-j|)$ before and after photon pumpings, respectively.
(c) and (d)
Time averaged pair-field correlation functions in the double-pulse pumping at half filling.
The first pulse amplitudes are taken to be $A_1=0.2$ in (c) and $A_1=0.5$ in (a) and (d).
The second pulse amplitude is $A_2=0.6$ in (a), (c) and (d).
Dashed, dotted, and solid lines represent ${\bar P}(|i-j|)$ before the first pulse, between the first and second pulses,
and after the second pulse, respectively.
Finite size cluster of the $N=12$ sites with the open boundary condition is adopted.
Parameter values are chosen to be $U/t=6$, and $t'/t=1$.
The pairing characteristics are examined under the double pulse pumping.
Time dependences of the pair-field correlation functions, $P(|i-j|)$, for $|i-j|=3-5$ at half filling are plotted in Fig. <ref>(a).
As shown previously, $P(|i-j|)$ before the first pulse is of short ranged; $P(|i-j|>3)$ is less than 5$\%$ of $P(|i-j|=1)$.
After the first pulse pumping, coherent oscillations appear in $P(|i-j|)$ for all $|i-j|$ and their maximum amplitudes of the oscillations are larger than 10$\%$ of $P(|i-j|=1)$ before the first pumping.
After the second pulse pumping, suppressions of the oscillation amplitudes are remarkably seen in $P(|i-j|)$ for $|i-j|=4$ and $5$.
We analyze this characteristic change in the pair-field correlation function by introducing an averaged absolute value of $P(|i-j|)$
after the first and second pulse pumpings defined by
\begin{align}
{\bar P}(|i-j|)=\frac{1}{\Delta t} \int_{t_0}^{t_0+\Delta t} dt | P(|i-j|) | ,
\end{align}
where the parameters $(t_0, \Delta t)$ are chosen to be $(0, 15)$ in the case after the first pulse, and $(15, 35)$ in the case after the second pulse.
Results are shown in Figs. <ref>(c) and (d) for the large and small $A_1$s, respectively.
For comparison, we show in Fig. <ref>(b) the calculated results in the single pulse pumping case at $x_h = 0$ and $1/6$ where we take $(t_0, \Delta t)=(15, 35)$.
After the first pulse pumping
(see the blue dotted lines in Figs. <ref>(c) and (d)), ${\bar P}(|i-j|)$ for $|i-j| \ge 3$ increases and the correlation becomes long ranged.
These results are similar to that in the single pumping case at $x_h=0$ shown by the red line in Fig. <ref>(b).
The stronger the first pump amplitudes are, the larger the change in ${\bar P}(|i-j|)$.
Results after the second pulse pumping are qualitatively different between the cases with the large and small $A_1$s;
${\bar P}(|i-j|)$ with $|i-j| \ge 3$ increase in the small $A_1$ case, but decrease in the large $A_1$.
In short, the second pulse promotes the pair correlation in the small $A_1$ case, and suppresses the pair coherence in the large $A_1$ case.
This change in ${\bar P}(|i-j|)$ by the second pulse pumping in the arge $A_1$ case is similar to that
by the single pumping in the hole doped case as shown in Fig. <ref>(b),
where the pair correlation becomes shortened by the pumping.
These results suggest a similarity of the chemically doped and photo-doped hole carriers.
§.§ Discussion and Summary
In this section, we discuss relations of the present theoretical calculations with the recent optical pump-pulse experiments in the ladder cuprates.
The photoinduced electronic state transition was observed by the time resolved optical spectroscopy in insulating Sr$_{14}$Cu$_{24}$O$_{41}$ in which the nominal valence of Cu ion is +2.25 and 0.25 hole per Cu ion exists. <cit.>
The insulating nature confirmed from the transport and optical measurements is attributable to the
hole carrier localization in the charge density wave state.
The pump pulse energy was tuned around the insulating gap energy (1.58eV).
After the photon pumping, the Drude like metallic state appears with 1ps and is maintained for more than 50ps.
The photoinduced metallic reflectivity increases with increasing the pump photon fluence.
These results are attributable to the photo doping carriers and collapse of the carrier localization.
Similar photoinduced transitions from an insulating state to a metallic state have been widely confirmed experimentally
in correlated insulating states <cit.>, such as La$_2$CuO$_4$.
This photoinduced metallic state is also demonstrated by the present calculations, although the carrier concentration is set to be $x_h$ which is different from that in Sr$_{14}$Cu$_{24}$O$_{41}$.
As shown in Fig. <ref>(a), just after the photon pulse pumping, a low energy spectral weight appears inside the insulating gap, which produces the metallic reflectivity spectra.
This characteristic photoinduced change of the electronic structure in the insulating state are insensitive to the anisotropy in the transfer integrals ($r_t$) as shown in Figs. <ref>(a) and (d),
and might be common properties in a wide class of insulating states realized by the electronic interactions. <cit.>
The time resolved optical experiments were also performed in the metallic ladder cuprates,
Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ with $x=10$. <cit.>
The pump photon energy was tuned at 1.58eV corresponding to the charge transfer excitation energy in the ladder plane.
Just after the photon pumping, the low energy optical reflectivity at 0.5eV shows a reduction, implying a suppression of an initial metallic character.
The experimental reflectivity shows a slow increasing after around 0.5ps, which is interpreted to be thermalization through relaxations to the lattice degree of freedom.
The present theoretical calculations provide a possible interpretation for this experimental observation of the photoinduced suppression of the metallic state.
We confirm this photoinduced suppression through not only the calculations of the pump-probe spectra, but also the calculations of the low energy component of the kinetic energy, the one-particle excitation spectra and others.
As shown in Fig. <ref>(b)-(f), we identify that the ladder lattice effect plays an essential roles on this suppression, in contrast to a photoinduced metallic state in the Mott insulating state.
The double pulse pumping reinforces a validity of the present scenario for the photoexcited state in
metallic ladder cuprates.
The time resolved spectroscopy under the double pulse pumping was performed experimentally in insulating Sr$_{14}$Cu$_{24}$O$_{41}$. <cit.>
The first pump pulse fluence was varied and the second pulse fluence was fixed, in the same way with the present theoretical calculation introduced in Sec. <ref>.
Increases of the reflectivity at 0.5eV were observed monotonically as function of the first pulse fluence.
On the other hand, the change in the reflectivity by the second pulse pumping qualitatively depends on the first pulse fluence; the reflectivity within 0.5ps after the second pulse pumping increases (decreases) in the cases of the weak (strong) first pulse fluence.
This tendency of the reflectivity change is well reproduced qualitatively by the present calculations shown in Fig. <ref>.
In the experiments, over 0.5ps after the second pulse pumping, the reflectivity starts to increase being independent of the second pulse fluence.
This is attributable to the relaxation effects to the lattice degree of freedom which is not taken into account in the present theoretical calculations.
In conclusion, we study photoinduced carrier dynamics in a correlated electron system on a two-leg ladder lattice. The ladder Hubbard model is analyzed by utilizing the numerical exact diagonalization method in finite size clusters based on the Lanczos algorithm.
Through the calculations of the transient low-energy kinetic energy, optical conductivity spectra, one-particle excitation spectra and others, we find that the initial metallic state is suppressed by the photon pulse pumping.
This is in contrast to the photoinduced metallic state in the Mott insulating state.
The ladder lattice effect plays an essential role on this photoinduced suppression of the metallic character.
Calculations of the double pulse pumping and comparison with the corresponding pump-probe experiments
convince us of the characteristic photoinduced excited states in this correlated electron system on the two-leg ladder lattice.
We thank S. Koshihara, Y. Okimoto, R. Fukaya , M. Naka, and J. Nasu for their helpful discussions.
This work was supported by KAKENHI from the Ministry of Education, Science and Culture of
Japan (No. 26287070). Some of the numerical calculations were performed using the supercomputing facilities at ISSP, the University of Tokyo.
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|
1511.00031
|
Departamento de Física, Univ. de Los Andes, Bogotá, Colombia.
Departamento de Física
Teórica II. Univ. Complutense. 28040 Madrid. Spain.
Departamento de Física, Univ. Nacional de Colombia, Bogotá, Colombia.
We consider the $O(N+1)/O(N)$ Non-Linear Sigma Model for large $N$ as an effective theory for low-energy QCD at finite temperature $T$, in the chiral limit. At $T=0$ this formulation provides a good description of scattering data in the scalar channel and generates dynamically the $f_0(500)$ pole, the pole position lying within experimental determinations. Previous $T=0$ results with this model are updated using newer analysis of pion scattering data. We calculate the pion scattering amplitude at finite $T$ and show that it satisfies exactly thermal unitarity, which had been assumed but not formally proven in previous works. We discuss the main differences with the $T=0$ result and we show that one can define a proper renormalization scheme with $T=0$ counterterms such that the renormalized amplitude can be chosen to depend only on a few parameters. Next, we analyze the behaviour of the $f_0(500)$ pole at finite $T$, which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the $f_0(500)$ state, in a second-order transition scenario and in accordance with lattice and theoretical analysis.
§ INTRODUCTION
The study of hadronic properties at finite temperature $T$ is one of the theoretical ingredients needed to understand the behaviour of matter created in Relativistic Heavy Ion Collision experiments, such as those in RHIC and LHC (ALICE). In particular, the QCD transition involving chiral symmetry restoration and deconfinement plays a crucial role, as it is clear from the many recent advances of lattice groups in the study of the phase diagram and other thermodynamical properties <cit.>.
For vanishing baryon chemical potential, the QCD transition is a crossover for 2+1 flavours with physical quark masses, the transition temperature being about $T_c\sim$ 150 -160 MeV. In the chiral limit (vanishing light quark masses for fixed strange mass) it is believed to become a second-order phase transition belonging to the universality class of the $O(4)$ model <cit.>. Lattice simulations also support this fact. Actually, in <cit.> it is shown that the lattice results are compatible with a $O(N)$-like restoration pattern in the chiral limit and for physical masses, by studying the scaling of different thermodynamical observables near $T_c$. The expected reduction in the transition temperature from the physical mass case to the chiral limit one based on those analysis is about 15-20$\%$, although subject to many lattice uncertainties <cit.> .
From the theoretical side, it is important to provide solid analysis of this chiral restoration pattern based on effective theories, given the limitations of perturbative QCD at those temperature scales. A simple model realization was historically the Linear Sigma Model (LSM) based on $O(4)\rightarrow O(3)$ spontaneous symmetry breaking <cit.>, where the $\sigma$-component of the $O(4)$ field acquires a thermal vacuum expectation value and mass, both of them vanish at the transition in the chiral limit, and $\pi-\sigma$ mesons degenerate as chiral partners. However, such a simple description is nowadays in conflict with observations: on the one hand, the $\sigma/f_0(500)$ broad resonance produced in pion-pion scattering and listed in the PDG <cit.> is not compatible with a particle-like state (see <cit.> for a recent review). On the other hand, to reproduce consistently pion data, the LSM requires working in a strong coupling regime, invalidating the perturbative description. Nevertheless, it is clear that the $\sigma/f_{0}(500)$ state must play an important role in chiral restoration, since it shares the quantum numbers of the QCD vacuum. Chiral symmetry restoration has also been studied within QCD inspired models like the Nambu-Jona-Lasinio or Gross-Neveu ones <cit.>.
A systematic and model-independent framework that takes into account the relevant light meson degrees of freedom and their interactions is Chiral Perturbation Theory (ChPT) <cit.>. The effective ChPT Lagrangian is constructed as a derivative and mass expansion ${\mathcal L}={\mathcal L}_{p^2}+{\mathcal L}_{p^4}+\dots$, where $p$ denotes generically a meson energy scale compared to the chiral scale $\Lambda_{\chi}\sim$ 1 GeV. The lowest order Lagrangian ${\mathcal L}_{p^2}$ is the Non-linear Sigma Model (NLSM). The use of energy expansions in chiral effective theories is also justified at finite temperature to describe Heavy Ion Physics. Pions are actually the most copiously produced particles after a Heavy Ion Collision and most of their properties from hadronization to thermal freeze-out can be reasonably described within the temperature range where these theories are applicable. Thus, the chiral restoring behaviour in terms of the quark condensate is qualitatively obtained within ChPT <cit.>. Moreover, the introduction of realistic pion interactions by demanding unitarity through the Inverse Amplitude Method (IAM) <cit.> extended at finite $T$ <cit.> improves ChPT, providing a more accurate description of several effects of interest in a Heavy-Ion environment, such as thermal resonances, transport coefficients and electromagnetic corrections <cit.>. This approach also provides a novel understanding of the role of the $\sigma/f_0(500)$ broad resonant state in chiral symmetry restoration, without having to deal with the typical LSM drawbacks. Thus, the unitarized $\pi\pi$ scattering amplitude within ChPT at finite temperature develops a $I=J=0$ thermal pole at $s_p= \left[M_p(T)-i \Gamma_p(T)/2\right]^2$, which for $T=0$ corresponds to the PDG state, and whose trajectory in the complex plane as $T$ varies shows some interesting features: the sudden drop of $M_p (T)$ towards the two-pion threshold can be interpreted in terms of chiral symmetry restoration, as opposed for instance to the $I=J=1$ $\rho$-channel where the mass drop is much softer. In addition, it has been recently shown <cit.> that the scalar susceptibility saturated with this $\sigma$-like state, with squared mass $M_S^2=M_p^2-\Gamma_p^2/4$, develops a maximum near $T_{c}$ compatible with lattice data, unlike the pure ChPT prediction which is monotonically increasing. Moreover, chiral partners in the scalar-pseudoscalar sector are understood through degeneration of correlators and susceptibilities <cit.>, something which is also directly seen in lattice data <cit.>. The role of the $f_0(500)$ state for chiral restoration could become more complicated if its possible tetraquark component is also considered at finite temperature <cit.>.
A crucial step in the unitarized approach is the condition of exact thermal unitarity for the partial waves, with a thermal space factor modified by the Bose-Einstein distribution function. This condition holds perturbatively in ChPT <cit.> and the unitarized amplitude is constructed by requiring thermal unitarity to all orders, based on the physical collision processes occurring in the thermal bath <cit.>. However, it is important to emphasize that thermal unitarity for the full amplitude was not formally proven in those works; in fact, that will be one of the relevant issues discussed in the present work.
Although the approaches based on effective theories in terms of the lightest mesons provide a good description of the physics involved, especially in what concerns the effect of the lightest resonances (as discussed above), a more accurate treatment near $T_{c}$ would require including heavier degrees of freedom. That is for instance the framework of the Hadron Resonance Gas, which describes the system through the statistical ensemble of all free states thermally available, and where corrections due to interactions and lattice masses can be also accounted for <cit.>. Effective chiral models including explicitly vector and axial-vector resonances have also been successfully used to depict several hadron thermal properties relevant for observables such as the dilepton and photon spectra and $\rho-a_1$ mixing/degeneration at the chiral transition <cit.>.
In this work, we will consider an alternative approach to the thermal pion scattering amplitude, namely the limit of large number of Nambu-Goldstone bosons $N$, or in other words, large number of light flavours with no strangeness, as treated before at $T=0$ in <cit.>. Previous large-$N$ analysis at $T\neq 0$ can be found in <cit.>. Within this framework, the lowest order chiral effective Lagrangian for low-energy QCD will be the $O(N+1)/O(N)$ NLSM, whose corresponding symmetry breaking pattern is $O(N+1)\rightarrow O(N)$. As we have just commented, the latter is believed to take place in chiral symmetry restoration for $N=3$, since $O(4)$ and $O(3)$ are respectively isomorphic to the isospin groups $SU_L(2)\otimes SU_R(2)$ and $SU_V(2)$. This technique has the advantage of allowing for a partial resummation of the scattering amplitude preserving many physical properties such as unitarity and the dynamical generation of the $f_0(500)$ pole, which will help us to shed more light on the chiral restoring issues discussed before. We will work in the chiral limit, since it simplifies considerably the analysis, besides enhancing chiral-restoring effects, as explained above. At this point, it is important to remark that massless pions remain massless at finite temperature <cit.>, unlike many other instances in thermal field theory where elementary massless excitations acquire mass in the thermal bath, like high-$T$ fermions <cit.>, gauge fields <cit.> including large-$N_f$ analysis <cit.> or when electromagnetic corrections are switched on <cit.>.
The study of the large-$N$ approach in low-energy QCD implies a simplification of the pion dynamics <cit.> without changing essential features such as analyticity, unitarity and the low-energy behaviour for pion scattering. This is fully accomplished when working in the functional formalism of the theory <cit.> so that the Lagrangian is built as $O(N+1)/O(N)=S^N$ covariant and $O(N+1)$ invariant (in the chiral limit). Furthermore, as we will see in detail here, this approach will allow to describe consistently the $f_0(500)$ state through its pole in the second Riemann sheet, where the parameters of the model are fitted to pion-pion scattering phase shift data. Thanks to the fact that the model is exactly unitary, we will be granted to go beyond the standard perturbative ChPT description for the scattering <cit.>, as a complementary description of unitarization methods such as the IAM.
An additional observation that makes this approach suitable for studying chiral restoring effects is that in order to reproduce correctly the $f_{0}(500)$ pole (linked to chiral restoration as mentioned before), the dominant contributions to $\pi\pi$ scattering are the loop diagrams from the leading order chiral Lagrangian, rather than the particular form of higher order terms needed to renormalize the amplitude <cit.>. Thus, the large-$N$ limit framework provides a resummation of the dominant loop contributions needed to maintain exact unitarity, so that the scalar pole can be correctly described.
With the above motivations kept in mind, we will analyze in this work elastic pion-pion scattering at finite temperature within the large-$N$ $O(N+1)/O(N)$ model in the chiral limit. We will show that at $T=0$ one gets reasonable values for the $f_{0}(500)$ pole from a fit to experimental data of a two-parameter partial wave. The extension to $T\neq 0$ includes a formal discussion of the renormalizability of the model, which as expected can be carried out in terms of $T=0$ counterterms, although with important subtleties to be taken into account. The important feature of exact unitarity is demonstrated, including thermal corrections, something that allows us to define the second-sheet pole. Having fixed the $T=0$ pole position, its $T$ dependence is obtained and it is shown that the results are compatible with a second-order chiral restoring phase transition, consistently with previous determinations and lattice data.
The paper is organized as follows: In section <ref> we introduce our large $N$ approach within the framework of the massless NLSM and work out the diagrammatic expansion for pion scattering, both at zero and at finite temperature; section <ref> is devoted to explain the renormalization procedure, for which technical details are relegated to Appendix <ref>. In section <ref> we perform the analysis of the $I=J=0$ partial wave, providing a fit to $T=0$ data and showing that the large-$N$ amplitude satisfies exactly unitarity at zero and finite temperature. The latter grants us to define the Riemann second-sheet pole corresponding to the $f_0(500)$ state, which we study in detail in section <ref>, paying special attention to its thermal evolution and the connection with chiral symmetry restoration. Our conclusions are presented in section <ref>.
§ PION SCATTERING AMPLITUDE IN THE $O(N+1)/O(N)$ NLSM
§.§ Diagrammatics at Zero Temperature
In a theory with spontaneous symmetry breaking $O(N+1)\rightarrow O(N)$, the coset space where the Nambu-Goldstone Bosons (NGB) are defined is the $N$-dimensional sphere $S^{N}=O(N+1)/O(N)$. In such a theory, the most general $O(N+1)$-invariant and $S^{N}$ covariant Lagrangian in the chiral limit can be obtained as a derivative expansion of the NGB modes, whose lowest order expression is given by the NLSM <cit.>:
\begin{equation}
\mathcal{L}_{NLSM}=\frac{1}{2}g_{ab}(\pif)\partial_{\mu}\pif^{a}\partial^{\mu}\pif^{b},
\label{NLSM}
\end{equation}
where $g_{ab}(\pif)$ is a metric in the $S^{N}$ manifold which is parametrized in the NGB $\pif_a$ coordinates as
\begin{equation}
\label{metric}
\end{equation}
with $\pif^2=\displaystyle\sum_{a=1}^{N}\pif_{a}\pif^{a}$.
As explained in <cit.>, the advantage of choosing a $S^{N}$ covariant formalism is that we can easily construct $O(N+1)$ invariant Lagrangians of higher order by properly contracting indices with the $g_{ab}$ metric. In addition, this formalism ensures the independence of the Green functions on NGB field reparametrizations, i.e., when changing coordinates in $S^N$. The covariance of the quantum model is guaranteed as long as we work in the Dimensional Regularization (DR) scheme with $D=4-\epsilon$, since the metric factor appearing in the NGB quantum measure $(\mathcal{D}\pif\sqrt{g})$, with $g$ the metric determinant, amounts to add to the Lagrangian a term proportional to $\delta^{D}(0)$ <cit.> which vanishes in DR <cit.>.
Although there is no need to invoke the LSM in the above construction, one can understand the NLSM as the kinetic part of the LSM when the vacuum constraint $\pif^2+\sigma^{2}=NF^{2}$ is imposed, where $v=\sqrt{N}F$ is the vacuum expectation value acquired by the $\sigma$ field at tree level. In that way, it is easy to understand the $N$-scaling of the $NF^2$ constant, where $F_{\pi}^{\,2}=NF^{2}$ is the pion decay constant at this order for the usual $N=3$ case <cit.>.
The Lagrangian (<ref>) provides the standard kinetic term for the NGB fields and along with it, an infinite set of self-interaction terms with an arbitrary even number of NGB. These interactions are obtained when the metric (<ref>) is expanded and written as a function of two field derivatives and powers of the pion field. Hence, to obtain the relevant Feynman rules for pion scattering, we can write the Lagrangian (<ref>) as follows:
\begin{equation}
\mathcal{L}_{NLSM}=\frac{1}{2}\partial_{\mu}\pif_{a}\partial^{\mu}\pif^{a}+\frac{1}{8NF^{2}}(\partial_{\mu}\pif^2)^{2}\left[1+\frac{\pif^2}{NF^{2}}+\left(\frac{\pif^2}{NF^{2}}\right)^{2}+\cdots\right].
\label{lagrangian1}
\end{equation}
Feynman rules and diagrams at tree level, where dashed lines correspond to multiple pairs of pion lines.
The first term in the latter expansion gives the standard kinetic Lagrangian and the rest are the $2n$-vertices with $n\geq 2$, represented in Fig. <ref>. The Feynman rule in momentum space for each vertex is $\frac{(p_A+p_B)\cdot(p_C+p_D)}{(NF^2)^{n-1}}\delta_{AB}\delta_{CD}\dots$ with $A,B$ and $C,D$ the isospin indices of all possible choices of four different lines in the diagram and where the dots indicate the rest of pair contractions, i.e., all products $\delta_{ij}$ with $ij$ the isospin indices of pairs different from $AB$ and $CD$. With these rules, we proceed to calculate the $\pi^{a}\pi^{b}\rightarrow \pi^{c}\pi^{d}$ scattering that, as customary, is parametrized as:
\begin{equation}
\label{generalT}
\end{equation}
Here, $s,t,u$ are the Mandelstam variables $s=(p_a+p_b)^2=(p_c+p_d)^2$, $ t=(p_a-p_c)^2$, $u=(p_a-p_d)^2$ and use of isospin and crossing symmetry has been made to parametrize the amplitude. We will denote $p=p_a+p_b$.
The dominant contribution to $A(s,t,u)$ in the large-$N$ limit comes from the diagrams showed in Fig. <ref>. Thus, when considering diagrams of arbitrary loop order with just the four pion vertex in Fig.<ref>, the dominant contribution of isospin flux is that where the
pairs of lines are chosen as $(p_A+p_B)\cdot(p_C+p_D)=(q_1+p-q_1)\cdot(q_2+p-q_2)=s$ for an internal vertex, i.e., with no attachments to any external line ($q_1$ and $q_2$ are the four-momenta of the loops attached to that vertex) and
$(p_A+p_B)\cdot(p_C+p_D)=p\cdot(q_1+p-q_1)=s$ for the external ones. In that way, an additional factor of $N$ is generated for every pair of vertices between a given loop, coming from a contraction $\delta_{ef}\delta^{ef}=N$, where $e$ ($f$) is one of the two free indices in the first (second) vertex. The result is a net factor $s/(NF^{2})$ for every vertex and an additional $N$ for every loop, so that the resulting amplitude is $\Od(1/N)$. Other loop contributions are subdominant according to this counting.
Zero-Temperature scattering amplitude.
For those vertices in Fig.<ref> with more than four pions, the only way to compensate the additional $(1/N)^{n-1}$ factors of the $2n$-pion vertex is to close $2n-4$ of them in tadpole-like contributions, giving rise to a $N^{n-2}$ factor, so that this contribution would count the same $1/N$ as the four-pion vertex. Those tadpole insertions correspond to fields sharing the same isospin index with no derivatives, since $\partial_\mu G(x)\vert_{x=0}=0$, with $G(x)$ the free pion propagator. At $T=0$, they actually vanish, since the tadpole contribution $G_{T=0}(x=0)=0$ in the chiral limit. That will be not the case at $T\neq0$, as we discuss in section <ref> and will become one of the main novelties of the present calculation. Finally, note also that in the chiral limit, the pion propagator is not corrected by loop effects and hence needs no renormalization <cit.>. For instance, a tadpole correction to the self-energy would require contracting two pion lines with the same isospin index to produce a $N$ factor, but that gives $\partial_\mu G(x)\vert_{x=0}=0$, and other contributions are non-dominant with respect to the tree level propagator, which is $\Od(1)$.
From the latter considerations, and after including the proper combinatoric factors, the $A(s,t,u)$ function in (<ref>) depends only on $s$ to leading order in $1/N$, which is actually one of the main simplifications of this approach, and is given by
\begin{equation}
\frac{s}{NF^{2}}\sum_{k=0}^\infty \left[\frac{s J(s)}{2F^2}\right]^k=
\frac{s}{NF^{2}[1-s\,J(s)/2F^{2}]},
\label{zerotamp}
\end{equation}
where $J(s)$ is the usual logarithmically-divergent loop integral that in the DR scheme reads <cit.>
\begin{equation}
J(s)=-i\int \frac{d^D q}{(2\pi)^D}\frac{1}{q^2}\frac{1}{(p-q)^2}=J_\epsilon(\mu)+\frac{1}{16\pi^2}\ln \left(\frac{\mu^{2}}{-s}\right)
\label{Js}
\end{equation}
and $J_\epsilon (\mu)$ contains the divergent part:
\begin{equation}
J_\epsilon(\mu)=-2\lambda(\mu)+\frac{1}{16\pi^2}=\frac{1}{16\pi^2}\left[\frac{2}{\epsilon}+\ln (4\pi)-\gamma+2-\ln\mu^2\right]+\Od(\epsilon).
\label{Jepsilon}
\end{equation}
Here $\lambda(\mu)=\frac{\Gamma(1-D/2)}{2(4\pi)^{D/2}}\mu^{D-4}$, $\gamma$ is Euler's constant and $\mu$ is the renormalization scale. Note that we follow the convention in <cit.> to define the pole contribution $\lambda(\mu)$ but we include the $1/(16\pi^2)$ contribution in the divergent part, unlike in <cit.>, in order to compare easily with previous large-$N$ works <cit.>.
We recall that for $s\in\IR$ and $s\geq0$ (i.e., above the two-pion threshold which in the chiral limit is at $s=0$) we can easily obtain the imaginary parts of the loop integral as the usual unitarity cut contribution (see section <ref>):
\begin{equation}
\im J(s+i0^+)= \frac{1}{16\pi} \qquad (s>0)
\label{imJ0}
\end{equation}
while $\im J(s)=0$ for $s<0$.
In section <ref> we will discuss the renormalization procedure implemented to absorb the divergent part (<ref>), but before that, let us explain the main distinctive features of the $T\neq 0$ calculation.
§.§ Diagrammatics at Finite Temperature
We will work within the imaginary-time formalism of Thermal Field Theory <cit.> so that the thermal scattering amplitude is understood as the analytic continuation to continuous energies of the corresponding four-point Green function after performing the loop Matsubara sums and applying the LSZ standard reduction formula for $T=0$ asymptotic states <cit.>.
Comparing with the analysis performed in the previous section, the first observation is that the absence of renormalization for the pion propagator remains the same at $T\neq 0$, so pions do not acquire effective thermal masses, following the same diagrammatic argument as before. However, there is an important difference with the $T=0$ case and is that the tadpole contribution is now different from zero in the chiral limit, namely <cit.>:
\begin{equation}
G_{T}(x=0)\equiv I_\beta=\tintq \frac{1}{\omega_n^2+\modq^2}=\frac{T^2}{12},
\label{tadpole}
\end{equation}
with the Matsubara frequencies $\omega_k=2\pi kT$, $k\in\IZ$.
This means that from now on, the diagrams coming from closing pairs of extra pion lines in the vertices with 6 or more legs in Fig.<ref> have to be considered. To accomplish this in an efficient way, we construct the effective thermal tadpole vertex given in Fig.<ref>, and we rebuild the scattering amplitude to all perturbative orders with the associated Feynman rule of the thermal vertex, something that we show schematically in Fig.<ref>.
Construction of the effective thermal tadpole vertex, where dashed lines correspond to multiple tadpole insertions.
Finite-Temperature scattering amplitude.
In addition, we have to take into account that the loop integral is also $T$-dependent, so that the thermal amplitude to leading order in $1/N$ is given by
\begin{equation}
A(p_{0},\modp;T)\equiv A(p;T)=\frac{s}{NF^{2}}\frac{f(I_{\beta})}{1-\frac{s}{2F^{2}}f(I_{\beta})J(p_0,\modp;T)},
\label{thermalampbare}
\end{equation}
which depends now separately on the space and time components of $p$ due to the loss of Lorentz covariance in the thermal bath. The vertex function reads
\begin{equation}
\end{equation}
and the finite-$T$ loop integral $J(p_0,\modp;T)$ is the analytic continuation of the external Matsubara frequency $i\omega_m\rightarrow p_0+i0^+$ of
\begin{equation}
J(i\omega_m,\modp;T)=\tintq \frac{1}{\omega_n^2+\vert\vec{q}\vert^2}\frac{1}{(\omega_n-\omega_m)^2+\vert \vec{p}-\vec{q}\vert^2},
\label{JT}
\end{equation}
where $J(p_0,\modp;T=0)=J(s)$ in (<ref>).
Explicit expressions for the above $J$ integral for arbitrary three-momentum $\vec{p}$ can be found for instance in <cit.>. Its UV divergent part is the same as for $T=0$, since Bose-Einstein factors regulate exponentially the UV behaviour, so that we will write in general
\begin{equation}
J(p;T)=J_\epsilon (\mu)+ J_{fin}(p;T;\mu),
\label{JTsepar}
\end{equation}
with $J_\epsilon (\mu)$ in (<ref>) and $J_{fin}(p;T;\mu)$ finite and whose scale dependence is contained only in the $T=0$ part, namely $J_{fin}(p;T=0;\mu)=\frac{1}{16\pi^2}\ln(-\mu^{2}/s)$.
In this work we will be interested only in calculations in the center of mass frame (corresponding to $\vec{p}=\vec{0}$), where partial waves are defined (see section <ref>) and moreover, we have <cit.>
\begin{align}
J_{fin}(p_0,\vec{0};T;\mu)=\frac{1}{16\pi^2}\ln \left(\frac{\mu^{2}}{-s}\right)
+\delta J(s;T); \label{Jthermal1}\\
\delta J(s;T)=\frac{1}{\pi^2}\int_0^\infty dy \frac{y \ n_B(y)}{4y^2-s},
\label{Jthermal2}
\end{align}
where $s=p_{0}^{\,2}$ and $n_B(x)$ is the usual Bose-Einstein distribution function
\[n_B(x)=\frac{1}{\exp(x/T)-1}.\]
Note that $\delta J(s;T)$ is UV finite ($y\rightarrow\infty$) thanks to the $n_B(y)$ term; besides, we can easily separate the real and imaginary parts of $\delta J(s;T)$ for $s\in\IR$ and $s>0$ by isolating the pole contribution at $y=\sqrt{s}/2=\vert p_0\vert/2$ in the integrand in (<ref>) as
\begin{align}
\re\delta J(s;T)= {\cal P} \frac{1}{\pi^2}\int_0^\infty dy \frac{y \ n_B(y)}{4y^2-s},\\
\im\delta J(s+i0^+;T)=\frac{1}{8\pi}n_B(\sqrt{s}/2) \qquad (s>0),
\label{imJther}
\end{align}
while for $s<0$ there is no pole in the integrand in (<ref>).
Finally, we recall that for $s\neq 0$, $\delta J(s;T)$ is IR finite ($y\rightarrow 0^+$) while for $s\rightarrow 0^+$, it diverges as $\delta J(s;T)\sim s^{-1/2}$ so that $s\delta J(s;T)$ (as it appears in the thermal amplitude (<ref>)) remains finite (and vanishing) in that limit.
§ RENORMALIZATION OF THE SCATTERING AMPLITUDE
We will first review the main points regarding the renormalization of the model in the $T=0$ case, as discussed in <cit.>. The scattering amplitude can be renormalized by choosing an appropriate (infinite) set of counterterm Lagrangians of higher orders in derivatives and summing their contribution to the amplitude to all orders. The philosophy behind this approach is to include only those Lagrangians needed to obtain a renormalized amplitude to leading order in $1/N$, although from the symmetry arguments explained above, many other operator structures are possible. Consistently, we can consider formally the couplings, or low-energy constants (LEC) of those additional Lagrangians to be suppressed in the $1/N$ counting. In the conventional ChPT approach <cit.> all possible terms are included to a given order in the derivative/momentum expansion and then the LEC are fixed with experimental data, although the predictions are limited to low energies. The energy applicability range can be enlarged when additional conditions such as unitarization are implemented, and then the LEC can take in general different values from the perturbative ones. Here we are considering a partial resummation of the series for the amplitude, namely the leading $1/N$ contribution, which in the end can be given in a finite form that depends only on a few parameters, to be fixed to experimental data. Nevertheless, there will be some important subtleties to be taken into account in the $T\neq 0$ case, as we will explain below, in order to ensure a renormalized amplitude with a $T=0$ renormalization scheme. We will discuss the main results in this section, while additional details are given in Appendix <ref>.
Let us consider, for instance, the possible counterterms Lagrangians to fourth-order in derivatives. It is clear that one of them satisfying the symmetry constraints would be just proportional to ${\mathcal L}_{NLSM}^2$:
\begin{equation}
{\cal L}_1=\frac{g_1}{2NF^4}\left[g_{ab}(\pif)\partial_{\mu}\pif^{a}\partial^{\mu}\pif^{b}\right]^2=\frac{g_1}{2NF^4}\left[\left(\partial_\mu\pif^a\partial^\mu\pif_a\right)^2+\Od\left(\frac{\pif^6}{N}\right)\right],
\label{L4}
\end{equation}
where the normalization proportional to the bare coupling $g_1$ has been conveniently chosen. At this order there is another term allowed, i.e. ${\cal L}'_1\sim\left[g_{ab}(\pif)\partial_{\mu}\pif^{a}\partial_{\nu}\pif^{b}\right]\left[g_{cd}(\pif)\partial^{\mu}\pif^{c}\partial^{\nu}\pif^{d}\right]$. The LEC multiplying the two allowed terms are the counterpart of the $l_1,l_2$ constants of ChPT <cit.>.
The main result at $T=0$ is that the scattering amplitude can be rendered finite by a set of infinite counterterm Lagrangians which to $\Od(\pif^4)$ have the form <cit.>
\begin{equation}
{\cal L}_k=(-1)^{k+1}2^{k-2}\frac{g_k}{NF^{2(k+1)}}\left[\partial_{\mu_1}\partial_{\mu_2}\cdots \partial_{\mu_k}\pif^a\partial^{\mu_1}\partial^{\mu_2}\cdots \partial^{\mu_k}\pif_a\partial^\nu \pif^b\partial_\nu\pif_b+\Od\left(\frac{\pif^6}{N}\right)\right].
\label{Lk}
\end{equation}
This reduces to (<ref>) for $k=1$. It is indeed always possible to find at each order an adequate contraction with the $g_{ab}$ metric that gives rise to the terms (<ref>), for instance $g_{ab}(\pif)g_{cd}(\pif)\partial_{\mu_1}\partial_{\mu_2}\cdots \partial_{\mu_k}\pif^a\partial^{\mu_1}\partial^{\mu_2}\cdots \partial^{\mu_k}\pif^b\partial^\nu \pif^c\partial_\nu\pif^d$. Consistently, the LEC multiplying other possible terms such as ${\cal L}'_{1}$ can be considered formally suppressed in the $1/N$ counting.
Insertions of counterterm Lagrangian vertices for the renormalization of the scattering amplitude
The dominant contributions of the new terms to the amplitude in the $1/N$ counting arise from all possible $g_k$ insertions in diagrams of the form showed in Fig.<ref>. It is actually not difficult to see that with the covariant structures discussed above, the $\Od(\pif^6)$ terms and higher in (<ref>) give subdominant contributions, which also holds at $T\neq 0$. As it is explained in Appendix <ref>, each Lagrangian (<ref>) insertion produces a $s^{k+1}$ power in the vertex at $T=0$. Thus, summing up all the possible $g_k$ insertions in the dominant loop diagrams in Fig.<ref> is equivalent to the following redefinition of the four-pion vertex <cit.>:
\begin{align}
&\frac{s}{NF^{2}}\rightarrow \frac{s}{NF^{2}}G_{0}(s), \nonumber \\
\label{4piren}
\end{align}
with $g_0=1$, which gives for the $T=0$ amplitude:
\begin{equation}
A(s)=\frac{s}{NF^2}\frac{G_{0}(s)}{1-\frac{s G_{0}(s)}{2F^{2}}J(s)},
\label{renormamp0}
\end{equation}
or equivalently,
\begin{equation}
\frac{1}{A(s)}=\frac{NF^2}{s}\left[\frac{1}{G_0(s)}-\frac{s J(s)}{2F^2}\right],
\label{invamp0}
\end{equation}
written in a more suitable way to implement its renormalization, as we discuss below.
Now, we can renormalize the bare divergent (and scale independent) LEC $g_{k}$ correctly to absorb order by order the loop divergences coming from the $J(s)$ function. Thus, we denote $g^{R}_{k}(\mu)$ for $k\geq 1$ the renormalized (and scale dependent) couplings that are renormalized in terms of the $g_j$ with $j=0,\dots,k$ (see details in Appendix <ref>). Equivalently, we define the renormalized function
\begin{equation}
\frac{1}{G_{R}(s;\mu)}=\frac{1}{G_{0}(s)}-\frac{s}{2F^2}J_\epsilon(\mu)
\label{renormG}
\end{equation}
which replaced in (<ref>) gives rise to the renormalized amplitude:
\begin{equation}
\label{renormamp1}
\end{equation}
where the subscript $R$ is merely added to emphasize that the amplitude is finite. Recall also that the renormalized amplitude is independent of the scale $\mu$, since it was so from the original expression (<ref>), the scale dependence of $J_\epsilon(\mu)$ being cancelled by that of the renormalized $g^{R}_{k}(\mu)$.
The function $G_{R}(s;\mu)$ can also be written as a formal series in powers of $s$ by expanding both sides of eq.(<ref>) using (<ref>), so that
\begin{equation}
\label{GRexp}
\end{equation}
Taking $g^{R}_{0}(\mu)=1$ would give the order-by-order renormalization of the $g_{k}$, presented explicitly in Appendix <ref> up to $\Od(s^3)$ (specifically (<ref>) and (<ref>)).
At $T\neq0$, from general grounds, we should be able to renormalize the amplitude with $T=0$ counterterms <cit.>. However, we notice that only with the renormalization of the four-pion vertex in (<ref>) and (<ref>) is not enough to renormalize the thermal amplitude in (<ref>) (even though the divergent part of the $J$ integral is the same as at $T=0$) unless powers of $f(I_{\beta})$ were attached to the counterterm Lagrangians, something that would violate the above mentioned $T\neq0$ renormalization principle. Actually, things become more complicated, since the Feynman rules arising from terms like (<ref>) are not as simple as the $T=0$ ones, which in particular means that a given diagram mixes different $s^{k}$ powers.
In Appendix <ref> we present a detailed analysis of the renormalization scheme that has to be applied to the $T\neq 0$ case. The main conclusion is that the thermal amplitude can be rendered finite with a $T=0$ renormalization where not only the four-point vertex (<ref>) is involved, but also all $2k+4$-pion vertices of the NLSM Lagrangian (<ref>) for arbitrary integer $k$ as follows:
\begin{equation}
\frac{s}{(NF^2)^{k+1}}\rightarrow \frac{s}{(NF^2)^{k+1}}G_{0}^{k+1} (s).
\end{equation}
The above renormalization, which can be understood either in terms of an addition of effective diagrams renormalizing the vertices or as a formal renormalization of the metric function at the Lagrangian level (see Appendix <ref>), amounts to the redefinition of the effective thermal vertex in Fig.<ref> as given in (<ref>) when the corresponding tadpole diagrams are summed up. In fact, one can also arrive to the same four-pion effective vertex renormalization in (<ref>) starting by redefining the thermal effective vertex with an unknown bare function of $s$ and $T$ that absorbs the divergent part of the $J$ integral in the total amplitude. Thus, we finally have for the thermal amplitude:
\begin{equation}
A(p;T)=\frac{s G_{0}(s)f[G_{0}(s) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_{0}(s)f[G_{0}(s) I_\beta]}{2F^2}J(p;T)}.
\end{equation}
After using exactly the same renormalization method given in (<ref>), we obtain a finite renormalized thermal amplitude given by
\begin{equation}
A_R(p;T)=\frac{s G_{R}(s;\mu)f[G_{R}(s;\mu) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_{R}(s;\mu)f[G_{R}(s;\mu) I_\beta]}{2F^2}J_{fin}(p;T;\mu)},
\label{renampT}
\end{equation}
where $J_{fin}$ is the finite part of the thermal $J$ function, defined in (<ref>).
Recall that our final finite thermal amplitude (<ref>) is also independent of the $\mu$ scale, as in the $T=0$ case, since the dependence in $J_{fin}$ cancels out that of $G_{R}(s;\mu)$ encoded by the finite renormalization constants $g_{i}^{R}(\mu)$ through (<ref>). We point out also that the renormalization scheme discussed here is the same as for $T=0$ and is of course consistent with the previous analysis of the scattering amplitude in the large-$N$ NLSM in <cit.>, but at $T=0$ it is enough to consider the four-point vertex renormalization (<ref>) because the $2k+4$ vertices with $k\geq 1$ simply do not show up in the scattering amplitude at leading $1/N$ order.
The renormalized thermal amplitude (<ref>) is one of our main results. As it happened in the $T=0$ case <cit.> the infinite couplings $g_i^R$ parametrize different choices of effective theories sharing basic properties such as renormalizability, the lowest order energy expansion (low-energy theorems) and unitarity (see section <ref>). Indeed, at $T=0$ one can for instance choose the $g_i^R$ to recover the amplitude of the LSM with explicit exchange of a scalar particle. An alternative approach, which is the one we will follow here, is to fix the scale and the renormalization conditions such as only a finite number of the $g_i^R$ are nonzero. In its simplest version, we can choose $g^R_{k\geq 1}(\mu)=0$. It is not difficult to see that this condition is compatible with the renormalization conditions of the $G_R$ function, namely (<ref>), and its corresponding Renormalization Group evolution <cit.> and leaves the thermal amplitude as dependent only on two free parameters, $\mu$ and $F$:
\begin{equation}
A_R(p;T)=\frac{s f[I_\beta]}{NF^2}\frac{1}{1-\frac{s f [I_\beta]}{2F^2}J_{fin}(p;T;\mu)}.
\label{renampTfinal}
\end{equation}
We will show in section <ref> that it is possible with this approach to fit scattering data fairly well, considering that this is a chiral-limit approach (the finite pion mass case has been analyzed with the same method in <cit.>). That will be enough for our present purposes, since our main goal is to show that with a $T=0$ amplitude which complies with the above physical requirements, we can obtain a thermal behaviour compatible with different theoretical and lattice expectations regarding chiral symmetry restoration, as we discuss in section <ref>.
It is also important to stress that following the guide principle that the $T=0$ renormalization should be enough to render a finite amplitude (proven perturbatively up to $\Od(s^3)$ in Appendix <ref>), the insertion of counterterms with bare renormalization constants $g_i$, the subsequent absorption of the divergent part of the loop integrals to define the $g^R_i$ and taking $g_{k>1}^R (\mu)=0$, would have been equivalent to take the thermal amplitude (<ref>) with $J$ replaced by $J_{fin}$. What we have derived here is an explicit construction of such a renormalization scheme.
To end this section and before proceeding with the analysis of partial waves and thermal poles, we provide a first result related to the pion decay constant at finite temperature. Taking the low-energy limit of the thermal amplitude from its general renormalized form (<ref>) gives simply:
\begin{equation}
A_{R}(p;T)= \frac{s}{NF^{2}}\frac{1}{1-\frac{I_{\beta}}{F^{2}}} + \Od(s^2/F^4),
\end{equation}
which we can compare with the low-energy expression of the scattering amplitude given by Weinberg's theorem <cit.> to define a $T$-dependent pion decay constant, namely,
\begin{equation*}
\end{equation*}
so that:
\begin{equation}
F^2_{\pi}(T)=N F^2\left(1-\frac{T^{2}}{12F^{2}}\right)=F^2_\pi(0)\left[1-\frac{T^{2}}{4F_{\pi}^{\,2}(0)}\right] (N=3).
\label{fpiT}
\end{equation}
where we have used that $F_{\pi}(0)=\sqrt{N}\,F$. The result (<ref>) coincides with the known ChPT result to $\Od(T^2)$ <cit.> and with the leading $N$ contribution studied in <cit.>, which are additional consistency checks of our present analysis. A more careful analysis of $F_\pi$ beyond $\Od(s)$ would require to analyze the residue of the axial-axial correlator <cit.>.
§ PARTIAL WAVE ANALYSIS AND UNITARITY
§.§ Fitting the $I=J=0$ phase shift at $T=0$
As discussed above, we will fix the undetermined constants in the scattering amplitude from experimental information. For that purpose and for the subsequent analysis of the $f_0(500)$ pole and chiral restoration, we will consider partial waves with well-defined values of total isospin $I$ and angular momentum $J$, which at $T=0$ are defined in the center of mass (COM) frame $\vec{p}=\vec{0}$:
\begin{equation}
\label{pwdef}
\end{equation}
where $P_J$ are Legendre polynomials, $\theta$ is the scattering angle and $T_I$ is a particular combination involving $A(s,t,u)$ (defined in (<ref>)) which gives the scattering amplitude at given isospin $I$, with $t(s,\cos\theta)$, $u(s,\cos\theta)$ in the COM frame. The large-$N$ analysis is specially adequate for the $I=J=0$ channel (see below), which on the other hand is the one we are interested in this work. In that case, we have:
\begin{equation}
\label{T0def}
\end{equation}
with $A_R(s)$ given for $T=0$ in (<ref>) and where we have made use of the fact that $A(s,t,u)$ depends only on $s$ to leading order in $1/N$ and consistently, we only take the leading order also for the partial wave combination (<ref>). The final result is that the $I=J=0$ partial wave is independent of $N$ (to leading order). At $T=0$ we have then:
\begin{equation}
a_{00}(s)=\frac{s}{32\pi F^2}\frac{G_{R}(s;\mu)}{1-\frac{s\,G_{R}(s;\mu)}{32\pi^{2}F^{2}}\ln\left(\frac{\mu^{2}}{-s}\right)}.
\label{a00T0}
\end{equation}
Recall that within the large-$N$ framework, the other possible isospin channels for pion scattering, namely $I=1$ and $I=2$, are subdominant, since they are proportional to $1/N$. This analysis is therefore particularly suited for $a_{00}$ <cit.> which for the thermal case is the most relevant one regarding chiral symmetry restoration from the point of view of thermodynamic quantities such as the scalar susceptibility, as explained above.
As discussed in section <ref>, we will work within the minimal approach for which $g_k^R(\mu)=0$ for $k\geq 1$ so that we end up with the $T=0$ partial wave in (<ref>) with $G_R=1$. Defining the phase shift as customary for elastic channels $a_{IJ}=\vert a_{IJ} \vert e^{i\delta_{IJ}}$, we can use this result to try to fit experimental phase shift data. There are several comments that are pertinent at this point: first, the choice of data sets is delicate because there have been several experiments over the years with results sometimes incompatible among them for this channel. In addition, we have to take into account that we are working in the chiral limit, so that we expect our amplitude to describe more naturally data sufficiently away from threshold, i.e., typically for large $\sqrt{s}/(2m_\pi)$. On the other hand, there is also a natural upper limit of applicability for $\sqrt{s}$, namely below the next resonance mass in this channel, which is the $f_0(980)$. This would need the inclusion of the strange sector. Thus, we have chosen as data for the fit the sets given by <cit.> in the $\sqrt{s}$ range 450-800 MeV. In addition, we consider also the parametrization of the scattering amplitude described in <cit.> in the same energy region. That parametrization provides a precise description of $\pi\pi$ scattering from a combined analysis based on dispersion relations and provides an accurate prediction of the $f_0(500)$ and $f_0(980)$ pole parameters. We do not include in the fit the recent (and also more precise) data of the NA48 experiment <cit.> which are very low-energy data below $\sqrt{s}\sim$ 400 MeV. Those low-energy data are very well described by the parametrization <cit.>. In the fit to the parametrization <cit.>, we select points with a 5 MeV energy interval and take into account the small uncertainties given in that paper.
Another important point is that in principle the values obtained for $F$ in our fits should not be far from the physical value of the pion decay constant $F_\pi=\sqrt{N}F$. In the chiral limit, $F_\pi\simeq$ 87 MeV <cit.>, so that $F$ would be around $F_\pi/\sqrt{3}\simeq$ 51 MeV. However, once again it is important to stress that we are forcing our chiral limit amplitude to fit data for massive pions and it is then not surprising that we need a higher value for $F$ since mass corrections increase the $F_\pi$ value, so that we are encoding in $F$ a great part of the uncertainty we have in our chiral limit analysis.
In any case, it is not the purpose of this work to provide a very precise fit to experimental data, as in other unitarized or dispersive approaches <cit.> which can even be compared to lattice results by suitable mass extrapolations <cit.>. After all, this is just a two-parameter fit in the chiral limit. We just need some reasonable reference values for the parameters such as we generate dynamically a pole in the second Riemann sheet with consistent values for the pole position, so that the thermal behaviour corresponds to a physically realistic situation.
In Fig.<ref> we show the $I=J=0$ phase shift as a function of the COM energy obtained with our large-$N$ amplitude with our best fits to Grayer data and to Peláez et al parametrization. The corresponding fit parameters are given in Table <ref>.
$I=J=0$ channel phase shift for different fits, as explained in main text. Peláez parametrization is given in <cit.>, NA48 low-energy data in <cit.> and Grayer data in <cit.>.
Parameters Grayer Peláez 1 Peláez 2
$F\pm\Delta F\text{ (MeV)}$ 63.16$\pm$1.62 65 (fixed) 75.98$\pm$ 0.16
$\mu\pm\Delta\mu\text{ (MeV)}$ 1523.35$\pm$143.34 1607.89$\pm$3.62 2763.51 $\pm$ 23.81
$R^{2}$ 0.9958 0.9951 0.9999
Parameters for the Grayer and Peláez data fits and their respective coefficients of determination.
The behaviour in this region is typically flat, compatible with having a wide resonance far from the real axis, in contrast with the $I=J=1$ channel where the presence of the $\rho(770)$ narrow resonance is clearly evident for real $s$ <cit.>. The fits are less sensitive to the value of $\mu$, which is natural taking into account that the dependence with that parameter is logarithmic. As commented above, the values for $F$ are rather high, compared with the expected value, which is a consequence of dealing with the chiral limit. In the case of the fit to the points in <cit.>, this effect is particularly notorious in the fit named “Peláez 2". However, we present the results of that fit anyway, because of its remarkably good accuracy to reproduce the parametrization <cit.>, even in the very-low energy region, where as commented the approach was not meant to be applicable. In that sense, this is the fit that describes better the scattering data for this channel, although the price to pay is an unnatural deviation in the pion decay constant. The chiral limit restriction, as well as other possible effects suppressed in the large-$N$ limit, such as the $t,u$ dependence of the amplitude, are encoded in that $F$ value. Alternatively, in the fit named “Peláez 1", we fix the value $F$=65 MeV such that we get a similar fit quality as the Grayer one, parametrized by $R^2$. Consequently, the values of $\mu$ obtained in those fits also remain close to each other. The uncertainties in $F$ and $\mu$ given in Table <ref> only include the error in the fit to the selected data, and are therefore clearly underestimates, taking into account the additional sources of uncertainty mentioned above. In this context, it is also useful to compare with the values obtained in <cit.> for a fit including mass corrections, giving $F=\text{55.41 MeV and }\mu=775\text{ MeV}$. We will denote the latter values as “standard values", whose corresponding curve with our chiral-limit amplitude is also depicted in Fig.<ref>. It is not surprising that this curve does not fit the data properly, because the parameters are taken from the massive case fit. Finally, to check the robustness of our approach we have also tried to fit the same data sets by including one nonzero additional parameter $g_1^R$ in $G_R$ so that we have now a three-parameter fit. The result is that the best fit yields values for $F$ and $\mu$ very close to those in Table <ref> with $g_1^R$ of the expected natural order for the LEC <cit.> but compatible with zero. This is a consistency check of this approach, reinforcing also the idea commented in the introduction that the $I=J=0$ channel is less sensitive to the LEC than to the loop effects.
§.§ Unitarity at zero and finite temperature and the $f_0(500)$ pole
At $T=0$ the unitarity condition for partial waves in elastic pion scattering reads $\im a_{IJ}(s+i0^+)=\sigma(s,m_\pi)\vert a_{IJ}(s) \vert^2$ for $s\geq 4m_\pi^2$ (two-pion threshold), where $\sigma$ is the two-pion phase space:
\begin{equation}
\sigma(s,m_\pi)=\sqrt{1-\frac{4m_\pi^2}{s}}.
\end{equation}
Equivalently, $T=0$ unitarity reads $\im \left[a_{IJ}^{-1}(s+i0^+)\right]=-\sigma(s,m_\pi)$. It is not difficult to see that the large-$N$ $a_{00}$ partial wave at $T=0$ given in (<ref>) satisfies this condition (recall that $G_R(s)$ is a real function) in the chiral limit:
\begin{equation}
\im\left[\frac{1}{a_{00}(s+i 0^+)}\right]=-\frac{1}{\pi}\im\left[\ln\left(\frac{\mu^2}{-s-i 0^+}\right)\right]=-1=-\sigma(s,0) \quad \mbox{for} \quad s\geq 0.
\end{equation}
Exact unitarity is one of the prominent features of the large-$N$ approach. Recall that in the standard ChPT series, unitarity holds only perturbatively order by order and demanding exact unitarity is what leads for instance to the IAM method.
What is even more interesting for our purposes is that there is a thermal unitarity relation which holds perturbatively in ChPT <cit.>, and is given by $\im a_{IJ}(s+i0^+;T)=\sigma_T(s,m_\pi)\vert a_{IJ}(s;T) \vert^2$ for $s\geq 4m_\pi^2$, where the partial waves at finite temperature are defined in the center of momentum frame $\vec{p}=\vec{0}$, i.e., the frame in which pions are at rest with the thermal bath, and where the thermal phase space $\sigma_T$ is:
\begin{equation}
\sigma_T(s,m_\pi)=\sigma(s,m_\pi)\left[1+2n_B\left(\frac{\sqrt{s}}{2}\right)\right].
\label{thermalphsp}
\end{equation}
The Bose-Einstein correction in (<ref>) can be interpreted as the difference of enhancement and absorption of scattering states in the thermal bath <cit.> . If this thermal perturbative relation is imposed to hold also for the full amplitude, one ends up with a unitarized thermal amplitude which gives rise to the $T$-dependence of the $f_0(500)$ and $\rho(770)$ thermal poles <cit.>.
An important result of the present work is that the thermal unitarity relation holds exactly for the large-$N$ scattering amplitude, thus providing theoretical support to the previously mentioned works on this subject. This can be readily checked from our previous results. From the definition of partial waves in (<ref>) and (<ref>), now with the thermal amplitude $A_R$
in (<ref>) at $\vec{p}=\vec{0}$, i.e., with $J_{fin}$ given by (<ref>)-(<ref>), we have:
\begin{equation}
a_{00}(s;T)=\frac{s G_{R}(s;\mu)f[G_{R}(s;\mu) I_\beta]}{32\pi F^{2}}\frac{1}{1-\frac{s G_{R}(s;\mu)f[G_{R}(s;\mu) I_\beta]}{32\pi^2F^2}\left[\ln\left(\frac{\mu^{2}}{-s}\right)+16\pi^2\delta J(s;T)\right]}.
\label{a00T}
\end{equation}
Using (<ref>), we get now:
\begin{equation*}
\text{Im}\left[\frac{1}{a_{00}(s+i0^+;T)}\right]=-\frac{1}{\pi}\left[\pi+16\pi^{2}\text{Im}\delta J(s;T)\right]=-\left[1+2n\left(\frac{\sqrt{s}}{2}\right)\right]=-\sigma_T(s,0),
\end{equation*}
which is the thermal unitarity relation.
Unitarity allows to define the Riemann second-sheet partial wave, both at $T=0$ and at $T\neq 0$, when the amplitude is continued analytically to the $s$ complex plane so that $\im a^{II}(s-i0^+)=\im a (s+i0^+)$ for $s>4m_\pi^2$. This is achieved by $a^{II}(s;T)=a(s;T)/\left[1-2i\sigma_T a(s;T)\right]$. The second-sheet amplitude presents poles which correspond to the physical resonances, which in the case of pion scattering are the $f_0(500)$ ($I=J=0$) and $\rho(770)$ ($I=J=1$). The $T$-dependent poles can be extracted numerically by searching for zeros of $1/a^{II}(s;T)$ in the $s$ complex plane. We denote the pole position as customary by $s_p(T)=\left[M_p(T)-i\Gamma_p(T)/2\right]^2$.
In the next section, we will give the detailed results of the thermal pole evolution within our present large-$N$ approach. Before that, in Table <ref> we give the values of the $T=0$ $f_0(500)$ pole, from the partial wave in (<ref>) taking $G_R=1$, with the different parameter sets of Table <ref> and in Fig.<ref> we provide the surface-level plots for those poles. For comparison, we also present the results of the IAM in the chiral limit, using the same LEC as in previous works <cit.>.
Fit $M_{P}(T=0)$ $\Gamma_{P}(T=0)$
Grayer 438.81 536.47
Peláez 1 452.42 546.26
Peláez 2 535.53 534.59
IAM 406.20 522.70
Standard 356.97 566.05
Values for masses and widths (in MeV) of the $f_{0}(500)$ pole at zero temperature.
Surface levels for $|a_{00}^{\,II}(s;T)|^{2}$ at $T=0$ with Peláez 1,2, Grayer and Standard fit parameters respectively. The elliptic regions show the positions of the pole in the upper half of the second Riemann sheet.
These values can be compared for instance with those obtained in the analysis <cit.> and given by $M_p=457^{+14}_{-13}$ MeV, $\Gamma_p=558^{+22}_{-14}$ MeV, compatible also with the PDG values $M_p\simeq$ 400-500 MeV, $\Gamma_p=400-700$ MeV <cit.>, with a large uncertainty and where the results of different analysis can be found. We refer also to the recent review <cit.> for updated results on the $f_0(500)$ pole parameters.
The values we obtain here are compatible with those typically quoted in the literature, which is remarkable given the uncertainties explained above related mostly to our chiral limit description. This is an important step in our analysis since we want our $T=0$ values for the pole to be as close as possible to realistic values, so that we can track its temperature evolution trustfully. In fact, we see that having already paid the price of increasing somewhat the value of $F$, the values for the mass and pole position are not far from those expected in the physical case. As a rule of thumb, we would expect that in the chiral limit, the pole mass would decrease (as it does the quark-like component of this state) and the pole width would increase by a phase-space argument. That is the case for instance for the IAM pole, which in the massive case is at $M_p=441.47$ MeV, $\Gamma_p=464.34$ MeV with the same LEC that give rise to the massless pole quoted in Table <ref>. This is also the reason why the results in Table <ref> for the “Standard" values, which correspond to a massive-pion fit, give in the chiral limit a smaller mass and higher width than the other large-$N$ fits.
§ THERMAL EVOLUTION OF THE POLE AND CONNECTION WITH CHIRAL SYMMETRY RESTORATION
From our previous discussion, we can now follow the temperature evolution of the $f_0(500)$ pole and compare with previous analysis. In addition, following the proposal in <cit.>, the thermal pole can be connected with chiral symmetry restoration (in the chiral limit) via the scalar susceptibility.
The results we obtain for the pole position parameters $M_p(T)$, $\Gamma_p(T)$ in the second Riemann sheet at finite $T$ from the thermal partial wave in (<ref>) (with $G_R=1$) are given in Figs. <ref> and <ref>. We also compare with the IAM approach in the chiral limit. A general tendency is observed regardless of the approach and the parameters, and is that the pole mass decreases with $T$ while the pole width increases. Thus, in the chiral limit at finite temperature, the $f_0(500)$ remains a wide resonance below the chiral transition. However, there are significant quantitative deviations when comparing different parameter sets, the results with the “Peláez 1" and “Grayer" fits and the IAM remaining reasonably close together.
Mass of the $f_{0}(500)$ pole as functions of temperature when considering different fits in the large-$N$ framework. We also compare with the IAM approach in the chiral limit.
Width of the $f_{0}(500)$ pole as functions of temperature when considering different fits and the IAM in the chiral limit.
What is more revealing is the behaviour of $M_S^2=M_p^2-\Gamma_p^2/4$. This is nothing but the real part of the self-energy of the effective scalar state exchanged in pion scattering. On the other hand, the scalar susceptibility $\chi_S(T)=-\partial\langle \bar q q \rangle/\partial m_q$, with $\langle \bar q q \rangle$ the quark condensate and $m_q$ the quark mass, is defined as the zero four-momentum scalar correlator and is saturated precisely by $M_S^2$, assuming that the real part of the self-energy does not vary much in momentum from $p^2=0$ to $p^2=s_p$ <cit.>. This is specially relevant close to the critical region where $M_S^2$ is expected to vanish, so that:
\begin{equation}
\frac{\chi_S(T)}{\chi_S(0)}=\frac{M_S^2(0)}{M_S^2(T)}=\frac{M_p^2(0)-\Gamma_p^2(0)/4}{M_p^2(T)-\Gamma_p^2(T)/4}.
\label{scalarsus}
\end{equation}
Moreover, in <cit.> it has been shown that using the IAM scalar $f_0(500)$ thermal pole to saturate the scalar susceptibility through (<ref>) generates precisely a unitarized version of $\chi_S$ which develops a maximum very close to the critical point predicted by lattice analysis, i.e. $T_c\sim$ 155 MeV, in the physical massive case where the transition is believed to be a crossover. In that case, the maximum comes from a combination of the dropping $M_p(T)$ behaviour and the $\Gamma_p(T)$ behaviour, which grows at low and moderate temperatures due to phase space increasing but drops near the transition where mass reduction is dominant. In our present approach in the chiral limit, $\Gamma_p(T)$ grows monotonically, so that the thermal phase space dominates over mass reduction in the temperature range of interest.
In the chiral limit the transition should be a second-order one, so that such maximum should become a pole, accompanied with a significant reduction in $T_c$. We show our results for $M_S^2(T)/M_S^2(0)$ in Fig.<ref>. A clear dropping behaviour vanishing at $T_c$ is observed, corresponding to a chiral restoration second-order continuous phase transition, according to our previous discussion. The values of $T_c$ obtained for different parameters are given in Table <ref>. We also compare with the IAM in the chiral limit, which shows a similar dropping behaviour, although qualitatively different in the intermediate temperature region.
Normalized scalar mass squared (inverse scalar susceptibility) as function of the temperature for different parameter sets and the IAM.
Parameter set $T_c$ (MeV)
Grayer 92.33
Peláez 1 96.00
Peláez 2 129.07
IAM 118.23
Standard 61.20
Values for the chiral critical temperature obtained for different parameter sets and the IAM.
Let us comment now on these results. A first interesting consistency check, from the formal point of view, is that the result for $T_c$ we obtain here is independent of $N$ for large $N$, since it is extracted from a $N$-independent quantity, namely the partial wave (<ref>). This is consistent with the $T_c$ extracted from the partition function, which to leading order in $1/N$ is also $N$-independent and is given by $T_c^2=12F^2$ in the chiral limit <cit.>. That happens also in other approaches such as ChPT, where the thermal loop corrections to the quark condensate increase proportionally to $N$ but are divided by $F_\pi^2\sim NF^2$ <cit.>. However, our numerical values for $T_c$ extracted in the way we have just discussed are remarkably closer to the range expected from lattice simulations than the large-$N$ value just mentioned. As commented in the introduction, phenomenologically we expect a $T_c$ value of about 80% of the massive case, namely around 120 MeV. In addition, the predictions from our large-$N$ approach are very close to the IAM one, which is formulated for $N=3$. Thus, with our approach we obtain results closer to the real $N=3$ world, even though they come from the leading order in $1/N$. Generically speaking, we would expect up to 30% uncertainties for $N=3$, coming from the neglected $1/N$ corrections, but we see that our results are even better than this. The key point to understand this is that, apart from the large-$N$ resummation, which incorporates important formal properties such as thermal unitarity, we have chosen our parameters to obtain reliable values for the phase shifts and pole at $T=0$, i.e., close to the physical case. In this sense, it is important to remark that getting $T=0$ pole values quite close to the physical (massive) ones, by increasing the $F$ value, does not imply that the $T$-evolution of the pole towards chiral restoration should be like the massive case, e.g. for $T_c$, since there are genuine massive thermal effects that we are neglecting when taking the chiral limit, like the combined dependence of thermal distribution functions on mass and temperature <cit.>. For that reason, we get $T_c$ values closer to the expected chiral limit ones. We also mention at this point that studies of the chiral phase transition based on Renormalization Group yield $T_c\simeq$ 100.7 MeV in the chiral limit <cit.>, also very close to our present analysis.
Fit $\gamma_\chi$ $R^{2}$
Grayer 0.875 0.99987
Peláez 1 0.938 0.99997
Peláez 2 0.919 0.99995
IAM 1.012 1
Standard 0.842 0.99728
Critical exponents for $\chi_S$ extracted from our results in Fig.<ref>
Another chiral-restoration property we can examine is the scaling law for the scalar susceptibility defined through (<ref>), i.e., calculate the critical exponent $\gamma$ determined as $\chi_S(T)/\chi_S(0)\sim (T_c-T)^{-\gamma_\chi}$ for $T\rightarrow T_c^-$. The results for the best fits are showed in Table <ref>. We can compare this analysis, on the one hand, with the exact result for the nonlinear $O(N)$ model for $N\rightarrow\infty$ in four dimensions, $\gamma_\chi^{O(\infty)4D}=1+\Od(1/N^2)$ <cit.>. On the other hand, the critical exponent of the $O(4)$ three-dimensional Heisenberg model, which lattice QCD results resemble within uncertainties <cit.>, is $\gamma_\chi^{O(4)3D}\simeq $ 0.54 for $T<T_c$ in the chiral limit <cit.>. Our results and the IAM one lie close to those values, providing then a consistency check of our approach to define the scalar susceptibility from the $f_0(500)$ pole.
§ CONCLUSIONS
We have studied pion scattering in the large-$N$ $O(N+1)/O(N)$ model at finite temperature in the chiral limit and its consequences regarding the $f_0(500)$ pole and chiral symmetry restoration. Our analysis gives rise to interesting theoretical and phenomenological results, consistent with previous analysis and lattice data.
After calculating the relevant Feynman diagrams, which include an effective thermal vertex from tadpole resummation, an important part of our work has been devoted to show that it is possible to find a renormalization scheme rendering the thermal amplitude finite with a $T=0$ renormalization of the corresponding vertices. This is a nontrivial extension of the $T=0$ renormalization of the scattering amplitude, since the breaking of Lorentz covariance in the thermal bath induces crossed terms between tadpole-like and $J_T$ loop functions. In the low-energy expansion of the model, up to $\Od(s^3)$, we have checked explicitly this renormalization scheme, providing a diagrammatic and Lagrangian interpretation.
Another relevant result is that the large-$N$ thermal amplitude satisfies exactly the thermal unitarity relation, imposed in previous works as a physical condition for the exact amplitude. Its low-energy properties are also preserved, being consistent for instance with the thermal dependence of the pion decay constant.
By a suitable choice of the low-energy constants, similarly to the $T=0$ case, compatible with the scale evolution of the renormalized couplings, we end up with a phenomenological unitary amplitude depending only on two parameters, $F$ and $\mu$. By fitting those parameters to experimental data in the $I=J=0$ channel, which is more reliable for data not very close to threshold in the elastic region, we reproduce the pole position of the $f_0(500)$ in the second Riemann sheet fairly consistently with PDG values and recent determinations. The chiral limit character of our approach implies a larger value for $F$ than phenomenologically expected, but it allows to obtain pole position parameters $M_p,\Gamma_p$ closer to the physical case. The fits to data are actually very good in the chosen region, precisely the most relevant energy range concerning this resonant state.
Once the $T=0$ pole has been fixed to physical values, we have studied its evolution with temperature. The $f_0(500)$ pole remains a wide state for all the temperature range of interest, the real and imaginary parts $M_p(T)$ and $\Gamma_p(T)$ behaving similarly to the IAM analysis, showing the signature of chiral restoration. In order to explore this further, we define a scalar susceptibility $\chi_S(T)$ saturated by the inverse of $M_S^2(T)=M_p^2(T)-\Gamma_p^2(T)/4$, corresponding to the real part of the scalar state self-energy at zero four-momentum, which diverges at a given $T_c$ with a power law, as it corresponds to a continuous second-order phase transition in the chiral limit. The values obtained for $T_c$, as well as the critical exponent of $\chi_S$, are consistent with those obtained with other analytical approaches, such as the IAM, and with lattice analysis, being compatible with a $O(4)$ scaling. The combination of the large-$N$ framework with the phenomenological features of the $f_0(500)$ pole allows to improve the predictions of previous approaches based on the partition function. Thus, we obtain a very reasonable description of the chiral restoration transition within this approach, given the different uncertainties involved, such as possible $1/N$ corrections near the physical $N=3$ case or the absence of heavier degrees of freedom, which should play an important role near the transition and improve our simple pion gas scenario.
§ ACKNOWLEDGMENTS
Work partially supported by the Spanish Research contracts FPA2011-27853-C02-02, FPA2014-53375-C2-2-P. We also acknowledge the support
of the EU FP7 HadronPhysics3 project, the Spanish Hadron Excelence Network (FIS2014-57026-REDT) and the UCM-Santander project GR3/14 910309. Santiago Cortés thanks Prof. José Rolando Roldán and the High Energy Physics group of Universidad de los Andes and COLCIENCIAS for financial support. We are also grateful to José Ramón Peláez and Jacobo Ruiz de Elvira for useful comments and for providing us with their parametrization values for the phase shift.
§ DETAILS OF THE RENORMALIZATION PROCEDURE
A Lagrangian of the form (<ref>) gives rise to the Feynman rule $2^k\left[(p_A\cdot p_B)^k (p_C\cdot p_D)+(p_A\cdot p_B) (p_C\cdot p_D)^k\right]\delta_{AB}\delta_{CD}$ where $p_{A,B,C,D}$ are the four-momenta of the four legs and $A,B,C,D$ their isospin indices. We will consider insertions of these counterterms in diagrams of the form depicted in Fig.<ref>, which will be the dominant ones in $1/N$. For those insertions, we will have to deal then with integrals of the type:
\begin{equation}
J_n(p,T)=\int_T d^D q\frac{\left[q\cdot(p-q)\right]^n}{q^2(p-q)^2}=\frac{1}{2^n}\int_T d^Dq \frac{\left[s-q^2-(p-q)^2\right]^n}{q^2(p-q)^2} \qquad (n=0,1,2\dots),
\label{Jn}
\end{equation}
where $\int_T d^Dq$ is short for $\displaystyle T\sum_n \int\frac{d^{D-1} \vec{q}}{(2\pi)^{D-1}}$, $q_0=i\omega_n$, $p_0=i\omega_m$ and $s=p^2$ (after analytic continuation). The case $n=0$ corresponds to $J_0=J(i\omega_m,\modp;T)$ in (<ref>).
First, consider the $T=0$ case. Since $\int dq q^\alpha=0$ in DR <cit.>, the only remaining terms after expanding the numerator in (<ref>) are the $s^n$ one and the contributions $\int\frac{q^{2j}}{(p-q)^2}=\int\frac{(p+q)^{2j}}{q^2}$ and $\int\frac{(p-q)^{2j}}{q^2}$. The latter vanish also in DR since $\int dq \frac{q_1^{N_1}\cdots q_D^{N_D}}{q^2}$ ($N_i$ even) is formally proportional to $\int dq \frac{q^N}{q^2}=0$, with $N=\sum_i N_i$ (using the standard parametrization for $1/q^2=\int_0^\infty d\lambda \exp\left(-\lambda q^2\right)$). Therefore, in that case we have simply:
\begin{equation}
J_n(s;T=0)=\left(\frac{s}{2}\right)^n J(s),
\label{JnT0}
\end{equation}
with $J(s)$ given in (<ref>).
Thus, at $T=0$, any $g_k$ insertion is proportional to $s^{k+1}$, regardless of the vertex being internal of external. As stated in the main text, this allows to renormalize the amplitude at every order. As an example, let us show here the diagrams contributing up to $\Od(s^3)$ at $T=0$, which are those showed in Fig.<ref> and where we have indicated the order of every diagram. Summing up these contributions, the amplitude is finite with the following renormalization of $g_1$, $g_2$:
\begin{align}
\label{reng1}\\
\label{reng2}
\end{align}
Diagrams up to $\Od(s^3)$ for the renormalized amplitude at $T=0$. We plot the different topological configurations contributing, so that diagram (e) is multiplied by two, corresponding to the possible vertex insertions of $g_1$.
When the $T=0$ amplitude is written in terms of the renormalized constants, it adopts the form (<ref>), where to this order $G_{R}(s;\mu)=1+g^{R}_{1}(\mu)\frac{s}{F^2}+g^{R}_{2}(\mu)\frac{s^2}{F^4}+\Od(s^3)$. As it was assured above, this is equivalent to renormalize the amplitude by the functional renormalization of the four-pion vertex given in eqns. (<ref>) and (<ref>).
At $T\neq 0$, there are additional complications that need to be analyzed in detail. First of all, the simple relation (<ref>) for the integrals $J_n$ in (<ref>) does no longer hold. Namely, for $n=1$ we get directly from (<ref>):
\begin{equation}
\label{J1}
\end{equation}
with $I_\beta$ the tadpole integral in (<ref>), while for $n=2$,
\begin{equation}
J_2(p,T)=\frac{1}{4}\left[s^2J_0+4sI_\beta+\int_T d^Dq \frac{(p+q)^2}{q^2}+\int_T d^D q \frac{(p-q)^2}{q^2}\right]=\frac{s}{4}\left[sJ_0+2I_\beta\right],
\label{J2}
\end{equation}
where we have used that in DR $\int_T d^Dq \ q^{2n}=\int_T d^Dq \ (p-q)^{2n}=0$ for $n=0,1,2,\dots$ (although not for any real $n$ as it happened in the $T=0$ case) since $\int d^{D-1} \ \vec{q} \modq^{2k}=0$ for $k=0,1,2,\dots$. Note also that in (<ref>), the two contributions $\int_T d^Dq \ \frac{p\cdot q}{q^2}$ cancel among them, and also independently by parity. However, this does not happen for $n\geq 3$, meaning that the $J_n$'s are not simply linear combinations of $J_0$ and $I_\beta$ as in the previous cases. For instance, for $n\geq 3$ the following integral contributes:
\begin{equation}
\int_T d^D q \frac{(p\cdot q)^2}{q^2}=p_\mu p_\nu I^{\mu\nu}(T)=-\omega_m^2 I^{00}(T)-\modp^2I_s(T),
\end{equation}
with $I^{\mu\nu}=\int_T d^D q \frac{q^\mu q^\nu}{q^2}$ and $I_s=\frac{1}{D-1}I^{j}_{\,j}$. At $T=0$, one has $I^{\mu\nu}=g^{\mu\nu}\frac{1}{D}\int d^D q=0$ in DR, but at $T\neq 0$ the timelike and spacelike contributions decouple, from the loss of Lorentz covariance in the thermal bath, and they are in general nonzero. Besides,
\begin{align}
I_s(T)=\frac{1}{D-1}\int_T d^Dq \modq^2 \int_0^\infty d\lambda \ e^{-\lambda (\omega_n^2+\modq^2)}=\frac{1}{(4\pi)^{\frac{D-1}{2}}}\frac{T}{2}\sum_{n=-\infty}^\infty \int_0^\infty d\lambda \ e^{-\lambda \omega_n^2} \lambda^{-1-\frac{D-1}{2}}\nonumber\\
=I_s(0)+\frac{1}{(4\pi)^{\frac{D}{2}}} \sum_{k=1}^\infty \int_0^\infty d\lambda \ \lambda^{-1-\frac{D}{2}} e^{-\frac{k^2}{4T^2\lambda}}=\frac{1}{2}g_0(0,T)=\frac{\pi^2}{90}T^4,
\label{Is}
\end{align}
where we have made use of the standard Feynman parametrization as well as Poisson's summation formula $\sum_n F(n)=\sum_k \int_{-\infty}^\infty dx F(x) \exp{(2\pi ikx)}$. The function $g_0(M,T)$ is defined in <cit.>. On the other hand, using again the DR properties, $I^{00}(T)=-(D-1)I_s (T)$. Note that $I_s(0)=0$ so that these pure thermal contributions would not give rise to new type of divergences, i.e. different from those coming from the standard loop integral in (<ref>).
Therefore, the Feynman rules at $T\neq 0$ for $g_k$ insertions change with respect to the $T=0$ ones. Namely, a $g_k$ insertion in the generic diagram of Fig.<ref> produces an integral of the type (<ref>) for the internal loop momenta $q$ and then is not equivalent to a simple $s^k$ power as for $T=0$.
One of the consequences of the above results is that when considering all the diagrams contributing to a given $s^k$ order, the $s$ and $T^2$ powers mix, so that a larger number of diagrams has to be considered. In Fig.<ref> we have displayed all the diagrams that would give $\Od(s^3)$ contributions, all of them including $g_1$ and $g_2$ insertions according to the results (<ref>) and (<ref>). The vertex with no $g_{1,2}$ insertions is the effective thermal vertex in Fig.<ref>. Attached to each diagram, we have indicated the different powers of $s^nI_\beta^m$ that it gives rise to.
Diagrams up to $\Od(s^3)$ for the renormalized amplitude at $T\neq0$. We plot the different topological configurations contributing, so that diagrams (e), (h), (j), (k), (l), (p) are multiplied by two, corresponding to the possible vertex insertions.
We have calculated all diagrams in Fig.<ref> with the Feynman rules discussed above. The result is that the amplitude to that order remains finite with the same $T=0$ renormalizations of $g_1$ and $g_2$ given in (<ref>)-(<ref>), which is a nontrivial consistency check. Furthermore, the analysis of the result reveals some interesting features that will shed light on the renormalization scheme to be followed in the general case.
First of all, we show the $\Od(s^2)$ calculation. In addition to diagrams (a), (b), (c) of Fig.<ref>, which are the counterparts of (a), (b), (c) in Fig.<ref> with the thermal vertex, we have to consider as well diagram (e) in Fig.<ref>, since it includes a $I_\beta s^2$ contribution, as well as diagram (g), whose $I_\beta^2 s^2$ part has to be taken into account. Altogether, we obtain for the amplitude at that order:
\begin{align}
A(p;T)&=\frac{s}{NF^2}f(I_\beta)\left\{1+\frac{s}{2F^2}f(I_\beta)\left[2g_1+J(p;T)\right]\right\}+\Od(s^3)\nonumber \\
\label{ampTs2}
\end{align}
where we have separated the loop integral into its divergent and finite parts according to (<ref>) and (<ref>) and we have used exactly the same renormalization of the $g_1$ constant as for $T=0$, namely (<ref>). The thermal amplitude in (<ref>) is explicitly finite and scale independent to this order. Moreover, note that we can write the thermal amplitude to that order in a form similar to the renormalized $T=0$ case in (<ref>) as follows:
\begin{align}
A(p;T)=\frac{s G_0(s)}{NF^2}\frac{1}{1-G_0(s)I_\beta/F^2}\frac{1}{1-\frac{s G_0(s)}{2F^2}\frac{1}{1-G_0(s)I_\beta/F^2}J(p;T)}+\Od(s^3)\nonumber\\=
\frac{s G_0(s)f[G_0(s) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_0(s)f[G_0(s) I_\beta]}{2F^2}J(p;T)}+\Od(s^3),
\end{align}
with $G_0(s)$ given by the same expression as in the $T=0$ analysis (<ref>) to this order, i.e., $G_0(s)=1+g_1(s/F^2) +\Od(s^2)$. Thus,
\begin{align}
\frac{1}{A(p;T)}=&\frac{NF^2}{s}\left[\frac{1}{G_0(s)f[G_0(s) I_\beta]}-\frac{s J(p;T)}{2F^2}\right]=\frac{NF^2}{s}\left[\frac{1}{G_0(s)}-\frac{I_\beta}{F^2}-\frac{s J(p;T)}{2F^2}\right]
\nonumber\\
=&\frac{NF^2}{s}\left[\frac{1}{G_R(s;\mu)}-\frac{I_\beta}{F^2}-\frac{s J_{fin}(p;T;\mu)}{2F^2}\right]\nonumber\\
\Rightarrow& A_R(p;T)=\frac{s G_R(s;\mu)f[G_R(s;\mu) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_R(s;\mu)f[G_R(s;\mu) I_\beta]}{2F^2}J_{fin}(p;T;\mu)}+\Od(s^3).
\end{align}
Here $G_R$ is written in terms of $G_{0}$ as in (<ref>), so that $G_{R}(s)=1+g_{1}^{R}(\mu)(s/F^2) +\Od(s^2)$, which renders the amplitude finite.
The same structure is obtained when we calculate up to $\Od(s^3)$ and then we take into account the corresponding contributions from the diagrams in Fig.<ref>. Now we obtain
\begin{align}
\nonumber\\+&\left.\frac{s^2}{F^4}f^2(I_\beta)\left[g_1J(p;T)+\frac{1}{4}J^2(p;T)+g_2\left(1-\frac{I_\beta}{F^2}\right)+g_1^2\frac{I_\beta}{F^2}\right]\right\}+\Od(s^4)\nonumber \\=&
\frac{s}{NF^2}f(I_\beta)\left\{1+\frac{s}{2F^2}f(I_\beta)\left[2g_1^R(\mu)+J_{fin}(p;T;\mu)\right]\right.\nonumber\\
+&\left.\frac{s^2}{F^4}f^2(I_\beta)\left[g^R_1 (\mu)J_{fin}(p;T;\mu)+\frac{1}{4}J_{fin}^2(p;T;\mu)+g_2^R(\mu)\left(1-\frac{I_\beta}{F^2}\right)+\left[g_1^R(\mu)\right]^2\frac{I_\beta}{F^2}\right]\right\}+\Od(s^4),
\label{ampTs3}
\end{align}
which is again finite with the renormalizations (<ref>) and (<ref>) and can also be written as
\begin{align}
A(p;T)=&\frac{s G_0(s)f[G_0(s) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_0(s)f[G_0(s) I_\beta]}{2F^2}J(p;T)}+\Od(s^4)
\nonumber\\\Rightarrow& A_R(p;T)=\frac{s G_R(s;\mu)f[G_R(s;\mu) I_\beta]}{NF^2}\frac{1}{1-\frac{s G_R(s;\mu)f[G_R(s;\mu) I_\beta]}{2F^2}J_{fin}(p;T;\mu)}+\Od(s^4),
\end{align}
with $G_{0}(s)=1+g_{1}(s/F^2)+g_{2}(s/F^2)^2+\Od(s^3)$ and $G_{R}(s;\mu)=1+g_{1}^{R}(\mu)(s/F^2)+g_{2}^{R}(\mu)(s/F^2)^2+\Od(s^3)$.
Therefore, from the previous expressions we observe that the $T\neq 0$ renormalization is equivalent to the following renormalization of the four-pion thermal effective vertex:
\begin{equation}
\frac{s}{NF^2}f(I_\beta)\rightarrow \frac{s}{NF^2} G_0(s) f\left[G_0(s) I_\beta\right].
\label{4effren}
\end{equation}
What is interesting for our purposes is that the renormalization given in (<ref>) can actually be achieved by a $T=0$ renormalization of each of the $2n$-pion vertices in the original Lagrangian by assigning them a $G_0^{n-1}(s)$ factor in momentum space, as displayed in Table <ref>, thus generalizing the $4\pif$ vertex renormalization in (<ref>).
Vertex Lagrangian Bare rule Renormalized rule
4 pions $\displaystyle -\frac{\pif^2\square\pif^2}{8NF^2}$ $\displaystyle\frac{s}{NF^2}$ $\displaystyle\frac{sG_{0}(s)}{NF^2}$
6 pions $\displaystyle-\frac{\left(\pif^2\right)^2\square\pif^2}{16 (NF^{2})^{2}}$ $\displaystyle\frac{s}{(NF^{2})^{2}} I_\beta$ $\displaystyle\frac{sG_{0}^{\,2}(s)}{(NF^{2})^{2}} I_\beta$
$2k+4$ pions $\displaystyle -\frac{\left(\pif^2\right)^{k+1}\square\pif^2}{8(k+1)(NF^{2})^{k+1}}$ $\displaystyle\frac{s}{(NF^{2})^{k+1}} I_\beta^k$ $\displaystyle\frac{sG_{0}^{\,k+1}(s)}{(NF^{2})^{k+1}}I_\beta^k$
Feynman rules renormalization for the interaction vertices of the Lagrangian (<ref>) for $\pi\pi$ scattering at large $N$.
It is actually possible to trace the origin of this renormalization scheme in terms of the contributing diagrams. Consider for instance diagrams with just one $g_1$ insertion. At $\Od(s^2)$, one has to sum the contributions to the amplitude from diagrams (c), (e) and (g) in Fig.<ref>, namely,
\begin{equation}
\frac{g_1 s^2}{NF^4}\left[1+\frac{2x}{1-x}+\frac{x^2}{(1-x)^2}\right]=\frac{g_1 s^2}{NF^4}\sum_{k=1}^\infty k x^{k-1},
\label{renint1g1}
\end{equation}
with $x=I_\beta/F^2$. Now, the infinite contributions in the sum in (<ref>) amount to four-point diagrams with $k-1$ tadpole contractions ($k=1,2,\dots$) i.e., like the diagrams in Fig.<ref> with a $kg_1 \frac{s^2}{NF^4}$ vertex. But we can interpret each of those diagrams as the contribution to the scattering amplitude of a multiplicative renormalization of the $(2+2k)$-pion vertex of the Lagrangian (<ref>) given by $\frac{s}{(NF^2)^k}\rightarrow \frac{s}{(NF^2)^k} \left[1+kg_1\frac{s}{F^2}\right]+\Od(s^3)=\frac{s}{(NF^2)^k} G_0^k(s)+\Od(s^3)$, for which closing $2(k-1)$ lines gives $(NI_\beta)^{k-1}$ to leading order in $N$. This is precisely the rule given in Table <ref>.
We have checked that we find the same rule by analyzing the remaining $g_i$ insertions in the graphs in Fig.<ref>. Specifically, the $\Od(s^3)$ contribution with one $g_1$ insertion (diagrams (e), (g), (h) and (k) ) give the renormalization rule in Table <ref> for diagrams with the one-loop $J$ function, the $\Od(s^3)$ with one $g_2$ insertion (diagrams (j) and (n)) give the linear part in $g_2$ of the $G_{0}^{k}(s)$ contribution which we have just analyzed above to $\Od(s^2)$, while the $\Od(s^3)$ with two $g_1$ insertions (diagrams (i), (l), (m), (o), (p), (q)) reproduce precisely the $g_{1}^{2}$ part of those $G_0^k(s)$ terms. Therefore, in this way we are able to reinterpret all $g_{i}$ insertions in Fig.<ref> giving $s^nI_\beta^m$ mixed powers with $n=2,3$ and $m=1,2,3,4$, in terms of the $T$-independent renormalization scheme in Table <ref> and (<ref>), from the contributing diagrams without mixed terms. For higher insertions, we would need higher order diagrams with respect to those of Fig. <ref>, i.e., up to $\Od(s^4)$.
The crucial conclusion is that following the above renormalization scheme also in the general nonperturbative case, namely starting from the full amplitude (<ref>), yields a finite scattering amplitude with a $T=0$ renormalization, as discussed in the main text.
Finally, let us comment about this renormalization scheme from the point of view of the effective Lagrangian. For that purpose, Let us write down the expansion of the NLSM Lagrangian (<ref>) as
\begin{align}
\mathcal{L}&=\frac{1}{2}g_{ab}(\pif)\partial_{\mu}\pif^{a}\partial^{\mu}\pif^{b}=\frac{1}{2}\left(\delta_{ab}+\frac{1}{NF^{2}}\frac{\pif_{a}\pif_{b}}{1-\pif^2/NF^{2}}\right)\partial_{\mu}\pif^{a}\partial^{\mu}\pif^{b} \notag \\
\notag \\
\label{laglogform}
\end{align}
where we have used
\begin{align*}
(\pif^2)^{j}(\partial_{\mu}\pif^2)(\partial^{\mu}\pif^2)&=\frac{1}{j+1}\left\{\partial_{\mu}\left[(\pif^2)^{j+1}\partial^{\mu}(\pif^2)\right]-(\pif^2)^{j+1}\square\pif^2\right\}; \\
\partial_{\mu}\pif_{a}\partial^{\mu}\pif^{a}&=\partial_{\mu}(\pif_{a}\partial^{\mu}\pif^{a})-\frac{1}{2}\square\pif^2,
\end{align*}
as well as integration by parts.
From the previous expression, we see that the Feynman rules for the renormalized Lagrangian listed in Table <ref>, as long as $\pi\pi$ scattering in the large-$N$ limit is concerned, are equivalent to replace:
\begin{equation}
(\pif^2)^{k+1}\square\pif^2\rightarrow (\pif^2)^{k+1}G_0^{k+1}(-\square)\square\pif^2.
\label{replag}
\end{equation}
Therefore, the renormalization scheme we are analyzing here is very natural in the sense that every $\pif^2$ power in the expansion is renormalized with the same power of the $G_0$ function, acting as showed in (<ref>). Since those powers come from the same metric covariant function, as indicated in (<ref>), this is consistent with the introduction of the counterterm Lagrangians in a covariant fashion as we have discussed in section <ref>.
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|
1511.00045
|
Department of Chemistry, University of California, Irvine, CA 92697
The Milestoning method has achieved great success in the calculation of equilibrium kinetic properties such as rate constants from molecular dynamics simulations. The goal of this work is to advance Milestoning into the realm of non-equilibrium statistical mechanics, in particular, the calculation of time correlation functions. In order to accomplish this, we introduce a novel methodology for obtaining flux through a given milestone configuration as a function of both time and initial configuration, and build upon it with a novel formalism describing autocorrelation for Brownian motion in a discrete configuration space. The method is then applied to three different test systems: a harmonic oscillator, which we solve analytically, a two well potential, which is solved numerically, and an atomistic molecular dynamics simulation of alanine dipeptide.
Valid PACS appear here
§ INTRODUCTION
The calculation of time correlation functions from time series measurements made along molecular dynamics trajectories plays the same central role in kinetics as calculating partition functions from sets of molecular configurations and their respective energies in the realm of thermodynamics. To put the magnitude of this task into perspective, consider a simple system where 100 different configurations are possible, and a transition between any pair of these configurations is possible. In this simple system, there are over $1.7 \times 10^{13}$ different 10 step trajectories possible (100 choose 10) without even considering the fact that the same series of 10 configurations can occur with different transition times which makes the number of possible trajectories proliferate even further! All important experimental properties can be calculated from time correlation functions measured from molecular dynamics simulations, but these effects are typically only measurable on timescales which are out of reach for brute force molecular dynamics. An example would be calculating RDCs (Residual Dipole Couplings) from NMR experiments from bond vector time correlation functions. The challenge of and demand for calculating kinetic properties from molecular dynamics simulations have caused it to become a major growth area in chemical physics <cit.> <cit.>, leading to the development of several methods, spanning from early treatments using transition state theory (TST) <cit.> <cit.>, to more recently, transition path sampling (TPS) <cit.>, transition path theory (TPT) <cit.>, and transition interface sampling (TiS) <cit.>. A common strategy in measuring kinetics in molecular dynamics simulations is the measurement of fluxes of trajectories through hyperplanes in phase space or configuration space <cit.> <cit.> . More recently, the use of the hyperplanes in the Milestoning method has been generalized to subdividing phase space into Voronoi cells, where the milestones exist as the interfaces between cells <cit.>. Thus far, Milestoning has been used to calculate many useful properties, such as equilibrium flux values through the set of milestones, rate constants <cit.>, and other equilibrium properties such as mean first passage times between states <cit.>, but the method has never before been used to calculate non-equilibrium dynamical objects such as time correlation functions. In our first paper, Advancements in Milestoning I, we introduced a methodology for rapid calculation of transition time density functions between milestone hyperplanes, the central objects of milestoning calculations, by artificially pushing the system toward the target milestone and then re-weighting the distribution to recover the true transition time distribution <cit.>. In this paper, we venture into this realm by introducing a method for calculating time correlation functions from milestoning data. In order to calculate autocorrelation from milestoning data, not only must we know the equilibrium flux values through each interface, we must also know the flux through each interface as a function of time and initial configuration. For this reason, it was necessary that we also introduce our stochastic path integral approach to calculating the time-dependent fluxes, in addition to the methodology for calculating time correlation functions from these time-dependent fluxes.
§ THEORY
§.§ Milestoning Theory
A more in-depth overview of milestoning theory can be found in our first paper <cit.>, or in <cit.>, but let us review a few of the key premises upon which our method for calculating time correlation functions hinge. The quantity of most fundamental importance in milestoning is the flux through a given milestone, for which the equation is <cit.>:
\begin{equation}
\label{m}
%q_{\alpha}(t) = p_{\alpha}\delta(t - 0^+) + \displaystyle\sum\limits_{\beta}\int_0^t q_{\beta}(t')K_{\beta\alpha}(t - t')dt'
P_s(t) = \int_0^t Q_s(t')\left[ 1-\int_0^{t-t'}K_s(\tau)d\tau \right]dt' , \nonumber
\end{equation}
\begin{equation}
Q_s(t) = 2 \delta(t)P_s(0) + \int_0^t Q_{s\pm1}(t'')K^{\mp}_{s\pm1}(t-t'')dt''
\end{equation}
where $P_{s}(t)$ is the probability of being at milestone $s$ at time $t$, (or, more specifically, arriving at time $t'$ and not leaving before time $t$ <cit.>), and $Q_{s}(t)$ is the probability of a transition to milestone $s$ at time t. $K_s(\tau)$ indicates the probability of transitioning out of milestone $s$ given an incubation time of $\tau$, thus $\int_0^{t-t'}K_s(\tau)d\tau$ is the probability of an exit from milestone $s$ anytime between $0$ and $t-t'$, which makes $1-\int_0^{t-t'}K_s(\tau)d\tau$ the probability of there not being an exit from milestone $s$ over that same time period. Since the probability of two independent events happening concurrently is the product of the two events, the equation for $P_s(t)$ is simply integrating the concurrent probabilities of arriving at milestone $s$ and not leaving over the time frame from time $0$ to $t$. Turning our attention towards the meaning of the first term, $Q_{s}(t)$, $2 \delta(t)P_s(0)$, simply represents the probability that the system is already occupying milestone $s$ at time $t = 0$, where the factor of 2 is present since the $\delta$-function is centered at zero, meaning only half of its area would be counted without this factor. $Q_{s\pm1}(t'')$ is the probability that the system transitioned into one of the two milestones adjacent to $s$ at an earlier time $t''$. $K^{\mp}_{s\pm1}(t-t'')$ is the probability of a transition from milestones $s\pm 1$ into milestone $s$. Thus the second term of the second line of equation 14 is another concurrent probability: the probability of the system entering an adjacent milestone at an earlier time, and then transitioning into milestone $s$ between time $t$ and $0$. It is important to note that all functions $P_s(t)$ and $Q_s(t)$ are calculated using the respective values of $K_s(\tau)$ between adjacent milestones, thus the set of $K_s(\tau)$ between all milestones of interest contains all the information needed to calculate kinetics using the milestoning method. It is also important to note that a $K$ function between two milestones $x = A$ and $x = B$, $K_{AB}(\tau)$, is simply a probability distribution representing the lifetime for the system remaining in state $A$ before transitioning to state $B$.
§.§ Time Correlation from Milestoning Data
This approach aims to glean the time correlation function $C(t)$ of an observable from Milestoning data. The key insight into this method is the approximation of the continuous configuration space, which we define as $x$, as a discrete space of milestone configurations. Although the formalism presented below requires that the equilibrium distribution of configurations occupied, $f(x)$, is known, any successful Milestoning simulation yields the equilibrium flux through the set of milestones, and so this set of fluxes will serve as the equilibrium distribution of configurations in our discrete space. For the sake of clarity of notation, we will be limiting our derivation to observables which are a function of configuration $x$, but it should be noted that all developments presented herein can be easily generalized to observables which are a function of both position and velocity by considering our variable $x$ as a phase space coordinate. We begin with the usual definition for a time correlation function for time-ordered measurements of an observable that is a function of configuration, $A(x;t)$, arising from the equilibrium distribution of configurations, $f(x)$:
\begin{equation}
C(t) = \langle A(x,0)A(x,t) \rangle = \int A(x_0, 0)A(x, t) f(x) dx
\label{coft}
\end{equation}
where time t is the lag time between measurements. For time $t = 0$ the time correlation function has the lower limit $C(0) = \int A(x_0,0)A(x_0,0) f(x) dx = \langle A^2 \rangle$, the variance. On the opposite extreme, given an infinite relaxation time, the mean value of $x$ at time $t$ will be equivalent to the mean at equilibrium, $\lim_{t \to \infty} \langle A(x,t) \rangle = \int A(x) f(x) dx$, which implies: $\lim_{t \to \infty} C(t) = \int A(x) \left( \int A(x) f(x) dx \right) f(x) dx = \int A(x) f(x) dx \int A(x) f(x) dx = \langle A \rangle^2$
So far, we have only discussed equilibrium probability distributions in configuration space, which we defined as $f(x)$, but let us now consider a time-dependent probability density function of configuration, which is a function of initial configuration $x(0)$. Keep in mind that time-dependent probability density functions such as these are the solutions to Fokker-Planck equations. Let us define this probability density function as $g(x, t ; x_0, 0)$, and express its mean value as a function of time and initial configuration, $\langle x(t, x_0) \rangle$, in the following manner:
\begin{equation}
\langle x(t, x_0) \rangle = \int x g(x, t ; x_0, 0) dx
\label{xg}
\end{equation}
Following suit, the expectation value of our observable $A$ as a function of time can be written as:
\begin{equation}
\langle A(x, t ; x_0, 0) \rangle = \int A(x) g(x, t ; x_0, 0) dx
\label{xg}
\end{equation}
We can now substitute $\langle A(x, t ; x_0, 0) \rangle$ for $A(x ,t)$ in the definition of a time correlation function:
\begin{equation}
C(t) = \int A(x) \left( \int A(x) g(x, t ; x_0, 0) dx \right) f(x) dx
\label{int1}
\end{equation}
As stated earlier in this section, our aim is to coarse grain the continuous configuration space of $x$ into a discrete space of milestone configurations, from which we can calculate a time correlation function. Our first step in constructing this model will be to approximate the outermost integral in $x$ with a sum over a discrete set of configurations $\{x_i\}$ multiplied by the equilibrium probability of finding the system in the configuration $i$. If we define the probability of the system being in configuration $x_i$ at time $t$ given an initial configuration $x_0$ as $P_i(t ; x_0)$, then given that our system will reach equilibrium at infinite time regardless of initial configuration, the equilibrium probability can be expressed as $P_i(\infty)$. Thus we arrive at our first discrete approximation of time correlation:
\begin{equation}
C(t) \approx \sum_i A(x_i) P_i(\infty) \left( \int A(x) g(x,x(0), t) dx \right)
\label{intr}
\end{equation}
Our next task is to approximate the remaining integral in the equation with a sum over milestone states. Equation <ref> gives us an expression for the mean value of $A(x)$ in a continuous space, given an amount of time elapsed $t$ and an initial configuration $x_0$. Now consider the case where $x$ can only occupy discrete values from the set $\left\{ x_s \right\}$. In this case, the integral in equation <ref> is replaced by a sum in a weighted average expression where each discrete value of $x_i$ multiplied by its statistical weight as a function of time:
\begin{equation}
\int A(x) g(x,x(0), t) dx \approx \sum_s A(x_s) P_s(t | x_0)
\end{equation}
Next, we substitute this weighted sum approximation into equation <ref>:
\begin{equation}
C(t) = \sum_i \left( A(x_i) P_i(\infty) \sum_s A(x_s) P_s(t | x_0) \right)
\label{geq}
\end{equation}
Note that we have now arrived at a complete expression for a discrete approximation of time correlation, with the assumption that $P_s(t | x_0)$ and $P_i(\infty)$ can be obtained from milestoning calculations. Since the set of equilibrium fluxes, $P_i(\infty)$, have been calculated from milestoning simulations since the beginning, and we will introduce a novel method for calculating $P_s(t | x_0)$ from milestoning simulations in the Random Walk / Path Integral Methodology subsection later in the article, we are able to demonstrate that time correlation can indeed be calculated from Milestoning simulations.
§ ANALYTICAL SOLUTION FOR 1D HARMONIC OSCILLATOR
In this section, we demonstrate the effectiveness of equation <ref> in approximating the time correlation function for diffusion in a harmonic potential, for which there is an analytical solution. Our potential is defined as $V(x) = \frac{1}{2} k x^2$, and it's equilibrium distribution in $x$ is the Boltzmann distribution, $f(x) = e^{-\beta V(x)}$. The closed form expression for the time-dependent probability distribution for diffusion in a harmonic well is <cit.>:
\begin{equation}
p(x, t | x_0, 0) = \frac{1}{ \sqrt{2 \pi k_BT S(t)/k}} \exp \left[ -\frac{\left( x - x_0e^{-2t/\bar{\tau}} \right)^2}{2 k_BT S(t)/k} \right]
\label{anap}
\end{equation}
where $S(t) = 1-e^{-4t/\bar{\tau}}$ and $\bar{\tau} = 2k_BT / kD$.
Given this analytical expression for $p(x, t, | x_0, 0)$, we can obtain an analytical expression for $C(t)$ by substituting $p(x, t, | x_0, 0)$ into equation <ref> for $g(x, x_i(0), t)$ and integrating. This yields the exact time correlation function $C(t)$ for diffusion in a harmonic potential:
\begin{equation}
C(t) = \frac{2 \sqrt{\pi } e^{-\frac{2 t}{\bar{\tau} }}}{\left(\frac{k}{k_B T}\right){}^{3/2} \sqrt{\frac{k_B T
\left(1-e^{-\frac{4 t}{\bar{\tau} }}\right)}{k}} \sqrt{\frac{k \left(\coth \left(\frac{2 t}{\bar{\tau}
}\right)+1\right)}{k_B T}}}
\label{aCt}
\end{equation}
Alternatively, we can apply equation <ref>, and obtain a general closed form expression for approximating $C(t)$ by summing over a discrete configuration space of $N$ milestones rather than integrating over a continuous one:
\begin{multline}
C(t) = \frac{1}{\sqrt{\frac{2 \pi k_B T \left(1-e^{-\frac{4 t}{\bar{\tau}} }\right)}{k}}} \\
\sum _{i=1}^N x_i P_i(\infty) \Delta x
\sum _{j=1}^N \left( x_j Q_{ji}(t) \Delta x + x_i Q_{ii}(t) \Delta x \right) \\
\label{gaCt}
\end{multline}
\begin{multline}
Q_{ji}(t) = \exp \left( -\frac{k \left(\coth
\left(\frac{2 t}{\bar{\tau} }\right)-1\right) \left(x_i-x_j e^{\frac{2 t}{\bar{\tau} }}\right)^2}{4 k_B T} \right) \\
Q_{ii}(t) = \exp \left( -\frac{x_i^2 k \tanh
\left(\frac{t}{\bar{\tau} }\right)}{2 k_B T} \right)
\label{trans}
\end{multline}
and $\Delta x$ is the distance between the evenly spaced milestones. $Q_{ji}(t)$ represents the discrete time-dependent probability density as a function of time that our system is in configuration $x_i$ at time $t$, given that the system was in state $x_j$ at time $t=0$. Likewise, $Q_{ii}(t)$ is the discrete probability density as a function of time that our system is still in configuration $x_i$ at time $t$ if it started in configuration $x_i$ at time $t=0$. Thinking in terms of the assumption of Markov statistics for transitions between milestones inherent to the Milestoning method, it makes sense that these probabilities are added given that we are interested in the outcome of finding our system in configuration $x_i$ whether it was already there, or it arrived there from another configuration.
The most straightforward and intuitive way to compare equations <ref> and <ref> is to plot them. In figure <ref>, we can compare the exact time correlation function for diffusion in a harmonic potential (with parameters $\beta = .35, k = 5, \text{ and } D = .2857$) with the approximate $C(t)$ generated using equation <ref>. Discretizing the space to three milestones is clearly too coarse of an approximation, but the gain in accuracy in going from 6 to 9 milestones is quite modest. As one might expect, the discrete approximation of the time correlation function is most accurate for long times and least accurate for $C(0)$. It turns out that this sacrifice in accuracy is a meager one because $C(0)$ is always available from Milestoning data because it is equivalent to the sum approximation of the variance in configuration space at equilibrium, $\sum_{i = 1}^N x_i^2 P_i(\infty)$. This will be leveraged to our advantage in the following section.
figure=cOftHO.pdf, width=3.5in
This figure shows the approximate time correlation functions calculated using equation <ref> for 3, 6, and 9 milestones overlaid on top of the exact analytical function $C(t)$.
§ NUMERICAL DEMONSTRATION
§.§ 1D Fokker-Planck Diffusion on a Bistable Potential
In order to further validate the approach of calculating time correlation functions using the nested sum in equation <ref> in a discrete configuration space to approximate integrating equation <ref> in continuous conformation space, the method was applied to a simple two well potential of equation $y = (x-1)^2(x+1)^2$, where the time evolution of the probability density function in configuration space was calculated using a Fokker-Planck formalism:
\begin{equation}
\frac{\partial \rho(x,t)}{\partial t} = D \frac{\partial^2 \rho(x,t)}{\partial x^2} + \frac{D}{k_{B}T}\frac{\partial}{\partial x}(\rho(x,t)\frac{\partial V}{\partial x})
\label{fp}
\end{equation}
By repeatedly solving equation <ref> numerically with the using the Mathematica software package <cit.>, using a normalized Gaussian distribution centered at the various $x_i(0)$ values as the initial condition, the manifolds $g(x, x_i(0), t)$ were obtained for each of the $10$ milestone configurations $x_i$ in the set $\{ -2, -1.6, ..., 1.6, 2 \}$. These manifolds were then used to find $C(t)$ using both the intermediate method described by equation <ref> (shown as red circles in figure <ref>) as well as our fully developed discrete method described by equation <ref> (shown as blue circles in figure <ref>). In the case of the equation <ref>, the integral $\int x g(x,x(0),t) dx$ was numerically integrated directly, while in the case of equation <ref>, the manifold $g(x,x(0),t)$ was used to obtain values of $P_i(x(0),t)$ by multiplying $g(x,x(0),t) \Delta x$, similar to the transformation from equation <ref> to equation <ref>, but in reverse. The results are shown superimposed over a plot of the time correlation function for the system obtained in the traditional manner by running $10^9$ steps of langevin dynamics and then calculating the time correlation function over this one long trajectory using the equation:
\begin{equation}
C(t) = \frac{1}{n - t} \sum_{i=1}^{n-t} x_i x_{t+i}
\label{trad}
\end{equation}
We would like to point out that, as we alluded to in the previous section, the data point for $C(0)$ is the only portion of the time correlation function approximated using equation <ref> with any appreciable error. In practice, the data point for $C(0)$ can always be replaced with the value obtained from the sum $C(0) = \sum_i x_i^2 P_i(\infty)$ (shown as the green ring in figure <ref>), due to the fact that the set of equilibrium probabilities, $P_i(\infty)$ are always known from Milestoning simulations.
figure=fokkerCoft, width=3.5in
This plot demonstrates a successful implementation of our method for approximating time correlation functions in continuous space by summing over time dependent joint probabilities of transitions between discrete states, as obtained in Milestoning simulations. The red rings mark the data points from implementing equation <ref>, the blue data points indicate the positions where the full nested sum approximation of equation <ref> was implemented, and the green ring is the data point for $C(0)$ calculated from equilibrium probabilities which is used to replace the value of $C(0)$ generated using equation <ref>. The data is shown superimposed over the time correlation function $C(t)$, represented by a solid black line, calculated using the traditional method of equation <ref>.
§.§ Random Walk / Path Integral Methodology
In order to make use of the formalism for obtaining autocorrelation in a discrete configuration space, as introduced in the Theory section, we require an expression for $P_s(t | x_i(0))$, i.e. the probability that our system is in configuration $s$ at time $t$, given that it was in configuration $i$ at time $t = 0$. Since previous implementations of the milestoning method have been “based on iterative determination of stationary flux vectors at milestones" <cit.>, and not the determination of non-equilibrium time dependent fluxes given some initial configuration, it was necessary to devise a methodology for obtaining the function $P_s(t | x_r(0))$ from milestoning data. In the case of diffusive systems which can be described using a Fokker-Planck formalism (eq. <ref>), the Fokker-Planck equation can be solved for a manifold $\rho(x,t)$ which represents a probability density of configurations evolving in time, where the distribution at time $t = 0$ is the distribution dictated by the initial condition and the distribution as $t \rightarrow \infty$ is equivalent to the equilibrium distribution in x. While this Fokker-Planck description can be directly solved for the time evolution of a probability density function of configurations (when tractable, as in figure <ref>), it is also possible to obtain the manifold $\rho(x,t)$ via a path integral approach using a large ensemble of trajectories generated using stochastic models such as Langevin dynamics. This equivalence was the inspiration behind the random walk / path integral method introduced in this section. There are some differences however, for example, instead of Langevin trajectories, we use random walks along the given set of milestones. Very long random walks, orders of magnitude longer than time scales accessible to molecular dynamics, can be quickly generated with minimal computational cost by taking advantage of two data sets which are already known in any milestoning calculation: the transition matrix K (essentially a Markov matrix) and the set of all $K_{AB}(\tau)$ functions, which are the probability density functions of transition times between milestone $A$ and milestone $B$. The $K_{AB}(\tau)$ functions are obtained by histogramming transition times between milestones, and each element $K_{ij}$ of the matrix K is obtained by integrating the distributions of transition times, $k_{ij}(\tau)$, over all time $\tau$ and then normalizing each row to impose the constraint that the system at state $i$ has probability 1 of transitioning to one of the states to which it is coupled ($j$). Since the matrix K gives the equilibrium transition probabilities between milestones, and the $k_{ij}$ functions are probability density functions for the transition time between connected milestones, these two pieces of information can be used to construct time-dependent random walks along a set of milestones. Each step taken from some current configuration $i$ is chosen by selecting between each possible coupled state $j$, weighted by the transition probabilities from K, next, the amount of time each selected transition from state $i$ to $j$ took is selected randomly from the distribution defined by $k_{ij}(\tau)$. In this manner, trajectories of arbitrary length in this discrete space can be very quickly generated in only the amount of CPU time necessary to select $2N$ random numbers, where $N$ is the desired number of steps in the random walk. Once a large set of these random walks is generated, they can be used to calculate discrete versions of the same $\rho(x,t)$ manifolds which would be obtained as the solutions to the Fokker-Planck equation (see figure <ref>). To elaborate on this, consider a single random walk along the milestone configurations. If, at each time step, we histogram the frequency with which our system has visited each milestone configuration up to that point in time into a normalized distribution, then we have constructed a discrete manifold in configuration space $x$ and time $t$ which represents the time evolution of the probability distribution of finding our system in a particular configuration for this particular realization of a random walk in our discrete configuration space. From here, it only remains to average the set of probability distributions generated from numerous manifestations of the random walk. An alternative approach to calculating time correlation functions from these random walks would be to “connect the dots” along the random walk using an interpolation method, and then use the traditional approach to numerically calculating time correlation, shown in equation <ref>, from the resulting continuous function, as shown in figure <ref>.
figure=surfacesContVsDisc.pdf, width=4.0in
This figure shows a graphical comparison between the time evolution of a discrete probability distribution for a set of 5 milestone configurations subjected to the two well 1D potential found in the Numerical Demonstration section using our random walk / path integral methodology (part A), and the manifold representing the time evolution of a continuous probability density function of configurations for the same two well system subjected to Fokker-Planck diffusion (part B). Part A is the set of probabilities as a function of time for the system being found at each milestone configuration, given that the system was in configuration $x = -1$ at time $t = 0$, and part B shows Fokker-Planck diffusion on the same two well system. Note that the random walk in part A began at the milestone located at $x = -1$2, thus we see a decay from $\{ P_1(0) = 0, P_2(0) = 1, P_3(0) = 0, P_4(0) = 0 , P_5(0)\}$ to the equilibrium distribution, the same way our initial continuous distribution, a normalized Gaussian centered at $-1$, decays to the equilibrium probability distribution predicted by the Bolzmann distribution for the two well potential, and both evolve in time on about the same time scale.
figure=1Dcoftfromdist.pdf, width=3.5in
Shown here are time correlation functions calculated using equation <ref>, where the conditional probability as a functions of time, $P_s(t | x(0))$, are calculated using our random walk / path integral methodology, represented graphically in figure <ref>A.
figure=1DcoftfromInterp.pdf, width=3.5in
Shown here are time correlation functions which were calculated by first generating one long random walk using the method introduced in this article, then linking each point in the trajectory using linear interpolation, and finally using equation <ref> to calculate $C(t)$.
§ APPLICATION TO CALCULATING LONG-TIME RDCS IN ATOMISTIC SIMULATIONS
§.§ Application of Discrete Space Time Correlation Methodology to the Alanine Dipeptide Bond Vector
In this section, we describe an application of our methodology to a molecular system. Shown in figure <ref> is the molecular structure of our system, alanine dipeptide. After constraining the nitrogen and carbon atoms labeled in yellow to remain fixed at their initial positions, Langevin dynamics at $T = 300 K$ was run for $4 \times 10^7$ time steps with a time step size of 0.001 ps for a total of 40 nanoseconds using the CHARMM molecular dynamics software package. As the molecular dynamics simulation ran, the orientation of the bond vector extending from the center of the labeled nitrogen atom to the center of the hydrogen atom indicated by the purple arrow in figure <ref> was recorded. Although this bond vector possesses three spatial degrees of freedom, it's orientation could be well approximated by a single rotational degree of freedom, as shown in figure <ref>. By counting the number of time steps between transitions from one milestone state to the next (shown graphically as the four colored planes in figure <ref>) over the course of the 40 nanosecond trajectory, probability distribution functions for the transition times between neighboring pairs were constructed as histograms to obtain the set of $k_{ij}(\tau)$ functions for each pair of neighboring milestone states. These $k_{ij}(\tau)$ functions were then used as the basis for the random walk / path integral approach described in the previous section. Thusly, the $P_s(t | x_0)$ functions necessary to calculate the time correlation function using equation <ref> were calculated by averaging 75,000 different time-dependent probability distribution functions which each resulted from some particular manifestation of the random walk. The time correlation functions of interest for this system are those which can be calculated using the Lipari-Szabo formalism <cit.>, as implemented by Xing and Andricioaei <cit.>, using the equation:
\begin{equation}
C(t) = \langle L_2(\textbf{u}(0)\textbf{u}(t)) \rangle
\label{L2}
\end{equation}
where $L_2(\textbf{u}(0)\textbf{u}(t))$ refers to plugging the scalar resulting from the dot product of time series measurements of the bond vector u into the second order Legendre polynomial. This motif of measuring the autocorrelation of this value is then applied to equation <ref> to yield the discrete space time correlation function relationship:
\begin{equation}
C(t) = \sum_i L_2\left[\sum_s (\textbf{u}_i(0) \cdot \textbf{u}_s) P_s(t | \textbf{u}_i(0)) \right] P_i(\infty)
\label{L2Coft}
\end{equation}
where the vectors $\textbf{u}_i$ represent the different possible values for the bond vector, given the coarse graining of the bond vector into a discrete space. The oscillatory and slower decay in correlation for the 4 milestone case is an effect of coarse graining the space. This is due to a loss in entropy in going from the continuous space to the discrete one, i.e. if only four possibilities exist for the position of the bond vector, the probability of pointing in the same direction as that of a previous time step increases compared to a system where 8 or more configurations are possible.
Notably, the oscillatory and slower decay in correlation for the 4 milestone case is an effect of coarse graining the space (the oscillations are reproduceable). This is due to a loss in entropy in going from the continuous space to the discrete one, i.e. if only four possibilities exist for the position of the bond vector, the probability of pointing in the same direction as that of a previous time step increases compared to a system where 8 or more configurations are possible.
figure=alaDipMol.jpg, width=2.9in
Shown in this figure is the alanine dipeptide molecule used as our model system. The two atoms shown in yellow were held fixed in space while the rest of the molecule was subjected to Langevin dynamics. The purple arrow gives the orientation of the bond vector which served as the measurable in our time correlation function calculations.
figure=arrowsForStones.jpg, width=3.5in
Shown here is a graphical representation of the four milestone configuration for measuring the time correlation function of the alanine dipeptide bond vector. Although the bond vector, shown as many thin, purple arrows, posses three degrees of freedom as it fluctuates in time, we are able to choose a frame of reference where the bulk of the motion is taking place as a rotation about the z-axis, shown as a thick green arrow. Using the four milestones, shown as the red, green, yellow, and blue planes, we can calculate transition time probability distributions between each pair of adjacent milestones.
figure=PsPlot.pdf, width=3.5in
This plot gives the probability of finding our system in each of the four milestone configurations as a function time, given that we began the simulation with our system in the configuration shown as the blue plane, using the same color scheme as in figure <ref>. The probability of being found in the blue milestone is equal to 1 at time $t = 0$ of course, but the plot range stops shy of $P_s(t) = 1$ in order to provide a more detailed view. Note that the probability of the system being in any of the other three milestone configurations is equal to zero at time $t = 0$, as expected. These functions were calculated using the methodology described in the Random Walk / Path Integral Methodology section. These functions contributed to the calculation of $C(t)$ shown in figure <ref>. Note that the probabilities converge to their equilibrium values on roughly the same timescale that $C(t)$ converges to its long time value.
figure=CoftAlaDip.jpg, width=3.7in
This figure shows the approximate time correlation functions calculated using equation <ref> superimposed over the true time correlation function, calculated using equation <ref>. The 4 milestone $C(t)$ function was calculated with the milestones placed 90 degrees apart as illustrated in figure <ref>, while the 8 milestone configuration was the same motif, only with 8 planes placed 45 degrees apart.
§ CONCLUDING DISCUSSION
We have demonstrated for the first time that time correlation functions for continuous processes can be approximated using equation <ref> to coarse grain the configuration space to a discrete one. Additionally, we have introduced a novel method for extending milestoning into non-equilibrium regimes by numerically calculating the time-dependent fluxes $P_s(t|x_i(0))$. The method consists of constructing random walks in the discrete configuration space, defined by a set of milestone configurations, from transition time probability density functions $k_{ij}(\tau)$ obtained using the milestoning method, followed by calculating time-dependent histograms of milestone states occupied using the stochastic path integral method described in the Random Walk / Path Integral Methodology section.
The time correlation function for the harmonic oscillator calculated analytically using our discretization method showed excellent agreement with the true time correlation function $C(t)$, also obtained analytically, for a harmonic oscillator. There was also an excellent agreement between the $C(t)$ calculated for a discrete configuration space for a bistable potential and the true autocorrelation function, where $P_s(t|x_i(0))$ was obtained by numerically solving a Fokker-Planck equation. We also obtained a promising result from applying the discretization method of equation <ref> in conjunction with the stochastic path integral method to an atomistic system. The autocorrelation function $C(t)$ for the bond vector calculated using the methods introduced herein showed a nice agreement with the true $C(t)$ calculated using equation <ref>. The limitations to the methods we have introduced appear to be limited to the challenges inherent to implementation of the milestoning method. A key advantage of our method is that the random walks between discrete configurations can be constructed at trivial computational cost, allowing for us to make predictions well into time regimes inaccessible to molecular dynamics simulations. We would like to note that, although the calculations described in this article were performed on systems where the observable of interest was constant along each milestone hyperplane, the method can easily be generalized for systems where the observable varies along each milestone hyperplane. In order to account for such observables, one must simply construct equilibrium probability distributions of the observable on each hyperplane, then select from this distribution at each time step of the random walk along the milestones. In other words, at each step, the algorithm must first choose the next step to take using the transition matrix, then select the transition time from the appropriate transition time distribution function, then select the value of the observable from the probability distribution describing the observable along that hyperplane. We feel that the methods introduced in this paper have the potential to allow for the calculation of experimental observables from molecular dynamics simulations that are currently unattainable by brute force long time simulations. The method presented herein could also be further enhanced by combining it with the enhanced sampling methodology introduced in the companion article to this paper, also found within this publication <cit.>.
§ ACKNOWLEDGMENTS
IA acknowledges funds from an NSF CAREER award (CHE-0548047).
|
1511.00599
|
Using all the observations from Rossi X-ray Timing Explorer for Z source GX 349+2,
we systematically carry out cross-correlation analysis between its soft and hard X-ray
light curves. During the observations from January 9 to January 29, 1998, GX 349+2
traced out the most extensive Z track on its hardness-intensity diagram, making a
comprehensive study of cross-correlation on the track. The positive correlations and
positively correlated time lags are detected throughout the Z track. Outside the Z
track, anti-correlations and anti-correlated time lags are found, but the anti-correlated
time lags are much longer than the positively correlated time lags, which might indicate
different mechanisms for producing the two types of time lags. We argue that neither the
short-term time lag models nor the truncated accretion disk model can account for the
long-term time lags in neutron star low mass X-ray binaries (NS-LMXBs). We suggest that
the extended accretion disk corona model could be an alternative model to explain the
long-term time lags detected in NS-LMXBs.
binaries: general — stars: individual (GX 349+2) — stars: neutron —
X-rays: binaries
§ INTRODUCTION
X-ray binaries (XRBs), consisting of a compact object and a companion star, can be
divided into low mass X$-$ray binaries (LMXBs) and high mass X$-$ray binaries (HMXBs)
according to the mass of the companion star. The compact object is either a
neutron star (NS) or a black hole (BH). Based on their spectral and timing properties,
NS-LMXBs could be classified as Z sources and atoll sources <cit.>.
The Z sources, with high luminosity close to the Eddington limit, trace out a Z-shape
track on their hardness-intensity diagrams (HIDs), which are divided into three
branches, called horizontal branch (HB), normal branch (NB), and flaring branch (FB),
respectively. Among the six confirmed Z sources, Cyg X-2, GX 5-1, and GX 340+0 are
called Cyg-like Z sources, and the other three Z sources, i.e. Sco X-1, GX 17+2, and
GX 349+2, are referred to as Sco-like Z sources. On the HIDs of atoll sources with
luminosity below $\sim$$10^{38}\ {\rm ergs\ s^{-1}}$, two main segments are seen,
which are called banana state (BS) and island state (IS), respectively; in the BS,
the variation of hardness is relatively small, while their luminosity spans a wide
range; however, in the IS, the variation of hardness is obvious, while the change
of luminosity is relatively small and they are with the lowest luminosity
<cit.>. Furthermore, the segment of BS is split into two subsections,
i.e. the lower banana (LB) with lower luminosities and the upper banana (UB) with
higher luminosities. In general, atoll sources are either in IS or in BS depending
on their luminosities. Generally, a Z source cannot become an atoll source and vice
versa. However, two peculiar NS-LMXBs, i.e. Cir X-1 and XTE J1701-462, show Z source
behaviors at relatively high luminosities, while they display atoll source behaviors
at low luminosities <cit.>.
The spectral and timing analysis are two main methods for studying XRBs. Among
the timing analyses, the cross$-$correlation analysis can be used to study the
relation of light curves in two different energy bands. By analyzing the
cross$-$correlation function (CCF) of light curves in two different energy bands,
the correlation and time lag between soft and hard X$-$ray light curves can be
obtained, which are useful and important to investigate the structures of accretion
disk and the mechanisms for producing X-rays. According to the behaviors of CCFs,
the correlations can be classified as anti$-$correlations, positive and ambiguous
correlations, respectively. Anti$-$correlations correspond to negative
cross$-$correlation coefficients (CCCs), while positive correlations correspond to
positive CCCs. If no obvious correlations are presented in the CCFs, such
correlations are called ambiguous correlations. Soft time lags mean that the lower
energy photons reach behind, while hard time lags imply that the lower energy
photons lead the higher energy photons. The cross$-$correlation analysis technique
has been used to analyze the correlations of X-rays and long-term time lags from
tens of seconds to over one thousand seconds in two energy bands of XRBs (e.g.
Choudhury & Rao 2004; Lei et al. 2008, 2013; Wang, et al. 2014). In addition to
the CCF method, the cross-spectral analysis is another important method for
analyzing time lags <cit.>, which has been widely used to investigate
the short-term time lags in the order of milliseconds in XRBs (e.g. van der Klis
et al. 1987; Miyamoto et al. 1988; Cui 1999; Qu, Yu & Li 2001; Qu et al. 2010a).
In the past several decades, various time lags have been detected in XRBs through
analyzing the data from X-ray satellites, of which RXTE has contributed
greatly (see review: Poutanen 2001).
GX 349+2, also known as Sco X-2, is a Sco-like Z source. Using the observations
from EXOSAT, <cit.> detected quasi-periodic oscillations (QPOs)
at $\sim$6 Hz in this source. Using observations from Rossi X-ray Timing
Explorer (RXTE), <cit.> investigated the evolution of QPOs along a
Z track on its HID and found QPOs at 3.3-5.8 Hz on the FB and at 11-54 Hz
on the NB. Also through analyzing data from RXTE, <cit.> detected
a twin kilohertz (kHz) QPOs with lower and upper frequencies of 712 Hz and 978 Hz,
respectively. In particular, the kHz QPOs were only found at the top of the NB.
Using the RXTE data for this source, <cit.> carried out the
spectral evolution on its Z track. They fitted the spectra in 2.5-25 keV using a
two-component model consisting of a disk blackbody and a Comptonized component
representing Comptonization in the central hot corona or the boundary layer,
which could act as the Eastern model <cit.>. However,
<cit.> used another two-component model consisting of a blackbody and
a cut-off power law representing Comptonization in an extended corona above the disk
to fit the spectra of the Sco-like Z sources, including GX 349+2. Using the data
from BeppoSAX, <cit.> studied its broadband spectra (0.1$-$200 keV)
and a hard tail was detected in this source. Moreover, they detected an absorption
edge at $\sim$9 keV in the spectra. Using the observation from XMM,
<cit.> performed spectral analysis of GX 349+2. They fitted the
continuum in the 0.7-10 keV energy range with the Eastern model
<cit.>. Significantly, <cit.> found several broad
emission features below 4 keV and a broader emission feature in the
Fe-${\rm K_{\alpha}}$ region in the spectra and proposed that these relativistic
lines are formed due to the reflection in the inner disk region that is
illuminated by the emission around the NS. The relativistic Fe-K emission line of
GX 349+2 was also detected in its Suzaku spectra <cit.>.
In this work, using all the data from the proportional counter array (PCA) on
board RXTE for GX 349+2, we systematically investigate the cross-correlation
correlations between soft and hard light curves of this source. We describe our
data analyses in section 2, present our results in section 3, discuss our results
in section 4, and, finally, give our conclusions in section 5.
§ DATA ANALYSIS
RXTE made 138 observations from 1996 to 2011 for Z source GX 349+2,
including 23 observations from January 9 to January 29, 1998, during which the
source evolved on the most extensive NB+FB tracks ever reported on its HID
<cit.>. With HEASOFT 6.11 and all the RXTE observations for
GX 349+2, we systematically perform cross-correlation analysis for this source.
In our analysis, only PCA data are needed. The PCA consists of five identical
proportional counter units (PCUs) in energy range 2$-$60 keV <cit.>.
We use the data during the intervals when all five PCUs were working. The PCA
Standard 2 mode data with bin size of 16 s are used to produce light curves with
RXTE FTOOLS SAEXTRCT. Then, with RUNPCABACKEST, a RXTE script, we
produce the background files from the bright background model
(pca$_{-}$bkgd$_{-}$cmbrightvle$_{-}$eMv20051128.mdl) provided by RXTE
team and thus produce the background light curves. Finally, applying LCMATH, a
XRONOS tool, we generate the background-subtracted light curves with various
energy bands. When extracting light curves, good time intervals (GTIs) are
restricted through inputting GTI files, which are created with the FTOOLS
MAKETIME obeying criteria: the earth elevation angle greater than $10^{\circ}$
and the pointing offset less than $0.02^{\circ}$.
To build the HID on which the source traced out the most extensive Z track,
following <cit.>, we define the hardness as the count rate ratio
between 8.7$-$19.7 keV and 6.2$-$8.7 keV energy bands, and the intensity as the
count rate in the 2.0$-$19.7 keV energy band. The produced HID is shown by
Figure <ref>. In order to investigate the evolution of cross-correlation
correlations along the Z track, we divide the track into 23 regions. For minimizing
the variation of count rate and meanwhile having enough observation time in each
region, these regions are produced obeying the criteria: the count rate variation
in each region is less than 1500 count $s^{-1}$ except the region No. 23; the
hardness variation in each of the regions in the NB and in the first half FB is
less than 0.025, while it is less than 0.5 for each region in the second half FB,
because, here, the points are scattered. Then, we study the evolution of
cross-correlation correlations between hard and soft X-ray light curves
along the Z track. Firstly, based on the values of hardness and count rate in
each region, we determine the relative and absolute time intervals of each region
with FTOOLS MAKETIME and TIMETRANS, respectively. Secondly, inputting the absolute
time intervals of each region when light curves are extracted, we produce the
soft and hard X-ray background-subtracted light curves of each region in 2-5 keV
and 16-30 kev energy bands, respectively. Thirdly, with XRONOS TOOL CROSSCOR,
we generate the CCF between the soft and hard background-subtracted light curves
of each region. To get the CCCs and time lags, we fit the CCFs with an inverted
Gaussian function at a 90% confidence level. The results of the 23 regions are
listed in Table <ref>. The total relative time length of each of the 23
regions spans from 1072 s to 8606 s and the absolute time intervals of each region
spread within 2 days.
As shown in Figure <ref>, the track consists of an extensive NB and an
elongated FB, respectively. Regions 1-10 constitute the NB, while regions 11-23
make up the FB. In order to describe the positions of the 23 regions on the
NB+FB tracks, the segment of regions 1-3 is called upper NB (UNB), the segment
of regions 4-6 is called middle NB (MNB), and the segment of regions 7-10 is
called lower FB (LFB); similarly, the segment of regions 11-15 is named lower
FB (LFB), the segment of regions 16-18 is named middle FB (MFB), and the segment
of regions 19-23 is named upper FB (UFB). The intensity as a function of time
during the HID is shown in Figure <ref>, in which the positions
of various time intervals on the HID are marked. As shown in
Figure <ref>, the two panels of the first row dominate the FB
positions, while the two panels of the second row and the left panel of the
third row are occupied by the NB positions, showing that the source
evolves from the FB to the NB on the NB+FB tracks; the right panel of the third
row and the left panel of the fourth row show the next evolutionary cycle from
the FB to the NB; the right panel of the fourth row shows the beginning of the
third evolutionary cycle. Therefore, the source regularly evolved on the HID
during each evolutionary cycle, which leads to that the data of each region
spread within two days. For Fourier analysis, unbroken sections of data are
needed. Certainly, for cross-correlation analysis continuous observations should
be better than broken observations. Here, although the observations within the
HID span 21 days, yet the data within each region do not spread across very
different observations, which ensures the validity of our results.
In addition, we perform cross-correlation analysis between soft and hard X-ray
light curves for all the observations outside the period during which the
source traced out the most extensive NB+FB tracks on its HID. Similarly, we
produce the soft and hard X-ray background-subtracted light curves of each
observation and then get its CCF, CCC, and time lag. It is noted that if
there are several segments in the light curves of an observation, we
produce the CCF of each segment.
§ RESULTS
In this work, using all the 138 RXTE observations from 1996 to 2011
for Z source GX 349+2, we systematically perform cross-correlation analysis
between 2-5 kev and 16-30 keV light curves with the CCF method. The source
traced out the most extensive Z track on the HID during the 23 observations
from January 1 to January 29, 1998. Among the 23 regions on the HID, positive
correlations and ambiguous correlations are detected in 18 and 5 regions,
respectively, while anti-correlations are not found throughout the Z track.
The HID positions of the 18 regions, the derived CCCs and time lags, and
hardness values are listed in Table <ref>. Figure <ref>
and Figure <ref> show two detected positive correlations and
one ambiguous correlation, respectively. Eight regions of the 18 regions with
detected positive correlations are assigned to the NB and other ten regions
are belonged to the FB. Among the five regions with ambiguous correlations,
the NB and FB host two and three regions, respectively. Fortunately, among
the observations outside the period during which the source traced out the
most extensive Z track on its HID, the anti-correlations are detected in ten
observations. The analysis results of these anti-correlations are listed in
Table <ref> and two representative anti-correlations are shown in
Figure <ref>. The derived anti-correlated time lags vary
between tens of seconds and thousands of seconds. Among the ten observations
with anti-correlations, hard and soft X-ray time lags are detected in four and
six observations, respectively. In one hundred observations outside the HID
period, positive correlations are detected, which are listed in Table <ref>,
while ambiguous correlations are found in five observations outside the HID
episode. The anti-correlations, positive correlations, and ambiguous correlations
are detected in 8.7%, 87%, and 4.3% of the total observations outside the HID
period, respectively. Comparing the time lags listed in
Tables <ref>-<ref>, one can see that the positively correlated
time lags vary from several seconds to tens of seconds, whereas the
anti-correlated time lags span a wide range from tens of seconds to thousands of
seconds, so the anti-correlated time lags of GX 349+2 are much larger than its
positively correlated time lags, which is consistent with what was found in
atoll source 4U 1735-44 and 4U 1608-52, as well as peculiar source XTE J1701-462
<cit.>. It should be informed that a small number of
short-term time lags less than one second are obtained in the positive
correlations of GX 349+2, as listed in Tables <ref> and <ref>.
§ DISCUSSION
§.§ The short-term time lags in XRBs
Through cross-spectral analysis, the short-term time lags ($<$1 s) between the
emissions of two adjacent X-ray energy bands have been detected in a few Galactic
black hole X-ray binaries (BHXBs) and NS-LMXBs. Among the BHXBs in which short-term
time lags were studied, the short-term time lag behaviors of Cyg X-1 were investigated
most deeply <cit.>.
The short-term time lags were also detected in microquasar GRS 1915+105
<cit.>, in GX 339-4 <cit.>, as well
as in black hole candidate GS 2013+338 and GRO J0422+32 <cit.>.
The short-term time lags with values of several milliseconds were also found in NS-LMXBs,
such as in Cyg X-2 and GX 5-1 <cit.> as well as in Cir X-1
<cit.>. Generally, the hard X-ray short-term time lags were detected in the
hard states of BHXBs, while both hard and soft X-ray short-term time lags were detected in
NS-LMXBs <cit.>.
Some models have been proposed to explain short-term time lags, but the observed various
short-term time lags cannot be explained by any single model. The cross-spectral analysis
technique is based on the Fourier transform, so the time lag spectrum, i.e. the correlation
between the short-term time lag and the frequency of Fourier transform can be obtained.
Usually, the short-term time lag is anti-correlated with Fourier frequency <cit.>. Among the
Fourier frequencies, some are the frequencies of QPOs, sequentially leading to that the
short-term time lag is also anti-correlated with QPO frequency. Since the short-term time
lag is in connection with the QPO parameter, it could be feasible to invoke the models
responsible for QPOs to account for the short-term time lags, such as the shot model
<cit.>. In the shot model, it is assumed that the gravitation energy of
accreting matter is transformed to thermal energy emitted as successive bursts, which
are called “shots” and responsible for QPOs; the different shot profiles or shot
distribution in different energy bands results in the time delays between the hard and
soft X-ray emissions <cit.>. Although
the shot model provided an explanation for the strong aperiodic variability of the flux
of accreting XRBs immediately after such variability was discovered and it was very
early applied to account for the short-term lags, yet this model inevitably encountered
difficulty when it was used to explain the large X-ray variability timescale
range <cit.> and the linear rms-flux relation
<cit.>. Since the shot model, various models have been proposed to
explain the QPOs in XRBs, such as the beat-frequency
models <cit.>, which were refuted by the fact that the
observed QPO peak separation is not a constant <cit.>, the Lense-Thirring
precession model <cit.>, and the propagating fluctuation model <cit.>,
etc. Using their propagating fluctuation model, <cit.> explained the
anti-correlation between the short-term time lag and the Fourier frequency well. There
may be a prospect of interpreting the short-term time lags observed in XRBs with the
help of these QPO models, whereas it is out of the scope of this paper.
Certainly, Comptonization is a common physical process for producing X-rays in XRBs, so
it is widely invoked to interpret the observed time lags in the order of milliseconds in
these sources <cit.>. In the
Comptonization model, the low-energy seed photons, emitted from a relatively cool region
such as the accretion disk, are inversely Comptonized by the energetic electrons from a
hot source such as a hot corona or the hot plasma near the compact objects, in which the
low-energy photons gain energy, while the high-energy electrons lose part of their energy,
leading to that high-energy photons undergo more inverse Compton scatterings and,
therefore, they escape later than the low-energy photons, resulting in hard
X-ray time lags <cit.>. Obviously, soft X-ray time lags cannot be explained by
this model. <cit.> proposed that the softening of shots could account for
soft time lags. In order to explain the time lags of GRS 1915+105, <cit.>
proposed a scenario in which a standard thin disk <cit.> coexists with a
central corona with two parts, an inner part with relatively high temperature and an outer
part with relatively low temperature. In the inner part, the inverse Compton scattering is
taken place, resulting in hard X-ray time lags. However, in the outer part, the disk seed
photons are Comptonized by the relatively cool electrons, in which the electrons gain energy,
while the seed photons lose energy, so the low-energy photons undergo more Compton
scatterings and escape later than high-energy photons, leading to soft X-ray time lags.
<cit.> invoked this scenario to explain the detected soft and hard X-ray time lags
less than ten milliseconds in Cir X-1. In addition to the two models, there are some other
models for explaining the short-term time lags of XRBs, such as the magnetic flare model
<cit.>. In this model, it is assumed that the X-rays are produced in compact
magnetic flares and the movement for magnetic loops to inflate and detach from the
accretion disk induces spectral evolution, which results in time lags corresponding to
the evolution timescales of the flares.
§.§ The long-term time lags in XRBs
Through CCF technique, the long-term time lags ranging from hundreds of seconds to
over one thousand seconds between hard and soft X-rays have been detected in a few
Galactic BH X-ray binaries (BHXBs). Anti-correlated long-term hard X-ray time lags
were first detected in the low/hard state of the high-mass BHXB Cyg X-3
<cit.>, then this kind of hard X-ray time lags were found in
microquasar GRS 1915+105 during its $\chi$ states <cit.>. It is noted
that the $\chi$ states of this microquasar are the closest analog to the low/hard
states of BHXBs. In addition, the similar long-term anti-correlated hard X-ray time
lags were also detected in another microquasar XTE J1550-564, but in its very high
state or steep power law state <cit.>. Interestingly, the long-term soft
X-ray time lags of microquasar GX 339-4 were found in its hard and soft intermediate
states <cit.>. It is noted that the spectral pivoting was shown during
the episodes when this kind of time lags were detected in these BHXBs
<cit.>. <cit.> proposed a
truncated accretion disk model to explain the detected anti-correlated long-term
hard X-ray time lags in BHXBs. In the scenario of this model, the optically thick
accretion disk, where the soft X-rays are emitted, is truncated far away from the
BH, while the truncated area, i.e. the high-temperature region between the inner
disk edge and the location near the BH, is full of Compton cloud which is
responsible for the hard X-ray emission. Any change in the disk will trigger
corresponding anti-correlated change in the Compton cloud in a viscous timescale,
during which the accreting matter flows from the optically thick accretion disk to
the truncated area, resulting in that the hard X-rays emitted in the Compton cloud
lag behind the soft X-rays emitted in the disk.
Through the CCF method, the anti-correlated long-term hard time lags from tens of
seconds to hundreds of seconds were also detected in a few NS-LMXBs. <cit.>
first detected anti-correlated long-term hard X-ray time lags in Z source Cyg X-2
and, meanwhile, anti-correlated long-term soft X-ray time lags were also found in
this source. <cit.> reported the similar anti-correlated long-term hard
and soft X-ray time lags detected in another Z source GX 5-1. Moreover, in atoll
source 4U 1735-44 and 4U 1608-52, <cit.> detected four types of
long-term X-ray time lags, i.e. anti-correlated long-term hard and soft X-ray time
lags and positively correlated long-term hard and soft X-ray time lags. <cit.>
systematically investigated the cross-correlation evolution of XTE J1701-462 on its
HIDs, detected the four types of long-term X-ray time lags, and found that its
cross-correlation behavior evolves with luminosity. The model of truncated accretion
disk was invoked to explain the detected long-term time lags in NS-LMXBs
<cit.>. However, we argue that the model of truncated
accretion disk cannot account for the detected long-term time lags in NS-LMXBs. Firstly,
this model can only account for the anti-correlated long-term hard X-ray time lags,
while anti-correlated long-term soft X-ray time lags are usually detected in NS-LMXBs.
Secondly, the truncated accretion disk model requires that an optically thick
accretion disk is far away from the compact object, while the accretion disk of
NS-LMXBs touches the NS <cit.>. Here, we argue that in
NS-LMXBs, the accretion disk touches the NS because of the absence of a magnetoshpere
around the NS in these systems. In the XRBs hosting a strongly magnetized NS with
surface magnetic field strength ${\rm B_0\sim10^{12}\ G}$, e.g. accretion-powered
X-ray binary pulsars, most of which are NS-HMXBs, a magnetosphere will be formed
around the NS <cit.>. In these XRBs, when the accreting matters approach the
magnetosphere, they will be channeled onto the polar caps of the NS along the magnetic
field lines, producing X-ray pulsations, which could be an evidence for a magnetosphere
in XRBs. Additionally, cyclotron lines are usually observed in the X-ray spectra of
accretion X-ray pulsars <cit.>, showing sufficient ${\rm B_0}$
to form a magnetosphere. Sequentially, in NS-HMXBs, the magnetic pressure from the
magnetosphere pushes the accretion disk outwards, leading to that the disk might be far
away from the NS. However, ${\rm B_0}$ of NS-LMXBs, including Z sources and atoll
sources, spans a range of ${\rm \sim10^{7}-10^{9}\ G}$ <cit.>, which is much smaller than that of strongly magnetized NSs, so that
it is unlikely for a magnetosphere to be formed in NS-LMXBs. Actually, the X-ray
pulsations have never been observed in Z sources and atoll sources except Aql X-1
<cit.> and cyclotron lines have never been observed in these sources too,
any of which conforms the absence of a magnetosphere in NS-LMXBs. In the case of the
absence of a magnetosphere, the accretion disk will reach the NS. Therefore, the
accretion disk of NS-LMXBs touches the NS.
§.§ The long-term time lags in GX 349+2
In this work, using all the 138 RXTE observations from 1996 to 2011 for Z
source GX 349+2, we systematically perform cross-correlation analysis between 2-5
kev and 16-30 keV light curves with CCF method. The results are listed in
Tables <ref>-<ref> and selectively shown by
Figures <ref>-<ref>.
As reviewed and pointed out above, the long-term hard X-ray time lags of BHXBs can
be explained by the truncated accretion disk model
<cit.>, but this model cannot account for
the long-term time lags of NS-LMXBs, because the accretion disk in NS-LMXBs actually
contacts with the NS <cit.>. Moreover, the
long-term time lags detected in XRBs, including the results that we obtain in this
work, are larger than dozens of seconds, even over one thousand seconds, while the
short-term time lags in XRBs spans in the range of milliseconds, so it is a feasible
assumption that the mechanisms responsible for the long-term X-ray time lags could
be very different from those models accounting for the short-term time lags in XRBs,
such as the shot model reviewed above. Besides, in general, the short-term time lags
are derived from two light curves in two adjacent energy bands, e.g. 2-5 keV vs. 5-13
keV for GRS 1915+105 <cit.> and 1.8-5.1 keV vs. 5.1-13.1 keV for Cir X-1
<cit.>, while the long-term time lags are usually derived from two light
curves with distant energy bands, e.g. 2-7 keV vs. 20-50 keV for Cyg X-3
<cit.> and 2-5 keV vs. 16-30 keV for GX 5-1 <cit.>, which
indicates that the X-rays responsible for the short-term time lags come from two
adjacent emission regions, even from the same area, while the the X-rays producing
long-term time lags might come from two regions which are far away each other.
Here, we invoke an extend accretion disk corona (ADC) model to explain the
long-term time lags detected in NS-LMXBs, including the long-term time lags of
GX 349+2 that we derive in this work. Analyzing the dip and non-dip spectra of
NS-LMXBs, <cit.> proposed a Birmingham model which consists
of a blackbody component interpreted as the emission from a point source, i.e. the
NS, and a power law component that might be resulted from the Comptonization of
thermal emission in an ADC above the accretion disk. Through dip ingress time
technique, <cit.> measured the radius of the ADC and developed the
Birmingham model into an extended ADC model. The measured radius of a thin, hot
corona above the accretion disk varies in the range of $\sim$(2–70)$\times10^4$ km.
Therefore, the corona is very extended and the disk is substantially covered by the
corona. In the extended ADC model, almost all soft X-ray photons from the accretion
disk are inversely Comptonized by the energetic electrons from the extended ADC, which
produces the observed hard X-rays, while the observed soft X-rays are interpreted
as the emission from the NS; the accretion disk is illuminated by the emission of
the NS, leading to the production of the extended ADC above the disk. This model
was successfully applied to Cyg-like Z sources <cit.>
and Sco-like Z sources (including GX 349+2) <cit.>, as well as atoll
sources <cit.>, so it could be a universal model for NS-LMXBs. Since
the extended ADC model is a unified model for NS-LMXBs, we try to interpret the
long-term time lags in NS-LMXBs with the help of this model. The extended ADC and
the NS are two independent emitting regions, which satisfies the request that the
hard and soft X-rays for long-term time lags are emitted from two distant regions,
as discussed above. In order to explain the long-term time lags detected in NS-LMXBs
in terms of the extended ADC model, we introduce two timescales. One is the
Comptonization timescale during which the disk seed photons are inversely Comptonized
by the high-energy electrons in the extended ADC, and another is the viscous timescale
in the order of hundreds of seconds <cit.>, during which the accreting matter
flows from the disk onto the NS. The hard X-ray time lags will be produced if the
Comptonization timescale is less than the viscous timescale, and, contrarily, the
soft X-ray time lags will be observed if the Comptonization timescale is larger than
the viscous timescale. It is noted that a minority of positively correlated short-term
time lags ($<$1 s) are listed in Tables <ref> and <ref>, which are derived
with the CCF method in our work. These short-term time lags cannot be explained by the
models reviewed in section 4.1, because those models are used to interpret the
short-term time lags produced in two adjacent energy intervals, while these short-term
time lags obtained in this work are derived from two distant energy intervals, i.e.
2-5 kev and 16-30 keV energy intervals. In the frame of the extended ADC model, we
propose that these short-term time lags will be observed under the circumstance that
the Comptonization timescale is comparable with the viscous timescale.
§ CONCLUSION
In this work, using all the RXTE observations for Z source GX 349+2, we
systematically perform cross-correlation analysis between soft and hard X-rays with
CCF method. Positive correlations and corresponding hard and soft X-ray long-term
time lags are detected throughout an extensive Z track on its HID. Anti-correlated
correlations and anti-correlated soft and hard X-ray long-term time lags are found
outside the HID. It is noted that in most observations outside the HID, positive
correlations and positively correlated hard and soft long-term X-ray time lags are
also obtained. We review the short-term time lags obtained with the Fourier
cross-spectral analysis in XRBs and some models responsible for the short-term time
lags. We also review the long-term hard X-ray time lags found in BHXBs, which can
be explained by the truncated accretion disk model. We argue that the long-term
X-ray time lags in NS-LMXBs cannot be interpreted by those models responsible for
the short-term time lags or the truncated accretion disk model. We invoke the
extended ADC model to explain the long-term X-ray time lags in NS-LMXBs, including
GX 349+2.
§ ACKNOWLEDGEMENTS
We thank the anonymous referee for her or his constructive comments and suggestions,
which helped us carry out this research and improve the presentation of this paper.
This research has made use of the data obtained through the High Energy Astrophysics
Science Archive Research Center (HEASARC) On-line Service, provided by NASA/Goddard
Space Flight Center (GSFC). This work is partially supported by the 2014 Project of
Xinjiang Uygur Autonomous Region of China for Flexibly Fetching in Upscale Talents,
National Key Basic Research Program of China (973 Program 2015CB857100), the Natural
Science Foundation of China under grant nos. 11173024 and 11203064, and the Program
of the Light in Chinese Western Region (LCWR) under grant no. XBBS 201121 provided
by Chinese Academy of Sciences (CAS).
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Zhang C. M., Chen L., Ma X., 2014, , 440, 3726
[Zhang, Strohmayer & Swank(1998)]Zhang1998Zhang W., Strohmayer T. E., Swank J. H.,
1998, , 500, L167
The results of cross-correlation analysis on the HID.
$^{a}$HID Region $^{b}$Position $^{c}$Live Time (s) $^{d}$CCC $^{e}$Time Lag (s) $^{f}$Hardness
1 UNB 2023 0.68$\pm$0.05 4.0$\pm$1.2 0.78
2 UNB 3936 0.71$\pm$0.13 -1.0$\pm$3.0 0.76
3 UNB 4624 0.41$\pm$0.09 9.4$\pm$6.4 0.75
4 MNB 3752 0.72$\pm$0.13 2.0$\pm$2.6 0.73
5 MNB 3152 0.64$\pm$0.56 0.4$\pm$1.5 0.72
6 MNB 3648 0.68$\pm$0.08 0.7$\pm$2.1 0.70
9 LNB 6112 0.27$\pm$0.05 -36.4$\pm$9.2 0.65
10 LNB 2432 0.53$\pm$1.07 -6.1$\pm$7.5 0.64
11 LFB 8606 0.62$\pm$0.09 -0.1$\pm$2.4 0.65
12 LFB 4960 0.74$\pm$0.09 -2.6$\pm$1.9 0.66
13 LFB 3560 0.82$\pm$0.08 -0.2$\pm$1.5 0.67
14 LFB 4032 0.53$\pm$0.07 1.3$\pm$3.7 0.68
15 LFB 4080 0.95$\pm$1.24 -3.8$\pm$11.1 0.69
16 MFB 3360 0.88$\pm$0.88 -3.6$\pm$8.1 0.70
17 MFB 3168 1.63$\pm$6.45 7.1$\pm$3.3 0.71
18 MFB 2224 0.79$\pm$0.14 0.1$\pm$2.9 0.72
22 UFB 1072 1.49$\pm$5.88 -5.7$\pm$9.4 0.81
23 UFB 1232 0.87$\pm$0.11 0.8$\pm$2.2 0.83
$^{a}$The regions in which positive correlations are detected.
$^{b}$The positions of the regions.
$^{c}$The total relative time length of each region.
$^{d}$The derived cross$-$correlation coefficients.
$^{e}$The derived time lags.
$^{f}$The hardness.
The results of anti-correlated correlations which are detected outside the period
of the HID.
$^{a}$ObsID $^{b}$Date $^{c}$CCC $^{d}$Time Lag (s)
30043-01-07-00 1998-10-09 -0.34$\pm$0.03 192$\pm$10
50017-01-02-00 2003-01-09 -0.21$\pm$0.01 1177$\pm$57
80105-05-01-00 2003-09-23 -0.72$\pm$0.02 -6107$\pm$46
90024-04-02-00 2004-04-17 -0.36$\pm$0.07 -77$\pm$5
90024-04-09-00 2004-08-02 -0.24$\pm$0.03 -327$\pm$16
90024-04-12-00 2004-09-22 -0.33$\pm$0.04 -66$\pm$10
90024-04-16-00 2005-01-08 -0.28$\pm$0.06 85$\pm$6
90024-04-21-00 2005-04-02 -0.24$\pm$0.05 258$\pm$20
93071-05-01-00 2008-09-17 -0.33$\pm$0.02 -127$\pm$15
93071-05-03-00 2008-09-26 -0.16$\pm$0.01 -25$\pm$27
$^{a}$The observations in which anti-correlated correlations are detected.
$^{b}$The observation dates.
$^{c}$The derived cross$-$correlation coefficients.
$^{d}$The derived time lags.
The results of positive correlations which are detected outside
the period of the HID.
$^{a}$ObsID $^{b}$Date $^{c}$CCC $^{d}$Time Lag (s)
10063-11-01-00 1996-09-07 0.62$\pm$0.04 4.8$\pm$5.1
10063-12-01-00 1996-09-06 0.69$\pm$0.01 7.6$\pm$23.2
30043-01-01-00 1998-09-29 0.50$\pm$0.03 1.1$\pm$46
30043-01-02-00 1998-09-29 0.89$\pm$0.01 -2.5$\pm$2.9
30043-01-03-00 1998-09-30 0.35$\pm$0.03 39.3$\pm$9.6
30043-01-04-00 1998-09-30 0.85$\pm$0.03 24.1$\pm$7.7
30043-01-05-00 1998-10-01 0.78$\pm$0.02 -29.5$\pm$5.7
30043-01-06-00 1998-10-01 0.85$\pm$0.03 8.6$\pm$2.9
30043-01-08-00 1998-10-02 0.91$\pm$0.02 6.8$\pm$4.5
30043-01-09-00 1998-10-03 0.77$\pm$0.03 -18.7$\pm$5.9
30043-01-10-00 1998-10-03 0.54$\pm$0.03 6.3$\pm$7.8
30043-01-11-00 1998-10-04 0.65$\pm$0.03 -18.0$\pm$14.1
30043-01-12-00 1998-10-04 0.84$\pm$0.02 0.8$\pm$3.9
30043-01-13-00 1998-10-05 0.53$\pm$0.06 2.2$\pm$4.8
30043-01-14-00 1998-10-05 0.64$\pm$0.02 -18.1$\pm$6.0
30043-01-15-00 1998-10-06 0.63$\pm$0.04 64.4$\pm$14.0
30043-01-16-00 1996-09-07 0.94$\pm$0.05 -18.7$\pm$16.7
30043-01-17-00 1998-10-07 0.54$\pm$0.03 38.1$\pm$5.9
30043-01-18-00 1998-10-08 0.69$\pm$0.06 -18.1$\pm$14.4
30043-01-19-00 1998-10-08 0.80$\pm$0.03 -3.4$\pm$6.7
30043-01-20-00 1998-10-09 0.51$\pm$0.06 -4.0$\pm$7.7
30043-01-21-00 1998-10-09 0.82$\pm$0.03 -7.8$\pm$5.1
30043-01-22-00 1998-10-10 0.84$\pm$0.05 -29.3$\pm$16.0
30043-01-23-00 1998-10-10 0.83$\pm$0.07 -3.6$\pm$4.73
30043-01-24-00 1998-10-11 0.88$\pm$0.03 -13.0$\pm$9.7
30043-01-25-00 1998-10-11 0.33$\pm$0.05 7.7$\pm$12.0
30043-01-26-00 1998-10-12 0.54$\pm$0.04 5.1$\pm$17.0
30043-01-27-00 1998-10-12 0.70$\pm$0.05 -5.4$\pm$10.8
30043-01-28-00 1998-10-13 0.86$\pm$0.05 0.6$\pm$5.8
30043-01-29-00 1998-10-13 0.42$\pm$0.06 2.2$\pm$4.8
30043-01-30-00 1998-10-14 0.32$\pm$0.05 -30.7$\pm$10.3
50017-01-01-00 2000-03-27 0.20$\pm$0.04 42.5$\pm$20.2
50017-01-01-02 2000-03-29 0.85$\pm$0.05 -2.6$\pm$4.0
50017-01-01-03 2000-03-29 0.78$\pm$0.02 -20.6$\pm$8.1
50017-01-01-04 2000-03-29 0.74$\pm$0.02 3.7$\pm$7.9
50017-01-01-06 2000-03-29 0.68$\pm$0.05 -16.2$\pm$8.3
50017-01-01-08 2000-03-30 0.57$\pm$0.06 6.7$\pm$4.2
50017-01-01-09 2000-03-31 0.62$\pm$0.05 -3.1$\pm$12.4
90022-03-01-00 2004-08-05 0.82$\pm$0.05 1.5$\pm$8.9
90022-03-02-00 2004-08-21 0.77$\pm$0.03 -6.8$\pm$9.2
90022-03-03-00 2004-08-28 0.73$\pm$0.01 -13.6$\pm$5.9
90022-03-04-00 2004-09-04 0.77$\pm$0.07 10.2$\pm$4.6
90022-03-04-01 2004-09-05 0.86$\pm$0.02 47.7$\pm$19.9
90022-03-04-02 2004-09-07 0.25$\pm$0.02 -13.6$\pm$10.1
90022-03-04-03 2004-09-07 0.37$\pm$0.05 -29.3$\pm$11.9
90022-03-05-00 2004-09-15 0.85$\pm$0.05 -26.6$\pm$12.6
90022-03-05-01 2004-09-14 0.81$\pm$0.03 9.5$\pm$7.4
90022-03-07-00 2005-10-05 0.61$\pm$0.07 -13.3$\pm$10.8
90022-03-07-01 2005-10-06 0.84$\pm$0.05 -8.2$\pm$15.1
90022-03-08-00 2005-10-07 0.82$\pm$0.06 -0.2$\pm$6.2
90022-03-08-01 2005-10-07 0.69$\pm$0.02 1.4$\pm$6.2
90022-03-09-00 2006-10-12 0.32$\pm$0.03 30.4$\pm$14.2
90024-04-01-00 2004-04-02 0.75$\pm$0.05 -4.1$\pm$4.2
$^{a}$ObsID $^{b}$Date $^{c}$CCC $^{d}$Time Lag (s)
90024-04-02-01 2004-04-18 0.96$\pm$0.11 0.3$\pm$6.3
90024-04-02-02 2004-04-22 0.91$\pm$0.17 -1.9$\pm$8.4
90024-04-03-00 2004-05-21 0.54$\pm$0.04 20.8$\pm$16.0
90024-04-04-00 2004-05-28 0.80$\pm$0.07 -0.6$\pm$5.2
90024-04-06-00 2004-06-14 0.85$\pm$0.06 6.6$\pm$8.6
90024-04-07-00 2004-07-01 0.37$\pm$0.06 102.8$\pm$14.6
90024-04-08-00 2004-07-16 0.75$\pm$0.07 -0.1$\pm$3.2
90024-04-10-00 2004-08-13 0.84$\pm$0.10 -3.9$\pm$3.2
90024-04-11-00 2004-09-09 0.27$\pm$0.11 -13.8$\pm$8.4
90024-04-13-00 2004-10-12 0.74$\pm$0.05 5.2$\pm$11.1
90024-04-14-00 2004-10-26 0.71$\pm$0.05 -26.1$\pm$9.1
90024-04-15-00 2004-11-13 0.53$\pm$0.05 1.5$\pm$6.7
90024-04-17-00 2005-01-26 0.67$\pm$0.08 25.1$\pm$11.0
90024-04-18-00 2005-02-15 0.69$\pm$0.06 18.8$\pm$10.4
90024-04-18-01 2005-02-12 0.79$\pm$0.08 -0.9$\pm$1.6
90024-04-18-02 2005-02-16 0.87$\pm$0.06 -23.7$\pm$7.9
90024-04-19-00 2005-02-24 0.81$\pm$0.09 5.6$\pm$6.8
90024-04-20-00 2005-03-16 0.42$\pm$0.10 -22.1$\pm$9.7
90024-04-22-00 2005-05-17 0.82$\pm$0.09 -13.7$\pm$8.3
90024-04-23-00 2005-05-25 0.87$\pm$0.10 -4.0$\pm$3.2
90024-04-25-00 2005-06-17 0.80$\pm$0.05 -48.4$\pm$11.8
90024-04-26-00 2005-07-11 0.24$\pm$0.05 1.6$\pm$10.3
90024-04-27-00 2005-07-19 0.21$\pm$0.04 -4.0$\pm$6.1
90024-04-28-00 2005-08-11 0.90$\pm$0.09 0.2$\pm$1.6
90024-04-29-00 2005-08-25 0.38$\pm$0.03 -49.1$\pm$10.4
90024-04-31-00 2005-09-21 0.76$\pm$0.07 -5.1$\pm$2.3
90024-04-32-00 2005-10-10 0.60$\pm$0.08 -0.8$\pm$2.4
90024-04-33-00 2005-10-17 0.37$\pm$0.04 15.0$\pm$11.7
90024-04-34-00 2005-11-02 0.90$\pm$0.05 -42.7$\pm$9.9
90024-04-35-00 2005-11-11 0.88$\pm$0.06 6.7$\pm$3.3
90024-04-36-00 2006-01-16 0.53$\pm$0.04 -32.7$\pm$6.3
90024-04-37-00 2006-02-13 0.91$\pm$0.14 -7.2$\pm$8.8
90024-04-38-00 2006-02-18 0.80$\pm$0.09 2.8$\pm$1.8
90024-04-39-00 2006-03-10 0.61$\pm$0.03 28.1$\pm$9.1
90024-04-42-00 2006-05-05 0.70$\pm$0.06 6.3$\pm$6.7
90024-04-43-00 2006-05-21 0.89$\pm$0.05 0.3$\pm$0.9
90024-04-44-00 2006-06-18 0.73$\pm$0.07 -11.4$\pm$3.8
90024-04-45-00 2006-07-17 0.65$\pm$0.05 -13.7$\pm$11.4
90024-04-46-00 2006-08-05 0.83$\pm$0.05 2.6$\pm$3.3
91152-01-01-00 2006-05-10 0.70$\pm$0.01 6.3$\pm$8.5
91152-01-01-01 2006-05-10 0.87$\pm$0.03 3.7$\pm$6.4
91152-01-02-00 2006-07-04 0.85$\pm$0.02 135.7$\pm$27.4
91152-01-03-00 2006-08-20 0.62$\pm$0.02 16.0$\pm$18.9
93406-07-01-00 2007-07-05 0.81$\pm$0.03 50.0$\pm$13.5
96378-04-01-00 2011-07-06 0.46$\pm$0.01 -286.7$\pm$40.1
96378-04-01-01 2011-07-06 0.87$\pm$0.05 -1.4$\pm$9.4
96378-04-02-00 2011-10-09 0.74$\pm$0.01 -76.8$\pm$19.6
$^{a}$The observations in which positive correlations are detected.
$^{b}$The observation dates.
$^{c}$The derived cross$-$correlation coefficients.
$^{d}$The derived time lags.
The HID of GX 349+2. Each point represents 16 s background-subtracted data. The
hardness is defined as the count rate ratio between 8.7-19.7 keV and 6.2-8.7 keV energy
bands, and the intensity as the count rate in the 2.0-19.7 keV energy band. In orde to
investigate cross-correlation evolution on the HID, the track of the extensive Z track is
divided into 23 regions (labeled 1, 2, ..., 23) that are used to group data for production
of CCF of each region.
The light curve during the period tracing out the HID. The positions of segments
on the HID are identified. The light curve shows that on the HID the source evolves from the
FB to the NB and it shows two evolutionary cycles and the beginning of the third
evolutionary cycle.
Left panels: the hard light curves (16-30 keV) and soft light
curves (2-5 kev) of two representative HID regions (1, 14) in which
positive correlations are detected. Right panels: the CCFs of the two regions.
The upper and low panels on the left show the hard X-ray light curve (16-30 keV)
and soft X-ray light curve (2-5 keV) of region 7 of the HID, respectively. The panel on the
right shows the CCF of the two light curves, where a typical ambiguous correlation is shown.
Two representative anti-correlated correlations which are detected in the epoch
outside the period tracing out the HID. Left panels: the hard X-ray light curves (16-30 keV) and
soft X-ray light curves (2-5 kev) of two observations in which anti-correlated correlations are
detected; right panels: the CCFs of the two observations.
|
1511.00303
|
Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
Protoplanetary disks fragment due to gravitational instability when there is enough mass for self-gravitation, described by the Toomre parameter, and when heat can be lost at a rate comparable to the local dynamical timescale, described by $t_{c}=\beta\Omega^{-1}$. Simulations of self-gravitating disks show that the cooling parameter has a rough critical value at $\beta_{\textnormal{crit}}=3$. When below $\beta_{\textnormal{crit}}$, gas overdensities will contract under their own gravity and fragment into bound objects while otherwise maintaining a steady state of gravitoturbulence. However, previous studies of the critical cooling parameter have found dependence on simulation resolution, indicating that the simulation of self-gravitating protoplanetary disks is not so straightforward. In particular, the simplicity of the cooling timescale $t_{c}$ prevents fragments from being disrupted by pressure support as temperatures rise. We alter the cooling law so that the cooling timescale is dependent on local surface density fluctuations, a means of incorporating optical depth effects into the local cooling of an object. For lower resolution simulations, this results in a lower critical cooling parameter and a disk more stable to gravitational stresses suggesting the formation of large gas giants planets in large, cool disks is generally suppressed by more realistic cooling. At our highest resolution however, the model becomes unstable to fragmentation for cooling timescales up to $\beta = 10$.
§ INTRODUCTION
Between the two major theories of planet formation, core accretion and gravitational instability (GI), only the latter shows a tendency to form gas giant planets in wide orbits around young stars. The core accretion scenario, whereby the coagulation of dust particles forms larger objects like planetesimals and eventually cores, is the dominant mode of planet formation <cit.>. It is particularly efficient at forming solid objects within $10$ au due to the amount of solid material and high collision rate <cit.>. However, it is a slow process in outer regions, forming planetary cores with masses $>10 M_{\oplus}$ at timescales longer than the lifetime of the disk ($10^{6}$ to $10^{7}$ years), whereas the timescale for generating dense gaseous clumps by gravitational instability is significantly shorter at around $1000$ years <cit.>. At distances beyond around 30 au, protoplanetary disks become gravitationally unstable when they accrete enough mass such that regions within the disk collapse due to self-gravity, creating fragments that can accumulate solid material to form cores of Jovian planets or possibly form low mass stars <cit.>. Thus discoveries of planets with large orbital radii <cit.> suggest that GI may be a feasible formation mechanism to form planets where core accretion has trouble operating, but not common enough to form planets at higher rates than via core accretion <cit.>.
Gravitational instability becomes a significant factor in the disk when enough cold gas is present for strong self-gravitation and the disk cools efficiently <cit.>. These two conditions are measured by the Toomre parameter $Q$ and a simple cooling relation $t_{c}$, respectively. The Toomre parameter represents a balance of the local centripetal and gravitational forces on a contracting fragment of the disk and is defined by <cit.> as:
\begin{equation} \label{eq:toomre}
Q = \frac{c_{s}\Omega}{\pi G\Sigma},
\end{equation}
where $\Omega = \sqrt{GM / R^{3}}$ is the orbital Keplerian frequency of the disk, which stabilizes the disk to large wavelength density perturbations and $c_{s}$ is the local speed of sound, which stabilizes the disk to shorter density perturbations <cit.>. Protoplanetary disks are generally considered thin, so the surface density is the vertically integrated density $\Sigma \approx \rho H$, where $H$ is the scale height of the disk. Regions of the disk with values below $Q \approx 1$ are unstable to axisymmetric perturbations and will contract into denser clumps, while a region with $Q > 1$ will remain stable to gravitational collapse <cit.>. When stable, a disk tends to stay close to $Q \approx 1$ as higher values mean there is more heat to be lost and the disk will cool faster and contract. Falling below $Q \approx 1$ results in strong shock heating which raises the temperature and returns the disk to marginal stability <cit.>. This gravitoturbulent situation arises when these effects balance each other and the simulation settles for several orbital periods.
When a fragment has formed it must remain bound under its own gravity and must remain cool enough so that a clump can collapse on a free-fall timescale shorter than shear can disrupt it <cit.>. Therefore a short cooling law
\begin{equation} \label{eq:cool}
t_{c} = \beta\Omega^{-1}
\end{equation}
is the second condition for creating an non-fragmenting disk, as a disk that can dump enough energy will form collapsed fragments which can withstand being torn apart by shocks and tidal shearing forces. Sufficient cooling has been an issue for the applicability of GI in the past <cit.>, but it has been shown that short enough cooling times <cit.>, represented in this paper by values below the critical value of $\beta_{\textnormal{crit}}=3$ <cit.>, it is possible to form gravitationally bound clumps that survive the shear of the disk.
<cit.> found $\beta_{\textnormal{crit}}$ changes with resolution, first for global smoothed particle hydrodynamic (SPH) simulations and later followed by local finite-difference shearing sheet models of <cit.>. These results would indicate that the critical cooling timescale may be significantly longer, up to $t_{c}=20\Omega^{-1}$, extending the GI formation domain to regions of the disk within 30 au (see Figure 2 of <cit.>). This conflicts with our current understanding of formation regimes of core accretion and gravitational instability, suggesting that GI may be more common than detected by current observations <cit.>. Additionally, this is a significant problem to the applicability of previous results as a resolution dependent solution implies that all previous studies at different resolutions produce different fragmentation conditions.
The widely used simple cooling law cools every location at the same rate, failing to account for the effects of optical depth on cooling efficiency or the increased strength of surface density fluctuations with resolution. To account for the varying optical depth with density, we alter the cooling to include a linear dependence on the local surface density in the disk. This is expected to suppress the formation of fragments from strong density fluctuations which will cool slower and be supported against collapse by higher internal temperature. Conversely, underdense regions will cool faster and prone to collapse, perhaps leading to convergence of the fragmentation boundary with resolution.
Other studies of thermodynamics on disk stability have focused on different methods, such as radiative cooling <cit.> and modifications to the equation of state <cit.> or have concentrated on different sources of heat such as irradiation from external sources <cit.>. The closest approach to what is carried out here, <cit.>, showed with two-dimensional finite difference shearing sheet simulations, a simple cooling scheme does not lead to a converged critical cooling parameter for fragmentation. <cit.> attempted to attain convergence by adjusting the numerical cooling and viscosity with a SPH code, but the critical cooling parameter identified therein does not agree with the results of both <cit.> and this paper.
The aim of this paper is to investigate the relation between the cooling law of the disk and the simulation resolution and their effect on formation conditions of gas giants planets. This means implementing a more realistic cooling relation per grid cell in local simulations, one dependent on the local surface density to mimic the change in optical depth for varying densities. This paper will proceed with an overview of the important equations in hydrodynamic simulations, the shearing sheet approximation and other theoretical considerations in Section <ref> followed by the specifics of the numerical setup in Section <ref>. Finally, we look at the results of simulations and how they differ from previous attempts in Section <ref> and discuss the limitations and implications in Section <ref>.
§ THEORY
To obtain high resolution simulations of disk dynamics the situation needs to be reduced to an numerically less expensive problem. Here we consider a fully gaseous disk with minimal magnetization, so the physical laws are the normal hydrodynamic (HD) equations. Since a very cool, thin disk is being considered, the equations and simulations here are only in radial and azimuthal directions. Like many astrophysical fluids, we treat the disk as a fluid with a high Reynolds number so we can describe its behavior in a disk around a young stellar object (YSO) using Eulers equation for the conservation of momentum and the typical equations for mass and energy conservation.
\begin{align}
& \frac{\partial\Sigma}{\partial t} + \nabla\cdot(\Sigma\mathbf{v}) = 0 \label{eq:massconserve} \\
& \frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v} = -\frac{\nabla P}{\Sigma} - \nabla\Phi \label{eq:momconserve} \\
& \frac{\partial\epsilon}{\partial t} + \nabla\cdot(\epsilon\mathbf{v}) = -P\nabla\cdot\mathbf{v}, \label{eq:energyconserve}
\end{align}
Since we are studying GI we consider the disk to be self-gravitating, defined by a potential $\Phi$ which is the solution to the Poisson equation for a razor-thin disk
\begin{equation} \label{eq:poisson}
\nabla^{2}\Phi=4\pi G\Sigma\delta(z),
\end{equation}
which is solved in Fourier space by transforming the surface density to find the potential at the scale of wavenumber $k$ and transforming the solution back into real space. The solution to the Poisson equation in Fourier space is
\begin{equation} \label{eq:gravpotential}
\Phi(k_{x}, k_{y}, t) = -\frac{2\pi G\Sigma(k_{x}, k_{y}, t)}{| \mathbf{k} |}.
\end{equation}
In these simulations we smooth self-gravity on the grid scale and do not limit small wavelength modes in the calculation of self-gravity.
Finally, we consider the gas in the disk to be ideal, with surface density $\Sigma$, internal energy $\epsilon$, and 2D specific heat ratio $\gamma$
\begin{equation} \label{eq:eos}
P = (\gamma - 1)\Sigma\epsilon.
\end{equation}
The selection of the specific heat ratio has an effect on the cooling rate required for fragmentation <cit.>. Higher values of $\gamma$ result in a stiffer equation of state that requires a lower cooling time for fragmentation. A ratio of $\gamma = 1.6$ is used here which compared to the value of $\gamma = 2$ by <cit.> might result in fragmentation at a slightly higher value of the critical cooling timescale $\beta_{\textnormal{crit}}$.
§.§ Shearing Sheet Model
From these initial equations we move to a local description of a small section of the disk using a sheering sheet approximation. The disk is modeled locally on a small radial-azimuthal patch of the disk, transforming the global cylindrical coordinates to local Cartesian coordinates co-rotating with the disk <cit.>. These assumptions allow for the modeling of the local properties of the disk while following the evolution of fragments that form when using periodic boundary conditions. This approximation ignores other properties of the disk such as accretion, non-local stresses and migration, which are saved for global simulations. Following this model, the relevant equations are similar to the conservation equations as above but with additional terms for the Coriolis effect $2\Omega\times\mathbf{u}$ and centripetal force $q\Omega v_{x}\mathbf{\hat{y}}$ as well as heating $H$ and cooling terms $C$ in the equation for conservation of energy. Additionally, the Pencil code used for these simulations uses entropy $s$ as the thermodynamic variable in the conservation of energy equation.
\begin{align}
\frac{\partial {\Sigma}}{\partial t} &- q\Omega x\frac{\partial {\Sigma}}{\partial y} + \nabla\cdot(\Sigma\mathbf{u}) = f_{D}(\Sigma) \label{eq:finalmassconserve} \\
\frac{\partial \mathbf{u}}{\partial t} &- q\Omega x\frac{\partial \mathbf{u}}{\partial y} + \mathbf{u}\cdot\nabla\mathbf{u} = \nonumber \\
& -\frac{\nabla P}{\Sigma} + q\Omega v_{x}\mathbf{\hat{y}} - 2\Omega\times\mathbf{u} - \nabla\Phi + f_{\nu}(\mathbf{u}) \label{eq:finalmomconserve} \\
\frac{\partial s}{\partial t} &-q\Omega x\frac{\partial s}{\partial y} + (\mathbf{u} \cdot \nabla)s = \nonumber \\
& \frac{1}{\Sigma T} \left( 2\Sigma\nu\mathbf{S}^{2} - \Lambda + f_{\chi}(s) \right) \label{eq:finalenergyconserve}
\end{align}
In these equations, $\mathbf{u} = (v_{x},v_{y}+q\Omega x )^{T}$ is the perturbed velocity in the disk due to the shear in the local box. Viscous heat is generated by $H = 2\Sigma\nu\mathbf{S}^{2}$, with rate-of-strain tensor $\mathbf{S}$, and radiated away by an approximation $\Lambda$ described in greater detail later in Section <ref>. The source terms $f_{D}(\Sigma)$, $f_{\nu}(\mathbf{u})$, $f_{\chi}(s)$ for hyperdiffusion, hyperviscosity, and hyperconductivity respectively are explicit terms to keep the solution well-behaved when shocks arise and will be expanded upon in Section <ref>.
§.§ The $\alpha$-parameter
An important disk parameter to be determined by our shearing sheet model is the $\alpha$ stress, a means to measuring and comparing the sources of turbulence in a disk <cit.>. This formalism relies on the assumptions that the disk is thin and that angular momentum is transported locally through a dimensionally defined viscosity $\nu = \alpha c_{s} H$ <cit.>. A thin disk means that the only non-vanishing term of the vertically integrated stress tensor $\mathbf{T}$ is
\begin{equation} \label{eq:stresstensor}
T_{r\phi} = -r\Sigma\nu\frac{d\Omega}{dr},
\end{equation}
which, when adding the above description of viscosity, becomes
\begin{equation} \label{eq:stresstensor2}
T_{r\phi} = -\alpha P\frac{d\ln\Omega}{d\ln r}.
\end{equation}
This shows the total stress in the disk comes only from the pressure and therefore viscosity is mostly local. Solving for $\alpha$ and defining the stress as the sum of the average gravitational and Reynolds stress tensors <cit.> gives:
\begin{equation} \label{eq:alpha}
\alpha = -\left(\frac{d\ln\Omega}{d\ln r}\right)^{-1}\frac{\langle G_{xy}\rangle + \langle H_{xy}\rangle}{\Sigma c_{s}^{2}},
\end{equation}
which allows us to calculate the $\alpha$ generated by the simulation, where
\begin{equation}
\langle G_{xy}\rangle = \int_{-\infty}^{\infty} \frac{g_{x} g_{y}}{4 \pi G} dz
\end{equation}
\begin{equation}
\langle H_{xy}\rangle = \langle \Sigma u_{x} u_{y} \rangle.
\end{equation}
The analytic expression of $\alpha$ based solely on input parameters $\gamma$ and $\beta$ is given by <cit.>:
\begin{equation} \label{eq:alphapara}
\alpha = \frac{4}{9}\frac{1}{\gamma(\gamma-1) t_{c}\Omega}
\end{equation}
which gives a prediction for the stress in a gravitoturbulent viscously heated disk. From equations (<ref>) and (<ref>) we are able to compare theoretical expectations of the gravitoturbulent state with the stresses generated by our simulations.
§.§ Heating and Cooling
For a disk to become unstable and its fragments to survive, it needs to cool fast enough to overcome the stabilizing effects of shocks and shearing motions. This means one needs a careful description of how the disk is treated thermodynamically; how and where heat is generated and released from the disk. The only active sources of heating come from viscous heating and shock dissipation, the first and third terms on the right side of equation (<ref>). Passive heating comes in the form of irradiation by the central star or other nearby stars, which limits the cooling from the surface.
A more physically accurate model currently in use is full radiative transfer, which typically requires three dimensional simulations with intricate opacities (see <cit.>, <cit.> and <cit.>). Simulating cooling by radiative transfer is complicated, requiring significant computational resources with results differing on whether there is enough cooling to form fragments by gravitational instability. The effect of varying opacities with a simple cooling timescale has been investigated by <cit.> but without looking into the effect of varying resolutions and the stronger surface density perturbations which arise with increasing resolution.
To use computational resources more efficiently, simulations in this paper are run with a cooling law such that each grid cell loses an amount of heat per time given by the cooling law in the form $\Lambda=U/t_{c}$
\begin{equation} \label{eq:coolinglaw2}
\Lambda = \frac{\Sigma (c_{\textnormal{s}}^{2} - c_{\textnormal{s,irr}}^{2})}{(\gamma -1) t_{c}},
\end{equation}
where $U=\Sigma (c_{\textnormal{s}}^{2} - c_{\textnormal{s,irr}}^{2}) / (\gamma -1)$ is the two-dimensional energy density with a non-zero background irradiation term $c_{\textnormal{s,irr}}^{2}$. A disk with no irradiation will allow the gas to cool to a lower temperature and thus have less support from gravitational collapse.
The simple cooling time is derived by the assumption that the cooling timescale $t_{c}$ is proportional to the shearing or dynamical timescale $\Omega^{-1}$. This means that $\beta = t_{c}\Omega$ will be near unity to allow for dense clumps to collapse before tidal disruption. This assumes however, that surface density fluctuations are small compared to the initial value $\Sigma_{0}$, but density fluctuations will become larger with increased resolution, making a constant cooling parameter less viable with higher resolutions.
Here we re-examine this approximation, providing a rationale for a cooling timescale which varies with surface density for each grid cell, cooling denser regions slower and less dense regions faster, offering a more realistic approach to disk cooling. In this way, cooling is not only more appropriate for the physical system, it also scales according to stronger surface density fluctuations with increased resolution.
To approximate the heat lost by the disk through radiation, we assume heat is lost at a rate per unit surface area according to the Stefan-Boltmann law
\begin{equation} \label{eq:boltzmann}
\Lambda(\Sigma,U,\Omega) = 2\sigma T_{e}^{4},
\end{equation}
with Stefan-Boltzmann constant $\sigma$ and effective disk temperature $T_{e}$. From the <cit.> treatment of an radiative transfer in a optically thick disk by the diffusion approximation, one can express the effective surface temperature in terms of the midplane temperature $T_{c}$ and the Rosseland mean optical depth $\tau_{\textnormal{R}}$ of the intervening disk material
\begin{equation} \label{eq:hubeny}
\end{equation}
This yields a heat loss relationship in terms of the midplane temperature and the passive heating due to irradiation from the star <cit.>
\begin{equation}
\Lambda=\frac{16}{3}\frac{\sigma (T_{c}^{4} - T_{\textnormal{irr}}^{4})}{\tau_{\textnormal{R}}}.
\end{equation}
From this and the internal energy $U$ one can write the cooling timescale
\begin{equation} \label{eq:completecool}
t_{c} = \frac{U}{\Lambda} = \frac{3}{16}\frac{U\tau_{\textnormal{R}}}{\sigma (T_{c}^{4}- T_{\textnormal{irr}}^{4})} \approx \frac{3}{16}\frac{U\Sigma\kappa}{\sigma (T_{c}^{4} - T_{\textnormal{irr}}^{4})},
\end{equation}
where the optical depth is estimated in the optically thick regime by $\tau_{\textnormal{R}} \approx \Sigma\kappa$. This approximation produces an additional surface density dependence of the cooling timescale compared to the standard cooling prescription. Since we are considering the cool region of the disk where opacities are dominated by ice grains, $\kappa$ has no additional $\Sigma$ dependence only a dependence on temperature which we ignore for now <cit.>.
Therefore, we approximate the cooling timescale in equation (<ref>) by including a linear dependence on surface density
\begin{equation} \label{eq:newcoolingtime}
t_{c} = \beta\left(\Sigma / \Sigma_{0}\right)\Omega^{-1}.
\end{equation}
This replaces equation (<ref>), which is specific to constant optical depths whereas equation (<ref>) will take into consideration the changing optical depths in the disk due to local surface densities over the course of the simulation. Previous studies using the simple cooling law (<ref>), notably <cit.>, have found a critical value of $\beta\simeq 3$ where disks tend to fragment for $\beta$ values below this critical value and will not fragment above it. It is important to note that this critical value is found for simulations with $1024 \times 1024$ grid cells, and will differ with changing resolution, which is the focus of this investigation.
§.§ Fragmentation
Gravitational instability does not necessarily result in fragmentation; if $1\leq Q\leq 2$ the disk will be unstable to nonaxisymmetric perturbations, or spiral arms, that will transport angular momentum, but not collapse into fragments, a so-called gravitoturbulent state <cit.>. Additionally, clumps may become overdense only to fall apart and allow a disk to settle to a steady state. Therefore, fragments need to both be able to form and survive disruption for a few orbits <cit.>. A fragment survives when its density is above the Roche limit, where the self-gravity of the fragment is sufficient to keep from being sheared apart by the tidal forces of the protostar<cit.>
\begin{equation}
\rho_{\textnormal{Roche}}=3.5 \frac{M_{*}}{R^3},
\end{equation}
where $M_{*}$ is the mass of the central star and $R$ is the radial distance between the fragment and star.
Since all presented simulations are local and take no consideration of any absolute central mass, one can use the Keplerian frequency $\Omega = \sqrt{GM / R^{3}}$ of the shearing box and the sound speed $c_{s}=H\Omega$ to formulate an expression for the Roche surface density in terms of simulation scale quantities and gravitational constant $G$
\begin{equation} \label{eq:rochedensity}
\Sigma_{\textnormal{Roche}}=7 \frac{c_{s}^2}{HG},
\end{equation}
where $\Sigma_{\textnormal{Roche}}=2\rho_{\textnormal{Roche}} H$ <cit.>. Fragments have formed when clump density is greater than the Roche surface density for more than a few cooling timescales $t_{\textnormal{c}}$.
§ NUMERICAL METHODS
The local simulation of a disk is conveniently handled by a finite difference, partial differential equation solver for compressible MHD equations. Used for my investigations is the Pencil Code[http://pencil-code.nordita.org/] <cit.>, which is chosen for its high-order numerical scheme and its modularity. As a finite difference code the simulation domain is divided into a grid of cells where physical quantities are calculated and advanced in discrete time-steps.
The length of the time-step is determined by the Courant criterion with Courant constant $C_{0} = 0.4$, which must be satisfied for convergence to be possible
\begin{equation} \label{eq:timestep}
\delta t = C_{0} min \left( \frac{\delta x}{|u_{x}|+c_{s}}, \frac{\delta y}{|u_{y}|+c_{s}} \right),
\end{equation}
which is calculated over the entire domain to calculate a single time-step to advance the entire system uniformly.
For a two-dimensional simulation this means that the derivatives for each basic physical quantity is calculated in each grid cell with an upwinding scheme to eliminate spurious Nyquist signals
\begin{multline} \label{eq:upwindedcalc}
f_{0}^{'} = \frac{-2f_{-3}+15f_{-2}-60f_{-1}+20f_{0}+30f_{1}-3f_{2}}{60\delta x} \\
- \frac{\delta x^{5} f^{(6)}}{60} = D^{(up,5)} + O(\delta x^{6}).
\end{multline}
The code then proceeds to the next time step determined by the Courant condition.
Shocks present problems in hydrodynamical simulations as discontinuities cannot be represented by high-order polynomials, leading to additional minimums and maximums. Therefore explicit dissipation terms are added to the conservation equations above to smooth out the waves so they do not hinder the performance of the simulation. These terms have two parts, one being a sixth-order hyper dissipation method and the second being a localized shock-capturing method, active in regions with large negative velocity divergences. The hyperdiffusion term looks like
\begin{equation} \label{eq:hyperdiff}
f_{D}(\Sigma) = \zeta_{D}(\nabla^{6}\Sigma),
\end{equation}
with analogous forms for both hyperviscosity $f_{\nu}(\mathbf{u})$ and hyperconductivity $f_{\chi}(s)$. The shock-capturing portions for each dissipative term are
\begin{align}
& f_{D}(\Sigma) = \zeta_{D}(\nabla^{2} \Sigma + \nabla\ln\zeta_{D}\cdot\nabla\Sigma) \label{eq:shockdiff} \\
& f_{\nu}(\mathbf{u}) = \zeta_{\nu} (\nabla (\nabla\cdot\mathbf{u}) + (\nabla\ln\Sigma + \nabla\ln\zeta_{\nu})\nabla\cdot\mathbf{u}) \label{eq:shockvisc} \\
& f_{\chi}(s) = \zeta_{\chi}(\nabla^{2} s + \nabla\ln\zeta_{\chi}\cdot\nabla s), \label{eq:shockconduc}
\end{align}
where the $\zeta$ term for each is analogous to the following viscous example
\begin{equation} \label{eq:shockterm}
\zeta_{\nu} = \nu_{sh}\langle max_{3}[(-\nabla\cdot\mathbf{u})_{+}] \rangle [min(\delta x, \delta y, \delta z)]^{2}.
\end{equation}
Logarithmic surface densities of two different outcomes of self-gravitating disks. On the left is the stable gravitoturbulent state where heating and cooling are in balance and produce density waves. On the right is the case where cooling is short enough to allow an overdense clump to survive tidal shearing and become a bound fragment.
§.§ Boundary Conditions
Typical of shearing sheet simulations, the boundary conditions are periodic along $y = 0$ and $y = L$ which means ghost zones are used to give conditions for grid cells near these boundaries <cit.>. Derivatives are not calculated in these regions, only the values of density, velocity, etc. so that physical properties and their derivatives maybe calculated near the borders.
\begin{equation} \label{eq:ybounds}
f(x, y, t) = f(x, y + L, t)
\end{equation}
The boundaries $x = 0$ and $x = L$ require a different boundary condition on account of the shear velocity $u_{y} = v_{y} + \frac{3}{2}\Omega x$ <cit.>.
\begin{equation} \label{eq:xbounds}
f(x, y, t) = f(x + L, y - \frac{3}{2}\Omega Lt, t)
\end{equation}
When calculating this over a shear periodic x-boundary, the displacement due to the shear is taken into account by shifting the entire y-direction to make the x-direction periodic before proceeding with the transform in the x-direction. After the calculation of the potential in Fourier space the process is reversed to get back to real space.
A plot of maximum surface density over time shows the change in the fragmentation behavior two identical simulations aside from the resolution. The $N=2048$ case should not fragment if the critical cooling criterion is $\beta=3$ according to <cit.>.
§.§ Initial Conditions
The Pencil Code uses dimensionless scale parameters for all physical values and constants. This helps keep numbers from getting too large or too small for the code to handle. The variables and constants can be scaled back to physical units after the computation is complete. The Keplerian frequency $\Omega = \sqrt{GM_{*}/R^{3}}$ is one of the more important parameters because it relates the shearing box to its surroundings, with $R$ the distance to the central massive object and $M_{*}$ being the mass of the central object.
The initial state of the disk is set so that the Toomre value throughout is on the borderline of stability and instability $Q=1$. By equation (<ref>), this condition is met by setting the gravitational constant $G$, the Keplerian orbital frequency $\Omega$, and the uniform surface density distribution $\Sigma_{0} = 1$. Furthermore, the sound speed $c_{s}$ is initially set to $\pi$ which means the constant background irradiation set by $c_{s,0}^{2}$ in equation (<ref>) is $\pi^{2}$. The physical length of the simulation needs to be longer than the critical wave length $\lambda_{\textnormal{crit}}=2 c_{s}^2/G\Sigma_{0}$ <cit.> and so the physical length of the shearing box is $L_{x} = L_{y} = \mathbf{(160/\pi)} H$ for all runs, which also ensures a resolution of one scale height by at least 10 grid cells. The ratio of specific heats is set to the 2D adiabatic case $\gamma = 1.6$, which maps to a 3D adiabatic index of between 1.6 and 1.9 depending on the self-gravitation in the disk <cit.>.
§ RESULTS
Simulations of self-gravitating disks have used simple cooling (<ref>) as a useful starting point to model disk cooling, but as a simple mathematical relationship between the cooling and dynamic timescales it does not remain consistent for changing resolutions because it does not consider that optical depths vary in the disk. Naturally the general inclination of these simulations is to push for higher resolution with the hope that the fragmentation boundary will eventually level off at some higher resolution, but this expectation is not realistic. Surface density fluctuations will continue to increase but without a physically motivated solution there is no way to adequately scale with resolution.
Here we will show that our initial setup is consistent with previous results, e.g. the fragmentation boundary varies with resolution and this boundary is $\simeq 3$ for the smallest number of grid cells considered $N_{x}=N_{y}=1024$. Finally we will show that changing the way the disk cools according to Section <ref> results in a disk that fragments at shorter cooling timescales, without demonstrating convergence.
§.§ Previous Results
Initial simulations run with $N_{x}=1024$ and $N_{y}=1024$ are established with a uniform surface density distribution, linear velocity shear and subsequently heated by shocks that develop from trailing density structures and cooled assuming large constant optical depth. At this resolution all simulations were adjusted to be consistent with the fragmentation criterion $\beta_{\textnormal{crit} = 3}$, in which case a simulation with a cooling parameter of $\beta = 10$ does not fragment. The simulation on the left of Figure <ref> demonstrates the steady state of such a disk with non-axisymmetric density structures and roughly constant overall Toomre stability.
The behavior of the gravitoturbulent simulation is shown as the solid red line in Figure <ref>, showing the maximum surface density to observe incidences of fragmentation. As discussed in <cit.>, initial small random velocities develop into non-linear fluctuations in surface density, velocity and gravitational potential before settling to a steady state in the non-fragmenting case, shown on the left of Figure <ref> or continuing to grow in the fragmenting case, shown on the right of Figure <ref>. As can be seen in the Figure <ref>, in the stable gravitoturbulent case clumps are continuously forming but are torn apart before they are allowed to reach Roche density where they will be able to withstand further disruption. The trailing density structures lead to a finite $\alpha$ viscosity. By equation (<ref>), $\alpha=0.046$ for disks with cooling criterion $\beta=10$, and is observed in the stable case.
In the gravitationally unstable case (solid blue line of Figure <ref>), fragments cool faster than they can be torn apart by tidal shear and collapse into one or more overdensities that may continue to grow or merge through collisions. Such behavior is expected for simulations with resolution $N=1024$ and cooling parameter $\beta \lesssim 3$, as a clump will be able to collapse to a compact density before being sheared apart over a few dynamical timescales. A disk is considered to fragment when it has surpassed the Roche surface density, indicated by the dashed lines for each simulation in Figure <ref>, as defined in Section <ref>.
The difference between the two simulations shown in Figures <ref> and <ref> is that the total number of grid cells is quadrupled, which should not lead to such a drastic shift towards fragmentation. This is consistent with simulations by <cit.> and <cit.>, which indicate the critical cooling criterion of <cit.> may be as high as $\beta_{\textnormal{crit}} = 10$. Thus a constant cooling parameter does not adequately scale with resolution and a new approach to cooling is needed to observe a convergent fragmentation boundary.
§.§ Results with adjusted cooling
The maximum surface densities of the two simulations shown in figure <ref>, but with cooling prescription changed to account for the varying optical depths in the disk. The dashed lines are the corresponding Roche densities which are a fragmentation threshold.
Maximum density evolution of the three simulations from Table <ref> with $\beta = 2$. Under the fragmentation criterion of <cit.> these simulations should fragment, but are instead gravitoturbulent with the change of cooling.
The results here use parameters identical to the simulations in the previous section, besides the modification to the cooling law described in Section <ref> by equation (<ref>). Figure <ref> shows the case where $\beta = 10$ is shown for four different resolutions, all of which show non-fragmentation. These simulations are a direct comparison to the two shown in Figure <ref> and in particular, the significant change between the behavior of the $N=1024$ and $N=2048$ cases shows the effect of altering the cooling prescription.
Simulations using the new cooling prescription.
Name Grid Cells ($N^2$) $\beta$ Fragmentation
hk10Q1 $512^2$ 10 No
hk2Q1 $512^2$ 2 No
hkp5Q1 $512^2$ 0.5 No
1k10Q1 $1024^2$ 10 No
1k5Q1 $1024^2$ 2 No
1kp5Q1 $1024^2$ 0.5 No
2k10Q1 $2048^2$ 10 No
2k5Q1 $2048^2$ 2 No
2kp5Q1 $2048^2$ 0.5 Yes
4k10Q1 $4096^2$ 10 Yes
4k2Q1 $4096^2$ 2 Yes
This is expected because defining the cooling according to equation (<ref>) creates an effective cooling time for each grid cell. When the disk cools according to the old cooling law $t_{c} = \beta \Omega^{-1}$, all regions in the disk lose the same amount of heat at the same timescale $t_{c}$. However, since one expects a clump to have a higher optical depth $\tau \approx \Sigma\kappa$ cooling efficiency should change from one location to another depending on the surface density. This motivates the alteration to the cooling timescale, which causes denser regions of the disk to have a higher optical depth and retain their heat, stabilizing to gravitational collapse. On the other hand, underdense regions will have a lower relative optical depth, cool faster and clump into dense structures easier.
Plot of cooling parameter $\beta$ against number of grid cells in one direction $N$ for all simulations run with the new cooling prescription from Table <ref>. The dashed line at $\beta_{\textnormal{crit}} = 3$ is the fragmentation boundary as defined by <cit.>. The dotted line is the proposed new fragmentation boundary based on the simulations carried out here.
Figure <ref> shows the total $\alpha$ stress in a disk with adjusted cooling and the calculated value matches the expectation of the analytical prediction (<ref>). At early times, non-axisymmetric perturbations become unstable and contract causing a small drop in the total value of $\alpha$ just after $t_{\textnormal{c}} = 40\Omega^{-1}$. After this initial burst which is dominated by the shocks and self-gravity of the formed density structures, cooling takes over and the simulation settles to the expected gravitoturbulent $\alpha$-value.
The convergent behavior continues to lower cooling parameter values as well, Figure <ref> showing the results of three simulations with different resolutions at $\beta =2$. Previously these simulations would have been expected to fragment, but here all show consistent steady gravitoturbulence. This begins to show the shift of the fragmentation boundary towards shorter cooling timescales. Figure <ref> shows the results of simulations at even short timescales with a new fragmentation boundary anticipated at around $\beta = 0.5$.
The calculated (solid black line) versus predicted value (solid red line) of $\alpha$ for simulation named 2k10Q1 which is a high resolution ($N=2048$) gravitoturbulent disk with adjusted cooling. Also plotted are the hydrodynamic and gravitational constituents of the $\alpha$ stress.
§ DISCUSSION
The results here show that the convergence issue of fragmentation in protoplanetary disks can not be approached by refining the numerical methods but by using more sophisticated physics. As simulations reach higher resolutions, they will likely need improved physical models to approach convergence. Additionally, the results here keep the formation regime of gravitational instability in the outer regions of disks where cooling times are sufficiently short. This is is contrary to other results which obtain convergence and see longer cooling timescale possible for the formation of planets, moving the formation region to shorter radii.
§.§ Convergence
Whereas the old cooling timescale showed drastic differences in fragmentation behavior between two resolutions as seen in Figure <ref>, the simulations here are consistent with each other over similar cooling timescales, cooling reaching a gravitoturbulent state at similar rates and settling at similar densities. This can be attributed to the sensitivity to density of the new cooling method employed here. When overdensities cool slower than other regions, they retain more heat and are more likely to be disrupted, decreasing the fragment density below the Roche density threshold and are suppressed from further fragmentation.
However this does not mean a new fragmentation boundary has been attained. Figure <ref> shows different fragmentation behavior for varying cooling times and resolutions and the lack of convergence is particularly noticeable in the cases where $\beta = 0.5$. The $N=1024$ simulation appears closer to fragmentation than the $N=512$ case and might be considered borderline fragmentation, where a clump surpasses the Roche density, but is sheared apart in less than an orbit ($2\pi\Omega^{-1}$) <cit.>. This is not the case for $N=512$ as a clump here never reaches Roche density and at the highest resolution studied ($N=2048$) the disk fragments. This may be due to the fact that assuming a simple linear relation in surface density does not fully capture the dependence of the cooling timescale on surface density. Also, as the cooling rate is a function of orbital frequency $\Omega$, surface density $\Sigma$ and temperature $T$ there may be an additional dependence on temperature that must be explored in the future.
<cit.> confirmed the non-convergence shown in SPH simulations with a finite difference code, which led to the assertion that fragmentation might be a stochastic process in circumstellar disks. The implication is that planet formation by GI is inevitable and the only reason that GI is not more prevalent is because the timescale for clumps in weakly cooling disks to achieve fragmentation is longer than the lifetime of the disk. This assumes that once fragmented, clumps do not fall apart and there is no process in the disk that could lead to the disruption of a successfully fragmenting clump. Our cooling implementation becomes weaker for a fragment as it increases in density, offering the necessary resistance to the stochastic formation of growing overdensities such that fragmentation is no longer an eventuality.
Consider the case of a clump which hovers very close to its Roche density, such as the solid blue line in figure <ref>. In this case the simulation forms a clump which should have a better chance of crossing the threshold into becoming a fragment and remaining so. However, even as it manages to form a clump which crosses this threshold once, it still returns to a gravitoturbulent state at simulation time $t = 50\Omega^{-1}$, a result of the local cooling time. For this reason, these simulations do not indicate fragmentation is a strictly stochastic process independent on the strength of fluctuations.
<cit.> and <cit.> have suggested that the lack of convergence may be an effect of the numerical setups used, however <cit.> did not find an issue with the artificial viscosity. <cit.> does however alter they way in which an SPH method cools, using kernel smoothing to spread released heat around to neighboring particles. At high densities this has a similar effect to the cooling used here, with clusters of particles able to share their heat among each other so that dense clumps retain heat and resist collapse. At lower densities cooling is unchanged and this shows in their resulting critical cooling criterion which increases to $7 \leq \beta \leq 9$ compared to the reduction of the criterion in this study. Additionally, the ability of this cooling modification to remain consistent with particle number is uncertain, as it introduces a parameter, smoothing length, which should be scaled with resolution.
<cit.> suggests that fragmentation is the result of the disk being unable to withstand the combined Reynolds and gravitational stresses which results in a fragmentation boundary at $\alpha\approx 0.1$. The simulations here do not support a fragmentation boundary at this value as some disks remain stable at values as high as $\alpha=1$. For simulations with the altered cooling time it is expected that the disk can remain stable to higher stresses because the localized cooling time stabilizes fragments and the disk as a whole.
§.§ Giant Planet Formation
The theories of how planets are formed are slowly starting to come to some agreement. Planetesimals formed by binary accretion of solid objects leads to the formation of rocky planets within 30 au of a young star with massive cores able to accumulate significant gaseous envelopes may become gas giant planets <cit.>. This does not explain how to form large gas planets in outer regions of the disk, but disk instability offers a niche formation mechanism that can form planets in these regions given the right conditions. Since these distant gas planets are not very common and planet formation by instability is not an easy process, it appears that for now this is a reasonable explanation.
A suggestion by <cit.> is that the non-convergence of the critical cooling parameter could lead to large gas giants forming at longer cooling timescales, which implies shorter orbital radii. While the effect of shifting the inner boundary of gravitational instability would be minimal in the case of a single system, the effect on a large population could be more significant. If such a shift becomes more significant it could affect whether the formation regime of gravitational instability is in fact as restricted as observations seem to indicate.
The fragmentation boundary using the cooling scheme presented here shows a significant change compared to that of <cit.> and others. This means that the planet formation by disk instability is more restrictive than previously thought and certainly not heading in the direction of longer cooling timescales. This keeps the formation region of gravitational instability in a narrow region where the dominant theory for planet formation cannot form planets fast enough by the current understanding <cit.>.
§.§ Limitations
In this investigation, only a small modification has been made to the physics of the circumstellar disk and should not be expected to be a final solution to the convergence issue regarding gravitational instability. There are still some drawbacks to this approach though, as numerous assumptions and simplifications were made for the sake of efficient computation of high resolution physics and these might influence the evolution of the disk simulation as well as the fragmentation criteria.
As local simulations, these results do not take into account global parameters like accretion or long-range interactions, offering only a limited view of the disk. Therefore, these simulations do not take into consideration the chance that fragments may migrate or consider how the disk got to that state <cit.>. Fortunately, this does not seem to have a significant effect on the results of fragmentation criteria, with both local and global simulations in agreement on fragmentation criteria in general in its previous implementation <cit.>.
Radiative transfer is a more physically complete description of the cooling in the disk, but due to its relation to realistic opacities and the need for an additional dimension for effective simulation makes it a complicated option. Implementing radiative transfer in addition to adding a vertical computational direction significantly increases the amount of processing power needed for resolutions similar to what is implemented here. Using a simple cooling timescale in 3D leads to the same fragmentation behavior as <cit.> <cit.>, but including radiative transfer has not been shown to lead to consistent fragmentation, with cooling by radiative transfer too slow to form fragments as in <cit.> and <cit.>, but not in <cit.>.
Questions have been raised about the razor-thin disk approximation and the calculation of self-gravity in such an approximation with <cit.> finding that the fragmentation boundary depends strongly on the gravitational smoothing used. Some studies of self-gravitating disks are smoothed on the grid scale (including those carried out here) which exaggerate the strength of self-gravity on small scales and may cause fragmentation at longer cooling timescales at higher resolutions and may be the cause of the non-convergence of the fragmentation boundary.
There are still some improvements which can be made to simulations which use simple cooling. A stronger dependence of the cooling timescale on surface density is possible due to an additional factor of $\Sigma$ in the energy density $U$ of equation (<ref>). While opacity remains independent of surface density at low temperatures, the temperature dependence varies greatly at low temperatures <cit.> and is difficult to model with this simple cooling prescription. For this reason we have only considered surface density in the cooling timescale and a more complete description would handle the changing temperature dependence of opacity.
§ CONCLUSIONS
The current understanding of planet formation is that core accretion is the dominant planet forming process. Core accretion shows the ability to form terrestrial and gaseous planets in a wide range of sizes in regions nearby the central star. But this mechanism does not explain the formation of a few gas giant planets that have formed at very large radii where core accretion takes far too long to occur before the gas in the disk is blown away. Thus, gravitational instability shows the ability to fill this niche by forming massive gas giant planets at radii beyond 50 au.
This picture of planet formation is generally well-formed, but recent results had suggested that it is not as clear as believed. Since gravitational instability showed to occur at shorter radii than expected, its formation region encroached on that of core accretion, possibly blurring the lines formation regions of the two mechanisms. We have carried out 2D hydrodynamic simulations of self-gravitating disks which show:
* At low resolutions, a cooling timescale with surface density dependency results in a disk more stable to fragmentation by self-gravity, with a critical cooling timescale around $\beta \approx 0.5$. This means no clumps had a local cooling time short enough to overcome disruption from tidal shear.
* The increased stability we find in our simulations suggests fragmentation preferentially in regions with short thermal relaxation times.
* At our highest resolution however, simulations fragment even for long cooling timescales (up to $\beta = 10$) indicating that our approach with a surface density dependent cooling timescale did not result in convergence of the fragmentation boundary.
* Many of the gravitoturbulent simulations are stable up to $\alpha=1$, with the gravitational stress component dominating the Reynolds component, and thus stable to very short cooling timescales.
For these reasons we have not found convergence of the fragmentation boundary by using a cooling timescale dependent on the surface density to mimic the effects of varying optical depth.
The authors thank the referee for their thorough and useful comments. We would like to thank the other members of the theory of planet and star formation group, in particular Andreas Schreiber, for their useful advice, patience and help with this project. We would also like to thank Ken Rice for his help in setting up the Pencil Code. Our simulations shown were run on the THEO cluster at the Rechenzentrum Garching (RZG) of the Max Planck Society and the JUQUEEN cluster of the Jülich Supercomputing Centre <cit.>.
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1511.00059
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College of Physical Science and Technology, Sichuan University, Chengdu 610064, China
College of Physical Science and Technology, Sichuan University, Chengdu 610064, China
College of Physical Science and Technology, Sichuan University, Chengdu 610064, China
School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below $10^{-5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels.
03.67.Lx, 03.67.Pp,
§ INTRODUCTION
In quantum computation and communication, quantum error correction (QEC) is necessary for preserving coherent states from noise and other unexpected interactions. Based on classic schemes using redundancy, Shor <cit.> has championed a strategy where a bit of quantum information is stored in an entanglement state of nine qubits. This scheme permits one to correct any error incurred by any of the nine qubits. For the same purpose, Steane <cit.> has proposed a protocol that uses seven quits. The five-qubit code was discovered by Bennett, DiVincenzo, Smolin and Wootters <cit.>, and independently by Laflamme, Miquel, Paz and Zurek <cit.>. The QEC conditions were proved independently by Bennett and co-authors <cit.>, and by Knill and Laflamme <cit.>. The protocols above with different quantum error correction codes (QECCs) can be viewed as active error correction. There are passive error avoiding techniques such as the decoherence-free subspaces <cit.> and noiseless subsystem <cit.>. Recently, it has been proved that all the active and passive QEC methods can be unified <cit.>.
With the known codes constructed in Refs. <cit.>, the standard QEC procedure is designed according to the principle of perfect correction for arbitrary single-qubit errors, where one postulates that single-qubit errors are the dominant terms in the noise process <cit.>. Recently, an optimization-based approach to QEC was explored. In each case, rather than correcting for arbitrary single-qubit errors, the error recovery scheme was adapted to model for the noise, with the goal to maximize the fidelity of the operation <cit.>. Robust channel-adapted QEC protocols, where the uncertainty of the noise channel is considered, have also been developed <cit.>.
For some tasks such as storing or transferring a qubit of information, a near-idealized channel is usually required. If the fidelity obtained from error correction is not high enough, the further increase in levels of concatenation is necessary. In the previous works <cit.>, the application of QEC with the concatenated code was discussed for the Pauli channel which includes the depolarizing channel as the most important example. In general, a QEC protocol contains three steps: encoding, error evolution, and decoding. When the concatenated code is used for encoding, there are two known methods for decoding: the widespread blockwise hard decoding technique and the optimal decoding using a message-passing algorithm <cit.>. As shown by Poulin, the Monte Carlo results using the five-qubit and Steane's code on depolarizing channel reveal significant advantages of message-passing algorithms. For the depolarizing channel, the concatenated five-qubit code is also more efficient than the concatenated seven-qubit code.
In the present work, we shall focus on the following questions: For an arbitrary noise model, instead of finding the optimal QEC protocol adapted to it, is it possible for us to construct a near-perfect channel with a error below $10^{-5}$ by performing a QEC procedure with the concatenated five-qubit code? In order to answer this question, two important methods developed in the previous works are applied here. At first, following the idea in Ref. <cit.>, we shall show that the blockwise decoding can be carried out in a way without performing the error syndrome, and the realization needs some additional quantum resources, required in the message-passing algorithms. The other method comes from the recent works where several general schemes have been developed for describing the exact performance of the QEC procedure based on a $N^l$-qubit concatenated code <cit.>. The main ideas in these schemes can be summarized in the following: First, the exact performance of the QEC with a $N^l$-qubit code is denoted by an effective Choi matrix; Secondly, the Choi matrix in the $l$-th level of concatenation is obtained by simulating the standard quantum process tomography (SQPT) <cit.>. Based on these results, the Choi matrix in the $(l+1)$-th level can be obtained in a similar way. The advantage of introducing the effective Choi matrix is clear: Instead of directly working in the $2^{N^{l}}$-dimensional Hilbert space, one could always simulate the error correction in each level of concatenation in a $2^N$-dimensional system.
For a general noise model, with five free parameters, our numerical simulation indicates that the concatenated five-qubit code is general: The QEC protocol with the concatenated five-qubit code, which is able to construct a near-perfect effective channel from the noise depolarizing channels, is also sufficient to complete the same task for other types of noise channels with the same channel fidelity.
The content of present work is organized as follows. In Sec. <ref>, we construct an error correction protocol with the five-qubit code <cit.>, and a chosen unitary transformation is shown to be sufficient for correcting the errors of the principle system. In Sec. <ref>, an explicit scheme is designed to obtain the effective Choi matrix. For three types of channels, the depolarizing channel, the amplitude damping channel and the bit-flip channel, the effective Choi matrices obtained by performing QEC with the concatenated five-qubit code is shown in Sec. <ref>. In Sec. <ref>, it is shown that the concatenated seven-qubit is not general. In Sec. <ref>, a noise model containing five free parameters is constructed, and we argue that the concatenated five-qubit code is general. Finally, we end our work with a short discussion.
§ UNITARY REALIZATION OF QUANTUM ERROR CORRECTION
The $N$-qubit code concatenated with itself $L$ times yields a $N^{L}$-qubit code, providing a better error resistance with increasing $L$. The direct simulation of the quantum dynamics, coding and encoding procedure require massive computation resources. By following the idea in Refs. <cit.>, the concept that the exact performance of QEC can be described by an effective channel, will make the calculation simplified.
Let us consider the QEC protocol with code in Ref. <cit.>,
\begin{eqnarray}
\vert 0_\mathcal{L}\rangle =&&\frac{1}{4}[\vert 00000\rangle+\vert 10010\rangle+\vert 01001\rangle+\vert 10100\rangle\nonumber\\
&&+\vert 01010\rangle-\vert 11011\rangle-\vert 00110\rangle-\vert 11000\rangle\nonumber\\
&&-\vert 11101\rangle-\vert 00011\rangle-\vert 11110\rangle-\vert 01111\rangle\nonumber\\
&&-\vert 10001\rangle-\vert 01100\rangle-\vert 10111\rangle+\vert 00101\rangle],\nonumber
\end{eqnarray}
\begin{eqnarray}
\vert 1_\mathcal{L}\rangle=&&\frac{1}{4}[\vert 11111\rangle+\vert 01101\rangle+\vert 10110\rangle+\vert 01011\rangle\nonumber\\
&&+\vert 10101\rangle-\vert 00100\rangle-\vert 11001\rangle-\vert 00111\rangle\nonumber\\
&&-\vert 00010\rangle-\vert 11100\rangle-\vert 00001\rangle-\vert 10000\rangle\nonumber\\
&&-\vert 01110\rangle-\vert10011\rangle-\vert 01000\rangle+\vert 11010\rangle].\nonumber
\end{eqnarray}
We use $H_{\mathcal{S}}$ for the two-dimensional principle system, where the basis vectors are denoted by $\vert 0\rangle$ and $\vert 1\rangle$, while the ancilla system system lies in a $2^4$-dimensional Hilbert space $H_{\mathcal{A}}$ with the basis $\{\vert a_m\rangle\}_{m=0,...,15}$.
The standard way to get the effective noise channel is depicted in Fig. <ref>(a). It contains the following steps.
(i) The encoding procedure can be realized with a unitary transformation $U$,
\begin{equation}
U\vert a_0\rangle\otimes \vert 0\rangle\rightarrow\vert 0_\mathcal{L}\rangle,
U\vert a_0\rangle\otimes \vert 1\rangle\rightarrow\vert 1_\mathcal{L}\rangle.
\end{equation}
(ii) The noise evolution is denoted by $\Lambda$. The five-qubit code above is designed to correct the set of single-qubit errors, $\{E_m\}_{m=0,1,...,15}$. Usually, $E_0$ is fixed to be identity operator $\hat{\texttt{I}}$, and each $E_m(m\neq 0)$ is one of the Pauli operators $\hat{\sigma}^i_j (i=1,...,5, j=x,y,z)$. With the logical codes, one could introduce a set of normalized states
\begin{equation}
\label{norsta}
\vert m,+\rangle=E_m\vert 0_\mathcal{L}\rangle, \vert m,-\rangle=E_m\vert
\end{equation}
Since the set of errors, $\{E_m\}_{m=0,...,15}$, could be perfectly corrected, there should be $\hat{P}_{\mathcal{C}}E^{\dagger}_mE_n\hat{P}_{\mathcal{C}}=\delta_{mn}\hat{P}_{\mathcal{C}}$,
where $\hat{P}_{\mathcal{C}}=\vert 0_\mathcal{L}\rangle\langle 0_\mathcal{L}\vert+ \vert
1_\mathcal{L}\vert$. Therefore, one may easily verify that $\{\vert m, \pm\rangle\}_{m=0}^{15}$ form an orthogonal basis.
(iii) With the denotation $\vert a_m , i\rangle=\vert a_m\rangle\otimes \vert i\rangle $, the recovery operation can be described by a process $\mathcal{R}$ such that $\mathcal{R}(\rho^{\mathcal{SA}})=
\sum_{m=0}^{15}R_{m}\rho^{\mathcal{SA}}R_{m}^{\dagger}$, where the Kraus operators $R_m$ are <cit.>,
\begin{equation}
R_m=\vert a_m, 0\rangle \langle m, +\vert+\vert a_m, 1\rangle\langle m, -\vert.
\end{equation}
(iv) The decoding is realized by $U^{\dagger}$, the Hermite conjugate of $U$, $UU^{\dagger}=\mathrm{I}^{\otimes 5}$, and the effective channel $\bar{\varepsilon}$ is
\begin{equation}
\tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr}_{\mathcal{A}}[\mathcal{U}^{\dagger}\circ \mathcal{R}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
\end{equation}
(a) The way of getting the effective channel from the standard QEC protocol including encoding, noise evolution, recovery and decoding. (b) The two processes, $\mathcal{R}\circ \mathcal{U}^{\dagger}$ and $\mathcal{U}^{\dagger}\circ \mathcal{\tilde{R}}$, are equivalent.
(c) When $U^{\dagger}$ for decoding is fixed according Eq. (6), the process $\mathcal{ \tilde{R}}$ can be decomposed as $\mathcal{\tilde{R}}_{\mathcal{SA}}=\mathcal{\tilde{R}}_{\mathcal{A}}\otimes \mathbb{I}_{\mathcal{S}}$. Certainly, it can be moved away. (d) Our protocol where the chosen unitary transformation is sufficient to correct the errors of the principle system. The errors of the ancilla system is left to be uncorrected.
The above is the standard QEC protocol, and the unitary transformation $U$ ($U^{\dagger}$) used for encoding (decoding) is not unique. In this work, however, the unitary transformation is fixed as
\begin{equation}
U\vert a_m\rangle\otimes \vert 0\rangle\rightarrow \vert m, +\rangle, U\vert a_m\rangle\otimes \vert 1\rangle\rightarrow \vert m, -\rangle,\end{equation}
or in the equivalent form
\begin{equation}
\label{udag}
U^{\dagger}\vert m, +\rangle\rightarrow\vert a_m\rangle\otimes \vert 0\rangle,U^{\dagger}\vert m, -\rangle\rightarrow\vert a_m\rangle\otimes \vert 1\rangle,
\end{equation}
Certainly, $\vert 0_\mathcal{L}\rangle =\vert 0,+\rangle, \vert 1_\mathcal{L}\rangle =\vert 0,-\rangle$. [It should be emphasized that the $U^{\dagger}$ introduced above is nothing else but the $U_2$ used in the Eq. (87) of the original work in Ref. <cit.>.] As it has been argued in Ref. <cit.>, the recovery process $\mathcal{R}$ is not a necessary step, since the $U^{\dagger}$ defined in Eq. (<ref>) is sufficient for correcting the errors of the principle system. One can observe that, the following two processes are equivalent
\begin{equation}
\mathcal{R}\circ \mathcal{U}^{\dagger}\equiv \mathcal{U}^{\dagger}\circ \mathcal{\tilde{R}},
\end{equation}
where the process $\mathcal{\tilde{R}}$ is defined as
\begin{equation}
\mathcal{\tilde{R}}=\mathcal{U}\circ \mathcal{R} \circ \mathcal{U}^{\dagger}.
\end{equation}
Furthermore, it can be expressed with a more explicit way, $\mathcal{\tilde{R}}(\rho^{\mathcal{SA}})=\sum_{m=0}^{15}\tilde{R}_m(\rho^{\mathcal{SA}})\tilde{R}_m^{\dagger}$,
where the Kraus operators $\tilde{R}_m$ take the form
By some simple algebra, one may get
$\tilde{R}_m=\vert a_0\rangle\langle a_m\vert\otimes \mathrm{I}$,
and one can easily verify that: After an arbitrary state $\rho^{\mathcal{SA}}$ of the jointed system is subjected to the process $\mathcal{\tilde{R}}$, the state of the principle system remains unchanged, say,
\begin{equation}
\rho^{\mathcal{S}}=\mathrm{Tr} _{\mathcal{A}}[\rho^{\mathcal{SA}}]=\mathrm{Tr} _{\mathcal{A}}[\mathcal{\tilde{R}}(\rho^{\mathcal{SA}})].
\end{equation}
According to this analysis, the process $\mathcal{\tilde{R}}$ can be moved away. It has been shown in Fig. <ref>(d) that a simplified protocol to obtain the effective channel is defined as,
\begin{equation}
\label{vare}
\tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr} _{\mathcal{A}}[\mathcal{U}^{\dagger}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
\end{equation}
§ THE STANDARD QUANTUM PROCESS TOMOGRAPHY
To get the complete information about the effective channel, we shall introduce a convenient tool where a bounded matrix in $H_{2}$ is related to a vector in the enlarged Hilbert space $H_{2}^{\otimes 2}$. Let $A$ be a bounded matrix in the $2$-dimensional Hilbert space $H_2$, with $A_{ij}=\langle i\vert A\vert j\rangle$
the matrix elements for it, and an isomorphism between $A$ and a $2^2$-dimensional vector $\vert A\rangle\rangle$ is defined as
\begin{equation}
\vert A\rangle\rangle =\sqrt{2} A\otimes \mathrm{I}_{2}\vert S_+\rangle=\sum_{i,j=0}^1 A_{ij}\vert ij\rangle,
\end{equation}
where $\vert S_+\rangle$ is the maximally entangled state for $H_{2}^{\otimes 2}$, and $\vert S_+ \rangle =\frac{1}{\sqrt{2}}\sum_{k=0}^{1}\vert kk\rangle$ with $\vert ij\rangle=\vert i\rangle\otimes \vert j\rangle$.
This isomorphism provides a one-to-one mapping between the matrix and its vector form.
For a quantum process $\varepsilon$, the Kraus operators $\{A_m\}$ can be described by a corresponding Choi matrix,
\begin{equation}
\label{choi}
\chi(\varepsilon)=\sum_{m}\vert A_m\rangle\rangle\langle\langle A_m\vert.
\end{equation}
Via the isomorphism above, this matrix can also be rewritten as
$ \chi(\varepsilon)=2\bar{\rho}$, with $\bar{\rho}=\varepsilon\otimes \mathrm{I}(\vert S_+\rangle\langle S_+\vert)$.
For the normalized state $\bar{\rho}$, Schumacher's entangling fidelity is defined as $F=\langle S_+\vert\bar{\rho}\vert S_+\rangle$, and it provides a measure for how well the entanglement is preserved by the quantum process $\varepsilon$ <cit.>. Certainly, one may calculate the entangling fidelity
\begin{equation}
\label{fed}
F(\varepsilon)=\frac{1}{2} \langle S_+\vert\chi(\varepsilon)\vert S_+\rangle.
\end{equation}
With $\chi$ known, one can derive the Kraus operators of $\varepsilon$. This can be completed through the following simple protocol: The eigenvalues $\lambda^m$ and the corresponding eigenvectors $\vert \Phi^m\rangle$ of $\chi$ can be easily calculated, say,
$\chi=\sum_m\lambda^m\vert \Phi^m\rangle\langle \Phi^m\vert$. Suppose that $\vert\Phi^m\rangle$ can be expanded as $\vert \Phi^m\rangle=\sum_{ij}c^m_{ij}\vert ij\rangle$, with $c^m_{ij}=\langle ij\vert\Phi^m\rangle$ the expanding coefficients. Then, the Kraus operators $A_m$ can be expressed as
\begin{equation}
\label{operator}
A_m=\sqrt{\lambda^m} \sum_{i,j=0}^1 c^m_{ij}\vert i\rangle\langle j\vert.
\end{equation}
With these operators, one may verify that the relation in Eq. (<ref>) is recovered.
The effective channel can be obtained via the performing the SQPT <cit.>. Here, it should be mentioned that the way of performing SQPT is not limited. In the present work, we shall apply the protocol presented in Ref. <cit.>. For convenience, a brief review of this protocol is organized in following: Introducing the set of operators, say, ${E}_{cd}=\vert c\rangle\langle d\vert (c,d=0,1)$, we take them as the inputs for the principle system, and for a given $E_{cd}$, the corresponding output is
\begin{equation}
\label{varep}
\tilde{\varepsilon}({{E}}_{cd })=\mathrm{Tr}_{\mathcal{A}}[\mathcal{U}^{\dagger}\circ \Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \vert c\rangle\langle d\vert)].
\end{equation}
Then, one may introduce the coefficients
\begin{equation}
\label{lambda}
\tilde{\lambda}_{ab;cd}=\langle a\vert \tilde{\varepsilon}({{E}}_{cd })\vert b\rangle,
\end{equation}
and $\tilde{\varepsilon}({{E}}_{cd })$ can be expanded as
${\tilde{\varepsilon}}({{E}}_{cd })=\sum_{a,b=0}^{1}\tilde{\lambda}_{ab;cd}\vert a\rangle\langle b\vert$.
The Choi matrix of the effective channel $\tilde{\varepsilon}$ in Eq. (<ref>), can be expanded as
\begin{equation}
\chi(\tilde{\varepsilon})=\sum_{a,b,c,d=0}^1\tilde{\chi}_{ab;cd} \vert ab\rangle\langle cd \vert,
\end{equation}
with $\tilde{\chi}_{ab;cd}$ its matrix elements,
\begin{equation}
\label{tchi}
\tilde{\chi}_{ab;cd}=\langle ab\vert \chi(\tilde{\varepsilon})\vert cd\rangle.
\end{equation}
It has been shown in Ref. <cit.> that the $\tilde{\chi}_{ab;cd}$ can be obtained in a simple way,
\begin{equation}
\label{mapping}
\tilde{\chi}_{ab;cd}=\tilde{\lambda}_{ac;bd}.
\end{equation}
§ EXACT PERFORMANCE OF THE CONCATENATED FIVE-QUBIT CODE
Now, we shall restrict our attention to the uncorrelated errors. We use $\varepsilon:\{A_m\}$ to denote the quantum process of each two-dimensional subsystem. Formally, $\Lambda=\varepsilon^{\otimes 5}$. Let us consider the lowest level of error correction for the depolarizing channel $\varepsilon_{\mathrm{DP}}$,
\begin{equation}
A_0=\sqrt{F_0}\hat{\texttt{I}}_2, A_i=\sqrt{\frac{1-F_0}{3}}\hat{\sigma}_i
\end{equation}
with $i=1,2,3$ and $F_0$ its channel fidelity. We shall take it as
an explicit example to show how the SQPT is completed.
(a) Let $\vert 0\rangle\langle 0\vert$ the input of the principle system. The corresponding output is denoted by $\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)$. With the equation
\[\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)=\mathrm{Tr} _{\mathcal{A}}[\mathcal{U}^{\dagger}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \vert 0\rangle\langle 0\vert],\]
one has
\begin{equation}
\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)=\left(
\begin{array}{cc}
a & 0 \\
0 & 1-a \\
\end{array}
\right),
\nonumber
\end{equation}
where $a=\frac{1}{81}(1+2F_0)^2(37-108F_0+144F_0^2-64F_0^3)$. Based on the definition in Eq. (<ref>), the four matrix elements, $\tilde{\lambda}_{ab;00}$ ($a,b=0,1$), are
\begin{equation}
\tilde{\lambda}_{00;00}=a, \tilde{\lambda}_{10;00}=\tilde{\lambda}_{01;00}=0, \tilde{\lambda}_{11;00}=1-a.\nonumber
\end{equation}
(b) Similarly with step (a), one may also obtain
\begin{eqnarray}
\tilde{\varepsilon}(\vert 0\rangle\langle 1\vert)&=&\left(
\begin{array}{cc}
0 & 2a-1 \\
0 & 0 \\
\end{array}
\right),\nonumber\\
\tilde{\varepsilon}(\vert 1\rangle\langle 0\vert)&=&\left(
\begin{array}{cc}
0 & 0 \\
2a-1 & 0 \\
\end{array}
\right),\nonumber\\
\tilde{\varepsilon}(\vert 1\rangle\langle 1\vert)&=&\left(
\begin{array}{cc}
1-a & 0 \\
0 & a \\
\end{array}
\right), \nonumber
\end{eqnarray}
and therefore,
\begin{eqnarray}
\tilde{\lambda}_{00;01}= \tilde{\lambda}_{10;01}=\tilde{\lambda}_{11;01}=0, \tilde{\lambda}_{01;01}=2a-1,\nonumber\\
\tilde{\lambda}_{00;10}= \tilde{\lambda}_{01;10}=\tilde{\lambda}_{11;10}=0, \tilde{\lambda}_{10;10}=2a-1,\nonumber\\
\tilde{\lambda}_{00;11}=1-a, \tilde{\lambda}_{10;11}=\tilde{\lambda}_{01;11}=0, \tilde{\lambda}_{11;11}=a.\nonumber
\end{eqnarray}
(c) With all the matrix elements, $\tilde{\lambda}_{ab;cd}$, the so-called matrix $\tilde{\lambda}$ can be orgnized as
\begin{equation}
\tilde{\lambda}=\left(
\begin{array}{cccc}
a & 0 & 0 & 1-a \\
0 & 2a-1& 0 & 0 \\
0 & 0 & 2a-1& 0 \\
1-a & 0 & 0 & a \\
\end{array}
\right).\nonumber
\end{equation}
According to the one-to-one relation in Eq. (<ref>), the effective Choi matrix $\tilde{\chi}$ is
\begin{equation}
\label{varchi}
\tilde{\chi}(\varepsilon_{\mathrm{DEP}})=\left(
\begin{array}{cccc}
a& 0 & 0 & 2a-1 \\
0 & 1-a & 0 & 0 \\
0 & 0 & 1-a & 0 \\
2a-1 & 0 & 0 & a \\
\end{array}
\right).
\end{equation}
(d) The entangling fidelity of the effective channel $\tilde{\chi}$ is denoted by $F_1$, and now Eq. (<ref>) can be rewritten as
\begin{equation}
\end{equation}
Based on it, there should be
\begin{equation}
\label{fed1}
\end{equation}
A parameter $p$ can be used to characterize the depolarizing channel, say, $A_0=\sqrt{1-3p/4}\hat{\texttt{I}}_2, A_i=\sqrt{p}/2\hat{\sigma}_i$, and with the relation $F_0=1-3p/4$, Eq. (<ref>) can be rewritten as
\begin{equation}
\end{equation}
This is the same as the result by Reimpell and Werner <cit.>. Furthermore, by requiring that $F_1\ge F_0$, we have the threshold $p<0.18$ (or $F_0>0.86$), the condition under which the five-qubit code works for the depolarizing channel.
(e) By some simple algebra, the eigenvalues $\lambda^m$ and the corresponding eigenvectors $\vert \Phi^m\rangle$ can be derived,
\begin{eqnarray}
\lambda^0&=&2F_1, \lambda^1=\lambda^2=\lambda^3=\frac{2(1-F_1)}{3},\nonumber\\
\vert\Phi^0\rangle&=&\frac{1}{\sqrt{2}}(\vert 00\rangle+\vert 11\rangle),\nonumber\\
\vert\Phi^1\rangle&=&\frac{1}{\sqrt{2}}(\vert 01\rangle+\vert 10\rangle),\nonumber\\
\vert\Phi^2\rangle&=&\frac{1}{\sqrt{2}}(\vert 01\rangle-\vert 10\rangle),\nonumber\\
\vert\Phi^3\rangle&=&\frac{1}{\sqrt{2}}(\vert 00\rangle-\vert 11\rangle).\nonumber
\end{eqnarray}
With Eq. (<ref>), one may verify that the effective Choi matrix can be also expressed by a set of Kraus operators $\tilde{A}_m$,
$\tilde{\chi}=\sum_m\vert \tilde{A}_m\rangle\rangle\langle\langle \tilde{A}_m\vert$, where
\begin{equation}
\tilde{A}_0=\sqrt{F_1}\hat{\texttt{I}}_2, \tilde{A}_i=\sqrt{\frac{1-F_1}{3}}\hat{\sigma}_i
\end{equation}
with $i=1,2,3$ and $F_1$ the entangling fidelity. Obviously, the effective channel
is also a depolarizing channel <cit.>.
(f) With the Kraus operators above, a new supper operator $\tilde{\varepsilon}$: $\tilde{\varepsilon}(\rho)=\sum \tilde{A}_m\rho \tilde{A}_m^{\dagger}$ can be defined. Let $\Lambda=\tilde{\varepsilon}^{\otimes 5}$ and follow the steps from (a) to (e), we can obtain the effective
channel by the $5^2$-qubit concatenated code. By repeating the argument above, we can get the effective channel from the $5^L$-qubit
concatenated code.
The protocol developed above, used for the depolarizing channel $ \tilde{\varepsilon}_{\mathrm{DEP}}$, can be easily generalized for other cases. For instance, the amplitude damping channel $\varepsilon_{\mathrm{AD}}$, has been widely discussed in previous works, and the Kraus operators are now
\begin{equation}
\label{damping1}
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{1-\gamma} \\
\end{array}
\right),A_1=\left(
\begin{array}{cc}
0 & \sqrt{\gamma} \\
0 & 0 \\
\end{array}
\right),\end{equation}
where $\gamma$ is the damping parameter. The entangling fidelity of it is $F_0=\frac{1}{4}(1+\sqrt{1-\gamma})^2$.
When the five-qubit code is applied for correcting the amplitude damping errors, the entangling fidelity of the effective channel is
\begin{eqnarray}
\label{damping1}
\end{eqnarray}
Another important case is the bit-flip channel $\varepsilon_{\mathrm{BF}}$ with
\begin{equation}
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right),
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right),
\end{equation}
where $F_0$ is the entangling fidelity.
For a reason which will be clear soon, here we first introduce the denotations,
$\varepsilon_{\mathrm{DP}}\equiv\varepsilon^{0}(\omega_0,F_0)$, $\varepsilon_{\mathrm{AD}}\equiv\varepsilon^{0}(\mathbb{\omega}_1, F_0)$ and $\varepsilon_{\mathrm{BF}}\equiv
\varepsilon^{0}(\omega_2, F_0)$, where $F_0$ is the fidelity of the initial channel without performing QEC. After performing QEC with the
$5^l$-qubit code, the effective channel is denoted by $\varepsilon^{l}(\omega, F_0)$ with the setting $(\mathbf{\omega}, F_0)$ indicating that the effective channel is originated from the initial $\varepsilon^0(\mathbf{\omega}, F_0)$.
With the Choi matrix $\chi(\varepsilon^{l}(\omega, F_0))$, the entangling fidelity of the effective channel can be calculated as
\begin{equation}
\label{Fedl}
F_{l}(\omega,F_0)=\frac{1}{2}\langle S_+\vert \chi(\varepsilon^{l}(\omega, F_0))\vert S_+\rangle.
\end{equation}
In the present work, a quantum channel is called near-perfect if the entangling fidelity has a value above $1-10^{-5}$. For a given initial channel $\varepsilon^{0}(\omega, F_0)$, we introduce the quantity $L_{(\omega,F_0)}$ and let $5^{L_{(\omega,F_0)}}$ be the minimum size of the concatenated code sufficient for constructing a near-perfect channel, $F_{L_{(\omega,F_0)}}\ge 1-10^{-5}$.
Correspondingly, a $5^{L_{(\omega,F_0)}}$-qubit code is said to be general if $L_{(\omega,F_0)}$ is independent of $\omega$, which is used for denoting the actual type of the error model.
When the concatenated code is applied, the effective $\chi(\varepsilon^{l}(\omega, F_0))$ ($\omega\neq \omega_0$) usually does not have an analytical form. Noting that $\chi(\varepsilon^{l}(\omega_0, F_0))$ always represents a depolarizing channel, we suppose that $\chi(\varepsilon^{l}(\omega, F_0))$ can be approximated by $\chi(\varepsilon^{l}(\omega_0, F_0))$, and the error of the approximation is characterized by the distance measure
\begin{equation}
\label{dist}
D_{l}(\omega, F_0)=\frac{1}{4}\vert \chi(\varepsilon^{l}(\mathbf{\omega}, F_0))-\chi(\varepsilon^{l}(\mathbf{\omega}_0, F_0))\vert,
\end{equation}
with $\vert A\vert=\sqrt{A^{\dagger}A}$ <cit.>. Especially, if $D_{l}(\omega, F_0)\ll 0$, the entangling fidelities of the two channels, $\varepsilon^{l}(\mathbf{\omega}, F_0)$ and $\varepsilon^{l}(\mathbf{\omega}_0, F_0)$, almost have the same value.
Under the condition that the entangling fidelity $F_0$ ($F_0\ge 0.86$) is fixed, besides the steps from (a) to (f) for getting the effective channel, we added another two ones:
(g) With the Choi matrix $\chi(\varepsilon^{1}(\mathbf{\omega}, F_0))$ corresponding to the effective channel $\varepsilon^{1}(\mathbf{\omega}, F_0)$, the entangling fidelity $F_1( \mathbf{\omega}, F_0)$ is decided by Eq. (<ref>), and the distance $D_{1}(\omega , F_0)$ is calculated according to Eq. (<ref>).
(h) With a simple program, the Kraus operators of the effective
channel can be decided, and the calculation for the effective channel in the second level starts from step (a).
The calculation for a given $\varepsilon(\mathbf{\omega}, F_0)$ can be terminated if
$F_l(\omega, F_0)\geq 1-10^{-5}$.
Based on the iterative protocol developed above, the effective channels in each level of the concatenation can be worked out. For the typical case
where $F_0=0.92$, as shown in Table 1, we have two observations that: (I) For all the possible channels, a perfect effective channel (with a error below $10^{-5}$) can be constructed by using the same concatenated five-qubit code;
The fidelity $F_{l}(\omega, F_0)$
$l$ $F_l(\omega_0)$ $F_l({\omega_1})$ $F_{l}(\omega_2)$
0 0.920 0.920 0.920
1 0.946665 0.946762 0.945639
2 0.974784 0.97487 0.973903
3 0.993991 0.99403 0.993576
4 0.999644 0.999648 0.999593
5 0.999999 0.999999 0.999998
(II) Meanwhile, the resulted effective channel can be approximated by the depolarizing channel. As shown in Table 2, the error of the approximation approaches to zero when $l$, the level of concatenation, is increased.
The error of the approximation.
$l$ $D_l(\omega_1, 0.92)$ $D_l(\omega_2, 0.92)$
0 $8.02\times 10^{-2}$ $5.33\times 10^{-2}$
1 $3.68\times 10^{-3}$ $1.79\times 10^{-2}$
2 $8.77\times 10^{-5}$ $2.04\times 10^{-3}$
3 $5.51\times 10^{-6}$ $2.92\times 10^{-5}$
4 $2.15\times 10^{-10}$ $6.29\times 10^{-9}$
5 $6.04\times 10^{-14}$ $1.46\times 10^{-13}$
The effective channel for other cases, where $F_0$ takes different values, have also been calculated. Our calculation indicates that the two observations, (I) and (II) above, are independent of the choice of $F_0$.
§ EXACT PERFORMANCE OF THE CONCATENATED SEVEN-QUBIT CODE
In this section, we will show that the concatenated seven-qubit code is not general, or in other word, the number of levels for the concatenation, which is necessary for constructing a near-perfect channel, is dependent on the actual type of the noise. For the seven-qubit code <cit.>, we can also define a unitary transformation $V$ in a $2^7$-dimensional Hilbert space for encoding, and its inverse $V^{\dagger}$ is applied for decoding.
Let $\vert 0_{\mathcal{L}}\rangle$ and $\vert 1_{L}\rangle$ be the logical codes, select a set of correctable errors$\{E_m\}_{m=0}^{63}$ including: The identity operator $ E_0=\mathrm{I}^{\otimes 7}$), all the rank-one Pauli operator $\sigma_i^{n}$ ($i=x,y,z$, $n=1,2,...,7)$, and a number of 44 rank-two operators like $\sigma^1_1\otimes \sigma^2_2$, $\sigma^1_3\otimes \sigma^5_2$, ..., etc., and with the definition
\[ \vert m,+\rangle=E_m\vert 0_{\mathcal{L}}\rangle, \vert m,-\rangle=E_m\vert 1_{\mathcal{L}}\rangle, \]
we find that the set of normalized vectors $\{\vert m, \pm\rangle\}_{m=0}^{63}$ form the basis of the $2^7$-dimensional Hilbert space.
Use $\{\vert a_m\rangle\}_{m=0}^{63}$ to denote the basis of ancilla system, and with the denotations
$\vert a_m, 0\rangle=\vert a_m \rangle\otimes \vert 0\rangle$ and $\vert a_m, 1\rangle=\vert a_m \rangle\otimes \vert 1\rangle$, the unitary transformation $V$ is
\[V=\sum_{m=0}^{63} \big(\vert m, +\rangle\langle a_m, 0\vert+ \vert m, -\rangle\langle a_m, 1\vert\big).\]
Finally, define $\Lambda=\varepsilon^{\otimes 7}$ and $\vert a_0\rangle=\vert 000000\rangle$, and the effective channel, which is obtained by performing
QEC procedure with the seven-qubit code, can be obtained as
\begin{equation}
\tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr} _{\mathcal{A}}[\mathcal{V}^{\dagger}\circ\Lambda\circ \mathcal{V}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
\end{equation}
To make sure that our program works in a perfect way, we consider a scenario where the seven-qubit code is applied for the amplitude damping in
Eq. (<ref>). The analytical expression for the entangling fidelity of the
effective channel is
\begin{eqnarray}
\label{damping2}
\end{eqnarray}
One can easily check that this result recovers the numerical one given in Ref. <cit.>, and the entangling fidelities of the effective channel when the five-qubit code and the seven-qubit code are used against the amplitude errors are compared in Fig. <ref>.
(Color online) The entangling fidelities of the effective channel when the five-qubit code and the seven-qubit code are applied against the amplitude damping errors. The exact function in Eq. (<ref>) is shown with the solid line while the one in Eq. (<ref>) for the seven-qubit code, is in the dash line.
The fidelity obtained for the concatenated Steane's code
1 $F_l(\varepsilon_{\mathrm{DEP}})$ $F_l(\varepsilon_{\mathrm{AD}})$ $F_l(\varepsilon_{\mathrm{BF}})$
0 0.94 0.94 0.94
1 0.952211 0.943496 0.943035
2 0.968897 0.950234 0.947904
3 0.986173 0.960975 0.955409
4 0.997048 0.975311 0.966146
5 0.99985 0.989674 0.979469
6 $\geqslant 1-10^{-5}$ 0.99811 0.99196
7 $\geqslant 1-10^{-5}$ 0.999935 0.998693
8 $\geqslant 1-10^{-5}$ $\geqslant 1-10^{-5}$ 0.99964
9 $\geqslant 1-10^{-5}$ $\geqslant 1-10^{-5}$ $\geqslant 1-10^{-5}$
As shown in the above section, one can get the exact performance of the QEC protocol based on the concatenated Steane code. For three types of noise models, the depolarizing channel $\varepsilon_{\mathrm{DEP}}$, the amplitude damping channel $\varepsilon_{\mathrm{AD}}$, and
the bit flip channel $\varepsilon_{\mathrm{BF}}$, the corresponding entangling fidelities $F_l$ have been calculated under the condition that the fidelity of the uncorrected channels is fixed to be $F_0=0.94$. Our results, listed in Table 3, demonstrate that the necessary numbers of the concatenation are dependend on the actual types of the error.
§ THE GENERAL NOISE MODEL
In this section, we shall show that the observation, which has been observed for the amplitude damping and bit flip channels, is general. Let $\{\bar{A}_m\}$ to be a set of Kraus-operators, introduce an arbitrary $2\times2$ unitary transformation,
\begin{equation}
\begin{array}{cc}
\cos\frac{\theta}{2} & \sin\frac{\theta}{2}\exp\{-i\phi\} \\
-\sin\frac{\theta}{2}\exp\{i\phi\} & \cos\frac{\theta}{2} \\
\end{array}
\right),\nonumber
\end{equation}
and another set of operators $\{A_m\}$ can be defined as
\begin{equation}
\label{Abar}
A_m=U_2(\theta,\phi)\bar{A}_m U^{\dagger}_2(\theta,\phi),
\end{equation}
where $\theta$ and $\phi$ are two free parameters.
Three free parameters, $\alpha$, $\beta$, and $\gamma$, are used to define the following four operators,
\begin{eqnarray}
\label{As}
\bar{A}_0=\left(
\begin{array}{cc}
\cos\alpha & 0 \\
0 & \sin\beta\cos\gamma \\
\end{array}
\right), \bar{A_1}=\left(
\begin{array}{cc}
0 & 0 \\
\sin\alpha\sin\gamma & 0 \\
\end{array}
\right),\nonumber\\
\bar{A}_2=\left(
\begin{array}{cc}
0 & \sin\beta\sin\gamma \\
0 & 0\\
\end{array} \right),\bar{A}_3=\left(
\begin{array}{cc}
\sin\alpha\cos\gamma & 0 \\
0 & \cos\beta \\
\end{array}
\right).\nonumber\\
\end{eqnarray}
Now, let us recall some discussions about using the Bloch sphere representation to describe the single-qubit channel <cit.>. With $\vec{r}$ the Bloch vector for an input state $\rho$, and $\vec{\bar{r}}$ for the output state, $\bar{\varepsilon}(\rho)=\sum_m\bar{A}_m\rho(\bar{A}_m)^{\dagger}$, on can obtain a map
\begin{equation}
\vec{r}\rightarrow\vec{\bar{r}}=M\vec{r}+\vec{\delta},
\end{equation}
where $M$ is a $3\times3$ real matrix, and $\vec{\delta}$ is a constant vector. This is an affine map, mapping the Bloch vector into itself. The set of operators in Eq. (<ref>) can be described as
\begin{equation}
\label{map}
\left(
\begin{array}{c}
\bar{r}_x \\
\bar{r}_y \\
\bar{r}_z \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\eta_{\bot} & 0 & 0 \\
0 & \eta_{\bot} & 0 \\
0 & 0 & \eta_z \\
\end{array}
\right)\left(
\begin{array}{c}
r_x \\
r_y \\
r_z \\
\end{array}
\right)+\left(
\begin{array}{c}
0 \\
0 \\
\delta_z \\
\end{array}
\right),
\end{equation}
with the coefficients
\begin{eqnarray}
\eta_{\bot}&=&\sin(\alpha+\beta)\cos\gamma,\nonumber\\
\eta_z&=&1-(\sin^2\alpha+\sin^2\beta)\sin^2\gamma,\\
\delta_z&=&(\sin^2\beta-\sin^2\alpha)\sin^2\gamma.\nonumber
\end{eqnarray}
This affine map can be roughly classified into the following two cases: the centered map with $\delta_z=0 $ (if $ \alpha=\beta$ ) and the non-centered one with $\delta_z\neq 0$.
The centered map of Eq. (<ref>) is equivalent with the Pauli-channel,
$\sqrt{p_0}\hat{\texttt{I}_2}, \sqrt{p_x}\hat{\sigma}_x,\sqrt{p_y}\hat{\sigma}_y,\sqrt{p_z}\hat{\sigma}_z$, and the parameters $p_i$ are given by
and $p_0=1-p_x-p_y-p_z$.
Certainly, it also contains the following two important situations: If $\alpha$ is fixed as
\begin{equation}
\cos\alpha=\frac{\cos\gamma+\sin\gamma}{\sqrt{2+\sin 2\gamma}},
\sin\alpha=\frac{1}{\sqrt{2+\sin 2\gamma}},\nonumber
\end{equation}
one can obtain a depolarizing channel, while for $\gamma=0$ and an arbitrary $\alpha$, we have the phase-flip channel.
With the $U_2(\theta,\phi)$ introduced above, our noise model should also contain the bit-flip channel and bit-phase flip channel. For the non-centered case, let $\alpha=0$ and $\beta=\frac{\pi}{2}$, and we can come to the amplitude damping channel,
\begin{equation}
\bar{A}_0=\left(
\begin{array}{cc}
1& 0 \\
0 & \cos\gamma \\
\end{array}
\right),\bar{A}_1=\left(
\begin{array}{cc}
0 & \sin\gamma \\
0 &0 \\
\end{array}
\right).\nonumber
\end{equation}
For the case where the constraint $\alpha+\beta=\frac{\pi}{2}$ holds, we have the so-called generalized amplitude damping channel <cit.>,
\begin{eqnarray}
\bar{A}_0=\cos\alpha\left(
\begin{array}{cc}
1 & 0 \\
0 & \cos\gamma \\
\end{array}
\right), \bar{A}_1=\sin\alpha\left(
\begin{array}{cc}
0 & 0 \\
\sin\gamma & 0 \\
\end{array}
\right),\nonumber\\
\bar{A}_2=\cos\alpha\left(
\begin{array}{cc}
0 & \sin\gamma \\
0 & 0 \\
\end{array}
\right),\bar{A}_3=\sin\alpha\left(
\begin{array}{cc}
\cos\gamma& 0 \\
0 & 1 \\
\end{array}
\right).\nonumber
\end{eqnarray}
From the discussions above, it can be seen that nearly all the noise channel listed in Ref. <cit.> are included here. Therefore, our noise model, defined in Eqs. (<ref>) and (<ref>), is general.
Now, let us consider a typical case where the entangling fidelity of the uncorrected channel is fixed as $ F_0=0.9$. For the depolarizing channel, using
the result in Eq. (<ref>),
we can get the entangling fidelities in each level of the concatenation:
\begin{eqnarray}
\label{0.9dep}
F_1(\omega_0)&=&0.920491, F_2(\omega_2)=0.947258, \nonumber \\
F_3(\omega_0)&=&0.975308, F_4(\omega_0)=0.9942310,\nonumber\\
F_5(\omega_0)&=&0.999714, F_6(\omega_0)=0.999999.
\end{eqnarray}
Therefore, to construct a near perfect channel from the depolarizing channel with entangling fidelity $F_0=0.9$, the five-qubit code should be concatenated with itself $L=6$ times. From Eq. (<ref>) and the known $F_l(\omega)$, the effective choi matrix
$\chi(\varepsilon^{1}(\omega_0, F_0))$ can be easily calculated.
Then, under the condition that $F_0$ is fixed, $F_0=0.9$, we can design a program to generate an arbitrary setting for the five free parameters introduced above. The generated channel is denoted by $\varepsilon^0(\mathbf{\omega}_1, F_0)$. Let $\Lambda= \varepsilon^0(\mathbf{\omega}_1, F_0)^{\otimes 5}$,
the effective channel in each $l$-th level of concatenation can be decided by following the same method as the amplitude channel in Sec. <ref>. As the result of this run of calculation, we can get the exact values $F_l(\omega_1, F_0)$ and $D_l(\omega_1, F_0)$ with $l=1,2,...,6$.
After the calculation of the first one is completed, another channel with a fidelity of $0.9$ will be generated and denoted by $\varepsilon^0(\mathbf{\omega}_2, F_0)$. Similarly, we have the results $F_l(\omega_2, F_0)$ and $D_l(\omega_2, F_0)$ with $l=1,2,...,6$.
Usually, in the $m$-th run of calculation, the generated channel is denoted by $\varepsilon^0(\mathbf{\omega}_m, F_0)$
with $F_0=0.9$. After performing QEC with the $5^6$-qubit concatenated code, we can get a series of exact values
$F_l(\omega_m, F_0)$ and $D_l(\omega_m, F_0)$ with $l=1,2,...,6$.
For a fixed value of $F_0$, there is about $M$ ($M\ge 10^5$) examples of noise channel that will be generated. Based on the numerical data in each level of concatenation, a distance measure $D_l^{\mathrm{max}}(F_0)$ can be defined to denote the maximum value of error for the approximation defined in Eq. (<ref>),
\begin{equation}
\label{dist1}
D_l^{\mathrm{max}}(F_0)=\max \{ D_{l}(\omega_m, F_0)\}, 1\leq l\leq 6, 0\leq m\leq M,
\end{equation}
and the minimum fidelity
\begin{equation}
\label{fedelity}
F_l^{\mathrm{min}}(F_0)=\min \{F_l(\omega_m,F_0)\}_{m=0}^{M}, 1\leq l\leq 6, 0\leq m\leq M.
\end{equation}
For the case $F_0=0.9$, our numerical calculation gives
\begin{eqnarray}
\label{fmin}
F_1^{\mathrm{min}}(F_0)=0.918540, F_2^{\mathrm{min}}(F_0)=0.945006,\nonumber\\
F_4^{\mathrm{min}}(F_0)=0.993281,\nonumber\\ F_5^{\mathrm{min}}(F_0)=0.999555, F_6^{\mathrm{min}}(F_0)=0.999998.
\end{eqnarray}
(Color online) Numerical results for the maximum distance measure $D_l^{\mathrm{max}}(F_0)$ defined in Eq. (<ref>).
The numerical results for $D^{\mathrm{max}}_l(F_0)$ are given in Fig. <ref>, where the initial fidelity take different values, 0.9, 0.92, 0.945, 0.97, 0.97, 0.992, and 0.9993. For each given initial fidelity, the values of the distance measure $D^{\mathrm{max}}_l(F_0)$ are given in the domain $0\leq l\leq L_{(\omega_0, F_0)}$. Our numerical results indicate that: In each level of the concatenation, the effective channel can be approximated by a corresponding depolarizing channel. The error of the approximation, which is characterized by the distance measure $D^{\mathrm{max}}_l(F_0)$, approaches zero as the level of concatenation is increased.
Based on the results in Eq. (<ref>) and Eq. (<ref>), one can observe that: (a) In each level of concatenation, $F_l(\omega_0)-F^{\mathrm{min}}_l\ge 0$. When $l$ is increased, this difference will approach to zero. (b) Since that $F_6^{\mathrm{min}}(F_0)\ge 1-10^{-5}$, the concatenated five-qubit code, which is able to construct a near perfect channel from the depolarizing channel, is also suitable for the arbitrary channel with the same initial fidelity as the depolarizing channel. For the other cases where the initial fidelity takes different values, 0.92, 0.945, 0.97, 0.97, 0.992,
our numerical results, which are depicted in Fig. <ref>, show that the properties (a) and (b) can be also observed.
Therefore, as a direct consequence of the observations, we argue that the concatenated five-qubit is general: Under the condition that the fidelity is fixed, a $5^{L_{(\omega_0, F_0)}}$-qubit code, which is the minimal-sized one for constructing a near-perfect qubit-channel from the depolarizing channels, can be used to complete the same task for the general noise channels with the same initial fidelity $F_0$.
(Color online) (a) For the general noise error model, the numerical results of the minimum fidelity $F_l^{\mathrm{min}}(F_0)$ defined in Eq. (<ref>) are depicted; (b) Numerical results for the entangling fidelity when the initial channels are fixed to be the depolarizing ones. The results show the generality of the five-qubit code. Though the numeral results look similar in the two viewgraphs, they are not strictly identical. More specifically, for the case $F_0=0.9$, the entangling fidelities and minimum fidelities in each level of the concatenation given in Eq. (<ref>) and Eq. (<ref>) respectively, are not strictly the same.
§ DISCUSSIONS AND REMARKS
Our present work is based on the assumption that the noise of quantum channel comes from the interaction between the physical qubit and the environment. The apparatus, which are used for state preparation, encoding and decoding, are supposed to be perfect.
The main results of this work are: (I) A simple and explicit scheme is developed to get the effective Choi matrix resulted by the QEC procedure with the concatenated five-qubit code. (II) Based on this scheme, we have shown that the QEC procedure, which is optimal for the noise depolarizing channels, also offers an efficient way to construct a near-perfect channel, where the noise of the physical qubit is a general one including the amplitude damping. Within a general noise model, though not a strict proof, our numerical results indicate that the concatenated five-qubit code is general: To construct an effective channel with a error below $10^{-5}$, the necessary number of the levels for concatenation is decided by the fidelity of the initial channels and it does not depend on the actual types of noise models.
Naturally, there is still an open question: Is the concatenated five-qubit general for an arbitrary noise channel? For the single-qubit case, such a trace-preserving model requires about twelve parameters. Under the constraint that the fidelity is fixed, how to effectively generate an arbitrary setting for all these parameters is still an unsolved problem. From the results of the present work, we guess that the concatenated five-qubit is general. Our argument is based on the following two facts.
First, a general QEC protocol does exist, and in the work of Horodeckis' <cit.>, the so-called twirling procedure has been introduced. Performing twirling on an arbitrary channel $\varepsilon(\omega, F_0)$, one may obtain a depolarizing channel $\varepsilon(\omega_0, F_0)$ with the unchanged channel fidelity, and before performing QEC with the concatenated five-qubit code, all the initial (unknown) channels may be transferred into the depolarizing channels through twirling. We may call the so-constructed QEC protocol, which is general for the arbitrary noise models, the term twirling-assisted QEC scheme. As a basic property, the effective channel in each level of concatenation should always be a depolarizing channel.
Second, within the general noise model, our numerical results indicate that the QEC protocol based on the concatenated five-qubit code can be viewed as an approximate realization of the twirling-assisted QEC scheme. As shown, the effective channel in each level can be approximated by the depolarizing channel while the error of the approximation approaches to zero as the level of concatenation is increased.
The blockwise decoding protocol, which is suboptimal, is used in our work. From the work of Poulin <cit.>, it is known that the message-passing decoding algorithm is optimal, and by jointing the twirling stage and the message-passing decoding protocol together, one can have an optimal way to construct a near perfect channel for the arbitrary noise. However, from the results in present work, it is shown that twirling is not a necessary step if the blockwise decoding is applied. Actually, one may guess that the twirling is also not necessary for the QEC with the message-passing decoding algorithm. In other word, the optimal QEC protocol for the depolarizing channel, which has been developed by Poulin, may also be general. We expect this guess could lead to further theoretical or experimental consequences.
§ ACKNOWLEDGEMENTS
The authors are grateful to the referee for very helpful comments. This work was partially supported by the National Natural Science Foundation of China under the Grant No. 11405136, and the Fundamental Research Funds for the Central Universities of China A0920502051411-56.
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|
1511.00003
|
We study the cut-off phenomenon for a family of stochastic small perturbations of a one dimensional dynamical system. We will focus in a semiflow of a deterministic differential equation which is perturbed by adding to the dynamics a white noise of small variance. Under suitable hypothesis on the potential we will prove that the family of perturbed stochastic differential equations present a profile cut-off phenomenon with respect to the total variation distance. We also prove a local cut-off phenomenon in a neighborhood of the local minima (metastable states) of multi-well potential.
Stochastic Small Perturbations of Dynamical SystemsStochastic Small Perturbations of Dynamical Systems
§ INTRODUCTION
In the last decades intense research has been devoted to the study of dynamical systems subjected to random perturbations. Considerable effort has been dedicated to investigate exit times and exit locations from given domains and how they relate to the respective deterministic dynamical system. The theory of large deviations provides the usual mathematical framework for tackling these problems in case of Gaussian perturbations. This theory sets up the precise time scales for transitions of non degenerate stochastic systems between certain regimes. The theory of random dynamical systems, on the other hand, assigns Lyapunov exponents to linear random dynamical systems. These are the exponential growth rates as time grows large for fixed intensities of the underlying noise. For details see M. Freidling & A. Wentzell <cit.>, <cit.>, <cit.>, M. Day <cit.>, <cit.> and W. Siegert <cit.>.
We will study the relation to the respective deterministic dynamical systems from a different point of view.
We study the asymptotically behavior or the so-called cut-off phenomenon for a family of stochastic small perturbations of a given dynamical system.
We will focus on the semiflow of a deterministic differential equation which is perturbed by adding to the dynamics a white noise perturbations.
Under suitable hypotheses on the vector field (coercivity assumption) we will prove that the
one parameter family of perturbed stochastic differential equations presents a profile cut-off in the sense of the definition of cut-off given by
J. Barrera & B. Ycart <cit.>.
The term “cut-off” was introduced by D. Aldous and P. Diaconis in <cit.> in the early eighties to describe the phenomenon of abrupt convergence of Markov chains introduced as models of shuffling cards. Since the appearance of <cit.> many families of stochastic processes have been shown to have similiar properties. For a good introduction to the different definitions of cut-off and the evolution of the concept in discrete time, see
J. Barrera & B. Ycart
<cit.> and
P. Diaconis
<cit.>. In <cit.>, L. Saloff-Coste gives an extensive list of random walks for which the phenomenon occurs. Now, it us a well studied feature of Markov processes.
What does the “cut-off” phenomenon mean?
It refers to an asymptotically drastic convergence of a family of stochastic processes labeled by some parameter.
Before a certain “cut-off time” those processes stay far from equilibrium in the sense that a suitable distance in some sense between the distribution at time $t$ and the equilibrium measure is far from $0$;
after a deterministic time “the cut-off time” the distance decays exponentially fast to zero.
The term “cut-off” is naturally associated to such an “all/nothing” or “1/0 behavior”, but it has the drawback of being used with other meanings in statistical mechanics and theoretical physics.
Alternative names have been proposed, including threshold phenomenon and abrupt convergence.
Consider a one parameter family of stochastic processes in continuous time $\{x^{\epsilon}\}_{\epsilon>0}$ indexed by $\epsilon>0$,
$x^{\epsilon}:=\{x^{\epsilon}_t\}_{t\geq 0}$, each one converging to a asymptotic distribution $\mu^{\epsilon}$ when $t$ goes to infinity. Let us denote by $d_{\epsilon}(t)$ the distance between the distribution at time $t$ of the $\epsilon$-th processes, $\mathbb{P}\left(x^\epsilon_t\in \cdot\right)$, and its asymptotic distribution as $t\rightarrow +\infty$, $\mu^\epsilon$, where the “distance” can be taken to being the total variation, separation, Hellinger, relative entropy, Wasserstein, $L^{p}$ distances, etc. Following J. Barrera & B. Ycart <cit.>, the cut-off phenomenon for $\{x^\epsilon\}_{\epsilon >0}$ can be expressed at three increasingly sharp levels. Let us denoted by $M$ the diameter of the respective metric space of probability measures in which we are working. In general, $M$ could be infinite. In our case, we will focus on the total variation distance so $M=1$.
The family $\{x^{\epsilon}\}_{\epsilon>0}$ has a cut-off at $\{t_{\epsilon}\}_{\epsilon>0}$ if $t_{\epsilon} \rightarrow +\infty$ when $\epsilon \rightarrow 0$ and
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0 }{d_{\epsilon}(ct_{\epsilon})}= \left\{ \begin{array}{lcc}
M & if & 0 < c < 1, \\
\\ 0 & if & c>1. \\
\end{array}
\right.
\end{eqnarray*}
The family $\{x^{\epsilon}\}_{\epsilon>0}$
has a window cut-off at
$\{\left(t_{\epsilon}, w_{\epsilon}\right)\}_{\epsilon>0}$, if $t_{\epsilon} \rightarrow +\infty$ when $\epsilon \rightarrow 0$, $w_{\epsilon}=o\left(t_{\epsilon}\right)$ and
\begin{eqnarray*}
\lim\limits_{c \rightarrow -\infty}{\liminf\limits_{\epsilon\rightarrow 0}
\lim\limits_{c \rightarrow +\infty}{\limsup\limits_{\epsilon\rightarrow 0}
\end{eqnarray*}
The family $\{x^{\epsilon}\}_{\epsilon>0}$ has profile cut-off at
$\{\left(t_{\epsilon}, w_{\epsilon}\right)\}_{\epsilon>0}$ with profile $G$, if $t_{\epsilon} \rightarrow +\infty$ when $\epsilon \rightarrow 0$, $w_{\epsilon}=o\left(t_{\epsilon}\right)$,
\begin{eqnarray*}
G(c):=\lim\limits_{\epsilon \rightarrow 0}{d_{\epsilon}(t_{\epsilon}+cw_{\epsilon})}
\end{eqnarray*}
exists for all $c\in \mathbb{R}$ and
\begin{eqnarray*}
\lim\limits_{c \rightarrow -\infty}{G(c)}&=&M,\\
\lim\limits_{c \rightarrow +\infty}{G(c)}&=&0.
\end{eqnarray*}
We also give a mathematical description of the phenomenon of metastability when the potential is a double well potential with some smooth conditions and certain increase rate at infinity.
According to the initials conditions the deterministic trajectories associated to the differential equation (<ref>) converge to the local minima of the potential $V$ or stay in its local minima.
Therefore, no transition between different domains of attraction is possible. This situation becomes different if we perturb the deterministic differential equation (<ref>) by a small aditive noise whose presence allows transitions between the potential wells. Depending on the initial conditions of the system and the properties of the noise certain potential wells may be reached only on appropriated long time scales or stay unvisited.
The phenomenon of metastability roughly speaking means that for different time scales and initial conditions the system may reach different local statistical equilibria.
Dynamical systems subject to small Gaussian perturbations have been studied extensively, for details see <cit.>.
The theory of large deviations allows to solve the exit problem from the domain of attraction of a stable point.
It turns out that the mean exit time is exponentially large in the small noise parameter, and its logarithmic rate is proportional to the height of the potential barrier the trajectories have to overcome.
Consequently, for a multi-well potential one can obtain a series of exponentially non-equivalent time scales given by the wells mean exit times.
Moreover, one can prove that the normalised exit times are exponentially distributed and have a memoryless property. For details see <cit.> for Gaussian perturbations and <cit.> for Lévy-driven diffusions.
This material will be organized as follows. Section <ref> describes the model and states the main results besides establishing the basic notation. Section <ref> provides the results for a linear approximations which is an essential tool in order to obtain the main results. Section <ref>
gives the ingredients in order to obtain the main results and provides the proof of the main results.
Section <ref> studies a kind of local cut-off phenomenon in a neighborhood of the local minima (metastable states) of multi-well potential.
The Appendix is divided in three section as follows:
Section <ref> gives elementary properties for the total variation distances of Normal distributions. Section <ref> provides the proofs that we do not proof in Section <ref> and Section
<ref> in order to the lecture be fluent.
Section <ref> gives some useful results that we use along of this material.
§ STOCHASTIC PERTURBATIONS: ONE DIMENSIONAL CASE
On this section, let $x_0 \in \mathbb{R} \setminus \{0\}$ be fixed and let us consider the semiflow $\{\psi_t\}_{t \geq 0}$ associated to the solution of the following deterministic differential equation,
\begin{eqnarray}\label{dde1}
\end{eqnarray}
for $t \geq 0$. The hypothesis made in Theorem <ref> on the potential $V$ guarantees existence and uniqueness of solutions of (<ref>), as well as all the other (stochastic or deterministic) equations defined below.
Let us establish some basic notation.
Let us take $\mu\in \mathbb{R}$ and let $\sigma^2\in ]0,+\infty[$ be fixed numbers.
We denote by $\mathcal{N}{\left(\mu,\sigma^2\right)}$ the Normal distribution with mean $\mu$ and variance $\sigma^2$. Given two probability measures $\mathbb{P}$ and $\mathbb{Q}$ which are defined in the same measurable space $\left(\Omega,\mathcal{F}\right)$,
we denote the total variation distance between $\mathbb{P}$ and $\mathbb{Q}$ by
$$\norm{\mathbb{P}-\mathbb{Q}}:=\sup\limits_{A\in \mathcal{F}}{|\mu(A)-\nu(A)|}.$$
Along this paper we always consider $\epsilon>0$.
Our main Theorem in the one dimensional case is the following:
Let $V:\mathbb{R}\rightarrow \mathbb{R}$ be an one dimensional potential
that satisfies the following:
$i)$ $V\in \mathcal{C}^3$.
$ii)$ $V(0)=0$.
$iii)$ $V^{\prime}(x)=0$ if only if $x=0$.
$iv)$ There exists $\delta>0$ such that
$V^{\prime\prime}(x)\geq \delta$ for every $x\in \R$.
Let us consider the family of processes indexed by
$\epsilon > 0$, $x^{\epsilon}=\{ {x^{\epsilon}_t} \}_{t \geq 0}$
which are given by the the semiflow of the following stochastic differential equation,
\begin{eqnarray*}
\end{eqnarray*}
for $t \geq 0$, where $x_0$ is a deterministic initial condition in $\mathbb{R}\setminus \{0\}$ and
$\{W_t\}_{t \geq 0}$ is a standard Brownian motion. This family
presents profile cut-off in the sense of
the Definition <ref> with respect to the total variation distance when $\epsilon$ goes to zero.
The profile function $G:\mathbb{R}\rightarrow \mathbb{R}$ is given by
\begin{equation*}
G(b):=\norm{\N{\left(\tilde{c} e^{-b},1\right)}-\N{\left(0,1\right)}},
\end{equation*}
where $\tilde{c}$ is the nonzero constant given by
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\psi_t}&=:&\tilde{c}.
\end{eqnarray*}
The cut-off time $t_{\epsilon}$ and window time $w_{\epsilon}$ are given by
\begin{eqnarray*}
t_{\epsilon}&:=&\frac{1}{2V^{\prime\prime}(0)}\left[\ln\left(\frac{1}{\epsilon}\right)+\ln\left(2V^{\prime\prime}(0)\right) \right],\\
\end{eqnarray*}
where $\delta_\epsilon=\epsilon^{\gamma}$ for some $\gamma\in ]0,1[$.
This Theorem will be proved at the end of the section <ref>.
§ THE LINEARIZED CASE
As an important intermediate step, we prove profile cut-off for a family of processes satisfying a linear, non-homogeneous stochastic differential equation which we will define bellow.
This result holds for a more general class of potentials that Theorem <ref>, which we define as follows.
We say that $V$ is a regular potential if
$V:\mathbb{R}\rightarrow \mathbb{R}$ satisfies
$a)$ $V$ is $\mathcal{C}^3$.
$b)$ $V(0)=0$.
$c)$ $V^{\prime}(x)=0$ iff $x=0$.
$d)$ $V^{\prime\prime}(0)>0$.
$e)$ $\lim\limits_{|x|\rightarrow +\infty}{V(x)}=+\infty$.
In order to prove Theorem <ref> we will prove the analogous result for a “linear approximations” of the potential $V$.
Let us consider the family of processes indexed by
$\epsilon > 0$, $y^{\epsilon}=\{ {y^{\epsilon}_t} \}_{t \geq 0}$
which are given by the solution of the following linear stochastic differential equation,
\begin{equation}
\label{linearapprox}
\begin{array}{rcl}
\end{array}
\end{equation}
for $t \geq 0$, where $y_0$ is a deterministic initial condition in $\R\setminus\{0\}$,
$\{W_t\}_{t \geq 0}$ is a standard Brownian motion, $V$ is a regular potential and $\{\psi_t\}_{t\geq 0}$ is given by the solution of the deterministic differential equation (<ref>).
This family
presents profile cut-off in the sense of
the Definition <ref>
with respect to the total variation distance when $\epsilon$ goes to zero.
The profile function $G:\mathbb{R}\rightarrow \mathbb{R}$ is given by
\begin{eqnarray*}
G(b)&:=&\norm{\N{\left(c e^{-b},1\right)}-\N{\left(0,1\right)}},
\end{eqnarray*}
where $c$ is the nonzero constant given by
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\Phi_t}&=:&c,
\end{eqnarray*}
where $\Phi=\{\Phi_{t}\}_{t\geq 0}$ is the fundamental solution of the non autonomous system
\begin{eqnarray*}
d\Phi_{t}=-V^{\prime\prime}{(\psi_t)}\Phi_t dt
\end{eqnarray*}
for every $t\geq 0$ with initial condition $\Phi_{0}=1$.
The cut-off time $t_{\epsilon}$ and window time $w_{\epsilon}$ are given by
\begin{eqnarray*}
t_{\epsilon}&:=&\frac{1}{2V^{\prime\prime}(0)}\left[\ln\left(\frac{1}{\epsilon}\right)+\ln\left(2V^{\prime\prime}(0) y^2_0\right) \right],\\
\end{eqnarray*}
Notice that choosing $V(x) = \frac{\alpha x^2}{2}$ for every $x\in \mathbb{R}$, where $\alpha>0$ is a fixed constant, we see that the Ornstein-Uhlenbeck process presents profile cut-off.
In order to prove Theorem <ref>, we will find the qualitative behavior of the semiflow $\psi=\{\psi_t\}_{t\geq 0}$ at infinity.
The following lemma tells us the asymptotic behavior of the expectation and variance of the “linear approximations”.
Leu us assume the hypothesis of Theorem <ref>.
$i)$ $\lim\limits_{t\rightarrow +\infty}{\psi_t}=0$.
$ii)$ $\lim\limits_{t\rightarrow +\infty}{\Phi_t}=0$.
$iii)$ There exist constants $c\not=0$ and $\tilde{c}\not=0$ such that
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\Phi_t}&=&c,
\end{eqnarray*}
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\psi_t}&=&\tilde{c},
\end{eqnarray*}
where $\Phi=\{\Phi_{t}\}_{t\geq 0}$ is the fundamental solution of the non autonomous system
\begin{eqnarray*}
d\Phi_{t}=-V^{\prime\prime}{(\psi_t)}\Phi_t dt
\end{eqnarray*}
for every $t\geq 0$ with initial condition $\Phi_{0}=1$.
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{\Phi_t^2 \int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}}=\frac{1}{2V^{\prime\prime}{(0)}}.
\end{eqnarray*}
For the proof of this lemma, see Lemma <ref>.
For the items $i)$ and $ii)$ in the Lemma <ref>, we do not need the assumption that $V\in \mathcal{C}^3$; we need less regularity, $V\in \mathcal{C}^2$ is enough.
The following lemma characterizes the distribution of the “linear approximations".
Under the hypothesis of Theorem <ref>, we have
\begin{eqnarray}\label{gaussfor}
\y=\Phi_{t}y_0+\sqrt{\epsilon} \Phi_{t}\int\limits_{0}^{t}{\frac{1}{\Phi(s)}d{W_{s}}}
\end{eqnarray}
for every $t\geq 0$,
where $\Phi=\{\Phi_{t}\}_{t\geq 0}$ is the fundamental solution of the non autonomous system
\begin{eqnarray*}
d\Phi_{t}=-V^{\prime\prime}{(\psi_t)}\Phi_t dt
\end{eqnarray*}
for every $t\geq 0$ with initial condition $\Phi_{0}=1$.
It follows from Itô's formula. For details check <cit.>.
Using the decomposition (<ref>) of the process $y^{\epsilon}$ into a deterministic part and a mean-zero martingale and using Itô's isometry for Wiener's integral, we obtain
\begin{eqnarray*}
\mathbb{E}\left[\y\right]&=& \Phi_t y_0,\\
\mathbb{V}\left[\y\right]&=&\epsilon \Phi_t^2 \int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}.
\end{eqnarray*}
By Lemma <ref>, we have that for each $\epsilon>0$ and $t>0$ fixed, $\y$ is a random variable with Normal distribution
with mean
\begin{eqnarray*}
\nu^{\epsilon}_{t}&:=&\Phi_t y_0
\end{eqnarray*}
and variance
\begin{eqnarray*}
\eta^{\epsilon}_{t}&:=&\epsilon \Phi_t^2 \int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}.
\end{eqnarray*}
Let us assume the hypothesis of Theorem <ref> and let $\epsilon>0$ be fixed. Then the random variable $\y$ converges in distribution as $t \to \infty$ to a Gaussian random variable $\N^{\epsilon}$ with mean zero
and variance $\frac{\epsilon}{2V^{\prime\prime}{(0)}}$.
It follows from the item $ii)$ and item $iv)$ of Lemma <ref>.
Now, we have all the tools in order to prove Theorem <ref>.
For each $\epsilon>0$ and $t>0$, we define
\begin{eqnarray*}
\end{eqnarray*}
\begin{eqnarray*}
\norm{\N{\left(\sqrt{\frac{2V^{\prime\prime}{(0)}}{\epsilon}}y_0 \Phi_t,1\right)}-\N{\left(0,1\right)}}.
\end{eqnarray*}
Using triangle's inequality and Lemma <ref>, for each $\epsilon>0$ and $t>0$ we obtain
\begin{eqnarray*}
d^{\epsilon}{(t)}&\leq & D^{\epsilon}{(t)}+
\norm{\N{\left(0,{2V^{\prime\prime}(0)}{\Phi_t}^2 I_t\right)}-\N{\left(0,1\right)}}
\end{eqnarray*}
\begin{eqnarray*}
D^{\epsilon}{(t)}&\leq & d^{\epsilon}{(t)}+
\norm{\N{\left(0,{2V^{\prime\prime}(0)}{\Phi_t}^2 I_t\right)}-\N{\left(0,1\right)}},
\end{eqnarray*}
where $I_t=\int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}$. Therefore,
\begin{eqnarray*}
|d^{\epsilon}{(t)}-D^{\epsilon}{(t)}| &\leq &
\norm{\N{\left(0,{2V^{\prime\prime}(0)}{\Phi_t}^2 I_t\right)}-\N{\left(0,1\right)}}.
\end{eqnarray*}
For each $\epsilon>0$ let us define
\begin{eqnarray*}
\end{eqnarray*}
\begin{eqnarray*}
\end{eqnarray*}
For every $b\in \R$, we define $t_{\epsilon}(b)=t_{\epsilon}+bw_{\epsilon}$. We take $\epsilon_b>0$ such that $t_{\epsilon}(b)>0$ for every $0<\epsilon<\epsilon_b$.
Using Lemma <ref>, we obtain
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}|d^{\epsilon}{(t_{\epsilon}(b))}-D^{\epsilon}{(t_{\epsilon}(b))}|&=&0
\end{eqnarray*}
for every $b\in \R$. Let us consider the function $G:\R\rightarrow [0,1]$ defined by
\begin{eqnarray*}
\end{eqnarray*}
where $c\not=0$ is the constant of item $iii)$ in Lemma <ref>.
Observe that
\begin{eqnarray*}
\norm{\N{\left(e^{V^{\prime\prime}(0) t_{\epsilon}(b)}\Phi_{t_{\epsilon}(b)}e^{-b},1\right)}-\N{(0,1)}}
\end{eqnarray*}
for every $b\in \R$ and $0<\epsilon<\epsilon_b$. Therefore, by item $iii)$ of Lemma <ref> and by Lemma <ref>,
we have
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}D^{\epsilon}{(t_{\epsilon}(b))}=G(b)
\end{eqnarray*}
for every $b\in \R$. By Lemma <ref>, we have
$\lim\limits_{b\rightarrow +\infty}{G(b)}=0$ and $\lim\limits_{b\rightarrow -\infty}{G(b)}=1$.
Consequently, the theorem is proved.
Let us consider the process $y=\{ {y_t} \}_{t \geq 0}$
which is given by the solution of the following linear stochastic differential equation,
\begin{eqnarray*}
\end{eqnarray*}
for $t \geq 0$, where
$\{W_t\}_{t \geq 0}$ is a standard Brownian motion, $V$ is a regular potential and $\{\psi_t\}_{t\geq 0}$ is given by the solution of the deterministic differential equation (<ref>).
For every $\epsilon>0$ fixed, let us define $z^{\epsilon}_t=\psi_t+\sqrt{\epsilon}y_t$ for every $t\geq 0$.
Then the family $\{z^{\epsilon}\}_{\epsilon>0}$
presents profile cut-off in the sense of Definition <ref> with respect to the total variation distance when $\epsilon$ goes to zero.
The profile function $G:\mathbb{R}\rightarrow \mathbb{R}$ is given by
\begin{eqnarray*}
G(b)&:=&\norm{\N{\left(\tilde{c} e^{-b},1\right)}-\N{\left(0,1\right)}},
\end{eqnarray*}
where $\tilde{c}$ is the nonzero constant given by
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\psi_t}&=:&\tilde{c}.
\end{eqnarray*}
and the cut-off time $t_{\epsilon}$ and window time $w_{\epsilon}$ are given by
\begin{eqnarray*}
t_{\epsilon}&:=&\frac{1}{2V^{\prime\prime}(0)}\left[\ln\left(\frac{1}{\epsilon}\right)+\ln\left(2V^{\prime\prime}(0)\right) \right],\\
\end{eqnarray*}
The proof of Theorem <ref> can be adapted in order to prove this corollary in a straight-forward way, so we omit it.
In what follows, we call the processes $\{z^\epsilon\}_{\epsilon>0}$ the “linear approximations”.
The constants $c$ and $\tilde{c}$ obtained in the item $iii)$ of Lemma <ref> depend on the initial condition of the semiflow $\psi=\{\psi_t\}_{t\geq 0}$.
Theorem <ref> and Corollary <ref> remain true without altering the cut-off time and the profile function if we take as window time $w^{\prime}_{\epsilon}=w_{\epsilon}+\delta_\epsilon$ for each $\epsilon>0$, where $\{\delta_{\epsilon}\}_{\epsilon>0}$ is any sequence of real numbers such that
$\lim\limits_{\epsilon\rightarrow 0}{\delta_{\epsilon}}=0$.
§ THE GRADIENT CASE
From now on and up to the end of this section we will use the following notations and names.
$a)$ The stochastic process $x^{\epsilon}:=\left\{x^{\epsilon}_t\right\}_{t\geq 0}$ defined in Theorem <ref> is called the Itô diffusion.
$a)$ The semiflow $\psi:=\left\{\psi_t \right\}_{t\geq 0}$ defined by the differential equation (<ref>) is called the zeroth order approximation of $x^{\epsilon}$.
$c)$ The stochastic Markov process $z^{\epsilon}:=\left\{z^{\epsilon}_t\right\}_{t\geq 0}$ defined in Corollary <ref> is called the first order approximation of $x^{\epsilon}$.
The following lemma will give us the existence of a stationary probability measure for the Itô diffusion
$x^{\epsilon}=\left\{x^{\epsilon}_t\right\}_{t\geq 0}$.
Let $V$ be a regular potential and for every $\epsilon>0$, let us consider the Itô diffusion $x^{\epsilon}=\{\x\}_{t\geq 0}$ which is given by the following stochastic differential equation,
\begin{eqnarray*}
\end{eqnarray*}
for $t \geq 0$, where $x_0$ is a deterministic initial condition in $\mathbb{R}\setminus \{0\}$ and
$\{W_t\}_{t \geq 0}$ is a standard Brownian motion. Let us assume that
\begin{eqnarray*}
\lim\limits_{|x|\rightarrow +\infty}{\left |V^{\prime}(x)\right |}&=&+\infty.
\end{eqnarray*}
Then, for every $\epsilon>0$ fixed, when $t\rightarrow +\infty$ the probability distribution of $x^{\epsilon}_t$, $\mathbb{P}(x^{\epsilon}_t\in \cdot)$ converges in distribution to the stationary probability measure $\mu^{\epsilon}$ given by
\begin{eqnarray*}
\mu^{\epsilon}(dx)&=&\frac{e^{-\frac{2}{\epsilon}V(x)}dx}{M^{\epsilon}},
\end{eqnarray*}
where $M^{\epsilon}=\int\limits_{\mathbb{R}}{e^{-\frac{2}{\epsilon}V(z)}dz}$.
For the proof of this lemma and further considerations, see <cit.> and <cit.>.
Now we will restrict our potential to the class of coercive regular potentials.
Let $V$ be a regular potential. We say that $V$ is a coercive regular potential if there exists $\delta>0$ such that $V^{\prime\prime}(x)\geq \delta$ for every $x\in \mathbb{R}$.
In the class of coercive regular potentials, we restrict ourselves to the class of potentials with bounded second and third derivatives which we call smooth coercive regular potentials.
Let $V$ be a coercive regular potential. We say that $V$ is a smooth coercive regular potential if
\begin{eqnarray*}
\kappa_2:=\|V^{\prime\prime}\|_{\infty}:=\sup\limits_{x\in\R}{|V^{\prime\prime}(x)|}<+\infty,
\end{eqnarray*}
\begin{eqnarray*}
\kappa_3:=\|V^{\prime\prime\prime}\|_{\infty}:=\sup\limits_{x\in\R}{|V^{\prime\prime\prime}(x)|}<+\infty.
\end{eqnarray*}
The following lemma tells us that the stationary probability measure of the Itô diffusion $\{x^{\epsilon}_t\}_{t\geq 0}$ is well approximated in total variation distance by the Normal distribution with mean zero and variance $\frac{\epsilon}{2V^{\prime\prime}(0)}$.
Let $V$ be a coercive regular potential,
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}\norm{\mu^{\epsilon}-\N^{\epsilon}}=0,
\end{eqnarray*}
where $\N^{\epsilon}$ is a Normal distribution with mean zero and variance $\frac{\epsilon}{2V^{\prime\prime}(0)}$.
Let $0<\eta<V^{\prime\prime}(0)$ be fixed. By Lemma <ref>, the $\mu^{\epsilon}(dx)=\frac{e^{-\frac{2}{\epsilon}V(x)}dx}{M^{\epsilon}}$ is a well defined probability measure
on $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$.
\begin{eqnarray*}
\norm{\mu^{\epsilon}-\N^{\epsilon}}&=&\frac{1}{2}\int\limits_{\mathbb{R}}
\end{eqnarray*}
$M^{\epsilon}=\int\limits_{\mathbb{R}}{e^{-\frac{2}{\epsilon}V(x)}dx}$ and
By triangle's inequality, we have
\begin{eqnarray*}
\norm{\mu^{\epsilon}-\N^{\epsilon}}&\leq &\frac{1}{2}\int\limits_{\mathbb{R}}
\frac{1}{2}\int\limits_{\mathbb{R}}
&=&\frac{\left|M^{\epsilon}-N^{\epsilon} \right|}{2N^{\epsilon}}+
\frac{1}{2N^{\epsilon}}\int\limits_{\mathbb{R}}
&\leq &
\frac{1}{N^{\epsilon}}\int\limits_{\mathbb{R}}
\end{eqnarray*}
Recall that $V$ is a coercive regular potential, so there exists $\delta>0$ such that $V^{\prime\prime}(x)\geq \delta>0$ for every $x\in \R$. Then, it follows that
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\frac{1}{N^{\epsilon}}\int\limits_{\{x\in \mathbb{R}:|x|\geq \beta\}}
\end{eqnarray*}
for every $\beta>0$.
By the continuity of $V^{\prime \prime }$ at zero, there exists $\delta_{\eta}>0$ such that
\begin{eqnarray*}
\end{eqnarray*}
for every $|x|<\delta_{\eta}$.
Also, by Taylor's Theorem, we have
$V^{\prime\prime}(x)=\frac{V^{\prime\prime}(\xi_x)x^2}{2}$ for every $|x|<\delta_{\eta}$ where $|\xi_x|< |x|$.
\[
\begin{split}
\frac{1}{N^{\epsilon}}\int\limits_{-\delta_{\eta}}^{\delta_{\eta}}
\Big|{e^{-\frac{2}{\epsilon}V(x)}}{}-&{e^{-\frac{2}{\epsilon}\frac{V^{\prime\prime}(0)x^2}{2}}}{}\Big|dx
= \frac{1}{N^{\epsilon}}\int\limits_{-\delta_{\eta}}^{\delta_{\eta}}
\left|
\right|dx\\
&\leq {\frac{1}{\epsilon N^{\epsilon}}\int\limits_{-\delta_{\eta}}^{\delta_{\eta}}
{x^2e^{-\frac{2}{\epsilon}\frac{\delta x^2}{2}}}
}\left| V^{\prime\prime}(\xi_x)-V^{\prime\prime}(0) \right|dx}\\
&\leq {\frac{\eta}{\epsilon N^{\epsilon}}\int\limits_{-\delta_{\eta}}^{\delta_{\eta}}
{x^2e^{-\frac{2}{\epsilon}\frac{\delta x^2}{2}}}
}dx} \leq {\frac{\eta\sqrt{V^{\prime\prime}(0)}}{\sqrt{\pi}(2\delta)^{\frac{3}{2}}}\int\limits_{-\delta_{\eta}\sqrt{\frac{2\delta}{\epsilon}}}^{\delta_{\eta}\sqrt{\frac{2\delta}{\epsilon}}}
&\leq {\frac{\eta\sqrt{V^{\prime\prime}(0)}}{\sqrt{\pi}(2\delta)^{\frac{3}{2}}}\int\limits_{\R}
\end{split}
\]
Consequently, first taking $\epsilon\rightarrow 0$ and then $\eta\rightarrow 0$ we obtain the result.
The following proposition will give us a quantitative estimation of the distance of the paths between the Itô diffusion and the zeroth order and first order approximations.
Let us assume that $V$ is a smooth coercive regular potential. Let us denote
$B_t=\sup\limits_{0\leq s\leq t}{|W_s|}$ for every $t\geq 0$.
$i)$ For every $\epsilon>0$ and $t\geq 0$, we have $\left|\x-\psi_t\right|\leq \sqrt{\epsilon}B_t(\kappa_{2}t+1)$. We call this estimate the zeroth order estimate.
$ii)$ For every $\epsilon>0$ and $t\geq 0$, it follows that $\left|\x-\psi_t-\sqrt{\epsilon}y_t\right|\leq {\epsilon}B^2_t \kappa_3(\kappa_{2}t+1)^2t$.
We call this estimate the first order estimate.
First we prove item $i)$.
Let $\epsilon>0$ and $t\geq 0$ be fixed. It follows that
\begin{eqnarray*}
\x-\psi_t &=&-\int\limits_{0}^{t}{\left(V^{\prime}(x^{\epsilon}_s)-V^{\prime}(\psi_s)\right)ds}+\sqrt{\epsilon}W_t\\
&=& -\int\limits_{0}^{t}{V^{\prime\prime}(\theta^{\epsilon}_s)\left(x^{\epsilon}_s-\psi_s\right)ds}+\sqrt{\epsilon}W_t\\
&=& -\sqrt{\epsilon}\int\limits_{0}^{t}{V^{\prime\prime}(\theta^{\epsilon}_s)W_s
\end{eqnarray*}
where the second equality follows from the Mean Value Theorem,
$\theta_s^\epsilon$ is between the minimum of $\psi_s$ and $x_s^\epsilon$ and the maximum of $\psi_s$ and $x_s^\epsilon$, and the third equality follows from the variation of parameters method. Therefore, using Gronwall's inequality we obtain,
$\left|\x-\psi_t\right|\leq \sqrt{\epsilon}B_t(\kappa_{2}t+1)$.
Now we prove item $ii)$.
Again, let $\epsilon>0$ and $t\geq 0$ be fixed. It follows that
\begin{eqnarray*}
\x-\psi_t -\sqrt{\epsilon}y_t&=& -\int\limits_{0}^{t}{\left[V^{\prime}(x^{\epsilon}_s)-V^{\prime}(\psi_s)-V^{\prime\prime}(\psi_s)\sqrt{\epsilon}y_s\right]ds}\\
&=& -\int\limits_{0}^{t}{\left[V^{\prime\prime}(\theta^{\epsilon}_s)\left(x^{\epsilon}_s-\psi_s\right)-V^{\prime\prime}(\psi_s)\sqrt{\epsilon}y_s\right]ds}\\
&=& -\int\limits_{0}^{t}{V^{\prime\prime}(\psi_s)(x^{\epsilon}_s-\psi_s-\sqrt{\epsilon}y_t)ds}-\\
\end{eqnarray*}
where the second equality comes from the Mean Value Theorem, $\theta_s^\epsilon$ is between the minimum of $\psi_s$ and $x_s^\epsilon$ and the maximum of $\psi_s$ and $x_s^\epsilon$. Let us define
Again, using the Mean Value Theorem and the zeroth order estimate already proved, we have
\begin{eqnarray*}
|e_t|\leq \int\limits_{0}^{t}{\kappa_3(x^{\epsilon}_s-\psi_s)^2ds}\leq \epsilon B^2_t\kappa_3(\kappa_{2}t+1)^2t
\end{eqnarray*}
for every $t\geq 0$.
Consequently, using the
variation of parameters method and Gronwall's inequality we obtain
\begin{eqnarray*}
\left|\x-\psi_t-\sqrt{\epsilon}y_t\right| &\leq & \epsilon B^2_t \kappa_3(\kappa_{2}t+1)^3t.
\end{eqnarray*}
The next proposition will allows us to prove that the total variation distance of two first order approximations with (random or deterministic) initial conditions that are close enough is negligible. In order to do that, we will need to keep track of the initial condition of the solution of various equations. Let $X$ be a random variable in $\mathbb R$ and let $T>0$. Let $\{\psi_{t}(X)\}_{t \geq 0}$ denote the solution of
\begin{eqnarray*}
d{\psi_t(X)}&=&- V^\prime(\psi_t(X))dt,\\
\psi_0(X)&=&X.
\end{eqnarray*}
Let $\{y_t(X,T)\}_{t \geq 0}$ be the solution of the stochastic differential equation
\begin{eqnarray*}
\end{eqnarray*}
and define $\{z^\epsilon_t(X,T)\}_{t\geq0}$ as $z^\epsilon_t(X,T) := \psi_t(X) + \sqrt \epsilon y_t(X,T)$. In what follows, we will always take $T = t_\epsilon(b) := t_\epsilon + b w_\epsilon>0$ for every $\epsilon>0$ small enough, so we will omit it from the notation.
Let us assume that $V$ is a smooth coercive regular potential. Take $\delta_{\epsilon}:=\epsilon^{\gamma}$, where $0<\gamma<1$.
Let us denote by $z^{\epsilon}(X):=\{z^{\epsilon}_t(X)\}_{t\geq 0}$ the first order approximation with initial random condition $X$.
Then, for every $b\in \R$ it follows that
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\norm{
\end{eqnarray*}
where for each $\epsilon>0$, $t_{\epsilon}$ and $w_{\epsilon}$ are defined in Corollary <ref> and for each $b\in \R$, $\epsilon_b>0$ is small enough so that $t_{\epsilon}(b):=t_{\epsilon}+bw_{\epsilon}> 0$ for every $0<\epsilon<\epsilon_b$.
By Itô's formula we obtain
\begin{eqnarray*}
\Phi_{b\delta_{\epsilon}}x^{\epsilon}_{t_{\epsilon}(b)}+\sqrt{\epsilon} \Phi_{b\delta_{\epsilon}}\int\limits_{0}^{b\delta_{\epsilon}}{\frac{1}{\Phi(s)}d{\left(W_{t_{\epsilon}(b)+s}-W_{t_{\epsilon}(b)}\right)}},
\end{eqnarray*}
\begin{eqnarray*}
\Phi_{b\delta_{\epsilon}}z^{\epsilon}_{t_{\epsilon}(b)}+\sqrt{\epsilon} \Phi_{b\delta_{\epsilon}}\int\limits_{0}^{b\delta_{\epsilon}}{\frac{1}{\Phi(s)}d{\left(W_{t_{\epsilon}(b)+s}-W_{t_{\epsilon}(b)}\right)}}
\end{eqnarray*}
for every $0<\epsilon<\epsilon_b$,
where $\Phi=\{\Phi_{t}\}_{t\geq 0}$ is the fundamental solution of the non-autonomous system
\begin{eqnarray*}
d\Phi_{t}=-V^{\prime\prime}{(\psi_t+{t}_\epsilon(b))}\Phi_t dt
\end{eqnarray*}
for every $t\geq 0$ with initial condition $\Phi_{0}=1$.
Applying Lemma <ref> with $X:=\Phi_{b\delta_{\epsilon}}x^{\epsilon}_{t_{\epsilon}(b)}$, $Y:=\Phi_{b\delta_{\epsilon}}z^{\epsilon}_{t_{\epsilon}(b)}$ and $Z:=\sqrt{\epsilon} \Phi_{b\delta_{\epsilon}}\int\limits_{0}^{b\delta_{\epsilon}}{\frac{1}{\Phi(s)}d{\left(W_{t_{\epsilon}(b)+s}-W_{t_{\epsilon}(b)}\right)}}$,
$\mathcal{G}=\sigma\left(X,Y\right)$ and $(\Omega,\mathcal{F},\mathbb{P})$ the canonical probability space of the Brownian motion, we obtain
\begin{eqnarray*}
\norm{
z^{\epsilon}_{b\delta_{\epsilon}}\left(x^{\epsilon}_{t_{\epsilon}(b)}\right)-z^{\epsilon}_{b\delta_{\epsilon}}\left(z^{\epsilon}_{t_{\epsilon}(b)}\right)}&\leq &
\frac{1}{\sqrt{{2\pi}{\epsilon}\int\limits_{0}^{b\delta_{\epsilon}}{\left(\frac{1}{\Phi(s)}\right)^2d{{s}}}}}\mathbb{E}{\left[\left|x^{\epsilon}_{t_{\epsilon}(b)}-z^{\epsilon}_{t_{\epsilon}(b)}\right|\right]}.
\end{eqnarray*}
Using Proposition <ref>, we obtain
\begin{eqnarray*}
\norm{
z^{\epsilon}_{b\delta_{\epsilon}}\left(x^{\epsilon}_{t_{\epsilon}(b)}\right)-z^{\epsilon}_{b\delta_{\epsilon}}\left(z^{\epsilon}_{t_{\epsilon}(b)}\right)}&\leq &
\sqrt{\frac{{\epsilon}}{{{2\pi}{}\int\limits_{0}^{b\delta_{\epsilon}}{\left(\frac{1}{\Phi(s)}\right)^2d{{s}}}}}} \times \\
\mathbb{E}{\left[B^2_{t_{\epsilon}(b)}\right]}.
\end{eqnarray*}
Using the fact that for each $\epsilon>0$, $\delta_{\epsilon}=\epsilon^{\gamma}$ for some $0<\gamma<1$, $\Phi_0=1$, the Intermediate Value Theorem for integrals and Lemma <ref> we obtain the result.
The following proposition will permit us to change the probability measure in a small interval of time in order to compare the total variation distance of the Itô diffusion and the first order approximation with a random initial condition.
Let us assume the same hypothesis of Proposition <ref>. Then for each $b\in \mathbb{R}$
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\norm{
\end{eqnarray*}
where $\delta_\epsilon=\epsilon^{\gamma}$ for some $\gamma>0$.
We will use Cameron-Martin-Girsanov Theorem and Novikov's Theorem. For the precise statements of these theorems we use here, see <cit.> and <cit.>.
Let $\epsilon>0$, $t\geq 0$ and $b\in \R$ be fixed. Let us define $\gamma^{\epsilon}_{t}:=\frac{V^{\prime}\left(x^{\epsilon}_t\right)}{\sqrt{\epsilon}}$ and
Then, for every $\epsilon>0$ and $t>0$ it follows that
\begin{eqnarray*}
\left(\gamma^{\epsilon}_{t}\right)^2
&\leq &2\kappa^2_2\frac{\left(x^{\epsilon}_t-\psi_t\right)^2}{{\epsilon}}+
&\leq & 4\kappa^2_2 B^2_t\left(\kappa_2 t^2+1\right)+
\end{eqnarray*}
\begin{eqnarray*}
\left(\Gamma^{\epsilon}_{t}\right)^2
&\leq &2\kappa^2_2\left(y_t\right)^2+
&\leq & 4\kappa^2_2 B^2_t\left(\kappa_2 t^2+1\right)+
\end{eqnarray*}
Let us define $I^{\epsilon}(b):=\left[t_{\epsilon}(b),t_{\epsilon}(b)+b\delta_{\epsilon}\right]$. Then, for every $\epsilon>0$ it follows that
\begin{eqnarray*}
\int\limits_{I(\epsilon)}\left(\gamma^{\epsilon}_{t}\right)^2dt
&\leq & 4b\kappa^2_2 \delta_{\epsilon}\left(\kappa_2 \left(t_{\epsilon}(b)+b\delta_{\epsilon}\right)^2+1\right)\sup\limits_{t\in I^{\epsilon}(b)}{B^2_t}+
2b\kappa^2_2 \delta_{\epsilon}\frac{\sup\limits_{t\in I^{\epsilon}(b)}\left(\psi_t\right)^2}{{\epsilon}}.\\
\end{eqnarray*}
\begin{eqnarray*}
\int\limits_{I(\epsilon)}\left(\Gamma^{\epsilon}_{t}\right)^2dt
&\leq & 4b\kappa^2_2 \delta_{\epsilon}\left(\kappa_2 \left(t_{\epsilon}(b)+b\delta_{\epsilon}\right)^2+1\right)\sup\limits_{t\in I^{\epsilon}(b)}{B^2_t}+
2b\kappa^2_2 \delta_{\epsilon}\frac{\sup\limits_{t\in I^{\epsilon}(b)}\left(\psi_t\right)^2}{{\epsilon}}.\\
\end{eqnarray*}
Using Lemma <ref>, there exists a constant $c>0$ such that
\begin{eqnarray*}
\int\limits_{I(\epsilon)}\left(\gamma^{\epsilon}_{t}\right)^2dt
&\leq & 4b\kappa^2_2 \delta_{\epsilon}\left(\kappa_2 \left(t_{\epsilon}(b)+b\delta_{\epsilon}\right)^2+1\right)\sup\limits_{t\in I^{\epsilon}(b)}{B^2_t}+
2bc\kappa^2_2 \delta_{\epsilon}
\end{eqnarray*}
\begin{eqnarray*}
\int\limits_{I(\epsilon)}\left(\Gamma^{\epsilon}_{t}\right)^2dt
&\leq & 4b\kappa^2_2 \delta_{\epsilon}\left(\kappa_2 \left(t_{\epsilon}(b)+b\delta_{\epsilon}\right)^2+1\right)\sup\limits_{t\in I^{\epsilon}(b)}{B^2_t}+
2bc\kappa^2_2 \delta_{\epsilon}
\end{eqnarray*}
for $\epsilon>0$ small enough. Consequently, for any constant $\rho>0$ it follows that
\begin{eqnarray*}
\mathbb{E}\left\{\exp\left[\rho\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\gamma^{\epsilon}_s\right)^2 ds}\right]\right\}<+\infty
\end{eqnarray*}
\begin{eqnarray*}
\mathbb{E}\left\{\exp\left[\rho\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\Gamma^{\epsilon}_s\right)^2 ds}\right]\right\}<+\infty
\end{eqnarray*}
for $\epsilon>0$ small enough.
From Novikov's Theorem it follows that
\begin{eqnarray*}
\frac{d\mathbb{P}^1_{t_{\epsilon}(b)+b\delta_{\epsilon}}}{d\mathbb{P}_{t_{\epsilon}(b)+b\delta_{\epsilon}}}&:=&
\exp\left\{\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\gamma^{\epsilon}_s dW_s}-\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\gamma^{\epsilon}_s\right)^2 ds}
\right\},\\
\frac{d\mathbb{P}^2_{t_{\epsilon}(b)+b\delta_{\epsilon}}}{d\mathbb{P}_{t_{\epsilon}(b)+b\delta_{\epsilon}}}&:=&
\exp\left\{\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\Gamma^{\epsilon}_s dW_s}-\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s\right)^2 ds}
\right\},
\end{eqnarray*}
are well defined and they define true probability measures
$\mathbb{P}^{i}_{t_{\epsilon}(b)+b\delta_{\epsilon}}$, $i\in\{1,2\}$.
From now on and up to the end of this proof we will use the notation
$\mathbb{P}^{i}:=\mathbb{P}^{i}_{t_{\epsilon}(b)+b\delta_{\epsilon}}$, $i\in\{1,2\}$ and
Under the probability measure $\mathbb{P}^1$,
$W^1_t:=W_t-\int\limits_{t_{\epsilon}(b)}^{t}{\gamma^{\epsilon}_s ds}$ is a Brownian motion on the time interval $t_{\epsilon}(b) \leq t\leq t_{\epsilon}(b)+b\delta_{\epsilon}$.
Also, under the probability measure $\mathbb{P}^2$,
$W^2_t:=W_t-\int\limits_{t_{\epsilon}(b)}^{t}{\Gamma^{\epsilon}_s ds}$ is a Brownian motion on the time interval $t_{\epsilon}(b) \leq t\leq t_{\epsilon}(b)+b\delta_{\epsilon}$.
\begin{eqnarray*}
\frac{d\mathbb{P}^1}{d\mathbb{P}^2}&=&
\frac{\exp\left\{\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\gamma^{\epsilon}_s dW_s}-\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\gamma^{\epsilon}_s\right)^2 ds}
\right\}}
{\Gamma^{\epsilon}_s dW_s}-\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s\right)^2 ds}
\right\}}\\
{\left(\gamma^{\epsilon}_s-\Gamma^{\epsilon}_s\right) dW_s}-\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\left(\gamma^{\epsilon}_s\right)^2-\left(\Gamma^{\epsilon}_s\right)^2\right) ds}
\right\}\\
{\left(\gamma^{\epsilon}_s-\Gamma^{\epsilon}_s\right) dW^1_s}+\frac{1}{2}
\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}
\right\}.
\end{eqnarray*}
By Pinsker's inequality and the mean-zero martingale property of the stochastic integral, we have for every
$t_{\epsilon}(b)\leq t\leq t_{\epsilon}(b)+b\delta_{\epsilon}$
\begin{eqnarray*}
&\leq &
\mathbb{E}_{\mathbb{P}^1}\left[\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right]\\
&= & \mathbb{E}_{\mathbb{P}}\left[\frac{d\mathbb{P}^1}{d\mathbb{P}}\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right].
\end{eqnarray*}
By Cauchy-Schwarz's inequality and the mean-one Doléans exponential martingale
property, we have
\begin{eqnarray*}
\mathbb{E}_{\mathbb{P}}\left[\frac{d\mathbb{P}^1}{d\mathbb{P}}\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right] &\leq &
\sqrt{
\mathbb{E}_{\mathbb{P}}\left[\exp\left\{\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\gamma^{\epsilon}_s\right)^2ds}\right\}
\left(\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right)^2\right]
&\leq &
\sqrt{
\mathbb{E}_{\mathbb{P}}
\left[\exp\left\{2\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\gamma^{\epsilon}_s\right)^2ds}\right\}\right]}\times\\
\sqrt{
\mathbb{E}_{\mathbb{P}}
\left[\left(\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right)^4\right]}
\end{eqnarray*}
It follows for $\epsilon>0$ small enough that
\begin{eqnarray*}
\exp\left\{\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\gamma^{\epsilon}_s\right)^2ds}\right\}&\leq & \exp\left\{
4b\kappa^2_2 \delta_{\epsilon}\left(\kappa_2 \left(t_{\epsilon}(b)+b\delta_{\epsilon}\right)^2+1\right)\sup\limits_{t\in I^{\epsilon}(b)}{B^2_t}+
2bc\kappa^2_2 \delta_{\epsilon}
\right\},
\end{eqnarray*}
where the last expression is $\mathbb{P}$-integrable for $\epsilon>0$ small enough.
Using the scaling property of Brownian motion and the distribution of the maximum of the Brownian motion in a compact interval, the last inequality implies that
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}
\mathbb{E}_{\mathbb{P}}
\left[
\exp\left\{\rho\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}{\left(\gamma^{\epsilon}_s\right)^2ds}\right\}\right]&=&1.
\end{eqnarray*}
for any constant $\rho>0$.
Also, it is true that
\begin{eqnarray*}
\left(\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right)^4 &\leq & \left(b\delta_{\epsilon}\sup\limits_{s\in I^{\epsilon}(b)}{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2}\right)^4\\
&\leq & Cb^4\delta^4_{\epsilon}\left(\sup\limits_{s\in I^{\epsilon}(b)}{\frac{\left(x^{\epsilon}_s-\psi_s\right)^{16}}{{\epsilon^4}}}
+\sup\limits_{s\in I^{\epsilon}(b)}{\frac{\left|x^{\epsilon}_s-\psi_s-\sqrt{\epsilon}y_t\right|^8}{{\epsilon^4}}}
\right),
\end{eqnarray*}
where $C=C(\kappa_2,\kappa_3)>0$ is a constant.
Using the last inequality and Proposition <ref>, we obtain that
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}\mathbb{E}_{\mathbb{P}}
\left[\left(\int\limits_{t_{\epsilon}(b)}^{t_{\epsilon}(b)+b\delta_{\epsilon}}
{\left(\Gamma^{\epsilon}_s-\gamma^{\epsilon}_s\right)^2 ds}\right)^4\right]&=&0.
\end{eqnarray*}
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\norm{
\end{eqnarray*}
Now we have all the tools in order to prove our result for the class of smooth coercive regular potentials.
Let $V$ a smooth coercive regular potential.
Let us consider the family $x^{\epsilon}=\{ {x^{\epsilon}_t} \}_{t \geq 0}$
given by the the semiflow of the following stochastic differential equation,
\begin{eqnarray*}
\end{eqnarray*}
for $t \geq 0$, where $x_0$ is a deterministic initial condition in $\mathbb{R}\setminus \{0\}$ and
$\{W_t\}_{t \geq 0}$ is a standard Brownian motion. This family presents profile cut-off in the sense of
the Definition <ref> with respect to the total variation distance when $\epsilon$ goes to zero.
The profile function $G:\mathbb{R}\rightarrow \mathbb{R}$ is given by
\begin{eqnarray*}
G(b)&:=&\norm{\N{\left(\tilde{c} e^{-b},1\right)}-\N{\left(0,1\right)}},
\end{eqnarray*}
where $\tilde{c}$ is the nonzero constant given by
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\psi_t}&=&:\tilde{c}.
\end{eqnarray*}
and the cut-off time $t_{\epsilon}$ and window time $w_{\epsilon}$ are given by
\begin{eqnarray*}
t_{\epsilon}&:=&\frac{1}{2V^{\prime\prime}(0)}\left(\ln\left(\frac{1}{\epsilon}\right)+\ln\left(2V^{\prime\prime}(0)\right) \right),\\
\end{eqnarray*}
where $\delta_\epsilon=\epsilon^{\gamma}$ for some $\gamma\in ]0,1[$.
Let $\epsilon>0$ and $t>0$ be fixed. We define
\begin{eqnarray*}
\end{eqnarray*}
\begin{eqnarray*}
\end{eqnarray*}
where $\mu^{\epsilon}$ and $\N^{\epsilon}$ are given by Lemma
<ref> and Lemma <ref>. For each $b\in \R$, take $\epsilon_b>0$ such that
b\delta_{\epsilon}> 0$ for every $0<\epsilon<\epsilon_b$.
By Corollary <ref> and Remark <ref> we know that for each $b \in \R$
\begin{eqnarray}\label{cutlin}
\lim\limits_{\epsilon\rightarrow 0}{d^{\epsilon}\left({t}^*_{\epsilon}(b)\right)}&=&G(b).
\end{eqnarray}
By definition,
\begin{eqnarray*}
\norm{
\end{eqnarray*}
Using Proposition <ref>, Proposition <ref>, Lemma <ref>, the relation (<ref>) and the item $i)$ of Lemma <ref>, we have $\limsup\limits_{\epsilon\rightarrow 0}{D^{\epsilon}{({t}^*_{\epsilon}(b))}}\leq G(b)$. In order to obtain the converse inequality, we observe that
\begin{eqnarray*}
&\leq &\norm{
\end{eqnarray*}
Again, using Proposition <ref>, Proposition <ref>, Lemma <ref>, the relation (<ref>) and the item $ii)$ of Lemma <ref> we have $\liminf\limits_{\epsilon\rightarrow0}{D^{\epsilon}{({t}^*_{\epsilon}(b))}}\geq G(b)$.
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow0}{D^{\epsilon}{({t}^*_{\epsilon}(b))}}=G(b).
\end{eqnarray*}
The following proposition will permit us to approximate a coercive regular potential by a smooth coercive regular potential.
Let us assume that $V$ is a coercive regular potential.
For every $M\in ]0,+\infty[$, there exists
a smooth coercive regular potential $V_M(x)$ which is an approximation of $V$ in the following way:
$V_M(x)=V(x)$ for every $|x|\leq \sqrt{2} M$.
By coercivity hypothesis there exists $\delta>0$ such that
$V^{\prime\prime}(x)\geq \delta$ for every $x\in \R$.
Let $g \in\mathcal{C}^{\infty}\left(\mathbb{R},[0,1]\right)$ be an increasing function
such that $g(u)=0$ for $u\leq \frac{1}{2}$ and $g(u)=1$ if $u\geq 1$. Let $M\in[1,\infty[$ be a fixed number.
Let $R_M:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by
\begin{eqnarray*}
\end{eqnarray*}
Since $V\in\mathcal{C}^3\left(\R,\R\right)$ and $g \in\mathcal{C}^{\infty}\left(\mathbb{R},[0,1]\right)$,
we have $R_M\in\mathcal{C}^{1}\left(\mathbb{R},\R\right)$.
We also have that $R_M(x)=V^{\prime \prime}(x)$ for every $|x|\leq \sqrt{2}M$, $R_M(x)=\delta$ for every $|x|\geq 2M$,
$R_M(x)\geq \delta$ for every $x\in \mathbb{R}$, $\|R_M\|_{\infty}<+\infty$ and $\|R^{\prime}_M\|_{\infty}<+\infty$.
Let us define $S_M(x):=\int\limits_{0}^{x}{R_M(y)dy}$ for every $x\in \mathbb{R}$ and let us define
Then $V_M$ is a smooth $\delta$-coercive regular potential such that
$V_M(x)=V(x)$ for every $|x|\leq \sqrt{2}M$.
The next proposition will tell us that the approximation of the coercive regular potential by a smooth coercive regular potential also implies an approximation in the total variation distance of the invariant measures associated to the potential $V$ and $V_M$
and the total variation distance for the processes at the “cut-off time" associated to the potentials $V$ and $V_M$.
Let $V$ be a coercive regular potential and for every $M>0$ let $V_M$ be the approximation of $V$ obtained from Proposition <ref>.
Let $x^{\epsilon,M}=\left\{x^{\epsilon,M}_{t}\right\}_{t\geq 0}$ be the Itô diffusion associated to the smooth coercive potential $V_M$
and let $\mu^{\epsilon,M}$ be the invariant probability measure associated to the stochastic process
$x^{\epsilon,M}$ defined in Lemma <ref>.
Let us denote by $x^{\epsilon}=\left\{x^{\epsilon}_{t}\right\}_{t\geq 0}$ the Itô diffusion associated to the coercive potential $V$
and let us denote by $\mu^{\epsilon}$ the invariant probability measure associated to the stochastic process
$x^{\epsilon}$ defined in Lemma <ref>.
It follows that
$i)$ For every $M>0$
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\norm{\mu^{\epsilon}-\mu^{\epsilon,M}}}
\end{eqnarray*}
Using the same notation as in Theorem <ref>, for each $b\in \R$, take $\epsilon_b>0$ such that
b\delta_{\epsilon}> 0$ for every $0<\epsilon<\epsilon_b$, where $\delta_\epsilon=\epsilon^\gamma$ for some $\gamma>0$.
\begin{eqnarray*}
{\lim\limits_{\epsilon\rightarrow 0}{\norm{x^{\epsilon}_{{t}^*_{\epsilon}(b)}-x^{\epsilon,M}_{{t}^*_{\epsilon}(b)}}}}&=&0
\end{eqnarray*}
for every $M>|x_0|$ and every $b\in \R$.
Let us prove item $i)$.
Notice that $V''_M(0)=V''(0)$. By triangle's inequality and Lemma <ref>, we have
\begin{eqnarray*}
{\norm{\mu^{\epsilon}-\mu^{\epsilon,M}}}&\leq & {\norm{\mu^{\epsilon}-\N^{\epsilon}}}
\end{eqnarray*}
Taking $\epsilon\rightarrow 0$ and using Lemma <ref> we obtain
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{\norm{\mu^{\epsilon}-\mu^{\epsilon,M}}}
for every $M>0$.
Now let us prove item $ii)$. Let $\epsilon>0$ and $M>|x_0|>0$ be fixed.
Let us define
$\tau^{\epsilon,M}:=\inf\left\{s\geq 0: \left|x^{\epsilon,M}_s\right|> M\right\}$. By the variational definition of total variation distance in terms of couplings
\begin{eqnarray*}
{{\norm{x^{\epsilon}_{{t}^*_{\epsilon}(b)}-x^{\epsilon,M}_{{t}^*_{\epsilon}(b)}}}}&\leq & {
{\mathbb{P}_{x_0}\left(\tau^{\epsilon,M}\leq {t}^*_{\epsilon}(b) \right)}}.
\end{eqnarray*}
Let us define $\sigma^{\epsilon,M}:=\inf\left\{s\geq 0: |x^{\epsilon,M}_s-\psi^{M}_s|>M-|x_0| \right\}$, where
is the semiflow associated to the autonomous differential equation,
\begin{eqnarray*}
\end{eqnarray*}
for every $t\geq 0$ and $\psi^{M}_0:=x_0$.
Using the coercivity hypothesis of $V_M$, we see that the semiflow $\psi^M$ is decreasing in norm, and $|\psi^{M}_t|\leq |x_0|$ for every $t\geq 0$.
In particular,
$\sigma^{\epsilon,M}\leq \tau^{\epsilon,M}$. Consequently.
$\mathbb{P}_{x_0}\left(\tau^{\epsilon,M}\leq {t}^*_{\epsilon}(b) \right)\leq \mathbb{P}_{x_0}\left(\sigma^{\epsilon,M}\leq {t}^*_{\epsilon}(b) \right)$.
Therefore, it is enough to prove that
$\lim\limits_{\epsilon\rightarrow 0}\mathbb{P}_{x_0}\left(\sigma^{\epsilon,M}> {t}^*_{\epsilon}(b) \right)=1$.
For every $s\geq 0$, let us define $z^{\epsilon,M}_s:=\frac{x^{\epsilon,M}_s-\psi^M_s}{\sqrt{\epsilon}}$.
Then, $\sigma^{\epsilon,M}=\inf\left\{s\geq 0: |z^{\epsilon,M}_s|>\frac{M-|x_0|}{\sqrt{\epsilon}} \right\}$.
We note that
\begin{eqnarray*}
\mathbb{P}_{x_0}\left(\sigma^{\epsilon,M}\geq {t}^*_{\epsilon}(b) \right)&=&\mathbb{P}_{x_0}\left(\sup\limits_{0\leq s \leq {t}^*_{\epsilon}(b)}{\left|z^{\epsilon,M}_s\right|}\leq \frac{M-|x_0|}{\sqrt{\epsilon}} \right).
\end{eqnarray*}
Let us define $c_M:=M-|x_0|>0$. We have
\begin{eqnarray*}
\mathbb{P}_{x_0}\left(\sup\limits_{0\leq s \leq {t}^*_{\epsilon}(b)}{\left|z^{\epsilon,M}_s\right|}> \frac{c_M}{\sqrt{\epsilon}} \right)&=&
\mathbb{P}_{x_0}\left(\sup\limits_{0\leq s \leq {t}^*_{\epsilon}(b)}{\left(z^{\epsilon,M}_s\right)^2}> \frac{c^2_M}{\epsilon} \right).
\end{eqnarray*}
Using Itô's formula and the coercivity of $V_M$, we have
\begin{eqnarray*}
\left(z^{\epsilon,M}_t\right)^2\leq t+\Pi^{\epsilon,M}_t
\end{eqnarray*}
for every $t\geq 0$, where the process $\Pi^{\epsilon,M}_t:=2\int\limits_{0}^{t}{z^{\epsilon,M}_sdW_s}$ is a martingale. Then,
\begin{eqnarray*}
\mathbb{E}\left[\left(z^{\epsilon,M}_t\right)^2\right] \leq t
\end{eqnarray*}
for every $t\geq 0$. Using Itô's isometry, we obtain
\begin{eqnarray*}
\mathbb{E}\left[\left(\Pi^{\epsilon,M}_t\right)^2\right] \leq 2t^2
\end{eqnarray*}
for every $t\geq 0$. Let us take $\epsilon_{M,b}>0$ such that for every $0<\epsilon<\epsilon_{M,b}$, we have
$c^2_M-\epsilon {t}^*_{\epsilon}(b)>0$.
Using Doob's inequality, we have
\begin{eqnarray*}
\mathbb{P}_{x_0}\left(\sup\limits_{0\leq s \leq {t}^*_{\epsilon}(b)}{\left(z^{\epsilon,M}_s\right)^2}> \frac{c^2_M}{\epsilon} \right)&\leq &
\mathbb{P}_{x_0}\left(\sup\limits_{0\leq s \leq {t}^*_{\epsilon}(b)}{\left|\Pi^{\epsilon,M}_s\right|}>
\frac{c^2_M-\epsilon {t}^*_{\epsilon}(b)}{\epsilon} \right)\\
& \leq &
\frac{\epsilon^2}{\left(c^2_M-\epsilon {t}^*_{\epsilon}(b)\right)^2}\mathbb{E}\left[\left(\Pi^{\epsilon,M}_{{t}^*_{\epsilon}(b)}\right)^2\right]\\
& \leq &
\frac{2\epsilon^2 \left({t}^*_{\epsilon}(b)\right)^2}{\left(c^2_M-\epsilon {t}^*_{\epsilon}(b)\right)^2}.
\end{eqnarray*}
Letting $\epsilon\rightarrow 0$ we obtain the desired limit.
Now, we are ready to prove Theorem <ref>. To stress the fact that Theorem <ref> is just a consequence of what we have proved up to here, let us state this as a Lemma:
Let $V_M$ be the approximation of $V$ obtained in Proposition <ref>.
Profile cut-off for $\{x^{\epsilon,M}_t\}_{t\geq 0}$ implies
profile cut-off for $\{x^{\epsilon}_t\}_{t\geq 0}$ with the same cut-off time, cut-off window and profile function.
Recall the notation introduced in Proposition <ref>. Let $\epsilon>0$ and $t>0$ be fixed.
Let us take $M>\max\left\{|x_0|,\|\psi\|_{\infty}\right\}$. We define
\begin{eqnarray*}
\end{eqnarray*}
\begin{eqnarray*}
\end{eqnarray*}
By triangle's inequality, we have
\begin{eqnarray*}
D^{\epsilon,M}{(t)}&\leq & \norm{x^{\epsilon,M}_t-x^{\epsilon}_t}+D^{\epsilon}(t)+\norm{\mu^{\epsilon}-\mu^{\epsilon,M}}.
\end{eqnarray*}
Recall that
$t^{}_{\epsilon}=\frac{1}{2V^{\prime\prime}(0)}\left(\ln\left(\frac{1}{\epsilon}\right)+\ln\left(2V^{\prime\prime}(0)\right) \right)$
$w^{\prime}_{\epsilon}=\frac{1}{V^{\prime\prime}(0)}+\delta_{\epsilon}$ respectively.
Let $b\in \R$ be fixed. Recall that
${t}^*_{\epsilon}(b)=t_{\epsilon}+bw^\prime_{\epsilon}$. Take $\epsilon_b>0$ such that for every $0<\epsilon<\epsilon_b$ we have,
\begin{eqnarray*}
&\leq & \norm{x^{\epsilon,M}_{{t}^*_{\epsilon}(b)}-x^{\epsilon}_{{t}^*_{\epsilon}(b)}}+D^{\epsilon}({t}^*_{\epsilon}(b))+\norm{\mu^{\epsilon}-\mu^{\epsilon,M}}.
\end{eqnarray*}
Therefore, using Proposition <ref> and Lemma <ref> we have
\begin{eqnarray*}
\limsup\limits_{\epsilon\rightarrow 0}{D^{\epsilon,M}{({t}^*_{\epsilon}(b))}}&\leq & \limsup\limits_{\epsilon\rightarrow 0}{D^{\epsilon}({t}^*_{\epsilon}(b))}.
\end{eqnarray*}
By Theorem <ref>, we know that
$\lim\limits_{\epsilon\rightarrow 0}{D^{\epsilon,M}{({t}^*_{\epsilon}(b))}}=G(b)$. Therefore
\begin{eqnarray*}
G(b)&\leq & \limsup\limits_{\epsilon\rightarrow 0}{D^{\epsilon}({t}^*_{\epsilon}(b))}.
\end{eqnarray*}
It also follows that
\begin{eqnarray*}
D^{\epsilon}{({t}^*_{\epsilon}(b))}&\leq & \norm{x^{\epsilon}_{{t}^*_{\epsilon}(b)}-x^{\epsilon,M}_{{t}^*_{\epsilon}(b)}}+D^{\epsilon,M}({t}^*_{\epsilon}(b))+\norm{\mu^{\epsilon,M}-\mu^{\epsilon}}.
\end{eqnarray*}
Therefore, using Lemma <ref>, Proposition <ref> and Theorem <ref> we have
\begin{eqnarray*}
\liminf\limits_{\epsilon\rightarrow 0}{D^{\epsilon}{({t}^*_{\epsilon}(b))}}&\leq & G(b).
\end{eqnarray*}
We conclude that
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}{D^{\epsilon}{({t}^*_{\epsilon}(b))}}&=& G(b).
\end{eqnarray*}
§ DOUBLE WELL POTENTIAL
We study the situation when the potential $V$ has only two wells of different depths. In this situation we can observe two statistical different regimes.
Firstly, if the horizon is shorter that the exit time from the shallow well, the system cannot leave the well where it has started, and therefore stays in the neighborhood of the well's local minimum.
Secondly, if the time horizon is longer that the exit time from the shallow well, the system has enough time to reach the deepest well from any starting point, and stays in a neighborhood of the global minimum. C. Kipnis and C. Newman in <cit.> proved the following metastability behavior: there is a time scale on which the dynamical system converges to a Markov two-state process with one absorbing state corresponding to the deep well. This time scale is given by the mean exit time from the shallow well.
Using the last fact we can observe the following:
Let us denote by $x{^-}$ the shallow well and by $x^+$ the deepest well.
In Theorem <ref> we prove that on the one-well potential case under the coercivity assumption we have profile cut-off phenomenon.
Given a deterministic initial condition $x_0$ in a small neighborhood of the well of $x^*$ where $x^* \in \{x^-,x^+\}$. Recall that $\{x^{\epsilon}_t\}_{t\geq 0}$ is the the following differential equation
\begin{eqnarray*}
\end{eqnarray*}
for $t \geq 0$.
Let us suppose that $V^{\prime\prime}(x^*)>0$.
We have that the exit time from the well associated to the local minimum $x^*$ is exponentially large and the time of the cut-off time obtained in the Theorem <ref> is much smallest.
Consequently, we have abrupt convergence to a “kind local asymptotic distribution"; that is we have a kind of local cut-off phenomenon with respect to the following distance:
\begin{eqnarray*}
\end{eqnarray*}
with time cut-off $t_\epsilon=\frac{1}{2V^{\prime\prime}(x^*)}\ln\left(\frac{1}{\epsilon}\right)$, window cut-off $w_\epsilon=\frac{1}{V^{\prime\prime}(x^*)}+\epsilon^\gamma$ for some
$\gamma\in ]0,1[$ and profile function $G:\mathbb{R}\rightarrow [0,1]$ given by
\begin{eqnarray*}
G(b)&:=&\norm{\N{\left(c(x^*) e^{-b},1\right)}-\N{\left(0,1\right)}},
\end{eqnarray*}
where $c(x^*)$ is the nonzero constant given by
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(x^*)}t}\Phi_t}&=&c(x^*).
\end{eqnarray*}
By the same facts, the last local cut-off phenomenon can also extend for a multi-well potential.
§ PROPERTIES OF THE TOTAL VARIATION DISTANCE OF NORMAL DISTRIBUTION
Let $\{\mu,\tilde{\mu}\} \subset \mathbb{R}$ and $\{\sigma^2,\tilde{\sigma}^2\}\subset ]0,+\infty[$ be fixed numbers.
$i)$ For any constant $c\not= 0$ we have
\begin{eqnarray*}
\norm{\N{\left(c\mu,{c}^2\sigma^2 \right)}-\N{\left(c\tilde{\mu},{c}^2\tilde{\sigma}^2\right)}}&=&
\norm{\N{\left(\mu,\sigma^2\right)}-\N{\left(\tilde{\mu},\tilde{\sigma}^2\right)}}.
\end{eqnarray*}
\begin{eqnarray*}
\norm{\N{\left(\mu,\sigma^2 \right)}-\N{\left(\tilde{\mu},\tilde{\sigma}^2 \right)}}&=&
\norm{\N{\left(|\mu-\tilde{\mu}|,\sigma^2\right)}-\N{\left(0,\tilde{\sigma}^2 \right)}}.
\end{eqnarray*}
This is done using the characterization of the total variation distance between two probability measures which are absolutely continuous with respect to the Lebesgue measure on $\left(\mathbb{R},\mathcal{B}\left(\R\right)\right)$ and using the Change of Variable Theorem.
Let $\mu\in \R$ then
\begin{eqnarray*}
\norm{\N{\left(\mu,1 \right)}-\N{\left(0,1\right)}}=\frac{2}{\sqrt{2\pi}}\int\limits_{0}^{\nicefrac{|\mu|}{2}}{e^{-\frac{x^2}{2}}dx} \leq \frac{|\mu|}{\sqrt{2\pi}}.
\end{eqnarray*}
Also, this is done using the characterization of the total variation distance between two probability measures which are absolutely continuous with respect to the Lebesgue measure on $\left(\mathbb{R},\mathcal{B}\left(\R\right)\right)$ and an straightforward calculations.
Let $\{\mu_{\epsilon}\}_{\epsilon>0}\subset \R$ be a sequence such that $\lim\limits_{\epsilon\rightarrow 0}{\mu_{\epsilon}}=\mu\in \R$. Then
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}\norm{\N{\left(\mu_{\epsilon},1 \right)}-\N{\left(0,1\right)}}=\norm{\N{\left(\mu,1\right)}-\N{\left(0,1\right)}}.
\end{eqnarray*}
This is done using triangle inequality, the item $ii)$ of Lemma <ref>, Lemma <ref> and the
Lemma <ref>.
Let $\{\sigma^2_{\epsilon}\}_{\epsilon>0}\subset ]0,+\infty[$ be a sequence such that $\lim\limits_{\epsilon\rightarrow 0}{\sigma^2_{\epsilon}}=\sigma^2\in ]0,+\infty[$. Then
\begin{eqnarray*}
\lim\limits_{\epsilon\rightarrow 0}\norm{\N{\left(0,\sigma^2_{\epsilon} \right)}-\N{\left(0,\sigma^2\right)}}&=&0.
\end{eqnarray*}
This is done using the item $i)$ of Lemma <ref>, the characterization of the total variation distance between two probability measures which are absolutely continuous with respect to the Lebesgue measure on $\left(\mathbb{R},\mathcal{B}\left(\R\right)\right)$
and an straightforward calculations.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset \mathcal{F}$ be a sub-sigma algebra of $\mathcal{F}$.
Let $X,Y,Z:(\Omega,\mathcal{F})\rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be random variables such that $X$ and $Y$ are $\mathcal{G}$ measurables and $X,Y,Z\in L^{1}\left(\Omega,\mathcal{F},\mathbb{P}\right)$. Let us consider the following random variables $X^{*}=X+Z$ and $Y^{*}=Y+Z$. Let us suppose that for some $\sigma^2>0$ we have
$\mathbb{P}\left[X^{*}\in F\left|\right.\mathcal{G}\right]=\mathbb{P}\left[\mathcal{G}(X,\sigma^2)\in F\right]$ and
$\mathbb{P}\left[Y^{*}\in F\left|\right.\mathcal{G}\right]=\mathbb{P}\left[\mathcal{G}(Y,\sigma^2)\in F\right]$
for every $F\in \mathcal{F}$.
\begin{eqnarray*}
\norm{X^{*}-Y^{*}}&\leq & \frac{1}{\sqrt{2\pi}\sigma}\mathbb{E}{\left[\left|X-Y\right|\right]}.
\end{eqnarray*}
Using the the properties of conditional expectation, the item $i)$, item $ii)$ of Lemma <ref> and Lemma <ref>, we have
\begin{eqnarray*}
\norm{X^{*}-Y^{*}}&=&\sup\limits_{F\in \mathcal{F}}
{\left|\mathbb{E}\left[\mathbbm{1}_{\left(X^* \in F \right)}-\mathbbm{1}_{\left(Y^* \in F\right)} \right]\right|}\\
&\leq &
\sup\limits_{F\in \mathcal{F}}
{\mathbb{E}\left[ \left| \mathbb{E}\left[\mathbbm{1}_{\left(X^* \in F\right)}-\mathbbm{1}_{\left(Y^* \in F\right)}\left|\right.\mathcal{G} \right] \right| \right]}\\
&\leq &
\sup\limits_{F\in \mathcal{F}}
{\mathbb{E}\left[ \left| \mathbb{P}\left(\N(X,\sigma^2)\in F\right)-\mathbb{P}\left(\N(Y,\sigma^2)\in F\right) \right| \right]}\\
&\leq &
\sup\limits_{F\in \mathcal{F}}
{\mathbb{E}\left[ \frac{1}{\sqrt{2\pi}\sigma}\left| X-Y \right| \right]}\\
\frac{1}{\sqrt{2\pi}\sigma}\mathbb{E}{\left[\left|X-Y\right|\right]}.
\end{eqnarray*}
§ QUALITATIVE AND QUANTITATIVE BEHAVIOR
Let us assume the hypothesis of Theorem <ref>.
$i)$ $\lim\limits_{t\rightarrow +\infty}{\psi_t}=0$.
$ii)$ $\lim\limits_{t\rightarrow +\infty}{\Phi_t}=0$.
$iii)$ There exist constants $c\not=0$ and $\tilde{c}\not=0$ such that
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\Phi_t}&=&c,
\end{eqnarray*}
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{e^{V^{\prime\prime}{(0)}t}\psi_t}&=&\tilde{c},
\end{eqnarray*}
where $\Phi=\{\Phi_{t}\}_{t\geq 0}$ is the fundamental solution of the non-autonomous system
\begin{eqnarray*}
d\Phi_{t}=-V^{\prime\prime}{(\psi_t)}\Phi_t dt
\end{eqnarray*}
for every $t\geq 0$ with initial condition $\Phi_{0}=1$.
$iv)$ $\lim\limits_{t\rightarrow +\infty}{\Phi_t^2 \int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}}=\frac{1}{2V^{\prime\prime}{(0)}}$.
$i)$ By our assumptioms $V^{\prime}(0)=0$, $V^{\prime\prime}(0)>0$ and $V^{\prime}(x)\not=0$ if $x\not=0$. Therefore, the unique critical point zero is
asymptotically stable, so there exists an open neighboorhood $N_0$ of zero such that for every $\psi_0\in N_0$. It follows that
$\psi_t$ goes to zero as $t$ goes to infinity.
Let us consider that $\psi_0\not \in N_{0}$ and $K:=V^{-1}\left([0,V(\psi_0)]\right)$. Then $\psi_t\in K$ for every $t\geq 0$.
Also, $K$ is a compact set because of $\lim\limits_{|x|\rightarrow +\infty}{V(x)}=+\infty$.
Because $K$ is bounded, then
there exist $r>0$ such that $K\subset B(0,r)$ where we denote $B(0,r):=\{x\in \mathbb{R}: |x|<r\}$
and $\overline{B(0,r)}:=\{x\in \mathbb{R}: |x|\leq r\}$ so we
we can choose $N_0$ small enough such that $N_0\subset B(0,r)\subset \overline{B(0,r)}$. Let us
call $\hat{K}:=\overline{B(0,r)}$ then $\psi_t \in \hat{K}$ for every $t\geq 0$.
Let us define $\delta:=\inf\limits_{x\in \hat{K}\setminus N_0}{\left(V^{\prime}(x)\right)^2}>0$.
Let us suppose that
$\psi_t\not\in N_{0}$ for every $t\geq 0$, then $dV(\psi_t)=-\left(V^{\prime}(\psi_t)\right)^2\leq -\delta$ for every $t\geq 0$. Therefore, $0\leq t\leq \frac{V(\psi_0)}{\delta}$ which is a contradiction. Consequently, there exists $\tau>0$ such that $\psi_{\tau}\in N_0$ and consequently, $\psi_t$ goes to zero as $t$ goes to infinity.
$ii)$ By our assumptions it follows that $\Phi_t=\frac{V^{\prime}(\psi_t)}{V^{\prime}(\psi_0)}$ for every $t\geq 0$, where $\psi_0=x_0\not=0$. So by item $i)$ and continuity of $V^{\prime}$ we have
$\lim\limits_{t\rightarrow \infty}{\Phi_t}=\frac{V^{\prime}(0)}{V^{\prime}(\psi_0)}=0$.
$iii)$ Let us define $H(z)=\left(\frac{V^{\prime\prime}(0)}{V^{\prime}(z)}-\frac{1}{z}\right)\mathbbm{1}_{\{z\not=0\}}
where $\mathbbm{1}_{A}$ denotes the indicator function of the set $A\subset \R$.
Let us define $h:\R\rightarrow \R$ by
\begin{eqnarray*}
\end{eqnarray*}
Since $H$ is everywhere continuous, then it follows that $h$ is well defined.
Let us define $\Psi_t:=h(\psi_t)$ for every $t\geq 0$, then $d\Psi_t=-V^{\prime\prime}{(0)}\Psi_t dt$ for every $t\geq 0$
and $\Psi_0=h(\psi_0)$. Therefore,
\begin{eqnarray}\label{ayuda}
\psi_t \exp{\left(V^{\prime\prime}(0)t\right)}&=&h(\psi_0)\exp{\left(-\int\limits_{0}^{\psi_t}{H(z)dz}\right)}
\end{eqnarray}
for every $t\geq 0$. By Intermediate Value Theorem, for every $t\geq 0$ there exists $\xi_t\in ]\min\{0,\psi_t\},\max\{0,\psi_t\}[$
such that $V^{\prime}(\psi_t)=V^{\prime\prime}(\xi_t)\psi_t$. Because of relation (<ref>), we see that
\begin{eqnarray}\label{ayuda2}
V^{\prime}(\psi_t) \exp{\left(V^{\prime\prime}(0)t\right)}&=&V^{\prime\prime}(\xi_t)h(\psi_0)\exp{\left(-\int\limits_{0}^{\psi_t}{H(z)dz}\right)}
\end{eqnarray}
for every $t\geq 0$. Consequently, by the relation (<ref>) and item $ii)$, we have
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{V^{\prime}(\psi_t) \exp{\left(V^{\prime\prime}(0)t\right)}}&=&
\end{eqnarray*}
Because $sgn(h(x))=sgn(x)$ for every $x\not=0$, then
$iv)$ By item $ii)$, we have
\begin{eqnarray*}
{\Phi_t^2 \int\limits_{0}^{t}{\left(\frac{1}{\Phi_s}\right)^2d{{s}}}}&=&
{(V^{\prime}(\psi_t))^2 \int\limits_{0}^{t}{\left(\frac{1}{V^{\prime}(\psi_s)}\right)^2d{{s}}}}
\end{eqnarray*}
for each $t\geq 0$. By item $iii)$ and for each $0<\epsilon<c^2$, we have
\begin{eqnarray*}
\limsup\limits_{t\rightarrow +\infty}{(V^{\prime}(\psi_t))^2 \int\limits_{0}^{t}{\left(\frac{1}{V^{\prime}(\psi_s)}\right)^2d{{s}}}}
&\leq & \left(\frac{c^2+\epsilon}{c^2-\epsilon}\right) \frac{1}{2V^{\prime\prime}(0)},\\
\liminf\limits_{t\rightarrow +\infty}{(V^{\prime}(\psi_t))^2 \int\limits_{0}^{t}{\left(\frac{1}{V^{\prime}(\psi_s)}\right)^2d{{s}}}}
&\geq & \left(\frac{c^2-\epsilon}{c^2+\epsilon}\right) \frac{1}{2V^{\prime\prime}(0)}.\\
\end{eqnarray*}
Letting $\epsilon\rightarrow 0$, we obtain
\begin{eqnarray*}
\lim\limits_{t\rightarrow +\infty}{(V^{\prime}(\psi_t))^2 \int\limits_{0}^{t}{\left(\frac{1}{V^{\prime}(\psi_s)}\right)^2d{{s}}}}
&= & \frac{1}{2V^{\prime\prime}(0)}.\\
\end{eqnarray*}
Let us assume the hypothesis of Theorem <ref>. Let us follow the same notation as in the proof of Theorem <ref>.
It follows that
\begin{eqnarray*}\lim\limits_{\epsilon\rightarrow 0}{\frac{\sup\limits_{{t}_{\epsilon}(b)\leq t\leq {t}^*_{\epsilon}(b)}
{|\psi_{t}}|}{\sqrt{\epsilon}}}=\rho(b)\in ]0,+\infty[.
\end{eqnarray*}
for every $b\in \R$.
By continuity we have
\begin{eqnarray*}
\frac{\sup\limits_{{t}_{\epsilon}(b)\leq t\leq {t}^*_{\epsilon}(b)}
{|\psi_{t}}|}{\sqrt{\epsilon}}&=& \frac{
\end{eqnarray*}
for some $\tilde{t}\in [{t}_{\epsilon}(b),{t}^*_{\epsilon}(b)]$.
Then, using the following relation and Lemma <ref>, it becomes straightforward.
\begin{eqnarray*}
\frac{{|\psi_{\tilde{t}}}|}{\sqrt{\epsilon}}&=&e^{V^{\prime\prime}(0)\tilde{t}} |\psi_{\tilde{t}}|\frac{e^{-V^{\prime\prime}(0)\tilde{t}}}{\sqrt{\epsilon}}.
\end{eqnarray*}
§ TOOLS
$\lim\limits_{\epsilon\rightarrow 0}{\epsilon^{\alpha}\left(\ln\left(\frac{1}{\epsilon}\right)\right)^{\beta}}=0$ for every $\alpha>0$ and $\beta>0$.
Let $\{a_{\epsilon}\}_{\epsilon>0}\subset \R$ and $\{b_{\epsilon}\}_{\epsilon>0}\subset \R$ be sequences such that
$\lim\limits_{\epsilon\rightarrow 0}{b_{\epsilon}}=b\in \R$. Then
$i)$ $\limsup\limits_{\epsilon\rightarrow 0}{\left(a_{\epsilon}+b_{\epsilon}\right)}=\limsup\limits_{\epsilon\rightarrow 0}{a_{\epsilon}}+b$.
$ii)$ $\liminf\limits_{\epsilon\rightarrow 0}{\left(a_{\epsilon}+b_{\epsilon}\right)}=\liminf\limits_{\epsilon\rightarrow 0}{a_{\epsilon}}+b$.
Let $\mu$ and $\nu$ be two probability measures define in the measurable space $\left(\Omega,\mathcal{F}\right)$. Then,
\begin{eqnarray*}
\norm{\mu-\nu}^2\leq 2\mathcal{H}\left(\mu\left|\right.\nu\right),
\end{eqnarray*}
where $\mathcal{H}\left(\mu\left|\right.\nu\right)$ is the Kullback information of $\mu$ respect to $\nu$ and it is defined as follows: if $\mu\ll\nu$ then take the Radon-Nikodym derivative $f=\frac{d\mu}{d\nu}$ and define
$\mathcal{H}\left(\mu\left|\right.\nu\right):=\int\limits_{\Omega}{f\ln(f)d\nu}$, in the case $\mu\not\ll\nu$ let us define $\mathcal{H}\left(\mu\left|\right.\nu\right):=+\infty$.
For details check <cit.>.
FI1 Avner Friedman, Stochastic differential equations and applications, Probability and Mathematical Statistics, Volume $1$, $1975$.
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KU Hui-Hsiung Kuo, Introduction to Stochastic Integration, Springer, $2006$.
KS Ioannis Karatzas & Steven Shreve Brownian Motion and Stochastic Calculus, Springer, $2004$.
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|
1511.00191
|
In this paper we shall establish an existence and uniqueness result
for solutions of multidimensional, time dependent, stochastic differential
equations driven simultaneously by a multidimensional fractional Brownian
motion with Hurst parameter $H>1/2$ and a multidimensional standard
Brownian motion under a weaker condition than the Lipschitz one.
Keywords: Fractional Brownian motion, stochastic differential equations, weak and strong solution,
Bihari's type lemma.
§ INTRODUCTION
The fractional Brownian motion (fBm for short) $B^{H}=\{B^{H}(t) , t\in [0,T]\}$ with Hurst parameter $H\in (0,1)$ is a Gaussian self-similar process with stationary increments.
This process was introduced by Kolmogorov <cit.> and studied by Mandelbrot and Van Ness in <cit.>, where a stochastic integral representation in terms of a standard
Brownian motion (Bm for short) was established. The parameter
$H$ is called Hurst index from the
statistical analysis, developed by the climatologist Hurst <cit.>. The self-similarity and stationary increments properties make the fBm an appropriate model for many applications in diverse fields from biology to finance. From the properties of the fBm it follows that, for every $\alpha >0$
\mathbb{E}\left(|B^H(t)-B^H(s)|^{\alpha}\right) = \mathbb{E}\left(|B^H(1)|^{\alpha}\right)|t-s|^{\alpha H}.
As a consequence of the Kolmogorov continuity theorem, we deduce that there exists a version of the fBm $B^H$ which is a continuous process and whose paths are $\gamma$-Hölder continuous for every $\gamma <H$. Therefore, the fBm with Hurst parameter $H\neq \frac12$ is not a semimartingale and then the Itô approach to the construction of stochastic integrals with respect to fBm is not valid. Two main approaches have been used in the literature to define stochastic integrals with respect to fBm with Hurst parameter $H$. Pathwise Riemann-Stieltjes stochastic integrals can be defined using Young's integral <cit.> in the case $H>\frac 12$. When $H\in (\frac14, \frac12)$, the rough path analysis introduced by Lyons <cit.> is a suitable method to construct pathwise stochastic integrals.
A second approach to develop a stochastic calculus with respect to the fBm is based on the techniques of Malliavin calculus. The divergence operator, which is the adjoint of the derivative operator, can be regarded as a stochastic integral, which coincides with the limit of Riemann sums constructed using the Wick product.
This idea has been developed by
Decreusefond and Üstünel <cit.>, Carmona, Coutin and Montseny <cit.>, Alòs, Mazet and Nualart <cit.>, Alòs and Nualart <cit.> and Hu <cit.>, among others. The integral constructed by this method has zero mean.
Let $T>0$ be a fixed time and $\big(\Omega,\mathcal{F},(\mbox{\ensuremath{\mathcal{F}}}_{t})_{t\in[0,T]},P\big)$
be a given filtered complete probability space with $\left(\mbox{\ensuremath{\mathcal{F}}}_{t}\right)_{t\in[0,T]}$
being a filtration that satisfies the usual hypotheses.
The aim of this paper is to study
the following stochastic differential equation (SDE for short) on $\mathbb{R}^{n}$
\begin{equation}
X(t)=x_{0}+\int_{0}^{t}b(s,X(s))\, ds+\int_{0}^{t}\sigma_{W}(s,X(s))dW(s)+\int_{0}^{t}\sigma_{H}(s,X(s))dB^{H}(s),\label{eq:1-1}
\end{equation}
where $t\in\left[0,T\right]$, $x_{0}\in\mathbb{R}^{n},$ $W$ is
a $m$-dimensional standard $\mbox{\ensuremath{\mathcal{F}}}_{t}$-Bm
and $B^{H}$ a $d$-dimensional $\mbox{\ensuremath{\mathcal{F}}}_{t}$-adapted
fBm. The main difficulty when considering Equation (<ref>) lies in the fact that both stochastic integrals are dealt in different ways. However, the integral with respect to the Bm is an Itô integral, while
the integral with respect to the fBm has to be understood in the pathwise sense. Mixing the two integrals makes things difficult, forcing to consider very smooth coefficients to prove existence and uniqueness of solution to Equation (<ref>).
It is well known that, under suitable assumptions on the coefficients
$b,\sigma_{W},\sigma_{H}$ (see below), the Equation (<ref>)
has a unique solution which is $(H-\varepsilon)$-Hölder continuous,
for all $\varepsilon>0$. This result was first considered in <cit.>, where unique solvability was proved for time-independent
coefficients and zero drift. Later, in <cit.>, existence
of solution to (<ref>) was proved under less restrictive assumptions,
but only locally, i.e. up to a random time. In <cit.>, global existence and uniqueness of solution to the Equation (<ref>)
was established under the assumption that $W$ and $B^{H}$ are independent.
The latter result was obtained in <cit.> without the independence
assumption. We stress on the fact that all these works consider the Lipschitz case. It should be noted,
in addition, that the Lipschitz condition is the most used to
establish the pathwise uniqueness for ordinary and SDEs via the Gronwall lemma.
Thus, the following question appears naturally: are there any weaker conditions than the
Lipschitz continuity under which the SDE (<ref>) has a unique strong solution?
In order to answer the above question our approach is to prove that the Euler's polygonal approximations converge uniformly in $t\in [0,T]$, in probability, to a process, which we show to be the strong solution. The basic tools are the pathwise uniqueness for the SDE (<ref>), tightness of the sequence of the laws of Euler's approximations and the Skorokhod's embedding theorem. It is important to note that the linear growth condition and the continuity of the coefficients are sufficient for the convergence of the Stieltjes and Itô integrals. However, the integral with respect the fBm needs more regularity. To prove the convergence in probability we use an elementary result due to Gyongy and Krylov <cit.> which highlights the famous result of Yamada and Watanabe saying that pathwise uniqueness implies uniqueness in law. It is worth mentioning that the pathwise uniqueness property for the SDE (<ref>) is obtained under weak assumption than the Lipschitz condition. More precisely our conditions are based on the modulus of continuity of the coefficients that achieve pathwise
uniqueness using Bihari's type lemma.
It should be noted that such conditions are considered by many authors for the existence and uniqueness of solutions of different kind of equations where the Bihari's lemma is the cornerstone in the proof of these results.
The article is organized as follows. In Section <ref>, we state our assumptions on the coefficients $b$, $\sigma_W$ and $\sigma_H$ of Equation (<ref>), recall briefly the deterministic fractional calculus in order to define the integral with respect to fBm and introduce proper normed spaces. In addition, we give the definition of strong, weak solution and pathwise uniqueness of Equation (<ref>). In Section <ref>, the pathwise uniqueness property for the solutions of Equation (<ref>) is proved (see Theorem <ref> below). Finally, in Section <ref>, we define the Euler approximations sequence and prove that it is tight. Moreover, we show that these approximations converge in probability to a process which turns out to be a strong solution of the SDE (<ref>), cf. Theorem <ref> below. In the Appendix, we recall some technical results which play a great
role in this work. We also show a version of Bihari's lemma which will be used in the proof of pathwise uniqueness to SDE (<ref>).
§ PRELIMINARIES
Throughout this paper we assume that the coefficients $b, \sigma_W$ and $\sigma_H$, which are continuous, satisfy, for all $x,y\in\mathbb{R}^{n}$ and $t\in\left[0,T\right]$, the
following hypotheses $\mathbf{(H.1)}$ and $\mathbf{(H.2)}$:
Hypothesis $\mathbf{(H.1)}$. The functions $b$ and $\sigma_W$ have a linear growth and satisfy suitable modulus of continuity with respect to the variable $x$ uniformly in $t$.
Hypothesis $\mathbf{(H.1)}$ means that $b$ and $\sigma_W$ satisfy
\begin{eqnarray*}
({\mathbf{H.1.1}}) & & |b(t,x)|\leq K(1+|x|),\\
({\mathbf{H.1.2}}) & & |b(t,x)-b(t,y)|^{2}\leq\varrho\big(|x-y|^{2}\big)\\
({\mathbf{H.1.3}}) & & |\sigma_{W}(t,x)|\leq K(1+|x|),\\
({\mathbf{H.1.4}}) & & |\sigma_{W}(t,x)-\sigma_{W}(t,y)|^{2}\leq\varrho\big(|x-y|^{2}\big),
\end{eqnarray*}
\[
\begin{array}{l}
\mathbf{(H.1.1)}\left|\, b(t,x)\right|\leq K\left(1+\left|x\right|\right),\\
\\
\mathbf{(H.1.2)}\,\left|b(t,x)-b(t,y)\right|^{2}\leq\varrho\left(\left|x-y\right|^{2}\right)
\\ \\
\mathbf{(H.1.3)}\,\left|\sigma_{W}(t,x)\right|\leq K\left(1+\left|x\right|\right),\\
\\
\mathbf{(H.1.4)}\,\left|\sigma_{W}(t,x)-\sigma_{W}(t,y)\right|^{2}\leq\varrho\left(\left|x-y\right|^{2}\right),
\end{array}
\]
$\varrho$ is a concave increasing function from $\mathbb{R}_{+}$
to $\mathbb{R}_{+}$ such that $\varrho(0)=0$, $\varrho(u)>0$ for
$u>0$ and for some $q>1$ we have
\begin{equation}
\int_{0^{+}}\dfrac{du}{\varrho^{q}(u^{1/q})}=\infty.\label{eq:rho est}
\end{equation}
Hypothesis $\mathbf{(H.2)}$. The function $\sigma_H$ is continuously differentiable in the second variable $x$. Its derivative, with respect to $x$, is bounded, Lipschitz with respect to the same variable uniformly with respect to the first variable $t$. Moreover, both $\sigma_H$ and its derivative are $\beta$-Hölder with respect to the first variable $t$ uniformly with respect to the second variable.
Hypothesis $\mathbf{(H.2)}$ means that $\sigma_H$ and its derivative satisfy
\begin{eqnarray*}
({\mathbf{H.2.1}}) & & \left|\partial_{x_{i}}\sigma_{H}(t,x)\right|\leq K \\
({\mathbf{H.2.2}}) & & \left|\partial_{x_{i}}\sigma_{H}(t,x)-\partial_{x_{i}}\sigma_{H}(t,y)\right|\leq K\left|x-y\right| \\
({\mathbf{H.2.3}}) & & \left|\sigma_{H}(t,x)-\sigma_{H}(s,x)\right|+\left|\partial_{x_{i}}\sigma_{H}(t,x)-\partial_{x_{i}}\sigma_{H}(s,x)\right|\leq K\left|s-t\right|^{\beta}.
\end{eqnarray*}
Let us give two examples of such function $\varrho$. Let $q>1$ and $\delta$
be sufficiently small. Define
\begin{eqnarray*}
\varrho_{1}(u)&:=&\left\{ \begin{array}{ll}
u\log^{1/q}(u^{-1}), & \,\,\,0\leq u\leq\delta \\ \\
\delta\log^{1/q}(\delta^{-1})+\varrho'_1(\delta_{-})(u-\delta), & \,\,\, u>\delta.
\end{array}\right. \\ \\ \\
\varrho_{2}(u)&:=&\left\{ \begin{array}{ll}
u\log^{1/q}(u^{-1})\log^{1/q}\left(\log(u^{-1})\right), & \,\,\,0\leq u\leq\delta\\
\\
\delta\log^{1/q}(\delta^{-1})\log^{1/q}\left(\log(\delta^{-1})\right)+
\varrho'_2(\delta_{-})(u-\delta), & \,\,\, u>\delta.
\end{array}\right.
\end{eqnarray*}
\[
\varrho_{1}(u)=\left\{ \begin{array}{ll}
u\log^{1/q}(u^{-1}), & \,\,\,0\leq u\leq\delta\\
\\
\delta\log^{1/q}(\delta^{-1})+\varrho'_1(\delta_{-})(u-\delta), & \,\,\, u>\delta.
\end{array}\right.
\]
\[
\varrho_{2}(u)=\left\{ \begin{array}{ll}
u\log^{1/q}(u^{-1})\log^{1/q}\left(\log(u^{-1})\right), & \,\,\,0\leq u\leq\delta\\
\\
\delta\log^{1/q}(\delta^{-1})\log^{1/q}\left(\log(\delta^{-1})\right)+
\varrho'_2(\delta_{-})(u-\delta), & \,\,\, u>\delta.
\end{array}\right.
\]
It is easy to see that, for $i=1,2$, the function $\varrho_{i}$ is
concave nondecreasing function satisfying (<ref>).
We begin by a brief review of the deterministic fractional calculus.
We start with the definition of the integral with respect to fBm as a generalized
Lebesgue-Stieltjes integral, following the work of Zähle <cit.>. We fix $\alpha\in(0,1)$.
The Weyl-Marchaud derivatives of $f:[a,b]\longrightarrow\mathbb{R}^{n}$ are given
\[
D_{a+}^{\alpha}f(x)=\dfrac{1}{\Gamma(1-\alpha)}\left(\dfrac{f(x)}{\left(x-a\right)^{\alpha}}+\alpha\int_{a}^{x}\dfrac{f(x)-f(y)}{\left(x-y\right)^{\alpha+1}}\, dy\right)
\]
\[
D_{b-}^{\alpha}f(x)=\dfrac{\left(-1\right)^{\alpha}}{\Gamma(1-\alpha)}\left(\dfrac{f(x)}{\left(b-x\right)^{\alpha}}+\alpha\int_{x}^{b}\dfrac{f(x)-f(y)}{\left(y-x\right)^{\alpha+1}}\, dy\right)1\!\!1_{\left(a,b\right)}(x),
\]
where $\Gamma(\alpha) =\int_0^{\infty} t^{\alpha -1} e^{-t}dt$ is the Gamma function.
Assuming that $D_{a+}^{\alpha}f_{a+}\in L^{1}[a,b]$ and $D_{b-}^{1-\alpha}g_{b-}\in L^{\infty}[a,b]$,
where $g_{b-}(x)=g(x)-g(b-)$, the generalized (fractional) Lebesgue-Stieltjes
integral of $f$ with respect to $g$ is defined as
\begin{equation}
\int_{a}^{b}f\, dg:=(-1)^{\alpha}\int_{a}^{b}\, D_{a+}^{\alpha}f(x)\, D_{b-}^{1-\alpha}g_{b-}(x)\, dx.\label{eq:frac int}
\end{equation}
If $a\leq c<d\leq b$ then we have
\[
\int_{c}^{d}f\, dg=\int_{a}^{b}1\!\!1_{(c,d)}f\, dg.
\]
It follows from the Hölder continuity of $B^{H}$ that $D_{b-}^{1-\alpha}B_{b-}^{H}\in L^{\infty}[a,b]$
almost surely (a.s. for short). Then, for a function $f$ with $D_{a+}^{\alpha}f\in L^{1}[a,b]$,
we can define the integral with respect to $B^{H}$ through (<ref>).
Let $0<\alpha<1/2$ and $\mu\in(0,1]$. We will consider the following normed spaces:
* $C^{\mu}$ is the space of $\mu$-Hölder continuous functions
$f:[0,T]\rightarrow\mathbb{R}^{d}$, equipped with the norm
\[
\|f\|_{\mu}:=\|f\|_{\infty}+\underset{0\leq s<t\leq T}{\sup}\dfrac{\left|f(t)-f(s)\right|}{\left(t-s\right)^{\mu}}<\infty,
\]
\[
\|f\|_{\infty}:=\underset{0\leq t\leq T}{\sup}\left|f(t)\right|.
\]
* $C_{0}^{\mu}$ denotes the space of $\mu$-Hölder continuous functions $f:[0,T]\longrightarrow \mathbb{R}^d$
such that
\[
\lim_{\varepsilon\rightarrow0}\left(\sup_{0<|t-s|<\varepsilon}\frac{|f(t)-f(s)|}{(t-s)^{\mu}}\right)=0.
\]
We note that $C_{0}^{\mu}$ is complete and separable with respect to the norm $\| \cdot \|_{\mu}$.
* $W_{0}^{\alpha,\infty}$ is the space of measurable functions $f:[0,T]\longrightarrow\mathbb{R}^{d}$
such that
\[
\|f\|_{\alpha,\infty}:=\underset{0\leq t\leq T}{\sup}\|f\|_{\alpha,t}<\infty,
\]
\[
\|f\|_{\alpha,t}:=\left|f(t)\right|+\int_{0}^{t}\dfrac{\left|f(t)-f(s)\right|}{\left(t-s\right)^{\alpha+1}}\, ds.
\]
* Finally, $W_{T}^{1-\alpha,\infty}$ denotes the space of measurable
functions $f:[0,T]\longrightarrow\mathbb{R}^{m}$ such that
\[
\|f\|_{1-\alpha,\infty,T}:=\underset{0\leq t\leq T}{\sup}\|f\|_{1-\alpha,\infty,t}<\infty,
\]
\[
\| f\|_{1-\alpha,\infty,t}:=\sup_{0\leq u<v<t}\left(\frac{\left|f(v)-f(u)\right|}{\left(v-u\right)^{1-\alpha}}+\int_{u}^{v}\dfrac{\left|f(y)-f(u)\right|}{\left(y-u\right)^{2-\alpha}}dy\right).
\]
Hence, it is clear that
\[
\underset{0\leq u<v<t}{\sup}\left|D_{v-}^{1-\alpha}B_{v-}^{H}(u)\right|\leq\dfrac{1}{\Gamma(\alpha)}\| B^{H} \|_{1-\alpha,\infty,t}<\infty,
\]
where the last inequality is a consequence of that fact that the random variable $\| B^{H}\|_{1-\alpha,\infty,t}$
has moments of all orders, see Lemma 7.5 in Nualart and Rascanu <cit.>.
Thus, the stochastic integral with respect to the fBm admits the following estimate
\begin{equation}
\left|\int_{0}^{t}f(s)\, dB^{H}(s)\right|
\leq \frac{1}{\Gamma(\alpha)}\| B^{H}\|_{1-\alpha,\infty,t}\| f\|_{\alpha,1,t}, \label{frac int est}
\end{equation}
\[
\| f \|_{\alpha,1,t}:=\int_{0}^{t}\dfrac{\left|f(s)\right|}{s^{\alpha}}\,ds+\int_{0}^{t}\int_{0}^{s}\dfrac{\left|f(s)-f(y)\right|}{\left(s-y\right)^{\alpha+1}}\, dy\,ds.
\]
We give the definition of strong and weak solution as well as pathwise uniqueness for Equation (<ref>).
By a strong solution of Equation (<ref>) we mean an $\mathcal{F}_{t}$-adapted
continuous process $X(t),t\in[0,T]$ such that there exists an increasing
sequence of stopping times $(T_{R})_{R>0}$ satisfying $\lim_{R\rightarrow\infty}T_{R}=T$
a.s. and for any $R>0$, we have
* $
\sup_{t\in[0,T]}\mathbb{E}\left[\|X(t\wedge T_{R})\|_{\alpha,t}^{2}\right]<\infty.\label{eq:C1}
* The equation
\begin{eqnarray}
X(t\wedge T_{R}) &=& x_{0}+\int_{0}^{t\wedge T_{R}}b\big(s,X(s)\big)\, ds+\int_{0}^{t\wedge T_{R}}\sigma_{W}\big(s,X(s)\big)\, dW(s)\nonumber \\
& & +\int_{0}^{t\wedge T_{R}}\sigma_{H}\big(s,X(s)\big)\, dB^{H}(s),\label{eq:C2}
\end{eqnarray}
holds a.s..
By a weak solution of Equation (<ref>) we mean a triplet $(X, W, B^H)$, $\big(\Omega,\mathcal{F}, P\big)$ and $(\mathcal{F}_{t})_{t\in[0,T]}$, such that
* $\big(\Omega,\mathcal{F}, P\big)$ is a probability space, and $(\mathcal{F}_{t})_{t\in[0,T]}$ is a filtration, of sub-$\sigma$-algebra of $\mathcal{F}$, satisfying the usual conditions.
* $W= (W_t, \mathcal{F}_{t})_{t\in[0,T]}$ is a Bm, $B^H= (B^H_t)_{t\in[0,T]}$ is a fBm and $X= (X_t, \mathcal{F}_{t})_{t\in[0,T]}$ is a continuous and $\mathcal{F}_{t}$-adapted process satisfying a.s. the Equation (<ref>) for some increasing
sequence of stopping times $(T_{R})_{R>0}$ such that $\lim_{R\rightarrow\infty}T_{R}=T$
We say that pathwise uniqueness holds for Equation (<ref>)
if, whenever $(X,W,B^{H})$ and $(\tilde{X},W,B^{H})$
are two weak solutions of Equation (<ref>) defined on the
same probability space $\big(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},P\big)$
then $X$ and $\tilde{X}$ are indistinguishable.
Under assumption $(H)$ the SDE <ref> has a unique solution
in the class of processes satisfying
\begin{equation}
\underset{t\in\left[0,T\right]}{\sup}\|X_{t}\|_{\alpha}\leq C\qquad a.s.\label{eq:norm estim}
\end{equation}
Under assumption we have
\[
\underset{0\leq s\leq t}{\sup}\mathbb{E}\left(\|X_{t}\|_{\alpha}^{2N}\right)\leq K_{N},
\]
for all $t\in[0,T]$ and for all $N\in\mathbb{N}$.
§ PATHWISE UNIQUENESS
In this section we investigate the pathwise uniqueness of a solution for Equation (<ref>), cf. Theorem <ref> below, where we make use of the so-called Bihari's type lemma (see Lemma <ref> in Appendix).
Let $X$ be a solution of Equation (<ref>).
For $R>0$, we define the following stopping time
\[
T_{R} := \inf\Big\{ t\geq0,\left\Vert B^{H}\right\Vert _{1-\alpha,\infty,t}\geq R\Big\} \wedge T,
\]
For every positive constant $R$, we define the stochastic processes $X_R$ by
\[
X_{R}(t):=X(t\wedge T_{R}),\quad t\in [0,T].
\]
Then it is easy to see that the following equation
\begin{eqnarray*}
X_{R}(t) & =x_{0}+ & \int_{0}^{t\wedge T_{R}}b(s,X(s))\, ds+\int_{0}^{t\wedge T_{R}}\sigma_{W}(s,X(s))dW(s)\\
\\
& & +\int_{0}^{t\wedge T_{R}}\sigma_{H}(s,X(s))dB^{H}(s)
\end{eqnarray*}
holds almost surely. We have the following Lemma.
For any integer $N\geq1$ and $R>0$, there exists a positive constant $C_{N}$ such that
\[
\sup_{t\in[0,T]}\mathbb{E}\left[\|X_{R}\|_{\alpha,t}^{2N}\right]\leq C_{N}R^{2N}.
\]
Along the proof $C_N$ will denote a generic positive constant, which may vary from line to line and may depend on $N$ and other parameters of the problem. It follows from the convexity of $x^{2N}$ that
\begin{eqnarray*}
\mathbb{E}\big[\|X_{R}\|_{\alpha,t}^{2N}\big] & \leq & C_{N}\left\{ \left|x_{0}\right|^{2N}+\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}b(s,X(s))\, ds\right\|_{\alpha,t}^{2N}\right]\right.\\
& & +\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}\sigma_{W}(s,X(s))\, dW(s)\right\|_{\alpha,t}^{2N}\right]\\
& & +\left.\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}\sigma_{H}(s,X(s))\, dB^{H}(s)\right\|_{\alpha,t}^{2N}\right]\right\} \\
& = & C_{N}\big(\left|x_{0}\right|^{2N}+A_{1}+A_{2}+A_{3}\big).
\end{eqnarray*}
Furthermore we have
\begin{eqnarray*}
&&\left\|\int_{0}^{.\wedge T_{R}}b(s,X(s))\, ds\right\|_{\alpha,t} \\ & &\leq \int_{0}^{t\wedge T_{R}}\left|b(s,X(s))\right|\, ds+\int_{0}^{t}(t-s)^{-\alpha-1}\int_{s\wedge T_{R}}^{t\wedge T_{R}}\left|b(u,X(u))\right|\, du\, ds\\
\\
&& \leq \int_{0}^{t}\left|b(s\wedge T_{R},X(s\wedge T_{R}))\right|\, ds\\
& & +\int_{0}^{t}(t-s)^{-\alpha-1}\int_{s}^{t}\left|b(u\wedge T_{R},X(u\wedge T_{R}))\right|\, du\, ds\\
\\
& &\leq \int_{0}^{t}\left|b(s\wedge T_{R},X(s\wedge T_{R}))\right|\, ds\\
& & +\dfrac{1}{\alpha}\int_{0}^{t}(t-r)^{-\alpha}\left|b(r\wedge T_{R},X(r\wedge T_{R}))\right|\, dr\\
\\
& &\leq C_{\alpha,T}\int_{0}^{t}(t-r)^{-\alpha}\left|b(r\wedge T_{R},X(r\wedge T_{R}))\right|\, dr
\end{eqnarray*}
where $C_{\alpha,T}$ is a constant depending on $\alpha$ and $T$.
Using the linear growth assumption in (H.1.1), Hölder's inequality and the fact that $\alpha <\frac12$, we obtain
\begin{eqnarray*}
A_{1} & \leq & C_{N}\,\mathbb{E}\left[\left(1+\int_{0}^{t}\dfrac{\left|X_{R}(s)\right|}{(t-s)^{\alpha}}\, ds\right)^{2N}\right]\\
& \leq & C_{N}\,\mathbb{E}\left[\left(1+\int_{0}^{t}\left|X_{R}(s)\right|^{2}\, ds\right)^{N}\right]\\
& \leq & C_{N}\,\left(1+\int_{0}^{t}\mathbb{E}\left[\left|X_{R}(s)\right|^{2N}\right]\, ds\right).
\end{eqnarray*}
We have also that
\begin{eqnarray*}
A_{2} & \leq & C_{N}\mathbb{E}\left[\left|\int_{0}^{t\wedge T_{R}}\sigma_{W}(s,X(s))dW(s)\right|^{2N}\right]
\\ &&+C_{N}\mathbb{E}\left[\left(\int_{0}^{t}{(t-s)^{-\alpha-1}}{\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(u,X(u))\, dW(u)\right|}\, ds\right)^{2N}\right]\\
\\
& = & A_{21}+A_{22}.
\end{eqnarray*}
For $A_{21}$, using the linear growth assumption in $\mathbf{(H.1.3)}$, the Burkhölder
and Hölder inequalities, we obtain
\begin{eqnarray*}
A_{21} & \leq & C_{N}\mathbb{E}\left[\int_{0}^{t\wedge T_{R}}\left|\sigma_{W}(s,X(s))\right|^{2N}ds\right]\\
\\
& \leq & C_{N}\mathbb{E}\left[\int_{0}^{t}\left|\sigma_{W}(s\wedge T_{R},X(s\wedge T_{R}))\right|^{2N}ds\right]\\
\\
& \leq & C_{N}\left(1+\int_{0}^{t}\mathbb{E}\left[\left|X_{R}(s)\right|^{2N}\right]ds\right)
\end{eqnarray*}
For $A_{22}$, again the Burkhölder and Hölder inequalities give
\begin{eqnarray*}
A_{22} & \leq & C_{N}\left(\int_{0}^{t}\dfrac{ds}{(t-s)^{\alpha+\frac{1}{2}}}\right)^{2N-1}\int_{0}^{t}{(t-s)^{-\alpha-\frac{1}{2}-N}}{\mathbb{E}\left[\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(u,X(u))\, dW(u)\right|^{2N}\right]}ds\\
& \leq & C_{N}\dint_{0}^{t}{(t-s)^{-\alpha-\frac{3}{2}}}{\mathbb{E}\left[\int_{s\wedge T_{R}}^{t\wedge T_{R}}\left|\sigma_{W}(u,X(u))\right|^{2N}\, du\right]} ds\\
& \leq & C_{N}\int_{0}^{t}{(t-s)^{-\alpha-\frac{3}{2}}}{\mathbb{E}\left[\int_{s}^{t}\left|\sigma_{W}(u\wedge T_{R},X(u\wedge T_{R}))\right|^{2N}\, du\right]}\, ds.
\end{eqnarray*}
Applying now Fubini’s theorem and using the growth
assumption in $\mathbf{(H.1.3)}$, we obtain
\[
A_{22}\leq C_{N}\left(\int_{0}^{t}(t-s)^{-\alpha-\frac{1}{2}}\left(1+\mathbb{E}\left[\left|X_{R}(s)\right|^{2N}\right]\right)\, ds\right).
\]
\[
A_{2}\leq C_{N}\left(1+\int_{0}^{t}(t-s)^{-\alpha-\frac{1}{2}}\mathbb{E}\left[\left|X_{R}(s)\right|^{2N}\right]\, ds\right).
\]
Let us remark that, for $t\in [0,T]$, we have
\begin{equation}
\int_{0}^{t\wedge T_{R}}\sigma_{H}(s,X(s))\, dB^{H}(s)=\int_{0}^{t}\sigma_{H}(s\wedge T_{R},X(s\wedge T_{R}))\, dB^{H}(s\wedge T_{R}).\label{stopfbmint}
\end{equation}
Then it follows from Proposition <ref> (jj), in the Appendix,
\[
A_{3}\leq C_{N}R^{2N}\int_{0}^{t}\left((t-s)^{-2\alpha}+s^{-\alpha}\right)\left(1+\mathbb{E}\left[\|X_{R}\|_{\alpha,s}^{2N}\right]\right)\, ds.
\]
Putting all the estimates obtained for $A_{1}$, $A_{2}$
and $A_{3}$ together, we obtain
\begin{equation}
\mathbb{E}\left[\|X_{R}\|_{\alpha,t}^{2N}\right] \leq C_{N}\left|x_{0}\right|^{2N}+C_{N}(1+R^{2N}) \int_{0}^{t}\varphi(t,s)\mathbb{E}\left[\|X_{R}\|_{\alpha,s}^{2N}\right]\, ds, \label{eq:est esp-1}
\end{equation}
\begin{eqnarray*}
\varphi(t,s) & := & s^{-\alpha}+(t-s)^{-\alpha-1/2}.
\end{eqnarray*}
Therefore, since the right hand side of Equation (<ref>)
is an increasing function of $t$, we have
\begin{eqnarray*}
\sup_{0\leq s\leq t}\mathbb{E}\big[\|X_{R}\|_{s}^{2N}\big] & \leq & C_{N}\left|x_{0}\right|^{2N}+C_{N}\big(1+R^{2N}\big) \int_{0}^{t}\varphi(s,t)\sup_{0\leq u\leq s}\mathbb{E}\big[\|X_{R}\|_{\alpha,u}^{2N}\big]\, ds.
\end{eqnarray*}
As a consequence, by the Gronwall type lemma (Lemma 7.6 in <cit.>),
we deduce the desired estimate.
Let $X$ and $Y$ be two solutions of Equation (<ref>) defined on the
same probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t\in[0,T]}),P)$.
For $M>0$, we define the following stopping time
\[
\tau_{M} := \inf\big\{t:\|X\|_{\alpha,t} \vee \|Y\|_{\alpha,t}>M \big\}\wedge T.
\]
Now for every positive constants $R$ and $M$, we define the stochastic processes $X_{R,M}$ (resp. $Y_{R,M}$) by
\[
X_{R,M}(t):=X(t\wedge T_{R}\wedge\tau_{M}),\quad t\in [0,T],
\]
(resp. $ Y_{R,M}(t):=Y(t\wedge T_{R}\wedge\tau_{M}),\quad t\in [0,T] $).
Under Hypotheses $\mathbf{(H.1)}$ and $\mathbf{(H.2)}$, there exists a positive constant $C_{R,M}$
such that for $t\in[0,T]$,
\begin{multline}
\mathbb{E}\left[\|X_{R,M}-Y_{R,M}\|_{\alpha,t}^{2}\right] \\
\leq C_{R,M}\int_{0}^{t}\varphi(s,t)\Big[\mathbb{E}\left[\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right]
\label{eq:differ est}
+\varrho\left(\mathbb{E}\left[\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right]\right)\Big]\, ds.
\end{multline}
The proof of this result is long and technical. It is divided into
several parts. First we have
\begin{eqnarray*}
X_{R,M}(t)-Y_{R,M} (t)& = & \int_{0}^{t\wedge T_{R}\wedge\tau_{M}}(b(s,X(s))-b(s,Y(s)))\, ds\\
& &+\int_{0}^{t\wedge T_{R}\wedge\tau_{M}}(\sigma_{W}(s,X(s))-\sigma_{W}(s,Y(s)))\, dW(s)\\
& & +\int_{0}^{t\wedge T_{R}\wedge\tau_{M}}(\sigma_{H}(s,X(s))-\sigma_{H}(s,Y(s)))\, dB^{H}(s)\\
\\
& = & B_{1}(t\wedge T_{R}\wedge\tau_{M})+B_{2}(t\wedge T_{R}\wedge\tau_{M})+B_{3}(t\wedge T_{R}\wedge\tau_{M}).
\end{eqnarray*}
It follows that
\begin{eqnarray*}
&&\|X_{R,M}-Y_{R,M}\|_{\alpha,t}^{2}\\&& \leq 3\left(\|B_{1}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2} +\|B_{2}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2} +\|B_{3}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2} \right).
\end{eqnarray*}
We have to estimated $\|B_{i}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2}$, $i\in\{1,2,3\}$. For the sake of conciseness, we define
$$\Delta(f)(s)=f(s,X(s))-f(s,Y(s)), \quad f\in \{b,\sigma_{W},\sigma_{H}\}.$$
Step 1: $B_1$. Using simple estimations it is easy to see that
\begin{eqnarray*}
\|B_{1}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t} &\leq & \int_{0}^{t\wedge T_{R}\wedge\tau_{M}}\left|\Delta(b)(s)\right|\, ds\\
& +& \int_{0}^{t}(t-s)^{-\alpha-1}\int_{s\wedge T_{R}\wedge\tau_{M}}^{t\wedge T_{R}\wedge\tau_{M}}\left|\Delta(b)(u)\right|\, du\, ds
\\
& \leq & \int_{0}^{t}\left|\Delta(b)(s\wedge T_{R}\wedge\tau_{M})\right|\, ds \\
& +&\int_{0}^{t}(t-s)^{-\alpha-1}\int_{s}^{t}\left|\Delta(b)(u\wedge T_{R}\wedge\tau_{M})\right|\, du\, ds\\
& \leq &
C_{\alpha, T}\int_{0}^{t}(t-r)^{-\alpha}\left|\Delta(b)(r\wedge T_{R}\wedge\tau_{M})\right|\, dr.
\end{eqnarray*}
We use the fact that $\alpha<\frac{1}{2}$, Hölder inequality and hypothesis $\mathbf{(H.1.2)}$
to obtain
\begin{eqnarray*}
\|B_{1}(.\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2}
& \leq & C_{\alpha, T}^{2}\int_{0}^{t}\dfrac{\left|\Delta(b)(s\wedge T_{R}\wedge\tau_{M})\right|^{2}}{\left(t-s\right)^{\alpha}}\, ds\\
& \leq & C_{\alpha, T}^{2}\int_{0}^{t}\dfrac{\varrho\left(|X_{R,M}(s)-Y_{R,M}(s)|^{2}\right)}{\left(t-s\right)^{\alpha}}\, ds\\
& \leq & C_{\alpha, T}^{2}\int_{0}^{t}\varphi(s,t)\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right)\, ds.
\end{eqnarray*}
Step 2: $B_3$.
If $1-H<\alpha<\min\left(\beta,1/2\right)$ , we have from Proposition
4.3 in <cit.> (see Proposition <ref> (ii)
in the Appendix)
\begin{multline*}
\|B_{3}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2} \leq C R^{2} \bigg(\int_{0}^{t}\left((t-s)^{-2\alpha}+s^{-\alpha}\right)\|\Delta(\sigma_{H})(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,s}\, ds\bigg)^{2}.
\end{multline*}
Now using the assumptions $\mathbf{(H.2)}$ and Lemma 7.1 in Nualart
Rascanu <cit.> we obtain
\begin{eqnarray*}
& & \left|\sigma_{H}(t,x_{1})-\sigma_{H}(s,x_{2})-\sigma_{H}(t,y_{1})+\sigma_{H}(s,y_{2})\right| \\
& \leq & K\left|x_{1}-x_{2}-y_{1}+y_{2}\right|+K\left|x_{1}-y_{1}\right|\left|t-s\right|^{\beta}\\
& & +K\left|x_{1}-y_{1}\right|\left(\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|\right).
\end{eqnarray*}
\begin{eqnarray*}
& &\bigg|\sigma_{H}(t\wedge T_{R}\wedge\tau_{M},X_{R,M}(t))-\sigma_{H}(s\wedge T_{R}\wedge\tau_{M},X_{R,M}(s))\\
\\& &-\sigma_{H}(t\wedge T_{R}\wedge\tau_{M},Y_{R,M}(t))+\sigma_{H}(s\wedge T_{R}\wedge\tau_{M},Y_{R,M}(s))\bigg| \\
\\ &\leq & K\Big[\left|X_{R,M}(t)-X_{R,M}(s)-Y_{R,M}(t)+Y_{R,M}(s)\right|+K\left|X_{R,M}(t)-Y_{R,M}(t)\right|\left|t-s\right|^{\beta}\\
\\ & & +\left|X_{R,M}(t)-Y_{R,M}(t)\right|\left(\left|X_{R,M}(t)-X_{R,M}(s)\right|+\left|Y_{R,M}(t)-Y_{R,M}(s)\right|\right)\Big].
\end{eqnarray*}
Thus we have
\begin{eqnarray*}
& &\|\Delta(\sigma_{H})(.\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t} \\
\\
& \leq & K\left[\left|X_{R,M}(t)-Y_{R,M}(t)\right|+\int_{0}^{t}\dfrac{\left|X_{R,M}(t)-X_{R,M}(s)-Y_{R,M}(t)+Y_{R,M}(s)\right|}{\left(t-s\right)^{\alpha+1}}\, ds\right.\\
& & +\left|X_{R,M}(t)-Y_{R,M}(t)\right|\left(\int_{0}^{t}\dfrac{ds}{\left(t-s\right)^{\alpha-\beta+1}}+\int_{0}^{t}\dfrac{\left|X_{R,M}(t)-X_{R,M}(s)\right|}{\left(t-s\right)^{\alpha+1}}\, ds\right.\\
& & \qquad\qquad\qquad\qquad\qquad\qquad+\left.\left.\int_{0}^{t}\dfrac{\left|Y_{R,M}(t)-Y_{R,M}(s)\right|}{\left(t-s\right)^{\alpha+1}}\, ds\right)\right].
\end{eqnarray*}
Now it is easy to see that
\begin{eqnarray*}
& & \|B_{3}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2} \\
& \leq &CR^{2}\int_{0}^{t}\left((t-s)^{-2\alpha}+s^{-\alpha}\right)\left(1+\|X_{R,M}\|_{\alpha,s}^{2}+\|Y_{R,M}\|_{\alpha,s}^{2}\right)\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\, ds\\
& \leq & CR^{2}M^{2}\int_{0}^{t}\varphi(s,t)\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\, ds.
\end{eqnarray*}
Spet 3: $B_2$.
Till now we have made estimates for pathwise integrals.
As $B_{2}$ is a stochastic integral we need to use martingale type
inequality. First we have
\[
\|B_{2}(\cdot\wedge T_{R}\wedge\tau_{M})\|_{\alpha,t}^{2}
\leq 2\left(|B_{2}(t\wedge T_{R}\wedge\tau_{M})|^{2}+\big(\tilde{B}_2(t)\big)^{2}\right),
\]
\[
\tilde{B}_{2}(t):=\int_{0}^{t}\frac{\left|\int_{s\wedge T_{R}\wedge\tau_{M}}^{t\wedge T_{R}\wedge\tau_{M}}\Delta(\sigma_{W})(u)\, dW(u)\right|}{(t-s)^{\alpha+1}}\, ds.
\]
It then follows from Burkhölder inequality and assumption $\mathbf{(H.1.4)}$
\begin{eqnarray*}
\mathbb{E}\big(|B_{2}(t\wedge T_{R}\wedge\tau_{M})|^{2}\big)
&\leq & \mathbb{E}\left(\int_{0}^{t}|\Delta(\sigma_{W})(s\wedge T_{R}\wedge\tau_{M})|^{2}\, ds\right)\\
&\leq & C\,\mathbb{E}\left(\int_{0}^{t}\varrho\left(|X_{R,M}(s)-Y_{R,M}(s)|^{2}\right)\, ds\right)\\
&\leq & C\,\mathbb{E}\left(\int_{0}^{t}\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right)\, ds\right).
\end{eqnarray*}
$\tilde{B_{2}}$: Using Hölder's inequality and Fubini's theorem we have
\begin{eqnarray*}
\mathbb{E}\left[\big|\tilde{B}_{2}(t)\big|^{2}\right]
& \leq & C\,\mathbb{E}\left[\int_{0}^{t}{(t-s)^{-\frac{3}{2}-\alpha}}{\left|\int_{s\wedge T_{R}\wedge\tau_{M}}^{t\wedge T_{R}\wedge\tau_{M}}\Delta(\sigma_{W})(u)\, dW(u)\right|^{2}}\, ds\right]\\
\\
& \leq & C\int_{0}^{t}{(t-s)^{-\frac{3}{2}-\alpha}}{\mathbb{E}\left[\left|\int_{s\wedge T_{R}\wedge\tau_{M}}^{t\wedge T_{R}\wedge\tau_{M}}\Delta(\sigma_{W})(u)\, dW(u)\right|^{2}\right]} ds.
\end{eqnarray*}
Using the same techniques as in the estimation of $I_{2}$ we have
\begin{eqnarray*}
\mathbb{E}\left[\left|\int_{s\wedge T_{R}\wedge\tau_{M}}^{t\wedge T_{R}\wedge\tau_{M}}\Delta(\sigma_{W})(u)\, dW(u)\right|^{2}\right]
& \leq & \mathbb{E}\left[\int_{s}^{t}|\Delta(\sigma_{W})(u\wedge T_{R}\wedge\tau_{M})|^{2}\, du\right]\\
& \leq & C\,\mathbb{E}\left[\int_{s}^{t}\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,u}^{2}\right)du\right].
\end{eqnarray*}
Then, it follows that
\begin{eqnarray*}
\mathbb{E}\left[\big|\tilde{B}_{2}(t)\big|^{2}\right]
& \leq & C\int_{0}^{t}{(t-s)^{-\frac{3}{2}-\alpha}}{\mathbb{E}\left[\int_{s}^{t}\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,u}^{2}\right)du\right]}ds.
\end{eqnarray*}
\[
\mathbb{E}\big[\|B_{2}\|_{\alpha,t}^{2}\big]\\
\leq C\int_{0}^{t}\varphi(s,t)\mathbb{E}\left[\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right)\right]ds.
\]
Step 4:
Combining all estimates, leads to
\begin{eqnarray*}
&& \mathbb{E}\left[\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right]
\\ && \leq
C_{M,R}\int_{0}^{t}\varphi(s,t)\mathbb{E}\Big[\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2} +\varrho\left(\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right)\Big]ds.
\end{eqnarray*}
Since $\varrho$ is concave, Jensen's inequality gives
\begin{eqnarray*}
\mathbb{E}\left[\|X_{R,M}-Y_{R,M}\|_{\alpha,s}^{2}\right]
\\ && \leq
\end{eqnarray*}
This concludes the proof.
Let $1-H<\alpha<\min\left(\beta,1/2\right)$.
Then, under hypotheses $\mathbf{(H.1)}$ and $\mathbf{(H.2)}$, the pathwise uniqueness property
holds for Equation (<ref>).
It is simple to see that the function $\tilde{\varrho}(u)=u+\varrho(u)$
is a concave increasing function from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$
such that $\tilde{\varrho}(0)=0$ and $\tilde{\varrho}(u)>0$ for
$u>0$. On the other hand, we have $\varrho(u)\geq\varrho(1)u$ for
$0\leq u\leq1$. Then
\[
\int_{0^{+}}\dfrac{du}{\tilde{\varrho}^{q}(u^{1/q})}\geq\left(\dfrac{\varrho(1)}{1+\varrho(1)}\right)^{q}\int_{0^{+}}\dfrac{du}{\varrho^{q}(u^{1/q})}=\infty.
\]
Therefore, the condition (<ref>) is satisfied for the
function $\tilde{\varrho}$. Consequently, we can apply Lemma <ref>
in the Appendix to the inequality (<ref>) to obtain
\[
\|X_{R,M}-Y_{R,M}\|_{\alpha,t}^{2}=0,\,\, a.s.
\]
This implies $X(t)=Y(t)$ a.s. for all $t< T_{R}\wedge\tau_{M}$.
By letting $M\rightarrow\infty$ we get, by Lemma <ref>,
$X(t)=Y(t)\,\, a.s.$ for all $t<T_{R}$. Using the that fact that the random variable $\| B^{H}\|_{1-\alpha,\infty,t}$
has moments of all orders, see Lemma 7.5 in Nualart and Rascanu <cit.>, it is not difficult that almost surely $T_{R} = T$ for $R$ large enough. This concludes the proof.
§ EULER APPROXIMATION SCHEME
In this section, we apply the Euler approximation procedure in order to obtain a weak solution of
Equation (<ref>). Under the condition that pathwise uniqueness holds for Equation (<ref>) we prove that the Euler approximation converges to a process which is a strong solution of the SDE (<ref>), see Theorem <ref> below.
Let $0=t_{0}^{n}<t_{1}^{n}<\cdots<t_{i}^{n}<\cdots<t_{n}^{n}=T$
be a sequence of partitions of $[0,T]$ such that
\[
\underset{0\leq i\leq n-1}{\sup}\left|t_{i+1}^{n}-t_{i}^{n}\right|\rightarrow 0,\quad \mbox{as}\quad n\rightarrow\infty.
\]
We define Euler's approximations as the
process $X^{n}$, $n\in\mathbb{N}$, satisfying
\begin{eqnarray}
X^{n}(t)& = & x_{0}+\int_{0}^{t}b(k_{n}(s),X(k_{n}(s)))\, ds+\int_{0}^{t}\sigma_{W}(k_{n}(s),X(k_{n}(s)))\, dW(s)\nonumber \\
& &+\int_{0}^{t}\sigma_{H}(k_{n}(s),X(k_{n}(s)))\, dB^{H}(s),
\label{eq:app sch}
\end{eqnarray}
where $k_{n}(t):=t_{i}^{n}$ if $t\in\left[t_{i}^{n},t_{i+1}^{n}\right)$ and $t\in [0,T]$.
For every positive constant $R$ we define the family of stochastic
processes by
\[
X_{R}^{n}(t):=X^{n}(t\wedge T_{R}), \quad t\in [0,T].
\]
Then it is easy to see that the process $X_{R}^{n}$
satisfies, a.s., the following
\begin{eqnarray*}
X_{R}^{n}(t) & =x_{0}+ & \int_{0}^{t\wedge T_{R}}b(k_{n}(s),X^{n}(k_{n}(s)))\, ds+\int_{0}^{t\wedge T_{R}}\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\, dW(s)\\
& & +\int_{0}^{t\wedge T_{R}}\sigma_{H}(k_{n}(s),X^{n}(k_{n}(s)))\, dB^{H}(s).
\end{eqnarray*}
We obtain for any integer $N\geq1$
Suppose that Assumptions $\mathbf{(H.1)}$ and $\mathbf{(H.2)}$ hold. Then, for all $n\in\mathbb{N}$, $N\in\mathbb{N}^*$ and $R>0$, there exists a positive constant $C_{N,R}$ such that
\begin{equation}
\sup_{t\in[0,T]}\mathbb{E}\left[\|X_{R}^{n}\|_{\alpha,t}^{2N}\right]\leq C_{N,R}. \label{app est}
\end{equation}
Moreover, we also have for all $s,t\in[0,T]$,
\begin{equation}
\mathbb{E}\left[\left|X_{R}^{n}(t)-X_{R}^{n}(s)\right|^{2N}\right]\leq C_{N,R}\left|t-s\right|^{N}. \label{app inc est}
\end{equation}
It follows from the convexity of $x^{2N}$ that
\begin{eqnarray*}
\mathbb{E}\left[\|X_{R}^{n}\|_{\alpha,t}^{2N}\right] & \leq & C_{N}\left\{ \left|x_{0}\right|^{2N}+\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}b(k_{n}(s),X^{n}(k_{n}(s)))\, ds\right\|_{\alpha,t}^{2N}\right]\right.\\
& & +\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))dW(s)\right\|_{\alpha,t}^{2N}\right]\\
& & +\left.\mathbb{E}\left[\left\|\int_{0}^{\cdot\wedge T_{R}}\sigma_{H}(k_{n}(s),X^{n}(k_{n}(s)))dB^{H}(s)\right\|_{\alpha,t}^{2N}\right]\right\} \\
& = & C_{N}\left(\left|x_{0}\right|^{2N}+I_{1}+I_{2}+I_{3}\right).
\end{eqnarray*}
Using the same estimations as in the proof of Lemma <ref>, we obtain
\begin{eqnarray*}
I_{1} & \leq & C_{N}\left(1+\int_{0}^{t}\mathbb{E}\left[\left|X^{n}(k_{n}(s)\wedge T_{R})\right|^{2N}\right]ds\right)\\
& \leq & C_{N}\left(1+\int_{0}^{t}\mathbb{E}\left[\left|X_{R}^{n}(k_{n}(s))\right|^{2N}\right]ds\right).
\end{eqnarray*}
\begin{eqnarray*}
I_{2} & \leq & C_{N}\mathbb{E}\left[\left|\int_{0}^{t\wedge T_{R}}\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\,dW(s)\right|^{2N}\right] \\
& & + C_{N}\mathbb{E}\left[\left(\int_{0}^{t}{(t-s)^{-\alpha-1}}{\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\, dW(u)\right|}\, ds\right)^{2N}\right]\\
\\
& = & I_{21}+I_{22}.
\end{eqnarray*}
For $I_{21}$, using the linear growth assumption in $\mathbf{(H.1.3)}$, Burkhölder's
and Hölder's inequalities, we obtain
\begin{eqnarray*}
I_{21} & \leq & C_{N}\mathbb{E}\left[\int_{0}^{t\wedge T_{R}}\left|\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\right|^{2N}ds\right]\\
\\
& \leq & C_{N}\mathbb{E}\left[\int_{0}^{t}\left|\sigma_{W}(k_{n}(s)\wedge T_{R},X^{n}(k_{n}(s)\wedge T_{R}))\right|^{2N}ds\right]\\
\\
& \leq & C_{N}\left(1+\int_{0}^{t}\mathbb{E}\left[\left|X_{R}^{n}(k_{n}(s))\right|^{2N}\right]\, ds\right).
\end{eqnarray*}
For $I_{22}$ , again the Burkhölder and Hölder inequalities give
\begin{eqnarray*}
I_{22} &
\leq & C_{N}\left(\int_{0}^{t}\dfrac{ds}{(t-s)^{\alpha+\frac{1}{2}}}\right)^{2N-1} \\
&& \times \int_{0}^{t}{(t-s)^{-\alpha-\frac{1}{2}-N}}{\mathbb{E}\left[\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\, dW(u)\right|^{2N}\right]}ds\\
& \leq & C_{N}\int_{0}^{t}{(t-s)^{-\alpha-\frac{3}{2}}}{\mathbb{E}\left[\int_{s\wedge T_{R}}^{t\wedge T_{R}}\left|\sigma_{W}(k_{n}(s),X^{n}(k_{n}(s)))\right|^{2N}du\right]} ds\\
& \leq & C_{N}\int_{0}^{t}(t-s)^{-\alpha-\frac{3}{2}}}{\mathbb{E}\left[\int_{s}^{t}\left|\sigma_{W}(k_{n}(u)\wedge T_{R},X^{n}(k_{n}(u)\wedge T_{R}))\right|^{2N}du\right]ds.
\end{eqnarray*}
Applying now Fubini\textquoteright s theorem and using the growth
assumption in $\mathbf{(H.1.3)}$, we obtain
\[
I_{22}\leq C_{N}\int_{0}^{t}(t-s)^{-\alpha-\frac{1}{2}}\left(1+\mathbb{E}\left[\left|X_{R}^{n}(k_{n}(s))\right|^{2N}\right]\right)ds.
\]
\[
I_{2}\leq C_{N}\left(1+\int_{0}^{t}(t-s)^{-\alpha-\frac{1}{2}}\mathbb{E}\left[\left|X_{R}^{n}(k_{n}(s))\right|^{2N}\right]ds\right).
\]
Let us remark that
\begin{eqnarray}
& &\int_{0}^{t\wedge T_{R}}\sigma_{H}(k_{n}(s)),X^{n}(k_{n}(s))))\, dB^{H}(s) \label{frac int stopped} \\
&& = \int_{0}^{t}\sigma_{H}(k_{n}(s)\wedge T_{R},X^{n}(k_{n}(s)\wedge T_{R}))\, dB^{H}(s\wedge T_{R})\nonumber\\\notag
& & = \int_{0}^{t}\sigma_{H}(k_{n}(s)\wedge T_{R},X_{R}^{n}(k_{n}(s)))\, dB^{H}(s\wedge T_{R})
\end{eqnarray}
Using (\ref{frac int stopped}) and Proposition \ref{prop:Nua Ras 2} (jj) in the Appendix we obtain
\[
I_{3}\leq C_{N}R^{2N}\left(\int_{0}^{t} \left((t-s)^{-2\alpha}+s^{-\alpha}\right) \big(1+\mathbb{E}\left[\|X_{R}^{n}(k_{n}(\cdot))\|_{\alpha,s}\right]\big)ds\right)^{2N}.
\]
By Hölder's inequality we have
\[
I_{3}\leq C_{N}R^{2N}\int_{0}^{t} \varphi(s,t) \left(1+\mathbb{E}\left[\|X_{R}^{n}(k_{n}(\cdot))\|_{\alpha,s}^{2N}\right]\right)\, ds.
\]
Putting all the estimates obtained for $I_{1}$, $I_{2}$ and $I_{3}$ together, we obtain
\begin{equation} \label{eq:est esp}
\mathbb{E}\left[\|X_{R}^{n}\|_{\alpha,t}^{2N}\right] \leq C_{N}\left|x_{0}\right|^{2N}+C_{N}\left(1+R^{2N}\right)
\int_{0}^{t}\varphi(s,t)\mathbb{E}\left[\|X_{R}^{n}(k_{n}(\cdot))\|_{\alpha,s}^{2N}\right]\, ds.
\end{equation}
Therefore, since the right hand side of Equation (\ref{eq:est esp})
is an increasing function of $t$, we have
\begin{equation*}
\sup_{0\leq s\leq t}\mathbb{E}\left[\|X_{R}^{n}\|_{\alpha,s}^{2N}\right] \leq C_{N}\left|x_{0}\right|^{2N}+C_{N}\left(1+R^{2N}\right)\\
\int_{0}^{t}\varphi(s,t)\mathbb{E}\left[\underset{0\leq u\leq s}{\sup}\|X_{R}^{n}\|_{\alpha,u}^{2N}\right]\, ds.
\end{equation*}
As a consequence, by the Gronwall type lemma (cf. Lemma 7.6 in \cite{NR}), we deduce the first estimate (\ref{app est}) of the lemma. Let us now prove the second estimate (\ref{app inc est}). We have
\begin{eqnarray*}
&& X_{R}^{n}(t)- X_{R}^{n}(s) \\ && = \int_{s\wedge T_{R}}^{t\wedge T_{R}}b(k_{n}(r),X^{n}(k_{n}(r)))\, dr+\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(k_{n}(r),X^{n}(k_{n}(r)))\, dW(r)\\
& & \quad+\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{H}(k_{n}(r),X^{n}(k_{n}(r)))\, dB^{H}(r).
\end{eqnarray*}
\begin{eqnarray*}
\mathbb{E}\left[\left|X_{R}^{n}(t)-X_{R}^{n}(s)\right|^{2N}\right] & \leq & C_{N}\left\{ \mathbb{E}\left[\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}b(k_{n}(r),X^{n}(k_{n}(r)))\, dr\right|^{2N}\right]\right.\\
& & +\mathbb{E}\left[\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{W}(k_{n}(r),X^{n}(k_{n}(r)))\, dW(r)\right|^{2N}\right]\\
& & +\left.\mathbb{E}\left[\left|\int_{s\wedge T_{R}}^{t\wedge T_{R}}\sigma_{H}(k_{n}(r),X^{n}(k_{n}(r)))\, dB^{H}(r)\right|^{2N}\right]\right\} \\
& = & C_{N}\left(J_{1}+J_{2}+J_{3}\right).
\end{eqnarray*}
Applying H{\"o}lder's inequality, the growth assumption ({\bf{H.1.1}}) and (\ref{app est}), we have
\begin{eqnarray*}
J_{1} & \leq & \mathbb{E}\left[\left(\int_{s}^{t}\left|b(k_{n}(r)\wedge T_{R},X^{n}(k_{n}(r)\wedge T_{R}))\right|\, dr\right)^{2N}\right]\\
& \leq & C_{N} (t-s)^{2N-1}\int_{s}^{t}\mathbb{E}\left[\left|b(k_{n}(r)\wedge T_{R},X_{R}^{n}(k_{n}(r)))\right|^{2N}\right] dr\\
& \leq & C_{N} (t-s)^{2N}.
\end{eqnarray*}
By the H{\"o}lder and Burkh{\"o}lder inequalities and using (\ref{app est}), we obtain
\begin{eqnarray*}
J_{2} & \leq & C_{N} (t-s)^{N-1}\mathbb{E}\left[\int_{s\wedge T_{R}}^{t\wedge T_{R}}\left|\sigma_{W}(k_{n}(r),X^{n}(k_{n}(r)))\right|^{2N}\, dr\right] \\
& \leq & C_{N} (t-s)^{N-1}\mathbb{E}\left[\int_{s}^{t}\left|\sigma_{W}(k_{n}(r)\wedge T_{R},X_{R}^{n}(k_{n}(r)))\right|^{2N}\, dr\right] \\
& \leq & C_{N} (t-s)^{N}.
\end{eqnarray*}
Let us note that we obtain from (\ref{frac int est}) and the H{\"o}lder inequality
\[
\left|\int_{s}^{t}f(u)\, dB^{H}(u)\right|^{2N}\leq C_{N}R^{2N}(t-s)^{2N(1-\alpha)+2\alpha-1}\int_{s}^{t}\dfrac{\|f(r)\|_{\alpha}^{2N}}{(r-s)^{2\alpha}}\, dr.
\]
Combining this estimate and (\ref{frac int stopped}) we obtain
\begin{eqnarray*}
J_{3} & \leq & C_{N}R^{2N}(t-s)^{2N(1-\alpha)+2\alpha-1}\mathbb{E}\left[\int_{s}^{t}\dfrac{\|\sigma_{H}(k_{n}(r)\wedge T_{R},X_{R}^{n}(k_{n}(r)))\|_{\alpha}^{2N}}{(r-s)^{2\alpha}}\, dr\right].
\end{eqnarray*}
Using the Hölder inequality, assumption ({\bf{H.2}}) and (\ref{app est}), we arrive at
\begin{eqnarray*}
J_{3} & \leq & C_{N}R^{2N}(t-s)^{2N(1-\alpha)+2\alpha-1}\mathbb{E}\left[\int_{s}^{t}\dfrac{1+\|X_{R}^{n}(k_{n}(r)))\|_{\alpha}^{2N}}{(r-s)^{2\alpha}}\, dr\right] \\ \\
& \leq & C_{N} (t-s)^{N}.
\end{eqnarray*}
All these estimates allow us to obtain
\[
\mathbb{E}\left[\left|X_{R}^{n}(t)-X_{R}^{n}(s)\right|^{2N}\right]\leq C_{N,R}\left|t-s\right|^{N}.
\]
The proof of Lemma \ref{lem:tight} is then completed.
\end{proof}
Now we are able to give the convergence result.
\begin{thm}\label{thm:Eulerstrong}
Assume that $\sigma_{W}$ and $b$ are continuous satisfying the linear
growth condition. Suppose moreover that $\sigma_{H}$ satisfies the assumption $\mathbf{(H.2)}$
and that for Equation (\ref{eq:1-1}) the pathwise uniqueness holds.
Then Euler's approximations $X^{n}(t)$ converge to a process $X(t)$
in probability, uniformly in $t$ in $[0,T]$. Furthermore $X(t)$
is the unique strong solution of Equation (\ref{eq:1-1}). \end{thm}
\begin{proof}
Fix $\eta<1/2$. We have from (\ref{app inc est}) in Lemma \ref{lem:tight} that $X_{R}^{n}$
is weakly relatively compact in $C_{0}^{\eta}$ for every $R$. We
want to deduce from this the weak compactness in $C_{0}^{\eta}$ of
$X^{n}$. Clearly it suffices to show that
\[
\limsup_{R\rightarrow\infty}P\left[T_{R}\leq T\right]=0.
\]
This is a consequence of that fact that the random variable $\left\Vert B^{H}\right\Vert _{1-\alpha,\infty,t}$
has moments of all orders (see Lemma 7.5 in \cite{NR}). We now
take two subsequences $X^{l},X^{m}$ of the Euler's approximations
$X^{n}$. Then obviously $\left(X^{l},X^{m}\right)$ is a tight family
of processes in $C_{0}^{\eta}\times C_{0}^{\eta}$. By Skorokhod's
embedding theorem there exist a probability space $\big(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{P}\big)$
and a sequence $\big(\tilde{X}^{l,n},\tilde{X}^{m,n},\tilde{B}^{n},\tilde{W}^{n}\big)$
with values in $C_{0}^{\eta}$ such that
\begin{enumerate}
\item The law of $\big(\tilde{X}^{l,n},\tilde{X}^{m,n},\tilde{B}^{n},\tilde{W}^{n}\big)$
and $\left(X^{l},X^{m},B^{H},W\right)$ coincide for every $n\in\mathbb{N}$.
\item There exist a subsequence $\big(\tilde{X}^{l(j)},\tilde{X}^{m(j)},\tilde{B}^{n(j)},\tilde{W}^{n(j)}\big)$
converging in $C_{0}^{\eta}$ to $\big(\hat{X},\hat{Y},\hat{B},\hat{W}\big)$
uniformly in $t$, $\tilde{P}$ a.s., that is
\[
\lim_{j\rightarrow\infty}\Big(\|\tilde{X}^{m(j)}-\hat{X}\|_{\eta}+\|\tilde{X}^{\textcolor{magenta}{l}(j)}-\hat{Y}\|_{\eta}+\|\tilde{B}^{n(j)}-\hat{B}\|_{\eta}+\|\tilde{W}^{n(j)}-\hat{W}\|_{\eta}\Big)=0.
\]
\end{enumerate}
We obtain from Lemma 3.1 in G\"{y}ongy and Krylov \cite{GK} and the convergence
of integrals with respect to fBms (5.7) in
Guerra and Nualart \cite{GN} that
\begin{eqnarray*}
\lim_{j\rightarrow\infty} \int_{0}^{t}b\big(k_{l(j)}(s),\tilde{X}^{l(j)}(k_{l(j)}(s))\big)\, ds &=& \int_{0}^{t}b\big(s,\hat{X}(s)\big)\, ds\\
\lim_{j\rightarrow\infty} \int_{0}^{t}\sigma_{W}\big(k_{l(j)}(s),\tilde{X}^{l(j)}(k_{l(j)}(s))\big)\,d\tilde{W}^{n(j)}(s) &=& \int_{0}^{t}\sigma_{W}\big(s,\hat{X}(s)\big)\,d\hat{W}(s)\\
\lim_{j\rightarrow\infty} \int_{0}^{t}\sigma_{H}\big(k_{l(j)}(s),\tilde{X}^{l(j)}(k_{l(j)}(s))\big)\,d\tilde{B}^{n(j)}(s) &=& \int_{0}^{t}\sigma_{H}\big(s,\hat{X}(s)\big)\,d\hat{B}(s),
\end{eqnarray*}
\begin{eqnarray*}
\lim_{j\rightarrow\infty} \int_{0}^{t}b\big(k_{m(j)}(s),\tilde{X}^{m(j)}(k_{m(j)}(s))\big)\, ds &=& \int_{0}^{t}b\big(s,\hat{Y}(s)\big)\, ds\\
\lim_{j\rightarrow\infty} \int_{0}^{t}\sigma_{W}\big(k_{m(j)}(s),\tilde{X}^{m(j)}(k_{m(j)}(s))\big)\, d\tilde{W}^{n(j)}(s) &=& \int_{0}^{t}\sigma_{W}\big(s,\hat{Y}(s)\big)\, d\hat{W}(s)\\
\lim_{j\rightarrow\infty} \int_{0}^{t}\sigma_{H}\big(k_{m(j)}(s),\tilde{X}^{m(j)}(k_{m(j)}(s))\big)\, d\tilde{B}^{n(j)}(s) &=& \int_{0}^{t}\sigma_{H}\big(s,\hat{Y}(s)\big)\, d\hat{B}(s),
\end{eqnarray*}
in probability, and uniformly in $t\in[0,T]$. Therefore, the processes
$\hat{X},\hat{Y}$ satisfy the same SDE~\eqref{eq:1-1}, on $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{P})$,
with the driving noises $\hat{W}$, $\hat{B}$ and the initial condition
$x_{0}$ on the time interval $[0,\hat{T}_{R})$ with
\[
\hat{T}_{R}:=\inf\big\{ t\geq0,\| \hat{B}\| _{1-\alpha,\infty,t}\geq R\big\} \wedge T,\quad R>0.
\]
Again, as above, we have a.s.~$\hat{T}_{R}=T$ for all $R$
large enough. So that $\hat{X},\hat{Y}$ satisfy the same SDE~\eqref{eq:1-1}, on $[0,T]$. Then by pathwise
uniqueness, we conclude that $\hat{X}(t)=\tilde{Y}(t)$ for all $t\in[0,T]$
$\tilde{P}$ a.s.. Hence, by applying Lemma \ref{lem:Gyo Kry}
in the Appendix we obtain the convergence of Euler's approximations
$X^{n}(t)$ to a process $X(t)$ in probability, uniformly in $t$
in $[0,T]$. Therefore, $\left\{ X(t),\, t\in[0,T]\right\} $ satisfy Equation (\ref{eq:1-1}).
\end{proof}
As a consequence we obtain the following existence result.
\begin{thm}
Assume that $b$, $\sigma_{W}$ and $\sigma_{H}$ satisfy the hypotheses $\mathbf{(H.1)}-\mathbf{(H.2)}$.
If $1-H<\alpha<\min\left(\beta/2,1\right)$, then the Equation (\ref{eq:1-1}) has a unique strong solution.
\end{thm}
\section*{Appendix}
In this appendix, we recall some results which play a great
role in this work. We also show a technical lemma that have been used in the proof of pathwise uniqueness.
We begin with some a priori estimates from the paper of Nualart and Rascanu \cite{NR}.
\begin{prop}
\label{prop:Nua Ras 1}We have
\begin{multline*}
(i)\,\|\int_{0}^{.}f(s)\, ds\|_{\alpha,t} \leq C\int_{0}^{t}\dfrac{\left|f(s)\right|}{(t-s)^{\alpha}}\, ds.\\
(ii)\,\|\int_{0}^{.}f(s)\, dB^{H}(s)\|_{\alpha,t} \leq C\left\Vert B^{H}\right\Vert _{1-\alpha,\infty,t}\int_{0}^{t}\left((t-s)^{-2\alpha}+s^{-\alpha}\right)\|f\|_{\alpha,s}\, ds.
\end{multline*}
\end{prop}
Moreover, under the linear growth assumption, we have from Nualart and Rascanu \cite{NR}, the following
\begin{prop}
\label{prop:Nua Ras 2}Assume $(H.1)$ and $(H.2)$. The following
estimates hold
\begin{multline*}
(j)\,\|\int_{0}^{.}b(s,f(s))\, ds\|_{\alpha,t} \leq C\left(\int_{0}^{t}\dfrac{\left|f(s)\right|}{(t-s)^{\alpha}}\, ds+1\right)\\
(jj)\,\|\int_{0}^{.}\sigma_{H}(s,f(s))\, dB^{H}(s)\|_{\alpha,t} \leq C\left\Vert B^{H}\right\Vert _{1-\alpha,\infty,t}\int_{0}^{t}\left((t-s)^{-2\alpha}+s^{-\alpha}\right)\left(1+\|f\|_{\alpha,s}\right)\, ds
\end{multline*}
\end{prop}
We recall the following characterization of the convergence in probability
in term of weak convergence, see G\"{y}ongy and Krylov \textcolor{red}{\cite{GK}}.
\begin{lem}
\label{lem:Gyo Kry}Let $(Z_{n})_{n\in\mathbb{N}}$ be a sequence
of random elements in a Polish space $(\mathcal{E},d)$ equipped with
the Borel $\sigma$-algebra. Then $(Z_{n})_{n\in\mathbb{N}}$ converges
in probability to an $\mathcal{E}$-valued random element if and only
if for every pair of subsequences $(Z_{m})_{m\in\mathbb{N}}$ and
$(Z_{k})_{k\in\mathbb{N}}$ there exists a subsequence $(Z_{m(p)},Z_{k(p)})_{p\in\mathbb{N}}$
converging weakly to a random element $v$ supported on the diagonal
$\left\{ (x,y)\in\mathcal{E}\times\mathcal{E}:x=y\right\} $.
\end{lem}
Finally, let us give a version of the Bihari's lemma.
\begin{lem}
\label{lem:Bihari}Let $1/2<\alpha<1$ and $c\geq0$ be fixed and
$f:\left[0,\infty\right)\longrightarrow\left[0,\infty\right)$ be
a continuous function such that
\[
f(t)\leq a+bt^{\alpha}\int_{0}^{t}(t-s)^{-\alpha}s^{-\alpha}\varrho\left(f(s)\right)\, ds.
\]
$\varrho$ is a concave increasing function from $\mathbb{R}_{+}$
to $\mathbb{R}_{+}$ such that $\varrho(0)=0$, $\varrho(u)>0$ for
$u>0$ and satisfying \eqref{eq:rho est} for some $q>1$. Then for any $1<p<2$ such that $\alpha<1/p$ and $q>1$ with $1/p+1/q=1$
we have
\[
f(t)\leq\left[F^{-1}\left(F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha,p}^{q/p}t^{q\left((1/p)-\alpha\right)+1}\right)\right]^{1/q},
\]
for all $t\in\left[0,T\right]$ such that
\[
F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha,p}^{q/p}t^{q\left((1/p)-\alpha\right)+1}\in Dom(F^{-1}),
\]
\[
F(x)=\int_{1}^{x}\dfrac{du}{\varrho^{q}(u^{1/q})},\quad for\, x\geq0,
\]
and $F^{-1}$ is the inverse function of $F$. In particular, if moreover,
$a=0$ then $f(t)=0$ for all $0<t<T$.\end{lem}
\begin{proof}
Let $1<p<2$ such that $\alpha<1/p$. Using the H{\"o}lder inequality
we obtain
\[
f(t)\leq a+bt^{\alpha}\left(\int_{0}^{t}(t-s)^{-p\alpha}s^{-p\alpha}\, ds\right)^{1/p}\left(\int_{0}^{t}\varrho^{q}\left(f(s)\right)\, ds\right)^{1/q}
\]
For the first integral, using $s=tu$, we have the estimate
\[
\int_{0}^{t}(t-s)^{-p\alpha}s^{-p\alpha}\, ds=t^{1-2p\alpha}\int_{0}^{1}(1-u)^{-p\alpha}u^{-p\alpha}\, du=C_{\alpha,p}t^{1-2p\alpha}
\]
where $C_{\alpha,p}=B\left(1-p\alpha,1-p\alpha\right)$ is the beta function. It follows
\[
f(t)\leq a+b\, C_{\alpha,p}^{1/p}t^{(1/p)-\alpha}\left(\int_{0}^{t}\varrho^{q}\left(f(s)\right)\, ds\right)^{1/q}.
\]
This yields
\[
f^{q}(t)\leq2^{q-1}a^{q}+2^{q-1}b^{q}\, C_{\alpha,p}^{q/p}t^{q\left((1/p)-\alpha\right)}\int_{0}^{t}\varrho^{q}\left(f(s)\right)\, ds.
\]
Then it follows from Bihari's Lemma, see \cite{Bihari56}, that
\[
f(t)\leq\left[F^{-1}\left(F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha,p}^{q/p}t^{q\left((1/p)-\alpha\right)+1}\right)
\right]^{1/q},
\]
for all such $t\in\left[0,T\right]$ such that
\[
F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha,p}^{q/p}t^{q\left((1/p)-\alpha\right)+1}\in \mathrm{Dom}(F^{-1}).
\]
Now, it is simple to see from (\ref{eq:rho est}) that if $a=0$ then
$f(t)=0$ for $t\in[0,T]$.
\end{proof}
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|
1511.00281
|
Search for a light dark sector particle at LHCb
Andrea Mauri
A search is presented for a hidden-sector boson, $\chi$, produced in the decay $B^0 \rightarrow K^* (892)^0 \chi$, with $K^* (892)^0 \rightarrow K^+ \pi^-$ and $\chi \rightarrow \mu^+ \mu^-$ . The search is performed using a $pp$-collision data sample collected at $\sqrt{s}=7$ and 8 TeV with the LHCb detector, corresponding to integrated luminosities of 1 and 2 fb$^{-1}$ respectively.
No significant signal is observed in the mass range $214 \le m_\chi \le 4350$ MeV, and upper limits are placed on the branching fraction product $\mathcal{B}(B^0 \rightarrow K^* (892)^0 \chi) \times \mathcal{B}(\chi \rightarrow \mu^+ \mu^- )$ as a function of the mass and lifetime of the $\chi$ boson. These limits place the most stringent constraints to date on many theories that predict the existence of additional low-mass dark bosons.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Most extensions of the Standard Model (SM) that address the problem of the existence of Dark Matter, postulate the existence of a hidden sector, see for example the review in Ref. <cit.>.
Particles of the hidden sector are singlets with respect to the SM gauge number, however they can interact with SM particles via kinetic mixing. In this analysis a search for a light scalar particle (dark scalar boson, $\chi$) belonging to the secluded sector and mixing with Higgs boson is performed.
Concrete examples of such models are theories where such a $\chi$ field was responsible for an inflationary period in the early universe <cit.>, and the associated inflaton particle is expected to have a mass in the range $270 < m(\chi) < 1800$ MeV.
Another class of models invokes the axial-vector portal <cit.> in theories of dark matter that seek to address the cosmic-ray anomalies, and to explain the suppression of charge-parity (CP) violation in strong interactions <cit.>.
These theories postulate an additional fundamental symmetry, the spontaneous breaking of which results in a particle called the axion <cit.>. The energy scale, $f(\chi)$, at which the symmetry is broken lies in the range $1 \lesssim f(\chi) \lesssim 3$ TeV <cit.>.
Feynman diagram for the decay $B^0 \rightarrow K^{*0} \chi$, with $\chi \rightarrow \mu^+ \mu^-$.
§ SEARCH FOR $B^0 \RIGHTARROW K^* (892)^0 \CHI (\RIGHTARROW \MU^+ \MU^-)$
The decay $B^0 \rightarrow K^{*0} \chi$, with $K^{*0} \rightarrow K^+ \pi^-$ and $\chi \rightarrow \mu^+ \mu^-$ is studied to search for such a hidden-sector particle.
An enhanced sensitivity to hidden-sector bosons arises because the $b \rightarrow s$ transition is mediated by a top quark loop at leading
order (Fig.<ref>). Therefore, a $\chi$ boson with $2m(\mu) < m(\chi) < m(B^0) - m(K^{*0})$ and a sizable top quark coupling (obtained via mixing with the Higgs sector), could be produced at a substantial rate in such decays.
Similar searches have been performed in the past by B-factories <cit.>, they were the most stringent direct constraints on a light scalar dark boson.
Their exclusion limits on the coupling (i.e. mixing angle) between the Higgs and the dark boson field lie between $7 \times 10^{-4}$ and $5 \times 10^{-3}$, with the most sensitive region just below the $J/\psi$ threshold <cit.>.
This search is performed with the full Run I dataset collected with the LHCb detector corresponding to an integrated luminosity of $3.0 \mbox{ fb}^{-1}$.
§ SELECTION AND STRATEGY
Depending on the strength of the mixing with the Higgs boson and its mass, the particle $\chi$ can decay in a secondary vertex, displaced from the $B^0 \rightarrow K^{*0} \chi$ decay vertex.
In order to increase the sensitivity, two regions of reconstructed di-muon lifetime, $\tau (\mu^+ \mu^- )$, are defined for each $m(\chi)$ considered in the search: a prompt region, $|\tau (\mu^+ \mu^- )| < 3\sigma[\tau (\mu^+ \mu^- )]$, and a displaced region, $\tau (\mu^+ \mu^- ) > 3\sigma [\tau (\mu^+ \mu^- )]$, where $\sigma [\tau (\mu^+ \mu^-)]$ is the lifetime resolution.
When setting a limit on the branching fraction the two regions are combined as a joint likelihood, $\mathcal{L} = \mathcal{L}^{prompt} \cdot \mathcal{L}^{displaced}$.
These two regions correspond to the two possible scenarios: the former is sensitive to short lifetime dark boson, it is characterized by high reconstruction efficiency but it is highly contaminated by the irreducible SM background $B^0 \rightarrow K^{*0} \mu^+ \mu^-$; the latter suffers of lower reconstruction efficieny but offers a very clear signature thanks to lower background yields.
A multivariate selection is applied to reduce the background, the uBoost algorithm <cit.> is employed to ensure that the performance is nearly independent of $m(\chi)$ and $\tau(\chi)$. The inputs to the algorithm include $B^0$ transverse momentum, various topological features of the decay, the muon identification quality, and isolation criteria.
Only candidates with invariant mass $m(B^0)$ within 50 MeV of the known $B^0$ mass are selected.
Then, the reconstructed $m(B^0)$ is constrained to its known value to improve the resolution of the dimuon mass, that results to be less than 8 MeV over the entire $m(\mu^+ \mu^-)$ range, and as small as $2\mbox{ MeV}$ below 220 MeV.
The strategy described in Ref. <cit.> is adopted: the $m(\mu^+ \mu^-)$ distribution is scanned for an excess of $\chi$ signal candidates over the expected background.
Since all the theoretical models predict the dark boson $\chi$ to have negligible width compared to the detector resolution, the signal window is entirely determinated by the di-muon mass resolution and is defined to be $\pm 2 \sigma[m(\mu^+ \mu^-)]$ around the tested mass.
The step sizes in $m(\chi)$ are $\sigma[m(\mu^+ \mu^- )]/2$.
In order to avoid experimenter bias, all aspects of the search are fixed without examining the selected $B^0 \rightarrow K^{*0} \chi$ candidates.
Narrow resonances are vetoed by excluding the regions near the $\omega$, $\phi$, $J/\psi$, $\psi(2S)$ and $\psi(3770)$ resonances. These regions are removed in both the prompt and displaced samples.
Distribution of $m(\mu^+ \mu^- )$ in the (black) prompt and (red) displaced regions. The shaded bands denote regions where no search is performed due to (possible) resonance contributions. The $J/\psi$, $\psi(2S)$ and $\psi(3770)$ peaks are suppressed to better display the search region.
Upper limit on the (left-axis) ratio of branching fractions $\mathcal{B}(B^0 \rightarrow K^{*0} \chi(\mu^+ \mu^- ))/\mathcal{B}(B^0 \rightarrow K^{*0} \mu^+ \mu^- )$, where the $B^0 \rightarrow K^{*0} \mu^+ \mu^-$ decay has $1.1 < m^2 (\mu^+ \mu^- ) < 6.0$ GeV$^2$ and (right-axis) on $\mathcal{B}(B^0 \rightarrow K^{*0} \chi(\mu^+ \mu^- ))$ as a function of the dimuon mass. The limits are given at 95% confidence level. Limits are presented for three different lifetimes of the dark boson. The sparseness of the data leads to rapid fluctuations in the limits. The relative limits for $\tau < 10$ ps are between $0.005-0.05$ except near $2m(\mu)$.
§ RESULTS AND EXCLUSION LIMITS
Figure <ref> shows the $m(\mu^+ \mu^-)$ distributions for the number of observed candidates in both the prompt and displaced regions.
The observation is consistent with the background only hypothesis with a $p$-value of about 80%, therefore an upper limit on $\mathcal{B}(B^0 \rightarrow K^{*0} \chi(\rightarrow \mu^+ \mu^-))$ is set.
Figure <ref> shows the upper limits both on the absolute branching fraction $\mathcal{B}(B^0 \rightarrow K^{*0} \chi(\mu^+ \mu^-))$ and on the relative ratio to the normalization channel $\mathcal{B}(B^0 \rightarrow K^{*0} \mu^+ \mu^-)$ in the $1.1 < m^2 (\mu^+ \mu^-) < 6.0 $ GeV$^2$ region. Limits are set at the 95% confidence level (CL) for several values of $\tau (\chi)$.
The limits become less stringent for higher values of $\tau (\chi)$, as the probability of the $\chi$ boson decaying within the LHCb's silicon vertex detector decreases.
Figure <ref> shows the interpretation of the exclusion limit in term of two benchmark models: the inflaton model of Ref. <cit.>, which only considers $m(\chi) < 1$ GeV, and the axion model of Ref. <cit.>.
In the first case, constraints are placed on the mixing angle between the Higgs and inflaton fields, $\theta$, which exclude most of the previously allowed region.
For the latter, exclusion regions are set in the limit of large ratio of Higgs-doublet vacuum expectation values, $\tan \beta \gtrsim 3$, for charged-Higgs masses m$(h) = 1$ and 10 TeV.
The branching fraction of the axion into hadrons varies greatly in different models, the results for two extreme cases are shown: $\mathcal{B}(\chi \rightarrow hadrons) = 0$ and 0.99.
Exclusion regions at 95% CL: (left) constraints on the inflaton model of Ref. <cit.>; (right) constraints on the axion model of Ref. <cit.>. The regions excluded by the theory <cit.> and by the CHARM experiment <cit.> are also shown.
§ CONCLUSION
In summary, a search is performed for light scalar dark boson in the decay $B^0 \rightarrow K^{*0} \chi(\rightarrow \mu^+ \mu^-)$ using $pp$-collision data collected at 7 and 8 TeV.
No evidence of signal is observed, and upper limits are placed on $\mathcal{B}(B^0 \rightarrow K^{*0} \chi) \times \mathcal{B}(\chi \rightarrow \mu^+ \mu^-)$.
This is the most sensitive search to date over the entire accessible mass range and stringent constraints are placed on theories that predict the existence of additional scalar or axial-vector fields.
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|
1511.00112
|
In the paper, we proposed a novel algorithm dedicated to adaptive video streaming based on HTTP. The algorithm employs a hybrid play-out strategy which combines two popular approaches: an estimation of network bandwidth and a control of a player buffer. The proposed algorithm was implemented in two versions which differ in the method of handling fluctuations of network throughput.
The proposed hybrid algorithm was evaluated against solutions which base their play-out strategy purely on bandwidth or buffer level assessment. The comparison was performed in an environment which emulated two systems: a Wi-Fi network with a single immobile node and HSPA (High Speed Packet Access) network with a mobile node. The evaluation shows that the hybrid approach in most cases achieves better results compared to its competitors, being able to stream the video more smoothly without unnecessary bit-rate switches. However, in certain network conditions, this score is traded for a worse throughput utilisation compared to other play-out strategies.
Keywords: Video streaming, Adaptive video, Performance evaluation
§ INTRODUCTION
During the past years, web based video sharing services like YouTube, Hulu or Dailymotion have become very popular. The users of YouTube, which allows for the distribution of user-produced multimedia content, alone request millions of videos every day <cit.>. Consequently, popularity of this kind results in a drastic shift in Internet traffic statistic, which reports increase in traffic from Web-based video sharing services. It is estimated that traffic will account for about 80% to 90% of the global Internet traffic in a next few years, according to the recent report published by Cisco <cit.>.
Video streaming in the above mentioned services is either web-based or HTTP-based; therefore, being transported using TCP. HTTP and TCP are general purpose protocols and were not primarily designed for streaming of multimedia. Thus, attempts are being made to adapt delivery of multimedia content to the Internet environment. One of such attempts tries to introduce an additional layer of application control to transmitted video traffic. Since TCP is designed to deliver data at the highest available transmission rate, it may sometimes be reasonable for a sender to provide additional flow control if it is not strictly necessary for application data to reach a receiver as fast as the TCP would otherwise allow. Therefore, an application may limit the rate at which the data is transmitted, and, if the video bit-rate is lower than the end-to-end available bandwidth, the traffic characteristic will not resemble the characteristics of a standard TCP flow. Furthermore, modern video players implement stream-switching (or multi bit-rate): the content, which is stored at the web server, is encoded at different bit-rate levels, then an adaptation algorithm selects the video level, which is to be streamed, based on a state of a network environment or on a state of a video player.
When the judgment is based on the state of the network environment, the video client estimates how fast the server can deliver video (i.e. the available capacity), e.g. by measuring arrival rate of video data. Then the client choses the video bit-rate which corresponds to the estimated network throughput. If it selects a video bit-rate that is too high, the viewer will experience re-buffering events, i.e. a playback will be suspended because the data transmission will not keep pace with a video bit-rate. If it picks a video bit-rate that is too low, a viewer will experience suboptimal video quality and part of the network bandwidth will be wasted.
When the decision is based on the state of the video player, the algorithm tries directly observe and control the playback buffer instead of estimating network capacity, believing that the buffer occupancy contains a lot of information and is a controllable variable. It is assumed that the buffer occupancy reflects the end-to-end system capacity, including current load conditions of the network and its rate of change reflects the mismatch between the network throughput and the requested video bit-rate.
In this work, we propose a new hybrid algorithm which combines the two above approaches: the bandwidth estimation and the buffer control. We assume that the hybrid solution will exploit strengths of both approaches and it will avoid their weaknesses. We compare performance of the hybrid solution with the two above described approaches subjecting them to variable network throughput, which is commonly encountered in wireless and mobile networks. During the performance evaluation of the algorithms, we measure how often the player buffer runs out, how long it takes to re-start the video transmission, how efficiently the algorithm utilises available network throughput, and finally, how often the player switches between different video rates.
We conduct this performance study using an emulation model. The emulation approach allows us to methodologically explore the behaviour of the examined system over a wide range of parameter settings, which would be a challenging task if we conducted such experiments only on a real-network. Simultaneously, as the emulation is performed in a laboratory environment, we are able to preserve much of the network realism because we conduct experiments using real hardware and software, which allows us to maintain a high level of accuracy for the obtained results.
§ THEORETICAL BACKGROUND
§.§ Application level flow control
One of the popular video transmission method is a progressive download, which is simply a transfer of a video file from an HTTP server to a client where the client may begin playback of the file before the download is completed. However, as the HTTP server progressively sends (streams) the whole video content to the client and usually does not take into account how much of the data has been already sent in advance, an abundance of data can overwhelm the video player and lead to a large amount of unused bytes if a user interrupts the video play-out <cit.>. To avoid such undesirable situation, the video file is divided into chunks of fixed length and the server pushes them to the client at a rate little higher than the video-bit rate of the transmitted content. As a result, the transmitted traffic creates an ON-OFF pattern, where ON and OFF periods have constant length.
The extension of this idea is an adaptive streaming, which offers more flexibility when a network environment is less stable, e.g. in wireless mobile networks. With this approach, it is possible to switch the media bit rate (and hence the quality) after each chunk is downloaded and adapt it to the current network conditions <cit.>. This technique has commenced the development of a new generation of HTTP-based streaming applications which implement client-side play-out algorithms, trying to deliver a continuous stream of video data to end users by mitigating unfavourable network conditions.
In this approach, a video stream is also divided into segments, but this time they are encoded in multiple quality levels, called representations, as drafted in Fig. <ref>. The algorithm deciding which segment should be requested in order to optimize the viewing experience is a main component and a major challenge in adaptive streaming systems because the client has to properly estimate, and sometimes even predict, network conditions, e.g. the dynamic of available throughput. Furthermore, the client has also to control a filling level of its local buffer in order to avoid underflows resulting in playback interruptions.
Architecture of a video adaptive system based on HTTP
The stream-switching technique is employed today less or more in some proprietary video players, among others in Apple HTTP-based streaming <cit.>, Microsoft IIS Smooth Streaming <cit.> or Adobe Dynamic Streaming <cit.>. Moreover, the technique is also adopted by the Dynamic Adaptive Streaming over HTTP (DASH), which is a new MPEG standard pursuing the interoperability between devices and servers of various vendors <cit.>.
§.§ Bandwidth estimation algorithm
As it was already mentioned, the bandwidth estimation algorithm tries to adjust a bit-rate of video to the measured network throughput. As the cited in the section <ref> adaptive video systems are proprietary and their owners do not give many details about the employed algorithms, we used the adaptive streaming algorithm, with some small modifications, which is implemented in the open-source software described in <cit.>. The approach is relatively simple and its main points were summarised in Algorithm <ref>. The examined algorithm calls a function which measures average network throughput $n_b$ in a certain time window $\Delta T$. This window can be considered as a parameter of the algorithm and it may be stretched or shortened in order to optimise streamed video quality. When the video bit-rate $v_b$, which is needed for a smooth video play-out, is lower than the computed average network throughput $n_b$ reduced by $\Delta L$, line <ref>, the algorithm reports that the video quality level $q$ may be increased, i.e. the chunk download module may ask the server for bigger chunks, encoded in higher quality. When the throughput is not sufficient for the given level of video quality, line <ref>, the opposite situation takes place: the quality level $q$ is decreased and the download module is instructed to obtain chunks of poorer quality what simultaneously demands less network throughput. The parameter $\Delta L$ marks a region of network throughput for which there is no need to switch the quality to a higher level. As a result, the parameter plays a stabilising role and prevents switching the quality levels too frequently, which could have a negative impact on the overall video quality perceived by users. The constants $Q_{\max}$ and $Q_{\min}$ define a range of available levels of the quality.
$q$ – current video quality level
$v_b$ – video bit rate
$\Delta T$ – time window for measurement of network throughput
$n_b \gets $ getNetworkBandwidth($\Delta T$)
$v_b < n_b - \Delta L$
$q < Q_{\max}$ $q \gets q + 1$ $v_b > n_b$
$q > Q_{\min}$ $q\gets q - 1$ $q$
Adaptation based on bandwidth estimation
The rate adaptation algorithm might work fairly well for the case when the player does not share a connection with other flows, network resources are stable and do not fluctuate. Since capacity is measured using an average of recent throughput, the estimate is typically not the same as the true current available capacity. This mismatch results in undesirable behaviour of the streaming algorithms which can be both too conservative and too aggressive. Recent studies have reported many examples of an inaccurate bandwidth estimation while a video client competed against another video client, or against long-lived TCP flow <cit.>. In other studies, e.g. in <cit.>, it was observed that competing streams behaved instable and unfair among each other what led to significant video quality variation over time. Therefore, some research works, e.g. <cit.><cit.>, try to improve the algorithm by predicting the future bandwidth, while the others, e.g. <cit.><cit.><cit.>, propose an algorithm based on measurement of buffer occupancy.
§.§ Buffer reactive algorithm
The basic idea of the buffer-based reactive algorithm is to select a video bit-rate based on the amount of data that is available in the buffer of a player. Thus, when the buffer reaches a certain level, the system is allowed to increase the quality. Similarly, when draining the buffer, the selected quality is reduced if the buffer shrinks below the threshold. Hence, the quality no longer depends directly on bandwidth availability, and during network outages, buffer under-runs and play-out interruptions may be avoided.
As it was mentioned, the reactive algorithm upgrades the quality once the buffer duration reaches certain chosen thresholds. However with the increasing quality, the video bit-rate rises non-linearly, therefore it is not practical for the buffer threshold to depend directly on the amount of data accumulated in the buffer measured in bytes, but rather on the amount of data measured in video frames, which may be translated into number of seconds of video preloaded in the player buffer.
Hence, the approach presented in Algorithm <ref> conserves current video bit rate $q$ as long as the buffer occupancy $b$ remains within the range denoted by $B_{\min}$, line <ref> and $B_{\max}$, line <ref>. This buffer range plays a role of a cushion which absorbs rate oscillations. If either of these high or low limits are hit, the rate is switched up or down respectively.
$b$ – current buffer occupancy [s]
$B_{\max} < b$
$q < Q_{\max}$ $q \gets q + 1$ $B_{\min} < b $
$q > Q_{\min}$ $q\gets q - 1$ $q$
Adaptation based on a level of a player buffer
According to <cit.>, buffer reactive algorithms perform fine in many cases, but sometimes they have tendency to too frequent oscillation between video bit rates. Therefore, the authors recommend optimisation of the buffer range confined by $B_{\min}$ and $B_{\max}$ and its adjustment to current network conditions.
§.§ Hybrid algorithm
In order to overcome the drawbacks of the solutions presented in Algorithms <ref> and <ref>, we joined their functionality obtaining a hybrid solution, proposed in Algorithm <ref>, which utilises all information available to the bandwidth estimation based and buffer reactive algorithms. When switching the video bit-rate up, the algorithm is cautions and takes into account both the buffer occupancy and the network throughput, refer to the line <ref>.
Furthermore, to protect users from video-bit rate oscillations, we added a countermeasure in line <ref> in which we check if the number of video-bit rate switches $s$ during the last period $\Delta T$ is within the specified limit $S_{\max}$. When network fluctuations are frequent, the induced by the algorithm changes of the bit-rate have a tendency to cease, making the algorithm insensitive to a variable network environment for time dependent on $S_{\max}$ and $\Delta T$. In order to avoid buffer under-runs, this condition is only verified during the change to a higher video bit-rate.
We assume that the changes of the video bit-rate are counted and updated in a separate code. There also no reasons for both the time windows used for bandwidth measurement and counting of bit-rate switches to be the same length $\Delta T$.
$q$ – current video quality level
$v_b$ – video bit rate
$b$ – current buffer occupancy
$\Delta T$ – time window for measurement of network throughput
$n_b$ – measured network bandwidth in the period $\Delta T$
$s$ – number of video-bit rate switches during the period $\Delta T$
$v_b < n_b - \Delta L$ $B_{\max} < b$
$s < S_{\max}$
$q < Q_{\max}$ $q \gets q + 1$
$v_b > n_b$ $B_{\min} < b$
$q > Q_{\min}$ $q \gets q - 1$
Hybrid adaptation based on bandwidth and buffer occupancy estimation
§ METHODOLOGY
§.§ Quality measures
From the user's perspective, the key performance characteristic of a network is the QoS of received multimedia content. As the video is transmitted through reliable TCP, no data will be lost. However, there may be play-out interruptions caused by either bandwidth fluctuations or long delays due to retransmissions after packet loss. Furthermore, when reduced network throughput is lower than the playback rate and the buffer will drain, the video playback will pause and wait for new video data. A user expects that delays resulting from content buffering will be minimized and will not occur during a normal video play. Any play-out interruptions are annoying to end users and should be taken into account when estimating the quality of experience (QoE). The QoE is based on popular subjective methods reflecting human perception, as a user is usually not interested in performance measures such as packet loss probability or received throughput, but mainly in the current quality of the received content. However, the quality assessment is time-consuming and cannot be done in real time; therefore, we concentrate on these parameters which we believe impact the QoE at most. We rely, among others, on objective measurement methods, introduced by us in <cit.><cit.>, which for the assessment takes into account video interruptions and its total stalling time.
The first measurement of the application QoE takes into account relative total stalling time (ST) experienced by a user. Assuming that the video clip is divided into $i$ chunks, each of them has length $\Delta t_i$ we define:
\begin{equation}
\text{ST}=\sum_i (\Delta t'_i - \Delta t_i),
\label{eq:SR}
\end{equation}
where $\Delta t'_i$ was the time needed in the reality to play-out the $i$th video chunk and we assume that $t'_i \geq t_i$. It is desirable to minimize the value of the ST by a network operator or a service provider.
The application QoS defined in Eq. (<ref>) did not differentiate between the cases in which a user can experience one long stalling period or several shorter stalling periods. Thus in our analysis, we also use a second, complementary measurement which quantifies the number of re-buffering events (RE) associated with every stalling period:
\begin{equation}
\text{RE}=\sum_i \text{sgn} (\Delta t'_i - \Delta t_i).
\label{eq:RE}
\end{equation}
In practice, if any re-buffering events occur, they will take place before the play-out of the $i$th video chunk when a player waits for its download as presented in Fig. <ref>.
(0,0) rectangle +(2,1);
[dashed] (2,0) –+(1,0);
at (2.4,-.5) interruption;
(3,0) rectangle +(2,1);
(5,0) rectangle +(2,1);
(7,0) rectangle +(2,1);
[<->] (9,-2) –+(4,0);
at (11,-1.7) $\Delta t'_i$;
[<->] (11,-1) –+(2,0);
at (12,-0.7) $\Delta t_i$;
(11,0) rectangle +(2,1);
[dotted] (9, 0) – (9,-2);
[dotted] (11, 0) – (11,-1);
[dotted] (13, 0) – (13,-2);
[dashed] (9, 0) – +(2,0);
Re-buffering events during a play-out of video chunks
In Eq. (<ref>), we exclude an initial buffering event which takes place at the beginning of video play, which is used to accommodate throughput variability or inter-packet jitters happening at the beginning of a video play.
The measurements defined in Eqs. (<ref>) and (<ref>) could quite good characterise performance of non-adaptive video system. However in our case, we can imagine an algorithm that will play-out the stream at the minimum available bit-rate, thus minimising the values of Eqs. (<ref>) and (<ref>), what will lead to a relatively low video quality and poor utilisation of available network throughput. Hence, we introduce a third measurement which assesses how effectively the algorithm utilises available network resources
\begin{equation}
\text{TE}=\frac{\sum_i (q_i/Q_i) \Delta t_i}{\sum_i \Delta t_i}.
\label{eq:BE}
\end{equation}
Eq. (<ref>) computes the relation between a quality level $q$ played by an examined algorithm to a theoretical quality level $Q$ which is possible to achieve for given network conditions. The computations take place within discrete time units $\Delta t_i$, into which the video clip is divided, and then are averaged through the duration $\sum_i \Delta t_i$ of the video clip.
The play-out algorithm may try to maximise the measurement presented in Eq. (<ref>) by adjusting the play-out quality to the given network conditions as frequent as it is possible. Such behaviour will result in rapid oscillations of the video quality, what will be negatively perceived by users <cit.>.
For this reason, we introduce the last measurement, which counts the total number of quality switches (SN) during a video play-out
\begin{equation}
\text{SN}=\sum_i |q_{i+1}-q_{i}|.
\label{eq:SN}
\end{equation}
The design goal of a play-out algorithm is to simultaneously minimise values of the measurements defined in Eqs. (<ref>), (<ref>), (<ref>), and maximise the value of the measurement defined in Eq. (<ref>).
§.§ Laboratory set-up
In order to capture performance of the adaptive play-out algorithms, we prepared a test environment emulating standard Internet connections encountered in Wi-Fi and HSPA networks. The environment consists of: a web server, video players, a network emulator and a measurement module implemented in the video player, as shown in Fig. <ref>.
The role of the web server plays Apache <cit.>, which stores the video clips as a set of chunks. As the video player, we chose VLC media player with the DASH plug-in <cit.>. Both the player and the plug-in have an open-source code, thus it was possible for us to manipulate and completely change the adaptation logic without affecting the other components. As a consequence, the plug-in allowed us to implement and integrate Algorithms <ref>, <ref> and <ref> and compare their performance.
As the network environment model, we used the network emulation node based on the built-in Linux Kernel module netem <cit.>. The module is capable of altering network QoS parameters such as network bandwidth or its delays; thus, it allows to test data transmission in different network environments.
Laboratory environment employed for the experiments
We transmitted several video files, acquired from <cit.> and presented in Table <ref>, through the simulation environment with variable network throughput. The bandwidth traces were obtained from measurements conducted in WiFi and HSPA networks. For this purpose, we implemented a custom made analysis tool that transferred data at a fixed, configurable rate using UDP packets. Every transmitted packet had its sequence number, enabling the receiver to precisely detect packet loss, and a time-stamp showing when the package left the receiver. The packets were sent at a fixed rate, and the packet reception rate was logged. The tests measured the download performance as it seems to be more important than upload performance for a one-way video streaming scenario.
Video clips used during the experiments
Name Genre Bitrate levels
Big Buck Bunny animation 6*150 kbit/s – 320x240,
300 kbit/s – 480x360,
600 kbit/s – 854x480,
1.2 Mbit/s – 1280x720,
2.5 Mbit/s – 1920x1080
Elephants Dream animation
Red Bull Playstreets sport
The Swiss Account sport
Valkaama movie
Of Forest and Men movie
The captured log was used as a template for the bandwidth shaper implemented in the mentioned earlier netem module. In addition to bandwidth throttling, the netem module also adds a delay of 20 ms to WI-FI connection and 100 ms to HSPA connection to emulate the average latency which was experienced and measured during gathering of the throughput traces. In order to obtain desirable, average network throughputs, which were used in the experiments, the traces were rescaled. Thus, having identical bandwidth trace we were able to perform a quite fair and realistic comparison of the play-out algorithms.
We believe that the above described methodology provides an attractive middle ground between simulation and real network experiments. To a large degree, the emulator should be able to maintain the repeatability, reconfigurability, isolation from production networks, and manageability of simulation while preserving the support for real video adaptive applications.
Using our laboratory environment, we compared the three presented solutions with the parameters specified in Table <ref>. The hybrid solution was applied in two versions: the base one, which tries to take advantage of network conditions in order to increase video quality without taking into account the buffer state of a player; and the adaptive one, which is more conservative and increases video quality after taking not only the network state and the buffer state, but also frequency of previous bit-rate switches. Each compared algorithm played first 600 s of every video clips presented in Table <ref>.
Algorithms and their parameters used in the experiments
Algorithm Parameters
Bandwidth est. (Alg. <ref>) $\Delta T=4 s$, $\Delta L=0.25n_b$
Buffer reactive (Alg. <ref>) $B_{\min}=3 s$, $B_{\max}=7 s$
Hybrid, basic (Alg. <ref>) $\Delta T$, $\Delta L$, $B_{\min}$ – as above, $\Delta T_s=10s$, $B_{\max}=0 s$, $S_{\max}=\inf$
Hybrid, adaptive (Alg. <ref>) $\Delta T$, $\Delta L$, $B_{\min}$, $B_{\max}$, $\Delta T_s$ – as above, $B_{\max}=7 s$, $S_{\max}=10$
§ RESULTS
The output of the experiments performed in the Wi-Fi environment is presented in Fig. <ref>. The network throughput oscillates between about 2450 kbps and 2700 kbps with an average set to 2600 kbps; however, one must notice that due to TCP/IP and other protocols overhead the effective throughout available for the video streaming is a few percent lower.
As the effective network throughput fluctuates near the highest available in the experiment video bit-rate, the play-out algorithm based on bandwidth estimation quite regularly switches between 1200 kbps and 2500 kbps. The switches are correlated with the local minima of the network throughput.
The buffer reactive algorithm starts from the lowest available bit rate and, as the buffer is being filled with data, gradually increases the quality, reaching 2500 kbps in about 80th s of the playback. Then, the playback quality starts to switch between 2500 kbps and 1200 kbps; nevertheless, the oscillations are less regular and more frequent compared to the bandwidth estimation algorithm.
Because the hybrid algorithm on the beginning of the play-out measures available throughput, it is able to start with a higher quality level compared to its buffer reactive competitor. Furthermore, the algorithm loses the highest quality rate less frequently, mostly in the times, when the available network throughput achieves local minima. A visual assessment may lead to a conclusion that in this particular experiment, the hybrid algorithm obtains better performance than its competitors, at least taking into account bandwidth effectiveness, which was defined in Eq. (<ref>).
The adaptive version of the hybrid algorithm starts its playback similarly to the buffer reactive algorithm: from the lowest quality, then gradually reaching the maximum. Because of this gradual improvement of quality, the algorithm is able to maintain the streaming at 2500 kbps a bit longer compared to its base version. Additionally, the algorithm spends more time streaming 1200 kbps video compared to its base version, which is a result of the condition defined in line <ref> of Algorithm <ref>, which takes into account both network bandwidth and the buffer state of a player into consideration when switching the quality rate to higher level. This leads to a drop in throughput efficiency of the algorithm, while the number of bit-rate switches is comparable to the base version of the algorithm.
Transient comparison of the play-out algorithms in an emulated Wi-Fi environment. Average bandwidth set to 2600 kbps, video clip Big Buck Bunny (see Table <ref>)
The same experiments were performed for the rest of the movies listed in Table <ref> and were extended to the cases, where the network throughput was set in average to and . These values were chosen to a certain extent arbitrarily, however we took into account that in the first case, the effective network throughput falls in the middle of 600 kbps and 1200 kbps levels of the available video bit-rate; and in the second case, the effective network throughput is theoretically a little above the fifth defined quality level of 1200 kbps.
The averages of the relative stalling time (ST), defined in Eq. (<ref>), are similar for all examined algorithms, as it was presented in Fig. <ref>. For the throughput set to , the average ST is between about 3 s and 5 s. With an increase of the throughput, the average ST decreases, taking values between about 3 s and 4 s for the throughput set to , and between about 2 s and 3.5 s for the throughput set to . The main observable difference among the algorithms is that the bandwidth estimation approach has a little higher variation compared to the other solutions, especially for throughput set to .
The higher variability of the bit-rate in the case of the bandwidth estimation algorithm may be explained when we examine the number of re-buffering events (RE) defined in Eq. (<ref>) and presented in Fig. <ref>. During the video streaming at the lowest network throughput, the algorithm experienced at least once a re-buffering event, which influenced also the ST. The re-buffering was probably caused by an overlap of two unfavourable factors: an overoptimistic estimation of the network throughput and a subsequent burst of the bit-rate in the transmitted variable bit-rate video stream. Other algorithms did not experience buffer under-runs.
The average throughput efficiency, defined in Eq. (<ref>), for all algorithms but the adaptive version of the hybrid are comparable, and in most cases ranges between 65% and 80%, as it was presented in Fig. <ref>. As expected from the transient analysis of quality traces presented in Fig. <ref>, the TE for the adaptive hybrid solution is clearly lower, achieving about 65% for the average throughput set to or , and less than 60% for the throughput set to .
The frequency of the bit-rate switching is the highest for the buffer reactive algorithm: from more than 30 switches for the throughput set to to about 25 switches for the throughput set to and , as shown in Fig. <ref>. However, we must note that the cautious increase of the bit-rate in the first minute of the play-out, see Fig. <ref>, influences negatively the algorithm score. The bandwidth reactive solution achieves significantly better results, ranging from about 15 switches for network throughput, through about 12 for , and less than 10 for throughput. The hybrid approaches achieve the best results, experiencing between 6 and 12 switches in the case of the base version of the algorithms, and from 7 to 13 switches for the adaptive version of the algorithm. Simultaneously, these scores have the lowest variation. Similarly to the buffer reactive solution, the score of the adaptive version of the hybrid approach is negatively biased due to the gradual increase of the video bit-rate on the beginning of the play-out.
[Stalling events]
[Number of switches]
Comparison of the play-out algorithms in an emulated Wi-Fi environment
The next series of the experiments were performed in an emulated mobile network environment based on captured traces from an HSPA system. Similarly to the Wi-Fi scenarios, on the beginning we examine the behaviour of the play-out algorithm for a network with an average throughput set to 2600 kbps; however, with much wider range of its oscillations, stretching from a level where the network totally collapses (being the result of e.g. signal fading) and throughput drops to zero kbps, to situations where the throughput is higher than 5000 kbps.
The beginning of the play-out, the network throughput is relatively high, therefore, the algorithm based on bandwidth estimation is able to serve video above its average quality. Nevertheless, with deteriorating network conditions, the algorithm rapidly decreases the transmission quality, what quite often ends up with congestions – undesired condition not met during the experiments in the Wi-Fi network. After the network collapses, its throughput has a tendency to shoot up, what in consequence translates to a rapid increase of the play-out quality. As a result, the quality record has a fairly high range of oscillations in short periods of time what may negatively influence end users' perception.
Similarly to the behaviour of the algorithm based on bandwidth estimation, the buffer reactive algorithm also reacts nervously to the rapid throughput fluctuations. The video quality jumps several times from 1200 kbps onto 2500 kbps, but the algorithm usually persists in serving the video at the highest level only for a relatively short period of time. Contrary to the previous algorithm, the buffer reactive solution handles better network collapses. The first and second network slips, which happen in about 75th s and 175th s respectively, remain almost unnoticed to users. In other similar critical situations, the algorithm struggles to keep up continuity of the play-out, however at the cost of reduction of its bit-rate to the lowest possible level. Generally, compared to the bandwidth estimation approach, the oscillation range of the video quality is a bit tighter, although the rate of the quality switches remains higher.
Compared to its two predecessors, the hybrid algorithm responds more calmly to the throughput fluctuation. The algorithm is able to maintain the highest quality level longer and avoids so frequent quality switches as the buffer reactive algorithm. Simultaneously, the hybrid solution handles better the throughput falls than the play-out regulated by the bandwidth estimation technique, being able to escape from stalls during the streaming and sudden plunges of the video quality.
The introduction of additional hedging against frequent bit-rate switches leads to a smoother quality trace, although we may still observe a few needless tries of quality improvement. The smoother quality comes at a certain price: it is traded for a lower average bit-rate what translates to significantly poorer efficiency of network throughput usage.
Transient comparison of the play-out algorithms in an emulated wireless mobile environment. Average bandwidth set to 2600 kbps, video clip Big Buck Bunny (see Table <ref>)
[Stalling events]
[Number of switches]
Comparison of the play-out algorithms in an emulated wireless mobile environment
Similarly to the examination of the algorithms in the Wi-Fi environment, we extended our analysis to the rest of the movies from Table <ref>. The network throughput was set in average to , and , which are the same values as in the case of the Wi-Fi experiment, in order to make the comparison of the algorithms easier in these two environments.
The ST is roughly comparable for all algorithms, as it was pictured in Fig. <ref>. There are some noticeable differences for the throughput set to ; although, after closer examination, their absolute values are of about several seconds what should not have much influence on end users' experience. As expected, with the increasing network throughput, the ST decreases, dropping from about 9 s - 13 s in the case of throughput, to about 3 s - 5 s in the case of throughput.
For the average throughput set to , the bandwidth assessment algorithm experiences up to 3 stalling events, depending on the video clip played. The buffer reactive algorithm experiences in average 1 stalling event. The hybrid solution in its base form experiences up to 1 stalling event while its adaptive version is free from stallings. With the increasing throughput, the probability of stalling is lower, however the bandwidth estimation approach still has some problems with a smooth play-out. Also for the throughput set to , the buffer reactive algorithm sporadically experiences breaks during the play-out. In contrast, both hybrid solutions are able to deliver the video without interruptions.
When it comes to the assessment of throughput efficiency, the base version of the hybrid algorithm clearly outperforms the rest when the throughput is set to , as shown in Fig. <ref>. When the throughput increases, the hybrid solution loses its advantage over the competitors. The buffer assessment approach notes slightly better result for throughput set to compared to the bandwidth one; nonetheless, when the throughput rises to or , its efficiency drops below the efficiency of the bandwidth assessment approach. The adaptive version of the hybrid algorithm obtains the lowest efficiency from all compared solutions.
Because the mobile network has more dynamical fluctuation of its throughput, the efficiency of the algorithms operating in this environment is in general about 20% worse compared to these examined in the Wi-Fi environment, see Fig. <ref>.
The bandwidth assessment approach clearly under-performs in the SN experienced during the play-out. For the throughput set to , the SN reaches nearly 70, which is about 30% higher than the second score of 45 switches for the buffer reactive algorithm. The hybrid solution achieves less than 45 switches for its base version and less than 25 switches for its adaptive version. The increase of the network throughput to reduces the SN for the bandwidth algorithm to about 50 switches, which is however still not enough to outperform the buffer reactive and hybrid algorithms, which achieve even better results than in the case of throughput. Further rise of the throughput to , aligns the results for bandwidth and buffer reactive algorithms to about 50 switches, while the hybrid algorithms still achieve significantly better score.
The evaluation shows that the hybrid approach in most cases achieves better results compared to its competitors.
In the Wi-Fi environment, where the network conditions are relatively stable and the throughput fluctuation has relatively low amplitude, the differences between the examined algorithms are mainly visible when we take into account efficiency of utilisation of network throughput and frequency of video bit-rate switches. The base hybrid strategy obtained in some scenarios about about 10% better throughput utilisation compared to the solutions based on buffer or bandwidth assessment, thereby it is able to play video of better quality. Simultaneously, the base hybrid solution obtains no worse results than its competitors in other performance measurements.
When we take into account the mobile network, the hybrid strategy is even more dominant. Except achieving significantly lower switching of the video-bit rate, the hybrid approach is free from under-runs of a player buffer which causes a video clip to stall in a middle of a play-out, what takes place in the case of two other strategies. Compared to the aggressive hybrid strategy, its adaptive version achieves usually better results when taking into account the stability of the play-out, especially in the mobile environment. However, this score is traded for significantly worse throughput utilisation compared to other strategies.
§ CONCLUSIONS
In the paper we proposed a novel algorithm dedicated for an adaptive streaming based on HTTP. The algorithm employs a hybrid play-out strategy which combines two popular approaches: a bandwidth estimation and a buffer control. As a consequence, we assumed that the hybrid solution will exploit strengths of both approaches and it will avoid their weaknesses. The proposed algorithm was implemented in two versions which differ in the method of handling throughput fluctuations. The first, base version tries to aggressively exploit any favourable network conditions in order to increase the bit-rate of played video. The second version is equipped with an adaptive accent: it increases the streamed bit-rate more carefully, taking into account the buffer state of a player and throughput variability.
The evaluation shows that in the mobile networks, where network throughput has lots of variability, the hybrid approach achieves better performance compared to traditional solutions based on bandwidth estimation or buffer assessment. The advantages of hybrid solution are less visible in networks where changes in network throughput have lower amplitude. Such a result is a consequence of the construction of the examined algorithms. Single parameters which describe video systems, like network throughput or amount of buffered video, tend to strictly depend on network conditions. The more variable the conditions are, the more variable the parameters. When the algorithms take into account only a one of these parameters, naturally the quality of the transmission will strictly depend on the current network condition described by this parameter. Therefore, an introduction of additional components which the algorithm takes into account when making decisions about a bit-rate adaptation, stabilises and improves a quality of a play-out. In the proposed solution, the algorithm considers two components and gives them equal weight. However, one can study scenarios, with multiple components describing the state of a video system, which may not only include network throughput or buffer level, but also their pace of change, deviations, averages etc. Some of these components may have priorities, which may be dependent on the characteristic of a network environment.
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1511.00326
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I would like to sincerely thank my supervisor Dr. Zdravko Botev for his guidance, tutelage and intellectual discussions he has provided throughout my Honours year. I am grateful for him taking time to share his interesting research ideas that has perked my interests. Also, I must express my gratitude to my colleagues for accompanying me on the same journey in 2015. I would also like to thank the School of Mathematics and Statistics at UNSW for the enriching experience over the last 5 years. Finally, I would like to thank my family for their encouragement and support.
We consider the problem of accurately measuring the credit risk of a portfolio consisting of loss exposures such as loans, bonds and other financial assets. We are particularly interested in the probability of large portfolio losses. We describe the popular models in the credit risk framework including factor models and copula models. To this end, we revisit the most efficient probability estimation algorithms within current copula credit risk literature, namely importance sampling. We illustrate the workings and developments of these algorithms for large portfolio loss probability estimation and quantile estimation. We then propose a modification to the dynamic splitting method which allows application to the credit risk models described. Our proposed algorithm for the unbiased estimation of rare-event probabilities, exploits the quasi-monotonic property of functions to embed a static simulation problem within a time-dependent Markov process. A study of our proposed algorithm is then conducted through numerical experiments with its performance benchmarked against current popular importance sampling algorithms.
Keywords: Rare-event probability estimation; Monte Carlo methods; Importance sampling; Splitting Method; Markov Processes
CHAPTER: INTRODUCTION
Simulation from intractable multidimensional distributions and estimates of corresponding real-valued quantities are hallmark problems in Monte Carlo methods; (see <cit.>). In this chapter we examine these two related problems more closely as they frequently arise in Monte Carlo applications.
The first problem is to simulate from the conditional density in $\mathbb{R}^d$
\begin{equation}
\label{cond}
f^*(\bx)= \frac{1}{\ell} f(\bx)\bb I\{S(\bx)\geq\gamma\},\quad \bx=(x_1,\ldots,x_d)^\top,
\end{equation}
where we assume that: $S: \mathbb{R}^d\rightarrow\mathbb{R}$ is a real-valued function, which we refer to as an importance function; $f$ is a density function on $\mathbb{R}^d$ such that $X_1,\ldots,X_d$ are independent; $\gamma$ is a real parameter, and
\begin{equation}
\label{ell}
\ell=\Pm_f(S(\bX)\geq\gamma),\quad \bX\sim f
\end{equation}
is a normalizing constant. $S$ can be interpreted as a measure of performance of $\bX$ with $\gamma$ being a loss threshold which, if exceeded, triggers an event whose associated probability we wish to estimate. To estimate the probability of this event, we must also solve a second problem, and that is to estimate the normalizing constant $\ell$ accurately and efficiently. $\ell$ can be interpreted the probability of default for a financial institution or a probability of ruin for an insurance company if $S(\bX)$ is the aggregate losses or claims incurred.
Note that, despite the specification of (<ref>), the definition of the conditional density function $f^*$ comes as a special case of any complex high-dimensional density function $\tilde f(\bx)=\breve f(\bx)/\mathcal{Z}$ on $\mathbb{R}^d$ with a known or unknown normalizing constant $\mathcal{Z}$. In other words, without loss of generality, the conditional density (<ref>) includes many models arising in Bayesian inference and statistics, econometrics and finance.
In the unlikely case that $\ell$ is not a rare-event probability (that is, it is not too small; say, larger than $10^{-4}$), one can simulate from (<ref>) exactly with the acceptance-rejection algorithm. That is, we simulate $\bX\sim f$ until $S(\bX)\geq\gamma$ is satisfied. For the simulated $\bX$ for which this condition is satisfied, we accept this as a sample from the conditional distribution (<ref>). Unfortunately, more often than not, $\ell$ is a rare-event probability and in such cases the only practicable approach to simulate from (<ref>) is approximate Markov Chain Monte Carlo (MCMC) sampling as described in <cit.>.
In this thesis, we utilize Monte Carlo methods to estimate these probabilities of the form <ref>. In particular, we focus on a classical Monte Carlo technique, Dynamic Splitting (DS), (see <cit.>). In the original formulation of dynamic splitting, the state space of a Markov process is decomposed into nested subsets so that the rare event is represented as the intersection of sequentially decreasing event subsets. Within each subset the sample paths of the Markov process are split into multiple copies; the rationale behind this is to promote and capture more occurrences of the rare event. As a result, the probability of the rare event is calculated as the product of conditional probabilities. Note that by splitting the Markov process into more copies, it allows more accurate estimation of each conditional probability.
The remaining chapters of the thesis are organized as follows. In Chapter <ref>, we cover essential background knowledge of Monte Carlo methods and describe algorithms for the efficient estimation of $\ell$, namely Importance Sampling and Cross-Entropy. In Chapter <ref>, we introduce and survey the framework of copula credit risk models in current literature with worked examples of the algorithms presented in Chapter <ref>. Further, in Chapter 4, we illustrate the workings of our proposed algorithm through numerical experiments benchmark its performance against existing robust Monte Carlo estimators. Moreover, critical analysis of advantages and limitations of the proposed DS algorithm is
given. Finally, in Chapter 5, we provide concluding remarks and directions for future research. To preserve the flow of the thesis, we delegate verifications and proofs of key results to the appendix.
CHAPTER: BACKGROUND ON MONTE CARLO METHODS
The distribution of losses $F_L$ is the structure of interest in credit risk modelling. There is often no closed form for $F_L$ making direct calculations of $\ell(l)$ for a given $l$, or $l$ for a given $\ell$. To estimate these measures we require a method to draw independent and identically distributed (iid) samples of $L$ from $F_L$ and a method to estimate probabilities and quantiles given an iid sample $L_1, \dots , L_N$, particularly those in the upper tail of $F_L$ such as the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). The purpose of this chapter is to describe the estimators of probabilities and quantiles using Monte Carlo methods, namely Importance Sampling.
§ EFFICIENCY
In this section, we describe the idea of efficiency as we require a criteria to benchmark performance of estimators illustrated in this paper. For a rare-event estimator, this is often measured by its relative error (RE)<cit.>. This is the normalized standard deviation of the estimator. Suppose we have an unbiased estimator $\hat{\ell}$ of the rare-event probability defined as
\[
\begin{split}
\ell(\gamma) & = \bb P(S(\bX)\geq \gamma)\\
&=\bb E[\bb I\{S(\bX)\geq \gamma\}]
\end{split}
\]
where $f$ is a probability density function (pdf),$S$ is a real-valued function, $\bX$ is a random vector and $\gamma$ is a threshold parameter. The event of interest is $\{S(\bX)\geq \gamma\}$ occurring under $f$. RE is defined as
\[
RE(\hat{\ell})=\frac{\sqrt{\Var(\hat{\ell})}}{\ell\sqrt{N} }
\]
where $N$ is the number of iid estimates of $\ell$.
§ CRUDE MONTE CARLO
Crude Monte Carlo (CMC) <cit.> approximates the cumulative distribution function (cdf) $F_L$ with the empirical distribution function (edf) $\hat{F}_L$. Let $L_1, \dots , L_N$, be an iid sample of size $N$, then $\hat{F}_L$ defined as follows
\[
\hat{\ell} (\gamma) = \frac{1}{N} \sum_{k=1}^N \bb I (L_k \geq \gamma).
\]
We define $\hat{v}_\alpha$ as the CMC estimate of the $\alpha$-VaR or equivalently the $\alpha$-quantile. By using the above definition, CMC sets $\hat{v}_\alpha$ as the solution to the problem
\[
\hat{v}_\alpha = \inf \{\gamma:\hat{\ell}(\gamma) \leq 1-\alpha\}.
\]
As a result, we can also obtain $\hat{v}_\alpha$ by sorting the $\{L_k\}$ in ascending order and taking the $\lceil \alpha N\rceil$-th largest value. Once $\hat{v}_\alpha$ is known, we can calculate the CMC estimator for the CVaR as follows
\[
\hat{c}_\alpha = \frac{1}{N(1-\alpha)} \sum_{k=1}^N L_k \bb I (L_k \geq \hat{v}_\alpha).
\]
It is well-known that the VaR of a portfolio is not additive, that is, it is not the sum of the VaR of sub-portfolios or individual losses as it is not the sum of independent random variables. Nevertheless,<cit.> provides some insight into the asymptotic distribution of these estimators through the following central limit theorems.
[Central Limit Theorems for CMC estimators]
If $L$ is positive and continuously differentiable density $f_L$ around $v_\alpha$ and $\bb E [L^2] < \infty$, then as $N\rightarrow \infty$
\[
\sqrt{N}(\hat{v}_\alpha - v_\alpha) \xrightarrow{d} \mathsf{N}\left(0,\frac{\alpha(1-\alpha)}{f_L(v_\alpha)^2}\right),
\]
\[
\sqrt{N}(\hat{c}_\alpha - c_\alpha) \xrightarrow{d} \mathsf{N}\left(0,\frac{\Var(L\bb I (L>v_\alpha))}{(1-\alpha)^2}\right).
\]
§ IMPORTANCE SAMPLING
Our aim is to efficiently estimate $\ell(\gamma)$ for a given $\gamma$, or $\gamma$ for a given $\ell$. CMC estimators heavily rely on the computational power of a large sample size to achieve accuracy. However, if $\ell$ is a rare-event probability say, smaller than $10^{-4}$, then generating large samples is costly to simulate large values of $\gamma$. Much of the work on simulation methods has been done on variance reduction algorithms to improve the efficiency of the CMC estimator. We now present an algorithm that is particularly well-suited to rare-event problems, Importance Sampling (IS). IS has been shown to accurately and efficiently sample from the upper tail of a general loss distribution, and hence reduce the variance of $\hat{\ell}$ <cit.>.
Suppose that $L$ can be simulated under another density $g_L$, which we denote as the IS density. Define,
\[
W(\gamma) = \frac{f_L(\gamma)}{g_L(\gamma)}
\]
as the likelihood ratio obtained by the change in probability measures. Note that
\begin{align*}
\bb P(L > \gamma) &= \bb E_f[\bb I \{L > \gamma\}]\\
&= \bb E_g [W(L) \bb I \{L > \gamma\}]
\end{align*}
where $\bb E_f$ and $\bb E_g$ denote expectations with respect to pdf $f_L$ and $g_L$ respectively. As we are interested in the upper tail of loss distribution, we can thus estimate tail probabilities as follows
\[
\hat{\ell} = \frac{1}{N} \sum_{k=1}^N W(L_k) \bb I \{L_k > \gamma\}.
\]
<cit.> suggests the IS approach to quantile estimation by defining the IS cdf as
\[
\begin{split}
\hat{F}_L^{IS} (\gamma) &= 1 - \hat{\ell} \\
&=1-\frac{1}{N} \sum_{k=1}^N W(L_k) \bb I \{L_k > \gamma\}.
\end{split}
\]
With this definition, the IS estimators for the VaR and CVaR are given by
\[
\hat{v}_\alpha^{IS} = \inf \{\gamma\hat{F}_L^{IS} \geq \alpha\},
\]
\[
\hat{c}_\alpha^{IS} = \frac{1}{N(1-\alpha)} \sum_{k=1}^N W(L_k)L_k \bb I \{L_k \geq \hat{v}_\alpha^{IS}\}.
\]
Similar to the CMC estimators, <cit.> show that the IS estimators asymptotically follow Normal distributions and give the following central limit theorems.
[Central Limit Theorems for IS estimators]
If $L$ is positive and continuously differentiable density $f_L$ around $v_\alpha$ and there exists $\epsilon>0$ and $p>2$ such that $W(\gamma)$ is bounded for all $\gamma \in (v_\alpha-\epsilon,v_\alpha+\epsilon)$ and $\bb E_g [\bb I (L\geq v_\alpha-\epsilon)(W(L))^p]<\infty$, then as $N\rightarrow \infty$
\[
\sqrt{N}(\hat{v}_\alpha^{IS} - v_\alpha) \xrightarrow{d} \mathsf{N}\left(0,\frac{\Var_{g}(W(L))\bb I (L\geq v_\alpha))}{f_L(v_\alpha)^2}\right),
\]
\[
\sqrt{N}(\hat{c}_\alpha^{IS} - c_\alpha) \xrightarrow{d} \mathsf{N}\left(0,\frac{\Var_{g}(W(L))L\bb I (L>v_\alpha))}{(1-\alpha)^2}\right).
\]
where $\Var_g$ denotes the variance under the pdf $g_L$. Note that $g_L$ is not known so the above theorems only describe attractive properties of a good choice of $g_L$. A good choice for the IS pdf depends on the distribution of $\v X$, properties of the set $\{S(\bX)\geq\gamma\}$ and more importantly, the tail behaviour of $S(\bx)$. A light-tailed random variable $X$ is defined as one which has a finite moment generating function (mgf), that is $\bb E[e^{\theta X}]<\infty$ for $\theta>0$. <cit.> suggests that a good IS pdf $g_L$ in a light-tailed setting is an exponentially twisted pdf derived from $f_L$, defined as
\[
g_L(\gamma) = \frac{\exp (\theta \gamma)f_L(\gamma)}{\bb E[e^{\theta L}] }.
\]
Here $\bb E[e^{\theta L}]$ acts as the normalizing constant to ensure $g_L$ is a density. The likelihood ratio of an exponentially twisted pdf is thus given by
\begin{equation}
\label{LLR}
W(\gamma) = E[e^{\theta L}]\exp (-\theta \gamma).
\end{equation}
Attractive properties of likelihood ratios of this form are described in <cit.>.
§.§ Adaptive Importance Sampling
Adaptive importance sampling methods aim to avoid theoretical complications and computational issues in rare-event probability estimation by deriving parameters for an optimal IS density by using sub-samples of sample data. As described previously, there is often no closed form for $f_L$ so we represent the portfolio loss in the form $L=S(\bX)$ and seek to parameterize a prespecified IS density. As this is discussed within a credit risk framework, our interest is to simulate from a density conditional on the event $\{S(\bX)\geq \gamma\}$ where $\gamma$ is often chosen to be large loss threshold. If an initial sample $\bX_1,\ldots,\bX_M$ can be generated directly from the optimal IS density, that is, the zero-variance density $g^*(\bx)=f(\bx|S(\bx)\geq \gamma)$ then the parameters can be computed to approximate $g^*$. A popular and versatile adaptive importance sampling method is Cross-Entropy (CE)<cit.>. The goal of CE is to specify a density $g$ `close' to $g^*$ so that both would behave similarly and give reasonably accurate IS estimators. We consider the family of distributions $\mathcal{G}=\{g(\v x; \v v)\}$ where $\v v$ denotes a vector of parameters. A convenient measure of the difference between two densities $g_1$ and $g_2$ is the Kullback-Leibler divergence also known as the cross-entropy distance:
\[
\mathcal{D}(g_1,g_2)=\int g_1(\bx)\log \frac{g_1(\bx)}{g_2(\bx)}\mathrm{d}\bx.
\]
Every density in $\mathcal{G}$ can be represented as $g(\cdot;\v v)$ for some $\v v$, so we obtain the optimal IS density by solving the following problem:
\[
\v v^*_{CE}=\argmin_{\v v} \mathcal{D}(g^*,g(\cdot;\v v)).
\]
This is equivalent <cit.> to solving
\[
\v v^*_{CE}=\argmax_{\v v} \bb E[ f(\bX)\bb I\{S(\bX)\geq \gamma\}\log g(\bX;\v v)].
\]
This problem does not typically have an explicit solution. Instead we can estimate $\v v^*_{CE}$ by solving the following problem:
\[
\hat{\v v}^*_{CE}=\argmax_{\v v} \frac{1}{N} \sum_{i=1}^N \bb I\{S(\bX_i)\geq \gamma\}\log g(\bX_i;\v v)]
\]
where $\bX_1,\ldots.\bX_N$ are simulated from $f$. This is further simplified if we are able to draw approximately from $g^*$, reducing the problem to
\[
\hat{\v v}^*_{CE}=\argmax_{\v v} \frac{1}{N} \sum_{i=1}^N \log g(\bX_i;\v v)]
\]
where $\bX_1,\ldots.\bX_N$ are drawn approximately from $g^*$.
Suppose we want to sample approximately from the zero-variance density $g^*(\bx)=f(\bx|S(\bx)>v_\alpha)$. Given a generated sample of portfolio losses $L_1,\ldots,L_N$ with corresponding vectors $\bX_{L_1},\ldots,\bX_{L_N}$ from $f$, we can order the portfolio losses in ascending order as $L_{(1)}\leq\cdots\leq L_{(N)}$ and choose $L_{(\lceil\alpha N\rceil)},\ldots,L_{(N)}$ with corresponding vectors $\bX_{L_{(\lceil\alpha N\rceil)}},\ldots,\bX_{L_{(N)}}$ as an approximate sample from $g^*$. With this approximate sample, we can use standard maximum likelihood estimation to parameterize a prespecified density $g$ which approximates $g^*$. The approximate sampling algorithm can be summarized as follows.
: Sampling approximately from $g^*$
distribution of $\bX$ $f$; importance function $S$; sample size $N$; loss threshold $x$
$t\leftarrow t + 1$
Simulate $\bX_t$ from $f$
$B_k\leftarrow \bb I\{X_k> x_k\}$ for $k=1,\ldots,d$
Generate the portfolio loss $L_t \leftarrow S(\bX_t)$
$t = N$
Sort the losses $\v L\leftarrow (L_{[1]},\ldots,L_{[N]})^\top$
$L_{(\lceil\alpha N\rceil)},\ldots,L_{(N)}$
$\bX_{L_{(\lceil\alpha N\rceil)}},\ldots,\bX_{L_{(N)}}$
CHAPTER: INTRODUCTION TO COPULA CREDIT RISK MODELS
§ SUMMARY OF THE MODEL
The model can be summarized as follows.
\[
L=c_1 B_1+\cdots+c_d B_d= \v c^\top \v B
\]
be the total loss incurred by a portfolio of $d$ obligors, where
$c_k$ is the loss incurred from the $k$-th obligor and $B_i\sim \mathsf{B}(P_k)$ is a Bernoulli random variable indicating whether the $k$-th obligor has defaulted. The distribution of the column vector $\v B$ is implicitly defined under factor models and copula models which are described later in this chapter. For now, we note that dependence of default events is captured in the model through the default probabilities $\{P_k\}$ which contain a set of common factors, say $\Psi$, that affect all obligors. Conditional on $\Psi$, the problem is simplified to modelling the sum of independent Bernoulli variables $\{B_k\}$ scaled by the losses $\{c_k\}$. These models have popular applications in finance <cit.>, particularly in the valuation of credit risk such as the estimation of Value-at-Risk for a given confidence level $\alpha$. Popular values for $\alpha$ are $0.95$, $0.99$ and $0.995$ since the loss values of interest lie in the upper tail of the loss distribution. The purpose of this chapter is to describe the most popular credit risk models, factor models and copula models, with applications of the algorithms presented in Chapter <ref>.
§ FACTOR MODELS
In factor models, $\v B$ are defined as
\[
B_k\idef \bb I\{X_k>x_k\},\quad k=1,\ldots,d
\]
where $\{x_k\}$ are given fixed thresholds and $\v X$ has a continuous joint density $f(\v x)$.
For example, $\v X\sim\mathsf{N}(\v \mu, \m\Sigma)$ for some mean vector $\v \mu$ and covariance matrix $\m\Sigma$. Note that $\m \Sigma$ can be singular.
§.§ Gaussian factor model
A popular Gaussian factor model for $\v X$ is
\begin{equation}
\label{GFM}
X_k=a_{k1} Z_1+\cdots+a_{km} Z_m+b_k \epsilon_k,\quad k=1,\ldots,d
\end{equation}
$Z_1,\cdots,Z_m\simiid \mathsf{N}(0,1)$ are the so called systematic risk factors which affect all obligors;
$a_{k1},\cdots,a_{kd}$ are default factor loadings for the $k$-th obligor with
\[
a_{k1}^2+\cdots+a_{km}^2\leq 1,
\]
\[
b_k=\sqrt{1-(a_{k1}^2+\cdots+a_{km}^2 )}.
\]
and $\epsilon_k\sim\mathsf{N}(0,1)$ is risk specific to the $k$-th obligor. We thus have the marginal distribution $X_k\sim \mathsf{N}(0,1)$. We can write
\[
\v X= \m A \v Z +\mathrm{diag}(\v b )\v \epsilon ,
\]
where $\m A$ is an $d\times m$ matrix.
Here, probabilities are conditionally independent on $\v Z$, that is $\Psi = \v Z$, the default probability of the $k$-th obligor $P_k(\v Z)$ is
\begin{equation}
\label{Gcon}
\begin{split}
P_k(\v Z) &= \bb P (X_k >x_k|\v Z=\v z)\\
&=\bb P \left(\epsilon_k >\frac{ x_k-(a_{k1} z_1+\cdots+a_{km} z_m)}{b_k}\right)\\
&=\Phi \left( \frac{(a_{k1} z_1+\cdots+a_{km} z_m)-x_k}{b_k}\right).
\end{split}
\end{equation}
§.§.§ Crude Monte Carlo
Suppose that the factor loadings matrix $\m A$ and default thresholds $\v x$ are known. For a chosen sample size $N$ and confidence level $\alpha$, the CMC algorithm is as follows.
: Generating $L$ under a Gaussian factor model using CMC
factor loadings $\m A$; cost vector $\v c$; default thresholds $\v x$; sample size $N$; confidence level $\alpha$; loss threshold $\gamma$
$t\leftarrow t + 1$
Simulate $\v Z \sim\mathsf{N}(\v 0,\m I_m)$
Simulate $\v\epsilon\sim\mathsf{N}(\v 0,\m I_d)$
$\v X\leftarrow \m A \v Z +\mathrm{diag}(\v b )\v \epsilon$
$B_k\leftarrow \bb I\{X_k> x_k\}$ for $k=1,\ldots,d$
$L_t \leftarrow \v c^\top \v B$
$t = N$
Sort the losses $\v L\leftarrow (L_{[1]},\ldots,L_{[N]})^\top$
$\hat{\ell} (\gamma) \leftarrow \frac{1}{N} \sum_{k=1}^N \bb I (L_k > \gamma)$
$\hat{v}_\alpha \leftarrow L_{\lceil \alpha \times N\rceil}$
§.§.§ Importance Sampling
Suppose that the conditional default probabilities $\v P(\v Z)$ are known, <cit.> describes an IS algorithm which changes the default indicators to $B_k \sim \mathsf{B}(P_k(\v Z))$ before applying exponential twisting to the conditional probabilities $P_k(\v Z)$ as follows
\begin{equation}
\label{exptwist}
P_{k,\theta}(\v Z)=\frac{P_k(\v Z) e^{\theta c_k }}{1+P_k(\v Z) (e^{\theta c_k}-1)}.
\end{equation}
By applying this change in probability measure, conditional default probabilities are increased if $\theta>0$. Exponential twisting are well known for the significant variance reduction in range of contexts <cit.>. It is difficult to apply exponential twisting directly on $f_L$ so it is applied on the default indicators $B_k$ instead and this can be shown to give equivalent results <cit.>. Conditional on $\Phi=\v Z$, $L$ becomes a sum of independent scaled Bernoulli random variables $c_k B_k$. Now each $c_k B_k$ has mgf
\[
\bb E[e^{\theta c_k B_k}]=(1-P_k(\v Z))+P_k(\v Z)e^{\theta c_k } <\infty
\]
so $\{c_k B_k\}$ are light-tailed with a good IS pdf being an exponential twisted density obtained by applying (<ref>). It can easily be verified that this leads to the likelihood ratio for $L$ given by (<ref>). The algorithm, which <cit.> calls the one-step algorithm, can be summarized as follows. Given $\v Z$ and $\{x_k\}$, we calculate the conditional probabilities $P_k(\v Z)$ and apply exponential twisting to each probability. This generates an exponential twisted density for each random variable $c_k B_k$; the product of these densities forms the exponentially twisted density $g_L$. The likelihood ratio is then used to estimate upper tail probabilities from which quantiles in the tail can then be estimated. The one-step IS algorithm is as follows.
: Generating $L$ using one-step IS algorithm
factor loadings $\m A$; cost vector $\v c$; default thresholds $\v x$; sample size $N$; confidence level $\alpha$; loss threshold $\gamma$
$t\leftarrow t + 1$
Simulate $\v Z \sim\mathsf{N}(\v 0,\m I_m)$
Calculate vector $\v P(\v Z)$ with $P_k (\v Z) \leftarrow \Phi\left(\frac{\sum_i \m A_{ki} Z_i-x_k}{b_k}\right)$
$\v c^\top \v P(\v Z) \geq \gamma$
$\theta \leftarrow 0$
$\theta$ is assigned to be the solution to
\[
\gamma=%\sum_{k=1}^m\frac{\partial}{\partial\theta} \log(1+P_k(\v Z) (\exp(\theta c_k) -1))
\sum_{k=1}^m\frac{P_k(\v Z) c_k\exp(\theta c_k)}{1+P_k(\v Z) (\exp(\theta c_k) -1)}
\]
$P_{k,\theta}(\v Z)\leftarrow \frac{P_k(\v Z) \exp(\theta c_k)}{1+P_k(\v Z)(\exp(\theta c_k)-1)}$
Simulate $B_k\sim \mathsf{B}(P_{k,\theta}(\v Z))$, independently
Set $L_t \leftarrow \v c^\top \v B$ and
\[
W(L_t) \leftarrow \exp\left(-\theta L_t+\sum_{k=1}^m \log\left(1+P_k(\v Z)(\exp(\theta c_k)-1)\right) \right)
\]
$\hat{\ell} (\gamma)\leftarrow \frac{1}{N} \sum_{t=1}^N \bb I \{L_t > \gamma\} W(L_t) $
$\hat{v}_\alpha^{IS} \leftarrow \inf \{\gamma:\hat{F}_L^{IS} (\gamma) \geq \alpha\}$
The efficiency of the algorithm depends on the level of dependence between obligors. <cit.> states that when dependence is weak, such as the case of the Gaussian factor model, increasing conditional default probabilities by exponential twisting already reduces variance in the estimators of tail probabilities and quantiles. The algorithm aims to simulate large values for $L$ centred around a carefully chosen loss threshold $\gamma$ that lies in the tail of $f_L$. This is done by solving for the unique value of $\theta$ as described in the algorithm and applying the exponential twisting with this value of $\theta$. An attractive property of exponential twisted pdf $g_L$, that can be easily verified, is that for each simulated $L$ we have
\[
\bb E_g[L|\v Z] \geq \gamma,
\]
which comes a result of solving for $\theta$ such that
\[
\gamma=\sum_{k=1}^m\frac {P_k(\v Z) c_k\exp(\theta c_k)}{1+P_k(\v Z) (\exp(\theta c_k) -1)}.
\]
Hence, the algorithm can sample from the tail of $f_L$ if we chose our loss threshold $\gamma$ to be a quantile in the tail, such as $\hat{v}_{0.95}$, which can be initially estimated with CMC. With this sampling, small portfolio losses are now rare events while large portfolio losses are now frequent.
The vector of likelihood ratios, $\v W$, can be used to estimate upper tail probabilities. After sorting the simulated losses in ascending order, the IS estimator $\hat{v}_\alpha^{IS}$ can be computed as $\mathrm{VaR}=L_{(j)}$ where $j$ is the solution to
\[
\min \limits_j \left(\frac{1}{N}\sum_{k=j}^N W(L_{(k)}) \leq 1 - \alpha \right)
\]
<cit.> states a further extension to form a two-step algorithm. This is motivated by the variance decomposition of the estimator $\hat{\ell}$
\[
\Var(\hat{\ell})= \bb E\left(\Var(\hat{\ell|\v P})\right)+\Var\left(\bb E(\hat{\ell}|\v P)\right).
\]
The one-step IS algorithm minimizes the variability of $\hat{\ell}$, that is, it minimizes $\Var(\hat{\ell|\v P})$. The two-step IS algorithm aims to minimize $\Var\left(\bb E(\hat{\ell}|\v P)\right)$. This is equivalent to minimizing the variance of the CMC estimator $\hat{q}$ <cit.> of
\[
q = \bb P(L>\gamma|\v P(\v Z)).
\]
The corresponding zero-variance density $g^*$ <cit.> is given by
\[
g^*_{\v Z}(\v z) \propto \bb P(L>\gamma|\v P(\v Z))f_{\v Z}(\v z).
\]
However, we must note that the normalizing constant $\ell$ is the same constant we wish to estimate so this is not a practical IS density. Nevertheless, this provides a direction in the searching for a good IS density. A common approach to by applying the one-step algorithm after a change of measure for $\v Z$. In particular, we change the mean of $\v Z$. The underlying rationale of this approach is to generate more defaults by shifting the mean of the factors $\v Z$ by increasing each of its components. This leads to high values for the default factor loadings $\{X_k\}$ which are more likely to exceed the thresholds $\{ x_k\}$. The challenge in the two-step algorithm is to describe a suitable IS distribution $g_Z$. <cit.> and <cit.> propose using a Normal distribution $\mathsf{N}(\v \mu, \m I_m)$ with the same mode as optimal pdf $g_Z^{*}$. The mode $\v \mu^*$ is also the mean of the Normal distribution and provided that we have loss threshold $x$, is given as the solution to the following problem
\begin{equation}
\label{shiftfactorsoptim}
\v \mu^* = \argmax \limits_{\v z} \mathbb{P}\left(L>\gamma|\v Z = \v z \right)\exp {(-\frac{1}{2}\v z^\top \v z)}.
\end{equation}
Hence the two-step algorithm applies a change in distribution to the factors $\v Z$ and simulates $\v Z \sim \mathsf{N}(\v \mu^*,\m I_m)$. The difficulty now lies in solving (<ref>). <cit.> states several approximations to simplify this problem. We focus on the constant approximation and the tail bound approximation as it provides the most convenient way to combine IS applied on the probabilities $P_k(\v Z)$ with IS applied on the factors $\v Z$. For details on other approximations used in solving (<ref>), interested readers may refer to <cit.>.
The constant approximation involves replacing $L$ with $\bb E[L|\v Z = \v z]$ and $\mathbb{P} (L>\gamma|\v Z = \v z)$ with $\bb I(\bb E[L|\v Z = \v z]>\gamma)$. This approximation replaces $\mathbb{P}\left(L>\gamma|\v Z = \v z \right)$ with a constant and so (<ref>) becomes
\begin{equation}
\label{constapprox}
\argmin \limits_{\v z} \{\v z^\top \v z: \bb E[L|\v Z = \v z]>\gamma \}.
\end{equation}
The tail bound approximation is an approach which aims to approximate $\mathbb{P}\left(L>\gamma|\v Z = \v z \right)$ by its upper bound. It then proceeds by maximising this upper bound which in turn, could maximise the probability $\mathbb{P}\left(L>\gamma|\v Z = \v z \right)$. Using this approximation, (<ref>) becomes
\begin{equation}
\label{tailbound}
\argmax \limits_{\v z} \left\{\sum_{k=1}^m \log[1+P_k(\v Z)(\exp(\theta c_k)-1)]-\theta \gamma-\frac{1}{2}\v z^\top \v z\right\}.
\end{equation}
The two-step algorithm for dependent obligors is as follows.
: Generating $L$ using Glasserman and Li's two-step algorithm
factor loadings $\m A$; cost vector $\v c$; default thresholds $\v x$; sample size $N$; confidence level $\alpha$; loss threshold $\gamma$; shifted mean vector $\v \mu^*$
$t\leftarrow t + 1$
Simulate $\v Z \sim\mathsf{N}(\v \mu ,\m I_m)$
Calculate vector $\v P(\v Z)$ with $P_k (\v Z) \leftarrow \Phi\left(\frac{\sum_j \m A_{k,j} Z_j-x_k}{b_k}\right)$
$\v c^\top \v P(\v Z) \geq \gamma$
$\theta \leftarrow 0$
$\theta$ is assigned to be the solution to
\[
\gamma=%\sum_{k=1}^m\frac{\partial}{\partial\theta} \log(1+P_k (\v Z) (\exp(\theta c_k) -1))
\sum_{k=1}^m\frac{P_k (\v Z) c_k\exp(\theta c_k)}{1+P_k (\v Z) (\exp(\theta c_k) -1)}
\]
$ P_{k,\theta} (\v Z)\leftarrow \frac{P_k (\v Z) \exp(\theta c_k)}{1+P_k (\v Z)(\exp(\theta c_k)-1)}$
$B_k\leftarrow \mathsf{Ber}(P_{k,\theta})$, independently
Set $L_t \leftarrow \v c^\top \v B$ and
\[
W(L_t) \leftarrow \exp\left(-\theta L+\sum_{k=1}^m \log\left(1+P_k (\v Z)(\exp(\theta c_k)-1)\right)\right)\exp \left(-\v\mu^{*\top} \v Z + \frac{\v \mu^{*\top} \v \mu^*}{2} \right)
\]
$\hat{\ell} (\gamma) \leftarrow \frac{1}{N} \sum_{t=1}^N \bb I (L_t > \gamma) W(L_t) $
$\hat{v}_\alpha^{IS} \leftarrow \inf \{\gamma:\hat{F}_L^{IS}(\gamma) \geq \alpha\}$
§.§ $t$ Factor Model
The $t$ factor model <cit.> differs from the Gaussian factor model as the factors now have a multivariate $t$ distribution rather than a multivariate Normal distribution. Following the desired properties and notation from (<ref>), $\v X$ generated from a $t$ factor model usually has the representation
\[
X_k= \sqrt{\frac{r}{V}} \left( a_{k1} Z_1+\cdots+a_{km} Z_m+b_k \epsilon_k \right) ,\quad k=1,\ldots,d
\]
where $Z_1,\cdots,Z_m\simiid \mathsf{N}(0,1)$ and $V\sim \mathsf{\chi}^{2} (r)$. Note $X_k$ is in the form
\[
\]
where $Z\sim \mathsf{N}(0,1)$ and $V\sim \mathsf{\chi}^{2} (r)$. This implies that we have the marginal distribution $X_k\sim \mathsf{T} (r)$, which we denote as a $t$ distribution with $r$ degrees of freedom.
Here, probabilities are conditionally independent on $\v Z = \v z$ and $V=v$, that is $\Psi = \{\v Z,V\}$, the default probability of the $k$-th obligor $P_k(\v Z,V)$ is
\begin{equation}
\label{tcon}
\begin{split}
P_k(\v Z,V) &=\bb P (X_k >x_k|\v Z=\v z, V=v)\\
&= \bb P \left(\epsilon_k >\frac{ \sqrt{\frac{v}{r}} x_k-(a_{k1} z_1+\cdots+a_{km} z_m)}{b_k}\right)\\
&=\Phi \left( \frac{(a_{k1} z_1+\cdots+a_{km} z_m)-\sqrt{\frac{v}{r}}x_k}{b_k}\right).
\end{split}
\end{equation}
§.§.§ Crude Monte Carlo
Suppose that the factor loadings matrix $\m A$ and default thresholds $\v x$ are known. For a chosen sample size $N$, confidence level $\alpha$ and degrees of freedom $r$, the CMC algorithm is as follows
: Generating $L$ under a $t$ factor model using CMC
factor loadings $\m A$; cost vector $\v c$; default thresholds $\v x$; sample size $N$; confidence level $\alpha$; loss threshold $\gamma$; degrees of freedom $r$
$k\leftarrow k + 1$
Simulate $V \sim\mathsf{\chi}^2(r)$
Simulate $\v Z \sim\mathsf{N}(\v 0,\m I_m)$
Simulate $\v\epsilon\sim\mathsf{N}(\v 0,\m I_d)$
$\v X\leftarrow \sqrt{\frac{r}{V}}\left(\m A \v Z +\mathrm{diag}(\v b )\v \epsilon\right)$
$B_i\leftarrow \bb I\{X_i> x_i\}$ for $i=1,\ldots,d$
$L_k \leftarrow \v c^\top \v B$
$k = N$
Sort the losses $\v L\leftarrow (L_{[1]},\ldots,L_{[N]})^\top$
$\hat{\ell} (\gamma) = \frac{1}{N} \sum_{k=1}^N \bb I (L_k > \gamma)$
$\hat{v}_\alpha\leftarrow L_{\lceil \alpha \times N\rceil}$
§.§.§ Importance Sampling
From the well-founded IS framework under the Gaussian factor model, much of the literature consider exponential twisting to the $t$ factors by first conditioning on $V$. Under $t$ factor models we note that the factors $\v Z$ contribute little to the occurrence of defaults as opposed to the case of Gaussian factor models. Rather, it is the value of $V$ that plays a much bigger role as a common shock factor. <cit.> applies exponential twisting to $V$, with the twisting parameter $\theta_V$ found as the solution to a linearly constrained optimization problem. The approach makes use of the observation that, conditional on $V$, one can apply the same IS algorithm as in <cit.> with modified thresholds for each obligor. By asymptotically optimal results, it forms an approximate zero-variance IS pdf for $V$ and implicitly applies the constant approximation through the constrained sets specified in the optimization problem when solving for the twisting parameter $\theta_V$. The algorithm is however, computationally expensive as it requires solving the optimization problem multiple times per sample. <cit.> suggests combining stratified sampling with IS to reduce this cost.
For a general single factor model where $V$ need not follow a Chi-squared or Gamma distribution, <cit.> presents two IS algorithms which apply exponential twisting on $V$ and $W=\frac{1}{V}$ with the twisting parameter $\theta$ found by solving optimization problems on the uniform upper bound of the likelihood ratio estimator in a similar manner to the tail bound approximation. Another recent advancement applies a shift in factors as described in the two-step IS algorithm and adjusting the degrees of freedom of $V$ to maintain independence <cit.>.
It can be shown that for $\lambda^2=\frac{V}{r}$ where $V\sim \mathsf{\chi}^{2} (r)$ then $\lambda^2 \sim \mathsf{G}(\frac{r}{2},\frac{r}{2})$. <cit.> utilizes ordered values of $\{ \frac{X_k}{x_k} \}$ and applies the cross-entropy method to efficiently sample large loss probabilities from a general $t$ copula $m$ factor model. Conditional on $\v Z$ and $\v \epsilon$, we can arrange the order statistics of $\{ \frac{X_k}{x_k} \}$ with corresponding costs $\{c_i\}$. The event $\{L>x\}$ occurs when $\lambda < \frac{X_{(i)}}{x_{(i)}}$ where $i=\min\{j:\sum_{k=j+1}^d c_{(k)}\leq \gamma\}$ so we can write
\[
\begin{split}
\bb P(L>x|Z,\v \epsilon)&=\bb P\left(\lambda < \frac{X_{(i)}}{x_{(i)}}\Bigg|Z,\v \epsilon\right)\\
&=F_\mathsf{G} \left(\frac{X_{(i)}^2}{x_{(i)}^2}\Bigg|Z,\v \epsilon\right).
\end{split}
\]
With this formulation, the cross-entropy method is applied to choose an optimal IS density $g^*(\v Z, \v \epsilon;\v v^*)$ which belongs to the parametric density family $\mathcal{F}$ defined as
\[
\begin{split}
\mathcal{F} =\left \{f(\v Z, \v \epsilon;\v v)=\prod_{j=1}^mf(Z_j; \mu_{Z},\sigma^2_{Z})\prod_{k=1}^d f(\epsilon_k; \mu_{\v \epsilon},\sigma^2_{\v \epsilon}) \right \}
\end{split}
\]
where $\v v^*=(\mu^*_{Z},\m \sigma^{2*}_{Z},\mu^*_{\v \epsilon}, \sigma^{2*}_{\v \epsilon})$ and $\v v=(\mu_{Z},\m \sigma^{2}_{Z},\mu_{\v \epsilon}, \sigma^{2}_{\v \epsilon})$. This formulation gives the following IS estimator
\[
\hat{\ell}=\frac{1}{N} \sum_{k=1}^N F_\mathsf{G} \left(\frac{X_{(i)}^2}{x_{(i)}^2}\Bigg|\v Z^*_k,\v \epsilon^*_k\right) \frac{f(\v Z^*_k,\v \epsilon^*_k;\v v)}{g^*(\v Z^*_k,\v \epsilon^*_k;\v v^*)}.
\]
§.§ Numerical Example: Gaussian and $t$ Factor Models
We illustrate CMC and IS with an example from <cit.> and <cit.> where we apply the two-step IS algorithm for the Gaussian and CE for the $t$ factor model. Here we consider a portfolio of size $d =1000$ under a $m=21$ factor model with costs, marginal probabilities and thresholds as follows.
\[
\begin{split}
c_k &=\left( \lceil \frac{5k}{d} \rceil \right)^2,\\
P_k &= 0.01 \times \left( 1 + \sin \left( \frac{16\pi k}{d}\right)\right),\\
x_{\mathsf{G},k} &= \Phi^{-1}(1-P_k),\\
x_{\mathsf{T}(r),k} &= F^{-1}_{\mathsf{T}(r)}(1-P_k), \quad k=1,\ldots,d
\end{split}
\]
where $x_{\mathsf{G},k}$ and $x_{\mathsf{T}(r),k}$ are the default thresholds for the $k$-th obligor under the Gaussian and $t$ factor model with $r=3$ degrees of freedom respectively.
The factor loadings matrix $\m A$ has the block structure
\[
\m A =
\begin{pmatrix}
\v r
\begin{bmatrix}
\v f & & \\
& \ddots & \\
& & \v f
\end{bmatrix}
\begin{array}{c}
\m G \\
\vdots \\
\m G
\end{array}
\end{pmatrix},
\quad \textrm{with $\m G=
\begin{pmatrix}
\v g & & \\
& \ddots & \\
& & \v g
\end{pmatrix}$},
\]
where $\v r$ is a column vector of $1000$ entries, all equal to $0.8$; $\v f$ is a column vector of $100$ entries, all equal to $0.4$; $\m G$ is a $100\times10$ matrix with $\v g$ a column vector of $10$ entries, all equal to $0.4$. The conditional probabilities used in the two-step IS algorithm are calculated as in (<ref>).
Estimation of risk measures for a Gaussian factor model
$\alpha$ $N$ $\hat{\ell}^{IS}$ $RE(\%)$ $\hat{v}_\alpha$ $\hat{v}_\alpha^{IS}$ $\hat{c}_\alpha$ $\hat{c}_\alpha^{IS}$
$0.95$ $10^4$ $0.0482$ $1.08$ $530$ $548$ $1607$ $1650$
$0.99$ $10^5$ $0.01$ $0.59$ $2310$ $2361$ $3720$ $3862$
$0.995$ $10^5$ $0.005$ $0.63$ $3376$ $3039$ $4863$ $5585$
Estimation of risk measures for a $t$ factor model
$\alpha$ $N$ $\hat{\ell}^{CE}$ $RE(\%)$ $\hat{v}_\alpha$ $\hat{v}_\alpha^{CE}$ $\hat{c}_\alpha$ $\hat{c}_\alpha^{CE}$
$0.95$ $10^4$ $0.0480$ $1.26$ $388$ $352$ $2144$ $1934$
$0.99$ $10^5$ $0.0106$ $0.48$ $2934$ $3072$ $4733$ $5171$
$0.995$ $10^5$ $0.0060$ $0.77$ $4272$ $4684$ $5931$ $6539$
We first use CMC to generate a sample of size $N$ for our initial estimates $\hat{v}_\alpha$ and $\hat{c}_\alpha$. We proceed to apply the two-step IS algorithm where we set $\hat{v}_\alpha$ as the loss threshold $\gamma$. In effect, we aim to simulate around the $\alpha$-VaR for the significance level specified in the above table. To apply the two-step IS algorithm, we have used the tail bound approximation to find the shifted mean vector for the risk factors $\v \mu^*$. Note that the sample size has been increased for higher levels of $\alpha$ to allow for generation of more loss values in the upper tail of the loss distribution.
After obtaining our parameter estimates for the two-step IS algorithm and CE, we have run $10$ iterations of both methods; with each iteration generating an elite sample of size $10^4$ to calculate $\hat{\ell}^{IS}$ and $\hat{\ell}^{CE}$. The mean and relative error of the $10$ values of $\hat{\ell}^{IS}$ and $\hat{\ell}^{CE}$ are calculated with the results shown in Table <ref>. From Table <ref>, we can see that the mean of the values $\{\hat{\ell}^{IS}\}$ and $\{\hat{\ell}^{CE}\}$ are close to the desired values of $1-\alpha$ with RE of $0.59\%$ to $1.08\%$ and $0.48\%$ to $1.26\%$ respectively.
§ COPULA MODELS
A copula is defined as a multivariate distribution in the form
\[
C\left(u_1, \dots, u_d \right) = \mathbb{P} \left(U_1 \leq u_1, \dots, U_d \leq u_d \right),
\]
where $U_1,\cdots ,U_d$ are marginal uniformly distributed variables. Now $U_1,\cdots ,U_d$ can be written with respect to random variables $X_1,\cdots,X_n$ with marginal distributions $F_1,\cdots,F_d$. Thus we have
\[
\left(U_1, \dots, U_d \right) = \left(F_1(X_1),\dots ,F_d(X_d) \right).
\]
Hence the dependency of $\{X_k\}$ can be described individually through their marginal distributions by setting
\[
\left(X_1, \dots, X_d \right) = \left(F_{1}^{-1}(U_1),\dots ,F_{n}^{-1}(U_d) \right).
\]
We focus our discussion on a popular class of copulas known as Archimedean Copulas. Archimedean copulas have the following form
\[
C\left(u_1, \dots, u_d \right) = \psi ^{-1} \left( \psi(u_1) + \cdots + \psi(u_d) \right),
\]
where the function $\psi : [0,1] \rightarrow [0,\infty]$ is strictly decreasing with $\psi (0)=\infty$, $\psi (1)=0$ and its inverse $\psi ^{-1}$ monotonic. This class of copulas includes the Gumbel copula, where $\psi_{\eta}(u)=\left(-\log u \right)^{\eta}$, and the Clayton copula, where $\psi_{\eta} (u) = \frac{1}{\eta} (u^{-\eta} - 1)$. Note that the Gumbel copula has dependence in the upper tail while the Clayton copula has dependence in the lower tail.
In the case of the Archimedean copulas, we can simulate the requisite
vector $\v U=(U_1,\ldots,U_d)$ as follows.
First, simulate $\Lambda\geq 0$ from the distribution $F_\Lambda(\lambda)$, where
the pdf $f_\Lambda$ has the Laplace transform
\[
\int_0^\infty f_\Lambda(\lambda)\exp(-u\lambda) \m d \lambda=\psi^{-1}(u),\quad u\geq 0.
\]
Note that $\psi^{-1}(u)$, with $\psi^{-1}(0)=1$ and $\psi^{-1}(\infty)=0$, is then a decreasing completely monotone function, as required for the copula definition.
Then, given $\Lambda$, simulate $E_1,\ldots,E_n\simiid \mathsf{Exp}(1)$ and output
\[
(U_1,\ldots,U_d)=\left(\psi^{-1}\left(\frac{E_1}{\Lambda}\right),\ldots, \psi^{-1}\left(\frac{E_d}{\Lambda}\right)\right),
\]
\[
(X_1,\ldots,X_d)=\left(F_1^{-1}\left(\psi^{-1}\left(\frac{E_1}{\Lambda}\right)\right),\ldots, F_d^{-1}\left(\psi^{-1}\left(\frac{E_d}{\Lambda}\right)\right)\right),
\]
Here, probabilities are conditionally independent on $\Lambda = \lambda$, that is $\Psi = \Lambda$, the default probability of the $k$-th obligor $P_k(\Lambda)$ is
\[
\begin{split}
P_k(\Lambda) &= \bb P (X_k>x_k|\Lambda=\lambda)\\
&=\bb P (E_k<\lambda \psi(F_k(x_k))) \quad \textrm{ since $\psi$ is invertible and decreasing}\\
&=1-\exp (\lambda \psi(F_k(x_k))).
\end{split}
\]
where $\{x_k\}$ are fixed thresholds similar to those present in factor models. Note that the nature of thresholds in copula models are often default times <cit.> and although this is also possible to model under factor models, the thresholds under factor models are more easily interpreted as economic and market risk thresholds.
§.§ Crude Monte Carlo
Suppose that the distribution $F_\Lambda$ is known. For a chosen sample size $N$ and confidence level $\alpha$, the CMC algorithm is as follows
: Generating $L$ under a Archimedean copula model using CMC
cost vector $\v c$; default thresholds $\v x$, sample size $N$; confidence level $\alpha$; loss threshold $x$; distribution $F_\Lambda$
$t\leftarrow t + 1$
Simulate $\Lambda \sim F_\Lambda$
Simulate $E_i \sim\mathsf{Exp}(1)$ for $i=1,\ldots,d$
$\v U\leftarrow \frac{\v E}{\Lambda}$
$X_k\leftarrow F_k^{-1}(U_k)$ for $k=1,\ldots,d$
$B_k\leftarrow \bb I\{X_k> x_k\}$ for $k=1,\ldots,d$
$L_t \leftarrow \v c^\top \v B$
$t = N$
Sort the losses $\v L\leftarrow (L_{[1]},\ldots,L_{[N]})^\top$
$\hat{\ell} (x) = \frac{1}{N} \sum_{k=1}^N \bb I (L_k > x)$
$\hat{v}_\alpha\leftarrow L_{\lceil \alpha \times N\rceil}$
§.§ Importance Sampling
Under an Archimedean copula model, $\{X_k\}$ are independent conditional on $\Lambda=\lambda$ with probabilities
\[
P_k(X_k>x_k) = 1-\exp (\lambda \psi(F_k(x_k))).
\]
Thus, we can apply the one-step IS algorithm similar to the case of the Gaussian factor model by now generating the default indicators as $B_k \sim \mathsf{B}(P_k(\Lambda))$. We then proceed by applying exponential twisting to the probabilities $P_k(\Lambda)$ with twisting parameter $\theta$, computing $L$ and the likelihood ratio
\[
W(L)=\exp\left(-\theta L+\sum_{k=1}^m \log[1+P_k(\v Z)(\exp(\theta c_k)-1)] \right)
\]
for a sample size $N$. Our IS estimator for $\ell$ remains unchanged
\[
\hat{\ell} (\gamma) = \frac{1}{N} \sum_{t=1}^N \bb I (L_t > \gamma) W(L_t).
\]
§.§ Numerical Example: a Clayton Copula Model
We now illustrate CMC and IS for a Clayton copula model with marginal $\mathsf{Exp}(1)$ factors. The copula parameter $\eta$, default indicators and costs are given as follows
\[
\begin{split}
\eta &=5.5,\\
B_k &= \bb I (X_k>3),\\
c_k &=1 \quad k=1,\ldots,d.
\end{split}
\]
Estimation of risk measures for a Clayton copula model
$\alpha$ $N$ $\hat{\ell}^{IS}$ $RE(\%)$ $\hat{v}_\alpha$ $\hat{v}_\alpha^{IS}$ $\hat{c}_\alpha$ $\hat{c}_\alpha^{IS}$
$0.95$ $10^4$ $0.0476$ $5.30$ $55$ $55$ $94$ $92$
$0.99$ $10^5$ $0.0093$ $2.92$ $121$ $119$ $162$ $158$
$0.995$ $10^5$ $0.0041$ $5.16$ $154$ $146$ $197$ $187$
We first use CMC to first generate a sample of size $N$ and used to give the estimates $\hat{v}_\alpha$ and $\hat{c}_\alpha$. We proceed to apply the one-step IS algorithm where we use $\hat{v}_\alpha$ as the loss threshold $\gamma$. In effect, we aim to simulate around the $\alpha$-VaR for the significance level specified in the above table. Note that the sample size has once again, been increased for higher levels of $\alpha$ to allow for generation of more loss values in the upper tail of the loss distribution. $10$ iterations of the one-step algorithm are then run; with each iteration generating an elite sample of size $10^4$ to calculate $\hat{\ell}^{IS}$. The mean and relative error of the $10$ values of $\hat{\ell}^{IS}$ are calculated with the results shown in Table <ref>. From Table <ref>, we can see that although the mean of the values $\{\hat{\ell}^{IS}\}$ are close to the desired values of $1-\alpha$, the relative error of the one-step IS algorithm is $2.92\%$ to $5.30\%$ as opposed to $0.59\%$ to $1.08\%$ from the two-step IS algorithm. From this, we can see that the shift in factors for the Gaussian factor does indeed provide greater variance reduction for $\{\hat{\ell}^{IS}\}$. We now wish to compare this performance with our proposed algorithm, which we refer to as the Dynamic Splitting Method.
CHAPTER: SPLITTING SIMULATION
We now assume that $S$ is a quasi-monotone function, that is, $x_i\leq y_i,\; i=1,\ldots,d$ implies $S(\bx)\leq S(\by)$.
In other words, a quasi-monotone function is one in which an increase in one of its components cannot reduce its value. It turns out that a large number of applied models either possess this property, or can be transformed into models possessing it. In the absence of any special properties of $f$ or $S$ in (<ref>), there is little hope that one can do better than MCMC.
The purpose of this chapter is to show how we can simulate efficiently from (<ref>) whenever the importance function $S$ is a quasi-monotone function. $S$ is a quasi-monotone function if $x_i\leq y_i,\; i=1,\ldots,d$, implies $S(\bx)\leq S(\by)$. In other words, a quasi-monotone function is one in which an increase in one of its components cannot reduce its value. It turns out that a large number of applied models either possess this property, or can be transformed into models possessing it.
The proposed algorithm is an ingenious application of the DS algorithm for simulation of Markov processes conditional on a rare event <cit.>. In its original form, DS cannot be applied to simulate from (<ref>), because there is no underlying Markov process that we can split. Our algorithm can be viewed as a way of transforming the problem of simulation from (<ref>) to one which involves the simulation of a Markov process. The underlying idea is to embed the static density (<ref>) within a continuous time Markov process in such a way that, at a particular instant of time, the Markov process has the exact same density as (<ref>).
Given this embedding, we can then apply the original splitting method of <cit.>.
We emphasize that, unlike generalized splitting
<cit.> and
subset simulation <cit.>, the classical splitting algorithm does not employ MCMC sampling.
§ CLASSICAL SPLITTING METHOD
In classical splitting, we consider a Markov process $\{\bX_t,t\geq0\}$ with the importance function $S$ over a state space $\scX$. It is assumed that $S(\bX_0)=0$ and that for any threshold $\gamma>0$, there are unique entry times to the sets $\{S(\bX_t)\geq \gamma\}$ and $\{S(\bX_t)\leq 0\}$ respectively given by
\[
\tau_\gamma=\min\{t:S(\bX(t))\geq\gamma\},
\]
\[
\tau_0=\min\{t:S(\bX(t))\leq 0\}.
\]
Note that $\tau_0$ only exists in the absence of the quasi-monotonic property in $S$, which is the case in classical splitting. Here the probability of interest is $\ell=\bb P(\tau_\gamma <\tau_0)$; the probability that the process reaches the threshold $\gamma$ before reaching $0$. Hence, $\ell$ depends on the distribution of $\bX_0$. Suppose there exists thresholds $\gamma_1$ and $\gamma_2$ such that $\gamma_2>\gamma_1$. The cornerstone of the splitting method is the observation that the process must first reach $\gamma_1$ to reach $\gamma_2$, and thus, giving us a sequence of nested event subsets
\[
E_{\gamma_2} =\{ \tau_{\gamma_2} <\tau_0 \} \subset E_{\gamma_1} =\{ \tau_{\gamma_1} <\tau_0 \}.
\]
Hence $\ell = \bb P(E_{\gamma_2}\gvn E_{\gamma_1})\bb P(E_{\gamma_1})$, a product of conditional probabilities. This can similarly be extended to more threshold levels, say $0=\gamma_0<\gamma_1\cdots<\gamma_L=\gamma$ giving us
\[
E_{\gamma_0} \supseteq E_{\gamma_1}\supseteq \cdots \supseteq E_{\gamma_L}.
\]
Let $c_i=\bb P(E_{\gamma_i}\gvn E_{\gamma_{i-1}})$ then we have $\ell=\prod_{i=1}^Lc_i$. Each $c_i$ is then estimated as follows.
Define $\scX_i=\{\bX_{t_i}:S(\bX_{t_i})\geq \gamma_i\}$, that is, the set of states in the Markov process that reach threshold $\gamma_i$ at time $t_i$ for $t_1, \cdots ,t_L$. At each $t_{i} \in \{t_1, \cdots ,t_L \}$, we run $s_i$ copies of the Markov process $\{\bX_{t_{i-1}}\}$ for each $\bX_{t_{i-1}} \in \scX_{i-1}$; giving $s_i|\scX_{i-1}|$ copies of $\{\bX_{t_{i}}\}$ and the corresponding evolved process $\{S(\bX_{t_{i}})\}$. Each copy of $\{\bX_{t_{i-1}}\}$ is run until $\{S(\bX_{t_i})\}$ either reaches the set $\{S(\bX_{t_i})\geq \gamma_i\}$ or $\{S(\bX_{t_i})\leq 0\}$ with ending state $\bX_{t_{i}}$. Each state $\bX_{t_i}$ that reaches the set $\{S(\bX_{t_i})\geq \gamma_i\}$ before the set $\{S(\bX_{t_i})\leq 0\}$, referred to as an entrance state, is then stored as the set $\scX_i$. It is from these states that the next iteration of the splitting method will begin from; giving rise to $s_{i+1}|\scX_i|$ sample paths. An unbiased estimate of $c_i$ is given by $\hat{c}_i=\frac{|\scX_i|}{s_i|\scX_{i-1}|}$. It is clear that $|\scX_i|$ is dependent on the entrance states $\scX_{i-1}$. Despite this dependence, <cit.> notes that the following estimate remains unbiased
\[
\begin{split}
\hat{\ell}&=\prod_{i=1}^L\frac{|\scX_i|}{s_i|\scX_{i-1}|}\\
\end{split}
\]
Note we begin splitting at $t_1$ rather than $t_0$. In the above framework, the integer-valued splitting factors $\{s_i\}$ are distinct and thus may vary for each $t_{i} \in \{t_1, \cdots ,t_L \}$. This need not be the case as a predetermined splitting factor $s_i=s$ for all $i$, may be used; this version of the algorithm is referred to as Fixed Factor Splitting. We will use this in combination with Fixed Effort Splitting described later for our results in Chapter <ref>. The splitting process is repeated for all $t_{i} \in \{t_1, \cdots ,t_L \}$ with $s_i|\scX_{i-1}|$ being the simulation effort at $t_i$. Potential problems when using the splitting method are large growths in the simulation effort and inefficiency. These problems come from inappropriate choices in the number of levels $L$, intermediate threshold levels $\{\gamma_1,\ldots,\gamma_{L-1}\}$ and splitting factors $\{s_1,\ldots,s_L\}$. <cit.> notes that ideally, levels should be chosen in such a way that the conditional probabilities $\{c_i\}$ can be easily estimated with CMC. Under the assumption of independence between the computational cost and time from running the Markov process, the total simulation effort is a random variable with expected value
\[
\begin{split}
\sum_{i=1}^L s_i\bb E [|\scX_{i-1}|]&=\sum_{i=1}^L s_i N_0 \prod_{j=1}^{i-1}c_js_j\\
&=N_0\sum_{i=1}^L \frac{1}{c_i} \prod_{j=1}^ic_js_j
\end{split}
\]
If $c_js_j>1$ for all $j$, the simulation effort would become large as it increases with the number of levels $L$. This phenomenon is referred to as an explosion in <cit.>. If $c_js_j<1$ for all $j$, most sample paths will not reach the threshold levels $\gamma_j$ and as a consequence, the algorithm will be inefficient. Thus, an ideal choice is $s_j=\frac{1}{c_j}$ for all $j$. An alternative approach to avoiding explosions is Fixed Effort Splitting, where the simulation effort is fixed for all $t_{i} \in \{t_1, \cdots ,t_L \}$ to say N, giving us a corresponding estimator
\[
\hat{\ell}_{\mathrm{FE}}=\prod_{i=1}^L \frac{|\scX_i|}{N}.
\]
Our results in Section <ref> will be based on this approach. Now that a basic description of the classical splitting method is given, we present the workings of our proposed algorithm including the embedding of the static density $\eqref{cond}$ in the Markov process $\bX_t$.
§ THE DYNAMIC SPLITTING METHOD FOR STATIC PROBLEMS
We first require a way to apply the classical splitting method of <cit.> to the problem of sampling from (<ref>). To achieve this, we induce an artificial Markov process whose paths we can then repeatedly split to encourage entry into a desired rare-event set, say $\{\bX: S(\bX)\geq\gamma\}$ .
A suitable Markov process for our algorithm is the multivariate Lévy subordinator <cit.>. A $d$-dimensional Lévy subordinator $\{\bLam(t),t\in \mathbb{R}_+\}$ with $\bLam(0)=0$ is an almost surely increasing stochastic process on a probability space $(\Omega,\Pm,\mathcal{F})$ with a continuous index set on $\mathbb{R}_+$ and with a continuous state space
$\mathbb{R}^d$ defined by the following properties:
(1) the increments of $\{\bLam(t)\}$ are stationary and non-negative, that is, $(\bLam(t+s)-\bLam(t))\geq \mathbf{0}$ has the same distribution as $\bLam(s)\geq\mathbf{0}$ for all $t,s\geq 0$; we denote the density of this stationary distribution as $\nu_s(\v\lambda)$;
(2) the increments of $\{\bLam(t)\}$ are independent, that is, $\bLam(t_i)-\bLam(t_{i-1}),\; i=1,2,\ldots$ are independent for any
$0\leq t_0<t_1<t_2<\cdots$; and
(3) for any $\epsilon>0$, we have $\Pm(\|\bLam(t+s)-\bLam(t)\|\geq\epsilon)=0$ as $s\downarrow 0$. The distributional properties of the Lévy subordinator are characterized by the characteristic exponent which is expressed as the logarithm of the characteristic function of the random column vector $\bLam(1)$:
\begin{equation*}
\log\Em[\exp(\i \v s^\top \bLam(1))]=\i \v s^\top\v\mu+\int_{\mathbb{R}^d}\left(\exp(\i \v s^\top \bx)-1-\i \v s^\top \bx\,\bb I_{\{\|\bx\|\leq 1\}}\right) \nu(\di \bx)\;,\quad \v s\in\mathbb{R}^d,
\end{equation*}
for some $\v\mu\in \mathbb{R}^d$, and measure $\nu$
such that $\nu(\{\mathbf{0}\})=0$ and $\int_{\mathbb{R}^d}\min\{1,\|\bx\|^2\}\nu(\di \bx)<\infty$.
One of the simplest multivariate subordinators we can have is the gamma process with independent components <cit.> in which the components $\Lambda_1(t),\ldots,\Lambda_d(t)$ of vector $\bLam(t)=(\Lambda_1(t),\ldots,\Lambda_d(t))^\top$
are independent one-dimensional subordinators with characteristic exponent
$\log((1-i s)^{-1})$. In other words, each $\Lambda(t)$ has gamma distribution with shape parameter $t$ and scale parameter $1$, which denote by $\mathsf{G}(t,1)$. Note that other multivariate subordinators are possible as long as they possess the quasi-monotonicity property, but the Gamma process with independent components will suffice for our illustrations.
[Simulating Gamma process]
Consider simulating a Gamma process at distinct times.
: Simulating Gamma Process at distinct times
Intermediate times $0=t_0<t_1<t_2<\cdots<t_L=1$
$\bLam(t_0)\leftarrow 0$
Simulate $\bLam^*(t_i-t_{i-1})$ independently from subordinator distribution.
$\bLam(t_{i})\leftarrow \bLam(t_{i-1})+\bLam^*(t_i-t_{i-1})$
Simulation of 1D Gamma processes
Simulation of a 3D Gamma process
Now that we have defined one of the simplest continuous-time processes (the gamma subordinator above),
we can proceed to embed the distribution of $\bX=(X_1,\ldots,X_d)^\top$ in (<ref>) within a
continuous time process as follows. Let $F_k(x)$ be the cdf of $X_k$ for $k=1,\ldots,d$. Define the vector $\bX(t)=(X_1(t),\ldots,X_d(t))^\top$ through the random variables
\begin{equation}
\label{embedtransform-dec}
\end{equation}
With this formulation, the non-decreasing property of the gamma subordinator implies that we will have an almost surely partial (and in fact total) ordering for the vectors $\bX(t)\succ\bX(t+s)$, meaning that $X_i(t)\geq X_i(t+s)$ for all $i$ and $s,t>0$. In addition, it can be easily verified that at time $t=1$ each $X_k(1)$ has the desired distribution $\Pm(X_k(1)\leq x)=F_k(x)$. This observation is what connects our induced Markov process to the static distribution of $\bX$. In fact, we can view the realization of the original vector $\bX$ as a snapshot of the state of a multivariate continuous-time process $\{\bX(t),t\geq 0\}$ at the instant $t=1$. Notice that $\Lambda_k(0)=0$ so we have $X_k(0)=\infty$. Consequently, we begin simulations in the set $\{S(\bX\geq\gamma)\}$ and evolve the gamma subordinator towards the set $\{S(\bX<\gamma)\}$ as a result of the ordering for the vectors $\bX(t)\succ\bX(t+s)$ for all $i$ and $s,t>0$. As this property suggests, we will refer to (<ref>) as a monotonically decreasing embedding transformation. Our numerical experiment results will be based on this transformation.
The above formulation now has a number of implications. First, the quasi-monotonicity of the importance function $S(\bX)$ implies that the exit time exists and is unique
\begin{equation}
\label{exit time}
\tau_\gamma=\sup\{t:S(\bX(t))\geq\gamma\}
\end{equation}
and in fact
$\Pm(S(\bX(t))\geq \gamma)=\Pm(\tau_\gamma>t).$
Second, if
is any sequence of increasing times, the state space can be decomposed into the decreasing sequence of events:
\[
\mathbb{R}^d\equiv \{S(\bX(t_0))\geq \gamma\}\supseteq \{S(\bX(t_1))\geq \gamma\}\cdots\supseteq \{S(\bX(t_{L}))\geq \gamma\}
\]
and therefore we have the following decomposition of (<ref>)
\begin{align*}
\ell=\Pm(S(\bX(1))\geq \gamma)&=\prod_{i=1}^L\Pm(S(\bX(t_{i}))\geq \gamma \gvn S(\bX(t_{i-1}))\geq \gamma)\\
&=\prod_{i=1}^L\Pm(\tau_{\gamma}>t_i\gvn \tau_{\gamma}>t_{i-1})\,.
\end{align*}
At each intermediate point in time $t_{i} \in \{t_1, \cdots ,t_{L-1} \}$ for $i=1,\dots,L-1$, we consider $s$ splits in $s^{i-1}$ sample paths of the gamma subordinator for each component of $\bLam (t_i)$. From this, it is clear the estimation of each $\Pm(\tau_{\gamma}>t_i\gvn \tau_{\gamma}>t_{i-1})$ can be done empirically as
\[
\Pm(\tau_{\gamma}>t_i\gvn \tau_{\gamma}>t_{i-1})= \frac{|\scX_i|}{s|\scX_{i-1}|}.
\]
Note that since we begin the splitting of the sample paths at $t=t_1$ so we have
\[
\Pm(\tau_{\gamma}>t_1\gvn \tau_{\gamma}>t_{0})= \frac{|\scX_1|}{|\scX_{0}|}.
\]
$\ell$ will thus be calculated as
\[
\begin{split}
\ell&=\frac{|\scX_1|}{|\scX_{0}|}\prod_{i=1}^{L-1}\frac{|\scX_{i+1}|}{s|\scX_{i}|}\\
&=\frac{|\scX_{L}|}{s^{L-1}} \quad \textrm{if $|\scX_{0}|=1$}.
\end{split}
\]
For completeness, we will also define the monotonically increasing embedding transformation. We consider the vector $\bX(t)=(X_1(t),\ldots,X_d(t))^\top$ with random variables is defined as
\begin{equation}
\label{embedtransform-inc}
\end{equation}
This transformation utilizes the observation that for any Uniformly distributed random variable $U$, $1-U$ is also uniform. This comes from the observation that $\Lambda_k(1) \sim \mathsf{Exp}(1)$ so its corresponding cdf value $1-\exp(-\Lambda_k(1))$ is Uniformly distributed and thus, $\exp(-\Lambda_k(1))$ is also Uniformly distributed. It can be easily verified that (<ref>) will also provide us with the desired distribution $\bb P(X_k(1)\leq x)=F_k(x)$. However, the consequences of this transformation differs from those of (<ref>).
The non-decreasing property of the Gamma subordinator will now lead to increases in $X_k(t)$ as $t$ increases. As a result, we will now have the partial or total ordering for the vectors $\bX(t)\prec\bX(t+s)$, that is, $X_i(t)\leq X_i(t+s)$ for all $i$ and $s,t>0$. As $\Lambda_k(0)=0$ we will have $X_k(0)=\min \{-\infty,0\}$ depending on the marginal distribution $F_k$. We will assume $X_k(0)=0$ for our purposes as we aim our discussion to the models described in Chapter <ref>. We note that all random variables that require this transformation in these models are indeed non-negative. Under this transformation we have $S(\bX(0))=0$ and begin the Markov process in the set $\{S(\bX(t))< \gamma\}$. We wish to estimate $\ell$ by modelling the entry time to the set $\{S(\bX(t))\geq \gamma\}$
\[
\tau_\gamma=\min\{t:S(\bX(t))\geq\gamma\},
\]
Hence, under the transformation (<ref>) we arrive at the classical splitting method as described in section <ref> with the modification that $\tau_0$ does not exist as a consequence of the quasi-monotonicity of the importance function and that we now seek
\begin{align*}
\ell = \bb P(S(\bX(1))\geq \gamma) &=\prod_{i=1}^L\Pm(S(\bX(t_{i}))\geq \gamma \gvn S(\bX(t_{i-1}))\geq \gamma)\\
&=\prod_{i=1}^L\Pm(\tau_{\gamma}<t_i\gvn \tau_{\gamma}<t_{i-1}).
\end{align*}
We can summarize the fixed factor dynamic splitting algorithm as follows.
Splitting of paths
: Fixed Factor Dynamic Splitting, returning $W$, an unbiased estimate of $\ell$
Splitting factor $s$ and intermediate times
Generate $\bLam(t_1)=(\Lambda_1(t_1),\ldots,\Lambda_{d}(t_1))^\top$ from the
subordinator distribution.
$ S(\bLam(t_1)) > \gamma$
$\scX_1 \leftarrow \{\bLam(t_1)\}$
$W\g 0$
$W\g 0$
$\scX_i \g \emptyset$
For $k=1,\ldots,d$ sample independently
\[
\Lambda_k^*(t_i-t_{i-1})\sim\mathsf{G}(t_i-t_{i-1},1).
\]
$\bLam(t_{i})\g \bLam(t_{i-1})+\bLam^*(t_i-t_{i-1})$
$ S(\bLam(t_i)) > \gamma$
add $\bLam(t_i)$ to $\scX_i$
$W \g |\scX_{L}| / s^{L-1}$ as an unbiased estimate.
In the above formulation, the splitting factor $s$ is chosen arbitrarily but under two idealizing assumptions, a near optimal value can be chosen for $s$ (see Appendix <ref>).
As noted previously, <ref> has the risk of explosions so we implement a fixed effort variant of the above algorithm to avoid this. At each $t_i$, we fix the simulation effort to $s$ splits for a randomly chosen $\bLam(t_{i-1}) \in \scX_{i-1}$. The fixed effort dynamic splitting algorithm can be summarized as follows.
: Fixed Effort Dynamic Splitting, returning $W$, an unbiased estimate of $\ell$
Total sample at each level $s$; intermediate times
Generate $\bLam(t_1)=(\Lambda_1(t_1),\ldots,\Lambda_{d}(t_1))^\top$ from the
subordinator distribution.
$ S(\bLam(t_1)) > \gamma$
Add $\bLam(t_1)$ to $\scX_1$
$W\g 0$
$\scX_i \g \emptyset$
Let $\bLam(t_{i-1})$ be a randomly chosen member of $\scX_{i-1}$.
For $k=1,\ldots,d$ sample independently
\[
\Lambda_k^*(t_i-t_{i-1})\sim\mathsf{G}(t_i-t_{i-1},1).
\]
$\bLam(t_{i})\g \bLam(t_{i-1})+\bLam^*(t_i-t_{i-1})$
$ S(\bLam(t_i)) > \gamma$
add $\bLam(t_i)$ to $\scX_i$
$W \leftarrow \prod_{i=1}^L |\scX_i|/s^L$ as an estimate of $\ell$.
§ NUMERICAL EXPERIMENTS
We now illustrate the performance of dynamic splitting (DS) in the credit risk models described in Chapter <ref> through numerical experiments. For these results, we first verify that the importance function $S(\bX)=L(\bX)$ is indeed quasi-monotonic and specify suitable transformation (<ref>) and (<ref>) to embed the static random variables $\{X_k\}$ into time-dependent random variables $\{X_k(t)\}$. We first note that all of these models have the form
\[
\begin{split}
&=\v c^\top \v B(\bX)
\end{split}
\]
where the $k$-th component of $\v B$ is generated as $\bb I(X_k>x_k)$. An increase in one of the components of $\bX$ will lead to the value of $L(\bX)$ either staying the same or increased due to the activation of the corresponding indicator variable. It is clear that the function value of $L(\bX)$ will not be reduced so the importance function $S(\bX)=L(\bX)$ does indeed possess the quasi-monotonicity property required for our algorithm. We now present the results of the application of our algorithm to the Gaussian factor model, $t$ factor model and Clayton copula model. In the following results, we have applied two-step IS for the Gaussian factor model, the CE-based estimator from <cit.> for the $t$ factor model and one-step IS for the Clayton copula module. We note that the threshold $\gamma$ is chosen with an initial run of IS before our splitting algorithm is applied.
In the following results, $\gamma$ is the $\alpha$-quantile of the loss distribution estimated with IS from Chapter <ref> while $\hat{\ell}^{IS}$ and $\hat{\ell}^{DS}$ are the probabilities of the a loss exceeding $\gamma$. Hence, ideally we expect $\alpha+\hat{\ell}^{IS}$ and $\alpha+\hat{\ell}^{DS}$ to both equal to $1$. Disregarding the initial simulation effort for $\gamma$, all algorithms implemented below have been constrained to a simulation effort of $10^5$ and the same number of runs $R=10$. For IS and CE estimators, this is simply $NR$ where $N=10^4$ is the sample size in each run of the IS algorithm. For the DS estimator, it is $sTR$ where $s=1000$ is the splitting factor and $T=10$ is the number of intermediate time levels. Note, we have made appropriate adjustments to $s$ and $T$ to match the simulation effort of both algorithms in each comparison below.
§.§ Factor Models
We illustrate the performance of the splitting algorithm with factor models. We assume that $\m A$ has the matrix structure given in our worked example
\[
\m A =
\begin{pmatrix}
\v r
\begin{bmatrix}
\v f & & \\
& \ddots & \\
& & \v f
\end{bmatrix}
\begin{array}{c}
\m G \\
\vdots \\
\m G
\end{array}
\end{pmatrix},
\quad \textrm{with $\m G=
\begin{pmatrix}
\v g & & \\
& \ddots & \\
& & \v g
\end{pmatrix}$},
\]
where $\v r$ is a column vector of $1000$ entries, all equal to $0.8$; $\v f$ is a column vector of $100$ entries, all equal to $0.4$; $\m G$ is a $100\times10$ matrix with $\v g$ a column vector of $10$ entries, all equal to $0.4$. $\v b$ is calculated as
\[
b_k=\sqrt{1-(a_{k1}^2+\cdots+a_{km}^2 )}
\]
§.§.§ Gaussian Factor Model
We have
\[
X_k=a_{k1} Z_1+\cdots+a_{km} Z_m+b_k\epsilon_k
\]
where $Z_1,\cdots,Z_m\simiid \mathsf{N}(0,1)$ and $\epsilon_k\sim \mathsf{N}(0,1)$. By using (<ref>) we have
\[
X_k(t)=a_{k1} \Phi^{-1}(\Lambda_1(t))+\cdots+a_{km} \Phi^{-1}(\Lambda_m(t))+b_k\Phi^{-1}(\Lambda_{m+k}(t)).
\]
Here we require gamma processes for $m$ standard Normal random variables which are the systematic risk factors $\v Z$ and $d$ standard Normal random variables for the obligor-specific risk factors $\v \epsilon$.
Estimation of $\ell$ for a Gaussian factor model
$\alpha$ $\gamma$ $\hat{\ell}^{IS}$ $\hat{\ell}^{DS}$ $RE^{IS}(\%)$ $RE^{DS}(\%)$
$0.95$ $548$ $0.0493$ $0.0497$ $0.73$ $4.35$
$0.99$ $2361$ $0.0098$ $0.0103$ $0.60$ $6.82$
$0.995$ $3039$ $0.0062$ $0.0056$ $0.58$ $7.28$
We can see that both $\alpha+\hat{\ell}^{IS}$ and $\alpha+\hat{\ell}^{DS}$ do indeed equal to $1$ approximately. However, the relative error of the DS estimator has a much larger relative error $RE^{DS}$ than the relative error for the two-step IS estimator $RE^{IS}$. Thus, in this study the DS estimator does not perform as efficiently as the two-step IS estimator.
§.§.§ $t$ Factor Model
We have
\[
\begin{split}
X_k&=\sqrt{\frac{r}{V}}\left(a_{k1} Z_1+\cdots+a_{km} Z_m+b_k\epsilon_k\right)\\
&=\sqrt{\frac{1}{G}}\left(a_{k1} Z_1+\cdots+a_{km} Z_m+b_k\epsilon_k\right)
\end{split}
\]
where $Z_1,\cdots,Z_m\simiid \mathsf{N}(0,1)$, $\epsilon_k\sim \mathsf{N}(0,1)$ and $G\sim \mathsf{G}(\frac{v}{2},\frac{v}{2})$. By using (<ref>) and (<ref>) we have
\[
X_k(t)=\frac{\left(a_{k1} \Phi^{-1}(\exp(-\Lambda_1(t)))+\cdots+a_{km} \Phi^{-1}(\exp(-\Lambda_m(t)))+b_k\Phi^{-1}(\exp(-\Lambda_{m+k}(t)))\right)}{\sqrt{F_\mathsf{G}^{-1}(1-\exp(-\Lambda_{d+m}(t)))}}.
\]
Here we require gamma processes for $m$ standard Normal random variables which are the systematic risk factors $\v Z$, $d$ standard Normal random variables for the obligor-specific risk factors $\v \epsilon$ and another for the common random variable $G$.
Estimation of $\ell$ for a $t$ factor model
$\alpha$ $\gamma$ $\hat{\ell}^{CE}$ $\hat{\ell}^{DS}$ $RE^{CE}(\%)$ $RE^{DS}(\%)$
$0.95$ $352$ $0.0500$ $0.0478$ $0.36$ $3.43$
$0.99$ $3072$ $0.0100$ $0.0096$ $1.00$ $6.55$
$0.995$ $4684$ $0.0050$ $0.0047$ $0.72$ $5.76$
The performance of the CE estimator for the $t$ factor model has similar performance efficiency as that of the two-step IS estimator for the Gaussian factor model. This supports our theoretical motivations of these estimators since both estimators aim to significantly reduce the variance of $\hat{\ell}$ (see Chapter <ref>) where CE algorithm aims to sample approximately from the zero-variance density by learning near optimal parameters for the likelihood function while two-step IS reduces the variance via exponential twisting and change in parameter measures. Hence, we expect the relative error of both estimators to be quite small as supported by the above empirical results. We observe that $\alpha+\hat{\ell}^{IS}$ and $\alpha+\hat{\ell}^{DS}$ equal to $1$ approximately. However, the relative error of the DS estimator remains larger than the relative error for the CE estimator $RE^{CE}$. Thus, in this study the DS estimator does not perform as efficiently as the one-step IS estimator.
§.§ Clayton Copula Model
In this example, we consider the variables $\{X_k\}$ with marginal exponential distributions $\mathsf{Exp}(1)$. Note that $\{X_k\}$ are static variables computed as
\[
\begin{split}
X_k &= F_k^{-1}(U_k)\\
\end{split}
\]
where $F_k$ is the marginal distribution of $X_k$, $U_k$ is a uniform variable generated from the Clayton copula, $G\sim \mathsf{G}(\frac{1}{\eta},1)$ and $\psi^{-1}(t)=(1+\eta t)^{\frac{1}{\eta}}$. Let $F_\eta$ be the cdf of the $\mathsf{G}(\frac{1}{\eta},1)$ then an appropriate embedding transformation is
\[
\]
where $\Lambda_k(t),\Lambda_{d+1}(t)\sim \mathsf{G}(t,1)$. Here we require gamma processes for $d$ independent $\mathsf{Exp}(1)$ random variables and another for the common random variable $G$.
Estimation of $\ell$ for a Clayton copula model
$\alpha$ $\gamma$ $\hat{\ell}^{IS}$ $\hat{\ell}^{DS}$ $RE^{IS}(\%)$ $RE^{DS}(\%)$
$0.95$ $55$ $0.0493$ $0.0513$ $1.83$ $4.87$
$0.99$ $119$ $0.0094$ $0.0099$ $4.14$ $6.67$
$0.995$ $146$ $0.0049$ $0.0047$ $5.49$ $8.14$
Once again, we can see that $\alpha+\hat{\ell}^{IS}$ and $\alpha+\hat{\ell}^{DS}$ equal to $1$ approximately. However, the relative error of the DS estimator is still larger than the relative error for the one-step IS estimator $RE^{IS}$. Thus, in this study the DS estimator does not perform as efficiently as the one-step IS estimator.
§ CRITICAL ANALYSIS OF THE DYNAMIC SPLITTING METHOD
In this section, we give a critical analysis of the proposed DS method when applied to rare-event probability estimation. The proposed DS method has advantages in terms of simplicity, versatility and simulation effort. Conversely, the DS approach can be inapplicable if the problem does not possess a quasi-monotonic importance function $S$. Advantages:
* Interpretability $-$ In the DS method, we estimate $\ell$ by computing the product of sequential conditional probabilities. Each conditional probability can be interpreted as a survival probability from the current time level to the next time level. This gives us a simple interpretation to our estimate of $\ell$ as a survival probability from the starting time to the terminal time.
* Versatility $-$ One of the biggest strengths of the proposed DS method is that it is versatile as it can generate any set of continuous random variables $\bX$ by using the inverse-transform method. In particular, we can generate sequential values of each variable such that $\{S(\bX)\geq \gamma\}$. This is possible as the monotonicity of the Gamma process is preserved from under the embedding transformation, thus allowing us generate the set $\{\bX:S(\bX)\geq \gamma\}$. We also note that unlike IS algorithms, DS does not require the computation of likelihood ratios and can be applied on any continuous pdf including heavy-tailed distributions. This is a major advantage for DS when compared to the one-step and two-step IS algorithms which are exclusively applicable to light-tailed distributions such as the Normal distribution. The same argument cannot hold against CE as the only main restriction in CE is that it only considers densities within the family of distributions which is not a strong assumption in itself.
* Simulation effort $-$ A natural question that would arise is the simulation effort and computational cost of DS since it simulates split paths of the Markov chain over many time levels. The danger of explosions can be avoided by a fixed effort implementation and paths $\bX$ for which $\{\bX:S(\bX)< \gamma\}$ are not simulated further. We also note that unlike generalized splitting or particle methods, our proposed DS method does not require approximate MCMC sampling at each level of splitting. These observations ensure that the implementation of our proposed DS method would not be computationally expensive.
* Quasi-monotonicity $-$ Our proposed DS algorithm hinges on the existence of a quasi-monotonic importance function $S$. If the problem of interest has an importance function $S$ that does not possess this property then we cannot apply DS as the absence of quasi-monotonicity in $S$ implies we can no longer apply the embedding transformations to the problem and the decomposition of <ref> into nested subsets no longer holds. Consequently, there is no connection between the static distributions of interest and a time-dependent Markov process, making DS inapplicable.
* Efficiency $-$ As shown by the results of numerical experiments earlier in this chapter, we can see that the RE of the DS method is much larger than current hallmark rare-event probability estimation algorithms such two-step IS and CE based algorithms <cit.>. This would be attributed to the decomposition of time levels to estimate each conditional probability to be inadequate form of variance reduction in comparison to two-step IS and CE, both of which seek to obtain a density with reduced variance for efficient estimation of $\ell$.
CHAPTER: CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
We have developed and illustrated a splitting method which is simple and effective to estimate rare event probabilities in the framework of popular copula credit risk models. This method is designed to estimate tail probabilities by decomposing the state space of the risk variables into nested subsets so that the rare event can be expressed as an intersection of these subsets. This decomposition allows us to achieve greater accuracy in estimating each conditional probability than estimating the rare probability itself. We have also shown that despite the inapplicability of the classical splitting method on static problems, one can always find an appropriate embedding transformation such that the static density being modelled can be taken as a snapshot of a continuous time Markov process at a particular instance in time. The illustrations of importance sampling estimators in this thesis, namely exponential twisting, have relied on conditional independence of the model to simplify the problem to a sum of scaled Bernoulli random variables and attractive forms of the likelihood ratio. Our proposed modification to the dynamic splitting algorithm does not rely on conditional independence, rather it relies on the quasi-monotonicity property of the importance function. A large number of models either possess this property or can achieve it through a transformation. Hence, although the algorithm does not perform as efficiently as current hallmark importance sampling estimators, it is a versatile method that is applicable to any static problem for which random variables required in the model have prespecified distributions.
A possible direction for future research include developing more efficient adaptive dynamic splitting algorithms and variance reduction techniques to dynamic splitting, perhaps by applying exponential twisting to each candidate sample path that will be split. The latter is based on the observation that each sample path that is split in fact Binomial distributed with $s$ trials, that is $\mathsf{Bin}(s,c)$. Possible practical applications for future work may include financial and insurance risk management. This is motivated by the observation that the splitting framework can be interpreted as a model for survival probabilities which is analogous to survival probabilities of insurance policyholders and insurers; and also default and bankruptcy rates of financial institutions. Specifically, it may be useful in the pricing of insurance products as well as risk and solvency capital allocation where the VaR and CVaR are possible risk measures.
CHAPTER: MONTE CARLO METHODS
§ ONE-STEP IS ALGORITHM: SHIFT IN PROBABILITIES
We simulate $L$ under the following model.
\[
L=c_1 B_1+\cdots+c_m B_m= \v c^\top \v B
\]
be the total loss incurred by a portfolio of $m$ obligors, where
$c_i$ is the loss incurred from the $i$-th obligor and $B_i\sim \mathsf{B}(P_i)$ is a Bernoulli random variable indicating whether the $i$-th obligor has defaulted. The distribution of the vector $\v B$ is implicitly
defined via
\[
B_k\idef \bb I\{X_k>x_k\},\quad k=1,\ldots,m
\]
where $\{x_k\}$ are given fixed thresholds and $\v X$ has a continuous joint density $f(\v x)$. Thus, the likehihood of the total loss $L$ is given by
\[
f_{L} (l) = \prod_{k=1}^m P_k^{b_k} (1-P_k)^{1-b_k}
\]
To simulate from the upper tail of the distribution, an exponential twist is applied to the probabilities $\{ P_i\}$ for $\theta \geq 0$ as the first step in importance sampling. It is defined as follows
\[
P_{k,\theta}=\frac{P_k e^{\theta c_k }}{1+P_k (e^{\theta c_k}-1)}.
\]
This change in probability measure results in a density that simulates higher values of $L$ with default indicators now generated by $B_i\sim \mathsf{B}(\tilde P_i)$. Note that the notation $L$ and $B_k$ have been preserved as the desired likelihood ratio should be described under the notation of the original distribution. As the probability of a particular default outcome is being compared across two densities, the likelihood ratio aims to describe the ratio of probabilities for the same outcome. The likelihood of $L$ under the exponentially twisted density $g_L$ is given by
\[
g_{ L} ( l) = \prod_{k=1}^m \tilde P_k^{ b_k} (1-\tilde P_k)^{1- b_k}.
\]
Hence the likelihood ratio $W$ is given by
\[
\begin{split}
W( l ) &= \frac{f_{ L}(l)}{g_{ L}(l)}\\
&= \prod_{k=1}^m \left(\frac{P_k}{\tilde P_k}\right)^{b_k} \left( \frac{1-P_k}{1-\tilde P_k} \right)^{1-b_k} \\
&=\prod_{k=1}^m \left(\frac{1+P_k (e^{\theta c_k}-1)}{e^{\theta c_k}}\right)^{b_k} (1+P_k (e^{\theta c_k}-1))^{1-b_k}\\
&=\prod_{k=1}^m \left(1+P_k (e^{\theta c_k}-1)\right) e^{-\theta b_k c_k}\\
&=\exp\left(-\theta l + \sum_{k=1}^m \log\left(1+P_k(\exp(\theta c_k)-1)\right)\right)
\end{split}
\]
§ TWO-STEP IS ALGORITHM: SHIFT IN FACTORS
As the second step in importance given in <cit.>, the mean of the factors $\v Z$ are shifted from $\v 0$ to $\v \mu$. Assuming at Normal copula, this means we now have $\v Z \sim\mathsf{N}(\v \mu,\m I_d)$ instead of $\v Z \sim\mathsf{N}(\v 0,\m I_d)$. Hence, the change in mean must also form part of the estimator. We note that the normalization constants of Normal densities are unchanged by a shift in the mean so the likelihood ratio is given by
\[
\frac{\exp{(-\frac{1}{2}\v Z^\top \v Z)}}{\exp{(-\frac{1}{2}\v {(\mu - Z)}^\top \v {(\mu - Z)})}}= \exp{(\frac{1}{2}\v \mu^\top \v \mu - \v \mu^\top \v Z)}
\]
§ ARCHIMEDEAN COPULA SAMPLING ALGORITHM
To verify that the algorithm in <cit.> draws $\v U=(U_1, \cdots , U_d)$ from an Archimedean copula, we must show that
\[
P(\v U\leq \v u) = \psi^{-1}\left(\sum_{i=1}^d\psi(u_i)\right),
\]
\[
(U_1,\ldots,U_d)=\left(\psi^{-1}\left(\frac{E_1}{\Lambda}\right),\ldots, \psi^{-1}\left(\frac{E_d}{\Lambda}\right)\right).
\]
The probability of $\v U$ can be written as
\[
\begin{split}
\bb P(\v U\leq \v u)&=\bb P( U_1\leq u_1,\ldots,U_d\leq u_d)\\
&=\bb P( E_1\geq \Lambda\psi(u_1),\ldots,E_d\geq \Lambda\psi(u_d))\quad \textrm{ since $\psi$ is invertible and decreasing}\\
&=\bb E_\Lambda\bb P( E_1\geq \Lambda\psi(u_1),\ldots,E_d\geq \Lambda\psi(u_d)|\Lambda = \lambda)\\
&=\bb E_\Lambda\prod_i\bb P( E_i\geq \lambda\psi(u_i))\\
&=\bb E_\Lambda \prod_i\exp(-\lambda \psi(u_i))\\
&=\bb E_\Lambda\exp\left(-\lambda \sum_{i=1}^d\psi(u_i)\right)\\
&=\int_0^\infty \exp\left(-\lambda \sum_{i=1}^d\psi(u_i)\right) f_\Lambda(\lambda) \m d \lambda\\
&=\psi^{-1}\left(\sum_{i=1}^d\psi(u_i)\right),\qquad \textrm{as required}.
\end{split}
\]
CHAPTER: DYNAMIC SPLITTING
§ EMBEDDING TRANSFORMATIONS
We now verify that the embedding transformations (<ref>) and (<ref>) do indeed have the desired distribution at $t=1$. For (<ref>) we have
\[
\begin{split}
\bb P (X_k(1)\leq x) &= \bb P (\exp(-\Lambda_k(1))\leq F_k(x))\\
&=\bb P (\Lambda_k(1)\geq -\log (F_k(x)))\\
&=\exp (\log (F_k(x)))\\
&= F_k(x).
\end{split}
\]
For (<ref>) we have
\[
\begin{split}
\bb P (X_k(1)\leq x) &= \bb P (1-\exp(-\Lambda_k(1))\leq F_k(x))\\
&=\bb P (\exp(-\Lambda_k(1))\geq 1-F_k(x))\\
&=\bb P (\Lambda_k(1)\leq -\log(1-F_k(x)))\\
&= F_k(x).
\end{split}
\]
Hence both transformations yield the desired distribution $F_k(x)$ at $t=1$ as required.
§ IDEAL CASE ANALYSIS OF FIXED FACTOR SPLITTING ALGORITHM
We now present an analysis of the performance of the Fixed Factor Splitting algorithm under an ideal assumption. The assumption is that the time levels $t_{i} \in \{t_1, \cdots ,t_{L} \}$ are selected such that the conditional probabilities $c=\bb P(S(\bX(t_i)) \gvn S(\bX(t_{i-1}))$ are exactly, rather than approximately, equal to $s$ for all $i$.
Let $N_i=|\scX_i|$ be the random number of states in the set $\scX_i=\{X(t_i):S(X(t_i))\geq \gamma\}$. At time $t_0=0$ we have $N_1=1$ as the algorithm begins with a single path for the Markov process $\bLam(t_1)$. If we denote the number of states in $\scX_{i+1}$ that are generated from the $j$-th state from $\scX_{i}$ by $Q_{j,i}$ then we have the branching process recursion
\[
N_{i+1} = Q_{1,i}+Q_{2,i}+\cdots+Q_{N_t,i}
\]
where it is clear that $Q_{j,i}\sim \mathsf{Bin}(s,c)$, that is, a Binomial distributed random variable with probability $c$. Thus, we have $\bb E [Q_{j,i}]=sc=1$ and $\Var(Q_{j,i})=sc(1-c)=1-c$. By standard branching process arguments <cit.> we have $\bb E[N_i]=1$ and $\Var (N_i)=(i-1)(1-c)$ for $1<i<L$. Hence, for the unbiased estimator $W=\frac{|\scX_L|}{s^{L-1}}=\frac{N_L}{s^{L-1}}$ we have $\bb E[W]=\ell=\frac{1}{s^{L-1}}$ and $\Var(W)=\frac{(L-1)(1-c)}{s^{2L-2}}$ with $\log(\ell)=(L-1)\log(s)$.
An estimator $\hat{\ell}$ of $\ell$ is logarithmically efficient <cit.> if the following condition holds:
\[
\limsup_{\ell\downarrow 0} \left|\frac{\log(\Var(\hat{\ell}))}{\log(\ell^2)}\right|\geq 1.
\]
For the logarithmic efficiency criterion we have
\[
\begin{split}
\lim_{\ell\downarrow 0} \left|\frac{\log(\Var(W))}{\log(\ell^2)}\right|&=\lim_{\ell\downarrow 0} \left|\frac{\log(L-1)+\log(1-c)-(2L-2)\log(s)}{(2L-2)\log(s)}\right|\\
&=\lim_{s\uparrow \infty} \left|\frac{\log(L-1)+\log(1-c)-(2L-2)\log(s)}{(2L-2)\log(s)}\right|\\
\end{split}
\]
Therefore, under the idealized assumption the estimator $W$ is logarithmically efficient. Note that the simulation effort, starting from $t_1$, is a random variable $s\sum_{i=1}^L N_i$ with expected value $s(L-1)$. The expected relative time variance product <cit.> is thus given by
\[
\begin{split}
\frac{\Var(W)}{\ell^2}s(L-1)&=(L-1)(1-c)s(L-1)\\
&=\left(\frac{\log\ell}{\log (s)}\right)^2(s-1),
\end{split}
\]
which is minimized as a function of $s$ for $s>1$ at $s=4.92155363$ or $s=5$ when constrained on the integers.
§ ADAPTIVE DYNAMIC SPLITTING
The description of the dynamic splitting method described up to this point has allowed arbitrary choices for the intermediate time levels $t_{i} \in \{t_1, \cdots ,t_{L-1} \}$. We now describe a pilot algorithm to select optimal values for the intermediate time levels $\{t_{i}\}$. The motivation behind this selection is to ensure that the conditional probabilities $\Pm(S(\bLam(t_{i}))\geq \gamma \gvn S(\bLam(t_{i-1}))\geq \gamma)$ are not too small and not rare-event probabilities so that they can be easily estimated with CMC. <cit.> notes that this formulation will in fact lead to biased estimates of $\ell$ and complications in the computation of the variance of $\hat{\ell}$. As a consequence, the relative error of the estimator can be difficult to compute, thereby limiting comparisons and benchmarks against other Monte Carlo estimators. Thus, the Fixed Effort Splitting algorithm is recommended as it will lead to unbiased estimates for $\ell$ and readily available estimates of variance by running the algorithm several times independently. The following algorithm implements a simple procedure similar to numerical root-finding by utilizing the Gamma Bridge Sampling algorithm.
: Adaptive Dynamic Splitting
start time $t_s$; end time $t_e$; endpoints $\bLam(t_s)\leq \bLam(t_e)$; total sample at each level $s$; time tolerance $\epsilon_t$; proportion tolerance $\epsilon_p$
$\scT \leftarrow\emptyset$
$\bLam(t_l)\leftarrow \bLam(t_s)$
$\bLam(t_u)\leftarrow \bLam(t_e)$
$t_m\leftarrow \frac{t_s+t_e}{2}$
Generate $\bLam_j(t_m)=(\Lambda_1(t_m),\ldots,\Lambda_{d}(t_m))^\top$ using Gamma Bridge Sampling with endpoints $\bLam(t_l)\leq \bLam(t_u)$
$i\leftarrow 1$
$|\frac{1}{s}\sum_{j=1}^s \bb I\{S(\bLam_j(t_m))\geq \gamma\}-\rho|>\epsilon_p$
$\frac{1}{s}\sum_{j=1}^s \bb I\{S(\bLam(t_m)\geq \gamma\}-\rho>0$
$\bLam(t_l)\leftarrow \bLam(t_m)$
$t_m \leftarrow \frac{t_m+t_u}{2}$
$\frac{1}{s}\sum_{j=1}^s \bb I\{S(\bLam(t_m)\geq \gamma\}-\rho<0$
$\bLam(t_u)\leftarrow \bLam(t_m)$
$t_m \leftarrow \frac{t_l+t_m}{2}$
Generate $\bLam_j(t_m)=(\Lambda_1(t_m),\ldots,\Lambda_{d}(t_m))^\top$ using Gamma Bridge Sampling with endpoints $\bLam(t_l)\leq \bLam(t_u)$
$t_i\leftarrow t_m$
add $t_i$ to $\scT$
$i\leftarrow i+1$
: Gamma bridge sampling
Endpoints $\bLam(t_l)\leq \bLam(t_u)$ and $t\in(t_l,t_u)$.
$B_k\sim \mathsf{Beta}(t-t_l,t_u-t)$, independently
$\Lambda_k(t)\g\Lambda_k(t_l)+ (\Lambda_k(t_u)-\Lambda_k(t_l))B_k$
|
1511.00218
|
$^1$Quantum Physics Section, Kyushu Institute of Technology, 1-1 Sensui-cho, Tobata, Kitakyushu, Fukuoka, 804-8550, Japan
$^2$Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany
$^3$Department of Computer Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
$^4$Centre de Physique Théorique, École polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau, France
We present ab initio $GW$ plus cumulant-expansion calculations for an organic compound (TMTSF)$_2$PF$_6$ and a transition-metal oxide SrVO$_3$. These materials exhibit characteristic low-energy band structures around the Fermi level, which bring about interesting low-energy properties; the low-energy bands near the Fermi level are isolated from the other bands and, in the isolated bands, unusually low-energy plasmon excitations occur. To study the effect of this low-energy-plasmon fluctuation on the electronic structure, we calculate spectral functions and photoemission spectra using the ab initio cumulant expansion of the Green's function based on the $GW$ self-energy. We found that the low-energy plasmon fluctuation leads to an appreciable renormalization of the low-energy bands and a transfer of the spectral weight into the incoherent part, thus resulting in an agreement with experimental photoemission data.
71.15.Mb, 71.45.Gm, 71.20.Rv,71.20.Be
§ INTRODUCTION
The understanding of low-energy electronic structures and excitations in real materials is an important subject of condensed-matter physics and material science. Interesting phenomena such as a non-Fermi-liquid behavior and unconventional superconductivity are caused by the instability of electronic structures near the Fermi level. A common feature is often found in the band structures showing such phenomena; isolated bands appear near the Fermi level. The width of these isolated bands is typically the order of 1 eV, which is comparable to local electronic interactions. Thus, in these isolated bands, the kinetic and potential energies compete with each other, and the competition is often discussed within a local-interaction approximation, as in the Hubbard model. In real materials, however, there exist various elementally excitations not described by the local electronic interaction. The plasmon in metallic systems or the exciton in insulating systems are well known examples of such nonlocal excitations, which result from the long-range Coulomb interaction. In the above-mentioned isolated-band systems, the plasmon excitation can occur in this band, and its energy scale can be very small (of the order of 1 eV), which is comparable to the bandwidth and the size of the local Coulomb interaction.
In this study, we investigate the effect of the low-energy-plasmon fluctuation on the electronic structure of real isolated-band systems from first principles. For this purpose, we choose two materials, a quasi-one dimensional organic conductor (TMTSF)$_2$PF$_6$ (Ref. Bechgaard,Ishiguro-Yamaji-Saito,Kuroki,Ishibashi), where TMTSF stands for tetramethyltetraselenafulvalene, and a three-dimensional perovskite transition-metal oxide SrVO$_3$ (Ref. LDA+DMFT-1, LDA+DMFT-2, LDA+DMFT-3). These materials are typical isolated-band systems and are studied as benchmark materials of the correlated metal, where the local Coulomb-interaction effect on the electronic properties is investigated with much interest. <cit.> In the present work, we focus on the low-energy plasmon effect on the electronic structure. <cit.> Through the comparison between theoretical and experimental results on the plasmon-related properties and spectral functions, we verify low-energy plasmon effects on the electronic structure of the real system.
The organic conductor (TMTSF)$_2$PF$_6$ is a representative quasi-one-dimensional material, <cit.> and basically behaves as a good metallic conductor. <cit.> At low temperature (around 12 K), it undergoes a transition to a spin-density-wave phase. <cit.> In the high-temperature metallic region, photoemission spectroscopy has observed small spectral weight near the Fermi level. <cit.> From this observation and the quasi-one-dimensional nature, the origin of the renormalization has been discussed in view of the Tomonaga-Luttinger liquid. <cit.> On the other hand, this material exhibits clear low-energy plasma edges around 0.1-1 eV in the reflectance spectra. <cit.> Therefore, this plasmon excitation would also be a prominent renormalization factor of the electronic structure.
The transition-metal oxide SrVO$_3$ is another well known correlated metal. <cit.> Many high-resolution photoemission measurements <cit.> including the bulk-sensitive version <cit.> were performed and clarified a strong renormalization of the low-energy isolated band and the satellite peak just below this band. The origin has actively been discussed in terms of the local electronic correlation. <cit.> On the other hand, the reflectance and electron-energy-loss spectra have clarified low-energy plasmon excitations around 1.4 eV. <cit.> In density-functional band structure, SrVO$_3$ has isolated bands of the $t_{2g}$ orbitals around the Fermi level, whose bandwidth is about 2.7 eV. <cit.> Also, the constrained random phase approximation gives an estimate of the local electronic interaction of $\sim$2-3 eV. <cit.> Thus, the energy scale of the experimentally observed plasmon excitation is comparable to the bandwidth and the local Coulomb interaction. Hence, the low-energy plasmon fluctuation would certainly be relevant to the low-energy properties.
To study how the plasmon excitation affects the electronic structure, we perform ab initio calculations based on the $GW$ approximation. <cit.> The $GW$ calculation considers the self-energy effect due to the plasmon fluctuation and describes properly quasiparticle energies in the valence region. On the other hand, the description for the plasmon satellite in the spectral function is known to be less accurate. <cit.> In order to improve this deficiency, ab initio $GW$ plus cumulant ($GW$+$C$) calculations have recently been performed. <cit.> The accuracy of the $GW$+$C$ method has been verified in bulk silicon <cit.> and simple metals, <cit.> where the satellite property is satisfactorily improved. In the present study, we apply the $GW$+$C$ method to the study of the above mentioned isolated-band systems, and show that the low-energy plasmon fluctuations modify substantially the low-energy electronic structure.
The present paper is organized as follows. In Sec. II, we describe the $GW$ and $GW$+$C$ methods to calculate dielectric and spectral properties. Computational details and results for (TMTSF)$_2$PF$_6$ and SrVO$_3$ are given in Sec. III. We also discuss the comparison between theory and experiment, focusing on the renormalization of the electronic structure due to the plasmon excitation. The summary is given in Sec. IV.
§ METHOD
In this section, we describe ab initio $GW$ and $GW+C$ methods. The latter is a post-$GW$ treatment and uses the self-energies calculated with the $GW$ approximation. Below, we first describe details of the $GW$ calculation.
§.§ GW approximation
The non-interacting Green's function is written as
\begin{eqnarray}
G_0({\bf r,r'},\omega) = \sum_{\alpha{\bf k}} \frac{\phi_{\alpha{\bf k}}({\bf r}) \phi^{*}_{\alpha{\bf k}}({\bf r'})} {\omega-\epsilon_{\alpha{\bf k}}+i \delta {\rm sgn}(\epsilon_{\alpha{\bf k}}-\mu)},
\end{eqnarray}
where ${\bf k}$ and $\omega$ are wavevector and frequency, respectively. $\phi_{\alpha{\bf k}}({\bf r})$ and $\epsilon_{\alpha{\bf k}}$ are the Kohn-Sham (KS) wavefunction and its energy, respectively, and $\mu$ is the Fermi level of the KS system. The $\delta$ parameter is chosen to be a small positive value to stabilize numerical calculations.
A polarization function of a type $-i G_0 G_0$ is written in a matrix form in the plane wave basis as
\begin{eqnarray}
\chi_{{\bf GG'}}({\bf q},\omega)&=&2\sum_{{\bf k}}\sum^{unocc}_{\alpha}\sum^{occ}_{\beta} M_{\alpha\beta}^{{\bf G}}({\bf k+q,k}) \nonumber \\
&\times&M_{\alpha\beta}^{{\bf G'}}({\bf k+q,k})^{*} X_{\alpha{\bf k+q},\beta{\bf k}}(\omega) \label{eq:chi}
\end{eqnarray}
with $M_{\alpha\beta}^{{\bf G}}({\bf k+q,k})=\langle\phi_{\alpha{\bf k+q}}|e^{i({\bf q+G})\cdot{\bf r}}|\phi_{\beta{\bf k}}\rangle$ and
\begin{eqnarray}
X_{\alpha{\bf k+q},\beta{\bf k}}(\omega)\!=\! \frac{1}{\omega\!-\!\epsilon_{\alpha{\bf k\!+\!q}}\!+\!\epsilon_{\beta{\bf k}}\!+\!i\delta}\!-\!
\frac{1}{\omega\!+\!\epsilon_{\alpha{\bf k\!+\!q}}\!-\!\epsilon_{\beta{\bf k}}\!-\!i\delta}, \nonumber \\
\end{eqnarray}
where ${\bf G}$ is a reciprocal lattice vector. With this polarization function, the symmetrized dielectric matrix in reciprocal space is defined by
\begin{eqnarray}\
\epsilon_{{\bf G\!G'}}\!({\bf q},\omega)\!=\!\delta_{{\bf G\!G'}}\!-\!\frac{4\pi}{V}\frac{1}{|{\bf q\!+\!G}|}\chi_{{\bf G\!G'}}({\bf q},\!\omega)\frac{1}{|{\bf q\!+\!G'}|} \label{symeps}
\end{eqnarray}
with $V$ being the crystal volume. In the ${\bf q+G}\to0$ limit, the dielectric matrix in Eq. (<ref>) is expressed by <cit.>
\begin{eqnarray}
\epsilon_{{\bf GG'}}(0,\omega)
= \left\{
\begin{array}{@{\,\,\,\,\,\,\,}l@{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}l}
\displaystyle
1-\frac{4\pi}{V} 2\sum_{{\bf k}}\sum^{unocc}_{\alpha}\sum^{occ}_{\beta}
\Biggl|\frac{p^{\mu}_{\alpha\beta}({\bf k})}
{\epsilon_{\alpha{\bf k}}-\epsilon_{\beta{\bf k}}}\Biggr|^2
X_{\alpha{\bf k},\beta{\bf k}}(\omega)
& \mbox{(${\bf G}={\bf G'}={\bf 0}$),}
\\[+20pt]
\displaystyle
2\sum_{{\bf k}}\sum^{unocc}_{\alpha}\sum^{occ}_{\beta}
\Biggl(\frac{p^{\mu}_{\alpha\beta}({\bf k})}
{\epsilon_{\alpha{\bf k}}-\epsilon_{\beta{\bf k}}}\Biggr)
\frac{M_{\alpha\beta}^{{\bf G'}}({\bf k,k})^*}{|{\bf G'}|}
X_{\alpha{\bf k},\beta{\bf k}}(\omega)
& \mbox{(${\bf G=0},{\bf G'\ne 0}$),}
\\[+20pt]
\displaystyle
2\sum_{{\bf k}}\sum^{unocc}_{\alpha}\sum^{occ}_{\beta}
\frac{M_{\alpha\beta}^{{\bf G}}({\bf k,k})}{|{\bf G}|}
\Biggl(\frac{p^{\mu}_{\alpha\beta}({\bf k})}
{\epsilon_{\alpha{\bf k}}-\epsilon_{\beta{\bf k}}}\Biggr)^*
X_{\alpha{\bf k},\beta{\bf k}}(\omega)
& \mbox{($ {\bf G \ne 0}, {\bf G'=0}$),}
\\[+20pt]
\displaystyle
\delta_{{\bf GG'}}-\frac{4\pi}{V} \frac{1}{|{\bf G}|}
\chi_{{\bf GG'}}({\bf 0},\omega)\frac{1}{|{\bf G'}|}
& \mbox{($otherwise$)}
\end{array}
\right.
\label{eq0}
\end{eqnarray}
where ${\bf q}$ approaches zero along the Cartesian $\mu$ direction and $p_{\alpha\beta{\bf k}}^{\mu}$ is a matrix element of a momentum as
\begin{eqnarray}
p_{\alpha\beta{\bf k}}^{\mu}=-i\langle\phi_{\alpha{\bf k}}|\frac{\partial}{\partial x_{\mu}}+[V_{NL},x_{\mu}]|\phi_{\beta{\bf k}}\rangle,
\label{p_ij}
\end{eqnarray}
with $V_{NL}$ being the nonlocal part of the pseudopotential. In the first line of Eq. (<ref>), the last term on the right hand side is the Drude term of the intraband transitions around the Fermi level, <cit.> where
\begin{eqnarray}
\omega_{pl,\mu\nu}=\sqrt{\frac{8\pi}{\Omega N} \sum_{\alpha{\bf k}} p_{\alpha\alpha{\bf k}}^{\mu} p_{\alpha\alpha{\bf k}}^{\nu} \delta(\epsilon_{\alpha {\bf k}}-\mu}) \label{wpl}
\end{eqnarray}
is the bare plasma frequency. The other terms in Eq. (<ref>) result from the interband transitions.
We next describe the calculation of the self-energy. The operator of the exchange self-energy is defined by
\begin{eqnarray}
\Sigma^X({\bf r,r'})\!=\!i\!\int \frac{d\omega}{2\pi} G_0({\bf r,r'},\omega) v({\bf r,r'}),
\label{GW_SIGMAX}
\end{eqnarray}
where $v({\bf r,r'})=1/|{\bf r-r'}|$ is the bare Coulomb interaction. In practice, we use an attenuation Coulomb interaction $\tilde{v}({\bf r,r'})=\Theta(R_c-|{\bf r-r'}|)/|{\bf r-r'}|$ with a cutoff $R_c$ instead of $v$ to treat the integrable singularities in the bare Coulomb interaction. <cit.> The matrix element of the exchange self-energy is thus
\begin{eqnarray}
\Sigma^{X}_{\alpha\beta{\bf k}}
&=& \int\!d{\bf r} \int\!d{\bf r'}\!\phi_{\alpha{\bf k}}^*({\bf r})\Sigma^X({\bf r,r'})\phi_{\beta{\bf k}}({\bf r'}) \nonumber \\
&=& \Sigma^{X,body}_{\alpha\beta{\bf k}}+\Sigma^{X,head}_{\alpha\beta{\bf k}}, \label{Sxb+Sxh}
\end{eqnarray}
where $\Sigma^{X,body}$ and $\Sigma^{X,head}$ are the “body” and “head” components of the exchange self-energy, respectively. The former body matrix element is expressed as
\begin{eqnarray}
\Sigma^{X,body}_{\alpha\beta{\bf k}} &=& \frac{4\pi}{V}\sum_{{\bf qG}n}{}^{\prime} M_{\alpha n}^{{\bf G}}({\bf k,\!k\!-\!q}) M_{\beta n}^{{\bf G}}({\bf k,\!k\!-\!q})^* \nonumber \\
&\times& \frac{1-\cos(|{\bf q\!+\!G}|R_c)}{|{\bf q\!+\!G}|^2} \tilde{\theta}(\mu-\epsilon_{n{\bf k-q}}) \label{Sx}
\end{eqnarray}
\begin{eqnarray}
\tilde{\theta}(\mu-\epsilon)=\frac{1}{\pi}\arctan \Bigl( \frac{\mu-\epsilon}{\delta}\Bigr)+\frac{1}{2}. \label{theta}
\end{eqnarray}
The prime in the sum of Eq. (<ref>) represents the summation excluding the contribution of the head term of ${\bf q+G=0}$. Corresponding to the replacement of $v$ with $\tilde{v}$, the related Fourier transform is modified from $\frac{1}{|{\bf q+G}|}$ to $\frac{\sqrt{1-\cos(|{\bf q+G}|R_c)}}{|{\bf q+G}|}$ in Eq. (<ref>). Accordingly, the body matrix element in Eq. (<ref>) is supplemented by the head component in Eq. (<ref>)
\begin{eqnarray}
\Sigma^{X,head}_{\alpha\beta{\bf k}}=\frac{2\pi}{V} R_c^2 \delta_{\alpha\beta} \theta(\mu-\epsilon_{\alpha{\bf k}}).
\end{eqnarray}
The operator of the correlation self-energy is defined by
\begin{eqnarray}
\Sigma^C({\bf r,\!r'},\!\omega)\!=\!i\!\int\!\frac{d\omega'}{2\pi} G_0({\bf r,\!r'},\!\omega\!+\!\omega') W_C({\bf r,\!r'},\!\omega'),
\label{GW_SIGMA}
\end{eqnarray}
where $W_C(\omega)=W(\omega)-v$ is the correlation part of the symmetrized screened Coulomb interaction $W(\omega)=v^{\frac{1}{2}}\epsilon^{-1}(\omega)v^{\frac{1}{2}}$. Here, $\epsilon^{-1}$ is the inverse dielectric function, which is calculated by inverting the symmetrized dielectric matrix in Eqs. (<ref>) and (<ref>). For a practical calculation of the matrix element of $\Sigma^C({\bf r,r'},\omega)$ in Eq. (<ref>), we introduce the following model screened interaction <cit.>
\begin{eqnarray}
\tilde{W}_C({\bf r,r'},\omega_i)=\sum_{j} b_{ij}a_j({\bf r,r'})\label{modelW}
\end{eqnarray}
\begin{eqnarray}
b_{ij}=\frac{1}{\omega_i-z_j}-\frac{1}{\omega_i+z_j}. \label{pole-z}
\end{eqnarray}
In Eq. (<ref>), the real frequency $\omega$ is discretized into $\omega_i$, and $z_j$ and $a_j({\bf r,r'})$ are the pole and amplitude of the model interactions, respectively. The matrix element $b_{ij}$ comprises a square matrix [see Sec. III(A)]. Since the frequency-dependent part $\tilde{W}_C$ is decoupled from the amplitude one, the frequency integral in $iG_0\tilde{W}_C$ can be analytically performed. The matrix element of $\Sigma^C(\omega)$ consists of the body and head components as follows:
\begin{eqnarray}
\Sigma^C_{\alpha\beta{\bf k}}(\omega)=\Sigma^{C,body}_{\alpha\beta{\bf k}}(\omega)+\Sigma^{C,head}_{\alpha\beta{\bf k}}(\omega). \label{Scb+Sch}
\end{eqnarray}
The body matrix element in the above is given by
\begin{eqnarray}
\Sigma^{C,body}_{\alpha\beta{\bf k}}(\omega)\!=\!\sum_{jn{\bf q}}{}^{\prime}
\frac{\langle\phi_{\alpha{\bf k}}\phi_{n{\bf k-q}}|a_j|\phi_{n{\bf k-q}}\phi_{\beta{\bf k}}\rangle}
{\omega\!-\!\epsilon_{n{\bf k-q}}\!-\!(z_j\!-\!i\delta){\rm sgn}(\epsilon_{n{\bf k-q}}\!-\!\mu)}, \label{eq:SGM} \nonumber \\
\end{eqnarray}
where the numerator is given by
\begin{eqnarray}
\langle&\phi_{\alpha{\bf k}}&\phi_{n{\bf k-q}}|a_j|\phi_{n{\bf k-q}}\phi_{\beta{\bf k}}\rangle \nonumber \\
&=& \sum_{i} \bigl({\bf b}^{-1}\bigr)_{ji}
\langle\phi_{\alpha{\bf k}}\phi_{n{\bf k-q}}|W_C(\omega_i)|\phi_{n{\bf k-q}}\phi_{\beta{\bf k}}\rangle
\end{eqnarray}
\begin{eqnarray}
\langle&\phi_{\alpha{\bf k}}&\phi_{n{\bf k-q}}|W_C(\omega_i)|\phi_{n{\bf k-q}}\phi_{\beta{\bf k}}\rangle \nonumber \\
&=& \frac{4\pi}{V} \sum_{{\bf GG'}}{}^{\prime} \frac{M_{\alpha n}^{{\bf G}}({\bf k,k-q})\sqrt{1-\cos(|{\bf q+G}|R_c)}}{|{\bf q+G}|} \nonumber \\
&\times& \Bigl(\epsilon^{-1}_{{\bf GG'}}({\bf q},\omega_i)-\delta_{{\bf GG'}}\Bigr) \nonumber \\
&\times& \frac{M_{\beta n}^{{\bf G'}}({\bf k,k-q})^*\sqrt{1-\cos(|{\bf q+G'}|R_c)}}{|{\bf q+G'}|}. \label{Wc}
\end{eqnarray}
Note that, in the practical calculation, the frequency $\omega$ for the self-energy in Eq. (<ref>) is distinguished from the frequency $\omega_i$ for the screened interaction in Eq. (<ref>). The body matrix element $\Sigma^{C,body}_{\alpha\beta{\bf k}}(\omega)$ in Eq. (<ref>) is supplemented with the head component in Eq. (<ref>)
\begin{eqnarray}
\Sigma^{C,head}_{\alpha\beta{\bf k}}(\omega)\!=\!\frac{2\pi R_c^2}{V}\delta_{\alpha\beta}\sum_{j} \frac{g_j}{\omega\!-\!\epsilon_{\alpha{\bf k}}\!-\!(z_j\!-\!i\delta){\rm sgn}(\epsilon_{\alpha{\bf k}}\!-\!\mu)} \nonumber \\
\end{eqnarray}
\begin{eqnarray}
g_j=\sum_{i} \bigl({\bf b}^{-1}\bigr)_{ji} \Bigl( \epsilon_{\bf 00}^{-1}({\bf 0},\omega_i)-1 \Bigr).
\end{eqnarray}
With these ingredients, the spectral function is calculated via the Wannier-interpolation method (see below). The spectral function at an arbitrary ${\bf k}$ is
\begin{eqnarray}
A({\bf k},\omega)=\frac{1}{\pi} \sum_{\alpha} \Bigl| {\rm Im} \frac{1}{\omega-({\cal E}_{\alpha {\bf k}}(\omega)+\Delta)} \Bigr|, \label{Akw}
\end{eqnarray}
where ${\cal E}_{\alpha{\bf k}}(\omega)$ is obtained by diagonalizing non-symmetric complex matrix in the Wannier basis
\begin{eqnarray}
{\cal H}_{ij}({\bf k},\omega)=h_{ij}({\bf k})+\Sigma_{ij}({\bf k},\omega), \label{hij}
\end{eqnarray}
where $h_{ij}({\bf k})$ is the Fourier transform of the KS Hamiltonian matrix in the Wannier basis as
\begin{eqnarray}
h_{ij}({\bf k})=\sum_{{\bf R}} h_{ij{\bf R}} e^{i{\bf kR}}
\end{eqnarray}
\begin{eqnarray}
h_{ij{\bf R}} = \frac{1}{N} \sum_{{\bf k'}\alpha} \langle w_{i{\bf 0}}|\phi_{\alpha{\bf k'}} \rangle \epsilon_{\alpha{\bf k'}} \langle \phi_{\alpha{\bf k'}}|w_{j{\bf 0}} \rangle e^{i{\bf k'R}}.
\end{eqnarray}
Here, ${\bf k'}$ is a $k$ point in the regular mesh and $N$ is the total number of the $k$ points in the regular mesh. Also, $|w_{i{\bf R}}\rangle$ is the $i$th Wannier orbital at the lattice point ${\bf R}$, and the transform $\langle\phi_{\alpha{\bf k'}}|w_{i{\bf 0}}\rangle$ is obtained in the Wannier-function-generation routine. $\Sigma_{ij}({\bf k},\omega)$ in Eq. (<ref>) is the Fourier transform of the self-energy in the Wannier basis as
\begin{eqnarray}
\Sigma_{ij}({\bf k},\omega)=\sum_{{\bf R}} \Sigma_{ij{\bf R}}(\omega) e^{i{\bf kR}} \label{Sijkw}
\end{eqnarray}
\begin{eqnarray}
\Sigma_{ij{\bf R}}(\omega)\!=\!\frac{1}{N}\!\!\sum_{{\bf k'}\alpha\beta}\!\langle w_{i{\bf 0}}|\phi_{\alpha{\bf k'}}\rangle\!\Sigma_{\alpha\beta{\bf k'}}(\omega)\!\langle \phi_{\beta{\bf k'}}|w_{j{\bf 0}}\rangle e^{i{\bf k'R}}.
\nonumber \\ \label{SijRw}
\end{eqnarray}
The matrix element $\Sigma_{\alpha\beta{\bf k'}}(\omega)$ is defined by
\begin{eqnarray}
\Sigma_{\alpha\beta{\bf k'}}(\omega)=\langle\phi_{\alpha{\bf k'}}|\Sigma^{X}+\Sigma^{C}(\omega)-V^{xc}|\phi_{\beta{\bf k'}}\rangle \label{Sigma-od}
\end{eqnarray}
with $V^{xc}$ being the exchange-correlation potential in density-functional theory.
In Eq. (<ref>), the energy shift $\Delta$ is introduced to correct the mismatch of the Fermi level between the KS and one-shot $GW$ systems. This parameter is determined from the equation on the spectral norm
\begin{eqnarray}
\frac{2}{N_{{\bf k}}}\sum_{{\bf k}} \int_{-\infty}^{\mu} A({\bf k},\omega)d\omega=N_{{\rm elec}}, \label{Norm}
\end{eqnarray}
where $N_{{\rm elec}}$ is the total number of electrons in the system and $N_{{\bf k}}$ is the total number of sampling $k$ points after the interpolation. Note that $\mu$ is set to the Fermi level for the KS system.
The flow of the calculation is as follows: We first perform density-functional calculations to obtain the band structures and the Wannier functions for bands associated with the self-energy calculations. We then calculate the self-energies for the irreducible $k$-points $\{{\bf \bar{k}}\}$ in a regular mesh, including band off-diagonal terms as $\Sigma_{\alpha\beta{\bf \bar{k}}}(\omega)=\Sigma^{X}_{\alpha\beta{\bf \bar{k}}}+\Sigma^{C}_{\alpha\beta{\bf \bar{k}}}(\omega)-V^{xc}_{\alpha\beta{\bf \bar{k}}}$ in the selected energy region. The self-energies at a $k'$ point symmetrically equivalent to ${\bf \bar{k}}$ are the same as $\Sigma_{\alpha\beta{\bf \bar{k}}}(\omega)$, but, in the time-reversal symmetry case, $\Sigma_{\alpha\beta{\bf k'}}(\omega)$ is obtained by $\Sigma_{\alpha\beta{\bf k'}}(\omega)=\Sigma_{\beta\alpha,{\bf -\bar{k}}}(\omega)$. Then, we transform $\Sigma_{\alpha\beta{\bf k'}}(\omega)$ in Eq. (<ref>) to the Wannier representation $\Sigma_{ij{\bf R}}(\omega)$ with Eq. (<ref>). With these data, we evaluate the self-energy at an arbitrary $\bf{ k}$ via Eq. (<ref>). Note that the calculated spectral function includes band-off-diagonal effects, which is discussed in Appendix A. Finally, we calculate the spectral function of Eq. (<ref>) considering the energy shift $\Delta$ in Eqs. (<ref>) and (<ref>).
§.§ GW+cumulant expansion method
The $GW+C$ approach is a theory beyond the $GW$ approximation, <cit.> which is based on systematic diagrammatic expansions. This approach is suitable for dealing with long-range correlations, i.e., various types of the plasmon-fluctuation diagrams not included in the usual $GW$ diagram. In the initial stage of the study, it was applied to a system of core electrons interacting with a plasmon field. <cit.> Currently, ab initio $GW$+$C$ calculations have been known to give a better description for satellites due to the plasmon excitation. <cit.>
The Green's function with the cumulant expansion is defined in the time domain by <cit.>
\begin{eqnarray}
G_{\alpha{\rm k}}(t)=i \Theta(-t) e^{-i\epsilon_{\alpha{\bf k}}t+C_{\alpha{\bf k}}^h(t)}-i \Theta(t) e^{-i\epsilon_{\alpha{\bf k}}t+C_{\alpha{\bf k}}^p(t)}. \nonumber \\
\end{eqnarray}
where $\epsilon_{\alpha{\bf k}}<\mu$ for the first term on the right hand side and $\epsilon_{\alpha{\bf k}}>\mu$ for the second term. The $C_{\alpha{\bf k}}^h(t)$ and $C_{\alpha{\bf k}}^p(t)$ are the cumulants for the hole and particle states, respectively. The spectral function is calculated by the Fourier transform as
\begin{eqnarray}
A({\bf k},\omega)
&=& \frac{1}{\pi}\sum_{\alpha}{\rm Im} \int_{-\infty}^{\infty} dt
e^{i\omega t} G_{\alpha{\bf k}}(t) \nonumber \\
&=& A^h({\bf k},\omega)+A^p({\bf k},\omega), \label{AkwGWC}
\end{eqnarray}
which consists of the hole $A^h({\bf k},\omega)$ and particle $A^p({\bf k},\omega)$ contributions.
The spectral function for the hole part is written as
\begin{eqnarray}
A^h({\bf k},\omega)\!=\!\frac{1}{\pi}\!\sum_{\alpha}^{{\rm occ}}{\rm Im}i\!\int_{-\infty}^{0}\!dt e^{i\omega t} e^{-i \epsilon_{\alpha{\bf k}}t}
e^{C_{\alpha{\bf k}}^{h}(t)}\!, \label{AhkwGWC}
\end{eqnarray}
where the band sum is taken over the occupied states. To the lowest order in the screened interaction $W$, the cumulant is obtained by <cit.>
\begin{eqnarray}
C_{\alpha{\bf k}}^h(t)=i\int_{t}^{\infty}dt'\int_{t'}^{\infty}d\tau e^{i\epsilon_{\alpha{\bf k}} \tau}\Sigma_{\alpha{\bf k}}(\tau), \label{Chdef}
\end{eqnarray}
where $\Sigma=\Sigma^{X}+\Sigma^{C}-V^{xc}$.
In the present study, the cumulant is expanded around the quasiparticle energy $E_{\alpha{\bf k}}$, <cit.> which is a solution of
\begin{eqnarray}
E_{\alpha{\bf k}}=\epsilon_{\alpha{\bf k}}+{\rm Re}\Sigma_{\alpha{\bf k}}(E_{\alpha{\bf k}}) +\Delta,
\end{eqnarray}
where $\Delta$ is an energy shift to correct a mismatch of the Fermi level between the KS and one-shot $GW$+$C$ systems.
By considering the Fourier transform of $\Sigma(\tau)$ in Eq. (<ref>) and the spectral representation of $\Sigma(\omega)$, and after some manipulation, <cit.> the expression of the cumulant is obtained, which consists of the quasiparticle and satellite parts as
\begin{eqnarray}
C_{\alpha{\bf k}}^{h}(E_{\alpha{\bf k}},t)=C_{\alpha{\bf k}}^{h,qp}(E_{\alpha{\bf k}},t)+C_{\alpha{\bf k}}^{h,s}(E_{\alpha{\bf k}},t) \label{Ctoth}
\end{eqnarray}
\begin{eqnarray}
C_{\alpha{\bf k}}^{h,qp}(E_{\alpha{\bf k}},t)\!=\!-i(\Sigma_{\alpha{\bf k}}(E_{\alpha{\bf k}})\!+\!\Delta)t\!+\!\frac{\partial \Sigma_{\alpha{\bf k}}^h(E_{\alpha{\bf k}})}{\partial \omega} \label{Cqph}
\end{eqnarray}
\begin{eqnarray}
C_{\alpha{\bf k}}^{h,s}(E_{\alpha{\bf k}},t)=\frac{1}{\pi} \int_{-\infty}^{\mu} d\omega' \frac{e^{i(E_{\alpha{\bf k}}-\omega'-i\delta)t}}{(E_{\alpha{\bf k}}-\omega'-i\delta)^2} {\rm Im} \Sigma_{\alpha{\bf k}}(\omega'). \nonumber \\ \label{Csh}
\end{eqnarray}
Note that $t$ is negative for the hole part. To show the expansion point explicitly, we add $E_{\alpha{\bf k}}$ as an index in the cumulant Eqs. (<ref>), (<ref>), and (<ref>). Within the one-shot calculation, the position of the cumulant expansion may be taken at the non-interacting energy $\epsilon_{\alpha{\bf k}}$. <cit.> By taking the expansion point at $E_{\alpha{\bf k}}$, the results may include some sort of the self-consistency effect.
The hole self-energy in Eq. (<ref>) is defined by
\begin{eqnarray}
\Sigma_{\alpha{\bf k}}^h(\omega)=\frac{1}{\pi}\int_{\infty}^{\mu} d\omega' \frac{{\rm Im}\Sigma_{\alpha{\bf k}}(\omega')}{\omega-\omega'-i\delta}.
\end{eqnarray}
It should be noted that the derivative $\frac{\partial\Sigma_{\alpha{\bf k}}^h(E_{\alpha{\bf k}})}{\partial\omega}$ in Eq. (<ref>) is related to the $t$=0 component of the satellite cumulant in Eq. (<ref>) as <cit.>
\begin{eqnarray}
\frac{\partial\Sigma_{\alpha{\bf k}}^h(E_{\alpha{\bf k}})}{\partial\omega}
&=& -C_{\alpha{\bf k}}^{h,s}(E_{\alpha{\bf k}},0) \nonumber \\
&=& -\frac{1}{\pi} \int_{-\infty}^{\mu} d\omega' \frac{{\rm Im}\Sigma_{\alpha{\bf k}}(\omega')}{(E_{\alpha{\bf k}}-\omega'-i\delta)^2}. \label{del-eq-C}
\end{eqnarray}
This expression is practically used for the evaluation of the derivative; we avoid the numerical calculation of the derivative by a finite difference and use the integral expression on the right hand side of Eq. (<ref>) for the derivative, since we find that the latter treatment is numerically more stable. The stable calculation of the derivative is important in keeping the sum rule on the $GW+C$ spectrum.
In the practical calculation, we divide $A^{h}({\bf k},\omega)$ in Eq. (<ref>) into the two parts for stable calculations as <cit.>
\begin{eqnarray}
A^{h}({\bf k},\omega)\!=\!A^{h,qp}({\bf k},\omega)\!+\!A^{h,qp}({\bf k},\omega)\!\ast\!A^{h,s}({\bf k},\omega). \label{ahkw}
\end{eqnarray}
The first quasiparticle term on the right hand side can be calculated analytically as
\begin{eqnarray}
A^{h,qp}({\bf k},\omega)\!=\!\frac{1}{\pi}\sum_{\alpha}^{{\rm occ}}e^{-\gamma_{\alpha{\bf k}}} \frac{\eta_{\alpha{\bf k}}\cos\beta_{\alpha{\bf k}}-(\omega-E_{\alpha{\bf k}})\sin\beta_{\alpha{\bf k}} }{(\omega-E_{\alpha{\bf k}})^2+\eta_{\alpha{\bf k}}^2}, \nonumber \\
\end{eqnarray}
\begin{eqnarray}
\eta_{\alpha{\bf k}}&=&{\rm Im}\Sigma_{\alpha{\bf k}}(E_{\alpha{\bf k}})+\delta, \label{ImSGMh} \\
\gamma_{\alpha{\bf k}}&=&-{\rm Re}\frac{\partial\Sigma_{\alpha{\bf k}}^h(E_{\alpha{\bf k}})}{\partial \omega}, \\
\beta_{\alpha{\bf k}}&=&-{\rm Im}\frac{\partial\Sigma_{\alpha{\bf k}}^h(E_{\alpha{\bf k}})}{\partial \omega}.
\end{eqnarray}
The latter convolution term in Eq. (<ref>) is calculated via the numerical integration as
\begin{eqnarray}
A^{h,qp}({\bf k},\omega)\ast A^{h,s}({\bf k},\omega) =\frac{1}{\pi} \sum_{\alpha}^{{\rm occ}} {\rm Im} i \int_{-\infty}^{0} dt e^{i\omega t} e^{-i \epsilon_{\alpha{\bf k}t}} e^{C_{\alpha{\bf k}}^{h,qp}(E_{\alpha{\bf k}},t)} \bigl(e^{C_{\alpha{\bf k}}^{h,s}(E_{\alpha{\bf k}},t)}-1\bigr), \label{Ash}
\end{eqnarray}
where the integrand in the right-hand side decays to zero rapidly.
The particle part of the spectral function in Eq. (<ref>) is given by
\begin{eqnarray}
A^p({\bf k},\omega)\!=\!\frac{1}{\pi}\!\sum_{\alpha}^{{\rm unocc}}{\rm Im}i\!\int_{0}^{\infty}\!dt e^{i\omega t} e^{-i \epsilon_{\alpha{\bf k}t}}
e^{C_{\alpha{\bf k}}^{p}(E_{\alpha{\bf k}},t)} \label{ApkwGWC}
\end{eqnarray}
\begin{eqnarray}
C_{\alpha{\bf k}}^{p}(E_{\alpha{\bf k}},t)=C_{\alpha{\bf k}}^{p,qp}(E_{\alpha{\bf k}},t)+C_{\alpha{\bf k}}^{p,s}(E_{\alpha{\bf k}},t).
\end{eqnarray}
In this case $t$ is positive. The band sum in Eq. (<ref>) runs over the unoccupied states. The $C_{\alpha{\bf k}}^{p,qp}(E_{\alpha{\bf k}},t)$ and $C_{\alpha{\bf k}}^{p,s}(E_{\alpha{\bf k}},t)$ are given by
\begin{eqnarray}
C_{\alpha{\bf k}}^{p,qp}(E_{\alpha{\bf k}},t)\!=\!-i(\Sigma_{\alpha{\bf k}}(E_{\alpha{\bf k}})\!+\!\Delta)t\!+\!\frac{\partial\Sigma_{\alpha{\bf k}}^p(E_{\alpha{\bf k}})}{\partial\omega}\! \label{Cqpp}
\end{eqnarray}
\begin{eqnarray}
C_{\alpha{\bf k}}^{p,s}(E_{\alpha{\bf k}},t)=\frac{-1}{\pi} \int_{\mu}^{\infty} d\omega' \frac{e^{i(E_{\alpha{\bf k}}-\omega'+i\delta)t}}{(E_{\alpha{\bf k}}-\omega'+i\delta)^2} {\rm Im} \Sigma_{\alpha{\bf k}}(\omega'), \label{Csp} \nonumber \\
\end{eqnarray}
The $\Sigma^p$ in Eq. (<ref>) is the particle self-energy as
\begin{eqnarray}
\Sigma_{\alpha{\bf k}}^p(\omega)=\frac{-1}{\pi}\int_{\mu}^{\infty} d\omega' \frac{{\rm Im}\Sigma_{\alpha{\bf k}}(\omega')}{\omega-\omega'+i\delta}
\end{eqnarray}
Note that the self-energies in the above equations are defined as the causal one, so that the imaginary part of the self-energy is positive for $\omega<\mu$ and negative for $\omega>\mu$. In the particle part, $\frac{\partial\Sigma_{\alpha{\bf k}}^p(E_{\alpha{\bf k}})}{\partial\omega}=-C_{\alpha{\bf k}}^{p,s}(E_{\alpha{\bf k}},0)$ holds similarly to Eq. (<ref>). The contribution from the quasiparticle part to $A^{p}({\bf k},\omega)$ is
\begin{eqnarray}
A^{p,qp}({\bf k},\omega)=\frac{-1}{\pi}\sum_{\alpha}^{{\rm unocc}} e^{-\gamma_{\alpha{\bf k}}} \frac{\eta_{\alpha{\bf k}}\cos\beta_{\alpha{\bf k}}-(\omega-E_{\alpha{\bf k}})\sin\beta_{\alpha{\bf k}} }{(\omega-E_{\alpha{\bf k}})^2+\eta_{\alpha{\bf k}}^2} \nonumber \\
\end{eqnarray}
\begin{eqnarray}
\eta_{\alpha{\bf k}}&=&{\rm Im}\Sigma_{\alpha{\bf k}}(E_{\alpha{\bf k}})-\delta, \label{ImSGMp} \\
\gamma_{\alpha{\bf k}}&=&-{\rm Re}\frac{\partial\Sigma_{\alpha{\bf k}}^p(E_{\alpha{\bf k}})}{\partial \omega}, \\
\beta_{\alpha{\bf k}}&=&-{\rm Im}\frac{\partial\Sigma_{\alpha{\bf k}}^p(E_{\alpha{\bf k}})}{\partial\omega},
\end{eqnarray}
and the convolution-part contribution is
\begin{eqnarray}
A^{p,qp}({\bf k},\omega)\ast A^{p,s}({\bf k},\omega) =\frac{1}{\pi} \sum_{\alpha}^{{\rm unocc}}{\rm Im} i \int_{0}^{\infty} dt e^{i\omega t} e^{-i \epsilon_{\alpha{\bf k}t}} e^{C_{\alpha{\bf k}}^{p,qp}(E_{\alpha{\bf k}},t)} \bigl(e^{C_{\alpha{\bf k}}^{p,s}(E_{\alpha{\bf k}},t)}-1\bigr). \label{Asp}
\end{eqnarray}
§ RESULTS AND DISCUSSIONS
§.§ Calculation condition
Density-functional calculations were performed with Tokyo Ab-initio Program Package <cit.> with plane-wave basis sets, where we employed norm-conserving pseudopotentials <cit.> and generalized gradient approximation (GGA) for the exchange-correlation potential. <cit.> Maximally localized Wannier functions <cit.> were used for the interpolation of the self-energy.
For the atomic coordinates of (TMTSF)$_2$PF$_6$, the experimental structure obtained by a neutron measurement <cit.> at 20 K was adopted. The cutoff energies in wavefunction and in charge densities are 36 Ry and 144 Ry, respectively, and a 15$\times$15$\times$3 $k$-point sampling was employed. The cutoff for the polarization function in Eq. (<ref>) was set to be 3 Ry, and 200 bands were considered, which covers an energy range from the bottom of the occupied states near $-$30 eV to the top of the unoccupied states near 15 eV, where 0 eV is the Fermi level. The frequency grid of the polarization was taken up to $\omega_{max}$ = 86 eV in a double-logarithmic form, for which we sampled 99 energy-points for [0.1 eV: 43 eV] and 10 points for [43 eV: 86 eV] with initial grid set to be 0.0 eV. The $k$ sum over BZ in Eqs. (<ref>), (<ref>), and (<ref>) was evaluated by the generalized tetrahedron method. <cit.> The Drude term of $\omega=0$ is evaluated at the slightly shifted frequency $\omega=10^{-10}$ (au). The self-energy in Eq. (<ref>) was calculated for the frequency range [$-\Omega_{max}$ eV: $\Omega_{max}$ eV], where $\Omega_{max}$ was set to be 100 eV. In the practical calculation, we sampled 450 points for [$-$100 eV: $-$10 eV] with the interval of 0.2 eV, 1000 points for [$-$10 eV: 10 eV] with the 0.02 eV interval, and 450 points for [10 eV: 100 eV] with the 0.2 eV interval. With this energy range, the high-frequency tail of the self-energy is sufficiently small. This convergence is important to preserve the norm of the spectral function.
For SrVO$_3$, band calculations were performed for the idealized simple cubic structure, where the lattice parameter was set to be $a$=3.84 Å. The cutoff energies for wavefunction and charge densities are 49 Ry and 196 Ry, respectively, and an 11$\times$11$\times$11 $k$-point sampling was employed. The cutoff energy for the polarization function was set to be 10 Ry and 130 bands were considered, which cover from the bottom of the occupied states near $-$20 eV to the top of the unoccupied states near 90 eV. The frequency range of the polarization function was taken to be $\omega_{max}=220$ eV, where finer logarithmic sampling was applied to [0.1 eV: 110 eV] with 189 points and coarser sampling was done for [110 eV: 220 eV] with 10 points, with initial grid set to be 0.0 eV. In the self-energy calculation, $\Omega_{max}$ was set to be 200 eV, thus, the frequency dependence of the self-energy was calculated for [$-$200 eV: 200 eV], where we sampled 200 points for [$-$200 eV: $-$40 eV] with the 0.8 eV interval, 1600 points for [$-$40 eV: 40 eV] with the 0.05 eV interval, and 200 points for [40 eV: 200 eV] with the 0.8 eV interval.
In the fitting of the model screened interaction [Eqs. (<ref>) and (<ref>)], the positions of the poles in the model interaction are set as follows: <cit.>
\begin{eqnarray}
\end{eqnarray}
with $\Delta_i=\omega_{i+1}-\omega_i$. The total number of {$z_i$} is the same as that of the frequency grid {$\omega_i$} for the polarization function. We checked that the ab initio screened interaction $W_C(\omega)$ are satisfactorily reproduced by this model function $\tilde{W}_C(\omega)$. The broadening $\delta$ in Eqs. (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) was set to be 0.02 eV for (TMTSF)$_2$PF$_6$ and 0.05 eV for SrVO$_3$. The value of $\delta$ is desirable to be small enough, but the lower-bound in the practical calculation is determined by the resolution of the band dispersion, depending primary on the $k$-mesh density. Also, the cutoff $R_c$ in the attenuation potential [Eqs. (<ref>) and (<ref>)] is 28.27 Å for (TMTSF)$_2$PF$_6$ and 11.53 Å for SrVO$_3$, respectively. For (TMTSF)$_2$PF$_6$, the shift $\Delta$ in the spectral function $A({\bf k}, \omega)$ in Eq. (<ref>) was found to be 1.06 eV for the $GW$ calculation and 0.90 eV for the $GW$+$C$ one, respectively. For SrVO$_3$, $\Delta$ = 2.1 eV for the $GW$ and 2.35 eV for the $GW$+$C$.
In $GW$+$C$, two numerical integrals on time and frequency appear [Eqs. (<ref>) and (<ref>) for the time integral and Eqs. (<ref>) and (<ref>) for frequency integral], which must be treated carefully. We performed time integrals in Eq. (<ref>) numerically for the range [$-t_{max}$ au: 0 au] and those in Eq. (<ref>) for [0 au: $t_{max}$ au], where $t_{max}$ is 50 (au). The total number of the time grid $N_t$ is 50000, with the interval $\Delta t=t_{max}\big/N_t=0.001$ au. Note that $\Omega_{max}\Delta t \ll 1$ is necessary to reproduce the norm of the spectral weight correctly. The frequency integral in Eqs. (<ref>) and (<ref>) was numerically evaluated with the Simpson's formula for the interval $\Delta\omega$ divided into 21 subintervals, which is also important in obtaining the correct time dependence of the satellite cumulant. Also, in the $k$ integration to obtain the $GW+C$ density of states $A(\omega)$, the random $k$-point sampling was performed to improve the statistical average, where the Wannier interpolation technique was efficiently applied. With this condition, we obtained well converged spectra.
§.§ Density-functional band structure
Figure <ref> shows calculated GGA band structures of (TMTSF)$_2$PF$_6$ [panel (a)] and SrVO$_3$ [(b)]. In both systems, isolated bands are found around the Fermi level. In (TMTSF)$_2$PF$_6$, the isolated bands consist of highest-occupied molecular orbital (HOMO) of two molecules in the unit cell. [For detailed atomic geometry of (TMTSF)$_2$PF$_6$, refer to Refs. Kuroki and TMTSF-plasmon-Nakamura.] We call them the “HOMO" bands which are shown with the green-dotted curves. In addition, in this figure, “HOMO$-$1" and “HOMO$-$2" bands are shown by blue- and black-dotted curves, respectively. <cit.> In SrVO$_3$, the isolated bands (green-dotted curves) around the Fermi level are formed by the $t_{2g}$ orbitals of the vanadium atom and bands around [$-$7 eV: $-$2 eV] come from the oxygen-$p$ orbitals (blue-dotted curves). <cit.>
(Color online) Density-functional GGA band structures (solid red curves) of (a) (TMTSF)$_2$PF$_6$ and (b) SrVO$_3$. The Fermi level is at zero energy. In the panel (a), the green-dotted, blue-dotted, and black-dotted curves denote the HOMO, HOMO$-1$, and HOMO$-2$ bands, respectively. In (b), the green-dotted and blue-dotted curves correspond to the $t_{2g}$ and O$_{p}$ bands, respectively.
§.§ Low-energy plasmon excitation
To confirm the low-energy plasmon excitation in the above isolated-band systems, we calculated the reflectance spectra with the random-phase approximation
\begin{eqnarray}
R_{\mu\mu}(\omega)=\Biggl| \frac{1-\sqrt{\epsilon_{\mu\mu}^{-1}(\omega)}}{1+\sqrt{\epsilon_{\mu\mu}^{-1}(\omega)}} \Biggr|,
\end{eqnarray}
where $\epsilon_{\mu\mu}^{-1}(\omega)$ is obtained by inverse of the dielectric matrix in Eq. (<ref>). Figure <ref> (a) is the result for (TMTSF)$_2$PF$_6$, where dark-red and light-green colors represent the results in the light polarization of $E\|a$ and $E\|b'$, respectively, and the $a$ axis ($a$$\perp$$b'$) is the one-dimensional conducting axis. The calculated results (solid curves) are compared with experimental results (circles). We see that the theoretical plasma edges satisfactorily reproduce the experimental ones around 0.8 eV for $E\|a$ and 0.1-0.2 eV for $E\|b'$, which indicates that the present scheme correctly captures the low-energy plasmon excitation. The panel (b) shows the result for SrVO$_3$. We again see a reasonable agreement between the theory and experiment for the plasma edge (1.8 eV for theory and 1.4 eV for the experiment).
(Color online) Ab initio reflectivity $R(\omega)$ (solid curve) based on the random-phase approximation and experimental one (open circles) of (a) (TMTSF)$_2$PF$_6$ measured at 25 K and (b) SrVO$_3$ measured at room temperature. The experimental data are taken from Ref. TMTSF-R-Dressel for (TMTSF)$_2$PF$_6$ and Ref. R-SVO for SrVO$_3$. In (TMTSF)$_2$PF$_6$, the results for $E\|a$ and $E\|b'$ are displayed by dark red and light green, respectively.
TABLE <ref> summarizes parameters characterizing low-energy electronic structures of the two isolated-band systems, i.e., the HOMO bands of (TMTSF)$_2$PF$_6$ and the $t_{2g}$ bands of SrVO$_3$. The table includes the calculated bare plasma frequency $\omega_{pl}$ in Eq. (<ref>), bandwidth $W$, and effective local-interaction parameter $U-V$ with $U$ and $V$ being onsite and nearest-neighbor interactions, respectively, calculated with the constrained random phase approximation. <cit.> We note that $\omega_{pl}$ is by definition different from the plasma edge in the reflectance spectra in Fig. <ref>; the latter energies are lowered from the bare $\omega_{pl}$ by the presence of the individual electronic excitations. We see that $\omega_{pl}$ has the same size as $W$ and $U-V$, indicating that energy scale of the plasmon excitation would compete with those of kinetic and local electronic-interaction energies of electrons in the isolated band. The long-range interaction and related plasmon fluctuations would clearly be important for electronic structure.
List of parameters for bare plasma frequency $\omega_{pl}$, bandwidth $W$, onsite interaction $U$, nearest-neighbor interaction $V$, and $U-V$ for (TMTSF)$_2$PF$_6$ and SrVO$_3$. The interaction parameters are calculated with the constrained random phase approximation. <cit.> The parameters of (TMTSF)$_2$PF$_6$ are calculated for the HOMO bands, and those of SrVO$_3$ are evaluated for the $t_{2g}$ bands. The unit is eV.
$\omega_{pl}$ $W$ $U$ $V$ $U-V$
2*(TMTSF)$_2$PF$_6$ 1.25 ($E\|a$) 2*1.26
2*2.02 2*0.94 2*1.08
0.20 ($E\|b'$)
SrVO$_3$ 3.54 2.55 3.48 0.79 2.69
§.§ Spectral function of (TMTSF)$_2$PF$_6$
To study effects of the low-energy plasmon excitation in the isolated bands on the electronic structure, we calculated a spectral function $A({\bf k},\omega)$ for the HOMO bands of (TMTSF)$_2$PF$_6$. Figure <ref> displays the calculated spectra, where the panels (a) and (b) are the $GW$ result via Eq. (<ref>) and the $GW$+$C$ one via Eq. (<ref>), respectively. For comparison, the GGA band structure is superposed with blue-solid curves. In the $GW$ spectrum, clear incoherent peaks appear; along the Y-$\Gamma$ line, the spectral intensities of plasmaron states <cit.> emerge about 1 eV above (below) the unoccupied (occupied) part of the HOMO bands. <cit.> Also, along the X-M line, the spectra are more broadened and spread in the range from $-$1.5 to 0 eV. Interestingly, these sharp plasmaron peaks do not appear in the $GW$+$C$ spectrum. Instead, the $GW$+$C$ spectrum exhibits a broad incoherent structure throughout BZ. This is because the $GW$+$C$ treatment further takes into account the long-range correlation effect <cit.> or various types of the self-energy diagram involving the plasmon fluctuation, which is not included in the standard $GW$ calculations.
In the panel (c), density of states
\begin{eqnarray}
A(\omega)=\int_{{\rm BZ}} A({\bf k},\omega) d{\bf k}
\end{eqnarray}
is shown, where the $GW$+$C$, $GW$, and GGA results are plotted by red-solid, blue-dotted, and thin-black-solid curves, respectively. Compared to the GGA spectrum, the $GW$ and $GW+C$ spectra show an appreciable band renormalization around the Fermi level by the plasmon excitation. We again confirm that the distinct plasmon satellite (plasmaron) around $-$2 eV and +1 eV in the $GW$ spectrum disappears in the $GW$+$C$ spectrum.
(Color online) Spectral function for the HOMO bands of (TMTSF)$_2$PF$_6$ calculated with (a) $GW$ approximation and (b) $GW+C$ method. Blue-solid curves are the GGA results. The Fermi level is at zero energy. The colorbar is in linear scale. (c) Comparison of the density of states among the $GW+C$ (red-solid curve), $GW$ (blue-dotted curve), and GGA (black-thin curve) results.
Figure <ref> is the comparison between theoretical photo-emission spectra
\begin{eqnarray}
A(\omega<\mu)=\int_{{\rm BZ}} A({\bf k},\omega<\mu) d{\bf k}
\end{eqnarray}
and experimental one (green open circles) obtained with HeII radiation ($h\nu$=40.8 eV) at 50 K (Ref. TMTSF-PES-1). Thick-red-solid, blue-dotted, and thin-black-solid curves are the $GW$+$C$, $GW$, and GGA results, respectively. The spectra were calculated for the HOMO, HOMO$-1$, and HOMO$-2$ bands to cover the energy range measured in the experiment. We see that the GGA spectrum around the Fermi level is largely reduced in the $GW$ and $GW$+$C$ spectra by the self-energy effect due to the plasmon excitation. The $GW$+$C$ spectrum agrees with the overall profile of the experimental spectrum better than the $GW$ result; the broader spectrum is obtained in the $GW+C$ result around $-2\sim-$3 eV than in the $GW$ one. The discrepancy in the level position in this region between the theory and experiment arises probably from the level underestimation of the flat GGA HOMO$-$1 band [see Fig. <ref> (a)].
(Color online) (a) Ab initio photo-emission spectra and experimental one measured with the HeII radiation at 50 K (green open circles) (Ref. TMTSF-PES-1). The spectra are calculated for the HOMO, HOMO$-1$, and HOMO$-2$ bands. Red-thick-solid, blue-dotted, and black-thin-solid curves denote the $GW$+$C$, $GW$, and GGA results, respectively. A Lorentzian broadening of 0.02 eV is applied to the calculated spectra.
§.§ Spectral function of SrVO$_3$
Next, we consider the low-energy plasmon-fluctuation effect on the electronic structure of the transition-metal oxide SrVO$_3$. Figure <ref> shows the spectral function calculated for the $t_{2g}$ and O$_p$ bands, where the panels (a) and (b) show the $GW$ and $GW$+$C$ results, respectively. The GGA bands are depicted with blue-solid curves. The low-energy plasmon satellite emerges around 1 eV above (below) the unoccupied (occupied) part of the $t_{2g}$ bands, but the intensity is weaker than that of (TMTSF)$_2$PF$_6$. Similarly to the TMTSF case, the $GW$+$C$ for SrVO$_3$ makes the plasmon satellite broader than the $GW$ result. On the O$_p$ bands, the self-energy effect is appreciable; the imaginary part of the self-energy, which is related to the lifetime of the quasiparticle states, becomes large as the binding energy is apart from the Fermi level. Thus, the spectrum of the O$_p$ band becomes rather broad.
The panel (c) shows the comparison of calculated density of states. We see that the GGA spectrum in the $t_{2g}$ and O$_p$ bands is largely renormalized in the $GW$ and $GW+C$ spectra. The shape of the $GW$ spectrum resembles the $GW$+$C$ one, though the latter is somewhat more broadened.
(Color online) Spectral function for the $t_{2g}$ and O$_p$ bands of SrVO$_3$ calculated with (a) $GW$ approximation and (b) $GW+C$ method. Blue-solid curves are the GGA results. The Fermi level is at zero energy. The colorbar is in linear scale. (c) Comparison of the density of states among $GW$+$C$ (red-solid curve), $GW$ (blue-dotted curve), and GGA (black-thin curve) results.
Figure <ref> compares theoretical photo-emission spectra with experimental ones taken from Refs.PES-SVO-60 and PES-SVO-900. Light-green circles and black dots are the experimental photoemission spectra with the photon energies $h\nu\sim$ 60 eV (Ref. PES-SVO-60) and $h\nu\sim$ 900 eV (Ref. PES-SVO-900), respectively. On the overall profile, the ab initio $GW$+$C$ spectrum reasonably reproduces the experimental results such as the relative intensity between the $t_{2g}$ and O$_p$ bands. <cit.> On the other hand, we see that the theoretical incoherent intensity around $-1\sim-$2 eV is weaker than the experimental one.
(Color online) (a) Ab initio photo-emission spectra and experimental ones. Red-thick-solid, blue-dotted, and black-thin-solid curves are the $GW$+$C$, $GW$, and GGA results, respectively. In the experiment, light-green circles and black dots denote the photoemission spectra with the photon energies $h\nu\sim$ 60 eV (Ref. PES-SVO-60) and $h\nu\sim$ 900 eV (Ref. PES-SVO-900), respectively. A Lorentzian broadening of 0.05 eV is applied to the calculated spectra.
In Fig. <ref>, we compare theoretical spectra of an unoccupied region in the $t_{2g}$ bands
\begin{eqnarray}
A(\omega>\mu)=\int_{{\rm BZ}} A({\bf k},\omega>\mu) d{\bf k}
\end{eqnarray}
with experimental one taken from the soft x-ray absorption spectrum (green circles) (Ref. XAS-SVO). The effect of the low-energy plasmon excitation on the electronic structure appears to be stronger in the unoccupied region than in the occupied one. <cit.> The ab initio $GW$+$C$ spectrum resembles the experimental spectrum more closely than the $GW$ result; the cumulant expansion reduces the quasiparticle intensity and shifts the plasmon satellite to a lower energy. We note that the theoretical spectra do not include the contribution from the $e_g$ states. Also, in SrVO$_3$, since the local-interaction effect competes with the plasmon excitation, the view of the competition of the several factors would be important for quantitative understanding, which remains to be explored.
(Color online) Comparison between ab initio spectra and experimental one (green open circles) for the unoccupied region of the $t_{2g}$ bands. The experimental data are taken from the soft x-ray absorption spectrum (Ref. XAS-SVO). Red-thick-solid, blue-dotted, and black-thin-solid curves are the $GW$+$C$, $GW$, and GGA results, respectively. A Lorentzian broadening of 0.05 eV is applied to the calculated spectra.
§ CONCLUSION
We have performed ab initio $GW$ plus cumulant-expansion calculations for an organic conductor (TMTSF)$_2$PF$_6$ and a transition-metal oxide SrVO$_3$ to study the low-energy plasmon-fluctuation effect on the electronic structure. The bands around the Fermi level of these materials are isolated from the other bands, and the low-energy plasmon excitations derived from these isolated bands exist. Our calculated reflectance spectra well identify the experimental low-energy plasmon peaks. By calculating the cumulant-expanded Green's function based on the $GW$ approximation to the self-energy, we have simulated spectral functions and compare them with photoemission data. We have found agreements between them, indicating that the low-energy plasmon excitation certainly affects the low-energy electronic structure; it reduces the quasiparticle spectral weight around the Fermi level and leads to the weight transfer to the satellite parts. This effect was found to be more or less appreciable in (TMTSF)$_2$PF$_6$ than in SrVO$_3$. In particular, in (TMTSF)$_2$PF$_6$, the spectrum at the standard $GW$ level exhibits a clear plasmaron state, but considering the plasmon-fluctuation effects not treated in the standard $GW$ calculation leads to the disappearance of the state in the $GW+C$. Since the low-energy isolated-band structure is commonly found in various materials, the low-energy plasmon effect pursued in the present work can provide a basis for understanding the electronic structure of real systems. The recent progress in the photoemission experiment for the correlated materials (Refs. TMTSF-ARPES-1,TMTSF-ARPES-2,TMTSF-ARPES-3,SVO-ARPES-1,SVO-ARPES-2,Q1D-PES-1,Q1D-PES-2,Q1D-ARPES-1,Q1D-ARPES-2,Q1D-ARPES-3) requires a concomitant progress on the theoretical side and the present ab initio many-body calculations would provide firm basis for them.
In the present study, we have focused on the long-range correlation and treated it effectively with the cumulant-expansion method, while the short-ranged correlation, which is appropriately described by the $T$-matrix framework, is neglected. This is a future challenge which remains to be explored.
In addition, recent photoemission spectroscopy reveals appreciable differences in the electronic structure between the bulk and thin film systems. <cit.> The electronic structure of the surface is very sensitive to the atomic configurations at the surface <cit.> and therefore careful analyses for the atomic structure and its effect on electronic structure are required. The ab initio calculations for the surface effect are clearly important for the deep understanding of the spectroscopy of the real materials. This is also a future challenge.
We would like to thank Norikazu Tomita and Teppei Yoshida for useful discussions. Calculations were done at Supercomputer center at Institute for Solid State Physics, University of Tokyo. This work was supported by Grants-in-Aid for Scientific Research (No. 22740215, 22104010, 23110708, 23340095, 23510120, 25800200) from MEXT, Japan, and Consolidator Grant CORRELMAT of the European Research Council (project number 617196).
§ THE OFF-DIAGONAL EFFECT ON THE SPECTRAL FUNCTION
We briefly describe effects of band-off-diagonal terms of the self-energy on the spectral functions. With neglecting the band-off-diagonal terms, the spectral function is calculated as
\begin{eqnarray}
A({\bf k},\omega)=\frac{1}{\pi} \sum_{\alpha} \Biggl| {\rm Im} \frac{1}{\omega-(\epsilon_{\alpha {\bf k}} + \Sigma_{\alpha {\bf k}} (\omega) + \Delta)} \Biggr|, \label{Akwdiag}
\end{eqnarray}
where the matrix element of the self-energy with respect to the KS state $|\phi_{\alpha{\bf k}}\rangle$
\begin{eqnarray}
\Sigma_{\alpha{\bf k}}(\omega)=\langle\phi_{\alpha{\bf k}}|\Sigma^{X}+\Sigma^{C}(\omega)-V^{xc}|\phi_{\alpha{\bf k}}\rangle \label{Sigma}
\end{eqnarray}
is the diagonal term of Eq. (<ref>). The $\Delta$ in Eq. (<ref>) is the energy shift to correct the mismatch of the Fermi level between the initial and final states [see Eq. (<ref>)].
Figure <ref> displays the GW spectral function of SrVO$_3$ (a) without and (b) with band off-diagonal terms of self-energy. Also, the panel (c) shows the comparison between the calculated density of states. We do not see discernible difference between the two; the band off-diagonal terms of the self-energy is negligible. This is because, in SrVO$_3$, the $d$ orbitals of the V atom are well localized and the hybridization with O$_p$ orbital is small. Similarly to SrVO$_3$, the band-off-diagonal effect on the spectral function is found to be very small in (TMTSF)$_2$PF$_6$.
(Color online) Spectral functions for $t_{2g}$ and O$_p$ bands of SrVO$_3$ (a) without and (b) with the band off-diagonal matrix elements of the self-energy. Blue-solid curves are the GGA result. The Fermi level is at zero energy. The colorbar is in linear scale. (c) Comparison of the density of states without (dark-red-solid curve) and with (light-green-dotted curve) the off-diagonal terms.
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\begin{eqnarray}
\end{eqnarray}
where $G_0$ and $C^h$ are the non-interacting Green's function and the cumulant, respectively, and we adopt the compressed notation $1\equiv({\bf r},t)$, etc. The Green's function based on the Dyson equation is
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\end{eqnarray}
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G_0(1,2)C^h(1,2)=\int d(34) G_0(1,3)\Sigma(3,4)G_0(4,2). \label{Ch12}
\end{eqnarray}
From this equation, the time dependence of the cumulant is determined. In the practical calculation, the $GW$ approximation is employed in the self-energy in the right hand side in Eq. (<ref>).
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1511.00422
|
Holroyd]Alexander E. Holroyd
Alexander E. Holroyd, Microsoft Research, Redmond, WA 98052, USA.
<http://research.microsoft.com/ holroyd>
Levine]Lionel Levine
Lionel Levine, Cornell University, Ithaca, NY 14853,
USA. <http://www.math.cornell.edu/ levine>
Winkler]Peter Winkler
Peter Winkler, Dartmouth College, Hanover, NH 03755, USA.
<http://math.dartmouth.edu/ pw>
The second author is supported by NSF grant
DMS-1455272 and a Sloan Fellowship. The third author is supported by NSF
grant DMS-1162172.
An abelian processor is an automaton whose output is independent of the order
of its inputs. Bond and Levine have proved that a network of abelian
processors performs the same computation regardless of processing order
(subject only to a halting condition). We prove that any finite abelian
processor can be emulated by a network of certain very simple abelian
processors, which we call gates. The most fundamental gate is a toppler, which absorbs input particles until their number exceeds some given
threshold, at which point it topples, emitting one particle and returning to
its initial state. With the exception of an adder gate, which simply
combines two streams of particles, each of our gates has only one input wire.
Our results can be reformulated in terms of the functions computed by
processors, and one consequence is that any increasing function from $\N^k$
to $\N^\ell$ that is the sum of a linear function and a periodic function can
be expressed in terms of floors of quotients by integers, and addition.
§ INTRODUCTION
Consider a network of finite-state automata, each with a finite input and
output alphabet. What can such a network reliably compute if the wires
connecting its components are subject to unpredictable delays? The networks
we will consider have a finite set of $k$ input wires and $\ell$ output
wires. Even these are subject to delays, so the network computes a function
$\N^k \to \N^\ell$: The input is a $k$-tuple of natural numbers ($\N =
\{0,1,2,\ldots\}$) indicating how many letters are fed along each input wire,
and the output is an $\ell$-tuple indicating how many letters are emitted
along each output wire.
The essential issue such a network must overcome is that the order in which
input letters arrive at a node must not affect the output. To address this
issue, Bond and Levine <cit.>, following Dhar <cit.>,
proposed the class of abelian networks. These are networks each of
whose components is a special type of finite automaton called an
abelian processor. Certain abelian networks such as sandpile
<cit.> and rotor <cit.> networks produce intricate fractal
outputs from a simple input. From the point of view of computational
complexity, predicting the final state of a sandpile on a simple undirected graph can be done in polynomial time <cit.>, and in fact this problem is $\P$-complete <cit.>. Computing the sum of two elements in the sandpile group is also $\P$-complete <cit.>.
But on finite directed multigraphs, deciding whether a sandpile will halt is already $\NP$-complete <cit.>. Analogous problems on infinite graphs are undecidable:
An abelian network whose underlying graph is $\Z^2$, or a sandpile network whose underlying graph is the product of $\Z^2$ with a finite path, can emulate a Turing machine <cit.>.
The following definition is equivalent to that in <cit.> but simpler to
A processor with input alphabet $A$,
output alphabet $B$ and state space $Q$ is a collection of transition maps and output maps
\[ t_i : Q \to Q \quad\text{and}\quad o_i : Q \to \N^B \]
indexed by $i \in A$.
The processor is abelian if
\begin{equation}\label{commute}
t_i t_j = t_j t_i \quad\text{and}\quad
o_i + o_j t_i = o_j + o_i t_j
\end{equation}
for all $i,j \in A$. The interpretation is that if the processor receives
input letter $i$ while in state $q$, then it transitions to state $t_i(q)$
and outputs $o_i(q)$. The first equation in (<ref>) above asserts
that the processor moves to the same state after receiving two letters,
regardless of their order. The second guarantees that it produces the same
output. The processor is called finite if both the alphabets $A$, $B$
and the state space $Q$ are finite. In this paper, all abelian processors
are assumed to be finite and to come with a distinguished starting state
$q^\start$ that can access all states: each $q \in Q$ can be obtained by a
composition of a finite sequence of transition maps $t_i$ applied to
We say that an abelian processor computes the function
$ F : \N^A \to \N^B $ if inputting $\xx_a$ letters $a$ for each $a \in A$
results in the output of $(F(\xx))_b$ letters $b$ for each $b \in B$. Our
convention that the various inputs and outputs are represented by different
letters is useful for notational purposes. An alternative viewpoint would be
to regard all inputs and outputs as consisting of indistinguishable
“particles”, whose roles are determined by which input or output wire they
pass along.
An abelian network is a directed graph with an abelian processor located
at each node, with outputs feeding into inputs according to the graph
structure, and some inputs and outputs designated as input and output wires
for the entire network. (We give a more formal definition below in
§<ref>.) An abelian network can compute a function as
follows. We start by feeding some number of letters along each input wire.
Then, at each step, we choose any processor that has at least one letter
waiting at one of its inputs, and process that letter, resulting in a new
state of that processor, and perhaps some letters emitted from its outputs.
If after finitely many steps all remaining letters are located on the output
wires of the network, then we say that the computation halts.
The following is a central result of <cit.>,
generalizing the “abelian property” of Dhar <cit.>
and Diaconis and Fulton <cit.> (see
<cit.> for further background). Provided the
computation halts, it does so
regardless of the choice of processing order. Moreover,
the letters on the output wires and the final states of the processors are also
independent of the processing order. Thus, a network that halts on
all inputs computes a function from $\N^k$ to $\N^\ell$
(where $k$ and $\ell$ are the numbers of input and output
wires respectively). This function is itself of a form
that can be computed by some abelian processor, and we say
that the network emulates this processor.
The main goal of this paper is to prove a result in the opposite direction.
Just as any boolean function $\{0,1\}^A \to \{0,1\}^B$ can be
computed by a circuit of AND, OR and NOT gates, we show that any function $\N^A\to\N^B$
computed by an abelian processor can be computed by a network of simple
abelian logic gates. Furthermore (as in the boolean case), the
network can be made directed acyclic, which is to say that the graph has
no directed cycles. We will define our gates immediately after stating the
main theorems.
Any finite abelian processor can be emulated by a
finite, directed acyclic network of , , , and .
If the processor satisfies certain additional conditions, then some gates are
not needed. An abelian processor is called bounded if the range of the
function that it computes is a finite subset of $\N^B$.
Any bounded finite abelian processor can
be emulated by a finite, directed acyclic network of , , and .
An abelian processor $\Proc$ is called recurrent if for every pair of
states $q,q'$ there is a finite sequence of input letters that causes it to
transition from $q$ to $q'$. An abelian processor that is not recurrent is
called transient.
Any recurrent finite abelian processor can be
emulated by a finite, directed acyclic network of , and .
§.§ The gates
3|c|Single state
[adder] (a) ;
(a) edge[->] ++(1,0);
(a) edge[<-] ++(-1,-.6);
(a) edge[<-] ++(-1,.6);
$(x,y)\mapsto x+y$
[splitter] (a) ;
(a) edge[->] ++(1,.6);
(a) edge[->] ++(1,-.6);
(a) edge[<-] ++(-1,0);
$x\mapsto (x,x)$
$(\lambda \geq 2$) [abelian]
[toppler] (a) $\lambda$;
[<-] (a.west) – ++(-1,0);
[->] (a.east) – ++(1,0);
$\displaystyle x\mapsto \Bigl\lfloor \frac x\lambda \Bigr\rfloor$
primed ($1 \leq q < \lambda$)
[toppler,prime=$q$] (a) $\lambda$;
[<-] (a.west) – ++(-1,0);
[->] (a.east) – ++(1,0);
$\displaystyle x\mapsto \Bigl\lfloor \frac {x+q}\lambda \Bigr\rfloor$
[delayer] (a) ;
[<-] (a.west) – ++(-1,0);
[->] (a.east) – ++(1,0);
$\begin{aligned}x&\mapsto \max(x-1,0)\end{aligned}$
[presink] (a) ;
[<-] (a.west) – ++(-1,0);
[->] (a.east) – ++(1,0);
$\begin{aligned}x&\mapsto \min(x,1)\end{aligned}$
Abelian gates and the functions they compute.
Table <ref> lists our abelian logic gates, along
with the symbols we will use when illustrating networks. A
splitter has one incoming edge, two outgoing edges,
and a single internal state. When it receives a letter, it
sends one letter along each outgoing edge. On the other
hand, an adder has two incoming edges, one outgoing
edge, and again a single internal state. For each letter
received on either input, it emits one letter. The rest of
our gates each have just one input and one output.
For integer $\lambda \geq 2$, a $\lambda$-toppler has
internal states $0,1,\ldots,\lambda{-}1$. If it receives a
letter while in state $q<\lambda{-}1$, it transitions to
state $q{+}1$ and sends nothing. If it receives a letter
while in state $\lambda{-}1$, it “topples": it transitions
to state $0$ and emits one letter. A $\lambda$-toppler
that begins in state $0$ computes the function $x \mapsto
\floor{x/\lambda}$; if begun in state $q>0$ it computes the
function $x \mapsto \floor{(x{+}q)/\lambda}$. A toppler is
called unprimed if its initial state is $0$, and
primed otherwise.
The above gates are all recurrent. Finally, we have two transient gates whose
behaviors are complementary to one another. A delayer has two internal
states $0,1$. If it receives an input letter while in state $0$, it moves
permanently to state $1$, emitting nothing. In state $1$ it sends out one
letter for every letter it receives. Thus, begun it state $0$, it computes
the function $x \mapsto \max(x{-}1,0)=(x-1)^+$. A presink has two
internal states $0,1$. If it receives a letter while in state $0$, it
transitions permanently to state $1$ and emits one letter. All subsequent
inputs are ignored. From initial state $0$ it computes $x \mapsto
\min(x,1)=\ind[x>0]$.
The topplers form an infinite family indexed by the
parameter $\lambda \geq 2$. If we allow our network to
have feedback (i.e., drop the requirement that it be
directed acyclic) then we need only the case $\lambda=2$,
and in particular our palette of gates is reduced to a
finite set. Feedback also allows us to eliminate one
further gate, the delayer.
For any $\lambda\geq 3$, a $\lambda$- can
be emulated by a finite abelian network of
, and $2$-. So can a .
The toppler is a very close relative of the two most extensively studied
abeilan processors: the sandpile node and the rotor router node (see
e.g. <cit.>). Specifically, for a node of degree $k$, either of
these is easily emulated by $k$ suitably primed topplers in parallel, as in
<ref>. (Sandpiles and rotors are typically considered on undirected
graphs, in which case the $k$ inputs and $k$ outputs of a vertex are both
routed along its $k$ incident edges). Rotor aggregation <cit.> can also
be emulated, by inserting a delayer into the network for the rotor.
[splitter,right= 1cm of a](s);
[toppler,right of=s,prime=1](t2)$3$;
[toppler,above of=t2,prime=0](t1)$3$;
[toppler,below of=t2,prime=2](t3)$3$;
[<-] (a)–++(-1,0);
[<-] (a)–++(-1,-1);
[<-] (a)–++(-1,1);
[->] (a)–(s);
[->] (s) edge[bend left=20](t1.west);
[->] (s) edge(t2);
[->] (s) edge[bend right=20](t3.west);
[->] (t1)–++(1,0);
[->] (t2)–++(1,0);
[->] (t3)–++(1,0);
Emulating a rotor of degree $3$ with topplers.
To emulate a sandpile node, prime the three topplers identically
(and optionally combine them into one toppler preceding a splitter).
For a rotor aggregation node, insert a
delayer between the adder and the splitter.
§.§ Unary input
A processor has unary input if its input alphabet $A$
has size $1$ (so that it computes a function
$\N\to\N^\ell$). It is easy to see from the definition
that any finite-state processor with unary input is
automatically abelian. Indeed, the same holds for any
processor with exchangeable inputs, i.e. one whose
transition maps and output maps are identical for each
input letter. (Such a processor can be emulated by adding
all its inputs and feeding them into a unary-input
processor). Note that all our gates have unary input
except for the adder, which has exchangeable inputs.
Theorems <ref>–<ref>
become rather straightforward if we restrict to unary-input
processors. (See Lemmas <ref> and
<ref>.) Our main contribution is that
unary-input gates (and adders) suffice to emulate processors
with any number of inputs. (In contrast, elementary
considerations will show that there is no loss of
generality in restricting to processors with unary
output; see Lemma <ref>.)
§.§ Function classes
Left: the graph of a ZILP function $f:\N^2\to\N$. The height of a bar
gives the value of the function, and the origin is at the front of the
picture. The periodic component has periods $4$ and $5$ in the two
coordinates, as indicated by the highlighted bars; the linear part has slopes $2/4$
and $4/5$ respectively. Right: A ZILEP function comprising the same “recurrent part"
together with added “transient margins".
An important preliminary step in the proofs of
<ref> will be to
characterize the functions that can be computed by abelian processors (as
well as by the bounded and recurrent varieties). The characterizations turn
out to be quite simple. A function $F:\N^k\to\N^\ell$ is computed by some
finite abelian processor if and only if: (i) it maps the zero vector
$\zero\in\N^k$ to $\zero\in\N^\ell$; (ii) it is (weakly) increasing; and
(iii) it can be expressed as a linear function plus an eventually
periodic function (see <ref> for precise meanings). We call a
function satisfying (i)–(iii) ZILEP (zero at zero, increasing, linear
plus eventually periodic). On the other hand, a function is computed by some
recurrent finite abelian processor if it is ZILP:
eventually periodic is replaced with periodic. <ref>
shows examples of ZILP and ZILEP functions of two variables, illustrating
some of the difficulties to be overcome in computing them by networks.
Our main theorems may be recast in terms of functions rather than processors.
Table <ref> summarizes our main results from this perspective.
(A function is $\N$-linear if it is linear and takes values in $\N^\ell$
for some $\ell$; $\Q$-linear is defined similarly). For instance, the
following is a straightforward consequence of <ref>.
Recurrent abelian functions Let $\mathcal{R}$ be
the smallest set of functions $F:\N^k \to \N$ containing
the constant function $1$ and the coordinate functions
$x_1, \ldots, x_k$, and closed under addition and
compositions of the form $F \mapsto \floor{F/\lambda}$ for
integer $\lambda \geq 2$. Then $\mathcal{R}$ is the set of
all increasing functions $\N^k \to \N$ expressable as $L+P$
where $L, P : \N^k \to \Q$ with $L$ linear and $P$
Components L + P Theorem(s)
(splitter and adder only) $\N$-linear + zero Lemma <ref>
presink, delayer $\N$-linear + eventually constant <ref>, <ref>
$\lambda$-toppler $\Q$-linear + periodic <ref>, <ref>
$\lambda$-toppler, presink, delayer $\Q$-linear + eventually periodic <ref>, <ref>
Four different classes of abelian network.
The second column indicates the class of increasing functions $\N^k \to \N^\ell$ computable by a finite, directed acyclic abelian network whose components are splitters, adders and the gates listed in the first column.
§.§ Outline of article
Section <ref> identifies the classes of functions computable by
abelian processors, as described above, and formalizes the definitions and
claims relating to abelian networks. Section <ref> contains a few
elementary reductions including the proof of Proposition <ref>.
The core of the paper is Sections <ref>, <ref>
and <ref>, which are devoted respectively to the proofs of
Theorems <ref>, <ref>
and <ref>. The first and last of these are by induction
on the number of inputs to the processor, and the last is by far the hardest.
A recurring theme in the proofs is meagerization, which amounts to use
of the easily verified identity
\begin{equation} \label{e.meagerization} x =
\floor{\frac{x}{m}} + \floor{\frac{x+1}{m}} + \cdots +
\floor{\frac{x+m-1}{m}} \end{equation}
for positive
integers $x$ and $m$. A key step in the proof of the general emulation result,
Theorem <ref>, is the introduction of a
ZILP function that computes the minimum of its $n$
arguments provided they are not too far apart. That this
function in turn can be emulated follows from the recurrent
case, Theorem <ref>.
In Section <ref> we show that no gates can
be omitted from our list. We conclude by posing some open
problems in Section <ref>.
§ FUNCTIONS COMPUTED BY ABELIAN PROCESSORS AND NETWORKS
In preparation for the proofs of the main results about emulation, we begin
by identifying the classes of functions that need to be computed.
§.§ Abelian processors
If $\Proc$ is an abelian processor with input alphabet $A$, and $w = i_1
\cdots i_\ell$ is a word with letters in $A$, then we define the transition
and output maps corresponding to the word:
\[ t_w := t_{i_\ell} \cdots t_{i_1}; \]
\[ o_w := o_{i_1} + o_{i_2} t_{i_1} + o_{i_3}t_{i_2}t_{i_1}
+ \cdots + o_{i_\ell} t_{i_{\ell-1}} \cdots t_{i_1}. \]
For any words $w,w'$ such that $w'$ is a permutation of $w$, we have $t_w =
t_{w'}$ and
$o_w = o_{w'}$.
This follows from the definition of an abelian processor, by induction on the length of $w$.
The function $f=f_{\Proc}$ computed by an abelian processor $\Proc$ with
initial state $q^\start$ is given by
\[ f(\xx) = o_{w(\xx)}(q^\start),\qquad \xx\in\N^A, \]
where $w(\xx)$ is
any word that contains $x_i$ copies of the letter $i$ for all $i \in A$. We
denote vectors by boldface lower-case letters, and their coordinates by the
corresponding lightface letter, subscripted.
Let $f=f_{\Proc}$. If $t_\yy(q^\start) =
t_{\yy'}(q^\start)$ then for any $\xx,\yy,\yy' \in \N^k$
\[ f(\xx+\yy) - f(\yy) = f(\xx+\yy') - f(\yy'). \]
Since $\yy$ and $\yy'$ leave $\Proc$ in the same state, subsequent inputs have the same effect.
A function
$f : \N^k \to \N^\ell$ is (weakly) increasing if $\xx \leq \yy$ implies
$f(\xx) \leq f(\yy)$ where $\leq$ denotes the coordinatewise partial
ordering. A function $P : \N^k \to \Q^\ell$ is periodic if there is a
subgroup $\Lambda \subset \Z^k$ of finite index such that $P(\xx) = P(\yy)$
whenever $\xx-\yy \in \Lambda$. A function $P$ is eventually periodic if
there exist $\lambda_1, \ldots, \lambda_k \geq 1$ and $r_1,\ldots,r_k$ such
\[ P(\xx) = P(\xx+\lambda_i \basis_i) \]
for all $i=1,\ldots,k$ and all
$\xx \in \N^k$ such that $x_i \geq r_i$. Here $\basis_i$ is the $i$th
standard basis vector.
Let $k, \ell \geq 1$. A function $f: \N^k \to \N^\ell$ can be computed by a
finite abelian processor if and only if $f$ satisfies all of the following.
* $f(\zero)=0$.
* $f$ is increasing.
* $f = L+P$ for a linear function $L$ and an eventually periodic function $P$.
As mentioned earlier, we call a function satisfying (i)–(iii) ZILEP.
Any $f = f_{\Proc}$ trivially satisfies $f(\zero)=0$. To see that $f$ is
increasing, given $\xx \leq \yy$ there are words $w(\xx)$ and $w(\yy)$ (where
the number of occurrences of letter $i$ in $w(\zz)$ is $z_i$) for which
$w(\xx)$ is a prefix of $w(\yy)$. Then
\[ o_{w(\xx)} + o_{u} t_{w(\xx)} = o_{w(\yy)}. \]
Since the second term of the left is nonnegative, $o_{w(\xx)}(q) \leq o_{w(\yy)}(q)$.
To prove (iii), note that since $Q$ is finite, some power of $t_i$ is idempotent, that is
\[ t_i^{2\lambda_i} = t_i^{\lambda_i} \]
for some $\lambda_i \geq 1$. Let $L : \N^k \to \N^\ell$ be the linear function sending
\[ \lambda_i \basis_i \mapsto f(2\lambda_i \basis_i) - f(\lambda_i
\basis_i) \]
for each $i=1,\ldots,k$. Now we apply Lemma <ref> with
$\yy=\lambda_i \basis_i$ and $\yy'=2\yy$
to get
\begin{equation} \label{e.eventually} f(\zz+\lambda_i \basis_i) - f(\zz)
= f(2\lambda_i \basis_i) - f(\lambda_i \basis_i) \qquad \text{for all } \zz \geq \lambda_i \basis_i, \end{equation}
which shows that $f-L$ is eventually periodic. Thus $f$ satisfies (i)–(iii).
Conversely, given an increasing $f = L+P$, define an equivalence relation on
$\N^k$ by $\yy \equiv \yy'$ if $f(\yy+\zz) - f(\yy) = f(\yy'+\zz)-f(\yy')$
for all $\zz \in \N^k$. If $L$ is linear and $P$ is eventually periodic, then
there are only finitely many equivalence classes: if $y_i \geq r_i +
\lambda_i$ then $\yy \equiv \yy - \lambda_i \basis_i$, so any $\yy \in \N^k$
is equivalent to some element of the cuboid $[0,\lambda_1+r_1] \times \cdots
\times [0,\lambda_k+r_k]$.
Now consider the abelian processor on the finite state space $\N^k / \equiv$
with $t_i(\xx) = \xx+ \basis_i$ and $o_i(\xx) = f(\xx+\basis_i) - f(\xx)$.
Note that $t_i$ and $o_i$ are well-defined. With initial state $\zero$, this
processor computes $f$.
§.§ Recurrent abelian processors
Recall that an abelian processor is called recurrent if for any states
$q,q'\in Q$ there exists $\xx \in \N^k$ such that $q'=t_{\xx}(q)$. Since we
assume that every state is accessible from the initial state $q^0$, this is
equivalent to the assertion that for every $q \in Q$ and $\yy \in \N^k$ there
exists $\zz \in \N^k$ such that $q = t_{\yy+\zz}(q)$. Our next result
differs from Theorem <ref> in only two words: recurrent
has been added and eventually has been removed! As mentioned earlier,
we call a function satisfying (i)–(iii) below ZILP.
Let $k, \ell \geq 1$. A function $f: \N^k \to \N^\ell$
can be computed by a recurrent finite abelian processor if and only if $f$
satisfies all of the following.
* $f(\zero)=0$.
* $f$ is increasing.
* $f = L+P$ for a linear function $L$ and a periodic function $P$.
By Theorem <ref>, $f$ satisfies (i) and (ii) and $f=L+P$ with
$L$ linear and $P$ eventually periodic. To prove (iii) we must show
that equation (<ref>) holds for all $\zz \in \N^k$.
By recurrence, for any $\yy \in \N^k$ and any $i \in A$
there exists $\yy' \geq \lambda_i \basis_i$ such that $t_{\yy'}(q)=t_{\yy}(q)$.
Now taking $\xx = \lambda_i \basis_i$ in Lemma <ref>, the linear terms cancel, leaving
\[ P(\yy + \lambda_i \basis_i) - P(\yy) = P(\yy' + \lambda_i \basis_i) - P(\yy'). \]
The right side vanishes since $P$ is eventually periodic. Since $\yy \in \N^k$ was arbitrary, $P$ is in fact periodic.
Conversely, given an increasing $f=L+P$, we define an abelian processor
$\Proc$ on state space $\N^k / \equiv$ as in the proof of
Theorem <ref>. If $L$ is linear and $P$ is periodic, then for
each $i=1,\ldots,k$ we have $\yy \equiv \yy - \lambda_i \basis_i$ whenever
$y_i \geq \lambda_i$. Now given any $\xx, \yy \in \N^k$ we find $\xx' \equiv
\xx$ with $\xx' \geq \yy$, so $\Proc$ is recurrent.
§.§ Abelian networks
An abelian network $\Net$ is a directed multigraph
$G=(V,E)$ along with specified pairwise disjoint sets
$I,O,T\subset E$ of input, output and trash
edges respectively. These edges are dangling: the input
edges have no tail, while the output and trash edges have
no head. The trash edges are for discarding unwanted
Each node $v \in V$ is labeled with an abelian processor $\Proc_v$ whose
input alphabet equals the set of incoming edges to $v$ and whose output
alphabet is the set of outgoing edges from $v$. In this paper, all abelian
networks are assumed finite: $G$ is a finite graph and each $\Proc_v$ is a
finite processor.
An abelian network operates as follows. Its total state is given by the internal
states $(q_v)_{v\in V}$ of all its processors $\Proc_v$, together with a vector
$\xx=(x_e)_{e\in E}\in \N^E$ that indicates the number of letters sitting on
each edge, waiting to be processed. Initially, $\xx$ is supported on the set
of input edges $I$. At each step, any non-output non-trash edge $e$ with $x_e>0$ is
chosen, and a letter is fed into the processor at its endnode $v$. Thus,
$x_e$ is decreased by $1$, the state of $\Proc_v$ is updated from $q_v$ to
$t_e(q_v)$, and $\xx$ is increased by $o_e(q_v)$ (interpreted as a vector in
$\N^E$ supported on the outgoing edges from $v$). Here $t$ and $o$ are the
maps associated to $\Proc_v$. The sequence of choices of the edges $e$ at
successive steps is called a legal execution. The execution is said to
halt if, after some finite number of steps, $\xx$ is supported on the
set of output and trash edges (so that there are no letters left to process).
The following facts are proved in <cit.>. Fixing the
initial internal states $\qq^\start=(q_v^\start)_{v\in V}$ and an input
vector $\xx\in\N^I$, if some execution halts then all legal executions halt.
In the latter case, the final states of the processors and the final output
vector do not depend on the choice of legal execution. Moreover, suppose for
a given $\qq^\start$ that the network halts on all input vectors.
Then the final output vector depends only on the input vector, so the abelian
network computes a function $\N^{I} \to \N^{O}$. If a network $\Net$ and a
processor $\Proc$ compute the same function, then we say that $\Net$
emulates $\Proc$.
If a finite abelian network
halts on all inputs, then it emulates some finite abelian processor.
We can regard the entire network as a processor, with its state given by the
vector of internal states $\qq=(q_v)_{v\in V}$. Its transition and output
maps are determined by feeding in a single input letter, performing any legal
execution until it halts, and observing the resulting state and output
letters. Feeding in two input letters and using (a special case of) the
insensitivity to execution order stated above, we see that the relations
(<ref>) hold, so the processor is abelian.
The abelian networks that halt on all inputs are characterized in <cit.>: they are those for which a certain matrix called the
production matrix has Perron-Frobenius eigenvalue strictly less than $1$. An
abelian network is called directed acyclic if its graph $G$ has no
directed cycles; such a network trivially halts on all inputs. This paper is
mostly concerned with directed acyclic networks, together with some networks
with certain limited types of feedback; all of them halt on all inputs.
§.§ Recurrent abelian networks
The next lemma follows from <cit.>, but
we include a proof for the sake of completeness. A
processor is called immutable if it has just one
state, and mutable otherwise. Among the abelian logic
gates in Table <ref>, splitters and adders are
immutable; topplers, delayers and presinks are mutable.
A directed acyclic network $\Net$ of recurrent processors emulates a recurrent processor.
We proceed by induction on the number $m$ of mutable processors in $\Net$. In
the case $m=0$, the network $\Net$ has only one state, so it is trivially
In the case $m=1$, the network consists of an immutable piece $\ProcI$
feeding into a recurrent processor $\Proc$ which feeds into an immutable
piece $\ProcJ$. Also $\ProcI$ can feed directly into $\ProcJ$. Writing
$(I_1,I_2)$ and $J$ for the functions computed by $\ProcI$ and $\ProcJ$
respectively, we have
\[ f_{\Net,q}(\xx) = J( f_{\Proc,q}(I_1(\xx))+I_2(\xx)). \]
$\ProcI$ and $\ProcJ$ are immutable, $I_1, I_2, J$ are linear functions. By
Theorem <ref> the right side is the sum of a linear and a
periodic function since $f_{\Proc,q}$ is.
For the inductive step, suppose $m \geq 1$. Since $\Net$ is
directed acyclic, it has a mutable processor $\Proc$ such
that no other mutable processor feeds into anything
upstream of $\Proc$. If $\Net$ has $k$ inputs, we can
regard the remainder $\Net - \Proc$ as a network with
$m{-}1$ mutable processors and $k{+}k'$ inputs, where $k'$
is the number of edges from $\Proc$ to $\Net-\Proc$. For
each state $q$ of $\Net$, the function $f = f_{\Net,q}$ has
the form
\[ f(\xx) = g(\xx, h(\xx)) + j(\xx) \]
where $g : \N^{k+k'} \to \N^\ell$
is the function computed by $\Net-\Proc$ in initial state $q$; and $h: \N^k
\to \N^{k'}$ and $j: \N^k \to \N^\ell$ are the functions sent by $\Proc$ in
initial state $q$ to $\Net-\Proc$ and the output of $\Net$, respectively.
By Theorem <ref> and the inductive hypothesis, each of $g,h,j$ is
the sum of a periodic and a linear function. Writing $g(\yy) = P(\yy) + \bb
\cdot \yy$ and $h(\xx) = Q(\xx) + \cc \cdot \xx$ we have
\[ f(\xx) = P(\xx, Q(\xx)+ \cc \cdot \xx) + \bb \cdot (\xx, Q(\xx) + \cc \cdot \xx) + j(\xx). \]
The first term is periodic and the second is a linear function plus a
periodic function. Since $q$ is arbitrary the proof is complete by
Theorem <ref>.
§.§ Varying the initial state
We remark that the emulation claims of our main theorems can be strengthened
slightly, in the following sense. Our definition of a processor $\Proc$
included a designated initial state $q^0$, but one may instead consider
starting $\Proc$ from any state $q\in Q$, and it may compute a different
function from each $q$. All of these functions can be computed by the
same network $\Net$, by varying the internal states of the gates in
$\Net$. To set up the network $\Net$ to compute the function $f_{\Proc,q}$,
we simply choose an input vector $\xx$ that causes $\Proc$ to transition from
$q^0$ to $q$, then feed $\xx$ to $\Net$ and observe the resulting gate
states. In the recurrent case, this amounts to adjusting the priming of
topplers. In the transient case, a “used” delayer can be replaced with a
wire, while a used presink becomes a trash edge.
§.§ Splitter-adder networks
In this section we show that splitter-adder networks
compute precisely the $\N$-linear functions. Using this, we
will see how Theorem <ref> implies
Let $k, \ell \geq 1$. The function $f :
\N^k \to \N^\ell$ can be computed by a network of splitters
and adders if and only if $f(\xx)=L\xx$ for some
nonnegative integer $\ell \times k$ matrix $L$.
If a network of splitters and adders has a directed cycle, then it does not
halt on all inputs, and so does not “compute a function” according to our
definition. If the network is directed acyclic then by
<ref> it computes a ZILP function. Since the network
is immutable, the periodic part of any linear + periodic decomposition must
be zero. Conversely, consider a network of $k$ splitters $\ProcS_i$ and
$\ell$ adders $\ProcA_j$, with $L_{ji}$ edges from $\ProcS_i$ to $\ProcA_j$.
Feed each input $i$ into $\ProcS_i$, and feed each $\ProcA_j$ into output
Given a function $F \in \mathcal{R}$, the function $\xx
\mapsto F(\xx)-F(\zero)$ can be computed by a finite,
directed acyclic network of splitters, adders and (possibly
primed) topplers.
By <ref> any such network emulates a recurrent
finite abelian processor, so $F$ has the desired form by
Theorem <ref>.
Conversely if $F=L+P$ with $L$ linear and $P$ periodic,
then $F(\xx)-F(\zero)$ is computable by a finite directed
acyclic network of splitters, adders and topplers by
Theorem <ref>. We induct on the number of
topplers to show that $F \in \mathcal{R}$. In the base case
there are no topplers, $\Net$ is a splitter-adder network,
so by Lemma <ref>, $F$ is an $\N$-linear function
of its inputs $x_1, \ldots, x_k$.
Assume now that at least one component of $\Net$ is a toppler. Since $\Net$
is directed acyclic, there is a toppler $\ProcT$ such that no other toppler
is downstream of $\ProcT$. Write $\ProcD$ for the portion of $\Net$
downstream of $\ProcT$, and $\ProcU = \Net - \ProcT - \ProcD$ for the
remainder of the network. Suppose $\ProcU$ sends outputs $r,s,\uu$
respectively to the output of $\Net$, to $\ProcT$, and to $\ProcD$; and that
the toppler $\ProcT$ sends output $t$ to $\ProcD$.
The downstream part $\ProcD$ consists of only splitters and adders, so it
computes a linear function
\[ L(t,\uu) = at+ \bb \cdot \uu\]
for some $a \in \N$ and $\bb \in \N^j$,
where $j$ is the number of edges from $\ProcU$ to $\ProcD$. The total output
of $\Net$ is
\[ F(\xx) - F(\zero) = r + L(t,\uu) = r + a\floor{\frac{s}{\lambda}} +
\bb \cdot \uu. \]
Each of $r,s,\uu$ is a function of the input $\xx =
(x_1,\ldots,x_k)$. By induction, $r$ and $s$ and each $u_i$ belongs to the
class $\mathcal{R}$, so $F$ does as well.
§ BASIC REDUCTIONS
In this section we describe some elementary network reductions.
Processor $\Proc$ emulates processor $\Proc'$ in initial state $q'$ if
there exists an initial state $q$ such that
\[ f_{\Proc,q} = f_{\Proc',q'}. \]
§.§ Multi-way splitters and adders
An $n$-splitter computes the function $\N \to \N^n$ sending $x \mapsto
(x,\ldots,x)$. It is emulated by a directed binary tree of $n-1$ splitters
with the input node at the root and the $n$ output nodes at the leaves.
Similarly, an $n$-adder computes the function $\N^n \to \N$ given by
$(x_1,\ldots,x_n)\mapsto x_1+\cdots+x_n$. It is emulated by a tree of $n-1$
§.§ The power of feedback
To emulate an unprimed $\lambda$-toppler, let $r = \ceiling{\log_2 \lambda}$ and let
\[ 2^r - \lambda = \sum_{i=0}^{r-2} b_i 2^i, \qquad
b_i \in \{0,1\} \]
be the binary representation of $2^r -
\lambda$. Consider $r$ $2$-topplers
$H_0,H_1,\ldots,H_{r-1}$ in series: the input node is
$H_0$, and each $H_i$ feeds into $H_{i+1}$ for $0\leq i <
r{-}1$. For $i < r-1$ the $2$-toppler $H_i$ is primed with
$b_i$. The last $2$-toppler $H_{r-1}$ is unprimed, and
feeds into an $s$-splitter ($s = 1+\sum_{i=0}^{r-1} b_i$)
which feeds one letter each into the output node $o$ and
the nodes $H_i$ such that $b_i=1$. This network repeatedly
counts in binary from $2^r - \lambda$ to $2^r -1$, and it
sends output precisely when it transitions from $2^r - 1$
back to $2^r - \lambda$. Hence, it emulates a
$\lambda$-toppler. See Figure <ref> for
We can emulate a $q$-primed $\lambda$-toppler using the same network, but with
different initial states for its $2$-topplers. The required states are simply those that result from feeding $q$ input letters into the network described above.
A delayer is constructed by splitting the output of a
$2$-toppler and adding one branch back in as input to the
$2$-toppler (Figure <ref>).
§.§ Primed topplers
The following shows that we can also do without primed topplers (at the
expense of using a transient gate: the presink).
A primed $\lambda$-toppler can be emulated by a directed acyclic network
comprising an unprimed $\lambda$-toppler, adders, splitters, and a presink.
See Figure <ref>. For $0\leq q<\lambda$ we have
$\lfloor(x+q)/\lambda\rfloor=\lfloor(x+q(x-1)^+)/\lambda\rfloor$, so we can
emulate a $q$-primed $\lambda$-toppler by splitting the input, feeding it
into a presink, and adding $q$ copies of the result into the original input
before sending it to an unprimed $\lambda$-toppler.
[toppler] (a) $3$;
[toppler,prime=$1$] (a3) $2$;
[toppler] (b3) [right of=a3] $2$;
[splitter] (s) [right=.5cm of b3] ;
[adder] (p) [left=5mm of a3] ;
[<-] (p.west) – ++(-.5,0);
[->] (s.east) – ++(.5,0);
(p) edge[->] (a3) (a3) edge[->] (b3) (b3) edge[->] (s)
(s) edge[->,bend left=50] (p);
[toppler] (a) $4$;
[toppler] (a) $2$;
[toppler] (b) [right of=a] $2$;
[<-] (a.west) – ++(-.5,0);
[->] (b.east) – ++(.5,0);
(a) edge[->] (b);
[toppler] (a) $5$;
[toppler,prime=$1$] (a) $2$;
[adder] (p2) [right=5mm of a] ;
[toppler,prime=$1$] (b) [right=5mm of p2] $2$;
[toppler] (c) [right of=b] $2$;
[splitter] (s) [right=.5cm of c] ;
[adder] (p) [left=5mm of a] ;
[<-] (p.west) – ++(-.5,0);
[->] (s.east) – ++(.5,0);
[->] (p) – (a);
[->] (a) – (p2);
[->] (p2) – (b);
[->] (b) – (c);
[->] (c) – (s);
(s) edge[->,bend left=50] (p)
(s) edge[->,bend left=50] (p2);
[delayer] (a) ;
[toppler] (a) $2$;
[splitter] (s) [right=.5cm of a] ;
[adder] (p) [left=5mm of a] ;
[<-] (p.west) – ++(-.5,0);
[->] (s.east) – ++(.5,0);
(p) edge[->] (a) (a) edge[->] (s)
(s) edge[->,bend left=50] (p);
Emulating a $3$-toppler, $4$-toppler, $5$-toppler
and delayer by networks of $2$-topplers.
[toppler,prime=$4$] (a) $\lambda$;
[toppler] (a) $\lambda$;
[adder,outer sep=3pt] (p) [left of= a] ;
[splitter] (s) [left of =p] ;
[presink] (x) [below=10mm of s] ;
[splitter] (xs) [right of=x] ;
[<-] (s.west) – ++(-.5,0);
[->] (a.east) – ++(.5,0);
(s) edge[->] (p) (p) edge[->] (a);
(s) edge[->] (x) (x) edge[->,bend right=20] (xs)
(xs) edge[->,bend left=55] (p)
(xs) edge[->,bend left=20] (p)
(xs) edge[->,bend right=20] (p)
(xs) edge[->,bend right=55] (p);
Emulating a primed toppler with an unprimed
§.§ Reduction to unary output
Let $\Proc$ be an abelian processor that computes $f : \N^k \to \N^\ell$. If
$\ell=1$ then we say that $\Proc$ has unary output. All of the logic
gates in Table <ref> have unary output with the exception of the
splitter. The next lemma shows that, for rather trivial reasons, it is
enough to emulate processors with unary output.
Any abelian processor can be emulated by a directed
acyclic network of splitters and processors with unary output.
Let $\Proc$ compute $f = (f_1,\ldots,f_\ell) : \N^k \to \N^\ell$. By ignoring
all outputs of $\Proc$ except the $j$th, we obtain an abelian processor
$\Proc_j$ that computes $f_j$. Each $\Proc_j$ has unary output, and $\Proc$
is emulated by a network that sends each input into an $\ell$-splitter that
feeds into $\Proc_1, \ldots, \Proc_\ell$ (Figure <ref>).
[block,minimum height=1cm,minimum width=1cm] (P1) $\Proc_1$;
[block,minimum height=1cm, minimum width=1cm,below of=P1] (P2) $\Proc_2$;
[splitter,left=2cm of P1] (a1) ;
[splitter,left=2cm of P2] (a2) ;
[left=1cm of a1] (aa1) $a_1$;
[left=1cm of a2] (aa2) $a_2$;
(aa1) edge (a1)
(aa2) edge (a2)
(a1) edge[bend left=20] (P1)
(a1) edge[bend right=20] (P2)
(a2) edge[bend left=20] (P1)
(a2) edge[bend right=20] (P2);
(P1.east) edge[->] ++(1,0)
(P2.east) edge[->] ++(1,0);
Emulating a $2$-output abelian processor with two unary-output processors.
In the subsequent proofs we can thus assume that the processors to be
emulated have unary output. By a $k$-ary processor we mean one with $k$
inputs. A $1$-ary processor is sometimes called unary.
§ THE RECURRENT CASE
In this section we prove Theorem <ref>. By
<ref> we may assume that the recurrent processor to be
emulated has unary output. We will proceed by induction on the number of
§.§ Unary case
We start with the unary case (i.e. one input), which will
form the base of our induction. An alternative would be to
start the induction with the trivial case of zero inputs,
but the simplicity of the unary case is illustrative.
By Theorem <ref>, a recurrent unary processor
computes an increasing function $f: \N \to \N$ of the form
$f(x) = cx + P(x)$, where $c \in \Q_{\geq 0}$ and $P: \N
\to \Q$ is periodic.
Let $\Proc$ be a recurrent unary processor that computes $f(x) = cx + P(x)$,
where $P$ is periodic of period $\lambda$. Then $\Proc$ can be emulated by a
network of adders, splitters and (suitably primed) $\lambda$-topplers.
Observe first that $c\lambda$ is an integer: since
$f(0)=0$, we have $P(\lambda) = P(0)=0$ thus $f(\lambda) =
c\lambda \in \N$. We construct a network of $c\lambda$
parallel $\lambda$-topplers as follows: the (unary) input
is split (by a $c\lambda$-splitter) into $c\lambda$
streams, each of which feeds into a separate
$\lambda$-toppler. The outputs of the topplers are then
combined (by a $c\lambda$-adder) to a single output
(Figure <ref>).
After $m\lambda$ letters are input to this network, $m \in \N$, each toppler
will return to its original state having output $m$ letters, for a total
output of $m \times c\lambda = c(m\lambda)$; thus the network does compute
$cx+Q(x)$ where $Q$ has period $\lambda$ or some divisor of $\lambda$. To
force $Q=P$ it suffices to choose the initial state $q$ in such a way that
the network's output for $x = 1,2,\dots,\lambda$ matches
$f(1),\dots,f(\lambda)$. This is easily done by setting $d_i =
f(x)-f(x{-}1)$, and for each $i$ with $1 \le i \le \lambda$, starting $d_i$
topplers in state $\lambda{-}i$.
Figure <ref> illustrates the network constructed to compute the
function $f = \frac34 x + P(x)$ where $P$ has period 4 with $P(0)=0$,
$P(1)=\frac14$, $P(2)=\frac64$ and $P(3)=\frac34$. The values of $f$ begin
$0,1,3,3,3,4,6,6,6,7,9,9,9,\dots$. The “I/O diagram" of $f$ is shown at the
top of the figure; dots represent input letters and bars are output letters;
the unfilled circle represents the initial state.
[circ] at (0,0) ;
in 1,...,11
[dot] at (,0) ; ;
in 1,1.9,2.1
ıin 0,4,8
(+ı-.5,-.7) – ++(0,1.4); ;
[toppler,prime=$3$] (a) $4$;
[toppler, below of=a,prime=$2$] (b) $4$;
[toppler, below of=b,prime=$2$] (c) $4$;
[splitter,left of=b] (i) ;
[adder, right of=b] (o) ;
(i) edge[bend left=35] (a) edge (b) edge[bend right=35] (c);
(o) edge[bend right=35] (a) edge (b) edge[bend left=35] (c);
(o) edge[->] ++(1,0);
(i) edge[<-] ++(-1,0);
Emulating a unary processor with a network of primed topplers.
§.§ Reduction to the meager case
A recurrent $k$-ary processor computes a function $f: \N^k
\to \N$ of the form $f(\xx)=\bb\cdot \xx + P(\xx)$ where
$P$ is periodic.
A recurrent processor is nondegenerate if $b_i \neq 0$
for all $i$.
Note that if $b_i=0$ then $f(\xx)$ does not depend on the
coordinate $x_i$. In this case, by
Theorem <ref> there is a finite $(k{-}1)$-ary
recurrent processor that computes $f$.
Denote the lattice of periodicity of $P$ by $\Lambda
\subset \Z^k$. Let $\lambda_i$ be the smallest positive
integer such that $\lambda_i \basis_i \in \Lambda$. For
the purposes of the forthcoming induction, we focus on the
last coordinate.
We say that a $k$-ary processor $\Proc$ is meager if
$f_\Proc(\lambda_k \basis_k)=1$.
Note that if $\Proc$ is meager then for all $\xx \in \N^k$ we have
\[ f_\Proc(\xx+\lambda_k \basis_k) = f_\Proc(\xx) + 1. \]
Next we
emulate a nondegenerate recurrent processor by a network of meager
Let $\Proc$ be a nondegenerate recurrent $k$-ary processor
and let $m = f_\Proc(\lambda_k \basis_k) = \lambda_k b_k$.
Then $\Proc$
can be emulated by a network of $m-1$ splitters, $m-1$ adders, and $m$ meager
recurrent $k$-ary processors.
For each $j=0,\ldots,m-1$ consider the function
\[ f_j(\xx) = \floor{ \frac{f(\xx)+j}{m} }. \]
We claim that $f_j=f_{\Proc_j}$ for some recurrent processor $\Proc_j$. One
way to prove this is to use Theorem <ref>, checking from the
above formula that since $f$ is ZILP, $f_j$ is also ZILP. Another route is
to note that $f_j$ is computed by a network in which the output of $\Proc$ is
fed into a $j$-primed $m$-toppler. By <ref>, $f_j$ is therefore
computed by some recurrent processor. (Note however that this network itself
will not help us to emulate $\Proc$ using gates, since it contains $\Proc$!)
<ref> illustrates an example of the reduction.
Now we use the meagerization identity (<ref>):
\[ f= \floor{\frac{f}{m}} + \floor{\frac{f+1}{m}} + \cdots +
\floor{\frac{f+m-1}{m}} \]
Thus, $\Proc$ is emulated by an $m$-splitter that
feeds into $\Proc_0,\ldots,\Proc_{m-1}$, with the results fed into an
$m$-adder. It remains to check that each $\Proc_j$ is meager.
We have
\[ f_j(\xx + \lambda_k \basis_k)
= \floor{ \frac{f(\xx+\lambda_k \basis_k)+j}{m}} =
\floor{ \frac{f(\xx) + \lambda_k b_k+j}{m}} = f_j(\xx)+1. \qedhere \]
The functions $f_j$ in the proof of Lemma <ref> satisfy for $i=2,\ldots,k$
\[ f_j(\xx + m \lambda_i \basis_i)
%= \floor{ \frac{f(\xx+m \lambda_i \basis_i)+j}{m}}
= \floor{ \frac{f(\xx) + m\lambda_i b_i+j}{m}} = f_j(\xx)+\lambda_i b_i \]
\[ f_j(\xx) = \bb' \cdot \xx + P_j(\xx) \]
with $P_j$ is periodic modulo $\lambda'_1 \Z \times \cdots \times \lambda'_k \Z$, where
$\lambda'_1 = \lambda_1$ and $\lambda'_i = m\lambda_i$ for $i=2,\ldots,k$ and
$\lambda'_1 b'_1 = 1$ and $\lambda'_i b'_i = \lambda_i b_i$ for $i=2,\ldots,k$.
-̊>̊ ̊+̊+̊(̊1̊,̊0̊)̊
-> ++(.5,0)
-̣>̣ ̣+̣+̣(̣0̣,̣-̣1̣)̣
-> ++(0,-.5)
[xstep=5,ystep=4,dotted] (-.5,-.5) grid (10.5,8.5);
ıin 0,5
ȷin 0,4
in 0,...,5
in 0,...,4
[dot] at (,) ; ;
(0,2.5) ̣̣;
(0,3.5) ;̊
(1.5,4) ;
(2.5,1.5) ;
(2.5,1.5) ̊̊;
(2.5,4) ̣̊;
(3.5,4) ̣̣;
(4.5,4) ;
[style=double, double distance=2pt,line cap=rect] (1.5,3.5) ̣̣;̊
[circ] at (0,0);
-̊>̊ ̊+̊+̊(̊1̊,̊0̊)̊
-> ++(.5,0)
-̣>̣ ̣+̣+̣(̣0̣,̣-̣1̣)̣
-> ++(0,-.5)
-> ++(.8,0)
-> ++(1.2,0)
-> ++(0,-.8)
-> ++(0,-1.2)
-> ++(.2,-.2)
[xstep=5,ystep=4,dotted] (-.5,-.5) grid (10.5,8.5);
ıin 0,1
in 0,...,5
in 0,...,4
[dot] at (,) ; ;
[hili] (0,2.5) ̣̣;
(0,3.5) ;
(1.5,4) ̣̣;
(2.5,4) -> ++(0,-.6) ;
[hili] (3.5,4) ;
(4.5,4) ;
ıin 0,1
in 0,...,5
in 0,...,4
[dot] at (,) ; ;
(0,2.5) ̣̣;
(0,3.5) ;
[hili] (1.5,4) ̣̣;
(2.5,4) -> ++(0,-.6) ;
(3.5,4) ;
(4.5,4) ;
[circ] at (0,0);
Left: Example state diagram of a recurrent binary
processor $\Proc$ with $\lambda = (4,5)$ and
$b=(\frac12,\frac45)$. (The function is the same as the
one in <ref> (left)). A dot with coordinates
$\xx=(x_1,x_2)$ represents the state of the processor after
it has received input $\xx$. (The initial state $(0,0)$ is
an unfilled circle.) Each solid contour line between two
adjacent dots indicates that a letter is emitted when
making that transition. Right: The highlighted
contours form the state diagram of the corresponding meager
processor $\Proc_3$, obtained by keeping every fourth
contour (starting from the first) of the left picture. The
vertical period is still $5$, although the horizontal
period has increased.
§.§ Reducing the alphabet size
Now we come to the main reduction.
Let $\Proc$ be a meager recurrent $k$-ary processor
satisfying $f_\Proc(\xx+\lambda_k \basis_k) = f_\Proc(\xx)+1$. Then $\Proc$
can be emulated by a network of a recurrent $(k{-}1)$-ary processor, a
$\lambda_k$-toppler, and an adder.
Let $\Proc$ compute $f$. By Theorem <ref>, $f$ is ZILP. Its
representation as a linear plus a periodic function makes sense as a function
on all of $\Z^k$. Now consider the increasing function
\[ g(x_1,\ldots,x_{k-1}) = -c - \min\{x_k \in \Z \,:\, f(x_1,\ldots,x_k)
\geq 0 \}. \]
where $c = -\min \{x_k \in \Z \,:\, f(0,\ldots,0,x_k) \geq 0\}$. Note that
$g$ is an increasing function of $(x_1,\ldots,x_{k-1}) \in \N^{k-1}$, and
$g(\zero) = 0$.
If $\ld \in \lambda_1 \Z \times \cdots \times \lambda_{k-1}
\Z\times\{0\} $, then
\begin{align*} g(\xx+\ld) &= -c -\min \{x_k \in \Z \,:\,
+ \bb \cdot \ld \geq 0\} \\
&= -c -\min \{x_k \in \Z \,:\, f( x_1,\ldots,x_{k-1},x_k + \lambda_k (\bb \cdot \ld)) \geq 0\} \\
&= g(\xx) + \lambda_k (\bb \cdot \ld)
\end{align*}
where the second equality holds because $\Proc$ is meager. Hence $g$ is ZILP.
Let $\ProcQ$ be the $(k-1)$-ary processor that computes $g$ (which exists by <ref>).
Note that for any integer $j$ we have that $f(x_1,\ldots,x_k) \geq j$ if and
only if $f(\xx-j\lambda_k \basis_k) \geq 0$, which in turn happens if and
only if $g(x_1,\ldots,x_{k-1})+x_k +c \geq j\lambda_k$. Hence
\[ f(x_1,\ldots,x_k) = \floor{
\frac{g(x_1,\ldots,x_{k-1})+x_k+c}{\lambda_k}}.
%= \floor{1 + \frac{x_1 - \min\{ x \,:\, f(x,x_2,\ldots,x_k) \geq 1 \} }{\lambda_1}}.
\]
The definition of $c$ gives that $0\leq c <\lambda_k$, since $f(\zero)=0$
and $f(-\lambda_k \basis_k)=-1$.
So $\Proc$ is emulated by the network that feeds the last input letter $a_k$
into a $\lambda_k$-toppler $\ProcT$ primed with $c$, and $a_1,\ldots,a_{k-1}$
into $\ProcQ$ which feeds into $\ProcT$.
[block,minimum height=2cm,minimum width=1cm] (Q) $\ProcQ$;
[<-] (Q.130) – ++(-1,0) node[left] (a1) [1.3em]$x_{1}$;
[<-] (Q.230) – ++(-1,0) node[left] (ak1) [1.3em]$x_{k-1}$;
(Q.180) – ++(-1,0) node[left] 1ex[1.4em]$\vdots$;
(ak) [below= 1cm of ak1] [1em]$x_{k}$;
[adder] (p) [right=3cm of ak1] ;
[toppler] (b) [right=.5cm of p,prime=$c$] $\lambda_k$;
(ak) edge[->, bend right=20] (p)
(Q.east) edge[->, bend left=10] (p)
(p) edge[->] (b)
(b.east) edge[->] ++(1,0);
Emulating a meager recurrent $k$-ary processor via a recurrent $(k{-}1)$-ary processor.
Now we can prove the main result in the recurrent case.
Let $\Proc$ be a recurrent abelian processor to be emulated. By
<ref> we can assume that it has unary output. We proceed by
induction on the number of inputs $k$. The base case $k=1$ is
<ref>. For $k>1$, we first use <ref> to emulate the
processor by a network of meager $k$-ary processors. Then we replace each of
these with a network of $(k-1)$-ary processors, by <ref>,
and then apply the inductive hypothesis to each of these.
§.§ The number of gates
How many gates do our networks use? For simplicity, consider the case of a
recurrent $k$-ary abelian processor with $\lambda_i=2$ and $b_i=1/2$ for all
$i$. It is not difficult to check that our construction uses $O(c^k)$ gates
as $k\to\infty$ for some $c$. In fact, a counting argument shows that
exponential growth with $k$ is unavoidable, as follows. Consider networks of
only adders, splitters, and $2$-topplers, but suppose that we allow feedback
(so that a $\lambda$-toppler can be replaced with $O(\log \lambda)$ gates, by
<ref>). The number of networks with at most $n$ gates is at most
$n^{c' n}$ for some $c'$ (we choose the type of each gate, together with the
matching of inputs to outputs). On the other hand, the number of different
ZILP functions $f$ that can be computed by a processor of the above-mentioned
form is at least $2^{\binom{k}{\lfloor k/2\rfloor}}$, since we may choose an
arbitrary value $f(\xx)\in\{0,1\}$ for each of the $\binom{k}{\lfloor
k/2\rfloor}$ elements $\xx$ of the middle layer $\{\xx\in\{0,1\}^k: \sum_i
x_i =\lfloor k/2\rfloor\}$ of the hypercube. If all $k$-ary processors can
be emulated with at most $n$ gates then $n^{c'n}>2^{\binom{k}{\lfloor
k/2\rfloor}}$. It follows easily that some such processor requires at least
$C^k$ gates, for some fixed $C>1$.
If we consider the dependence on the quantities $\lambda_i$ and $b_i$ as well
as $k$, our construction apparently leaves more room for improvement
in terms of the number of gates, since repeated meagerization tends to
increase the periods $\lambda_i$. One might also investigate whether there
is an interesting theory of $k$-ary functions that can be computed with only
polynomially many gates as a function of $k$.
Our construction of networks emulating transient processors
(<ref>) will be much less efficient than the recurrent case,
since the induction will rely on a ZILP function of a potentially large
number of arguments (<ref>) that is emulated by appeal to
Theorem <ref>. It would be of interest to reduce the
number of gates here.
§ THE BOUNDED CASE
In this section we prove <ref>. Moreover,
we identify the class of functions computable without
Let $f : \N^k \to \{0,1\}$ be increasing with $f(\zero)=0$.
There is a directed acyclic network of adders, splitters, presinks and
delayers that computes $f$.
Let $M$ be the set of $\mm \in \N^k$ that are minimal (in the coordinate
partial order) in $f^{-1}(1)$. By Dickson's Lemma <cit.>, $M$ is finite;
and $f(\xx)=1$ if and only if $\xx \ge \mm$ for some $\mm \in M$. Thus
\[
f(\xx) = \bigvee_{\mm \in M} \bigwedge_{i \in A} \one[x_i \ge m_i].
\]
The function $\one[x_i \ge m_i]$ is computed by $m_i-1$ delayers in series
followed by a presink. The minimum ($\wedge$) or maximum ($\vee$) of a pair
of boolean ($\{0,1\}$-valued) inputs is computed by adding the inputs and
feeding the result into a delayer or a presink respectively. The minimum or
maximum of any finite set of boolean inputs is computed by repeated pairwise
operations. See <ref>. The lemma follows.
[delayer] (d1) ;
[delayer,right of=d1] (d2) ;
[delayer,right of=d2] (d3) ;
[presink,right of=d3] (s) ;
[->] (d1) – (d2);
[->] (d2) – (d3);
[->] (d3) – (s);
[->] (s) – ++(1,0);
[<-] (d1) – ++(-1,0);
(b1) $b_{1}$;
[below=1mm of b1] (x1) ;
[below=1mm of x1] (b2) $b_{2}$;
[below=1mm of b2] (x2) ;
[below=1mm of x2] (b3) $b_{3}$;
[adder,right=1cm of x1] (a1) ;
[adder,right=3cm of b2] (a2) ;
[delayer,right=.5cm of a1] (d1) ;
[delayer,right=.5cm of a2] (d2) ;
(b1) edge[->,bend left=20] (a1);
(b2) edge[->,bend right=20] (a1);
(b3) edge[->,bend right=20] (a2);
(a1) edge[->] (d1);
(d1) edge[->,bend left=20] (a2);
(a2) edge[->] (d2);
(d2) edge[->] ++(1,0);
(b1) $b_{1}$;
[below=1mm of b1] (x1);
[below=1mm of x1] (b2) $b_{2}$;
[below=1mm of b2] (x2);
[below=1mm of x2] (b3) $b_{3}$;
[adder,right=1cm of x1] (a1);
[adder,right=3cm of b2] (a2);
[presink,right=.5cm of a1] (d1);
[presink,right=.5cm of a2] (d2);
(b1) edge[->,bend left=20] (a1);
(b2) edge[->,bend right=20] (a1);
(b3) edge[->,bend right=20] (a2);
(a1) edge[->] (d1);
(d1) edge[->,bend left=20] (a2);
(a2) edge[->] (d2);
(d2) edge[->] ++(1,0);
Networks computing $\one[x \geq 4]$, and
$\min(b_1,b_2,b_3)$ and $\max(b_1,b_2,b_3)$ for boolean
inputs $b_i\in\{0,1\}$.
Suppose $f : \N^k \to \N$ is increasing and bounded
with $f(\zero)=0$. Then there is a directed acyclic network of adders,
splitters, presinks and delayers that computes $f$.
By Lemma <ref>, for each $j \in \N$ there is a network of the desired
type that computes $\xx \mapsto \ind [ f(\xx) > j ]$. If $f$ is bounded by
$J$, then $f(\xx) = \sum_{j=0}^{J-1} \ind [f(\xx)>j]$, so we add the outputs
of these $J$ networks.
Let $\Proc$ be a bounded abelian processor. By <ref> we can
assume that it has unary output. The function that it computes is increasing
and bounded, and maps $\zero$ to $0$. Therefore, apply <ref>.
What is the class of all functions computable by a network of adders,
splitters presinks and delayers? Let us call a function $P : \N^k \to
\N^\ell$ eventually constant if it is eventually periodic with all
periods $1$; that is, there exist $r_1, \ldots, r_k \in \N$ such that $P(\xx)
= P(\xx+\basis_i)$ whenever $\xx_i \geq r_i$. (Note the relatively weak
meaning of this term – such a function may admit multiple limits as some
arguments tend to infinity while the others are held constant).
Let $k \geq 1$. A function $f : \N^k \to \N$ can be
computed by a finite, directed acyclic network of adders, splitters, presinks
and delayers if and only if it satisfies all of the following.
* $f(\zero)=0$.
* $f$ is increasing.
* $f = L+P$ for an $\N$-linear function $L$ and an eventually constant function $P$.
Let $\Net$ be such a network, and let it compute $f$. Since
adders and splitters are immutable, and presinks and
delayers become immutable after receiving one input, the
internal state of $\Net$ can change only a bounded number
of times. In fact, for each $i=1,\ldots,k$ we have $t_i^{r}
= t_i^{r+1}$ where $r$ is the total number of presinks and
delayers in $\Net$. Letting $b_i := f((r+1)\basis_i) -
f(r\basis_i)$, it follows from Lemma <ref>
\begin{equation} \label{e.eventuallylinear} f(\xx+\basis_i) - f(\xx) = b_i \end{equation}
whenever $x_i \geq r$. Note that $b_i \in \N$. Letting
$ P(\xx) := f(\xx) - \bb \cdot \xx $,
we find that $P(\xx+\basis_i) = P(\xx)$ whenever $x_i \geq r$, so $P$ is eventually constant.
Conversely, suppose that $f$ satisfies (i)-(iii). Write $f=L+P$ for a linear
function $L(\xx) = \bb \cdot \xx$ with $\bb \in \N^k$, and an eventually
constant function $P$. Then there exist $r_1, \ldots, r_k$ such that
(<ref>) holds for all $i=1,\ldots,k$ and all $\xx \in
\N^k$ such that $x_i \geq r_i$. In particular, the function
\[ g(\xx) := f(\xx) - \sum_{i=1}^k b_i (x_i -r_i)^+ \]
is ZILP and
bounded. By Theorem <ref> there is a network $\Net$ of
adders, splitters, presinks and delayers that computes $g$. To compute $f$,
feed each input $x_i$ into a splitter which feeds into $\Net$ and into an
$r_i$-delayer followed by a $b_i$-splitter.
§ THE GENERAL CASE
In this section we prove <ref>. As in
the recurrent case, the proof will be by induction on the
number of inputs, $k$, of the abelian processor. We
identify $\N^k$ with $\N^{k-1} \times \N$, and write
$(\yy,z) = \yy + z\basis_k$. Meagerization will again play
a crucial role. A major new ingredient is “interleaving of
§.§ The unary case
As before, we first prove the case of unary input, although
an alternative would be to start the induction with the
trivial zero-input processor.
Any abelian processor with unary input and output can be
emulated by a directed acyclic network of adders,
splitters, topplers, presinks and delayers.
Let the processor $\Proc$ compute $F:\N\to\N$. Since $F$
is ZILEP, it is linear plus periodic when the argument is
sufficiently large; thus, there exists $R\in\N$ such that
the function $G$ given by
$$G(x):=F(x+R)-F(R),\qquad x\in\N$$
is ZILP. We have
$$F(x)=G\bigl((x-R)^+\bigr)+\sum_{i=0}^{R-1} \bigl[F(i+1)-F(i)\bigr]\,\ind[x> i]$$
as is easily checked by considering two cases: when $x\leq R$ the first term
vanishes and the second telescopes; when $x\geq R$, the second term is $F(R)$
and we use the definition of $G$.
By <ref>, the function $G$ can be computed by a network of
adders, splitters and topplers. Now to compute $F$, we feed the input $x$
into $R$ delayers in series. For each $0\leq i<R$, the output after $i$ of
them is also split off and fed to a delayer, to give $\ind[x>i]$ (as in the
proof of <ref>); this is split into $F(i+1)-F(i)$ copies, while
the output $(x-R)^+$ of the last delayer is fed into a network emulating
$\ProcG$, and all the results are added. See <ref> for an
[splitter] (s1) ;
[delayer,right of=s1] (d1) ;
[splitter,right of=d1] (s2);
[delayer,right of=s2] (d2);
[right=5mm of d2] (dd) $\cdots$;
[splitter,right=5mm of dd] (sr);
[delayer,right of=sr] (dr);
[presink,below of=s1] (p1);
[presink,below of=s2] (p2);
[presink,below of=sr] (pr);
[block,below of=dr,minimum width=1cm,minimum height=1cm] (g)$\ProcG$;
[adder,below of= g](a);
[splitter,below of=p2] (ss2);
[<-] (s1)–++(-1,0);
[->] (s1)–(d1);
[->] (d1)–(s2);
[->] (s2)–(d2);
[->] (sr)–(dr);
[->] (s1)–(p1);
[->] (s2)–(p2);
[->] (sr)–(pr);
[->] (dr)–(g);
[->] (g)–(a);
[->] (pr)–(a);
[->] (p2)–(ss2);
[->] (ss2) edge[bend right=15] (a);
[->] (ss2) edge[bend left=15] (a);
[->] (ss2) edge (a);
[->] (a)–++(1,0);
Emulating a transient unary processor. (In this example, the difference
$F(i+1)-F(i)$ takes values $0,3,\ldots,1$ for $i=0,1,\ldots,R-1$).
§.§ Two-layer functions
We now proceed with a simple case of the inductive step, which
provides a prototype for the main argument, and which will also be used as a
step in the main argument.
Let $\Proc$ be a $k$-ary abelian processor that computes a function $F$, and
suppose that
\begin{equation}\label{constant-after-1}
F(\yy,z)=F(\yy,z') \quad\text{if }z,z'\geq 1.
\end{equation}
Then $\Proc$ can be emulated by a network of topplers, presinks, and
$(k-1)$-ary abelian processors.
$$W:=\sup_{\yy\in \N^{k-1}} F(\yy,1)-F(\yy,0),$$
and note that $W<\infty$, because the difference inside the supremum is an
eventually periodic function of $\yy$, and is thus bounded. If $W=0$, then
$F$ is constant in $z$, and therefore $\Proc$ can be emulated by a single
$(k-1)$-ary processor.
Suppose $W\geq 1$. We reduce to the case $W=1$ by the meagerization identity
(<ref>). Specifically, we express $F$ as $\sum_{i=0}^{W-1}
F_i$, where $F_i:=\lfloor(F+i)/W\rfloor$. Each $F_i$ is ZILEP (this can be
checked directly, or by <ref>, since $F_i$ is computed by feeding
the output of $F$ into a toppler). Each $F_i$ satisfies the condition
(<ref>), but now has $F_i(\yy,1)-F_i(\yy,0)\leq 1$ for all
$\yy$, as promised. If we can find a network to compute each $F_i$ then the
results can be fed to an adder to compute $F$.
Now we assume that $W=1$. Define the two $(k-1)$-ary functions (“layers”):
\begin{align*}
f_1(\yy)&:=F(\yy,1)-u,\qquad \text{where }u:=F(\zero,1).
\end{align*}
Each of $f_0,f_1$ is ZILEP. Therefore, by <ref>, they are computed by suitable
$(k-1)$-ary abelian processors $\Proc_0,\Proc_1$. Note that since $W=1$, we
have $u\in\{0,1\}$. We now claim that
\begin{equation}\label{2-layer}
\frac{f_0(\yy)+f_1(\yy)+u+\ind[z>0]}{2}\biggr\rfloor.
\end{equation}
Once this is proved, the lemma follows: we split $\yy$ and feed it to both
$\Proc_0$ and $\Proc_1$, while feeding $z$ into a presink. The three outputs
are added and fed into a primed $2$-toppler in initial state $u$. See
[block,minimum height=1cm,minimum width=1cm] (p0) ${f_0}$;
[below of=p0,block,minimum height=1cm,minimum width=1cm] (p1) ${f_1}$;
[below of=p1,presink] (o);
(p0) – (p1) node[pos=.5,left=1.5cm,splitter] (s);
[right of=p1,adder] (a);
[right of=a,toppler,prime=$u$] (t)$2$;
[<-] (s) – ++(-1,0) node[left] (y)$\yy$;
[below of=y] (z)$z$;
[<-] (o) – (z);
[->] (s) to[bend left=20] (p0);
[->] (s) to[bend right=20] (p1);
[->] (p0) to[bend left=20] (a);
[->] (o) to[bend right=20] (a);
[->] (p1) to (a);
[->] (a) to (t);
[->] (t) – ++(1,0);
Inductive step for emulating a two-layer function.
(Here the solid disk represents $k-1$ parallel splitters that
split each of the $k-1$ entries of the vector $\yy$ into two.)
It suffices to check (<ref>) for $z=0$ and $z=1$, since both sides
are constant in $z\geq 1$. Write $\Delta(\yy)=F(\yy,1)-F(\yy,0)$, so that
$\Delta(\yy)\in\{0,1\}$ for each $\yy$. For $z=0$, the right side of
(<ref>) is
\frac{2F(\yy,0)+\Delta(\yy)}{2}
\bigg\rfloor
On the other hand, for $z=1$ we obtain
\frac{2F(\yy,1)+(1-\Delta(\yy))}{2}
\bigg\rfloor
as required.
§.§ A pseudo-minimum and interleaving
The proof of <ref> will follows similar lines to the
proof above, but is considerably more intricate. Again we will start by
using the megearization identity to reduce to a simpler case. The last step
of the above proof can be interpreted as relying on the fact that $\lfloor
(a+b)/2\rfloor=\min\{a,b\}$ if $a$ and $b$ are integers with $|a-b|\leq 1$.
We need a generalization of this fact involving the minimum of $n$ arguments.
The minimum function $(x_1,\ldots,x_n) \mapsto \min\{x_1,\ldots,x_n\}$ itself
is increasing but only piecewise linear. Since it has unbounded
difference with any linear function, it cannot be expressed as the sum of a
linear and an eventually periodic function, and thus cannot be computed by a
finite abelian processor. The next proposition states, however, that there
exists a ZILP function that agrees with $\min$ near the diagonal. For the
proof, it will be convenient to extend the domain of the function from $\N^n$
to $\Z^n$. <ref> implies that the restriction of such a
function to $\N^n$ can be computed by a recurrent abelian network of gates.
Fix $n\geq 1$. There exists an increasing
function $M:\Z^n\to\Z$ with the following properties:
* $M(\xx+n^2 \basis_j)= M(\xx)+n$, for all
$\xx\in\Z^n$ and $1\leq j\leq n$;
* if $\xx\in\Z^n$ is such that $\max_j x_j-\min_j
x_j\leq n-1$ then $M(\xx)=\min_j x_j$.
The case $n=1$ of the above result is trivial, since we can take $M$ to be
the identity. When $n=2$ we can take $M(\xx)=\lfloor(x_1+x_2)/2\rfloor$
(which satisfies the stronger periodicity condition $M(\xx+2 \basis_j)=
M(\xx)+1$ than (i)). The result is much less obvious for $n\geq 3$. Our proof
is essentially by brute force. Our $M$ will in addition be symmetric in the
We start by defining a function $\pre$ that satisfies
the given conditions but is not defined everywhere. Then
we will fill in the missing values. Let
$$\K:=\bigl\{x\in\Z^n: \textstyle\min_j x_j=0,\; \max_j x_j\leq n{-}1\bigr\}
=[0,n{-}1]^n\setminus [1,n{-}1]^n.$$
(This is the set on which
(ii) requires $M$ to be $0$.) Write
$\11=(1,\ldots,1)\in\Z^n$. Let $\pre:\Z^n\to\Z\cup\{\un\}$
be given by
\begin{equation}\label{partial-f}
\pre(\xx)=\begin{cases}
n\sum_ju_j +s &\text{if } \xx\in
\K+n^2 \uu + s\11
\quad\text{for some }\uu\in\Z^n,\; s\in\Z,\\
\un&\text{otherwise.}
\end{cases}
\end{equation}
Here the symbol $\un$ means “undefined”. See <ref> for an
(-.1,-.1) rectangle (7.1,7.1);
[green] (0,0)–(2,0)–(2,1)–(1,1)–(1,2)–(0,2)–cycle;
(0,0) grid (7,7);
in 0,...,6
at (+.5,+.5) ;
at (+1.5,+.5) ;
at (+.5,+1.5) ;
at (+2.5,-1.5) ;
at (+3.5,-1.5) ;
at (+2.5,-.5) ;
at (-1.5,+2.5) ;
at (-1.5,+3.5) ;
at (-.5,+2.5) ;
Part of the function $\pre$ when $n=2$.
The origin is at the bottom left, and the region $\K$ is shaded.
We first check that the above definition is
self-consistent. Suppose that $\xx\in \K+n^2\uu+s\11$ and
$\xx\in \K+ n^2\vv+t\11$; we need to check that the
assigned values agree. First suppose that $\uu{-}\vv$ does
not have all coordinates equal. Then
\bigl\|n^2(\uu{-}\vv)+(s{-}t)\11\bigr\|_\infty \geq \frac
since two coordinates of $n^2(\uu{-}\vv)$ differ
by at least $n^2$, and the same quantity $s{-}t$ is added to
each. This gives a contradiction since $\K$ has
$\|\cdot\|_\infty$-diameter $n{-}1<n^2/2$. Therefore,
$\uu{-}\vv$ has all coordinates equal, i.e. $\uu{-}\vv=w\11$
for some $w\in\Z$. Since $\K+a\11$ and $\K+b\11$ are
disjoint for $a\neq b$, we must have
$n^2\uu+s\11=n^2\vv+t\11$, so $n^2 w=t{-}s$. But then the
two values assigned to $\pre(\xx)$ by (<ref>) are
$n\sum_j u_j+s$ and $n(\sum_j u_j - nw)+t$, which are
Next observe that $\pre$ satisfies an analogue of (i).
\begin{equation}\label{f-per}
\pre(\xx)\neq \un \quad\text{implies}\quad \pre(\xx+n^2\vv)=\pre(\xx)+n\sum_jv_j.
\end{equation}
This is immediate from (<ref>). Note also that
$\pre$ satisfies (ii), i.e.
\begin{equation}\label{f-diag}
\max_j x_j-\min_j x_j\leq n{-}1 \quad\text{implies}\quad
\pre(\xx)=\min_j x_j,
\end{equation}
since the assumption is equivalent to $\xx\in \K+(\min_j
The key claim is that $\pre$ is increasing where it is
\begin{equation}\label{f-mono}
\xx\leq\yy\text{ and } \pre(\xx)\neq\un\neq \pre(\yy)
\quad\text{imply}\quad
\pre(\xx)\leq \pre(\yy).
\end{equation}
To prove this, suppose that $\xx\in
\K+n^2\uu+s\11$ and $\yy\in \K+ n^2\vv+t\11$ satisfy
$\xx\leq \yy$. If $\uu{-}\vv$ has all coordinates equal then
we again write $\uu{-}\vv=w\11$, so $\yy\in
\K+n^2\uu+(t-n^2w)\11$. For $a<b$, no element of $\K+a\11$
is $\geq$ any element of $\K+b\11$. Therefore $\xx\leq\yy$
implies $s\leq t-n^2 w$, which yields $\pre(\xx)\leq
\pre(\yy)$ in this case. Now suppose $\ww:=\uu{-}\vv$ does
not have all coordinates equal, and write
$\overline{w}=n^{-1} \sum_j w_j$. Since $\00\leq \K\leq
(n{-}1)\11\leq n\11$,
$$n^2\uu+s\11 \leq \xx\leq\yy \leq n^2\vv+t\11+n\11,$$
which gives $n^2\ww \leq (t-s+n)\11$. Suppose for a
contradiction that $\pre(\xx)>\pre(\yy)$, which is to say
$n\sum_j u_j+s>n\sum_j v_j+t$, i.e. $t-s<n^2\overline{w}$.
Combined with the previous inequality, this gives
$n^2\ww<(n+n^2\overline{w})\11$ (where $<$ denotes strict
inequality in all coordinates). That is
$$w_j-\overline{w}<\frac{1}{n},\qquad 1\leq j\leq n$$
which is impossible by the assumption on $\ww$. Thus (<ref>) is proved.
Now we fill in the gaps: define $M$ by
\sup\bigl\{\pre(\zz):\zz\leq \xx\text{ and }\pre(\zz)\neq\un\bigr\},
where the supremum is $-\infty$ if the set is empty and
$+\infty$ if it is unbounded above. (But these
possibilities will in fact be ruled out below).
If $\xx\leq\yy$ then the set in the definition of
$M(\xx)$ is contained in that for $M(\yy)$. So
$M$ is increasing. If $\pre(\xx)\neq\un$ then taking
$\zz=\xx$ and using (<ref>) gives
$M(\xx)=\pre(\xx)$. In particular $M$ satisfies (ii)
by (<ref>). It is easily seen that for every $\xx$
there exist $\uu\leq\xx\leq\vv$ such that
$\pre(\uu)\neq\un\neq \pre(\vv)$. Therefore monotonicity
of $M$ shows that $M(\xx)$ is finite. Finally, the
definition of $M$ and (<ref>) immediately imply
that $F$ satisfies the same equality as $\pre$ in
(<ref>) (now for all $\xx$), which is (i).
The above result will be applied as follows.
Interleaving Fix $n,k \geq 1$. Let $F : \N^{k-1} \times \N
\to \N$ be an increasing function satisfying
\begin{equation} \label{e.benign} F(\yy,z) \leq F(\yy,z{+}1) \leq F(\yy,z)+1. \end{equation}
for all $\yy \in \N^{k-1}$ and all $z \in \N$. Then
\[ F(\yy,z) = M \bigl( F(\yy,z_0), \ldots, F(\yy,z_{n-1}) \bigr) \]
where $M$ is the function from <ref>, and
\[ z_i =z_i(z):= n \floor{\frac{z{+}n{-}i{-}1}{n}} + i. \]
[magenta] (0,0)–(1,0)–++(0,4)–++(4,0)–++(0,4)–++(4,0) node[right](z0)$z_0$;
[orange] (0,1)–(2,1)–++(0,4)–++(4,0)–++(0,4)–++(3,0) node[right](z1)$z_1$;
[green!70!black] (0,2)–(3,2)–++(0,4)–++(4,0)–++(0,4)–++(2,0) node[right](z2)$z_2$;
[blue] (0,3)–(4,3)–++(0,4)–++(4,0)–++(0,4)–++(1,0) node[right](z3)$z_3$;
[black,->] (.5,-.2)–(10.5,-.2) node[right](z)$z$;
[black,<-] (.5,11.5)–(.5,-.2);
The interleaving functions $z_i(z)$ for $n=4$.
The idea is that the function $(\yy,z)\mapsto
F(\yy,z_i(z))$ appearing in <ref>
picks out every $n$th layer of $F$, starting from the $i$th
(with each such layer repeated $n$ times after an
appropriate initial offset). <ref>
illustrates the functions $z_i$. The lemma says that we can
recover $F$ from these functions by “interleaving” their
layers, thus reducing the computation of $F$ to potentially
simpler functions. Note that the functions $(\yy,z)\mapsto
F(\yy,z_i(z))$ do not necessarily map $\zero$ to $0$ (even
if $F$ does), and so cannot themselves be computed by
abelian processors. We will address this issue with
appropriate adjustments (akin to the use of the quantity
$u$ in the proof of <ref>) when we apply the
lemma in the proof of <ref>.
As $i$ ranges from $0$ to $n{-}1$, note that $z_i$ takes on each of the
values $z,z{+}1,\ldots,z{+}n{-}1$ exactly once. Thus, the increasing
rearrangement of $(F(\yy,z_i))_{i=0}^{n-1}$ is $(F(\yy,z{+}j))_{j=0}^{n-1}$.
By (<ref>) it follows that
\[ M((F(\yy,z_i))_{i=0}^{n-1}) = \min (F(\yy,z_i))_{i=0}^{n-1} = F(\yy,z). \qedhere \]
§.§ Proof of main result
By Lemma <ref> we may assume that the
processor $\Proc$ to be emulated has unary output. Suppose
that $\Proc$ computes $F: \N^k \to \N$, and recall from
<ref> that $F$ is ZILEP. We will use
induction on $k$, with Lemma <ref> providing
the base case. We therefore suppose that $k\geq 2$ and
focus on the $k$th coordinate. Suppose that
\begin{equation}\label{def-slr}
F(\yy,z+L)=F(\yy,z)+SL\quad\text{for all }\yy\in\N^{k-1}\text{ and }z\geq
\end{equation}
We call $L$ the period, $S$ the slope, and $R$ the margin (the
width of the non-periodic part) of $F$ with respect to the $k$th coordinate.
(In the notation of <ref> we can take $L=\lambda_k$, $S=b_k$ and
Motivated by the proof of <ref>, we also consider the parameter
\begin{equation}\label{def-w}
W:=\sup_{(\yy,z)\in \N^{k}} F(\yy,z+1)-F(\yy,z),
\end{equation}
which we call the roughness of $F$. Since the difference inside the
supremum is an eventually periodic function of $(\yy,z)$, we have $W<\infty$.
If $W=0$ then $F$ does not depend on the $k$th coordinate, so we are done by
induction. Assuming now that $W>0$, we will first reduce to functions satisfying
(<ref>) and (<ref>) with parameters satisfying one of the
Case 0: $W=1$, $L=R=n$, $S=0$;
Case 1: $W=1$, $L=R=n$, $S=1/n$,
where in both cases, $n$ is a positive integer.
Reduction to Case 0. Suppose that the original function $F$ has slope
$S=0$. We use the meagerization identity (<ref>) to express
$F$ as the sum $\sum_{j=0}^{W-1} F_j$ where $F_j= \floor{(F+j)/W}$. Each
$F_j$ is ZILEP by <ref> or <ref>. We will emulate
each $F_j$ separately and use an adder. Note that $F_j$ has roughness $1$.
Moreover, since $S=0$ we have
\begin{equation} \label{e.poopsout} F_j(\yy,z)= F_j(\yy,R) \quad\text{for
all }z\geq R.\end{equation}
Therefore we can set $n=R$,
and $F_j$ satisfies (<ref>) and (<ref>) with the claimed
parameters for case 0.
Reduction to Case 1. Suppose on the other hand that $F$ has slope $S>0$. We
use the meagerization identity (<ref>) to express $F$ as
$\sum_{j=0}^{Sn-1} F_j$ where
$$n := \Bigl\lfloor\frac{R}{WL}\Bigr\rfloor WL; \qquad
\frac{F+j}{Sn}\Bigr\rfloor.$$
We will again emulate each $F_j$ separately and add
them. Note that since $Sn \geq \max(R,SL)$, each $F_j$ has roughness $1$.
Next we check that $n$ can be taken as both the period and the margin for
each $F_j$. Since $L$ divides $n$ and $n \geq R$, we have
\[ F_j (\yy,z{+}n) = \floor{\frac{F(\yy,z) + Sn + j}{Sn}} = F_j(\yy,z) +1
\]
for all $\yy \in \N^{k-1}$ and all $z \geq n$. Finally, $F_j$ has slope
$1/n$ as desired.
Inductive step.
Assume now that $F$ satisfies (<ref>) and (<ref>) with
parameters as given in case 0 or case 1 in the above table. Also assume
that the result of the theorem holds for all processors with $k-1$ inputs.
We will apply the interleaving result, <ref>, rewritten in terms of functions
that map $\zero$ to $0$. To this end, define for $i=0,\ldots, n-1$,
\begin{align*}
\zeta_i(z)&:=\Bigl\lfloor\frac{z+n-i-1}{n}\Bigr\rfloor, &z\in\N;\\
u_i&:=F(\zero,i);& \\
G_i(\yy,\zeta)&:=F(\yy,n\zeta+i)-u_i, &\yy\in \N^{k-1},\;\zeta\in\N;
\intertext{and}
M_\uu(\vv)&:=M(\vv+\uu), & \vv\in\N^n,
\end{align*}
where $\uu:=(u_0,\ldots,u_{n-1})\in \Z^n$ and $M$ is the function from
<ref>. Then we have
\begin{equation}\label{central-equation}
\end{equation}
This follows
from <ref>: the condition (<ref>) is satisfied because
Note that $u_0=0$ and $u_{i+1}-u_{i}\leq 1$ (also because
$W=1$), so by <ref> we have
$M_\uu(\zero)=M(\uu)=0$. Moreover, the function $M_\uu$ is
increasing and periodic, so by <ref> there is
an recurrent abelian processor $\ProcM_\uu$ that computes
it, and by <ref> there exists a network
(of topplers, adders and splitters) that emulates
$\ProcM_\uu$. Also note that the function $z\mapsto
\zeta_i(z)$ is computed by a primed toppler.
For each $i$, the function $G_i$ is increasing, and can be expressed as a
linear plus an eventually periodic function (by <ref>). And we have
So our task is reduced to finding a network to compute $G_i$. For all $\yy$
and $\zeta\geq 1$ we have
0&\text{in case 0}\\
1&\text{in case 1}.
\end{cases}
Thus, in case 0, $G_i$ is ZILEP and satisfies the condition
(<ref>), so by <ref> it can be computed by a
network of gates and $(k-1)$-ary processors. By the inductive hypothesis,
each $(k-1)$-ary processor can be replaced with a network of gates that
emulates it.
On the other hand, in case 1 we can write
for a function $H_i$ (which can be written
$H_i(\yy,\zeta):=G_i(\yy,\ind[\zeta>0])$), that is ZILEP
and satisfies (<ref>). By
<ref> and the inductive hypothesis, $H_i$ can be
computed by a network of gates. Thus, we can compute $G_i$
by feeding $\zeta$ into a splitter, sending one output to a
delayer and the other to a processor $\ProcH_i$ that
computes $H_i$, and adding the results. See <ref>.
[block,minimum height=1cm,minimum width=1cm] (h) $\ProcG_i$;
[block,minimum height=1cm,minimum width=1cm] (h) $\ProcH_i$;
[delayer,below of=h](d);
(h) – (d) node[pos=.5,right=1.5cm,adder] (a);
[splitter,left of=d](s);
[<-] (s) – ++(-1,0) node[left](z)$\zeta$;
[above of=z] (y)$\yy$;
[->] (s) to (h);
[->] (y) to (h);
[->] (s) to (d);
[->] (h) to[bend left=20] (a);
[->] (d) to[bend right=20] (a);
[->] (a)–++(1,0);
The additional reduction in case 1.
Finally, (<ref>) and <ref>
show how to complete the emulation of $F$ in either case:
$z$ is split and fed into various primed topplers that
compute the functions $\zeta_i(z)$, which are combined with
$\yy$ and fed into networks emulating the $\ProcG_i$. The
results are combined using $\ProcM_\uu$.
[block,minimum height=1cm,minimum width=1cm] (p0) $\ProcG_0$;
[toppler,below left of=p0,xshift=-3mm,inner sep=0pt,prime=$n-1$] (t0)$n$;
[block,below of=p0,minimum height=1cm,minimum width=1cm] (p1) $\ProcG_1$;
[toppler,below left of=p1,xshift=-3mm,inner sep=0pt,prime=$n-2$] (t1)$n$;
[below=2mm of p1] (ddd)$\vdots$;
[block,below= 5mm of ddd,minimum height=1cm,minimum width=1cm] (pn) $\ProcG_n$;
[toppler,below left of=pn,xshift=-3mm,prime=$0$] (tn)$n$;
[block,right=1.3cm of p1,minimum height=2cm,minimum width=1cm](m)$\ProcM_\uu$;
[splitter,left=4cm of p1] (sy);
[splitter,left=4cm of pn] (sz);
[<-] (sy)–++(-1,0)node[left] (y)$\yy$;
[<-] (sz)–++(-1,0)node[left] (z)$z$;
[->] (m)–++(1,0);
[->] (sy)to[bend left=30](p0);
[->] (sy)to(p1);
[->] (sy)to[bend right=30](pn);
[->] (sz)to[bend left=30](t0.west);
[->] (sz)to(t1.west);
[->] (sz)to[bend right=30](tn.west);
[->] (p0)to(m);
[->] (p1)–(m);
[->] (pn)to(m);
[->] (t0)–(p0);
[->] (t1)–(p1);
[->] (tn)–(pn);
The main inductive step.
§ NECESSITY OF ALL GATES
In this section we study the classes of functions computable by various
subsets of the abelian logic gates in Table <ref>. The following
observation will be useful: A function on $\N^k$ can be decomposed in at most
one way as the sum of a linear and an eventually periodic function. Indeed,
the difference of two linear functions is either zero or unbounded on $\N^k$,
so if
\[ L_1 + P_1 = L_2 + P_2 \]
for some linear functions $L_1, L_2$ and some
eventually periodic functions $P_1, P_2$, then $L_1 - L_2$ is bounded on
$\N^k$ and hence $L_1 = L_2$, which in turn implies $P_1 = P_2$.
§.§ Necessity of infinitely many component types
We have seen that $2$-topplers, splitters and adders suffice to emulate any
finite recurrent abelian processor if feedback is permitted. The goal of this
section is to show that no finite list of components will suffice to emulate
all finite recurrent processors by a directed acyclic network.
The exponent of a recurrent abelian processor is the smallest
positive integer $m$ such that inputting $m$ copies of any letter acts as the
identity: $t_i^m(q)=q$ for all recurrent states $q$ and all input letters
Let $\Net$ be a finite, directed acyclic network of
recurrent abelian components. The exponent of $\Net$ divides the product of
the exponents of its components.
Induct on the number of components. Since $\Net$ is directed acyclic, it has
at least one component $\Proc$ such that no other component feeds into
$\Proc$. Let $m$ be the exponent of $\Proc$, and let $M$ be the product of
the exponents of all components of $\Net$. For any letter $i$ in the input
alphabet of $\Proc$, if we input $M$ letters $i$ to $\Proc$, then $\Proc$
returns to its initial recurrent state and outputs a nonnegative integer
multiple of $M/m$ letters of each type. By induction, the exponent of the
remaining network $\Net - \Proc$ is divisible by $M/m$, so all other
processors also return to their initial recurrent states.
The product of exponents in the above lemma can be improved slightly to the quantity
\[ M = \mathrm{lcm} \{ m_\gamma | \gamma : I \to O \}, \]
the least
common multiple over all paths $\gamma$ from an input node to an output node,
of the product $m_\gamma$ of the exponents of the components along $\gamma$.
Lemma <ref>
remains true even if we do not require the components to be recurrent.
The proof in this case uses that any recurrent state of $\Net$ is also
locally recurrent <cit.>.
Let $\Net$ be a finite, directed acyclic network of
recurrent abelian components that emulates a $\lambda$-toppler. Then
$\lambda$ divides the exponent of $\Net$.
If $m$ is the exponent of $\Net$, then $x \mapsto F_{\Net}(mx)$ is a linear
function. Equating the linear parts of the $L+P$ decomposition of $\Net$ and
the $\lambda$-toppler, we obtain
\[ \frac{F_{\Net}(mx)}{m} = \frac{x}{\lambda} \]
for all $x \in \N$. Setting $x=1$ gives $\lambda$ divides $m$.
Lemmas <ref> and <ref> immediately imply the following.
Let $\mathcal{L}$ be any finite list of finite recurrent abelian processors.
There exists $p \in \N$ such that a finite, directed acyclic network of
components from $\mathcal{L}$ cannot emulate a $p$-toppler.
Let $p$ be a prime that does not divide the exponent of any member of $\mathcal{L}$.
§.§ Necessity of primed topplers in the recurrent case.
A directed acyclic network of adders, splitters and unprimed topplers
computes a function $L+P$ with $L$ linear and $P$ periodic with $P \leq 0$.
The inequality follows from converting each toppler $\floor{x/\lambda}$ into
its linear part $x/\lambda$. Recall however that we can do away with primed
topplers if we allow presinks (<ref>).
§.§ Necessity of delayers and presinks.
Lemma <ref> implies that a directed acyclic network of recurrent
components is itself recurrent, so at least one transient gate is needed in
order to emulate an arbitrary finite abelian processor. But why do we have
two transient gates, the delayer and the presink? In this section we
will show that neither can be used along with recurrent components to emulate
the other.
If $G : \N \to \N$ is both ZILP and bounded, then $G \equiv 0$.
Write $G = L+P$ for $L$ linear and $P$ periodic. In particular, $P$ is
bounded, so if $G$ is bounded then $L$ is both bounded and linear, hence
zero. But then $G=P$, and the only increasing periodic function is the zero
Let $\Net$ be a finite directed acyclic network of recurrent components and
delayers. Then $\Net$ cannot emulate a presink.
Let $A$ be the total alphabet of $\Net$, and let $F = F_\Net : \N^A \to \N$.
Let $D \subset A$ the set of incoming edges to the delayers. Note that
inputting $\11_D$ converts all delayers to wires, and has no other effect (in
particular, no output is produced: $F(\11_D)=0$). The resulting network with
delayers converted to wires is recurrent by Lemma <ref>, so the
\[ \til F (\xx) := F(\xx + \11_D) \]
is ZILP by Theorem <ref>.
Now suppose for a contradiction that $\Net$ emulates a presink; that is, for
some letter $a \in A$ we have $F(n \11_a) = \one \{n > 0\}$. Then the
\[ G(n) := F(n \11_a + \11_D) \]
is bounded (by $1+ \max_q
F_{\Net,q}(\11_D)$, where the maximum is over the finitely many states $q$ of
$\Net$). Since $G$ is the restriction of the ZILP function $\til F$ to a
coordinate ray, $G$ is ZILP, which implies $G \equiv 0$ by
Lemma <ref>. But $G(1) \geq F(\11_a) = 1$, which gives the
required contradiction.
The proof shows a bit more: If $\Net$ is a directed acyclic network of
recurrent components and delayers, then $F_\Net$ is either zero or unbounded
along any coordinate ray.
If $G : \N \to \N$ is ZILP, say $G=L+P$ with $L$
linear and $P$ periodic, then $G(x) = L(x)$ for infinitely many $x$.
Since $G(0)=L(0)=0$ we have $P(0)=0$. Since $P$ is periodic, $P(x)=0$ for infinitely many $x$.
Let $\Net$ be a finite, directed acyclic network of
recurrent components and presinks. Then $\Net$ cannot
emulate a delayer.
Let $A$ be the total alphabet of $\Net$, and let $F =
F_\Net : \N^A \to \N$. Let $S \subset A$ the set of
incoming edges to the presinks. Note that inputting $\11_S$
converts all presinks to sinks. However, unlike the input
$\11_D$ of the previous proposition, the input $\11_S$ may
have other effects: It may change the states of other
components, and may produce a nonzero output $F(\11_S)$.
Denote by $\qq^0$ the initial state of $\Net$ and by
$\qq^1$ the state resulting from input $\11_S$. The
resulting network $\ProcR$ with presinks converted to sinks
is recurrent by Lemma <ref>, so the function
\[ F_{\ProcR, \qq^1} (\xx) = F(\xx + \11_S) - F(\11_S) \]
is ZILP by Theorem <ref>.
Now we relate $F_{\ProcR, \qq^1}$ to $F_{\ProcR, \qq^0}$.
Since $\ProcR$ is recurrent, there is an input $\uu \in
\N^A$ such that inputting $\uu$ to $\ProcR$ in state
$\qq^1$ results in state $\qq^0$. Since converting presinks
to sinks without changing the states of any other
components cannot increase the output, we have
\begin{align*} F(\xx) = F_{\Net,\qq^0}(\xx) &\geq F_{\ProcR, \qq^0}(\xx) \\
&= F_{\ProcR, \qq^1}(\xx+\uu) - F_{\ProcR, \qq^1}(\uu) \\
&= F(\xx+\uu+\11_S) - F(\uu+\11_S). \end{align*}
Finally, suppose for a contradiction that $\Net$ emulates a
delayer; that is, for some letter $a \in A$ we have $F(n
\11_a) = (n-1)^+$. Then the function
\[ G(n) := F(n \11_a + \uu+\11_S) - F(\uu+\11_S), \]
ZILP with linear part $L(n)=n$. By
Lemma <ref>, $G(n)=n$ for infinitely
many $n$. This yields the required contradiction, since $n
> F(n \11_a) \geq G(n)$ for all $n \geq 1$.
A directed acyclic network of recurrent components
and delayers, if it sends any output at all, must be
nondegenerate: if it computes $f(x) = Lx + P(x)$ then
either $f \equiv 0$ or $L>0$. The proof is by induction on
the number of components: if some component sends output
then it must receive input, so by the inductive hypothesis
it can be made to receive an arbitrarily large input and
hence send an arbitrarily large output. So a presink
cannot be emulated by such a network.
§ OPEN PROBLEMS
§.§ Floor depth
Let us define the floor depth of a ZILP function
as the minimum number of nested floor functions in a formula for it. More
precisely, let $\mathcal{R}_0$ be the set of $\N$-affine functions $\N^k \to
\N$, and for $n \geq 1$ let $\mathcal{R}_n$ be the smallest set of functions
closed under addition and containing all functions of the form
$\floor{f/\lambda}$ for $f \in \mathcal{R}_{n-1}$ and positive integer
$\lambda$. The floor depth of $f$ is defined as the smallest $n$ such that
$f \in \mathcal{R}_n$.
If $f$ is computed by a directed acyclic network of splitters, adders and
topplers, then the proof of Corollary <ref> in
Section <ref> shows that the floor depth of $f$ is at most
the maximum number of topplers on a directed path in the network. Hence, by
the construction of the emulating network in Section <ref>, every
ZILP function $\N^k \to \N$ has floor depth at most $k$.
Is this sharp?
§.§ Unprimed topplers
What class of functions $\N^k \to \N$ can be computed by a
directed acyclic network of splitters, adders and unprimed
§.§ Conservative gates
Call a finite abelian processor conservative if, in the matrix of the
linear part of the function it computes, each column sums to $1$. We can
think of the input and output letters of such a processor as
indistinguishable physical objects (balls) that are conserved. An
internal state represents a configuration of (a bounded number of) balls
stored inside the processor. (Splitters and topplers are not conservative:
splitters create balls while topplers consume them.) A finite network of
conservative abelian processors with no trash edges emulates a single
conservative processor (provided the network halts). Find a minimal set of
conservative gates that allow any finite conservative abelian processor to be
§.§ Gates with infinite state space
Each of the following functions $\N^2 \to \N$
\begin{align*}
(x,y) &\mapsto \min(x,y) \\
(x,y) &\mapsto \max(x,y) \\
(x,y) &\mapsto xy \end{align*}
can be computed by an abelian processor
with an infinite state space. In the case of $\min$ and $\max$ the state
space $\N$ suffices, with transition function $t_{(x,y)}(q) = q+x-y$. The
product $(x,y) \mapsto xy$ requires state space $\N^2$,
as well as unbounded output: for example, when it receives input $\basis_1$
in state $(x,y)$ it transitions to state $(x+1,y)$ and outputs $y$ letters.
What class of functions can be computed by an abelian network (with or
without feedback) whose components are finite abelian processors and a
designated subset of the above three? Such functions have an $L+P$
decomposition where the $L$ part is piecewise linear, polynomial or piecewise
polynomial (Table <ref>). Integrality becomes an interesting
constraint in this setting. For example, the function $x \mapsto (x^2+x)/2$
is $\N$-valued for $x \in \N$. It can obviously be computed using a
$2$-toppler, adder and multiplier, but can it be computed by a network of
immutable components and multipliers?
L P
linear zero
piecewise linear eventually constant
polynomial periodic
piecewise polynomial eventually periodic
Sixteen ($4\times 4$) types of $L+P$ decomposition
for an increasing function $\N^k \to \N^\ell$.
§ ACKNOWLEDGMENTS
We thank Ben Bond, Sergey Fomin and Jeffrey Lagarias for
inspiring conversations, and Swee Hong Chan for carefully reading an early draft.
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1511.00097
|
Schrödinger operators exhibiting a spectral transition]
Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition
Department of Mathematics, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic
Nuclear Physics Institute, Academy of Sciences of the Czech Republic,
Hlavní 130, 25068 Řež near Prague, Czech Republic
Nuclear Physics Institute, Academy of Sciences of the Czech Republic,
Hlavní 130, 25068 Řež near Prague, Czech Republic
Doppler Institute, Czech Technical University, Břehová 7, 11519 Prague, Czech Republic
Institute of Analysis, Karlsruhe
Institute of Technology, Englerstr. 2, 76131 Karlsruhe, Germany
Nuclear Physics Institute, Academy of Sciences of the Czech Republic,
Hlavní 130, 25068 Řež near Prague, Czech Republic
Doppler Institute, Czech Technical University, Břehová 7, 11519 Prague, Czech Republic
We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $\lambda$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $\lambda$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.
Keywords: Schrödinger operator, eigenvalue estimates, spectral transition
J. Phys. A.: Math. Theor.
§ INTRODUCTION
One of the problems which attracted attention recently concerns Schrödinger operators with potentials dependent on a parameter which exhibit a sudden spectral transition when the value of the parameter passes a critical value. The potential is typically unbounded from below and has narrow channels through which the particle can `escape to infinity' in the supercritical situation. Possibly the best know example of this type is the so-called Smilansky model <cit.> and its regular version <cit.>. Another example, which will be the main subject of this paper, is a modification of the well-known potential $|xy|^p$ in $\R^2$ obtained by adding a rotationally symmetric negative component which becomes stronger with the growing radius, see (<ref>) below. Recall that without the negative component this potential and its modifications serves to demonstrate the possibility of a purely discrete spectrum in the situation when the classically a!
llowed volume of the phase space is infinite <cit.>.
The mechanism of the spectral transition comes from the balance between the negative part of the potential and the positive contribution to the energy coming from the transverse confinement to a channel narrowing towards infinity. This means that the behavior of the two potential components at large distances from the origin must be properly correlated. In our case this is achieved by considering the following class of operators,
\begin{equation} \label{operator}
L_p(\lambda)\,:\; L_p(\lambda)\psi= -\Delta\psi + \left( |xy|^p - \lambda (x^2+y^2)^{p/(p+2)} \right)\psi\,, \quad p\ge 1\,,
\end{equation}
on $L^2(\R^2)$, where $(x,y)$ in $\R^2$ are the Cartesian coordinates $(x,y)$ in $\R^2$ and the non-negative parameter $\lambda$ in the second term of the potential will serve to control the transition. Note that $\frac{2p}{p+2}<2$, and consequently, the operator (<ref>) is essentially self-adjoint on $C_0^\infty(\R^2)$ by Faris-Lavine theorem – cf. <cit.>, Thms. X.28 and X.38; in the following the symbol $L_p(\lambda)$ will always mean its closure.
We have found already some properties of these operators in <cit.>, our aim here is to present a deeper spectral analysis. To describe what is know we need the (an)harmonic oscillator Hamiltonian on line,
\begin{equation}\label{tildeH}
H_p : H_p u = -u^{\prime\prime} + |t|^p u
\end{equation}
on $L^2(\mathbb{R})$ with the standard domain, more exactly, its principal eigenvalue $\gamma_p$; since the potential has a mirror symmetry and the ground state is even, we can equivalently consider the `cut' (an)harmonic oscillator on $L^2(\mathbb{R}_+)$ with Neumann condition at $t = 0$. The eigenvalue is known exactly for $p = 2$ where it equals one as well as for $p\to\infty$ where the potential becomes an infinitely deep rectangular well of width two and $\gamma_\infty = \frac{1}{4}\pi^2$. It is easy to see that the function $p\mapsto \gamma_p$ is continuous and positive on the interval $[1,\infty)$; a numerical solution shows that it reaches the minimum value $\gamma_p \approx 0.998995$ at $p \approx 1.788$.
In the paper <cit.> we have shown that the spectral transition occurs at the value $\lambda_\mathrm{crit}=\gamma_p\:$: the spectrum of $L_p(\lambda)$ is purely discrete and below bounded for $\lambda< \lambda_\mathrm{crit}$, remaining below bounded for $\lambda= \lambda_\mathrm{crit}$, while for $\lambda>\lambda_\mathrm{crit}$ it becomes unbounded from below. We have also derived there crude bounds on eigenvalue sums in the subcritical case. In the present work we are going to establish first that for $\lambda>\lambda_{\mathrm{crit}}$ the spectrum of $L_p(\lambda)$ covers the whole real line. Next we shall analyze in more detail the critical case, $\lambda=\lambda_{\mathrm{crit}}$, showing that one has
\sigma_{\mathrm{ess}}(L_p(\lambda_{\mathrm{crit}}))=[0, \infty)\,.
The question of existence of a negative discrete spectrum is addressed numerically. We show that there a range of values of $p$ for which the critical operator $L_p(\gamma_p)$ has a single negative eigenvalue. Finally, we return to the subcritical case and establish Lieb-Thirring-type bounds to eigenvalue moments.
§ SUPERCRITICAL CASE
As indicated, our first main result is the following.
For any $\lambda>\gamma_p$ we have $\sigma(L_p(\lambda))=\mathbb{R}$.
To demonstrate that any real number $\mu$ belongs to essential spectrum of operator $L_p$ we are going to use Weyl's criterion: we have to find a sequence $\{\psi_k\}_{k=1}^\infty\subset D(L_p)$ such that $\|\psi_k\|=1$ which contains no convergent subsequence and
\|L_p\psi_k-\mu\psi_k\|\to 0\quad\text{as}\quad k\to\infty\,.
For the sake of clarity let us first show that $0\in\sigma_{\mathrm{ess}}(L_p)$. We define
\begin{equation}\label{sequence}
\psi_k(x,y):=\frac{1}{k^{1/(p+2)}}\, h_p\left(x y^{p/(p+2)}\right)\,\e^{i\beta y^{(2p+2)/(p+2)}}\chi\left(\frac{y}{k}\right)\,,
\end{equation}
where $h_p$ is the ground state eigenfunction of $H_p$, $\:\chi$ is a smooth function with $\mathrm{supp}\,\chi\subset[1, 2]$ satisfying $\int_1^2\chi^2(z)\,\mathrm{d}z=1$, and $\beta>0$ will be chosen later. We note that for a given $k$ one can achieve that $\|\psi_k\|_{L^2(\mathbb{R}^2)} \ge\frac{1}{2^{p/(p+2)}}$ as the following estimates show,
\begin{eqnarray}
\lefteqn{\int_{\mathbb{R}^2}\left|\frac{1}{k^{1/(p+2)}}\,h_p(x y^{p/(p+2)})\,\e^{i\beta y^{(2p+2)/(p+2)}}\chi\left(\frac{y}{k}\right)
\right|^2\,\mathrm{d}x\,\mathrm{d}y} \nonumber \\ && =\frac{1}{k^{2/(p+2)}}\int_k^{2k}\int_{\mathbb{R}}
\left|h_p(xy^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}x\,\mathrm{d}y \nonumber \\ && =\frac{1}{k^{2/(p+2)}}\int_k^{2k}\int_{\mathbb{R}}\frac{1}{y^{p/(p+2)}}\left|h_p(t)\,\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}t\,\mathrm{d}y
\nonumber \\ && =\frac{1}{k^{2/(p+2)}}\int_{\mathbb{R}}|h_p(t)|^2\,\mathrm{d}t\,\int_k^{2k}\frac{1}{y^{p/( p+2)}}\left|\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}y \nonumber \\ && =\frac{1}{k^{2/(p+2)}}\int_k^{2k}\frac{1}{y^{p/(p+2)}}\left|\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}y \nonumber \\ && \label{firstpart} \ge\frac{1}{2^{p/(p+2)}}\int_1^2|\chi(z)|^2\,\mathrm{d}z=\frac{1}{2^{p/(p+2)}}\,.
\end{eqnarray}
Our next aim is to show that for any positive $\varepsilon$ one can find $k=k(\varepsilon)$ such that $\|L_p\psi_k\|_{L^2(\mathbb{R}^2)}^2 <\varepsilon$ holds. By a straightforward calculation one gets
\frac{\partial^2\psi_k}{\partial x^2}=\frac{1}{k^{1/(p+2)}}\, y^{2p/(p+2)}\, h_p''(x y^{p/(p+2)})\,\e^{i\beta y^{(2p+2)/(p+2)}}\,\chi\left(\frac{y}{k}\right)
\begin{eqnarray}
\lefteqn{\frac{\partial^2\psi_k}{\partial y^2}
=\frac{1}{k^{1/(p+2)}}\,\e^{i\beta y^{(2p+2)/(p+2)}}\biggl(-\frac{2px}{(p+2)^2}\, y^{-(p+4)/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)}
\nonumber \\ && +\frac{p^2 x^2}{(p+2)^2}\, y^{-4/(p+2)}\,h_p''(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)
\nonumber \\ && +\frac{i p (4p+4) \beta x}{(p+2)^2}\,y^{(p-2)/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\nonumber \\ &&
+\frac{2px}{k( p+2)}y^{-2/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right)
\nonumber \\ && +\frac{i \beta(2p+2)p}{(p+2)^2}\, y^{-2/(p+2)}\, h_p(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)
\nonumber \\ &&
+\frac{2i \beta(2p+2)}{(p+2)k}y^{p/(p+2)}\, h_p(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right)+\frac{1}{k^2}\,h_p(x y^{p/(p+2)})\,\chi''\left(\frac{y}{k}\right)\biggr)\nonumber \\ && -\frac{ \beta^2(2p+2)^2}{(p+2)^2}\,y^{2p/(p+2)}\,h_p(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right). \label{calculations}
\end{eqnarray}
Our aim is to show that choosing $k$ sufficiently large one can make most terms at the right-hand side of (<ref>) as small as we wish. Changing the integration variables, we get for the first term the following estimate,
\begin{eqnarray*}
\lefteqn{\int_{\mathbb{R}^2}\left|\frac{x}{k^{1/(p+2)}\, y^{(p+4)/(p+2)}}\,h_p'(x y^{p/(p+2)})\,\e^{i\beta y^{(2p+2)/(p+2)}}\,\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}x\,\mathrm{d}y} \\ && =\frac{1}{k^{2/(p+2)}}\int_k^{2k}\int_{\mathbb{R}}
\left|\frac{x}{y^{(p+4)/(p+2)}}\,h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}x\,\mathrm{d}y
\\ && =\frac{1}{k^{2/(p+2)}}\int_k^{2k}\frac{1}{y^{(5p+8)/(p+2)}}
\left|\chi\left(\frac{y}{k}\right)\right|^2\,\mathrm{d}y\,\int_{\mathbb{R}}t^2\,|h_p'(t)|^2\,\mathrm{d}t \\ && \le\frac{1}{k^4}
\int_1^2|\chi(z)|^2\mathrm{d}z\,\int_{\mathbb{R}}t^2\,|h_p'(t)|^2\,\mathrm{d}t\,,
\end{eqnarray*}
where the right-hand side tends to zero as $k\to\infty$. In the same way we establish that for large enough $k$ all the terms in (<ref>) except the last one can be made small. The last term is not small, what is important that it asymptotically compensates with the negative part of the potential; using the same technique one can prove that for large $k$ the integral
\frac{1}{k^{2/(p+2)}}\int_{\mathbb{R}^2}\biggl((x^2+y^2)^{p/(p+2)}-y^{2p/(p+2)}\biggr)^2 h_p^2(x y^{p/(p+2)})\,\chi^2\left(\frac{y}{k}\right)\,\mathrm{d}x\,\mathrm{d}y
is small again as small as we wish. Consequently, for any fixed $\varepsilon>0$ one can choose $k$ large enough such that
\begin{eqnarray*}
\lefteqn{\int_{\mathbb{R}^2}|L_p\psi_k|^2(x,y)\,\mathrm{d}x\,\mathrm{d}y} \\ &&
=\int_{\mathbb{R}^2}\left|-\frac{\partial^2\psi_k}{\partial x^2}-
\frac{\partial^2\psi_k}{\partial y^2}+|x y|^p\psi_k-\lambda (x^2+y^2)^{p/(p+2)}\psi_k\right|^2\,
\mathrm{d}x\,\mathrm{d}y \\ && \le\frac{1}{k^{2/(p+2)}}\int_{k}^{2k}\int_{\mathbb{R}}\biggl|y^{2p/(p+2)}\, h''_p(x y^{p/(p+2)})\chi\left(\frac{y}{k}\right) \\ && -\frac{\beta^2 (2p+2)^2}{(p+2)^2}\, y^{2p/(p+2)}\, h_p(x y^{p/(p+2)})\chi\left(\frac{y}
{k}\right) \\ && -|x y|^p\,h_p(x y^{p/(p+2)})\chi\left(\frac{y}{k}\right)
+\lambda y^{2p/(p+2)}\, h(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\biggr|^2\,\mathrm{d}x\,\mathrm{d}y+\varepsilon \\ &&
=\frac{1}{k^{2/(p+2)}}\int_{k}^{2k}\int_{\mathbb{R}}\biggl|y^{2p/(p+2)}\biggl(h_p''(x y^{p/(p+2)})-|x y^{p/(p+2)}|^p\, h_p(x y^{p/(p+2)}) \\ && -\frac{\beta^2 (2p+2)^2}{(p+2)^2}\, h_p(x y^{p/(p+2)})+\lambda h_p(x y^{p/(p+2)})\biggr)\chi\left(\frac{y}{k}\right)\biggr|^2\,\mathrm{d}x\,\mathrm{d}y+\varepsilon\,.
\end{eqnarray*}
Combining this result with the fact that $H_p h_p= \gamma_p h_p$ and choosing
\begin{equation} \label{beta-super}
\beta=\frac{(p+2)}{2p+2}\sqrt{\lambda-\gamma_p}
\end{equation}
we get
\begin{equation}
\label{final}
\int_{\mathbb{R}^2}|L_p\psi_k|^2(x,y)\,\mathrm{d}x\,\mathrm{d}y\le\varepsilon\,.
\end{equation}
To complete this part of the proof we fix a sequence $\{\varepsilon_j\}_{j=1}^\infty$ such that $\varepsilon_j\searrow0$ holds as $j\to\infty$ and to any $j$ we construct a function $\psi_{k(\varepsilon_j)}$ such that the supports for different $j$'s do not intersect each other; this can be achieved by choosing each next $k(\varepsilon_j)$ large enough. The norms of $L_p\psi_{k(\varepsilon_j)}$ satisfy the inequality (<ref>) with $\varepsilon_j$ on the right-hand side, and by construction the sequence $\psi_{k(\varepsilon_j)}$ converges weakly to zero; this yields the sought Weyl sequence for zero energy.
Passing now to an arbitrary nonzero real number $\mu$ we can use the same procedure replacing the above functions $\psi_k$ by
\begin{equation} \label{superweyl}
\psi_k(x,y)=\frac{1}{k^{1/(p+2)}}\, h_p(x y^{p/(p+2)})\,\e^{i\epsilon_\mu(y)}\, \chi\left(\frac{y}{k}\right)\,,
\end{equation}
\epsilon_\mu(y):= \displaystyle{\int_{\frac{|\mu|^{(p+2)/2p}(p+2)^{(p+2)/p}}{(2p+2)^{(p+2)/p}\beta^{(p+2)/p}}}^y\sqrt{\frac{(2p+2)^2 \beta^2}{(p+2)^2}\, t^{2p/(p+2)}+\mu}\:\mathrm{d}t}\,,
and furthermore, the functions $h_p,\,\chi$ and the number $\beta$ are the same way as above. The second derivatives of those functions are
\frac{\partial^2\psi_k}{\partial x^2}=\frac{1}{k^{1/(p+2)}}\, y^{2p/(p+2)}\, h_p''(x y^{p/(p+2)})\,\e^{i\epsilon_\mu(y)}\,
\chi\left(\frac{y}{k}\right)
\begin{eqnarray*}
\lefteqn{\frac{\partial^2\psi_k}{\partial y^2}
=\frac{1}{k^{1/(p+2)}}\,\e^{i\epsilon_\mu(y)}\biggl(\frac{-2p x}{(p+2)^2}\, y^{-(p+4)/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)} \\ && \hspace{-1.5em} +\frac{p^2 x^2}{(p+2)^2}\,y^{-4/(p+2)}\,h_p''(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)+\frac{2p x}{k(p+2)}\,y^{-2/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right) \\ && \hspace{-1.5em}
+i p\,\frac{(2p+2)^2}{(p+2)^3}\,\beta^2 y^{(p-2)/(p+2)}\left(\frac{(2p+2)^2\beta^2}{(p+2)^2}\,y^{2p/(p+2)}+\mu\right)^{-1/2} \,h_p(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right) \\ && \hspace{-1.5em} +\frac{2i p x}{(p+2)}\,y^{-2/(p+2)}\left(\frac{(2p+2)^2\beta^2}{(p+2)^2}\,y^{2p/(p+2)}+\mu\right)^{1/2}h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right) \\ && \hspace{-1.5em}
+\frac{2i}{k}\left(\frac{(2p+2)^2\beta^2}{(p+2)^2}\,y^{2p/(p+2)}+\mu\right)^{1/2}\, h_p(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right) \\ && \hspace{-1.5em} +\frac{1}{k^2}\,h_p(x y^{p/(p+2)})\,\chi''\left(\frac{y}{k}\right)
-\left(\frac{(2p+2)^2\beta^2}{(p+2)^2}\,y^{2p/(p+2)}+\mu\right) h_p(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\biggr).
\end{eqnarray*}
It is not difficult to check that for any positive $\varepsilon$ one choose a number $k$ large enough to ensure that the inequality
\begin{eqnarray*}
\lefteqn{\biggl\|\frac{\partial^2\psi_k}{\partial y^2} \,\e^{-i\epsilon_\mu(y)}
+\mu\psi_k \,\e^{-i\epsilon_\mu(y)}} \\ && - \e^{-i\beta y^{(2p+2)/(p+2)}}\frac{\partial^2}{\partial y^2}\biggl(\psi_k\, \e^{-i\epsilon_\mu(y)+i\beta y^{(2p+2)/(p+2)}}\biggr)\biggr\|_{L^2(\mathbb{R}^2)}
\end{eqnarray*}
holds. Using further the identity
\frac{\partial^2\psi_k}{\partial x^2}\,\e^{-i\epsilon_\mu(y)}= e^{-i\beta y^{(2p+2)/(p+2)}}
\frac{\partial^2}{\partial x^2}\bigl(\psi_k\,\e^{-i\epsilon_\mu(y)+i\beta y^{(2p+2)/(p+2)}}\bigr)
we arrive at the estimate
\begin{eqnarray*}
\lefteqn{\|L_p\psi_k-\mu\psi_k\|_{L^2(\mathbb{R}^2)}=\bigg\|(L_p\psi_k) e^{-i\epsilon_\mu(y)}
-\mu\psi_k \,e^{-i\epsilon_\mu(y)}\biggr\|_{L^2(\mathbb{R}^2)} } \\ && <\biggl\| \e^{i\beta y^{(2p+2)/(p+2)}}L_p\biggl(\psi_k \,\e^{-i\epsilon_\mu(y)+i\beta y^{(2p+2)/(p+2)}}\biggr)\biggr\|_{L^2(\mathbb{R}^2)}+\varepsilon\,;
\end{eqnarray*}
now we can use the result of the first part of proof to establish the claim.
§ CRITICAL CASE
Let us now pass to the case when the parameter value is critical, in other words, consider the operator $L_p(\gamma_p)=-\Delta +(|x y|^p-\gamma_p(x^2+y^2)^{p/(p+2)}),\:p\ge1$, on $L^2(\mathbb{R}^2)$. We shall consider the positive and negative spectrum separately.
§.§ The essential spectrum
First we are going to show that the discreteness is lost in the positive halfline once the coupling constant reaches the critical value.
The essential spectrum of $L_p(\gamma_p)$ contains the interval $[0, \infty)$.
The argument is similar to that used in the proof of Theorem<ref>, hence we present it briefly with emphasis on the differences. As before we check first that $0\in\sigma_{\mathrm{ess}}(L_p)$ by constructing a Weyl sequence, which is now of the form
\psi_k(x,y):=\frac{1}{k^{1/(p+2)}}\, h_p\left(x y^{p/(p+2)}\right)\,\chi\left(\frac{y}{k}\right)
with $h_p$ and $\chi$ the same as before. As this nothing but (<ref>) with $\beta=0$, not surprisingly in view of (<ref>) we can repeat the reasoning with the involved expressions appropriately simplified.
Passing now to an arbitrary nonnegative number $\mu$ we replace (<ref>) by
\psi_k(x,y)=\frac{1}{k^{1/(p+2)}}\, h_p(x y^{p/(p+2)})\,\e^{i \eta_\mu (y)}\, \chi\left(\frac{y}{k}\right)\,,
where the functions $h_p,\,\chi$ are again the same way as above and $(\eta^\prime_\mu(y))^2=\mu$. This can be achieved for any $\mu\ge0$ t by choosing $\eta_\mu(y)=\sqrt{\mu} y$, note that the classically allowed region is now the whole halfline instead of the interval entering the definition of $\epsilon_\mu(y)$ above. The second derivatives of the functions $\psi_k$ obtained in this way are
\frac{\partial^2\psi_k}{\partial x^2}=\frac{1}{k^{1/(p+2)}}\, y^{2p/(p+2)}\, h_p''(x y^{p/(p+2)})\,\e^{i \sqrt{\mu} y}\,
\chi\left(\frac{y}{k}\right)
\begin{eqnarray*}
\lefteqn{\frac{\partial^2\psi_k}{\partial y^2}
=\frac{1}{k^{1/(p+2)}}\,\e^{i \sqrt{\mu} y}\biggl(\frac{-2p x}{(p+2)^2}\, y^{-(p+4)/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)} \\ && \hspace{-1.5em} +\frac{p^2 x^2}{(p+2)^2}\,y^{-4/(p+2)}\,h_p''(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)+\frac{2p x}{k(p+2)}\,y^{-2/(p+2)}\,h_p'(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right) \\ && \hspace{-1.5em}
+\frac{2i \sqrt{\mu} p x}{(p+2)}\,y^{-2/(p+2)}h_p'(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right) \\ && \hspace{-1.5em}
+\frac{2i \sqrt{\mu}}{k}\, h_p(x y^{p/(p+2)})\,\chi'\left(\frac{y}{k}\right) \\ && \hspace{-1.5em} +\frac{1}{k^2}\,h_p(x y^{p/(p+2)})\,\chi''\left(\frac{y}{k}\right)
-\mu h_p(x y^{p/(p+2)})\,\chi\left(\frac{y}{k}\right)\biggr).
\end{eqnarray*}
One finds easily that for any positive $\varepsilon$ and $k$ large enough we have
\biggl\|\frac{\partial^2\psi_k}{\partial y^2} \,\e^{-i \sqrt{\mu} y}
+\mu\psi_k \,\e^{-i \sqrt{\mu} y} - \frac{\partial^2}{\partial y^2}\biggl(\psi_k\, \e^{-i \sqrt{\mu} y}\biggr)\biggr\|_{L^2(\mathbb{R}^2)}<\varepsilon
and using further the trivial identity $\frac{\partial^2\psi_k}{\partial x^2}\,\e^{-i \sqrt{\mu} y}= \frac{\partial^2}{\partial x^2}\bigl(\psi_k\,\e^{-i \sqrt{\mu} y}\bigr)$ we arrive at
\|L_p\psi_k-\mu\psi_k\|_{L^2(\mathbb{R}^2)}=\Big\|\big(L_p\psi_k
-\mu\psi_k\big) \,e^{-i \sqrt{\mu} y}\Big\|_{L^2(\mathbb{R}^2)} <\Big\| L_p\Big(\psi_k \,\e^{-i \sqrt{\mu} y}\bigr)\Big\|_{L^2(\mathbb{R}^2)}+\varepsilon
and the result of the first part of proof allows us to establish the claim.
§.§ Discreteness of the negative spectrum
Next we are going to show that the inclusion $\sigma_\mathrm{ess}(L_p(\gamma_p)) \supset [0, \infty)$ established in Theorem <ref> is in fact an equality.
The negative spectrum of $L_p(\gamma_p),\:p\ge1$, is discrete.
By the minimax principle it is sufficient to estimate $L_p$ from below by a self-adjoint operator with a purely discrete negative spectrum. To construct such a lower bound we employ a bracketing argument, imposing additional Neumann conditions at the rectangles $G_n=\{-\alpha_{n+1}<x<\alpha_{n+1}\}\times \{\alpha_n< y<\alpha_{n+1}\}$, $\:\widetilde{G}_n=\{-\alpha_{n+1}<x<\alpha_{n+1}\}\times \{-\alpha_{n+1}< y<-\alpha_n\}$, $\:Q_n=\{\alpha_n< x<\alpha_{n+1}\}\times \{-\alpha_n<y<\alpha_n\}$, and $\widetilde{Q}_n=\{-\alpha_{n+1}< x<-\alpha_n\}\times \{-\alpha_n<y<\alpha_n\},\: n=1,2,\ldots$, together with central square $G_0=(-\alpha_1, \alpha_1)^2$ – cf. Fig. 1. Here $\{\alpha_n\}_{n=1}^\infty$ is a monotone sequence such that $\alpha_n\to\infty$ as $n\to\infty$ which will be specified later. In this way we obtain a direct sum of operators with Neumann boundary conditions at the rectangle boundaries which we denote as
The Neumann bracketing scheme
L^{(1)}_{n, p}=L_p|_{G_n}, \quad \widetilde{L}^{(1)}_{n, p}=L_p|_{\widetilde{G}_n},\quad L^{(2)}_{n, p}=L_p|_{Q_n}, \quad \widetilde{L}^{(2)}_{n, p}=L_p|_{\widetilde{Q}_n}
and $L^0_p=L_p|_{G_0}$. It is obvious that the spectra of $L_{n, p}^{(i)},\;\tilde{L}^{(i)}_{n, p},\; i=1, 2$, and $L_p^0$ are purely discrete, hence one needs to check that $\underline{\lim}_{n\to\infty}\,\inf\,\sigma\big(L^{(i)}_{n,p}\big)\ge 0$ and $\underline{\lim}_{n\to\infty}\,\inf\, \big(\sigma(\tilde{L}^{(i)}_{n, p}\big)\ge 0$ holds for $i=1,2$, since then the spectra of all the direct sums $\bigoplus_{n=1}^\infty L^{(i)}_{n, p}$ and $\bigoplus_{n=1}^\infty \tilde{L}^{(i)}_{n, p},\; i=1,2$, below any fixed negative number contain a finite number of eigenvalues, the multiplicity taken into account, which implies the sought claim.
Furthermore, the goal will be achieved if we estimate $L^{(i)}_{n, p},\; \tilde{L}_{n, p}^{(i)},\; i=1,2$, from below by operators with separated variables and prove the analogous limiting relations for them. We use the lower bounds
\begin{equation}\label{Hnp1}
H^{(1)}_n\psi =-\Delta \psi+ (\alpha_n^p |x|^p-\gamma_p(x^2+\alpha_{n+1}^2)^{p/(p+2)})\psi
\end{equation}
on $L^2(-\alpha_{n+1}, \alpha_{n+1}) \otimes L^2(\alpha_n, \alpha_{n+1})$ with the boundary conditions
\left. \frac{\partial\psi}{\partial x}\right|_{x=-\alpha_{n+1}}= \left.
\frac{\partial\psi}{\partial x}\right|_{x=\alpha_{n+1}}=0 \,,
\left. \frac{\partial\psi}{\partial y}\right|_{y=\alpha_n}= \left.
\frac{\partial\psi}{\partial y}\right|_{y=\alpha_{n+1}}=0 \,.
It is clear that the spectra of $H^{(1)}_{n, p},\;n=1,2,\ldots$, are purely discrete; we are going to check that
\begin{equation}\label{lim}
\underline{\lim}_{n\to\infty}\inf\,\sigma\big(H_{n, p}^{(1)}\big)\ge 0\,.
\end{equation}
Since the lowest Neumann eigenvalue of $-\frac{\mathrm{d}^2}{\mathrm{d}y^2}$ on the interval is zero corresponding to a constant eigenfunction, the problem reduces to analysis of the operator $h^{(1)}_{n, p}=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\alpha_n^p |x|^p-\gamma_p (x^2+ \alpha_{n+1}^2)^{p/(p+2)}$ on $L^2(-\alpha_{n+1}, \alpha_{n+1})$. Using a simple scaling transformation, one can check that $h_{n, p}^{(1)}$ is unitarily equivalent to
\begin{equation}\label{un.eq.}
\end{equation}
on the interval $\big(-\alpha_{n+1}\,\alpha_n^{p/(p+2)}, \alpha_{n+1}\,\alpha_n^{p/(p+2)}\big]$ with Neumann boundary conditions at its endpoints. To proceed we need to specify the sequence $\{\alpha_n\}$. Let us assume that
\begin{equation}\label{Decay}
\alpha_{n+1}^{2p/(p+2)}-\alpha_n^{2p/(p+2)}\to 0\quad\text{as}\quad n\to\infty\,.
\end{equation}
Combining this assumption with the inequality
\begin{eqnarray*}
\lefteqn{\frac{\gamma_p}{\alpha_n^{2p/(p+2)}}\left(\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}+\alpha_{n+1}^2\right)^{p/(p+2)}
-\alpha_{n+1}^{2p/(p+2)}\right)} \\ && \le\frac{\gamma_p}{\alpha_n^{2p/(p+2)}}\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}\right)^{p/(p+2)} \le\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}\: (|x|^p+1)\,,
\end{eqnarray*}
we infer that
\begin{eqnarray}
\lefteqn{h_{n,p}^{(2)}=\alpha_n^{2p/(p+2)}\,\biggl(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}
+|x|^p-\frac{\gamma_p\,\alpha_{n+1}^{2p/(p+2)}}{\alpha_n^{2p/(p+2)}}} \nonumber \\ && \quad
-\frac{\gamma_p}{\alpha_n^{2p/(p+2)}}\biggl(\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}+\alpha_{n+1}^2\right)^{p/(p+2)} -\alpha_{n+1}^{2p/(p+2)}\biggr)\biggr) \nonumber \\ && \ge\alpha_n^{2p/(p+2)} \Bigg(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\left(1-\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}\right)|x|^p \nonumber \\ && \quad -\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}
-\frac{\gamma_p \alpha_{n+1}^{2p/(p+2)}}{\alpha_n^{2p/(p+2)}}\Bigg) \nonumber \\ && \ge
\alpha_n^{2p/(p+2)} \Bigg(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}
+\Bigg(1-\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}\Bigg)|x|^p-\gamma_p \nonumber \\ && \quad -\gamma_p\Bigg(\frac{\alpha_{n+1}^{2p/(p+2)}
-\alpha_n^{2p/(p+2)}}{\alpha_n^{2p/(p+2)}}\Bigg)\Bigg) +o(1) \nonumber \\ && \ge\alpha_n^{2p/(p+2)}\left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\left(1-\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}\right)|x|^p-\gamma_p\right)+o(1)\nonumber \\ && \label{As.}\ge\alpha_n^{2p/(p+2)}\left(1-\frac{\gamma_p}{\alpha_n^{4p(p+1)/(p+2)^2}}\right) \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+|x|^p-\gamma_p\right)+o(1)\,,
\end{eqnarray}
where the corresponding Neumann (an)harmonic oscillator is restricted to the interval
(-\alpha_{n+1}\,\alpha_n^{p/(p+2)}, \alpha_{n+1}\,\alpha_n^{p/(p+2)})\,.
Next we need to establish the following lemma.
Let $l_{k, p}=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+|x|^p$ be the Neumann operator defined on the interval $[-k, k],\;k>0$. Then
\begin{equation}\label{hnp}
\inf\,\sigma\left(l_{k, p}\right)\ge\gamma_p+o\left(\frac{1}{k^{p/2}}\right) \;\quad\text{as}\quad k\to\infty\,.
\end{equation}
The relation (<ref>) is certainly valid if $\inf \sigma\left(l_{k, p}\right)\ge\gamma_p$ holds for all $k$ from some number on. Assume thus that
we have $\inf \sigma\left(l_{k, p}\right)<\gamma_p$ for infinitely many numbers $k$. Let $\psi_{k, p}$ be the normalized ground-state eigenfunction of $l_{k, p}$. We fix a positive $\delta$ and check that
\begin{eqnarray}
\int_{-k}^{-k+1}\left(|\psi_{k, p}'|^2+|x|^p|\psi_{k, p}|^2\right)\,\mathrm{d}x<\delta\,, \nonumber
\\[-.7em]\label{eqn}\\[-.7em]
\int_{k-1}^k\left(|\psi_{k, p}'|^2+|x|^p|\psi_{k, p}|^2\right)\,\mathrm{d}x<\delta\,. \nonumber
\end{eqnarray}
Indeed, suppose that at least one of inequalities (<ref>) does not hold, then
\begin{equation}\label{middle}
\int_{-k+1}^{k-1}\left(|\psi_{k, p}'|^2+|x|^p|\psi_{k, p}|^2\right)\,\mathrm{d}x<\gamma_p-\delta\,.
\end{equation}
Since $\psi_{k, p}$ is by assumption the ground-state eigenfunction of $l_{k, p}$, we have
\inf \sigma (l_{k, p})=\int_{-k}^k\left(|\psi_{k, p}'|^2+|x|^p|\psi_{k, p}|^2\right)\,\mathrm{d}x \le\int_{-1}^1\left(|\phi'|^2+|x|^p|\phi|^2\right)\,\mathrm{d}x
for all $k\ge 1$ and any normalized function $\phi$ from the domain of the operator, in particular, for any $\phi$ from the class $C_0^\infty(-1,1)$ such that $\int_{-1}^1|\phi|^2\,\mathrm{d}x=1$. Consequently, for large enough $k$ there must exist points $x_{k, p}^{(1)}\in(-k+1, -k+2)$ and $x_{k, p}^{(2)}\in(k-2, k-1)$ such that
\psi_{k, p}\left(x_{k, p}^{(1)}\right)=\mathcal{O}\Big(\frac{1}{k^{p/2}}\Big)\quad\text{and}\quad \psi_{k, p}\left(x_{k, p}^{(2)}\right)=\mathcal{O}\Big(\frac{1}{k^{p/2}}\Big)\;\quad\text{as}\quad k\to\infty\,.
Next we construct a function $\varphi_{k, p}$ on semi-infinite intervals $(-\infty, x_{k, p}^{(1)})$ and $(x_{k, p}^{(2)},
\infty)$ in such a way that
g_{k, p}(x):=\psi_{k, p}(x)\chi_{(x_{k, p}^{(1)}, x_{k, p}^{(2)})}(x)+\varphi_{k, p}(x)\chi_{(-\infty, x_{k, p}^{(1)})\cup(x_{k, p}^{(2)}, \infty)}(x)\in\mathcal{H}^{1}(\mathbb{R})
\begin{equation}\label{condition}
\int_{-\infty}^{x_{k, p}^{(1)}}\left(|\varphi_{k, p}'|^2+|x|^p|\varphi_{k, p}|^2\right)\,\mathrm{d}x+\int_{x_{k, p}^{(2)}}^\infty\left(|\varphi_{k, p}'|^2+|x|^p|\varphi_{k, p}|^2\right)\,\mathrm{d}x =\mathcal{O}\Big(\frac{1}{k^{p/2}}\Big)\,;
\end{equation}
this can be always achieved, one can take, e.g., the function decreasing linearly with respect to $|x-x_{k, p}^{(j)}|$ from the values $\psi_{k, p}\left(x_{k, p}^{(j)}\right),\; j=1,2,$ to zero. By virtue of (<ref>) and (<ref>) we then have
\int_{\mathbb{R}}\left(|g_{k, p}|^2+|x|^p|g_{k, p}|^2\right)\,\mathrm{d}x<\gamma_p-\delta+\mathcal{O}\left(\frac{1}{k^{p/2}}\right)<\gamma_p
for large enough $k$, however, this is in contradiction with the fact that $\gamma_p$ is the ground-state eigenvalue of $l_{k,p}$. This proves the validity of (<ref>).
Having established the validity of inequalities (<ref>) we infer from them that there are points $y^{(1)}_{k, p}\in(-k, -k+1)$ and $y_{k, p}^{(2)}\in(k-1, k)$ such that
\psi_{k, p}(y^{(j)}_{k, p})=\mathcal{O}\Big(\frac{\delta}{k^{p/2}}\Big)\,, \quad j=1,2\,.
Now we repeat the argument and construct a function $\tilde{\varphi}_{k, p}$ on the semi-infinite intervals $(-\infty, y^{(1)}_{k, p})$ and $(y^{(2)}_{k, p}, \infty)$ in such a way that
$$\tilde{g}_{k, p}(x):=\psi_{k, p}(x)\chi_{(y_{k, p}^{(1)}, y_{k, p}^{(2)})}(x)+\tilde{\varphi}(x)\chi_{(-\infty, y_{k, p}^{(1)})\cup(y_{k, p}^{(2)}, \infty)}(x)\in\mathcal{H}^{1}(\mathbb{R})
\int_{-\infty}^{y_{k, p}^{(1)}}\left(|\tilde{\varphi}_{k, p}'|^2+|x|^p|\tilde{\varphi}_{k, p}|^2\right)\,\mathrm{d}x+\int_{y_{k, p}^{(2)}}^\infty\left(|\tilde{\varphi}_{k, p}'|^2+|x|^p|\tilde{\varphi}_{k, p}|^2\right)\,\mathrm{d}x=\mathcal{O}\Big(\frac{\delta}{k^{p/2}}\Big)\,.
Using the last relation one finds that
\int_{\mathbb{R}}|\tilde{g}_{k, p}'|^2\,\mathrm{d}x+\int_{\mathbb{R}}|x|^p|\tilde{g}_{k, p}|^2\,\mathrm{d}x<\inf\,\sigma\left(l_{k, p}\right)+ \mathcal{O}\Big(\frac{\delta}{k^{p/2}}\Big)\,.
However, $\gamma_p$ is the ground-state eigenvalue,
\int_{\mathbb{R}}|\tilde{g}_{k, p}'|^2\,\mathrm{d}x+\int_{\mathbb{R}}|x|^p|\tilde{g}_{k, p}|^2\,\mathrm{d}x\ge\gamma_p\,,
which in combination with above inequality gives
\inf\,\sigma\left(l_{k, p}\right)>\gamma_p- \mathcal{O}\Big(\frac{\delta}{k^{p/2}}\Big)\,,
proving the claim of the lemma.
It follows from Lemma <ref> that the right-hand side of the estimate (<ref>) behaves asymptotically as
o\left(\frac{1}{\alpha_n^{p(p+1)/(p+2)}}\right) \alpha_n^{2p/(p+2)}+o(1)
which can be made arbitrarily small by choosing $n$ is large enough; this is what we needed to conclude the proof of Theorem <ref>.
We know from <cit.> that the critical operator $L_p(\gamma_p)$ is bounded from below. Estimating separately the contributions to the respective quadratic form coming from the regions $\{(x,y):\: |y|\ge 1\}$, $\{(x,y):\: |x|\ge 1\,, |y|\le 1\}$, and the central square $(-1,1)^2$, we can derive a lower bound to the threshold of the negative spectrum in terms of spectral properties of the one-dimensional operators with the symbol
with $z\ge1$. As such a bound is not simple and does not provide any significant insight, however, we are not going to present it here.
§.§ Existence of the negative spectrum: a numerical indication
Theorem <ref> tells us that the spectrum in the negative halfline can be discrete only, and as we have remarked above one can find a lower estimate to its threshold, however, neither of these results implies anything about the negative spectrum existence. Now we are going address this question numerically and provide an evidence of the discrete spectrum nontriviality.
We considering first the operator $L_2(\gamma_2)$ — recall that $\gamma_2=1$ — and impose a cutoff at a circle of radius $R$ circled at the origin with Dirichlet and Neumann boundary condition, and find the corresponding first and second eigenvalue using the Finite Element Method. The result is shown on Fig. <ref>.
The eigenvalues $E_j,\: j=1,2$, of the critical operator with p=2 as functions of the cutoff radius $R$. The blue and red curves correspond to the Neumann and Dirichlet boundary, respectively.
We see, in particular, that the lowest Dirichlet eigenvalue is for $R\gtrsim 7$ practically independent of the cutoff radius and negative which by an elementary bracketing argument indicates that $L_2(1)$ has a negative eigenvalue. Furthermore, the difference between the Dirichlet and Neumann eigenvalue becomes negligible for large enough $R$ which shows that true ground-state eigenvalue in this case is $E\approx -0.18365$. For the second eigenvalue the DN gap also squeezes, although much slower and the Neumann eigenvalue is positive which hints that the discrete spectrum consists of a single point.
The ground-state eigenfunction for $p=2$, view from the top.
The Finite Element Method allows us also to compute the ground-state eigenfunction as shown on Fig. <ref>. The result is practically independent of the boundary condition used which is understandable since the function has an exponential falloff and the influence of the boundary is negligible for large enough $R$.
By continuity, the ground-state eigenvalue of $L_p(\lambda)$ exists in the vicinity of the point $p=2$; one is naturally interested what one can say about a broader range of the parameter.
Positivity of $L_p(\lambda)$ as a function of $\lambda$ and $p$.
To this aim we plot in the left part of Fig. <ref> the lowest eigenvalue of the cut-off operator as the function of $p$ and the coupling constant. The right part shows the zero-energy cut of the surface in which the shaded region indicates the part of the $(\lambda,p)$ plane where the lowest eigenvalue of the cut-off operator is positive, as compared to $\lambda_\mathrm{crit}=\gamma_p$. The two curves meet at $p\approx 20.392$ corresponding to $\lambda_\mathrm{crit}\approx 1.563$. Up to this value, it is thus reasonable to expect that a negative eigenvalue exists. For higher values of $p$ the numerical accuracy is a demanding problem, we nevertheless conjecture that at least the Dirichlet region operator, $p=\infty$, is positive. Fig. <ref> also provides an idea of how the spectral threshold of $L_p(\lambda)$ depends on the coupling constant.
§ SUBCRITICAL CASE, EIGENVALUE ESTIMATES
Let us finally pass to the subcritical case, $\lambda<\gamma_p$. According to <cit.> the operator $L_p(\lambda)$ has in this case a purely discrete spectrum. In the mentioned paper a crude bound on eigenvalue sums was established for small values of the coupling constant $\lambda$. We are going derive now a substantially stronger result, an estimate on eigenvalue moments valid for any $\lambda<\gamma_p$. More specifically, let $\mu_1<\mu_2\le\mu_3\le\cdots$ be the set of ordered eigenvalues of (<ref>); we are looking for bounds of the quantities $\sum_{j=1}^\infty (\Lambda-\mu_j)_+^\sigma$ for fixed numbers $\Lambda$ and $\sigma$. This is the contents of the following theorem.
Let $\lambda<\gamma_p$, then for any $\Lambda\ge0$ and $\sigma\ge3/2$ the following trace inequality holds,
\begin{eqnarray}\label{LTbound}
\lefteqn{\hspace{-4em} \mathrm{tr} \left(\Lambda-L_p(\lambda)\right)_+^\sigma} \\&& \nonumber \hspace{-3em} \le C_{p,\sigma}\frac{\left(\Lambda+1\right)^{\sigma+(p+1)/p}}{(\gamma_p-\lambda)^{\sigma+(p+1)/p}}
\left(\left|\ln\left(\frac{\Lambda+1}{\gamma_p-\lambda}\right)\right|+1\right)+C_{p, \sigma}\, C_\lambda^2\left(\Lambda+C_\lambda^{2p/(p+2)}\right)^{\sigma+1}\!,
\end{eqnarray}
where the constant $C_{p, \sigma}$ depends on $p$ and $\sigma$ only and
By the minimax principle it is sufficient to estimate $L_p$ from below by a self-adjoint operator with a purely discrete spectrum for which the moments in question can be calculated. To construct such a lower bound we again employ a bracketing, imposing additional Neumann conditions at the rectagles $G_n,\: \widetilde{G}_n$, and $Q_n,\:\widetilde{Q}_n$ introduced in the proof of Theorem <ref>. The sequence $\{\alpha_n\}_{n=1}^\infty$ is monotonically increasing by construction; we assume again that $\alpha_n\to\infty$ and that the rectangles get asymptotically thinner according to (<ref>), i.e.
\alpha_{n+1}^{2p/(p+2)}-\alpha_n^{2p/(p+2)}\to 0\quad\text{as}\quad n\to\infty\,.
Then, as before, we obtain a direct sum of operators $L^{(1)}_{n, p}, \,\widetilde{L}^{(1)}_{n, p},\,\,L^{(2)}_{n, p}, \,\,\widetilde{L}^{(2)}_{n, p}$ and $L_p^0$. We are going to find the eigenvalue momentum estimates for those.
Let us start from $L_{n, p}^{(1)},\,n=1, 2,\ldots$. We again find a lower bound using the operator $H^{(1)}_n$ given by (<ref>), the spectrum of which is the sum of two one-dimensional operators. Since the spectrum of one-dimensional Neumann operator $-\frac{\mathrm{d}^2}{\mathrm{d}y^2}$ on the interval $(\alpha_n, \alpha_{n+1})$ is discrete and simple with the eigenvalues $\left\{\frac{\pi^2 k^2}{(\alpha_{n+1} -\alpha_n)^2}\right\}_{k=0}^\infty$, the problem reduces to analysis of the operator $h^{(1)}_{n, p}=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\alpha_n^p |x|^p-\lambda (x^2+\alpha_{n+1}^2)^{p/(p+2)}$ on $L^2(-\alpha_{n+1}, \alpha_{n+1})$ which is unitarily equivalent to (<ref>).
To proceed we put $\kappa:=\frac{\gamma_p-\lambda}{2(\gamma_p+\lambda+2)}$ and assume that the edge coordinates satisfy
\begin{eqnarray}\label{decay1}
\alpha_1\ge\left(\frac{\lambda}{\kappa}\right)^{(p+2)^2/(4p(p+1))}\,,\\
\label{decay2}\alpha_{n+1}^{2p/(p+2)}-\alpha_n^{2p/(p+2)}<\frac{\kappa}{\lambda}\,.
\end{eqnarray}
Using then the fact that $(a+b)^q\le a^q+b^q$ holds for any positive numbers $a,\,b$ and $q<1$, in combination with (<ref>), we arrive at the inequalities
\begin{eqnarray*}
\lefteqn{\frac{\lambda}{\alpha_n^{2p/(p+2)}}\left(\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}+\alpha_{n+1}^2\right)^{p/(p+2)}-\alpha_{n+1}^{2p/(p+2)}\right)} \\ && \le\frac{\lambda}{\alpha_n^{2p/(p+2)}}\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}\right)^{p/(p+2)} \le\kappa (|x|^p+1)\,.
\end{eqnarray*}
Next, by virtue of (<ref>) and above estimate, we have
\begin{eqnarray}
\lefteqn{\nonumber h_{n,p}^{(2)}(\lambda)=\alpha_n^{2p/(p+2)}\,\biggl(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+|x|^p
-\frac{\lambda\,\alpha_{n+1}^{2p/(p+2)}}{\alpha_n^{2p/(p+2)}}} \\ && \nonumber \quad -\frac{\lambda}{\alpha_n^{2p/(p+2)}}\biggl(\left(\frac{x^2}{\alpha_n^{2p/(p+2)}}+\alpha_{n+1}^2\right)^{p/(p+2)}
-\alpha_{n+1}^{2p/(p+2)}\biggr)\biggr)\\ && \nonumber \ge\alpha_n^{2p/(p+2)} \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+(1-\kappa)|x|^p-\kappa-\frac{\lambda \alpha_{n+1}^{2p/(p+2)}}{\alpha_n^{2p/(p+2)}}\right)
\\ && \nonumber =\alpha_n^{2p/(p+2)} \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+(1-\kappa)|x|^p-\kappa-\frac{\lambda \left(\alpha_{n+1}^{2p/(p+2)}-\alpha_n^{2p/(p+2)}\right)}{\alpha_n^{2p/(p+2)}}-\lambda\right)
\\ && \label{as.}\ge (1-\kappa) \alpha_n^{2p/(p+2)}\,\left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+|x|^p-\lambda^\prime-\kappa^\prime\right)-\kappa\,,
\end{eqnarray}
where $\kappa^\prime:=\frac{\kappa}{1-\kappa},\,\lambda^\prime:=\frac{\lambda}{1-\kappa}$, and the corresponding Neumann (an)harmonic oscillator is defined on the interval
\begin{equation}\label{interval}
(-\alpha_{n+1}\,\alpha_n^{p/(p+2)}, \alpha_{n+1}\,\alpha_n^{p/(p+2)})\,.
\end{equation}
It follows from Lemma <ref> that if the interval (<ref>) is large enough, which can be achieved by choosing
\begin{equation}\label{new assump.}
\alpha_2 \alpha_1^{p/(p+2)}>\alpha_1^{2(p+1)/(p+2)}>K_{0, p}
\end{equation}
with a large enough $K_{0, p}$, we have the estimate
\begin{equation}\label{inf}
h_{n, p}^{(2)}\ge(1-\kappa)\alpha_n^{2p/(p+2)}\left(\gamma_p-\frac{1}{\alpha_{n+1}^{p/2} \alpha_n^{p^2/(2(p+2))}}-\lambda^\prime-\kappa^\prime\right)-\kappa\,.
\end{equation}
Our aim is now to show that by choosing a suitable sequence $\{\alpha_n\}_{n=1}^\infty$ we can achieve that for any $n\ge1$ the following estimate holds,
\begin{equation}\label{inf1}
\inf \sigma\left(h_{n, p}^{(2)}\right)\ge(1-\kappa) \alpha_n^{2p/(p+2)}\frac{(\gamma_p-\lambda)}{2}-\kappa\,.
\end{equation}
This is ensured, for instance, if
\begin{equation}\label{alpha_1guarant.}
\alpha_1\ge \left(\frac{2(1-\kappa)}{\gamma_p-\lambda-\kappa (\gamma_p+\lambda+2)}\right)^{(p+2)/(p(p+1))}\,.
\end{equation}
Combining (<ref>), (<ref>) and (<ref>) we thus choose
\begin{eqnarray}\label{alpha2}
\lefteqn{\hspace{-5.5em} \alpha_1=1} \\ && \nonumber \hspace{-5em} +\left[\max\left\{\!K_{0, p}^{(p+2)/(2(p+1))},
\left(\frac{2(1-\kappa)}{\gamma_p-\lambda-\kappa(\gamma_p+\lambda+2)}\right)^{(p+2)/(p(p+1))}\!\!\!,
\left(\frac{\lambda}{\kappa}\right)^{(p+2)^2/(4p(p+1))}\right\}\right]\!,
\end{eqnarray}
where $[\cdot]$ means the entire part. Let us now return to the eigenvalue momentum estimates. One has
\begin{equation}\label{eigenvalue momentum}
\mathrm{tr} \left(\Lambda-h_{n, p}^{(2)}\right)_+^\sigma
=\inf\,\sigma\left(h_{n, p}^{(2)}-\Lambda\right)_-^\sigma+\mathrm{tr}^{\prime}\left(h_{n, p}^{(2)}-\Lambda\right)_-^\sigma\,,
\end{equation}
where $\mathrm{tr}^\prime$ the summation which yields the corresponding eigenvalue moment in which the ground state is not taken into account. Using next inequalities (<ref>), (<ref>), in combination with version of Lieb-Thiring inequality suitable for our purpose <cit.>), we infer from (<ref>) that for any positive $\Lambda,\,\sigma\ge3/2$ and $n\ge1$ one has
\begin{eqnarray}
\lefteqn{\hspace{-5.5em}\nonumber \mathrm{tr} \left(\Lambda-h_{n, p}^{(2)}\right)_+^\sigma\le\left(\Lambda+\kappa\right)^\sigma} \\ && \hspace{-5em}\nonumber +(1-\kappa)^\sigma\,\alpha_n^{2p\sigma/(p+2)}\,L_{\sigma, 1}^{\mathrm{cl}} \int_{-\alpha_{n+1} \alpha_n^{2p/(p+2)}}^{ \alpha_{n+1} \alpha_n^{2p/(p+2)}}\left(\frac{\Lambda+\kappa}{(1-\kappa)\,\alpha_n^{2p/(p+2)}}
-|x|^p+\lambda^\prime+\kappa^\prime\right)_+^{\sigma+1/2}\,\mathrm{d}x \\ && \nonumber \hspace{-5em}
\le\left(\Lambda+\kappa\right)^\sigma+(1-\kappa)^\sigma\,\alpha_n^{2p\sigma/(p+2)}\,L_{\sigma, 1}^{\mathrm{cl}} \int_{\mathbb{R}}\left(\frac{\Lambda+\kappa}{(1-\kappa)\,\alpha_n^{2p/(p+2)}}-|x|^p+\lambda^\prime+\kappa^\prime\right)_+^{\sigma+1/2} \mathrm{d}x
\\ && \hspace{-5em} \label{L.T.h_n} \le\left(\Lambda+\kappa\right)^\sigma+2 \alpha_n^{2p\sigma/(p+2)}\,L_{\sigma, 1}^{\mathrm{cl}}\left(\frac{\Lambda+\kappa}{(1-\kappa) \alpha_n^{2p/(p+2)}}+\lambda^\prime+\kappa^\prime\right)^{\sigma+(p+2)/(2p)}\,.
\end{eqnarray}
We further restrict the choice of the sequence $\{\alpha_n\}_{n=1}^\infty$ demanding
\begin{equation}\label{decay*}
\alpha_{n+1}-\alpha_n<\pi\left(\Lambda-\inf\,\sigma(h_{n, p}^{(2)})(\lambda)\right)_+^{-1/2}\,;
\end{equation}
this allows us to write the following estimate
\begin{eqnarray}\label{Ln}
\lefteqn{\mathrm{tr} \left(\Lambda-\bigoplus_{n=1}^\infty L_{n, p}^{(1)}\right)_+^\sigma\le\mathrm{tr} \left(\Lambda-\bigoplus_{n=1}^\infty H_n^{(1)}\right)_+^\sigma} \\ && \nonumber
\le\sum_{n=1}^\infty\sum_{k=0}^\infty\mathrm{tr} \left(\Lambda-\frac{\pi^2 k^2}{(\alpha_{n+1}-\alpha_n)^2}-h_{n, p}^{(2)}\right)_+^\sigma \le\sum_{n=1}^\infty\mathrm{tr} \left(\Lambda-h_{n, p}^{(2)}\right)_+^\sigma\,.
\end{eqnarray}
Using next the fact that $\inf \sigma\left(L_{n, p}^{(1)}\right)\ge \inf \sigma\left(h_{n, p}^{(2)}\right)$ in combination with estimates (<ref>), (<ref>), and (<ref>) one gets
\begin{eqnarray}
\lefteqn{\hspace{-5.5em} \label{tr.est.} \mathrm{tr} \left(\Lambda-\bigoplus_{n=1}^\infty L_{n,p}^{(1)}\right)_+^\sigma \le\sum_{(\gamma_p-\lambda)\alpha_n^{2p/(p+2)}<\frac{2(\Lambda+\kappa)}{1-\kappa}}\left(\Lambda+\kappa\right)^\sigma}
\\ && \hspace{-5em} \nonumber +2 L_{\sigma, 1}^{\mathrm{cl}}\sum_{(\gamma_p-\lambda) \alpha_n^{2p/(p+2)}<\frac{2(\Lambda+\kappa)}{1-\kappa}}\alpha_n^{2p\sigma/(p+2)}\,\biggl(\frac{\Lambda+\kappa}{(1-\kappa) \alpha_n^{2p/(p+2)}}+\lambda^\prime+\kappa^\prime\biggr)^{\sigma+(p+2)/(2p)} \\ && \hspace{-5em} \nonumber
\le\sum_{(\gamma_p-\lambda)\alpha_n^{2p/(p+2)}<\frac{2(\Lambda+\kappa)}{1-\kappa}}\left(\Lambda+\kappa\right)^\sigma
\\ && \hspace{-5em} \nonumber +\frac{2 L_{\sigma, 1}^{\mathrm{cl}}}{(1-\kappa)^{\sigma+(p+2)/(2p)}}\,\sum_{(\gamma_p-\lambda) \alpha_n^{2p/(p+2)}<\frac{2(\Lambda+\kappa)}{1-\kappa}}\alpha_n^{2p\sigma/(p+2)}\,
\left(\frac{\Lambda+\kappa}{\alpha_n^{2p/(p+2)}}+\lambda+\kappa\right)^{\sigma+(p+2)/(2p)} \\ && \hspace{-5em} \nonumber \le\left(\Lambda+\kappa\right)^\sigma \#\left\{\alpha_n<\left(\frac{2(\Lambda+\kappa)}{(1-\kappa)(\gamma_p-\lambda)}\right)^{(p+2)/(2p)}\right\}
\\ && \hspace{-5em} \nonumber +\frac{2L_{\sigma,1}^{\mathrm{cl}}\,(\lambda+1+\kappa)^{\sigma+(p+2)/(2p)}}{(1-\kappa)^{\sigma+(p+2)/(2p)}}\,
\\ && \hspace{-5em} \nonumber +\frac{2 L_{\sigma, 1}^{\mathrm{cl}}\,(\lambda+1+\kappa)^{\sigma+(p+2)/(2p)}}{(1-\kappa)^{\sigma+(p+2)/(2p)}}\,
\sum_{(\Lambda+\kappa)^{(p+2)/(2p)}<\alpha_n<\frac{1}{(\gamma_p-\lambda)^{(p+2)/(2p)}}\left(\!
\frac{2(\Lambda+\kappa)}{1-\kappa}\right)^{(p+2)/(2p)}}\alpha_n^{2p\sigma/(p+2)}\,,
\end{eqnarray}
where $\#\{\cdot\}$ means the cardinality of the corresponding set.
Using the same technique one obtains estimates for operators $\widetilde{L}_{n, p}^{(1)},\,L_{n, p}^{(2)},\,\widetilde{L}_{n, p}^{(2)}$ analogous to (<ref>). Finally, the operator $L_p^0$ can be estimated from below by
H_{0, p}=-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}\quad\text{on}\quad G_0
with Neumann conditions at the boundary $\partial G_0$. The spectrum of $H_{0, p}$ is
\left\{\frac{\pi^2 k^2}{4\alpha_1^2}+\frac{\pi^2 m^2}{4\alpha_1^2}-2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}\right\}_{k, m=0}^\infty\,,
and therefore
\begin{eqnarray}
\lefteqn{ \hspace{-5.5em} \nonumber \mathrm{tr}\left(\Lambda-H_{0, p}\right)_+^\sigma\le\sum_{k, m=0}^\infty\left(\Lambda+2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}-\frac{\pi^2 k^2}{4\alpha_1^2}-\frac{\pi^2 m^2}{4\alpha_1^2}\right)_+^\sigma} \\ && \nonumber \hspace{-5em} \le\left(\Lambda+2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}\right)^\sigma \\ && \nonumber \hspace{-5em} \times \sum_{k=0}^{2\alpha_1\sqrt{\Lambda+2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}}/\pi}\left(\frac{2 \alpha_1}{\pi}\biggl(\Lambda+2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}-\frac{\pi^2 k^2}{4\alpha_1^2}\right)^{1/2}+1\biggr) \\ && \hspace{-5em} \label{L_0} \le\left(\frac{2\alpha_1}{\pi}\left(\Lambda+2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}\right)^{1/2}+1\right)^2\,\left(\Lambda+
2^{p/(p+2)}\lambda \alpha_1^{2p/(p+2)}\right)^\sigma\,.
\end{eqnarray}
Consider now $\alpha_1$ is defined in (<ref>) and to any $\nu=\alpha_1, \alpha_1+1, \alpha_1+2, \ldots\,$ define a finite sequence of numbers by $\beta_k(\nu)=\nu+\frac{k}{\left[\nu^{p/(p+2)} \ln\nu\right]},\,k=0, 1,\ldots,\left[\nu^{p/(p+2)}\,\ln\nu\right]-1$. This allows us to construct a sequence $\{\alpha_n\}_{n=1}^\infty$ of the rectangle edge coordinates using the following prescription: the first term is given by (<ref>) and the further ones are $\alpha_2=\beta_1(\alpha_1),\,\ldots,\alpha_{\left[\alpha_1^{p/(p+2)}\,\ln \alpha_1\right]} =\beta_{\left[\alpha_1^{p/(p+2)}(\alpha_1)\,\ln \alpha_1\right]-1},\,\,\alpha_{\left[\alpha_1^{p/(p+2)}\,\ln \alpha_1\right]+1}=\beta_0(\alpha_1+1), \ldots\,$, etc., where $[\cdot]$ as usual denotes the entire part. With this choice of $\{\alpha_n\}_{n=1}^\infty$, one can check that the right-hand side of (<ref>) is not larger than
\begin{eqnarray}
\nonumber C_{p, \sigma}\biggl(\frac{\left(\Lambda+\kappa\right)^\sigma}{(\gamma_p-\lambda)^\sigma}\,\max\left\{0,\:
\frac{(\Lambda+\kappa)^{(p+1)/p}}{(\gamma_p-\lambda)^{(p+1)/p}}\,\ln\left(\frac{2(\Lambda+\kappa)}{(1-\kappa)(\gamma_p-\lambda)}\right)\right\}
\\\label{final.in.} \quad +\left(\Lambda+\kappa\right)^{\sigma+1/2+1/p}\,\max\left\{0,\:
\left(\Lambda+\kappa\right)^{1/2}\,\ln\left(\Lambda+\kappa\right)\right\}\biggr)
\end{eqnarray}
with a constant depending on $p$ and $\sigma$ only. On the other hand, the right-hand side of (<ref>) is not larger than
\tilde{C}_{p, \sigma}\alpha_1^2\left(\Lambda+\alpha_1^{2p/(p+2)}\right)^{\sigma+1}
with another constant $\tilde{C}_{p, \sigma}$. In this way the theorem is established.
§ ACKNOWLEDGMENTS
We are obliged to Ari Laptev for a useful discussion. The research has been supported by the Czech Science Foundation (GAČR) within the project 14-06818S. D.B. acknowledges the support of the University of Ostrava and the project “Support of Research in the Moravian-Silesian Region 2013”.
The research of A.K. is supported by the German Research Foundation through CRC 1173 “Wave phenomena: analysis and numerics”.
§ REFERENCES
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D. Barseghyan, P. Exner, A regular version of Smilansky model, J. Math. Phys. 55 (2014), 042194 (13pp)
B. Camus, N. Rautenberg, Higher dimensional nonclassical eigenvalue asymptotics, J. Math. Phys. 56 (2015), 021506 (14 pp).
W.D. Evans. M. Solomyak, Smilansky's model of irreversible quantum graphs: I. The absolutely continuous spectrum, II. The point spectrum, J. Phys. A: Math. Gen. 38 (2005), 4611–4627, 7661–7675.
P. Exner, D. Barseghyan, Spectral estimates for a class of Schrödinger operators with infinite phase space and potential unbounded from below,
J. Phys. A: Math. Theor. 45 (2012), 075204 (14pp).
L. Geisinger, T. Weidl, Sharp spectral estimates in domains of infinite volume, Rev. Math. Phys. 23 (2011), 615–641.
I. Guarneri, Irreversible behaviour and collapse of wave packets in
a quantum system with point interactions, J. Phys. A: Math. Theor. 44 (2011), 485304
O. Mickelin, Lieb-Thirring inequalities for generalized magnetic fields, Bull. Math. Sci. (2015), to appear; doi: 10.1007/s13373-015-0067-9.
S. Naboko, M. Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph Proc. Lond. Math. Soc. 92 (2006), 251–272.
M. Reed, B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis. Self-Adjointness, Academic Press, New York 1975
B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983), 209–220.
U. Smilansky, Irreversible quantum graphs, Waves Random Media 14 (2004), 143–153.
M. Solomyak, On a differential operator appearing in the theory of irreversible quantum graphs, Waves Random Media 14 (2004), 173–185.
|
1511.00368
|
Department of Mathematics, South China University of Technology,
GuangZhou 510640, China
We study the quantum separability problem by using general symmetric informationally complete
measurements and derive separability criteria for arbitrary high dimensional bipartite systems of a $d_1$-dimensional subsystem and a $d_2$-dimensional subsystem and multipartite systems of multipartite-level subsystems. These criteria are of more effective and wider application range than previous criteria. They provide experimental implementation in detecting entanglement of unknown quantum states.
03.67.Mn, 03.67.Hk
§ INTRODUCTION
The detection of entanglement is one of the most fundamental and attractive tasks in quantum information theory and quantum information processing. It is well-known that quantum entanglement enables numerous applications ranging from quantum cryptography to quantum computing(see reviews <cit.> and the references therein). For pure quantum states, there have been numerous criteria to distinguish quantum entangled states from the separable ones. For the general mixed states, there has been considerable effort to analyze the separability. But till now we don't have an operational criterion of separability. There have been some necessary criterion for separability, such as Bell inequality <cit.>, positive partial transposition criterion <cit.>, realignment criterion <cit.>, covariance matrix criterion <cit.>,and correlation matrix criterion <cit.>, entanglement witness <cit.>.
Although numerous mathematical tools have been employed in entanglement detection of given known quantum states, experimental implementation of entanglement detection for unknown quantum states has fewer results <cit.>. The authors <cit.> connected the separability criteria to mutually unbiased bases (MUBs) <cit.> in two-qudit, multipartite and continuous-variable quantum systems. Based on the correlation
functions, the criterion uses local measurements only, and can be implemented experimentally. In Ref. <cit.>, Chen et al. generalized such idea and proposed a separability criteria for two-qudit states by using mutually unbiased measurements (MUMs) <cit.>. It is shown that the criterion based on MUMs is more effective than the criterion based on MUBs and for isotropic states this criterion becomes both necessary and sufficient. After that, Liu et al. <cit.> derived separability criteria for arbitrary high-dimensional bipartite and multipartite systems using sets of MUMs.
Besides mutually unbiased bases, another intriguing topic in quantum information theorem is the symmetric informationally complete positive operator-valued measurements (SIC-POVMs) <cit.>. Most of the literature on SIC-POVMs focus on rank $1$ SIC-POVMs (all the POVM elements are proportional to rank $1$ projectors). Such rank $1$ SIC-POVMs have been shown analytically to exist in dimensions $d=1,\cdots,16,19,24,28,35,48$, and numerically for all dimensions $d\leq67$ (see <cit.> and references therein). However, despite the enormous effort of the last years, it is still not known if rank $1$ SIC-POVMs exist in all finite dimensions. In <cit.>, the author introduced the concept of general SIC-POVMs in which the elements need not to be of rank one, and showed that such general SIC-POVMs exist in all finite dimensions. And then Gour and Kalev <cit.> constructed the set of all general SIC-POVMs from the generalized Gell-Mann matrices. Recently, in Ref. <cit.>, the authors used the general SIC-POVMs to derive separability criteria for arbitrary $d$-dimensional bipartite and multipartite systems.
In this paper, we study separability problem via general SIC-POVMs and propose some criteria for the separability of arbitrary high dimensional bipartite systems and multipartite systems of multi-lever subsyetems.
§ SICS AND GENERAL SICS
A POVM $\{P_{j}\}$ with $d^2$ rank one operators acting on $\mathbb{C}^{d}$ is symmetric informationally complete, if
(1)$P_{j}=\frac{1}{d}|\phi_{j}\rangle\langle\phi_{j}|, j=1,2,\cdots,d^2$,
where the vectors $|\phi_{j}\rangle$ satisfy $|\langle\phi_{j}|\phi_{k}\rangle|^{2}=\frac{1}{d+1},j\neq k
$, and $I$ is the identity matrix. The existence of SIC-POVMs in arbitrary dimension d is an open problem. Only in a number of low dimensional cases, the existence of SIC-POVMs has been proved analytically, and numerically for all dimensions up to 67(see <cit.> and the references therein).
A set of $d^{2}$ positive-semidefinite operators $\{{P}_{\alpha}\}_{\alpha=1}^{d^{2}}$ on $\mathbb{C}_{d}$ is said to be a general SIC measurement, if
where $\alpha,\beta=1,2,\cdots,d^{2}$, $\alpha\neq\beta$, $I$ is the identity operator, and the parameter $a$ satifies
$\frac{1}{d^{3}}<a\leq\frac{1}{d^{2}}$. Morever $a=\frac{1}{d^{2}}$ if and only if $P_{\alpha}$ are rank one, which gives rise to a SIC-POVM.
Moreover,like the mutually unbiased measurements, in Ref. <cit.>, the authors
explicitly constructed general symmetric informationally complete measurements for arbitrary dimensional spaces. Let $\{F_{\alpha}\}_{\alpha=1}^{d^2-1}$ be an orthonormal basis of a real vector space of dimensional $d^{2}-1$, satisfying $\mathrm{Tr}(F_{\alpha}F_{\beta})=\delta_{\alpha,\beta}$, $\alpha,\beta=1,2\cdots,d^{2}-1$.
Define $F=\sum_{\alpha=1}^{d^{2}-1}F_{\alpha}$, then the $d^{2}$ operators
(1)$P_{\alpha}=\frac{1}{d^{2}}I+t[F-d(d+1)F_{\alpha}], \alpha=1,2,\cdots,d^{2}-1$,
form a general SIC-POVM measurement. Here $t$ should be chosen such that $P_{\alpha}\geq0$, and corresponding to the construction of general SIC-POVMs, the parameter $a$ is given by
The entanglement detection based SIC-POVMs has been briefly discussed in Ref.<cit.>, but the method is subject to the existence of SIC-POVMs. However these general symmetric informationally complete measurements do exist for arbitrary dimension $d$, and have many useful applications in quantum information theory. And in Ref. <cit.>, based on the calculation of the so-called index of the coincidence, the author derived a number of uncertainty relation inequalities via general SIC-POVMs measurements. In addition, for the given SIC-POVM $\mathcal{P}=\{P_{j}\}$ on $\mathbb{C}^{d}$ and the density matrix $\rho$, the author <cit.> proposed a equality about the index of the coincidence $\mathcal{C}(\mathcal{P}|\rho)$, that is,
\begin{equation}
\mathcal{C}(\mathcal{P}|\rho)=\frac{(ad^{3}-1)\mathrm{Tr}(\rho^{2})+d(1-ad)}{d(d^{2}-1)}
\end{equation}
\mathcal{C}(\mathcal{P}|\rho)=\sum_{j=1}^{d^{2}}[\mathrm{Tr}(P_{j}\rho)]^{2}
At the same time, when the density matrix $\rho$ is pure, $\mathcal{C}(\mathcal{P}|\rho)=\frac{ad^{2}+1}{d(d+1)}$.
§ GENERAL SIC-POVMS BASED SEPARABILITY CRITERION
In Ref.<cit.>, the authors used the general SIC-POVMs to propose separability criteria for arbitrary $d$-dimensional bipartite and multipartite systems. Here we will give the criterion about arbitrary high dimensional bipartite and multipartite systems.
Suppose $\rho$ is a density matrix in $\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}$. Let
$\mathcal{P}=\{P_{j}\}_{j=1}^{d_{1}^{2}}$ and
$\mathcal{Q}=\{Q_{k}\}_{k=1}^{d_{2}^{2}}$ be any two sets of general SIC-POVMs on
$\mathbb{C}^{d_{1}}$ and $\mathbb{C}^{d_{2}}$ with parameters $a_{1},a_{2}$, respectively. Define
\{P_{n_{j}}\}\subseteq \{P_{j}\}\\
\{Q_{n_{j}}\}\subseteq \{Q_{j}\} \end{array}}\sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]
Here $d=min\{(d_{1})^{2},(d_{2})^{2}\}$. If $\rho$ is separable, then
[Proof]. Assume that $\rho=\Sigma_{k}p_{k}\rho_{k},\sum_{k}p_{k}=1$, $I(\rho_{k})=\sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho_{k}]$, where $\rho_{k}=|\phi\rangle\langle\phi|\otimes|\psi\rangle\langle\psi|$. From the linearity of the trace function, we need only to consider pure separable states $\rho_{k}$. So we have
\begin{eqnarray*}
I(\rho_{k}) & = & \sum_{j=1}^{d}\mathrm{Tr}(P_{n_{j}}\bigotimes Q_{n_{j}}\rho_{k})\\
& = & \sum_{j=1}^{d}\mathrm{Tr}(P_{n_{j}}|\phi\rangle\langle\phi|)\mathrm{Tr}(Q_{n_{j}}|\psi\rangle\langle\psi|)\\
& \leq &
\frac{1}{2}\sum_{j=1}^{d}\{[\mathrm{Tr}(P_{n_{j}}|\phi\rangle\langle\phi|)]^{2}+[\mathrm{Tr}(Q_{n_{j}}|\psi\rangle\langle\psi|)]^{2}\}\\
& \leq &
\frac{1}{2}[\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}+\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}].
\end{eqnarray*}
Then we can get
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
J(\rho) &=& \max_{\begin{array}{c}
\{P_{n_{j}}\}\subseteq \{P_{j}\}\\
\{Q_{n_{j}}\}\subseteq \{Q_{j}\} \end{array}}\sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho] \\
&=& \max\sum_{k}p_{k}I(\rho_{k})\\
& \leq &
\frac{1}{2}[\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}+\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}]
\end{eqnarray*}
So $J(\rho)\leq\frac{1}{2}[\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}+\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}]$. $\Box$
It is worthy to note that the criteria in Ref. <cit.> is the corollary of Theore $1$. In fact, if $d_{1}=d_{2}=d$, and $\{P_{j}\}_{j=1}^{d_{1}^{2}}$ and $\{Q_{k}\}_{k=1}^{d_{2}^{2}}$ are any two sets of general SIC-POVMs on $\mathbb{C}_{d}$, with the same parameter $a$, then by Theorem $1$ there is
\{P_{n_{j}}\}\subseteq \{P_{j}\}\\
\{Q_{n_{j}}\}\subseteq \{Q_{j}\} \end{array}}\sum_{j=1}^{d^{2}}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]\leq\frac{ad^{2}+1}{d(d+1)},
which is the desired result. Therefore ,the criterion in Ref. <cit.> is the special case of our criterion of Theorem $1$. And Our criteria is also both necessary and sufficient for the separability of isotropic states. Unlike the criterion based on SIC-POVMs in Ref. <cit.>, our criteria work perfectly for any dimensional $d$.
By using the Cauchy-Schwarz inequality, we can obtain stronger bound than in Theorem $1$ .
Let $\rho$ be a density matrix in $\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}$,
$\mathcal{P}=\{P_{j}\}_{j=1}^{d_{1}^{2}}$ and
$\mathcal{Q}=\{Q_{j}\}_{j=1}^{d_{2}^{2}}$ be any two sets of general SIC-POVMs on
$\mathbb{C}^{d_{1}}$ and $\mathbb{C}^{d_{2}}$ with parameters $a_{1},a_{2}$. Define
\{P_{n_{j}}\}\subseteq \{P_{j}\}\\
\{Q_{n_{j}}\}\subseteq \{Q_{j}\} \end{array}}\sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]
Here $d=min\{(d_{1})^{2},(d_{2})^{2}\}$, If $\rho$ is separable, then $J(\rho)\leq\sqrt{\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}}\sqrt{\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}}.$
[Proof]. Let $I(\rho)=\sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]$. As theorem $1$, we need only to consider pure separable state
$\rho=|\phi\rangle\langle\phi|\otimes|\psi\rangle\langle\psi|$, since $\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]$ is a linear function. Then we have
\begin{eqnarray*}
I(\rho) & = & \sum_{j=1}^{d}\mathrm{Tr}[(P_{n_{j}}\bigotimes Q_{n_{j}})\rho]\\
& = & \sum_{j=1}^{d}[\mathrm{Tr}(P_{n_{j}}|\phi\rangle\langle\phi|)][\mathrm{Tr}(Q_{n_{j}}|\psi\rangle\langle\psi|)]\\
& \leq &
\sqrt{\sum_{j=1}^{d}[\mathrm{Tr}(P_{n_{j}}|\phi\rangle\langle\phi|)]^{2}}\sqrt{\sum_{j=1}^{d}[\mathrm{Tr}(Q_{n_{j}}|\psi\rangle\langle\psi|)]^{2}}\\
& \leq &
\sqrt{\sum_{j=1}^{(d_{1})^{2}}[\mathrm{Tr}(P_{n_{j}}|\phi\rangle\langle\phi|)]^{2}}\sqrt{\sum_{j=1}^{(d_{2})^{2}}[\mathrm{Tr}(Q_{n_{j}}|\psi\rangle\langle\psi|)]^{2}}\\
& = &
\sqrt{\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}}\sqrt{\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}}
\end{eqnarray*}
Here we use the Cauchy-Schwarz inequality and the equality $(1)$. Then for any density matrix, we can have
The bound in Theorem $2$ is lower than that in Theorem $1$ since $\sqrt{\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}}\sqrt{\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}}\leq \frac{1}{2}[\frac{a_{1}d_{1}^{2}+1}{d_{1}(d_{1}+1)}+\frac{a_{2}d_{2}^{2}+1}{d_{2}(d_{2}+1)}]$.
It is characteristic for multipartite systems that the definition of entanglement is not unique. For the reason, we can discuss it with the the notions of $k$-partite entanglement or $k$-nonseparability for given partition and unfixed partition,respectively <cit.>. A pure state $\mathbf{|\phi\rangle}$ of a $n$-partite system is called $k$-separable if it can be written as a tensor product of $k$ vectors, i.e. $\mathbf{|\phi\rangle}=|\phi\rangle_{1}\bigotimes|\phi\rangle_{2}\bigotimes\cdots\bigotimes|\phi\rangle_{k}$.
States that are $n$-separable do not contain any entanglement and are called fully separable. In addition, those states whose entanglement ranges over all $n$ parties are called genuine multipartite entangled states. The generalization to mixed states is direct: A mixed state is called $k$-separable if it can be written as a convex combination of $k$-separable states $\rho=\sum_{k}p_{k}\rho_{k}$, where $\rho_{k}$ is $k$-separable pure states. In the following, we will give two criteria for multipartite systems and then argue $k$-nonseparability for a given partition of $n$-partite system.
Suppose $\rho$ is a density matrix in $\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}\bigotimes\cdots\bigotimes\mathbb{C}^{d_{m}}$,
and $\{\mathcal{P}^{(i)}\}$ are $m$ sets of general SIC-POVMs on
$\mathbb{C}^{d_{i}}$ with parameters $a_{i},i=1,2,\cdots,m$,
where$\{\mathcal{P}^{(i)}\}=\{P_{j}^{(i)}\}_{j=1}^{d_{i}^{2}}$. Define
J(\rho)=\max_{\{P_{n_{j}}^{(i)}\}\subseteq\{\mathcal{P}_{j}^{(i)}\}}\sum_{j=1}^{d}\mathrm{Tr}(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)}\rho) $$
Here $d=min\{(d_{1})^{2},(d_{2})^{2},\cdots,(d_{m})^{2}\}$. If $\rho$ is fully separable, then
[Proof]. Let $\rho=\sum _{k}p_{k}P_{k}$, with $\sum_{k}p_{k}=1$, be a fully separable density matrix, where $\rho_{k}=\bigotimes_{i=1}^{m}|\phi_{ik}\rangle\langle\phi_{ik}|$. Since
\begin{eqnarray*}
\sum_{j=1}^{d}\mathrm{Tr}[(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)})\rho_{k}]
& = & \sum_{j=1}^{d}\mathrm{Tr}[(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)})(\bigotimes_{i=1}^{m}|\phi_{ik}\rangle\langle\phi_{ik}|)]\\
& = & \sum_{j=1}^{d}[\prod_{i=1}^{m}\mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{ik}\rangle\langle\phi_{ik}|)]\\
& \leq &
\sum_{j=1}^{d}\{\frac{\sum_{i=1}^{m}[\mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{ik}\rangle\langle\phi_{ik}|)]^{2}}{m}\}^{\frac{m}{2}}\\
& \leq &
\sum_{j=1}^{d}\frac{\sum_{i=1}^{m}[\mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{ik}\rangle\langle\phi_{ik}|)]^{2}}{m}\\
& = &
\sum_{i=1}^{m}\sum_{j=1}^{d}\frac{[\mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{ik}\rangle\langle\phi_{ik}|)]^{2}}{m}\\
& \leq &
\frac{1}{m}\sum_{i=1}^{m}[\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}]
\end{eqnarray*}
where we use the inequality $x_{1}x_{2}\cdots x_{n}\leq[\frac{\sum_{i=1}^{n}(x_{i})^{2}}{n}]^{\frac{n}{2}}$,where $x_{i}\geq0,i=1,2,\cdots n$ <cit.>
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
\sum_{j=1}^{d}\mathrm{Tr}(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)}\rho) &=& \sum_{k}\sum_{j=1}^{d}p_{k}\mathrm{Tr}(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)}\rho_{k}) \\
&\leq & \frac{1}{m}\sum_{i=1}^{m}[\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}]
\end{eqnarray*}
Finally, we can get $J(\rho)\leq\frac{1}{m}\sum_{i=1}^{m}[\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}]$.$\Box$
Assume that $\rho$ is a density matrix in $\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}\bigotimes\cdots\bigotimes\mathbb{C}^{d_{m}}$,
and $\{P_{j}^{(i)}\}_{j=1}^{(d_{i})^{2}}$ are $m$ sets of general SIC-POVMs on
$\mathbb{C}^{d_{i}}$ with parameters $a_{i},i=1,2,\cdots,m$.
J(\rho)=\max_{\{P_{n_{j}}^{(i)}\}\subseteq\{\mathcal{P}_{j}^{(i)}\}}\sum_{j=1}^{d}\mathrm{Tr}(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)}\rho) $$
Here $d=min\{(d_{1})^{2},(d_{2})^{2},\cdots,(d_{m})^{2}\}$. If $\rho$ is fully separable, then
J(\rho)\leq \min_{i\neq j}\sqrt{\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}} \sqrt{\frac{a_{j}d_{j}^{2}+1}{d_{j}(d_{j}+1)}}
[Proof]. Let $\rho=\sum _{k}p_{k}P_{k}$ be a fully separable pure state,where $\sum _{k}p_{k}=1$.
\begin{eqnarray*}
I(\rho) & = &
\sum_{j=1}^{d} \sum_{k} p_{k} \mathrm{Tr}[(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)})P_{k}]\\
& = &
\sum_k \sum_{j=1}^{d} p_{k} \mathrm{Tr}[(\bigotimes_{i=1}^{m}P_{n_{j}}^{(i)})(\bigotimes_{i=1}^{m}|\phi_{i}\rangle\langle\phi_{i}|)]\\
&= &
\sum_k \sum_{j=1}^{d} p_k \mathrm{Tr}[\bigotimes_{i=1}^{m} (P_{n_{j}}^{(i)}|\phi_{i}\rangle\langle\phi_{i}|)]\\
&= &
\sum_k \sum_{j=1}^{d} p_k \prod_{i=1}^{m} \mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{i}\rangle\langle\phi_{i}|) \\
&\leq &
\sum_k \sum_{j=1}^{d} p_k \mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{i}\rangle\langle\phi_{i}|)
\mathrm{Tr}(P_{n_{j}}^{(i')}|\phi_{i'}\rangle\langle\phi_{i'}|)
\end{eqnarray*}
Then using the Cauchy-Schwarz inequality, we can get
I(\rho)\leq \sqrt{\sum_{j=1}^{d}[\mathrm{Tr}(P_{n_{j}}^{(i)}|\phi_{i}\rangle\langle\phi_{i}|)]^2}
\sqrt{\sum_{j=1}^{d}[\mathrm{Tr}(P_{n_{j}}^{(i')}|\phi_{i'}\rangle\langle\phi_{i'}|)]^2}
So using the equality $(1)$, we finally get $
J(\rho)= \max I(\rho)\leq \min_{i\neq j}\sqrt{\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}} \sqrt{\frac{a_{j}d_{j}^{2}+1}{d_{j}(d_{j}+1)}}
$ $\Box$
For Theorem $3$ and Theorem $4$, we don't require the subsystems with the same dimension, so we can use them straightforward to detect $k$-nonseparable states with respect to a fixed partition. For an $n$-partite state $\rho$ in $\mathbb{C}^{1}\bigotimes\mathbb{C}^{2}\bigotimes\cdots\bigotimes\mathbb{C}^{n}=\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}\bigotimes\cdots\bigotimes\mathbb{C}^{d_{k}}$, if there are sets of $k$ general SIC-POVMs $\{\mathcal{P}^{(i)}\}$ on
$\mathbb{C}^{d_{i}}$ with parameters $a_{i}$ such that
\sum_{j=1}^{d^{2}}\mathrm{Tr}(\bigotimes_{i=1}^{k}P_{n_{j}}^{(i)}\rho)>\frac{1}{k}\sum_{i=1}^{k}\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}
\sum_{j=1}^{d^{2}}\mathrm{Tr}(\bigotimes_{i=1}^{k}P_{n_{j}}^{(i)}\rho)>
\min_{1\leq i\neq j\leq k} \sqrt{\frac{a_{i}d_{i}^{2}+1}{d_{i}(d_{i}+1)}}\sqrt{\frac{a_{j}d_{j}^{2}+1}{d_{j}(d_{j}+1)}} $$
for some $\{P_{n_{j}}^{(i)}\}_{j=1}^{d^2}\subseteq\{\mathcal{P}^{(i)}\}$, then $\rho$ is $k$-nonseparable in $\mathbb{C}^{d_{1}}\bigotimes\mathbb{C}^{d_{2}}\bigotimes\cdots\bigotimes\mathbb{C}^{d_{k}}$, where $d=min\{d_{1},d_{2},\cdots,d_{k}\}$ and $i=1,2,\cdots,k$.
Our criteria are better than the previous ones <cit.>. First, Comparing with <cit.> and <cit.>, our criteria are suitable for arbitrary dimension $d$ because the general SIC-POVMs do exist for arbitrary dimension $d$. Second, we can detect entanglement more wider and effective than the authors <cit.> and <cit.>. The criteria in this paper could detect the separability of arbitrary high dimensional bipartite systems and multipartite systems of different dimensions, but the criteria <cit.> are not. To Ref. <cit.>, we just need less joint local measurements, reducing the complexity of experimental implementation.
§ CONCLUSION AND DISCUSSIONS
We have studied the separability problem via general SIC-POVMs and have presented separability criteria for the separability of arbitrary high dimensional bipartite systems of a $d_1$-dimensional subsystem and a $d_2$-dimensional subsystem and multipartite systems of multipartite-level subsystems. For isotropic states, our criteria are both necessary
and sufficient. It detects all the entangled isotropic states
of arbitrary dimension $d$. The powerfulness of our criterion
is due to that these criteria are more effective and wider applications range than previous criteria and comparing with the criteria based on mutually unbiased measurements, our criteria require less local measurements. For multipartite systems, we can detect the $k$-nonseparability of $n$-partite and high dimensional systems. It would be interesting to study the separability criteria of multipartite systems with different dimensions via complete set of general SIC-POVMs.
This work is supported by the NSFC through Grants No.11475178 and No.11571119. And we would like
to thank Prof. Shao-Ming Fei at School of Mathematical Sciences, Capital Normal University for helpful discussion.
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The ATLAS Collaboration
G. Aad$^{\rm 85}$,
B. Abbott$^{\rm 113}$,
J. Abdallah$^{\rm 151}$,
O. Abdinov$^{\rm 11}$,
R. Aben$^{\rm 107}$,
M. Abolins$^{\rm 90}$,
O.S. AbouZeid$^{\rm 158}$,
H. Abramowicz$^{\rm 153}$,
H. Abreu$^{\rm 152}$,
R. Abreu$^{\rm 116}$,
Y. Abulaiti$^{\rm 146a,146b}$,
B.S. Acharya$^{\rm 164a,164b}$$^{,a}$,
L. Adamczyk$^{\rm 38a}$,
D.L. Adams$^{\rm 25}$,
J. Adelman$^{\rm 108}$,
S. Adomeit$^{\rm 100}$,
T. Adye$^{\rm 131}$,
A.A. Affolder$^{\rm 74}$,
T. Agatonovic-Jovin$^{\rm 13}$,
J. Agricola$^{\rm 54}$,
J.A. Aguilar-Saavedra$^{\rm 126a,126f}$,
S.P. Ahlen$^{\rm 22}$,
F. Ahmadov$^{\rm 65}$$^{,b}$,
G. Aielli$^{\rm 133a,133b}$,
H. Akerstedt$^{\rm 146a,146b}$,
T.P.A. Åkesson$^{\rm 81}$,
A.V. Akimov$^{\rm 96}$,
G.L. Alberghi$^{\rm 20a,20b}$,
J. Albert$^{\rm 169}$,
S. Albrand$^{\rm 55}$,
M.J. Alconada Verzini$^{\rm 71}$,
M. Aleksa$^{\rm 30}$,
I.N. Aleksandrov$^{\rm 65}$,
C. Alexa$^{\rm 26b}$,
G. Alexander$^{\rm 153}$,
T. Alexopoulos$^{\rm 10}$,
M. Alhroob$^{\rm 113}$,
G. Alimonti$^{\rm 91a}$,
L. Alio$^{\rm 85}$,
J. Alison$^{\rm 31}$,
S.P. Alkire$^{\rm 35}$,
B.M.M. Allbrooke$^{\rm 149}$,
P.P. Allport$^{\rm 18}$,
A. Aloisio$^{\rm 104a,104b}$,
A. Alonso$^{\rm 36}$,
F. Alonso$^{\rm 71}$,
C. Alpigiani$^{\rm 138}$,
A. Altheimer$^{\rm 35}$,
B. Alvarez Gonzalez$^{\rm 30}$,
D. Álvarez Piqueras$^{\rm 167}$,
M.G. Alviggi$^{\rm 104a,104b}$,
B.T. Amadio$^{\rm 15}$,
K. Amako$^{\rm 66}$,
Y. Amaral Coutinho$^{\rm 24a}$,
C. Amelung$^{\rm 23}$,
D. Amidei$^{\rm 89}$,
S.P. Amor Dos Santos$^{\rm 126a,126c}$,
A. Amorim$^{\rm 126a,126b}$,
S. Amoroso$^{\rm 48}$,
N. Amram$^{\rm 153}$,
G. Amundsen$^{\rm 23}$,
C. Anastopoulos$^{\rm 139}$,
L.S. Ancu$^{\rm 49}$,
N. Andari$^{\rm 108}$,
T. Andeen$^{\rm 35}$,
C.F. Anders$^{\rm 58b}$,
G. Anders$^{\rm 30}$,
J.K. Anders$^{\rm 74}$,
K.J. Anderson$^{\rm 31}$,
A. Andreazza$^{\rm 91a,91b}$,
V. Andrei$^{\rm 58a}$,
S. Angelidakis$^{\rm 9}$,
I. Angelozzi$^{\rm 107}$,
P. Anger$^{\rm 44}$,
A. Angerami$^{\rm 35}$,
F. Anghinolfi$^{\rm 30}$,
A.V. Anisenkov$^{\rm 109}$$^{,c}$,
N. Anjos$^{\rm 12}$,
A. Annovi$^{\rm 124a,124b}$,
M. Antonelli$^{\rm 47}$,
A. Antonov$^{\rm 98}$,
J. Antos$^{\rm 144b}$,
F. Anulli$^{\rm 132a}$,
M. Aoki$^{\rm 66}$,
L. Aperio Bella$^{\rm 18}$,
G. Arabidze$^{\rm 90}$,
Y. Arai$^{\rm 66}$,
J.P. Araque$^{\rm 126a}$,
A.T.H. Arce$^{\rm 45}$,
F.A. Arduh$^{\rm 71}$,
J-F. Arguin$^{\rm 95}$,
S. Argyropoulos$^{\rm 63}$,
M. Arik$^{\rm 19a}$,
A.J. Armbruster$^{\rm 30}$,
O. Arnaez$^{\rm 30}$,
H. Arnold$^{\rm 48}$,
M. Arratia$^{\rm 28}$,
O. Arslan$^{\rm 21}$,
A. Artamonov$^{\rm 97}$,
G. Artoni$^{\rm 23}$,
S. Asai$^{\rm 155}$,
N. Asbah$^{\rm 42}$,
A. Ashkenazi$^{\rm 153}$,
B. Åsman$^{\rm 146a,146b}$,
L. Asquith$^{\rm 149}$,
K. Assamagan$^{\rm 25}$,
R. Astalos$^{\rm 144a}$,
M. Atkinson$^{\rm 165}$,
N.B. Atlay$^{\rm 141}$,
K. Augsten$^{\rm 128}$,
M. Aurousseau$^{\rm 145b}$,
G. Avolio$^{\rm 30}$,
B. Axen$^{\rm 15}$,
M.K. Ayoub$^{\rm 117}$,
G. Azuelos$^{\rm 95}$$^{,d}$,
M.A. Baak$^{\rm 30}$,
A.E. Baas$^{\rm 58a}$,
M.J. Baca$^{\rm 18}$,
C. Bacci$^{\rm 134a,134b}$,
H. Bachacou$^{\rm 136}$,
K. Bachas$^{\rm 154}$,
M. Backes$^{\rm 30}$,
M. Backhaus$^{\rm 30}$,
P. Bagiacchi$^{\rm 132a,132b}$,
P. Bagnaia$^{\rm 132a,132b}$,
Y. Bai$^{\rm 33a}$,
T. Bain$^{\rm 35}$,
J.T. Baines$^{\rm 131}$,
O.K. Baker$^{\rm 176}$,
E.M. Baldin$^{\rm 109}$$^{,c}$,
P. Balek$^{\rm 129}$,
T. Balestri$^{\rm 148}$,
F. Balli$^{\rm 84}$,
W.K. Balunas$^{\rm 122}$,
E. Banas$^{\rm 39}$,
Sw. Banerjee$^{\rm 173}$,
A.A.E. Bannoura$^{\rm 175}$,
L. Barak$^{\rm 30}$,
E.L. Barberio$^{\rm 88}$,
D. Barberis$^{\rm 50a,50b}$,
M. Barbero$^{\rm 85}$,
T. Barillari$^{\rm 101}$,
M. Barisonzi$^{\rm 164a,164b}$,
T. Barklow$^{\rm 143}$,
N. Barlow$^{\rm 28}$,
S.L. Barnes$^{\rm 84}$,
B.M. Barnett$^{\rm 131}$,
R.M. Barnett$^{\rm 15}$,
Z. Barnovska$^{\rm 5}$,
A. Baroncelli$^{\rm 134a}$,
G. Barone$^{\rm 23}$,
A.J. Barr$^{\rm 120}$,
F. Barreiro$^{\rm 82}$,
J. Barreiro Guimarães da Costa$^{\rm 57}$,
R. Bartoldus$^{\rm 143}$,
A.E. Barton$^{\rm 72}$,
P. Bartos$^{\rm 144a}$,
A. Basalaev$^{\rm 123}$,
A. Bassalat$^{\rm 117}$,
A. Basye$^{\rm 165}$,
R.L. Bates$^{\rm 53}$,
S.J. Batista$^{\rm 158}$,
J.R. Batley$^{\rm 28}$,
M. Battaglia$^{\rm 137}$,
M. Bauce$^{\rm 132a,132b}$,
F. Bauer$^{\rm 136}$,
H.S. Bawa$^{\rm 143}$$^{,e}$,
J.B. Beacham$^{\rm 111}$,
M.D. Beattie$^{\rm 72}$,
T. Beau$^{\rm 80}$,
P.H. Beauchemin$^{\rm 161}$,
R. Beccherle$^{\rm 124a,124b}$,
P. Bechtle$^{\rm 21}$,
H.P. Beck$^{\rm 17}$$^{,f}$,
K. Becker$^{\rm 120}$,
M. Becker$^{\rm 83}$,
M. Beckingham$^{\rm 170}$,
C. Becot$^{\rm 117}$,
A.J. Beddall$^{\rm 19b}$,
A. Beddall$^{\rm 19b}$,
V.A. Bednyakov$^{\rm 65}$,
C.P. Bee$^{\rm 148}$,
L.J. Beemster$^{\rm 107}$,
T.A. Beermann$^{\rm 30}$,
M. Begel$^{\rm 25}$,
J.K. Behr$^{\rm 120}$,
C. Belanger-Champagne$^{\rm 87}$,
W.H. Bell$^{\rm 49}$,
G. Bella$^{\rm 153}$,
L. Bellagamba$^{\rm 20a}$,
A. Bellerive$^{\rm 29}$,
M. Bellomo$^{\rm 86}$,
K. Belotskiy$^{\rm 98}$,
O. Beltramello$^{\rm 30}$,
O. Benary$^{\rm 153}$,
D. Benchekroun$^{\rm 135a}$,
M. Bender$^{\rm 100}$,
K. Bendtz$^{\rm 146a,146b}$,
N. Benekos$^{\rm 10}$,
Y. Benhammou$^{\rm 153}$,
E. Benhar Noccioli$^{\rm 49}$,
J.A. Benitez Garcia$^{\rm 159b}$,
D.P. Benjamin$^{\rm 45}$,
J.R. Bensinger$^{\rm 23}$,
S. Bentvelsen$^{\rm 107}$,
L. Beresford$^{\rm 120}$,
M. Beretta$^{\rm 47}$,
D. Berge$^{\rm 107}$,
E. Bergeaas Kuutmann$^{\rm 166}$,
N. Berger$^{\rm 5}$,
F. Berghaus$^{\rm 169}$,
J. Beringer$^{\rm 15}$,
C. Bernard$^{\rm 22}$,
N.R. Bernard$^{\rm 86}$,
C. Bernius$^{\rm 110}$,
F.U. Bernlochner$^{\rm 21}$,
T. Berry$^{\rm 77}$,
P. Berta$^{\rm 129}$,
C. Bertella$^{\rm 83}$,
G. Bertoli$^{\rm 146a,146b}$,
F. Bertolucci$^{\rm 124a,124b}$,
C. Bertsche$^{\rm 113}$,
D. Bertsche$^{\rm 113}$,
M.I. Besana$^{\rm 91a}$,
G.J. Besjes$^{\rm 36}$,
O. Bessidskaia Bylund$^{\rm 146a,146b}$,
M. Bessner$^{\rm 42}$,
N. Besson$^{\rm 136}$,
C. Betancourt$^{\rm 48}$,
S. Bethke$^{\rm 101}$,
A.J. Bevan$^{\rm 76}$,
W. Bhimji$^{\rm 15}$,
R.M. Bianchi$^{\rm 125}$,
L. Bianchini$^{\rm 23}$,
M. Bianco$^{\rm 30}$,
O. Biebel$^{\rm 100}$,
D. Biedermann$^{\rm 16}$,
S.P. Bieniek$^{\rm 78}$,
N.V. Biesuz$^{\rm 124a,124b}$,
M. Biglietti$^{\rm 134a}$,
J. Bilbao De Mendizabal$^{\rm 49}$,
H. Bilokon$^{\rm 47}$,
M. Bindi$^{\rm 54}$,
S. Binet$^{\rm 117}$,
A. Bingul$^{\rm 19b}$,
C. Bini$^{\rm 132a,132b}$,
S. Biondi$^{\rm 20a,20b}$,
D.M. Bjergaard$^{\rm 45}$,
C.W. Black$^{\rm 150}$,
J.E. Black$^{\rm 143}$,
K.M. Black$^{\rm 22}$,
D. Blackburn$^{\rm 138}$,
R.E. Blair$^{\rm 6}$,
J.-B. Blanchard$^{\rm 136}$,
J.E. Blanco$^{\rm 77}$,
T. Blazek$^{\rm 144a}$,
I. Bloch$^{\rm 42}$,
C. Blocker$^{\rm 23}$,
W. Blum$^{\rm 83}$$^{,*}$,
U. Blumenschein$^{\rm 54}$,
S. Blunier$^{\rm 32a}$,
G.J. Bobbink$^{\rm 107}$,
V.S. Bobrovnikov$^{\rm 109}$$^{,c}$,
S.S. Bocchetta$^{\rm 81}$,
A. Bocci$^{\rm 45}$,
C. Bock$^{\rm 100}$,
M. Boehler$^{\rm 48}$,
J.A. Bogaerts$^{\rm 30}$,
D. Bogavac$^{\rm 13}$,
A.G. Bogdanchikov$^{\rm 109}$,
C. Bohm$^{\rm 146a}$,
V. Boisvert$^{\rm 77}$,
T. Bold$^{\rm 38a}$,
V. Boldea$^{\rm 26b}$,
A.S. Boldyrev$^{\rm 99}$,
M. Bomben$^{\rm 80}$,
M. Bona$^{\rm 76}$,
M. Boonekamp$^{\rm 136}$,
A. Borisov$^{\rm 130}$,
G. Borissov$^{\rm 72}$,
S. Borroni$^{\rm 42}$,
J. Bortfeldt$^{\rm 100}$,
V. Bortolotto$^{\rm 60a,60b,60c}$,
K. Bos$^{\rm 107}$,
D. Boscherini$^{\rm 20a}$,
M. Bosman$^{\rm 12}$,
J. Boudreau$^{\rm 125}$,
J. Bouffard$^{\rm 2}$,
E.V. Bouhova-Thacker$^{\rm 72}$,
D. Boumediene$^{\rm 34}$,
C. Bourdarios$^{\rm 117}$,
N. Bousson$^{\rm 114}$,
S.K. Boutle$^{\rm 53}$,
A. Boveia$^{\rm 30}$,
J. Boyd$^{\rm 30}$,
I.R. Boyko$^{\rm 65}$,
I. Bozic$^{\rm 13}$,
J. Bracinik$^{\rm 18}$,
A. Brandt$^{\rm 8}$,
G. Brandt$^{\rm 54}$,
O. Brandt$^{\rm 58a}$,
U. Bratzler$^{\rm 156}$,
B. Brau$^{\rm 86}$,
J.E. Brau$^{\rm 116}$,
H.M. Braun$^{\rm 175}$$^{,*}$,
W.D. Breaden Madden$^{\rm 53}$,
K. Brendlinger$^{\rm 122}$,
A.J. Brennan$^{\rm 88}$,
L. Brenner$^{\rm 107}$,
R. Brenner$^{\rm 166}$,
S. Bressler$^{\rm 172}$,
T.M. Bristow$^{\rm 46}$,
D. Britton$^{\rm 53}$,
D. Britzger$^{\rm 42}$,
F.M. Brochu$^{\rm 28}$,
I. Brock$^{\rm 21}$,
R. Brock$^{\rm 90}$,
J. Bronner$^{\rm 101}$,
G. Brooijmans$^{\rm 35}$,
T. Brooks$^{\rm 77}$,
W.K. Brooks$^{\rm 32b}$,
J. Brosamer$^{\rm 15}$,
E. Brost$^{\rm 116}$,
P.A. Bruckman de Renstrom$^{\rm 39}$,
D. Bruncko$^{\rm 144b}$,
R. Bruneliere$^{\rm 48}$,
A. Bruni$^{\rm 20a}$,
G. Bruni$^{\rm 20a}$,
M. Bruschi$^{\rm 20a}$,
N. Bruscino$^{\rm 21}$,
L. Bryngemark$^{\rm 81}$,
T. Buanes$^{\rm 14}$,
Q. Buat$^{\rm 142}$,
P. Buchholz$^{\rm 141}$,
A.G. Buckley$^{\rm 53}$,
S.I. Buda$^{\rm 26b}$,
I.A. Budagov$^{\rm 65}$,
F. Buehrer$^{\rm 48}$,
L. Bugge$^{\rm 119}$,
M.K. Bugge$^{\rm 119}$,
O. Bulekov$^{\rm 98}$,
D. Bullock$^{\rm 8}$,
H. Burckhart$^{\rm 30}$,
S. Burdin$^{\rm 74}$,
C.D. Burgard$^{\rm 48}$,
B. Burghgrave$^{\rm 108}$,
S. Burke$^{\rm 131}$,
I. Burmeister$^{\rm 43}$,
E. Busato$^{\rm 34}$,
D. Büscher$^{\rm 48}$,
V. Büscher$^{\rm 83}$,
P. Bussey$^{\rm 53}$,
J.M. Butler$^{\rm 22}$,
A.I. Butt$^{\rm 3}$,
C.M. Buttar$^{\rm 53}$,
J.M. Butterworth$^{\rm 78}$,
P. Butti$^{\rm 107}$,
W. Buttinger$^{\rm 25}$,
A. Buzatu$^{\rm 53}$,
A.R. Buzykaev$^{\rm 109}$$^{,c}$,
S. Cabrera Urbán$^{\rm 167}$,
D. Caforio$^{\rm 128}$,
V.M. Cairo$^{\rm 37a,37b}$,
O. Cakir$^{\rm 4a}$,
N. Calace$^{\rm 49}$,
P. Calafiura$^{\rm 15}$,
A. Calandri$^{\rm 136}$,
G. Calderini$^{\rm 80}$,
P. Calfayan$^{\rm 100}$,
L.P. Caloba$^{\rm 24a}$,
D. Calvet$^{\rm 34}$,
S. Calvet$^{\rm 34}$,
R. Camacho Toro$^{\rm 31}$,
S. Camarda$^{\rm 42}$,
P. Camarri$^{\rm 133a,133b}$,
D. Cameron$^{\rm 119}$,
R. Caminal Armadans$^{\rm 165}$,
S. Campana$^{\rm 30}$,
M. Campanelli$^{\rm 78}$,
A. Campoverde$^{\rm 148}$,
V. Canale$^{\rm 104a,104b}$,
A. Canepa$^{\rm 159a}$,
M. Cano Bret$^{\rm 33e}$,
J. Cantero$^{\rm 82}$,
R. Cantrill$^{\rm 126a}$,
T. Cao$^{\rm 40}$,
M.D.M. Capeans Garrido$^{\rm 30}$,
I. Caprini$^{\rm 26b}$,
M. Caprini$^{\rm 26b}$,
M. Capua$^{\rm 37a,37b}$,
R. Caputo$^{\rm 83}$,
R.M. Carbone$^{\rm 35}$,
R. Cardarelli$^{\rm 133a}$,
F. Cardillo$^{\rm 48}$,
T. Carli$^{\rm 30}$,
G. Carlino$^{\rm 104a}$,
L. Carminati$^{\rm 91a,91b}$,
S. Caron$^{\rm 106}$,
E. Carquin$^{\rm 32a}$,
G.D. Carrillo-Montoya$^{\rm 30}$,
J.R. Carter$^{\rm 28}$,
J. Carvalho$^{\rm 126a,126c}$,
D. Casadei$^{\rm 78}$,
M.P. Casado$^{\rm 12}$,
M. Casolino$^{\rm 12}$,
E. Castaneda-Miranda$^{\rm 145a}$,
A. Castelli$^{\rm 107}$,
V. Castillo Gimenez$^{\rm 167}$,
N.F. Castro$^{\rm 126a}$$^{,g}$,
P. Catastini$^{\rm 57}$,
A. Catinaccio$^{\rm 30}$,
J.R. Catmore$^{\rm 119}$,
A. Cattai$^{\rm 30}$,
J. Caudron$^{\rm 83}$,
V. Cavaliere$^{\rm 165}$,
D. Cavalli$^{\rm 91a}$,
M. Cavalli-Sforza$^{\rm 12}$,
V. Cavasinni$^{\rm 124a,124b}$,
F. Ceradini$^{\rm 134a,134b}$,
B.C. Cerio$^{\rm 45}$,
K. Cerny$^{\rm 129}$,
A.S. Cerqueira$^{\rm 24b}$,
A. Cerri$^{\rm 149}$,
L. Cerrito$^{\rm 76}$,
F. Cerutti$^{\rm 15}$,
M. Cerv$^{\rm 30}$,
A. Cervelli$^{\rm 17}$,
S.A. Cetin$^{\rm 19c}$,
A. Chafaq$^{\rm 135a}$,
D. Chakraborty$^{\rm 108}$,
I. Chalupkova$^{\rm 129}$,
Y.L. Chan$^{\rm 60a}$,
P. Chang$^{\rm 165}$,
J.D. Chapman$^{\rm 28}$,
D.G. Charlton$^{\rm 18}$,
C.C. Chau$^{\rm 158}$,
C.A. Chavez Barajas$^{\rm 149}$,
S. Cheatham$^{\rm 152}$,
A. Chegwidden$^{\rm 90}$,
S. Chekanov$^{\rm 6}$,
S.V. Chekulaev$^{\rm 159a}$,
G.A. Chelkov$^{\rm 65}$$^{,h}$,
M.A. Chelstowska$^{\rm 89}$,
C. Chen$^{\rm 64}$,
H. Chen$^{\rm 25}$,
K. Chen$^{\rm 148}$,
L. Chen$^{\rm 33d}$$^{,i}$,
S. Chen$^{\rm 33c}$,
S. Chen$^{\rm 155}$,
X. Chen$^{\rm 33f}$,
Y. Chen$^{\rm 67}$,
H.C. Cheng$^{\rm 89}$,
Y. Cheng$^{\rm 31}$,
A. Cheplakov$^{\rm 65}$,
E. Cheremushkina$^{\rm 130}$,
R. Cherkaoui El Moursli$^{\rm 135e}$,
V. Chernyatin$^{\rm 25}$$^{,*}$,
E. Cheu$^{\rm 7}$,
L. Chevalier$^{\rm 136}$,
V. Chiarella$^{\rm 47}$,
G. Chiarelli$^{\rm 124a,124b}$,
G. Chiodini$^{\rm 73a}$,
A.S. Chisholm$^{\rm 18}$,
R.T. Chislett$^{\rm 78}$,
A. Chitan$^{\rm 26b}$,
M.V. Chizhov$^{\rm 65}$,
K. Choi$^{\rm 61}$,
S. Chouridou$^{\rm 9}$,
B.K.B. Chow$^{\rm 100}$,
V. Christodoulou$^{\rm 78}$,
D. Chromek-Burckhart$^{\rm 30}$,
J. Chudoba$^{\rm 127}$,
A.J. Chuinard$^{\rm 87}$,
J.J. Chwastowski$^{\rm 39}$,
L. Chytka$^{\rm 115}$,
G. Ciapetti$^{\rm 132a,132b}$,
A.K. Ciftci$^{\rm 4a}$,
D. Cinca$^{\rm 53}$,
V. Cindro$^{\rm 75}$,
I.A. Cioara$^{\rm 21}$,
A. Ciocio$^{\rm 15}$,
F. Cirotto$^{\rm 104a,104b}$,
Z.H. Citron$^{\rm 172}$,
M. Ciubancan$^{\rm 26b}$,
A. Clark$^{\rm 49}$,
B.L. Clark$^{\rm 57}$,
P.J. Clark$^{\rm 46}$,
R.N. Clarke$^{\rm 15}$,
C. Clement$^{\rm 146a,146b}$,
Y. Coadou$^{\rm 85}$,
M. Cobal$^{\rm 164a,164c}$,
A. Coccaro$^{\rm 49}$,
J. Cochran$^{\rm 64}$,
L. Coffey$^{\rm 23}$,
J.G. Cogan$^{\rm 143}$,
L. Colasurdo$^{\rm 106}$,
B. Cole$^{\rm 35}$,
S. Cole$^{\rm 108}$,
A.P. Colijn$^{\rm 107}$,
J. Collot$^{\rm 55}$,
T. Colombo$^{\rm 58c}$,
G. Compostella$^{\rm 101}$,
P. Conde Muiño$^{\rm 126a,126b}$,
E. Coniavitis$^{\rm 48}$,
S.H. Connell$^{\rm 145b}$,
I.A. Connelly$^{\rm 77}$,
V. Consorti$^{\rm 48}$,
S. Constantinescu$^{\rm 26b}$,
C. Conta$^{\rm 121a,121b}$,
G. Conti$^{\rm 30}$,
F. Conventi$^{\rm 104a}$$^{,j}$,
M. Cooke$^{\rm 15}$,
B.D. Cooper$^{\rm 78}$,
A.M. Cooper-Sarkar$^{\rm 120}$,
T. Cornelissen$^{\rm 175}$,
M. Corradi$^{\rm 20a}$,
F. Corriveau$^{\rm 87}$$^{,k}$,
A. Corso-Radu$^{\rm 163}$,
A. Cortes-Gonzalez$^{\rm 12}$,
G. Cortiana$^{\rm 101}$,
G. Costa$^{\rm 91a}$,
M.J. Costa$^{\rm 167}$,
D. Costanzo$^{\rm 139}$,
D. Côté$^{\rm 8}$,
G. Cottin$^{\rm 28}$,
G. Cowan$^{\rm 77}$,
B.E. Cox$^{\rm 84}$,
K. Cranmer$^{\rm 110}$,
G. Cree$^{\rm 29}$,
S. Crépé-Renaudin$^{\rm 55}$,
F. Crescioli$^{\rm 80}$,
W.A. Cribbs$^{\rm 146a,146b}$,
M. Crispin Ortuzar$^{\rm 120}$,
M. Cristinziani$^{\rm 21}$,
V. Croft$^{\rm 106}$,
G. Crosetti$^{\rm 37a,37b}$,
T. Cuhadar Donszelmann$^{\rm 139}$,
J. Cummings$^{\rm 176}$,
M. Curatolo$^{\rm 47}$,
J. Cúth$^{\rm 83}$,
C. Cuthbert$^{\rm 150}$,
H. Czirr$^{\rm 141}$,
P. Czodrowski$^{\rm 3}$,
S. D'Auria$^{\rm 53}$,
M. D'Onofrio$^{\rm 74}$,
M.J. Da Cunha Sargedas De Sousa$^{\rm 126a,126b}$,
C. Da Via$^{\rm 84}$,
W. Dabrowski$^{\rm 38a}$,
A. Dafinca$^{\rm 120}$,
T. Dai$^{\rm 89}$,
O. Dale$^{\rm 14}$,
F. Dallaire$^{\rm 95}$,
C. Dallapiccola$^{\rm 86}$,
M. Dam$^{\rm 36}$,
J.R. Dandoy$^{\rm 31}$,
N.P. Dang$^{\rm 48}$,
A.C. Daniells$^{\rm 18}$,
M. Danninger$^{\rm 168}$,
M. Dano Hoffmann$^{\rm 136}$,
V. Dao$^{\rm 48}$,
G. Darbo$^{\rm 50a}$,
S. Darmora$^{\rm 8}$,
J. Dassoulas$^{\rm 3}$,
A. Dattagupta$^{\rm 61}$,
W. Davey$^{\rm 21}$,
C. David$^{\rm 169}$,
T. Davidek$^{\rm 129}$,
E. Davies$^{\rm 120}$$^{,l}$,
M. Davies$^{\rm 153}$,
P. Davison$^{\rm 78}$,
Y. Davygora$^{\rm 58a}$,
E. Dawe$^{\rm 88}$,
I. Dawson$^{\rm 139}$,
R.K. Daya-Ishmukhametova$^{\rm 86}$,
K. De$^{\rm 8}$,
R. de Asmundis$^{\rm 104a}$,
A. De Benedetti$^{\rm 113}$,
S. De Castro$^{\rm 20a,20b}$,
S. De Cecco$^{\rm 80}$,
N. De Groot$^{\rm 106}$,
P. de Jong$^{\rm 107}$,
H. De la Torre$^{\rm 82}$,
F. De Lorenzi$^{\rm 64}$,
D. De Pedis$^{\rm 132a}$,
A. De Salvo$^{\rm 132a}$,
U. De Sanctis$^{\rm 149}$,
A. De Santo$^{\rm 149}$,
J.B. De Vivie De Regie$^{\rm 117}$,
W.J. Dearnaley$^{\rm 72}$,
R. Debbe$^{\rm 25}$,
C. Debenedetti$^{\rm 137}$,
D.V. Dedovich$^{\rm 65}$,
I. Deigaard$^{\rm 107}$,
J. Del Peso$^{\rm 82}$,
T. Del Prete$^{\rm 124a,124b}$,
D. Delgove$^{\rm 117}$,
F. Deliot$^{\rm 136}$,
C.M. Delitzsch$^{\rm 49}$,
M. Deliyergiyev$^{\rm 75}$,
A. Dell'Acqua$^{\rm 30}$,
L. Dell'Asta$^{\rm 22}$,
M. Dell'Orso$^{\rm 124a,124b}$,
M. Della Pietra$^{\rm 104a}$$^{,j}$,
D. della Volpe$^{\rm 49}$,
M. Delmastro$^{\rm 5}$,
P.A. Delsart$^{\rm 55}$,
C. Deluca$^{\rm 107}$,
D.A. DeMarco$^{\rm 158}$,
S. Demers$^{\rm 176}$,
M. Demichev$^{\rm 65}$,
A. Demilly$^{\rm 80}$,
S.P. Denisov$^{\rm 130}$,
D. Derendarz$^{\rm 39}$,
J.E. Derkaoui$^{\rm 135d}$,
F. Derue$^{\rm 80}$,
P. Dervan$^{\rm 74}$,
K. Desch$^{\rm 21}$,
C. Deterre$^{\rm 42}$,
K. Dette$^{\rm 43}$,
P.O. Deviveiros$^{\rm 30}$,
A. Dewhurst$^{\rm 131}$,
S. Dhaliwal$^{\rm 23}$,
A. Di Ciaccio$^{\rm 133a,133b}$,
L. Di Ciaccio$^{\rm 5}$,
A. Di Domenico$^{\rm 132a,132b}$,
C. Di Donato$^{\rm 104a,104b}$,
A. Di Girolamo$^{\rm 30}$,
B. Di Girolamo$^{\rm 30}$,
A. Di Mattia$^{\rm 152}$,
B. Di Micco$^{\rm 134a,134b}$,
R. Di Nardo$^{\rm 47}$,
A. Di Simone$^{\rm 48}$,
R. Di Sipio$^{\rm 158}$,
D. Di Valentino$^{\rm 29}$,
C. Diaconu$^{\rm 85}$,
M. Diamond$^{\rm 158}$,
F.A. Dias$^{\rm 46}$,
M.A. Diaz$^{\rm 32a}$,
E.B. Diehl$^{\rm 89}$,
J. Dietrich$^{\rm 16}$,
S. Diglio$^{\rm 85}$,
A. Dimitrievska$^{\rm 13}$,
J. Dingfelder$^{\rm 21}$,
P. Dita$^{\rm 26b}$,
S. Dita$^{\rm 26b}$,
F. Dittus$^{\rm 30}$,
F. Djama$^{\rm 85}$,
T. Djobava$^{\rm 51b}$,
J.I. Djuvsland$^{\rm 58a}$,
M.A.B. do Vale$^{\rm 24c}$,
D. Dobos$^{\rm 30}$,
M. Dobre$^{\rm 26b}$,
C. Doglioni$^{\rm 81}$,
T. Dohmae$^{\rm 155}$,
J. Dolejsi$^{\rm 129}$,
Z. Dolezal$^{\rm 129}$,
B.A. Dolgoshein$^{\rm 98}$$^{,*}$,
M. Donadelli$^{\rm 24d}$,
S. Donati$^{\rm 124a,124b}$,
P. Dondero$^{\rm 121a,121b}$,
J. Donini$^{\rm 34}$,
J. Dopke$^{\rm 131}$,
A. Doria$^{\rm 104a}$,
M.T. Dova$^{\rm 71}$,
A.T. Doyle$^{\rm 53}$,
E. Drechsler$^{\rm 54}$,
M. Dris$^{\rm 10}$,
E. Dubreuil$^{\rm 34}$,
E. Duchovni$^{\rm 172}$,
G. Duckeck$^{\rm 100}$,
O.A. Ducu$^{\rm 26b,85}$,
D. Duda$^{\rm 107}$,
A. Dudarev$^{\rm 30}$,
L. Duflot$^{\rm 117}$,
L. Duguid$^{\rm 77}$,
M. Dührssen$^{\rm 30}$,
M. Dunford$^{\rm 58a}$,
H. Duran Yildiz$^{\rm 4a}$,
M. Düren$^{\rm 52}$,
A. Durglishvili$^{\rm 51b}$,
D. Duschinger$^{\rm 44}$,
B. Dutta$^{\rm 42}$,
M. Dyndal$^{\rm 38a}$,
C. Eckardt$^{\rm 42}$,
K.M. Ecker$^{\rm 101}$,
R.C. Edgar$^{\rm 89}$,
W. Edson$^{\rm 2}$,
N.C. Edwards$^{\rm 46}$,
W. Ehrenfeld$^{\rm 21}$,
T. Eifert$^{\rm 30}$,
G. Eigen$^{\rm 14}$,
K. Einsweiler$^{\rm 15}$,
T. Ekelof$^{\rm 166}$,
M. El Kacimi$^{\rm 135c}$,
M. Ellert$^{\rm 166}$,
S. Elles$^{\rm 5}$,
F. Ellinghaus$^{\rm 175}$,
A.A. Elliot$^{\rm 169}$,
N. Ellis$^{\rm 30}$,
J. Elmsheuser$^{\rm 100}$,
M. Elsing$^{\rm 30}$,
D. Emeliyanov$^{\rm 131}$,
Y. Enari$^{\rm 155}$,
O.C. Endner$^{\rm 83}$,
M. Endo$^{\rm 118}$,
J. Erdmann$^{\rm 43}$,
A. Ereditato$^{\rm 17}$,
G. Ernis$^{\rm 175}$,
J. Ernst$^{\rm 2}$,
M. Ernst$^{\rm 25}$,
S. Errede$^{\rm 165}$,
E. Ertel$^{\rm 83}$,
M. Escalier$^{\rm 117}$,
H. Esch$^{\rm 43}$,
C. Escobar$^{\rm 125}$,
B. Esposito$^{\rm 47}$,
A.I. Etienvre$^{\rm 136}$,
E. Etzion$^{\rm 153}$,
H. Evans$^{\rm 61}$,
A. Ezhilov$^{\rm 123}$,
L. Fabbri$^{\rm 20a,20b}$,
G. Facini$^{\rm 31}$,
R.M. Fakhrutdinov$^{\rm 130}$,
S. Falciano$^{\rm 132a}$,
R.J. Falla$^{\rm 78}$,
J. Faltova$^{\rm 129}$,
Y. Fang$^{\rm 33a}$,
M. Fanti$^{\rm 91a,91b}$,
A. Farbin$^{\rm 8}$,
A. Farilla$^{\rm 134a}$,
T. Farooque$^{\rm 12}$,
S. Farrell$^{\rm 15}$,
S.M. Farrington$^{\rm 170}$,
P. Farthouat$^{\rm 30}$,
F. Fassi$^{\rm 135e}$,
P. Fassnacht$^{\rm 30}$,
D. Fassouliotis$^{\rm 9}$,
M. Faucci Giannelli$^{\rm 77}$,
A. Favareto$^{\rm 50a,50b}$,
L. Fayard$^{\rm 117}$,
O.L. Fedin$^{\rm 123}$$^{,m}$,
W. Fedorko$^{\rm 168}$,
S. Feigl$^{\rm 30}$,
L. Feligioni$^{\rm 85}$,
C. Feng$^{\rm 33d}$,
E.J. Feng$^{\rm 30}$,
H. Feng$^{\rm 89}$,
A.B. Fenyuk$^{\rm 130}$,
L. Feremenga$^{\rm 8}$,
P. Fernandez Martinez$^{\rm 167}$,
S. Fernandez Perez$^{\rm 30}$,
J. Ferrando$^{\rm 53}$,
A. Ferrari$^{\rm 166}$,
P. Ferrari$^{\rm 107}$,
R. Ferrari$^{\rm 121a}$,
D.E. Ferreira de Lima$^{\rm 53}$,
A. Ferrer$^{\rm 167}$,
D. Ferrere$^{\rm 49}$,
C. Ferretti$^{\rm 89}$,
A. Ferretto Parodi$^{\rm 50a,50b}$,
M. Fiascaris$^{\rm 31}$,
F. Fiedler$^{\rm 83}$,
A. Filipčič$^{\rm 75}$,
M. Filipuzzi$^{\rm 42}$,
F. Filthaut$^{\rm 106}$,
M. Fincke-Keeler$^{\rm 169}$,
K.D. Finelli$^{\rm 150}$,
M.C.N. Fiolhais$^{\rm 126a,126c}$,
L. Fiorini$^{\rm 167}$,
A. Firan$^{\rm 40}$,
A. Fischer$^{\rm 2}$,
C. Fischer$^{\rm 12}$,
J. Fischer$^{\rm 175}$,
W.C. Fisher$^{\rm 90}$,
N. Flaschel$^{\rm 42}$,
I. Fleck$^{\rm 141}$,
P. Fleischmann$^{\rm 89}$,
G.T. Fletcher$^{\rm 139}$,
G. Fletcher$^{\rm 76}$,
R.R.M. Fletcher$^{\rm 122}$,
T. Flick$^{\rm 175}$,
A. Floderus$^{\rm 81}$,
L.R. Flores Castillo$^{\rm 60a}$,
M.J. Flowerdew$^{\rm 101}$,
A. Formica$^{\rm 136}$,
A. Forti$^{\rm 84}$,
D. Fournier$^{\rm 117}$,
H. Fox$^{\rm 72}$,
S. Fracchia$^{\rm 12}$,
P. Francavilla$^{\rm 80}$,
M. Franchini$^{\rm 20a,20b}$,
D. Francis$^{\rm 30}$,
L. Franconi$^{\rm 119}$,
M. Franklin$^{\rm 57}$,
M. Frate$^{\rm 163}$,
M. Fraternali$^{\rm 121a,121b}$,
D. Freeborn$^{\rm 78}$,
S.T. French$^{\rm 28}$,
F. Friedrich$^{\rm 44}$,
D. Froidevaux$^{\rm 30}$,
J.A. Frost$^{\rm 120}$,
C. Fukunaga$^{\rm 156}$,
E. Fullana Torregrosa$^{\rm 83}$,
B.G. Fulsom$^{\rm 143}$,
T. Fusayasu$^{\rm 102}$,
J. Fuster$^{\rm 167}$,
C. Gabaldon$^{\rm 55}$,
O. Gabizon$^{\rm 175}$,
A. Gabrielli$^{\rm 20a,20b}$,
A. Gabrielli$^{\rm 15}$,
G.P. Gach$^{\rm 18}$,
S. Gadatsch$^{\rm 30}$,
S. Gadomski$^{\rm 49}$,
G. Gagliardi$^{\rm 50a,50b}$,
P. Gagnon$^{\rm 61}$,
C. Galea$^{\rm 106}$,
B. Galhardo$^{\rm 126a,126c}$,
E.J. Gallas$^{\rm 120}$,
B.J. Gallop$^{\rm 131}$,
P. Gallus$^{\rm 128}$,
G. Galster$^{\rm 36}$,
K.K. Gan$^{\rm 111}$,
J. Gao$^{\rm 33b,85}$,
Y. Gao$^{\rm 46}$,
Y.S. Gao$^{\rm 143}$$^{,e}$,
F.M. Garay Walls$^{\rm 46}$,
F. Garberson$^{\rm 176}$,
C. García$^{\rm 167}$,
J.E. García Navarro$^{\rm 167}$,
M. Garcia-Sciveres$^{\rm 15}$,
R.W. Gardner$^{\rm 31}$,
N. Garelli$^{\rm 143}$,
V. Garonne$^{\rm 119}$,
C. Gatti$^{\rm 47}$,
A. Gaudiello$^{\rm 50a,50b}$,
G. Gaudio$^{\rm 121a}$,
B. Gaur$^{\rm 141}$,
L. Gauthier$^{\rm 95}$,
P. Gauzzi$^{\rm 132a,132b}$,
I.L. Gavrilenko$^{\rm 96}$,
C. Gay$^{\rm 168}$,
G. Gaycken$^{\rm 21}$,
E.N. Gazis$^{\rm 10}$,
P. Ge$^{\rm 33d}$,
Z. Gecse$^{\rm 168}$,
C.N.P. Gee$^{\rm 131}$,
Ch. Geich-Gimbel$^{\rm 21}$,
M.P. Geisler$^{\rm 58a}$,
C. Gemme$^{\rm 50a}$,
M.H. Genest$^{\rm 55}$,
S. Gentile$^{\rm 132a,132b}$,
M. George$^{\rm 54}$,
S. George$^{\rm 77}$,
D. Gerbaudo$^{\rm 163}$,
A. Gershon$^{\rm 153}$,
S. Ghasemi$^{\rm 141}$,
H. Ghazlane$^{\rm 135b}$,
B. Giacobbe$^{\rm 20a}$,
S. Giagu$^{\rm 132a,132b}$,
V. Giangiobbe$^{\rm 12}$,
P. Giannetti$^{\rm 124a,124b}$,
B. Gibbard$^{\rm 25}$,
S.M. Gibson$^{\rm 77}$,
M. Gignac$^{\rm 168}$,
M. Gilchriese$^{\rm 15}$,
T.P.S. Gillam$^{\rm 28}$,
D. Gillberg$^{\rm 30}$,
G. Gilles$^{\rm 34}$,
D.M. Gingrich$^{\rm 3}$$^{,d}$,
N. Giokaris$^{\rm 9}$,
M.P. Giordani$^{\rm 164a,164c}$,
F.M. Giorgi$^{\rm 20a}$,
F.M. Giorgi$^{\rm 16}$,
P.F. Giraud$^{\rm 136}$,
P. Giromini$^{\rm 47}$,
D. Giugni$^{\rm 91a}$,
C. Giuliani$^{\rm 101}$,
M. Giulini$^{\rm 58b}$,
B.K. Gjelsten$^{\rm 119}$,
S. Gkaitatzis$^{\rm 154}$,
I. Gkialas$^{\rm 154}$,
E.L. Gkougkousis$^{\rm 117}$,
L.K. Gladilin$^{\rm 99}$,
C. Glasman$^{\rm 82}$,
J. Glatzer$^{\rm 30}$,
P.C.F. Glaysher$^{\rm 46}$,
A. Glazov$^{\rm 42}$,
M. Goblirsch-Kolb$^{\rm 101}$,
J.R. Goddard$^{\rm 76}$,
J. Godlewski$^{\rm 39}$,
S. Goldfarb$^{\rm 89}$,
T. Golling$^{\rm 49}$,
D. Golubkov$^{\rm 130}$,
A. Gomes$^{\rm 126a,126b,126d}$,
R. Gonçalo$^{\rm 126a}$,
J. Goncalves Pinto Firmino Da Costa$^{\rm 136}$,
L. Gonella$^{\rm 21}$,
S. González de la Hoz$^{\rm 167}$,
G. Gonzalez Parra$^{\rm 12}$,
S. Gonzalez-Sevilla$^{\rm 49}$,
L. Goossens$^{\rm 30}$,
P.A. Gorbounov$^{\rm 97}$,
H.A. Gordon$^{\rm 25}$,
I. Gorelov$^{\rm 105}$,
B. Gorini$^{\rm 30}$,
E. Gorini$^{\rm 73a,73b}$,
A. Gorišek$^{\rm 75}$,
E. Gornicki$^{\rm 39}$,
A.T. Goshaw$^{\rm 45}$,
C. Gössling$^{\rm 43}$,
M.I. Gostkin$^{\rm 65}$,
D. Goujdami$^{\rm 135c}$,
A.G. Goussiou$^{\rm 138}$,
N. Govender$^{\rm 145b}$,
E. Gozani$^{\rm 152}$,
H.M.X. Grabas$^{\rm 137}$,
L. Graber$^{\rm 54}$,
I. Grabowska-Bold$^{\rm 38a}$,
P.O.J. Gradin$^{\rm 166}$,
P. Grafström$^{\rm 20a,20b}$,
K-J. Grahn$^{\rm 42}$,
J. Gramling$^{\rm 49}$,
E. Gramstad$^{\rm 119}$,
S. Grancagnolo$^{\rm 16}$,
V. Gratchev$^{\rm 123}$,
H.M. Gray$^{\rm 30}$,
E. Graziani$^{\rm 134a}$,
Z.D. Greenwood$^{\rm 79}$$^{,n}$,
C. Grefe$^{\rm 21}$,
K. Gregersen$^{\rm 78}$,
I.M. Gregor$^{\rm 42}$,
P. Grenier$^{\rm 143}$,
J. Griffiths$^{\rm 8}$,
A.A. Grillo$^{\rm 137}$,
K. Grimm$^{\rm 72}$,
S. Grinstein$^{\rm 12}$$^{,o}$,
Ph. Gris$^{\rm 34}$,
J.-F. Grivaz$^{\rm 117}$,
J.P. Grohs$^{\rm 44}$,
A. Grohsjean$^{\rm 42}$,
E. Gross$^{\rm 172}$,
J. Grosse-Knetter$^{\rm 54}$,
G.C. Grossi$^{\rm 79}$,
Z.J. Grout$^{\rm 149}$,
L. Guan$^{\rm 89}$,
J. Guenther$^{\rm 128}$,
F. Guescini$^{\rm 49}$,
D. Guest$^{\rm 163}$,
O. Gueta$^{\rm 153}$,
E. Guido$^{\rm 50a,50b}$,
T. Guillemin$^{\rm 117}$,
S. Guindon$^{\rm 2}$,
U. Gul$^{\rm 53}$,
C. Gumpert$^{\rm 44}$,
J. Guo$^{\rm 33e}$,
Y. Guo$^{\rm 33b}$$^{,p}$,
S. Gupta$^{\rm 120}$,
G. Gustavino$^{\rm 132a,132b}$,
P. Gutierrez$^{\rm 113}$,
N.G. Gutierrez Ortiz$^{\rm 78}$,
C. Gutschow$^{\rm 44}$,
C. Guyot$^{\rm 136}$,
C. Gwenlan$^{\rm 120}$,
C.B. Gwilliam$^{\rm 74}$,
A. Haas$^{\rm 110}$,
C. Haber$^{\rm 15}$,
H.K. Hadavand$^{\rm 8}$,
N. Haddad$^{\rm 135e}$,
P. Haefner$^{\rm 21}$,
S. Hageböck$^{\rm 21}$,
Z. Hajduk$^{\rm 39}$,
H. Hakobyan$^{\rm 177}$,
M. Haleem$^{\rm 42}$,
J. Haley$^{\rm 114}$,
D. Hall$^{\rm 120}$,
G. Halladjian$^{\rm 90}$,
G.D. Hallewell$^{\rm 85}$,
K. Hamacher$^{\rm 175}$,
P. Hamal$^{\rm 115}$,
K. Hamano$^{\rm 169}$,
A. Hamilton$^{\rm 145a}$,
G.N. Hamity$^{\rm 139}$,
P.G. Hamnett$^{\rm 42}$,
L. Han$^{\rm 33b}$,
K. Hanagaki$^{\rm 66}$$^{,q}$,
K. Hanawa$^{\rm 155}$,
M. Hance$^{\rm 137}$,
B. Haney$^{\rm 122}$,
P. Hanke$^{\rm 58a}$,
R. Hanna$^{\rm 136}$,
J.B. Hansen$^{\rm 36}$,
J.D. Hansen$^{\rm 36}$,
M.C. Hansen$^{\rm 21}$,
P.H. Hansen$^{\rm 36}$,
K. Hara$^{\rm 160}$,
A.S. Hard$^{\rm 173}$,
T. Harenberg$^{\rm 175}$,
F. Hariri$^{\rm 117}$,
S. Harkusha$^{\rm 92}$,
R.D. Harrington$^{\rm 46}$,
P.F. Harrison$^{\rm 170}$,
F. Hartjes$^{\rm 107}$,
M. Hasegawa$^{\rm 67}$,
Y. Hasegawa$^{\rm 140}$,
A. Hasib$^{\rm 113}$,
S. Hassani$^{\rm 136}$,
S. Haug$^{\rm 17}$,
R. Hauser$^{\rm 90}$,
L. Hauswald$^{\rm 44}$,
M. Havranek$^{\rm 127}$,
C.M. Hawkes$^{\rm 18}$,
R.J. Hawkings$^{\rm 30}$,
A.D. Hawkins$^{\rm 81}$,
T. Hayashi$^{\rm 160}$,
D. Hayden$^{\rm 90}$,
C.P. Hays$^{\rm 120}$,
J.M. Hays$^{\rm 76}$,
H.S. Hayward$^{\rm 74}$,
S.J. Haywood$^{\rm 131}$,
S.J. Head$^{\rm 18}$,
T. Heck$^{\rm 83}$,
V. Hedberg$^{\rm 81}$,
L. Heelan$^{\rm 8}$,
S. Heim$^{\rm 122}$,
T. Heim$^{\rm 175}$,
B. Heinemann$^{\rm 15}$,
L. Heinrich$^{\rm 110}$,
J. Hejbal$^{\rm 127}$,
L. Helary$^{\rm 22}$,
S. Hellman$^{\rm 146a,146b}$,
D. Hellmich$^{\rm 21}$,
C. Helsens$^{\rm 12}$,
J. Henderson$^{\rm 120}$,
R.C.W. Henderson$^{\rm 72}$,
Y. Heng$^{\rm 173}$,
C. Hengler$^{\rm 42}$,
S. Henkelmann$^{\rm 168}$,
A. Henrichs$^{\rm 176}$,
A.M. Henriques Correia$^{\rm 30}$,
S. Henrot-Versille$^{\rm 117}$,
G.H. Herbert$^{\rm 16}$,
Y. Hernández Jiménez$^{\rm 167}$,
G. Herten$^{\rm 48}$,
R. Hertenberger$^{\rm 100}$,
L. Hervas$^{\rm 30}$,
G.G. Hesketh$^{\rm 78}$,
N.P. Hessey$^{\rm 107}$,
J.W. Hetherly$^{\rm 40}$,
R. Hickling$^{\rm 76}$,
E. Higón-Rodriguez$^{\rm 167}$,
E. Hill$^{\rm 169}$,
J.C. Hill$^{\rm 28}$,
K.H. Hiller$^{\rm 42}$,
S.J. Hillier$^{\rm 18}$,
I. Hinchliffe$^{\rm 15}$,
E. Hines$^{\rm 122}$,
R.R. Hinman$^{\rm 15}$,
M. Hirose$^{\rm 157}$,
D. Hirschbuehl$^{\rm 175}$,
J. Hobbs$^{\rm 148}$,
N. Hod$^{\rm 107}$,
M.C. Hodgkinson$^{\rm 139}$,
P. Hodgson$^{\rm 139}$,
A. Hoecker$^{\rm 30}$,
M.R. Hoeferkamp$^{\rm 105}$,
F. Hoenig$^{\rm 100}$,
M. Hohlfeld$^{\rm 83}$,
D. Hohn$^{\rm 21}$,
T.R. Holmes$^{\rm 15}$,
M. Homann$^{\rm 43}$,
T.M. Hong$^{\rm 125}$,
W.H. Hopkins$^{\rm 116}$,
Y. Horii$^{\rm 103}$,
A.J. Horton$^{\rm 142}$,
J-Y. Hostachy$^{\rm 55}$,
S. Hou$^{\rm 151}$,
A. Hoummada$^{\rm 135a}$,
J. Howard$^{\rm 120}$,
J. Howarth$^{\rm 42}$,
M. Hrabovsky$^{\rm 115}$,
I. Hristova$^{\rm 16}$,
J. Hrivnac$^{\rm 117}$,
T. Hryn'ova$^{\rm 5}$,
A. Hrynevich$^{\rm 93}$,
C. Hsu$^{\rm 145c}$,
P.J. Hsu$^{\rm 151}$$^{,r}$,
S.-C. Hsu$^{\rm 138}$,
D. Hu$^{\rm 35}$,
Q. Hu$^{\rm 33b}$,
X. Hu$^{\rm 89}$,
Y. Huang$^{\rm 42}$,
Z. Hubacek$^{\rm 128}$,
F. Hubaut$^{\rm 85}$,
F. Huegging$^{\rm 21}$,
T.B. Huffman$^{\rm 120}$,
E.W. Hughes$^{\rm 35}$,
G. Hughes$^{\rm 72}$,
M. Huhtinen$^{\rm 30}$,
T.A. Hülsing$^{\rm 83}$,
N. Huseynov$^{\rm 65}$$^{,b}$,
J. Huston$^{\rm 90}$,
J. Huth$^{\rm 57}$,
G. Iacobucci$^{\rm 49}$,
G. Iakovidis$^{\rm 25}$,
I. Ibragimov$^{\rm 141}$,
L. Iconomidou-Fayard$^{\rm 117}$,
E. Ideal$^{\rm 176}$,
Z. Idrissi$^{\rm 135e}$,
P. Iengo$^{\rm 30}$,
O. Igonkina$^{\rm 107}$,
T. Iizawa$^{\rm 171}$,
Y. Ikegami$^{\rm 66}$,
K. Ikematsu$^{\rm 141}$,
M. Ikeno$^{\rm 66}$,
Y. Ilchenko$^{\rm 31}$$^{,s}$,
D. Iliadis$^{\rm 154}$,
N. Ilic$^{\rm 143}$,
T. Ince$^{\rm 101}$,
G. Introzzi$^{\rm 121a,121b}$,
P. Ioannou$^{\rm 9}$,
M. Iodice$^{\rm 134a}$,
K. Iordanidou$^{\rm 35}$,
V. Ippolito$^{\rm 57}$,
A. Irles Quiles$^{\rm 167}$,
C. Isaksson$^{\rm 166}$,
M. Ishino$^{\rm 68}$,
M. Ishitsuka$^{\rm 157}$,
R. Ishmukhametov$^{\rm 111}$,
C. Issever$^{\rm 120}$,
S. Istin$^{\rm 19a}$,
J.M. Iturbe Ponce$^{\rm 84}$,
R. Iuppa$^{\rm 133a,133b}$,
J. Ivarsson$^{\rm 81}$,
W. Iwanski$^{\rm 39}$,
H. Iwasaki$^{\rm 66}$,
J.M. Izen$^{\rm 41}$,
V. Izzo$^{\rm 104a}$,
S. Jabbar$^{\rm 3}$,
B. Jackson$^{\rm 122}$,
M. Jackson$^{\rm 74}$,
P. Jackson$^{\rm 1}$,
M.R. Jaekel$^{\rm 30}$,
V. Jain$^{\rm 2}$,
K. Jakobs$^{\rm 48}$,
S. Jakobsen$^{\rm 30}$,
T. Jakoubek$^{\rm 127}$,
J. Jakubek$^{\rm 128}$,
D.O. Jamin$^{\rm 114}$,
D.K. Jana$^{\rm 79}$,
E. Jansen$^{\rm 78}$,
R. Jansky$^{\rm 62}$,
J. Janssen$^{\rm 21}$,
M. Janus$^{\rm 54}$,
G. Jarlskog$^{\rm 81}$,
N. Javadov$^{\rm 65}$$^{,b}$,
T. Javůrek$^{\rm 48}$,
L. Jeanty$^{\rm 15}$,
J. Jejelava$^{\rm 51a}$$^{,t}$,
G.-Y. Jeng$^{\rm 150}$,
D. Jennens$^{\rm 88}$,
P. Jenni$^{\rm 48}$$^{,u}$,
J. Jentzsch$^{\rm 43}$,
C. Jeske$^{\rm 170}$,
S. Jézéquel$^{\rm 5}$,
H. Ji$^{\rm 173}$,
J. Jia$^{\rm 148}$,
Y. Jiang$^{\rm 33b}$,
S. Jiggins$^{\rm 78}$,
J. Jimenez Pena$^{\rm 167}$,
S. Jin$^{\rm 33a}$,
A. Jinaru$^{\rm 26b}$,
O. Jinnouchi$^{\rm 157}$,
M.D. Joergensen$^{\rm 36}$,
P. Johansson$^{\rm 139}$,
K.A. Johns$^{\rm 7}$,
W.J. Johnson$^{\rm 138}$,
K. Jon-And$^{\rm 146a,146b}$,
G. Jones$^{\rm 170}$,
R.W.L. Jones$^{\rm 72}$,
T.J. Jones$^{\rm 74}$,
J. Jongmanns$^{\rm 58a}$,
P.M. Jorge$^{\rm 126a,126b}$,
K.D. Joshi$^{\rm 84}$,
J. Jovicevic$^{\rm 159a}$,
X. Ju$^{\rm 173}$,
P. Jussel$^{\rm 62}$,
A. Juste Rozas$^{\rm 12}$$^{,o}$,
M. Kaci$^{\rm 167}$,
A. Kaczmarska$^{\rm 39}$,
M. Kado$^{\rm 117}$,
H. Kagan$^{\rm 111}$,
M. Kagan$^{\rm 143}$,
S.J. Kahn$^{\rm 85}$,
E. Kajomovitz$^{\rm 45}$,
C.W. Kalderon$^{\rm 120}$,
S. Kama$^{\rm 40}$,
A. Kamenshchikov$^{\rm 130}$,
N. Kanaya$^{\rm 155}$,
S. Kaneti$^{\rm 28}$,
V.A. Kantserov$^{\rm 98}$,
J. Kanzaki$^{\rm 66}$,
B. Kaplan$^{\rm 110}$,
L.S. Kaplan$^{\rm 173}$,
A. Kapliy$^{\rm 31}$,
D. Kar$^{\rm 145c}$,
K. Karakostas$^{\rm 10}$,
A. Karamaoun$^{\rm 3}$,
N. Karastathis$^{\rm 10,107}$,
M.J. Kareem$^{\rm 54}$,
E. Karentzos$^{\rm 10}$,
M. Karnevskiy$^{\rm 83}$,
S.N. Karpov$^{\rm 65}$,
Z.M. Karpova$^{\rm 65}$,
K. Karthik$^{\rm 110}$,
V. Kartvelishvili$^{\rm 72}$,
A.N. Karyukhin$^{\rm 130}$,
K. Kasahara$^{\rm 160}$,
L. Kashif$^{\rm 173}$,
R.D. Kass$^{\rm 111}$,
A. Kastanas$^{\rm 14}$,
Y. Kataoka$^{\rm 155}$,
C. Kato$^{\rm 155}$,
A. Katre$^{\rm 49}$,
J. Katzy$^{\rm 42}$,
K. Kawade$^{\rm 103}$,
K. Kawagoe$^{\rm 70}$,
T. Kawamoto$^{\rm 155}$,
G. Kawamura$^{\rm 54}$,
S. Kazama$^{\rm 155}$,
V.F. Kazanin$^{\rm 109}$$^{,c}$,
R. Keeler$^{\rm 169}$,
R. Kehoe$^{\rm 40}$,
J.S. Keller$^{\rm 42}$,
J.J. Kempster$^{\rm 77}$,
H. Keoshkerian$^{\rm 84}$,
O. Kepka$^{\rm 127}$,
B.P. Kerševan$^{\rm 75}$,
S. Kersten$^{\rm 175}$,
R.A. Keyes$^{\rm 87}$,
F. Khalil-zada$^{\rm 11}$,
H. Khandanyan$^{\rm 146a,146b}$,
A. Khanov$^{\rm 114}$,
A.G. Kharlamov$^{\rm 109}$$^{,c}$,
T.J. Khoo$^{\rm 28}$,
V. Khovanskiy$^{\rm 97}$,
E. Khramov$^{\rm 65}$,
J. Khubua$^{\rm 51b}$$^{,v}$,
S. Kido$^{\rm 67}$,
H.Y. Kim$^{\rm 8}$,
S.H. Kim$^{\rm 160}$,
Y.K. Kim$^{\rm 31}$,
N. Kimura$^{\rm 154}$,
O.M. Kind$^{\rm 16}$,
B.T. King$^{\rm 74}$,
M. King$^{\rm 167}$,
S.B. King$^{\rm 168}$,
J. Kirk$^{\rm 131}$,
A.E. Kiryunin$^{\rm 101}$,
T. Kishimoto$^{\rm 67}$,
D. Kisielewska$^{\rm 38a}$,
F. Kiss$^{\rm 48}$,
K. Kiuchi$^{\rm 160}$,
O. Kivernyk$^{\rm 136}$,
E. Kladiva$^{\rm 144b}$,
M.H. Klein$^{\rm 35}$,
M. Klein$^{\rm 74}$,
U. Klein$^{\rm 74}$,
K. Kleinknecht$^{\rm 83}$,
P. Klimek$^{\rm 146a,146b}$,
A. Klimentov$^{\rm 25}$,
R. Klingenberg$^{\rm 43}$,
J.A. Klinger$^{\rm 139}$,
T. Klioutchnikova$^{\rm 30}$,
E.-E. Kluge$^{\rm 58a}$,
P. Kluit$^{\rm 107}$,
S. Kluth$^{\rm 101}$,
J. Knapik$^{\rm 39}$,
E. Kneringer$^{\rm 62}$,
E.B.F.G. Knoops$^{\rm 85}$,
A. Knue$^{\rm 53}$,
A. Kobayashi$^{\rm 155}$,
D. Kobayashi$^{\rm 157}$,
T. Kobayashi$^{\rm 155}$,
M. Kobel$^{\rm 44}$,
M. Kocian$^{\rm 143}$,
P. Kodys$^{\rm 129}$,
T. Koffas$^{\rm 29}$,
E. Koffeman$^{\rm 107}$,
L.A. Kogan$^{\rm 120}$,
S. Kohlmann$^{\rm 175}$,
Z. Kohout$^{\rm 128}$,
T. Kohriki$^{\rm 66}$,
T. Koi$^{\rm 143}$,
H. Kolanoski$^{\rm 16}$,
M. Kolb$^{\rm 58b}$,
I. Koletsou$^{\rm 5}$,
A.A. Komar$^{\rm 96}$$^{,*}$,
Y. Komori$^{\rm 155}$,
T. Kondo$^{\rm 66}$,
N. Kondrashova$^{\rm 42}$,
K. Köneke$^{\rm 48}$,
A.C. König$^{\rm 106}$,
T. Kono$^{\rm 66}$,
R. Konoplich$^{\rm 110}$$^{,w}$,
N. Konstantinidis$^{\rm 78}$,
R. Kopeliansky$^{\rm 152}$,
S. Koperny$^{\rm 38a}$,
L. Köpke$^{\rm 83}$,
A.K. Kopp$^{\rm 48}$,
K. Korcyl$^{\rm 39}$,
K. Kordas$^{\rm 154}$,
A. Korn$^{\rm 78}$,
A.A. Korol$^{\rm 109}$$^{,c}$,
I. Korolkov$^{\rm 12}$,
E.V. Korolkova$^{\rm 139}$,
O. Kortner$^{\rm 101}$,
S. Kortner$^{\rm 101}$,
T. Kosek$^{\rm 129}$,
V.V. Kostyukhin$^{\rm 21}$,
V.M. Kotov$^{\rm 65}$,
A. Kotwal$^{\rm 45}$,
A. Kourkoumeli-Charalampidi$^{\rm 154}$,
C. Kourkoumelis$^{\rm 9}$,
V. Kouskoura$^{\rm 25}$,
A. Koutsman$^{\rm 159a}$,
R. Kowalewski$^{\rm 169}$,
T.Z. Kowalski$^{\rm 38a}$,
W. Kozanecki$^{\rm 136}$,
A.S. Kozhin$^{\rm 130}$,
V.A. Kramarenko$^{\rm 99}$,
G. Kramberger$^{\rm 75}$,
D. Krasnopevtsev$^{\rm 98}$,
M.W. Krasny$^{\rm 80}$,
A. Krasznahorkay$^{\rm 30}$,
J.K. Kraus$^{\rm 21}$,
A. Kravchenko$^{\rm 25}$,
S. Kreiss$^{\rm 110}$,
M. Kretz$^{\rm 58c}$,
J. Kretzschmar$^{\rm 74}$,
K. Kreutzfeldt$^{\rm 52}$,
P. Krieger$^{\rm 158}$,
K. Krizka$^{\rm 31}$,
K. Kroeninger$^{\rm 43}$,
H. Kroha$^{\rm 101}$,
J. Kroll$^{\rm 122}$,
J. Kroseberg$^{\rm 21}$,
J. Krstic$^{\rm 13}$,
U. Kruchonak$^{\rm 65}$,
H. Krüger$^{\rm 21}$,
N. Krumnack$^{\rm 64}$,
A. Kruse$^{\rm 173}$,
M.C. Kruse$^{\rm 45}$,
M. Kruskal$^{\rm 22}$,
T. Kubota$^{\rm 88}$,
H. Kucuk$^{\rm 78}$,
S. Kuday$^{\rm 4b}$,
S. Kuehn$^{\rm 48}$,
A. Kugel$^{\rm 58c}$,
F. Kuger$^{\rm 174}$,
A. Kuhl$^{\rm 137}$,
T. Kuhl$^{\rm 42}$,
V. Kukhtin$^{\rm 65}$,
R. Kukla$^{\rm 136}$,
Y. Kulchitsky$^{\rm 92}$,
S. Kuleshov$^{\rm 32b}$,
M. Kuna$^{\rm 132a,132b}$,
T. Kunigo$^{\rm 68}$,
A. Kupco$^{\rm 127}$,
H. Kurashige$^{\rm 67}$,
Y.A. Kurochkin$^{\rm 92}$,
V. Kus$^{\rm 127}$,
E.S. Kuwertz$^{\rm 169}$,
M. Kuze$^{\rm 157}$,
J. Kvita$^{\rm 115}$,
T. Kwan$^{\rm 169}$,
D. Kyriazopoulos$^{\rm 139}$,
A. La Rosa$^{\rm 137}$,
J.L. La Rosa Navarro$^{\rm 24d}$,
L. La Rotonda$^{\rm 37a,37b}$,
C. Lacasta$^{\rm 167}$,
F. Lacava$^{\rm 132a,132b}$,
J. Lacey$^{\rm 29}$,
H. Lacker$^{\rm 16}$,
D. Lacour$^{\rm 80}$,
V.R. Lacuesta$^{\rm 167}$,
E. Ladygin$^{\rm 65}$,
R. Lafaye$^{\rm 5}$,
B. Laforge$^{\rm 80}$,
T. Lagouri$^{\rm 176}$,
S. Lai$^{\rm 54}$,
L. Lambourne$^{\rm 78}$,
S. Lammers$^{\rm 61}$,
C.L. Lampen$^{\rm 7}$,
W. Lampl$^{\rm 7}$,
E. Lançon$^{\rm 136}$,
U. Landgraf$^{\rm 48}$,
M.P.J. Landon$^{\rm 76}$,
V.S. Lang$^{\rm 58a}$,
J.C. Lange$^{\rm 12}$,
A.J. Lankford$^{\rm 163}$,
F. Lanni$^{\rm 25}$,
K. Lantzsch$^{\rm 21}$,
A. Lanza$^{\rm 121a}$,
S. Laplace$^{\rm 80}$,
C. Lapoire$^{\rm 30}$,
J.F. Laporte$^{\rm 136}$,
T. Lari$^{\rm 91a}$,
F. Lasagni Manghi$^{\rm 20a,20b}$,
M. Lassnig$^{\rm 30}$,
P. Laurelli$^{\rm 47}$,
W. Lavrijsen$^{\rm 15}$,
A.T. Law$^{\rm 137}$,
P. Laycock$^{\rm 74}$,
T. Lazovich$^{\rm 57}$,
O. Le Dortz$^{\rm 80}$,
E. Le Guirriec$^{\rm 85}$,
E. Le Menedeu$^{\rm 12}$,
M. LeBlanc$^{\rm 169}$,
T. LeCompte$^{\rm 6}$,
F. Ledroit-Guillon$^{\rm 55}$,
C.A. Lee$^{\rm 145a}$,
S.C. Lee$^{\rm 151}$,
L. Lee$^{\rm 1}$,
G. Lefebvre$^{\rm 80}$,
M. Lefebvre$^{\rm 169}$,
F. Legger$^{\rm 100}$,
C. Leggett$^{\rm 15}$,
A. Lehan$^{\rm 74}$,
G. Lehmann Miotto$^{\rm 30}$,
X. Lei$^{\rm 7}$,
W.A. Leight$^{\rm 29}$,
A. Leisos$^{\rm 154}$$^{,x}$,
A.G. Leister$^{\rm 176}$,
M.A.L. Leite$^{\rm 24d}$,
R. Leitner$^{\rm 129}$,
D. Lellouch$^{\rm 172}$,
B. Lemmer$^{\rm 54}$,
K.J.C. Leney$^{\rm 78}$,
T. Lenz$^{\rm 21}$,
B. Lenzi$^{\rm 30}$,
R. Leone$^{\rm 7}$,
S. Leone$^{\rm 124a,124b}$,
C. Leonidopoulos$^{\rm 46}$,
S. Leontsinis$^{\rm 10}$,
C. Leroy$^{\rm 95}$,
C.G. Lester$^{\rm 28}$,
M. Levchenko$^{\rm 123}$,
J. Levêque$^{\rm 5}$,
D. Levin$^{\rm 89}$,
L.J. Levinson$^{\rm 172}$,
M. Levy$^{\rm 18}$,
A. Lewis$^{\rm 120}$,
A.M. Leyko$^{\rm 21}$,
M. Leyton$^{\rm 41}$,
B. Li$^{\rm 33b}$$^{,y}$,
H. Li$^{\rm 148}$,
H.L. Li$^{\rm 31}$,
L. Li$^{\rm 45}$,
L. Li$^{\rm 33e}$,
S. Li$^{\rm 45}$,
X. Li$^{\rm 84}$,
Y. Li$^{\rm 33c}$$^{,z}$,
Z. Liang$^{\rm 137}$,
H. Liao$^{\rm 34}$,
B. Liberti$^{\rm 133a}$,
A. Liblong$^{\rm 158}$,
P. Lichard$^{\rm 30}$,
K. Lie$^{\rm 165}$,
J. Liebal$^{\rm 21}$,
W. Liebig$^{\rm 14}$,
C. Limbach$^{\rm 21}$,
A. Limosani$^{\rm 150}$,
S.C. Lin$^{\rm 151}$$^{,aa}$,
T.H. Lin$^{\rm 83}$,
F. Linde$^{\rm 107}$,
B.E. Lindquist$^{\rm 148}$,
J.T. Linnemann$^{\rm 90}$,
E. Lipeles$^{\rm 122}$,
A. Lipniacka$^{\rm 14}$,
M. Lisovyi$^{\rm 58b}$,
T.M. Liss$^{\rm 165}$,
D. Lissauer$^{\rm 25}$,
A. Lister$^{\rm 168}$,
A.M. Litke$^{\rm 137}$,
B. Liu$^{\rm 151}$$^{,ab}$,
D. Liu$^{\rm 151}$,
H. Liu$^{\rm 89}$,
J. Liu$^{\rm 85}$,
J.B. Liu$^{\rm 33b}$,
K. Liu$^{\rm 85}$,
L. Liu$^{\rm 165}$,
M. Liu$^{\rm 45}$,
M. Liu$^{\rm 33b}$,
Y. Liu$^{\rm 33b}$,
M. Livan$^{\rm 121a,121b}$,
A. Lleres$^{\rm 55}$,
J. Llorente Merino$^{\rm 82}$,
S.L. Lloyd$^{\rm 76}$,
F. Lo Sterzo$^{\rm 151}$,
E. Lobodzinska$^{\rm 42}$,
P. Loch$^{\rm 7}$,
W.S. Lockman$^{\rm 137}$,
F.K. Loebinger$^{\rm 84}$,
A.E. Loevschall-Jensen$^{\rm 36}$,
K.M. Loew$^{\rm 23}$,
A. Loginov$^{\rm 176}$,
T. Lohse$^{\rm 16}$,
K. Lohwasser$^{\rm 42}$,
M. Lokajicek$^{\rm 127}$,
B.A. Long$^{\rm 22}$,
J.D. Long$^{\rm 165}$,
R.E. Long$^{\rm 72}$,
K.A. Looper$^{\rm 111}$,
L. Lopes$^{\rm 126a}$,
D. Lopez Mateos$^{\rm 57}$,
B. Lopez Paredes$^{\rm 139}$,
I. Lopez Paz$^{\rm 12}$,
J. Lorenz$^{\rm 100}$,
N. Lorenzo Martinez$^{\rm 61}$,
M. Losada$^{\rm 162}$,
P.J. Lösel$^{\rm 100}$,
X. Lou$^{\rm 33a}$,
A. Lounis$^{\rm 117}$,
J. Love$^{\rm 6}$,
P.A. Love$^{\rm 72}$,
H. Lu$^{\rm 60a}$,
N. Lu$^{\rm 89}$,
H.J. Lubatti$^{\rm 138}$,
C. Luci$^{\rm 132a,132b}$,
A. Lucotte$^{\rm 55}$,
C. Luedtke$^{\rm 48}$,
F. Luehring$^{\rm 61}$,
W. Lukas$^{\rm 62}$,
L. Luminari$^{\rm 132a}$,
O. Lundberg$^{\rm 146a,146b}$,
B. Lund-Jensen$^{\rm 147}$,
D. Lynn$^{\rm 25}$,
R. Lysak$^{\rm 127}$,
E. Lytken$^{\rm 81}$,
H. Ma$^{\rm 25}$,
L.L. Ma$^{\rm 33d}$,
G. Maccarrone$^{\rm 47}$,
A. Macchiolo$^{\rm 101}$,
C.M. Macdonald$^{\rm 139}$,
B. Maček$^{\rm 75}$,
J. Machado Miguens$^{\rm 122,126b}$,
D. Macina$^{\rm 30}$,
D. Madaffari$^{\rm 85}$,
R. Madar$^{\rm 34}$,
H.J. Maddocks$^{\rm 72}$,
W.F. Mader$^{\rm 44}$,
A. Madsen$^{\rm 166}$,
J. Maeda$^{\rm 67}$,
S. Maeland$^{\rm 14}$,
T. Maeno$^{\rm 25}$,
A. Maevskiy$^{\rm 99}$,
E. Magradze$^{\rm 54}$,
K. Mahboubi$^{\rm 48}$,
J. Mahlstedt$^{\rm 107}$,
C. Maiani$^{\rm 136}$,
C. Maidantchik$^{\rm 24a}$,
A.A. Maier$^{\rm 101}$,
T. Maier$^{\rm 100}$,
A. Maio$^{\rm 126a,126b,126d}$,
S. Majewski$^{\rm 116}$,
Y. Makida$^{\rm 66}$,
N. Makovec$^{\rm 117}$,
B. Malaescu$^{\rm 80}$,
Pa. Malecki$^{\rm 39}$,
V.P. Maleev$^{\rm 123}$,
F. Malek$^{\rm 55}$,
U. Mallik$^{\rm 63}$,
D. Malon$^{\rm 6}$,
C. Malone$^{\rm 143}$,
S. Maltezos$^{\rm 10}$,
V.M. Malyshev$^{\rm 109}$,
S. Malyukov$^{\rm 30}$,
J. Mamuzic$^{\rm 42}$,
G. Mancini$^{\rm 47}$,
B. Mandelli$^{\rm 30}$,
L. Mandelli$^{\rm 91a}$,
I. Mandić$^{\rm 75}$,
R. Mandrysch$^{\rm 63}$,
J. Maneira$^{\rm 126a,126b}$,
A. Manfredini$^{\rm 101}$,
L. Manhaes de Andrade Filho$^{\rm 24b}$,
J. Manjarres Ramos$^{\rm 159b}$,
A. Mann$^{\rm 100}$,
A. Manousakis-Katsikakis$^{\rm 9}$,
B. Mansoulie$^{\rm 136}$,
R. Mantifel$^{\rm 87}$,
M. Mantoani$^{\rm 54}$,
L. Mapelli$^{\rm 30}$,
L. March$^{\rm 145c}$,
G. Marchiori$^{\rm 80}$,
M. Marcisovsky$^{\rm 127}$,
C.P. Marino$^{\rm 169}$,
M. Marjanovic$^{\rm 13}$,
D.E. Marley$^{\rm 89}$,
F. Marroquim$^{\rm 24a}$,
S.P. Marsden$^{\rm 84}$,
Z. Marshall$^{\rm 15}$,
L.F. Marti$^{\rm 17}$,
S. Marti-Garcia$^{\rm 167}$,
B. Martin$^{\rm 90}$,
T.A. Martin$^{\rm 170}$,
V.J. Martin$^{\rm 46}$,
B. Martin dit Latour$^{\rm 14}$,
M. Martinez$^{\rm 12}$$^{,o}$,
S. Martin-Haugh$^{\rm 131}$,
V.S. Martoiu$^{\rm 26b}$,
A.C. Martyniuk$^{\rm 78}$,
M. Marx$^{\rm 138}$,
F. Marzano$^{\rm 132a}$,
A. Marzin$^{\rm 30}$,
L. Masetti$^{\rm 83}$,
T. Mashimo$^{\rm 155}$,
R. Mashinistov$^{\rm 96}$,
J. Masik$^{\rm 84}$,
A.L. Maslennikov$^{\rm 109}$$^{,c}$,
I. Massa$^{\rm 20a,20b}$,
L. Massa$^{\rm 20a,20b}$,
P. Mastrandrea$^{\rm 5}$,
A. Mastroberardino$^{\rm 37a,37b}$,
T. Masubuchi$^{\rm 155}$,
P. Mättig$^{\rm 175}$,
J. Mattmann$^{\rm 83}$,
J. Maurer$^{\rm 26b}$,
S.J. Maxfield$^{\rm 74}$,
D.A. Maximov$^{\rm 109}$$^{,c}$,
R. Mazini$^{\rm 151}$,
S.M. Mazza$^{\rm 91a,91b}$,
G. Mc Goldrick$^{\rm 158}$,
S.P. Mc Kee$^{\rm 89}$,
A. McCarn$^{\rm 89}$,
R.L. McCarthy$^{\rm 148}$,
T.G. McCarthy$^{\rm 29}$,
N.A. McCubbin$^{\rm 131}$,
K.W. McFarlane$^{\rm 56}$$^{,*}$,
J.A. Mcfayden$^{\rm 78}$,
G. Mchedlidze$^{\rm 54}$,
S.J. McMahon$^{\rm 131}$,
R.A. McPherson$^{\rm 169}$$^{,k}$,
M. Medinnis$^{\rm 42}$,
S. Meehan$^{\rm 145a}$,
S. Mehlhase$^{\rm 100}$,
A. Mehta$^{\rm 74}$,
K. Meier$^{\rm 58a}$,
C. Meineck$^{\rm 100}$,
B. Meirose$^{\rm 41}$,
B.R. Mellado Garcia$^{\rm 145c}$,
F. Meloni$^{\rm 17}$,
A. Mengarelli$^{\rm 20a,20b}$,
S. Menke$^{\rm 101}$,
E. Meoni$^{\rm 161}$,
K.M. Mercurio$^{\rm 57}$,
S. Mergelmeyer$^{\rm 21}$,
P. Mermod$^{\rm 49}$,
L. Merola$^{\rm 104a,104b}$,
C. Meroni$^{\rm 91a}$,
F.S. Merritt$^{\rm 31}$,
A. Messina$^{\rm 132a,132b}$,
J. Metcalfe$^{\rm 25}$,
A.S. Mete$^{\rm 163}$,
C. Meyer$^{\rm 83}$,
C. Meyer$^{\rm 122}$,
J-P. Meyer$^{\rm 136}$,
J. Meyer$^{\rm 107}$,
H. Meyer Zu Theenhausen$^{\rm 58a}$,
R.P. Middleton$^{\rm 131}$,
S. Miglioranzi$^{\rm 164a,164c}$,
L. Mijović$^{\rm 21}$,
G. Mikenberg$^{\rm 172}$,
M. Mikestikova$^{\rm 127}$,
M. Mikuž$^{\rm 75}$,
M. Milesi$^{\rm 88}$,
A. Milic$^{\rm 30}$,
D.W. Miller$^{\rm 31}$,
C. Mills$^{\rm 46}$,
A. Milov$^{\rm 172}$,
D.A. Milstead$^{\rm 146a,146b}$,
A.A. Minaenko$^{\rm 130}$,
Y. Minami$^{\rm 155}$,
I.A. Minashvili$^{\rm 65}$,
A.I. Mincer$^{\rm 110}$,
B. Mindur$^{\rm 38a}$,
M. Mineev$^{\rm 65}$,
Y. Ming$^{\rm 173}$,
L.M. Mir$^{\rm 12}$,
K.P. Mistry$^{\rm 122}$,
T. Mitani$^{\rm 171}$,
J. Mitrevski$^{\rm 100}$,
V.A. Mitsou$^{\rm 167}$,
A. Miucci$^{\rm 49}$,
P.S. Miyagawa$^{\rm 139}$,
J.U. Mjörnmark$^{\rm 81}$,
T. Moa$^{\rm 146a,146b}$,
K. Mochizuki$^{\rm 85}$,
S. Mohapatra$^{\rm 35}$,
W. Mohr$^{\rm 48}$,
S. Molander$^{\rm 146a,146b}$,
R. Moles-Valls$^{\rm 21}$,
R. Monden$^{\rm 68}$,
K. Mönig$^{\rm 42}$,
C. Monini$^{\rm 55}$,
J. Monk$^{\rm 36}$,
E. Monnier$^{\rm 85}$,
A. Montalbano$^{\rm 148}$,
J. Montejo Berlingen$^{\rm 12}$,
F. Monticelli$^{\rm 71}$,
S. Monzani$^{\rm 132a,132b}$,
R.W. Moore$^{\rm 3}$,
N. Morange$^{\rm 117}$,
D. Moreno$^{\rm 162}$,
M. Moreno Llácer$^{\rm 54}$,
P. Morettini$^{\rm 50a}$,
D. Mori$^{\rm 142}$,
T. Mori$^{\rm 155}$,
M. Morii$^{\rm 57}$,
M. Morinaga$^{\rm 155}$,
V. Morisbak$^{\rm 119}$,
S. Moritz$^{\rm 83}$,
A.K. Morley$^{\rm 150}$,
G. Mornacchi$^{\rm 30}$,
J.D. Morris$^{\rm 76}$,
S.S. Mortensen$^{\rm 36}$,
A. Morton$^{\rm 53}$,
L. Morvaj$^{\rm 103}$,
M. Mosidze$^{\rm 51b}$,
J. Moss$^{\rm 143}$,
K. Motohashi$^{\rm 157}$,
R. Mount$^{\rm 143}$,
E. Mountricha$^{\rm 25}$,
S.V. Mouraviev$^{\rm 96}$$^{,*}$,
E.J.W. Moyse$^{\rm 86}$,
S. Muanza$^{\rm 85}$,
R.D. Mudd$^{\rm 18}$,
F. Mueller$^{\rm 101}$,
J. Mueller$^{\rm 125}$,
R.S.P. Mueller$^{\rm 100}$,
T. Mueller$^{\rm 28}$,
D. Muenstermann$^{\rm 49}$,
P. Mullen$^{\rm 53}$,
G.A. Mullier$^{\rm 17}$,
J.A. Murillo Quijada$^{\rm 18}$,
W.J. Murray$^{\rm 170,131}$,
H. Musheghyan$^{\rm 54}$,
E. Musto$^{\rm 152}$,
A.G. Myagkov$^{\rm 130}$$^{,ac}$,
M. Myska$^{\rm 128}$,
B.P. Nachman$^{\rm 143}$,
O. Nackenhorst$^{\rm 54}$,
J. Nadal$^{\rm 54}$,
K. Nagai$^{\rm 120}$,
R. Nagai$^{\rm 157}$,
Y. Nagai$^{\rm 85}$,
K. Nagano$^{\rm 66}$,
A. Nagarkar$^{\rm 111}$,
Y. Nagasaka$^{\rm 59}$,
K. Nagata$^{\rm 160}$,
M. Nagel$^{\rm 101}$,
E. Nagy$^{\rm 85}$,
A.M. Nairz$^{\rm 30}$,
Y. Nakahama$^{\rm 30}$,
K. Nakamura$^{\rm 66}$,
T. Nakamura$^{\rm 155}$,
I. Nakano$^{\rm 112}$,
H. Namasivayam$^{\rm 41}$,
R.F. Naranjo Garcia$^{\rm 42}$,
R. Narayan$^{\rm 31}$,
D.I. Narrias Villar$^{\rm 58a}$,
T. Naumann$^{\rm 42}$,
G. Navarro$^{\rm 162}$,
R. Nayyar$^{\rm 7}$,
H.A. Neal$^{\rm 89}$,
P.Yu. Nechaeva$^{\rm 96}$,
T.J. Neep$^{\rm 84}$,
P.D. Nef$^{\rm 143}$,
A. Negri$^{\rm 121a,121b}$,
M. Negrini$^{\rm 20a}$,
S. Nektarijevic$^{\rm 106}$,
C. Nellist$^{\rm 117}$,
A. Nelson$^{\rm 163}$,
S. Nemecek$^{\rm 127}$,
P. Nemethy$^{\rm 110}$,
A.A. Nepomuceno$^{\rm 24a}$,
M. Nessi$^{\rm 30}$$^{,ad}$,
M.S. Neubauer$^{\rm 165}$,
M. Neumann$^{\rm 175}$,
R.M. Neves$^{\rm 110}$,
P. Nevski$^{\rm 25}$,
P.R. Newman$^{\rm 18}$,
D.H. Nguyen$^{\rm 6}$,
R.B. Nickerson$^{\rm 120}$,
R. Nicolaidou$^{\rm 136}$,
B. Nicquevert$^{\rm 30}$,
J. Nielsen$^{\rm 137}$,
N. Nikiforou$^{\rm 35}$,
A. Nikiforov$^{\rm 16}$,
V. Nikolaenko$^{\rm 130}$$^{,ac}$,
I. Nikolic-Audit$^{\rm 80}$,
K. Nikolopoulos$^{\rm 18}$,
J.K. Nilsen$^{\rm 119}$,
P. Nilsson$^{\rm 25}$,
Y. Ninomiya$^{\rm 155}$,
A. Nisati$^{\rm 132a}$,
R. Nisius$^{\rm 101}$,
T. Nobe$^{\rm 155}$,
M. Nomachi$^{\rm 118}$,
I. Nomidis$^{\rm 29}$,
T. Nooney$^{\rm 76}$,
S. Norberg$^{\rm 113}$,
M. Nordberg$^{\rm 30}$,
O. Novgorodova$^{\rm 44}$,
S. Nowak$^{\rm 101}$,
M. Nozaki$^{\rm 66}$,
L. Nozka$^{\rm 115}$,
K. Ntekas$^{\rm 10}$,
G. Nunes Hanninger$^{\rm 88}$,
T. Nunnemann$^{\rm 100}$,
E. Nurse$^{\rm 78}$,
F. Nuti$^{\rm 88}$,
B.J. O'Brien$^{\rm 46}$,
F. O'grady$^{\rm 7}$,
D.C. O'Neil$^{\rm 142}$,
V. O'Shea$^{\rm 53}$,
F.G. Oakham$^{\rm 29}$$^{,d}$,
H. Oberlack$^{\rm 101}$,
T. Obermann$^{\rm 21}$,
J. Ocariz$^{\rm 80}$,
A. Ochi$^{\rm 67}$,
I. Ochoa$^{\rm 35}$,
J.P. Ochoa-Ricoux$^{\rm 32a}$,
S. Oda$^{\rm 70}$,
S. Odaka$^{\rm 66}$,
H. Ogren$^{\rm 61}$,
A. Oh$^{\rm 84}$,
S.H. Oh$^{\rm 45}$,
C.C. Ohm$^{\rm 15}$,
H. Ohman$^{\rm 166}$,
H. Oide$^{\rm 30}$,
W. Okamura$^{\rm 118}$,
H. Okawa$^{\rm 160}$,
Y. Okumura$^{\rm 31}$,
T. Okuyama$^{\rm 66}$,
A. Olariu$^{\rm 26b}$,
S.A. Olivares Pino$^{\rm 46}$,
D. Oliveira Damazio$^{\rm 25}$,
A. Olszewski$^{\rm 39}$,
J. Olszowska$^{\rm 39}$,
A. Onofre$^{\rm 126a,126e}$,
K. Onogi$^{\rm 103}$,
P.U.E. Onyisi$^{\rm 31}$$^{,s}$,
C.J. Oram$^{\rm 159a}$,
M.J. Oreglia$^{\rm 31}$,
Y. Oren$^{\rm 153}$,
D. Orestano$^{\rm 134a,134b}$,
N. Orlando$^{\rm 154}$,
C. Oropeza Barrera$^{\rm 53}$,
R.S. Orr$^{\rm 158}$,
B. Osculati$^{\rm 50a,50b}$,
R. Ospanov$^{\rm 84}$,
G. Otero y Garzon$^{\rm 27}$,
H. Otono$^{\rm 70}$,
M. Ouchrif$^{\rm 135d}$,
F. Ould-Saada$^{\rm 119}$,
A. Ouraou$^{\rm 136}$,
K.P. Oussoren$^{\rm 107}$,
Q. Ouyang$^{\rm 33a}$,
A. Ovcharova$^{\rm 15}$,
M. Owen$^{\rm 53}$,
R.E. Owen$^{\rm 18}$,
V.E. Ozcan$^{\rm 19a}$,
N. Ozturk$^{\rm 8}$,
K. Pachal$^{\rm 142}$,
A. Pacheco Pages$^{\rm 12}$,
C. Padilla Aranda$^{\rm 12}$,
M. Pagáčová$^{\rm 48}$,
S. Pagan Griso$^{\rm 15}$,
E. Paganis$^{\rm 139}$,
F. Paige$^{\rm 25}$,
P. Pais$^{\rm 86}$,
K. Pajchel$^{\rm 119}$,
G. Palacino$^{\rm 159b}$,
S. Palestini$^{\rm 30}$,
M. Palka$^{\rm 38b}$,
D. Pallin$^{\rm 34}$,
A. Palma$^{\rm 126a,126b}$,
Y.B. Pan$^{\rm 173}$,
E.St. Panagiotopoulou$^{\rm 10}$,
C.E. Pandini$^{\rm 80}$,
J.G. Panduro Vazquez$^{\rm 77}$,
P. Pani$^{\rm 146a,146b}$,
S. Panitkin$^{\rm 25}$,
D. Pantea$^{\rm 26b}$,
L. Paolozzi$^{\rm 49}$,
Th.D. Papadopoulou$^{\rm 10}$,
K. Papageorgiou$^{\rm 154}$,
A. Paramonov$^{\rm 6}$,
D. Paredes Hernandez$^{\rm 154}$,
M.A. Parker$^{\rm 28}$,
K.A. Parker$^{\rm 139}$,
F. Parodi$^{\rm 50a,50b}$,
J.A. Parsons$^{\rm 35}$,
U. Parzefall$^{\rm 48}$,
E. Pasqualucci$^{\rm 132a}$,
S. Passaggio$^{\rm 50a}$,
F. Pastore$^{\rm 134a,134b}$$^{,*}$,
Fr. Pastore$^{\rm 77}$,
G. Pásztor$^{\rm 29}$,
S. Pataraia$^{\rm 175}$,
N.D. Patel$^{\rm 150}$,
J.R. Pater$^{\rm 84}$,
T. Pauly$^{\rm 30}$,
J. Pearce$^{\rm 169}$,
B. Pearson$^{\rm 113}$,
L.E. Pedersen$^{\rm 36}$,
M. Pedersen$^{\rm 119}$,
S. Pedraza Lopez$^{\rm 167}$,
R. Pedro$^{\rm 126a,126b}$,
S.V. Peleganchuk$^{\rm 109}$$^{,c}$,
D. Pelikan$^{\rm 166}$,
O. Penc$^{\rm 127}$,
C. Peng$^{\rm 33a}$,
H. Peng$^{\rm 33b}$,
B. Penning$^{\rm 31}$,
J. Penwell$^{\rm 61}$,
D.V. Perepelitsa$^{\rm 25}$,
E. Perez Codina$^{\rm 159a}$,
M.T. Pérez García-Estañ$^{\rm 167}$,
L. Perini$^{\rm 91a,91b}$,
H. Pernegger$^{\rm 30}$,
S. Perrella$^{\rm 104a,104b}$,
R. Peschke$^{\rm 42}$,
V.D. Peshekhonov$^{\rm 65}$,
K. Peters$^{\rm 30}$,
R.F.Y. Peters$^{\rm 84}$,
B.A. Petersen$^{\rm 30}$,
T.C. Petersen$^{\rm 36}$,
E. Petit$^{\rm 42}$,
A. Petridis$^{\rm 1}$,
C. Petridou$^{\rm 154}$,
P. Petroff$^{\rm 117}$,
E. Petrolo$^{\rm 132a}$,
F. Petrucci$^{\rm 134a,134b}$,
N.E. Pettersson$^{\rm 157}$,
R. Pezoa$^{\rm 32b}$,
P.W. Phillips$^{\rm 131}$,
G. Piacquadio$^{\rm 143}$,
E. Pianori$^{\rm 170}$,
A. Picazio$^{\rm 49}$,
E. Piccaro$^{\rm 76}$,
M. Piccinini$^{\rm 20a,20b}$,
M.A. Pickering$^{\rm 120}$,
R. Piegaia$^{\rm 27}$,
D.T. Pignotti$^{\rm 111}$,
J.E. Pilcher$^{\rm 31}$,
A.D. Pilkington$^{\rm 84}$,
A.W.J. Pin$^{\rm 84}$,
J. Pina$^{\rm 126a,126b,126d}$,
M. Pinamonti$^{\rm 164a,164c}$$^{,ae}$,
J.L. Pinfold$^{\rm 3}$,
A. Pingel$^{\rm 36}$,
S. Pires$^{\rm 80}$,
H. Pirumov$^{\rm 42}$,
M. Pitt$^{\rm 172}$,
C. Pizio$^{\rm 91a,91b}$,
L. Plazak$^{\rm 144a}$,
M.-A. Pleier$^{\rm 25}$,
V. Pleskot$^{\rm 129}$,
E. Plotnikova$^{\rm 65}$,
P. Plucinski$^{\rm 146a,146b}$,
D. Pluth$^{\rm 64}$,
R. Poettgen$^{\rm 146a,146b}$,
L. Poggioli$^{\rm 117}$,
D. Pohl$^{\rm 21}$,
G. Polesello$^{\rm 121a}$,
A. Poley$^{\rm 42}$,
A. Policicchio$^{\rm 37a,37b}$,
R. Polifka$^{\rm 158}$,
A. Polini$^{\rm 20a}$,
C.S. Pollard$^{\rm 53}$,
V. Polychronakos$^{\rm 25}$,
K. Pommès$^{\rm 30}$,
L. Pontecorvo$^{\rm 132a}$,
B.G. Pope$^{\rm 90}$,
G.A. Popeneciu$^{\rm 26c}$,
D.S. Popovic$^{\rm 13}$,
A. Poppleton$^{\rm 30}$,
S. Pospisil$^{\rm 128}$,
K. Potamianos$^{\rm 15}$,
I.N. Potrap$^{\rm 65}$,
C.J. Potter$^{\rm 149}$,
C.T. Potter$^{\rm 116}$,
G. Poulard$^{\rm 30}$,
J. Poveda$^{\rm 30}$,
V. Pozdnyakov$^{\rm 65}$,
P. Pralavorio$^{\rm 85}$,
A. Pranko$^{\rm 15}$,
S. Prasad$^{\rm 30}$,
S. Prell$^{\rm 64}$,
D. Price$^{\rm 84}$,
L.E. Price$^{\rm 6}$,
M. Primavera$^{\rm 73a}$,
S. Prince$^{\rm 87}$,
M. Proissl$^{\rm 46}$,
K. Prokofiev$^{\rm 60c}$,
F. Prokoshin$^{\rm 32b}$,
E. Protopapadaki$^{\rm 136}$,
S. Protopopescu$^{\rm 25}$,
J. Proudfoot$^{\rm 6}$,
M. Przybycien$^{\rm 38a}$,
E. Ptacek$^{\rm 116}$,
D. Puddu$^{\rm 134a,134b}$,
E. Pueschel$^{\rm 86}$,
D. Puldon$^{\rm 148}$,
M. Purohit$^{\rm 25}$$^{,af}$,
P. Puzo$^{\rm 117}$,
J. Qian$^{\rm 89}$,
G. Qin$^{\rm 53}$,
Y. Qin$^{\rm 84}$,
A. Quadt$^{\rm 54}$,
D.R. Quarrie$^{\rm 15}$,
W.B. Quayle$^{\rm 164a,164b}$,
M. Queitsch-Maitland$^{\rm 84}$,
D. Quilty$^{\rm 53}$,
S. Raddum$^{\rm 119}$,
V. Radeka$^{\rm 25}$,
V. Radescu$^{\rm 42}$,
S.K. Radhakrishnan$^{\rm 148}$,
P. Radloff$^{\rm 116}$,
P. Rados$^{\rm 88}$,
F. Ragusa$^{\rm 91a,91b}$,
G. Rahal$^{\rm 178}$,
S. Rajagopalan$^{\rm 25}$,
M. Rammensee$^{\rm 30}$,
C. Rangel-Smith$^{\rm 166}$,
F. Rauscher$^{\rm 100}$,
S. Rave$^{\rm 83}$,
T. Ravenscroft$^{\rm 53}$,
M. Raymond$^{\rm 30}$,
A.L. Read$^{\rm 119}$,
N.P. Readioff$^{\rm 74}$,
D.M. Rebuzzi$^{\rm 121a,121b}$,
A. Redelbach$^{\rm 174}$,
G. Redlinger$^{\rm 25}$,
R. Reece$^{\rm 137}$,
K. Reeves$^{\rm 41}$,
L. Rehnisch$^{\rm 16}$,
J. Reichert$^{\rm 122}$,
H. Reisin$^{\rm 27}$,
C. Rembser$^{\rm 30}$,
H. Ren$^{\rm 33a}$,
A. Renaud$^{\rm 117}$,
M. Rescigno$^{\rm 132a}$,
S. Resconi$^{\rm 91a}$,
O.L. Rezanova$^{\rm 109}$$^{,c}$,
P. Reznicek$^{\rm 129}$,
R. Rezvani$^{\rm 95}$,
R. Richter$^{\rm 101}$,
S. Richter$^{\rm 78}$,
E. Richter-Was$^{\rm 38b}$,
O. Ricken$^{\rm 21}$,
M. Ridel$^{\rm 80}$,
P. Rieck$^{\rm 16}$,
C.J. Riegel$^{\rm 175}$,
J. Rieger$^{\rm 54}$,
O. Rifki$^{\rm 113}$,
M. Rijssenbeek$^{\rm 148}$,
A. Rimoldi$^{\rm 121a,121b}$,
L. Rinaldi$^{\rm 20a}$,
B. Ristić$^{\rm 49}$,
E. Ritsch$^{\rm 30}$,
I. Riu$^{\rm 12}$,
F. Rizatdinova$^{\rm 114}$,
E. Rizvi$^{\rm 76}$,
S.H. Robertson$^{\rm 87}$$^{,k}$,
A. Robichaud-Veronneau$^{\rm 87}$,
D. Robinson$^{\rm 28}$,
J.E.M. Robinson$^{\rm 42}$,
A. Robson$^{\rm 53}$,
C. Roda$^{\rm 124a,124b}$,
S. Roe$^{\rm 30}$,
O. Røhne$^{\rm 119}$,
S. Rolli$^{\rm 161}$,
A. Romaniouk$^{\rm 98}$,
M. Romano$^{\rm 20a,20b}$,
S.M. Romano Saez$^{\rm 34}$,
E. Romero Adam$^{\rm 167}$,
N. Rompotis$^{\rm 138}$,
M. Ronzani$^{\rm 48}$,
L. Roos$^{\rm 80}$,
E. Ros$^{\rm 167}$,
S. Rosati$^{\rm 132a}$,
K. Rosbach$^{\rm 48}$,
P. Rose$^{\rm 137}$,
P.L. Rosendahl$^{\rm 14}$,
O. Rosenthal$^{\rm 141}$,
V. Rossetti$^{\rm 146a,146b}$,
E. Rossi$^{\rm 104a,104b}$,
L.P. Rossi$^{\rm 50a}$,
J.H.N. Rosten$^{\rm 28}$,
R. Rosten$^{\rm 138}$,
M. Rotaru$^{\rm 26b}$,
I. Roth$^{\rm 172}$,
J. Rothberg$^{\rm 138}$,
D. Rousseau$^{\rm 117}$,
C.R. Royon$^{\rm 136}$,
A. Rozanov$^{\rm 85}$,
Y. Rozen$^{\rm 152}$,
X. Ruan$^{\rm 145c}$,
F. Rubbo$^{\rm 143}$,
I. Rubinskiy$^{\rm 42}$,
V.I. Rud$^{\rm 99}$,
C. Rudolph$^{\rm 44}$,
M.S. Rudolph$^{\rm 158}$,
F. Rühr$^{\rm 48}$,
A. Ruiz-Martinez$^{\rm 30}$,
Z. Rurikova$^{\rm 48}$,
N.A. Rusakovich$^{\rm 65}$,
A. Ruschke$^{\rm 100}$,
H.L. Russell$^{\rm 138}$,
J.P. Rutherfoord$^{\rm 7}$,
N. Ruthmann$^{\rm 30}$,
Y.F. Ryabov$^{\rm 123}$,
M. Rybar$^{\rm 165}$,
G. Rybkin$^{\rm 117}$,
N.C. Ryder$^{\rm 120}$,
A.F. Saavedra$^{\rm 150}$,
G. Sabato$^{\rm 107}$,
S. Sacerdoti$^{\rm 27}$,
A. Saddique$^{\rm 3}$,
H.F-W. Sadrozinski$^{\rm 137}$,
R. Sadykov$^{\rm 65}$,
F. Safai Tehrani$^{\rm 132a}$,
P. Saha$^{\rm 108}$,
M. Sahinsoy$^{\rm 58a}$,
M. Saimpert$^{\rm 136}$,
T. Saito$^{\rm 155}$,
H. Sakamoto$^{\rm 155}$,
Y. Sakurai$^{\rm 171}$,
G. Salamanna$^{\rm 134a,134b}$,
A. Salamon$^{\rm 133a}$,
J.E. Salazar Loyola$^{\rm 32b}$,
M. Saleem$^{\rm 113}$,
D. Salek$^{\rm 107}$,
P.H. Sales De Bruin$^{\rm 138}$,
D. Salihagic$^{\rm 101}$,
A. Salnikov$^{\rm 143}$,
J. Salt$^{\rm 167}$,
D. Salvatore$^{\rm 37a,37b}$,
F. Salvatore$^{\rm 149}$,
A. Salvucci$^{\rm 60a}$,
A. Salzburger$^{\rm 30}$,
D. Sammel$^{\rm 48}$,
D. Sampsonidis$^{\rm 154}$,
A. Sanchez$^{\rm 104a,104b}$,
J. Sánchez$^{\rm 167}$,
V. Sanchez Martinez$^{\rm 167}$,
H. Sandaker$^{\rm 119}$,
R.L. Sandbach$^{\rm 76}$,
H.G. Sander$^{\rm 83}$,
M.P. Sanders$^{\rm 100}$,
M. Sandhoff$^{\rm 175}$,
C. Sandoval$^{\rm 162}$,
R. Sandstroem$^{\rm 101}$,
D.P.C. Sankey$^{\rm 131}$,
M. Sannino$^{\rm 50a,50b}$,
A. Sansoni$^{\rm 47}$,
C. Santoni$^{\rm 34}$,
R. Santonico$^{\rm 133a,133b}$,
H. Santos$^{\rm 126a}$,
I. Santoyo Castillo$^{\rm 149}$,
K. Sapp$^{\rm 125}$,
A. Sapronov$^{\rm 65}$,
J.G. Saraiva$^{\rm 126a,126d}$,
B. Sarrazin$^{\rm 21}$,
O. Sasaki$^{\rm 66}$,
Y. Sasaki$^{\rm 155}$,
K. Sato$^{\rm 160}$,
G. Sauvage$^{\rm 5}$$^{,*}$,
E. Sauvan$^{\rm 5}$,
G. Savage$^{\rm 77}$,
P. Savard$^{\rm 158}$$^{,d}$,
C. Sawyer$^{\rm 131}$,
L. Sawyer$^{\rm 79}$$^{,n}$,
J. Saxon$^{\rm 31}$,
C. Sbarra$^{\rm 20a}$,
A. Sbrizzi$^{\rm 20a,20b}$,
T. Scanlon$^{\rm 78}$,
D.A. Scannicchio$^{\rm 163}$,
M. Scarcella$^{\rm 150}$,
V. Scarfone$^{\rm 37a,37b}$,
J. Schaarschmidt$^{\rm 172}$,
P. Schacht$^{\rm 101}$,
D. Schaefer$^{\rm 30}$,
R. Schaefer$^{\rm 42}$,
J. Schaeffer$^{\rm 83}$,
S. Schaepe$^{\rm 21}$,
S. Schaetzel$^{\rm 58b}$,
U. Schäfer$^{\rm 83}$,
A.C. Schaffer$^{\rm 117}$,
D. Schaile$^{\rm 100}$,
R.D. Schamberger$^{\rm 148}$,
V. Scharf$^{\rm 58a}$,
V.A. Schegelsky$^{\rm 123}$,
D. Scheirich$^{\rm 129}$,
M. Schernau$^{\rm 163}$,
C. Schiavi$^{\rm 50a,50b}$,
C. Schillo$^{\rm 48}$,
M. Schioppa$^{\rm 37a,37b}$,
S. Schlenker$^{\rm 30}$,
K. Schmieden$^{\rm 30}$,
C. Schmitt$^{\rm 83}$,
S. Schmitt$^{\rm 58b}$,
S. Schmitt$^{\rm 42}$,
B. Schneider$^{\rm 159a}$,
Y.J. Schnellbach$^{\rm 74}$,
U. Schnoor$^{\rm 44}$,
L. Schoeffel$^{\rm 136}$,
A. Schoening$^{\rm 58b}$,
B.D. Schoenrock$^{\rm 90}$,
E. Schopf$^{\rm 21}$,
A.L.S. Schorlemmer$^{\rm 54}$,
M. Schott$^{\rm 83}$,
D. Schouten$^{\rm 159a}$,
J. Schovancova$^{\rm 8}$,
S. Schramm$^{\rm 49}$,
M. Schreyer$^{\rm 174}$,
N. Schuh$^{\rm 83}$,
M.J. Schultens$^{\rm 21}$,
H.-C. Schultz-Coulon$^{\rm 58a}$,
H. Schulz$^{\rm 16}$,
M. Schumacher$^{\rm 48}$,
B.A. Schumm$^{\rm 137}$,
Ph. Schune$^{\rm 136}$,
C. Schwanenberger$^{\rm 84}$,
A. Schwartzman$^{\rm 143}$,
T.A. Schwarz$^{\rm 89}$,
Ph. Schwegler$^{\rm 101}$,
H. Schweiger$^{\rm 84}$,
Ph. Schwemling$^{\rm 136}$,
R. Schwienhorst$^{\rm 90}$,
J. Schwindling$^{\rm 136}$,
T. Schwindt$^{\rm 21}$,
F.G. Sciacca$^{\rm 17}$,
E. Scifo$^{\rm 117}$,
G. Sciolla$^{\rm 23}$,
F. Scuri$^{\rm 124a,124b}$,
F. Scutti$^{\rm 21}$,
J. Searcy$^{\rm 89}$,
G. Sedov$^{\rm 42}$,
E. Sedykh$^{\rm 123}$,
P. Seema$^{\rm 21}$,
S.C. Seidel$^{\rm 105}$,
A. Seiden$^{\rm 137}$,
F. Seifert$^{\rm 128}$,
J.M. Seixas$^{\rm 24a}$,
G. Sekhniaidze$^{\rm 104a}$,
K. Sekhon$^{\rm 89}$,
S.J. Sekula$^{\rm 40}$,
D.M. Seliverstov$^{\rm 123}$$^{,*}$,
N. Semprini-Cesari$^{\rm 20a,20b}$,
C. Serfon$^{\rm 30}$,
L. Serin$^{\rm 117}$,
L. Serkin$^{\rm 164a,164b}$,
T. Serre$^{\rm 85}$,
M. Sessa$^{\rm 134a,134b}$,
R. Seuster$^{\rm 159a}$,
H. Severini$^{\rm 113}$,
T. Sfiligoj$^{\rm 75}$,
F. Sforza$^{\rm 30}$,
A. Sfyrla$^{\rm 30}$,
E. Shabalina$^{\rm 54}$,
M. Shamim$^{\rm 116}$,
L.Y. Shan$^{\rm 33a}$,
R. Shang$^{\rm 165}$,
J.T. Shank$^{\rm 22}$,
M. Shapiro$^{\rm 15}$,
P.B. Shatalov$^{\rm 97}$,
K. Shaw$^{\rm 164a,164b}$,
S.M. Shaw$^{\rm 84}$,
A. Shcherbakova$^{\rm 146a,146b}$,
C.Y. Shehu$^{\rm 149}$,
P. Sherwood$^{\rm 78}$,
L. Shi$^{\rm 151}$$^{,ag}$,
S. Shimizu$^{\rm 67}$,
C.O. Shimmin$^{\rm 163}$,
M. Shimojima$^{\rm 102}$,
M. Shiyakova$^{\rm 65}$,
A. Shmeleva$^{\rm 96}$,
D. Shoaleh Saadi$^{\rm 95}$,
M.J. Shochet$^{\rm 31}$,
S. Shojaii$^{\rm 91a,91b}$,
S. Shrestha$^{\rm 111}$,
E. Shulga$^{\rm 98}$,
M.A. Shupe$^{\rm 7}$,
S. Shushkevich$^{\rm 42}$,
P. Sicho$^{\rm 127}$,
P.E. Sidebo$^{\rm 147}$,
O. Sidiropoulou$^{\rm 174}$,
D. Sidorov$^{\rm 114}$,
A. Sidoti$^{\rm 20a,20b}$,
F. Siegert$^{\rm 44}$,
Dj. Sijacki$^{\rm 13}$,
J. Silva$^{\rm 126a,126d}$,
Y. Silver$^{\rm 153}$,
S.B. Silverstein$^{\rm 146a}$,
V. Simak$^{\rm 128}$,
O. Simard$^{\rm 5}$,
Lj. Simic$^{\rm 13}$,
S. Simion$^{\rm 117}$,
E. Simioni$^{\rm 83}$,
B. Simmons$^{\rm 78}$,
D. Simon$^{\rm 34}$,
P. Sinervo$^{\rm 158}$,
N.B. Sinev$^{\rm 116}$,
M. Sioli$^{\rm 20a,20b}$,
G. Siragusa$^{\rm 174}$,
A.N. Sisakyan$^{\rm 65}$$^{,*}$,
S.Yu. Sivoklokov$^{\rm 99}$,
J. Sjölin$^{\rm 146a,146b}$,
T.B. Sjursen$^{\rm 14}$,
M.B. Skinner$^{\rm 72}$,
H.P. Skottowe$^{\rm 57}$,
P. Skubic$^{\rm 113}$,
M. Slater$^{\rm 18}$,
T. Slavicek$^{\rm 128}$,
M. Slawinska$^{\rm 107}$,
K. Sliwa$^{\rm 161}$,
V. Smakhtin$^{\rm 172}$,
B.H. Smart$^{\rm 46}$,
L. Smestad$^{\rm 14}$,
S.Yu. Smirnov$^{\rm 98}$,
Y. Smirnov$^{\rm 98}$,
L.N. Smirnova$^{\rm 99}$$^{,ah}$,
O. Smirnova$^{\rm 81}$,
M.N.K. Smith$^{\rm 35}$,
R.W. Smith$^{\rm 35}$,
M. Smizanska$^{\rm 72}$,
K. Smolek$^{\rm 128}$,
A.A. Snesarev$^{\rm 96}$,
G. Snidero$^{\rm 76}$,
S. Snyder$^{\rm 25}$,
R. Sobie$^{\rm 169}$$^{,k}$,
F. Socher$^{\rm 44}$,
A. Soffer$^{\rm 153}$,
D.A. Soh$^{\rm 151}$$^{,ag}$,
G. Sokhrannyi$^{\rm 75}$,
C.A. Solans$^{\rm 30}$,
M. Solar$^{\rm 128}$,
J. Solc$^{\rm 128}$,
E.Yu. Soldatov$^{\rm 98}$,
U. Soldevila$^{\rm 167}$,
A.A. Solodkov$^{\rm 130}$,
A. Soloshenko$^{\rm 65}$,
O.V. Solovyanov$^{\rm 130}$,
V. Solovyev$^{\rm 123}$,
P. Sommer$^{\rm 48}$,
H.Y. Song$^{\rm 33b}$$^{,y}$,
N. Soni$^{\rm 1}$,
A. Sood$^{\rm 15}$,
A. Sopczak$^{\rm 128}$,
B. Sopko$^{\rm 128}$,
V. Sopko$^{\rm 128}$,
V. Sorin$^{\rm 12}$,
D. Sosa$^{\rm 58b}$,
M. Sosebee$^{\rm 8}$,
C.L. Sotiropoulou$^{\rm 124a,124b}$,
R. Soualah$^{\rm 164a,164c}$,
A.M. Soukharev$^{\rm 109}$$^{,c}$,
D. South$^{\rm 42}$,
B.C. Sowden$^{\rm 77}$,
S. Spagnolo$^{\rm 73a,73b}$,
M. Spalla$^{\rm 124a,124b}$,
M. Spangenberg$^{\rm 170}$,
F. Spanò$^{\rm 77}$,
W.R. Spearman$^{\rm 57}$,
D. Sperlich$^{\rm 16}$,
F. Spettel$^{\rm 101}$,
R. Spighi$^{\rm 20a}$,
G. Spigo$^{\rm 30}$,
L.A. Spiller$^{\rm 88}$,
M. Spousta$^{\rm 129}$,
R.D. St. Denis$^{\rm 53}$$^{,*}$,
A. Stabile$^{\rm 91a}$,
S. Staerz$^{\rm 44}$,
J. Stahlman$^{\rm 122}$,
R. Stamen$^{\rm 58a}$,
S. Stamm$^{\rm 16}$,
E. Stanecka$^{\rm 39}$,
C. Stanescu$^{\rm 134a}$,
M. Stanescu-Bellu$^{\rm 42}$,
M.M. Stanitzki$^{\rm 42}$,
S. Stapnes$^{\rm 119}$,
E.A. Starchenko$^{\rm 130}$,
J. Stark$^{\rm 55}$,
P. Staroba$^{\rm 127}$,
P. Starovoitov$^{\rm 58a}$,
R. Staszewski$^{\rm 39}$,
P. Steinberg$^{\rm 25}$,
B. Stelzer$^{\rm 142}$,
H.J. Stelzer$^{\rm 30}$,
O. Stelzer-Chilton$^{\rm 159a}$,
H. Stenzel$^{\rm 52}$,
G.A. Stewart$^{\rm 53}$,
J.A. Stillings$^{\rm 21}$,
M.C. Stockton$^{\rm 87}$,
M. Stoebe$^{\rm 87}$,
G. Stoicea$^{\rm 26b}$,
P. Stolte$^{\rm 54}$,
S. Stonjek$^{\rm 101}$,
A.R. Stradling$^{\rm 8}$,
A. Straessner$^{\rm 44}$,
M.E. Stramaglia$^{\rm 17}$,
J. Strandberg$^{\rm 147}$,
S. Strandberg$^{\rm 146a,146b}$,
A. Strandlie$^{\rm 119}$,
E. Strauss$^{\rm 143}$,
M. Strauss$^{\rm 113}$,
P. Strizenec$^{\rm 144b}$,
R. Ströhmer$^{\rm 174}$,
D.M. Strom$^{\rm 116}$,
R. Stroynowski$^{\rm 40}$,
A. Strubig$^{\rm 106}$,
S.A. Stucci$^{\rm 17}$,
B. Stugu$^{\rm 14}$,
N.A. Styles$^{\rm 42}$,
D. Su$^{\rm 143}$,
J. Su$^{\rm 125}$,
R. Subramaniam$^{\rm 79}$,
A. Succurro$^{\rm 12}$,
S. Suchek$^{\rm 58a}$,
Y. Sugaya$^{\rm 118}$,
M. Suk$^{\rm 128}$,
V.V. Sulin$^{\rm 96}$,
S. Sultansoy$^{\rm 4c}$,
T. Sumida$^{\rm 68}$,
S. Sun$^{\rm 57}$,
X. Sun$^{\rm 33a}$,
J.E. Sundermann$^{\rm 48}$,
K. Suruliz$^{\rm 149}$,
G. Susinno$^{\rm 37a,37b}$,
M.R. Sutton$^{\rm 149}$,
S. Suzuki$^{\rm 66}$,
M. Svatos$^{\rm 127}$,
M. Swiatlowski$^{\rm 143}$,
I. Sykora$^{\rm 144a}$,
T. Sykora$^{\rm 129}$,
D. Ta$^{\rm 48}$,
C. Taccini$^{\rm 134a,134b}$,
K. Tackmann$^{\rm 42}$,
J. Taenzer$^{\rm 158}$,
A. Taffard$^{\rm 163}$,
R. Tafirout$^{\rm 159a}$,
N. Taiblum$^{\rm 153}$,
H. Takai$^{\rm 25}$,
R. Takashima$^{\rm 69}$,
H. Takeda$^{\rm 67}$,
T. Takeshita$^{\rm 140}$,
Y. Takubo$^{\rm 66}$,
M. Talby$^{\rm 85}$,
A.A. Talyshev$^{\rm 109}$$^{,c}$,
J.Y.C. Tam$^{\rm 174}$,
K.G. Tan$^{\rm 88}$,
J. Tanaka$^{\rm 155}$,
R. Tanaka$^{\rm 117}$,
S. Tanaka$^{\rm 66}$,
B.B. Tannenwald$^{\rm 111}$,
N. Tannoury$^{\rm 21}$,
S. Tapia Araya$^{\rm 32b}$,
S. Tapprogge$^{\rm 83}$,
S. Tarem$^{\rm 152}$,
F. Tarrade$^{\rm 29}$,
G.F. Tartarelli$^{\rm 91a}$,
P. Tas$^{\rm 129}$,
M. Tasevsky$^{\rm 127}$,
T. Tashiro$^{\rm 68}$,
E. Tassi$^{\rm 37a,37b}$,
A. Tavares Delgado$^{\rm 126a,126b}$,
Y. Tayalati$^{\rm 135d}$,
F.E. Taylor$^{\rm 94}$,
G.N. Taylor$^{\rm 88}$,
P.T.E. Taylor$^{\rm 88}$,
W. Taylor$^{\rm 159b}$,
F.A. Teischinger$^{\rm 30}$,
M. Teixeira Dias Castanheira$^{\rm 76}$,
P. Teixeira-Dias$^{\rm 77}$,
K.K. Temming$^{\rm 48}$,
D. Temple$^{\rm 142}$,
H. Ten Kate$^{\rm 30}$,
P.K. Teng$^{\rm 151}$,
J.J. Teoh$^{\rm 118}$,
F. Tepel$^{\rm 175}$,
S. Terada$^{\rm 66}$,
K. Terashi$^{\rm 155}$,
J. Terron$^{\rm 82}$,
S. Terzo$^{\rm 101}$,
M. Testa$^{\rm 47}$,
R.J. Teuscher$^{\rm 158}$$^{,k}$,
T. Theveneaux-Pelzer$^{\rm 34}$,
J.P. Thomas$^{\rm 18}$,
J. Thomas-Wilsker$^{\rm 77}$,
E.N. Thompson$^{\rm 35}$,
P.D. Thompson$^{\rm 18}$,
R.J. Thompson$^{\rm 84}$,
A.S. Thompson$^{\rm 53}$,
L.A. Thomsen$^{\rm 176}$,
E. Thomson$^{\rm 122}$,
M. Thomson$^{\rm 28}$,
R.P. Thun$^{\rm 89}$$^{,*}$,
M.J. Tibbetts$^{\rm 15}$,
R.E. Ticse Torres$^{\rm 85}$,
V.O. Tikhomirov$^{\rm 96}$$^{,ai}$,
Yu.A. Tikhonov$^{\rm 109}$$^{,c}$,
S. Timoshenko$^{\rm 98}$,
E. Tiouchichine$^{\rm 85}$,
P. Tipton$^{\rm 176}$,
S. Tisserant$^{\rm 85}$,
K. Todome$^{\rm 157}$,
T. Todorov$^{\rm 5}$$^{,*}$,
S. Todorova-Nova$^{\rm 129}$,
J. Tojo$^{\rm 70}$,
S. Tokár$^{\rm 144a}$,
K. Tokushuku$^{\rm 66}$,
K. Tollefson$^{\rm 90}$,
E. Tolley$^{\rm 57}$,
L. Tomlinson$^{\rm 84}$,
M. Tomoto$^{\rm 103}$,
L. Tompkins$^{\rm 143}$$^{,aj}$,
K. Toms$^{\rm 105}$,
E. Torrence$^{\rm 116}$,
H. Torres$^{\rm 142}$,
E. Torró Pastor$^{\rm 138}$,
J. Toth$^{\rm 85}$$^{,ak}$,
F. Touchard$^{\rm 85}$,
D.R. Tovey$^{\rm 139}$,
T. Trefzger$^{\rm 174}$,
L. Tremblet$^{\rm 30}$,
A. Tricoli$^{\rm 30}$,
I.M. Trigger$^{\rm 159a}$,
S. Trincaz-Duvoid$^{\rm 80}$,
M.F. Tripiana$^{\rm 12}$,
W. Trischuk$^{\rm 158}$,
B. Trocmé$^{\rm 55}$,
C. Troncon$^{\rm 91a}$,
M. Trottier-McDonald$^{\rm 15}$,
M. Trovatelli$^{\rm 169}$,
L. Truong$^{\rm 164a,164c}$,
M. Trzebinski$^{\rm 39}$,
A. Trzupek$^{\rm 39}$,
C. Tsarouchas$^{\rm 30}$,
J.C-L. Tseng$^{\rm 120}$,
P.V. Tsiareshka$^{\rm 92}$,
D. Tsionou$^{\rm 154}$,
G. Tsipolitis$^{\rm 10}$,
N. Tsirintanis$^{\rm 9}$,
S. Tsiskaridze$^{\rm 12}$,
V. Tsiskaridze$^{\rm 48}$,
E.G. Tskhadadze$^{\rm 51a}$,
K.M. Tsui$^{\rm 60a}$,
I.I. Tsukerman$^{\rm 97}$,
V. Tsulaia$^{\rm 15}$,
S. Tsuno$^{\rm 66}$,
D. Tsybychev$^{\rm 148}$,
A. Tudorache$^{\rm 26b}$,
V. Tudorache$^{\rm 26b}$,
A.N. Tuna$^{\rm 57}$,
S.A. Tupputi$^{\rm 20a,20b}$,
S. Turchikhin$^{\rm 99}$$^{,ah}$,
D. Turecek$^{\rm 128}$,
R. Turra$^{\rm 91a,91b}$,
A.J. Turvey$^{\rm 40}$,
P.M. Tuts$^{\rm 35}$,
A. Tykhonov$^{\rm 49}$,
M. Tylmad$^{\rm 146a,146b}$,
M. Tyndel$^{\rm 131}$,
I. Ueda$^{\rm 155}$,
R. Ueno$^{\rm 29}$,
M. Ughetto$^{\rm 146a,146b}$,
M. Ugland$^{\rm 14}$,
F. Ukegawa$^{\rm 160}$,
G. Unal$^{\rm 30}$,
A. Undrus$^{\rm 25}$,
G. Unel$^{\rm 163}$,
F.C. Ungaro$^{\rm 48}$,
Y. Unno$^{\rm 66}$,
C. Unverdorben$^{\rm 100}$,
J. Urban$^{\rm 144b}$,
P. Urquijo$^{\rm 88}$,
P. Urrejola$^{\rm 83}$,
G. Usai$^{\rm 8}$,
A. Usanova$^{\rm 62}$,
L. Vacavant$^{\rm 85}$,
V. Vacek$^{\rm 128}$,
B. Vachon$^{\rm 87}$,
C. Valderanis$^{\rm 83}$,
N. Valencic$^{\rm 107}$,
S. Valentinetti$^{\rm 20a,20b}$,
A. Valero$^{\rm 167}$,
L. Valery$^{\rm 12}$,
S. Valkar$^{\rm 129}$,
S. Vallecorsa$^{\rm 49}$,
J.A. Valls Ferrer$^{\rm 167}$,
W. Van Den Wollenberg$^{\rm 107}$,
P.C. Van Der Deijl$^{\rm 107}$,
R. van der Geer$^{\rm 107}$,
H. van der Graaf$^{\rm 107}$,
N. van Eldik$^{\rm 152}$,
P. van Gemmeren$^{\rm 6}$,
J. Van Nieuwkoop$^{\rm 142}$,
I. van Vulpen$^{\rm 107}$,
M.C. van Woerden$^{\rm 30}$,
M. Vanadia$^{\rm 132a,132b}$,
W. Vandelli$^{\rm 30}$,
R. Vanguri$^{\rm 122}$,
A. Vaniachine$^{\rm 6}$,
F. Vannucci$^{\rm 80}$,
G. Vardanyan$^{\rm 177}$,
R. Vari$^{\rm 132a}$,
E.W. Varnes$^{\rm 7}$,
T. Varol$^{\rm 40}$,
D. Varouchas$^{\rm 80}$,
A. Vartapetian$^{\rm 8}$,
K.E. Varvell$^{\rm 150}$,
F. Vazeille$^{\rm 34}$,
T. Vazquez Schroeder$^{\rm 87}$,
J. Veatch$^{\rm 7}$,
L.M. Veloce$^{\rm 158}$,
F. Veloso$^{\rm 126a,126c}$,
T. Velz$^{\rm 21}$,
S. Veneziano$^{\rm 132a}$,
A. Ventura$^{\rm 73a,73b}$,
D. Ventura$^{\rm 86}$,
M. Venturi$^{\rm 169}$,
N. Venturi$^{\rm 158}$,
A. Venturini$^{\rm 23}$,
V. Vercesi$^{\rm 121a}$,
M. Verducci$^{\rm 132a,132b}$,
W. Verkerke$^{\rm 107}$,
J.C. Vermeulen$^{\rm 107}$,
A. Vest$^{\rm 44}$,
M.C. Vetterli$^{\rm 142}$$^{,d}$,
O. Viazlo$^{\rm 81}$,
I. Vichou$^{\rm 165}$,
T. Vickey$^{\rm 139}$,
O.E. Vickey Boeriu$^{\rm 139}$,
G.H.A. Viehhauser$^{\rm 120}$,
S. Viel$^{\rm 15}$,
R. Vigne$^{\rm 62}$,
M. Villa$^{\rm 20a,20b}$,
M. Villaplana Perez$^{\rm 91a,91b}$,
E. Vilucchi$^{\rm 47}$,
M.G. Vincter$^{\rm 29}$,
V.B. Vinogradov$^{\rm 65}$,
I. Vivarelli$^{\rm 149}$,
F. Vives Vaque$^{\rm 3}$,
S. Vlachos$^{\rm 10}$,
D. Vladoiu$^{\rm 100}$,
M. Vlasak$^{\rm 128}$,
M. Vogel$^{\rm 32a}$,
P. Vokac$^{\rm 128}$,
G. Volpi$^{\rm 124a,124b}$,
M. Volpi$^{\rm 88}$,
H. von der Schmitt$^{\rm 101}$,
H. von Radziewski$^{\rm 48}$,
E. von Toerne$^{\rm 21}$,
V. Vorobel$^{\rm 129}$,
K. Vorobev$^{\rm 98}$,
M. Vos$^{\rm 167}$,
R. Voss$^{\rm 30}$,
J.H. Vossebeld$^{\rm 74}$,
N. Vranjes$^{\rm 13}$,
M. Vranjes Milosavljevic$^{\rm 13}$,
V. Vrba$^{\rm 127}$,
M. Vreeswijk$^{\rm 107}$,
R. Vuillermet$^{\rm 30}$,
I. Vukotic$^{\rm 31}$,
Z. Vykydal$^{\rm 128}$,
P. Wagner$^{\rm 21}$,
W. Wagner$^{\rm 175}$,
H. Wahlberg$^{\rm 71}$,
S. Wahrmund$^{\rm 44}$,
J. Wakabayashi$^{\rm 103}$,
J. Walder$^{\rm 72}$,
R. Walker$^{\rm 100}$,
W. Walkowiak$^{\rm 141}$,
C. Wang$^{\rm 151}$,
F. Wang$^{\rm 173}$,
H. Wang$^{\rm 15}$,
H. Wang$^{\rm 40}$,
J. Wang$^{\rm 42}$,
J. Wang$^{\rm 150}$,
K. Wang$^{\rm 87}$,
R. Wang$^{\rm 6}$,
S.M. Wang$^{\rm 151}$,
T. Wang$^{\rm 21}$,
T. Wang$^{\rm 35}$,
X. Wang$^{\rm 176}$,
C. Wanotayaroj$^{\rm 116}$,
A. Warburton$^{\rm 87}$,
C.P. Ward$^{\rm 28}$,
D.R. Wardrope$^{\rm 78}$,
A. Washbrook$^{\rm 46}$,
C. Wasicki$^{\rm 42}$,
P.M. Watkins$^{\rm 18}$,
A.T. Watson$^{\rm 18}$,
I.J. Watson$^{\rm 150}$,
M.F. Watson$^{\rm 18}$,
G. Watts$^{\rm 138}$,
S. Watts$^{\rm 84}$,
B.M. Waugh$^{\rm 78}$,
S. Webb$^{\rm 84}$,
M.S. Weber$^{\rm 17}$,
S.W. Weber$^{\rm 174}$,
J.S. Webster$^{\rm 31}$,
A.R. Weidberg$^{\rm 120}$,
B. Weinert$^{\rm 61}$,
J. Weingarten$^{\rm 54}$,
C. Weiser$^{\rm 48}$,
H. Weits$^{\rm 107}$,
P.S. Wells$^{\rm 30}$,
T. Wenaus$^{\rm 25}$,
T. Wengler$^{\rm 30}$,
S. Wenig$^{\rm 30}$,
N. Wermes$^{\rm 21}$,
M. Werner$^{\rm 48}$,
P. Werner$^{\rm 30}$,
M. Wessels$^{\rm 58a}$,
J. Wetter$^{\rm 161}$,
K. Whalen$^{\rm 116}$,
A.M. Wharton$^{\rm 72}$,
A. White$^{\rm 8}$,
M.J. White$^{\rm 1}$,
R. White$^{\rm 32b}$,
S. White$^{\rm 124a,124b}$,
D. Whiteson$^{\rm 163}$,
F.J. Wickens$^{\rm 131}$,
W. Wiedenmann$^{\rm 173}$,
M. Wielers$^{\rm 131}$,
P. Wienemann$^{\rm 21}$,
C. Wiglesworth$^{\rm 36}$,
L.A.M. Wiik-Fuchs$^{\rm 21}$,
A. Wildauer$^{\rm 101}$,
H.G. Wilkens$^{\rm 30}$,
H.H. Williams$^{\rm 122}$,
S. Williams$^{\rm 107}$,
C. Willis$^{\rm 90}$,
S. Willocq$^{\rm 86}$,
A. Wilson$^{\rm 89}$,
J.A. Wilson$^{\rm 18}$,
I. Wingerter-Seez$^{\rm 5}$,
F. Winklmeier$^{\rm 116}$,
B.T. Winter$^{\rm 21}$,
M. Wittgen$^{\rm 143}$,
J. Wittkowski$^{\rm 100}$,
S.J. Wollstadt$^{\rm 83}$,
M.W. Wolter$^{\rm 39}$,
H. Wolters$^{\rm 126a,126c}$,
B.K. Wosiek$^{\rm 39}$,
J. Wotschack$^{\rm 30}$,
M.J. Woudstra$^{\rm 84}$,
K.W. Wozniak$^{\rm 39}$,
M. Wu$^{\rm 55}$,
M. Wu$^{\rm 31}$,
S.L. Wu$^{\rm 173}$,
X. Wu$^{\rm 49}$,
Y. Wu$^{\rm 89}$,
T.R. Wyatt$^{\rm 84}$,
B.M. Wynne$^{\rm 46}$,
S. Xella$^{\rm 36}$,
D. Xu$^{\rm 33a}$,
L. Xu$^{\rm 25}$,
B. Yabsley$^{\rm 150}$,
S. Yacoob$^{\rm 145a}$,
R. Yakabe$^{\rm 67}$,
M. Yamada$^{\rm 66}$,
D. Yamaguchi$^{\rm 157}$,
Y. Yamaguchi$^{\rm 118}$,
A. Yamamoto$^{\rm 66}$,
S. Yamamoto$^{\rm 155}$,
T. Yamanaka$^{\rm 155}$,
K. Yamauchi$^{\rm 103}$,
Y. Yamazaki$^{\rm 67}$,
Z. Yan$^{\rm 22}$,
H. Yang$^{\rm 33e}$,
H. Yang$^{\rm 173}$,
Y. Yang$^{\rm 151}$,
W-M. Yao$^{\rm 15}$,
Y.C. Yap$^{\rm 80}$,
Y. Yasu$^{\rm 66}$,
E. Yatsenko$^{\rm 5}$,
K.H. Yau Wong$^{\rm 21}$,
J. Ye$^{\rm 40}$,
S. Ye$^{\rm 25}$,
I. Yeletskikh$^{\rm 65}$,
A.L. Yen$^{\rm 57}$,
E. Yildirim$^{\rm 42}$,
K. Yorita$^{\rm 171}$,
R. Yoshida$^{\rm 6}$,
K. Yoshihara$^{\rm 122}$,
C. Young$^{\rm 143}$,
C.J.S. Young$^{\rm 30}$,
S. Youssef$^{\rm 22}$,
D.R. Yu$^{\rm 15}$,
J. Yu$^{\rm 8}$,
J.M. Yu$^{\rm 89}$,
J. Yu$^{\rm 114}$,
L. Yuan$^{\rm 67}$,
S.P.Y. Yuen$^{\rm 21}$,
A. Yurkewicz$^{\rm 108}$,
I. Yusuff$^{\rm 28}$$^{,al}$,
B. Zabinski$^{\rm 39}$,
R. Zaidan$^{\rm 63}$,
A.M. Zaitsev$^{\rm 130}$$^{,ac}$,
J. Zalieckas$^{\rm 14}$,
A. Zaman$^{\rm 148}$,
S. Zambito$^{\rm 57}$,
L. Zanello$^{\rm 132a,132b}$,
D. Zanzi$^{\rm 88}$,
C. Zeitnitz$^{\rm 175}$,
M. Zeman$^{\rm 128}$,
A. Zemla$^{\rm 38a}$,
Q. Zeng$^{\rm 143}$,
K. Zengel$^{\rm 23}$,
O. Zenin$^{\rm 130}$,
T. Ženiš$^{\rm 144a}$,
D. Zerwas$^{\rm 117}$,
D. Zhang$^{\rm 89}$,
F. Zhang$^{\rm 173}$,
G. Zhang$^{\rm 33b}$,
H. Zhang$^{\rm 33c}$,
J. Zhang$^{\rm 6}$,
L. Zhang$^{\rm 48}$,
R. Zhang$^{\rm 33b}$$^{,i}$,
X. Zhang$^{\rm 33d}$,
Z. Zhang$^{\rm 117}$,
X. Zhao$^{\rm 40}$,
Y. Zhao$^{\rm 33d,117}$,
Z. Zhao$^{\rm 33b}$,
A. Zhemchugov$^{\rm 65}$,
J. Zhong$^{\rm 120}$,
B. Zhou$^{\rm 89}$,
C. Zhou$^{\rm 45}$,
L. Zhou$^{\rm 35}$,
L. Zhou$^{\rm 40}$,
M. Zhou$^{\rm 148}$,
N. Zhou$^{\rm 33f}$,
C.G. Zhu$^{\rm 33d}$,
H. Zhu$^{\rm 33a}$,
J. Zhu$^{\rm 89}$,
Y. Zhu$^{\rm 33b}$,
X. Zhuang$^{\rm 33a}$,
K. Zhukov$^{\rm 96}$,
A. Zibell$^{\rm 174}$,
D. Zieminska$^{\rm 61}$,
N.I. Zimine$^{\rm 65}$,
C. Zimmermann$^{\rm 83}$,
S. Zimmermann$^{\rm 48}$,
Z. Zinonos$^{\rm 54}$,
M. Zinser$^{\rm 83}$,
M. Ziolkowski$^{\rm 141}$,
L. Živković$^{\rm 13}$,
G. Zobernig$^{\rm 173}$,
A. Zoccoli$^{\rm 20a,20b}$,
M. zur Nedden$^{\rm 16}$,
G. Zurzolo$^{\rm 104a,104b}$,
L. Zwalinski$^{\rm 30}$.
$^{1}$ Department of Physics, University of Adelaide, Adelaide, Australia
$^{2}$ Physics Department, SUNY Albany, Albany NY, United States of America
$^{3}$ Department of Physics, University of Alberta, Edmonton AB, Canada
$^{4}$ $^{(a)}$ Department of Physics, Ankara University, Ankara; $^{(b)}$ Istanbul Aydin University, Istanbul; $^{(c)}$ Division of Physics, TOBB University of Economics and Technology, Ankara, Turkey
$^{5}$ LAPP, CNRS/IN2P3 and Université Savoie Mont Blanc, Annecy-le-Vieux, France
$^{6}$ High Energy Physics Division, Argonne National Laboratory, Argonne IL, United States of America
$^{7}$ Department of Physics, University of Arizona, Tucson AZ, United States of America
$^{8}$ Department of Physics, The University of Texas at Arlington, Arlington TX, United States of America
$^{9}$ Physics Department, University of Athens, Athens, Greece
$^{10}$ Physics Department, National Technical University of Athens, Zografou, Greece
$^{11}$ Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan
$^{12}$ Institut de Física d'Altes Energies and Departament de Física de la Universitat Autònoma de Barcelona, Barcelona, Spain
$^{13}$ Institute of Physics, University of Belgrade, Belgrade, Serbia
$^{14}$ Department for Physics and Technology, University of Bergen, Bergen, Norway
$^{15}$ Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA, United States of America
$^{16}$ Department of Physics, Humboldt University, Berlin, Germany
$^{17}$ Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland
$^{18}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
$^{19}$ $^{(a)}$ Department of Physics, Bogazici University, Istanbul; $^{(b)}$ Department of Physics Engineering, Gaziantep University, Gaziantep; $^{(c)}$ Department of Physics, Dogus University, Istanbul, Turkey
$^{20}$ $^{(a)}$ INFN Sezione di Bologna; $^{(b)}$ Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy
$^{21}$ Physikalisches Institut, University of Bonn, Bonn, Germany
$^{22}$ Department of Physics, Boston University, Boston MA, United States of America
$^{23}$ Department of Physics, Brandeis University, Waltham MA, United States of America
$^{24}$ $^{(a)}$ Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; $^{(b)}$ Electrical Circuits Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora; $^{(c)}$ Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei; $^{(d)}$ Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil
$^{25}$ Physics Department, Brookhaven National Laboratory, Upton NY, United States of America
$^{26}$ $^{(a)}$ Transilvania University of Brasov, Brasov, Romania; $^{(b)}$ National Institute of Physics and Nuclear Engineering, Bucharest; $^{(c)}$ National Institute for Research and Development of Isotopic and Molecular Technologies, Physics Department, Cluj Napoca; $^{(d)}$ University Politehnica Bucharest, Bucharest; $^{(e)}$ West University in Timisoara, Timisoara, Romania
$^{27}$ Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina
$^{28}$ Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
$^{29}$ Department of Physics, Carleton University, Ottawa ON, Canada
$^{30}$ CERN, Geneva, Switzerland
$^{31}$ Enrico Fermi Institute, University of Chicago, Chicago IL, United States of America
$^{32}$ $^{(a)}$ Departamento de Física, Pontificia Universidad Católica de Chile, Santiago; $^{(b)}$ Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile
$^{33}$ $^{(a)}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; $^{(b)}$ Department of Modern Physics, University of Science and Technology of China, Anhui; $^{(c)}$ Department of Physics, Nanjing University, Jiangsu; $^{(d)}$ School of Physics, Shandong University, Shandong; $^{(e)}$ Department of Physics and Astronomy, Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai; $^{(f)}$ Physics Department, Tsinghua University, Beijing 100084, China
$^{34}$ Laboratoire de Physique Corpusculaire, Clermont Université and Université Blaise Pascal and CNRS/IN2P3, Clermont-Ferrand, France
$^{35}$ Nevis Laboratory, Columbia University, Irvington NY, United States of America
$^{36}$ Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark
$^{37}$ $^{(a)}$ INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati; $^{(b)}$ Dipartimento di Fisica, Università della Calabria, Rende, Italy
$^{38}$ $^{(a)}$ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow; $^{(b)}$ Marian Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland
$^{39}$ Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland
$^{40}$ Physics Department, Southern Methodist University, Dallas TX, United States of America
$^{41}$ Physics Department, University of Texas at Dallas, Richardson TX, United States of America
$^{42}$ DESY, Hamburg and Zeuthen, Germany
$^{43}$ Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany
$^{44}$ Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany
$^{45}$ Department of Physics, Duke University, Durham NC, United States of America
$^{46}$ SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
$^{47}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy
$^{48}$ Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg, Germany
$^{49}$ Section de Physique, Université de Genève, Geneva, Switzerland
$^{50}$ $^{(a)}$ INFN Sezione di Genova; $^{(b)}$ Dipartimento di Fisica, Università di Genova, Genova, Italy
$^{51}$ $^{(a)}$ E. Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi; $^{(b)}$ High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia
$^{52}$ II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany
$^{53}$ SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
$^{54}$ II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany
$^{55}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, Grenoble, France
$^{56}$ Department of Physics, Hampton University, Hampton VA, United States of America
$^{57}$ Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA, United States of America
$^{58}$ $^{(a)}$ Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(b)}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(c)}$ ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany
$^{59}$ Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan
$^{60}$ $^{(a)}$ Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong; $^{(b)}$ Department of Physics, The University of Hong Kong, Hong Kong; $^{(c)}$ Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
$^{61}$ Department of Physics, Indiana University, Bloomington IN, United States of America
$^{62}$ Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria
$^{63}$ University of Iowa, Iowa City IA, United States of America
$^{64}$ Department of Physics and Astronomy, Iowa State University, Ames IA, United States of America
$^{65}$ Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia
$^{66}$ KEK, High Energy Accelerator Research Organization, Tsukuba, Japan
$^{67}$ Graduate School of Science, Kobe University, Kobe, Japan
$^{68}$ Faculty of Science, Kyoto University, Kyoto, Japan
$^{69}$ Kyoto University of Education, Kyoto, Japan
$^{70}$ Department of Physics, Kyushu University, Fukuoka, Japan
$^{71}$ Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina
$^{72}$ Physics Department, Lancaster University, Lancaster, United Kingdom
$^{73}$ $^{(a)}$ INFN Sezione di Lecce; $^{(b)}$ Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy
$^{74}$ Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
$^{75}$ Department of Physics, Jožef Stefan Institute and University of Ljubljana, Ljubljana, Slovenia
$^{76}$ School of Physics and Astronomy, Queen Mary University of London, London, United Kingdom
$^{77}$ Department of Physics, Royal Holloway University of London, Surrey, United Kingdom
$^{78}$ Department of Physics and Astronomy, University College London, London, United Kingdom
$^{79}$ Louisiana Tech University, Ruston LA, United States of America
$^{80}$ Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France
$^{81}$ Fysiska institutionen, Lunds universitet, Lund, Sweden
$^{82}$ Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid, Madrid, Spain
$^{83}$ Institut für Physik, Universität Mainz, Mainz, Germany
$^{84}$ School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom
$^{85}$ CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France
$^{86}$ Department of Physics, University of Massachusetts, Amherst MA, United States of America
$^{87}$ Department of Physics, McGill University, Montreal QC, Canada
$^{88}$ School of Physics, University of Melbourne, Victoria, Australia
$^{89}$ Department of Physics, The University of Michigan, Ann Arbor MI, United States of America
$^{90}$ Department of Physics and Astronomy, Michigan State University, East Lansing MI, United States of America
$^{91}$ $^{(a)}$ INFN Sezione di Milano; $^{(b)}$ Dipartimento di Fisica, Università di Milano, Milano, Italy
$^{92}$ B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Republic of Belarus
$^{93}$ National Scientific and Educational Centre for Particle and High Energy Physics, Minsk, Republic of Belarus
$^{94}$ Department of Physics, Massachusetts Institute of Technology, Cambridge MA, United States of America
$^{95}$ Group of Particle Physics, University of Montreal, Montreal QC, Canada
$^{96}$ P.N. Lebedev Institute of Physics, Academy of Sciences, Moscow, Russia
$^{97}$ Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia
$^{98}$ National Research Nuclear University MEPhI, Moscow, Russia
$^{99}$ D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow, Russia
$^{100}$ Fakultät für Physik, Ludwig-Maximilians-Universität München, München, Germany
$^{101}$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München, Germany
$^{102}$ Nagasaki Institute of Applied Science, Nagasaki, Japan
$^{103}$ Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya University, Nagoya, Japan
$^{104}$ $^{(a)}$ INFN Sezione di Napoli; $^{(b)}$ Dipartimento di Fisica, Università di Napoli, Napoli, Italy
$^{105}$ Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, United States of America
$^{106}$ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands
$^{107}$ Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, Netherlands
$^{108}$ Department of Physics, Northern Illinois University, DeKalb IL, United States of America
$^{109}$ Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, Russia
$^{110}$ Department of Physics, New York University, New York NY, United States of America
$^{111}$ Ohio State University, Columbus OH, United States of America
$^{112}$ Faculty of Science, Okayama University, Okayama, Japan
$^{113}$ Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK, United States of America
$^{114}$ Department of Physics, Oklahoma State University, Stillwater OK, United States of America
$^{115}$ Palacký University, RCPTM, Olomouc, Czech Republic
$^{116}$ Center for High Energy Physics, University of Oregon, Eugene OR, United States of America
$^{117}$ LAL, Université Paris-Sud and CNRS/IN2P3, Orsay, France
$^{118}$ Graduate School of Science, Osaka University, Osaka, Japan
$^{119}$ Department of Physics, University of Oslo, Oslo, Norway
$^{120}$ Department of Physics, Oxford University, Oxford, United Kingdom
$^{121}$ $^{(a)}$ INFN Sezione di Pavia; $^{(b)}$ Dipartimento di Fisica, Università di Pavia, Pavia, Italy
$^{122}$ Department of Physics, University of Pennsylvania, Philadelphia PA, United States of America
$^{123}$ National Research Centre "Kurchatov Institute" B.P.Konstantinov Petersburg Nuclear Physics Institute, St. Petersburg, Russia
$^{124}$ $^{(a)}$ INFN Sezione di Pisa; $^{(b)}$ Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy
$^{125}$ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA, United States of America
$^{126}$ $^{(a)}$ Laboratório de Instrumentação e Física Experimental de Partículas - LIP, Lisboa; $^{(b)}$ Faculdade de Ciências, Universidade de Lisboa, Lisboa; $^{(c)}$ Department of Physics, University of Coimbra, Coimbra; $^{(d)}$ Centro de Física Nuclear da Universidade de Lisboa, Lisboa; $^{(e)}$ Departamento de Fisica, Universidade do Minho, Braga; $^{(f)}$ Departamento de Fisica Teorica y del Cosmos and CAFPE, Universidad de Granada, Granada (Spain); $^{(g)}$ Dep Fisica and CEFITEC of Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal
$^{127}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic
$^{128}$ Czech Technical University in Prague, Praha, Czech Republic
$^{129}$ Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic
$^{130}$ State Research Center Institute for High Energy Physics (Protvino), NRC KI,Russia, Russia
$^{131}$ Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom
$^{132}$ $^{(a)}$ INFN Sezione di Roma; $^{(b)}$ Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italy
$^{133}$ $^{(a)}$ INFN Sezione di Roma Tor Vergata; $^{(b)}$ Dipartimento di Fisica, Università di Roma Tor Vergata, Roma, Italy
$^{134}$ $^{(a)}$ INFN Sezione di Roma Tre; $^{(b)}$ Dipartimento di Matematica e Fisica, Università Roma Tre, Roma, Italy
$^{135}$ $^{(a)}$ Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies - Université Hassan II, Casablanca; $^{(b)}$ Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat; $^{(c)}$ Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; $^{(d)}$ Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; $^{(e)}$ Faculté des sciences, Université Mohammed V, Rabat, Morocco
$^{136}$ DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat à l'Energie Atomique et aux Energies Alternatives), Gif-sur-Yvette, France
$^{137}$ Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, United States of America
$^{138}$ Department of Physics, University of Washington, Seattle WA, United States of America
$^{139}$ Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom
$^{140}$ Department of Physics, Shinshu University, Nagano, Japan
$^{141}$ Fachbereich Physik, Universität Siegen, Siegen, Germany
$^{142}$ Department of Physics, Simon Fraser University, Burnaby BC, Canada
$^{143}$ SLAC National Accelerator Laboratory, Stanford CA, United States of America
$^{144}$ $^{(a)}$ Faculty of Mathematics, Physics & Informatics, Comenius University, Bratislava; $^{(b)}$ Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic
$^{145}$ $^{(a)}$ Department of Physics, University of Cape Town, Cape Town; $^{(b)}$ Department of Physics, University of Johannesburg, Johannesburg; $^{(c)}$ School of Physics, University of the Witwatersrand, Johannesburg, South Africa
$^{146}$ $^{(a)}$ Department of Physics, Stockholm University; $^{(b)}$ The Oskar Klein Centre, Stockholm, Sweden
$^{147}$ Physics Department, Royal Institute of Technology, Stockholm, Sweden
$^{148}$ Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America
$^{149}$ Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom
$^{150}$ School of Physics, University of Sydney, Sydney, Australia
$^{151}$ Institute of Physics, Academia Sinica, Taipei, Taiwan
$^{152}$ Department of Physics, Technion: Israel Institute of Technology, Haifa, Israel
$^{153}$ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel
$^{154}$ Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
$^{155}$ International Center for Elementary Particle Physics and Department of Physics, The University of Tokyo, Tokyo, Japan
$^{156}$ Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo, Japan
$^{157}$ Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
$^{158}$ Department of Physics, University of Toronto, Toronto ON, Canada
$^{159}$ $^{(a)}$ TRIUMF, Vancouver BC; $^{(b)}$ Department of Physics and Astronomy, York University, Toronto ON, Canada
$^{160}$ Faculty of Pure and Applied Sciences, and Center for Integrated Research in Fundamental Science and Engineering, University of Tsukuba, Tsukuba, Japan
$^{161}$ Department of Physics and Astronomy, Tufts University, Medford MA, United States of America
$^{162}$ Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia
$^{163}$ Department of Physics and Astronomy, University of California Irvine, Irvine CA, United States of America
$^{164}$ $^{(a)}$ INFN Gruppo Collegato di Udine, Sezione di Trieste, Udine; $^{(b)}$ ICTP, Trieste; $^{(c)}$ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Udine, Italy
$^{165}$ Department of Physics, University of Illinois, Urbana IL, United States of America
$^{166}$ Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden
$^{167}$ Instituto de Física Corpuscular (IFIC) and Departamento de Física Atómica, Molecular y Nuclear and Departamento de Ingeniería Electrónica and Instituto de Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC, Valencia, Spain
$^{168}$ Department of Physics, University of British Columbia, Vancouver BC, Canada
$^{169}$ Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada
$^{170}$ Department of Physics, University of Warwick, Coventry, United Kingdom
$^{171}$ Waseda University, Tokyo, Japan
$^{172}$ Department of Particle Physics, The Weizmann Institute of Science, Rehovot, Israel
$^{173}$ Department of Physics, University of Wisconsin, Madison WI, United States of America
$^{174}$ Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany
$^{175}$ Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany
$^{176}$ Department of Physics, Yale University, New Haven CT, United States of America
$^{177}$ Yerevan Physics Institute, Yerevan, Armenia
$^{178}$ Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), Villeurbanne, France
$^{a}$ Also at Department of Physics, King's College London, London, United Kingdom
$^{b}$ Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan
$^{c}$ Also at Novosibirsk State University, Novosibirsk, Russia
$^{d}$ Also at TRIUMF, Vancouver BC, Canada
$^{e}$ Also at Department of Physics, California State University, Fresno CA, United States of America
$^{f}$ Also at Department of Physics, University of Fribourg, Fribourg, Switzerland
$^{g}$ Also at Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Portugal
$^{h}$ Also at Tomsk State University, Tomsk, Russia
$^{i}$ Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France
$^{j}$ Also at Universita di Napoli Parthenope, Napoli, Italy
$^{k}$ Also at Institute of Particle Physics (IPP), Canada
$^{l}$ Also at Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom
$^{m}$ Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg, Russia
$^{n}$ Also at Louisiana Tech University, Ruston LA, United States of America
$^{o}$ Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona, Spain
$^{p}$ Also at Department of Physics, The University of Michigan, Ann Arbor MI, United States of America
$^{q}$ Also at Graduate School of Science, Osaka University, Osaka, Japan
$^{r}$ Also at Department of Physics, National Tsing Hua University, Taiwan
$^{s}$ Also at Department of Physics, The University of Texas at Austin, Austin TX, United States of America
$^{t}$ Also at Institute of Theoretical Physics, Ilia State University, Tbilisi, Georgia
$^{u}$ Also at CERN, Geneva, Switzerland
$^{v}$ Also at Georgian Technical University (GTU),Tbilisi, Georgia
$^{w}$ Also at Manhattan College, New York NY, United States of America
$^{x}$ Also at Hellenic Open University, Patras, Greece
$^{y}$ Also at Institute of Physics, Academia Sinica, Taipei, Taiwan
$^{z}$ Also at LAL, Université Paris-Sud and CNRS/IN2P3, Orsay, France
$^{aa}$ Also at Academia Sinica Grid Computing, Institute of Physics, Academia Sinica, Taipei, Taiwan
$^{ab}$ Also at School of Physics, Shandong University, Shandong, China
$^{ac}$ Also at Moscow Institute of Physics and Technology State University, Dolgoprudny, Russia
$^{ad}$ Also at Section de Physique, Université de Genève, Geneva, Switzerland
$^{ae}$ Also at International School for Advanced Studies (SISSA), Trieste, Italy
$^{af}$ Also at Department of Physics and Astronomy, University of South Carolina, Columbia SC, United States of America
$^{ag}$ Also at School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China
$^{ah}$ Also at Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow, Russia
$^{ai}$ Also at National Research Nuclear University MEPhI, Moscow, Russia
$^{aj}$ Also at Department of Physics, Stanford University, Stanford CA, United States of America
$^{ak}$ Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary
$^{al}$ Also at University of Malaya, Department of Physics, Kuala Lumpur, Malaysia
$^{*}$ Deceased
|
1511.00502
|
September 12, 2025
A $6.8 \ {\mathrm nb^{-1}}$ sample of $pp$ collision data collected under
low-luminosity conditions
at $\sqrt{s} = 7$ by the ATLAS detector at the Large Hadron Collider is
used to study diffractive dijet production. Events containing at least two jets
with $p_\mathrm{T} > 20$ are selected and analysed in terms of
variables which discriminate between diffractive and non-diffractive processes.
Cross sections are measured differentially in $\detaf$, the size of the
observable forward region of pseudorapidity which is devoid of hadronic
activity, and in an estimator, $\xiepz$, of the fractional momentum loss
of the proton assuming single diffractive dissociation ($pp \rightarrow pX$).
Model comparisons indicate a
dominant non-diffractive contribution up to moderately large
$\detaf$ and small $\xiepz$, with a diffractive contribution which
is significant at the highest $\detaf$ and the lowest $\xiepz$. The
rapidity-gap survival probability is estimated from comparisons of the data in
this latter region with predictions based on diffractive parton distribution
12 October 2015
Hardeep Bansil, Oldrich Kepka, Vlastimil Kus, Paul
Newman, Marek Tasevsky
Lauren Tompkins (chair), Christopher Meyer, Christophe
Royon, Andrew Hamiltion
|
1511.00502
|
September 12, 2025
Phys. Lett. B
A $6.8 \ {\rm nb^{-1}}$ sample of $pp$ collision data collected under
low-luminosity conditions
at $\sqrt{s} = 7$ by the ATLAS detector at the Large Hadron Collider is
used to study diffractive dijet production. Events containing at least two jets
with $p_\mathrm{T} > 20$ are selected and analysed in terms of
variables which discriminate between diffractive and non-diffractive processes.
Cross sections are measured differentially in $\detaf$, the size of the
observable forward region of pseudorapidity which is devoid of hadronic
activity, and in an estimator, $\xiepz$, of the fractional momentum loss
of the proton assuming single diffractive dissociation ($pp \rightarrow pX$).
Model comparisons indicate a
dominant non-diffractive contribution up to moderately large
$\detaf$ and small $\xiepz$, with a diffractive contribution which
is significant at the highest $\detaf$ and the lowest $\xiepz$. The
rapidity-gap survival probability is estimated from comparisons of the data in
this latter region with predictions based on diffractive parton distribution
|
1511.00198
|
[* Corresponding author.]
[E-mail address: [email protected] (X.D. Cao), [email protected] (C. Mortici) ]
[2010 Mathematics Subject Classification
: 11Y65, 33B15, 11A55, 41A20, 11J70]
[Key words and phrases: Mathieu series; Alternating Mathieu series; Continued fraction; Rational series; Simple closed form; Gamma function; Multiple-correction method; Rate of approximation.
[This work is supported by
the National Natural Science Foundation of China (Grant No.11171344) and the Natural
Science Foundation of Beijing (Grant No.1112010).]
Xiaodong Cao:
Department of Mathematics and Physics,
Beijing Institute of Petro-Chemical Technology,
Beijing, 102617, P. R. China
Cristinel Mortici:
Dept. of Mathematics,
Valahia University of Târgovişte, Bd. Unirii 18, 130082 Targovişte, Romania
Academy of Romanian Scientists, Splaiul Independenţei nr. 54, 050094 Bucharest, Romania
E-mail addresses: [email protected]]
The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.
§ INTRODUCTION
Let the general term of an infinite series have the form
\begin{align}
\end{align}
where $P_l(n), Q_m(n)$ are polynomials of degree $l$ and $m$ with real coefficients, respectively. Finding the sum of a rational series $\sum u_n$ in simple closed form is a very important research area, see, e.g.,
R.L. Graham, D.E. Knuth and O. Patashnik <cit.>, M. Petkovs̆ek, H.S. Wilf and D. Zeilberger <cit.>, H.S. Wilf <cit.> and references therein. Throughout the paper, the simple closed form always means a rational function with real coefficients. Otherwise, we expect that a continued fraction representation for the series may be discovered, see Chapter 12 in Berndt <cit.>, L. Lorentzen and H. Waadeland <cit.>, or A. Cuyt, V.B. Petersen, B. Verdonk, H. Waadeland, W.B. Jones <cit.>. The main purpose of this paper is to investigate a kind of fundamental rational series in a unified setting, which contains some mathematical constants and series, such as Catalan constant, $\zeta(2)$, Apéry number $\zeta(3)$, the Mathieu series and the alternating Mathieu series, etc. The Mathieu series was introduced by Émile Leonard Mathieu in his book <cit.>, which is defined by
\begin{align}
S(r):=\sum_{m=1}^{\infty}\frac{2m}{(m^2+r^2)^2},\quad (r>0),\label{Mathieu-Series}
\end{align}
while the alternating Mathieu series is given as follows
\begin{align}
\tilde{S}(r):=\sum_{m=1}^{\infty}(-1)^{m-1}\frac{2m}{(m^2+r^2)^2}.
\quad (r>0).\label{Alternating Mathieu-Series}
\end{align}
The Mathieu series has important applications in science, such as in the theory of elasticity of solid bodies <cit.>, or in the problem of the rectangular
plate <cit.> and it is closely related to the Riemann Zeta function $\zeta$ <cit.>. Moreover, A. Jakimovski and D.C. Russell <cit.> showed that an
extended form of the Mathieu series plays a role in examining Mercerian theorems for Cesáro summability. For the research history of the Mathieu series and the related series, readers interested may refer to R. Frontczak <cit.>,
G.V. Milovanović and T.K. Pogány <cit.>, C. Mortici <cit.>, T.K. Pogány, H.M. Srivastava and Z̆. Tomovski <cit.>, and references therein.
The paper is organized as follows. In Sec. 2, we gives several notations and definitions for later use. In Sec. 3, based on the work in <cit.>, we shall formulate a systematical method to look for either a simple closed form solution or the fastest possible finite continued fraction approximation solution for a class of linear difference equation of order one, sometimes we may guess further its continued fraction solution. In order to prove our new “conjectures", in Sec. 4 we shall prepare some important tools, which are two of the famous Ramanujan's continued fraction formulas for the quotient of the gamma functions. In Sec. 5 and 6, we shall investigate the rational series $\sum\frac{1}{Q_l(n)}$ for $l=2,3$, respectively, where $Q_l(x)$ is a polynomial of degree $l$ in $x$. In Sec. 7, we shall continue to study two extended Mathieu series. In Sec. 8, we shall give two applications of main results in Sec. 7. For example, we establish first a new representation for the alternating Mathieu series in the form of a linear combination of two continued fractions. In the last section, we analyze the related perspective of research in this direction.
§ NOTATION AND DEFINITION
Throughout the paper, we use the digamma notation $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$. The set $\mathbb{Z}\backslash\mathbb{N}$ means $\{0,-1,-2,\ldots\}$. The notation
$P_k(x)$ (or $Q_k(x)$) means a polynomial of degree $k$
in $x$, while $U(x)$ (or $V(x)$) denotes a rational function in $x$ . We shall use the $\Phi(k;x)$ to denote a polynomial of degree $k$ in $x$ with the leading coefficient equals one, which may be different at each occurrence. Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 0}$ be two sequences of real or complex numbers. The generalized continued fraction
\begin{align}
\tau=b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\ddots}}=b_0+
\begin{array}{ccccc}
a_1 && a_2 & \\
\cline{1-1}\cline{3-3}\cline{5-5}
b_1 & + & b_2 & + \cdots
\end{array}
\left(\frac{a_n}{b_n}\right)
\end{align}
is defined as the limit of the $n$th approximant
\begin{align}
\frac{A_n}{B_n}=b_0+\K_{k=1}^{n}\left(\frac{a_k}{b_k}\right)
\end{align}
as $n$ tends to infinity. In line with Ramanujan we adopt the convention that if $a_N=0$ and $a_n\neq 0$ for all $n<N$, then the
continued fraction $\tau$ terminates and has the value
\begin{align}
\tau=\frac{A_{N-1}}{B_{N-1}}=
\end{align}
This has the advantage that the continued fraction is always well defined. For the continued fraction theory, see L. Lorentzen and H. Waadeland <cit.>, and A. Cuyt, V.B. Petersen, B. Verdonk, H. Waadeland, W.B. Jones <cit.>, or other classical books therein.
For the sake of convenience, the sum $\sum_{j=n_1}^{n_2}a_j$ or the continued fraction $\K_{j=n_1}^{n_2}\left(\frac{c_j}{d_j}\right)$ for $n_2<n_1$ is stipulated to be zero.
In order to describe our method clearly, Let us recall three definitions introduced in <cit.>.
Definition 1.
Let $f(x)$ be a function defined on $(x_0,+\infty)$ for some real $x_0$. We assume that there exists a fixed positive number $\mu$ and a constant $c\neq 0$ such that $\lim_{x\rightarrow +\infty}x^{\mu}f(x)=c$. We define
\begin{align}
\mathrm{R}(f(x)):=\mu,
\end{align}
where $\mu$ is the exponent of $x^{\mu}$. For convenience, $\mathrm{R}(0)$ is stipulated to be infinity. Hence, $\mathrm{R}(f(x))$ characterizes the rate of convergence for $f(x)$ as $x$ tends to infinity.
Definition 2. Let $c_0\neq 0$, and $x$ be a free variable. Let $(a_n)_{n=1}^{\infty}$, $(b_n)_{n=1}^{\infty}$ and $(c_n)_{n=1}^{\infty}$ be three real sequences. The formal continued fraction
\begin{align}
\frac{c_0}{\Phi(\nu;x)+\K_{n=1}^{\infty}\left(\frac{a_n}{x+b_n}
\right)}
\end{align}
is said to be a Type-I continued fraction, i.e. when $n\ge 2$ the $n$-th partial denominator is a linear function in $x$ . While,
\begin{align}
\frac{c_0}{\Phi(\nu;x)+\K_{n=1}^{\infty}\left(\frac{a_n}{x^2+b_n x+c_n}\right)}
\end{align}
is said to be a Type-II continued fraction, i.e. when $n\ge 2$ the $n$-th partial denominator is a polynomial of degree 2 in $x$.
The Type-I and Type-II are two kinds of fundamental structures we often meet. Similarly, we may define other type continued fractions. Certainly, there also exists the hybrid-type continued fractions. In this paper, we shall not discuss the involved problems. It should be remarked that for the formal power series solution of the equation (<ref>) below, its structure is unique. However, the structure of the formal continued fraction solution is more complicated, which may be a Type-I or Type-II, or a hybrid-type. This is the main motivation that we introduce the definitions of a Type-I, Type-II, or other type to classify the fastest possible continued fraction solution.
Definition 3. If the sequence $(b_n)_{n=1}^{\infty}$ is a constant sequence $(b)_{n=1}^{\infty}$ in the Type-I ( or Type-II)
continued fraction, we call the number $\omega=b$ (or $\omega=\frac b2$) the $\mathrm{MC}$-point for the corresponding continued fraction.
We use $\hat{x}=x+\omega$ to denote the $\mathrm{MC}$-shift of $x$.
If there exists the $\mathrm{MC}$-point, we have the following simplified form
\begin{align}
\frac{c_0}{\Phi_1(\nu;\hat{x})+\K_{n=1}^{\infty}\left(\frac{a_n}
\mbox{or}\quad
\frac{c_0}{\Phi_1(\nu;\hat{x})+\K_{n=1}^{\infty}\left(\frac{a_n}
\end{align}
where $d_n=c_n-\frac{b^2}{4}.$
§ THE MULTIPLE-CORRECTION METHOD AND THE CONTINUED FRACTION SOLUTION OF LINEAR DIFFERENCE EQUATION OF ORDER ONE
Let $U(x)$ and $V(x)$ be rational functions in $x$. We consider the following linear difference equation of order one
\begin{align}
y(x)-U(x)y(x+1)-V(x)=0.\label{Difference Equation}
\end{align}
We are concern with the continued fraction solution of the difference equation (<ref>) when $x$ tends to infinity (or for “large" $x$). One of our purpose is to try to look for the fastest possible finite continued
fraction approximation solution or guess the formal continued fraction solution.
In the other words, we are looking for the solution (or approximation solution) of the equation (<ref>)
in the formal continued fraction space, which contains the subspace
of the rational function fields. Our method may be described as the following five steps.
(Step 1) Let us develop further the previous multiple-correction method formulated in <cit.>.
In fact, the
multiple-correction method is a recursive algorithm, and one of its advantages is that by repeating correction-process we always can accelerate the rate of approximation. More precisely, every non-zero coefficient plays an important role in accelerating the rate of approximation. For the sake of completeness, we shall give a description in details. The multiple-correction method consists of the following several steps.
(Step 1-1) The initial-correction. The choice of initial-correction is
vital. Determine the initial-correction $\mathrm{MC}_0(x)=\frac{c_0}{\Phi_0(\nu;x)}$ (or $\mathrm{MC}_0(x)=c_0\Phi_0(\nu;x)$) such that
\begin{align}
\frac{c_0}{\Phi_0(\nu;x+1)}-V(x)
\right)\\
\max_{c, \Phi(\nu;x)}\mathrm{R}
\left(\frac{c}{\Phi(\nu;x)}-U(x)
\frac{c}{\Phi(\nu;x+1)}-V(x)\right).\nonumber
\end{align}
In the second case, we can modify the approach above as follows
\begin{align}
\right)\\
\max_{c, \Phi(\nu;x)}\mathrm{R}
\left(c~\Phi(\nu;x)-U(x)c~\Phi(\nu;x+1)
\end{align}
In the sequel, we only give the exact expressions for the first case. For the second case, the correction-functions can be constructed mutatis mutandis as for the first case.
After determining the initial-correction, we define the initial-correction error function $E_0(x)$ by
\begin{align}
\end{align}
Find $\mathrm{R}(E_0(x))$. If $E_0(x)\equiv 0$, then the difference equation (<ref>) has a simple closed form solution $\mathrm{MC}_0(x)$.
Now we explain how to determine all the coefficients in $\mathrm{MC}_0(x)$. Firstly, we try to look for $c_0$ and $\nu$, which satisfy the following condition
\begin{align}
\max_{c,l}\mathrm{R}
\left(\frac{c}{x^l}-U(x)\frac{c}{(x+1)^l}
\end{align}
Secondly, If $\nu>0$, then we take $\mathrm{MC}_0(x)$ in the form $\frac{c_0}{\Phi_0(\nu;x)}$. Otherwise, we choose $\mathrm{MC}_0(x)=c_0 \Phi_0(-\nu;x)$. Thirdly, we may determine other coefficients in $\mathrm{MC}_0(x)$ by successively solving a linear equation.
(Step 1-2) The first-correction. If there exists a real number $\kappa_1$ such that
\begin{align}
\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+\frac{\kappa_1}{x}}
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1}}-V(x)
\right)
\end{align}
then we take the first-correction $\mathrm{MC}_1(x)=\frac{\kappa_1}{x+\lambda_1}$ with
\begin{align}
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda_1}}
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda}}-V(x)\right).
\nonumber
\end{align}
Otherwise, we take the first-correction $\mathrm{MC}_1(x)=\frac{\kappa_1}{x^2+\lambda_{1,1}x
+\lambda_{1,2}}$ such that
\begin{align}
\mathrm{R}
+\frac{\kappa_1}{x^2+\lambda_{1,1} x+\lambda_{1,2}}}
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{(x+1)^2+\lambda_{1,1} (x+1)+\lambda_{1,2}}}-V(x)
\right)\\
\max_{\kappa,\lambda_1,\lambda_2}\mathrm{R}
\left(\frac{c_0}{\Phi_0(\nu;x)
+\frac{\kappa}{x^2+\lambda_1 x+\lambda_2}}
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa}{(x+1)^2+\lambda_1 (x+1)+\lambda_2}}-V(x)
\right).\nonumber
\end{align}
If $\kappa_1=0$, we need to stop the correction-process, which means that the rate of approximation can not be further improved only by making use of Type-I or Type-II continued fraction structure. In the other words, in order to improve the rate of approximation, we have to choose a more general continued fraction structure instead of it. More precisely, we take first $\mathrm{MC}_1(x)=\frac{\kappa_1}{x^j}$.
Then we need to begin from $j=1$ and try step by step. Once we have found that the convergence rate can be improved for the first positive integer, say $j_0$, we use $\Phi(j_0; n)$ to replace $n^{j_0}$ immediately, and determine all the corresponding coefficients of the polynomial $\Phi(j_0; n)$, which is the main new idea introduced in <cit.>. Lastly, we choose $\mathrm{MC}_1(x)=\frac{\kappa_1}{\Phi(j_0; n)}$. In what follows,
we only describe our method for the structures of the Type-I
and Type-II continued fraction approximation.
Now we define the first-correction error function $E_1(x)$ by
\begin{align}
\end{align}
Find $\mathrm{R}(E_1(x))$. If $E_1(x)\equiv 0$, then the difference equation (<ref>) has a simple closed form solution
\begin{align}
\frac{c_0}{\Phi_0(\nu;x)
\begin{array}{cc}
c_0& \\
\cline{1-1}
\Phi_0(\nu;x) & +
\end{array}\mathrm{MC}_1(x).
\end{align}
(Step 1-3) The second-correction to the $k$th-correction. If $\mathrm{MC}_1(x)$ has the form
Type-I, we take the second-correction
\begin{align}
\mathrm{MC}_2(x)=\frac{\kappa_1}
\begin{array}{ccc}
\kappa_1& &\kappa_2\\
\cline{1-1}\cline{3-3}
x+\lambda_1 & +&x+\lambda_2
\end{array}
\end{align}
which satisfies
\begin{align}
\max_{\kappa,\lambda}\mathrm{R}
\left(\frac{c_0}{\Phi_0(\nu;x)
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda_1
\right).
\end{align}
Similarly to the first-correction, if $\kappa_2=0$, we stop the correction-process.
If $\mathrm{MC}_1(x)$ has the form Type-II, we take the second-correction
\begin{align}
\mathrm{MC}_2(x)=\frac{\kappa_1}
\end{align}
such that
\begin{align}
\left(\frac{c_0}{\Phi_0(\nu;x)
{x^2+\lambda_{1,1}x+\lambda_{1,2}+\frac{\kappa}{x^2+\lambda_1 x+\lambda_2}}}\right.\\
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}
+\lambda_1 (x+1)+\lambda_2}}}-V(x)
\right).\nonumber
\end{align}
If $\kappa_2=0$, we also need to stop the correction-process.
Now we define the second-correction error function $E_2(x)$ by
\begin{align}
\end{align}
If $E_2(x)\equiv 0$, then the difference equation (<ref>) has a simple closed form solution
\begin{align}
\frac{c_0}{\Phi_0(\nu;x)
\begin{array}{cc}
c_0& \\
\cline{1-1}
\Phi_0(\nu;x) & +
\end{array}\mathrm{MC}_2(x).
\end{align}
If we can continue the above correction-process to determine the $k$th-correction function $\mathrm{MC}_k(x)$ until some $k^*$ you want, then one may use a recurrence relation to determine the $k$th-correction $\mathrm{MC}_k(x)$. More precisely, in the case of Type-I we choose
\begin{align}
\mathrm{MC}_k(x)=\K_{j=1}^{k}
\left(\frac{\kappa_j}{x+\lambda_j}\right)
& \kappa_k \\
\cline{2-2}
+ & x+\lambda_k
\end{array}
\end{align}
such that
\begin{align}
\max_{\kappa,\lambda}\mathrm{R}
\left(\frac{c_0}{\Phi_0(\nu;x)
\frac{c_0}{\Phi_0(\nu;x+1)+G(\kappa,\lambda;x+1)}
\end{align}
\begin{align*}
& \kappa \\
\cline{2-2}
+ & x+\lambda
\end{array}=
\begin{array}{ccccc}
\kappa_1 & & \kappa_{k-1} & & \kappa \\
\cline{1-1}\cline{3-3}\cline{5-5}
x+\lambda_1 & +\cdots+ & x+\lambda_{k-1} & + &x+\lambda
\end{array}.
\end{align*}
While, in the case of Type-II we take
\begin{align}
\mathrm{MC}_k(x)=\K_{j=1}^{k}\left(\frac{\kappa_j}
\end{align}
which satisfies
\begin{align}
\max_{\kappa,\lambda_{1},\lambda_{2}}\mathrm{R}
\left(\frac{c_0}{\Phi_0(\nu;x)
\frac{c_0}{\Phi_0(\nu;x+1)+H(\kappa,\lambda_1,\lambda_2;x+1)}
\end{align}
\begin{align*}
\mathrm{MC}_{k-1}(x)\begin{array}{cc}
& \kappa \\
\cline{2-2}
+ & x^2+\lambda_1 x+\lambda_2
\end{array}\\
\kappa_1 & & \kappa_{k-1} & & \kappa \\
\cline{1-1}\cline{3-3}\cline{5-5}
x^2+\lambda_{1,1}x+\lambda_{1,2} & +\cdots+ & x^2+\lambda_{k-1,1}x+\lambda_{k-1,2} & + &x^2+\lambda_1 x+\lambda_2
\end{array}.
\end{align*}
In both cases, if $\kappa_k=0$, we have to stop the correction-process.
Now we define the $k$th-correction error function $E_k(x)$ by
\begin{align}
\end{align}
Find $\mathrm{R}(E_k(x))$. Lastly, if $E_k(x)\equiv 0$, then the difference equation (<ref>) has a simple closed form solution
\begin{align}
\frac{c_0}{\Phi_0(\nu;x)
%c_0& \\
%\Phi_0(\nu;x) & +
\end{align}
For the reader's convenience, we would like to give the complete Mathematica program for determining
all the coefficients in $\mathrm{MC}_k(x)$.
(1). First, let the function $Ek[x]$ be defined by (<ref>).
(2). Then we manipulate the following Mathematica command to
expand $Ek[x]$ into a power series in terms of $1/x$:
\begin{align}
\text{Normal}[\text{Series}
\text{/.}~ x\rightarrow
1/u, \{u,0,l_k\}]]\text{/.}~ u\rightarrow 1/x~(\text{// Simplify}).
\label{Ek-Mathematica-Program}
\end{align}
We remark that the variable $l_k$ needs to be suitable chosen according to the different functions and $k$. Another approach is that putting the whole thing over a common denominator such that $\mathrm{R}\left(E_k(x)\right)$
is strictly decreasing function of $k$. We may manipulate Mathematica commands “Together" and “Collect" to achieve them.
(3). Taking out the first some coefficients in the above power series, then we enforce them to be zero, and finally solve the related coefficients successively.
Actually, once we have found $\mathrm{MC}_k(x)$, (<ref>) can be used again to determine $\mathrm{R}\left(E_k(x)\right)$. In addition, we can apply it to check the general term formula for $\mathrm{MC}_k(x)$.
A lot of experiment results show that $\mathrm{R}\left(E_k(x)\right)\ge\mathrm{R}\left(E_0(x)\right)+2k$ in the case of Type-I, and $\mathrm{R}\left(E_k(x)\right)\ge\mathrm{R}\left(E_0(x)\right)+4k$ in the case of Type-II, respectively. We also note that the $k$th-approximation solution
\frac{c_0}{\Phi_0(\nu;x)
of the equation (<ref>) may be written in the form $\frac{P(l_1;x)}{Q(l_2;x)}$ with $l_2=\nu+k$ or $l_2=\nu+2k$, respectively, which explains that our method provides indeed an effective approach for the approximation solution problem of the equation (<ref>). We think that it should be the best possible rational approximation solution.
(Step 2) Find the general term formula of the $k$th-correction. Here we often use some tools in number theory, difference equation, etc.
If one can not find the general term formulas of both the $n$-th partial numerator and denominator, then only the finite continued fraction approximation solution can be provided. For instance, the BBP-type series of some mathematical constants like $\pi$, Catalan constant, $\pi^2$, etc. (e.g. see <cit.>). At the same times, it predicts that finding a continued fraction representation started started from this series seems “hopeless", one should replace other series expressions to try again and again, or find a linear combination solution of several continued fractions. Perhaps, an unexpected surprise will happen !
(Step 3) If we are lucky, we find that $E_{k^*}(x)\equiv 0$ for some integer $k^*$, then we attain a simple closed form (or a finite continued fraction) solution of the difference equation (<ref>). Now we give two examples to illustrate our method.
Example 1 Consider the following equation
\begin{align}
y(x)-y(x+1)=\frac{12 x^4-1}{(4 x^4+1 )^2}.\label{Example-1}
\end{align}
As $\mathrm{R}\left(\frac{12 x^4-1}{(4 x^4+1 )^2}\right)=4$, firstly, we look for the exponent $\nu$ and the $c_0$ such that $\nu, c_0$ satisfy the following condition
\begin{align*}
\max_{\nu,c}\mathrm{R}\left(\frac{c}{x^{\nu}}
-\frac{c}{(x+1)^{\nu}}-\frac{12 x^4-1}{(4 x^4+1 )^2}\right).
\end{align*}
It is not difficult to see that $\nu=3$. Then
we put three functions in the expression above over a common denominator, then let the first coefficient $-12 + 48 c$ in the numerator to be zero. In this way, we find $c_0=\frac 14$. Now we
choose the first-correction $\mathrm{MC}_0(x)=\frac{c_0}{\Phi_0(3;x)}
=\frac{1/4}{x^3+b_1x^2+b_2x+b_3}$ such that $b_1, b_2$ and $b_3$ satisfy the following condition
\begin{align*}
\max_{b_1,b_2,b_3}\mathrm{R}\left(\mathrm{MC}_0(x)
-\mathrm{MC}_0(x+1)-\frac{12 x^4-1}{(4 x^4+1 )^2}\right).
\end{align*}
By (<ref>) (also see example 3 below), we obtain that $b_1=-\frac 32, b_2=\frac 54, b_3=-\frac 38$. Then we
take $\mathrm{MC}_1(x)=\frac{\kappa_1}{x+\lambda_1}$ such that $\kappa_1, \lambda_1$ satisfy the following condition
\begin{align*}
\max_{\kappa,\lambda}\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)
\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa}{x+1+\lambda}}-\frac{12 x^4-1}{(4 x^4+1 )^2}\right).
\end{align*}
By using of (<ref>) again, we find that $\kappa_1=\frac{1}{16}$ and $\lambda_1=-\frac 12$.
Finally, we check directly that the following finite continued fraction
\begin{align}
\frac{\frac 14}{x^3 -\frac 32 x^2 +\frac 54 x -\frac 38
+ \frac{\frac{1}{16}}{x -\frac 12}}
\end{align}
is a solution of the equation (<ref>). For this example, it suffices for us to work for the initial-correction and the first-correction. We can also use Mathematica command “RSolve" to verify it again. Further, we note that after simplifying the solution above, the final closed form solution $ y(x)=\frac{-1 + 2 x}{2 (1 - 2 x + 2 x^2)^2}$ satisfies (<ref>) for all real $x$. Lastly, by the telescoping method, we find that
\begin{align}
\sum_{n=1}^{\infty}\frac{12 n^4-1}{(4 n^4+1 )^2}=y(1)=\frac 12.
\end{align}
Example 2
It seems it impossible to treat this problem by “RSolve" command of Mathematica software, because its a very huge of computations. By making use of our method, we can find that the following function
\begin{align}
\frac{3/16}{x^7 -\frac 72x^6 + \frac{91}{12}x^5 -\frac{245}{24}x^4 + \frac{1561}{144}x^3 -\frac{749}{
96}x^2 +\frac{9451}{1728}x -\frac{5845}{3456} +\mathrm{MC}_5(x) } \end{align}
is a solution of the equation
\begin{align}
y(x)-y(x+1)- \frac{1 - 480 x^4 + 8736 x^8 - 21504 x^{12} + 5376 x^{16}}{(1 + 4 x^4)^6}=0,
\end{align}
\begin{align*}
\mathrm{MC}_5(x)=\K_{j=1}^{5}\left(\frac{\kappa_j}{x-\frac 12}\right),\quad \kappa_1 = \frac{41041}{20736}, \kappa_2 = -\frac{1024}{1353}, \kappa_3 = \frac{243}{41041}, \kappa_4 = -\frac{451}{4368}, \kappa_5 = \frac{1}{48}.
\end{align*}
Certainly, the final simple closed form solution may be simplified as
$$\frac{(-1 + 2 x) (-1 - 2 x + 2 x^2) (1 - 6 x + 6 x^2)}{2 (1 - 2 x +2 x^2)^6}.$$
Hence, in a certain sense, our method may be viewed
as a supplement
of Gosper's algorithm (see Chapter 5 of Petkovsek et al. <cit.>, and some Exercises: 1 (c), 3 (b), 3 (f) in <cit.>), as our approach does not need to do factors
of polynomials, all things is that we only solve several linear
equations. We shall give some more examples in Sec. 5–7 below.
(Step 4) Based on Step 2, construct a formal continued fraction solution of of the difference equation (<ref>), then propose a reasonable conjecture. For instance, the continued fraction representation for many mathematical constants such as Catalan constant, $\zeta(2)$, Apéry number $\zeta(3)$, the Mathieu series, the alternating Mathieu series, etc. can be guessed by our method, some of them are new. Lastly, with the help of continued fraction theory and hypergeometric series, etc, we
try to prove and extend the assertion.
(Step 5) Based on Step 3, one may construct a simple closed form solution of some other equation (<ref>) by making use of the theory of the linear difference equation, for example, such as the following two properties:
(i) Let $f_1(x)$ and $f_2(x)$ be two rational function in $x$. If $y_j(x)~(j=1,2)$ is a simple closed form solution of the equation
\begin{align}
\end{align}
respectively, then for all $c_1,c_2\in \mathbb{R}$, $c_1y_1(x)+c_2y_2(x)$ is also a simple closed form solution of the equation
\begin{align}
\end{align}
(ii) Given a rational function $f(x)$. If the equation
\begin{align}
\end{align}
has a simple closed form solution, so is
\begin{align}
\end{align}
In fact, Example 1 and 2 were constructed by property (ii) and the (<ref>) in Theorem 9 with $(p,q,s,r)=(1,0,0,1/4)$. All calculations in this work were performed by using of Mathematica version 8.0.
Now we use the Mathieu series to illustrate how to guess its continued fraction representation.
Example 3 Let us consider the equation
\begin{align}
\end{align}
As $\mathrm{R}\left(\frac{2x}{(x^2+r^2)^2}\right)=3$, it is not hard to see that we should choose the initial-correction $\mathrm{MC}_0(x)$ in the form $\mathrm{MC}_0(x)=\frac{c_0}{x^2+d_1 x+d_2}$. We manipulate Mathematica software to expand $E_0(x)$ into a power series in terms of $1/x$
\begin{align}
=&\frac{-2 + 2 c_0}{x^3} - \frac{
3 (c_0 + c_0 d_1)}{x^4} +\frac{4 c_0 + 6 c_0 d_1 + 4 c_0 d_1^2 - 4 c_0 d_2 + 4 r^2}{x^5}+O\left(\frac{1}{x^6}\right).\nonumber
\end{align}
We enforce the first three coefficients to be zero, and find
\begin{align}
c_0=1,\quad d_1=-1,\quad d_2=\frac{1 + 2 r^2}{2}.
\end{align}
Note that the solution is unique ! By Mathematica software again, one may check that $\mathrm{R}\left(E_0(x)\right)=7$. Repeating the process above several times, one observes that the $k$th-correction $\mathrm{MC}_k(x)$ is a Type-II and the $\mathrm{MC}$-point
$\omega=-\frac 12$. As the detail is quite similar to the initial-correction, here we only list final computation results as follows
\begin{align}
\mathrm{MC}_k(x)=\K_{j=1}^{k}\left(\frac{\kappa_j}
\end{align}
\begin{align*}
&\kappa_1=-\frac{1}{12} (1 + 4 r^2),\quad \lambda_1=\frac{5 + 4 r^2}{4},\\
&\kappa_2=-\frac{16}{15} (1 + r^2),\quad \lambda_2=\frac{13 + 4 r^2}{4},\\
&\kappa_3=-\frac{81}{140} (9 + 4 r^2),\quad \lambda_3=\frac{25 + 4 r^2}{4},\\
&\kappa_4=-\frac{256}{63} (4 + r^2),\quad \lambda_4=\frac{41 + 4 r^2}{4},\\
&\kappa_5=-\frac{625}{396} (25 + 4 r^2),\quad \lambda_5=\frac{61 + 4 r^2}{4},\\
&\kappa_6=-\frac{1296}{143} (9 + r^2),\quad \lambda_6=\frac{85 + 4 r^2}{4},\\
&\kappa_7=-\frac{2401}{780} (49 + 4 r^2),\quad \lambda_7=\frac{113 + 4 r^2}{4}.
\end{align*}
Just as did in Sec. 8 of <cit.>, by careful data analysis and further checking, we may guess that the following formal continued fraction
\begin{align}
\frac{1}{\left(x-\frac 12\right)^2+\frac 14\left(1+4r^2\right)+\K_{n=1}^{\infty}
\left(\frac{\kappa_n}{\left(x-\frac 12\right)^2+\lambda_n}\right)}
\end{align}
should be a solution of the equation (<ref>), where
\begin{align}
\kappa_n= -\frac{n^4\left(n^2 + 4 r^2\right)}{4 (2 n - 1) (2 n + 1)},\quad\lambda_n=\frac 14 (2 n^2 + 2 n + 1 + 4 r^2).
\end{align}
Finally, applying the conjecture above, (<ref>) and the telescoping method, we could conjecture further a continued fraction formula for the Mathieu series, which was already proved in <cit.>. Also see Example 8 in Sec. 7 below.
As a briefly summary of this section, we stress that the order of priority for our method is as follows:
The best situation is to look for a simple closed form solution, which is a finite continued fraction; The next one is to find a continued fraction solution; The third is to find a linear combination solution
of several continued fractions; The last one is
to find a finite continued fraction approximation solution as you want, some examples, see <cit.>.
On one hand, in order to determine all the related coefficients, we often use an appropriate symbolic computation software, which needs a huge of computations. On the other hand, the exact expression at each occurrence also takes a lot of space. Moreover, in order to guess the continued fraction formula, we have to do a lot of additional works. All theorems in Sec. 5 to 7 are built on experimental results described in this section. Hence, we shall focus on the rigorous proof of all conjectures, and omit the related details for guessing these formulas. Readers interested may refer to Sec. 6 and 8 in reference <cit.>.
§ SOME PRELIMINARY LEMMAS
In this section, we shall prepare some lemmas for later use. The main lemmas are two of Ramanujan's continued fraction formulas involving the quotient of the gamma functions (see <cit.>).
Let $x,m,$ and $n$ be complex. We define
\begin{align}
Q=Q(x,m,n):=\frac{\Gamma\left(\frac 12(x+m-n+1)\right)\Gamma\left(\frac 12(x-m+n+1)\right)}{\Gamma\left(\frac 12(x+m+n+1)\right)\Gamma\left(\frac 12(x-m-n+1)\right)}.
\end{align}
If either $m$ or $n$ is an integer or if $\Re x>0$, then
\begin{align}
\frac{1-Q}{1+Q}=\frac{mn}{x+\K_{j=1}^{\infty}
\left(\frac{(m^2-j^2)(n^2-j^2)}{
\end{align}
This is Entry 33 of Berndt <cit.>. For its research history, see p. 156 in <cit.>.
Let $x,l,m$, and $n$ denote complex numbers. We define
\begin{align}
=:&\frac{\Gamma\left(\frac 12(x+l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m-n+1)\right)\Gamma\left(\frac 12(x-l-m+n+1)\right)}{\Gamma\left(\frac 12(x-l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m+n+1)\right)\Gamma\left(\frac 12(x+l+m-n+1)\right)}.\nonumber
\end{align}
Then if either $l,m$, or $n$ is an integer or if $\Re x>0$,
\begin{align}
\frac{1-P}{1+P}=\frac{2lmn}{x^2-l^2-m^2-n^2+1+\K_{j=1}^{\infty}
\left(
\frac{4(l^2-j^2)(m^2-j^2)(n^2-j^2)}{(2j+1)\left(x^2-l^2-m^2-n^2
\end{align}
This is Entry 35 of B. C. Berndt <cit.>, which was claimed first by Ramanujan <cit.>. The first published proof was provided by Watson <cit.>. For the full proof of
Lemma 2, we refer the reader to L. Lorentzen's paper <cit.>.
$b_0+\K\left(a_n/b_n\right)\approx d_0+\K\left(c_n/d_n\right)$ if and only if there exists a sequence $\{r_n\}$ of complex numbers with $r_0=1,r_n\neq 0$ for all $n\in \mathbb{N}$, such that
\begin{align}
d_0=b_0,\quad c_n=r_{n-1}r_n a_n,\quad d_n=r_n b_n\quad \mbox{for all
$n\in \mathbb{N}$.}
\end{align}
See Theorem 9 in L. Lorentzen, H. Waadeland <cit.> .
§ THE RATIONAL SERIES $\SUM\FRAC{1}{N^2+AN+B}$
Let $a,b\in\mathbb{R}$. Given the following infinite series
\begin{align}
\end{align}
where $n_0$ is a suitable non-negative integer such that
$n^2+an+b\neq 0$
for all integer $n\ge n_0$. It is a natural question when the coefficient
$a$ and $b$ satisfy the conditions, one can get a simple closed
form for the sum of $S(a,b)$. It seems that the results in this section are not very new. However, our purpose is to treat it in a unified setting.
Firstly, we shall study the following difference equation
\begin{align}
y(x)-y(x+1)=\frac{1}{x^2+a x +b}.%\quad x>x_0,
\end{align}
Let $a, b\in \mathbb{R}$, and the formal continued fraction $F(a,b;x)$ or shortly $F(x)$, be defined by
\begin{align}
\left(\frac{\kappa_n}{x+\omega}\right)
\end{align}
\begin{align}
\omega=\frac{a-1}{2},\quad\kappa_n=\frac{n^2(n^2 + 4 b - a^2)}{4 (2 n - 1) (2 n + 1)}.\label{omega-1}
\end{align}
We assume that $x\notin\{q+\alpha: q\in\mathbb{Z}\backslash\mathbb{N},~ {\alpha}^2+a{\alpha} +b=0, ~\alpha\in\mathbb{C} \}$. If either $\sqrt{a^2-4b}\in\mathbb{N}$ or $\Re x>-\omega$, then
\begin{align}
\frac{1}{x^2+a x+b}.\label{Difference Equation-1}
\end{align}
We shall discuss the following three cases.
(Case 1)
Assume $b<\frac{a^2}{4}$.
By Lemma 1 with
$(x,m)=\left(2(x+\omega),\sqrt{a^2-4b}\right)$, under the conditions of Theorem 1 we have
\begin{align}
\frac{1-Q}{1+Q}=\frac{ n\sqrt{a^2-4b}}{2(x+\omega)+\K_{j=1}^{\infty}\left(
\frac{(a^2-4b-j^2)(n^2-j^2)}{2(2j+1)\left(x+\omega\right)}\right)}.
\end{align}
Dividing both sides by $ n\sqrt{a^2-4b}$ and letting $n$ tend to zero, on the right side, we deduce that
\begin{align}
\frac{1}{2(x+\omega)+\K_{j=1}^{\infty}\left(
\frac{j^2(j^2+4b-a^2)}{2(2j+1)\left(x+\omega\right)}\right)}.
\end{align}
The following classical representation is well-known (e.g., see <cit.>)
\begin{align}
\psi(z+1)=-\gamma+\sum_{k=1}^{\infty}\left(\frac 1k-\frac{1}{k+z}\right),\quad (z\neq -1,-2,-3,\ldots),\label{PolyGamma-Exprssion}
\end{align}
where $\gamma$ denotes Euler-Mascheroni constant. In the sequel, we shall use this formula several times, usually without comment.
On the other hand, from the definition of $Q$, we see easily that $\lim_{n\rightarrow 0}Q=1$. A direct calculation with the use of L'Hospital's rule gives
\begin{align}
&\frac{1}{\sqrt{a^2-4b}}\lim_{n\rightarrow 0}\frac{1-Q}{n(1+Q)}
=\frac{1}{\sqrt{a^2-4b}}\lim_{n\rightarrow 0}\frac{1}{1+Q}
\lim_{n\rightarrow 0}\frac{1-Q}{n}
=\frac{1}{2\sqrt{a^2-4b}}\lim_{n\rightarrow 0}\frac {\partial}{\partial n}(1-Q)
\\
\left\{-\psi\left(x+\frac{a-\sqrt{a^2-4b}}{2}\right)
=&\frac{1}{2}\sum_{k=0}^{\infty}\frac{1}{(k+x+\frac a2)^2-\frac{a^2-4b}{4}}.\nonumber
\end{align}
\begin{align*}
\sum_{k=0}^{\infty}\frac{1}{(k+x+\frac a2)^2-\frac{a^2-4b}{4}}
\frac{j^2(j^2+4b-a^2)}{2(2j+1)\left(x+\omega\right)}\right)}
\end{align*}
where we used Lemma 3 in the last equality.
Under the conditions of Theorem 1, it is not difficult to prove that
\begin{align}
F(x)-F(x+1)=&\sum_{k=0}^{\infty}\frac{1}{(k+x+\frac a2)^2-\frac{a^2-4b}{4}}-\sum_{k=0}^{\infty}\frac{1}{(k+x+1+\frac a2)^2-\frac{a^2-4b}{4}}\\
=&\frac{1}{(x+\frac a2)^2-\frac{a^2-4b}{4}}
=\frac{1}{x^2+a x+b}.\nonumber
\end{align}
Hence the identity (<ref>) is true in this case.
(Case 2) Suppose $b>\frac{a^2}{4}$, it follows from Lemma 1 with $(x,m)=\left(2(x+\omega),\sqrt{4b-a^2}~i\right)$ that
\begin{align}
\frac{1-Q}{1+Q}=\frac{ n\sqrt{4b-a^2}~i}{2(x+\omega)+\K_{j=1}^{\infty}\left(
\frac{(a^2-4b-j^2)(n^2-j^2)}{2(2j+1)\left(x+\omega\right)}
\right)}.
\end{align}
By using of L'Hospital's rule, we deduce that
\begin{align}
&\frac{1}{\sqrt{4b-a^2}~i}\lim_{n\rightarrow 0}\frac{1-Q}{n(1+Q)}\\
=&\frac{1}{\sqrt{4b-a^2}~i}\lim_{n\rightarrow 0}\frac{1}{1+Q}
\lim_{n\rightarrow 0}\frac{1-Q}{n}
=\frac{1}{2\sqrt{4b-a^2}~i}\lim_{n\rightarrow 0}\frac {\partial}{\partial n}(1-Q)\nonumber
\\
\left(-\psi\left(x+\frac{a-\sqrt{4b-a^2}~i}{2}\right)
=&\frac{1}{2}\sum_{k=0}^{\infty}\frac{1}{(k+x+\frac a2)^2+\frac{4b-a^2}{4}}.\nonumber
\end{align}
Quite similarly to Case 1, Theorem 1 holds true in Case 2.
(Case 3) $b=\frac{a^2}{4}$. In this case, it follows from Lemma 1 with $x=2(x+\omega)$ that
\begin{align}
\lim_{m\rightarrow 0}\frac 1m\lim_{n\rightarrow 0}
\frac{1-Q}{n(1+Q)}=&\lim_{m\rightarrow 0}\lim_{n\rightarrow 0}
\frac{1}{2(x+\omega)+\K_{j=1}^{\infty}\left(
\frac{(m^2-j^2)(n^2-j^2)}{2(2j+1)\left(x+\omega\right)}\right)}\\
\frac{j^4}{2(2j+1)\left(x+\omega\right)}\right)}=\frac 12 F(x).\nonumber
\end{align}
By making use of L'Hospital's rule twice, we find that
\begin{align}
&\lim_{m\rightarrow 0}\frac 1m\lim_{n\rightarrow 0}
\frac{1-Q}{n(1+Q)}=\lim_{m\rightarrow 0}\frac 1m\left(
\lim_{n\rightarrow 0}\frac{1}{1+Q}\lim_{n\rightarrow 0}
\frac{1-Q}{n}
\right)\\
=&\frac 12\lim_{m\rightarrow 0}\frac 1m\left\{-\psi\left(x+\frac{a-m}{2}\right)
=&\frac 12\psi'(x+\frac a2)=\frac 12\sum_{k=1}^{\infty}\frac{1}{(k-1+x+\frac{a}{2})^2}.\nonumber
\end{align}
Applying the similar argument as the proof of Case 1, one has
\begin{align}
-\sum_{k=1}^{\infty}\frac{1}{(k+x+\frac{a}{2})^2}\label{Theorem-1-Case 3}\\
=&\frac{1}{(x+\frac a2)^2}=\frac{1}{x^2+a x+b}.\nonumber
\end{align}
We remark that combining Case 1, Case 2 and a limiting process for (<ref>) (i.e. let $b$ tend to $\frac{a^2}{4}$), the (<ref>) may be proved easily. This completes the proof of Theorem 1. With the notations of Theorem 1, let $n_0$ is a non-negative integer such that $n_0>\left\{\alpha:~\alpha\in\{\frac {-a\pm\sqrt{a^2-4b}}{2}\}~\text{and}~\alpha\in\mathbb{Z}\right\}$. If either $\sqrt{a^2-4b}\in \mathbb{N}$ or $n_0>\frac{-a+1}{2}$, then
\begin{align}
\sum_{n=n_0}^{\infty}\frac{1}{n^2+an+b}=F(n_0).
\end{align}
In particular, if
\begin{align}
b\in \left\{\frac{a^2-k^2}{4}: a\in \mathbb{R}, k\in \mathbb{N}\right\},
\end{align}
and $n_0>\max\left\{\alpha: \alpha\in\{-1,\frac {-a\pm k}{2}\}~\text{and}~\alpha\in\mathbb{Z}\right\}$, then
\begin{align}
\sum_{n=n_0}^{\infty}\frac{1}{n^2+an+b}=
\frac{1}{n_0+\frac{a-1}{2}+\K_{n=1}^{k-1}
\left(\frac{\frac{n^2(n^2 -k^2)}{4 (2 n - 1) (2 n + 1)}}{n_0+\frac{a-1}{2}}\right)
\end{align}
It follows readily from Theorem 1 and the telescoping method.
Example 4. We let $k=3$, $b=\frac{a^2-9}{4}$, $\omega=\frac{a-1}{2}$, and
\begin{align}
\left(\frac{\kappa_n}{x+\omega}\right)
=\frac{2 (-1 - 12 x + 12 x^2 - 6 a + 12 x a +
3 a^2)}{3 (-1 + 2 x + a) (-3 - 4 x + 4 x^2 -
2 a + 4 x a + a^2)}.\nonumber
\end{align}
We can check directly that if $x\neq \frac{1-a}{2}+q, \frac{-a\pm 3}{2}+q$, $q\in \{0,-1\}$, then
\begin{align}
F_a(x)-F_a(x+1)=\frac{1}{x^2+a x+\frac{a^2-9}{4}}.\label{Example 1-1}
\end{align}
Let $p$ be prime and set $a=\sqrt{p}$, then the following series is an irrational number
\begin{align}
\sum_{n=1}^{\infty}\frac{1}{n^2+\sqrt {p} n+\frac {p-9}{4}}=F_{\sqrt {p}}(1)=\frac{6 (p - 1) - 2 \sqrt{p} (3 p - 19)}{3 (-9 + 10 p - p^2)}.\label{Example 1-2}
\end{align}
For instance, $
F_{\sqrt {2}}(1)=\frac{2(3 + 13 \sqrt{2})}{21}$ is irrational. As a by-product of Theorem 2, one may employ the infinite series in (<ref>) to construct many irrational numbers . Moreover, we can check (<ref>) and (<ref>) by applying of Mathematica commands “RSolve" and “Sum", respectively.
Example 5. Let $p,q\in\mathbb{N}$ with $p>q$ and $(p,q)=1$, and $r>0$, then the following rational series
\begin{align}
\sum_{n=1}^{\infty}\frac{1}{(p n+q)^2},\quad \sum_{n=1}^{\infty}\frac{1}{n^2+r^2}
\end{align}
can be written as a continued fraction expansion. For complex $r$, Ramanujan even deduced a exact expression for the second series above, see Entry 24(i) and (ii) in <cit.>. From Whittaker and Watson's text <cit.>, one has
\begin{align}
\frac{1}{e^x-1}=\frac 1x-\frac 12+\sum_{m=1}^{\infty}\frac{2x}{x^2+4\pi ^2m^2}.
\end{align}
However, our approach is different from their methods.
§ THE RATIONAL SERIES $\SUM\FRAC{1}{Q_3(N)}$
Let $a,b,c\in\mathbb{R}$ and $Q_3(x)=x^3+a x^2+b x+c$. Similarly to the previous section, we first study the following difference equation
of order one
\begin{align}
y(x)-y(x+1)=\frac{1}{Q_3(x)}.\label{Difference Equation-2}
\end{align}
From the fundamental theorem of algebra, a polynomial of degree three with real coefficients may be expressed as
\begin{align}
Q_3(x)=(x+t)(x^2+r x+s),
\end{align}
here $r,s,t\in\mathbb{R}$. If the discriminant $\Delta=r^2-4s\ge 0$ for the last polynomial of degree two, then we further write it in the form
\begin{align}
Q_3(x)=(x+t)(x+\alpha)(x+\beta),\quad \alpha,\beta\in \mathbb{R}.
\end{align}
If $\alpha=\beta=t$, then it reduces to Entry 32 (iii) in Berndt <cit.>, also see Case 3 in the proof of Theorem 3 below. Otherwise, without loss of the generality, we assume $\alpha\neq \beta$. In which case, it follows from Theorem 1 that for
$\Re x>\max\{-\frac{\alpha+t-1}{2},-\frac{\beta+t-1}{2}\}$
\begin{align}
\frac{1}{(x+t)(x+\alpha)}-\frac{1}{(x+t)(x+\beta)}\right)
\label{trivial assertion-1}\\
=&\frac{1}{\beta-\alpha}\left\{(F(t+\alpha,t\alpha;x)-F(t+\alpha,t \alpha;x+1))
-(F(t+\beta,t \beta;x)-F(t+\beta,t \beta;x+1))\right\},\nonumber
\end{align}
where $F(a,b;x)$ is given as (<ref>). By Theorem 1, it is not hard to get the following assertion.
Example 6. Let $\upsilon\in \mathbb{R}$, $k\in \mathbb{Z}\backslash\{0,1\}$ and $d\in \mathbb{Z}\backslash\{0\}$. If $x>\max\{-\upsilon, -\upsilon-d, -\upsilon-k d\}$, then the following equation
\begin{align}
y(x)-y(x+1)=\frac{1}{(x+\upsilon)(x+\upsilon+d)(x+\upsilon+k d)},\label{Example -4}
\end{align}
has a simple closed form solution.
If $\Delta=r^2-4s<0$, to the best knowledge of authors, up to now
very little has been established except in the form of complex function.
A lot of experiment results show that the structure of the continued fraction solution for the equation (<ref>) may be the type I or type II or other type according to the various conditions of the parameters $a,b$ and $c$. In this section, we shall apply our method to find new results for two special classes, and then give further remarks.
§.§ For the case of $c=\frac{-2 a^3 + 9 a b}{27}$
Let $a, b\in\mathbb{R}$ and $c=\frac{-2 a^3 + 9 a b}{27}$. Let the formal continued fraction $G_1(x)$ be defined by
\begin{align}
G_1(x):=\frac{1/2}{(x+\omega)^2+\frac{3 - 2 a^2 + 6 b}{12}+\K_{n=1}^{\infty}
\left(\frac{\kappa_n}{(x+\omega)^2+\lambda_n}\right)},
\end{align}
where $\omega=\frac{2 a-3}{6}$ and
\begin{align}
\kappa_n=-\frac{n^2\left(-3 n^2 + a^2 - 3 b\right)^2}{6^2 (2 n -1)
(2 n + 1)},\quad
\lambda_n=\frac{3 - 2 a^2 + 6 b}{12} + \frac{ n + n^2}{2}.
\end{align}
Assume that $x\notin\{q+\alpha: q\in\mathbb{Z}\backslash\mathbb{N},~ {\alpha}^3+a{\alpha}^2 +b\alpha +c=0,~\alpha\in\mathbb{C} \}$. If either $\sqrt{(a^2-3 b)/3})\in\mathbb{N} $ or $\Re x>-\omega$, then
\begin{align}
G_1(x)-G_1(x+1)=\frac{1}{x^3 + a x^2 + b x + c}.\label{Difference Equation-3}
\end{align}
It is not difficult to verify that
\begin{align}
x^3 + a x^2 + b x + c
=\frac{1}{27} (a + 3 x) (-2 a^2 + 9 b + 6 a x + 9 x^2),
\end{align}
and the last polynomial of degree $2$ above has the discriminant
\begin{align}
\Delta=(6a)^2-4\cdot 9(-2 a^2 + 9 b)=108 (a^2 - 3 b)\begin{cases}
\geq 0,\quad \mbox{if $b\leq \frac{a^2}{3}$},\\
<0,\quad \mbox{otherwise.}
\end{cases}
\end{align}
We shall consider three cases.
(Case 1) $a^2-3 b>0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{(a^2-3 b)/3},\sqrt{(a^2-3 b)/3})$ and dividing both sides by $2 n(a^2-3 b)/3$, under the conditions of Theorem 5 we obtain that
\begin{align}
\frac{3}{2(a^2-3 b)}\frac{1-P}{n(1+P)}
=\frac{1}{4(x+\omega)^2-n^2-\frac{2(a^2-3 b)}{3}+1+\K_{j=1}^{\infty}\left(
\frac{4\left(\frac{a^2-3b}{3}-j^2\right)^2(n^2-j^2)}{(2j+1)
\left(4(x+\omega)^2-n^2-\frac{2(a^2-3 b)}{3}
\end{align}
Now let $n$ tend to zero. On the right side, we arrive at
\begin{align}
\frac{1}{4(x+\omega)^2-\frac{2(a^2-3 b)}{3}+1+\K_{j=1}^{\infty}\left(
\frac{-4j^2\left(\frac{a^2-3b}{3}-j^2\right)^2}{(2j+1)
\left(4(x+\omega)^2-\frac{2(a^2-3 b)}{3}
=\frac 12 G_1(x).
\end{align}
On the other hand, from the definition of $P$, we observe easily that $\lim_{n\rightarrow 0}P=1$. A direct calculation with the use of L'Hospital's rule gives
\begin{align}
&\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}=\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}
\frac {\partial}{\partial n}(1-P)
\\
=&\psi\left( x+\frac a3\right)-\frac 12\psi\left(x+\frac a3+\sqrt{\frac{a^2-3b}{3}}\right)-\frac 12\psi\left(x+\frac a3-\sqrt{\frac{a^2-3b}{3}}\right)\nonumber\\
=&\frac 12\sum_{k=0}^{\infty}\left(-\frac{2}{k+x+a/3}+\frac{1}{k+x+a/3+
\sqrt{\frac{a^2-3b}{3}}}+\frac{1}{k+x+a/3-
\sqrt{\frac{a^2-3b}{3}}}
\right)\nonumber\\
=&\frac{a^2-3b}{3}\sum_{k=0}^{\infty}\frac{1}{(k + x+\frac a3)\left((k + x+\frac a3)^2+3b-a^2\right)}.\nonumber
\end{align}
\begin{align}
=&\sum_{k=0}^{\infty}\frac{1}{( k + x+\frac a3)\left(( k + x+
\frac a3)^2-a^2+3b\right)}-\sum_{k=0}^{\infty}\frac{1}{(k + x+1+\frac a3)\left(( k + x+1+\frac a3)^2-a^2+3b\right)}\nonumber\\
=&\frac{1}{x^3 + a x^2 + b x + c}.\nonumber
\end{align}
This completes the proof of Theorem 3 in this case.
(Case 2) $a^2-3 b<0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{(3 b-a^2)/3}~i,\sqrt{(3 b-a^2)/3}~i)$ and dividing both sides by $2 n(a^2-3 b)/3$, similarly to Case 1, we also have
\begin{align}
&\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}=\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}
\frac {\partial}{\partial n}(1-P)\\
=&\psi\left( x+\frac a3\right)-\frac 12\psi\left(x+\frac a3+\sqrt{\frac{3b-a^2}{3}}~i\right)-\frac 12\psi\left(x+\frac a3-\sqrt{\frac{3b-a^2}{3}}~i\right)\nonumber\\
=&\frac 12\sum_{k=0}^{\infty}\left(-\frac{2}{k+x+a/3}+\frac{1}{k+x+a/3+
\sqrt{\frac{3b-a^2}{3}}~i}+\frac{1}{k+x+a/3-
\sqrt{\frac{3b-a^2}{3}}~i}
\right)\nonumber\\
=&\frac{a^2-3b}{3}\sum_{k=0}^{\infty}\frac{1}{(k + x+\frac a3)\left((k + x+\frac a3)^2+3b-a^2\right)}.\nonumber
\end{align}
Hence, Theorem 3 is true in Case 2.
(Case 3) $a^2-3 b=0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),n, n)$, dividing both sides by $2n^3$,
and employing L'Hospital's rule three times, then
\begin{align}
\lim_{n\rightarrow 0}\frac{1-P}{n^3(1+P)}=&
\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n^3}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}\frac{1}{3n^2}
\frac {\partial}{\partial n}(1-P)
\\
\frac a3\right)=\sum_{k=0}^{\infty}\frac{1}{\left(k+x+
\frac a3\right)^3}.\nonumber
\end{align}
Now Theorem 3 follows from the trivial equality
\begin{align*}
x^3 + a x^2 + b x + c=\left(x+
\frac a3\right)^3.
\end{align*}
Lastly, combining three cases above will finish the proof of Theorem 3. With the notations of Theorem 3, let $n_1$ be a non-negative integer such that $n_1>\max\left\{\alpha:~\alpha\in\{-\frac a3, -\frac a3\pm\sqrt{\frac{a^2-3b}{3}}\}~\text{and}~\alpha\in \mathbb{Z}\right\}$. If
either $\sqrt{(a^2-3 b)/3}\in\mathbb{N} $ or $n_1>-\omega$ , then
\begin{align}
\sum_{n=n_1}^{\infty}\frac{1}{n^3+an^2+bn +\frac{-2 a^3 + 9 a b}{27}}=G_1(n_1).
\end{align}
In particular, if
\begin{align}
b\in\mathfrak{D}_1=\{\frac{a^2}{3}-k^2: k\in \mathbb{N}, a\in\mathbb{R}\},
\end{align}
and $n_1>\max\left\{\alpha:~\alpha\in\{-1, -\frac a3, -\frac a3\pm k\}~\text{and}~\alpha\in \mathbb{Z}\right\}$, then
\begin{align}
\sum_{n=n_1}^{\infty}\frac{1}{n^3+an^2+bn +\frac{-2 a^3 + 9 a b}{27}}=\frac{1/2}{(n_1+\omega)^2+\frac{1-2k^2}{4}+\K_{n=1}^{k-1}
\left(\frac{-\frac{n^2(k^2-n^2)^2}{4(2n-1)(2n+1)}}
\end{align}
It follows from Theorem 3 and the telescoping method.
§.§ For the infinite series
Let $u, v\in\mathbb{R}$. Let the formal continued fraction $G_2(x)$ be defined by
\begin{align}
\left(\frac{\kappa_n}{(x+\omega)^2+\lambda_n}\right)},
\end{align}
\begin{align}
&\omega=-\frac 12+u-v,\quad\lambda_n=\frac{2n^2 + 2n + 1}{4}-\frac{(u-v)^2}{8},\\
&\kappa_n=-\frac{n^2 \left(- n^2 + \left(\frac{ u-v}{2}\right)^2\right)^2 }{4 (2 n - 1) (2 n + 1)}.
\end{align}
Let $x\notin\left\{\alpha +q: q\in\mathbb{Z}\backslash\mathbb{N}, \alpha\in\{-u,-v, -\frac{u+v}{2} \}\right\}$. If either $\frac{ u-v}{2}\in\mathbb{Z}\backslash\{0\}$ or $\Re x>-\omega$, then
\begin{align}
\frac{1}{(x+u)\left(x+\frac{u+v}{2}\right)
\left(x+v\right)} .\label{Difference Equation-4}
\end{align}
In what follows, we always assume $u\neq v$, otherwise it turns to Case 3 in the proof of Theorem 3. We set $t=\frac{(u - v)^2}{8}$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\frac{u-v}{2},\frac{u-v}{2})$ and dividing both sides by $\frac{n(u-v)^2}{2}$, under the conditions of Theorem 5 we get that
\begin{align}
\frac{2}{(u-v)^2}\frac{1-P}{n(1+P)}
\frac{4\left(\left(\frac{u-v}{2}\right)^2-j^2\right)^2(n^2-j^2)}{(2j+1)
\left((2x+2\omega)^2-n^2-4t
\end{align}
Now let $n$ tend to zero. On the right side, we arrive at
\begin{align}
\frac{1}{(2x+2\omega)^2-4t+1+\K_{j=1}^{\infty}\left(
\frac{-4j^2\left(\left(\frac{u-v}{2}\right)^2-j^2\right)^2}{(2j+1)
\left((2x+2\omega)^2-4t
+2j^2+2j+1\right)}\right)}=\frac 12 G_2(x).
\end{align}
On the other hand,
a direct calculation with the use of L'Hospital's rule deduces
\begin{align}
&\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}=\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}
\frac {\partial}{\partial n}(1-P)\\
=&-\frac 12\psi\left( x+u\right)-\frac 12\psi\left(x+v\right)+\psi\left(x+
\frac{u+v}{2}\right)\nonumber\\
=&\frac 12\sum_{k=0}^{\infty}\left(\frac{1}{k+x+u}+\frac{1}{k+x+v}
\right)\nonumber\\
\left(k+x+\frac{u+v}{2}\right)\left(k+x+v\right)
\end{align}
\begin{align}
\left(k+x+\frac{u+v}{2}\right)\left(k+x+v\right)}
\nonumber\\
\left(k+1+x+\frac{u+v}{2}\right)\left(k+1+x+v\right)}
\nonumber\\
\left(x+\frac{u+v}{2}\right)\left(x+v\right)}.\nonumber
\end{align}
This completes the proof of Theorem 5. With the notations of Theorem 5, let $n_2$ be a non-negative integer such that $n_2>\max\left\{\alpha:~\alpha\in\{-u, -v,-(u+v)/2\}~\text{and}~\alpha\in\mathbb{Z}\right\}$. If either $\frac{u-v}{2}\in\mathbb{Z}\backslash\{0\}$ or $n_2>\frac 12-u+v$, then
\begin{align}
\sum_{n=n_2}^{\infty}\frac{1}{(n+u)\left(n+\frac{u+v}{2}\right)
\left(n+v\right)}=G_2(n_2).
\end{align}
In particular, if
\begin{align}
u\in\mathfrak{D}_2=\{v\pm 2k: k\in \mathbb{N}, w\in\mathbb{R}\},
\end{align}
and $n_2>\max\{\alpha:~\alpha\in\{-1,-v, -v\mp 2k\}~\text{and}~\alpha\in\mathbb{Z}\}$, then
\begin{align}
\left(n+v\right)}=\sum_{n=n_2}^{\infty}\frac{1}{(n+v\pm 2k)\left(n+v\pm k\right)
\left(n+v\right)}\\
=&\frac{1/2}{(n_2-\frac 12\pm 2k)^2+\frac{1-2k^2}{4}+\K_{n=1}^{k-1}
\left(\frac{-\frac{n^2(k^2-n^2)^2}{4(2n-1)(2n+1)}}{(n_2-\frac 12\pm 2k)^2+\frac{2n^2+2n+1}{4}-\frac{k^2}{2}}\right)}.\nonumber
\end{align}
It follows from Theorem 5 and the telescoping method at once.
§.§ Some remarks
(1) Except the two cases that the polynomial $x^3+ax^2+bx+c$ satisfies the condition of Theorem 3 or 5, for other cases,
the structure of the continued fraction solution of the equation (<ref>) is not a Type-II.
(2) Let $r>0$. Taking $a=0$ and $b=r^2$ in Theorem 4, one may obtain a continued fraction representation for the infinite series
\begin{align}
\sum_{n=1}^{\infty}\frac{1}{n(n^2+r^2)}.
\end{align}
When $r\in\mathbb{Q}\backslash\{0\}$, to the best knowledge of authors, the irrationality of the series above remains unproven.
If $u,v,w\in \mathbb{R}$ with $u\neq v$ and $w\neq 0$ (In fact, if $u=v$, it may be treated by Theorem 3), one has
\begin{align}
\frac{1}{(x+u)\left((x+v)^2+w^2\right)}=&
\frac{1}{(x+u)(x+v-w~i)(x+v+w~i)}\\
=&\frac {1}{2w~i}\left(
\frac{1}{(x+u)(x+v-w~i)}-\frac{1}{(x+u)(x+v+w~i)}
\right).\nonumber
\end{align}
For the following difference equation
\begin{align}
\label{Complex-Case}
\end{align}
similarly to (<ref>), by making use of Mathematica software, authors have checked that if
$\Re \left(x+\frac{u+v-1}{2}\right)>0$, then
\begin{align}
\frac {1}{2w~i}\left\{F(u+v-w~i,u(v-w~i);x)-F(u+v+w~i,u(v+w~i);x)\right\}
\end{align}
is also a solution of the equation (<ref>). Here $F(a,b;x)$ is defined as (<ref>). For the summation of rational series by means of polygamma functions, please see Section 6.8 in <cit.>.
§ TWO EXTENDED MATHIEU SERIES
For a rational series with the general term $u(n)$ in the form of
$u(n)=P_1(n)/P_4(n)$, where $P_j(x)$ is a polynomial of degree $j$ in $x$ with real coefficients, the question become very complexity. Hence in this section, we shall study only two kind of extended Mathieu series.
§.§ The rational series $\sum\frac{2n+a}{(n^2+a n+b_1)(n^2+a n+b_2)}$
In this subsection we shall study first the following difference equation of order one
\begin{align*}
\frac{2x+a}{(x^2+a x+b_1)(x^2+a x+b_2)},
\end{align*}
and the results may is stated as follows.
Let $a,b_1,b_2\in\mathbb{R}$, and the formal continued fraction $H_1(a,b_1,b_2;x)$ (or shortly $H_1(x)$) be defined by
\begin{align}
\frac{\kappa_n}{\left(x+\omega\right)^2+\lambda_n}\right)
\end{align}
\begin{align}
&\omega=\frac{a-1}{2}, \quad \kappa_n=\frac{n^2 \left(- (b1 - b2)^2+\left(a^2 - 2 (b1 + b2)\right) n^2 - n^4 \right)}{4 (2 n - 1) (2 n + 1)} ,\\
&\lambda_n=\frac{2 n^2 + 2 n + 1+2b_1+2b_2-a^2}{4}.
\end{align}
We assume $x\notin\{q+\alpha: q\in\mathbb{Z}\backslash\mathbb{N},~ ({\alpha}^2+a{\alpha} +b_1)({\alpha}^2+a{\alpha} +b_2)=0, ~\alpha\in\mathbb{C} \}$. If either one of $\sqrt{\frac{a^2 - 2 (b1 + b2)\pm\sqrt{(a^2-4b_1)(a^2-4b_2)}}{2}}$ is a positive integer, or $\Re x>-\omega$, then
\begin{align}
\frac{2x+a}{(x^2+a x+b_1)(x^2+a x+b_2)}.
\end{align}
In this and next subsection, when $t<0$, we shall use the convention $\sqrt{t}=\sqrt{-t}~i$. It is not difficult to prove that
\begin{align}
(b1 - b2)^2-\left(a^2 - 2 (b1 + b2)\right) n^2 + n^4
\left(\frac{\beta-\sqrt{\Delta_1}}{2}-n^2\right).
\end{align}
where $\beta=a^2 - 2 (b1 + b2)$ and $\Delta_1=(a^2-4b_1)(a^2-4b_2)$.
Since the proof of Theorem 7 is quite similar to that of Theorem 5 or Theorem 9 below, we only give its outline. We shall discuss the following nine cases.
(Case 1) $\Delta_1>0$ and $\beta-\sqrt{\Delta_1}>0$.
We take $(x,l,m)=\left(2(x+\omega),\sqrt{\frac{\beta-\sqrt{\Delta_1}}{2}},
\sqrt{\frac{\beta+\sqrt{\Delta_1}}{2}}\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 2) $\Delta_1>0$, $\beta+\sqrt{\Delta_1}>0$, and $\beta-\sqrt{\Delta_1}<0$.
We take $(x,l,m)=\left(2(x+\omega),\sqrt{\frac{-\beta+\sqrt{\Delta_1}}{2}}~i,
\sqrt{\frac{\beta+\sqrt{\Delta_1}}{2}}\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 3) $\Delta_1>0$ and $\beta+\sqrt{\Delta_1}<0$.
We take $(x,l,m)=\left(2(x+\omega),\sqrt{\frac{-\beta+\sqrt{\Delta_1}}{2}}~i,
\sqrt{\frac{-\beta-\sqrt{\Delta_1}}{2}}~i\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 4) $\Delta_1>0$ and $\beta-\sqrt{\Delta_1}=0$.
We take $(x,l)=\left(2(x+\omega),\sqrt{\frac{\beta+\sqrt{\Delta_1}}{2}}
\right)$ in Lemma 2, then let $m$ and $n$ tend to zero, successively.
(Case 5) $\Delta_1>0$ and $\beta+\sqrt{\Delta_1}=0$.
We take $(x,l)=\left(2(x+\omega),\sqrt{\frac{-\beta
+\sqrt{\Delta_1}}{2}}~i\right)$ in Lemma 2, then let $m$ and $n$ tend to zero, successively.
(Case 6) $\Delta_1<0$. We take $(x,l,m)=\left(2(x+\omega),\sqrt{\frac{\beta-\sqrt{-\Delta_1}~i}{2}},
\sqrt{\frac{\beta+\sqrt{-\Delta_1}~i}{2}}\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 7) $\Delta_1=0$ and $\beta>0$. We take $(x,l,m)=\left(2(x+\omega),\sqrt{\frac{\beta}{2}},
\sqrt{\frac{\beta}{2}}\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 8) $\Delta_1=0$ and $\beta<0$. We take $(x,l,m)=\left(2(x+\omega),\sqrt{-\frac{\beta}{2}}~i,
\sqrt{-\frac{\beta}{2}}~i\right)$ in Lemma 2, then let $n$ tend to zero.
(Case 9) $\Delta_1=0$ and $\beta=0$.
In this case, we have $b_1=b_2$. It is same as Case 3 in Theorem 3.
Finally, combining the nine cases above will finish the proof of Theorem 7. With the notations of Theorem 7, let $n_1$ be a non-negative integer such that $n_1>\max\limits_{\alpha}\{\alpha:~(\alpha^2+a\alpha+b_1)
(\alpha^2+a\alpha+b_2)=0,\alpha\in\mathbb{Z}\}$. If either
one of $\sqrt{\frac{a^2 - 2 (b1 + b2)\pm\sqrt{(a^2-4b_1)(a^2-4b_2)}}{2}}$ is a positive integer, or $n_1>\frac{-a+1}{2}$, then
\begin{align}
\sum_{n=n_1}^{\infty}
\frac{2n+a}{(n^2+a n+b_1)(n^2+a n+b_2)}=H_1(n_1)\label{AN extended MS-1}.
\end{align}
In particular, if
\begin{align}
a\in\left\{\pm\sqrt{k^2 +2 (b1 + b2)+\frac{(b_1-b_2)^2}{k^2}}\in\mathbb{R}:
~b_1,b_2\in\mathbb{R}, k\in\mathbb{N}
\right\},
\end{align}
and $n_1>\max\limits_{\alpha}\{-1, \alpha:~(\alpha^2+a\alpha+b_1)
(\alpha^2+a\alpha+b_2)=0, \alpha\in\mathbb{Z}\}$, then
\begin{align}
\sum_{n=n_1}^{\infty}
\frac{2n+a}{(n^2+a n+b_1)(n^2+a n+b_2)}=\frac{1}{\left(n_1+\omega\right)^2
\frac{\kappa_n}{\left(n_1+\omega\right)^2+\lambda_n}\right)
}\label{AN extended MS-1-1}.
\end{align}
Further, if $b_1=b_2=b$, $b\in \{\frac{a^2-k^2}{4}:~a\in \mathbb{R}, k\in\mathbb{N}\}$ and $n_1>\max\left\{\alpha:~\alpha\in\{-1,\frac{-a\pm k}{2}\}~\text{and}~\alpha\in\mathbb{Z}\right\}$, then
\begin{align}
\sum_{n=n_1}^{\infty}\frac{2n+a}{(n^2+a n+b)^2}=\frac{1}{\left(n_1+\frac{a-1}{2}\right)^2+\frac{1-k^2}
\frac{\frac{n^4\left(k^2-n^2\right)}{4 (2 n - 1) (2 n + 1)}}{\left(n_1+\frac{a-1}{2}\right)^2+\frac{2 n^2 + 2 n + 1-k^2}{4}}\right)
}.\label{AN extended MS-1-2}
\end{align}
Applying Theorem 7 and the telescoping method will finish the proof of Theorem 8.Example 7 Taking $k=2, a=\sqrt{41}/2, b_1=2$ and $b_2=1$ in (<ref>), we find that the function
\begin{align}
\frac{12 - \sqrt{41} - 4 x + 2 \sqrt{41} x + 4 x^2}{35 - 5 \sqrt{41} - 61 x + 12 \sqrt{41} x + 65 x^2 - 6 \sqrt{41} x^2 - 8 x^3 +
4 \sqrt{41} x^3 + 4 x^4}
\end{align}
is a simple closed form solution of the following equation
\begin{align}
\frac{2x+a}{(x^2+a x+b_1)(x^2+a x+b_2)}.
\end{align}
§.§ The rational series $\sum\frac{2(p n + q)}{(p n + q)^4 + s (p n + q)^2 + r}$
Firstly, we shall study the following difference equation of order one
\begin{align}
y(x)-y(x+1)=\frac{2(p x + q)}{(p x + q)^4 + s (p x + q)^2 + r}.
\end{align}
Let $p, q, r, s\in\mathbb{R}$ with $p>0$. We define $H_2(p,q,r,s;x)$ (or shortly $H_2(x)$) by
\begin{align}
H_2(p,q,r,s;x):=\frac{\frac{1}{ p^3}}{(x+\omega)^2+\frac 14+\frac{ s}{2p^2}+\K_{n=1}^{\infty}\left(\frac{\kappa_n}
\end{align}
\begin{align*}
\omega = -\frac 12+\frac{q}{ p},\quad \kappa_n=\frac{n^2 \left(- n^4 - \frac{2s}{p^2} n^2 +\frac{ 4 r- s^2}{p^4}\right)}{
2^2 (2 n - 1) (2 n + 1)},\quad\lambda_n=\frac{2 n^2 + 2 n + 1}{4} + \frac{s}{2 p^2}.
\end{align*}
Let $x\notin\{l+\alpha: l\in\mathbb{Z}\backslash\mathbb{N},~ (p\alpha+q)^4+s(p\alpha+q)^2 +r=0, ~\alpha\in\mathbb{C} \}$. If either one of $\sqrt{-2 \sqrt{r} - s}/p$, $\sqrt{2 \sqrt{r} - s}/p$ is a positive integer, or $\Re x>-\omega$, then
\begin{align}
H_2(x)-H_2(x+1)=\frac{2(p x + q)}{(p x + q)^4 + s (p x + q)^2 + r}.\label{Difference Equation-6}
\end{align}
Firstly, we note that
\begin{align}
n^4 +\frac{2s}{p^2} n^2 +\frac{s^2 -4 r }{p^4}
\left(\frac{-s-2\sqrt{r}}{p^2}-n^2\right).
\end{align}
We shall discuss seven cases.
(Case 1) $r\ge 0$ and $ -2 \sqrt{r} - s> 0$.
In this case, we have $ 2 \sqrt{r} - s> 0$.
Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{2 \sqrt{r} - s}/p,\sqrt{-2 \sqrt{r} - s}/p)$ and dividing both sides by $2n \sqrt{s^2-4r}/p^2$, we assume that the conditions of Theorem 9 hold, then
\begin{align}
\frac{p^2}{2 \sqrt{s^2-4r}}\frac{1-P}{n(1+P)}
\frac{4\left((2 \sqrt{r} - s)/p^2-j^2\right)\left((-2 \sqrt{r} - s)/p^2-j^2\right)(n^2-j^2)}{(2j+1)
\left(\left(2(x+\omega)\right)^2-n^2+2s/p^2
\frac{4\left(j^4+2sj^2/p^2+(s^2-4r)/p^4\right)(n^2-j^2)}{(2j+1)
\left(\left(2(x+\omega)\right)^2-n^2+2s/p^2
\end{align}
Now let $n$ tend to zero. On the right side, we arrive at
\begin{align}
\frac{1}{\left(2(x+\omega)\right)^2+2s/p^2+1
\frac{-4\left(j^4+2sj^2/p^2+(s^2-4r)/p^4\right)j^2}{(2j+1)
\left(\left(2(x+\omega)\right)^2+2s/p^2
+2j^2+2j+1\right)}\right)}=\frac {p^3}{4} H_2(x).
\end{align}
On the other hand, from the definition of $P$, it is easy to see that $\lim_{n\rightarrow 0}P=1$. A direct calculation with the use of L'Hospital's rule gives
\begin{align}
&\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}=\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}
\frac {\partial}{\partial n}(1-P)\\
=&\frac{1}{2 }\left\{-\psi\left(x+\frac qp+\frac {1}{2p}(-\sqrt{-2 \sqrt{r} - s}-\sqrt{2 \sqrt{r} - s})\right)+\psi\left(x+\frac qp+\frac{1}{2p}(\sqrt{-2 \sqrt{r} - s}-\sqrt{2 \sqrt{r} - s})\right)\right.\nonumber\\
&\left.+\psi\left(x+\frac qp+\frac{1}{2p}(-\sqrt{-2 \sqrt{r} - s}+\sqrt{2 \sqrt{r} - s})\right)-\psi\left(x+\frac qp+\frac{1}{2p}(\sqrt{-2 \sqrt{r} - s}+\sqrt{2 \sqrt{r} - s})\right)\right\}\nonumber\\
=&\frac 12\sum_{k=0}^{\infty}\left(\frac{1}{k+x+\frac qp+\frac {1}{2p}(-\sqrt{-2 \sqrt{r} - s}-\sqrt{2 \sqrt{r} - s})}-\frac{1}{k+x+\frac qp+\frac {1}{2p}(\sqrt{-2 \sqrt{r} - s}-\sqrt{2 \sqrt{r} - s})}\right.\nonumber\\
&\left.\quad -\frac{1}{k+x+\frac qp+\frac{1}{2p} (-\sqrt{-2 \sqrt{r} - s}+\sqrt{2 \sqrt{r} - s})}+\frac{1}{k+x+\frac qp+\frac{1}{2p}(\sqrt{-2 \sqrt{r} - s}+\sqrt{2 \sqrt{r} - s})}\right)\nonumber\\
\nonumber
\end{align}
\begin{align}
\end{align}
This proves (<ref>) in Case 1.
(Case 2) $r\ge 0$, $2 \sqrt{r} - s> 0$, and
$ -2 \sqrt{r} - s<0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{2 \sqrt{r} - s}/p,\sqrt{2 \sqrt{r} + s}~i/p)$, and then let $n$ tend to zero. The proof is very similar to that of Case 1, we omit the detail here.
(Case 3) $r\ge 0$ and $ 2 \sqrt{r} - s< 0$. In this case, we have $ -2 \sqrt{r} - s<0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{-2 \sqrt{r} + s}~i/p,\sqrt{2 \sqrt{r} + s}~i/p)$, and let $n$ tend to zero. The proof is very similar to that of Case 1.
(Case 4) $r> 0$ and $ s=2 \sqrt{r}$. Applying Lemma 2 with $(x,l)=(2(x+\omega),\sqrt{4 \sqrt{r}}~i/p)$ and dividing both sides by $2m n \sqrt{4 \sqrt{r}}~i/p$, we have
\begin{align}
&\frac{p}{4 r^{1/4}~i}\frac{1-P}{m n(1+P)}\\
=&\frac{1}{\left(2(x+\omega)\right)^2-m^2-n^2+4 \sqrt{r}/p^2+1
\frac{-4\left(4 \sqrt{r}/p^2+j^2\right)(m^2-j^2)(n^2-j^2)}{(2j+1)
\left(\left(2(x+\omega)\right)^2-m^2-n^2+4 \sqrt{r}/p^2
\end{align}
We let $n$ and $m$ tend to zero, successively. On the right side, one has
\begin{align}
\frac{1}{\left(2(x+\omega)\right)^2+4 \sqrt{r}/p^2+1
\frac{-4\left(4 \sqrt{r}/p^2+j^2\right)j^4}{(2j+1)
\left(\left(2(x+\omega)\right)^2+4 \sqrt{r}/p^2
\end{align}
By using of L'Hospital's rule, we deduce that
\begin{align}
&\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}=\lim_{n\rightarrow 0}
\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n}
\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}
\frac {\partial}{\partial n}(1-P)\\
=&\frac{1}{2 }\left\{-\psi\left(-\frac m2+\frac{q-r^{1/4}~i}{p}+x\right)
+\psi\left(\frac m2+\frac{q-r^{1/4}~i}{p}+x\right)\right.\nonumber\\
&\left.\quad+\psi\left(-\frac m2+\frac{q+r^{1/4}~i}{p}+x\right)
-\psi\left(\frac m2+\frac{q+r^{1/4}~i}{p}+x\right)
\right\}.\nonumber
\end{align}
By making use of L'Hospital's rule again, we find that
\begin{align}
&\lim_{m\rightarrow 0}\lim_{n\rightarrow 0}\frac{1-P}{mn(1+P)}
=\lim_{m\rightarrow 0}\frac 1m \left(\lim_{n\rightarrow 0}\frac{1-P}{n(1+P)}\right)\\
=&\frac 12\left\{\psi'\left(x+\frac{q-r^{1/4}~i}{p}\right)
=&\frac 12\sum_{k=0}^{\infty}\frac{1}{\left(k+x+\frac{q-r^{1/4}~i}{p}
\right)^2}
-\frac 12\sum_{k=0}^{\infty}\frac{1}{\left(k+x+\frac{q+r^{1/4}~i}{p}
\right)^2}\nonumber\\
=&2p^2 r^{1/4}~i\sum_{k=0}^{\infty}\frac{\left(p(k+x)+q\right)}
=&p^2 r^{1/4}~i\sum_{k=0}^{\infty}
\frac{2\left(p(k+x)+q\right)}{\left(p(k+x)+q\right)^4
\end{align}
Following the same argument as Case 1, we find that Theorem 9 holds in this case.
(Case 5) $r> 0$ and $ s=-2 \sqrt{r}$. Applying Lemma 2 with $(x,l)=(2(x+\omega),\sqrt{4 \sqrt{r}}/p)$, and let $n$ and $m$ tend to zero, successively. The proof is very similar to that of Case 4.
(Case 6) $r<0$. Applying Lemma 2 with $(x,l,m)=(2(x+\omega),\sqrt{2 \sqrt{-r}~i - s}/p,\sqrt{-2 \sqrt{-r}~i - s}/p)$, and let $n$ tend to zero. The proof is very similar to that of Case 1.
(Case 7) $r=s=0$. By the case 3 in the proof of Theorem 3 with $a=3q/p$, we deduce that Theorem 9 holds true.
Finally, combining Case 1 to 7 will finish the proof of Theorem 9. With the conditions of Theorem 9, let $n_2$ be a non-negative integer such that $n_2>\max\limits_{\alpha}\{\alpha:~ (p\alpha+q)^4+s(p\alpha+q)^2 +r=0,~\alpha\in\mathbb{Z}\}$, and either one of $\sqrt{-2 \sqrt{r} - s}/p$, $\sqrt{2 \sqrt{r} - s}/p$ is a positive integer, or $n_2>\omega$, then
\begin{align}
\sum_{n=n_2}^{\infty}\frac{2(pn+q)}{(pn+q)^4+s(p n+q)^2+r}=H_2(p,q,s,r;n_2).\label{AN extended MS-2}
\end{align}
In particular, let
\begin{align}
r=\frac{(p^2k^2 + s)^2}{4},\quad k\in\mathbb{N}.\label{AN extended MS-2-1}
\end{align}
Assume that $n_2>\max\limits_{\alpha}\{-1, \alpha:~ (p\alpha+q)^4+s(p\alpha+q)^2 +r=0,~\alpha\in\mathbb{Z}\}$, then
\begin{align}
\sum_{n=n_2}^{\infty}\frac{2(pn+q)}{(pn+q)^4+s (pn+q)^2+r}
\frac{1}{\left(n_2-\frac 12+\frac qp\right)^2+\frac{1 }{4}+\frac{s}{2p^2}+\K_{n=1}^{k-1}\left(
\frac{\frac{n^2 (n^2+k^2+2s/{p^2})(k^2-n^2)}{4 (2 n - 1)
(2 n + 1)}}{\left(n_2-\frac 12+\frac qp\right)^2+\frac{2 n^2+ 2 n +1}
%, &\mbox{if $p^2k^2+s\ge 0$}.\\
%\frac{1}{\left(n_1-\frac 12+\frac qp\right)^2+\frac{1 }{4}
%\frac{\frac{n^2 (n^2+k^2+2s/{p^2})(k^2-n^2)}{4 (2 n - 1)
%(2 n + 1)}}{\left(n_1-\frac 12+\frac qp\right)^2+\frac{2 n^2+ 2 n
%+1 }{4}+\frac{s}{2p^2}}\right)
%}, &\mbox{othewise}.\nonumber
\end{align}
It follows from Theorem 9 and the telescoping method readily.Example 8 Taking $(p,q,s,r)=(1,0,2r^2,r^4)$ in Theorem 10 (or $(a,b_1,b_2)=(0,r^2,r^2)$ in (<ref>)), the series in (<ref>) become the Mathieu series. Hence, for $l\ge 1$
\begin{align}
\end{align}
Example 9 Let $\lambda$ be real. Consider the infinite series
\begin{align}
T(\lambda)=\sum_{n=1}^{\infty}\frac{2n}{n^4+\lambda n^2+\frac{ (4+\lambda)^2}{4}}.
\end{align}
We take $k=2$ and $(p,q,s)=(1,0,\lambda)$ in (<ref>), then
\begin{align}
H_2\left(1,0,\lambda, \frac{ (4+\lambda)^2}{4};x\right)=\frac{2 (3 - 2 x + 2 x^2 + \lambda)}{(2 + 2 x^2 + \lambda) (4 - 4 x + 2 x^2 +\lambda)},
\end{align}
and for $\lambda\neq -2, -4$
\begin{align}
T(\lambda)=H_2\left(1,0,\lambda, \frac{ (4+\lambda)^2}{4};1\right)=\frac{2 (3 + \lambda)}{(2 + \lambda) (4 + \lambda)}.
\end{align}
Note that
\begin{align}
T(\sqrt{2})=\frac{2 (3 + \sqrt{2})}{10 + 6 \sqrt{2}}=\frac{9 - 4 \sqrt{2}}{7},
\end{align}
hence $T(\sqrt{2})$ is an irrational number. In fact, the assertion above may be checked directly by hands.
§ TWO APPLICATIONS OF THEOREM 8 AND 10
§.§ The alternating Mathieu series
Let the formal continued fraction $H_1(a,b_1,b_2;x)$ be defined by (<ref>). For all positive integer $k_1$ and $k_2$, we have
\begin{align}
\tilde{S}(r)=&\frac 18\sum_{m=0}^{k_1-1}\frac{2m+1}{\left(m^2+m+\frac {1+r^2}{4}\right)^2}-\frac 18\sum_{m=0}^{k_2-1}\frac{2m}{\left(m^2+(\frac r2)^2\right)^2}
\label{A-Mathiew-CF}\\
\frac 18 H_1\left(1,\frac {1+r^2}{4},\frac {1+r^2}{4};k_1\right)-\frac 18 H_1\left(0,\frac {r^2}{4},\frac {r^2}{4};k_2\right).\nonumber
\end{align}
In particular,
\begin{align}
\tilde{S}(r)=\frac{2}{(1+r^2)^2}+\frac 18 H_1\left(1,\frac {1+r^2}{4},\frac {1+r^2}{4};1\right)-\frac 18 H_1\left(0,\frac {r^2}{4},\frac {r^2}{4};1\right).\label{A-Mathiew-CF-1}
\end{align}
From the definition of the alternating Mathieu series in (<ref>), we rewrite it into two parts
\begin{align}
\tilde{S}(r)=&\sum_{m=1}^{\infty}(-1)^{m-1}\frac{2m}{(m^2+r^2)^2}
=\sum_{n=0}^{\infty}\frac{2\cdot (2n+1)}{\left((2n+1)^2+r^2\right)^2}-\sum_{n=1}^{\infty}
\frac{2\cdot 2n}{\left((2n)^2+r^2\right)^2}\\
=&\frac 18\sum_{n=0}^{\infty}\frac{2n+1}{\left(n^2+n+\frac {1+r^2}{4}\right)^2}-\frac 18\sum_{n=0}^{\infty}\frac{2n}{\left(n^2+(\frac r2)^2\right)^2}.\nonumber
\end{align}
Applying Theorem 8 with $(a,b_1,b_2)=(1,\frac {1+r^2}{4},\frac {1+r^2}{4})$ and $(a,b_1,b_2)=(0,\frac {r^2}{4},\frac {r^2}{4})$, respectively, we get the desired assertion. Finally, on taking $k_1=k_2=1$, (<ref>) follows from (<ref>) readily.
§.§ For the numbers $\mathbb{M}_2^{(m,j)}$
Let $m\in\mathbb{N}$ and $j\in\{1,2,\ldots,m-1\}$ with $(m,j)=1$. P. J. Szablowski <cit.> introduced the numbers
\begin{align}
\mathbb{M}_k^{(m,j)}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(mn+j)^k}.
\end{align}
Notice that $\mathbb{M}_k^{(1,1)}=\sum_{j=1}^{\infty}
(-1)^{j-1}/j^k$ and $\mathbb{M}_1^{(2,1)}=\pi/4$. The number $\mathbb{M}_2^{(2,1)}=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{(2j+1)^2}$
is Catalan constant $\mathbb{K}$, which is one of those classical constants whose irrationality and transcendence remain unproven. Two continued fraction formulas for Catalan constant can be found in Berndt <cit.>. W. Zudilin <cit.> obtained new one, also see Cuyt et al. <cit.>. When $k=1$, the continued fraction representation of the numbers $\mathbb{M}_1^{(m,j)}$ could be obtained from Theorem 2 easily. When $k=3$, quite similarly to the alternating Mathieu series in Theorem 11, we may use Theorem 4 to write $\mathbb{M}_3^{(m,j)}$ into a linear combination of two continued fractions. Now we state the main result as follows.
Let $\omega=\frac{2 j - 3m}{4 m}$, and the formal continued fraction $CF_2(m,j;x)$ be defined by
\begin{align}
\K_{n=1}^{\infty}\left(\frac{-\frac{n^4}{16}}{(x+\omega)^2
+\frac{8 n^2 + 8 n + 3}{16}}\right)}.
\end{align}
We let $a=\frac jm - \frac 12, b=-\frac{j}{4 m} + \frac{j^2}{4 m^2}$. For all positive integer $l$, then
\begin{align}
\mathbb{M}_2^{(m,j)}=\frac{1}{j^2}-\frac{1}{8m^2}
\sum_{n=1}^{l-1}\frac{2n+b}{(n^2+an+b)^2}
\end{align}
In particular,
\begin{align}
\mathbb{M}_2^{(m,j)}=\frac{1}{j^2}-\frac{1}{8m^2}\frac{1}
\K_{n=1}^{\infty}\left(\frac{-\frac{n^4}{16}}
{\left(\frac{2j+m}{4m}\right)^2+\frac{8 n^2 + 8 n + 3}{16}}\right)}.
\end{align}
First, we note that the equalities $0<\frac{3m-2j}{4m}<1$ always hold. Following the same argument as Theorem 11, we also have
\begin{align}
\mathbb{M}_2^{(m,j)}=\frac{1}{j^2}-\frac{1}{8m^2}
\sum_{n=1}^{\infty}\frac{2n+b}{(n^2+an+b)^2}.
\end{align}
It is elementary to check that
\begin{align*}
\kappa_n=\frac{n^4 (a^2 - 4 b - n^2)}{4 (2 n - 1) (2 n + 1)}=-\frac{n^4}{16},\quad
\lambda_n=\frac{2 n^2 + 2 n + 1 + 4 b - a^2}{4}=\frac{8 n^2 +
8 n + 3}{16}.
\end{align*}
Now Theorem 12 follows easily from Theorem 8 .
§ CONCLUSIONS
From the above discussion, we observe that for a specific rational series, the multiple-correction method provides a useful tool for finding a simple closed form solution, testing and guessing the continued fraction representation, or getting the fastest possible finite continued fraction approximation, etc. So our method should help advance the approximation theory, the theory of continued fraction and the generalized hypergeometric function, etc. Furthermore, probably these continued fraction formulas could be used to study the irrationality, transcendence of the involved series.
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Experimental results show that,
or in the problem of the rectangular
plate [3] and it is closely related to the Riemann Zeta function [4]. Moreover, Jakimovski and Russell [5] showed that an
extended form of the Mathieu series plays a role in examining Mercerian theorems for Cesàro summability.
Slightly modifications should be done
The question is quite analogous to the question of indefinite integration in finite terms.
Our formulation is in agreement with all computations so far
|
1511.00540
|
Solving constraint satisfaction problems (CSPs) is a notoriously expensive computational task.
Recently, it has been proposed that efficient stochastic solvers can be obtained through appropriately configured spiking neural networks performing Markov Chain Monte Carlo (MCMC) sampling.
The possibility to run such models on massively parallel, low-power neuromorphic hardware holds great promise; however, previously proposed networks are based on probabilistically spiking neurons, and thus rely on random number generators or external noise sources to achieve the necessary stochasticity, leading to significant overhead in the implementation.
Here we show how stochasticity can be achieved by implementing deterministic models of integrate and fire neurons using subthreshold analog circuits that are affected by thermal noise.
We present an efficient implementation of spike-based CSP solvers using a reconfigurable neural network VLSI device, and the device's intrinsic noise as a source of randomness.
To illustrate the overall concept, we implement a generic Sudoku solver based on our approach and demonstrate its operation.
We establish a link between the neuron parameters and the system dynamics, allowing for a simple temperature control mechanism.
§ INTRODUCTION
Constraint satisfaction problems (CSPs) include some of the most prominent problems in science and engineering.
A CSP is defined by a set of variables and a set of conditions (constraints) on those variables that need to be satisfied simultaneously.
Solutions to a given CSP typically form a vanishingly small subset of an exponentially large search space, rendering this type of problem NP-hard in the general case.
Many CSP solvers therefore involve problem-specific heuristics to avoid searching the entire space.
Alternatively, one can consider a massively parallel system of simple computing elements that tackle different parts of the problem simultaneously, and discover a global solution through communicating local search mechanisms.
Here, we explore event-based neural hardware as a substrate of computation, however, there are various other approaches, such as quantum annealing <cit.>, special cellular automata hardware <cit.>, or oscillator networks <cit.>.
The idea of using recurrent neural networks for finding solutions to computationally hard problems goes back to Hinton <cit.> and Hopfield <cit.>.
While the deterministic Hopfield network fails in the general case because of local minima it can get caught in, the probabilistic Boltzmann machine proposed by Hinton and Sejnowski <cit.> overcomes this limitation by sampling its states from a probability distribution rather than evolving along a deterministic trajectory.
It has been shown recently, how such stochastic samplers can be implemented in networks of spiking neurons <cit.>.
The resulting neural sampling framework has been applied to constraint satisfaction problems by Jonke et al. <cit.>, demonstrating
advantages of the spiking network approach over standard Gibbs sampling in certain cases.
The main technological advantage of using such neural samplers in practical applications lies in the ability to implement spiking neurons efficiently in neuromorphic hardware.
However, the models proposed in <cit.> are difficult to directly transfer onto spiking VLSI neurons, because they require individual neurons to emit spikes probabilistically, sampling from a probability distribution of a specific form. This is not easily implementable in an electronic circuit without explicit sources of noise, or random number generators.
Here we show, for the first time, how a similar sampling mechanism can be implemented in a system of analog electronic integrate and fire neurons.
To avoid the additional cost of implementing dedicated noise sources on chip, we propose a mechanism that makes use of the small amount of (thermal) noise that is present in any analog electronic system to achieve the desired stochastic network dynamics.
We demonstrate these principles using a standalone system, based on a reconfigurable neuromorphic processor <cit.>, comprising a configurable network of adaptive exponential integrate and fire neurons and synapse circuits with biophysically realistic dynamics <cit.>.
Once programmed for a given CSP, the system will output streams of candidate solutions.
Our experimental results demonstrate that these samples predominantly represent configurations with no or few violated constraints.
§ SOLVING CSPS WITH SPIKING NEURAL NETWORKS
Solving an NP-complete decision problem stochastically through sampling means transforming it into an NP-hard optimization problem, and solving it using some kind of annealing mechanism rather than sophisticated algorithmics.
Thereby, a cost function is formulated such that the solutions to a given problem are transformed into optima.
It is not clear whether the corresponding optimization problem is easier to solve in general, however, typically the conversion can be done in polynomial time, and in some cases the optimization problem can be parallelized more easily or more efficiently than the decision problem.
Many types of problems can be transformed to simple graph structures with little effort <cit.>, and thus almost naturally map onto a network of nodes that interact through positive and negative links.
In contrast to most conventional algorithmic solvers, the optimization-based approach does not depend on problem-specific heuristics, and can thus be regarded as more general.
As a first step, we outline how arbitrary discrete CSPs can be mapped onto a network of neurons.
We consider constraint satisfaction problems defined by a set of $n$ discrete variables $\{x_1,\ldots,x_n\}$ on finite domains and a set of constraints, each linking several of those variables, e.g. $(x_i=a \lor x_j=b) \land x_i \neq x_j$ etc.
Without loss of generality, any such problem can be expressed in terms of binary variables by using a one-hot scheme, i.e. by representing each discrete variable $x_i$ as a vector of binary variables $(x_{i,k})_k$, where exactly one is active at any time, $x_i=a \Leftrightarrow x_{i,a}=1,~x_{i,k \neq a}=0$.
Furthermore, a CSP can be written in conjunctive normal form, i.e. in the form $\land_i \left(\lor_j l_{ij}\right)$, where the $l_{ij}$ are literals (binary variables or their negations).
In a network of spiking neurons, following the models introduced in <cit.>, the state of each variable is represented by the spiking activity of multiple cells.
There is one cell per value a variable can assume, and a cell is called active at time $t$ if it has emitted a spike within a certain time window $[t-\tau_\mathrm{bin},t]$.
The neural sampling framework <cit.> describes how a network of stochastically spiking cells can generate samples from a Boltzmann distribution, where the samples are represented by network states.
A problem in conjunctive normal form can be mapped to an Ising model <cit.> and therefore be solved by the spiking network sampling from the corresponding Boltzmann distribution.
Alternatively, special network motifs implementing the OR relation between several variables can be used to achieve a smaller and more efficient network <cit.>.
The energy function which defines the probability distribution of outputs, i.e. the sampled states of all variables, is designed such that solutions to the CSP occur at particularly high rates.
Note that such a sampler does not know whether its current state represents a solution and will continue exploring the state space.
However, potential solutions can be validated by a secondary mechanism in polynomial time, for example by testing their conformity with the pairwise interactions between cells in the network.
Illustration of the Sudoku solver network. Colored dots represent neurons, lines crossing several dots represent all-to-all inhibitory connections between them.
A variable \(x\in\{1,2,3,4\}\) is one-hot encoded and its value represented by activity of one of four cells, which mutually inhibit each other (left). The condition \(x_i \neq x_j\) can be implemented by setting negative connections between cells representing the same value (middle).
The whole Sudoku network can be implemented using constraints of the type \(x_i \neq x_j\) (right; only a subset of the 64 cells comprising the $4\times4\times4$ cube is shown).
A $9\times9$ Sudoku solver implemented based on the same scheme would require $9^3=729$ neurons.
To illustrate our experimental results we implemented a $4\times4$ Sudoku solver on spiking analog hardware. A reduced version of the standard $9\times9$ Sudoku problem was chosen due to the limited number of neurons available on the device.
In this example problem, there are 16 variables $x_i \in \{1,2,3,4\},\,i=1,\ldots,16$ that are aligned in a $4\times4$ grid and restricted by the constraints that no two variables in a row, in a column, or in a $2\times2$ quadrant must assume the same value.
As described above, this can be written in binary form, by introducing four binary variables for each variable $x_i$, whereby exactly one of them must be true at any time.
Whenever one cell becomes active it shuts off all the others representing the same variable for a certain period by providing a strong inhibitory post-synaptic potential (IPSP) of duration $\tau_\mathrm{inh}$.
On the other hand, all cells in the array receive a constant excitatory input current, such that one of them will spike if none are active, ensuring that the respective variable is in a defined state at any time.
The activity of a single cell is limited by the refractory period, inactivating a cell for a duration $\tau_\mathrm{ref}$ immediately after a spike is emitted.
The implementation of the Sudoku solver is illustrated in <ref>.
The constraint $x_i \neq x_j$, which exists for any two variables of the same row, column, or $2\times2$ quadrant, is implemented by specifying inhibitory interactions between cells representing the same value (<ref> middle), such that the two variables cannot assume the same value at the same time.
§ STOCHASTIC DYNAMICS IN AN ANALOG VLSI NEURAL NETWORK
In this section we introduce a simple neuron model that can be used to describe our analog VLSI implementation of neural sampling.
In principle, the stochastic spiking neurons used in previous work <cit.> can be approximated by integrate-and-fire neurons, which are injected large amounts of noise, as proposed by <cit.>.
This approach, however, requires an independent noise source for every cell, and therefore cannot easily be implemented directly in hardware.
Instead, we propose a mechanism that is based on conventional deterministic neuron models, and becomes stochastic through slight jitter in the
duration of temporally extended pulses in analog VLSI.
Such small (thermal) fluctuations are inherent to any analog electronic system, and thus can be exploited in analog hardware implementations of the proposed model.
Abstract neuron model.
EPSPs (blue) are provided to a cell at times $t_1$ and $t_2$, triggering spikes and subsequent state transition of the cell into refractory period (red).
Due to the variability in the duration of the refractory period and post-synaptic potentials, sometimes one or the other is longer, even though on average, they are roughly of the same length.
For instance, the EPSP provided at $t_2$ lasts slightly longer than the refractory period triggered by the spike at $t_2$, leading to a second spike at $t_3$.
Equivalent effects are observed for IPSPs (functionally, the refractory period is equivalent to an IPSP in our model).
On the right, measurements of refractory period duration from the actual hardware are shown, indicating an approximately Gaussian distribution.
To illustrate our model, we assume simple leaky integrate-and-fire neurons that produce an output spike and remain in refractory period for a duration $\tau_\mathrm{ref}$ when their membrane potential crosses a threshold $\Theta$.
A spike in one cell triggers an excitatory or inhibitory postsynaptic potential (PSP) at synapses connecting it to other cells.
In the simplest case, which is also at the core of previous models <cit.>, this can be thought of as a rectangular signal of duration $\tau_\mathrm{inh}$ or $\tau_\mathrm{exc}$, respectively.
We make the assumption that the magnitudes of these signals are large enough to either trigger a spike in the target cell almost instantaneously (for excitatory inputs),
or silence the cell completely (for inhibitory inputs), such that additional excitatory inputs have no effect.
Note that the refractory period can be thought of as a strong inhibitory input of a cell to itself.
The stochasticity in our system is then introduced by small amounts of noise in the duration of those PSPs and refractory periods.
As a consequence, we can regard $\tau_\mathrm{inh}$, $\tau_\mathrm{exc}$, and $\tau_\mathrm{ref}$ as mean values, and in practice the durations are jittered around those values, as shown in <ref>.
As an example, assume two neurons that are coupled through inhibitory connections and are driven by a constant external current, and assume further that $\tau_\mathrm{ref}\ll\tau_\mathrm{inh}$.
In this circuit, whichever cell became active first would keep inhibiting the other cell. This is due to the short refractory period, which lets the active neuron spike again before the IPSP it provides to the other cell ends.
This network would end up in a local optimum, and would never explore the other possible state where the second cell is active.
If, however, the refractory period $\tau_\mathrm{ref}$ and the inhibitory pulse width $\tau_\mathrm{inh}$ are of similar size, small amounts of noise in the analog system, leading to jitter in the duration of both pulses, will sometimes cause the inhibitory PSP to be longer than the refractory period, and vice versa. Such a system could indeed explore all possible states.
This mechanism can be considered a kind of noise amplification or, alternatively the system can be regarded as being close to a critical point, where vanishingly small fluctuations can lead to dramatically different behavior.
Intuitively, longer refractory periods cause more explorative behavior, whereas short refractory periods let the network settle into local energy minima.
Note that all time constants in the system are defined relative to each other, and the relation to real time is of little relevance.
Thus, extending the refractory period is equivalent to shortening the PSPs, i.e. weakening the links between nodes.
In that sense, the refractory period can be regarded as a kind of temperature, that could be used in an annealing schedule to steer the dynamics of the network.
The assumptions made to construct our model are fulfilled in intrinsically noisy analog neural hardware which can, with this method, be configured such that very small fluctuations in the electronic signals can lead to large deviations in the network dynamics.
The jitter in the refractory period or PSP durations is thereby introduced by thermal fluctuations in the analog signals representing those variables.
As signals in analog electronics are affected by fabrication-induced variability, in practice systematic deviations will be observed between PSPs of different synapses.
In order for the sampling mechanism to function properly under these conditions, the variability in PSP duration due to thermal noise needs to be at least of the same order as the variability due to fabrication-induced mismatch.
If this is not the case, the distribution gets biased and might not represent the problem accurately enough. Note that this is not necessarily problematic as long as the minima are conserved.
For our experimental setup, we used the programmable neuromorphic device described in <cit.>, comprising 256 integrate and fire neurons and 128k programmable synapses, to implement a network that evolves in real time and produces a stream of output events that can be interpreted as states of the system.
The network described in <ref> was programmed into the hardware by setting the respective inhibitory connections and run by injecting small amounts of direct current into each cell.
The mean values of the tunable time constants $\tau_\mathrm{ref}$ and $\tau_\mathrm{inh}$ were set to approximately 100 ms for all cells.
The sampling rate at which the network states were evaluated was set to 10 Hz, such that the network activity was binned over 100 ms for each sample.
While the system could be run much faster, the time constants were set to relatively large values to allow for visualization of the network evolution in real time.
§ EXPERIMENTAL RESULTS
Sudoku solver on an analog VLSI chip.
The panels show representative spiking activity of a subset of the neurons used in the Sudoku solver implementation (bottom), and the temporal evolution of the number of constraints violated (top) over a period of 20 s.
A binary sample vector is acquired by binning the network activity over 100 ms and assigning a 1 or 0 to each cell, depending on whether it has spiked in the given interval or not.
Three example states are shown, where a red cell indicates that the respective variable violates one or more constraints.
The system frequently converges to states that represent solutions to the problem (0 constraints violated), but, due to the noise it is able to escape from those energy minima.
Note that the solutions found at around 2 s and 16 s are not identical (3 and 4 are swapped).
<Ref> shows representative spiking activity of the Sudoku solver network implemented and running on an analog VLSI chip. The system occasionally converges to states solving the problem (0 constraints violated), however, it is also able to escape from those local optima and explore other possible states.
The “temperature” or “exploration rate”, can be controlled by tuning the neuron parameters, i.e. the IPSP duration or the length of the refractory period.
As expected from our considerations above, the system can be constrained to lower energy regions by lowering the temperature parameter, i.e. decreasing the refractory period.
Empirical analysis of the samples generated by the hardware CSP solver.
The plots show histograms of the number of constraints violated, based on 10 mins of spiking activity of the Sudoku network.
Solid lines are least-squares fits to the data.
Slightly varying one of the network parameters (the mean refractory period, in this case) leads to fundamentally different behavior of the system.
If the refractory period is longer than the IPSP, the frequency of states of a certain energy decreases exponentially with the number of constraints violated.
For shorter refractory periods, the system samples almost exclusively from very low energy states, and the distribution can be described by a double exponential function.
For these measurements, $\tau_\mathrm{inh}$ was fixed to $\approx110$ ms, while $\tau_\mathrm{ref}$ was set to $\approx100$ ms (red) or $\approx120$ ms (blue).
<Ref> shows the distribution of the number of constraints violated, i.e. the measure of “energy” that we intend to minimize.
As expected, we observe fundamentally different behavior for the case where the refractory period is larger than the IPSP, compared to the case where it is smaller. This leads to a phase transition-like phenomenon when this threshold is crossed.
For longer refractory periods the distribution can be well fitted by an exponential function, indicating a strong concentration around the low energy states.
If the refractory period is shorter than the IPSP, however, the distribution is even more concentrated around zero, and can be approximated by a double exponential function.
As shown in <ref>, a similar effect can be observed in the distribution of energy jumps, i.e. the difference in the number of constraints violated between one state and the next. Good fits are again obtained by exponential and a double exponential functions, respectively.
This indicates that, similar to a Gibbs sampler, the system predominantly switches between states of similar energy, rather than doing high energy jumps.
§ CONCLUSION
We present the first analog VLSI implementation of a CSP solver based on neural sampling with spiking neurons.
Our contribution is a simple neuron model that achieves stochasticity without the external noise sources required by previous approaches, but instead exploits small variations and noise in the duration of temporally extended signals.
The empirical results obtained from an analog neuromorphic processor demonstrate the function and performance of the proposed mechanism.
While the hardware system used in our experiments is deliberately slowed down to operate at timescales similar to real neurons, our approach could, without restrictions, be used with much faster hardware to solve computationally hard problems quickly and efficiently.
We can empirically relate the duration of the refractory period, or alternatively the duration of PSPs, to a temperature parameter, such that those time constants could be varied in an annealing schedule to control the temperature.
We regard our work as a proof-of-concept and further research is required to optimize performance and analyze theoretical properties of the model.
Empirical analysis of the dynamics of the analog VLSI Sudoku solver.
The plots show the distribution of energy jumps, i.e. the difference in the number of constraints violated between two consecutive states for different network parameters (values as in <ref>).
§ ACKNOWLEDGMENT
We thank Wolfgang Maass and our colleagues at the Institute of Neuroinformatics for fruitful discussions.
The research was supported by the Swiss National Science Foundation Grant 200021_146608 and by the EU ERC Grant “neuroP” (257219).
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1511.00144
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#1#2#3 #3_#1^#2 -#3
Ihara's lemma for some unitary groups
Universitť Paris 13, Sorbonne Paris Citť
LAGA, CNRS, UMR 7539
F-93430, Villetaneuse (France)
PerCoLaTor: ANR-14-CE25
L'auteur remercie l'ANR pour son soutien dans le cadre du projet PerCoLaTor 14-CE25.
Le rťsultat principal de ce travail est la preuve de certaines instances de ce que depuis les
travaux de Clozel-Harris-Taylor sur Sato-Tate, on appelle le lemme d'Ihara. Ces instances sont liťes
aux hypothŤses
restrictives portťes par l'idťal maximal de l'algŤbre de Hecke par lequel on localise l'espace
des formes automorphes modulo $l$ du groupe unitaire considťrť. La stratťgie consiste ŗ transfťrer
cet ťnoncť en une propriťtť similaire portant sur la cohomologie d'une variťtť de Shimura de type
Kottwitz-Harris-Taylor que l'on montre en prouvant que la cohomologie, localisť en cet idťal maximal,
d'un faisceau pervers d'Harris-Taylor est sans torsion. On donne enfin quelques ťnoncťs d'augmentation
du niveau.
The main result of this paper is the proof of some new instances of the so called Ihara's lemma as it was
introduced in their work on Sato-Tate by Clozel-Harris-Taylor. These new cases are linked to the particular
hypothesis on the maximal ideal of the Hecke algebra by which we localize the space of automorphic
forms modulo $l$ of the unitary group. The strategy consists to go through the same property of
genericness of subspace of the localized cohomology of some Kottwitz-Harris-Taylor Shimura variety,
which relies on the freeness of the localized cohomology groups of Harris-Taylor's perverse sheaves.
Finally we prove some instances of level raising.
11F70, 11F80, 11F85, 11G18, 20C08
Shimura varieties, torsion in the cohomology, maximal ideal of the Hecke algebra,
localized cohomology, galois representation
headings arabic
§ INTRODUCTION
Soit $F=F^+E$ un corps CM avec $E/\Qm$ quadratique imaginaire. Pour $\overline B$ une algŤbre ŗ
division centrale sur $F$ de dimension $d^2$ munie d'une involution de seconde espŤce $*$ et
$\beta \in \overline B^{*=-1}$, on considŤre le groupe des similitudes $\overline G/\Qm$ dťfini pour toute
$\Qm$-algŤbre $R$ par
$$\overline G(R):= \{ (\lambda,g) \in R^\times \times (\overline B^{op} \otimes_\Qm R)^\times \hbox{ tel que }
gg^{\sharp_\beta}=\lambda \}$$
avec $\overline B^{op}=\overline B \otimes_{F,c} F$ oý $c=*_{|F}$ est la conjugaison complexe et $\sharp_\beta$ l'involution
$x \mapsto x^{\sharp_\beta}=\beta x^* \beta^{-1}$. On note $\overline G_0$ le groupe des similitudes
associť. Pour $p=uu^c$ un premier dťcomposť dans $E$, on a
$$\overline G(\Qm_p) \simeq \Qm_p^\times \times \prod_{w | u} (\overline B^{op}_v)^\times$$
oý $w$ dťcrit les places de $F$ au dessus de $u$. On suppose en outre que:
* $\overline G_0(\Rm)$ est compact,
* si $x$ est une place de $\Qm$ qui n'est pas dťcomposťe dans $E$ alors $G(\Qm_x)$ est
* qu'il existe une place $v_0$ de $F$ au dessus de $u$ telle que $\overline B_{v_0} \simeq D_{v_0,d}$
est l'algŤbre ŗ division centrale sur le complťtť $F_{v_0}$ de $F$ ŗ la place $v_0$,
d'invariant $\frac{1}{d}$.
Soit ŗ prťsent $l$ un nombre premier distinct de $p$ et $S$ un ensemble fini de premiers de $\Qm$
contenant $l$: on note $\spl$ (resp. $\spl^S$) l'ensemble des places finies de $F$ (resp. et $p_w:=w_{|\Qm} \not \in S$)
telles que $p_w$ est dťcomposť dans $E$ et le facteur de $\overline G(\Qm_{p_w})$
correspondant ŗ $w$ soit isomorphe ŗ $GL_d(F_w)$. Soit alors
$$\Tm_S=\Zm_l [ T_{w,1},\cdots,T_{w,d}:~w \in \spl^S ]$$
l'algŤbre de Hecke associťe aux opťrateurs de Hecke non ramifiťs aux places
$w \in \spl^S$, oý $T_{w,i}$ est la fonction caractťristique de
$$GL_d(\OC_w) \diag(\overbrace{\varpi_w,\cdots,\varpi_w}^{i}, \overbrace{1,\cdots,1}^{d-i} )
GL_d(\OC_w) \subset GL_d(F_w).$$
Pour $\mathfrak m$ un idťal maximal de nature automorphe
de $\Tm_S$ et $w \in \spl^S$, on note
$$P_{\mathfrak{m},w}(X):=\sum_{i=0}^d(-1)^i q_w^{\frac{i(i-1)}{2}} \overline{T_{w,i}} X^{d-i} \in \overline
\Fm_l[X]$$
le polynŰme de Hecke associť ŗ $\mathfrak m$ et
S_{\mathfrak{m}}(w) := \bigl \{ \lambda \in \Tm_I/\mathfrak m \simeq \overline \Fm_l \hbox{ tel que }
P_{\mathfrak{m},w}(\lambda)=0 \bigr \} ,$$
le multi-ensemble des paramŤtres de Satake modulo $l$ en $w$ associťs ŗ $\mathfrak m$.
Les idťaux premiers minimaux
de $\Tm_S$ sont ceux au dessus de l'idťal nul de $\Zm_l$ et sont donc en
bijection naturelle avec les idťaux premiers de $\Tm_S \otimes_{\Zm_l} \Qm_l$. Ainsi pour
$\widetilde{\mathfrak m} \subset \mathfrak m$ un tel idťal premier minimal, $(\Tm_S \otimes_{\Zm_l} \Qm_l)
/\widetilde{\mathfrak m}$ est une extension finie $K_{\widetilde{\mathfrak m}}$ de $\Qm_l$.
ņ un idťal premier minimal $\widetilde{\mathfrak m}$ cohomologique est associť une classe d'ťquivalence
proche $\Pi_{\widetilde{\mathfrak m}}$ au sens de <cit.>, i.e. un ensemble fini de reprťsentations
automorphes irrťductibles $\Pi$ non ramifiťes aux places $w \not \in S$ telles que la rťduction modulo $l$
de ses paramŤtres de Satake est le multi-ensemble $S_{\mathfrak m}(w)$. On note
$$\rho_{\widetilde{\mathfrak m}}:G_F:=\gal(\bar F/F) \longrightarrow GL_d(\overline \Qm_l)$$
la reprťsentation galoisienne associťe ŗ un tel $\Pi$ d'aprŤs
<cit.> et <cit.>, laquelle est nťcessairement bien dťfinie ŗ isomorphisme prŤs d'aprŤs le thťorŤme
de Cebotarev. Les valeurs propres de $\frob_w$ de la rťduction modulo $l$,
$\overline \rho_{\mathfrak m}: G_F \longrightarrow GL_d(\overline \Fm_l)$
sont alors donnťes par le multi-ensemble $S_{\mathfrak m}(w)$. La version du
lemme d'Ihara pour $\overline G$ est la conjecture suivante.
(Lemme d'Ihara)
* $\overline U$ un sous-groupe ouvert compact de $\overline G(\Am)$ tel que pour tout place
$w \in \spl^S$, sa composante locale $\overline U_w$ est le sous-groupe compact maximal
* $w_0 \in \spl^S$ et
* $\mathfrak m$ un idťal maximal de $\Tm_S$ tel que
$\overline \rho_{\mathfrak m}$ est absolument irrťductible.
Si $\bar \pi$ est une sous-reprťsentation
irrťductible de $\mathcal C^\oo(\overline G(\Qm) \backslash \overline G(\Am)/ \overline U^{w_0},
\overline \Fm_l)_{\mathfrak m}$, oý $\overline U=\overline U_{w_0} \overline U^{w_0}$,
alors la composante locale $\bar \pi_{w_0}$ de $\bar \pi$ en $w_0$ est gťnťrique.
cette version en dimension supťrieure du classique lemme d'Ihara pour $GL_2$ est due ŗ
Clozel-Harris-Taylor dans leur preuve de la conjecture de Sato-Tate . Pour contourner cette conjecture,
Taylor inventa, peu aprŤs, ce que dťsormais on appelle Ihara avoidance
, qui lui permit donc de faire l'automorphie nťcessaire pour prouver la conjecture de Sato-Tate.
Malgrť cela l'obtention du lemme d'Ihara soulŤve toujours un trŤs vif intťrÍt, cf. par exemple les travaux
de Clozel et Thorne <cit.>, ou ceux d'Emerton et Helm <cit.>.
Dans cet article, nous allons prouver cette conjecture sous les conditions restrictives suivantes portant
sur $\mathfrak m$.
* ($H_\leadsto$): Il existe une place $v_1 \in \spl$ telle que
la restriction $\overline \rho_{\mathfrak m,v_1}$ de $\overline \rho_{\mathfrak m}$ au groupe de
dťcomposition en $v_1$ est irrťductible,
* ($H_{tor}$): Il existe $w_1 \in \spl^S$ tel que $S_{\mathfrak m}(w_1)$ ne contient aucun
de la forme $\{ \alpha,q_{w_1} \alpha \}$ oý $q_{w_1}$ est le cardinal du corps rťsiduel de $F_{w_1}$,
* ($H_{combi}$): L'image de $\overline \rho_{\mathfrak m,w_0}$ dans son groupe de Grothendieck, est sans
La stratťgie consiste ŗ considťrer la cohomologie de la tour de variťtťs de Shimura $X_{\IC}$
associťe au groupe des similitudes $G$ oý $G/\Qm$ est tel que
* $G(\Am^{\oo})=\overline G(\Am^{\oo,p}) \times \Bigl ( \Qm_{p_{v_0}}^\times GL_d(F_{v_0}) \times
\prod_{\atop{w | u}{w \neq v_0}} (\overline B_{w}^{op})^\times \Bigr )$,
* $G(\Rm)$ a pour signatures $(1,d-1) \times (0,d) \times \cdots \times (0,d)$.
Sur $\overline \Qm_l$,
* la cohomologie de $X_\IC$ se calcule ŗ l'aide de la suite spectrale des cycles
ťvanescents ŗ la place $v_0$ sous la forme $E_1^{p,q} \Rightarrow
H^{p+q}(X_{\IC,\bar \eta_{v_0}},\overline \Qm_l)$ oý les $E_1^{p,q}$ sont nuls en dehors du cadran
$p \geq 0$ et $q\geq 0$.
* Gr‚ce ŗ ($H_{tor}$) la localisation en $\widetilde{\mathfrak m}$
des $E_1^{p,q}$ est concentrťe sur la droite $p+q=d-1$. Le sous-espace donnť par $E_1^{d-1,0}$
correspond ŗ une contribution
des points supersinguliers qui se dťcrit ŗ l'aide de l'espace des formes automorphes
pour $\overline G(\Am)$.
On en dťduit alors, sur $\overline \Qm_l$, que $\CC^\oo(\overline G(\Qm) \backslash \overline G(\Am)/
\bar U^{w_0},\overline \Qm_l)_{\widetilde{\mathfrak m}}$ fournit un sous-espace de
$H^{d-1}(X_{U^{w_0},\bar \eta_{v_0}},\overline \Qm_l)_{\widetilde{\mathfrak m}}$.
Le principe est ainsi de reprendre
cet argument sur $\overline \Zm_l$ et de rťduire modulo $l$. Pour ce faire on dťmontre,
thťorŤme <ref>, que
les localisťs $(E_1^{p,q})_{\mathfrak m}$ sont sans torsion et donc encore concentrťs sur la droite
$p+q=d-1$, de sorte que, proposition <ref>, on peut construire
une sous-reprťsentation irrťductible $\pi$
du localisť $H^{d-1}(X_U,\overline \Fm_l)_{\mathfrak m}$ en $\mathfrak m$
de $H^{d-1}(X_U,\overline \Fm_l)$ telle que $\pi^{\oo,v_0} \simeq \bar \pi^{\oo,v_0}$. Le principe
est d'utiliser ($H_{tor}$) pour s'assurer que la partie localisťe
de la cohomologie de toute la variťtť de Shimura est, d'aprŤs <cit.>, sans torsion et donner
corps ŗ l'idťe suivante: s'il y avait de la torsion dans un des $(E_1^{p,q})_{\mathfrak m}$ alors il y en
aurait dans l'aboutissement.
On est ainsi ramener ŗ un ťnoncť du type lemme d'Ihara mais portant ŗ prťsent
sur la cohomologie d'une variťtť de Shimura.
La question est alors plus aisťment manipulable en utilisant la gťomťtrie de la fibre spťcial en $w_0$
de $X_U$ ŗ travers les propriťtťs suivantes:
* la contribution de la strate supersinguliŤre ne peut donner que des gťnťriques en sous-espace:
le rťsultat est la plupart du temps trivial sauf quand les reprťsentations de Steinberg gťnťralisťes
portťes par cette strate ont une rťduction modulo $l$ qui n'est pas irrťductible. On utilise alors
le contrŰle des rťseaux donnť dans <cit.> et <cit.>.
* Les strates de Newton non supersinguliŤres sont gťomťtriquement induites.
En particulier la cohomologie de ces strates non supersinguliŤres est une induite parabolique et il
est bien connu qu'en gťnťral les induites paraboliques ne sont pas semi-simples.
Sur $\overline \Qm_l$
et donc plus encore sur $\overline \Fm_l$, toute reprťsentation irrťductible de $GL_d(F_{w_0})$,
et donc en particulier les non gťnťriques, peut s'obtenir comme une induite parabolique, cependant
quand on regarde aussi l'action galoisienne, on remarque, cf. le lemme <ref> et la remarque
qui le suit,
que certains couples $\rho \otimes \sigma$ de reprťsentations irrťductibles de $GL_d(F_{w_0}) \times
W_{w_0}$ n'apparaissent pas et qu'on a en outre des restrictions supplťmentaires quand on regarde
les sous-espaces des induites paraboliques donnťes par la cohomologie des systŤmes locaux
d'Harris-Taylor calculant les $E_1^{p,q}$, cf. le lemme <ref>. Le principe de la preuve
est alors le suivant:
* on raisonne par l'absurde en supposant qu'une non gťnťrique en $w_0$ intervient
comme sous-espace.
* ($H_\leadsto$) permet alors de construire une reprťsentation irrťductible $\rho \otimes \sigma$
de $GL_d(F_{w_0}) \times W_{w_0}$ avec $\rho$ non gťnťrique qui, comme dans le lemme
<ref>, ne peut pas Ítre un sous-espace des induites paraboliques
construites par la cohomologie des faisceaux pervers d'Harris-Taylor alors qu'il est
un sous-espace de la cohomologie globale.
La contradiction dťcoule alors de la filtration de la cohomologie globale, donnťe par la suite
spectrale des cycles ťvanescents puisqu'aucun des graduťs ne possŤde $\rho \otimes \sigma$
comme sous-espace. Dans tous les arguments mentionnťs plus haut, l'hypothŤse
($H_{combi}$) est nťcessaire, ŗ la fois pour les arguments de libertť de la cohomologie,
de non simplicitť des induites et au final pour repťrer les $\rho \otimes \sigma$ nous amenant
ŗ la contradiction.
La stratťgie de la dťmonstration permet aussi d'obtenir des ťnoncťs d'augmentation du niveau, lesquels
doivent normalement se dťduire des travaux de Taylor et de son ťcole.
Par exemple dans le cas $d=2$, supposons qu'il existe $\widetilde{\mathfrak m}$ tel que
(a) pour tout idťal premier $\widetilde{\mathfrak m} \subset \mathfrak m$, la composante locale
en $w_0$ de $\Pi_{\widetilde{\mathfrak m}}$ est non ramifiťe,
(b) qu'il existe un tel idťal premier avec $\Pi_{\widetilde{\mathfrak m},w_0} \simeq
\chi_{w_0,1} \times \chi_{w_0,2}$ tel que
$$\chi_{w_0,1} \chi_{w_0,2}^{-1}=\nu: ~ \alpha \in F_{w_0}^\times \mapsto q_{w_0}^{-\val (\alpha)}$$
(c) et que l'ordre de $q_{w_0}$ modulo $l \geq 3$ est ťgal ŗ $2$.
De (a), en utilisant la suite spectrale des cycles ťvanescents ŗ la place $w_0$, on en dťduit que
$$H^1(X_U,\overline \Fm_l)_{\mathfrak m}=H^1(X_{U,\bar s_{w_0}}^{=1},\Psi(\overline \Fm_l))_{\mathfrak m}$$
oý, $\Psi(\overline \Fm_l)$ dťsigne le complexe des cycles ťvanescents ŗ coefficients dans $\overline
\Fm_m$ et $X_{U,\bar s_{w_0}}^{=1}$ est la fibre spťciale gťomťtrique de $X_U$ en $w_0$ privťe de ses
points supersinguliers. Classiquement cette cohomologie est une induite parabolique et d'aprŤs (b) et
(c), la seule reprťsentation irrťductible non dťgťnťrťe sous-quotient de la rťduction modulo $l$
de $\chi_{w_0,1} \times \chi_{w_0,1} \nu$ est cuspidale et ne peut donc pas Ítre une sous-reprťsentation
de $H^1(X_U,\overline \Fm_l)$. Ainsi, d'aprŤs le lemme d'Ihara,
pour $\mathfrak m$ vťrifiant les points (b) et (c) ci-dessus, le point
(a) doit Ítre faux, autrement dit il existe un $\widetilde{\mathfrak m} \subset \mathfrak m$ tel que
$\Pi_{\widetilde{\mathfrak m},w_0}$ est une reprťsentation de Steinberg tordue par un caractŤre.
En d'autres termes, on a une propriťtť d'augmentation du niveau.
Nous prouverons un rťsultat similaire, plus prťcisťment,
pour un idťal premier $\widetilde{\mathfrak m} \subset \mathfrak m$, la rťduction modulo $l$ de
$\Pi_{\widetilde{\mathfrak m},w_0}$ qui ne dťpend que de $\mathfrak m$,
a un support supercuspidal $\SC_{w_0}(\mathfrak m)$ qui, par hypothŤse sur $\mathfrak m$,
est sans multiplicitť. On dťcompose alors
$$\SC_{w_0}(\mathfrak m)=\coprod_{\varrho} \SC_{\varrho}(\mathfrak m)$$
selon les classes d'ťquivalence inertielle $\varrho$ de $\overline \Fm_l$-reprťsentations irrťductibles supercuspidales d'un
$GL_{g(\varrho)}(F_{w_0})$ pour $1 \leq g(\varrho) \leq d$. Pour chaque $\varrho$ on note alors
$$l_1(\mathfrak m,\varrho) \leq \cdots \leq l_{r(\mathfrak m,\varrho)}(\mathfrak m,\varrho)$$
de sorte que $S_{\varrho}(\mathfrak m)$ puisse s'ťcrire comme la rťunion de $r(\varrho)$
segments de Zelevinsky non liťs
$$[\varrho \nu^k_i,\bar \rho \nu^{k+l_i(\mathfrak m,\varrho)}]=
\bigl \{ \varrho \nu^k,\varrho \nu^{k+1},\cdots, \varrho \nu^{k+l_i(\mathfrak m,\varrho)} \bigr \}.$$
Il existe un idťal premier $\widetilde{\mathfrak m} \subset \mathfrak m$ tel que
$$\Pi_{\widetilde{\mathfrak m},w_0} \simeq \bigtimes_{\varrho \in \scusp_{w_0}}
\Pi_{\widetilde{\mathfrak m},w_0}(\varrho)$$
oý $\scusp_{w_0}$ dťsigne l'ensemble des classes d'ťquivalence inertielles des
$\overline \Fm_l$-reprťsentations irrťductibles de $GL_n(F_{w_0})$
pour $n \geq 1$ et oý pour tout $\varrho \in \scusp_{w_0}$, il existe des reprťsentations irrťductibles
cuspidales $\pi_1(\varrho),\cdots, \pi_{r(\mathfrak m,\varrho)}(\varrho)$ de $GL_{g(\varrho)}(F_v)$ telles
$$\Pi_{\widetilde{\mathfrak m},w_0} \simeq \st_{l_1(\mathfrak m,\varrho)}(\pi_1(\varrho)) \times
\cdots \times \st_{l_{r(\mathfrak m,\varrho)}(\mathfrak m,\varrho)}(\pi_{r(\mathfrak m,\varrho)}(\varrho)).$$
§ GŤOMŤTRIE DES VARIŤTŤS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR
§.§ Donnťes de Shimura
On fixe un nombre premier $l$ relatif aux coefficients dont on prend la
cohomologie. Par la suite les diffťrents nombres premiers qui interviendront, seront toujours
supposťs Ítre distincts de $l$.
Soit $F=F^+ E$ un corps CM avec $E/\Qm$ quadratique imaginaire, dont on fixe
un plongement rťel $\tau:F^+ \hookrightarrow \Rm$ et tel que $l$ est non ramifiť dans $E$.
Pour $v$ une place de $F$, on notera $F_v$ le complťtť du localisť de $F$ en $v$,
$\OC_v$ son anneau des entiers de $F_v$, $\varpi_v$ une uniformisante et
$q_v$ le cardinal du corps rťsiduel $\kappa(v)=\OC_v/(\varpi_v)$.
On note $\nu$ le caractŤre de $F_v^\times$ dťfini par
$\alpha \mapsto q_v^{-\val(\alpha)}$.
Soit $B$ une algŤbre ŗ
division centrale sur $F$ de dimension $d^2$ telle qu'en toute place $x$ de $F$,
$B_x$ est soit dťcomposťe soit une algŤbre ŗ division et on suppose $B$
munie d'une involution de
seconde espŤce $*$ telle que $*_{|F}$ est la conjugaison complexe $c$. Pour
$\beta \in B^{*=-1}$, on note $\sharp_\beta$ l'involution $x \mapsto x^{\sharp_\beta}=\beta x^*
\beta^{-1}$ et $G/\Qm$ le groupe de similitudes, notť $G_\tau$ dans <cit.>, dťfini
pour toute $\Qm$-algŤbre $R$ par
G(R) \simeq \{ (\lambda,g) \in R^\times \times (B^{op} \otimes_\Qm R)^\times \hbox{ tel que }
gg^{\sharp_\beta}=\lambda \}
avec $B^{op}=B \otimes_{F,c} F$.
Si $x$ est une place de $\Qm$ dťcomposťe $x=yy^c$ dans $E$ alors
$$G(\Qm_x) \simeq (B_y^{op})^\times \times \Qm_x^\times \simeq \Qm_x^\times \times
\prod_{z_i} (B_{z_i}^{op})^\times,$$
oý, en identifiant les places de $F^+$ au dessus de $x$ avec les places de $F$ au dessus de $y$,
$x=\prod_i z_i$ dans $F^+$.
Dans <cit.>, les auteurs justifient l'existence d'un $G$ comme ci-dessus tel qu'en outre:
* si $x$ est une place de $\Qm$ qui n'est pas dťcomposťe dans $E$ alors
$G(\Qm_x)$ est quasi-dťployť;
* les invariants de $G(\Rm)$ sont $(1,d-1)$ pour le plongement $\tau$ et $(0,d)$ pour les
On fixe ŗ prťsent un nombre premier $p=uu^c$ dťcomposť dans $E$ tel qu'il existe une
place $v$ de $F$ au dessus de $u$ telle que
$$(B_v^{op})^\times \simeq GL_d(F_v).$$
On note abusivement
$$G(\Am^{\oo,v})=G(\Am^{\oo,p}) \times \Qm_p^\times \times
\prod_{\atop{w|u}{w \neq v}} (B_w^{op})^\times.$$
par rapport aux notations de l'introduction, la place $v$ sera, selon les cas, soit $v_0$, $v_1$
ou $w_0$.
Pour tout sous-groupe compact $U^p$ de $G(\Am^{\oo,p})$ et $m=(m_w)_{w|u} \in \Zm_{\geq 0}^r$,
on pose
$$U_v(m)=U^p \times \Zm_p^\times \times \prod_{w|u}
\ker \bigl ( \OC_{B_{w}}^\times \longto (\OC_{B_{w}}/\PC_{w}^{m_w})^\times \bigr ).$$
Soit $\IC_v$ l'ensemble des sous-groupes compacts ouverts assez petits [tel qu'il existe
une place $x \neq v$ pour laquelle la projection de $U^v$ sur $G(\Qm_x)$ ne contienne aucun
ťlťment d'ordre
fini autre que l'identitť, cf. <cit.> bas de la page 90] de $G(\Am^\oo)$, de la forme $U_v(m)$.
Pour $m$ comme ci-dessus, on a une application
$$m_v: \IC_v \longrightarrow \Nm.$$
Pour un tel $I \in \IC_v$, soit
$$X_I \longrightarrow \spec \OC_v$$
la variťtť de Shimura de <cit.> dite de Kottwitz-Harris-Taylor associťe ŗ $G$.
$X_I$ est un schťma projectif sur $\spec \OC_v$ tel
que $(X_I)_{I \in \IC_v}$ forme un systŤme projectif dont les morphismes de transition
sont finis et plats. Quand $m_v=m_v'$ alors $X_{U^p(m)} \longrightarrow X_{U^p(m')}$ est ťtale.
Ce systŤme projectif est par ailleurs muni d'une action de $G(\Am^\oo) \times \Zm$ telle que l'action
d'un ťlťment $w_v$ du groupe de Weil $W_v$ de $F_v$ est donnťe par celle de $-\deg (w_v) \in \Zm$,
oý $\deg=\val \circ \art^{-1}$ oý $\art^{-1}:W_v^{ab} \simeq F_v^\times$ est
l'isomorphisme d'Artin qui envoie les Frobenius gťomťtriques sur les uniformisantes.
(cf. <cit.> 1.3)
Pour $I \in \IC_v$, on note:
* $X_{I,s_v}$ (resp. $X_{I,\bar s_v}:=X_{I,s_v} \times \spec \overline \Fm_p$)
la fibre spťciale (resp. gťomťtrique) de $X_I$ en $v$ et $X_{I,\eta_v}$ (resp. $X_{I,\bar \eta_v}$)
la fibre gťnťrique (resp. gťomťtrique).
* Pour tout $1 \leq h \leq d$, $X_{I,\bar s_v}^{\geq h}$ (resp. $X_{I,\bar s_v}^{=h}$)
dťsigne la strate fermťe (resp. ouverte) de Newton de hauteur $h$, i.e. le sous-schťma dont
la partie connexe du groupe de Barsotti-Tate en chacun de ses points gťomťtriques
est de rang $\geq h$ (resp. ťgal ŗ $h$).
pour tout $1 \leq h \leq d$, la strate de Newton de hauteur $h$ est
de pure dimension $d-h$. La strate ouverte est en outre gťomťtriquement induite au sens
oý il existe un systŤme projectif
$$X_{\IC_v,\bar s_v,1}^{=h}:=\bigl (X_{I,\bar s_v,1}^{=h} \bigr )_{I \in \IC_v}$$
de sous-schťmas fermťs de
$X_{\IC_v,\bar s_v}^{=h}:=\bigl ( X_{I,\bar s_v}^{=h} \bigr )_{I \in \IC_v}$ muni d'une action
par correspondance de $P_{h,d}(F_v)$ tel que
$$X_{\IC_v,\bar s_v}^{=h}=X_{\IC_v,\bar s_v,1}^{=h} \times_{P_{h,d}(F_v)} GL_d(F_v).$$
Pour tout $I \in \IC_v$, nous utiliserons les notations suivantes:
$$i^h:X^{\geq h}_{I,\bar s} \hookrightarrow X^{\geq 1}_{I,\bar s}, \quad
j^{\geq h}: X^{=h}_{I,\bar s} \hookrightarrow X^{\geq h}_{I,\bar s}$$
ainsi que $j^{=h}=i^h \circ j^{\geq h}$.
§.§ SystŤmes locaux d'Harris-Taylor
Pour $g$ un diviseur de $d=sg$ et $\pi_v$ une $\overline \Qm_l$-reprťsentation cuspidale
irrťductible de $GL_g(F_v)$, l'induite parabolique
$$\pi_v\{ \frac{1-s}{2} \} \times \pi_v \{\frac{3-s}{2} \} \times \cdots \times \pi_v \{\frac{s-3}{2} \} \times
\pi_v \{ \frac{s-1}{2} \}$$
* un unique quotient irrťductible notť $\st_s(\pi_v)$; c'est une reprťsentation de Steinberg gťnťralisťe.
* une unique sous-reprťsentation irrťductible notťe $\speh_s(\pi_v)$; c'est une reprťsentation de Speh gťnťralisťe.
* si $\pi_v$ est un caractŤre $\chi_v$ de $F_v^\times$ alors $\speh_s(\chi_v)$ est le
caractŤre de $GL_d(F_v)$ donnť par $\chi_v \circ \det$.
* Une $\overline \Qm_l$-reprťsentation irrťductible de $GL_d(F_v)$ sera dite essentiellement
de carrť intťgrable
si elle est de la forme $\st_s(\pi_v)$ pour un diviseur $g$ de $d=sg$ et $\pi_v$ une reprťsentation
irrťductible cuspidale de $GL_g(F_v)$.
La correspondance de Jacquet-Langlands locale, sur $\overline \Qm_l$, est une bijection entre les classes
d'ťquivalences des reprťsentations irrťductibles essentiellement de carrť intťgrable de $GL_d(F_v)$ et
les reprťsentations irrťductibles du groupe des inversibles $D_{v,d}^\times$ de l'algŤbre ŗ division
centrale $D_{v,d}$ sur $F_v$ d'invariant $\frac{1}{d}$.
On note $\pi_v[s]_D$ la reprťsentation irrťductible de $D_{v,d}^\times$ associťe ŗ $\st_s(\pi_v^\vee)$
par la correspondance de Jacquet-Langlands locale.
Rappelons que dans <cit.>, les auteurs, via les variťtťs d'Igusa de premiŤre et seconde
espŤce, associent ŗ toute reprťsentation $\rho_v$ de $\DC_{v,h}^\times$, oý $\DC_{v,h}$
est l'ordre maximal de $D_{v,h}$,
un systŤme local $\LC(\rho_v)_1$ sur $X^{=h}_{\IC,\bar s_v,1}$ dont on note
$\LC(\rho_v)$ la version induite sur $X^{=h}_{\IC_v,\bar s_v}$.
D'aprŤs <cit.> 1.4.2, le systŤme local $\LC(\rho_v)_1$ est muni d'une action
par correspondances de
$$G(\Am^{\oo,p}) \times \Qm_p^\times \times P_{h,d}(F_v) \times
\prod_{i=2}^r (B_{v_i}^{op})^\times \times \Zm$$
qui d'aprŤs <cit.> p.136, se factorise par $G^{(h)}(\Am^\oo)/\DC_{F_v,h}^\times$ via
\begin{equation} \label{eq-action-tordue}
(g^{\oo,p},g_{p,0},c,g_v,g_{v_i},k) \mapsto (g^{p,\oo},g_{p,0}q^{k-v (\det g_v^c)},
g_v^{et},g_{v_i}, \delta).
\end{equation}
* $G^{(h)}(\Am^\oo):=G(\Am^{\oo,p}) \times \Qm_p^\times \times GL_{d-h}(F_v) \times
\prod_{i=2}^r (B_{v_i}^{op})^\times \times D_{F_v,h}^\times$,
* $g_v=\left ( \begin{array}{cc} g_v^c & * \\ 0 & g_v^{et} \end{array} \right )$ et
* $\delta \in D_{v,h}^\times$ est tel que $v(\rn (\delta))=k+v(\det g_v^c)$.
on prendra l'habitude d'identifier
$D_{v,h}^\times/\DC_{v,h}^\times$ avec $\Zm$ via la norme rťduite.
Pour $\pi_v$ une $\overline \Qm_l$-reprťsentation irrťductible cuspidale de $GL_g(F_v)$
et $t \geq 1$, on note
$$\HT_{\overline \Qm_l}(\pi_v,t,\Pi_t)(n):=\LC_{\overline \Qm_l}(\rho_{v,t})[d-tg] \otimes \Pi_t
\otimes \Xi^{\frac{tg-d}{2}-n} \otimes \Lm(\pi_v )$$
* $\Lm$ est la correspondance Langlands sur $F_v$, ŗ dualitť prŤs,
* $\Xi:\frac{1}{2} \Zm \longrightarrow \overline \Zm_l^\times$ est dťfinie par
* $\rho_{v,t}$ est un sous-quotient irrťductible quelconque de $(\pi_v[t]_D)_{|\DC_{v,tg}^\times}$
oý $\DC_{v,tg}$ est l'ordre maximal de $D_{v,tg}$;
* $GL_h(F_v)$ agit diagonalement sur $\Pi_t$ et sur $\LC(\pi_v,t) \otimes \Xi^{\frac{tg-d}{2}-n}$
via son quotient $GL_h(F_v) \twoheadrightarrow \Zm$,
* le groupe de Weil $W_v$ en $v$ agit diagonalement sur $\Lm(\pi_v)$ et le facteur
$\Xi^{\frac{tg-d}{2}-n}$ via l'application $\deg: W_v \twoheadrightarrow \Zm$ qui envoie les
frobenius gťomťtriques sur $1$.
Une $\overline \Zm_l$-version entiŤre sera notťe $\HT_\Gamma(\pi_v,t,\Pi_t)(n)$ oý $\Gamma$
dťsigne un rťseau stable, par forcťment sous la forme d'un produit tensoriel et la version sans $\Gamma$
dťsignera une $\overline \Zm_l$-structure quelconque.
Enfin on utilisera les notations $HT_{\Gamma,1}(\pi_v,t,\Pi_t)$
pour les versions non induites.
$\HT(\pi_v,t,\Pi_t)(n)$ ne dťpend pas du choix de $\rho_{v,t}$.
On utilisera, comme dans <cit.>, parfois la notation
en remplaÁant dans la formule prťcťdente $\LC(\rho_{v,t})$ par $\LC(\pi_v[t]_D)$ oý
$\pi_v[t]_D$ est vue, par restriction, comme une reprťsentation de $\DC_{v,tg}^\times$: on notera
qu'il ne dťpend que de la classe d'ťquivalence inertielle de $\pi_v$. Prťcisons cela dans le cas oý
$\Pi_t=\st_t(\pi_v)$, alors pour $\chi:\Zm \longrightarrow \overline \Qm_l^\times$,
les faisceaux pervers $\HT(\pi_v,t,\st_t(\pi_v))$ et $\HT(\pi_v \otimes \chi,t,\st_t(\pi_v \otimes \chi))$,
munis de leurs actions par correspondances, sont isomorphes.
On notera plus simplement
$$\HT(\pi_v,t):=\HT(\pi_v,t,\mathds 1_{tg})$$
oý $\mathds 1_{tg}$ dťsigne la reprťsentation triviale de $GL_{tg}(F_v)$.
Pour $\pi_v$ une reprťsentation irrťductible cuspidale de $GL_g(F_v)$ et $1 \leq t \leq
s:= \lfloor \frac{d}{g} \rfloor$, le $\overline \Qm_l$-faisceau pervers d'Harris Taylor
est, cf. <cit.> dťfinition 2.1.3,
$$\PF(\pi_v,t):=j^{=tg}_{!*} \HT(\pi_v,t,\st_t(\pi_v)).$$
Avec les notations prťcťdentes, si $\rho \otimes \sigma$ est un sous-$GL_d(F_v) \times W_v$-quotient
irrťductible de $H^i(X_{\IC,\bar s_v},\PF(\pi_v,t) \otimes_{\overline \Zm_l} \overline \Fm_l)$ alors
* $\sigma$ est un constituant irrťductible de la rťduction modulo $l$ de $\Lm(\pi_v \otimes \chi_v)$,
oý $\chi_v$ est un caractŤre non ramifiť de $F_v$, et
* $\rho$ est un sous-quotient irrťductible de la rťduction modulo $l$ d'une induite de la forme
$\st_t(\pi_v \otimes \chi_v) \times \psi_v$ oý $\psi_v$ est une reprťsentation irrťductible
entiŤre de $GL_{d-tg}(F_v)$.
Le rťsultat dťcoule, cf. la formule donnťe en (<ref>), de la description des actions.
le lemme prťcťdent sera utilisť de maniŤre cruciale dans les preuves du thťorŤme <ref>
et du lemme d'Ihara du <ref>. C'est l'un des ingrťdients principal qui explique,
sous ($H_{combi}$), pourquoi la
traduction du lemme d'Ihara sur la cohomologie des variťtťs de Shimura du <ref> est
plus facilement abordable. Plus prťcisťment afin de distinguer les sous-espaces on utilise
le fait que les strates non supersinguliŤres sont induites avec le lemme <ref>
qui est une version induite du lemme prťcťdent.
§.§ Faisceau pervers des cycles ťvanescents
Rappelons que pour $X$ un $\Fm_p$-schťma et
$\Lambda=\overline \Qm_l,\overline \Zm_l,\overline \Fm_l$, la $t$-structure usuelle sur
la catťgorie dťrivťe $D^b_c(X,\Lambda)$ est dťfinie par:
A \in \lexp p D^{\leq 0}(X,\Lambda)
\Leftrightarrow \forall x \in X,~h^k i_x^* A=0,~\forall k >- \dim \overline{\{ x \} } \\
A \in \lexp p D^{\geq 0}(X,\Lambda) \Leftrightarrow \forall x \in X,~h^k i_x^! A=0,~\forall k <- \dim \overline{\{ x \} }
\end{array}$$
oý $i_x:\spec \kappa(x) \hookrightarrow X$. On note alors $\lexp p \CC(X,\Lambda)$ le cœur de cette $t$-structure
et $\lexp p \hi^i$ les foncteurs cohomologiques associťs.
Pour $\Lambda$ un corps, cette $t$-structure est autoduale pour la dualitť de Verdier.
Pour $\Lambda=\overline \Zm_l$, on peut munir la catťgorie abťlienne $\overline \Zm_l$-linťaire
$\lexp p \CC(X,\Lambda)$ d'une thťorie de torsion $(\TC,\FC)$ oý
$\TC$ (resp. $\FC$) est la sous-catťgorie
pleine des objets de torsion $T$ (resp. libres $F$) , i.e. tels que $l^N 1_T$ est nul pour $N$ assez grand (resp. $l.1_F$ est un monomorphisme). Soit alors
\lexp {+} \DC^{\leq 0}(X,\overline \Zm_l):= \{ A \in \DC^{\leq 1}(X,\overline \Zm_l):~
\lexp p h^1(A) \in \TC \} \\
\lexp {+} \DC^{\geq 0}(X,\overline \Zm_l):= \{ A \in \DC^{\geq 0}(X,\overline \Zm_l):~
\lexp p h^0(A) \in \FC \} \\
\end{array}$$
la $t$-structure duale de cœur $\lexp {+} \CC(X,\overline \Zm_l)$ muni de sa thťorie de torsion
$(\FC,\TC[-1])$ duale de celle de $\lexp p \CC(X,\overline \Zm_l)$.
(cf. <cit.> 1.3)
Soit $\FC(X,\Lambda):=\lexp p \CC(X,\Lambda) \cap \lexp {p+} \CC(X,\Lambda)$
la catťgorie quasi-abťlienne des faisceaux pervers libres sur $X$ ŗ coefficients dans
Pour tout $I \in \IC_v$, le complexe
$\Psi_{I,v,\Lambda}:=R\Psi_{\overline \eta_v,I}(\Lambda)[d-1](\frac{d-1}{2})$ sur $X_{I,\bar s_v}$
est alors un objet de de $\FC(X_{I,\bar s_v},\Lambda)$, i.e. un
faisceau pervers libre dit des cycles ťvanescents. Pour $I$ variant dans $\IC_v$, on obtient,
au sens de la dťfinition 1.3.6 de <cit.>,
un $W_v$-faisceau pervers de Hecke notť $\Psi_{\IC_v,\Lambda}$ ou plus simplement $\Psi_{\IC_v}$
dans le cas oý $\Lambda=\overline \Zm_l$.
Rappelons, cf. <cit.> 2.4, que la restriction
$\Bigl ( \Psi_{\IC_v,\Lambda} \Bigr )_{|X^{=h}_{\IC,\bar s_v}}$ du faisceau pervers des
cycles ťvanescents ŗ la strate $X^{=h}_{\IC_v,\bar s_v}$, est munie d'une action de
$$(D_{v,h}^\times)^0:=\ker \Bigl ( \val \circ \rn: D_{v,h}^\times \longrightarrow \Zm \Bigr )$$
et de $\varpi_v^\Zm$ que l'on voit plongť dans $F_v^\times \subset D_{v,h}^\times$.
Pour tout $h \geq 1$, on note
$\RC_{ \overline \Fm_l}(h)$ l'ensemble des classes d'ťquivalence des
$\overline \Fm_l$-reprťsentations irrťductibles de $D_{v,h}^\times$ dont
le caractŤre central est trivial sur $\varpi^\Zm \subset K^\times$.
Pour $\bar \tau \in \RC_{ \overline \Fm_l}(h)$, la sous-catťgorie
$\CC_{\bar \tau} \subset \rep_{\Zm_l^{nr}}^\oo(D_{v,d}^\times)$
formťe des objets dont tous les $\Zm_l^{nr}\DC_{v,d}^\times$-sous-quotients
irrťductibles sont isomorphes ŗ un sous-quotient de $\bar \tau_{|\DC_{v,d}^\times}$,
est facteur direct dans $\rep_{\Zm_l^{nr}}^\oo(D_{v,d}^\times)$ de sorte que
toute $\Zm_l^{nr}$-reprťsentation $V_{\Zm_l^{nr}}$ de $D_{v,d}^\times$
se dťcompose en une somme directe
\begin{equation} \label{eq-tau-iso}
V_{\Zm_l^{nr}} \simeq \bigoplus_{\bar \tau \in \RC_{\overline \Fm_l}(h)} V_{\Zm_{l,\bar \tau}^{nr}}
\end{equation}
oý $V_{\Zm_{l,\bar \tau}^{nr}}$ est un objet de $\CC_{\bar \tau}$.
(cf. <cit.> proposition IV.2.2 et le 2.4 de <cit.>)
On a un isomorphisme $G(\Am^{\oo,v}) \times P_{h,d-h}(F_v) \times W_v$-ťquivariant[Noter
le dťcalage $[d-1]$ dans la dťfinition de $\Psi_{\IC,\overline \Zm_l}$.]
$$\ind_{(D_{v,h}^\times)^0 \varpi_v^\Zm}^{D_{v,h}^\times}
\Bigl ( \hi^{h-d-i} \Psi_{\IC,\overline \Zm_l} \Bigr )_{|X^{=h}_{\IC,\bar s}} \simeq
\bigoplus_{\bar \tau \in \RC_{\overline \Fm_l}(h)} \LC_{\overline \Zm_l}(\UC_{\bar \tau,\Nm}^{h-1-i})$$
oý $\LC_{\overline \Zm_l}(\UC^{h-1}_{\bar \tau,\Nm})$ est le systŤme local
associť ŗ la $D_{v,h}^\times$-reprťsentation[La correspondance entre le systŤme indexť par
$\IC$ et $\Nm$ est donnťe par l'application $m_1$ de <ref>.]
$\UC^\bullet_{\bar \tau,\Nm}={\displaystyle \lim_{\rightarrow} ~\UC^\bullet_{\bar \tau,n}}$
oý $\UC_{\bar \tau,n}^\bullet$ est le $\bar \tau$-facteur isotypique de la $D^\times_{v,h}$-reprťsentation
admissible $\UC^\bullet_n:=H^\bullet (\MC_{LT,n}^{h/F_v},\overline \Zm_l)$ obtenue comme
la cohomologie de la fibre gťnťrique gťomťtrique
$$\MC_{LT,n}^{h/F_v}:=\MC_{LT,h,n} \hat \otimes_{\hat F_v^{nr}} \hat{\overline F_v}$$
du schťma formel de Lubin-Tate reprťsentant les classes d'isomorphismes des dťformations par
quasi-isogťnies du $\OC_v$-module formel de hauteur $h$ et de dimension $1$.
Pour $L \in \FC(X,\Lambda)$, on considŤre le diagramme suivant
& L \ar[drr]^{\can_{*,L}} \\
\lexp {p+} j_! j^* L \ar[ur]^{\can_{!,L}}
\ar@{->>}[r]|-{+} & \lexp p j_{!*}j^* L \ar@{^{(}->>}[r]_+ &
\lexp {p+} j_{!*} j^* L \ar@{^{(}->}[r] & \lexp p j_*j^* L
oý la ligne du bas est la factorisation canonique de
$\lexp {p+} j_! j^* L \longrightarrow \lexp {p} j_* j^* L$
et les flŤches $\can_{!,L}$ et $\can_{*,L}$ donnťes par adjonction. Rappelons que les flŤches
aux extrťmitťs (resp. du milieu) de la ligne du bas, sont strictes (resp. est un bimorphisme),
i.e. la flŤche canonique de la coimage vers l'image est un isomorphisme (resp. la flŤche est
un monomorphisme et un ťpimorphisme).
Si $X$ est en outre muni d'une stratification
$\SF=\{ X=X^{\geq 1} \supset X^{\geq 2} \supset \cdots \supset X^{\geq d} \}$,
en utilisant des notations similaires ŗ celles de <ref>, on dťfinit dans
<cit.>, une filtration dite de stratification de $L$
$$0=\Fil^{0}_{\SF,!}(L) \subset \Fil^1_{\SF,!}(L) \subset \Fil^2_{\SF,!}(L) \cdots \subset
\Fil^{d-1}_{\SF,!}(L) \subset \Fil^d_{\SF,!}(L)=L,$$
oý pour $1 \leq r \leq d-1$, on pose
$\Fil^r_{\SF,!}(L):=\im_\FC \Bigl ( j^{1 \leq r}_! j^{1 \leq r,*} L \longrightarrow L \Bigr )$. Par construction
tous les graduťs $\gr_{\SF,!}^r(L)$ sont libres.
Dans <cit.> proposition 2.3.3, on construit par ailleurs,
de faÁon fonctorielle, la filtration $\SF$-exhaustive de stratification de tout objet $L$ de
$$0=\Fill^{-2^{e-1}}_{\SF,!}(L) \subset \Fill^{-2^{e-1}+1}_{\SF,!}(L) \subset \cdots \subset
\Fill^0_{\SF,!}(L) \subset \cdots \subset \Fill^{2^{e-1}-1}_{\SF,!}(L)=L,$$
dont tous les graduťs $\grr_{\SF,!}^r(L)$
sont libres et, aprŤs tensorisation par $\overline \Qm_l$, des extensions intermťdiaires
de systŤmes locaux irrťductibles.
§.§ Structures entiŤres
Dans ce paragraphe nous allons rappeler quelques uns des rťsultats de <cit.>
concernant la filtration de stratification de $\Psi_{\IC_v}$. La premiŤre ťtape consiste ŗ le dťcomposer
selon ses $\varrho$-types au sens suivant.
(cf. <cit.> proposition 3.2.2)
Il existe une dťcomposition
$$\Psi_{\IC_v} \simeq \bigoplus_{1 \leq g \leq d}\bigoplus_{\varrho \in \scusp_v(g)} \Psi_{\IC_v,\varrho}$$
* $\scusp_v(g)$ dťsigne l'ensemble des classes d'ťquivalences inertielles
des $\overline \Fm_l$-reprťsentation irrťductibles supercuspidales de $GL_g(F_v)$;
* $\Psi_{\IC_v,\varrho}$ est un faisceau pervers de type
$\varrho$ au sens oý tous les constituants irrťductibles de
$\Psi_{\IC_v,\varrho} \otimes_{\overline \Zm_l} \overline \Qm_l$
sont des extensions intermťdiaires de systŤmes locaux $\HT(\pi_v,t,\Pi)(n)$ oý
la rťduction modulo $l$ de $\pi_v$ a pour support supercuspidal un segment
relativement ŗ $\varrho$.
on rappelle qu'une reprťsentation irrťductible $\pi_v$ de $GL_n(F_v)$ est dite cuspidale
(resp. supercuspidale)
si elle n'est pas un sous-quotient (resp. un quotient) d'une induite parabolique propre.
Sur $\overline \Qm_l$, ces deux notions coÔncident et d'aprŤs <cit.> III.5.10,
la rťduction modulo $l$ d'une $\overline \Qm_l$-reprťsentation irrťductible cuspidale entiŤre
est irrťductible cuspidale, mais pas nťcessairement supercuspidale auquel cas son support
supercuspidal est un segment.
Un des rťsultats clefs de <cit.> concernant la filtration de stratification
$\Fil^\bullet_{\SF,!}(\Psi_{\IC_v,\varrho})$ est le fait que les morphismes d'adjonction
$$j^{=r}_! j^{=r,*} \gr^r_{\SF,!}(\Psi_{\IC_v}) \longrightarrow \gr^r_{\SF,!}(\Psi_{\IC_v})$$
sont surjectifs dans $\lexp p \CC$.
Pour dťcrire plus prťcisťment les $\gr^r_{\SF,!}(\Psi_{\IC_v})$, commenÁons par quelques rappels sur
la rťduction modulo $l$ d'une reprťsentation de Steinberg $\st_s(\pi_v)$, pour $\pi$ irrťductible cuspidale
de $GL_g(F_v)$ dont la rťduction modulo $l$, notťe $\varrho$, est supercuspidale. Cette rťduction
n'est en gťnťral, pas irrťductible mais contient toujours une unique reprťsentation irrťductible non
dťgťnťrťe notťe $\st_s(\varrho)$. Si $\epsilon(\varrho)$ est le cardinal
de la droite de Zelevinsky de $\varrho$, on note suivant <cit.> p.51,
$$m(\varrho)=\left\{ \begin{array}{ll} \epsilon(\varrho), & \hbox{si } \epsilon(\varrho)>1; \\ l, & \hbox{sinon.} \end{array} \right.$$
Alors $\st_s(\varrho)$ est cuspidale si et seulement si $s=1$ ou $m(\varrho)l^k$ pour $k \in \Nm$: on le note
$$\varrho_k:=\st_{m(\varrho)l^k}(\varrho), \qquad k \in \Nm$$
et $\varrho_{-1}:=\varrho$. La description complŤte de la rťduction modulo $l$ de $\st_s(\pi)$ est donnťe
dans <cit.>. Dans <cit.>, on construit des rťseaux stables des $\st_s(\pi)$
dits rťseaux d'induction. Parmi ceux-ci, il en existe un notť
dont la rťduction modulo $l$
admet comme unique sous-espace irrťductible $\st_s(\varrho)$, qui est donc gťnťrique.
Une $\overline \Qm_l$ reprťsentation $\pi$ irrťductible entiŤre et cuspidale dont la rťduction modulo $l$
est isomorphe ŗ $\varrho_k$ est dite de $\varrho$-type $k$. On note
l'ensemble des classes d'ťquivalences de ces reprťsentations de $GL_{g_k}(F_v)$ avec donc
$$g_{-1}(\varrho)=g \quad \hbox{ et } \quad \forall 0 \leq i \leq s(\varrho),~
g_i(\varrho)=m(\varrho)l^i g,$$
oý on a notť $s(\varrho)$ la plus grande puissance de $l$ divisant $\frac{d}{m(\varrho)g}$.
Revenons ŗ prťsent ŗ la description des $\gr^r_{\SF,!}(\Psi_{\IC_v,\varrho})$. Par construction il
est ŗ support dans $X^{\geq r}_{\IC,\bar s_v}$ nul si $g$ ne divise par $r$ et sinon
$$%\ind_{(D_{v,tg}^\times)^0 \varpi_v^\Zm}^{D_{v,tg}^\times} \Bigl (
j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho}) \otimes_{\overline \Zm_l} \overline \Qm_l
\simeq \bigoplus_{\atop{i=-1}{t_ig_i(\varrho)= tg}}^{s(\varrho)}
\bigoplus_{\pi_v \in \scusp_i(\varrho)} \HT(\pi_v,t_i,\st_{t_i}(\pi_v)) (\frac{1-t_i}{2}).$$
Considťrons alors la $\varrho$-filtration naÔve de
$$\Fil_{\varrho,s(\varrho),\overline \Qm_l} (\Psi,tg) \subset \cdots \subset
\Fil_{\varrho,-1,\overline \Qm_l} (\Psi,tg) =j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho})
\otimes_{\overline \Zm_l} \overline \Qm_l$$
$$%\ind_{(D_{v,tg}^\times)^0 \varpi_v^\Zm}^{D_{v,tg}^\times} \Bigl (
\Fil_{\varrho,k,\overline \Qm_l} (\Psi,tg) \Bigr )
\simeq \bigoplus_{\atop{i=k}{t_ig_i(\varrho) = tg}}^{s(\varrho)}
\bigoplus_{\pi_v \in \scusp_i(\varrho)} \HT(\pi_v,t_i,\st_{t_i}(\pi_v)) (\frac{1-t_i}{2}),$$
ainsi que la filtration associťe de $j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho})$
\Fil_{\varrho,k} (\Psi,tg) \ar@{^{(}-->}[r] \ar@{^{(}-->}[d] & \Fil_{\varrho,k,\overline \Qm_l} (\Psi,tg)
\ar@{^{(}->}[d] \\
j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho}) \ar@{^{(}->}[r] &
j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho}) \otimes_{\overline \Zm_l} \overline \Qm_l.
Pour $k=-1,\cdots,s(\varrho)$, les graduťs $\gr_{\varrho,k} (\Psi,tg)$ associťs
sont alors sans torsion et de $\varrho$-type $k$.
On filtre ŗ nouveau chacun de ces graduťs en sťparant, sur $\overline \Qm_l$, les diffťrents
$\pi_v \in \scusp_k(\varrho)$ et on note
$$(0)=\Fil_{\varrho}^0(\Psi,tg) \subset \Fil_{\varrho}^1(\Psi,tg) \subset \cdots \subset
\Fil_{\varrho}^r(\Psi,tg)=j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho})$$
la filtration ainsi obtenue. On considŤre alors la filtration de $\gr^{tg}_{\SF,!}(\Psi_{\IC_v,\varrho})$
construite comme suit:
* on prend l'image de
$$j^{=tg}_! \Fil_{\varrho}^1(\Psi,tg) \longrightarrow \gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho})$$
et on note $\gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho})^1$ le quotient libre;
* comme $j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho})^1 \simeq \Bigl (
j^{=tg,*} \gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho}) \Bigr ) / \Fil_{\varrho}^1(\Psi,tg)$, on prend l'image de
$$j^{=tg}_! \Bigl ( \Fil_{\varrho}^2(\Psi,tg)/ \Fil_{\varrho,\pi_v}^1(\Psi,tg) \Bigr )
\longrightarrow \gr^{tg}_{\SF,!}(\Psi_{\IC,\varrho})^1$$
et ainsi de suite.
* On filtre enfin chacune des images considťrťes ŗ l'aide d'une filtration de stratification exhaustive.
D'aprŤs <cit.>, les graduťs obtenus sont, sur $\overline \Qm_l$ et en utilisant les notations
de <ref>, de la forme
$\PF(\pi_v,t)(\frac{s-1-2k}{2})$ pour $k=0,\cdots,s-1$. Par construction et d'aprŤs <cit.>,
la filtration ci-avant vťrifie les propriťtťs suivantes:
* si deux graduťs ne diffŤrent, sur $\overline \Qm_l$, que par leur poids, alors
celui de plus petit poids possŤde un indice plus petit que l'autre
(cf. <cit.> ou <cit.>);
* Étant donnť un graduť de poids minimal parmi tous les autres graduťs qui lui sont, sur
$\overline \Qm_l$, isomorphes
au poids prŤs, les graduťs d'indice strictement plus petit sont
* soit tous ŗ supports dans une strate de dimension strictement plus petite,
* soit ŗ support dans la mÍme strate et de $\varrho$-type infťrieur ou ťgal.
* Dans le cas de mÍme support et de $\varrho$-type ťgal, on peut modifier la filtration prťcťdente
pour ťchanger l'ordre des deux graduťs considťrťs.
* Enfin, d'aprŤs le rťsultat principal de <cit.>, tant que
$\lfloor \frac{d}{g} \rfloor +1 \leq m(\varrho)$, pour tout $\pi_v$ de type $\varrho$,
il n'y a ŗ isomorphisme prŤs qu'une notion d'extension intermťdiaire, i.e. avec les notations de
<cit.>, $\lexp p j^{=tg}_{!*} \HT(\pi_v,t,\Pi_t)(n) \simeq \lexp {p+} j^{=tg}_{!*}
\HT(\pi_v,t,\Pi_t)(n)$.
dans la suite les hypothŤses faites sur l'idťal $\mathfrak m$ feront que les systŤmes locaux
d'Harris-Taylor considťrťs, vťrifieront tous la condition du dernier tiret ci-avant.
§ COHOMOLOGIE ENTIŤRE
§.§ Localisation en un idťal non pseudo-Eisenstein
Pour $I \in \IC$, on note
* $\spl(I)$ l'ensemble des places $w$ de $F$ telles que
* $w$ ne divise pas le niveau $I$ et
* $p_w:=w_{|\Qm}$ est dťcomposť dans $F$ et distinct de $l$, avec
$$G(\Qm_{p_w}) \simeq \Qm_{p_w}^\times \times GL_d(F_w) \times \prod_{i=2}^r (B_{w_i}^{op})^\times,$$
oý $p_w=w.\prod_{i=2}^r w_i$ dans $F^+$.
* Pour $i=0,\cdots, d$, on note $T_{w,i}$ la fonction caractťristique de
$$GL_d(\OC_w) \diag(\overbrace{\varpi_w,\cdots,\varpi_w}^{i}, \overbrace{1,\cdots,1}^{d-i} )
GL_d(\OC_w) \subset GL_d(F_w),$$
oý $\varpi_w$ dťsigne une uniformisante de l'anneau des entiers $\OC_w$ de $F_w$ et on note
$$\Tm_I:=\Zm_l \bigl [T_{w,i}:~w \in \spl(I) \hbox{ et } i=1,\cdots,d \bigr ],$$
l'algŤbre de Hecke associťe ŗ $\spl(I)$.
* Pour $\mathfrak m$ un idťal maximal de $\Tm_I$ et $w \in \spl(I)$, on note
$$P_{\mathfrak{m},w}(X):=\sum_{i=0}^d(-1)^i q_w^{\frac{i(i-1)}{2}} \overline{T_{w,i}} X^{d-i} \in \overline
\Fm_l[X]$$
le polynŰme de Hecke associť ŗ $\mathfrak m$ et
S_{\mathfrak{m}}(w) := \bigl \{ \lambda \in \Tm_I/\mathfrak m \simeq \overline \Fm_l \hbox{ tel que }
P_{\mathfrak{m},w}(\lambda)=0 \bigr \} ,$$
le multi-ensemble des paramŤtres de Satake modulo $l$ en $w$ associťs ŗ $\mathfrak m$,
i.e. on garde en mťmoire la multiplicitť des paramŤtres.
On dit d'un idťal premier minimal $\widetilde{\mathfrak m}$ de $\Tm_I$ qu'il est cohomologique s'il
existe une $\overline \Qm_l$-reprťsentation automorphe cohomologique $\Pi$ de $G(\Am)$
possťdant des vecteurs non nuls fixes sous $I$ et telle que pour tout $w \in \spl(I)$, les paramŤtres
de Satake de $\Pi_{p_w}$ sont donnťs par les racines du polynŰme de Hecke
$\sum_{i=0}^d(-1)^i q_w^{\frac{i(i-1)}{2}} T_{w,i} X^{d-i} \in K_{\widetilde{\mathfrak m}}[X]$ oý
$K_{\widetilde{\mathfrak m}}:=(\Tm_I \otimes_{\Zm_l} \Qm_l)/\widetilde{\mathfrak m}$
est une extension finie de $\Qm_l$. Un tel $\Pi$ n'est pas
nťcessairement unique mais dťfinit une unique classe d'ťquivalence proche au sens de <cit.> que
l'on notera $\Pi_{\widetilde{\mathfrak m}}$. On note
$$\rho_{\widetilde{\mathfrak m},\overline \Qm_l}:\gal(\bar F/F) \longrightarrow GL_d(\overline \Qm_l)$$
la reprťsentation galoisienne associťe ŗ un tel $\Pi$ d'aprŤs
<cit.> et <cit.>, laquelle est, d'aprŤs le thťorŤme de Cebotarev,
dťfinie ŗ isomorphisme sur $K_{\widetilde{\mathfrak m}}$, i.e.
$\rho_{\widetilde{\mathfrak m},\overline \Qm_l} \simeq \rho_{\widetilde{\mathfrak m}}
\otimes_{K_{\widetilde{\mathfrak m}}} \overline \Qm_l$.
Étant donnť un idťal maximal $\mathfrak m$ de $\Tm_I$, pour tout idťal premier
minimal $\widetilde{\mathfrak m} \subset \mathfrak m$, la rťduction modulo
$\varpi_{\widetilde{\mathfrak m}}$ de
$\rho_{\widetilde{\mathfrak m}}$, bien dťfinie ŗ semi-simplification prŤs, ne dťpend que de $\mathfrak m$,
on note $\overline \rho_{\mathfrak m}: G_F \longrightarrow GL_d(\overline \Fm_l)$ son extension des
scalaires ŗ $\overline \Fm_l$: ses paramŤtres
de Satake modulo $l$ en $w \in \spl(I)$ sont donnťs par le multi-ensemble
$S_{\mathfrak m}(w)$.
Dans la suite, on fixe un idťal maximal $\mathfrak m$ de $\Tm_I$ qui vťrifie une des conditions (1) et (2)
(1) Il existe $w_1 \in \spl(I)$ tel que $S_{\mathfrak{m}}(w_1)$ ne contient aucun
sous-multi-ensemble de la forme $\{ \alpha, q_{w_1} \alpha \}$ oý $q_{w_1}$ est le cardinal
du corps rťsiduel en $w_1$;
(2) On a $l \geq d+2$ et $\mathfrak m$ vťrifie l'un des deux conditions suivantes:
* soit $\overline \rho_{\mathfrak m}$ est induit d'un caractŤre de $G_K$ pour $K/F$ une extension
galoisienne cyclique;
* soit $l \geq d$ et $SL_n(k) \subset \overline \rho_{\mathfrak m}(G_F) \subset \overline \Fm_l^\times
GL_n(k)$ pour un sous-corps $k \subset \overline \Fm_l$.
(cf. <cit.>)
Pour $\mathfrak m$ comme ci-dessus, les localisťs
$H^i(X_{I,\bar \eta},\overline \Zm_l)_{\mathfrak m}$, en l'idťal $\mathfrak m$
des groupes de cohomologie $X_I$, sont sans torsion.
§.§ Reprťsentations elliptiques et libertť de la cohomologie
Rappelons que $X_I$ ťtant propre, pour tout place $w$ de $F$, on a un isomorphisme
$$H^{d-1+i}(X_{\IC_v,\bar \eta_v},\overline \Zm_l) \simeq H^i(X_{I,\bar s_v},\Psi_{\IC_v})$$
qui est $G(\Am^\oo) \times W_v$ ťquivariant. En utilisant la filtration de stratification de
$\Psi_{\IC,v}$, on a une suite spectrale qui ŗ partir des groupes de cohomologie des
$\PF(\pi_v,t)$ calcule la cohomologie de $X_{\IC,\bar \eta_v}$. Sur $\overline \Qm_l$, l'ťtude
de cette suite spectrale est menťe dans <cit.> et aprŤs localisation par
$\mathfrak m$, comme les
$$H^i(X_{\IC,\bar s_v},\PF(\pi_v,t))_{\mathfrak m} \otimes_{\overline \Zm_l} \overline \Qm_l$$
sont concentrťs en degrť $0$, cette suite spectrale dťgťnŤre clairement en $E_1$. Le but de ce
paragraphe est de reprendre cette ťtude sur $\overline \Fm_l$ et de montrer qu'ŗ nouveau les
$$H^i(X_{\IC,\bar s_v},\PF(\pi_v,t))_{\mathfrak m} \otimes_{\overline \Zm_l} \overline \Fm_l$$
sont concentrťs en degrť $0$, et donc qu'il n'y a pas de torsion dans les
$H^i(X_{\IC,\bar s_v},\PF(\pi_v,t))_{\mathfrak m}$.
Pour ce faire, rappelons les notations du 1.2 de <cit.>. Pour tout $t \geq 0$, on note
$$\Gamma^t:=\Bigl \{ (a_1,\cdots,a_r,\epsilon_1,\cdots,\epsilon_r) \in \Nm^r \times \{ \pm \}^r:~
r \geq 1,~\sum_{i=1}^r a_i=t \Bigr \}.$$
Un ťlťment de $\Gamma^t$ sera notť sous la forme $(\overleftarrow{a_1}, \cdots , \overrightarrow{a_r})$ oý
pour tout $i$ la flŤche au dessus de $a_i$ est $\overleftarrow{a_i}$ (resp.
$\overrightarrow{a_i}$) si $\epsilon_i=-$ (resp. $\epsilon_i=+$). On considŤre alors sur $\Gamma^t$ la
relation d'ťquivalence induite par les identifications suivantes:
$$(\cdots,\overleftarrow{a},\overleftarrow{b},\cdots)=(\cdots,\overleftarrow{a+b},\cdots), \quad
$(\cdots, \overleftarrow{0},\cdots)=(\cdots,\overrightarrow{0},\cdots)$.
On notera $\overrightarrow{\Gamma}^t$ l'ensemble
quotient dont les ťlťments seront notťs $[\overleftarrow{a_1},\cdots,\overrightarrow{a_r}]$.
pour $t>0$, dans toute classe $[\overleftarrow{a_1},\cdots,\overrightarrow{a_k}]
\in \overrightarrow{\Gamma}^t$, il existe un unique ťlťment, dit rťduit, de
$(a_1,\cdots,a_r,\epsilon_1,\cdots,\epsilon_r) \in \Gamma^t$ tel que
* pour tout $1 \leq i \leq r$, on ait $a_i>0$;
* pour tout $1 \leq i < r$, $\epsilon_i \epsilon_{i+1}=-$.
ņ tout reprťsentant rťduit $(a_1,\cdots,a_r,\epsilon_1,\cdots,\epsilon_r)$ d'une classe de
$[\overleftarrow{a_1},\cdots,\overrightarrow{a_k}] \in \overrightarrow{\Gamma}^t$, on associe
$$\SC\Bigl ( [\overleftarrow{a_1},\cdots,\overrightarrow{a_k}] \Bigr )$$
comme le sous-ensemble des permutations $\sigma$ de $\{ 0,\cdots,t-1 \}$ telles que pour tout
$1 \leq i \leq r$ avec $\epsilon_i=+$ (resp. $\epsilon_i=-$) et pour
tout $a_1+\cdots + a_{i-1} \leq k < k' \leq a_1+ \cdots +a_i$ alors $\sigma^{-1}(k) < \sigma^{-1}(k')$
(resp. $\sigma^{-1}(k) > \sigma^{-1}(k')$).
Dans le cas oý sous les mÍmes conditions, on demande inversement $\sigma^{-1}(k)> \sigma^{-1}(k')$
(resp. $\sigma^{-1}(k) < \sigma^{-1}(k')$), l'ensemble obtenu sera notť $\SC^{op}
\Bigl ( [\overleftarrow{a_1},\cdots,\overrightarrow{a_k}] \Bigr )$.
(cf. <cit.> 2)
Soit $g$ un diviseur de $d=sg$ et $\pi$ une reprťsentation cuspidale irrťductible de $GL_g(F_w)$.
Il existe une bijection
$$[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}] \in \overrightarrow{\Gamma}^{s-1}
\mapsto [\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi}$$
dans l'ensemble des sous-quotients irrťductibles de l'induite
$$\pi \{ \frac{1-s}{2} \} \times \pi \{ \frac{3-s}{2} \} \times \cdots \times \pi \{ \frac{s-1}{2} \}$$
caractťrisťe par la propriťtť suivante
$$J_{P_{g,2g,\cdots,sg}}([\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi})=\sum_{\sigma \in
\SC \Bigl ( [\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}] \Bigr ) } \pi \{ \frac{1-s}{2}+\sigma(0) \}
\otimes \cdots \otimes \pi\{ \frac{1-s}{2}+\sigma(s-1) \},$$
ou encore par
$$J_{P^{op}_{g,2g,\cdots,sg}}([\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi})=\sum_{\sigma \in
\SC^{op} \Bigl ( [\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}] \Bigr ) } \pi \{ \frac{1-s}{2}+\sigma(0) \}
\otimes \cdots \otimes \pi\{ \frac{1-s}{2}+\sigma(s-1) \}.$$
avec ces notations $\st_s(\pi)$ (resp. $\speh_s(\pi)$) est $[\overleftarrow{s-1}]$
(resp. $[\overrightarrow{s-1}]$).
(cf. <cit.>)
Soit $\psi$ une $\overline \Qm_l$-reprťsentation irrťductible telle que le support cuspidal de l'induite
$[\overleftarrow{t-1}]_{\pi} \times \psi$ soit un segment de Zelevinsky
$[\pi \{ \frac{a}{2} \},\pi \{ \frac{a+s-1}{2}]$ contenant strictement $[\pi \{ \frac{1-t}{2} \},\pi \{ \frac{t-1}{2} \} ]$.
Si $a < 1-t$ (resp. si $a+s-1 > t-1$), alors, avec les notations de la proposition
<ref>, tout sous-espace irrťductible de $[\overleftarrow{t-1}]_{\pi} \times \psi$
est de la forme $[\cdots,\overleftarrow{1},\overleftarrow{t-1},\cdots]_{\pi \{ \frac{1-s-a}{2} \} }$
(resp. $[\cdots,\overleftarrow{t-1},\overrightarrow{1},\cdots]_{\pi \{ \frac{1-s-a}{2} }$)
oý le support cuspidal du $\overleftarrow{t-1}$ dans l'ťcriture ci-avant est ťgal ŗ
celui $[\pi \{ \frac{1-t}{2} \},\pi \{ \frac{t-1}{2} \} ]$.
Soit $\pi$ une $\overline \Qm_l$-reprťsentation irrťductible cuspidale entiŤre de $GL_g(F_w)$ dont
le cardinal de la droite de Zelevinsky de sa rťduction modulo $l$ est supťrieur ou ťgal
ŗ $s$. Dans ces conditions,
(i) pour tout $[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}] \in
\overrightarrow{\Gamma}^{s-1}$, les sous-quotients irrťductibles la rťduction modulo $l$ des
$[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi}$ pour
$[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]$ dťcrivant $\overrightarrow{\Gamma}^{s-1}$
sont disjoints deux ŗ deux.
(ii) Avec les notations de la proposition prťcťdente, tout sous espace irrťductible
de la rťduction modulo $l$ de $[\overleftarrow{t-1}]_{\pi} \times \psi$ est la rťduction modulo $l$
d'une reprťsentation de la forme donnťe par la proposition prťcťdente.
dans le cas oý le cardinal de la droite de Zelevinsky du lemme ci-avant est $>s$, la rťduction
modulo $l$ de tout $[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi}$ est irrťductible.
En cas d'ťgalitť, seul $[\overleftarrow{s-1}]_\pi$ admet une rťduction modulo $l$ rťductible qui est
alors de longueur $2$ avec un unique constituant cuspidal.
(i) Par hypothŤse sur $\pi$, toutes les reprťsentations considťrťes ont le mÍme support cuspidal et ont
donc tous une image non nulle par $J_{P_{g,2g,\cdots,sg}}$. Le rťsultat dťcoule alors directement
* de la commutation des foncteurs de Jacquet avec la rťduction modulo $l$,
* du fait que ces rťductions modulo $l$ des $[\overleftarrow{a_1}, \cdots , \overrightarrow{a_r}]_{\pi}$
n'admettent aucun sous-quotient irrťductible cuspidal sauf ťventuellement
$[\overleftarrow{s-1}]_\pi$ dans le cas oý la droite de Zelevinsky de $r_l(\pi)$
est de cardinal $s$,
* et du fait que les $r_l(\pi) \{ \frac{1-s}{2}+k \}$ sont distincts deux ŗ deux pour $0 \leq k < s$
de sorte que l'image par $J_{P_{g,2g,\cdots,sg}}$ de
$\pi \{ \frac{1-s}{2} \} \times \pi \{ \frac{3-s}{2} \} \times \cdots \times \pi \{ \frac{s-1}{2} \}$
est sans multiplicitť.
(ii) Le rťsultat dťcoule de la proposition prťcťdente en utilisant:
* la rťduction au cas unipotent de
$GL_s$ d'aprŤs <cit.> IV.6.3;
* comme la droite de Zelevinsky de la rťduction modulo $l$ de $\pi$
est de cardinal $\geq s$, l'ordre de $q$ modulo $l$ est $>s$,
* i.e. $l$ est banal relativement ŗ $GL_s(\Fm_q)$ et qu'alors le bloc unipotent sur $\overline \Fm_l$
est ťquivalent ŗ celui sur $\overline \Qm_l$.
Fixons ŗ prťsent un idťal maximal $\mathfrak m$ de $\Tm_I$ tel que l'hypothŤse suivante est vťrifiťe:
($H_{combi}$): il existe une place $w_0 \in \spl(I)$ telle que l'image de $\bar \rho_{\mathfrak m,w_0}$
dans son groupe de Grothendieck est sans multiplicitťs.
On ťcrit $I$ sous la forme $U_{w_0}(m)$ avec $m=(m_w)_{w|u_0} \in \Nm^r$ et oý $u_0$ est la
restriction de $w_0$ ŗ $E$.
Pour tout $n \in \Nm$, soit $m^{w_0}(n)$ le $r$-uplet obtenu ŗ partir
de $n$ en remplaÁant $m_{w_0}$ par $n$. Alors pour $P:=(P_n)_{n \in \Nm}$
un faisceau pervers de Hecke sur la tour
$\bigl ( X_{U_{w_0}(m^{w_0}(n)),\bar s_{w_0}} \bigr )_{n \in \Nm}$, on note
$$H^i(X_{I(w_0),\bar s_{w_0}},P):= \lim_{\atop{\longrightarrow}{n \in \Nm}}
H^i(X_{U_{w_0}(m^{w_0}(n)),\bar s_{w_0}},P_n).$$
Avec les notations prťcťdentes et sous l'hypothŤse ($H_{combi}$), on suppose en outre que
pour tout $i$, le
$\overline \Zm_l$-module $H^i(X_{I(w_0),\bar \eta},\overline \Zm_l)_{\mathfrak m}$ est sans torsion.
Alors pour tout $\overline \Zm_l$-systŤme local d'Harris-Taylor $\HT(\pi_{w_0},t)$, la torsion de
$H^i(X_{I(w_0),\bar s_{w_0}},\lexp p j^{=tg}_{!*} \HT(\pi_{w_0},t))_{\mathfrak m}$
est nulle pour tout $i$.
Notons $\scusp_{w_0}(\mathfrak m)$ l'ensemble des classes d'ťquivalences inertielles des
$\overline \Fm_l$-reprťsentations irrťductibles supercuspidales du support supercuspidal de la rťduction
modulo $l$ de la composante locale en $w_0$ d'une reprťsentation $\Pi \in \Pi_{\mathfrak m}$ de la
classe d'ťquivalence proche associťe ŗ $\mathfrak m$. La suite spectrale des cycles ťvanescents
localisťe en $\mathfrak m$ s'ťcrit
H^i(X_{I(w_0),\bar \eta_{w_0}},\overline \Zm_l[d-1])_{\mathfrak m} & \simeq H^i(X_{I(w_0),\bar s_{w_0}},
\bigoplus_{\varrho \in \scusp_{w_0}(\mathfrak m)} \Psi_{I,w_0,\varrho})_{\mathfrak m} \\ & \simeq
\bigoplus_{\varrho \in \scusp_{w_0}(\mathfrak m)} H^i(X_{I(w_0),\bar s_{w_0}},
\Psi_{I,w_0,\varrho})_{\mathfrak m},
\end{array}$$
de sorte que pour tout $\varrho \in \scusp_{w_0}(\mathfrak m)$ le localisť en
$\mathfrak m$ de la cohomologie de $\Psi_{I(w_0),\varrho}$ est concentrťe en degrť mťdian et sans torsion.
Soit alors $\varrho \in \scusp_{w_0}(\mathfrak m)$, la filtration de stratification exhaustive de
$\Psi_{I(w_0),\varrho}$ raffinťe par celle de son $\varrho$-type comme au <ref>, permet
de calculer, via une suite spectrale, la cohomologie de $\Psi_{I(w_0),\varrho}$ localisťe
en $\mathfrak m$
ŗ l'aide de celle des faisceaux pervers d'Harris-Taylor $\PF(\pi_{w_0},t)$ pour $\pi_{w_0}$ irrťductible
cuspidale de type $\varrho$.
On raisonne par l'absurde,
la stratťgie consistant alors ŗ montrer que le localisť en $\mathfrak m$ de la cohomologie
de $\Psi_{I(w_0),\varrho}$ aurait de la torsion, ce qui n'est pas d'aprŤs nos hypothŤses.
Considťrons tout d'abord le cas des $\pi_{w_0}$ de $\varrho$-type $-1$, i.e. tels
que $r_l(\pi_{w_0})$ est supercuspidale de la forme $\varrho \{ \frac{r}{2} \}$ pour $r \in \Zm$.
Soit alors $t_0$ minimal tel qu'il existe $\pi_{w_0}$ de $\varrho$-type $-1$ et $i \neq 0$ pour lesquels
$$H^{i} \bigl (X_{I(w_0),\bar s_{w_0}},\PF(\pi_{w_0},t_0) \otimes_{\overline \Zm_l} \overline \Fm_l
\bigr )_{\mathfrak m} \neq (0).$$
D'aprŤs le rťsultat principal de
<cit.>, le dual de Verdier $D (\PF(\pi_{w_0},t))$ est isomorphe ŗ
$\PF(\pi_{w_0}^\vee,t)$, i.e. les $p$ et $p+$-extensions intermťdiaires sont les mÍmes
de sorte que $\PF(\pi_{w_0},t_0) \otimes_{\overline \Zm_l} \overline \Fm_l$ est une
$\overline \Fm_l$-extension intermťdiaire; en particulier quitte ŗ considťrer $\pi_{w_0}^\vee$
de type $\varrho^\vee$, on peut supposer $i_0<0$ que l'on prendra minimal. Ainsi pour tout
$\pi_{w_0}'$ de $\varrho$-type $-1$, on a
$$H^{i} \bigl (X_{I(w_0),\bar s_{w_0}},\PF(\pi_{w_0},t)
\otimes_{\overline \Zm_l} \overline \Fm_l \bigr )_{\mathfrak m} = \left \{ \begin{array}{ll}
(0) & \hbox{si } t< t_0 \hbox{ ou } i<i_0 \\
\hbox{non nul} & \hbox{si } t=t_0 \hbox{ et } i=i_0<0. \end{array} \right.$$
D'aprŤs le lemme <ref>, un $GL_d(F_{w_0}) \times W_{w_0}$
constituant irrťductible d'un des
$H^{i} \bigl (X_{I(w_0),\bar s_{w_0}},\PF(\pi_{w_0},t)(\frac{t_0-1}{2}-k)
\otimes_{\overline \Zm_l} \overline \Fm_l \bigr )_{\mathfrak m}$
est ŗ prendre parmi ceux de la rťduction modulo $l$ de
$$\Bigl ( [\overleftarrow{a_1}, \cdots , \overrightarrow{a_{i-1}}, \overleftrightarrow{1},
\overleftarrow{t_0-1}, \overleftrightarrow{1}, \overrightarrow{a_{i+1}},
\cdots \overrightarrow{a_r}]_{\pi}
\times \Upsilon_{w_0} \Bigr ) \otimes \LC(\pi \{ \frac{\delta}{2} \} )$$
* le $\overleftrightarrow{1}$ avant (resp. aprŤs) le $\overleftarrow{t_0-1}$ est autorisť sous ses
deux formes $\overleftarrow{1}$ et $\overrightarrow{1}$ si $\sum_{j=1}^{i-1} a_i$ (resp.
$\sum_{j=i+1}^r a_i$) est strictement positive: on notera en passant $a_i=t_0+1$
* la rťduction modulo $l$ de $\pi$ est de la forme $\varrho \{ \frac{\delta_0}{2} \}$,
* $\Upsilon_{w_0}$ est une $\overline \Qm_l$-reprťsentation irrťductible entiŤre dont
la rťduction modulo $l$ a un support supercuspidal qui n'est pas reliť ŗ celui de
$ [\overleftarrow{a_1}, \cdots , \overrightarrow{a_{i-1}}, \overleftrightarrow{1},
\overleftarrow{t_0-1}, \overleftrightarrow{1}, \overrightarrow{a_{i+1}},
\cdots \overrightarrow{a_r}]_{\varrho}$,
* si on note $\Bigl \{ \pi \{ \frac{\alpha}{2} \}, \pi \{ \frac{\alpha}{2} +1 \}, \cdots,
\pi \{ \frac{\alpha}{2}+t_0-1 \} \Bigr \}$ le support supercuspidal associť ŗ la flŤche
$\overleftarrow{t_0-1}$ dans $[\overleftarrow{a_1}, \cdots , \overleftarrow{t_0-1},\cdots
\overrightarrow{a_r}]_{\pi}$ alors $\pi \{\frac{\alpha}{2} + k \}=\pi \{ \frac{\delta}{2} \}$.
Considťrons alors un $\overline \Fm_l[GL_d(F_{w_0})\times W_{w_0}]$-module irrťductible
$(\tau \times \psi_{w_0}) \otimes \LC(\pi \{ \frac{\delta}{2} \} )$ avec $\delta$ comme ci-dessus, oý:
* $\psi_{w_0}$ est un sous-quotient irrťductible quelconque de la rťduction modulo $l$ de
* $\tau$ est un sous-quotient irrťductible de la rťduction modulo $l$ de
$$[\overleftarrow{a_1}, \cdots , \overrightarrow{a_{i-1}}, \overrightarrow{1},
\overleftarrow{t_0-1}, \overrightarrow{1}, \overrightarrow{a_{i+1}},
\cdots \overrightarrow{a_r}]_{\pi}$$
qui, d'aprŤs le lemme prťcťdent, ne sera jamais un constituant de la rťduction modulo $l$ d'un autre
$[\overleftarrow{a'_1}, \cdots,\overrightarrow{a'_{r'}}]_{\pi}$ avec $\sum_{j=1}^{r'} a'_j=\sum_{j=1}^r a_j$.
Dans le cas oý $\forall j \neq i,~ a_j =0$, et si la rťduction modulo $l$ de
$[\overleftarrow{t_0-1}]_{\pi_{w_0}}$ est rťductible, on prend pour $\tau$ un sous-quotient non gťnťrique.
Montrons que ce constituant irrťductible est alors un sous-quotient de
$$H^{i_0}(X_{I(w_0),\bar s_{w_0}},\Psi_{I(w_0),\varrho} \otimes_{\overline \Zm_l}
\overline \Fm_l)_{\mathfrak m} \simeq
H^{i_0}(X_{I(w_0),\bar s_{w_0}},\Psi_{I(w_0),\varrho})_{\mathfrak m} \otimes_{\overline \Zm_l}
\overline \Fm_l,$$
ce qui sera contradictoire puisque ($H_{tor}$) implique que la cohomologie de $\Psi_{I(w_0),\varrho}$
est sans torsion concentrťe en degrť mťdian.
Avec les notations prťcťdentes on considŤre $k=0$,
i.e. la cohomologie de $\PF(\pi_{w_0},t_0)(\frac{t_0-1}{2})$ de sorte que $\delta=\alpha$.
Notons $\Fill_{\SF,!}^{k-1}(\Psi_{I(w_0),\varrho}) \subset \Fill_{\SF,!}^{k}(\Psi_{I(w_0),\varrho})
\subset \Psi_{I(w_0),\varrho}$ les indices de la filtration exhaustive
de stratification considťrťe plus haut, tels que
\Fill_{\SF,!}^k(\Psi_{I(w_0),\varrho})/\Fill_{\SF,!}^{k-1}(\Psi_{I(w_0),\varrho}) \simeq
\PF(\pi_{w_0},t_0)(t-1).$$
Par construction de la filtration et notamment en modifiant l'ordre dans le raffinement par le $\varrho$-type,
les faisceaux pervers d'Harris-Taylor dans $\Fill_{\SF,!}^{k-1}(\Psi_{I(w_0),\varrho})$
sont de la forme
$\PC(\pi'_{w_0},t')$ oý $\pi'_{w_0}$ de $\varrho$-type $-1$ avec $t'>t_0$, cf. la fin du <ref>.
En particulier si $\sum_{j \neq i} a_i=0$ alors les
$H^{i_0+1}(X_{I(w_0),\bar s_{w_0}},\Fill_{\SF,!}^{k-1}(\Psi_{I(w_0),\varrho})
\otimes_{\overline \Zm_l} \overline \Fm_l)_{\mathfrak m}$ sont nuls et sinon il est filtrť
par des sous-quotients irrťductibles de la rťduction modulo $l$ de
$$\Bigl ( [\overleftarrow{a_1}, \cdots , \overrightarrow{a_{i-1}}, \overleftrightarrow{1},
\overleftarrow{t'-1}, \overleftrightarrow{1}, \overrightarrow{a_{i+1}},
\cdots \overrightarrow{a_r}]_{\pi'}
\times \psi_{w_0} \Bigr ) \otimes \LC(\pi' \{ \frac{\delta'}{2} \} )$$
avec $t'>t_0$ et oý $\pi' \{\frac{\delta'}{2} \}$ est une des cuspidales du support cuspidal du
$\overleftarrow{t'-1}$ dans l'ťcriture ci-dessus. On remarque alors que la contrainte $t'>t_0$
ne nous permet jamais de retrouver le $\tau$ de la rťduction modulo $l$ de
$[\overleftarrow{a_1}, \cdots , \overrightarrow{a_{i-1}}, \overrightarrow{1},
\overleftarrow{t_0-1}, \overrightarrow{1}, \overrightarrow{a_{i+1}},
\cdots \overrightarrow{a_r}]_{\pi}$.
En ce qui concerne les $H^{i_0-1}(X_{I(w_0),\bar s_{w_0}},\Psi_{I(w_0),\varrho}/
\Fill_{\SF,!}^k(\Psi_{I(w_0),\varrho}))_{\mathfrak m}\otimes_{\overline \Zm_l} \overline \Fm_l$, on raisonne
de mÍme:
* les constituants $\PC(\pi'_{w_0},t')$ oý $\pi'_{w_0}$ est de $\varrho$-type $-1$ et avec $t'>t_0$,
ne permettent pas, comme prťcťdemment, de retrouver la bonne reprťsentation;
* ceux pour $\pi'_{w_0}$ de $\varrho$-type $-1$ avec $t' < t_0$ (resp. $t'=t_0$)
sont nuls par minimalitť de $t_0$ (resp. de $i_0$);
* les $\pi'_{w_0}$ qui sont de $\varrho$-type $\geq 0$, ont une contribution nulle d'aprŤs
($H_{combi}$). Le seul cas non couvert pas ($H_{combi}$) est celui oý $\pi'_{w_0}$ est de
$\varrho$-type $0$ avec $t_0=g_0$, $t'=1$ et $\sum_{j \neq i} a_j=0$: mais alors comme on a pris
$\tau$ non gťnťrique, il ne peut pas Ítre obtenu dans la cohomologie de
$\PF(\pi'_{w_0},1)$ laquelle s'ťcrit toujours sous la forme $\rho_0 \times \psi'$.
Ainsi donc, on obtient un sous-quotient non nul d'un $H^{i_0}(X_{I(w_0),\bar s_{w_0}},\Psi_{I(w_0),\varrho}
\otimes_{\overline \Zm_l} \overline \Fm_l)_{\mathfrak m}$, avec $i_0<0$ ce qui n'est pas, d'oý la
§.§ Du lemme d'Ihara ŗ la cohomologie
…tant donnť un faisceau de Hecke ŗ support dans les points supersinguliers,
ses groupes de cohomologie $H^i$ son nuls pour tout $i \neq 0$. En ce qui concerne son $H^0$,
rappelons que $X_{\IC,\bar s_{v_0}}^{=d}$ peut s'ťcrire comme une rťunion disjointe
de schťmas de dimension $0$
$$X_{\IC,\bar s_{v_0}}^{=d}=\coprod_{i \in \ker^1(\Qm,G)} X_{\IC,\bar s_{v_0},i}^{=d}.$$
Un faisceau de Hecke $\FC_{\IC,i}$ sur $X_{\IC,\bar s_{v_0},i}^{=d}$ a sa fibre en une tour $z_i$ de points
supersinguliers munie d'une action de $\overline G(\Qm) \times GL_d(F_{v_0})^0$ oý
$GL_d(F_{v_0})^0$ est le noyau de la valuation du dťterminant de sorte que,
cf. <cit.> proposition 5.1.1, en tant que $G(\Am^\oo)\simeq \overline G(\Am^{\oo,v_0})
\times GL_d(F_{v_0})$-module, on a
$$H^0( X_{\IC,\bar s_{v_0},i}^{=d},\FC_{\IC,i}) \simeq
\ind_{\overline G(\Qm)}^{\overline G(\Am^{\oo,v_0}) \times \Zm} z_i^* \FC_{\IC,i}$$
avec $\delta \in \overline G(\Qm) \mapsto (\delta^{\oo,v_0},\val \circ \rn( \delta_{v_0}))
\in \overline G(\Am^{\oo,v_0}) \times \Zm$ et
oý l'action de $g_{v_0} \in GL_d(F_{v_0})$ est donnťe par celle de
$(g_0^{-\val \det g_{v_0}} g_{v_0},\val \det g_{v_0}) \in GL_d(F_{v_0})^0 \times \Zm$
oý $g_0 \in GL_d(F_{v_0})$ est un ťlťment fixť tel que $\val \det g_0=1$.
En outre, cf. <cit.> corollaire 5.1.2, si $z_i^* \FC_\IC$ est munie d'une action
du noyau $(D_{v_0,d}^\times)^0$ de la valuation de la norme rťduite qui est compatible ŗ l'action de
$\overline G(\Qm) \hookrightarrow D_{v_0,d}^\times$, alors en tant que $G(\Am^\oo)$-module,
pour $I=U_{v_0}(m)$, on a
\begin{equation} \label{h0-ss}
H^0(X_{I(v_0),\bar s_{v_0},i}^{=d},\FC_{\IC,i}) \simeq \CC^\oo(\overline G(\Qm) \backslash
\overline G(\Am^{\oo})/I^{v_0},\Lambda)
\otimes_{D_{v_0,d}^\times} \ind_{(D_{v_0,d}^\times)^0}^{D_{v_0,d}^\times} z_i^* \FC_{\IC,i}
\end{equation}
Soit $\bar \pi$ une sous-$\overline \Fm_l$-reprťsentation irrťductible de
$\mathcal C^\oo(\overline G(\Qm) \backslash \overline G(\Am)/I^{v_0},\overline \Fm_l)$.
On choisit alors une $\overline \Qm_l$-reprťsentation irrťductible cuspidale entiŤre $\pi_{v_0}$
de $GL_g(F_{v_0})$ pour $g$ divisant
$d=sg$, telle que
* la rťduction modulo $l$ de $\pi_{v_0}$, notťe $\varrho$, est supercuspidale, et
* la composante locale $\bar \pi_{v_0}$ de $\bar \pi$ en $v_0$
est la rťduction modulo $l$ de $\pi_{v_0}[s]_D$.
Alors $\bar\pi^{\oo,v_0}$ est une sous-reprťsentation de la rťduction modulo $l$ de
$$H^0 \bigl ( X_{I(v_0),\bar s_{v_0}},\HT_{\overline \Fm_l}(\pi_{v_0}^\vee,s) \bigr )=
H^0 \bigl ( X_{I(v_0),\bar s_{v_0}},\HT_{\overline \Zm_l}(\pi_{v_0}^\vee,s) \bigr ) \otimes_{\overline \Zm_l}
\overline \Fm_l.$$
$\pi_{v_0}$ ťtant de $\varrho$-type $-1$, on notera que tous les rťseaux stables de
$\HT_{\overline \Zm_l}(\pi_{v_0}^\vee,s)$ sont isomorphes.
On a $\bar \pi \subset \mathcal C^\oo(\overline G(\Qm) \backslash \overline G(\Am)/I^{v_0},
\overline \Fm_l)$ et donc $\bar \pi^{v_0} \subset \mathcal C^\oo(\overline G(\Qm) \backslash
\overline G(\Am)/I^{v_0},\overline \Fm_l) \otimes \bar \pi_{v_0}^\vee$.
Par ailleurs comme $\HT(\pi_{v_0}^\vee,s)$ est supportť aux points supersinguliers, son $H^0$
est sans torsion et
$$H^0 \bigl (X_{I(v_0),\bar s_{v_0}},\HT(\pi_{v_0}^\vee,s) \bigr ) \otimes_{\overline \Zm_l}
\overline \Fm_l \simeq
H^0 \Bigl ( X_{I(v_0),\bar s_{v_0}}, \HT(\pi_{v_0}^\vee,s)
\otimes_{\overline \Zm_l} \overline \Fm_l \Bigr ).$$
Le rťsultat dťcoule alors directement de la formule (<ref>).
* $I$ un niveau fini et $S$ un ensemble fini de places contenant les places $w$ de $F$ telles que $I_w$ n'est pas maximal;
* $\mathfrak m$ un idťal maximal de $\Tm_S$ tel qu'il existe $w_1 \in \spl^S$ tel que
$S_{\mathfrak m}(w_1)$ ne contient aucun sous-multi-ensemble de la forme $\{ \alpha,q_{w_1} \alpha \}$;
* $\bar \pi$ une sous-$\overline \Fm_l$-reprťsentation irrťductible de
$\mathcal C^\oo(\overline G(\Qm) \backslash \overline G(\Am)/I^{v_0},\overline \Fm_l)_{\mathfrak m}$.
Alors $\bar \pi^{\oo,v_0}$ est une sous-$\overline \Fm_l$-reprťsentation irrťductible de
$H^{d-1}(X_{I(v_0),\bar \eta_{v_0}},\overline \Fm_l)_{\mathfrak m}$.
Dans <cit.>, nous avons montrť que les faisceaux de cohomologie de $\Psi_{I(v_0)}$
ťtaient sans torsion. De la description de ceux-ci, on en dťduit alors que s'il existe $i$ tel que
$H^i(X_{I(v_0),\bar s_{v_0}},\Psi_{I(v_0),\varrho})_{\mathfrak m}$ est non nul alors nťcessairement
la composante locale $\bar \pi_{v_0}$ de $\bar \pi$ doit Ítre la rťduction modulo $l$ de
$\pi_{v_0}[s]_D$ oý $\pi_{v_0}$ est une $\overline \Qm_l$-reprťsentation irrťductible entiŤre
de $GL_g(F_{v_0})$ pour $d=sg$, de $\varrho$-type $-1$.
Fixons alors une telle reprťsentation $\pi_{v_0}$.
Des propriťtťs de la filtration de stratification exhaustive raffinťe par les $\varrho$-types
et en utilisant que $\pi_{v_0}$ est de $\rho$-type $-1$,
on en dťduit que $\PF_\Gamma(\pi_{v_0},s)(\frac{s-1}{2})$ est un sous-faisceau pervers de
$\Psi_{I(v_0),\varrho}$ pour un certain rťseau stable $\Gamma$.
Comme par ailleurs, les hypothŤses sur $\mathfrak m$ impliquent, cf. le thťorŤme
<ref>, que le localisť en $\mathfrak m$
de la cohomologie d'un graduť quelconque de $\Psi_{I(v_0)}$ est sans torsion et concentrťe en degrť
mťdian, on en dťduit que $H^0(X_{I(v_0),\bar s_{v_0}},\PF_\Gamma(\pi_{v_0},s)(\frac{s-1}{2}))$
est un sous-espace de $H^{d-1}(X_{I(v_0),\bar \eta_{v_0}},\overline \Zm_l)$.
Si on ne s'intťresse pas ŗ la reprťsentation portťe en
$v_0$, tous les rťseaux $\Gamma$ sont isomorphes, i.e. $\PF(\pi_{v_0},s)$ est une somme directe finie
de $\HT(\pi_{v_0},t)$ de sorte que le rťsultat dťcoule alors du lemme prťcťdent.
§ SUR LA COHOMOLOGIE MODULO $L$
§.§ Un lemme combinatoire
Soit comme dans l'introduction un idťal maximal $\mathfrak m$ de $\Tm_S$ vťrifiant les hypothŤses
($H_{tor}$) et ($H_{combi}$) de sorte qu'en particulier
l'image de $\overline \rho_{\mathfrak m,w_0}$ dans son groupe de Grothendieck est sans
multiplicitťs. On ťcrit le support supercuspidal $\SC(\mathfrak m)$ associť ŗ
$\overline \rho_{\mathfrak m,w_0}$ par la correspondance de Langlands, sous la forme
$$\SC(\mathfrak m)=\coprod_{\varrho \in \scusp_{w_0}} \SC_{\varrho}(\mathfrak m)$$
oý $\scusp_{w_0}$ dťsigne l'ensemble des classes d'ťquivalence inertielles des $\overline \Fm_l$-reprťsentations
irrťductibles supercuspidales d'un $GL_{g(\varrho)}(F_{w_0})$ pour $1 \leq g(\varrho) \leq d$.
Pour chaque $\varrho$ on note alors
$$l_1(\mathfrak m,\varrho) \geq \cdots \geq l_{r(\mathfrak m,\varrho)}(\mathfrak m,\varrho)$$
de sorte que $S_{\varrho}(\mathfrak m)$ puisse s'ťcrire comme la rťunion de $r(\mathfrak m,\varrho)$
segments de Zelevinsky non liťs
$$[\varrho \nu^{k_i},\varrho \nu^{k_i+l_i(\mathfrak m,\varrho)}]=
\bigl \{ \varrho \nu^{k_i},\varrho \nu^{k_i+1},\cdots, \varrho \nu^{k_i+l_i(\mathfrak m,\varrho)} \bigr \}.$$
Pour $\bar \pi$ un sous-quotient irrťductible de $\CC^\oo(\overline G(\Qm) \backslash
\overline G(\Am)/U^{v_0},\overline \Fm_l)_{\mathfrak m}$. On ťcrit sa composante locale en $w_0$
sous la forme $\bar \pi_{w_0} \simeq \bigtimes_{\varrho \in \scusp_{w_0}} \bar \pi(\varrho)$
oý le support supercuspidal de $\bar \pi(\varrho)$ est $\SC_\varrho(\mathfrak m)$.
Soit $\bar \pi$ un sous-quotient
de la rťduction modulo $l$ d'une reprťsentation de la forme
$$[\cdots,\overleftarrow{t-1},\overrightarrow{1},\cdots]_{\pi'_{w_0}} \times \psi, \quad
\hbox{(resp. } [\cdots,\overleftarrow{1},\overleftarrow{t-1},\cdots]_{\pi'_{w_0}} \times \psi)$$
oý $\psi$ est une reprťsentation quelconque de sorte que la rťduction modulo $l$ de la reprťsentation
ci-avant ait pour support supercuspidal $\SC_{\varrho}(\mathfrak m)$. On note $\varrho \{ \delta \}$
la cuspidale ŗ l'extrťmitť droite de $\overrightarrow{1}$ (resp. de $\overleftarrow{1}$)
dans l'ťcriture prťcťdente.
Alors pour tout faisceau pervers $\PF(\pi_{w_0},t')(\frac{1-t'+2k}{2})$ pour $0 \leq k \leq t'-1$, et pour tout
rťseau stable, la rťduction modulo $l$ de $H^i(X_{I(w_0),\bar s_{w_0}},
\PF_\Gamma(\pi_{w_0},t') (\frac{1-t'+2k}{2}))_{\mathfrak m}$ n'admet pas un sous-espace de la forme
$$\bigl ( \bigtimes_{\varrho \neq \varrho' \in \scusp_{w_0}} \bar \pi(\varrho') \bigr ) \times
\bar \pi \times \Lm(\varrho \{ \delta \} ).$$
D'aprŤs <ref>, les $H^i(X_{I(w_0),\bar s_{w_0}}, \PF_\Gamma(\pi_{w_0},t')
(\frac{t'-1-2k}{2}))_{\mathfrak m}$ sont sans torsion, de sorte que leurs rťduction modulo $l$
se calcule ŗ partir de la description de la $\overline \Qm_l$-cohomologie donnťe dans
<cit.>. En particulier pour obtenir $\Lm(\varrho \{ \delta \} )$, il faut que la rťduction
modulo $l$ de $\pi_{w_0}$ soit dans la droite de Zelevinsky de $\varrho$. Plus prťcisťment
rappelons que, d'aprŤs <cit.>, la $\overline \Qm_l$-cohomologie de
$\PF_\Gamma(\pi_{w_0},t') (\frac{t'-1}{2})$ est de la forme
$\bigl ( \st_{t'}(\pi_{w_0} \{ \alpha \} ) \times ? \bigr ) \otimes \Lm(\pi_{w_0} \{ \frac{1-t'}{2}+ \alpha \} )$, autrement dit l'action galoisienne fixe la cuspidale ŗ gauche du support cuspidal
$\Bigl \{ \pi_{w_0} \{ \frac{1-t'}{2}+\alpha \}, \cdots ,\pi_{w_0} \{ \frac{t'-1}{2} + \alpha \} \Bigl \}$
de $\st_{t'}(\pi_{w_0} \{ \alpha \} )$. On utilise alors l'hypothŤse sur $\mathfrak m$ que le support
supercuspidal de $\SC(\mathfrak m)$ est sans multiplicitť.
* Pour $k>0$, il dťcoule de ce qui prťcŤde que la rťduction modulo $l$ de
$[\cdots,\overleftarrow{t-1},\overrightarrow{1},\cdots]_{\pi'_{w_0}}$ doit Ítre obtenu comme un
sous-quotient de la rťduction modulo $l$ de $[\cdots,\overleftarrow{t'-1},\cdots]_{\pi_{w_0}}$
oý dťsormais la supercuspidale $\varrho \{ \delta \}$ est dans le support cuspidal de la partie
$\overleftarrow{t'-1}$ ce qui n'est pas.
* Pour $k=0$, d'aprŤs le lemme prťcťdent
$H^i(X_{I(w_0),\bar s_{w_0}}, \PF_\Gamma(\pi_{w_0},t') (\frac{t'-1}{2}))_{\mathfrak m}$
s'ťcrit comme une induite $[\overleftarrow{t'-1}]_{\pi_{w_0}\{ \alpha \} } \times \psi$
oý $\varrho_0 \{ \delta \}$
est la supercuspidale ŗ gauche du support de la rťduction modulo $l$ de
$[\overleftarrow{t'-1}]_{\pi_{w_0}\{ \alpha \} }$. Mais alors, d'aprŤs le lemme <ref>,
un sous-espace irrťductible de la
rťduction modulo $l$ d'un tel groupe de cohomologie est nťcessairement de la forme
$[\cdots,\overleftarrow{1},\cdots]_{\varrho} \times \psi'$ oý $\varrho \{ \delta \}$
est la cuspidale ŗ droite de $\overleftarrow{1}$.
Le cas respectif se traite de maniŤre strictement similaire.
§.§ Rťseaux globaux
Par hypothŤse, il existe $v_1 \in \spl$ telle que $\overline \rho_{\mathfrak m,v_1}$ est irrťductible
de sorte que pour toute reprťsentation irrťductible $\Pi \in \Pi_{\mathfrak m}$, sa composante locale
en $v_1$ est cuspidale et sa rťduction modulo $l$, $\varrho$ est supercuspidale. En particulier on a
$$H^{d-1+i}(X_{I,\bar \eta_{v_1}},\overline \Zm_l)_{\mathfrak m} \simeq H^i(X_{I,\bar s_{v_1}},
\Psi_{I,\varrho})_{\mathfrak m}.$$
Comme $\varrho$ est une reprťsentation irrťductible supercuspidale, $\Psi_{I,\varrho}$
est un systŤme local
concentrť sur $X^{=d}_{U,\bar s_{v_1}}$ tel que pour tout point supersingulier $z$
$$ \ind_{(D_{v_1,d}^\times)^0}^{D_{v_1,d}^\times} z^* \Psi_{\IC,\varrho} \otimes_{\overline \Zm_l}
\overline \Qm_l \simeq \bigoplus_{\pi_{v_1} \in \scusp_{-1}(\varrho)} \pi_{v_1} \otimes†\pi_{v_1}[1]_D
\otimes \Lm(\pi_{v_1}).$$
Soit $\Pi^{\oo} \otimes \Lm(\Pi_{v_1})$ une sous-reprťsentation irrťductible de
$H^{d-1}(X_{\IC,\bar \eta_{v_1}},\overline \Qm_l)_{\mathfrak m}$. Alors le rťseau
$$H^{d-1}(X_{\IC,\bar \eta_{v_1}},\overline \Zm_l)_{\mathfrak m} \cap \Bigl (
\Pi^{\oo} \otimes \Lm(\Pi_{v_1}) \Bigr )$$
est un produit tensoriel $\Gamma_G \otimes \Gamma_W$ d'un rťseau stable $\Gamma_G$
de $\Pi^{\oo}$ par un rťseau stable $\Gamma_W$ de $\Lm(\Pi_{v_1})$.
Comme la rťduction modulo $l$ de
$\Pi_{v_1} \otimes \Pi_{v_1}[1]_D \otimes \Lm(\Pi_{v_1})$ est irrťductible, il ne possŤde,
ŗ isomorphisme prŤs, qu'un unique rťseau stable de sorte que
$$\Gamma_{GDW} := \Bigl ( \ind_{(D_{v_1,d}^\times)^0}^{D_{v_1,d}^\times} z^* \Psi_{\IC,\rho} \Bigr ) \cap
\Bigl (\Pi_{v_1} \otimes \Pi_{v_1}[1]_D \otimes \Lm(\Pi_{v_1}) \Bigr )$$
est isomorphe au produit tensoriel
$\Gamma_G \otimes \Gamma_D \otimes \Gamma_W$ des rťseaux stables de respectivement
$\Pi_{v_1}$, $\Pi_{v_1}[1]_D$ et $\Lm(\Pi_{v_1})$.
Revenons au calcul de la cohomologie de $X_U$ via la suite spectrale des cycles ťvanescents: on a
$$H^{d-1+i}(X_{\IC,\bar \eta_{v_1}},\overline \Zm_l)_{\mathfrak m} \simeq H^i(X_{\IC,\bar s_{v_1}},
\Psi_{\IC,\varrho})$$
et d'aprŤs (<ref>)
H^0(\bar X_{\IC,\bar s_{v_1},i}^{=d},\Psi_{\IC,\varrho}) & \simeq \CC^\oo
(\overline G(\Qm) \backslash \overline G(\Am^{\oo}),\Lambda)
\otimes_{D_{v_1,d}^\times} \bigl ( \ind_{(D_{v_1,d}^\times)^0}^{D_{v_1,d}^\times} z^* \Psi_{\IC,\varrho} \bigr ) \\
& \simeq \CC^\oo (\overline G(\Qm) \backslash \overline G(\Am^{\oo}),\Lambda)
\otimes_{D_{v_1,d}^\times} \bigl ( \Gamma_G \otimes \Gamma_D \otimes \Gamma_W \bigr ),
\end{array}$$
d'oý le rťsultat.
comme la cohomologie est donnťe par les points supersinguliers, l'ťnoncť est aussi valable ŗ niveau
fini en considťrant l'action de $\Tm_I \times W$.
§.§ Rťseaux locaux
Soit $\PF_\Gamma(\pi_{w_0},t)$ une $\overline \Zm_l$-structure entiŤre d'un faisceau pervers
d'Harris-Taylor tel qu'il existe $i$ vťrifiant
$$H^i(X_{I(w_0),\bar s_{w_0}},\PF_\Gamma(\pi_{w_0},t))_{\mathfrak m} \neq (0).$$
(i) soit $r_l(\pi_{w_0})$ est supercuspidal notť $\varrho$ avec
$$l_1(\mathfrak m,\varrho) + \cdots + l_{r(\mathfrak m,\varrho)}(\mathfrak m,\varrho) < m(\varrho)$$
et alors le rťseau stable du systŤme local
$j^{=tg,*} \PF_\Gamma(\pi_{w_0},t)$ est un produit tensoriel $\Gamma_G \otimes \Gamma_D \otimes
\Gamma_W$ d'un rťseau stable $\Gamma_G$ (resp. $\Gamma_D$, resp. $\Gamma_W$) de
$\st_t(\pi_{w_0})$ (resp. de $\HT(\pi_{w_0},t)$, resp. de $\Lm(\pi_{w_0})$).
(ii) Sinon $r(\mathfrak m,\varrho)=1$ et
* soit $r_l(\pi_{w_0})$ est supercuspidale avec $t=l_1(\mathfrak m,\varrho)=m(\varrho)$,
* soit $t=1$ et $r_l(\pi_{w_0}) \simeq \varrho_0$.
Notons tout d'abord que, d'aprŤs <ref>, comme
$H^i(X_{I(w_0),\bar s_{w_0}},\PF_{\Gamma}(\pi_{w_0},t))_{\mathfrak m}$ est sans torsion, s'il est
non nul alors $t \leq \max \{ l_1(\mathfrak m,\varrho),\cdots,
l_{r(\mathfrak m,\varrho)}(\mathfrak m,\varrho) \}$.
a) Si le support supercuspidal
de la rťduction modulo $l$ de $\st_t(\pi_{w_0})$ n'est pas ťgal ŗ toute la droite de Zelevinsky
d'une supercuspidale alors $r_l(\pi_{w_0})$ est supercuspidale et les
rťductions modulo $l$ de $\st_t(\pi_{w_0})$, $\pi_{w_0}[t]_D$ et $\Lm(\pi_{w_0})$ sont irrťductibles.
Dans ce cas tous les rťseaux stables de $j^{=tg,*} \PF(\pi_{w_0},t)$ sont homothťtiques et
donc isomorphes ŗ un produit tensoriel de rťseaux stables. On se retrouve ainsi dans le
cas dťcrit par le (ii) de l'ťnoncť.
b) Si le support supercuspidal de $r_l(\st_t(\pi_{w_0}))$ est ťgal ŗ toute la droite de Zelevinsky
d'une cuspidale $\varrho$ alors, comme par hypothŤse $S_\varrho(\mathfrak m)$ est
sans multiplicitť, on a nťcessairement $r(\mathfrak m,\varrho)=1$ avec
* soit $r_l(\pi_{w_0})$ supercuspidale et alors $t=l_1(\mathfrak m,\varrho)$,
* soit $t=1$ et $r_l(\pi_{w_0}) \simeq \varrho_0$.
§.§ Preuve du lemme d'Ihara
Soit $\mathfrak m$ un idťal maximal de $\Tm_S$ vťrifiant les hypothŤse ($H_\leadsto$), ($H_{tor}$) et ($H_{combi}$) de
l'introduction. On reprend les notations du <ref> avec $r(\mathfrak m,\varrho)$
et les $l_i(\mathfrak m,\varrho)$ pour $i=1,\cdots, r(\mathfrak m,\varrho)$.
Dans la situation du premier tiret du cas (ii) du lemme <ref>, i.e. oý $r(\mathfrak m,\varrho)=1$
et avec $\pi_{w_0}$ une $\overline \Qm_l$-reprťsentation irrťductible cuspidale de $GL_g(F_{w_0})$
dont la rťduction modulo $l$ est $\varrho$, on note
$\PF_\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho)) (\frac{1-l_1(\mathfrak m,\varrho)}{2})$ le rťseau
du faisceau pervers d'Harris-Taylor donnťe par la filtration de stratification exhaustive de
$\Psi_{\IC,w_0}(\overline \Zm_l)$. Si
$$H^0(X_{I(w_0),\bar s_{w_0}},\PF\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho))_{\mathfrak m}
(\frac{1-l_1(\mathfrak m,\varrho)}{2})) \neq (0),$$
alors tout $\overline \Fm_l[GL_d(F_{w_0})]$-sous-espace irrťductible de sa rťduction modulo $l$ est de la forme
$\bar \pi_{\varrho} \times \bar \pi^{\varrho}$ oý
* $\bar \pi_{\varrho}$ est une reprťsentation irrťductible gťnťrique de support supercuspidal
un segment de Zelevinsky relatif ŗ $\varrho$ et
* $\bar \pi^{\varrho}$ a un support supercuspidal disjoint de la droite de Zelevinsky associťe ŗ
Dans le cas oý la rťduction modulo $l$ de $\st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$ est irrťductible,
elle est gťnťrique et le rťsultat est immťdiat. Le seul cas ŗ traiter est celui oý cette rťduction
modulo $l$ est de longueur $2$ avec un unique constituant gťnťrique, notť $\varrho_0$ au
<ref>. Il s'agit alors de montrer
que le rťseau stable de $\PF_\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho))
(\frac{l_1(\mathfrak m,\varrho)-1}{2})$ est un produit de rťseaux stables et que celui sur
$\st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$ est tel que sa rťduction modulo $l$ admet la gťnťrique
comme unique sous-espace irrťductible.
Pour ce faire, rappelons que d'aprŤs le rťsultat principal de <cit.>, le faisceau pervers
entier $\PF_\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho)) (\frac{l_1(\mathfrak m,\varrho)-1}{2})$
s'obtient aussi en considťrant la filtration de stratification exhaustive de $j^{=g}_! \HT_{\overline \Zm_l}
(\pi_{w_0},1,\pi_{w_0})$ dont la structure entiŤre est unique ŗ isomorphisme prŤs et donc sous la
forme d'un produit tensoriel. La structure entiŤre cherchťe se dťduit alors de l'ťtude de la suite
spectrale calculant la fibre en un point gťomťtrique de $X^{=l_1(\mathfrak m,\varrho)}_{\IC,\bar s_{w_0}}$
du faisceau de cohomologie $h^{l_1(\mathfrak m,\varrho)g-d} j^{=g}_! \HT(\pi_{w_0},1,\pi_{w_0})$,
laquelle est trivialement nulle. Dans <cit.>, on montre que cette suite spectrale est sans torsion
de sorte qu'on peut la lire sur $\overline \Qm_l$ telle qu'elle est dťcrite dans <cit.>.
En particulier la structure entiŤre de
$\PF_\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho)) (\frac{l_1(\mathfrak m,\varrho)-1}{2})$
se lit dans le faisceau de cohomologie d'indice $l_1(\mathfrak m,\varrho)g-d$
de l'extension intťrmťdiaire de $\HT_{\Gamma'}(\pi_{w_0},l_1(\mathfrak m,\varrho)-1, \Pi)$
oý $\Pi=\pi_{w_0}\{ \frac{2-l_1(\mathfrak m,\varrho)}{2} \} \times
\st_{l_1(\mathfrak m,\varrho)-1}(\pi_{w_0} \{ \frac{1}{2} \}$ et oý le rťseau $\Gamma'$ est un
produit tensoriel. Son faisceau de cohomologie d'indice $l_1(\mathfrak m,\varrho)g-d$
est alors l'induite parabolique d'un rťseau produit tensoriel de $\HT(\pi_{w_0},l_1(\mathfrak m,\varrho),1)
\otimes \st_{l_1(\mathfrak m,\varrho)-1}(\pi_{w_0} \{ \frac{1}{2} \} \otimes \Lm(\pi_{w_0})$, ŗ des dťcalages
par le poids prŤs. Comme tous les rťseaux stables de $\HT(\pi_{w_0},l_1(\mathfrak m,\varrho),1)$
sont isomorphes et donc sous la forme d'un produit tensoriel, on en dťduit que le rťseau de
$\PF_\Gamma(\pi_{w_0},l_1(\mathfrak m,\varrho)) (\frac{l_1(\mathfrak m,\varrho)-1}{2})$
est aussi un produit tensoriel.
Enfin le rťseau stable associť ŗ $\st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$ doit Ítre tel que la flŤche
$$\pi_{w_0}\{ \frac{2-l_1(\mathfrak m,\varrho)}{2} \} \times \st_{l_1(\mathfrak m,\varrho)-1}(\pi_{w_0}
\{ \frac{1}{2} \} \longrightarrow \st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$$
soit surjective, ce qui impose, cf. aussi <cit.>, que l'unique gťnťrique,
isomorphe ŗ la cuspidale $\varrho_0$, de la rťduction
modulo $l$ de $\st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$ ne peut pas Ítre un quotient irrťductible
de la rťduction modulo $l$ du rťseau stable et que donc il en est l'unique sous-espace
dans loc. cit., le rťseau stable de $\st_{l_1(\mathfrak m,\varrho)}(\pi_{w_0})$
construit dans la preuve prťcťdente, est appelť rťseau d'induction et notť
$RI_-(\pi_{w_0}, l_1(\mathfrak m,\varrho))$, cf. aussi le dťbut du <ref>.
Considťrons une sous-reprťsentation irrťductible $\bar \pi$ de $\CC^\oo(\overline G(\Qm) \backslash
\overline G(\Am) / \overline U^{w_0},\overline \Fm_l)_{\mathfrak m}$ de sorte que d'aprŤs les propositions
<ref> et <ref>, $\bar \pi^{\oo,v_0} \otimes \Lm(\bar \pi)$
est une sous-$\overline \Fm_l$-reprťsentation irrťductible
de $H^{d-1}(X_{I(w_0),\bar \eta_{v_0}},\overline \Fm_l)_{\mathfrak m}$ oý $I^{w_0,v_0}=\bar U^{w_0,v_0}$
comme au <ref>.
Supposons que $\bar \pi_{w_0}$ n'est pas gťnťrique, de sorte qu'elle s'obtient comme un
sous-quotient de la rťduction modulo $l$ d'une reprťsentation de la forme $[\cdots, \overleftarrow{a},
\overrightarrow{b},\cdots]_{\pi_{w_0}} \times \psi$ avec $b>0$ et oý le support supercuspidal
de $r_l(\psi)$ n'est pas liť ŗ celui de $[\cdots, \overleftarrow{a},\overrightarrow{b},\cdots]_{\varrho}$
oý $\varrho=r_l(\pi_{w_0})$. Ainsi il existe une reprťsentation
irrťductible supercuspidale $\varrho \{ \delta \}$ du support supercuspidal de la rťduction modulo $l$ de
$[\cdots, \overleftarrow{a},\overrightarrow{b},\cdots]_{\pi_{w_0}}$ telle que
$$[\cdots, \overleftarrow{a}, \overrightarrow{b},\cdots]_{\varrho} \times r_l(\psi) \otimes
\Lm(\varrho \{ \delta \} )$$
a un sous-espace irrťductible qui est un sous-espace de
$H^{d-1}(X_{U(w_0),\bar \eta_{v_0}},\overline \Fm_l)_{\mathfrak m}$.
En utilisant une filtration de stratification exhaustive du faisceau pervers des cycles ťvanescents en
$w_0$, et comme la cohomologie des faisceaux pervers d'Harris-Taylor est sans torsion,
cette reprťsentation est un sous-espace irrťductible d'un
$H^0(X_{I(w_0),\bar s_{w_0}},\PF_\Gamma(\pi'_{w_0},t) (\frac{1-t+2k}{2}))_{\mathfrak m}
\otimes_{\overline \Zm_l} \overline \Fm_l$ avec $0 \leq k \leq t-1$. Or ce dernier espace est de la forme
$r_l(\st_t(\pi'_{w_0} \{ \delta+k \})) \times \psi' \otimes \Lm(\varrho \{ \delta \}$ oý le support supercuspidal de
$r_l(\st_t(\pi'_{w_0} \{ \delta+k \}))$ est contenu dans celui de $[\cdots, \overleftarrow{a},
\overrightarrow{b},\cdots]_{\varrho}$. Si ces supports supercuspidaux sont ťgaux, et comme d'aprŤs le lemme prťcťdent
$r(\mathfrak m,\varrho)>1$, on en dťduit que la rťduction modulo $l$ de $\st_t(\pi'_{w_0} \{ \delta+k \})$
est irrťductible et est donc nťcessairement gťnťrique, ce qui ne convient pas.
Ainsi donc, d'aprŤs le lemme <ref>, il existe une sous-reprťsentation irrťductible de
$H^{d-1}(X_{I(w_0),\bar \eta_{v_0}},\overline \Fm_l)_{\mathfrak m}$ dont la composante en $w_0$ est de
la forme
$$\bigl ( \bigtimes_{\varrho \neq \varrho' \in \scusp_{w_0}} \bar \pi(\varrho') \bigr ) \times
\pi'(\varrho) \times \Lm(\varrho \{ \delta' \} )$$
avec $\pi'(\varrho)$ comme dans le lemme <ref> et oý $\varrho(\delta')$
est une supercuspidale du support de la partie $\overleftarrow{t-1}$ dans l'ťcriture de $\pi'(\varrho)$.
Or d'aprŤs la proposition <ref>,
$$\bigl ( \bigtimes_{\varrho \neq \varrho' \in \scusp_{w_0}} \bar \pi(\varrho') \bigr ) \times
\pi'(\varrho) \times \Lm(\varrho \{ \delta \} )$$
oý $\varrho \{ \delta \}$ est la cuspidale de l'extrťmitť droite de $\overrightarrow{1}$
(resp. $\overleftarrow{1}$) dans l'ťcriture prťcťdente, doit aussi Ítre un sous-espace irrťductible
de $H^{d-1}(X_{I(w_0),\bar \eta_{v_0}},\overline \Fm_l)_{\mathfrak m}$, et donc, via la filtration
de stratification exhaustive, d'un
$H^i(X_{I(w_0),\bar s_{w_0}}, \PF_\Gamma(\pi_{w_0},t') (\frac{1-t'+2k}{2}))_{\mathfrak m}$
pour $0 \leq k \leq t'-1$, ce qui n'est pas d'aprŤs le lemme <ref>.
§.§ Augmentation du niveau
On reprend les notations prťcťdentes,
$$\SC_{w_0}(\mathfrak m)=\coprod_{\varrho} \SC_{\varrho}(\mathfrak m)$$
oý $\varrho$ dťcrit les classes d'ťquivalence inertielles de $\overline \Fm_l$-reprťsentations
irrťductibles supercuspidales d'un
$GL_{g(\varrho)}(F_{w_0})$ pour $1 \leq g(\varrho) \leq d$. Pour chaque $\varrho$ on note
$$l_1(\varrho) \leq \cdots \leq l_{r(\varrho)}(\varrho)$$
de sorte que $S_{\varrho}(\mathfrak m)$ puisse s'ťcrire comme la rťunion de $r(\varrho)$
segments de Zelevinsky non liťs
$$[\varrho \nu^k_i,\bar \rho \nu^{k+l_i(\varrho)}]=\bigl \{ \varrho \nu^k,\varrho \nu^{k+1},\cdots,
\varrho \nu^{k+l_i(\varrho)} \bigr \}.$$
Il existe un idťal premier $\widetilde{\mathfrak m} \subset \mathfrak m$ tel que
$$\Pi_{\widetilde{\mathfrak m},w_0} \simeq \bigtimes_{\varrho \in \scusp_{w_0}} \Pi_{\widetilde{\mathfrak m},w_0}(\varrho)$$
oý pour tout $\varrho \in \scusp_{w_0}$, il existe des reprťsentations irrťductibles cuspidales
$\pi_1(\varrho),\cdots, \pi_{r(\varrho)}(\varrho)$ de $GL_{g(\varrho}(F_v)$ telles que
$$\Pi_{\widetilde{\mathfrak m},w_0}(\varrho) \simeq \st_{l_1(\varrho)}(\pi_1(\varrho)) \times \cdots \times
\st_{l_{r(\varrho)}}(\pi_{r(\varrho)}(\varrho)).$$
On raisonne par l'absurde en supposant qu'il existe un $\varrho_0$ pour lequel aucun des idťaux premiers contenus
dans $\mathfrak m$ ne vťrifie la conclusion de la proposition. Soit alors un idťal premier
$\widetilde{\mathfrak m} \subset \mathfrak m$: le facteur $\Pi_{\widetilde{\mathfrak m},w_0}(\varrho_0)$
est de la forme
$$\Pi_{\widetilde{\mathfrak m},w_0}(\varrho_0) \simeq \st_{l_1(\varrho_0)}(\pi_1(\varrho_0)) \times
\cdots \times \st_{l_k(\varrho_0)}(\pi_{k}(\varrho_0)) \times \st_{l'_1}(\pi'_1(\varrho_0)) \times
\cdots \times \st_{l'_r}(\pi'_{r}(\varrho_0)),$$
avec $l_{k+1}(\varrho_0)>l'_1 \geq \cdots \geq l'_r$. On choisit $\widetilde{\mathfrak m}$ de sorte
que la partition $(l_1(\varrho_0) \geq \cdots \geq l_k(\varrho_0) \geq l'_1 \geq \cdots \geq l'_r)$ ainsi
obtenue, soit maximale au sens de l'ordre de Bruhat.
La rťduction modulo $l$ de $H^{d-1}(X_{I(w_0),\bar \eta},\overline \Zm_l)_{\mathfrak m} \cap \bigl (
\Pi_{\widetilde{\mathfrak m}}^\oo \otimes \Lm(\Pi_{\widetilde{\mathfrak m}} \bigr )$
fournit alors, d'aprŤs la proposition <ref>, un sous-espace de
$H^{d-1}(X_{I(w_0),\bar \eta},\overline \Fm_l)_{\mathfrak m}$ de la forme $\bar \pi^{\oo} \otimes
\overline \rho_{\mathfrak m}$ avec $\bar \pi^\oo$ irrťductible de composante locale $\bar \pi_{w_0}$
en $w_0$ s'ťcrivant sous
la forme $\bar \pi_{w_0}= \bigtimes_{\varrho \in \scusp_{w_0}} \bar \pi_{w_0,\varrho}$ avec
$$\bar \pi_{w_0,\varrho_0} \simeq
\Bigl ( \st_{l_1(\varrho_0)}(\varrho_0 \nu^{\delta_1}) \times \cdots \times
\st_{l_k(\varrho_0)}(\varrho_0 \nu^{\delta_k}) \times \bar \pi_1 \times \cdots \times \bar \pi_t \Bigr ),$$
oý $\bar \pi_1 \times \cdots \times \bar \pi_t$ est un sous-quotient irrťductible de la rťduction modulo $l$
de $\st_{l'_1}(\pi'_1(\varrho_0)) \times \cdots \times \st_{l'_r}(\pi'_{r}(\varrho_0))$.
par dťfinition des $l_i(\varrho)$ et comme $\SC(\mathfrak m)$ est sans multiplicitť, la rťduction
modulo $l$ de $\st_{l_1(\varrho_0)}(\varrho_0 \nu^{\delta_1}) \times \cdots \times
\st_{l_k(\varrho_0)}(\varrho_0 \nu^{\delta_k})$ est irrťductible, tout comme $\bar \pi_{w_0,\varrho_0}$.
La cohomologie ťtant concentrťe en degrť mťdian et sans torsion, on en dťduit que
$H^{d-1}(X_{I(w_0),\bar \eta},\overline \Fm_l)_{\mathfrak m}$ admet un
sous espace de la forme
$r_l(\Pi^{\oo,w_0}) \otimes\pi_{w_0} \otimes \bar \rho_{\mathfrak m}$. Considťrons alors une filtration
$$\Fil_0(\bar \rho_{\mathfrak m}) \subset \cdots \subset \Fil_r(\bar \rho_{\mathfrak m})=
\bar \rho_{\mathfrak m}$$
dont les graduťs sont irrťductibles et donc de la forme $\varrho \nu^k$. Par hypothŤse, tous les graduťs ont
des poids distincts modulo $l$, ce qui permet de parler du graduť associť ŗ $\varrho \nu^k$.
Avec les notations prťcťdentes,
pour $i=1,\cdots,t$, soit $\alpha_i$ tel que le support supercuspidal de la rťduction modulo $l$ de
$\st_{l'_i}(\pi'_i(\varrho))$ est celui du segment $[\varrho \nu^{\alpha_i},\varrho \nu^{\alpha_i+l'_i-1}]$.
Considťrons le graduť $\gr_k(\bar \rho_{\mathfrak m})$
associť ŗ $\varrho \nu^{\alpha_1}$. On note
$H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m,k-1}$
l'image de $r_l(\Pi^{\oo,w_0}) \otimes \pi_{w_0} \otimes \Fil_{k-1}(\bar \rho_{\mathfrak m})$
et $H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m,/k}$ le quotient
$H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m}/H^{d-1}(X_{U,\bar \eta_{w_0}},
\overline \Fm_l)_{\mathfrak m,k-1}$. On a par ailleurs une filtration
$$\Fil^\bullet (H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m})$$
obtenue ŗ partir de la filtration de stratification exhaustive de $\Psi_{\IC,\varrho}$ et dont les graduťs
sont la rťduction modulo $l$ de la cohomologie d'un rťseau stable d'un
$HT(\pi_{w_0},t,\st_t(\pi_{w_0}))(\frac{1-s+2k}{2})$ pour $\pi_{w_0}$ de $\varrho$-type.
Cette filtration induit alors une filtration
$\Fil^\bullet(H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m,/k})$
de $H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m,/k}$. Comme par hypothŤse,
$\bar \rho_{\mathfrak m}$ est sans multiplicitťs, ces graduťs sont, en tant que
$\overline\Fm_l$-reprťsentation de $W_v$, isotypiques pour un $\varrho \nu^\alpha$ avec des $\alpha$
distincts pour chacun de ces graduťs. En particulier l'image de $\gr_k(\bar \rho_{\mathfrak m})$
se factorise par un seul de ces graduťs et on en dťduit que $\bar \pi_1 \times \cdots \times \bar \pi_t$
est un sous-espace d'une induite de la forme
$\st_{l'_1}(\varrho \nu^{\alpha_1+\frac{l'_1-1}{2}}) \times \bar \pi$.
Par hypothŤse il existe un indice $i'$ tel que le support supercuspidal de la rťduction modulo $l$ de
$\st_{l'_{i'}}(\pi'_{i'}(\varrho))$ est liť ŗ celui de $\st_{l'_1}(\pi'_1(\varrho))$ de sorte que d'aprŤs
le lemme <ref>, $H^{d-1}(X_{I(w_0),\bar \eta_{w_0}},\overline \Fm_l)_{\mathfrak m}$
admettrait un sous-espace non dťgťnťrť (comme dans le lemme <ref>), ce qui
n'est pas d'aprŤs la version du lemme d'Ihara prouvťe au paragraphe prťcťdent.
|
1511.00287
|
Neutrino Flux Studies at NO$\nu$A
Kuldeep Kaur Maan$^{1}$, Hongyue Duyang$^{2}$, Sanjib Ratan Mishra$^{2}$ $\&$ Vipin Bhatnagar$^{1}$
(for the NO$\nu$A Collaboration)
We present the systematic-error study of the neutrino flux in the NO$\nu$A experiment.
Systematic errors on the flux at the near detector (ND), far detector (FD), and
the ratio FD/ND, due to the beam-transport and hadro-production
are estimated. Prospects of constraining the $\nu_{\mu}$ and $\nu_{e}$
flux using data from ND are outlined.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The NuMI Off-axis $\nu_{e}$ Appearance (NO$\nu$A) experiment is composed of two functionally identical
detectors. NO$\nu$A is designed to address a broad range of open questions in the neutrino sector through precision measurements of $\nu_{\mu} \rightarrow \nu_{e}$, $\bar{\nu}_{\mu}\rightarrow \bar{\nu}_{e}$, $\nu_{\mu} \rightarrow \nu_{\mu}$ and $\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{\mu}$ oscillations including neutrino mass hierarchy, CP violation in the neutrino sector.
To minimize the systematic errors, a functionally identical Near Detector (ND 0.3kt) is placed close to the neutrino source, while the far detector (FD 14kt) is located 810km from the source,
observes the oscillated beam.
For all of the oscillation measurements, NOvA takes
advantage of a two-detector configuration to mitigate
uncertainties in neutrino flux, neutrino cross sections,
and event selection efficiencies. NO$\nu$A uses Fermilab's NuMI beam line as its neutrino source.
This paper focuses on the estimating the systematic errors on the NOvA flux, and constraining
these errors using the ND-measurements.
§ DETECTORS
NOvA detectors, with a human figure shown for scale. The FD differs from the ND only in the length of its PVC cells and the number of layers present.
The NO$\nu$A detectors are situated 14 mrad off the NuMI beam axis, so they are exposed to a relatively narrow band of neutrino energies centered at 2 GeV. The NO$\nu$A detectors are
largely active ( 65%) and highly segmented detectors composed of low-Z tracking calorimeters. The segmentation and the overall mechanical structure of the detectors are provided by a lattice of PVC cells, as shown in Figure <ref>. The dimension of the PVC cells is 4$\times$6 cm$^2$.
Each layer is 0.15 X0 (radiation- length) thick. Each plane is composed of individual cells instrumented with 1-sided readout using avalanche photodiodes (APD). Each layer in the detectors is oriented orthogonally to adjacent ones to provide 3D event reconstruction, a cut-away view of the PVC cellular structure. The ND(FD), where the cells are 4.2 m (15.5 m) long,
is composed of 192 (896) planes.
§ NUMI BEAM LINE
Schematic of the NuMI Beam: Shown are the primary proton-C collision, the $\pi^{+}$, $\pi^{-}$, K$^{+}$, K$^{-}$, and K$_{L}^{0}$ mesons that are the primary progenitor of neutrinos, the focusing beam elements, and secondary/tertiary sources of neutrinos.
$\nu$ mode: horns focus positives
$\bar{\nu}$ mode: horns focus negatives
The schematic of the NuMI beamline is shown in Figure <ref>.
A 120 GeV proton-beam from the Main Injector is impinged upon a graphite target.
Secondary particles produced from the p-C interaction are focused by two horns where a strong magnetic field is present. Of all these secondary particles, most important are pions and kaons, because they are the dominant source of neutrinos. After being focused, they are left free to decay in a decay pipe. At the end of the decay pipe, the hadrons are absorbed in a hadron-absorber<cit.>. Due to Off-axis position of detector, the beam is rich in pure $\nu_{\mu}$ in neutrino mode as shown in Figure <ref> and $\bar\nu_{\mu}$ in the antineutrino mode, see Figure <ref>.
§ MOTIVATION FOR THE BEAM-SYSTEMATICS
Oscillated and Un$\mbox{-}$Oscillated spectrum at FD
The sensitivity of the oscillation studies critically depends upon the precise prediction of the
ratio of the unoscillated to oscillated flux, $\nu_{\mu}$, $\bar{\nu_{\mu}}$, $\nu_{e}+\bar{\nu_{e}}$ in FD/ND(E$_{\nu}$) as shown in Figure <ref>. Uncertainties in FD/ND come from the proton-nucleon hadro-production and the beam transport simulation. Needed are data-driven methods to constrain the uncertainties. The most important data are the NO$\nu$A-ND data. Other constraints include MINOS, NDOS (Near Detector Prototype On Surface) data, and the hadro-production data (MIPP, NA49...)<cit.>.
§ BEAM SYSTEMATIC UNCERTAINTIES
Neutrino flux prediction based solely upon MC is not precise. Large uncertainties
associated with the proton-nucleon hadro-production processes in primary and
secondary/tertiary targets induce a large uncertainty ($\approx$20-25%) in the neutrino flux.
However measurements and discoveries of the elements of the neutrino
mixing matrix critically depend upon the precision with which one can predict the
neutrino flux ratio at the far detector (FD) with respect to the near detector (ND) as a
function of the neutrino energy ($E_{\nu}$) and the $\nu_{e}$/$\nu_{\mu}$ flux ratio. Poor measurements of the secondary meson production in p-Nucleus collision,
contribute to the flux error. The mesons include
$\pi^{\pm}$, K$^{\pm}$, and K$^{0}$, produced in the 120 GeV p-C collision.
Additional errors are due to the beam-transport simulation.
Beam Simulation is based upon
FLUGG 2009.4 Flugg 2009-3d<cit.> and Fluka (2011.2b.6) as standard Monte Carlo.
§.§ Beam Transport Systematics
This study includes variations in parameters associated with the beam transport
and presenting the variations in the flux at ND, FD, and (FD/ND) as a function
of neutrino energy<cit.><cit.>.
For the beam simulation, the nominal parameters are: Flugg 2009-3d and Fluka (2011.2b.6), Forward Horn Current, nominal Horn Current 200kA, linear BField distribution. Beam spot size 1.1mm, PEANUT generator turned on for all energies<cit.>.
Left: ratio of $\nu$ flux with variants, $\pm$1kA shift, blue (+1kA) and red (-1kA), to nominal $\nu$ flux (200kA) at NO$\nu$A ND. Right: ratio of $\nu$ flux with variants,$\pm$1kA shift, blue (+1kA) and red (-1kA), to nominal $\nu$ flux (200kA) at NO$\nu$A FD. ).
Double-Ratio $\frac{\Phi_{FD/ND}(variant)}{\Phi_{FD/ND}(Std)}$ for +1kA shift (blue) and -1kA shift (red)
$\delta(\%)$ $\nu_{\mu}$ and Energy(Mean and RMS) at NO$\nu$A ND (1-3)GeV
Shift $\delta(\%)$ at ND $\delta(\%)$ at FD
Std 0.00 0.00
+1kA Horn Current -0.20 -0.16
-1kA Horn Current 0.16 0.10
Horn1 +2mm X $\&$ Y -0.44 -0.39
Horn1 -2mm X $\&$Y -1.70 -1.76
Horn2 +2mm X $\&$ -0.51 -0.47
Horn2 -2mm X $\&$ Y 0.37 0.30
Exp Magnetic Field -4.30 -4.32
BmPosX 0.5mm -0.66 -0.68
BmPosX -0.5mm 0.26 0.24
BmPosY 0.5mm 0.13 0.18
BmPosY -0.5mm -0.35 -0.45
BmSpotSize +0.2mm X $\&$ Y -0.77 -0.81
BmSpotSize -0.2mm X $\&$ Y 0.29 0.29
TarPos +2mm in Z -0.08 -0.09
FTFP$\_$BERT -3.65 -3.76
Flux Systematics Variants for Beam transport: The following variations are considered:
$\bullet$ Horn Current shifted by $\pm$kA w.r.t nominal
$\bullet$ Beam spot size shifted by $\pm$.2mm both in X and Y w.r.t nominal
$\bullet$ Horn1 $\&$ Horn2 position shifted by $\pm$2mm w.r.t nominal
$\bullet$ Target position shifted by +2mm shift w.r.t nominal
$\bullet$ Beam position on the target shifted by $\pm.5$mm in X $\&$ Y separately
$\bullet$ B-field modeling changed to an exponential magnetic field (0.77cm skin depth) in the horn skin.
In the following we show only one sample flux-error calculation due to the variation in
the horn current, shown in Figure <ref> and Figure <ref>. Similar set of
calculation is performed for each of the errors listed above. The fractional variations in the
yield of $\nu_\mu$ at ND and FD due to the various error-conditions are presented
in table <ref><cit.>.
§.§ Uncertainties in hadron production based on NA49 data
One key systematic uncertainty is the uncertainty on the simulation of the production of pions and kaons off the carbon target because the yield and kinematics of the
pions and kaons coming off the target can alter the abundance and energy of
focused pions and kaons that decay to produce muon and electron neutrinos
observed at the NO$\nu$A detectors.
The core concept is to vary hadro-production parameters within reasonable limits in a
physically justifiable way, to use the shifted hadron production to create shifted-flux distributions,
and then use the shifted-flux
distributions to generate a covariance matrix.
Alternative hadron production parameterizations are created
around a best fit (BMPT<cit.>) to a FLUKA simulation of the NA49 target; the resulting
error covers the difference between the Fluka-MC and NA49<cit.>.
Invariant differential cross section for particular x$_{F}$ and as a function
of p$_{T}$ for Pions (left) and Kaons (right) produced in p+C collisions at 158 GeV/c beam momentum. Data is shown in solid black, MC in solid light red,
the parameterization of the MC as a light red dotted line, and the 1 $\&$ 2 sigma
spread in alternative parameterizations of the MC is shown as a pair of light
red bands.
The square root of the diagonal elements of the covariance matrix describing the hadron production uncertainty on the beam $\nu_{\mu}$ flux at the ND and FD.
The error band represents a ±1 sigma shift of all beam systematics: including NA49 Hadroproduction Uncertainty, Spot size, Beam position on the target (X/Y), Target position, Horn current, Horn positions, $\&$ the modeling of horn$'$s B-field.
The difference between nominal and shifted parameterizations is used to create
weights in $P_T$ and $x_F$ of hadrons produced off the NuMI target, which, then,
can be used to re-weight the NO$\nu$A Near and Far Detector neutrino spectra, see
Figure <ref> and Figure <ref>.
Beam Transport Errors, including NA49 Hadroproduction Uncertainty on Reconstructed neutrino energy[GeV] in NO$\nu$A ND $\&$ FD for 6e20 POT as shown in Figure <ref>.
§ CONSTRAINTS USING ND DATA
E$_{\nu_{\mu}}$ at ND
E$_{\nu_{e}}$ at ND
We have shown systematic uncertainties from the beam transport and hadron production.
These predictions need to be further constrained by the ND Data.
Since 97$\%$ of $\nu_{\mu}$ at the ND are from
$\pi\rightarrow\nu_{\mu} + \mu$. We plan to use neutrino data in 1$\mbox{-}$3 GeV
to constrain the pion yield, and use $E_\nu \geq 5$ GeV to constrain K$^{+}$ yield.
§ SUMMARY:
We present the flux systematic errors arising from the uncertainties in the beam transport $\&$ hadro-production MC model. For beam transport parameters $\delta(\%)$ for $\nu_{\mu}, \bar{\nu_{\mu}}, \nu_{e}, \bar{\nu_{e}}$ is $\approx$3$\%$ for ND and FD(1$\mbox{-}$3)GeV, Energy variation for$\nu_{\mu}, \bar{\nu_{\mu}}, \nu_{e}, \bar{\nu_{e}}$ $\approx$1$\%$ for ND and FD(1$\mbox{-}$3GeV). Combined uncertainties using hadroproction with beam transport parameters at ND is $\pm$23.9$\%$ and at ND is $\pm$20.9$\%$<cit.>. The flux prediction can be made more precise by using the
constraints provided by the neutrino spectra from ND, as we plan to do in the future.
|
1511.00312
|
On the
Asymptotic Integration of a
System of Linear Differential
Equations with Oscillatory
Decreasing Coefficients
14ptV.Sh. Burd and
V.A. Karakulin
18ptAbstact. A system of linear differential
equations with oscillatory
decreasing coefficients is considered.
The coefficients has the form
$t^{-\alpha}a(t)$, $\alpha>0$,
where $a(t)$ is trigonometric
polynomial with an arbitrary set
of frequencies. The asymptotic
behavior of the solutions of
this system as $t\to\infty$ is
studied. We construct an invertible
(for sufficiently large $t$)
change of variables that takes
the original system to a system
not containing oscillatory
coefficients in its principal
part. The study of the asymptotic
behavior of the solutions of the
transformed system is a simpler
problem. As an example, the
following equation is considered:
\frac{d^2x}{dt^2}+\left(1+\frac{\sin\lambda t}
where $\lambda$ and $\alpha$,
$0<\alpha\le 1$, are real numbers.
14ptThe stability and asymptotic
behavior of the solutions of
linear systems
of differential equations
\frac{dx}{dt}=Ax+B(t)x,
where $A$ is a constant matrix, and matrix
$B(t)$ is small in a certain
sense when $t\to\infty$ has
been studied
by many authors
(see [1-6]) as well as the
Shtokalo [12-13] studied
of the stability of
solutions of the following
system of differential equations
\frac{dx}{dt}=Ax+
\varepsilon B(t)x.
\eqno(1)
Here $\varepsilon>0$ is a small
$A$ is a constant square matrix,
$B(t)$ is a square matrix whose
elements are
trigonometric polynomials
$b_{kl}(t)$ $(k,l =1,...,m)$ in the
b_{kl}(t)=\sum_{j=1}^m b_{j}^{kl}
e^{i\lambda_j t};
Matrices with such elements are
called matrices of class
The mean of a matrix from $\Sigma$
is a constant matrix
that consists of the constant
$(\lambda_j=0)$ of the elements
of the matrix.
Using the Bogoliubov averaged
method, Shtokalo transformed
system (1) into a system with
constant coefficients depending
on the parameter $\varepsilon$
up to terms of any order in
smallnes in $\varepsilon$.
We also mention that the method
averaging in the first
was utilized in [14,16]
for studying
the asymptotic behavior of
solutions of a particular class of systems of equations
with oscillatory decreasing
In the present paper we adopt
the method
of Shtokalo
for the problem of asymptotic
integration of
systems of linear differential
equations with oscillatory
decreasing coefficients.
We consider the following
system of
differential equations
in $n$-dimensional space
${\cal R}^n$
\frac{dx}{dt}=\left\{A_0+
\sum_{j=1}^k\frac{1}{t^{j\alpha}}
A_j(t)\right\} x+
\frac{1}{t^{(1+\delta)}}F(t)x.
\eqno(2)
Here, $A_0$ is a constant
$n\times n$ matrix, and
$A_1(t),A_2(t),..,A_k(t)$ are
$n\times n$ matrices
that belong to $\Sigma$.
We shall assume that matrix $A_0$
is in Jordan canonical form,
a real number $\alpha$ and
a positive integer
$k$ satisfy
$0<k\alpha\le 1<(k+1)
\alpha,~\delta>0$. The square
matrix $F(t)$ satisfies
||F(t)||\le C<\infty
for $t_0\le t<\infty$, where
is some matrix norm in
${\cal R}^n$.
We are concerned with the behavior
of solutions
of system (2) when $t\to\infty$.
Let us construct an invertible
change of variables (for
sufficiently large
$t$, $t>t^*>t_0$)
that would transform system
(2) into the simpler system
\frac{dy}{dt}=\left\{\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}y+
\frac{1}{t^{
\quad\varepsilon>0,\quad t>t^*.
\eqno(3)
$A_0,A_1,..,A_k$ are constants
square matrices
(moreover, $A_0$ is the same
matrix as in (1)),
and, the matrix $G(t)$ has the
properties as the matrix
$F(t)$ in system (2).
Without loss of generality,
we can assume that
all eigenvalues of the matrix
$A_0$ are real.
Indeed, if matrix $A_0$ has
complex eigenvalues,
then we can make a change of
variables in (2)
where $R$ is diagonal matrix
composed of the imaginary
parts of eigenvalues of the
$A_0$. This change of variable
with coefficients, which are
bounded in
$t$, $t\in (-\infty,\infty)$,
transforms the matrix
$A_0$ into the matrix $A_0-iR$, which only has real eigenvalues.
We shall try to choose an
change of variables
(for sufficiently large $t$),
in the form
\frac{1}{t^{j\alpha}}Y_j(t)
\right\}y, \eqno(4)
to transform system (2) into
system (3),
where $Y_0(t)=I$ is the identity
matrix, and,
$Y_1(t),..,Y_k(t)$ are
$n\times n$ matrices
that belong to $\Sigma$ and have
zero mean value.
By substituting (4) into (2),
and, replacing
$\frac{dy}{dt}$ by the right
hand side of (3) we obtain
\begin{array}{l}
\left\{\sum_{j=0}^k\frac{1}
\left\{\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}
\left\{\sum_{j=0}^k\frac{1}
\right\}G(t)y+
\frac{1}{t^{(1+\alpha)}}W(t)y+
\left\{\sum_{j=1}^k\frac{1}
\right\}y=\\
\frac{1}{t^{j\alpha}}A_j(t)
\right\}\left\{
\sum_{j=0}^k\frac{1}{t^{j\alpha}}
\frac{1}{t^{(1+\delta)}}U(t)y,
\end{array}\eqno(5)
\right\},\eqno(6)
\frac{1}{t^{j\alpha}}Y_j(t)
\right\}. \eqno(7)
Equating the summands that
$t^{-j\alpha}$ $(j=1,\dots,k)$
in the left and the right hand
sides of (5) yields a system
of $k$ linear matrix
differential equation with
constant coefficients
\begin{array}{l}
\frac{dY_j(t)}{dt}-A_0Y_j(t)+
\sum_{l=0}^{j-1}Y_l(t)A_{j-l},
\quad (j={1,\dots,k}).
\end{array}\eqno(8)
The solvability of system (8)
was studied [11]. We
represent $Y_j(t)$ as a finite
Y_j(t)=\sum_{\lambda\ne 0}y_
{\lambda}^je^{i\lambda t},
where $y_{\lambda}^j$ are
constants $n\times n$
matrices, and, obtain matrix
Since all the eigenvalues of $A$ are real,
the matrix equations have
unique solutions for $\lambda\ne 0$
(see, for instance, [16,17].
On each of the $k$ steps of
the solution process
we determine the matrix $A_j$
from the condition that the
right hand side
of (8) has a zero mean value.
In particular, for $j=1$
\frac{dY_1(t)}{dt}-A_0Y_1(t)+
where $A_1$ is the mean value of
the matrix $A_1(t)$.
Relation (5) implies the
following result.
Theorem 1. System
(2), for sufficiently large $t$,
can be transformed using a change of variables
(4) into a system
\frac{dy}{dt}=\left\{\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}y
where $\varepsilon>0$, and
$||G(t)||\le C_1<\infty$ .
Proof. Substituting (4)
into (2), we obtain
\begin{array}{l}\left\{
\sum_{j=0}^k\frac{1}
\sum_{j=1}^k\frac{1}
\sum_{j=0}^k\frac{1}{t^{j\alpha}}
\right\}y-
\frac{1}{t^{(1+\alpha)}}
\frac{1}{t^{(1+\delta)}}U(t)y,
\end{array}
where $W(t)$ and $U(t)$ are
by (6) and (7) respectively.
The last relation can be
rewritten as
\begin{array}{l}\left\{
\sum_{j=0}^k\frac{1}{t^
\left\{\frac{dy}{dt} -\left\{
\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}y
\right\}=\\
\left\{\sum_{j=0}^k\frac{1}
\left\{\sum_{j=1}^k\frac{1}
\right\}y-\\
\left\{\sum_{j=0}^k\frac{1}{t^
U(t)y. \end{array}
Due to (8) we get
\begin{array}{l}\left\{
\sum_{j=0}^k\frac{1}
\left\{\frac{dy}{dt}-\left\{
\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}
\frac{1}{t^{(1+\alpha)}}W(t)y+
\frac{1}{t^{(1+\delta)}}U(t)y,
\end{array}\eqno(9)
where elements of the matrix
$S(t)$ can be represented as
and $a_j(t)$ are trigonometric
polynomials. Therefore,
\frac{1}{t^{(1+k)\alpha}}S(t)-
\frac{1}{t^{(1+\alpha)}}W(t)+
\frac{1}{t^{(1+\delta)}}U(t)=
\frac{1}{t^{(1+\varepsilon)}}R(t),
\eqno(10)
where $\varepsilon>0$ and
$R(t)$ satisfies
||R(t)||\le C_2<\infty.
The identity (9), for
sufficiently large $t$, can be
\frac{dy}{dt}=\left\{\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}y
\frac{1}{t^{(1+\varepsilon)}}
\left\{\sum_{j=0}^k\frac{1}
This relation and (10) yield
the assertion of the theorem.
The main part of system (3)
\frac{dy}{dt}=\left\{\sum_{j=0}^k
\frac{1}{t^{j\alpha}}A_j\right\}y
does not have oscillating
coefficients. This makes it
simpler than the original
system (2).
In particular, the Fundamental
Theorem of Levinson on asymptotic
behavior of solutions of
linear systems of differential
equations (see [1.2], as well as
[8,9]), readily yields the following
resolt for system (3).
Theorem 2. Let us
first nonzero matrix among the
matrices $A_j$, $j=0,1,\dots,k$
be the matrix $A_l$.
Let the matrix $A_l$ have distinct
eigenvalues. Then the fundamental matrix
of system (3) has the following
\Lambda(s)ds,\quad t>t^*,\quad
t\to \infty,
where $P$ is a matrix composed
of the eigenvectors of the matrix
$A_l$, and $\Lambda(t)$ is a
diagonal matrix
whose elements are eigenvalues
of the matrix
$\sum_{j=l}^k t^{-j\alpha}A_j$.
To prove this theorem we just
have to observe
that the system of differential
\frac{dx}{dt}=\frac{1}{t^l}A_lx+
\sum_{j=l+1}\frac{1}{t^j}A_jx
can be transformed into
\frac{dx}{d\tau}=\frac{1}{1-l}[A_l+
\sum_{j=l+1}\frac{1}{t^{j-l}}A_j]x
using the change of variables
As an example we consider an
equation of an adiabatic oscillator
\frac{d^2y}{dt^2}+(1+\frac{1}
{t^{\alpha}}\sin \lambda t)y=0,
\eqno(11)
where $\lambda$, $\alpha$ are
real numbers,
and $0<\alpha\le 1$.
The problem of asymptotic integration of
the equation (11) has been
studied in [5,6, 18–20].
In particular, asymptotics of
solutions for
$\frac{1}{2} \le \alpha \le 1$
were obtained.
The method that we proposed in
this chapter
can be used to obtain
(in a simple manner)
all known results on asymptotics
of solutions
of equation (11) as well as to
establish new results.
Let us pass from equation (11)
to the
system of
equations $(x=(x_1,x_2))$ using a
change of variables
y=x_1 \cos t + x_2 \sin t, \quad
y'= -x_1 \sin t + x_2 \cos t.
\eqno(12)
We obtain the system
\frac{dx}{dt}=\frac{1}
\eqno(13)
It is convenient to rewrite
the matrix $A(t)$ in complex
form as
\aligned
\bar a_1e^{-i(\lambda+2)t}+
\bar a_2e^{-i(\lambda-2)t}+
a_3e^{i\lambda t}+\bar a_3
e^{-i\lambda t},
\endaligned
\left(\begin{array}{cc}
-1 & i\\
i & 1
\end{array}\right),\quad
\left(\begin{array}{cc}
1 & i\\
i &-1
\end{array}\right),\quad
\left(\begin{array}{cc}0&-2i
\\2i&0
\end{array}\right),
and the matrices $\bar a_1$,
$\bar a_2$, $\bar a_3$ are complex
conjugates to the matrices
$a_1,a_2,a_3$ respectively.
The values of $\alpha$ and
significantly affect the behavior of solutions
of system (13). We denote by
$R(t)$ a $2\times 2$ matrix that satisfies
||R(t)||\le C_3 <\infty.
for all $t$.
First, assume
$\frac{1}{2}<\alpha\le 1$.
For $\lambda\ne\pm 2$
system (3) becomes
\frac{dy}{dt}=\frac{1}
\varepsilon>0.
Therefore, it is easy to see
(taking into consideration
change (12)),
that the fundamental system of
of equation (11) for
$\frac{1}{2}<\alpha\le 1$,
$\lambda\ne\pm 2$ as $t\to\infty$
has the form
x_1= \cos t +o(1),\quad x_2=
\sin t +o(1),
x'_1= -\sin t +o(1),\quad x'_2=
\cos t + o(1).
We shall represent the fundamental system
of solutions of equation (11)
as a matrix with rows
$x_1,x_2$ and $ x'_1,x'_2$.
Now assume $\lambda=\pm2$.
specifically let $\lambda=2$.
Then system (3) becomes
\aligned
\quad\varepsilon>0.
\endaligned
a_2 +\bar a_2=
\frac{1}{4}\left
1 & 0\\
0 & -1
\end{array}\right).
Theorem 2 implies that for
$t\to\infty$ the fundamental matrix
of system (11) has the following
\begin{array}{cc}
\frac{1}{4s^{\alpha}}ds
\right) & 0 \\
0 & exp\left(-\int_{t^*}^
\right)
\end{array}\right)
\left[I+o(1)\right].
Therefore, for $\alpha=1,~\lambda=2$ we obtain
the fundamental system of solutions
of equation (11) as
\begin{array}{l}\left
t^{\frac{1}{4}}\cos t &
t^{-\frac{1}{4}}\sin t\\
-t^{\frac{1}{4}}\sin t &
t^{-\frac{1}{4}}\cos t
\end{array}\right)
\left[I+o(1)\right],
\end{array}
while for $1/2<\alpha<1,
for $t\to\infty$ we get
\begin{array}{l}\left(\begin{array}{cc}
{4(1-\alpha)}\right)\cos t &
{4(1-\alpha)}\right)\sin t \\
{4(1-\alpha)}\right)\sin t &
{4(1-\alpha)}\right)\cos t
\end{array}\right)
\left[I+o(1)\right].
\end{array}
We note that, for $\lambda=\pm 2$,
and $\frac{1}{2}<\alpha\le 1$,
equation (11)
has unbounded solutions. Moreover,
for $\alpha=1$, the solutions
have a polynomial growth,
while, for $\alpha\ne 1$, they
grow exponentially.
We now assume
\le\frac{1}{2}$.
In this case a change of variables
(4) transforms (13) into
\frac{dy}{dt}=\frac{1}
\quad\varepsilon>0.
If $\lambda\ne\pm 2,\pm 1$, then
the matrix $A_1$ is zero, and
matrix $A_2$ has the form
A_2 = i\left[\frac{1}{{\lambda + 2}}
(a_1\bar a_1 - \bar a_1
\frac{1}{{\lambda -2}}(a_2
\bar a_2 -\bar a_2 a_2)
\bar a_3 -\bar a_3
a_3)\right]. \eqno(14)
Computing $A_2$ yields
\left(\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}\right).
The system
\frac{dy}{dt}=\frac{1}{t^
can be integrated. We obtain
that, for $t\to\infty$, the
fundamental system of solutions
of equation (11) with
$\lambda\ne\pm 2,\pm 1$,
has the form
\left(\begin{array}{cc}
\cos(t+\gamma\ln t) &\sin(t+
\gamma\ln t)\\
-\sin(t+\gamma\ln t) & \cos
(t+\gamma\ln t)
\end{array}\right)[I+o(1)],
where $\gamma=\frac{1}{4
(\lambda^2-4)}$. For
and $\lambda\ne\pm 2,\pm 1$,
the fundamental system of solutions
of equation (11) has the form
\left(\begin{array}{cc}
\cos\left(\frac{t^{1-2\alpha}}
\right) &
\sin\left(\frac{t^{1-2\alpha}}
\right)\\
\right) &
\cos\left(\frac{t^{1-2\alpha}}
\right)
\end{array}\right)[I+o(1)]
as $t\to\infty$.
We now assume
\lambda=1$. In this case
$A_1$ is zero, and $A_2$ is determined by
iA_2=-\frac{1}{3}a_1\bar a_1
+\frac{1}{3}\bar a_1a_1-
\bar a_2a_2+a_2\bar a_2
-a_3\bar a_3 +\bar a_3
a_2a_3+a_3a_2-\bar a_2
\bar a_3+\bar a_3
\bar a_2 .
A simple calculation yields
\left(\begin{array}{cc}
0 & -5\\
-1 & 0
\end{array}\right).
The corresponding system (3)
has the following form
\frac{dy}{dt}=\frac{1}
\varepsilon}}R(t)y,\quad
\varepsilon>0.
By integrating the system
\frac{dy}{dt}=\frac{1}
we obtain its fundamental matrix
-\sqrt 5 t^\varrho & \sqrt 5
t^\varrho & t^{-\varrho}
\end{array}\right),
where $\varrho=\frac{\sqrt 5}
{24}$. Then the fundamental
system of solutions of equation
(11) for
$\lambda=1$, and $t\to\infty$,
has the form
\left(\begin{array}{cc}
t^{\varrho}\sin(t-\beta) & t^{-\varrho}\sin(t+\beta)\cr
t^{\varrho}\cos(t-\beta) &
\end{array}\right)[I+o(1)],
\varrho=\frac{\sqrt 5}{24},
\quad \beta=arctg\sqrt{5},\quad
\eqno(15)
If $\frac{1}{3}<\alpha<\frac{1}{2}$ and $\lambda=1$
we have the system
\frac{dy}{dt}=\frac{1}
\quad\varepsilon>0.
Using Theorem 2 we obtain
the asymptotics of the fundamental matrix of this system,
and then, using the change
(12), the asymptotics of
the fundamental system of solutions of equation (1.7.12) for
$\frac{1}{3}<\alpha<\frac{1}{2}$, $\lambda=1$,
and $t\to\infty$:
\left(\begin{array}{cc}
\sin (t-\beta) &
{1-2\alpha})\sin (t+\beta)
\cr
{1-2\alpha})\cos (t-\beta)
& exp(-\varrho\frac{t^{1-2\alpha}}
{1-2\alpha})\cos (t+\beta)
\end{array}\right)[I+o(1)],
where $\varrho$ are $\beta$
are defined by (15).
Thus, for $\alpha=\frac{1}{2}$
and $\lambda=1$
we observe a polynomial growth
of solutions, while for
and $\lambda=1$ the
solutions grow exponentially.
Now let $\alpha=\frac{1}{2}$
and $\lambda=2$. Simple calculations
show that
\frac{1}{4} & 0\\
0 & -\frac{1}{4}
\end{array}\right),\quad
0 & -\frac{1}{64}\\
\frac{1}{64} & 0
\end{array}\right).
Therefore, we get a system
\frac{dy}{dt}=\frac{1}{t^{\frac{1}
\frac{1}{t^{1+
\varepsilon}}R(t)y,\quad
\varepsilon>0. \eqno(16)
We compute the eigenvalues of
the matrix
\frac{1}{t^{\frac{1}{2}}} A_1
integrate them, and, using
Theorem 2,
we obtain the asymptotics of the
fundamental matrix of system
Next we find the fundamental system of
solutions of equation (11) for $\alpha=\frac{1}{2},\,
\lambda=2$ and $t\to\infty$:
\left(\begin{array}{cc}
exp(\phi(t))\cos t &
exp-(\phi(t))\sin t\\
-exp(\phi(t))\sin t &
exp-(\phi(t))\cos t
\end{array}\right)[I+o(1)],
where $\phi(t)=\frac{1}{2}
\sqrt{t}$.
For $\frac{1}{3}<\alpha<
\frac{1}{2},~\lambda=2$
instead of (16) we get a system
\frac{dy}{dt}=\frac{1}{t^{\alpha}}
\frac{1}{t^{1+\varepsilon}}
where $\varepsilon>0$, with the same matrices $A_1$, $A_2$.
Therefore, it is straightforward to write the asymptotics
of the fundamental system of solutions
of equation (11) for
$\frac{1}{3}<\alpha<\frac{1}{2}$, $\lambda=2$, and $t\to\infty$.
Finally, let
$\lambda\ne\pm 1$, and
$\lambda\ne\pm 2$.
Then, it turns out that
$A_1$ is zero, and $A_2$ is
defined by (14).
Matrix $A_3$ differs from
zero only if
Assume $\lambda=\frac{2}{3}$.
System (3) then becomes
\frac{dy}{dt}=\frac{1}{t^{\frac{2}
{3}}}A_2 y+\frac{1}{t}A_3y +
\frac{1}{t^
where $\varepsilon>0$ and the matrices $A_2$ and $A_3$
are defined by
0 & -\frac{9}{128}\\
\frac{9}{128} & 0
\end{array}\right),\quad
-\frac{27}{1024} & 0\\
0 & \frac{27}{1024}
\end{array}\right).
We compute the eigenvalues of
the matrix
\frac{1}{t^{\frac{2}{3}}}A_2+
\frac{1}{t}A_3.
These eigenvalues have zero real parts,
for sufficiently large $t$.
Further, using the same
scheme as before
we find the asymptotics of
the fundamental
system of solutions of
equation (11).
We only note that the solutions of equation
(11) are bounded for
\lambda=\pm\frac{2}{3}$ as
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|
1511.00479
|
In two recent papers it is argued that the “proton radius puzzle”
can be explained by truncating the electron scattering data to low
momentum transfer and fit the rms radius in the low momentum expansion
of the form factor. It is shown that this procedure is inconsistent
and violates the Fourier theorem. The puzzle cannot be explained in
this way.
The “proton radius puzzle” is the difference of the rms radius $R_p
= \langle r^2 \rangle^{1/2}$ as determined from elastic electron
scattering and as derived from a very precise Lamb shift measurement
of muonic hydrogen. The electron scattering result $R^e_p =
\unit[0.877(5)]{fm}$
<cit.> deviates
from the muonic result $R^\mu_p = \unit[0.8409(4)]{fm}$ by
$\unit[0.036(5)]{fm}$ <cit.> or 7 standard
In two recent papers Keith Griffioen, Carl Carlson, and Sarah Maddox
<cit.> and Douglas W. Higinbotham, et al.<cit.> conjecture that this difference, sometimes
called the “proton radius puzzle”, could be resolved by just
restricting the analysis of the electron scattering data to the data
at low momentum transfer $Q^2$. In ref. <cit.> the
new data of Bernauer et al. for $Q^2 < \unit[0.02]{(GeV/c)^2}$
<cit.> are used, whereas in
ref. <cit.> the old data, i.e. before 1980, for
$Q^2 < \unit[0.03]{(GeV/c)^2}$ <cit.> are
analyzed. In both analyses radii are extracted which appear close to
the one derived from the muonic Lamb shift within relatively large
errors. In both papers the statistical significance of these results
are in focus, however, it is overlooked that the approach of limiting
the data sets to small $Q^2$ is not in accord with basic facts of form
factors and rms radii derived from them as will be shown in the
The charge distribution of a nucleus and its form factor are connected
by the 3-dimensional Fourier transforms
\begin{equation}
\rho(r) = \frac{1}{(2 \pi \hbar \,c)^3} \int d^3Q \, G(Q^2) \,
\exp(\frac{i}{\hbar \,c} \, \vec{Q} \cdot \vec{r} )
\label{eq:rho}
\end{equation}
and its inversion
\begin{equation}
G(Q^2) = \int d^3r \, \rho(r) \, \exp(- \frac{i}{\hbar \,c} \, \,
\vec{Q} \cdot \vec{r})
\label{eq:G}
\end{equation}
This implies that either $G(Q^2)$ or $\rho(r)$ is known and integrated
over the full range of $0 \leqslant Q^2 \leqslant \infty$ or $0
\leqslant r \leqslant \infty$, respectively. Experimentally the
minimum and maximum of $Q^2$ are limited and $G(Q^2)$ cannot be
determined over the full range. Therefore, one has to use models for
$\rho(r)$ for nuclei, or for $G(Q^2)$ for the proton. However, it is
mandatory to ensure that the models used are consistent with the
Fourier theorem given by eqs. (<ref>) and (<ref>).
The method used in refs. <cit.>
is well known since the early days of electron scattering and uses the
expansion of eq. (<ref>):
\begin{equation}
G(Q^2) = 1- \frac{1}{6} \langle r^2 \rangle Q^2 + \frac{1}{120}
\langle r^4 \rangle Q^4 - \frac{1}{5040} \langle r^6 \rangle Q^6
\label{eq:exp}
\end{equation}
\begin{equation}
\langle r^n \rangle = 4\pi \int r^2 dr \, \rho(r) \, r^n .
\label{eq:rms}
\end{equation}
In the two papers the rms radius in eq. (<ref>) is determined
by fitting the truncated data basis for low $Q^2$ by
eq. (<ref>) to order $Q^2$ (linear), $Q^4$ (quadratic), and
$Q^6$ (cubic) <cit.> and order $Q^2$
<cit.>. It is a well known fact though, that a
sharp truncation in the coordinate space (e.g. a uniformly charged
sphere) produces an oscillating behaviour in momentum space. The same
is true for a sharp truncation in momentum space. A “saw tooth”
like form factor as given by eq. (<ref>) corresponds to the
charge distribution as given by eq. (<ref>), i.e. produces
an oscillating behaviour in coordinate space.
\begin{equation}
G(q)=\left(1-\frac{q^2 R^2}{6 (\hbar c)^2}\right) \Theta
\left(\frac{6 (\hbar c)^2}{R^2}-q^2\right)
\label{eq:sawG}
\end{equation}
The form factor $G(Q^2)$ for the “saw tooth” model.
\begin{equation}
\rho(r)=-\frac{\left(2 r^2-R^2\right) \sin \left(\frac{\D \sqrt{6}\,
r}{\D R}\right)+\sqrt{6} \,r R \cos \left(\frac{\D \sqrt{6}\,
r}{\D R}\right)}{2 \pi ^2 r^5}
\label{eq:sawrho}
\end{equation}
The charge distribution $\rho(r)$ for the “saw tooth” model.
\rho(r) \hbox{ label}
& \D \frac{\langle r^2 \rangle}{\hbox{fm}^2}
& \D \frac{\langle r^4 \rangle}{\hbox{fm}^4}
& \D \frac{\langle r^6 \rangle}{\hbox{fm}^6}
& \D \frac{\langle r^1 \rangle_{(2)}}{\hbox{fm}}
& \D \frac{\langle r^3 \rangle_{(2)}}{\hbox{fm}^3} \nonumber \\[7pt] \hline
\hbox{Exponential~} & 0.7056 & 1.2447 & 4.0985 & 1.0609 & 2.2457 \nonumber \\
\hbox{Gaussian} & 0.7056 & 0.8298 & 1.3662 & 1.0945 & 2.0594 \nonumber \\
\hbox{Uniform} & 0.7056 & 0.5927 & 0.5421 & 1.1154 & 1.9433 \nonumber \\
%\hbox{Yukawa I} & 0.7056 & 2.9872 & 31.617 & 0.9449 & 2.9032 \nonumber \\
%\hbox{Yukawa II} & 0.7056 & 1.6596 & 8.197 & 1.0288 & 2.4197 \nonumber \\
\hbox{Saw tooth} & 0.7056 & 0 & 0 & 0.7277 & 0.6162 \nonumber \\
\hbox{Experiment} & 0.774 & 2.59 & 29.8 &
1.085 & 2.85 \nonumber \\
\hbox{(stat.)(syst.)\,err.} & (8) & (19)(04) & (7.6)(12.6) & (3) & (8) \nonumber
\end{array}$$
Moments of the indicated charge distributions also including
the first and third Zemach moments. The moments are given for an rms
radius of $\unit[0.840]{fm}$. Experimental values are taken from
<cit.>. All charge distributions are equivalent to
simple form factor models which are evidently at variance with the
experimentally observed form factor of the proton. The saw tooth model
demonstrates clearly the impossibility to neglect the high $Q^2$
For this form factor all values $\langle r^n \rangle$ for $n\geqslant
4$ obtained with the derivatives of eq. (<ref>) are zero. It
produces also unreasonably small values for the Zemach moments (see
Tab. <ref>) inconsistent with both electron scattering and
spectroscopy <cit.>. This is of
course not observed and unphysical as there is no known mechanism that
could produce oscillating charges at large distances in the proton.
However, this pathological situation is within the statistical errors
of the fits of Carl Carlson <cit.> to the truncated
data range taking $\langle r^2 \rangle$ and $\langle r^4 \rangle$ as
fit parameters and with Higinbotham et al. <cit.>
putting unjustifiably $\langle r^4 \rangle \equiv 0$.
In ref. <cit.> the problem is realized and its
influence is determined for three of the charge distribution in the
Tab. <ref>: exponential, Gaussian and uniformly charged
sphere. But these are just very crude models. We know more about the
proton. Two independent studies of the world data have been published
by John Arrington et al. <cit.>
and Jan Bernauer et al. <cit.> based
on the measured form factors up to large $Q^2 \leqslant
\unit[10]{(GeV/c)^2}$. It should be unnecessary to state that models like
the Gaussian or homogeneously charged sphere are excluded for the
proton since the work of Robert Hofstadter <cit.>.
The extracted radius in ref. <cit.> depends
strongly on the $Q^4$ term, the curvature of $G(Q^2)$ at small $Q^2$:
higher values $\langle r^4 \rangle$ correlate with larger radii
$\langle r^2 \rangle$. The fits over the whole $Q^2$ range by Bernauer
et al. <cit.> indeed find a strong
curvature. In Kraus et al. <cit.> it has been shown that a
low-order fit including a fit of the curvature to a truncated data
set is not reliable.
It is noted that the form factor and charge distributions going into
the rms radius in the two independent studies
are automatically fulfilling the Fourier relation, since they are
determined from one consistent fit over the full $Q^2$ range. The two
papers <cit.> using the
truncated $G(Q^2)$ disregard this consistency. Griffioen et al.<cit.> insert the rms radius fitted for small $Q^2$
into a model assumed to be valid for large $Q^2 \geqslant
\unit[0.02]{(GeV/c)^2}$. However, this form factor is not in agreement
with the cited measurements
Higinbotham et al. <cit.> neglect the $Q^4$ term
completely though they fit the data of the larger region $Q^2
\leqslant \unit[0.03]{(GeV/c)^2}$. Yet, their reasoning is erroneous on
several aspects. First, we know that the true form factor has a
finite curvature, so using an F-test to decide about the significance
of the $Q^4$-term in the expansion of eq. (<ref>) is not
justified. Any hypothesis test in classical statistics is
based on a very important assumption: one has to know the true
function. It may be the case that we do not know the precise true
functions for the form factors of the proton, but the truncated
polynomial of eq. (<ref>) can definitely be excluded. A
p-value or a significance level calculated with a wrong model
assumption is not valid.
Second, large off-diagonal elements in the covariance matrix are not a
“problem”. On the contrary, if one fits a polynomial expansion like
the one in eq. (<ref>) one will always end up with highly
correlated parameters. It is also always possible to construct an
orthogonal functional basis (ref. <cit.> recommends the
Forsythe method for polynomials) where the resulting covariance matrix
of the parameters is indeed diagonal. For a quantitative discussion of
the correlation it is useful to calculate the correlation matrix:
\qquad\hbox{with}\,-1\le\rho_{ij}\le 1$$
where the $\sigma_{ij}$ are the elements of the covariance matrix and
$\sigma_{i}=\sqrt{\sigma_{ii}}$. For the covariance matrix in
<cit.>, eq. (5) therein, one gets
$\rho_{23}=-0.95$ and therefore a strong negative correlation between
the linear and the quadratic parameter. Taking into account the factor
$-1/6$ of eq. (<ref>) one gets a large positive correlation
between the quadratic parameter and the extracted radius. The authors
of <cit.> have shown themselves that if they
artificially reduce the quadratic term to zero, they reduce the radius
from $R_p=\unit[0.875]{fm}$ to $\unit[0.840]{fm}$ therefore giving a
very strong reason not to neglect the quadratic term.
But the reasoning in ref. <cit.> has a more
serious problem still. In their F-test two regression models are compared,
where one model, the order 2 polynomial, includes the second, linear,
model. It is clear that the order 2 polynomial will always fit the
data better than the linear model, unless the quadratic term becomes
zero and both models are identical. So, Higinbotham et al. are
rejecting a model not because it gives a worse fit but because the fit
is not “significantly” better. Moreover, fitting the data well is
only a precursor to the more important goal: getting a robust estimate
of the rms radius. As we have argued in the previous paragraph, the
extracted radius changes dramatically when the order of the polynomial
model is reduced from quadratic to linear. This makes the quadratic
term very significant for the extraction of the radius.
In addition, their two parameter fits are not influenced by the form
factor at large $Q^2$. Therefore, their considerations of this form
factor in the second part of their paper are in view of refs.<cit.> not only
misplaced, but wrong.
A numerically precise calculation with the charge distribution derived
from eq. (<ref>) and based on the data and fits of refs.<cit.> of moments with $n \leqslant
6$ and Zemach moments is published in ref. <cit.>.
One may ask why the method of small $Q^2$ expansion was relatively
successful for nuclei. This is due to the fact that the short range
nuclear force produces to a good approximation a uniformly charged
sphere and one can derive the $R_A \propto A^{1/3}$ dependence of the
rms radius. This is a good model for nuclei. However, for the charge
distribution of the proton we do not yet have a good model and
consequently the form factor and the rms radius are only derivable
from a fit over a sufficiently wide $Q^2$ range as performed in refs.<cit.>.
Ingo Sick has investigated the minimal $Q^2$ required in a recent
paper <cit.>.
The approach of the two papers is, nonetheless, not only wrong in
analytical terms it also misunderstands the statistical evaluation of
physics data. The truncated data set represents just one statistical
sample. The
$\chi^2$ evaluation has originally nothing to do with finding a
optimal “model/theory function” by fitting. The minimal sum of the
weighted squares of deviations of the data from a model function
should be distinguished from $\chi^2$ and we call it $M^2$. If one
knows the model function with certainty - this includes the knowledge
of the parameters - there is no fitting and $\chi^2 = M^2$
representing a test of statistical pureness of the data (Pearson
test). In Physics, however, one has neither a certain model/theory
function nor the certainty that the sample is statistically pure. One
has therefore to deal with a mixed and dirty situation. It may very
well be that the fits of eq. (<ref>) with $\langle r^n
\rangle$ as free fit parameters are giving a small $M^2$, but this
cannot be interpreted as a value of the $\chi^2$ distribution and
consequently as measure of the significance of the fit. The sample is
not “true” but just one of the possible statistical
fluctuations. Since $M^2$ is not a $\chi^2$ an estimate of errors from
such equating is an approximation. The physics constraints discussed
above have to be realized even if the $M^2$ gets worse. It just means
that the sample is not following so closely the model expectations
that the $M^2$ is absolutely minimal. Therefore it is really light
hearted to neglect the information contained in 80% of the data and
believe that this is a valid approach. A similar remark holds for the
Fisher-Snedecor test variable in the F-test. It is recommended to
study the landmark book of Frederick James <cit.> which was
exactly meant as an educational means for CERN users in 1970 to get
out off the over simplified application of statistics in physics. It
served as the basis for the chapters about statistics in the Review of
Particle Physics of the Particle Data Group <cit.>
which serve as the standard in particle physics.
In the second part of the papers of Griffioen et al.<cit.> and Higinbotham et al.<cit.> the “continued fraction expansion” for
the form factor $G(Q^2)$ is tried as an alternative to the many
ansätze in refs.
yielding a radius in accord with the muonic hydrogen measurement
albeit with relatively large error. The statistical evaluation is
based on a markedly worse normalized $\chi^2/dof$. (We continue to
call it $\chi^2$ since people are so used to it.) It has to be noted
that the “continued fraction expansion” of $G(Q^2)$ is the only one
giving a small rms radius and was excluded from the analysis of
refs. <cit.> since it was to stiff
to fit the data. It is not better justified by any theoretical
argument than the others used, and, therefore, the “model error”
assigned to the rms radius had to include all models with a
sufficiently good $\chi^2$ disfavoring the small radius. A detailed
discussion of the correct statistical analysis of fits to the new
Mainz cross section data (including the constant fraction expansion)
will be part of a forthcoming paper <cit.>.
In summary, a low order expansion of the form factors has to be
consistent with our knowledge of the shape at large $Q^2$ derived from
experiments over the 50 years since the work of Robert Hofstadter. A
fit to a truncated $Q^2$-range data set cannot be used to extract a
robust value for the radius since it neglects this knowledge. The full
$Q^2$ range of $G(Q^2)$ has to be used to be able to determine the
mandatory knowledge of $\rho(r)$. Since the two papers are neglecting
this requirement they do not explain the “proton radius puzzle”.
It is worth noting that the realization of the importance of the full
form factor also limits the conjectures of spikes or bumps at very low
$Q^2$ where no electron scattering measurements are possible. Any
structure there must introduce significant long range contribution to
the charge distribution.
Acknowledgments This work was supported by the Collaborative
Research Center 1044 and the State of Rhineland-Palatinate.
|
1511.00452
|
Many two-sided matching markets, from labor markets to school choice programs, use a clearinghouse based on the applicant-proposing deferred acceptance algorithm, which is well known to be strategy-proof for the applicants. Nonetheless, a growing amount of empirical evidence reveals that applicants misrepresent their preferences when this mechanism is used.
This paper shows that no mechanism that implements a stable matching is obviously strategy-proof for any side of the market, a stronger incentive property than strategy-proofness that was introduced by <cit.>. A stable mechanism that is obviously strategy-proof for applicants is introduced for the case in which agents on the other side have acyclical preferences.
§ INTRODUCTION
A number of labor markets and school admission programs that can be viewed as two-sided matching markets use centralized mechanisms to match agents on both sides of the market (or agents on one side of the market and objects on the other side of the market).
One important criterion in the design of such mechanisms is stability <cit.>, requiring that no two agents, one from each side of the market, prefer each other over the partners with whom they are matched. Another highly desired property is strategy-proofness, which alleviates agents' incentives to behave strategically.[See also <cit.>, which finds that non-strategy-proof mechanisms favor sophisticated players over more naïve players.]
Indeed, many clearinghouses have adopted in recent years the remarkable deferred acceptance (DA) mechanism <cit.>,[Examples include the National Resident Matching Program <cit.>, as well as school choice programs in Boston <cit.> and New York <cit.> <cit.>.] which finds a stable matching and is strategy-proof for one side of the market, namely the proposing side in the DA algorithm <cit.>.[This mechanism is also approximately strategy-proof for all participants in the market <cit.>.],[Indeed, removing the incentives to “game the system" was a key factor in the city of Boston's decision to replace its school assignment mechanism in 2005 <cit.>.] Interestingly, although participants are advised that it is in their best interest to state their true preferences, empirical evidence suggests that a significant fraction nonetheless attempt to strategically misreport their true preferences <cit.>; this was observed in experiments <cit.>, in surveys <cit.>, and in the field <cit.>. This paper asks whether one can implement the deferred acceptance outcome via a mechanism whose description makes its strategy-proofness more apparent. Toward this goal, we adopt the notion of obvious strategy-proofness, an incentive property introduced by <cit.> that is stronger than strategy-proofness.
<cit.> formulated the idea that it is “easier to be convinced” of the strategy-proofness of some mechanisms over others.
He introduces, and characterizes, the class of obviously strategy-proof
mechanisms. He shows that, roughly speaking,
obviously strategy-proof mechanisms are those whose strategy-proofness can be proved even
under a cognitively limited proof model that does not allow for contingent reasoning.[For instance, this notion separates sealed-bid second-price auctions from ascending auctions (where bidders only need to decide at any given moment whether to quit or not) and provides a possible explanation as to why more subjects have been reported to behave insincerely in the former than in the latter <cit.>.]
In his paper, studies whether various well-known auction and assignment mechanisms with attractive revenue or welfare properties for one side of the market
can be implemented in an obviously strategy-proof manner. Whether one may implement stable matchings in an obviously strategy-proof manner remained an open problem.
For the purpose of this paper, we adopt the Gale and Shapley (1962) one-to-one matching market with men and women to represent two-sided matching markets; our main results naturally extend to many-to-one markets such as labor markets and school choice programs.
When women's preferences over men are perfectly aligned, the unique stable matching may be recovered via serial dictatorship, where men, in their ranked order, choose their partners. In this case, a sequential implementation of such serial dictatorship is obviously strategy-proof. (This follows from <cit.>, who shows that in a two-sided assignment market with agents and objects, serial dictatorship, when implemented sequentially, is obviously strategy-proof.[Since, after selecting an object, the agent quits the game, no contingent reasoning is needed in order to verify that she must ask for her favorite unallocated object. However, serial dictatorship (the same strategy-proof social choice rule), when implemented by having each agent simultaneously submit a ranking over all objects in advance, is not obviously strategy-proof. This example and the example in <ref> both demonstrate that whereas strategy-proofness is a property of the social choice rule,
obvious strategy-proofness is a property of the mechanism implementing the social choice rule.])
Generalizing to allow for weaker forms of alignment of women's preferences, we
show that if women's preferences are acyclical <cit.>,[A preference profile for a woman over men is cyclical if there are
three men $a,b,c$ and two women $x,y$ such that $a \succ_x b \succ_x c \succ_y a$.] then the men-optimal stable matching can be implemented via an obviously strategy-proof mechanism. While the obvious truthfulness of the basic questions that we use to construct this implementation (questions of the form “do you prefer $x$ the most out of all currently unmatched women?”) draws from the same intuition upon which the serial dictatorship mechanism is based, the questions are considerably more flexible, and the order of the questions more subtle.
The main finding of this paper is that for general preferences, no mechanism that implements the men-optimal stable matching (or any other stable matching) is obviously strategy-proof for men.
We first prove this impossibility in a specifically crafted matching market with 3 women and 3 men, in which women have fixed (cyclical) commonly known preferences
and men have unrestricted private preferences. It is then shown that for the impossibility to hold in any market, it is sufficient for some $3$ women to have this structure of preferences over some $3$ men.
Moreover, the same result holds even if women's preferences are privately known.
An immediate implication of these results is that in a large market, in which women's preferences are drawn independently and uniformly at random, with high probability no implementation of any stable mechanism is obviously strategy-proof for all men (or even for most men).
These results apply to school choice settings even when schools are not strategic and have commonly known priorities over students. For example, unless schools' priorities over students are sufficiently aligned, no mechanism that is stable with respect to students' preferences and schools' priorities is obviously strategy-proof for students.
This paper sheds more light on fundamental
differences between two-sided market mechanisms that aim to implement a two-sided notion such as stability, and closely related two-sided market mechanisms that aim to implement some efficiency notion for one of the sides of the market.
First, as noted, in assignment markets there exists an obviously strategy-proof ex-post efficient mechanism
(serial dictatorship). Second, a variety of ascending auctions, from familiar multi-item auctions
<cit.> to recently proposed clock auctions
<cit.>, maximize welfare or revenue and are obviously strategy-proof,
despite the latter's being based on deferred acceptance principles. In contrast, this paper shows that
there is no way to achieve stability that is obviously strategy-proof for either side of the market.
Obvious strategy-proofness was introduced by <cit.>, who studies this property extensively in mechanisms with monetary transfers. In settings without transfers, <cit.> studies this property in implementations of serial dictatorship and top trading cycles. Several papers further study this property in different settings. Closely related is <cit.>, who studies two-sided markets with agents and objects and asks for which priorities for objects one can implement in an obviously strategy-proof manner the Pareto-efficient top trading cycles algorithm. Farther technically from our paper but closely related in spirit, <cit.> constructively characterize Pareto-efficient social choice rules that admit obviously strategy-proof implementations in popular domains (object assignment, single-peaked preferences, and combinatorial auctions). <cit.> characterize general obviously strategy-proof mechanisms without transfers under a “richness” assumption on the preferences domain, and characterize the sequential version of random serial dictatorship under such an assumption via a natural set of axioms that includes obvious strategy-proofness. It is worth noting that all three of these papers utilize machinery and observations that originated in this paper.
The paper is organized as follows. <ref> provides the model and background, including the definition of obvious strategy-proofness in matching markets.
<ref> presents special cases for which an obviously strategy-proof implementation of the men-optimal stable matching exists.
<ref> provides the main impossibility result. <ref> presents corollaries in a model where women also have private preferences. <ref> concludes.
§ PRELIMINARIES
§.§ Two-sided matching with one strategic side
For the bulk of our analysis it will be sufficient to consider two-sided markets in which only one side of the market is strategic. We begin by defining the notions of matching and strategy-proofness in such markets.
In a two-sided matching market, the participants are partitioned into a finite set of men $M$ and a finite set of women $W$.
A preference list (for some man $m$) over $W$ is a totally ordered subset of $W$ (if some woman $w$ does not appear on the preference list, we think of her as being unacceptable to $m$).
Denote the set of all preference lists over $W$ by $\prefs(W)$.
A preference profile $\bar{p}=(p_m)_{m\in M}$ for $M$ over $W$ is a specification of a preference list $p_m$ over $W$ for each man $m\in M$. (So the set of all preference profiles for $M$ over $W$ is $\prefs(W)^M$.)
Given a preference list $p_m$ for some man $m$,
we write $w\succ_m w'$ to denote that man $m$ strictly prefers woman $w$ over woman $w'$, (i.e., either woman $w$ is ranked higher than $w'$ on $m$'s preference list, or $w$ appears on this list while $w'$ does not),
and write $w\succeq_m w'$ if it is not the case that $w' \succ_m w$.
A matching between $M$ and $W$ is a one-to-one mapping between a subset of $M$ and a subset of $W$.
Denote the set of all matchings between $M$ and $W$ by $\matchings$. Given a matching $\matching$ between $M$ and $W$, for a participant $a\in M\cup W$ we write $\matching_a$ to denote $a$'s match in $\matching$, or write $\matching_a=a$ if $a$ is unmatched.
A (one-side-querying) matching rule is a function $C:\prefs(W)^M\rightarrow\matchings$, from preference profiles for $M$ over $W$ to matchings between $M$ and $W$.
A matching rule $C$ is said to be strategy-proof for a man $m$ if for every preference profile $\bar{p}=(p_m)_{m\in M}\in\prefs(W)^M$ and for every (alternate) preference list $p'_m\in \prefs(W)$, it is the case that $C_m(\bar{p}) \succeq_m C_m(p_m',\bar{p}_{-m})$ according to $p_m$.[As is customary, $(p_m',\bar{p}_{-m})$ denotes the preference profile obtained from $\bar{p}$ by setting the preference list of $m$ to be $p'_m$.]
$C$ is said to be strategy-proof if it is strategy-proof for every man.
§.§ Obvious strategy-proofness
This section briefly describes the notion of obvious strategy-proofness, developed in great generality by <cit.>. We rephrase these notions for the special case of deterministic matching mechanisms with finite preference and outcome sets. For ease of presentation, attention is restricted to mechanisms under perfect information; however, the results in this
paper still hold (mutatis mutandis) via the same proofs for the general definitions of <cit.>.[Readers who are familiar with the general definitions of <cit.> may easily verify that if a randomized stable obviously strategy-proof (OSP) mechanism exists, then derandomizing it by fixing in advance each choice of nature to some choice made with positive probability yields a deterministic stable OSP mechanism. Furthermore, if some stable mechanism is OSP under partial information, then it is also OSP under perfect information.]
Whereas strategy-proofness is a property of a given matching rule, obvious strategy-proofness is a property of a specific implementation, via a specific mechanism, of such a matching rule. A mechanism implements a matching rule by specifying, roughly speaking, an extensive-form game tree that implements the standard-form game associated (where strategies coincide with preference lists) with the matching rule, where each action at each node of the extensive-form game tree corresponds to some set of possible preference lists for the acting participant. We now formalize this definition.
A one-side-querying extensive-form) matching mechanism for $M$ over $W$ consists of:
* A rooted tree $T$.
* A map $X:L(T)\rightarrow\matchings$ from the leaves of $T$ to matchings between $M$ and $W$.
* A map $Q:V(T)\setminus L(T)\rightarrow M$, from internal nodes of $T$ to $M$.
* A map $A:E(T)\rightarrow 2^{\prefs(W)}$, from edges of $T$ to predicates over $\prefs(W)$, such that all of the following hold:
* Each such predicate must match at least one element in $\prefs(W)$.
* The predicates corresponding to edges outgoing from the same node are disjoint.
* The disjunction (i.e., set union) of all predicates corresponding to edges outgoing from a node $n$ equals the predicate corresponding to the last edge outgoing from a node labeled $Q(n)$ along the path from the root to $n$, or to the predicate matching all elements of $\prefs(W)$ if no such edge exists.
A preference profile $\bar{p}\in\prefs(W)^M$ is said to pass through a node $n \in V(T)$ if, for each edge $e$ along the path from the root of $T$ to $n$, it is the case that $p_{Q(n')}\in A(e)$, where $n'$ is the source node of $e$. That is, the nodes through which $\bar{p}$ passes are the nodes of the path that starts from the root of $T$ and follows, from each internal node $n'$ that it reaches, the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Given an extensive-form matching mechanism $\impl$, we denote by $C^{\impl}$, called the matching rule implemented by $\impl$, the (one-side-querying) matching rule mapping a preference profile $\bar{p}\in\prefs(W)^M$ to the matching $X(n)$, where $n$ is the unique leaf through which $\bar{p}$ passes. Equivalently, $n$ is the node in $T$ obtained by traversing $T$ from its root, and from each internal node $n'$ that is reached, following the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Two preference lists $p,p'\in\prefs(W)$ are said to diverge at a node $n\in V(T)$ if there exist two distinct edges $e,e'$ outgoing from $n$ such that $p\in A(e)$ and $p'\in A(e')$.
Let $\impl$ be an extensive-form matching mechanism.
* $\impl$ is said to be obviously strategy-proof (OSP) for a man $m\in M$ if for every node $n$ with $Q(n)=m$ and for every
$\bar{p}=(p_{m'})_{m'\in M}\in\prefs(W)^M$ and $\bar{p}'=(p'_{m'})_{m'\in M}\in\prefs(W)^M$ that both pass through $n$ such that $p_m$ and $p'_m$ diverge at $n$, it is the case that $C^{\impl}_m(\bar{p}) \succeq_m C^{\impl}_m(\bar{p}')$ according to $p_m$. In other words, the worst possible outcome for $m$ when acting truthfully (i.e., according to $p_m$) at $n$ is no worse than the best possible outcome for $m$ when misrepresenting his preference list to be $p'_m$ at $n$.
* $\impl$ is said to be obviously strategy-proof (OSP) if it is obviously strategy-proof for every man $m\in M$.
<cit.> shows that obviously strategy-proof mechanisms are, in a precise sense, mechanisms that can shown to implement strategy-proof rules under a cognitively limited proof model that does not allow for contingent reasoning.
To observe how strategy-proofness of the matching rule $C^\impl$ for a man $m\in M$ is indeed a weaker condition than obvious strategy-proofness of the mechanism $\impl$ for $m$, note that the matching rule $C^{\impl}$ is strategy-proof for $m$ if and only if for every node $n$ with $Q(n)=m$ and for every $\bar{p}=(p_m)_{m\in M}\in\prefs(W)^M$ that passes through $n$ and for every $p'_m\in\prefs(W)$ that diverges from $p_m$ at $n$,[These conditions imply that $(p'_m,\bar{p}_{-m})$ also passes through $n$.] it is the case that $C^{\impl}_m(\bar{p})\succeq_m C^{\impl}_m(p'_m,\bar{p}_{-m})$ according to $p_m$.[We emphasize that this rephrased definition is equivalent to the definition of strategy-proofness of the matching rule $C^{\impl}$ that is given in <ref>, however it is not equivalent to standard definition of strategy-proofness of the extensive-form game underlying the mechanism $\impl$, which would allow each man to condition the type he is “pretending to be” under any strategy on the information revealed by other men in preceding nodes. Once we move to the realm of obvious strategy-proofness, the restriction on each strategy to always consistently “pretend to be” of the same type is inconsequential, as the definition of OSP considers the case in which other men may play different types when the man in question acts truthfully or deviates. It is for this reason that we have chosen to implicitly define a strategy in the extensive-form game underlying $\impl$ to be restricted to consistently “pretending to be” of the same type. This somewhat nonstandard implicit definition of a strategy considerably simplifies notation throughout this paper (by considering only consistent behavior on behalf of every agent) without changing the mathematical meaning of obvious strategy-proofness (or of strategy-proofness of a matching rule) and without limiting the generality of our results.]
A (one-side-querying) matching rule $C:\prefs(W)^M\rightarrow\matchings$ is said to be OSP-implementable if $C=C^{\impl}$ for some obviously strategy-proof matching mechanism $\impl$. In this case, we say that $\impl$ OSP-implements $C$.
§.§ Stability
We proceed to describe a simplified version of stability in matching markets as introduced by <cit.>.
While, as stated in <ref>, for the bulk of our analysis it is sufficient to consider markets in which only men are strategic, to define the notion of stability one must consider not only preferences for the (strategic) men, but also preferences (sometimes called priorities) for the (nonstrategic) women. Women's preference lists and preference profiles are defined analogously with those of men.
We continue to denote a preference profile for men by $\bar{p}=(p_m)_{m\in M}\in\prefs(W)^M$, while denoting a preference profile for women by $\bar{q}=(q_w)_{m\in M}\in\prefs(M)^W$.
Let $\bar{p}$ and $\bar{q}$ be preference profiles of men and women respectively. A matching $\matching$ is said to be unstable with respect to $\bar{p}$ and $\bar{q}$ if there exist a man $m$ and a woman $w$ each preferring the other over the partner matched to them by $\matching$, or if some participant $a\in M\cup W$ is matched with some other participant not on $a$'s preference list. A matching that is not unstable is said to be stable.
<cit.> showed that a stable matching exists with respect to every pair of preference profiles and, furthermore, that for every pair of preference profiles there exists an $M$-optimal stable matching, i.e., a stable matching such that each man weakly prefers his match in this stable matching over his match in any other stable matching.
We now relate the concept of stability to the (one-side-querying) matching rules and mechanisms defined in the previous sections.
Let $\bar{q}\in\prefs(M)^W$ be a preference profile for $W$ over $M$.
A (one-side-querying) matching rule $C$ is said to be $\bar{q}$-stable if for every preference profile $\bar{p}\in\prefs(W)^M$ for $M$ over $W$, the matching $C(\bar{p})$ is stable with respect to $\bar{p}$ and $\bar{q}$. A (one-side-querying) matching mechanism is said to be $\bar{q}$-stable if the matching rule that it implements is $\bar{q}$-stable.
We denote by $C^{\bar{q}}:\prefs(W)^M\rightarrow\matchings$ the $M$-optimal stable matching rule, i.e., the (one-side-querying, $\bar{q}$-stable) matching rule mapping each preference profile for men $\bar{p}$ to the $M$-optimal stable matching with respect to $\bar{p}$ and $\bar{q}$.
It is well known that $C^{\bar{q}}$
is strategy-proof for all men <cit.>.
Moreover, no other matching rule is strategy-proof for all men <cit.>.[For a more general result, see <cit.>.]
In the notation of this paper:
For every preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$, no $\bar{q}$-stable matching rule $C\ne C^{\bar{q}}$ is strategy-proof.
In this paper, we ask whether $C^{\bar{q}}$ is not only strategy-proof, but also OSP-implementable. (As it is the unique strategy-proof $\bar{q}$-stable matching rule, it is the only candidate for OSP-implementability.)
§ OSP-IMPLEMENTABLE SPECIAL CASES
Before stating our main impossibility result, we first present a few special cases in which $C^{\bar{q}}$, the $M$-optimal stable matching rule for a fixed women's preference profile $\bar{q}$, is in fact OSP-implementable. These are the first known OSP mechanisms without transfers that are not dictatorial.[All OSP mechanisms that are surveyed in the end of the introduction are based upon the query structure of the mechanisms of this sec:special-cases.]
For simplicity, we describe all of these cases under the assumption that the market is balanced (i.e., that $|W|=|M|$) and that all preference lists are full (i.e., that each participant prefers being matched to anyone over being unmatched); generalizing each of the below cases for unbalanced markets or for preference lists for men that are not full is straightforward.[Indeed, asking any man whether he prefers being unmatched over being matched with any (remaining not-yet-matched) woman never violates obvious strategy-proofness.]
The first case we consider is that in which women's preferences are perfectly aligned.
[$C^{\bar{q}}$ is OSP-implementable when women's preferences are perfectly aligned]
Let $q\in\prefs(M)$ and let $\bar{q}=(q)_{w\in W}$ be the preference profile for $W$ over $M$ in which all women share the same preference list $q$. $C^{\bar{q}}$ is OSP-implementable by the following serial dictatorship mechanism: ask the man most preferred according to $\bar{q}$ which woman he prefers most, and assign that woman to this man (in all leaves of the subtree corresponding to this response), ask the man second-most preferred according to $\bar{q}$ which woman he prefers most out of those not yet assigned to any man, and assign that woman to this man (in all leaves of the subtree corresponding to this response), etc. This mechanism can be shown to be OSP by the same reasoning that <cit.> uses to show that serial dictatorship is OSP.
Another noteworthy example is that of arbitrary preferences in a very small matching market.
[$C^{\bar{q}}$ is OSP-implementable when $|M|=|W|=2$]
When $|M|=|W|=2$, $C^{\bar{q}}$ is OSP-implementable for every preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$. Indeed, let $M=\{a,b\}$ and $W=\{x,y\}$. If $q_x=q_y$, then $C^{\bar{q}}$ is OSP-implementable as explained in <ref>. Otherwise, without loss of generality $a\succ_x b$ and $b\succ_y a$; for this case, <ref> describes an OSP mechanism that implements $C^{\bar{q}}$.
[node distance=3.5cm]
(q1) [internal] $a$;
(q2) [internal, right of=q1] $b$;
(a1) [leaf, below of=q1] $a \Leftarrow x$ $b \Leftarrow y$;
(a2) [leaf, below of=q2] $a \Leftarrow x$ $b \Leftarrow y$;
(a3) [leaf, right of=q2] $a \Leftarrow y$ $b \Leftarrow x$;
[arrow] (q1) – node[anchor=west] $x \succ_a y$ (a1);
[arrow] (q1) – node[anchor=north] $y \succ_a x$ (q2);
[arrow] (q2) – node[anchor=west] $y \succ_b x$ (a2);
[arrow] (q2) – node[anchor=north] $x \succ_b y$ (a3);
An OSP mechanism that implements $C^{\bar{q}}$ for $|W|=|M|=2$ and for $\bar{q}$ where $a\succ_x b$ and $b\succ_y a$. (The notation, e.g., $a\Leftarrow x$, indicates that $x$ is matched to $a$ in the matching corresponding to that leaf of the mechanism tree.)
The preference profiles in <ref> are special cases of the class of acyclical preference profiles, whose structure was defined by <cit.>.
A preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$ is said to be cyclical if there exist $a,b,c\in M$ and $x,y\in W$ such that $a \succ_x b \succ_x c \succ_y a$.
If $\bar{q}$ is not cyclical, then it is said to be acyclical.
<cit.> shows that acyclicality of $\bar{q}$ is necessary and sufficient for $C^{\bar{q}}$ to be group strategy-proof (and not merely weakly group strategy-proof) and Pareto efficient.
We now generalize <ref> by showing that acyclicality of $\bar{q}$ (as in both of these perfectly-aligned) is sufficient for $C^{\bar{q}}$ to be also OSP-implementable. Much like the implementations in <ref>, the strategy-proofness of the OSP implementation that emerges for acyclical preferences is far easier to understand that that of the standard deferred-acceptance implementation, thus showcasing the usefulness of obvious strategy-proofness in identifying easy-to-understand implementations. In each mechanism step, either a single man is given free pick out of all remaining $w\in W$, or two men are each given first priority over some subset of $W$ (i.e., free pick if his favorite remaining $w\in W$ is there), and second priority over the rest (i.e,. free pick out of all other remaining $w\in W$ except the one chosen by the other man if the latter invoked his first priority).
$C^{\bar{q}}$ is OSP-implementable for every acyclical preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$.
We prove the result by induction over $|M|=|W|$. By acyclicality, at most two men are ranked by some woman as her top choice. If only one such man $m\in M$ exists, then he is ranked by all women as their top choice—in this case, similarly to <ref>, we ask this man for his top choice $w \in W$, assign her to him, and then continue by induction (finding in an OSP manner the $M$-optimal stable matching between $M\setminus\{m\}$ and $W\setminus\{w\}$). Otherwise, there are precisely two men $a\in M$ and $b\in M$ who are ranked by some woman as her top choice. By acyclicality, each woman either has $a$ as her top choice and $b$ as her second-best choice, or vice versa.[This is reminiscent of the priorities of the first two agents in bipolar serially dictatorial rules <cit.>, which are indeed included in the analysis of <ref> as a special case.] We conclude somewhat similarly to <ref>: for each woman $w\in W$ that prefers $a$ most, we ask $a$ whether he prefers $w$ most; if so, we assign $w$ to $a$ and continue by induction. Otherwise, for each woman $w\in W$ that prefers $b$ most, we ask $b$ whether he prefers $w$ most; if so, we assign $w$ to $b$ and continue by induction. Otherwise, we ask each of $a$ and $b$ for his top choice, assign each of them his top choice, and continue by induction.
To see that this implementation is OSP, consider a man $m\in M$ who is asked by this mechanism whether a woman $w\in W$ is his top choice (among the remaining women). If $m$ really does prefer $w$ most, then answering truthfully matches him to $w$, which he weakly prefers over any outcome that occurs if he is not truthful. Similarly, if $m$ does not prefer $w$ most, then answering truthfully may get $m$ a more preferred choice, but also assures $m$ that if he does not get such a preferred choice, then he would still be able to choose to get matched to $w$ (he would do so if he fails to get his top choice, and $w$ is his second-best); so, any outcome that results from truthfulness is weakly preferred by $m$ over any outcome that results from nontruthfulness in this case as well.
We conclude this section by noting, however, that acyclicality of $\bar{q}$ is not a necessary condition for OSP-implementability of $C^{\bar{q}}$, as demonstrated by the following example.
[OSP-implementable $C^{\bar{q}}$ with cyclical $\bar{q}$]
Let $M=\{a,b,c\}$ and $W=\{x,y,z\}$. We claim that $C^{\bar{q}}$, for the following cyclical preference profile $\bar{q}$ for $W$ over $M$ (where each woman prefers being matched to any man over being unmatched), is OSP-implementable:
\[\begin{array}{ccccc}
a & \succ_x & b & \succ_x& c \\
a & \succ_y & c & \succ_y & b \\
b & \succ_z & a & \succ_z & c.
\end{array}\]
We begin by noting that $\bar{q}$ is indeed cyclical, as $a \succ_y c \succ_y b \succ_z a$. We now note that the following mechanism OSP-implements $C^{\bar{q}}$:
* Ask $a$ whether he prefers $x$ the most; if so, assign $x$ to $a$ and continue as in <ref> (finding in an OSP manner the $M$-optimal stable matching between $\{y,z\}$ and $\{b,c\}$).
* Ask $a$ whether he prefers $y$ the most; if so, assign $y$ to $a$ and continue as in <ref>. (Otherwise, we deduce that 1) $a$ prefers $z$ the most and therefore 2) $c$ will not end up being matched to $z$.)
* Ask $b$ whether he prefers $z$ the most; if so, assign $z$ to $b$ and continue as in <ref>.
* Ask $b$ whether he prefers $x$ the most; if so, assign $x$ to $b$, $z$ to $a$, and $y$ to $c$. (Otherwise, we deduce that $b$ prefers $y$ the most.)
* Ask $c$ whether he prefers $x$ over $y$. If so, assign $x$ to $c$, $y$ to $b$, and $z$ to $a$. (Otherwise, we deduce that $b$ will not end up being matched to $y$.)
* Ask $b$ whether he prefers $z$ over $x$. Assign $b$ to his preferred choice between $z$ and $x$ and continue as in <ref>.
Nonetheless, as we show in the next section, when there are more than $2$ participants on each side and women's preferences are sufficiently unaligned, $C^{\bar{q}}$ is not OSP-implementable.
§ IMPOSSIBILITY RESULT FOR GENERAL PREFERENCES
We now present our main impossibility result.
If $|M|\ge3$ and $|W|\ge3$, then there exists a preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$, such that
no $\bar{q}$-stable (one-side-querying) matching rule is OSP-implementable.
Observe that <ref> applies to any $\bar{q}$-stable (one-side-querying) matching rule, and not only to the $M$-optimal stable matching rule $C^{\bar{q}}$.
Before proving the result, we first prove a special case that cleanly demonstrates the construction underlying our proof.
For $|M|=|W|=3$, there exists a preference profile $\bar{q}\in\prefs(M)^W$ for $W$ over $M$ such that
no $\bar{q}$-stable (one-side-querying) matching rule is OSP-implementable.
Let $M=\{a,b,c\}$ and $W=\{x,y,z\}$. Let $\bar{q}$ be the following preference profile (where each woman prefers being matched to any man over being unmatched):
\begin{equation}\label{tricyclical}
\begin{array}{ccccc}
a & \succ_x & b & \succ_x& c \\
b & \succ_y & c & \succ_y & a \\
c & \succ_z & a & \succ_z & b.
\end{array}
\end{equation}
Assume for contradiction that an OSP mechanism $\impl$ that implements a $\bar{q}$-stable matching rule $C^{\impl}$ exists. Therefore, $C^{\impl}$ is strategy-proof, and so, by <ref>, $C^{\impl}=C^{\bar{q}}$.
In order to reach a contradiction by showing that such a mechanism (that OSP-implements $C^{\bar{q}}$) cannot possibly exist, we dramatically restrict the domain of preferences of all men, which results in a simpler mechanism, where the contradiction can be identified in a less cumbersome manner. We define:
$p^1_a \eqdef z \succ y \succ x$ $p^1_b \eqdef x \succ z \succ y$ $p^1_c \eqdef y \succ x \succ z$
$p^2_a \eqdef y \succ x \succ z$ $p^2_b \eqdef z \succ y \succ x$ $p^2_c \eqdef x \succ z \succ y$,
and set $\prefs_a\eqdef\{p^1_a,p^2_a\}$, $\prefs_b\eqdef\{p^1_b,p^2_b\}$, and $\prefs_c\eqdef\{p^1_c,p^2_c\}$.
Following the “pruning” technique in <cit.>, we note that if we “prune” the tree of $\impl$ by replacing, for each edge $e$, the predicate $A(e)$ with the conjunction (i.e., set intersection) of $A(e)$ with the predicate matching all elements of $\prefs_{Q(n)}$, where $n$ is the source node of $e$, and by consequently deleting all edges $e$ for which $A(e)=\bot$,[The standard notation $\bot$ stands for “false” (mnemonic: an upside-down “true” $\top$), i.e., the predicate that matches nothing, so an edge for which $A(e)=\bot$ will never be followed.] we obtain, in a precise sense, a mechanism that implements $C^{\bar{q}}$ where the preference list of each man $m\in M$ is a priori restricted to be in $\prefs_m$.[The definition of mechanisms and OSP when the domain of preferences is restricted extends naturally from that given in <ref> for unrestricted preferences.
The interested reader is referred to Appendix <ref> for precise details.]
By a proposition in <cit.>, since the original mechanism $\impl$ is OSP, so is the pruned mechanism as well.
Let $n$ be the earliest (i.e., closest to the root) node in the pruned tree that has more than one outgoing edge (such a node clearly exists, since $C^{\impl}=C^{\bar{q}}$ is not constant over $\prefs_a\times\prefs_b\times\prefs_c$). By symmetry of $\bar{q},\prefs_a,\prefs_b,\prefs_c$, without loss of generality $Q(n)=a$. By definition of pruning, it must be the case that $n$ has two outgoing edges, one labeled $p^1_a$, and the other labeled $p^2_a$. We claim that the mechanism of the pruned tree is in fact not OSP. Indeed, for $p_a=p^2_a$ (the “true preferences”), $p_b=p^2_b$, and $p_c=p^1_c$, we have that $C_a^{\impl}(\bar{p})=C_a^{\bar{q}}(\bar{p})=x$, yet for $p'_a=p^1_a$ (a “possible manipulation”), $p'_b=p^1_b$, and $p'_c=p^2_c$, we have that $C_a^{\impl}(\bar{p}')=C_a^{\bar{q}}(\bar{p}')=y$, even though $C_a^{\impl}(\bar{p}')=y\succ_a x = C_a^{\impl}(\bar{p})$ according to $p_a$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and $p_a$ and $p_a'$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
The not-osp follows from a reduction to <ref>. Indeed, let $a,b,c$ be three distinct men and let $x,y,z$ be three distinct women. Let $\bar{q}\in\prefs(W)^M$ be a preference profile such that the preferences of $x,y,z$ satisfy <ref> with respect to $a,b,c$ (with arbitrary preferences over all other men), and with arbitrary preferences for all other women. Assume for contradiction that a $\bar{q}$-stable OSP mechanism $\impl$ exists.
We prune (see the proof of <ref> for an explanation of pruning) the tree of $\impl$ such that the only possible preference lists for $a,b,c$ are those in which they prefer each of $x,y,z,$ over all other women, and the only possible preference list for all other men is empty.[Alternatively, one could set for all other men arbitrary preference lists that do not contain $x,y,z$.] Let $\bar{q}'$ be the preference profile given in <ref>; the resulting (pruned) mechanism is a $\bar{q}'$-stable matching mechanism for $a,b,c$ over $x,y,z$,[Formally, it is a matching mechanism for $W$ over $M$ with respect to the pruned preferences, but can be shown to always leave all participants but $a,b,c$ and $x,y,z,$ unmatched, and so can be thought of as a matching mechanism for $a,b,c$ over $x,y,z$.] and so, by <ref>, it is not OSP; therefore, by the same proposition in <cit.> that is used in <ref>, neither is $\impl$.
As <ref> shows, it is enough that some three women have preferences that satisfy <ref> with respect to some three men in order for obvious strategy-proofness to be unattainable. This implies that obvious strategy-proofness in also unattainable in large random markets with high probability.
If $|M|\ge3$ and $|W|\ge3$, then as $|M|+|W|$ grows, we have for a randomly drawn preference profile $\bar{q}\sim U\bigl(\prefs(M)^W\bigr)$ for $W$ over $M$ that:[This result also holds, with the same proof, if $\bar{q}$ is drawn uniformly at random from the set of all full preferences (i.e., where each woman prefers being matched to any man over being unmatched).]
* With high probability no $\bar{q}$-stable (one-side-querying) matching rule is OSP-implementable.
* For every three distinct men $a,b,c\in M$, as $|W|$ grows, with high probability no $\bar{q}$-stable (one-side-querying) matching mechanism is OSP for $a$, $b$, and $c$.
If $|M|\le\poly\bigl(|W|\bigr)$, then with high probability no $\bar{q}$-stable (one-side-querying) matching mechanism is OSP for more than two men.
<ref> follows from an argument similar to the one in the proof of <ref>. Indeed, our proof of <ref> in fact shows that if $\bar{q}$ satisfies <ref> with respect to three men $a,b,c$ and three women $x,y,z$, then no $\bar{q}$-stable matching mechanism is OSP for $a$, $b$, and $c$. For <ref>, for instance, we note that for a fixed triplet of distinct men $a,b,c\in M$, the probability that <ref> is not satisfied by $\bar{q}$ with respect to $a,b,c$ and any three women $x,y,z$ decreases exponentially with $|W|$, while the number of triplets of men increases polynomially with $|M|$.
We conclude this sec:impossibility by noting that while the aesthetic preference profile defined in <ref> is sufficient for proving <ref> and even <ref>, it is by no means the unique preference profile that eludes an obviously strategy-proof implementation, even when $|M|=|W|=3$. Indeed, <ref> in <ref> gives an additional example of such a preference profile, which could be described as “less cyclical,” in some sense.[While the proof of <ref> also follows a pruning argument, the reasoning is more involved than in the proof given for <ref> above.]
In this context, it is worth noting that following up on our paper, <cit.> gives a necessary and sufficient condition, “weak acyclicality” (weaker, indeed, than acyclicality as defined in <ref>), on the preferences of objects in the (Pareto efficient, not necessarily stable) top trading cycles algorithm for this algorithm to be OSP-implementable for the agents. The example given in <ref> also demonstrates that 's condition does not suffice for the existence of an OSP-implementable stable mechanism.
A comparison of the respective preference profiles used for the positive result of <ref> and the negative result of <ref>,
noting that the former is obtained by taking the latter and arguably making it “more aligned” by modifying the preference list of woman $x$ to equal that of woman $y$,
suggests that an analogous succinct “maximal domain” characterization of preference profiles that admit OSP-implementable stable mechanisms may be delicate, and obtaining it may be challenging.
§ MATCHING WITH TWO STRATEGIC SIDES
So far, this paper has studied two-sided matching markets in which only men are strategic and women's preference lists are commonly known. This allowed us to ask questions such as, for which preference profiles of women
one can OSP-implement the $M$-optimal stable matching rule? This setting is furthermore practically relevant in school choice where, for example, schools do not act strategically but have priorities over students.
Our analysis, however, also immediately yields that when both men and women behave strategically, no stable matching mechanism is OSP-implementable. To formalize this result, we introduce a few definitions.
A two-sides-querying matching rule is a function $C:\prefs(W)^M\times\prefs(M)^W\rightarrow\matchings$, from preference profiles for both men and women to a matching between $M$ and $W$.
A two-sides-querying matching rule $C$ is stable if for any preference profiles $\bar{p}$ and $\bar{q}$ for men and women, $C(\bar{p},\bar{q})$ is stable with respect to $\bar{p}$ and $\bar{q}$. A two-sides-querying matching mechanism[The definition of mechanisms and OSP for markets where both sides are strategic extends naturally from that given in <ref> for markets where only one side is strategic.
The interested reader is referred to Appendix <ref> for precise details.]
is stable if the two-sides-querying matching rule that it implements is stable.
<ref> implies the following impossibility result for two-sides-querying matching mechanisms:
If $|M|\ge3$ and $|W|\ge3$, then no stable two-sides-querying matching rule is OSP-implementable for $M$. Moreover, no stable two-sides-querying matching mechanism is OSP for more than two men.
As with <ref>, we note that <ref> applies to any stable two-sides-querying matching rule, and not only to the $M$-optimal two-side-querying stable matching rule (i.e., the two-sides-querying matching rule that maps each pair of preference profiles to the corresponding $M$-optimal stable matching).
Similarly, <ref> implies the following possibility result for two-sides-querying matching mechanisms:
If $|M|=2$, then the two-sides-querying $M$-optimal stable matching rule is OSP-implementable (by first querying the women, and then, given their preferences, continuing as in <ref>).
A precise argument that relates the results for markets with one strategic side and those for markets with two strategic sides is given in <ref>.
§ CONCLUSION
The main finding of this paper is that no stable matching mechanism is obviously strategy-proof for the participants even on one of the sides of the market. This suggests that there may not be any alternative general way to describe the deferred acceptance procedure that makes its strategy-proofness more apparent, implying that strategic mistakes observed in practice <cit.> may not be avoidable by better explaining the mechanism. This highlights the importance of gaining the trust of the agents who participate in stable mechanisms, so that they both act as advised (even when it is hard to verify that no strategic opportunities exist) and are assured that the social planner will not deviate from the prescribed procedure after preferences are elicited.
For the case in which women's preferences are acyclical, we describe an OSP mechanism that implements the men-optimal stable matching. As may be expected, the strategy-proofness of this OSP implementation is easier to understand than that of deferred acceptance. It is interesting to compare and contrast this mechanism with OSP mechanisms for auctions. In binary allocation problems, such as private-value auctions with unit demand, procurement auctions with unit supply, and binary public good problems, <cit.> shows that in every OSP mechanism, each buyer chooses, roughly speaking, between a fixed option (i.e., quitting) and a “moving” option that is worsening over time (i.e., its price is increasing). In contrast, in the OSP mechanism that we construct for the men-optimal stable matching with acyclical women's preferences, each man $m$ either is assigned his (current) top choice or chooses between a fixed option (i.e., being unmatched) and a “moving” option that is improving over time: choosing any woman who prefers $m$ most among all remaining (yet-to-be-matched) men. This novel construction has come to be utilized by various OSP implementations, such as all of those that are surveyed in the end of the introduction.
Bridging the negative and positive results via an exact, succinct characterization of how aligned the preference profile of the proposed-to side needs to be in order to support an obviously strategy-proof implementation remains an open question. A comparison of the respective preference profiles used for the positive result of <ref> and the negative result of <ref> (in <ref>) suggests that such a succinct “maximal domain” characterization may be delicate, and obtaining it may be challenging.[While a technical challenge, we find it unlikely that resolving this problem will yield interesting economic insights.]
Interestingly, while deferred acceptance is (even weakly group) strategy-proof and has an ascending flavor similar to that of ascending unit-demand auctions or clock auctions (which are all obviously strategy-proof), deferred acceptance is in fact not OSP-implementable.
It seems that the fact that stability is a two-sided objective (concerning the preferences of agents on both sides of the market), in contrast with maximizing efficiency or welfare for one side, increases the difficulty of employing strategic reasoning over stable mechanisms. In this context, it is worth noting a line of work <cit.> that highlights a similar message in terms of complexity rather than strategic reasoning, by showing that the communication complexity of finding (or even verifying) an approximately stable matching is significantly higher than the communication complexity of approximate welfare maximization for one of the sides of the market <cit.>.
Indeed, in more than one way, stability is not an “obvious" objective.
§ MECHANISMS WITH RESTRICTED DOMAINS
In this restricted-domain-mechanisms, we explicitly adapt the definitions in <ref> to a restricted domain of preferences, as used in the proof of <ref>. The differences from the definitions in <ref> are marked with an underscore. We emphasize that these definitions, like those in <ref>, are also a special case of the definitions in <cit.>. For every $m\in M$, fix a subset $\prefs_m\subseteq\prefs(W)$. Furthermore, define $\prefs\eqdef\bigtimes_{m\in M}\prefs_m$.
A (one-side-querying extensive-form) matching mechanism for $M$ over $W$ with respect to $\prefs$ consists of:
* A rooted tree $T$.
* A map $X:L(T)\rightarrow\matchings(M,W)$ from the leaves of $T$ to matchings between $M$ and $W$.
* A map $Q:V(T)\setminus L(T)\rightarrow M$, from internal nodes of $T$ to $M$.
* A map $A:E(T)\rightarrow 2^{\prefs(W)}$, from edges of $T$ to predicates over $\prefs(W)$, such that all of the following hold:
* Each such predicate must match at least one element in $\prefs(W)$.
* The predicates corresponding to edges outgoing from the same node are disjoint.
* The disjunction (i.e., set union) of all predicates corresponding to edges outgoing from a node $n$ equals the predicate corresponding to the last edge outgoing from a node labeled $Q(n)$ along the path from the root to $n$, or to the predicate matching all elements of $\uline{\prefs_{Q(n)}}$ if no such edge exists.[In particular, this implies that the predicates corresponding to edges outgoing from a node $n$ are predicates over $\uline{\prefs_{Q(n)}}$.]
A preference profile $\bar{p}\in\uline{\prefs}$ is said to pass through a node $n \in V(T)$ if, for each edge $e$ along the path from the root of $T$ to $n$, it is the case that $p_{Q(n')}\in A(e)$, where $n'$ is the source node of $e$. That is, the nodes through which $\bar{p}$ passes are the nodes of the path that starts from the root of $T$ and follows, from each internal node $n'$ that it reaches, the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Given an extensive-form matching mechanism $\impl$ with respect to $\prefs$, we denote by $C^{\impl}$, called the matching rule implemented by $\impl$, the (one-side-querying) matching rule mapping a preference profile $\bar{p}\in\uline{\prefs}$ to the matching $X(n)$, where $n$ is the unique leaf through which $\bar{p}$ passes. Equivalently, $n$ is the node in $T$ obtained by traversing $T$ from its root, and from each internal node $n'$ that is reached, following the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Two preference lists $p,p'\in\prefs(W)$ are said to diverge at a node $n\in V(T)$ if there exist two distinct edges $e,e'$ outgoing from $n$ such that $p\in A(e)$ and $p'\in A(e')$.[In particular, this implies that $p,p'\in\uline{\prefs_{Q(n)}}$.]
Let $\impl$ be an extensive-form matching mechanism with respect to $\prefs$ .
* $\impl$ is said to be obviously strategy-proof (OSP) for a man $m\in M$ if for every node $n$ with $Q(n)=m$ and for every
$\bar{p}=(p_{m'})_{m'\in M}\in\uline{\prefs}$ and $\bar{p}'=(p'_{m'})_{m'\in M}\in\uline{\prefs}$ that both pass through $n$ such that $p_m$ and $p'_m$ diverge at $n$, it is the case that $C^{\impl}_m(\bar{p}) \succeq_m C^{\impl}_m(\bar{p}')$ according to $p_m$. In other words, the worst possible outcome for $m$ when acting truthfully (i.e., according to $p_m$) at $n$ is no worse than the best possible outcome for $m$ when misrepresenting his preference list to be $p'_m$ at $n$.
* $\impl$ is said to be obviously strategy-proof (OSP) if it is obviously strategy-proof for every man $m\in M$.
§ A "LESS CYCLICAL" NON-OSP-IMPLEMENTABLE EXAMPLE
In this app:not-osp-less-cyclical, we give an additional example of a preference profile $\bar{q}\in\prefs(M)^W$,
for three women over three men, for which no $\bar{q}$-stable matching rule is OSP-implementable.
This preference profile could be described, in some sense, as “less cyclical” than the one used above to drive the proof of the results of <ref>.
(Indeed, as noted above, this non-OSP-implementable preference profile is obtained by taking the OSP-implementable preference profile from <ref> and arguably making it “more aligned” by modifying the preference list of woman $x$ to equal that of woman $y$.)
While, similarly to the proof of <ref>, we show the impossibility of OSP-implementation of this example via a pruning argument, the reasoning in this argument is more involved than in the one in the proof given for <ref> in <ref>.
For $|M|=|W|=3$, no OSP mechanism implements a $\bar{q}$-stable (one-side-querying) matching rule, for the following preference profile $\bar{q}\in\prefs(M)^W$ for $M$ over $W$ (where each woman prefers being matched to any man over being unmatched):
\[
\begin{array}{ccccc}
a & \succ_x & c & \succ_x& b \\
a & \succ_y & c & \succ_y & b \\
b & \succ_z & a & \succ_z & c.
\end{array}
\]
The proof starts similarly to that of <ref>.
Let $M=\{a,b,c\}$ and $W=\{x,y,z\}$. Let $\bar{q}$ be the above preference profile, and assume for contradiction that an OSP mechanism $\impl$ that implements a $\bar{q}$-stable matching rule $C^{\impl}$ exists. Therefore, $C^{\impl}$ is strategy-proof, and so, by <ref>, $C^{\impl}=C^{\bar{q}}$.
In order to reach a contradiction we dramatically restrict the domain of preferences of all men, however in this proof to a slightly richer domain than in the proof of <ref>. We define:
$p^1_a \eqdef z \succ x \succ y$ $p^1_b \eqdef y \succ z \succ x$ $p^1_c \eqdef x \succ y \succ z$
$p^2_a \eqdef z \succ y \succ x$ $p^2_b \eqdef x \succ z \succ y$ $p^2_c \eqdef y \succ x \succ z$,
$p^3_b \eqdef x \succ y \succ z $
and set $\prefs_a\eqdef\{p^1_a,p^2_a\}$, $\prefs_b\eqdef\{p^1_b,p^2_b,p^3_b\}$, and $\prefs_c\eqdef\{p^1_c,p^2_c\}$.
Following a proof technique in <cit.>, we prune (see the proof of <ref> for more details) the tree of $\impl$ according to $\prefs_a,\prefs_b,\prefs_c$, to obtain a mechanism that implements $C^{\bar{q}}$ where the preference list of each man $m\in M$ is a priori restricted to be in $\prefs_m$.
By a proposition in <cit.>, since the original mechanism $\impl$ is OSP, so is the pruned mechanism as well.
Let $n$ be the earliest (i.e., closest to the root) node in the pruned tree that has more than one outgoing edge (such a node clearly exists, since $C^{\impl}=C^{\bar{q}}$ is not constant over $\prefs_a\times\prefs_b\times\prefs_c$). While the lack of symmetry of $\bar{q}$ does requires a slightly longer argument compared to the proof of <ref> to complete this proof (reasoning by cases according to $Q(n)$ below), what makes the reasoning in this argument more involved (see the reasoning in the case $Q(n)=b$ below) than in its counterpart in the proof of <ref> is the fact that we have left possible three preference lists for man $b$.[To our knowledge, the first instance of an impossibility-by-pruning proof with more than two possible preference lists/types for any of the agents is in an impossibility result for OSP-implementation of combinatorial auctions in <cit.>. While that paper is much newer than any other result in our paper, the first draft of that proof predated the proof given in this app:not-osp-less-cyclical.] We conclude the proof by reasoning by cases according to the identity of $Q(n)$, in each case obtaining a contradiction by showing that the pruned tree is in fact not OSP.
By definition of pruning, it must be the case that $n$ has two outgoing edges, one labeled $p^1_a$, and the other labeled $p^2_a$.
In this case, for $p_a=p^1_a$ (the “true preferences”), $p_b=p^1_b$, and $p_c=p^2_c$, we have that $C_a^{\impl}(\bar{p})=C_a^{\bar{q}}(\bar{p})=x$, yet for $p'_a=p^2_a$ (a “possible manipulation”), $p'_b=p^2_b$, and $p'_c=p^2_c$, we have that $C_a^{\impl}(\bar{p}')=C_a^{\bar{q}}(\bar{p}')=z$, even though $C_a^{\impl}(\bar{p}')=z\succ_a x = C_a^{\impl}(\bar{p})$ according to $p_a$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and $p_a$ and $p_a'$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
By definition of pruning, it must be the case that $n$ has two outgoing edges, one labeled $p^1_c$, and the other labeled $p^2_c$.
In this case, for $p_c=p^1_c$ (the “true preferences”), $p_a=p^1_a$, and $p_b=p^2_b$, we have that $C_c^{\impl}(\bar{p})=C_c^{\bar{q}}(\bar{p})=y$, yet for $p'_c=p^2_c$ (a “possible manipulation”), $p'_a=p^2_a$, and $p'_b=p^1_b$, we have that $C_c^{\impl}(\bar{p}')=C_c^{\bar{q}}(\bar{p}')=x$, even though $C_c^{\impl}(\bar{p}')=x\succ_c y = C_c^{\impl}(\bar{p})$ according to $p_c$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and $p_c$ and $p_c'$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
By definition of pruning, it must be the case that $n$ has at least two outgoing edges, and therefore has at least one edge labeled by a singleton preference list $p^i_b$. We prove this case by reasoning by subcases according to the value of $i$.
In this case, for $p_b=p^i_b=p^1_b$ (the “true preferences”), $p_a=p^1_a$, and $p_c=p^2_c$, we have that $C_b^{\impl}(\bar{p})=C_b^{\bar{q}}(\bar{p})=z$, yet for $p'_b=p^3_b$ (a “possible manipulation”), $p'_a=p^1_a$, and $p'_c=p^1_c$, we have that $C_b^{\impl}(\bar{p}')=C_b^{\bar{q}}(\bar{p}')=y$, even though $C_b^{\impl}(\bar{p}')=y\succ_b z = C_b^{\impl}(\bar{p})$ according to $p_b$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and since $i=1$ we have that $p_b=p^i_b$ and $p_b'\ne p^i_b$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
In this case, for $p_b=p^i_b=p^2_b$ (the “true preferences”), $p_a=p^2_a$, and $p_c=p^1_c$, we have that $C_b^{\impl}(\bar{p})=C_b^{\bar{q}}(\bar{p})=z$, yet for $p'_b=p^3_b$ (a “possible manipulation”), $p'_a=p^1_a$, and $p'_c=p^2_c$, we have that $C_b^{\impl}(\bar{p}')=C_b^{\bar{q}}(\bar{p}')=x$, even though $C_b^{\impl}(\bar{p}')=x\succ_b z = C_b^{\impl}(\bar{p})$ according to $p_b$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and since $i=2$ we have that $p_b=p^i_b$ and $p_b'\ne p^i_b$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
In this case, for $p_b=p^i_b=p^3_b$ (the “true preferences”), $p_a=p^1_a$, and $p_c=p^1_c$, we have that $C_b^{\impl}(\bar{p})=C_b^{\bar{q}}(\bar{p})=y$, yet for $p'_b=p^2_b$ (a “possible manipulation”), $p'_a=p^1_a$, and $p'_c=p^2_c$, we have that $C_b^{\impl}(\bar{p}')=C_b^{\bar{q}}(\bar{p}')=x$, even though $C_b^{\impl}(\bar{p}')=x\succ_b y = C_b^{\impl}(\bar{p})$ according to $p_b$ (by definition of $n$, both $\bar{p}$ and $\bar{p}'$ pass through $n$, and since $i=3$ we have that $p_b=p^i_b$ and $p_b'\ne p^i_b$ diverge at $n$), and so the mechanism of the pruned tree indeed is not OSP — a contradiction.
§ TWO-SIDES-QUERYING MECHANISMS
In this two-sided-mechanisms, we explicitly adapt the definitions in <ref> for two-sides-querying mechanisms, where the (strategic) participants include not only the men but also the women, as in <ref>. The differences from the definitions in <ref> are marked with an underscore. We emphasize that these definitions, like those in <ref>, are also a special case of the definitions in <cit.>. Define $\prefs\eqdef\prefs(W)^M\times\prefs(M)^W$. For every two-sided preference profile $\bar{r}=(\bar{p},\bar{q})\in\prefs$, we write $r_m=p_m$ for every $m\in M$ and $r_w=q_w$ for every $w\in W$.
A two-sides-querying (extensive-form) matching mechanism for $M$ and $W$ consists of:
* A rooted tree $T$.
* A map $X:L(T)\rightarrow\matchings(M,W)$ from the leaves of $T$ to matchings between $M$ and $W$.
* A map $Q:V(T)\setminus L(T)\rightarrow M\uline{\,\cup\,W}$, from internal nodes of $T$ to participants $M\uline{\,\cup\,W}$.
* A map $A:E(T)\rightarrow 2^{\prefs(W)}\uline{\,\cup\,2^{\prefs(M)}}$, from edges of $T$ to predicates over $\prefs(W)$ or over $\prefs(M)$, such that all of the following hold:
* Each such predicate must match at least one element in $\prefs(W)$ if $Q(n)\in M$ and at least one element in $\prefs(M)$ if $Q(n)\in W$.
* The predicates corresponding to edges outgoing from the same node are disjoint.
* The disjunction (i.e., set union) of all predicates corresponding to edges outgoing from a node $n$ equals the predicate corresponding to the last edge outgoing from a node labeled $Q(n)$ along the path from the root to $n$, or, if no such edge exists, to the predicate matching all elements of $\prefs(W)$ if $Q(n)\in M$ and all elements of $\prefs(M)$ if $Q(n)\in W$.[In particular, this implies that the predicates corresponding to edges outgoing from a node $n$ are predicates over $\prefs(W)$ if $Q(n)\in M$ and over $\prefs(M)$ if $Q(n)\in W$.]
A two-sides-querying preference profile $\bar{r}\in\uline{\prefs}$ is said to pass through a node $n \in V(T)$ if, for each edge $e$ along the path from the root of $T$ to $n$, it is the case that $r_{Q(n')}\in A(e)$, where $n'$ is the source node of $e$. That is, the nodes through which $\bar{r}$ passes are the nodes of the path that starts from the root of $T$ and follows, from each internal node $n'$ that it reaches, the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Given a two-sides-querying extensive-form matching mechanism $\impl$, we denote by $C^{\impl}$, called the two-sides-querying matching rule implemented by $\impl$, the two-sides-querying matching rule mapping a two-sides-querying preference profile $\bar{r}\in\uline{\prefs}$ to the matching $X(n)$, where $n$ is the unique leaf through which $\bar{r}$ passes. Equivalently, $n$ is the node in $T$ obtained by traversing $T$ from its root, and from each internal node $n'$ that is reached, following the unique outgoing edge whose predicate matches the preference list of $Q(n')$.
Two preference lists $r,r'\in\prefs(W)\uline{\,\cup\,\prefs(M)}$ are said to diverge at a node $n\in V(T)$ if there exist two distinct edges $e,e'$ outgoing from $n$ such that $r\in A(e)$ and $r'\in A(e')$.[In particular, this implies that $r,r'\in\prefs(W)$ if $Q(n)\in M$ and that $r,r'\in\prefs(M)$ if $Q(n)\in W$.]
Let $\impl$ be a two-sides-querying extensive-form matching mechanism.
$\impl$ is said to be obviously strategy-proof (OSP) for a participant $a\in M\uline{\,\cup\,W}$ if for every node $n$ with $Q(n)=a$ and for every
$\bar{r},\bar{r}'\in\uline{\prefs}$ that both pass through $n$ such that $p_a$ and $p'_a$ diverge at $n$, it is the case that $C^{\impl}_a(\bar{r}) \succeq_a C^{\impl}_a(\bar{r}')$ according to $r_a$. In other words, the worst possible outcome for $a$ when acting truthfully (i.e., according to $r_a$) at $n$ is no worse than the best possible outcome for $a$ when misrepresenting his or her preference list to be $r'_a$ at $n$.
A two-sides-querying matching rule $C:\uline{\prefs}\rightarrow\matchings(M,W)$ is said to be OSP-implementable for a set of participants $A\subseteq M\uline{\,\cup\,W}$ if $C=C^{\impl}$ for some two-sides-querying matching mechanism $\impl$ that is OSP for (every participant in) $A$.
§ FROM ONE STRATEGIC SIDE TO TWO STRATEGIC SIDES
The next one-vs-two allows us to obtain results in the two-strategic-sides model from the results obtained in the one-strategic-side model (as alluded to in the discussion opening <ref>, the converse is not as immediate, e.g., neither <ref> nor <ref> is an immediate corollary of results that are naturally stated for two-sides-querying mechanisms/matching rules). Indeed, <ref> both follow via this one-vs-two from the respective analogous results for one-side-querying mechanisms/matching rules.
For every $M'\subseteq M$, there exists a stable two-sides-querying matching mechanism that is OSP for $M'$ if and only if for every $\bar{q}\in\prefs(W)^M$ there exists a $\bar{q}$-stable one-side-querying matching mechanism that is OSP for $M'$.
$\Rightarrow$: Assume that there exists a stable two-sides-querying matching mechanism $\impl$ that is OSP for $M'$, and let $\bar{q}\in\prefs(W)^M$. We prune (see the proof of <ref> for an explanation of pruning) the tree of $\impl$ such that the women's preference profile is fixed to be $\bar{q}$. The resulting (pruned) mechanism is a one-side-querying matching mechanism that is $\bar{q}$-stable and (by the same proposition in <cit.> that is used in <ref>) OSP for $M'$, as required.
$\Leftarrow$: Assume that for every $\bar{q}\in\prefs(M)^W$ there exists a $\bar{q}$-stable one-side-querying matching mechanism $\impl^{\bar{q}}$ that is OSP for $M'$. We construct a stable two-sides-querying matching mechanism $\impl$ as follows: first ask all women, in some order, for all of their preference lists; the leaves of the tree so far are thus in one-to-one correspondence with preference profiles $\bar{q}\in\prefs(M)^W$ that pass through them. Next, at each “interim leaf” $n^{\bar{q}}$ corresponding to a preference profile $\bar{q}\in\prefs(M)^W$ (that passes through it), construct a subtree that is identical to the tree of $\impl^{\bar{q}}$, with $n^{\bar{q}}$ as its root. It is straightforward to verify that the fact that each $\impl^{\bar{q}}$ is $\bar{q}$-stable and OSP for $M'$ implies that $\impl$ is stable and OSP for $M'$.
|
1511.00018
|
We use three-dimensional magnetohydrodynamic (MHD) simulations to investigate
the quasi-equilibrium states of galactic disks regulated by star formation
feedback. We incorporate effects from massive-star feedback via time-varying
heating rates and supernova (SN) explosions. We find that the disks in our
simulations rapidly approach a quasi-steady state that satisfies vertical
dynamical equilibrium. The star formation rate (SFR) surface density
self-adjusts to provide the total momentum flux (pressure) in the vertical
direction that matches the weight of the gas. We quantify feedback efficiency
by measuring feedback yields, $\eta_c\equiv P_c/\Sigma_{\rm SFR}$ (in suitable
units), for each pressure component. The turbulent and thermal feedback yields
are the same for HD and MHD simulations, $\eta_{\rm th}\sim 1$ and $\eta_{\rm
turb}\sim 4$, consistent with the theoretical expectations. In MHD simulations,
turbulent magnetic fields are rapidly generated by turbulence, and saturate at
a level corresponding to $\eta_{\rm mag,t}\sim 1$. The presence of magnetic
fields enhances the total feedback yield and therefore reduces the SFR, since
the same vertical support can be supplied at a smaller SFR. We suggest further
numerical calibrations and observational tests in terms of the feedback yields.
§ INTRODUCTION
“What determines the SFR in galaxies?” In order to answer this long-standing,
fundamental question, a correlation between the SFR and the gas content has
been extensively explored. Among many studies since the pioneering work by
<cit.>, <cit.> presents a well-defined power-law
relationship between total gas surface density ($\Sigma$) and the SFR surface
density ($\Sigma_{\rm SFR}$) for galaxies as a whole, $\Sigma_{\rm SFR}\propto
\Sigma^{1+p}$ with $p=0.4$. This observed correlation was soon widely accepted
as the “Kennicutt-Schmidt law” (KS law) and used as a star formation recipe for
large scale galaxy formation and cosmological simulations.
The observed power-law index with $p=0.4$ of the KS law makes it tempting to
infer simple dimensional relationship, $\Sigma_{\rm SFR}=\Sigma/t_{\rm dep}$,
with the gas depletion time related to the gas free-fall time, $t_{\rm
dep}\propto t_{\rm ff}\sim(G\rho)^{-1/2}$. With a fixed gas scale height, this
relation would imply $p=0.5$, close to the observed value. Many theoretical
studies based on this simple argument have been investigated, with low star
formation efficiency per free-fall time $\epsilon_{\rm ff}\equiv t_{\rm
ff}/t_{\rm dep}\sim 1\%$ (e.g., <cit.>). On scales of molecular clouds,
the low efficiency has been attributed to the broad density probability
distribution function generated by supersonic turbulence (e.g., <cit.>
and references therein), but it is unclear whether this picture can be simply
extended to large scales ($>$0.1-1 kpc).
Moreover, recent high-resolution observations of nearby galaxies reveal more
complex correlations (<cit.>). In particular, the simple power-law
relation between $\Sigma_{\rm SFR}$ and $\Sigma$ fails at low surface density
regime ($\Sigma < 10 \; M_{\odot}\; {\rm pc}^{-2}$), where the gas is
predominantly atomic. Rather, the power-law index becomes steeper and/or varies
from one galaxy to another. The SFR shows a tighter correlation with the
stellar surface density $\Sigma_*$ or its combination with $\Sigma$ (e.g.,
The increasing complexities of the observed KS law at low-$\Sigma$ regime
implies that $\Sigma$ is not the only control parameter of the star formation.
In this article, we describe a fundamental correlation based on physical
causality between the SFR surface density and the total pressure ($P_{\rm
tot}$). We in <ref> and <ref> respectively
summarise the theory from <cit.> and simulations from
Schematic diagram of the equilibrium theory. (a) Energy/momentum
equilibrium sets the total pressure in response to the SFR, $P_{\rm
tot}\sim\eta\Sigma_{\rm SFR}$. (b) Vertical dynamical equilibrium constrains
the total momentum flux (pressure), $P_{\rm tot}\sim P_{\rm DE}$, and hence the
SFR, $\Sigma_{\rm SFR}\sim P_{\rm DE}/\eta$.
§ THEORY
The interstellar medium (ISM) disk in an equilibrium state should satisfy
vertical force balance between gravity and pressure gradients, which can be
directly derived from the momentum equation of MHD.[ Although the
resulting equation would be essentially the same with so called
hydrostatic equilibrium with an effective (total) sound speed, we
prefer to term this vertical dynamical equilibrium since the ISM is
highly dynamic and equilibrium holds only in an average sense.] In the
integrated form, the vertical dynamical equilibrium can be written as
$\mathcal{W}=\Delta P_{\rm tot}$, a balance between the total weight of gas and
the momentum flux differences across the gas disk.
The other condition to satisfy is the energy/momentum equilibrium between gain
from star formation feedback and loss in the dissipative ISM. Since cooling
and turbulence dissipation time scales are typically short compared to
dynamical time scales, continuous injection of energy and momentum is
necessary to heat gas and drive turbulence. The far-UV radiation from massive
young stars is the major heating source in the atomic ISM via the photoelectric
effect onto grains. The momentum injection from SNe is the dominant source of the
turbulence driving. Because the energy and momentum from stellar radiation
and SNe fundamentally derive from nuclear processes, SF feedback is a highly
efficient way to balance losses in the ISM.
Figure <ref> shows a schematic diagram for (a) energy/momentum equilibrium
and (b) vertical dynamical equilibrium as well as the connection between two
equilibria. Since the ISM mostly gains energy/momentum through the star
formation feedback, the total pressure (momentum flux) is set by the SFR
surface density. The total pressure determined by Figure <ref>(a) provides
the vertical support in Figure <ref>(b), which should match with the
dynamical equilibrium pressure $P_{\rm DE}$ (or the weight of the gas
The equilibrium is naturally a stable one. For example, if the SFR gets higher
than equilibrium, enhanced thermal heating and turbulence driving set the total
pressure higher than the equilibrium level. The disk becomes thermally and
dynamically hotter, vertically expanding and dispersing cold, dense clumps to
quench further star formation. On the other hand, when the SFR drops below of
the level of equilibrium, reduced feedback makes the disk thermally and
dynamically cold and susceptible to gravitational collapse, forming more stars.
Therefore, the SFR is self-regulated to satisfy both equilibria shown in
Figure <ref>.
In order to quantify this process, we define the feedback yield of any
pressure component “$c$” (= thermal, turbulent, or magnetic) in
suitable units as
\begin{equation}
\eta_c \equiv \frac{P_{c,3}}{\Sigma_{\rm SFR,-3}},
\end{equation}
where $P_{c,3}\equiv P_{c}/(10^3 k_B {\rm \; cm}^{-3}{\rm \; K})$ and
$\Sigma_{\rm SFR,-3}\equiv \Sigma_{\rm SFR}/(10^{-3} \; M_{\odot}{\rm \;
pc}^{-2} {\rm \; Myr}^{-1})$. The feedback yield can be considered as a
energy/momentum conversion efficiency of the star formation feedback, depending
on the detailed thermal and dynamical processes in the ISM. To first order, we
can simply connect the thermal and turbulent pressures with the SFR surface
density linearly. Our adopted cooling and heating formalism gives $\eta_{\rm
th}=1.2$, and the momentum feedback prescription with specific momentum
injection per star formation $(p_*/m_*)=3000 {\rm \;km/s}$ gives $\eta_{\rm
§ NUMERICAL SIMULATIONS
Utilizing the Athena code (<cit.>), we run three-dimensional
simulations for the local patch of galactic disks including optically thin
cooling, galactic differential rotation, self-gravity, vertical external
gravity, and magnetic fields. We apply a spatially-constant, time-varying
heating rate $\Gamma\propto \Sigma_{\rm SFR}$, and also momentum injection from
SNe $\propto \Sigma_{\rm SFR}$. While in some other recent simulations, the
SFR and SN rate is pre-specified and constant in time, for all of our models
the time-dependent SFR and SN rate are self-consistently set by
self-gravitating localized collapse.
Our simulations achieve a quasi-steady state after a few vertical oscillation
times (less than one orbit time). We confirm vertical dynamical equilibrium
using horizontally and temporally averaged vertical profiles that are converged
for different initial and boundary conditions as well as numerical resolutions.
For a wide range of disk conditions such that $0.1<\Sigma_{\rm SFR,-3}<10$, the
equilibrium thermal and turbulent pressures give consistent feedback yields,
$\eta_{\rm th}=1.3\Sigma_{\rm SFR,-3}^{0.14}$ and $\eta_{\rm turb}=
4.3\Sigma_{\rm SFR,-3}^{0.11}$, respectively.
In MHD simulations, the time scales to reach a quasi-steady state depend on the
initial magnetizations. For initial magnetic energy varying by two orders of
magnitude, however, saturated states converge to the same asymptote for the
turbulent magnetic fields, whose energy is about a half of the turbulent
kinetic energy. The final turbulent magnetic fields provide additional vertical
support that is directly related to the turbulent pressure and hence the SFR,
giving rise to $\eta_{\rm mag,t}\sim1$ for solar neighborhood models. Since we
fix disk parameters for MHD models, further investigation is necessary to
calibrate the detailed dependence of $\eta_{\rm mag,t}$ on $\Sigma_{\rm SFR}$.
§ CONCLUDING REMARKS
We have developed a theory for self-regulation of the SFR based on
energy/momentum equilibrium and vertical dynamical equilibrium, and confirmed
and calibrated this theory with numerical simulations. The former equilibrium
gives rise to the correlation between $\Sigma_{\rm SFR}$ and $P_{\rm tot}$
owing to the physical causality; the higher/lower SFR causes higher/lower
energy density and momentum flux ($P_{\rm tot}=\eta\Sigma_{\rm SFR}$). The
latter equilibrium sets $\Sigma_{\rm SFR}$ based on the requirement $P_{\rm
tot}=P_{\rm DE}$. Therefore, correlations between $\Sigma_{\rm SFR}$ and
galactic properties are caused by dependences embedded in $P_{\rm DE}$ such
\begin{equation}\label{eq:PDE}
P_{\rm DE}\equiv \mathcal{W} \approx \frac{\pi G \Sigma^2}{2}+
\Sigma\sigma_z(2G\rho_{\rm sd})^{1/2}
\end{equation}
where $\rho_{\rm sd}$ is the midplane density of stars plus dark matter and
$\sigma_z$ is total vertical velocity dispersion. The observed complexities in
the KS law for the low-$\Sigma$ regime naturally arise when the second term in
RHS of Equation (<ref>) dominates.
Lastly, we suggest further numerical calibrations and observational tests of
the equilibrium theory. Theorists who include any form of star formation
feedback can calibrate $\eta_c$ from a “$P_c$-$\Sigma_{\rm SFR}$” plot for
each measured pressure (turbulent, thermal, magnetic) in their simulations. It
could be interesting tu check consistency and/or to investigate differences
among simulations with different setups, including comparing global vs.
local models. Additional calibrations of other components such as radiation
pressure and cosmic ray pressure would enable comparison of the relative
importance of each component.
It is difficult to measure the total pressure directly from observations even
in solar neighborhood. However, the dynamical equilibrium pressure can be
determined from direct observables (such as those from <cit.>) with
proper assumptions.[Please refer to the following link for
a practical guide: http://www.astro.princeton.edu/$\sim$cgkim/to_observers.html] From the
“$P_{\rm DE}$-$\Sigma_{\rm SFR}$” observed plot, one can measure total
feedback yield $\eta=P_{\rm DE,3}/\Sigma_{\rm SFR,-3}$. This can be compared
with the sum of the theoretical values $\eta_c$ as a test of the
self-regulation theory, also constraining the dominant sources of SF feedback.
[Bigiel (2008)]Big08
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|
1511.00440
|
There are increasingly demands on building large-scale distributed system for training machine learning models, such as massive topic models or deep neuron networks.
The system design is quite complicated that it must takes into consideration both machine learning factors and system factors, let alone their interleaving interaction.
Instead of designing the system from scratch, we believe that it can be built on top of existing distributed data-parallel systems such as Apache Spark,
which provides distributed data abstraction and expressive APIs to simplify the parallel programming and simultaneously hides the system complexities such as task scheduling and fault tolerance.
Another advantage is that the entire training pipeline, from training data preparation to model training even model inference, can be programmed in one job and executed in the same framework.
This paper presents our recent efforts, , an efficient and scalable Collapsed Gibbs Sampling (CGS) system for Latent Dirichlet Allocation (LDA) training,
which is thought to be challenging that both data parallelism and model parallelism are required because of the Big sampling data with up to billions of documents and Big model size with up to trillions of parameters.
combines both algorithm level improvements and system level optimizations.
It first presents a novel CGS algorithm that balances the time complexity, model accuracy and parallelization flexibility.
The input corpus in is represented as a directed graph and model parameters are annotated as the corresponding vertex attributes.
The distributed training is parallelized by partitioning the graph that in each iteration it first applies CGS step for all partitions in parallel, followed by synchronizing the computed model each other.
In this way, both data parallelism and model parallelism are achieved by converting them to graph parallelism.
We revisited the tradeoff between system efficiency and model accuracy and presented approximations such as unsynchronized model, sparse model initialization and “converged” token exclusion.
is built on GraphX in Spark that provides distributed data abstraction (RDD) and expressive APIs to simplify the programming efforts and simultaneously hides the system complexities.
This enables us to implement other CGS algorithm with a few lines of code change.
To better fit in distributed data-parallel framework and achieve comparable performance with contemporary systems,
we also presented several system level optimizations to push the performance limit.
was evaluated it against web-scale corpus, and the result indicates that
can achieve about much better performance than other CGS algorithm we implemented, and simultaneously achieve better model accuracy.
The experiments also demonstrates the effectiveness of presented techniques.
The state-of-art topic modelling systems are carefully designed from scratch, built with native languages and run in a dedicated environment.
Instead, we consider this challenge in the context of building topic models on distributed data-parallel system that is widely adopted by big data computing in industry,
thereby the entire training pipeline can be programmed and executed in the same framework, including training data preparation, model training and model inference.
In this paper, we model the collapsed Gibbs sampling of topic model training as a graph computing problem, where the input corpus is represented as a directed graph that
an edge exists from word vertex to document vertex if that word is occurred in the document.
The model parameter word-topic array and doc-topic array are attached as the corresponding vertex attributes, and the current topic of an edge is annotated as edge attribute.
The Gibbs sampling process is essentially to sample a new topic for each edge and update the model parameter accordingly.
The distributed training is executed by partitioning the graph that in each iteration it first applies Gibbs sampling on partitions in parallel, followed by synchronizing the computed model parameter lastly.
In this paper, we present that aims to build a large scalability, high efficiency while still achieves accuracy.
In this paper, we presented multiple optimizations and proper tradeoffs to push the limit, such as graph partitioning to balance network communication and load balance,
a bounded asynchronous data-parallel scheme that reads staled model parameter locally during sampling and only synchronizes the global state at the end of an iteration,
and approximation by sparse model initialization and early-excluding the edges that rarely changes topic.
We also presented several low-level optimizations, such as multi-threading to fully utilize CPU resource, and customized data structure to keep low memory footprint.
We implement on GraphX in Apache Spark, and evaluate it against web-scale corpora with billions of documents, and millions of words,
that can achieve comparable scale and performance with state-of-art topic model trainer.
The experiments also demonstrates the effectiveness of presented techniques.
§ INTRODUCTION
Topic models provide a way to aggregate vocabulary from a document corpus to form latent “topics”. In particular,
Latent Dirichlet Allocation (LDA) <cit.> is one of the most popular models <cit.> that
has rich applications in web mining, from News clustering, hot topics mining, search intent mining even to user interests profiling.
Collapsed Gibbs Sampling (CGS) is the most commonly used algorithm in LDA
that samples the latent variables for a word occurrence (token) by integrating out the Dirichlet priors.
However, the training with massive corpus is challenging because of high time and space complexity.
Consider a typical web-scale application with millions of documents and words and with thousands of topics, there are billions of parameters.
No single machine can hold such Big corpus data nor Big model size, which motivates a scalable and efficient way of distributing the computation across multiple machines.
Significant progresses were achieved recently in pushing CGS algorithm into limit,
The time complexity is largely reduced from $O(K)$ (standard LDA), to $O(K_w)$ (SparseLDA <cit.>), $O(K_d)$ (AliasLDA <cit.>), and even $O(1)$ in LightLDA <cit.>.
On the other hand, different topic modelling systems were designed but with different parallelization choice.
PLDA <cit.>, AD-LDA <cit.>, and Peacock <cit.> were build on MPI <cit.>/OpenMP <cit.> primitives <cit.>
that provides low-level communication library and APIs to express parallelism.
Yahoo!LDA <cit.> introduces a parameter server abstraction <cit.> and let
each machine put its latest update to server and query the server to retrieve recent updates from other machines.
PLDA+ <cit.> makes use of data placement and pipeline processing to greatly reduce communication time.
LightLDA <cit.> was build on Petuum <cit.> that is also a parameter server but with stale synchronous parallel (SSP) consistency model <cit.>.
However, an efficient and scalable solution should combine the innovations from both algorithm side and system side.
Contemporary topic modelling systems either focus on algorithm side <cit.> that has different sampling methods,
or focus on system side <cit.>.
These separation efforts makes it hard to port new algorithms on old systems.
Up to now, there is still no a general system that support all different CGS algorithms, let alone a system that supports different models.
Recently, LightLDA <cit.> is the first trial that integrates both algorithm improvement and system optimization.
However, it conflates the learning algorithm and system logic together, which makes it hard to extend.
Contemporary systems are considered as that they are almost designed from scratch, programmed in native language and run in a dedicated environment.
They repeatedly address the same system challenges, lose generality due to deep customization and are hardly to debug and extend by couple learning and system together.
In this paper, we consider an alternative ( that bets on existing distributed data-parallel systems <cit.> and do not need to consider the system complexities such as scheduling, communication and fault tolerance. Another benefit is that entire learning pipeline, from feature engineering to model training, can be programmed in the same framework,
considering that data-parallel system has already been widely adopted in industry for feature engineering from Big raw data.
Hadoop Mahout <cit.> and Spark <cit.> MLlib <cit.> have validated such generalized approach,
such as SparkLDA <cit.> and the official one in MLlib <cit.>.
However, they are considered to be performed and scaled poorly that may be 10100X slower than customized systems.
In this paper, we address the performance concern and try to prove that generalized approach can still achieve comparable or even better efficiency and scalability with customized systems.
reflects our latest efforts that builds an efficient and scalable CGS system on distributed data-parallel platform.
combines both algorithm level improvements and system level optimizations.
It first presents a novel CGS algorithm that balances the time complexity, model accuracy and parallelization flexibility.
The input corpus in is represented as a directed graph and model parameters are annotated as the corresponding vertex attributes.
The distributed training is parallelized by partitioning the graph that in each iteration it first applies CGS step for all partitions in parallel, followed by synchronizing the computed model each other.
In this way, both data parallelism and model parallelism are achieved by converting them to graph parallelism.
We further revisited the tradeoff between system efficiency and model accuracy and presented approximations such as unsynchronized model, sparse model initialization and “converged” token exclusion.
To better fit in distributed data-parallel framework,
we also presented several system level optimizations to push the performance limit.
is built on GraphX in Spark that provides distributed data abstraction (RDD) and expressive APIs to simplify the programming efforts and simultaneously hides the system complexities.
Such generalization approach enables us to implement other CGS algorithms with only a few lines of code change,
specifically, SparseLDA and LightLDA are implemented as evaluation baseline.
The comparison with them against NYTtimes and one real web-scale dataset with about 3 billions tokens shows
that can achieve about 2X-6X better performance than LightLDA and about 14X speedup than SparseLDA, and simultaneously achieve better model accuracy.
The effectiveness of presented techniques is also evaluated.
And we also conduct scalability experiment against another bigger Bing web chunk data with about 50 billions tokens and run it a multi-tenancy production environment,
the result indicates has good scalability and industry-strength quality.
§ BACKGROUND AND RELATED WORK
This section describes the necessary background, including LDA and the corresponding Collapsed Gibbs Sampling (CGS) training algorithm, as well as the description of
state-of-art distributed data-parallel system, Apache Spark and its library GraphX where is built.
§.§ LDA
In LDA, each of $D$ documents is modeled as a
mixture over $K$ latent topics, each being a multi-nomial distribution over $W$ vocabulary
words. In order to generate a new document $d$, LDA first draw a mixing proportion $\theta_{k|d}$ from
a Dirichlet with parameter $\alpha$. For the $w$th word in the document, a topic assignment $z_{wd}$ is
drawn with topic $k$ chosen with probability $\theta_{k|d}$. Then word $x_{dw}$ is drawn from the $z_{dw}$th topic,
with $x_{dw}$ taking on value $w$ with probability $\phi_{w|k}$, where $\phi_{w|k}$ is drawn from a Dirichlet prior
with parameter $\beta$. Finally, the generative process is below:
\begin{equation}
\theta_{k|d} \sim Dir(\alpha), \phi_{w|k} \sim Dir(\beta), z_{dw} \sim \theta_{k|d}, x_{dw} \sim \phi_{w|z_{dw}}
\label{f:1}
\end{equation}
where $Dir(\alpha)$ represents the Dirichlet distribution.
§.§ Collapsed Gibbs Sampling Algorithm
Given the observed words $x = {x_{dw}}$, the task of Bayesian inference for LDA is to compute the
posterior distribution over the latent topic assignments $z = {z_{dw}}$, the mixing proportions $\theta_{k|d}$ and the topics $\phi_{w|k}$.
Approximate inference for LDA can be performed either using variational methods <cit.> or Markov chain Monte Carlo (MCMC) methods <cit.>.
In the MCMC context, the usual procedure is to integrate out the mixtures $\theta$ and topics $\phi$ in Formula <ref> and just sample the latent variables $z$, which exhibits fast convergence.
This procedure is called Collapsed Gibbs Sampling (CGS), where the conditional
probability of $z_{dw}$ is computed as follows:
\begin{equation}
p(z_{dw}=k|z^{\neg{dw}},x_{dw},\alpha,\beta) \propto \frac{N_{x_{w|k}}^{\neg{dw}} + \beta}{W\beta + N_k^{\neg{dw}}} (N_{k|d}^{\neg{dw}} + \alpha)
\label{f:2}
\end{equation}
where the superscript $\neg{dw}$ means the corresponding topic sampled last time is excluded in the count
values, $N_{k|d}$ denotes the number of tokens in document $d$ assigned to topic $k$, $N_{w|k}$ denotes
the number of tokens with word $w$ assigned to topic $k$, and $N_k = \sum_wn_{w|k}$.
Note that $p_k \geq 0$ is unnormalized.
Implementations of topic models typically use symmetric Dirichlet priors with fixed concentration parameters.
However, Wallach, etc. <cit.> found that an asymmetric Dirichlet prior
over the document-topic distributions ($\alpha$) has substantial advantages over a symmetric prior,
and introduce another hyper-parameter $\alpha'$ to approximate that asymmetric prior as $K\alpha \frac{N_k + \frac{\alpha'}{K}}{\sum_kN_k+\alpha'}$: by that CGS sampling formula can be rewritten as:
\begin{equation}
p(z_{dw}=k|...) \propto \frac{N_{w|k}^{\neg{dw}} + \beta}{W\beta + N_k^{\neg{dw}}} (N_{k|d}^{\neg{dw}} + K\alpha \frac{N_k + \frac{\alpha'}{K}}{\sum_kN_k+\alpha'})
\label{f:3}
\end{equation}
Algorithm <ref> describes the standard CGS algorithm [We skipped the for-each-occurrence loop between line 4 and line 5.]. Note that the processing order of for-loops in line 3 and line 4 can be interchanged. There are two steps in CGS sampling, first is step that computes the sampling probability of each topic $k$ ($K$-dimensional discrete distribution in total),
followed by step that draws a sample $z$ of topic such that $P_r(z = t) \sim p_t$.
The time complexity is $O(D*W*K)$.
Similarly, the space complexity is also extremely high that the storage of input corpus would be $O(D*W)$,
and with $O(W*K)$ for word-topic matrix, $O(D*K)$ for document-topic matrix.
each epoch $e$
each document $d$
each word $w$
each topic $k$
p($k$) = $\frac{N_{w|k}^{\neg{dw}} + \beta}{W\beta + N_k^{\neg{dw}}} (N_{k|d}^{\neg{dw}} + \alpha \frac{N_k + \frac{\alpha'}{K}}{\sum_kN_k+\alpha'})$
$t$ = TopicSampling(p($k$))
update $N_{t|d}$, $N_{t|k}$ and $N_k$ accordingly
Serial standard CGS algorithm.
Review of multinomial sampling
There are two steps in CGS sampling, first is step that computes the sampling probability of each topic $k$ ($K$-dimensional discrete distribution in total),
followed by step that draws a sample $z$ of topic such that $P_r(z = t) \sim p_t$. There are four
four different sampling algorithms can be applied to draw a sample $z$ of topic such that $P_r(z = t) \sim p_t$.
* LSearch: Linear search on $p$. Constructing: Compute the normalization constant $C_K = \sum_kp_k$. Sampling: First generate $u = uniform(c_K)$, a uniform random number in $[0, c_K)$, and perform a linear search to find $z = min{t: (\sum_{s \leq k}p_s) > u}$.
* BSearch: Binary search on c = cumsum(p). Constructing: Compute $c = cumsum(p)$ such that $c_t = \sum_{s\leq t}p_s$. Sampling: First generate the cumulated sum $u = uniform(c_K)$ and perform a binary search on
$c$ to find $z = min{t:c_t} > u$.
* Alias method. Constructing: Construct an Alias table <cit.> for $p$, which contains two arrays of length $K$: and . See <cit.> for a linear time construction scheme. Sampling: First generate $u = uniform(K)$, $j = \lfloor u \rfloor$, and
\begin{equation}
z = \left\{
\begin{array}{lcl}
{j + 1} &\text{if} &(u - j) \leq prob[j+1] \\
{alias[j+1]} &\text{otherwise}
\end{array}
\right.
\end{equation}
* F+ Tree. F+ tree is first introduced for weighted sampling without replacement in <cit.>, and is used for CGS sampling in F+Nomad LDA <cit.>. Each leaf node corresponds to a dimension t and stores $p_k$ as its value, and each internal node stores the sum of the values of all of its leaf descendants. The corresponding updating and sampling algorithms can be referred to F+Nomad LDA paper.
Table <ref> lists the comparison of the time/space requirements of each of the above sampling methods.
LSearch BSearch Alias Table F+ Tree
Data structure array cdf dense array tree
Space $O(K)$ $O(K)$ $O(K)$ $O(K)$
Construct time $O(K)$ $O(K)$ $O(K)$ $O(K)$
Construct space $O(1)$ $O(1)$ $O(K)$ $O(1)$
Sample time $O(K)$ $O(log(K))$ $O(1)$ $O(log(K))$
Update time $O(1)$ $O(K)$ $O(K)$ $O(log(K))$
Comparison of samplers.
§.§ Spark
Spark is a fast and general engine for large-scale data processing, which was open-sourced as a top-level Apache project at 2013.
It grew fast with a mature community, and is becoming the de-facto big computing engine that has been widely adopted by industry.
Two types of applications that current computing frameworks handle inefficiently can be benefited from RDDs:
iterative algorithms and interactive data mining tools. In both cases, keeping data in memory can improve performance by an order of magnitude.
With the characteristics with that can run programs up to 100x faster than Hadoop MapReduce in memory, or 10x faster on disk,
that supports application written in Java, Scala, Python and R,
that combines SQL and DataFrames, Spark streaming, GraphX for graph computing and MLlib for machine learning,
and that includes Hadoop, Mesos, standalone or the cloud,
Spark grew fast with a mature community, and is becoming the de-facto big computing engine that has been widely adopted by industry.
RDD abstraction.
Spark improves the distributed data-parallel systems such as MapReduce, Hadoop, Dryad by providing a Resilient Distributed Datasets (RDDs) abstraction,
which is an efficient, general-purpose and fault-tolerant abstraction for sharing data in cluster applications.
Essentially, RDD represents an immutable, partitioned collection of elements that can be operated on in parallel.
The RDD element can be any type, from primitive types to complex classes.
RDDs is immutable, and is offered with APIs that support coarse-grained transformations that transform RDDs and actions that return result.
lets them recover data efficiently using lineage that tracks how to re-compute lost data from previous RDDs.
Users can explicitly cache an RDD in memory or disk across machines
and reuse it in successive computing.
In addition, Spark supports two restricted types of shared variables, accumulator that workers can only “add” to using an associative operation and only the driver can read,
and broadcast variable that create a “broadcast variable” object that wraps the value and ensures that it is only copied to each worker once.
Spark core itself is written in Scala language, and each RDD is represented by a Scala object.
High-level languages such as Java, Python and R are also supported in SparkR as a light-weight frontend.
Therefore, developer can easily write learning applications like single-box environments.
To use Spark, developers write a driver program that implements the high-level control flow of their application
and launches various RDD operations in parallel. These operations are invoked by passing a function to apply on
a RDD. Driver is responsible for scheduling the tasks and coordinating the worker execution.
Machine learning.
It is unsurprised that there are already progresses on learning in Spark.
MLlib is SparkˇŻs machine learning (ML) library. Its goal is to simplify practical machine learning programming, by provides high-level representations on top of RDDs.
MLlib consists of common learning algorithms and utilities, including classification, regression, clustering, collaborative filtering, dimensionality reduction,
as well as lower-level optimization primitives and higher-level pipeline APIs.
It is noteworthy that there already are two LDA implementations, including expectation-maximization (EM) <cit.> on the likelihood function and variational inference based online training.
Besides data/model parallelism, graph parallelism is also inherent to learning algorithm and is widely used to parallelize the training process.
GraphX extends the Spark RDD by introducing a new abstraction: a directed graph with properties attached to each vertex and edge.
To support graph computation, GraphX exposes a set of fundamental operators (e.g., subgraph, joinVertices, and aggregateMessages) as well as an optimized variant of the Pregel API.
In addition, GraphX includes a growing collection of graph algorithms and builders to simplify graph analytics tasks.
§ DESIGN
In this section, we first describe the serial CGS algorithm in that has different formula decomposition,
then followed by the comparison with existing approaches.
§.§ Sampling approach.
Another dimension in design space is the choice of decomposition of Formula <ref>, which costs large proportion of execution time in one iteration.
Different formula decompositions have different sampling characteristics.
There are three major considerations in : 1). whether the decomposed part is loop invariant or with negligible change?
For example, $\frac{\alpha_k*\beta}{N_k+W\beta}$ is loop invariant while $N_{k|d} * N_{w|k}$ changes significantly.
2). whether the decomposed part is sparse with respect to topic $k$? Sparse part has less computing complexity as well as less memory consumption.
For example, $N_{w|k}*\alpha$ is sparse since $N_{w|k}$ is sparse, and the computing complexity is $O(K_w)$.
3). whether or not the approximation is permitted in computing topic probability that does not compromise sampling accuracy?
It is reasonable to the approximation on formula parts with less value proportion would have less deviation errors in total.
For instance, $N_{k|d} * N_{w|k}$ has largest value, while $\alpha_k * \beta$ is the smallest.
It is thus unnecessary to compute the less important part every time, including $N_{k|d}*\beta$ and $N_{w|k}*\alpha$.
chooses a different decomposition with $\frac{\alpha_k*\beta}{N_k+W\beta} + \frac{N_{wk}*\alpha_k}{N_k+W\beta} + \frac{N_{kd}*(N_{wk}+\beta)}{N_k+W\beta}$, which has the following benefits compared with other approaches:
* $\frac{\alpha_k*\beta}{N_k+W\beta}$ is only computed once and reused afterwards in an iteration.
And an alias table <cit.>, $gTable$, is created accordingly, thus $O(1)$ sampling complexity is achieved.
Approximation happens here since $N_k$ changes for each sampling, that is why SparseLDA adopts linear search based sampler which has $O(K)$ sampling complexity.
* $\frac{N_{wk}*\alpha_k}{N_k+W\beta}$ is also approximated that it is pre-computed and reused for the same word ($w$).
Similarly, the alias table ($wSparse$) is created accordingly.
The lifecycle of this alias table in is reduced with word-by-word process order, that all tokens of the same word are grouped and sampled together.
Recall that the corresponding topic ($k$) sampled last time should be excluded in the count values ($N_{w|k}$) for current sampling, i.e, $N_{w|k}$ should be subtracted by one for that $k$.
However, such subtraction is skipped since there is no information on which topic should be subtracted during the pre-computing.
We apply remedy by resampling with a probability of $\frac{1}{N_{w|k}}$ if the sampled topic is equal to the topic sampled last time. This is especially useful when $N_{w|k}$ is small that is close to 1.
* Only $\frac{N_{kd}*(N_{wk}+\beta)}{N_k+W\beta}$ is computed for each token with $O(K_d)$ time complexity.
And a cumulative distribution function (CDF) is created and the corresponding sampling complexity is $O(logK_d)$.
It is noteworthy that it is only computed once for different occurrences of the same word in the same document.
Thereby similarly, this did not subtract 1 for $N_{k|d}$ and $N_{w|k}$, thus the $1*\beta$ should be excluded from $N_{k|d}*\beta$ and $N_{k|d} + N_{w|k} - 1$ should be excluded from $N_{k|d}*N_{w|k}$.
Therefore, resampling is applied for remedy with probability of $\frac{1}{N_{k|d}} + \frac{N_{k|d} + N_{w|k} - 1}{N_{k|d}*N_{w|k}}$ if the sampled topic is equal to the topic sampled last time
[We actually compute this decomposed part with 1 is subtracted if only one occurrence for the same (word, document) pair.].
Hybrid decomposition.
$\frac{\alpha*\beta}{N_k+W\beta} + \frac{N_{wk}*\alpha}{N_k+W\beta} + \frac{N_{kd}*(N_{wk}+\beta)}{N_k+W\beta}$ is better than
$\frac{\alpha*\beta}{N_k+W\beta} + \frac{N_{kd}*\beta}{N_k+W\beta} + N_{wk}(\frac{N_{kd}+\alpha}{N_k+W\beta})$ (used in SparseLDA <cit.>),
since decomposed part $N_{wk}(\frac{N_{kd}+\alpha}{N_k+W\beta})$ in latter one has complexity of $O(K_w)$, which is worse than $O(K_d)$ in former one,
consider that word-topic array is generally more dense than document-topic array.
However, the long-tail words may have less occurrences than the document length, thus the corresponding word-topic array may be more sparse that $K_w < K_d$.
We further provide a hybrid sampling approach, , that alternates the formula decomposition between them,
that we choose the former one for tokens with more sparse document-topic array;
otherwise, we choose the latter one for tokens with more sparse document-topic array.
Note that $\frac{N_{kd}*\beta}{N_k+W\beta}$ would have significant change, we adopt F+ tree based sampler that has $O(logK_d)$ complexity.
To minimize the lifecycle of alias table that corresponds to the second decomposed part, the tokens should be grouped according to vertex that has larger degree, and tokens in a group are processed together.
§.§ Algorithm
The specific serial algorithm of CGS training in is described in Algorithm <ref>.
We skipped the algorithm for Hrbrid that is with a natural extension.
Compared with standard CGS that has $O(K)$ complexity, significantly reduces the complexity into $O(min(K_d, K_w))$.
There are multiple factors that constitute large design space when consider to parallelize the CGS across multiple machines.
for (each epoc $e$)
for (each topic $k \in K$)
$gDense \gets \frac{\alpha_k*\beta}{N_k+W\beta}$
$gTable \gets createAliasTable(gDense)$
for (each word $w \in W$)
for (each topic $k \in K_w$)
$wSpase \gets \frac{N_{w|k}*\alpha_k}{N_k+W\beta}$
$wTable \gets createAliasTable(wSparse)$
for (each edge $e = (w, d_i) \in E, d_i \in D$)
for (each topic $k \in K_d$)
$dSparse \gets \frac{N_{k|d}*(N_{w|k}+\beta)}{N_k+W\beta}$
$dTable \gets createAliasTable(dSparse)$
for (each token $t \in {e_{dw}}$)
$z_t \gets sample(gTable, wTable, dSparse)$
Update $N_{k|d}, N_{w|k}, N_k$
The single-box CGS training algorithm in .
Input: The set of edges $E$; the set of words $W$ and documents $D$ in a partition $P$.
Output: Sample new topic for each edge and update the model state.
each epoc $e$
each topic $k \in K$
$gDense \gets \frac{\alpha_k*\beta}{N_k+W\beta}$
$gTable \gets createAliasTable(gDense)$
each word $w \in W$
each topic $k \in K_w$
$wSpase \gets \frac{N_{w|k}*\alpha_k}{N_k+W\beta}$
$wTable \gets createAliasTable(wSparse)$
each edge $e = (w, d_i) \in E, d_i \in D$
each topic $k \in K_d$
$dSparse \gets \frac{N_{k|d}*(N_{w|k}+\beta)}{N_k+W\beta}$
$dTable \gets createAliasTable(dSparse)$
each token $t \in {e_{dw}}$
$z_t \gets sample(gTable, wTable, dSparse)$
Update $N_{k|d}, N_{w|k}$ and $N_k$
The single-box CGS training algorithm in .
§.§ Related work in CGS algorithm
Hybrid AliasLDA
$\frac{\alpha_k*\beta}{N_k+W\beta} + \frac{N_{wk}*\alpha_k}{N_k+W\beta} + \frac{N_{kd}*(N_{wk}+\beta)}{N_k+W\beta}$
$\frac{\alpha_k*\beta}{N_k+W\beta} + \frac{N_{kd}*\beta}{N_k+W\beta} + N_{wk}(\frac{N_{kd}+\alpha_k}{N_k+W\beta})$
$\alpha_k(\frac{N_{wk}+\beta}{N_k+W\beta}) + N_{kd}(\frac{N_{wk}+\beta}{N_k+W\beta})$
Sampler Alias Alias CDF Alias Alias CDF Alias Alias
Fresh no no yes no no yes no yes
Computing $O(1)$ $O(1)$ $O(K_d)$ $O(1)$ $O(1)$ $O(min(K_d,K_w))$ $O(1)$ $O(K_d)$
Sampling $O(1)$ $O(1)$ $O(logK_d)$ $O(1)$ $O(1)$ $O(min(logK_d,logK_w))$ $O(\#MH)$ $O(\#MH)$
Process Order Word-by-Word Doc-by-Doc Doc-by-Doc
Approximation yes yes no
LightLDA F+LDA F+LDA SparseLDA
$\frac{N_{wk}+\beta}{N_k+W\beta}\enspace * \enspace N_{kd}+\alpha$
$\alpha(\frac{N_{wk}+\beta}{N_k+W\beta}) + N_{kd}(\frac{N_{wk}+\beta}{N_k+W\beta})$
$\beta(\frac{N_{kd}+\alpha}{N_k+W\beta}) + N_{wk}(\frac{N_{kd}+\alpha}{N_k+W\beta})$
$\frac{\alpha*\beta}{N_k+W\beta} + \frac{N_{kd}*\beta}{N_k+W\beta} + N_{wk}(\frac{N_{kd}+\alpha}{N_k+W\beta})$
Sampler Alias Alias F+tree BSearch F+Tree BSearch LSearch LSearch LSearch
Fresh no yes yes yes yes yes yes yes yes
Computing $O(1)$ $O(1)$ $O(logK)\qquad O(K_d)$ $O(logK)\qquad O(K_w)$ $O(1)$ $O(1)$ $O(K_w)$
Sampling $O(\#MH)$ $O(\#MH)$ $O(logK)\qquad O(logK_d)$ $O(logK)\qquad O(logK_w)$ $O(K)$ $O(K_d)$ $O(K_w)$
Process Order Word-by-Word Word-by-Word Doc-by-Doc Doc-by-Doc
Approximation no no no yes
Comparison of different LDA sampling approaches. Note that two decompositions are alternately used in Hybrid, another one is listed in .
Table <ref> depicts the detailed summary on comparison among different CGS approaches. Besides the difference in decomposition,
this table also list the difference on which is used, whether it is that the formula is computed for each token, whether is applied,
the corresponding if computing is needed, the , and the applied in CGS step.
SparseLDA <cit.> is the first sampling method which considered decomposing $p_k$ into a sum of sparse vectors and a dense vector.
In particular, it considers a three-term decomposition as:
$\frac{\alpha*\beta}{N_k+W\beta} + \frac{N_{kd}*\beta}{N_k+W\beta} + N_{wk}(\frac{N_{kd}+\alpha}{N_k+W\beta})$,
where the first term is dense, the second term is sparse
with $K_d$ non-zeros, and the third term is sparse with $K_w$ non-zeros.
As SparseLDA follows the document-by-document sequence, very few elements will be changed for the first two terms at each CGS step.
Linear search <cit.> (LSearch) is applied to all of these three terms in both SparseLDA implementations (Yahoo!LDA <cit.> and Mallet LDA <cit.>),
which makes the sampling complexity with $O(K)$, $O(K_d)$ and $O(K_w)$ for these three terms, respectively.
AliasLDA <cit.> considers the following decomposition of p:
$\alpha(\frac{N_{wk}+\beta}{N_k+W\beta}) + N_{kd}(\frac{N_{wk}+\beta}{N_k+W\beta})$,
Instead of the exact multinomial sampling, AliasLDA considers a proposal distribution $q_k$ with a very efficient generation routine
and performs a series of Metropolis-Hasting (MH) steps using this proposal to simulate the true distribution $p_k$.
In particular, the proposal distribution is constructed using the latest second term and a stale version of the first term.
For both terms, Alias method is applied. $\#MH$ steps decides the quality of
the sampling results. The overall amortized cost for each CGS step is $O(K_d + \#MH)$. Note the initialization cost
$O(K)$ for the first term can be amortized. Therefore, AliasLDA reduces the amortized cost of each step to $O(K_d)$.
LightLDA <cit.> is a recently proposed approach that develops an Metropolis Hastings sampler with different constructed proposals that
combines the document-proposal ($N_{k|d} + \alpha$) and word-proposal ($\frac{N_{wk}+\beta}{N_k+W\beta}$) into a “cycle proposal”.
The word-proposal is pre-computed and Alias method is applied, thus the $O(K)$ complexity is amortized to be $O(1)$.
And $O(1)$ complexity is also achieved by a lookup table with document length that stores the corresponding topic for its word occurrences.
F+LDA <cit.> has two variants that with different decompositions. The document-by-document process order has the decomposition as:
$\beta(\frac{N_{k|d}+\alpha}{N_k+W\beta}) + N_{w|k}(\frac{N_{k|d}+\alpha}{N_k+W\beta})$,
For each document, a sample step only changes two topic counter for $N_{k|d}$ in the first term, so they adopts F+tree sampling for the first term that with $O(logK)$ complexity for both updating and sampling.
Correspondingly, different word in the same document has different $N_{w|k}$ for the second term, and the complexity is $O(K_w)$.
Similarly, the word-by-word process order has opposite decomposition and properties:
$\alpha(\frac{N_{w|k}+\beta}{N_k+W\beta}) + N_{k|d}(\frac{N_{w|k}+\beta}{N_k+W\beta})$,
It is worth to compare with LightLDA, since the complexity of is better than other alternatives but could be worse than LightLDA <cit.>.
First, LightLDA needs an extra lookup table that stores the correspondence of a token and its sampled topic (analogous to edge and its attribute in ).
Instead of directly read $N_{k|d}$, LightLDA samples the lookup table to simulate $N_{k|d}$, thus the complexity is reduced from $O(K_d)$ to $O(1)$.
However, tt requires data is partitioned in a document-wise way, otherwise the sample result would be inaccurate.
This limits the exploration of better partition approaches (Section <ref>).
Second, a MH-step will compute the true probability (Formula <ref>) of the sampled topic, $O(1)$ complexity can only be achieved when dense vector or hash table is used,
this will result in high memory consumption.
Otherwise, MH-step could result in $O(max(K_w, K_d))$ complexity to get value from $N_{w|k}$ and $N_{k|d}$ if the sparse data structure is used.
§ PARALLELIZATION
In this section, we first discuss the parallelization design in , followed by other useful utilities provided.
§.§ Parallelization design in
Consider a typical web-scale application with 100 millions of documents and words, and with a large number of topics (typically on the order of thousands), where there are almost trillions of parameters. No single machine can hold the entire Big corpus data nor the Big model size.
This made single machine solution impossible, which motivates a scalable and efficient way of distributing the computation across multiple machines.
However, the design is challenging that a typical web-scale LDA training requires both data parallelism and model parallelism and involves hundreds of machines.
In this section, we will discuss multiple design dimensions to parallelize the across multiple machines.
Graph based data and model representation.
In stead of represented the data (input corpus) and model (word-topic and document-topic) as matrix [Note that both data and model are sparse.],
represents data as a directed bipartite graph that is the dual representation of sparse matrix.
Figure <ref> depicts the graph representation of a corpus with three words ($w_1,w_2,w_3$) and documents ($d_1,d_2,d_3$).
The graph has two kinds of vertices, word vertices and document vertices.
An edge exists from word vertex to document vertex only if that word is occurred in the document.
This corpus in LDA is a natural graph <cit.> like many other natural language processing problems, where the graph have highly skewed power-law degree distributions.
This graph representation can be naturally mapped to in GraphX.
Note that the corpus graph is treated as directed graph just because in GraphX is directed, the direction is actually meaningless.
The edges in GraphX is grouped in a partition according to the source vertex ID (word vertex).
The model parameter word-topic matrix is split in a word-wise fashion that each word vertex is attached with the corresponding word-topic array ($N_{w|k}$) as attributes.
Similarly, the document-topic matrix is also split in a document-wise fashion and each document vertex is attached with the corresponding document-topic array ($N_{k|d}$).
Both word-topic and document-topic array are sparse that not all topics are sampled,
and they are becoming more and more sparse as the training converged.
Relatively speaking, a long-tail word may have more sparse word-topic array than a hot word;
and document-topic array maybe more sparse than word-topic array since a word may have more occurred tokens than the average document length.
The current topic ($Z_{dw}$) of a word occurrence (token) $w_{dw}$ is annotated as the corresponding edge attribute.
It is noteworthy that the edge attribute is an array since there may be multiple occurrences of the same word in one document.
And the global state $N_k = \sum_dN_{k|d} = \sum_wN_{w|k}$ records the total number of tokens (edges) been sampled as topic $k$, which
is computed by aggregating the $N_{k|d}$ from all document vertices (or $N_{w|k}$ from all word vertices).
Graph based CGS abstraction.
Partition approach.
The distributed parallelism is achieved by partitioning the graph into multiple partitions (described in next paragraph), and workers apply CGS process in Algorithm <ref> for all partitions in parallel,
followed by synchronizing the model state at the end of iteration. In this way, data parallelism and model parallelism are achieved <cit.>, where the model is also partitioned and distributed across workers.
Partition strategy that determines how to partition the corpus and model plays crucial impact on system performance.
The improper partition would result in load imbalance and large network communication.
Compared with (sparse-)matrix based representation that can only be partitioned in a “rectangle” way, graph has more freedom for partitioning choice.
There are two partition strategies, that tries to evenly assign the vertices to machines by cutting the edges.
<cit.> that tries to evenly assign the edges to machines by cutting the vertices.
PowerGraph <cit.> pointed out that vertex-cut can achieve better performance than edge-cut, especially for power-law graphs.
And the workload of a machine in vertex-cut is determined by the number of edges located in that machine, and the total communication cost is proportional to the number of mirrors of the vertices.
However, the power-law distributions in corpus graph makes the partitioning challenging <cit.>.
GraphX currently only supports vertex-cut method, and provides three partitioning approaches:
that assigns edges to partitions by hashing the source and destination vertex IDs;
that assigns edges to partitions using only the source/destionation vertex ID, collocating edges with the same source/destionation;
and that assigns edges to partitions using a 2D (“rectangle”) partitioning of the sparse edge adjacency matrix, guranteeing a $2*\sqrt{numParts}$ bound on the number of vertex replication.
Xie, .etc <cit.> presented degree-based hashing (DBH) partition method
that can achieve lower communication cost than existing methods and can simultaneously guarantee good workload balance.
The theoretical bounds on the communication cost and workload balance can also be derived.
DBH is also vertex-cut that it first applies randomized hash function to evenly assign vertices to partitions,
then assigns an edge ($(v_i, v_j)$) to partition that contains its source or destination whose degree is less.
In other words, the vertex with larger degree is cut that it would have multiple replicas.
DBH shares the same insights with PowerLyra <cit.> that locality matters for low-degree vertex thus it places all edges related to this vertex together,
while parallelism matters for high-degree vertex thus it favors to cut high-degree vertex.
However, DBH only considers the relative size between source degree and destination degree,
without considering their absolute value.
Consider the case where both source and destination degree are small (smaller than a threshold value), it is not reasonable to still correspond the edge to vertex with lower degree, but should be the vertex with higher degree.
In , we improved DBH as DBH+, and the algorithm is listed in Algorithm <ref>.
Input: The set of edges $E$; the set of vertices $V$; the number of machines $p$.
Output: The assignment $P(e) \in [p] partitions$ for each edge $e$.
each $v \in V$
$P(v) = hash(v)$
count the degree $d_i$ for each $i \in V$ in parallel
each $e = (v_i,v_j) \in E$
$max(d_i,d_j) < threshold$
$d_i \leq d_j$
$P(e) = P(d_j)$
$d_i \leq d_j$
$P(e) = P(d_i)$
$P(e) = P(d_j)$
DBH: an improved Degree-based hashing (DBH) algorithm.
We actually explored many other partitioning strategies, such as different greedy algorithms <cit.> and iterative algorithm <cit.>.
Synchronization approach.
After partitioning and distributing the data/model, the remaining thing to consider is the synchronization among machines.
In theory, the update of topic assignment $Z_{dw}$ can not be performed concurrently with the update
of any other topic assignment $Z_{d'w'}$, with conflicts on $N_k$ and possible conflicts on $N_{w|k}$ or $N_{k|d}$
The conflicts must be guaranteed using locks [Topic level parallelism exists that the topic probability computing for each topic ($p(k)$) can be parallelized without any locks.], which is costly and is hardly implemented in distributed environment.
The good news is that this dependence is weak, given the typically large number of word tokens compared to the number of machines.
If two processors are concurrently sampling, but sampling different words in different documents
(i.e., $w_{dw} \neq w{d'w'}$, then concurrent sampling will be very close to sequential sampling that the only affected state is $N_k$.
This means that we can still achieve convergence by relaxing these locks.
There are three possible design choices, either only $N_{k|d}$ or $N_{w|k}$ is strictly synchronized, or none of them are not synchronized.
This is more suitable for distributed data-parallel processing that communication happens only across the stage boundary.
AD-LDA <cit.>, SparseLDA <cit.>, AliasLDA <cit.> and SparkLDA <cit.> permit the asynchronized update on $N_{w|k}$ and $N_k$.
supports all these three synchronization approaches.
Unlike SparkLDA <ref> and LightLDA <ref> that further block a partition into the mini-batches in a “conjugated” way [If the partition is document-wise, then the mini-batch is word-wise, or vice versa.], and synchronization happens across mini-batches to reduce the state staleness.
The first two can be achieved by choosing method that either all edges corresponding to a word or a document are located in the same partition,
where $N_{k|d}$ or $N_{w|k}$ is not synchronized.
However, would result in data severely skew, that even the number of documents is even distributed but the number of words is not, or vice versa.
Therefore, further aggressively relaxes the dependency that both model states are updated independently,
this asynchronization approach enables us to choose any possible partition methods that has better load balance with less network communication, such as DBHPlus presented above.
Furthermore, even inside a partition, the CGS process result is only used to update the edge attribute ($Z_{dw}$), and only update the vertex attribute ($N_{k|d}$, $N_{w|k}$) at the end,
i.e., line 21 in Algorithm <ref> is moved to at the epoch. This will largely reduce the lock cost if multi-threaded is enabled inside a partition.
The side effect is that $N_{k|d}$, $N_{w|k}$, and $N_k$ are stale with value computed in last iteration.
why they would not affect convergence?
workflow in an iteration.
In conclusion, the CGS workflow of one iteration in is illustrated in Figure <ref>. There are five steps:
1). driver broadcasts $N_k$ to all workers. 2). the vertex master ships the model state ($N_{k|d}$ or $N_{w|k}$) to all of the corresponding vertex slaves.
3). workers apply Algorithm <ref> in parallel and update the $N_{k|d}$ or $N_{w|k}$ locally at the end.
4). at the end of an iteration, vertex master aggregates $N_{k|d}$ or $N_{w|k}$ all local updates from workers.
5). driver aggregates $N_k = \sum_wN_{w|k}$ from all word master vertices. Note that we do not aggregate $N_k = \sum_dN_{k|d}$ since the number of documents may be 100+ times larger than word number.
§.§ Related work in parallelization
Recently, there are many efforts towards to build distributed topic modelling system.
They have different partition approaches and synchronization approaches. Almost all works <cit.> partition the corpus in a document-wise way.
Instead, permits any kind of partition methods.
No parallelization work is found to have strictly synchronization semantic that at least $N_k$ is not synchronized.
The above works all achieve the synchronization on $N_{k|d}$ since they choose document-wise partitioning.
They further processes one document partition in a word-by-word order, and synchronization cross-machine happens once a mini-batch with certain workload is completed.
However, SparkLDA and F+LDA schedule the tasks that only one task is working on a word mini-batch that they have no conflict on $N_{w|k}$.
§.§ Utilities
supports both log likelihood and perplexity as the metric to evaluate the model convergence.
also supports flexible termination condition, which can be based on a given number of training iteration or the perplexity value.
Besides the core of model training, provides useful utilities in entire lifecyle.
Incremental training.
Model can be saved in the middle of training rather than waiting the end of the training. In this way,
users can terminate the training if they think the model is converged.
Incremental training that the model can be initialized by a pre-trained model is also supported.
This is useful when users are not satisfied the trained model and continue the model training.
The model re-training could be equipped with better hyper-parameters, or new training data, etc.
Merge duplicated topics.
Frequent words often dominate more than one topics, and the learned
topics are similar to each other. These similar topics are duplicated <cit.>.
adopts the asymmetric Dirichlet prior <cit.> that automatically combine similar topics into
one large topic, rather than splitting topics more uniformly by symmetric priors.
This is especially useful when the number of topic number is large ($\leq 10^6$).
Besides, we also cluster topic duplicates if their L1-distance is below a threshold. The lower L1-distance threshold means that we would
remove more duplicates from large number of topics.
Model inference.
also supports model inference besides model training, that inferences the topic distribution over the given a document with the same CGS process.
The inference process can use the same tricks presented in (Section <ref>).
To accelerate the model inference in online applications like search engine and online advertising systems to
predict latent semantics of new user queries in real-time (usually in milliseconds) from large number of topics,
we adopts RT-LDA <cit.> that replace the sampling operation in Equation <ref> by the max operation,
which makes RT-LDA significantly faster than standard inference equation, but still with similar perplexity.
§ OPTIMIZATIONS
Data-parallel system helps to simplify the programming efforts, hide the system complexity and integrate with entire training pipeline.
However, it would result in sub-optimal performance compared with customized approaches.
In this section, we describe the techniques that help to achieve comparable performance and scalability.
including approximated training, network I/O reduction and several low-level optimizations.
§.§ Approximated training
As proved in asynchronized update that sampling can work on staled model state, CGS training could tolerate a certain degree of approximation.
In this section, we will present two important approximations to make better tradeoffs on efficiency and model accuracy, including sparse model initialization and “converge” edges exclusion from sampling.
Sparse model initialization.
It is well known that the execution time per iteration decreases by degrees as training makes progress,
and the first several iterations are always the performance and scalability bottleneck,
i.e., the training would succeed if it passes the first iteration.
A figure to show the time per iteration distribution.
Usually, CGS training is initialized by first randomly sampling a topic for each token with equal topic probability, and initiating the model state $N_{k|d}$ and $N_{w|k}$ by aggregating the topic distribution for each word and document, respectively.
However, such would result in relatively dense topic distribution for word, especially for hot words that occurred in most of the documents.
As a consequence, this dense word-topic distribution takes more storage, memory consumption, network I/O (step 2 in Figure <ref>) and more computing complexity.
Can we initialize the model with a sparse word-topic distribution but still achieve the similar convergence speed and accuracy?
To validate the assumption, presents two approaches that demonstrate much better performance in the first several iterations and comparable or even better convergence and accuracy (See Figure <ref>).
* The first approaches is to directly sparsify word-topic array ($N_{w|k}$). Assume that there are $T$ tokens of word $w$ in the corpus.
Given total topics $K$ and the sparsity degree $deg \ll 1$, it first randomly samples $deg*K$ topic set $S$ from $K$ topics, then randomly samples a topic from $k \in S$ for each token of that word with equal probability, and updates $N_{k|d}$ and $N_{w|k}$ such that $\sum_kN_{w|k} = T$. Such sparse initialization would largely relieve the performance burden in the first iteration.
However, it has side effects on model accuracy. The good side is that it reduces the possibility to allocate the same topic for two words that should be with different topic, since their topic overlapping probability is reduced due to sparse initialization; on the contrary, the bad side is that this also reduces the possibility if the two words should be with the same topic.
Our evaluation indicates that the CGS process still converges and gradually recovers the side effect of sparse initialization.
This optimization is essentially to gradually amortize the cost of first iteration to the following iterations.
We further neutralize the side effect by increasing the $\beta$ value in decomposed part $N_{k|d}*(N_{w|k} + \beta)$ for those topics that are not assigned during initialization.
* The second approach is to sparsify document-topic array ($N_{k|d}$) with the same method, that thus indirectly results in sparse word-topic array.
“Converged” token exclusion.
We observed that different tokens are with different convergence rate. However, the CGS process still be applied normally without differentiation.
We present “converged” that excludes tokens from CGS process that have been converged,
which will largely reduce the workload per iteration, especially as for later iterations that almost tokens are converged (See Figure <ref>).
The question is how to identify a token is converged, and how to neutralize the possible side effect?
We treat a token is converged if current sampled topic is the same as topic sampled in last iteration.
To reduce the side effect, we do not exclude the converged token directly, instead, they are still sampled with a probability.
Such probability considers how many iterations a token has not been processed ($i$) and how many times it was processed but with the same sampled topic ($t$).
Both $i$ and $t$ are zeroed for clearing once the sampled topic is different.
Thereby, the probability is $2^{i-t}$ that the probability has positive correlation with $i$ but has negative correlation with $t$.
We also support user to configure this optimization to be enabled only after certain iterations.
overhead due to extra storage cost.
§.§ Network I/O reduction via delta aggregation
Network I/O still matters that a significant portion of time is spend in average, especially for large scale execution with large number of partitions.
As shown in Figure <ref>, there are only four steps (except step 3) involved with network I/O, where the size of $N_k$ in step 1 is negligible, and
the size of $N_{k|d}$ and $N_{w|k}$ (step 2, 3 and 5) has already been reduced by sparse initialization.
In this section, we describe techniques to further reduce the network I/O in step 4 that each vertex slaves sends its locally aggregated $N_{k|d}$ and $N_{w|k}$ to master.
The $N_{k|d}$ to be transferred are locally aggregated from $Z_{dw}$ from all tokens of document $d$, and the same is for $N_{w|k}$.
Obviously, they are positively correlated with the number of tokens per document and per word, respectively.
With the same insight as “converged” token exclusion, high proportion of tokens are converged without topic change, we present
delta aggregation that only the topic of changed tokens is aggregated in local and transferred to master.
Therefore, the network I/O would be largely reduced as the model becomes converged.
This requires to store the old topic sampled last time, other than new topic sampled currently, which doubles the attribute size in edge.
Moreover, the effectiveness would be offset by “converged” token exclusion.
Thereby, we will disable this optimization if token exclusion is enabled.
However, it is worth noting that unlike token exclusion, this optimization would not affect the model accuracy.
§.§ Low-level optimizations
Besides the design principle, the performance also lies in the detail.
In this section, we introduce several low-level optimizations that are proved to be generally effective, including efficient data structure that exploits the sparsity and redundant computing elimination.
Sparse data structure.
The right choice on data structure is crucial for performance. Here we discuss three different choices that exploits the inherent sparsity in word-topic array $N_{w|k}$ and document-topic array $N_{k|d}$,
including and that provided in MLlib, as well as our new proposed .
DenseVector is represented as an array, and SparseVector is represented by an index array that records the indices of non-empty elements and an value array that records the corresponding values. For instance,
a vector $(1, 0, 0, 0, 0, 3)$ can be represented in dense format as $[1, 0, 0, 0, 3]$
in sparse format as $(6, [0, 5], [1, 3])$, where $6$ is the size of the vector.
Compared with dense vector, it is more memory efficient if vector has large sparsity,
but with increased cost for operations such as search that the complexity is increased from $O(1)$ to $O(log(length))$.
Note that it would result in more memory space for vector with less sparsity.
The tipping point is when the sparsity is 0.5 that only half of the elements are empty, where the total length of index and value array is equal to the original vector length.
Instead, we provide a new sparse vector representation, called , that also includes a value array as and a different index array.
The index array is composed of $(s, n)$ pairs where $s$ records the starting index of an empty sequence and $n$ records the number of non-empty elements before position $s$. For example,
$(1, 0, 0, 0, 0, 3)$ is represented as $(6, [(1, 1)], [1, 3])$.
Figure <ref> describes how to get value from CompactVector, given the original index $x$.
The time complexity is $O{logN}$ where $N$ is the number of empty sequences, i.e., $N$ is the number of non-empty sequences, thus $N$ is less than the number of non-empty elements $E$, since
sequence is composed of at least one element.
Thereby, the time complexity is lower than it in SparseLDA that is with $O(logE)$
In addition, the size of CompactVector could be smaller specifically when $\frac{E}{N} \geq 2$, consider that CompactVector represents a sequence with two (both $s$ and $n$) data element.
The disadvantage of CompactVector is that the insertion is much costly with $O(N)$ complexity.
The right choice should tradeoff between the space requirement and computing cost.
Generally, is more suitable for scenarios where space is critical and almost operations are read;
is suitable for vectors with large sparsity; and is suitable for dense vector with many write operations, since array in Scala can be updated in place while the others are immutable that a new operation is required each time the value is changed or a new value is inserted.
Take the computing of $N_{k|d}*(N_{w|k} + \beta)$ as an example, $(N_{w|k}$ is read given the topic $k$ where $N_{k|d}$ is non-zero. Such read is with $O(logK_w)$,
which increases the complexity of probability computing from $O(K_d)$ to $O(K_d*logK_w)$, thus chooses to convert $N_{w|k}$ from sparse vector.
Input: $CV = (len, index, value)$, original index $x$.
Output: the value indexed at $x$.
GetValue($CV$, $x$)
$(s_i, n_i), (s_j, n_j) \gets BSearch(x, CV)$
assert($s_i \leq x \leq s_j$)
$x \neq s_i \; \&\& \; x \neq s_k$
$x \geq s_j - (n_j - n_i)$
$d = x - (s_j - (n_j - n_i))$
$value[n_i + d]$
Get value from CompactVector.
Alias table.
We use alias table, $gTable$ for $\frac{\alpha*\beta}{N_k+W\beta}$ and $wTable$ for $\frac{N_{wk}*\alpha}{N_k+W\beta}$, to avoid re-computation cost and save the sampling complexity to $O(1)$.
However, the time complexity to build alias table is $O(K)$ and $O(K_w)$, respectively. Moreover, Each word vertex has a $wTable$ , which requires more memory space.
We reduce the memory consumption by processing the tokens in word-by-word fashion that reduces the lifecycle of $wTable$ thus unused $wTable$ would be freed (GC).
To reduce the creation cost, we further refine the algorithm presented in AliasLDA <cit.>.
First, we only maintain the $H$ queue that keeps the topic information ($(k, p_k)$) that is with higher probability than the average $\frac{1}{K_w}$, and do not maintain the $L$ queue described in in AliasLDA.
Instead, we directly insert the topic information with lower probability into the bin of alias table in a sequential way.
Second, when create alias table for $N_{k|d}$ (used in LightLDA), the probability (count) is integer, but the average probability would be float that is the result of dividing the sum by $K_d$.
Consequently, the split probability in a bin should be float. Instead, we first multiply $K_d$ for each individual probability, therefore, both the average and the split probability are also integer.
In this way, we avoid the costly divide operation, and simultaneously save the space.
Redundant computing elimination.
There are many redundant computing in CGS formula <ref>. For instance, $\frac{1}{N_k+W\beta}$ will be used many times during entire iteration,
thus we can pre-compute it first and re-use the result later [Note that $N_k$ is constant with value computed in last iteration.]
The following code snippet (Algorithm <ref>) depicts how we decompose the computation and eliminate the redundancy.
Besides redundancy elimination, the multiplication between scalar and vector (denoted as $.*$) enables the instruction level parallelism that uses vectorization via SIMD instructions.
Lastly, this is also CPU cache friendly. For example, only $N_k$ is accessed to compute $t1$, and only $N_{w|k}$ is accessed to compute $wSparse$.
It is worth to note that such concept can also be applied to other CGS decompositions.
$t1$ = $\frac{1}{N_k+W\beta}$ // vector
$t2$ = $\frac{K\alpha}{N+\alpha'}$ // constant
$t3$ = $(\frac{\alpha'}{K} - W\beta)$ // constant
$t4$ = $\frac{\alpha_k}{N_k+W\beta}$ = $\frac{K\alpha \frac{N_k + \frac{\alpha'}{K}}{N+\alpha'}}{{N_k+W\beta}}$ = $\frac{K\alpha}{N+\alpha'}$*$\frac{N_k+\frac{\alpha'}{K}}{N_k+W\beta}$ = $\frac{K\alpha}{N+\alpha'}$*$(1+\frac{\frac{\alpha'}{K} - W\beta}{N_k+W\beta})$ = $\frac{K\alpha}{N+\alpha'}$ + $\frac{\frac{K\alpha}{N+\alpha'}(\frac{\alpha'}{K} - W\beta)}{N_k+W\beta}$
= $t2$ .+ ($t2*t3$).*$t1$ // vectorization
$t5$ = $\beta$.*$t1$ // vectorization
$gDense$ = $\frac{\alpha_k*\beta}{N_k+W\beta}$ = $\beta$.*$t4$ // vectorization
(foreach $w \in W$)
$wSparse$ = $\frac{N_{w|k}*\alpha_k}{N_k+W\beta}$ = $N_{w|k}$*$t4(k)$ // $N_{w|k} \neq 0$
$t6$ = $\frac{N_{w|k}+\beta}{N_k+W\beta}$ = $t5$ + $N_{w|k}$*$t1(k)$ // $N_{w|k} \neq 0$
(foreach $d \in D$)
$dSparse$ = $N_{k|d}(\frac{N_{w|k}+\beta}{N_k+W\beta})$ = $N_{k|d}$*$t6(k)$ // $N_{k|d} \neq 0$
Redundant computing elimination.
$t1$ = $\frac{1}{N_k+W\beta}$
$t2$ = $\frac{K\alpha}{N+\alpha'}$
$t3$ = $(\frac{\alpha'}{K} - W\beta)$
$t4$ = $\frac{\alpha_k}{N_k+W\beta}$ = $\frac{K\alpha \frac{N_k + \frac{\alpha'}{K}}{N+\alpha'}}{{N_k+W\beta}}$ = $\frac{K\alpha}{N+\alpha'}$*$\frac{N_k+\frac{\alpha'}{K}}{N_k+W\beta}$ = $\frac{K\alpha}{N+\alpha'}$*$(1+\frac{\frac{\alpha'}{K} - W\beta}{N_k+W\beta})$ = $\frac{K\alpha}{N+\alpha'}$ + $\frac{\frac{K\alpha}{N+\alpha'}(\frac{\alpha'}{K} - W\beta)}{N_k+W\beta}$ = $t2$ .+ ($t2*t3$).*$t1$
$t5$ = $\beta$.*$t1$
$gDense$ = $\frac{\alpha_k*\beta}{N_k+W\beta}$ = $\beta$.*$t4$
each $w \in W$
$wSparse$ = $\frac{N_{w|k}*\alpha_k}{N_k+W\beta}$ = $N_{w|k}$*$t4(k)$ // $N_{w|k} \neq 0$
$t6$ = $\frac{N_{w|k}+\beta}{N_k+W\beta}$ = $t5$ + $N_{w|k}$*$t1(k)$ // $N_{w|k} \neq 0$
each $d \in D$
$dSparse$ = $N_{k|d}(\frac{N_{w|k}+\beta}{N_k+W\beta})$ = $N_{k|d}$*$t6(k)$ // $N_{k|d} \neq 0$
Redundant computing elimination.
also implements several other optimizations that are generally beneficial.
First, tries to reuse the same generated random number as far as possible to avoid cost of random number generation.
For example, there are three random number generations, the first is used to locate $gTable$, $wTable$ or $gSparse$, the second is for locate the specific bin in alias table, and the
last one is to locate the high or low region in a bin. Apparently, the last two can use the same random number.
Second, we can pre-generate $n$ random numbers for $n$ different tokens in the same $(d, w)$ pair, in this way, the CDF sampling cost is reduced from $O(n*logK_d)$ to $logn*logK_d$ that only $logn$ passes of CDF binary search are required.
Lastly, also implements optimization to exploit the difference between hot and long-tail word, as described in LightLDA <cit.>
§ IMPLEMENTATION
We encountered scalability issues due to the inherent inefficiency of managed language (Scala) and framework cost of GraphX.
The implementation must balance resource (CPU, network and memory) usage that no resource is the bottleneck and all are fully utilized.
Memory is the major bottleneck to when we first try to scale out . Data-parallel system like Spark is designed to process one partition per core and the whole partition must be loaded in memory.
“Out of memory” occurs frequently if many partitions (we have 16-32 cores per machine) loaded at the same time.
The dilemma lies in that if too many partitions would reduce the memory consumption but with the increased network I/O.
We observed that these partitions in a machine may share common data, such as the same word or document may exist in multiple partitions, thus the same as the corresponding word-topic or document-topic array.
Instead, we load less partitions at one time and use multi-thread computing in a partition to fully utilize the CPU cores.
More specifically, edges is sorted queued in word-by-word order in a partition (already done by GraphX), and work-stealing with word granularity is adopted among multiple threads that once a thread completed all edges of one word, it will get all edges of the first word from edge queue. This achieve relative good load balance. The more fine-grained edge granularity is also feasible that the edges of the same word are processed in parallel, where
$N_{w|k}$ is further shared among threads.
Besides CGS processing (step 3 in Figure <ref>), we re-implement some GraphX APIs (except shuffling operator) to make them multi-threaded,
such as (step 2) and (step 4).
Actually, we abandon that updates vertex attribute with value aggregated from edges,
and constraints that the edge attribute type must be the same as with vertex attribute, thus costly type conversion happened.
Instead, we directly operate on the $Graph$ data structures (e.g., edge array and vertex index array in EdgePartition, vertex array and routing table in VertexPartition).
This will significantly reduce memory costs and scale 10X up than original GraphX implementation.
Many GraphX APIs would create many intermediate objects that raise higher memory costs and the overhead of garbage collection.
For example, to represent the bin of alias table ($(i,h,p_h)$), we use three arrays with primitive type instead of one array with Scala to avoid the boxing/unboxing overhead.
Once we fixed the memory limitation via multi-threading, we found that high CPU cost on RDD decompression and deserialization that is processed by a single thread.
Therefore, we prefer to configure RDD in uncompressed and deserialized format.
Besides the optimizations presented in Section <ref>, the avoid of boxing/unboxing and generation of closures can also reduce the CPU cost.
The relax on memory burden can enable more partitions, but this will increase the network I/O. We adopt Kryo serialization library in Spark
that is significantly faster and more compact than default Java serialization.
Shuffling cost is largely reduced when Kryo is enabled for serializing shuffling data.
§ EVALUATION
This section describes the evaluation to demonstrate the effectiveness and efficiency of .
§.§ Evaluation design
Dateset Tokens Words Docs $T/D$
NYTimes 99,542,125 101,636 299,752 332
BingWebC1Mon 3,150,765,984 302,098 16,422,424 192
BingWebC320G 54,059,670,863 4,780,428 406,038,204 133
Three different datasets used in evaluation.
We use 3 different datasets, including a small sized NYTimes <cit.> (about 520MB), a medium sized one month web chunk data indexed by Bing News (about 17GB),
and a large scale Bing web chunk data (320G). They are all pre-processed and saved as format.
The detailed information is listed in Table <ref>.
Evaluation design.
The evaluation aims to evaluate: 1). the algorithm effectiveness and efficiency in compared with LightLDA who represents the state of art.
2). the scalability of that varies topic number, dataset size and number of machines.
3). the effectiveness of proposed techniques in ().
Cluster configuration.
We have two Spark clusters with different scale. The small one is in lab environment and has 10 homogeneous computing nodes are connected via 40Gbps
Infiniband network and each node has 16 2.40GHz Intel(R) Xeon(R) CPU E5-2665 cores and 128GB memory. There are
1 driver configured with 5GB memory and 10 workers configured with 100GB memory.
The experiments against NYTimes and BingWebC1Mon are conducted in this small cluster, where NYTimes is partitioned into
20 partitions and each partition has 8 threads, BingWebC1Mon has 80 partitions and each one has 2 threads.
The large Spark cluster is deployed on a multi-tenancy data center managed by Yarn <cit.> that the resource is not always guaranteed.
An executor is configured to have 20GB memory and 14 cores.
The scalability experiments against BingWebC320G are conducted in this cluster.
§.§ Effectiveness and efficiency of CGS algorithm in
Execution time comparison between ZenLDA and LightLDA.
Log-likelihood comparison between ZenLDA and LightLDA.
To compared with LightLDA, we also implemented it with the same framework with 8 Metropolis-Hasting steps.
We excluded sparse initialization and toke exclusion, and applied the same optimizations described in Section <ref> to LightLDA, and the only difference is the algorithm.
The comparison is against NYTimes and BingWebC1Mon datasets with 1,000 and 10,000 topics, respectively. Both $\alpha$ and $\beta$ are 0.01.
Both execution time per iteration and log-likelihood per iteration are compared, and the result is shown in Figure <ref> and Figure <ref>, respectively.
Note that it excludes the log-likelihood computing time, and the spikes in Figure <ref> stems from full GC in JVM.
We can see significant execution reduction (2-6X speedup) and better model accuracy than LightLDA for all experiments.
Different datasets has different speedup, the larger the dataset is, the more speedup achieved (2X in NYTimes and about 4-6X in BingWebC1Mon).
There is no obvious speedup difference when topics varies from $1,000$ to $10,000$.
The experiments also show that as the number of topics increased, the performance is still almost the same in $\name$ but increases a little (about from 13s to 17s in NYTimes and from 220s to 250s) in LightLDA.
Performance is slowdown with sub-linearity if dataset is increased.
The performance result is a little bit “surprising”, consider that LightLDA has $O(1)$ perplexity.
As discussed in Section <ref>,
the MH-step in LightLDA would be more costly due to the computation of true probability, which requires $O(max(logK_w,logK_d)$ complexity since $N_{k|d}$ and $N_{w|k}$ in our implementation is sparse thus with $O(logK_w)$ or $O(logK_d)$ complexity to read the value. And there are $\#MH$ (8 in our implementation) MH-steps. As a comparison, the complexity in can be as low as $min(K_d, K_w)$.
With respect to log-likelihood, outperforms LightLDA, and even more significant as the number of topics increased.
This may be due to facts that the asymmetric prior is used in and the proposal distribution in LightLDA is an approximation of the true probability.
[Note that we cannot directly compare the result with Figure 13 and 14 in LightLDA paper, since we double confirmed with LightLDA author that they used different log likelihood formula we used
($llh = \sum_w{log\sum_k{\frac{N_{k|d}+\alpha_k}{N_d+K\alpha_k}*\frac{N_{w|k}+\beta_w}{N_k+W\beta_w}}}, \alpha_k = \frac{N_k+\alpha'}{N+K\alpha'}$).]
We also compared with other algorithms such as SparseLDA (we implemented in the same framework), and the EM based implementation in MLLib [We cannot compare SparkLDA since it is not open-sourced. But we believe will win since SparkLDA uses standard CGS algorithm.].
SpasreLDA is much slower that we did not run full length of SparseLDA but only with the first 15 iterations. It spends about 27,707 seconds (10,000) while only needs 1,907 seconds.
And the EM algorithm in MLlib even cannot finish the first iteration against BingWebC1Mon dataset with errors reported 1 hour later.
§.§ Scalability
Execution time change curve as executor number varies.
The scalability experiments are conducted against the largest dataset and run on our multi-tenancy data center.
Figure <ref> indicates that can support super large dataset in acceptable time.
When 2X more executors (containers in Yarn) joined in (240 VS 120), the performance is almost linearly speedup.
As we continue to add more executors (360), the performance can still be improved, but with less speedup due to the network I/O becomes larger.
Execution time change curve as topic number varies.
We also evaluated the performance when topic number varies.
The experiment is conducted against BingWebC1Mon with 1,000, 10,000 and 100,000 topics, respectively.
Their training time of first 50 iterations is shown in Figure <ref>.
When $K=10,000$, the average time per iteration is only increased a little bit compared with $K=1,000$.
Even with 100X more topics ($K=100,000$), the time is only increased by about 3X.
§.§ Optimization evaluation
Log-likelihood comparison among different initializations.
Sampling time comparison among different initializations.
(a) Change rate of token's topic assignments.
(b) Sampling time with or without “converged” token exclusion.
(c) Log-likelihood with or without “converged” token exclusion.
Sparse initialization.
Figure <ref> shows the log-likelihood of different initialization strategies.
Specially, we further split log-likelihood into word log-likelihood and doc log-likelihood.
With respect to accuracy, sparse initialization of word-topic distribution () can even achieve better total and word log-likelihood, but with worse document log-likelihood.
In contrast, sparsifying document-topic distribution() only achieves better doc log-likelihood in the first several iterations, and
is “dragged” to normal distribution as random initialization.
With respect to performance, Figure <ref> shows that both SparseWord and SparseDoc make the sampling time faster than random initialization
at the first several iterations. This is helpful to reduce the scalability bottleneck.
However, it gradually increases to normal performance as random initialization as we expected, and even higher in SparseWord because of the increased $K_d$ (the worse document log-likelihood, the dense document-topic distribution).
“Converged" token exclusion.
chooses to turn on this optimization after the 30th iteration.
Both sampling time (exclude time on shuffling) and log-likelihood are compared.
The result shown in Figure <ref> and Figure <ref> indicates that “converged" token exclusion technics can achieve about
50% faster in later iterations, without hurting the log-likelihood much.
Figure <ref> explains the underlying reason that the changing rate of topic assignment decreases as the iteration increases, with only about 22% remained at the end.
This figures also demonstrates that delta aggregation (Section <ref>) can largely reduce the network I/O.
The speedup is not strictly align with the change rate since the sample rate also considers the other factors.
Redundant computing elimination.
We only evaluate the effectiveness of “redundant computing elimination” and skip other low-level optimizations that are hard to separated out.
The result in Figure <ref> shows that the sampling is faster with about 11% improvements.
Sampling time with or without redundant computing elimination.
§ DISCUSSION AND FUTURE WORK
This section describes several points that has crucial impacts on model accuracy and system performance, and they are out of the scope of this paper and remain future work.
Hyper-Parameter tuning.
There are three hyper-parameters, Dirichlet priors ($\alpha$ and $\beta$), and the number of topics ($K$).
They will affect the perplexity, and how to get best configuration is still an art.
First, in we tried asymmetric prior topic specific $\beta_k$ to offset the side effect of sparse initialization, like asymmetric prior $\alpha_k$ that aims to improve the model robustness.
However, the impact of asymmetric priors that are word or document specific and how to set the proper asymmetric prior are still unknown and remain future work.
Second, prefers larger topics at first and deduplicates topics by merging similar topics. However, too large number of topics than needed would result in the inefficient statistical inference <cit.>.
Lastly, like many other systems, there are several heuristics involved in , such as
how to set the sparsity in initialization, how to set the right sampling rate for “converged” token exclusion, how to dynamically enable them, as well as how to identify them and how to set the certain threshold is still manual work.
Graph partitioning.
Conventional graph partitioning algorithms usually assume that the network I/O introduced by cutting different vertices is the same.
However, this assumption does not remain true in LDA training. For example,
Given a vertex with less degrees but more dense word-topic and document-topic distribution, the partition strategy that cuts that vertex may introduce more the network I/O in step 2 and 4 of (Figure <ref>).
Besides, consider the situation where “converged” edge exclusion is enabled, the number of active edges would gradually decrease. This will conversely affect the partitioning approach that
an approach may get good load balance at first, but it gradually becomes imbalance as training proceeds.
Currently, GraphX does not permit different typed attributes for different vertices. This prohibits us to efficiently exploit the difference between hot words and long tail words.
It would also be interesting to theoretically analyze the impact of sparse initialization and “converged” token exclusion, and how to systematically neutralize their side effect remains future work.
§ CONCLUSION
In this paper, we present that proves to be an efficient and scalable collapsed Gibbs sampling system for LDA model on distributed data-parallel platform.
This reflects our belief that build distributed machine learning system is not only feasible and beneficial, but also efficient and scalable.
comes from combined innovations from both algorithm side and system side, and both are indispensable to achieve the goal.
A clear abstraction like RDD in Spark can accelerate the research on both sides, respectively.
We will continue this methodology and add more and more models in the future.
|
1511.00572
|
$\Lambda$-CDM type Heckmann - Suchuking model and union 2.1 compilation
G. K. Goswami$^1$, R. N. Dewangan$^2$ & Anil Kumar Yadav$^3$
$^{1, 2} $Department of Mathematics, Kalyan P. G. College, Bhilai - 490006, India
Email: [email protected]
$^3$ Department of Physics, United College of Engineering & Research,
Greater Noida - 201306, India
Email: [email protected]
In this paper, we have investigated $\Lambda$-CDM type cosmological model in
Heckmann-Schucking space-time, by using 287 high red shift ($ .3 \leq z \leq 1.4$ ) SN Ia
data of observed absolute magnitude along with their
possible error from Union 2.1 compilation. We have used $\chi^{2}$ test
to compare Union 2.1 compilation observed data and
corresponding theoretical values of apparent magnitude $(m)$. It is found that the best fit value
for $(\Omega_{m})_0$, $(\Omega_{\Lambda})_0$ and $(\Omega_{\sigma})_0$ are $0.2940$, $0.7058$ and $0.0002$ respectively
and the derived model represents the features of accelerating universe which is consistent with recent
astrophysical observations.
Key words: Dark energy, $\Lambda$-CDM cosmology and deceleration parameter.
PACS: 98.80.Es, 98.80-k
§ INTRODUCTION
Wilkinson Microwave Anisotropy Probe (WMAP)[25] and
Hubble Key Project (HKP)[9] explored that our universe is nearly flat.
This has given concept of two component density parameters $\Omega_{m}$ and $\Omega_{\Lambda}$ ,
which are related through
\begin{equation}\label{eq1}
\Omega_{\Lambda}+\Omega_{m}=1.
\end{equation}
Equation (1) is obtained by solving Einstein's Field
Equations with cosmological constant for FRW cosmological model,
which represent a spatially homogeneous and isotropic accelerating expanding flat universe. One can see the details of
$\Lambda$-CDM model in Refs. [1-18].
Luminosity distance $(D_{L})$ in $\Lambda$- CDM model is as follows:
\begin{equation}
\label{eq2}
\end{equation}
\begin{equation}\label{eq3}
\end{equation}
The luminosity distance $(D_{L})$ is associated with absolute and
apparent magnitudes by the following equation
\begin{equation}\label{eq4}
\end{equation}
To get the absolute magnitude $M$ of a supernova, we consider a supernova 1992P at
low-redshift z = 0.026 with m = 16.08.
Equations (<ref>) and (<ref>) read as
\begin{equation}
\label{eq5}
\end{equation}
These equations (<ref>)-(<ref>) produce the following expression for absolute
magnitude $m$.
\begin{equation}
\end{equation}
In the last decade of $20^{th}$ century, Riess et al. [20] and Perlmutter et al. [18]
found that the present
values of $\Omega_{m}$ and $\Omega_{\Lambda}$ are nearly $0.29$ and
$0.71$ respectively. Perlmutter et al. had used only 60 SN Ia low red shift data set while in the
present analysis, we have used 287 high red shift data out of 500 SN Ia data set as reported in ref. [26].
The recent SN Ia observations, BOSS, WMAP and Plank result for CMB anisotropy [12] give more precise
value of cosmological parameters. After publication of WMAP data, we notice that today, there is considerable evidence in support of
anisotropic model of universe. On the theoretical front, Misner [16] has investigated
an anisotropic phase of universe, which turns into isotropic one. The authors of ref. [17]
have investigated the accelerating model of universe with anisotropic EOS parameter and have also shown that
the present SN Ia data permits large anisotropy. Recently DE models with variable EOS parameter in anisotropic
space-time have been studied by Yadav and Yadav [27], Yadav et al [28,29],
Akarsu and Kilinc [3], Yadav [30], Saha and Yadav [22] and Pradhan [19].
In Ref. [10], we have presented a $\Lambda$-CDM type cosmological model in spatially
homogeneous and anisotropic Heckmann-Schucking space-time given by
\begin{equation}\label{metric}
ds^{2}= c^{2}dt^{2}- A^{2}dx^{2}-B^{2}dy^{2}-C^{2}dz^{2},
\end{equation}
Where $A(t)$, $B(t)$ and $C(t)$ are scale factors along $x$, $y$ and $z$ axes. In the literature, metric (<ref>)
is also named as Bianchi type I [31].
We consider energy momentum tensor for a perfect fluid i.e.
\begin{equation}\label{emt}
\end{equation}
Where $g_{ij}u^{i}u^{j}=1$ and $u^{i}$ is the 4-velocity vector.
In co-moving co-ordinates
\begin{equation}
u^{\alpha}=0,~~~~~~~~~\alpha=1,2, 3.
\end{equation}
The Einstein field equations are
\begin{equation}
\label{efe}
R_{ij}-\frac{1}{2}Rg_{ij} + \Lambda g_{ij}= -\frac{8\pi
\end{equation}
Choosing co-moving coordinates, the field equations (<ref>), for the line element (<ref>), read as
\begin{equation}
\label{fe1}
\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{B}\dot{C}}{BC}=-\frac{8\pi G}{c^{2}} p+\Lambda
\end{equation}
\begin{equation}\label{fe2}
\frac{\ddot{A}}{A}+\frac{\ddot{C}}{C}+\frac{\dot{A}\dot{C}}{AC}=- \frac{8\pi G}{c^{2}}p+\Lambda
\end{equation}
\begin{equation}\label{fe3}
\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}=- \frac{8\pi G}{c^{2}}p+\Lambda
\end{equation}
\begin{equation}\label{fe4}
\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{C}\dot{A}}{AC}= \frac{8\pi G}{c^{2}}\rho+\Lambda
\end{equation}
Solving equations (<ref>) - (<ref>) by approach given in ref. [10], one can obtain the following
relation for scale factors
\begin{equation}
\label{sf1}
B = AD(t),
\end{equation}
\begin{equation}
\label{sf2}
C = AD(t)^{-1},
\end{equation}
\begin{equation}
\label{sf3}
D(t) = exp\left[\int\frac{K}{A^{3}}\right],
\end{equation}
Where $K$ is the constant of integration.
In view of equations (<ref>) - (<ref>), equations (<ref>) - (<ref>), read as
\begin{equation}\label{fe5}
G}{3c^{2}}\Bigl(p-\frac{\Lambda c^{4}}{8\pi
G}+\frac{K^{2}c^{2}}{8\pi GA^{6}}\Bigr).
\end{equation}
\begin{equation}\label{fe6}
G}{3c^{2}}\Bigl(\rho+\frac{\Lambda c^{4}}{8\pi
G}+\frac{K^{2}c^{2}}{8\pi GA^{6}}\Bigr).
\end{equation}
Now, we assume that the cosmological constant $\Lambda$ and the
term due to anisotropy also act like energies with densities and
pressures as
$$\rho_{\Lambda}=\frac{\Lambda c^{4}}{8\pi G}\hspace{.5in}
\rho_{\sigma}=\frac{K^{2}c^{2}}{8\pi GA^{6}}$$
\begin{equation}
p_{\Lambda}=-\frac{\Lambda c^{4}}{8\pi G}\hspace{.5in}
p_{\sigma}=\frac{K^{2}c^{2}}{8\pi GA^{6}}.
\end{equation}
It can be easily verified that energy conservation law holds
separately for $\rho_{\Lambda}$ and $\rho_\sigma$ i.e.
The equations of state for matter, $ \sigma$ and $\Lambda$ energies are
read as
\begin{equation}
\end{equation}
where $\omega_m = 0$ for matter in form of dust,
$\omega_m=\frac{1}{3}$ for matter in form of radiation. There are
certain more values of $\omega_m$ for matter in different forms
during the course of evolution of the universe.
\begin{equation}
\end{equation}
$$p_{\Lambda}+ \rho_{\Lambda}=0,$$
where $\omega_{m}$, $\omega_{\Lambda}$ and $\omega_{\sigma}$ are equation of state parameters of matter,
$\Lambda$ and $\sigma$.
In the literature, $\Lambda$-CDM model is described by FLRW metric. In the derived model, equations (<ref>) and
(<ref>) represent the field equation of FLRW metric that is why the derived model is $\Lambda$-CDM type. For flat
$\Lambda$-CDM model, we have
\begin{equation}
\label{eq1}
\Omega_{\Lambda}+\Omega_{m}+ \Omega_{\sigma} = 1,
\end{equation}
where $
\Omega_{m}=\frac{\rho_{m}}{\rho_{c}}
=\frac{(\Omega_m)_0H^2_0(1+z)^3}{H^2}$, $
\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{c}}
=\frac{(\Omega_\Lambda)_0H^2_0}{H^2}$ and
$\Omega_{\sigma}$ is the anisotropic energy density which is taken very small for present analysis. The recent
observation of CMB support the existence of anisotropy in early phase of evolution of universe. Therefore, it
make sense to consider the model of universe with an anisotropic background in the presence of cosmological term $\Lambda$.
The expression for luminosity distance $(D_{L})$ and apparent magnitude $(m)$
are as follows
\begin{equation}
\end{equation}
\begin{equation}
m =16.08 +5log_{10}(\frac{(1+z)}{.026}\int_{0}^{z}\frac{dz}
\end{equation}
The purpose of the present work is to compute the $\Omega_{\Lambda}$ and
the $\Omega_{m}$ of the universe in light of Union $2.1$ compilation in anisotropic space-time which is
different from our previous work [10]. In our previous work [10], we used old SN Ia data
to compute the physical parameters at present epoch. Here the anisotropic energy density $(\Omega_{\sigma})$ is taken
to be very small i. e.
$\Omega_{\sigma}$ = 0.0002. In order to obtain $\Omega_{\Lambda}$
and $\Omega_{\sigma}$, we have considered high red-shift SN Ia supernova data of observed $m$
along with their possible error from Union $2.1$
compilation and have obtained corresponding theoretical values of
$m$ for various $\Omega_m$, ranging in between $0$ and $1$. The paper is organized as follows.
In section 2, we have estimated the
values of $\Omega_{\Lambda}$ and $\Omega_{m}$. Section 3 deals with the matter and dark energy densities and
estimation of present age of universe. Finally the results are displayed in section 4.
§ ESTIMATION OF $(\OMEGA_{M})_{0}$ AND $(\OMEGA_{\LAMBDA})_{0}$ FROM 287 SN IA DATA SET
For the sake of comparison of theoretical value of $\Omega_{m}$ with observational values, we compute the
$\chi^{2}$ value as following:
$ $
$A =\overset{287} {\underset{i=1}{\sum}}\frac{\left[\left(m\right)_{ob}-\left(m\right)_{th}\right]^{2}}
$B =\overset{287} {\underset{i=1}{\sum}}\frac{\left[\left(m\right)_{ob}-\left(m\right)_{th}\right]}
$ $
$C =\overset{287} {\underset{i=1}{\sum}}\frac{1}{\sigma_{i}^{2}}$.
Here the summations are taken over data sets of observed and theoretical values of apparent magnitudes of 287 supernovae.
Based on the above expressions, we obtain
the following Table-1 which describes the various values of $\chi^{2}$
against values of $\Omega_{m}$ ranging between $0$ to $1$
$\Omega_{m}$ $\chi_{SN}^{2}$
5522.1000 19.24076655 0.1000
50002.1000 174.22334495 0.2000
4833.7000 16.84216028 0.2800
4799.8000 16.72404181 0.2900
4799.2000 16.72195122 0.2940
4799.3000 16.72229965 0.9998
4799.2000 16.72195122 0.2960
4799.3000 16.72229965 0.9996
5276.7000 18.38571429
Table: 1
From Table-1, we find that for minimum value of $\chi^2$, the best fit present values of $\Omega_{m}$ and
$\Omega_{\Lambda}$ are $0.2940$ and $0.7058$ respectively.
In order to compare the various theoretical value of luminosity distance
$D_L$ corresponding to different values of $\Omega_{m}$ with observational values, we
compute again $\chi^{2}$ values as following:
$ $
$ $
Here the summations are over data sets of observed and theoretical values of luminosity distances of 287 supernovae.
Based on the above expressions , we obtain
the following Table-2 which describes the various values of $\chi^{2}$
against values of $\Omega_{m}$ ranging between $0$ to $1$
$\Omega_{m}$ $\chi_{SN}^{2}$
225.5970 0.78605226 0.1000
204.7956 0.71357352 0.2000
198.0609 0.69010767 0.2800
196.7042 0.68538049 0.2900
196.6806 0.68529826 0.2910
196.6797 0.68529512 0.2920
196.9790 0.68633798 0.2930
196.6785 0.68529094 0.2940
196.6783 0.68529024 0.2960
196.6786 0.68529129 0.2970
196.6791 0.68529303 0.3000
196.6821 0.68530348 0.9998
215.6214 0.75129408
Table: 2
From Table-2, we find that for minimum value of $\chi^2$, the best fit present values of $\Omega_{m}$ and
$\Omega_{\Lambda}$ are presented again as $(\Omega_{m})_0$ = $0.2940$ and $(\Omega_{\Lambda})_0$ = $0.7058$ respectively.
§ SOME PHYSICAL PARAMETERS OF THE UNIVERSE
§.§ Density of the universe and Hubble's constant
$ $
The energy density at present is given by
\begin{equation}
\label{eq7}
(\rho_{i})_{0}=\frac{3c^{2}H_{0}^{2}}{8\pi G}(\Omega_{i})_{0},
\end{equation}
where i stands for different types of energies such as matter energy,
dark energy etc.
Taking, $(\Omega_{m})_{0}$ = 0.2940, $H_{0}=72km/sec./Mpc$.
The current value of dust energy $(\rho_{m})_{0}$ for flat
universe is given by
\begin{equation}
\label{eq8} (\rho_{m})_{0}= 0.5527 h^2_0\times10^{-29}gm/cm^{3}.
\end{equation}
The current value of dark energy $(\rho_{\Lambda})_{0}$ read as
\begin{equation}
\label{eq9} (\rho_{\Lambda})_{0}=\frac{3c^{2}H_{0}^{2}}{8\pi
G}(\Omega_{\Lambda})_{0}= 1.3269 h^2_0\times10^{-29}gm/cm^{3},
\end{equation}
Where $(\Omega_{\Lambda})_{0}$= 0.7058.
The expression for Hubble parameter is given by
\begin{equation}
\label{eq10}
\end{equation}
\begin{equation}
\label{eq11}
\end{equation}
§.§ Age of the universe
The present age of the universe is obtained as follows
\begin{equation}\label{eqt}
\int^{\infty}_0\frac{dz}{H_0 (1+z)\sqrt{\bigl[(\Omega_m)_0(1+z)^3+(\Omega_\sigma)_0(1+z)^6+(\Omega_\Lambda)_0\bigr]}},
\end{equation}
\begin{equation}\label{eqt1}
\int^{\infty}_0\frac{dz}{H_0 (1+z)\sqrt{\bigl[(\Omega_m)_0(1+z)^3+(\Omega_\sigma)_0(1+z)^6+(\Omega_\Lambda)_0\bigr]}}.
\end{equation}
From equation (<ref>), one can easily obtain $t_0\rightarrow 0.9388H_0^{-1}$ for high
redshift and $(\Omega_\Lambda)_0=0.7058$. This means that the present age of
the universe is 12.7534 Gyrs $\sim$13 Gyrs for $\Lambda$ dominated
universe. From WMAP data, the empirical value of present age of
universe is $13.73_{-.17}^{+.13}Gyrs$ which is closed to
present age of universe, estimated in the this paper.
§.§ Deceleration parameter $q$:
The deceleration
parameter is given by
\begin{equation}
q = \frac{3}{2}\Bigl( \frac{(\Omega_m)_o (1+z)^3 +
2(\Omega_\sigma)_o (1+z)^6}{(\Omega_m)_o
(1+z)^3+(\Omega_\Lambda)_o + (\Omega_\sigma)_o
\end{equation}
Since in the derived model, the best fit values of
$(\Omega_{m})_{0}$, $(\Omega_{\Lambda})_{0}$ and
$(\Omega_{\sigma})_{0}$ are 0.2940, 0.7058 and 0.0002 respectively
hence we compute the present value of deceleration parameter for
derived $\Lambda$-CDM universe by putting $z = 0$ in eq. (19). The
present value of DP comes to
\begin{equation}
q_{0} = -0.5584.
\end{equation}
Also it is evident that the universe had entered in the accelerating phase
at $z\thicksim0.6805\backsimeq t\thicksim 0.4442 H^{-1}_0\thicksim 6.0337\times10^9 yrs$ in
the past before from now.
§ RESULT AND DISCUSSION
In the present work, $\Lambda$-CDM type cosmological model
in Heckmann-Suchuking space-time has been investigated.
We summarize our work by presenting the following table which
displays the values of cosmological parameters at present.
Cosmological Parameters Values at Present
$(\Omega_\Lambda)_0$ .7058
$(\Omega_m)_0$ .2940
$(\Omega_\sigma)_0$ .0002
$(q)_0$ -0.5584
$\small{(\rho_{m})_{0}}$ $0.5527 h^2_0\times10^{-29}gm/cm^{3}$
$\small{(\rho_{\Lambda}})_{0}$ $1.3269h^2_0\times10^{-29}gm/cm^{3}$
$\small{Age~ of~ the~ universe}$ $12.7534~Gyrs $
The figures 1 and 2 shows how the observed values of apparent magnitudes
and luminosity distances come close to the theoretical graphs for
$(\Omega_\Lambda)_0 = 0.7058$, $(\Omega_m)_0 = 0.2940$ and
$(\Omega_\sigma)_0 = 0.0002$. Figures 3 and 4 shows the
dependence of Hubble's constant with
red shift and scale factors. Figure 5 shows that the time tends to a definite value for large
redshift which in turn determines the age of universe.
The various figures validates that the best fit value for energy parameters corresponding to matter and
dark energy are $0.2940$ and $0.7058$.
As a final comment, we note that the
present model represents the features of accelerating universe. The present value of DP is found $-0.5584$ which
is consistence with modern astrophysical observations.
Fig1: Apparent magnitude $(m)$ versus redshift $(z)$
Fig2: Luminosity distance $(D_{L})$ versus redshift $(z)$
Fig3: Hubble constant $\left(\frac{H}{H_{0}}\right)$
versus redshift $(z)$
Fig4: Hubble constant $\left(\frac{H}{H_{0}}\right)$
versus scale factor
Fig5: Time versus redshift $(z)$
§ ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee for valuable comments to improve the quality of manuscript. One of us G. K. G. is thankful to IUCAA, Pune, India for providing facility and support where part of this work was carried out during a visit.
This work is supported by the CGCOST Research Project 789/CGCOST/MRP/14.
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|
1511.00536
|
I suggest that stars introduce mass and density scales that lead
to `naturalness' in the Universe. Namely, two ratios of order
unity. (1) The combination of the stellar mass scale,
$M_\ast(c,\hbar,G, m_p, m_e, e, \dots)$, with the Planck mass,
$M_{\rm Pl}$, and the Chandrasekhar mass leads to a ratio of order
unity that reads $N_{{\rm Pl \ast}} \equiv {M_{\rm Pl}}/({M_\ast}
m^2_p )^{1/3} \simeq 0.15 - 3$, where $m_p$ is the proton mass.
(2) The ratio of the density scale, $\rho_{\rm D \ast}(c,\hbar,G,
m_p, m_e, e, \dots) \equiv ( G ~ \tau^2_{\rm nuc \ast} ) ^{-1}$,
introduced by the nuclear life time of stars, $\tau_{\rm nuc
\ast}$, to the density of the dark energy, $\rho_{\Lambda}$, is
$N_{\lambda \ast} = {\rho_{\Lambda}}/{\rho_{\rm D \ast}}
\approx 10^{-7} - 10^{5}$.
Although the range is large, it is critically much smaller than
the 123 orders of magnitude usually referred to when
$\rho_{\Lambda}$ is compered to the Planck density. In the pure
fundamental particles domain there is no naturalness; either
naturalness does not exist or there is a need for a new physics or
new particles. The `Astrophysical Naturalness' offers a third
possibility: stars introduce the combinations of, or relations
among, known fundamental quantities that lead to naturalness.
§ INTRODUCTION
The naturalness topic is nicely summarized by Natalie Wolchover in
an article from May 2013 in Quanta
I here discuss two points as listed in the talk “Where are we
heading?” given by Nathan Seiberg in
(1) “Why doesn't dimensional analysis work? All dimensionless
numbers should be of order one”; (2) “The cosmological constant
is quartically divergent - it is fine tuned to 120 decimal
My answer to the first point is that in astrophysics dimensional
analysis does work when stars are considered as fundamental
entities. This answers the second question as well. If the nuclear
lifetime of stars is taken to be a dynamical time of the Universe,
then naturalness emerges from the observed cosmological constant.
No fine tuning is required.
Many relations among microscopic quantities and their relations
with macroscopic quantities are discussed by <cit.>
who try to explain these relations, or else they refer to the
anthropic principle to account for some of the relations. A more
recent study was conducted by <cit.>. Here I
do not repeat the explanations in those two papers. I simply take
stars to provide the relations among the many physical constants
and particles properties, and show that naturalness emerges from
the relations introduced by stars. One might refer to relations
introduced by stars as coincidental (e.g.,
), but in the present essay I prefer to
refer to these relations as naturalness. My goal is to suggest a
third option to treat naturalness, as explained in the last
This essay does not discover anything new, but rather suggests to
include stars as `fundamental entities' when considering
naturalness in our Universe. As naturalness was discussed in talks
and popular articles, I use them as references. I also limit the
discussion to two commonly discussed quantities in relation to
naturalness, the Planck mass and the cosmological constant (dark
energy). Many other relations and coincidences can be found in the
papers by <cit.> and <cit.>. I
will not touch the question of multiverse which is often connected
to the values of fundamental quantities, e.g., <cit.>
and <cit.>.
§ THE CHANDRASEKHAR MASS
The Planck mass that starts the discussion on naturalness is
defined as
\begin{equation}
M_{\rm Pl}= \left( \frac{\hbar c}{G} \right)^{1/2} = 2.177 \times
10^{-5} \g.
\label{eq:mpl}
\end{equation}
It is many orders of magnitude above the mass of the Higgs boson
and all other fundamental particles. If one constrains himself to
the particle world, no naturalness exists (e.g.
). Let us add stars.
We start with the Chandrasekhar mass limit $M_{\rm Ch}$. This is
the maximum mass where a degenerate electron gas can support a
body against gravity. The electrons are relativistic at this mass
limit, and the expression reads
\begin{equation}
M_{\rm Ch}=K_1 \left( \frac{Z}{A} \right)^2
\left( \frac{M_{\rm Pl}}{m_p} \right)^2 M_{\rm Pl} = K_1 \left( \frac{Z}{A} \right)^2 M_{\rm BCh} ,
\label{eq:mch}
\end{equation}
where $m_p=1.673 \times 10^{-24} \g$ is the proton mass, and $Z$
and $A$ are the atomic number and atomic mass number,
respectively, of the element(s) composing the white dwarf (the
ratio $Z/A$ is the mean number of electrons per nucleon in the
white dwarf). The constant $K_1 \simeq 3.1$ is composed of pure
numbers (no physical constants), and $K_1 (Z/A)^2 \simeq 0.8$ for
white dwarfs in nature where $Z=0.5A$. The last equality defines
what I term the bare Chandrasekhar mass
\begin{equation}
M_{\rm BCh} \equiv \left( \frac{M_{\rm Pl}}{m_p} \right)^2
M_{\rm Pl} = \alpha_G^{-1} M_{\rm Pl} = 1.85 M_\odot ,
\label{eq:bmch}
\end{equation}
where $\alpha_G=Gm^2_p/\hbar c=5.9\times 10^{-39}$ is the
gravitational fine structure constant, which is also used to
express $M_{\rm BCh}$, e.g., <cit.>.
§ NATURALNESS WITH STARS
The mass of stars is determined by the requirement that hydrogen
burns to helium. From below it is limited by brown dwarfs, where
the star cannot compress and heat enough to ignite hydrogen. The
minimum mass for a star is $M_\ast> 0.08 M_\odot$. The maximum
stellar mass of hundreds solar masses is not well determined, but
radiation pressure limits the upper mass (e.g.,
). Interestingly, the
Chandrasekhar mass sits more or less in the center of the stellar
mass range in logarithmic scale (e.g., )
\begin{equation}
N_{{\rm M \ast}} \equiv \frac{M_{\rm BCh}}{M_\ast} = \left(
\frac{M_{\rm Pl}}{m_p} \right)^2 \frac{M_{\rm Pl}}{M_\ast} \simeq
0.01 - 20.
\label{eq:nm}
\end{equation}
In the logarithmic scale the range of this ratio is approximately
$-2$ to $1.4$, much-much smaller than the 17 orders of magnitude
difference between the mass of the Higgs boson and the Planck
mass. Moreover, if the ratio is with the Planck mass rather than
$M_{\rm BCh}$, then the ratio is closer to unity, as it reads
\begin{equation}
N_{{\rm Pl \ast}} \equiv \frac{M_{\rm Pl}}{\left( M_\ast m^2_p
\right)^{1/3}} \simeq 0.15 - 3.
\label{eq:npl}
\end{equation}
It is important to emphasize that the mass of stars is determined
by the requirement that hydrogen experience thermonuclear burning
to helium. The Chandrasekhar mass is determined from the pressure
that a degenerate electrons gas can hold against gravity. Nothing
demands them to be equal. But they are. Namely, the ratio of the
Chandrasekhar mass, that is composed of the Planck and the proton
masses, to stellar mass is or order unity. This is
Of course, the properties of stars are determined by the
properties of the four fundamental forces, as all of them are
involved in the nuclear burning and stellar structure, and the
properties of the particles involved. The question is what
combination of the fundamental constants of the forces and of the
particles' properties gives two quantities whose ratio is $\approx
1$? The answer here is that stars form this combination as
\begin{equation}
M_\ast=M_\ast(c,\hbar,G, m_p, m_e, e, {\rm Forces~of~nature,}
\dots),
\label{eq:Mstarf}
\end{equation}
hence give us the naturalness in the Universe. This is expressed in
equations (<ref>) and (<ref>).
In other words, much as the proton `forms' a combination from the
properties of the quarks and the electric and color forces to give
a mass, the proton mass $m_p$, so do stars. But stars build a much
more complicated combination, and with many more, of the
fundamental constants and forces, and the output of this relation
is not quantized, but it is rather a continuous function.
I note that <cit.> try to show that $N_{{\rm Pl
\ast}} \sim 1$ is expected. However, they had to use numbers from
more complicated calculations than just order of magnitude
estimates. They specifically use the nuclear burning temperature
of hydrogen, $T_H$, and take a factor of $q \sim 10^{-2}$ in the
expression $k T_H = q m_e c^2$. <cit.>
consider the ratio between the maximum stellar mass and the
Chandrasekhar mass, and take the extra (external) factor that
comes from observations and detailed calculations to be the ideal
gas pressure to total pressure ratio $\beta$. In setting the lower
stellar mass limit <cit.> take another extra
parameter to get the burning temperature of hydrogen. The
parameter is the ratio of the Gamow energy to $k_B T/3$, which
they set equal to 5. That is, it is not trivial to express stellar
properties from fundamental particles and physical constants. My
approach is different. I avoid these extra parameters. I take
stars to simply provide the relations among the different
There is also the demand that the baryonic density in the Universe
be high enough for stars to form in the first place (e.g.,
). A related natural ratio is
discussed in section <ref>
§ STELLAR EXPLOSION ENERGY
The naturalness has several implications. One of them is that
regular stars can lead to white dwarfs with a mass close to and
above the Chandrasekhar mass. White dwarfs with that mass or
above, and iron cores of massive stars with that mass, explode
eventually as a supernova. White dwarfs explode as thermonuclear
supernovae where carbon and oxygen burn to nickel; cores of
massive stars explode as core-collapse supernovae where a neutron
star is formed. The typical kinetic energy of the ejected gas in
supernovae, $\approx 10^{51} \erg$, can be derived from
fundamental quantities.
The radius of an idealized white dwarf supported by a degenerate
non-relativistic electrons gas is given by
\begin{equation}
R_{\rm WD}= K_2
\frac {\hbar^2} {G ~m_e ~m^{5/3}_p ~M^{1/3}_{\rm WD}}
\left( \frac{Z}{A}\right)^{5/3},
\label{eq:rwd}
\end{equation}
where $M_{\rm WD}$ is the white dwarf mass and the constant $K_2
\approx 1$ is composed of pure numbers. For other forms of this
expression for the white dwarf radius see
<cit.>. Although for a white dwarf at the
Chandrasekhar mass the electrons gas is degenerate, I nonetheless
substitute the bare Chandrasekhar mass $M_{\rm BCh}$ in equation
(<ref>) to obtain an estimate for the bare white dwarf
\begin{equation}
R_{\rm BWD} \equiv
\frac {\hbar^2}{G ~m_e ~m^{5/3}_p ~M_{\rm BCh}^{1/3}}
= \frac {G}{m_e c^2} \frac{M^3_{\rm Pl}}{m_p} = 5000 \km. % 5026 km
\label{eq:rmch}
\end{equation}
Due to the factor $(Z/A)^{5/3}$ the real radius is smaller by a
factor of $\approx 3$. We can define the bare gravitational-energy
of the bare white dwarf as
\begin{equation}
E_{\rm BCh} \equiv \frac{G M^2_{\rm BCh}}{R_{\rm BWD}} =
\left( \frac{M_{\rm Pl}}{m_p} \right)^3 m_e c^2
= 1.8 \times 10^{51} \erg .
\label{eq:Emch}
\end{equation}
This is the typical kinetic energy of the mass ejected in
supernova explosions, either massive stars (core collapse
supernovae) or white dwarfs (Type Ia supernovae). Simply the
explosion energy is of the order of the binding energy.
Accurate calculations give lower binding energy values to
exploding white dwarfs and collapsing cores by a factor of
several. This is because the internal energy has a positive value.
The explosion kinetic energy is then several times the binding
energy. But this does not change the argument.
The factor $( {M_{\rm Pl}}/{m_p} )^3$ is the number of nucleons in
the white dwarf, so that the binding energy per nucleon is
$\approx m_e c^2$. This is also the order of magnitude of the
nuclear energy released per nucleon when carbon and oxygen burn to
nickel. This nuclear energy is the energy source of type Ia
supernovae. Indeed, about $20-60 \%$ of the white dwarf burns to
nickel during a type Ia supernova.
When a core of a massive star collapses to a neutron star it
releases a total energy of $\approx {\rm few} \times 10^{53}
\erg$. This energy comes from the final radius of the neutron star
which is determined from nuclear repulsive forces acting against
gravity. Most of this energy is carried out by neutrinos (and
anti-neutrinos) of the three kinds.
§ THE NATURALNESS OF DARK ENERGY
The usual approach to search for naturalness is to compare the
observed density of the dark energy $\rho_{\Lambda}= 7 \times
10^{-30} \g \cm^{-3}$ with the Planck density $\rho_{\rm
Pl}(c,\hbar,G)=M_{\rm Pl} l^{-3}_{\rm Pl}= 5.155\times 10^{93} \g
\cm^{-3}$, where $l_{\rm Pl}=(\hbar G/c^3)^{1/2}$ is the Planck
length. This gives an `unnatural' ratio of
$U=\rho_{\Lambda}/\rho_{\rm Pl}(c,\hbar,G)= 10^{123}$. We in
astrophysics are not accustomed to such astronomical numbers. This
`unnatural' ratio is referred to as the cosmological constant
problem (e.g., ). A density function to
replace $M_{\rm Pl} l^{-3}_{\rm Pl}$ is required.
As we saw in previous sections, stars introduce a (complicated)
combination of the fundamental quantities to give a mass ratio of
order unity, that is, an astrophysical mass naturalness (equation
<ref>). Stars also introduce some typical time scales, like
their dynamical time scale, thermal time scale, and nuclear life
time. I take here the nuclear time scale which is the life time
over which a star evolves, as I compare the quantity with the dark
energy that is related to the evolution of the Universe.
Stars spend most of their nuclear lives burning hydrogen to
helium. The nuclear life time of stars depends mainly on the
initial mass of the star, with $\tau_{\rm nuc \ast}(0.1M_\odot)
\approx 10^{13} \yr$, $\tau_{\rm nuc \ast}(M_{\rm BCh}) \approx
10^{9} \yr$, and $\tau_{\rm nuc \ast}(M \ga 10M_\odot) \approx
10^{7} \yr$. I now equate this nuclear time scale to a dynamical
time scale $t_{D\ast} \equiv \tau_{\rm nuc \ast} = (G \rho_{\rm D
\ast})^{-1/2}$ associated with a density
\begin{equation}
\rho_{\rm D \ast} \equiv \left( G ~ \tau^2_{\rm nuc \ast} \right)
^{-1} \approx 10^{-34} - 10^{-22} \g \cm^{-3}.
\label{eq:rhod}
\end{equation}
This density comes for the nuclear life time of stars that depends
on many fundamental parameters. Namely,
\begin{equation}
\rho_{\rm D \ast}=\rho_{\rm D \ast}(c,\hbar,G, m_p, m_e, e, {\rm Forces~of~nature,} \dots)
\label{eq:rhodf}
\end{equation}
The point here is that stars introduce the basic relation among
these fundamental quantities.
The second natural number defined in this essay is therefore
\begin{equation}
N_{\lambda \ast} \equiv \frac{\rho_{\Lambda}}{\rho_{\rm D \ast}}
\approx 10^{-7} - 10^{5}.
\label{eq:nlambda}
\end{equation}
Although the range is large, it is critically much smaller than
the 123 orders of magnitude usually referred to when
$\rho_{\Lambda}$ is compered to `natural density'. Moreover, a
ratio of unity sits just near the center of this range.
I conclude that stars introduce a nuclear time scale, whose
associated dynamical time scale leads to a density about equal to
the dark energy density. Again, the nuclear time scale of stars is
determined by a complicated relation of fundamental quantities,
constants and particle properties. Stars combine the fundamental
quantities to lead to a naturalness.
It is important to emphasize that the approach here is different
than the question “Why does the cosmological constant (dark
energy) became significant only recently? (e.g.,
). Namely, why the age of the universe is
about equal to the dynamical time associated with the density of
the cosmological density? The approach here also differs from
coincidental identities that are related to the present age or
size of the Universe (e.g., ).
In the present approach the age of the universe has no importance
at all. The same argument presented here holds as soon as hydrogen
becomes the main element in the universe; the first minute of the
universe, at an age of $10^{-16}$ times the present Universe age.
The same argument will be true when the universe be $10^{16}$
times its present age (as long as the dark energy density stays
It is true that if the cosmological constant (dark energy) had
been much larger, stars would not have formed (see, e.g.,
, and also for other time scales
involving the cosmological constant). The value of primordial
density fluctuations is also related to the question of star
formation <cit.>. But here I don't examine these
questions; I look for ratios of order unity, i.e., naturalness.
§ SUMMARY
The naturalness question I studied here can be posed as follows:
“What is the combination of the fundamental constants and
particle properties that leads to a ratio of two values that is of
order unity?” In the present essay I showed that stars introduce
these combinations that give what might be termed “Astrophysical
Stars introduce the stellar mass given in equation
(<ref>) that leads to the natural relation
(<ref>), or (<ref>). Stars also introduce a nuclear
timescale. If this time scale is associated with a dynamical time
scale, then a density $\rho_{\rm D \ast}$ given by a very
complicated relation (eq. <ref>) is defined. The density
leads to the natural relation (<ref>).
Nathan Seiberg summarizes his talk by a diagram that leaves two
basic options, ($i$) abandon naturalness, or ($ii$) go beyond
known physics/particles to find naturalness. Here I take a third
option which is basically to add stars as a basic entity in our
Universe, much as the proton is a composite particle. This brings
out naturalness in a beautiful way, at least in the eyes of an
The arguments presented here are not the anthropic principle,
e.g., as presented by <cit.>. <cit.>
list the necessity of stars to form in order to have life. I
differ here in two respects. (1) I treat stars on the same level
as I treat baryons. I do not require that the properties of
protons allow stars as <cit.> do. I simply treat
stars as I treat baryons (although definitely stars are more
complicated and composed of baryons). Both baryons and stars are
composite entities that exist in the Universe. They appear on the
same level in equations (<ref>) and (<ref>). (2) The
arguments presented here do not require the presence of carbon in
the universe. All arguments here apply if nuclear reactions would
have ended with helium. As well, life requires some chemical
properties. The arguments presented here don't involve chemistry
at all. For example, even if all stars were much hotter and the
strong UV radiation would prevent life, the arguments presented
here still hold.
I thank my graduate students for their patient to listen to my
arguments on naturalness.
[Burrows & Ostriker(2014)]BurrowsOstriker2014 Burrows, A. S., &
Ostriker, J. P. 2014, Proceedings of the National Academy of
Science, 111, 2409
[Carr & Rees(1979)]CarrRees1979 Carr, B. J., & Rees, M. J. 1979, , 278, 605
[Carroll(2002)]Carroll2002 Carroll, S. 2002, KITP: Colloquium Series, 2
[Dine(2015)]Dine2015 Dine, M. 2015, Annual Review of
Nuclear and Particle Science, 65, 18001
[Garriga et al.(2000)]Garrigaetal2000 Garriga, J., Livio, M.,
& Vilenkin, A. 2000, , 61, 023503
[Livio & Rees(2005)]LivioRees2005 Livio, M., & Rees, M. J.2005, Science, 309, 1022
[Weinberg(2005)]Weinberg2005 Weinberg, S. 2005,
See e-mail from the Journal on January 6.
https://www.quantamagazine.org/20130524-is-nature-unnatural/ taken
from Scientific American
>From this talk by Nati Seiberg
I take these sentences: "Why doesn't dimensional analysis work?
All dimensionless numbers should be of order one. " ** It works
with stars!! ** "The cosmological constant is quartically
divergent - it is fine tuned to 120 decimal points. " ** No fine
tune when a star is a natural entity of the universe **
Life is a natural consequences of naturalness-it is not the
anthropic principle! The stars are basic entities, not life. Much
as the existence of the proton of the electron are not the
anthropic principle.
|
1511.00037
|
6in!KN-spaces, infinite root stacks, and the profinite homotopy type of log schemes]
Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes
Department of Mathematical Sciences
George Mason University
4400 University Drive, MS: 3F2
Exploratory Hall
Fairfax, Virginia 22030
Mathematisches Institut
Endenicher Allee 60
53115 Bonn
Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver, BC, V6T 1Z2
Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver, BC, V6T 1Z2
[2010]MSC Primary: 14F35, 55P60; Secondary: 55U35
For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space <cit.>. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack <cit.>. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C},$ these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite étale homotopy type of its infinite root stack.
§ INTRODUCTION
Log schemes are an enlargement of the category of schemes due to Fontaine, Illusie and Kato <cit.>. The resulting variant of algebraic geometry, “logarithmic geometry”, has applications in a variety of contexts ranging from moduli theory to arithmetic and enumerative geometry (see <cit.> for a recent survey).
In the past years there have been several attempts to capture the “log” aspect of these objects and translate it into a more familiar terrain. In the complex analytic case, Kato and Nakayama introduced in <cit.> a topological space $X_\log$ (where $X$ is a log analytic space), which may be interpreted as the “underlying topological space” of $X$, and over which, in some cases, one can write a comparison between logarithmic de Rham cohomology and ordinary singular cohomology. In a different direction, for a log scheme $X,$ Kato
introduced two sites, the Kummer-flat site $X_{Kfl}$ and the Kummer-étale site $X_{Ket}$, that are analogous to the small fppf and étale site of a scheme, and were used later by Hagihara and Nizioł <cit.> to study the K-theory of log schemes.
Recently in <cit.>, the fourth author together with Vistoli introduced and studied a third incarnation of the “log aspect” of a log structure, namely the infinite root stack $\sqrt[\infty]{X}$, and used it to reinterpret Kato's Kummer sites and link them to parabolic sheaves on $X$. This stack is defined as the limit of an inverse system of algebraic stacks $\sqrt[\infty]{X}=\varprojlim_n \sqrt[n]{X}$, parameterizing $n$-th roots of the log structure of $X$.
The infinite root stack can be thought of as an “algebraic incarnation” of the Kato-Nakayama space: if $X$ is a log scheme locally of finite type over $\bC$, both $X_\log$ and $\sqrt[\infty]{X}$ have a map to $X$. The fiber of $X_\log\to X_\an$ over a point $x\in X_\an$ is homeomorphic to $(S^1)^r$, where $r$ is the rank of the log structure at $x$. For all $n,$ the reduced fiber of $\sqrt[n]{X}\to X$ over the corresponding closed point of $X$ is equivalent to the classifying stack $\class \left(\mathbb{Z}/{n\mathbb{Z}}\right)^r$ (for the same $r$). Regarding the infinite root stack not as the limit $\varprojlim_n \sqrt[n]{X},$ but instead as the diagram of stacks
$$n \mapsto \sqrt[n]{X},$$
i.e. as a pro-object or “formal limit,” yields then that the reduced fiber of $\sqrt[\infty]{X}\to X$ is the diagram of stacks,
$$n \mapsto \class \left(\mathbb{Z}/{n\mathbb{Z}}\right)^r,$$
which regarded as a pro-object is simply $\class \widehat{\mathbb{Z}}^r\simeq \widehat{\class{\mathbb{Z}}^r}$, the profinite completion of $\left(S^1\right)^r.$
In this paper we formalize this analogy and prove a comparison result between the profinite completions of $X_\log$ and $\sqrt[\infty]{X}$ for a fine saturated log scheme $X$ locally of finite type over $\bC$. Furthermore, we put this result in a wider circle of ideas, centered around the concept of the profinite homotopy type of a log scheme.
Our approach relies in a crucial way on a careful reworking of the foundations of the theory of topological stacks and profinite completions within the framework of $\infty$-categories <cit.>. This allows us to have greater technical control than earlier and more limited treatments, and plays an important role in the proof of our main result.
In the second half on the paper we construct a comparison map between $X_\log$ and $\sqrt[\infty]{X}$ and show that it is induces an equivalence between their profinite completions. The proof involves an analysis of the local geometry of log schemes, and a local-to-global argument which reduces the statement to a local computation. Next, we review the main ideas in the paper in greater detail.
§.§ Topological stacks and profinite completions of homotopy types
The first ingredient that we need in order to compare $X_\log$ and $\sqrt[\infty]{X}$ is the notion of a topological stack <cit.> associated with an algebraic stack. This is an extension of the analytification functor defined on schemes and algebraic spaces, that equips algebraic stacks with a topological counterpart, and allows one for example to talk about their homotopy type. Given an algebraic stack $\mathcal{X}$ locally of finite type over $\bC$, let us denote by $\mathcal{X}_{top}$ its “underlying topological stack”. This formalism allows us to carry over $\sqrt[\infty]{X}$ to the topological world, where $X_\log$ lives.
The second ingredient we need is a functorial way of associating to a topological stack its homotopy type. Although this is in principle accomplished in <cit.> and <cit.>, the construction is a bit complicated and it is difficult to notice the nice formal properties this functor has from the construction. We instead construct a functor $\Pi_\i$ associating to a topological stack $\cX$ its fundamental $\i$-groupoid. The source of this functor is a suitable $\i$-category of higher stacks on topological spaces, and the target is the $\i$-category $\cS$ of spaces. Using the language and machinery of $\i$-categories makes the construction and functoriality of $\Pi_\i$ entirely transparent; it is the unique colimit preserving functor which sends each space $T$ to its weak homotopy type.
The third ingredient we need is a way of associating to a space its profinite completion. Combining this with the functor $\Pi_\i$ gives a way of associating to a topological stack a profinite homotopy type. The notion of profinite completion of homotopy types is originally due to Artin and Mazur <cit.>. Profinite homotopy types have since played many important roles in mathematics, perhaps most famously in relation to the Adams conjecture from algebraic topology <cit.>. A more modern exposition using model categories is given in <cit.>; however the notion of profinite completion is a bit complicated in this framework. Finally, Lurie briefly introduces an $\i$-categorical model for profinite homotopy types in <cit.>, which has recently been shown to be equivalent to Quick's model in <cit.> (and also to a special case of Isaksen's). The advantage of Lurie's framework is that the definition of profinite spaces and the notion of profinite completion become very simple. A $\pi$-finite space is a space $X$ with finitely many connected components, and finitely many homotopy groups, all of whom are finite, and a profinite space is simply a pro-object in the $\i$-category of $\pi$-finite spaces. The profinite completion functor
$$\widehat{\left(\blank\right)}:\cS \to \Profs$$
from the $\i$-category of spaces to the $\i$-category of profinite spaces preserves colimits, and composing this functor with $\Pi_\i$ gives a colimit preserving functor $\Pip$ which assigns a topological stack its profinite homotopy type. This property is used in an essential way in the proof of our main theorem. Using this machinery, we are able to derive some non-trivial properties of profinite spaces that are used in a crucial way to prove our main result; in particular we show that profinite spaces can be glued along hypercovers (Lemma <ref>).
§.§ The comparison map and the equivalence of profinite completions
Our main result states:
[see Theorem <ref>]
Let $X$ be a fine saturated log scheme locally of finite type over $\bC$.
Then there is a canonical map of pro-topological stacks
\Phi_X: X_{log} \rightarrow \sqrt[\infty]{X}_{top}
that induces an equivalence upon profinite completion
$$\Pip\left(X_{\log}\right) \stackrel{\sim}{\longlongrightarrow} \Pip\left(\sqrt[\infty]{X}_{top} \right).$$
This theorem makes precise the idea that the infinite root stack is an algebraic incarnation of the Kato-Nakayama space, and that it completely captures the “profinite homotopy type” (à la Artin-Mazur) of the corresponding log scheme.
The construction of the comparison map $\Phi_X$ is first performed étale locally on $X$ where there is a global chart for the log structure, and then globalized by descent. The local construction uses the quotient stack description of the root stacks, that reduces the problem of finding a map to constructing a (topological) torsor on $X_\log$ with an equivariant map to a certain space.
This permits the construction of $\Phi_X$ as a canonical morphism of pro-topological stacks over $X_{an}$:
\xymatrix{
X_{log} \ar[rr]^{\Phi_X} \ar[dr]_-{\pi_{log}} & & \sqrt[\infty]{X}_{top} \ar[dl]^-{\pi_\infty} \\
& X_{an}. & }
The jump patterns of the fibers of $\pi_{log}$ and $\pi_\infty$ reflect the way in which the rank of the
log structure varies over $X_{an}$. More formally, the log structure defines a canonical stratification on $X_{an}$ called the “rank stratification”, which makes $X_{log}$ and $\sqrt[\infty]{X}_{top}$ into stratified fibrations. After profinite completion,
the fibers of $\pi_{log}$ and $\pi_\infty$ on each stratum become equivalent: indeed they are equivalent
to real tori of dimension $n$, and to the (pro-)classifying stacks $\class \widehat{\mathbb{Z}}^n$. The fact that the fibers of $\pi_{log}$ and $\pi_\infty$ are profinite homotopy equivalent was in fact our initial intuition as to why the main result should be true.
Extracting from this fiber-wise statement a proof that
$\Phi_X$ induces an equivalence of profinite homotopy types requires a local-to-global argument that makes full
use of the $\infty$-categorical framework developed in the first half of the paper.
The Kato-Nakayama space models the topology of log schemes, but its applicability is limited to schemes over the complex numbers. Our results suggest
that the infinite root stack encodes all the topological information of log schemes (or at least its profinite completion) in a way that is exempt from this limitation. More precisely, if $X$ is a log scheme locally of finite type over $\bC$, there are three natural candidates for its “profinite homotopy type”: the profinite completion of the Kato-Nakayama space $X_\log$, the profinite étale homotopy type of $\sqrt[\infty]{X}$ and the profinite completion of the (pro-)topological stack $\sqrt[\infty]{X}_\topst$. Theorem <ref> and Theorem <ref> (proved in <cit.>)
imply that these three constructions give the same result.
This justifies the definition of the profinite homotopy type for a log scheme $X$, even outside of the complex case, as the profinite étale homotopy type of its infinite root stack $\sqrt[\infty]{X}$.
Another possible approach to this would be to define the homotopy type of a log scheme via Kato's Kummer-étale topos (see <cit.>). As proved in <cit.>, this topos is equivalent to an appropriately defined small étale topos of the infinite root stack. It is not immediate, however, to link the resulting profinite homotopy time and the one that we define in the present paper. We plan to address this point in future work.
We believe that our results hold in the framework of log analytic spaces as well. Even though root stacks of those have not been considered anywhere yet, the construction and results about them that we use in the present paper should carry through without difficulty, using some notion of “analytic stacks” instead of algebraic ones.
In recent unpublished work, Howell and Vologodsky give a definition of the motive of a log schemes inside Voevodsky's triangulated category of motives. Based on our results we
expect that infinite root stacks should provide
an alternative encoding of the motive of log schemes, or a profinite approximation of it. It is an interesting question to explore possible connections between these two viewpoints.
§.§ Description of content
The paper is structured as follows.
In the first two sections we develop the framework necessary to associate profinite homotopy types to (pro-)algebraic and topological stacks. Along the way, in Section <ref> we prove an interesting result (Theorem <ref>) which expresses the homotopy type of the Kato-Nakayama space of a log scheme as the classifying space of a natural category.
As a first step towards the main theorem, we construct in Section <ref> (Proposition <ref>) a canonical map of pro-topological stacks
\begin{equation}\label{eq:map}
\Phi_X\colon X_\log\to \sqrt[\infty]{X}_{\topst}
\end{equation}
by exploiting the local quotient stack presentations of the root stacks $\sqrt[n]{X}$, and gluing the resulting maps.
Section <ref> contains results about the topology of the Kato-Nakayama space and the topological infinite root stack that we use in an essential way in the proof of our main result.
In Section <ref>, we give the proof of Theorem <ref>: we show that the canonical map (<ref>)
induces an equivalence after profinite completion. The proof is based on a local-to-global analysis: we use a suitable hypercover $U^\bullet$ of $X_{an}$ constructed in Section <ref> to reduce the question to the restriction of the map $\Phi_X$ to each element of this hypercover. We then use the results about the topology of the Kato-Nakayama space and the topological infinite root stack proven in the same section to reduce to showing that the map induces a profinite homotopy equivalence along fibers. This concludes the proof.
Finally, in Section <ref> we make some remarks about the definition of the profinite homotopy type of a general log scheme.
In Appendix <ref>, we gather definitions and facts that we use throughout the paper about log schemes, the analytification functor, the Kato-Nakayama space, root stacks, and topological stacks.
In particular, in (<ref>), we carefully construct the “rank stratification” of $X$ (and $X_\an$), over which the characteristic monoid $\overline{M}$ of the log structure is locally constant.
§.§ Acknowledgements
All of the authors would like to thank their respective home institutions for their support.
We are also happy to thank Kai Behrend, Thomas Goodwillie, Marc Hoyois, Jacob Lurie, Thomas Nikolaus, Behrang Noohi, Gereon Quick, Angelo Vistoli, and Kirsten Wickelgren for useful conversations.
We are grateful to the anonymous referee for a careful reading and useful comments, in particular for pointing out the short proof of Proposition <ref>.
§.§ Notations and conventions
We will always work over a field $k$, which will almost always be the complex numbers $\bC$. In particular all our log schemes will be fine and saturated, and locally of finite type over $\bC$, unless otherwise stated.
If $P$ is a monoid we denote by $P^\gp$ the associated group. Our monoids will typically be integral, finitely generated, saturated and sharp (hence torsion-free). A monoid $P$ with these properties has a distinguished “generating set”, consisting of all its indecomposable elements. This gives a presentation of any such monoid $P$ through generators and relations.
If $F$ is a sheaf of sets on the small étale site of a scheme, its “stalks” will always be stalks on geometric points.
By an $\i$-category, we mean a quasicategory or inner-Kan complex. These are a model for $\left(\i,1\right)$-categories. We will follow very closely the notational conventions and terminology from <cit.>, and refer the reader to the index and notational index in op. cit. One slight deviation from the notational conventions just mentioned that will be made is that for $C$ and $D$ objects of an $\i$-category $\sC,$ we will denote by $\Hom_{\sC}\left(C,D\right)$ the space of morphisms from $C$ to $D$ in $\sC,$ rather than using the notation $\mbox{Map}_{\sC}\left(C,D\right),$ in order to highlight the analogy with classical category theory. A very brief heuristic introduction to $\i$-categories can be found in Appendix A of <cit.>. See also <cit.>.
§ PROFINITE HOMOTOPY TYPES
In this section we will introduce the $\i$-categorical model for profinite spaces that we will use in this article. This $\i$-category is introduced in <cit.>; a profinite space will succinctly be a pro-object in the $\i$-category of $\pi$-finite spaces. This notion is equivalent to the notion of profinite space introduced by Quick in <cit.> (see <cit.>), but the machinery and language of $\i$-categories is much more convenient to work with. Most importantly, the notion of profinite completion becomes completely transparent in this set up, and it is left-adjoint to the canonical inclusion of profinite spaces into pro-spaces, and hence in particular preserves all colimits. We use this fact in an essential way in the proof of our main result, and we do not know how to prove the analogous fact about profinite completion in any other formalism.
We start first by reviewing the notion of ind-objects and pro-objects.
We will interchangeably use the notation $\cS$ and $\iGpd$ for the $\i$-category of spaces, and the $\i$-category of $\i$-groupoids. These two $\i$-categories are one and the same, and we will use the different notations solely to emphasize in what way we are viewing the objects.
Recall that for $\sD$ a small category, the category of ind-objects is essentially the category obtained from $\sD$ by freely adjoining formal filtered colimits. This construction carries over for $\i$-categories. Moreover, if $\sD$ is an essentially small $\i$-category, the $\i$-category of ind-objects in $\sD,$ $\Ind\left(\sD\right),$ admits a canonical functor
$$j:\sD \to \Ind\left(\sD\right)$$
satisfying the following universal property:
For every $\i$-category $\sE$ which admits small filtered colimits, composition with $j$ induces an equivalence of $\i$-categories
$$\Fun_{\mathit{filt.}}\left(\Ind\left(\sD\right),\sE\right) \to \Fun\left(\sD,\sE\right),$$
where $\Fun_{\mathit{filt.}}\left(\Ind\left(\sD\right), \sE \right)$ denotes the $\i$-category of all functors $\Ind\left(\sD\right) \to \sE$ which preserve filtered colimits.
A more concrete description of the $\i$-category $\Ind\left(\sD\right)$ is as follows. First, recall the following proposition:
Denote by $\Pshi\left(\sD\right)$ the $\i$-category of $\i$-presheaves on $\sD,$ that is, the functor category
Let $\sD$ be an essentially small $\i$-category and let $F:\sD^{op} \to \iGpd$ be an $\i$-presheaf. Then the following conditions are equivalent:
i) The associated right fibration
$$\int_\sD F \to \sD$$
classified by $F$ has $\int_\sD F$ a filtered $\i$-category.
ii) There exists a small filtered $\i$-category $\cJ$ and a functor
$$f:\cJ \to \sD$$
such that $F$ is the colimit of the composite
$$\cJ \stackrel{f}{\longrightarrow} \sD \stackrel{y}{\hookrightarrow} \Pshi\left(\sD\right)$$
(where $y$ denotes the Yoneda embedding).
and if $\sD$ has finite colimits, $i)$ and $ii)$ are equivalent to:
iii) $F$ is left exact (i.e. preserves finite limits).
The $\i$-category $\Ind\left(\sD\right)$ may be described as the full subcategory of $\Pshi\left(\sD\right)$ satisfying the equivalent conditions $i)$ and $ii)$ (or $iii)$ if $\sD$ has finite colimits). In particular, this implies that $j$ is full and faithful, since it is a restriction of the Yoneda embedding. In a nutshell $\Pshi\left(\sD\right)$ is the $\i$-category obtained from $\sD$ by freely adjoining formal colimits, and $ii)$ above states that $\Ind\left(\sD\right)$ is the full subcategory thereof on those formal colimits of objects in $\sD$ which are filtered colimits.
The notion of a pro-object is dual to that of an ind-object; it is a formal cofiltered limit. By definition, the $\i$-category of pro-objects of an essentially small $\i$-category $\sD$ is
$$\Pro\left(\sD\right)\defeq \Ind\left(\sD^{op}\right)^{op}.$$
If $\sD$ has small limits, we see that $\Pro\left(\sD\right)$ can be described as the full subcategory of $\Fun\left(\sD,\iGpd\right)^{op}$ on those functors
$$F:\sD \to \iGpd$$
such that $F$ preserves finite limits. Since this definition makes sense even when $\sD$ is not essentially small, we make the following definition, due to Lurie:
If $\sE$ is any accessible $\i$-category with finite limits, then we define the $\i$-category of pro-objects of $\sE,$ $\Pro\left(\sE\right),$ to be the full subcategory of $\Fun\left(\sE,\iGpd\right)^{op}$ on those functors $F:\sE \to \iGpd$ which are accessible and preserve finite limits.
If $\sE$ is any accessible $\i$-category and $E$ is an object of $\sE,$ then the functor
$$\Hom\left(E,\blank\right):\sE \to \iGpd$$
co-represented by $E$ is accessible and preserves all limits. This induces a fully faithful functor
$$\sE \stackrel{j}{\hookrightarrow} \Pro\left(\sE\right).$$
The functor $j$ satisfies the following universal property:
If $\cD$ is any $\i$-category admitting small cofiltered limits, then composition with $j$ induces an equivalence of $\i$-categories
\begin{equation}\label{eq:pro}
\Fun_{\mathit{co-filt.}}\left(\Pro\left(\sE\right),\sD\right) \to \Fun\left(\sE,\sD\right),
\end{equation}
where $\Fun_{\mathit{co-filt.}}\left(\Pro\left(\sE\right),\sD\right)$ is the full subcategory of $\Fun\left(\Pro\left(\sE\right),\sD\right)$ spanned by those functors which preserve small cofiltered limits, see <cit.>.
If $\sC$ is any (not necessarily accessible) $\i$-category, there always exists an $\i$-category $\Pro\left(\sC\right)$ satisfying the universal property (<ref>). This is a special case of <cit.>.
Let $\sE$ be any accessible $\i$-category which is not necessarily essentially small. Let $\cU$ be the Grothendieck universe of small sets and let $\cV$ be a Grothendieck universe such that $\cU \in \cV,$ so that we may regard $\cV$ as the Grothendieck universe of large sets. Let $\widehat{\operatorname{Gpd}}_\i$ denote the $\i$-category of $\i$-groupoids in the universe $\cV.$ By the proof of <cit.>, it follows that the essential image of the composition
$$\Pro\left(\sE\right) \hookrightarrow \Fun\left(\sE,\iGpd\right)^{op} \hookrightarrow \Fun\left(\sE,\widehat{\operatorname{Gpd}}_\i\right)^{op}$$
consists of those functors $F:\sE \to \widehat{\operatorname{Gpd}}_\i$ for which there exists a small filtered $\i$-category $\cJ$ and a functor
$$f:\cJ \to \sE^{op}$$
such that $F$ is the colimit of the composite
$$\cJ \stackrel{f}{\longrightarrow} \sE^{op} \hookrightarrow \Fun\left(\sE,\widehat{\operatorname{Gpd}}_\i\right).$$
In light of Remark <ref>, any object $X$ of $\Pro\left(\sE\right),$ for $\sE$ an accessible $\i$-category, can be written as a cofiltered limit of a diagram of the form
$$F:\cI \to \sE \stackrel{j}{\hookrightarrow} \Pro\left(\sE\right),$$
or in more informal notation,
$$X = \underset{i \in \cI} \lim X_i.$$
Unwinding the definitions, we see that if $Y= \underset{j \in \cJ} \lim Y_j$ is another such object of $\Pro\left(\sE\right),$ then the usual formula for the morphism space holds:
$$\Hom_{\Pro\left(\sE\right)}\left(X,Y\right) \simeq \underset{j \in \cJ} \lim \underset{i \in \cI} \colim \Hom_{\sE}\left(X_i,Y_j\right).$$
Now suppose that $\sE$ has a terminal object $1.$ Then
$$\Hom_{\Pro\left(\sE\right)}\left(X,j\left(1\right)\right) \simeq \underset{i \in \cI} \colim \Hom_{\sE}\left(X_i,1\right).$$
Notice that each space $\Hom_{\sE}\left(X_i,1\right)$ is contractible since $1$ is terminal, and since $\left(-2\right)$-truncated objects (i.e. terminal objects) are closed under filtered colimits in $\cS$ by <cit.>, it follows that $\Hom_{\Pro\left(\sE\right)}\left(X,j\left(1\right)\right)$ itself is a contractible space, and hence we conclude that $j\left(1\right)$ is a terminal object.
Let $\sE=\cS$ be the $\i$-category of spaces. Then the $\i$-category of pro-spaces, $\Pro\left(\cS\right),$ can be identified with the opposite category of functors $F:\cS \to \cS$ such that $F$ is accessible and left exact. Notice that any space $X$ gives rise to a pro-space
$$\Hom\left(X,\blank\right):\cS \to \cS$$
which moreover preserves all limits. Moreover if $F:\cS \to \cS$ is any functor which preserves all limits, then by the Adjoint Functor theorem for $\i$-categories (<cit.>), $F$ must have a right adjoint $G$, and is moreover accessible by Proposition 5.4.7.7 of op. cit. This then implies that
\begin{eqnarray*}
\Hom\left(G\left(*\right),X\right) &\simeq& \Hom\left(*,F\left(X\right)\right)\\
&\simeq& F\left(X\right).
\end{eqnarray*}
Hence $F\simeq j\left(G\left(*\right)\right).$ We conclude that the essential image of
$$j:\cS \hookrightarrow \Pro\left(\cS\right)$$
is precisely those $\i$-functors $\cS \to \cS$ which preserve all small limits.
The functor
$$T:\Pro\left(\cS\right) \stackrel{\Hom\left(j\left(*\right),\blank\right)}{\longlonglongrightarrow} \cS$$
is right adjoint to the canonical inclusion $j:\cS \to \Pro\left(\cS\right).$
By Remark <ref>, $\Pro\left(\cS\right)^{op}$ may be identified with a subcategory of $\Fun\left(\cS,\widehat{\operatorname{Gpd}}_\i\right)$ of large $\i$-co-presheaves, and since limits commute with limits, this subcategory is stable under small limits. Note that this implies that $\Pro\left(\cS\right)$ is cocomplete. Since the Yoneda embedding into large $\i$-presheaves
$$\cS^{op} \stackrel{y}{\hookrightarrow} \LPshi\left(\cS\right)$$
preserves small limits, it follows that
$$j:\cS \hookrightarrow \Pro\left(\cS\right)$$
preserves small colimits. Since $\cS\simeq \Pshi\left(1\right),$ where $1$ is the terminal $\i$-category, and since $\Pro\left(\cS\right)$ is cocomplete, one has by <cit.> that $j\simeq \operatorname{Lan}_{y_1} \left(t\right)$ where $y_1$ is the Yoneda embedding $1 \to \cS$ and $t:1 \to \Pro\left(\cS\right)$ is the functor picking out the object $j\left(*\right).$ It follows immediately from the Yoneda lemma that $\Hom\left(j\left(*\right),\blank\right)$ is right adjoint to $\operatorname{Lan}_{y_1}\left(t\right).$
Let $P:\cS \to \cS$ be a pro-space. By <cit.>, since $P$ is accessible, it follows that the associated left fibration
$$\int_\cS P$$
is accessible, and hence has a small cofinal subcategory
$$r:\cC_P \hookrightarrow \int_\cS P,$$
and $P$ may be identified with the limit of the composite
$$\cC_P \stackrel{r}{\hookrightarrow} \int_\cS P \stackrel{\pi_P}{\longlongrightarrow} \cS \stackrel{j}{\hookrightarrow} \Pro\left(\cS\right).$$
We claim that
$$T\left(P\right)\simeq \lim \pi_P \circ r.$$
\begin{eqnarray*}
T\left(P\right) &=& \Hom\left(j\left(*\right),P\right)\\
&\simeq& \Hom\left(j\left(*\right),\lim j \circ \pi_P \circ r\right)\\
&\simeq& \lim \Hom\left(j\left(*\right), j\circ \pi_P \circ r\right)\\
&\simeq& \lim \Hom\left(*,\pi_P \circ r\right)\\
&\simeq& \lim \pi_P \circ r.
\end{eqnarray*}
By the same proof, if one has $P$ presented as a cofiltered limit $P = \lim j\left(X_\alpha\right)$ of spaces, then $T\left(P\right) \simeq \lim X_\alpha.$ In fact, this holds more generally by the following proposition:
Let $\sC$ be an accessible $\i$-category which admits small filtered limits. Then the canonical inclusion
$$j:\sC \hookrightarrow \Pro\left(\sC\right)$$
has a right adjoint $T$ and if $F:\cI \to \sC$ is a cofiltered diagram corresponding an object in $\Pro\left(\sC\right),$ then $T\left(F\right)= \lim F.$
By Remark <ref>, composition with
$$j:\sC \hookrightarrow \Pro\left(\sC\right)$$
induces an equivalence of $\i$-categories
$$\Fun_{\mathit{co-filt.}}\left(\Pro\left(\sC\right),\sC\right) \to \Fun\left(\sC,\sC\right),$$
so we can find a functor $T:\Pro\left(\sC\right) \to \sC$ and an equivalence
$$\eta:id_\sC \stackrel{\sim}{\longrightarrow} T \circ j.$$
Let $Z$ be an arbitrary object of $\Pro\left(\sC\right),$ then we can write $Z = \underset{i \in \cI} \lim j\left(X_i\right).$ First note that since $\eta$ is an equivalence and $T$ preserves cofiltered limits (by definition), we have that for such a $Z,$
$$T\left(Z\right) \simeq \underset{i \in \cI} \lim X_i.$$
This shows that $T$ has the desired properties on pro-objects. Let us now show that $T$ is a right adjoint to $j$. Let $C$ be an object of $\sC,$ then we have
\begin{eqnarray*}
\Hom_\sC\left(D,T\left(Z\right)\right) &\simeq& \Hom_\sC\left(D,\underset{i \in \cI} \lim X_i\right)\\
&\simeq& \underset{i \in \cI} \lim \Hom_\sC\left(D,X_i\right),
\end{eqnarray*}
and since $j$ is fully faithful, we have for each $i$
$$\Hom_\sC\left(D,X_i\right)\simeq \Hom_{\Prof\left(\sC\right)}\left(j\left(D\right),j\left(X_i\right)\right).$$
It follows then that
\begin{eqnarray*}
\Hom_\sC\left(D,T\left(Z\right)\right) &\simeq& \underset{i \in \cI} \lim \Hom_{\Prof\left(\sC\right)}\left(j\left(D\right),j\left(X_i\right)\right)\\
&\simeq& \Hom_{\Prof\left(\sC\right)}\left(j\left(D\right),\underset{i \in \cI} \lim j\left(X_i\right)\right)\\
&=& \Hom_{\Prof\left(\sC\right)}\left(j\left(D\right),Z\right).
\end{eqnarray*}
A space $X$ in $\cS$ is $\pi$-finite if all of its homotopy groups are finite, it has only finitely many non-trivial homotopy groups, and finitely many connected components.
Let $\sfc$ denote the full subcategory of the $\i$-category $\cS$ on the $\pi$-finite spaces. $\sfc$ is essentially small and idempotent complete (and hence accessible). The $\i$-category of profinite spaces is defined to be the $\i$-category
$$\Profs \defeq \Pro\left(\sfc\right).$$
Let $V$ be a $\pi$-finite space. Note that $V$ is $n$-truncated for some $n,$ since it has only finitely many homotopy groups. The associated profinite space $j\left(V\right)$ is also $n$-truncated.
Let $X=\underset{i \in \cI} \lim X_i$ be a profinite space. Then by Remark <ref>, we have that
$$\Hom_{\Profs}\left(X,j\left(V\right)\right) \simeq \underset{i \in \cI} \colim \Hom_{\cS^{fc}}\left(X_i,V\right).$$
Each space $\Hom_{\cS^{fc}}\left(X_i,V\right)$ is $n$-truncated since $V$ is, and $n$-truncated spaces are stable under filtered colimits by <cit.>, so it follows that $\Hom_{\Profs}\left(X,j\left(V\right)\right)$ is also $n$-truncated.
The assignment $\sC \mapsto \Pro\left(\sC\right)$ is functorial among accessible $\i$-categories with finite limits. Given a functor $f:\sC \to \sD,$ the composite
$$\sC \stackrel{f}{\longrightarrow} \sD \stackrel{j}{\hookrightarrow}\Pro\left(\sD\right)$$
corresponds to an object of the $\i$-category $\Fun\left(\sC,\Pro\left(\sD\right)\right),$ which by Remark <ref> is equivalent to the $\i$-category $\Fun_{\mathit{co-filt.}}\left(\Pro\left(\sC\right),\Pro\left(\sD\right)\right).$ Hence, one gets an induced functor
$$\Pro\left(f\right):\Pro\left(\sC\right) \to \Pro\left(\sD\right)$$
which preserves cofiltered limits. Moreover, $\Pro\left(f\right)$ is fully faithful if $f$ is. If $f$ happens to be accessible and left exact, then there is an induced functor in the opposite direction, given by
\begin{eqnarray*}
f^*:\Pro\left(\sD\right) &\to& \Pro\left(\sC\right)\\
\left(\sD \stackrel{F}{\longrightarrow} \iGpd\right) &\mapsto& \left(\sC \stackrel{f}{\longrightarrow}\sD \stackrel{F}{\longrightarrow} \iGpd\right),
\end{eqnarray*}
and $f^*$ is left adjoint to $\Pro\left(f\right).$ See <cit.> (but note there is a typo, since $f^*$ is in fact a left adjoint, not a right adjoint).
The canonical inclusion $i:\sfc \hookrightarrow \cS$ induces a fully faithful embedding
$$\Pro\left(i\right):\Prof\left(\cS\right) \hookrightarrow \Pro\left(\cS\right)$$
of profinite spaces into pro-spaces. Moreover, $i$ is accessible and preserves finite limits, hence the above functor has a left adjoint
$$i^*:\Pro\left(\cS\right) \to \Prof\left(\cS\right).$$
This functor sends a pro-space $P$ to its profinite completion.
We denote by $\widehat{\left(\blank\right)}$ the composite
$$\cS \stackrel{j}{\hookrightarrow} \Pro\left(\cS\right) \stackrel{i^*}{\longrightarrow} \Prof\left(\cS\right)$$
and call it the profinite completion functor. Concretely, if
$X$ is a space in $\cS,$ then $\widehat{X}$ corresponds to the composite
$$\sfc \stackrel{i}{\hookrightarrow} \cS \stackrel{\Hom\left(X,\blank\right)}{\longlonglongrightarrow} \cS.$$
This functor has a right adjoint given by the composite
$$\Prof\left(\cS\right) \stackrel{\Pro\left(i\right)}{\longlonghookrightarrow} \Pro\left(\cS\right) \stackrel{T}{\longrightarrow} \cS.$$
We will denote this right adjoint simply by $U$.
We will sometimes abuse notation and denote the profinite completion of a pro-space $Y$ also by $\widehat{Y}$ rather than $i^*Y,$ when no confusion will arise.
§.§ The relationship with profinite groups
In this subsection, we will touch briefly upon the relationship between profinite groups and profinite spaces. Recall the notion of profinite completion of a group. A profinite group is a pro-object in the category of finite groups. Equivalently, a profinite group is a group object in profinite sets, see <cit.>.
Denote by $i:FinGp \hookrightarrow Gp$ the fully faithful inclusion of the category of finite groups into the category of groups. The composite
$$Gp \hookrightarrow \Pro\left(Gp\right) \stackrel{i^*}{\longlongrightarrow} \Pro\left(FinGp\right)\simeq Gp\left(\Pro\left(FinSet\right)\right)$$
is the functor assigning a group its profinite completion. We also denote this functor by $\widehat{\left(\blank\right)}$ when no confusion will arise. Recall that the profinite completion of a group has a classical concrete description as follows: Let $G$ be a group, then its profinite completion is the limit $\underset{N} \lim j\left(G/N\right),$ where $N$ ranges over all the finite index normal subgroups of $G.$
Similarly, denote by $i_{ab}:FinAbGp \hookrightarrow AbGp$ the fully faithful inclusion of the category of finite abelian groups into the category of abelian groups. By the analogous construction to the above, there is an induced profinite completion functor
$$\widehat{\left(\blank\right)}_{ab}:AbGp \to \Pro\left(FinAbGp\right).$$
It can be described classically by the same formula as in the non-abelian case. If
$$\phi:AbGp \hookrightarrow Gp$$
is the canonical inclusion of abelian groups into groups, it follows that the following diagram commutes up to canonical natural equivalence:
$$\xymatrix@C=2.5cm{AbGp \ar[r]^-{\widehat{\left(\blank\right)}_{ab}} \ar[d]_-{\phi} & \Pro\left(FinAbGp\right) \ar[d]^-{\Pro\left(\phi\right)}\\Gp \ar[r]^-{\widehat{\left(\blank\right)}} & \Pro\left(FinGp\right).}$$
By <cit.>, there is a canonical equivalence of categories
$$\Pro\left(FinAbGp\right) \simeq AbGp\left(\Pro\left(FinSet\right)\right),$$
between the category of pro-objects in finite abelian groups and the category of abelian group objects in profinite sets. In particular, finite coproducts (direct sums) in $\Pro\left(FinAbGp\right)$ coincide with finite products. Since $\widehat{\left(\blank\right)}_{ab}$ is a left adjoint, it preserves direct sums, and by Remark <ref>, $\Pro\left(\phi\right)$ is a right adjoint (since $\phi$ preserves finite limits), so $\Pro\left(\phi\right)$ preserves products. It follows that the composite
$$\widehat{\left(\blank\right)} \circ \phi:AbGp \to \Pro\left(FinGp\right)$$
preserves finite products.
Let $k$ be a non-negative integer. Then there is a canonical isomorphism of profinite groups
$$\widehat{\mathbb{Z}^k} \cong \widehat{\mathbb{Z}}^k.$$
We now note a recent result which compares the $\i$-categorical model for profinite spaces just presented with the model categorical approach developed by Quick in <cit.>:
The $\i$-category associated to the model category presented in <cit.> is equivalent to $\Profs.$
The details of Quick's model category need not concern us here, but we cite the above theorem in order to freely use results of <cit.> about profinite spaces.
Let $k$ be a non-negative integer. There is a canonical equivalence of profinite spaces
$$\widehat{B\left(\mathbb{Z}^k\right)} \simeq B\left(\widehat{\mathbb{Z}}^k\right).$$
Since $\mathbb{Z}^k$ is a finitely generated free abelian group, it is good in the sense of Serre in <cit.>. It follows from <cit.> and Theorem <ref> that the canonical map
$$\widehat{B\left(\mathbb{Z}^k\right)} \to B\left(\widehat{\mathbb{Z}^k}\right)$$
is an equivalence of profinite spaces. The result now follows from Corollary <ref>.
The following lemma will be used in an essential way several times in this paper:
Let $f:\Delta \to \sC$ be a cosimplicial diagram and suppose that $\sC$ is an $\left(n+1,1\right)$-category, i.e. an $\i$-category whose mapping spaces are all $n$-truncated. Then, provided both limits exist, the canonical map
$$\lim f \to \lim \left(f|_{\Delta{\le n}}\right)$$
is an equivalence.
Let $\sC$ be an arbitrary $\left(\infty,n+1\right)$-category. Notice that for any diagram $f:\Delta \to \sC,$ and any object $C$ of $\sC,$ we have
$$\Hom\left(C,\underset{k \in \Delta} \lim f(k)\right) \simeq \underset{k \in \Delta} \lim \Hom\left(C, f(k)\right),$$
and since $\sC$ is an $\left(\infty,n+1\right)$-category, each $\Hom\left(C, f(k)\right)$ is an $n$-truncated space. Therefore the general case follows from the case when $\sC$ is the full subcategory $\cS^{\le n}$ of $\cS$ on the $n$-truncated spaces.
By <cit.>, to prove the lemma for the special case $\sC=\cS^{\le n},$ it suffices to prove the corresponding statement about homotopy limits in the Quillen model structure on the category of compactly generated spaces $\mathbf{CG}$, since the associated $\i$-category is $\cS.$
Suppose that
$$X^\bullet:\Delta \to \mathbf{CG}$$
is a cosimplicial space which is fibrant with respect to the projective model structure on $\Fun\left(\Delta,\mathbf{CG}\right)$ (with respect to the Quillen model structure on $\mathbf{CG}$), i.e., the diagram $X^\bullet$ consists entirely of Serre fibrations. Then the homotopy limit of $X^\bullet$ may be computed as $\operatorname{Tot}\left(X\right),$ and moreover, $\operatorname{Tot}\left(X\right)$ can be written as the (homotopy) limit of a tower of fibrations
$$\ldots \to \operatorname{Tot}\left(X\right)_k \to \operatorname{Tot}\left(X\right)_{k-1}\to \ldots \to \operatorname{Tot}\left(X\right)_1\to \operatorname{Tot}\left(X\right)_0=X,$$
where each $\operatorname{Tot}\left(X\right)_k$ is a model for the homotopy limit of $X|_{\Delta_{\le k}}.$ Moreover, the (homotopy) fiber of each map
$$\operatorname{Tot}\left(X\right)_k \to \operatorname{Tot}\left(X\right)_{k-1}$$
is homotopy equivalent to the $k$-fold loop space $\Omega^k\left(M^k\left(X^\bullet\right)\right),$ where
$$M^k X^\bullet = \underset{{{ [k+1]} \twoheadrightarrow {[j]}\atop j \leq k}} \lim X^j$$
is the $k$-th matching object of $X^\bullet$ (see e.g. the introduction of <cit.>).
Now, let us assume that each $X_k$ is $n$-truncated. Then as $X^\bullet$ is fibrant, the diagram involved in the limit above consists entirely of fibrations, so the limit is a homotopy limit, hence each matching object is also $n$-truncated (since $n$-truncated objects are stable under limits in $\cS$ by <cit.>). It follows then that each homotopy fiber
$$\operatorname{Tot}\left(X\right)_k \to \operatorname{Tot}\left(X\right)_{k-1}$$
is weakly contractible for $k > n,$ and hence the natural map
$$\holim X^\bullet= \operatorname{Tot}\left(X\right) \to \operatorname{Tot}\left(X\right)_n=\holim X^\bullet|_{\Delta_{\le n}}$$
is a weak homotopy equivalence.
Let $\underset{i \in \cI} \lim G_i$ be a pro-object in the category of finite groups, or equivalently, a group object in $\Pro\left(\mathit{FinSet}\right).$ Consider the profinite space
$$B\left(\underset{i} \lim G_i\right)\defeq \colim\left[\ldots \underset{i} \lim G_i \times \underset{i} \lim G_i \rrrarrow \underset{i} \lim G_i \rightrightarrows *\right],$$
where the colimit is computed in $\Profs,$ and $*$ denotes the terminal profinite space. In more detail, the diagram whose colimit is being taken is the simplicial diagram which is the Čech nerve of the unique map $\underset{i} \lim G_i \to *$ in $\Profs.$ Consider for each $i$ the object in $\cS$
$$B\left(G_i\right)\defeq \colim\left[\ldots G_i \times G_i \rrrarrow G_i \rightrightarrows *\right],$$
i.e. the colimit in $\cS$ of the Čech nerve of $G_i.$ Then these spaces assemble into a profinite space $\underset{i} \lim B\left(G_i\right),$ and we have a canonical equivalence
$$B\left(\underset{i} \lim G_i\right) \simeq \underset{i} \lim B\left(G_i\right)$$
in $\Profs.$
It suffices to prove that for each $\pi$-finite space $V,$ we have an equivalence
$$\Hom_{\Profs}\left(B\left(\underset{i} \lim G_i\right),j\left(V\right)\right) \simeq \Hom_{\Profs}\left(\underset{i} \lim B\left(G_i\right),j\left(V\right)\right).$$
Recall that by Proposition <ref> $j\left(V\right)$ is $n$-truncated for some $n.$ As such, we have
\begin{eqnarray*}
\Hom_{\Profs}\left(B\left(\underset{i} \lim G_i\right),j\left(V\right)\right) &\simeq& \Hom_{\Profs}\left(\underset{\Delta^{op}}\colim N\left(\underset{i} \lim G_i\right),j\left(V\right)\right)\\
&\simeq& \underset{\Delta}\lim \Hom_{\Profs}\left(N\left(\underset{i} \lim G_i\right),j\left(V\right)\right)\\
&\simeq& \underset{\Delta_{\le n}} \lim \Hom_{\Profs}\left(N\left(\underset{i} \lim G_i\right),j\left(V\right)\right),
\end{eqnarray*}
that last equivalence following from Lemma <ref>. Expanding this out we get
$$\resizebox{6in}{!}{$\underset{\Delta_{\le n}} \lim \left[\Hom_{\Profs}\left(1,j\left(V\right)\right) \rightrightarrows \Hom_{\Profs}\left(\underset{i} \lim G_i,j\left(V\right)\right) \rrrarrow \Hom_{\Profs}\left(\left(\underset{i} \lim G_i\right)^2,j\left(V\right)\right) \ldots \Hom_{\Profs}\left(\left(\underset{i} \lim G_i\right)^n,j\left(V\right)\right)\right]$}$$
which is equivalent to
$$\resizebox{6in}{!}{$\underset{\Delta_{\le n}} \lim \left[\Hom_{\cS}\left(*,V\right) \rightrightarrows \underset{i} \colim \Hom_{\cS}\left(G_i,j\left(V\right)\right) \rrrarrow \underset{i} \colim \Hom_{\cS}\left(G_i^2,j\left(V\right)\right) \ldots \underset{i} \colim \Hom_{\cS}\left(G_i^n,j\left(V\right)\right)\right]$}$$
and since by <cit.> finite limits commute with filtered colimits in $\cS,$ we get
$$\resizebox{6in}{!}{$ \underset{i} \colim \underset{\Delta_{\le n}} \lim \left[\Hom_{\cS}\left(*,V\right) \rightrightarrows\Hom_{\cS}\left(G_i,V\right) \rrrarrow \Hom_{\cS}\left(G_i^2,j\left(V\right)\right) \ldots\Hom_{\cS}\left(G_i^n,V\right)\right].$}$$
Now since $j\left(V\right)$ is $n$-truncated by Proposition <ref>, it follows from Lemma <ref> that we can rewrite this as
$$\resizebox{6in}{!}{$ \underset{i} \colim \underset{\Delta} \lim \left[\Hom_{\cS}\left(*,V\right) \rightrightarrows\Hom_{\cS}\left(G_i,V\right) \rrrarrow \Hom_{\cS}\left(G_i^2,j\left(V\right)\right) \ldots\Hom_{\cS}\left(G_i^n,V\right) \ldots\right]$}$$
which is equivalent to
\begin{eqnarray*}
\underset{i} \colim \Hom_{\cS}\left( \underset{\Delta^{op}} \colim N\left(G_i\right),V\right) &\simeq& \underset{i} \colim \Hom_{\cS}\left( B\left(G_i\right),V\right)\\
&\simeq& \Hom_{\Profs}\left(\underset{i} \lim B\left(G_i\right),j\left(V\right)\right). {\qedhere}
\end{eqnarray*}
§ THE HOMOTOPY TYPE OF TOPOLOGICAL STACKS
In this section we use the formalism of $\i$-categories to produce two important constructions necessary for our paper. Firstly, we extend the construction of analytification, which sends a complex variety to its underlying topological space with the complex analytic topology, to a colimit-preserving functor
$$\left(\blank\right)_{top}:\Shi\left(\Aff,\mbox{\'et}\right) \to \Hshi\left(\TopC\right)$$
from the $\i$-category of $\i$-sheaves over the étale site of affine schemes of finite type over $\mathbb{C}$ to the $\i$-category of hypersheaves over an appropriate topological site. This functor, in particular, sends an Artin stack locally of finite type over $\mathbb{C}$ to its underlying topological stack in the sense of Noohi <cit.>. Using this functor, one associates to the infinite root stack $\sqrt[\i]{X}$ of a log-scheme a pro-topological stack $\sqrt[\i]{X}_{top}.$ In Section <ref>, we produce a map
\begin{equation}\label{eq:Knmap}
X_{log} \to \sqrt[\i]{X}_{top}
\end{equation}
from the Kato-Nakayama space to the underlying (pro-)topological stack of the infinite root stack. The main result of the paper is that this map is a profinite homotopy equivalence, but to make sense of such a statement, one first needs to associate to each of these objects a (pro-)homotopy type, in a functorial way. To achieve this, the second construction we produce is a colimit-preserving functor
$$\Pi_\i:\Hshi\left(\TopC\right) \to \cS$$
which sends every topological space $X$ to its underlying homotopy type, and sends every topological stack to its homotopy type in the sense of Noohi <cit.>. Using this construction and the map (<ref>), one has an induced map in $\Pro\left(\cS\right)$
$$\Pi_\i\left(X_{log}\right) \to \Pi_\i\left(\sqrt[\i]{X}_{top}\right)$$
which we prove in Section <ref> to become an equivalence after applying the profinite completion functor, i.e. the map (<ref>) is a profinite homotopy equivalence.
§.§ The underlying topological stack of an algebraic stack
Let $\Top$ be the category of topological spaces and let $\TopCs$ denote the full subcategory of $\Top$ of all contractible and locally contractible spaces which are homeomorphic to a subspace of $\mathbb{R}^n$ for some $n.$ Note that $\TopCs$ is essentially small. Denote by $\TopC$ the following subcategory of topological spaces:
$\bullet$ A topological space $T$ is in $\TopC$ if $T$ has an open cover $\left(U_\alpha \hookrightarrow T\right)_\alpha$ such that each $U_\alpha$ is an object of $\TopCs.$
We use the subscript $\mathbb{C}$ to highlight the fact that $\TopC$
will serve as the target of the analytification functor from the category of algebraic spaces over $\mathbb{C}$. Note that the objects of $\TopC$ are closed under taking open subspaces. As such, it makes sense to equip $\TopC$ with the Grothendieck topology generated by open covers. Denote by $\Hshi\left(\TopC\right)$ the $\i$-topos of hypersheaves on $\TopC,$ i.e. the hypercompletion of the $\i$-topos of $\i$-sheaves. There is also a natural structure of a Grothendieck site on $\TopCs$ as follows:
$\bullet$ Let $T$ be a space in $\TopCs.$ A covering family of $T$ consists of an open cover $\left(U_\alpha \hookrightarrow T\right)$ such that each $U_\alpha$ is in $\TopCs$.
Note that every open cover of $T$ can be refined by such a cover. We denote the associated $\i$-topos of hypersheaves by $\Hshi\left(\TopCs\right).$ By the Comparison Lemma of <cit.> III, we have that restriction along the canonical inclusion
$$\TopCs \hookrightarrow \TopC$$
induces an equivalence between their respective categories of sheaves of sets. It then follows from <cit.> and <cit.> that this lifts to an equivalence
$$\Hshi\left(\TopC\right) \stackrel{\sim}{\longrightarrow} \Hshi\left(\TopCs\right),$$
and in particular, $\Hshi\left(\TopC\right)$ is an $\i$-topos (which is not a priori clear for sites which are not essentially small).
Denote by $\Aff$ the category of affine schemes of finite type over $\mathbb{C}.$ Note that it is a small category with finite limits. Denote by
$$\left(\blank\right)_{an}:\Aff \to \Top$$
the functor associating to such an affine scheme its space of $\mathbb{C}$-points, equipped with the analytic topology. The above functor preserves finite limits, and is the restriction of a functor defined for all algebraic spaces locally of finite type over $\mathbb{C},$ see <cit.>. Note also that if $V$ is a scheme which is separated and locally of finite type,
then $V_{an}$ is locally (over any affine) a triangulated space by <cit.>, so in particular $V_{an}$ is locally contractible. Also observe that $V_{an}$ is locally cut-out of $\mathbb{C}^n$ by polynomials, so it follows that $V_{an}$ is in $\TopC.$
Consequently $\left(\blank\right)_{an}$ restricts to a functor
$$\left(\blank\right)_{an}:\Aff \to \TopC,$$
which preserves finite limits.
Note that the category $\Aff$ can be equipped with the Grothendieck topology generated by étale covering families. Denote the associated $\i$-topos of $\i$-sheaves on this site by $\Shi\left(\Aff,\mbox{\'et}\right).$
The following theorem is an extension of <cit.>:
The functor
$$\left(\blank\right)_{an}:\Aff \to \TopC$$
lifts to a left exact colimit preserving functor
$$\left(\blank\right)_{top}:\Shi\left(\Aff,\mbox{\'et}\right) \to \Hshi\left(\TopC\right).$$
Note that the image under $\left(\blank\right)_{an}$ of an étale map is a local homeomorphism. Also note that if
$$S \to T$$
is a local homeomorphism and $T$ is in $\TopC,$ so is $S.$ Furthermore, since the inclusion of any open subspace of a topological space is a local homeomorphism, and since any cover by local homeomorphisms can be refined by a cover by open subspaces, it follows that open covers and local homeomorphisms generate the same Grothendieck topology on $\TopC.$ It follows that any $\i$-sheaf on $\TopC$, so in particular any hypersheaf, satisfies descent with respect to covers by local homeomorphisms. The result now follows from <cit.>.
Denote by $Y$ the Yoneda embedding
$$Y:\TopC \hookrightarrow \Hshi\left(\TopC\right)$$
and denote by $y$ the Yoneda embedding
$$y:\Aff \hookrightarrow \Shi\left(\Aff,\mbox{\'et}\right).$$
Explicitly, $\left(\blank\right)_{top}$ is the left Kan extension of $Y \circ \left(\blank\right)_{an}$ along $y,$
$$\Lan_y\left[Y\circ \left(\blank\right)_{an}\right]:\Shi\left(\Aff,\mbox{\'et}\right) \to \Hshi\left(\TopC\right),$$
or more concretely, it is the unique colimit preserving functor such that for a representable $y\left(X\right),$ i.e. an affine scheme,
$$y\left(X\right)_{top}\cong Y\left(X_{an}\right).$$
By the proof of Theorem <ref>, we see that given any hypersheaf $F$ on $\TopC,$ the functor
$$F \circ \left(\blank\right)_{an}$$
is an $\i$-sheaf on $\left(\Aff,\mbox{\'et}\right),$ i.e. we have a well-defined functor
$$\left(\blank\right)_{an}^{*}:\Hshi\left(\TopC\right) \to \Shi\left(\Aff,\mbox{\'et}\right).$$
The functor $\left(\blank\right)_{top}$ is left adjoint to $\left(\blank\right)_{an}^{*}$.
Since $\left(\blank\right)_{top}$ is colimit preserving, it follows from <cit.> that it has a right adjoint. Let us denote the right adjoint by $R.$ By the Yoneda lemma, we have that if $F$ is a hypersheaf $F$ on $\TopC,$ then $R\left(F\right)$ is the $\i$-sheaf on $\left(\Aff,\mbox{\'et}\right)$ such that if $X$ is an affine scheme,
\begin{eqnarray*}
R\left(F\right)\left(X\right) &\simeq& \Hom\left(y\left(X\right),R\left(F\right)\right)\\
&\simeq& \Hom\left(\left(y\left(X\right)\right)_{top},F\right)\\
&\simeq& \Hom\left(Y\left(X_{an}\right),F\right)\\
&\simeq& F\left(X_{an}\right).
\end{eqnarray*}
The adjoint pair $\left(\blank\right)_{top} \dashv \left(\blank\right)_{an}^{*}$ assemble into a geometric morphism of $\i$-topoi
$$f:\Hshi\left(\TopC\right) \to \Shi\left(\Aff,\mbox{\'et}\right),$$
with direct image functor
and inverse image functor
Let $\As$ denote the category of algebraic spaces locally of finite type over $\mathbb{C}.$ Equip $\As$ with the étale topology. Then restriction along the canonical inclusion
$$\Aff \hookrightarrow \As$$
induces an equivalence of $\i$-categories
$$\Shi\left(\As,\mbox{\'et}\right) \stackrel{\sim}{\longrightarrow} \Shi\left(\Aff,\mbox{\'et}\right).$$
The inclusion satisfies the conditions of the Comparison Lemma of <cit.> III, so we have an induced equivalence
$$\Sh\left(\As,\mbox{\'et}\right) \stackrel{\sim}{\longrightarrow} \Sh\left(\Aff,\mbox{\'et}\right)$$
between sheaves of sets. Since both sites have finite limits, the result now follows from <cit.>.
Let $X$ be any algebraic space locally of finite type over $\mathbb{C}.$ Then $X_{top} \cong X_{an}.$
Let $\cU$ be the Grothendieck universe of small sets and let $\cV$ be the Grothendieck universe of large sets with $\cU \in \cV.$ Denote by $\LiGpd$ the $\i$-category of large $\i$-groupoids, and denote by $\widehat{\Hshi}\left(\TopC\right)$ the $\i$-category of hypersheaves on $\TopC$ with values in $\LiGpd,$ and similarly let $\widehat{\Shi}\left(\As,\mbox{\'et}\right)$ denote the $\i$-category of sheaves on the étale site of algebraic spaces with values in $\LiGpd$. Then by same proof as Theorem <ref>, by left Kan extension there is a $\mathcal{V}$-small colimit preserving functor
$$L:\widehat{\Shi}\left(\As,\mbox{\'et}\right) \to \widehat{\Hshi}\left(\TopC\right)$$
such that for all representable sheaves $y\left(P\right)$ on $\left(\As,\mbox{\'et}\right),$
$$L\left(y\left(P\right)\right)\cong Y\left(P_{an}\right).$$
By <cit.>, both inclusions
$$\Hshi\left(\TopC\right) \hookrightarrow \widehat{\Hshi}\left(\TopC\right)$$
$$\Shi\left(\Aff,\mbox{\'et}\right)\hookrightarrow \widehat{\Shi}\left(\Aff,\mbox{\'et}\right)$$
preserve $\cU$-small colimits. Hence both composites
$$\Shi\left(\Aff,\mbox{\'et}\right)\hookrightarrow \widehat{\Shi}\left(\Aff,\mbox{\'et}\right)\simeq \widehat{\Shi}\left(\As,\mbox{\'et}\right) \stackrel{L}{\longrightarrow} \widehat{\Hshi}\left(\TopC\right)$$
$$\Shi\left(\Aff,\mbox{\'et}\right) \stackrel{\left(\blank\right)_{top}}{\longlongrightarrow} \Hshi\left(\TopC\right) \hookrightarrow \widehat{\Hshi}\left(\TopC\right)$$
are $\mathcal{cU}$-small colimit preserving, and agree up to equivalence on every representable $y\left(X\right)$, for $X$ an affine scheme. It follows from <cit.> that both compositions must in fact be equivalent. However, the inclusion
$$\Shi\left(\Aff,\mbox{\'et}\right)\hookrightarrow \widehat{\Shi}\left(\Aff,\mbox{\'et}\right)\simeq \widehat{\Shi}\left(\As,\mbox{\'et}\right)$$
carries an algebraic space $P$ to its representable sheaf $y\left(P\right)$. The result now follows.
The following lemma follows immediately from the fact that $\left(\blank\right)_{top}$ preserves finite limits:
Let $\cG$ be a groupoid object in sheaves of sets on the étale site $\left(\Aff,\mbox{\'et}\right)$. Then applying $\left(\blank\right)_{top}$ level-wise produces a groupoid object in sheaves of sets on $\TopC$, denoted by $\cG_{top}$. Moreover, if the original groupoid $\cG$ is a groupoid object in algebraic spaces, then $\cG_{top}$ is degree-wise representable, i.e. a topological groupoid.
Let $\cG$ be a groupoid object in sheaves of sets on the étale site $\left(\Aff,\mbox{\'et}\right).$ Denote by $\left[\cG\right]$ its associated stack of torsors, and denote by $\left[\cG_{top}\right]$ the stack of groupoids on $\TopC$ associated to $\cG_{top},$ i.e. stack on $\TopC$ of principal $\cG_{top}$-bundles. Then $\left[\cG\right]_{top} \simeq \left[\cG_{top}\right].$
The stack $\left[\cG\right]$ is the stackification of the presheaf of groupoids $\tilde y\left(\cG\right)$ which sends an affine scheme $X$ to the groupoid $\cG\left(X\right).$ Denote by $N\left(\cG\right)$ the simplicial presheaf which is the nerve of this presheaf of groupoids. Consider the diagram
$$\Delta^{op} \stackrel{N\left(\cG\right)}{\longlongrightarrow} \Psh\left(\Aff,\mathit{Set}\right) \hookrightarrow \Psh\left(\Aff,\iGpd\right).$$
We claim that the colimit of the above functor is $\tilde y\left(\cG\right).$ Since colimits are computed object-wise, it suffices to show that if $\cH$ is any discrete groupoid, then $N\left(\cH\right)$ is the homotopy colimit of the diagram
$$\Delta^{op} \stackrel{N\left(\cH\right)}{\longlongrightarrow} \mathit{Set} \hookrightarrow \mathit{Set}^{{\Delta}^{op}},$$
which follows easily from the well-known fact that the homotopy colimit of a simplicial diagram of simplicial sets can be computed by taking the diagonal. It follows then that $\left[\cG\right]$ is the colimit of the diagram
$$\Delta^{op} \stackrel{N\left(\cG\right)}{\longlongrightarrow} \Sh\left(\Aff,\mbox{\'et}\right) \hookrightarrow \Shi\left(\Aff,\mbox{\'et}\right),$$
since $\i$-sheafification preserves colimits, as it is a left adjoint. By the same argument, we have that $\left[\cG_{top}\right]$ is the colimit of the diagram
$$\Delta^{op} \stackrel{N\left(\cG_{top}\right)}{\longlongrightarrow} \Sh\left(\TopC\right) \hookrightarrow \Shi\left(\TopC\right).$$
Notice that for all $n$ we have
The result now follows from the fact that $\left(\blank\right)_{top}$ preserves colimits.
A topological stack is a stack on $\TopC$ of the form $\left[\cG\right]$ for $\cG$ a groupoid object in $\TopC.$ Denote the associated $\left(2,1\right)$-category of topological stacks by $\mathfrak{TopSt}.$
In the literature, typically there is no restriction on a topological stack to come from a topological groupoid which is locally contractible, and such a stack is represented by its functor of points on the Grothendieck site of all topological spaces. However, the $\left(2,1\right)$-category of topological stacks in the sense we defined above embeds fully faithfully into the larger $\left(2,1\right)$-category of all topological stacks in the classical sense.
The functor
$$\left(\blank\right)_{top}:\Shi\left(\Aff,\mbox{\'et}\right) \to \Hshi\left(\TopC\right)$$
restricts to a left exact functor
$$\left(\blank\right)_{top}:\mathbf{A}\!\mathfrak{lgSt}^{LFT}_{\mathbb{C}} \to \mathfrak{TopSt}$$
from Artin stacks locally of finite type over $\mathbb{C}$ to topological stacks.
Up to the identification mentioned in Remark <ref>, the construction in the above corollary agrees with that of Noohi in <cit.>.
§.§ The fundamental infinity-groupoid of a stack
The following proposition will allow us to talk about homotopy types of topological stacks.
There is a colimit preserving functor
$$\overline{L}:\Hshi\left(\TopC^s\right) \to \cS$$
sending every representable sheaf $y\left(T\right)$ for $T$ a space in $\TopC^s$ to its weak homotopy type.
The proof is essentially the same as <cit.>. By Lemma 3.1 in op. cit., there is a functor
$$\TopC^s \hookrightarrow \Top \stackrel{h}{\longrightarrow} \cS$$
assigning to each space $T$ its associated weak homotopy type. Denote this functor by $\pi$. Since $\TopC^s$ is essentially small, by left Kan extension there is a colimit preserving functor
$$\Lan_Y \pi:\Pshi\left(\TopC^s\right) \to \cS$$
sending every representable presheaf $Y\left(T\right)$ to the underlying weak homotopy type of $T$. It follows from the Yoneda lemma that this functor has a right adjoint $R_\pi$ which sends an $\i$-groupoid $Z$ to the $\i$-presheaf
$$R_\pi\left(Z\right):T \mapsto \Hom\left(\pi\left(T\right),Z\right).$$
We claim that $R_\pi\left(Z\right)$ is a hypersheaf. To see this, it suffices to observe that if
$$U^{\bullet}:\Delta^{op} \to \TopC^s/T$$
is a hypercover of $T$ with respect to the coverage of contractible open coverings, then the colimit of the composite
$$\Delta^{op} \stackrel{U^\bullet}{\longlongrightarrow} \TopC^s/T \to \TopC^s \stackrel{\pi}{\longrightarrow} \cS$$
is $\pi\left(T\right),$ which follows from <cit.>. We thus have that $\Lan_y \pi$ and $R_\pi$ restrict to an adjunction
$$\overline{L} \dashv \Delta$$
between $\Hshi\left(\TopC^s\right)$ and $\cS,$ so in particular, $\overline{L}$ preserves colimits.
Let $\cG$ be an $\i$-groupoid. Denote by $\Delta\left(\cG\right)$ the constant presheaf on $\TopC^s.$ Then $\Delta\left(\cG\right)$ is a hypersheaf.
Following the proof of the above theorem, we have that $R_\pi\left(\cG\right)$ is a hypersheaf. Moreover, for each space $T$ in $\TopC^s,$ we have that
\begin{eqnarray*}
R_\pi\left(\cG\right)\left(T\right) &\simeq& \Hom\left(Y\left(T\right),R_\pi\left(\cG\right)\right)\\
&\simeq& \Hom\left(\overline{L}\left(Y\left(T\right)\right),\cG\right)\\
&\simeq& \Hom\left(*,\cG\right)\\
&\simeq& \cG,
\end{eqnarray*}
since each such $T$ is in fact contractible.
The $\i$-category $\cS$ of spaces is the terminal $\i$-topos. In particular, if $\sC$ is any $\i$-category equipped with a Grothendieck topology, then the unique geometric morphism
$$\Shi\left(\sC\right) \to \cS$$
has as direct image functor the global sections functor
$$\Gamma:\Shi\left(\sC\right) \to \cS$$
defined by $\Gamma\left(F\right)=\Hom\left(1,F\right),$ which is the same as $F\left(1\right)$ if $\sC$ has a terminal object. The inverse image functor is given by
$$\Delta:\cS \to \Shi\left(\sC\right)$$
and it sends an $\i$-groupoid $\cG$ to the sheafification of the constant presheaf with value $\cG$. Similarly, the unique geometric morphism
$$\Hshi\left(\sC\right) \to \cS$$
has its direct image functor $\Gamma$ given by the same construction as for $\i$-sheaves, and the inverse image functor $\Delta$ assigns an $\i$-groupoid $\cG$ the hypersheafification of the constant presheaf with values $\cG$. In either case we have $\Delta \dashv \Gamma.$ In particular, Corollary <ref> implies that for the $\i$-topos $\Hshi\left(\TopC^s\right)$ we have a triple of adjunctions:
$$\overline{L} \dashv \Delta \dashv \Gamma.$$
Although we will not prove it here, there is in fact a further right adjoint to $\Gamma,$ $coDisc \vdash \Gamma$ and moreover the quadruple
$$\overline{L} \dashv \Delta \dashv \Gamma \dashv coDisc$$
exhibits $\Hshi\left(\TopC^s\right)$ as a cohesive $\i$-topos in the sense of <cit.>.
The composite
$$\Hshi\left(\TopC\right) \stackrel{\sim}{\longrightarrow} \Hshi\left(\TopC^s\right) \stackrel{\overline{L}}{\longrightarrow} \cS$$
is colimit preserving and sends a representable sheaf $Y\left(X\right)$, for $X$ in $\TopC$, to its underlying weak homotopy type.
By <cit.>, there is a functor
$$\TopC \hookrightarrow \Top \stackrel{h}{\longrightarrow} \cS$$
assigning to each space $X$ its associated weak homotopy type. Denote this functor by $\Pi$. By exactly the same proof as Proposition <ref>, by using that $\TopC$ is $\cV$-small, with $\cV$ the Grothendieck universe of large sets, we construct a colimit preserving functor
$$\mathbb{L}:\widehat{\Hshi}\left(\TopC\right) \to \widehat{\cS},$$
where $\widehat{\cS}$ is the $\i$-category of large spaces (or large $\i$-groupoids), which sends every representable sheaf $Y\left(X\right)$ to its underlying weak homotopy type. The rest of the proof is analogous to that of Proposition <ref>.
We denote the composite from Proposition <ref> by
$$\Pi_\i:\Hshi\left(\TopC\right) \to \cS.$$
For $F$ a hypersheaf on $\TopC,$ we call $\Pi_\i\left(F\right)$ its fundamental $\i$-groupoid.
In light of Remark <ref>, we have that $\Pi_\i \dashv \Delta \dashv \Gamma,$ where $\Gamma$ is global sections, and $\Delta$ assigns an $\i$-groupoid $\cG$ the hypersheafification of the constant presheaf with value $\cG$. In particular, we have a formula for $\Delta\left(\cG\right),$ namely, if $X$ is a space in $\TopC,$
$$\Delta\left(\cG\right)\left(X\right)\simeq \Hom\left(\Pi_\i\left(X\right),\cG\right),$$
that is, the space of maps from the homotopy type of $X$ to $\cG.$
The following proposition may be seen as an extension of the results of <cit.>:
Let $\cG$ be a groupoid object in $\TopC$ and denote by $\left[\cG\right]$ denote the associated stack of groupoids on $\TopC$, i.e. the stack of principal $\cG$-bundles. Then $\Pi_\i\left(\left[\cG\right]\right)$ has the same weak homotopy type as
- the fat geometric realization of the simplicial space arising as the topologically enriched nerve of $\cG.$
We know that $\left[\cG\right]$ is the colimit in $\Hshi\left(\TopC\right)$ of the diagram
$$\Delta^{op} \stackrel{N\left(\cG\right)}{\longlongrightarrow} \TopC \stackrel{Y}{\hookrightarrow} \Hshi\left(\TopC\right)$$
(as in the proof of Proposition <ref>). The result now follows from Proposition <ref> and <cit.>
Let $F$ be a hypersheaf on $\TopC^s.$ Then $\overline{L}\left(F\right)$ is the colimit of $F,$ i.e. the colimit of the diagram
$$F:\left(\TopC^s\right)^{op} \to \cS.$$
By the proof of Proposition <ref>, $\overline{L}$ factors as the composition
$$\Hshi\left(\TopC^s\right) \hookrightarrow \Pshi\left(\TopC^s\right) \stackrel{\Lan_Y \pi}{\longlonglongrightarrow} \cS=\iGpd.$$
Note however that every space in $\TopC^s$ is contractible, so the canonical morphism $\pi \to t,$ to the terminal functor
$$t:\TopC^s \to \iGpd$$
(i.e. the functor with constant value the one point set), is an equivalence, and hence $\Lan_Y \pi$ is left adjoint to the constant functor $t^*$ which sends an $\i$-groupoid $\cG$ to the constant presheaf with value $\cG$. Since $\Pshi\left(\TopC^s\right)=\Fun\left(\left(\TopC^s\right)^{op},\iGpd\right),$ the result now follows from the universal property of $\colim\left(\blank\right).$
Let $F$ be a hypersheaf on $\TopC$. Then $\Pi_\i\left(F\right)$ is the colimit of $F|_{\TopC^s}.$
§.§ The profinite homotopy type of a (pro-)stack
Let us define the profinite version of the homotopy type of a stack.
We denote the composite
$$\Hshi\left(\TopC\right) \stackrel{\Pi_\i}{\longlongrightarrow} \cS \stackrel{\widehat{\left(\blank\right)}}{\longlongrightarrow} \Profs$$
by $\widehat{\Pi}_\i.$ For $F$ a hypersheaf on $\TopC,$ we call $\widehat{\Pi}_\i\left(F\right)$ its profinite fundamental $\i$-groupoid or simply its profinite homotopy type.
Let us extend the constructions of this section to pro-objects. Note that the functor
$$\left(\blank\right)_{top}:\Shi\left(\Aff,\mbox{\'et}\right) \to \Hshi\left(\TopC\right)$$
extends to a functor on pro-objects, which by abuse of notation, we will denote by the same symbol
$$\left(\blank\right)_{top}:\Pro\left(\Shi\left(\Aff,\mbox{\'et}\right)\right) \to \Pro\left(\Hshi\left(\TopC\right)\right).$$
We now describe how to define the profinite homotopy type of a pro-object in $\Hshi\left(\TopC\right)$. First, we may extend the profinite fundamental $\i$-groupoid functor on $\Hshi\left(\TopC\right)$ to pro-objects. This can be achieved easily by the universal property of $\Pro\left(\Hshi\left(\TopC\right)\right).$ Indeed, consider the functor
$$\widehat{\Pi}_\i:\Hshi\left(\TopC\right) \to \Profs$$
and denote its unique cofiltered limit preserving extension, by abuse of notation, again by
$$\widehat{\Pi}_\i:\Pro\left(\Hshi\left(\TopC\right)\right) \to \Profs.$$
Unwinding the definitions, we see that if $\underset{i \in \cI} \lim \cY^i$ is a pro-object in hypersheaves on $\TopC,$ then its profinite homotopy type is
$$\widehat{\Pi}_\i\left(\underset{i \in \cI} \lim \cY^i\right)=\underset{i \in \cI} \lim \widehat{\Pi}_\i\left(\cY^i\right).$$
§.§ The homotopy type of Kato-Nakayama spaces
In this subsection, we will give a formula expressing the homotopy type of the Kato-Nakayama space of a log scheme in terms of algebro-geometric data. We first start by reviewing a functorial approach to Kato-Nakayama spaces which is due to Kato, Illusie and Nakayama. Let $ \left( X, M, \alpha \right)$ be a log scheme, and let $X_{an}$ be the analytification of $ X$, which is an object of $\TopC$.
Consider the slice category $\TopC/ X_{an}.$ If $\left(T \stackrel{p}{\rightarrow} X_\an\right)$ is an object in $\TopC/ X_{an}$, one can pullback $M$ to
$T$ and take the section-wise group completion. In this way we obtain a sheaf of abelian groups on $T$ that we denote $p^*M^{\gp}$. Note that $p^*M^{\gp}$ contains $p^*\cO_X^\times$ as a sub-sheaf of abelian groups.
Let $G$ be any abelian topological group. If $T$ is a topological space, we denote $\underline{G}_T$ the sheaf on $T$ of continuous maps to $G$ equipped with the group structure coming from addition in $G$. Note that we have
We denote by $F_{log}$ the presheaf of sets on $\TopC/ X_{an}$ that is defined on objects by the following assignment:
\left(T \stackrel{p}{\rightarrow} X_{an}\right) \mapsto \left\{ \text{morphisms of sheaves } s: p^*M^{\gp} \rightarrow \underline{S}^1_T \text{ such that } s(f)=\frac{f}{|f|} \text{ for } f\in \cO_X^\times \right\}.
The presheaf $F_{log}$ is represented by $X_{log}$.
Since $X_{log}$ is an object of $\TopC,$ the functor $F_{\log}$ completely determines $X_{log}.$ Moreover, we can use this functorial description to give an expression for the homotopy type of $X_{log},$ as we will now show.
Denote by $\sC_{KN}\left(X\right)$ the following category:
the objects consist of triples $\left(T,p,s\right)$ where
* $T$ is a topological space in $\TopC^s,$
* $p$ is a continuous map $p:T \to X_{\an},$
* and $s$ is a morphism of sheaves of abelian groups
$$s: p^*M^{\gp} \rightarrow \underline{S}^1_T$$
such that $s(f)=\frac{f}{|f|}$ for $f\in \cO_X^\times$.
The morphisms $\left(T,p,s\right) \to \left(S,q,r\right)$ are continuous maps $f:T \to S$ such that $f^*\left(r\right)=s.$
Let $X$ be a log scheme. The weak homotopy type of the Kato-Nakayama space is that of $B\sC_{KN}\left(X\right).$
The reader may notice that $\sC_{KN}\left(X\right)$ is nothing but the Grothendieck construction
$$\int_{\TopC^s} \left(F_{\log}|_{\TopCs/X_{an}}\right).$$
Notice also that
$$\TopC^s/X_{an} \to \TopC^s$$
is the Grothendieck construction of $Y\left(X_{an}\right)|_{\TopC^s}$ (where $Y$ denotes the Yoneda embedding) i.e. the corresponding fibered category. Now, there is a canonical equivalence of categories
$$\Sh\left(\TopCs/X_{an}\right)\simeq \Sh\left(\TopCs\right)/ Y\left(X_{an}\right)|_{\TopC^s},$$
and it follows that $\int_{\TopC^s} \left(F_{\log}|_{\TopCs/X_{an}}\right)$ is equivalent to the Grothendieck construction of $Y\left(X_{log}\right).$ By Proposition <ref>, we have that $\Pi_\i\left(Y\left(X_{log}\right)\right)$ is the weak homotopy type of $X_{log}.$ The result now follows from Corollary <ref> and <cit.>.
Let $X$ be a log scheme. The profinite homotopy type of its Kato-Nakayama space $X_{log}$ is that of the profinite completion of $B\sC_{KN}\left(X\right).$
§ CONSTRUCTION OF THE MAP
In all that follows $X$ will be a fine and saturated log scheme over $\bC$ that is locally of finite type. See Appendix <ref> for a condensed introduction to the main concepts and notations that we will use in this section. Our goal is to prove the following proposition:
There is a canonical morphism of pro-topological stacks
$$\Phi_X\colon X_\log\to (\sqrt[\infty]{X})_{\topst}.$$
Later (Section <ref>) we will show that this map induces a weak equivalence of profinite homotopy types. The proof of Proposition <ref> will take up the rest of this section.
Our strategy will be to construct the morphism $\Phi_X$ étale locally on $X$, where the log structure has a Kato chart, and then to show that the locally defined morphisms glue together to give a global one.
Step 1 (local case): first let us assume that $X\to\Spec \bC[P]$ is a Kato chart for $X$, where $P$ is a fine saturated sharp monoid. In this case everything is very explicit: as explained in Section <ref> in the appendix, there is an isomorphism $\sqrt[n]{X}\simeq [X_n/\mu_n(P)]$, where $X_n=X\times_{\Spec \bC[P]} \Spec \bC[\frac{1}{n}P]$, the group $\mu_n(P)$ is the Cartier dual of the cokernel of $P^\gp\to\frac{1}{n}P^\gp$, and the action on $X_n$ is induced by the natural one on $\Spec \bC[\frac{1}{n}P]$.
By following Noohi's construction (see <ref>) we see that $\sqrt[n]{X}_\topst$ is canonically isomorphic to the quotient $[(X_n)_\an/\mu_n(P)_\an]$, where $\mu_n(P)_\an\cong (\bZ/n)^r$. Note that the finite morphism $\Spec \bC[\frac{1}{n}P]\to\Spec \bC[P]$ is étale on the open torus $\Spec \bC[P^\gp]\subseteq \Spec \bC[P]$, and ramified exactly on the complement.
Now let us construct a morphism of topological stacks $X_\log\to \sqrt[n]{X}_\topst$. By the quotient stack description of the target, this is equivalent to giving a $\mu_n(P)_\an$-torsor (i.e. principal bundle) on $X_\log$, together with a $\mu_n(P)_\an$-equivariant map to $(X_n)_\an$.
Let us look at a couple of examples first.
Let $X$ be the standard log point $\Spec \bC$ with log structure given by $\bN\oplus \bC^\times\to \bC$ sending $(n,a)$ to $0^n\cdot a$. Then $X_\log\cong S^1$, and $\sqrt[n]{X}_\topst \simeq \class (\bZ/n)$. In this case the morphism $S^1\to \class (\bZ/n)$ corresponds to the $(\bZ/n)$-torsor $S^1\to S^1$ defined by $z\mapsto z^n$.
Let $X$ be $\bA^1$ with the divisorial log structure at the origin. Then $X_\log \cong \bR_{\geq 0}\times S^1$ and $\sqrt[n]{X}_\topst \simeq [\bC /(\bZ/n)]$, where the morphism $[\bC /(\bZ/n)]\to (\bA^1)_\an=\bC$ is induced by $z\mapsto z^n$, and $(\bZ/n)$ acts by roots of unity.
In this case the map $\bR_{\geq 0}\times S^1\to [\bC /(\bZ/n)]$ corresponds to the $(\bZ/n)$-torsor $\bR_{\geq 0}\times S^1\to \bR_{\geq 0}\times S^1$ defined by $(a,b)\mapsto (a^n,b^n)$ and the equivariant map $\bR_{\geq 0}\times S^1\to \bC$ given by $(a,b)\mapsto a\cdot b$.
Note that the map $\bR_{\geq 0}\times S^1\to \bR_{\geq 0}\times S^1$ coincides with $z\mapsto z^n$ outside of the “origin” $\{0\}\times S^1$, and this is étale even on the algebraic side. Over the “origin”, it is precisely the presence of the $S^1$ introduced by the Kato-Nakayama construction that allows the map to be a $(\bZ/n)$-torsor. This is what happens in general (see also <cit.>).
Consider the map $\phi_\log \colon (X_n)_\log\to X_\log$ induced by the morphism of log schemes $\phi\colon X_n\to X$. The map $\phi_\log$ is a $\mu_n(P)_\an$-torsor, and the projection $(X_n)_\log \to (X_n)_\an$ is a $\mu_n(P)_\an$-equivariant map.
Note (Definition <ref>) that if $P$ is a monoid, $\bC(P)$ will denote the log analytic space $(\Spec \bC[P])_\an$ with the induced natural log structure.
The action of $\mu_n(P)$ on $\Spec \bC[\frac{1}{n}P]$ induces an action on $X_n$, and the map $X_n\to X$ is invariant. Consequently we have an induced action of $\mu_n(P)_\an$ on $(X_n)_\log$, and the map $\phi_\log \colon (X_n)_\log\to X_\log$ is invariant.
Moreover, since taking $(\blank)_\log$ commutes with strict base change (see Proposition <ref>), we have a cartesian diagram
\xymatrix{
(X_n)_\log\ar[r]\ar[d]_{\phi_\log} & \bC(\frac{1}{n}P)_\log\ar[d]^{\phi_{P,\log}}\\
X_\log\ar[r] & \bC(P)_\log
and because the action of $\mu_n(P)_\an$ on $(X_n)_\log$ comes from the one on $\bC(\frac{1}{n}P)_\log$, it suffices to prove the statement for the right-hand column.
Similarly, in order to verify that $(X_n)_\log \to (X_n)_\an$ is $\mu_n(P)_\an$-equivariant we are reduced to checking that $\bC(\frac{1}{n}P)_\log\to \bC(\frac{1}{n}P)$ is $\mu_n(P)_\an$-equivariant.
Now note that $\mu_n(P)_\an$ is precisely the kernel of the map $\Hom(\frac{1}{n}P,S^1)\to \Hom(P,S^1)$, So that the action of $\mu_n(P)_\an$ on $\Hom(\frac{1}{n}P,S^1)$ is free and transitive. It is also not hard to check that there are local sections (note that $\Hom(P,S^1)=\Hom(P^\gp,S^1)\cong (S^1)^k$ non-canonically), so that the map is a $\mu_n(P)_\an$-torsor.
Furthermore, $\phi_{P,\log}\colon \bC(\frac{1}{n}P)_\log\to \bC(P)_\log$ is the restriction map $\Hom(\frac{1}{n}P, \bR_{\geq 0}\times S^1)\to \Hom(P, \bR_{\geq 0}\times S^1)$, and this is the product of the two maps $\Hom(\frac{1}{n}P, \bR_{\geq 0})\to \Hom(P, \bR_{\geq 0})$ (which is a homeomorphism) and $\Hom(\frac{1}{n}P, S^1)\to \Hom(P, S^1)$. The action of $\mu_n(P)_\an$ is trivial on the factor $\Hom(\frac{1}{n}P, \bR_{\geq 0})$ and the one given by the aforementioned inclusion as a subgroup, on the factor $\Hom(\frac{1}{n}P, S^1)$. Consequently, $\phi_{P,\log}$ is a $\mu_n(P)_\an$-torsor for the natural action, as required.
The map $\bC(\frac{1}{n}P)_\log\to \bC(\frac{1}{n}P)$ coincides with the map $\Hom(\frac{1}{n}P, \bR_{\geq 0}\times S^1)\to \Hom(\frac{1}{n}P,\bC)$ induced by the natural map $\bR_{\geq 0}\times S^1\to \bC$, and therefore it is manifestly $\Hom(\frac{1}{n}P,S^1)$-equivariant, and in particular $\mu_n(P)_\an$-equivariant.
The previous proposition gives a morphism of pro-topological stacks $\Phi_{n,P} \colon X_\log \to \sqrt[n]{X}_\topst$. It is clear from the construction that if $n\mid m$, then the diagram
\xymatrix{
X_\log\ar[r]^<<<<{\Phi_{m,P}}\ar[rd]_{\Phi_{n,P}} & \sqrt[m]{X}_\topst\ar[d]\\
& \sqrt[n]{X}_\topst
is (2-)commutative, so we obtain a morphism $(\Phi_X)_P\colon X_\log\to (\sqrt[\infty]{X})_{\topst}$ of pro-topological stacks.
Step 2 (compatibility of the local constructions): let us extend this local construction to a global one. The idea is of course to use descent and glue the local constructions, and intuitively, one would expect that these local maps patch together to define a global one without incident. However, writing down all the necessary 2-categorical coherences gets pretty technical quickly, and it is much cleaner to use the machinery of $\infty$-categories.
We will need some preliminary lemmas and constructions.
Let $X$ be a fine saturated log scheme over a field $k$ with two Kato charts $X\to \Spec k[P]$ and $X\to \Spec k[Q]$ for the log structure. Then for every geometric point $x$ of $X$, after passing to an étale neighborhood of $x$, there is a third chart $X\to \Spec k[R]$ with maps of monoids $P\to R$ and $Q\to R$ inducing a commutative diagram
\xymatrix{
& &\Spec k[P] \\
X\ar[r]\ar@/^1.5pc/[rru]\ar@/_1.5pc/[rrd] &\Spec k[R]\ar[ru]\ar[rd] & \\
& & \Spec k[Q].
We can take $R=\overline{M}_x$. There is a chart with monoid $R$ in an étale neighborhood of $x$ by <ref>, and we have maps $P\to R$ and $Q\to R$ that induce a commutative diagram as in the statement, possibly after further localization.
Now let us define a category $\mathfrak{I}$ of étale open subsets of $X$ with a global chart: objects are triples $(\phi\colon U\to X,P,f)$ where $\phi\colon U\to X$ is étale, $P$ is a fine saturated sharp monoid and $f\colon U\to \Spec \bC[P]$ is a chart for the log structure on $U$ (pulled back via $\phi$).
A morphism $(\phi\colon U\to X, P, f) \to (\psi\colon V\to X, Q, g)$ is given by a (necessarily étale) map $U\to V$ over $X$ and a morphism $Q\to P$, such that the diagram
\xymatrix{
U\ar[r]^<<<<<f\ar[d] & \Spec \bC[P]\ar[d]\\
V\ar[r]^<<<<<g & \Spec \bC[Q]
is commutative.
We have two (lax) functors $(\blank)_\log$ and $(\sqrt[n]{\blank})_\topst\colon \mathfrak{I}\to \Topst/X_\an$, as follows:
for each $A=(\phi\colon U\to X, P, f) \in \mathfrak{I}$ we get, via strict pullback through the chart morphism, a local model for the Kato-Nakayama space $X_\log^A$ (over $U$) and one for the $n$-th root stack $\sqrt[n]{X}_\topst^A$. We set $A_\log=X_\log^A$ and $\sqrt[n]{A}_\topst=\sqrt[n]{X}_\topst^A$. The maps to $X_\an$ are given by the composites of the projections to $U_\an$ and the local homeomorphism $U_\an\to X_\an$. The action of these two functors on morphisms is clear.
The construction in the local case (i.e. Step 1 above) gives an assignment, for each $A\in\mathfrak{I}$, of a morphism of topological stacks $\alpha^n_A\colon A_\log\to \sqrt[n]{A}_\topst$.
The family $(\alpha^n_A)$ gives a lax natural transformation
$$\alpha^n:(\blank)_\log \Rightarrow (\sqrt[n]{\blank})_\topst,$$
in the sense of <cit.>.
By translating the definition, in the present case this means the following: if $a\colon A=(\phi\colon U\to X, P, f)\to (\psi\colon V\to X, Q, g)=B$ is a morphism in $\mathfrak{I}$, then the diagram
\xymatrix{
A_\log\ar[r]^<<<<{\alpha^n_A}\ar[d] & \sqrt[n]{A}_\topst \ar[d]\ar@{}[ld]^(.30){}="a"^(.70){}="b" \ar@{=>}_{\alpha^n(a)} "a";"b"\\
B_\log\ar[r]^<<<<{\alpha^n_B} & \sqrt[n]{B}_\topst
2-commutes, and the 2-cells $\alpha^n(a)$ satisfy a compatibility condition.
This follows from the fact that the morphism $a=(U\to V, Q\to P)$ gives a commutative diagram
\xymatrix@=18px{
& (U_n)_\log \ar[rr]\ar[ld]\ar[dd]|\hole & & (U_n)_\an \ar[ld]\\
(V_n)_\log\ar[rr]\ar[dd] & & (V_n)_\an & \\
& X_\log^A \ar[ld] & & \\
X_\log^B & & &
between the two objects corresponding to the functors $\alpha^n_A$ and $\alpha^n_B$. This gives a canonical natural transformation that makes the diagram
\xymatrix@C=75pt{
X_\log^A\ar[d]\ar[r]^<<<<<<<<<<<<<{\alpha^n_A} & [(U_n)_\an/\mu_n(P)_\an]=\sqrt[n]{X}_\topst^A \ar[d] \ar@{}[ld]^(.40){}="a"^(.60){}="b" \ar@{=>}_{\alpha^n(a)} "a";"b"\\
X_\log^B \ar[r]^<<<<<<<<<<<<<{\alpha^n_B}& [(V_n)_\an/\mu_n(Q)_\an]=\sqrt[n]{X}^B_\topst.
2-commutative, and this is the required diagram.
Now if $C=(\eta\colon W\to X, R, h)$ is a third object of $\mathfrak{I}$ with a morphism $b\colon B\to C$ in $\mathfrak{I}$, then the fact that the diagram
\[
\begin{tikzcd}[row sep=small, column sep=small]
& & (U_n)_\log \ar{rrr}\ar{ld}\ar{ddd} & & & (U_n)_\an \ar{ld}\\
& (V_n)_\log\ar[crossing over]{rrr}\ar{ddd}\ar{dl} & & & (V_n)_\an\ar{dl} & \\
(W_n)_\log\ar[crossing over]{rrr}\ar{ddd} & & &(W_n)_\an &\\
& & X_\log^A \ar{ld} & & & \\
& X_\log^B\ar{dl} & & & &\\
X_\log^C & & & & &
\end{tikzcd}
\]
commutes implies that the composite of the two 2-cells $\alpha^n(b)$ and $\alpha^n(a)$ is equal to $\alpha^n(b\circ a)$.
By composition with the natural functor
$$\Topst/X_\an \hookrightarrow \Hshi\left(\TopC\right)/X_{an}$$
to hypersheaves on $\TopC$ (see Section <ref>) and by abuse of notation we get a natural transformation of functors of $\i$-categories:
$$ \xygraph{!{0;(5,0):(0,0.18)::}
{\mathfrak{I}}="a" [r] {\Hshi\left(\TopC\right)/X_{an}.}="b"
"l" [d(.3)] [r(0.1)] :@{=>}^{\alpha^n} [d(.5)]} $$
Step 3 (the global case): we will now use the natural transformation $\alpha^n$ above to construct a global map
$$\Phi_X:X_{log} \to \sqrt[n]{X}.$$
We will first need a crucial lemma:
Let $\iota:\mathfrak{I} \to \Hshi\left(\TopC\right)$ be the functor sending a triple $\left(\phi:U \to X,P,f\right)$ to $U_{an}.$ Then the canonical map $\colim \iota \to X_{an}$ is an equivalence.
Before proving the above lemma, we will show how we may use this lemma to produce the global morphism we seek. The key idea is the following basic fact about $\i$-topoi:
Let $\underset{i \in I} \colim A_i \to B$ be a morphism in an $\i$-topos $\cE,$ and let $C \to B$ be another morphism. Then the canonical map
$$\underset{i \in I} \colim\left(C \times_B A_i\right) \to C \times_B \underset{i \in I} \colim A_i$$
is an equivalence.
The above fact is standard and is an immediate consequence of the fact that any $\i$-topos is locally Cartesian closed.
Let us now see how we may complete the construction. Suppose we know that the canonical map $\colim \iota \to X_{an}$ is an equivalence. We can write this informally as
$$\underset{\left(\phi:U \to X,P,f\right)} \colim \mspace{5mu} U_{an} \stackrel{\sim}{\longrightarrow} X_{an}.$$
Consider the morphism $X_{log} \to X_{an}.$ Then since colimits are universal we have that the following is a pullback diagram:
$$\xymatrix{\underset{\left(\phi:U \to X,P,f\right)} \colim U_{an} \times_{X_{an}} X_{log} \ar[r] \ar[d] & X_{log} \ar[d]\\
\underset{\left(\phi:U \to X,P,f\right)} \colim U_{an} \ar[r]^-{\sim} & X_{an}.}$$
It follows that top map
$$\underset{\left(\phi:U \to X,P,f\right)} \colim U_{an} \times_{X_{an}} X_{log} \to X_{log}$$
is also an equivalence. However, notice that we have a canonical identification
$$U_{an} \times_{X_{an}} X_{log} \cong U_{log},$$
$$X_{log} \simeq \underset{\left(\phi:U \to X,P,f\right)} \colim U_{log} = \colim \left(\blank\right)_{log}.$$
By a completely analogous argument, one sees that
$$\sqrt[n]{X}_{top} \simeq \underset{\left(\phi:U \to X,P,f\right)} \colim \sqrt[n]{U}_{top} = \colim \sqrt[n]{\blank}_{top}.$$
For each $n$, the global map is then defined to be
$$\colim \alpha^n:\colim \left(\blank\right)_{log} \to \colim \sqrt[n]{\blank}_{top}.$$
Just as in the local case, one easily sees that the maps
$$\colim \alpha^n:X_{log} \to \sqrt[n]{X}_{top}$$
assemble into a morphism of pro-objects
$$\Phi_X:X_{log} \to \sqrt[\i]{X}_{top}.$$
It is immediate from the construction that this map agrees locally with the map constructed in Step 1. In the next sections we will prove that $\Phi_X$ induces an equivalence of profinite spaces.
To finish the proof of the existence of the above map, it suffices to prove Lemma <ref>. Without further ado, we present the proof below.
Equip $\mathfrak{I}$ with the following Grothendieck topology: A collection of morphisms
$$\left( \left(\phi_i:U_i \to X,P_i,f_i\right) \to \left(\phi:U \to X,P,f\right)\right)_i$$
will be a covering family if the induced family
$$\left(U_i \to U\right)_i$$
is an étale covering family. Note that there is a canonical morphism of sites
$$F: \mathfrak{I} \to X_{\et}$$
from $\mathfrak{I}$ to the small étale site of $X.$ Moreover, by Lemma <ref>, one easily checks that $F$ satisfies the conditions of the Comparison Lemma of <cit.>, so the induced geometric morphism of topoi
$$\Sh\left(\mathfrak{I}\right) \to \Sh\left(X_{\et}\right)$$
is an equivalence. It then follows
from <cit.> and <cit.> that the induced geometric morphism between the respective $\i$-topoi of hypersheaves
$$\Hshi\left(\mathfrak{I}\right) \to \Hshi\left(X_{\et}\right)$$
is an equivalence. By Remark <ref>, the analytification functor is the inverse image part of a geometric morphism
$$f:\Hshi\left(\TopC\right) \to \Shi\left(\Aff,\mbox{\'et}\right).$$
By <cit.>, there is an induced geometric morphism
$$\tilde f:\Hshi\left(\TopC\right) \to \Hshi\left(\Aff,\mbox{\'et}\right).$$
By left Kan extension of the canonical functor
$$X_{\et} \to \Hshi\left(\Aff,\mbox{\'et}\right)/X$$
which sends each étale open $U \to X$ to itself, one produces a colimit preserving functor
$$\omega:\Hshi\left(X_{\et}\right) \to \Hshi\left(\Aff,\mbox{\'et}\right)/X.$$
Consider the composite
$$\resizebox{6.5in}{!}{$\Hshi\left(\mathfrak{I}\right) \simeq \Hshi\left(X_{\et}\right) \stackrel{\omega}{\longrightarrow} \Hshi\left(\Aff,\mbox{\'et}\right)/X \to \Hshi\left(\Aff,\mbox{\'et}\right) \stackrel{\tilde f^*}{\longrightarrow} \Hshi\left(\TopC\right),$}$$
where $\Hshi\left(\Aff,\mbox{\'et}\right)/X \to \Hshi\left(\Aff,\mbox{\'et}\right)$ is the canonical projection. Denote the composite by $\Theta.$ The functor $\Theta$ is colimit preserving as it is the composite of colimit preserving functors, and unwinding definitions, one sees that the composite
$$\mathfrak{I} \stackrel{y}{\longrightarrow} \Hshi\left(\mathfrak{I}\right) \stackrel{\Theta}{\longrightarrow} \Hshi\left(\TopC\right)$$
is canonically equivalent to $\iota.$ It follows that there is a canonical equivalence
$$\colim \iota \simeq \Theta\left(\colim y \right).$$
But $y$ is strongly generating, so by the proof of <cit.>, the colimit of $y$ is the terminal object. Unwinding the definitions, one sees that the terminal object gets sent to $X_{an}$ by $\Theta.$ This completes the proof.
§ THE TOPOLOGY OF LOG SCHEMES
This section contains preliminaries about some topological properties of fine saturated log schemes locally of finite type over $\bC$, the Kato-Nakayama space and the root stacks.
§.§ Stratified fibrations
The following proposition is a consequence of the material in Section <ref>.
Recall that if $X$ is a fine saturated log scheme locally of finite typer over $\bC$ there is a stratification $\cR=\{R_n\}_{n\in\bN}$ of $X$, the rank stratification (Definition <ref>), given by $R_n=\{x\in X\mid \rank_\bZ \overline{M}_{\overline{x}}^\gp\geq n \}$.
The Kato-Nakayama space $X_\log$, the topological root stacks $\sqrt[m]{X}_\topst$ and the topological infinite root stack $\sqrt[\infty]{X}_\topst$ are stratified fibrations over $X_\an$ with respect to the stratification $\cR$, i.e. they are fibrations over the strata $(S_n)_\an=(R_{n}\setminus R_{n+1})_\an$ of the stratification $\cR_\an$.
All constructions are compatible with arbitrary base change along strict morphisms, so
X_\log|_{(S_n)_\an}\cong (S_n)_\log
\sqrt[m]{X}|_{S_n}\simeq \sqrt[m]{S_n}
where $m$ can be $\infty$, and $S_n$ has the log structure pulled back from $X$. It suffices then to show that the two maps $(S_n)_\log\to (S_n)_\an$ and $(\sqrt[m]{S_n})_\topst\to (S_n)_\an$ are fibrations over $S_n$.
Let us cover $(S_n)_\an$ with open subsets over which the sheaf $\overline{M}$ is constant, and recall that by definition of $S_n$ it will have rank $n$. We can choose such opens in order to have a cartesian diagram
\xymatrix{
(S_n)_\log\ar[r]\ar[d] & (\Spec k[P])_\log \ar[d]\\
(S_n)_\an\ar[r] & (\Spec k[P])_\an
over each of them, where the bottom horizontal arrow sends everything to the vertex $v_P$ (as in the proof of <ref>). It follows that $(S_n)_\log\cong (S^1)^n\times (S_n)_\an$, and that the map $(S_n)_\log\to (S_n)_\an$ is identified with the projection. The factor $(S^1)^n$ is the fiber of the map $ (\Spec k[P])_\log \to (\Spec k[P])_\an$ over the point $v_P$.
The analogous diagram
\xymatrix{
(\sqrt[m]{S_n})_\topst \ar[r]\ar[d] & \sqrt[m]{\Spec k[P]}_\topst \ar[d]\\
(S_n)_\an\ar[r] & (\Spec k[P])_\an
shows the same conclusion for root stacks. In this case we get an isomorphism $(\sqrt[m]{S_n})_\topst \simeq \cX \times (S_n)_\an$, where $\cX$ is the fiber of the map $ \sqrt[m]{\Spec k[P]}_\topst \to (\Spec k[P])_\an$ over the vertex $v_P$.
We need a similar (local) statement for groupoid presentations of the root stacks.
Take $x\in X$, and an open étale neighborhood $U\to $X of $x$ where there is a global chart $U\to \Spec \bC[P]$ for the log structure, where $P$ is fine, saturated and sharp. Then we have a quotient stack presentation for the topological $n$-th root stack $\sqrt[n]{U}_\topst \simeq (\sqrt[n]{X}|_U)_\topst $ for every $n$ (see the discussion preceding Proposition <ref>).
Let us denote by $\bG(n)$ the simplicial topological space associated with this quotient presentation. There are compatible maps $\bG(m)\to \bG(n)$ whenever $n\mid m$, and the whole system gives a (simplicial) presentation for the topological infinite root stack $\sqrt[\infty]{U}_\topst$.
Explicitly, the simplicial space $\bG(n)$ is obtained from the action of $\mu_n(P)$ on the scheme $U_n= U \times_{\Spec \bC[P]} \Spec \bC[\frac{1}{n}P]$ (see the local description of the root stacks in <ref>), so that
$$\bG(n)_k\cong (U_n\times \mu_n(P)\times\cdots\times \mu_n(P))_\an$$
where there are $k$ copies of $\mu_n(P)$, and the map $\bG(n)_k\to U_\an$ is the composite of the projection to $(U_n)_\an$ followed by the map $(U_n)_\an\to U_\an$.
Every $x\in U_\an$ has arbitrarily small neighborhoods over which for every $n$ and $k$ the map $\bG(n)_k\to U_\an$ is a product over $U_\an \cap (S_r)_\an$, where $x\in (S_r)_\an$
In particular, for every $n$ and $k$ the topological space $\bG(n)_k$ is a stratified fibration over $U_\an$.
Note first of all that since the map $U\to \Spec \bC[P]$ is strict, the rank stratification of $\Spec \bC[P]$ with its natural log structure is pulled back to the rank stratification of $U$, in the obvious sense.
Moreover from the cartesian diagram
\xymatrix{
(U_n)_\an\ar[r]\ar[d] & \bC(\frac{1}{n}P) \ar[d]\\
U_\an \ar[r] & \bC(P)
and from the fact that $\bG(n)_k\to U_\an$ is the projection $\bG(n)_k\cong (U_n\times \mu_n(P)\times\cdots\times \mu_n(P))_\an\to (U_n)_\an$ followed by $(U_n)_\an\to U_\an$, we see that it suffices to prove that the map $\pi\colon \bC(\frac{1}{n}P)=(\Spec \bC[\frac{1}{n}P])_\an\to \bC(P)=(\Spec \bC[P])_\an$ is a stratified fibration. The proof will show that for a stratum $S$ we can find an open subset $V\subseteq \bC(P)$ such that the map $\pi$ is a product over $V\cap S$ for every $n$.
Let us pick $\phi\in \bC(P)=\Hom(P,\bC)$, and call $p_1,\hdots,p_l$ the (finitely many) indecomposable elements of $P$ (cfr. <cit.>). Assume (by reordering) that the first $h$ of those get sent to $0$ by $\phi$, and the last ones are sent to non-zero complex numbers. Call $r$ the rank of the group associated to the quotient $\sfrac{P}{\langle p_i\mid i=h+1,\hdots, l}\rangle$ (i.e. the rank of the log structure of $\bC(P)$ at $\phi$).
The stratum of the rank stratification of $\bC(P)$ to which $\phi$ belongs will be then $S_r$, the set of points of $\bC(P)$ where the log structure has rank exactly equal to $r$. It's clear that $\phi$ actually belongs to the open subset $S_\phi$ of $S_r$ consisting of the morphisms $\psi \in \Hom(P,\bC)$ such that $\psi(p_i)=0$ for $1\leq i\leq h$ and $\psi(p_i)\neq 0$ for $h< i\leq l$.
Note also that the same condition on images of indecomposables of $\frac{1}{n}P$ will determine a subset $S'_\phi\subseteq \bC(\frac{1}{n}P)=\Hom(\frac{1}{n}P,\bC)$ (of those morphisms such that the image of $\frac{p_i}{n}$ is zero exactly if $1\leq i\leq h$), that a moment's reflection will show to be exactly the preimage $\pi^{-1}(S_\phi)$. Let us check that we can choose a neighborhood of $\phi$ in $\bC(P)$ over which the restriction of $\pi\colon \pi^{-1}(S_\phi)\to S_\phi$ is a product.
For each $i=h+1,\hdots, l$ let us choose a small open disk $D_i$ around $\phi(p_i)$ in $\bC$ that does not contain the origin, and for $i=1,\hdots, h$ let $D_i$ be a small open disk around the origin. These define an open neighborhood $W$ of $\phi$ in $\bC(P)$, made up of those functions $\psi$ such that $\psi(p_i)\in D_i$ for every $i$.
Let us also choose an $n$-th root $\sqrt[n]{\phi(p_i)}$ of the non-zero complex number $\phi(p_i)$ for $i=h+1,\hdots, l$. There are a finite number of such choices, and there is a subset of those choices for which the homomorphism $\frac{1}{n}P\to \bC$ given by $\frac{p_i}{n}\mapsto \sqrt[n]{\phi(p_i)}$ is well-defined (note that this assignment might not give a well-defined homomorphism due to the relations among the indecomposable elements of the monoid $P$). Let us call $A$ this set of “good” choices.
Any element of $A$ determines for each $i=h+1,\hdots, l$ an $n$-th root function $\sqrt[n]{-}_i$ defined on the small disk $D_i$. Let us define a map $W\cap S_\phi \to \pi^{-1}(W\cap S_\phi)\subseteq \Hom(\frac{1}{n}P,\bC)$ by sending $\psi$ to the morphism defined by $\frac{p_i}{n}\mapsto \sqrt[n]{\psi(p_i)}_i$. This is a section of the projection $\pi^{-1}(W\cap S_\phi)\to W\cap S_\phi$, and one can check that this induces a homeomorphism $W\cap S_\phi \times A\cong \pi^{-1}(W\cap S_\phi)$, where $A$ is seen as a discrete set. We leave the details to the reader.
These arguments are uniform in $n\in \bN$, so the open subset $W$ that we identified will work for any $n$.
§.§ A system of open neighborhoods for $X_{an}$
In this subsection we will prove the following crucial lemma:
For all
$x \in X_{an}$ there exists a fundamental system of contractible analytic open neighborhoods $\mathcal{U}_x$ of $x$ with global charts $f\colon U\to (\Spec \bC[P])_{an}$ for $U\in \mathcal{U}_x$, such that:
* the map $f$ sends $x$ into the vertex of $(\Spec \bC[P])_{an}$ (i.e. the maximal ideal generated by all non-zero elements of $P$), and
* each of the
following maps is a weak homotopy equivalence:
\left(X_{log}\right)_x \to X_{log}|_U
$$\left(\mathbb{G}(n)_i\right)_x \to \left(\mathbb{G}(n)_i\right)|_U$$
where $\{\bG(n)\}_{n\in \bN}$ is the family of topological groupoid presentations for the topological $n$-th root stack coming from the chart $f$, as in Proposition <ref>.
First of all we review some standard facts on triangulations and open covers.
Let $M$ be a topological space equipped with a triangulation $\cT$. Denote $\cV_\cT$ the set of vertices of $\cT$.
If $f$ is a simplex of
$\cT$, we denote $s(f)$ the union of the relative interiors of the
simplices of $\cT$ that contain $f$. We call $s(f)$ the star of $f$.
Note that $s(f)$ is a contractible open subset of $M$. If $v$ is a vertex of $\cT$, we set $U_v\defeq s(v)$. The star of a simplex $f$ is naturally stratified by the simplices containing $f$: the strata
are the relative interiors of the simplices containing $f$.
We say that a subspace of $\bR^n$ is a cone if it is invariant under the action
of $\bR_{>0}$ by rescaling. We say that a cone is linear if it can be expressed as an intersection of linear spaces and linear half-spaces.
Let $v$ be in $\cV_\cT$. Then there exists an $N \in \bN$ such that $U_v$ can be embedded as a linear cone in
$\bR^N$. Further we can choose this embedding in such a way that, for all simplices $f$ containing $v$,
$s(f) \subset U_v$ is mapped to a linear subcone.
Let $v_1 \dots v_N$ be the one dimensional simplices that contain $v$, and let $e_1, \dots, e_N$ be the standard basis of $\bR^N$. If $I$ is a subset of
$\{1, \dots, N\}$ we denote
O_I \defeq \{ \Sigma_{i \in I} \alpha_i e_i | \alpha_i \geq 0\} \subset \bR^N.
Every simplex $\Delta$ containing $v$ determines a subset
$I_\Delta$ of $\{1, \dots, N\}$ in the following way: $i$ belongs to $I_\Delta$ if and only if $\Delta$ contains $v_i$. We obtain an embedding of $U_v$ into $\bR^ N$ by considering a piecewise linear homeomorphism
U_v \simeq \bigcup_{v \in \Delta} O_{I_\Delta}.
This embedding has all the properties claimed by the lemma.
Let $x$ be in $M$, and let $f$ be the lowest dimensional
simplex such that $x$ belongs to $f$.
Then there exists a system of open neighborhoods $\cU_x$ of $x$ such that all $U$ in $\cU_x$ have the following properties:
* $U$ is contractible.
* $U$ does not intersect simplices of $\cT$ that do not contain $f$.
Let $v$ be a vertex incident to $f$. By Lemma <ref> the open neighborhood
$U_v$ can be embedded as a linear cone in $\bR^n$ in such a way that $s(f) \subset U_v$ is a linear subcone. Equip $\bR^n$ with a Euclidean metric. Then $\cU_x$ can be obtained by intersecting $s(f)$ with a system of open neighborhoods given by open balls in $\bR^n$ centered at $x$.
Next we turn to the log scheme $X$. Let $x$ be in $X_{an}$. Since we are interested in constructing a system of open neighborhoods for $x$ we can assume, by étale localizing around $x$, that
* $X$ is affine, and
* that we have a global chart $f\colon X\to \Spec \bC[P]$, where $P=\overline{M}_x$ (see <ref>), which sends $x$ to the vertex of $\Spec \bC[P]$.
The fact that $X$ is affine is key in order to produce triangulations, which we do in lemma
By Lemma <ref> the log structure determines a stratification $\cR_X$ of $X$.
There exists a triangulation $\cT_X$ of $X$ that refines $\cR_X$.
The existence of triangulations refining stratifications of affine schemes goes back to Lojasiewicz <cit.>. See also Shiota's work <cit.> for a more recent reference.
By Lemma <ref>, the triangulation $\cT_X$ gives us a system of open neighborhoods $\cU_x$ of $x$ in $X_{an}$ satisfying the two
properties stated there. We claim that $\cU_x$ has all the properties required by lemma <ref>. Note that, since we assumed without loss of generality that $X$ is affine and that has a global chart to $ \Spec \bC[P]$ sending $x$ to the vertex of $\Spec \bC[P]$, we only need to prove that property $(3)$ holds. We do this next.
The following lemma was proved in <cit.>.
Let $W_1$ and $W_2$ be locally compact and locally contractible Hausdorff spaces. Let $p : W_1 \rightarrow W_2$ be a continuous map,
and let $K_2 \subset W_2$ be a closed deformation retract. Suppose that the restriction
$ p^{-1}(W_2 -K_2) \rightarrow
W_2 - K_2$ is homeomorphic to the projection from a product
$F \times (W_2 - K_2) \rightarrow W_2 - K_2$.
Then $K_1 \defeq p^{-1}(K_2)$ is a deformation retract of $W_1$.
We will actually need a slight variant of Lemma <ref>. Assume that $W_2 - K_2$ decomposes as a finite disjoint union of $m$ components, that we denote $(W_2-K_2)_i$,
W_2 - K_2 = \bigcup_{i=1}^{i=m} (W_2-K_2)_i .
Then the claim still holds if the restriction $ p^{-1}(W_2 -K_2) \rightarrow
W_2 - K_2$ is homeomorphic to the projection from a disjoint union of products
\bigcup_{i=1}^{i=m} F_i \times (W_2 - K_2)_i \rightarrow \bigcup_{i=1}^{i=m}(W_2 - K_2)_i .
This stronger statement is proved exactly as Lemma <ref>: and in fact, follows from it through an induction on the number of connected components of $W_2 - K_2$.
We conclude the proof of Lemma <ref> by showing that the following proposition holds.
For all $U$ in $\cU_x$ each of the
following maps is a weak homotopy equivalence:
\left(X_{log}\right)_x \to X_{log}|_U, \quad \left(\mathbb{G}(n)_i\right)_x \to \left(\mathbb{G}(n)_i\right)|_U
where $\{\bG(n)\}_{n\in \bN}$ is the family of topological groupoid presentations for the topological $n$-th root stack coming from the chart $f$.
The proof is the same for both $X_{log}$ and $\mathbb{G}(n)_i$. The argument relies exclusively on the fact that $X_{log}$ and $\mathbb{G}(n)_i$ give stratified fibrations on $X_{an}$ with respect to the stratification $\cR_X$.
To avoid repetition, we prove the statement only for $X_{log}$ but the argument remains valid if we substitute $\mathbb{G}(n)_i$ in all occurences of $X_{log}$.
Let $f$ be the lowest dimensional simplex of $\cT_X$
such that $x$ lies on $f$. Recall from the proof of Lemma <ref> that, in order to define $\cU_x$, we pick a vertex $v$ of the triangulation $\cT_X$ that is incident to $f$. By construction, $U$ is an open subset of
$U_v$. Thus $U$ carries a stratification which is obtained by restricting to it the stratification on $U_v$ by the simplices containing $v$.
For all $k \in \bN$, denote $U_k \subset U$ the $k$-skeleton of $U$: that is, $U_k$ is the union of the strata of dimension less than or equal to $k$. Note that if $k < dim(f)$, $U_k$ is empty, and is contractible if $dim(f) \leq k$. Further if $dim(f) \leq k \leq k'$,
$U_k$ is a strong deformation retract of $U_{k'}$. Indeed both
$U_k$ and $U_{k'}$ are CW complexes (up to compactifying), and any contractible subcomplex of a contractible
CW complex is a strong deformation retract, see e.g. <cit.>.
We prove next that if $dim(f) \leq k-1$, the map
X_{log}|_{U_{k-1}} \rightarrow X_{log}|_{U_k}
is a deformation retract. Note that $U_k - U_{k-1}$ is equal to the disjoint union of $k$-dimensional strata. That is $U_k - U_{k-1}$ can be
written as a disjoint union of $m$ components
U_k - U_{k-1} = \bigcup_{i=1}^{i=m} (U_k - U_{k-1})_i .
The restriction of the map
$X_{log}|_U \rightarrow X_{an}|_U$ to each stratum of $U$ is a principal bundle. Indeed, the stratification on $U$ is finer that the restriction to $U$ of $\cR_X$. Further, it is a trivializable principal bundle,
since the strata are paracompact Hausdorff and contractible.
Thus the restriction
X_{log}|_{U_k - U_{k-1}} \rightarrow U_k - U_{k-1}
is homeomorphic to a projection from a disjoint union of products
X_{log}|_{U_k - U_{k-1}} \simeq \bigcup_{i=1}^{i=m} F_i \times (U_k - U_{k-1})_i \rightarrow \bigcup_{i=1}^{i=m} (U_k - U_{k-1})_i .
We have showed that the map $U_{k-1} \rightarrow U_k$ is a deformation retract.
We apply Lemma <ref>, or rather the variant that was discussed immediately after the statement of Lemma <ref>, (note that $X_\log$ is locally compact Hausdorff and locally contractible by Proposition <ref>),
and deduce that the map
X_{log}|_{U_{k-1}} \rightarrow X_{log}|_{U_k}
is also a deformation retract, as we claimed.
There exists a $N \in \bN$ such that $U_N = U$. By applying recursively the retractions that we have constructed in the previous paragraph, we obtain a deformation retract
X_{log}|_{U_{dim(f)}} \rightarrow X_{log}|_U .
By property $(2)$ of Lemma <ref>,
$U_{dim(f)}$ is connected. Further it is contractible and paracompact, and thus $X_{log}|_{U_{dim(f)}}$ is homeomorphic to a product $F \times {U_{dim(f)}}$. This implies that there are homotopy equivalences
(X_{log})|_x \simeq F \times \{x \} \stackrel{\sim}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}} F \times {U_{dim(f)}} \simeq X_{log}|_{U_{dim(f)}}
and this concludes the proof.
§ THE EQUIVALENCE
Finally, in this section we will prove the main result of this paper, namely that there is an equivalence
$$\Pip\left(\Phi_X\right)\colon \Pip\left(X_\log\right)\to \Pip\left(\sqrt[\infty]{X}_{\topst}\right)$$
of profinite spaces, where $\Pip$ is the “profinite homotopy type” functor defined in <ref>, and $\Phi_X$ is the morphism of pro-topological stacks constructed in Section <ref>.
The main idea is to use the basis of open subsets constructed in Lemma <ref> to produce a suitable hypercover of $X_{an}$ and to use this to reduce to checking that one has a profinite homotopy equivalence along fibers. First, we will need a few more technical lemmas.
The following lemma makes precise in what way one can glue profinite spaces together using hypercovers:
Let $\cX$ be a hypersheaf in $\Hshi\left(\TopC\right).$ Let $\cI$ be a cofiltered $\i$-category and let
$$f_{\bullet}:\cI \to \Hshi\left(\TopC\right)/\cX$$
be an $\cI$-indexed pro-system with associated pro-object $\underset{i \in \cI} \lim \left(f_i:\cY^i \to \cX\right).$
$$U^\bullet:\Delta^{op} \to \Hshi\left(\TopC\right)/\cX$$
be a hypercover of $\cX.$ For each $i,$ denote by $f_i^*U^\bullet$ the pullback of the hypercover $U^\bullet$ to a hypercover of $\cY^i.$ Consider the underlying pro-object $\underset{i \in \cI} \lim \cY^i$ in $\Hshi\left(\TopC\right).$ Then there is a canonical equivalence of profinite spaces
$$\Pip\left(\underset{i \in \cI} \lim \cY^i\right) \simeq \underset{n \in \Delta^{op}} \colim \left[\Pip\left(\underset{i \in \cI} \lim f_i^*U^n\right)\right],$$
$$\widehat{\Pi}_\i:\Pro\left(\Hshi\left(\TopC\right)\right) \to \Profs$$
is the functor constructed in Section <ref>.
It suffices to show that for every $\pi$-finite space $V,$ there is a canonical equivalence
$$\Hom_{\Profs}\left(\underset{n \in \Delta^{op}} \colim \left[\Pip\left(\underset{i \in \cI} \lim f_i^*U^n\right)\right],j\left(V\right)\right) \simeq \Hom_{\Profs}\left(\Pip\left(\underset{i \in \cI} \lim \cY^i\right),j\left(V\right)\right)$$
which is natural in $V.$ We have that
$$\Hom_{\Profs}\left(\underset{n \in \Delta^{op}} \colim \left[\Pip\left(\underset{i \in \cI} \lim f_i^*U^n\right)\right],j\left(V\right)\right) \simeq \underset{n \in \Delta} \lim \left[\underset{ i \in \cI^{op}} \colim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)\right].$$
Notice that $V$ is $k$-truncated for some $k$, and hence so is $j\left(V\right)$ by Proposition <ref>. Since filtered colimits of $k$-truncated spaces are $k$-truncated, it follows that for all $n,$
$$\underset{i \in \cI^{op}}\colim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)$$
is $k$-truncated. By Lemma <ref>, it then follows that
$$\Hom_{\Profs}\left(\underset{n \in \Delta^{op}} \colim \left[\Pip\left(\underset{i \in \cI} \lim f_i^*U^n\right)\right],j\left(V\right)\right) \simeq \underset{n \in \Delta_{\le k}} \lim \left[\underset{ i \in \cI^{op}} \colim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)\right].$$
By using that filtered colimits commute with finite limits, we then have that this is in turn equivalent to
$$\underset{ i \in \cI^{op}} \colim \left[\underset{n \in \Delta_{\le k}} \lim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)\right].$$
Again by Lemma <ref> this is equivalent to
$$\underset{ i \in \cI^{op}} \colim \left[\underset{n \in \Delta} \lim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)\right].$$
Finally, we have the following string of natural equivalences:
\begin{eqnarray*}
\underset{ i \in \cI^{op}} \colim \left[\underset{n \in \Delta} \lim \Hom_{\cS}\left(\Pi_\i f_i^*U^n,V\right)\right] &\simeq& \underset{ i \in \cI^{op}} \colim \Hom_{\cS}\left(\underset{n \in \Delta^{op}} \colim \Pi_\i f_i^*U^n,V\right)\\
&\simeq& \underset{ i \in \cI^{op}} \colim \Hom_{\cS}\left(\Pi_\i \underset{n \in \Delta^{op}} \colim f_i^*U^n,V\right)\\
&\simeq& \underset{ i \in \cI^{op}} \colim \Hom_{\cS}\left(\Pi_\i\cY^i,V\right)\\
&\simeq& \Hom_{\Profs}\left(\Pip\left(\underset{i \in \cI} \lim \cY^i\right),j\left(V\right)\right).
\end{eqnarray*}
Let $X$ be a log scheme. Denote by $\cU$ the basis of contractible open subsets of $X_{an}$ given by Lemma <ref>.
There is a hypercover
$$U^\bullet:\Delta^{op} \to \TopC/X_{an}$$
such that for all $n,$ the map $U^n \to X_{an}$ is isomorphic to the coproduct of inclusions of open neighborhoods in the basis $\cU,$ and all the structure maps are local homeomorphisms.
Using standard techniques, since $\cU$ is a basis for the topology of $X_{an}$ we can construct a split hypercover satisfying the above by induction (cfr. <cit.>).
The image under the Yoneda embedding of the hypercover of topological spaces $U^\bullet$ just constructed is a hypercover of $Y\left(X_{an}\right)$ in the $\i$-topos $\Hshi\left(\TopC\right).$ We will abuse notation by identifying the two.
We now prove our main result:
Let $X$ be a fine saturated log scheme locally of finite type over $\bC$. The induced map
$$\Pip\left(\Phi_X\right)\colon \Pip\left(X_{\log}\right) \stackrel{\sim}{\longlongrightarrow} \Pip\left(\sqrt[\infty]{X}_{top} \right)$$
is an equivalence of profinite spaces.
Consider now the hypercover $U^\bullet$ of $X_{an}$ just constructed. Then each $U^n=\underset{\alpha} \coprod V_{\alpha}$ where each $V_\alpha$ is in $\cU.$ Let us restrict to one such $V=V_\alpha.$ Since $V$ is in $\cU,$ there exists an $x \in V$ such that $\left(X_{log}\right)_x \to X_{log}|_V$ is a weak homotopy equivalence, and such that there is a Kato chart $U \to \Spec \bC[P]$, with $U \to X$ étale, such that $U_{an} \to X_{an}$ admits a section $\sigma$ over $V,$ and with the property that the composite
$$V \stackrel{\sigma}{\longrightarrow} U_{an} \to (\Spec \bC[P])_{an}$$
carries $x$ to the vertex point of the toric variety $\Spec \bC[P].$ Let us fix this $x$, and call it the center of $V$. Suppose that the monoid $P$ has rank $k,$ then the log structure at $x$ also has rank $k$. Moreover, the fiber of the map
$$V_n\defeq V \times_{(\Spec \bC[P])_{an}} \left(\Spec \bC\left[\frac{1}{n}P\right]\right)_{an} \to V$$
over $x$ consists of a single point (see <cit.>).
Let us fix an $n$, then we have that
$$\sqrt[n]{X}_{top}|_{V}\simeq \left[\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n\right]=\left[V_n/\left(\mathbb{Z}/n\mathbb{Z}\right)^k\right].$$
Hence our groupoid presentation $\mathbb{G}\left(n\right)$ for $\sqrt[n]{X}_{top}|_{V}$ guaranteed by Proposition <ref> is the topological action groupoid $\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n.$ This groupoid admits a continuous functor to $V$ (viewing $V$ as a topological groupoid with only identity arrows) which on objects is simply the canonical map $V_n \to V.$ Similarly, regard the one-point space $*$ also as a topological groupoid, and consider the canonical map
$$* \to V$$
picking out $x.$ Since $V$ and $*$ have no non-identity arrows, the lax fibered product of topological groupoids
$$* \times^{\left(2,1\right)}_{V} \left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n\right)$$
is equivalent to the strict fibered product
$$* \times_{V} \left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n\right)$$
which is canonically equivalent to the action groupoid
$$\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes \left(V_n\right)_x,$$
where $\left(V_n\right)_x$ is the fiber over $V_n \to V.$ Since this fiber consists of a single point, we conclude that the lax fibered product may be identified with $\left(\mathbb{Z}/n\mathbb{Z}\right)^k,$ where we are identifying the group $\left(\mathbb{Z}/n\mathbb{Z}\right)^k$ with its associated $1$-object groupoid.
Consider the continuous functor of topological groupoids
$$\left(\mathbb{Z}/n\mathbb{Z}\right)^k\simeq * \times^{\left(2,1\right)}_{V} \left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n\right) \to \left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n.$$
This induces a map of simplicial topological spaces between their simplicially enriched nerves
$$N\left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k\right) \to N\left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V_n\right).$$
By Lemma <ref>, this map is degree-wise a weak homotopy equivalence. It follows from Proposition <ref> and <cit.> that the induced map
$$B\left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k\right) \simeq \Pi_\i\left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes *\right) \to \Pi_\i\left(\left[\left(\mathbb{Z}/n\mathbb{Z}\right)^k \ltimes V\right]\right) \simeq \Pi_\i\left(\sqrt[n]{X}_{top}|_{V}\right)$$
is an equivalence in $\cS.$ Since the topological groupoid presentations for $\sqrt[n]{X}_{top}$ constructed in Section <ref> are compatible with the natural maps $\sqrt[m]{X}_{top} \to \sqrt[n]{X}_{top}$ when $n\mid m$, it follows that we have a natural identification of
$$\Pip\left(\sqrt[\i]{X}_{top}|_{V}\right)\simeq \underset{n} \lim B\left(\left(\mathbb{Z}/n\mathbb{Z}\right)^k\right)$$
in $\Profs.$ Consider the pro-system of finite groups
$$n \mapsto \left(\mathbb{Z}/n\mathbb{Z}\right)^k.$$
This is the $k^{th}$ Cartesian power of the pro-system
$$n \mapsto \left(\mathbb{Z}/n\mathbb{Z}\right),$$
which is simply $\widehat{\mathbb{Z}}.$ By Proposition <ref>, it follows that
$$\Pip\left(\sqrt[\i]{X}_{top}|_{V}\right)\simeq B\left(\widehat{\mathbb{Z}}^k\right),$$
and hence by Proposition <ref>, we have that
$$\Pip\left(\sqrt[\i]{X}_{top}|_{V}\right) \simeq B\left(\widehat{\mathbb{Z}^k}\right).$$
We also have that
$$\left(X_{log}\right)_x \cong \left(S^1\right)^k.$$
It follows that
$$\Pi_\i \left(X_{log}|_V\right) \simeq \Pi_\i\left(S^1\right)^k\simeq B\left(\mathbb{Z}^k\right),$$
and so
$$\Pip\left(X_{log}|_V\right) \simeq \widehat{B\left(\mathbb{Z}^k\right)}.$$
Since $\mathbb{Z}^k$ is a finitely generated free abelian group, it is good in the sense of Serre in <cit.>. It follows from <cit.> and Theorem <ref> that the canonical map
$$\widehat{B\left(\mathbb{Z}^k\right)} \to B\left(\widehat{\mathbb{Z}^k}\right)$$
is an equivalence of profinite spaces, hence
$$\Pip\left(X_{log}|_V\right) \simeq B\left(\widehat{\mathbb{Z}^k}\right).$$
It now follows that
$$\Pip\left(\sqrt[\i]{X}_{top}|_{V}\right)\simeq \Pip\left(X_{log}|_V\right),$$
which is a local version of our statement.
Now let us globalize using the hypercover $U^\bullet$. For each $n$, denote by $q_n$ the natural map
$$q_n:\sqrt[n]{X}_{top} \to X_{an}.$$
Since $\Pip$ preserves colimits, it follows that the induced map
$$\underset{l \in \Delta^{op}}\colim \Pip\circ \tau^*U^l \to \underset{l \in \Delta^{op}}\colim \left( \Pip\circ \underset{n} \lim q_n^*U^l\right)$$
is an equivalence of profinite spaces, where $\tau$ is the canonical map $\tau:X_{log} \to X_{an}.$ However,
$$\underset{l \in \Delta^{op}}\colim \Pip\circ \tau^*U^l \simeq \Pip\left(\underset{l \in \Delta^{op}}\colim \tau^*U^l\right)\simeq \Pip\left(X_{log}\right),$$
since $\tau^*U^\bullet$ is a hypercover of $X_{log}.$ Finally, by Lemma <ref>,
$$ \underset{l \in \Delta^{op}}\colim \left( \Pip\circ \underset{n} \lim q_n^*U^l\right)\simeq \Pip\left( \underset{n } \lim \sqrt[n]{X}_\topst\right)=\Pip\left(\sqrt[\i]{X}_{top}\right).$$
§ THE PROFINITE HOMOTOPY TYPE OF A LOG SCHEME
We conclude this paper by defining the profinite homotopy type of an arbitrary log scheme over a ground ring $k,$ by using the notion of étale homotopy type.
Étale homotopy theory, as originally introduced by Artin and Mazur in <cit.>, is a way of associating to a suitably nice scheme a pro-homotopy type. In this seminal work they proved a generalized Riemann existence theorem:
Let $X$ be scheme of finite type over $\mathbb{C},$ then the profinite completion of the étale homotopy type of $X$ agrees with the profinite completion of $X_{an}$.
In light of the above theorem, the étale homotopy type of a complex scheme of finite type gives a way of accessing homotopical information about its analytic topology by using only algebro-geometric information, and for a setting where the analytic topology is not available, such as a scheme over an arbitrary base, the profinite completion of its étale homotopy type serves as a suitable replacement.
In the original work of Artin and Mazur, for $X$ a locally Noetherian scheme, one associates a pro-object in the homotopy category of spaces $\mbox{Ho}\left(\cS\right).$ This definition was later refined by Friedlander in <cit.> to produce a pro-object in the category $\Set^{\Delta^{op}}$ of simplicial sets, and a generalized Riemann existence theorem is also proven in this context. In recent work of Lurie <cit.>, the étale homotopy type of an arbitrary higher Deligne-Mumford stack is defined by using shape theory to produce an object in the $\i$-category $\Pro\left(\cS\right)$ (in fact the definition in op. cit. is for spectral Deligne-Mumford stacks - analogues of Deligne-Mumford stacks for algebraic geometry over $\mathbb{E}_\infty$-rings), and Hoyois has recently proven that up to profinite completion, this definition agrees with that of Friedlander for a classical locally Noetherian scheme in <cit.>. See also recent work of Barnea, Harpaz, and Horel in <cit.>.
In recent work of the first author <cit.>, the étale homotopy type of an arbitrary higher stack on the étale site of affine $k$-schemes is defined, and is shown to agree with the definition of Lurie when restricted to higher Deligne-Mumford stacks. In particular, there is shown to be a functor
$$\widehat{\Pi}^{\et}_\i:\Shi\left(\Affk,\mbox{\'et}\right) \to \Profs$$
associating to a higher stack $\cX$ on the étale site of affine $k$-schemes of finite type a profinite space $\widehat{\Pi}^{\et}_\i\left(\cX\right)$ called its profinite homotopy type, and an even more generalized Riemann existence theorem is proven:
Let $\cX$ be higher stack on affine schemes of finite type over $\mathbb{C},$ then there is a canonical equivalence of profinite spaces
$$\widehat{\Pi}^{\et}_\i\left(\cX\right)\simeq \Pip\left(\cX_{top}\right)$$
between its profinite étale homotopy type and the profinite homotopy type of its underlying topological stack $\cX_{top}$ in the sense of Theorem <ref>.
Now let $X$ be a log scheme locally of finite type over $\bC.$ Its infinite root stack $\sqrt[\i]{X}$ is a pro-object in $\Shi\left(\Aff,\mbox{\'et}\right).$ Notice that the functor $\widehat{\Pi}^{\et}_\i$ canonically extends to a functor
$$\widehat{\Pi}^{\et}_\i:\Pro\left(\Shi\left(\Affk,\mbox{\'et}\right)\right) \to \Profs.$$
In light of the above theorem, we conclude that there is a canonical equivalence of profinite spaces
$$\widehat{\Pi}^{\et}_\i\left(\sqrt[\i]{X}\right) \simeq \Pip\left(\sqrt[\i]{X}_{top}\right)$$
between the profinite étale homotopy type of the infinite root stack $\sqrt[\i]{X}$ and the profinite homotopy type of the underlying topological stack of the infinite root stack $\sqrt[\i]{X}_{top}.$ Combining this with Theorem <ref> yields the following theorem.
Let $X$ be a log scheme locally of finite type over $\bC.$ Then the following three profinite spaces are canonically equivalent:
i) The profinite completion $\widehat{X_{log}}$ of its Kato-Nakayama space.
ii) The profinite homotopy type $\Pip\left(\sqrt[\i]{X}_{top}\right)$ of the underlying topological stack of its infinite root stack $\sqrt[\i]{X}.$
iii) The profinite étale homotopy type $\widehat{\Pi}^{\et}_\i\left(\sqrt[\i]{X}\right)$ of its infinite root stack $\sqrt[\i]{X}.$
In light of the above theorem, we make the following definition:
Let $X$ be a log scheme over a ground ring $k.$ Then the profinite homotopy type of $X$ is the profinite étale homotopy type of its infinite root stack $\sqrt[\i]{X}.$
In this appendix we gather some definitions and results about log schemes, analytification, the Kato-Nakayama space, root stacks and topological stacks.
§.§ Log schemes
Log (short for “logarithmic”) schemes were first defined and studied systematically in <cit.>. A modern introduction (with a view towards moduli theory) can be found in <cit.>.
We will give definitions and facts in the algebraic category, but we will apply them to the complex-analytic context as well. The only difference is that instead of the étale topology we will be using the analytic topology.
A log scheme is a scheme $X$ with a sheaf of monoids $M$ on the small étale site $X_{\et}$ and a homomorphism $\alpha\colon M\to \cO_X$ of sheaves of monoids, where $\cO_X$ is seen as a monoid with respect to multiplication of regular functions, such that $\alpha$ induces an isomorphism
\alpha|_{\alpha^{-1}(\cO^\times_X)}\colon \alpha^{-1}(\cO^\times_X)\to \cO^\times_X.
Note that the last condition gives us a canonical embedding $\cO_X^\times\hookrightarrow M$ as a subsheaf of groups.
We denote a log scheme by $(X,M,\alpha)$ or sometimes simply by $X$.
* Any scheme $X$ is a log scheme with $M=\cO_X^\times$ and $\alpha$ the inclusion. This is the trivial log structure on $X$.
* Any effective Cartier divisor $D\subseteq X$ induces a log structure, by taking $M$ to be the subsheaf of $\cO_X$ given by functions that are invertible outside of $D$.
* If $P$ is a monoid, the spectrum of the monoid algebra $X_P\defeq \Spec k[P]$ has a natural log structure. The sheaf $M$ is obtained by considering the natural map $P\to k[P]=\Gamma(\cO_{X_P})$ and taking the “associated log structure” (see below for a few more details).
Log structures can be pulled back and pushed forward along morphisms of schemes. In particular
* any open subscheme of a log scheme can be equipped with the restriction of the log structure,
* if we have a morphism of schemes $f\colon X\to \Spec k[P]$ we get an induced log structure on $X$. This happens in the following way: $f$ gives a morphism of monoids $P\to \cO_X(X)$, that induces $\widetilde{\alpha}\colon\underline{P}\to \cO_X$ where $\underline{P}$ is the constant sheaf. It is typically not true that $\widetilde{\alpha}$ induces an isomorphism between $\widetilde{\alpha}^{-1}\cO^\times_X$ and $\cO^\times_X$, but there is a procedure to fix the behaviour of the units, and this produces a log structure $\alpha\colon M\to \cO_X$. See <cit.> for details.
We remark that in the situation of the last bullet, the quotient $M/\cO_X^\times$ is obtained from $\underline{P}$ by locally “killing the sections of $\underline{P}$ that become invertible in $\cO_X$”, so in particular all the stalks of $M/\cO_X^\times$ are quotients of the monoid $P$.
We consider only coherent log structures, which are those that, étale locally, come by pullback from the spectrum of the monoid algebra of a monoid.
A log scheme $X$ is quasi-coherent if there is an étale cover $U_i$ of $X$, monoids $P_i$ and morphisms of log schemes $f_i\colon U_i\to \Spec k[P_i]$ that are strict, i.e. the log structure on $U_i$ is pulled back from $\Spec k[P_i]$ via $f_i$. The monoid $P_i$ and the map $f_i$ are a chart for the log structure over $U_i$.
A log scheme $X$ is coherent (resp. fine, resp. fine and saturated) if the monoids $P_i$ above can be taken finitely generated (resp. finitely generated and integral, resp. finitely generated, integral and saturated).
A morphism such as $f_i$ in the definition above that identifies the pullback of the log structure on the target with the one of the source will be called strict.
We are interested only in fine and saturated log schemes.
Let $X$ be a fine saturated log scheme and $x$ a geometric point. Then there exists an étale neighborhood $U$ of $x$ over which there is a chart for the log structure with monoid $P=(M/\cO_X^\times)_x$.
This says in particular that if $X$ is fine and saturated, we can locally find charts with $P$ fine, saturated and sharp.
The quotient sheaf $\overline{M}=M/\cO^\times_X$ is called the characteristic sheaf of the log structure. Taking the quotient (in an appropriate sense) by $\cO^\times_X$ of the map $\alpha$, we get an alternative definition of a (quasi-integral) log scheme, introduced in <cit.>.
Let us denote by $\Div_X$ the fibered category over $X_{\et}$ whose objects over $U\to X$ are pairs $(L,s)$ where $L$ is an invertible sheaf of $\cO_U$-modules on $U$ and $s$ is a global section. This is a symmetric monoidal fibered category, where the monoidal operation is given by tensor product.
A log scheme is a scheme $X$ together with a sheaf of monoids $A$ and a symmetric monoidal functor $L\colon A\to \Div_X$ with trivial kernel.
The phrasing “trivial kernel” in the definition means that if a section $a$ is such that $L(a)$ is isomorphic to $(\cO_X,1)$ in $\Div_X$, then $a=0$.
Given a (quasi-integral) log scheme $(X,M,\alpha)$, by taking the “stacky quotient” of $\alpha\colon M\to \cO_X$ by $\cO^\times_X$ we get the functor $L\colon A=\overline{M}=[M/\cO^\times_X]\to [\cO_X/\cO_X^\times]=\Div_X$. Quasi-integrality ensures that the quotient $[M/\cO^\times_X]$ is actually a sheaf. Of course integral log structures are quasi-integral. See <cit.> for details.
One can give a notion of charts in this context as well. For many purposes these two notions of chart can be used indifferently. We mostly use charts as in the first definition above. These are called “Kato charts” in <cit.>.
A fist approximation of how one should “visualize” a log scheme is by thinking about the stalks of the sheaf $\overline{M}$. This sheaf is locally constant on a stratification of $X$ (see Proposition <ref>) and the stalks are fine saturated sharp monoids. Of course this disregards the particular extension $M$ of $\overline{M}$ by $\cO_X^\times$ and the map $\alpha$ (or equivalently the functor $L$), so it is indeed just a crude approximation.
§.§ Analytification
We are mainly concerned with log schemes locally of finite type over $\bC$, and with their analytifications.
Recall that if $X$ is a scheme locally of finite typer over $\bC$, the associated analytic space $X_\an$ is defined as a set as the $\bC$ points $X(\bC)=X(\Spec\bC)$ of $X$. This has an “analytic” topology coming from the local embeddings into $\bC^n$. Moreover this construction extends also to algebraic spaces locally of finite type over $\bC$ (see <cit.>).
If $X$ is a log scheme locally of finite type over $\bC$, the analytication $X_\an$ inherits a log structure, because of the relationship between the étale topos of $X$ and the analytic topos of $X_\an$. An étale morphism $X\to Y$ induces a local homeomorphism $X_\an\to Y_\an$, that consequently has local sections in the analytic topology. This gives a functor from the étale site of $X$ to the analytic site of $X_\an$, and induces a morphism of topoi. The log structure on $X_\an$ is obtained via this functor. Thus,
in what follows every time something holds étale locally for the log scheme $X$, it will hold analytically locally for the log analytic space $X_\an$.
We will use this without further mention, and we will use the same letter to denote the sheaf of monoids $M$ on $X$ and the induced one on $X_\an$. This should cause no real confusion.
For a monoid $P$ we denote by $\bC(P)$ the analytification of the spectrum of the monoid algebra $\Spec \bC[P]$.
As sets we have $\bC(P)=\Hom(P,\bC)$, the set of homomorphisms of monoids, where $\bC$ is given the multiplicative structure.
A basis of opens of $\bC(P)$ (where $P$ is fine, saturated and sharp) can be described as follows: call $p_1,\hdots, p_k$ the indecomposable elements of $P$ (see <cit.>), and choose open disks $D_i$ in the complex plane $\bC$. Then the set of homomorphisms $\phi\in \Hom(P,\bC)$ such that $\phi(p_i)\in D_i$ is open in $\bC(P)$.
Letting the disks $D_i$ vary we get a basis for the open subsets of $\bC(P)$.
Analytification commutes with finite limits.
We will need the following result on the topological properties of analytifications of schemes locally of finite type over $\bC$. As references, we point out <cit.>.
Let $X$ be an affine scheme of finite type over $\bC$ and $Y\subseteq X$ be a closed subscheme. Then there exist compatible triangulations of $X_\an$ and $Y_\an$, realizing $Y_\an$ as a subcomplex.
We can apply this iteratively to a stratification, to get compatible triangulations of the ambient affine scheme and of all the (closed) strata.
§.§ Kato-Nakayama space
From now on all log schemes will be fine and saturated unless we specify otherwise. Just for this subsection, $X$ will denote an analytic space rather than a scheme.
The Kato-Nakayama space $X_\log$ of a log analytic space $X$ (for example of the form $Y_\an$ for some log scheme $Y$ locally of finite type over $\bC$) is a topological space introduced in <cit.>.
The idea is to define a topological space that “embodies” the log structure of $X$ in a topological way (i.e. without using the sheaf of monoids, but only “points”).
What comes out is a topological space $X_\log$ (that also comes with a natural sheaf of rings, but we do not use this in the present work) with a continuous map $\tau\colon X_\log\to X$ that is proper and surjective. Moreover if $U\subseteq X$ is the trivial locus of the log structure (the largest open subset over which $\cO_X^\times\hookrightarrow M$ is an isomorphism), the open embedding $i\colon U\to X$ factors through $\tau$, so that $X_\log$ can be considered as a “ relative compactification” of the open immersion $i$.
Let us denote by $p^\dagger$ the log analytic space given by the point $\mathrm{pt}=(\Spec \bC)_\an$ with monoid $M=\bR_{\geq 0}\times S^1$, and map $\alpha\colon M\to \bC$ described by $(r,a)\mapsto r\cdot a$. Note that this log structure is not integral.
As a set we have $X_\log=\Hom(p^\dagger,X)$, the set of morphisms of log analytic spaces from the log point $p^\dagger$ to $X$. By unraveling this one can also write
$$X_\log=\left\{(x,c) \mid c \colon M^\gp_x \to S^1 \mbox{ is a group hom. such that } c(f)=\frac{f}{|f|} \mbox{ for all } f\in \cO_{X,x}^\times\right\}.$$
In particular one can see that $\bC(P)_\log=\Hom(p^\dagger,\bC(P))=\Hom(P,\bR_{\geq 0}\times S^1)$, and the projection $\tau\colon \bC(P)_\log\to \bC(P)$ is given by post-composition with $\bR_{\geq 0}\times S^1\to \bC$.
Note that from the above description $\bC(P)_\log$ has a natural topology, that by means of local charts for the log structure gives a topology on $X_\log$ in general <cit.>.
From the description one sees easily that for $x\in X_\an$, the fiber $\tau^{-1}(x)$ is homeomorphic to $(S^1)^r$ where $r$ is the rank of the stalk $\overline{M}_x$, defined to be the rank of the free abelian group $\overline{M}_x^\gp$.
The construction of the Kato-Nakayama space is clearly functorial, and is also compatible with strict base change.
Let $f\colon X\to Y$ be a strict morphism of fine saturated log analytic spaces. Then the diagram of topological spaces
\xymatrix{
X_\log\ar[r] \ar[d]& Y_\log\ar[d]\\
X\ar[r] & Y
is cartesian.
The description of $X_\log$ as a set can actually be enhanced to a description of its functor of points (see Section <ref>).
We now prove that following proposition.
For any log scheme $X,$ the Kato-Nakayama space $X_{log}$ is locally Hausdorff, locally contractible and locally compact.
We will start by assuming that $X$ is affine and has a global chart $X\to \Spec \bC[P]$ for a fine saturated sharp monoid $P$, and will prove that $X_\log$ is locally compact, Hausdorff and locally contractible.
This implies the conclusion for arbitrary $X$.
Note that since $f\colon X\to \Spec \bC[P]$ is strict, there is a Cartesian diagram of topological spaces
\xymatrix{
X_\log\ar[r]\ar[d] & \bC(P)_\log\ar[d]^\tau\\
X_\an\ar[r]^{f_\an} & \bC(P).
Our proof will be as follows: we note that $X_\an$ and $\bC(P)$ are semialgebraic, and the map $X_\an\to \bC(P)$ is a semialgebraic function (this part of the diagram is even algebraic). We will check that $\bC(P)_\log$ is semialgebraic, and that the projection to $\bC(P)$ is a semialgebraic function.
After we do that, it will follow that $X_\log$ is semialgebraic as well (being the inverse image of the diagonal $\bC(P)\subseteq \bC(P)\times \bC(P)$, a semialgebraic set, through a semialgebraic map $(f_\an,\tau) \colon X_\an\times \bC(P)_\log\to \bC(P)\times\bC(P)$, see <cit.>), hence triangulable (by the results of <cit.>), and any triangulable locally semialgebraic set is locally compact, Hausdorff and locally contractible <cit.>.
The space $\bC(P)_\log$ is semialgebraic, and the projection $\bC(P)_\log \to \bC(P)$ is a semialgebraic map.
We will check this by writing out these spaces explicitly. Let $p_i$ be a finite set of generators for $P$ (for example the indecomposable elements), and assume to have a finite number of relations that present the monoid $P$, of the form $\sum_j r_{ij} p_j=\sum_j s_{ij}p_j$. Say there's $k$ generators and $h$ relations.
Then we have a map $\bC(P)=\Hom(P,\bC) \to \bC^k$ given by $\phi\mapsto (\phi(p_i))$. This is an embedding, and the closed image is the Zariski closed subset with equations $\prod_j (z_j)^{r_{ij}}=\prod_j (z_j)^{s_{ij}}$ obtained from the $h$ relations of the chosen presentation of $P$, and $(z_j)$ are the coordinates of $\bC^k$.
In the exact same way we have a map $\bC(P)_\log=\Hom(P,\bR_{\geq 0}\times S^1)\to (\bR_{\geq 0}\times S^1)^k$ given by $\psi\mapsto (\psi(p_i))$. To describe the image, let us note that we have $\bR_{\geq 0}\times S^1\subseteq \bR^3$ in a natural way, as a semialgebraic subset. If we denote by $(\zeta_j)$ the “coordinates” of $(\bR_{\geq 0}\times S^1)^k$, then the (isomorphic) image of $\bC(P)_\log$ is again described by the equations $\prod_j (\zeta_j)^{r_{ij}}=\prod_j (\zeta_j)^{s_{ij}}$, so it is semialgebraic (the equations translate into algebraic equations on $(\bR^3)^k$).
Of course the diagram
\xymatrix{
\bC(P)_\log \ar[r]\ar[d] & (\bR_{\geq 0}\times S^1)^k \ar[d]\\
\bC(P) \ar[r] & \bC^k
From this, it suffices to check that the map $(\bR_{\geq 0}\times S^1)^k\to \bC^k$ is semialgebraic, and this is easy: in coordinates (where we see $(\bR_{\geq 0}\times S^1)^k\subseteq (\bR^3)^k$ and $\bC^k\cong (\bR^2)^k$) it is given by $(a_i,b_i,c_i)\mapsto (a_i\cdot b_i,a_i\cdot c_i)$.
§.§ Root stacks
Root stacks of log schemes were introduced in <cit.>. The infinite root stack, an inverse limit of the ones with finitely generated weight system, is the subject of <cit.>. We briefly recall the functorial definition and the groupoid presentations coming from local charts.
Let us fix a natural number $n$ and a log scheme $X$ with log structure $L\colon A\to \Div_X$. We can consider a sheaf $\frac{1}{n}A$ of “fractions” of sections of $A$: the sections of $\frac{1}{n}A$ are formal fractions $\frac{a}{n}$ where $a$ is a section of $A$. There is a natural inclusion $i_n\colon A\to \frac{1}{n}A$.
Note that $\frac{1}{n}A$ is isomorphic to $A$ via $a\mapsto \frac{a}{n}$. Through this isomorphism, the inclusion $i_n$ corresponds to multiplication by $n \colon A\to A$. The fact that this map is injective follows from torsion-freeness of stalks of $A$, which are fine saturated sharp monoids.
The $n$-th root stack $\sqrt[n]{X}$ of the log scheme $X$ is the stack over $\Sch,$ the category of schemes (with the étale topology), whose functor of points sends a scheme $T$ to the groupoid whose objects are pairs $(\phi, N, a)$ where $\phi\colon T\to X$ is a morphism of schemes, $N\colon \frac{1}{n}\phi^*A\to \Div_X$ is a symmetric monoidal functor with trivial kernel and $a$ is a natural isomorphism between $\phi^*L$ and the composite $N\circ i_n$.
\xymatrix@=.5cm@R=.35cm{ \phi^*A\ar[rr]\ar[dd] & \ar@{=>}[d]!<-3ex,0ex>^a& \Div_X\\ & & \\ \frac{1}{n}\phi^*A\ar[uurr] & & }
Morphisms are defined in the obvious way.
In other words the $n$-th root stack parametrizes extensions of the symmetric monoidal functor $L\colon A\to \Div_X$ to the sheaf $\frac{1}{n}A$. The pair $(N,a)$ in the definition above could be called an “$n$-th root” of the log structure $L\colon A\to \Div_X$.
Every time $n\mid m$ there is a morphism $\sqrt[m]{X}\to \sqrt[n]{X}$, and by letting $n$ and $m$ vary, these maps give an inverse system of stacks over $\Sch$.
The infinite root stack $\sqrt[\infty]{X}$ of the log scheme $X$ is the pro-algebraic stack $(\sqrt[n]{X})_{n\in\bN}$.
In <cit.> the infinite root stack is defined as the actual limit of the inverse system in the $2$-category of fibered categories, but in the present paper it will always be the pro-object. We remark that the two contain the same information, since by the results of <cit.> the limit of the system of $n$-th root stacks recovers the log scheme completely, and hence recovers the pro-object as well.
The $n$-th root stack $\sqrt[n]{X}$ is a tame Artin stack with coarse moduli space $X$.
Moreover there are presentations of $\sqrt[n]{X}$ for each $n$ that assemble into a pro-object in groupoids in schemes, and can be regarded as a presentation of the pro-object $\sqrt[\infty]{X}$. This follows from the following local descriptions as quotient stacks <cit.>.
Let us fix a monoid $P$, and let us denote by $C_n$ the cokernel of the injective map $P^\gp\to \frac{1}{n}P^\gp$. Furthermore denote by $\mu_n(P)$ the Cartier dual of $C_n$. This acts on the monoid algebra $\Spec k[\frac{1}{n}P]$ ($k$ here is some base field, but this works the same way over $\bZ$).
If $X$ is a log scheme with a global chart $X\to \Spec k[P]$, then there is a cartesian diagram
\xymatrix{
\sqrt[n]{X}\ar[r]\ar[d] & [\Spec k[\frac{1}{n}P]/\mu_n(P)]\ar[d]\\
X \ar[r] & \Spec k[P]
presenting $\sqrt[n]{X}$ as a quotient stack $[X_n/\mu_n(P)]$, where $X_n=X\times_{\Spec k[P]} \Spec k[\frac{1}{n}P]$.
As we mentioned, these quotient stack presentations are all compatible, in the sense that they give a pro-object in groupoids in schemes $(X_n\times \mu_n(P)\rightrightarrows X_n)_{n\in\bN}$, that can be seen as a groupoid presentation of $\sqrt[\infty]{X}$.
If $X$ does not have a global chart we cover it with étale opens $U_i$ where there is a chart with monoid $P_i$
and assemble together the corresponding groupoid presentations.
The $n$-th root stack $\sqrt[n]{X}$ is a tame Artin stack, and is Deligne–Mumford when we are over a field of characteristic $0$.
§.§ Topological stacks
The main reference for this section is <cit.>.
The two preceding subsections were about the objects that we would like to compare, namely the Kato-Nakayama space and the infinite root stack of a log scheme locally of finite type over $\bC$. Note that the former is of topological nature, and the latter is algebraic. In order to find a map between them, we carry over the root stacks to the topological side.
One can talk about stacks over any Grothendieck site. Algebraic stacks (a.k.a. Artin stacks) are stacks on the category of schemes over a base with the étale topology[Sometimes, rather than working with the étale topology, one defines algebraic stacks with the fppf topology. However, the resulting $2$-category of stacks is the same, cfr. <cit.>.] that admit a representable smooth epimorphism from a scheme and whose diagonal is representable by algebraic spaces (and often one imposes some conditions on this diagonal morphism, like being quasi-compact or locally of finite type). Equivalently, one can describe algebraic stacks as stacks of (étale) torsors for certain groupoid objects in algebraic spaces, whose structure maps are smooth.
If instead of schemes over a base with the étale topology we start from topological spaces with the étale topology (where covers are local homeomorphisms), and we require a representable epimorphism from a topological space, we obtain the theory of topological stacks[In <cit.>, Noohi demands further conditions for such a stack to be called a topological stack, however in subsequent papers (e.g. <cit.>), he relaxes these conditions to the ones just described.]. Such a stack will always have diagonal representable by a topological space. As on the algebraic side, a topological stack can be defined through a groupoid presentation: a topological stack is a stack of principal $\cG$-bundles for $\cG$ a topological groupoid, and much of the basic yoga that one learns when working with algebraic stacks carries over in close analogy in this context.
In particular if $G$ is a topological group acting on a space $X$, the functor of points of the quotient stack $[X/G]$ is described as principal $G$-bundles (the topological analogue of $G$-torsors) with an equivariant map to $X$. In the same fashion, if $R\rightrightarrows U$ is a topological groupoid, one can characterize the associated stack $[U/R]$ as the stack of principal bundles for this groupoid.
There is a procedure to produce a topological stack starting from an algebraic one, that extends the analytication functor. We apply this in particular to the $n$-th root stacks of a log scheme.
Denote by $\Algst$ the $2$-category of algebraic stack locally of finite type over $\bC$ and by $\Topst$ the $2$-category of topological stacks.
There is a functor of $2$-categories
$$\left(\blank\right)_{top}:\mathbf{A}\!\mathfrak{lgSt}^{LFT}_{\mathbb{C}} \to \mathfrak{TopSt}$$
that associates a topological stack to an algebraic stack locally of finite type over $\mathbb{C}.$
In Section <ref>, we extend Noohi's results to produce a left exact colimit preserving functor from $\i$-sheaves (a.k.a. stacks of $\i$-groupoids) on the algebraic étale site, to hypersheaves on a suitable topological site. See Theorem <ref> and Corollary <ref>.
This functor has several nice properties. We point out the ones that we use:
1. If $X$ is a scheme (or algebraic space) locally of finite type over $\bC$, then $X_\topst\simeq X_\an$ is the analytification
2. The functor $\left(\blank\right)_{top}$ preserves all finite limits (i.e. is left exact).
3. The preceding properties give us a procedure for calculating $\cX_{top}$ for an algebraic stack $\cX$. If $R\rightrightarrows U$ is a groupoid presentation of $\cX$ where $R$ and $U$ are locally of finite type and the maps are smooth, then by the first property we can apply the analytification functor to the diagram, and by the second one, this will result in another groupoid, namely the groupoid in topological spaces $R_\an\rightrightarrows U_\an$. The topological stack $\cX_\topst$ is then the associated stack $[U_\an/R_\an]$.
In particular if $\cX=[U/G]$ for an action of an algebraic group locally of finite type $G$ on a scheme locally of finite type $X$, we have $\cX_\topst=[U_\an/G_\an]$.
Let $X$ be a log scheme locally of finite type over $\bC$. The topological $n$-th root stack of $X$ is the topological stack $\sqrt[n]{X}_\topst$.
As for the algebraic ones, the topological root stacks form an inverse system. The pro-topological stack $\sqrt[\infty]{X}_\topst\defeq (\sqrt[n]{X}_\topst)_{n\in\bN}$ is the topological infinite root stack of $X$.
§.§ The rank stratification
In this section we will prove that the characteristic sheaf $\overline{M}$ is locally constant on a stratification over the log scheme $X$. This is used in the main body of this article to prove that the Kato-Nakayama space and the infinite root stack are “stratified fibrations” over $X$, and that the map that we construct between them induces an equivalence of profinite completions.
The results of this part are probably known to experts, and we are including them because of the lack of a suitable reference.
By a stratification of a topological space $T$ we mean a collection of closed subsets $\cS=\{S_i\subseteq T\}_{i\in I}$ where $I$ is partially ordered, and the following are satisfied:
* if $i\leq j$, then $S_i\subseteq S_j$, and
* the stratification is locally finite: every point $t\in T$ has an open neighborhood $U$ such that only finitely many of the intersections $U\cap S_i$ are non-empty.
The locally closed subsets $S_j\setminus S_i$ will be called the strata of the stratification.
If in the above definition $T$ is the underlying topological space of a scheme $X$ and each $S_i$ is Zariski closed, we will say that $\cS$ is an algebraic stratification of the scheme $X$. Note that an algebraic stratification on $X$ will induce a stratification on the analytification $X_\an$.
Let $T$ be a topological space equipped with a stratification $\cS$, and let $f\colon T'\to T$ be a morphism, where $T'$ is a topological space or stack. We will say that $f$ is a stratified fibration with respect to $\cS$ if the restrictions of $f$ to the strata of $\cS$ are fibrations (in our case, this will always mean “locally the projection from a product”).
Now let $X$ be a log scheme locally of finite type over a field $k$. We will describe an algebraic stratification of $X$ over which the sheaf $\overline{M}$ is locally constant.
The basic idea is that we are stratifying by the rank of the stalks $\overline{M}_x^\gp$ of the sheaf of abelian groups $\overline{M}^\gp$.
The sheaf $\overline{M}^\gp$ is a constructible sheaf of $\bZ$-modules <cit.>. This means that (Zariski locally) there is a decomposition of $X$ into locally closed subsets over which $\overline{M}^\gp$ is a locally constant sheaf.
If $\xi$ is a generalization of $\eta$ in $X$, meaning that $\eta \in \overline{\{\xi\}}$, then there is a natural morphism of the stalks $\overline{M}_{\overline{\eta}}\to \overline{M}_{\overline{\xi}}$, and this is surjective (more specifically, it is a quotient by a face).
This last lemma follows from Proposition <ref> and from the explicit description of the stalks of the monoid $\overline{M}$ of the log structure obtained from a chart, see Remark <ref>.
In particular the rank “only jumps up in closed subsets”, i.e. for every $n\in\bN$ the subset $R_n$ of points of $X$ where the rank of the group $\overline{M}_{\overline{x}}^\gp$ is $\geq n$ is closed: it is constructible by Lemma <ref>, and stable under specialization by Lemma <ref>, so it is closed. Note also that $R_{n+1}\subseteq R_n$.
The rank stratification of a log scheme $X$ is the algebraic stratification $\cR=\{R_n\}_{n\in\bN}$, where
R_n=\{x\in X\mid \rank_\bZ \overline{M}_{\overline{x}}^\gp\geq n \}.
We will denote the strata by $S_n \defeq R_n\setminus R_{n+1}$.
For example, $R_0=X$ and the complement $X\setminus R_1$ is the open subset of $X$ where the log structure is trivial (which might be empty). In general $S_n$ is the locally closed subset of $X$ over which the rank of $\overline{M}_{\overline{x}}^\gp$ is equal to $n$.
We claim that both sheaves $\overline{M}$ and $\overline{M}^\gp$ are locally constant on the strata $S_n$.
To check this, let's describe the canonical log structure $\overline{M}_P \to \Div_{X_P}$ on $X_P=\Spec k[P]$ in more detail: the log structure is induced by the morphism of monoids $P\to k[P]$, which gives a morphism of sheaves of monoids $\underline{P} \to \cO_{X_P}$ (here $\underline{P}$ denotes the constant sheaf), from which we get the sheaf $\overline{M}_P$ by killing the preimage of the units in $\cO_{X_P}$. More precisely, denote by $\{p_i\}_{i\in I}$ the finitely many indecomposable elements of the fine saturated monoid $P$; these are generators of $P$. For a geometric point $x \to X_P$ call $S\subseteq I$ the subset of indices such that the image of $t^{p_i}\in k[P]$ is invertible in the residue field $k({x})$. Then the the stalk $(\overline{M}_P)_{x}$ is the quotient $\sfrac{P}{\langle p_i\mid i \in S\rangle}$.
In particular we note the following:
The only point $x$ of $X_P$ where the stalk $(\overline{M}_P)_{{\overline{x}}}$ has rank $n=\rank_\bZ P^\gp$ is the “vertex” $v_P$, the point given by the maximal ideal $\langle t^{p_i}\mid i \in I \rangle$ generated by the variables corresponding to the indecomposable elements of $P$.
The point $v_P$ is also sometimes referred to as the “torus-fixed point”.
Since $P^\gp\cong \bZ^n$ for some $n$, as soon as at least one of the indecomposable elements $p_i$ is killed, the rank will drop at least by $1$. The only point in which no indecomposable is killed is exactly the maximal ideal generated by all the $t^{p_i}$.
For every $n$ and every point $x$ of $S_n=R_n\setminus R_{n-1}$ there is an étale neighborhood $U\to S_n$ of $x$ such that the sheaves $\overline{M}|_{S_n}$ and $\overline{M}^\gp|_{S_n}$ are constant sheaves.
If we equip $R_n$ with the reduced subscheme structure, it is a (fine saturated) log scheme with the log structure pulled back from $X$, and the same is true for the open subset $S_n\subseteq R_n$. Consequently there is an étale neighborhood $U\to S_n$ of $x$ and a chart $U\to \Spec k[P]$ for the induced log structure on $U$, where $P=\overline{M}_{\overline{x}}$ (Proposition <ref>). If $\overline{M}_P$ is the sheaf of monoids for the canonical log structure on $\Spec k[P]$, there is exactly one point where the stalk has rank $n=\rank \; P$ (=$\rank_\bZ P^\gp$), corresponding to the vertex $v_P$ (Lemma <ref>).
This implies (since over $U$ the rank of the stalks of $\overline{M}$ is always $n$) that the morphism $U\to \Spec k[P]$ sends everything to $v_P$, and in turn that the sheaf $\overline{M}|_U$, being a pullback from $\Spec k[P]$, is constant. This implies that $\overline{M}^\gp|_U$ is constant as well, and concludes the proof.
Note that if $k=\bC$, the algebraic stratification of $X$ we just constructed induces a stratification of the analytification $X_\an$, and the sheaves $\overline{M}$ and $\overline{M}^\gp$ of the log analytic space are locally constant over the strata.
|
1511.00466
|
TUM]Shubhangi Guptacor1
[cor1]We gratefully acknowledge the support for the first author by the German Research Foundation (DFG) through project no. WO 671/11-1.
TUM,UB]Barbara Wohlmuth
IWS]Rainer Helmig
[TUM]Chair for Numerical Mathematics,
Technical University Munich,
Boltzmannstraße 3, 85748 Garching bei München, Germany
[IWS]Dept. of Hydromechanics and Modelling of Hydrosystems,
University of Stuttgart,
Pfaffenwaldring 61,
70569 Stuttgart,
[UB]Department of Mathematics,
University of Bergen,
Realfagbygget, Allégt. 41, Bergen, Norway
We present an extrapolation-based semi-implicit multirate time stepping (MRT) scheme and a compound-fast MRT scheme
for a naturally partitioned, multi-time-scale hydro-geomechanical hydrate reservoir model.
We evaluate the performance of the two MRT methods compared to an iteratively coupled solution scheme and discuss their advantages and disadvantages.
The performance of the two MRT methods is evaluated in terms of speed-up and accuracy by comparison to an iteratively coupled solution scheme.
We observe that the extrapolation-based semi-implicit method gives a higher speed-up but is strongly dependent on the relative time scales of the latent (slow) and active (fast) components.
On the other hand, the compound-fast method is more robust and less sensitive to the relative time scales, but gives lower speed up as compared to the semi-implicit method,
especially when the relative time scales of the active and latent components are comparable.
Multirate time stepping semi-implict multirate method compound-fast multirate method hydro-geomechanical model methane hydrate reservoir Differential Algebraic Equations (DAE)
§ INTRODUCTION
Methane hydrates are crystalline solids formed when water molecules form a cage-like structure and trap a large number of methane molecules within.
Methane hydrates are thermodynamically stable under conditions of low temperature and high pressure and occur naturally in permafrost regions or sub-seafloor soils.
If warmed or depressurized, methane hydrates destabilize and dissociate into water and methane gas.
Natural gas hydrates are considered to be a promising energy resource.
It is widely believed that the energy content of methane occurring in hydrate form is immense, possibly even exceeding the combined energy content of all other conventional fossil fuels <cit.>.
Therefore, the development of multiphysics models and numerical codes for coupled hydro-thermo-chemo-geo-mechanical processes are of particular interest
for evaluating future technologies for gas extraction from methane hydrate reservoirs and for making detailed risk quantification from the inherent geohazards.
A mathematical model describing the hydromechanical processes in a subsurface methane hydrate system has been presented in our earlier work (Gupta et al. (2015)) <cit.>.
The governing PDE's are summarised in Table <ref> along with some selected closing and constitutive relationships.
This system of PDEs can be decomposed into two sub-classes of models, the flow and transport model comprising the mass and energy balance equations
for the phases occupying the pore spaces in the hydrate formation, i.e., equations (<ref>), (<ref>), (<ref>), and (<ref>),
and the geomechanical model, comprising the momentum balance equation (<ref>) for the soil-hydrate composite phase, also referred as the solid-skeleton.
The model accounts for the effects of the geomechanics on the flow model through the adjustment of the affected reaction and hydraulic properties by scaling the properties with functions of total porosity $\phi$.
In this sense, the soil-phase mass balance (Eqn.(<ref>)), which solves for the total porosity, can be seen as mortar between these two sub-models.
For simplicity, we eliminate the PDE (<ref>) by approximating the total porosity as a function of the volumetric strain
by assuming that the hydrate-coated soil grains are relatively incompressible compared to the solid skeleton $\epsilon_v$ <cit.>.
In <cit.>, we have presented an iteratively coupled solution strategy where, the hydrate reservoir model is decomposed into flow model and geomechanical model as described above,
and is solved iteratively for a given time-step by exchanging shared state variable values between the flow and the geomechaniccal models through a block Gauss-Seidel solution scheme.
At each iteration step, the flow and the geomechanical models solve their corresponding subsystems of equations separately.
This iteratively coupled solution scheme greatly reduces the computational effort as compared to a monolithic fully implicit scheme.
However, the dynamics of the flow and geomechanical models evolve at different time scales <cit.>.
We know a priori that the ground deformations manifest at a much slower rate as compared to the flow and transport processes.
Since the refinement of the time-mesh is controlled by the dynamics of active (or the fast) components,
solving the latent (or the slow) components at this fine time-mesh results in a lot of unnecessary computational work.
The computation can be made cheaper if the slow components are solved on a coarse time-mesh and the fast components on a fine time-mesh.
Such time stepping methods are called Multi-Rate Time-stepping (MRT) methods.
The concept of MRT methods was introduced for systems of differential equations (ODEs and DAEs) in such studies as <cit.>,
and some recent results are presented in <cit.>.
MRT methods for hyperbolic conservative laws are developed in <cit.> and for parabolic equations in <cit.>.
A review of the MRT methods developed over the last two decades can be found in <cit.>.
The application of MRT methods, especially the Implicit-Explicit methods (IMEX), is becoming increasingly popular in the PDE community.
Some of the recent extensions of these methods to application areas of coupled free and porous media flows, air pollution modelling, multi-scale fluid-solid interaction, among others, can be found in
In this article we present two MRT algorithms for our hydro-geomechanical hydrate reservoir model.
The first MRT algorithm is based on a semi-implicit 'fastest-first' approach and the second is based on a compound-fast approach.
We evaluate the performance of these MRT schemes in comparison to the iteratively coupled solution scheme,
and discuss the advantages and disadvantages of the two MRT methods with respect to the multi-time-scale hydro-geomechanical problems.
To understand the stability of these and related MRT methods in general, the reader is refered to <cit.>.
Summary of the mathematical model
3c Governing equations:
Mass balance eqn. for each mobile component $\kappa=CH_4,H_2O$ $\sum\limits_\alpha \partial_t \left( \phi \rho_\alpha S_\alpha \chi_\alpha^\kappa \right)$
$ + \sum\limits_\alpha \nabla \cdot \left( \phi \rho_\alpha S_\alpha \chi_\alpha^\kappa \myvec{v}_{\alpha,t} \right)$
$ = \sum\limits_\alpha \nabla \cdot \left( \phi S_\alpha \myvec{J}_{\alpha}^\kappa \right)$
$ + \dot g^\kappa + \sum\limits_\alpha \dot q_{\alpha,m}^\kappa$
Mass balance eqn. for hydrate phase $\partial_t \left( \phi \rho_h S_h \right)$
$ + \nabla \cdot \left( \phi \rho_h S_h \myvec{v}_{h,t} \right)$
$ = \dot g^h $
Mass balance eqn. for soil phase $\partial_t \left[\left(1-\phi\right) \rho_s \right]$
$ + \nabla \cdot \left[\left(1-\phi\right) \rho_s \myvec{v}_{s} \right]$
$ = 0 $
Energy balance eqn. $\partial_t \left[ \left(1-\phi\right) \rho_s u_s + \sum\limits_\beta \left( \phi \rho_\beta S_\beta u_\beta \right) \right]$
$ + \sum\limits_\alpha \nabla \cdot \left( \phi \rho_\alpha S_\alpha \chi_\alpha^\kappa \myvec{v}_{\alpha,t} h_\alpha \right)$
$ = \nabla \cdot k^c_{eff} \nabla T$
$ + \dot Q^h + \sum\limits_\alpha \left( \dot q_{\alpha,m}^\kappa h_\alpha \right) $
momentum balance eqn. for hydrate-soil composite $ \nabla \cdot \myvec{\tilde\sigma} + \rho_{sh} \myvec{g} = 0 $
3c Closing and constitutive relationships:
Closure relationships 2L12cm$P_g - P_w = P_c\left(S_{we}\right)$ , $\sum\limits_\beta S_\beta = 1$ , $\forall \kappa$: $\sum\limits_\alpha \chi_\alpha^\kappa = 1$
Phase velocities 2L12cm $\phi S_\beta \myvec{v}_{\beta,t} = \myvec{v}_{\beta} + \phi S_\beta \myvec{v}_{s} $ ,
$\myvec{v}_\alpha = \kappa \dfrac{k_{r,\alpha}}{\mu_\alpha} \left( \nabla P_\alpha - \rho_\alpha \myvec{g} \right) $ ,
$\myvec{v}_h = 0$ ,
$\myvec{v}_s = \partial_t \myvec{u}$
Diffusive solute flux 2L12cm$\myvec{J}_\alpha^\kappa = - \tau D^\alpha \left( \rho_\alpha \nabla \chi_\alpha^\kappa \right)$
Stress-strain relationship 2L12cm $\myvec{\tilde\sigma} = 2 G_{sh} \myvec{\tilde\epsilon} + \lambda_{sh} \left( tr\ \myvec{\tilde\epsilon} \right) \myvec{\tilde I} + \alpha_{biot} P_{eff} \myvec{\tilde I}$ ,
$\myvec{\tilde\epsilon} = \frac{1}{2}\left( \nabla \myvec{u} + \nabla^T \myvec{u} \right)$
Reaction kinetics 2L12cm $\dot g^{CH_4} = k^{r} M_{g} A_{rs} \left( P_{eqb}(T) - P_g \right) $ ,
$\dot g^{H_2O} = \dfrac{N_h M_{w}}{M_{g}} \dot g^{CH_4} $ ,
$\dot g^{h} = \dfrac{M_{h}}{M_{g}} \dot g^{CH_4} $ ,
$\dot Q^{h} = \dfrac{\dot g^h}{M_h}\left( B_1 - \dfrac{B_2}{T} \right) $
3c Description:
3L16cm$\alpha = g,w$ denotes the mobile gas and liquid phases respectively. The gas phase contains methane gas and water vapour. The liquid phase contains water and dissolved methane.
Subscript $s$ denotes the soil phase, $h$ denotes the hydrate phase, and $sh$ denotes the soil-hydrate composite phase.
$\beta = g,w,h$ denotes all the phases that occupy the pore space.
$\kappa = CH_4,H_2O$ denotes the molecular components.
Primary variables 2L12cmGas pressure $P_g$, water phase saturation $S_w$, hydrate phase saturation $S_h$, total porosity $\phi$, temprature $T$, and soil displacement $\myvec{u}$.
Seconday variables 2L12cmGas phase saturation $S_g$, water phase pressure $P_w$, mole concentrations $\chi_\alpha^\kappa$, relative phase velocities $\myvec{v}_\beta$,
soil phase velocity $\myvec{v}_s$, total stress $\myvec{\tilde\sigma}$, strain $\myvec{\tilde\epsilon}$.
Other variables 2L12cmMethane generation rate $\dot g^{CH_4}$, water generation rate $\dot g^{H_2O}$, hydrate dissociation rate $\dot g^{h}$,
heat of dissociation $\dot Q^{h}$, effective pore pressure $P_{eff}$.
Material properties 2L12cmPhase density $\rho_\gamma$, capillary pressure $P_c$, intrinsic permeability $\kappa$, relative permeability $k_{r,\alpha}$, phase viscosity $\mu_\alpha$,
binary diffuion cofficient $D^\alpha$, tortuosiy $\tau$, phase entalpy $h_\alpha$, phase internal heat $u_{\gamma}$, effective thermal conductivity $k^c_{eff}$,
lame's parameters $(G_{sh},\lambda_{sh})$, Biot's consant $\alpha_{biot}$,
reaction rate $k^r$, reaction surface area $A_{rs}$, hydrate equilibium pressure $P_{eqb}$, molar mass $M_\beta$, hydration number $N_h$.
Other material properties 2L12cmCompressibility $K_\gamma$, Bulk modulus $\gamma$ $B_{\gamma}$, Young's modulus of soil-hydrate composite phase $E_{sh}$,
Poisson ratio of soil-hydrate composite phase $\nu_{sh}$, thermal conductivities $k^c_{\gamma}$, specific heat capacities $Cp_\alpha,Cv_\gamma$.
§ MULTIRATE TIME STEPPING ALGORITHM
Let the vectors $\myvec{X}_F(t):\mathbb{R}\rightarrow \mathbb{R}^{d_f}$ and $\myvec{X}_G(t):\mathbb{R}\rightarrow \mathbb{R}^{d_g}$ denote the time-dependent discrete-in-space approximations to the
primary variables of the Flow model (i.e. $P_g,S_w,S_h,T$) and the Geomechanical model (i.e. $\myvec{u}$) respectively (see Table <ref> for definitions of the variables).
We will refer to $\myvec{X}_F$ as the active components and $\myvec{X}_G$ as the latent components.
Further, let $\myvec{F}:\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^{d_f}\times\mathbb{R}^{d_g}\rightarrow \mathbb{R}^{d_f}$
and $\myvec{G}:\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^{d_f}\times\mathbb{R}^{d_g}\rightarrow \mathbb{R}^{d_g}$
denote the spatial discretization operators for the Flow and the Geomechanical models respectively.
Here, $d$ is the dimension of the space domain.
In our numerical scheme, the operator $\myvec{F}$ is obtained by discretizing PDEs (<ref>-<ref>,<ref>) using the cell-centered finite volume method,
and the operator $\myvec{G}$ is obtained by discretizing PDE (<ref>) using the Galerkin finite element method.
However, the MRT methods described in this paper are independent of the methods used for spatial discretization.
The spatial discretization of the PDEs (<ref>-<ref>) governing our hydro-geomechanical model leads to a semi-discrete problem of the following form:
\begin{align}
&\text{For $t\in[0,T]$, given the initial conditions } \notag\\
&\quad \myvec{X}_F\left(t=0\right) = \myvec{X}_F^0 \ \text{and, } \myvec{X}_G\left(t=0\right) = \myvec{X}_G^0 \ , \notag\\
&\text{find solutions for $\myvec{X}_F$ and $\myvec{X}_G$ which satisfy}\notag \\
&\partial_t \myvec{X}_F = \myvec{F}\left( t, \myvec{x}, \myvec{X}_F, \myvec{X}_G \right)
\label{eqn:activeSystem}
\\
&\quad \ \ \myvec{0} = \myvec{G}\left( t, \myvec{x}, \myvec{X}_F, \myvec{X}_G \right)
\label{eqn:latentSystem}
\end{align}
Eqn.(<ref>) is the active ODE (Ordinary Differential Equation) system and Eqn.(<ref>) is the latent AE (Algebraic Equations) system.
Together, they form a naturally partitioned multi-scale DAE (Differential Algebraic Equations) system.
Each part of the partitioned DAE system is marched in time on an independent time-mesh which depends on it's own activity.
Here, activity of a component refers to the time-scale at which the dynamics of the governing equation for that component evolves.
We assume that the activity of the components does not vary in space, i.e., that the components evolve on the same time-scale throughout the spatial domain.
For the latent system, we define a coarse time-mesh $\{ T_n, 0 \leq n \leq N \}$ with time step sizes $\{H_n = T_n - T_{n-1}, 0 < n \leq N \}$.
We will refer to this as the macro-grid, and the time step from $T_{n-1}$ to $T_n$ as the macro-step.
For the active system, we define a refined time-mesh $\{ t_{n,k} \ , 0 \leq n < N, 0 \leq k \leq m \}$ with time step sizes $\{ h_{n,k} = t_{n,k} - t_{n,k-1}, 0 \leq n < N, 0 < k \leq m \}$
and multirate factor $m$.
We will refer to this as the micro-grid, and the time step from $t_{n-1,k-1}$ to $t_{n-1,k}$ for each $k=[1,...,m]$ as the micro-steps.
The two time-meshes are synchronized, which implies that for all n, $T_n = t_{n,0} = t_{n-1,m}$ (See Fig. <ref>).
Time-mesh for active and latent components
All MRT methods have the basic property that the time integration can proceed from synchronization level $n$ to $n+1$ only when all the components, slow and fast, have made their resepective macro and micro steps
and have synchronized at the level $n$.
For marching the DAE system (<ref>,<ref>) forward in time from $T_{n-1}$ to $T_n$,
the active ODE (<ref>) is integrated on the micro grid $t_{n-1,k}$ using the implicit Euler method for each micro step,
while the latent AE (<ref>) is evaluated directly at the macro grid point $T_n$ using the solution of the active ODE at $t_{n-1,m}$.
The two MRT algorithms that we will discuss differ in how the latent components are approximated on the micro grid for solving the active ODE.
In the semi-implicit MRT method (Algorithm 1), we first make the $m$ micro steps for the active components from $t_{n-1,0}$ to $t_{n-1,m}$.
The values of the latent components needed on the micro grid, i.e. ${\myvec{X}_G}_{n-1,k}$, for making the micro steps are approximated by means of extrapolation.
In our scheme, we construct a polynomial function of order $p$ for extrapolation using the values of $\myvec{X}_G$ evaluated at $p+1$ previous macro grid points,
i.e., $T_{n-1},...,T_{n-(p+1)}$.
We then make the final macro step to evaluate the latent component at $T_n$.
For $m=1$, this method essentially becomes a decoupled sequential solution scheme, which by itself is faster than the iteratively coupled solution scheme.
For $m\geq 1$, all systems are solved only once on their respective time-meshes,
and the coefficients of the extrapolation function also need evaluation only once per macro step,
thus requiring very little computational effort.
ALGORITHM 1: Semi-implicit MRT method
STEP 1: Extrapolation macro step Extrapolate ${\myvec{X}_G}_{n-1,k}$ at each $k = [1...m]$ on the fine time-mesh using the $p+1$ old step values of ${\myvec{X}_G}$
computed at $T_{n-1},...,T_{n-(p+1)}$:
\begin{align}\label{eqn:alg1-extrapolation}
&{\myvec{\tilde{X}{}}_{G}}_{n-1,k} = {\myvec{X}_{G}}_{n-1} + \sum \limits_{j=1}^{p} A_j \left( t_{n-1,k} - T_{n-1} \right)^{j}
% \\
% &\text{where, }
% \notag
% \\
% &A_2 = \frac{1}{H_{n-2}}\left(\frac{ {\myvec{X}_G}_{n-1} - {\myvec{X}_G}_{n-2} }{ H_{n-1} } - \frac{ {\myvec{X}_G}_{n-1} - {\myvec{X}_G}_{n-3} }{ H_{n-1} + H_{n-2} } \right)\notag
% \\
% &A_1 = \frac{ {\myvec{X}_G}_{n-1} - {\myvec{X}_G}_{n-2} }{ H_{n-1} } + A_2 \ H_{n-1} \notag
% \left( {\myvec{X}_G}_{n-1}, {\myvec{X}_G}_{n-2}, {\myvec{X}_G}_{n-3} \right)
\end{align}
STEP 2: Micro-steps Solve for ${\myvec{X}_F}_{n-1,k}$ at each $k = [1,...,m]$ on the fine time-mesh using the implicit Euler method:
\begin{align}\label{alg:1-2}
&{\myvec{X}_F}_{n-1,k} = {\myvec{X}_F}_{n-1,k-1} \notag \\
&+ h_{n-1,k} \ \myvec{F}\left( t_{n-1,k} \ , \ \myvec{x} \ , \ {\myvec{X}_F}_{n-1,k} \ , \ {\myvec{\tilde{X}{}}_G}_{n-1,k} \right)
\end{align}
STEP 3: Macro-step Solve for ${\myvec{X}_G}_{n}$:
\begin{align}
\myvec{G} \left( T_{n} \ , \ \myvec{x} \ , \ {\myvec{X}_F}_{n-1,m}\ , \ {\myvec{X}_G}_{n} \right) = 0.
\end{align}
In the compound-fast MRT method (Algorithm 2), we first make a predictor macro step to get an approximate value of the latent component at $T_n$.
In this step, we integrate the active ODE on the macro grid from $T_{n-1}$ to $T_n$ with a relaxed stopping criteria for the Newton slover.
This gives a rough approximation of $\myvec{X}_F$ at $T_n$ (denoted by ${\myvec{\tilde{X}{}}_F}_{n}$),
which is then used to solve the latent AE to get an approximate value of $\myvec{X}_G$ at $T_n$ (denoted by ${\myvec{\tilde{X}{}}_G}_{n}$).
We then make the micro steps to integrate the active ODE from $t_{n-1,0}$ to $t_{n-1,m}$.
The values of the latent components needed on the micro grid, i.e. ${\myvec{X}_G}_{n-1,k}$, for making the micro steps are approximated by means of linear interpolation (refer Eqn.(<ref>)).
In the final step, called the corrector macro step, we solve the latent AE once more at the macro grid point $T_n$ to correct (improve) the solution from the predictor step.
If the ODE becomes unsolvable on the macro grid and the predictor step fails, then, in our simulator we reduce the value of $H_n$ by half and attempt the predictor step once again.
ALGORITHM 2: Compound-fast MRT method
STEP 1: Predictor macro step (or compound step)
Integrate active ODE (<ref>) with large step size $H_n$. Relax the stopping criteria for the Newton solver to get a rough approximation at $T_n$, ${\myvec{\tilde{X}{}}_F}_{n}$:
\begin{align}\label{alg:2-1}
{\myvec{\tilde{X}{}}_F}_{n} &= {\myvec{X}_F}_{n-1} + H_{n} \cdot \myvec{F}\left( T_{n} \ , \ \myvec{x} \ , \ {\myvec{\tilde{X}{}}_F}_{n} \ , \ {\myvec{X}_G}_{n-1} \right)
\end{align}
Use ${\myvec{\tilde{X}{}}_F}_{n}$ to predict ${\myvec{\tilde{X}{}}_G}_{n}$:
\begin{align}
\myvec{G} \left( T_{n} \ , \ \myvec{x} \ , \ {\myvec{\tilde{X}{}}_F}_{n}\ , \ {\myvec{\tilde{X}{}}_G}_{n} \right) = 0.
\end{align}
STEP 2: Micro-steps
Solve for ${\myvec{X}_F}_{n-1,k}$ at each $k = [1,...,m]$ on the fine time-mesh using implicit Euler method:
\begin{align}\label{alg:2-2}
&{\myvec{X}_F}_{n-1,k} = {\myvec{X}_F}_{n-1,k-1} \notag \\
&+ h_{n-1,k} \ \myvec{F}\left( t_{n-1,k} \ , \ \myvec{x} \ , \ {\myvec{X}_F}_{n-1,k} \ , \ {\myvec{\tilde{X}{}}_G}_{n-1,k} \right)
\end{align}
where, ${\myvec{\tilde{X}{}}_G}_{n-1,k}$ are the linearly interpolated values of $\myvec{X}_G$ at $t_{n-1,k}$:
\begin{align}\label{eqn:alg2-interpolation}
{\myvec{\tilde{X}{}}_G}_{n-1,k} = {\myvec{X}_G}_{n-1} + \left( {\myvec{\tilde{X}{}}_G}_{n} - {\myvec{X}_G}_{n-1} \right) \sum\limits _{i=1}^k \frac{h_{n-1,k}}{H_n}
\end{align}
STEP 3: Corrector macro step
Solve for ${\myvec{X}_G}_{n}$:
\begin{align}
\myvec{G} \left( T_{n} \ , \ \myvec{x} \ , \ {\myvec{X}_F}_{n-1,m}\ , \ {\myvec{\tilde{X}{}}_{G}}_{n} + \Delta {\myvec{X}_{G}}_{n} \right) = 0.
\end{align}
Both the MRT algorithms are implemented in the C++ based DUNE PDELab framework <cit.> as an extension to our hydro-geomechanical hydrate reservoir simulator.
This code is capable of solving problems in 1D, 2D and 3D domains. The MRT methods discussed in this section are however, independent of the dimension of the space domain.
§ NUMERICAL EXAMPLES
We now present two model problems to test the performance of the MRT algorithms presented in Section <ref>.
The first problem considers $1D$ consolidation in a depressurized methane hydrate sample.
The second problem considers a relatively complex $3D$ example where we simulate the hydro-geomechanical processes in a subsurface hydrate reservoir
which is destabilized by depressurization using a vertically placed low pressure gas well.
§.§ Test 1
§.§ Test 2
§ CONCLUSIONS
For multi time scale hydro-geomechanical subsurface flow problems, the multirate time stepping methods provide a significant speed up as compared to fully coupled or decoupled (iterative or sequential) schemes,
especially for $3D$ problems, provided the model can be partitioned into sufficiently weakly coupled subsystems having distinctly different time scales.
In our case, we deal with subsurface hydrate reservoirs where the mathematical model is naturally partitioned into the active flow system and latent geomechanical system.
The stability of the extrapolation-based semi-implicit method is sensitive to the activity of the latent component,
while the compound-fast method is fairly independent of the activity of the latent component.
If the difference in the time scales between the active and latent components is comparable, then the semi-implicit MRT is more attractive as it gives a higher speed up.
It must, however, be kept in mind that this method is only conditionally stable for higher-order extrapolation and the extrapolation errors tend to accumulate with increasing $m$.
It is therefore necessary to keep the choice of $m$ small.
On the other hand, if the difference in the time scale between the active and latent components is large, then the compound-fast MRT method is more suitable as it is stable for arbitrarily large values of $m$,
provided that the active system is solvable in the predictor step.
Another important consideration is whether the relative activities are expected to vary over time.
For problems where the activity of the latent component in particular fluctuates in time, it is important that the MRT method be insensitive to the the activity of the latent component, thus
making compound-fast MRT methods more attractive in such cases.
Many extensions of these MRT methods are possible in our hydrate reservoir simulator, for example, accounting for local variations in time scales of each component over the space domain, and,
stabilization of the semi-implicit MRT method, etc., which can make the MRT methods more attractive for solving large scale problems, especially in $3D$, more efficiently.
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1511.00273
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We consider inference for the parameters of a linear model when the covariates are random and the relationship between response and covariates is possibly non-linear. Conventional inference methods such as z intervals perform poorly in these cases. We propose a double bootstrap-based calibrated percentile method, perc-cal, as a general-purpose CI method which performs very well relative to alternative methods in challenging situations such as these. The superior performance of perc-cal is demonstrated by a thorough, full-factorial design synthetic data study as well as a real data example involving the length of criminal sentences. We also provide theoretical justification for the perc-cal method under mild conditions. The method is implemented in the R package `perccal', available through CRAN and coded primarily in C++, to make it easier for practitioners to use.
Keywords: Confidence intervals; Edgeworth expansion; Second-order correctness; Resampling
§ INTRODUCTION
In many applied settings, practitioners would like to make interpretable statements such as the expected average difference in a response variable, $Y$, associated with a unit difference in a covariate of interest, $X_j$, controlling for all other predictors. In situations like these, practitioners often run linear regressions despite the fact that the true but unobservable relationship between $Y$ and $X_j$'s may be non-linear. Doing so may be sensible when the utility of being able to make more interpretable statements such as the one above outweighs the cost of possible model bias, which may hard to discern (particularly in multivariate settings). An important challenge is how practitioners can produce valid inference upon their estimates of the true population-level best linear approximation for the relationship between predictor and response in these settings. We denote the target of interest by $\bbeta$ (for more detail, please see Section <ref>). Our aim with this paper is to help practitioners perform better inference for $\bbeta$ in these situations.
<cit.> call much-needed attention to this issue, showing that when the relationship between response $Y$ and covariates $\vX=(1,X_1,\ldots,X_p)^T$ is truly non-linear with noise that is possibly heteroskedastic, and when $\vX$ is itself random, standard linear model theory standard errors are asymptotically invalid. They show that the “sandwich estimator” of standard error does provide asymptotically correct inference for the population slopes, even when non-linearity and heteroskedasticity are present, and $\vX$ is random.
While the sandwich estimator may provide asymptotically valid inference, practitioners will also be understandably interested in better understanding how the finite sample performance of various methods of inference compare, as well as the asymptotic properties of those methods. We will show that in our setting, empirical coverage of population regression slopes deteriorates considerably for all traditional confidence interval methods. The primary contribution of this paper is to shine new light on these issues, proposing and studying an inference method which is convincingly superior to the sandwich estimator, and making a very fast implementation of this proposed method accessible to practitioners.
We propose a double bootstrap-based calibrated percentile method, perc-cal. The seminal work of Peter Hall shows the advantages of double bootstrap approach in classical settings involving population means. Population slopes as defined here are a more complex, non-linear object. For example, <cit.> studies univariate data without model misspecification. The methods he uses, then, need to be augmented with additional material about Edgeworth expansions that is adapted from <cit.>. For the first time, we prove in Section <ref> and in the appendix that even when $Y$ and $\vX$ have a non-linear joint distribution, and $\vX$ is random, that under relatively mild regularity conditions the rate of coverage error of perc-cal for two-sided confidence intervals of the best linear population slopes between a response variable $Y$ and $p$-dimensional covariates $\vX$ is $\mathcal{O}(n^{-2})$. In contrast, conventional methods achieve a rate of coverage error of $\mathcal{O}(n^{-1})$. We then show in a Monte Carlo study that perc-cal performs better than traditional confidence interval methods, including the BCa method (<cit.>), and other Sandwich-based estimators discussed in <cit.> and <cit.>. Our study is similar in structure to the simulation study that was performed in <cit.>, but modified to study a very wide variety of misspecified mean functions. We follow up this synthetic simulation study with a real data example involving a criminal sentencing dataset, and show that perc-cal once again performs satisfactorily. We have released an R package, `perccal' (<cit.>), available through CRAN and coded primarily in C++, so that practitioners may benefit from a fast implementation of perc-cal for their own analyses.
We argue the combination of theoretical and empirical justification presented in this paper supports the claim that perc-cal is a reliable confidence interval methodology that performs well in general, even in the presence of relatively severe model misspecification. The remainder of the paper is organized as follows. Section <ref> provides a review of confidence interval methods. Section <ref> presents the theoretical results. Section <ref> compares the performance of perc-cal with that of other often more commonly used confidence interval estimators in synthetic and real data settings. Section <ref> provides concluding remarks, and an Appendix gives all of the proofs.
§ LITERATURE REVIEW
§.§ Review of Bootstrap Confidence Intervals
There is a very wide variety of bootstrap methods that have been proposed in the literature to compute $(1-\alpha)$ confidence intervals. These methods include Efron's percentile method (<cit.>, page 146), Hall's percentile approach (<cit.>, page 7), and Hall's percentile-$t$ method (<cit.>, page 937). Other forms of bootstrap CIs include symmetric CIs (<cit.>, page 108) and short bootstrap confidence intervals (<cit.>, page 114). In general, performance of these methods depends upon the properties of the data generating process and/or the sample size. We are primarily interested in confidence interval methods that assume much less about the true underlying data generating process, which is usually unknown and often not well behaved in real data applications, making these methods less relevant to the work which follows.
Hall advocates the use of pivotal bootstrap statistics because pivotal bootstrap statistics have higher asymptotic accuracy when the limiting distributions are indeed pivotal (<cit.>, page 83). We emphasize that Hall's preference for pivotal bootstrap statistics, and much of the discussion regarding the relative merits of various confidence interval methods, are based on the asymptotic properties of these methods. When the sample size is small, these asymptotic considerations do not necessarily reflect the empirical performance of these methods. For example, Hall cautioned that “our criticism of the percentile method and our preference for percentile-$t$ lose much of their force when a stable estimate of $\sigma^2$ is not available” (<cit.> page 930). Simulation studies that reinforce this include <cit.>.
Another class of confidence intervals may be formed by replacing the standard error estimator in the standard z or t interval with a so-called `Sandwich' estimator (<cit.>), or one of the many extensions of the Sandwich estimator (<cit.>). A comprehensive review of Sandwich estimators can be found in <cit.>, and we will compare these methods with our proposed method in Section <ref>.
§.§ Review of Iterative Bootstrap Confidence Intervals
The idea of the iterative bootstrap (or double-bootstrap) was first introduced in <cit.>. The improvement on coverage probability of CIs was first analyzed in <cit.> and later discussed in more detail in <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>. A comprehensive review can be found in Section 3.11 in <cit.> (see also <cit.>, page 268). In general, the iterative bootstrap provides more accurate coverage probability at the cost of more computing.
To fix ideas, in this section we shall introduce the proposed double-bootstrap confidence interval method in a univariate case with generic notations. We will extend this procedure to the regression setting in Section <ref>. We assume that we observe $Z_1,\ldots,Z_m \stackrel{iid}{\sim} F$ for some distribution $F$. Let $\theta=\theta(F)$ be a parameter of our interest. We will estimate $\theta$ through the empirical distribution $\hat{F}(z)={1 \over m}\sum_{i=1}^m I(Z_i \le z).$ The estimator is denoted by $\hat{\theta} = \theta(\hat{F})=\theta(Z_1,\ldots,Z_m)$. The construction of the confidence interval is illustrated in Figure <ref> and is described as follows.
* For chosen bootstrap sample size $B_1$, obtain bootstrap samples $(\Z_1^{*}, \ldots, \Z_{B_1}^{*})$. Each $\Z_j^*$ consists of $m$ i.i.d. samples with replacement from $\hat{F}$. For chosen bootstrap sample size $B_2$, obtain double bootstrap samples corresponding to all bootstrap samples, $(\Z_{1,1}^{**}, \ldots, \Z_{1,B_2}^{**}, \Z_{2,1}^{**}, \ldots, \Z_{2,B_2}^{**}, \ldots, \Z_{B_1,1}^{**}, \ldots, \Z_{B_1,B_2}^{**})$ in the same manner as in the first-level bootstrap. Denote the empirical distributions by $\hat{F}_j^*$'s, $j=1,\ldots, B_1$, and $\hat{F}_{j,k}^{**}$'s, $j=1,\ldots, B_1, k=1,\ldots,B_2$, respectively.
* Obtain parameter estimates corresponding to the observed sample, $\hat{\theta}=\theta(\hat{F})$, all bootstrap samples, $(\hat{\theta}_1^{*}, \ldots, \hat{\theta}_{B_1}^{*})$ with $\hat{\theta}_{j}^{*}=\theta(\hat{F}_j^{*})$ and all double bootstrap samples corresponding to all bootstrap samples, $(\hat{\theta}_{1,1}^{**}, \ldots, \hat{\theta}_{1,B_2}^{**}, \hat{\theta}_{2,1}^{**}, \ldots, \hat{\theta}_{2,B_2}^{**}, \ldots, \hat{\theta}_{B_1,1}^{**}, \ldots, \hat{\theta}_{B_1,B_2}^{**})$ with $\hat{\theta}_{j,k}^{**}=\theta(\hat{F}_{j,k}^{**})$.
* Form $B_1$ double-bootstrap histograms $\hat{\btheta}_1^{**}, \ldots, \hat{\btheta}_{B_1}^{**}$, where each histogram $\hat{\btheta}_j^{**}$ is comprised of all $B_2$ double bootstrap estimates $(\hat{\theta}^{**}_{j,1}, \ldots, \hat{\theta}^{**}_{j,B_2})$ corresponding to the $j$th bootstrap sample and estimate, $\Z_j^*$ and $\hat{\theta}_j$, respectively, $j \in \{1, 2, \ldots, B_1 \}$.
* Find the largest $\hat{\lambda}$ such that $1/2<\hat{\lambda}<1$ and that $\hat{\theta}$ lies in the $1-\hat{\lambda}$ percentile and the $\hat{\lambda}$ percentile of the histograms $1-\alpha$ proportion of the time.
* $\hat{\theta}$ lies between the $(1-\hat{\lambda},\hat{\lambda})$ percentiles of the second-level bootstrap distributions $1-\alpha$ proportion of the time. Therefore our perc-cal $(1-\alpha)$ interval for $\theta$ is equal to the $(1-\hat{\lambda},\hat{\lambda})$ percentiles of the first-level bootstrap distribution, $[\hat{\btheta}^*_{(1-\hat{\lambda})}, \hat{\btheta}^*_{(\hat{\lambda})}]$.
perc-cal diagram
For a $(1-\alpha)$ left-sided perc-cal confidence interval for $\theta$, the only change in the procedure is in Step 4, where one uses the histograms to find the smallest $\hat{\lambda}$ such that $\hat{\theta}$ lies below the $\hat{\lambda}$ percentile of the histograms $1-\alpha$ percent of the time. In what follows, we shall refer the two-sided perc-cal interval as $\mathcal{I}_2=[\hat{\btheta}^*_{(1-\hat{\lambda})},\hat{\btheta}^*_{(\hat{\lambda})}]$ and the one-sided perc-cal interval as $\mathcal{I}_1=(\infty,\hat{\btheta}^*_{(\hat{\lambda})}].$
A similar double-bootstrap confidence interval is the double-bootstrap-$t$ method which uses the second-level bootstrap to calibrate the coefficient of the bootstrap standard deviation estimate. In practice, both methods can be applied. Hall commented in his book (<cit.>, page 142) that “either of the two percentile methods could be used, although the `other percentile method' seems to give better results in simulations, for reasons that are not clear to us.” Here “the `other percentile method”' refers to confidence intervals $\mathcal{I}_1$ and $\mathcal{I}_2$. Our simulation studies in Section <ref> demonstrate the same phenomenon. We have observed through simulation that the performance of double-bootstrap-$t$ can be erratic at times due to the instability which arises by relying upon a statistic which has a double bootstrap estimated standard error in its denominator. Some samples will inevitably have very small or even degenerate bootstrap standard errors, making this statistic very large. This issue is particularly acute when sample sizes are small.
Research on optimizing the trade-off between the number of simulations, $B_1$ and $B_2$, in double-bootstrap and the CI accuracy can be found in <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, among many others. In particular, <cit.> study the asymptotic convergence rate of the coverage probability involving $B_1$ and $B_2$, and suggest an adaptive method to optimize the choice of $B_2$ as a function of $B_1$. They note that $B_1$ should be to set to a larger number (for example, $B_1=1000$), and $B_2$ equal to a lesser value. In all simulations which follow, we set $B_1=B_2=2000$. Since the computation of perc-cal is reasonably efficient as discussed in Section <ref>, we do not optimize the number of bootstrap samples further but note that further performance gains for robust linear regression inference are a promising area for future research (in particular, with respect to $B_2$).
§.§ Review of Bootstrap Applications in Conventional Linear Models
Bootstrap in linear models is studied in Section 4.3 in <cit.>. Hall refers the fixed design case the “regression model” and the random design case the “correlation model.” Bootstrap estimation and confidence intervals for the slopes, as well as simultaneous confidence bands, are described.
Since the seminal paper of <cit.>, the bootstrap has been widely used in regression models because of its robustness to the sample distributions. A review of bootstrap methods in economics can be found in <cit.>. <cit.> consider bootstrapping the sandwich estimator for the standard error when the observations are dependent and heterogeneous. Bootstrap applications under other types of model misspecifications are recently considered in <cit.> and <cit.>. In this paper, we focus on a different case when observations of $(Y,\vX)$ are i.i.d. but the joint distribution is assumption-lean – we elaborate upon this further in the next section.
§.§ The Assumption-Lean Framework and Double-Bootstrap Applications
Conventional linear models assume $\E[Y|\vX]=\vX \bbeta$ for some $(p+1)$-vector $\bbeta$ as regression coefficients, so that $Y$ depends on $\vX$ only through a linear function. While this is commonly assumed in the bootstrap literature, we may not want to require it when performing inference in real data settings because the true relationship may not be linear. Moreover, as first noted in <cit.>, a non-linear relationship between $Y$ and $\vX$ and randomness in $\vX$ can lead to serious bias in the estimation of standard errors. <cit.> reviewed this problem, and proposed an “assumption-lean” framework for inference in regressions. In this framework, no assumption is made regarding the relationship between $Y$ and $\vX$. The only assumptions are on the existence of certain moments of the joint distribution of $\vV=(X_1,\ldots,X_p,Y)^T$. This consideration makes the model very general and thus widely applicable. Readers are referred to <cit.> for more details.
Even though a linear relationship between $\E[Y|\vX]$ and $\vX$ is not assumed in an assumption-lean framework, the slope coefficients that are estimated are always well-defined through a population least-squares consideration: the population least-squares coefficients $\bbeta$ minimize squared error risk over all possible linear combinations of $\vX$:
\begin{equation}\label{eq:bbeta}
\bbeta=argmin_{\b} \E\|Y-\b^T \vX\|_2^2 = \E[\vX\vX^T]^{-1} \E[Y \vX].
\end{equation}
This definition of coefficients $\bbeta$ is meaningful in addition to being well defined under minimal assumptions: $\bbeta$ provides us with the best linear approximation from $\vX$ to $Y$, whether or not $\vX$ and $Y$ are linearly related to one another. This setup allows for situations including random $\vX$, non-Normality, non-linearity and heteroskedasticity and we show later that the proposed perc-cal method provides better empirical coverage of the true population least-squares coefficients $\bbeta$ on average over a wide variety of data generating processes, even if those data generating processes involve random $\vX$, non-linearity in $E(Y|\vX)$ and/or heteroskedasticity. In contrast, previous research on the double bootstrap has studied functions that are linear or approximately linear.
To estimate $\bbeta$, denote the i.i.d. observations of $\vV$ by $\vV_1,\ldots,\vV_n$ and denote the $n \times (p+1)$ matrix with rows $\vV_1,\ldots,\vV_n$ by $\V$. Denote the distribution of $\V$ by $G$. The multivariate empirical distribution of $\vV$ is then $\hat{G}(\vV)=\hat{G}(x_1,\ldots,x_p,y)={1 \over n}\sum_{i=1}^n I(X_{1,i} \le x_1,\ldots,X_{p,i} \le x_p,Y_i \le y).$ The least squares estimate for $\bbeta$ defined in (<ref>) is
\begin{equation}\label{eq:bbeta.hat}
\hbbeta = (\X^T\X)^{-1} \X^T \Y,
\end{equation}
where $\X$ is the $n \times (p+1)$ matrix and $\Y$ is the $n \times 1$ vector containing the i.i.d. observations of $\vX$ and $Y$ respectively. Note that each estimate $\hat{\beta}_j$ of $\beta_j$ can be written as a function of $\hat{G}$, $\hat{\beta}_j = \beta_j (\hat{G}).$
The perc-cal confidence intervals for each of the slopes $\beta_j$ in $\bbeta$ are constructed similarly as described in Section <ref>. We use the pairs bootstrap first proposed by <cit.> and create $B_1$ i.i.d. bootstrap samples, $\V_k^*$, where each matrix $\V_k^*$ consists $n$ i.i.d. samples with replacement from $\hat{G}$. From these samples, one can create the empirical distribution $\hat{G}_k^*.$ To find the proper calibration of $\mathcal{I}_1$ or $\mathcal{I}_2$ (as defined in Section <ref>), we sample $B_2$ i.i.d. pairs bootstraps $\V_{k,h}^{**}$ with empirical distributions $\hat{G}_{k,h}^{**}$. The other steps in the construction are identical to those in Section <ref> with $\theta(\cdot)$ replaced by $\beta_j(\cdot)$ with respective empirical distributions as arguments. We note here that although confidence intervals can be built through empirical process theory <cit.>, the accuracy is usually not as good as the proposed perc-cal method, as will be discussed in the next section. While <cit.> studies bootstrap confidence bands under a nonparametric regression setting, we are unaware of any existing general bootstrap theory that yields general results for iterated bootstrap confidence intervals for a non-linear function of expectations of non-linear functions.
§ ASYMPTOTIC THEORY
In this section, we discuss the theoretical properties of perc-cal. The following theorem describes the accuracy on the coverage probability of perc-cal confidence intervals.
Consider $n$ i.i.d. observations of the $(p+1)$-dimensional random vector $\vV=(X_1,\ldots,X_p,Y)^T.$ Denote the vector of $Y$ and the continuous $X_j$'s by $\vV_C$ and that of the discrete $X_j$'s by $\vV_D$. Suppose that
* $\exists \ C_0>0$, such that the minimal eigenvalue of the covariance matrix $Var(\vV)$ is larger than $C_0$.
* The moment generating function of the random variable $\|\vV\|_2$ exists for all $t \in \RR$: $\E[e^{t\|\vV\|_2}]< \infty$ for all $t \in \RR$.
* Conditional on every single value of $\vV_D$, the joint distribution of $\vV_C$ is absolutely continuous with respect to the Lebesgue measure.
Consider the population least squares parameter defined in (<ref>) whose estimate is the sample least squares defined in (<ref>). Then for each $1 \le j \le p$, the $(1-\alpha)$ perc-cal CI for $\beta_j$ described in Section <ref> and Section <ref> have the coverage probabilities
\begin{eqnarray}
\P(\bbeta_j \in \mathcal{I}_1) &=& 1-\alpha+O(n^{-1}).\\
\P(\bbeta_j \in \mathcal{I}_2) &=& 1-\alpha+O(n^{-2}).
\end{eqnarray}
The proof of Theorem <ref> is in Section <ref> in the Appendix. It uses techniques of the Edgeworth expansion as in Section 3.11.3 of <cit.>, with a focus on the assumption-lean regression model setting where the dependence between $Y$ and $X_j$'s are not necessarily linear. We are not aware of prior investigation of the performance of the double bootstrap confidence intervals under this situation. Moreover, the results in <cit.> and <cit.> accommodates only $X_j$'s satisfying Cramèr's condition, which excludes distributions such as the Poisson distribution. Discrete distributions for $X_j$ are often encountered in models involving categorical predictors, and the data example we study in Section <ref> involves such covariates. Through a conditioning argument in <cit.>, we are able to show that the same performance is enjoyed by a wider class of mixed discrete and continuous $X_j$'s.
Our results show that the coverage probability is $1-\alpha+O(n^{-2})$. Note that for other construction approaches of confidence intervals such as from the sandwich estimator or from the empirical process theory, the resulting one-sided confidence intervals often have a coverage probability of $1-\alpha +O(n^{-1/2})$, and two-sided ones often have a coverage probability of $1-\alpha+O(n^{-1})$ (see Section 3.5.4 and Section 3.5.5 in <cit.>). Thus, the double bootstrap method provides better coverage.
§ NUMERICAL STUDIES
In this section, we study the performance of perc-cal compared to alternative (often more common) methods for forming confidence intervals, including other double bootstrap methods. We first compare perc-cal to these other methods using simulated data under a very wide variety of true data generating processes. We then illustrate our approach in a real data example. We will see that perc-cal performs very satisfactorily in general.
§.§ Synthetic Simulation Study
§.§.§ Design: Synthetic Simulation
We compare the performance of perc-cal with 10 other methods that are commonly used for constructing confidence intervals:
1. Standard normal interval: z (<cit.>, pp. 168).
2-6. Five sandwich variants: sand1, sand2, sand3, sand4 and sand5 (<cit.> provides a review of these methods, denoted there by H1, H2, H3, HC4 and HC5).
7. Hall's Studentized interval: stud (<cit.>).
8. Hall's “bootstrap-t” method: boot-t (<cit.>, pp. 160-162).
9. Efron's BCa interval: BCa (<cit.>).
10. Single percentile method: perc (<cit.>, pp. 170).
We consider a very wide range of underlying true data generating models, to obtain a more general understanding for how these confidence interval methods compare against one another in a wide variety of data settings, for large sample sizes as well as small. The data generating models represent a full factorial design of the following 48 factors, after excluding non-denerate combinations (i.e., combinations for which the conditional mean is not finite):
* Simple regression - one predictor, $Y = \beta_0 + \beta_1 X + \epsilon$
* Sample size $n$ = 32, 64, 128, 256
* Relationships between Y and X: (1) $Y = X + e$; (2) $Y = \exp(X) + e$; (3) $Y = X^3 + e$.
* Distribution of X: (1) $X \sim \mathcal{N}(0,1)$, (2) $X \sim \exp(\mathcal{N}(0,1))$.
* Noise: $\epsilon \sim (1) N(0,1); (2) \ |X|*\mathcal{N}(0,1); (3) \ \exp(\mathcal{N}(0,1)).$
In each of the above cases, we use 2000 first and second-level bootstrap samples for all bootstrap methods ($B_1=B_2=2000$). We obtain empirical coverage figures for the slope coefficient in the regression, $\beta_1$. Results are averaged over 500 replications to reduce the empirical standard error of the resulting intervals to below 1.5% on average across scenarios and methods. We present the results for a target coverage of 90%, without loss of generality (results for a target coverage of 95% are qualitatively the same).
§.§.§ Results
To more easily visualize the performance of many methods under many different scenarios, we begin with a coverage scatterplot in Figure <ref>. We rank sort the 48 scenarios by the empirical coverage proportion of perc-cal, in ascending order. This ordered list determines the scenario number associated with each scenario (i.e., scenario number 1 represents the scenario in which perc-cal's empirical coverage was the smallest, while scenario number 48 represents the scenario in which perc-cal's empirical coverage was the largest). On the x-axis, we provide the scenario number. On the y-axis, we provide the empirical coverage proportion of $\beta_1$ using all methods. We exclude sand1, sand2, sand3, and sand4 but include sand5, because sand5 has a better empirical coverage proportion than the other sandwich estimators. We add a horizontal line to the graph at the desired target coverage level of 90%.
Scatterplot of Empirical Coverage Proportion of Methods by Scenario Number – 90% Target Coverage
In general, none of the methods were “perfect” in the sense of always providing coverage at or above the target level of coverage. All noticeably undercover in particular cases and in these cases, perc-cal's relative performance is generally noticeably strong. Specifically, perc-cal achieves coverage less than target in 31 of the 48 scenarios. In 27 of those 31 below-target scenarios, perc-cal achieved a higher empirical coverage than all other alternative methods. In the remaining 4 of the 31 below-target scenarios where perc-cal was not the best-performing method, its absolute performance was close to target, averaging to 88.7%, and only 1 scenario with under 88% coverage. While perc-cal also had the highest empirical average coverage across all scenarios, we believe this robustness to challenging scenarios is more important to practitioners, who may take comfort in the fact that when perc-cal undercovers it almost always outperforms alternative methods, and in all other scenarios, it achieves or exceeds the desired coverage.
Across scenarios, while perc-cal provided the most consistent empirical coverage, sand5 was itself generally superior to the other alternative methods, including BCa, which has favorable asymptotic properties. perc-cal achieved a mean absolute deviation from target coverage of only 3.8 percentage points, versus 5.8 and 8.9 percentage points for sand5 and BCa, respectively. Restricting our attention to just the scenarios in which perc-cal achieved empirical coverage below 90%, the corresponding mean absolute deviation statistics were 5, 7.7, and 10.4, respectively. While it is perhaps no surprise that traditional methods such at the z interval fare so poorly because they typically assume a fixed-X setting, the relative performance of these methods which do not make such assumptions is more interesting.
perc-cal's improved coverage came at the cost of modestly longer interval lengths – across these 48 scenarios, perc-cal had an average interval length of 1.39, at the upper end (but not the top) of other methods – excluding the poor performance of z and boot-t, these other methods had interval lengths between 0.99 and 1.49, averaging to 1.16. While these other methods have interval lengths that are approximately 16% smaller on average, it would not be acceptable to a practitioner for this shortness to come at the expense of falling below desired target coverage. Only when target coverage is achieved do considerations like average interval length become a primary concern, and 16% longer intervals seems like a reasonable price for the “insurance” provided by perc-cal.
§.§ Real Data Example: Criminal Sentencing Dataset
§.§.§ Design: Real Data Example
We turn now to an example of how well perc-cal performs in practice, on real data. In this section, we compare perc-cal to other methods on a criminal sentencing dataset. This dataset contains information regarding criminal sentencings in a large state from January 1st 2002 to December 31st 2007. There are a total of 119,983 offenses in the dataset, stemming from a variety of crimes – murder (899 cases), violent crime (80,402 cases), sex-related crime (7,496 cases), property-related crime (92,743 cases), firearm-related crime (15,326 cases) and drug crime (93,506 cases). An individual offense can involve multiple types of crime, and an offender's case can involve multiple charges of each type of crime.
Our modeling objective is to form marginal confidence intervals for the slope coefficients of a linear regression. The response variable of our regression is the number of days of jail time an offender must serve (log-transformed), which we predict with the following 8 covariates:
* race: Binary variable for the race of the offender (1 if white, 0 if non-white).
* seriousness: A numerical variable scaled to lie between 0 and 10 indicating the severity of the crime. A larger number denotes a more serious crime.
* age: Age of offender at the time of the offense.
* race: The percent of the neighborhood that is not of Caucasian ethnicity in the offender's home zip code.
* in-state: Binary variable for whether the offender committed the crime in his/her home state (1 if in offender’s home state, 0 otherwise).
* juvenile: Binary variable for whether the offender had at any point committed a crime as a juvenile (1 if yes, 0 otherwise). 18% of all offenses involved offenders who had committed a crime as a child.
* prior-jaildays: Number of days of jail time the offender had previously served.
* age-firstcrime: The age of the offender when the offender was charged with his/her first crime as an adult.
This is truly a random X setting because the predictors themselves are stochastic, coming to us from an unknown distribution. $\vX$ is stochastic, the relationship between $\vX$ and $Y$ is unknown and possible non-linear, and error may have heteroskedastic variance. These results are not meant to be a complete study of the issue, but rather are presented to illustrate the potential of our methodology.
We first run a linear regression upon the full dataset containing all 119,983 offenses. We treat the coefficients as if this is a population-level regression. We then proceed as if we do not have the full dataset and instead only have the ability to observe random subsets of the dataset of size 500 – large, but not so large that all coefficient estimates are over-powered. We study the empirical coverage performance of confidence intervals formed using the methods in the simulation exercise over repeated realizations which are obtained through random subsamples of size 500.
§.§.§ Results: Real Data Example
Linear regression across the full dataset has an $R^2$ of 16.9% with 6 of the 8 predictors coming up as significant. We then take repeated random subsamples of size 500 from this population of offenses and treat these subsamples as if they were the observed dataset. Presupposing that each crime represents an $iid$ draw, this framework allows us to compare and contrast the empirical coverage performance of confidence interval methods.
In Figure <ref>, we present the empirical coverage for each of our predictors when we form 90% confidence intervals. The y-axis of the plot below represents the empirical coverage over 10,000 realizations for each of the methods in question (i.e., 90% empirical coverage for a particular method implies that 9,000 of the 10,000 realizations had confidence intervals for that method which contained the true but unknown population-level parameters). Along the x-axis, we have the predictors listed above. We include a bold horizontal line at the target level of empirical coverage of 90%. The standard error associated with the coverages presented below average to .002 across scenarios, predictors and methods.
Scatterplot of coverage proportion of methods – 90% Target Coverage
There are a number of inferences that we can draw from the above chart:
* All methods generally perform as expected, with empirical coverage proportions generally falling between 85% and 92%.
* perc-cal is the only method that consistently achieves empirical coverage over 90%. All other methods, including sand5, where unable to do so.
* prior-jaildays appears to be the predictor with the most disappointing empirical coverage. All methods except for perc-cal do not achieve 90% empirical coverage. The average empirical coverage of prior-jaildays for all non-perc-cal methods was 87.0%.
* There is also considerable disparity in the ability of various methods to cover the coefficients associated with the intercept term and the in-state covariate. Although the BCa method has near-90% empirical coverage of the in-state super-population coefficient, its coverage is less satisfactory for the seriousness and prior-jaildays covariates.
When we plot the relationship of jail length (log transformed) against prior total jail length in Figure <ref>, adjusted for all of the other covariates in the super-population, we see an almost bi-modal relationship.
Jail Length (log transformed) versus Previous Total Jail Length, Adjusted for Other Predictors
It is clear from the plot in Figure <ref> that the highly misspecified relationship between $Y$ and $\vX$ is likely to be driving the large disparity (and general deterioration) in coverage performance across the various non-perc-cal confidence interval methods. Overall, these results support the notion that perc-cal is a good all-purpose confidence interval method, and that all other methods, while performing well for some of the covariates, do not perform well for all of the covariates as was the case for perc-cal. The results assuming target coverage of 95% are qualitatively the same as the results presented above.
§ DISCUSSION AND CONCLUDING REMARKS
If perc-cal performs so well relative to alternative more popular CI methods, why is it not used more in practice? We believe the use of double bootstrap methods in general have not been widely adopted primarily because of their computational cost. Although it is true that double bootstrap methods in general and perc-cal in particular require more computation, the computational burden of these procedures is far less problematic than in the past because of current computational advances. For example, the rise of grid computing has greatly facilitated parallel computation. Because perc-cal is trivially parallelizable, it is relatively straightforward to compute all second-level bootstrap calculations in parallel, allowing researchers to compute perc-cal at a computational “cost” that is on the order of a single bootstrap. Furthermore, the perceived computational cost of double bootstrap methods may be inflated due to the inefficiency with which the calculations are carried out in popular statistical programming languages, most notably R – the very same calculations are orders of magnitude faster in lower level languages, such as C++. The rising popularity and adoption of packages integrating R with C++ (<cit.>) can greatly reduce the cost of double bootstrap methods for practitioners performing data analysis in R who do not know C++. In the spirit of this, the R package we have created allows users to compute perc-cal intervals in R efficiently using C++ code via Rcpp. We are optimistic that the use of double bootstrap methods will only increase further as the cost of computing declines further over the next 10 years.
We have restricted our attention to equal-tailed intervals for all methods considered here. It is natural and certainly possible to extend our approach to compute the shortest unequal-tailed interval, even if other methods cannot or would not, because of the symmetry of the asymptotic distribution underlying those alternative methods. At the same time, this advantage should not be over-stated – for example, one may be forced so far into the tails of the bootstrap distribution that a considerably larger number of first and second-level bootstrap samples are required. Because this is not the focus of our paper, we do not pursue it further here.
The asymptotic theory we developed, examined, and compared the more traditional percentile and “bootstrap-t” methods to their double bootstrap analogs in our “assumption lean” setting. We did not study the asymptotic properties of alternative confidence interval methods in our setting. Although it would be interesting to do so, there are a very wide range of methods in the literature, making systematic theoretical study impractical.
In summary, randomness in $\vX$, non-linearity in the relationship between $Y$ and $\vX$, and heteroskedasticity “conspire” against classical inference in a regression setting (<cit.>), particularly when the sample size is small. We have shown that in theory, the percentile-calibrated method perc-cal provides very satisfactory empirical coverage – the asymptotic rate of coverage error under mild regularity conditions for a two-sided confidence interval of the best linear approximation between $Y$ and $\vX$ is $\mathcal{O}(1/n^2)$. Furthermore, perc-cal performs very well in practice, both in synthetic and real data settings. We believe that perc-cal is a good general-purpose CI method and merits consideration when confidence intervals are needed in applied settings by practitioners.
§ PROOF OF THEOREM <REF>
The proof consists of two parts. The first part shows the existence of the Edgeworth expansion of the pivoting quantity. The second part derives the asymptotic order of the error term by using the first terms in this expansion and the Cornish-Fisher expansion, which can be regarded as the inverse of the Edgeworth expansion.
Part I: Existence of Edgeworth Expansion.
Under three assumptions on the joint distribution in which part of the variables can be discrete, the validity of the Edgeworth expansion was shown in <cit.> through a conditioning argument. In this section, we shall show the existence of the Edgeworth expansion of the pivoting quantity for the confidence intervals by checking these three assumptions.
We note first that by (<ref>), $\bbeta_{(p+1)\times 1}$ can be written as a smooth function of the moments $\E[X_jY]$'s and $\E[X_{j_1}^{k_1}X_{j_2}^{k_2}]$'s, where $1 \le j, \le p$, $1 \le j_1 <j_2 \le p$, $k_1, k_2 \ge 0,$ and $k_1+k_2 \le 2.$ Moreover, by the Central Limit Theorem, the asymptotic distribution of the least square estimate $\hbbeta$ in (<ref>) is given by
\begin{equation}\label{eq:clt}
\sqrt{n}\left(\hbbeta - \bbeta\right) \rightarrow \cN\left(\0, \E[\vX \vX^T]^{-1}\E[(Y-\vX^T\bbeta)^2\vX \vX^T]\E[\vX \vX^T]^{-1}\right).
\end{equation}
Starting with the general case when $p \ge 4$, we note that the $(p+1)\times 1$ vector of the asymptotic variances of each $\sqrt{n}(\hbeta_j-\beta_j)$, $1 \le j \le p$, is again a smooth function of the moments $\E[X_{j_1}^{k_1}X_{j_2}^{k_2}X_{j_3}^{k_3}X_{j_4}^{k_4}]$'s, $\E[X_{j_5}^{k_5}X_{j_6}^{k_6}X_{j_7}^{k_7}Y]$'s, and $\E[X_{j_8}^{k_8}X_{j_9}^{k_9}Y^2]$'s, where $1 \le j_1 < j_2<j_3<j_4 \le p$, $1 \le j_5 < j_6<j_7 \le p$, $1 \le j_8 < j_9 \le p$, $k_1,k_2,\ldots,k_9 \ge 0$, $k_1 + k_2+k_3+k_4 \le 4$, $k_5+k_6+k_7 \le 3$, and $k_8+k_9 \le 2.$ Note that these moments for the asymptotic mean and variance of $\hbbeta$ can all be consistently estimated by their corresponding sample moments.
We now collect all related monomials of $X_j$ and $Y$ into a random vector $\W_p$ such that
\begin{equation}
\W_p =( \{X_{j_1}^{k_1}X_{j_2}^{k_2}X_{j_3}^{k_3}X_{j_4}^{k_4}\}, \{X_{j_5}^{k_5}X_{j_6}^{k_6}X_{j_7}^{k_7}Y\},\{X_{j_8}^{k_8}X_{j_9}^{k_9}Y^2\})^T,
\end{equation}
where $1 \le j_1 < j_2<j_3<j_4 \le p$, $1 \le j_5 < j_6<j_7 \le p$, $1 \le j_8 < j_9 \le p$, $k_1,k_2,\ldots,k_9 \ge 0$, $k_1 + k_2+k_3+k_4 \le 4$, $k_5+k_6+k_7 \le 3$, and $k_8+k_9 \le 2.$ For $p \geq 4$, the dimension of $\W_p$ is
$$d_p={p \choose 4}{8 \choose 4}+{p \choose 3}{6 \choose 3}+{p \choose 2}{4 \choose 2}={1 \over 12}p(p-1)(35p^2-135p+166).$$
For $1 \le p \le 3$, we can take a similar approach to write out $\W_p$.
* For $p=1$, $\W_1=(\{X_{1}^{k_1}\},\{X_{1}^{k_2}Y\},\{X_{1}^{k_3}Y^2\})^T$ where $0\le k_1 \le 4$, $0 \le k_2 \le 3$, and $0 \le k_3 \le 2.$ The dimension of $\W_1$ is $d_1=12.$
* For $p=2,$ $\W_{2}=( \{X_{1}^{k_1}X_{2}^{k_2}\},\{X_{1}^{k_3}X_{2}^{k_4}Y\},\{X_{1}^{k_5}X_{2}^{k_6}Y^2\})^T$ where $k_1 + k_2 \le 4$, $k_3+k_4 \le 3$, and $k_5+k_6 \le 2.$ The dimension of $\W_2$ is $d_2=31.$
* For $p=3,$ $\W_3 =( \{X_{1}^{k_1}X_{2}^{k_2}X_{3}^{k_3}\}, \{X_{1}^{k_4}X_{2}^{k_5}X_{3}^{k_6}Y\},\{X_{j_7}^{k_7}X_{j_8}^{k_8}Y^2\})^T$,
where $1 \le j_7 < j_8 \le 3$, $k_1,k_2,\ldots,k_9 \ge 0$, $k_1 + k_2+k_3 \le 4$, $k_4+k_5+k_6 \le 3$, and $k_7+k_8 \le 2.$ The dimension of $\W_3$ is $d_3=73$.
With $\W_p$ defined, we can write $\bbeta_{(p+1)\times 1}=\bbeta(\E[\W_p])$. We can also write the vector of the asymptotic variances of $\sqrt{n}(\hbeta_j-\beta_j)$, $1 \le j \le p$, by $\bsigma^2_{(p+1)\times 1} =\bsigma^2(\E[\W_p])$. From the $n$ i.i.d. samples of $\vV=(X_1,\ldots,X_p,Y)^T$, we can form the $\W_p$'s as $\W_{p,1},\ldots,\W_{p,n}$. The estimates of $\bbeta$ and $\bsigma^2$ can thus be written as $\hbbeta = \bbeta(\bar{\W}_p)$ and $\hbsigma^2=\bsigma^2(\bar{\W}_p)$ respectively, where $\bar{\W}_p={1 \over n}\sum_{i=1}^n\W_{p,i}$.
To check the three assumptions in <cit.>, we first denote the random vector of discrete variables in $\W_p$ by $\W_D$ with dimension $d_D$ and the random vector of the rest of the variables in $\W_p$ by $\W_C$ with dimension $d_C$. We shall use the same subscripts $D$ and $C$ for other related quantities to distinguish discrete variables from others. Note that in $\W_C$, some variables are products of discrete and continuous variables. The distributions of these product variables may not be absolutely continuous with respect to the Lebesgue measure. Nonetheless, the characteristic function of $\W_C$ still exists, and the proof in <cit.> still holds with this relaxation. Thus, we continue to check these three conditions.
Assumptions $1(i), 1(ii), 1(iii), 3(i),$ and $3(ii)$ pertain to the cumulants of $\W_p$. Under the existence of the moment generating function of $\vV$, these assumptions are satisfied.
To check Assumptions $2(i)$ and $2(ii)$, note that the pivoting quantity for the confidence intervals for $\bbeta$ is $(\hbbeta-\bbeta)\circ \bsigma^{\circ (-1)} $, where $\circ$ denotes the Hadamard product. Let $\tilde{\W}_p=\bar{\W}_p-\E[\W_p]$ be the centered version of $\bar{W}_p$. The function $g(\cdot)$ in <cit.> for regression can be written as
\begin{equation}
g(\tilde{\W}_p) = (\bbeta(\tilde{\W}_p+\E[\W_p])-\bbeta(\E[\W_p]))\circ \bsigma(\E[\W_p])^{\circ (-1)}.
\end{equation}
It is easy to see that $g(\0)=\0.$ Furthermore, it can be shown that since the distribution of $\vV$ is non-degenerate, the derivatives of $g$ with respect to $\tilde{\W}_C$ exist, are continuous in a neighborhood of $\0$, and the $d_C \times (p+1)$ matrix of the derivatives has full rank.
To check Assumptions $1(iv)$ and $1(v)$, we first note that the derivatives of the characteristic functions are bounded by appropriate moments, which all exist under the assumptions of Theorem <ref>. Therefore, we only need to show that the characteristic function of $\W_p$ is bounded away from $1$. To show this, we decompose this characteristic function by conditioning on $\W_D$, i.e.,
\begin{equation}
\E[e^{i\t^T\W_p}] = \sum_{\w_D} e^{i\t_D^T\w_D}\E[e^{i\t_C^T\W_C}|\W_D=\w_D],
\end{equation}
where the dimensions of $\t$, $\t_D$, and $\t_C$ are $d_p$, $d_D$, and $d_C$ respectively. To verify Assumptions $1(iv)$ and $1(v)$, it now suffices to show that $|\E[e^{i\t_C^T\W_C}|\W_D=\w_D]|<1$ for each $\w_D$. This proof is patterned after the arguments in Section 2.4 of <cit.> (page 65 to 67).
Denote the joint density of $\vV_C$ condition on $w_D$ by $f_{w_D}(\x_C,y): \RR^{p_C} \rightarrow \RR.$ We first approximate this density through simple functions. Given $\epsilon>0,$ let $f_1(\x_C,y)=\sum_{m=1}^M c_m I((\x_C,y) \in \S_m)$, where $c_m$'s are appropriate constants, $\S_m$'s are appropriate rectangular prisms, and $1 \le m \le M$ for some appropriate $M>0 $ such that $\int_{\RR^{p_C}} |f_{w_D}(\x_C,y)-f_1(\x_C,y)| d \x_C d y < \epsilon.$ We can then focus on showing that $ \lim_{\|\t_C\|_2 \rightarrow \infty} | \int_{\S_m} e^{i \t_C^T \w_C} d \x_C dy| \rightarrow 0$ for each $m$.
Let $\t_C=\t_C(u),$ where $u$ is an index that diverges to infinity. Through a subsequence argument, we can assume without loss of generality that for some $1 \le h \le d_C$, and with $s(h,\tilde{h}, u)= \t_{\tilde{h}}(u)/\t_h(u)$, the limit $s(h,\tilde{h})=\limsup_{u \rightarrow \infty} s(h,\tilde{h},u)$ exists for $1 \le \tilde{h} \le d_C$, and $|s(h,\tilde{h})| \le 1. $ We now can take $\s=\{s(h,\tilde{h},u)\}$ and write the real part of the integral $\int_{\S_m} e^{i \t_C^T \w_C} d \x_C dy$ as $ \int_{\S_m} \cos \{t_h \s^T \w_C\} d \x dy$. Note that each entry in $\w_C$ is a monomial of $x_j$'s and $y$. Thus, $\s^T \w_C$ is a polynomial of $x_j$'s and $y$. We can now take transformations of variables to show that $ \int_{\S_m} \cos \{t_h \s^T \w_C\} d \x dy = O(t_h^{-1})$. Similarly, the imaginary part of the integral can be shown to be $O(t_h^{-1})$. These facts entail that the magnitude of the conditional characteristic function is strictly less than 1 when $\|\t_C\|_2$ is large, which completes the proof.
Part II: The Asymptotic Accuracy of Double Bootstrap CIs
With the existence of the Edgeworth expansion, we develop the asymptotic accuracy of the two-sided double-bootstrap CI for regression. In this section, we use $\theta_0$ to denote a generic $\beta_j$ and use $\hat{\theta}$ to denote the corresponding $\hat{\beta}_j$. We show here only the proof for the two-sided perc-cal confidence intervals $\mathcal{I}_1$. The one-sided case for $\mathcal{I}_1$ is proved in a similar (and easier) manner. The techniques used in this proof are patterned after those in Section 3.11 in <cit.> but are reorganized for readability and included so that our analysis is self-contained.
Consider the distribution of $\hat{A}(\bar{\W}^*) = {\hat{\theta}^*-\hat{\theta} \over \hat{\sigma}}$, where $\bar{\W}^*$ is the bootstrap version of $\bar{\W}_p$. Note that for any $0<\gamma<1$, the quantile estimate $\hat{v}_\gamma$ satisfies
\begin{equation}\label{eq:quant_v}
\P(\sqrt{n} (\hat{\theta}^*-\hat{\theta})/\hat{\sigma} \le \hat{v}_{\gamma}|\W_{p,1},\ldots,\W_{p,n})=\gamma.
\end{equation}
Due to the existence of the Edgeworth expansion as described in Part I, we can write $\hat{v}_\gamma$ in the standard normal quantile $z_\gamma$ through the Cornish-Fisher expansion:
\begin{equation}
\hat{v}_\gamma = z_\gamma + n^{-1/2}\hat{p}_1 (z_\gamma)+n^{-1} \hat{p}_2 (z_\gamma) + O_p(n^{-3/2})
\end{equation}
where $\hat{p}_1$ and $\hat{p}_2$ are polynomials whose coefficients are sample estimates of that of $p_1$ and $p_2,$ and the coefficients of $p_1$ and $p_2$ depend only on the moments of $\W_p$. Given the condition that all moments of $\vV$ exist, all of these estimates are root-$n$ consistent.
Now consider the quantile $\hat{w}_\lambda$ in the bootstrap distribution of $\hat{\theta}^*$ such that
\begin{equation}\label{eq:quant_w}
\P(\hat{\theta}^* \le \hat{w}_\lambda | \W_{p,1},\ldots,\W_{p,n}) = \lambda.
\end{equation}
By comparing (<ref>) and (<ref>) with $\gamma=\lambda$, we see
\begin{equation}\label{eq:w_lambda}
\hat{w}_\lambda=\hat{\theta}+ n^{-1/2}\hat{\sigma} \hat{v}_\lambda = \hat{\theta}+ n^{-1/2} \hat{\sigma}(z_\lambda + n^{-1/2}\hat{p}_1 (z_\lambda)+n^{-1} \hat{p}_2 (z_\lambda) + O_p(n^{-3/2}))
\end{equation}
Thus, by Proposition 3.1 in <cit.> (page 102), we have
\begin{equation}
\begin{split}
& \P(\theta_0 \in (-\infty, \hat{w}_\lambda))\\
=& \P(\theta_0 \le \hat{\theta}+ n^{-1/2} \hat{\sigma}(z_\lambda + n^{-1/2}\hat{p}_1 (z_\lambda)+n^{-1} \hat{p}_2 (z_\lambda) + O_p(n^{-3/2})))\\
=& \lambda +n^{-1/2}r_1(z_\lambda)\phi(z_\lambda) + n^{-1}r_2(z_\lambda)\phi(z_\lambda) +O(n^{-3/2})
\end{split}
\end{equation}
where $\phi$ is the density of the standard normal distribution, and $r_1$ and $r_2$ are even and odd polynomials whose coefficients can be root-$n$ consistently estimated.
Let $\xi=2(1-\alpha/2-\lambda)$ and $\lambda=1-\alpha/2+\xi/2.$ To find a proper $\lambda$ for the perc-cal interval $\mathcal{I}_2$ is now to find $\xi$ such that
\begin{equation}\label{eq:quant_t}
\P(\theta_0 \in (\hat{w}_{1-\lambda},\hat{w}_{\lambda})) =\P(\theta_0 \in (\hat{w}_{\alpha/2-\xi/2},\hat{w}_{1-\alpha/2+\xi/2})) = 1-\alpha.
\end{equation}
Note that the coverage probability of a two-sided CI can be written as
\begin{equation}
\begin{split}
& \P(\theta_0 \in (\hat{w}_{1-\lambda}, \hat{w}_\lambda)) \\
=& \P(\theta_0 \leq \hat{w}_{\lambda}) - \P(\theta_0 \leq (\hat{w}_{1-\lambda}))\\
=& 2\lambda-1+ 2 n^{-1}r_2(z_\lambda) \phi(z_\lambda) +O(n^{-2}).\\
=& 1-\alpha+\xi + 2n^{-1}r_2 (z_{1-\alpha/2+\xi/2})\phi(z_{1-\alpha/2+\xi/2})+O(n^{-2})
\end{split}
\end{equation}
The cancellation of the $O(n^{-1/2})$ term due to that $-z_{1-\lambda}=z_\lambda$ and that $r_1$ is an even polynomial is crucial for the improvement in double-bootstrap. To achieve the accuracy of the coverage in Theorem <ref>, we would like to choose $\xi$ such that
\begin{equation}
\xi=-2n^{-1}r_2 (z_{1-\alpha/2+\xi/2})\phi(z_{1-\alpha/2+\xi/2})+O(n^{-2})
\end{equation}
Now consider the second-level bootstrap, in which we calibrate $\hat{\xi}$ for $\hat{\lambda}=1-\alpha/2+\hat{\xi}/2$ in the perc-cal intervals. Through a similar argument for the first-level bootstrap, we see that the calibrated $\hat{\xi}$ satisfies that
\begin{equation}
\hat{\xi}=-2n^{-1}\hat{r}_2 (z_{1-\alpha/2+\hat{\xi}/2})\phi(z_{1-\alpha/2+\hat{\xi}/2})+O_p(n^{-2})
\end{equation}
so that
\begin{equation}
\hat{\xi}-\xi =O_p(n^{-3/2}).
\end{equation}
Finally, consider the coverage probability of the double-bootstrap CI $(\hat{w}_{\alpha/2-\hat{\xi}/2}, \hat{w}_{1-\alpha/2+\hat{\xi}/2}).$ Note that by (<ref>), the Taylor expansion
\begin{equation}
z_{\gamma+\epsilon}=z_{\gamma}+\epsilon \phi(z_{\gamma})^{-1} + O(\epsilon^2),
\end{equation}
and the derivations for (3.36) in <cit.>, we have
\begin{equation}
\begin{split}
& \P(\theta_0 \in (-\infty, \hat{w}_{1-\alpha/2+\hat{\xi}/2})) \\
=& \P(\sqrt{n}(\hat{\theta}-\theta_0)/\hat{\sigma}>-z_{1-\alpha/2+\hat{\xi}/2}-n^{-1/2}\hat{p}_1 (z_{1-\alpha/2+\hat{\xi}/2})-n^{-1}\hat{p}_2(z_{1-\alpha/2+\hat{\xi}/2}) +\ldots))\\
=& \P(\sqrt{n}(\hat{\theta}-\theta_0)/\hat{\sigma}>-z_{1-\alpha/2+\xi/2}-n^{-1/2}\hat{p}_1(z_{1-\alpha/2+\xi/2})-{1 \over 2}(\hat{\xi}-\xi)\phi(z_{1-\alpha/2})^{-1} -\\
&n^{-1} \hat{p}_2(z_{1-\alpha/2+\xi/2})+O_p(n^{-2}))\\
% =& \P(\sqrt{n}(\hat{\theta}-\theta_0)/\hat{\sigma}>-z_{1-\alpha/2+\xi/2}-n^{-1/2}\hat{p}_1(z_{1-\alpha/2+\xi/2})-{1 \over 2}(\hat{\xi}-\xi)\phi(z_{1-\alpha/2})^{-1}) +O(n^{-2})\\
=& \P(\sqrt{n}(\hat{\theta}-\theta_0)/\hat{\sigma}>-z_{1-\alpha/2+\xi/2}-n^{-1/2}\hat{p}_1(z_{1-\alpha/2+\xi/2})-n^{-1}\hat{p}_2(z_{1-\alpha/2+\xi/2})+\ldots+\\
& {1 \over 2}(\hat{\xi}-\xi)\phi(z_{1-\alpha/2})^{-1}) +O(n^{-2})\\
=& \P(\theta_0 < \hat{w}_{1-\alpha/2+\xi/2})+n^{-3/2} b z_{1-\alpha/2}\phi(z_{1-\alpha/2})+O(n^{-2})\\
\end{split}
\end{equation}
where the constant $b$ is defined through
\begin{equation}
\E[\sqrt{n}(\hat{\theta}-\theta_0)/\hat{\sigma} n^{3/2}(\hat{\xi}-\xi)/2]=b+O(n^{-1}).
\end{equation}
The $O(n^{-1})$ term is derived as in equation (3.35) in <cit.> (page 100). Similarly,
\begin{equation}
\P(\theta_0 \in (-\infty, \hat{w}_{\alpha/2-\hat{\xi}/2}))= \P(\theta_0 \in (-\infty,\hat{w}_{\alpha/2-\xi/2}))-n^{-3/2} b z_{\alpha/2}\phi(z_{\alpha/2})+O(n^{-2})
\end{equation}
\begin{equation}
\begin{split}
& \P(\theta_0 \in (\hat{w}_{\alpha/2-\hat{\xi}/2}, \hat{w}_{1-\alpha/2+\hat{\xi}/2})) \\
=& \P(\theta_0 \in (-\infty, \hat{w}_{1-\alpha/2+\hat{\xi}/2})) -\P(\theta_0 \in (-\infty, \hat{w}_{\alpha/2-\hat{\xi}/2}))\\
=&\P(\theta_0 \in (-\infty,\hat{w}_{1-\alpha/2+\xi/2}))+n^{-3/2} b z_{1-\alpha/2}\phi(z_{1-\alpha/2})+O(n^{-2}) \\
&- (\P(\theta_0 \in (-\infty,\hat{w}_{\alpha/2-\xi/2}))-n^{-3/2} b z_{\alpha/2}\phi(z_{\alpha/2})+O(n^{-2}))\\
\end{split}
\end{equation}
which concludes our proof.
|
1511.00078
|
$^a$Department of Physics, Visva-Bharati, Santiniketan 731235,
The nature of single particle classical phase space trajectories in
Rindler space with non-hermitian $PT$-symmetric Hamiltonian
have been studied both in the relativistic as well as in the non-relativistic scenarios. It has been shown that
in the relativistic scenario, both positional coordinates and the corresponding canonical momenta are real in
nature and diverges with time. Whereas the phase space trajectories are a set of hyperbolas in Rindler space. On the
other hand in the non-relativistic approximation the spatial coordinates are complex in nature, whereas the
corresponding canonical momenta of the particle are purely imaginary. In this case the phase space trajectories are
quite simple in nature. But the spatial coordinates are restricted in the negative region only.
§ INTRODUCTION
Exactly like the Lorentz transformations of space time coordinates in
the inertial frame <cit.>, the Rindler coordinate
transformations are for the uniformly accelerated frame of
references <cit.>.
From the references <cit.>, it can very easily be
shown that the Rindler
coordinate transformations are given by:
\begin{eqnarray}
ct&=&\left (\frac{c^2}{\alpha}+x^\prime\right )\sinh\left (\frac{\alpha t^\prime}
{c}\right ) ~~{\rm{and}}~~ \nonumber \\
x&=&\left (\frac{c^2}{\alpha}+x^\prime\right )\cosh\left (\frac{\alpha t^\prime}
{c}\right )
\end{eqnarray}
Hence it is a matter of simple algebra to prove that the inverse
transformations are given by:
\begin{equation}
ct^\prime=\frac{c^2}{2\alpha}\ln\left (\frac{x+ct}{x-ct}\right )
~~{\rm{and}}~~ x^\prime=(x^2-(ct)^2)^{1/2}-\frac{c^2}{\alpha}
\end{equation}
Here $\alpha$ indicates the uniform acceleration of the frame. Hence it can
very easily be shown from eqns.(1) and (2) that the square of the
four-line element changes from
\begin{eqnarray}
ds^2&=&d(ct)^2-dx^2-dy^2-dz^2 ~~{\rm{to}}~~\nonumber \\ ds^2&=&\left
(1+\frac{\alpha x^\prime}{c^2}\right)^2d(ct^\prime)^2-{dx^\prime}^2
\end{eqnarray}
where the former line element is in the Minkowski space.
Hence the metric in the Rindler space can be written as
\begin{equation}
g^{\mu\nu}={\rm{diag}}\left (\left (1+\frac{\alpha x}{c^2}\right
)^2,-1,-1,-1\right )
\end{equation}
whereas in the Minkowski space-time we have the usual form
\begin{equation}
g^{\mu\nu}={\rm{diag}}(+1, -1, -1, -1)
\end{equation}
It is therefore quite obvious that the Rindler space is also flat. The only difference from the Minkowski space is
that the frame of the observer is moving with uniform acceleration.
It has been noticed from the literature survey,
that the principle of equivalence plays an important role in obtaining the
Rindler coordinates in the uniformly accelerated frame of reference.
According to this principle an accelerated frame in absence
of gravity is equivalent to a frame at rest in presence of a
gravity. Therefore in the present scenario, $\alpha$ may be treated
to be the strength of
constant gravitational field for a frame at rest.
Now from the relativistic dynamics of special theory
of relativity <cit.>, the action integral is given by
\begin{equation}
S=-\alpha_0 c \int_a^b ds\equiv \int_a^b Ldt
\end{equation}
where $\alpha_0=-m_0 c$ <cit.> and $m_0$ is the
rest mass of the particle and $c$ is the speed of light in vacuum.
The Lagrangian of the particle may be written as
\begin{equation}
L=-m_0c^2\left [\left ( 1+\frac{\alpha x}{c^2}\right )^2 -\frac{v^2}{c^2}
\right ]^{1/2}
\end{equation}
where $\vec v$ is the three velocity vector. Hence the three
momentum of the particle is given by
\begin{equation}
\vec p=\frac{\partial L}{\partial \vec v}, ~~ {\rm{or}}
\end{equation}
\begin{equation}
\vec p=\frac{m_0\vec v}{\left [ \left (1+\frac{\alpha x}{c^2} \right )^2
-\frac{v^2}{c^2} \right ]^{1/2}}
\end{equation}
Then from the definition, the Hamiltonian of the particle may be written as
\begin{equation}
H=\vec p.\vec v-L ~~ {\rm{or}}
\end{equation}
\begin{equation}
H=m_0c^2 \left (1+\frac{\alpha x}{c^2}\right ) \left (1+
\frac{p^2}{m_0^2c^2}\right )^{1/2}
\end{equation}
Hence it can very easily be shown that in the non-relativistic approximation, the Hamiltonian is given by
H=\left (1+\frac{\alpha x}{c^2}\right ) \left (\frac{p^2}{2m_0}+m_0c^2 \right ) \eqno(11a)
In the classical level, the quantities $H$, $x$ and $p$ are treated
as dynamical variables.
Further, it can very easily be verified that in the quantum mechanical scenario where these quantities are
considered to be
operators, the Hamiltonian $H$ is not
hermitian. However the energy eigen spectrum for the Schrödinger equation has been observed to be real
<cit.>. This is
found to be solely because of the fact that $H$ is $PT$-invariant. Now it is well know
that $PxP^{-1}=-x$, $P p P^{-1}=-p$, whereas, $TpT^{-1}=-p$ and
$P\alpha P^{-1}=-\alpha$
but $T\alpha T^{-1}=\alpha$, therefore it is a matter of simple
algebra to show that $PT~H~(PT)^{-1}=H^{PT}=H$. As has been shown by several
authors <cit.> that if $H$ is $PT$-invariant, then the energy
eigen values will
be real. Here $P$ and $T$ are respectively the parity and the time
reversal operators. Further if the Hamiltonian is $PT$ symmetric,
then $H$ and $PT$ should have common eigen states. In <cit.> we
have noticed that the solution of the Schr$\ddot{\rm{o}}$dinger
equation is obtained in terms of the variable $u=1+\alpha x/c^2$,
which is $PT$-symmetric. Hence any function, e.g.,
Whittaker function $M_{k,\mu}(u)$ or Associated Laguerre function
$L_m^n(u)$, the solution of the Schrödinger equation are $PT$-symmetric. These polynomials are also the eigen
functions of the operator $PT$.
Of course with the replacement of hermiticity
of the Hamiltonian with the $PT$-symmetry, we have not
discarded the important quantum mechanical key features of the system
described by this Hamiltonian and also kept the canonical
quantization rule invariant, i.e., $TiT^{-1}=-i$.
This point was
also discussed in an elaborate manner in reference <cit.> and in some of the references cited there.
In this article we have investigated the time evolution for both the space and the momentum coordinates of the
particle moving in Rindler space. We have considered both the relativistic and the non-relativistic form of the
Rindler Hamiltonian (eqns.(11) and (11a) respectively). Hence we
shall also obtain the classical phase space trajectories
for the particle in the Rindler space. We have noticed that in the relativistic scenario, both the spatial
and the momentum coordinates are real in nature and diverge as $t
\longrightarrow \infty$. For both the variables the
time dependencies are extremely simple. Hence we have obtained classical trajectories $p(x)$ by eliminating
the time dependent part.
However, in the non-relativistic approximation, the spatial coordinates are quite complex in nature, whereas the momentum
coordinates are purely imaginary. Since the mathematical form of the
phase space trajectories are quite complicated,
we have
obtained $p(x)$ numerically in the non-relativistic scenario.
In the first part of this article, we have considered the relativistic picture and obtained the phase space
trajectories, whereas in the second part, the
classical phase space structure is obtained for non-relativistic case.
To the best of our knowledge such studies have not been done before
§ RELATIVISTIC PICTURE
The classical Hamilton's equation of motion for the particle is
given by <cit.>
\begin{equation}
\dot{x}=[H,x]_{p.x} ~~{\rm{and}}~~~ \dot{p}=[H,p]_{p,x}
\end{equation}
where $[H,f]_{p,x}$
is the Poisson bracket and is defined by <cit.>
\begin{equation}
[f,g]_{p,x}=\frac{\partial f}{\partial p}\frac{\partial g}{\partial
x} -\frac{\partial f}{\partial x}\frac{\partial g}{\partial p}
\end{equation}
In this case $f=x$ or $p$.
In eqn.(12) the dots indicate the derivative with respect to time. Now
using the relativistic version of Rindler Hamiltonian from eqn.(11), the explicit form of the
equations of motion are given by
\begin{equation}
\dot{x}=\left (1+\frac{\alpha x}{c^2}\right ) \frac{pc^2} {(p^2c^2+
m_0^2c^4)^{1/2}} ~~{\rm{and}}~~ \dot{p}=-\frac{\alpha}{c} (p^2 c^2+
\end{equation}
The parametric form of expressions for $x$ and $p$ represent the time evolution of
spatial coordinate and the corresponding canonical momentum. The
analytical expressions for time evolution for both the quantities
can be obtained
after integrating these coupled equations and are given by
\begin{equation}
x=\frac{c^2}{\alpha} [C_0 \cosh(\omega t-\phi)-1] ~~{\rm{and}}~~
~\sinh(\omega t-\phi)
\end{equation}
where $C_0$ and $\phi$ are the integration constants, which are real in
nature and $\omega=\alpha /c$
is the
frequency defined for some kind of quanta in <cit.>. Hence
eliminating the time coordinate, we can write
\begin{equation}
\left ( 1+\frac{\alpha x}{c^2}\right )^2\frac{1}{C_0^2}
\end{equation}
This is the mathematical form of the set of
classical trajectories of the particle in
the phase space. Or in other wards,
these set of hyperbolas are the classical trajectories of the particle
in the Rindler space. This is consistent with the hyperbolic motion of the
particle in a uniformly accelerated frame.
These set of hyperbolic equations can also be written as
\begin{equation}
p^2= m_0^2c^2 \left (\frac{2\alpha x}{c^2}\right )
\left ( 1+\frac{x\omega}{2c}\right )
\end{equation}
It is quite obvious from the parametric form of the variation of $x$
and $p$ with time that both the quantities are unbound. This is also
reflected from the nature of phase space trajectories as shown in
fig.(1) for the scaled $x$ and $p$. The scaling factors are
$\alpha/c^2$ for $x$ and $(m_0c)^{-1}$ for $p$. For the sake of
illustration, we have chosen the arbitrary constant $C_0=1$.
In this figure
we have also taken both the scaling factors identically
equal to unity. Then obviously eqn.(16) reduces to
\[
\]
We shall get the
other set of trajectories by choosing different values for the scaling
factors. It is obvious that in this case the centre of the hyperbola
is at $(-1, 0)$.
Therefore with the increase of $\alpha$, the centre $\longrightarrow
(0,0)$. Further the vertices for this particular hyperbolic curve are at
$(0,0)$ and $(-2,0)$. The second one is in scaled form. Therefore for the gravitational field
$\alpha$ large enough, both the vertices coincide at the centre
$(0,0)$. It is also obvious that for very large values of
$\alpha$, these two curves touch each other at $(0,0)$.
We have therefore noticed that the phase space trajectories are unbound and consistent with the motion of the
particle in Rindler space.
§ NON-RELATIVISTIC PICTURE
We next consider the non-relativistic form of Rindler Hamiltonian given by eqn.(11a). Now following
eqn.(12), the equations of motion for the particle in Rindler space
in the non-relativistic approximation are given by
\begin{equation}
\dot{x}=\left ( 1+ \frac{\alpha x}{c^2}\right ) \frac{p}{m_0}
~~{\rm{and}}~~ \dot{p}=
-\frac{\alpha}{c^2}\left ( \frac{p^2}{2m_0} + m_0c^2\right )
\end{equation}
On integrating the second one we have
\begin{equation}
p=i2^{1/2} m_0c \cot\left (\frac{2^{1/2}\omega t+\phi}{2}\right )=ip_I
\end{equation}
The particle momentum is therefore purely imaginary in nature with
its real part $p_R=0$. Here $\phi$ is a real constant
phase. Next evaluating the first integral analytically, we have
\begin{eqnarray}
x&=&\frac{c}{\omega} \left [ -1 +\cos\left \{\ln\left ( \sin^2\left (\frac{2^{1/2}\omega t-\phi}{2}\right )\right )
\right \} \right ]
\nonumber \\ &+& i\frac{c}{\omega}\left [ \sin\left \{\ln \left (\sin^2\left (\frac{2^{1/2}\omega t-\phi}{2} \right
) \right ) \right \} \right ]=x_R+ix_I
\end{eqnarray}
The spatial part is therefore complex in nature, where the real part
\begin{equation}
x_R=\frac{c}{\omega} \left [ -1 +\cos\left \{\ln\left ( \sin^2\left (\frac{2^{1/2}\omega t-\phi}{2}\right )\right )
\right \} \right ]
\end{equation}
and the corresponding imaginary part is given by
\begin{equation}
x_I=\frac{c}{\omega}\left [ \sin\left \{\ln \left (\sin^2\left (\frac{2^{1/2}\omega t-\phi}{2} \right
) \right ) \right \} \right ]
\end{equation}
Here again eliminating the time part, we have the mathematical form
of phase space trajectories for the imaginary parts only
\begin{equation}
p_I=2^{1/2} m_0c \frac{\left [1-\exp\left \{ \sin^{-1} \left (\frac{\omega}{c}x_I\right ) \right \} \right ]^{1/2}}
{\exp\left\{\frac{1}{2}\sin^{-1}\left ( \frac{\omega}{c}x_I\right )\right \} }
\end{equation}
Which gives the phase space trajectories of the particle in the Rindler space in non-relativistic scenario. It
should be noted here that since the real part of the particle momentum is zero, we have considered the imaginary parts
only. Since $p_I$ is real, therefore $\mid \omega x_I/c\mid \leq 1$, i.e.,
can not have all possible values.
In fig.(2) we have plotted the scaled $x_R$, i.e. $(\omega x_R/c)$ with
scaled time $(\omega t/2^{1/2})$ for $\phi=0$. Since the constant
phase $\phi$ is completely arbitrary, for the sake of illustration we
have chosen it to be zero. In
this diagram the scaling factors are also taken to be unity. Now if
we consider variation of the scaling factors, the
qualitative nature of the graphs will not change but there will be
quantitative changes.
In fig.(3) we have plotted the scaled $x_I$, i.e., $(\omega x_I/c)$ with scaled time
$(\omega t/2^{1/2})$ for $\phi=0$. In
this case also same type of changes as has been mentioned for $x_R$ will
be observed.
In fig.(4) we have plotted the scaled $p_I$, which is actually
$(p_I/2^{1/2}m_0c)$ with scaled time $(\omega t/2^{1/2})$ for $\phi=0$. In
this case also the scaling factors are exactly equal to one. Further
the same
kind of variation as mentioned above will be observed for $p_I$ with
the change of scaling parameters.
Finally in fig.(5) the phase space trajectory for scaled $x_I$ and scaled $p_I$ is shown
Since the physically accepted domain for scaled $x_I$ is from $-1$ to $0$, we have shown in figs.(6) and (7) the
plot of scaled $x_I$ and scaled $p_I$ with scaled time.
§ CONCLUSION
Finally in conclusion we would like to mention that to the best of our knowledge this is the first time the phase
space trajectories are obtained in Rindler space using non-hermitian $PT$-symmetric Hamiltonian.
In the relativistic case the trajectories can be represented by a set of hyperbolas. Whereas in the non-relativistic
picture, particle momenta are purely imaginary and the space coordinates are complex in nature. The variation of
real and imaginary parts of space coordinates are quite complicated. Further, the phase space is restricted within
the domain
of negative $x$-values. The imaginary part of particle momentum has
been observed to change with time in a discrete manner in
this region.
If we consider the Rindler Hamiltonian in the form
\begin{equation}
H=\left (1+\frac{\alpha x}{c^2}\right )\frac{p^2}{2m}
\end{equation}
then it is a matter of simple algebra to show that
\begin{equation}
1+\frac{\omega x}{c}= t\exp(2m) ~{\rm{and}}~ p= \frac{2mc}{\omega t}
\end{equation}
Hence redifining $1+\omega x/c$ as new $x$ and $2mc/(\omega p)\exp(2m)$ as new $1/p$, we have
$xp=1$, which gives the phase space trajectories in Rindler space. The trajectories are rectangular hyperbola.
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Butterworth-Heimenann, Oxford, (1975).
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Quantum Field Theory in Curved Space, Cambridge University Press,
Cambridge, (1982).
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Sanchez C.L., Revista Mexican De Fisika 52, 70, (2006).
R32 M. Socolovsky, Annales de la
Foundation Louis de Broglie 39, 1, (2014).
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R5 Domingo J Louis-Martinez, Class. Quantum Grav.,
28, 036004, (2011).
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9, 307, (1992).
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30, 1550182 (2015).
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Phase space trajectories for the relativistic scenario with the scaling parameters equal to unity
Variation of scaled $x_R$ with scaled time
Variation of scaled $x_I$ with scaled time
Variation of scaled $p_I$ with scaled time
Phase space trajectories for the non-relativistic scenario
with the scaling parameters equal to unity
Temporal variation of $x_I$ in physically acceptable domain
Temporal variation of $p_I$ in physically acceptable domain
|
1511.00449
|
address1]D. Ramos-López
address1]Miguel Ángel Sánchez-Granero
address3]Manuel Fernández-Martínez
address1,address2]A. Martínez–Finkelshteincor1
[cor1]Corresponding author.
[address1]Department of Mathematics,
University of Almería, Spain
[address2]Instituto Carlos I de Física Teórica y Computacional,
Granada University, Spain
[address3]University Center of Defense at the Spanish Air Force Academy, MDE-UPCT, Santiago de la Ribera, Murcia, Spain
A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numerically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for
a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface.
Interpolation Numerical condition Zernike polynomials Lebesgue constants
§ INTRODUCTION
Zernike or circular polynomials <cit.> constitute a set of basis functions, very popular in optics and in optical engineering, especially appropriate to express wavefront data due to their connection with classical aberrations. Some of their applications in optics include optical engineering <cit.>, aberrometry of the human eye <cit.>, corneal surface modeling <cit.> and other topics <cit.> in optics and ophthalmology. Due to their optical properties and pervasiveness, Zernike polynomials are included in the ANSI standard to report eye aberrations <cit.>.
In almost every practical application, an optical surface or wavefront is sampled at a finite set of points, followed by a fit of the collected data by a linear combination of the Zernike basis with the purpose to determine the corresponding coefficients (sometimes called modes). This is normally done by means of the standard technique of linear least squares, which reduces to interpolation when the size of the data and the dimension of the basis match.
The process is often ill-conditioned, as its stability strongly depends on an adequate choice of the sampling nodes. Analogous situation arises, for instance, when the Zernike basis is used to fit the slopes of the wavefront in a Shack-Hartmann device <cit.>. Their partial derivatives are used in a least squares fit and if the sampled nodes are not chosen carefully, the resulting Zernike modes might be totally inaccurate.
Different sampling patterns can be found in <cit.>. Some of them are based on random or pseudo-random points drawn according to a probability distribution on the disk. Other schemes include regular or quasi-regular grids, such as squared or hexagonal Cartesian grids, regular polar grids or hexapolar grids, that cover the surface of the disk more or less uniformly. However, these sampling patterns generally produce an ill-conditioned collocation matrix even for moderate Zernike polynomial orders. This issue was addressed in <cit.> for the reconstruction of the wavefront from its slopes, putting forward a spiral arrangement as the best best-performing sampling pattern for this problem. However, the numerical results show that even this pattern is not totally satisfactory for the elevation data.
The main goal of this paper is to discuss in a certain sense optimal patterns for sampling and interpolation on the disk using the basis of the Zernike polynomials for moderate degrees, used in practical applications in optics and ophthalmology. They render well-conditioned collocation matrices and provide numerically stable surface reconstruction, starting from both the elevation and the slope data.
§ METHODS
§.§ Zernike polynomials
Zernike polynomials are usually defined in polar coordinates using the double-index notation (see e.g. <cit.>),
\begin{equation}%\label{eq:1}
Z_n^m(\rho,\theta) =
\begin{cases}
\gamma_n^m R_n^{|m|}(\rho) \cos(m\theta), &\text{if } m\geq 0, \\
\gamma_n^m R_n^{|m|}(\rho) \sin(|m|\theta), &\text{if } m< 0,
\end{cases}
\end{equation}
where $n\geq 0$, $|m|\leq n $, and $n-m$ is even, $\gamma_n^m$ are normalization constants, and the radial part $R_n^m$ is
R_n^{|m|}(\rho)=\sum_{s=0}^{(n-|m|)/2}\frac{(-1)^s (n-s)!}{s!((n+|m|)/2-s)! ((n-|m|)/2-s)!}\rho^{n-2s},
which can be expressed in terms of shifted Jacobi polynomials $P_n^{(0, m-n)}$. Alternatively, a single index notation is used, and the conversion from $Z_n^m$ to $Z_j$ is made by the formula <cit.>
\begin{equation}
\label{formulaJ}
j =\frac{n (n+ 2) + m}{2}\in \N \cup \{0\}.
\end{equation}
Functions $Z_n^m$ are actually polynomials in Cartesian coordinates $(x,y)$. Index $n$ is called the radial order of $Z_n^m$. It is easy to see that the number of distinct Zernike polynomials of radial order $\leq n$ is
\begin{equation}
\label{defN}
\end{equation}
which matches the dimension of algebraic polynomials in two variables of total degree $\leq n$. In fact, Zernike polynomials are a complete polynomial set on the unit disk $\D=\{(x,y)\in \R^2: x^2+y^2\leq 1\}$, orthogonal with respect to the area measure on the disk.
The normalization factor $\gamma_n^m$ for simplicity can be set to $1$; however, with $\gamma_n^m = \sqrt{ (2-\delta_{0,m})(n+1)}$, where $\delta$ is the Kronecker delta, the set of polynomials is orthonormal:
\iint_{\D}Z_n^m(\rho,\theta)Z_r^s(\rho,\theta)\rho \,d\rho d\theta=\delta_{n,r}\delta_{m,s}.
In what follows, notation $Z_n^m$ stands for the orthonormal Zernike polynomials.
The standard Fourier theory can be easily extended to the circular polynomials. In particular, any function $W\in L^2$ defined over $\D$ can be represented as
\begin{equation}
\label{Fourierseries}
W(\rho,\theta) = \sum_{m,n} c_{n}^m\, Z_n^m(\rho,\theta), \quad c_{n}^m = \iint_{\D} W(\rho,\theta) Z_n^m(\rho,\theta) \rho\, d\rho d\theta.
\end{equation}
However, in real applications function $W$ is sampled only at a discrete (and finite) set of nodes, and its Zernike (Fourier) coefficients $ c_{n}^m$ must be recovered using only this available information. A common and statistically meaningful procedure seeks a solution in the (weighted) least squares sense to a linear system whose matrix (known as the collocation matrix) consists of evaluations of the Zernike basis in the given set of points or nodes (see e.g. <cit.>). In the case when the number of nodes matches the dimension of the polynomial subspace used for approximation (called “critical sampling” in <cit.>), the problem boils down to the polynomial interpolation of the function $W$ at a given set of nodes.
It is worth mentioning an alternative approach, called hyperinterpolation <cit.>, where $W$ is approximated
by truncations of the series in (<ref>), but the Fourier coefficients $c_{n}^m$ are computed using quadrature formulas evaluated at the discrete set of nodes. The analysis of this method lies beyond the scope of this paper.
§.§ Goodness of the sampling patterns
It is well known that
in the multivariate case the unisolvence of the interpolation problem for arbitrary nodes is not guaranteed, so not every sampling pattern is acceptable (several configurations of interpolation points on the disk that guarantee unisolvence are well-known and can be found in the literature, see e.g. <cit.>). Moreover, the error in approximating a function by its interpolating polynomial depends on the interpolation nodes: standard upper bounds for the error are based on the so-called Lebesgue constants corresponding to these nodes, which give the norm of the interpolation as a projection operator onto the polynomial subspace (see e.g. <cit.>).
Thus, if we are interested in the set of interpolation points which gives the smallest possible upper bound
on the interpolation error in an arbitrary continuous function, an optimal choice
of interpolation points (at least, in this sense) is given by those which minimize the Lebesgue constant.
The asymptotic theory of these interpolation sets is rather well understood. For instance, as it was established in <cit.>, the order of growth of the Lebesgue constants on the disk for algebraic polynomials of total degree $\leq n$ is $\geq \mathcal O(\sqrt{n})$.
Moreover, the sub-exponential growth of the Lebesgue constants implies, among other facts, the weak-* convergence of the nodes counting measure to the (pluripotential theory) equilibrium measure of the disk, given by
$(2\pi \sqrt{1-x^2-y^2})^{-1}dxdy$ (see <cit.> and <cit.>).
Closely related is the notion of admissible and weakly admissible meshes (see <cit.> and <cit.>), namely sequence of discrete subsets $\mathcal A_n$ of a compact set $K$ such that
\sup_p \, \frac{\max \{| p(x)|: x\in K\}}{\max \{| p(x)|: x\in \mathcal A_n\}}
over the polynomials of degree $\leq n$ is either uniformly bounded (admissible) or grows at most polynomially in $n$ (weakly admissible).
Minimizing the Lebesgue constant amounts to solving a large scale non-linear optimization problem, to which the true solution is not explicitly known, even in the case of univariate interpolation. In fact, no explicit examples of multivariate interpolation sets providing an at most polynomial rate of growth of the Lebesgue constants are currently available. In the case of the square $[0,1]^2$, the most popular nodes are given by the so-called Padua points <cit.>, but their construction seems to be hardly generalizable to other sets. For the disk, no explicit configurations of interpolation points obeying the order of growth of $ \mathcal O(\sqrt{n})$ are known. Good candidates are points minimizing certain energy on the disk, such as Leja (giving a polynomial growth of the Lebesgue constants, as established in <cit.>) or Fekete points <cit.>. The so-called Bos arrays (see <cit.>, as well as <cit.> and <cit.>) provide a polynomial growth of the Lebesgue constants, which usually suffices for practical applications.
The Lebesgue constants for several other unisolvent configurations have been numerically analyzed in <cit.>.
However, these criteria of optimality of nodes do not take into account the numerical aspects of the interpolation. For instance, if we have a polynomial basis fixed, interpolation boils down to finding the coefficients of the expansion of the interpolating polynomial in terms of this basis, which is equivalent to solving a linear system given by the so-called collocation matrix. It is well know that the actual choice of the basis greatly influences the accuracy of the solution to this problem.
This is the situation with the actual practice in ophthalmology and the visual science: the polynomial basis is usually given a priori by the orthonormal Zernike polynomials (see e.g. <cit.> for other bases). Assuming this as the starting hypothesis, we analyze the dependence of the numerical stability of the associated linear system from the selection of the nodes.
In theory, unisolvence of the interpolation problem is equivalent to the regularity of the corresponding collocation matrix.
However, from the practical point of view invertibility of the collocation matrix is not sufficient, since ill-conditioned problems are numerically infeasible. The numerical conditioning of a system of linear equations can be measured by the condition number $\kappa(A)$ of the system matrix $A$ with respect to inversion, see e.g. <cit.>. Roughly speaking, $\kappa(A)\approx 10^s$ means a possible loss of about $s$ digits of accuracy in the solution of the linear system. In particular, when working in the IEEE double precision with matrices with condition numbers of order $10^{16}$, all the significant digits may be lost.
Our goal is to put forward a construction of a Bos array of interpolation nodes on the unit disk for which the corresponding collocation matrix built from Zernike polynomials is particularly well-conditioned, at least for moderate degrees. Motivated essentially by basic applications in visual sciences, we do not tackle the asymptotic problem here.
We will also make a comparison with the recently introduced spiral sampling <cit.> for the reconstruction of the wavefront from its slopes, which allowes for a stable use of the Zernike polynomials up to radial order 15, approximately (see Figure <ref>, right, where the spiral pattern is depicted for radial orders 9 and 12, respectively).
Concentric sampling patterns for Zernike polynomials (left) and spiral sampling pattern, as defined in <cit.> (right), for Zernike polynomials of radial order 9 (top) and 12 (bottom).
§.§ Interpolation nodes with low condition numbers
We minimize the condition number of the collocation matrix for a pattern of nodes on concentric circles (the Bos arrays), for which unisolvence has been established (see <cit.>, as well as <cit.>). Namely, given the maximal radial order $n$ of Zernike polynomials, we choose
1 \geq r_1> r_2 >\dots > r_k\geq 0, \quad k=k(n) = \left\lfloor \frac{n}{2}\right \rfloor+1
radii (where $\lfloor \cdot \rfloor$ is the floor operator); on the $i$-th circle with center at the origin and radius $r_i$ we place
n_i = 2 n+5-4i
equally spaced nodes. Notice that
\sum_{i=1}^k n_i = \frac{(n+1)(n+2)}{2} = N,
namely, the number of Zernike polynomials of radial order $\leq n$. With this rule, the $k$-th (innermost) circle contains $1$ node if $n$ is even, and $3$ nodes otherwise (see Figure <ref>, left). Observe that we have not prescribed the exact position of the nodes (only that they are equally spaced). The latter condition is relevant: there are examples in <cit.> showing that the interpolation problem is not poised for arbitrary points on the circle.
For the sake of precision, let us set the nodes on the circle of radius $r_i$ having the arguments $2\pi t/n_i$, $t=0, \dots, n_i-1$. However, we will see that the actual positions of the nodes on the respective circles have a relative relevance.
If we denote by $P_i$, $i=1, \dots, N$, the nodes constructed according to this rule, and if $Z_j$ are the Zernike polynomials enumerated using formula (<ref>), then the collocation matrix takes the form
\begin{equation}
\label{defA_n}
A_n=\left( Z_{j-1}(P_i)\right)_{i,j=1}^N,
\end{equation}
and we are interested in its condition number $\kappa_2(A_n)$, which can be computed as the ratio of the largest to the smallest singular values of $A_n$ (see <cit.>). Since the singular values are invariant by permutation of rows and columns of $A_n$, the value of $\kappa_2(A_n)$ is independent of the order of the nodes $P_i$ and of the Zernike polynomials $Z_j$.
In the proposed scheme, the relevant parameters at hand are the radii $r_i$. We consider the problem of
\begin{equation}
\label{minimization}
\min \{\kappa_2(A_n):\, 1 > r_1> r_2 >\dots > r_k\geq 0\};
\end{equation}
the optimal values of $r_j$'s were found using simulated annealing followed by the algorithms for non-linear optimization implemented in the Optimization Toolbox of Matlab, achieving at least 4 digits of precision, which is sufficient for interpolation with Zernike polynomials of radial order $\leq 30$. The constraint $r_1<1$ is for a practical reason: in actual measurements the data in the periphery is usually much less reliable.
A similar problem, but minimizing the Lebesgue constants for the nodes $\{P_i\}$, was considered in <cit.>. In <cit.>, the radii were studied in connection with the notion of spherical orthogonality for multivariate polynomials, and values of $r_j$'s, given by the zeros of certain Gegenbauer orthogonal polynomials, were analyzed experimentally.
In <cit.>, along with the Lebesgue constants, the condition numbers $\kappa_\infty(A_n)=\|A_n\|_\infty \|A_n^{-1}\|_\infty$ were used to optimize the radii.
The nodes described below, found as the solution to the problem (<ref>), differ from those obtained in <cit.> and <cit.>.
So far, we have not been able to associate the optimal values of $r_j$'s with any known set. Instead, we will prescribe the quasi-optimal radii $r_j$
using the following formula, obtained by the least square fitting of the optimal radii:
\begin{equation}
\label{recCheb}
r_j=r_j(n) = 1.1565 \, \zeta_{j,n} - 0.76535 \, \zeta_{j,n}^2 + 0.60517 \, \zeta_{j,n}^3, \quad
\end{equation}
where $\zeta_{j,n}$ are zeros of the $(n+1)$-st Chebyshev polynomial of the first kind,
\zeta_{j,n} = \cos\left( \frac{(2j-1) \pi}{2(n+1) }\right), \quad j=1, \dots , k = \left\lfloor \frac{n}{2}\right \rfloor+1.
Given these radii, the interpolation nodes $P_j$ are defined in polar coordinates as
\begin{equation}
\label{nodes}
\left( r_j, 2\pi \frac{s_j-1}{n_j} \right), \quad j=1, \dots, k(n), \quad s_j=1, \dots, n_j.
\end{equation}
We have also studied numerically the behavior of the Lebesgue constants corresponding to our interpolation nodes $\{P_i\}$. Recall <cit.> that the Lebesgue constant $\Lambda_n$ corresponding to the polynomial interpolation with total degree $\leq n$ is the maximum over the disk $\D$ of the function
\begin{equation}
\label{lebesgue1}
\ell(x,y)=\sum_{i=1}^N |\ell_i(x,y)|,
\end{equation}
where $\ell_i(x,y) $ is the $i$-th basic Lagrange polynomial characterized by
\ell_i(P_j) =\delta_{ij}$. In particular,
\begin{equation}
\label{lebesgue2}
\ell_i(x,y)= \frac{1}{\det A_n}\, \det \left( A_n^{(i)}(x,y)\right),
\end{equation}
where $A_n$ was defined in (<ref>), and $A_n^{(i)}(x,y)$ is obtained from $A_n$ by replacing the $i$-th row with $\left( Z_{j-1}(x,y)\right)_{j=1}^N$.
It is worth mentioning that other nodes can be obtained following <cit.>, where greedy optimization algorithms to compute a “good” set of nodes for multivariate polynomial interpolation are used.
§ EXPERIMENTAL RESULTS
We have carried out numerical results in order to assess the performance of several sampling patterns. The random, hexagonal, hexapolar, square and the spiral pattern proposed by Navarro et al. were compared in <cit.>. The results of our analysis corroborate the main conclusion of <cit.>, i.e., that the spiral sampling outperforms the rest of the patterns discussed therein. However, as it could be inferred already from <cit.>, the concentric configurations can give better results, at least in terms of the numerical conditioning of the collocation matrix $A_n$.
In this paper, we compare the spiral sampling pattern from <cit.> with the interpolation scheme, given by (<ref>)–(<ref>), and to which we refer as the optimal concentric sampling, or the OCS, for short.
To make a proper comparison between the spiral sampling and the OCS, a variety of experiments were carried out. In a wide range of Zernike radial orders, from 1 to 30, the condition numbers $\kappa_2$ of the collocation matrix (<ref>) corresponding to both sampling methods were calculated, and a summary of the results is available both in Table <ref> and in Figure <ref>, left. We see that $\kappa_2$ for the spiral pattern have a reasonable behavior for low radial orders $n$, but grow large for higher degrees[One of the several possible explanations can lie in the intrinsic structure of the spiral points: the radial density of the sampling is almost constant when approaching the border of the disk. Thus, neither they correlate well with the increasingly oscillatory behavior of the Zernike polynomials close to the boundary, nor they approximate appropriately the pluripotential theory equilibrium measure of the disk.], being greater than $10^6$ for $n=15$. In comparison, observe that the maximum condition number for the OCS, corresponding to a radial order of 30, is less than 100. In order to stay within this frame with the spiral sampling, we must restrict ourselves to Zernike polynomials of radial order not greater than 6.
$\mathbf{n}$ $\mathbf{[N]}$
$\kappa_2$ for spiral sampling $\kappa_2$ for OCS
10 [66] $5.8 \times 10^{4}$ 3.2
15 [136] $2.4 \times 10^{6}$ 5.7
20 [231] $2.7 \times 10^{8}$ 11.3
22 [276] $7.0 \times 10^{15}$ 15.2
27 [406] $1.9 \times 10^{16}$ 32.8
30 [496] $3.0 \times 10^{17}$ 53.3
Dependence of the condition number $\kappa_2$ on the maximal Zernike radial orders $n$ and on the total number of polynomials $N$, both for the spiral sampling and for the optimal concentric sampling (OCS).
Left: condition number $\kappa_2$ of the collocation matrix (<ref>) as a function of the radial order, for both the spiral pattern and the OCS. Right: range plot of the RMS error of the recovered Zernike modes with respect to the exact coefficients.
We know that the high condition number of the collocation matrix $A_n$ in (<ref>) has a direct impact on the accuracy of the solution of the interpolation problem. For illustration, a synthetic wavefront given by a vector of 496 randomly generated Zernike coefficients between -1 and 1 was sampled according to both schemes. These samples were used as the interpolation data for recovery of the original Zernike coefficients by solving the system with the collocation matrix (<ref>). This procedure was repeated 100 times and the mean and standard deviation of the RMS error distribution for the coefficients were computed, as depicted in Figure <ref>, right.
The RMS of $10^{-5}$ can be considered as a threshold above which the recovered coefficients are unreliable (less than $5$ accurate decimal digits). Observe the correspondence between both graphs in Figure <ref>.
As it was explained, formula (<ref>) renders an approximation to the optimal radii $r_i(n)$, found numerically. The impact of replacing the optimal values by this approximation on the numerical condition of the matrix (<ref>) can be appreciated in Figure <ref>; we can see that the values of $r_i(n)$ are quite robust: the approximation hardly affects the condition number, at least for radial orders $\leq 30$ (we have computed numerically $r_i(n)$ for $n\leq 30$).
The effect of using the approximated values of the radii $r_i(n)$ given by (<ref>), instead of the optimal ones computed numerically, on the condition number of the matrix (<ref>).
The nodes found in <cit.> are prescribed by (<ref>), but the radii are given by the formula
r_j=1-\left( \frac{2(j-1)}{n} \right)^a, \quad j=1, \dots, k(n),
where the exponent $1<a<2$ is found experimentally and can depend on $n$; the value $a=1.46$ is said to give reasonable results for all degrees $n$. Numerical experiments show that these nodes give moderate condition numbers $\kappa_\infty(A_n)$, although the nodes (<ref>)–(<ref>) outperform them, using as the criterion the values of either $\kappa_2(A_n)$ or $\kappa_\infty(A_n)$.
Recall that the OCS interpolation nodes $\{P_i\}$ were built assuming that each ring has one node with argument $0$ (i.e., aligned with the positive $OX$ semi-axis), and then optimizing the radii $r_j$, see (<ref>)–(<ref>). A curious fact, already observed before (see e.g. <cit.>) is that the relative location of the equidistant nodes on each ring has a relatively low impact on the condition number of the collocation matrix.
The singular values of $A_n$, used for the computation of $\kappa_2$, can be computed as the square roots of the eigenvalues of the matrix $B=B_n=A_n^T A_n$, where $A_n^T$ is the matrix transpose of $A_n$. Observe that the elements $b_{ij}$ of $B=(b_{ij})$ are given by
b_{ij} = \sum_{s=1}^N Z_{i-1}(P_s) Z_{j-1}(P_s) =\sum_{m=1}^k \sum_{\|P_s\|=r_m} Z_{i-1}(P_s) Z_{j-1}(P_s),
where $\|P\|$ is the distance of the node $P\in \R^2$ to the origin. Observe that for $\ell=1, \dots, k(n)$,
\sum_{\|P_s\|=r_\ell} Z_{i-1}(P_s) Z_{j-1}(P_s)=\gamma_{i-1}\gamma_{j-1} R_{i-1}(r_\ell) R_{j-1}(r_\ell)\Omega_{ij}(\ell),
\Omega_{ij}(\ell)= \sum_{s=1}^{n_\ell} f_i \left(|m_i | \theta^\ell_s(\alpha_{\ell}) \right) f_j \left(|m_i |\theta^\ell_s(\alpha_{\ell}) \right), \quad \theta^\ell_s(\alpha_{\ell})=\alpha_{\ell} + \frac{2\pi(s-1)}{n_\ell} ,
%f_{i,s} g_{j,s}
where $n_\ell = 2 n+5-4\ell $, functions $f_{i}$ are either $\sin$ or $\cos$, and $0\leq \alpha_{\ell} < 2\pi/n_{\ell}$ is a parameter. The dependence of $m_i$ from $i$ is given by (<ref>). By standard trigonometric formulas,
f_i \left(|m_i | \theta^\ell_s(\alpha_{\ell}) \right) f_j \left(|m_i | \theta^\ell_s(\alpha_{\ell}) \right) = \pm h \left((|m_i| - |m_j|) \theta^\ell_s(\alpha_{\ell}) \right) \pm h \left( (|m_i| +|m_j|) \theta^\ell_s(\alpha_{\ell}) \right) ,
where $h=\frac{1}{2}\cos$ if $f_i=f_j$, and $h=\frac{1}{2}\sin$ otherwise. Consequently,
\Omega_{ij}(\ell)= \pm \sum_{s=1}^{n_\ell} h \left((|m_i| - |m_j|) \theta^\ell_s(\alpha_{\ell}) \right) \pm \sum_{s=1}^{n_\ell} h \left((|m_i| + |m_j|) \theta^\ell_s(\alpha_{\ell}) \right).
In particular, let $\ell=1$; values $||m_i| \pm |m_j||\leq 2n$ are not divisible by $n_1=2n+1$ except for $||m_i| \pm |m_j||=0$, and straightforward calculations show that
\sum_{s=1}^{n_1} h \left((|m_i| \pm |m_j|) \theta^1_s(\alpha_{\ell}) \right) =\begin{cases}
\frac{n_1}{2}, & \text{if } m_i= m_j, \\
0, & \text{otherwise,}
\end{cases}
and in any case, $\Omega_{ij}(1)$ is independent of $\alpha_{\ell}$. In other words, the singular values of $A_n$ (and hence, the condition number $\kappa_2(A_n)$) do not depend on the exact location of the nodes $P_s$, equally spaced on the outermost ring.
The considerations above are supported by some numerical experiments. First, for the maximal radial order of Zernike polynomials equal to $n=25$ (respectively, $n=30$), equally spaced nodes were placed on $k=13$ (respectively, $k=16$) concentric circles with radii given by (<ref>). Then the nodes on one of these circles were rotated preserving the equally spaced structure, and the maximal condition number $\kappa_2$ was computed. This procedure was repeated for each ring; the results are illustrated in Figure <ref>.
Maximal change in the condition number when a single circle of radius $r_j$ is rotated, $1\leq j \leq k(n)$, for radial orders $n=25$ and $n=30$.
Since in practice the precise placement of the interpolation nodes is difficult to guarantee, another criterion of usability of the OCS pattern is its stability to global perturbations of the circles and nodes. In Figure <ref> we illustrate the result of two independent experiments for radial orders $n=20, 25$ and $30$ of Zernike polynomials. The curves, marked with 'o', reflect the sensitivity of $\kappa_2$ to the variation of the optimal radii $r_j$, while the curves with 'x' show the change in $\kappa_2$ when individual nodes are randomly perturbed. It is worth pointing out that a study, similar in spirit, shows that weakly admissible meshes for the disk, unisolvent for polynomial interpolation, remain unisolvent and weakly admissible under small perturbations, see <cit.>.
Change in the condition number when the radii of circles are perturbed ('o') and when individual nodes are perturbed ('x'), for radial orders 20, 25 and 30.
We have also studied the behavior of the Lebesgue constants corresponding to the OCS interpolation nodes $\{P_i\}$, using the formulas (<ref>)–(<ref>). Obviously, straightforward numerical maximization of (<ref>) is a formidable task. A more efficient procedure was described recently in <cit.>: it uses an alternative way to assemble the Lagrange fundamental polynomials (following <cit.>), along with replacing maximization of $\ell(x,y)$ over the whole disk by its maximization on an admissible mesh. We have used both the ideas from <cit.> and the direct evaluation of (<ref>) in a fine grid (definitely, a much slower procedure). The results are illustrated in Figure <ref>.
We see that in this case the dependence of $\Lambda_n$ from the dimension of the interpolation space $N$ is roughly linear, which is still far away from the theoretically optimal $\mathcal O(\sqrt{n})$ from <cit.>, but of the same order than the values obtained in <cit.>. The Lebesgue numbers corresponding to the OCS interpolation nodes are also larger but comparable to those reported in
<cit.> (see Table 3 and Figure 8 therein).
Lebesgue constants (vertical axis) as function of the order $n$ (left) and of the dimension of the interpolation space $N$ (right) for the case of the OCS.
Formula (<ref>) shows that the squares of the optimal radii are (asymptotically) uniformly distributed on $[0,1]$ with respect to the function
\begin{equation}
\label{defG}
G(x)=p\left(\sin \left(\frac{ \pi x}{2} \right)\right), \quad p(x)=\left(1.1565 \, x - 0.76535 \, x^2 + 0.60517 \, x^3\right)^2.
\end{equation}
$G$ is a strictly increasing function on $[0,1]$, with $G(0)=0$, and $G(1)=0.992654<1$.
As it follows from <cit.> (see also <cit.>), the interpolation nodes are asymptotically optimal (in the sense made more precise therein) if
L(G):=\int_0^1 x^2 \log\left( G(x)\right) \, dx + 2 \int_0^1 \int_x^1 x \log\left( G(y)-G(x)\right)\, dy dx = - 2/3.
Numerical integration shows that for $G$ given in (<ref>), $L(G)= -0.681567$, which is slightly smaller.
The two examples in <cit.> are
G_1(x)=\sin^2 \left(\frac{ \pi x}{2} \right)\quad \text{and} \quad G_2(x)=1-(x^2-1)^2,
for which $L(G_1)=-0.680609$ and $L(G_2)=-0.675676$, respectively. Notice that $G_1$ can be written in the form (<ref>) with $p(x)= x^2$.
As it was mentioned earlier, the wavefront sensors recover the actual wavefront from sampling its slopes at a finite number of points. Mathematically it boils down to solving systems of linear equations where the collocation matrix is built from the partial derivatives of the Zernike polynomials evaluated at the given nodes. This means, in particular, that each Zernike polynomial provides two rows of that matrix (corresponding to its partial derivatives), and that the first (constant) polynomial plays no role since its derivatives vanish. These considerations oblige to remove one sampling node for each pattern, in order to make the size of the polynomial basis match the size of the sample. We chose to remove in each case the innermost node, although the results were very similar if any other sampling node was removed instead. The condition numbers of these collocation matrices of partial derivatives have been plotted in Figure <ref> for both sampling patterns.
In contrast to the results obtained for the matrix (<ref>), now the condition numbers corresponding to the spiral and the OCS are similar, and in both cases remain below $10^4$. We can observe also some differences. For low orders (up to 13), the spiral sampling produces a relatively smaller condition number, although the difference is not significant. However, the condition number for the optimal concentric pattern grows slower with the radial order, so that for radial orders above 13 we get better results than with the spiral sampling. This difference can amount to about one order of magnitude for radial orders between 25 and 30.
Condition number of the collocation matrix, built from partial derivatives of the Zernike polynomials, as a function of the radial order, for both sampling patterns, namely, OCS and the spiral sampling.
§ CONCLUSIONS
The sampling nodes for polynomial interpolation, in particular using the Zernike basis, are crucial for an accurate reconstruction of an optical surface or wavefront from the sampled data. The optimal concentric sampling, defined by formulas (<ref>)–(<ref>), provides in many senses a quasi-optimal choice: this set is unisolvent (poised) for interpolation and exhibits a very moderate growth of both the condition numbers $\kappa_2$ and of the Lebesgue constants. This set is also fairly stable: as we have seen, small perturbations in the values of the radii $r_j$ or in the location of the individual nodes $P_j$ have no considerable influence on the condition number $\kappa_2$ of the collocation matrix. Since in general the precise placement of the interpolation nodes is difficult to guarantee, this fact is very relevant for the practical applicability of the interpolation scheme.
We have analyzed also the influence of rotation of the equally spaced nodes along each individual ring, showing that we can choose the nodes on every circle independently from the others without affecting $\kappa_2$ considerably.
We can conclude that the optimal concentric sampling described in this paper not only renders a significant improvement in the accuracy of the recovery of the Zernike coefficients for low orders, but allows also the possibility of using higher radial orders (of total degree 30 and even higher), which is not practical with other sampling patterns.
§ ACKNOWLEDGEMENTS
The first (DRL) and the fourth (AMF) authors were partially supported by MICINN of Spain and by the
European Regional Development Fund (ERDF) under grant
MTM2011-28952-C02-01, by Junta de Andalucía (Excellence Grant P11-FQM-7276 and the research group
FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the
University of Almería.
This work was completed during a visit of AMF to the Department of Mathematics of the Vanderbilt University. He acknowledges the hospitality of the hosting department, as well as a partial support of the Spanish Ministry of Education, Culture and Sports through the travel grant PRX14/00037.
The second author (MASG) acknowledges the partial support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
The third author (MFM) especially acknowledges the valuable support provided by Centro Universitario de la Defensa en la Academia General del Aire de San Javier (Murcia, Spain).
We are also indebted to the referee of this paper for several bibliographical references and helpful comments.
§ REFERENCES
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1511.00256
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Present Address: Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India
We have investigated the physical properties of a pyrochlore hafnate Nd$_2$Hf$_2$O$_7$ using ac magnetic susceptibility $\chi_{\rm ac}(T)$, dc magnetic susceptibility $\chi(T)$, isothermal magnetization $M(H)$ and heat capacity $C_{\rm p}(T)$ measurements, and determined the magnetic ground state by neutron powder diffraction study. An upturn is observed below 6 K in $C_{\rm p}(T)/T$, however both $C_{\rm p}(T)$ and $\chi(T)$ do not show any clear anomaly down to 2 K. The $\chi_{\rm ac}(T)$ shows a well pronounced anomaly indicating an antiferromagnetic transition at $T_{\rm N}= 0.55$ K. The long range antiferromagnetic ordering is confirmed by neutron diffraction. The refinement of neutron diffraction pattern reveals an all-in/all-out antiferromagnetic structure, where for successive tetrahedra, the four Nd$^{3+}$ magnetic moments point alternatively all-into or all-out-of the tetrahedron, with an ordering wavevector k = (0, 0, 0) and an ordered state magnetic moment of $m = 0.62(1)\,\mu_{\rm B}$/Nd at 0.1 K. The ordered moment is strongly reduced reflecting strong quantum fluctuations in ordered state.
75.25.-j, 75.50.Ee, 75.40.Cx, 75.40.Gb
§ INTRODUCTION
The observations of spin-ice behavior and magnetic monopoles in frustrated 227 rare earth pyrochlores, such as in Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$, have created tremendous research interests in these materials <cit.>.
In these pyrochlores the magnetic ions sit at the vertices of the corner-sharing tetrahedra (see Fig. <ref>), and in this topology of the lattice, under the action of the crystal electric field (CEF), the ferromagnetic interaction between the moments becomes frustrated <cit.>. The spin-ice behavior of Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$ is a nice illustration of such CEF dictated frustration of the ferromagnetic dipolar interaction. It gives rise to an almost classical Ising spin system, forcing the magnetic moments at the corners of the tetrahedra to point along the local cubic $\langle 111 \rangle$ directions i.e., along the tetrahedron axes such that the moments can point only either towards the center or away from the center of each tetrahedra. Under these conditions the ferromagnetic exchange energy of an individual tetrahedron is minimized by “two-in/two-out" spin configuration <cit.> referred as the “ice rule" by analogy with the disordered configurations of protons in water ice. With this “two-in/two-out" spin configuration, a Pauling residual entropy is found even in the $T = 0$ limit <cit.>.
A delicate competition and balance between the magnetic exchange, CEF and dipolar interactions lead to a variety of very rich and unconventional low-temperature magnetic and thermodynamic properties in these frustrated 227 pyrochlore materials, for example observation of Dirac strings in spin ice Dy$_2$Ti$_2$O$_7$ <cit.>, spin-liquid behavior in Tb$_2$Ti$_2$O$_7$ <cit.>, and a Higgs transition in quantum spin-ice Yb$_2$Ti$_2$O$_7$ <cit.>. While the ferromagnetic dipolar interaction is frustrated by the Ising anisotropy, exchange interactions are antiferromagnetic and in the case they are stronger than the dipolar interactions, can result in a long range antiferromagnetic ordering as has been observed in several of 227 pyrochlore compounds <cit.>.
The Heisenberg antiferromagnet Gd$_2$Ti$_2$O$_7$ exhibits long range magnetic ordering below $T_{\rm N} \approx 1.1$ K accompanied by another magnetic transition near 0.7 K and additional magnetic field induced transitions <cit.>. The $XY$-antiferromagnet Er$_2$Ti$_2$O$_7$ with the moments constrained to the local $\langle 111 \rangle$ planes orders below $T_{\rm N} \approx 1.2$ K <cit.> where the mechanism driving the ordering is suggested to be order-by-disorder <cit.>. The long range antiferromagnetic orderings of Ir$^{4+}$ ($5d^{5}$, $S=1/2$) moments in iridate pyrochlores $R_2$Ir$_2$O$_7$ ($R$ = Nd–Yb) have recently attracted attention for the associated metal-insulator transition <cit.>. Eu$_2$Ir$_2$O$_7$ with Eu in Eu$^{3+}$ ($J=0$) exhibits an antiferromagnetic ordering accompanied with a metal-insulator transition at 120 K <cit.>. Magnetic structure determination using resonant x-ray diffraction revealed an all-in/all-out antiferromagnetic structure of Ir$^{4+}$ in this compound with a propagation vector k = (0, 0, 0) <cit.>. In Nd$_2$Ir$_2$O$_7$ which exhibits a metal-insulator transition at 33 K, ordering of both Nd$^{3+}$ and Ir$^{4+}$ moments has been suggested by neutron diffraction (ND) and muon spin relaxation measurements with an all-in/all-out magnetic structure <cit.>.
In our effort to search for novel 227 rare earth pyrochlores, we have investigated the physical properties of a pyrochlore hafnate Nd$_2$Hf$_2$O$_7$ having Nd$^{3+}$ ($4f^{3}$, $^4I_{9/2}$) as magnetic ion with $S=3/2$, $L=6$ and $J=9/2$. This compound has recently been studied for its promising dielectric properties (high dielectric constant) <cit.>, however its magnetic properties have not been investigated. Ubic et al. <cit.> suggested presence of small disorder in oxygen sublattice of Nd$_2$Hf$_2$O$_7$, however, latter investigations by Karthik et al. <cit.> revealed a well-ordered pyrochlore structure. Our x-ray and neutron powder diffraction data confirm the ordered cubic $Fd\bar{3}m$ pyrochlore structure.
Here we report the results of ac magnetic susceptibility $\chi_{\rm ac}$, dc magnetic susceptibility $\chi$, isothermal magnetization $M$ and heat capacity $C_{\rm p}$ measurements on Nd$_2$Hf$_2$O$_7$ as a function of temperature $T$ and magnetic field $H$. The $\chi_{\rm ac}(T)$ data which is measured down to 0.2 K show evidence for an antiferromagnetic transition at $T_{\rm N}= 0.55$ K. The antiferromagnetic long range ordering is further confirmed by neutron powder diffraction which reveals an all-in/all-out magnetic structure with an ordered moment of $0.62(1)\,\mu_{\rm B}$/Nd at 0.1 K and propagation wavevector k = (0, 0, 0). Very recently Nd$_2$Zr$_2$O$_7$ has also been identified to order antiferromagnetically below 0.3 K with an all-in/all-out magnetic structure <cit.>. Investigations on the single crystal Nd$_2$Zr$_2$O$_7$ have revealed local $\langle 111 \rangle$ Ising anisotropy <cit.> for the ground state of Nd$^{3+}$. Our $M(H)$ and $\chi(T)$ data consistently comply with the Ising nature of ground state of Nd$^{3+}$ in Nd$_2$Hf$_2$O$_7$ with an effective spin $S = 1/2$ and effective $g$-factor $g_{zz} = 5.01(3)$.
An interesting aspect of the rare earth pyrochlores with Kramers doublet (like Nd$^{3+}$) having well separated ground state and first excited state is that the ground state properties can be described by an effective pseudo-spin $S = 1/2$. Of particular interests are the Kramers doublet systems with total angular momenta $J = 9/2$ (Nd$^{3+}$) and 15/2 (Dy$^{3+}$ and Er$^{3+}$) for which Huang et al. <cit.> showed that under specific conditions they may behave like `dipolar-octupolar' doublets. For a dipolar-octupolar doublet the $x$ and $z$ components of pseudo-spin operator transform like the $z$ component of a magnetic dipole whereas the $y$ component transforms like a component of a magnetic octupolar tensor <cit.>. The fact that the pseudo-spin ground state of Nd$^{3+}$ with $J=9/2$ can behave like dipolar-octupolar doublet makes Nd$_2$Hf$_2$O$_7$ an interesting system for further investigations.
(Color online) (a) An illustration of corner-sharing tetrahedra formed by Nd atoms for the face centered cubic (space group $Fd\bar{3}m$) pyrochlore structure of Nd$_2$Hf$_2$O$_7$. (b) Projection of atomic arrangements onto the $ac$ plane.
§ EXPERIMENTAL DETAILS
A polycrystalline sample of Nd$_2$Hf$_2$O$_7$ was synthesized by solid state reaction method using Nd$_2$O$_3$ (99.99%) and HfO$_2$ (99.95%) from Alfa Aesar. A stoichiomteric mixture of finely ground Nd$_2$O$_3$ and HfO$_2$ was first fired at 1300 $^{\circ}$C for 50 h, with two more succesive grindings and firings at 1400 $^{\circ}$C and 1500 $^{\circ}$C, each for 50 h. The finely ground mixture was then pressed into a pellet and fired at 1550 $^{\circ}$C for 80 h. We also synthesized the nonmagnetic reference compound La$_2$Hf$_2$O$_7$ using La$_2$O$_3$ (99.999%, Alfa Aesar) and HfO$_2$ with similar grinding and firing sequence however the last firing after pelletizing was done at 1500 $^{\circ}$C. An agate mortar and pestle was used for grinding with a grinding of approximately half an hour to achieve homogeneity, and a zirconium oxide crucible was used for firing. The grinding and heat treatment of the samples were done in air.
The crystal structure and the quality of the samples were checked by room temperature powder x-ray diffraction (XRD, Brucker). A commercial superconducting quantum interference device (SQUID) magnetometer (MPMS, Quantum Design Inc.) was used for dc magnetic measurements at Mag Lab, Helmholtz-Zentrum Berlin (HZB). The ac susceptibility was collected at temperatures down to 200 mK using an adiabatic demagnetization cooler where the sample is located in the center of mutual inductance. The ac susceptibility was measured using a LR700 mutual inductance bridge. As a magnetocaloric active substance for the sub-K temperature range, an Fe$^{3+}$ salt was used. The demagnetization stage was precooled in a physical property measurement system (PPMS, Quantum Design Inc.) at Helmholtz-Zentrum Dresden-Rossendorf. Heat capacity was also measured using PPMS by means of the adiabatic-relaxation technique down to 2 K at Mag Lab, HZB.
The neutron diffraction measurements were carried out using the D20 powder neutron diffractometer at the Institute Laue Langevin, Grenoble, France. A thin-walled copper can (diameter 10 mm) was used to mount the powdered sample. Low temperatures down to 0.1 K were achieved by cooling the sample in a dilution fridge. High intensity ND data were collected at few selected temperatures between 0.1 K and 0.6 K. Incident neutron beam of wavelength $\lambda = 2.41$ Å was used for these measurements and counted for 4 hours at each of the temperatures. The XRD and ND data were refined using the package FullProf suite <cit.>.
§ CRYSTALLOGRAPHY
(Color online) Powder x-ray diffraction pattern of Nd$_2$Hf$_2$O$_7$ recorded at room temperature. The solid line through the experimental points is the Rietveld refinement profile calculated for the ${\rm Eu_2Zr_2O_7}$-type face centered cubic (space group $Fd\bar{3}m$) pyrochlore structure. The short vertical bars mark the Bragg-peak positions. The lowermost curve represents the difference between the experimental and calculated intensities.
Figure <ref> shows the Rietveld refinement of powder XRD pattern of Nd$_2$Hf$_2$O$_7$ collected at room temperature. The refinement confirms the ${\rm Eu_2Zr_2O_7}$-type face-centered cubic (space group $Fd\bar{3}m$) pyrochlore structure of Nd$_2$Hf$_2$O$_7$. All the observed peaks are indexed, thus revealing the single phase nature of the sample. The crystallographic parameters obtained from the structural refinement of the room temperature XRD are listed in Table <ref> along with those obtained from the refinement of the neutron diffraction data recorded at 0.6 K. The parameters $a$ and $x_{\rm O1}$ agree well with the literature values <cit.>. Because of the limitations of laboratory based x-ray measurements it is not possible to obtain any reliable information about the oxygen vacancy or Nd-Hf site mixing. Our neutron data which were collected at low Q are also of not much help in resolving these issues because of the small number of nuclear Bragg peaks detected and almost equal scattering lengths for Nd ($0.7690 \times 10^{-12}$ cm) and Hf ($0.7770\times 10^{-12}$ cm).
Refined crystallographic parameters and agreement factors obtained from the structural Rietveld refinement of room temperature (RT) powder XRD data and 0.6 K neutron powder diffraction data for Nd$_2$Hf$_2$O$_7$. The Wyckoff positions of Nd, Hf, O1 and O2 atoms in space group $Fd\bar{3}m$ are 16d (1/2,1/2,1/2), 16c (0,0,0), 48f ($x_{\rm O1}$,1/8,1/8) and 8b (3/8,3/8,3/8), respectively. The atomic coordinate $x_{\rm O1}$ is listed below.
XRD (RT) ND (0.6 K)
Lattice parameters
$a$ (Å) 10.6389(1) 10.573(1)
$V_{\rm cell}$ (Å$^{3}$) 1204.19(1) 1182.0(2)
Atomic coordinate
$x_{\rm O1}$ 0.3317(9) 0.3340(5)
Refinement quality
$\chi^2$ 2.86 1.98
$R_{\rm p}$ (%) 3.69 5.97
$R_{\rm wp}$ (%) 5.27 10.70
$R_{\rm Bragg}$ (%) 5.19 6.48
The pyrochlore structure of Nd$_2$Hf$_2$O$_7$ is shown in Fig. <ref>. In this structure the Nd atoms form corner-shared tetrahedra, as shown in Fig. <ref>(a), the center of each tetrahedron is occupied by an O atom. The atomic arrangements viewed along the crystallographic $b$ direction (projected in $ac$ plane) are shown in Fig. <ref>(b). The Nd$^{3+}$ occupy 16d (1/2,1/2,1/2) sites and the Hf$^{4+}$ occupy 16c (0,0,0) positions, whereas the O$^{2-}$ occupy two sites: O1 in 48f ($x_{\rm O1}$,1/8,1/8) and O2 in 8b (3/8,3/8,3/8) positions, and the formula unit can be viewed as ${\rm Nd_2Hf_2O(1)_6O(2)}$. The atoms sitting at both the 16d and 16c sites form (separately) three-dimensional networks of corner-shared tetrahedra leading to two distinct pyrochlore sublattices [only the tetrahedra formed by 16d site atoms (Nd here) are shown in Fig. <ref>(a)]. The Nd atoms are eight-fold coordinated (by 6 O1 and 2 O2) and the Hf atoms are six-fold cordinated (by 6 O1) <cit.>.
The La$_2$Hf$_2$O$_7$ also forms in the same face-centered cubic pyrochlore structure with parameters $a = 10.7731(1)$ Å and $x_{\rm O1} = 0.3308(9)$ in agreement with the reported values <cit.>. The single phase nature of sample was inferred from the refinement of XRD data (not shown).
§ MAGNETIZATION AND MAGNETIC SUSCEPTIBILITY
(Color online) Zero-field-cooled magnetic susceptibility $\chi$ of Nd$_2$Hf$_2$O$_7$ as a function of temperature $T$ for $2~{\rm K} \leq T \leq 25$ K measured in magnetic fields $H= 0.01$ T and 1.0 T. Inset: Inverse magnetic susceptibility $\chi^{-1}(T)$ for $2~{\rm K} \le T \leq 30$ K in $H = 1.0$ T. The solid red line is the fit of the $\chi^{-1}(T)$ data by the Curie-Weiss law in $10~{\rm K} \leq T \leq 30$ K and the dashed line is an extrapolation. All data pertain to per mole of Nd$_2$Hf$_2$O$_7$.
Zero-field-cooled dc $\chi(T)$ data of Nd$_2$Hf$_2$O$_7$ measured in $H = 0.01$ T and 1.0 T are shown in Fig. <ref>. The $\chi(T)$ data do not show any anomaly and remain paramagnetic at $T \geq 2$ K. The high temperature $\chi(T)$ data follow Curie-Weiss behavior, $\chi(T) = C/(T-\theta_{\rm p})$. The linear fit of $\chi^{-1}(T)$ for the range 100 K $\leq T\leq$ 300 K gives the Curie constant $C = 1.32(1)$ emu K/mol Nd and Weiss temperature $\theta_{\rm p} = -29.9(7)$ K. The $C$ value gives an effective moment $\mu_{\rm eff} = 3.25\, \mu_{\rm B}$/Nd according to the relation $C= N_{\rm A} \mu_{\rm eff}^2/3 k_{\rm B}$ where $N_{\rm A}$ is the Avogadro number and $k_{\rm B}$ is the Boltzmann constant. The obtained $\mu_{\rm eff}$ is little smaller than the theoretically expected value of effective moment for $^4$I$_{9/2}$ ground state of Nd$^{3+}$ ions ($\mu_{\rm eff} = g_J\sqrt{J(J+1)} = 3.62 \, \mu_{\rm B}$ for $g_J = 8/11$ and $J = 9/2$).
As estimated in the next section, the first excited crystal field level is situated at about 230 K, therefore because of the thermal population from CEF at $T \geq 100$ K, the above analysis of $\chi(T)$ data does not give the correct estimate of $\theta_{\rm p}$ or $\mu_{\rm eff}$ for the Ising ground state. Therefore we fit the $\chi(T)$ data at $T$ below 30 K using the modified Curie-Weiss behavior $\chi(T) = \chi_0 + C/(T-\theta_{\rm p})$ where temperature independent term $\chi_0$ is added to account for the Van Vleck contribution. The fit of $\chi(T)$ by this modified Curie-Weiss law is shown by solid line in the inset of Fig. <ref> plotted as the inverse of susceptibility $\chi^{-1}(T)$. In order to avoid the effect of short-range magnetic interactions at $T < 10$ K as evident from the heat capacity data discussed in next section, we fit the $\chi^{-1}(T)$ data in 10 K $\leq T\leq$ 30 K which yields $\chi_0 = 3.04(4) \times 10^{-3}$ emu/mol Nd, $C = 0.752(2)$ emu K/mol Nd and $\theta_{\rm p} = +0.17(2)$ K. When the $\chi(T)$ data are corrected for demagnetization effects by roughly approximating the sample to be spherical in shape we obtain $\chi_0 = 3.04(4) \times 10^{-3}$ emu/mol Nd, $C = 0.753(2)$ emu K/mol Nd and $\theta_{\rm p} = +0.24(2)$ K. Thus we see that $\theta_{\rm p}$ is positive which would imply a weak ferromagnetic coupling among Nd spins. A positive $\theta_{\rm p}$ was also found in the case of Nd$_2$Zr$_2$O$_7$ <cit.>. The $C$ value gives $\mu_{\rm eff} \approx 2.45\, \mu_{\rm B}$/Nd for the Ising ground state of Nd$_2$Hf$_2$O$_7$.
(Color online) Isothermal magnetization $M$ (per Nd ion) of Nd$_2$Hf$_2$O$_7$ as a function of applied magnetic field $H$ for $0 \leq H \leq 5$ T measured at the indicated temperatures. The solid curves are the fits of $M(H)$ data by Eq. (<ref>) with an effective longitudinal $g$-factor $g_{zz}= 5.01(3)$.
The isothermal $M(H)$ data of Nd$_2$Hf$_2$O$_7$ at $T= 2$ K, 5 K and 10 K are shown in Fig. <ref>. At 2 K initially $M$ increases rapidly and is linear in $H$ for $H\leq 0.5$ T above which $M(H)$ shows nonlinear behavior and tends towards saturation with a magnetization value of $M \approx 1.2\, \mu_{\rm B}$/Nd at 5 T which is much lower than the theoretical saturation magnetization $M_{\rm sat} = g_J J\,\mu_{\rm B} = 3.27\,\mu_{\rm B}$/Nd for free Nd$^{3+}$ ions ($g_J = 8/11$ and $J = 9/2$). The observed $M$ is only about 37% of the free ion theoretical saturation value and reflects substantial single-ion anisotropy as would one expect for a local $\langle 111 \rangle$ Ising anisotropic system. With increasing $T$ the linear regime of $M(H)$ extends over a large field range, although at a more gradual rate.
As the first excited state is well separated ($\sim230$ K) from the ground state, the low temperature magnetic properties of Kramers ground doublet of Nd$^{3+}$ can be described by an effective spin $S = 1/2$. For a Kramers doublet of dipolar-octupolar type (Nd$^{3+}$) the transverse $g$-factor is found to be zero, i.e. $g_{\bot} = 0$. For an effective spin-half doublet ground state system with local $\langle 111 \rangle$ Ising anisotropy assuming $g_{\bot} = 0$ and $g_{||}=g_{zz}$ the powder- and thermally-averaged magnetization is given by <cit.>
\begin{equation}
\langle M \rangle = \frac{(k_{\rm B} T)^2}{g_{zz}\mu_{\rm B} H^2 S} \int_0^{g_{zz}\mu_{\rm B} H S/k_{\rm B} T} x \tanh(x) \,dx
\label{MH-Ising}
\end{equation}
where $x = g_{zz}\mu_{\rm B} H S/k_{\rm B} T$. For a pure $m_J = \pm 9/2$ doublet $g_{zz} = 2 g_J J = 6.54$. However, due to the mixing of the $m_J$ states by the crystal field the effective $g$-factor is different from 6.54 and can be determined by fitting the $M(H)$ data which is the only adjustible parameter in Eq. (<ref>). The fits of $M(H)$ data at $T= 2$ K, 5 K and 10 K (solid curves in Fig. <ref>) yield $g_{zz}= 5.01(3)$. It is seen from Fig. <ref> that the $M(H)$ data are reasonably well described by Eq. (<ref>). The $g_{zz} = 5.01(3)$ obtained this way is lower than that of a pure $m_J = \pm 9/2$ doublet. This reduction of $g_{zz}$ possibly suggests an admixture of other $m_J$ terms in the ground state <cit.>. For Nd$_2$Zr$_2$O$_7$ $g_{zz}$ is found to be 4.793 <cit.> and 5.30(6) <cit.>. The effective $g_{zz} = 5.01(3)$ with an effective $S = 1/2$ suggests an Ising moment of $g_{zz} S \, \mu_{\rm B} = 2.50 \, \mu_{\rm B}$/Nd. For a powder sample the effective moment is related to $g$-factor as $\mu_{\rm eff} = (\sqrt 3/2)\overline{g}\,\mu_{\rm B} $, where $\overline{g}^2 = (g_{||}^2+2 g_{\bot}^2)/3$, which for $g_{\bot} = 0$ and $g_{||}= 5.01(3)$ gives $\mu_{\rm eff} = 2.50 \, \mu_{\rm B}$/Nd in agreement with the above inferred value of $ 2.45\, \mu_{\rm B}$/Nd from the fit of $\chi(T)$ data. Thus the $M(H)$ and $\chi(T)$ data consistently follow the Ising behavior.
(Color online) The temperature $T$ dependence of real $\chi'$ and imaginary $\chi''$ parts of ac magnetic susceptibility $\chi_{\rm ac}$ of Nd$_2$Hf$_2$O$_7$ measured in different dc magnetic fields at 16 Hz. Inset: An expanded view of $\chi''(T)$ in $H=0$. All data pertain to per mole of Nd$_2$Hf$_2$O$_7$. Because of the uncertainty in calculating the sample filling factor in the ac coil-set, the conversion to units of `emu/mol' for $y$-scale has accuracy of about 10%.
The real $\chi'$ and imaginary $\chi''$ parts of ac magnetic susceptibility $\chi_{\rm ac}$ of Nd$_2$Hf$_2$O$_7$ measured in $H\leq 0.30$ T and 16 Hz are shown in Fig. <ref> for low temperatures $T\leq 1.5$ K. In $H = 0$ the $\chi'(T)$ data show a pronounced peak at 0.55 K indicating a magnetic phase transition. Further, with increasing field the peak position shifts towards lower temperatures. This behavior is a characteristic of an antiferromagnetic phase transition. A weak anomaly with a similar $H$ dependence is also observed in imaginary part of ac susceptibility $\chi''(T)$. The $\chi''(T)$ is much smaller in magnitude than the $\chi'(T)$. Due to weak signal at 16 Hz the signal-to-noise ratio for $\chi''(T)$ is very poor and data appear quite noisy. As can be seen from the inset of Fig. <ref>, despite the noisy data an anomaly near 0.55 K is also visible in the $\chi''(T)$ data in $H=0$. Thus the $\chi_{\rm ac}(T)$ data indicate a long range antiferrromagnetic ordering of Nd$^{3+}$ at the Néel temperature $T_{\rm N} = 0.55$ K as is confirmed by the neutron diffraction study discussed below in Sec. <ref>. Further it is seen that the application of field also causes a decrease in $\chi'$ at $T > T_{\rm N}$ likely due to the effect of field on short range magnetic correlations above $T_{\rm N} $.
§ HEAT CAPACITY
(Color online) (a) Heat capacity $C_{\rm p}$ of Nd$_2$Hf$_2$O$_7$ and nonmagnetic reference La$_2$Hf$_2$O$_7$ as a function of temperature $T$ for 1.8 K $\leq T \leq$ 300 K measured in zero field. The solid curves are the fits by Debye+Einstein models of lattice heat capacity (plus crystal field contribution for the case of Nd$_2$Hf$_2$O$_7$). Inset: Expanded view of $C_{\rm p}(T)$ data over $1.8~{\rm K} \leq T \leq 12$ K for Nd$_2$Hf$_2$O$_7$. (b) $C_{\rm p}/T$ versus $T^2$ plot for Nd$_2$Hf$_2$O$_7$ for $T \leq 17$ K. The solid line is the fit to $C_{\rm p}/T = \beta T^2 +\delta T^4 $ in $10~{\rm K} \leq T \leq 16$ K and the dashed lines are extrapolations. Inset: Magnetic contribution to heat capacity $C_{\rm mag}(T)$ for Nd$_2$Hf$_2$O$_7$. The solid curve represents the crystal field contribution to heat capacity as discussed in text.
The $C_{\rm p}(T)$ data of Nd$_2$Hf$_2$O$_7$ and nonmagnetic reference compound La$_2$Hf$_2$O$_7$ are shown in Fig. <ref>(a) for 1.8 K $\leq T \leq$ 300 K. Consistent with the $\chi(T)$ data, the $C_{\rm p}(T)$ data of Nd$_2$Hf$_2$O$_7$ do not show any anomaly at $T\geq1.8$ K. However, the low-$T$ $C_{\rm p}(T)$ at $T \leq 6$ K reveals an upturn as shown in the inset of Fig. <ref>(a). This reflects the onset of short range magnetic correlation well above the antiferromagnetic transtion at $T_{\rm N} = 0.55$ K. The low-$T$ $C_{\rm p}(T)$ data (above 6 K) are well described by $C_{\rm p}(T) = \gamma T + \beta T^{3} + \delta T^{5}$, with the coefficient of electronic heat capacity $\gamma = 0$ which reflects an insulating ground state in Nd$_2$Hf$_2$O$_7$. A fit to the $C_{\rm p}/T$ versus $T^2$ plot by $C_{\rm p}/T = \beta T^2 + \delta T^{4}$ over $9.5~{\rm K} \leq T \leq 16$ K as shown by the solid magneta line in Fig. <ref>(b) gives $\beta= 2.58(5) \times 10^{-4}$ J/mole K$^{4}$ and $\delta = 1.38(3) \times 10^{-6}$ J/mole K$^{6}$. The Debye temperature $\Theta_{\rm D} = 436(4)$ K is estimated from $\beta$ using the relation $\Theta_{\rm D} = (12 \pi^{4} n R /{5 \beta} )^{1/3}$ where $n=11$ is the number of atoms per formula unit and $R$ is the molar gas constant.
The $C_{\rm p} \approx 230$ J/mol K at 300 K [Fig. <ref>(a)] is much lower than the expected high-$T$ limit Dulong-Petit value $C_{\rm V} = 3nR = 33R \approx 274.4$ J/mol K which is consistent with the high $\Theta_{\rm D}$ value. The $\Theta_{\rm D}$ in 227 pyrochlore is found to be highly temperature dependent, for Dy$_2$Ti$_2$O$_7$ the low-$T$ $C_{\rm p}(T)$ yields a $\Theta_{\rm D}$ of 295 K whereas the high-$T$ $C_{\rm p}(T)$ gives much higher $\Theta_{\rm D} = 722(8)$ K <cit.>. A better estimate of $\Theta_{\rm D}$ can be obtained from fitting the $C_{\rm p}(T)$ data by a combination of the Debye and Einstein models of lattice heat capacity. A fit of $C_{\rm p}(T)$ data of La$_2$Hf$_2$O$_7$ by Debye+Einstein models in 1.8 K $\leq T \leq$ 300 K gives $\Theta_{\rm D} = 792(5)$ K and Einstein temperature $\Theta_{\rm E} = 163(2)$ K. The fit is shown by the solid olive curve in Fig. <ref>(a) which is obtained with 66% weight to Debye term and 34% to Einstein term. Further details about fitting heat capacity data by Debye+Einstein models can be found in Ref. <cit.>.
On the other hand for Nd$_2$Hf$_2$O$_7$ we have an additional magnetic contribution due to crystal electric field. The magnetic contribution to heat capacity $C_{\rm mag}(T)$ for Nd$_2$Hf$_2$O$_7$ is shown in the inset of Fig. <ref>(b) which was obtained by subtracting the lattice heat capacity (equivalent to $C_{\rm p}(T)$ of La$_2$Hf$_2$O$_7$) from the heat capacity of Nd$_2$Hf$_2$O$_7$. A correction for the small formula mass difference of the two compounds was employed <cit.>. As can be seen from the inset of Fig. <ref>(b), $C_{\rm mag}(T)$ is noisy at high-$T$ (due to the limitations in the sensitivity of our experimental setup), nevertheless the basic feature of the data is quite evident. A broad Schottky type anomaly (due to crystal electric field) centered around 120 K is seen in $C_{\rm mag}(T)$. We analyzed $C_{\rm mag}(T)$ to extract the CEF levels and found that $C_{\rm mag}(T)$ is well represented by a doublet ground state lying below a doublet excited state at 229(6) K and a quasi-quartet (two closely situated doublets) at 460(9) K. The crystal field contribution to heat capacity calculated according to this CEF level scheme is shown by the solid red curve in the inset of Fig. <ref>(b). A nice agreement is seen between $C_{\rm mag}(T)$ and the CEF model fit. However, because of large noise in high-$T$ data the precise determination of the splitting energy for higher excited states is not possible. For Nd$_2$Zr$_2$O$_7$ a splitting energy of 250–270 K between the ground state doublet and first excited doublet has been found <cit.>. Similar to the present compound, two closely situated doublets at 388 K and 400 K have also been found from inelastic neutron scattering study on Nd$_2$Zr$_2$O$_7$ <cit.>.
The fit of $C_{\rm p}(T)$ data of Nd$_2$Hf$_2$O$_7$ by CEF+Debye+Einstein models in 1.8 K $\leq T \leq$ 300 K shown by the solid blue curve in Fig. <ref>(a) yields $\Theta_{\rm D} = 785(6)$ K and Einstein temperature $\Theta_{\rm E} = 162(2)$ K with 66% weight to Debye term and 34% to Einstein term. The value of $\Theta_{\rm D} = 785(6)$ K obtained this way is much higher than the $\Theta_{\rm D} = 436(4)$ K estimated above from $\beta$.
§ NEUTRON DIFFRACTION
(Color online) (a) Neutron diffraction (ND) pattern of Nd$_2$Hf$_2$O$_7$ recorded at 0.6 K. The solid line through the experimental points is the calculated pattern by considering the ${\rm Eu_2Zr_2O_7}$-type face centered cubic (space group $Fd\bar{3}m$) pyrochlore nuclear structure. The short vertical bars mark the Bragg peak positions of primary phase [upper row (olive)] and sample holder Cu [lower row (orange)]. The lowermost curve represents the difference between the experimental and calculated patterns. The two most intense peaks marked with stars ($\star$) belong to the sample holder. Inset: Expanded view of ND pattern between 35$^\circ$–45$^\circ$ and comparison of ND patterns at 0.6 K and 0.1 K to highlight the presence of magnetic scattering (marked with #). (b) Magnetic diffraction pattern at 0.1 K (after subtracting 0.6 K nuclear pattern) together with the calculated magnetic refinement pattern. The region where the sample holder contribution dominates is excluded. The difference between the experimental and calculated intensities is shown by the blue curve at the bottom. The Miller indices ($hk\ell$) of the strongest magnetic Bragg peaks are indicated.
(Color online) A two dimensional view of all-in/all-out magnetic structure of Nd$_2$Hf$_2$O$_7$ along with the three dimensional view of corner-shared `all-in' and `all-out' tetrahedra. The arrows denote the odered Nd$^{3+}$ moment directions, pointing towards or away from the center of tetrahedra. The two dimensional representation of crystal structure can be seen in Fig. <ref>(b) and the three dimensional view of corner-shared tetrahedra in Fig. <ref>(a).
The neutron diffraction data collected at 0.6 K are shown in Fig. <ref>(a) together with the calculated pattern for the nuclear structure of Nd$_2$Hf$_2$O$_7$. The crystallographic parameters are listed in Table <ref> and are consistent with the parameters from room temperature XRD. This also confirms that there is no structural transition down to 0.6 K. At 0.1 K, weak and noticeable additional intensities in the diffraction pattern confim the long range magnetic ordering. A comparison of the ND data collected at 0.6 K ($> T_{\rm N}$) and 0.1 K ($< T_{\rm N}$) is shown in the inset of Fig. <ref>(a) for $2\theta$ range 35$^\circ$–45$^\circ$. An enhancement of the two nuclear peaks (2 2 0) and (3 1 1) can be seen (marked with symbol #). This additional intensity corresponds to the most prominent magnetic peaks in the ordered state. From the difference between the ND patterns recorded at 0.1 K and 0.6 K [Fig. <ref>(b)] a multitude of magnetic Bragg peaks can be clearly visible. The appearance of an additional magnetic Bragg peak (4 2 0) at $59.6^\circ$ confirms the antiferromagnetic ordering. All magnetic Bragg peaks are well indexed by the propagation wavevector k = (0, 0, 0). The propagation wavevector k = (0, 0, 0) was also found to index the magnetic Bragg peaks in Nd$_2$Zr$_2$O$_7$ <cit.>.
In order to determine the magnetic structure compatible with the space group symmetry we carried out representational analysis using the program BASIREPS from the FullProf package <cit.>. The symmetry analysis for the propagation vector k = (0, 0, 0) and space group $Fd\bar{3}m$ yielded four nonzero irreducible representations (IRs) for the magnetic Nd(16d) site: 1 one-dimensional ($\Gamma_3^1$), 1 two-dimensional ($\Gamma_6^2$) and two three-dimensional ($\Gamma_8^3$, $\Gamma_{10}^3$) for the little group. The magnetic representation $\Gamma_{\rm mag\,Nd}$ is thus composed of four IRs as
\begin{equation}
\Gamma_{\rm mag\,Nd} = 1\,\Gamma_3^1 + 1\,\Gamma_6^2 + 1\,\Gamma_8^3 + 2\, \Gamma_{10}^3.
\label{eq:IRs}
\end{equation}
The IRs $\Gamma_3^1$, $\Gamma_6^2$ and $\Gamma_8^3$ enters only once in magnetic decomposition whereas $\Gamma_{10}^3$ is repeated twice. The basis vectors (BVs) of these IRs are listed in Table <ref>. While the BVs vector of $\Gamma_6^2$ consist of both real and imaginary components, the BVs for $\Gamma_3^1$, $\Gamma_8^3$ and $\Gamma_{10}^3$ have only real components. As listed in Table <ref>, $\Gamma_3^1$ has one BV, $\Gamma_6^2$ has two, $\Gamma_8^3$ has three and $\Gamma_{10}^3$ has six BVs.
Nonzero irreducible representations (IRs) and associated basis vectors $\psi_\nu$ for Nd(16d) site in space group $Fd\bar{3}m$ with propagation vector k = (0, 0, 0) for Nd$_2$Hf$_2$O$_7$ obtained from the representational analysis using the program BASIREPS. The atoms of the nonprimitive basis are defined according to Nd1: (0.50, 0.50, 0.50); Nd2: (0.25, $-$0.25, 1.00); Nd3: ($-$0.25, 1.00, 0.25); Nd4: (1.00, 0.25, $-$0.25).
IRs $\psi_\nu$ component Nd1 Nd2 Nd3 Nd4
$\Gamma_3^1$ $\psi_1$ Real (1 1 1) ($-$1 $-$1 1) ($-$1 1 $-$1) (1 $-$1 $-$1)
$\Gamma_6^2$ $\psi_1$ Real (1 $-$0.5 $-$0.5) ($-$1 0.5 $-$0.5) ($-$1 $-$0.5 0.5) (1 0.5 0.5)
Imaginary (0 $-$0.87 0.87) (0 0.87 0.87) (0 $-$0.87 $-$0.87) (0 0.87 $-$0.87)
$\psi_2$ Real ($-$0.5 1 $-$0.5) (0.5 $-$1 $-$0.5) (0.5 1 0.5) ($-$0.5 $-$1 0.5)
Imaginary (0.87 0 $-$0.87) ($-$0.87 0 $-$0.87) ($-$0.87 0 0.87) (0.87 0 0.87)
$\Gamma_8^3$ $\psi_1$ Real (1 $-$1 0) ($-$1 1 0) (1 1 0) ($-$1 $-$1 0)
$\psi_2$ Real (0 1 $-$1) (0 1 1) (0 $-$1 $-$1) (0 $-$1 1)
$\psi_3$ Real ($-$1 0 1) ($-$1 0 $-$1) (1 0 -1) (1 0 1)
$\Gamma_{10}^3$ $\psi_1$ Real (1 1 0) ($-$1 $-$1 0) (1 $-$1 0) ($-$1 1 0)
$\psi_2$ Real (0 0 1) (0 0 1) (0 0 1) (0 0 1)
$\psi_3$ Real (0 1 1) (0 1 $-$1) (0 $-$1 1) (0 $-$1 $-$1)
$\psi_4$ Real (1 0 0) (1 0 0) (1 0 0) (1 0 0)
$\psi_5$ Real (1 0 1) (1 0 $-$1) ($-$1 0 $-$1) ($-$1 0 1)
$\psi_6$ Real (0 1 0) (0 1 0) (0 1 0) (0 1 0)
Out of the above four IRs the best refinement of the magnetic diffraction pattern is obtained for $\Gamma_3^1$ (with a magnetic $R$ factor of 10.1%) which corresponds to the all-in/all-out spin configuration. For the refinement, the scale factor was fixed to the value obtained from the nuclear structure refinement at 0.6 K. Only the coefficient of one basis vector of $\Gamma_3^1$ was the refinable parameter. The refinement of the magnetic diffraction pattern at 0.1 K with the all-in/all-out type model as shown in Fig. <ref>(b) gives an ordered moment of $m = 0.62(1) \,\mu_{\rm B}$/Nd. The all-in/all-out magnetic structure of Nd$_2$Hf$_2$O$_7$ is illustrated in Fig. <ref>. The magnetic structure is comprised of alternating `all-in' and `all-out' units of corner-shared tetrahedra, where each tetrahedral unit consists of four Nd$^{3+}$ magnetic moments at the vertices of the tetrahedra all pointing either towards the center (all-in) or away from the center (all-out) of tetrahedra as illustrated in the lower panel of Fig. <ref>.
(Color online) Temperature $T$ dependence of the ordered moment $m$ (per Nd ion) obtained from the refinement of neutron powder diffraction data at various temperatures. The solid curve represents the fit according to $m = m_0 (1 - T/T_{\rm N})^{\beta}$ for $T_{\rm N} = 0.53(1)$ K and $\beta=0.34(5)$ in .
The $T$ dependence of the ordered moment $m$ obtained from the refinement of ND patterns at different temperatures is shown in Fig. <ref>. The $m(T)$ data above $T= 0.4$ K are reasonably described by $m = m_0 (1 - T/T_{\rm N})^{\beta}$. The fit for 0.4 K $\leq T \leq T_{\rm N}$ is shown by the solid curve in Fig. <ref> which gives $T_{\rm N} = 0.53(1)$ K and $\beta=0.34(5)$. The critical exponent $\beta$ is close to $ \beta \approx 0.33$ for a three-dimensional Ising system <cit.>. We also notice that $m(T)$ at $T \leq 0.4$ K is almost independent of $T$ with a value of $\sim 0.62\,\mu_{\rm B}$/Nd which is quite unusual. The origin of this unusual flattening of $m(T)$ is not clear and requires further investigation. Because of the anisotropic nature one would expect a gapped magnon spectrum in this compound, however a small energy gap will not be sufficient to explain the observed $T$ dependence of $m(T)$ at $T \leq 0.4$ K.
For an effective $S = 1/2$ with effective $g_{zz} = 5.01(3)$ one would expect an ordered moment of $g_{zz} S = 2.50 \, \mu_{\rm B}$/Nd. However, the obtained ordered moment $0.62(1)\,\mu_{\rm B}$/Nd at 0.1 K is much smaller than this value. The strongly reduced value of ordered moment reflects the presence of strong quantum fluctuations persisting deep into the ordered state down to 0.1 K. In a pyrochlore material with local $\langle 111 \rangle$ Ising anisotropy the magnetic ground state strongly depends on the relative strength of the dipolar and magnetic exchange interactions <cit.>. Spin ice behavior is observed when the ferromagnetic dipolar interaction dominates over the antiferromagnetic exchange. On the other hand a dominating antiferromagnetic exchange stabilizes all-in/all-out long range magnetic order. We estimate the nearest neighbor dipole-dipole interaction $D_{\rm nn}$ using our effective moment $\mu_{\rm eff} = 2.45\, \mu_{\rm B}$/Nd and unit cell lattice parameter $a = 10.6389(1)$ Å as <cit.>
\begin{equation}
D_{\rm nn} = \frac{5}{3}\left(\frac{\mu_0}{4\pi}\right) \frac{\mu_{\rm eff}^2}{r_{\rm nn}^3} \approx 0.12~{\rm K}
\label{eq:Dnn}
\end{equation}
where $\mu_0$ is the magnetic permeability of vacuum and $r_{\rm nn }= (a/4)\surd 2$ is the nearest neighbor distance. Following Siddharthan et al. <cit.> a rough estimate of nearest neighbor exchange interaction between the $\langle 111 \rangle$ Ising moments $J_{\rm nn}$ can be made by fitting the $\chi(T)$ data with a high-temperature series expansion up to order $(1/T^2)$, i.e. by $\chi(T) = (C_1/T)[1+C_2/T]$ where $C_2$ can be decomposed as a sum of exchange and dipolar interactions, $C_2 = (6S^2/4)[2.18 D_{\rm nn} + 2.67 J_{\rm nn}]$. The fit of $\chi(T)$ data with this expression in 10 K $\leq T\leq 30$ K gives $C_2 = -0.67(5)$ K. Thus using the above estimated $D_{\rm nn} \approx 0.12$ K, from $C_2$ we obtain $J_{\rm nn} \approx -0.77$ K. Though this estimate of $J_{\rm nn}$ is not very accurate, it clearly shows that the antiferromagnetic $J_{\rm nn}$ dominates over the dipolar $D_{\rm nn}$, eventually leading to a long range ordered ground state in Nd$_2$Hf$_2$O$_7$ with an all-in/all-out magnetic structure. A better estimation of $J_{\rm nn}$ is desired to check if the ratio $J_{\rm nn}/D_{\rm nn}$ complies with the phase diagram of Ising pyrochlore magnets which predicts a long range antiferromagnetic ordering for $J_{\rm nn}/D_{\rm nn} < -0.91$ <cit.>. We would also like to point out that because of the octupolar tensor component for Nd$^{3+}$ <cit.>, the estimate of $D_{\rm nn}$ using Eq. (<ref>), which only accounts for the dipolar component, may also not be very precise.
§ SUMMARY AND CONCLUSIONS
The physical properties of a pyrochlore hafnate Nd$_2$Hf$_2$O$_7$ have been investigated by $\chi_{\rm ac}(T)$, $\chi(T)$, $M(H)$ and $C_{\rm p}(T)$ measurements. Evidence of an antiferromagnetic transition below $T_{\rm N} =0.55$ K is seen from the $T$ dependence of $\chi_{\rm ac}$ measured down to 0.2 K. The analysis of $M(H)$ data indicate a local $\langle 111 \rangle$ Ising anisotrpic behavior with an effective longitudinal $g$-factor of 5.01(3) for the pseudo spin-half Kramers doublet ground state of Nd$^{3+}$. The low-$T$ $\chi(T)$ reveals an effective moment of $2.45\, \mu_{\rm B}$/Nd for the Ising ground state and a positive $\theta_{\rm p}$ reflecting ferromagnetic coupling between the Nd spins, though the compound orders antiferromagnetically. The $C_{\rm p}(T)$ data show the presence of short range correlations well above the antiferromagnetic ordering. The crystal field analysis of $C_{\rm mag}(T)$ suggests the splitting energy between the ground state doublet and first excited state doublet to be about 230 K.
Magnetic structure determination by neutron powder diffraction confirmed the long range antiferromagnetic ordering with a magnetic propagation wavevector k = (0, 0, 0). The Nd$^{3+}$ moments are found to adopt an all-in/all-out structure with the four magnetic moments at the vertices of the tetrahedra pointing alternatively either all-into or all-out-of the centers of the neighboring tetrahedra. The ordered state magnetic moment of Nd$^{3+}$ $m = 0.62(1)\,\mu_{\rm B}$/Nd at 0.1 K is highly reduced compared to the expected Ising moment value of $2.50\,\mu_{\rm B}$/Nd with an effective spin $S = 1/2$ and $g_{zz} = 5.01(3)$ Kramers doublet ground state, reflecting the presence of strong quantum fluctuations. The unusual reduction of ordered moment and presence of strong quantum fluctations could be due to the dipolar-octupolar nature of Kramers doublet ground state of Nd$^{3+}$ <cit.>, which however remains to be confirmed by further theoretical and experimental works.
We acknowledge Helmholtz Gemeinschaft for funding via the Helmholtz Virtual Institute (Project No. VH-VI-521).
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|
1511.00087
|
Department of Physics, Applied Optics Beijing Area
Major Laboratory, Beijing normal University, Beijing 100875, China
We present an efficient proposal for error-rejecting quantum
computing with quantum dots (QD) embedded in single-sided optical
microcavities based on the interface between the circularly
polarized photon and QDs. An almost unity fidelity of the quantum
entangling gate (EG) can be implemented with a detectable error that
leads to a recycling EG procedure, which improves further the
efficiency of our proposal along with the robustness to the errors
involved in imperfect input-output processes. Meanwhile, we discuss
the performance of our proposal for the EG on two solid-state spins
with currently achieved experiment parameters, showing that it is
feasible with current experimental technology. It provides a
promising building block for solid-state quantum computing and
quantum networks.
§ INTRODUCTION
Compared with the traditional computer, quantum computing <cit.>
can factor an $n$-bit integer with the magical Shor algorithm
<cit.>, exponentially faster than the best-known classical
algorithms. It can also run the famous quantum search algorithm, the
Grover algorithm <cit.> or the optimal Long algorithm
<cit.>, for unsorted database search, which requires
O($\sqrt{N}$) operations only, rather than O($N$) operations
involved in its classical counterpart. Both circuit-based quantum
computing and the measurement-based one require quantum entangling
gates. That is, the ability to entangle the quantum bits (qubits)
is an essential building block in the construction of a quantum
computer <cit.>. Since the early quantum entangling gate (EG)
for single atoms was designed with the assistance of a high-Q
optical cavity <cit.>, more and more attention has been
paid to the entangling operation between stationary qubits
The previous EG between two stationary qubits is implemented by
various methods that resort to different interactions, i.e., the
coherent control of the direct qubit-qubit interaction, the
indirect interaction meditated with high-Q optical cavities
or the controllable exchange interaction involved in the solid-state
spin systems <cit.>. The typical absence of a
heralding measurement in the EG resulting from the direct or
indirect qubit-qubit interaction will lead to some ambiguous error,
such as the one originating from the photon loss as a result of the
cavity decay or the radiative deexcitation of the stationary dipole.
These proposals could work successfully under the condition that the
amount of noise involved in these EGs is less than a small threshold
value <cit.>. It will be more physical-resource
consuming and largely increase the complexity of the target quantum
system when performing scalable quantum computing with EGs of a
higher error probability <cit.>. However,
with a layered quantum-computer architecture, the resources
required for error correction will become manageable when the
physical error rate is about an order of magnitude below the
threshold value of the chosen code <cit.>.
An alternative strategy exploits a measurement on the auxiliary
photonic qubits that entangle with the corresponding stationary
qubits to project the target stationary system into an entangled
state, which constitutes a quantum EG of high fidelity. Meanwhile,
its success is heralded by the detection of photons
Its fidelity does not suffer from the photon loss noise, and it is
relatively robust to the variation of the system parameters. Since
these special schemes involve the optical Bell-state measurement
(BSM) assisted by linear optical elements, they can succeed with the
maximal efficiency of $1/2$ in the ideal situation
<cit.>. Besides the nondeterministic efficiency,
the two photons for the BSM are required to be indistinguishable in
all degrees of freedom except for the one used to encode the quantum
information <cit.>. Therefore, if the polarization degree
of freedom of photons is used to encode the photonic qubit, it will
result in a smooth degradation in the performance of these EGs when
the photonic spectral differences and the practical experimental
control of the arrival time of the photons are considered.
It is well known that solid-state spin systems offer a promising
candidate for the realization of quantum computing
<cit.>. Its solid-state nature
combined with nanofabrication techniques provide a relatively
simple way to incorporate the spins into optical microcavities and
allow for the generation of arrays of solid qubits
<cit.>. One attractive type of
solid-state spin system is the electron spin in a quantum dot (QD)
<cit.>. Not only does it provide easy ways
of optical initialization, single-qubit manipulation, and readout,
but also it processes a long coherence time of the electron spin in
QDs, which is typically around several microseconds when spin echo
techniques are used <cit.>. Most existing
quantum computing schemes based on single photons and single spins
in QDs are performed in a strong coupling regime as a result of
cavity quantum electrodynamics (QED)
<cit.>. However, the strong
coupling regime remains a challenge and some EG proposals for QDs
in a low-Q cavity are proposed at the price of a decrease in the
fidelity and the efficiency of the EG
<cit.>. Meanwhile, the
solid-state spins used in quantum computing are supposed to be
homogeneous and the inhomogeneity of the spins will further decrease
the feasibility of the proposed solid-state spin quantum computing
In this article, we propose a robust proposal for the quantum EG
between two QDs embedded in single-sided microcavities
<cit.>. It is a practical proposal for performing efficient
solid-state quantum computing that overcomes the existing
limitations, since the fidelity of the EG for two QDs is always
towards unity and the efficiency of the EG can, in principle, also
approach unity. Compared with previous EGs for QDs based on cavity
QED <cit.>, the
present scheme also has several other advantages. First, it does not
require the strong-coupling limit and can work efficiently in low-Q
cavities or even in the regime of resonance scattering
<cit.> where the modified spontaneous emission parameter
of QDs coupled resonantly to a microcavity is matched to that in
the bulk dielectric. Second, the success of our EG is heralded
with fidelity larger than $0.99$, and it is signaled by the
detection of a photon of orthogonal polarization as a result of
cavity QED <cit.>, where only one effective input-output
process is involved in single-sided cavities, similar to the one in
a nitrogen-vacancy center coupled to two-sided cavities
<cit.>. This is the origin of our high efficiency rather
than the maximal efficiency $1/8$ based on two-sided cavities.
Third, the imperfect reflection of the cavity due to the deviation
from the ideal conditions, i.e., the nonzero photonic bandwidth, the
finite coupling rate between the QD and cavity mode, and the
mismatch between the incident photon and the cavity mode, can only
lead to photon loss or a click on either vertical detector rather
than a decrease in the fidelity of the EG. Meanwhile, the QD
subsystem will be collapsed into the original state when one of
vertical detectors clicks, and we can input another probe photon to
restart the EG directly without any re-preparation of the QDs, which
makes our proposal more efficient than others. With our EGs, one can
implement universal quantum computing, including both the one-way
quantum computing and the circuit one <cit.>.
§ ERROR-REJECTING ENTANGLING GATE FOR TWO QDS IN LOW-Q OPTICAL MICROCAVITIES
Let us consider a quantum system consisting of a singly charged
self-assembled In(Ga)As QD embedded in a single-sided micropillar
cavity <cit.>. The quantization axis
$z$ is chosen along the growth direction of the QD and is also
parallel to the light propagation direction, shown in
Fig. <ref>(a). The dipole transition associated with the
negatively charged QD is strictly governed by Pauli's exclusion
principle <cit.>, shown in Fig. <ref>(b). The single
electron ground states have $J_z=\pm1/2$, denoted
$|\!\uparrow\rangle$ and $|\!\downarrow\rangle$, respectively, and
the optical excited states are the trion states
($X^-=\{|\!\!\uparrow\downarrow\Uparrow\rangle$ or
$|\!\!\uparrow\downarrow\Downarrow\rangle\}$) consisting of two
antisymmetric electrons in the singlet state
and one hole with $J_z=\pm3/2$ ($|\!\Uparrow\rangle$ and
$|\!\Downarrow\rangle$). The dipole-allowed transitions between the
ground state and the trion state are
along with the absorbtion of a right-handed circularly polarized
photon $|R\rangle$ and a left-handed one $|L\rangle$, respectively,
while the crossing transitions are dipole forbidden <cit.>.
The spin-dependent transitions for negatively charged
exciton $X^-$. (a) A singly charged QD inside a single-sided optical
micropillar cavity. (b) The relative energy levels and the optical
transitions of a QD.
When a circularly polarized probe photon is launched into the
single-sided cavity, it will be reflected by the cavity with a
spin-dependent reflection coefficient $r_j(\omega)$
<cit.>. The dynamic process can be
represented by Heisenberg equations for the cavity field operator
$\hat{a}$ and dipole operator $\hat{\sigma}_-$ in the interaction
picture <cit.>,
\begin{equation} \label{eq1} % Eq_1
\begin{split}
\frac{d \hat{a}}{dt} &= -\left[i(\omega_c\!-\!\omega) \!+\!
\frac{\kappa}{2}\!+\!\frac{\kappa_s}{2}\right] \hat{a}
-g\hat{\sigma}_{-} \!-\!\sqrt{\kappa}\, \hat{a}_{in}\!+\!\hat{R}, \\
\frac{d\hat{\sigma}_-}{dt} &= -\left[i(\omega_{X^-}\!-\! \omega)
\!+\!
\frac{\gamma}{2}\right] \hat{\sigma}_{-} \!-\! g\hat{\sigma}_z \, \hat{a} +\!\hat{N},%\\
\end{split}
\end{equation}
where $\omega_{X^-}$, $\omega_{c}$, and $\omega$ are the frequencies
of the dipole transition, the cavity resonance, and the probe
photon, respectively. $\hat{R}$ and $\hat{N}$ are noise operators
which help to preserve the desired commutation relations. The
parameter $g$ is the coupling strength between $X^-$ and the cavity
mode. $\kappa$ describes the coupling to the input and output ports,
while $\kappa_s$ and $\gamma$ represent the cavity leakage rate and
the trion $X^-$ decay rate, respectively. In the weak excitation
limit where the QD dominantly occupies the ground state, assisted
by the standard cavity input-output theory $\hat{a}_{out} =
\hat{a}_{in} \!+\!\sqrt{\kappa}\, \hat{a}$ <cit.>, one can
obtain the spin-dependent reflection coefficient
\begin{eqnarray} % Eq_2
\label{rcoe}
\end{eqnarray}
Here the subscript $j$ is used to discriminate the case that the
polarized probe photon agrees with the trion transition ($j=1$) and
feels a QD-cavity coupled system and the case that the polarized photon
decouples from the trion transition ($j=0$) and feels an empty
Suppose the electron spin $s$ of a QD is initialized to
+\beta|\!\downarrow\rangle_s$, with $|\alpha|^2+|\beta|^2=1$. When
the input photon is in the polarized state
the photon reflected by the cavity directly due to the mismatching
between the incident probe photonic field and the cavity mode, or
reflected by the desired cavity-QD system, together with the QD,
evolves into an unnormalized state
\begin{eqnarray} % Eq_3
\end{eqnarray}
Here $\eta_{in}$ is the probability amplitude of the photon
reflected by the desired cavity-QD system
<cit.>. If one rewrites $|\Phi\rangle_{H}$ with
the linear-polarization basis $\{|H\rangle \equiv
\frac{1}{\sqrt{2}}(|R\rangle+|L\rangle)$, $|V\rangle \equiv
\frac{1}{\sqrt{2}}(|R\rangle-|L\rangle)\}$, one can get the system
composed of the photon p and the electron spin $s$ evolving
into a partially entangled hybrid state,
\begin{eqnarray} % Eq_4
|\Phi\rangle_{H_0}\!\!&=&\!\!\! \left[\frac{\eta_{in}}{2}
\!+\!\sqrt{1\!-\!\eta^2_{in}}\,\right]\!\!(\alpha|\!\uparrow\rangle+\beta|\!\downarrow\rangle)_s\otimes|V\rangle_{p} \nonumber\\
\label{eqfaithful}
\end{eqnarray}
Here the photon p is partially entangled with the electron
spin $s$, and one can determine the state of the spin according to
the outcome of the measurement on photon p. In detail, the
detection of an $|H\rangle_{p}$ photon leads to a phase-flip
operation on spin $s$. Alternatively, the detection of a
$|V\rangle_{p}$ photon signals an error and results in an unchanged
electron spin $s$, no matter where the error originates (the
mismatch between the incident field and the cavity mode, the low-Q
cavity, or the detuning). For simplicity, we can take
$\eta_{in}\equiv1$ below, since it will not affect the dominant
performance of our EG protocol, and can only reduce the efficiency
of our protocol by the amount of $1-\eta_{in}^2$. Meanwhile, the
output state of the combined hybrid system composed of the spin $s$
and the probe photon $p$ only depends on the combined coefficients
$r_1-r_0$ or $r_1+r_0$ of the cavity-QD system, while it is
independent of the particular parameters that affect the reflection
coefficients $r_j$, shown in Eq. (<ref>). Therefore, the output
states of two individual inhomogeneous electron spins embedded in
different optical microcavities along with their respective probe
photons could be amended to be the same one by utilizing an
adjustable beam splitter <cit.>. The negative
effect of the inhomogeneity of the solid-state spins could be
eliminated formally, which leads to the same result as in
homogeneous cavity-QD systems <cit.>.
The schematic setup of the EG. BS represents a $50:50$ beam
splitter. PBS is the polarizing beam splitter that transmits
$|H\rangle$ photons and reflects $|V\rangle$ photons. BS$^a_i$
denotes the beam splitter with adjustable reflection coefficient
$r^a_i$, i.e., $r^a_1=r^a_2=1$ is utilized for two identical
cavity-QD systems; otherwise, $|r^a_1 r^a_2|<1$ that might lead to
the click of single-photon detector $D'_i$ and restart the recycling
procedure before a phase-flip operation on spin $s_i$.
With the faithful process described above, we can construct an
error-rejecting EG, shown in Fig. <ref>, for two identical
electron spins $s_1$ and $s_2$ (the reflection coefficients
$r_i^a=1$ of the adjustable beam splitter BS$^a_i$ are adopted),
which will collapse spins $s_1$ and $s_2$ into a state with a
deterministic parity after the entangling process. Suppose the
electron $s_i$ ($i=1,2$) is initially in the state
with $|\alpha_i|^2+|\beta_i|^2=1$. One probe photon p in
launched into the import of the EG passes through the beam splitter
(BS$_1$), and it will be reflected by either the left cavity
containing the electron spin $s_1$ or the right one containing
$s_2$. The unnormalized state of the hybrid system composed of the
photon p and the electron spins $s_1$ and $s_2$ after being
reflected by the cavities evolves into
\begin{eqnarray} % Eq_5
\!+\!\beta_1|\!\downarrow\rangle)_{s_1} (\alpha_2|\!\uparrow\rangle
\!+\!\beta_2|\!\downarrow\rangle)_{s_2} \;\;\;\;\;\;\nonumber\\
&&\!\! \otimes(|V\rangle_{p_1}\!+\!|V\rangle_{p_2})
\!+\!(r_1\!-\!r_0)\big[(\alpha_1|\!\uparrow\rangle\!-\!\beta_1|\!\downarrow\rangle)_{s_1}\nonumber\\
\otimes(\alpha_2|\!\uparrow\rangle\!+\!\beta_2|\!\downarrow\rangle)_{s_2}|H\rangle_{p_1}
\!+\!(\alpha_1|\!\uparrow\rangle\!+\!\beta_1|\!\downarrow\rangle)_{s_1}\nonumber\\
&&\!\! \otimes(\alpha_2|\!\uparrow\rangle
\!-\!\beta_2|\!\downarrow\rangle)_{s_2}|H\rangle_{p_2}\big]\Big\}.
\label{Removeerror}
\end{eqnarray}
Here the subscripts $p_1$ and $p_2$ denote photon components that
occupy the left path and the right path, respectively. When the
photon is in the horizonal polarized state $|H\rangle_{p_1}$ or
$|H\rangle_{p_2}$, the two different spatial modes of photon
p are combined on the BS$_2$. The interference of
$|H\rangle_{p_1}$ and $|H\rangle_{p_2}$ modes will collapse the
hybrid system into
\begin{eqnarray} % Eq_6
\big[\!(\alpha_1\alpha_2|\!\uparrow\rangle_{s_1}|\!\uparrow\rangle_{s_2}
\!\!\!-\!\beta_1\beta_2|\!\downarrow\rangle_{s_1}|\!\downarrow\rangle_{s_2}\!)
\!\!\!-\!\beta_1\alpha_2|\!\downarrow\rangle_{s_1}
\label{pcdfaith}
\end{eqnarray}
Upon a click of the detector $D_3$ or $D_4$, the EG is completed and
the electron-spin system $s_1s_2$ is projected into a subspace with
a deterministic parity, which is independent of the reflection
coefficients $r_j$, since $r_j$ only appears as a global coefficient
in Eq. (<ref>). In detail, when the photon detector $D_3$
clicks, the spins $s_1s_2$ collapse into the even-parity entangled
state of the form
\begin{eqnarray} % Eq_7
\label{pcdE}
\end{eqnarray}
When the photon detector $D_4$ clicks, the spins $s_1s_2$ are
projected into the odd-parity entangled state of the following form
\begin{eqnarray} % Eq_8
\label{pcdfaith0}
\end{eqnarray}
Both states $|\Phi\rangle_{E}$ and $|\Phi\rangle_{O}$ keep the
information of the initial state. Therefore, the coefficient
$\alpha_i$ and $\beta_j$ could be the state of other QD spins
that are entangled with $s_1$ and $s_2$, which makes the EG
effective for constructing cluster states in the next section. The
total probability that either $D_3$ or $D_4$ detects one photon of
horizonal polarization is $\eta_{_H}$:
\begin{eqnarray} % Eq_9
\eta_{_H}=\frac{|r_1-r_0|^2}{4}. \label{etav}
\end{eqnarray}
Here $\eta_{_H}$ equals the efficiency of the EG without recycling
The first term on the right-hand side of Eq. (<ref>)
contains the vertical polarization component $|V\rangle_{p_1}$
($|V\rangle_{p_2}$) and it will lead to a click on the photon
detector $D_1$ ($D_2$). In this time, the state of the electron
spins $s_1 s_2$ is projected into
$|\Phi\rangle_{s_1}\otimes|\Phi\rangle_{s_2}$, exactly identical to
the original one without any interaction between the spins and the
photon p, which takes place with probability $\eta_{_V}$:
\begin{eqnarray} % Eq_10
\eta_{_V}=\frac{|r_1+r_0|^2}{4}. \label{etah}
\end{eqnarray}
Here $\eta_{_V}$ equals the heralded error efficiency of the EG,
and the electron spins $s_1 s_2$, in this case, could be directly
used in the recycling EG procedure.
In a word, one can obtain two kinds of useful results with our EG setup.
When only one probe photon is exploited, the probabilities of heralded success or failure of
the EG are $\eta_H$ or $\eta_V$, respectively. When the heralded error of EG takes place, a
$|V\rangle$ polarized photon is detected and the state of the spin
subsystem has not been changed. One can input another probe photon
p$'$ in state
to repeat the EG process until a horizonal photon $|H\rangle$ is
detected by $D_3$ or $D_4$. This procedure will project the spin
system $s_1s_2$ into an even-parity subspace or an odd-parity one
eventually. By taking the recycling procedure into account, the
total success probability $\eta_{_S}$ of our error-rejecting EG is
\begin{eqnarray} % Eq_11
\eta_{_S}=\frac{|r_1-r_0|^2}{4-|r_1+r_0|^2}, \label{etaS}
\end{eqnarray}
which is state independent, resulting in a more efficient quantum
computing <cit.>. Note that each recycling procedure is
conditioned on a click of either vertical detector $D_1$ or $D_2$,
and it should be stopped when photon loss takes place.
Subsequently, one has to reinitialize the spins before performing a
new EG operation on the spins.
§ CLUSTER STATE GENERATION WITH OUR EG FOR MEASUREMENT-BASED ONE-WAY QUANTUM COMPUTING
Our error-rejecting EG can be used directly to implement the
one-way quantum computing <cit.> based on QDs
embedded in optical cavities. In the following, we demonstrate that
our EG can be used to construct the two-dimensional (2D) QD
cluster state <cit.>, which constitutes the
base of one-way quantum computing on solid-state spins.
Suppose there are $j+1$ QD electron spins $\{s_1, s_2, \dots{},
s_{j}\}$ and $ s_{j+1}$, and $s_{j+1}$ is initialized to be the
and the first $j$ spins are initially in the one-dimensional (1D)
cluster state of the form
\begin{eqnarray} % Eq_12
|\psi_{j}\rangle &=&
\cdots\;\;\;\;\;\;\;\;\;\nonumber\\ && \otimes(|\!\uparrow\rangle_{j-1}
\label{cluster1dj}
\end{eqnarray}
with the phase flip operator
To increase the length of the 1D cluster state, an error-rejecting
EG for spins $s_j$ and $s_{j+1}$ is applied. When the EG fails, the
state of spin $s_j$ is ambiguous and a state measurement on $s_j$
with basis $ \{|\!\uparrow\rangle, |\!\downarrow\rangle \}$ will
collapse the remaining spins into a 1D cluster state of $j-1$
qubits, with or without a $\hat{Z}_{j-1}$ feedback operation. When
the EG succeeds in the case that is heralded by the click of photon
detector $D_3$, the $j+1$ spins will be projected into
\begin{eqnarray} % Eq_13
|\psi'_{j+1}\rangle &=&
\nonumber\;\;\;\;\\ &&
\end{eqnarray}
which could be transformed into the 1D cluster state similar to
that in Eq. (<ref>) of length $j+1$ by a Hadamard
operation $\hat{H}_{j}$ [$\hat{H}$ completes the following
performed on spin $j$. If the success of the EG for $s_j$ and
$s_{j+1}$ is signaled by the click of $D_4$, a local operation
$\hat{H}_{j+1}\hat{X}_{j+1}$ (here the spin-flip operator
on spin $s_{j+1}$ could also evolve the ${j+1}$ spins into the
desired 1D cluster state.
This procedure of cluster growth discussed above could be used to
generate a larger cluster, since the efficiency of our
error-rejecting EG $\eta_s>0.5$ and can, in principle, approach
unity. To speed up the cluster generation process, some shorter
clusters could be prepared in parallel and then be connected
together to generate the longer one
<cit.>. In the following, we
introduce an efficient cluster-connecting proposal. Suppose the two
1D clusters M and N available are, respectively, of
lengths $m$ and $n$,
\begin{eqnarray} % Eq_14
|\psi_{m}\rangle \!\!&=&\!\!
\nonumber\\
|\psi_{n}\rangle\!\! &=& \!\! (|\!\uparrow_{N}\rangle_1+|\!\downarrow_{N}\rangle_1{}\hat{Z}_{N_2})\cdots(|\!\uparrow_{N}\rangle_{n-1}
\nonumber\\
\label{clustermn}
\end{eqnarray}
Before performing EG on $M_m$ and $N_{1}$, a phase-flip operation
$\hat{Z}_{M_m}$ is applied on spin $M_m$. The success of the EG
heralded by the click of photon detector $D_3$ will project the
entire spin system into
\begin{eqnarray} % Eq_15
|\psi^n_{m}\rangle\! \! &=&\! \!
\nonumber\\
\otimes\hat{Z}_{M_m})(|\!\uparrow_{M}\rangle_{m}|\!\uparrow_{N}\rangle_1
\cdots(|\!\uparrow_{N}\rangle_{n-1}
\nonumber\\
&&+|\downarrow_{N}\rangle_{n-1}{}\hat{Z}_{N_n})(|\!\uparrow_{N}\rangle_{n} +|\!\downarrow_{N}\rangle_{n}).
\label{clustermn1}
\end{eqnarray}
An additional Hadamard operation $\hat{H}$ on $M_m$ will evolve the
spin system into a 1D cluster $|\psi_{m+n}\rangle$ of $m+n$ qubits.
As for the case that the success of the EG is signaled by a click of
detector $D_4$, a local single-qubit operation $\hat{H}\hat{X}$ on
$M_m$ can also evolve the $m+n$ spins into the 1D cluster
The cluster-connecting procedure based on parity measurement above
is similar to that used in linear optical quantum computing
<cit.>, whereas both the outcome of the EG operation and
the feedback operations after the EG
are quite different. It generates a 1D cluster of $m+n$
qubit, rather than $m+n-1$ in linear optical quantum computing
<cit.> or in previous schemes for solid-state spins
<cit.> in which the outcomes of the EG can only lead
to an odd parity and the cluster connecting procedure is completed
by a spin measurement later. One can also perform a cross-linking
between linear chains to construct a 2D cluster similar to the
previous schemes
<cit.>, which means
our EG can be used to complete universal one-way quantum computing
The efficiency of the EG vs different parameters with
$\omega_{X^-}/\omega_c=1$ and $\gamma/\kappa=0.1$: (a)
$(\omega_c-\omega)/\kappa=0$, $C=1/4$; (b)
$(\omega_c-\omega)/\kappa=0$, $C=1$; (c)
$(\omega_c-\omega)/\kappa=\gamma/\kappa$, $C=1/4$; and (d)
$(\omega_c-\omega)/\kappa=\gamma/\kappa$, $C=1$.
§ PERFORMANCE OF OUR ERROR-REJECTING EG WITH CURRENT EXPERIMENTAL PARAMETERS
The total success probability $\eta_{_S}$ together with $\eta_{_H}$
and $\eta_{_V}$ of our EG are shown in Fig. <ref> as a
function of the side leakage $\kappa/\kappa_s$ with the
cooperativity $C\equiv g^2/\gamma\kappa_{_T}$,
$\kappa_{_T}\equiv\kappa_s+\kappa$, and $\gamma/\kappa=0.1$
<cit.>. We tune the transition frequency $\omega_{X^-}$
of the QD to be resonant to that of the cavity,
$\omega_{X^-}/\omega_{c}=1$ <cit.>. When the probe
photon is also resonant to cavity [see Figs. <ref> (a) and
(b)], $\eta^r_S=0.255$ and $\eta^p_S=0.559$ can be achieved in the
regime of resonance scattering with $C=1/4$ and the Purcell regime
with $C=1$, respectively, for $\kappa/\kappa_s=13$
<cit.>. When the probe photon detunes from
the trion transition by $(\omega_c-\omega)/\kappa=\gamma/\kappa$,
shown in Figs. <ref> (c) and (d), $\eta^r_S=0.194$ and
$\eta^p_S=0.538$ can be achieved for the same remaining parameters,
and the contribution from the recycling procedure $\eta_{_V}$
increases. Furthermore, the EG could enjoy a higher efficiency with
a lower side leakage and a higher cooperativity $C$, which can be
achieved by utilizing adiabatic cavities with smaller pillar
diameters <cit.>. In other words, the
near -unity efficiency of the error-rejecting EG can be achieved
when the deep Purcell regime with low side leakage is available, and
we can easily attribute this improvement of the efficiency to the
enhancement of the photon into the cavity mode.
In the above discussion, we can get an efficient error-rejecting EG
for QDs with the perfect spin qubit and the monochromatic
($\delta$-function-like) single photon wavepacket. In fact, every
single photon pulse is of finite linewidth, i.e., a polarized photon
of pulse shape in Gaussian function
$f(\omega)=exp(-\omega/\Delta)^2/(\sqrt{\pi}\Delta)$ with bandwidth
$\Delta$. This finite-linewidth character usually introduces some
additional infidelity in the previous EG protocols
while it has little harmful effect on the fidelity of our EG. When
one constitutes our EG with a polarized single-photon pulse $p$ of
Gaussian shape, $|\psi_p\rangle=\frac{1}{\sqrt{2}}\int{}d\omega
|0\rangle$, where $\hat{a}^{\dagger}_k(\omega)$ is the creation
operator of a $ k$-polarized photon with frequency $\omega$, the
state of the hybrid system composed of the photon $p$ and electron
spins $s_1$ and $s_2$ just before photon detection process, shown
in Eq. (<ref>), will be modified to
\begin{eqnarray} % Eq_17
\Big\{\!(\alpha_1\alpha_2|\!\uparrow\rangle_{s_1}|\!\uparrow\rangle_{s_2}\nonumber\\
\!\!+\!\!(\alpha_1\beta_2|\!\uparrow\rangle_{s_1}|\!\downarrow\rangle_{s_2}\nonumber\\
\label{pcdfaithm}
\end{eqnarray}
where $
\hat{a}^{\dagger}_H(\omega)=\frac{1}{\sqrt{2}}[\hat{a}^{\dagger}_R(\omega)+\hat{a}^{\dagger}_L(\omega)]$.
Upon the click of photon detector $D_3$ or $D_4$, one can still
complete the EG by projecting spins $s_1$ and $s_2$ into a subspace
of determined parity, as one does with a monochromatic photon wave
packet. One can find that the fidelity of our EG is independent of
the finite linewidth of the photon pulse, since the
frequency-dependent reflection coefficients $r_j(\omega)$ appear
only in the global coefficient.
In fact, the effects of dephasing and decay of electron spins
will affect the performance of the EG. The time needed for the
coherent control of single electron spin in QDs is on the scale of
picoseconds <cit.> and the cavity photon time is tens of
picoseconds when the cavity $Q$-factor is about $ 1\times 10^4
-1\times 10^5$ <cit.>. Therefore, it is the spontaneous
emission lifetime of a QD, which is about $1$ ns, that sets the
upper limit for the fidelity of the EG. Meanwhile, the electron
spin coherence time of $10$ ns has been achieved at zero magnetic
field <cit.>, and it could be extended to several
microseconds if the all-optical spin echo technique is exploited
<cit.>. The ratio of the decoherence time of electron
spins to the operation time needed to complete the EG can exceed
$1\times 10^3$, and thus the fidelity of the EG proposal will be
larger than $0.99$ when taking into account the dephasing process of
the electron spin, which suggests the strong promise of electron
spin in QDs for scalable quantum computing.
§ DISCUSSION AND SUMMARY
Our scheme of error-rejecting EG can work efficiently with almost
unity fidelity in the strong coupling regime, $g>\kappa_T, \gamma$,
the Purcell regime, $C>1/4$, or even the resonantly scattering
regime, $C=1/4$. It is robust to the imperfections involved in the
practical input-output process, i.e., the nonzero bandwidth, QD or
cavity decay, and the finite coupling $g/\kappa$, since the fidelity
of our EG is independent of the reflective coefficients
$r_j(\omega)$ and thus independent of the cooperativity $C$, which
is far different from other schemes that depends on $C$
<cit.>. The
original low fidelity or error items originating from the practical
input-output process are converted into a relatively lower
efficiency in our EG. Fortunately, the low fidelity or error items
trigger the single-photon detector $D_1$ or $D_2$, which can be
used to improve the efficiency of the EG by introducing the
recycling procedure. In fact, our recycling procedure can contribute
little when perfect circular birefringence ($C\gg1$ and
$\omega_{X^{\!\!-}}$$=\omega_c=\omega$ ) is available, since the
efficiency $\eta_H$ of the EG without recycling procedure
approaches unity in this situation. Although our proposal is
detailed with the QD-cavity system, it could also be implemented
with solid-state spin coupled to a photonic crystal waveguide
The previous EG performed in a resonantly scattering regime, $C=1/4$
and $\omega_{X^{\!\!-}}$$=\omega_c=\omega$ could also be completed
with high fidelity, since a reflectivity $r_1=0$ in the resonantly
scattering regime could be automatically eliminated, and only the
photon that decouples the electron spin could be reflected.
Therefore, one can entangle two spins by subsequently probing the
two spin-cavity systems with a linear polarized photon or entangle
two linear polarized photons by subsequently importing them into a
spin-cavity system, where the even-parity subspace of the spin
system or the photon system could be easily picked out, since the
odd-parity case will inevitably be signaled by photon loss
<cit.>, and the corresponding efficiency of the EG is
$0.25$. It is quite different from our EG where the linear polarized
photon, after being reflected by the QD-cavity system, in the ideal
case $C\gg1$ and $\omega_{X^{\!\!-}}$$=\omega_c=\omega$, is
supposed to change its polarization into the orthogonal polarization
and exert a phase-flip operation on the spin. The interference of
the photon after being reflected by two cavities in superposition
can project the two spins into either even-parity subspace or
odd-parity subspace in a heralded way.
The error-rejecting EG only involves one effective input-output
process, which makes our scheme more efficient than others since
the practical input-output coupling $\eta_{in}<1$
<cit.>. In this situation, the probe photon
can be reflected directly by the cavity, and it is harmful and will
reduce the fidelity of the entangling process in the other schemes
However, it can only lead to a decrease of the efficiency of our
EG, since the state of the photon reflected directly by the
microcavity together with that of the spins will be kept unchanged.
In other words, the photon reflected directly by the microcavity is
still in $|V\rangle$ polarization and it will trigger the
single-photon detector $D_1$ or $D_2$, which signals the restarting
of the EG. This makes the EG different from the one based on a
double-sided cavity where the photon is encoded in its Fock state
<cit.>. The photon loss during the EG process owing to the
inefficiency of the single-photon detector or cavity absorbtion will
decrease the efficiency of the EG, but it does not affect the
fidelity of our EG since both the success of the EG and the
restarting of the EG are signaled by a click of single photon
In conclusion, we have proposed an efficient error-rejecting EG
proposal for two electron spins of QDs embedded in low-$Q$ optical
microcavities. With our error-rejecting EGs, a cluster-state
connection scheme could be completed efficiently. Under the
practical experimental condition, the EG could be performed well
with almost unity fidelity and an efficiency of $\eta_s>0.53$ for
$C=1$. We believe the EG could provide a promising building block
for solid-state scalable quantum computing and quantum networks in
the future.
§ ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of
China under Grants No. 11474026 and No. 11674033, and the
Fundamental Research Funds for the Central Universities under Grant
No. 2015KJJCA01.
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|
1511.00199
| |
1511.00199
|
\begin{align*}
c_{{1}}=& {\frac {c_{{2}}c_{{7}}-c_{{5}}c_{{9}}+c_{{8}}c_{{6}}}{c_{{8}}}},
c_{{2}}= c_{{2}},
c_{{3}}= {\frac {-c_{{9}}c_{{2}}c_{{7}}+c_{{5}}{c_{{9}}}^{2}-c_{{9}}c_{{8}}c_{{6}}+c_{{6}}c_{{2}}c_{{8}}}{c_{{8}} \left( c_{{5}}-c_{{7}} \right) }},
\\
c_{{4}}= {\frac {c_{{5}}c_{{2}}c_{{7}}-{c_{{5}}}^{2}c_{{9}}+c_{{5}}c_{{8}}c_{{6}}
-c_{{7}}c_{{6}}c_{{8}}}{c_{{8}} \left( c_{{2}}-c_{{9}} \right) }},
c_{{5}}= c_{{5}},
c_{{6}}= c_{{6}},
c_{{7}}= c_{{7}},
c_{{8}}= c_{{8}},
c_{{9}}= c_{{9}}
\\
c_{{1}}=& {\frac {c_{{9}}c_{{7}}}{c_{{8}}}},
c_{{2}}= c_{{9}},
c_{{3}}={\frac {{c_{{9}}}^{2}}{c_{{8}}}},
c_{{5}}= c_{{5}},
c_{{6}}= {\frac {c_{{5}}c_{{9}}}{c_{{8}}}},
c_{{8}}= c_{{8}},
c_{{9}}= c_{{9}}
\\
c_{{1}}=& c_{{6}},
\\
c_{{1}}= & {\frac {c_{{5}}c_{{2}}}{c_{{8}}}},
c_{{4}}={\frac {{c_{{5}}}^{2}}{c_{{8}}}},
c_{{6}}={\frac {c_{{5}}c_{{9}}}{c_{{8}}}},
\\
c_{{1}}=& c_{{1}},
c_{{2}}={\frac {c_{{5}}c_{{9}}}{c_{{7}}}},
c_{{3}}={\frac {c_{{9}} \left( c_{{1}}c_{{5}}c_{{7}}-c_{{4}}c_{{5}}c_{{9}}
+c_{{1}}{c_{{7}}}^{2} \right)}{{c_{{7}}}^{3}}},
\\
c_{{6}}={\frac {c_{{1}}c_{{5}}c_{{7}}
-c_{{4}}c_{{5}}c_{{9}} +c_{{4}}c_{{9}}c_{{7}}}{{c_{{7}}}^{2}}},
\\
c_{{1}}=& c_{{1}},
\\
c_{{1}}=& {\frac {c_{{4}}c_{{2}}}{c_{{5}}}},
c_{{3}}={\frac {c_{{6}}c_{{2}}}{c_{{5}}}},
\\
c_{{1}}=& {\frac {c_{{6}}c_{{2}}}{c_{{9}}}},
\\
c_{{1}}=& c_{{1}},
c_{{9}}=0 \\
c_{{1}}=& c_{{1}},
\end{align*}
|
1511.00604
|
The dynamics of symplectic gauge fields provides a consistent framework
for fundamental interactions based on
spin three gauge fields.
One remarkable property is that symplectic gauge fields
only have minimal couplings with gravitational fields and not with any other field of
the Standard Model.
Interactions with ordinary matter and radiation can only arise from radiative corrections.
In spite of the gauge nature of symplectic
fields they acquire a mass by the Coleman-Weinberg mechanism which generates
Higgs-like mass terms where
the gravitational field is playing the role of a Higgs field.
Massive symplectic gauge fields
weakly interacting with ordinary matter are natural candidates for the
dark matter component of the Universe.
§ INTRODUCTION
In the Standard Model all fundamental interactions are described by gauge theories.
In the Einstein theory of General Relativity (GR) the gravitational interaction is also formulated in
terms of a gauge field. Although there are significant differences between both
theories, mainly due to the strong connection of GR with the structure
of space-time, the fact that both theories are gauge theories helped to consolidate the gauge
paradigm where all fundamental interactions are described by gauge fields.
The search of new physics beyond the Standard Model is supported by
astrophysical and cosmological evidences of the existence of a new type of
invisible matter with unknown interacting properties. The search for new types of interactions
following the gauge principle suggest to explore the possibility of gauge theories with
higher spin <cit.>. The pathologies associated to interactions based on
massless particles with helicities
higher than two <cit.>
-<cit.> provided an argument to explain why
this kind of interactions are not observed in Nature. Nevertheless, the challenge is so
interesting that there have been numerous attempts to give a physical meaning to gauge
theories of higher helicity fields. Free massless fields with arbitrary helicity (or its generalizations)
do exist in any dimension. In fact, Wigner's theory of covariant representations of the Poincaré group,
provides a general theory of free massless gauge fields <cit.>. Massless fields
with integer helicity are described by transverse, symmetric traceless
tensor fields with some equivalence relations which are reminiscent of gauge transformations
<cit.>. The application of BRST methods to the consistency analysis
of generalized gauge theories boosted the attempts to extend the analysis of free massless gauge fields
to interacting theories from a new viewpoint <cit.>
-<cit.>. The consistency of
the BRST approach requires an infinite tower of higher helicities <cit.>,
-<cit.> in
close analogy with string theory. However, even in that case it was impossible to
show the consistency of
the interacting theory <cit.>,<cit.>
The appearance of new string dualities introduced new approaches based on
five-dimensional theories on anti-de Sitter backgrounds <cit.>
In such a scheme the approach to higher spin fields acquired a new perspective <cit.>
In this paper we explore a different approach to higher spin gauge fields based on
a gauge theory of symplectic fields <cit.>.
In this approach gauge fields are symplectic connections and since their covariant derivatives are
non-trivial only for fields of spin higher than two,
they are minimally decoupled from the Standard Model physics and only interact with gravitational fields.
This special characteristic promoted these fields as excellent candidates for the invisible dark matter
component of the Universe.
§ SYMPLECTIC GAUGE FIELDS
Let us consider a symplectic form $\omega$ in a four-dimensional space-time[The theory can be generalized for
arbitrary even dimensional space-times], i.e.
a regular antisymmetric tensor field $\omega_{\mu\nu}=-\omega_{\mu\nu}$ which is closed $d \omega=0$.
The symplectic form $\omega$ can be considered as the antisymmetric component of a generalized space-time metric
in the sense first considered by R. Forster (formerly known as R. Bach) and developed by
Schrödinger and Einstein in the context of unified field theories. It can also be considered
as a background electromagnetic field $\omega_{\mu\nu}= \partial_\mu A_\nu-
\partial_\nu A_\mu$ with non-trivial topological density $\epsilon^{\mu_1\mu_2\mu_3\mu_4} \omega_{\mu_1\mu_2}\omega_{\mu_3\mu_4}(x)\neq0$ .
A symplectic gauge field is by definition a linear connection
which preserves the symplectic form, i.e. the covariant derivative
of $\omega$
\begin{equation} D_\mu \omega =0\,\end{equation}
In local coordinates
\begin{equation}
\ \partial_\mu \omega_{\nu \sigma }-\Gamma_{\mu \sigma }^\alpha\, \omega_{\alpha \nu}+\Gamma_{\mu \nu}^\alpha \, \omega_{\alpha
\sigma}=0,
\label{simpconn}
\end{equation}
where $\Gamma_{\mu \sigma }^\alpha$ are the local components of the symplectic
gauge field. Although, gravitational fields
are also defined in a similar manner as the linear connections that preserve the space-time metric symmetric $g$,
the contrast between both types of fields is very important as we will see below.
The gauge symmetry is given by space-time transformations which leave the symplectic form invariant (symplectomorphisms). They are canonical transformations whose infinitesimal generators are
given in local coordinates by vector fields of the form
\begin{equation} \xi_\mu= \partial_\mu \phi,\end{equation}
where $\phi$ is any scalar field. By using canonical transformations it is always possible
to find local coordinates, Darboux coordinates, where $\omega$ becomes a constant form
0 & \I \cr
-\I & 0
\end{pmatrix}.$$
In those coordinates, $\partial_\mu \omega=0$ and
\begin{equation}
\Gamma_{\mu \sigma }^\alpha\, \omega_{\alpha \nu}=\Gamma_{\mu \nu}^\alpha \, \omega_{\alpha\sigma}
\label{cuatro}
\end{equation}
If we impose the vanishing of the torsion as in the case of a Levi-Cività metric connection, we have
\begin{equation}
\ \Gamma_{\mu\nu}^\alpha =\Gamma_{\nu\mu}^\alpha.
\label{cinco}
\end{equation}
The components of a torsionless symplectic gauge field
in Darboux coordinates
\begin{equation}T_{\nu\mu\sigma}=\Gamma_{\mu \nu}^\alpha \, \omega_{\alpha \sigma} \hspace{.7cm}{\ }\end{equation}
define a 3-covariant symmetric tensor
\begin{equation}T_{\nu\mu\sigma}= T_{\mu\nu\sigma}=T_{\nu\sigma\mu}=T_{\mu\sigma\nu}=T_{\sigma \nu\mu}=T_{\sigma\mu\nu }.\end{equation}
Thus, the space of torsionless symplectic gauge fields <cit.>
can be identified with the space of 3-covariant symmetric tensors. This space of symplectic gauge fields
is infinite dimensional in contrast with the space of Riemannian gauge fields where the Levi-Cività
connection is unique for any Riemannian metric.
The curvature tensor $R^\alpha_{\beta \mu\nu}$,
\begin{equation} R^\alpha_{\ \beta \mu\nu}=\partial_\mu \Gamma_{\beta \nu }^\alpha -\partial_\nu \Gamma_{\beta \mu }^\alpha+\Gamma_{ \nu \beta}^\sigma\Gamma_{ \mu \sigma}^\alpha-\Gamma_{ \mu \beta}^\sigma\Gamma_{ \nu \sigma}^\alpha\end{equation}
defines by contraction with $\omega$ a (0,4)-tensor
\begin{equation}\ R_{\alpha \beta \mu\nu}= \omega_{\alpha \sigma }R^\sigma_{\ \beta \mu\nu},\end{equation}
with interesting symmetry properties
\begin{eqnarray*}\ \phantom{\Big(} R_{\alpha \beta \mu\nu}&=& - R_{\alpha \beta \nu \mu}= R_{\beta \alpha\mu\nu}, \\
\ \phantom{\Big(}R_{(\alpha \beta \mu\nu)}&=&R_{\alpha \beta \mu\nu}+R_{\nu \alpha \beta \mu}+R_{\mu\nu \alpha \beta}+R_{\beta \mu\nu\alpha }=0.\end{eqnarray*}
The permutation symmetries of this tensor are characterized by the Young tableau
which is in contrast with that of the standard Riemannian tensor
A symplectic Ricci tensor can also defined by
\begin{equation}
\label{ricci}
R_{\beta\nu}=\omega^{\mu\alpha}R_{\alpha \beta \mu\nu},\end{equation}
and is symmetric
\begin{equation}\ R_{\nu \mu}= R_{\mu\nu}, \end{equation}
like the Riemannian Ricci tensor.
However, there is no scalar symplectic curvature because the
contraction of the Ricci tensor with the symplectic form vanishes.
§ SYMPLECTIC FIELD THEORY
The simplest dynamics for symplectic gauge fields
is defined by the action
\begin{equation}
\!\!S(\Gamma,\omega)\!=\!\frac1{2{\alpha_0}^2}\!\int\! d^4 x\ R^{\alpha \beta \mu\nu} R_{\alpha \beta \mu\nu} + \frac{{\ \theta}}{32
\pi^2 }\!\int\! d^4 x \left(R_{\alpha \beta \mu\nu } R^{ \alpha\beta\mu\nu}\!-\!2 R_{\mu\nu} R^{\mu\nu} \right),
\label{seis}
\end{equation}
which only involves the curvature tensors
\begin{equation}
{ R}^{\alpha \beta \mu\nu} = \omega^{\alpha\alpha'}\omega^{\beta\beta'}\omega^{\mu\mu'}\omega^{\nu\nu'}R_{\alpha' \beta'
\mu'\nu'}\qquad { R}^{\mu\nu} = \omega^{\mu\mu'}\omega^{\nu\nu'}R_{ \mu'\nu'}
\end{equation}
and the symplectic form $\omega$.
The second term of (<ref>) is proportional to the Pontryagin class of the
manifold which has a topological meaning and does not contribute to
the classical dynamics.
The metric independence of (<ref>) implies that the dynamics of the
symplectic fields is completely decoupled from gravity.
The action (<ref>) is the most general
metric independent action of symplectic fields with
quadratic dependence in the curvature tensor <cit.>.
Although one could add an extra term
proportional to the square of the Ricci tensor ricci, it turns out that such a term is not independent
of the other two terms of the action seis.
Thus, the extra term can be absorbed by shifting the couplings $\alpha_0$ and $\theta$.
The theory is invariant under symplectomorphisms, i.e. canonical transformations.
Symplectic gauge fields, however, transform as
\begin{equation}
T_{\mu\nu\sigma}^\prime=T_{\mu\nu\sigma}+D_\mu D_\nu\partial_\sigma \phi.
\label{siete}
\end{equation}
under symplectomorphisms, where $D_\mu=\partial_\mu+ \Gamma^\sigma_{\mu\nu}$.
The invariance of the action (<ref>) under these transformations implies
the existence of an infinity of zeromodes.
The field theory governed by (<ref>) is very interesting from a geometrical viewpoint <cit.>,
but from a quantum field theory perspective it presents many pathologies. The Cauchy problem is highly degenerated
as it is pointed out by the existence of
many zero modes in quadratic terms which are
not associated to any known gauge symmetry. Apart from the zero modes
associated to the symplectic gauge symmetry (<ref>) there are nine extra
zeromodes. The remaining non-null modes of the quadratic variation of the action on a trivial
$T=0$ background are of the form
$$\textstyle \ \ \ \ \ \ \ \ -\frac1{3}p^2\, , \ \ \ \pm \frac{\sqrt{2}}{3} p^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm {(double\ degenerated)}$$
$$\textstyle \frac1{3} p^2\ \mathrm {}
, \ \ \ - \frac2{3} p^2 \mathrm{},
\ \ \ \pm \frac{{1}}{\sqrt{3}} p^2\ \ \ \ \ \ \ \ \ \mathrm {(non\ degenerated),} $$
where $p_\mu$ are the momentum of Fourier modes in Darboux coordinates.
Although the eigenvalues of the quadratic terms of the action are $SO(4)$ rotation
invariant the corresponding eigenfunctions $T_{\mu\nu\sigma}$ are not
invariant under Euclidean or Poincaré transformations. This is due to the
background symplectic form $\omega_{\mu\nu}$ which
introduces a phase space structure in the space-time which is not compatible
with Euclidean or Poincaré symmetries.
Moreover, the quadratic terms of the action are not
definite positive as a consequence of the symplectic structure.
This implies that the Gaussian projection defines a theory with ghost fields
which is not unitary quantum field theory .
Poincaré symmetry can be recovered if we consider a generalization of the action where the symplectic
form $\omega$ becomes a full-fledged dynamical field.
A natural choice is to introduce a kinetic term for the symplectic form
$$\frac1{2{\rm e}^2}\int d^4 x\ \omega^{\mu\nu}\omega_{\mu\nu},$$
with $\omega_{\mu\nu}= \partial_\mu A_\nu-
\partial_\nu A_\mu$. But, because of the identity $ \omega^{\mu\nu}\omega_{\mu\nu}=4$
the integrand is constant and there is no dynamical content as the trivial motion equations
point out.
The only non-trivial possibility is to include terms with tensorial contractions which
involve the space-time metric (i.e. coupling to gravity). In this framework it is possible to recover
Poincaré invariance in a Minkowskian metric background.
§ INTERACTION WITH GRAVITY
Let us consider a different theory of the symplectic gauge fields interacting with the
space-time metric $g$
\begin{equation}S_0(\Gamma,\omega,g)=\frac1{2
}\int d^4 x\ \sqrt{g} g^{\mu\mu'}g^{\nu\nu'}\omega_{\mu'\nu'}\omega_{\mu\nu}.\end{equation}
Instead of imposing the restriction to the symplectic gauge fields that preserve the symplectic form $\omega$
simpconn, we introduce the constraint in a softer way via a Lagrange multiplier term in the action
\begin{equation}
S_0'(\Gamma,\omega,g)=\frac1{2{\rm \alpha_0}^2}\int d^4 x\ \sqrt{g}\ g^{\gamma\gamma'}g^{\mu\mu'}g^{\nu\nu'}D_{\gamma'}\omega_{\mu'\nu'}D_\gamma\omega_{\mu\nu}.
\label{constr}
\end{equation}
The strong symplectic condition, $D_\gamma\,\omega_{\mu\nu}=0$,
is recovered in the weak coupling limit $\alpha_0\to 0$
The main interaction of symplectic fields with gravity can be introduced by contracting
indices of the curvature tensor with the space-time metric instead of only using the symplectic form, e.g.
\begin{equation}
S_1(\Gamma,\omega,g)={{ \alpha}^2}\int d^4x\, \sqrt{\ g}\,
g^{\alpha\alpha'}g^{\beta\beta'}g^{\mu\mu'}g^{\nu\nu'}R_{\alpha' \beta' \mu'\nu'}R_{\alpha \beta \mu\nu}+ \cdots
\end{equation}
However, integration over symplectic forms can generate new
local terms in the effective action and the renormalizability condition requires to
consider all possible relevant couplings which do not violate any fundamental gauge symmetry.
Since the symplectic gauge fields generically do
not preserve the space-time metric
\begin{equation}
D_\sigma\, {g_{\mu\nu}}\neq 0,
\end{equation}
marginally relevant terms of the form
\begin{equation} S_1'(\Gamma,\omega,g)= {{ \alpha_1}^2}\int d^4x\, \sqrt{\ g}\,
|D_\sigma D_\delta \,{ g_{\mu\nu}}|^2+ \cdots\end{equation}
should also be considered because there is no symmetry preventing its
appearance as radiative corrections.
In summary, one has to include all renormalizable possible independent couplings between gravitational field
and the symplectic gauge field. There are only six independent types of renormalizable interaction terms
\begin{equation}
DD{ g}\, DD{ g}, \
D{ g}\, D{g}\, DD{ g}, \
D{ g}\, D{g}\, D{ g}\, D{ g},\
R\, R, \
R\, DD{g}, \
R\, D{g}\, D{ g},
\label{interaction2}
\end{equation}
because all others can be expressed as linear combinations of these
terms <cit.>. However, the different contractions of the Lorentz indices
give rise to 78 different interaction terms involving 78 independent dimensionless
couplings $\alpha_1,\cdots, \alpha_{78}$: twenty two ($\alpha_{1}\dots \alpha_{22}$) of the type $ DD{ g}\ DD{ g}$, six ($\alpha_{23}\dots \alpha_{28}$)
of the type $D{ g}\ D{ g}\ D{ g}\ D{ g}$ and
fifty ($\alpha_{29}\dots \alpha_{78}$) of the type $ D{ g}\ D{ g}\ DD{ g}$.
The complete list of these terms is given by equations 22-50 in appendix A.
The corresponding theory is renormalizable. In particular, the effective
action generated by integrating out the symplectic form $\omega$ in the action $S_1$
gets non-trivial contributions to all seventy eight $\alpha$ couplings of symplectic fields with gravity.
In fact, these corrections are logarithmically divergent and the coefficients of
the corresponding contributions to the beta functions are displayed in Table 1 in appendix A.
We remark that some of the beta functions are positive and some others are negative.
This means that not all of them will be relevant in the full-fledged quantum theory. However,
the above calculations have not taken into account the radiative corrections due to
symplectic gauge field fluctuations. This calculation is beyond the scope of this paper, but it is
crucial to elucidate which couplings of the theory are finally relevant.
The above calculations show that the symplectic field theory is a renormalizable quantum field
theory, however, the appearance of four order derivative terms in the action introduces some
ghost components in the symplectic gauge theory. The absence of a larger gauge
symmetry means that unitary is not guaranteed.
§ SYMPLECTIC FIELDS AND DARK MATTER
Symplectic gauge fields as linear connections cannot interact by minimal couplings with
scalar fields, because the minimal coupling in this case reduces to
$D_\mu\phi=\partial_\mu\phi$. A similar effect
arises in the interaction with fermions. The gauge group of symplectic connections is $GL(4, \R)$ and only
the trivial representation of this group acts on spinors, i.e. there is no analogue of spin connection
for symplectic gauge fields, then $\Dsl\psi=\dsl\psi$.
Thus, the minimal coupling of symplectic gauge fields
to Standard Model particles can only be possible with gauge particles: the photon or the
intermediate gauge bosons $W^\pm$ and $Z$. However, due to the intrinsic gauge character of these particles
this coupling is not possible. The torsionless character of symplectic gauge fields is responsible for the
decoupling also of vector potentials. Indeed,
\begin{equation}F_{\mu\nu}= \partial_\mu A_\nu-\partial_\nu A_\mu + \Gamma_{\mu\nu}^\sigma A_\sigma
- \Gamma_{\nu\mu}^\sigma A_\sigma=\partial_\mu A_\nu-\partial_\nu A_\mu.\end{equation}
Thus, symplectic gauge fields cannot minimally interact
with any particle of the Standard Model. They can
only minimally couple to gravitation, whenever $D_\gamma g_{\mu\nu}\neq 0$. If the corresponding quanta
were massive particles, they will be natural candidates for
the dark matter component of the Universe and indeed, this is what happens. In the standard $\Lambda$CDM cosmological model dark matter is usually assumed to be fermionic matter. However, a bosonic component could solve some dark matter puzzles as we shall discuss below.
However, some non-minimal couplings of symplectic gauge fields with ordinary matter like
$\phantom{,}\phi^\dagger \, \partial_\nu \phi\, D_{\mu}\, g^{\mu\nu}\phantom{,}$, $|\phi|^2\, D_{\mu} D_{\nu}\, g^{\mu\nu}$ or $\bar\psi \gamma_\nu\, \psi D_\mu g^{\mu\nu}$
can arise as radiative corrections. However, the genuine interacting terms of symplectic gauge fields
with gravitation interaction2 are invariant under the signature flip transformation,
\begin{equation}
g_{\mu\nu}\to -g_{\mu\nu},
\end{equation}
which changes the signature of the metric tensor $g_{\mu\nu}$ from $(1,3)$
to $(3,1)$. This symmetry acts as a custodial symmetry which prevents the appearance of
non-minimal coupling between ordinary matter and symplectic gauge fields.
Although the coupling of symplectic gauge fields to the symplectic field $\omega$ constr breaks
the signature flip symmetry, the effects of such a symmetry breaking only affect
the couplings between ordinary matter and symplectic gauge fields
via radiative corrections at two-loop level.
The breaking of signature flip symmetry also affects the couplings of gravity to symplectic gauge fields
via one-loop corrections. We have assumed until now
that these couplings are dimensionless, however, radiative corrections generate terms of the form
\begin{equation} S_1''(\Gamma,\omega,g)=\frac1{2{ \alpha_m}^2}\int d^4x\, \sqrt{\ g}\,
|D_\sigma \,{\ g_{\mu\nu}}|^2+ \cdots.\end{equation}
Since the metric $g$ is not preserved by symplectic gauge fields nothing prevents
the appearance of these terms with mass square dimension.
Indeed, such radiative corrections appear in the form
\begin{eqnarray}\label{efectivamasa}
S_{\text{Higgs}}''&=&\int d^4x\ D_{\gamma_1} g_{\mu_1\nu_1} D_{\gamma_2} g_{\mu_2\nu_2}\ \left(\frac{1}{48}\
g^{\gamma_1\mu_1}g^{\nu_1\mu_2}g^{\gamma_2\nu_2}+\right. \nonumber\\
&&{}\left. \frac{5}{16}\ g^{\gamma_1\gamma_2}g^{\mu_1\mu_2}g^{\nu_1\nu_2}-\frac{3}{16}\
g^{\gamma_1\mu_2}g^{\mu_1\gamma_2}g^{\nu_1\nu_2} \right) I_2,
\end{eqnarray}
of quadratic divergent terms, with
\begin{equation}
I_2= \int \frac{1}{(2\pi)^4}\frac{d^4r}{r^2}.
\label{integralcuaddivergente}
\end{equation}
Thus, such terms must be included
in the bare action to ensure the renormalizability of the theory. Now, in
a Minkowski background (i.e. $g_{\mu\nu}=\eta_{\mu\nu} $) these terms
provide a real mass terms for the spin three gauge fields because then
\begin{equation}
S_{\text{Higgs}}''\approx \frac1{2{ \alpha_m}^2}\int d^4x\, \, \widetilde{T}^{\mu\nu\sigma} T_{\mu\nu\sigma}
\label{mass}
\end{equation}
i.e., although symplectic gauge fields were in principle related to massless
particles, they acquire a mass from quantum radiative corrections in
Minkowski space-time metric backgrounds. The phenomenon is reminiscent of the Coleman-Weinberg
mechanism of generation of mass for conformal scalar electrodynamics.
The way symplectic gauge fields $T_{\mu\nu\sigma} $ acquire a mass is also
reminiscent of the Higgs mechanism with the gravitational field playing the role of the Higgs
Conversely, the alternative mechanism where symplectic fields condensate into a non-trivial
value and provides a mass terms for the graviton is also possible but not physically
realistic because a non-trivial expectation value of such a field will break Lorentz
invariance which is quite unlikely to happen. As a consequence the graviton remains
massless but the symplectic fields become massive.
In a similar manner radiative corrections generate at two-loop level new interacting terms involving
symplectic fields and Higgs fields of the form
\begin{equation}
S_{\text{Higgs}}'''(\Gamma,\phi)=\frac1{2{ \alpha_h}^2}\int d^4x\, \sqrt{\ g}\,|\phi|^2
|D_\sigma \,{\ g_{\mu\nu}}|^2 +\cdots,
\end{equation}
which in a Minkowski background
provide real mass terms for the symplectic gauge fields like in equation mass
\begin{equation}
S_{\text{Higgs}}'''\approx \frac{|v|^2}{2{ \alpha_h}^2}\int d^4x\, \, \widetilde{T}^{\mu\nu\sigma} T_{\mu\nu\sigma},
\label{higgs3}
\end{equation}
where $v=<\!\phi\!> $ is the vacuum expectation value of the Higgs field.
The Higgs contribution to the mass of the symplectic gauge fields higgs3 is similar
to the mass terms of the other particles of the Standard Model. The only difference is that the mass term of
symplectic gauge fields has an extra mass contribution due to radiative corrections of symplectic fields.
§ DISCUSSION
The Standard Model sector of the Universe contains a large variety of particles. It is then
envisageable that
the dark matter sector is also made of more than one type of
particles. The characteristics of spin
three massive gauge particles associated to symplectic gauge fields
suggest that they are natural candidates as components of dark matter.
The mass of these gauge particles is only dictated by
the coupling to gravitation which means that generically it can be
large enough to provide a relevant component
of the cold dark matter. On the other hand, the bosonic character of the new particles
could explain the smooth behavior of the central dark matter density in galaxy halos
-<cit.> and it could give rise to
bosonic condensates which provide interesting scenarios for dwarf galaxies <cit.>
Since the only primary interaction of symplectic gauge fields involves gravitational
fields the effect of the new interaction can be mimicked
by a effective theory of gravitation. The results obtained via integration of symplectic gauge fields
yield an effective action which is highly non-local and it will only become local in the infinite
mass limit of symplectic gauge fields. In that case one gets back the standard gravitational action with
extra $R^2$ terms. However, the physical interpretation of the effective theory is very subtle because
the calculation is highly dependent on the background space-time metric.
There are metric backgrounds where the Higgs
mechanism provides a mass to symplectic gauge fields, and metric backgrounds without such a
mass generating mechanism. In the latter case the symplectic gauge fields contain massless particles. Thus,
the theory provides scenarios which interpolate between hot and cold dark matter scenarios
depending on the gravitational background. This chameleonic property
of symplectic gauge fields is very attractive and deserves further exploration.
§ ACKNOWLEDGMENTS
We thank J.M. Muñoz-Castañeda for discussions. J. A. acknowledges financial support
from U.S. Department of Energy Grant No. DE-SC0009932. M.A. has been
partially supported by Spanish DGIID-DGA Grant No. 2015-E24/2, Spanish
MINECO Grants No. FPA2012-35453 and No. CPAN-CSD2007-00042 and
European Cooperation in Science and Technology COST Action MP1405 QSPACE.
§ RENORMALIZATION
The 78 independent dimensionless couplings of the symplectic gauge fields to gravity
can be obtained by using Tensorial and FeynCalc packages of Mathematica.
There are three different types of terms: twenty two of the type $DD{ g}\ DD{ g}$,
\begin{equation}
\begin{array}{lll}
S'_{22}&=&\displaystyle \int\!\! d^4x \sqrt{g}\ (D_{\tau_1} D_{\gamma_1} g_{\mu_1 \nu_1})
\ (D_{\tau_2} D_{\gamma_2} g_{\mu_2 \nu_2})
\hfill\\
&\Big[& \displaystyle
{\alpha}_{1}\ \, g^{ {\mu_1} {\nu_1}}\ g^{ {\mu_2} {\nu_2}} \
g^{ {\tau_1} {\gamma_1}}\
g^{ {\tau_2} {\gamma_2}} +%&
\displaystyle
\alpha_{2}\ \, \ g^{ {\mu_1} {\nu_1}} \ g^{ {\tau_1} {\gamma_1}} \ g^{ {\gamma_2} {\nu_2}}
\ g^{ {\tau_2} {\mu_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{3}\ \, \ g^{ {\mu_1} {\tau_2}} \ g^{ {\mu_2} {\nu_2}} \ g^{ {\nu_1} {\gamma_2}}
\ g^{ {\tau_1} {\gamma_1}}
\alpha_{4}\ \, \ g^{ {\mu_1} {\tau_2}} \ g^{ {\nu_1} {\mu_2}} \ g^{ {\tau_1} {\gamma_1}}
\ g^{ {\gamma_2} {\nu_2}}\\
&+& \!\!\! \phantom{[}\displaystyle \alpha_{5}\ \, \ g^{ {\mu_1} {\gamma_2}} \ g^{ {\nu_1} {\mu_2}} \ g^{ {\tau_1} {\gamma_1}}
\ g^{ {\tau_2} {\nu_2}}
\alpha_{6}\ \, \ g^{ {\mu_1} {\mu_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\tau_1} {\gamma_1}}
\ g^{ {\tau_2} {\gamma_2}}\\
&+& \!\!\! \phantom{[}\displaystyle \alpha_{7}\ \, \ g^{ {\gamma_1} {\nu_1}} \ g^{ {\tau_1} {\mu_1}} \ g^{ {\gamma_2} {\nu_2}}
\ g^{ {\tau_2} {\mu_2}}
+ \alpha_{8}\ \, \ g^{ {\mu_2} {\nu_2}} \ g^{ {\nu_1} {\gamma_2}} \ g^{ {\gamma_1} {\tau_2}}
\ g^{ {\tau_1} {\mu_1}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{9}\ \, \ g^{ {\nu_1} {\mu_2}} \ g^{ {\gamma_1} {\tau_2}} \ g^{ {\tau_1} {\mu_1}}
\ g^{ {\gamma_2} {\nu_2}}
+ \alpha_{10}\, \ g^{ {\nu_1} {\mu_2}} \ g^{ {\gamma_1} {\gamma_2}} \ g^{ {\tau_1} {\mu_1}}
\ g^{ {\tau_2} {\nu_2}}\\
&+& \!\!\! \phantom{[}\displaystyle \alpha_{11}\, \ g^{ {\nu_1} {\tau_2}} \ g^{ {\gamma_1} {\mu_2}} \ g^{ {\tau_1} {\mu_1}}
\ g^{ {\gamma_2} {\nu_2}}
+ \alpha_{12}\, \ g^{ {\nu_1} {\gamma_2}} \ g^{ {\gamma_1} {\mu_2}} \ g^{ {\tau_1} {\mu_1}}
\ g^{ {\tau_2} {\nu_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{13}\, \ g^{ {\mu_2} {\nu_2}} \ g^{ {\nu_1} {\gamma_2}} \ g^{ {\gamma_1} {\mu_1}}
\ g^{ {\tau_1} {\tau_2}}
+ \alpha_{14}\, \ g^{ {\nu_1} {\mu_2}} \ g^{ {\gamma_1} {\mu_1}} \ g^{ {\tau_1} {\tau_2}}
\ g^{ {\gamma_2} {\nu_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{15}\, \ g^{ {\mu_1} {\nu_1}} \ g^{ {\mu_2} {\nu_2}} \ g^{ {\gamma_1} {\gamma_2}}
\ g^{ {\tau_1} {\tau_2}}
+ \alpha_{16}\, \ g^{ {\mu_1} {\mu_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\gamma_1} {\gamma_2}}
\ g^{ {\tau_1} {\tau_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{17}\, \ g^{ {\mu_1} {\gamma_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\gamma_1} {\mu_2}}
\ g^{ {\tau_1} {\tau_2}}
+ \alpha_{18}\, \ g^{ {\mu_1} {\mu_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\gamma_1} {\tau_2}}
\ g^{ {\tau_1} {\gamma_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{19}\, \ g^{ {\mu_1} {\tau_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\gamma_1} {\mu_2}}
\ g^{ {\tau_1} {\gamma_2}}
+ \alpha_{20}\, \ g^{ {\nu_1} {\tau_2}} \ g^{ {\gamma_1} {\mu_1}} \ g^{ {\tau_1} {\mu_2}}
\ g^{ {\gamma_2} {\nu_2}}\\
&+& \!\!\! \phantom{[} \displaystyle \alpha_{21}\, \ g^{ {\mu_1} {\tau_2}} \ g^{ {\nu_1} {\nu_2}} \ g^{ {\gamma_1} {\gamma_2}}
\ g^{ {\tau_1} {\mu_2}}
+ {a}_{22}\, \ g^{ {\mu_1} {\tau_2}} g^{ {\nu_1} {\gamma_2}} g^{ {\gamma_1} {\nu_2}}
\ g^{ {\tau_1} {\mu_2}}\Big],
\end{array}
\label{22}
\end{equation}
of the type $D{ g}\ D{ g}\ D{ g}\ D{ g}$,
\begin{equation}
\begin{array}{lll}
S'_{6}=\displaystyle \int d^4x &&\!\!\!\!\!\!\!\!\!\!\!\!\!\sqrt{g}\ (D_{\gamma_1}\ g_{\mu_1\nu_1})\ (D_{\gamma_2}\
g_{\mu_2\nu_2})\ (D_{\gamma_3}\ g_{\mu_3\nu_3})\ (D_{\gamma_4}\ g_{\mu_4\nu_4})\\
&\Big[&\alpha_{23}\ \ g^{\mu_2 {\gamma_3}} \ g^{\nu_1 {\gamma_2}} \ g^{\nu_2\mu_3}
\ g^{\nu_3\mu_4} \ g^{ {\gamma_1}\mu_1} \ g^{ {\gamma_4}\nu_4}\\
\alpha_{24}\ \ g^{\mu_2 {\gamma_3}} \ g^{\mu_3 {\gamma_4}} \ g^{\nu_1 {\gamma_2}}
\ g^{\nu_2\mu_4} \ g^{\nu_3\nu_4} \ g^{ {\gamma_1}\mu_1}\\
\alpha_{25}\ \ g^{\mu_2\mu_3} \ g^{\nu_1 {\gamma_2}} \ g^{\nu_2\mu_4}
\ g^{ {\gamma_1}\mu_1} \ g^{ {\gamma_3}\nu_3} \ g^{ {\gamma_4}\nu_4}\\
\alpha_{26}\ \ g^{\nu_1\mu_2} \ g^{\nu_3\mu_4} \ g^{ {\gamma_1}\mu_1}
\ g^{ {\gamma_2}\nu_2} \ g^{ {\gamma_3}\mu_3} \ g^{ {\gamma_4}\nu_4}\\
\alpha_{27}\ \ g^{\nu_1\mu_2} \ g^{\nu_2\mu_3} \ g^{\nu_3\mu_4}
\ g^{ {\gamma_1}\mu_1} \ g^{ {\gamma_2} {\gamma_3}} \ g^{ {\gamma_4}\nu_4}\\
\alpha_{28}\ \ g^{\nu_1\mu_2} \ g^{\nu_2\mu_4} \ g^{\nu_3\nu_4}
\ g^{ {\gamma_1}\mu_1} \ g^{ {\gamma_2}\mu_3} \ g^{ {\gamma_3} {\gamma_4}}
\Big],
\end{array}
\label{6}
\end{equation}
and fifty of the type $ D{ g}\ D{ g}\ DD{ g}$,
\begin{equation}
\begin{array}{lllll}
\!\!\! S'_{50}&=&\displaystyle \int d^4x\ \sqrt{g}\ (D_{\gamma_1} g_{\mu_1 \nu_1}) \,(D_{\gamma_2} g_{\mu_2 \nu_2})&\!\!\! \!\!&\!\!\!\!(D_{\tau_3} D_{\gamma_3} g_{\mu_3 \nu_3}) \\
\alpha_{29} \ g^{ {\mu_2} {\tau_3}} g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\gamma_2}}
g^{ {\nu_2} {\gamma_3}} g^{ {\gamma_1} {\mu_1}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{30}\ \ g^{ {\mu_2} {\tau_3}} g^{ {\nu_1} {\gamma_2}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_3} {\nu_3}}\\
& + & \!\! \alpha_{31}\ g^{ {\mu_2} {\gamma_3}} g^{ {\nu_1} {\gamma_2}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\tau_3} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{32}\ \ g^{ {\mu_2} {\mu_3}} g^{ {\nu_1} {\gamma_2}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\tau_3} {\gamma_3}}
\\
& + & \!\! \alpha_{33}\ g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\mu_2}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\nu_2}} g^{ {\tau_3} {\gamma_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{34}\ \ g^{ {\nu_1} {\mu_2}} g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\nu_2}}
g^{ {\gamma_3} {\nu_3}} g^{ {\tau_3} {\mu_3}}\\
& +& \!\! \alpha_{35}\ g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\tau_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{36}\ \ g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\mu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\tau_3}} g^{ {\gamma_3} {\nu_3}} \\
& +&\!\! \alpha_{37}\ g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\mu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\gamma_3}} g^{ {\tau_3} {\nu_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+& \!\!\!\!\!\!\!\! \alpha_{38}\ g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\tau_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_3}} g^{ {\gamma_3} {\nu_3}}\\
& +&\!\! \alpha_{39}\ g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\gamma_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_3}} g^{ {\tau_3} {\nu_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{40}\ g^{ {\nu_1} {\mu_2}} g^{ {\nu_2} {\nu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_3}} g^{ {\tau_3} {\gamma_3}} \\
& +& \!\! \alpha_{41}\ g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\mu_2}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+&\!\!\!\!\!\!\!\! \alpha_{42}\ g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\mu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_2}} g^{ {\gamma_3} {\nu_3}} \\
& +& \!\! \alpha_{43}\ g^{ {\mu_2} {\mu_3}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\gamma_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{44}\ g^{ {\mu_2} {\gamma_3}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\mu_3}} \\
& +&\!\! \alpha_{45}\ g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\mu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_2}} g^{ {\tau_3} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{46}\ g^{ {\mu_2} {\mu_3}} g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\tau_3}} \\
& +& \!\! \alpha_{47}\ g^{ {\mu_2} {\tau_3}} g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\mu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{48}\ g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}} g^{ {\gamma_1} {\mu_1}}
g^{ {\gamma_2} {\mu_2}} g^{ {\tau_3} {\gamma_3}}\\
& +&\!\! \alpha_{49}\ g^{ {\mu_2} {\gamma_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\tau_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+& \!\!\!\!\!\!\!\! \alpha_{50}\ g^{ {\mu_2} {\tau_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\gamma_3}} \\
& +&\!\! \alpha_{51}\ g^{ {\mu_2} {\tau_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_1}} g^{ {\gamma_2} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{52}\ g^{ {\mu_1} {\mu_2}} g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\tau_3}}
g^{ {\nu_2} {\gamma_3}} g^{ {\gamma_1} {\gamma_2}} \\
& +&\!\! \alpha_{53}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\gamma_2}} g^{ {\gamma_3} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{54}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\gamma_2}} g^{ {\tau_3} {\nu_3}} \\
& +& \!\! \alpha_{55}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\gamma_2}} g^{ {\tau_3} {\gamma_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{56} \,\, g^{ {\mu_1} {\tau_3}} g^{ {\mu_2} {\mu_3}} g^{ {\nu_1} {\gamma_3}}
g^{ {\nu_2} {\nu_3}} g^{ {\gamma_1} {\gamma_2}} \\
& +&\!\! \alpha_{57}\ g^{ {\mu_1} {\tau_3}} g^{ {\mu_2} {\gamma_3}} g^{ {\nu_1} {\mu_3}}
g^{ {\nu_2} {\nu_3}} g^{ {\gamma_1} {\gamma_2}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{58}\ g^{ {\mu_1} {\gamma_2}} g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\nu_2}}
g^{ {\gamma_1} {\mu_2}} g^{ {\tau_3} {\gamma_3}}\\
& +&\!\! \alpha_{59}\ g^{ {\mu_1} {\gamma_2}} g^{ {\nu_1} {\nu_2}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_3} {\nu_3}} g^{ {\tau_3} {\mu_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{60}\ g^{ {\mu_1} {\gamma_2}} g^{ {\mu_3} {\nu_3}} g^{ {\nu_1} {\tau_3}}
g^{ {\nu_2} {\gamma_3}} g^{ {\gamma_1} {\mu_2}} \\
& +& \!\! \alpha_{61}\ g^{ {\mu_1} {\gamma_2}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_3} {\nu_3}}
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{62}\ g^{ {\mu_1} {\gamma_2}} g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\mu_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\tau_3} {\nu_3}} \\
& +&\!\! \alpha_{63}\ g^{ {\mu_1} {\gamma_2}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\tau_3} {\gamma_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{64}\ g^{ {\mu_1} {\nu_2}} g^{ {\nu_1} {\tau_3}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_2} {\mu_3}} g^{ {\gamma_3} {\nu_3}}\\
& +&\!\! \alpha_{65}\ g^{ {\mu_1} {\nu_2}} g^{ {\nu_1} {\gamma_3}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_2} {\mu_3}} g^{ {\tau_3} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{66}\ g^{ {\mu_1} {\nu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_2} {\tau_3}} g^{ {\gamma_3} {\nu_3}}\\
& +& \!\! \alpha_{67}\ g^{ {\mu_1} {\nu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_2} {\gamma_3}} g^{ {\tau_3} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{68}\ g^{ {\mu_1} {\nu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\gamma_1} {\mu_2}}
g^{ {\gamma_2} {\nu_3}} g^{ {\tau_3} {\gamma_3}} \\
& +& \!\! \alpha_{69}\ g^{ {\mu_1} {\tau_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_2} {\gamma_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +& \!\!\!\!\!\!\!\! \alpha_{70}\ g^{ {\mu_1} {\tau_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_2} {\nu_3}}\\
& +& \!\! \alpha_{71}\ g^{ {\mu_1} {\gamma_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_2} {\tau_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{72}\ g^{ {\mu_1} {\gamma_3}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\tau_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_2} {\nu_3}}\\
& +&\!\! \alpha_{73}\ g^{ {\mu_1} {\mu_3}} g^{ {\nu_1} {\nu_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_2}} g^{ {\gamma_2} {\tau_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{74}\ g^{ {\mu_1} {\tau_3}} g^{ {\mu_2} {\nu_3}} g^{ {\nu_1} {\gamma_3}}
g^{ {\gamma_1} {\nu_2}} g^{ {\gamma_2} {\mu_3}} \\
& +&\!\! \alpha_{75}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\gamma_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\tau_3}} g^{ {\gamma_2} {\mu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{76}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\nu_3}}
g^{ {\gamma_1} {\tau_3}} g^{ {\gamma_2} {\gamma_3}}\\
& +&\!\! \alpha_{77}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\mu_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\tau_3}} g^{ {\gamma_2} {\nu_3}}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +&\!\!\!\!\!\!\!\! \alpha_{78}\ g^{ {\mu_1} {\mu_2}} g^{ {\nu_1} {\tau_3}} g^{ {\nu_2} {\gamma_3}}
g^{ {\gamma_1} {\mu_3}} g^{ {\gamma_2} {\nu_3}}\!
\Big].
\label{50}
\end{array}
\end{equation}
Integration over the symplectic fields $\omega$ in the action $S_1$
generates logarithmically divergent contributions to all
$\alpha$ couplings. The coefficients of these divergent terms can be identified with the coefficients of the
beta functions of $\alpha$ couplings displayed in the Table 1.
\beta_{{1}} = -\frac{11}{320} $
$ %\displaystyle
\beta_{{2}} =\hfill -\frac{9019}{15360} $
\beta_{{3}} = \hfill \frac{1103}{6144} $
$ %\displaystyle
\beta_{{4}} = \hfill -\frac{569}{3840} $
$ %\displaystyle
\beta_{{5}} = \hfill -\frac{221}{640} $
$ %\displaystyle
\beta_{{6}} = \hfill \frac{151}{1536} $
$ %\displaystyle
\beta_{{7}} = \hfill -\frac{811}{7680} $
$ %\displaystyle
\beta_{{8}} = \hfill -\frac{2173}{7680} $
$ %\displaystyle
\beta_{{9}} = \hfill \frac{569}{3840} $
$ %\displaystyle
\beta_{{10}} = \hfill -\frac{481}{3840} $
$ %\displaystyle
\beta_{{11}} = \hfill -\frac{1}{24} $
$ %\displaystyle
\beta_{{12}} = \hfill \frac{509}{3840} $
$ %\displaystyle
\beta_{{13}} = \hfill \frac{1733}{2560} $
$ %\displaystyle
\beta_{{14}} = \hfill -\frac{1}{32} $
$ %\displaystyle
\beta_{{15}} = \hfill -\frac{959}{10240} $
$ %\displaystyle
\beta_{{16}} = \hfill \frac{5}{96} $
$ %\displaystyle
\beta_{{17}} = \hfill -\frac{5}{96} $
$ %\displaystyle
\beta_{{18}} = \hfill -\frac{125}{512} $
$ %\displaystyle
\beta_{{19}} = \hfill \frac{983}{1920} $
$ %\displaystyle
\beta_{{20}} = \hfill \frac{811}{7680} $
$ %\displaystyle
\beta_{{21}} = \hfill \frac{161}{3840} $
$ %\displaystyle
\beta_{{22}} = \hfill -\frac{143}{1280} $
$ %\displaystyle
\beta_{{23}} = \hfill \frac{353}{1024} $
$ %\displaystyle
\beta_{{24}} = \hfill -\frac{77}{7680} $
$ %\displaystyle
\beta_{{25}} = \hfill -\frac{6691}{30720} $
$ %\displaystyle
\beta_{{26}} = \hfill -\frac{1}{80} $
$ %\displaystyle
\beta_{{27}} = \hfill -\frac{601}{1280} $
$ %\displaystyle
\beta_{{28}} = \hfill \frac{1981}{1920} $
$ %\displaystyle
\beta_{{29}} = \hfill \frac{3299}{10240} $
$ %\displaystyle
\beta_{{30}} = \hfill \frac{121}{480} $
$ %\displaystyle
\beta_{{31}} = \hfill \frac{8977}{15360} $
$ %\displaystyle
\beta_{{32}} = \hfill \frac{151}{1280} $
$ %\displaystyle
\beta_{{33}} = \hfill \frac{2283}{5120} $
$ %\displaystyle
\beta_{{34}} = \hfill \frac{1083}{2560} $
$ %\displaystyle
\beta_{{35}} = \hfill \frac{293}{3840} $
$ %\displaystyle
\beta_{{36}} = \hfill -\frac{13}{960} $
$ %\displaystyle
\beta_{{37}} = \hfill -\frac{1447}{3840} $
$ %\displaystyle
\beta_{{38}} = \hfill \frac{4909}{15360} $
$ %\displaystyle
\beta_{{39}} = \hfill -\frac{15437}{30720} $
$ %\displaystyle
\beta_{{40}} = \hfill -\frac{169}{960} $
$ %\displaystyle
\beta_{{41}} = \hfill -\frac{1807}{15360} $
$ %\displaystyle
\beta_{{42}} = \hfill -\frac{95}{256} $
$ %\displaystyle
\beta_{{43}} = \hfill \frac{187}{7680} $
$ %\displaystyle
\beta_{{44}} = \hfill \frac{121}{160} $
$ %\displaystyle
\beta_{{45}} = \hfill -\frac{8459}{7680} $
$ %\displaystyle
\beta_{{46}} = \hfill -\frac{101}{384} $
$ %\displaystyle
\beta_{{47}} = \hfill -\frac{89}{96} $
$ %\displaystyle
\beta_{{48}} = \hfill -\frac{769}{7680} $
$ %\displaystyle
\beta_{{49}} = \hfill \frac{167}{15360} $
$ %\displaystyle
\beta_{{50}} = \hfill \frac{8647}{30720} $
$ %\displaystyle
\beta_{{51}} = \hfill \frac{349}{3840} $
$ %\displaystyle
\beta_{{52}} = \hfill \frac{2449}{3840} $
$ %\displaystyle
\beta_{{53}} = \hfill \frac{3323}{15360} $
$ %\displaystyle
\beta_{{54}} = \hfill -\frac{6377}{15360} $
$ %\displaystyle
\beta_{{55}} = \hfill \frac{271}{320} $
$ %\displaystyle
\beta_{{56}} = \hfill \frac{1921}{3072} $
$ %\displaystyle
\beta_{{57}} = \hfill -\frac{3407}{15360} $
$ %\displaystyle
\beta_{{58}} = \hfill -\frac{457}{768} $
$ %\displaystyle
\beta_{{59}} = \hfill \frac{629}{15360} $
$ %\displaystyle
\beta_{{60}} = \hfill -\frac{57}{512} $
$ %\displaystyle
\beta_{{61}} = \hfill \frac{61}{640} $
$ %\displaystyle
\beta_{{62}} = \hfill \frac{1453}{7680} $
$ %\displaystyle
\beta_{{63}} = \hfill -\frac{695}{3072} $
$ %\displaystyle
\beta_{{64}} = \hfill -\frac{55}{64} $
$ %\displaystyle
\beta_{{65}} = \hfill -\frac{33}{2560} $
$ %\displaystyle
\beta_{{66}} = \hfill -\frac{513}{5120} $
$ %\displaystyle
\beta_{{67}} = \hfill \frac{73}{960} $
$ %\displaystyle
\beta_{{68}} = \hfill -\frac{351}{2560} $
$ %\displaystyle
\beta_{{69}} = \hfill \frac{1}{2} $
$ %\displaystyle
\beta_{{70}} = \hfill \frac{203}{960} $
$ %\displaystyle
\beta_{{71}} = \hfill \frac{7253}{15360} $
$ %\displaystyle
\beta_{{72}} = \hfill \frac{1753}{15360} $
$ %\displaystyle
\beta_{{73}} = \hfill -\frac{15}{64} $
$ %\displaystyle
\beta_{{74}} = \hfill -\frac{6607}{15360} $
$ %\displaystyle
\beta_{{75}} = \hfill -\frac{1417}{1920} $
$ %\displaystyle
\beta_{{76}} = \hfill \frac{2603}{7680} $
$ %\displaystyle
\beta_{{77}} = \hfill -\frac{329}{640} $
$ %\displaystyle
\beta_{{78}} = \hfill \frac{1309}{2560} $
Beta function coefficients of gravitational $\alpha$ couplings of symplectic gauge fields
The fact that no new couplings are generated by one loop diagrams
points out the renormalizable character of the theory.
The couplings whose beta function coefficients are listed in Table 1
are the only dimensionless renormalized couplings of the theory.
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|
1511.00590
|
A matching $M$ of a graph $G$ is maximal if it is not a proper subset of any other
matching in $G$. Maximal matchings are much less known and researched than
their maximum and perfect counterparts. In particular, almost nothing is known about
their enumerative properties. In this paper
we present the recurrences and generating functions for the sequences
enumerating maximal matchings in two classes of chemically interesting
linear polymers: polyspiro
chains and benzenoid chains. We also analyze the asymptotic behavior of those
sequences and determine the extremal cases.
Keywords: maximal matching; benzenoid chain; polyspiro chain
§ INTRODUCTION
A matching in a graph is a collection of its edges such that no two edges
in this collection have a vertex in common.
Matchings in graphs serve as successful models of many phenomena in engineering,
natural and social sciences. A strong initial impetus to their
study came from the chemistry of benzenoid compounds after it was observed
that the stability of benzenoid compounds is related to the existence and
the number of
perfect matchings in the corresponding graphs. That observation gave rise to a
number of enumerative results that were accumulated over the course of several
decades; we refer the reader to monograph <cit.> for a survey.
Further motivation came from
the statistical mechanics via the Kasteleyn's solution of the dimer
problem <cit.> and its applications to evaluations of partition
functions for a given value of temperature. In both cases, the matchings under
consideration are perfect, i.e., their edges are collectively incident to
all vertices of $G$. It is clear that perfect matchings are as large as
possible and that no other matching in $G$ can be “larger” than a perfect
one. It turns out that in all other applications we are also interested
mostly in large matchings.
Basically, there are two ways to quantify the largeness of a matching. One way,
by using the number of edges, gives rise to the idea of maximum
matchings. Maximum matchings are well researched and well understood; there is
a well developed structural theory and enumerative results are abundant. The
classical monograph by Lóvasz and Plummer <cit.> is an excellent
reference for all aspects of the theory.
An alternative way is to say that a matching is large if no other matching
contains it as a proper subset; this gives rise to the concept of
maximal matchings. Every maximum matching is also maximal, but the
opposite is usually not true. Unlike their maximum counterparts, maximal
matchings can have different cardinalities (unless the graph is
equimatchable; see <cit.>) and the recurrences used for
their enumeration are essentially non-local. As a consequence, maximal matchings
are much less understood then the maximum ones. There is nothing analogous
to the structural theory of maximum matchings and the enumerative results
are scarce and scattered through the literature <cit.>.
In spite of their obscurity, maximal matchings are natural models for several
problems connected with adsorption of dimers on a structured substrate and
block-allocation of a sequential resource. One can find them also in the
context of polymerization of organic molecules, as witnessed by an early
paper of Flory <cit.>. A probabilistic approach to the same problem
can be found in <cit.>. We refer the reader to papers
<cit.> for some structural and enumerative
results on those models. In this paper our goal is to further the line of
research of reference <cit.> by considering graphs with more
complicated connectivity patterns and richer structure of basic units.
We provide enumerative and extremal results on maximal matchings
in two classes of linear polymers of chemical interest: the polyspiro chains
and benzenoid chains. We extablish the recurrences and generating
functions for the enumerating sequences of maximal matchings in three classes
of uniform polyspiro chains and use the obtained results to determine the
asymptotic behavior and to find the extremal chains. Further, we also
enumerate maximal matchings in three classes of benzenoid chains and show
that one of them is extremal with respect to the number of maximal matchings.
Our results show that maximal matchings behave in a radically different
way that the perfect matchings; chains rich in maximal matchings are poor
in perfect matchings and vice versa. We end by comparing our results with
enumerative results for other type of structures in similar polymers and
by discussing some possible directions of future research.
§ PRELIMINARIES
Our terminology and notations are mostly standard and taken from
<cit.>. All graphs $G$ considered in this paper will be finite and simple, with vertex set $V(G)$ and edge set $E(G)$. For a subset of vertices $S$ of $V(G)$, we make use of the notation $G - S$ (or $G-v$ if $S=\{v\}$) to denote the subgraph of $G$ obtained by deleting the vertices of $S$ and all edges incident to them. For a graph $G$ and subset of edges $X$ of $G$, we use the notation $G\setminus X$ (or $G\setminus e$ if $X=\{e\}$) to denote the subgraph of $G$ obtained by deleting the endpoints of the edges in $X$ as well as all incident edges to these endpoints.
A matching $M$ in G is a set of edges of $G$ such that no two edges from $M$ have a vertex in common. The number of edges in $M$ is called its size. A matching in $G$ with the largest possible size is called a maximum matching. If a matching in $G$ is not a subset of a larger matching of $G$, it is called a maximal matching. Let $\Psi(G)$ denote the number of maximal matchings of $G$.
In this paper we are mainly concerned with counting maximal matchings in two classes of linear polymers (or facsiagraphs, <cit.>) with simple connectivity patterns. The first class are $6$-uniform cactus chains. Chain cacti are in chemical literature known as polyspiro chains.
A cactus graph is a connected graph in which no edge lies in more than one cycle. Consequently, each block of a cactus graph is either an edge or a cycle. If all blocks of a cactus $G$ are cycles of the same length $m$, the cactus is $m$-uniform.
A hexagonal cactus is a 6-uniform cactus, i.e., a cactus in which every block is a hexagon. A vertex shared by two or more hexagons is called a cut-vertex. If each hexagon of a hexagonal cactus $G$ has at most two cut-vertices, and each cut-vertex is shared by exactly two hexagons, we say that $G$ is a chain hexagonal cactus. The number of hexagons is called the length of the chain. An example of a chain hexagonal cactus is shown in Figure <ref>.
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A chain hexagonal cactus of length 6.
Furthermore, any chain hexagonal cactus of length greater than one has exactly two hexagons with only one cut-vertex; such hexagons are called terminal and all other hexagons with two cut-vertices are called internal.
Internal hexagons can be one of three types depending upon the distance between its cut-vertices: in an ortho-hexagon cut vertices are adjacent, in a meta-hexagon they are at distance two, and in a para-hexagon cut-vertices are at distance three. The terminology is borrowed from the theory of benzenoid hydrocarbons; see <cit.> for more details. These give rise to the following three types of hexagonal cactus chains of length $n$: the unique chain whose internal hexagons are all para-hexagons is $P_n$, the unique chain whose internal hexagons are all meta-hexagons is $M_n$, and the unique chain whose internal hexagons are all ortho-hexagons is $O_n$.
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The hexagonal cactus chains $P_n$, $M_n$, and $O_n$.
The other class of unbranched polymers we consider are benzenoid chains. A benzenoid system is a is a connected, plane graph without cut-vertices in which all faces, except the unbounded one, are hexagons. Two hexagonal faces are either disjoint or they share exactly one common edge (adjacent hexagons). A vertex of a benzenoid system belongs to at most three hexagonal faces and the benzenoid system is called catacondensed if it does not posses such a vertex. If no hexagon in a catacondensed benzenoid is adjacent to three other hexagons, we say that the benzenoid is a chain see Figure <ref>.
The number of hexagons in a benzenoid chain is called its length. In each benzenoid chain there are exactly two hexagons adjacent to one other hexagon; those two hexagons are called terminal, while any other hexagons are called interior. An interior hexagon has two vertices of degree 2. If these two vertices are not adjacent, then hexagon is called straight. If the two vertices are adjacent, then the hexagon is called kinky.
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A benzenoid chain of length 6.
If all $n-2$ interior hexagons of a benzenoid chain with $n$ hexagons are straight, we call the chain a polyacene and denote it by $L_n$. If all interior hexagons are kinky, the chain is called a polyphenacene. Since the number of perfect matchings in a polyphenacene of length $n$ is equal to the $(n + 2)$-nd Fibonacci number $F_{n+2}$, these chains are also known as fibonacenes <cit.>. We consider two specific families of polyphenacenes depicted in Figure <ref>: the zig-zag polyphenacene, $Z_n$, and helicene, $H_n$.
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The polyacene, zig-zag polyphenacene, and helicene chains.
§ CHAIN HEXAGONAL CACTI
§.§ Generating functions
In this section, we obtain ordinary generating functions for the number of maximal matchings in the hexagonal chain cacti $P_n$, $M_n$, and $O_n$. To do this, we first find recursions for the number of maximal matchings using auxiliary graphs (initial conditions are obtained by direct counting). These recursions can be verified via casework. By introducing generating functions for the number of maximal matchings in each auxiliary graph, the recursions can be transformed into a solvable system of equations in terms of unknown generating functions. Finally, we solve this system of equations for the desired generating function. We omit the details of most of these computations.
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Auxiliary graphs for $P_n$.
Let $p_n$ be the number of maximal matchings in $P_n$ and $p^i_n$ be the number of maximal matchings in the auxiliary graph $P^i_n$ in Figure <ref>. Then
$(i)$ $p_n = 2p^1_{n-1} + p_{n-1}$,
$(ii)$ $p^1 _n = p^2_{n} + p^3_{n-1}$,
$(iii)$ $p^2 _n = p^3_{n-1} + 2p^1_{n-1}$,
$(iv)$ $p^3 _n = p_n + 2p^3_{n-1}$,
with the initial conditions $p_0=1$, $p^1_0=2$, $p^2_0=1$, and $p^3_0=3$.
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Let $m_n$ be the number of maximal matchings in $M_n$ and $m^i_n$ be the number of maximal matchings in the auxiliary graph $M^i_n$ in Figure <ref>. Then
$(i)$ $m_n = 2m^1_{n-1} + m_{n-1}$,
$(ii)$ $m^1 _n = m^2_{n} + m^3_{n-1}$,
$(iii)$ $m^2 _n = m^3_{n-1} + m^1_{n-1} + m^2_{n-1} + m_{n-1}$,
$(iv)$ $m^3 _n = 2m^3_{n-1} + m^1_{n-1} + m^2_{n-1} + m_{n-1} + m^2_{n}$,
with the initial conditions $m_0=1,$ $m^1_0=2,$ $m^2_0=1,$ and $m^3_0=3$.
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Let $o_n$ be the number of maximal matchings in $O_n$ and $o^i_n$ be the number of maximal matchings in the auxiliary graph $O^i_n$ in Figure <ref>. Then
$(i)$ $o_n = 2o^1_{n-1} + o_{n-1}$,
$(ii)$ $o^1 _n = o^2_{n} + o^3_{n-1}$,
$(iii)$ $o^2 _n = o^3_{n-1} + o^2_{n-1} + o_{n-1} + 2o^3_{n-2}$,
$(iv)$ $o^3 _n = o_n + o^3_{n-1} + o^2_n$,
with the initial conditions $o_0=1$, $o^1_0=2$, $o^2_0=1$, $o^2_1=7$, and $o^3_0=3$.
Let $P(x)$, $M(x)$, and $O(x)$ be the ordinary generating functions for the sequences $p_n$, $m_n$, and $o_n$, respectively. Then
P(x) = \frac{1 + 4x^2}{1 - 5x +4x^2-4x^3},
M(x) = \frac{1 - x - 2x^2}{1-6x+3x^2-2x^3},
O(x) = \frac{1 + x + x^2}{1 - 4x - 4x^2 - x^3}.
Since $P(x)$, $M(x)$, and $O(x)$ are rational functions, we can conclude that the numbers $p_n$, $m_n$, and $o_n$ each satisfy a third order linear recurrence with constant coefficients. The initial conditions can be verified by direct computations.
$(i)$ $p_n = 5p_{n-1}-4p_{n-2}+4p_{n-3}$
with initial conditions $p_0 = 1$, $p_1 = 5$, $p_2 = 25$,
$(ii)$ $m_n = 6m_{n-1} - 3m_{n-2} + 2m_{n-3}$
with initial conditions $m_0 = 1$, $m_1 = 5$, $m_2 = 25$,
$(iii)$ $o_n = 4o_{n-1} + 4o_{n-2} + o_{n-3}$
with initial conditions $o_0 = 1$, $o_1 = 5$, $o_2 = 25$.
None of the obtained sequences appear in The On-Line Encyclopedia of Integer Sequences <cit.>.
Now we can apply a version of Darboux's theorem to deduce the asymptotic behavior of the sequences $p_n$, $m_n$, and $o_n$. We refer the reader to any of standard books on generating functions, such as <cit.> for more information on these techniques.
Let $f(x) = \sum _{n=0}^\infty a_nx^n$ denote the ordinary generating function of a sequence $a_n$. If $f(x)$ can be written as
f(x) = \left( 1 - \frac{x}{w} \right)^\alpha g(x),
where $w$ is the smallest modulus singularity of $f$ and $g$ is analytic at $w$, then
a_n \sim \frac{g(w)}{\Gamma (-\alpha)} w^{-n}n^{-\alpha - 1}.
Here $\Gamma (x)$ denotes the gamma function.
$(i)$ $p_n \sim 1.37804 \cdot 4.28428^n$,
$(ii)$ $m_n \sim 0.81408 \cdot 5.52233^n$,
$(iii)$ $o_n \sim 1.05177 \cdot 4.86454^n$.
The characteristic equations of the three recurrences can be solved exactly,
but the resulting formulas tend to be too cumbersome to be of any use.
The equation for meta-chains, however, allows a compact formula for the
smallest (and the only) positive root: it is equal to $\frac{1}{2}(1 +
\sqrt[3]3 - \sqrt [3]9)$.
The obtained asymptotics suggest that meta-chains could be the richest and
para-chains the poorest in maximal matchings among all polyspiro chains of the
same length. In the next subsection we prove that this is, indeed, the case.
§.§ Extremal structures
Let $G_n$ be a hexagonal cactus of length $n$. Then
\Psi (P_n) \le \Psi (G_n) \le \Psi (M_n).
Let $G_m$ be an arbitrary hexagonal cactus of length $m$. Observe that we can always draw $G_m$ as in Figure <ref>, where $h_m$ is a terminal hexagon and the hexagon adjacent to the left of $h_{m-1}$ may attach at any of the vertices $b, a, k, j,$ or $i$. Let us assume the hexagons of $G_m$ are labeled $h_1, \ldots, h_m$ according to their ordering in Figure <ref> where ($h_1$ is the other terminal hexagon).
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A terminal hexagon, $h_m$, and its adjacent hexagon, $h_{m-1}$, in the hexagonal chain cactus $G_m$.
In what follows, for $1 \le \ell, p \le m$ let $H_\ell$ be the subgraph of $G_m$ induced by the vertices of the hexagons $h_1, \ldots, h_\ell$ and let $H_{\ell,p}$ denote the subgraph of $G_m$ induced by the vertices of the two hexagons $h_\ell$ and $h_p$. We will need the following lemmas. The proof of the first lemma is immediate.
If $H$ is a subgraph of the graph $G$, then $\Psi(H) \le \Psi (G)$.
Any maximal matching in $G_m$ must contain exactly one of the edges $cb, cd, ch,$ or $ci$, or the maximal matching must contain all the edges $ab, de, ji,$ and $hg$.
Take a maximal matching $M$ in $G_m$. For sake of contradiction, suppose that $M$ does not contain any of the edges $cb$, $cd$, $ch$, or $ci$ and that $M$ does not contain all of the edges $ab$, $de$, $ji$, and $hg$. Then at least one of the edges $ab$, $de$, $ji$, and $hg$ is missing, say $ab$. Since $ab$ is not in $M$, then we can add the edge $bc$ to $M$, which is a contradiction to the fact that $M$ is a maximal matching. The lemma follows.
For the subgraph $H_{m-1}$ of $G_m$, at least one of the following holds:
$(i)$ $2 \cdot \Psi (H_{m-1} - \{b,c\}) \ge \Psi (H_{m-1} - c)$
$(ii)$ $2 \cdot \Psi (H_{m-1} - \{c,i\}) \ge \Psi (H_{m-1} - c)$
The proof depends on where the hexagon $h_{m-2}$ attaches to $h_{m-1}$. By symmetry, suppose that $h_{m-2}$ attaches at either $i, j,$ or $k$ (the case $a,b,k$ is similar). Consider a maximal matching of $H_{m-1}-c$. If such a matching contains the edge $ab$, then the remaining edges give a maximal matching of $H_{m-1} - \{a,b,c\}$. If a maximal matching does not contain the edge $ab$, then the matching must also be maximal in the graph $H_{m-1} - \{b,c\}$. Thus by Lemma <ref> we have
\begin{align*}
\Psi (H_{m-1} - \{c\}) &= \Psi (H_{m-1} - \{a,b,c\}) + \Psi (H_{m-1} - \{b,c\}) \\
&\le 2 \cdot \Psi (H_{m-1} - \{b,c\}).
\end{align*}
Take a hexagonal cactus $C$ of length $n-1$. Let us set $m=n-1$ and suppose that $C$ is drawn as in Figure <ref> with vertices labeled as such, so that we may refer to this picture to aid this proof. We consider three cases of extending $C$ by an $n$th hexagon $h_n$.
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The hexagonal cactus CP.
Case 1. The hexagon $h_n$ attaches in the para position to the vertex $f$ and let us denote the resulting graph by $CP$, see Figure <ref>. To compute $\Psi (CP)$ we make use of Lemma <ref>. Consider maximal matchings in $CP$ containing the edge $bc$. The remaining edges of the matching must be a maximal matching of $H_{n-2}-\{b,c\}$ and a maximal matching of $H_{n-1,n}-c$. By direct counting, we find that $\Psi (H_{n-1,n}-c) = 11$ and hence, the number of maximal matchings containing the edge $bc$ is $11\cdot \Psi(H_{n-2}-\{b,c\})$. We count the maximal matchings containing the edges $ci, cd,$ or $ch$ as well as the maximal matchings containing all the edges $ab, de, ji,$ and $hg$ similarly, to obtain
\begin{align*}
\Psi (CP) = & 11(\Psi(H_{n-2}-\{b,c\}) + \Psi(H_{n-2}-\{c,i\})) + 20\cdot \Psi (H_{n-2} - c) \\
&+ 5\cdot \Psi (H_{n-2}-\{a,b,c,i,j\}).
\end{align*}
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The hexagonal cactus CM.
Case 2. The hexagon $h_n$ attaches in the meta position to the vertex $e$ and let us denote the resulting graph by $CM$, see Figure <ref>. Counting similarly to Case 1 above we obtain
\begin{align*}
\Psi (CM) = & 17(\Psi(H_{n-2}-\{b,c\}) + \Psi(H_{n-2}-\{c,i\})) + 22\cdot \Psi (H_{n-2} - c) \\
&+ 3\cdot \Psi (H_{n-2}-\{a,b,c,i,j\}).
\end{align*}
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The hexagonal cactus CO.
Case 3. The hexagon $h_n$ attaches in the ortho position to the vertex $d$ and let us denote the resulting graph by $CO$, see Figure <ref>. Counting as in Cases 1 and 2,
\begin{align*}
\Psi (CO) = & 15(\Psi(H_{n-2}-\{b,c\}) + \Psi(H_{n-2}-\{c,i\})) + 18\cdot \Psi (H_{n-2} - c)\\
&+ 3\cdot \Psi (H_{n-2}-\{a,b,c,i,j\}).
\end{align*}
Now $\Psi (CM) \ge \Psi (CO)$ follows immediately by comparing terms. By Lemma <ref>, we have $\Psi (H_{n-2} - c) \ge \Psi (H_{n-2}-\{a,b,c,i,j\})$ and by comparing the remaining terms we see that $\Psi (CM) \ge \Psi (CP)$. The preceding shows that attaching a hexagon in the meta position yields the most maximal matchings, implying
\Psi (G_n) \le \Psi (M_n)
as desired.
To get the remaining inequality of our theorem, we need only show that $\Psi (CO) \ge \Psi (CP)$. Now we must have either $(i)$ or $(ii)$ of Lemma <ref>, say $(i)$ holds. Then $4\cdot \Psi(H_{n-2}-\{b,c\}) \ge 2 \cdot \Psi (H_{n-2} - c)$ and by Lemma <ref> we have $\Psi(H_{n-2}-\{c,i\}) \ge \Psi (H_{n-2}-\{a,b,c,i,j\})$, showing that
\begin{align}
\Psi (CO) &\ge 11\Psi(H_{n-2}-\{b,c\}) + 13\Psi(H_{n-2}-\{c,i\}) + 20\cdot \Psi (H_{n-2} - c) \notag \\
&+ 5\cdot \Psi (H_{n-2}-\{a,b,c,i,j\}). \label{ineqCO}
\end{align}
Now by comparing the terms of $\Psi (CP)$ with the inequality $(\ref{ineqCO})$, it follows that $\Psi (CO) \ge \Psi (CP)$, which completes the proof.
It is instructive to compare the above results with the corresponding results
for all matchings and for independent sets from reference <cit.>
(Theorems 3.23 and 4.14, respectively). It can be seen that with respect
to the richest chains, the number of
maximal matchings behaves more like the number of independent sets than the
number of all matchings. A possible explanation might be the fact that
maximal matchings in any graph $G$ are in a bijective correspondence with
nice independent sets in $G$. (A set of vertices $S$ is nice if
$G-S$ has a perfect matching.)
§ BENZENOID CHAINS
§.§ Generating functions
Now we turn our attention to benzenoid chains. Here the connectivity increases
to two, and one can expect that this will result in longer recurrences, as
indicated in <cit.>. This is, indeed, the case.
Using the same techniques outlined in subsection <ref>, we obtain ordinary generating functions for the number of maximal matchings in the benzenoid chains $L_n$, $Z_n$, and $H_n$.
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Auxiliary graphs for $L_n$.
Let $\ell_n$ be the number of maximal matchings in $L_n$ and $\ell^i_n$ be the number of maximal matchings in the auxiliary graph $L^i_n$ in Figure <ref>. Then
$(i)$ $\ell _n = \ell ^1 _{n-1} + \ell _{n-1} + 2\ell ^2 _{n-2}$,
$(ii)$ $\ell ^1 _n = 2 \ell ^1 _{n-1} + \ell _{n-1} + 2\ell ^3 _{n-1}$,
$(iii)$ $\ell ^2 _n = \ell ^3 _n + \ell ^1 _{n-1} + \ell ^3 _{n-1}$,
$(iv)$ $\ell ^3 _n = \ell ^1 _{n-1} + \ell _{n-1} + \ell ^3 _{n-1} + \ell ^2 _{n-2} + \ell ^1 _{n-2} + \ell ^3 _{n-2}$,
with the initial conditions $\ell_0=1$, $\ell_1=5$, $\ell ^1_0=2$, $\ell ^2_0=3$, $\ell ^3_0=2$, and $\ell ^3_1=7$.
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Let $z_n$ be the number of maximal matchings in $Z_n$ and $z^i_n$ be the number of maximal matchings in the auxiliary graph $Z^i_n$ in Figure <ref>. Then
$(i)$ $z _n = z^1_{n-1} + z^2 _{n-1} + z^3_{n-2}$,
$(ii)$ $z ^1 _n = 2z^2_{n-1} + z^4_{n-2} + z^5_{n-1} + z^3_{n-2} + z^2_{n-2}$,
$(iii)$ $z ^2 _n = z_n + z^5_{n-1} + z_{n-1}$,
$(iv)$ $z ^3 _n = 2z^2_{n-1} + z^3_{n-1} + z^1_{n-1} + z^5_{n-1}$,
$(v)$ $z ^4 _n = z_n + z^5_{n-1} + z_{n-1} + z^2_{n-1} + z^3_{n-1}$,
$(vi)$ $z ^5 _n = z^5_{n-1} + z^4_{n-2} + z^2_{n-1} + z^3_{n-2} + z_{n-1}$,
with the initial conditions $z_0=1$, $z_1=5$, $z^1_0=2$, $z^1_1=9$, $z^2_0=2$, $z^3_0=3$, $z^4_0=4$, $z^5_0=2$, and $z^5_1=7$.
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Let $z_n$ be the number of maximal matchings in $Z_n$ and $z^i_n$ be the number of maximal matchings in the auxiliary graph $Z^i_n$ in Figure <ref>. Then
$(i)$ $h _n = h_{n-1} + h^1_{n-1} + h^2_{n-2} + h^3_{n-2}$,
$(ii)$ $h ^1 _n = 2h^4_{n-1} + h^5_{n-1} + h^3_{n-2} + 2h^4_{n-2} + h^5_{n-2}$,
$(iii)$ $h ^2 _n = h^3_{n-1} + 2h^4_{n-1} + 2h^4_{n-2} + 2h^3_{n-2} + h^5_{n-2}$,
$(iv)$ $h ^3 _n = h^5_n + h_n$,
$(v)$ $h ^4 _n = h_n + h^2_{n-1}$,
$(vi)$ $h ^5 _n = h^2_{n-1} + h^4_{n-1} + h^1_{n-1}$,
with the initial conditions $h_0=1$, $h_1=5$, $h^1_0=2$, $h^1_1=9$, $h^2_0=3$, $h^2_1=11$, $h^3_0=3$, $h^4_0=2$, and $h^5_0=2$.
Let $L(x)$, $Z(x)$, and $H(x)$ be the ordinary generating functions for the sequences $\ell_n$, $z_n$, and $h_n$, respectively. Then
L(x) = \frac{1 + x - x^3}{1 - 4x - x^4 - x^5},
Z(x) = \frac{1 + 2x + 4x^2 + 4x^3 + 6x^4 + 4x^5 + x^6}{1 - 3x - x^2 - 6x^3 - 7x^4 - 7x^5 - 5x^6 - x^7},
H(x) = \frac{1 + 4x + 8x^2 + 8x^3 + 7x^4 + 4x^5 + 2x^6}{1 - x - 7x^2 - 12x^3 - 6x^4 - 7x^5 - 4x^6 - 2x^7}.
Since $L(x)$, $Z(x)$, and $H(x)$ are rational functions, we can examine their denominators to obtain linear recurrences for the sequences $\ell _n$, $z_n$, and $h_n$. The initial conditions can be verified by direct computations.
$(i)$ $\ell_n = 4\ell _{n-1} + \ell _{n-4} + \ell _{n-5}$
with initial conditions $\ell _0 = 1$, $\ell _1 = 5$, $\ell _2 = 20$, $\ell _3 = 79$, and $\ell _4 = 317$,
$(ii)$ $z_n = 3z_{n-1} + z_{n-2} + 6z_{n-3} +7z_{n-4} + 7z_{n-5} + 5z_{n-6} + z_{n-7}$
with initial conditions $z _0 = 1$, $z _1 = 5$, $z _2 = 20$, $z _3 = 75$, $z _4 = 288$, $z _5 = 1105$, and $z _6 = 4234$,
$(iii)$ $h_n = h_{n-1} + 7h_{n-2} + 12h_{n-3} + 6h_{n-4} + 7h_{n-5} + 4h_{n-6} + 2h_{n-7}$
with initial conditions $h _0 = 1$, $h _1 = 5$, $h _2 = 20$, $h _3 = 75$, $h _4 = 288$, $h _5 = 1094$, and $h _6 = 4171$.
Again we can use Darboux's Theorem to deduce the asymptotics of the sequences $\ell_n$, $z_n$, and $h_n$. The smallest modulus singularity of $L(x)$ is approximately $x=0.248804$. Hence, the asymptotic behavior of $\ell _n$ is given by $\ell_n \sim 4.01923^{n+1}$ for large $n$. Similarly, we deduce that $z_n \sim 3.83256^{n+1}$ and $h_n \sim 3.81063^{n+1}$ for large $n$.
§.§ Extremal structure
In this subsection, we prove the linear polyacene has most maximal matchings among all benzenoid chains of the same length.
Let $G_n$ be a benzenoid chain of length $n$. Then
\Psi (G_n) \le \Psi (L_n).
Let $G_m$ be an arbitrary benzenoid chain of length $m$. Observe that we can always draw $G_m$ as in Figure <ref>, where $h_m$ is a terminal hexagon and the hexagon adjacent to the left of $h_{m-1}$ may attach at any of the edges $f, g,$ or $h$. Let us assume the hexagons of $G_m$ are labeled $h_1, \ldots, h_m$ according to their ordering in Figure <ref> where ($h_1$ is the other terminal hexagon).
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A terminal hexagon, $h_m$, and its adjacent hexagon, $h_{m-1}$, in the benzenoid chain $G_m$.
In what follows, let us adopt all of the same notation introduced in section <ref>. We also make use of Lemma <ref> introduced previously, since this holds for arbitrary graphs.
Any maximal matching of $G_m$ must contain at least one of the edges $a, b, c, d$ or $e$. Moreover, any maximal matching of $G_m$ contains exactly one of these edges, or contains exactly one of the following pairs of edges: $a$ and $e$, $a$ and $d$, $b$ and $e$, or $b$ and $d$.
Take a maximal matching $M$. For sake of contradiction, suppose $M$ contains none of the edges $a, b, c, d$ or $e$. Then we could add the edge $c$ to $M$, which is a contradiction to $M$ being a maximal matching. Hence at least one of the edges $a, b, c, d$ or $e$. The remaining part of the lemma follows by considering which pairs of edges can belong to the same matching.
Take a benzenoid chain $B$ of length $n-1$. Let us set $m=n-1$ and suppose that $B$ is drawn as in Figure <ref> with edges labeled as such, so that we may refer to this picture to aid this proof. We consider two cases of extending $B$ by an $n$th hexagon $h_n$.
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The benzenoid chain BL.
Case 1. The hexagon $h_n$ attaches in the linear position to the edge $y$ and let us denote the resulting graph by $BL$, see Figure <ref>. To compute $\Psi (BL)$ we make use of Lemma <ref> and count matchings based on which of the edges $a, b, c, d, e$ are saturated. Of the possibilities in Lemma <ref>, consider the maximal matchings of $BL$ containing only the edge $a$. Such a matching must also contain the edges $f$ and $z$, else this matching would contain one of the other edges $d$ or $e$. The remaining edges of the matching must be a maximal matching of $H_{n-2}\setminus \{a,f\}$ and a maximal matching of $H_{n-1,n}\setminus z$. By directly counting, we find that $\Psi (H_{n-1,n}\setminus z) = 4$ and hence, the number of maximal matchings containing only the edge $a$ is $4\cdot \Psi(H_{n-2}\setminus \{a,f\})$. We count the remaining cases from Lemma <ref> similarly. We note that a $H_{n-1}\setminus c$ is used to count maximal matchings containing the edges $b$ or $d$, since these edges do not belong to the subgraph $H_{n-2}$. For example, the number of maximal matchings containing only the edge $b$ is $3\cdot \Psi(H_{n-2}\setminus \{c,f\})$. Thus
\begin{align*}
\Psi (BL) =\ &4\cdot \Psi(H_{n-2}\setminus \{a,f\}) + 3\cdot \Psi(H_{n-2}\setminus \{c,f\})+14\cdot \Psi(H_{n-2}\setminus c) \\
&+4\cdot \Psi(H_{n-2}\setminus \{e,h\})+3\cdot \Psi(H_{n-2}\setminus \{c,h\})+9\cdot \Psi(H_{n-2}\setminus \{a,e\})\\
&+7\cdot \Psi(H_{n-2}\setminus \{a,c\})+7\cdot \Psi(H_{n-2}\setminus\{c,e\}).
\end{align*}
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Case 2. The hexagon $h_n$ attaches in the kinky position to the edge $z$ and let us denote the resulting graph by $BK$, see Figure <ref>. Counting as in Case 1 above we obtain
\begin{align*}
\Psi (BK) =\ &6\cdot \Psi(H_{n-2}\setminus \{a,f\}) + 5\cdot \Psi(H_{n-2}\setminus \{c,f\})+12\cdot \Psi(H_{n-2}\setminus c) \\
&+5\cdot \Psi(H_{n-2}\setminus \{e,h\})+3\cdot \Psi(H_{n-2}\setminus \{c,h\})+8\cdot \Psi(H_{n-2}\setminus \{a,e\})\\
&+5\cdot \Psi(H_{n-2}\setminus \{a,c\})+7\cdot \Psi(H_{n-2}\setminus\{c,e\}).
\end{align*}
Now considering the terms in $\Psi (BL)$, by Lemma <ref> we have
\begin{align*}
\Psi(H_{n-2}\setminus \{a,c\}) &\ge \Psi(H_{n-2}\setminus \{a,f\}), \\
\Psi(H_{n-2}\setminus \{c\}) &\ge \Psi(H_{n-2}\setminus \{c,f\}), and\\
\Psi(H_{n-2}\setminus \{a,e\}) &\ge \Psi(H_{n-2}\setminus \{e,h\}),
\end{align*}
implying that
\begin{align*}
\Psi (BL) \ge &6\cdot \Psi(H_{n-2}\setminus \{a,f\}) + 5\cdot \Psi(H_{n-2}\setminus \{c,f\})+12\cdot \Psi(H_{n-2}\setminus c) \\
&+5\cdot \Psi(H_{n-2}\setminus \{e,h\})+3\cdot \Psi(H_{n-2}\setminus \{c,h\})+8\cdot \Psi(H_{n-2}\setminus \{a,e\})\\
&+5\cdot \Psi(H_{n-2}\setminus \{a,c\})+7\cdot \Psi(H_{n-2}\setminus\{c,e\}) \\
&\ge \Psi (BK).
\end{align*}
The above proves that attaching a hexagon linearly gives more maximal matchings than attaching a hexagon in the kinky position. The inequality stated in the theorem follows.
Again, we can see that the number of maximal matchings follows the same
pattern as the number of independent sets, contrary to the number of all and
of perfect matchings. While the last two increase with the number of kinky
hexagons, the number of maximal matchings decreases. Further, unlike the
number of perfect matchings which does not discriminate between left and right
kinks, the number of maximal matchings seems to be sensitive to the
direction of successive turns. It seems that the helicenes have the smallest
number of maximal matchings among all benzenoid chains of the same length.
§ FURTHER DEVELOPMENTS
In this last section we list some unresolved problems and indicate some
possible directions of future research. We start by stating a conjecture
about the extremal benzenoid chains.
Let $B_n$ be a benzenoid chain of length $n$. Then $\Psi (H_n) \leq
\Psi (B_n)$.
Now we turn to some structural properties. The cardinality of any smallest
maximal matching in $G$ is called the saturation number of $G$. The
saturation number is of interest in the context of random sequential
adsorption, since it gives the information on the worst possible case of
clogging the substrate; see <cit.> for a discussion and
<cit.> for some specific cases. However, it is not
enough to know the size of the worst possible case; it is also imprtant to
know how (un)likely is it to happen. This brings us back to enumerative
problems, since the answer to this question depends on the ability to count
maximal matchings of a given size. A neat way to handle information about
maximal matchings of different sizes is to use the maximal matching
polynomial. It was introduced in <cit.> and some of its basic
properties were established there. There are, however, many open questions
about this polynomial. For example, for ordinary (generating) matching
polynomials <cit.> we know that their coefficients are
log-concave. Is this valid also for maximal matching polynomials? We have
computed maximal matching polynomials explicitly for several families of
graphs, and we have enumerated maximal matchings in several other families.
So far, no counterexample has been found, but the proof still eludes us.
Another interesting thing to do would be to look at the dynamic aspect of
the problem, emulating the approach of Flory <cit.>.
Finally, it would be interesting to extend our results on other classes of
graphs, such as rotagraphs, branching polymers, composite graphs and
finite portions of various lattices.
§ ACKNOWLEDGEMENTS
This work was partially supported by a SPARC Graduate Research Grant from the
Office of the Vice President for Research at the University of South Carolina
and also supported in part by the NSF DMS under contract 1300547. It was
supported in part by Croatian Science Foundation under
the project 8481 (BioAmpMode).
V. Andova, F. Kardoš, R. Škrekovski,
Sandwiching saturation number of fullerene graphs,
MATCH Commun. Math. Comput. Chem. 73 (2015) 501–518.
E. A. Bender, S. Gill Williamson,
Foundations of Combinatorics with Applications,
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|
1511.00065
|
Dedicated to Professor Alain Chenciner on his 70th birthday
§ INTRODUCTION
In economic theory, and in optimal control, it has been customary to discount
future gains at a constant rate $\delta>0$. If an individual with utility
function $u\left( c\right) $ has the choice between several streams of
consumption $c\left( t\right) $, $0\leq t$, he or she will choose the one
which maximises the present value, given by:
\begin{equation}
\int_{0}^{\infty}u\left( c\left( t\right) \right) e^{-\delta t}dt
\label{a1}%
\end{equation}
That future gains should be discounted is well grounded in fact. On the one
hand, humans prefer to enjoy goods sooner than later (and to suffer bads later
than sooner), as every child-rearing parent knows. On the other hand, it is
also a reflection of our own mortality: 10 years from now, I may simply no
longer be around to enjoy whatever I have been promised. These are two good
reasons why people are willing to pay a little bit extra to hasten the
delivery date, or will require compensation for postponement, which is the
essence of discounting.
On the other hand, there is no reason why the discount rate should be
constant, i.e. why the discount factor should be an exponential $e^{-\delta
t}$. The practice probably arises from the compound interest formula
$\lim_{\varepsilon\rightarrow0}\left( 1-\varepsilon\delta\right)
^{t/\varepsilon}=e^{-\delta t}$, when a constant interest rate $\delta$ is
assumed, but even in finance, interest rates vary with the horizon: long-term
rates can be widely different from short-term ones. As for economics, there is
by now a huge amount of evidence that individuals use higher discount rates
for the near future than for the long-term (see <cit.> for a review
up to 2002). There is also an aggregation problem: in a society where
individuals use constant (but different) discount rates, the collective
discount rate may be non-constant (see <cit.>). So the present value
formula (<ref>) should be replaced by the more general one:
\begin{equation}
\int_{0}^{\infty}u\left( c\left( t\right) \right) h\left( t\right) dt
\label{a2}%
\end{equation}
where $h\ $ is a decreasing function, with $h\left( 0\right) =1$.
But then a new problem arises, which is now well recognized in economic
theory, but to our knowledge has not yet received the attention it deserves in
control theory. It is the problem of time-inconsistency, which runs as
follows. Suppose the decision-maker has the choice between two streams of
consumption $c_{1}\left( t\right) $ and $c_{2}\left( t\right) $, starting
at time $T>0$. At time $t=0$, he or she finds $c_{1}\left( t\right) $ yields
the highest present value:
\begin{equation}
\int_{T}^{\infty}u\left( c_{1}\left( t\right) \right) h\left( t\right)
dt>\int_{T}^{\infty}u\left( c_{2}\left( t\right) \right) h\left(
t\right) dt. \label{a3}%
\end{equation}
He or she then chooses $c_{1}\left( t\right) $. When time $T$ is reached,
the present values are now:
\begin{equation}
\int_{T}^{\infty}u\left( c_{1}\left( t\right) \right) h\left( t-T\right)
dt\text{ \ and}\ \ \int_{T}^{\infty}u\left( c_{2}\left( t\right) \right)
h\left( t-T\right) dt \label{a4}%
\end{equation}
If $h\left( t\right) =e^{-\delta t}$, then the ordering found at time $t=0$
will persist at time $t=T$. Indeed:
\[
\int_{T}^{\infty}u\left( c\left( t\right) \right) e^{-\delta\left(
t-T\right) }dt=e^{\delta T}\int_{T}^{\infty}u\left( c\left( t\right)
\right) e^{-\delta t}dt
\]
so that the two terms in (<ref>) are proportional to the two terms in
(<ref>). However, this is a peculiarity of the exponential function, and it
is not to be expected with more general discount rates. The decision-maker
then faces a basic rationality problem: what should he or she do ? To be more
specific, assume the state $k\left( t\right) $ is related to the control
$c\left( t\right) $ by the dynamics:
\begin{align}
\frac{dk}{dt} & =f\left( k\right) -c\left( t\right) ,\ \ k\left(
0\right) =k_{0}\label{a7}\\
c\left( t\right) & \geq0,\ \ k\left( t\right) \geq0 \label{a8}%
\end{align}
and the decision-maker is interested in maximising $\left( \ref{a2}\right)
$. How should he or she behave ?
In the exponential case, when $h\left( t\right) =e^{-\delta t}$, the answer
is to pick the optimal solution: if it is optimal at time $t=0$, it will still
be optimal at all times $T>0$ (this, by the way, is the content of the dynamic
programming principle). But in the non-exponential case, the notion of
optimality changes with time: each observer, from time $t=0$ on, has his or
her own optimal solution. No one agrees on what the optimal solution is, so
optimality no longer provides an answer to the decision-making process, and
one must look for other concepts to describe rational behaviour.
A clear requirement for rationality is that any strategy put forward be
implementable. Suppose a Markov strategy $c=\sigma\left( k\right) $ is put
forward at time $t=0$. If it is to be followed at all later times $t>0$, then
it must be the case that the decision-maker at that time finds no incentive to
deviate. More precisely, if he/she assumes that at all later times the
strategy (closed-loop feedback) $c=\sigma\left( k\right) $ will be applied,
then he/she should find it in his/her interest to apply $\sigma$ as well. In
other words, $\sigma$ should be a subgame-perfect Nash equilibrium of the
leader-follower game played by the successive decision-makers. This idea has
been introduced by Phelps (<cit.>, <cit.>) in models with
discrete time (see <cit.> and <cit.> for further
developments), and adapted by Karp (<cit.>, <cit.>), and by Ekeland
and Lazrak (<cit.>, <cit.>, <cit.>, <cit.>) to the case of
continuous time.
In this paper, we will follow the approach by Ekeland and Lazrak. It consists
of introducing a value function $V\left( k\right) $, which is very similar
to the value function in optimal control, and of showing that it satisfies a
functional-differential equation which is reminescent of the
Hamilton-Jacobi-Bellman (HJB) of optimal control. Conversely, any solution of
that equation with suitable boundary conditions will give us an equilibrium strategy.
In the work by Ekeland and Lazrak, this approach was applied to (<ref>),
with $h\left( t\right) =\alpha\exp\left( -r_{1}t\right) +\left(
1-\alpha\right) \exp\left( -r_{2}t\right) $, and it was showed that the
corresponding problem had a continuum of equilibrium strategies. In the
present paper, in view of applications to economics, and of the mathematical
interest, we aim to extend the analysis to the more general case:
\begin{equation}
\left( 1-\alpha\right) \int_{0}^{\infty}e^{-r_{1}t}u\left( c\left(
t\right) ,k\left( t\right) \right) dt+\alpha\int_{0}^{\infty}e^{-r_{2}%
t}U\left( c\left( t\right) ,k\left( t\right) \right) dt \label{a5}%
\end{equation}
As a by-product of our analysis, we will treat the problem:
\begin{equation}
\left( 1-\alpha\right) \delta\int_{0}^{\infty}e^{-\delta t}u\left( c\left(
t\right) ,k\left( t\right) \right) dt+\alpha\lim_{t\rightarrow\infty
}U\left( k\left( t\right) ,c\left( t\right) \right) \label{a6}%
\end{equation}
which was introduced by Chichilnisky (see <cit.>, <cit.>) to model
sustainable development. Note that, if $u\left( c,k\right) \geq0$ and:
\[
\sup\left\{ U\left( c,k\right) \ |\ c>0,\ k>0,\ c=f\left( k\right)
\right\} =\infty
\]
then maximising (<ref>) under the dynamics (<ref>), (<ref>) leads to
the value $+\infty$, so that optimisation is clearly not an answer to the
problem. Instead, we find equilibrium strategies. To our knowledge, this is an
entirely new result. We show that there is a continuum of such
strategies. More precisely, there is a continuum of points $k_{\infty}$ which
can be realized as the long-term level of capital by an equilibrium strategy.
This support, however, is one-sided, that is, $k_{\infty}$ can be reached only
from the initial level of capitals $k_{0}$ lying on its left (or from its
right). To our knowledge, this is the first time such strategies have been identified.
The structure of the paper is as follows. In the next section, we consider the
problem of maximising
\begin{equation}
\int_{0}^{\infty}e^{-\delta t}u\left( c\left( t\right) ,k\left( t\right)
\right) dt \label{a9}%
\end{equation}
under the dynamics (<ref>), (<ref>), and we show that it has a solution.
On the way, we introduce the corresponding HJB equation, and we show that it
has a $C^{2}$ solution. Next, we define equilibrium strategies. With each such
strategy we associate a value function $V\left( k\right) $, and we show that
it satisfies an integro-differential equation which generalizes the HJB
equation, and we prove a verification theorem: any solution of this equation
with suitable boundary conditions gives an equilibrium strategy. We show that
the trajectories satisfy an integro-differential equation which generalizes
the classical Euler-Lagrange equations, and we connect the Ekeland-Lazrak
approach with the Karp approach.
In section 4, we apply the theory to problem (<ref>), and show that it has
a continuum of equilibrium strategies, thereby extending the results of
<cit.>. It should be noted that the equations for $V\left( k\right) $
are given in implicit form, that is, they cannot be solved with respect to
$V^{\prime}\left( k\right) $, so that finding a $C^{2}$ solution requires
special techniques (first a blow-up, and then the central manifold theorem).
We then consider the criterion:
\begin{equation}
\left( 1-\alpha\right) \delta\int_{0}^{\infty}e^{-\delta t}u\left( c\left(
t\right) ,k\left( t\right) \right) dt+\alpha r\int_{0}^{\infty}%
e^{-rt}U\left( c\left( t\right) ,k\left( t\right) \right) dt \label{a10}%
\end{equation}
which belongs to the class (<ref>) and we let $r\rightarrow0$. In the
limit, we get equilibrium strategies for the Chichilnisky problem (<ref>),
which we describe explicitly.
§ THE RAMSEY PROBLEM
This is the classical model for economic growth, originating with the seminal
paper of Ramsey <cit.> in 1928, and developed by Cass <cit.>,
Koopmans <cit.> and many others (see <cit.> for a modern
exposition). We are given a point $k_{0}>0$ and a concave continuous function
$f$ on $[0,\ \infty)$, which is $C^{\infty}$ on $]0,\ \infty\lbrack$ and
satisfies the Inada conditions:
\begin{equation}
\label{Inada}\lim_{x\rightarrow0}f^{\prime}\left( x\right) =+\infty
,\ \ \lim_{x\rightarrow\infty}f^{\prime}\left( x\right) \leq0.
\end{equation}
A capital-consumption path $\left( c\left( t\right) ,k\left( t\right)
\right) ,\ t\geq0$, is admissible if:
\begin{align}
k\left( t\right) & >0\text{ and }c\left( t\right) >0\text{ for all
\frac{dk}{dt} & =f\left( k\right) -c,\ \ k\left( 0\right) =k_{0},
\label{24}%
\end{align}
The set of all admissible paths starting from $k_{0}$ (i.e, such that
$k\left( 0\right) =k_{0}$) will be denoted by $\mathcal{A}\left(
k_{0}\right) $.
We are given a number $\delta>0$ and another concave, increasing function $u$
on $]0,\ \infty)$, which is $C^{\infty}$ on the interior, with $u^{\prime
\prime}\left( c\right) >0$ everywhere. We introduce the following criterion
on $\mathcal{A}\left( k_{0}\right) $:
\begin{equation}
I\left( c,k\right) =\int_{0}^{\infty}u(c(t),k(t))e^{-\delta t}dt, \label{18}%
\end{equation}
and we consider the optimization problem:
\begin{equation}
\sup\left\{ I\left( c,k\right) \ |\ \left( c,k\right) \in\mathcal{A}%
\left( k_{0}\right) \right\} . \label{19}%
\end{equation}
The Euler-Lagrange equation is given by:
\begin{equation}
u_{11}^{\prime\prime}\frac{dc}{dt}=\left( \delta-f^{\prime}\left( k\right)
\right) u_{1}^{\prime}-u_{2}^{\prime}-\left( f\left( k\right) -c\right)
u_{12}^{\prime\prime}. \label{21}%
\end{equation}
Equation (<ref>), together with equation (<ref>), constitute a system of
two first-order ODEs for the unknown functions $\left( c\left( t\right)
,k\left( t\right) \right) $. In the particular case when $u=u\left(
c\right) $ does not depend on $k$, the last equation simplifies to:
\[
u_{11}^{\prime\prime}\frac{dc}{dt}=\left( \delta-f^{\prime}\left( k\right)
\right) u_{1}^{\prime}%
\]
and the (<ref>), (<ref>) gives rise to a well-known phase diagram, with
a hyperbolic stationary point $\left( c_{\infty},k_{\infty}\right) $
characterized by $f^{\prime}\left( k_{\infty}\right) =\delta$ and $f\left(
k_{\infty}\right) =c_{\infty}$. The optimal solution of the Ramsey problem in
that case then is the solution of (<ref>), (<ref>) which converges to
$\left( c_{\infty},k_{\infty}\right) $ (see <cit.> for instance).
In the general case where $u\left( c,k\right) $ depends on $k$, the
situation is not as simple, and to our knowledge has not been investigated.
Stationary points $\left( c_{\infty},k_{\infty}\right) $ of the dynamics (if
any) are given by:
\begin{align}
c_{\infty}-f\left( k_{\infty}\right) & =0\label{50}\\
\left( \delta-f^{\prime}\left( k_{\infty}\right) \right) u_{1}^{\prime
}\left( f\left( k_{\infty}\right) ,k_{\infty}\right) -u_{2}^{\prime
}\left( f\left( k_{\infty}\right) ,k_{\infty}\right) & =0. \label{51}%
\end{align}
To prove the existence of an optimal strategy, we do not use the Euler
equation. We use the Hamilton-Jacobi-Bellman (HJB) equation instead. Introduce
the optimal value as a function of the initial point:
\[
V\left( k_{0}\right) :=\sup\left\{ I\left( c,k\right) \ |\ \left(
c,k\right) \ \in\mathcal{A}\left( k_{0}\right) \right\}
\]
If there is an optimal solution $\left( c\left( t\right) ,k\left(
t\right) \right) $, and it converges to $\left( c_{\infty},k_{\infty
}\right) $ when $t\rightarrow\infty$, then, substituting in (<ref>), we
must have:
\begin{equation}
V\left( k_{\infty}\right) =\int_{0}^{\infty}e^{-\delta t}u\left( c_{\infty
},k_{\infty}\right) dt=\frac{1}{\delta}u\left( c_{\infty},k_{\infty}\right)
=\frac{1}{\delta}u\left( f\left( k_{\infty}\right) ,k_{\infty}\right)
\label{20a}%
\end{equation}
If $V\left( k\right) $ is $C^{1}$, it satisfies the HJB equation,
\begin{equation}
\delta V(k)=\max_{c}\{u(c,k)+(f(k)-c){V}^{\prime}(k)\}. \label{20}%
\end{equation}
Conversely, suppose the HJB equation has a $C^{2}$ solution satisfying
(<ref>) for some $\left( c_{\infty},k_{\infty}\right) $, and define a
strategy $c=\sigma\left( k\right) $ by:
\begin{equation}
u_{1}^{\prime}\left( \sigma\left( k\right) ,k\right) =V^{\prime}\left(
k\right) \label{20b}%
\end{equation}
Suppose moreover that the solution of:
\begin{equation}
\frac{dk}{dt}=f\left( k\right) -\sigma\left( k\right) ,\ \ k\left(
0\right) =k_{0}, \label{20c}%
\end{equation}
converges to $k_{\infty}$ for all initial points $k_{0}$. Then $\sigma\left(
k\right) $ is an optimal solution of the generalized Ramsey problem (<ref>)
This is the so-called verification theorem, which is classical (see
<cit.>). We need $V$ to be $C^{2}$, so that $\sigma$ is defined. Since
$u_{1}^{\prime\prime}>0$, we can use the implicit function theorem on equation
(<ref>) to define $\sigma$. If $V$ is $C^{2}$, then $\sigma$ is $C^{1}$,
and the initial-value problem (<ref>) has a unique solution. Note also
that everything is local: the functions $V$ and $\sigma$ are defined in some
neighbourhood of $k_{\infty}$ only, and the initial value $k_{0}$ is assumed
to belong to that neighbourhood of $k_{\infty}$.
So, to prove the (local) existence of an optimal strategy $\sigma\left(
k\right) $, we have to prove that the HJB equation has a $C^{2}$ solution $V$
with $V\left( k_{\infty}\right) =\delta^{-1}u\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) $, and that the corresponding path $k\left(
t\right) $ converges to $k_{\infty}$. Then the right-hand side of (<ref>)
converges to $0$, so that $c\left( t\right) =\sigma\left( k\left(
t\right) \right) $ converges to $\sigma\left( k_{\infty}\right) =f\left(
k_{\infty}\right) $. This is the content of the following two results
Suppose there is some $k_{\infty}>0$ satisfying (<ref>) and:
\begin{equation}
^{\prime\prime}\right) -\frac{u_{2}^{\prime}}{u_{1}^{\prime}}u_{12}%
^{\prime\prime}<0 \label{27}%
\end{equation}
(all values to be taken at $k_{\infty}$ and $c_{\infty}=f\left( k_{\infty
}\right) $). Then there is an optimal strategy $c=\sigma\left( k\right) $
converging to $k_{\infty}$
If $u=u\left( c\right) $ does not depend on $k$, then $u_{2}^{\prime}=0$.
Equation (<ref>) becomes $f^{\prime}\left( k_{\infty}\right) =\delta$,
which defines $k_{\infty}$ uniquely because of the Inada conditions
(<ref>) on $f$, and condition (<ref>) becomes $u_{1}^{\prime}\left(
c_{\infty}\right) f^{\prime\prime}\left( k_{\infty}\right) <0$, which is
satisfied automatically. So there is an optimal strategy in that case, and one
can even show that it is globally defined. For $u$ depending on $k$, however,
the situation is different.
We follow the method of <cit.>. By the inverse function theorem, the
equation $u_{1}^{\prime}(c,k))=p$ defines $c$ as a $C^{1}$ function of $p$ and
\begin{equation}
c=\varphi(p,k)\text{, }u_{1}^{\prime}(\varphi(p,k),k))=p \label{c.of.k}%
\end{equation}
We rewrite (<ref>) as a Pfaff system:
\begin{align}
dV & =pdk,\label{paff1}\\
p(f(k)-\varphi(p,k))+u(\varphi(p,k),k) & =\delta V, \label{paff2}%
\end{align}
and we seek a $C^{2}$ solution $V$ satisfying:
\begin{equation}
\label{paff.initial}%
\end{equation}
Differentiating (<ref>) leads to:
\begin{equation}
\delta dV=(f(k)-\varphi(p,k))dp+pf^{\prime}(k)dk+u_{2}^{\prime}(\varphi
(p,k),k)dk \label{delta.dv}%
\end{equation}
Plugging (<ref>) into (<ref>), we get:
\begin{equation}
(p,k),k))dk. \label{system.tmp}%
\end{equation}
Introducing an auxiliary variable $t$, we rewrite this as a system of two ODES
for two functions $p\left( t\right) $ and $k\left( t\right) $:
\begin{equation}%
\begin{array}
\frac{dk}{dt}=f(k)-\varphi(p,k)\\
\frac{dp}{dt}=\delta{p}-pf^{\prime}(k)-u_{2}^{\prime}(\varphi(p,k),k)
\end{array}
\label{ODE}%
\end{equation}
with the initial condition
\[
\]
By (<ref>), we must have
\[
)=u_{1}^{\prime}(c_{\infty},k_{\infty})=u_{1}^{\prime}(f\left( k_{\infty
}\right) ,k_{\infty})
\]
Differentiating (<ref>) with respect to $p$ and $k$ respectively, we
derive the formulas:
\begin{align}
\varphi_{1}^{\prime}(p_{\infty},k_{\infty}) & =\frac{1}{u_{11}^{\prime
\prime}(\varphi(p_{\infty},k_{\infty}),k_{\infty})}=\frac{1}{u_{11}%
\varphi_{2}^{\prime}(p_{\infty},k_{\infty}) & =-\frac{u_{11}^{\prime\prime
\label{i2'.infty}%
\end{align}
We can now linearizd (<ref>) at the point $\left( p_{\infty},k_{\infty
}\right) $. We get:
\begin{equation}
\frac{d}{dt}\left(
\begin{array}
\end{array}
\right) =A_{\infty}\left(
\begin{array}
\end{array}
\right) . \nonumber\label{linearized.system}%
\end{equation}
where the constant matrix $A_{\infty}$ is given by:
\[
\begin{array}
f^{\prime}+\frac{u_{12}^{\prime\prime}}{u_{11}^{\prime\prime}}, & -\frac
^{\prime\prime})^{2}}{u_{11}^{\prime\prime}}, & \delta-f^{\prime}-\frac
\end{array}
\right) .
\]
and all the values are to be taken at $\left( k_{\infty},p_{\infty}\right)
$. The characteristic polynomial is:
\[
\]
where we used $f_{\infty}^{\prime}+\frac{u_{2\infty}^{\prime}}{u_{1\infty
}^{\prime}}=\delta$ by (<ref>). Because of assumption (<ref>), it has
two real roots with different signs, $\lambda_{+}>0$ and $\lambda_{-}<0$. Thus
$(k_{\infty},p_{\infty})$ is a hyperbolic fixed point of (<ref>), with a
stable $C^{\infty}$-manifold $\mathcal{S}$ which corresponds to $\lambda_{-}$
and an unstable $C^{\infty}$-manifold $\mathcal{U}$ which corresponds to
$\lambda_{+}$. Choose a smooth parametrization $({k}_{s}(x),{p}_{s}(x))$ for
the curve $\mathcal{S}$. The tangent at the fixed point is:
\[
\frac{d{p}_{s}}{d{k}_{s}}(k_{\infty})=u_{11}^{\prime\prime}(f_{\infty}%
\]
and plugging $k=k_{s}(x),\ p=p_{s}(x)$ into equation (<ref>), we get
\[
\]
Moreover, differentiating (<ref>), and using (<ref>) again, we find:
\[
\frac{d{V}_{s}}{d{k}_{s}}(k_{\infty})=u_{1}^{\prime}(f(k_{\infty}),k_{\infty
\]
It follows that the curve in parametric form $x\rightarrow\left( k\left(
x\right) ,V\left( x\right) \right) $ is in fact the graph of a function
$V\left( k\right) $ which solves HJB and satisfies (<ref>). By
(<ref>), we have
\begin{equation}
\frac{d^{2}{V}_{s}}{d{k_{s}}^{2}}(k_{\infty})=\frac{d{p}_{s}}{d{k_{s}}%
_{-})+u_{12\infty}^{\prime\prime}. \label{v''}%
\end{equation}
Thus $V_{s}(k)$ is $C^{2}$ in $k$ at $k_{\infty}$, and the $C^{2}$ property at
other points near $k_{\infty}$ follows from (<ref>).
It remains to show that the strategy $\sigma$ defined by $u_{1}^{\prime
}(\sigma\left( k\right) ,k)=V^{\prime}\left( k\right) $ converges to
$\left( f\left( k_{\infty}\right) ,k_{\infty}\right) $. We rewrite
$\sigma$ as:
\[
\sigma(k)=\varphi(V^{\prime}(k),k).
\]
Linearizing the equation
\[
\frac{dk}{dt}=f(k)-c=f(k)-\varphi(V^{\prime}(k),k)
\]
\begin{equation}
\frac{d(k-k_{\infty})}{dt}=\lambda_{-}(k-k_{\infty})\nonumber
\end{equation}
and this concludes the proof.
Let us show how to deduce the Euler equation (<ref>) from the HJB equation
(<ref>) and the optimal strategy (<ref>). Setting $c\left( t\right)
:=\sigma\left( k\left( t\right) \right) $, differentiating (<ref>) with
respect to $k$, and applying the envelope theorem, we get:
\[
\delta V^{\prime}(k)=u_{2}^{\prime}\left( c,k\right) +f^{\prime}\left(
k\right) V^{\prime}\left( k\right) +\left( f\left( k\right) -c\right)
V^{\prime\prime}\left( k\right) ,
\]
and hence, noting that $\frac{dk}{dt}=f\left( k\right) -c\left( t\right)
\[
\left( \delta-f^{\prime}\left( k\right) \right) V^{\prime}\left(
k\right) -u_{2}^{\prime}(c,k)=\left( f\left( k\right) -c\right) \frac
{d}{dk}V^{\prime}\left( k\right) =\frac{d}{dt}V^{\prime}\left( k\right) .
\]
Replacing $V^{\prime}\left( k\right) $ by $u_{1}^{\prime}(\sigma\left(
k\right) ,k)=u_{1}^{\prime}(c,k)$, we get:
\[
\left( \delta-f^{\prime}\left( k\right) \right) u_{1}^{\prime}%
\]
which is precisely the Euler equation.
§ TIME-INCONSISTENCY.
§.§ Equilibrium strategies
We consider the intertemporal decision problem (as it seen at time $t=0$)
\begin{equation}
J\left( c,k\right) =\int_{0}^{\infty}\left[ h\left( t\right) u\left(
c\left( t\right) ,k\left( t\right) \right) +H\left( t\right) U\left(
c\left( t\right) ,k\left( t\right) \right) \right] dt \label{model1}%
\end{equation}
under the dynamics described by (<ref>) and (<ref>). Here $h$ and $H$
are discount factors, i.e. $C^{\infty}$ non-increasing functions on
$[0,\ \infty)$, such that $h\left( 0\right) =H\left( 0\right) =1$ and
$h\left( \infty\right) =H\left( \infty\right) =0$, while $u$ and $U$ are
utility functions. They are assumed to be $C^{\infty}$ on $]0,\ \infty)^{2}$,
with $u^{\prime\prime}<0\,$ and $U^{\prime\prime}<0$ everywhere. We shall
also assume that they decay exponentially, so that there is some $\rho>0$ and
some $T>0$ such that $h\left( t\right) <e^{-\rho t}$ and $H\left( t\right)
<e^{-\rho t}$ for $t\geq T$.
Because of time-inconsistency, the decision problem can no longer be seen as
an optimization problem. There is no way for the decision-maker at time $0$ to
achieve what is, from her point of view, the first-best solution of the
problem, and she must turn to a second-best policy: the best she can do is to
guess what her successor are planning to do, and then to plan her own
consumption $c(0)$ accordingly. In other word, we will be looking for a
subgame-perfect equilibrium of the leader-follower game played by successive generations.
The equilibrium policy was described in <cit.> for the case when the
criterion (<ref>) did not include the second term, and $u$ did not
depend on $k$. We will extend this analysis to the present situation, and then
compare it with the approach in <cit.>.
A Markov strategy $c=\sigma\left( k\right) $ is convergent if there
is a point $k_{\infty}$ and a neighbourhood $\mathcal{N}$ of $k_{\infty}$ such
that, for every $k_{0}\in\mathcal{N}$ the solution $k(t)$ of (<ref>)
converges to $k_{\infty}$ (and so $c\left( t\right) =\sigma\left( k\left(
t\right) \right) $ converges to $c_{\infty}=f\left( k_{\infty}\right) $).
If this is the case, we shall say that $\left( c_{\infty},k_{\infty}\right)
$ is supported by $\sigma$.
Let us suppose that a convergent Markov strategy $\sigma$ has been announced
and is the public knowledge $\sigma$. The decision-maker at $T$ has capital
stock $k_{T}$. If all future decision-maker apply the strategy $\sigma$, the
resulting future capital stock flow $k(t)$ obeys:
\begin{equation}
\frac{dk}{dt}=f\left( k\right) -\sigma\left( k\right) ,\ k\left(
T\right) =k_{T},\ T\leq t \label{equation.of.kb}%
\end{equation}
Since every decision-maker faces the same problem (with different stock
levels) it is enough to take $T=0$. Suppose the decision-maker at time $0$
holds power for $0\leq t<\varepsilon$, and expects all later decision-makers
to apply the strategy $\sigma$. He or she then explores whether it is in his
or her interest to apply the strategy $\sigma$, that is, to play $c_{0}%
=\sigma\left( k_{0}\right) $ for $0\leq t<\varepsilon$. If he or she applies
the constant control $c$ for $0\leq t\leq\varepsilon$.
Suppose the constant control $c$ is use on $0\leq t\leq\varepsilon$. The
immediate utility flow during $[0,\varepsilon]$ is $[u(c,k_{0})+U(c,k_{0}%
)]\varepsilon+o(\varepsilon)$ where $o(\varepsilon)$ is a higher order term of
$\varepsilon$. At time $\varepsilon$, the resulting capital will be
$k_{0}+(f(k_{0})-c)\varepsilon+o(\varepsilon)$. From then on, the strategy
$\sigma$ will be applied, which results in a capital stock $k_{c}$ satisfying
(we omit higher-order terms):
\begin{equation}
\frac{dk_{c}}{dt}=f\left( k_{c}\right) -\sigma\left( k_{c}\right)
,\ k_{c}\left( \varepsilon\right) =k_{0}+(f(k_{0})-c)\varepsilon
,\ t\geq\varepsilon\label{kc}%
\end{equation}
The capital stock $k_{c}$ can be written as
\begin{equation}
k_{c}(t)=k_{0}(t)+k_{1}(t)\varepsilon, \label{kc.kb.ki}%
\end{equation}
where $k_{0}\left( t\right) $ is the unperturbed solution, and $k_{1}\left(
t\right) $ is given by the linearized equation:
\begin{align}
\frac{dk_{0}}{dt} & =f\left( k_{0}\right) -\sigma\left( k_{0}\right)
,\text{ }k_{0}\left( 0\right) =k_{0}\label{a12}\\
\frac{dk_{1}}{dt} & =\left( f^{\prime}\left( k_{0}\right) -\sigma
^{\prime}\left( k_{0}\right) \right) k_{1},\ \ k_{1}\left( 0\right)
=\sigma\left( k_{0}\right) -c\ \label{a11}%
\end{align}
Evaluating the integral (<ref>) we get:
\[%
\begin{array}
J\left( \varepsilon\right) =u(c,k_{0})\varepsilon+\int_{\varepsilon}%
^{\infty}h(s)u(\sigma(k_{0}(t)+\varepsilon k_{1}(t)),k_{0}(t)+\varepsilon
\quad\quad\quad\quad\ +U(c,k_{0})\varepsilon+\int_{\varepsilon}^{\infty
}H(s)U(\sigma(k_{0}(t)+\varepsilon k_{1}(t)),k_{0}(t)+\varepsilon k_{1}(t))dt
\end{array}
\]
Letting $\varepsilon\rightarrow0$, so that commitment span of the
decision-maker vanishes, we get:
\[
\lim_{\varepsilon\rightarrow0}\frac{1}{\varepsilon}\left( J\left(
\varepsilon\right) -J\left( 0\right) \right) ={P}(k_{0},\sigma,c)
\]
\begin{align}
P(k_{0},\sigma,c) & =u(c,k_{0})-u(\sigma(k_{0}),k_{0})+(U(c,k_{0}%
& +\int_{0}^{\infty}h(t)u_{1}^{\prime}(\sigma(k_{0}(t)),k_{0}(t))\sigma
& +\int_{0}^{\infty}H(t)U_{1}^{\prime}(\sigma(k_{0}(t)),k_{0}(t))\sigma
\end{align}
A convergent Markov strategy $\sigma$ is an equilibrium if we have:
\[
\max_{c}P(k,\sigma,c)=P(k,\sigma,\sigma\left( k\right) ),\text{ \ }\forall
\]
§.§ The HJB approach
We now characterizes the equilibrium strategy. We write $k_{c}(t)=\mathcal{K}%
(t;k_{0},\sigma)$ where $\mathcal{K}$ is the flow associated with the
differential equation (<ref>). We also define a function $\varphi\,$ by:
\begin{align*}
u_{1}^{\prime}(\varphi(x,k),k)+U_{1}^{\prime}\left( \varphi(x,k),k\right)
& =x,\\
\varphi(u_{1}^{\prime}(c,k)+U_{1}^{\prime}\left( c,k),k\right) & =c
\end{align*}
Since $u_{1}^{\prime\prime}$ and $U_{1}^{\prime\prime}$ are both negative, the
function $\varphi$ is well-defined by the implicit function theorem.
Let $\sigma$ be an equilibrium strategy. The function:
\begin{equation}
(t;k_{0},\sigma)),\mathcal{K}(t;k_{0},\sigma))dt \label{value.function}%
\end{equation}
satisfies the integral equation:
\begin{equation}%
\begin{array}
V(k_{0})=\int_{0}^{\infty}h(t)u(\varphi\circ V^{\prime}(\mathcal{K}%
(t;k_{0},\varphi\circ V^{\prime})),\mathcal{K}(t;k_{0},\varphi\circ V^{\prime
\quad\quad\quad\ \ +\int_{0}^{\infty}H(t)U(\varphi\circ V^{\prime}%
(\mathcal{K}(t;k_{0},\varphi\circ V^{\prime})),\mathcal{K}(t;k_{0}%
,\varphi\circ V^{\prime}))dt
\end{array}
\tag{IE}\label{IE}%
\end{equation}
and the instantaneous optimality condition
\begin{equation}
\label{instantaneous.condition}%
\end{equation}
Conversely, suppose a function $V$ is twice continuously differentiable,
satisfies (IE), and the strategy $\sigma(k_{0}):=\varphi(V^{\prime}%
(k_{0}),k_{0})$ is convergent. Then $\sigma$ is an equilibrium strategy.
For the sake of convenience, we have shortened $\sigma(k_{0}):=\varphi
(V^{\prime}(k_{0}),k_{0})$ to $\sigma=\varphi\circ V^{\prime}$.
Since the system is autonomous, we have:
\begin{equation}
\mathcal{K}(s;\mathcal{K}(t;k_{0},\sigma),\sigma)=\mathcal{K}(s+t;k_{0}%
,\sigma). \label{property.1.of.K}%
\end{equation}
Next, denote the fundamental solution of the linearized equation of
(<ref>) at $k_{0}$ by $\mathcal{R}(k_{0};t)$ so that:
\begin{align*}
k_{1}(t) & =\mathcal{R}(k_{0};t)(\sigma(k_{0})-c)\\
\frac{d\mathcal{R}}{dt} & =(f^{\prime}(\mathcal{K}(t;k_{0},\sigma
\mathcal{R}(k_{0};0)=I,
\end{align*}
$\mathcal{R}$ and $\mathcal{K}$ are related by:
\begin{align}
\frac{\partial\mathcal{K}(t;k_{0},\sigma)}{\partial k_{0}} & =\mathcal{R}%
\frac{\partial\mathcal{K}(t;k_{0},\sigma)}{\partial t} & =f(\mathcal{K}%
(t;k_{0},\sigma))-\sigma(\mathcal{K}(t;k_{0},\sigma)). \label{property.3.of.K}%
\end{align}
Let us now turn to the first part of the theorem. Differentiating
(<ref>)with respect to $k_{0}$:
\begin{align}
V^{\prime}(k_{0}) & =\int_{0}^{\infty}h(t)u_{1}^{\prime}(\sigma
& +\int_{0}^{\infty}h(t)u_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{1}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
,\sigma)),\mathcal{K}(t;k_{0},\sigma))\mathcal{R}(k_{0};t)dt. \label{c5}%
\end{align}
Substituting $k_{0}(t)=\mathcal{K}(t;k_{0},\sigma)$ and $k_{1}(t)$ in the
definition of $P$, we get:
\begin{align}
P(k_{0},\sigma,c) & =\lbrack u(c,k_{0})+U(c,k_{0})]-[u(\sigma(k_{0}%
& +\int_{0}^{\infty}h(t)u_{1}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}h(t)u_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{1}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
\end{align}
Since $u$ and $U$ are strictly concave and differentiable with respect to $c$,
the necessary and sufficient condition to maximize $P(k_{0},\sigma,c)$ with
respect to $c$ is that the derivative vanishes at $c=\sigma(k_{0})$, that is:
\begin{align*}
u_{1}^{\prime}(\sigma(k_{0}),k_{0})+U_{1}^{\prime}(\sigma(k_{0}),k_{0})= &
\int_{0}^{\infty}h(t)u_{1}^{\prime}(\sigma(\mathcal{K}(t;k_{0},\sigma
& +\int_{0}^{\infty}h(t)u_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{1}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
& +\int_{0}^{\infty}H(t)U_{2}^{\prime}(\sigma(\mathcal{K}(t;k_{0}%
\end{align*}
The right-hand side is precisely $V^{\prime}(k_{0})$, as we wanted. Therefore,
the equilibrium strategy satisfies
\[
\]
and we have $\sigma(k_{0})=\varphi(V^{\prime}(k_{0}),k_{0})$. Substituting
back into equation (<ref>), we get the functional equation
(IE). This prove the first part of the theorem (necessity). We refer to
<cit.> for the second part (sufficiency).
The following theorem gives an alternative characterization, the differential
equation, which resembles the usual HJB equation from the calculus of variation.
Let $V$ be a $C^{2}$ function such that the strategy
$\sigma=\varphi\circ V^{\prime}$ converges to $\bar{k}$. Then $V$ satisfies
the integral equation (<ref>) if and only if it satisfies the following
integro-differential equation
\begin{equation}%
\begin{array}
(t;k_{0},\varphi\circ V^{\prime})),\mathcal{K}(t;k_{0},\varphi\circ V^{\prime
(t;k_{0},\varphi\circ V^{\prime})),\mathcal{K}(t;k_{0},\varphi\circ V^{\prime
\end{array}
\tag{DE}\label{DE}%
\end{equation}
together with the boundary condition
\begin{equation}
{k})\int_{0}^{\infty}H(t)dt. \tag{BC}\label{BC}%
\end{equation}
Introduce the function $\phi$ defined by
\begin{equation}
\phi(k_{0})=V(k_{0})-\int_{0}^{\infty}h(t)u(\sigma(\mathcal{K}(t;k_{0}%
\end{equation}
where $\sigma(k_{0})=\varphi(V^{\prime}(k_{0}),k_{0})$. Consider the value
$\psi(t,k_{0})$ of $\phi$ along the trajectory $t\rightarrow\mathcal{K}%
(t;k_{0},\sigma)$ originating from $k_{0}$ at time $0$, that is
\begin{align*}
\psi(t,k_{0}) & =\phi(\mathcal{K}(t;k_{0},\sigma))\\
& =V(\mathcal{K}(t;k_{0},\sigma))-\int_{t}^{\infty}h(s-t)u(\sigma
& \quad-\int_{t}^{\infty}H(s-t)U(\sigma(\mathcal{K}(\sigma;s,k_{0}%
\end{align*}
Then the derivative of $\psi$ with respect to $t$ is given by:
\begin{align}
\frac{\partial\psi(t,k_{0})}{\partial{t}}= & V^{\prime}(\mathcal{K}%
\circ{V^{\prime}}(\mathcal{K}_{t}),\mathcal{K}_{t})\nonumber\\
& +\int_{0}^{\infty}h^{\prime}(s)u(\varphi\circ{V^{\prime}}(\mathcal{K}%
& +\int_{0}^{\infty}H^{\prime}(s)U(\varphi\circ{V^{\prime}}(\mathcal{K}%
\end{align}
where we have denoted $\mathcal{K}(t;k_{0},\varphi\circ V^{\prime})$ by
$\mathcal{K}_{t}$ for convenience. If (<ref>) holds, then the right hand
side is identically zero along the trajectory, so that $\psi(t,k_{0})$ does
not depend on $t$, thus $\psi(s,k_{0})=\psi(t,k_{0})$ for all $s,t\geq0$.
Letting $t\rightarrow{\infty}$ in the definition of $\psi$, we get
\begin{align}
\psi(s,k_{0}) & =\lim_{t\rightarrow{\infty}}\psi(t,k_{0})\nonumber\\
& =V(\bar{k})-\int_{0}^{\infty}h(s)u(\sigma(\bar{k}),\bar{k})ds-\int%
\end{align}
and hence, if (BC) holds, then $\psi=\phi\equiv0$ and so equation (IE) holds.
Conversely, if $V(k)$ satisfies equation (IE), then the same lines of
reasoning shows that equation (DE) and the boundary condition (BC) are satisfied.
§.§ The Euler equations
To obtain the Euler-Lagrange-like equation, we differentiate the both side of
(<ref>) with respect to $k_{0}$. We get:
\begin{align*}
& -\int_{0}^{\infty}h^{\prime}(t)u_{1}^{\prime}(\sigma(\mathcal{K}%
(t;k_{0},\sigma))\frac{\partial\mathcal{K}(t;k_{0},\sigma)}{\partial k_{0}%
& -\int_{0}^{\infty}h^{\prime}(t)u_{2}^{\prime}(\sigma(\mathcal{K}%
(t;k_{0},\sigma)}{\partial k_{0}}dt\\
& -\int_{0}^{\infty}H^{\prime}(t)U_{1}^{\prime}(\sigma(\mathcal{K}%
(t;k_{0},\sigma))\frac{\partial\mathcal{K}(t;k_{0},\sigma)}{\partial k_{0}%
& -\int_{0}^{\infty}H^{\prime}(t)U_{2}^{\prime}(\sigma(\mathcal{K}%
(t;k_{0},\sigma)}{\partial k_{0}}dt\\
& =[u_{1}^{\prime}(\sigma(k_{0}),k_{0})+U_{1}^{\prime}(\sigma(k_{0}%
& \quad+V^{\prime}(k_{0})f^{\prime}(k_{0})-V^{\prime}(k_{0})\sigma^{\prime
\end{align*}
Plugging $k_{0}=k(t)$, $\sigma(k(t))=c(t)$ and using
(<ref>) to cancel the first and the fifth terms,
together with (<ref>) and (<ref>), we have
\begin{align*}
& -\int_{0}^{\infty}h^{\prime}(s)u_{1}^{\prime}(\sigma(\mathcal{K}%
(s;k(t),\sigma))\mathcal{R}(k\left( t\right) ,s)ds\\
& -\int_{0}^{\infty}h^{\prime}(s)u_{2}^{\prime}(\sigma(\mathcal{K}%
(s;k(t),\sigma)),\mathcal{K}(s;k(t),\sigma))\mathcal{R}(k\left( t\right)
& -\int_{0}^{\infty}H^{\prime}(s)U_{1}^{\prime}(\sigma(\mathcal{K}%
(s;k(t),\sigma))\mathcal{R}(k\left( t\right) ,s)ds\\
& -\int_{0}^{\infty}H^{\prime}(s)U_{2}^{\prime}(\sigma(\mathcal{K}%
(s;k(t),\sigma)),\mathcal{K}(s;k(t),\sigma))\mathcal{R}(k\left( t\right)
& =u_{2}^{\prime}(c(t),k(t))+U_{2}^{\prime}(c(t),k(t))+V^{\prime
& =u_{2}^{\prime}(c(t),k(t))+U_{2}^{\prime}(c(t),k(t))+[u_{1}^{\prime
& \quad+\frac{d}{dt}[u_{1}^{\prime}(c(t),k(t))+U_{1}^{\prime}(c(t),k(t))],
\end{align*}
where to get the last two terms in the right hand side, we have used
(<ref>) again.
We have $\mathcal{K}(s;k(t),\sigma)=k(s+t)$, and $\mathcal{R}(k\left(
t\right) ;s)=\mathcal{R}(k_{0};s+t)$
\begin{align}
\label{sigma'.of.kc}\sigma^{\prime}\left( k\left( t\right) \right) &
\mathcal{R}(k_{0};s+t) & =\exp(f^{\prime}(k\left( t\right) )-\frac
\end{align}
Writing this into the preceding equations, we finally get:
\begin{align}
& -\int_{t}^{\infty}h^{\prime}(s-t)\left[ u_{1}^{\prime}(c(s),k(s))\gamma
\left( s\right) +u_{2}^{\prime}(c(s),k(s))\right] e^{f^{\prime
}(k(s))-\gamma\left( s\right) }ds\label{a15}\\
& -\int_{t}^{\infty}H^{\prime}(s-t)\left[ U_{1}^{\prime}(c(s),k(s))\gamma
\left( s\right) +U_{2}^{\prime}(c(s),k(s))\right] e^{f^{\prime
}(k(s))-\gamma\left( s\right) }ds\label{a16}\\
& =u_{2}^{\prime}(c(t),k(t))+U_{2}^{\prime}(c(t),k(t))+\left[ u_{1}^{\prime
}(c(t),k(t))+U_{1}^{\prime}(c(t),k(t))\right] f^{\prime}(k(t))\label{a17}\\
& +\frac{d}{dt}[u_{1}^{\prime}(c(t),k(t))+U_{1}^{\prime}(c(t),k(t))]
\label{a18}%
\end{align}
\begin{equation}
\gamma\left( s\right) :=\frac{1}{f(k(s))-c(s)}\frac{dc}{ds} \label{a19}%
\end{equation}
which is the Euler-Lagrange-like equation for the time-inconsistent case.
§.§ The control theory approach
Karp <cit.>, <cit.> has developed a different method to deal with
time-inconsistency. In this section, we connect his results with ours.
Defining $V(k_{0})$ as above, we must have:
\begin{equation}
V(k_{0})=\max_{c}\left\{ u(c,k_{0})+U(c,k_{0})]\varepsilon+\int_{\varepsilon
(t))]dt\right\} \label{karp1}%
\end{equation}
On the other hand, we have
\[
\]
\begin{equation}
\begin{array}
\lbrack u(c,k_{0})+U(c,k_{0})]\varepsilon+V(k_{c}(\varepsilon))\\
\end{array}
\right\} \nonumber\label{karp3}%
\end{equation}
Letting $\varepsilon\rightarrow0$, we have
\begin{align}
& -\int_{0}^{\infty}h^{\prime}(t)u(\sigma(k(t)),k(t))dt-\int_{0}^{\infty
& =\max_{c}\{u(c,k_{0})+U(c,k_{0})+V^{\prime}(k_{0})(f(k_{0})-c)\}. \label{K}%
\end{align}
This equation was first obtained by Karp <cit.>, <cit.>. This is the
HJB equation for a certain control problem, which he terms the auxiliary
control problem. We now show that this approach is equivalent to the preceding
one. approach by Ekeland-Lazrak. In the right hand side of (<ref>), the
maximum is attained at:
\[
c=\ \arg\max_{c}\{u(c,k_{0})+U(c,k_{0})+V^{\prime}(k_{0})(f(k_{0}%
\]
thus we have
\begin{align*}
& -\int_{0}^{\infty}h^{\prime}(t)u(\sigma(k(t)),k(t))dt-\int_{0}^{\infty
& =u(\varphi(V^{\prime}(k_{0}),k_{0}),k_{0})+U(\varphi(V^{\prime}%
\end{align*}
which is exactly the same as (DE)
§ THE BIEXPONENTIAL CASE
§.§ The equations
In this section, we consider the biexponential criterion
\begin{equation}
\lambda\int_{0}^{\infty}e^{-\delta_{1}s}u(c(s),k(s))ds+(1-\lambda)\int%
_{0}^{\infty}e^{-\delta_{2}s}U(c(s),k(s))ds. \label{biexponential.criterion}%
\end{equation}
Without loss of generality, we assume that:
\[
\delta_{1}>\delta_{2}%
\]
If $\lambda=0$ or $1$, then this is just the Ramsey criterion (<ref>). Thus
we are interested in the case $0<\lambda<1$. Dividing by $\lambda$, we find
that (<ref>) is a special case of (<ref>),
where $h(t)=e^{-\delta_{1}t}$, $H(t)=e^{-\delta_{2}t}$ and $U$ is replaced by
$\frac{1-\lambda}{\lambda}U$. So all the results of the preceding section hold.
§.§.§ The HJB-type equations
Given an equilibrium strategy $\sigma_{\lambda}$, introduce the two functions:
\begin{align}
V_{\lambda}\left( k\right) & :=\int_{0}^{\infty}e^{-\delta_{1}t}%
u(\sigma_{\lambda}\left( k(t)\right) ,k(t))dt+\frac{1-\lambda}{\lambda}%
\int_{0}^{\infty}e^{-\delta_{2}t}U(\sigma_{\lambda}\left( k(t)\right)
W_{\lambda}\left( k\right) & :=\int_{0}^{\infty}e^{-\delta_{1}t}%
u(\sigma_{\lambda}\left( k(t)\right) ,k(t))dt-\frac{1-\lambda}{\lambda}%
\int_{0}^{\infty}e^{-\delta_{2}t}U(\sigma_{\lambda}\left( k(t)\right)
,k(t))dt \label{26a}%
\end{align}
In Proposition <ref>, we prove that the HJB-type
equation (DE) reduces to a system of two ODEs for $V_{\lambda}$ and
\begin{align}
\left( f-\varphi_{\lambda}\left( V_{\lambda}^{\prime}\right) \right)
V_{\lambda}^{\prime}+u\left( V_{\lambda}^{\prime}\right) +\frac{1-\lambda
}{\lambda}U\left( V_{\lambda}^{\prime}\right) = & \frac{\delta_{1}%
\left( f-\varphi_{\lambda}\left( V_{\lambda}^{\prime}\right) \right)
W_{\lambda}^{\prime}+u\left( V_{\lambda}^{\prime}\right) -\frac{1-\lambda
}{\lambda}U\left( V_{\lambda}^{\prime}\right) = & \frac{\delta_{1}%
\label{36}%
\end{align}
where $\varphi_{\lambda}\left( V_{\lambda}^{\prime}\right) $ and $u\left(
V_{\lambda}^{\prime}\right) $ denote the functions $k\rightarrow
\varphi_{\lambda}\left( V_{\lambda}^{\prime}\left( k\right) ,k\right) $
and $k\rightarrow u\left( \varphi_{\lambda}\left( V^{\prime}\right)
,k\right) $. Recall that $\varphi_{\lambda}$ is defined by the equivalent
\begin{align}
U_{1}^{\prime}\left( \varphi_{\lambda}(x,k),k\right) & =x,\label{39}\\
\varphi_{\lambda}(u_{1}^{\prime}(c,k)+\frac{1-\lambda}{\lambda}U_{1}^{\prime
}\left( c,k),k\right) & =c \label{40}%
\end{align}
Similarly, the boundary condition (BC) becomes:
\begin{align}
V_{\lambda}\left( k_{\infty}\right) & =\frac{1}{\delta_{1}}u\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) +\frac{1-\lambda}{\lambda
}\frac{1}{\delta_{2}}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right)
W_{\lambda}\left( k_{\infty}\right) & =\frac{1}{\delta_{1}}u\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) -\frac{1-\lambda}{\lambda
}\frac{1}{\delta_{2}}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right)
. \label{38}%
\end{align}
If $\lambda=1$, we find the usual HJB equation (<ref>) for $V$ with the
boundary condition (<ref>).
Suppose that $\sigma_{\lambda}$ is an
equilibrium strategy such that $k\left( t\right) $ converges to $k_{\infty}%
$. Then $V_{\lambda}$ and $W_{\lambda}$ defined by (<ref>) and (<ref>)
satisfy the equations (<ref>) and (<ref>) with the boundary conditions
(<ref>) and (<ref>). Conversely, suppose there is a point $k_{\infty}$,
a $C^{2}$ function $V_{\lambda}$ and a $C^{1}$ function $W_{\lambda}$, both
defined on some open neighbourhood of $k_{\infty}$, satisfying the equations
(<ref>) and (<ref>) with the boundary conditions (<ref>) and
(<ref>); suppose moreover that strategy $\sigma_{\lambda}\left( k\right)
:=\varphi_{\lambda}\left( V_{\lambda}^{\prime}\left( k\right) ,k\right) $
converges to $\left( f\left( k_{\infty}\right) ,k_{\infty}\right) $. Then
$\sigma_{\lambda}$ is an equilibrium strategy.
Let us draw the reader's attention to the fact that $V_{\lambda}$ must be
$C^{2}$ while $W_{\lambda}$ needs only be $C^{1}$.
Let us simplify the notation. Write $\sigma$ instead of $\sigma_{\lambda}$and
\[
a=\frac{\delta_{1}+\delta_{2}}{2},\ \ b=\frac{\delta_{1}-\delta_{2}}{2}%
\]
Arguing as in Theorem <ref>, we obtain (<ref>) and
(<ref>) by differentiating (<ref>) and (<ref>). The
boundary conditions (<ref>) and (<ref>) follow from setting $k\left(
t\right) =k_{\infty}$ and $c\left( t\right) =\sigma\left( k_{\infty
}\right) $ in (<ref>) and (<ref>).
Conversely suppose $v_{1}$ and $w_{1}$ satisfy (<ref>), (<ref>) and
(<ref>), (<ref>), and suppose the strategy $\sigma_{1}=\varphi_{\lambda
}\circ{v_{1}^{\prime}}$ converges to $k$. Consider the following functions
\begin{align}
v_{2}(k_{0}) = & \int_{0}^{\infty}e^{-\delta_{1}{t}}u(\sigma_{1}%
& +\frac{1-\lambda}{\lambda}\int_{0}^{\infty}e^{-\delta_{2}t}U(\sigma
w_{2}(k_{0}) = & \int_{0}^{\infty}e^{-\delta_{1}{t}}u(\sigma_{1}%
& -\frac{1-\lambda}{\lambda}\displaystyle\int_{0}^{\infty}e^{-\delta_{2}%
\end{align}
Arguing as in Theorem <ref>, we find that $v_{2}$ and $w_{2}$ also
satisfy (<ref>), (<ref>) and (<ref>), (<ref>). Setting
then we get
\begin{equation}%
\begin{array}
\left( f\left( k\right) -\sigma\left( k\right) \right) v_{3}^{\prime
}\left( k\right) +u\left( \sigma\left( k\right) ,k\right) +\frac
{1-\lambda}{\lambda}U\left( \sigma\left( k\right) ,k\right) =av_{3}\left(
k\right) +bw_{3}\left( k\right) \\
\left( f\left( k\right) -\sigma\left( k\right) \right) w_{3}^{\prime
}\left( k\right) +u\left( \sigma\left( k\right) ,k\right) +\frac
{1-\lambda}{\lambda}U\left( \sigma\left( k\right) ,k\right) =bv_{3}\left(
k\right) +aw_{3}\left( k\right)
\end{array}
\label{a28}%
\end{equation}
with the boundary conditions
\begin{equation}%
\begin{array}
v_{3}\left( k_{\infty}\right) =0\\
w_{3}\left( k_{\infty}\right) =0
\end{array}
\label{a29}%
\end{equation}
Obviously, $v_{3}=w_{3}=0$ is a solution. Lemma <ref> below shows that
We need to show that it is the only one, so that $v_{1}=v_{2}$ and
$w_{1}=w_{2}$. This is the desired result.
If $(v_{3},w_{3})$ is a pair of continuous functions on a
neighborhood $k_{\infty}$, continuously differentiable for $k\neq k_{\infty}$,
and which solve (<ref>) with boundary conditions (<ref>), then
Set $f(k)-\sigma_{1}(k)=\xi(k)$, then $\xi(k)\rightarrow0$ as $k\rightarrow
k_{\infty}$. The system (<ref>) can be rewritten as:
\begin{equation}
\begin{array}
\xi v_{3}^{\prime}\\
\xi w_{3}^{\prime}%
\end{array}
\right) }={\left(
\begin{array}
a & b\\
b & a
\end{array}
\right) }{\left(
\begin{array}
\end{array}
\right) }.\nonumber
\end{equation}
\[
\begin{array}
\end{array}
\right) }={\left(
\begin{array}
\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}}\\
-\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}}%
\end{array}
\right) }{\left(
\begin{array}
\end{array}
\right) },
\]
we have:
\begin{equation}
\xi{\left(
\begin{array}
\end{array}
\right) }={\left(
\begin{array}
\delta_{1} & 0\\
0 & \delta_{2}%
\end{array}
\right) }{\left(
\begin{array}
\end{array}
\right) }. \nonumber\label{Matrix2}%
\end{equation}
The first equation yields $V(k)=V(k_{0})\exp\int_{k_{0}}^{k}\frac{\delta_{1}%
}{\xi(u)}du$. Without loss of generality, we can assume that $k_{0}<k_{\infty
Let $S=\{k\ |\ \xi(k)=0,\ k_{0}\leq k\leq k_{\infty}\}$, where $k_{0}$ is the
initial stock. We consider the following two cases:
First case:$\ S=\{k_{\infty}\}$. Then $\xi(k)>0$ for $k\in\lbrack
k_{0},k_{\infty})$. So $\frac{dk}{dt}=\xi(k)>0$, and:
\[
\]
It follows that $V(k)=0$ for all $k$.
Second case:$\ S\backslash\{k_{\infty}\}\neq\emptyset$. Then $S$ is a
closed set and $V(k)=\frac{\xi(k)V^{\prime}(k)}{\delta_{1}}=0$ in $S$. The
complement of $S$ is a countable union of disjoint intervals, and $\xi$
vanishes at the endpoints of each interval. Arguing as above it follows that
$V\left( k\right) $ vanishes for $k_{0}\leq k\leq k_{\infty}$.
The same argument holds for $W\left( k\right) $. This concludes the proof.
§.§.§ The Euler-type equations
The Euler-type equations (<ref>) to (<ref>) can be reduced to a
non-autonomous system of four first-order ODEs. It is better to work directly
on (<ref>) and (<ref>) to get an autonomous system. Proceeding as in the
end of Section 2, we differentiate (<ref>) and (<ref>) w.r.t. $k$,
and notice that $(f-\sigma)V_{\lambda}''=\frac{dk}{dt}\frac{V_{\lambda}'}{dk}=\frac{d}{dt}V_{\lambda}'$,
we get:
\begin{align*}
\frac{\delta_{1}+\delta_{2}}{2}V_{\lambda}^{\prime}+\frac{\delta_{1}%
-\delta_{2}}{2}W_{\lambda}^{\prime} & =\frac{d}{dt}V_{\lambda}^{\prime
\frac{\delta_{1}-\delta_{2}}{2}V_{\lambda}^{\prime}+\frac{\delta_{1}%
+\delta_{2}}{2}W_{\lambda}^{\prime} & =\frac{d}{dt}W_{\lambda}^{\prime
\end{align*}
where $c=\varphi\left( V_{\lambda}^{\prime}\left( k\right) ,k\right) $ and
$\sigma^{\prime}$ is given by (<ref>). Using formula (<ref>),
and setting $w\left( t\right) :=W_{\lambda}^{\prime}\left( k\left(
t\right) \right) $, this becomes:
\begin{align}
\frac{\delta_{1}+\delta_{2}}{2}\left(u_{1}^{\prime
}+\frac{1-\lambda}{\lambda}U_{1}^{\prime}\right) +\frac{\delta_{1}%
-\delta_{2}}{2}w & =\frac{d}{dt}\left(u_{1}^{\prime
\nonumber\\
}+\frac{1-\lambda}{\lambda}U_{1}^{\prime}\right)\sigma' +u_{2}^{\prime
\nonumber\\
\frac{\delta_{1}-\delta_{2}}{2}\left(u_{1}^{\prime
}+\frac{1-\lambda}{\lambda}U_{1}^{\prime}\right) +\frac{\delta_{1}+\delta_{2}}%
{2}w & =\frac{dw}{dt}+(f'-\sigma')w
\nonumber\\
& \quad+\left(u_{1}^{\prime}-\frac{1-\lambda}{\lambda}U_{1}^{\prime}\right)\sigma^{\prime}
\end{align}
and hence
\begin{align}
(\frac{\delta_{1}+\delta_{2}}{2}-f^{\prime})\left( \lambda u_{1}^{\prime
}+\left( 1-\lambda\right) U_{1}^{\prime}\right) +\frac{\delta_{1}%
-\delta_{2}}{2}\lambda w & =\frac{d}{dt}\left( \lambda u_{1}^{\prime
}+\left( 1-\lambda\right) U_{1}^{\prime}\right) +\lambda u_{2}^{\prime
}+\left( 1-\lambda\right) U_{2}^{\prime},\label{42}\\
\frac{\delta_{1}-\delta_{2}}{2}\left( \lambda u_{1}^{\prime}+\left(
1-\lambda\right) U_{1}^{\prime}\right) +(\frac{\delta_{1}+\delta_{2}}%
{2}-f^{\prime})\lambda w & =\lambda\frac{dw}{dt} +(-\lambda w+\lambda
u_{1}^{\prime}-\left( 1-\lambda\right) U_{1}^{\prime})\sigma^{\prime
& \quad+\lambda u_{2}^{\prime}-\left( 1-\lambda\right) U_{2}^{\prime}.
\label{43}%
\end{align}
to which we should add:
\begin{equation}
\frac{dk}{dt}=f\left( k\right) -c \label{44}%
\end{equation}
The system (<ref>) to (<ref>) is a system of three first-order ODEs for
the unknown functions $k\left( t\right) ,c\left( t\right) $ and $w\left(
t\right) $. Note that for $\lambda=0$, (<ref>) and (<ref>) reduce to
two copies of the usual Euler-Lagrange equation. For $\lambda=1$, taking
$w\left( t\right) =u_{1}^{\prime}\left( c\left( t\right) ,k\left(
t\right) \right) $ gives us again two copies of the same equation.
§.§.§ The control theory approach
Equation (<ref>) is the HJB equation for the following:
\begin{align}
& \max_{c\left( .\right) }\int_{0}^{\infty}e^{-rt}\left[ u\left( c\left(
t\right) ,k\left( t\right) \right) +\frac{1-\lambda}{\lambda}U\left(
c\left( t\right) ,k\left( t\right) \right) -K\left( k\left( t\right)
\right) \right] dt\label{45}\\
& \frac{dk}{dt}=f\left( k\right) -c\left( t\right) ,\ \ k\left(
0\right) =k_{0} \label{46}%
\end{align}
where we seek a feedback $\sigma\left( k\right) $ such that:
\begin{equation}
K(k_{0})=\left\{ (\delta-r)\int_{0}^{\infty}e^{-\delta t}u(\sigma\left(
k\left( t\right) \right) ,k\left( t\right) )dt\ |\
\begin{array}
\frac{dk}{dt}=f\left( k\right) -\sigma\left( k\right) \\
k\left( 0\right) =k_{0}%
\end{array}
\right\} \label{47}%
\end{equation}
Solving (<ref>), (<ref>) under the constraint (<ref>) is a
fixed-point problem for the feedback $c=\sigma\left( k\right) $.
§.§ Solving the boundary-value problem
Define a function $\overline{g}_{\lambda}(k)$ by:
\begin{equation}
\overline{g}_{\lambda}(k):=\frac{\lambda\delta_{1}{u_{1}^{\prime}}\left(
f\left( k\right) ,k\right) +(1-\lambda)\delta_{2}U_{1}^{\prime}\left(
f\left( k\right) ,k\right) }{\lambda u_{1}^{\prime}\left( f\left(
k\right) ,k\right) +(1-\lambda)U_{1}^{\prime}\left( f\left( k\right)
,k\right) }-\frac{\lambda u_{2}^{\prime}\left( f\left( k\right) ,k\right)
+(1-\lambda)U_{2}^{\prime}\left( f\left( k\right) ,k\right) }{\lambda
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +(1-\lambda)U_{1}^{\prime
}\left( f\left( k\right) ,k\right) } \label{bar.g}%
\end{equation}
Assume $\overline{g}_{\lambda}(k)\neq0$. If there
is some $k_{\infty}$ such that
\begin{equation}
f^{\prime}(k_{\infty})\ne\overline{g}_{\lambda}(k_{\infty}), \label{53}%
\end{equation}
then the equations (<ref>) and (<ref>) with the boundary conditions
(<ref>) and (<ref>) have a solution $\left( V,W\right) $ near the
point $k_{\infty}$ with $V$ of class $C^{2}$ and $W$ of class $C^{1.}$
Proof. We adapt the argument in <cit.> to the
present situation. We note that the boundary-value problem (<ref>),
(<ref>), (<ref>), (<ref>) cannot be reduced to a standard
initial-value problem for the pair $\left( V_{\lambda},W_{\lambda}\right) $.
To see that, rewrite equation (<ref>) as follows:
\[
\left( f\left( k\right) -\varphi\left( V_{\lambda}^{\prime},k\right)
\right) V_{\lambda}^{\prime}+u(\varphi\left( V_{\lambda}^{\prime},k\right)
,k)+\frac{1-\lambda}{\lambda}U(\varphi\left( V_{\lambda}^{\prime},k\right)
\]
Since the function $c\rightarrow u(c,k)+\frac{1-\lambda}{\lambda}U(c,k)$ is
concave, the function:
\[
c\rightarrow u(c,k)+\frac{1-\lambda}{\lambda}U(c,k)-cx
\]
attains its maximum at the point $c=\varphi_{\lambda}\left( x,k\right) $
defined by (<ref>). We set:
\begin{equation}
u_{\lambda}^{\ast}\left( x,k\right) =\max_{c}\left\{ u(c,k)+\frac
{1-\lambda}{\lambda}U(c,k)-cx\right\} =u\left( \varphi_{\lambda}\left(
x,k\right) ,k\right) +\frac{1-\lambda}{\lambda}U\left( \varphi_{\lambda
}\left( x,k\right) ,k\right) -\varphi_{\lambda}\left( x,k\right) x
\label{52}%
\end{equation}
The function $x\rightarrow u_{\lambda}^{\ast}\left( x,k\right) $ is convex,
and the equation (<ref>) becomes:
\begin{equation}
f\left( k\right) V_{\lambda}^{\prime}+u_{\lambda}^{\ast}\left( V_{\lambda
}^{\prime},k\right) =aV_{\lambda}+bW_{\lambda}. \label{41}%
\end{equation}
This is an equation for $V_{\lambda}^{\prime}$. From the basic duality results
in convex analysis (see for instance <cit.>), we find that:
\[
\min_{y}\left\{ f\left( k\right) y+u_{\lambda}^{\ast}\left( y,k\right)
\right\} =u(f\left( k\right) ,k)+\frac{1-\lambda}{\lambda}U(f\left(
k\right) ,k).
\]
Note that $f(k)y+u_{\lambda}^{\ast}\left( y,k\right) $ is convex in $y$ with
minimal value $u(f(k),k)+\frac{1-\lambda}{\lambda}U(f(k),k)$. Then (<ref>),
considered as an equation for $V_{\lambda}^{\prime}$, has two solutions if:
\[
aV_{\lambda}\left( k\right) +bW_{\lambda}\left( k\right) >u(f\left(
k\right) ,k)+\frac{1-\lambda}{\lambda}U(f\left( k\right) ,k),
\]
and no solutions if the inverse inequality holds. If we have equality:
\[
aV_{\lambda}\left( k\right) +bW_{\lambda}\left( k\right) =u(f\left(
k\right) ,k)+\frac{1-\lambda}{\lambda}U(f\left( k\right) ,k)
\]
then equation (<ref>) has precisely one solution, namely:
\[
V_{\lambda}^{\prime}=u_{1}^{\prime}(f\left( k\right) ,k)+\frac{1-\lambda
}{\lambda}U_{1}^{\prime}(f\left( k\right) ,k)
\]
At the point $k_{\infty}$, with the values (<ref>), (<ref>), we find
\[
aV_{\lambda}\left( k_{\infty}\right) +bW_{\lambda}\left( k\right)
=u\left( f\left( k_{\infty}\right) ,k_{\infty}\right) +\frac{1-\lambda
}{\lambda}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) ,
\]
so that we are exactly on the boundary case. Equation (<ref>) has precisely
one solution, by (<ref>) below, which satisfies
\[
V_{\lambda}^{\prime}\left( k_{\infty}\right) =u_{1}^{\prime}(f\left(
k_{\infty}\right) ,k_{\infty})+\frac{1-\lambda}{\lambda}U_{1}^{\prime
}(f\left( k_{\infty}\right) ,k_{\infty}),
\]
but it is degenerate:
\[
\frac{\partial}{\partial y}\left[ f\left( k_{\infty}\right) y+u_{\lambda
}^{\ast}\left( y,k_{\infty}\right) \right] |_{y=V_{\lambda}^{\prime}\left(
k_{\infty}\right) }=0,
\]
so that equation (<ref>) cannot be written in the form $V_{\lambda}%
^{\prime}=\psi\left( k,V_{\lambda}\right) $. It is an implicit differential
equation, and special techniques are needed to solve it. The difficulty is
further increased by the fact that we need only $V$ (but not $W$) to be
The proof of Theorem <ref> proceeds in several steps.
Step 1: Changing the unknown functions from $\left( V_{\lambda
}\left( k\right) ,W_{\lambda}\left( k\right) \right) $ to
$\left( V_{\lambda}\left( k\right) ,\mu\left( k\right) \right) $
Using (<ref>), we find that the equation (<ref>) is equivalent to
\[
F\left( V^{\prime}\left( k\right) ,k\right) =\mu(k)
\]
\begin{align*}
F\left( x,k\right) & :=u_{\lambda}^{\ast}\left( x,k\right) +xf\left(
k\right) -u(f(k),k)-\frac{1-\lambda}{\lambda}U(f(k),k)\\
\mu(k) & :=aV_{\lambda}(k)+bW_{\lambda}(k)-u(f(k),k)-\frac{1-\lambda
\end{align*}
Note that $x\rightarrow F\left( x,k\right) $ is a linear perturbation of the
convex function $x\rightarrow u_{\lambda}^{\ast}\left( x,k\right) $. To
shorten the notations, let us set:
\begin{equation}
\label{y.star}y^{\ast}\left( k\right) :=u_{1}^{\prime}(f\left( k\right)
,k)+\frac{1-\lambda}{\lambda}U_{1}^{\prime}(f\left( k\right) ,k)
\end{equation}
As we pointed out, the equation
\[
F(x+y^{\ast}\left( k\right) ,k)=\mu
\]
in the variable $x$ has two solutions $x_{-}\left( k,\mu\right)
<0<x_{+}\left( k,\mu\right) $ for $\mu(k)>0$, none for $\mu(k)<0$ and a
single solution $x=0$ for $\mu(k)=0$.
From the equation (<ref>), we have
\begin{equation}
}-u\left( \sigma_{\lambda}\left( k\right) ,k\right) +\frac{1-\lambda
}{\lambda}U\left( \sigma_{\lambda}\left( k\right) ,k\right) }{aV_{\lambda
}+bW_{\lambda}-u\left( \sigma_{\lambda}\left( k\right) ,k\right)
-\frac{1-\lambda}{\lambda}U\left( \sigma_{\lambda}\left( k\right)
,k\right) } \nonumber\label{w'}%
\end{equation}
Differentiating $\mu(k)$ with respect to $k$ yields
\begin{align}
\frac{d\mu}{dk} & =V_{\lambda}^{\prime}\frac{(a^{2}+b^{2})V_{\lambda
}+2abW_{\lambda}-\delta_{1}{u}\left( \sigma_{\lambda}\left( k\right)
,k\right) -\frac{(1-\lambda)\delta_{2}}{\lambda}U\left( \sigma_{\lambda
}\left( k\right) ,k\right) }{aV_{\lambda}+bW_{\lambda}-u\left(
\sigma_{\lambda}\left( k\right) ,k\right) -\frac{1-\lambda}{\lambda
}U\left( \sigma_{\lambda}\left( k\right) ,k\right) }\nonumber\\
& -y^{\ast}f^{\prime}-u_{2}^{\prime}\left( f\left( k\right) ,k\right)
-\frac{1-\lambda}{\lambda}U_{2}^{\prime}\left( f\left( k\right) ,k\right)
\nonumber
\end{align}
We now take $(V_{\lambda}(k),\mu(k))$ as our new unknown functions. They
satisfy the equations:
\begin{align*}
\frac{dV}{dk} & =y^{\ast}\left( k\right) +x\left( k,\mu\left( k\right)
\right) ,\\
\frac{d\mu}{dk} & =\left( y^{\ast}+x\left( k,\mu\left( k\right) \right)
\right) \frac{(a^{2}+b^{2})V_{\lambda}+2abW_{\lambda}-\delta_{1}{u}\left(
\sigma_{\lambda}\left( k\right) ,k\right) -\frac{(1-\lambda)\delta_{2}%
}{\lambda}U\left( \sigma_{\lambda}\left( k\right) ,k\right) }{aV_{\lambda
}+bW_{\lambda}-u\left( \sigma_{\lambda}\left( k\right) ,k\right)
-\frac{1-\lambda}{\lambda}U\left( \sigma_{\lambda}\left( k\right)
,k\right) }\\
& \quad-y^{\ast}f^{\prime}-u_{2}^{\prime}\left( f\left( k\right)
,k\right) -\frac{1-\lambda}{\lambda}U_{2}^{\prime}\left( f\left( k\right)
,k\right) .
\end{align*}
In fact, according to which determination $\left( x\left( k,\mu\left(
k\right) \right) \right) $ is chosen, $x_{+}\left( k,\mu\left( k\right)
\right) $ or $x_{-}\left( k,\mu\left( k\right) \right) $, these equations
define two distinct dynamical systems on the region $\mu>0$.
Step 3: Taking $x$ instead of $k$ as the independent
To get rid of the indetermination, we pick $x$ instead of $k$ as the
independent variable. We shorten our notation by setting $x(k)=x(\mu(k),k)$.
We get:
\begin{equation}
\frac{dk}{dx}=\frac{f(k)-\varphi_{\lambda}(y^{\ast}+x,k)}{D(x,k,V_{\lambda
},W_{\lambda})}\left[ aV_{\lambda}+bW_{\lambda}-u(\varphi_{\lambda}(y^{\ast
+x,k),k)\right] \label{dk.dx}%
\end{equation}
where (here $\varphi_{\lambda}$ stands for $\varphi_{\lambda}(y^{\ast}+x,k)$)
\begin{align*}
D(x,k,V_{\lambda},W_{\lambda}) & =(y^{\ast}+x)\left[ (a^{2}+b^{2}%
)V_{\lambda}+2abW_{\lambda}-\delta_{1}{u}\left( \sigma_{\lambda}\left(
k\right) ,k\right) -\frac{(1-\lambda)\delta_{2}}{\lambda}U\left(
\sigma_{\lambda}\left( k\right) ,k\right) \right] \\
& \quad-A(aV_{\lambda}+bW_{\lambda}-u(\varphi_{\lambda},k)-\frac{1-\lambda
A(x,k,V_{\lambda},W_{\lambda}) & =(f-\varphi_{\lambda})\left[ \left(
u_{11}^{\prime\prime}\left( f\left( k\right) ,k\right) +\frac{1-\lambda
}{\lambda}U_{11}^{\prime\prime}\left( f\left( k\right) ,k\right) \right)
f^{\prime}+u_{12}^{\prime\prime}\left( f\left( k\right) ,k\right) \right.
\\
& \quad\left. +\frac{1-\lambda}{\lambda}U_{12}^{\prime\prime}\left(
f\left( k\right) ,k\right) \right] \ +\left( y^{\ast}+x\right)
\end{align*}
Further more, we have
\begin{equation}
\frac{dV_{\lambda}}{dx}=\frac{dV_{\lambda}}{dk}\frac{dk}{dx}=(y^{\ast}%
_{\lambda},k)]. \label{dvdx}%
\end{equation}
Step 4: Rescaling the time
We introduce a new variable $s$ such that $D(x,k,V_{\lambda},W_{\lambda
})ds=dx$, then the system becomes
\[%
\begin{array}
\frac{dx}{ds} & = & D(x,k,V_{\lambda},W_{\lambda})\\
\frac{dk}{dx} & = & \left( f(k)-\varphi_{\lambda}\right) \left[
{1-\lambda}{\lambda}U(\varphi_{\lambda}(y^{\ast}+x,k),k)\right] \\
\frac{dV_{\lambda}}{dx} & = & (y^{\ast}+x)\left( f(k)-\varphi_{\lambda
}(y^{\ast}+x,k)\right) [aV_{\lambda}+bW_{\lambda}-u(\varphi_{\lambda
\end{array}
\]
We now eliminate $W_{\lambda}$, to get an equation in $\left( x,k,V\right) $
only. From the equation of $\mu\left( k\right) =F\left( x,k\right) $, we
\[
\]
The dynamics of $\left( x\left( s\right) ,k\left( s\right) ,V_{\lambda
}\left( s\right) \right) $ are given by:
\begin{equation}%
\begin{array}
\frac{dx}{ds} & = & \tilde{D}(x,k,V_{\lambda})\\
\frac{dk}{dx} & = & \left( f(k)-\varphi_{\lambda}\right) \left[ F(y^{\ast
}(k)+x,k)+u(f(k),k)+\frac{1-\lambda}{\lambda}U(f(k),k)\right. \\
& & \left. -u(\varphi_{\lambda}(y^{\ast}+x,k),k)-\frac{1-\lambda}{\lambda
}U(\varphi_{\lambda}(y^{\ast}+x,k),k)\right] \\
\frac{dV_{\lambda}}{dx} & = & (y^{\ast}+x)\left( f(k)-\varphi_{\lambda
}(y^{\ast}+x,k)\right) [F(y^{\ast}(k)+x,k)+u(f(k),k)+\frac{1-\lambda}%
& & -u(\varphi_{\lambda}(y^{\ast}+x,k),k)-\frac{1-\lambda}{\lambda}%
\end{array}
\label{sx}%
\end{equation}
where $\tilde{D}(x,k,V_{\lambda})=$ $D(x,k,V_{\lambda},W_{\lambda})$. For
$\tilde{D}$ to be $C^{2}$, we need $f$ to be $C^{3}$ and $u,U$ to be $C^{4}$.
Step 5: Linearizing the system
As we already noted, if $k=k_{\infty}$, then $\mu(k_{\infty})=0$ and
$F(y^{\ast}(k_{\infty})+x,k_{\infty})=\mu(k_{\infty})=0$ has only one solution
$x=0$. Set $v_{\infty}=V_{\lambda}(k_{\infty})$. We consider the system near
the point $(x,k,V_{\lambda})=(0,k_{\infty},v_{\infty})$. For simplicity, we
\[
\]
Computing the linearized system at $(0,k_{\infty},v_{\infty})$, we find:
\begin{equation}
\frac{d}{ds}{\left(
\begin{array}
\end{array}
\right) }={\left(
\begin{array}
a_{\infty} & b_{\infty} & c_{\infty}\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right) }{\left(
\begin{array}
\end{array}
\right) }. \nonumber\label{linearized. system.1}%
\end{equation}
\begin{align*}
a_{\infty} & =y^{\ast}(k_{\infty})^{2}\frac{\partial\varphi_{\lambda}%
}{\partial y}\left( y^{\ast}(k_{\infty}),k_{\infty}\right) (f^{\prime
\overline{g}_{\lambda}(k_{\infty}) & =\frac{\lambda\delta_{1}{u_{1}^{\prime
}}\left( f\left( k_{\infty}\right) ,k_{\infty}\right) +(1-\lambda
)\delta_{2}U_{1}^{\prime}\left( f\left( k_{\infty}\right) ,k_{\infty
}\right) } {\lambda u_{1}^{\prime}\left( f\left( k_{\infty}\right)
,k_{\infty}\right) +(1-\lambda)U_{1}^{\prime}\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) }\\
& \quad-\frac{\lambda u_{2}^{\prime}\left( f\left( k_{\infty}\right)
,k_{\infty}\right) +(1-\lambda)U_{2}^{\prime}\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) } {\lambda u_{1}^{\prime}\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) +(1-\lambda)U_{1}^{\prime}\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) }%
\end{align*}
By the concavity assumptions, we find:
\begin{align*}
& y_{\infty}=u_{1\infty}^{\prime}+\frac{1-\lambda}{\lambda}U_{1\infty
& u_{11}^{\prime}(f(k_{\infty}),k_{\infty})+\frac{1-\lambda}{\lambda}%
\frac{\partial\varphi_{\lambda}}{\partial y}\left( y^{\ast}(k_{\infty
}),k_{\infty}\right) & =\left( u_{1}^{\prime\prime}(f(k_{\infty
}),k_{\infty})\right) ^{-1}<0
\end{align*}
For any $k_{\infty}$ satisfying (<ref>) with $\overline{g}_{\lambda
}(k_{\infty})-f^{\prime}(k_{\infty})\neq0$, we then have $a_{\infty}\neq0$.
\begin{equation}
\tilde{x}=x+\frac{b_{\infty}}{a_{\infty}}(k-k_{\infty})+\frac{c_{\infty}%
}{a_{\infty}}(V_{\lambda}-v_{\infty}), \label{tilde.x}%
\end{equation}
we transform the system (<ref>) for $\left( \tilde{x}\left( s\right)
,k\left( s\right) ,V_{\lambda}\left( s\right) \right) $. The
linearization at the origin is given by:
\begin{equation}
\frac{d}{ds}{\left(
\begin{array}
\tilde{x}\\
\end{array}
\right) }={\left(
\begin{array}
})-f^{\prime}(k_{\infty})) & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right) }{\left(
\begin{array}
\tilde{x}\\
\end{array}
\right) }. \nonumber\label{linearized. system.2}%
\end{equation}
Step 6: Applying the center manifold theorem
By the center manifold theorem (cf. Theorem 1 of <cit.>), there exist an
$\epsilon>0$, and a map $h(k,V_{\lambda})$, defined in a neighborhood
}-\epsilon,v_{\infty}+\epsilon)$ of $(k_{\infty},v_{\infty})$ such that
\[
\]
and the manifold $\mathcal{M}$ defined by
\[
\mathcal{M}=\left\{ \left( h(k,V_{\lambda})-\frac{b_{\infty}}{a_{\infty}%
}),k,V_{\lambda}\right) \ |\ (k,V_{\lambda})\in\mathcal{O}\right\} ,
\]
is invariant under the flow associated to the system (<ref>). The map $h$
and the central manifold $\mathcal{M}$ are $C^{2}$, and $\mathcal{M}$ is
two-dimensional and tangent to the critical plane defined by $\tilde{x}=0$.
If $k=k_{\infty}$ and $V_{\lambda}=v_{\infty}$, then $x=0$, and $h(k_{\infty
We are interested in the solutions which lie on the central manifold
$\mathcal{M}$. Writing:
\[
\]
in the equation $\frac{dV}{dk}=y^{\ast}\left( k\right) +x$, we get
\begin{equation}
\left\{
\begin{array}
\frac{dV_{\lambda}}{dk} & = & u_{1}^{\prime}\left( f\left( k\right)
,k\right) +\frac{1-\lambda}{\lambda}U_{1}^{\prime}\left( f\left( k\right)
,k\right) +h\left( k,V_{\lambda}\right) -\frac{b_{\infty}}{a_{\infty}%
}\left( k-k_{\infty}\right) -\frac{c_{\infty}}{a_{\infty}}\left(
V_{\lambda}-v_{\infty}\right) ,\\
V_{\lambda}\left( k_{\infty}\right) & = & v_{\infty} \label{vk}%
\end{array}
\right.
\end{equation}
which can be viewed as eliminating the variable $s$ from the second and third
equations of the system (<ref>). Since $a_{\infty}\neq0$ and the right hand
side of the first equation of (<ref>) is continuously differentiable in
$\mathcal{O}_{1}=(k_{\infty}-\epsilon,k_{\infty}+\epsilon)$, therefore, is
locally Lipschiz continuous. By $\frac{d{V_{\lambda}}}{d{k}}|_{k=k_{\infty}%
}=y_{\infty}\neq0$ which follows from (<ref>) and the first equation of
(<ref>), the nonconstant solution of this initial-value problem exist in
$\mathcal{O}_{1}$ which we denote by $V_{\lambda}(k)=\zeta(k)$ ,where
$\zeta(k_{\infty})=v_{\infty}$ and $\zeta\in{C}^{2}(\mathcal{O}_{1})$ if
$h\in{C}^{2}(\mathcal{O})$. Substituting $V_{\lambda}(k)=\zeta(k)$ into
{c_{\infty}}{a_{\infty}}(V_{\lambda}-v_{\infty})$ yields
\[
\]
\[
\mu(k)=F(u_{1}^{\prime}(f(k),k)+\frac{1-\lambda}{\lambda}U_{1}^{\prime
\]
\[
\]
is also $C^{2}$, so we have found a $C^{2}$ solution of the system
(<ref>),(<ref>) with boundary conditions (<ref>),(<ref>).
§.§ The existence of equilibrium strategies
Introduce another function $\underline{g}_{\lambda}(k)$ defined by:
\begin{equation}
\underline{g}_{\lambda}(k)=\frac{\lambda{u}_{1}^{\prime}\left( f\left(
k\right) ,k\right) +(1-\lambda)U_{1}^{\prime}\left( f\left( k\right)
,k\right) }{\frac{\lambda}{\delta_{1}}u_{1}^{\prime}\left( f\left(
k\right) ,k\right) +\frac{1-\lambda}{\delta_{2}}U_{1}^{\prime}\left(
f\left( k\right) ,k\right) }-\frac{\lambda\delta_{2}u_{2}^{\prime}\left(
f\left( k\right) ,k\right) +\left( 1-\lambda\right) \delta_{1}%
U_{2}^{\prime}\left( f\left( k\right) ,k\right) }{\lambda\delta_{2}%
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\left( 1-\lambda\right)
\delta_{1}U_{1}^{\prime}\left( f\left( k\right) ,k\right) }
\label{under.g}%
\end{equation}
Suppose $f^{\prime}\left( k_{\infty}\right) $ lies between
$\underline{g}_{\lambda}(k_{\infty})$ and $\bar{g}_{\lambda}(k_{\infty})$.
Then there exists an equilibrium strategy converging to $k_{\infty}$
Comparing (<ref>) and (<ref>), we find that:
\begin{equation}
\overline{g}_{\lambda}(k)-\underline{g}_{\lambda}(k)=\lambda(1-\lambda
)(\delta_{1}-\delta_{2})\frac{\left( \delta_{1}-\delta_{2}\right)
U_{2}^{\prime}}{(\lambda u_{1}^{\prime}+(1-\lambda)U_{1}^{\prime}%
)(\lambda\delta_{2}u_{1}^{\prime}+\left( 1-\lambda\right) \delta_{1}%
\end{equation}
So the sign of $\overline{g}_{\lambda}(k)-\underline{g}_{\lambda}(k)$ is the
sign of $\left( \delta_{1}-\delta_{2}\right) u_{1}^{\prime}U_{1}^{\prime
\[%
\begin{array}
\delta_{1}-\delta_{2}>\frac{u_{2}^{\prime}}{u_{1}^{\prime}}-\frac
\delta_{1}-\delta_{2}<\frac{u_{2}^{\prime}}{u_{1}^{\prime}}-\frac
\end{array}
\]
Let us give some examples:
Example 1: $u\left( c\right) =U\left( c\right) $
In this case, $u=U$ and they do not depend on $k$. We get:
\begin{align*}
\overline{g}_{\lambda}(k) & =\lambda\delta_{1}+(1-\lambda)\delta_{2}\\
\underline{g}_{\lambda}(k) & =\frac{1}{\frac{\lambda}{\delta_{1}}%
+\frac{\left( 1-\lambda\right) }{\delta_{2}}}%
\end{align*}
These formulas do not depend on the the utility function $u\left( c\right)
$. They were first derived in <cit.> for the special case $u\left(
c\right) =\ln c$. Note that $\lambda\delta_{1}+(1-\lambda)\delta_{2}$ is the
arithmetic mean, and $\left( \frac{\lambda}{\delta_{1}}+\frac{\left(
1-\lambda\right) }{\delta_{2}}\right) ^{-1}$ is the geometric mean.
Example 2: $u\left( c,k\right) =U\left( c,k\right) $
In that case, the criterion (<ref>) becomes:
\begin{equation}
\int_{0}^{\infty}\left( \lambda e^{-\delta_{1}t}+\left( 1-\lambda\right)
e^{-\delta_{2}t}\right) u\left( c\left( t\right) ,k\left( t\right)
\right) dt \label{c1}%
\end{equation}
This problem was studied in <cit.> for $u\left( c,k\right) =U\left(
c,k\right) =\ln c$. In the case at hand, (<ref>), we find:
\begin{align*}
\overline{g}_{\lambda}(k) & =\lambda\delta_{1}+(1-\lambda)\delta_{2}%
-\frac{u_{2}^{\prime}\left( f\left( k\right) ,k\right) }{u_{1}^{\prime
}\left( f\left( k\right) ,k\right) }\\
\underline{g}_{\lambda}(k) & =\frac{1}{\frac{\lambda}{\delta_{1}}%
+\frac{\left( 1-\lambda\right) }{\delta_{2}}}-\frac{u_{2}^{\prime}\left(
f\left( k\right) ,k\right) }{u_{1}^{\prime}\left( f\left( k\right)
,k\right) }%
\end{align*}
These are the same as the preceding ones, with the corrective term
Example 3: $u=u\left( c\right) \ $and $U=U\left( c\right) $
In that case, the criterion (<ref>) becomes:
\[
\lambda\int_{0}^{\infty}e^{-\delta_{1}t}u\left( c\left( t\right) \right)
dt+\left( 1-\lambda\right) \int_{0}^{\infty}e^{-\delta_{2}t}U\left(
c\left( t\right) \right) dt
\]
We find:
\begin{align*}
\overline{g}_{\lambda}(k) & =\frac{\lambda\delta_{1}{u_{1}^{\prime}}\left(
f\left( k\right) \right) +(1-\lambda)\delta_{2}U_{1}^{\prime}\left(
f\left( k\right) \right) }{\lambda u_{1}^{\prime}\left( f\left( k\right)
\right) +(1-\lambda)U_{1}^{\prime}\left( f\left( k\right) \right) }\\
\underline{g}_{\lambda}(k) & =\frac{\lambda{u}_{1}^{\prime}\left( f\left(
k\right) \right) +(1-\lambda)U_{1}^{\prime}\left( f\left( k\right)
\right) }{\frac{\lambda}{\delta_{1}}u_{1}^{\prime}\left( f\left( k\right)
\right) +\frac{\left( 1-\lambda\right) }{\delta_{2}}U_{1}^{\prime}\left(
f\left( k\right) \right) }%
\end{align*}
Here again we find $\underline{g}_{\lambda}(k)<\overline{g}_{\lambda}(k)$.
We now proceed to the proof of Theorem
<ref>. We apply Theorem <ref>, and we denote by
$V_{\lambda}$ and $W_{\lambda}$ the solution of equations (<ref>) and
(<ref>) with the boundary conditions (<ref>) and (<ref>). Set
$\sigma_{\lambda}\left( k\right) :=\varphi_{\lambda}\left( V_{\lambda
}^{\prime}\left( k\right) ,k\right) $, where $\varphi_{\lambda}$ is defined
by (<ref>) or (<ref>). We will now show that the corresponding
trajectory $k\left( t\right) $, defined by (<ref>), converges
to $k_{\infty}$. By Proposition <ref>, this will
prove that $\sigma_{\lambda}$ is an equilibrium strategy.
We need the value of $V_{\lambda}^{\prime}(k_{\infty})$ and $\sigma_{\lambda
}^{\prime}(k_{\infty})$. Differentiating (<ref>), we find:
\begin{align}
V_{\lambda}^{\prime}(k_{\infty})f(k_{\infty}) & -V_{\lambda}^{\prime
& +\frac{1-\lambda}{\lambda}U(\varphi_{\lambda}(V_{\lambda}^{\prime
}(k_{\infty}),k_{\infty}),k_{\infty}) =u(f(k_{\infty}),k_{\infty}%
\end{align}
thus $F(V_{\lambda}^{\prime}(k_{\infty}),k_{\infty})=0$ and hence:
\begin{equation}
}(f(k_{\infty}),k_{\infty}). \label{Vr'}%
\end{equation}
To compute $\sigma_{\lambda}^{\prime}(k_{\infty})$, we consider (<ref>),
the integrated form of $V_{\lambda}^{\prime}(k)$ where we substitute
$h(t)=e^{-\delta t}$ and $H\left( t\right) =e^{-\delta_{2}t}$, and replace
$U$ with $\left( 1-\lambda\right) \lambda^{-1}$. Evaluating (<ref>) at
$k=k_{\infty}$, we get:
\begin{align}
V_{\lambda}^{\prime}(k_{\infty}) & =\frac{u_{1}^{\prime}(f(k_{\infty
& +\frac{1-\lambda}{\lambda}\frac{U_{1}^{\prime}(f(k_{\infty}),k_{\infty
}(k_{\infty})}. \label{c6}%
\end{align}
Comparing (<ref>) with (<ref>), we can solve for $\sigma_{\lambda
}^{\prime}(k_{\infty})$. We get:
\begin{equation}
\sigma_{\lambda}^{\prime}(k_{\infty})=f^{\prime}(k_{\infty})-\delta_{1}%
\delta_{2}\frac{(\frac{1}{\delta_{1}}u_{1}^{\prime}(f(k_{\infty}),k_{\infty
})-f^{\prime}(k_{\infty}))}, \label{c7}%
\end{equation}
Linearizing the equation of motion $\frac{dk}{dt}=f(k)-\sigma_{\lambda}(k)$ at
$k=k_{\infty}$ yields
\[
\frac{dk}{dt}=(f^{\prime}(k_{\infty})-\sigma_{\lambda}^{\prime}(k_{\infty
\]
Then $k$ converges to $k_{\infty}$ if $f^{\prime}(k_{\infty})-\sigma_{\lambda
}^{\prime}(k_{\infty})<0$. Comparing with (<ref>), and writing in the
expressions of $\underline{g}_{\lambda}(k_{\infty})$ and $\overline
{g}_{\lambda}(k_{\infty})$ yields:
\[
\delta_{1}\delta_{2}\frac{(\frac{1}{\delta_{1}}u_{1}^{\prime}(f(k_{\infty
\]
Because $u$ and $U$ are increasing with respect to their first variable, the
first factor in the numerator and denominator are positive. We are left with:
\[
\frac{\underline{g}_{\lambda}(k_{\infty})-f^{\prime}(k_{\infty})}{\overline
\]
and the proof is complete.
§ THE CHICHILNISKY CRITERION
In two influential papers <cit.>, <cit.>, Chichilnisky has proposed
an axiomatic approach to sustainable development, based on the twin ideas that
there should be no dictatorship of the present and no dictatorship of the
future. She suggests to use the following criterion
\begin{equation}
I_{\alpha}\left( c\left( \cdot\right) ,k\left( \cdot\right) \right)
:=\delta\int_{0}^{\infty}u(c(t),k(t))e^{-\delta t}dt+\alpha\lim_{t\rightarrow
\infty}U(c(t),k(t)), \label{C-criterion}%
\end{equation}
The coefficient $\delta>0$ in front of the integral will make later formulas simpler.
Suppose $u\left( c,k\right) \geq0$ and
\[
\sup\left\{ U\left( c,k\right) \ |\ c>0,\ k>0,\ c=f\left( k\right)
\right\} =\infty.
\]
Then, for any $\alpha>0$,
\[
\sup\left\{ I_{\alpha}\left( c\left( \cdot\right) ,k\left( \cdot\right)
\right) \ |\ \left( c\left( \cdot\right) ,k\left( \cdot\right) \right)
\in\mathcal{A}\left( k_{0}\right) \right\} =\infty,
\]
where $\mathcal{A}(k_{0})$ is defined below Definition 1.
For fixed $k_{0}>0$, pick any $A>0$, and choose some constants $\left(
c_{1},k_{1}\right) $ such that $c_{1}=f\left( k_{1}\right) $ and $U\left(
c_{1},k_{1}\right) >A\alpha^{-1}$. Then choose some $c_{0}\in(0,c_{1})$ such
that $f\left( k_{0}\right) -c_{0}>0$. With every $T>0$, we associate the
path $\left( c_{T}\left( t\right) ,k_{T}\left( t\right) \right) $
defined by:
\[
c_{T}\left( t\right) =\left\{
\begin{array}
c_{0}\text{ for }0\leq t\leq T,\\
c_{1}\text{ for }T\leq t,
\end{array}
\right.
\]
and denote the consumption-capital pair by $(c_{T},k_{T})$.
Since $f$ is increasing, we have $\frac{dk}{dt}\geq f\left( k_{0}\right)
-c_{0}$ for all $t$, so eventually $k\left( t\right) $ will reach the value
$k_{1}$. Choose for $T$ the first time when $k\left( t\right) =k_{1}$.
Writing this into the criterion, and remembering $c_{1}=f\left( k_{1}\right)
$, we find:
\[
I_{\alpha}\left( c_{T}\left( \cdot\right) ,k_{T}\left( \cdot\right)
\right) \geq\alpha\lim_{t\rightarrow\infty} U\left( c_{T}(t),k_{T}%
(t)\right) =\alpha U\left( c_{1},k_{1}\right) \geq A.
\]
This result shows that very often it is not possible to optimize $I_{\alpha}$.
Even when it is, there is the time-inconsistency problem: successive
decision-makers will not agree on what the optimal solution is. This is seen
most easily by considering the following criterion:
\begin{equation}
\left( 1-\alpha\right) \delta\int_{0}^{\infty}u(c(t),k(t))e^{-\delta
t}dt+\alpha{r}\int_{0}^{\infty}U(c(t),k(t))e^{-rt}dt. \label{E(r)-criterion}%
\end{equation}
When $r\rightarrow0$, the last term converges to $\alpha U\left( c_{\infty
},k_{\infty}\right) $, so that (<ref>) converges to
(<ref>). On the other hand, Criterion (<ref>) is a
special case of the biexponential criterion (<ref>)
with $\delta_{1}=\delta$, $\delta_{2}=r$ , and:
\begin{equation}
\frac{\alpha r}{\left( 1-\alpha\right) \delta}=\frac{1-\lambda}{\lambda
},\ \lambda=\frac{\left( 1-\alpha\right) \delta}{\alpha r+\left(
1-\alpha\right) \delta},\ 1-\lambda=\frac{\alpha r}{\alpha r+\left(
1-\alpha\right) \delta} \label{f}%
\end{equation}
So, for each $r>0$, the criterion (<ref>) gives rise to a
time-inconsistent problem. Using the results in the preceding section, we find
a continuum of equilibrium strategies. Substituting (<ref>) into
(<ref>) and (<ref>), we find the corresponding values:
\begin{align*}
\overline{g}_{r}(k) & =\frac{\left( 1-\alpha\right) \delta^{2}%
{u_{1}^{\prime}}\left( f\left( k\right) ,k\right) +\alpha r^{2}%
U_{1}^{\prime}\left( f\left( k\right) ,k\right) }{\left( 1-\alpha\right)
\delta u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha
rU_{1}^{\prime}\left( f\left( k\right) ,k\right) }-\frac{\left(
1-\alpha\right) \delta u_{2}^{\prime}\left( f\left( k\right) ,k\right)
+\alpha rU_{2}^{\prime}\left( f\left( k\right) ,k\right) }{\left(
1-\alpha\right) \delta u_{1}^{\prime}\left( f\left( k\right) ,k\right)
+\alpha rU_{1}^{\prime}\left( f\left( k\right) ,k\right) }\\
\underline{g}_{r}(k) & =\frac{\left( 1-\alpha\right) \delta{u}%
_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha rU_{1}^{\prime
}\left( f\left( k\right) ,k\right) }{\left( 1-\alpha\right)
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{1}^{\prime
}\left( f\left( k\right) ,k\right) }-\frac{\left( 1-\alpha\right)
u_{2}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{2}^{\prime
}\left( f\left( k\right) ,k\right) }{\left( 1-\alpha\right)
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{1}^{\prime
}\left( f\left( k\right) ,k\right) }%
\end{align*}
The equations (<ref>), (<ref>) become:
\begin{align}
\left( f-\varphi_{r}\left( V_{r}^{\prime}\right) \right) V_{r}^{\prime
}+u\left( \varphi_{r}\right) +\frac{\alpha r}{\left( 1-\alpha\right)
\delta}U\left( V_{r}^{\prime}\right) = & \frac{\delta+r}{2}V_{r}%
\left( f-\varphi_{\lambda}\left( V_{r}^{\prime}\right) \right)
W_{r}^{\prime}+u\left( \varphi_{\lambda}\right) +\frac{\alpha r}{\left(
1-\alpha\right) \delta}U\left( V_{\lambda}^{\prime}\right) = &
\frac{\delta-r}{2}V_{r}+\frac{\delta+r}{2}W_{r} \label{f2}%
\end{align}
where $\varphi_{r}\left( x,k\right) $ is defined by:
\[
u_{1}^{\prime}(\varphi_{r}(x,k),k)+\frac{\alpha r}{\left( 1-\alpha\right)
\delta}U_{1}^{\prime}\left( \varphi_{r}(x,k),k\right) =x
\]
and the boundary conditions (<ref>), (<ref>):
\begin{align}
V_{r}\left( k_{\infty}\right) & =\frac{1}{\delta}u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) +\frac{\alpha}{\left( 1-\alpha\right)
\delta}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) ,\label{f3}\\
W_{r}\left( k_{\infty}\right) & =\frac{1}{\delta}u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) -\frac{\alpha}{\left( 1-\alpha\right)
\delta}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) . \label{f4}%
\end{align}
The equilibrium strategy is given by $\sigma_{r}\left( k\right) =\varphi
_{r}\left( V_{r}^{\prime}\left( k\right) ,k\right) $, and the
corresponding trajectory converges to $k_{\infty}$. Note that these strategies
are defined locally. More precisely, denote by $]k_{r}^{-},\ \ k_{r}^{+}[$ the
maximal interval of existence of the solution $\left( V_{r},W_{r}\right) $
of the ODE (<ref>), (<ref>) with the boundary condition (<ref>),
(<ref>), so that $k_{r}^{-}<k_{\infty}<\ k_{r}^{+}$.
We shall now solve the Chichilnisky problem by setting $r=0$ in the preceding
equations. We have:
\begin{align}
\overline{g}_{0}(k) & =\delta-\frac{u_{2}^{\prime}\left( f\left( k\right)
,k\right) }{u_{1}^{\prime}\left( f\left( k\right) ,k\right) }%
\underline{g}_{0}(k) & =\frac{\left( 1-\alpha\right) \delta{u}_{1}%
^{\prime}\left( f\left( k\right) ,k\right) }{\left( 1-\alpha\right)
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{1}^{\prime
}\left( f\left( k\right) ,k\right) }-\frac{\left( 1-\alpha\right)
u_{2}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{2}^{\prime
}\left( f\left( k\right) ,k\right) }{\left( 1-\alpha\right)
u_{1}^{\prime}\left( f\left( k\right) ,k\right) +\alpha U_{1}^{\prime
}\left( f\left( k\right) ,k\right) } \label{55a}%
\end{align}
The pair $\left( V_{0},W_{0}\right) $ has to solve the following
boundary-value problem:
\begin{align}
& \left( f\left( k\right) -\varphi_{0}\left( V_{0}^{\prime},k\right)
\right) V_{0}^{\prime}+u(\varphi_{0}\left( V_{0}^{\prime},k\right)
,k)=\frac{\delta}{2}\left( V_{0}+W_{0}\right) ,\label{70}\\
& \left( f\left( k\right) -\varphi_{0}\left( V_{0}^{\prime},k\right)
\right) W_{0}^{\prime}+u(\varphi_{0}\left( V_{0}^{\prime},k\right)
,k)=\frac{\delta}{2}\left( V_{0}+W_{0}\right) ,\label{71}\\
& V_{0}\left( k_{\infty}\right) =\frac{1}{\delta}u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) +\frac{\alpha}{\left( 1-\alpha\right)
\delta}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) ,\label{72}\\
& W_{0}\left( k_{\infty}\right) =\frac{1}{\delta}u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) -\frac{\alpha}{\left( 1-\alpha\right)
\delta}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) \label{73}%
\end{align}
\begin{equation}
}(c,k),k)=c. \label{74}%
\end{equation}
Substracting (<ref>) from (<ref>), we get:
\[
\left( f\left( k\right) -\varphi_{0}\left( V_{0}^{\prime},k\right)
\right) \left( V_{0}^{\prime}-W_{0}^{\prime}\right) =0.
\]
Similarly to Lemma 9, it follows that $V_{0}^{\prime}-W_{0}^{\prime}=0$, and
so $V_{0}-W_{0}$ is a constant, namely:
\[
W_{0}\left( k\right) =V_{0}\left( k\right) +W_{0}\left( k_{\infty
}\right) -V_{0}\left( k_{\infty}\right) =V_{0}\left( k\right)
-2\frac{\alpha}{\left( 1-\alpha\right) \delta}U\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) .
\]
Writing this in (<ref>) and (<ref>), we find that $V_{0}\left(
k\right) $ is a solution of the boundary-value problem:
\begin{align*}
& \left( f\left( k\right) -\varphi_{0}\left( V_{0}^{\prime},k\right)
\right) V_{0}^{\prime}+u(\varphi_{0}\left( V_{0}^{\prime},k\right) ,k)
=\delta V_{0}-\frac{\alpha}{1-\alpha}U\left( f\left( k_{\infty}\right)
,k_{\infty}\right) ,\\
& V_{0}\left( k_{\infty}\right) =\frac{1}{\delta}u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) +\frac{\alpha}{\left( 1-\alpha\right)
\delta}U\left( f\left( k_{\infty}\right) ,k_{\infty}\right) .
\end{align*}
Setting $V:=V_{0}-\frac{\alpha}{\left( 1-\alpha\right) \delta}U\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) $, we see that $V$ is a
solution of the boundary-value problem:
\begin{align}
\left( f\left( k\right) -\varphi_{0}\left( V^{\prime},k\right) \right)
V^{\prime}+u(\varphi_{0}\left( V^{\prime},k\right) ,k) & =\delta
V\left( k_{\infty}\right) & =\frac{1}{\delta}u\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) . \label{82}%
\end{align}
This problem has been studied in <cit.> (Section 2.4, case 2) when
$u\left( c,k\right) $ does not depend on $k$. In the general case, we have:
\begin{equation}
f^{\prime}\left( k_{\infty}\right) \neq\delta-\frac{u_{2}^{\prime}\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) }{u_{1}^{\prime}\left(
f\left( k_{\infty}\right) ,k_{\infty}\right) } \label{inequality.1}%
\end{equation}
the problem (<ref>), (<ref>) has two solutions $V_{1}$ and $V_{2}$,
defined on some non-empty half-interval $[k_{\infty},\ k_{\infty}+a)$ or
$(k_{\infty}-a,k_{\infty}]$. Both are $C^{1}$ on the half-interval, $C^{2}$ on
the interior, and have the same derivative at $k_{\infty}$, given by:
\[
\varphi_{0}\left( V_{i}^{\prime}\left( k_{\infty}\right) ,k_{\infty
}\right) =c_{\infty}=f\left( k_{\infty}\right) ,\quad i=1,2.
\]
From now on, write $\varphi$ instead of $\varphi_{0}$. Rewrite (<ref>) as a
Pfaff system:
\begin{align}
dV & =pdk,\label{e.paff1}\\
p(f(k)-\varphi(p,k))+u(\varphi(p,k),k) & =\delta V \label{e.paff2}%
\end{align}
Differentiating (<ref>) leads to:
\begin{equation}
\delta dV=(f(k)-\varphi(p,k))dp+pf^{\prime}(k)dk+u_{2}^{\prime}(\varphi
\end{equation}
where we used (<ref>). Together with (<ref>), this yields
\begin{equation}
(p,k),k)]dk. \label{e.system.tmp}%
\end{equation}
We have to investigate this system near the point $k=k_{\infty}$ and
$V=V_{\infty}$. Writing (<ref>) at this point, we get:
\begin{equation}
})=\delta V_{\infty} \label{e1}%
\end{equation}
which has to be solved for $p$. Note that, because of (<ref>), we have:
\begin{equation}
\min_{p}\{(f(k_{\infty})-\varphi(p,k_{\infty}))p+u(\varphi(p,k_{\infty
}),k_{\infty})\}=u(f(k_{\infty}),k_{\infty}) \label{e2}%
\end{equation}
In the case at hand, we have $\delta V_{\infty}=u\left( f\left( k_{\infty
}\right) ,k_{\infty}\right) $, but for the sake of completeness, and to have
a full description of the phase space in the $\left( k,V\right) $ plane, we
will first investigate the cases $\delta V_{\infty}<u(f(k_{\infty}),k_{\infty
})$ and $\delta V_{\infty}>u(f(k_{\infty}),k_{\infty})$.
Case 0: $\delta V_{\infty}<u(f(k_{\infty}),k_{\infty})$
Because of (<ref>), equation (<ref>) has no solution. So there are no
solutions going through $\left( k_{\infty},V_{\infty}\right) .$
Case 1. $\delta V_{\infty}>u(f(k_{\infty}),k_{\infty})$.
Equation (<ref>) has two distinct solutions $p_{1}\neq p_{2}$. Note that
neither $p_{1}$ nor $p_{2}$ minimize the left hand side so $f(k_{\infty
})-\varphi(p_{i},k_{\infty})\neq0$ for $i=1,2$. We may therefore consider the
initial value problem:
\[
\frac{dp}{dk}=\frac{\delta{p}-pf^{\prime}(k)-u_{2}^{\prime}(\varphi
(p,k),k)}{f(k)-\varphi(p,k)},\quad p(k_{\infty})=p_{i}.
\]
It has a well-defined smooth solution $p_{i}(k)$, defined in a neighborhood of
$k_{\infty}$. We then define a function $V_{i}$ by:
\[
\]
Notice that $V_{i}(k_{\infty})=V_{\infty}$, so $V_{i}(k)$ solves the initial
value problem (<ref>), (<ref>), with $V_{i}^{\prime}(k_{\infty})=p_{i}$.
Taking $i=1,2$, we find two solutions $V_{1}(k)$ and $V_{2}(k)$ of the same
initial value problem, with $V_{1}^{\prime}(k_{\infty})\neq V_{2}^{\prime
}\left( k_{\infty}\right) $.
Case 2. $\delta V_{\infty}=u(f(k_{\infty}),k_{\infty})$ and
Equation (<ref>) then has a single solution $p_{0}$, and we have
$f(k_{\infty})=\varphi(p_{0},k_{\infty})$, so that:
\[
\]
In this case, we shall use the same system (<ref>)-(<ref>),
but we will take $p$ instead of $k$ as the independent variable. We consider
the initial value problem
\[
\frac{dk}{dp}=\frac{f(k)-\varphi(p,k)}{p(\delta-f^{\prime}(k)-\frac
{u_{2}^{\prime}(\varphi(p,k),k)}{p})},\quad k(p_{0})=k_{\infty}.
\]
It has a $C^{2}$ solution $k(p)$, defined in a neighborhood of $p=p_{0}$. We
associate with it a curve in the phase space $(k,V)$, defined in parametric
form by the equations
\begin{align*}
k & =k(p),\\
V & =\frac{1}{\delta}[(f(k)-\varphi(p,k))p+u(\varphi(p,k),k)].
\end{align*}
Then $V$ is also $C^{2}$ with respect to $p$ near $p=p_{0}$. Moreover, we
\begin{align*}
\frac{dV}{dp} & =\frac{1}{\delta}[(f(k)-\varphi(p,k)-\varphi_{1}^{\prime
& +\frac{1}{\delta}[(f^{\prime}(k)-\varphi_{2}^{\prime}(p,k))p+u_{1}^{\prime
& =\frac{f(k)-\varphi(p,k)}{\delta-f^{\prime}(k)-\frac{u_{2}^{\prime}%
\end{align*}
Since $f(k_{\infty})=\varphi(p_{0},k_{\infty})$, we obtain
\begin{align*}
\frac{dk}{dp}(p_{0}) & =0,\\
\frac{dV}{dp}(p_{0}) & =0.
\end{align*}
This shows that the parametric curve $p\rightarrow\left( k\left( p\right)
,V\left( p\right) \right) $ in the $\left( k,V\right) $-plane has a cusp
at $\left( k_{\infty},V_{\infty}\right) $. To find the type of the cusp, we
compute the second order derivatives with respect to $p$. We find:
\begin{align}
\frac{d^{2}k}{dp^{2}}(p_{0}) & =\frac{-\varphi_{1}^{\prime}(p_{0},k_{\infty
\frac{d^{2}V}{dp^{2}}(p_{0}) & =\frac{-\varphi_{1}^{\prime}(p_{0},k_{\infty
}),k_{\infty})}}\neq0, \label{e4}%
\end{align}
This shows that the cusp has two branches, with a common tangent between them.
The branches extend on the left of $k_{\infty}$ if the left-hand side of
(<ref>) is negative, that is, if:
\begin{equation}
\delta-f^{\prime}(k_{\infty})-\frac{u_{2}^{\prime}(f(k_{\infty}),k_{\infty}%
)}{u_{1}^{\prime}(f(k_{\infty}),k_{\infty})}<0 \label{g}%
\end{equation}
and on the right if it is positive.The slope $m$ of the common tangent is
given by
\[
\]
and it is also the one-sided derivative of $V$ at $k_{\infty}$. Note for
future use that because $p_{0}=u_{1}^{\prime}(f(k_{\infty}),k_{\infty})$ we
\[
\frac{d}{dk}\frac{1}{\delta}u(f(k),k)|_{k=k_{\infty}}=\frac{1}{\delta}\left(
}(f(k_{\infty}),k_{\infty}\right) ,
\]
so that $m=p_{0}$ does not coincide with the tangent of the curve $\delta
V=u\left( f\left( k\right) \right) $ at $k=k_{\infty}$.
Case 3: $\delta V_{\infty}=u(f(k_{\infty}),k_{\infty})$ and
This is the case which was investigated in Theorem <ref>. We have shown
that there exists a $C^{2}$ solution $V\left( k\right) $ on a neighbourhood
of $k_{\infty}\,.$
This concludes the proof of Proposition <ref> (in fact, we only need Case 2).
Let us summarize these results. The curve $\Gamma=\left\{ \left( k,V\right)
\ |\ \delta V=u\left( f\left( k\right) ,k\right) \right\} \,$ separates
the plane in two regions.
* The region below the curve corresponds to Case 0: there are no solution there.
* The region above the curve corresponds to Case 1: through each point
$\left( k_{\infty},V_{\infty}\right) $ there are two smooth solutions
intersecting transversally.
* If $\left( k_{\infty},V_{\infty}\right) $ is on the curve, but
{u_{1}^{\prime}(f(k_{\infty}),k_{\infty})}\neq\delta$, we are in Case 2. There
are two $C^{1}$ solutions defined only on one side of $k_{\infty}$. They are
tangent at $\left( k_{\infty},V_{\infty}\right) $, and transversal to
* If $\left( k_{\infty},V_{\infty}\right) $ is on the curve, and
{u_{1}^{\prime}(f(k_{\infty}),k_{\infty})}=\delta$, we are in Case 3: there is
a $C^{2}$ solution defined on a neighbourhood of $k_{\infty}$.the
Figure 1 gives the phase diagram in the $\left( k,V\right) $ plane:
Figure 1. The illustration of the solutions.
Proposition <ref> gives us two solutions, $V_{i},i=1,2$. Each of them
gives rise to an strategy $\sigma_{i}$ through the formula $\sigma_{i}\left(
k\right) =\varphi_{0}\left( V_{i}^{\prime}\left( k\right) ,k\right) $
with $\varphi_{0}$ defined by (<ref>). The strategy $\sigma_{i}$ is $C^{0}$
on the half-interval, $C^{1}$ on its interior, with $\sigma_{i}\left(
k_{\infty}\right) =f\left( k_{\infty}\right) $.
One, and only one, of the strategies $\sigma_{1}$ and $\sigma_{2}$, converges
to $k_{\infty}$.
Consider the Euler-Lagrange equation (<ref>) for the Ramsey problem. The
phase diagram is given in Figure 2, where $\underline{k}$ is defined by
Figure 2. The phase diagram for the Euler equation.
If $k_{\infty}\neq\underline{k}$, there is one trajectory $\mathcal{T}$ going
through $\left( k_{\infty},u\left( f\left( k_{\infty}\right) ,k_{\infty
}\right) \right) $. The point $\left( k_{\infty},u\left( f\left(
k_{\infty}\right) ,k_{\infty}\right) \right) $ separates it into two
branches, the upper one and the lower one. One of them goes to $\left(
k_{\infty},u\left( f\left( k_{\infty}\right) ,k_{\infty}\right) \right)
$, and the other one leaves it. These two branches are also the trajectories
associated with the two strategies $\sigma_{1}$ and $\sigma_{2}$, so one of
them converges and the other diverges.
Since we are looking for a strategy which converges to $k_{\infty}$, we pick
the strategy $\sigma_{i}$, and the solution $V_{i}$, associated with the
branch oriented towards $k_{\infty}$. We will denote them by $\sigma$ and $V$.
We have proved the following:
Suppose $f^{\prime}\left( k_{\infty}\right) $ lies between $\overline{g}%
_{0}(k_{\infty})$ and $\underline{g}_{0}(k_{\infty})$. If $\underline{g}%
_{0}(k_{\infty})>\overline{g}_{0}(k_{\infty})$, then there exists an
equilibrium strategy $\sigma\left( k\right) $, defined on some interval
$]k_{\infty}-\kappa,\ k_{\infty}]\,$, which converges to $k_{\infty}$. It is
continuous on the interval, $C^{1}$ on its interior, with $\sigma\left(
k_{\infty}\right) =f\left( k_{\infty}\right) $. If If $\underline{g}%
_{0}(k_{\infty})<\overline{g}_{0}(k_{\infty})$, there exists an equilibrium
strategy with the same properties, defined on some interval $[k_{\infty
},\ k_{\infty}+\kappa\lbrack$.
Note that one of the boundary values for $f^{\prime}\left( k_{\infty}\right)
$, namely $\overline{g}_{0}(k_{\infty})$, corresponds to the solution of the
Ramsey problem ($\alpha=0$). Indeed, the equation $f^{\prime}\left(
k_{\infty}\right) =\overline{g}_{0}(k_{\infty})$ coincides with equation
We have thus identified a class of equilibrium strategies for the Chichilnisky
problem. They are one-sided, except when $f^{\prime}\left( k_{\infty}\right)
=\overline{g}_{0}(k_{\infty})$, where we can apply Theorem <ref> to get a
strategy defined on a neighbourhood of $k_{\infty}$. For every other value of
$k_{\infty}$ satisfying (<ref>) and (<ref>), the function $V(k)$ and
the strategy $\sigma\left( k\right) $ defined by $u_{1}^{\prime}\left(
\sigma\left( k\right) ,k\right) =V^{\prime}\left( k\right) $ are defined
only on one of the two half-intervals limited by $k_{\infty}$. Suppose for
instance it is the right one, $[k_{\infty},\ k_{\infty}+\kappa\lbrack$. Then,
if $k_{\infty}\leq k_{0}<k_{\infty}+\kappa$, the equilibrium strategy will
bring $k_{0}$ to $k_{\infty}$ in finite time and stay there.
To our knowledge, this is the first time equilibrium strategies have been
found for the Chichilinisky criterion. Their economic interpretation, and
their detailed study, will be the subject of forthcoming work.
Let us give some examples.
Example 1: $u\left( c\right) =U\left( c\right) $
Neither depends on $k$, and we have:
\begin{align*}
\overline{g}_{0}(k) & =\delta,\\
\underline{g}_{0}(k) & =\left( 1-\alpha\right) \delta
\end{align*}
We have $\underline{g}_{0}(k)<\overline{g}_{0}(k)$, so the equilibrium
strategy exists only on the right hand side of $\underline{g}_{0}(k)$. The
existence condition is:
\[
\left( 1-\alpha\right) \delta<f^{\prime}(k_{\infty})<\delta
\]
and the equilibrium strategy $\sigma$ is defined on $[k_{\infty}%
,\ \infty\lbrack$. We denote $f^{\prime-1}(\delta)$ and $f^{\prime
-1}((1-\alpha)\delta)$ by $\underline{k}$ and $\overline{k}$ respectively.
There are three cases, depending on the position of the initial point $k_{0}$:
* If $k_{0}>\overline{k}$, then, for any $k_{\infty}\in]\underline{k}%
,\ \overline{k}[$, there exists an equilibrium strategy starting from $k_{0}$
and converging to $k_{\infty}$.
* if $\underline{k}<k_{0}<\overline{k}$, then, for any $k_{\infty}%
\in]\underline{k},\ k_{0}[$, there exists an equilibrium strategy starting
from $k_{0}$ and converging to $k_{\infty}$.
* if $k_{0}<\underline{k}$, the only equilibrium strategy starting from
$k_{0}$ is the optimal strategy for the Ramsey problem (that is, for the case
$\alpha=0$ ) which converges to the level $\underline{k}$ where $f^{\prime
Example 2: $u\left( c,k\right) =U\left( c,k\right) $
We find:
\begin{align*}
\overline{g}_{0}(k) & =\delta-\frac{u_{2}^{\prime}(f(k),k)}{u_{1}^{\prime
}\left( f\left( k\right) ,k\right) }\\
\underline{g}_{0}(k) & =\left( 1-\alpha\right) \delta-\frac{u_{2}^{\prime
}(f(k),k)}{u_{1}^{\prime}\left( f\left( k\right) ,k\right) }%
\end{align*}
The existence condition is:
\[
\left( 1-\alpha\right) \delta-\frac{u_{2}^{\prime}(f(k),k)}{u_{1}^{\prime
}\left( f\left( k\right) ,k\right) }<f^{\prime}(k_{\infty})<\delta
-\frac{u_{2}^{\prime}(f(k),k)}{u_{1}^{\prime}\left( f\left( k\right)
,k\right) }%
\]
and the equilibrium strategy $\sigma$ is defined on $[k_{\infty},\ k_{\infty
}+\kappa\lbrack$ for some $\kappa>0$. The situation is similar to the
preceding one, bearing in mind that now the strategy $\sigma$ may be defined
locally only
Example 3: $u=u\left( c\right) \,$ and $U=U\left( k\right) $
In that case, we find:
\begin{align*}
\overline{g}_{0}(k) & =\delta,\\
\underline{g}_{0}(k) & =\delta-\frac{\alpha}{\left( 1-\alpha\right) }%
\frac{U_{2}^{\prime}\left( f\left( k\right) ,k\right) }{u_{1}^{\prime
}\left( f\left( k\right) ,k\right) }%
\end{align*}
There are two subcases:
* if $\frac{U_{2}^{\prime}\left( f\left( k\right) ,k\right) }
{u_{1}^{\prime}\left( f\left( k\right) ,k\right) }>0 $, then
$\underline{g}_{0}(k)<\overline{g}_{0}(k)$. The equilibrium strategy then is
defined on the right hand side of $k_{\infty}$, as in the preceding cases.
* if $\frac{U_{2}^{\prime}\left( f\left( k\right) ,k\right) }
{u_{1}^{\prime}\left( f\left( k\right) ,k\right) }<0 $. The equilibrium
strategy then is defined on the left hand side of $k_{\infty}$.
Remember that $u_{1}^{\prime}>0$. If $U_{2}^{\prime}>0$, that is, if capital
is beneficial, we are in the first case. The second case corresponds to the
case when capital is detrimental, for instance if accumulating capital
accumulates pollution. In that case, the situation is inverted with respect to
the preceding examples: if the initial stock of capital is small enough, one
may stop capital accumulation at a level $k_{\infty}$ smaller than the level
$\underline{k}\ $where $f^{\prime}(\underline{k})=
\delta-\frac{\alpha}{\left( 1-\alpha\right) }\frac{U_{2}^{\prime}\left(
f\left( k\right) ,k\right) }{u_{1}^{\prime}\left( f\left( k\right)
,k\right) }$. But there is no turning back: one can never achieve a
stationary level $k_{\infty}<k_{0}$. The situation is described in Figure 3
Figure 3. The case when capital pollutes.
Acknowledgements. The authors thank sincerely Professors Larry Karp
and Gerd Asheim for generously sharing their knowledge and ideas with them,
and pointing out several useful references. Qinglong Zhou and Yiming Long
would like to express their thanks to Professors Jean-Pierre Bourguignon, Jean
Dolbeault, Olivier Glass and Ivar Ekeland as well as the IHES and the CEREMADE
(Université Paris Dauphine) for supporting Qinglong's visits to IHES and
CEREMADE from February to April of 2012, and their warm hospitalities during
this time.
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|
1511.00171
|
Know the single-receptor sensing limit? Think again.
Gerardo Aquino$^{1}$, Ned S. Wingreen$^{2}$, and Robert G. Endres$^{1,\ast}$
1 Department of Life Sciences & Centre for Integrative Systems Biology and
Bioinformatics, London, United Kingdom
2 Department of Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA
$\ast$ E-mail: [email protected]
§ ABSTRACT
How cells reliably infer information about their environment is a fundamentally important question. While sensing and
signaling generally start with cell-surface receptors, the degree of accuracy with which a cell can measure external
ligand concentration with even the simplest device - a single receptor - is surprisingly hard to pin down. Recent studies
provide conflicting results for the fundamental physical limits. Comparison is made difficult as different studies either suggest
different readout mechanisms of the ligand-receptor occupancy, or differ on how ligand diffusion is implemented.
Here we critically analyse these studies and present a unifying perspective on the limits of sensing, with wide-ranging
biological implications.
§ INTRODUCTION
In 1977, physicists Howard Berg and Edward Purcell published their results on the
fundamental biological problem of sensing <cit.>. The question they
addressed was how accurately a biological cell, viewed as a tiny measurement device,
can sense its chemical environment using cell-surface receptors. The
paper is not only highly cited, but, more importantly, a large fraction of the
citations stems from the last ten years, demonstrating how far ahead of its time the study was.
In essence, the message of the paper was simple: sensing in the
microscopic world boils down to counting molecules, which arrive at
the cell surface by diffusion. Humans encounter a
similar limit when we try to see in the near dark as our photoreceptors count
single photons <cit.>. Berg and Purcell's paper has influenced many
fields of quantitative biology, including nutrient scavenging
<cit.>, mating <cit.>, signal transduction
<cit.>, gene regulation <cit.>, cell division
<cit.>, and embryonic development <cit.>.
While there is no disagreement on the importance of knowing the fundamental
physical limits of sensing, there has been disagreement on what this limit is, even
for a single receptor. The analysis here interprets and unifies these studies
to yield a coherent picture of the limits of sensing.
§ OVERVIEW
To introduce the topic and to build intuition, we follow Berg
and Purcell <cit.> and begin with simple models for measuring ligand
concentration $c_0$. The first is the Perfect Monitor <cit.>.
This model assumes a permeable sphere of radius $a$,
capable of counting the number of molecules $N$ inside its volume (Fig. <ref>a).
For concreteness, the sphere might represent a bacterial cell. Since the molecules
diffuse independently, finding a molecule in one small
volume element is independent of finding another one in a different small volume element,
and so the number of molecules $N$ will be Poisson distributed. Since for the Poisson distribution
the variance equals the mean, i.e. $\delta N^2=\bar N$ (omitting ensemble-averaging
brackets for simplicity of notation), we obtain for a single measurement (“snapshot”)
\begin{equation}
\frac{\delta c^2}{c_0^2}=\frac{\delta N^2}{\bar N^2}=\frac{1}{\bar N}=
\frac{1}{c_0V},\label{Eq1}
\end{equation}
where $c_0$ is a fixed, given ligand concentration and $V$ is the volume of the monitoring sphere.
However, if we assume the Perfect Monitor has some time $T$ available to make
a measurement, the uncertainty in the estimate of the true ligand concentration can be
further reduced. In time $T$, the Perfect Monitor can make approximately $M\sim T/\tau_D$ statistically
independent measurements, where $\tau_D\sim a^2/D$ is the diffusive turnover time for the
molecules inside the sphere. This leads to the reduced uncertainty
\begin{equation}
\frac{\delta c^2}{c_0^2}=\frac{1}{M\bar N}=
\frac{1}{(T/\tau_D)c_0V}\sim\frac{1}{Dac_0T},\label{Eq2}
\end{equation}
where we neglect prefactors for this heuristic derivation. (The exact result is $3/(5\pi Dac_0T)$,
which can be derived by considering autocorrelations of the molecules inside the volume <cit.>.)
Simple measurement devices for concentration. (a) The Perfect Monitor
is permeable to ligand molecules and estimates the concentration $c_0$ by
counting the molecules in its volume during time $T$. (b) The Perfect Absorber estimates
the ligand concentration from the number of molecules incident on its surface
during time $T$.
However, the Perfect Monitor is not the best one can do. A more accurate estimate
can be made if each ligand molecule is only measured once rather than being
allowed to diffuse in and out of the sphere. Thus, we consider
a perfectly absorbing sphere <cit.>,
estimating concentration from the number of absorbed ligand molecules
$N_T$ in time $T$, and find (Fig. <ref>b)
\begin{equation}
\frac{\delta c^2}{c_0^2}=\frac{1}{N_T}=\frac{1}{4\pi Dac_0T}<\frac{3}{5\pi Dac_0T}\label{Eq3}
\end{equation}
This Perfect Absorber is thus more accurate than the Perfect
Monitor (and even more so for spatial gradient sensing by almost a factor of 10)
<cit.>. This result contrasts with Berg and Purcell's
original suggestion that rebinding previously measured ligand molecules
does not increase the uncertainty in measurement <cit.>.
However, one of their many key insights was that a sphere with many absorbing
patches for ligand is nearly as good at sensing as a fully absorbing sphere, making room
for multiple receptor types with different ligand specificity without sacrificing
much accuracy.
Measuring ligand concentration with a single receptor. (a) A
receptor binds ligand with rate $k_+c_0$ when unbound, and unbinds
ligand when bound with rate $k_-$. (b) Time series of receptor
occupancy during time interval $T$. Berg and Purcell considered the accuracy
obtained by taking the average (dashed horizontal line).
§ SINGLE RECEPTOR WITHOUT LIGAND REBINDING
The single receptor is the simplest measurement device and thus needs to be
thoroughly understood. Unfortunately, different approaches to estimating its sensing
accuracy have resulted in significant discrepancies. We first disregard the effects of
diffusion and rebinding of previously bound ligands, and just consider ligand binding and unbinding.
Consider the receptor shown in Fig. <ref>a, which binds
ligand with rate $k_+c_0$ when unbound and unbinds ligand with rate $k_-$ when bound. The
probability of being bound is then $p=c_0/(c_0 +K_D)$ with $K_D=k_-/k_+$
the ligand dissociation constant. A potential time series of receptor occupancy
$\Gamma(t)$ during time $T$ is illustrated in Fig. <ref>b. Berg and Purcell argued
that the best a cell can do to estimate the ligand concentration is to average the
occupancy $\Gamma(t)$ over time. For such an average, the variance $\delta \Gamma^2$ was
derived from the autocorrelations of occupancy, leading to
the relative uncertainty in estimating the ligand concentration
\begin{equation}
\frac{\delta c^2}{c_0^2}=\left(c_0\frac{\partial p}
{\partial c}\right)^{-2}\delta \Gamma^2\label{Eq4},
\end{equation}
where the derivative $\partial p/\partial c$ is the gain or amplification. Naively, one could
be tempted to set $\delta \Gamma^2=p(1-p)$ equal to the variance of a
Bernoulli random variable (binomial trials). However, this would correspond to the
uncertainty in the concentration estimate following a single instantaneous observation
of the state of the receptor, or equivalently to the frequency integral of the noise
power spectrum $\delta \Gamma^2=\int d\omega /(2\pi) S_\Gamma(\omega)$
(see Supplementary Information for details). This snapshot limit can be improved
if we assume $T{>\!\!>}1/(k_+c_0)+1/k_-$, i.e.
that the receptor is allowed to average over a time $T$ much larger than the correlation
time of ligand binding and unbinding. In this case, one can take the low-frequency limit
$\delta \Gamma^2\approx S_\Gamma(\omega=0)/T$ instead, and Eq. <ref> leads to
the Berg-Purcell limit for a single receptor
\begin{equation}
\frac{\delta c^2}{c_0^2}=\frac{2\tau_b}{T p}=\frac{2}{\bar N}
\rightarrow\frac{1}{2Dac_0(1-p)T}\label{Eq5},
\end{equation}
where $\tau_b=1/k_-$ is the average duration of a bound interval. The simple
formulation as $2/\bar N$ follows because the average number of binding and unbinding
events in time $T$ is $\bar N=T/(\tau_b+\tau_u)$, where $\tau_u=(k_+c_0)^{-1}$ is the
average duration of an unbound interval. The final result in Eq. <ref> follows from
detailed balance for diffusion-limited binding.
But is it true that averaging receptor occupancy is the best way to estimate concentration?
More recently a limit lower than Eq. <ref> was found by applying
maximum-likelihood estimation to a time series $\Gamma(t)$ of receptor
occupancy <cit.>. Here the probability $P(\Gamma,c)$ of observing a time series
$\Gamma$ is maximised with respect to the concentration $c$. The resulting
best estimate of the concentration comes only from the unbound intervals, since
only they depend on the rate of binding and thus on the ligand concentration. To obtain
a lower limit on the uncertainty the Cramér-Rao bound <cit.>
can be used, leading to
\begin{equation}
\frac{\delta c^2}{c_0^2}\geq\frac{1}{c_0^2I(c_0)}\rightarrow\frac{1}{N}\label{Eq6},
\end{equation}
where $I(c_0)=-\partial^2\ln(P)/\partial c^2$ is the Fisher information evaluated at $c_0$
and averaged over all trajectories with the same $N$ (when employing maximum-likelihood
estimation it is easier to work with a fixed number of binding/unbinding events $N$
than a fixed time $T$). The limit on the right-hand side of Eq. <ref>
is obtained for long time series for which the inequality becomes an equality.
Note, however,
that a slightly sharper bound $1/(N-2)$ can be obtained when using a further improved
estimator (see Supplementary Information).
Eq. <ref> shows that the uncertainty in Eq. <ref> can be reduced
by a factor of two. This is because only unbound intervals carry information about
the ligand concentration. In contrast, the bound intervals only increase the
uncertainty and hence are discarded by the maximum-likelihood procedure.
What does the maximum-likelihood result imply about tuning receptor parameters to minimise the
uncertainty? The minimal uncertainty is obtained for $N_\text{max}$, the maximal
number of binding events provided by very fast unbinding ($k_-\rightarrow \infty$).
This ideal limit corresponds to the Perfect Absorber from Eq. <ref> as every binding
event is counted. (However, the increased accuracy comes at the expense of
specificity as any ligand molecule dissociates immediately and
hence different ligand types cannot be differentiated.) Maximum-likelihood estimation
can also be extended to ramp sensing (temporal gradients) <cit.>
and multiple receptors <cit.>.
Three schemes of receptor readout. (a) Integrating receptor,
which signals while ligand bound (left) <cit.>. For example,
the active receptor might phosphorylate a protein with concentration
$n$. The concentration of the phosphorylated protein is $n_p$ (right).
(b) Alternatively, the receptor may signal in generic bursts
at onset of ligand binding (left) <cit.>. This scheme can be
implemented by an energy-driven cycle of $L$ active/bound receptor
conformations, which reduces variability (right) <cit.>.
(c) A receptor could also retain a memory of previous binding and
unbinding events, potentially improving its accuracy of sensing
Adding a downstream signaling molecule cannot increase the accuracy of sensing,
in fact this only adds noise.
For example, consider an integrating receptor à la Berg and Purcell,
which signals while being ligand bound (Fig. <ref>a) <cit.>.
In this simple network a downstream signaling molecule with concentration $n$ is
phosphorylated by ligand-bound receptors with the phosphorylated concentration given by
$n_p$ with lifetime $\tau$ (beyond this time the protein converts back to the
unphosphorylated form). Now, instead of taking the snapshot limit, i.e. the total
variance $\delta n_p^2$, we time average to reduce the uncertainty. Specifically, let us assume a
long averaging time, that is $T>\!\!>\tau>\!\!>1/(k_+c_0)+1/k_-$, allowing us to use again the
low-frequency limit of the corresponding power spectrum.
We then obtain (see Supplementary Information for details)
\begin{eqnarray}
\frac{\delta c^2}{c_0^2}&=&\left[{\frac{2}{\bar N_\tau}}+
{\frac{2}{\bar n(1-p)^2}}\right]
&=&\underbrace{\frac{2}{\bar N}}_\text{BP limit}+
\underbrace{\left[\frac{2}{\bar n(1-p)^2}\right]}_\text{Poisson-like}
\underbrace{\frac{\tau}{T}}_\text{time ave}.\label{Eq7b}
\end{eqnarray}
Eq. <ref> shows that by integrating receptor output one cannot do better than the
Berg-Purcell limit, given by the first term.
The second term represents additional Poisson-like noise from number fluctuations of the
signaling molecule due to imperfect averaging <cit.>. For $T{>\!\!>}\tau$ the
Berg-Purcell limit is approached from time averaging this noise. While we focus here
on averaging in time of stationary stimuli, non-stationary ligand concentrations
may be more accurately sensed via non-uniform time averaging, requiring appropriately
designed signaling cascades <cit.>.
There has been some confusion about whether sensing actually costs energy.
On the one hand, C. H. Bennett pointed out long ago that sensing does not need to cost
if done reversibly (and hence extremely slowly) <cit.>. On the other hand, cells obviously
consume energy, e.g. using ATP to phosphorylate proteins. In other words, what energy cost is
actually necessary for performing a measurement? As stressed by the authors in <cit.>
the process of sensing in terms of ligand-receptor binding does not need to cost energy if done
using an equilibrium receptor in the spirit of Berg and Purcell. However, to accurately infer the external
ligand concentration, the cell needs to time average, which cannot be done without consuming
energy. This is in line with the Landauer erasure principle <cit.>, which
predicts a lower theoretical limit of energy consumption of a computation. In essence, to keep a
record of the past for averaging, old information needs to be erased and time-reversal symmetry
broken <cit.>. Time averaging can be implemented by phosphorylation of a
downstream protein: when the receptor is bound it phosphorylates and when unbound it
dephosphorylates. Since these are energetically driven reactions the reverse reaction, e.g.
dephosphorylation by a bound receptor is extremely unlikely, and time averaging is very
efficient. The issue of the cost was avoided in Berg and Purcell's analysis by providing an effective averaging
time $T$ without specifying how this averaging is achieved.
How is the maximum-likelihood result, Eq. <ref>, useful? The maximum-likelihood result
makes interesting predictions about sophisticated sensing strategies cells
might employ. For example, to implement maximum likelihood in the fast unbinding limit a receptor should only signal upon a ligand-binding event
as illustrate in Fig. <ref>b (thin arrows), rather than continuously signaling
while ligand is bound (see <cit.> for further discussion). How can the cell
achieve such short and well-defined signaling durations? Reducing variability and
achieving determinism requires energy consumption and irreversible cycles <cit.>.
Examples may include ligand-gated ion channels <cit.> and single-photon
responses in rhodopsin of rod cells <cit.>.
Maximum likelihood provides another valuable insight - it shows that information
from an estimate and memory from a prior are equivalent, and both can
contribute to lowering the uncertainty (Fig. <ref>c). This kind of
receptor “learning” from past estimates can be implemented using the
Bayesian Cramér-Rao bound for the uncertainty. Using prior
information $I(\lambda)$, one obtains <cit.>
\begin{equation}
\frac{\delta c^2}{c_0^2}=-\frac{1/c_0^2}
{\underbrace{I(c_0)}_\text{Fisher info.}+
\underbrace{I(\lambda)}_\text{prior}}=\frac{1}{2N}\label{Eq8},
\end{equation}
assuming the prior had variance $1/N$, identical to the actual measurement.
Importantly, memory can even help in fluctuating environments if a filtering scheme
is implemented by the cell <cit.>: if the environment fluctuates weakly
and/or with long temporal correlations, memory improves precision significantly.
If, on the other hand, the environment fluctuates very strongly and/or without any
correlations, the cell can still rely on the current measurement (and disregard memory).
A form of memory is implemented by receptor methylation in
bacterial chemotaxis <cit.>, and in principle memory could be implemented by
any slow process in the cell, e.g. the expression of LacY permease in enzyme
induction in the lac system <cit.>, or the remodelling of the actin cortex
in eukaryotic chemotaxis <cit.>.
§ SINGLE RECEPTOR WITH LIGAND REBINDING
So far we have neglected the possibility of rebinding by previously bound ligands.
In fact, the role of ligand rebinding in the
accuracy of sensing is a tricky issue, because rebinding can
introduce non-trivial correlations between binding events. In practice, these
correlations can only be included approximately in analytical calculations, and so the
question is how to proceed. Originally Berg and Purcell made the
reasonable suggestion that a molecule that fails to bind to a receptor may
return to the receptor by diffusion and rebind, and that this effect
may be included by considering diffusion-limited binding with a
renormalised receptor size <cit.>. However, the question is how to
formally separate ligand binding and unbinding from ligand
Bialek and Setayeshgar addressed this problem by coupling ligand-receptor binding
and unbinding to the diffusion equation <cit.>.
Assuming that the averaging time is long compared to the typical binding and
unbinding time, the low-frequency limit can be used. This results in
\begin{equation}
\frac{\delta c^2}{c_0^2}={\frac{2}{k_+c_0(1-p)T}}
+{\frac{1}{\pi Dac_0T}}\label{Eq9}
\end{equation}
with $a$ now the size of the receptor. Equation <ref> indicates noise
contributions from two independent sources. According to Ref. <cit.>,
the first term represents binding and unbinding noise and depends on the rate
parameters, while the second depends on diffusion and was interpreted as
a Berg-Purcell-like noise floor. However, we argue for a different interpretation:
For diffusion-limited binding, the first term in Eq. <ref> is not zero, but
rather $k_+$ needs to be set to a Kramer-like expression, which is proportional
to the diffusion constant <cit.> and an Arrhenius factor at most equal to one
<cit.>. Due to their dependence on the diffusion constant, both terms can be
combined <cit.>. Indeed, for
diffusion-limited binding it is the first, not the second term of Eq. <ref> that captures the
Berg-Purcell limit. Since Berg and Purcell did not consider rebinding by diffusion, the
second term constitutes increased noise due to a rebinding correction that does not arise in Berg and Purcell's
derivation <cit.>. Bialek and Setayeshgar also applied their method to multiple
receptors, and showed that the second term can introduce correlations among receptors,
as ligand unbinding at one receptor can lead to re-binding at another nearby receptor
<cit.>. Hence, while multiple independent receptors
allow for spatial averaging <cit.>, mutual rebinding among different receptors
by diffusion increases the uncertainty of sensing.
More recently, Kaizu et al. readdressed this problem <cit.> by applying a
formalism developed by Agmon and Szabo for diffusion-influenced reactions <cit.>.
By calculating survival probabilities of bimolecular reactions with a number of
simplifying assumptions (see below), they obtained for the relative uncertainty
of a single receptor
\begin{equation}
\frac{\delta c^2}{c_0^2}={\frac{2}{k_+c_0(1-p)T}}
+{\frac{1}{2\pi Dac_0(1-p)T}}\label{Eq10}.
\end{equation}
Similar to Bialek and Setayesghar, there are two noise contributions with
the first terms in Eq. <ref> and <ref> formally identical. The second term is, however, different.
While the lost factor 2 in the second term in Eq. <ref> can be traced to different definitions of the
receptor geometry (cubic in Eq. <ref> and spherical in Eq. <ref>), the
factor $1-p$ in Eq. <ref> is missing from Eq. <ref>. Due to this factor, both terms of the
uncertainty in Eq. <ref> diverge
if the receptor is fully bound on average ($p=1$), while in Eq. <ref> only the
first term diverges. Unlike Eq. <ref> Kaizu et al. made the additional assumption that
during a bound interval the external ligand equilibrates. As a result, an unbound ligand molecule
cannot diffuse away and rebind at a later time with another ligand bound in between. However, using
exact simulations they showed that such delayed rebinding is only a minor effect under biologically relevant
conditions. Hence, as the factor $1-p$ in the second term of Eq. <ref> also appears
in the Berg-and-Purcell limit, Eq. <ref>, Kaizu et al. argue that their result is more
accurate than Eq. <ref>.
However, we propose a slightly different interpretation of Eq. <ref>. Similar
to <cit.> we suggest that the second term is not the Berg-Purcell limit (Eq. <ref>)
for diffusion-limited binding since the first term captures the Berg-Purcell limit <cit.>.
As described above, for diffusion-limited binding the first term cannot be neglected.
This aside, how may the factor $1-p$ in
the second term be interpreted? In Kaizu et al.'s derivation, diffusion means that a ligand molecule
enters the ligand pocket of a receptor without actually binding (hence the factor $1-p$ in the second
term since they assume only an unbound receptor can be approached by a ligand molecule).
In Bialek and Setayeshgar's derivation, no such factor appears as the second term describes fluctuations
in ligand concentration simply in the vicinity of the receptor. Can further insight into the effects of diffusion be
obtained by yet an alternative method?
Maximum-likelihood estimation can also be applied to a receptor with ligand diffusion, albeit
only in a special case. The probability of observing a time series
of receptor occupancy of $N$ binding and unbinding events can be formally written down even with
diffusion <cit.>. However, the rate of binding will depend on the current ligand concentration,
which is influenced by the history of all previous binding and unbinding events (even
before the first recorded binding event). To estimate the uncertainty, the Cramér-Rao
bound can be applied but cannot be evaluated exactly. Nevertheless, for fast diffusion
or slow binding an approximate expression can be derived for both 2D and 3D
(see Supplementary Information)
\begin{equation}
\frac{\delta c^2}{c_0^2}\approx\frac{1}{N}\left(1+2\frac{\Delta c}{c_0}\right)
\left\{\begin{array}{ll}
\frac{\ln(4\pi D/(k_+c_0a^2))}{2\pi Dc_0(1-p_{1/2})T} & \mbox{for 2D}\\
\frac{1}{\pi Dac_0(1-p_{1/2})T} & \mbox{for 3D}
\end{array}\right.\label{Eq11}.
\end{equation}
In Eq. <ref> the average local “excess” ligand concentration $\Delta c$
due to previous binding and unbinding events is $k_+c_0/(4\pi D)\cdot\ln[4\pi D/(k_+c_aa^2)]$ in 2D and
$k_+c_0/(2\pi Da)$ in 3D (for the derivation half occupancy $p_{1/2}=1/2$ is required).
The ratios in the excess concentration reflect the competition between rebinding and diffusion.
As expected, in 2D this concentration decays more slowly to zero with increasing diffusion constant
than in 3D, and also the spatial dependence on the receptor size is weaker in 2D than in 3D.
Coming back to the different receptor models with diffusion, the first term of
Eq. <ref> produces exactly half the uncertainty of the first terms of Bialek and Setayeshgar
(Eq. <ref>) and Kaizu et al. (Eq. <ref>) by utilizing only the unbound time intervals.
However, due to factor $1-p$ the second term of Eq. <ref> resembles the second term of
Kaizu et al. (both Eq. <ref> and Eq. <ref> for 3D use a spherical receptor). This
suggests that Kaizu et al. is the correct result for the accuracy of sensing by time averaging,
while Eq. <ref> is the more accurate result when using maximum-likelihood estimation.
Comparison of decision-making algorithms and fixed-time algorithms.
(a) Decision-making algorithms: Wald algorithm (solid curve) has lower uncertainty than
fixed-time log-likelihood ratio estimation based on the Neyman-Pearson (NP) lemma (dashed
curve). Uncertainty is calculated by converting decision error into variance.
(b) Uncertainty estimates based on direct measurement of ligand concentration in a fixed amount of
time: Maximum-likelihood (ML) estimation (solid curve) based on the Cramér-Rao bound of
Fisher information has only half the uncertainty of the Berg-Purcell (BP) limit (dashed curve)
for the standard error of the mean concentration. For further details see Supplementary
§ SINGLE RECEPTOR AS A DECISION MAKER
All the above approaches considered the accuracy based on a fixed measurement time (or number
of binding and unbinding events). However, similar to humans, cells might follow a different strategy
and approach a problem from a decision-making perspective <cit.>:
either deciding based on existing information or waiting to accumulate more data.
Recently, Siggia and Vergassola considered decision making in the context of cells,
proposing that the above maximum-likelihood estimate can further be improved in this way <cit.>.
The simplest implementation of a decision-making strategy is the so-called Wald algorithm <cit.>.
For a single receptor, the Wald algorithm requires calculating the ratio $R$ of the likelihoods of
the time series of binding and unbinding events of a receptor $\Gamma$ (data), conditioned to either
of two hypothesised values of external ligand concentration
\begin{equation}
R=\frac{P(\Gamma|c_1)}{P(\Gamma|c_2)}. \label{EqR}
\end{equation}
The cell then concludes that the ligand concentration is $c_1$ if $R\geq H_1$, that the ligand concentration
is $c_2$ if $R\leq H_2$, or keeps collecting data if $H_1<R<H_2$. $H_1$ and $H_2$ are thresholds
that set the probability of error, i.e. concluding the concentration is $c_1$ if the true concentration is
$c_2$ and vice versa.
This algorithm, by not having a fixed-time constraint, can be shown to be optimal, i.e. the average time
to make a decision between the two options is shorter than provided by any other algorithm with the
same accuracy (decision-error probability).
How can decision making be compared with maximum-likelihood estimation and the Berg-Purcell limit?
Siggia and Vergassola suggested a fixed-time log-likelihood-ratio estimation à la Eq. <ref>
based on the Neyman-Pearson lemma. Due to the fixed-time constraint the Neyman-Pearson
algorithm is in spirit similar to maximum-likelihood estimation. Siggia and Vergassola showed
that the Wald algorithm leads to a shorter decision-making time, on average, than the Neyman-Pearson
algorithm, and so suggested that the Wald algorithm provides the ultimate limit for sensing.
The result for the Wald algorithm indeed shares properties with the maximum-likelihood estimate and the
Berg-Purcell limit. All three reveal a dependence of the measurement (decision)
time on the inverse of the square of the difference of concentration (i.e. $\Delta c^2$). Although
no decision making is involved in maximum-likelihood estimation or the Berg-Purcell limit,
one can still conclude that concentrations $c_1$ and $c_2$ can be distinguished if the measurement
uncertainty is smaller than the difference $\delta c^2 < (c_2 -c_1)^2$, and that, assuming either
$c_1$ or $c_2$ as the true value, an incorrect decision occurs if measurement returns a value closer
to the wrong concentration. This way a decision error can be converted into a type of measurement
uncertainty and vice versa (see Supplementary Information for details).
Fig. 4a shows the thus derived uncertainty in measuring a ligand concentration by
the Wald algorithm and the Neyman-Pearson lemma as a function of average measurement time. For comparison the
maximum-likelihood estimate and the Berg-Purcell limit are shown in Fig. 4b.
However, since the fixed-time likelihood algorithms (Neyman-Pearson lemma
and maximum-likelihood estimate) do not agree, it is difficult to directly compare Wald with the Berg-Purcell
limit. After all, Wald and Neyman-Pearson algorithms are about hypothesis testing and discrimination,
while time averaging (Berg-Purcell) and maximum likelihood are about estimation.
What types of algorithm are cells actually implementing? Consider chemotaxis in the bacterium
Escherichia coli as a prototypical example of chemical sensing. Downstream signaling,
especially slow motor switching, could provide a time scale for Berg-Purcell-type time averaging.
In contrast, biological systems with hysteresis, that is two different thresholds for activation
and deactivation of the downstream pathway, may implement a type of decision-making algorithm.
The classical example is the lactose utilisation system in E. coli, which can be stimulated
by the non-metabolisable 'gratuitous' inducer TMG <cit.>. When TMG is high enough
enzymes of the lac system become induced. Once induced, however, the TMG level must be reduced below
a much lower threshold in order to uninduce the lac system.
§ OUTLOOK
While the question of the physical limits of sensing has been around for decades, only over the
last few years has the importance of this question become clear and its predictions testable by
quantitative experiments <cit.>. While current work is mostly about chemical sensing,
the limits of sensing other stimuli, such as substrate stiffness during durotaxis
(or temperature, pH, particles, and combinations of them, etc.) may be next.
For such measurements, the role of domain size and spatial dimension are interesting
questions. Measurements are often done inside a cell, on 2D surfaces, or along 1D DNA
molecules, and
correlations due to rebinding depend on these parameters <cit.>.
The question of the limits of sensing has also opened up completely new directions, including the
role of active, energy-consuming sensing strategies <cit.>, and
hence the importance of nonequilibrium-physical processes in cell biology. This then connects to
the Landauer limit of information erasure and cellular computation in general <cit.>.
In this area, important questions are linking information theory, statistical inference,
and thermodynamics e.g. in order to produce generalized second laws <cit.>. Additionally, analysis
may move away from only considering receptors to considering receptors and their downstream signaling pathways,
and questions of optimal resource allocation in such pathways emerge <cit.>.
Other areas of study have started to benefit from this work as well, such as gene regulation.
For instance, why do cells often use bursty frequency modulation of gene expression under stress
and in development <cit.>? This may either reflect a need for accurately sensing and
monitoring chemical cues, or simply enhance robustness, e.g. similar to when information is
transmitted between neurons by action potentials. The questions whether cells sense at the physical
limit and if so, how they reach it, and how to design experiments to answer these questions will
occupy us for a while.
§ ACKNOWLEDGMENTS
GA and RGE thankfully acknowledge financial support by the Leverhulme-Trust Grant N. RPG-181.
RGE was also supported by the European Research Council Starting-Grant N. 280492-PPHPI.
NSW was supported by National Science Foundation Grant PHY-1305525. We also
would like to thank an anonymous referee for his valuable comments on the Cramér-Rao bound.
|
1511.00595
|
James Franck Institute and the Department of Physics at the University of Chicago
In this letter we analyze an optical Fabry-Pérot resonator as a time-periodic driving of the (2D) optical field repeatedly traversing the resonator, uncovering that resonator twist produces a synthetic magnetic field applied to the light within the resonator, while mirror aberrations produce relativistic dynamics, anharmonic trapping, and spacetime curvature. We develop a Floquet formalism to compute the effective Hamiltonian for the 2D field, generalizing the idea that the intra-cavity optical field corresponds to an ensemble of non-interacting, massive, harmonically trapped particles. This work illuminates the extraordinary potential of optical resonators for exploring the physics of quantum fluids in gauge fields and exotic space-times.
Time-periodic modulation is under active development both theoretically and experimentally as a tool for Hamiltonian engineering in platforms ranging from cold atoms in optical lattices <cit.> to microwave photons in arrays of superconducting resonators <cit.> and electrons in solids <cit.>. By imposing external fields which couple states of different energies and symmetries, modulation enables time-reversal symmetry breaking and the introduction of synthetic gauge fields <cit.>, as well as manipulation of interactions <cit.>.
In parallel, there is an aggressive effort to explore optical modes coupled to matter as a platform for quantum manybody phenomenology. Single- <cit.> and multi- <cit.> mode optical resonators, as well as photonic crystal structures <cit.> are under investigation to induce long-range interactions between atoms; near-planar resonator/exciton heterostructures <cit.> and quantum fluids <cit.> have been employed to study interacting quantum fluids <cit.>; arrays of microwave resonators coupled to superconducting qubits <cit.> have been harnessed as a model Bose-Hubbard system; and Rydberg Electromagnetically Induced Transparency (rEIT) in trapped atomic gases has recently been demonstrated as a platform for studying 1D quantum dynamics of strongly interacting photons <cit.>.
Here we formalize a new approach to photonic quantum materials based upon exotic optical resonators; following up on our prior work describing Rydberg-dressed photons in a near-degenerate optical resonator as interacting, massive, harmonically trapped 2D particles in synthetic magnetic fields <cit.>, we now provide a more detailed framework for designing the resonators and characterizing the resulting single-particle photonic Hamiltonian dynamics.
Our approach begins in the ray-optics picture where, assuming round-trip ray-trajectories are nearly closed, round-trip propagation may be coarse-grained using a Floquet formalism to provide an effective 2D time-continuous Hamiltonian for the photon (Section <ref>, and Figure <ref>). We will quantize this Hamiltonian (Section <ref>), leading to a wave-optics view of the resulting physics and the appearance of longitudinal modes due to the energy periodicity of the Floquet formalism. In Section <ref> we will classify all of the terms in this Hamiltonian: an inertial mass tensor, harmonic confinement tensor, and a synthetic magnetic field, as well as gauge (non-physical) degrees of freedom. In Section <ref> we consider what happens when the coarse-graining breaks down because ray-trajectories are not nearly closed, and explore a way to recover a simple Hamiltonian picture if the trajectories nearly close after multiple round-trips.
To illustrate the techniques developed in the preceding sections, we next consider several different resonator geometries (Section <ref>), focusing in particular on the symmetric two-mirror resonator. We distinguish between mechanical- and canonical- ray momentum, and show that while photons in near-planar cavities exhibit a positive mass, those in near-concentric cavities exhibit a negative mass. We then briefly analyze twisted resonators, which introduce synthetic magnetic fields for photons.
Finally, we explore the impact of mirror aberrations and non-paraxial optics on the photonic Hamiltonian (Section <ref>). We show that these corrections provide a route to arbitrary potentials and dispersion relations for resonator photons, along with a path to photonic dynamics on curved spatial manifolds.
Schematic Three-Mirror Fabry-Pérot Resonator. A single ray (thin red line) is followed over many-roundtrips through the resonator. The intersection pattern of this ray (red spheres) in a chosen transverse plane (gray polygon) of the resonator traces out a stroboscopic evolution corresponding, in this case, to a massive, harmonically trapped particle, and its image reflected across the origin. These dynamics may be formally understood using the Floquet formalism described in this work.
§ FLOQUET FORMALISM FOR RAYS IN OPTICAL RESONATORS
A paraxial optical resonator may be characterized by an ABCD <cit.> matrix $\bm{M}$, describing the round trip evolution of all light rays in a given transverse plane of the resonator. In particular, the ray described by $V \equiv \begin{pmatrix}\bm{x}\\\bm{s}\end{pmatrix}$, where $\bm{x}$ is the (2D) transverse location of the ray, and $\bm{s}$ its slope, becomes $\bm{M}V$ under round-trip propagation. This describes a discrete linear transformation in phase space, and suggests that such stroboscopic dynamics (see Figure <ref>) are equivalent to periodically sampled continuous evolution under a quadratic time invariant Hamiltonian.
To develop a Hamiltonian formalism describing the continuous evolution of the ray within a particular transverse plane we must first convert the slope $\bm{s}$ into a momentum $\bm{p}$ which is canonically conjugate to $\bm{x}$. This momentum is $\bm{p}=\hbar k \bm{s}$, with $k\equiv2\pi/\lambda$ and $\lambda$ the optical wavelength. We may thus define a phase-space state-vector $\mu\equiv \begin{pmatrix}\bm{x}\\\bm{p}\end{pmatrix}$, and a round-trip propagation matrix $\bm{B}=\bm{\beta}\bm{M}\bm{\beta^{-1}}$, with $\bm{\beta}\equiv \begin{pmatrix}\bm{I}_2&&\bm{0}\\\bm{0}&&\hbar k\bm{I}_2\end{pmatrix}$ and $\bm{I}_2$ the 2x2 identity matrix. The same round-trip propagation matrix applies to $\bm{x}$ and $\bm{p}$ as operators in paraxial wave optics <cit.>.
Noting that round-trip propagation requires a time $T_{rt}=L_{rt}/c$, where $L_{rt}$ is the total round-trip length along the resonator axis, and $c$ is the speed of light, we may now view $\bm{B}$ as a stroboscopic time evolution operator: $\mu(t+T_{rt})=\bm{B}\mu(t)$. If $\mu$ is to be described by continuous evolution under a general (time-invariant) quadratic Hamiltonian,
\begin{equation}\label{eqn:Qdef}
H\equiv\frac{1}{2}\mu^\intercal \bm{G}^\intercal \bm{Q} \mu
\end{equation}
with symmetric $\bm{G}^\intercal \bm{Q}$ and $ \bm{G}=\begin{pmatrix}0&&\bm{I_2}\\-\bm{I_2}&&0\end{pmatrix}$, then Hamilton's equations imply $\frac{d\mu}{dt}=\bm{Q}\mu$. The same result follows using the canonical commutation relations and Heisenberg equations of motion for $\bm{x}$ and $\bm{p}$ as operators.
We may now integrate these equations of motion: $\mu(t+\tau)=\exp(\bm{Q}\tau)\mu(t)$. Using $\tau=T_{rt}$ and solving for $\bm{Q}$, we arrive at $\bm{Q}=\frac{c}{L_{rt}}(\log(\bm{B})-2\pi \bm{I} i l)$, where $l \in \mathbb{Z}$, and $\bm{I}$ is the identity matrix. Substituting for $\bm{Q}$ in (<ref>) yields the effective Floquet Hamiltonian:
\begin{equation}\label{eqn:Hdef}
H = \frac{c}{L_{rt}} \left[\frac{1}{2}\begin{pmatrix} \bm{p}^\intercal && -\bm{x}^\intercal \end{pmatrix} (\log{\bm{\beta M \beta^{-1}}}) \begin{pmatrix} \bm{x} \\ \bm{p} \end{pmatrix}-i\pi l\cdot\sum_{i=1}^{2}[x_i,p_i]\right]
\end{equation}
Here we employ the standard definition of the matrix logarithm as the inverse of the matrix exponential, which is itself defined in terms of its Taylor series. Since $\bm{\beta M \beta^{-1}}$ has only eigenvalues of unit modulus, the logarithm will be purely imaginary and due to the branch cut in its domain will be defined only modulo $2\pi i$. So long as $\bm{x}$ and $\bm{p}$ commute (as they do in the ray-optics limit), this term drops out of the Hamiltonian, leaving:
\begin{equation}\label{eqn:Hray}
H_{classical/ray}= \frac{c}{L_{rt}} \left[\frac{1}{2}\begin{pmatrix} \bm{p}^\intercal && -\bm{x}^\intercal \end{pmatrix} (\log{\bm{\beta M \beta^{-1}}}) \begin{pmatrix} \bm{x} \\ \bm{p} \end{pmatrix}\right]
\end{equation}
§ QUANTUM MECHANICAL TREATMENT
Quantizing the Hamiltonian (eqn <ref>) turns the $\bm{x}$'s and $\bm{p}$'s into non-commuting operators. In this case, noting that $[x_i,p_j]=i\hbar\delta_{ij}$, the Hamiltonian becomes:
\begin{equation}\label{eqn:Hwave}
H_{quantum/wave}= \frac{c}{L_{rt}} \left[\frac{1}{2}\begin{pmatrix} \bm{p}^\intercal && -\bm{x}^\intercal \end{pmatrix} (\log{\bm{\beta M \beta^{-1}}}) \begin{pmatrix} \bm{x} \\ \bm{p} \end{pmatrix}\right]+\frac{\hbar c}{L_{rt}}2\pi\cdot l
\end{equation}
The additional $\frac{\hbar c}{L_{rt}}2\pi\cdot l$ in the energy reflects the fact that we are considering a Floquet Hamiltonian <cit.>; the periodic influence of the mirrors on the optical field means that the eigen-frequencies are only defined up to the inverse round-trip time; this is analogous to the quasi-momentum being defined only up to the lattice spacing in a crystal (see Figure <ref>). It is interesting to note that this energy periodicity corresponds to the cavity free-spectral range, and the $l$ quantum number is actually the familiar longitudinal mode index.
Single-Pass Near-Degenerate Resonators. fig:Classical1Pass Poincaré Section in a plane transverse to the propagation axis of a near-planar near-degenerate optical resonator. The dots indicate discrete transits through the reference plane, with their size reflecting the transit number. The solid curve is the corresponding coarse-grained dynamics in a symmetric harmonic trap. fig:QM1Pass Floquet Energy Spectrum for a near-planar, near-degenerate optical resonator. Each near-degenerate manifold is built entirely from states with the same longitudinal quantum number, with a transverse mode-spacing corresponding to the trapping frequency. The energy spacing between Floquet manifolds is given by Planck's constant times the resonator free spectral range, $c/L_{rt}.$
§ DECOMPOSING A GENERAL QUADRATIC HAMILTONIAN
A general paraxial Fabry-Pérot may include non-commuting non-planar reflections and mirror astigmatism, reflected in a near-arbitrary 4x4 ABCD matrix $\bm{M}$, corresponding to a Floquet Hamiltonian $H_{Floquet}= \frac{c}{L_{rt}} \left[\frac{1}{2}\begin{pmatrix} \bm{p}^\intercal && -\bm{x}^\intercal \end{pmatrix} (\log{\bm{\beta M \beta^{-1}}}) \begin{pmatrix} \bm{x} \\ \bm{p} \end{pmatrix}\right]$. We would like to be able to ascertain, for such an arbitrary resonator, what types of dynamics we can engineer, and for a particular resonator, what we have engineered.
Because this Hamiltonian must be Hermitian it has only 10 independent parameters, and may be decomposed in the following convenient and physically illuminating way:
\begin{equation}\label{eqn:FullDecomp}
H=\frac{1}{2}(\bm{p}-\beta_k\bm{\sigma}^k\cdot \bm{x})^\intercal\bm{m}_{eff}^{-1}(\bm{p}-\beta_k\bm{\sigma}^k\cdot \bm{x})+
\frac{1}{2}\bm{x}^\intercal\bm{\omega}_{trap}^{\intercal}\bm{m}_{eff}^{-1}\bm{\omega}_{trap}\bm{x}
\end{equation}
$\bm{\sigma}^k\equiv[\bm{I},\bm{\sigma}^x,\bm{\sigma}^y,\bm{\sigma}^z]$, $\beta_k\equiv[\delta,\Delta_\times,-iB_z/2,\Delta_{+}]$, $\bm{R}_\phi\equiv\begin{pmatrix}\cos{\phi}&&\sin{\phi}\\-\sin{\phi}&&\cos{\phi}\end{pmatrix}$, and the Pauli matrices operate on the real-space vector $\bm{x}$.
The significance and sources of the various terms:
* $\bm{m}_{eff}$ and $\bm{\omega}_{trap}$ are the transverse effective mass and trapping frequencies of the particle, respectively, arising from the interplay of cavity length and mirror curvature; they become anisotropic (with axes rotated by $\theta_{m,t}$, and fractional difference $\epsilon_{m,t}$) in the presence of mirror astigmatism, often caused by off-axis reflection <cit.> from otherwise spherical mirrors.
* The $\beta_k$'s are the remaining 4 degrees of freedom, and parameterize the gauge potential arising from common-mode defocus ($\delta$), rotated differential defocus ($\Delta_{+}, \Delta_\times$), and resonator twist ($B_z$). Defining the vector potential $\bm{A}\equiv\beta_k\bm{\sigma}^k\cdot \bm{x}$, we find $\bm{\nabla}\times\bm{A}=B_z\hat{\bm{z}}$, indicating that the rest of the terms ($\delta, \Delta_+,\Delta_\times$) may be gauged away via $\bm{A}\rightarrow\bm{A}-\bm{\nabla}f$ for $f=\frac{1}{2}\delta(x^2+y^2)+\Delta_\times x y+\frac{1}{2}\Delta_+(x^2-y^2)$. Thus the only physically significant term is $B_z$, the magnetic field induced by twist.
The recipe for going from an arbitrary resonator geometry to the physical parameters of the space in which the trapped photons live is to: (1) compute a round-trip 4x4 ABCD matrix for the resonator geometry under consideration, (2) from this compute a Floquet Hamiltonian, and finally (3) decompose this Hamiltonian into the physically significant parameters.
§ NEAR-DEGENERACY AFTER MULTIPLE ROUND-TRIPS
The stroboscopic time evolution under the round-trip ray matrix becomes indistinguishable from continuous time evolution in the limit where the transverse dynamics are slow compared to the longitudinal dynamics. In this limit, the frequency splittings between the quantized transverse modes described by the first term in Eqn. (<ref>) become small compared to the splitting between the longitudinal modes described by the second term in (<ref>), and the cavity is said to be nearly degenerate. In such a cavity, rays return to near their original location after each round trip, and the first term in (<ref>) describes the slow precession of the rays after many round trips.
It is often the case that while the ray does not pass near its original phase-space location after a single round-trip, it may do so after several round trips (see Figure <ref>). In the wave-optics picture, such a resonator exhibits a Floquet spectrum where near-degeneracy arises from incrementing both longitudinal and transverse quantum numbers in the appropriate proportion (see Figure <ref>).
Multi-Pass Near-Degenerate Resonators. fig:Classical2Pass Poincaré Section in a plane transverse to the propagation axis of a near-confocal near-degenerate optical resonator. The dots indicate discrete transits through the reference plane, with their (decreasing) size reflecting the transit number. The red and blue solid curves reflect the corresponding coarse-grained dynamics for even- and odd- transit numbers. fig:QM2Pass Floquet Energy Spectrum for a near-confocal, near-degenerate optical resonator. Each near-degenerate manifold is built from states with many different longitudinal quantum numbers. The energy spacing between Floquet manifolds is given by half of the resonator free spectral range.
More formally, we can write $\bm{M}_{multi}=\bm{M}^s$ and $L_{multi}=s\cdot L_{rt}$; replacing $\bm{M}\rightarrow\bm{M}_{multi}$, $L_{rt}\rightarrow L_{multi}$ in the equations from the preceding sections provides the resulting effective Hamiltonian, with the caveat that the ray appears in every plane with multiple images mirroring its dynamics.
Some typical examples of this phenomenon are (1) the near-confocal resonator, which provides the low energy dynamics of a massive particle in a harmonic trap after two round-trips ($s$=2), and (2) the astigmatism-compensated twisted resonator, which provides the low energy dynamics of a massive particle in a magnetic field after a twist-controllable number of round trips. In the former case, the image rays are reflections across the resonator's longitudinal axis. In the latter case, the image rays are rotated about the resonator's axis.
§ EXAMPLES OF SIMPLE RESONATORS
§.§ Symmetric Two-Mirror Fabry-Pérot in the Focal Plane
Here we consider a two-mirror symmetric Fabry-Pérot resonator, with length L, and mirror radii of curvature R. The round-trip ABCD matrix for a single transverse direction, in the central plane of the cavity (a distance L/2 from each mirror) is (defining $g\equiv1-L/R$):
\begin{equation}
\bm{M}_{focal}=
\begin{pmatrix}
-1+2g^2 && g R (1-g^2)\\
-\frac{4g}{R} && -1+2g^2
\end{pmatrix}
\end{equation}
The matrix logarithm may be obtained using a similarity transform to a rotation matrix, giving
(for $-1\leq g\leq 1$ as required for resonator stability):
\begin{equation}
\bm{Q}_{focal}=\frac{c}{2L}
\begin{pmatrix}
0&&\frac{R\alpha}{2\hbar k}\theta\\
-\frac{2\hbar k}{R\alpha}\theta && 0
\end{pmatrix}
\end{equation}
with $\alpha\equiv\sqrt{1-g^2}$, $\theta\equiv\cos^{-1}(-1+2g^2)$.
Thus the Floquet Hamiltonian is given by:
$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2$, where $\omega\equiv\frac{c}{2L}\theta$, and $m\equiv\frac{2\hbar k}{\frac{c}{2L}\alpha R \theta}$.
Quantizing and diagonalizing this Hamiltonian gives a quantum harmonic oscillator $H=\hbar \omega (a^\dagger a+1/2)$ with energy-level spacing $\hbar\omega$, harmonic oscillator lengths $x_0=\sqrt{\frac{\hbar}{m\omega}}=\sqrt{\frac{R\lambda}{4\pi}\sqrt{1-g^2}}$, $p_0=\sqrt{m\hbar\omega}=\sqrt{\frac{4\pi\hbar^2}{\alpha\lambda R}}$, raising operator $a^\dagger\equiv\frac{1}{\sqrt{2}}(\frac{\hat{x}}{x_0}+i\frac{\hat{p}}{p_0})$ and Hermite-Gauss eigenstates $\psi_n=\frac{1}{\sqrt{\sqrt{\pi}2^{n}n!x_0}}e^{-\frac{x^2}{2x_0^2}}H_n(\frac{x}{x_0})$.
These results are consistent with the standard expressions for the two-mirror Fabry-Pérot <cit.>, where the transverse mode spacing is $\hbar\omega$, and the $1/e^2$ waist of the lowest mode is $w_0=\sqrt{\frac{R\lambda}{2\pi}\sqrt{1-g^2}}=x_0\sqrt{2}$; the factor of $\sqrt{2}$ arises from the different conventions for optical mode waist and harmonic oscillator length.
§.§ Symmetric Two-Mirror Fabry-Pérot out of the focal plane
One might be inclined to ask about the impact upon the transverse Hamiltonian of considering a plane other than the focal plane of the resonator. We will work this out backwards first, using knowledge of the resonator eigenmodes and scalar diffraction theory, and then applying the full machinery of the Floquet formalism.
Clearly the eigen-energies of the resonator cannot change (since the eigenstates of the paraxial wave equation are solutions over the full 3D resonator). Furthermore, we know from scalar diffraction theory <cit.> that the impact of diffraction on the mode-functions is (1) a radial rescaling according to $w(z)=w_0\sqrt{1+(\frac{z}{z_r})^2}$; (2) a quadratic wavefront curvature of the form $e^{-i\frac{k x^2}{2\mathcal{R}(z)}}$, for $\mathcal{R}(z)\equiv z \left[1+(\frac{z_r}{z})^2\right]$; and (3) a mode-dependent Gouy phase shift $\zeta_n(z)\equiv n\cdot\tan^{-1}\frac{z}{z_r}$. Here the Rayleigh range is defined by $z_r\equiv\frac{\pi w_0^2}{\lambda}$.
The Gouy phase may be gauged away through a trivial pre-factor on the wavefunction, and we are left with a Hamiltonian system with uniformly spaced eigenvalues and Hermite-Gauss eigenfunctions $\tilde{\psi}_n=\frac{1}{\sqrt{\sqrt{\pi}2^{n-1/2}n!w(z)}}e^{(\frac{-1}{w(z)^2}-\frac{i k}{2\mathcal{R}(z)})x^2}H_n(\frac{x\sqrt{2}}{w(z)})$.
The question, then, is what Hamiltonian has these mode-functions? We would know (a quantum-harmonic oscillator Hamiltonian $H_{QHO}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2$, were it not for the wave-front curvature term. We can remove this term by a unitary transformation $U=e^{i\frac{k x^2}{2 \mathcal{R}(z)}}$. The resulting Hamiltonian is $H=UH_{QHO}U^{\dagger}=\frac{(p+\hbar k x/\mathcal{R}(z))^2}{2m}+\frac{1}{2}m\omega^2 x^2$.
We now work forwards, arriving at this Hamiltonian via Floquet techniques. The round-trip ray-matrix for the same two-mirror symmetric Fabry-Pérot resonator considered previously, but for a plane located at $z=\frac{\epsilon}{2}L$ from the resonator focal plane, is:
\begin{equation}
\bm{M}_{nonfocal}=
\begin{pmatrix}
-1+2g(g-\epsilon(1-g)) && g R (1-g^2+(1-g)^2\epsilon^2)\\
-\frac{4g}{R} && -1+2g^2+2g\epsilon(1-g)
\end{pmatrix}
\end{equation}
A bit of arithmetic yields:
\begin{equation}
\bm{Q}_{nonfocal}=\bm{Q}_{focal}+\frac{c}{2L}\epsilon\theta\sqrt{\frac{1-g}{1+g}}
\begin{pmatrix}
1&&(1-g)\epsilon\frac{R}{2\hbar k}\\
\end{pmatrix}
\end{equation}
The off-diagonal correction term in $\bm{Q}_{nonfocal}$ modifies the photon mass, as it impacts only $\frac{\partial^2 H}{\partial p^2}$; it corresponds to a change in the mode-waist due to diffraction. More interesting are the diagonal corrections to $\bm{Q}_{nonfocal}$: $\bm{Q}_{focal}$ lacks any such terms, which correspond to a $\frac{\partial^2 H}{\partial p \partial x}$ term and reflect a term in the Hamiltonian proportional to $xp$. Following the calculation through, we arrive at:
$H=\frac{(p+b x)^2}{2m}+\frac{1}{2}m\omega^2 x^2$, where $\omega=\frac{c}{2L}\theta$,
$m=\frac{\hbar k}{\frac{c}{4L}R\theta\alpha}\left[\frac{1}{1+\frac{1-g}{1+g}\epsilon^2}\right]$,
$b=\frac{2\hbar k \epsilon}{(1+g)R+(1-g)R \epsilon^2}=\frac{\hbar k}{\mathcal{R}(z)}$ for $\mathcal{R}(z)$ defined as above. This expression coincides with our expectation from the paraxial wave equation, and indeed, the trap frequency does not depend upon defocus.
In short: outside of the focal plane of the resonator, the canonical momentum of the ray (corresponding to its slope as it propagates along the cavity axis) is no longer proportional to the mechanical momentum of the ray (the rate at which it moves in the 2D transverse plane under consideration). Instead, there is an additive correction which is linear in the position, reflecting the wave-front curvature.
§.§ Near-Concentric vs Near-Planar Fabry-Pérot
A separation of timescales between the cavity round-trip time and the harmonic oscillator period requires tuning the cavity geometry near a degeneracy point, where at least one of the transverse mode frequencies becomes much smaller than the cavity free spectral range. Both near-planar ($L\ll R$) and near-concentric ($L\approx 2R$) cavities exhibit such a near-degeneracy. One must be cautious in defining “near-degenerate,” however, because while the ratio of the transverse- to longitudinal- mode spacing goes to zero in both cases, the transverse spacing itself only approaches zero when the appropriate parameter is tuned: the mirror radius of curvature in the near-planar case, and the cavity-length in the near-concentric case.
The trap frequency and mass in the near-planar cavity are: $\omega_{trap}\approx\frac{c}{\sqrt{LR/2}}$, $m\approx\frac{\hbar k}{c}$ . In the near-concentric case they are: $\omega_{trap}\approx\frac{2c}{L}\sqrt{1-\frac{L}{2R}}$, $m\approx -\frac{\frac{\hbar k}{c}}{1-L/2R}$.
Note that:
* In the near-planar case the mass remains finite, and the trap frequency approaches zero only if “planarity” is approached by increasing mirror radius of curvature to infinity rather than by reducing cavity length zero;
* In the near-concentric case the trap frequency goes to zero and the mass diverges, no matter how one approaches degeneracy (by adjusting resonator length, or mirror curvature);
* The photon mass is negative in the near-concentric case. This reflects the fact that the direction of ray propagation out of the plane is opposite to the direction of motion of the particle within the plane (canonical and mechanical momenta are opposite), due to an inversion from the refocusing of the cavity mirrors.
§.§ Twisted Resonators
The simplest resonators that exhibit synthetic magnetic fields for the photons traveling within them are (a) four-mirror resonators that do not reside in a plane, and (b) three-mirror resonators with astigmatic mirrors that are twisted with respect to one another. What these resonator geometries have in common is a helicity to the round-trip manipulation of the photon trajectory, producing dynamics akin to a Floquet topological insulator <cit.>. Because (a) is easier to realize experimentally, it is the path that we will explore here (see Figure <ref>).
Schematic non-planar resonator. $\theta$ is the cavity opening angle, and $l$ is the on-axis length. The out-of-plane reflections result in an image rotation, akin to a pair of dove-prisms, on each round-trip through the resonator. This rotation is equivalent to Coriolis and centrifugal couplings in the Floquet Hamiltonian, the former of which may in turn be interpreted as a uniform synthetic magnetic field for the cavity photons.
We consider a four-mirror resonator where the mirrors do not all reside in a plane, but where all mirrors are curvature-less (“planar”), to keep the analysis simple. As shown in the figure at right, the resonator geometry consists of an opening angle $\theta$, and a principal arm length $l$. To analyze such a resonator requires 4x4 ABCD matrices and careful transformation of coordinate bases at each reflection. The outcome is:
$\bm{M}=\begin{pmatrix}1&&l\\0&&1\end{pmatrix}\otimes \bm{R}(\phi)$, where $\bm{R}(\phi)$ is a 2D rotation through an angle $\phi$ given by: $\cos{\phi}\equiv\frac{1}{16}(3+8\cos{\theta}+12\cos{2\theta}-8\cos{3\theta}+\cos{4\theta})\approx \theta^2$ for small $\theta$.
It is thus apparent that the resonator rotates the coordinate axes on each round-trip; the Floquet Hamiltonian is:
\begin{equation}
\label{eqn:HTwist}
\begin{split}
H_{Floquet}&=\frac{1}{\sqrt{1+2\tan^2{\frac{\phi}{2}}}}\left[\frac{1}{2}\frac{c}{\hbar k}(p_x^2+p_y^2)+\frac{c}{2 l} \phi (y p_x- x p_y)\right]=\left[\frac{p^2}{2 m_{eff}}-\frac{q B_{eff}}{2 m_{eff}}L_z\right]\\
&=\frac{(\bm{p}-\frac{1}{2}q B_{eff}\hat{\bm{z}}\times\bm{r})^2}{2m_{eff}}-\frac{1}{2}m_{eff}\omega_{trap}^2 r^2
\end{split}
\end{equation}
with an effective mass of $m_{eff}=\frac{\hbar k}{c}\sqrt{1+2\tan^2{\frac{\phi}{2}}}$, and rotation giving rise to an effective magnetic field $q B_{eff}=\frac{\hbar k}{l}\phi$ and corresponding centrifugal anti-trapping with frequency $\omega_{trap}=\frac{1}{\sqrt{1+2\tan^2\frac{\phi}{2}}}\frac{c}{2 l}\phi$. The magnetic length corresponding to the synthetic field is $l_B\equiv\sqrt{\frac{\hbar}{\left|qB\right|}}=\sqrt{\frac{l \lambda}{2 \pi \phi}}$. Note that the effective photon charge $q$ and synthetic magnetic field $B_{eff}$ are individually meaningless (and arbitrary); only the product $q B_{eff}$ is well defined.
In practice mirror curvature is essential to compensate the centrifugal anti-trapping; analysis of curved-mirror non-planar resonators in the presence of non-normal-reflection-induced astigmatism is beyond the scope of this work, and will be presented in a separate publication.
§ HIGHER-ORDER PERTURBATIONS TO THE RESONATOR
Thus far we have analyzed the Hamiltonian that results from light trapped within a resonator composed entirely of quadratic optics, with paraxial (quadratic) propagation between these optics. It is well-known that resonator mirrors are measurably imperfect <cit.>, both because they are spherical rather than parabolic and due to wavefront errors. Furthermore, the propagation of optical fields only approximately obeys the paraxial wave equation, with corrections arising at the same order as spherical aberration on the mirrors $\phi_{guoy}\propto\frac{\lambda}{R_{mirror}}$. Such corrections already become relevant even for low-order modes in moderate finesse $F\sim10^4$ resonators (for 1cm ROC mirrors), and become increasingly important for higher order modes. There are a several calculations of the resonator modes in the presence of such corrections <cit.>; here we instead compute the impact of such higher-order terms on the Floquet Hamiltonian.
§.§ Impact of Non-Quadratic Optics on Trapping Potential, Single-Particle Dispersion, and Spatial Curvature
Consider an arbitrary lens providing a position dependent phase shift $\alpha(x)$ in a transverse plane that is a longitudinal distance $z$ from the plane where the Floquet Hamiltonian is defined. If the lens is weak enough that it couples only within Floquet bands, but not between them, its impact may be written as a perturbation to the Floquet Hamiltonian itself. In fact, we rely upon such corrections to truncate the Floquet energy spectrum before the Floquet bands overlap.
The simplest case is $z=0$; a lens in the plane where the Floquet Hamiltonian is defined (henceforth the “Floquet plane”). In this situation the phase shift $\alpha(x)$ per round trip corresponds to an energy (and thus effective potential) $\frac{\hbar c}{L_{rt}}\alpha(\hat{x})$; the lens directly acts as a potential for the cavity photons. Note that $\alpha(\hat{x})$ is evaluated using the Taylor series expansion of $\alpha(x)$.
If the lens were placed within the cavity in a Fourier plane, the phase shift would be dependent upon the momentum in the Floquet plane, and the corresponding Hamiltonian term would be $\frac{\hbar c}{L_{rt}}\alpha(\frac{f}{\hbar k}\hat{p})$, where $f$ is effective focal length of the real-to-Fourier space imaging. Thus in this case, the lens acts to control the dispersion of the photons in the Floquet plane.
It is natural to ask what happens if the lens is placed between real- and Fourier- planes; We will find, in accordance with our ray-optics expectation, that the correction is $\frac{\hbar c}{L_{rt}}\alpha(\hat{x}+\frac{z}{\hbar k}\hat{p})$
We now perform the simplest version of the aforementioned calculation: for an arbitrary lens a distance $z$ from our Floquet plane, in one transverse dimension. More sophisticated calculations in two transverse dimensions with an arbitrary ABCD matrix in-between, are simply extensions of this technique.
Consider the Hamiltonian for an arbitrary lens in the plane z, which produces a round-trip phase-shift of $\alpha(x)$. We can compute its expansion in the plane at $z=0$ by inserting identity operators:
\begin{equation}
\begin{split}
H_{lens}&=\frac{\hbar c}{L_{rt}}\alpha(\hat{x};z)=\frac{\hbar c}{L_{rt}}\int \! \alpha(x) \left|x;z\right\rangle\left\langle x;z\right| \, \mathrm{d}x\\
&=\frac{\hbar c}{L_{rt}}\int \! \alpha(x) \left|x_1;z=0\right\rangle\left\langle x_1;z=0 | x;z\right\rangle\left\langle x;z | x_2;z=0\right\rangle\left\langle x_2;z=0\right| \, \mathrm{d}x \,\mathrm{d}x_1 \, \mathrm{d}x_2
\end{split}
\end{equation}
We now relate localized excitations in the different planes via the free-space Green-function in the paraxial (Fresnel) approximation <cit.>: $\left\langle x_1;z=0 | x;z \right \rangle=\frac{e^{i k z}}{\sqrt{i\lambda z}}e^{\frac{i k }{2 z}(x-x_1)^2}$.
\begin{equation}
H_{lens}=\frac{\hbar c}{L_{rt}}\frac{1}{\lambda z}\int \! \alpha(x) e^{\frac{i k }{2 z}(x-x_1)^2}e^{\frac{-i k }{2 z}(x-x_2)^2} \left | x_2;z=0\right\rangle\left\langle x_2;z=0\right| \, \mathrm{d}x \,\mathrm{d}x_1 \, \mathrm{d}x_2
\end{equation}
redefining $x_j\rightarrow x_j+x$ we have:
\begin{equation}
H_{lens}=\frac{\hbar c}{L_{rt}}\frac{1}{\lambda z}\int \mathrm{d}x_1 \, \mathrm{d}x_2 e^{\frac{i k }{2 z}(x_1^2-x_2^2)}\int\mathrm{d}x\alpha(x) \left | x_1+x;z=0\right\rangle\left\langle x_2+x;z=0\right|
\end{equation}
and identifying:
\begin{equation}
\int\mathrm{d}x\left| x_1+x;z=0\right\rangle\alpha(x)\left\langle x_2+x;z=0\right|=e^{i\hat{p} x_1}\alpha(\hat{x})e^{-i\hat{p}x_2}=e^{i \hat{p}(x_1-x_2)}\alpha(\hat{x}+x_2)
\end{equation}
\begin{equation}
H_{lens}=\frac{\hbar c}{L_{rt}}\frac{1}{\lambda z}\int \mathrm{d}x_1\mathrm{d}x_2 e^{\frac{i k}{2 z}(x_1^2-x_2^2)}e^{i\hat{p}(x_1-x_2)}\alpha(\hat{x}+x_2)
\end{equation}
Performing the $x_1$ integral yields:
\begin{equation}
\begin{split}
H_{lens}&=\frac{\hbar c}{L_{rt}}\frac{\sqrt{i\lambda z}}{\lambda z}\int\mathrm{d}x_2 e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}\alpha(\hat{x}+x_2)\\
&=\frac{\hbar c}{L_{rt}}\frac{1}{\sqrt{-i\lambda z}}\int\mathrm{d}x_2 e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}\alpha(\hat{x}+x_2)e^{\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2} \\
&=\frac{\hbar c}{L_{rt}}\frac{1}{\sqrt{-i\lambda z}}\int\mathrm{d}x_2 \alpha\left(e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}\hat{x}e^{\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}+x_2\right)e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}\\
&=\frac{\hbar c}{L_{rt}}\frac{1}{\sqrt{-i \lambda z}}\alpha(\hat{x}+\frac{z}{\hbar k}\hat{p})\int \mathrm{d}x_2 e^{-\frac{i k}{2 z}(x_2-\frac{z}{\hbar k}\hat{p})^2}\\
&=\frac{\hbar c}{L_{rt}}\alpha(\hat{x}+\frac{z}{\hbar k}\hat{p})
\end{split}
\end{equation}
Where we have (1) inserted an identity operator; (2) required that $\alpha(x)$ be analytic; (3) used the Baker-Campbell-Hausdorf formula; and (4) performed the remaining Gaussian integration. In two transverse dimensions it may be shown that an arbitrary lens produces a correction to the Hamiltonian: $\frac{\hbar c}{L_{rt}}\alpha(\bm{\hat{x}}+\frac{\bm{\hat{p}}}{\hbar k})$.
We now consider the simple case of two quartic lenses $\alpha(x)=\beta x^4$ placed symmetrically around z=0: $\alpha(\bm{x}-\mu\bm{p})+\alpha(\bm{x}+\mu\bm{p})$. The resulting Hamiltonian contains terms quartic in $\bm{x}$, which we view as a quartic confining potential, terms quartic in $\bm{p}$, which we view as quartic single-particle dispersion, and those quadratic in both $\bm{x}$ and $\bm{p}$ corresponding to manifold curvature. Matching terms in the (non-relativistic) geodesic equation yields a scalar curvature $R=\frac{2r^2/r_0^2}{\left(1+(\frac{r}{r_0})^2\right)^2\left(1+3(\frac{r}{r_0})^2\right)^2}$, where $r_0^2\equiv\frac{1}{\beta\mu^2}$, from a metric:
\begin{equation}
%\mathrm{d}s^2=\mathrm{d}x^2+\mathrm{d}y^2+\frac{(4 x y \mathrm{d}x\mathrm{d}y+\mathrm{d}x^2 (3 x^2+y^2)+\mathrm{d}y^2 (3 y^2+x^2))}{r_0^2}-c^2\mathrm{d}t^2
\mathrm{d}s^2=(\mathrm{d}x^2+\mathrm{d}y^2)\frac{x^2+y^2}{r_0^2}+\frac{2(x\,\mathrm{d}x+y\,\mathrm{d}y)^2}{r_0^2}-c^2\mathrm{d}t^2
\end{equation}
Note that an arbitrary perturbation in a single plane may always be understood (through a linear canonical transformation) as a real-space potential- two such perturbations (in different planes) are required to generate curvature that cannot be trivially removed through such a generalized coordinate transformation.
One might be inclined to attempt to draw a parallel to a Trotterized cold atom implementation, where atoms are allowed to evolve in a harmonic trap, and then a quarter of a trap period later, when they are in momentum space, an optical potential is briefly applied to them with the hope that it would provide a effective momentum-dependent force (when viewed another quarter trap cycle later). A simple calculation reveals that this does not work, because the atoms evolve back to real space in a way that depends upon the optical potential applied to them. The key to this idea working for photons where it fails for atoms is that the potential may be weak enough that it does not appreciably impact the photons within a single round-trip (it does not mix different Floquet/longitudinal manifolds), but is still strong enough to substantially change the transverse dynamics within a single near-degenerate Floquet manifold; in short, the cavity photons live simultaneously in real space, momentum space, and everywhere in-between.
§.§ Beyond Paraxial Optics
Taking the scalar wave equation $c^2\nabla^2\psi=\partial_t^2\psi$, and substituting $\psi=\phi e^{i k (z-c t)}$ yields:
$(\nabla_\perp^2+2 i k\partial_z+\partial_z^2)\phi=0$.
At lowest order in $\frac{\partial_z}{k}$ this reduces to the paraxial wave equation: $(\nabla_\perp^2+2 i k \partial_z)\phi=0$.
Iterating this approximate solution for the second $z$ derivative yields a first Born correction: $(2 i k \partial_z+\nabla_\perp^2-\frac{(\nabla_\perp^2)^2}{4 k^2})\phi=0$.
This expression says that, up to a constant offset, the phase acquired from propagation along an arm of the cavity of length $l$, is $k_z l =\frac{1}{2}k l \left[(\frac{p}{\hbar k})^2+\frac{1}{4}(\frac{p}{\hbar k})^4\right]$. The first (quadratic) term gives rise to the kinetic dynamics we have been studying throughout this work, and the second (quartic) term is a new correction. Note that this expression is equivalent to a Taylor expansion of $k_z l = l \sqrt{(\hbar k)^2-p^2}$ to quartic order.
Because the ray momentum is transformed after each mirror reflection, the total non-paraxial correction, arising from the term in each arm of the cavity, is:
\begin{equation}
H_{non-paraxial}=\frac{\hbar c}{L_{rt}}\int\mathrm{d}z\frac{(\nabla_\perp^2)^2}{8 k^3}=\frac{c}{8 (\hbar k)^3}\sum_j\epsilon_j(\bm{D}_j\hat{\bm{p}}+\hbar k \bm{C}_j \hat{\bm{x}})^4
\end{equation}
Here we have employed ABCD matrices that move from the Floquet plane to the region between the $j^{th}$ and $(j+1)^{st}$ mirrors and $\epsilon_j$, the fraction of the path length between those two mirrors.
We thus see that the lowest order correction to paraxial optics introduces a quartic potential, quartic dispersion, and cross terms (including manifold curvature), akin to an out-of-focus quartic lens.
§ CONCLUSION
In this paper, we have harnessed the fact that an optical resonator may be viewed as a periodic drive applied to a 2D optical field to develop a Hamiltonian formalism for understanding photonic dynamics in such resonators. This approach applies both within the paraxial, quadratic approximation, where it results in arbitrary quadratic Hamiltonians tunable through resonator geometry, and to perturbations which extend beyond the paraxial limit and produce exotic photon traps, dispersions, and manifold curvatures. This work points to fascinating studies of wave dynamics on curved manifolds, and in conjunction with Rydberg EIT to induce interactions between photons, an exciting route to strongly correlated photonic quantum materials, including those in the presence of synthetic gauge fields and manifold curvature.
§ ACKNOWLEDGEMENTS
We would like to thank Brandon Anderson, Michael Levin, and Paul Weigmann for fruitful discussions. This work was supported by AFOSR grant FA9550-13-1-0166, and DARPA grant D13AP00053.
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1511.00446
|
In this paper, we investigate an energy efficiency (EE) maximization problem in multi-user multiple input single output downlink channels.
The optimization problem in this system model is difficult to solve in general, since it is in non-convex fractional form.
Hence, conventional algorithms have addressed the problem in an iterative manner for each channel realization, which leads to high computational complexity.
To tackle this complexity issue, we propose a new simple method by utilizing the fact that the EE maximization is identical to the spectral efficiency (SE) maximization for the region of the power below a certain transmit power referred to as saturation power.
In order to calculate the saturation power, we first introduce upper and lower bounds of the EE performance by adopting a maximal ratio transmission beamforming strategy.
Then, we propose an efficient way to compute the saturation power for the EE maximization problem.
Once we determine the saturation power corresponding to the maximum EE in advance, we can solve the EE maximization problem with SE maximization schemes with low complexity.
The derived saturation power is parameterized by employing random matrix theory, which relies only on the second order channel statistics.
Hence, this approach requires much lower computational complexity compared to a conventional scheme which exploits instantaneous channel state information, and provides insight on the saturation power.
Numerical results validate that the proposed algorithm achieves near optimal EE performance with significantly reduced complexity.
§ INTRODUCTION
The material in this paper was presented in part at the
IEEE International Conference on Communications (ICC), London, UK, June 2015 <cit.>.
Exponentially increasing service demands for wireless communications have mainly required huch higher transmission rate, which leads to increased energy consumption <cit.>.
Recently, the energy consumption has been regarded as a crucial parameter when designing wireless networks, since low energy efficient transmission has a negative impact on the environment and hamper sustainable development.
Thus, from the perspective of green communications, energy efficiency (EE) has received a lot of attentions for future wireless communication systems <cit.>.
The EE is defined as the ratio of the sum rate to the total power consumption measured in bit/Joule.
Many researches have addressed EE solutions for various system model scenarios <cit.>.
In <cit.>, the EE problem was formulated by exploiting dirty paper coding and the uplink-downlink duality in broadcasting channels.
While this work presented a performance upper bound for broadcasting channels, many practical constraints exist due to high complexity.
For general scenarios with inter-user interference (IUI), the optimization problem for EE remains non-convex, and thus it is difficult and more challenging to solve.
Recently, EE schemes based on linear beamforming were studied for multiple-input single-output (MISO) interfering broadcasting channels <cit.>.
By transforming the fractional programming into linear programming <cit.> and applying the weighted minimum mean square error (WMMSE) approach in <cit.>, a local optimal solution was obtained in <cit.>.
However, this algorithm solved the EE problem in an iterative manner for each channel realization that gives rise to high computational complexity.
Moreover, it is difficult to get insight on the system performance without resorting to Monte Carlo simulations.
To tackle these issues mentioned above, we investigate a simple and practical EE maximization scheme in multi-user (MU) MISO downlink channels.
First, we observe that the EE value obtained from the EE maximization problem is saturated at a certain transmit power, which will be referred to as saturation power.
Then, the problem of the EE maximization becomes identical to that of the spectral efficiency (SE) maximization for the region below.
As a result, the EE maximization problem can simply be computed from the SE maximization by identifying the saturation power.
However, the optimum saturation power for the EE maximization scheme in considered system models is difficult to compute.
Hence, we first attempt to derive lower and upper bounds of the EE performance by applying maximal ratio transmission (MRT) beamforming.
Then, the saturation power of the lower and upper bounds of the EE are presented in closed form by employing random matrix theory <cit.>.
Here, based on the relationship of the derived saturation power and the EE performance, we can efficiently determine the saturation power by exploiting an interpolation method.
It is noted that the optimal saturation power is bounded by the derived saturation power for the lower and upper bounds of the EE.
Consequently, utilizing the derived saturation power, we can solve the EE problem efficiently by only adopting the SE maximization scheme.
Numerical results validate that the proposed algorithm achieves near optimal EE performance with much lower complexity.
The rest of the paper is comprised as follows: Section II presents a system model and the problem formulation.
In Section III, the relationship between EE and SE is described briefly.
Then, we derive the saturation power based on large system analysis and suggest a simplified scheme for the EE maximization utilizing the derived saturation power in Section IV.
From the simulation results in Section V, we confirm the validity of the proposed method. Finally, this paper is terminated with conclusions in Section VI.
Throughout the paper, we adopt lowercase and uppercase boldface letters for vectors and matrices, respectively. The superscript $(\cdot)^H$ stands for conjugate transpose. In addition, $||\cdot||$ and tr$(\cdot)$ represent Euclidean 2-norm and trace, respectively.
Also, $\bold{I}_d$ denotes an identity matrix of size $d$. A set of $N$ dimensional complex column vectors is expressed by $\mathbb{C}^N$.
§ SYSTEM MODEL
In this paper, we consider an MU-MISO channel with bandwidth $W$ where a base station (BS) equipped with $M$ transmit antennas serves $N$ users with a single antenna.
Then, the received signal $y_k$ at the $k$-th user ($k = 1, \cdots, N$) is given by
y_k = √(p_k)h_k^H v_k s_k + ∑_j ≠k√(p_j)h_k^H v_j s_j + n_kwhere $p_k$ is the transmit power consumed by the $k$-th user satisfying $\sum_{k=1}^N p_k \leq P$ [Watt/Hz] in order to satisfy BS transmit power constraint $PW$, $\bold{h}_k \in \mathbb{C}^{M}$ defines the flat fading channel vector from the BS to the $k$-th user with the coherence time $T$, $\bold{v}_k$ means the beamforming vector for the $k$-th user with $||\bold{v}_k||^2 = 1$, $s_k \sim \mathcal{CN}(0,1)$ represents the complex data symbol intended for the $k$-th user, and $n_k \sim \mathcal{CN}(0,\sigma^2)$ stands for the additive white Gaussian noise at the $k$-th user.
For notational conveniences, we denote $\{\bold{p}\}$ and $\{\bold{v}\}$ as a set of all transmit power values and beamforming vectors, respectively.
Then, the individual rate of the $k$-th user is computed as
R_k({p}, {v}) = log(1 + SINR_k({p}, {v}))where $\text{SINR}_k(\{\bold{p}\}, \{\bold{v}\})$ indicates the individual signal-to-interference-plus-noise-ratio (SINR) for the $k$-th user as
SINR_k({p}, {v}) = |h_k^H v_k|^2 p_k/∑_j ≠k|h_k^H v_j|^2 p_j + 𝒩_0.Here, $\mathcal{N}_0$ represents $\mathcal{N}_0 = \sigma^2/W$.
During a time-frequency block $TW$, the total amount of the transmitted information is given by
TW∑_klog_2(1 + SINR_k). [bits]From an EE point of view, we consider the power consumption for a BS <cit.>, where the total power consumption during the time-frequency block $TW$ is modeled as
P_T({p}) = TW(ξ∑_k p_k ||v_k||^2 + P_const). [Joule]
Here, $\xi \geq 1$ stands for an inefficiency of the power amplifier and $P_{const}$ equals $P_{const} = MP_c + P_o$ where $P_c$ is defined as $P_c = \frac{P_c^{'}}{W}$ with $P_c^{'}$ being the constant circuit power consumption proportional to the number of radio frequency chains, and $P_o$ means $P_o = \frac{P_o^{'}}{W}$ with $P_o^{'}$ indicating the static power at the BS which is independent of the number of transmit antennas.
Then, the EE is defined as the ratio of the sum rate to the total power consumption
EE({p}, {v}) = ∑_k R_k({p}, {v})/P_T({p}).Therefore, the EE maximization problem can be formulated by
max_{p}, {v} EE({p}, {v})
s.t. ∑_k=1^N p_k ≤P.
It is noted that problem (<ref>) is non-convex because of coupled interference and the fractional form, and thus computing a solution of the problem is quite complicated.
In <cit.>, a local optimal solution of the EE for interfering broadcasting channels was obtained by two layer optimization adopting a linear subtractive form.
However, it should be solved in an iterative manner for each channel realization, which gives rise to high computational complexity.
In what follows, we focus on a simple algorithm which can solve the EE maximization with reduced complexity.
§ PROPERTIES OF ENERGY EFFICIENCY
In this section, we investigate the characteristics of the EE.
Based on the properties of the EE described in this section, the derivation of the saturation power is triggered to optimize the EE performance in a simple manner.
It is interesting to note that the EE performance is saturated once the total transmit power exceeds a certain point, which we call saturation power.
Then, the maximization of the EE becomes identical to that of the SE for the region below the saturation power.
To explain this, we consider a simple EE model for the transmit power $P$ as
EE(P) = R(P)/P+P_static
where $R(P) = \text{log}(1+P)$ and $P_{static}$ indicates the static power consumption term.
From this EE expression (<ref>), the optimal transmit power $P_{EE}$ which maximizes the EE can be calculated in closed form as <cit.>
P_EE = exp(𝒲_0(P_static-1/e)+1)-1where $\mathcal{W}_0(\cdot)$ denotes the principal branch of the Lambert W function defined as the inverse function of $f(x) = xe^x$.
For the transmit power region below $P_{EE}$, full transmit power should be applied to achieve the maximal performance of the EE.
This is due to the fact that when the total transmit power is fully consumed at the region below the saturation power, the consumed power becomes constant which does not affect the EE optimization.
In this case, the considered problem is equivalent to the sum rate maximization.
This suggests that transmitting the maximum available power is most energy efficient at this region.
In the same way, the SE performance can be maximized at the same region because the rate $R(P)$ is monotonically increasing function with respect to $P$.
In contrast, for the region above the saturation power, consuming full power at the BS degrades the EE performance, since a sum rate gain cannot compensate for the increased power consumption in the EE.
In Figure 1, we illustrate the performance curves of the SE and EE for equation (<ref>) with respect to the transmit power $P$.
In this example, the saturation power $P_{EE}$ is shown to be about 2 dB.
It can be observed that the SE maximization is identical to the EE maximization when the transmit power $P$ is smaller than the saturation power $P_{EE}$.
Also, $P_{EE}$ corresponds to the power which yields the maximal SE.
In Figure 1 (a), the EE scheme achieves the same rate as the SE scheme for $P \leq P_{EE}$, while the rate of the EE scheme becomes saturated for $P \geq P_{EE}$, since the EE scheme fixes the power at $P_{EE}$ to maximize the EE.
Meanwhile, the SE algorithm always transmits at full power for maximizing the SE even after the saturation power $P_{EE}$.
In Figure 1 (b), the SE scheme exhibits a performance loss in terms of of the EE because a gain of the rate cannot make up for the impact of the increased power consumption as mentioned before.
Comparison of SE and EE for equation (<ref>) in terms of the transmit power $P$ (a) SE performance and (b) EE performance
§ DERIVATION OF SATURATION POWER
In this section, motivated by properties of the EE shown in Section III, we first focus on determining the saturation power which starts to yield the saturated EE performance.
Unfortunately, the EE solution in <cit.> requires iterative methods, and thus it is not possible to obtain the saturation power in closed form.
Alternatively, we address lower and upper bounds of the EE performance to allow simple computations of the saturation power.
From the derived lower and upper bounds of the EE, we can identify the saturation power corresponding to each bound of the EE in a closed form solution.
Then, by utilizing the determined saturation power, we propose a new EE maximization scheme only adopting the SE maximization method with reduced complexity.
For mathematical tractability, we assume that the transmit power of each user $p_k$ for $k = 1, \cdots, N$ is equally allocated.
§.§ Saturation Power for a Lower Bound of EE
First, we start with obtaining a lower bound of the EE performance.
One simple precoding which can serve as a lower bound is MRT beamforming which employs $\bold{v}_{k,MRT} = \frac{\bold{h}_k}{||\bold{h}_k||}$.
In this case, the EE for MRT $\eta_{MRT}$ with equal power allocation can be expressed as
η_MRT = ∑_k=1^N log(1+SINR_k,MRT)/ξP + P_const
where $\text{SINR}_{k,MRT}$ is given by
SINR_k,MRT = |h_k^Hv_k,MRT|^2P/N/∑_j ≠k|h_k^Hv_j,MRT|^2P/N + 𝒩_0 .
It is clear that $\text{SINR}_{k,MRT}$ changes for every channel realizations.
To avoid calculating these instantaneous channel gains, we apply random matrix theory in (<ref>).
It is worth noting that in the asymptotic regime, the channel gain values become deterministic which depends only on the second order channel statistics and the randomness according to instantaneous channel realizations disappears.
Although the parameters are obtained in the large system limit, we will show in the simulation section that this approximation is well matched even for small dimensions.
To derive large system results, we will utilize the lemma in <cit.> which assumes that the number of users $N$ and the number of transmit antennas $M$ grow large with $\frac{N}{M}$ at a fixed ratio.
We emphasize that the asymptotic analysis is used only to derive deterministic channel gain values, and the system which we consider in this paper has a finite dimension.
From <cit.>, we can calculate the deterministic value for both the desired signal power $|\bold{h}_k^H\bold{v}_{k,MRT}|^2$ and the interference signal power $|\bold{h}_k^H\bold{v}_{j,MRT}|^2$ in (<ref>).
First, the desired signal power for MRT is written by
|h_k^Hv_k,MRT|^2=|h_k^H I_M h_k|^2/||h_k||^2 = h_k^H I_M h_k.Then, using the trace lemma in <cit.>, we have
h_k^H I_M h_k - tr(I_M) 0.
Also, the interference signal power for MRT is determined in a similar manner as
|h_k^Hv_j,MRT|^2=|h_k^H I_M h_j|^2/||h_j||^2 = h_k^H I_M h_jh_j^H I_Mh_k/h_j^H h_j.
After some mathematical manipulations, the numerator and denominator terms for the interference signal power in (<ref>) converge almost surely as
h_k^H I_M h_jh_j^H I_Mh_k - tr(I_M^2) 0,
h_j^H h_j - tr(I_M) 0.Then, the interference signal power $|\bold{h}_k^H\bold{v}_{j,MRT}|^2$ is given by
h_k^H I_M h_jh_j^H I_Mh_j/h_j^H h_j - 1 0.
By replacing the deterministic channel gain values in (<ref>) and (<ref>) into $\text{SINR}_{k,MRT}$, the asymptotic EE for MRT $\eta_{MRT}^\circ$ is presented as
η_MRT^∘= ∑_k=1^N log(1+SINR_k,MRT^∘)/ξP + P_const
where $\text{SINR}_{k,MRT}^\circ$ is denoted as
SINR_k,MRT^∘= M P/(N-1)P + N𝒩_0.We can note that $\text{SINR}_{k,MRT}^\circ$ is a function of $P$ and is no longer dependent on channel realizations.
As a result, $\eta_{MRT}^\circ$ can be identified only based on given system configurations.
To obtain the saturation power for a lower bound of the EE, we utilize equation (<ref>) for the following theorem.
A lower bound of $\eta_{MRT}^\circ$, defined by $\eta_{LB}$, is expressed by
η_LB = N M P/(ξP + P_const){(N + M -1) P + N𝒩_0}.Then, the saturation power $P_{LB}$ corresponding to the EE lower bound $\eta_{LB}$ is computed by
P_LB = √(N 𝒩_0 P_const/ξ(N + M -1)).
The numerator term of (<ref>) is reformulated by
N log(1+ M P/(N-1) P+ N𝒩_0) = N log(N+M-1) P+ N𝒩_0/(N-1) P + N𝒩_0
= -N log(1 - M P/(N+M-1) P + N𝒩_0).
From the fact that the term $\frac{M P}{(N+M-1) P + N\mathcal{N}_0}$ is smaller than 1, equation (<ref>) can be bounded by adopting the relationship $\log(1+x) \leq x $ for $|x| < 1$ as
-N log(1 - M P/(N+M-1) P + N𝒩_0) ≥N M P/(N+M-1) P + N𝒩_0 ≜R_LB.
Consequently, the lower bounded EE $\eta_{LB}$ is presented by
η_LB = R_LB/ξP + P_const.
Then, the saturation power $P_{LB}$ can be determined by differentiating $\eta_{LB}$ in (<ref>) with respect to $P$.
Thus, $P_{LB}$ which maximizes $\eta_{LB}$ is calculated from the following equation as
(N+M-1)ξP_LB^2 = N𝒩_0 P_const.From this equation, we arrive at (<ref>)
According to the result in Theorem 1, the saturation power for a lower bound of EE is obtained with a closed form expression (<ref>).
§.§ Saturation Power for an Upper Bound of EE
In the previous subsection, a lower bound for the EE is found by applying MRT with equal power allocation.
Now, we derive an upper bound of the EE by ignoring the effect of IUI.
Then, the EE with no IUI, $\eta_{no-IUI}$, can be given by
η_no-IUI = ∑_k=1^N log(1+|h_k^Hv_k,MRT|^2/𝒩_0P/N)/ξP + P_const.
When the IUI is not considered, the numerator term of $\eta_{no-IUI}$ is maximized by the MRT beamforming because the beam is aligned with the channel for the intended user.
Therefore, the EE performance is upper bounded by $\eta_{no-IUI}$.
By employing the large system analysis as in Section IV-A, the numerator of $\eta_{no-IUI}$ in (<ref>) is presented as
log(1+|h_k^Hv_k,MRT|^2/𝒩_0P/N) - R_UB 0
where $R_{UB}$ is defined as $R_{UB} \triangleq N \log\left(1+\frac{M}{N \mathcal{N}_0}P\right)$.
Then, an asymptotic upper bound of the EE, denoted by $\eta_{UB}$, is expressed as
η_UB = R_UB/ξP + P_const.
To compute the saturation power for the EE upper bound $\eta_{UB}$, we address the following theorem.
The saturation power $P_{UB}$ which maximizes $\eta_{UB}$ is written by
P_UB = N 𝒩_0/M[exp(1 + 𝒲_0(1/e(M P_const/N 𝒩_0 ξ-1)))-1].
By differentiating $\eta_{UB}$ with respect to the total transmit power $P$, it follows
d η_UB/d P = N/(ζP + P_const)^2 [ M/N𝒩_0ξP + P_const/1+M/N𝒩_0P - ξlog(1+M/N𝒩_0P) ].
For the saturation power $P_{UB}$ corresponding to $\eta_{UB}$, setting equation (<ref>) to zero yields
M/N 𝒩_0(ξP_UB + P_const) - ξslogs = 0where $s =1+\frac{M}{N \mathcal{N}_0}P_{UB}$.
Then, we have
M P_const/N 𝒩_0 ξ - 1 = s(logs - 1).With the Lambert W function, this form can be solved by a closed form expression as
logs = 1 + 𝒲_0(1/e(M P_const/N 𝒩_0 ξ-1)).Since $\mathcal{W}\left(\frac{1}{e}\left(\frac{M P_{const}}{N \mathcal{N}_0 \xi}-1\right)\right) \geq -1$, which is guaranteed by $s \geq 1$ for $P_{UB} \geq 0$, the principal branch of the Lambert W function $\mathcal{W}_0$ is selected.
From this equation, we can reach the saturation power $P_{UB}$ in (<ref>).
§.§ Relationship among $P_{LB}$, $P_{UB}$, and the optimal saturation power $P^*$
From the previous subsections, the saturation power for lower and upper bounds of the EE has been derived.
In this subsection, we address the property of the optimal saturation power denoted as $P^*$ related to $P_{LB}$ and $P_{UB}$.
As already mentioned, the average sum rate of the SE maximization scheme denoted by $R_{SE}$ is clearly bounded between $R_{LB}$ in (<ref>) and $R_{UB}$ in (<ref>).
From this, we will show that the optimal saturation power $P^*$ lies between $P_{LB}$ and $P_{UB}$ by adopting the linear parametric programming approach.
It can be seen that the optimization problem (<ref>) belongs to fractional programming.
Hence, this problem can be transformed into parametric programming as in <cit.>.
We consider the following equivalent form of the fractional program in (<ref>) as
max_{p}, {v}, λ∈ℝ λ
s.t. ∑_k R_k({p}, {v}) - λP_T({p}) ≥0.For a given parameter $\lambda$, it is noted that the optimization problem is referred to as a feasibility problem in $\{\bold{p}\}$ and $\{\bold{v}\}$.
Therefore, the optimal value of the parameter $\lambda$ can be found by using a bisection method for the feasibility problem at each step of the algorithm <cit.>.
Defining a function $F(\lambda)$ as
F(λ) = max_{p}, {v} ∑_k R_k({p},{v}) - λP_T({p}),it is obvious that $F(\lambda)$ is convex and strictly decreasing in $\lambda$.
Moreover, this is regarded as bi-criterion optimization such that $\sum_k R_k(\{\bold{p}\},\{\bold{v}\})$ is maximized while $P_T(\{\bold{p}\})$ is minimized.
The parameter $\lambda$ determines the relative weight of the total power consumption $P_T(\{\bold{p}\})$.
For this bi-criterion problem, the set of Pareto-optimal values is called the optimal trade-off curve <cit.>.
As presented in <cit.>, solving problem (<ref>) is equivalent to finding the root of the nonlinear function $F(\lambda)$, i.e., $F(\lambda)=0$.
In other words, $\lambda$ means the slope of the tangent for the trade-off curve and the optimal $\lambda$ denoted by $\lambda^*$ occurs when $F(\lambda^*) = 0$.
Trade-off curves between sum rate and total consumed power with $P_c' = 30$ dBm and $P_o' = 40$ dBm
Then, by exploiting these properties of the linear parametric programming, we can address the relationship among $P_{LB}$, $P_{UB}$, and $P^*$.
From the trade-off curves in Figure 2, we can identify the slope of the tangent for $R_{LB}, R_{SE}$, and $R_{UB}$.
In this figure, the trade-off curve is illustrated between the sum rate and the total power consumption in (<ref>) with $N=M=3$ and $P_{const} = 13$.
We can see that the SE performance $R_{SE}$ is certainly bounded as $R_{LB} < R_{SE} < R_{UB}$ for all $P_T$.
Here, $P_{T,LB}$, $P_{T}^*$, and $P_{T,UB}$ denote the total power consumption at the BS for $R_{LB}$, $R_{SE}$, and $R_{UB}$, respectively.
It is noted that the optimal power consumption for each scheme is defined as a contact point with each sum rate curve and the corresponding tangent.
Then, the saturation power is calculated by subtracting $P_{const}$ from the optimal power consumption, e.g., $P_{LB} = P_{T,LB} - P_{const}$.
As shown in the figure, the saturation power for each scheme has the relationship of $P_{LB} < P^* < P_{UB}$.
Accordingly, the achieved EE values which equal the slope of the tangent for each sum rate are given as $\eta_{LB}(P_{LB}) < \eta_{SE}(P^*) < \eta_{UB}(P_{UB})$ where $\eta_{SE}(P^*) = \frac{R_{SE}(P^*)}{\zeta P^* + P_{const}}$ represents the maximum EE value for $\eta_{SE}$ corresponding to the saturation power $P^*$.
Therefore, utilizing this property, we quantify the optimal saturation power $P^*$ as a function of $P_{LB}$ and $P_{UB}$ in the following.
§.§ Proposed EE Scheme based on the Saturation Power
From the derived saturation power for lower and upper bounds of the EE, we will determine the saturation power $P^*$.
To this end, we adopt an interpolation method in a similar way to <cit.>.
First, the maximum value of $\eta_{SE}$ is defined by $\gamma_{SE} = \eta_{SE}(P^*)$.
Also, the maximum EE value for the lower and upper bounds $\gamma_{LB}$ and $\gamma_{UB}$ are denoted by $\eta_{LB}(P_{LB})$ and $\eta_{UB}(P_{UB})$, respectively.
Then, the proposed saturation power $P_{prop}$ can be computed as a point between $P_{LB}$ and $P_{UB}$ by exploiting the relationship between the saturation power and the corresponding EE value for each bound.
For instance, if $\gamma_{SE}$ is close to $\gamma_{UB}$, the saturation power $P^*$ should be set near $P_{UB}$.
Therefore, we consider the following relation on the difference between various bounds addressed in Section IV-C as
After some mathematical manipulations, equation (<ref>) is expressed as
P_UB-P_prop = G(P_prop-P_LB),where the EE gap $G$ represents
G = γ_UB-γ_SE/γ_SE-γ_LB.
Then, the proposed saturation power $P_{prop}$ can be calculated as
P_prop = ωP_LB + (1-ω) P_UB
where $\omega = \frac{G}{1+G}$ means the weight factor between $P_{LB}$ and $P_{UB}$.
In fact, it is not possible to obtain $G$ directly since $\gamma_{SE}$ is unknown.
Therefore, in order to compute the EE gap $G$, we consider regularized zero forcing (RZF) beamforming <cit.>.
The relation between $\gamma_{SE}$ and the maximum EE performance for RZF $\gamma_{RZF}$ can be formulated by $\gamma_{SE} = \beta \gamma_{RZF}$, where a constant $\beta > 1$ accounts for the performance gain of $\eta_{SE}$ over $\eta_{RZF}$.
In what follows, we determine $\gamma_{RZF}$ which will be used in calculating $G$ in (<ref>) for a given $\beta$.
By adopting random matrix theory in <cit.>, the asymptotic EE performance for RZF $\eta_{RZF}^\circ$ is expressed as
η_RZF^∘= ∑_k=1^N log(1+SINR_k,RZF^∘)/ξ∑_k=1^N p_k + P_constwhere $\text{SINR}_{k,RZF}^\circ$ is obtained as
SINR_k,RZF^∘= (m_k^∘)^2/Γ_k^∘+ Ψ^∘/ρ(1+m_k^∘)^2.Here, the deterministic equivalent values $m_k^\circ$, $\Gamma_k^\circ$, and $\Psi^\circ$ are derived in <cit.> for RZF and $\rho = P /\mathcal{N}_0$.
These parameters are affected by $p_k$ for $\forall k$.
Assuming equal power allocation with uncorrelated channels, the deterministic equivalent for all users has the same value, i.e., $m_k^\circ = m^\circ$ and $\Gamma_k^\circ = \Gamma^\circ$.
Then, the $\eta_{RZF}^\circ$ with equal power allocation denoted by $\eta_{RZF}$ can be calculated as
η_RZF = Nlog(1+(m^∘)^2 P/Γ^∘P + Ψ^∘(1+m^∘)^2𝒩_0)/ξP + P_const.To determine the maximum EE for RZF $\gamma_{RZF}$, $\eta_{RZF}$ is differentiated with respect to $P$ and is set to zero as
dη_RZF/d P =N (m^∘)^2 A/{((m^∘)^2+Γ^∘)P+A}(Γ^∘P+A)(ξP+P_const)- N log(1+(m^∘)^2 P/Γ^∘P+ A)ξ/(ξP+ P_const)^2 = 0
where $A = \Psi^\circ(1+m^\circ)^2 \mathcal{N}_0$.
Here, we define the function $f(P)$ to identify the saturation power $P_{RZF}$ for $\eta_{RZF}$ as
f(P) = log(1 + (m^∘)^2 P/Γ^∘P + A) - (m^∘)^2 A(P + P_const/ξ)/{((m^∘)^2+Γ^∘)P + A}(Γ^∘P + A).
It is interesting to note that the function $f(P)$ is monotonically increasing with respect to $P$ and the equation $f(P) = 0$ has a unique solution.
Also, $f(P)$ converges to $-\frac{(m^\circ)^2 P_{const}}{\zeta A}$ for $P \rightarrow 0$ and $\log(1 + (m^\circ)^2/\Gamma^\circ)$ for $P \rightarrow \infty$.
Therefore, the EE saturation power $P_{RZF}$ can be computed simply by one dimensional search and the maximum EE performance can be obtained as $\gamma_{RZF} = \eta_{RZF}(P_{RZF})$.
Consequently, after $\gamma_{SE}$ is replaced by $\beta \gamma_{RZF}$, we can determine the saturation power $P_{prop}$ in (<ref>).
Now, we propose a simplified EE maximization scheme by utilizing the derived saturation power $P_{prop}$.
In the simplified scheme, a solution of the SE maximization problem is adopted for the original EE maximization problem.
First, when the available transmit power $P$ is less than $P_{prop}$, the SE maximization scheme is conducted with full power $P$ to maximize the EE performance.
On the contrary, if $P$ is greater than $P_{prop}$, the fixed transmit power $P_{prop}$ is used with the SE maximization method.
In summary, after the saturation power $P_{prop}$ is calculated in (<ref>), the SE maximization algorithm in <cit.> is processed with the transmit power given by $\min(P_{prop},P)$ to generate a beamforming solution.
Next, we briefly address the computational complexity.
The structure of the EE algorithm in <cit.> is comprised by the outer layer and the inner layer optimization.
The outer layer searches for the EE parameter $\eta$, while the inner layer solves the non-fractional subtractive problem for a given $\eta$ computed at the outer layer.
Thus, the inner layer algorithm should be executed whenever the EE parameter is updated at the outer layer, and this causes high computational complexity.
The complexity of the algorithm in <cit.> is similar to that of the inner layer part in <cit.>.
It is noted that the SE maximization algorithm in <cit.> is processed only once in the proposed EE scheme.
Moreover, when determining the saturation power $P_{prop}$, we have $P_{LB}$ and $P_{UB}$ in closed form which depends only on the second order channel statistics and the estimation of $\gamma_{SE}$ needs simple one-dimensional search.
Hence, the complexity of our proposed algorithm is much lower than that of the EE algorithm in <cit.>.
The computation time of the algorithm in <cit.> is contingent on the convergence threshold $\delta$ for the outer layer and it takes about ten times higher than that of the proposed scheme with $\delta = 10^{-3}$, while the EE performance of the proposed scheme is quite close to that of the algorithm in <cit.>.
§ NUMERICAL RESULTS
Comparison of the saturation power with $P_c' = 30$ dBm and $P_o' = 40$ dBm
In this section, we verify the validity of our proposed method through Monte Carlo simulations.
The numbers of users $N$ and transmit antennas $M$ are equal to 3 unless specified otherwise.
Also, we adopt the bandwidth $W = 20$ MHz, the noise spectral density $\mathcal{N}_0 = -174$ dBm/Hz, noise figure $N_F = 7$ dB and the inefficiency of the power amplifier $\xi = 1$.
The circuit power per antenna $P_c'$ and the static power consumed at the BS $P_o'$ are set to 30 dBm and 40 dBm, respectively.
First, we evaluate the saturation power derived in Section IV in Figure 3.
Here, regular and inverted triangles mean the maximum EE for (<ref>) and (<ref>), respectively, and
$P_{LB}$ and $P_{UB}$ are computed as the derived saturation power corresponding to these maximum points shown in (<ref>) and (<ref>).
Also, star and rectangular marks denote the EE performance of the SE maximization scheme $\eta_{SE}$ obtained by the saturation power $P^*$ and $P_{prop}$, respectively.
It is noted that the saturation power for the lower and upper bounds of the EE is the same as the values calculated by (<ref>) and (<ref>), respectively.
Comparison of the saturation power with $P_c' = 40$ dBm and $P_o' = 50$ dBm
Moreover, $P_{prop}$ is quite well matched with the true saturation power $P^*$.
This demonstrates that our approach of determining the saturation power generates an accurate estimate of the actual saturation power.
When computing $P_{prop}$, we find that $\beta = 1.3$ achieves the maximum EE performance through numerical simulations.
Note that the optimal $\beta$ may change when system parameters vary.
Nevertheless, it can be observed that the proposed saturation power $P_{prop}$ using the fixed $\beta$ yields performance nearly identical to that of $P^*$ for various conditions.
Figure 4 exhibits the comparison of the saturation power for $P_c' = 40$ dBm and $P_o' = 50$ dBm.
Again in this figure, the saturation power derived by (<ref>) and (<ref>) match well with high accuracy.
In this case, $P^*$ is slightly larger than $P_{prop}$.
Despite the gap between the saturation power, the EE performance corresponding to $P_{prop}$ is very close to the maximum EE in <cit.>.
Moreover, the average EE performance $\eta_{UB}$ and $\eta_{MRT}^\circ$ obtained from the large system analysis are quite close to that of $\eta_{no-IUI}$ and $\eta_{MRT}$ for the finite system case, respectively.
Therefore, we can conclude that even for a system with finite dimension, the analysis of the EE performance with the large system limit provides an accurate approximation.
EE performance of the proposed algorithm with $P_c' = 30$ dBm and $P_o' = 40$ dBm
Next, we validate the EE performance of the proposed scheme based on the derived saturation power in Figure 5 with $N=M$.
In this figure, it is observed that the EE performance becomes larger when $M$ and $N$ are increased from 2 to 4.
Note that compared to the EE maximization algorithm in <cit.>, almost the same EE performance is achieved by the proposed method with $P_{prop}$ which utilizes the SE maximization scheme with much reduced complexity.
Also, in Figure 6, we demonstrate the EE performance for $P_c' = 40$ dBm and $P_o' = 50$ dBm.
EE performance of the proposed algorithm with $P_c' = 40$ dBm and $P_o' = 50$ dBm
It is remarkable that the proposed scheme with $P_{prop}$ produces the EE performance quite close to the optimal EE solution in <cit.> for different configurations.
Furthermore, the derived saturation power gives insight for the BS power designs in terms of the EE.
The impact of the EE performance and the saturation power with respect to constant power consumption $P_{const}$
Finally, we exhibit the effect of the EE performance with respect to the constant power consumption term $P_{const}$.
Comparing Figures 5 and 6, when $P_{const}$ increases, the saturation power for achieving the maximum EE also becomes large while the performance of the EE is decreased.
In Figure 7, this phenomenon is illustrated by the trade-off curve of the sum rate and the total consumption power.
In the plot, the curves with circular and rectangular marks denote the sum rate for the SE maximization scheme with $P_{const} = 5$ and 15, respectively.
We can see that for a larger $P_{const}$, the trade-off curve is shifted to the right.
Then, the optimal slope of the tangent which accounts for the performance of the EE becomes small.
On the contrary, the required saturation power is increased to achieve the optimal slope of the tangent.
From these results, we confirm that reducing the amount of $P_{const}$ has a main impact on improving the performance of the EE and saving the transmit power consumption.
§ CONCLUSIONS
In this paper, we have proposed a simple scheme to solve the EE maximization problem for MU-MISO channels.
Leveraging the relationship between EE and SE, the EE is maximized by only utilizing the SE maximization scheme based on the saturation power.
From large system analysis, we have determined the saturation power corresponding to the maximal EE in closed form by exploiting the property between lower and upper bounds of the EE.
This asymptotic result provides insight into the saturation power of the EE for various system configurations.
As a result, the proposed EE scheme makes it possible to provide solutions for the EE maximization efficiently.
It is noted that a performance loss of the proposed scheme is quite small compared to the optimal EE maximization scheme in <cit.>, and the computational complexity of the proposed scheme is significantly reduced.
Also, the simulations demonstrate that the asymptotic results are well matched even for the finite system case.
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1511.00150
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Wireless Physical Layer Identification (WPLI) system aims at identifying or classifying authorized devices based on the unique Radio Frequency Fingerprints (RFFs) extracted from their radio frequency signals at the physical layer. Current works of WPLI focus on demonstrating system feasibility based on experimental error performance of WPLI with a fixed number of users. While an important question remains to be answered: what's the user number that WPLI can accommodate using different RFFs and receiving equipment. The user capacity of the WPLI can be a major concern for practical system designers and can also be a key metric to evaluate the classification performance of WPLI. In this work, we establish a theoretical understanding on user capacity of WPLI in an information-theoretic perspective. We apply information-theoretic modeling on RFF features of WPLI. An information-theoretic approach is consequently proposed based on mutual information between RFF and user identity to characterize the user capacity of WPLI. Based on this theoretical tool, the achievable user capacity of WPLI is characterized under practical constrains of off-the-shelf receiving devices. Field experiments on classification error performance are conducted for the validation of the information-theoretic user capacity characterization.
§ INTRODUCTION
Wireless Physical-layer Identification (WPLI) is a promising wireless security solution. Since the software-level device identities (e.g., IP or MAC address) can be manipulated, the physical layer feature cannot be modified without significant efforts. The physical layer features are extracted from signal by WPLI to form the radio frequency fingerprints (RFFs) which are rooted in the hardware imperfections of analog (radio) circuitry at the transmitter device <cit.>. Fig.<ref> illustrates the processing procedures of WPLI and typical application scenarios. The signals are obtained by the identification system through an acquisition setup to acquire signals from devices. A feature extraction module is then to obtain selected kinds of identification-relevant feature from the identification signal to form a fingerprint. A fingerprint matcher compares the fingerprints with
reference fingerprints stored in database using dimensionality reduction classification technique. The identities are classified and assigned to devices. Two application scenarios are involved in WPLI: (i) identification scenario is between all unauthorized imposters and the whole authorized users. (ii) classification scenario is the N-class identification between all authorized users within this network <cit.>.
Typical logic procedures and application scenarios in WPLI.
The RFFs are rooted in the hardware imperfections of the transmitter device <cit.>, which include the nonlinearity of RF front-end system <cit.>, <cit.>, <cit.>, clock jitter <cit.>, distortions due to modulator sub-circuit <cit.>, etc. The classification procedure of WPLI is to use RFF features to estimate the user identities. However, due to channel effects, in-band hardware noise, and resolution errors of extraction algorithms, the features extracted by receiver become random variables with certain distribution which brings the uncertainty to the classification results.
If more user classes are kept adding into WPLI, the distributions of features of different identities are more likely to overlap. Hence the uncertainty between feature and identity is increased, resulting the increase of classification errors. Current works in WPLI research area, mainly focus on demonstrating feasibility of system with the classification error performance of a fixed-number network measured by high quality receiving equipment. For instance, in <cit.>, 50 COTS Tmote Sky nodes and an oscilloscope are utilized to achieve a high sampling rate (4GS/s). In <cit.>, 138 Network Interface Cards (NIC) are measured and a vector signal analyzer are used as the receiver. In <cit.>, 54 Universal Mobile Telecommunications System (UMTS) user equipment (UE) devices and a signal spectrum analyzer are utilized. To our best knowledge, no existing works have analyzed the user number that WPLI can accommodate using different RFFs and receiving equipment within certain performance, i.e., the user capacity of the WPLI.
Since the existing research analyses are mainly conducted in different experiment scenarios, the results of which cannot have the repeatable accuracy due to different individual experiment setup. To this end, in this paper, information-theoretic analyses are utilized to provide the theoretical tool, which can be universally applied for various types of WPLI. This theoretical tool can be a fundamental approach to describe the uncertainty between feature and user class identity in WPLI. Specifically, entropy can be used as a measure of the uncertainty on the values taken by a feature member. Meanwhile mutual information can be seen as the reduction in the uncertainty of one feature member due to the knowledge of user class identity. <cit.>. Moreover, the classification error performance is restricted by the uncertainty remains in the feature member after the reduction of mutual information <cit.>. Based on the relations of uncertainty, the key factors in WPLI, including feature, class identity, classification error performance, and user capacity, can be jointly analyzed. However, to date, no existing work in the research field of WPLI has fully covered this research direction.
In this paper, we establish a theoretical understanding of user capacity of WPLI in an information-theoretic perspective. Based on mutual information of RFF, an information-theoretic approach is established to characterize the user capacity of WPLI. The RFF feature of WPLI is modeled according to the signal processing procedures of WPLI. The mutual information between RFF feature member and user identity is calculated. The ensemble mutual information (EMI) between RFF and identity is obtained using an approximation calculation. The user capacity of WPLI is derived using the EMI and class identity entropy. To illustrate the usage of this theoretical tool, we use a experiment-based approach to calculate the mutual information and then to derive the achievable user capacity under practical constrains of different application cases. Experiments on classification error performance of a practical system are also conducted to validate the user capacity characterization for each application case setting.
§ INFORMATION-THEORETIC ANALYSES OF USER CAPACITY
In this section, we firstly provide the information-theoretic modeling of RFF feature according to the processing procedures. The mutual information between fingerprinting feature member and user identity is modeled. The model to calculate ensemble mutual information between RFF and identity is then given. Finally, the user capacity is derived using ensemble mutual information and entropy.
§.§ Modeling of RFF feature
The beginning of the RRF classification procedure is the Analog-to-Digital Converter (ADC) sampling procedure of received signal. This signal can be either baseband signal or passband signal resulting in different signal type are utilized to extract the fingerprints which are basedband preamble <cit.> or passband transient signal respectively <cit.>. The ADC sampling procedure can be modeled as,
\begin{gather}
\label{eq: ADC}
\textstyle s[n]=s(nT_s)+\eta+\xi_{ADC},
\end{gather}
where $s(t)$ is the received analog signal, $s[n], n\in\mathbf{N}$ is the sampled digital signal, $\mathbf{N}$ is the set of all sampled digital signals in this round of identification. $\eta$ is the in-band AWGN noise which can be measured by the receiver. $\xi_{ADC}$ is the random ADC quantization error, for Q-bit ADC quantization and input dynamic range $U$ Vp-p, the maximum quantization error is $ \delta_{ADC}=2^{-Q}U$.
After ADC sampling procedure, the signal sequence then go through the signal acquisition procedure to extract the valid part of the signal i.e., the preamble or transient part. The next procedure is feature extraction to obtain the fingerprinting feature from the signal, which can be modeled as,
\begin{gather}
\label{eq: feature}
\textstyle \mathbf{X}_{1:M}=Feature(\mathbf{S}_{1:N}),
\end{gather}
where $\mathbf{S}$ is $N$ point raw signal data vector set, $\mathbf{X}$ is the fingerprint feature set with the feature dimensionality $M$. The feature dimensionality depends on the feature selection approach. For instance, the spectral feature in <cit.> is high-dimensional feature of which the $M$ is the number of FFT points. In <cit.>, two single-dimensional features, TIE error and average signal power, are utilized as a combined feature. While in <cit.>, a combined low-dimensional feature of frequency error, I/Q offset, magnitude error and phase error are used as fingerprinting feature. Since the multi-dimensional feature are widely applied in the feature selection, dimensionality reduction techniques are applied to reduce the computation burden and find more discriminant subspaces which highlight the relevant features that may be hidden in noise <cit.>. Currently typical dimensionality reduction techniques in machine learning are applied for WPLI classification scenario, including PCA <cit.>, Fisher LDA <cit.>, Maximum Mutual Information (MMI) <cit.> etc.
§.§ Mutual information between RFF and identity
To derive the user capacity $N_C$, we firstly calculate the mutual information between RFF feature and its identity. Specifically, variable $X$ is the one single dimensional component of the feature vector i.e., $X\in \mathbf{X}$. Y denotes the user class identity of this RFF feature. The value of $X$ and $Y$ varies for each testing RFF sample received by WPLI. Consequently, the number of $X$ values, $N_X$, equals the number of all the test samples received by WPLI. The entropy of feature values can be calculated as, $ H(X) =-\sum_{i=1}^{N_X}p(x_i)\log(p(x_i)) $. While the number of $Y$, $N_Y$ is the number of users connected to WPLI. The classification procedure of WPLI is to utilize larges samples of $X$ with different identities to decide the user identity $Y$. Hence the conditional entropy, which describe the uncertainty remaining in $X$ after obtained the outcome of $Y$, can be calculated as $H(X|Y) =-\sum_{j=1}^{N_Y} p(y_j) \sum_{i=1}^{N_X}p(x_i|y_i)\log(p(x_i)|y_i)$. The mutual information between $X$ and $Y$ can be finally derived as,
\begin{align}
\label{eq: MI}
\textstyle I(X;Y)&=I(Y;X) =H(X)-H(X|Y)
\\\textstyle &=\sum_{j=1}^{N_Y} \sum_{i=1}^{N_X} p(x_i y_i)\log(\frac{p(x_i y_i)}{p(x_i)p(y_j)}). \notag
\end{align}
For instance, if the whole signal spectrum is used as feature $\mathbf{X}$ for RFF, each value of frequency can be seen as the one variable $X$ for the spectral feature. $x_i$ is the specific magnitude value of each frequency point is of the $i$th RFF sample. Hence mutual information between each frequency point and identity can be measured and calculated using large number of tests. The specific measure and calculation approach which will be detailed discussed using an application case in section <ref>. To have a clear understanding of mutual information between feature member and identity, in Fig.<ref> we present the two PSDs of signal preambles of two Micaz sensor nodes and corresponding mutual information between each frequency point and signal identity. The difference of spectrums of different devices is the reason that spectral feature can be utilized to classify the identities. From the figures, we can see that the frequency points which have larger differences in spectrum also show larger mutual information value with their identity. Just as the discussion in <cit.> and <cit.>, mutual information is a significant metric to characterize the relevance of the feature to its identity.
(a) Preamble PSDs of Micaz sensor nodes. (b) Mutual information between spectral feature member and identity.
If only single dimensional feature is utilized to form a RFF, the mutual information between RFF and identity can already be calculated using equation (<ref>). However, as our previous discussion, even single dimensional features are combined to form a multi-dimensional feature $\mathbf{X}$ to form a RFF. Hence the Ensemble Mutual Information (EMI) between ensemble feature and class identity, $I(\mathbf{X};Y)$, needs to be calculated to characterize the relation between the ensemble RFF and the identity. In <cit.>, a definition for the ensemble mutual information is given. However, as the increasing of dimensionality of ensemble feature, exponential number of possible variable values will make the calculation for this approach infeasible in practice. In <cit.>, the EMI between high-dimensional feature and class identities is given as,
\begin{align}
\label{eq: EMI}
\textstyle I(\mathbf{X};\it Y) &=\sum_Y \int p(\mathbf{x}, y) \log \frac{p(\mathbf{x}, y)}{p(\mathbf{x}) p(y)} d\mathbf{x}
\\ \textstyle &=\sum_Y p(y) \mathbb{E}_{\mathbf{x}| y} \left [ \log \frac{p(\mathbf{x}| y)}{p(\mathbf{x})} \right ], \notag
\end{align}
The pdfs $p(\mathbf{x}| y)$, $p(\mathbf{x})$ and the conditional expectation $\mathbb{E}_{\mathbf{x}| y} $ can be approaximatedly calculated using nonparametrical Kernel Density Estimator (KDE) with $K(.)$ as the kernel <cit.>,
\begin{align}
\label{eq: EMI_app}
\textstyle I(\bf{X};\it Y) & \approx\sum_Y \frac{p(y)}{N_Y} \sum_{j=1}^{N_Y} \log \frac{(1/N_Y)\sum_{i=1}^{N_Y} K(\mathbf{x}_j^y-\mathbf{x}_i^y)}{(1/N_\mathbf{x})\sum_{i=1}^{N_Y} K(\mathbf{x}_j^y-\mathbf{x}_i)}
\\ \textstyle &\approx \sum_Y \frac{p(y)}{N_Y} \sum_{j=1}^{N_Y} \log \left [ \frac{\bar{\varphi}^T(\mathbf{x}_j)\bar{\Lambda} \mathbf{\bar{\mu}}_y }{\bar{\varphi}^T(\mathbf{x}_j)\bar{\Lambda} \mathbf{\bar{\mu}}} \right ] \notag
\end{align}
where the kernel $K(.)$ can be calculated with the eigenvectors $\mathbf{\varphi(x)}$ and the eigenmatrix $\mathbf{\bar{\Phi}_x}=[\mathbf{\varphi(x)}_1, ...,\mathbf{\varphi(x)}_N ]$, $ \mathbf{\bar{\mu}}_y=(1/N_Y)\mathbf{\bar{\Phi}_x}\mathbf{m}_y$ is the average eigenvector for class $y$, $ \mathbf{\bar{\mu}}=(1/N_\mathbf{x})\mathbf{\bar{\Phi}_x}\mathbf{1}$ is the average eigenvector for all the training samples, and $N_\mathbf{x}$ is the number of all RFF feature samples.
§.§ User capacity of WPLI
The WPLI system finally assign the identity of testing RFF to the class with minimal feature distance scores between reference RFFs. After a large number of sample tests, the WPLI classification performance can be evaluated using average classification error rate as the metric <cit.> which can be denoted as $P_e$. With the EMI $I(\mathbf{X};Y)$ obtained, the important property of mutual information related to classification error rate $P_e$ can be utilized to derive the user capacity of WPLI. In <cit.>, the information-theoretic bounds for classification error rate are given in details using Fano's Inequality (note that the inequality is valid for three or more classes scenario). Hence the classification error rate of WPLI can be bounded as,
\begin{align}
\label{eq: bounds}
\textstyle \frac{H(Y)-I(\mathbf{X};Y)-H(P_e)}{\log(N_Y-1)}
\leq P_e
\leq \frac{1}{2} \left (H(Y)-I(\mathbf{X};Y) \right).
\end{align}
These bounds determine that no classifier can possibly achieve better than error lower bound and also there exists a classifier that can achieve at least error upper bound. The bounds are restricted by two terms. One is the ensemble mutual information between feature and identity $I(\mathbf{X};Y)$, the other is class identity entropy $H(Y)$. Considering a specific scenario of WPLI, the stable mutual information between feature and identity, can be measured with a large number of test samples <cit.> and the EMI can be calculated using equation (<ref>). Hence the bound error rate can be bounded by the class identity entropy which is directly related to the user number $N_Y$ of WPLI. Considering an equal-identity-probability WPLI system, i.e., $H(Y)=\log(N_Y)$, the upper-bound user capacity can then be derived as,
\begin{gather}
\label{eq: up_UC}
\textstyle N_C= \max(\mathbf{N}_Y) | \frac{\log(N_Y)-I(\mathbf{X};Y)-H(\lambda)}{\log(N_Y-1)}\leq\lambda,
\end{gather}
where $N_C$ is the user capacity, $\mathbf{N}_Y$ is the set of all possible user number, $Y$ is the user identity, and $\lambda$ is the performance threshold for classification error rate $P_e$. By far, the theoretical tool to derive the user capacity of WPLI is given.
§ APPLICATION CASE STUDY ON USER CAPACITY UNDER PRACTICAL CONSTRAINS
To apply this theoretical tool, we conduct an application case study to illustrate its usage for a specific type of WPLI. We use a experiment-based approach to calculate the mutual information between feature and identity. Then the achievable user capacity of this type of WPLI is derived under practical constrains of different application case settings. Moreover, the effects of key system parameters on user capacity are evaluated and analyzed.
§.§ Case overview
The most existing works try to present the best performance with the high quality receiving equipment. Differently, we try to derive the user capacity and evaluate the system feasibility using the existing approach under practical constrains of off-the-shelf devices. The details of this application case are given as,
* Feature selection: FFT spectrum of baseband preamble <cit.>.
* Transmitter: Micaz, Imote2 and TelosB sensor nodes (3 typical models with the same ZigBee Protocol radio chip).
* Receiver: USRP N210 with SBX daughter board (14-bit ADC).
* Sampling rates: 2M$\sim$10MS/s
* Number of FFT points: 64$\sim$2048p.
* Communication channel: indoor AWGN channel (SNR=20$\sim$30dB).
* Number of transmitters (user class identities): 40 in all.
* Number of signal samples: 2000 samples per class.
§.§ User capacity characterization
After the one-time collection of the raw signals from sensor nodes, the training samples of RFF can be obtained. We calculate the ensemble mutual information for each RFF and its class identiy using the nonparametrical Kernel Density Estimation approach in equation (<ref>). With the EMI obtained, the next step is to characterize the user capacity using equation (<ref>). Subsequently, we try to derive the user capacity under different constrains of key parameters in the user capacity modeling and RFF forming, including number of training transmitters $N_Y$, in-band AWGN noise level, ADC quantization bits $Q$, number of FFT points $N_{FFT}$ and sampling rate of receiver $f_s$. we set the parameters due to different typical application scenarios of WPLI and derive the user capacity with targeted performance as the following figures. In each case, we present the ensemble mutual information (EMI) (blue curve) together with user capacity under 1% and 10% classification error rate i.e., $N_C|P_e\leq1\%$ (black curve) and $N_C|P_e\leq10\%$ (red curve).
§.§.§ Effects of number of training transmitters
Since we obtain the raw RFF samples from limited number of transmitters and limited number of RFF samples to characterize the user capacity, firstly, we characterize user capacity results using RFF samples from 3 to 40 training transmitters to find out what's the least number of class identities we need to characterize a stable user capacity for this system. We collect the raw RFF samples in a typical system setting as $f_s=4MS/s$, $N_{FFT}=512p$, and $SNR=24dB$. In Fig. <ref>, the user capacity is presented from 3 to 25 training transmitters together with the EMI we obtained from these samples. The EMI directly related to classification error rate and user capacity, the relation of which can be easily observed in the figures. In the beginning, when the number of training transmitters is too small, the obtained EMI is also small which results in the user capacity is near to $N_Y$. As the increasing of $N_Y$, the results are increased unstably. After the number of training transmitters is larger than 18, the characterization result becomes a stable and reliable result which is $13|P_e\leq1\%$ and $22|P_e\leq10\%$. Hence in order to characterize the user capacity we at least should use 13 nodes to collect the raw training RFF samples.
User capacity under different numbers of training transmitters.
§.§.§ Effects of RFF noise level
We firstly present user capacity due to various the noise effects in Fig.<ref>. Here, we set the other parameters as $f_s=4MS/s$, $N_{FFT}=512p$. As the modeling in equation (<ref>), the noise level of RFF feature is mainly contributed by the ADC quantization error and in-band AWGN noise. The noise level significantly affects the classification performance of WPLI resulting in the decrease of user capacity we finally obtained. We fix the ADC quantization bit $Q=14$bits and simulate the noise feature value within SNR$=0\sim 28$dB. The user capacity results are presented in Fig.<ref>. The EMI and user capacity are decreased synchronously as the AWGN SNR level decreases. It should be noted that, according to equation (<ref>), the user capacity we obtained is the upper bound for all classification methods and classifiers using all these RFF samples. Hence in high SNR situations, the typical classification procedure of WPLI can easily achieve the user capacity quite accurately. However, in the extremely low SNR scenarios, it is hard to use a single method or feature to achieve the upper bound of user capacity. Hence more combined features extracted from RFF for multiple classifiers should be utilized to achieve the upper bound of user capacity, as the work in <cit.>. Moreover, in the low SNR scenarios, the error in signal acquisition procedure of WPLI can also contribute to worsen the classification performance <cit.>, which is out of the scope of this paper and can be discussed in future works.
Then we fixe the SNR$=29$dB and simulate the feature value of ADC quantization error within $Q=6\sim14bits$. The corresponding user capacity results are presented in Fig.<ref>. The user capacity becomes stable when the ADC quantization bits are increased to 10 bits. In practical applications, the effects of AWGN noise level usually more significant than the effects of ADC quantization error, while the effects of ADC quantization error can be significant when AWGN SNR level is very high. Here we only can simulate the feature value for 14 bits quantization due to the constrains of USRP daughter board, while in practical, 16 or more quantization bits can also be found in higher standard equipment. With the development of device resolutions, the effects can be kept to minimal,
(a) User capacity under different RFF noise levels. (b) User capacity under different ADC quantization bits.
§.§.§ Effects of number of FFT points
Here we present user capacity due to various number of FFT points within the same sampling rate setting. The number of FFT points $N_{FFT}$ is the key parameter to form the spectral RFF which decides the resolution of the spectral feature $\mathbf{X}$ in equation (<ref>) and consequently reflects the distribution of RFF feature. The specific modeling about the number of FFT points of spectral feature can be found in <cit.>. In Fig.<ref>, we present the preamble spectrum obtained with two different number of FFT points to present difference of resolution. Here, we set the other parameters as $f_s=8$MS/s, SNR$=29$dB, $Q=14$bits. We simulate the result within number of FFT points $N_{FFT}=64\sim1024$p. The user capacity results are presented in Fig.<ref>. From the results, we can see the larger number of FFT points can increase the EMI of feature and improve the performance and user capacity. However, when the resolution is accurate to some extend, the improvement of performance is not so significant. Since the increase of FFT points can cause greater computation burden for WPLI, here involves a trade off for system designer.
Preamble spectrum of signal under different number of FFT points.
User capacity under different numbers of FFT points.
Preamble spectrum of signal under different sampling rates.
User capacity under different sampling rates.
§.§.§ Effects of sampling rate
Here we present user capacity due to various sampling rates of receiver with the same frequency resolution, which are the key parameters to determine the bandwidth of the spectral feature $\mathbf{X}$ in equation (<ref>) and consequently reflects its distribution. In Fig.<ref>, we present the preamble spectrum obtained with two different sampling rates to present difference of spectrum bandwidth. In the case $f_s=2$MS/s, the spectrum bandwidth covers the main lobe of signal PSD. While the higher sampling rates can cover more side lobe information of signal PSD which are more beneficial for WPLI performance. However, the choice of sampling rates also involves a trade off that with the increasing of bandwidth, the bandwidth of noise is also increased which can result in the decreasing of signal SNR which worsen the WPLI performance. This phenomenon can be observed in the user capacity characterization and also experimental validations. Here, we set the other parameters as, SNR$=29$dB for $f_s=8$Ms/s, $Q=14$bits. We simulate the result within different sampling rates, $f_s=2\sim10$MS/s, with the same spectrum resolution ( $N_{FFT}=512p$ for $f_s=4 Ms/s$), the user capacity results of which are presented in Fig.<ref>. It can be inferred that when low-noise devices are applied in high SNR scenarios, a higher sampling rate can be applied for WPLI system. While, for the low-SNR scenarios where low quality devices are applied, a sampling rate which tightly covers the main lobe of spectrum should be chosen.
§ EXPERIMENTAL VALIDATIONS FOR USER CAPACITY
In this section, we conduct field experiments on classification error performance according to the different case setting to validate the user capacity characterization in section <ref>. We still utilize the equipments in section <ref>. We use newly collected 1000 thousand samples per class to train the LDA training matrix and another 1000 samples per class to be tested for classification.
For classification procedure of WPLI, we select the Fisher LDA <cit.> as the feature dimensionality reduction technique and the Mahalanobis distance as the distance metric <cit.>. We set a large LDA subspace dimensionality $\kappa=150$ despite the computation time in order to achieve the optimal classification performance .
We present the classification error performance of WPLI with selected user number near the upper-bound user capacity we obtained. Hence if the classification error rate is larger than the threshold error rate i.e., $P_e|N_Y > \lambda$, when the user number is larger than the user capacity, i.e., $N_Y > N_C|\lambda$, the user capacity is proved. Meanwhile, we also present the classification error performance when user number is near to the user capacity bound i.e., $N_Y \leq N_C$, to show the tightness of this bound. The classification results are shown in the following figures where the x-axis is the number of test samples, the y-axis is the minimal distance score between test sample and its reference, and the z-axis is the identity number assigned to the test samples. Besides the color of each sample is to present its true identity which can help the reader to compare classified identities of test samples.
§.§ Effects of RFF noise level
Here, we use the experiment results to validate the user capacity characterization for RFF noise level case. Because the ADC quantization bits is impossible to change for given hardware setting, we can only fix the ADC quantization bits to Q$=14$bit according to USRP daughter board setting. As the case setting in Fig.<ref>, we conduct the experiments at SNR=26dB, where the 1% error rate user capacity is 13, i.e., $N_C=13|P_e\leq1\%$. In Fig.<ref>, classification results of 13 classes, are shown of which the classification error rate is $P_e=0.91\%|N_Y=13$. While in Fig.<ref>, the classification results of 14 classes are shown of which the classification error rate is $P_e=1.09\%|N_Y=14$. Hence we can see the user capacity characterization at this point is validated accurately. Similarly, we change SNR situation and threshold error rate to see validate another point of our user capacity curves. In Fig.<ref>, when SNR=22dB, the user capacity is 12 with 1% error rate i.e., $N_C=12|P_e\leq1\%$. In Fig.<ref>, classification results of 11 classes, are shown of which $P_e=0.75\%|N_Y=11$. While in Fig.<ref>, the classification results of 12 classes are shown of which $P_e=1.79\%|N_Y=12$. The user capacity characterization at this case is slightly larger than the experimental result. As we discuss in the previous section, with the decrease of SNR level, single classifier and single feature selection are not enough achieve the upper bound user capacity.
SNR=26dB, $f_s=4MS/s$, $N_{FFT}=512p$. (a) Classification results for 13 classes. (b) Classification results for 14 classes.
SNR=22dB, $f_s=4MS/s$, $N_{FFT}=512p$. (a) Classification results for 11 classes. (b) Classification results for 12 classes.
§.§ Effects of number of FFT points
We use the experiment results to validate the user capacity characterization for number of FFT points case. We conduct the experiments as the case setting for Fig.<ref>. As the user capacity characterization under $N_{FFT}=64p$ is ,$N_C=2|P_e\leq10\%$. In Fig.<ref>, the classification results of 3 classes are shown with $P_e=15.67\%|N_Y=3$ which is out of user capacity. For $N_{FFT}=256p$, $N_C=13|P_e\leq1\%$. In Fig.<ref>, the classification results of 14 classes are shown with $P_e=1.25\%|N_Y=14$ which is still out of user capacity. For $N_{FFT}=1024p$, $N_C=15|P_e\leq1\%$. In Fig.<ref>, the classification results of 15 classes are shown with $P_e=0.47\%|N_Y=15$ which is within user capacity. Hence the experimental results match the discussion in previous section very well.
SNR=29dB , $f_s=8MS/s$. (a) Classification results for 3 classes, $N_{FFT}=64$. (b) Classification results 14 classes, $N_{FFT}=128$. (c) Classification results for 15 classes, $N_{FFT}=1024$.
SNR=29dB (8MS/s). (a) Classification results for 17 classes, $f_s=2$MS/s. (b) Classification results 17 classes, $f_s=6$MS/s. (c) Classification results for 12 classes, $f_s=10$MS/s.
§.§ Effects of sampling rate
We use the experiment results to validate the user capacity characterization for sampling rate case. We conduct the experiments as the case setting for Fig.<ref>. As the user capacity characterization under $f_s=2$MS/s is, $N_C=16|P_e\leq1\%$. In Fig.<ref>, the classification results of 17 classes are shown with $P_e=1.17\%|N_Y=17$ which is out of user capacity. For $f_s=6$MS/s, $N_C=17|P_e\leq1\%$. In Fig.<ref>, the classification results of 17 classes are shown with $P_e=0.58\%|N_Y=17$ which is within the capacity. For $f_s=10$MS/s, $N_C=11|P_e\leq1\%$. In Fig.<ref>, the classification results of 12 classes under $f_s=10$MS/s are shown with $P_e=1.51\%|N_Y=12$, which is out of the capacity. The analyses in our previous discussion is also validated.
§ CONCLUSION
In this work, we establish a theoretical understanding on user capacity of Wireless Physical-layer Identification (WPLI) in an information-theoretic perspective unprecedentedly. Specifically, Radio Frequency Fingerprint (RFF) features of WPLI are analyzed in an information-theoretic perspective, including feature selection, extraction, noise level, resolution, and bandwidth, which advance the understanding of RFF feature. We then propose an information-theoretic approach based on mutual information between RFF and user identity to characterize the user capacity of WPLI. Using this theoretical tool, the achievable user capacity under practical constrains of a WPLI system is characterized with data collected by off-the-shelf receiving devices. Various experiments on classification error performance of a practical system are conducted to validate the accuracy and tightness of the information-theoretic user capacity characterization.
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1511.00135
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Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Many giant exoplanets in close orbits have observed radii which exceed theoretical predictions. One suggested explanation for this discrepancy is heat deposited deep inside the atmospheres of these “hot Jupiters”. Here, we study extended power sources which distribute heat from the photosphere to the deep interior of the planet. Our analytical treatment is a generalization of a previous analysis of localized “point sources”. We model the deposition profile as a power law in the optical depth and find that planetary cooling and contraction halt when the internal luminosity (i.e. cooling rate) of the planet drops below the heat deposited in the planet's convective region.
A slowdown in the evolutionary cooling prior to equilibrium is possible only for sources which do not extend to the planet's center.
We estimate the Ohmic dissipation resulting from the interaction between the atmospheric winds and the planet's magnetic field, and apply our analytical model to Ohmically heated planets. Our model can account for the observed radii of most inflated planets which have equilibrium temperatures $\approx 1500\textrm{ K}-2500\textrm{ K}$, and are inflated to a radius $\approx 1.6 R_J$. However, some extremely inflated planets remain unexplained by our model.
We also argue that Ohmically inflated planets have already reached their equilibrium phase, and no longer contract. Following Wu & Lithwick who argued that Ohmic heating could only suspend and not reverse contraction, we calculate the time it takes Ohmic heating to re-inflate a cold planet to its equilibrium configuration.
We find that while it is possible to re-inflate a cold planet, the re-inflation timescales are longer by a factor of $\approx 30$ than the cooling time.
§ INTRODUCTION
The discovery of extra-solar planets in the last two decades has been accompanied with a variety of surprises, which challenge standard planetary formation and evolution theories that were originally inspired by our solar system. One of these mysteries is the detection of close-orbit planets with radii as large as $\sim 2 R_J$, where $R_J$ is the radius of Jupiter <cit.>.
Theoretical evolution models predict that isolated gas giants older than $\sim 1 \textrm{ Gyr}$ cool and contract to a radius of about $1.0 R_J$ <cit.>. The proximity of the observed planets to their parent stars imposes a strong stellar irradiation that induces a deep radiative envelope at the outer edge of the otherwise fully convective planets <cit.>. This radiative layer slows down the evolutionary cooling of the planet compared with an isolated one, resulting in a higher bulk entropy and radius at a given age <cit.>.
However, at least some of the observed hot Jupiters have radii which exceed the theoretical predictions, even with stellar irradiation taken into account <cit.>.
There have been several suggested explanations to the radius discrepancy <cit.>, varying from enhanced atmospheric opacities <cit.>, suppression of convective
heat loss <cit.>, or an extra power source inside the planet's atmosphere. In this work we focus on the effects of an extra power source in the atmosphere. Possible heat sources include tidal dissipation due to orbital eccentricity <cit.>,
“thermal tides” <cit.>,
Ohmic heating <cit.>, turbulent mixing <cit.>, and dissipation of kinetic energy of the atmospheric circulation <cit.>.
The influence of the additional heat on the planet's cooling history (and therefore on its radius) increases with its deposition depth inside the atmosphere <cit.>. In a previous work we gave an intuitive analytic description of the effects of additional power sources on hot Jupiters, which explains this result and reproduces the numerical survey of <cit.>. However, and previous numerical works focus on the specific case of localized “point-source” energy deposition. In the current work we study a more general scenario, in which the deposited heat is distributed over a range of depths in the atmosphere <cit.>.
The outline of the paper is as follows. We summarize the results of for localized power sources in Section <ref>, and generalize them to extended sources in Section <ref>. In Section <ref> we apply our model to the specific case of Ohmic dissipation, and quantitatively estimate the radii of Ohmically heated hot Jupiters, with a comparison to observations. In Section <ref> we discuss the possibility of re-inflating a cold planet. Our main conclusions are summarized in Section <ref>.
§ POINT SOURCE ENERGY DEPOSITION
In we analyzed the effects of localized heat sources deep in the atmosphere on the cooling rate of irradiated gas giants. We summarize the analysis here because a similar technique is used in Section <ref>. Our model is one-dimensional and does not differentiate between the day
and night sides of the planet <cit.>. We first discuss the effects of stellar irradiation, and then incorporate an additional heat source.
§.§ Irradiated Planets
Irradiated planets, in contrast with isolated ones, develop a deep radiative outer layer which governs the convective heat loss rate (i.e., internal luminosity) of the interior <cit.>. The radiative-convective transition and the internal luminosity are best analyzed in the $[\tau, U]$ plane, with
\begin{equation}\label{eq:tau}
\tau(r)=\int_r^\infty{\kappa\rho dr'}
\end{equation}
denoting the optical depth at radius $r$, and $U\equiv a_{\rm rad}T^4$ is the radiative energy density. The density, temperature, and opacity are denoted by $\rho$, $T$, and $\kappa$, respectively, and $a_{\rm rad}$ is the radiation constant.
Assuming power-law opacities $\kappa\propto\rho^a T^b$ <cit.>, the convective interior is described by
\begin{equation}\label{eq:U_con}
\frac{U}{U_c}=\left(\frac{\tau}{\tau_c}\right)^\beta,
\end{equation}
with $U_c\equiv a_{\rm rad} T_c^4 $ denoting the central radiation energy density, determined by the central temperature $T_c$, $\tau_c\sim\kappa_c\rho_c R$, with $\kappa_c$ denoting the estimate for the central opacity, if the power-law opacity could be extrapolated to the center, $\rho_c$ denoting the central density, and $R$ is the planet's radius. The power $\beta$ is given by
\begin{equation}\label{eq:beta}
\beta=\frac{4}{b+1+n(a+1)},
\end{equation}
where $n$ is the polytropic index.
The radiative envelope is characterized by the equilibrium temperature on the planet surface (i.e., photosphere)
\begin{equation}\label{eq:Teq}
T_{\rm eq}=(1-A)^{1/4} T_\sun\left(\frac{R_\sun}{2D}\right)^{1/2},
\end{equation}
with $T_\sun$ and $R_\sun$ denoting the stellar temperature and radius, respectively, and where $D$ and $A$ are the planet's orbital distance and albedo, respectively <cit.>. This equilibrium temperature defines an energy density of $U_{\rm eq}\equiv a_{\rm rad}T_{\rm eq}^4$, and luminosity $L_{\rm eq}\equiv 4\pi R^2\sigma_{\rm SB} T_{\rm eq}^4$, where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant. The radiative profile is given by the diffusion approximation (valid for $\tau\gg 1$)
\begin{equation}\label{eq:U_rad}
U=U_{\rm eq}+\frac{3}{c}\frac{L_{\rm int}}{4\pi R^2}\tau,
\end{equation}
where $c$ is the speed of light and $L_{\rm int}$ is the internal luminosity.
According to the Schwarzschild criterion, convective instability develops when the radiative temperature profile is steeper than the adiabatic one. By differentiating Equations (<ref>) and (<ref>) we find that a profile with $\beta>1$ is fully radiative, while for $\beta<1$ convection sets in at the radiative-convective boundary, located at an optical depth of
\begin{equation}\label{eq:irradiated}
\tau_{\rm rad}\sim\frac{L_{\rm eq}}{L_{\rm int}}\sim\tau_c\left(\frac{U_{\rm eq}}{U_c}\right)^{1/\beta},
\end{equation}
where the radiative energy density is $U_{\rm eq}/(1-\beta)$. Equation (<ref>) shows that increasing stellar irradiation decreases the internal luminosity and deepens the penetration of the radiative layer, which is isothermal to within a factor of $(1-\beta)^{-1/4}$ <cit.>.
These results can be also obtained graphically by drawing a radiative tangent with a slope $3L_{\rm int}/(4\pi R^2 c)$ from the point $[0,U_{\rm eq}]$ to the convective profile .
§.§ Power Deposition
In we parametrized an energy point-source with its power $L_{\rm dep}$ and some deposition optical depth $\tau_{\rm dep}$. This source alters the radiative profile of Equation (<ref>)
\begin{equation}\label{eq:deposit_rad}
\frac{dU}{d\tau}=\frac{3}{4\pi R^2c}\cdot
\begin{cases}
L_{\rm tot}\equiv L_{\rm int}+L_{\rm dep} & \tau<\tau_{\rm dep}
\\[1.5ex]
L_{\rm int} & \tau>\tau_{\rm dep}
\end{cases},
\end{equation}
with $L_{\rm tot}$ denoting the total luminosity for $\tau<\tau_{\rm dep}$.
If we focus on intense deposition $L_{\rm dep}\gg L_{\rm int}$, then we may approximate $L_{\rm tot}\approx L_{\rm dep}$, and by analogy with Equation (<ref>), convection sets in at $\tau_b\sim L_{\rm eq}/L_{\rm dep}$ (we reserve the notation $\tau_{\rm rad}$ to the inner radiative-convective transition, as discussed below). In the regime $L_{\rm dep}\tau_{\rm dep}/L_{\rm eq}\gtrsim 1$ a convective layer appears between $\tau_b$ and $\tau_{\rm dep}$, which we distinguish as the secondary convective region. The internal luminosity is found by drawing a radiative tangent from $[\tau_{\rm dep},U(\tau_{\rm dep})]$ to the main interior convective profile, with the transition point denoted by $\tau_{\rm rad}$. This is equivalent to drawing a tangent from $[0,U_{\rm iso}]$, with
\begin{equation}\label{eq:U_eq_eff_big}
\frac{U_{\rm iso}}{U_{\rm eq}}\approx\frac{1}{1-\beta}\frac{U(\tau_{\rm dep})}{U(\tau_b)}\sim\left(\frac{L_{\rm dep}\tau_{\rm dep}}{L_{\rm eq}}\right)^\beta,
\end{equation}
where $U(\tau_{\rm dep})$ is adiabatically related to $U(\tau_b)$. Therefore, the results of Section <ref> are reproduced, but with $U_{\rm iso}$ instead of $U_{\rm eq}$ (note the change in notation of the deep isotherm from $U_{\rm eq}^{\rm eff}$ in to $U_{\rm iso}$ here). Combining Equations (<ref>) and (<ref>) shows that the internal luminosity is reduced from a value of $L_{\rm int}^0$ without heat sources to
\begin{equation}\label{eq:short}
\frac{L_{\rm int}}{L_{\rm int}^0}\sim\left(1+\frac{L_{\rm dep}\tau_{\rm dep}}{L_{\rm eq}}\right)^{-(1-\beta)},
\end{equation}
where we have interpolated with the weak heating regime ($L_{\rm dep}\tau_{\rm dep}/L_{\rm eq}\lesssim 1$). The conclusion is that additional heat sources slow the cooling rate if deposited deep enough.
§.§ Effect of Heating on Planet Radius
The evolution of a planet's central temperature with time is determined by its internal luminosity:
\begin{equation}\label{eq:cooling}
L_{\rm int}\sim-k_{\rm B}\frac{M}{m_p}\frac{dT_c}{dt},
\end{equation}
where $k_{\rm B}$ is the Boltzmann constant, $m_p$ is the proton mass, $M$ is the mass of the planet, and $t$ denotes time.
The radius of a planet can be determined directly by its central temperature.
The relation between the radius increase relative to the zero-temperature radius $\Delta R\approx R-0.9R_J$ and the central temperature can be derived either by using a linear approximation for $\Delta R\ll R_J$
\begin{equation}\label{eq:delta_r_t}
\Delta R\sim\frac{k_{\rm B}T_c}{m_pg},
\end{equation}
with $g\approx 10^3\textrm{ cm}\textrm{ s}^{-2}$ denoting the surface gravity <cit.>, or by using a numerical radius-central-temperature curve <cit.>.
At an age of $\sim 1\textrm{ Gyr}$, isolated Jupiter-mass planets reach a radius of $R\approx 1.0 R_J$ <cit.>.
As explained in Section <ref>, stellar irradiation slows down the planetary contraction, allowing more inflated planets at the same age. Specifically, hot Jupiters irradiated by an equilibrium temperature of $T_{\rm eq}\approx 1500\textrm{ K}$ are expected to reach a radius of $R\approx 1.3 R_J$ at this age <cit.>.
As explained in Section <ref>, deep heat deposition slows down the cooling rate even more, resulting in an additional radius inflation at a given age. Moreover, deep deposition raises the final (equilibrium) central temperature of the planet (which is roughly equal to the temperature at the inner radiative-convective boundary; see Appendix <ref>, specifically Figure <ref>) from $T_{\rm eq}$ to $T_{\rm iso}\equiv (U_{\rm iso}/a_{\rm rad})^{1/4}$, given by Equation (<ref>). When the planet reaches this equilibrium temperature, cooling and contraction stop entirely, and an enlarged equilibrium radius is retained.
§ POWER-LAW ENERGY DEPOSITION
We now generalize the results of Section <ref> to account for an extended source that spans a broad range in optical depth
\begin{equation}\label{eq:power_dep}
L_{\rm dep}(\tau)=\epsilon L_{\rm eq}\tau^{-\alpha}\qquad 1\leq\tau\leq\min{(\tau_{\rm cut},\tau_c}),
\end{equation}
where $L_{\rm dep}(\tau)$ denotes the heat deposited deeper than $\tau$, and $\epsilon$ is the total heat which is deposited below the photosphere (at $\tau\sim 1$), measured in units of the incident stellar irradiation <cit.>.
We assume $\epsilon\ll 1$ and $\alpha>0$, consistent with many studies that invoke conversion of a portion of the stellar irradiation into heat deposited deeper inside the atmosphere <cit.>.
The heat deposition mechanism may have a cut-off at some $\tau_{\rm cut}$ or continue all the way to the center $\tau_c$. In this work we focus on the Ohmic heating mechanism (see Section <ref>), which extends to the planet's deep interior <cit.>. The consequences of a cut-off at $\tau_{\rm cut}<\tau_c$ are discussed in Appendix <ref>.
Schematic temperature profile (logarithmic scale) of a hot Jupiter with an energy deposition that extends to its center. The equilibrium state (solid black line) is characterized by an equilibrium central temperature $T_c^\infty$. A hot-Jupiter profile with $T_c>T_c^\infty$ which has not yet reached equilibrium (dashed blue line) is also plotted. The structure of the planet consists of an outer radiative, nearly isothermal region, and a convective interior. Typical values of the temperature are provided.
The profile in the outer radiative layer follows Equation (<ref>), which reduces to
\begin{equation}\label{eq:rad_power}
\frac{dU}{d\tau}=\frac{3}{4\pi R^2c}\epsilon L_{\rm eq}\tau^{-\alpha},
\end{equation}
as long as $L_{\rm dep}(\tau)>L_{\rm int}$.
For $\alpha<1$ (see Appendix <ref> for other cases), and $\tau\gg 1$ (where the diffusion approximation holds) integration of Equation (<ref>) from the photosphere inward yields
\begin{equation}\label{eq:u_tau_hat}
U=U_{\rm eq}+\frac{3}{c}\frac{\epsilon L_{\rm eq}}{4\pi R^2}\frac{\tau^{1-\alpha}}{1-\alpha}.
\end{equation}
Therefore, the radiative profile is linear in $\tau^{1-\alpha}$ and some of the results of Section <ref> can be reproduced by considering the $[\tau^{1-\alpha},U]$ plane instead of $[\tau,U]$. The convective profile is given by
\begin{equation}\label{eq:power_conv}
\end{equation}
with $\beta/(1-\alpha)$ playing the role of $\beta$ in the analogy with Section <ref>.
We focus on heating profiles which are too flat (decline too gradually with depth) to support a radiative temperature profile $\alpha<1-\beta$ (relevant for Ohmic heating, as discussed in Section <ref>; see Appendix <ref> for other values of $\alpha$). In this case, by analogy with Section <ref>, convection appears at $\tau_b\sim L_{\rm eq}/L_{\rm dep}(\tau_b)$, or
\begin{equation}\label{eq:power_tau_b}
\tau_b\sim\epsilon^{-1/(1-\alpha)},
\end{equation}
and radiation energy density $U_{\rm eq}/U=1-\beta/(1-\alpha)$.
From $\tau_b$ the convective region continues to the planet's center $\tau=\tau_c$, reaching a central radiation energy density of
\begin{equation}\label{eq:u_final}
\frac{U_c}{U_{\rm eq}}\sim\left(\frac{\tau_c}{\tau_b}\right)^\beta\sim\left(\tau_c\epsilon^{1/(1-\alpha)}\right)^\beta.
\end{equation}
Thus, the energy deposition dictates an equilibrium central temperature of
\begin{equation}\label{eq:t_final}
\frac{T_c^\infty}{T_{\rm eq}}\sim\left(\tau_c\epsilon^{1/(1-\alpha)}\right)^{\beta/4},
\end{equation}
and according to Section <ref>, a final planet radius of $\Delta R^\infty\sim k_{\rm B}T_c^\infty/m_pg$, at which evolutionary cooling stops. A schematic profile of a planet in this equilibrium state is given in Figure <ref>.
For a planet with a central temperature $T_c>T_c^\infty$, it is easy to see from Equations (<ref>), (<ref>), and from Figure <ref> that $\tau_{\rm rad}<\tau_b$ and that $L_{\rm dep}(\tau_{\rm rad})<L_{\rm int}$. Therefore, the radiative-convective transition and the internal luminosity in this case are unaffected by the energy deposition, and are determined as in Section <ref>.
In summary, deep heat deposition, which extends to the planet's interior, does not slow down the cooling rate of the planet, but rather imposes a high equilibrium central temperature (and radius), at which the planet stops evolving entirely <cit.>. The cooling history of the planet until it reaches this equilibrium state is unaffected by the deposited energy (see Appendix <ref> for a more general discussion). Using Equation (<ref>), we find that the planet cools down, and $\tau_{\rm rad}$ increases (see Section <ref>), until
\begin{equation}\label{eq:crit_tau_rad}
\epsilon\tau_{\rm rad}^{1-\alpha}\gtrsim 1,
\end{equation}
or, equivalently, $L_{\rm dep}(\tau_{\rm rad})\gtrsim L_{\rm int}$, meaning that the deposited heat in the convective region exceeds the internal luminosity . Similar results are found by <cit.>. Since the radiative-convective boundary of $\sim 1\textrm{ Gyr}$ old irradiated planets with a typical equilibrium temperature of $T_{\rm eq}\approx 2\cdot 10^3\textrm{ K}$ lies at an optical depth of $\tau_{\rm rad}\approx 10^5$ in the absence of power deposition <cit.>, the critical efficiency required to inflate observed planets can be roughly estimated as $\epsilon\gtrsim 10^{-5(1-\alpha)}$.
§ APPLICATION TO OHMIC HEATING
We now apply the results of Section <ref> to the Ohmic heating mechanism <cit.>.
§.§ Atmospheric Winds and Induced Currents
Schematic current surface density $J$ field line representation. $J$ is given by the solution to Equation (<ref>), under the assumption of a dipolar magnetic field, and a constant velocity zonal wind, confined to a shallow wind zone <cit.>. The width of the wind zone is exaggerated. Due to the $l=2$, $m=0$ symmetry, only the first quadrant of a meridional plane is displayed. Below the wind zone the radial and tangential components are comparable, since $\nabla\cdot(\sigma\nabla\Phi)=0$, while at the surface the radial component vanishes.
The electric current surface density $\vec{J}$ is related to the planet's magnetic field $\vec{B}$ and to the atmospheric wind velocity $\vec{v}$ through Ohm's law
\begin{equation}\label{eq:ohm_law}
\vec{J}=\sigma\left(-\nabla\Phi+\frac{\vec{v}}{c}\times \vec{B}\right),
\end{equation}
with $\sigma$ denoting the conductivity, and $\Phi$ the induced electric potential. For a simple approximate model, we follow <cit.> and assume a constant-velocity wind, confined to a shallow wind-zone with thickness determined by the isothermal atmosphere scale height $H=k_{\rm B}T_{\rm eq}/m_pg\approx 10^8\textrm{ cm}\ll R$. <cit.> denote the wind-zone depth by an arbitrary $z_{\rm wind}$ (for which they choose fiducial values $\sim 10^8\textrm{ cm}$), but as we show below, $z_{\rm wind}\sim H$ <cit.>. The potential $\Phi$ and the current surface density $\vec{J}$ are found by applying the continuity equation in steady state $\nabla\cdot \vec{J}=0$ to Equation (<ref>). The solution to this model is described in detail by <cit.>, and is characterized by a current density magnitude of
\begin{equation}\label{eq:j_size}
J_0 & R-r<H \\
J_0\frac{H}{R}\left(\frac{r}{R}\right)^{l-1}\sim J_0\frac{H}{R} & R-r>H
\end{cases},
\end{equation}
\begin{equation}\label{eq:j0}
\end{equation}
evaluated in the wind zone and with a discontinuous drop of order $H/R$ over the edge of the wind zone. This current drop is the result of the outer boundary condition (the radial component of the current vanishes at $r=R$) and the solution to Equation (<ref>) below the wind zone $\nabla\cdot(\sigma\nabla\Phi)=0$, with an $l=2$ symmetry imposed by the $\vec{v}\times \vec{B}$ term <cit.>, though this result can be generalized to other geometries. In Figure <ref> we present a schematic plot of the current field lines, which must form closed loops (since $\nabla\cdot \vec{J}=0$). The $H/R$ current drop is intuitively understood due to the folding of the current lines inside the wind zone, in order to eliminate their radial component at the surface (note that the wind zone depth is exaggerated in Figure <ref>). Alternatively, the drop can be understood by noting that the condition $\nabla\cdot\vec{J}=0$ leads to $J_r/H\sim J_{\theta}/R$ (with $J_\theta$, $J_r$ denoting the tangential and radial components of the current) in the wind zone and that $J_r$ transitions continuously below the wind zone, where it is comparable in magnitude to the tangential component. Since $J(r)\propto(r/R)^{l-1}$ below the wind zone, the current surface density is roughly constant there (while the pressure varies by orders of magnitude) as long as the radius is not much smaller than $R$. The decrease of the current, and therefore the Ohmic dissipation, at $r\ll R$ is irrelevant for the planet's inflation, because the density, and therefore the temperature, approach their maximal (central) values at $r\sim R/2$ in a polytropic profile <cit.>. We note that our two layer model is very similar to the three layer analytical model of <cit.>. Instead of a discontinuous jump in the conductivity with depth (between their middle and inner layers), we use a smooth power-law, as described in Section <ref>, which better represents numerical conductivity profiles calculated in previous studies <cit.>, including <cit.> itself.
The acceleration of a fluid element due to the Lorentz force is given by
\begin{equation}\label{eq:lorentz}
\vec{f}=\frac{1}{\rho}\vec{J}\times\frac{\vec{B}}{c}.
\end{equation}
By combining Equations (<ref>) and (<ref>) we find the magnetic drag deceleration
\begin{equation}\label{eq:magnetic_drag}
\vec{f}=-\frac{B^2}{\rho c^2}\sigma\vec{v}.
\end{equation}
Following <cit.> and <cit.>, we estimate the magnetic field of the planet using the Elsasser number criterion <cit.>, which arises from a balance between the Lorentz and Coriolis forces at the core
\begin{equation}\label{eq:elsasser}
\frac{B^2}{\rho_c c^2}=\frac{\Omega}{\sigma_c},
\end{equation}
where the Lorentz force is given by Equation (<ref>), but with the conductivity $\sigma_c$ and density $\rho_c$ of the planet's interior instead of the atmosphere. $\Omega\approx 10^{-5}\textrm{ s}^{-1}$ denotes the rotation frequency of the planet, which is equal to its orbital frequency since close-in planets are tidally locked <cit.>.
The atmospheric flow velocity $v$ should be determined by global circulation models, coupled with magnetic fields <cit.>.
For example, one effect which is taken into account in these simulations, but neglected in our following analysis, is the induced (by currents) magnetic field, which should be added to the planet's dipolar field.
Nonetheless, we make a rough order of magnitude estimate here, following <cit.>. The winds are driven by a horizontal temperature difference $\Delta T\lesssim T_{\rm eq}$ between the day and night sides of the tidally locked planet, leading to a forcing acceleration of $(c_s^2/R)(\Delta T/T_{\rm eq})$, with $c_s\sim(k_{\rm B}T_{\rm eq}/m_p)^{1/2}$ denoting the speed of sound in the atmosphere. This forcing is balanced by both the Coriolis force <cit.> and the magnetic drag <cit.>
\begin{equation}\label{eq:wind_vel}
\frac{c_s^2}{R}\frac{\Delta T}{T_{\rm eq}}=\Omega v\left(1+\frac{\sigma}{\sigma_c}\frac{\rho_c}{\rho}\right),
\end{equation}
where the magnetic drag is given by Equations (<ref>) and (<ref>). Unlike <cit.>, we neglect the nonlinear advective term $v\nabla v\sim v^2/R$ with respect to the Coriolis force, since the Rossby number is $\textrm{Ro}\sim v/(\Omega R)<1$, as evident from Equation (<ref>) <cit.>. Note that ${\rm Ro}\sim 1$ only when the magnetic drag is negligible, and the temperature difference is maximal $\Delta T\approx T_{\rm eq}$, since (by coincidence) the sound speed and rotation velocity are similar $c_s\sim\Omega R\sim 10^5\textrm{ cm s}^{-1}$. Therefore, a low Rossby number approximation is more adequate for the general case ($\Delta T\le T_{\rm eq}$, and with magnetic drag included). A more careful analysis takes into account the dependence of the Rossby number (the ratio between the advective and Coriolis terms) on the latitude $\phi$: $\textrm{Ro}=v/(2\Omega R\sin\phi)$. Evidently, our low Rossby number approximation is valid except for a narrow ring around the equator $\phi\to 0$, therefore providing a reasonable estimate for the average atmospheric behavior. An alternative analysis, relevant close to the equator, and in which the advective term is dominant, is presented in Appendix <ref>. As seen in Appendix <ref>, both methods lead to qualitatively similar results <cit.>.
Equation (<ref>) indicates the existence of two regimes: an unmagnetized regime, where the forcing is balanced by the Coriolis force, and a magnetized regime, in which the magnetic drag is the balancing force. The transition between the regimes is when $\sigma/\sigma_c\sim\rho/\rho_c$. Due to the sharp increase of the conductivity with temperature (see Section <ref>), the magnetized regime corresponds to high equilibrium temperatures. Using the conductivities calculated by <cit.>, we estimate the transition equilibrium temperature at $T_{\rm eq}\approx 1500\textrm{ K}$, inside the observationally relevant range <cit.>.
The temperature difference $\Delta T$ is determined, in the simplest analysis, by the ratio of advective to radiative timescales <cit.>, with higher-order effects taken into account by more comprehensive treatments <cit.>. Explicitly, we write a diffusion equation for the change of $\delta T\equiv T-T_{\rm eq}$ with time and optical depth $\tau$, taking into account the atmospheric thermal inertia (and therefore the radiative timescale)
\begin{equation}\label{eq:diffusion}
\frac{\partial \delta T}{\partial t}=\kappa m_p\frac{\sigma_{\rm SB} T_{\rm eq}^3}{k_{\rm B}}\frac{\partial^2\delta T}{\partial\tau^2}.
\end{equation}
Note that by writing the diffusion equation using the optical depth, instead of the spatial coordinate, we take into account the variation of the diffusion coefficient with depth.
We assume a solution of the form $\delta T=\Delta Te^{i\omega(\tau)t-k(\tau)\tau}$, where $\omega(\tau)$ and $k(\tau)$ are power laws, and the periodic temporal dependence is determined by the advective timescale between the day and night sides $\omega^{-1}\equiv R/v$. We solve equations (<ref>) and (<ref>) together, and find the decay of the day-night temperature difference $\Delta T$ with optical depth, up to a logarithmic factor.
\begin{equation}\label{eq:delta_t}
\frac{\Delta T}{T_{\rm eq}}=\frac{R^2\sigma_{\rm SB}T_{\rm eq}^4\Omega}{c_s^4}\frac{\kappa}{\tau^2}\left(1+\frac{\sigma}{\sigma_c}\frac{\rho_c}{\rho}\right).
\end{equation}
Using Equation (<ref>) and atmospheric opacities from <cit.> and <cit.>, we find that the condition $\Delta T\sim T_{\rm eq}$ is satisfied near the photosphere ($\tau\sim 1$) for our fiducial $T_{\rm eq}\approx 2\cdot 10^3\textrm{ K}$ <cit.>. This result explains observed day-night temperature differences <cit.>, which are smaller than $T_{\rm eq}$, but only by an order of unity factor. <cit.> and <cit.> obtain the same result using a simpler prescription for the radiative timescale, which is different by a factor of $\tau$ and valid only for $\tau=1$, coinciding with Equation (<ref>) in this case. Our factor of $\tau^2$, as well as a more intuitive approach to deriving Equation (<ref>), is also obtained by explicitly writing the radiative timescale at an optical depth $\tau$
\begin{equation}
t_{\rm rad}(\tau)=\frac{E}{L}\sim\frac{\frac{R^2\tau}{\kappa m_p}k_{\rm B}T_{\rm eq}}{R^2\sigma_{\rm SB} T_{\rm eq}^4/\tau}=\frac{c_s^2\tau^2}{\kappa\sigma_{\rm SB} T_{\rm eq}^4},
\end{equation}
where one factor of $\tau$ <cit.> is due to the mass, and therefore the energy, of a layer of thickness $\tau$, while a second factor of $\tau$ is due to the luminosity through this optical depth $L\sim R^2\sigma_{\rm SB}T_{\rm eq}^4/\tau$ <cit.>.
Interestingly, Equation (<ref>) indicates that for low temperatures (and therefore, in the unmagnetized regime) the relative day-night temperature difference at the photosphere ($\tau\sim 1$) falls rapidly with decreasing temperatures $\Delta T/T_{\rm eq}\propto T_{\rm eq}^5$, where we neglect the weak dependence of the photospheric opacity on the temperature, and substitute $\Omega\propto T_{\rm eq}^3$ (though at large enough separations the planets may not be tidally locked). Consequently, we predict that warm Jupiters, with $T_{\rm eq}\lesssim 10^3\textrm{ K}$ will have very small day-night temperature differences, compared to hot Jupiters, with $T_{\rm eq}\gtrsim 10^3\textrm{ K}$, which are in the saturated regime $\Delta T\sim T_{\rm eq}$ of Equation (<ref>). This prediction, which is robust to the Rossby number regime, as seen by Equation (<ref>), can be tested against observations <cit.>.
Using Equations (<ref>) and (<ref>), we find the decay of the velocity with depth $v\propto\Delta T\propto\kappa/\tau^2\propto P^{-5/2}$, with $P$ denoting the pressure, and utilizing the relation $P/g\sim\tau/\kappa$ (we neglect the mild change of $\sigma/\rho$ with depth in the atmosphere for the magnetized case).
The velocity drops as a power law in pressure, which increases exponentially with a scale height $H$, allowing us to replace the discontinuous drop in the current surface density in the <cit.> model with a continuous drop from an outer $J=J_0$, given by Equation (<ref>), to a roughly constant $J=J_0(H/R)$ in the interior.
The Ohmic dissipation per unit mass is given by
\begin{equation}\label{eq:ohmic}
\frac{dL}{dm}=\frac{J^2}{\rho\sigma},
\end{equation}
implying a drop of order $(H/R)^2$ in the dissipated power.
Since <cit.> choose fiducial values $z_{\rm wind}\sim 10^8\textrm{ cm}\sim H$, we predict a similar decrease in the Ohmic dissipation, without introducing the arbitrary $z_{\rm wind}$ parameter.
In contrast to <cit.> and to this work, the model of <cit.> does not predict a drop in the current at the edge of the wind zone. This results in their overestimation of the deposition at depth, though it is partially balanced by their slightly steeper heating profile (see Section <ref>).
§.§ Ohmic Deposition as a Power Law
Since the current density outside the wind zone is roughly constant, using Equation (<ref>), the Ohmic dissipation power-law is determined by $dL/dm\propto 1/(\rho\sigma)$.
The electric conductivity in the outer layers of the planet is determined by the ionization level of alkali metals, with Potassium dominating the results, due to its low ionization energy <cit.>. <cit.> calculated the Potassium ionization level using the Saha equation, and arrived at the scaling of the conductivity with pressure and temperature. For a simple power-law estimate, we evaluate their scaling (their Equation A3) at the radiative convective boundary as $\sigma\propto T^{7.5}\rho^{-0.5}$ (the exponential dependence on the temperature is approximated as a power law with index $={\rm d}\ln\sigma(T)/{\rm d}\ln T$, evaluated at the radiative-convective boundary), and obtain the relation $\sigma\propto P^{0.8}$, where we used $P\propto T^{n+1}$, with $n\approx 5$ from the equation of state of <cit.>. This polytropic equation of state is a good approximation in the relevant temperature range, as seen, for example, in Figure 2 of <cit.>, who also incorporate the <cit.> equation of state. We note that in we used a different $n\approx 2$, relevant for lower radiative-convective boundary temperatures that characterize less irradiated and unheated planets (the results of depend weakly on $n$). Although our modeling of the conductivity as a power-law is an ad-hoc simplifying approximation, realistic conductivities exhibit a (roughly) power-law dependence on the pressure level inside the planet <cit.>.
Using Equation (<ref>) and $\rho\propto P^{n/(n+1)}$, we find that the specific Ohmic dissipation scales as $dL/dm\propto P^{-1.6}$. This scaling is in between the somewhat steeper scaling of <cit.> and the flatter scaling of <cit.>. The accumulated luminosity therefore scales as $L\propto P^{-0.6}$ (since pressure is linear in mass at the outer edge of the planet).
In order to find $\alpha$ we relate the pressure to the optical depth $P\propto T^{n+1}\propto\tau^{\beta(n+1)/4}$. We estimate $\beta\approx 0.35$ using Equation (<ref>) and the opacities of <cit.>, in the vicinity of the radiative-convective boundary . The resulting power law is $\alpha\approx 0.3$, which is in the $\alpha<1-\beta$ regime (see Section <ref> and Appendix <ref>).
§.§ Ohmic Dissipation Efficiency
In this section we estimate the efficiency $\epsilon$ in the Ohmic dissipation scenario, following the arguments of <cit.> and <cit.>. As we discuss below (see Figure <ref>), this efficiency is dominated by the dissipation in the wind zone, and is higher than the effective efficiency $\epsilon_{\rm eff}$ (which is related to $\epsilon$ below) used to evaluate the deep energy deposition.
Using Equations (<ref>) and (<ref>), the efficiency, which is defined as the dissipated energy rate in units of the stellar irradiation, is given by
\begin{equation}\label{eq:epsilon_sigma}
\epsilon=\frac{\left(v/c\right)^2\sigma B^2H}{\sigma_{\rm SB}T_{\rm eq}^4}=\frac{\left(\sigma/\sigma_c\right)\rho_cv^2H\Omega}{\sigma_{\rm SB}T_{\rm eq}^4},
\end{equation}
where we have eliminated the magnetic field using Equation (<ref>).
This result can also be understood by dividing the kinetic energy by the magnetic drag's stopping time $(\rho c^2/B^2)\sigma^{-1}$, which is obtained from Equation (<ref>).
It is instructive to consider the variation of the dominant term in the efficiency $\sigma v^2$ with conductivity, while assuming all other parameters constant. From Equation (<ref>) we find
\begin{equation}\label{eq:epsilon_power_law_sigma}
\sigma v^2\propto\begin{cases}
\sigma & \sigma<\sigma_m \\[0.5ex]
\sigma^{-1} & \sigma>\sigma_m
\end{cases},
\end{equation}
with $\sigma_m\sim\sigma_c(\rho/\rho_c)\sim 10^9\textrm{ s}^{-1}$ denoting the transition between the magnetized and unmagnetized regimes <cit.>. The maximal efficiency, obtained at $\sigma_m$ is
\begin{equation}\label{eq:epsilon_max}
\epsilon_{\rm max}=\frac{\rho v^2H\Omega}{\sigma_{\rm SB}T_{\rm eq}^4}=\frac{\rho c_s^4H/\Omega}{4R^2\sigma_{\rm SB}T_{\rm eq}^4}\approx\frac{1}{4\tau_0}\approx 0.3,
\end{equation}
with $\tau_0\sim 1$ denoting the optical depth where the day-night temperature difference falls below order unity, which is obtained by setting $\Delta T/T_{\rm eq}=1$ in Equation (<ref>), and with the density $\rho$ given by the condition $\tau_0\sim\kappa\rho H$.
This maximal efficiency is similar to <cit.>, who neglected the Coriolis force. <cit.> also decoupled the magnetic field strength from the rotation of the planet (and therefore the equilibrium temperature). In our approach, on the other hand, the magnetic field is related to $\Omega\propto T_{\rm eq}^3$ through the Elssaser number condition, as described above. By taking this relation into account, incorporating the scaling of conductivity $\sigma$ with temperature from Section <ref>, and considering the dependence of all other variables: $H$, $\rho$, and $c_s$ on the temperature, Equations (<ref>) and (<ref>) yield
\begin{equation}\label{eq:epsilon_power_law}
\epsilon\propto\begin{cases}
T_{\rm eq}^3 & T_{\rm eq}<T_m \\[1.5ex]
T_{\rm eq}^{-6} & T_{\rm eq}>T_m
\end{cases},
\end{equation}
with $T_m\approx 1500\textrm{ K}$ denoting the transition between the magnetized and unmagnetized regimes. Equation (<ref>) has the same qualitative behavior as the more illustrative Equation (<ref>), which demonstrates the dependence of the efficiency on the conductivity $\sigma$ (which is the dominant factor, due to its sharp increase with temperature).
Schematic Ohmic heating profile (logarithmic scale) of a hot Jupiter.
The profile (solid black line) is given by the integrated power, deposited below optical depth $\tau$, in units of the incident stellar irradiation. The profile is characterized by three distinct regions: a constant velocity wind zone up to $\tau_0\sim 1$, a velocity drop from $\tau_0$ to $\tau_1$, and a constant current surface density $J$ region in the interior. An effective power law heating profile which defines $\epsilon_{\rm eff}$ (dashed blue line) is also plotted.
The efficiency $\epsilon$ above denotes the total dissipated energy rate in units of the stellar irradiation. However, the formalism of Section <ref> assumes a single power-law deposition profile, while the actual heating function is a broken power-law with three segments, as shown in Figure <ref>. Nevertheless, we can use Section <ref>, by replacing $\epsilon$ with $\epsilon_{\rm eff}$. The first segment, characterized by a flat deposition due to the increase of conductivity with depth (since $L\propto J^2/\sigma$, $J\propto\sigma v$ in the wind zone, and $v$ is constant for $\tau<\tau_0$), extends to $\tau_0\sim 1$. Beyond $\tau_0$, the velocity drops as $v\propto P^{-5/2}$, so $J\propto P^{-1.7}\propto\tau^{-0.9}$ until $\tau_1=\tau_0(H/R)^{-1.1}$, beyond which $J$ is roughly constant. Combining these factors, we find $\epsilon_{\rm eff}\approx\epsilon\tau_0^\alpha(H/R)^2\approx 2\cdot 10^{-3} \epsilon$.
§.§ Implications for Planet Inflation
We estimate an effective critical heating efficiency of $\epsilon_{\rm eff}\approx 10^{-4}$, required to inflate observed ($\approx 3$ Gyr old with $T_{\rm eq}\approx 2\cdot 10^3\textrm{ K}$) hot Jupiters, by substituting $\alpha\approx 0.3$ and $\tau_{\rm rad}\approx 10^5$ in Equation (<ref>). By taking into account the translation between $\epsilon_{\rm eff}$ and $\epsilon$ in the Ohmic scenario (see Section <ref>), our estimate for the actual critical efficiency is $\epsilon\approx 5\%$, consistent with <cit.> and <cit.>. However, our results agree with <cit.> only because the absence of an electric-current drop in their model is balanced by a steeper heating profile.
More concretely, Equation (<ref>) predicts an equilibrium central temperature of
\begin{equation}\label{eq:t_final_ohm}
\frac{T_c^\infty}{T_{\rm eq}}\sim\left[\tau_m\left(\frac{T_{\rm eq}}{T_m}\right)^b\epsilon_{\rm eff}^{1/(1-\alpha)}\right]^{\beta/(4-\beta b)},
\end{equation}
where we normalize the relation $\tau_c/\tau_m=(T_c/T_m)^b$ to the temperature of the magnetic transition $T_m$ and to the corresponding optical depth $\tau_m\approx 10^{10}$. As discussed in Section <ref>, this temperature implies an equilibrium radius of
\begin{equation}\label{eq:eq_radius}
\Delta R^\infty\approx 0.3 R_J\left(\frac{\epsilon}{5\%}\right)^{0.3}\left(\frac{T_{\rm eq}}{1500\textrm{ K}}\right)^3,
\end{equation}
where we substitute $\beta=0.35$, $b=7$ from , and with the transition between $\epsilon$ and $\epsilon_{\rm eff}$ accounted for.
The planet's radius as a function of time, before the planet reaches equilibrium is given in :
\begin{equation}\label{eq:rad_cooling}
\Delta R(t)\approx 0.2 R_J\left(\frac{T_{\rm eq}}{1500\textrm{ K}}\right)^{0.25}\left(\frac{t}{5\textrm{ Gyr}}\right)^{-0.25}.
\end{equation}
Comparison of Equations (<ref>) and (<ref>) shows that $\epsilon\gtrsim 5\%$ can explain the majority of observed radius discrepancies. In Figure <ref> we present an estimate of the radii of hot Jupiters which are Ohmically heated with a constant efficiency of 3%. Our analytical model roughly reproduces the numerical results of <cit.> in this scenario, with differences explained by the flatter heating profile and the absence of coupling between the wind zone depth and the temperature in <cit.>.
However, the ad-hoc assumption of some constant efficiency is inadequate for the Ohmic dissipation mechanism, as explained in Section <ref> and <cit.>. For this reason, we also present in Figure <ref> a more comprehensive, variable efficiency model, according to Equations (<ref>) and (<ref>).
This variable $\epsilon$ model predicts that Ohmic dissipation inflates planets with equilibrium temperatures $\gtrsim 1500\textrm{ K}$ to a radius $\gtrsim 1.5 R_J$.
These results are in agreement with most of the observations, with a few extremely bloated planets remaining unexplained by Ohmic dissipation <cit.>.
Specifically, the variable efficiency may explain the excess of observed radius anomalies at $T_{\rm eq}\gtrsim 1500\textrm{ K}$ <cit.>.
Radius inflation of 3 Gyr old Jupiter-mass planets due to Ohmic dissipation, as a function of their equilibrium temperature. The radius is given by the maximum of the equilibrium radius imposed by the heat deposition, according to Equation (<ref>), and the radius which cooling under the influence of stellar irradiation predicts (solid black line), according to Equation (<ref>). The constant efficiency curve (dashed blue line) is given by $\epsilon=3\%$, while the variable efficiency curve (dot-dashed red line) is given by Equation (<ref>) with $\epsilon_{\rm max}=0.3$, and $T_m=1500\textrm{ K}$. The magenta points, taken from the exoplanet.eu database, correspond to observed 0.5-2.0 $M_J$ planets.
In this work we limited ourselves, for simplicity, to roughly Jupiter-mass planets. Gas giants with a different mass, but still in the regime $M\sim M_J$, are located near the inversion of the zero temperature radius-mass relation , and their density can therefore be modeled approximately as $\rho\propto M/R^3\propto M$. By inserting this relation in Equations (<ref>) and (<ref>) we estimate $\Delta R^\infty\propto M^{(a+1)\beta/(4-\beta b)-1}=M^{-0.6}$, with $a=0.5$ . This result is similar to the fit of <cit.>, and can explain the large radii (up to $\approx 1.8 R_J$) of many low-mass ($\approx 0.4 M_J$) inflated planets (with the small decrease in the zero-temperature radius taken into account). However, some of the most inflated hot Jupiters depicted in Figure <ref> have a large mass $M\gtrsim 0.8M_J$ <cit.>, and therefore exceed the predictions of our model.
§.§ Comparison with Previous Works
In this section we summarize the main qualitative differences between this work and previous studies of the Ohmic heating mechanism.
The total dissipated power, given by the efficiency $\epsilon$, depends on the speed of atmospheric flows. <cit.> and <cit.> assume fiducial speeds of $\sim 1\textrm{ km s}^{-1}$, which result in $\epsilon\approx 3\%$. However, <cit.> advocate for a discontinuous drop of order $(z_{\rm wind}/R)^2\sim 10^{-3}$ in the dissipated power over the wind zone edge, which is absent in the model of <cit.>.
In this work we studied the decay of atmospheric winds with depth, by comparing the advective and radiative timescales. In contrast to previous studies <cit.> our treatment is valid for optical depths above unity. We found that the winds decay as a power law with pressure, allowing us to replace the discontinuity of <cit.> with a continuous drop, and to relate the wind zone depth $z_{\rm wind}$ <cit.> to the atmospheric scale height.
In contrast to <cit.>, we do not choose a constant fiducial wind speed, and therefore a constant efficiency, but rather calculate the wind speed by comparing the thermal forcing (due to the day-night temperature difference) to the Coriolis force (relevant for $T_{\rm eq}\lesssim 1500\textrm{ K}$) and to the magnetic drag (relevant for $T_{\rm eq}\gtrsim 1500\textrm{ K}$). This analysis leads us to replace the constant $\epsilon$ with an efficiency which rises to a maximum of $\epsilon\approx 0.3$ at $T_{\rm eq}\approx 1500\textrm{ K}$, and then drops at higher equilibrium temperatures due to the reduction of wind speeds by the magnetic drag. Our variable $\epsilon$ model is similar to <cit.>, with two main differences: we consider the balance between the thermal forcing and the Coriolis force instead of the nonlinear advective term $v\nabla v$ (our low Rossby number approximation is more appropriate in this case, see Section <ref>), and we do not treat the planet's magnetic field as a free parameter, but rather couple it to the planet's rotation rate, and therefore to its equilibrium temperature. Despite these differences, our model qualitatively reproduces the results of <cit.>, as seen by comparing our Equation (<ref>) to their Figure 4.
Another new ingredient in our work is the analytic translation of a given heat dissipation efficiency to an inflated planet radius. Most previous studies calculate the Ohmically heated planet evolution using stellar evolution codes, with the exception of <cit.>, who numerically integrate a simplified model based on <cit.>. In this work, however, we calculate the planet's evolution using a generalization of a simple analytic theory, derived in . This analytical approach provides a broader understanding of the scaling laws governing hot-Jupiter inflation.
One interesting example of an insight gained by our analytical model is the qualitative shape of the $R(T_{\rm eq})$ curve plotted in Figure <ref>. Specifically, how come the inflation increases with temperature, while the heating efficiency $\epsilon(T_{\rm eq})$ drops due to the magnetic drag at high temperatures, as explained above? The answer is obtained by considering Equation (<ref>), which indicates that the inflation is determined by two competing factors. While the efficiency decreases, the increasing ${\rm H}^-$ opacity pushes the radiative-convective boundary (in the final equilibrium state) to lower pressures (since $P/g\sim\tau/\kappa$), raising the temperatures of the inner adiabat (this effect is represented by the $T_{\rm eq}^b$ term in the equation).
Quantitatively, due to its high maximal efficiency $\epsilon_{\rm max}\approx 0.3$, our model predicts somewhat more inflated planets in the range $1500\textrm{ K}\leq T_{\rm eq}\leq 2000\textrm{ K}$, compared with the constant $\epsilon=3\%$ model of <cit.>, as seen by comparing our Figure <ref> and their Figure 5. However, the differences between the two models are modest ($\approx 0.1 R_J$), due to the relatively weak dependence of the inflated radius on the efficiency $\Delta R^\infty\propto\epsilon^{0.3}$, the sharp drop of the efficiency at $T_{\rm eq}\gtrsim 1500\textrm{ K}$, evident from Equation (<ref>), and the somewhat flatter heating profile of <cit.>. The model of <cit.>, on the other hand, predicts higher inflations, due to the absence of a drop in their heating profile (see Figure <ref>). Nonetheless, by comparing our Figure <ref> to their Figures 6-8, we find that the differences are partially compensated by the lower efficiencies $\epsilon\leq 5\%$ and the somewhat steeper heating profile of <cit.>, and amount to $\approx 0.3R_J$ for $1M_J$ planets.
§ PLANET RE-INFLATION
In the previous sections we assumed that planets cool and contract from high temperatures (and therefore large radii) under the influence of both stellar irradiation and additional power deposition (e.g. Ohmic dissipation). However, another possible scenario (due to migration on long timescales, for example) involves dissipation mechanisms that come into play only once the planet has already cooled and contracted to a relatively small radius <cit.>.
It is clear from the discussion in Section <ref> that the final equilibrium temperature profile of the planet is given by Figure <ref>, with central temperature $T_c^\infty$, even if the planet had initially $T_c<T_c^\infty$ (the general equilibrium profile, in case the energy deposition does not reach the center, is given in Appendix <ref>). However, although planets cool down (and contract) or heat up (and expand) to the same temperatures (and radii), imposed by the stellar irradiation and the heat deposition, we show below that the timescales to reach equilibrium are different. In Figure <ref> we present a schematic plot of the reheating (and therefore re-inflation) of a planet from an initial central temperature $T_c$ (assuming the planet has cooled for a few Gyr in the absence of power deposition) to a final equilibrium central temperature $T_c^\infty>T_c$, imposed by the energy deposition. As seen in Figure <ref>, the planet heats up from the outside in. This result, which is also evident from numerical calculations by <cit.>, is due to both the increase in heat capacity and final temperature with depth and the decrease in deposition with depth, so the heating rate is $\propto\tau^{-(1+\alpha+\beta/4)}$. This outside-in heating implies that models, which exploit the entire heat deposited in the convective region to heat up the planet, overestimate the reheating rate <cit.>.
Schematic temperature profile (logarithmic scale) of a hot Jupiter with an energy deposition that extends to its center. The equilibrium state (solid black line) is characterized by an equilibrium central temperature $T_c^\infty$. A hot-Jupiter profile with an initial $T_c<T_c^\infty$ (dashed blue line) is also plotted. Two intermediate stages (dotted red lines) show the reheating (re-inflation) of the planet from the initial to the equilibrium phase. Typical values of the temperature are provided.
The time to heat the planet's center, which determines the re-inflation timescale is given by
\begin{equation}\label{eq:t_heat}
t_{\rm heat}\sim\frac{\frac{M}{m_p}k_{\rm B}T_c^\infty}{L_{\rm dep}(\tau_c)}.
\end{equation}
It is instructive to compare this timescale to the cooling timescale of initially hot planets to the same equilibrium central temperature. Due to the decrease of the internal luminosity with central temperature, the cooling timescale is also determined by the final equilibrium temperature $T_c^\infty$, as seen by combining Equations (<ref>) and (<ref>)
\begin{equation}\label{eq:t_cool}
t_{\rm cool}\sim\frac{\frac{M}{m_p}k_{\rm B}T_c^\infty}{L_{\rm int}}=
\frac{\frac{M}{m_p}k_{\rm B}T_c^\infty}{L_{\rm dep}(\tau_{\rm rad})},
\end{equation}
where the last equality is due to the condition $L_{\rm int}=L_{\rm dep}(\tau_{\rm rad})$, which is fulfilled in the final cooling stage (see Section <ref> and Figure <ref>). By combining Equations (<ref>) and (<ref>) we find
\begin{equation}\label{eq:time_ratio}
\frac{t_{\rm heat}}{t_{\rm cool}}\sim\frac{L_{\rm dep}(\tau_{\rm rad})}{L_{\rm dep}(\tau_c)}=\left(\frac{\tau_c}{\tau_{\rm rad}}\right)^\alpha\approx 30,
\end{equation}
with the numerical value calculated using $\tau_c=10^{10}$, $\tau_{\rm rad}=10^5$, and $\alpha=0.3$. Since the typical cooling time to a radius of $1.3R_J$ is $\sim 1\textrm{ Gyr}$ (see, e.g., Figure <ref>), the long heating timescales of $\sim 30\textrm{ Gyr}$ (for $\epsilon=5\%$, which matches inflation to $1.3R_J$) imply that planets can only be mildly re-inflated with Ohmic dissipation (up to about $0.2 R_J$), and that they do not reach their final equilibrium temperature <cit.>.
§ CONCLUSIONS
In this work we analytically studied the effects of additional power sources on the radius of irradiated giant gas planets. The additional heat sources halt the evolutionary cooling and contraction of gas giants, implying a large final equilibrium radius. A slowdown in the evolutionary cooling prior to equilibrium is possible only for sources which do not extend to the planet's center.
We generalized our previous work , which was confined to localized point sources, to treat sources that extend from the photosphere to the deep interior of the planet. We parametrized such a heat source by the total power it deposits below the photosphere $\epsilon L_{\rm eq}$, and by the logarithmic decay rate of the deposited power with optical depth $\alpha>0$. Implicitly, we assumed a heating profile $L_{\rm dep}(\tau)=\epsilon L_{\rm eq}\tau^{-\alpha}$, with $L_{\rm dep}(\tau)$ denoting the accumulated heat deposited below an optical depth $\tau$. Motivated by previous studies <cit.>, we measured the total heat with respect to the incident stellar irradiation $L_{\rm eq}$ and adopted the efficiency parameter $\epsilon$, which was assumed small $\epsilon <1$.
We generalized the technique used in and showed that planetary cooling and contraction stop when the internal luminosity (i.e. cooling rate) drops below the heat deposited in the convective region $L_{\rm int}\lesssim L_{\rm dep}(\tau_{\rm rad})$ <cit.>. This condition defines a threshold efficiency $\epsilon\gtrsim\tau_{\rm rad}^{-(1-\alpha)}$, required to explain the inflation of observed hot Jupiters, where $\tau_{\rm rad}\approx 10^5$ is the optical depth of the radiative-convective boundary (of $\sim 1$ Gyr old planets with an equilibrium temperature of $T_{\rm eq}\approx 2\cdot 10^3\textrm{ K}$) in the absence of heat deposition, and only flat enough heating profiles $\alpha<1$ have an impact.
The method presented in this work reproduces previous numerical results while providing simple intuition and it may be used to study the effects of any power source on the radius and structure of hot Jupiters. Combining the model with observational correlations <cit.> may reveal the nature of the additional heat deposition mechanism, if exists, and provide a step toward solving the observed radius anomalies.
For a quantitative example, we focused on the suggested Ohmic dissipation mechanism, which stems from the interaction of atmospheric winds with the planet's magnetic field <cit.>.
This mechanism can be described by a power law $\alpha\approx 0.3$ in the planet's interior, and a reduction of $\approx 5\cdot 10^2$ in the efficiency, due to the slimness of the wind zone.
We therefore found that the threshold efficiency in this case is $\approx 5\%$, in accordance with previous numerical studies.
Assuming a constant efficiency of 3%, we estimated inflated radii which are similar to the numerical predictions of <cit.>. However, we challenged the assumption of a constant efficiency, made in previous studies, and examined the correlation between the efficiency and the equilibrium temperature. We found that the efficiency rises with temperature, due to the increase in electrical conductivity, to a maximum of $\approx 0.3$ at $T_{\rm eq}\approx 1500\textrm{ K}$, and then drops due to the magnetic drag <cit.>. As a result, we are able to explain the concentration of radius anomalies around this temperature <cit.>, and to account for the radii of most inflated hot Jupiters, which are in the range $\approx 1500\textrm{ K}-2500\textrm{ K}$ and reach $\approx 1.6 R_J$. In addition, we argue that if these planets are indeed inflated by the Ohmic mechanism then they have already reached their final equilibrium state <cit.>, and that the energy deposition must have suspended their contraction, and could not have re-inflated them from a smaller radius, since re-inflation timescales are too long <cit.>. Nonetheless, some extremely inflated planets have radii which exceed the predictions of our model.
In contrast to most previous studies, we did not introduce any free parameters to model the wind zone, but rather related the wind velocity, and therefore the amount of dissipated heat, to the strength of the magnetic field and to the equilibrium temperature. This procedure, combined with the generalized technique from , enabled us to estimate the observational correlations expected in an Ohmic heating scenario, and to compare them with observations <cit.>. Although our main conclusions are robust, the exact shape of the radius-equilibrium-temperature curve should be studied with more detailed simulations, due to the approximate nature of our assessments.
This research was partially supported by ISF, ISA and iCore grants.
We thank Oded Aharonson, Konstantin Batygin, Peter Goldreich, Yohai Kaspi, Thaddeus Komacek, Yoram Lithwick, Adam Showman, and David J. Stevenson for insightful discussions. We also thank the anonymous referee for valuable comments, which improved the paper.
§ GENERAL POWER-LAW ENERGY DEPOSITION
In Section <ref> we analyzed the effects of extended heat deposition, parametrized by a power-law heating profile which extends from the surface to the interior of a hot Jupiter. However, in Section <ref> we confined the discussion to heating profiles with a cumulative power index $\alpha<1-\beta$, which extend to the planet's center. In this section we relax both constrains, and address more general sources, which may have steeper profiles and a cut-off at some $\tau_{\rm cut}<\tau_c$.
We now consider different values of $\alpha$, with a distinction made by the value of $\beta/(1-\alpha)$.
Case I: $\alpha<1-\beta$. In this case, as explained in Section <ref>, a convective region emerges at $\tau_b$, given by Equation (<ref>). The convective region continues until $\tau_{\rm cut}$, reaching an energy density of
\begin{equation}\label{eq:power_u_eq}
\frac{U_{\rm iso}}{U_{\rm eq}}\sim\left(\frac{\tau_{\rm cut}}{\tau_b}\right)^\beta\sim\left(\tau_{\rm cut}\epsilon^{1/(1-\alpha)}\right)^\beta,
\end{equation}
which is derived similarly to Equations (<ref>) and (<ref>). This secondary convective region is connected to the main interior convective region with a radiative tangent, and $L_{\rm int}$ is found using $U_{\rm iso}$ in the same manner as in Section <ref>. A schematic temperature profile which clarifies the alternating radiative-convective structure, which is analogous to and to Section <ref>, is given in Figure <ref>.
Case II: $1-\beta<\alpha<1$. In this case there is no transition to a secondary convective region. Rather, the radiative profile of Equation (<ref>) continues up to $\tau_{\rm cut}$, reaching
\begin{equation}\label{eq:u_eq_eff_case2}
\frac{U_{\rm iso}}{U_{\rm eq}}\approx 1+\epsilon\tau_{\rm cut}^{1-\alpha}\approx\epsilon\tau_{\rm cut}^{1-\alpha},
\end{equation}
with the last approximation made for the significant heating regime.
Case III: $\alpha>1$. In this case, according to Equation (<ref>), the radiative profile is governed by low optical depths and
\begin{equation}
U=U_{\rm eq}+\frac{3}{\alpha-1}\frac{\epsilon L_{\rm eq}}{4\pi R^2c}
\end{equation}
for $\tau\gg 1$. It is easy to verify that there is no secondary convective region in this case for $\epsilon\ll 1$. Therefore, the radiation energy density of the deep isotherm is
\begin{equation}
\frac{U_{\rm iso}}{U_{\rm eq}}\approx 1+\epsilon\approx 1,
\end{equation}
regardless of $\tau_{\rm cut}$. We conclude that heating with $\epsilon\ll 1$ is unable to significantly effect the planetary cooling for $\alpha>1$.
Schematic temperature profile (logarithmic scale) of a hot Jupiter with an energy deposition that corresponds to Case I, i.e., $\alpha<1-\beta$ (see text). The structure of the planet is characterized by two radiative, nearly isothermal, regions (solid black lines) and two convective regions (dashed blue lines): the main convective interior, and an induced exterior secondary convective zone. Typical values of the temperature are provided.
Schematic temperature profiles (logarithmic scale) of hot Jupiters with an energy deposition with a cut-off at $\tau_{\rm cut}<\tau_c$. The lower bound on the outer temperature profile, set by a combination of the stellar irradiation and heat deposition (solid black line), is connected to the internal boundary condition (temperature $T_c$ at $\tau_c$). Planet profiles (dashed blue lines) are given for decreasing central temperatures, which correspond to the different evolutionary stages 1-4 in the text.
As in Section <ref>, the decrease in the internal luminosity is given by
\begin{equation}\label{eq:l_u_eq}
\frac{L_{\rm int}}{L_{\rm int}^0}=\left(\frac{U_{\rm iso}}{U_{\rm eq}}\right)^{-(1-\beta)/\beta}.
\end{equation}
From both Equations (<ref>) and (<ref>) we find the critical criterion for a significant effect on the cooling rate of the planet, provided that $\alpha<1$:
\begin{equation}\label{eq:crit}
\epsilon\tau_{\rm cut}^{1-\alpha}\gtrsim 1.
\end{equation}
As discussed in Section <ref>, and seen in Figure <ref>, an additional requirement is that $\tau_b<\tau_{\rm rad}$ (for Case I, with a similar analog for Case II), with $\tau_{\rm rad}$ denoting the radiative-convective transition in the absence of heat deposition. This requirement is satisfied by the condition of Equation (<ref>) for $\tau_{\rm cut}<\tau_{\rm rad}$. On the other hand, if the deposition is intense or deep enough, so that $U_{\rm iso}\sim U_c$, then the planet reaches equilibrium and cooling stops entirely, as explained in .
A more general analysis, presented schematically in Figure <ref>, shows that for a given heating profile with a cut-off at $\tau_{\rm cut}<\tau_c$, the planet evolves through 4 distinct stages, with Equation (<ref>) relevant only to Stage 3:
Stage 1: isolation. For very high central temperatures, the planet is fully convective (since $\tau_{\rm rad}<1$), and its cooling rate is unaffected by the stellar irradiation or by the heat deposition .
Stage 2: irradiation. As the planet cools, it develops a radiative envelope (which thickens with time) and its internal luminosity is determined by the stellar irradiation (see Section <ref>) but is unaffected by the heat deposition (since $\tau_{\rm rad}<\tau_b$).
Stage 3: deposition. At even lower central temperatures (when $\tau_{\rm rad}>\tau_b$), the planet's cooling rate is reduced by the heat deposition, according to Equation (<ref>). This stage is also depicted in Figure <ref>.
Stage 4: equilibrium. When the planet reaches $T_c=T_{\rm iso}$, evolutionary cooling stops entirely, and the planet reaches its final state.
In the special case of a heating profile without a cut-off ($\tau_{\rm cut}=\tau_c$), the planet skips the intermediate Stage 3 and transitions directly from Stage 2 (cooling unaffected by deposition) to the equilibrium state. This transition is explained in Section <ref>, and is evident from Figures <ref> and <ref>.
Essentially, the equilibrium central temperature $T_c^\infty$, imposed by the heat deposition and introduced in Section <ref>, is a special case (for $\tau_{\rm cut}=\tau_c$) of the deep isotherm temperature $T_{\rm iso}$.
Combining Equations (<ref>), (<ref>), and (<ref>), we find the effect of heating on the central temperature (and radius) at a given age, during the relevant Stage 3, when the cooling is influenced by the heat deposition
\begin{equation}\label{eq:t_c}
\Delta R(t)\propto T_c(t)\propto\left(\frac{U_{\rm iso}}{U_{\rm eq}}\right)^{(1-\beta)/(4-\beta-\beta b)}\approx\left(\frac{U_{\rm iso}}{U_{\rm eq}}\right)^{0.5},
\end{equation}
with $\beta=0.35$ and $b=7$ estimated in , and with the ratio $U_{\rm iso}/U_{\rm eq}$ given by Equation (<ref>) or (<ref>).
§ HIGH ROSSBY NUMBER REGIME
In Section <ref> we adopted a low Rossby number approximation ${\rm Ro}\ll 1$, in which the advective term $v\nabla v\sim v^2/R$ in the force balance equation is negligible when compared to the Coriolis acceleration $2\Omega v\sin\phi$. Although it is justified for the atmosphere on average, this approximation breaks down close to the equator ($\phi=0$). In this section we reanalyze the atmospheric wind velocity derivation in the high Rossby number limit ${\rm Ro}\gg 1$, relevant for the equator, and compare the results to the conclusions of Section <ref>.
In the high $\rm{Ro}$ case, the force balance Equation (<ref>) is replaced with
\begin{equation}\label{eq:wind_vel_high_ro}
\frac{c_s^2}{R}\frac{\Delta T}{T_{\rm eq}}=\Omega v\left(\frac{v}{\Omega R}+\frac{\sigma}{\sigma_c}\frac{\rho_c}{\rho}\right),
\end{equation}
where the advective term replaces the (now negligible) Coriolis term. Equation (<ref>) indicates that $v<c_s$, and therefore ${\rm Ro}=v/(2\Omega R\sin\phi)<1$, except for the equator (since $c_s\sim\Omega R$; see Section <ref>). This understanding, together with a similar insight from the low Rossby number Equation (<ref>), self-consistently justifies our low Ro approximation for the atmosphere on average (excluding the equator). We see again, as in the low Ro case, that due to the strong dependence of the conductivity on the temperature, the magnetized regime corresponds to high equilibrium temperatures. In addition, Equation (<ref>), which demonstrates the dependence of the heating efficiency $\epsilon\propto\sigma v^2$ on the conductivity (the dominant parameter for an intuitive understanding, due to its strong dependence on the temperature), is clearly valid in the high Ro limit as well, so $\epsilon\propto\sigma$ for low conductivities, while $\epsilon\propto\sigma^{-1}$ for high conductivities <cit.>.
The decay of the velocity and day-night temperature difference with depth is calculated similarly to Section <ref>, with Equation (<ref>) replacing Equation (<ref>). By combining Equation (<ref>) with the diffusion Equation (<ref>), we find that for the unmagnetized regime (the magnetized regime is indifferent to Ro)
\begin{equation}\label{eq:delta_t_high_ro}
\frac{v}{c_s}=\left(\frac{\Delta T}{T_{\rm eq}}\right)^{1/2}=\frac{R\sigma_{\rm SB}T_{\rm eq}^4}{c_s^3}\frac{\kappa}{\tau^2}.
\end{equation}
Equation (<ref>), which is the high Ro version of Equation (<ref>), shows that the velocity and temperature difference decay with depth in the high Ro regime as well. Quantitatively, we find that the decay of the velocity with depth $v\propto\kappa/\tau^2$ is the same in both regimes (low and high Ro), as can be immediately understood from Equation (<ref>).
We conclude that the high Ro regime, relevant for a narrow strip around the equator, exhibits the same qualitative behavior as the low Ro regime, with some of the quantitative results reproduced as well. Nevertheless, some of the specific power-law scalings with the equilibrium temperature change in this regime.
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1511.00222
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In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3–manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3–manifolds that have the same set of surfaces.
§ INTRODUCTION
The geodesic length spectrum of a Riemannian manifold $M$ is a basic invariant that has been well-studied due to its connection with the geometric and analytic structure of $M$. For instance, when $M$ has negative sectional curvature, there is a strong relationship between this spectrum and the eigenvalue spectrum of the Laplace–Beltrami operator (see <cit.>,<cit.>), and the latter is well known to determine basic geometric/topological invariants like the dimension, volume, and total scalar curvature of $M$.
In this article, we focus on variations of the surface analog of the geodesic length spectrum for closed, orientable, hyperbolic 3–manifolds introduced by the authors in <cit.> (see also <cit.>, <cit.>, and <cit.>). We take this theme further and study the full surface spectrum (or set) of such manifolds (see 2 for definitions) which loosely takes into account all of the $\pi_1$–injective surface subgroups of the fundamental group of $M$. Our main result can be informally stated as follows (see Theorem <ref> for the precise statement).
For any closed, orientable, hyperbolic $3$–manifold $M$, there are at most finitely many non-isometric closed, orientable, hyperbolic 3–manifolds with the same surface set as $M$. Furthermore, all such manifolds are commensurable.
For the eigenvalue or geodesic length spectra, many commensurability
and finiteness results have been established. The second author
<cit.> proved that isospectral (i.e. the same eigenvalue spectra)
or length isospectral (i.e. the same geodesic length spectra)
arithmetic hyperbolic 2–manifolds are
commensurable. Chinburg–Hamilton–Long–Reid <cit.>
proved an identical result for arithmetic hyperbolic
3–manifolds. Prasad–Rapinchuk <cit.> determined when
these commensurability rigidity results hold for general,
arithmetic, locally symmetric orbifolds, proving that in many settings
the commensurability class of the manifold is determined by the
eigenvalue or geodesic length spectra. It was already known that the
commensurability class is not always determined by these spectra as
Lubotzky–Samuel–Vishne <cit.> produced higher rank,
arithmetic, locally symmetric incommensurable isospectral examples
prior to the work of Prasad and Rapinchuk. In <cit.>, the
authors proved a result similar to Theorem <ref>. Namely, if
$M_1,M_2$ are arithmetic hyperbolic 3–manifolds that contain a totally geodesic
surface, and have the same set of totally
geodesic surfaces, then they are
commensurable. Meyer <cit.> established
a higher dimensional analog for certain classes of arithmetic
hyperbolic $n$–manifolds. It is worth emphasizing that our present
work differs from all the above works in one important and fundamental
way. Namely, we do not impose an arithmetic assumption.
In <cit.>, examples of non-isometric, closed, hyperbolic 3–manifolds with the same spectra of totally geodesic surfaces were constructed (see also <cit.> and <cit.>). Those methods can be employed to also produce arbitrarily large finite sets of non-isometric closed hyperbolic 3–manifolds $\set{M_j}$ that pairwise have the same totally geodesic surface spectra (the spectra can be ensured to be infinite as well). However, it is unknown if an infinite set of such manifolds can exist. In particular, the totally geodesic surface analog of our finiteness result is unknown. Finally, for the full surface spectrum, there are no known examples of non-isometric hyperbolic 3–manifolds $M_1,M_2$ with the same full surface spectrum.
§ NOTATION AND PRELIMINARIES
Throughout, $M={\bf H}^3/\Gamma$ will be a closed, orientable,
hyperbolic 3–manifold and $\Sigma_g$ will denote the closed
orientable surface of genus $g$. It was proved by Thurston <cit.> that the number of $\Gamma$–conjugacy classes of subgroups
of $\Gamma$ isomorphic to $\pi_1(\Sigma_g)$ is finite. A breakthrough
was provided by Kahn
and Markovic <cit.> who proved that this number is non-zero for
certain values of $g$. Furthermore, they then
went onto provide estimates for these
numbers in <cit.> (building on previous work of Masters
We need to refine this discussion somewhat. For each discrete,
faithful representation $\rho\colon \pi_1(\Sigma_g) \to \PSL(2,\bf
C)$, we refer to the image $\Delta_\rho$ as a Kleinian surface
group. For each $\Delta_\rho$, let $\ell_\rho(M)$ denote the number
of $\Gamma$–conjugacy classes of subgroups $\Delta < \Gamma$ that are
$\PSL(2,\bf C)$–conjugate to $\Delta_\rho$. Typically the value of
$\ell_\rho(M)$ will be zero (e.g. for those $\Delta_\rho$ that contain
an element with transcendental trace), but for those that are non-zero
we can define the full surface spectrum of $M$ to be the collection of
such pairs $(\Delta_\rho,\ell_\rho(M))$. Specifically, the full
surface spectrum of $M$ is the set ${\cal S}(M) =
\{(\Delta_\rho,\ell_\rho(M))~:\ell_\rho(M) \ne 0\}$. Additionally, we
define the surface set of $M$ to be the set $\mathrm{S}(M) =
\{\Delta_\rho~:\ell_\rho(M) \ne 0\}$. The case when $\Delta_\rho$ is
Fuchsian was studied in <cit.> and gives rise to an associated
spectrum that we denote here by $\mathrm{S}_{Fuc}(M)$ and call the
genus spectrum.
In this note, particular emphasis will be placed upon
those Kleinian surface groups $\Delta_\rho$ corresponding to
virtual fiber subgroups of $\Gamma$. By the work of Bonahon <cit.>
and Thurston <cit.> (and more generally the solution to the Tameness
Conjecture by Agol <cit.> and Calegari–Gabai <cit.>), these
virtual fiber subgroups of $\Gamma$ are precisely those $\Delta_\rho$
that are finitely generated, geometrically infinite subgroups of
$\Gamma$. Since being geometrically infinite depends only on
$\Delta_\rho$ and not on the ambient group $\Gamma$, these surface
subgroups provide an important subclass of surface subgroups that can be
used to control the topology of 3–manifolds. For future reference,
let us denote the associated spectrum for this subclass of surface
subgroups by
\[ \mathrm{S}_{vf}(M) = \{\Delta_\rho\in \mathrm{S}(M)~:\Delta_\rho~\hbox{is a virtual fiber subgroup}\}. \]
Essential in our work is the groundbreaking work of Agol
<cit.> and the aforementioned work of Kahn–Markovic
<cit.>. We summarize their collective work in the following
Let $M={\bf H}^3/\Gamma$ be a closed, orientable, hyperbolic 3–manifold. Then
(a) $\mathrm{S}(M)\neq \emptyset$.
(b) $\mathrm{S}_{vf}(M)\neq \emptyset$.
(c) $\mathrm{S}_{vf}(M)$ contains infinitely many elements $F_\rho$ that
are not commensurable and in particular have arbitrarily large genus.
Proof: Given the preamble to the statement of the theorem, the only part that needs comment is (c). By <cit.> there is a finite sheeted cover $M_0 \rightarrow M$ such that $b_1(M_0)\geq 2$ and $M_0$ is fibered. In particular, by <cit.>, $M_0$ is fibered in infinitely many different ways. Indeed, it follows from <cit.> that we can find fibered surfaces of arbitrarily large genus occuring as integral lattice points in the (open) cone over a top dimensional face of the Thurston norm ball. Since the degree of the cover $M_0\rightarrow M$ is finite and the fibers have arbitrarily large genus, projecting these fibers back to $M$ provides infinitely many incommensurable virtual fibers.
§ PROOF OF THEOREM <REF>
We now state the precise version of Theorem <ref> that we will prove in this section.
If $M$ is a closed, orientable, hyperbolic 3–manifold, then the set
\[ \mathcal{S}_M = \set{N~:\mathcal{S}(M) = \mathcal{S}(N)} \]
is finite. Moreover, if $N \in \mathcal{S}_M$, then $M,N$ are commensurable.
As noted above, since being a virtual fiber depends only on $\Delta_\rho$ and not on the ambient manifolds, if $\mathcal{S}(M) = \mathcal{S}(N)$, then $\mathrm{S}_{vf}(M) = \mathrm{S}_{vf}(N)$. In particular, to prove Theorem <ref>, it suffices to prove the following result.
If $M$ is a closed, orientable, hyperbolic 3–manifold, then the set
\[ \mathcal{S}_{M,vf} = \set{N~:~\mathrm{S}_{vf}(M) = \mathrm{S}_{vf}(N)} \]
is finite. Moreover, if $N \in \mathcal{S}_{M,vf}$, then $M,N$ are commensurable.
Proof of Theorem <ref>: We first prove
that if $\mathrm{S}_{vf}(M) = \mathrm{S}_{vf}(N)$, then $M,N$ are
commensurable. To that end, let $\Delta=\Delta_\rho$ denote a common
virtual fiber subgroup and set $g$ to be the genus of $\Delta$. Since
$\Delta$ is a virtual fiber, we can find pseudo-Anosov maps
$\phi,\psi\colon \Sigma_g \longrightarrow \Sigma_g$ so that
$M_{\phi}\rightarrow M$, $M_{\psi}\rightarrow N$ are finite sheeted
covers and $\pi_1(M_{\phi}), \pi_1(M_{\psi})$ have a common fiber
group $\Delta$. Associated to the fiber group $\Delta$ is a unique
pair of ending laminations $\nu^{\pm}$ in the projective measured
lamination space of $\Sigma_g$ which are left invariant by $\phi,
\psi$ (see <cit.>). As a result, there exist integers $r,s$ such
that the mapping classes $\phi,\psi$ satisfy $\phi^r =
\psi^s$. Consequently, the bundles $M_{\phi^r}$ and $M_{\psi^s}$ are
isometric. In particular, we have
\[ \xymatrix{ & M_{\phi^r} \cong M_{\psi^s} \ar[rdd]^{\mathrm{finite}} \ar[ldd]_{\mathrm{finite}} & \\ & & \\ M_\phi \ar[dd]_{\mathrm{finite}} & & M_\psi \ar[dd]^{\mathrm{finite}} \\ & & \\ M & & N} \]
and thus conclude that $M,N$ are commensurable.
It remains to establish the finiteness of $\mathcal{S}_{M,vf}$.
We will argue by contradiction, and to that end, we assume that there are
infinitely many non-isometric $M_i={\bf H}^3/\Gamma_i$, $i=1,2,
\ldots$ with $\mathrm{S}_{vf}(M)=\mathrm{S}_{vf}(M_i)$ for all $i$. We
will prove that for $i \geq i_0$, the groups $\Gamma_i$ have
uniformly bounded rank. We will then show that for an even larger $i_1$,
the groups $\Gamma_i$ for $i\geq i_1$ must have rank larger than
this uniform bound. Towards that goal, we first assert that the volumes
of the manifolds $M_i$ must be unbounded. Specifically, we have the
following general lemma.
The set of volumes for any infinite set $\set{M_i}$ of commensurable, finite volume, hyperbolic $3$–manifolds is unbounded.
Proof: We split into two cases depending on whether
the manifolds are arithmetic or not. Note that since arithmeticity is
a commensurability invariant, either all of the $M_i$ are arithmetic
or all of the $M_i$ are non-arithmetic. If the $M_i$ are arithmetic,
the assertion follows from work of Borel <cit.> since there are
only finitely many arithmetic hyperbolic 3–manifolds of bounded
volume. If the $M_i$ are non-arithmetic, by work of Margulis
<cit.>, there is a unique maximal lattice in the common
commensurability class that contains all of the $\Gamma_i$ as finite
index subgroups. In particular, all the $M_i$ cover the fixed closed
hyperbolic 3–orbifold $Q$ associated to this unique maximal
lattice. Since $Q$ has only finitely many degree $d$ covers for any
$d$, the covering degrees must go to infinity. Consequently, the
volumes cannot be bounded in this case
We further note that since the manifolds $M_i$ are all commensurable,
there is also a uniform lower bound of the injectivity radii of the
$M_i$. This is easily proved using
the arithmetic/non-arithmetic dichotomy once again.
For future reference, we denote this lower bound by $s$.
Thus we can assume that we have a sequence of manifolds
$M_i$ with injectivity radius $s$ and whose volumes get arbitrarily large.
We now show how to use this to bound the
ranks of the groups $\Gamma_i$ for $i$ sufficiently large.
Towards that goal, set
$\Delta_0$ to be a common, minimal genus, virtual, fiber group in
$\Gamma_i$, and set $g$ to be this common, minimal genus. In order to
control the ranks of the groups $\Gamma_i$, we will utilize a
quantitative virtual fibering result of Soma <cit.>. To
state his result, let
$\mathrm{InjRad}(M), \mathrm{Vol}(M)$ denote the injectivity radius
and volume of $M$, respectively, and set
$d_1(g,s) = {4s(g-1)\over{\sinh^2(s/2)}}$.
If $M$ is a closed, orientable, hyperbolic 3–manifold with
\[ \mathrm{InjRad}(M) \geq s~\textrm{and}~ \mathrm{Vol}(M) \geq 2\pi d_1(g,s)\sinh^2(d_1(g,s)+1), \]
then any immersed virtual fiber in $M$ of genus $g$ is embedded.
Theorem <ref> in tandem with the above conditions on
$\mathrm{InjRad}(M_i),\mathrm{Vol}(M)$ implies that there is $i_{g,s}
\in\mathbf{N}$ such that if $i \geq i_{g,s}$, the virtual fiber group
$\Delta_0$ corresponds to an embedded incompressible surface
of genus $g$ in $M_i$. This incompressible surface greatly limits the structural
possibilities for the manifolds $M_i$. Specifically, $M_i$ is either a
fiber bundle over the circle with fiber group $\Delta_0$, or $M_i$ is
the union of two twisted $I$–bundles. Moreover, in the latter case,
we have a double cover $N_i\rightarrow M_i$ such that $N_i$ is a fiber
bundle with fiber group $\Delta_0$ (see <cit.>).
We now leverage the above fiber bundle structure to obtain bounds for the rank of $\Gamma_i$ for $i$ sufficiently large. The rank of $\Gamma_i$ will be denoted by $\mathrm{Rank}(\Gamma_i)$.
There exists $i_0 \geq i_{g,s}$ such that if $i \geq i_0$, then $g +1 \leq \mathrm{Rank}(\Gamma_i) \leq 2g+2$.
Proof: We assume throughout that $i \geq i_{g,s}$. Let
$\mathcal{I}_1$ to be the set of $i \geq i_{g,s}$ such that $M_i$ is a
fiber bundle with fiber group $\Delta_0$ and let $\mathcal{I}_2$ to be
the set of $i \geq i_{g,s}$ such that $M_i$ is double covered by $N_i$
where $N_i$ is a fiber bundle with fiber group $\Delta_0$. We first
consider $\set{M_i}_{i \in \mathcal{I}_1}$. We know from the proof of
the commensurability invariance of $\mathrm{S}_{vf}$ that each $M_i$
must have the form $M_{\phi^{r_i}}$ for some pseudo-Anosov element
$\phi$. Applying Souto <cit.>, there exist $i' \in
\mathbf{N}$ such that $\mathrm{Rank}(\Gamma_i)=2g+1$ for all $i \geq
i'$. Next, we consider $\set{M_i}_{i \in \mathcal{I}_2}$ and apply the
above argument to $N_i$. We obtain $i'' \in \mathbf{N}$ such that
$\mathrm{Rank}(\pi_1(N_i)) = 2g+1$ for all $i \geq i''$. As $N_i$ is a
double cover of $M_i$, we can adjoin one element to $\pi_1(N_i)$ to
generate $\pi_1(M_i)$. Therefore, $\mathrm{Rank}(\Gamma_i) \leq 2g+2$
for all $i \geq i''$.
Now, set $i_0 = \max\set{i',i''}$ and note that
$\mathrm{Rank}(\Gamma_i) \leq 2g+2$ for all $i \geq i_0$. For the
lower bound, by the Nielsen–Schreier, we have
$\mathrm{Rank}(\pi_1(N_i)) \leq 2\mathrm{Rank}(\Gamma_i) - 1$ for all
$i \geq i_0$ and $i \in \mathcal{I}_2$. In particular, $g +1 \leq
\mathrm{Rank}(\Gamma_i)$ for all $i \geq
We now use Lemma <ref> to complete the proof of Theorem
<ref>. By Theorem <ref> (c), we can find
incommensurable virtual fiber subgroups of arbitrarily large
genus. Choosing a virtual fiber subgroup $\Delta_1$ of genus $g_1$
with $2g+2 < g_1 + 1$ and repeating the above argument, we obtain an
integer $i_1 \geq i_{g_1,s}$ such that $g_1 + 1 \leq
\mathrm{Rank}(\Gamma_i)$ for all $i \geq i_1$. For all $i \geq
\max\set{i_0,i_1}$, we must have $g_1 + 1 \leq \mathrm{Rank}(\Gamma_i)
\leq 2g+ 2 < g_1 +1$, a contradiction. Hence $\mathcal{S}_{M,vf}$ is
finite as required.
Remarks:(1) In the proof of Lemma <ref> we
could also have used <cit.> for both the bundle case and the union
of two twisted I-bundles. However, the setting of <cit.> is more
appropriate in this case (i.e. commensurable manifolds), and only a
mild extension is needed for us to handle the union of two twisted
I-bundles. Hence the reason for not using <cit.> in this case. In
4, we will need to use <cit.>.
(2) As noted in the introduction, we do not know if there
exists a pair of non-isometric, closed, orientable, hyperbolic
3–manifolds $M_1,M_2$ with $\mathcal{S}(M_1) =
\mathcal{S}(M_2)$. Since being either a virtual fiber or Fuchsian
depends only on $\Delta_\rho$ and not the ambient manifold, such a
pair would also satisfy both $\mathrm{S}_{vf}(M_1) =
\mathrm{S}_{vf}(M_2)$, $\mathrm{S}_{Fuc}(M_1) =
\mathrm{S}_{Fuc}(M_2)$. Examples where the latter equality holds were
constructed in <cit.> using a variation of the
method of Sunada <cit.> for
constructing isospectral and length isospectral manifolds. That method
does not seem well-suited for also arranging equality between virtual
fibers. As with the full spectrum, we do not presently know if there
exists a pair of non-isometric, closed, hyperbolic 3–manifolds
$M_1,M_2$ with $\mathrm{S}_{vf}(M_1) = \mathrm{S}_{vf}(M_2)$.
§ A CONJECTURAL STRENGTHENING FOR ARITHMETIC HYPERBOLIC 3–MANIFOLDS
In this section we deal with closed, arithmetic, hyperbolic 3–manifolds, and prove a stronger result (conjecturally) that involves only topological data. We refer the reader to <cit.> for background on arithmetic hyperbolic 3–manifolds. Let us define the topological virtual fiber set of $M$ to be the set
\[ \mathrm{S}_{tvf}(M) = \{\Delta~:\Delta~\hbox{is isomorphic to a virtual fiber subgroup}\}. \]
Our strengthening relies on the following conjecture often referred to as the short geodesic conjecture.
Let $M$ be a closed, orientable, arithmetic, hyperbolic 3–manifold. Then there is a constant $C>0$ (independent of $M$) so that the length of the shortest closed geodesic in $M$ is at least $C$.
Assuming this conjecture, we establish the following result.
Assuming Conjecture <ref> there are at most finitely many closed orientable arithmetic hyperbolic 3–manifolds $M_1, M_2\ldots M_n$ so that $\mathrm{S}_{tvf}(M_i)=\mathrm{S}_{tvf}(M_j)$ for each $i,j$.
Proof: The proof of Theorem <ref> is similar to the proof of Theorem <ref> and is done by contradiction. If there is an infinite sequence of such manifolds $M_i$, by Borel <cit.> their volumes are unbounded and Conjecture <ref> implies that the injectivity radii are bounded from below. Choosing a minimal genus (topological) virtual fiber in each $M_i$ and applying Theorem <ref>, it follows that for sufficiently large $i$, $M_i$ is either a genus $g$ fiber bundle or a union of two twisted $I$–bundles which is double covered by a genus $g$ fiber bundle. We now apply
Biringer's extension of <cit.>, namely <cit.>. That allows us to get control of the rank as in the proof of Lemma <ref>, and in particular, following the argument in the proof of Lemma <ref> leads to a similar contradiction on ranks as used in the proof of Theorem
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I. Biringer, Geometry and rank of fibered $3$–manifolds, Algebr. Geom. Topol. 9 (2009), 277–292.
F. Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986), 71–158.
A. Borel, Commensurability classes and volumes of hyperbolic $3$–manifolds, Ann. Scuola Norm. Sup. Pisa 8 (1981) 1–33.
D. Calegari, D. Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006), 385–446.
T. Chinburg, E. Hamilton, D. D. Long, A. W. Reid, Geodesics and commensurability classes of arithmetic hyperbolic $3$–manifolds, Duke Math. J. 145 (2008), 25–44.
J. J. Duistermaat, V. W. Guillemin, The spectrum of positive elliptic operators and periodic geodesics, Invent. Math. 29 (1975), 39–79.
R. Gangolli, The length spectra of some compact manifolds, J. Diff. Geom. 12 (1977), 403–424.
J. Hempel, $3$–Manifolds, Ann. of Math. Studies 86, Princeton University (1976).
J. Kahn, V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012), 1127–1190.
J. Kahn, V. Markovic, Counting essential surfaces in a closed hyperbolic three-manifold, Geom. Topol. 16 (2012), 601–624.
B. Linowitz, J. S. Meyer, Systolic surfaces of arithmetic hyperbolic 3–manifolds, arXiv:math/1506.08341.
A. Lubotzky, B. Samuels, U. Vishne, Division algebras and noncommensurable isospectral manifolds, Duke Math. J. 135 (2006), 361–379.
C. Maclachlan, A. W. Reid, The Arithmetic of Hyperbolic $3$–Manifolds, Graduate Texts in Mathematics, 219, Springer-Verlag (2003).
G. Margulis, Discrete Subgroups of Semi-simple Lie Groups, Ergeb. der Math. 17, Springer-Verlag (1989).
J. D. Masters, Counting immersed surfaces in hyperbolic $3$–manifolds, Algebr. Geom. Topol. 5 (2005), 835–864.
D. B. McReynolds, Geometric spectra and commensurability, Canad. J. Math. 67 (2015), 184–197.
D. B. McReynolds, J. S. Meyer, M. Stover, Constructing geometrically equivalent hyperbolic orbifolds, arXiv:math/1507.06708.
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orbifolds, arXiv:math/1505.03643.
G. Prasad, A. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. Inst. Hautes Études Sci. 109 (2009), 113–184.
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Department of Mathematics
Purdue University
West Lafayette, IN 47906
email: [email protected]
Department of Mathematics
University of Texas
Austin, TX 78712
email: [email protected]
Proof of Theorem <ref>: We first prove
that if $\mathrm{S}_{vf}(M) = \mathrm{S}_{vf}(N)$, then $M,N$ are
commensurable. To that end, let $\Delta=\Delta_\rho$ denote a common
virtual fiber subgroup and set $g$ to be the genus of $\Delta$. Since
$\Delta$ is a virtual fiber, we can find pseudo-Anosov maps
$\phi,\psi\colon \Sigma_g \longrightarrow \Sigma_g$ so that
$M_{\phi}\rightarrow M$, $M_{\psi}\rightarrow N$ are finite sheeted
covers and $\pi_1(M_{\phi}), \pi_1(M_{\psi})$ have a common fiber
group $\Delta$. Associated to the fiber group $\Delta$ is a unique
pair of ending laminations $\nu^{\pm}$ in the projective measured
lamination space of $\Sigma_g$ which are left invariant by $\phi,
\psi$ (see <cit.>). As a result, there exist integers $r,s$ such
that the mapping classes $\phi,\psi$ satisfy $\phi^r =
\psi^s$. Consequently, the bundles $M_{\phi^r}$ and $M_{\psi^s}$ are
isometric. In particular, we have
\[ \xymatrix{ & M_{\phi^r} \cong M_{\psi^s} \ar[rdd]^{\mathrm{finite}} \ar[ldd]_{\mathrm{finite}} & \\ & & \\ M_\phi \ar[dd]_{\mathrm{finite}} & & M_\psi \ar[dd]^{\mathrm{finite}} \\ & & \\ M & & N} \]
and thus conclude that $M,N$ are commensurable.
It remains to establish the finiteness of $\mathcal{S}_{M,vf}$.
We will argue by contradiction, and to that end, we assume that there are
infinitely many non-isometric $M_i={\bf H}^3/\Gamma_i$, $i=1,2,
\ldots$ with $\mathrm{S}_{vf}(M)=\mathrm{S}_{vf}(M_i)$ for all $i$. We
will prove that for $i \geq i_0$, the groups $\Gamma_i$ have
uniformly bounded rank. We will then show that for an even larger $i_1$,
the groups $\Gamma_i$ for $i\geq i_1$ must have rank larger than
this uniform bound. Towards that goal, we first assert that the volumes
of the manifolds $M_i$ must be unbounded. Specifically, we have the
following general lemma.
The set of volumes for any infinite set $\set{M_i}$ of commensurable, finite volume, hyperbolic $3$–manifolds is unbounded.
Proof: We split into two cases depending on whether
the manifolds are arithmetic or not. Note that since arithmeticity is
a commensurability invariant, either all of the $M_i$ are arithmetic
or all of the $M_i$ are non-arithmetic. If the $M_i$ are arithmetic,
the assertion follows from work of Borel <cit.> since there are
only finitely many arithmetic hyperbolic 3–manifolds of bounded
volume. If the $M_i$ are non-arithmetic, by work of Margulis
<cit.>, there is a unique maximal lattice in the common
commensurability class that contains all of the $\Gamma_i$ as finite
index subgroups. In particular, all the $M_i$ cover the fixed closed
hyperbolic 3–orbifold $Q$ associated to this unique maximal
lattice. Since $Q$ has only finitely many degree $d$ covers for any
$d$, the covering degrees must go to infinity. Consequently, the
volumes cannot be bounded in this case
We further note that since the manifolds $M_i$ are all commensurable,
there is also a uniform lower bound of the injectivity radii of the
$M_i$. This is easily proved using
the arithmetic/non-arithmetic dichotomy once again.
For future reference, we denote this lower bound by $s$.
Thus we can assume that we have a sequence of manifolds
$M_i$ with injectivity radius $s$ and whose volumes get arbitrarily large.
We now show how to use this to bound the
ranks of the groups $\Gamma_i$ for $i$ sufficiently large.
To that end, set
$\Delta_0$ to be a common, minimal genus, virtual, fiber group in
$\Gamma_i$, and set $g$ to be this common, minimal genus. In order to
control the ranks of the groups $\Gamma_i$, we will utilize a
quantitative virtual fibering result of Soma <cit.>. To
state his result, let
$\mathrm{InjRad}(M), \mathrm{Vol}(M)$ denote the injectivity radius
and volume of $M$, respectively, and set
$d_1(g,s) = {4s(g-1)\over{\sinh^2(s/2)}}$.
If $M$ is a closed, orientable, hyperbolic 3–manifold with
\[ \mathrm{InjRad}(M) \geq s~\textrm{and}~ \mathrm{Vol}(M) \geq 2\pi d_1(g,s)\sinh^2(d_1(g,s)+1), \]
then any immersed virtual fiber in $M$ of genus $g$ is embedded.
Theorem <ref> in tandem with the above conditions on
$\mathrm{InjRad}(M_i),\mathrm{Vol}(M)$ implies that there is $i_{g,s}
\in\mathbf{N}$ such that if $i \geq i_{g,s}$, the virtual fiber group
$\Delta_0$ corresponds to an embedded incompressible surface
of genus $g$ in $M_i$. This incompressible surface greatly limits the structural
possibilities for the manifolds $M_i$. Specifically, $M_i$ is either a
fiber bundle over the circle with fiber group $\Delta_0$, or $M_i$ is
the union of two twisted $I$–bundles. Moreover, in the latter case,
we have a double cover $N_i\rightarrow M_i$ such that $N_i$ is a fiber
bundle with fiber group $\Delta_0$ (see <cit.>).
We now leverage the above fiber bundle structure to obtain bounds for the rank of $\Gamma_i$ for $i$ sufficiently large. The rank of $\Gamma_i$ will be denoted by $\mathrm{Rank}(\Gamma_i)$.
There exists $i_0 \geq i_{g,s}$ such that if $i \geq i_0$, then $g +1 \leq \mathrm{Rank}(\Gamma_i) \leq 2g+2$.
Proof: We assume throughout that $i \geq i_{g,s}$. Set
$\mathcal{I}_1$ to be the set of $i \geq i_{g,s}$ such that $M_i$ is a
fiber bundle with fiber group $\Delta_0$ and set $\mathcal{I}_2$ to be
the set of $i \geq i_{g,s}$ such that $M_i$ is double covered by $N_i$
where $N_i$ is a fiber bundle with fiber group $\Delta_0$. We first
consider $\set{M_i}_{i \in \mathcal{I}_1}$. We know from the proof of
the commensurability invariance of $\mathrm{S}_{vf}$ that each $M_i$
must have the form $M_{\phi^{r_i}}$ for some pseudo-Anosov element
$\phi$. Applying Souto <cit.>, there exist $i' \in
\mathbf{N}$ such that $\mathrm{Rank}(\Gamma_i)=2g+1$ for all $i \geq
i'$. Next, we consider $\set{M_i}_{i \in \mathcal{I}_2}$ and apply the
above argument to $N_i$. We obtain $i'' \in \mathbf{N}$ such that
$\mathrm{Rank}(\pi_1(N_i)) = 2g+1$ for all $i \geq i''$. As $N_i$ is a
double cover of $M_i$, we can adjoin one element to $\pi_1(N_i)$ to
generate $\pi_1(M_i)$. Therefore, $\mathrm{Rank}(\Gamma_i) \leq 2g+2$
for all $i \geq i''$.
Now, set $i_0 = \max\set{i',i''}$ and note that
$\mathrm{Rank}(\Gamma_i) \leq 2g+2$ for all $i \geq i_0$. For the
lower bound, by the Nielsen–Schreier, we have
$\mathrm{Rank}(\pi_1(N_i)) \leq 2\mathrm{Rank}(\Gamma_i) - 1$ for all
$i \geq i_0$ and $i \in \mathcal{I}_2$. In particular, $g +1 \leq
\mathrm{Rank}(\Gamma_i)$ for all $i \geq
We now use Lemma <ref> to complete the proof of Theorem
<ref>. By Theorem <ref> (c), we can find
incommensurable virtual fiber subgroups of arbitrarily large
genus. Choosing a virtual fiber subgroup $\Delta_1$ of genus $g_1$
with $2g+2 < g_1 + 1$ and repeating the above argument, we obtain an
integer $i_1 \geq i_{g_1,s}$ such that $g_1 + 1 \leq
\mathrm{Rank}(\Gamma_i)$ for all $i \geq i_1$. For all $i \geq
\max\set{i_0,i_1}$, we must have $g_1 + 1 \leq \mathrm{Rank}(\Gamma_i)
\leq 2g+ 2 < g_1 +1$, a contradiction. Hence $\mathcal{S}_{M,vf}$ is
finite as required.
Remarks:(1) In the proof of Lemma <ref> we
could also have used <cit.> for both the bundle case and the union
of two twisted I-bundles. However, the setting of <cit.> is more
appropriate in this case (i.e. commensurable manifolds), and only a
mild extension is needed for us to handle the union of two twisted
I-bundles. Hence the reason for not using <cit.> in this case. In
4, we will need to use <cit.>.
(2) As noted in the introduction, we do not know if there
exists a pair of non-isometric, closed, orientable, hyperbolic
3–manifolds $M_1,M_2$ with $\mathcal{S}(M_1) =
\mathcal{S}(M_2)$. Since being either a virtual fiber or Fuchsian
depends only on $\Delta_\rho$ and not the ambient manifold, such a
pair would also satisfy both $\mathrm{S}_{vf}(M_1) =
\mathrm{S}_{vf}(M_2)$, $\mathrm{S}_{Fuc}(M_1) =
\mathrm{S}_{Fuc}(M_2)$. Examples where the latter equality holds were
constructed in <cit.> using a variation of the
method of Sunada <cit.> for
constructing isospectral and length isospectral manifolds. That method
does not seem well-suited for also arranging equality between virtual
fibers. As with the full spectrum, we do not presently know if there
exists a pair of non-isometric, closed, hyperbolic 3–manifolds
$M_1,M_2$ with $\mathrm{S}_{vf}(M_1) = \mathrm{S}_{vf}(M_2)$.
§ A CONJECTURAL STRENGTHENING FOR ARITHMETIC HYPERBOLIC 3–MANIFOLDS
In this section we deal with closed, arithmetic, hyperbolic 3–manifolds, and prove a stronger result (conjecturally) that involves only topological data. We refer the reader to <cit.> for background on arithmetic hyperbolic 3–manifolds. Let us define the topological virtual fiber set of $M$ to be the set
\[ \mathrm{S}_{tvf}(M) = \{\Delta~:\Delta~\hbox{is isomorphic to a virtual fiber subgroup}\}. \]
Our strengthening relies on the following conjecture often referred to as the short geodesic conjecture.
Let $M$ be a closed, orientable, arithmetic, hyperbolic 3–manifold. Then there is a constant $C>0$ (independent of $M$) so that the length of the shortest closed geodesic in $M$ is at least $C$.
Assuming this conjecture, we establish the following result.
Assuming Conjecture <ref> there are at most finitely many closed orientable arithmetic hyperbolic 3–manifolds $M_1, M_2\ldots M_n$ so that $\mathrm{S}_{tvf}(M_i)=\mathrm{S}_{tvf}(M_j)$ for each $i,j$.
Proof: The proof of Theorem <ref> is similar to the proof of Theorem <ref> and is done by contradiction. If there is an infinite sequence of such manifolds $M_i$, by Borel <cit.> their volumes are unbounded and Conjecture <ref> implies that the injectivity radii are bounded from below. Choosing a minimal genus (topological) virtual fiber in each $M_i$ and applying Theorem <ref>, it follows that for sufficiently large $i$, $M_i$ is either a genus $g$ fiber bundle or a union of two twisted $I$–bundles which is double covered by a genus $g$ fiber bundle. We now apply
Biringer's extension of <cit.>, namely <cit.>.
This allows us to
get control of the rank as in the proof of Lemma <ref>, and
in particular, following the argument
in the proof of Lemma <ref> leads to a similar
contradiction on ranks.
|
1511.00184
|
Pointer Race Freedom
FIT BUT Technical Report Series
Frédéric Haziza, Lukáš Holík, Roland Meyer, and Sebastian Wolff
Technical Report No. FIT-TR-2015-05
Faculty of Information Technology, Brno University of Technology
Last modified: September 12, 2025
We propose a novel notion of pointer race for concurrent programs manipulating a shared heap.
A pointer race is an access to a memory address which was freed, and it is out of the accessor's control whether or not the cell has been re-allocated.
We establish two results.
(1) Under the assumption of pointer race freedom,
it is sound to verify a program running under explicit memory management as if it was running with garbage collection.
Even the requirement of pointer race freedom itself can be verified under the garbage-collected semantics.
We then prove analogues of the theorems for a stronger notion of pointer race
needed to cope with performance-critical code purposely using racy comparisons and even racy dereferences of pointers.
As a practical contribution, we apply our results to optimize a thread-modular analysis under explicit memory management.
Our experiments confirm a speed-up of up to two orders of magnitude.
§ INTRODUCTION
Today, one of the main challenges in verification is the analysis of concurrent programs that manipulate a shared heap.
The numerous interleavings among the threads make it hard to predict the dynamic evolution of the heap.
This is even more true if explicit memory management has to be taken into account.
With garbage collection as in Java, an allocation request results in a fresh address that was not being pointed to.
The address is hence known to be owned by the allocating thread.
With explicit memory management as in C, this ownership guarantee does not hold.
An address may be re-allocated as soon as it has been freed, even if there are still pointers to it.
This missing ownership significantly complicates reasoning against the memory-managed semantics.
In the present paper, we carefully investigate the relationship between the memory-managed semantics and the garbage-collected semantics.
We show that the difference only becomes apparent if there are programming errors of a particular form that we refer to as pointer races.
A pointer race is a situation where a thread uses a pointer that has been freed before.
We establish two theorems.
First, if the memory-managed semantics is free from pointer races, then it coincides with the garbage-collected semantics.
Second, whether or not the memory-managed semantics contains a pointer race can be checked with the garbage-collected semantics.
The developed semantic understanding helps to optimize program analyses.
We show that the more complicated verification of the memory-managed semantics can often be reduced to an analysis of the simpler garbage-collected semantics — by applying the following policy:
check under garbage collection whether the program is pointer race free.
If there are pointer races, tell the programmer about these potential bugs.
If there are no pointer races, rely on the garbage-collected semantics in all further analyses.
In thread-modular reasoning,
one of the motivations for our work,
restricting to the garbage-collected semantics allows us to use a smaller abstract domain
and an optimized fixed point computation.
Particularly, it removes the need to correlate the local states of threads,
and it restricts the possibilities of how threads can .
We illustrate the idea of pointer race freedom on Treiber's stack <cit.>, a lock-free implementation of a concurrent stack that provides the following methods:
$//$ global variables: $\Topp$
$\mathit{void}: \mathit{push}(v)$
\begin{align}
\quad\,
& \dsel{\node}:=v;\\
& \mathtt{repeat}\\
&\quad \topp:= \Topp;\\
&\quad \psel{\node}:=\topp;\\
&\mathtt{until}\ \mathtt{cas}(\Topp, \topp, \node);
\end{align}
$\mathit{bool}: \mathit{pop}(\&v)$
\begin{align}
\qquad
&\quad \topp := \Topp;\\
&\quad \mathtt{if}\ (\topp = \mathtt{null})\ \mathtt{return}\ \mathit{false};\\
&\quad \node := \psel{\topp};\label{popsprf}\\
&\mathtt{until}\ \cas(\Topp, \topp, \node);\\
&v := \topp.data;\\
&\freeof{\topp};\ \mathtt{return}\ \mathit{true};
\end{align}
This code is correct (i.e. linearizable and pops return the latest value pushed) in the presence of garbage collection, but it is incorrect under explicit memory management.
The memory-managed semantics suffers from a problem known as ABA, which indeed is related to a pointer race.
The problem arises as follows.
Some thread $\athread$ executing pop
sets its local variable $\topp$ to the global top of the stack $\Topp$, say address $\anadr$.
The variable $\node$ is assigned the second topmost address $\anadrp$.
While $\athread$ executes pop, another thread frees address $\anadr$ with a pop.
Since it has been freed, address $\anadr$ can be re-allocated and pushed, becoming the top of the stack again.
However, the stack might have grown in between the free and the re-allocation.
As a consequence, $\anadrp$ is no longer the second node from the top.
Thread $\athread$ now executes the $\cas$ (atomic compare-and-swap).
The command first tests $\Topp = \topp$ (to check for consistency of the program state: has the top of the stack moved?).
The test passes since $\Topp$ has come back to $\anadr$ due to the re-allocation.
Thread $\athread$ then redirects $\Topp$ to $\node$.
This is a pointer race: $\athread$ relies on the variable $\topp$ where the address was freed, and the re-allocation was not under $\athread$'s control.
At the same time, this causes an error.
If $\node$ no longer points to the second address from the top, moving $\Topp$ loses stack content.
Performance-critical implementations often intentionally make use of pointer races and employ other mechanisms to protect themselves from harmful effects due to accidental re-allocations.
The corrected version of Treiber's stack <cit.> for example equips every pointer with a version counter logging the updates.
Pointer assignments then assign the address together with the value of the associated version counter,
and the counters are taken into account in the comparisons within $\cas$.
That is, the $\cas(\Topp, \topp, \node)$ command atomically executes the following code:
\begin{align*}
&\mathtt{if}\ (\Topp = \topp\ \wedge\ \Topp.\mathtt{version} = \topp.\mathtt{version})\ \{\\
&\quad\Topp:=\node;\ \Topp.\mathtt{version}:=\topp.\mathtt{version}+1;\ \mathtt{return}\ \mathit{true}; \\
&\}\ else\ \{\ \mathtt{return}\ \mathit{false};\ \}
\end{align*}
This makes the $\cas$ from Example <ref> fail and prevents stack corruption.
Another pointer race occurs when the $\mathtt{pop}$ in Line (<ref>) dereferences a freed pointer.
With version counters, this is harmless.
Our basic theory, however, would consider the comparison as well as the dereference pointer races, deeming the corrected version of Treiber's stack buggy.
To cope with performance-critical applications
that implement version counters or techniques such as
hazard pointers <cit.>, reference counting <cit.>, or grace periods <cit.>, we strengthen the notion of pointer race.
We let it tolerate assertions on freed pointers and dereferences of freed pointers where the value obtained by the dereference does not visibly influence the computation
(e.g., it is assigned to a dead variable).
To analyse programs that are only free from strong pointer races, the garbage-collected semantics is no longer sufficient.
We define a more general ownership-respecting semantics by imposing an ownership discipline on top of the memory-managed semantics.
With this semantics, we are able to show the following analogues of the above results.
First, if the program is free from strong pointer races (SPRF) under the memory-managed semantics, then the memory-managed semantics coincides with the ownership-respecting semantics.
Second, the memory-managed semantics is SPRF if and only if the ownership-respecting semantics is SPRF.
As a last contribution, we show how to apply our theory to optimize thread-modular reasoning.
The idea of thread-modular analysis is to buy efficiency by abstracting from the relationship between the local states of individual threads.
The loss of precision, however, is often too severe.
For instance, any inductive invariant strong enough to show memory safety of Treiber's stack must correlate the local states of threads.
Thread-modular analyses must compensate this loss of precision.
Under garbage collection, an efficient way used e.g. in <cit.> is keeping as a part of the local state of each thread information about the ownership of memory addresses.
A thread owns an allocated address.
No other thread can access it until it enters the shared part of the heap.
Unfortunately, this exclusivity cannot be guaranteed under the memory-managed semantics.
Addresses can be re-allocated with pointers of other threads still pointing to them.
Works such as <cit.> therefore correlate the local states of threads by more expensive means (cf. Section <ref>), for which they pay by severely decreased scalability.
We apply our theory to put back ownership information into thread-modular reasoning under explicit memory management.
We measure the impact of our technique on the method of <cit.> when used to prove linearizability of programs such as Treiber's stack or Michael & Scott's lock-free queue under explicit memory management.
We report on resource savings of about two orders of magnitude.
We claim the following contributions, where $\mmsem{\aprog}$ denotes the memory-managed semantics, $\resmmsem{\aprog}$ the ownership-respecting semantics, and $\gcsem{\aprog}$ the garbage-collected semantics of program $\aprog$.
(1) We define a notion of pointer race freedom (PRF) and an equivalence $\heapequiv$ among computations such that the following two results hold.
(1.1) If $\mmsem{\aprog}$ is PRF, then $\mmsem{\aprog}\heapequiv\gcsem{\aprog}$.
(1.2) $\mmsem{\aprog}$ is PRF if and only if $\gcsem{\aprog}$ is PRF.
(2) We define a notion of strong pointer race freedom (SPRF) and an ownership-respecting semantics $\resmmsem{\aprog}$ such that the following two results hold.
(2.1) If $\mmsem{\aprog}$ is SPRF, then $\mmsem{\aprog}=\resmmsem{\aprog}$.
(2.2) $\mmsem{\aprog}$ is SPRF if and only if $\resmmsem\aprog$ is SPRF.
Using the Results (2.1) and (2.2), we optimize the recent thread-modular analysis <cit.> by a use of ownership and report on an experimental evaluation.
The Results (2.1) and (2.2) give less guarantees than (1.1) and (1.2) and hence allow for less simplifications of program analyses.
On the other hand, the stronger notion of pointer race makes (2.1) and (2.2) applicable to a wider class of programs which would be racy in the original sense (which is the case for our most challenging benchmarks).
Finally, we note that our results are not only relevant for concurrent programs but apply to sequential programs as well.
The point in the definition of pointer race freedom is to guarantee the following: the execution does not depend on whether a malloc has re-allocated an address, possibly with other pointers still pointing to it, or it has allocated a fresh address. However, it is mainly reasoning about concurrent programs where we see a motivation to strive for such guarantees.
Related Work
Our work was inspired by the data race freedom (DRF) guarantee <cit.>.
The DRF guarantee can be understood as a contract between hardware architects and programming language designers.
If the program is DRF under sequential consistency (SC), then the semantics on the actual architecture will coincide with SC.
We split the analogue of the statement into two, coincidence ($\mmsem{\aprog}$ PRF implies $\mmsem{\aprog}\heapequiv\gcsem{\aprog}$) and means of checking ($\mmsem{\aprog}$ PRF iff $\gcsem{\aprog}$ PRF).
There are works that weaken the DRF requirement while still admitting efficient analyses <cit.>.
Our notion of strong pointer races is along this line.
The closest related work is <cit.>.
Gotsman et al. study re-allocation under explicit memory management.
The authors focus on lock-free data structures implemented with hazard pointers, read-copy-update, or epoch-based reclamation.
The key observation is that all three techniques rely on a common synchronization pattern called grace periods.
Within a grace period of a cell $\anadr$ and a thread $\athread$, the thread can safely access the cell without having to fear a free command.
The authors give thread-modular reasoning principles for grace periods and show that they lead to elegant and scalable proofs.
The relationship with our work is as follows.
If grace periods are respected, then the program is guaranteed to be SPRF (there are equality checks on freed addresses).
Hence, using Theorem <ref> in this work,
it is sufficient to verify lock-free algorithms under the ownership-respecting semantics.
Interestingly, Gotsman et al. had an intuitive idea of pointer races without making the notion precise (quote: ...potentially harmful race between threads accessing nodes and those trying to reclaim them is avoided <cit.>).
Moreover, they did not study the implications of race freedom on the semantics, which is the main interest of this paper.
We stress that our approach does not make assumptions about the synchronization strategy.
Finally, Gotsman et al. do not consider the problem of checking the synchronization scheme required by grace periods.
We show that PRF and SPRF can actually be checked on simpler semantics.
Data refinement in the presence of low-level memory operation is studied in <cit.>.
The work defines a notion of substitutability that only requires a refinement of error-free computations.
In particular, there is no need to refine computations that dereference dangling pointers.
In our terms, these dereferences yield pointer races.
We consider <cit.> as supporting our requirement for (S)PRF.
The practical motivation of our work,
thread-modular analysis <cit.>, has already been discussed.
We note the adaptation to heap-manipulating programs <cit.>.
Interesting is also the combination with separation logic from <cit.> (which uses ownership to improve precision).
There are other works studying shape analysis and thread-modular analysis.
As these fields are only a part of the motivation, we do not provide a full overview.
§ HEAP-MANIPULATING PROGRAMS
§.§.§ Syntax
We consider concurrent heap-manipulating programs, defined to be sets of threads $\aprog = \set{\athread_1, \athread_2, \ldots}$ from a set $\threads$.
We do not assume finiteness of programs.
This ensures our results carry over to programs with a parametric number of threads.
Threads $\athread$ are ordinary while-programs operating on data and pointer variables.
Data variables are denoted by $\advar, \advarp\in\dvars$.
For pointer variables, we use $\apvar, \apvarp\in\pvars$.
We assume $\dvars\cap \pvars = \emptyset$ and obey this typing.
Pointer variables come with selectors $\pselarg{\apvar}{1},\ldots, \pselarg{\apvar}{n}$ and $\dsel{\apvar}$ for finitely many pointer fields and one data field (for simplicity; the generalization to arbitrary data fields is straightforward).
We use $\apt$ to refer to pointers $\apvar$ and $\psel{\apvar}$.
Similarly, by $\adt$ we mean data variables $\advar$ and the corresponding selectors $\dsel{\apvar}$.
Pointer and data variables are either local to a thread, indicated by $\apvar,\advar\in\localof{\athread}$, or they are shared among the threads in the program.
We use $\shared$ for the set of all shared variables.
The commands $\acom\in \coms$ employed in our while-language are
\begin{align*}
\acond&::=\phantom{\bnf} \apvar = \apvarp\bnf \advar = \advarp\bnf \neg \acond\\
\acom&::=\phantom{\bnf} \assert\ \acond \bnf \apvar:= \malloc \bnf \freeof{\apvar}\\
&\phantom{::=}\bnf \apvarp := \psel{\apvar}\bnf \psel{\apvar}:=\apvarp\bnf \apvar:=\apvarp\\
&\phantom{::=}\bnf \advar:=\dsel{\apvar}\bnf
\dsel{\apvar}:=\advar\bnf \advar:=\opof{\advar_1,\ldots, \advar_n}\ .
\end{align*}
Pointer variables are allocated with $\apvar := \malloc$ and freed via $\freeof{\apvar}$.
Pointers and data variables can be used in assignments.
These assignments are subject to typing: we only assign pointers to pointers and data to data.
Moreover, a thread only uses shared variables and its own local variables.
To compute on data variables, we support operations $\op$ that are not specified further.
We only assume that the program comes with a data domain $(\dom, \ops)$ so that its operations $\op$ stem from $\ops$.
We support assertions that depend on equalities and inequalities among pointers and data variables.
Like in if and while commands, we require assertions to have complements: if a control location has a command $\assert\ \acond$, then it also has a command $\assert\ \neg \acond$.
We use as a running example the program in Example <ref>, Treiber's stack <cit.>.
§.§.§ Semantics
A heap is defined over a set of addresses $\adr$ that contains the distinguished element $\segval$.
Value $\segval$ indicates that a pointer has not yet been assigned a cell and thus its data and next selectors cannot be accessed.
Such an access would result in a segfault.
A heap gives the valuation of pointer variables $\pvars\nrightarrow \adr$,
the valuation of the next selector functions $\adr\nrightarrow \adr$, the valuation of the data variables $\dvars\nrightarrow \dom$, and the valuation of the data selector fields $\adr\nrightarrow \dom$.
In Section <ref>, we will restrict heaps to a subset of so-called valid pointers.
To handle such restrictions, it is convenient to let heaps evaluate expressions $\psel{\anadr}$ rather than next functions.
Moreover, with the use of restrictions valuation functions will typically be partial.
Let $\pexp:=\pvars\disunion \setcond{\psel{\anadr}}{\anadr\in\adr\setminus\set{\segval}\text{ and } \mathtt{next}\text{ a selector}}$ be the set of pointer expressions and $\dexp:=\dvars\disunion \setcond{\dsel{\anadr}}{\anadr\in\adr\setminus\set{\segval}}$ be the set of data expressions.
A heap is a pair
$\aheap = (\apval, \adval)$
with $\apval:\pexp\nrightarrow \adr$ the valuation of the pointer expressions and $\adval:\dexp\nrightarrow \dom$ the valuation of the data expressions.
We use $\apexp$ and $\adexp$ for a pointer and a data expression, and also write $\aheap(\apexp)$ or $\aheap(\adexp)$.
The valuation functions are clear from the expression.
The addresses inside the heap that are actually in use are
\begin{align*}
\adrof{\aheap}:=(\domof{\apval}\cup \rangeof{\apval}\cup \domof{\adval})\cap \adr.
\end{align*}
Here, we use $\set{\psel{\anadr}}\cap \adr :=\set{\anadr}$ and similar for data selectors.
We model heap modifications with updates $[\apexp\mapsto \anadr]$ and $[\adexp\mapsto \advalue]$ from the set $\updates$.
Update $[\apexp\mapsto \anadr]$ turns the partial function $\apval$ into the new partial function $\apval[\apexp\mapsto \anadr]$ with $\domof{\apval[\apexp\mapsto\anadr]}:=\domof{\apval}\cup\set{\apexp}$.
It is defined by $\apval[\apexp\mapsto\anadr](\apexpp):=\apval(\apexpp)$ if $\apexpp\neq \apexp$, and $\apval[\apexp\mapsto\anadr](\apexp):=\anadr$.
We also write $\aheap[\apexp\mapsto \anadr]$ since the valuation that is altered is clear from the update.
We define three semantics for concurrent heap-manipulating programs.
All three are in terms of computations, sequences of actions from
$\actions\ :=\ \threads\times \coms \times \updates$.
An action $\anact = (\athread, \acom, \anup)$ consist of a thread $\athread$, a command $\acom$ executed in the thread, and an update $\anup$.
By $\threadof{\anact}:=\athread$, $\comof{\anact}:=\acom$, and $\updateof{\anact}:=\anup$ we access the thread, the command, and the update in $\anact$.
To make the heap resulting from a computation $\tau\in\actions^*$ explicit, we define $\heapcomput{\varepsilon}:=(\emptyset, \emptyset)$ and
$\heapcomput{\tau.\anact} := \heapcomput{\tau}[\updateof{\anact}]$.
So we modify the current heap with the update required by the last action.
The garbage-collected semantics and the memory-managed semantics only differ on allocations.
We define a strict form of garbage collection that never re-allocates a cell.
With this, we do not have to define unreachable parts of the heap that should be garbage collected.
We only model computations that are free from segfaults.
This means a transition accessing next and data selectors is enabled only if the corresponding pointer is assigned a cell.
Formally, the garbage-collected semantics of a program $\aprog$, denoted by $\gcsem{\aprog}$, is a set of computations in $\actions^*$.
The definition is inductive.
In the base case, we have single actions $(\bot, \bot, [\apval, \adval])\in\gcsem{\aprog}$ with
$\apval:\pvars\rightarrow\set{\segval}$ and
$\adval:\dvars\rightarrow \dom$ arbitrary.
No pointer variable is mapped to a cell and the data variables contain arbitrary values.
In the induction step, consider $\tau\in\gcsem{\aprog}$ where thread $\athread$ is ready to execute command $\acom$.
Then $\tau.(\athread, \acom, \anup)\in\gcsem{\aprog}$, provided one of the following rules holds.
(Asgn) Let $\acom$ be $\psel{\apvar}:=\apvarp$, $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$, $\heapcomputof{\tau}{\apvarp}=\anadrp$.
We set $\anup=[\psel{\anadr}\mapsto\anadrp]$.
The remaining assignments are similar.
(Asrt) Let $\acom$ be $\assert\ \apvar=\apvarp$.
The precondition is $\heapcomputof{\tau}{\apvar}=\heapcomputof{\tau}{\apvarp}$.
There are no updates, $\anup=\emptyset$.
The assertion with a negated condition is defined analogously.
A special case occurs if $\heapcomputof{\tau}{\apvar}$ or $\heapcomputof{\tau}{\apvarp}$ is $\segval$.
Then the assert and its negation will pass.
Intuitively, undefined pointers hold arbitrary values.
Our development does not depend on this modeling choice.
(Free) If $\acom$ is $\freeof{\apvar}$, there are no constraints and no updates.
(Malloc1) Let $\acom$ be $\apvar:=\malloc$, $\anadr\notin\adrof{\heapcomput{\tau}}$, and $\advalue\in \dom$.
Then we define $\anup=[\apvar\mapsto \anadr, \dsel{\anadr}\mapsto\advalue,\setcond{\psel{\anadr}\mapsto\segval}{\text{for every selector $\mathtt{next}$}}]$. The rule only allocates cells that have not been used in the computation.
Such a cell holds an arbitrary data value and the next selectors have not yet been allocated.
With explicit memory management, we can re-allocate a cell as soon as it has been freed.
Formally, the memory-managed semantics $\mmsem{\aprog}\subseteq \actions^*$ is defined like $\gcsem{\aprog}$ but has a second allocation rule:
(Malloc2) Let $\acom$ be $\apvar:=\malloc$ and $\anadr\in \freedof{\tau}$.
Then $\anup=[\apvar\mapsto \anadr]$.
Note that (Malloc2) does not alter the selectors of address $\anadr$.
The set $\freedof{\tau}$ contains the addresses that have been allocated in $\tau$ and freed afterwards.
The definition is by induction.
In the base case, we have $\freedof{\varepsilon}:= \emptyset$.
The step case is
\begin{align*}
\freedof{\tau.(\athread, \freeof{\apvar}, \anup)}&:=\freedof{\tau}\cup \set{\anadr},&&\text{if $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$}\\
\freedof{\tau.(\athread, \apvar:=\malloc, \anup)}&:=\freedof{\tau}\setminus \set{\anadr},&&\text{if $\malloc$ returns $\anadr$}\\
\freedof{\tau.(\athread, \anact, \anup)}&:=\freedof{\tau},&&\text{otherwise}.
\end{align*}
§ POINTER RACE FREEDOM
We show that for well-behaved programs the garbage-collected semantics coincides with the memory-managed semantics.
Well-behaved means there is no computation where one pointer frees a cell and later a dangling pointer accesses this cell.
We call such a situation a pointer race, referring to the fact that the free command and the access are not synchronized, for otherwise the access should have been avoided.
To apply this equivalence, we continue to show how to reduce the check for pointer race freedom itself to the garbage-collected semantics.
§.§ PRF Guarantee
The definition of pointer races relies on a notion of validity for pointer expressions.
To capture the situation sketched above, a pointer is invalidated if the cell it points to is freed.
A pointer race is now an access to an invalid pointer.
The definition of validity requires care when we pass pointers.
Consider an assignment $\apvar:=\psel{\apvarp}$ where $\apvarp$ points to $\anadr$ and $\psel{\anadr}$ points to $\anadrp$.
Then $\apvar$ becomes a valid pointer to $\anadrp$ only if both $\apvarp$ and $\psel{\anadr}$ were valid.
In Definition <ref>, we use $\apexp$ to uniformly refer to $\apvar$ and $\psel{\anadr}$ on the left-hand side of assignments.
In particular, we evaluate pointer variables $\apvar$ to $\heapcomputof{\tau}{\apvar}=\anadr$ and write $\psel{\anadr}:=\apvarp$ for the assignment $\psel{\apvar}:=\apvarp$.
The valid pointer expressions in a computation $\tau\in\mmsem{\aprog}$, denoted by $\validof{\tau}\subseteq \pexp$, are defined inductively by $\validof{\varepsilon}:= \pexp$ and
\begin{align*}
\validof{\tau.(\athread, \apvar:=\psel{\apvarp}, \anup)} &:= \validof{\tau}\cup\set{\apvar},&&\text{if }\apvarp\in \validof{\tau}\wedge \psel{\heapcomputof{\tau}{\apvarp}}\in\validof{\tau}\\
\validof{\tau.(\athread, \apvar:=\psel{\apvarp}, \anup)} &:= \validof{\tau}\setminus\set{\apvar},&&\text{if }
\apvarp\notin \validof{\tau}\vee \psel{\heapcomputof{\tau}{\apvarp}}\notin\validof{\tau}\\
\validof{\tau.(\athread, \apexp:=\apvarp, \anup)} &:= \validof{\tau}\cup\set{\apexp},&&\text{if }\apvarp\in \validof{\tau}\\
\validof{\tau.(\athread, \apexp:=\apvarp, \anup)} &:= \validof{\tau}\setminus\set{\apexp},&&\text{if }\apvarp\notin \validof{\tau}\\
\validof{\tau.(\athread, \freeof{\apvar}, \anup)}&:=\validof{\tau}\setminus \invalidof{\anadr},&&\text{if }\anadr=\heapcomputof{\tau}{\apvar}\\
\validof{\tau.(\athread, \apvar:=\malloc, \anup)}&:=\validof{\tau}\cup\set{\apvar},\\
\validof{\tau.(\athread, \anact, \anup)}&:=\validof{\tau},&&\text{otherwise.}
\end{align*}
If $\anadr\neq \segval$, then $\invalidof{\anadr}:=\setcond{\apexp}{\heapcomputof{\tau}{\apexp}=\anadr}\cup \set{\pselarg{\anadr}{1},\ldots, \pselarg{\anadr}{n}}$.
If $\anadr= \segval$, then $\invalidof{\anadr}:=\emptyset$.
When we pass a valid pointer, this validates the receiver (adds it to $\validof\tau$).
When we pass an invalid pointer, this invalidates the receiver.
As a result, only some selectors of an address may be valid.
When we free an address $\anadr\neq \segval$, all expressions that point to $\anadr$ as well as all next selectors of $\anadr$ become invalid.
This has the effect of isolating $\anadr$ so that the address behaves like a fresh one for valid pointers.
A malloc validates the respective pointer but does not validate the next selectors of the allocated address.
A computation $\tau.(\athread, \acom, \anup)\in\mmsem{\aprog}$ is called a pointer race (PR), if $\acom$ is
(i) a command containing $\dsel{\apvar}$ or $\psel{\apvar}$ or $\freeof{\apvar}$ with $\apvar\notin \validof{\tau}$, or
(ii) an assertion containing $\apvar\notin \validof{\tau}$.
The last action of a PR is said to raise a PR.
A set of computations is pointer race free (PRF) if it does not contain a PR.
In Example <ref>,
the discussed comparison $\Topp=\topp$ within $\cas$ raises a PR since $\topp$ is invalid.
It is worth noting that we can still pass around freed addresses without raising a PR.
This means the memory-managed and the garbage-collected semantics will not yield isomorphic heaps, but only yield isomorphic heaps on the valid pointers.
We now define the notion of isomorphism among heaps $\aheap$.
A function $\funa:\adrof{\aheap}\rightarrow \adr$ is an address mapping, if $\funa(\anadr)=\segval$ if and only if $\anadr=\segval$.
Every address mapping induces a function $\fune:\domof{\aheap}\rightarrow\pexp\cup \dexp$ on the pointer and data expressions inside the heap by
() := () :=
() :=()
() :=().
Pointer and data variables are mapped identically.
Pointers on the heap $\psel{\anadr}$ are mapped to $\psel{\funa(\anadr)}$ as defined by the address mapping, and similar for the data.
Two heaps $\aheap_1$ and $\aheap_2$ with $\aheap_i=(\apval_i, \adval_i)$ are isomorphic,
denoted by $\aheap_1\heapiso \aheap_2$, if there is a bijective address mapping
$\anisoa: \adrof{\aheap_1}\rightarrow \adrof{\aheap_2}$ where the induced
$\anisoe: \domof{\aheap_1}\rightarrow \domof{\aheap_2}$ is again bijective and satisfies
(_1()) = _2(()) _1() = _2(()).
To prove a correspondence between the two semantics, we restrict heaps to the valid pointers.
The restriction operation keeps the data selectors for all addresses that remain.
To be more precise, let $\aheap=(\apval, \adval)$ and $P\subseteq \pexp$.
The restriction of $\aheap$ to $P$ is the new heap
$\restrict{\aheap}{P}:=(\restrict{\apval}{P}, \restrict{\adval}{D})$ with
\begin{align*}
D:=\dvars\cup \setcond{\dsel{\anadr}}{\exists \apexp\in \domof{\apval}\cap P:\apval(\apexp)=\anadr}\ .
\end{align*}
Restriction and update enjoy a pleasant interplay with isomorphism.
Let $\aheap_1\heapiso \aheap_2$ via $\anisoa$ and let $P\subseteq \pexp$.
\begin{align}
\restrict{\aheap_1}{P}&\heapiso \restrict{\aheap_2}{\anisoe(P)}\label{Equation:HeapIsoRestrict}\\
\aheap_1[\psel{\anadr}\mapsto\anadrp]&\heapiso \aheap_2[\psel{\anadr'}\mapsto \anadrp']\label{Equation:HeapIsoModifyPointer}\\
\aheap_1[\dsel{\anadr}\mapsto d]&\heapiso \aheap_2[\dsel{\anadr'}\mapsto d]\label{Equation:HeapIsoModifyData}.
\end{align}
Isomorphisms (<ref>) and (<ref>) have a side condition.
If $\anadr\in\adrof{\aheap_1}$ then $\anadr'=\anisoa(\anadr)$.
If $\anadr\notin\adrof{\aheap_1}$ then $\anadr'\notin\adrof{\aheap_2}$,
and similar for $\anadrp$.
Two computations are heap equivalent, if their sequences of actions coincide when projected to the threads and commands, and if the resulting heaps are isomorphic on the valid part.
We use $\downarrow$ for projection.
Computations $\tau, \sigma\in\mmsem{\aprog}$ are heap-equivalent, $\tau\heapequiv \sigma$, if
\begin{align*}
\project{\tau}{\threads\times\coms}\ =\ \project{\sigma}{\threads\times\coms}\qquad
\text{and}\qquad \restrict{\heapcomput{\tau}}{\validof{\tau}}&\heapiso\ \restrict{\heapcomput{\sigma}}{\validof{\sigma}}\ .
\end{align*}
We also write $\mmsem{\aprog}\heapequiv\gcsem{\aprog}$ to state that for every computation $\tau\in\mmsem{\aprog}$, there is a computation $\sigma\in\gcsem{\aprog}$ with $\tau\heapequiv \sigma$, and vice versa.
We are now ready to state the PRF guarantee.
The idea is to consider pointer races programming errors.
If a program has pointer races, the programmer should be warned.
If the program is PRF, further analyses can rely on the garbage-collected semantics:
If $\mmsem{\aprog}$ is PRF, then $\mmsem{\aprog}\heapequiv \gcsem{\aprog}$.
The memory-managed semantics of Treiber's stack suffers from the ABA-problem while the garbage-collected semantics does not.
The two are not heap-equivalent.
By Theorem <ref>, the difference is due to a PR.
One such race is discussed in Example <ref>.
In the proof of Theorem <ref>,
the inclusion from right to left always holds.
The reverse direction needs information about the freed addresses:
if an address has been freed, it no longer occurs in the valid part of the heap —
provided the computation is PRF.
Assume $\tau\in \mmsem{\aprog}$ is PRF.
Then $\freedof{\tau}\cap \adrof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}=\emptyset$.
Lemma <ref> and <ref> allow us to prove
Proposition <ref>.
The result implies the missing direction of Theorem <ref> and will also be helpful later on.
Consider $\tau\in\mmsem{\aprog}$ PRF. Then there is $\sigma\in\gcsem{\aprog}$ with $\sigma\comequiv \tau$.
To apply Theorem <ref>, one has to prove $\mmsem{\aprog}$ PRF.
We develop a technique for this.
§.§ Checking PRF
We show that checking pointer race freedom for the memory-managed semantics can be reduced to checking pointer race freedom for the garbage-collected semantics.
The key argument is that the earliest possible PR always lie in the garbage-collected semantics.
Technically, we consider a shortest PR in the memory-managed semantics and from this construct a PR in the garbage-collected semantics.
$\mmsem{\aprog}$ is PRF if and only if $\gcsem{\aprog}$ is PRF.
To illustrate the result, the pointer race in Example <ref> belongs to the memory-managed semantics.
Under garbage collection, there is a similar computation which does not re-allocate $\anadr$.
Freeing $\anadr$ still renders $\topp$ invalid
and, as before, leads to a PR in $\cas$.
The proof of Theorem <ref> applies Proposition <ref> to mimic the shortest racy computation up to the last action.
To mimic the action that raises the PR, we need the fact that an invalid pointer variable does not hold $\segval$, as stated in the following lemma.
Consider a PRF computation $\sigma\in\gcsem{\aprog}$.
(i) If $\apvar\notin\validof{\sigma}$, then
$\heapcomputof{\sigma}{\apvar}\neq \segval$.
(ii) If $\apexp\in\validof{\sigma}$, $\heapcomputof{\sigma}{\apexp}=\anadr\neq \segval$, and $\psel{\anadr}\notin\validof{\sigma}$, then $\heapcomputof{\sigma}{\psel{\anadr}}\neq \segval$.
While the completeness proof of Theorem <ref> is non-trivial, checking PRF for $\gcsem{\aprog}$ is an easy task.
One instruments the given program $\aprog$ to a new program $\aprog'$ as follows: $\aprog'$ tags every address that is freed and checks whether a tagged address is dereferenced, freed, or used in an assertion.
In this case, $\aprog'$ enters a distinguished goal state.
$\gcsem{\aprog}$ is PRF if and only if $\gcsem{\aprog'}$ cannot reach the goal state.
For the correctness proof, we only need to observe that under garbage collection the invalid pointers are precisely the pointers to the freed cells.
Let $\sigma\in\gcsem{\aprog}$ and $\heapcomputof{\sigma}{\apexp}=\anadr\neq \segval$.
Then $\apexp\notin\validof{\sigma}$ iff $\anadr\in\freedof{\sigma}$.
The lemma does not hold for the memory-managed semantics.
Moreover, the statement turns Lemma <ref>, which can be read as an implication, into an equivalence.
Namely, Lemma <ref> says that if a pointer has been freed, then it cannot be valid.
Under the assumtpions of Lemma <ref>, it also holds that if a pointer is not valid, then it has been freed.
§ STRONG POINTER RACE FREEDOM
The programing style in which a correct program should be pointer race free
counts on the following policy:
a memory address is freed only if it is not meant to be touched until its re-allocation, by any means possible.
This simplified treatment of dynamic memory is practical in common programing tasks,
but the authors of performance-critical applications are often forced to employ subtler techniques.
For example, the version of Treiber's stack equipped with version counters to prevent ABA under explicit memory management contains two violations of the simple policy, both of which are pointer races.
(1) The $\cas$ may compare invalid pointers.
This could potentially lead to ABA, but the programmer prevents the harmful effect of re-allocation using version counters, which make the $\cas$ fail.
(2) The command $\node:=\psel\topp$ in Line (<ref>) of $\mathtt{pop}$ may dereference the $\mathtt{next}$ field of a freed (and therefore invalid) pointer.
This is actually correct only under the assumption
that neither the environment nor any thread of the program itself may redirect a once valid pointer outside the accessible memory (otherwise the dereference could lead to a segfault).
The value obtained by the dereference may again be influenced by that the address was re-allocated.
The reason for why this is fine is that the subsequent $\cas$ is bound to fail, which makes $\node$ a dead variable — its value does not matter.
In both cases, the programmer only prevents side effects of an accidental re-allocation.
He uses a subtler policy and frees an address only if
its content is not meant to be of any relevance any more.
Invalid addresses can still be compared, and their pointer fields can even be dereferenced unless the obtained value influences the control.
§.§ SPRF Guarantee
We introduce a stronger notion of pointer race that expresses the above subtler policy. In the definition, we will call strongly invalid the pointer expressions that
have obtained their value from dereferencing an invalid/freed pointer.
The set of strongly invalid expressions in $\tau\in\mmsem{\aprog}$, denoted by $\sinvalidof{\tau}\subseteq \pexp\cup \dexp$, is defined inductively by $\sinvalidof{\varepsilon}:= \emptyset$ and
\begin{align*}
\sinvalidof{\tau.(\athread, \apvar:=\psel{\apvarp}, \anup)} &:= \sinvalidof{\tau}\cup\set{\apvar},&&\text{if }\apvarp\not\in \validof{\tau}\\
\sinvalidof{\tau.(\athread, \apexp:=\apvarp, \anup)} &:= \sinvalidof{\tau}\cup\set{\apexp},&&\text{if }\apvarp\in \sinvalidof{\tau}\\
\sinvalidof{\tau.(\athread, \advar:=\dsel{\apvarp}, \anup)} &:= \sinvalidof{\tau}\cup\set{\advar},&&\text{if }\apvarp\not\in \validof{\tau}\\
\sinvalidof{\tau.(\athread, \adexp:=\advar, \anup)} &:= \sinvalidof{\tau}\cup\set{\adexp},&&\text{if }\advar\in \sinvalidof{\tau}\\
\sinvalidof{\tau.\anact} &:= \sinvalidof{\tau}\setminus\validof{\tau.\anact}, &&\text{in all other cases.}
\end{align*}
The value obtained by dereferencing a freed pointer may depend on
actions of other threads that point to the cell due to re-allocation.
However, by assuming that a once valid pointer can never be set to $\segval$,
we obtain a guarantee that the actions of other threads cannot prevent the dereference itself from being executed (they cannot make it segfault).
Assigning the uncontrolled value to a local variable is therefore not harmful.
We only wish to prevent a correct computation from being influenced by that value.
We thus define incorrect/racy any attempt to compare or dereference the value.
Then, besides allowing for the creation of strongly invalid pointers, the notion of strong pointer race strengthens PR by tolerating comparisons of invalid pointers.
A computation $\tau.(\athread, \acom, \anup)\in\mmsem{\aprog}$
is a strong pointer race (SPR), if the command $\acom$ is one of the following:
(i) $\psel{\apvar}:=\apvarp$ or $\dsel{\apvar}:=\advar$ or $\freeof{\apvar}$ with $\apvar\notin\validof{\tau}$
(ii) an assertion containing $\apvar$ or $\advar$ in $\sinvalidof\tau$
(iii) a command containing $\psel{\apvar}$ or $\dsel{\apvar}$ where $\apvar\in\sinvalidof\tau$.
The last action of an SPR raises an SPR.
A set of computations is strong pointer race free (SPRF) if it does not contain an SPR.
An SPR can be seen in Example <ref> as a continuation of the race ending at $\cas$.
The subsequent $\topp:=\free$ raises an SPR as $\topp$ is invalid.
The implementation corrected with version counters is SPRF.
Theorems <ref> and <ref> no longer hold for strong pointer race freedom.
It is not possible to verify $\mmsem\aprog$ modulo SPRF by analysing $\gcsem \aprog$.
The reason is that the garbage-collected semantics does not cover SPRF computations that compare or dereference invalid pointers.
To formulate a sound analogy of the theorems, we have to replace $\gcsem .$ by a more powerful semantics.
This, however, comes with a trade-off.
The new semantics should still be amenable to efficient thread-modular reasoning.
The idea of our new semantics $\resmmsem{\aprog}$ is to introduce the concept of ownership to the memory-managed semantics, and show that SPRF computations stick to it.
Unlike with garbage collection, we cannot use a naive notion of ownership that guarantees the owner exclusive access to an address.
This is too strong a guarantee.
In $\mmsem\aprog$,
other threads may still have access to an owned address via invalid pointers.
Instead, we design ownership such that dangling pointers are not allowed to influence the owner.
The computation will thus proceed as if the owner had allocated a fresh address.
To this end, we let a thread own an allocated address until one of the two events happen: either (1) the address is published, that is, it enters the shared part of the heap (which consists of addresses reached from shared variables by following valid pointers and of freed addresses), or
(2) the address is compromised, that is,
the owner finds out that the cell is not fresh by comparing it with an invalid pointer.
Taking away ownership in this situation is needed since the owner can now change its behavior based on the re-allocation.
The owner may also spread the information about the re-allocation among the other threads and change their behavior, too.
It can thus no longer be guaranteed that the computation will continue as if a fresh address had been allocated.
For $\tau\in\mmsem{\aprog}$ and a thread $\athread$,
we define the set of addresses owned by $\athread$, denoted by $\ownedof \athread \tau$, as $\ownedof \athread{\varepsilon}:=\emptyset$ and
\begin{align*}
\ownedof \athread{\tau.(\athread,\apvar:=\malloc,\anup)}&\!:=\!
\ownedof \athread{\tau}\!\cup\!\set{\anadr}, \!\!\!\!\!&&\text{if } \apvar\in \localof{\athread}\text{ and $\malloc$ returns $\anadr$}\\
\ownedof \athread{\tau.(\athread,\freeof{\apvar},\emptyset)}&\!:=\! \ownedof\athread{\tau}\!\setminus\! \set{\heapcomputof{\tau}{\apvar}}, \!\!\!\!\!&&\text{if }\apvar\in\validof{\tau}\\
\ownedof \athread{\tau.(\athread,\apvar:=\apvarp,[\apvar\mapsto\anadr])}&\!:=\!
\ownedof \athread{\tau}\!\setminus\!\set{\anadr}, \!\!\!\!\!&&\text{if } \apvar\in \shared\wedge\apvarp\in\validof{\tau}\\
\ownedof \athread{\tau.(\athread,\apvar:=\psel{\apvarp},[\apvar\mapsto\anadr])}&\!:=\!
\ownedof \athread{\tau}\!\setminus\!\set{\anadr}, \!\!\!\!\!&&\text{if } \apvar\in \shared\wedge\apvarp,\psel{\heapcomputof{\tau}{\apvarp}}\in\validof{\tau}\\
\ownedof \athread{\tau.(\cdot,\apvar:=\psel\apvarp,[\apvar\mapsto \anadr])}&\!:=\! \ownedof \athread{\tau}\!\setminus\!\set{\anadr}, \!\!\!\!\!&&\text{if } \heapcomputof{\tau}{\apvarp}\!\!\not\in\!\ownedof\athread\tau\wedge \apvarp,\psel{\heapcomputof{\tau}{\apvarp}}\!\!\in\!\validof{\tau}\\
\ownedof \athread{\tau.(\athread,\assert\ \apvar=\apvarp,\emptyset)}&\!:=\! \ownedof \athread{\tau}\!\setminus\!\set{\heapcomputof{\tau}{\apvar}}, \!\!\!\!\!&&\text{if } \apvar\notin\validof{\tau}\vee \apvarp\notin\validof{\tau}\\
\ownedof {\athread}{\tau.\anact} &\!:=\! \ownedof {\athread}{\tau}, \!\!\!\!\!&&\text{in all other cases}.
\end{align*}
The first four cases of losing ownership are due to publishing, the last case is due to the address being compromised by comparing with an invalid pointer.
The following lemma states the intuitive fact that an owned address cannot be pointed to by a valid shared variable or by a valid local variable of another thread, since such a configuration can be achieved only by publishing the address.
Let $\tau\in\mmsem{\aprog}$ and $\apvar\in\validof{\tau}$ with $\heapcomputof{\tau}{\apvar}\in\ownedof{\athread}{\tau}$.
Then $\apvar\in\localof{\athread}$.
We now define ownership violations as precisely those situations in which the fact that an owned address was re-allocated while an invalid pointer was still pointing to it influences the computation.
the address is freed or its content is altered due to
an access via a pointer of another thread or a shared pointer.
A computation $\tau.(\athread, \acom, \anup)\in\mmsem{\aprog}$ violates ownership, if $\acom$ is one of the following
\begin{align*}
\psel{\apvarp}:=\apvar,\quad\dsel{\apvarp}:=\advar,\quad\text{or}\quad \freeof{\apvarp},
\end{align*}
where $\heapcomputof{\tau}{\apvarp}\in\ownedof{\athread'}{\tau}$ and ($\athread'\neq \athread$ or $\apvarp\in\shared$).
The last action of a computation violating ownership is called an ownership violation
and a computation which does not violate ownership respects ownership.
We define the ownership-respecting semantics $\resmmsem \aprog$ as those computations of $\mmsem \aprog$ that respect ownership.
The following lemma shows that SPRF computations respect ownership.
If $\tau.(\athread, \acom, \anact)\in\mmsem{\aprog}$ violates ownership, then it is an SPR.
The proof of
Lemma <ref> (c.f. Appendix <ref>)
is immediate from Lemma <ref> and the definitions of ownership violation and strong pointer race.
The lemma implies the main result of this section: the memory-managed semantics coincides with the ownership-respecting semantics modulo SPRF (c.f. Appendix <ref>).
If $\mmsem{\aprog}$ is SPRF, then $\mmsem{\aprog}=\resmmsem{\aprog}$.
§.§ Checking SPRF
This section establishes that checking SPRF may be done in the ownership-respecting semantics.
In other words, if $\mmsem{\aprog}$ has an SPR, then there is also one in $\resmmsem{\aprog}$.
This result, perhaps much less intuitively expected
than the symmetrical result of Section <ref>,
is particularly useful for optimizing thread-modular analysis of lock-free programs (cf. Section <ref>).
Its proof depends on a subtle interplay of ownership and validity.
Let $\ownpof{\tau}$ be the owning pointers, pointers in $\heapcomput{\tau}$ to addresses that are owned by threads and the next fields of addresses owned by threads.
To be included in $\ownpof{\tau}$, the pointers have to be valid.
A set of pointers $O\subseteq \ownpof{\tau}$ is coherent if for all $\apexp,\apexpp\in\ownpof{\tau}$ with the same target or source address (in case of $\psel{\anadr}$ or $\dsel{\anadr}$) we have $\apexp\in O$ if and only if $\apexpp\in O$.
Lemma <ref> below establishes the following fact.
For every computation that respects ownership, there is another one that coincides with it but assigns fresh cells to some of the owning pointers.
To be more precise, given a computation $\tau\in\resmmsem{\aprog}$ and a coherent set of owning pointers $O\subseteq \ownpof{\tau}$, we can find another computation $\tau'\in\resmmsem{\aprog}$ where the resulting heap coincides with $\heapcomput{\tau}$ except for $O$.
These pointers are assigned fresh addresses.
The proof of Lemma <ref> is nontrivial and can be found in Appendix <ref>.
Consider $\tau\in\resmmsem{\aprog}$ SPRF and $O\subseteq \ownpof{\tau}$ a coherent set.
There is $\tau'\in\resmmsem{\aprog}$ and an address mapping $\funa:\adrof{O}\rightarrow \adr$ that satisfy the following:
(1) τ× = τ'× τ ⊆τ' (4)
(2) τ∖O = τ'∖(O) τ' =(τ∖O) ∪ (O) (5)
(3) ττ τ'τ'by∪
τ ∩τ'(O)=∅. (6)
In this lemma, function $\funa$ specifies the new addresses that $\tau'$ assigns to the owning expressions in $O$. These new addresses are fresh by Point (6).
Point (1) says that $\tau$ and $\tau'$ are the same up to the particular addresses they manipulate, and
Point (2) says that the reached states $\heapcomput\tau$ and $\heapcomput{\tau'}$ are the same up to the pointers touched by $\funa$.
Point (3) states that the valid pointers of $\heapcomput\tau$ stay valid or become valid $\fune$-images of the originals.
Point (5) says that also the owned pointers of $\heapcomput\tau$ remain the same or become $\fune$-images of the originals.
Finally, Point (4) says that $\heapcomput{\tau'}$ re-allocates less cells.
Lemma <ref> is a cornerstone in the proof of the main result in this section, namely that SPRF is equivalent for the memory-managed and the ownership-respecting semantics.
$\mmsem{\aprog}$ is SPRF if and only if $\resmmsem{\aprog}$ is SPRF.
If $\mmsem{\aprog}$ is SPRF, by $\resmmsem{\aprog}\subseteq \mmsem{\aprog}$ this carries over to the ownership-respecting semantics.
For the reverse direction, assume $\mmsem{\aprog}$ has an SPR.
In this case, there is a shortest computation $\tau.\anact\in\mmsem{\aprog}$ where $\anact$ raises an SPR.
In case $\tau.\anact\in\resmmsem{\aprog}$, we obtain the same SPR in the ownership-respecting semantics.
Assume $\tau.\anact\notin\resmmsem{\aprog}$.
We first argue that $\anact$ violates ownership.
By prefix closure, $\tau\in\mmsem{\aprog}$.
By minimality, $\tau$ is SPRF.
Since ownership violations are SPR by Lemma <ref>, $\tau$ does not contain any, $\tau\in \resmmsem{\aprog}$.
Hence, if $\anact$ respected ownership we could extend $\tau$ to the computation $\tau.\anact\in\resmmsem{\aprog}$ — a contradiction to our assumption.
We turn this ownership violation in the memory-managed semantics into an SPR in the ownership-respecting semantics.
To this end, we construct a new computation $\tau'.\anact'\in\resmmsem{\aprog}$ that mimics $\tau.\anact$, respects ownership, but suffers from SPR.
Since $\tau.\anact$ is an ownership violation, $\anact$ takes the form $(\athread, \acom, \anup)$ with $\acom$ being
\begin{align*}
\psel{\apvarp}:=\apvar,\quad \dsel{\apvarp}:=\advar,\quad \text{or}\quad \freeof{\apvarp}.
\end{align*}
Here, $\heapcomputof{\tau}{\apvarp}\in\ownedof{\athread'}{\tau}$ and ($\athread'\neq \athread$ or $\apvarp\in\shared$).
Since the address is owned, Lemma <ref> implies $\apvarp\notin\validof{\tau}$.
As a first step towards the new computation, we construct $\tau'$.
Let $O:=\ownpof{\tau}$ be the (coherent) set of all owning pointers in all threads (with $\apvarp\notin O$).
With this choice of $O$, we apply Lemma <ref>.
It returns $\tau'\in\resmmsem{\aprog}$ with $\project{\tau'}{\threads\times\coms}\ =\ \project{\tau}{\threads\times\coms}$ and
\begin{align*}
\restrict{\heapcomput{\tau'}}{\pexp\setminus\fune(O)}\ =\ \restrict{\heapcomput{\tau}}{\pexp\setminus O}
\quad\text{and}\quad
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}\ \heapiso\ \restrict{\heapcomput{\tau}}{\validof{\tau}}.
\end{align*}
Address $\heapcomputof{\tau'}{\apvarp}$ is not owned by any thread.
This follows from
$$\ownpof{\tau'}=(\ownpof{\tau}\setminus O)\ \cup\ \fune(O) = \fune(O)$$
and $q\not\in\fune(O)$.
Finally, $\apvarp\notin\validof{\tau'}$
by the isomorphism $\restrict{\heapcomput{\tau'}}{\validof{\tau'}}\ \heapiso\ \restrict{\heapcomput{\tau}}{\validof{\tau}}
As a last step, we mimic $\anact=(\athread, \acom, \anup)$ by an action $\anact'=(\athread, \acom, \anup')$.
If $\acom$ is $\freeof{\apvarp}$, then we free the invalid pointer $\apvarp\notin\validof{\tau'}$ and obtain an SPR in $\resmmsem{\aprog}$.
Assume $\acom$ is an assignment $\psel{\apvarp}:=\apvar$ (the case of $\dsel{\apvarp}:=\advar$ is similar).
Since $\anact$ is enabled after $\tau$ and $\heapcomputof{\tau'}{\apvarp}=\heapcomputof{\tau}{\apvarp}$, we have $\heapcomputof{\tau'}{\apvarp}\neq\segval$.
Hence, the command is also enabled after $\tau'$.
Since $\apvarp\notin\validof{\tau'}$, the assignment is again to an invalid pointer.
It is thus an SPR according by Definition <ref>.(i).
§ IMPROVING THREAD-MODULAR ANALYSES
We now describe how the theory developed so far can be used to increase the efficiency of thread-modular analyses of pointer programs under explicit memory management.
Thread-modular reasoning abstracts a program state into a set of states of individual threads.
A thread's state consists of the local state, the part of the heap reachable from the local variables, and the shared state, the heap reachable from the shared variables.
The analysis saturates the set of reachable thread states by a fixpoint computation.
Every step in this computation creates new thread states out of the existing ones by applying the following two rules.
* Sequential step: a thread's state is modified by an action of this thread.
* Interference: a state of a victim thread is changed by an action of another, interfering thread.
This is accounted for by creating combined two-threads states from existing pairs of states of the victim and the interferer thread.
The states that are combined have to agree on the shared part.
The combined state is constructed by deciding which addresses in the two local states coincide.
It is then observed how an action of the interferer changes the state of the victim within the combined state.
Pure thread-modular reasoning does not keep any information about what thread states can appear simultaneously during a computation and what identities can possibly hold between addresses of local states of threads.
This brings efficiency, but also easily leads to false positives.
To see this, consider in Treiber's stack a state $s$ of a thread that is just about to perform the $\cas$ in .
Variable $\node$ points to an address $\anadr$ allocated in the first line of , $\Topp$, $\topp$, and $\psel\node$ are at the top of the stack.
Consider an interference step where the states $s_v$ of the victim and $s_i$ of the interferer are isomorphic to $s$,
with $\node$ pointing to the newly allocated addresses $\anadr_v$ and $\anadr_i$, respectively.
Since the shared states conincide, the interference is triggered.
The combination must account for all possible equalities among the local variables.
Hence, there is a combined state with $\anadr_v = \anadr_i$, which does not occur in reality.
This is a crucial imprecision, which leads to false positives.
Namely, the interferer's $\cas$ succeeds, resulting in the new victim's state $s_v'$ with $\Topp$ on $\anadr_i$ (which is equal to $\anadr_v$).
The victim's $\cas$ then fails, and the thread continues with the commands $\topp:=\Topp;\psel\node:=\topp$.
This results in $\psel{\anadr_v}$ pointing back to $\anadr_v$, and a loss of the stack content.
Methods based on thread-modular reasoning must prevent such false positives by maintaining the necessary information about correlations of local states.
An efficient technique commonly used under garbage collection is based on ownership:
a thread's state records that $\anadr$ has just been allocated and hence no other thread can access the address, until it enters the shared state.
This is enough to prevent false positives such as the one described above.
Namely, the addresses $\anadr_i$ and $\anadr_v$ are owned by the respective threads and therefore they cannot be equal.
Interference may then safely ignore the problematic case when $\anadr_v = \anadr_i$.
Moreover, besides the increased precision, the ability to avoid interference steps due to ownership significantly improves the overall efficiency.
This technique was used for instance to prove safety (and linearizability) of Treiber's stack and other subtle lock-free algorithms in <cit.>.
Under explicit memory management, ownership of this form cannot be guaranteed.
Addresses can be freed and re-allocated while still being pointed to.
Other techniques must be used to correlate the local states of threads.
The solution chosen in <cit.> is to replace the states of individual threads by states of pairs of threads.
Precision is thus restored at the cost of
an almost quadratic blow-up of the abstract domain
that in turn manifests itself in a severe decrease of scalability.
§.§ Pointer Race Freedom Saves Ownership
Using the results from Sections <ref> and <ref>,
we show how to apply the ownership-based optimization of thread-modular reasoning to the memory-managed semantics.
To this end, we split the verification effort into two phases.
Depending on the notion of pointer race freedom,
we first check whether the program under scrutiny is (S)PRF.
If the check fails, we report pointer races as potential errors to the developer.
If the check succeeds, the second phase verifies the property of interest (here, linearizability) assuming (S)PRF.
When the notion of PRF from Section <ref> is used,
the second verification phase can be performed in the garbage-collected semantics due to Theorem <ref>.
This allows us to apply the ownership-based optimization discussed above.
Moreover, Theorem <ref> says that the first PR has to appear in the garbage-collected semantics.
Hence, even the first phase, checking PRF, can rely on garbage collection and ownership.
The PRF check itself is simple.
Validity of pointers is kept as a part of the individual thread states and updated at every sequential and interference step.
Based on this, every computation step is checked for raising a PR according to Definition <ref>.
Our experiments suggest that the overhead caused by the recorded validity information is low.
For SPRF, we proceed analogously.
Due to the Theorems <ref> and <ref>, checking SPRF in the first phase and property verification in the second phase can both be done in the ownership-respecting semantics.
The SPRF check is similar to the PRF check.
Validity of pointers together with an information about strong invalidity is kept as a part of a thread's state, and every step is checked for raising an SPR according to Definition <ref>.
The surprising good news is that both phases can again use the ownership-based optimization.
That is, also in the ownership-respecting semantics, interferences on the owned memory addresses can be skipped.
We argue that this is sound.
Due to Lemma <ref>,
if a thread $\athread$ owns an address $\anadr$, other threads may access $\anadr$ only via invalid pointers.
* modifications of $\anadr$ by $\athread$ need not be considered as an interference step for other threads.
Indeed, if a thread $\athread'\neq \athread$ was influenced by such a modification ($\athread'$ reads a next or the data field of $\anadr$),
then the corresponding variable of $\athread'$ would become strongly invalid, Definition <ref>.
Hence, either this variable is never used in an assertion or in a dereference again (it is effectively dead),
or the first use raises an SPR, Cases (ii) and (iii) in Definition <ref>.
* In turn, in the ownership-respecting semantics, another thread $\athread'$ cannot make changes to $\anadr$, by Definition <ref> of ownership violations.
This means we can also avoid the step where $\athread'$ interferes with the victim $\athread$.
§.§ Experimental Results
To substantiate our claim for a more efficient analysis with practical experiments, we implemented the thread-modular analysis from <cit.> in a prototype tool.
This analysis is challenging for three reasons: it checks linearizability as a non-trivial requirement, it handles an unbounded number of threads, and it supports an unbounded heap.
Our tool covers the garbage-collected semantics, the new ownership-respecting semantics of Section <ref>, and the memory-managed semantics.
For the former two, we use the abstract domain where local states refer to single threads.
Moreover, we support the ownership-based pruning of interference steps from Section <ref>.
For the memory-managed semantics, to restore precision as discussed above, the abstract domain needs local states with pairs of threads.
Rather than running two phases, our tool combines the PRF check and the actual analysis.
We tested our implementation on lock-free data structures from the literature and verified linearizability following the approach in <cit.>.
Experimental results for thread-modular reasoning using different memory semantics.
time in seconds
explored state count
sequential step count
interference step count
pruned interferences
correctness established
5Single lock stack
GC 0.053 328 941 3276 10160 yes
OWN 0.21 703 1913 6983 22678 yes
GC$^-$ 0.20 507 1243 19321 – yes
OWN$^-$ 0.60 950 2474 38117 – yes
MM$^-$ 5.34 16117 25472 183388 – yes
5Single lock queue
GC 0.034 199 588 738 5718 yes
OWN 0.56 520 1336 734 31200 yes
GC$^-$ 0.19 331 778 9539 – yes
OWN$^-$ 2.52 790 1963 65025 – yes
MM$^-$ 31.7 27499 60263 442306 – yes
5Treiber's lock free stack (with version counters) <cit.>
GC 0.052 269 779 3516 15379 yes
OWN 2.36 744 2637 43261 95398 yes
GC$^-$ 0.16 296 837 11530 – yes
OWN$^-$ 4.21 746 2158 73478 – yes
MM$^-$ 602 116776 322057 7920186 – yes
5Michael & Scott's lock free queue <cit.> (with hints)
GC 2.52 3134 6607 46838 1237012 yes
OWN 10564 19553 43305 6678240 20747559 yes
GC$^-$ 9.08 3309 7753 187349 – yes
OWN$^-$ 51046 31329 64234 35477171 – yes
MM$^-$ aborted $\ge\,$69000 $\ge\,$90000 – – false positive
The experimental results are listed in Table <ref>.
The experiments were conducted on an Intel Xeon E5-2650 v3 running at 2.3 GHz.
The table includes the following:
* runtime taken to establish correctness,
* number of explored thread states (i.e. size of the search space),
* number of sequential steps,
* number of interference steps,
* number of interference steps that have been pruned by the ownership-based optimization, and
* the result of the analysis, i.e. whether or not correctness could be established.
For a comparison, we also include the results with the ownership-based optimization turned off (suffix $^-$).
Recall that the optimization does not apply to the memory-managed semantics.
We elaborate on our findings.
Our experiments confirm the usefulness of pointer race freedom.
When equipped with pruning (OWN), the ownership-respecting semantics provides a speed-up of two orders of magnitude for Treiber's stack and the single lock data structures compared to the memory-managed semantics (MM$^-$).
The size of the explored state space is close to the one for the garbage-collected semantics (GC) and up to two orders of magnitude smaller than the one for explicit memory management.
We also stress tested our tool by purposely inserting pointer races, for example, by discarding the version counters.
In all cases, the tool was able to detect those races.
For Michael & Scott's queue we had to provide hints in order to eliminate certain patterns of false positives.
This is due to an imprecision that results from joins over a large number of states (we are using the joined representation of states from <cit.> based on Cartesian abstraction).
Those hints are sufficient for the analysis relying on the ownership-respecting semantics to establish correctness.
The memory-manged semantics produces more false positives, the elimination of which would require more hinting, as also witnessed by the implementation of <cit.>.
Regarding the stress tests from above, note that we ran those tests with the same hints and were still able to find the purposely inserted bugs.
§ CONCLUSION
We have conducted a semantic study on the relationship between concurrent heap-manipulating programs running under explicit memory management and under garbage collection.
We proposed the notion of pointer race that captures the difference between the two semantics and characterizes common synchronizations errors
similar to the well-known data races.
We proved that the verification of pointer race free programs under explicit memory management can be reduced to the easier verification under garbage collection.
We showed an analogous result with a stronger notion of pointer race proposed to fit performance critical (e.g. lock-free) implementations, which are intentionally racy in our original sense.
Our results are particularly useful in thread-modular analysis under explicit memory-management.
We showed that they allow us to apply an ownership-based optimization available before only under garbage-collection.
Using this optimization, our prototype was able to verify lock-free algorithms like Treiber's stack and Micheal & Scott's queue for the memory-managed semantics with a performance gain of up to two orders of magnitude.
§ MISSING DETAILS
The intended behavior of Treiber's stack is as follows, see Figure <ref>.
Upon a push, the corresponding thread allocates a new cell using a local pointer variable $\node$ and sets the given value.
In a loop, the thread now tries to alter the top of stack.
It sets a local pointer variable $\topp$ to the old top of stack stored in the global pointer variable $\Topp$.
Then it redirects the next selector of $\node$ to the old top of stack.
If no concurrent execution of a push or a pop has interefered, $\Topp$ and $\topp$ still point to the same cell and the thread atomically sets $\Topp$ to $\node$.
To be precise, the compare-and-swap command $\cas$ atomically checks the equality $\Topp=\topp$ and, in case it holds, assigns to $\Topp$ the value of $\node$ and returns true.
If the values differ, the command returns false.
We decided not to add $\cas$ to the set of commands to keep our instruction set small.
The theory can be extended to cover $\cas$.
The pop method also creates a local copy $\topp$ of the global top of stack.
It checks whether the stack is empty and, in case, returns negatively.
Otherwise, the method copies the new top of stack $\psel{\topp}$ into the local variable $\node$ and atomically moves the global top of stack $\Topp$ to $\node$.
Now the thread executing pop can access the value of the cell, free $\topp$, and return.
cell = [rectangle split,rectangle split horizontal, rectangle split parts=2,draw,rounded corners,text width=0.3cm,text height=0.2cm]
next = [circle,minimum size=0.15cm,inner sep=0pt,fill=black,draw=black]
[cell] (a)$a$two;
(nr)[above left of=a, node distance=1.2cm](1);
[next](apointer)[right of=a, node distance=0.3cm] ;
(top) [above of=a,node distance=0.7cm]$\topp$;
(Top) [below of=a,node distance=0.7cm]$\Topp$;
[cell] (other)[right of=a, node distance=1.5cm]two;
[next](otherpointer)[right of=other, node distance=0.3cm] ;
(node) [above of=other,node distance=0.7cm]$\node$;
[](dots)[right of=other, node distance=1.3cm] $\cdots$;
[cell] (a)$a$two;
(nr)[above left of=a, node distance=1.2cm](2);
[next](apointer)[right of=a, node distance=0.3cm] ;
(top) [above of=a,node distance=0.7cm]$\topp^{\dagger}$;
[cell] (other)[right of=a, node distance=1.5cm]two;
(Top) [below of=other,node distance=0.7cm]$\Topp$;
[next](otherpointer)[right of=other, node distance=0.3cm] ;
(node) [above of=other,node distance=0.7cm]$\node$;
[](dots)[right of=other, node distance=1.3cm] $\cdots$;
[cell] (a)$a$two;
(nr)[above left of=a, node distance=1.2cm](3);
[next](apointer)[right of=a, node distance=0.3cm] ;
(top) [above of=a,node distance=0.7cm]$\topp^{\dagger}$;
[cell] (b)[below of=a, node distance=0.8cm]$b$two;
[next](bpointer)[right of=b, node distance=0.3cm] ;
[cell] (other)[right of=a, node distance=1.5cm]two;
(Top) [below of=b,node distance=0.7cm]$\Topp$;
[next](otherpointer)[right of=other, node distance=0.3cm] ;
(node) [above of=other,node distance=0.7cm]$\node$;
[](dots)[right of=other, node distance=1.3cm] $\cdots$;
[cell] (a)$a$two;
(nr)[above left of=a, node distance=1.5cm](4);
[next](apointer)[right of=a, node distance=0.3cm] ;
(top) [above of=a,node distance=0.7cm]$\topp^{\dagger}$;
[cell] (b)[below of=a, node distance=0.8cm]$b$two;
[next](bpointer)[right of=b, node distance=0.3cm] ;
(Top) [left of=a,node distance=1.2cm]$\Topp$;
[cell] (other)[right of=a, node distance=1.5cm]two;
[next](otherpointer)[right of=other, node distance=0.3cm] ;
(node) [above of=other,node distance=0.7cm]$\node$;
[](dots)[right of=other, node distance=1.3cm] $\cdots$;
[cell] (a)$a$two;
(nr)[above left of=a, node distance=1.2cm](5);
[next](apointer)[right of=a, node distance=0.3cm] ;
(top) [above of=a,node distance=0.7cm]$\topp^{\dagger}$;
[cell] (b)[below of=a, node distance=0.8cm]$b$two;
[next](bpointer)[right of=b, node distance=0.3cm] ;
[cell] (other)[right of=a, node distance=1.5cm]two;
[next](otherpointer)[right of=other, node distance=0.3cm] ;
(Top) [below right of=other,node distance=0.9cm]$\Topp$;
(node) [above of=other,node distance=0.7cm]$\node$;
[](dots)[right of=other, node distance=1.3cm] $\cdots$;
ABA-problem in Treiber's stack.
§ PROOFS IN SECTION <REF>
Throughout the appendix, we refer to the two equations for heap isomorphism as compatibility requirements.
We consider the first claim.
Let $\aheap_1\heapiso\aheap_2$ via $\anisoa:\adrof{\aheap_1}\rightarrow \adrof{\aheap_2}$.
Let $A:=\adrof{\restrict{\aheap_1}{P}}$.
The task is to show that $\anisoa':=\restrict{\anisoa}{A}:A\rightarrow\anisoa(A)$
defines an isomorphism between
$\restrict{\aheap_1}{P}$ and $\restrict{\aheap_2}{\anisoe(P)}$.
To this end, it is sufficient to show that $\anisoa'$ induces a bijection $\anisoe'$ between $\domof{\restrict{\aheap_1}{P}}$ and $\domof{\restrict{\aheap_2}{\anisoe(P)}}$ that satisfies the compatibility requirements for an isomorphism.
From this we derive that $\anisoa'$ is a bijection between the addresses.
Function $\anisoe'$ is total since $\anisoa'$ maps all addresses in
$\domof{\restrict{\apval_1}{P}}\cap \adr$ and in $\domof{\restrict{\adval_1}{D}}\cap \adr$.
The function is injective as $\anisoe$ is.
In the case of pointer expressions, surjectivity means for every $\apexpp\in \domof{\restrict{\apval_2}{\anisoe(P)}}$ there is $\apexp\in \domof{\restrict{\apval_1}{P}}$ with $\anisoe(\apexp)=\apexpp$.
Since $\apexpp\in \domof{\restrict{\apval_2}{\anisoe(P)}}$, we have $\apexpp\in \domof{\apval_2}$.
Since $\anisoa$ is a heap isomorphism, there is $\apexp\in \domof{\apval_1}$ with $\anisoe(\apexp)=\apexpp$.
Moreover, since $\apexpp\in \anisoe(P)$, we have $\apexp\in P$.
Together, $\apexp\in \domof{\restrict{\apval_1}{P}}$.
Surjectivity for data expressions is similar.
The compatibility required for an isomorphism holds as it holds for $\anisoe$.
Consider now the second claim and assume $\anadr\in\adrof{\aheap_1}$ but $\anadrp\notin\adrof{\aheap_1}$.
Let $\anisoa$ be the isomorphism between $\aheap_1$ and $\aheap_2$.
We extend the function by $\anadrp\mapsto\anadrp'$ and restrict it to the new domain $\adrof{\aheap_1[\psel{\anadr}\mapsto\anadrp]}$.
Note that we indeed may lose the address that $\psel{\anadr}$ was pointing to
so that a restriction is necessary:
\begin{align*}
\anisoa':=\restrict{(\anisoa\cup\set{\anadrp\mapsto\anadrp'})}{\adrof{\aheap_1[\psel{\anadr}\mapsto\anadrp]}}.
\end{align*}
We first check that $\anisoa'$ is a function.
This holds as $\anadrp\notin\adrof{\aheap_1}=\domof{\anisoa}$.
To show that $\anisoa'$ is a bijection between the addresses, it is again sufficient to show that the induced function on pointer and data expressions $\anisoe'$ is a bijection satisfying the requirements for an isomorphism.
The induced function is total as $\anisoa'$ maps all addresses in
In particular does $\anisoa'$ extend $\anisoa$ and hence map $\adrof{\aheap_1}$
containing $\anadr$.
The induced function is injective essentially as $\anisoe$ is.
To be precise, if $\psel{\anadr}$ was not in the domain of $\aheap_1$, then $\psel{\anisoa(\anadr)}$ was not in the domain of $\aheap_2$ since $\anisoe$ is a bijection.
We can therefore safely add this mapping.
It remains to show that the function is surjective.
Consider $\apexpp\in \domof{\aheap_2[\psel{\anadr'}\mapsto\anadrp']}$ with $\anadr'=\anisoa(\anadr)$.
If $\apexpp=\psel{\anadr'}$, then $\apexpp=\anisoe'(\psel{\anadr})$.
If we have $\apexpp\in \domof{\aheap_2}\setminus\set{\psel{\anadr'}}$, then
there is $\apexp\in \domof{\aheap_1}$ so that $\apexpp=\anisoe(\apexp)$.
This $\apexp$ exists as $\anisoe$ is a bijection between the old domains.
To check the compatibility requirements, let $\apval_1$ be the pointer valuation in $\aheap_1[\psel{\anadr}\mapsto\anadrp]$ and let $\apval_2$ be the valuation in $\aheap_2[\psel{\anadr'}\mapsto\anadrp']$. Then
\begin{align*}
\anisoa'(\apval_1(\psel{\anadr})) &= \anisoa'(\anadrp) \\
&= \anadrp'\\
&= \apval_2(\psel{\anadr'})
= \apval_2(\anisoe'(\psel{\anadr})).
\end{align*}
For the remaining pointers, the requirement holds by the fact that $\anisoa$ was a heap isomorphism.
Assume $\anadr\in \freedof{\tau}$.
To show that $\anadr\notin \adrof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}$,
we show that (i) no pointer $\apexp$ to $\anadr$ is valid (in $\validof{\tau}$) and (ii) no selector $\psel{\anadr}$ is valid.
Since the restriction $\restrict{\heapcomput{\tau}}{\validof{\tau}}$ only keeps selectors $\dsel{\anadr}$ for valid pointers to $\anadr$, Argument (i) also removes $\anadr$ from $\domof{\adval_{\tau}}$.
For (i), we show that $\heapcomputof{\tau}{\apexp}=\anadr$ implies $\apexp\notin\validof{\tau}$.
Since $\anadr\in\freedof{\tau}$, there was no malloc after the free of the address.
Hence, there are two cases.
Either $\apexp$ learned about $\anadr$ before or after the address was freed.
In the former case, $\apexp$ is invalidated by the free of address $\anadr$.
In the latter case, $\apexp$ can only learn about $\anadr$ from an invalid pointer.
This renders $\apexp$ invalid, too.
We show (ii), all selectors $\psel{\anadr}$ are invalid.
These selectors were declared invalid at the moment address $\anadr$ was freed.
The only way to validate $\psel{\anadr}$ is via an assignment to it.
This assignment is forbidden as computation $\tau$ is assumed to be PRF.
Indeed, with Argument (i), all pointers to $\anadr$ are invalid and hence accessing the next selector will result in a pointer race according to Definition <ref>(i).
We proceed by induction.
In the base case, all pointers are valid and there is nothing to prove.
Assume the claim holds for $\sigma$ and consider $\sigma.\anact\in\gcsem{\aprog}$ PRF.
The main task is to carefully consider the assignments.
There are five cases.
Case $\psel{\apvar}:=\apvarp$ Let $\heapcomputof{\sigma}{\apvar}=\anadr\neq \segval$. We have $\anadr\neq \segval$ by enabledness.
(1) If $\apvarp$ is invalid, it points to $\anadrp\neq \segval$ by the induction hypothesis(i). If $\apvar$ was valid, claim(ii) now holds for $\psel{\anadr}$.
(2) If $\apvarp$ is valid pointing to $\anadrp$ which may be $\segval$, then $\psel{\anadr}$ will be valid.
If $\anadrp\neq \segval$, the next selectors of $\anadrp$ behave as required by the induction hypothesis(ii) on $\apvarp$.
If $\anadrp=\segval$, claim(ii) is trivial for $\psel{\anadr}$.
Case $\apvar:=\apvarp$ $\phantom{Text}$
If $\apvarp$ is invalid, it points to $\anadr\neq \segval$ by the induction hypothesis(i).
This proves claim(i) for $\apvar$.
If $\apvarp$ is valid, then $\apvar$ will become valid.
If $\heapcomputof{\sigma}{\apvarp}=\segval$, claim(ii) trivially holds for $\apvar$.
If $\heapcomputof{\sigma}{\apvarp}=\anadrp\neq \segval$, the statement about the invalid next selectors of $\anadrp$ carries over to $\apvar$ by the induction hypothesis(ii) on $\apvarp$.
Case $\apvar:=\psel{\apvarp}$
Let $\heapcomputof{\sigma}{\apvarp}=\anadrp\neq \segval$. Again, we have $\anadrp\neq \segval$ by enabledness.
If $\apvarp$ is invalid, this is again a pointer race.
This means $\apvarp$ is valid.
If $\heapcomputof{\sigma}{\psel{\anadrp}}=\segval$, by the induction hypothesis(ii) expression $\psel{\anadrp}$ has to be valid.
Then $\apvar$ becomes valid and points to $\segval$.
In this case, claim(ii) trivially holds.
Otherwise, $\heapcomputof{\sigma}{\psel{\anadrp}}=\anadrpp\neq \segval$.
If $\psel{\anadrp}$ is invalid, $\apvar$ becomes invalid and claim(i) holds.
If $\psel{\anadrp}$ is valid, claim(ii) about the next selectors of $\anadrpp$ holds for $\apvar$ by the induction hypothesis(ii) on $\psel{\anadrp}$.
Consider a free command $\freeof{\apvar}$ with $\heapcomputof{\sigma}{\apvar}=\anadr\neq \segval$.
This invalidates all pointers to $\anadr$ and claim(i) and claim(ii) hold.
Consider a malloc $\apvar:=\malloc$.
This returns a fresh cell $f$ where all next selectors $\psel{f}$ are valid.
Hence, claim(ii) trivially holds.
We proceed by induction on the length of $\tau$.
The base case of single actions setting data variables to arbitrary values is trivial.
In the induction step, assume for $\tau$ we have the heap-equivalent computation $\sigma$.
The fact that the sequences of commands coincide for $\tau$ and $\sigma$ means we can assume the resulting control states to coincide.
This allows us to always choose the same next command in both semantics.
We therefore focus on the heap content and do a case distinction along the transition rule that leads to $\tau'$.
In the following, let $\anisoa:\adrof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}\rightarrow \adrof{\restrict{\heapcomput{\sigma}}{\validof{\sigma}}}$ be the isomorphism between $\restrict{\heapcomput{\tau}}{\validof{\tau}}$ and $\restrict{\heapcomput{\sigma}}{\validof{\sigma}}$.
Case (Malloc2) Let $\tau'=\tau.(\athread, \apvar:=\malloc, [\apvar\mapsto \anadr])$ with $\anadr\in \freedof{\tau}$.
It can be shown that the value of $\dsel{\anadr}$ is defined for addresses $\anadr$ that have been freed.
Let it be $\heapcomputof{\tau}{\dsel{\anadr}}=d$.
We now have
\begin{align}
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}
=&\ \restrict{\heapcomput{\tau}[\apvar\mapsto \anadr]}{\validof{\tau}\cup\set{\apvar}} \notag\\
%=&\ \restrict{\heapcomput{\tau}}{(\validof{\tau}\cup\set{\apvar})}[\apvar\mapsto \anadr] \notag\\
%=&\ (\restrict{\heapcomput{\tau}}{(\validof{\tau}\setminus\set{\dsel{\anadr}})})
%[\apvar\mapsto \anadr,\dsel{\anadr}\mapsto d] \notag\\
=&\ (\restrict{\heapcomput{\tau}}{\validof{\tau}})[\apvar\mapsto \anadr,\dsel{\anadr}\mapsto d].\label{Equation:Malloc2FormTau}
\end{align}
The first equation is by the definition of $\tau'$ and $\validof{\tau'}$.
To understand the second equation, note that $\anadr\notin\adrof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}$ by $\anadr\in\freedof{\tau}$ and Lemma <ref>.
This means in
$\restrict{\heapcomput{\tau}[\apvar\mapsto \anadr]}{\validof{\tau}\cup\set{\apvar}}$, pointer $\apvar$ is the only reference to $\anadr$.
Since we have a reference to $\anadr$, value $\dsel{\anadr}$ is defined in $\restrict{\heapcomput{\tau}[\apvar\mapsto \anadr]}{\validof{\tau}\cup\set{\apvar}}$.
So if we push the restriction over the update, we have to preserve definedness of $\dsel{\anadr}$.
Therefore, we add $\dsel{\anadr}\mapsto d$ to the update.
To mimic the command with garbage collection, we apply
Rule (Malloc1) and get $\sigma':=\sigma.(\athread, \apvar:=\malloc, [\apvar\mapsto f, \dsel{f}\mapsto d, \setcond{\psel{f}\mapsto\segval}{\text{for every selector $\texttt{next}$}}])$.
Note that we allocate a fresh address $f\notin \adrof{\heapcomput{\sigma}}$.
The transition rule allows us to select an arbitrary value.
We choose the value of $\dsel{\anadr}$ in $\heapcomput{\tau}$.
The next selectors are yet undefined.
We have
\begin{align}
\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}
=&\ \restrict{\heapcomput{\sigma}[\apvar\mapsto f, \dsel{f}\mapsto d, \setcond{\psel{f}\mapsto\segval}{\text{for every $\texttt{next}$}}]}{\validof{\sigma}\cup\set{\apvar}}\notag\\
=&\ (\restrict{\heapcomput{\sigma}}{\validof{\sigma}})[\apvar\mapsto f, \dsel{f}\mapsto d].\label{Equation:Malloc2FormSigma}
\end{align}
The first equation is again by the definition of $\sigma'$ and of $\validof{\sigma'}$.
The second equation preserves the assignments $\apvar\mapsto f$ and $\dsel{f}\mapsto\advalue$.
Since we overwrite the value of $\apvar$, there is no need to keep $\apvar$ with the valid pointers when we push the restriction inside.
Since $f$ is fresh, the pointers $\psel{f}$ are not contained in $\validof{\sigma}$.
The restriction removes the corresponding assignments to $\segval$.
To see that $\restrict{\heapcomput{\tau'}}{\validof{\tau'}}$ and $\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}$ are isomorphic, we first note that
\begin{align*}
\restrict{\heapcomput{\tau}}{\validof{\tau}}\heapiso
\restrict{\heapcomput{\sigma}}{\validof{\sigma}}
\end{align*}
by the induction hypothesis.
We already argued for $\anadr\notin\adrof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}$. Similarly, $f\notin\adrof{\restrict{\heapcomput{\sigma}}{\validof{\sigma}}}$.
This allows us to apply Lemma <ref>(<ref>), more precisely a variant of Case (<ref>) where the next pointer is replaced by $\apvar$:
\begin{align*}
(\restrict{\heapcomput{\sigma}}{\validof{\sigma}})[\apvar\mapsto f].
\end{align*}
The isomorphism for this new heap maps $\anadr$ to $f$.
This allows us to apply Lemma <ref>(<ref>) and get
\begin{align*}
(\restrict{\heapcomput{\tau}}{\validof{\tau}})[\apvar\mapsto\anadr][\dsel{\anadr}\mapsto d]\heapiso
(\restrict{\heapcomput{\sigma}}{\validof{\sigma}})[\apvar\mapsto f][\dsel{f}\mapsto d].
\end{align*}
With Equations (<ref>) and (<ref>), this is the desired
\begin{align*}
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}\heapiso \restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}.
\end{align*}
Case (Free) Let $\tau'=\tau.(\athread, \freeof{\apvar}, \emptyset)$.
Since $\tau'$ is assumed to be PRF, we get $\apvar\in \validof{\tau}$.
The more complex case is that $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$.
The pointers that remain valid after the free are
\begin{align*}
\validof{\tau'}&=\validof{\tau}\setminus \invalidof{\anadr}.
\end{align*}
Recall that $\invalidof{\anadr}:=\setcond{\apexp}{\heapcomputof{\tau}{\apexp}=\anadr}\cup \set{\pselarg{\anadr}{1},\ldots, \pselarg{\anadr}{n}}$.
\begin{align}
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}
=&\ \restrict{\heapcomput{\tau}}{\validof{\tau}\setminus \invalidof{\anadr}} \label{Equation:FreeFormTau}
\end{align}
To mimic the command with garbage collection, we also free pointer $\apvar$ in $\sigma$ and obtain $\sigma':=\sigma.(\athread, \freeof{\apvar}, \emptyset)$.
Since $\apvar$ is defined in $\restrict{\heapcomput{\tau}}{\validof{\tau}}$, heap isomorphism requires
$\apval_{\sigma}(\apvar)=\anisoa(\apval_{\tau}(\apvar))=\anisoa(\anadr)\neq \segval$. We have $\anisoa(\anadr)\neq\segval$ as $\anadr\neq \segval$ and $\anisoa$ is an address mapping.
The pointers that remain valid after the free are
\begin{align*}
\validof{\sigma'}&= \validof{\sigma}\setminus \invalidof{\anisoa(\anadr)}.
\end{align*}
As in the case of $\tau'$, we obtain
\begin{align}
\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}
= \restrict{\heapcomput{\sigma}}{\validof{\sigma}\setminus \invalidof{\anisoa(\anadr)}}. \label{Equation:FreeFormSigma}
\end{align}
For the isomorphism, we first show that
\begin{align}
\anisoe(\validof{\tau}\setminus{\invalidof{\anadr}})=\validof{\sigma}\setminus \invalidof{\anisoa(\anadr)}. \label{Equation:IsoInvalid}
\end{align}
To prove the inclusion from left to right, consider $\anisoe(\apexp)$ with $\apexp\in \validof{\tau}$ and $\apexp\notin \invalidof{\anadr}$.
Since $\apexp\in \validof{\tau}$, we have that $\anisoe(\apexp)\in\validof{\sigma}$.
This holds since $\anisoe$ defines a bijection between $\domof{\restrict{\heapcomput{\tau}}{\validof{\tau}}}$ and $\domof{\restrict{\heapcomput{\sigma}}{\validof{\sigma}}}$.
To see that $\anisoe(\apexp)\notin \invalidof{\anisoa(\anadr)}$, assume for the sake of contradiction that it was in the set.
This either means $\anisoe(\apexp)$ points to $\anisoa(\anadr)$ or it is a selector of $\anisoa(\anadr)$.
Consider the former case.
Then we have
\begin{align*}
\anisoa(\anadr)=\heapcomputof{\sigma}{\anisoe(\apexp)} = \anisoa(\heapcomputof{\tau}{\apexp}).
\end{align*}
The second equation holds by the fact that $\anisoa$ is a heap isomorphism.
Together, we get $
\heapcomputof{\tau}{\apexp}= \anadr$.
This contradicts the fact that $\apexp\notin\invalidof{\anadr}$.
The reverse inclusion is along similar lines.
We establish the desired isomorphism as follows:
\begin{align*}
&\ \restrict{\heapcomput{\tau'}}{\validof{\tau'}}\\
\text{Equation~\eqref{Equation:FreeFormTau}}=&\ \restrict{\heapcomput{\tau}}{\validof{\tau}\setminus\invalidof{\anadr}}\\
=&\ \restrict{(\restrict{\heapcomput{\tau}}{\validof{\tau}})}{\validof{\tau}\setminus\invalidof{\anadr}}\\
\text{Ind. hypothesis, Equation~\eqref{Equation:IsoInvalid}, Lemma~\ref{Lemma:HeapIsoAlgebra}\eqref{Equation:HeapIsoRestrict}}\heapiso&\
\restrict{(\restrict{\heapcomput{\sigma}}{\validof{\sigma}})}{\validof{\sigma}\setminus\invalidof{\anisoa(\anadr)}}\\
=&\ \restrict{\heapcomput{\sigma}}{\validof{\sigma}\setminus\invalidof{\anisoa(\anadr)}}\\
\text{Equation~\eqref{Equation:FreeFormSigma}}=&\ \restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}\ .
\end{align*}
Case (Asgn) valid Consider $\tau'=\tau.(\athread, \psel{\apvar}:=\apvarp, [\psel{\anadr}\mapsto \anadrp])$.
Since the assignment is enabled, we have $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$.
Since the computation is PRF, we have $\apvar\in\validof{\tau}$.
Pointer $\apvarp$ has value $\heapcomputof{\tau}{\apvarp}=\anadrp$, which may be $\segval$.
Assume $\apvarp\in \validof{\tau}$.
In this case, we have
\begin{align*}
\validof{\tau'}&=\validof{\tau}\cup\set{\psel{\anadr}}
\end{align*}
and hence
\begin{align}
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}
=&\ \restrict{\heapcomput{\tau}[\psel{\anadr}\mapsto \anadrp]}{\validof{\tau}\cup\set{\psel{\anadr}}} \notag\\
=&\ (\restrict{\heapcomput{\tau}}{\validof{\tau}})[\psel{\anadr}\mapsto \anadrp].\label{Equation:AsgnValidFormTau}
\end{align}
The first equality is by definition of $\tau'$ and $\validof{\tau'}$.
The second equality uses the fact that $\apvarp$ is valid and points to $\anadrp$.
This means we preserve $\dsel{\anadrp}$ in $\restrict{\heapcomput{\tau}}{\validof{\tau}}$ (provided $\anadrp\neq\segval$) and only have to adapt the mapping of $\psel{\anadr}$. The situation may be contrasted with the case of (Malloc2) where we had to maintain $\dsel{\anadr}$.
To mimic the command, observe that $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$, $\apvar\in\validof{\tau}$, and $\restrict{\heapcomput{\tau}}{\validof{\tau}}\heapiso\restrict{\heapcomput{\sigma}}{\validof{\sigma}}$.
Together, this yields
$\heapcomputof{\sigma}{\apvar}=\anisoa(\anadr)\neq \segval$ and allows us to dereference the address.
Again due to isomorphism, $\apvarp$ has to be valid in $\sigma$ and $\heapcomputof{\sigma}{\apvarp}=\anisoa(\anadrp)$.
We thus get
$\sigma':=\sigma.(\athread, \psel{\apvar}:=\apvarp, [\psel{\anisoa(\anadr)}\mapsto
\anisoa(\anadrp)])$.
By definition,
\begin{align*}
\validof{\sigma'}&=\validof{\sigma}\cup\set{\psel{\anisoa(\anadr)}}
\end{align*}
and with the same argument as for $\tau'$
\begin{align}
\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}
=&\ \restrict{\heapcomput{\sigma}[\psel{\anisoa(\anadr)}\mapsto \anisoa(\anadrp)]}{\validof{\sigma}\cup\set{\psel{\anisoa(\anadr)}}} \notag\\
=&\ (\restrict{\heapcomput{\sigma}}{\validof{\sigma}})[\psel{\anisoa(\anadr)}\mapsto \anisoa(\anadrp)].\label{Equation:AsgnValidFormSigma}
\end{align}
The desired isomorphism $\restrict{\heapcomput{\tau'}}{\validof{\tau'}}\heapiso\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}$ now follows with Lemma <ref>(<ref>) in combination with the above Equations (<ref>) and (<ref>).
Case (Asgn) invalid Let $\tau'=\tau.(\athread, \psel{\apvar}:=\apvarp, [\psel{\anadr}\mapsto \anadrp])$.
As in the previous case, by enabledness $\heapcomputof{\tau}{\apvar}=\anadr\neq \segval$ and by PRF $\apvar\in\validof{\tau}$.
Again $\heapcomputof{\tau}{\apvarp}=\anadrp$ may be $\segval$.
We now assume $\apvarp\notin \validof{\tau}$.
This gives
\begin{align*}
\validof{\tau'}&=\validof{\tau}\setminus\set{\psel{\anadr}}.
\end{align*}
As a result, we have
\begin{align}
\restrict{\heapcomput{\tau'}}{\validof{\tau'}}
=&\ \restrict{\heapcomput{\tau}[\psel{\anadr}\mapsto \anadrp]}{\validof{\tau}\setminus\set{\psel{\anadr}}} \notag\\
=&\ \restrict{\heapcomput{\tau}}{\validof{\tau}\setminus\set{\psel{\anadr}}}.\label{Equation:AsgnInvalidFormTau}
\end{align}
The first equality is by definition.
For the second equality, note that $\psel{\anadr}$ may already be defined in $\heapcomput{\tau}$.
Therefore, we have to remove the pointer explicitly also from this heap.
To mimic the command, we again deduce
$\heapcomputof{\sigma}{\apvar}=\anisoa(\anadr)\neq \segval$.
Since $\apvarp$ is not valid in $\tau$, it cannot be valid in $\sigma$ due to the isomorphism between $\restrict{\heapcomput{\tau}}{\validof{\tau}}$ and $\restrict{\heapcomput{\sigma}}{\validof{\sigma}}$.
Let the value be $\heapcomputof{\sigma}{\apvarp}=\anadrpp$.
We thus obtain the computation
$\sigma':=\sigma.(\athread, \psel{\apvar}:=\apvarp, [\psel{\anisoa(\anadr)}\mapsto
\anadrpp])$.
Like in the case of $\tau$, we have
\begin{align*}
\validof{\sigma'}&=\validof{\sigma}\setminus\set{\psel{\anisoa(\anadr)}}
\end{align*}
and hence
\begin{align}
\restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}
=&\ \restrict{\heapcomput{\sigma}[\psel{\anisoa(\anadr)}\mapsto \anadrpp]}{\validof{\sigma}\setminus\set{\psel{\anisoa(\anadr)}}} \notag\\
=&\ \restrict{\heapcomput{\sigma}}{\validof{\sigma}\setminus\set{\psel{\anisoa(\anadr)}}}.\label{Equation:AsgnInvalidFormSigma}
\end{align}
We derive the desired isomorphism with Lemma <ref>(<ref>) in combination with Equations (<ref>) and (<ref>):
\begin{align*}
&\ \restrict{\heapcomput{\tau'}}{\validof{\tau'}}\\
\text{Equation~\eqref{Equation:AsgnInvalidFormTau}}=&\ \restrict{\heapcomput{\tau}}{\validof{\tau}\setminus\set{\psel{\anadr}}}\\
\restrict{(\restrict{\heapcomput{\tau}}{\validof{\tau}})}{\validof{\tau}\setminus\set{\psel{\anadr}}}\\
\text{Ind. hypothesis, Lemma~\ref{Lemma:HeapIsoAlgebra}\eqref{Equation:HeapIsoRestrict}} \heapiso&\
\restrict{(\restrict{\heapcomput{\sigma}}{\validof{\sigma}})}{\validof{\sigma}\setminus\set{\psel{\anisoa(\anadr)}}}\\
=&\ \restrict{\heapcomput{\sigma}}{\validof{\sigma}\setminus\set{\psel{\anisoa(\anadr)}}}\\
\text{Equation~\eqref{Equation:AsgnInvalidFormSigma}}=&\ \restrict{\heapcomput{\sigma'}}{\validof{\sigma'}}.
\end{align*}
The implication from left to right is due to the fact that $\mmsem{\aprog}\supseteq \gcsem{\aprog}$.
For the reverse implication, we assume $\mmsem{\aprog}$ has a pointer race and from this construction a pointer race in $\gcsem{\aprog}$.
If the memory-managed semantics has a pointer race, then it has a shortest one.
Let it be $\tau.\anact\in\mmsem{\aprog}$ with $\anact$ an access to an invalid pointer.
We remove $\anact$ and obtain $\tau\in\mmsem{\aprog}$.
Membership holds as the memory-managed semantics is prefix-closed.
As $\tau$ is shorter than $\tau.\anact$, it is PRF.
This allows us to apply Proposition <ref>: There is a computation $\sigma\in\gcsem{\aprog}$ in the garbage-collected semantics with $\sigma\heapequiv \tau$.
Note that $\sigma$ is again PRF by minimality of $\tau.\anact$ and $\sigma\in \mmsem{\aprog}$.
By definition of heap equivalence, the commands in $\tau$ and $\sigma$ coincide.
This means they lead to the same control location.
So the two semantics are, up to enabledness, ready to execute the same next command.
We moreover have $\restrict{\heapcomput{\tau}}{\validof{\tau}}\heapiso\restrict{\heapcomput{\sigma}}{\validof{\sigma}}$.
We now show how to mimic $\anact$ in the garbage-collected semantics in a way that also raises a pointer race.
Case Free Let $\anact=(\athread, \freeof{\apvar}, \emptyset)$.
Since $\tau.\anact$ is a pointer race, we have $\apvar\notin \validof{\tau}$.
Since $\anisoe$ defines a bijection between the pointers in
$\restrict{\heapcomput{\tau}}{\validof{\tau}}$ and in
$\restrict{\heapcomput{\sigma}}{\validof{\sigma}}$, we conclude that $\apvar\notin \validof{\sigma}$.
This means computation
\begin{align*}
\sigma.(\athread, \freeof{\apvar}, \emptyset)\in\gcsem{\aprog}\ .
\end{align*}
is also a pointer race — as required.
Case Assignment Let $\anact=(\athread, \acom, \anup)$ where $\acom$ is $\apvarp:=\psel{\apvar}$ with $\apvar\notin \validof{\tau}$.
As before, we conclude that $\apvar\notin\validof{\sigma}$.
With Lemma <ref>(i), we obtain $\heapcomputof{\sigma}{\apvar}=\anadr\neq \segval$.
This means we are able to dereference the address and
can execute command $\acom$ in the garbage-collected semantics:
\begin{align*}
\sigma.(\athread, \acom, \anup')\in \gcsem{\aprog}\ .
\end{align*}
The update may differ due to the use of invalid pointers.
However, the computation will again use $\psel{\apvar}$ and, since $\apvar\notin\validof{\sigma}$, will again be racy.
Case Assertion Let $\anact=(\athread, \assert\ \acond, \emptyset)$ where $\acond$ contains $\apvar$ with $\apvar\notin\validof{\tau}$.
As before, we derive $\apvar\notin \validof{\sigma}$.
We are not guaranteed that the valuations in $\heapcomput{\sigma}$ and in $\heapcomput{\tau}$ coincide.
The definition of programs, however, ensures assert commands have complements.
This means if $\assert\ \acond$ is not enabled after $\sigma$, then $\assert\ \neg \acond$ will be ready for execution.
We thus have
\begin{align*}
\sigma.(\athread, \assert\ (\neg)\acond, \emptyset)\in \gcsem{\aprog}\ .
\end{align*}
Since condition $\acond$ coincides, it will again make use of $\apvar$ with $\apvar\notin\validof{\sigma}$.
This means the computation is again racy.
If $\anadr\in\freedof{\sigma}$, then the pointers to it cannot be valid by Lemma <ref>.
Assume that $\anadr\notin\freedof{\sigma}$.
In the presence of garbage collection, the set $\freedof{\sigma}$ monotonically increases as $\sigma$ gets longer.
This means $\anadr$ has not been freed throughout the computation.
We now show that there cannot be an invalid pointer to $\anadr$.
There are two ways of creating an invalid pointer to $\anadr$:
Either by assigning it an invalid pointer $\apexp$ or by freeing the address.
In particular would the first (in a shortest prefix) invalid $\apexp$ have to stem from a free on $\anadr$.
Since the address has never seen a free, there is no invalid pointer to it.
Proposition <ref> follows from the following characterization of PRF under garbage collection.
$\gcsem{\aprog}$ is PRF if and only if there is no $\sigma_1.\anact_1.\sigma_2.\anact_2\in \gcsem{\aprog}$ so that $\comof{\anact_1}$ is $\freeof{\apvar}$ with $\heapcomputof{\sigma_1}{\apvar}=\anadr\neq \segval$ and
$\comof{\anact_2}$ involves $\psel{\apvarp}, \dsel{\apvarp}$, $\freeof{\apvarp}$, or is an assertion with $\apvarp$, and $\heapcomputof{\sigma_1.\anact_1.\sigma_2}{\apvarp}=\anadr$.
For the only-if, we show the contrapositive.
Note that $\anadr\in\freedof{\sigma_1.\anact_1.\sigma_2}$ by monotonicity of $\freedof{\sigma}$ under garbage collection.
With $\heapcomputof{\sigma_1.\anact_1.\sigma_2}{\apvarp}=\anadr$ and Lemma <ref>, we conclude $\apvarp\notin\validof{\sigma_1.\anact_1.\sigma_2}$.
Moreover, pointer variable $\apvarp$ is used in a way that raises a pointer race.
For the if-direction, we again reason by contraposition. Consider a shortest pointer race $\sigma.\anact\in\gcsem{\aprog}$.
Then there is an invalid pointer $\apvarp\notin\validof{\sigma}$ that is used in $\anact$ in a way that raises a pointer race.
Since computation $\sigma$ is shorter than $\sigma.\anact$, it is PRF. By Lemma <ref>(i), we have $\heapcomputof{\sigma}{\apvarp}=\anadr\neq \segval$.
By Lemma <ref>, we conclude that $\anadr$ has been freed somewhere in $\sigma$.
§ PROOFS IN SECTION <REF>
We show the contrapositive and assume that $\heapcomputof{\tau}{\apvar}\in\ownedof{\athread}{\tau}$ but (i) $\apvar\in\shared$ or (ii) $\apvar\in\localof{\athread'}$ with $\athread'\neq\athread$.
From this we derive $\apvar\notin\validof{\tau}$.
Consider Case (i).
If the owning thread had passed the address via a valid pointer to $\apvar$ (potentially transitively via other public pointers, but we refrain from doing this case distinction), then $\athread$ would have lost ownership of the cell.
As a consequence, either (i.i) $\athread$ never passed the address to $\apvar$ or (i.ii) it did so via an invalid pointer or an invalid next selector.
In the former Case (i.i), $\apvar$ is a dangling pointer to a cell that has been re-allocated, which in particular means $\apvar\notin\validof{\tau}$.
In the latter Case (i.ii), the invalid right-hand side $\apt$ of the assignment $\apvar:=\apt$ will have rendered $\apvar$ invalid.
Consider now Case (ii).
We note that threads do not assign their local pointers to the local pointers of other threads.
Therefore, the only way $\athread'$ could point to an owned cell of $\athread$ is by (ii.i) being a dangling pointer or (ii.ii) having received the reference from a shared pointer.
In the former Case (ii.i), we immediately have $\apvar\notin\validof{\tau}$ like in Case (i.i).
In the latter Case (ii.ii), the argumentation from Case (i) shows that the shared pointer has to be invalid.
As a consequence, also $\apvar$ that receives the content of the shared pointer becomes invalid.
If $\tau.(\athread, \acom, \anact)\in\mmsem{\aprog}$ violates ownership, then $\acom$ is an assignment as in Definition <ref>.
Let the variable be $\apvarp$ with $\heapcomputof{\tau}{\apvarp}\in\ownedof{\athread'}{\tau}$ and ($\athread'\neq \athread$ or $\apvarp\in\shared$).
By Lemma <ref>, the pointer cannot be valid, $\apvarp\notin\validof{\tau}$.
Combined, we obtain the definition of SPR, Definition <ref>.(i).
The ownership-respecting semantics $\resmmsem{\aprog}$ is a subset of the memory-managed semantics $\mmsem{\aprog}$, so the inclusion from right to left holds without precondition.
For the reverse inclusion, assume $\mmsem{\aprog}$ is not included in $\resmmsem{\aprog}$.
Then there is $\tau\in\mmsem{\aprog}$ that violates ownership.
By Lemma <ref>, $\tau$ is an SPR.
This contradicts $\mmsem{\aprog}$ SPRF.
For the sake of contradiction, assume that a dirty trace from $\mmsem\aprog$ is not a SPR.
Without loss of generality, assume that the trace is of the form $\tau.\anact$ with $\tau$ clean.
By the definition of a dirty access,
there are two threads $\athread\neq\athread'$ and a pointer $\apvar$
such that
$\aheap_\tau(\apvar)\in\ownedof\tau{\athread'}$ and $\athread$
dereferences or frees $\apvar$.
By Lemma <ref>, $\apvar$ cannot be valid at this moment, which makes the dereference a SPR.
The left-to-right implication is trivial.
The right-to-left implication can be shown by contradiction.
Assume that $\resmmsem {\aprog}$ does not admit SPR and $\mmsem{\aprog}$ contains a SPRs.
Let $\tau.\anact$ be a shortest such SPR in $\mmsem{\aprog}$.
By Lemma <ref>, $\tau$ is clean.
Hence we have a trace $\tau\in\resmmsem {\aprog}$ which admits a SPR; contradiction.
The inclusion
$\mmsem{\aprog}\supseteq\resmmsem {\aprog}$
is trivial. The inclusion
$\mmsem{\aprog}\subseteq\resmmsem {\aprog}$ is shown by contradiction:
if $\mmsem{\aprog}$ contains a dirty trace, then, by Lemma <ref>,
the trace is a SPR,
which contradicts that $\mmsem{\aprog}$ is SPR free.
* $\project{\tau}{\threads\times\coms}\ =\ \project{\tau'}{\threads\times\coms}$
* $\restrict{\heapcomput{\tau}}{\pexp\setminus O}\ =\ \restrict{\heapcomput{\tau'}}{\pexp\setminus \fune(O)}$
* $\restrict{\heapcomput{\tau}}{\validof{\tau}}\ \heapiso\ \restrict{\heapcomput{\tau'}}{\validof{\tau'}}$.
* $\freedof{\tau}\subseteq\freedof{\tau'}$
* $\ownpof{\tau'}=(\ownpof{\tau}\setminus O)\ \cup\ \fune(O)$.
* $\adrof{\restrict{\heapcomput{\tau'}}{\fune(O)}}\cap\adrof{\heapcomput{\tau}}=\emptyset$.
We proceed by induction on the length of the computation.
In the base case, we have single actions that set data variables arbitrarily.
We mimic them identically.
In the induction step, consider the SPRF computation $\tau.\anact\in\resmmsem{\aprog}$ and assume we are given $O\subseteq \ownpof{\tau.\anact}$.
We invoke the induction hypothesis depending on $\anact$.
Since we will always execute the same command, Requirement (1) will trivially hold and we rephrain from commenting on it.
Case (Malloc2) Consider $\anact = (\athread, \apvar:=\malloc, [\apvar\mapsto\anadr])$, which means we re-allocate an address that has been freed.
With this assignment, address $\anadr$ is owned by thread $\athread$ and $\apvar$ is an owning pointer, $\anadr\in \ownedof{\athread}{\tau.\anact}$ and $\ownpof{\tau.\anact}=\ownpof{\tau}\cup\set{\apvar}$.
Since we only turn $\apvar$ into an owning pointer, we can safely use $O':=O\setminus\set{\apvar}\subseteq\ownpof{\tau}$ to invoke the induction hypothesis for $\tau$.
The hypothesis returns a computation $\tau'$.
Since the sequences of commands coindice for $\tau$ and for $\tau'$, the threads reach the same control locations and hence also in $\tau'$ thread $\athread$ is ready to execute a malloc.
The result of the allocation will depend on whether or not $\apvar$ belongs to the given set $O$:
$\apvar\notin O$
We again allocate the address $\anadr$ with (Malloc2).
The transition is enabled after $\tau'$ by $\anadr\in\freedof{\tau}\subseteq\freedof{\tau'}$.
It remains to check (2) to (6).
Concerning Requirement (2), the only pointer outside $O$ that we change is $\apvar$, and we set it consistently to $\anadr$ in both $\tau.\anact$ and in $\tau'.\anact$.
The isomorphism in Requirement (3) is also fine by
\begin{align*}
\project{\heapcomput{\tau.\anact}}{\validof{\tau.\anact}}
\end{align*}
where we preserve $\dsel{\anadr}$ as in Proposition <ref>.
For Requirement (4), we consistently remove $\anadr$ from the set of freed addresses in $\tau.\anact$ and in $\tau'.\anact$.
For the owning pointers in Requirement (5) we have
\begin{align*}
\ownpof{\tau'.\anact}
=&\ \ownpof{\tau'}\cup\set{\apvar}\\
=&\ (\ownpof{\tau}\setminus O')\ \cup\ \fune(O')\cup\set{\apvar}\\
=&\ ((\ownpof{\tau}\cup\set{\apvar})\setminus O)\ \cup\ \fune(O)\\
=&\ (\ownpof{\tau.\anact}\setminus O)\ \cup\ \fune(O).
\end{align*}
The first equation is by definition, the second is the hypothesis for $\tau$ and $O'$, the third equation holds by $O'=O$ as $\apvar$ is assumed not to belong to $O$, and the last equation is again by definition.
Concerning freshness of the owning pointers, Requirement (6),
we note $O'=O$.
This gives
\begin{align*}
\heapcomputof{\tau'.\anact}{\fune(O)}
= \heapcomput{\tau'}[\apvar\mapsto\anadr](\fune(O'))
= \heapcomputof{\tau'}{\fune(O')}.
\end{align*}
The latter equality is because $\apvar\notin O'$ and hence $\apvar\notin\fune(O')$.
We have $\apvar\notin\fune(O')$ as only $\fune(\apvar)=\apvar$.
Moreover, $\adrof{\heapcomput{\tau.\anact}}=\adrof{\heapcomput{\tau}}\cup\set{\anadr}=\adrof{\heapcomput{\tau}}$.
The latter equality holds because $\dsel{\anadr}$ is defined in $\heapcomput{\tau}$ and we re-allocate the address.
The hypothesis, $\adrof{\heapcomput{\tau}}\cap\heapcomputof{\tau'}{\fune(O')}=\emptyset$, combined with the previous argumentation yields $\adrof{\heapcomput{\tau.\anact}}\cap\heapcomputof{\tau'.\anact}{\fune(O)}=\emptyset$.
$\apvar\in O$
We allocate a fresh address using (Malloc1) and $\anact'=(\athread, \apvar:=\malloc, \anup)$ with $\anup=[\apvar\mapsto \anadrp, \dsel{\anadrp}\mapsto\advalue,\setcond{\psel{\anadrp}\mapsto\segval}{\text{for every selector $\mathtt{next}$}}]$.
Requirement (2) holds as we only change a pointer in $O$.
Requirement (3) is as in the proof of Proposition <ref>.
Requirement (4) holds by
\begin{align*}
\freedof{\tau.\anact}\ =\ \freedof{\tau}\setminus\set{\anadr}\ \subseteq\ \freedof{\tau'}\ =\ \freedof{\tau'.\anact'},
\end{align*}
where the inclusion is due to the hypothesis.
Concerning the address mapping, we set it to $\funa\disunion\set{\anadr\mapsto\anadrp}$.
It remains to check that this is a function, which means $\anadr\notin\domof{\funa}=\adrof{O}$.
We have $\anadr\in\freedof{\tau}$, hence there is no valid pointer to this address and no valid next selector defined at this address by Lemma <ref>.
Since all pointers in $O$ are valid, the claim follows.
Actually, Lemma <ref> assumes the computation to be PRF, but an inspection of the proof shows that it continues to hold for SPRF computations.
For Requirement (5), we have
\begin{align*}
\ownpof{\tau'.\anact'}&=\ownpof{\tau'}\cup\set{\apvar}\\
&=(\ownpof{\tau}\setminus O')\cup \fune(O')\cup\set{\apvar}\\
&=(\ownpof{\tau}\setminus O')\cup \fune(O)\\
&=(\ownpof{\tau.\anact}\setminus O)\cup \fune(O).
\end{align*}
The first equation is by definition, the second invokes the hypothesis for $\tau$ and $O'$.
The third equation uses the fact that $\fune(\apvar)=\apvar$.
In the last equation, we add $\apvar$ to $\ownpof{\tau}$ and $O'$.
For Requirement (6), $
\adrof{\heapcomput{\tau.\anact}}=\adrof{\heapcomput{\tau}}\cup\set{\anadr}=\adrof{\heapcomput{\tau}}$.
The latter equation holds because $\anadr$ is re-allocated and thus $\dsel{\anadr}$ is defined in $\heapcomput{\tau}$.
Moreover, we have
\begin{align*}
\heapcomputof{\tau'.\anact'}{\fune(O)}
= \heapcomput{\tau'}[\apvar\mapsto\anadrp](\fune(O))
= \heapcomput{\tau'}(\fune(O'))\cup\set{\anadrp}.
\end{align*}
An application of the induction hypothesis gives $\adrof{\heapcomput{\tau}}\cap\heapcomputof{\tau'}{\fune(O')}=\emptyset$.
Since $\anadrp$ is fresh, the required disjointness $\adrof{\heapcomput{\tau.\anact}}\cap\heapcomputof{\tau'.\anact}{\fune(O)}=\emptyset$ holds.
Case (Malloc1) Consider the case that $\apvar$ allocates a fresh address $\anadr$ using (Malloc1).
Like in the previous case, we invoke the induction hypothesis for $\tau$ with $O':=O\setminus\set{\apvar}$ and obtain $\tau'$.
We can safely assume that $\anadr$ has not been allocated in $\tau'$.
Indeed, in cases where $\tau'$ deviates from $\tau$, it can allocate fresh addresses different from $\anadr$.
We mimic the allocation of $\apvar$ in $\tau'$ with another invocation to (Malloc1).
Depending on whether $\apvar\notin O$ or $\apvar\in O$, the mimicking allocation also selects $\anadr$ or it selects a fresh $\anadrp$, respectively.
If $\apvar\in O$, the address mapping is extended by $\anadr\mapsto\anadrp$.
The Requirements (2) to (4) are immediate.
Requirement (5) is checked like in the previous case.
Requirement (6) is by the induction hypothesis and the fact that $\anadrp$ is chosen fresh.
Case (Free) Consider $\anact = (\athread, \freeof{\apvar}, \emptyset)$.
We note that $O\subseteq \ownpof{\tau}$ since $\ownpof{\tau.\anact}=\ownpof{\tau}\setminus\invalidof{\heapcomputof{\tau}{\apvar}}$.
This allows us to invoke the hypothesis and obtain the computation $\tau'$.
After $\tau'$ we are ready to execute the same action $\anact$ as in the computation $\tau$.
To establish $\tau'.\anact\in \resmmsem{\aprog}$, we have to show that the computation respects ownership.
Towards a contradiction, assume $\heapcomputof{\tau'}{\apvar}\in \ownedof{\tau'}{\athread'}$ with $\athread'\neq \athread$ or $\apvar$ shared.
With Lemma <ref>, we have that $\apvar$ cannot be valid in $\tau'$.
Since the valid pointers in $\tau$ and $\tau'$ coincide by Requirement (3) in the induction hypothesis, we have that $\apvar$ was not valid in $\tau$.
But this in turn means that action $\anact$ frees an invalid pointer in $\tau$, which raises an SPR.
A contradiction to the assumption that $\tau.\anact$ is SPRF.
It remains to check the guarantees (2) to (6) required by the induction.
For Requirement (2) note that $\freeof{\apvar}$ does not execute any updates.
Hence $\heapcomputof{\tau.\anact}=\heapcomputof{\tau}$ and similarly for $\tau'$.
Hence it remains to apply the induction hypothesis as follows:
\begin{align*}
\restrict{\heapcomput{\tau.act}}{\pexp\setminus O}
&= \restrict{\heapcomput{\tau}}{\pexp\setminus O} \\
&= \restrict{\heapcomput{\tau'}}{\pexp\setminus\fune(O)} \\
&= \restrict{\heapcomput{\tau'.act}}{\pexp\setminus\fune(O)}.
\end{align*}
Requirement (3) is like in Proposition <ref>.
For Requirement (4), note that the induction hypothesis guarantees $\heapcomputof{\tau}{\apvar}=\heapcomputof{\tau'}{\apvar}$ because $\apvar\notin O$.
Moreover, the hypothesis gives $\freedof{\tau}\subseteq\freedof{\tau'}$.
Together, this yields
\begin{align*}
\freedof{\tau.\anact}=\freedof{\tau}\cup\set{\heapcomputof{\tau}{\apvar}}\subseteq \freedof{\tau'}\cup\set{\heapcomputof{\tau'}{\apvar}}=\freedof{\tau'.\anact}.
\end{align*}
Concerning Property (5), we note that
\begin{align*}
\ownpof{\tau'.\anact}
&=((\ownpof{\tau}\setminus O)\cup \fune(O))\setminus\invalidof{\heapcomputof{\tau'}{\apvar}}\\
&=((\ownpof{\tau}\setminus O)\cup \fune(O))\setminus\invalidof{\heapcomputof{\tau}{\apvar}}\\
&=((\ownpof{\tau}\setminus O)\setminus\invalidof{\heapcomputof{\tau}{\apvar}})\cup \fune(O)\\
&=((\ownpof{\tau}\setminus\invalidof{\heapcomputof{\tau}{\apvar}})\setminus O)\cup \fune(O)\\
&=(\ownpof{\tau.\anact}\setminus O)\cup \fune(O).
\end{align*}
The first equality is by the definition of owning pointer, more precisely, the requirement that they have to be valid.
The second equation is the induction hypothesis for $\tau$ and $O$.
For the third equality, we show $\invalidof{\heapcomputof{\tau}{\apvar}}=\invalidof{\heapcomputof{\tau'}{\apvar}}$.
We argue for $\subseteq$.
We have $\apvar\notin O$.
Since $O$ is chosen coherent, for every other pointer $\apexp=\apvarp$ or $\apexp=\psel{\anadr}$ to $\heapcomputof{\tau}{\apvar}$, we have $\apexp\notin O$.
With the induction hypothesis, Requirement (2), all these pointers $\apexp$ are mapped identically,
\begin{align*}
\heapcomputof{\tau'}{\apexp}=\heapcomputof{\tau}{\apexp}=\heapcomputof{\tau}{\apvar}=\heapcomputof{\tau'}{\apvar}.
\end{align*}
So $\apexp\in\invalidof{\heapcomputof{\tau'}{\apvar}}$.
For the next pointers $\psel{\heapcomputof{\tau}{\apvar}}$ that are turned invalid by the free in $\tau$, membership in $\invalidof{\heapcomputof{\tau'}{\apvar}}$ is by $\heapcomputof{\tau}{\apvar}=\heapcomputof{\tau'}{\apvar}$.
For the reverse inclusion $\supseteq$, we argue towards a contradiction and assume that it does not hold.
Certainly, every next pointer $\psel{\heapcomputof{\tau'}{\apvar}}$ is also in $\invalidof{\heapcomputof{\tau}{\apvar}}$ by $\heapcomputof{\tau'}{\apvar}=\heapcomputof{\tau}{\apvar}$.
So there is a pointer $\apexp$ to $\heapcomputof{\tau'}{\apvar}$ that is not in $\invalidof{\heapcomputof{\tau}{\apvar}}$.
Since by the hypothesis, Requirement (2), the heaps coincide except for $O$ and $\fune(O)$, this other pointer has to belong to $\fune(O)$.
But then $\apexp$ cannot point to $\heapcomputof{\tau'}{\apvar}=\heapcomputof{\tau}{\apvar}$, because the address also belongs to $\adrof{\heapcomput{\tau}}$.
This would violate Requirement (6) in the induction hypothesis.
For the fourth equation, we argue that $\fune(O)\cap
\invalidof{\heapcomputof{\tau}{\apvar}}=\emptyset$.
Assume there was a pointer $\apexp\in\fune(O)$ with $\apexp\in \invalidof{\heapcomputof{\tau}{\apvar}}$.
Then $\apexp$ points to $\heapcomputof{\tau}{\apvar}$ or it has the shape $\psel{\heapcomputof{\tau}{\apvar}}$.
In both cases, $\apexp\notin O$ since $\apvar\notin O$ and $O$ is chosen coherent.
In the former case, we get $\heapcomputof{\tau'}{\apexp}=\heapcomputof{\tau}{\apexp}$ by Requirement (2) in the hypothesis.
This contradicts the disjointness in Requirement (6) in the hypothesis.
In the latter case, we have $\heapcomputof{\tau'}{\apvar}=\heapcomputof{\tau}{\apvar}$, and hence $\heapcomputof{\tau}{\apvar}\in \heapcomputof{\tau'}{\fune(O)}$ also violates the disjointness in Requirement (6) in the induction hypothesis.
The fifth equation is set theory.
The last equation is the definition of the set of owning pointers.
Requirement (6) follows from the induction hypothesis and the fact that the free does not change the heap.
Case (Asgn) owned Consider an assignment $\anact=(\athread, \psel{\apvar}:=\apvarp, [\psel{\anadr}\mapsto \anadrp])$.
We consider the case that $\psel{\anadr}\in O$.
To invoke the hypothesis, we choose the largest coherent subset $O'\subseteq O$ with $O'\subseteq \ownpof{\tau}$.
Let the resulting computation be $\tau'$.
As a first step, we argue that $\apvarp\in O'$.
We have $\ownpof{\tau.\anact}=\ownpof{\tau}\cup\set{\psel{\anadr}}$.
Hence, $O'=O\setminus\set{\psel{\anadr}}$ if $\heapcomputof{\tau}{\psel{\anadr}}$ is not the target or source address of a pointer in $O$, and $O'=O$ otherwise.
We have $\apvarp\in O$ since $\heapcomputof{\tau.\anact}{\apvarp}=\heapcomputof{\tau.\anact}{\psel{\anadr}}$, $\psel{\anadr}\in O$, and $O$ is coherent.
Moreover, $\apvarp\in\ownpof{\tau}$, for otherwise $\psel{\anadr}$ would not have become an owning pointer to $\anadrp$ after the assignment.
Hence, $\apvarp\in O'$.
To mimic the command, note that by the induction hypothesis we are in the same control state and thus ready to execute the assigment.
To make sure the assignment is enabled, we check that $\heapcomputof{\tau'}{\apvar}\neq \segval$.
This holds by the hypothesis, Requirements (2) and (3).
Indeed, if $\apvar\notin O'$, then the address is mapped identically, and we get $\heapcomputof{\tau'}{\apvar}=\anadr=\heapcomputof{\tau}{\apvar}$.
If $\apvar\in O'$, then the address is mapped by $\funa$.
Since $\funa$ is an address mapping, it only maps $\segval$ to $\segval$, and hence $\anadr\neq\segval$ to $\heapcomputof{\tau'}{\apvar}=\funa(\anadr)\neq \segval$.
With this argument, we obtain $\tau'.\anact'$ with $\anact':=(\athread, \psel{\apvar}:=\apvarp, [\heapcomputof{\tau'}{\apvar}\mapsto\funa(\anadrp)])$.
To establish $\tau'.\anact'\in\resmmsem{\aprog}$, we have to show that the assignment in $\anact'$ respects ownership.
Towards a contradiction, assume $\heapcomputof{\tau'}{\apvar}\in \ownedof{\tau'}{\athread'}$ with $\athread'\neq \athread$ or $\apvar$ shared.
By Lemma <ref>, $\apvar$ cannot be valid in $\tau'$.
Since the valid pointers in $\tau'$ and in $\tau$ coincide, Requirement (3) in the induction hypothesis, $\apvar$ cannot be valid in $\tau$.
By Definition <ref>, the assignment $\psel{\apvar}:=\apvarp$ would raise an SPR — in contradiction to the assumption that $\tau.\anact$ is SPRF.
We now show that the new computation satisfies the Requirements (2) to (6).
For Requirement (2), we obtain equality by the induction hypothesis.
Indeed, we may only add $\psel{\anadr}$ to $O$ if it did not belong to $O'$ before the assignment.
Requirement (3) is as in Proposition <ref>.
The freed addresses do not change by the assignment, therefore, for Requirement (4) there is nothing to do.
For Requirement (5), we note that
\begin{align*}
\ownpof{\tau'.\anact'}&=\ownpof{\tau'}\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}\\
&=(\ownpof{\tau}\setminus O')\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}.
\end{align*}
There are two cases.
Assume $O'=O$ or, phrased differently,
$\psel{\heapcomputof{\tau}{\apvar}}\in O'$.
Then we have $\psel{\heapcomputof{\tau'}{\apvar}}\in \fune(O')$
by Requirement (2) in the hypothesis. Thus,
\begin{align*}
&(\ownpof{\tau}\setminus O')\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}\\
=&((\ownpof{\tau}\cup\set{\psel{\heapcomputof{\tau}{\apvar}}})\setminus (O'\cup\set{\psel{\heapcomputof{\tau}{\apvar}}}))\ \cup\ \fune(O')\\
=&(\ownpof{\tau.\anact}\setminus O)\cup \fune(O).
\end{align*}
The first equation is set theory and the fact that $\psel{\heapcomputof{\tau'}{\apvar}}\in \fune(O')$.
The second equation is by definition of $\ownpof{\tau.\anact}$, the fact that $\psel{\heapcomputof{\tau}{\apvar}}\in O'$, and by $O'=O$.
Assume $O'=O\setminus\set{\psel{\heapcomputof{\tau}{\apvar}}}$.
For the induction step, we update $\funa$ by mapping $\heapcomputof{\tau}{\apvar}$ identically.
We have $\heapcomputof{\tau'}{\apvar}=\heapcomputof{\tau}{\apvar}$, for otherwise $\apvar\in O'$ and by coherence $\psel{\heapcomputof{\tau}{\apvar}}\in O'$.
From this, we derive
\begin{align*}
&(\ownpof{\tau}\setminus O')\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}\\
=&((\ownpof{\tau}\cup\set{\psel{\heapcomputof{\tau}{\apvar}}})\setminus (O'\cup \set{\psel{\heapcomputof{\tau}{\apvar}}}))\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}\\
=&(\ownpof{\tau.\anact}\setminus O)\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}\\
=&(\ownpof{\tau.\anact}\setminus O)\cup \fune(O).
\end{align*}
We have $\fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}}=
\fune(O')\cup\set{\psel{\heapcomputof{\tau}{\apvar}}}=\fune(O)$
by the choice of the address function.
We do a case distinction according to whether or not $\psel{\heapcomputof{\tau'}{\apvar}}$ is disjoint with $\fune(O')$.
In case $\psel{\heapcomputof{\tau'}{\apvar}}\notin\fune(O')$, we have $\heapcomputof{\tau'}{\apvar}=\heapcomputof{\tau}{\apvar}$ by Requirement (2) from the induction hypothesis.
Hence we get
\begin{align*}
& (\ownpof{\tau}\setminus O')\cup \fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}} \\
=~& ((\ownpof{\tau}\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}})\setminus O')\cup \fune(O') \\
=~& (\ownpof{\tau.\anact}\setminus O')\cup \fune(O').
\end{align*}
From this we can establish the claim similar to the proof of as in Case (Malloc2).
In the case that $\psel{\heapcomputof{\tau'}{\apvar}}\in\fune(O')$, we have $\fune(O')\cup\set{\psel{\heapcomputof{\tau'}{\apvar}}} = \fune(O')$.
Requirement (6) is easier to check.
Case (Asgn) not owned Consider an assignment $\anact=(\athread, \psel{\apvar}:=\apvarp, [\psel{\anadr}\mapsto \anadrp])$.
Assume $\psel{\apvar}\not\in O$.
Since $O$ is coherent for $\tau.act$, we also have $\apvarp\not\in O$.
We invoke the induction hypothesis for $\tau$ with $O$.
We can do so since $O$ is also coherent for $\tau$ by the definition of owned addresses.
The hypothesis yields some $\tau'\in\resmmsem{\aprog}$ and some address mapping $\funa$ satisfying Requirements (1) to (6).
First, we argue that $\anact$ is enabled after $\tau'$.
Therefore, we have to show that $\heapcomputof{\tau'}{\apvarp}\not=\segval$.
The argument is as in the previous case ((Asgn) owned).
We can now establish Requirements (2) to (6).
Requirement (2) follows from $\psel{\apvar}\not\in O$, $\apvarp\not\in O$,
$\fune({\psel{\apvar}})\not\in\fune(O)$, $\fune({\apvarp})\not\in\fune(O)$ and the induction hypothesis.
Hence, we can derive the following:
\begin{align*}
\restrict{\heapcomput{\tau.act}}{\pexp\setminus O}
&= \restrict{\heapcomput{\tau}[\psel{\anadr}\mapsto \anadrp]}{\pexp\setminus O} \\
&= \restrict{\heapcomput{\tau}}{\pexp\setminus O}[\psel{\anadr}\mapsto \anadrp] \\
&= \restrict{\heapcomput{\tau'}}{\pexp\setminus\fune(O)}[\psel{\anadr}\mapsto \anadrp] \\
&= \restrict{\heapcomput{\tau'}[\psel{\anadr}\mapsto \anadrp]}{\pexp\setminus\fune(O)} \\
&= \restrict{\heapcomput{\tau'.act}}{\pexp\setminus\fune(O)}.
\end{align*}
Requirement (3) is as in Proposition <ref>.
The freed addresses do not change by the assignment, therefore, Requirement (4) holds, too.
For Requirement (5), we observe that $\ownpof{\tau.\anact}=\ownpof{\tau}\cup X$ with $X=\ownpof{\tau.\anact}\setminus\ownpof{\tau}$, by the definition of ownership.
The analogue holds for $\ownpof{\tau'.\anact}$.
Moreover, $X\cap O=\emptyset$.
Hence, we can establish Requirement (5) as follows:
\begin{align*}
\ownpof{\tau'.\anact}
&= \ownpof{\tau'} \cup X \\
&= (\ownpof{\tau}\setminus O)\cup\fune(O)\cup X \\
&= ((\ownpof{\tau}\cup X)\setminus O)\cup\fune(O) \\
&= (\ownpof{\tau.\anact}\setminus O)\cup\fune(O).
\end{align*}
Requirement (6) follows from the fact that $\psel{\apvar}\not\in\fune(O)$.
Case (Asrt) Consider $\anact = (\athread, \assert\ \apvar=\apvarp, \emptyset)$ with $\heapcomputof{\tau}{\apvar}=\anadr$.
By enabledness we have $\anadr=\anadrp \vee \anadr=\segval \vee \anadrp=\segval$.
We invoke the induction hypothesis on $\tau$ with $O$.
This yields some $\tau'\in\resmmsem{\aprog}$ and some address mapping $\funa$ satisfying Requirements (1) to (6).
We now prove that $\anact$ is enabled in $\tau'$, i.e. $\tau'.\anact\in\resmmsem{\aprog}$.
Towards a contradiction, assume that $\anact$ is not enabled in $\tau'$.
Therefore, let $\heapcomputof{\tau'}{\apvar}=\anadr'$ and $\heapcomputof{\tau'}{\apvarp}=\anadrp'$.
Since $\anact$ is not enabled, we have $\anadr'\not=\anadrp'$ with $\anadr'\not=\segval$ and $\anadrp'\not=\segval$.
Moreover, we can conclude that $\apvar,\apvarp\not\in\validof{\tau'}$ since $\restrict{\heapcomput{\tau}}{\validof{\tau}}\heapiso\restrict{\heapcomput{\tau'}}{\validof{\tau'}}$ by induction hypothesis.
From this we get $\apvar,\apvarp\not\in O$ as $O$ may only contain valid pointers by definition.
Hence, we come up with the following equalities due to Requirement (2):
\begin{align*}
= \restrict{\heapcomput{\tau'}}{\pexp\setminus \fune(O)}(\apvar)
= \restrict{\heapcomput{\tau}}{\pexp\setminus O}(\apvar)
= a, \\
= \restrict{\heapcomput{\tau'}}{\pexp\setminus \fune(O)}(\apvarp)
= \restrict{\heapcomput{\tau}}{\pexp\setminus O}(\apvarp)
= b. \\
\end{align*}
Since $\segval\not=\anadr'\not=\anadrp'\not=\segval$ by assumption, we conclude $\segval\not=\anadr\not=\anadrp\not=\segval$.
This contradicts enabledness of $\anact$ in $\tau$.
Hence, we have proven that $\anact$ is indeed enabled in $\tau'$.
It remains to establish Requirements (2) to (6).
For Requirement (5), consider the two sets $X$ and $X'$ which contain exactly those pointer expressions which $\athread$ loses ownership of by executing the assertion in $\tau$ and $\tau'$, respectively.
Formally, $X$ and $X'$ are defined as
\begin{align*}
X := \ownpof{\tau} \setminus \ownpof{\tau.\anact} &&
X' := \ownpof{\tau'} \setminus \ownpof{\tau'.\anact}.
\end{align*}
By the definition of owning pointers, this is equivalent to
\begin{align*}
X &= \{\, \apexp ~|~ \apexp\in\validof{\tau} \wedge \heapcomput{\tau}(\apexp)=\heapcomput{\tau}(\apvar)\\&\qquad\qquad~ \wedge \heapcomput{\tau}(\apvar)\in\ownpof{\tau} \wedge \heapcomput{\tau}(\apvar)\not\in\ownpof{\tau.\anact} \,\}, \\
X' &= \{\, \apexp ~|~ \apexp\in\validof{\tau'} \wedge \heapcomput{\tau'}(\apexp)=\heapcomput{\tau'}(\apvar)\\&\qquad\qquad~ \wedge \heapcomput{\tau'}(\apvar)\in\ownpof{\tau'} \wedge \heapcomput{\tau'}(\apvar)\not\in\ownpof{\tau'.\anact} \,\}.
\end{align*}
Now, we can easily state that both $X\cap O=\emptyset$ and $X'\cap O=\emptyset$ hold.
Hence, Requirement (2) from the induction hypothesis gives us $\heapcomput{\tau}(\apvar)=\heapcomput{\tau'}(\apvar)$.
This ultimately implies that $X=X'$.
With this equality at hand, we can now establish Requirement (5) as follows:
\begin{align*}
\ownpof{\tau'.\anact}
&= \ownpof{\tau'} \setminus X' \\
&= \ownpof{\tau'} \setminus X \\
&= ((\ownpof{\tau}\setminus O) \cup \fune(O)) \setminus X \\
&= ((\ownpof{\tau}\setminus X)\setminus O) \cup \fune(O) \\
&= (\ownpof{\tau.\anact}\setminus O) \cup \fune(O)
\end{align*}
For the remaining Requirements note that $\heapcomput{\tau.\anact}=\heapcomput{\tau}$, $\validof{\tau.\anact}=\validof{\tau}$ and $\freedof{\tau.\anact}=\freedof{\tau}$ hold.
Furthermore, the analogues for $\tau'$ hold, too.
To prove the remaining Requirements it is now sufficient to apply the above equalities and invoke the induction hypothesis.
§ EVALUATION DETAILS
This section provides additional information about the experiments discussed in Section <ref>.
Figures <ref> and <ref> give the implementation of the single lock data structures coarse stack and coarse queue.
Moreover, Table <ref> provides experimental results for our stress tests.
Those test were conducted using the ownership-respecting semantics.
We tested whether or not our tool is able to detected purposely inserted bugs.
For each linearisation point we executed a test where we moved it to an erroneous position: once to late and once to early.
A description of the correct linearisation points can be found in Table <ref>).
In addition we swapped some assignments.
In Treiber's stack (Figure <ref>) we moved the in before the statement reading the value from the node to be freed.
In Michael&Scott's queue (Figure <ref>) we moved the statement reading the value to be returned by after the following .
Both swapped statements result in unsafe behavior as potentially freed cells are accessed.
Coarse Stack
struct Node
data_type data;
Node* next;
Node* ToS;
void init()
ToS = NULL;
void push(data_type val)
Node* node = new Node();
node->data = val;
node->next = ToS;
ToS = node;
bool pop(data_type dst)
Node* node;
node = ToS;
if (node != NULL)
ToS = node->next;
if (node == NULL)
return false;
dst = node->data;
delete node;
Coarse Queue
struct Node
data_type data;
Node* next;
Node* Head, Tail;
void init()
Head = new Node();
Tail = Head;
void enq(data_type val)
Node* node = new Node();
node->data = val;
node->next = NULL;
Tail->next = node;
Tail = node;
bool deq(data_type dst)
Node* node = Head;
Node* next = Head->next;
if (next == NULL)
return false;
// read data inside
// the atomic block
// to ensure that
// no other thread
// frees "next" in
// between
dst = next->data;
Head = next;
delete node;
Treiber's lock-free stack with linearisation points.
struct pointer_t Node* ptr; int age;
struct Node data_type data; pointer_t next;
pointer_t ToS;
void init()
ToS.ptr = NULL;
void push(data_type val)
pointer_t node;
node.ptr = new Node();
node.ptr->data = val;
while (true)
pointer_t top = ToS;
node.ptr->next = top;
if (DWCAS(ToS, top, node)) // @1
bool pop(data_type dst)
while (true)
pointer_t top = ToS; // @2
if (top.ptr == NULL)
return false;
pointer_t node = top.ptr->next;
if (DWCAS(ToS, top, node)) // @3
dst = top.ptr->data;
delete top.ptr;
bool DWCAS(pointer_t dst, pointer_t cmp, pointer_t src)
if (dst.ptr == cmp.ptr dst.age == cmp.age)
dst.ptr = src.ptr;
dst.age = cmp.age + 1;
return true;
else return false;
Michael&Scott's lock-free queue with linearisation points.
struct pointer_t Node* ptr; int age;
struct Node data_type data; pointer_t next;
pointer_t Head, Tail;
void init()
Head = new Node();
Head.ptr->next = NULL;
Tail = Head;
void enq(data_type val)
pointer_t node;
node.ptr = new Node();
node.ptr->data = val;
node.ptr->next = NULL;
while (true)
pointer_t tail = Tail;
pointer_t next = tail.ptr->next;
if (tail == Tail)
if (tail.age == Tail.age)
if (next.ptr == NULL)
if (DWCAS(tail.next, next, node)) // @1
else DWCAS(Tail, tail, next);
DWCAS(Tail, tail, node);
bool deq(data_type dst)
while (true)
pointer_t head = Head;
pointer_t tail = Tail;
pointer_t next = head.ptr->next; // @2
if (head == Head)
if (head.age == Head.age)
if (head.ptr == tail.ptr)
if (next.ptr == NULL)
return false;
DWCAS(Tail, tail, next);
dst = next.ptr->data;
if (DWCAS(Head, head, next)) // @3
return true;
Linearisation points.
Program Linearisation Point Description
3Treiber's stack
reading global top of stack pointer in
3Michael&Scott's queue
in adding new node to the tail
reading next field of head of queue in
in moving global head of queue pointer
Experimental results for erroneous programs.
Test case Time in seconds Detected defect
6Treiber's stack, bad linearisation point
, early 0.05 value loss
, late 0.07 value out of thin air
, early 0.08 multiple linearisation events emitted
, late 0.05 value loss
, early 0.02 value duplication
, late 0.02 value out of thin air
Treiber's stack, swapped statements
0.001 returned value stems from freed cell
Treiber's stack, age fields discarded
0.001 strong pointer race detected
6Michael&Scott's queue (with false-positive prevention), bad linearisation point
, early 170 fifo property violated
, late 2.7 value out of thin air
, early 4.1 multiple linearisation events emitted
, late 5.1 value loss
, early 0.2 duplicate output
, late 0.4 duplicate output
Swapped Statements, Michael&Scott's queue
3.78 returned value stems from freed cell
Michael&Scott's queue, age fields discarded
0.13 strong pointer race detected
|
1511.00584
|
$\star$Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
1Department of Astronomy and Astrophysics, University of Toronto, 50 St George Street, Toronto, Ontario M5S 3H4, Canada
2Department of Physics, McGill University, 3600 rue University, Montréal, Québec H3A 2T8, Canada
3Institute of Astronomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, UK
4Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
5Laboratoire AIM, IRFU/Service d'Astrophysique - CEA/DSM - CNRS - Universit Paris Diderot, Bat. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
6Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
7NASA Herschel Science Center, IPAC, 770 South Wilson Avenue, Pasadena, CA 91125, USA
We present a five-band Herschel study (100–500 ) of three galaxy clusters at $z\sim1.2$ from the Spitzer Adaptation of the Red-Sequence Cluster Survey (SpARCS). With a sample of 120 spectroscopically-confirmed cluster members, we investigate the role of environment on galaxy properties utilizing the projected cluster phase space (line-of-sight velocity versus clustercentric radius), which probes the time-averaged galaxy density to which a galaxy has been exposed. We divide cluster galaxies into phase-space bins of , tracing a sequence of accretion histories in phase space. Stacking optically star-forming cluster members on the Herschel maps, we measure average infrared star formation rates, and, for the first time in high-redshift galaxy clusters, dust temperatures for dynamically distinct galaxy populations—namely, recent infalls and those that were accreted onto the cluster at an earlier epoch. Proceeding from the infalling to virialized (central) regions of phase space, we find a steady decrease in the specific star formation rate and increase in the stellar age of star-forming cluster galaxies. We perform a probability analysis to investigate all acceptable infrared spectral energy distributions within the full parameter space and measure a $\sim4\sigma$ drop in the average dust temperature of cluster galaxies in an intermediate phase-space bin, compared to an otherwise flat trend with phase space. We suggest one plausible quenching mechanism which may be consistent with these trends, invoking ram-pressure stripping of the warmer dust for galaxies within this intermediate accretion phase.
§ INTRODUCTION
In the framework of hierarchical structure formation, galaxy clusters continually build up their mass over time, preferentially accreting matter along the cosmic filaments. Thus, in the most basic picture, galaxy clusters consist of hundreds of galaxies belonging to one of two populations: an older collection of galaxies that may have formed in situ; and a younger population that was accreted over cosmic time. This signifies the potential of clusters as laboratories with which to gauge differences between galaxies formed in distinct environments—the foundation of galaxy evolution studies. However, it also necessitates an accurate definition for environment, as a cluster observed at a single redshift contains galaxies that have been exposed to different density environments dictated by their accretion time onto the cluster.
The significance of clusters as nurseries for galaxy transformations is substantiated by the many correlations between environment and galaxy properties, such as star formation rate (SFR), age, color and morphology <cit.>. However, there also exists a strong covariance between stellar mass and environment, as massive galaxies are found in increasingly denser regions <cit.>. The key to disentangling this covariance requires systematically mapping out trends with each parameter, while fixing the other, over cosmic time. Indeed, once the star-formation history is accounted for, either through stellar age or mass, many of the aforementioned environmental trends appear to weaken <cit.>.
Further elucidation of the processes that govern galaxy evolution relies primarily on two criteria: the extension of cluster surveys beyond $z\sim1$, and a coherent definition of environment. Indeed, it is now clear that the peak epoch of star formation occurred at $1<z<3$ <cit.>. Moreover, the fraction of star-forming galaxies within clusters increases with redshift, as seen in optical <cit.> and infrared studies <cit.> out to $z\sim1$, and may even rise with increasing galaxy density in clusters beyond $z\sim1.5$ <cit.>. This evolution in star-forming galaxies appears to be dominated by the infalling population as it mimics the changes in the coeval field population <cit.>. As such, a proper definition of environment that isolates the recently accreted infalling population from the older in-situ population is crucial to accurately assess the effect of environmental quenching.
A galaxy's path taken through a cluster, and thus its exposure to different density environments, is encoded in its orbital history. Unfortunately, this is not directly observable as we are limited to a single projected snapshot in time. Recent simulations, however, have shown that cluster phase space—member galaxies' line-of-sight velocity relative to the cluster versus clustercentric radius—can help to circumvent this problem as it is sensitive to the time since galaxy infall <cit.>. Moreover, distinct regions in phase space can isolate different satellite populations <cit.>.
Utilizing the Millennium Simulation from <cit.>, <cit.> trace out galaxy accretion histories for orbiting galaxies of 30 massive clusters as a function of phase space (see figure 3 in ). These diagrams reveal the distinct trumpet- (or chevron-) shaped loci occupied by the older, virialized population with respect to the more recently accreted infall population (see also ). This latter population consists of galaxies that have not yet entered into the virial cluster radius, those that have already reached pericenter and are on their way back out, known as back-splash galaxies <cit.>, and everything in between. Thus, a phase-space analysis for environment can effectively account for distinctive cluster populations and alleviate some of the projection effects that bias the traditional probes for environment: clustercentric radius and local density.
Many recent studies have exploited phase space to further study galaxy properties, such as morphologies <cit.>, AGN distribution <cit.>, post-starburst distribution <cit.>, optical colors <cit.>, star formation activity <cit.>, HI gas <cit.>, and quenching timescales <cit.>. In <cit.>, we parameterized the accretion history of a $z=0.871$ cluster with lines of constant in phase space to study environmental effects on 24-detected cluster galaxies. These lines roughly delineate the trumpet-shaped caustic regions that trace the expected orbital velocities within massive clusters <cit.>. Once we group galaxies according to their accretion history in <cit.>, we find an environmental dependence on stellar age and specific star formation rate (SSFR) for star-forming galaxies, rather than the flatter trend seen with clustercentric radius and galaxy density <cit.>. The parameterization of thus has the potential to expose environmental processes that could otherwise be hidden due to the mixing of infall histories.
In this paper, we extend the phase-space analysis to three spectroscopically-confirmed clusters at $z\sim1.2$. The improvements to <cit.> are primarily two-fold: we consider a more statistically significant sample of 123 cluster members (48 of which are optically star-forming) and we measure more robust SFRs through full coverage of the thermal portion of the spectral energy distribution with five-band Herschel photometry.
In Section <ref> we describe the sample and the Herschel observations. We discuss our stacking analysis in Section <ref>, and the results from the phase-space analysis in Section <ref>. We propose a plausible quenching scenario in Section <ref>, with our final remarks in Section <ref>. We assume a standard cosmology throughout the paper with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\textup{M}}=0.3$, $\Omega_{\Lambda}=0.7$. Stellar masses and SFRs are based on a Chabrier initial mass function <cit.>.
§ OBSERVATIONS AND DATA REDUCTION
§.§ A $z\sim1.2$ Sample from SpARCS/GCLASS
The three clusters in this study derive from the Spitzer Adaptation of the Red-Sequence Cluster Survey (SpARCS) <cit.>, which utilizes an infrared color technique to pinpoint the red sequence of cluster galaxies <cit.> out to $z\sim1.6$. Specifically, SpARCS locates over-densities of red-sequence cluster galaxies using a $z-3.6$ color selection which brackets the 4000 Å-break at $z>1$. This simultaneously traces dense cluster regions while providing a photometric redshift for the cluster from the color of the red-sequence. Additionally, the three $z\sim1.2$ clusters were selected as part of an ambitious spectroscopic follow-up survey, the Gemini Cluster Astrophysics Spectroscopic Survey (GCLASS; ). Utilizing the Gemini Multi-Object Spectrograph (GMOS), GCLASS has successfully obtained redshifts for $\sim$800 galaxies within ten rich cluster fields from $0.85<z<1.3$, including $\sim400$ cluster members.
To optimize the number of cluster redshifts, GCLASS targets were prioritized based primarily on three criteria: small clustercentric radius; proximity in observed $z^\prime - 3.6$ color to the cluster red-sequence; and 3.6 flux. A 3.6 flux-selected sample is advantageous at $z\sim1$ as it probes rest-frame $H$ band and is thus similar to a stellar-mass limited sample. The color selection was broader at $z\sim1.2$ to account for the bluer rest-frame probed by the red-sequence color. This helps to eliminate any potential selection biases by spanning the colors of both red and blue cluster members, while excluding obvious background/foreground galaxies. Additional priority was also given to galaxies off the red-sequence with 24-MIPS detections to select dusty star-forming cluster member candidates without any specific color cut. Cluster members are defined as sources within 1500 km s$^{-1}$ of the cluster velocity dispersion, yielding a final sample of 122 cluster members with secure redshifts within the three $z\sim1.2$ clusters presented here. Cluster properties, including M$_{200}$, R$_{200}$, and $\sigma_v$ (G. Wilson et al., in preparation) are listed in Table <ref>, and spectroscopic details, target selection, survey completeness, and stellar masses are described in detail in <cit.>.
The Herschel-GCLASS sample at $z\sim1.2$.
1cSpARCS Name
1c# of
1c(km s$^{-1}$)
J161641+554513 EN1-349 1.1555 16 16 41.232 $+$55 45 25.708 1.7$^{+0.7}_{-0.7}$ 0.74$^{+0.09}_{-0.12}$ 660$^{+80}_{-110}$ 42
J163435+402151 EN2-111 1.1771 16 34 35.402 $+$40 21 51.588 3.1$^{+1.3}_{-1.1}$ 0.89$^{+0.11}_{-0.12}$ 810$^{+100}_{-110}$ 43
J163852+403843 EN2-119 1.1958 16 38 51.625 $+$40 38 42.893 0.77$^{+0.31}_{-0.40}$ 0.56$^{+0.06}_{-0.12}$ 510$^{+60}_{-110}$ 37
§.§ Herschel-PACS Imaging
We observed with the Photodetector Array Camera and Spectrometer (PACS) instrument <cit.> aboard the Herschel Space Observatory <cit.> over 5$\times$5 arcmin around each cluster using the medium speed scan (20 arcsec/s) in array mode with homogenous coverage. We divided each cluster into two astronomical observation requests (AORs) with alternating scan directions of $45^\circ$ and $135^\circ$ to further maximize homogeneity (OBSIDs 1342247340, 1342247341 for EN1-349; 1342248661, 1342248662 for EN2-111; and 1342248631, 1342248632 for EN2-119). The final maps consist of 7.2 hours of integration over each field at 100 and 160 .
We reduce each map with the Unimap pipeline <cit.>, which employs a generalized least squares map-making technique to help reduce the ubiquitous $1/f$ noise. We first produce Level 1 products from the raw AORs using the automatic pipeline in the hipe v12.0 environment <cit.>. We then use UniHipe to convert these products into fits files appropriate to input into Unimap. We run Unimap with the standard parameters and project onto final pixel sizes of 1.6 and 3.2 for 100 and 160, respectively. In Figure <ref> we show the maps for each cluster field in both channels, over-plotted with symbols denoting the positions of spectroscopically confirmed star-forming cluster members.
§.§ Herschel-SPIRE Data
We download raw archival data from the Herschel Science Archive for the Spectral and Photometric Imaging Receiver (SPIRE; ) for the Elais North-1 and Elais North-2 fields, which contain our three clusters, and reduce the data ourselves. The observations derive from the Herschel Multi-tiered Extragalactic Survey (HerMES), the largest program carried out with Herschel, covering $\sim$70 deg$^2$ <cit.>. Both fields belong to the level-6 tier of the survey, meaning they are wide-field but relatively shallow depth, and were observed in parallel mode with PACS and SPIRE. The relevant obsids are 1342228450/1342228354 for Elais North-1 and 1342214712/1342226997 for Elais North-2.
In general, we follow the default pipeline for reducing large SPIRE maps in parallel mode, with a few modifications when necessary. Processing each AOR separately, the pipeline first loops over the scans, correcting for various artifacts, including jumps in the thermistor timelines, cosmic ray glitches, low-temperature noise drifts, and pointing calibrations. We then merge the orthogonal scan directions from each AOR.
The next step involves making a correction for residual offsets between detectors, which can lead to stripes in the map. The SPIRE Destriper module has been found to produce optimal results <cit.>. This algorithm iteratively removes the offsets by adjusting the baseline removal with either a simple median or polynomial fit of a specified degree. We use a zeroth-order polynomial for the Elais North-2 field, and a first-order polynomial for the Elais North-1 field, which was found to have significant stripes with the zeroth-order fit. While a higher-order polynomial has been found to produce a tilted background in some cases, visual inspection of the resulting map revealed it was a robust solution, producing no varying extended emission. Finally, the baseline-removed scans are passed to the map maker; again we use the default naive mapper, which projects each bolometer signal onto a sky pixel, and creates a flux density map by dividing the total signal map by the coverage map.
Herschel-PACS maps at 100 (left column) and 160 (right column) for EN1-349 (top row), EN2-111 (middle row) and EN2-119 (bottom row). We plot spectroscopically confirmed cluster members with [OII] emission in blue and cluster members without [OII] emission in red. In green, we additionally highlight foreground and background galaxies (i.e., non-cluster galaxies) that have high-quality spectroscopic redshifts and are individually detected in the PACS maps.
§ ANALYSIS
Since most of the cluster galaxies are undetected in the PACS and SPIRE maps, we perform a stacking analysis (see Sections <ref> and <ref>) to study the average properties of cluster members. We utilize the extensive spectroscopy available over all three clusters (122 members), but focus on non-passive galaxies (defined by the presence of [OII] emission), as we are primarily interested in environmental effects on star formation activity. This is a reasonable selection given that GCLASS spectroscopy in $z\sim1$ galaxy clusters is not significantly biased against dusty star formation, as seen through an agreement of the total SFR per unit cluster mass measured separately by the [OII] equivalent width and 24 flux <cit.>. Moreover, we note that 78% of 24-detected cluster members have [OII] emission over the three $z\sim1.2$ clusters.
After removing quiescent galaxies from the sample, we are left with 57 star-forming galaxies. Through visual inspection, we further remove 9 sources that are heavily contaminated by a bright (at 100 or 160), nearby field galaxy, yielding 9, 20, and 19 star-forming cluster galaxies in EN1-349, EN2-111, and EN2-119, respectively. We further divide the sources into four phase-space bins, defined as as described in <cit.>, with 12 galaxies per bin.
§.§ PACS Stacking
For each bin, we stack the PACS maps at 100 and 160 combined over all three clusters, using the following approach. We extract a thumbnail image around each source, large enough to cover the recommended annulus for sky estimation (see Section <ref>), yielding a radius of 26 arcsec for 100 and 30 arcsec for 160 . We mask the inner pixels of the thumbnails (6 and 12 arcsec), just slightly more than the recommended radius for aperture photometry at each wavelength, and perform a $3\sigma$ clipping on the remaining pixels to remove any bright sources within the extracted map, thereby cleaning the sky annulus region. We also remove any $2\sigma$ outliers that neighbor the $3\sigma$ pixels. The clipped map is then weighted by the inverse of the variance taken from a matched thumbnail of the noise map (the standard deviation of the naive map). We combine all subsequent thumbnails to make a cube for each bin, where each layer represents the clipped, weighted map with the inner masking now removed. Along the cube dimension, we perform a trimmed average: we first discard the minimum and maximum pixel values along all layers in the cube (not including the clipped values), pixel by pixel, and calculate the mean in the remaining pixels. Finally, we normalize by the total of the inverse variance associated with the pixels that contribute to the trimmed mean. The resulting map, a flattened cube that has been averaged (with trimming) pixel by pixel with each layer inverse weighted and $3\sigma$ clipped, represents our stacked image.
§.§ PACS Aperture Photometry
We perform aperture photometry on the final stacked images within the hipe v12.0 environment, using the recommended radii for faint sources: 5.6, 20, and 25 arcsec (10.5, 24, and 28 arcsec) for source flux, inner sky annulus, and sky annulus, respectively at 100 (160 ). The sky estimate is determined from an algorithm adapted from daophot which iteratively computes the mean and standard deviation of the provided sky pixels, each time removing possible outliers, and is ultimately subtracted from the source flux.
The aperture radii are below the nominal FWHM of the beam at each wavelength, which helps to reduce contamination from any residual flux from neighboring sources that was not removed during stacking. We must then apply an aperture correction to properly account for the missing flux, calculated with version 7 of the responsivity function in each band: 0.57 for 5.6 arcsec at 100 and 0.64 for 10.5 arcsec at 160 . We bootstrap the sources 1000 times in each phase-space bin and calculate the standard deviation on the mean to determine the errors on the binned fluxes.
§.§ SPIRE Stacking
The large beam at SPIRE wavelengths (18.2, 24.9, and 36.3 arcsec FWHM at 250, 350, and 500 , respectively) renders traditional stacking analyses difficult. This is due to the confusion-limited nature of the maps, which is typically reached when the source density surpasses $\sim$0.02–0.03 beam$^{-1}$ <cit.>; this is found to occur at 18.7, 18.4, and 13.2 mJy (at 40 beams per source) with ascending SPIRE wavelengths <cit.>. As a result, a single flux peak in the map is likely to have contributions from a sea of fainter, unresolved sources and disentangling these fluxes can be problematic. Moreover, this effect amplifies when one considers intrinsically correlated populations, as they are expected to be clustered on large beam scales.
Monte Carlo simulations can circumvent this problem by correcting for the bias measured in stacked fluxes. However, an even cleaner approach lies in fitting the stacked fluxes of correlated populations simultaneously. An algorithm called simstack developed specifically for SPIRE maps by <cit.> exploits the latter technique and has been made publicly available. It was designed specifically to deconvolve the flux from inherently clustered populations (e.g., the CIB), and simulations found it to be an unbiased estimator compared to traditional stacking methods.
The general method relies on using positional priors from a deeper, less-confused image (e.g., 24 ), separating potentially clustered populations into individual lists, and fitting the flux at all positions simultaneously. This effectively allows for a deconvolution of the flux contribution from multiple sources within a beam (assuming that all these sources were detected in the prior catalog).
More specifically, a “hits map" is created for each list of grouped sources, where pixels are assigned an integer value corresponding to the number of sources that fall within it. These maps are smoothed with the FWHM of the beam and mean-subtracted. For each list, a vector is populated with the values in the mean-subtracted smoothed map at all pixels of interest around each source, combined from all lists. All vectors are then passed to the fitting routine of the functional form
\begin{equation}
M_j = \sum_{\alpha}S_{\alpha}C_{\alpha j},
\end{equation}
where $M_j$ corresponds to the real map values in $j$ pixels of interest, $C_{\alpha j}$ is the beam-convolved mean-subtracted values in the pixels of interest for each list $\alpha$, and $S_{\alpha}$ is the stacked flux in list $\alpha$. This process is iterated until the resulting $\chi^2$ is minimized, yielding a simultaneous stacked flux for each list. The entire procedure is subsequently repeated for each SPIRE wavelength.
As we aim to combine sources from all three clusters into each phase-space bin, we slightly alter the public code to suit our needs. Specifically, we create a separate beam-convolved mean-subtracted “hits map" for each field, and then merge the resulting vectors together (along with merged vectors of real data) to pass to the fitting routine.
In order to optimize the functionality of simstack for reducing the bias due to beam-size clustering, we simultaneously pass prior catalogs to the fitting routine that could contribute any source confusion (in addition to our spectroscopic catalogs). MIPS-24 <cit.> is the conventional prior catalog to use for SPIRE for two reasons: it correlates well with far-infrared wavelengths, and a large fraction of 24 sources are resolved <cit.>. We use deep MIPS maps for each field from a Guaranteed Time Observer program (proposal ID 50161) with exposures of 1200 seconds per pixel. The catalogs are complete down to $\sim70$ . Our final simultaneously-stacked bins include: (1) four phase-space bins with star-forming ([OII]-detected) cluster members; (2) a single bin containing all remaining cluster members without [OII] emission; (3) two separate catalogs of field galaxies within the redshift range $1.10<z<1.21$, one containing [OII]-detected galaxies, and one with passive galaxies; and (4) all 24 -detected sources above $3\sigma$. This latter list has been purged of any cluster members and field galaxies that have already been included in the previous bins. Error bars are computed from the standard deviation on 1000 bootstrapped means for each bin, stacking all catalogs listed above simultaneously.
§.§ Possible AGN Contamination
Recent studies have shown that high-redshift clusters harbor a higher fraction of AGN than their local counterparts <cit.>, and can reach up to 10% at $z\sim1.25$ <cit.>. AGN heat their surrounding dusty torus, which radiates monotonically in the mid-infrared with a power-law spectrum. Thus, Spitzer-IRAC colors can purge AGN from star-forming galaxies which display the 1.6 stellar bump <cit.>. We exploit this technique to identify any cluster AGN in our sample using the color criteria from <cit.>. The IRAC photometry for GCLASS is described in <cit.>. Only one cluster member with [OII]-emission falls into the AGN wedge, and is still consistent with the star-forming region within 1$\sigma$. It also lies well outside the revised bounds proposed by <cit.> and <cit.>, which further remove high-redshift star-forming interlopers. We therefore choose to keep the source in the sample, as its infrared luminosity is unlikely to be dominated solely by an AGN.
§ PHASE-SPACE RESULTS
In total, we utilize 48 spectroscopically-confirmed, star-forming cluster galaxies with [OII] emission. Here, we extend the phase-space analysis from <cit.> to higher-redshifts and utilize more extensive photometry. Since we have stacked the Herschel fluxes, we aim to create phase-space bins with equal numbers of galaxies and therefore deviate slightly from the values used in <cit.>. Our final bin delineations are as follows: $(r/r_{200})\times(\Delta v/\sigma_v) <0.20$ (central bin); $0.20<(r/r_{200})\times(\Delta v/\sigma_v) <0.64$ (intermediate bin); $0.64<(r/r_{200})\times(\Delta v/\sigma_v) <1.35$ (recently accreted bin); and $(r/r_{200})\times(\Delta v/\sigma_v) >1.35$ (infalling bin). The resulting bins are shown in Figure <ref>.
The phase space (line-of-sight velocity versus clustercentric radius, normalized by the cluster velocity dispersion and r$_{200}$, respectively) for all cluster galaxies at $z\sim1.2$. The gray points are galaxies without any discernible [OII] emission, while the larger colored points all have [OII] emission. They are color-coded based on their position in phase space, with delineations of constant shown by the black lines.
§.§ Stellar Age and Mass in Phase Space
We test the efficacy of using phase space as an accretion history sequence by plotting the strength of the 4000 Å-break as a function of $\log[(r/r_{200})\times(\Delta v/\sigma_v)]$ for all cluster members in Figure <ref> (star-forming cluster members are highlighted with a box). There is a general trend toward lower 4000 Å-break depths for higher values of . A Spearman's $\rho$ test reveals a mild correlation of $-0.38$, with a high probability ($>99\%$) of rejecting the null hypothesis of no correlation. This signifies that galaxies that were accreted most recently to the cluster have had a more recent episode of star formation. After dividing the cluster galaxies into two stellar mass bins, defined by the median mass, the trend persists only for the lower mass bin with a mild correlation coefficient of $-0.32$ at high significance ($99\%$). The null hypothesis is not rejected for the higher mass galaxies, suggesting that the trend is primarily driven by the low-mass galaxies. While the two mass bins themselves are wide, covering over an order of magnitude of stellar mass, the median mass in each bin remains roughly constant as a function of phase space, changing by less than 0.4 dex over each bin. This indicates that the relationship between the 4000 Å-break and phase space is not driven entirely by mass segregation, and the cluster environment may play a role at fixed (lower) stellar mass. We note that 74% of the [OII]-detected cluster galaxies are within this lower mass range, and therefore primarily follow this correlation. In order to visually highlight the correlations present, we apply a two-dimensional kernel density estimator to all the data points, and to the lower mass galaxies. We smooth the data using a Gaussian kernel with the width given by Silverman's rule, proportional to the standard deviation along both axes. The FWHM of the Gaussian kernel in $\log[(r/r_{200})\times(\Delta v/\sigma_v)]$ and $D_n(4000)$ is $[0.76, 0.38]$ and $[0.60, 0.30]$ for all galaxies and lower mass galaxies, respectively. Regions representing 68% and 95% of the kernel-convolved surface density are shown by the gray and and green contours, for all galaxies and the lower mass galaxies, respectively.
The strength of the 4000 Åbreak as a function of $\log[(r/r_{200})\times(\Delta v/\sigma_v)]$ for all cluster members. The green and purple circles represent galaxies with lower and higher stellar masses, respectively. The open squares denote star-forming galaxies detected with [OII] emission. The dashed vertical lines show the bin delineations we use in . The gray contours represent the 68% and 95% smoothed surface density regions after applying a two-dimensional kernel density estimator to all the points, while the green contours represent the lower mass galaxies only. A typical error bar is shown in the upper right corner.
§.§ Infrared Spectral Energy Distributions
With broad characterization of the thermal portion of the spectral energy distribution, we can fit modified blackbodies to the stacked fluxes to estimate the dust temperature and infrared luminosity of each population. We adopt the modified blackbody given by:
\begin{equation}
S_{\nu}\ = A[1-e^{-\tau_\nu}]B_{\nu}(T_d)
\label{eqn:modbb}
\end{equation}
where $A$ represents an amplitude parameter, $B_{\nu}(T_d)$ is the Planck function, $\tau_\nu = (\nu/\nu_0)^{\beta}$, and we assume a crossover frequency of $\nu_0=3$ THz <cit.>.
We keep the slope of the emissivity, $\beta$, fixed to a typical value of 1.7—with the accepted range of values between 1.5–2.0 <cit.>. We note that fixing $\beta$ at either limit does not alter the resulting dust temperatures beyond their 1$\sigma$ uncertainties. There is also a strong degeneracy between dust temperature and redshift, as both parameters alter the location of the thermal peak in a similar manner: colder temperatures and increasing redshift shift the peak to longer wavelengths and attenuated fluxes. However, stacking on spectroscopically confirmed star-forming members removes this problem as we no longer need to fit for the redshift, reducing the final errors on the dust temperature.
Rather than using a predefined fitting routine, we explore a wide range of parameter values for both the amplitude and dust temperature. We create a grid of linearly-spaced dust temperatures from 10–100 K and logarithmically-spaced amplitudes from 0.01–1000. For each pair of parameters ($A, T_d$), we compute the $\chi^2$ value between the model corresponding to these parameters and our stacked fluxes, summing over the five wavelengths.
In Figure <ref>, we plot the stacked fluxes and resulting best-fit modified blackbody from $\chi^2$-minimization for each of the phase-space bins (top panel) and star-forming field galaxies (bottom panel). The intermediate phase-space bin peaks at longer wavelengths (i.e., colder dust temperatures) compared to the other bins.
Modified blackbody fits to the thermal portion of the spectral energy distribution using five wavelengths from stacked Herschel-PACS (100 and 160 ) and Herschel-SPIRE (250, 350, and 500 ) fluxes, shown as solid circles. The best fit for each phase-space bin is represented by the solid line. The orange, purple, blue, and maroon colors correspond to the ascending phase-space bins: $(r/r_{200})\times(\Delta v/\sigma_v) <0.20$; $0.20<(r/r_{200})\times(\Delta v/\sigma_v) <0.64$; $0.64<(r/r_{200})\times(\Delta v/\sigma_v) <1.35$; and $(r/r_{200})\times(\Delta v/\sigma_v) >1.35$, respectively. The bottom panel is the same, except for star-forming field galaxies over $1.10<z<1.21$ (green curve). Uncertainties are measured from the standard deviation of 1000 bootstrapped stacked fluxes. The points have been offset slightly in observed wavelength to avoid overcrowding. There is a clear difference in the shape of the SED for the intermediate (purple) bin, which peaks at colder dust temperatures.
From the 2D grid of $\chi^2$ values, we further define a probability distribution, given by $P(A, T_d) \propto \exp(-\chi^2(A, T_d)/2)$. This allows for a more accurate representation of the uncertainties given the shape of the degeneracy between $A$ and $T_d$. Figure <ref> displays the 2D parameter space with contours which enclose the 68% and 95% surfaces of these probabilities. We then calculate mean dust temperatures for each of the four phase-space bins (middle panel) and the field sample (lower panel), weighting by the probability distributions. The reported values are given by, $\langle T_{d}\rangle = \int dA\, dT_d\ P(A, T_d)\, T_d$. We similarly derive errors using the variance, $\sigma = \sqrt{\langle T_{d}^2\rangle - \langle T_{d}\rangle^2}$. The 1D distributions for dust temperature, after marginalizing over the amplitude, are shown in the top panel of Figure <ref>.
The full parameter space of modified blackbody amplitude and dust temperature for the four phase-space bins (middle panel) and field sample (lower panel). The gray-scale background corresponds to values of infrared luminosity, given each amplitude and temperature. The contours represent the 68% and 95% likelihoods for each phase-space bin. The upper panel shows the 1D probabilities of dust temperature, marginalized over all amplitudes. The listed mean dust temperatures and SSFRs (derived from $\langle L_{\rm{IR}}\rangle$) are calculated using the normalized probability distribution as a weight. The resulting error bars are given by the rms. The intermediate phase-space (purple) bin occupies a completely separate region in parameter space compared to the earliest accreted (orange) bin.
We also attempt to calculate the best fit SED for a bin of all cluster members without [OII]-emission. The stacked fluxes in this bin are all consistent with 0 mJy at the 1$\sigma$ level. The concern is that if we see emission in this bin, it could be due to increased confusion in the cluster center from un-detected sources as the cluster source density rises. This could artificially boost emission in the other bins as well, especially the central phase-space bin where the source density is highest. This problem is somewhat mitigated by stacking all cluster members and a catalog of 24 prior positions simultaneously, but only accounts for galaxies detected above $3\sigma$, not a population below the noise (at 24). We are unable to constrain the 68% and 95% contours in the 2D parameter space for this bin (i.e., the modified blackbody template is not a good fit), and therefore conclude that higher confusion in the cluster is not significantly altering our results.
The infrared luminosities are obtained by integrating the modified blackbody over rest-frame 8–1000 , and then weighted by the probability distribution. Infrared luminosities derived from fits to spectral energy distributions that span the peak of the thermal emission should provide robust estimates of the bolometric SFR within galaxies <cit.>, as they trace dust emission both directly and indirectly related to star formation. The former tracer includes any dust that is directly heated from ultraviolet radiation from young stars, while the latter describes the inherent association between dust and gas—the raw fuel for star formation. The SFRs are calculated using the relation from <cit.>, and adjusted to Chabrier IMF with a factor of 1.65 <cit.>.
While the two highest phase-space bins and the field bin are entirely consistent within 1$\sigma$, there is a clear break in the full parameter space between the lowest phase-space bin (orange curve) and the intermediate bin (purple curve), at a level $>2\sigma$. The derived properties and fit parameters are listed in Table <ref>.
Weighted mean and best fit values for stacked star-forming cluster members in phase-space bins.
1cPhase Space
1c$\rm T_{\rm best\ fit}$
1c$\langle\rm T_{\rm dust}\rangle$
1c$\langle\rm L_{\rm IR}\rangle$
1c$\langle\rm SFR\rangle$
1c$\langle\rm log(SSFR)\rangle$
$0.00<r\times v<0.20$ orange 1.179 2.1 58 64$\pm$17 9.6$\pm$4.3 10$\pm$4.5 $-10.1^{+0.16}_{-0.26}$
$0.20<r\times v<0.64$ purple 1.179 0.8 18 20$\pm$4.0 6.6$\pm$1.5 6.9$\pm$1.6 $-9.68^{+0.09}_{-0.11}$
$0.64<r\times v<1.35$ blue 1.178 1.2 44 45$\pm$7.5 24$\pm$5.0 25$\pm$5.2 $-9.40^{+0.08}_{-0.10}$
$1.35<r\times v<3.00$ maroon 1.178 1.6 43 44$\pm$9.6 17$\pm$5.3 17$\pm$5.5 $-9.30^{+0.12}_{-0.17}$
field green 1.133 2.6 31 31$\pm$4.0 17$\pm$3.6 17$\pm$3.8 $-9.34^{+0.09}_{-0.11}$
§.§ Running Average
Given the nature of stacking, it is possible that one galaxy could significantly alter the stacked properties and lead to the dip in the intermediate bin and/or high dust temperatures in the earliest accreted bin. We test this by performing a running bin average, where each subsequent bin (sorted by phase-space values) replaces the lowest galaxy with the next galaxy in the list, yielding 36 (non-independent) bins in total. The full probability analysis is executed for each running bin, with the weighted mean dust temperature plotted in Figure <ref>. There is a smooth decline in dust temperature from the lowest phase-space bin toward the intermediate bin. This suggests that the drop in dust temperature is due to the overall population of galaxies as they move toward higher values, rather than a single outlier galaxy which would manifest as a sharp drop in the running mean.
We also note that our sample of [OII] emitters contains two of the three brightest cluster galaxies (BCGs) with the lowest values of . This is not surprising given that beyond $z\sim1$ BCGs are more likely to contain star formation as seen at 24 (Webb et al., submitted). Again, we can confirm these two BCGs are not single-handedly augmenting the dust temperature in the earliest accreted bin as there is a steady change in the running average. In fact, the third point in Figure <ref> roughly corresponds to the dust temperature without the two BCGs (though technically this bin also includes two additional sources that are normally in the intermediate bin since the number of sources in each bin remains at a constant 12). If we adopt this value to compare to the intermediate bin, the significance of the change in dust temperature between the intermediate (purple) and earliest accreted (orange) bins remains unchanged at $\sim2.7\sigma$ (see <ref>).
The weighted mean dust temperature as a function of the mid bin from the running average. The galaxies are sorted by ascending values of and each bin is shifted by one galaxy, for 12 binned galaxies in total. The four independent bins from our analysis are highlighted with their appropriate colors from Figure <ref>. Arbitrary values of are plotted as vertical gray lines, showing the last bin that contains galaxies below that value of . The lack of a sharp transition between any of the bins signifies that there are no single galaxies responsible for the trend.
§.§ Properties of Star-forming Cluster Members at $z\sim1.2$ as a function of Phase Space
Our main results are highlighted together in Figure <ref>, which displays various star-forming galaxy properties as a function of cluster phase space (i.e., accretion history), and out to the field. In the upper panel, we again plot the strength of the 4000 Å break versus , but only for star-forming cluster members this time. The break is measured on the stacked spectra in each phase-space bin, with uncertainties from 1000 bootstrap resamplings in each bin. There is a monotonic decrease in the strength of the break toward cluster members that are infalling and/or most recently accreted (blue and maroon points), both of which are also consistent with the field value (green circle).
In the two middle panels of Figure <ref>, we plot the infrared luminosity-derived SFRs and SSFRs as a function of phase-space bin. There exists two discrete levels of star formation: $\sim20\,$ for the field and recently accreted populations, and a drop to $\sim10\,$ for the intermediate and earliest accreted populations. After dividing by the average stellar mass in each bin, we find a decline in SSFR with accretion history, with a 0.8 dex drop between infalling galaxies and those accreted at earlier times. This confirms our work in <cit.> with MIPS galaxies in a $z=0.872$ cluster, but now with a larger sample of star-forming galaxies, more robust star-formation rates, and at a higher redshift of $z=1.2$. In open circles, we also plot the average star formation divided by the total stellar mass of all members in each bin (with and without [OII] detections). This is more akin to measuring the fraction of star-forming galaxies. Again, we see a sharp transition to the intermediate phase-space bin. Given that star-forming field galaxies from $1.1<z<1.2$ have levels of star formation consistent with the recently accreted and infalling populations, this could be suggestive of environment playing an active role in the quenching of star formation, but only after a delayed time in phase space. This delayed period could represent a period of constant SFR after the initial infall time, a long fading time after a mechanism begins to quench star formation, or a combination thereof. However, we need to further control for stellar mass to disentangle any effects from mass-quenching in order to elucidate the precise role of environment. We do note, however, that in the field at $z\sim1$ there is only a $\sim0.2$ dex drop in the SSFR of main-sequence star-forming galaxies as a function of stellar mass over our entire mass range. We plot this as a dashed black line, calculating the SSFR for star-forming galaxies with the highest and lowest stellar masses in our sample, using the $z=1$ field relation from <cit.>. To allow for the greatest degree of mass segregation in the cluster, we plot the SSFR corresponding to the lowest (highest) stellar mass at the largest (smallest) values of , maroon and orange bins, respectively. We normalize the SSFR at low stellar masses to equal that of the maroon bin to illustrate the expected drop in SSFR as a function of assuming the largest possible range of stellar mass (2.1 dex). In reality, the median stellar mass only varies by 0.5 dex from high to low values of , and this effect is even smaller. It therefore seems unlikely that mass segregation could fully account for the 0.8 dex drop we see as a function of accretion history, but probably exaggerates the trend.
The bottom panel shows the weighted mean dust temperature for the dynamically distinct galaxy populations. There is a $3.5\sigma$ drop in dust temperature in the intermediate phase-space (purple) bin (20 K) compared to the average of the recently accreted (blue and maroon) bins, and a subsequent $2.6\sigma$ rise to 64 K to the earliest accreted galaxies (orange). The field population is consistent to within 1.9$\sigma$ of the two infalling galaxy bins. If we make the simple assumption that there is a flat trend in dust temperature as a function of phase space and exclude the intermediate bin, we find the best fit amplitude to the remaining three cluster bins of $47\pm5.6$ K. This yields a $4.0\sigma$ deviation between the intermediate phase-space bin and an otherwise flat temperature distribution.
The depth of the 4000 Å break (upper panel), SFR and SSFR (middle panels), and dust temperature (lower panel) for star-forming galaxies as a function of phase space (solid points). The corresponding field values are shown in the right panel (green circles). The 4000 Å break is measured on stacked spectra, with uncertainties estimated from 1000 bootstrapped resamplings in each bin. The lower three panels are weighted means measured from the full probability distribution (see Figure <ref>), with rms uncertainties. The open circles represent the SFR divided by the total stellar mass from [OII] and non-[OII] cluster members. The dashed black line in the SSFR panel corresponds to the change expected in SSFR over the stellar mass range given the $z=1$ field trend from <cit.>.
§ DISCUSSION
Combining the best fit SEDs in Figure <ref>, the running mean from Figure <ref>, and the phase-space trends in Figure <ref>, the most simplistic view of the data, moving from infalling to virialized (central) regions, is a removal of the warm dust component ($\lambda\lesssim100$ rest-frame), and a subsequent reheating of the cold dust ($\lambda\gtrsim100$ rest-frame). We attempt to interpret this trend through the interplay between gas and dust within galaxies.
§.§ Multi-component Dust and Gas Phases in the Infrared
The gamut of dust grain sizes gives rise to many features in the infrared regime, and each dust component is sensitive to a particular heating mechanism. The full infrared SED is thus a composite of modified blackbodies each described by a dust temperature. Many studies have attempted to disentangle the dust temperature components with specific gas phases in the interstellar medium <cit.>.
In the rest-wavelength regime of our sample, the dust is primarily composed of three temperature phases. Warm dust ($T\sim40$ K) peaking at $\sim40$–100 is associated with the younger stellar population, and therefore probes ionized gas around star-forming regions. The atomic hydrogen in the galactic disk comprises cool dust of 20–30 K that emits at 100 . The coldest dust ($T\sim15$ K) traces quiescent molecular clouds, peaking at $\sim200$. It is now becoming evident that the evolved stellar population also plays a role in heating the large grains associated with the cooler dust at $\lambda>160$ <cit.>. Nevertheless, many recent studies have found that far-infrared wavelengths also correlate with direct tracers of star formation and gas, indicating that the cold dust is at least partially heated from younger stars, or probes the density of gas that fuels star formation <cit.>.
We note that, ideally, a multi-temperature modified blackbody is required to fully understand the contributions from each component. However, due to lack of available photometry at high-redshift, we risk overfitting the data and therefore must rely on an average dust temperature. Moreover, there is a degeneracy between estimates of warm and cold dust temperatures from a two-temperature model <cit.>, which could further confuse any interpretation. This therefore warrants caution in comparing our absolute dust temperatures and SFRs to other studies with multiple dust components, and instead focus on the relative differences within our own sample. Nevertheless, our measured dust temperatures are consistent with other studies that adopt a single-temperature modified blackbody. For example, <cit.> and <cit.> measure temperatures of $15\lesssim T\lesssim60$ K for field galaxies out to $z\sim2.8$, with galaxies in lower-redshift clusters ($z<0.25$) displaying a similar range of values from $10<T<70$ K though mostly concentrated between $20-45$ K <cit.>.
§.§ A Simple Interpretation for Quenching
We attempt to explain the above star formation and dust temperature trends with a simple interpretation of the possible quenching mechanisms at work, in light of the various dust phases. Starting from the cluster infall regions moving to the intermediate (purple) phase-space bin, we see a $\sim3.5\sigma$ drop in dust temperature and a sharp decline in the star-formation rate. The SED of the intermediate (purple) bin in Figure <ref> displays a striking lack of warm and cool dust components at rest-frame wavelengths shorter than 100, with only the coldest dust (represented by the SPIRE emission) dominating. In general, some cold dust is heated by the interstellar radiation field, thereby complicating SFR estimates from luminosity-weighted blackbodies. However, there is recent evidence that a substantial fraction of cold dust is also heated by ongoing star formation, given a strong trend between the cold dust temperature and SFR normalized by the dust mass surface density <cit.>.
Progressing to the central (orange) phase-space bin, we find another change in the mean dust temperature, climbing to 64 K from the intermediate bin at a level of 2.6$\sigma$ (but only a 1.1$\sigma$ change from the two infalling bins). The central (orange) SED in Figure <ref> is devoid of the coldest dust emission at longer wavelengths, and displays scaled-down warm dust emission at rest-frame $\lambda<100$. While the SFR remains roughly constant in these two interior bins (orange and purple), the SSFR drops an additional 0.4 dex.
Many mechanisms have been invoked to explain the evolution of cluster galaxies as they fall into the cluster potential, such as strangulation <cit.>, ram-pressure stripping <cit.>, galaxy harassment <cit.>, and viscous stripping <cit.>. Ram-pressure stripping is a more violent process, operating deep within the cluster potential <cit.> and occurring when high-velocity galaxies move through the dense intracluster medium. The intracluster medium (ICM) exerts a strong dynamical pressure on these galaxies, capable of stripping the cold disk of gas, thereby directly exhausting the fuel for star formation. Molecular clouds, however, are impenetrable to the effects of stripping due to their higher densities. Ram-pressure stripping is thus thought to have a suspended, then rapid ($\lesssim100$ Myr), effect on star formation <cit.>; molecular clouds already in existence can still form stars, but the removal of atomic disk gas prevents further production of molecular hydrogen. Star formation thus ceases on the scale of a molecular cloud lifetime (approximately tens of millions of years; ).
The phase-space trends in Figure <ref> seem consistent with a delayed timescale for cluster-specific processes (or a long combination of delay plus fading time), as both the SFR and dust temperature remain constant until $ <1.0$. At this point, many of the galaxies would have likely encountered the denser regions of the ICM, either on their first infall or on their way back out. We note that statistically, the intermediate (purple) phase-space bin also contains a higher percentage of backsplash galaxies <cit.> than the other bins using the regions of <cit.>. Many of the galaxies in this intermediate (purple) bin (either those approaching pericenter or a backsplash population) are thus susceptible to ram-pressure stripping, which can effectively remove the entire HI component in the disk, along with the hot halo gas. The dust could be stripped concurrently as spiral galaxies are likely to contain diffuse ionized gas in their halo that is associated with the presence of dust <cit.>. Moreover, observational evidence for dust stripping in galaxies with likely ongoing ram-pressure stripping has been detected <cit.>. Therefore, the warmer dust component could be removed with the gas, along with the cool dust associated with the HI <cit.>. As the molecular clouds are left unperturbed, the remaining dust would be the coldest component associated with dense quiescent molecular clouds. This is corroborated in the intermediate (purple) SED, where only the coldest dust emission remains, and there is a rapid drop to low dust temperatures in this phase-space region. Over time, these surviving molecular clouds could eventually fragment and form additional stars. This is consistent with the residual star formation in the central (orange) phase-space bin, and the reheating of the cold dust as the new stars ionize the gas and heat the surrounding dust. Our results thus seem consistent with a scenario in which the cluster has little effect on star-forming galaxies until they reach a sphere of influence, possibly within higher ICM densities where it is conducive to ram-pressure stripping. At this point, there is a rapid drop in dust temperature and SFR.
Granted, we are intentionally selecting star-forming galaxies as determined by [OII] emission. We thus should expect to find some level of dusty star-formation in each phase-space bin. The more interesting observation is that we see evidence for a change in the level of star formation activity and dust temperature as a function of time-averaged density (i.e., phase space). Moreover, this seems consistent with ram-pressure stripping only occurring at later accretion times in the intermediate (purple) phase-space bin. If many of these galaxies are backsplash or close to pericenter, it suggests that roughly one cluster crossing is required before any quenching effects are observed. This corresponds to $\sim1$ Gyr in our cluster sample.
§.§ Dust Properties and Quenching of Cluster Galaxies in the Literature
With the advent of Herschel, many studies have now begun to investigate the dust properties of galaxies as a function of environment, for example, the Herschel Virgo Cluster Survey <cit.>, the Herschel Reference Survey <cit.>, the Local Cluster Substructure Survey <cit.> and the Herschel-Astrophysical Terahertz Large Area Survey <cit.>, albeit all at lower redshifts than presented here. Almost all these studies have found a morphological <cit.> and/or environmental dependence <cit.> on the properties of dust. Specifically, extra-planar dust existing off the disk has been found to be coincident with stripped gas in cluster galaxies <cit.>, and the extent of dust in the disk is truncated in HI-deficient Virgo galaxies <cit.>. These results all seem to suggest that environmental mechanisms are capable of altering the distribution of dust and/or stripping it along with the gas; in most cases, ram-pressure stripping is invoked to explain the trends. Moreover, while <cit.> find dust stripping from the disk to occur in star-forming cluster galaxies, they find it to be less efficient than the removal of atomic gas. This can be explained with the presence of a less-extended dust disk that instead follows the more compact molecular gas phase, and is consistent with our findings of only the coldest dust remaining in the intermediate (purple) phase-space bin as a result of ram-pressure stripping.
Our simple interpretation of ram-pressure stripping as a viable quenching mechanism is compatible with other studies. <cit.> find a bimodal distribution of SSFRs, split between central and satellite galaxies in $z\simeq0$ group and cluster galaxies from the SDSS. They claim that environment has little effect on satellite galaxies until they cross the virial radius. They derive quenching timescales to explain this trend <cit.>, finding a plausible scenario in which satellite galaxies undergo delayed ($\sim$2–4 Gyr), then rapid ($<0.8$ Gyr) quenching, likely due to ram-pressure stripping. Similarly, Balogh et al. (submitted) propose both a long delay timescale and shorter fading time that are dependent on stellar mass, though suggest that overconsumption might be the primary cause for quenching of high SFRs in groups and clusters at $z=0.9$. In a large sample of clusters over $0.3<z<1.5$, <cit.> propose multiple mechanisms for altering SFRs, with strangulation working in higher-mass galaxies and ram-pressure stripping causing a quick transition in the fraction of star-forming galaxies within the clusters at all redshifts. Moreover, direct evidence for ram-pressure stripping—in the form of trailing HI gas tails—has previously been observed in the Virgo cluster <cit.> and in a merging cluster at $z=0.3$ <cit.>.
Further support for mechanisms which quench star formation rapidly after a delayed period emerges with the observed abundance of poststarburst galaxies in clusters compared to the field <cit.>. These galaxies exhibit strong Balmer absorption (e.g., H$\delta$) and weak emission lines (e.g., little-to-no [OII] emission), indicative of star formation that ended abruptly within the last few hundred million years <cit.>. A population of poststarburst galaxies in cluster cores is expected from ram-pressure stripping given the resulting removal of disk gas, a possible compression of molecular clouds from the shock, and a subsequent rapid decline in star formation on the order of molecular cloud lifetimes.
<cit.> have recently uncovered a correlation in the location of poststarburst galaxies with phase-space in SpARCS/GCLASS clusters, finding them preferentially located in an intermediate phase-space bin with higher line-of-sight velocities. Their star formation history is best reproduced with a model that adapts a long delay period ($\sim2$ Gyr) before a short (0.4 Gyr) quenching of star formation. <cit.> also find a segregation in phase space for HI-detected galaxies in a $z=0.2$ cluster. They infer the presence of a stripping region in phase-space at low radii and/or high-velocities where ram-pressure can sufficiently remove the HI gas and lead to a decay in SFRs. Our Herschel study corroborates these findings as we measure a trend in dust temperature that is consistent with a complete removal of all warm and cool (ISM) dust after first infall that can be explained with ram-pressure stripping.
In the work presented here, we have specifically selected galaxies with [OII] emission in order to expose environmental trends with star-forming galaxies; therefore, we are not likely observing a current poststarburst population. However, we could be witnessing the central and/or intermediate-bin galaxies during the delayed period sometime after ram-pressure stripping. While they do exhibit somewhat depressed levels of star-formation compared to their recently accreted counterparts, they certainly are not a quenched population. Perhaps these are the progenitors of poststarburst (i.e., recently quenched) galaxies before a rapid suppression of star formation. Some authors have even suggested that poststarburst galaxies could be the descendants of e(a) galaxies: dusty starbursts with both moderate emission lines and strong Balmer absorption <cit.>. These galaxies are thought to contain a multi-phase dust distribution that obscures young OB stars in HII regions while leaving A stars relatively unaffected <cit.>. We would thus be observing these galaxies in the calm before the storm—a delayed period of somewhat passive evolution, while there is still ample molecular gas available to fuel star formation before a more rapid quenching commences.
§ CONCLUSIONS
We have presented a Herschel study of star-forming galaxies in three $z\sim1.2$ clusters from the SpARCS-GCLASS sample. We stack PACS (100 and 160 ) and SPIRE (250, 350, and 500 ) maps at the location of galaxies with [OII] emission—the star-forming population—binned by their location in $(r/r_{200})\times(\Delta v/\sigma_v)$ phase space based on our previous study in <cit.>. We utilize bins with an equal number of 12 star-forming galaxies: $(r/r_{200})\times(\Delta v/\sigma_v) <0.20$ (central bin); $0.20<(r/r_{200})\times(\Delta v/\sigma_v) <0.64$ (intermediate bin); $0.64<(r/r_{200})\times(\Delta v/\sigma_v) <1.35$ (recently accreted bin); and $(r/r_{200})\times(\Delta v/\sigma_v) >1.35$ (infalling bin). This isolates the earliest accreted cluster galaxies from galaxies that have completed at least one passage through the cluster, and those that are currently infalling. We summarize our results as follows:
* We fit the thermal portion of the stacked spectral energy distribution for each phase-space bin, deriving weighted mean dust temperatures and integrated infrared luminosities using photometry from 100–500 in the observed frame, corresponding to 45–230 rest-frame. After converting the luminosities to SSFRs, we find a steady 0.8 dex decline from the infalling population towards virialized (central) galaxies in the core. This confirms the MIPS study of a $z=0.871$ cluster presented in <cit.>. The actual SFRs and the fraction of star-forming galaxies, however, display a rapid decline in the intermediate phase-space bin, suggesting a rapid quenching of cluster galaxies.
* There exists a $\sim$4.0$\sigma$ drop in the dust temperature for the intermediate phase-space bin, when compared to a flat trend that is fit to the infalling and central bins. Its full probability distribution favors cooler dust temperatures and occupies a distinct region in the 2D parameter space of temperature and amplitude for a modified blackbody. The recently accreted/infalling galaxies have both dust temperatures and SSFRs consistent with the $1.10<z<1.21$ field galaxies, suggesting they have not undergone any substantial evolution within the cluster yet.
* A running bin average of weighted dust temperatures shows a removal of warm dust moving inwards in phase space, from the infalling galaxies to the intermediate phase-space bin. There is then a steady rise in the dust temperature towards the virialized (central) galaxies.
* We propose a simple interpretation for quenching in which infalling galaxies remain unscathed until roughly one cluster crossing ($\sim1$ Gyr). At that point, they experience the violent stripping of all dust and gas components, except in the densest regions of quiescent molecular clouds which contain the coldest dust. As the surviving clouds form stars, there is a reheating of the coldest dust in the earliest accreted star-forming galaxies.
We emphasize that this last conclusion is just a plausible quenching model to explain the observed trends in star formation activity and dust temperatures as a function of phase-space environment. We have not attempted to account for mass segregation within the clusters, which could partially contribute to the trends. Further observations of the gas components are crucial to verify this claim; a detailed study of CO gas in cluster galaxies would provide insight into the available reservoir of molecular gas that fuels star formation. However, qualitatively, it illustrates the power in the parameterization of in phase space as a means for studying galaxy evolution in the context of time-averaged densities.
We thank the anonymous referee, whose comments improved the clarity of the manuscript. The authors would also like to thank numerous people for useful discussions, including Rachel Friesen, Suresh Sivanandam, Alexander van Engelen, and Marco Viero. This work is based in part on observations made with Herschel, a European Space Agency Cornerstone Mission with significant participation by NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech.
TMAW acknowledges the support of the NSERC Discovery Grant. HKCY is supported by the NSERC Discovery Grant and a Tier 1 Canada Research Chair. GW gratefully acknowledges support from NSF grants AST-0909198 and AST-1517863. RFJvdB acknowledges support from the European Research Council under FP7 grant number 340519.
Facilities: Herschel Space Observatory (PACS; SPIRE), Spitzer Space Telescope (MIPS; IRAC), Gemini (GMOS)
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1511.00238
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Indian Institute of Science Education and Research, Kolkata, Mohanpur, 741252, Nadia, West Bengal, India
Institut für Theoretische Physik, J.W.Goethe Universität, Max Von Laue-Straße 1, D-60438 Frankfurt Am Main, Germany
Institut für Theoretische Physik, J.W.Goethe Universität, Max Von Laue-Straße 1, D-60438 Frankfurt Am Main, Germany
Compact stars consisting of massless quark matter and fermionic dark matter are studied by solving the Tolman-Oppenheimer-Volkoff equations for two fluids separately. Dark matter is further investigated by incorporating inter-fermionic interactions among the dark matter particles. The properties of stars made of quark matter particles and self-interacting and free dark matter particles are explored by obtaining their mass-radius relations. The regions of stability for such a compact star are determined and it is demonstrated that the maximum stable total mass of such a star decreases approximately linearly with increasing dark matter fraction.
§ INTRODUCTION
A quark star is a hypothetical compact star and consists of self-bound strange quark matter (SQM) <cit.>. The existence of quark stars is controversial and its equation of state is also uncertain.
One of the popular models for the equation of state of the quark star is the so called MIT Bag model <cit.>. The model is often used for describing cold and massless (strange) quark matter <cit.>. Standard values for the MIT bag constant are around $B^{1/4}$ = 145 MeV as follows from fits to hadron masses <cit.>, which results in maximum masses of about 2.0$ M_{\odot}$ at a radius of about 11 km <cit.> , which are actually very close to the ones of realistic neutron star models.
Dark matter stars are modelled in our work as a free or self-interacting fermion gas at zero temperature. The possible candidates for dark matter particles are a type of fermion predicted in extensions of the standard model including supersymmetric particles, the neutralino, the gravitino and the axino <cit.>. In our discussion, we consider dark matter to be made of fermionic particles with a mass of 100 GeV, the classical WIMP mass scale. We assume that the dark matter particles cannot self-annihilate as in asymmetric dark matter (ADM) <cit.>.
Self-annihilating WIMP dark matter with masses above a few GeV accreted onto neutron stars may trigger a conversion of most of the star into a strange star <cit.> or the accreted dark matter may significantly affect the kinematical properties of the compact star <cit.>. Constraints on the properties of dark matter candidates can be obtained from stars which can accrete asymmetric dark matter in its lifetime and then collapse into a neutron star <cit.>. Constraints on the mass of dark matter candidates can also be obtained by the possible collapse of compact stars due to dark matter accretion <cit.>. The cooling process of compact objects can be affected by the capture of dark matter which can annihilate the star <cit.>. Recent studies have been done to explore compact stars with non self-annihilating dark matter to analyze the gravitational effects of dark matter on the stellar matter under intense conditions <cit.>. In these studies, masses of dark matter in the GeV range has been assumed. Studies have also been performed to investigate the compact objects formed due the admixture of neutron star matter and dark matter <cit.> leading to the possibilities for new stable solutions of compact stars with planet like masses. To study of the effect of dark matter on compact objects is therefore of great interest. If quark stars do exist in nature, they can also accumulate dark matter and hence their properties might change. This accumulation will lead to various changes in the mass-radius relation of a quark star which is studied in this work.
This paper is organised as follows: in section 2, we shortly discuss about the two fluid TOV equations. In section 3, we discuss the equations of state for both quark matter and fermionic dark matter and discuss the general scaling solutions for these stars. In section 4, we present the numerically obtained results (mass-radius relations) for quark matter stars by solving the TOV equations. In section 5, TOV equations are solved for dark matter composed of both strongly self-interacting and free fermions and their corresponding mass-radius relations are obtained. Section 6 is dedicated to the numerical solutions of two fluid TOV equations namely, dark matter and quark matter which are coupled together only by gravity. We demonstrate that the maximum mass of the quark star admixed with dark matter reduces due to the presence of dark matter and decreases in a linear fashion in case in strongly self-interacting dark matter fermions while for free dark matter, the maximum mass remains almost unaffected. Finally, in section 7, we summarize our findings and discuss our results.
Throughout the paper, we use natural units where c=$\hbar$=1, c being the speed of light and $\hbar$ is the reduced Planck's constant.
§ TWO FLUID TOLMANN-OPPENHEIMER-VOLKOFF EQUATIONS
Since our aim is to see the properties of a quark star admixed with dark matter, we need the TOV equations for two fluids admixed with each other. There will be a hydrostatic equilibrium condition for each of the two fluids and the fact that there is only gravitational interaction between them will be encoded in the metric describing the system. The two fluid TOV equations that we use here are <cit.>:
\begin{equation}
\begin{split}
\emph{\(\frac{dp_{1}}{dr}\)} &=-\emph{\(\frac{GM(r)\rho_{1}(r)}{r^{2}}\)}\left(1+\emph{\(\frac{p_{1}(r)}{\rho_{1}(r)}\)}\right)\times \\
& \left(1+4\pi r^{3} \emph{\(\frac{(p_{1}(r)+p_{2}(r))}{M(r)}\)}\right)\left(1-2G\emph{\(\frac{M(r)}{r}\)}\right)^{-1}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\emph{\(\frac{dp_{2}}{dr}\)} &=-\emph{\(\frac{GM(r)\rho_{2}(r)}{r^{2}}\)}\left(1+\emph{\(\frac{p_{2}(r)}{\rho_{2}(r) }\)}\right)\times\\
&\left(1+4\pi r^{3} \emph{\(\frac{(p_{1}(r)+p_{2}(r))}{M(r)}\)}\right)\left(1-2G\emph{\(\frac{M(r)}{r}\)}\right)^{-1}
\end{split}
\end{equation}
\begin{equation}
\emph{\(\frac{dM_{1}}{dr}\)}=4\pi r^{2} \rho_{1}(r)
\end{equation}
\begin{equation}
\emph{\(\frac{dM_{2}}{dr}\)}=4\pi r^{2} \rho_{2}(r)
\end{equation}
\begin{equation}
\end{equation}
Here M(r) represents the total mass at radius r, $p_{1}$, $p_{2}$, $\rho_{1}$ and $\rho_{2}$ are the pressures and densities of the fluids 1 and 2 respectively. We could separate out the hydrostatic equilibrium condition for the two stars into equations (1) and (2) because the interaction acts only through gravity and nothing else. The gravitational interaction is taken into account because of the fact that the mass that is considered in the equation M(R) is the total mass of both the fluids at radius r which means each of the fluid attracts the other gravitationally. The equations for the conservation of mass for the two fluids remains the same as that for individual fluids.
For solving the two fluid TOV equations, we need proper boundary conditions. $M_{1}(0)$ and $M_{2}(0)$ must be equal to zero at r=0. Central pressures for the two fluids are calculated from the central densities given as the initial condition using the respective equation of states for the two fluids. Then the two TOV equations are solved together simultaneously and we obtain either $R_{1}$ or $R_{2}$ as the radius of the complete star depending on which fluid ends up having a larger radius. The radius of the individual fluids occur at those points where the individual pressures drop down to zero.
§ EQUATIONS OF STATE FOR QUARK MATTER AND FREE AND SELF-INTERACTING DARK MATTER
The equations for state for quark matter is discussed using MIT Bag model. The EOS for free dark matter particles along with strongly self-interacting dark matter particles is briefly described using statistical mechanics of free and self-interacting fermions. Scaling relations for quark stars and dark matter stars is also discussed.
§.§ Equation of state for quark matter
The MIT Bag equation of state <cit.> is taken as the equation of state for quark matter in our work. In this model, the quarks are assumed to be made of free fermions constrained within a bag with a vaccumm pressure that keeps the particles within the bag. The MIT Bag equation of state is:
\begin{equation}
p = \emph{\(\frac{1}{3}\)}(\epsilon -4B)
\end{equation}
Here, p denotes the pressure, $\epsilon$ denotes the energy density and B is the Bag constant whose standard accepted values are around B1/4 = 145 MeV or B1/4= 200 MeV <cit.>. Note that the equation of state for a cold gas of interacting massless quarks within perturbative quantum chromodynamics can be approximated by a similar form of equation of state as the MIT Bag model <cit.>.
§.§ Equation of state for free and self-interacting dark matter fermions
Dark matter will be assumed to be made of fermions of mass 100 GeV. For a study of compact fermionic stars, we refer to some recent papers <cit.>. The equation of state for a gas of free fermions can be calculated via explicit expressions for energy density ($\epsilon$) and pressure (p) <cit.>
\begin{equation}\label{xx}
\small
\begin{split}
\epsilon &=\emph{\(\frac{1}{\pi^{2}}\)} \int_{0}^{k_{F}} k^2 \sqrt{m_f^2 + k^2} dk\\
\end{split}
\end{equation}
\begin{equation}\label{xx}
\small
\begin{split}
p &=\emph{\(\frac{1}{3\pi^{2}}\)} \int_{0}^{k_{F}} \emph{\(\frac{k^4}{\sqrt{m_f^2 + k^2}}\)} dk\\
\end{split}
\end{equation}
where z=$k_{F}$/$m_{f}$ is the dimensionless Fermi momentum.
Similarly, the interactions between the fermions is modelled by considering the simplest two-body interactions between fermions. The repulsion amongst the fermions constituting the dark matter star has been modelled by considering the interaction energy density to be proportional to $n^2$ <cit.> to the lowest order approximation, where $n$ is the number density of fermions. The resulting equation of state has been calculated in reference <cit.>:
\begin{equation}\label{xx}
\small
\begin{split}
\emph{\(\frac{\epsilon}{m_f^{4}}\)} &= \emph{\(\frac{1}{8\pi^{2}}\)}[(2z^3+z)\sqrt{1+z^2}-sinh^{-1}(z)] \\& + [\left(\emph{\(\frac{1}{3\pi^{2}}\)}\right)^2 y^2z^6]
\end{split}
\end{equation}
\begin{equation}\label{xx}
\small
\begin{split}
\emph{\(\frac{p}{m_f^{4}}\)} &=\emph{\(\frac{1}{24\pi^{2}}\)}[(2z^3-3z)\sqrt{1+z^2}+3sinh^{-1}(z)]\\ &+[\left(\emph{\(\frac{1}{3\pi^{2}}\)}\right)^2 y^2z^6]
\end{split}
\end{equation}
where z=$k_{F}$/$m_{f}$ is again the dimensionless Fermi momentum and y is the dimensionless interaction strength. Also, y=$m_{f}$/$m_{I}$ where $m_I$ is the scale of interaction.
The mass of the fermions $m_{f}$ used for self-interacting and free dark matter has been taken as 100 GeV and it is assumed that they don't self annihilate <cit.>. The value of the interaction strength y determines whether the self-interactions are weak or strong. For example, neutralinos, that form a candidate for WIMP dark matter and are in the mass range of 100 GeV <cit.> can have weak self-interactions with y $\sim$ 0.1 or they may be strongly self-interacting with y $\sim$ $10^3$ where the strong interaction scale corresponds to $\Lambda_{QCD}$ $\simeq$ 200 MeV. In our discussions, we would focus on two situations, one for free fermionic dark matter matter with y=0 and the other for strongly self-interacting dark matter with y=$10^3$. For a reference of self-interacting fermionic stars and the corresponding interaction strengths we refer to <cit.>.
§.§ Scaling relations for quark matter and dark matter
We generally scale dimensional quantities to dimensionless ones in order to represent any arbitrary mass configuration of a star in a single graph. From equation (6), it is clear that if we scale the energy density and pressure values by four times the Bag Constant (4B), the EOS reduces to a dimensionless form <cit.> the corresponding total mass and radius of the star would then be scaled by $\sqrt{4B}$. Similarly , for the dark matter fermionic particles, it is a natural choice to scale the pressure and energy density by the fermion mass $m_{f}^4$ which will again make the equations dimensionless. The scaling relations for quark matter are $\epsilon^{\prime}_{quark}$ = $\epsilon_{quark}$/(4B), $p^{\prime}_{quark}$ = $p_{quark}$/(4B), $M^{\prime}_{quark}$ = $M_{quark}$/(2$\sqrt B$), $R^{\prime}_{quark}$ = $R_{quark}$/(2$\sqrt B$). The corresponding relations for fermions are $\epsilon^{\prime}_{f}$ = $\epsilon_{f}$/$m_{f}^4$, $p^{\prime}_{f}$ = $p_{f}$/$m_{f}^4$, $M^{\prime}_{f}$ = $M_{f}$/a, $R^{\prime}_{f}$ = $R_{f}$/b where a = $M_{p}^{3}$/$m_{f}^{2}$ and b = $M_{p}$/$m_{f}^{2}$ where $M_{p}$ is the Planck Mass ( G = $M_{p}^{-2}$ ). For detailed derivations of the scaling relations we refer to <cit.>.
§ SOLVING TOV EQUATIONS FOR QUARK MATTER STAR
Numerical solutions to the mass-radius relations of quark stars can be found in literature <cit.>. In our nomenclature $M_{quark}$ and $R_{quark}$ represents the mass and radius of the quark star respectively. The curve is shown in Fig. 1 for two different Bag values.
Mass ($M_{quark}$) vs. Radius ($R_{quark}$) curve for quark stars for two different Bag values.
Upto a certain point mass increases with the radius reaching a maximum value of mass at a certain value of radius after which the mass starts decreasing, the star starts becoming unstable from this point.The maximum stable mass for $\textit{B}^{1/4}$ = 145 MeV is about 2.01 $ M_{\odot}$ and the corresponding radius of around 11 km. While $\textit{B}^{1/4}$=200 MeV gives a maximum mass of about 1.06 $M_{\odot}$ with the radius being around 5.8 km. Quark stars are incompressible stars and form a self bound system <cit.>.
§ SOLVING TOV EQUATION NUMERICALLY FOR FREE AND STRONGLY SELF-INTERACTING DARK MATTER PARTICLES
§.§ Solutions for free fermionic dark matter
We first consider dark matter made of free fermionic particles with 100 GeV mass. Single fluid TOV equations are solved taking (7) and (8) as the equation of state. $M_{dark}$ and $R_{dark}$ represents the mass and radius of the dark matter star composed of free fermions respectively. The resulting mass-radius curve is plotted in Fig. 2.
Plot of the mass ($M_{dark}$) vs. radius ($R_{dark}$) for a free gas of fermions of mass 100 GeV at zero temperature. This graph corresponds to the mass-radius curve of the dark matter composed of free fermionic particles.
From the graph, we see that the mass at first increases with a decrease in radius for increasing central energy density values, reaches a maximum and then starts decreasing. Stellar configurations to the right side of the maximum masse are stable whereas those on the left side are unstable. The maximum stable mass for the dark matter star made of free fermions comes out to be 6.27$\times$ $10^{-5}$ $M_{\odot}$ with a radius of 0.81 meters.
§.§ Solutions for strongly self-interacting dark matter
The TOV equations are solved for strongly self-interacting dark matter particles (y = $10^3$) of mass 100 GeV. The equation of state used are (9) and (10). $M_{int}$ and $R_{int}$ represents the mass and radius of the dark matter star composed of strongly self-interacting fermions respectively. The resulting mass-radius curve is plotted in Fig 3.
Plot of mass $M_{int}$ vs. radius $R_{int}$ for strongly self-interacting dark matter fermionic particles (y = $10^{3}$).
For strong self-interaction, the mass and radius are much larger compared to free fermions and hence the maximum mass and the minimum radius is about 1000 times larger. From the curve it is observed that for very low central densities of dark matter particles, i.e the tail of the graph, the rate of increase of mass with decreasing radius is much higher as compared to the free dark matter particle case discussed in the previous subsection. The maximum mass and the minimum radius for the self-interacting dark matter star turns out to be 2.67 $\times$ $10^{-2}$ $M_{\odot}$ and 0.189 km respectively, larger than for the non interacting case due to repulsive forces between the dark matter particles.
§ SOLUTION OF TOV EQUATION FOR AN ADMIXTURE OF QUARK MATTER AND DARK MATTER
Nomenclature used is $M_{quark}$ and $R_{quark}$ for the mass and radius of quark matter, $M_{dark}$ and $R_{dark}$ for the mass and radius of the star composed of free dark matter particles, $M_{int}$ and $R_{int}$ for strongly self-interacting dark matter star. $\epsilon_{0,quark}$, $\epsilon_{0,dark}$ and $\epsilon_{0,intdark}$ represents the central energy densities of quark matter, dark matter made of free fermions and dark matter made of strongly self-interacting fermions respectively.
§.§ Solution for combination of quark matter and free dark matter particles
Plot of mass ($M_{quark}$) vs. central energy density ($\epsilon_{0,quark}$) of quark matter for three different central energy densities of dark matter ($\epsilon_{0,dark}$).
The two fluid TOV equations (1), (2), (3) and (4) are solved for a mixture of quark matter with MIT Bag model by taking the bag value to be $B^{1/4}$= 145 MeV and dark matter composed of free fermionic particles of mass 100 GeV. We start with the initial given central energy densities for the two components and compute the corresponding central pressures using the EOS for the respective fluids (eqn. (6) for quark matter and (7) and (8) for dark matter).
Plot of radius ($R_{quark}$) vs. central energy density ($\epsilon_{0,quark}$) of quark matter for different central energy densities of dark matter ($\epsilon_{0,dark}$).
Plot of mass ($M_{dark}$) vs. central energy density ($\epsilon_{0,dark}$) of dark matter for different central energy densities of quark matter ($\epsilon_{0,quark}$).
Plot of radius ($R_{dark}$) vs. central energy density ($\epsilon_{0,dark}$) of dark matter for different central energy densities of quark matter ($\epsilon_{0,quark}$).
Mass ($M_{quark}$) vs. central energy density of quark matter ($\epsilon_{0,quark}$) are plotted for three different values of central density of dark matter ($\epsilon_{0,dark}$) each kept constant at a time (See Fig.4). From the plot, it is clear that as the central density of dark matter is increased in the mixture, the maximum mass of the quark matter still reaches to 2.005 $ M_{\odot}$ but now at higher central densities ($\epsilon_{0,quark}$) of quark matter after which the quark matter becomes unstable and the star would collapse. This behaviour can be explained via the fact that as ($\epsilon_{0,dark}$) increases, then within the stable branch of dark matter, the allowed mass of dark matter inside the quark star also increases which contributes to a greater gravitational pull, so, a much higher central quark energy density ($\epsilon_{0,quark}$) is needed to support the maximum possible mass against the greater gravitational pull. The maximum stable mass of the quark component ($M_{max,quark}$) is almost the same as pure quark star (2.01 $M_{\odot}$) because the maximum possible value of the dark matter mass is 6.27 . $10^{-5}$ $M_{\odot}$ (Section 5.A) , which is much less than 2.01 $M_{\odot}$, to cause a notable reduction in the quark matter mass. Fig. 5 shows the plot for the radius ($R_{quark}$) vs. the central energy density ($\epsilon_{0,quark}$) of quark matter for different values of $\epsilon_{0,dark}$. The figure shows that the maximum stable radius of the quark matter is independent of the amount of free fermionic dark matter present in the admixed star.
Contour plot with (\(\frac{\epsilon_{0,quark}}{4\textit{B}}\) , \(\frac{\epsilon_{0,dark}}{m_f^4}\) , $M_{total}$(in $M_{\odot}$) ) as the x, y and z axis respectively. The region marked as A represents the stable region for the formation of quark star admixed with dark matter while all the points in region B are unstable to such a formation.
After observing that the maximum possible mass of quark matter is hardly reduced in the presence of free fermionic dark matter particles of various central densities, it is essential to determine which configurations of the admixed star are stable. The plots for the profile of dark matter component is obtained by keeping $\epsilon_{0,quark}$ fixed and slowly varying $\epsilon_{0,dark}$ (Fig. 6 and 7). It is seen that the dark matter masses and radii are the same for varying $\epsilon_{0,quark}$ which is expected since dark matter is more compact than quark matter and is not affected very much by the presence of quarks.
Figs. 4, 5, 6 and 7 allow us to analyse the stability of the entire configuration. Since we realise from fig. 6 and 7 that $\epsilon_{0,dark}$ for which dark matter mass hits a maxima is the same for all $\epsilon_{0,quark}$, we at first mark those points where the quark matter becomes unstable i.e hits the maximum mass by doing the plots done in figures 4 and 5 for different $\epsilon_{0,dark}$. As we slowly increase $\epsilon_{0,dark}$ , the dark matter content inside the admixed star keeps on increasing and the radius of the dark matter keeps on decreasing. After a sufficiently large $\epsilon_{0,dark}$ , the dark matter mass content hits a maximum after which the dark matter mass decreases with further increase in $\epsilon_{0,dark}$. This is then the unstable branch for the dark matter . After this critical value of $\epsilon_{0,dark}$, no dark matter configurations are stable and hence quark matter and dark matter can't exist together since the dark matter would collapse into a black hole. Hence we expect that up to a certain value of $\epsilon_{0,dark}$ , the quark matter stable mass increases to reach 2.005 $M_{\odot}$ for sufficiently high $\epsilon_{0,quark}$, and after a critical $\epsilon_{0,dark}$, dark matter itself becomes unstable which leads to instability of the entire admixture of the dark matter and quark matter.
Next we study the configurations in the $\epsilon_{0,quark}$ - $\epsilon_{0,dark}$ plane. At first, $\epsilon_{0,dark}$ is kept fixed and $\epsilon_{0,quark}$ is slowly increased. The stable boundary is marked in the contour plot (Fig.8) by the maximum stable quark matter mass for increasing $\epsilon_{0,dark}$ which gives the line inclined at an angle in the contour plot. The sequences continue up to the points where the dark matter mass reaches its maxima. Above this value of $\epsilon_{0,dark}$, all configurations become unstable since dark matter itself becomes unstable. This leads to the boundary line that is almost parallel to the x-axis.
For a quark star admixed with dark matter made of free gas of fermionic particles, the maximum possible mass of the stable configuration is approximately $M_{total}$ $\sim$ 2.01 $M_{\odot}$ with a dark matter content of around 0.63 $\times$ $10^{-4}$ $M_{\odot}$ which has a small radius of about 0.80 meters while the quark matter extends much further to a radius of around 11 km.
§.§ Solution for combination of quark matter and strongly self-interacting dark matter
The two fluid TOV equations (eqns. (1), (2), (3), (4)) are solved now for massive dark matter fermions taken to be strongly self-interacting using the model discussed in section 5.B. The interaction strength y is taken to be $10^{3}$. The presence of self-interactions causes the maximum stable mass of a dark star to be increased from about $10^{-4}$ $M_{\odot}$ for the free fermionic case to about $10^{-2}$ $M_{\odot}$ for the strongly self-interacting case (Section 5.A and 5.B).
Plot for the mass of the quark component ($M_{quark}$) vs. the quark central density $\epsilon_{0,quark}$ for three different dark matter central densities ($\epsilon_{0,intdark}$). It is visible that as $\epsilon_{0,intdark}$ is increased, the maximum stable quark mass ($M_{quark,max}$) is reduced and is now attained at a higher $\epsilon_{0,quark}$
Plot for the radius of the quark component ($R_{quark}$) vs. the quark central density $\epsilon_{0,quark}$ for different dark matter central densities ($\epsilon_{0,intdark}$).
The plot for the mass of the quark component ($M_{quark}$) vs. the central energy density $\epsilon_{0,quark}$ of the quark component for different values of $\epsilon_{0,intdark}$ (Fig. 9) reveals that the maximum stable mass of the quark matter decreases with increasing central energy density of dark matter ($\epsilon_{0,intdark}$) within the stable branch of dark matter, though the decrease is very moderate. The maximum stable mass of the quark component at $\epsilon_{0,intdark}$ = $10^5$ MeV/$fm^3$ is 1.995 $M_{\odot}$ and this mass reduces to 1.937 $M_{\odot}$ at $\epsilon_{0,intdark}$ = $ 2 \times 10^6 $ MeV/$fm^3$. It is also observed just as in the previous section that the maximum quark mass ($M_{quark,max}$) is attained at a much higher value of $\epsilon_{0,quark}$, the reason being the same as described in free fermion case. The corresponding plot for the radius of the quark component vs. $\epsilon_{0,quark}$ for two different $\epsilon_{0,intdark}$ is shown in Fig. 10 which shows that the maximum radius of quark component also decreases with an increase in $\epsilon_{0,intdark}$ .
Plot for the mass of the dark component ($M_{int}$) vs. the dark central density $\epsilon_{0,intdark}$ for different quark matter central densities ($\epsilon_{0,quark}$). It is visible that the dark matter mass profile is not altered very much with changing $\epsilon_{0,quark}$
Plot for the radius of the dark component ($R_{int}$) vs. the dark central density $\epsilon_{0,intdark}$ for different quark matter central densities ($\epsilon_{0,quark}$).
Contour plot with (\(\frac{\epsilon_{0,quark}}{4\textit{B}}\) , \(\frac{\epsilon_{0,intdark}}{m_f^4}\) , $M_{total}$(in $M_{\odot}$) ) as the x, y and z axis respectively. The region marked as A represents the stable region for the formation of quark star admixed with dark matter while all the points in region B are unstable to such a formation. \(\frac{\epsilon_{0,quark}}{4\textit{B}}\) starts from 1.0 in the plot because energy density of quark matter can't be zero according to the equation of state (6). .
Keeping $\epsilon_{0,quark}$ constant and obtaining $M_{int}$ vs. $\epsilon_{0,intdark}$ and $R_{int}$ vs. $\epsilon_{0,intdark}$ gives the profile for dark matter present in the admixture (Fig. 11 and 12). It is evident from Figs. 11 and 12 that the dark matter mass and radius profile does not change much with increasing $\epsilon_{0,quark}$ since dark matter is much more compact than quark matter and its particles are also much more massive to be significantly affected by quark matter particles. Figs. 9, 10, 11 and 12 allow us to determine the stability of the quark matter star admixed with self-interacting dark matter. For the first two plots showing the dependence of the mass and radius of the quark matter vs. $\epsilon_{0,quark}$ tells us up to which point the quark star configuration would remain stable by noting the point of maxima of the mass and the radius. The next two plots , Fig. 11 and 12 allows us to determine up to which point the dark matter remains stable for varying $\epsilon_{0,quark}$ . It is evident from the graphs that the dark matter parameter profile is almost independent of $\epsilon_{0,quark}$. So, the $\epsilon_{0,intdark}$ at which the dark matter becomes unstable is the same for all $\epsilon_{0,quark}$.
The contour plot showing the dependence of the total mass of the entire star ($M_{total}$) on the dimensionless central energy densities of the two fluids (Fig. 13) reflects the decrease in the maximum stable total mass with increase in $\epsilon_{0,intdark}$.
The region of stability is marked in the contour diagram . The shape of the boundary is similar to the free case discussed before. The upper branch of the boundary line is an indicator of the $\epsilon_{0,intdark}$ after which the the dark matter component becomes unstable for a given $\epsilon_{0,quark}$. As a quick check, in the contour diagram, the plot converges to the appropriate mass limit for low $\epsilon_{0,intdark}$. For low $\epsilon_{0,intdark}$ say, $10^5$ MeV/$fm^3$, the maximum stable mass is $\sim$ 2.0 $M_{\odot}$ at a radius of about 11 km showing the convergence to pure quark star limit.
Plot showing the dependence of maximum total mass vs. the dark matter content for quark star admixed with dark matter. The slope of the plot is -3.62. It shows that the maximum total mass decreases with increasing dark matter content in the star. The red line shows the linear fit.
Fig. 14 shows the maximum total mass vs. the fraction of dark matter which can be fitted by a linear fit. The slope of the fit comes out to be about -3.62.
The reason for the decrease of the maximum total mass with increasing dark matter content is the increased gravitational force due to extra dark matter content which causes a collapse of the star. Using the linear fit in the Fig. 14, the equation for the dependence of the maximum total stable mass of the admixed star on the dark matter fraction present is given as:
\begin{equation}
\small
\frac{M_{tot,max}}{M_{\odot}} = 2.004- 3.62 \emph{\(\frac{M_{int}}{M_{tot,max}}\)}
\end{equation}
\begin{equation}
\small
\frac{M_{tot,max}}{M_{\odot}} = 2.004- 3.62 f
\end{equation}
where f is the fraction of self-interacting dark matter in the admixed star at the maximum stable total mass. The maximum allowed strongly self-interacting dark matter content is about 2.64 $\times$ $10^{-2}$ $M_{\odot}$ at a maximum total stable mass of about 1.95 $M_{\odot}$ which gives a maximum limit on the possible dark matter fraction $f_{max}$ $\simeq$ 0.014. Radio timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion lead to a precise pulsar mass measurement of 2.01 $\pm$ 0.04 $M_{\odot}$ which is by far the highest yet measured with this precision <cit.>. The maximum stable dark admixed quark star mass (1.95 $M_{\odot}$) falls slightly lower than the error limit of the highest measured pulsar mass.
§ SUMMARY AND DISCUSSIONS
Pure quark matter is studied by using the MIT Bag model <cit.>. Dark matter stars are studied here by considering the dark matter to be made up of fermionic particles of mass 100 GeV <cit.> with the assumption that these particles do not self-annihilate. The maximum stable mass of the dark matter star composed of strongly self-interacting particles is about 2.7$\times$ $10^{-2}$ $M_{\odot}$ at a radius of about 0.19 km while for free fermions, the mass is about 6.0$\times$ $10^{-5}$ $M_{\odot}$ at a radius of around 1 meter.
The complete dimensional two-fluid TOV equations equations are solved to study the behaviour of admixed quark matter with dark matter. First, the equations are solved for a mixture of quark matter and free dark matter. The maximum stable mass of the admixed star is almost the same as that for a pure quark star for increasing dark matter fraction within the star. As the content of dark matter is gradually increased in the admixed star, the dark matter reaches its maximum stable configuration after which no admixed star configuration remains stable since the dark matter component collapses to form a black hole. For a quark star admixed with dark matter made of free gas of particles, the maximum possible mass of the stable configuration is approximately $M_{total}$ $\sim$ 2.01 $M_{\odot}$ with a dark matter content of around 0.63 $\times$ $10^{-4}$ $M_{\odot}$. A reduction in the maximum stable total mass is noted in case of a quark star admixed with dark matter star composed of strongly self-interacting fermions. The decrease was from about 2.01 $ M_{\odot}$ for zero dark matter content inside the star to about 1.95 $ M_{\odot}$ for the maximum allowed mass of strongly interacting dark matter in the star. The maximum dark matter content is around 2.64 $\times$ $10^{-2}$ $M_{\odot}$ at a maximum stable total mass of about 1.95 $M_{\odot}$. The maximum stable total mass in case of strongly self-interacting dark matter is seen to reduce linearly with increasing dark matter fraction in the star. The maximum accretion rate of dark matter by the quark star can be estimated to be about \(\frac{M_{int,max}}{\tau}\) $\sim$ 2.03$\times$ $10^{-12}$ $M_{\odot}$ per year, where $\tau$ $\sim$ 1.3 $\times$ $10^{10}$ years is the estimate for the age of the universe and $M_{int,max}$ is the maximum possible self-interacting dark matter content in the quark star. If the accretion rate is higher than this, the quark star will collapse .
§ ACKNOWLEDGEMENTS
This work started started as a summer project of P.M at the Goethe University in Frankfurt and is supported by Deutscher Akademischer Austausch Dienst (DAAD). P.M thanks the Institute of Theoretical Physics for their hospitality. We thank Andreas Zacchi, Rainer Stiele and Chhanda Samanta for helpful discussions and a critical reading of the manuscript.
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1511.00080
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Department of Mathematics
University of Wisconsin – Eau Claire
Eau Claire, WI, USA
|
1511.00180
|
XLIM UMR 7252 , DMI, University of Limoges; CNRS
123, Avenue Albert Thomas, 87060 Limoges, France
Max Planck Institute for Informatics,
Saarbruecken, Germany
Fields Institute
222 College St, Toronto, ON M5T 3J1 Ontario, Canada
In this paper, we present an algorithm for computing a fundamental matrix of
formal solutions of completely integrable Pfaffian systems with normal
crossings in several variables. This algorithm is a generalization of a method
developed for the bivariate case based on a combination of several reduction
techniques and is partially[Our Maple package PfaffInt can be
downloaded at: <http://www.mjaroschek.com/pfaffian/> It contains
functionalities illustrated by examples for the splitting, column reduction,
rank reduction, and the computation of exponential parts of multivariate
completely integrable systems with normal crossings]implemented in the
computer algebra system Maple.
Linear systems of partial differential equations, Pfaffian systems, Formal
solutions, Rank reduction, Hukuhara-Turrittin's normal form, Normal crossings.
§ INTRODUCTION
Pfaffian systems arise in many applications <cit.>, including the studies
of aerospace, celestial mechanics <cit.>, and
statistics <cit.>. So far, the most important systems for applications
are those with so-called normal crossings <cit.>.
A univariate completely integrable Pfaffian system with normal crossings reduces
to a singular linear system of ordinary differential equations (ODS, in short),
which have been studied extensively (see <cit.> and references
therein). Moreover, unlike the general case of several variables considered
herein, algorithms to related problems leading to the computation of formal
solutions have been developed by various authors
(see <cit.> and references therein). The Maple
package Isolde <cit.> and Mathemagix package
Lindalg <cit.> are dedicated to the symbolic resolution of such
More recently, bivariate systems were treated by the first and third author of
this paper in <cit.>. This paper refines the results of the bivariate
case and generalizes them to treat the more general multivariate case.
To get an intuition of the kind of systems we consider, we informally study the
following simple bivariate completely integrable Pfaffian system with normal
crossings. A formal definition of these systems will be given in
Section <ref>.
Given the following bivariate system over the ring of formal power series in
$(x_1,x_2)$ with complex coefficients:
\begin{equation*}
\begin{cases}
x_1^{4} \pder{F}{x_1} = A_1 F = \left(\begin{matrix}
x_1^3 + x_1^2+x_2 & x_2^2 \\
-1 & x_1^3 + x_1^2 -x_2
\end{matrix}\right) F, \\
x_2^3 \pder{F}{x_2} = A_2 F= \left(\begin{matrix}
x_2^2 -2 x_2 -6 & x_2^3 \\
-2 x_2 & -3 x_2^2 -2 x_2 -6
\end{matrix}\right) F,
\end{cases}
\end{equation*}
we are interested in constructing the formal objects $F$ that satisfy the
system. The existence of a fundamental matrix of solutions and its general
form follows from well known theoretical results (see
Corollary <ref>). The proof, however, is not constructive. For
simplicity, we assume we already know that a fundamental matrix of formal
solutions in our particular case is of the form
\begin{equation} \label{sim:sol1} \Phi(x_1, x_2) x_1^{C_1}x_2^{C_2}
e^{q_1(x_1^{- 1/s_1})} e^{q_2(x_2^{- 1/s_2})}, \end{equation}
$\Phi(x_1, x_2)$ is a matrix with formal power series entries, $C_1$ and $C_2$
are matrices with entries in $\set C$, and $q_1$, $q_2$ are polynomials in
$\set C[z_1], \set C[z_2]$ respectively. We now want to determine
$ \Phi, C_1, C_2, q_1, q_2,s_1,s_2$. For this purpose, we use the algorithm
presented in <cit.>. The main idea is to compute one part
of the solution by considering an associated ODS in only one variable and then
use this information to compute the other parts of the solution by
transforming and decoupling the system into smaller and simpler systems:
* First, we construct two associated systems whose equations are derived
by setting either $x_1=0$ or $x_2=0$:
\[
\begin{cases}
x_1^{4} \pder{F}{x_1} = A_1 (x_1,0) \; F= \left(\begin{matrix}
x_1^3+ x_1^2 & 0 \\
-1 & x_1^3 + x_1^2
\end{matrix}\right) F,\\
x_2^3 \pder{F}{x_2} = A_2 (0, x_2) \; F= \left(\begin{matrix}
x_2^2 -2 x_2 -6 & x_2^3 \\
-2 x_2 & -3 x_2^2 -2 x_2 -6
\end{matrix}\right)F.
\end{cases}
\]
We show in Section <ref> that the formal invariants $q_1,
q_2,s_1$ and $s_2$ can be computed from these associated systems. Via
Isolde or Lindalg we compute $s_1=s_2=1$ and
$q_1(1/x_1)=\frac{-1}{x_1}$, $q_2(1/x_2)=(\frac{3}{x_2^2} + \frac{2}{x_2} ),$
and (<ref>) becomes
\[
\Phi (x_1, x_2) x_1^{{C}_1} x_2^{C_2} e^{\frac{-1}{x_1}} e^{\frac{3}{{x_2}^2}
+ \frac{2}{x_2}}.
\]
* Next, we apply the so-called eigenvalue shifting
$F = e^{\frac{-1}{x_1}} e^{\frac{3}{x_2^2} + \frac{2}{x_2}} G$ (for a new
unknown vector $G$), to facilitate the next step. The shifting
\[
\begin{cases}
x_1^{4} \pder{G}{x_1} = \left(\begin{matrix}
x_1^3+x_2 & x_2^2 \\
-1 & x_1^3 -x_2
\end{matrix}\right) G, \\
x_2^2 \pder{G}{x_2} = \left(\begin{matrix}
x_2 & x_2^2 \\
-2 & -3 x_2
\end{matrix}\right) G.
\end{cases}
\]
* After the eigenvalue-shifting we apply another transformation that
reduces the orders of the singularities in $x_1$ and $x_2$ to their minimal
integer values. By setting $G = T_1 H$ where
\[
x_2 x_1^3 & -x_2 \\
0 & 1
\end{matrix}\right),
\]
we get:
\begin{equation*}
\begin{cases}
x_1 \pder{H}{x_1} = \left(\begin{matrix}
-2 & 0 \\
-x_2 & 1
\end{matrix}\right) H, \\
x_2 \pder{H}{x_2} = \left(\begin{matrix}
-2 & 0\\
-2 x_1^3 & -1
\end{matrix}\right) H.
\end{cases}
\end{equation*}
* Finally, via some linear algebra (see <cit.> for
general cases) we compute the transformation
\[
1 & 0 \\
\frac{x_2}{3} + 2 x_1^3 &\hspace{0.2cm}-1
\end{matrix}\right),
\]
and setting $H=T_2 U$ results in the system
\[
\begin{cases}
x_1 \pder{U}{x_1} = C_1 U = \left(\begin{matrix} -2 & 0 \\ 0 &
1 \end{matrix}\right) U ,\\
x_2 \pder{U}{x_2} = C_2 U = \left(\begin{matrix} -2 & 0 \\ 0 &
-1 \end{matrix}\right)U.
\end{cases}
\]
We can now read off $C_1$ and $C_2$. We collect the applied
transformations and get a fundamental matrix of solutions:
\[
\underbrace{T_1 T_2}_{=:\Phi} x_1^{C_1} x_2^{C_2}e^{\frac{-1}{x_1}}
e^{\frac{3}{x_2^2} + \frac{2}{x_2}},
\]
where $C_1=\left(\begin{matrix} -2 & 0 \\ 0 & 1 \end{matrix}\right)$ and
$C_2=\left(\begin{matrix} -2 & 0 \\ 0 & -1 \end{matrix}\right)$.
Unlike this simple example, the steps of computation can be far more involved
and demand multiple levels of recursion. In order to generalize this algorithm
to more than two variables, the following nontrivial questions have to be
* Can the information on the formal invariants still be obtained from the
associated ODS systems?
* Can a rank reduction algorithm be developed without relying on properties
of principal ideal domains as in the univariate and bivariate case?
The results of <cit.> which have an immediate generalization to the
multivariate setting are refined herein and supported by fully transparent
proofs and illustrating figures (Theorem <ref> which answers
positively and unconditionally the first question, the structure of the main
algorithm described in Section <ref>, and
Theorem <ref>). However, the answer to the second question is more
elaborate (see Section <ref>) and requires the discussion of two
problems which are not discussed in <cit.>:
* The major obstacle to a generalization of the results on the
bivariate case lies in the process of finding integral relations among
generators of certain modules over the ring of multivariate power series. In
Section <ref>, we propose a solution that relaxes the condition
of working over a principal ideal domain to working over local rings and show
how to utilize Nakayama’s Lemma in the formal reduction process if the modules
under investigation are free.
* We discuss an algorithmic difficulty which arises as not all formal power
series under manipulation admit a finite representation, even if the input
Pfaffian system is given in a finite form (see
Example <ref>). Although this problem arises in the bivariate case
as well, it has not been addressed before (neither in <cit.> nor
in <cit.>). We provide a reasoning to check the correctness of our
We thus present the first comprehensive description of the state of the art
algorithmic approach for solving completely integrable Pfaffian systems with
normal crossings in the multivariate setting. Our investigation also involves
the multivariate versions of the transformations used classically in the
well-studied univariate case (e.g. shearing transformations in
Section <ref>, column reduction transformations in
Section <ref>, and properties of transformations in
Proposition <ref>). Not only does this discussion serve the manipulation
of such transformations within our proposed formal reduction, but it also plays
a role in future generalizations of many other algorithms available for
univariate systems (e.g. the alternative rank reduction algorithm of
Section <ref> and the notion of simple systems as suggested in
the conclusion).
This paper is divided as follows: In Section <ref>, we recall the
basic definitions and the necessary theory for our algorithm. This includes the
general form of the solutions, the notion of equivalence between systems, the
classification of singularities, and a description of the necessary
transformations whose generalization to the multivariate case is
straightforward. In Section <ref>, we give the general structure of
our proposed algorithm which relies on two major components: The first is
associating to our system a set of ODS's from which its formal invariants can be
efficiently derived. This is detailed in Section <ref>. The
second component is the rank reduction which we give in
Section <ref>. The main algorithm is then given in
Section <ref> before concluding in Section <ref>.
§ PRELIMINARIES
§.§ Completely Integrable Pfaffian Systems with Normal Crossings
The systems considered in this paper are those whose associated differential
form is a 1-form. More explicitly, let ${\rm R}:=\ps{1}{n}$ be the ring of
formal power series in $x_1, x_2,\dots, x_n$ over the field of complex numbers
$\set C$. A Pfaffian system with normal crossings is a system of linear partial
differential equations of the form
\begin{equation}
\label{eq:sys}
x_i^{p_i+1} \pder{F}{x_i} = A_iF, \quad 1 \leq i \leq n,
\end{equation}
where the $A_i$'s are $d \times d$ matrices with entries in $\rm R$. The
system (<ref>) is completely determined by the $A_i$'s and $p_i$'s and we
conveniently denote it by $[A]$. Each of the $p_i$'s is an integer and the
number $p’_i:=\max(0,p_i)$ is called the Poincaré rank of the $i^{th}$
component $A_i$. The $n$-tuple
$$ p := (p'_1, \dots, p'_n)$$
is called the Poincaré rank of the system $[A]$.
If $p_i \leq -1$ for every $i \in \{1, \dots, n\}$ then the origin is an
ordinary (non-singular) point of the system and the system is said to be
regular. In this paper, we tackle the rather more interesting singular
systems. The singular locus of a system with normal crossings is a union of
hyperplanes of coordinates $x_1 x_2 \dots x_n =0$. A Pfaffian system is called
completely integrable, if the following commutation rule holds for all
\begin{equation}
\label{eq:cond}
A_iA_j - A_jA_i = x_i^{p_i+1}\pder{A_j}{x_i} - x_j^{p_j+1}\pder{A_i}{x_j}.
\end{equation}
Subsequently, whenever we refer to a Pfaffian system, we assume it is a
completely integrable system with normal crossings. For the remainder of this
paper we once and for all fix a Pfaffian system $[A]$ for which all the $A_i$
are non-zero and there is at least one strictly positive $p_i$. All subsequent
definitions and theorems are stated in this setting, disregarding systems for
which the origin is an ordinary point.
§.§ Notations and Algebraic Structures
Our notations follow a set of guidelines in order to help the reader remember
the multitude of different objects involved in our work. Single letter
identifiers are usually chosen to be the initial letter of the mathematical term
attached to the referenced object, like $d$ for dimension and $\rm R$ for a
ring. For a vector $v$, its $i^{th}$ component is given by $v_i$ and for a
univariate power series $s$ the $i^{th}$ coefficient is denoted by $s_i$. We do
not distinguish between row and column vectors. Upper case letters are used for
algebraic structures, matrices and the unknown in a Pfaffian system. A family of
matrices is given with lower indices, e.g. $(M_{i,j})_{i,j\geq 0}$, and for a
matrix $M_{i,j}$, blocks are given with upper indices, e.g.
\[
M_{i,j}^{11} & M_{i,j}^{12}\\
M_{i,j}^{21} & M_{i,j}^{22}
\end{matrix}\right),
\]
where the size of the different blocks are clear from the context.
By $x$ we denote the collection of variables $x_1,\dots,x_n$ and we use
$\bar{x}_i$ to refer to the variables
One can expand the $A_i$ in system $[A]$ as a formal power series with respect
to $x_i$:
\begin{equation*}
x_i^{p_i +1 }\pder{F}{x_i} = (A_{i,0}+ A_{i,1} x_i +
A_{i,2} x_i^2 + \dots ) F,
\end{equation*}
where the $A_{i,j}$ are elements of $\set C[[\bar{x}_i]]$. We denote this ring
by ${\rm R}_{\bar{x}_i}$. The first coefficient $A_{i,0} = A(x_i=0)$ in such an
expansion can be regarded as non-zero without any loss of generality, otherwise
$p_i$ can be readjusted. We call $A_{i,0}$ the leading matrix coefficient of the
$i^{th}$ component.
Aside from the rings $\rm R$ and ${\rm R}_{\bar{x}_i}$, we will frequently have
to work in other algebraic structures. We denote by
${\rm K}:=\operatorname{Frac}({\rm R})$ (respectively ${\rm K}_{\bar{x}_i}$) the
fraction field of $\rm R$ (respectively ${\rm R}_{\bar{x}_i}$). Let $L$ be the set of
monomials given by,
$$ L = \{ x^{\beta} = x_1^{\beta_1} x_2^{\beta_2} \dots x_n^{\beta_n}\text{ for
} \beta = (\beta_1, \beta_2, \dots, \beta_n) \in \mathbb{N}^n \}. $$
Clearly, $L$ is closed under multiplication and contains the unit element. Then,
one can define ${\rm R}_{L} := L^{-1} \rm R$, the localization of $\rm R$ at
$L$, i.e. the ring of series with only finitely many terms having monomials of
strictly negative exponents. Unlike in the univariate case, $\rm K$ and
${\rm R}_L$ do not refer to the same algebraic structure, e.g. $(x_1+x_2)^{-1}$
is an element of $\rm K$ which is not an element of ${\rm R}_L$. In fact, there
exists no $\beta \in \mathbb{N}^2$ such that
$x^{\beta} (x_1+x_2)^{-1} \in \rm R$:
$(x_1+x_2)^{-1} = \frac{1/x_1}{1-(-x_2/x_1)} $ whose formal expansion with
respect to $x_2$ is $\sum_{i\geq 0}(-1)^ix_1^{-i-1}x_2^i$, which has infinitely
many poles in $x_1$. For a further characterization of $\rm K$, one may refer
to <cit.>.
Finally, in the sequel it will be necessary to introduce ramifications of the
form $x_i=t_i^{\alpha_i}$ for new variables $t_i$ and positive integers
$\alpha_i$. We will therefore write ${\rm R}_t$ for $\set C[[t_1,\dots,t_n]]$
and allow analogous notations for all structures introduced so far. The identity
and zero matrices of dimension $d$ are denoted by $I_d$ and $O_d$, and we set: $\rm{GL}_d (R) = \{ M \in\operatorname{Mat}_{d\times d}({\rm R})\; | \; \operatorname{det} (M) \; \text{is invertible in} \; \rm R \}.$
§.§ Equivalent Systems
As we have seen in the introductory example, we will make use of transformations
to bring a system into particular forms. Such a transformation acts on a
Pfaffian system as follows: A linear transformation (also called gauge
transformation) $F=T G$, where
$T \in GL_d({\rm K})$, applied to (<ref>) results in the system
\begin{equation}
\label{eq:equiv}
x_i^{\tilde{p}_i+1} \pder{G}{x_i} = \tilde{A}_i G, \quad 1 \leq i \leq n,
\end{equation}
\begin{equation}
\label{eq:gauge}
\frac{\tilde{A}_i }{x^{\tilde{p}_i +1}_i} = T^{-1} ( \frac{A_i}{
x^{p_i+1}_i} T - \pder{T}{x_i}), \quad 1 \leq i \leq n.
\end{equation}
We say that system (<ref>) is equivalent to
system (<ref>) and we write $T[A]:=[\tilde{A}]$. It can be easily
verified that complete integrability is inherited by an equivalent
system. Subsequently, to stay in the same class of systems under study, special
care will be taken so that the transformations used in our considerations do not
alter the normal crossings. In fact, a major difficulty within the symbolic
manipulation of system (<ref>) arises from (<ref>). It is
evident that any transformation alters all the components simultaneously. In
particular, the equivalent system does not necessarily inherit the normal
crossings even for very simple examples.
Consider the following completely integrable Pfaffian system with normal
crossings of Poincaré rank $(3,1)$:
\begin{equation*}
\begin{cases}
x_1^4 \pder{F}{x_1} = A_{1}(x_1, x_2)\; F = \left(\begin{matrix} x_1^3 +x_2 &
x_2^2 \\ -1 & -x_2 + x_1^3 \end{matrix}\right) F,\\
x_2^2 \pder{F}{x_2} = A_{2}(x_1, x_2)\; F = \left(\begin{matrix} x_2 & x_2^2
\\ -2 & -3x_2\end{matrix}\right) F.
\end{cases}
\end{equation*}
This system appears within the reduction of the system of Example <ref>
in the introduction. As we have seen, there exists a transformation which
drops $p_1$ to zero. This can also be attained by the transformation
\[F = \left(\begin{matrix} x_1^3 & -x_2^2 \\ 0 & x_2 \end{matrix}\right) G,\]
which is computed by the univariate-case Moser-based rank reduction algorithm,
upon regarding the first component as an ODS in $x_1$ and $x_2$ as a
transcendental constant. This results in the equivalent system
\[
\begin{cases}
\label{gaugePfaffian}
x_1x_2 \pder{G}{x_1} = \tilde{A}_{1} (x_1,x_2) \; G= \left(\begin{matrix} -2
x_2 & 0 \\ -1 & x_2 \end{matrix}\right) G,\\
x_2^3 \pder{G}{x_2}= \tilde{A}_{2}(x_1,x_2) \; G = \left(\begin{matrix}
-x_2^2 & 0 \\ -2 x_1^3 & -2 x_2^2 \end{matrix}\right) G.
\end{cases}
\]
We can see that such a transformation achieves the goal of reducing the
Poincaré rank of the first component. However, it alters the normal crossings
as it introduces the factor $x_2$ on the left hand side of the first
component. Moreover, it elevates the Poincaré rank of the second component.
In order to preserve the normal crossings, we restrict the class of
transformations that we use in our algorithm:
Let $T\in GL_d({\rm K})$. We say that the transformation $F= TG$ (respectively $T$) is weakly compatible with system $[A]$ if $T[A] :=\tilde{A}$ is
again a completely integrable Pfaffian system with normal crossings. In
particular, $\tilde{A}_i \in {\rm R}^{d \times d}$ for every
$i \in \{1, \dots, n\}.$
Clearly, any constant or unimodular invertible matrix is an example of such
In the sequel, we will also need to resort to transformations with stronger
Let $T\in GL_d({\rm K})$. We say that the transformation $F= TG$
(respectively $T$) is compatible with system $[A]$ if it is weakly
compatible with $[A]$ and the Poincaré rank of each individual component of
$T[A]$ does not exceed that of the respective component of $[A]$.
§.§ Fundamental Matrix of Formal Solutions
Before studying how to construct formal solutions to a given system, the
question arises if and how many solutions exist. The language of stable modules
over the ring of power series is used in <cit.> and <cit.> independently to establish the following theorem which gives an
answer to this question.
There exist strictly positive integers $\alpha_i$, $1 \leq i \leq n$, and an
invertible matrix $T \in {\rm R}^{d\times d}_t$ such that, upon setting
$x_i = t_i^{\alpha_i}$, the transformation $T(t)$ yields the following
equivalent system:
\[
t_i^{\alpha_i \tilde{p}_i+1} \pder{G}{t_i}= \tilde{A}_{i}(t_i) G, \quad 1 \leq
i \leq n,\]
\[ \tilde{A}_{i}(t_i) = \operatorname{Diag} (\tilde{A}^{11}_{i}(t_i),
\tilde{A}^{22}_{i}(t_i), \dots, \tilde{A}^{jj}_{i}(t_i)) ,\]
and for every $\ell \in \{ 1, \dots, j\}$ we have that
$\tilde{A}^{\ell \ell}_i(t_i) $ is a square matrix of dimension $d_\ell$ of the
\[ \tilde{A}^{ \ell \ell}_{i}(t_i) = w^{ \ell \ell}_{i} (t_i) I_{d_\ell} +
t_i^{\alpha_i \tilde{p}_i}( c^{ \ell \ell}_{i} I_{d_\ell} + N^{ \ell \ell}_{i}),\]
* $d_1 + d_2 + \dots + d_j = d$;
* $w^{ \ell \ell}_{i} (t_i) = \sum_{m=0}^{\alpha_i \tilde{p}_i -1} \lambda_{im \ell}
t_i^{m}$ is a polynomial in $t_i$, with coefficients in
* $c^{ \ell \ell}_{i} \in \mathbb{C}$ and $N^{ \ell \ell}_{i}$ is a constant
(with respect to all derivations $\partial/\partial t_i$) $d_\ell$-square
matrix having nilpotent upper triangular form;
* for any fixed $\ell \in \{ 1, \dots, j \}$, the matrices
${\{ N^{ \ell \ell}_{i}\}}_{i = 1, \dots, n}$ are permutables;
* for all $\ell \in \{ 1, \dots, j-1 \}$, there exists
$i \in \{ 1, \dots, n\}$ such that
\[ w^{ \ell \ell}_{i} (t_i) \neq w^{(\ell+1)(\ell+1)}_{i}
(t_i) \text{\qquad or\qquad} c^{ \ell \ell}_{i} - c^{(\ell+1)(\ell+1)}_{i} \not
\in \mathbb{Z}.\]
This theorem guarantees the existence of a transformation which takes
system (<ref>) to the so-called Hukuhara-Turrittin's normal form from
which the construction of a fundamental matrix of formal
solutions (<ref>) is straightforward. In fact, we have:
Given system (<ref>), a fundamental matrix of formal solutions exists
and is of the form
\begin{equation}
\label{eq:sol}
\Phi(x_1^{1/s_1}, \dots, x_n^{1/s_n} ) \prod_{i=1}^{n} x_{i}^{{C}_i}
\prod_{i=1}^{n} \operatorname{exp}(Q_i(x_i^{-1/s_i})),
\end{equation}
where $\Phi$ is an invertible matrix with entries in ${\rm R}_t$ and for each
$i \in \{1, \dots, n\}$ we have:
* $s_i$ is a positive integer;
* the diagonal matrix
\[
Q_i(x_i^{-1/s_i}) = \operatorname{Diag}\left(q_{i,1}(x_i^{-1/s_i}),
q_{i,2}(x_i^{-1/s_i}), \dots, q_{i,d}(x_i^{-1/s_i})\right)
\]
contains polynomials in $x_i^{-1/s_i}$ over $\set C$
without constant terms. We refer to $Q_i(x_i^{-1/s_i})$ as the
$x_i$-exponential part. Under the notations of
Theorem <ref>, it is obtained by formally integrating
$\frac{w_{i}^{\ell\ell}}{t_i^{\alpha_i \tilde{p}_i +1}}$;
* $C_i $ is a constant matrix which commutes with
A singular system $[A]$ is said to be regular singular whenever, for every
$i \in \{1, \dots , d \}$, $Q_i(x_i^{-1/s_i})$ is a zero matrix. Otherwise,
system (<ref>) is said to be irregular singular and the entries
of $Q_i(x_i^{-1/s_i})$, $1\leq i \leq n$, determine the main asymptotic behavior
of the actual solutions as $x_i \rightarrow 0$ in appropriately small sectorial
regions <cit.>.
Let $i \in \{1, \dots, n\}$. If $Q_i(x_i^{-1/s_i})$ is a nonzero matrix then we set $m_{i,j}$ to be the minimum order in $x_i$ within the terms
$q_{i,j}(x_i^{-1/s_i})$ for $1\leq j\leq d$. The $x_i$-formal exponential growth order
($x_i$-exponential order, in short) of $A_i$ is the rational number
$$\omega(A_i) = - \operatorname{min}_{1 \leq j \leq d} m_{i,j}.$$
The $n$-tuple of rational numbers
$\omega(A) = (\omega(A_1), \dots, \omega(A_n))$ then defines the exponential
order of system $[A]$. Otherwise, we set $\omega(A_i) =0$.
If two systems are equivalent then they have the same $x_i$-exponential parts,
and consequently the same $x_i$ exponential orders, for all $1 \leq i \leq n$,
under any transformation $T \in GL_d({\rm K})$.
[Example <ref> cont.]
From our investigations in the example of Section <ref>, we see that
for the given fundamental system of formal solutions, we have non-zero exponential
parts with $\omega(A_1)=1$ and $\omega(A_2)=2$ and so the system is irregular
singular (although $s_1 = s_2 =1$).
The above theoretical results on existence do not establish the formal reduction
itself, that is the algorithmic procedure which computes explicitly the
$\alpha_i$'s and a transformation which takes the system to a normal form that
allows the construction of such solutions. This will be our interest in the
following sections.
The computation of the formal invariants is a difficult task in the univariate
case <cit.>. However, we will prove in Section <ref> that
in the multivariate case, these invariants can be computed from associated
univariate systems. Unlike the univariate case, the main difficulties of the
algorithm lie in rank reduction. Before proceeding to describe the algorithms we
propose, we give a property of the transformations which can be deployed:
Consider a completely integrable Pfaffian system $[A]$ with normal
crossings. Let $T \in GL_d({\rm K})$ and set $T[A] = [\tilde{A}]$. If $T$ is a
transformation which is weakly compatible with system $[A]$ then
$T \in GL_d(R_L)$.
It follows from (<ref>) that
\pder{T}{x_i} = \frac{A_i}{
x^{p_i+1}_i} T - T \frac{\tilde{A}_i }{x^{\tilde{p}_i +1}_i} , \quad 1 \leq i \leq n.
Thus, we have (see, e.g. <cit.>):
\begin{equation}
\label{Tform11}
\pder {\det(T)}{x_i} = ( \frac{\operatorname{tr}(A_i)}{x^{p_i+1}} -
\frac{\operatorname{tr}(\tilde{A}_i)}{x^{\tilde{p}_i+1}}) \det(T) , \quad 1 \leq
i \leq n.
\end{equation}
Therefore, $\det(T)$ itself is a solution of a completely integrable Pfaffian
system with normal crossings. By Corollary <ref>, $\det(T)$ has the
form (<ref>). Since $T \in {\rm K}^{d \times d}$ then $\det(T)$ is free
of logarithmic and exponential terms. Hence, $\det(T)$ corresponds to a log-free
regular solution of (<ref>). Thus, $\det(T) \in {\rm R}_L$. The same
argument serves to prove that $\det(T^{-1})$ is an element of ${\rm R}_L$ as
well, upon remarking that $T^{-1}[\tilde{A}] = [A]$. Hence,
$\det(T)^{-1} \in {\rm R}_L$ and consequently $T \in R_L^{d \times d}$.
However, the converse of Proposition (<ref>) is not true, which
complicates the task of constructing adequate transformations in the reduction
process (see, e.g., Example <ref> or the shearing transformations of
Section <ref>).
§ STRUCTURE OF THE MAIN ALGORITHM
If one is only interested in the asymptotic behavior of the
solutions of system $[A]$, then one can compute the formal invariants from
associated univariate systems as we prove in Section <ref>.
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lower dim.] at (10.1,5.3) ;
(11.3,7.1) – (11.6,7.1) – (11.6,8.1) – (1.1,8.1);
(11.3,5.9) – (11.6,5.9) – (11.6,8.1);
[->] (1.1,8.1) edge (1.1,2.3);
[line width=1pt]
(9,0) rectangle (11.2,2.2) node[label=[align=center]Apply rank
first var.] at
(10.1,0.25) ;
[->] (11.3,1.1) edge (11.9,2.0);
[->] (11.3,1.1) edge (11.9,0.2);
[line width=1pt] (12,1.2) rectangle (14.2,3.4)
node[label=[align=center]$\geq$ 2 distinct
eigenvalues] at (13.1,1.6) ;
(13.1,3.5) – (13.1,8.5) – (7.1,8.5);
[->] (7.1,8.5) edge (7.1,7.7);
[line width=1pt]
(12,1.0) rectangle (14.2,-1.2) node[label=[align=center]Nilpotent] at
(13.1,-0.6) ;
[->] (14.3,-0.1) edge (14.9,-0.1);
[line width=1pt]
(15,1.0) rectangle (17.2,-1.2) node[label=[align=center]Compute
exp. order
in first var.] at
(16.1,-0.95) ;
[->] (17.3,-0.1) edge (17.9,-0.1);
[line width=1pt]
(18,1.0) rectangle (20.2,-1.2) node[label=[align=center]Apply rami-
fication in
first var.] at
(19.1,-0.95) ;
(19.1,-1.3) – (19.1,-1.9) – (10.1,-1.9);
[->] (10.1,-1.9) edge (10.1,-0.1);
Computing a fundamental matrix of formal solutions by working with one of
the components, e.g. the first component. The other components would follow
the chosen component in the uncoupling.
If the singularity is regular or one is interested in computing a full
fundamental matrix of formal solutions, as given by (<ref>), then, besides
computing these invariants, further involved steps are required, as illustrated
in Example <ref>. The recursive algorithm we propose generalizes that of
the univariate case given by the first author in <cit.>. At each level of
recursion with input $[A]$, we consider the leading matrix coefficients
$A_{i,0}=A_i(x_i=0)$ (we use both notation interchangeably) and distinguish
between three main cases:
* There exists at least one index $i \in \{ 1, \dots, n \}$ such that
$A_{i,0}$ has at least two distinct eigenvalues.
* All of the leading matrix coefficients have exactly one eigenvalue and
there exists at least one index $i \in \{ 1, \dots, n \}$ such that
$A_{i,0}$ has a nonzero eigenvalue.
* For all $i \in \{ 1, \dots, n \}$, $A_{i,0}$ is nilpotent.
In order to identify the properties of the eigenvalues of $A_{i}(x_i=0)$, it
suffices to consider the constant matrix $A_i(x=0)$ due to the following
well-known proposition (see, e.g., <cit.>
or <cit.> for a proof within the context of eigenrings):
The eigenvalues of $A_{i,0}$ , $1 \leq i \leq n$, belong to $\mathbb{C}$.
Then, based on the above classification, a linear or an exponential
transformation will be computed as described in the following subsections.
§.§ Distinct Eigenvalues: Uncoupling the System Into Systems
of Lower Dimensions
Whenever there exists an index $i \in\{1,\dots,n \}$ such that $A_{i,0}$ has at
least two distinct eigenvalues, the system can be uncoupled into subsystems of
lower dimensions as shown in Theorem <ref>. For a constructive proof,
one may refer to <cit.>.
Suppose that for some $i \in\{ 1,\dots,n \}$, the leading matrix coefficient
$A_{i,0}$ has at least two distinct eigenvalues. Then there exists a unique
transformation $T \in GL_d({\rm R})$ of the form
\[T{(x)} =\left(
\begin{matrix}
T^{11} & T^{12} \\
T^{21} & T^{22}\\
\end{matrix}\right)=\left(
\begin{matrix}
I_{d'} & T^{12}{(x)} \\
T^{21}{(x)} & I_{d-d'}\\
\end{matrix}\right),
\]
where $0< d'< d$, such that the transformation $F = T G$ yields
the equivalent system
\[x_{i}^{p_{i} +1} \pder{G}{x_i} = \left(
\begin{matrix}
\tilde{A}^{11}_{i}(x) & O\\
O & \tilde{A}^{22}_{i}(x)\\
\end{matrix}\right)G, \quad 1 \leq i \leq n .
\]
and $\tilde{A}^{11}_{i}(x),\tilde{A}^{22}_{i} (x)$, $i \in \{ 1, \dots, n\}$
are of dimensions $d'$ and $d-d'$ respectively.
The theorem can be restated by saying that if one of the components of the
system has a leading matrix coefficient with at least two distinct eigenvalues,
then it can be uncoupled. All of the other components are uncoupled
simultaneously. In the sequel, we aim to determine changes of the independent
variables $x_i$ (ramifications), and construct transformations, which will allow
the reduction of any input system to a system for which the leading matrix
coefficient of at least one of its components has at least two distinct
eigenvalues. This allows us to either arrive at a system with lower Poincaré
rank or uncouple it into several subsystems of lower dimensions. The
recursion stops whenever we arrive at regular singular ($p =(0, \dots, 0)$) or scalar
($d=1$) subsystems. The former have been already investigated in <cit.> and the resolution of the latter is straightforward.
We remark that, by Proposition <ref>, it suffices that there exists
$i \in \{1 , \dots, n\}$ such that the constant matrix
$A_{i}(x_1=0, \dots, x_{i}=0, \dots, x_n=0)$ has at least two distinct
§.§ Unique Eigenvalue: Shifting
For any $i\in\{1,\dots,n\}$ such that $A_{i,0}$ has a unique nonzero
eigenvalue $\gamma_{i} \in \mathbb{C}$, applying the so-called eigenvalue
\begin{equation*}
F = \operatorname{exp}\left(\int^{\text{\rlap{$x_{i}$}}}\gamma_{i}
z_{i}^{-p_{i}-1} dz_{i}\right) G,
\end{equation*}
yields a system $[\tilde{A}]$ whose ${i}^{th}$ component has a nilpotent leading
matrix coefficient:
\begin{eqnarray*}
x_i^{\tilde{p}_i+1} \pder{G}{x_i} = \tilde{A}_i{(x)} G, \quad \text{where}
\quad \tilde{A}_{i}{(x)} = A_{i}{(x)} - \gamma_{i} I_d .
\end{eqnarray*}
The other components of the system are not modified by this transformation which
is clearly compatible with system $[A]$.
Hence, due to the uncoupling and shifting, we can assume without
loss of generality that for all $i \in \{ 1, \dots, n\}$, the leading matrix
coefficients $A_{i,0}$ are nilpotent.
§.§ Nilpotency: Rank Reduction and Exponential Order
In the univariate case, $n=1$, the nilpotency of $A_{1,0}$ suggests at least one
of the following two steps, as proposed by the first author in <cit.>:
Rank reduction and computation of the exponential order $\omega(A_1)$. The
former reduces $p_1$ to its minimal integer value. It is possible that $p_1$
drops to zero, i.e. we arrive at a regular singular system, or that the leading matrix
coefficient of the resulting system has at least two distinct eigenvalues, in
which case we can again uncouple the system. Otherwise,
$\omega(A_1) = \ell / m$ is to be computed, where $\ell$ and $m$ are
coprime. Then, by setting $x_1 = t_1^{m}$ and applying rank reduction again, it
is proven that we arrive at a system whose leading matrix coefficient has two
distinct eigenvalues. Therefore, the system can be uncoupled (see
Figure 1).
The bivariate case, $n=2$, is studied by the first and third authors of this
paper in <cit.>. For rank reduction, the properties of principal ideal
domains were used. To determine the formal exponential order $\omega(A)$,
associated univariate systems were defined. In this paper, we show that on the
one hand, this approach to determine the formal exponential order remains valid
in the multivariate setting, as we will see in the next section. On the other
hand, the generalization of the rank reduction algorithm to the multivariate
case is nontrivial and is discussed in Section <ref>. The
multivariate formal reduction algorithm is then summed up in
Section <ref>.
§ COMPUTING THE FORMAL INVARIANTS
In the univariate case, where the system is given by a single matrix $A_1$,
$\omega(A_1)$ can be computed from the characteristic polynomial of $A_1$, i.e.$\det(\lambda I_d - A_1)$, based on the analysis of a Newton polygon associated
with the system <cit.>. In this section we show that one need
not search for a generalization of this algorithm to the multivariate case as
the formal invariants of $[A]$, i.e. the exponential parts and $\omega(A)$, can
be obtained from an associated univariate system. We do not only give a method
to retrieve these invariants but we also reduce computations to computations
with univariate rather than multivariate formal series.
Given a Pfaffian system $[A]$, we call the following the associated ODS
of $[A]$:
\begin{equation*}
x_i^{p_i+1} \frac{d}{dx_i} \mathcal{F}_i =
\mathcal{A}_{i}(x_i)\;\mathcal{F}_i, \quad 1 \leq i \leq n,
\end{equation*}
$\quad \mathcal{A}_{i}(x_i) := A (x_1=0, \dots, x_{i-1}=0, x_i, x_{i+1}=0,
\dots, x_n=0).$
For every $i \in \{ 1, \dots, n \}$, the $x_i$-exponential part of a
Pfaffian system is equal to the exponential part of the $i^{th}$
component of its associated ODS.
To establish this result, we rely on a triangular form weaker than the
Hukuhara-Turrittin's normal form given in Theorem <ref>. This
weaker form suffices to give insight into the computation of (<ref>).
The following theorem is an reformulation of a theorem which was first given
in <cit.> for the bivariate case, and then
generalized in <cit.> to the general multivariate case.
[scale=0.6, every node/.style=scale=0.6]
[line width=1pt]
(0,0) rectangle (1.8,1.8) node[label=[align=center]Input
system] at
(0.9,0.2) ;
[->] (1.9,1.9) edge (2.9,2.9);
[->] (1.9,-0.1) edge (2.9,-1.1);
[line width=1pt]
(3,3) rectangle (4.8,4.8) node[label=[align=center]First
component] at
(3.9,3.2) ;
[->] (4.9,3.9) edge (5.9,3.9);
[thick,dotted] (3.9,2.5) – (3.9,-0.7);
[line width=1pt]
(3,-3) rectangle (4.8,-1.2) node[label=[align=center]Last
component] at
(3.9,-2.8) ;
[->] (4.9,-2.1) edge (5.9,-2.1);
[line width=1pt] (6,3) rectangle (7.8,4.8)
ODS] at (6.9,3.1) ;
[->] (7.9,3.9) edge (8.9,3.9);
[thick,dotted] (6.9,2.5) – (6.9,-0.7);
[line width=1pt] (6,-3) rectangle (7.8,-1.2)
ODS] at (6.9,-2.9) ;
[->] (7.9,-2.1) edge (8.9,-2.1);
[line width=1pt] (9,3) rectangle (10.8,4.8)
node[label=[align=center]Exp. part
in first var.] at (9.9,3.3) ;
[thick,dotted] (9.9,2.5) – (9.9,-0.7);
[line width=1pt] (9,-3) rectangle (10.8,-1.2)
node[label=[align=center]Exp. part
in last var.] at (9.9,-2.7) ;
[->] (10.9,2.9) edge (11.9,1.9);
[->] (10.9,-1.1) edge (11.9,-0.1);
[line width=1pt]
(12,0) rectangle (13.8,1.8) node[label=[align=center]Exp. part] at
(12.9,0.4) ;
Computing the exponential part from associated ODS's
Consider the Pfaffian system $[A]$. There exists a positive integer $\alpha_1$,
and a transformation $T \in GL_d({\rm K_t})$ (where $x_1 = t_1^{\alpha_1}$ and
$x_i = t_i, \; 2 \leq i \leq n$), such that the transformation $F= T G$ yields
the equivalent system:
\begin{equation}
\label{eq:ger}
\begin{cases}
t_1^{\alpha_1 \hat{p}_1 + 1} \pder{G}{t_1} = \hat{A}_{1}(t_1, x_2, \dots, x_n)
\; G, \\[7pt] x_i^{ \hat{p}_i+1} \pder{G}{x_i}= \hat{A}_{i}(x_2, \dots, x_n)\; G,
\quad 2 \leq i \leq n,
\end{cases}
\end{equation}
Ã_1(t_1, x_2, …, x_n) =
Diag (Â^11_1, Â^22_1,
…, Â^jj_1),
Â_i(x_2, …, x_n) = Diag
(Â^11_i, Â^22_i, …, Â^jj_i), 2
≤i ≤n,
and for all $\ell \in \{1, \dots, j\}$ and $i \in \{2, \dots, n\}$ the entries
of $\hat{A}^{{\ell} {\ell} }_{{i}}$ lie in ${\rm R}_{\bar{x}_{1}}$. The
$\hat{A}_1^{\ell\ell}$'s are of the form
\[\hat{A}^{{\ell} {\ell} }_{1} = w^{{\ell} {\ell} }_{1} (t_1) I_{d_{\ell}
} + t_1^{\alpha_1 \hat{p}_1}(\hat{N}^{{\ell} {\ell} }_{1}(x_2, \dots, x_n) +
c^{{\ell} {\ell} }_{1} I_{d_{\ell}}),\]
* $d_1 + d_2 + \dots + d_j = d$;
* $w^{{\ell} {\ell} }_{1} (t_1)$ and $c^{{\ell} {\ell} }_{1}$ are as in
Theorem <ref>;
* If $\ell, \ell' \in \{ 1, \dots, j-1 \}$ and $\ell \neq \ell'$, then
$ w^{ \ell \ell}_{1} (t_1) \neq w^{ \ell' \ell'}_{1} (t_1)$ or
$c^{ \ell \ell}_{1} - c^{ \ell' \ell'}_{1} \not \in
\mathbb{Z}$;
* $\hat{N}^{{\ell} {\ell} }_{1} (x_2, \dots, x_n)$ is a nilpotent
$d_{\ell} $-square matrix whose entries lie in ${\rm R}_{\bar{x}_{1}}$.
Moreover, $T$ can be chosen as a product of transformations in
$GL_d({\rm R}_t)$ and transformations of the form
$\operatorname{Diag} (t_1^{\beta_1}, \dots, t_1^{\beta_d})$, where
$\beta_1, \dots, \beta_d$ are non-negative integers.$\hfill\qed$
Upon the change of
independent variable $x_1=t_1^{\alpha_1}$, the transformation $F = T G$ yields
system (<ref>) for which the first component is given by
\begin{equation*}
t_1^{\alpha_1 \hat{p}_1 + 1} \pder{G}{t_1} = \tilde{A}_{1}(t_1, x_2, \dots, x_n) \; G.
\end{equation*}
with the notations and properties as in Theorem <ref>. It
then follows from (<ref>) that
\begin{equation}
\label{relation}
t_1^{\alpha_1 {\hat{p}}_1 + 1}\pder{T}{t_1}= \alpha_1
A_{1}(x_1=t_1^{\alpha_1})\; T - T {\hat{A}}_{1}.
\end{equation}
Due to the particular choice of $T$ in Theorem <ref>, we
can set $x_i=0$, $2 \leq i \leq n$ in (<ref>). In particular, the
relation between the leading terms
𝒜_1(x_1=t_1^α_1) := A_1 (x_1 =
t_1^α_1, x_2=0,…,x_n=0),
𝒜̂_1 (t_1) :=
Â_1 (t_1, x_2=0, …, x_n=0),
𝒯 (t_1) :=
T(t_1, x_2=0,…,x_n=0),
is given by
\begin{equation*}
t_1^{\alpha_1 \hat{p}_1 + 1} \pder{\mathcal{T}}{t_1} = \alpha_1
{\mathcal{A}}_{1}(x_1=t_1^{\alpha_1})\; \mathcal{T} - \mathcal{T}
\end{equation*}
Hence, the systems given by $\alpha_1{\mathcal{A}}_{1}(x_1=t_1^{\alpha_1})$
(respectively ${\mathcal{A}}_{1}$) and ${\hat{\mathcal{A}}}_{1}$ are
equivalent. It follows that they have the same formal invariants. Clearly, the
same result can be obtained for any of the other components via permutation with
the first component. Noting that the $x_i$-exponential part is independent of
$\bar{x}_i$ completes the proof.
For univariate systems, the true Poincaré rank $p_{true}(A_1)$ is
defined as the smallest integer greater or equal than the exponential order $\omega(A_1)$ of
the system $[A_1]$. It is known that this integer coincides with the minimal value for
$p_1$ which can be obtained upon applying any non-ramified linear transformation
to $A_1$. With the help of Theorem <ref> we can establish the
analogous result for multivariate systems. We first give the following
Let $[A]$ be a Pfaffian system and for any $i\in\{1,\dots,n\}$, let
$p_{true}(A_i)$ be the minimal integer value which bounds the exponential order
in $x_i$, i.e.
\[
p_{true}(A_i) - 1 < \omega(A_i) \leq p_{true}(A_i).
\]
Then $p_{true}(A)=(p_{true}(A_1),\dots,p_{true}(A_n))$ is
called the true Poincaré rank of $A$.
It is shown by Deligne and van den Essen separately in <cit.>, in
the multivariate setting that a necessary and sufficient condition for system
$[A]$ to be regular singular is that each individual component $A_i$, considered
as a system of ordinary differential equations in $x_i$, with the remaining
variables held as transcendental constants, is regular singular. As a
consequence, system $[A]$ is regular singular if and only if its true
Poincaré rank is $(0,0, \dots,0)$. To test this regularity, algorithms
available for the univariate case of $n=1$ (e.g. <cit.>) can be
applied separately to each of the individual components. The following corollary
follows directly from Theorem <ref>, showing that the $i^{th}$
component of the true Poincaré rank of system $[A]$ is equal to the true
Poincaré rank of the $i^{th}$ associated (univariate) ODS.
For all $1\leq i\leq n$ we have
\[
p_{true}(A_i) =p_{true}(\mathcal{A}_i).
\]
For the proof, it suffices to remark that[Stronger bounds are given
in <cit.>]
\[p_{true}({\mathcal{A}}_{i}) -1 < \omega(A_i) = \omega({\mathcal{A}}_{i} ) \leq
From this Corollary it does not yet follow for the multivariate case, as in the
univariate case, that it is possible to apply a compatible transformation to
system $[A]$ such that all the $p_i$ simultaneously equal the $p_{true}(A_i)$. We
investigate this possibility in the next section.
In summary, the formal exponential order, the true Poincaré rank, and most
importantly the $Q_i$'s in (<ref>), can be obtained efficiently by
computations with univariate rather than multivariate series, making use of
existing algorithms and packages. As mentioned in the introduction, this
exponential part is of central importance in applications since it determines
the asymptotic behavior of the solution in the neighborhood of an irregular
singularity. To compute a full fundamental matrix of formal solutions, we still
have to determine suitable rank reduction transformations. Transformations which
reduce the rank of the associated systems do not suffice, since they are not
necessarily compatible. We therefore proceed to develop a multivariate rank
reduction algorithm.
§ RANK REDUCTION
In this section, we are interested in the rank reduction of Pfaffian systems,
more specifically, the explicit computation of a transformation which, given
system $[A]$, yields an equivalent system whose Poincaré rank is the true
Poincaré rank. We show that, under certain conditions, the true Poincaré
ranks of the subsystems of $[A]$ can be attained simultaneously via a
transformation compatible with $[A]$.
We first generalize Moser's reduction criterion <cit.> to multivariate
systems. We then establish an extension of the algorithm we gave
in <cit.> for the bivariate case to the multivariate setting. The main
problem in the treatment of multivariate systems is that the entries of the
$A_i$ do not necessarily lie in a principal ideal domain. This is a common
problem within the study of systems of functional equations. The same obstacle
arises in <cit.> and in the analogous theory of formal decomposition of
commuting partial linear difference operators established in <cit.>.
§.§ Generalized Moser's Criterion
For univariate systems, Moser's criterion characterizes systems for which rank
reduction is possible. To adapt this criterion to our setting, we
follow <cit.> and define the generalized Moser rank and the Moser
invariant of a system $[A]$ as the following $n$-tuples of rational numbers:
\begin{eqnarray*}
m (A) = (m (A_1) , \dots, m (A_n)), & \text{where} & m (A_{i}) =
\max\left(0, p_i + \frac{\operatorname{rank}(A_{i,0})}{d}\right),\\
\mu (A) = (\mu (A_1) , \dots, \mu (A_n)), & \text{where} & \mu (A_{i})
= \operatorname{min}(\{ m (T(A_i)) \mid T \in GL_d({\rm R}_L) \}).
\end{eqnarray*}
We remark that $\mu(A)$ is well-defined due to Corollary <ref>.
Consider the partial order $\leq$ on $\set Q^n$ for which $\ell < k$ holds if
and only if $\ell_j\leq k_j$ for all $1\leq j\leq n$ and there is at least one
index for which the inequality is strict. The system $[A]$ is called
reducible if $\mu(A) < m(A)$. Otherwise it is said to be irreducible.
In other words, system $[A]$ is irreducible whenever each of its components
is. In particular, it is easy to see from this definition that a system $[A]$ is
regular singular if and only if $\mu(A_i) \leq 1$ for all
$i \in \{ 1, \dots, n\}$, i.e. the true Poincaré rank is a zero $n$-tuple,
which coincides with Deligne's and van den Essen's criterion for regular
singular systems.
§.§ Main Theorems
With the help of compatible transformations and the criterion established in
Section <ref>, we study the rank reduction of some component
$A_i$ of $[A]$ which is given by (<ref>). We will see that rank reduction
can be carried out for each component independently without affecting the
individual Poincaré ranks of the other components. We fix
$i \in \{ 1, \dots, n\}$ and we recall that one can expand the components
of $[A]$ with respect to $x_i$. In particular, we have,
\begin{equation*}
x_i^{p_i +1 }\pder{F}{x_i} = A_i(x) F = (A_{i,0}({\bar{x}_i})+ A_{i,1}
({\bar{x}_i})x_i + A_{i,2}({\bar{x}_i}) x_i^2 + \dots ) F.
\end{equation*}
We set
\[r:= \operatorname{rank} (A_{i,0}).\]
For all $i$ we can assume without loss of generality that $A_{i,0}$ is not the
zero matrix and thus the reducibility of system $[A]$ coincides with the
existence of an equivalent system such that for some $i$ the rank of the
leading matrix coefficient
$\tilde{A}_{i,0}$ is less than $r$. We establish the
following theorem:
Consider a Pfaffian system $[A]$ and suppose that $m(A_i) > 1$ for some index
$i \in \{1, \dots, n\}$. If $\mu(A_i) < m(A_i)$ then the polynomial
\begin{equation}
\label{eq:theta}
\theta_{i} (\lambda) := {x_i}^{r} \det(\lambda I + \frac{A_{i,0}}{x_i} + A_{i,1}
\end{equation}
vanishes identically in $\lambda$.
Suppose that there exists a transformation $T(x) \in {\rm R}_L^{d \times d}$
which reduces $m(A_i)$ for some $i \in \{ 1, \dots, n\}$. That is, setting
$T[A] := [\tilde{A}]$, we have:
\begin{equation} \label{ppp} \operatorname{m}(\tilde{A}_{i}) <
\operatorname{m}(A_{i}) .\end{equation}
The $i^{th}$ component of
system $[A]$ can also be viewed as a system of ordinary differential equations
(ODS) in $x_i$ upon considering the $x_j$'s for $j \in \{1, \dots, n\}$,
$j \neq i$, to be transcendental constants. Hence, by (<ref>)
and <cit.>, $\theta_i(\lambda) = 0$.
Intuitively, the characteristic polynomial of $A_i/x_i$ is used
in (<ref>) to detect the true Poincaré rank of the $i^{th}$ component
via the valuation of $x_i$. It turns out that the valuation is only influenced
by $A_{i,0}$ and $A_{i,1}$. Even though the true Poincaré rank can be
determined from the associated ODS, the criterion is essential as it leads to
the construction of the transformation $T$.
The converse of Theorem <ref> also hold true under certain conditions
(see Theorem <ref>): If $\theta_i (\lambda)$ vanishes for some index
$i \in \{ 1, \dots, n\}$ then we can construct a compatible transformation
$T \in GL_d({\rm R}_L)$ which reduces $m(A_i)$. We will establish this result
and describe the steps of the algorithm after establishing a series of
intermediate ones. We will need two kinds of transformations, shearing
transformations and column reductions, which we explain in the next two
§.§ Shearing Transformation
Consider the expansion $A_i=\sum_{k=0}^\infty A_{i,k}x_i^k$ of $A_i$ with
respect to $x_i$ for a fixed $i\in\{1,\dots,n\}$. Shearing transformations are
polynomial transformations that, roughly speaking, are used to exchange blocks
between the $A_{i,k}$'s. The ones we consider here are of the form
\[S = \operatorname{Diag} (x_{i}^{\beta_1}, \dots , x_{i}^{\beta_d}),\]
with $\beta_j \in\{0,1\}$ for all $j \in\{1,\dots,d\}$. We illustrate the
shearing effect of such a transformation in an easy example.
We apply a shearing transformation to a univariate system $[A]$ given by
$x_1^{p_1+1} \frac{\partial}{\partial x_1} F = A_1 F$ where
$A_1 =\sum_{k=0}^\infty A_{1,k} x^k_1$, with
\[A_{1,0} = \left(\begin{matrix} 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -2 & 0 & 0 &
0 \\ 0 & 1 & 0 & 0\end{matrix}\right)\quad\text{and}\quad A_{1,1} =
\left(\begin{matrix} 4 & 9 & 2 & -5 \\ 8 & 9 & 0 & 0 \\ 8 & 6 & 2 & 4 \\ 5 & 6
& 3 & 3\end{matrix}\right) . \]
The transformation $S = \operatorname{Diag}(x_1,x_1,1,1)$ yields the equivalent
system $[\tilde{A}]$ given by
$$x_1^{p_1+1} \frac{\partial}{\partial x_1} F = \tilde{A_1} F \quad \text{where}
\quad \tilde{A_1} = S^{-1} A_1 S - x_1^p \operatorname{Diag}(1,1,0,0) .$$
As we are interested in the effect of $S$ on the first few terms of $A_1$, we
look into $S^{-1} A_1 S$ which exchanges the upper right and lower left
$2\times2$ blocks of the $A_1$ as exhibited in the following diagram:
[line width=1pt,pattern=section]
(0,0.5) rectangle (0.5,1);
[line width=1pt,pattern=section]
(0,0) rectangle (0.5,0.5);
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(0.5,0) rectangle (1,0.5) node[right] $x_1^0\quad\ +$;
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node[right] $x_1^3\quad\ +\dots$;
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(9.6,0.6) rectangle (10.1,1.1);
[->] (3.7,1.2) edge [out= 160, in= 30] (1,1.2);
[->] (6.7,1.2) edge [out= 160, in= 30] (4,1.2);
[->] (9.7,1.2) edge [out= 160, in= 30] (7,1.2);
[->] (0.2,-0.2) edge [out= -30, in= -160] (3.1,-0.2);
[->] (3.2,-0.2) edge [out= -30, in= -160] (6.1,-0.2);
[->] (6.2,-0.2) edge [out= -30, in= -160] (9.1,-0.2);
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(3.5,0) rectangle (4,0.5) node[right] $x_1^1\quad\ +$;
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(3.5,0.5) rectangle (4,1);
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(6,0) rectangle (6.5,0.5);
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(6.5,0) rectangle (7,0.5) node[right] $x_1^2\quad\ +$;
[line width=1pt,thickness=2pt, pattern=section3]
(6.5,0.5) rectangle (7,1);
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(9,0.5) rectangle (9.5,1);
[line width=1pt,thickness=0.5pt, pattern=section2]
(9,0) rectangle (9.5,0.5);
[line width=1pt,thickness=2pt, pattern=section3]
(9.5,0) rectangle (10,0.5) node[right] $x_1^3\quad\ +\dots$;
[line width=1pt]
(9.5,0.5) rectangle (10,1) node[pos=0.5] $*$;
Consequently, $A_{1,0}$ and $A_{1,1}$ become
\[\tilde{A}_{1,0} = \left(\begin{matrix} 1 & 2 & 2 & -5 \\ 0 & 0 & 0 & 0 \\ 0
& 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right),\ \ \tilde{A}_{1,1} =
\left(\begin{matrix} 4 & 9 & * & * \\ 8 & 9 & * & * \\ -2 & 0 & 2 & 4 \\ 0 &
1 & 3 & 3\end{matrix}\right).\]
Note that the lower left zero entries in $\tilde{A}_{1,0}$ come from
$A_{1,-1}$, which is a zero matrix. In return, the upper right block of
$A_{1,0}$ is sent to $A_{1,-1}$. Since it is a zero block, this transformation
does not introduce denominators. The upper right entries in $\tilde{A}_{1,1}$
come from $A_{1,2}$. With this transformation, we reduced the rank of
$A_{1,0}$ from $2$ to $1$.
More generally, let $[A]$ be a multivariate Pfaffian system. The transformation
$F = SG$, where $S$ is a shearing transformation in $x_i$, yields an equivalent
system with:
\[
\begin{cases}
\tilde{A}_{i} = S^{-1} A_{i} S - x_{i}^{p_{i}}\operatorname{Diag}({\beta_1}, \dots ,
{\beta_d}), \\
\tilde{A}_{j} = S^{-1} {A_{j}} S, \quad 1 \leq j \neq i \leq n , \end{cases}\]
\[S^{-1} {A_{j}} S = \left(\begin{matrix} A_{j, 11} & A_{j,12}
x_{i}^{\beta_2 - \beta_1} & \dots & A_{j, 1d} x_{i}^{\beta_d - \beta_1} \\[5pt]
A_{j, 21} x_{i}^{\beta_1 - \beta_2} & A_{j, 22} & \dots & A_{j, 2d}
x_{i}^{\beta_d- \beta_2} \\[5pt] \vdots & \vdots & \dots & \vdots
\\[5pt] A_{j, d1} x_{i}^{\beta_1 - \beta_d} & A_{j, d2} x_{i}^{\beta_2
- \beta_d} & \dots & A_{j, dd} \end{matrix}\right)\text{ for all }1\leq
j \leq n.\]
The shearing in Example <ref> reduced the rank of the leading matrix
coefficient and was compatible with the system, i.e. it did not introduce
undesired denominators of $x_{i}$, because of the column reduced form of
$A_{i,0}$. The input system is not always given in such a
form for $A_{i,0}$, and so we investigate in the following subsection how to
achieve it.
§.§ Column Reduction
To enable rank reduction, we alternate between the shearing transformation and
transformations which reduce some columns of a leading matrix coefficient to
zero. For this we discuss in this section the following problem.
(P) Given a square matrix
$A=[v_{1},\dots,v_{d}]\in\operatorname{Mat}_{d\times d}({\rm R})$ (where
$v_{i}$ denotes the $i$th column) of rank $r<d$ when considered as an element
of $\operatorname{Mat}_{d\times d}({\rm K})$, find a
transformation $T\in GL_d({\rm R})$ such that the last $d-r$ columns of
$TAT^{-1}$ are zero.
Before considering the algorithmic aspects,
we first discuss the existence of such a transformation. As the next example
shows, the desired transformation does not necessarily exist for any matrix $A$.
The matrix
\[
\left(\begin{matrix}
0 & x_1 & x_2\\
0 & 0 & 0\\
0 & 0 & 0
\end{matrix}\right)
\]
is obviously of rank 1 over $\rm K$. There is, however, no
transformation $T\in GL_3({\rm R})$ such that $TAT^{-1}$ contains only one
non-zero column.
Consider the finitely generated ${\rm R}^d$-submodule
$M:=\langle v_{1},\dots,v_{d}\rangle$. We call it the column module of $A$. In
order to construct a suitable transformation for bivariate Pfaffian systems (for
which the leading matrix coefficients are univariate) the authors
of <cit.> use the fact that $\set C[[x_1]]$ and $\set C[[x_2]]$ are
principal ideal domains and hence that every finitely generated submodule of a
free module over this ring is free. We generalize this for the multivariate case
by showing in Corollary <ref> that the freeness of the column module
$M$ is a necessary and sufficient condition for the existence of a
transformation that meets our requirements. This is a direct consequence of
Nakayama's Lemma for local rings.
Let $\rm R$ be a local ring, $\mathcal{M}$ its maximal ideal and let $M$ be a
finitely generated $\rm R$-module. Then $v_1,\dots,v_r\in M$ form a minimal
set of generators for $M$ if and only if their images
$\bar{v}_1,\dots,\bar{v}_r$ under the canonical homomorphism
$M\rightarrow M/\mathcal{M}M$ form a basis of the vector space
$M/\mathcal{M}M$ over the field ${\rm R}/\mathcal{M}$.
The central consequence of Theorem <ref> for us is that if $M$ is
free, a module basis of $M$ can be chosen among the columns of $A$. We adapt the
theorem to our situation to show that we can bring $A$ into a column-reduced
form if and only if its column module is free.
Let $A\in\operatorname{Mat}_{d\times d}({\rm R})$ be of rank $r$ over $\rm K$
and let $M$ be the module generated by the columns of $A$. If $M$ is free,
then there exists a subset $B$ of the columns in $A$ with $r$ elements such
that $B$ is a module basis of $M$. Furthermore, $B$ is also a $\rm K$-vector
space basis of the column space of $A$.
By Theorem <ref> we can find a basis $B=\{b_1,\dots,b_k\}$ of $M$
among the columns of $A$. By definition, the $b_i$ are linearly independent
over $\rm R$, so they are also linearly independent over
$\rm K$ (otherwise, multiplying a linear relation in
$\rm K$ with a common denominator yields a relation in
$\rm R$). Since $B$ is a basis of the column module, it also contains a
generating set of the $\rm K$-vector space generated by
the columns of $A$.
In theory, Corollary <ref> would allow the computation of a unimodular
column reduction transformation as required in $(P)$ simply via Gaussian
elimination. Assume we are given a matrix $A$ and already know a subset
$B=(b_1,\dots,b_r)$ of the columns of $A$ which forms a basis of the column
module. Let $v$ be a column vector of $A$ which is not in $B$. Then, since $B$
is a vector space basis, there exist $c_1,\dots,c_r\in {\rm K}$ such that
\[c_1b_1+\dots +c_rb_r=v.\]
By assumption, $B$ is also a module basis, so there also exist
$d_1,\dots,d_r\in {\rm R}$ with
\[d_1b_1+\dots +d_rb_r=v.\]
The $b_i$ are linearly independent, and therefore the cofactors of $v$ with
respect to $B$ are unique. It follows that $c_i=d_i$ for all $1\leq i \leq
The main algorithmic difficulty stems from the fact that not all formal power
series admit a finite representation, and even if the initial system is given in
a finite form, the splitting transformation as in Theorem <ref> does
not preserve finiteness. In particular, we face two main problems when working
with truncated power series:
$(P1)$ Detecting the correct rank and the linear independent columns of
$(P2)$ If we know the independent columns, a column reduction
transformation computed after truncation is not uniquely determined.
These computational problems arise for general multivariate and for bivariate
systems, but were not addressed in previous algorithmic works on this
topic <cit.>. Before we propose our resolution, we illustrate
both problems in the following example:
Consider the matrix
\[\left(\begin{matrix}
x & 0 & x^2 & x^2+x\\
0 & x & x & x\\
1 & 0 &0 & 1
\end{matrix}\right).
\]
Here, the first three columns $v_1,v_2,v_3$ are linearly
independent and generate the column module. A linear combination of the fourth
column $v_4$ is given by
\[1\cdot v_1 + 0\cdot v_2+1\cdot v_3 = v_4.\]
When truncating at order 1, the system is given as
\[\left(\begin{matrix}
x & 0 & 0 & x\\
0 & x & x & x\\
1 & 0 &0 & 1
\end{matrix}\right)
\]
The original rank cannot be determined from the truncated matrix. Furthermore,
even if we know that $v_1,v_2,v_3$ are linearly independent, there are several
linear combinations of the fourth column after truncation:
\[1\cdot v_1 + 0\cdot v_2+1\cdot v_3 = v_4.\]
\[1\cdot v_1 + 1\cdot v_2+0\cdot v_3 = v_4.\]
The cofactors of the second linear combination are not the truncated cofactors
of the first. It can not be extended with higher order terms to a suitable linear
combination over the formal power series ring without truncation.
We can solve both $(P1)$ and $(P2)$ with the help of minors of the original
system. Let $r$ be the rank of $A$. Then there exists a nonzero $r\times r$
submatrix $B$ of $A$ whose determinant is nonzero. Let $k$ be the order of the
determinant. If we take the truncated system
$\tilde{A} = A \operatorname{rem} x^{k+1}$, the same submatrix $\tilde{B}$ in
$\tilde{A}$ will have a non-zero determinant modulo $x^{k+1}$ and we can
therefore identify in $\tilde{A}$ which columns in $A$ are linearly
independent. This resolves $(P1)$ as long as the truncation order $k$ is chosen
big enough.
Next assume that for instance the first $r$ columns of $A$ are linearly
independent, i.e. we can choose $B$ such that its columns correspond to
$v_1,\dots,v_r$. Let $k$ be as above, $\ell$ be a positive integer and let $v$
be a column vector that is linearly dependent on the columns of $B$. Then there
exist $c_1,\dots,c_r\in\set C[[x_1,\dots,x_n]]$ such that
\[B\cdot(c_1,\dots,c_r)=v.\]
By Cramer's rule, we know that the $c_i$ are given by
\begin{equation}
\label{cramer}
\end{equation}
where $B_i$ is the matrix obtained by replacing the $i^{th}$ column of $B$ by
$v$. Rewriting Equation (<ref>) gives
\begin{equation}
\label{cramer2}
\det(B)c_i-\det(B_i)=0,
\end{equation}
and this equation allows the computation of $c_i$ by coefficient comparison. In
particular, we are guaranteed to obtain the correct $c_i$ up to order $\ell$ if
in (<ref>) we replace $B$ by $\tilde{B}$, its truncation at order
$\ell+k+1$, and $B_i$ by $\tilde{B}_i$, the truncation of $B_i$ at order
$\ell+k+1$. This resolves $(P2)$.
This approach is based on the fact that there is a truncation order $k$ such
that we can find a submatrix of maximal dimension with non-zero determinant. We
have to remark, however, that by the nature of formal power series, it is in
general not possible to tell a priori if a given truncation is high
enough. Furthermore, we emphasize that it is in general not
possible to draw a conclusion about the freeness of the column module from the
integral relations among the truncated column vectors, since any linear
combination of the form $c_1v_1+\dots+c_{d-1}v_{d-1}-v_d=0 \mod x^k$ can
require a non-unit cofactor $c_d$ for higher truncation orders. However, if no
integral relations can be found with the above method, also the column module
without truncation cannot be free. Both observations lead to the following
practical approach. The full algorithm is carried out with a given truncation
order. If the output is correct (compared to the invariant exponential part which can be obtained by Theorem <ref>), we are done. If not, we increase the
truncation order until we get a correct output or arrive at a point where no
integral relations can be found anymore. This procedure necessarily
terminates, since there exists a suitable truncation order.
One should note
that not every $\rm K$-vector space basis of the column space of $A$ is also a
module basis. So, in the worst case, $\binom{d}{r}$ submatrices have to be
tested to obtain a module basis.
§.§ Converse of Theorem <ref>
We consider again a multivariate system $[A]$ as in (<ref>). We fix
$i \in \{ 1, \dots, n \}$ and investigate the rank reduction of its ${i}^{th}$
component given by
\begin{equation}
\label{first}
x_{i}^{p_{i}+1} \pder{F}{x_i} = A_{i}F = (A_{i,0} + A_{i,1} x_{i} + A_{i,2}
x^2_{i} + A_{i,3} x^3_{i} + \dots )F ,
\end{equation}
where the matrices $A_{i,j}$ have their entries in ${\rm R}_{\bar{x}_i}$ and the
algebraic rank of $A_{i,0}$ is denoted by $r$. We recall that we defined
$\bar{x}_{i}:= (x_1, \dots, x_{i-1},x_{i+1}, \dots, x_n)$ and ${\rm R}_{\bar{x}_i}
:= \mathbb{C}[[\bar{x}_{i} ]]$.
The establishment of the converse of Theorem <ref> for the reduction in $x_i$ follows
essentially the steps of that of the bivariate case which was given
in <cit.>. The construction requires successive application of
transformations in $GL_d(R_{\bar{x}_i})$ and shearing transformations in
$x_i$. We remark that when applying a transformation
$T \in GL_d({\rm R}_{\bar{x}_{i}})$ on the $i^{th}$ component, (<ref>)
reduces to
$$ \tilde{A}_{i}= T^{-1} A_{i} T .$$
Given (<ref>), if the column module of ${A}_{i,0}$ is free, then one can
compute a transformation $U_1 \in GL_d({\rm R}_{\bar{x}_{i}})$ such that
$$U_1^{-1}{A}_{i,0} U_1 = \left(\begin{matrix} B^{11} & O
\\ B^{21} & O \end{matrix}\right)$$
has rank $r$, entries in ${\rm R}_{\bar{x}_i}$ and with diagonal blocks of
sizes $r\times r$ and $(d-r)\times(d-r)$ respectively. Let $v$ be the rank of
$B^{11}$. If also the column module of $B^{11}$ is free, then one
can compute a transformation $U_2 \in GL_r({\rm R}_{\bar{x}_i})$
such that
$$U_2^{-1}B^{11} U_2 = \left(\begin{matrix} E^{11} & O \\
E^{21} & O \end{matrix}\right)$$
has rank $v$, entries in ${\rm R}_{\bar{x}_i}$ and with diagonal blocks of
sizes $v\times v$ and $(r-v)\times(r-v)$ respectively. We set
$U : = \operatorname{Diag} (U_2,I_{d-r})\cdot U_1$. Then the leading
coefficient $\tilde{A}_{i,0}$ of the equivalent system $U[A_{i}]$ has the
following form:
\begin{equation} \label{gaussformPfaffian}
\tilde{A}_{i,0}
\;=\; \left(\begin{matrix}\tilde{A}_{i,0}^{11} & O& O\\[5pt]
\tilde{A}_{i,0}^{21} & O_{r-v} & O \\[5pt] \tilde{A}_{i,0}^{31} &
\tilde{A}_{i,0}^{32} & O_{d-r} \end{matrix}\right)\end{equation}
with diagonal blocks of sizes $v\times v$, $(r-v)\times(r-v)$ and
$(d-r)\times(d-r)$ respectively for some $0\leq v < r$ and where
\[
\left(\begin{matrix} \tilde{A}_{i,0}^{11} \\[5pt]
\tilde{A}_{i,0}^{21} \end{matrix}\right) \quad \text{and} \quad
\left(\begin{matrix}\tilde{A}_{i,0}^{11} & O \\[5pt] \tilde{A}_{i,0}^{21} & O
\\[5pt] \tilde{A}_{i,0}^{31} & \tilde{A}_{i,0}^{32} \end{matrix}\right)\]
are $r \times v$ and $d \times r$ matrices of full column ranks $v$ and $r$
respectively. Clearly, $U$ is compatible with
system $A$ since it is unimodular.
From now on, we assume that the leading coefficient $A_{i,0}$ of (<ref>)
is in form (<ref>). In particular, we require the
column module of $A_{i,0}$ and the column module of $B^{11}$ as given above to
be free. We then partition $A_{i,1}$ in accordance with $A_{i,0}$ and set
\begin{equation}
\label{glambdaformPfaffian}
G_{A_{i}} (\lambda) := \left(\begin{matrix} A_{i,0}^{11} & O & A_{i,1}^{13}
\\[5pt] A_{i,0}^{21} & O & A_{i,1}^{23} \\[5pt] A_{i,0}^{31} & A_{i,0}^{32} &
A_{i,1}^{33}+ \lambda I_{d-r}\end{matrix}\right) .\end{equation}
The polynomial $\det(G_{A_{i}} (\lambda))$ vanishes identically in
$\lambda$ if and only if $\theta_{i} (\lambda)$ given by (<ref>)
does. In fact, let
$D(x_i)= \operatorname{Diag}(x_i I_{r}, I_{d-r})$. Then we can write
${x_i}^{-1} A_i(x) = N(x) D^{-1}(x_i)$ where $N(x) \in {\rm R}^{d \times d}$, and
set $D_0 = D(x_i=0)$, $N_0= N(x_i=0)$. Then we have
\begin{eqnarray*} \det (G_{A_i}(\lambda)) & =& \det (N_0 + \lambda D_0) = \det
(N + \lambda D)|_{x_i=0} \\ & = & (\det(\frac{A}{x_i} + \lambda I_d) \det
(D))|_{x_i=0} \\ &=& (\det (\frac{A_{i,0}}{x_i} + A_{i,1} + \lambda I_d)
{x_i}^{r} )|_{x_i=0} = \theta_{i} (\lambda). \end{eqnarray*}
Moreover, $G_{A_{i}} (\lambda)$ has an additional important application within the
construction of a desired transformation as we show in the following proposition:
Suppose that $m(A_{i}) >1$ and $\det(G_{A_{i}} (\lambda))$ is identical to
zero. If the row module of $G_{A_{i}} (\lambda =0)$ is free then there exists a
transformation $Q({\bar{x}_i})$ in $GL_d({\rm R}_{\bar{x}_i})$ with
$\det(Q) = \pm 1$, compatible with system $A$, such that the matrix
$G_{\tilde{A}_{i}} (\lambda))$ has the form
\begin{equation}
\label{particularform3Pfaffian}
G_{\tilde{A}_{i}} (\lambda) = \left(\begin{matrix} A_{i,0}^{11} & O & U_1& U_2
\\[5pt] A_{i,0}^{21} & O & U_3 & U_4 \\[5pt] V_1 & V_2 & W_1 + \lambda I_{d- r
-\varrho} & W_2 \\[5pt] M_1 & O & M_3 & W_3 + \lambda
I_\varrho \end{matrix}\right) ,\end{equation}
where $0 \leq \varrho \leq d-r$,
rank( A_i,0^11 U_1
M_1 M_3 ) =
rank( A_i,0^11 U_1
U_3 ),
rank ( A_i,0^11 U_1
U_3 ) < r.
If the row module of $G_{A_{i}} (\lambda =0)$ is free then the transformation $Q ({\bar{x}_i})$ can be constructed as in the proof
of <cit.> for the bivariate case.
We remark that in the particular case of
$v=0$, (<ref>) is given by
\tilde{A}_{i,0} (\bar{x}_{i} ) = \left(\begin{matrix}
O_{r} & O \\[5pt]
\tilde{A}_{i,0}^{32} & O_{d-r} \end{matrix}\right) , \quad \text{with} \quad
\operatorname{rank}(\tilde{A}_{i,0}^{32}) = r .$$
Consequently, it can be easily verified that (<ref>) is
given by
G_{\tilde{A}_{i}} (\lambda) = \left(\begin{matrix} O_r & U_3 \\[5pt] V_2 & W_1 +
\lambda I_{d- r} \end{matrix}\right) , \quad \text{and} \quad \varrho = 0
. $$
If $m(A_{i}) >1$ and $\det(G_{A_{i}} (\lambda)) \equiv 0$ is as
in (<ref>) with
conditions (<ref>)
and (<ref>) satisfied, then the component $A_{i}$
of $A$ in (<ref>) is reducible and reduction can be carried out with the
shearing $F = S(x_i)\;G$ where
$$\begin{cases} S(x_i)=\operatorname{Diag}(x_i I_r , I_{d-r-\varrho}, x_i
I_\varrho) \quad \text{if } \varrho \neq 0 \\[5pt]
S(x_i)=\operatorname{Diag}(x_i I_r , I_{d-r}) \hspace{37px}
\text{otherwise.} \end{cases} $$
Furthermore, this shearing is compatible with system $A$.
Given system (<ref>). For any $j \in \{1, \dots, n\}$ we partition
$A_{j}$ according to (<ref>)
$$A_{j} = \left(\begin{matrix} A_{j}^{11} & A_{j}^{12} & A_{j}^{13}&
A_{j}^{14}\\[5pt] A_{j}^{21} & A_{j}^{22} & A_{j}^{23} & A_{j}^{24}\\[5pt]
A_{j}^{31} & A_{j}^{32} & A_{j}^{33} & A_{j}^{34}\\[5pt] A_{j}^{41} &
A_{j}^{42} & A_{j}^{43} & A_{j}^{44}\end{matrix}\right), \quad 1 \leq j \leq
n, $$
where $A_{j}^{11} , A_{j}^{22} , A_{j}^{33} , A_{j}^{44}$ are square matrices of
dimensions $v, r-v, d-r-\varrho,$ and $\varrho$ respectively. It is easy to
verify that the equivalent system $S[A] \equiv {\tilde{A}}$ given
by (<ref>) admits the form
Ã_i = (x_i A_i^11
x_i A_i^12 x_i^-1 A_i^13 x_i
x_i A_i^21 x_i A_i^22
x_i^-1 A_i^23 x_iA_i^24
x_i A_i^31
x_i A_i^32 x_i^-1 A_i^33 x_i A_i^34
x_i A_i^41 x_i A_i^42 x_i^-1
A_i^43 x_i A_i^44 ) - x_i^p_i
Diag (I_r, O_d-r-ϱ, I_ϱ)
Ã_j = ( x_j A_j^11 x_j A_j^12
x_j^-1 A_j^13 x_j A_j^14
A_j^21 x_j A_j^22 x_i^-1 A_j^23
x_j A_j^24
x_j A_j^31
x_j A_j^32 x_j^-1A_j^33 x_j A_j^34
x_j A_j^41 x_j A_j^42 x_j^-1A_j^43
x_j A_j^44 ) , 1 ≤j ≠i
Hence, the leading matrix coefficient of the equivalent
$i^{th}$-component is given by
$$ \tilde{A}_{i,0} (\bar{x}_{i}) = \left(\begin{matrix} A_{i,0}^{11}
& O & U_1 & O \\[5pt] A_{i,0}^{21} & O & U_3 & O \\[5pt] O& O &O& O \\[5pt]
M_1 & O & M_3 & O \end{matrix}\right) $$
where $\operatorname{rank}(\tilde{A}_{i,0}) < r$ since (<ref>)
and (<ref>) are satisfied.
It remains to prove the compatibility of $S(x_i)$ with the
system (<ref>), in particular, that the normal crossings are
preserved. It suffices to prove that the entries of
${A}_{j} , 1 \leq j \neq i \leq n, $ which will be multiplied by $x_{i}^{-1}$
upon applying $S(x_i)$, namely, the entries of $A_{j}^{13}, A_{j}^{23},$ and
$A_{j}^{43}$ are zero matrices modulo $x_i$ otherwise poles in $x_{i}$ will be
introduced. This can be restated as requiring $A_{j}^{13}({x}_{i} = 0),$
$A_{j}^{23}({x}_{i} = 0),$ and $A_{j}^{43}({x}_{i} = 0)$ to be zero
submatrices. This requirement is always satisfied due to the integrability
condition and the resulting equality, obtained by setting $x_i=0$, which we
restate here
\begin{equation}
\label{eq:cond0}
x_j^{p_j+1}\pder{A_{i,0}}{x_j}= A_{j}(x_i=0)\;A_{i,0} - A_{i,0} A_{j}(x_i=0) ,
\quad 1 \leq j \neq i \leq n .
\end{equation}
This equality induces a structure of $A_{j}(x_i=0)$ which depends on that of
$A_{i,0}$. Since $G_{A_{i}} (\lambda)$ is as in (<ref>),
then, before applying the shearing transformation,
$A_{i,0} ({\bar{x}}_{i})$ has the following form (<ref>) and
$A_{j} (x_i = 0)$ can be partitioned accordingly. So we have for
$1 \leq j \neq i \leq n$
\begin{eqnarray} \label{form0} A_{i,0}({\bar{x}}_{i}) &=&\left(\begin{matrix}
A_{i,0}^{11} & O & O & O \\[5pt] A_{i,0}^{21} & O_{(r-v)(r-v)} & O & O \\[5pt]
V_1 & V_2 & O_{(d-r-\varrho)(d-r-\varrho)} & O \\[5pt] M_1 & O& O & O_{\varrho
\varrho} \end{matrix}\right) ,\\[10pt]
\label{form1} A_{j}(x_i=0) &=& \left(\begin{matrix} A_{j}^{11}(x_i=0) &
A_{j}^{12}(x_i=0)& A_{j}^{13}(x_i=0)& A_{(j)}^{14}(x_i=0) \\[5pt]
A_{j}^{21}(x_i=0) & A_{j}^{22}(x_i=0) & A_{j}^{23}(x_i=0) &
A_{j}^{24}(x_i=0)\\[5pt] A_{j}^{31}(x_i=0) & A_{j}^{32}(x_i=0) &
A_{j}^{33}(x_i=0)& A_{j}^{34}(x_i=0)\\[5pt] A_{j}^{41}(x_i=0) &
A_{j}^{42}(x_i=0) & A_{j}^{43}(x_i=0) &
A_{j}^{44}(x_i=0) \end{matrix}\right). \end{eqnarray}
Inserting (<ref>)
and (<ref>) in (<ref>), one can obtain the desired results by
equating the entries of (<ref>). More explicitly, upon investigating
the entries in (Column 3), (Rows 1 and 2, Column 2), and (Row 4, Column 2), we
observe the following respectively:
* We have that
\[\left(\begin{matrix} A_{i,0}^{11} & O \\[5pt] A_{i,0}^{21} & O \\[5pt]
V_1 & V_2 \\[5pt] M_1 & O \end{matrix}\right) \cdot \left(\begin{matrix}
A_{j}^{13}(x_i=0) \\[5pt] A_{j}^{23}(x_i=0) \end{matrix}\right) = O_{n,
The former matrix is of full rank $r$ by construction thus $
\smash{\left(\begin{matrix} A_{j}^{13}(x_i=0) \\[5pt]
A_{j}^{23}(x_i=0) \end{matrix}\right)}$ is a zero matrix.
* We also get
\[\smash[t]{\left(\begin{matrix} A_{i,0}^{11} \\ A_{i,0}^{21} \end{matrix}\right)}
\cdot A_{j}^{12}(x_i=0) = O_{r, r-v}.\]
The former is of full rank $v$ by
construction thus $A_{j}^{12}(x_i=0)$ is a zero matrix.
* Finally, $A_{j}^{43}(x_i=0)\cdot V_2 - M_1 \cdot A_{j}^{12}(x_i=0)
=O_{\varrho, (r-v)}$. Since $A_{j}^{12}(x_i=0)$ is null and $V_2$ is of full
column rank $r-v$ by construction then $A_{j}^{43}(x_i=0)$ is a zero matrix as
This completes the proof.
We can thus establish the following theorem:
Consider a Pfaffian system $[A]$ and suppose that $m(A_i) > 1$ for some index $i \in \{1, \dots, n\}$. If $\theta_i (\lambda)$ given by (<ref>) vanishes for some index $i \in \{ 1, \dots, n\}$, then under the conditions required to attain (<ref>) and Proposition <ref>, we can
construct a compatible transformation $T \in GL_d({\rm R}_L)$ which reduces
$m(A_i)$ (and consequently $m(A)$). In this case, $T$ can be chosen to be a
product of transformations in $GL_d({\rm R}_{\bar{x}_i})$ and polynomial
transformations of the form
$\operatorname{Diag}(x_i^{\beta_1}, \dots, x_i^{\beta_d})$ where
$\beta_1, \dots, \beta_d$ are non-negative integers.
Under the required conditions, we can assume that $A_{i,0} $ has the
form (<ref>). Let $G_{A_{i}} (\lambda)$ be given by (<ref>). Then
$\det(G_{A_{i}} (\lambda))$ vanishes identically in $\lambda$ if and only
if $\theta_i(\lambda)$ does. Then the system $S[Q[A]]$ where
$S, Q$ are as in Propositions <ref> and <ref>
respectively, has the desired property.
For a given index $i\in\{1,\dots,n\}$, the algebraic rank of the leading matrix coefficient can be decreased as long as $\theta_{i} (\lambda) $ vanishes
identically in $\lambda$. In case the leading matrix coefficient eventually
reduces to a zero matrix, the Poincaré rank drops at least by one. This
process can be repeated until the Moser rank of system $[A]$ equals to its Moser
invariant. Due to the compatibility of $T$ in Theorem <ref>, rank
reduction can be applied to any of the components of $[A]$ without altering the
Moser rank of the others. Hence, by Corollary <ref>, the true Poincaré
rank of system $[A]$ can be attained by a successive application of the rank
reduction to each of its components.
Finally, we remark that the conditions of Theorem <ref> are always satisfied in the bivariate case $n=2$ of arbitrary dimension.
§.§ Examples
Consider the completely integrable Pfaffian system with normal crossings given by
\begin{equation*}
\begin{cases}
x_1^{2} \pder{F}{x_1} = A_1 F = \left(\begin{matrix}
(x_1 x_2 x_3 +1)(x_1-1)& x_3 (x_1 -1) \\
x_1 x_2 (1 -2x_1 + x_1 x_2 x_3 - x_1^2x_2x_3)& x_1 x_2 x_3 (1-x_1)
\end{matrix}\right) F, \\
x_2^3 \pder{F}{x_2} = A_2 F= \left(\begin{matrix}
(2+3 x_2) (x_1 x_2 x_3 +1) & x_3 (2+3x_2)\\
-x_1x_2(3x_1x_2^2x_3 + 2x_1 x_2 x_3 + x_2^2 + 3 x_2 +2) & -x_1 x_2 x_3 (2+3x_2)
\end{matrix}\right) F,\\
x_3 \pder{F}{x_3} = A_3 F= \left(\begin{matrix}
1 & 0 \\
-x_1 x_2 & 0
\end{matrix}\right) F .
\end{cases}
\end{equation*}
A fundamental matrix of formal solutions is given by:
\begin{equation} \label{exm3:sol} \Phi(x_1, x_2, x_3) x_1^{C_1}x_2^{C_2} x_3^{C_3}
e^{q_1(x_1^{- 1/s_1})} e^{q_2(x_2^{- 1/s_2})} e^{q_3(x_3^{- 1/s_3})}. \end{equation}
If we only seek to compute the exponential parts $q_1, q_2, q_3$ in (<ref>), then from the associated ODS, we compute:
$$ \begin{cases} s_1 =1 \; \text{and} \; q_1(x_1) = \frac{1}{x_1},\\ s_2 =1 \; \text{and} \; q_2(x_2) = \frac{-1 - 3x_2}{x_2^2} , \\
s_3 =1 \; \text{and} \; q_3(x_3) =0. \end{cases}$$
Furthermore, if we wish to compute a fundamental matrix of formal solutions,
then following the steps of our formal reduction algorithm, we look at the
leading coefficients of each of the three components of the given system. If one
of these coefficients has two distinct eigenvalues, the we can apply the
splitting lemma (Theorem <ref>). Indeed, since $A_{1,0}(x_2, x_3)$ has this property, we can
compute such a transformation $ F = T G$ up to any order. In particular, up to
order $10$, we compute:
$$ T = \begin{pmatrix} 1 & x_1^3 x_2^3 x_3^4 - x_1^2 x_2^2 x_3^3 + x_1 x_2 x_3^2 - x_3 \\ -x_1 x_2 & 1 \end{pmatrix} ,$$
which yields the following diagonalized system (up to order $10$):
\begin{equation*}
\begin{cases}
x_1^{2} \pder{G}{x_1} = A_1 G = \left(\begin{matrix}
x_1 -1 & 0 \\
0 & x_1^2 x_2 x_3 f(x)
\end{matrix}\right) G, \\
x_2^3 \pder{G}{x_2} = A_2 G= \left(\begin{matrix}
(2+3 x_2) & 0\\
0 & x_1 x_2^3 x_3 f(x)
\end{matrix}\right) G,\\
x_3 \pder{G}{x_3} = A_3 G= \left(\begin{matrix}
1 & 0 \\
0 & x_1 x_2 x_3 f(x)
\end{matrix}\right) G,
\end{cases}
\end{equation*}
with $f(x):=(x_1^2 x_2^2 x_3^2 - x_1 x_2 x_3 +1)$.
Hence, the system can be uncoupled into two subsystems of linear scalar
equations and integrated to construct $G$, and consequently, $F$.
An example of the reduction process for a system in two variables was given in
Example <ref>. However, examples of dimensions two and three do not cover
the richness of the techniques presented. So, to illustrate the full process, we
treat an example of dimension six. Due to the size of the system and the number
of necessary computation steps, we are not able to include it directly in this
paper. It is available in several formats at
§ FORMAL REDUCTION ALGORITHM
§.§ The Algorithm In Pseudo Code
We now give the full algorithm in pseudo-code and we refer to more
detailed descriptions within the article whenever necessary.
Throughout the article, we adopted the field of complex numbers $\set C$ as
the base field for the simplicity of the presentation. However, any computable
commutative field $K$ with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$ can
be considered instead. In this case, the restrictions on the extensions of the
base field discussed in <cit.> apply
as well and are taken into consideration within our Maple
Given system $[A]$, we discuss the eigenvalues of the leading matrix coefficients
$A_{i,0}$, $i \in \{ 1, \dots, n\}$, of its $n$ components. If for all of these
components uncoupling is unattainable, then we fix $i \in \{ 1, \dots, n\}$ and
proceed to compute the exponential order $\omega(A_i)$ from the associated ODS.
Suppose that $\omega(A_i)= \frac{{\ell} }{m}$ with ${\ell} ,m$ coprime positive
integers. One can then set $t = {x_i}^{1/m}$ (re-adjustment of the independent
variable), and perform again rank reduction to get an equivalent system whose
$i^{th}$ component has Poincaré rank equal to ${\ell}$ and leading matrix
coefficient with at least $m$ distinct eigenvalues. Consequently,
block-diagonalization can be re-applied to uncouple the $i^{th}$-component. By
Section <ref>, this uncoupling results in an uncoupling for the
whole system. As mentioned before, this procedure can be repeated until we
attain either a scalar system, i.e. a system whose $n$ components are scalar
equations, or a system whose Poincaré rank is given by $(0, \dots,0)$. The
former is trivial and effective algorithms are given for the latter
in <cit.>.
Algorithm 1: fmfs_pfaff($p,A$)
Input: $p=(p_1,\dots,p_n), A(x)=(A_1,\dots,A_n)$ of (<ref>).
Output: A fundamental matrix of formal solutions (<ref>).
${\{C_i\}}_{1 \leq i \leq n} \gets O_n$; ${\{Q_i\}}_{1 \leq i \leq n} \gets O_d$; $\Phi \gets I_d$
WHILE $d \neq 1$ or $p_{i} >0$ for some $i \in \{ 1, \dots , n\}$ DO
IF $A_{i,0}$ has at least two distinct eigenvalues
Split system $[A]$ as in Section <ref>; Update $\Phi$
Fmfs_pfaff $(p, \tilde{A}^{11})$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
Fmfs_pfaff $(p, \tilde{A}^{22})$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
ELSE IF $A_{i,0}$ has one non-zero eigenvalue
Update $Q_i$ from the eigenvalues of ${A}_{i,0}$
$A(x) \gets $ Follow Section <ref> ($A_{i,0}$ is now nilpotent)
fmfs_pfaff $(p, \tilde{A}(x))$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
Apply rank reduction of Section <ref>; Update
$\Phi$; Update $p$; Update $A_{i,0}$
IF $p_{i} >0$ and $A_{i,0}$ has at least two distinct eigenvalues
Split system as in Section <ref>; Update $\Phi$
fmfs_pfaff $(p, \tilde{A}^{11}(x))$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
fmfs_pfaff $(p, \tilde{A}^{22}(x))$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
ELSE IF $A_{i,0}$ has one non-zero eigenvalue
Update $Q_i$ from the eigenvalues of $A_{i,0}$
$A(x) \gets $ Follow Section <ref>; ($A_{i,0}$ is now nilpotent)
Fmfs_pfaff $(p, \tilde{A}(x))$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
Follow Section <ref> to compute $\omega(A_i) = \frac{{\ell} }{m}$
$x_{i} \gets {x_{i}}^m$
Apply rank reduction of Section <ref>
Update $\Phi$; Update $p$ ($p_{i} \gets {\ell} $); Update $A_{i,0}$
Update $Q_i$ from the eigenvalues of ${A}_{i,0}$
$A(x) \gets $ Follow Section <ref>; ($A_{i,0}$ is now nilpotent)
fmfs_pfaff $(p, A(x))$; Update $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$
END IF
END IF
END WHILE
RETURN $p$, $A$, $\Phi$, ${\{C_i\}}_{1 \leq i \leq n}$, ${\{Q_i\}}_{1 \leq i \leq n}$.
Algorithm 2: rankReduce($p,A$)
Input: $p_1\dots,p_n, A_1.\dots,A_n$ of (<ref>).
Output: $T(x) \in GL_d({\rm R}_L)$ and an irreducible equivalent system
{$T(A) $} whose Poincaré rank is its true
Poincaré rank and the rank of its leading coefficient
matrices is minimal ($\mu(T(A)) = m(T(A))$)
$T \leftarrow I_{d}$
FOR every $i$ from $1$ to $n$ DO
$T_i \leftarrow I_{d}$; $p_i \leftarrow$ Poincaré rank of $A_{i}$
$U(\bar{x_i}) \leftarrow $ yields the
form (<ref>)
IF $U(\bar{x_i})$ cannot be determined
ERROR “Column module not free.” END IF
$A_{i} \gets U^{-1} A_{i} U $; $T_i \leftarrow T_i U$
WHILE $\det (G_{A_{i}} (\lambda))=0$ and $p_i >0$ DO
$Q(\bar{x_i}), \varrho \leftarrow $
Proposition <ref>
IF $Q(\bar{x_i})$ cannot be determined
ERROR “Row module not free.” END IF
$S(x_i) \leftarrow $ Proposition <ref>
$P \leftarrow Q S$; $T_i \leftarrow T_i P$
$A_{i} \leftarrow P^{-1} A_{i} P - x_i^{p_i} S^{-1} \frac{\partial S}{\partial x_i}$
$p_i \leftarrow $ Poincaré rank of $A_{i}$
$U(\bar{x_i}) \leftarrow $ yields the form (<ref>)
$A_{i} \leftarrow U^{-1} A_{i} U $; $T_i \leftarrow T_i U$
END WHILE
FOR every $j \neq i$ from $1$ to $n$ DO
$A_{j} \leftarrow T_i^{-1} A_{j} T_i - x_j^{p_j+1} T_i^{-1} \pder{T_i}{x_j}$
END FOR
$T \leftarrow T T_i$
END FOR
RETURN ($T, p_1, \dots, p_n, A_1,\dots,A_n$).
§.§ An Alternative Rank Reduction Algorithm
In the case of univariate systems, Levelt's investigations of the existence of
stationary sequences of free lattices lead to an algorithm which reduces the
Poincaré rank to its minimal integer value <cit.>. This algorithm was
then generalized to the bivariate case by the first author of this paper et
al. in <cit.>. The theoretical basis of this algorithm differs
substantially from the algorithm given herein based on Moser's criterion. The
final result of both approaches however, i.e. the algorithm itself, is based on
applying column reductions and shearing transformations in both algorithms,
though in a different manner. In fact, the algorithms coincide for the
particular case of $\varrho =0$. The limitation in <cit.> within the
generalization to the multivariate case is in guaranteeing the freeness
conditions for the leading matrix coefficient $A_{i,0}$ as stated in
Section <ref>. The additional condition of the freeness of the row
module as in Proposition <ref> is not required. Since the linear algebra
problem is resolved in Section <ref>, this results in Algorithm $3$.
Although both algorithms have an identical cost <cit.>,
experimental results for the univariate case and certain bivariate systems
(singularly-perturbed linear differential systems) suggest that the
lattice-based algorithm complicates dramatically the coefficients of the system
under reduction, even if Moser's criterion is adjoined to avoid some unnecessary
computations <cit.>. Hence, Algorithm $2$ can be used as
long as the required freeness conditions hold. Nevertheless, if the freeness of
the row module of $G_{A_i}(\lambda=0)$ is not satisfied, then Algorithm $3$ can
be used as long as the column modules of $A_{i,0}$ and $B^{11}$ is free. There remains
however, the question on the equivalence of these conditions.
Algorithm 3: rankReduce_alt($p,A$)
Input: $p_1\dots,p_n, A_1.\dots,A_n$ of (<ref>).
Output: $T(x) \in GL_d({\rm R}_L)$ and an irreducible equivalent system
{$T(A) $} whose Poincaré rank is its true
Poincaré rank and the rank of its leading coefficient
matrices is minimal ($\mu(T(A)) = m(T(A))$)
$T \leftarrow I_{d}$
FOR every $i$ from $1$ to $n$ DO
$T_i \leftarrow I_{d}$; $p_i \leftarrow$ Poincaré rank of $A_{i}$
WHILE $j < d-1$ and $p_i>0$ DO
$U(\bar{x_i}) \leftarrow $ yields the form
IF $U(\bar{x_i})$ cannot be determined
ERROR “Column module not free.” END IF
$r = \operatorname{rank}(A_{i,0})$
$S(x_i) \leftarrow $ Proposition <ref> with $\varrho =0$ (i.e. $S(x_i) \gets \operatorname{Diag}(x_i I_r, I_{d-r})$)
$P \leftarrow U S$
$A_{i} \leftarrow P^{-1} A_{i} P - x_i^{p_i} S^{-1} \frac{\partial S}{\partial x_i}$
$\tilde{p}_i \leftarrow $ Poincaré rank of $A_{i}$
IF $\tilde{p}_i < p_i$ THEN
$j \leftarrow 0$
$j \leftarrow j+1$
END IF
$p_i \leftarrow \tilde{p}_i$
$T_i \leftarrow T_i P$
END WHILE
FOR every $j \neq i$ from $1$ to $n$ DO
$A_{j} \leftarrow T_i^{-1} A_{j} T_i - x_j^{p_j+1} T_i^{-1} \frac{\partial T_i}{\partial x_j}$
END FOR
$T \leftarrow TT_i$
END FOR
RETURN ($T, {A_{i}}_{\{1 \leq i \leq n\}}$).
§ CONCLUSION
In this article, we studied completely integrable Pfaffian systems with normal
crossings in several variables. We showed that one can associate a set of
univariate linear singular differential systems from which the formal invariants
of the former can be retrieved. This reduces computations to computations over a
univariate field via Isolde or Lindalg, and limits the numbers
of coefficients necessary for the computations. We then complemented our work
with a rank reduction algorithm based on generalizing Moser's criterion and the
algorithm given by Barkatou in <cit.>. The former is applicable to any
bivariate system. However, for multivariate systems, it demands that several explicitly
described conditions are met.
One field of investigation is the possibility of weakening the conditions required in the multivariate setting for the rank reduction. Another one is the adaptation of the techniques developed herein for rank reduction to generalize the notion of simple systems (see <cit.> for
$m=1$). This notion, in the univariate case, gives another approach to construct a basis of the space of regular
solutions <cit.>, and the obstacles encountered are similar to those in rank reduction. An additional field is to study closed form solutions <cit.>
in the light of associated ODS introduced in Section <ref>.
In future work we aim
to thoroughly study the theoretical complexity of the proposed algorithm as well
as give detailed information of the effects on the truncation of the input
system during the computation. This work is not straightforward and even in the
case of regular systems, it is
not studied in the existing work (i.e. <cit.> and references therein).
Systems arising from applications do not necessarily or directly fall into the
class of completely integrable Pfaffian systems with normal
crossings. Investigations in more general classes can be found
in <cit.> and references therein.
An additional field of investigation is the formal reduction in the difference
case using the approaches proposed herein. Praagman established
in <cit.> a formal decomposition of $m$ commuting partial linear
difference operators. This study was intended as an analog to that established
by Levelt, van den Essen, Gérard, Charrière, Deligne, and
others <cit.>.
§ ACKNOWLEDGMENTS
We are grateful to the anonymous referees whose comments and suggestions
improved the readability of this paper. The second author would also like to
thank the Technische Universität Wien, where he is employed by the time of
submission and supported by the ERC Starting Grant 2014 SYMCAR 639270. He
furthermore would like to thank Matteo Gallet for valuable discussions. The
third author would like to thank the University of Limoges since a part of this
work was done during the period of her doctoral studies at DMI.
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#1 1
A Two-Stage Penalized Least Squares Method for Constructing Large Systems of Structural Equations
We propose a two-stage penalized least squares method to build large systems of structural equations based on the instrumental variables view of the classical two-stage least squares method. We show that, with large numbers of endogenous and exogenous variables, the system can be constructed via consistent estimation of a set of conditional expectations at the first stage, and consistent selection of regulatory effects at the second stage. While the consistent estimation at the first stage can be obtained via the ridge regression, the adaptive lasso is employed at the second stage to achieve the consistent selection. The resultant estimates of regulatory effects enjoy the oracle properties. This method is computationally fast and allows for parallel implementation. We demonstrate its effectiveness via simulation studies and real data analysis.
Keywords: Graphical model; High-dimensional data; Reciprocal graphical model; Simultaneous equation model; Structural equation model.
§ INTRODUCTION
We consider a linear system with $p$ endogenous and $q$ exogenous variables. With a sample of $n$ observations from this system, we denote the observed values of endogenous and exogenous variables by $\mathbf{Y}_{n\times p} = (\mathbf{Y}_1, \cdots, \mathbf{Y}_p)$ and $\mathbf{X}_{n\times q} = (\mathbf{X}_1, \cdots, \mathbf{X}_q)$, respectively. The interactions among endogenous variables and the direct causal effects by exogenous variables can be described by a system of structural equations,
\begin{eqnarray} \label{Eqn-FullInfo}
\mathbf{Y}=\mathbf{Y}\boldsymbol{\Gamma} + \mathbf{X}\boldsymbol{\Psi} + \boldsymbol{\epsilon},
\end{eqnarray}
where the $p\times p$ matrix $\boldsymbol{\Gamma}$ has zero diagonal elements and contains regulatory effects, the $q\times p$ matrix $\boldsymbol{\Psi}$ contains causal effects, and $\mathbf{\boldsymbol{\epsilon}}$ is an $n\times p$ matrix of error terms. We assume that $\mathbf{X}$ and $\mathbf{\boldsymbol{\epsilon}}$ are independent of each other, and each component of $\mathbf{\boldsymbol{\epsilon}}$ is independently distributed as normal with zero mean while rows of $\mathbf{\boldsymbol{\epsilon}}$ are identically distributed.
With gene expression levels and genotypic values as endogenous and exogenous variables, respectively, the model (<ref>) has been used to represent a gene regulatory network with each equation modeling the regulatory genetic effects as well as the causal genomic effects from cis-eQTL (i.e., expression quantitative trait loci located within the regions of their target genes) on a given gene, see <cit.>, and <cit.>, among others. Genetical genomics experiments, which collect genome-wide gene expressions and genotypic values, have been widely undertaken to construct gene regulatory networks <cit.>. However, fitting a system of structural equations in (<ref>) to genetical genomics data for the purpose of revealing a whole-genome gene regulatory network is still hindered by lack of an effective statistical method which addresses issues brought by large numbers of endogenous and exogenous variables.
Several efforts have been made to construct the system (<ref>) with genetical genomics data. <cit.> proposed to use a genetic algorithm to search for genetic networks which minimize the Akaike Information Criterion (AIC; ), and <cit.> instead proposed to minimize the Bayesian Information Criterion (BIC; ) and its modification <cit.> for the optimal genetic networks. Both AIC and BIC are applicable to inferring networks for only a small number of endogenous variables. For a large system with many endogenous and exogenous variables, <cit.> proposed to maximize a penalized likelihood to construct a sparse system. However, it is computationally formidable to fit a large system based on the likelihood function of the complete model. <cit.> instead proposed to apply the adaptive lasso <cit.> to fitting each structural equation separately, and then recover the network relying on additional assumption on unique exogenous variables. However, <cit.> demonstrated its inferior performance via simulation studies, which is consistent with our conclusion.
Instead of the full information model specified in (<ref>), we seek to establish the large system via constructing a large number of limited information models, each for one endogenous variable <cit.>. For example, when the $k$-th endogenous variable is concerned, we focus on the $k$-th structural equation in (<ref>) which models the regulatory effects of other endogenous variables and direct causal effects of exogenous variables, and ignore the system structures contained in other structural equations, leading to the following limited-information model,
\begin{eqnarray} \label{Eqn-LimitedInfo}
\left\{\begin{array}{l} \mathbf{Y}_k = \mathbf{Y}_{-k}\boldsymbol{\gamma}_k + \mathbf{X}\boldsymbol{\psi}_k+\boldsymbol{\epsilon}_k,\\
\mathbf{Y}_{-k} = \mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} + \mathbf{\boldsymbol{\xi}}_{-k}. \end{array}\right.
\end{eqnarray}
Here $\mathbf{Y}_{-k}$ refers to $\mathbf{Y}$ excluding the $k$-th column, $\boldsymbol{\gamma}_k$ refers to the $k$-th column of $\boldsymbol{\Gamma}$ excluding the diagonal zero, and $\boldsymbol{\psi}_k$ and $\boldsymbol{\epsilon}_k$ refer to the $k$-th columns of $\boldsymbol{\Psi}$ and $\boldsymbol{\epsilon}$ respectively. The second part of the model (<ref>) is from the following reduced model by excluding the $k$-th reduced-form equation, with $\boldsymbol{\pi} = \boldsymbol{\Psi}(\mathbf{I}-\boldsymbol{\Gamma})^{-1}$ and $\boldsymbol{\xi} = \boldsymbol{\epsilon}(\mathbf{I}-\boldsymbol{\Gamma})^{-1}$,
\begin{eqnarray} \label{Eqn-ReducedForm}
\mathbf{Y} = \mathbf{X}\boldsymbol{\pi} + \boldsymbol{\xi}.
\end{eqnarray}
In a classical low-dimensional setting, applying the ordinary least squares method to the first equation in (<ref>) leads to underestimated $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_k$ due to correlated $\mathbf{Y}_{-k}$ and $\boldsymbol{\epsilon}_k$. Instead, the reduced-form equations in (<ref>) are fitted to obtain least squares estimator $\hat{\mathbf{\boldsymbol{\pi}}}_{-k}$ of $\mathbf{\boldsymbol{\pi}}_{-k}$, and least squares estimators of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_k$ are further obtained by regressing $\mathbf{Y}_k$ against $\hat{\mathbf{Y}}_{-k}=\mathbf{X}\hat{\mathbf{\boldsymbol{\pi}}}_{-k}$ and $\mathbf{X}$. This procedure is widely known as the two-stage least squares (2SLS) method which can produce consistent estimates of the parameters when the system is identifiable. The 2SLS estimator was originally proposed by <cit.> and, independently, <cit.>, and can be restated as the instrumental variables estimator <cit.>.
As in a typical genetical genomics experiment, we are interested in constructing a large system with the number of endogenous variables $p$ possibly larger than the sample size $n$. Such a high-dimensional and small sample size data set makes it infeasible to directly apply the 2SLS method. Indeed, $p\ge n$ may result in perfect fits of reduced-form equations at the first stage, which implies that we regress against the observed values of endogenous variables at the second stage and therefore obtain ordinary least squares estimates of the parameters. It is well known that such ordinary least squares estimates are inconsistent. Furthermore, constructing a large system demands, at the second stage, selecting regulatory endogenous variables among massive candidates, i.e., variable selection in fitting high-dimensional linear models.
In the setting of selecting instrumental variables (IVs) among a large number of candidates, $L_1$ regularized least squares estimators have been recently proposed to replace the ordinary least squares estimator at the first stage of 2SLS <cit.>. <cit.> applied lasso-based methods to select IVs and obtain consistent estimations at the first stage when the first stage is approximately sparse. For sparse instrumental variables models, <cit.> proposed to replace with lasso-based methods at both stages of 2SLS and <cit.> considered the representative $L_1$ regularization methods and a class of concave regularization methods for both stages. All of these methods assume that each endogenous variable is only associated to a relatively small set of exogenous variables, i.e., each row of $\boldsymbol{\pi}$ in (<ref>) only has a small set of nonzero components.
Here we consider to construct a general system of structural equations, which allows us to model nonrecursive or even cyclic relationships between endogenous variables. With the instrumental variables view of the two-stage approach, we observed that successful identification and consistent estimation of model parameters rely on consistent estimation of a set of conditional expectations which are optimal instruments. Therefore, establishing the system (<ref>) in a high-dimensional setting is contingent on obtaining consistent estimation of these conditional expectations at the first stage, and effectively selecting and estimating of regulatory effects out of a large number of candidates at the second stage. Accordingly, we propose a two-stage penalized least squares (2SPLS) method to fit regularized linear models at each stage, with $L_2$ regularized linear models at the first stage and $L_1$ regularized linear models at the second stage.
The proposed method addresses three challenging issues in constructing a large system of structural equations, i.e., memory capacity, computational time, and statistical power. First, the limited information models are considered to develop the algorithm. In this way, we avoid working with full information models which may consist of many subnetworks and involve a massive number of endogenous variables. Second, allowing us to fit one linear model for each endogenous variable at each stage makes the algorithm computationally fast. It also makes it feasible to parallelize the large number of model fittings at each stage. Finally, the oracle properties of the resultant estimates show that the proposed method can achieve optimal power in identifying and estimating regulatory effects. Furthermore, the efficient computation makes it feasible to use the bootstrap method to evaluate the significance of regulatory effects.
The rest of this paper is organized as follows. First, we state an identifiable model in Section <ref>. Section <ref> revisits the instrumental variables view on the classical 2SLS method, which motivates our development of the 2SPLS method in Section <ref>. In Section <ref>, we show that the estimates from 2SPLS have the oracle properties with the proof included in the Appendix. Simulation studies are carried out in Section <ref> to evaluate the performance of 2SPLS. An application to a real data set to infer a yeast gene regulatory network is presented in Section <ref>. We conclude this paper with a discussion in Section <ref>.
§ THE IDENTIFIABLE MODEL
We follow the practice of constructing system (<ref>) in analyzing genetical genomics data <cit.>, and assume that each endogenous variable is affected by a unique set of exogenous variables, that is, the structural equation in (<ref>) has known zero elements of $\boldsymbol{\psi}_k$. Explicitly, we use $\mathcal{S}_k$ to denote the set of row indices of known nonzero elements in $\boldsymbol{\psi}_k$. Then we have known sets $\mathcal{S}_k, k=1, 2, \cdots, p$, which dissect the set $\{1, 2, \cdots, q\}$. We explicitly state this assumption in the below.
Assumption A. $\mathcal{S}_k\ne\emptyset$ for $k=1,\cdots, p$, but $\mathcal{S}_j\cap\mathcal{S}_k=\emptyset$ as long as $j\ne k$.
The above assumption satisfies the rank condition <cit.>, which is a sufficient condition for model identification. Since each $\boldsymbol{\psi}_k$ has a set of known zero components, from this point forward we ignore them and rewrite the structural equation in the model (<ref>) as,
\begin{eqnarray} \label{Eqn-YkStructural}
\mathbf{Y}_k = \mathbf{Y}_{-k}\boldsymbol{\gamma}_k + \mathbf{X}_{\mathcal{S}_k}\boldsymbol{\psi}_{\mathcal{S}_k} + \boldsymbol{\epsilon}_k, \ \ \ \ \ \boldsymbol{\epsilon}_k\sim N(\mathbf{0}, \sigma_k^2 \mathbf{I}_n),
\end{eqnarray}
where $\mathbf{X}_{\mathcal{S}_k}$ refers to $\mathbf{X}$ including only columns indicated by $\mathcal{S}_k$, and $\boldsymbol{\psi}_{\mathcal{S}_k}$ refers to $\boldsymbol{\psi}_k$ including only elements indicated by $\mathcal{S}_k$.
§ THE INSTRUMENTAL VARIABLES VIEW OF THE TWO-STAGE LEAST SQUARES METHOD
Because $\mathbf{Y}_{-k}$ and $\boldsymbol{\epsilon}_k$ are correlated, fitting merely the model (<ref>) results in biased estimates of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$. However, the following two sets of variables are independent,
\begin{eqnarray*}
\left\{\begin{array}{l}
\mathbf{Z}_{-k} = E[\mathbf{Y}_{-k}|\mathbf{X}] = \mathbf{X}\boldsymbol{\pi}_{-k},\\
\boldsymbol{\varepsilon}_k = \boldsymbol{\epsilon}_k+\boldsymbol{\xi}_{-k}\boldsymbol{\gamma}_k. \end{array}\right.
\end{eqnarray*}
Consequently, consistent estimates of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$ can be obtained by applying least squares method to the following model,
\begin{eqnarray} \label{Eqn-IdealModel}
\mathbf{Y}_k = \mathbf{Z}_{-k}\boldsymbol{\gamma}_k + \mathbf{X}_{\mathcal{S}_k}\boldsymbol{\psi}_{\mathcal{S}_k} + \boldsymbol{\varepsilon}_k.
\end{eqnarray}
Observing $\mathbf{Y}_{-k}$ instead of $\mathbf{Z}_{-k} = E[\mathbf{Y}_{-k}|\mathbf{X}]$ naturally leads to application of the instrumental variables method <cit.>, that is, replacing $\mathbf{Z}_{-k} = \mathbf{X}\boldsymbol{\pi}_{-k}$ with its estimate $\hat{\mathbf{Z}}_{-k} = \mathbf{X}\hat{\boldsymbol{\pi}}_{-k}$ in fitting the linear model (<ref>). When a $\sqrt{n}$-consistent least squares estimator of $\boldsymbol{\pi}_{j}$ is obtained by fitting each equation in (<ref>) for $j=1, \cdots, p$, the resultant estimators of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$ are exactly the 2SLS estimators by <cit.> and <cit.>.
Suppose that the matrix $\mathbf{X}$ satisfies the assumption in the below. It is easy to prove that, in a low-dimensional setting, we can obtain consistent estimators for the model (<ref>) with any consistent estimate of $\boldsymbol{\pi}_{-k}$.
Assumption B. $n^{-1}\mathbf{X}^T\mathbf{X}\to\mathbf{C}$, where $\mathbf{C}$ is a positive definite matrix.
Suppose Assumptions A and B are satisfied for the system (<ref>) with fixed $p\ll n$ and $q\ll n$. When there exists a consistent estimator $\hat{\boldsymbol{\pi}}_{-k}$ of $\boldsymbol{\pi}_{-k}$, the ordinary least squares estimators of $(\boldsymbol{\gamma}_k, \boldsymbol{\psi}_{\mathcal{S}_k})$ obtained by regressing $\mathbf{Y}_k$ against $(\mathbf{X}\hat{\boldsymbol{\pi}}_{-k}, \mathbf{X}_{\mathcal{S}_k})$ are also consistent.
The above instrumental variables view implies that the conditional expectation $\mathbf{Z}_{-k} = E[\mathbf{Y}_{-k}|\mathbf{X}]$ serves as the optimal instrument for $\mathbf{Y}_{-k}$. Although, in a low-dimensional setting, any consistent estimator $\hat{\boldsymbol{\pi}}_{-k}$ leads to the instrument $\hat{\mathbf{Z}}_{-k} = \mathbf{X}\hat{\boldsymbol{\pi}}_{-k}$, an efficient estimate of $\boldsymbol{\pi}_{-k}$ should be used to produce efficient estimates of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$. In the following section, we build up on this view and construct the high-dimensional system (<ref>) by first fitting high-dimensional linear models to consistently estimate the conditional expectations of endogenous variables given exogenous variables.
§ THE TWO-STAGE PENALIZED LEAST SQUARES METHOD
To construct the limited-information model (<ref>), we can obtain consistent estimates of the conditional expectations of endogenous variables given exogenous variables by fitting high-dimensional linear models, and then conduct a high-dimensional variable selection following our view on the model (<ref>). Accordingly, we propose a two-stage penalized least squares (2SPLS) procedure to construct each model in (<ref>) so as to establish the large system (<ref>).
§.§ The Method
At the first stage, we use the ridge regression to fit each reduced-form equation in (<ref>) to obtain consistent estimates of the conditional expectations of endogenous variables given exogenous variables, that is, for each $j=1, 2, \cdots, p$, we obtain the ridge regression estimator of $\boldsymbol{\pi}_j$ by minimizing the following penalized sum of squares,
\begin{eqnarray} \label{Est-pi}
\|\mathbf{Y}_j-\mathbf{X}\boldsymbol{\pi}_j\|^2 + \tau_j ||\boldsymbol{\pi}_j||^2,
\end{eqnarray}
where $\tau_j>0$ is a tuning parameter that controls the strength of the penalty. The solution to the minimization problem is $\hat{\boldsymbol{\pi}}_j=(\mathbf{X}^T\mathbf{X}+\tau_j\mathbf{I})^{-1}\mathbf{X}^T\mathbf{Y}_j$, which leads to a consistent estimate of $\mathbf{Z}_j$,
\begin{eqnarray*} %\label{Est-Prediction}
\hat{\mathbf{Z}}_j = \mathbf{P}_{\tau_j} \mathbf{Y}_j,
\end{eqnarray*}
where $\mathbf{P}_{\tau_j} = \mathbf{X}(\mathbf{X}^T\mathbf{X}+\tau_j\mathbf{I})^{-1}\mathbf{X}^T$. With a proper choice of $\tau_j$, the ridge regression has a good estimation performance as shown in the next section.
At the second stage, we replace $\mathbf{Z}_{-k}$ with $\hat{\mathbf{Z}}_{-k}$ in model (<ref>) to derive estimates of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$, specifically, we minimize the following penalized error squares to obtain estimates of $\boldsymbol{\gamma}_k$ and $\boldsymbol{\psi}_{\mathcal{S}_k}$,
\begin{eqnarray} \label{Est-gammapsi}
\frac{1}{2} \|\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k-\mathbf{X}_{\mathcal{S}_k}\boldsymbol{\psi}_{\mathcal{S}_k}\|^2 + \lambda_n \boldsymbol{\omega}_{k}^T|\boldsymbol{\gamma}_k|,
\end{eqnarray}
where $|\boldsymbol{\gamma}_k|$ denotes componentwise absolute value of $\boldsymbol{\gamma}_k$, $\boldsymbol{\omega}_k$ is a known weight vector, and $\lambda_n>0$ is a tuning parameter.
Minimizing for $\boldsymbol{\psi}_{\mathcal{S}_k}$ in (<ref>) leads to
\begin{eqnarray*} %\label{Est-psi}
\hat{\boldsymbol{\psi}}_{\mathcal{S}_k}=(\mathbf{X}_{\mathcal{S}_k}^T \mathbf{X}_{\mathcal{S}_k})^{-1} \mathbf{X}_{\mathcal{S}_k}^T (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_{k}),
\end{eqnarray*}
where $\mathbf{X}_{\mathcal{S}_k}$ is usually of low dimension, and the above least squares estimator of $\boldsymbol{\psi}_{\mathcal{S}_k}$ is easy to obtain.
Plugging $\hat{\boldsymbol{\psi}}_{\mathcal{S}_k}$ into (<ref>), we can solve the following minimization problem to obtain an estimate of $\boldsymbol{\gamma}_k$,
\begin{eqnarray} \label{Est-gamma}
\hat{\boldsymbol{\gamma}}_{k} = \arg\min_{\boldsymbol{\gamma}_{k}} \left\{\frac{1}{2}(\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_{k})^T\mathbf{H}_k(\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_{k}) +\lambda_n \mathbf{\boldsymbol{\omega}}_{k}^T|\boldsymbol{\gamma}_k|\right\},
\end{eqnarray}
where $\mathbf{H}_k=\mathbf{I}-\mathbf{X}_{\mathcal{S}_k} (\mathbf{X}_{\mathcal{S}_k}^T \mathbf{X}_{\mathcal{S}_k})^{-1} \mathbf{X}_{\mathcal{S}_k}^T$, this is equivalent to a variable selection problem in regressing $\mathbf{H}_k\mathbf{Y}_k$ against high-dimensional $\mathbf{H}_k\hat{\mathbf{Z}}_{-k}$. We will resort to adaptive lasso to select nonzero components of $\boldsymbol{\gamma}_k$ and estimate them. Specifically, picking up a $\delta>0$ and obtaining $\tilde{\boldsymbol{\gamma}}_k$ as a $\sqrt{n}$-consistent estimate of $\boldsymbol{\gamma}_k$, we calculate the weight vector $\boldsymbol{\omega}_k$ with components inversely proportional to components of $|\tilde{\boldsymbol{\gamma}}_k|^\delta$. The above minimization problem (<ref>) is a convex optimization problem which is computationally efficient.
§.§ Tuning Parameter Selection
In this method, we need to select tuning parameters at each stage. At the first stage, we propose to choose each $\tau_j$ in (<ref>) by the method of generalized cross-validation (GCV; ), that is,
\[
\tau_j = \arg\min_{\tau>0} G_j(\tau) = \arg\min_{\tau>0}\frac{(\mathbf{Y}_j-\mathbf{P}_{\tau}\mathbf{Y}_j)^T(\mathbf{Y}_j-\mathbf{P}_{\tau}\mathbf{Y}_j)} {(n-\mathrm{tr}\{\mathbf{P}_{\tau}\})^2}.
\]
It is a rotation-invariant version of ordinary cross-validation, and leads to an approximately optimal estimate of the conditional expectation $\mathbf{Z}_j$. At the second stage, the tuning parameter $\lambda_n$ in (<ref>) is obtained via $K$-fold cross validation.
§ THEORETICAL PROPERTIES
As an extension of the classical 2SLS method to high dimensions, the proposed 2SPLS method also has some good theoretical properties. In this section, we will show that the 2SPLS estimates enjoy the oracle properties. As the second-stage estimation relies on the ridge estimates $\hat{\mathbf{Z}}_{-k}$ obtained from the first stage, we start with the theoretical properties of $\hat{\mathbf{Z}}_{-k}$.
As mentioned previously, each $\tau_j$ in (<ref>) is obtained by GCV. Interestingly, as stated by <cit.>, such a $\tau_j$ is closely related to the one minimizing
\[
T_j(\tau) = (\mathbf{Z}_j-\mathbf{P}_{\tau}\mathbf{Y}_j)^T(\mathbf{Z}_j-\mathbf{P}_{\tau}\mathbf{Y}_j).
\]
We have the following result similar to Theorem 2 of <cit.>.
Suppose that all components of $\boldsymbol{\pi}_j$ are i.i.d. with mean zero and variance $\sigma_{\boldsymbol{\pi}}^2$, then
\[
\arg\min_{\tau>0} E\left[E[G_j(\tau)|\boldsymbol{\pi}_j]\right] = \arg\min_{\tau>0} E\left[E[T_j(\tau)|\boldsymbol{\pi}_j]\right] = \frac{\sigma_{\boldsymbol{\xi}_j}^2}{\sigma_{\boldsymbol{\pi}}^2},
\]
where $\sigma_{\boldsymbol{\xi}_j}^2$ is the variance component of $\boldsymbol{\xi}_j$ in model (<ref>).
This theorem implies that the GCV estimate $\hat{\mathbf{Z}}_j = \mathbf{P}_{\tau_j}\mathbf{Y}_j$ is approximately the optimal estimate of the conditional expectation $\mathbf{Z}_j$; furthermore, as the optimal tuning parameter approximates a constant determined by the variance components ratio, we make the following assumption on $\tau_j$.
Assumption C. $\tau_j/\sqrt{n}\to 0$ as $n\to \infty$, for $j=1, \cdots, p$.
We then have the following properties on $\hat{\mathbf{Z}}_{-k}$.
For $k=1,\dots,p$, let $\mathbf{M}_k = \mathbf{\boldsymbol{\pi}}_{-k}^T (\mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k} \mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet})\mathbf{\boldsymbol{\pi}}_{-k}$ where each $\mathbf{C}_{\mathcal{S}_r\mathcal{S}_c}$ is a submatrix of $\mathbf{C}$ identified with row indices in $\mathcal{S}_r$ and column indices in $\mathcal{S}_c$ (the dot implies all rows or columns).
Then, under Assumptions A, B, and C,
a. $n^{-1}\hat{\mathbf{Z}}_{-k}^T\mathbf{H}_k\hat{\mathbf{Z}}_{-k}\to_p \mathbf{M}_k$, as $n\to\infty$;
b. $n^{-1/2} (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k)^T \mathbf{H}_k\hat{\mathbf{Z}}_{-k} \to_d N(\mathbf{0},\sigma_k^2\mathbf{M}_k)$, as $n\to\infty$.
Since $n^{-1}\mathbf{Z}_{-k}^T\mathbf{H}_k\mathbf{Z}_{-k}\to\mathbf{M}_k$, Theorem <ref>.a states that $\hat{\mathbf{Z}}_{-k}^T\mathbf{H}_k\hat{\mathbf{Z}}_{-k}$ is a good approximation to $\mathbf{Z}_{-k}^T\mathbf{H}_k\mathbf{Z}_{-k}$. On the other hand,
$\mathbf{H}_k(\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k)$ is the error term in regressing $\mathbf{H}_k\mathbf{Y}_k$ against $\mathbf{H}_k\hat{\mathbf{Z}}_{-k}$, and Theorem <ref>.b implies that $n^{-1}(\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k)^T \mathbf{H}_k\hat{\mathbf{Z}}_{-k} \to_d 0$. Thus $\hat{\mathbf{Z}}_{-k}$ results in regression errors with good properties, i.e., the error effects on the 2SPLS estimators will vanish when the sample size gets sufficiently large.
In summary, the above theorem indicates that $\hat{\mathbf{Z}}_{-k}$ behaves the same way as $\mathbf{Z}_{-k}$ asymptotically, which makes it reasonable to replace $\mathbf{Z}_{-k}$ with $\hat{\mathbf{Z}}_{-k}$ at the second stage. Denote the $j$-th elements of $\boldsymbol{\gamma}_{k}$ and $\hat{\boldsymbol{\gamma}}_{k}$ as $\gamma_{kj}$ and $\hat{\gamma}_{kj}$, respectively. Then, the properties of $\hat{\mathbf{Z}}_{-k}$ in Theorem <ref>, together with the oracle properties of the adaptive lasso, will lead to the following oracle properties of our proposed estimates.
(Oracle Properties)
Let $\mathcal{A}_k=\left\{j: \gamma_{kj}\neq 0\right\}$, $\hat{\mathcal{A}}_k=\left\{j:\hat{\gamma}_{kj}\neq 0\right\}$, and $\mathbf{M}_{k,\mathcal{A}_k}$ be the submatrix of $\mathbf{M}_k$ identified with both row and column indices in $\mathcal{A}_k$.
Suppose that $\lambda_n/\sqrt{n} \to 0$ and $\lambda_n n^{(\delta-1)/2} \to \infty$. Then, under Assumptions A, B, and C, the estimates from the proposed 2SPLS method satisfy the following properties,
a. Consistency in variable selection: $\lim_{n\rightarrow\infty} P(\hat{\mathcal{A}}_k=\mathcal{A}_k)=1$;
b. Asymptotic normality: $\sqrt{n}(\hat{\boldsymbol{\gamma}}_{k,\mathcal{A}_k}-\boldsymbol{\gamma}_{k,\mathcal{A}_k}) \to_d N(\mathbf{0},\sigma_k^2\mathbf{M}_{k,\mathcal{A}_k}^{-1})$, as $n\rightarrow\infty$.
It is worth mentioning that Theorem <ref> plays an essential role in establishing the oracle properties of 2SPLS. In fact, as long as the properties in Theorem <ref> hold true for the first-stage estimates of $\mathbf{Z}_{-k}$, the oracle properties can be expected from the adaptive lasso <cit.> at the second stage. On the other hand, we can also generalize the second-stage regularization to a wide class of regularization methods <cit.>, the theoretical properties, of which, can still be inherited due to the results in Theorem <ref>.
§ SIMULATION STUDIES
We conducted simulation studies to compare 2SPLS with the adaptive lasso based algorithm (AL) by <cit.>, and the sparsity-aware maximum likelihood algorithm (SML) by <cit.>. To investigate whether it is necessary to select instrumental variables at the first stage as proposed in <cit.>, <cit.>, and <cit.>, we also consider a method which replaces the ridge regression at the first stage of 2SPLS with the adaptive lasso, that is, the two-stage adaptive lasso (2SAL) method. Both acyclic networks and cyclic networks were simulated, each involving $300$ endogenous variables. Each endogenous variable was simulated to have, on average, one regulatory effect for sparse networks, or three regulatory effects for dense networks. The regulatory effects were independently simulated from a uniform distribution over $(-1,-0.5)\cup(0.5,1)$. To allow the use of AL and SML, every endogenous variable in the same network was simulated to have the same number (either one or three) of nonzero exogenous effects (EEs) by the exogenous variables, with all effects equal to one. Each exogenous variable was simulated to take values 0, 1 and 2 with probabilities 0.25, 0.5 and 0.25, respectively, emulating genotypes of an F2 cross in a genetical genomics experiment. All error terms were independently simulated from $N(0,0.1^2)$, and the sample size $n$ varied from $100$ to $1000$. For each network setup, we simulated 100 data sets and applied all four algorithms to calculate the power and false discovery rate (FDR).
For inferring acyclic networks, the power and FDR of the four different algorithms are plotted in Figure <ref>. 2SPLS has greater power than the other three algorithms to infer both sparse and dense acyclic networks when the sample size is small or moderate. When the sample size is large, 2SPLS, SML, and 2SAL are comparable for constructing both sparse and dense acyclic networks. In any case, AL has much lower power than other methods. Specifically, AL provides power as low as under $10\%$ when the sample size is not large, and its power is still under $50\%$ even when the sample size increases to $1000$. On the other hand, 2SPLS provides power over $80\%$ for small sample sizes, and over $90\%$ for moderate to large sample sizes.
a. Power of Sparse Networks
b. FDR of Sparse Networks
c. Power of Dense Networks
d. FDR of Dense Networks
Performance of 2SPLS, AL, SML, and 2SAL when identifying regulatory effects in acyclic networks with one EE or three EEs.
As shown in Figure <ref>, 2SPLS controls the FDR under $20\%$ except for the case which has three available EEs with small sample sizes ($n=100$). Although SML controls the FDR as low as under $5\%$ for sparse acyclic networks when the sample sizes are large, it reports large FDRs when the sample sizes are not large. For example, when the sample sizes are under 200, SML reports FDR over $40\%$ for dense acyclic networks. In general, both 2SPLS and SML outperform AL and 2SAL in terms of FDR. Only in the case when inferring sparse acyclic networks with one available EE from data sets of moderate or large sample sizes, AL and 2SAL report FDR lower than 2SPLS.
Plotted in Figure <ref> are the power and FDR of the four different algorithms when inferring cyclic networks. Similar to the results on acyclic networks, 2SPLS has greater power than SML and AL across all sample sizes and has lower FDR when the sample size is not large. 2SPLS has greater power than 2SAL in most scenarios and has much lower FDR than 2SAL except for the case when inferring sparse cyclic networks from data sets of large sample sizes.
SML provides power competitive to 2SPLS for sparse cyclic networks, but its power is much lower than that of 2SPLS for dense cyclic networks. Similar to the case of acyclic networks, SML reports much higher FDR for inferring dense networks from data sets with small sample sizes though it reports small FDR when the sample sizes are large. 2SAL reports the highest FDR, especially for networks with three available EEs.
Although not performing as well as 2SPLS, 2SAL reports competitive power to SML when inferring either acyclic or cyclic networks. For the acyclic sparse network with one EE, 2SAL can control FDR at a similar level to 2SPLS because each endogenous variable may be associated to a very small set of exogenous variables in (<ref>). However, we observed high FDR of 2SAL in Figure <ref>.b for the acyclic sparse network with three EEs which triples the average number of exogenous variables associated to each endogenous variable. The similar phenomenon of 2SAL appears in Figure <ref>.b for the cyclic sparse networks. The dense networks also triple the average number of regulatory effects for each endogenous variable, which implies an increased number of exogenous variables associated to each endogenous variable in (<ref>). Therefore, we unsurprisingly observed even higher FDR of 2SAL in Figure <ref>.d and Figure <ref>.d, where the FDR is as high as over 0.8. In summary, variable selection at the first stage seems work well when each endogenous variable is associated to a small set of exogenous variables in (<ref>), but may compromise the identification of regulatory effects at the second stage when the number of exogenous variables associated to an endogenous variable increases.
a. Power of Sparse Networks
b. FDR of Sparse Networks
c. Power of Dense Networks
d. FDR of Dense Networks
Performance of 2SPLS, AL, SML, and 2SAL when identifying regulatory effects in cyclic networks with one EE or three EEs.
Both 2SPLS and 2SAL are two-stage methods developed based on the limited-information model (<ref>), instead of the full-information model used by SML, leading to fast computation and potential implementation of parallel computing. To demonstrate the computational advantage of 2SPLS and 2SAL, we recorded the computing time of all algorithms when inferring the same networks from small data sets ($n=100$). Each algorithm analyzed the same data set using only one CPU in a server with Quad-Core AMD Opteron™ Processor 8380. Reported in Table <ref> are the running times of all four algorithms for inferring different networks. AL is the fastest although it performs with the least power. The running time of 2SPLS usually doubles or triples that of AL, but the computation time of 2SAL generally triples that of 2SPLS because 2SAL employed $K$-fold cross-validation to choose the tuning parameter at the first stage. SML is the slowest algorithm which generally takes more than 40 times longer than 2SPLS to infer different networks. In particular, SML is almost 200 times slower than 2SPLS when inferring acyclic sparse networks.
4cAcyclic 4cCyclic
2cSparse 2cDense 2cSparse 2cDense
1 EE 3 EEs 1 EE 3 EEs 1 EE 3 EEs 1 EE 3 EEs
2SPLS 1303 1332 1127 1112 1297 1337 1125 1165
AL 405 652 404 637 443 659 430 781
SML 258875 195739 58509 43118 49393 58716 67949 68081
2SAL 3239 4726 3398 5357 3135 4681 3686 5651
The running time (in seconds) of inferring networks from a data set with $n=100$.
The robustness of 2SPLS was also evaluated from different aspects: (i) its robustness to different noise levels by doubling or even quadrupling the error variance; (ii) its robustness to non-normality of error terms by simulating errors sampled from a t-distribution, i.e., $t(3)$; (iii) its robustness to uncertainty in the connections between exogenous and endogenous variables by simulating three exogenous effects for each endogenous variable (to emulate the genetical genomics experiment, the three exogenous variables are correlated with correlation coefficients at 0.8, and have effects at 1, 0.5, and -0.3, respectively) but including only one exogenous variable with the strongest estimated effects for each endogenous variable; (iv) its robustness to existence of hub nodes by simulating networks with six hub nodes having five regulatory effects on average while other endogenous variables having on average one regulatory effect for sparse networks, or three regulatory effects for dense networks. All networks include 300 endogenous variables, and the networks with errors following $N(0, 0.01)$ are the same as those shown in Figure <ref>. As shown in Figure <ref>, the 2SPLS method demonstrated robust power while the FDR was slightly affected when the error variance doubled. When the error variance quadrupled, a higher FDR was reported as expected. With errors from $t(3)$, we observed similar power and slightly increased FDR of 2SPLS, which confirms the robustness of 2SPLS to non-normality. The uncertainty in the connections between exogenous and endogenous variables had almost no effect on the power of 2SPLS, and only slightly increased the FDR in constructing sparse networks. The existence of hub nodes rarely affected construction of dense networks, but decreased the FDR in constructing sparse networks. Overall, the performance of 2SPLS is remarkable in demonstrating robustness under a variety of realistic data structures.
a. Power of Sparse Networks
b. FDR of Sparse Networks
c. Power of Dense Networks
d. FDR of Dense Networks
Performance of 2SPLS in robustness tests when identifying regulatory effects in acyclic networks with one EE.
§ REAL DATA ANALYSIS
We analyzed a yeast data set with 112 segregants from a cross between two strains BY4716 and RM11-la <cit.>. A total of 5,727 genes were measured for their expression values, and 2,956 markers were genotyped. Each marker within a genetic region (including 1kb upstream and downstream regions) was evaluated for its association with the corresponding gene expression, yielding 722 genes with marginally significant cis-eQTL ($p$-value $<0.05$). The set of cis-eQTL for each gene was filtered to control a pairwise correlation under $0.90$, and then further filtered to keep up to three cis-eQTL which have the strongest association with the corresponding gene expression.
With 112 observations of 722 endogenous variables and 732 exogenous variables, we applied 2SPLS to infer the gene regulatory network in yeast. The constructed network includes 7,300 regulatory effects in total. To evaluate the reliability of constructed gene regulations, we generated 10,000 bootstrap data sets (each with $n=112$) by randomly sampling the original data with replacement, and applied 2SPLS to each data set to infer the gene regulatory network. Among the 7,300 regulatory effects, 323 effects were repeatedly identified in more than 80% of the 10,000 data sets, and Figure <ref> shows the three largest subnetworks formed by these 323 effects. Specifically, the largest subnetwork consists of 22 endogenous variables and 26 regulatory effects, the second largest one includes 14 endogenous variables and 18 regulatory effects, and the third largest one has 11 endogenous variables and 16 regulatory effects.
Three gene regulatory subnetworks in yeast (the dotted, dashed, and solid arrows implied that the corresponding regulations were constructed respectively from over $80\%$, $90\%$, and $95\%$ of the bootstrap data sets).
A gene-enrichment analysis with DAVID <cit.> showed that the three subnetworks are enriched in different gene clusters (controlling $p$-values from Fisher's exact tests under $0.01$). A total of six gene clusters are enriched with genes from the first subnetwork, and four of them are related to either methylation or methyltransferase. Six of 22 genes in the first subnetwork are found in a gene cluster which is related to none-coding RNA processing. The second subnetwork is enriched in nine gene clusters. While three of the clusters are related to electron, one cluster includes half of the genes from the second subnetwork and is related to oxidation reduction. The third subnetwork is also enriched in nine different gene clusters, with seven clusters related to proteasome.
A total of 18 regulations were constructed from each of the 10,000 bootstrap data sets, and are shown in Figure <ref>. There are seven pairs of genes which regulate each other. It is interesting to observe that all regulatory genes up regulate the target genes except two genes, namely, YCL018W and YEL021W.
The yeast gene regulatory subnetworks constructed in each of 10,000 bootstrap data sets (with arrow- and bar-headed lines implying up and down regulations, respectively).
§ DISCUSSION
In a classical setting with small numbers of endogenous/exogenous variables, constructing a system of structural equations has been well studied since <cit.>. <cit.> first proposed to estimate the parameters of a single structural equation with the limited information maximum likelihood estimator. Later, <cit.> and <cit.> independently developed the 2SLS estimator, which is the simplest and most common estimation method for fitting a system of structural equations. However, genetical genomics experiments usually collect data in which both the number of endogenous variables and the number of exogenous variables can be very large, invalidating the classical methods for building gene regulatory networks. It is noteworthy that, although each structural equation modeling gene regulation has few exogenous variables, the genome-wide gene regulatory network consists of a large number of structural equations and therefore has a large number of exogenous variables.
The instrumental variables view of 2SLS sheds light on the consistency of 2SLS estimators which is guaranteed by good estimation of the conditional expectations of endogenous variables given exogenous variables. For large systems, we proposed to estimate these conditional expectations via ridge regression coupled with GCV so as to address possible overfitting issues brought by a large number of exogenous variables. We obtained approximately optimal estimation of these conditional expectations at the first stage. At the second stage, we could adopt results from high-dimensional variable selection, e.g., <cit.>, <cit.>, <cit.>, and <cit.>, to consistently identify and further estimate the regulatory effects of the endogenous variables. As a high-dimensional extension of the classical 2SLS method, the 2SPLS method is also computationally fast and easy to implement. As shown in constructing a genome-wide gene regulatory network of yeast, the high computational efficiency of 2SPLS allows us to employ the bootstrap method to calculate the $p$-values of the regulatory effects.
Our simulation studies show a seemingly counterintuitive result that our moment-based method 2SPLS provides higher power than the likelihood-based method SML, because the maximum likelihood method is usually the most efficient method, and dominates moment methods. However, as evidenced in <cit.> and <cit.> (p.134), 2SLS can perform better than the maximum likelihood method in small samples. Furthermore, SML is not a pure likelihood method but rather a penalized likelihood method, and 2SPLS is not a pure moment method but rather a penalized moment method. Therefore, the theoretical advantage of likelihood methods over moment methods may not carry over to comparing penalized likelihood methods versus penalized moment methods. In fact, SML uses an $L_1$ penalty to penalize nonzero regulatory effects, but 2SPLS employs an $L_2$ penalty on the regression coefficients of the reduced models at the first stage and an $L_1$ penalty on the regulatory effects at the second stage. We conjecture that the different choice of penalty terms may also distinguish the two different methods as shown in the advantage of the elastic net <cit.> over lasso <cit.>.
Although applicable to diverse fields, our development of 2SPLS is motivated by constructing gene regulatory networks using genetical genomics data. The algorithm is applicable to any population-based studies with either experimental crosses or natural populations. Assumption A means that each gene under investigation has at least one unique polymorphism from its cis-eQTL, which can be detected with classical eQTL mapping methods, e.g., <cit.>, <cit.>, and <cit.>. Trans-eQTL (i.e., eQTL outside the regions of their target genes) hold the key to our understanding of gene regulation because their indirect regulations are likely caused by interactions among genes. When the gene regulatory network is modeled with a system of structural equations, classical eQTL mapping methods essentially identify both cis-eQTL and trans-eQTL involved in each reduced-form equation in the reduced model (<ref>). Nonetheless, it is challenging, if not impossible, to recover a large system from the reduced model.
An alternative strategy to construct the whole system is to build undirected graphs first <cit.> and then locally orient the edges in the graphs <cit.>. While constructing a small network is much easier and more robust than constructing a large system, we here intend to construct large networks, such as whole-genome gene regulatory networks from genetical genomics data. Furthermore, application of the alternative strategy is contingent on whether the underlying system is composed of unconnected subsystems, because ignoring the regulatory effects from other genes outside a subset of genes may lead to false regulatory interaction <cit.>. Instead, 2SPLS allows to construct a subset of structural equations inside the whole system, ignoring many other structural equations. Therefore, we can apply 2SPLS to investigate the interactive regulation among a subset of genes as well as how these genes are regulated by others.
It is evidenced in different species that effects of trans-eQTL are weaker than that of cis-eQTL and trans-eQTL are more difficult to identify than cis-eQTL <cit.>. On the other hand, a system of structural equations modeling genome-wide gene regulation may induce a large number of trans-eQTL to each reduced-form equation in (<ref>). While constructing the system is contingent on the accuracy of predicting each endogenous variable on the basis of the corresponding reduced-form equation in (<ref>), the weak effects of a large number of trans-eQTL privilege the use of ridge regression at the first stage of 2SPLS for constructing gene regulatory networks <cit.>. By comparing 2SPLS with 2SAL, our simulation studies demonstrated the superiority of using ridge regression instead of the adaptive lasso at the first stage. In fact, when some genes have a relatively large number of trans-eQTL, selecting variables at the first stage may compromise the identification of regulatory effects at the second stage.
§ APPENDIX A: PROOF OF THEOREM <REF>
a. Since $\tau_j/\sqrt{n}\to 0$ for any $1\le j\le p$, the different choice of $\tau_j$ for each $j$ does not affect the following asymptotic property involving $\tau_j$,
\begin{eqnarray} \label{Asy-XXtau}
\end{eqnarray}
Without loss of generality, we assume $\tau_1=\tau_2=\dots=\tau_p=\tau$. Then $\hat{\mathbf{Z}}_{-k}=\mathbf{P}_\tau\mathbf{Y}_{-k}$.
\begin{eqnarray*}
\lefteqn{\frac{1}{n}\hat{\mathbf{Z}}_{-k}^T\mathbf{H}_k\hat{\mathbf{Z}}_{-k}}\\
%&=& \frac{1}{n} %(\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k})^T(\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k})\\
&=& \frac{1}{n} (\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}+\mathbf{\boldsymbol{\xi}}_{-k})^T \mathbf{P}_{\tau}^T \mathbf{H}_k \mathbf{P}_{\tau} (\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}+\mathbf{\boldsymbol{\xi}}_{-k}) \\
&=& \frac{1}{n} \mathbf{\boldsymbol{\pi}}_{-k}^T \mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} + \frac{1}{n} \mathbf{\boldsymbol{\xi}}_{-k}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}\\
& & + \frac{1}{n} \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k} + \frac{1}{n} \mathbf{\boldsymbol{\xi}}_{-k}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k}
\end{eqnarray*}
We will consider the asymptotic property of each of the above four terms.
First, $\frac{1}{n}\mathbf{X}^T\mathbf{X}\to\mathbf{C}$ implies that
\begin{eqnarray} \label{Asy-XHX}
\frac{1}{n} \mathbf{X}^T\mathbf{H}_k\mathbf{X} =
\frac{1}{n} \mathbf{X}^T \{\mathbf{I}-\mathbf{X}_{\mathcal{S}_k} (\mathbf{X}_{\mathcal{S}_k}^T \mathbf{X}_{\mathcal{S}_k})^{-1}\mathbf{X}_{\mathcal{S}_k}^T\}\mathbf{X} \to \mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k}\mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet}.
\end{eqnarray}
The above result and (<ref>) easily lead to the following result,
\begin{eqnarray} \label{Asy-Mk}
\lefteqn{\frac{1}{n} \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T \mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}} \nonumber\\
&=& \frac{1}{n} \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T \mathbf{X} (\mathbf{X}^T\mathbf{X}+\tau\mathbf{I})^{-1} \mathbf{X}^T\mathbf{H}_k\mathbf{X} (\mathbf{X}^T\mathbf{X}+\tau\mathbf{I})^{-1} \mathbf{X}^T\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} \nonumber \\
&\to& \mathbf{\boldsymbol{\pi}}_{-k}^T (\mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k}\mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet}) \mathbf{\boldsymbol{\pi}}_{-k} = \mathbf{M}_k.
\end{eqnarray}
The other three terms approaching to zero directly follows that $\frac{1}{n} \mathbf{\boldsymbol{\xi}}_{-k}^T\mathbf{X}\to_p\mathbf{0}$. Thus, $\frac{1}{n}\hat{\mathbf{Z}}_{-k}^T\mathbf{H}_k\hat{\mathbf{Z}}_{-k} \to_p \mathbf{M}_k$.
b. Since $\mathbf{H}_k(\mathbf{Y}_{k}-\mathbf{Y}_{-k}\boldsymbol{\gamma}_k)=\mathbf{H}_k\boldsymbol{\epsilon}_k$, we have
\begin{eqnarray*}
\lefteqn{\frac{1}{\sqrt{n}} (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k)^T \mathbf{H}_k \hat{\mathbf{Z}}_{-k}}\\
&=& \frac{1}{\sqrt{n}} \{(\mathbf{Y}_{k}-\mathbf{Y}_{-k}\boldsymbol{\gamma}_k) + (\mathbf{I}-\mathbf{P}_{\tau})\mathbf{Y}_{-k}\boldsymbol{\gamma}_k\}^T \mathbf{H}_k \hat{\mathbf{Z}}_{-k} \\
&=& \frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k} + \frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T \{(\mathbf{I}-\mathbf{P}_{\tau})\mathbf{Y}_{-k}\}^T \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k} \end{eqnarray*}
In the following, we will prove that the second term approaches to zero, and the first term asymptotically approaches to the required distribution, i.e.,
\begin{eqnarray} \label{Asy-EHPX}
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k} \to_d N(\mathbf{0},\sigma_k^2 \mathbf{M}_k).
\end{eqnarray}
We notice that
\[
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{P}_{\tau} \mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} \sim N(\mathbf{0}, \frac{\sigma_k^2}{n} \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}).
\]
Following (<ref>), we have
\begin{eqnarray} \label{Asy-EHPXPi}
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{P}_{\tau} \mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} \to_d N(\mathbf{0},\sigma_k^2 \mathbf{M}_k).
\end{eqnarray}
Because of (<ref>) and
\[
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T\mathbf{H}_k\mathbf{X} \sim N(\mathbf{0}, \frac{\sigma_k^2}{n}\mathbf{X}^T\mathbf{H}_k\mathbf{X}),
\]
we have
\[
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T\mathbf{H}_k\mathbf{X} \to_d N(\mathbf{0}, \sigma_k^2(\mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k} \mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet})).
\]
Since $\frac{1}{n}\boldsymbol{\xi}_{-k}^T\mathbf{X}\to_p \mathbf{0}$, we can apply Slutsky's theorem and obtain that
\begin{eqnarray*}
\frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k} = \frac{1}{\sqrt{n}} \boldsymbol{\epsilon}_k^T \mathbf{H}_k\mathbf{X}(\mathbf{X}^T\mathbf{X} + \tau\mathbf{I})^{-1}\mathbf{X}^T\mathbf{\boldsymbol{\xi}}_{-k} \to_p \mathbf{0}.
\end{eqnarray*}
Pooling the above result and (<ref>) leads to the asymptotic distribution in (<ref>).
To prove that the second term asymptotically approaches to zero, we further partition it as follows,
\begin{eqnarray*}
\lefteqn{\frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\{(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{Y}_{-k}\}^T\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k}}\\
&=& \frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} + \frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\boldsymbol{\xi}_{-k}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}\\
&& +\frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\pi}}_{-k}^T \mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\boldsymbol{\xi}_{-k} + \frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\boldsymbol{\xi}_{-k}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\boldsymbol{\xi}_{-k}.
\end{eqnarray*}
It suffices to prove each of these four parts asymptotically approaches to zero.
First, notice that
\[
\mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) = \tau(\mathbf{X}^T\mathbf{X}+\tau\mathbf{I})^{-1}\mathbf{X}^T,
\]
we have
\begin{eqnarray} \label{Asy-XIPHPXPi}
\lefteqn{\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}} \nonumber\\
&=& \frac{\tau}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\pi}}_{-k}^T (\mathbf{X}^T\mathbf{X}+\tau\mathbf{I})^{-1} \mathbf{X}^T\mathbf{H}_k\mathbf{X} (\mathbf{X}^T\mathbf{X}+\tau\mathbf{I})^{-1} \mathbf{X}^T\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} \to \mathbf{0},
\end{eqnarray}
which follows (<ref>) and that $\tau/\sqrt{n}\to 0$ as $n\to \infty$.
Because $\mathbf{C}_{\mathcal{S}_{k}\bullet}\mathbf{C}^{-1}\mathbf{C}_{\bullet\mathcal{S}_{k}} = \mathbf{C}_{\mathcal{S}_{k}\mathcal{S}_{k}}$, we have
\[
(\mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k} \mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet}) \mathbf{C}^{-1} (\mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k} \mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet}) = \mathbf{C}-\mathbf{C}_{\bullet\mathcal{S}_k} \mathbf{C}_{\mathcal{S}_k,\mathcal{S}_k}^{-1} \mathbf{C}_{\mathcal{S}_k\bullet},
\]
which implies that
\begin{eqnarray*}
\lefteqn{\frac{1}{n} \mathbf{X}^T \mathbf{P}_{\tau}^T \mathbf{H}_k^T (\mathbf{I}-\mathbf{P}_{\tau})^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}} \\
&=& \frac{1}{n} \mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X} - \frac{2}{n} \mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X} + \frac{1}{n} \mathbf{X}^T\mathbf{P}_{\tau}\mathbf{H}_k\mathbf{P}_{\tau}^2\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\to \mathbf{0}.
\end{eqnarray*}
Since Var$(\boldsymbol{\xi}_{-k}\boldsymbol{\gamma}_k)$ is proportional to an identity matrix, the above result leads to that
\[
\mathrm{Var}\left(\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k}\right)\to \mathbf{0},
\]
which implies that
\begin{eqnarray} \label{Asy-GXIPHPXPi}
\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{X}\mathbf{\boldsymbol{\pi}}_{-k} \to_p \mathbf{0}.
\end{eqnarray}
Similarly, we can prove that, for each $\boldsymbol{\xi}_{j}$,
\[
\mathrm{Var}\left(\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_j\right)\to \mathbf{0},
\]
which implies that
\begin{eqnarray} \label{Asy-GPiXIPHPX}
\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T \mathbf{\boldsymbol{\pi}}_{-k}^T\mathbf{X}^T (\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k} \to_p \mathbf{0}.
\end{eqnarray}
Note that
\begin{eqnarray*}
\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k} = \left\{\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k
\mathbf{X}\right\} \left\{(\mathbf{X}^T\mathbf{X}+\tau \mathbf{I})^{-1} \mathbf{X}^T\mathbf{\boldsymbol{\xi}}_{-k}\right\}.
\end{eqnarray*}
\begin{eqnarray*}
\frac{1}{n}\mathbf{X}^T \mathbf{H}_k (\mathbf{I}-\mathbf{P}_{\tau})
(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{X} \to \mathbf{0},
\end{eqnarray*}
we have
\[
\mathrm{Var}(\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k
\mathbf{X}) \to \mathbf{0}.
\]
\[
\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k
\mathbf{X} \to _p \mathbf{0},
\]
which, together with $(\mathbf{X}^T\mathbf{X}+\tau \mathbf{I})^{-1}\mathbf{X}^T\boldsymbol{\xi}_{-k}\to_p \mathbf{0}$, leads to that
\begin{eqnarray}\label{Asy-GXIPHPX}
\frac{1}{\sqrt{n}} \boldsymbol{\gamma}_k^T\mathbf{\boldsymbol{\xi}}_{-k}^T(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{H}_k\mathbf{P}_{\tau}\mathbf{\boldsymbol{\xi}}_{-k}\to_p \mathbf{0}.
\end{eqnarray}
Pooling (<ref>), (<ref>), (<ref>) and (<ref>), we have proved that
$\frac{1}{\sqrt{n}}\boldsymbol{\gamma}_k^T\{(\mathbf{I}-\mathbf{P}_{\tau}) \mathbf{Y}_{-k}\}^T\mathbf{H}_k\mathbf{P}_{\tau}\mathbf{Y}_{-k}\to_p \mathbf{0}$, which concludes our proof.
§ APPENDIX B: PROOF OF THEOREM <REF>
Let $\boldsymbol{\psi}_n{(\boldsymbol{\mu})} = \|\mathbf{H}_k\mathbf{Y}_k-\mathbf{H}_k\hat{\mathbf{Z}}_{-k}(\boldsymbol{\gamma}_k + \frac{\boldsymbol{\mu}}{\sqrt{n}})\|^2 + \lambda_n \mathbf{\boldsymbol{\omega}}_k^T |\boldsymbol{\gamma}_k+\frac{\boldsymbol{\mu}}{\sqrt{n}}|$. Let $\hat{\boldsymbol{\mu}} = \arg\min_{\boldsymbol{\mu}}\boldsymbol{\psi}_n(\boldsymbol{\mu})$, then $\hat{\boldsymbol{\gamma}}_k = \boldsymbol{\gamma}_k + \frac{\hat{\boldsymbol{\mu}}}{\sqrt{n}}$ or $\hat{\boldsymbol{\mu}} = \sqrt{n}(\hat{\boldsymbol{\gamma}}_k - \boldsymbol{\gamma}_k)$. Note that $\boldsymbol{\psi}_n(\boldsymbol{\mu})-\boldsymbol{\psi}_n(\mathbf{0})=V_n(\boldsymbol{\mu})$, where
\begin{eqnarray*}
V_n(\boldsymbol{\mu}) & = & \boldsymbol{\mu}^T(\frac{1}{n}\hat{\mathbf{Z}}_{-k}^T\mathbf{H}_k\hat{\mathbf{Z}}_{-k})\boldsymbol{\mu} - \frac{2}{\sqrt{n}}(\mathbf{Y}_{k} - \hat{\mathbf{Z}}_{-k} \boldsymbol{\gamma}_k)^T \mathbf{H}_k \hat{\mathbf{Z}}_{-k} \boldsymbol{\mu} \\
& & + \frac{\lambda_n}{\sqrt{n}} \mathbf{\boldsymbol{\omega}}_k^T\times \sqrt{n}(|\boldsymbol{\gamma}_k + \frac{\boldsymbol{\mu}}{\sqrt{n}}|-|\boldsymbol{\gamma}_k|).
\end{eqnarray*}
Denote the $j$-th elements of $\boldsymbol{\omega}_k$ and $\boldsymbol{\mu}$ as $\omega_{kj}$ and $\mu_j$, respectively.
If $\gamma_{kj}\neq 0$, then $\omega_{kj}\to_p |\gamma_{kj}|^{-\delta}$ and $\sqrt{n}(|\gamma_{kj}+\frac{\mu_j}{\sqrt{n}}|-|\gamma_{kj}|) \to_p \mu_j \mathrm{sign}({\gamma_{kj}})$. By Slutsky's theorem, we have $\frac{\lambda_n}{\sqrt{n}}\omega_{kj}\sqrt{n}(|\gamma_{kj}+\frac{\mu_j}{\sqrt{n}}|-|\gamma_{kj}|) \to_p 0$. If $\gamma_{kj}=0$, then $\sqrt{n}(|\gamma_{kj}+\frac{\mu_j}{\sqrt{n}}|-|\gamma_{kj}|)=|\mu_j|$ and $\frac{\lambda_n}{\sqrt{n}} \omega_{kj} = \frac{\lambda_n}{\sqrt{n}} n^{\delta/2}(|\sqrt{n}\tilde{\gamma}_{kj}|)^{-\delta}$, where $\sqrt{n}\tilde{\gamma}_{kj}=O_p{(1)}$. Thus,
\[
\frac{\lambda_n}{\sqrt{n}} \mathbf{\boldsymbol{\omega}}_k^T\times \sqrt{n}(|\boldsymbol{\gamma}_k + \frac{\boldsymbol{\mu}}{\sqrt{n}}|-|\boldsymbol{\gamma}_k|)
%\frac{\lambda_n}{\sqrt{n}} \omega_{kj}\sqrt{n}(|\gamma_{kj}+\frac{\mu_j}{\sqrt{n}}|-|\gamma_{kj}|)
\to_p \left\{\begin{array}{ll}
0, &\mathrm{if}\ \|\boldsymbol{\mu}_{\mathcal{A}_k^c}\|=0; \\
\infty, &\mathrm{otherwise}.
\end{array}\right.
\]
Hence, following Theorem <ref> and Slutsky's theorem, we see that $V_n(\boldsymbol{\mu})\to_d V(\boldsymbol{\mu})$ for every $\boldsymbol{\mu}$, where
\[
\left\{\begin{array}{ll}
\boldsymbol{\mu}_{\mathcal{A}_k}^T\mathbf{M}_{k,\mathcal{A}_k} \boldsymbol{\mu}_{\mathcal{A}_k}-2 \boldsymbol{\mu}_{\mathcal{A}_k}^T\mathbf{W}_{k,\mathcal{A}_k}, &\mathrm{if}\ \|\boldsymbol{\mu}_{\mathcal{A}_k^c}\|=0; \\
\infty, &\mathrm{otherwise}.
\end{array}\right.
\]
$V_n(\boldsymbol{\mu})$ is convex, and the unique minimizer of $V(\boldsymbol{\mu})$ is $(\mathbf{M}_{k,\mathcal{A}_k}^{-1}\mathbf{W}_{k,\mathcal{A}_k},\mathbf{0})^T$.
Following the epi-convergence results of <cit.> and <cit.>, we have
\begin{eqnarray*}
\left\{\begin{array}{l}\hat{\boldsymbol{\mu}}_{\mathcal{A}_k} \to_d \mathbf{M}_{k,\mathcal{A}_k}^{-1}\mathbf{W}_{k,\mathcal{A}_k},\\
\hat{\boldsymbol{\mu}}_{\mathcal{A}_k^c} \to_d \mathbf{0}. \end{array}\right.
\end{eqnarray*}
Since $\mathbf{W}_{k,\mathcal{A}_k} \sim N(\mathbf{0},\sigma_k^2\mathbf{M}_{k,\mathcal{A}_k})$, we indeed have proved the asymptotic normality.
Now we show the consistency in variable selection. $\forall j \in \mathcal{A}_k$, the asymptotic normality indicates that $\hat{\gamma}_{kj} \to_p \gamma_{kj}$, thus $P(j\in\hat{\mathcal{A}}_k) \to 1$. Then it suffices to show that $\forall j \notin \mathcal{A}_k$, $P(j\in\hat{\mathcal{A}}_k) \to 0$.
When $j\in\hat{\mathcal{A}}_k$, by the KKT normality conditions, we know that $\hat{\mathbf{Z}}_{j}^T \mathbf{H}_k (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\hat{\boldsymbol{\gamma}}_k) = \lambda_n \omega_{kj}$. Note that $\lambda_n\omega_{kj}/\sqrt{n} \to_p \infty$, whereas $\frac{1}{\sqrt{n}}\hat{\mathbf{Z}}_{j}^T \mathbf{H}_k (\mathbf{Y}_k - \hat{\mathbf{Z}}_{-k} \hat{\boldsymbol{\gamma}}_k) = \frac{1}{n} \hat{\mathbf{Z}}_{j}^T \mathbf{H}_k \hat{\mathbf{Z}}_{-k} \times \sqrt{n} (\boldsymbol{\gamma}_k-\hat{\boldsymbol{\gamma}}_k) + \frac{1}{\sqrt{n}} \hat{\mathbf{Z}}_{j}^T \mathbf{H}_k (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k}\boldsymbol{\gamma}_k)$. Following Theorem <ref> and the asymptotic normality, $\frac{1}{\sqrt{n}}\hat{\mathbf{Z}}_{j}^T \mathbf{H}_k (\mathbf{Y}_k - \hat{\mathbf{Z}}_{-k} \hat{\boldsymbol{\gamma}}_k)$ asymptotically follows a normal distribution. Thus, $P(j\in\hat{\mathcal{A}}_k) \leq P(\hat{\mathbf{Z}}_{j}^T \mathbf{H}_k (\mathbf{Y}_k-\hat{\mathbf{Z}}_{-k} \hat{\boldsymbol{\gamma}}_k) = \lambda_n\omega_{kj}) \to 0$. Then we have proved the consistency in variable selection.
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1511.00041
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We consider the problem of learning causal networks with interventions, when each intervention is limited in size under Pearl's Structural Equation Model with independent errors (SEM-IE). The objective is to minimize the number of experiments to discover the causal directions of all the edges in a causal graph. Previous work has focused on the use of separating systems for complete graphs for this task. We prove that any deterministic adaptive algorithm needs to be a separating system in order to learn complete graphs in the worst case. In addition, we present a novel separating system construction, whose size is close to optimal and is arguably simpler than previous work in combinatorics. We also develop a novel information theoretic lower bound on the number of interventions that applies in full generality, including for randomized adaptive learning algorithms.
For general chordal graphs, we derive worst case lower bounds on the number of interventions. Building on observations about induced trees, we give a new deterministic adaptive algorithm to learn directions on any chordal skeleton completely. In the worst case, our achievable scheme is an $\alpha$-approximation algorithm where $\alpha$ is the independence number of the graph. We also show that there exist graph classes for which the sufficient number of experiments is close to the lower bound. In the other extreme, there are graph classes for which the required number of experiments is multiplicatively $\alpha$ away from our lower bound.
In simulations, our algorithm almost always performs very close to the lower bound, while the approach based on separating systems for complete graphs is significantly worse for random chordal graphs.
§ INTRODUCTION
Causality is a fundamental concept in sciences and philosophy. The mathematical formulation of a theory of causality in a probabilistic sense has received significant attention recently (e.g. <cit.>). A formulation advocated by Pearl considers the structural equation models: In this framework, $X$ is a cause of $Y$, if $Y$ can be written as $f(X,E)$, for some deterministic function $f$ and some latent random variable $E$. Given two causally related variables $X$ and $Y$, it is not possible to infer whether $X$ causes $Y$ or $Y$ causes $X$ from random samples, unless certain assumptions are made on the distribution of $E$ and/or on $f$ <cit.>.
For more than two random variables, directed acyclic graphs (DAGs) are the most common tool used for representing causal relations.
For a given DAG $D=(V,E)$, the directed edge $(X,Y)\in E$ shows that $X$ is a cause of $Y$.
If we make no assumptions on the data generating process, the standard way of inferring the causal directions is by performing experiments, the so-called interventions. An intervention requires modifying the process that generates the random variables: The experimenter has to enforce values on the random variables. This process is different than conditioning as explained in detail in <cit.>.
The natural problem to consider is therefore minimizing the number of interventions required to learn a causal DAG.
Hauser et al. <cit.> developed an efficient algorithm that minimizes this number in the worst case. The algorithm is based on optimal coloring of chordal graphs and requires at most $\log \chi$ interventions to learn any causal graph where $\chi$ is the chromatic number of the chordal skeleton.
However, one important open problem appears when one also considers the size of the used interventions: Each intervention is an experiment where the scientist must force a set of variables to take random values. Unfortunately, the interventions obtained in
<cit.> can involve up to $n/2$ variables.
The simultaneous enforcing of many variables can be quite challenging in many applications: for example in biology, some variables may not be enforceable at all or may require complicated genomic interventions for each parameter.
In this paper, we consider the problem of learning a causal graph when intervention sizes are bounded by some parameter $k$.
The first work we are aware of for this problem is by Eberhardt et al. <cit.>, where he provided an achievable scheme.
Furthermore <cit.> shows that the set of interventions to fully identify a causal DAG must satisfy a specific set of combinatorial conditions called a separating system[A separating system is a $0$-$1$ matrix with $n$ distinct columns and each row has at most $k$ ones.], when the intervention size is not constrained or is 1. In <cit.>, with the assumption that the same holds true for any intervention size, Hyttinen et al. draw connections between causality and known separating system constructions.
One open problem is: If the learning algorithm is adaptive after each intervention, is a separating system still needed or can one do better? It was believed that adaptivity does not help in the worst case <cit.> and that one still needs a separating system.
Our Contributions:
We obtain several novel results for learning causal graphs with interventions bounded by size $k$.
The problem can be separated for the special case where the underlying undirected graph (the skeleton) is the complete graph and the more general case where the underlying undirected graph is chordal.
* For complete graph skeletons, we show that any adaptive deterministic algorithm needs a $(n,k)$ separating system. This implies that lower bounds for separating systems also hold for adaptive algorithms and resolves the previously mentioned open problem.
* We present a novel combinatorial construction of a separating system that is close to the previous lower bound. This simple construction may be of more general interest in combinatorics.
* Recently <cit.> showed that randomized adaptive algorithms need only $\log \log n$ interventions with high probability for the unbounded case. We extend this result and show that $O \left(\frac{n}{k} \log \log k \right)$ interventions of size bounded by $k$ suffice with high probability.
* We present a more general information theoretic lower bound of $\frac{n}{2k}$ to capture the performance of such randomized algorithms.
* We extend the lower bound for adaptive algorithms for general chordal graphs. We show that over all orientations, the number of experiments from a $(\chi(G),k)$ separating system is needed where $\chi(G)$ is the chromatic number of the skeleton graph.
* We show two extremal classes of graphs. For one of them, the interventions through $(\chi,k)$ separating system is sufficient. For the other class, we need $\frac{\alpha \left(\chi-1 \right)}{2k} \approx \frac{n}{2k}$ experiments in the worst case.
* We exploit the structural properties of chordal graphs to design a new deterministic adaptive algorithm that uses the idea of separating systems together with adaptability to Meek rules.
We simulate our new algorithm and empirically observe that it performs quite close to the $(\chi,k)$ separating system. Our algorithm requires much fewer interventions compared to $(n,k)$ separating systems.
§ BACKGROUND AND TERMINOLOGY
§.§ Essential graphs
A causal DAG $D=(V,E)$ is a directed acyclic graph where $V=\{x_1,x_2 \ldots x_n\}$ is a set of random variables and $(x,y) \in E$ is a directed edge if and only if $x$ is a direct cause of $y$. We adopt Pearl's structural equation model with independent errors (SEM-IE) in this work (see <cit.> for more details). Variables in $S \subseteq V$ cause $x_i$, if $x_i = f(\{x_j\}_{j \in S}, e_y )$ where $e_y$ is a random variable independent of all other variables.
The causal relations of $D$ imply a set of conditional independence (CI) relations between the variables. A conditional independence relation is of the following form: Given $Z$, the set $X$ and the set $Y$ are conditionally independent for some disjoint subsets of variables $X,Y,Z$. Due to this, causal DAGs are also called causal Bayesian networks. A set $V$ of variables is Bayesian with respect to a DAG $D$ if the joint probability distribution of $V$ can be factorized as a product of marginals of every variable conditioned on its parents.
All the CI relations that are learned statistically through observations can also be inferred from the Bayesian network using a graphical criterion called the d-separation <cit.> assuming that the distribution is faithful to the graph [Given Bayesian network, any CI relation implied by d-separation holds true. All the CIs implied by the distribution can be found using d-separation if the distribution is faithful. Faithfulness is a widely accepted assumption, since it is known that only a measure zero set of distributions are not faithful <cit.>.]. Two causal DAGs are said to be Markov equivalent if they encode the same set of CIs. Two causal DAGs are Markov equivalent if and only if they have the same skeleton[Skeleton of a DAG is the undirected graph obtained when directed edges are converted to undirected edges.] and the same immoralities[An induced subgraph on $X,Y,Z$ is an immorality if $X$ and $Y$ are disconnected, $X\rightarrow Z$ and $Z\leftarrow Y$. ]. The class of causal DAGs that encode the same set of CIs is called the Markov equivalence class.
We denote the Markov equivalence class of a DAG $D$ by $[D]$. The graph union[Graph union of two DAGs $D_1=(V,E_1)$ and $D_2=(V,E_2)$ with the same skeleton is a partially directed graph $D=(V,E)$, where $(v_a,v_b)\in E$ is undirected if the edges $(v_a,v_b)$ in $E_1$ and $E_2$ have different directions, and directed as $v_a\rightarrow v_b$ if the edges $(v_a,v_b)$ in $E_1$ and $E_2$ are both directed as $v_a\rightarrow v_b$.] of all DAGs in $[D]$ is called the essential graph of $D$. It is denoted $\ess(D)$.
$\ess(D)$ is always a chain graph
with chordal[An undirected graph is chordal if it has no induced cycle of length greater than $3$.] chain components [This means that $\ess(D)$ can be decomposed as a sequence of undirected chordal graphs $G_1,G_2 \ldots G_m$ (chain components) such that there is a directed edge from a vertex in $G_i$ to a vertex in $G_j$ only if $i<j$ ] <cit.>.
The $d$-separation criterion can be used to identify the skeleton and all the immoralities of the underlying causal DAG <cit.>. Additional edges can be identified using the fact that the underlying DAG is acyclic and there are no more immoralities. Meek derived $3$ local rules (Meek rules), introduced in <cit.>, to be recursively applied to identify every such additional edge (see Theorem 3 of <cit.>). The repeated application of Meek rules on this partially directed graph with identified immoralities until they can no longer be used yields the essential graph.
§.§ Interventions and Active Learning
Given a set of variables $V=\{x_1,...,x_n\}$, an intervention on a set $S \subset X$ of the variables is an experiment where the performer forces each variable $s \in S$ to take the value of another independent (from other variables) variable $u$, i.e., $s=u$. This operation, and how it affects the joint distribution is formalized by the do operator by Pearl <cit.>.
An intervention modifies the causal DAG $D$ as follows: The post intervention DAG $D_{\{S\}}$ is obtained by removing the connections of nodes in $S$ to their parents.
The size of an intervention $S$ is the number of intervened variables, i.e., $|S|$. Let $S^c$ denote the complement of the set $S$.
CI-based learning algorithms can be applied to $D_{\{S\}}$ to identify the set of removed edges, i.e. parents of $S$ <cit.>, and the remaining adjacent edges in the original skeleton are declared to be the children. Hence,
(R0) The orientations of the edges of the cut between $S$ and $S^c$ in the original DAG $D$ can be inferred.
Then, $4$ local Meek rules (introduced in <cit.>) are repeatedly applied to the original DAG $D$ with the new directions learnt from the cut to learn more till no more directed edges can be identified. Further application of CI-based algorithms on $D$ will reveal no more information. The Meek rules are given below:
(R1) ($a-b$) is oriented as ($a\rightarrow b$) if $\exists c$ s.t. $(c\rightarrow a)$ and $(c,b)\notin E$.
(R2) ($a-b$) is oriented as ($a\rightarrow b$) if $\exists c$ s.t. $(a\rightarrow c)$ and $(c\rightarrow b)$.
(R3) ($a-b$) is oriented as ($a\rightarrow b$) if $\exists c,d$ s.t. $(a-c)$,$(a-d)$,$(c\rightarrow b)$,$(d\rightarrow b)$ and $(c,d)\notin E$.
(R4) ($a-c$) is oriented as ($a\rightarrow c$) if $\exists b,d$ s.t. $(b\rightarrow c)$,$(a-d)$,$(a-b)$,$(d\rightarrow b)$ and $(c,d)\notin E$.
The concepts of essential graphs and Markov equivalence classes are extended in <cit.> to incorporate the role of interventions: Let $\mathcal{I}=\{I_1,I_2,...,I_m\}$, be a set of interventions and let the above process be followed after each intervention. Interventional Markov equivalence class ($\mathcal{I}$ equivalence) of a DAG is the set of DAGs that represent the same set of probability distributions obtained when the above process is applied after every intervention in $\mathcal{I}$. It is denoted by $[D]_{\mathcal{I}}$.
Similar to the observational case, $\mathcal{I}$ essential graph of a DAG $D$ is the graph union of all DAGs in the same $\mathcal{I}$ equivalence class; it is denoted by $\ess_{\mathcal{I}}(D)$.
We have the following sequence:
\begin{align}\label{eqn:learnseq}
D \rightarrow \mathrm{CI~learning} \rightarrow \mathrm{Meek~rules} \rightarrow \ess(D) \rightarrow I_1 \overset{a}\rightarrow \mathrm{learn~by~R0} \overset{b}\rightarrow
\mathrm{Meek~rules} & \nonumber \\
\rightarrow \ess_{\{I_1\}} (D) \rightarrow I_2 \ldots \rightarrow \ess_{\{I_1,I_2\}} (D) \ldots &
\end{align}
Therefore, after a set of interventions ${\cal I}$ has been performed, the essential graph
$\ess_{{\cal I}}(D)$ is a graph with some oriented edges that captures all the causal relations we have discovered so far, using
${\cal I}$. Before any interventions happened $\ess (D)$ captures the initially known causal directions.
It is known that $\ess_{{\cal I}}(D)$ is a chain graph with chordal chain components. Therefore when all the directed edges are removed, the graph becomes a set of disjoint chordal graphs.
§.§ Problem Definition
We are interested in the following question:
Given that all interventions in ${\cal I}$ are of size at most $k < n/2$ variables, i.e.,
for each intervention $I$, $\lvert I \rvert \leq k,\forall I\in\mathcal{I}$, minimize the number of interventions $\lvert {\cal I} \rvert$ such that the partially directed graph with all directions learned so far $\ess_{{\cal I}} (D) =D$.
The question is the design of an algorithm that computes the small set of interventions ${\cal I}$ given $\ess(D)$. Note, of course, that the unknown directions of the edges $D$ are not available to the algorithm. One can view the design of ${\cal I}$ as an active learning process to find $D$ from the essential graph $\ess(D)$. $\ess(D)$ is a chain graph with undirected chordal components and it is known that interventions on one chain components do not affect the discovery process of directed edges in the other components <cit.>. So we will assume that $\ess(D)$ is undirected and a chordal graph to start with. Our notion of algorithm does not consider the time complexity (of statistical algorithms involved) of steps $a$ and $b$ in (<ref>). Given $m$ interventions, we only consider efficiently computing $I_{m+1}$ using (possibly) the graph $\ess_{\{I_{1}, \ldots I_{m} \}}$. We consider the following three classes of algorithms:
* Non-adaptive algorithm: The choice of ${\cal I}$ is fixed prior to the discovery process.
* Adaptive algorithm: At every step $m$, the choice of $I_{m+1}$ is a deterministic function of $\ess_{\{I_{1}, \ldots I_{m} \}}(D)$.
* Randomized adaptive algorithm: At every step $m$, the choice of $I_{m+1}$ is a random function of $\ess_{\{I_{1}, \ldots I_{m} \}}(D)$.
The problem is different for complete graphs versus more general chordal graphs since rule R$1$ becomes applicable when the graph is not complete. Thus we give a separate treatment for each case. First, we provide algorithms for all three cases for learning the directions of complete graphs $\ess(D)= K_n$ (undirected complete graph) on $n$ vertices. Then, we generalize to chordal graph skeletons and provide a novel adaptive algorithm with upper and lower bounds on its performance.
The missing proofs of the results that follow can be found in the Appendix.
§ COMPLETE GRAPHS
In this section, we consider the case where the skeleton we start with, i.e. $\ess(D)$, is an undirected complete graph (denoted $K_n$). It is known that at any stage in (<ref>) starting from $\ess(D)$, rules R$1$, R$3$ and R$4$ do not apply. Further, the underlying DAG $D$ is a directed clique. The directed clique is characterized by an ordering $\sigma$ on $[1:n]$ such that, in the subgraph induced by $\sigma(i),\sigma(i+1) \ldots \sigma(n)$, $\sigma(i)$ has no incoming edges. Let $D$ be denoted by $\vec{K}_n(\sigma)$ for some ordering $\sigma$. Let $[1:n]$ denote the set $\{1,2 \ldots n\}$. We need the following results on a separating system for our first result regarding adaptive and non-adaptive algorithms for a complete graph.
§.§ Separating System
An $(n,k)$-separating system on an $n$ element set $[1:n]$ is a set of subsets ${\cal S}=\{S_1,S_2 \ldots S_m \}$ such that $\lvert S_i \rvert \leq k$ and for every pair $i,j$ there is a subset $S \in {\cal S}$ such that either $i \in S,~ j \notin S$ or $j \in S,~ i \notin S$. If a pair $i,j$ satisfies the above condition with respect to ${\cal S}$, then ${\cal S}$ is said to separate the pair $i,j$. Here, we consider the case when $k<n/2$
In <cit.>, Katona gave an $(n,k)$-separating system together with a lower bound on $\lvert \cal S\rvert$. In <cit.>, Wegener gave a simpler argument for the lower bound and also provided a tighter upper bound than the one in <cit.>. In this work, we give a different construction below where the separating system size is at most$\lceil \log_{\lceil n/k \rceil} n\rceil$ larger than the construction of Wegener. However, our construction has a simpler description.
There is a labeling procedure that produces distinct $\ell$ length labels for all elements in $[1:n]$ using letters from the integer alphabet $\{0,1 \ldots a \}$ where $\ell=\lceil \log_{a} n \rceil$. Further, in every digit (or position), any integer letter is used at most $\lceil n/a \rceil$ times.
Once we have a set of $n$ string labels as in Lemma <ref>, our separating system construction is straightforward.
Consider an alphabet ${\cal A}=[0:\lceil \frac{n}{k} \rceil]$ of size $\lceil \frac{n}{k} \rceil+1$ where $k<n/2$. Label every element of an $n$ element set using a distinct string of letters from ${\cal A}$ of length $ \ell= \lceil \log_{\lceil \frac{n}{k} \rceil} n \rceil$ using the procedure in Lemma <ref> with $a=\lceil \frac{n}{k} \rceil$. For every $1 \leq i \leq \ell$ and $1 \leq j \leq \lceil \frac{n}{k} \rceil $, choose the subset $S_{i,j}$ of vertices whose string's $i$-th letter is $j$. The set of all such subsets ${\cal S} = \{S_{i,j}\}$ is a $k$-separating system on $n$ elements and $\lvert {\cal S} \rvert \leq (\lceil \frac{n}{k} \rceil ) \lceil \log_{\lceil \frac{n}{k} \rceil} n \rceil $.
§.§ Adaptive algorithms: Equivalence to a Separating System
Consider any non-adaptive algorithm that designs a set of interventions ${\cal I}$, each of size at most $k$, to discover $\vec{K}_n(\sigma)$. ${\cal I}$ has to be a separating system in the worst case over all $\sigma$. This is already known. Now, we prove the necessity of a separating system for deterministic adaptive algorithms in the worst case.
Let there be an adaptive deterministic algorithm $A$ that designs the set of interventions ${\cal I}$ such that the final graph learnt $\ess_{{\cal I}}(D)= \vec{K}_n(\sigma)$ for any ground truth ordering $\sigma$ starting from the initial skeleton $\ess(D) =K_n$. Then, there exists a $\sigma$ such that $A$ designs an ${\cal I}$ which is a separating system.
The theorem above is independent of the individual intervention sizes. Therefore, we have the following theorem, which is a direct corollary of Theorem <ref>:
In the worst case over $\sigma$, any adaptive or a non-adaptive deterministic algorithm on the DAG $\vec{K}_n(\sigma)$ has to be such that $\frac{n}{k} \log_{\frac{ne}{k}} n \leq \lvert {\cal I} \rvert$. There is a feasible ${\cal I}$ with $\lvert {\cal I} \rvert \leq \lceil (\frac{n}{k} \rceil-1) \lceil \log_{\lceil \frac{n}{k} \rceil} n \rceil$
By Theorem <ref>, we need a separating system in the worst case and the lower and upper bounds are from <cit.>.
§.§ Randomized Adaptive Algorithms
In this section, we show that that total number of variable accesses to fully identify the complete causal DAG is $\Omega(n)$.
To fully identify a complete causal DAG $\vec{K}_n(\sigma)$ on $n$ variables using size-$k$ interventions, $\frac{n}{2k}$ interventions are necessary. Also, the total number of variables accessed is at least $\frac{n}{2}$.
The lower bound in Theorem <ref> is information theoretic. We now give a randomized algorithm that requires $O(\frac{n}{k}\log\log k)$ experiments in expectation. We provide a straightforward generalization of <cit.>, where the authors gave a randomized algorithm for unbounded intervention size.
Let $\ess(D)$ be $K_n$ and the experiment size $k=n^r$ for some $0<r<1$. Then there exists a randomized adaptive algorithm which designs an $\mathcal{I}$ such that $\ess_{\mathcal{I}}(D)=D$ with probability polynomial in $n$, and $\lvert\mathcal{I}\rvert=\mathcal{O}(\frac{n}{k}\log\log(k))$ in expectation.
§ GENERAL CHORDAL GRAPHS
In this section, we turn to interventions on a general DAG $G$. After the initial stages in (<ref>), $\ess(G)$ is a chain graph with chordal chain components. There are no further immoralities throughout the graph. In this work, we focus on one of the chordal chain components. Thus the DAG $D$ we work on is assumed to be a directed graph with no immoralities and whose skeleton $\ess(D)$ is chordal. We are interested in recovering $D$ from $\ess(D)$ using interventions of size at most $k$ following (<ref>).
§.§ Bounds for Chordal skeletons
We provide a lower bound for both adaptive and non-adaptive deterministic schemes for a chordal skeleton $\ess(D)$. Let $\chi \left( \ess(D) \right)$ be the coloring number of the given chordal graph. Since, chordal graphs are perfect, it is the same as the clique number.
Given a chordal $\ess(D)$, in the worst case over all DAGs $D$ (which has skeleton $\ess(D)$ and no immoralities), if every intervention is of size at most $k$, then $ \lvert {\cal I}\rvert \geq \frac{\chi \left(\ess(D)\right)}{k} \log_{\frac{\chi \left(\ess(D)\right)e}{k}} \chi \left(\ess(D)\right)$ for any adaptive and non-adaptive algorithm with $\ess_{\mathcal{I}}(D)=D$.
Upper bound: Clearly, the separating system based algorithm of Section <ref> can be applied to the vertices in the chordal skeleton $\ess(D)$ and it is possible to find all the directions. Thus, $ \lvert {\cal I} \rvert \leq \frac{n}{k} \log_{\lceil \frac{n}{k} \rceil} n \leq \frac{\alpha (\ess(D)) \chi(\ess(D)) }{k} \log_{\lceil \frac{n}{k} \rceil} n$. This with the lower bound implies an $\alpha$ approximation algorithm (since $\log_{\lceil \frac{n}{k} \rceil} n \leq \log_{\frac{\chi \left(\ess(D)\right)e}{k}} \chi \left(\ess(D)\right) $ , under a mild assumption $\chi(\ess(D)) \leq \frac{n}{e}$ ).
Remark: The separating system on $n$ nodes gives an $\alpha$ approximation. However, the new algorithm in Section <ref> exploits chordality and performs much better empirically. It is possible to show that our heuristic also has an $\alpha$ approximation guarantee but we skip that.
§.§ Two extreme counter examples
We provide two classes of chordal skeletons $G$: One for which the number of interventions close to the lower bound is sufficient and the other for which the number of interventions needed is very close to the upper bound.
There exists chordal skeletons such that for any algorithm with intervention size constraint $k$, the number of interventions $\lvert \cal I \rvert$ required is at least $\alpha \frac{(\chi-1)}{2k}$ where $\alpha$ and $\chi$ are the independence number and chromatic numbers respectively. There exists chordal graph classes such that $\lvert {\cal I} \rvert = \lceil \frac{\chi}{k} \rceil \lceil \log_{\lceil \frac{\chi}{k} \rceil} \chi \rceil$ is sufficient.
§.§ An Improved Algorithm using Meek Rules
In this section, we design an adaptive deterministic algorithm that anticipates Meek rule R$1$ usage along with the idea of a separating system. We evaluate this experimentally on random chordal graphs. First, we make a few observations on learning connected directed trees $T$ from the skeleton $\ess(T)$ (undirected trees are chordal) that do not have immoralities using Meek rule R$1$ where every intervention is of size $k=1$. Because the tree has no cycle, Meek rules R$2$-R$4$ do not apply.
Every node in a directed tree with no immoralities has at most one incoming edge. There is a root node with no incoming edges and intervening on that node alone identifies the whole tree using repeated application of rule R$1$.
If every intervention in ${\cal I}$ is of size at most $1$, learning all directions on a directed tree $T$ with no immoralities can be done adaptively with at most $\lvert {\cal I} \rvert \leq O(\log_2 n)$ where $n$ is the number of vertices in the tree. The algorithm runs in time $\mathrm{poly}(n)$.
Given any chordal graph and a valid coloring, the graph induced by any two color classes is a forest.
In the next section, we combine the above single intervention adaptive algorithm on directed trees which uses Meek rules, with that of the non-adaptive separating system approach.
§.§.§ Description of the algorithm
The key motivation behind the algorithm is that, a pair of color classes is a forest (Lemma <ref>). Choosing the right node to intervene leaves only a small subtree unlearnt as in the proof of Lemma <ref>. In subsequent steps, suitable nodes in the remaining subtrees could be chosen until all edges are learnt. We give a brief description of the algorithm below.
Let $G$ denote the initial undirected chordal skeleton $\ess(D)$ and let $\chi$ be its coloring number. Consider a $(\chi,k)$ separating system ${\cal S}=\{S_i\}$. To intervene on the actual graph, an intervention set $I_i$ corresponding to $S_i$ is chosen. We would like to intervene on a node of color $c \in S_i$.
Consider a node $v$ of color $c$. Now, we attach a score $P(v,c)$ as follows. For any color $c'\notin S_i$, consider the induced forest $F(c,c')$ on the color classes $c$ and $c'$ in $G$. Consider the tree $T(v,c,c')$ containing node $v$ in $F$. Let $d(v)$ be the degree of $v$ in $T$. Let $T_1, T_2, \ldots T_{d(v)}$ be the resulting disjoint trees after node $v$ is removed from $T$. If $v$ is intervened on, according to the proof of Lemma <ref>: a)
All edge directions in all trees $T_i$ except one of them would be learnt when applying Meek Rules and rule R$0$. b) All the directions from $v$ to all its neighbors would be found.
The score is taken to be the total number of edge directions guaranteed to be learnt in the worst case.
Therefore, the score $P(v)$ is:
P(v) = \sum \limits_{c': \lvert {c,c'} \bigcap \rvert =1} \left( \lvert T(c,c') \rvert - \max \limits_{ 1 \leq j \leq d(v)} \lvert T_{j} \rvert \right).
The node with the highest score among the color class $c$ is used for the intervention $I_i$. After intervening on $I_i$, all the edges whose directions are known through Meek Rules (by repeated application till nothing more can be learnt) and R$0$ are deleted from $G$. Once ${\cal S}$ is processed, we recolor the sparser graph $G$. We find a new ${\cal S}$ with the new chromatic number on $G$ and the above procedure is repeated. The exact hybrid algorithm is described in Algorithm <ref>.
Given an undirected choral skeleton $G$ of an underlying directed graph with no immoralities, Algorithm <ref> ends in finite time and it returns the correct underlying directed graph. The algorithm has runtime complexity polynomial in $n$.
Hybrid Algorithm using Meek rules with separating system
Input: Chordal Graph skeleton $G=(V,E)$ with no Immoralities.
Initialize $\vec{G}(V,E_d=\emptyset)$ with $n$ nodes and no directed edges. Initialize time $t=1$.
$E \neq \emptyset$
Color the chordal graph $G$ with $\chi$ colors. Standard algorithms exist to do it in linear time
Initialize color set ${\cal C}= \{1,2 \ldots \chi\}$. Form a $(\chi, \min (k,\lceil \chi/2 \rceil ))$ separating system ${\cal S}$ such that $\lvert S \rvert \leq k,~ \forall S \in {\cal S}$.
$i=1$ until $\lvert {\cal S} \rvert $
Initialize Intervention $I_t=\emptyset$.
$c \in S_i$ and every node $v$ in color class $c$
Consider $F(c,c')$, $T(c,c',v)$ and $\{T_j\}_{1}^{d(i)}$(as per definitions in Sec. <ref>).
Compute: $P(v,c) = \sum \limits_{c' \in {\cal C} \bigcap S_i^c} \lvert T(c,c',v) \rvert - \max \limits_{ 1 \leq j \leq d(i)} \lvert T_{j} \rvert $.
$k \leq \chi/2$
$ I_t = I_t \bigcup \limits_{c \in S_i} \{\mathop{\rm argmax} \limits_{v:P(v,c) \neq 0} P(v,c)\}$.
$ I_t = I_t \mathop{\cup}_{c \in S_i} \{\mathrm{First~}\frac{k}{\lceil \chi/2 \rceil} \mathrm{~nodes~}v\mathrm{~with~ largest~ nonzero~} P(v,c) \}$.
Apply R$0$ and Meek rules using $E_d$ and $E$ after intervention $I_t$. Add newly learnt directed edges to $E_d$ and delete them from $E$.
Remove all nodes which have degree $0$ in G.
§ SIMULATIONS
We simulate our new heuristic, namely Algorithm <ref>, on randomly generated chordal graphs and compare it with a naive algorithm that follows the intervention sets given by our $(n,k)$ separating system as in Theorem <ref>. Both algorithms apply R$0$ and Meek rules after each intervention according to (<ref>). We plot the following lower bounds: a) Information Theoretic LB of $\frac{\chi}{2k}$ b) Max. Clique Sep. Sys. Entropic LB which is the chromatic number based lower bound of Theorem <ref>. Moreover, we use two known $(\chi,k)$ separating system constructions for the maximum clique size as “references": The best known $(\chi,k)$ separating system is shown by the label Max. Clique Sep. Sys. Achievable LB and our new simpler separating system construction (Theorem <ref>) is shown by Our Construction Clique Sep. Sys. LB. As an upper bound, we use the size of the best known $(n,k)$ separating system (without any Meek rules) and is denoted Separating System UB.
Random generation of chordal graphs: Start with a random ordering $\sigma$ on the vertices. Consider every vertex starting from $\sigma(n)$. For each vertex $i$, $(j,i)\in E$ with probability inversely proportional to $\sigma(i)$ for every $j\in S_i$ where $S_i=\{v:\sigma^{-1}(v)<\sigma^{-1}(i)\}$. The proportionality constant is changed to adjust sparsity of the graph. After all such $j$ are considered, make $S_i\cap \textrm{ne}(i)$ a clique by adding edges respecting the ordering $\sigma$, where $\textrm{ne}(i)$ is the neighborhood of $i$. The resultant graph is a DAG and the corresponding skeleton is chordal. Also, $\sigma$ is a perfect elimination ordering.
$n$: no. of vertices, $k$: Intervention size bound. The number of experiments is compared between our heuristic and the naive algorithm based on the $(n,k)$ separating system on random chordal graphs. The red markers represent the sizes of $(\chi,k)$ separating system. Green circle markers and the cyan square markers for the same $\chi$ value correspond to the number of experiments required by our heuristic and the algorithm based on an $(n,k)$ separating system(Theorem <ref>), respectively, on the same set of chordal graphs. Note that, when $n=1000$ and $n=2000$, the naive algorithm requires on average about $130$ and $260$ (close to $n/k$) experiments respectively, while our algorithm requires at most $\sim40$ (orderwise close to $\chi/k =10$) when $\chi=100$.
Results: We are interested in comparing our algorithm and the naive one which depends on the $(n,k)$ separating system to the size of the $(\chi,k)$ separating system. The size of the $(\chi,k)$ separating system is roughly $\tilde{O}(\chi/k)$. Consider values around $\chi=100$ on the x-axis for the plots with $n=1000,k=10$ and $n=2000,k=10$. Note that, our algorithm performs very close to the size of the $(\chi,k)$ separating system, i.e. $\tilde{O}(\chi/k)$. In fact, it is always $<40$ in both cases while the average performance of naive algorithm goes from $130$ (close to $n/k=100$) to $260$ (close to $n/k=200$). The result points to this: For random chordal graphs, the structured tree search allows us to learn the edges in a number of experiments quite close to the lower bound based only on the maximum clique size and not $n$. The plots for $(n,k)=(500,10)$ and $(n,k)=(2000,20)$ are given in Appendix.
§ CONCLUSIONS
We have considered the problem of adaptively designing interventions of bounded size to learn a causal graph under Pearl's SEM-IE model. We proposed lower and upper bounds for the number of interventions needed in the worst case for various classes of algorithms, when the causal graph skeleton is complete. We developed lower and upper bounds on the minimum number of interventions required in the worst case for general graphs. We characterized two extremal graph classes such that the minimum number of interventions in one class is close to the lower bound and in the other class it is close to the upper bound. In the case of chordal skeletons, we proposed an algorithm that combines ideas for the complete graphs with the ones when the skeleton is a forest via application of Meek rules. Empirically, on randomly generated chordal graphs, our algorithm performs close to the lower bound and it outperforms the previous state of the art. Possible future work includes obtaining a tighter lower bound for chordal graphs that would possibly establish a tighter approximation guarantee for our algorithm.
§.§.§ Acknowledgments
Authors acknowledge the support from grants: NSF CCF 1344179, 1344364, 1407278, 1422549 and a ARO YIP award (W911NF-14-1-0258). We also thank Frederick Eberhardt for helpful discussions.
§ APPENDIX
§.§ Proof of Lemma <ref>
We describe a string labeling procedure as follows to label elements of the set $[1:n]$.
String Labelling:
Let $a>1$ be a positive integer. Let $x$ be the integer such that $a^{x} < n \leq a^{x+1}$. $x+1 = \lceil \log_{a} n \rceil$. Every element $j \in [1:n]$ is given a label $L(j)$ which is a string of integers of length $x+1$ drawn from the alphabet $\{0,1,2 \ldots a\}$ of size $a+1$. Let $n= p_d a^d+r_d$ and $n=p_{d-1}a^{d-1}+r_{d-1}$ for some integers $p_d,p_{d-1},r_{d},r_{d-1}$, where $r_d < a^d$ and $r_{d-1}<a^{d-1}$. Now, we describe the sequence of the $d$-th digit across the string labels of all elements from $1$ to $n$:
* Repeat $0$ $a^{d-1}$ times, repeat the next integer $1$ $a^{d-1}$ times and so on circularly [Circular means that after $a-1$ is completed, we start with $0$ again.] from $\{0,1 \ldots a-1 \}$ till $p_da^d$.
* After that, repeat $0$ $\lceil r_d/a \rceil$ times followed by $1$ $\lceil r_d/a \rceil$ times till we reach the $n$th position. Clearly, $n$-th integer in the sequence would not exceed $a-1$.
* Every integer occurring after the position $a^{d-1}p_{d-1}$ is increased by $1$.
From the three steps used to generate every digit, a straightforward calculation shows that every integer letter is repeated at most $\lceil n/a \rceil$ times in every digit $i$ in the string. Now, we would like to prove inductively that the labels are distinct for all $n$ elements. Let us assume the induction hypothesis: For all $n < a^{q+1}$, the labels are distinct. The base case of $q=0$ is easy to see. Then, we would like to show that for $a^{q+1} \leq n < a^{q+2}$, the labels are distinct.
Another way of looking at the labeling procedure is as follows. Let $n=a^{q+1}p+r$ with $r< a^{q+1}$. Divide the label matrix $L$ (of dimensions $(q+2) \times n$) into two parts, one $L_1$ consisting of the first $pa^{q+1}$ columns and the other $L_2$ consisting of the remaining columns. The first $q+1$ rows of $L_1$ is nothing but the string labels for all numbers from $0$ to $pa^{q+1}$ expressed in base $a$. For any row $i \leq \lceil \log_a r \rceil$ in the original matrix $L$ of labels, till the end of first $pa^{q+1}$ columns, the labeling procedure would be still in Step $1$. After that, one can take $r$ to be the new size of the set of elements to be labelled and then restart the procedure with this $r$. Therefore we have the following key observation: $L_2(1: \lceil \log_a r \rceil,: )$ (the matrix with first $\lceil \log_a r \rceil $ rows of $L_2$) is nothing but the label matrix for $r$ distinct elements from the above labeling procedure.
Since, $r < a^{q+1}$, by the induction hypothesis, the columns are distinct. Hence, any two columns in $L_2$ are distinct. Suppose the first $q+1$ rows of two columns $b$ and $c$ of $L_1$ are identical. These correspond to base $a$ expansion of $b-1$ and $c-1$. They are separated by at least $a^{q+1}+1$ columns. But the last row of columns $b$ and $p$ in $L_1$ has to be distinct because according to Step $2$ and Step $3$ of the labeling procedure, in the $q+2^{th}$ row, every integer is repeated at most $\lceil n/a \rceil \leq a^{q+1}$ times continuously, and only once. Therefore, any two columns in $L_1$ are distinct. The last row entries in $L_1$ are different from $L_2$ because of the addition in Step $3$. Therefore, all columns of $L$ are distinct. Hence, by induction, the result is shown.
§.§ Proof of Theorem <ref>
By Lemma <ref>, $i$th place has at most $\lceil \frac{n}{\lceil n/k\rceil} \rceil \leq k$ occurrences of symbol $j$. Therefore, $\lvert S_{i,j} \rvert \leq k$. Now, consider the pair of distinct elements $p,q \in [1:n]$. Since they are labelled distinctly (Lemma <ref>), there is at least one letter $i$ in their string labels where they differ. Suppose the distinct $i$th letters are $a,b \in {\cal A},~a \neq b $ and let us say $a \neq 0$ without loss of generality. Then, clearly the separation criterion is met by the subset $S_{i,a}$. This proves the claim.
§.§ Proof of Theorem <ref>
We construct a worst case $\sigma$ inductively. Before every step $m$, the adaptive algorithm deterministically chooses $I_m$ based on $\ess_{\{I_1,I_2 \ldots I_{m-1}\}}(K_n)$. Therefore, we will reveal a partial order $\sigma^{(m-1)}$ to satisfy the observations so far. Inductively for every $m$, we will make sure that after $I_m$ is chosen by the algorithm, further details about $\sigma$ can be revealed to form $\sigma^{(m)}$ such that after intervening on $I_2$ and then applying R$0$, we will make sure there is no opportunity to apply the rule $R2$. This would make sure that ${\cal I}$ is a separating system on $n$ elements.
Before intervention at any step $m$, let us `tag' every vertex $i$ using a subset $C_i^{(m-1)} \subseteq [1:m]$ such that $C_i^{(m-1)} = \{ p : i \in I_p ,~ p \leq m-1 \}$. $C_i^{(m-1)}$ contains indices of all those interventions that contain vertex $i$ before step $m$. Let ${\cal C}^{(m-1)}$ contain distinct elements of the multi-set $\{C_i^{(m-1)}\}$ .We will construct $\sigma$ partially such that it satisfies the following criterion always:
Inductive Hypothesis:
The partial order $\sigma^{(m-1)}$ is such that for any two elements $i,j$ with $C_i$ and $C_j$, $i$ and $j$ are incomparable if $C_i=C_j$ and comparable otherwise. This means the edges between the elements tagged with the same tag $C$ has not been revealed, and thus the relevant directed edges are not known by the algorithm.
Now, we briefly digress to argue that if we could construct $\sigma^{(1)}, \sigma^{(2)} \ldots $ satisfying such a property throughout, then clearly all vertices must be tagged differently otherwise the directions among the vertices that are tagged similarly cannot be learned by the algorithm. Therefore, the algorithm has not succeeded in its task. If all vertices are tagged differently, then it means it is a separating system.
Construction of $\sigma^{(m)}$: We now construct $\sigma^{(m)}$ that can be shown to satisfy the induction hypothesis before step $m+1$. Before step $m$ , consider the vertices in $C \in {\cal C}^{(m-1)}$ for any $C$. Let the current intervention be $I_m$ chosen by the deterministic algorithm. We make the following changes: Modify $\sigma^{(m-1)}$ such that vertices in $I_m \bigcap C$ come before $(I_m)^c \bigcap C$ in the partial order $\sigma^{(m)}$ (vertices inside either sets are still not ordered amongst themselves) in the ordering and clearly the directions between these two sets are revealed by R$0$. By the induction hypothesis for step $m$ and with the new tagging of vertices into ${\cal C}^{(m)}$, it is easy to see that only directions between distinct $C's$ in the new ${\cal C}^{(m)}$ have been revealed and all directions within a tag set $C$ are not revealed and all vertices in a tag set are contiguous in the ordering so far. We need to only show that rule R$2$ cannot reveal anymore edges amongst vertices in $C \in {\cal C}^{(m)}$ after the new $\sigma^{(m)}$ and intervention $I_m$. Suppose there are two vertices $i,j$ such that just after intervention $I_m$ and the modified $\sigma^{(m)}$, they are tagged identically and application of R$2$ reveals the direction between $i$ and $j$ before the next intervention. Then there has to be a vertex $k$ tagged differently from $i,j$ such that $j \rightarrow k$ and $k \rightarrow i$ are both known. But this implies that $j$ and $i$ are comparable in $\sigma^{(m)}$ leading to a contradiction. This implies the hypothesis holds for step $m+1$.
Base case: Trivially, the induction hypothesis holds for step $0$ where $\sigma^{(0)}$ leaves the entire set unordered.
§.§ Proof of Lemma <ref>
The proof is a direct obvious consequence of acyclicity, non-existence of immoralities and the definition of rule R1.
§.§ Proof of Lemma <ref>
By Lemma <ref>, it is sufficient for an algorithm to identify the root node of the tree. Suppose the root node is $b$ unknown to the algorithm. Every tree has a single vertex separator that partitions the tree into components each of which has size at most $\frac{2}{3}n$ <cit.>. Choose that vertex separator $a_1$ (it can be found in by removing every node and determining the components left). If it is a root node we stop here. Otherwise, its parent $p_1$ (if it is not) after application of rule R$0$ is identified. Let us consider component trees $T_1,T_2 \ldots T_k$ that result by removing node $a_1$. Let $T_1$ contain $p_1$. All directions in all other trees are known after repeated application of R$1$ on the original tree after R$0$ is applied. Directions in T$1$ will not be known. For the next step, $\ess(T_1)$ is the new skeleton which has no immoralities. Again, we find the best vertex separator $a_2$ and the process continues. This procedure will terminate at some step $j$ when $a_j =b$ or there is only one node left which should be $b$ by Lemma <ref>. Since the number of nodes reduce by about $1/3$ at least each time, and initially it can be at most $n$, this procedure terminates in at most $O(\log_2 n)$ steps.
§.§ Proof of Lemma <ref>
The graph induced by two colors classes in any graph is a bi-partite graph and bi-partite graphs do not have odd induced cycles. Since the graph and any induced subgraph is chordal, it implies the induced graph on a pair of color classes does not have a cycle. This proves the theorem.
§.§ Proof of Theorem <ref>
Assume $n$ is even for simplicity. We define a family of partial order $\sigma^{(p)}$ as follows: Group $i,i+1$ into $C_i$. Ordering among $i$ and $i+1$ is not revealed. But all the edges between $C_i$ and $C_j$ for any $j>i$ are directed from $C_i$ to $C_j$. Now, one has to design a set of interventions such that exactly one node among every $C_i$ is intervened on at least once. This is because, if neither $i$ nor $i+1$ in $C_i$ are intervened on, then the direction between $i$ and $i+1$ cannot be figured out by applying rule R$2$ on any other set of directions in the rest of the graph. Since the size of every intervention is at most $k$ and at least $n/2$ nodes need to be covered by intervention sets, the number of interventions required is at least $\frac{n}{2k}$.
§.§ Proof of Theorem <ref>
Separate $n$ vertices arbitrarily into $\frac{n}{k}$ disjoint subsets $C_i$ of size-$k$. Let the first $n/k$ interventions $\{I_1,I_2,...,I_{n/k}\}$ be such that $I_i(v)=1$ if and only if $v\in C_i$. This divides the problem of learning a clique of size $n$ into learning $n/k$ cliques of size $k$. Then, we can apply the clique learning algorithm in <cit.> as a black box to each of the $\frac{n}{k}$ blocks: Each block is learned with probability $k^{-c}$ after $\log c\log k$ experiments in expectation. For $k=cn^r$, choose $c>1/r-1$. Then the union bound over $n/k$ blocks yields probability polynomial in $n$. Since each block takes $\mathcal{O}(\log\log k)$ experiments, we need $\frac{n}{k}\mathcal{O}(\log\log k)$ experiments.
§.§ Proof of Theorem <ref>
We need the following definitions and some results before proving the theorem.
A perfect elimination ordering $\sigma_p=\{v_1,v_2 \ldots v_n\}$ on the vertices of an undirected chordal graph $G$ is such that for all $i$, the induced neighborhood of $v_i$ on the subgraph formed by $\{v_1,v_2 \ldots v_{i-1} \}$ is a clique.
(<cit.>) If all directions in the chordal graph are according to perfect elimination ordering (edges go only from vertices lower in the order to higher in the order), then there are no immoralities.
We make the following observation: Let the directions in a graph $D$ be oriented according to an ordering $\sigma$ on the vertices. If a clique comes first in the ordering, then the knowledge of edge directions in the rest of the graph, excluding that of the clique, cannot help at any stage of the intervention process on the clique; because all the edges are directed outwards from the clique and hence none of the Meek rules apply. This is because, if $a \rightarrow b$ is to be inferred by Meek rules from other known directions, then either there has to be a known edge direction into $a$ or $b$ before the inference step. So if one of the directed edges not from the clique was to help in the discovery process, either that edge has to be directed towards $a$ or $b$ (like in Meek rules R$1$, R$2$ and R$3$), or it has to be directed towards $c$ in another $c \rightarrow a$ (R$4$) which belongs to the clique. Both the above cases are not possible.
(<cit.>) Let $C$ be a maximum clique of an undirected chordal graph $\ess(D)$, then there is an underlying DAG $D$ on the chordal skeleton that is oriented according to a perfect elimination ordering (implying no immoralities), where the clique $C$ occurs first.
By Lemmas <ref>, <ref> and the observation above, given a chordal skeleton, we can construct a DAG on the skeleton with no immoralities such that the directions of the maximum clique in $D$ cannot be learned by using knowledge of the directions outside. This means that only the intervention sets $\{I_1 \bigcap C, I_2 \bigcap C \ldots\}$ matter for learning the directions on this clique. Therefore inference on the clique is isolated. Hence, all the lower bounds for the clique case transfer to this case and since the size of the largest clique is exactly the coloring number of the chordal skeleton, the theorem follows.
§.§ Proof of Theorem <ref>
Example with a feasible solution with $\lvert {\cal I} \rvert$ close to the lower bound:
Consider a graph $G$ that can be partitioned into a clique of size $\chi$ and an independent set $\alpha$. Such graphs are called split graphs and as $n \rightarrow \infty$, the fraction of split graphs to chordal graphs tends to $1$.
If $\ess(D)=G$ where $G$ is a split graph skeleton, it is enough to intervene only on the nodes in the clique and therefore the number of interventions that are needed is that for the clique. It is certainly possible to orient the edges in such a way so as to avoid immoralities, since the graph is chordal.
Example with $\lvert {\cal I} \rvert$ which needs to be close to the upper bound:
We construct a connected chordal skeleton with independent set $\alpha$ and clique size $\chi$ (also coloring number) such that it would require $\frac{\alpha (\chi-1)}{2k}$ interventions at least for any algorithm over a class of orientations.
Consider a line $L$ consisting of vertices $1,2 \ldots 2 \alpha$ such that every node $1<i<2\alpha$ is connected to $i-1$ and $i+1$. For, all $1 \leq p \leq \alpha$, consider a clique $C_p$ of size $\chi$ which only has nodes $2p-1,2p$ from the line $L$. Now assume that the actual orientation of the L is $1 \rightarrow 2 \ldots \rightarrow 2 \alpha$. In every clique, the orientation is partially specified as follows: In every clique $C_p$, all edges from node $2p-1$ are outgoing. It is very clear that this partial orientation excludes all immoralities. Further, each clique $C_p-\{2p-1\}$ can have any arbitrary orientation out of $\chi-1$ possible ones in the actual DAG. Now, even if all the specified directions are revealed to the algorithm, the algorithm has to intervene on all $\alpha$ disjoint cliques $\{C_p-\{2p-1\}\}_{p=1}^{\alpha}$ each of size $\chi-1$ and directions in one clique will not force directions on the others through any of the Meek rules or rule R$0$. Therefore, the lower bound of $\frac{\alpha(\chi-1)}{2}$ total node accesses (total number of nodes intervened) is implied by Theorem <ref>. Given every intervention is of size $k$, these chordal skeletons with the revealed partial order needs at least $\frac{\alpha(\chi-1)}{2k}$ more experiments.
§.§ Performance Comparison of Our Algorithm vs. Naive Scheme for $n=500,k=10$ and $n=2000,k=20$
$n$: no. of vertices, $k$: Intervention size bound. The number of experiments is compared between our heuristic and the naive algorithm based on the $(n,k)$ separating system on random chordal graphs. The red markers represent the sizes of $(\chi,k)$ separating system. Green circle markers and the cyan square markers for the same $\chi$ value correspond to the number of experiments required by our heuristic and the algorithm based on an $(n,k)$ separating system(Theorem <ref>), respectively, on the same set of chordal graphs. All four plots (including the ones in the main text) indicate that our algorithm requires number of experiments proportional to the clique number $\chi$, whereas naive separating system based algorithm requires experiments on the order of number of variables $n$.
§.§ Proof of Theorem <ref>
We provide the following justifications for the correctness of Algorithm <ref>.
* At line <ref> of the algorithm, when Meek rules and R$0$ are applied after every intervention, the intermediate graph $G$, with unlearned edges, will always be a disjoint union of chordal components (refer to (<ref>) and the comments below) and hence a chordal graph.
* The number of unlearned edges before and after the main while loop in Algorithm <ref> reduces by at least one. Every edge in $E$ is incident on two colors and one of the colors is always picked for processing because we use a separating system on the colors. Therefore, one node belonging to some edge has a positive score and is intervened on. The edge direction is learnt through rule R$0$. Therefore, the algorithm terminates.
* It identifies the correct $\vec{G}$ because every edge is inferred after some intervention $I_t$ by applying rule R$0$ and Meek rules as in (<ref>) both of which are correct.
* the algorithm has polynomial run time complexity because the main while loop ends in time $\lvert E \rvert$.
|
1511.00553
|
§ INTRODUCTION
The understanding of the QCD phase structure is
one of the basic goals of lattice QCD computations at non-zero temperature.
It has been noted by Pisarski and Wilczek quite a long time ago that the QCD phase structure
may depend on the number of light quark degrees of freedom <cit.>.
The QCD phase structure in the quark mass plane is summarized in the so-called Columbia plot <cit.>, i.e.
Fig. <ref>. The detailed description of the plot can be seen from e.g. Ref. <cit.>.
Schematic QCD phase structure with different values of quark
masses ($m_{u,d}$, $m_{s}$) at zero baryon number density.
In this proceedings we report on the updated studies of chiral phase structure
of (2+1)-flavor ($N_f$=2+1) and 3-flavor ($N_f$=3) QCD from the lattice QCD simulations with values of quark masses decreasing
towards the chiral limit along the yellow horizontal and diagonal arrows shown in Fig. <ref>, respectively.
The simulations have been performed using the Highly Improved Staggered Quarks on $N_\tau$=6 lattices.
We will mainly discuss the relative position of the tri-critical point $m_s^{tri}$ and the physical point $m_s^{phy}$, and the
estimate of the critical quark mass $m_c$ as sketched in Fig. <ref>.
Previous studies were reported in Ref. <cit.>.
§ LATTICE SETUP
We performed simulations of $N_f$=2+1 and 3 QCD with lattice
spacings corresponding to temporal lattice size $N_\tau=6$ using
the Highly Improved Staggered fermions (HISQ).
In the case of $N_f=2+1$ QCD the strange quark mass is chosen to be fixed to its physical value ($m^{phy}_s$)
and five values of light quark masses ($m_l$) that are varied in the range of 1/20$\gtrsim m_l/m^{phy}_s \gtrsim$1/80.
These values of quark masses correspond to the lightest pseudo Goldstone pion masses of about 160, 140, 110, 90 and 80 MeV
in the continuum limit. To ensure $m_\pi L \gtrsim 4$ simulations are performed
with $N_s$=24, 32 and 40 at light quark mass $m_l$=$m^{phy}_s/20$, $m^{phy}_s/40$ and $m^{phy}_s/60$, respectively.
At our lowest quark mass corresponding to lightest Goldstone pion mass of about 80 MeV we also performed simulations at two different volumes of
$N_s=48$ and 32. Simulation parameters are listed in the left table in Table <ref>.
In the case of $N_f=3$ QCD we performed simulations with 3 degenerate quarks in which
five different values of quark masses $am_q$ are varied from 0.0009375 to 0.0075
corresponding to pion masses in the region of $230 \gtrsim m_\pi \gtrsim 80$ MeV.
The aspect ratio $N_\sigma/N_\tau$ is generally 4, although other volumes i.e. $N_\sigma=16,12,10$ at $am_q=0.00375$
and $N_\sigma=16$ at $am_q=0.0009375$ have also been used. Simulation parameters are listed in the right table in Table <ref>.
$N_\sigma^3\times N_\tau$ $m_l/m^{phy}_s$ $m_{\pi}$ [MeV] average # of conf.
$24^3\times$ 6 1/20 160 1200
$12^3\times$ 6 1/27 140 1200
$16^3\times$ 6 1/27 140 1500
$20^3\times$ 6 1/27 140 1000
$24^3\times$ 6 1/27 140 1000
$32^3\times$ 6 1/27 140 1300
$32^3\times$ 6 1/40 110 1200
$40^3\times$ 6 1/60 90 1000
$32^3\times$ 6 1/80 80 1200
$48^3\times$ 6 1/80 80 900
$N_\sigma^3\times N_\tau$ $am_q$ $m_{\pi}$ [MeV] average # of conf.
$16^3\times$ 6 0.0075 230 1000
$24^3\times$ 6 0.00375 160 2000
$16^3\times$ 6 0.00375 160 2000
$12^3\times$ 6 0.00375 160 2000
$10^3\times$ 6 0.00375 160 1500
$24^3\times$ 6 0.0025 130 1300
$24^3\times$ 6 0.001875 110 1000
$24^3\times$ 6 0.00125 90 1000
$24^3\times$ 6 0.0009375 80 1500
$16^3\times$ 6 0.0009375 80 1500
Parameters of the numerical simulations for $N_f$=2+1 QCD (left) and $N_f=3$ QCD (right).
§ UNIVERSALITY CLASS NEAR CRITICAL LINES
Close to the chiral limit the chiral order parameter ($M$) and its
susceptibility ($\chi_M$) can be described by the universal properties of the chiral
transition <cit.>, i.e. so called magnetic equation of state (MEOS),
M(t,h)= h^1/δ f_G(z) + f_reg
, and χ_M(t,h) = ∂M/∂H=1/h_0 h^1/δ-1
f_χ(z) + χ_reg
t=1/t_0 T-T_c/T_c, and h= H/h_0=1/h_0 m_l-m_c/m_s.
$t$ and $h$ are reduced temperature and symmetry-breaking field, respectively.
They reflect the proximity of a system to a critical region.
Here $z=th^{-1/\beta\delta}$ is the scaling variable. The critical exponents $\beta$, $\delta$ and the scaling functions $f_G(z)$ and
$f_\chi(z)$ uniquely characterize the universality class of the QCD chiral
phase transition.
In the chiral limit of $N_f=2$ QCD the universality class
is believed to be that of the 3-d O(4) spin model [At the finite lattice cutoff for staggered fermions, as used in the current study, there is only one Goldstone boson in the
chiral limit, the relevant universality class is rather that of 3-d O(2) spin model.] and the critical quark mass $m_c=0$, while in the 3 degenerate flavor case
towards the chiral limit it would be that of a Z(2) spin model with the values of quark mass closer
to a presumably nonzero critical quark mass $m_c$. The parameters $t_0$, $h_0$, $T_c$ are non-universal and specific for a
theory and $T_c$ here is a fundamental quantity, i.e. the phase transition temperature of QCD.
These non-universal parameters can be determined by studying the universal behavior of the order parameter and its corresponding
susceptibility described in Eq. (<ref>).
§ CHIRAL PHASE TRANSITION OF $N_F$=2+1 QCD
In Fig. <ref> we show volume dependence of light quark chiral condensates $\langle\bar{\psi}\psi\rangle_l$ and disconnected chiral susceptibilities $\chi_{disc}$ at our lowest
light quark mass corresponding to $m_\pi=80 $ MeV in the continuum limit in the left and right plot, respectively.
The chiral condensate has minor volume dependence and there is no sudden jump seen in the temperature dependence of the chiral condensate.
Together with minor volume dependence seen from the disconnected susceptibility it is clear that in the investigated
quark mass window there does not exist a first order phase transition.
The mass dependence of chiral condensates and disconnected chiral susceptibilities
is shown in Fig. <ref>. As expected the chiral condensate decreases
and the peak height of the disconnected susceptibility becomes larger at a smaller value of quark mass, since
the system approaches to a phase transition as the light quark masses go to zero.
In the continuum 2-flavor QCD the connected quark susceptibility $\chi_{con}$ stays finite, however, for the staggered
quarks $\chi_{con}$ becomes divergent in the chiral limit. In Fig. <ref> we show the temperature dependence
of connected susceptibility $\chi_{con} $ and the ratio $\chi_{con}/\chi_{total}$ at various values of quark masses.
It can be seen that at the temperature from about 142 MeV to about 150 MeV the connected susceptibility increases
with temperature and becomes larger at smaller values of quark masses. This trend is similar to the disconnected susceptibility.
The ratio of the connected susceptibility to the total susceptibility on the other hand is almost temperature independent in this
temperature region and it is also independent on the quark mass at the smaller quark masses. Besides that the connected susceptibility is larger than the
disconnected susceptibility at a certain set of temperature and quark mass. This may indicate
that singular contributions dominate in both two quantities but have different fractions in the total singular contributions.
When looking into the ratios the only thing left is the relative factor which is independent of quark mass and temperature.
Volume dependences of light quark chiral condensates (left) and of disconnected chiral susceptibilities (right)
for the light quark mass $m_l=m_s^{phy}/80$ corresponding
to $m_\pi\simeq80~$MeV.
Quark mass dependences of light quark chiral condensates (left) and disconnected chiral susceptibilities (right).
Left: Temperature dependence of connected susceptibility at various values of quark masses.
Right: Ratio of the connected susceptibility
to the total susceptibility.
Left half panel: O(2) scaling analysis for all quark masses obtained from the scaling fits to the order parameter
$M=m_s\langle \bar\psi\psi\rangle_l N_\tau^4$ and $\chi_M=m_s^2\chi_{total}N_\tau^4$.
Right half panel: the resulting fraction of the contribution from the singular part in $M$ (left) and $\chi_M$ (right).
Same as the plots in the left half panel in Fig. 5 but using the Z(2) universality class.
To analyze the universal behavior of chiral phase transition in the chiral limit of two-flavor QCD we performed
a joint scaling fit to chiral condensates and total chiral susceptibilities at all values of quark masses available using the MEOS
according to Eq. (<ref>). The scaling functions used in the fits belong to the O(2) universality class.
For the regular part we used $f_{reg}=\left(a_0 + a_1 \frac{T-T_c}{T_c} \right) \frac{m_l}{m_s}$ and $\chi_{reg}=a_0 + a_1 \frac{T-T_c}{T_c}$.
The fitting results are shown in the two plots in the left hand side of Fig. <ref>. We can see that the MEOS provides
a good description to chiral condensates and the scaling curve passes through the data points of total chiral susceptibilities.
The deviation in the total susceptibility at large quark mass from the scaling curve could be due to the fact that
the regular contributions are larger there and one needs to include in the regular term the contributions from the
beyond-leading-order corrections of the quark mass, while the deviation from the data points at lowest quark mass
should be due to the poor statistics. We also show the fraction of the contribution from the singular part in the chiral condensate
and total chiral susceptibilities in the two plots in the right hand side of Fig. <ref>. It is expected that the contribution from the singular part
dominates over that from the regular part more at the smaller quark mass as the system approaches to the critical regime. In fact, the ratio
of $\chi_{sing}/\chi_{M}$ at $z\sim 0$, i.e. $T\sim T_c$ increases from $\sim 72$% at $m_\pi=160$ MeV
to $\sim 88$% at $m_\pi=80$ MeV.
Universality class $m_c/m_{s}^{phy}$ $T_c $ [MeV] $t_0$ $h_0$ $a_0$ $a_1$
O(2) N/A 145.6(1) 3.34(4)e-03 4.56(4)e-06 10.1(3) -384(10)
Z(2) -0.0077(2) 141.9(2) 1.5(2)e-04 3.54(8)e-06 1.7(5) –282(8)
Obtained non-universal parameters in $N_f=2+1$ QCD with O(2) and Z(2) universality classes.
In the previous O(2) scaling analysis we assumed that
the tri-critical point in the columbia plot is located below the physical point. It may
happen that tri-critical point is above the physical point and the system passes through the
critical line belonging to Z(2) universality class towards the chiral limit of 2 light quarks.
We thus have performed a scaling fit to both chiral condensates and chiral susceptibilities
using also Z(2) scaling functions. The fit includes an additional fit parameter $m_c$. From the fit results
shown in Fig. <ref> a very small (even negative) value of $m_c$ is obtained. This certainly suggests
that the Z(2) scaling does not work for the $N_f$=2+1 QCD. The other non-universal parameters
are also listed in Table <ref>.
Thus, this finding together
with that from the O(2) scaling analysis suggest that the tri-critical point is indeed below the tri-critical point, i.e. $m_s^{tri}<m_s^{phy}$.
§ CHIRAL PHASE STRUCTURE OF $N_F$=3 QCD
To investigate whether the system has reached the chiral first order phase transition region
we have performed lattice QCD simulations with various mass values of 3 degenerate quarks.
We show the volume dependence of chiral condensates
and disconnected susceptibilities at lowest quark mass $am_q=0.0009375$ in the left half panel of Fig. <ref>.
There is no discontinuity in the temperature dependence of chiral condensate and no linear increase of disconnected
chiral susceptibility in volume observed. This suggests that the system with investigated quark mass window, i.e. corresponding to $ 80 \lesssim m_\pi \lesssim 230$ MeV
is not yet in the chiral first order phase transition region.
Volume dependences of chiral condensates and disconnected chiral susceptibilities at $am_q$=0.0009375 (left half panel) and $am_q$=0.00375 (right half panel).
To estimate to what value of quark mass the first order chiral phase transition region extends we
assume the system is in the criticality window belonging to the Z(2) universality class. We here use the chiral condensate
as an approximation of the order parameter of the chiral phase transition of 3 degenerate favor QCD. Thus at the
pseudo-critical temperature, i.e. peak location of the disconnected chiral susceptibility, the inverse disconnected susceptibility
$T/\chi^{max}_{q,disc}$ is proportional to $(m-m_c)^{1-1/\delta}$. Thus an estimate of $m_c$ can be obtained by fitting a 2-parameter ansatz
to the inverse peak height of the disconnected susceptibility as can be read off from the left plot of Fig. <ref>. The resulting fit is shown in
the right plot of Fig. <ref> and the intercept of the fitting curve to the $x$ axis is the estimate of $am_c$. The grey band indicates the
uncertainties arising from the fit with and without taking the data point at largest quark mass into account. The data points obtained in the smaller volume
are shown as the black points which could make the estimate of $am_c$ smaller. Thus we obtained a upper bound for the critical quark mass, i.e. $am_c \leq0.00039(1)$ which
corresponds to a upper bound of the critical pion mass of 50 MeV. This is consistent with the results obtained from the calculations using stout fermions <cit.> and Wilson fermions
towards the continuum limit <cit.>. On the other hand the fit using the O(2) universality class, in which $m_c=0$, does not accommodate the data at all as seen from the dashed
line in the right plot in Fig. <ref>.
Left: Disconnected susceptibilities from the largest available volume. Right: Estimate of the critical quark mass $m_c$ according to the Z(2) universality class.
§ SUMMARY
We have studied the chiral phase structure in $N_f$=2+1 and 3 flavor QCD using lattice QCD simulations with various values of quark masses
towards chiral limit using HISQ fermions on $N_\tau=6$ lattices. Our results suggest that the location of the tri-critical point is below that of the physical point,
i.e. $m_s^{tri} < m_s^{phy}$. We found that the transition of $N_f=3$ QCD with bare quark mass $am=0.0009375$ corresponding $m_\pi=80$ MeV in the continuum limit is still
a crossover, and the point where the system starts to have a first order phase transition is estimated to be $am_c \leq 0.00039(1)$ correspoing to $m_\pi^c \lesssim 50 $ MeV.
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|
1511.00549
|
Institut für Theoretische Teilchenphysik und Kosmologie,
RWTH Aachen University, D-52056 Aachen, Germany
Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK
Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK
Cavendish-HEP-15/10, TTK-15-34
We present the first complete next-to-next-to-leading order (NNLO) QCD predictions for differential distributions in the top-quark pair production process at the LHC. Our results are derived from a fully differential partonic Monte Carlo calculation with stable top quarks which involves no approximations beyond the fixed-order truncation of the perturbation series. The NNLO corrections improve the agreement between existing LHC measurements [V. Khachatryan et al. (CMS Collaboration), Eur. Phys. J. C 75, 542 (2015)] and standard model predictions for the top-quark transverse momentum distribution, thus helping alleviate one long-standing discrepancy. The shape of the top-quark pair invariant mass distribution turns out to be stable with respect to radiative corrections beyond NLO which increases the value of this observable as a place to search for physics beyond the standard model. The results presented here provide essential input for parton distribution function fits, implementation of higher-order effects in Monte Carlo generators as well as top-quark mass and strong coupling determination.
§ INTRODUCTION
There is remarkable overall agreement between standard model (SM) predictions for top-quark pair production and LHC measurements. Measurements of the total inclusive cross section at 7, 8, and 13 TeV <cit.> agree well with next-to-next-to leading order (NNLO) QCD predictions <cit.>. Differential measurements of final state leptons and jets are generally well described by existing NLO QCD Monte Carlo (MC) generators. Concerning top-quark differential distributions, the description of the top-quark $\PT$ has long been in tension with data <cit.>; see also the latest differential measurements in the bulk <cit.> and boosted top <cit.> regions. The first 13 TeV measurements have just appeared <cit.> and they show similar results; i.e., MC predictions tend to be harder than data.
This “$\PT$ discrepancy" has long been a reason for concern. Since the top quark is not measured directly, but is inferred from its decay products, any discrepancy between top-quark-level data and SM prediction implies that, potentially, the MC generators used in unfolding the data may not be accurate enough in their description of top-quark processes. With the top quark being a main background in most searches for physics beyond the SM (BSM), any discrepancy in the SM top-quark description may potentially affect a broad class of processes at the LHC, including BSM searches and Higgs physics.
The main “suspects" contributing to such a discrepancy are higher order SM corrections to top-quark pair production and possible deficiencies in MC event generators. A goal of this work is to derive the NNLO QCD corrections to the top-quark $\PT$ spectrum at the LHC and establish if these corrections bridge the gap between LHC measurements, propagated back to top-quark level with current MC event generators, and SM predictions at the level of stable top quarks.
Normalized top-antitop $\PT$ distribution vs CMS lepton+jets data <cit.>. NNLO error band from scale variation only. The lower panel shows the ratios LO/NNLO, NLO/NNLO, and data/NNLO.
As in Fig. <ref> but for the top-antitop rapidity.
Our calculations are for the LHC at 8 TeV. They show that the NNLO QCD corrections to the top-quark $\PT$ spectrum are significant and must be taken into account for proper modeling of this observable. The effect of NNLO QCD correction is to soften the spectrum and bring it closer to the 8 TeV CMS data <cit.>. In addition to the top-quark $\PT$, all major top-quark pair differential distributions are studied as well.
§ DETAILS OF THE CALCULATION
In the context of our previous work on the top-quark forward-backward asymmetry at the Tevatron <cit.>, we have already preformed a complete differential calculation of NNLO QCD corrections to on-shell top-quark pair production. Unfortunately, our Tevatron setup turned out not to be sufficiently powerful to deal with the increased demands of the LHC configuration. One reason is that the cross-section is now dominated by gluon fusion instead of quark annihilation. The main cause lies, however, in the substantially higher collider energy, which raises the fraction of events with top quarks far away from threshold. For the latter, phase space integrals yield large logarithms of the ratio of the top-quark mass and the partonic center-of-mass energy. In consequence, the convergence rate of the numerical Monte Carlo integration is severely diminished.
The results presented in this Letter are obtained using a fresh complete implementation of the sector-improved residue subtraction scheme, Stripper <cit.>, in its four-dimensional formulation as developed in Ref. <cit.>. We note that the subtraction scheme relies on the known soft and collinear limits of tree-level and one-loop matrix elements <cit.>. It also exploits the singularity structure of one- and two-loop virtual amplitudes <cit.>. Its main strength consists in preserving process independence and generality without requiring intricate analytic integration. The price of the obvious advantage is a numerical (as opposed to analytical) cancellation of the poles in the dimensional regularization parameter.
The process specific matrix elements for top-quark pair production in the Born approximation were obtained using the software from Ref. <cit.>. We evaluated the four-point one-loop amplitudes ourselves, although they can also be found in Refs. <cit.>. The five-point one-loop amplitudes, on the other hand, were computed with a code used in the calculation of $pp \to t\t j$ at NLO <cit.>. Finally, the two-loop matrix elements were taken in the form of numerical values on a dense grid supplemented with threshold and high-energy expansions from Refs. <cit.>. Notice that some partial analytical results are also known at two loops <cit.>.
As for our setup, we use the top-quark pole mass $m_t=173.3\GeV$, the MSTW2008 parton distribution function (PDF) set <cit.>, and kinematics-independent scales with the central value $\mu_R=\mu_F=m_t$. The theoretical uncertainty is estimated with independent scale variation $\mu_R\neq\mu_F$ subject to the additional restriction $0.5<\mu_R/\mu_F<2$ <cit.>. The PDF uncertainty is not included. The above choice of scales, PDF set, and parameters is dictated mainly by reasons of backward compatibility with our previous work and the need for extensive checks at the level of intermediate and final results. In the future, we intend to consider various choices of running scales, PDF sets and errors as well as values of $m_t$.
We have checked that our calculation reproduces $\sigma_{\rm tot}$ from Refs. <cit.> for each value of $\mu_R,\mu_F$ with a precision around two per mil for the $\mathcal{O}(\alpha_s^4)$ contribution, which translates to about $2 \times 10^{-4}$ for the complete result. We have also verified the cancellation of infrared singularities in each histogram bin. At NLO, our calculation has been cross-checked with the MC generator MCFM <cit.>. The predicted NNLO $p_{T,t\bar t}$ distribution for nonvanishing transverse momentum is consistent with results for the NLO QCD corrections to $pp \to t\t j$ from Refs. <cit.> and agrees with an independent evaluation using Helac-Nlo <cit.>. The new software also reproduces our previous Tevatron results.
§ RESULTS
In the following we discuss the $\PTt, y_t, \Mtt$, and $\ytt$ differential distributions. We do not present the transverse momentum distribution of the top-quark pair since it can be obtained with readily available NLO tools applied to the $tt+j$ process. The $\PTt$ and $y_t$ distributions are assumed to be insensitive to the charge of the heavy quark; i.e., they are an average of the respective top- and antitop-quark distributions.
In Fig. <ref> we show the prediction for the normalized $\PTt$ distribution computed in LO, NLO, and NNLO QCD, and compared to the most recent CMS data <cit.>. The corresponding top-quark rapidity distribution is shown in Fig. <ref>. As explained in the previous section, PDF variation has not been included in these results (or in any other results shown in this Letter). For clarity, in Figs. <ref> and <ref> the scale variation is only shown for the NNLO correction. When computing various perturbative orders we always use PDFs of matching order.
No overflow events are included in any of the bins shown in this Letter. The normalizations of the distributions in Figs. <ref> and <ref> are derived in such a way that the integral over the bins shown in these figures yields unity. Because of a slight difference in the bins, we note a small mismatch with respect to the measurements we compare to: for the top-quark $\PT$ distribution CMS has one additional bin $400\GeV < \PT < 500\GeV$ (not shown in Fig. <ref>). This bin contributes only around 4 per mil to the normalization of the data and we neglect it in the comparison. The $y_t$ distribution computed by us extends to $|y_t|<2.6$. This last bin differs slightly from the corresponding CMS bin which extends to $|y_t|<2.5$. This mismatch is shown explicitly in Fig. <ref>.
Top-antitop $\PT$ distribution in LO, NLO and NNLO QCD. Error bands from scale variation only.
We observe that the inclusion of NNLO QCD corrections in the $\PTt$ distribution brings SM predictions closer to CMS data in all bins. In fact the two agree within errors in all bins but one (recall that the PDF error has not been included in Fig. <ref>). The case of the $y_t$ distribution is more intriguing; we observe in Fig. <ref> that the NNLO and NLO central values are essentially identical in the whole rapidity range (this is partly related to the size of the bins). Given the size of the data error, it does not appear that there is any notable tension between NNLO QCD and data. The apparent stability of this distribution with respect to NNLO radiative corrections will clearly make comparisons with future high-precision data very interesting.
We do not compare with the CMS data for the $\Mtt$ and $\ytt$ distributions since the mismatch in binning is more significant. Instead, in Figs. <ref> and <ref> we present the NNLO predictions for the absolute normalizations of these distributions. We stress that the bin sizes we present are significantly smaller than the ones in the existing experimental publications. This should make it possible to use our results in a variety of future experimental and theoretical analyses. For this reason, in Fig. <ref> we also present the absolute prediction for the top-quark $\PT$ distribution with much finer binning compared to the one in Fig. <ref>.
As in Fig. <ref> but for the top pair invariant mass.
In Figs. <ref>,<ref>, and <ref> we show the scale variation for each computed perturbative order, together with the NLO and NNLO $K$ factors. In all cases one observes a consistent reduction in scale variation with successive perturbative orders. Importantly, we also conclude that our scale variation procedure is reliable, since NNLO QCD corrections are typically contained within the NLO error bands (and to a lesser degree for NLO with respect to LO). We also notice that the NNLO corrections do not affect the shape of the $\Mtt$ distribution. The stability of this distribution with respect to higher-order corrections makes it, among others, an ideal place to search for BSM physics. It will be very interesting to check if this property is maintained with dynamic scales and if it extends to higher $\Mtt$.
The $K$ factors in Figs. <ref> and <ref> show a peculiar rise at low $\PTt$ and $\Mtt$, respectively, which is due to soft gluon and Coulomb threshold effects. We do not investigate them in detail in the present work; related past studies include Refs. <cit.>.
A feature of our calculation that needs to be addressed more extensively is the fact that we use fixed scales. Running scales are usually thought to be more appropriate for such a differential calculation. However, in this first work on the subject, we opt for the simplicity of fixed scales in order to perform checks with existing NNLO calculations. We intend to extend our result to dynamical scales, which typically involve the top transverse mass $\sqrt{p_T^2+m_t^2}$ and thus start to deviate from fixed scales at large $p_T$, in future publications. The result presented here, however, should not be affected substantially by such a change due to the limited kinematical range considered (for instance $\PTt<400\GeV$).
As in Fig. <ref> but for the top pair rapidity.
§ CONCLUSIONS
In this Letter, we present for the first time NNLO accurate differential distributions for top-quark pair production at the LHC at 8 TeV. It is easy to conclude from the shown $K$ factors that our calculation is of very high quality (i.e. MC errors are small). Our result is exact in the sense that it fully includes all partonic channels contributing to NNLO and, moreover, includes them completely (in particular, we do not resort to the leading color approximation).
Partial NNLO results have been computed by two groups <cit.>. At the level of the total inclusive cross section these results agree with our previous calculations <cit.>. Although highly desirable, a comparison at the differential level is not possible at present since in our current calculation we do not separate subsets of partonic reactions or implement the leading colour approximation. Additionally, various NNLO approximations exist in the literature <cit.>. A dedicated comparison with these approximate results would be valuable.
The results derived in this Letter would allow one to undertake a number of high-caliber phenomenological LHC analyses. Some examples are: validation of different implementations of higher-order effects in MC event generators, extraction of NNLO PDFs from LHC data, improved determination of the top-quark mass, and direct measurement of the running of $\alpha_S$ at high scales. Moreover, SM predictions with improved precision will enable a higher level of scrutiny of the SM with the help of LHC data as well as novel searches for BSM physics, possibly along the lines of Refs. <cit.>. Finally, this result will serve as the basis for future inclusion of top-quark decay <cit.>.
We thank Stefan Dittmaier for kindly providing us with his code for the evaluation of the one-loop virtual corrections. M.C. thanks Emmanuel College Cambridge for hospitality during the completion of this work. A.M. thanks Durham University for hospitality during the completion of this work.
The work of M.C. was supported in part by grants of the DFG and BMBF. The work of D.H. and A.M. is supported by the UK Science and Technology Facilities Council [Grants No. ST/L002760/1 and No. ST/K004883/1].
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1511.00134
|
The SiD Detector for the International Linear Collider
Andrew P. White(for the SiD Consortium)
The SiD Detector is one of two validated detector designs for the future International Linear Collider.
SiD features a compact, cost-constrained design for precision Higgs couplings determination, and other measurements, and sensitivity
to a wide range of possible new phenomena. A robust silicon vertex and tracking system, combined with a 5 Tesla
central solenoidal field, provides excellent momentum resolution. The highly granular calorimeter system is
optimized for Particle Flow application to achieve very good jet energy resolution over a wide range of energies.
Details of the proposed implementation of the SiD subsystems, as driven by the physics requirements, will be given.
The shared interaction point, push-pull mechanism, will be described, together with the estimated timeline for construction.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The ILC accelerator showing the location of the detectors.
The SiD Detector is one of two detector concepts developed for the future
International Linear Collider, to be located in Japan at the proposed Kitakami
site. The ILC accelerator is shown in Fig. <ref>.
The SiD detector is designed for a comprehensive program
of e+e- physics ranging from percent-level measurement of Higgs boson couplings to precision top quark
studies, to searches for new physics such as supersymmetry. The detector requirements
to address this physics program are given in Table <ref>. Many of the Higgs
processes require excellent jet energy performance which derives from the design and performance
of the tracking and calorimeter systems together with an efficient particle-flow algorithm.
The Higgs recoil mass measurement requires excellent charged particle momentum resolution,
while Higgs branching fraction measurements for heavy flavors rely on a precision vertex
detector. Finally, searches for new phenomena require a hermetic
Detector performance needed for key ILC physics measurements.
1lPhysics 1lMeasured 1lCritical 1lPhysical 1lRequired
1lProcess 1lQuantity 1lSystem 1lMagnitude 1lPerformance
Zhh Triple Higgs coupling Tracker Jet Energy
$Zh \rightarrow q\bar{q}b\bar{b}$ Higgs mass and Resolution
$Zh \rightarrow ZWW^*$ $B(h \rightarrow WW^*$) Calorimeter $\Delta E/E$ 3% to 4%
$\nu\overline{\nu} W^+W^-$ $\sigma(e^+e^- \rightarrow\nu\overline{\nu} W^+W^-)$
$Zh \rightarrow \ell^+ \ell^- X$ Higgs recoil mass $\mu$ detector Charged particle
$ \mu^+ \mu^- (\gamma)$ Luminosity weighted E$_{\rm {cm}}$ Tracker Momentum Resolution $5 \times 10^{-5} (GeV/c)^{-1}$
$Zh + h \nu \overline{\nu} \rightarrow \mu^+ \mu^- X$ BR($h \rightarrow \mu^+ \mu^-$) $\Delta p_t / p^2_t$
$Zh, h \rightarrow b \bar{b}, c\bar{c}, b \bar{b}, gg $ Higgs branching fractions Vertex Impact $5 \mu m \oplus $
b-quark charge asymmetry parameter $10\mu m /p\rm{(GeV/c)}sin^{3/2}\theta$
Tracker Momentum Resolution
SUSY, eg. $\tilde{\mu}$ decay $\tilde{\mu}$ mass Calorimeter Hermeticity
$\mu$ detector
§ THE SID DETECTOR CONCEPT DESIGN
The SiD Detector Concept.
SiD is a compact, cost-constrained detector designed to make precision measurements
and be sensitive to a wide range of new phenomena. The compact design is achieved
by the use of a high precision silicon vertexing and tracking system in combination with a
5 Tesla solenoidal magnetic field. The vertexing and tracking system offers excellent
charged particle momentum resolution and is live for single bunch crossings.
The calorimetry is optimized for jet energy resolution, based on a particle flow (PFA) approach,
with tracking calorimeters, compact showers in the electromagnetic section, and highly segmented,
longitudinally and transversely, electromagnetic and hadronic systems.
The iron flux return and muon identifier is a component of SiD self-shielding.
The complete detector system is designed for push-pull operation.
The main elements of the SiD detector are shown in Fig. <ref>.
§ SID VERTEXING AND TRACKING
The SiD vertex detector, Fig. <ref> (left), is an all-silicon system
consisting of five barrel layers, four disk layers, and three additional small
pixel disks in the forward region. The carbon fiber support structure is connected to
the beam tube in four places.
Half view of the SiD Vertex Detector.
SiD Main tracking system.
SiD tracking system - outer tracker (right) and vertex detector (left).
Various technologies are being considered for the vertex detector, ranging from
standard silicon diode pixels, through monolithic active pixels (as in the Chronopix <cit.> design ),
to vertically integrated 3-dimensional structures <cit.>.
Power management for the vertex detector will take advantage of the ILC beam time
structure to use pulsed power, and will use DC-DC conversion to avoid the needed
for high mass cables that would compromise the otherwise excellent low material
The main tracker is also an all-silicon system with the cylindrical barrel layers
closed at the ends by conical, annular disks, as shown in Fig. <ref> (right).
The main tracker layers are instrumented with silicon microstrip tiles read out
via the KPiX <cit.> ASIC which features a four-deep pipeline and single bunch
time stamping with readout occurring between bunch trains. Power pulsing and DC-DC conversion
allows the use of gas cooling and cable mass reduction.
Overall for the tracking system, better than 20% of a radiation length is achieved
over the angular range to within 10 degrees of the beam direction. A tracking efficiency
of at least 95% is achieved over a similar angular range for charged particles with
momenta above 1 GeV. The transverse momentum resolution for single muons is shown in
Fig. <ref>, which, for central tracks, exceeds the requirement in
Table <ref>.
Normalized transverse momentum resolution for single muons in SiD.
§ CALORIMETRY
SiD calorimetry is designed for the particle plow approach to improving jet energy
resolution. The goal is to achieve 3% or better jet energy resolution for jets above 100 GeV.
For a PFA the tracker and calorimeter must work together to ensure efficient and effective
association of charged tracks with the correct energy deposits in the calorimeter. This
implies the need for a high degree of transverse and longitudinal segmentation in the
calorimeters. The Moliere radius for the electromagnetic calorimeter should be minimized to
facilitate the separation of charged tracks and electromagnetic showers. Naturally, the entire
calorimeter system is located inside the volume of the solenoid - which, however, imposes
limitations on radial dimensions due to cost considerations.
The main elements of the calorimeter system are shown in Fig. <ref> (a).
Components of the SiD calorimeter system: purple - hadron calorimeter;
green - electromagnetic calorimeter.
Mechanics and components of the SiD electromagnetic calorimeter.
SiD Calorimeter system and detail of electromagnetic calorimeter.
§.§ Electromagnetic calorimeter
In addition to supporting the PFA requirements described above, the ECAL must
allow precise measurement of electrons and positrons from Bhabha scattering for
determination of electroweak couplings and for a component of the measurement of
the luminosity spectrum. The ECAL should also provide for efficient detection
and measurement of photons and pizero, for contributions to the jet energy resolution
and for reconstruction of $\tau$ decays.
The ECAL baseline design <cit.> features tungsten absorber plates and highly segmented silicon
sensor layers. The main elements of the design are shown in Fig. <ref> (b).
Each silicon sensor is divided into 1024 pixels, read out by one KPiX ASIC <cit.>.
Each ECAL layer features an aggressive design with only 1.25 mm gap between
successive tungsten plates - including the sensor, KPiX, flex cables and a
passive cooling system. A nine-layer prototype of the ECAL has been tested and
single and multiple electron tracks successfully recorded <cit.>.
There also is a MAPS (Monolithic Active Pixel) design for the ECAL, which has
very fine, 50$\times$50 micronsquare, silicon pixels, for which first generation
sensors have been tested.
§.§ Hadron calorimeter
The HCAL is designed for efficient and unambiguous identification of energy
deposits by charged particles, their association with the related tracks in
the tracking system, and the measurement of the energies of neutral particles.
Within the PFA approach, these functions demand fine transverse and longitudinal
segmentation, with the requirement (for the barrel sections) to keep the active
layer thickness to a minimum to control the cost of the radially external solenoid.
The baseline technology for the HCAL is resistive plate chambers (RPC) with steel
absorber plates <cit.> . The basic RPC design is shown in Fig. <ref> (a).
Design of the two-glass RPC for the SiD hadron calorimeter.
Events recorded in the digital hadron calorimeter prototype.
The RPC design and prototype results for the SiD Hadronic Calorimeter.
The baseline design has been implemented in a 38-layer prototype with transverse size
1$\times$1 m$^2$, large enough to contain hadronic showers. Fig. <ref> (b)
shows several examples of hadron showers, and a single muon track, recorded in the prototype which had
1 $\times$ 1 cm$^2$ readout pads.
Several other technologies are also being developed and considered for the SiD HCAL: Scintillator tiles,
GEM's (both foils and ThickGem), Micromegas, and variations on the RPC approach. Fig. <ref>
shows an example: a 1$\times$1 m$^2$ active layer of scintillator tiles with SiPM readout <cit.>.
Active layer from the scintillating tile-SiPM approach to the SiD HCAL.
§.§ Forward Calorimetry
Fig. <ref> shows the recently updated forward calorimeter and
luminosity calorimeter layout with the new common L* agreed by SiD and ILD.
The LumiCal will use small angle Bhabha scattering to determine the integrated
luminosity to better than one part per mil. The BeamCal will provide an instantaneous
measurement of the luminosity using beamstrahlung pairs, and will provide small-angle
coverage for physics searches.
The new forward region layout for SiD with L* of 4.1m.
§ MUON SYSTEM AND FLUX RETURN
The baseline technology for the muon system is long scintillator strips with wavelength
shifting fibers and SiPM readout. The roles of the muon system are to identify muons from
the interaction point efficiently, and, as a tail-catcher, to flag possible shower leakage through the
superconducting solenoid as a part of the particle flow algorithm input. Fig. <ref> (a)
shows prototype long strips and fibers underdevelopment.
The steel of the muon system acts as the flux return for the magnetic field from the superconducting
solenoid. There is a local site requirement of a maximum of 50 Gauss at 15m from the main detector axis.
Fig. <ref> (b) shows a recent design of the muon steel, with a 30 degree angle between the
barrel and endcap steel. This design limits the fringe field and satisfies the 50 Gauss requirement.
Muon system prototype with long scintillator strips with embedded wavelength shifting fibers.
New design of the SiD flux return steel.
Muon system steel and flux return, and scintillator prototype.
§ INSTALLATION
The single interaction region of the ILC requires a sharing between the two detector concepts
in a push-pull arrangement as shown in Fig. <ref>. An assembly and installation
procedure has been created. It is estimated to take a period of eight years to complete.
The SiD Detector, on the beam axis, and the ILD Detector in the push-pull configuration.
The more compact design of the SiD Detector results in a deeper platform as shown in black.
§ SUMMARY
The design of the SiD Detector Concept has been presented, with details of subsystems.
Physics studies with this design have shown that superb performance is expected on the
full range of ILC physics. Current detector development topics include a third generation
Chronopix and advanced 3-D for the vertex detector, a new silicon sensor for the ECAL, engineering
studies for a full-size scintillator-steel HCAL module, and overall optimization of the detector
design. The SiD Consortium remains open to new colleagues and to creative input to further
optimize or improve the detector design.
N. Sinev, talk at SiD Workshop, SLAC, January 2015.
R.Lipton, talk at SiD Workshop, SLAC, January 2015.
R.Frey at SiD Workshop, SLAC, January 2015.
D. Freytag et al., Linear Collider Power Distribution and Pulsing Workshop, Paris, 2011.
J.Repond, talk at LCWS 2013, U. Tokyo, November 2013.
B.Bilki et al., JINST 10 (2015) P04014.
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1511.00139
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As protein folding is a NP-complete problem, artificial intelligence tools like neural networks and genetic algorithms are used to attempt to predict the 3D shape of an amino acids sequence.
Underlying these attempts, it is supposed that this folding process is predictable.
However, to the best of our knowledge, this important assumption has been neither proven, nor studied.
In this paper the topological dynamic of protein folding is evaluated.
It is mathematically established that protein folding in 2D hydrophobic-hydrophilic (HP) square lattice model is chaotic as defined by Devaney.
Consequences for both structure prediction and biology are then outlined.
§ INTRODUCTION
Proteins, formed by a string of amino acids folding into a specific tridimentional shape, carry out the majority of functionality within an organism.
However, simulating perfectly the folding processes or molecular dynamic occurring in biology nature is indeed infeasible, due to the following reasons.
Firstly, the forces involved in the stability of the protein conformation are currently not modeled with enough accuracy <cit.>.
Indeed, we can even wonder if it is so realistic to hope finding one day an accurate model for this problem.
Secondly, due to an astronomically large number of possible 3D protein structures for a corresponding primary sequence of amino acids <cit.>: the computation capability required even for handling a moderately-sized folding transition exceeds drastically the capacity of the most powerful computers around the world.
Consequently, proteins structures are not exactly computed, but they are predicted.
As this Protein Structure Prediction (PSP) is a NP-complete problem <cit.>, prediction for optimal protein structures is principally performed using computational intelligence approaches such as genetic algorithms, ant colonies <cit.>, particle swarm, or neural networks.
Models of various resolutions are applied too, to tackle with the complexity of this problem.
In low resolution models, atoms into the same amino acid can for instance be considered as a same entity.
These low resolutions models are often used first to predict the backbone of the 3D conformation.
Then, high resolution models come next for further exploration.
Such a prediction strategy is commonly used in PSP softwares like ROSETTA <cit.> or TASSER.
In this paper, we demonstrate that protein folding is indeed unpredictable, that is, it is chaotic according to Devaney.
This well-known topological notion for a chaotic behavior is one of the most established mathematical definition of unpredictability for dynamical systems.
This proof has been achieved in the framework of protein structure prediction for a 2D hydrophobic-hydrophilic (HP) lattice model <cit.>.
This popular lattice model with low resolution focuses only upon hydrophobicity by separating the amino acids into two sets: hydrophobic (H) and hydrophilic (or polar P) <cit.>.
Numerous variations are proposed in the general HP model: 2D or 3D lattices, with square, cubic, triangular, or face-centered-cube shapes...
In our demonstration, we have chosen a 2D square lattice for easy understanding.
However the process still remains general and can be applied to high resolution models by using a more refined formulation.
After having established the chaotic behavior of the folding dynamic by using two different proofs, we will outline the consequences of this fact.
More precisely, we will focus on the following questions.
Firstly, is it possible to predict 3D protein structures using artificial intelligence tools if the folding process is chaotic ?
In other words, are genetic algorithms, neural networks, and so on, able to predict chaotic behaviors, at least as defined by Devaney (in this paper, we will only study neural networks) ?
The remainder of this paper is organized as follows.
In Section <ref>, basic notations and terminologies concerning both HP-model and Devaney's topological chaos are recalled.
Then in the next two sections the proofs of the chaotic behavior of protein folding dynamic are established.
The first proof is directly realized in the Devaney's context whereas the second one uses a previously proven result concerning chaotic iterations <cit.>.
Consequences of this unpredictable behavior are outlined in Section <ref>.
Among other things, it is regarded whether chaotic behaviors are harder to predict than “normal” behaviors or not.
Additionally, reasons explaining why a chaotic behavior unexpectedly leads to approximately one thousand categories of folds are proposed.
This paper ends by a conclusion section, in which our contribution is summarized and intended future work is presented.
§ BASIC RECALLS
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ the $i^{th}$ component of a vector $V$.
$f^{k}=f\circ...\circ f$ is the $k^{th}$ composition of a function $f$.
Finally, the following notation is used: $\llbracket1;N\rrbracket =\{1,2,\hdots,N\}$.
§.§ 2D hydrophilic-hydrophobic (HP) model
§.§.§ The HP model
In the HP model, hydrophobic interactions are supposed to dominate protein folding.
This model was formerly introduced by Dill, who consider in <cit.> that the protein core freeing up energy is formed by hydrophobic amino acids, whereas hydrophilic amino acids tend to move in the outer surface due to their affinity with the solvent (see Figure <ref>).
A protein conformation is then a self-avoiding walk (SAW) on a 2D or 3D lattice such that its energy $E$, depending on topological neighboring contacts between hydrophobic amino acids which are not contiguous in the primary structure, is minimal.
In other words, for an amino-acid sequence $P$ of length $\mathsf{N}$ and for the set $\mathcal{C}(P)$ of all SAW conformations of $P$, the chosen conformation will be $c^* = min \left\{E(c) \big/ c \in \mathcal{C}(P)\right\}$ <cit.>.
In that context and for a conformation $c$, $E(c)=-q$ where $q$ is equal to the number of topological hydrophobic neighbors.
For example, $E(c)=-5$ in Figure <ref>.
Hydrophilic-hydrophobic (HP) model (black squares are hydrophobic residues).
§.§.§ Protein Encoding
Additionally to the direct coordinate presentation, at least two other isomorphic encoding strategies for HP model are possible: relative encoding and absolute encoding.
In relative encoding <cit.>,
the move direction is defined relative to the direction of the previous move.
Alternatively, in absolute encoding <cit.>,
which is the encoding chosen in this paper,
the direct coordinate presentation is replaced by letters or numbers representing directions with respect to the lattice structure.
For a 2D absolute encoding, the permitted moves are: forward $\rightarrow$ (denoted by 1), backward $\leftarrow$ (2), up $\uparrow$ (3), and down $\downarrow$ (4).
A 2D conformation $c$ of $\mathsf{N}$ residues for a protein $P$ could then be $c \in \llbracket 1; 4 \rrbracket^{\mathsf{N}-2}$, as the initial move is always forward (1) <cit.>.
For example, in Figure <ref>, the 2D absolute encoding is (1)1144423322414 (starting from the upper left corner).
In that situation, $3^{\mathsf{N}-2}$ conformations are possible when considering $\mathsf{N}$ residues, even if some of them are invalid due to the SAW requirement.
§.§ Devaney's Chaotic Dynamical Systems
Let us now introduce the notion of chaos used in this document.
Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f$ on $\mathcal{X}$.
$f$ is said to be topologically transitive if, for any pair
of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that
$f^k(U) \cap V \neq \emptyset$.
An element (a point) $x$ is a periodic element (point) for
$f$ of period $n\in \mathds{N}^*,$ if $f^{n}(x)=x$.
$f$ is said to be regular on $(\mathcal{X}, \tau)$ if the set
of periodic points for $f$ is dense in $\mathcal{X}$: for any point
$x$ in $\mathcal{X}$, any neighborhood of $x$ contains at least one
periodic point.
$f$ is said to be chaotic on $(\mathcal{X},\tau)$ if $f$ is
regular and topologically transitive.
The chaos property is related to the notion of “sensitivity”, defined on a metric space $(\mathcal{X},d)$ by:
$f$ has sensitive dependence on initial conditions
if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$
and any neighborhood $V$ of $x$, there exist $y\in V$ and
$n \geq 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta
Indeed, Banks et al. have proven in <cit.> that when
$f$ is chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has
the property of sensitive dependence on initial conditions (this
property was formerly an element of the definition of chaos). To sum
up, quoting Devaney in <cit.>, a chaotic dynamical system “is
unpredictable because of the sensitive dependence on initial
conditions. It cannot be broken down or simplified into two subsystems
which do not interact because of topological transitivity. And in the
midst of this random behavior, we nevertheless have an element of
regularity”. Fundamentally different behaviors are consequently
possible and occur in an unpredictable way.
§ PROTEIN FOLDING IS CHAOTIC
We will now give a first proof of the chaotic behavior of the protein folding dynamic.
§.§ Initial Premises
Let us firstly introduce the preliminaries of our approach.
The primary structure of a given protein $p$ with $\mathsf{N}+1$ residues is coded by $1 1 \hdots 1$ ($\mathsf{N}$ times) in absolute encoding.
Its final 2D conformation has an absolute encoding equal to $1 C_1^* \hdots C_{\mathsf{N}-1}^*$, where $\forall i, C_i^* \in \llbracket 1, 4 \rrbracket$, is such that $E(C^*) = min \left\{E(C) \big/ C \in \mathcal{C}(p)\right\}$.
This final conformation depends on the repartition of hydrophilic and hydrophobic amino acids in the initial sequence.
Moreover, we suppose that, if the residue number $n+1$ is forward the residue number $n$ in absolute encoding ($\rightarrow$) and if a fold occurs after $n$, then the forward move can only by changed into up ($\uparrow$) or down ($\downarrow$).
That means, in our simplistic model, only rotations of $+\frac{\pi}{2}$ or $-\frac{\pi}{2}$ are possible.
Obviously, for a given residue that is supposed to be updated, only one of the two possibilities below can appear for its absolute move during a fold:
* $1 \longmapsto 4, 4 \longmapsto 2, 2 \longmapsto 3,$ or $ 3 \longmapsto 1$ for a fold in the clockwise direction, or
* $4 \longmapsto 1, 2 \longmapsto 4, 3 \longmapsto 2,$ or $1 \longmapsto 3$ for an anticlockwise.
This fact leads to the following definition:
The clockwise fold function is the function $f: \llbracket 1;4 \rrbracket \longrightarrow \llbracket 1;4 \rrbracket$ defined by:
\begin{array}{crcl}
& 1 & \longmapsto & 4 \\
& 2 & \longmapsto & 3 \\
& 3 & \longmapsto & 1 \\
& 4 & \longmapsto & 2 \\
\end{array}
Obviously the anticlockwise fold function is $f^{-1}$.
Thus at the $n^{th}$ folding time, a residue $k$ is chosen and its absolute move is changed by using either $f$ or $f^{-1}$. As a consequence, all of the absolute moves must be updated from the coordinate $k$ until the last one $\mathsf{N}$ by using the same folding function.
If the current conformation is $C=111444$, i.e., $\rightarrow\rightarrow\rightarrow\downarrow\downarrow\downarrow$, and if the third residue is chosen to fold by a rotation of $-\frac{\pi}{2}$ (mapping $f$), thus the new conformation will be:
$$(C_1,C_2,f(C_3),f(C_4),f(C_5),f(C_6)) = (1,1,4,2,2,2).$$
That is, $\rightarrow\rightarrow\downarrow\leftarrow\leftarrow\leftarrow$.
These considerations lead to the formalization of the next section.
§.§ Formalization and Notations
Let $\mathsf{N}+1$ be a fixed number of amino acids, where $\in\mathds{N}^*$.
We define
$$\mathcal{X}=\llbracket 1; 4 \rrbracket^\mathsf{N}\times \llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N}$$
as the phase space of the protein folding process.
An element $X=(C,F)$ of this dynamical folding space is constituted by:
* A conformation of the $\mathsf{N}+1$ residues in absolute encoding: $C=(C_1,\hdots, C_\mathsf{N}) \in \llbracket 1; 4 \rrbracket^\mathsf{N}$.
* A sequence $F \in \llbracket -\mathsf{N} ; \mathsf{N} \rrbracket^\mathds{N}$ of future folds, depending on hydrophobicity, such that when $F_i \in \llbracket -\mathsf{N}; \mathsf{N} \rrbracket$ is $k$, it means that it occurs:
* a fold after the $k-$th residue by a rotation of $-\frac{\pi}{2}$ (mapping $f$) at the $i-$th step, if $k = F_i >0$,
* no fold at time $i$ if $k=0$,
* a fold after the $|k|-$th residue by a rotation of $\frac{\pi}{2}$ (i.e., $f^{-1}$) at the $i-$th time, if $k<0$.
On this phase space, the protein folding dynamic can be formalized as follows.
Denote by $i$ the map that transforms a folding sequence into its first term (the first folding operation):
\begin{array}{lccl}
i:& \llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N} &\longrightarrow& \llbracket -\mathsf{N};\mathsf{N} \rrbracket,\\
& F &\longmapsto & F^0,
\end{array}$$
by $\sigma$ the shift function over $\llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N}$, that is to say,
\begin{array}{lccl}
\sigma :& \llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N} &\longrightarrow& \llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N},\\
& \left(F^k\right)_{k \in \mathds{N}} &\longmapsto & \left(F^{k+1}\right)_{k \in \mathds{N}},
\end{array}$$
and by $sign$ the function:
sign(x) = \left\{
\begin{array}{ll}
1 & \textrm{if } x>0,\\
0 & \textrm{if } x=0,\\
-1 & \textrm{else.}
\end{array}
\right.
The shift function removes the first folding operation from the sequence after it has been achieved once.
Consider now the map $G:\mathcal{X} \to \mathcal{X}$ defined by:
$$G\left((C,F)\right) = \left( f_{i(F)}(C);\sigma(F)\right)$$
where $\forall k \in \llbracket -\mathsf{N};\mathsf{N} \rrbracket$, $f_k: \llbracket 1;4 \rrbracket^\mathsf{N} \to \llbracket 1;4 \rrbracket^\mathsf{N}$ is defined by:
$f_k(C_1, ..., C_\mathsf{N}) =$
$ (C_1,... ,C_{|k|-1}, f^{sign(k)}(C_{|k|}),...,f^{sign(k)}(C_\mathsf{N})).$
Thus the folding process of a protein $P$ in the 2D HP square lattice model, with initial conformation equal to $(1,1, \hdots, 1)$ in absolute encoding, and a folding sequence equal to $(F^i)_{i \in \mathds{N}}$ provided by hydrophobic interactions, is defined by the following dynamical system over $\mathcal{X}$:
\left\{
\begin{array}{l}
X^0 = ((1,1,\hdots,1);F)\\
X^{n+1} = G(X^n), \forall n \in \mathds{N}.
\end{array}
\right.
In other words, at each step $n$, if $X^n=(C,F)$, we take the first folding operation to realize, that is $i(F) = F^0 \in \llbracket -\mathsf{N};\mathsf{N} \rrbracket$, we update the current conformation $C$ by rotating all of the residues coming after the $|i(F)|-$th one, which means that we replace the conformation $C$ with $f_{i(F)}(C)$.
Lastly, we remove this rotation (the first term $F^0$) from the folding sequence $F$: thus $F$ becomes $\sigma(F)$.
Let us reconsider example <ref>.
One iteration of its dynamical folding process can be described in this formalization by:
$\left((1,1,4,2,2,2),(F^1,F^2, \hdots)\right) =$ $G\left((1,1,1,4,4,4),(+3,F^1,F^2, \hdots)\right).$
A protein $P$ that has finished to fold, if such a protein exists, has the form $(C;(0,0,0,\hdots))$, where $C$ is the final 2D structure of $P$.
Such a formalization allows the study of proteins that never stop to fold, for instance due to their never ended interactions with the environment.
§.§ A Metric for the Folding Process
We define a metric $d$ over $\mathcal{X}=\llbracket 1;4 \rrbracket^\mathsf{N} \times \llbracket -\mathsf{N};\mathsf{N} \rrbracket^\mathds{N}$ by:
$$\displaystyle{d(X, \check{X}) = d_C(C, \check{C}) + d_F (F, \check{F}).}$$
where $\delta(a,b) =0$ if $a=b$, else $\delta(a,b)=1$, and
\left\{
\begin{array}{ll}
d_C(C, \check{C}) = & \displaystyle{\sum_{k=1}^\mathsf{N} \delta(C_k,\check{C}_k) 2^{\mathsf{N}-k}} \\
d_F (F, \check{F}) = & \displaystyle{\dfrac{9}{2 \mathsf{N}} \sum_{k=1}^\infty \dfrac{|F^k-\check{F}^k|}{10^k}}
\end{array}
\right.$$
This new distance for the dynamical description of the protein folding process in 2D HP square lattice model can be justified as follows.
The integral part of the distance between two points $X=(C,F)$ and $\check{X}=(\check{C},\check{F})$ of $\mathcal{X}$ measures the differences between the current 2D structures of $X$ and $\check{X}$.
More precisely, if $d_C(C,\check{C})$ is in $\llbracket 2^k,2^{k+1} \rrbracket$, then the first $k$ terms in the conformations $C$ and $C'$ (absolute encoding) are equal, whereas the $k+1^{th}$ terms differ.
The decimal part of $d(X,\check{X})$ will decrease when the duration the folding process will be similar increase.
More precisely, $F^k = \check{F}^k$ if and only if the $k+1^{th}$ digit of this decimal part is 0.
Lastly, $\frac{9}{\mathsf{N}}$ is just a normalization factor.
For instance, if we know where are the $\mathsf{N}+1$ residues of our protein $P$ in the lattice, and if we know what will be its $k$ next folding, then we are into the ball $\mathcal{B}(C,10^{-k})$, that is, very close to the point $(C,F)$ if $k$ is large.
In $X^0 = ((1,1, \hdots, 1), F)$, the folding sequence $F^0$ results, among other things, on the hydrophobic interactions between amino acids.
Indeed, it is this $F^0$ that is searched, when trying to predict the folding process of a given protein $P$.
That is to say, the error on $X^0$ measured by $d$ corresponds to our incapacity to determine exactly the whole future folding process $F$ of $P$.
Improving this prediction with better computational intelligence tools leads to the reduction of the distance between $X^0$ (what is looked for) and ${X'}^0$ (our approximation).
The question raised by this study is: even if we cannot have access with an infinite precision to all of the forces that participate to the folding process, i.e., even if we only know an approximation ${X'}^0$ of $X^0$, can we claim that the predicted conformation ${X'}^{n_1}$ still remains close to the true conformation ${X}^{n_2}$ ?
Or, on the contrary, do we have a chaotic behavior, a kind of butterfly effect that magnifies any error on the evaluation of the forces in presence ?
Raising such a question leads to the study of the dynamical behavior of the folding process.
To do so, we must firstly establish that $G$ is a continuous map on $(\mathcal{X},d)$.
§.§ Continuity of the Folding Operation
$G:\mathcal{X} \to \mathcal{X}$ is a continuous map.
We will use the sequential characterization of the continuity.
Let $(X^n)_{n \in \mathds{N}} = \left((C^n,F^n)\right)_{n \in \mathds{N}} \in \mathcal{X}^\mathds{N},$ such that $X^n \rightarrow X = (\check{C},\check{F})$.
We will show that $G\left(X^n\right) \rightarrow G(X)$.
Let us firstly remark that $\forall n \in \mathds{N}, F^n$ is a sequence: $F$ is thus a sequence of sequences.
On the one hand, as $X^n=(C^n,F^n) \rightarrow (\check{C},\check{F})$, we have $d_C\left(C^n,\check{C}\right) \rightarrow 0$, thus $\exists n_0 \in \mathds{N},$ $n \geqslant n_0$ $\Rightarrow d_C(C^n,\check{C})=0$. That is, $\forall n \geqslant n_0,$ $\forall k \in \llbracket 1;\mathsf{N} \rrbracket$, $\delta(C_k^n,C_k) = 0$, and so $C^n = \check{C}, \forall n \geqslant n_0.$
Additionally, $d_F(F^n,\check{F}) \rightarrow 0$, then $\exists n_1 \in \mathds{N},$ $d_F(F^n, \check{F}) \leqslant \frac{1}{10}$. As a consequence, $\exists n_1 \in \mathds{N},$ $\forall n \geqslant n_1$, the first term of the sequence $F^n$ is $\check{F}^0$: $i(F^n) = i(\check{F})$.
So, $\forall n \geqslant max(n_0,n_1),$ $f_{i(F^n)}\left(C^n\right) = f_{i\left(\check{F}\right)}\left(\check{C}\right)$, and then $f_{i(F^n)}\left(C^n\right)$ $\rightarrow$ $f_{i\left(\check{F}\right)}\left(\check{C}\right)$.
On the other hand, $\sigma(F^n) \rightarrow \sigma(F)$.
Indeed, $F^n \rightarrow F$ implies $\sum_{k=1}^{\infty} \frac{| \left(F^n\right)^k-\check{F}^k |}{10^k} \rightarrow 0$.
Thus, $\frac{1}{10} \sum_{k=1}^{\infty} \frac{| \left(F^n\right)^{k+1}-\check{F}^{k+1} |}{10^k} \rightarrow 0$, so $\sum_{k=1}^{\infty} \frac{| \sigma(F^n)^k-\sigma(\check{F})^k |}{10^k} \rightarrow 0$.
Finally, $\sigma(F^n) \rightarrow \sigma(\check{F}).$
To conclude, as $f_{i(F^n)}\left(C^n\right)$ $\rightarrow$ $f_{i\left(\check{F}\right)}\left(\check{C}\right)$ and $\sigma(F^n) \rightarrow \sigma(\check{F})$, we have $G\left(X^n\right) \rightarrow G(X)$.
It is now possible to study the chaotic behavior of the folding process.
§.§ Regularity of the Folding Operation
Let us firstly introduce the following notations: for $X = (C,F) \in \llbracket 1;4 \rrbracket^\mathsf{N}\times \llbracket 1;\mathsf{N} \rrbracket^\mathds{N}$, $\mathcal{C}(X) = C$ and $\mathcal{F}(X)=F$. We will now prove that,
For all $C,C'$ in $\llbracket 1;4 \rrbracket^\mathsf{N},$ there exist $ k_1, \hdots, k_\mathsf{N}$ in $\llbracket -\mathsf{N}; \mathsf{N} \rrbracket$ s.t. $G^\mathsf{N}\left(C,(k_1, \hdots, k_\mathsf{N},0,\hdots)\right) = \left(C',(0, 0, \hdots ) \right).$
We will prove this lemma by a mathematical induction on $\mathsf{N} \in \mathds{N}^*$.
For the base case $\mathsf{N}=1$, if $C_1 = C_1'$, then the result is satisfied with $k_1=0$. Else, either $C_1' = f_1(C_1)$ then $k_1=1$ holds, or $C_1' = f_1^{-1}(C_1)$ then $k_1=-1$.
Let us now suppose that the statement holds for some $\mathsf{N} \in \mathds{N}^*$. Let $C,C' \in \llbracket 1;4 \rrbracket^{\mathsf{N}+1}$. According to the inductive hypothesis, $\exists k_1, \hdots, k_\mathsf{N} \in \llbracket -\mathsf{N}, \mathsf{N} \rrbracket$ such that $\tilde{G}^{\mathsf{N}}\left( (C_1, \hdots, C_\mathsf{N}), (k_1, \hdots, k_\mathsf{N},0, \hdots) \right) = \left( (C_1', \hdots, C_\mathsf{N}'), (0, 0,\hdots) \right)$, where $\tilde{G}$ is the restriction of $G$ on its $\mathsf{N}-$th firsts variables. Let $x = \mathcal{C}\left(G^\mathsf{N}\left( (C_1, \hdots, C_\mathsf{N+1}), (k_1, \hdots, k_\mathsf{N+1},0, \hdots) \right)\right)_{\mathsf{N+1}}$. If $x=C_{\mathsf{N}+1}'$, then $k_{\mathsf{N}+1}=0$ holds. Else, either $f(x)=C_{\mathsf{N}+1}'$, and $k_{\mathsf{N}+1}=\mathsf{N}+1$ holds,
or $f^{-1}(x)=C_{\mathsf{N}+1}'$, and then $k_{\mathsf{N}+1}=-(\mathsf{N}+1$).
We can now prove that,
Protein folding is regular.
Let $X=(C,F) \in \mathcal{X}$ and $\varepsilon > 0$.
Define $k_0=-\lfloor log_{10} (\varepsilon) \rfloor$
and $\tilde{X}$ such that:
(1) $\mathcal{C}(\tilde{X}) = C$,
(2) $\forall k \leqslant k_0, \mathcal{F}\left(G^k(\tilde{X})\right) = \mathcal{F}\left(G^k(X)\right)$,
(3) $\forall i \in \llbracket 1; \mathsf{N} \rrbracket, \mathcal{F}\left(G^{k_0+i}(\tilde{X})\right) = k_i,$ and
(4) $\forall i \in \mathds{N}, \mathcal{F} \left(G^{k_0+\mathsf{N}+i+1}(\tilde{X})\right) = \mathcal{F}\left(G^i(\tilde{X})\right)$,
where $k_1, \hdots, k_n$ are integers given by lemma <ref> with $C=\mathcal{C}\left( G^{k_0}(X) \right), C'=\mathcal{C}(X)$. Such a $\tilde{X}$ is a periodic point for $G$ into the ball $\mathcal{B}(X,\varepsilon)$.
(1) and (2) are to make $\tilde{X}$ $\varepsilon-$close to $X$, (3) is for mapping $\mathcal{C}\left(G^{k_0}(\tilde{X})\right)$ into $C$ in at most $\mathsf{N}$ folding process.
Lastly, (4) is for the periodicity of the folding process.
§.§ Transitivity of the Folding Operation
Instead of proving the transitivity of $G$, which is required in the definition of chaos, we will establish its strong transitivity:
A dynamical system $\left( \mathcal{X}, f\right)$ is strongly transitive if $\forall x,y \in \mathcal{X},$ $\forall r > 0,$ $\exists z \in \mathcal{X},$ $d(z,x) \leqslant r \Rightarrow$ $\exists n \in \mathds{N}^*,$ $f^n(z)=y$.
Obviously, strong transitivity implies transitivity. Let us now prove that,
Protein folding is strongly transitive.
Let $X_A=(C_A,F_A)$, $X_B=(C_B, F_B)$ and $\varepsilon > 0$.
We will show that $X \in \mathcal{B}\left(X_A, \varepsilon\right)$ and $n \in \mathds{N}$ can be found such that $G^n(X)=X_B$.
Let $k_0 = - \lfloor log_{10} (\varepsilon ) \rfloor$ and $\check{X}=G^{k_0}(C_A,F_A)$, denoted by $\check{X}=(\check{C},\check{F})$.
According to lemma <ref> applied to $\check{C}$ and $C_B$, $\exists k_1, \hdots, k_\mathsf{N} \in \llbracket -\mathsf{N}, \mathsf{N} \rrbracket$ such that
$$G^\mathsf{N}\left( \check{C}, (k_1, \hdots, k_\mathsf{N},0,\hdots)\right) = \left(C_B, (0, \hdots )\right).$$
Let us define $X=(C,F)$ in the following way:
(1) $C=C_A$,
(2) $\forall k \leqslant k_0, F^k=F_A^k$,
(3) $\forall i \in \llbracket 1; \mathsf{N} \rrbracket, F^{k_0+i} = k_i$, and
(4) $\forall i \in \mathds{N}, F^{k_0+\mathsf{N}+i+1}=F_B^i$.
This point $X$ is thus an element of $\mathcal{B}(X_A,\varepsilon)$ (due to $1,2$), which is such that $G^{k_0+\mathsf{N}+1}(X) = X_B$ (by using $3,4$). As a consequence, $G$ is strongly transitive.
This property is very important. It shows among other things that being as close as possible of the true folding process, for instance by using a very large basis of knowledge and numerous levels of resolution, is not a guarantee of success. Indeed, for any possible conformation $c$, there is a prediction as good as possible of our considered protein, which leads to $c$.
§.§ Chaotic behavior of the folding process
As $G$ is regular and (strongly) transitive, we have:
The protein folding process $G$ is chaotic according to Devaney.
Consequently this process is highly sensitive to its initial condition.
In particular, even a minute difference on an intermediate conformation of the protein, in forces that act in the folding process, or in the position of an atom, can lead to enormous differences in its final conformation, even over fairly small timescales.
This is the so-called butterfly effect.
In particular, it seems very difficult to predict the 2D structure of a given protein by using the knowledge of the structure of similar proteins.
Let us finally remark that the whole 3D folding process with real torsion angles is obviously more complex than this 2D HP model.
Indeed, if the complete 3D folding process were predictable, then this simplistic version would be predictable too, as one of its particular cases.
As a conclusion, theoretically speaking, the folding process in unpredictable. Before studying some practical aspects of this unpredictability in Section <ref>, we will initiate a second proof of the chaotic behavior of this process.
§ OUTLINES OF A SECOND PROOF
§.§ Motivations
In this section a second proof of the chaotic behavior of the protein folding process is given.
It is proven that the folding dynamic can be modeled as chaotic iterations (CIs). CIs are a tool used in distributed computing and in the computer science security field <cit.> that has been established to be chaotic according to Devaney <cit.>.
This second proof is the occasion to introduce these CIs, which will be used at the end of this paper to study whether a chaotic behavior is really more difficult to learn with a neural network than a “normal” behavior.
§.§ Chaotic Iterations: Basic Recalls
Let us consider a system with a finite number $\mathsf{N} \in
\mathds{N}^*$ of elements (or cells), so that each cell has a
boolean state.
A sequence of length $\mathsf{N}$ of boolean
states of the cells corresponds to a particular state of the
A sequence, which elements are subsets of $\llbracket 1;\mathsf{N}
\rrbracket $, is called a strategy. The set of all strategies is denoted by $\mathbb{S}$ and the set $\mathds{B}$ is for $\{0,1\}$.
$f:\mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$ be
a function and $S\in \mathbb{S}$ be a strategy. The so-called
chaotic iterations (CIs) are defined by $x^0\in
\mathds{B}^{\mathsf{N}}$ and $\forall n\in \mathds{N}^{\ast }, \forall i\in
\llbracket1;\mathsf{N}\rrbracket ,$
\begin{array}{ll}
x_i^{n-1} & \text{ if } i \notin S^n \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if } i \in S^n.
\end{array}\right.
In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cells are
Let us remark that the term “chaotic”, in the name of these iterations, has a
priori no link with the mathematical theory of chaos recalled previously.
We will now recall that CIs can be written as a dynamical system, and characterize functions $f$ such that their CIs are chaotic according to Devaney <cit.>.
§.§ CIs and Devaney's chaos
Given a function
$f: \mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$,
define the function $F_{f}:$
$\llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}}\longrightarrow
\mathds{B}^{\mathsf{N}}$
such that
\begin{equation*}
F_{f}(k,E)=\left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta (k,j)}\right)_{j\in \llbracket1;\mathsf{N}\rrbracket},
\end{equation*}
where + and . are the boolean addition and product
operations, $\overline{x}$ is for the negation of $x$.
We have proven in <cit.> that chaotic iterations can be described by the following
dynamical system:
\begin{equation*}
\left\{
\begin{array}{l}
X^{0}\in \tilde{\mathcal{X}} \\
\end{array}
\right.
\end{equation*}
where $\tilde{G}_{f}\left( S,E\right) =\left( \sigma (S),F_{f}(i(S),E)\right)$, and $\tilde{\mathcal{X}}$ is a metric space for an ad hoc distance such that $\tilde{G}$ is continuous on $\mathcal{X}$ <cit.>.
Let now be given a configuration $x$. In what follows the configuration
$N(i,x) = (x_1,\ldots,\overline{x_i},\ldots,x_n)$ is obtained by
switching the $i-$th component of $x$. Intuitively, $x$ and $N(i,x)$
are neighbors. The chaotic iterations of the function $f$ can be
represented by the graph $\Gamma(f)$ defined below.
In the oriented graph of iterations $\Gamma(f)$, vertices are
configurations of $\mathds{B}^\mathsf{N}$, and there is an arc labeled
$i$ from $x$ to $N(i,x)$ iff $F_f(i,x)$ is $N(i,x)$.
We have proven in <cit.> that:
Functions $f :
\mathds{B}^{n} \to \mathds{B}^{n}$ such that $\tilde{G}_f$ is chaotic
according to Devaney, are functions such that the graph $\Gamma(f)$ is
strongly connected.
We will now show that the protein folding process can be modeled as chaotic iterations, and conclude the proof by using the theorem recalled above.
§.§ Protein Folding as Chaotic Iterations
The attempt to use chaotic iterations with a view to model protein folding can be justified as follows.
At each iteration, the same process is applied to the system (i.e., to the conformation), that is the folding operation.
Additionally, it is not a necessity that all of the residues fold at each iteration:
indeed it is possible that, at a given iteration, only some of these residues folds.
Such iterations, where not all the cells of the considered system are to be updated, are exactly the iterations modeled by CIs.
Indeed, the protein folding process with folding sequence $(F^n)_{n \in \mathds{N}}$ consists in the following chaotic iterations: $C^0 = (1,1, \hdots, 1)$ and,
C_{|i|}^{n+1} = \left\{
\begin{array}{ll}
C_{|i|}^n & \textrm{if } i \notin S^n,\\
f^{sign(i)}(C^n)_i & \textrm{else},
\end{array}
\right.
where the chaotic strategy is defined by $\forall n \in \mathds{N}, S^n = \llbracket -\mathsf{N}; \mathsf{N} \rrbracket \setminus \llbracket -F^n; F^n \rrbracket$.
Thus, to prove that the protein folding process is chaotic as defined by Devaney, is equivalent to prove that the graph of iterations of the CIs defined above is strongly connected. This last fact is obvious, as it is always possible to find a folding process that map any conformation $(c_1, \hdots, c_\mathsf{N}) \in \llbracket 1 ;4 \rrbracket^\mathsf{N}$ to any other $(c_1', \hdots, c_\mathsf{N}') \in \llbracket 1 ;4 \rrbracket^\mathsf{N}$ (this is lemma <ref>).
Let us finally remark that it is easy to study processes s.t. more than one fold occur per time unit, by using CIs.
This point will be deepened in a future work.
We will now investigate some consequences of the chaotic behavior of the folding process.
§ CONSEQUENCES
§.§ Is a chaotic behavior incompatible with approximately one thousand folds ?
Claiming that the protein folding process is chaotic seems to be contradictory with the fact that only approximately one thousand folds have been discovered this last decade.
The number of proteins that have an understood 3D structure increase largely year after year.
However the number of new categories of folds seems to be limited by a fixed value approximately equal to one thousand.
Indeed, there is no contradiction as a chaotic behavior does not forbid a certain form of order.
For example, seasons are not forbidden even if weather forecast has a non-intense chaotic behavior.
A same regularity appears in brains: even if hazard and chaos play an important rule in a microscopic scale, a statistical order appears in the complete neural network.
That is, a certain order can emerge from a chaotic behavior, even if it is not a rule of thumb.
More precisely, in our opinion these thousand folds can be related to basins of attractions or strange attractors of the dynamical system, objects that are well described by the mathematical theory of chaos.
Thus, it should be possible to determine all of the folds that can occur, by refining our model and looking for its basins of attractions with topological tools.
However, this assumption still remains to be more largely investigated.
§.§ Is Artificial Intelligence able to Predict Chaotic dynamic ?
§.§.§ Experimental Protocol
We will now wonder whether a chaotic behavior can be learned by a neural network or not.
These considerations have been formerly proposed in <cit.>.
We consider $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{N}^\mathsf{N}$, strategies of singletons ($\forall n \in \mathds{N}, S^n \in \llbracket 1; \mathsf{N} \rrbracket$), and a MLP which recognize $F_{f}$.
That means, for all $(k,x) \in \llbracket 1 ; \mathsf{N} \rrbracket \times
\mathds{B}^\mathsf{N}$, the response of the output layer to the input
$(k,x)$ is $F_{f}(k,x)$.
We thus connect the output layer to the input one as it is depicted in
Figure <ref>, leading to a global recurrent artificial neural network (ANN) working as follows <cit.>.
RNN modeling $F_{f}$
At the initialization stage, the ANN receives a boolean vector
$x^0\in\mathds{B}^\mathsf{N}$ as input state, and $S^0 \in \llbracket
1;\mathsf{N}\rrbracket$ in its input integer channel $i()$. Thus,
$x^1 = F_{f}(S^0, x^0)\in\mathds{B}^\mathsf{N}$ is computed by the
neural network. This state $x^1$ is published as an output. Additionally, $x^1$
is sent back to the input layer, to act as boolean state in the next
Finally, at iteration number $n$, the recurrent neural network receives
the state $x^n\in\mathds{B}^\mathsf{N}$ from its output layer and
$i\left(S^n\right) \in \llbracket 1;\mathsf{N}\rrbracket$ from its
input integer channel $i()$. It can thus calculate $x^{n+1} =
F_{f}(i\left(S^n\right), x^n)\in\mathds{B}^\mathsf{N}$, which will
be the new output of the network.
Obviously, this particular MLP produces exactly the same values than CIs with update function $f$.
That is, such MLPs are equivalent, when working with $i(s)$, to CIs with $f$ as update function <cit.> and strategy $S$.
Let us now introduce the two following functions:
$f_1(x_{1},x_2,x_3)=(\overline{x_{1}},\overline{x_{2}},\overline{x_{3}})$ and
It can easily be checked that these functions satisfy the hypothesis of Theorem <ref>, thus their CIs are chaotic according to Devaney.
Then when the MLP defined above learn to recognize $F_{f_1}$ or $F_{f_2}$, indeed it tries to learn these CIs, that is, a chaotic behavior as defined by Devaney <cit.>.
On the contrary, the function $g(x_{1},x_2,x_3)=(\overline{x_{1}},x_{2},x_{3})$
is such that $\Gamma(g_1)$ is not strongly connected. In this case, due to Theorem <ref>, the MLP does not learn a chaotic process.
We will now study the training process of functions $F_{f_1}$, $F_{f_2}$, and $F_{g}$ <cit.>, that is to say, the ability to learn one iteration of CIs.
§.§.§ Experimental results
For each neural network we have considered MLP architectures with one and two hidden layers, with in the first case different numbers of hidden neurons (sigmoidal activation).
Thus we will have different versions of a neural network modeling the same iteration function <cit.>.
Only the size and number of hidden layers may change, since the numbers of inputs and output neurons (linear activation) are fully specified by the function.
The neural networks are trained using the quasi-Newton L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) algorithm in combination with the Wolfe linear search <cit.>.
The training is performed until the learning error (MSE) is lower than a chosen threshold value ($10^{-2}$).
Results of some iteration functions learning, using different recurrent MLP architectures
4c|One hidden layer
2c||8 neurons 2|c|10 neurons
Function Mean Success Mean Success
epoch rate epoch rate
$f_1$ 82.21 100% 73.44 100%
$f_2$ 76.88 100% 59.84 100%
$g_1$ 36.24 100% 37.04 100%
4c|Two hidden layers: 8 and 4 neurons
2c|Mean epoch number 2|c|Success rate
$f_1$ 2c|203.68 2c|76%
$f_2$ 2c|135.54 2c|96%
$g_1$ 2c|72.56 2c|100%
Table <ref> gives for each considered neural network the mean number of epochs needed to learn one iteration in their ICs, and a success rate which reflects a successful training in less than 1000 epochs.
Both values are computed considering 25 trainings with random weights and biases initialization.
These results highlight several points <cit.>.
Firstly, the two hidden layer structure seems to be quite inadequate to learn chaotic behaviors.
Secondly, training networks so that they behave chaotically seems to be difficult for these simplistic functions only iterated one time, since they need in average more epochs to be correctly trained.
However, the correctness of this point needs to be further investigated.
At this point we can only claim that it is not completely evident that computational intelligence as neural networks are able to predict, with a good accuracy, protein folding.
To reinforce this belief, tools optimized to chaotic behaviors must be found – if such tools exist.
Similarly, there should be a link between the training difficulty and the “quality” of the disorder induced by a chaotic iteration function (their constants of sensitivity, expansivity, etc.), and this link must be found.
§ CONCLUSION
In this paper the topological dynamic of protein folding is evaluated.
More precisely, it is regarded whether this folding process is predictable or not.
It is achieved to determine if it is reasonable to think that computational intelligence as neural networks are able to predict the 3D shape of an amino acids sequence.
It is mathematically proven, by using two different ways, that protein folding in 2D hydrophobic-hydrophilic (HP) square lattice model is chaotic according to Devaney.
Consequences both for structure prediction and biology are then outlined.
In particular, the first comparison of the learning by neural networks of a chaotic behavior on the one hand, and of a more natural dynamic on the other hand, are outlined.
Obtained results tend to show that such chaotic behaviors are more difficult to learn than non-chaotic ones.
It is not our pretension to claim that it is impossible to predict chaotic behaviors as protein folding with computational intelligence.
Our opinion is just that this important point must now be regarded with attention.
In future work the dynamical behavior of the protein folding process will be more deeply studied, by using topological tools as expansivity, topological mixing, Knudsen and Li-Yorke notions of chaos, topological entropy, etc.
The quality and intensity of this chaotic behavior will then be evaluated.
Consequences both on folding prediction and on biology will then be regarded in detail.
Other molecular or genetic dynamics will be investigate by using mathematical topology, and other chaotic behaviors will be looked for (as neurons in the brain).
More specifically, various tools taken from the field of computational intelligence will be studied to determine if some of these tools are capable to predict behaviors that are chaotic.
It is highly possible that prediction depends both on the tool and on the chaos quality.
Moreover, the study presented in this paper will be extended to high resolution 3D models.
Impacts of the chaotic behavior of the protein folding process in biology will be regarded.
Finally, the links between this established chaotic behavior and stochastic models in gene expression, mutation, or in Evolution, will be investigated.
|
1511.00600
|
[email protected] Department of Physics,University of Trieste, Trieste, Italy.
[email protected], [email protected]
Pailan College of Management and Technology, Bengal Pailan Park, Kolkata-700 104, India.
Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan
In this paper, we study the effects which are produced by the interaction between a brane Universe and the bulk in which the Universe is embedded.
Taking into account the effects produced by the interaction between a brane Universe and the bulk, we derived the Equation of State (EoS) parameter $\omega_D$ for
three different models of Dark Energy (DE), i.e. the Holographic DE (HDE) model with infrared (IR) cut-off given by the Granda-Oliveros cut-off, the Modified Holographic Ricci DE (MHRDE) model and a DE model which is function of the Hubble parameter $H$ squared and to higher derivatives of $H$. Moreover, we have considered two different cases of scale factor (namely, the power law and the emergent ones). A nontrivial contribution of the DE is observed to be different from the standard matter fields confined to the brane. Such contribution has a monotonically decreasing behavior upon the evolution of the Universe for the emergent scenario of the scale factor, while monotonically increasing for the power-law form of the scale factor $a(t)$.
§ INTRODUCTION
The evidence that our Universe is experiencing a phase of expansion with accelerated rate has been well demonstrated by cosmological data obtained from different independent observations of SNeIa, Cosmic Microwave Background Radiation (CMBR) anistropies, X-ray experiments and Large Scale Structures (LSS) <cit.>. Three main ideas have been suggested to give a reasonable explanation to the present day observed accelerated expansion of our Universe:
the Cosmological Constant $\Lambda$ model, dark energy (DE) models and theories of modified gravity models. Thorough discussions on these three ideas are available in the reviews of <cit.>. The Cosmological Constant $\Lambda$, which has EoS parameter $\omega = p/\rho = -1$, represents the earliest and the simplest theoretical candidate suggested in order to give a plausible explaination to the observational evidences of the Universe's present day accelerated expansion. It is well-known, anyway, that there are two main problems associated with $\Lambda$: the fine-tuning and the cosmic coincidence problems. The former mainly asks why the vacuum energy density is so small (about an order of $10^{123}$ lower than what we can observe) while the latter asks why the vacuum energy and DM give a nearly equal contribution at the present epoch even if they had an evolution which is independent and they had also evolved from mass scales which are different (this fact represents a really strange coincidence if some internal connections between them are not taken into account). Till now, many attempts have been done in order to find a possible plausible explanation to the coincidence problem (see <cit.>).
The second idea suggested in order to possibly explain the observed accelerated expansion of the Universe involve DE models (reviewed in <cit.>). In relativistic cosmology, the cosmic acceleration we are able to observe can be described by the mean of a perfect fluid which pressure and energy density, indicated with $p$ and $\rho$, satisfy the relation given by $\rho + 3p < 0$. This kind of fluid is dubbed as Dark Energy (DE). The relation $\rho + 3p < 0$ also tells us that the EoS parameter of the fluid $\omega$ must be in agreement with the condition $\omega <-1/3$, while, from an observational point of view, it is a difficult work to constrain its exact value. The most direct evidence we have for the detection of DE is obtained from observations of supernovae of a type Ia (SNeIa) whose intrinsic luminosities can be safely considered practically uniform <cit.>.
If we assume that the DE idea is the right one in order to explain the present expansion of the Universe with accelerated rate, we must have that the largest amount of the total cosmic energy density $\rho_{tot}$ must be concentrated in the two Dark sectors, i.e. Dark Energy (DE) and Dark Matter (DM) which represent, according to recent cosmological observations, about the 70$\%$ and about the 25$\%$ of the total energy density $\rho_{tot}$ of the present day Universe <cit.>. Moreover, the ordinary baryonic matter we are able to observe with our scientific instruments contributes for only the 5$\%$ of $\rho_{tot}$. Furthermore, the radiation density gives a contribution to the total cosmic energy density which we can safely consider negligible. Many different models have been well studied in recent times to understand the exact nature of DE. Some of these models include tachyon, quintessence, k-essence, quintom, Chaplygin gas, Agegraphic DE (ADE), NADE and phantom. The various candidates of DE have been reviewed in <cit.>.
A model of DE, motivated by the holographic principle, was proposed by Li <cit.> and it has been further studied in the references like <cit.>. The energy density of HDE $\rho_D$ as follows:
\begin{eqnarray}
\rho_D = 3c^2 M_p^2 L^{-2}, \label{2}
\end{eqnarray}
with $c^2$ indicating a dimensionless constant parameter which which value $c$ is evinced by observational data: for a flat Universe (i.e. for $k=0$) it is obtained that $c=0.818_{-0.097}^{+0.113}$ and in the case of a
non-flat Universe (i.e. for $k=1$ or $k=-1$) it is obtained thar $c=0.815_{-0.139}^{+0.179}$ <cit.>. Chen et al. <cit.> used the HDE model in order to drive inflation in the early evolutionary phases of the Universe. Jamil et al. <cit.> studied the EoS parameter $\omega_D$ of the HDE model considering not a constant but a time-dependent Newton's gravitational constant, i.e. $G \equiv G\left( t \right)$; furthermore, they obtained that $\omega_D$ can be significantly modified in the low redshift limit.
Recently, the cosmic acceleration has been also well studied by imposing the concept of modification of gravity <cit.>. This new model of gravity (predicted by string/M theory) gives a very natural gravitational alternative to the idea of the presence of exotic components. The explanation of the phantom, non-phantom and quintom phases of the Universe can be well described using modified gravity theories without the necessity of the introduction of a negative kinetic term in DE models. The relevance of modified gravity models for the late acceleration of the Universe has been recently studied by many Authors. Some of the most famous and known models of modified gravity are represented by braneworld models, $f\left(T\right)$ modified gravity (where $T$ indicates the torsion scalar), $f \left(R\right)$ modified gravity (where $R$ indicates the Ricci scalar curvature), $f \left(G\right)$ modified gravity (where $G$ indicates the Gauss-Bonnet invariant which is defined as $G=R^2-4R_{\mu \nu}R^{\mu \nu} + R_{\mu \nu \lambda \sigma}R^{\mu \nu \lambda \sigma}$, with $R$ representing the Ricci scalar curvature, $R_{\mu \nu}$ representing the Ricci curvature tensor and $R_{\mu \nu \lambda \sigma}$ representing the Riemann curvature tensor), $f \left(R,T\right)$ modified gravity, $f \left(R,G\right)$ modified gravity, DGP model, DBI models, Horava-Lifshitz gravity and Brans-Dicke gravity. Modified theories of gravity have been reviewed in <cit.>.
Recently, the idea that our Universe is a brane which is embedded in
a higher-dimensional space obtained a lot of attention from scientific community <cit.>. The Friedmann equation on the brane
has some corrections with respect to the usual four-dimensional equation
<cit.>. Binetruy et al. <cit.> found a term $H\propto \rho$,
which is problematic from an observational point of view. The model is consistent if the tension on the brane and a
cosmological constant in the bulk are
considered. This leads to a cosmological version of the
Randall-Sundrum (RS) scenario of warped geometries <cit.>. Bruck et al. <cit.> considered an interaction between the bulk and the
brane, which can be considered as another non-trivial aspect of braneworld
theories. The main aim of this paper is to disclose the
effects produced by the energy exchange between the brane and the bulk on
the evolutionary history of the Universe by taking into account the flow of energy
onto (or away) from the brane. In this paper, we will focus our attention to three particular DE models, i.e the HDE model with IR cut-off given by the recently proposed Granda-Oliveros cut-off, the Modified Holographic DE (MHRDE) model and a DE model which is proportional to the Hubble parameter $H$ squared and to higher time derivatives of $H$. Moreover, we will consider two different scale factors, i.e. the power law and the emergent ones, in order to study the cosmological properties of the DE models in the Bulk-Brane interaction. Both the DE models and the scale factors considered will be described in details in the following Sections.
This study is motivated by the works of <cit.>. In an interaction between the bulk and the brane,
Setare <cit.> considered the holographic model of DE in
non-flat Universe under the assumption that the CDM energy density
on the brane is conserved while the HDE energy density
on the brane is not conserved because of to brane-bulk energy exchange.
Sheykhi <cit.> considered the agegraphic models of DE in the framework of a braneworld scenario with brane-bulk energy exchange
under the assumption that the adiabatic equation for the DM is satisfied but it is violated for the Agegraphic DE (ADE) model because of the energy exchange between the brane and the bulk.
In the paper of Sheykhi <cit.>, it was obtained that the
EoS parameter can evolve from the quintessence regime
to the phantom regime. Myung $\&$ Kim <cit.> introduced the brane-bulk interaction in order to discuss a limitation of the cosmological Cardy-Verlinde formula which is useful for the holographic description of brane cosmology. They also showed that if there is presence of the brane-bulk interaction, it is not possible to derive the entropy representation of the
first Friedmann equation.
Saridakis <cit.> studied a generalized version of the HDE model arguing that it must be taked into account in the maximally subspace of a cosmological model; moreover he showed that, in the framework of brane cosmology, it leads to a bulk HDE which transfers its holographic nature to the effective $4D$ DE.
Furthermore, Saridakis <cit.> applied the bulk HDE in general $5D$ two-brane models and he also extracted the Friedmann equation on the physical brane, showing that in the general moving-brane case the effective $4D$ HDE has a quintom-like behavior for a large parameter-space area of a simple solution subclass.
In this paper, we consider an interaction between the bulk and the brane, which represents a non-trivial aspect of
the braneworld theories. We also discuss the flow of energy onto or
away from the brane-Universe. We then apply this idea to a braneworld cosmology under the assumption that the DE energy density on
the brane is conserved, but the DE energy density on the brane is not conserved because of the brane-bulk energy exchange.
The plan of the paper is the following. In Section 2, we describe the main features of bulk-brane interaction. In Section 3, we describe the main features of the DE models considered in this paper; moreover, we derive the expression of the EoS parameter $\omega_D$ and the evolutionary form of the parameter $u$ (defined as $\frac{\chi}{\rho_m + \rho_D}$) for the DE models we are considering. In Section 4, we consider two different models of scale factors, (in particular, the power law and the emergent ones) in order to study the behavior of the expression of $\dot{u}$ derived in the previous Section. Finally, in Section 5, we write the Conclusion of this work.
§ BULK-BRANE ENERGY EXCHANGE
In this Section, we want to describe the main features of the bulk-brane interaction, introducing the main quantities useful for the following part of the work.
The bulk-brane action $S$ is given by the following expression <cit.>:
\begin{equation}\label{action}
S=\int d^5x\sqrt{-G}\left(\frac{R_5}{2\kappa_{5}^{2}}-\Lambda_5+L_{B}^{m}\right)+\int d^4x\sqrt{-g}(-\sigma+L_{b}^{m}),
\end{equation}
where $R_5$ represents the 5D curvature scalar, $\Lambda_5$ denotes the bulk cosmological constant, $\kappa_5$ stands for the 5D coupling constant, $\sigma$
indicates the brane tension, $G$ and $g$ denote the determinant of the 5D and of the 4D metric tensors, respectively while $L_{B}^{m}$ and $L_{b}^{m}$ are the matter Lagrangian in the bulk and the matter Lagrangian in the brane.
We here consider the cosmological solution with a metric given by
\begin{equation}\label{metric}
\end{equation}
where $\gamma_{ij}$ represents the metric for the maximally symmetric three-dimensional space.
The non-zero components of Einstein tensor are given by <cit.>:
\begin{eqnarray}\label{g00}
\left(\frac{a'}{a}-\frac{b'}{b}\right)\right]+\frac{kn^2}{b^2}\right\}~,\\
\label{gij}
\frac{b'}{b}\left(\frac{n'}{n}+\frac{2a'}{a}\right)+\frac{2a''}{a}+\frac{n''}{n}\right]+ \nonumber \\
-\frac{\ddot{b}}{b}\right]-k\gamma_{ij}~, \\
\label{g05}
\label{g55}
\left[\frac{\dot{a}}{a}\left(\frac{\dot{a}}{a}-\frac{\dot{n}}{n}\right)+\frac{\ddot{a}}{a}\right]
\end{eqnarray}
where $k$ denotes the curvature parameter of space which possible values are $k= 0, 1, -1$ which correspond, respectively, to a flat, a closed and an open Universe. Moreover, the primes and the dots indicate, respectively, a derivative with respect to the variable $y$ and a derivative with respect to the variable $t$.
The 4D braneworld Universe is assumed to be at $y=0$. The
Einstein equations are given by:
\begin{eqnarray}
\end{eqnarray}
where we have that the stress-energy momentum tensor $T_{\mu\nu}$ has both bulk and brane components and it can be also written as follows <cit.>:
\begin{equation}\label{stress}
\end{equation}
\begin{eqnarray}\label{sigmab}
\label{lambdab}
\label{mb}
\end{eqnarray}
where $p$ and $\rho$ represent, respectively, the total pressure and the total density on the brane.
By integrating Eqs. (<ref>) and (<ref>) with respect to the variable $y$ around the point $y = 0$ and assuming the $Z_2$ symmetry around the brane, we derive the following jump conditions:
\begin{eqnarray}\label{jump1}
\label{jump2}
\end{eqnarray}
The 2 subscripts + and - indicate, respectively, the sides corresponding to $y > 0$ and $y < 0$, which represent the two sides of the brane embedded in the bulk. Moreover, the subscript 0 indicates quantities which are evaluated at $y = 0$.
Starting from the results of Eqs. (<ref>) and (<ref>), we can obtain the following expressions:
\begin{eqnarray}
\frac{n_0'\dot{a}_0}{n_0a_0}+ \frac{a_0'\dot{b}_0}{a_0b_0}-\frac{\dot{a}_0'}{a_0} &=& \frac{\kappa _5^2}{3}T_{05},\\
\left[\frac{\dot{a}_0}{a_0}\left(\frac{\dot{a}_0}{a_0}-\frac{\dot{n}_0}{n_0}\right)+\frac{\ddot{a}_0}{a_0}\right]
-k\frac{b_0^2}{a_0^2}\right\} &=& -\kappa _5^2\Lambda_5 b_0^2 +\kappa_5^2T_{55},
\end{eqnarray}
where the terms $T_{05}$ and $T_{55}$ represent, respectively, the $05$ and $55$ components of $T_{\mu\nu}|_{m,b}$ when evaluated on the brane.
Moreover, using Eqs. (<ref>) and (<ref>), we obtain:
\begin{eqnarray}\label{field1}
\dot{\rho}+3\frac{\dot{a}_{0}}{a_{0}}(\rho+p)&=&-\frac{2n_{0}^{2}}{b_{0}}T^{0}_{5},\\
\label{field2}
\frac{1}{n_{0}^{2}}\left[\frac{\ddot{a}_{0}}{a_{0}}+\left(\frac{\dot{a}_{0}}{a_{0}}\right)^{2}
-\frac{\dot{a}_{0}\dot{n}_{0}}{a_{0}n_{0}}\right]&+&\frac{k}{a_{0}^{2}} \nonumber \\
&=&\frac{\kappa_{5}^{2}}{3}\left(\Lambda_{5}+\frac{\kappa_{5}^{2}\sigma^{2}}{6}\right) \nonumber \\
\end{eqnarray}
Considering an appropriate gauge with the coordinate frame $n_0 = b_0 = 1$, Eqs. (<ref>) and (<ref>) can be also expressed in the following equivalent forms:
\begin{eqnarray}
\dot{\rho} + 3H\left( 1+\omega \right)\rho &=& -2 T_5^0, \\
\left( \frac{\dot{a}}{a} \right)^2 &=& \Lambda -\frac{\kappa}{a^2} + \beta \rho^2 + 2\gamma \left(\rho + \chi \right), \\
\dot{\chi}+4H\chi &=&
2\left( \frac{\rho}{\sigma} +1 \right)T_5^0 - \frac{12}{\kappa_5^2}\frac{H}{\sigma}T_5^5,
\end{eqnarray}
where $\beta = \frac{\kappa_5^4}{36}$ and $\gamma = \frac{\sigma \kappa_5^4}{36}$. The effective 4D cosmological constant
$\Lambda$ on the brane, the bulk cosmological constant $\Lambda_5$, and the brane tension
$\sigma$ are well known to be constrained by the fine-tuning relation <cit.>:
\begin{eqnarray}
\Lambda=\frac{\kappa_5^{2}}{2}\left(\frac{1}{6}\kappa_5^{2}\sigma^{2}+\Lambda_{5}\right).
\end{eqnarray}
If we assume that the bulk matter (relative to bulk vacuum energy) is much less than the ratio of the brane matter to the brane vacuum energy, we can neglect the $T^{5}_{5}$ term: this can lead to the derivation of a solution that is largely independent of the bulk dynamics. If we take into account this approximation and we concentrate on the low-energy region with $\rho/\sigma\ll 1$, Eqs. (<ref>) and (<ref>) can be simplified, leading to the following system of equations:
\begin{eqnarray}
\label{field11} \dot{\rho}+3H(1+\omega)\rho &=&-2T^{0}_{5}=T, \\
\label{field22} H^{2}&=&\frac{8 \pi G_{4}}{3}(\rho+\chi)-\frac{k}{a^{2}}+\Lambda, \\
\label{field33} \dot{\chi}&+&4H\chi\approx 2T^{0}_{5}=-T.
\end{eqnarray}
The auxiliary field $\chi$ (which appear in Eqs. (<ref>) and (<ref>)) incorporates non-trivial contributions of DE which differ from the standard matter fields confined to the brane. Hence, with the energy exchange $T$ between the bulk and brane, the usual energy conservation is violated.
We shall denote the energy density of DE by $\rho_D$. Since we will consider two dark components in the Universe, namely, DM and DE, we will have $\rho=\rho_D+\rho_{m}$.
In the following Section, three different DE models are concerned, namely, the HDE model with Granda-Oliveros cut-off, the MHRDE model and the DE model proportional to the Hubble parameter $H$ squared and to higher time derivatives of $H$ in the framework of bulk brane interaction. It is accomplished by using some of the concepts introduced in this Section and two choices of the scale factor, namely the power law and the emergent ones.
§ MHRDE AND GO DE MODEL IN THE BULK-BRANE INTERACTION
We now want to give a description of the DE models considered in this work and to find some relevant cosmological quantities. We will also introduce some relevant equations which will be useful for the understanding of the work.
The bulk-brane interaction has
been studied for various aspects, where in particular the effective DE of the braneworld Universe is dynamical, as a result of the non-minimal coupling, which gives a mechanism for bulk-brane interaction through gravity
<cit.>. We assume here that the
adiabatic equation for the DM is satisfied, while it results to be
violated for DE due to the energy exchange between
the brane and the bulk <cit.>. Then, we obtain the following continuity equations:
\begin{eqnarray}\label{matter}
\dot{\rho}_{m}&+&3H\rho_{m}=0,\\
\label{energy}
\dot{\rho}_D&+&3H(1+\omega_D)\rho_D=T.
\end{eqnarray}
We define the fractional energy densities for DM, DE and $\chi$, respectively, as follows:
\begin{eqnarray}
\Omega_m &=& \frac{\rho_m}{\rho_{cr}}, \label{fracden1}\\
\Omega_D &=& \frac{\rho_D}{\rho_{cr}},\label{fracden2}\\
\Omega_{\chi} &=& \frac{\chi}{\rho_{cr}},\label{fracden3} \\
\Omega_k &=& \frac{k}{a^2H^2}.\label{fracden4}
\end{eqnarray}
The Planck data provide the values $\Omega_m \approx 0.3089$ and $\Omega_D \approx 0.6911$ at $68\%$ CL <cit.>.
The critical energy density $\rho_{cr}$ (i.e. the energy density required for flatness) is defined as follows:
\begin{eqnarray}
\rho_{cr} = \frac{3H^2}{8\pi G_4},
\end{eqnarray}
or, assuming units of $8\pi G_4=1$, as:
\begin{eqnarray}
\rho_{cr} = 3H^2. \label{rhocr}
\end{eqnarray}
Using the definition of $\rho_{cr}$ given in Eq. (<ref>), we can write the fractional energy densities given in Eqs. (<ref>), (<ref>) and (<ref>), respectively, as follows:
\begin{eqnarray}\label{Fractional}
\Omega_D&=&\frac{\rho_D}{3H^2}, \label{Fractional1}\\
\Omega_{m}&=&\frac{\rho_{m}}{3H^2}, \label{Fractional2}\\
\Omega_{\chi}&=&\frac{\chi}{3H^2}.\label{Fractional3}
\end{eqnarray}
The interaction between bulk and brane is given by the relation
$T=\Gamma \rho_D$, where the parameter $\Gamma$ represents the rate of
The Wilkinson Microwave
Anisotropy Probe (WMAP) satellite is well known to have measured the curvature parameter $\Omega_k$ in Eq. (<ref>), and, along with Baryon Acoustic Oscillation (BAO) and Hubble parameter measurement, it constrained the fractional energy density of the curvature parameter $k$ as $-0.0133 < \Omega_k < -0.0084$, in 95% CL <cit.>. Eq. (<ref>) for $\Omega_k$ is hence equal to zero in this context.
Considering the parameter $u=\frac{\chi}{\rho_D+\rho_{m}}$,
the above equations lead to <cit.>:
\begin{equation}\label{udot}
\dot{u}=\left(\frac{3Hu\Omega_D}{\Omega_D+\Omega_{m}}\right)\left[\omega_D-
\frac{1}{3}\left(\frac{\Omega_{m}}{\Omega_D}+1\right)-\frac{1+u}{u}\frac{\Gamma}{3H}\right].
\end{equation}
In this paper, we decided to consider the particular case corresponding to $\Lambda=0$. Furthermore,
following <cit.>, we have chosen the following expression for $\Gamma$:
\begin{equation}\label{gamma}
\Gamma=3b^2 (1+u)H,
\end{equation}
where $b^2$ represents a coupling parameter between DM and DE, also known as transfer strength <cit.>.
From the observational data of the Gold SNeIa samples, CMBR data obtained from the WMAP and Planck satellites and the Baryonic Acoustic Oscillations (BAO) obtained thanks to the Sloan Digital Sky Survey (SDSS), the coupling parameter between DM and DE is estimated to assume a small positive value, satisfying the requirement for solving the cosmic coincidence problem and the constraints given by the second law of thermodynamics <cit.>.
Cosmological observations of the CMBR anisotropies and of clusters of galaxies indicate that $b^2 < 0.025$ <cit.>. This evidence is in agreement with the fact that $b^2$ must be taken in the range of values [0,1] <cit.>, with $b^2 = 0$ representing the non-interacting FLRW model.
Using the definitions of the fractional energy densities given in Eqs. (<ref>), (<ref>) and (<ref>), we can rewrite the first Friedmann equation defined in Eq. (<ref>) as follows:
\begin{eqnarray}
\Omega_m + \Omega_D + \Omega_{\chi}= 1.\label{allfrac}
\end{eqnarray}
Eq. (<ref>) has the main property of relating all the fractional energy densities considered in this work.
Moreover, using Eqs. (<ref>), (<ref>) and (<ref>) along with the definition of $u$ and the relation $\rho_m + \rho_D = \Omega_m + \Omega_D$, we can easily obtain the following relation between the parameter $u$ and the fractional energy densities:
\begin{equation}
u=\frac{1-\Omega_D-\Omega_{m}}{\Omega_D+\Omega_m}. \label{ufra}
\end{equation}
We now want to introduce three different energy density models for DE, i.e. the HDE with Granda-Oliveros cut-off, the Modified Holographic Ricci DE (MHRDE) model and the DE model proportional to $H^2$ and to higher time derivatives of $H$. Before proceeding with calculations, we briefly describe these three models.
Recently, Granda $\&$ Oliveros introduced a new IR cut-off based on purely dimensional ground which includes a term proportional to $\dot{H}$ and one term proportional to $H^2$. This new IR cut-off is known as Granda-Oliveros (GO) scale, indicated with the symbol $L_{GO}$ and it is given by <cit.>:
\begin{equation}
L_{GO}=\left( \alpha H^{2}+\beta \dot{H}\right) ^{-1/2}, \label{lgo5}
\end{equation}
where $\alpha $ and $\beta $ represent two constant parameters. In the limiting case corresponding to $ \alpha = 2$ and $\beta = 1$, the GO scale $L_{GO}$ becomes proportional to the average radius of the Ricci scalar curvature (i.e., $L_{GO} \propto R^{-1/2}$) in the case the curvature parameter $k$ assume the value of zero (i.e. $k=0$), corresponding to a flat Universe.
Recently, Wang $\&$ Xu <cit.> have constrained the new HDE model in non-flat Universe using observational data. The best fit values of the two parameters $\left(\alpha, \beta \right)$ with their confidence levels they found are given by $\alpha = 0.8824^{+0.2180}_{-0.1163}(1\sigma)\,^{+0.2213}_{-0.1378}(2\sigma)$ and $\beta = 0.5016^{+0.0973}_{-0.0871}(1\sigma)\,^{+0.1247}_{-0.1102}(2\sigma)$ for non flat Universe, while for flat Universe they found that are $\alpha = 0.8502^{+0.0984}_{-0.0875}(1\sigma)\,^{+0.1299}_{-0.1064}(2\sigma)$ and $\beta = 0.4817^{+0.0842}_{-0.0773}(1\sigma)\,^{+0.1176}_{-0.0955}(2\sigma)$.
We decided to consider the GO scale $L_{GO}$ as infrared cut-off for some specific reasons. If the IR cut-off chosen is given by the particle horizon, the HDE model cannot produce
an expansion of the Universe with accelerated rate. If we consider as cut-off of the system the future event horizon, the HDE model has a causality problem.
The DE models which consider the GO scale $L_{GO}$ depend only on local quantities, then it is possible to avoid the causality problem, moreover it is also possible to obtain the accelerated phase of the Universe.
Granda $\&$ Oliveros considered that, since the origin of the HDE model is still not known exactly up to now, the consideration of the term with the time derivative of the Hubble parameter in the expression of the energy density of DE may be expected since this term appears in the curvature scalar and it has the right dimension.
The expression of the HDE energy density with $L_{GO}$ cut-off is given by:
\begin{equation}
\rho_{D_{GO}}= 3c^2\left( \alpha H^{2}+\beta \dot{H}\right). \label{lgo5-1}
\end{equation}
We must underline here that we are considering the Planck mass $M_p$ equal to one.
Contrary to the HDE model based on the event horizon, the DE models which consider the GO scale depend only on local quantities, then it is possible to avoid in this way the causality problem.
The second model we consider in this paper is the Modified Holographic Ricci DE (MHRDE) model, which is given by the following expression:
\begin{equation}
\rho_{D_{MHRDE}}= \frac{2}{\alpha - \beta} \left(\dot{H} + \frac{3\alpha}{2}H^2 \right), \label{mhrde}
\end{equation}
where $\alpha$ and $\beta$ are the model parameters. Hereupon, we shall denote by $A_{\tiny\circ}$ any quantity $A_{MHRDE}$ related to the MHRDE model. This DE model was studied for the non-interacting case in reference <cit.>, and Chimento et al. have analyzed this this type of DE in interaction with DM as
Chaplygin gas <cit.>.
In the limiting case corresponding to $\alpha = 4/3$ and $\beta =1$, the DE energy density model given in Eq. (<ref>) leads to the DE energy density with Ricci scalar curvature for a spatially
at FLRW space-time as IR cut-off.
The use of the MHRDE is motivated by the holographic principle since we can relate the DE with an UV cut-off for the vacuum energy with an IR scale such as the one
given by the Ricci scalar curvature $R$. Moreover, it is possible to proceed in a different way taking into account that the Ricci scalar curvature $R$ is a new kind of DE, for example, a geometric DE instead of evoking the holographic principle. Irrespective of the origin
of the DE component, it modifies the Friedmann equation leading to a second order differential equation for the scale factor.
In this work, we decided to consider also a DE energy density model which was recently proposed by Chen $\&$ Jing <cit.>. This new model is function of the Hubble parameter squared $H^2$ and of the first and second derivatives with respect to the cosmic time $t$ of the Hubble parameter $H$ and it is given by the following expression:
\begin{equation}
\rho_{D,higher}= 3c^2\left( \alpha H^{2}+\beta \dot{H} + \varrho \frac{\ddot{H}}{H}\right), \label{lgo5-1higher}
\end{equation}
where $\alpha$, $\beta$ and $\gamma$ represent three arbitrary dimensionless parameters. The inverse of the Hubble parameter, i.e. $H^{-1}$, is introduced in the first term of Eq. (<ref>) so that the dimensions of each of the three terms are the same.
The behavior and the main cosmological features of the DE energy density model defined in Eq. (<ref>) strongly depend on the three parameters of the model, i.e. $\alpha$, $\beta$ and $\gamma$.
Eq. (<ref>) can be considered as a generalization of two previously proposed energy density models of DE. In fact, in the limiting case corresponding to $\alpha = 0$, we recover the energy density of DE in the case the IR cut-off of the system is given by the Granda-Oliveros cut-off. Moreover, in the limiting case corresponding to $\alpha=0$, $\beta=1$ and $\gamma=2$, we obtain the expression of the energy density of DE with IR cut-off proportional to the average radius of the Ricci scalar (i.e., $L \propto R^{-1/2}$) in the case of curvature parameter $k$ assumes the value of zero ($k=0$).
Using the expressions of the energy densities of DE given in Eqs. (<ref>), (<ref>) and (<ref>) in Eq. (<ref>), we obtain the following expressions for $\Omega_{D_{GO}}$, $\Omega_{D_{\tiny\circ}}$ and $\Omega_{D,higher}$:
\begin{eqnarray}
\Omega_{D_{GO}}&=&\frac{\rho_{D_{GO}}}{3H^2}, \label{Fractional1GO}\\
\Omega_{D_{\tiny\circ}}&=&\frac{\rho_{D_{\tiny\circ}}}{3H^2}, \label{Fractional1MH}\\
\Omega_{D,higher}&=&\frac{\rho_{D,higher}}{3H^2}. \label{Fractional1higher}
\end{eqnarray}
The final expression of $\rho_m$ can be derived by first solving the continuity equation for $\rho_m$ given in Eq. (<ref>), yielding:
\begin{eqnarray}
\rho_m &=& \rho_{m0}a^{-3}, \label{rhoemme}
\end{eqnarray}
where $\rho_{m0}$ indicates the present day of the energy density of DM.
Using the expression of $\rho_{m}$ given in Eq. (<ref>), we can write the fractional energy density of DM as follows:
\begin{eqnarray}
\Omega_m = \frac{\rho_{m0}a^{-3}}{3H^2}.
\end{eqnarray}
We now want to find the final expressions of the EoS parameter $\omega_D$ and of $\dot{u}$ for all the DE models considered in this work.
Differentiating Eq. (<ref>) with respect to the cosmic time $t$ and using Eqs. (<ref>) - (<ref>), we obtain (considering units of $8\pi G_4=1$) the following expression of the time derivative of the Hubble parameter for flat Universe:
\begin{equation}\label{Hdot}
\dot{H}=-\frac{1}{6}\left[3\rho_D(1+\omega_D)+3\rho_m+4\chi\right].
\end{equation}
Moreover, using Eqs. (<ref>) and (<ref>) in Eqs. (<ref>), (<ref>) and (<ref>), we obtain the following expressions for the EoS parameters of the DE models we are dealing with:
\begin{eqnarray}
\omega_{D_{GO}} &=&\frac{2}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\frac{\chi}{\rho_{D_{GO}}} - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) +\left( \frac{2\alpha - 3\beta}{3\beta}
\right)\frac{\rho_m}{\rho_{D_{GO}}}, \label{eosgo} \\
\omega_{D_{\tiny\circ}} &=& \left( \alpha -1 \right)\left(\frac{\rho_m}{\rho_{D_{\tiny\circ}}} \right)+\left(\alpha - \frac{4}{3} \right)\frac{\chi}{\rho_{D_{\tiny\circ}}} +\beta -1, \label{eosmh} \\
\omega_{D,higher} &=&\frac{2}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\frac{\chi}{\rho_{D,higher}} - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) +\left( \frac{2\alpha - 3\beta}{3\beta}
\right)\frac{\rho_m}{\rho_{D,higher}} \nonumber \\
&&+ \frac{2\varrho}{\beta \rho_{D,higher}}\frac{\ddot{H}}{H} . \label{eoshigher}
\end{eqnarray}
Using the relation between $u$ and $\chi$ given by $u=\frac{\chi}{\rho_m + \rho_D}$, we can find the following expression for $\frac{\chi}{\rho_D}$:
\begin{eqnarray}
\frac{\chi}{\rho_D} = \frac{u\left( \rho_m + \rho_D \right)}{\rho_D} = u\left( 1 + \frac{\rho_m}{\rho_D} \right). \label{u}
\end{eqnarray}
Then, inserting Eq. (<ref>) in the expressions of the EoS parameters obtained in Eqs. (<ref>), (<ref>) and (<ref>), along with the relation $\frac{\rho_m}{\rho_D}=\frac{\Omega_m}{\Omega_D}$, we can rewrite Eqs. (<ref>), (<ref>) and (<ref>) as follows:
\begin{eqnarray}
\omega_{D_{GO}} &=&\frac{2u_{GO}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\left( 1 + \frac{\Omega_m}{\Omega_{D_{GO}}} \right) - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) +\left( \frac{2\alpha - 3\beta}{3\beta} \right)\frac{\Omega_m}{\Omega_{D_{GO}}}, \label{eosgo1}\\
\omega_{D_{\tiny\circ}} &=& \left( \alpha -1 \right)\left(\frac{\Omega_m}{\Omega_{D_{\tiny\circ}}} \right)+\left(\alpha - \frac{4}{3} \right)u\left( 1 + \frac{\Omega_m}{\Omega_{D_{\tiny\circ}}} \right) +\beta -1,\label{eosmh1}\\
\omega_{D,higher} &=&\frac{2u_{higher}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\left( 1 + \frac{\Omega_m}{\Omega_{D,higher}} \right) - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) \nonumber\\
&&+\left( \frac{2\alpha - 3\beta}{3\beta} \right)\frac{\Omega_m}{\Omega_{D,higher}}+ \frac{2\varrho}{3\beta \Omega_{D,higher}}\frac{\ddot{H}}{H^3} . \label{eoshigher1}
\end{eqnarray}
We must underline that in Eq. (<ref>) we used the main definition of $\Omega_{D,higher}$ given in Eq. (<ref>).
Moreover, using the relation $\Omega_D+\Omega_{m}=\left(1+u\right)^{-1}$ (which can be obtained from Eq. (<ref>)) in Eqs. (<ref>), (<ref>) and (<ref>), we can write:
\begin{eqnarray}
\omega_{D_{GO}} &=&\frac{2u_{GO}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\left[ \frac{1}{\left(1+u_{GO}\right)\Omega_{D_{GO}}} \right] - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) \nonumber\\
&& + \left( \frac{2\alpha -3\beta}{3\beta} \right)\left[ \frac{1}{\left(1+u_{GO}\right)\Omega_{D_{GO}}} -1 \right]\nonumber \\
&=& \left[\frac{2u_{GO}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2}{3}\frac{\alpha}{\beta}-1 \right]\left[ \frac{1}{\left(1+u_{GO}\right)\Omega_{D_{GO}}} \right] - \frac{2}{3c^2\beta}, \label{eosgo2}\\
\omega_{D_{\tiny\circ}} &=& \left( \alpha -1 \right)\left[\frac{1}{\left(1+u_{\tiny\circ}\right)\Omega_{D_{\tiny\circ}}-1} \right]+\left(\alpha - \frac{4}{3} \right)u\left( 1 + \frac{\Omega_m}{\Omega_{D_{\tiny\circ}}} \right) +\beta -1,\label{eosmh2}\\
\omega_{D,higher} &=&\frac{2u_{higher}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right)\left[ \frac{1}{\left(1+u_{higher}\right)\Omega_{D,higher}} \right] - \left( 1-\frac{2}{3}\frac{\alpha}{\beta} + \frac{2}{3c^2\beta} \right) \nonumber\\
&& + \left( \frac{2\alpha -3\beta}{3\beta} \right)\left[ \frac{1}{\left(1+u_{higher}\right)\Omega_{D,higher}} -1 \right] + \frac{2\varrho}{3\beta \Omega_{D,higher}}\frac{\ddot{H}}{H^3}\nonumber \\
&=& \left[\frac{2u_{higher}}{3}\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2}{3}\frac{\alpha}{\beta}-1 \right]\left[ \frac{1}{\left(1+u_{higher}\right)\Omega_{D,higher}} \right] - \frac{2}{3c^2\beta} \nonumber \\
&&+ \frac{2\varrho}{3\beta \Omega_{D,higher}}\frac{\ddot{H}}{H^3}. \label{eoshigher2}
\end{eqnarray}
Using Eqs. (<ref>) and (<ref>) along with the expression of $\Gamma$ we have chosen, we obtain the following expression for the time evolution of $u$ for the three different DE models we are dealing with:
\begin{eqnarray}
\dot{u}_{GO}&=&\left(\frac{3Hu_{GO}\Omega_{D_{GO}}}{\Omega_{D_{GO}}+\Omega_{m}}\right)\left[\omega_{D_{GO}}-
\frac{1}{3}\left(\frac{\Omega_{m} + \Omega_{D_{GO}}}{\Omega_{D_{GO}}}\right)-\frac{b^2\left(1+u_{GO}\right)^2}{u_{GO}}\right], \label{uasa1}\\
\dot{u}_{\tiny\circ}&=&\left(\frac{3Hu_{\tiny\circ}\Omega_{D_{\tiny\circ}}}{\Omega_{D_{\tiny\circ}}+\Omega_{m}}\right)\left[\omega_{D_{\tiny\circ}}-
\frac{1}{3}\left(\frac{\Omega_{m} + \Omega_{D_{\tiny\circ}}}{\Omega_{D_{\tiny\circ}}}\right) -\frac{b^2\left(1+u_{\tiny\circ}\right)^2}{u_{\tiny\circ}}\right], \label{uasa1-1}\\
\dot{u}_{higher}&=&\left(\frac{3Hu_{higher}\Omega_{D,higher}}{\Omega_{D,higher}+\Omega_{m}}\right)\left[\omega_{D,higher}-
\frac{1}{3}\left(\frac{\Omega_{m} + \Omega_{D,higher}}{\Omega_{D,higher}}\right)-\frac{b^2\left(1+u_{higher}\right)^2}{u_{higher}} \right]. \label{uasahigher}
\end{eqnarray}
Inserting the expressions of the EoS parameters obtained in Eqs. (<ref>), (<ref>) and (<ref>) into Eqs. (<ref>), (<ref>) and (<ref>) and using the relation $\Omega_D+\Omega_{m}=(1+u)^{-1}$, we obtain the following expressions for the three different DE models considered:
\begin{eqnarray}
\dot{u}_{GO} &=& \frac{3Hu_{GO}\left( 1+u_{GO} \right)}{\Omega_{D_{GO}}} \left\{ \left[ \frac{2}{3}u_{GO}\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right]\left[ \frac{1}{\left(1+u_{GO}\right)\Omega_{D_{GO}}}\right]\right. \nonumber \\
&&\left. -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{GO}\right)^2}{u_{GO}}\right\} , \label{uasanew1}\\
\dot{u} _{\tiny\circ}&=& \frac{3Hu_{\tiny\circ}\left( 1+u_{\tiny\circ} \right)}{\Omega_{D_{\tiny\circ}}} \left\{ \left(\alpha - \frac{4}{3}\right) \left( \frac{1}{\Omega_{D_{\tiny\circ}}}\right) +\beta - \alpha -\frac{b^2\left(1+u_{\tiny\circ}\right)^2}{u_{\tiny\circ}} \right\}, \label{uasanew2}\\
\dot{u}_{higher} &=& \frac{3Hu_{higher}\left( 1+u_{higher} \right)}{\Omega_{D,higher}} \left\{ \left[ \frac{2}{3}u_{higher}\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right]\times \right.\nonumber \\
&&\left. \left[ \frac{1}{\left(1+u_{higher}\right)\Omega_{D,higher}}\right] + \frac{2\varrho}{3\beta \Omega_{D,higher}}\frac{\ddot{H}}{H^3}\right. \nonumber \\
&&\left. -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{higher}\right)^2}{u_{higher}}\right\} . \label{uasanew3}
\end{eqnarray}
In the following Section, we will study the behavior of the evolutionary forms of $\dot{u}_{GO}$, $\dot{u}_{\tiny\circ}$ and $\dot{u}_{higher}$ obtained, respectively, in Eqs. (<ref>), (<ref>) and (<ref>) for two different choices of the scale factor, i.e. the power law and the emergent ones. Using the reconstructed expressions of $u$, we will use them in order to study the behavior of the EoS parameters for the three DE models we are considering and obtained, respectively, in Eqs. (<ref>), (<ref>) and (<ref>).
We must also emphasize that will find the final expression of the term $\frac{\ddot{H}}{\Omega_D H^3}$ according to the choice of the scale factor we will make.
§ SCALE FACTORS
In this Section, we want to study the behavior of the reconstructed expressions of $u$, determined from $\dot{u}_{GO}$, $\dot{u}_{\tiny\circ}$ and $\dot{u}_{higher}$ obtained, respectively, in Eqs. (<ref>), (<ref>) and (<ref>), for two different choices of the scale factor, i.e. the power law and the emergent ones.
In order to find the final expressions of $\dot{u}$ for the different choices of scale factor, we need to calculate the expressions of $\Omega_{D_{GO}}$, $\Omega_{D_{\tiny\circ}}$ and $\Omega_{D,higher}$ (defined, respectively, in Eqs. (<ref>), (<ref>) and (<ref>)) and $H$ for the relevant case of the scale factor (remembering that $H=\frac{\dot{a}}{a}$). We will then plot the reconstructed expressions of $u$ derived from $\dot{u}$ we obtained for some range of values of the parameters involved. Thanks to the reconstructed expression of $u$, we can plot the behavior of the EoS parameter $\omega_D$ for the relevant model and the specific scale factor.
§.§ Power Law form of the scale factor
We start the study of the different scale factors taking into account the power law scenario.
Following Setare <cit.>, we consider the power law case of the scale factor in the following form:
\begin{eqnarray}
a(t) = a_0\left(t_s -t\right)^n, \label{plscale}
\end{eqnarray}
where $a_0$, $t_s$ and $n$ are three constants. The term $t_s$ indicates the finite future singularity
time and the scale factor defined in Eq. (<ref>) is often used in scientific literature in order to check the type II (sudden singularity) or type IV (which corresponds to $\dot{H}$) for positive values of the power law index $n$.
We have that the derivative of the scale factor given in Eq. (<ref>) with respect to the cosmic time $t$ is given by:
\begin{eqnarray}
\dot{a}\left(t \right) = -na_0\left(t_s -t\right)^{n-1}. \label{plscale2}
\end{eqnarray}
Using the results of Eqs. (<ref>) and (<ref>), we obtain that the expression of the Hubble parameter and its first and second time derivatives are given, respectively, by:
\begin{eqnarray}
H &=& \frac{\dot{a}}{a}=-\frac{n}{t_s -t}, \label{Hpl} \\
\dot{H} &=& \frac{\dot{H}}{dt}=-\frac{n}{\left(t_s -t\right)^2}, \label{Hdotpl}\\
\ddot{H} &=&\frac{\ddot{H}}{dt^2}=-\frac{2n}{\left(t_s -t\right)^3} \label{Hddotpl}.
\end{eqnarray}
Using the expression of $H$ obtained in Eq. (<ref>) and the expressions of $\Omega_{D_{GO}}$, $\Omega_{D_{\tiny\circ}}$ and $\Omega_{higher}$, obtained inserting in Eqs. (<ref>), (<ref>) and (<ref>) the expressions of $\rho_{D_{GO}}$, $\rho_{D_{\tiny\circ}}$ and $\rho_{higher}$ defined in Eqs. (<ref>), (<ref>) and (<ref>) calculated for $H$, $\dot{H}$ and $\ddot{H}$ given in Eqs. (<ref>), (<ref>) and (<ref>), we derive the following expressions for $\dot{u}_{GO}$, $\dot{u}_{\tiny\circ}$ and $u_{higher}$:
\begin{eqnarray}
\dot{u}_{GO} &=& 3u_{GO}\left( 1+u_{GO} \right) \left[ \frac{n^2}{c^2 (t_s-t) (\alpha -n \beta )} \right]\times \nonumber \\
&&\left\{ \left[ \frac{2}{3}u\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right] \frac{n}{c^2 (-\alpha +n \beta )\left(1+u_{GO}\right)} \right. \nonumber \\
&&\left. -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{GO}\right)^2}{u_{GO}}\right\} , \label{ugopl}\\
\dot{u}_{\tiny\circ} &=& 3u_{\tiny\circ}\left( 1+u_{\tiny\circ} \right) \left \{ \frac{3n^2\left( -\alpha + \beta \right)}{\left( t_s -t \right)\left( 3n\alpha -2 \right)} \right\} \times \nonumber \\
&&\left\{ \left[\left(\alpha - \frac{4}{3}\right) \frac{3n (\alpha -\beta )}{-2+3 n \alpha } \right]+\beta - \alpha -\frac{b^2\left(1+u_{\tiny\circ}\right)^2}{u_{\tiny\circ}} \right\}, \label{umhpl}\\
\dot{u}_{higher} &=& 3u_{higher}\left( 1+u_{higher} \right) \left[ \frac{n^2}{c^2 (t_s-t) (\alpha -n \beta )} \right]\times \nonumber \\
&&\left\{ \left[ \frac{2}{3}u\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right] \frac{n}{c^2 (-\alpha +n \beta )\left(1+u_{higher}\right)} \right. \nonumber \\
&&\left. + \frac{4\varrho}{3c^2\beta \left[ n\left(n\alpha - \beta\right) + 2\varrho \right]} -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{higher}\right)^2}{u_{higher}}\right\} . \label{uhigherpl}
\end{eqnarray}
By using numerical integration, the evolution for $u_{GO}$, $u_{\tiny\circ}$ and $u_{higher}$ are depicted in Figures <ref>, <ref> and <ref>. For the case pertaining to the HDE model with GO cut-off, we considered three different cases, i.e. for $\beta = 4.4$ (plotted in red), $\beta= 4.6$ (plotted in green) and $\beta= 4.8$ (plotted in blue), while the other parameters involved have been chosen as $\alpha =4$, $n=1.4$, $c^2=0.818$, $b^2 = 0.025$ and $t_s=7$. It is worthwhile to emphasize that $u_{GO}$ has a monotonically increasing behavior for all the three cases considered.
For the case corresponding to the MHRDE model, three different cases are regarded, namely $\beta= 2.5$ (plotted in red), $\beta=3$ (plotted in green) and $\beta= 3.5$ (plotted in blue), while the other parameters involved have been chosen as $\alpha = 4$, $n= 1.4$, $c^2=0.818 $, $b^2 =0.025 $ and $t_s=7$. As for the previous case, an increasing profile of $u_{\tiny\circ}$ can be observed, for all the three cases considered.
For the model proportional to higher time derivatives of the Hubble parameter $H$, we have considered three different cases, corresponding to $\beta = 4.4$ (plotted in red), $\beta= 4.6$ (plotted in green) and $\beta= 4.8$ (plotted in blue), while the other parameters involved have been chosen as $\alpha =4$, $\varrho = 5$, $n=1.4$, $c^2=0.818$, $b^2 = 0.025$ and $t_s=7$. We can observe in Figure <ref> that $u_{higher}$ monotonically increases for all the cases considered, as also found for the other two DE model considered.
These increasing behaviors of $u_{GO}$, $u_{\tiny\circ}$ and $u_{higher}$ shown in Figures <ref>, <ref> and <ref> clearly indicate a non-trivial contribution of DE, contribution which increases with the temporal evolution of the Universe.
Plot of $u_{GO}$ for power-law scale factor against the time $t$. The increasing pattern indicates that the non-trivial contribution of DE increases with the evolution of the Universe.
Plot of $u_{\tiny\circ}$ for power-law scale factor against the time $t$. The increasing pattern indicates the non-trivial contribution of DE increases with the evolution of the Universe.
Plot of $u_{higher}$ for power-law scale factor against the time $t$. The increasing pattern indicates that the non-trivial contribution of DE increases with the evolution of the Universe.
Plot of $\omega_{D_{GO}}$ against the time $t$ for power-law scale factor. We observe a decreasing behavior for all cases considered.For $\beta = 4.4$ (plotted in red), $\omega_{D_{GO}}$ starts being $>-1$, then it decreases and it can eventually cross $\omega_D =-1$. For the other two cases, we obtain that $\omega_{D_{GO}}$ is always lower that $-1$.
Plot of $\omega_{D_{\tiny\circ}}$ against the time $t$ for the emergent scale factor. For all the cases considered, we have that $\omega_{D_{\tiny\circ}}$ has a slowly decreasing pattern and it is always greater than $-1$.
Plot of $\omega_{D,higher}$ against the time $t$ for power-law scale factor. For all the cases considered, we have that $\omega_{D,higher}$ has a decreasing pattern and it is always greater than $-1$.
Using the reconstructed expressions of $u_{GO}$, $u_{\tiny\circ}$ and $u_{higher}$ obtained, respectively, from Eqs. (<ref>), (<ref>) and (<ref>) and plotted in Figures <ref>, <ref> and <ref>, we can also derive and plot the profile of the EoS parameters obtained in Eqs. (<ref>), (<ref>) and (<ref>) for the three DE models concerned.
For the DE model with GO cut-off, we obtain that, for $\beta = 4.4$, the EoS parameter $\omega_{D_{GO}}$ starts being $>-1$, while with the passing of the time, it decreases and it asymptotically reaches the value $-1$ and can eventually cross it. For the other two cases, we obtain that $\omega_{D_{GO}}$ has a decreasing behavior, being always lower that $-1$.
Instead, for the MHRDE model, we obtain that the EoS parameter $\omega_{D_{\tiny\circ}}$ has a slowly decreasing behavior for all the three cases considered, staying always greater than $-1$.
For the model proportional to higher time derivatives of the Hubble parameter $H$, we observe a slowly decreasing behavior of the EoS parameter $\omega_{D,higher}$, with $\omega_{D,higher}>-1$ for the range of time considered. It is possible that, for sufficiently high time, the three models can cross the value $\omega_D=-1$.
We now consider some particular values of the parameters involved.
For the DE model with GO cut-off, we study the case corresponding to the Ricci scale, which is recovered for $\alpha = 2$ and $\beta =1$ (plotted in green) and we will also consider the case corresponding to $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red). Instead, for the MHRDE model, we will consider the case corresponding to the Ricci scale, which is recovered for $\alpha = 4/3$ and $\beta =1$. The values of the other parameters have been taken as the previous cases considered.
We can clearly see in Figure <ref> that, for both limiting cases, $u_{GO}$ has a decreasing behavior while $\omega_{D_{GO}}$ has a slowly increasing behavior. Moreover, we have that for the case pertaining to the Ricci scale, $\omega_{D_{GO}}$ is always greater than -1 while for the case with $\alpha = 0.8502$ and $\beta = 0.4817$ it is always lower than -1.
For the limiting case of the MHRDE, we observe that $u_{\tiny\circ}$ has an increasing behavior while $\omega_{D_{\tiny\circ}}$ slowly decreases, being always greater than -1.
Plot of $u_{GO}$ for power-law scale factor against the time $t$ for the limiting cases of $\alpha = 2$ and $\beta =1$ (plotted in green) and $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red).
Plot of $u_{\tiny\circ}$ for power-law scale factor against the time $t$ for the limiting case of $\alpha = 4/3$ and $\beta =1$.
Plot of $\omega_{D_{GO}}$ against the time $t$ for power-law scale factor for the limiting cases of $\alpha = 2$ and $\beta =1$ (plotted in green) and $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red).
Plot of $\omega_{D_{\tiny\circ}}$ against the time $t$ for the emergent scale factor for the limiting case of $\alpha = 4/3$ and $\beta =1$.
§.§ Scale factor pertaining to emergent scenario
We now consider the second scale factor chosen in this work, i.e. the emergent one.
This form of scale factor $a\left( t\right)$, as stated in <cit.> is given by:
\begin{eqnarray}
a\left( t \right)= a_0 \left(e^{\mu t}+\lambda \right)^m, \label{aeme}
\end{eqnarray}
where $a_0$, $\lambda$ , $\mu$ and $m$ represents four positive constant parameters. We can make some considerations about the values which can be assumed by the parameters present in Eq. (<ref>):
* if both $a$ and $m$ are negative defined, then the emergent scenario produces the Big Bang singularity at the infinity paste time, i.e. for $t = -\infty$
* $a_0$ must be a positive defined quantity if we want to have the scale factor of the emergent scenario as a positive defined quantity
* $a$ or $m$ must be positive defined in order to obtain an expanding model of the Universe
* $\lambda$ must be a positive defined quantity if we want to avoid singularities (like the Big Rip) at finite time $t$
Consequences of this choice are discussed in <cit.>.
The emergent scenario of the Universe in the framework of DE has been taken into account in many recent papers. Ghosh et al. <cit.> studied the Generalized Second Law of Thermodynamics (GSLT) for the emergent scenario of the Universe for some particular models of $f\left(T\right)$ modified gravity theory. Mukherjee et al. <cit.> studied a general context for an emergent Universe scenario and they derived that the emergent Universe scenarios do not represent isolated solutions but they can occur for different combinations of matter and radiation. del Campo et al. <cit.> considered the emergent model of scale factor in the framework of a self-interacting Jordan-Brans-Dicke modified gravity theory: they derived that this model is able to lead to a stable past eternal static solution which eventually is able to enter a phase where the stability is broken, which leads to a period of inflation.
The first time derivative of the scale factor for the emergent scenario given in Eq. (<ref>) is:
\begin{eqnarray}
\dot{a}\left( t \right) = a_0 m \mu e^{ \mu t}\left[ \lambda + e^{ \mu t } \right]^{m-1}. \label{dotaeme}
\end{eqnarray}
Using the definition of the scale factor given in Eq. (<ref>) along with its time derivative given in Eq. (<ref>), we can easily derive that the Hubble parameter $H$ and its first and second derivatives with respect to the cosmic time $t$ are given, respectively, by the following expressions:
\begin{eqnarray}
H &=& \frac{\dot{a}}{a}= \frac{e^{\mu t} m \mu }{e^{\mu t}+\lambda }, \label{heme}\\
\dot{H} &=& \frac{\dot{H}}{dt}= \frac{m\lambda \mu^2e^{ \mu t } }{\left[e^{ \mu t } + \lambda\right]^2}, \label{dotheme}\\
\ddot{H} &=& \frac{\ddot{H}}{dt^2} =\frac{m\lambda \mu^3e^{ \mu t }\left( \lambda - e^{ \mu t } \right) }{\left[e^{ \mu t } + \lambda\right]^3}\label{ddoteme}.
\end{eqnarray}
Using the expression of $H$ obtained in Eq. (<ref>) and the expressions of $\Omega_{D_{GO}}$, $\Omega_{D_{\tiny\circ}}$ and $\Omega_{higher}$, obtained inserting in Eqs. (<ref>), (<ref>) and (<ref>) the expressions of $\rho_{D_{GO}}$, $\rho_{D_{\tiny\circ}}$ and $\rho_{higher}$ defined in Eqs. (<ref>), (<ref>) and (<ref>) calculated for $H$, $\dot{H}$ and $\ddot{H}$ given in Eqs. (<ref>), (<ref>) and (<ref>), we derive the following expressions for $\dot{u}_{GO}$, $\dot{u}_{\tiny\circ}$ and $u_{higher}$:
\begin{eqnarray}
\dot{u}_{GO} &=& 3u_{GO}\left( 1+u_{GO} \right) \left[ \frac{e^{2 t \mu } m^2 \mu }{c^2 \left(e^{t \mu }+\lambda \right) \left(e^{t \mu } m \beta +\alpha \lambda \right)} \right] \times \nonumber \\
&&\left\{ \left[ \frac{2}{3}u\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right] \frac{e^{t \mu } m}{c^2 \left(e^{t \mu } m \beta +\alpha \lambda \right)\left(1+u_{GO}\right)} \right. \nonumber \\
&& \left. -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{GO}\right)^2}{u_{GO}}\right\}, \label{ugoeme}\\
\dot{u}_{\tiny\circ} &=& 3u_{\tiny\circ}\left( 1+u_{\tiny\circ} \right) \left\{ \frac{3e^{2\mu t}m^2 \mu \left( \alpha -\beta \right)}{\left( e^{\mu t} + \lambda \right) \left( 3 e^{t \mu } m \alpha +2 \lambda \right)} \right\}\times \nonumber \\
&&\left\{ \left[\left(\alpha - \frac{4}{3}\right)\frac{3e^{t \mu } m (\alpha -\beta )}{3 e^{t \mu } m \alpha +2 \lambda } \right]+\beta - \alpha -\frac{b^2\left(1+u_{\tiny\circ}\right)^2}{u_{\tiny\circ}} \right\}, \label{umheme}\\
\dot{u}_{higher} &=& 3u_{higher}\left( 1+u_{higher} \right) \left[ \frac{e^{2 t \mu } m^2 \mu }{c^2 \left(e^{t \mu }+\lambda \right) \left(e^{t \mu } m \beta +\alpha \lambda \right)} \right] \times \nonumber \\
&&\left\{ \left[ \frac{2}{3}u\left( \frac{\alpha - 2\beta}{\beta} \right) + \frac{2\alpha}{3\beta}-\frac{4}{3} \right] \frac{e^{t \mu } m}{c^2 \left(e^{t \mu } m \beta +\alpha \lambda \right)\left(1+u_{higher}\right)} \right. \nonumber \\
&& \left. -\frac{2\varrho \left(e^{t \mu } -\lambda \right)\lambda}{3c^2\beta\left[ m^2\alpha e^{2t \mu } +e^{t \mu }\left(m\beta - \varrho \right)\lambda + \varrho \lambda^2 \right]} -\frac{2}{3c^2\beta} -\frac{b^2\left(1+u_{higher}\right)^2}{u_{higher}}\right\}. \label{uhighereme}
\end{eqnarray}
As accomplished for the previous model studied, we use a numerical integration in order to obtain the evolutionary forms of $u_{GO}$, $u_{\tiny\circ}$ and $u_{higher}$ and we plot them, respectively, in Figures <ref>, <ref> <ref>.
For the to the HDE model with GO cut-off, three different cases have been considered, namely $\left\{\alpha =4, \beta=8\right\}$ (plotted in red), $\left\{\alpha = 5, \beta=5.8\right\}$ (plotted in green) and $\left\{\alpha =3,\beta=5 \right\}$ (plotted in blue), while the other parameters involved have been chosen as $m=0.03$, $\mu=6$, $\lambda = 5$, $c^2=0.818$ and $b^2=0.025$. We can clearly observe that $u_{GO}$ has a decreasing behavior for all the three cases considered.
For the case corresponding to the MHRDE model, we consider three different cases, $\left(\alpha=3,\beta=1.5\right)$ (plotted in red), $\left(\alpha=4,\beta=2.5\right)$ (plotted in green) and $\left(\alpha=6,\beta=4.5\right)$ (plotted in blue) , while the other parameters involved have been chosen as $m=0.03$, $\mu=6$, $\lambda = 5$, $c^2=0.818$ and $b^2=0.025$. Similarly to $u_{GO}$, $u_{\tiny\circ}$ has a decreasing behavior for all the three cases considered.
For the model proportional to higher time derivatives of the Hubble parameter $H$, we considered three different models, one with $\varrho = 3$ (plotted in red), one with $\varrho = 3.5$ (plotted in green) and one with $\varrho = 4$ (plotted in blue). The other parameters have been chosen as follows: $\alpha = 3.5$, $\beta =3$, $\mu = 1.1$, $\lambda = 5$, $m=5$, $c=0.818$ and $b^2 = 0.025$. We can observe in Figure <ref> that $u_{higher}$ has a decreasing behavior for all the cases considered.
Therefore, we conclude that we find a decreasing behavior for all the three DE models considered for all the range of values we considered.
Plot of $u_{GO}$ for scale factor emergent scenario. The decreasing pattern indicates that the non-trivial contribution of DE decreases with the evolution of the Universe.
Plot of $u_{{\tiny\circ}}$ for scale factor emergent scenario. The decreasing pattern indicates that the non-trivial contribution of DE decreases with the evolution of the Universe.
Plot of $u_{higher}$ for scale factor emergent scenario. The decreasing pattern indicates that the non-trivial contribution of DE decreases with the evolution of the Universe.
Plot of $\omega_{D_{GO}}$ for scale factor emergent scenario. $\omega_{D_{GO}}$ has a decreasing behavior for all the three cases considered.
Plot of $\omega_{D_{\tiny\circ}}$ for scale factor emergent scenario. $\omega_{D_{\tiny\circ}}$ can go beyond the phantom phase of the Universe in all cases.
Plot of $\omega_{D,higher}$ for scale factor emergent scenario. $\omega_{D,higher}$ has a decreasing behavior for all the three cases considered. Only the case with $\varrho = 4$ (which is plotted in blue) is able to cross $\omega_D =-1$, instead for the other two models we always have $\omega_D >-1$.
Using the expression of $u_{GO}$, $u_{\tiny\circ}$ and $u_{higher}$ obtained, respectively, from Eqs. (<ref>), (<ref>) and (<ref>) and plotted in Figures <ref>, <ref> and <ref>, we can also plot the EoS parameters for the three DE models considered in this paper derived in Eqs. (<ref>), (<ref>) and (<ref>). The EoS parameter of the DE model with GO cut-off $\omega_{D_{GO}}$ has a decreasing behavior, staying always in the region corresponding to $\omega_D>-1$. Moreover, $\omega_{D_{GO}}$ assumes a constant value of $\left[-0.3,-0.5\right]$ (according to the values of the parameters considered) for $t\approx 1.5$.
Studying the plot of $\omega_{D_{\tiny\circ}}$, we observe an increasing behavior of the EoS parameter for all the three cases considered. Therefore, we have that $\omega_{D_{\tiny\circ}}$ can go beyond the phantom phase of the Universe in all cases we considered.
For the case pertaining to the DE model proportional to $H^2$ and to higher time derivatives of the Hubble parameter $H$, we observe that all the cases considered have a decreasing behavior. Moreover, we observe that only the case with $\varrho = 4$ and plotted in blue can cross the line $\omega_D =-1$, while the other two models always stay in the region $\omega_D >-1$.
As done for the power law scale factor studied in the previous Section, we now consider some particular values of the parameters involved. For the DE model with GO cut-off, we study the case corresponding to the Ricci scale, which is recovered for $\alpha = 2$ and $\beta =1$ (plotted in green), and we also consider the case corresponding to $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red). Instead, for the MHRDE model, we consider the case corresponding to the Ricci scale, which is recovered for $\alpha = 4/3$ and $\beta =1$. The values of all the other parameters are taken as the previous cases considered.
Plot of $u_{GO}$ for scale factor emergent scenario against the time $t$ for the limiting cases of $\alpha = 2$ and $\beta =1$ (plotted in green) and $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red).
Plot of $u_{{\tiny\circ}}$ for scale factor emergent scenario against the time $t$ for the limiting case of $\alpha = 4/3$ and $\beta =1$.
Plot of $\omega_{D_{GO}}$ for scale factor emergent scenario against the time $t$ for the limiting cases of $\alpha = 2$ and $\beta =1$ (plotted in green) and $\alpha = 0.8502$ and $\beta = 0.4817$ (plotted in red).
Plot of $\omega_{D_{\tiny\circ}}$ for scale factor emergent scenario against the time $t$ for the limiting case of $\alpha = 4/3$ and $\beta =1$.
We can clearly observe in Figure <ref> that $u_{GO}$ has a decreasing behavior for both limiting cases considered. Moreover, for the case with $\alpha = 0.8502$ and $\beta = 0.4817$, $u_{GO}$ starts to assume a constant value for $t \approx 2$. Instead, the EoS parameter $\omega_{D_{GO}}$ has an initial increasing behavior for both case considered, becoming constant for $t \approx 1.4$. Moreover, for the case corresponding to the Ricci scale, we have $\omega_{D_{GO}}$ staying always greater than -1 while for the case with $\alpha = 0.8502$ and $\beta = 0.4817$, $\omega_{D_{GO}}$ is always lower than -1.
For the MHRDE model, we observe that $u_{{\tiny\circ}}$ has a decreasing behavior while the EoS parameter $\omega_{D_{\tiny\circ}}$ start with an increasing behavior and it becomes constant from $t \approx 1.8$, being always greater than -1.
§ CONCLUSION
In this work, we have investigated and studied the effects which are produced by the interaction between a brane Universe and the bulk in which the Universe is embedded. We have assumed that the adiabatic equation for the DM is satisfied, while it is violated for the DE due to the energy exchange between the brane and the bulk. Taking into account the effects of the interaction between a brane Universe and the bulk, we have obtained the EoS parameter for the interacting HDE model with Granda-Oliveros cut-off, the Modified Holographic Ricci DE (MHRDE) model and the DE model proportional to $H^2$ and to higher time derivatives of the Hubble parameter having their energy densities given by $\rho_{D_{GO}}= 3c^2\left( \alpha H^{2}+\beta \dot{H}\right)$, $\rho_{D_{\tiny\circ}}= \frac{2}{\alpha - \beta} \left(\dot{H} + \frac{3\alpha}{2}H^2 \right)$ and $\rho_{D,higher}= 3c^2\left( \alpha H^{2}+\beta \dot{H} + \varrho \frac{\ddot{H}}{H}\right)$, respectively. Moreover, we must underline that we are considering a flat Universe, then $k=0$.
We have considered two choices of scale factor, namely, the power-law and the emergent ones. The rate of interaction has been taken as $ \Gamma=3b^2 (1+u)H$. We observed that, for the model pertaining to the power law scale factor, the parameter $u$ has an increasing pattern for all the three DE energy density models considered while, for the scale factor pertaining to the emergent case, the parameter $u$ has a decreasing pattern for all the three DE energy density models considered. These observation are valid for all the values of the parameters considered.
We have also studied the behavior of the EoS parameter using the reconstructed parameter $u$. We first considered the model with power law scale factor. For the DE model with GO cut-off, we obtained that, for the case corresponding to $\beta = 4.4$, $\omega_{D_{GO}}$ starts being $>-1$, while with the passing of the time, it asymptotically reaches the point $-1$ and can eventually cross it. For the other two cases considered, we obtained that $\omega_{D_{GO}}$ has a decreasing behavior, being always lower that $-1$. Instead, for the MHRDE model, we obtained that $\omega_{D_{\tiny\circ}}$ has a slowly decreasing behavior for all the three cases considered, staying always greater than $-1$. Moreover, for the model proportional to higher derivatives of the Hubble parameter $H$, we obtain that $\omega_{D,higher}$ has a decreasing behavior for all the cases considered, staying always in the region $\omega_D>-1$.
Considering the case corresponding to the emergent scale factor, we obtained that $\omega_{D_{GO}}$ has a decreasing behavior, staying always in the region $\omega_D>-1$. Furthermore, $\omega_{D_{GO}}$ assumes a constant value in the region $\left[-0.3,-0.5\right]$ for $t\approx 1.5$. Studying the plot of $\omega_{D_{\tiny\circ}}$, we observed an increasing behavior of $\omega_{D_{\tiny\circ}}$ for all the three cases considered. Moreover, we have that $\omega_{D_{\tiny\circ}}$ can go beyond the phantom phase of the Universe in all cases. For the model proportional to higher derivatives of the Hubble parameter $H$, we obtain a decreasing behavior for $\omega_{D,higher}$ for all the cases considered. Furthermore, we have that only the case with $\varrho = 4$ (which is plotted in blue) can cross the phantom divide line corresponding to $\omega_D =-1$, instead the other two models always stay in the region $\omega_D >-1$.
We also considered the limiting cases corresponding to the Ricci scale for the interacting HDE model with Granda-Oliveros cut-off and for the Modified Holographic Ricci DE (MHRDE) and also the interacting HDE model with Granda-Oliveros cut-off for some particular values of the parameters $\alpha$ and $\beta$ (i.e. $\alpha = 0.8502$ and $\beta = 0.4817$) derived in a recent work of Wang $\&$ Xu.
For the case corresponding to the power law scale factor, we obtained that, for both limiting cases considered, $u_{GO}$ has a decreasing behavior while $\omega_{D_{GO}}$ has a slowly increasing behavior. Moreover, we have that for the case corresponding to the Ricci scale, $\omega_{D_{GO}}$ stays always greater than the value -1, while for the case with $\alpha = 0.8502$ and $\beta = 0.4817$ it is always lower than -1. For the limiting case of the MHRDE, we observe that $u_{\tiny\circ}$ has an increasing behavior while $\omega_{D_{\tiny\circ}}$ slowly decreases, being always greater than the value of -1.
For the case corresponding to the emergent scale factor, we can obtained that $u_{GO}$ has a decreasing behavior for both limiting cases considered. Moreover, for the case with $\alpha = 0.8502$ and $\beta = 0.4817$, $u_{GO}$ starts to assume a constant value for $t \approx 2$. Instead, we have that the EoS parameter $\omega_{D_{GO}}$ has an initial increasing behavior for both limiting cases considered, becoming constant for $t \approx 1.4$. Moreover, for the case corresponding to the Ricci scale, we have $\omega_{D_{GO}}$ staying always greater than -1 while for the case with $\alpha = 0.8502$ and $\beta = 0.4817$, $\omega_{D_{GO}}$ is always lower than -1.
For the MHRDE model, we observe that $u_{{\tiny\circ}}$ has a decreasing behavior while the EoS parameter $\omega_{D_{\tiny\circ}}$ start with an increasing behavior and it becomes constant from $t \approx 1.8$, being always greater than -1.
§ ACKNOWLEDGEMENT
SC acknowledges financial support from DST, Govt of India under project grant no. SR/FTP/PS-167/2011.
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1511.00004
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Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
Recently, Lechner, Hauke and Zoller <cit.> have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all pairwise connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise, and we analyze the code's performance using a belief-propagation decoding algorithm. Our observations may also apply to more general encoding schemes and noise models.
Quantum annealing <cit.> is a method for solving combinatorial optimization problems by using quantum adiabatic evolution to find the ground state of a classical spin glass.
Hoping to extend the reach of quantum annealing in practical devices, Lechner et al. <cit.> have proposed an elegant scheme, using only geometrically local interactions, for simulating a classical spin system with all-to-all pairwise connectivity.
Their scheme may be viewed as a classical low-density parity-check code (LDPC code) <cit.>; here we point out that the error-correcting power of this LDPC code makes the scheme highly robust against weakly correlated bit-flip noise. This observation also applies to other schemes for simulating spin systems based on LDPC codes.
The bipartite Forney-style factor graph (FFG) for the $[7,4,3]$ Hamming code, with linear constraint nodes denoted $\oplus$ and variable nodes denoted ($=$). Constraint nodes correspond to rows of the parity check matrix $H$ in eq.(<ref>), and variable nodes correspond to columns; an edge connects two nodes if a 1 appears in $H$ where the corresponding row and column meet.
Lechner et al. propose representing $N$ logical bits $\vec b =\{b_i, i = 1, 2, \dots, N\}$ using $K=\binom{N}{2}$ physical bits $\vec g=\{g_{ij}, 1\le i < j \le N\}$, where $g_{ij}$ encodes $b_i\oplus b_j$ and $\oplus$ denotes addition modulo $2$.
The $K$ physical variables obey $K-N+1$ independent linear constraints. Hence only $N{-1}$ physical variables are logically independent; we may, for example, choose the independent variables to be $\{g_{12}, g_{23}, g_{34}, \dots , g_{N{-}1,N}\}$.
The linear constraints may be chosen to be weight-3 parity checks.
If weight-4 constraints are also allowed then the parity checks can be chosen to be geometrically local in a two-dimensional array. Higher-dimensional versions of the scheme may also be constructed <cit.>; we will discuss only the two-dimensional coding scheme here, but the same ideas also apply in higher dimensions.
An LDPC code is a (classical) linear error-correcting code which can be represented as a sparse bipartite graph called a Forney-style factor graph (FFG), also known as a Tanner graph. To illustrate the FFG concept, Fig. <ref> shows the FFG for the $[7,4,3]$ Hamming code, which has parity check matrix [This choice for $H$ matches the FFG in Fig. 1, and also makes manifest the invariance of the code space under cyclic permutations of the bits.]
\begin{align}\label{eq:parity-check}
H = \left(
\begin{matrix}
\end{matrix}
\right).
\end{align}
The code's parity checks are the linear constraint nodes, denoted $\oplus$ in the graph, while the bits in the code block are the variable nodes, denoted ($=$). All lines connecting to a variable node have the same value (either 0 or 1), and all lines connecting to a constraint node are required to sum to 0 modulo 2. Thus each variable node corresponds to a column of $H$, each constraint node corresponds to row of $H$, and an edge of the FFG connects a variable node and constraint node if and only if $H$ has the entry $1$ in that row and column. The code has “low density” in the sense that each parity check has low Hamming weight, and correspondingly each constraint node is connected by edges of the FFG to a small number of variable nodes. The parity check matrix for a particular linear code can be chosen in many ways; hence there are many possible FFG presentations of the same code. Fig. <ref> shows one possible FFG for the LDPC code of the LHZ scheme. Later we will discuss another FFG for this code.
An FFG for the LDPC code of the LHZ scheme, based on Fig. 1d of Ref. <cit.>, shown for $N=5$. Here the variable nodes represent the physical spin variables $\{g_{ij}\}$, with $i\in[1,N-1]$ and $j\in [2,N]$. The geometrically local linear constraints ensure that certain closed loops of spins have even parity.
While $g_{ij}$ denotes the value of $b_i\oplus b_j$ in the ideal ground state of the classical spin glass, we use $g^{\prime}_{ij}$ to denote the (possibly noisy) readout of the corresponding physical variable after a run of the quantum annealing algorithm.
If the readout is not too noisy, we can exploit the redundancy of the LDPC code to recover the ideal value of $\{b_i\oplus b_j\}$ from the noisy readout $\vec{g}'$ with high success probability.
Given an error model, we can determine the conditional probability $p(\vec{g}'|\vec b)$ of observing $\vec{g}'$ given $\vec b$.
Assuming that each $\vec b$ has the same a priori probability, we decode $\vec{g}'$ by finding the most likely $\vec b$:
\begin{align}
\vec b_{\rm decoded} = {\rm MLE}(\vec{g}')= {\rm ArgMax}_{\vec b}\ p(\vec{g}'|\vec{b}),
\end{align}
where MLE means “maximum likelihood estimate.”
In fact, we can only recover the ideal $\vec b$ up to an overall global flip since one bit of information is already lost during encoding.
We adopt the simplifying assumption of independent and identically distributed (i.i.d.) noise: $g^{\prime}_{ij}$ is flipped from its ideal value $g_{ij}$ with probability $\varepsilon \leq 1/2$, and agrees with its ideal value with probability $1-\varepsilon$.
Though we do not necessarily expect this simple noise model to faithfully describe the errors arising from imperfect quantum annealing, our assumption follows the presentation of <cit.>. This model might be appropriate if, for example, the noise is dominated by measurement errors in the readout of the final state. It also allows us to estimate $p(\vec {g^{\prime}}|\vec b)$, either analytically or numerically. Exact MLE decoding is possible in principle, but has a very high computational cost. We will settle instead for decoding methods which are computationally feasible though not optimal.
There is a very simple error correction procedure for which we can easily estimate the probability of a decoding error.
For the purpose of decoding (say) $g_{12} \equiv b_1\oplus b_2$, we make use of the following $N{-}2$ weight-3 parity checks:
\begin{align}\label{eq:3termConstraints}
0= (12)\oplus (23)\oplus (13)= \dots =(12)\oplus (2N)\oplus (1N),
\end{align}
where we've used $(ij)$ as a shorthand for $g_{ij}$.
These checks provide us with $N{-}2$ independent ways to recover the logical value of $b_1\oplus b_2$, namely
\begin{align}
b_1\oplus b_2 = (13)\oplus(23)=(14)\oplus (24)= \dots =(1N)\oplus (2N).
\end{align}
(Of course, $g^{\prime}_{12}$ itself provides another independent way to recover $b_1\oplus b_2$, but to keep our analysis simple we will not make use of $g'_{12}$ here.)
Since $g^{\prime}_{ij}\neq g_{ij}$ with probability $\varepsilon$, each $g^{\prime}_{1j} \oplus g^{\prime}_{2j} \neq g_{ij}$ with probability
\begin{align}
\varepsilon^* := 2 \varepsilon (1 - \varepsilon) \leq 1/2.
\end{align}
Therefore, $g_{12}$ is protected by a length-$(N{-}2)$ classical repetition code with bits flipping independently with probability $\varepsilon^*$.
The probability of a majority vote decoding error can be estimated from the Chernoff bound:
\begin{align}\label{eq:chernoff}
p_{\rm fail} \le \exp\left(-2(N-2) \left(\frac{1}{2} - \varepsilon^* \right)^2\right).
\end{align}
This is not the tightest possible Chernoff bound, and using additional information such as the observed value of $g^{\prime}_{12}$ will only improve the success probability.
However, eq.(<ref>) already illustrates our main point:
the probability of a decoding error for any $b_i\oplus b_j$ decays exponentially with $N$.
A simple union bound constrains the probability with which any of the $N-1$ bits are decoded incorrectly:
\begin{align}\label{eq:errorDecrease}
p_{\rm fail}^{\rm total} \le (N-1) \exp\left(-2(N-2) \left(\frac{1}{2} - \varepsilon^* \right)^2\right).
\end{align}
Including $g^{\prime}_{12}$ in the decoding algorithm improves the accuracy of our estimate of $b_1\oplus b_2$, and including higher-weight parity checks such as $0=(12)\oplus (23) \oplus (34)\oplus (14)$ can yield further improvements.
Following a pragmatic approach to using such information, we have implemented belief propagation (BP) <cit.>, a fairly standard decoding heuristic for LDPC codes.
BP efficiently approximates MLE decoding when the constraint graph is a tree, and sometimes works well in cases where the graph contains closed loops. For an introductory account of FFGs and BP see Ref. <cit.>.
In BP, a marginal distribution is assigned to each variable, and updated during each iteration based on the values of neighboring variables on the FFG. Therefore, the implementation of BP depends not only on the code and the noise model, but also on how the code is represented by the FFG. For our implementation, rather than using the FFG in Fig. <ref>, with $\binom{N-1}{2} = O(N^2)$ constraint nodes, we use an FFG with $\binom{N}{3}=O(N^3)$ constraint nodes instead. For each triplet $(b_i,b_j,b_k)$ of logical bits, the corresponding constraint is
\begin{align}
0= (ij)\oplus (jk)\oplus (ik)
\end{align}
in the notation of eq.(<ref>). These constraints are highly redundant, and the larger number of constraints increases the cost of each BP iteration. On the other hand, this scheme has the advantage that it treats all variables symmetrically, and furthermore it includes all the constraints used in our majority voting scheme, which we have already seen has a noise threshold of $1/2$ for i.i.d. noise in the limit of large $N$, ensuring that BP will also converge to the correct answer in this limit. Our FFG is shown in Fig. <ref> for $N=4$, in which case the FFG is planar, with six variable nodes and four constraint nodes. For large values of $N$ the FFG is highly connected and hard to draw.
In a single iteration of BP, the marginal probability distributions assigned to the variables are updated by the following two-step procedure. In the first step, each constraint node sends a message to each of its neighboring variable nodes. For the edge of the FFG connecting constraint node $a$ to variable node $v$, this message, computed using the sum-product formula, is constraint node $a$'s guess regarding the marginal distribution for $v$, based on the marginal distributions for its other neighbors besides $v$. To be concrete, in the FFG for the LHZ code, let $g_{ij}(0)$ denote the probability that variable $g_{ij}$ has the value $0$, and let $g_{ij}(1)$ denote the probability that $g_{ij}=1$. The message sent by the constraint node $a=(12)\oplus(23)\oplus(13)$ to the variable node $v =(12)$ is
\begin{align}
\binom{g_{12}(0)}{ g_{12}(1)}_{a\to v} = \binom{g_{23}(0)g_{13}(0)+g_{23}(1)g_{13}(1)}{g_{23}(0)g_{13}(1)+g_{23}(1)g_{13}(0)},
\end{align}
where $\left( g_{12}\right)_{a\to v}$ denotes $a$'s guess. In the second step of the procedure, each variable node updates its marginal distribution by evaluating the normalized product of its previous a priori probability and all estimated probabilities passed by the neighboring constraint nodes. To be concrete, suppose that variable node $v$ is connected by edges to constraint nodes $a$ and $b$; then the updated probability distribution for variable node $v$ will be
\begin{align}
\binom{g_v(0)}{ g_v(1)}_{\rm updated}\propto \binom{g_v(0) g_{a\to v}(0) g_{b\to v }(0)}{g_v(1) g_{a\to v}(1) g_{b\to v }(1)},
\end{align}
up to normalization.
For an i.i.d. noise model with error probability $\varepsilon$, we assign initial distributions to each variable node by assuming that the observed value of $g_{ij}$ is correct with probability $1-\varepsilon$ and incorrect with probability $\varepsilon$. To decode, probabilities are updated repeatedly until they converge to stable values or until the decoding runtime has elapsed. Intuitively, a consistent neighborhood reduces the entropy of the local marginal distributions, whereas an inconsistent neighborhood may increase the entropy or even change a variable's most likely value. How inconsistencies are resolved is illustrated in Fig. <ref>, which depicts one iteration of BP for the LHZ code with $N=4$. There, the marginal distribution of one variable node is incompatible with the rest, and its updated distribution favors a flipped value, correcting the error.
[Prior probabilities]
[Sum of products summary]
[Posteriori probabilities]
One iteration of the BP realization for the case $\varepsilon=0.1$ and $N=4$. Each number shown is the probability $g_{ij}(0)$ that the associated node $(ij)$ has the value $0$.
(a) The prior distribution assuming each measured physical spin has the value 0, except for one spin in the lower right corner which has value 1. This value is incompatible with the rest, indicating a likely error.
(b) Values of $g_{jk}(0)$ passed from variable nodes $(=)$ to neighboring constraint nodes $\oplus$.
(c) Values of $\left(g_{ij}\right)_{a\to v}$, computed by the sum-product formula, passed from constraint nodes to neighboring variable nodes.
(d) Updated a posteriori values for $g_{ij}(0)$, calculated as the (normalized) product of received messages and prior probabilities.
For the LHZ code and i.i.d. noise the numerically estimated probability $p_{\rm fail}^{\rm total}$ of a decoding error is plotted in Fig. <ref> as a function of the error probability $\varepsilon$ and the number $N$ of encoded spins, for $N$ ranging from 2 to 40. As expected, we find that the failure probability falls steeply as $N$ increases if $\varepsilon$ is not too close to the threshold value $1/2$. Also as expected, $p_{\rm fail}^{\rm total}$ is substantially smaller than the crude estimate in eq.(<ref>).
Performance of iterative BP decoding algorithm.
The probability of a decoding error is plotted as a function of the number $N$ of encoded spins, for various values of the physical error probability $\varepsilon$. Each data point was obtained by averaging over 5000 noise realizations, and for each realization the BP algorithm was iterated five times, incorporating information about loops up to length $33=2^5+1$. The decoding performance is significantly better than for a single BP iteration, where only loops of size $\le 3$ are considered.
The logical error probability starts at $p_{\rm fail}^{\rm total}= \varepsilon$ for $N=2$ and rises with $N$ until the onset of exponential decay, which begins for a smaller value of $N$ than suggested by eq.(<ref>).
We conclude that the architecture proposed in <cit.>, and the decoding method proposed here, provide good protection against i.i.d. noise in the readout of the physical spins, assuming an error probability $\varepsilon$ for each physical spin which is independent of the total number $N$ of encoded spins. More generally, we expect powerful decoding strategies such as BP to enhance the performance of other quantum annealing schemes in which the simulated spins are the logical bits of an LDPC code. We note that BP and other related methods have also been used to solve combinatorial problems in a purely classical setting <cit.>. Perhaps sophisticated classical decoding strategies and quantum annealing, when used together, can solve problems which are beyond the reach of either method used alone.
To keep our analysis simple, we assumed an i.i.d. noise model for the physical spins, which might not be an accurate description of the noise in realistic quantum annealing. In fact, Albash et al. <cit.> have recently provided evidence that this noise model is inadequate, by investigating the performance of the LHZ scheme using simulated quantum annealing, a Monte Carlo method (using a classical computer) for approximating the behavior of a quantum annealing procedure. The output distributions in actual quantum annealing experiments have been found to agree reasonably well with simulated quantum annealing predictions, and the numerical results in Ref. <cit.> indicate that the LHZ scheme does not outperform the annealing architectures used in current experiments <cit.>, even after including a final decoding step.
Perhaps quantum error-correcting codes can be invoked to achieve further improvements in performance <cit.>, but so far no truly scalable scheme for quantum annealing has been proposed <cit.>.
How well the Lechner et al. architecture performs under realistic laboratory conditions is a question best addressed by experiments.
We thank W. Lechner and E. Crosson
for useful comments and discussions.
FP and JP gratefully acknowledge funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation, and by the Army Research Office.
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1511.00169
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authorlabel1]Mihaï BOSTAN
authorlabel1]Aurélie FINOT
authorlabel1]Maxime HAURAY
[authorlabel1]Institut de Mathématiques de Marseille I2M, Centre de Mathématiques et Informatique CMI, UMR CNRS 7373, 39 rue Frédéric Joliot Curie, 13453 Marseille Cedex 13
Nous étudions le régime du rayon de Larmor fini pour le système de Vlasov-Poisson, dans le cas où la longueur de Debye est égale au rayon de Larmor. Le champ magnétique est supposé uniforme. Nous restreignons l'étude de ce problème non linéaire au cas bi-dimensionnel. Nous obtenons le modèle limite en appliquant les méthodes de gyro-moyenne cf. <cit.>, <cit.>. Nous donnons l'expression explicite du champ d'advection effectif de l'équation de Vlasov, dans laquelle nous avons substitué le champ électrique auto-consistant, via la résolution de l'équation de Poisson moyennée à l'échelle cyclotronique. Nous mettons en évidence la structure hamiltonienne du modèle limite et présentons ses propriétés : conservations de la masse, de l'énergie cinétique, de l'énergie électrique, etc.
Abstract 0.5We study the finite Larmor radius regime for the Vlasov-Poisson system. The magnetic field is assumed to be uniform. We investigate this non linear problem in the two dimensional setting. We derive the limit model by appealing to gyro-average methods cf. <cit.>, <cit.>. We indicate the explicit expression of the effective advection field, entering the Vlasov equation, after substituting the self-consistent electric field, obtained by the resolution of the averaged (with respect to the cyclotronic time scale) Poisson equation. We emphasize the Hamiltonian structure of the limit model and present its properties : conservationss of the mass, kinetic energy, electric energy, etc.
§ ABRIDGED ENGLISH VERSION
Motivated by the magnetic confinement fusion, which is one of the main application in plasma physics, we analyse the dynamics of a population of charged particles, under the action of a strong uniform magnetic field. The goal of this note is to study the finite Larmor radius regime, that is, we assume that the particle distribution fluctuates at the Larmor radius scale along the orthogonal directions, with respect to the magnetic field <cit.>, <cit.>, <cit.>. To simplify, we consider the two dimensional setting, i.e., $x = (x_1, x_2), v = (v_1, v_2)$, with a magnetic field orthogonal to $x_1Ox_2$. We chose a regime such that
* The reference time $T$ is much larger than the cyclotronic period (strong magnetic field) i.e.,
\begin{equation}
\label{EquLoi1}
T \frac{q|\Be|}{m} \approx \frac{1}{\eps}, \mbox{ with } 0 < \eps << 1~;
\end{equation}
Notice that the above hypothesis writes also $\frac{TV}{\rho _L} \approx \frac{1}{\eps}$, where $V$ is the reference velocity (along the orthogonal directions), and $\rho _L$ is the typical Larmor radius.
* The kinetic energy is much larger than the potential energy
\begin{equation}
\label{EquLoi2}
\frac{m |V|^2}{q\phi } \approx \frac{1}{\eps}
\end{equation}
where $m$ is the particle mass, $q$ is the particle charge, and $\phi$ is the reference electric potential.
* The Larmor radius is of the same order as the Debye length i.e.,
\begin{equation}
\label{EquLoi3}
\lambda ^2 _D = \frac{\eps _0 \phi }{n q} \approx \rho ^2 _L.
\end{equation}
Here $\eps _0$ is the electric permittivity of the vacuum and $n$ is the charge concentration.
Accordingly, the presence density $\fe = \fe (t,x,v)$ and the electric potential $\phie$ satisfy the following Vlasov-Poisson system, up to some multiplicative constants, of order one
\begin{equation}
\label{Equ1}
\partial _t \fe + \frac{1}{\eps}( v \cdot \nabla _x \fe + \oc \vorth \cdot \nabla _v \fe ) - \nabla _x \phie \cdot \nabla _v \fe = 0,\;\;(t,x,v) \in \R_+ \times \R ^2 \times \R^2
\end{equation}
\begin{equation}
\label{Equ2}
- \Delta _x \phie = \rho ^\eps := \intv{\fe (t,x,v)},\;\;(t,x) \in \R _+ \times \R ^2
\end{equation}
\begin{equation}
\label{EquIC}
\fe (0,x,v) = \fin (x,v),\;\;(x,v) \in \R ^2 \times \R^2.
\end{equation}
Here $\oc$ stands for the rescaled cyclotronic frequency (the real cyclotronic frequency is $\omega _c ^\eps = \oc /\eps$, and the real cyclotronic period is $T_c ^\eps = \frac{2\pi}{\oc ^\eps} = \eps \frac{2\pi}{\oc} = \eps T_c$), and it is assumed constant (uniform magnetic field). For any $v = (v_1, v_2) \in \R^2$, we denote by $\vorth$ the vector $\vorth = (v_2, - v_1) \in \R^2$. We study the stability of the family $(\fe, \phie)_{\eps >0}$, when $\eps$ becomes small. The asymptotic behavior follows by filtering out the fast oscillations of the caracteristic equations for (<ref>). It is easily seen that the changes over one cyclotronic period of the quantities $\tx = x + \frac{\vorth}{\oc}$, $\tv = {\mathcal R}(\oc t/\eps)v$, are negligible. We expect that the family $\tfe (t, \tx, \tv) = \fe (t, \tx - {\mathcal R}(-\oc t/\eps) \tvorth /\oc, {\mathcal R}(- \oc t/\eps ) \tv)$ converges, as $\eps$ becomes small, toward some profile $\tf (t, \tx, \tv)$.
Let $\fin = \fin (x,v)$ be a non negative, smooth, compactly supported presence density and $(\fe, \phie)_{\eps >0}$ be the solutions of the Vlasov-Poisson sistem (<ref>), (<ref>), (<ref>). We denote by $\tf = \tf (t, \tx, \tv)$ the solution of
\begin{equation}
\label{EquAveVla}
\partial _t \tf + {\mathcal V}[\tf (t)] (\tx, \tv) \cdot \nabla _{\tx} \tf + {\mathcal A}[\tf (t)] (\tx, \tv) \cdot \nabla _{\tv} \tf = 0,\;\;(t, \tx, \tv) \in \R_+ \times \R ^2 \times \R^2
\end{equation}
with the initial condition
\begin{equation}
\label{EquAveIC}
\tf (0, \tx, \tv) = \fin \left (\tx - \frac{\tvorth}{\oc}, \tv \right ),\;\;(\tx, \tv) \in \R^2 \times \R^2
\end{equation}
where the velocity and acceleration vector fields ${\mathcal V}, {\mathcal A}$ are given by
\[
{\mathcal V}[\tf (t)] (\tx, \tv) = - \frac{^\bot \nabla _{\tx}}{\oc} \tphi [\tf (t)],\;\;{\mathcal A}[\tf (t)] (\tx, \tv) = \oc \;^\bot \nabla _{\tv} \tphi [\tf (t)]
\]
\[
\tphi [\tf(t)] (\tx, \tv) = - \frac{1}{2\pi} \inttytw{\left \{\ln \frac{|\tv - \tw|}{|\oc|} \;\ind{|\tx - \ty| \leq \frac{|\tv - \tw|}{|\oc|} }
+ \ln |\tx - \ty|\;\ind{|\tx - \ty| > \frac{|\tv - \tw|}{|\oc|} }\right \}
\tf (t, \ty, \tw)}.
\]
Therefore $\fe (t,x,v) - \tf(t, x + \vorth/\oc, {\mathcal R} (\oc t/\eps)v) = o(1)$ when $\eps \searrow 0$.
§ TRAJECTOIRES EFFECTIVES
Ce travail s'inscrit dans le cadre de la modélisation des plasmas de fusion. Nous concentrons notre étude au régime du rayon de Larmor fini pour le système de Vlasov-Poisson bi-dimensionnel. Sous les hypothèses (<ref>), (<ref>), (<ref>), ce régime est décrit par (<ref>), (<ref>), (<ref>). La méthode développée ici consiste à exprimer le potentiel électrique à l'aide de la solution fondamentale de l'opérateur de Laplace dans $\R^2$, puis insérer cette expression dans les trajectoires de l'équation de Vlasov. Nous obtenons alors, à l'aide des méthodes classiques de gyro-moyenne <cit.>, <cit.>, <cit.> les trajectoires limites et ainsi, les expressions effectives des champs vitesse et accélération de la nouvelle équation de Vlasov, décrivant le régime asymptotique considéré. Pour plus de détails sur les preuves de ces résultats, nous renvoyons à <cit.>.
Notons $e$ la solution fondamentale de l'opérateur de Laplace dans $\R^2$
\[
e(z) = - \frac{1}{2\pi} \ln |z|,\;\;z \in \R^2 \setminus \{0\}
\]
c'est-à-dire $- \Delta e = \delta _0$ dans ${\mathcal D}^\prime (\R^2)$. Le potentiel électrique, solution de l'équation de Poisson (<ref>), s'écrit donc
\begin{equation}
\label{Equ21}
\phie (t,x) = \inty{e(x-y) \rhoe (t,y) } = \intyw{e(x-y) \fe (t,y,w)}.
\end{equation}
Les équations caractéristiques de l'équation de transport (<ref>) sont données par
\begin{equation*}
\label{Equ22}
\frac{\dd \Xe}{\dd t} = \frac{\Ve (t)}{\eps},\;\;\frac{\dd \Ve }{\dd t} = \oc \frac{\Veorth (t)}{\eps} - \nabla _x \phie (t, \Xe (t)),\;\;(\Xe (0), \Ve(0)) = (x,v).
\end{equation*}
Nous cherchons des quantités qui varient peu sur une période cyclotronique. Plus exactement, à tout instant fixé $t>0$, on introduit le changement de coordonnées
\[
\tx = x + \frac{\vorth}{\oc},\;\;\tv = {\mathcal R} \left ( \frac{\oc t}{\eps} \right ) v
\]
où $\calR(\theta)$ désigne la rotation de $\R^2$ d'angle $\theta$. On vérifie aisément que le déterminant jacobien vaut $1$ et alors ces transformations préservent la mesure de Lebesgue de $\R^4$ i.e., $\dd\tv \dd \tx = \dd v \dd x$. En effet, $\tx$ est le centre du cercle de Larmor d'écrit par une particule passant par $x$ avec la vitesse $v$. Ce centre ne varie pas à l'échelle du mouvement rapide cyclotronique, correspondant à la fréquence cyclotronique $\frac{\oc}{\eps}$. Plus exactement on a
\begin{equation}
\label{Equ23}
\frac{\dd \tXe}{\dd t} = - \frac{^\bot \nabla _x \phie }{\oc} (t, \Xe (t)) = - \frac{^\bot \nabla _x \phie}{\oc} \left (t, \tXe (t) - \frac{\calR}{\oc}\left ( - \frac{\oc t}{\eps} \right ) {^\bot\tVe }(t)\right )
\end{equation}
\begin{equation}
\label{Equ24}
\frac{\dd \tVe}{\dd t} = - \calR \left ( \frac{\oc t}{\eps} \right )
\nabla _x \phie (t, \Xe (t)) = - \calR \left ( \frac{\oc t}{\eps} \right )\nabla _x \phie \left (t, \tXe (t) - \frac{\calR}{\oc}\left ( - \frac{\oc t}{\eps} \right ) {^\bot \tVe} (t)\right ).
\end{equation}
On souhaite remplacer le potentiel électrique par l'éxpression de (<ref>). On introduit les densités de présence en les coordonnées $(\tx, \tv)$
\[
\fe (t,x,v) = \tfe (t, \tx, \tv),\;\;\tx = x+ \frac{\vorth}{\oc}, \tv = \rotp{} v .
\]
Ainsi, (<ref>) conduit à
\begin{align*}
\phie \left (t, \tXe (t) - \frac{\calR}{\oc}\left ( - \frac{\oc t}{\eps} \right ) {^\bot \tVe }(t)\right ) = \inttytw{\!\!\!\!e \left ( \tXe (t) - \ty - \frac{\calR}{\oc}\left ( - \frac{\oc t}{\eps} \right ) ^\bot ( \tVe (t) - \tw) \right )\tfe (t, \ty, \tw)}
\end{align*}
et par conséquent, (<ref>), (<ref>) deviennent
\begin{equation}
\label{Equ25}
\frac{\dd \tXe }{\dd t} = - \frac{1}{\oc} \inttytw{{^\bot \nabla} e \left ( \tXe (t) - \ty - \frac{1}{\oc} \rotm{} ^\bot ( \tVe (t) - \tw) \right )\tfe (t, \ty, \tw)}
\end{equation}
\begin{equation}
\label{Equ26}
\frac{\dd \tVe }{\dd t} = - \rotp{} \inttytw{ \nabla e \left ( \tXe (t) - \ty - \frac{1}{\oc} \rotm{} ^\bot ( \tVe (t) - \tw) \right )\tfe (t, \ty, \tw)}.
\end{equation}
En prenant la moyenne de (<ref>) sur la période cyclotronique $[t, t+T_c ^\eps]$, avec $T_c ^\eps = \eps \frac{2\pi}{\oc}$, et en introduisant la variable rapide $s = (\tau - t)/\eps$, $\tau \in [t, t + T_c ^\eps]$, nous obtenons
\begin{align}
\label{Equ27}
& \frac{\tXe (t + T_c ^\eps) - \tXe (t)}{T_c ^\eps} = - \frac{1}{\oc T_c ^\eps}\int _t ^{t + T_c ^\eps} \!\!\!\!\inttytw{\!\!\!\!{^\bot \nabla }e \left( \tXe (\tau) - \ty - \calR \left (- \frac{\oc \tau}{\eps} \right ) \frac{^\bot ( \tVe (\tau) - \tw)}{\oc} \right ) \tfe(\tau)}\dd \tau \nonumber \\
& = - \frac{1}{\oc T_c} \int _0 ^{T_c} \inttytw{{^\bot \nabla }e \left( \tXe (t + \eps s) - \ty - {\mathcal R}\left (- \frac{\oc t }{\eps} - \oc s \right ) \frac{^\bot ( \tVe (t + \eps s) - \tw)}{\oc} \right ) \tfe(t + \eps s)} \dd s \nonumber \\
& \approx - \frac{^\bot \nabla _{\xi}}{\oc} \inttytw{
{\mathcal E}(\tXe (t) - \ty, \tVe (t) - \tw) \tfe(t, \ty, \tw)}\dd \theta
\end{align}
où la fonction ${\mathcal E}$ est définie par
\[
{\mathcal E} (\xi, \eta) = \avetpi{} e \left (\xi - \oc ^{-1} \calR (\theta){ ^\bot \eta} \right ) \;\dd \theta,\;\;\xi, \eta \in \R^2.
\]
Nous procédons de la manière identique pour obtenir, à partir de (<ref>)
\begin{align}
\label{Equ28}
& \frac{\tVe (t + T_c ^\eps) - \tVe (t)}{T_c ^\eps} = - \frac{1}{ T_c ^\eps}\int _t ^{t + T_c ^\eps} \!\!\!\!\!\!\calR \left ( \frac{\oc \tau}{\eps} \right ) \int{{\nabla }e \left( \tXe (\tau) - \ty - \calR \left (- \frac{\oc \tau}{\eps} \right ) \frac{^\bot ( \tVe (\tau) - \tw)}{\oc} \right ) \tfe(\tau)}\dd \tw \dd \ty\dd \tau \nonumber \\
& = \!- \frac{1}{T_c} \!\!\int _0 ^{T_c} \!\!\!\!\!\!\calR \left ( \frac{\oc t}{\eps} + \oc s\right )\!\!\!\int{\!\!{\nabla }e \left( \tXe (t + \eps s) - \ty - {\mathcal R}\left (- \frac{\oc t }{\eps} - \oc s \right ) \frac{^\bot ( \tVe (t + \eps s) - \tw)}{\oc} \right ) \tfe(t + \eps s)} \dd \tw \dd \ty\dd s \nonumber \\
& \approx \oc {^\bot \nabla _{\eta}} \inttytw{
{\mathcal E}(\tXe (t) - \ty, \tVe (t) - \tw )\tfe(t, \ty, \tw)}\dd \theta.
\end{align}
En passant à la limite dans (<ref>), (<ref>), quand $\eps \searrow 0$, nous obtenons les trajectoires après filtration du mouvement rapide cyclotronique
\begin{equation*}
\label{Equ29}
\frac{\dd \tX }{\dd t} = - \frac{ ^\bot \nabla _\xi }{\oc} \inttytw{{\mathcal E}(\tX (t) - \ty, \tV (t) - \tw)\tf(t)},\;\;\frac{\dd \tV }{\dd t} = \oc { ^\bot \nabla _\eta } \inttytw{{\mathcal E}(\tX (t) - \ty, \tV (t) - \tw)\tf(t)}
\end{equation*}
où $\tf = \lime \tfe$ est la distribution limite. Par la suite nous déterminons une expression pour ${\mathcal E}(\xi, \eta)$. Cela résulte de la propriété de la moyenne pour les fonctions harmoniques. En effet, si $|\xi | > |\eta |/|\oc|$, la fonction $z \to e(z)$ est harmonique dans l'ouvert $\R ^2 \setminus \{0\}$, contenant le disque fermé, de centre $\xi$ et de rayon $|\eta |/|\oc|$, et par conséquent nous avons, grâce à la formule de la moyenne
\[
{\mathcal E}(\xi, \eta) = e(\xi) = - \frac{1}{2\pi} \ln |\xi|,\;\;|\xi | > \frac{|\eta|}{|\oc|}.
\]
Plus exactement, on démontre cf. <cit.>
Pour tout $\xi, \eta \in \R^2$, nous avons
\[
\calE (\xi, \eta) = e \left ( \frac{\eta}{\oc}\right ) \ind{|\xi | \leq |\eta|/|\oc|} + e(\xi) \ind{|\xi | > |\eta|/|\oc|}
\]
\[
\nabla _\xi \calE (\xi, \eta)= \nabla e (\xi) \;\ind{|\xi | > |\eta|/|\oc|},\;\;\nabla _\eta \calE (\xi, \eta)= \oc ^{-1} \nabla e \left ( \frac{\eta}{\oc}\right ) \;\ind{|\xi | \leq |\eta|/|\oc|}\;\;\mbox{au sens des distributions}.
\]
§ LE MODÈLE LIMITE
Nous introduisons le potentiel électrique
\begin{align*}
\label{Equ45}
\tphi [\tf (t) ] (\tx, \tv) & = \inttytw{\calE(\tx - \ty, \tv - \tw) \tf(t, \ty, \tw)} \\
& = \inttytw{\left \{ e \left ( \frac{\tv - \tw}{\oc}\right ) \ind{|\tx - \ty | \leq |\tv - \tw|/|\oc|} + e(\tx - \ty) \;\ind{|\tx - \ty | > |\tv - \tw|/|\oc|} \right \}\tf(t, \ty, \tw)} \nonumber
\end{align*}
et les fonctions
\begin{equation*}
\label{Equ35}
{\mathcal V}[\tf(t)] (\tx, \tv) = - \frac{^\bot \nabla _\xi }{\oc} \inttytw{\calE (\tx - \ty, \tv - \tw) \tf(t,\ty, \tw)} = - \frac{^\bot \nabla _{\tx}}{\oc} \tphi [\tf (t)]
\end{equation*}
\begin{equation*}
\label{Equ36}
{\mathcal A}[\tf(t)] (\tx, \tv) = \oc {^\bot \nabla _\eta } \inttytw{\calE (\tx - \ty, \tv - \tw) \tf(t,\ty, \tw)} = \oc {^\bot \nabla _{\tv}} \tphi [\tf (t)].
\end{equation*}
En dérivant sous le signe intégral, il est également possible de représenter les champs vitesse et accélération sous la forme (cf. Lemme <ref>)
\begin{align}
\label{Equ37}
{\mathcal V}[\tf(t)] (\tx, \tv) & = - \frac{1}{\oc} \inttytw{{^\bot \nabla _\xi}\calE (\tx - \ty, \tv - \tw) \tf(t,\ty, \tw)} \\
& = - \frac{1}{\oc} \inttytw{{^\bot \nabla e} (\tx - \ty) \;\ind{|\tx - \ty | > |\tv - \tw|/|\oc|} \tf(t, \ty, \tw)} \nonumber
\end{align}
\begin{align}
\label{Equ38}
{\mathcal A}[\tf(t)] (\tx, \tv) & = \oc \inttytw{{^\bot \nabla _\eta}\calE (\tx - \ty, \tv - \tw) \tf(t,\ty, \tw)} \\
& = \oc \inttytw{{^\bot \nabla e}
\left ( \frac{\tv - \tw}{\oc}\right ) \ind{|\tx - \ty | \leq |\tv - \tw|/|\oc|} \tf(t, \ty, \tw)}. \nonumber
\end{align}
Les trajectoires limites sont déterminées par les champs vitesse et accélération ${\mathcal V}[\tf], {\mathcal A}[\tf]$
\begin{equation*}
\label{Equ46}
\frac{\dd \tX }{\dd t} = {\mathcal V}[\tf (t)] (\tX (t), \tV (t)),\;\;\frac{\dd \tV }{\dd t} = {\mathcal A}[\tf (t)] (\tX (t), \tV (t)).
\end{equation*}
Les densités de présence étant conservées le long des trajectoires, nous obtenons
\[
\tfe (t, \tXe (t), \tVe(t)) = \fe (t, \Xe (t), \Ve (t)) = f(0, x, v) = f(0, \tx - \oc ^{-1} \tvorth, \tv)
\]
et par conséquent la densité limite $\tf$ est solution de (<ref>), (<ref>). Notons que les équations caractéristiques limites forment un système hamiltonien, en les variables conjuguées $(\tx _2, \oc ^{-1} \tv_1)$ et $(\oc \tx_1, \tv_2)$
\[
\frac{\dd \tX_2}{\dd t} = \frac{\partial \tphi [\tf(t)]}{\partial (\oc \tx _1)}(\tX (t), \tV(t)),\;\;\frac{\dd (\oc ^{-1} \tV_1)}{\dd t} = \frac{\partial \tphi [\tf(t)]}{\partial \tv _2}(\tX (t), \tV(t))
\]
\[
\frac{\dd (\oc \tX_1)}{\dd t} = - \frac{\partial \tphi [\tf(t)]}{\partial \tx _2}(\tX (t), \tV(t)),\;\;\frac{\dd \tV_2}{\dd t} = -\frac{\partial \tphi [\tf(t)]}{\partial (\oc ^{-1} \tv _1)}(\tX (t), \tV(t)).
\]
§ QUELQUES PROPRIÉTÉS DU MODÈLE LIMITE
Les champs de vitesse et accélération étant à divergence nulle
\[
\dive _{\tx} {\mathcal V} [\tf (t)] = - \frac{1}{\oc} \dive _{\tx} \;^\bot \nabla _{\tx} \tphi [\tf (t)] = 0,\;\;\dive _{\tv} {\mathcal A} [\tf (t)] = \oc \dive _{\tv} \;^\bot \nabla _{\tv} \tphi [\tf (t)] = 0
\]
l'équation (<ref>) s'écrit aussi sous la forme conservative
\[
\partial _t \tf + \dive _{\tx} \{ \tf {\mathcal V}[\tf(t)]\} + \dive _{\tv} \{ \tf {\mathcal A}[\tf(t)]\} = 0.
\]
En particulier nous obtenons la conservation de la masse. Plus généralement, nous démontrons le résultat suivant.
Soit $\tf = \tf (t,\tx,\tv)$ la solution du problème (<ref>), (<ref>) et $\psi = \psi (\tx, \tv)$ une fonction intégrable par rapport à $\tf (0,\tx, \tv)\dd\tv \dd \tx = \fin (\tx - \oc ^{-1} {^\bot \tv}, \tv)\dd \tv \dd \tx$.
* Pour tout $t \in \R_+$ nous avons
\begin{align}
\label{Equ41}
& 2 \frac{\dd }{\dd t} \inttxtv{\psi (\tx, \tv) \tf (t, \tx, \tv) } = \int _{\R^2} \!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!
\tf(t,\ty,\tw)\tf (t, \tx, \tv) \\
& \times
\left [
\frac{1}{\oc} \left (\nabla _{\ty} \psi (\ty, \tw) - \nabla _{\tx} \psi (\tx, \tv) \right )\cdot {^\bot \nabla} e (\tx - \ty) \;\ind{|\tx - \ty| > |\tv - \tw|/|\oc|}
\right. \nonumber \\
& +
\left.
\left (\nabla _{\tv} \psi (\tx, \tv) - \nabla _{\tw} \psi (\ty, \tw) \right )\cdot {^\bot \nabla} e \left ( \frac{\tv - \tw}{\oc}\right ) \ind{|\tx - \ty| < |\tv - \tw|/|\oc|}
\right ]
\;\dd\tw \dd\ty \dd \tv \dd \tx. \nonumber
\end{align}
* En particulier, pour tout $t \in \R_+$ nous avons
\[
\inttxtv{\{1, \tx, \tv, |\tx |^2, |\tv |^2\}\tf (t, \tx, \tv)} = \inttxtv{\{1, \tx, \tv, |\tx |^2, |\tv |^2\}\fin (\tx - \oc ^{-1} \tvorth, \tv)}.
\]
* Pour tout $t \in \R_+$, nous obtenons, grâce aux formules de représentation (<ref>), (<ref>)
\begin{align*}
\frac{\dd }{\dd t}& \inttxtv{\psi (\tx, \tv) \tf (t, \tx, \tv) } =
\inttxtv{\psi(\tx, \tv) \partial _t \tf} \\
& = \inttxtv{\left [\nabla _{\tx} \psi \cdot {\mathcal V}[\tf(t)] + \nabla _{\tv}\psi \cdot {\mathcal A} [\tf(t)] \right ]\tf(t, \tx, \tv)} \\
& = - \frac{1}{\oc} \int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\nabla _{\tx} \psi (\tx, \tv) \cdot {^\bot \nabla} e (\tx - \ty) \;\ind{|\tx - \ty| > |\tv - \tw|/|\oc|} \tf(t,\ty,\tw) \tf(t,\tx,\tv) \;\dd \tw \dd\ty \dd \tv \dd \tx \\
& + \;\;\;\;\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\nabla _{\tv} \psi (\tx, \tv) \cdot {^\bot \nabla} e \left(\frac{\tv - \tw}{\oc}\right) \ind{|\tx - \ty| < |\tv - \tw|/|\oc|} \tf(t,\ty,\tw) \tf(t,\tx,\tv) \;\dd \tw \dd\ty \dd \tv \dd \tx.
\end{align*}
La formule (<ref>) résulte en interchangeant $(\tx, \tv)$ contre $(\ty, \tw)$, combiné à Fubini.
* Les conservations résultent facilement, par (<ref>) appliquée successivement aux fonctions $1, \tx, \tv, |\tx |^2, |\tv |^2$.
Etant donné que l'énergie cinétique est conservée, et comme on s'attend à ce que l'énergie globale soit conservée, nous devrions retrouver aussi la conservation de l'énergie électrique. Effectivement nous démontrons
Pour tout $t \in \R_+$ nous avons
\[
\frac{\dd }{\dd t} \frac{1}{2} \inttxtv{\tphi [\tf(t)] (\tx, \tv) \tf(t,\tx, \tv)} = 0.
\]
L'énergie électrique s'écrit sous la forme
\[
\frac{1}{2}\inttxtv{\tphi [\tf(t)] (\tx, \tv) \tf(t,\tx, \tv)} = \frac{1}{2}\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\int _{\R^2}\!\!\calE (\tx - \ty, \tv - \tw)\tf(t,\tx,\tv) \tf (t, \ty, \tw)\;\dd \tw \dd \ty \dd \tv \dd \tx
\]
et en utilisant la parité de $\calE (\xi, \eta)$ en les variables $\xi$ et $\eta$, nous obtenons facilement, par Fubini, que
\begin{align*}
\frac{\dd }{\dd t} \frac{1}{2} \inttxtv{\tphi [\tf(t)] (\tx, \tv) \tf(t,\tx, \tv)} & = \inttxtv{\tphi [\tf(t)](\tx, \tv) \partial _t \tf }\\
& = \inttxtv{\left [ \nabla _{\tx} \tphi [\tf (t)] \cdot {\mathcal V} [\tf(t)] + \nabla _{\tv} \tphi [\tf (t)] \cdot {\mathcal A} [\tf(t)] \right ] \tf } = 0.
\end{align*}
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1511.00095
|
$^1$ Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing normal University, Beijing 100875, China
$^2$ State Key Laboratory of Networking and Switching Technology,
Beijing University of Posts and Telecommunications, Beijing 100876,
Quantum repeater is one of the important building blocks for long
distance quantum communication network. The previous quantum
repeaters based on atomic ensembles and linear optical elements can
only be performed with a maximal success probability of 1/2 during
the entanglement creation and entanglement swapping procedures.
Meanwhile, the polarization noise during the entanglement
distribution process is harmful to the entangled channel created.
Here we introduce a general interface between a polarized photon and
an atomic ensemble trapped in a single-sided optical cavity, and
with which we propose a high-efficiency quantum repeater protocol in
which the robust entanglement distribution is accomplished by the
stable spatial-temporal entanglement and it can in principle create
the deterministic entanglement between neighboring atomic ensembles
in a heralded way as a result of cavity quantum electrodynamics.
Meanwhile, the simplified parity check gate makes the entanglement
swapping be completed with unity efficiency, other than 1/2 with
linear optics. We detail the performance of our protocol with
current experimental parameters and show its robustness to the
imperfections, i.e., detuning and coupling variation, involved in
the reflection process. These good features make it a useful
building block in long distance quantum communication.
03.67.Pp, 03.65.Ud, 03.67.Hk
§ INTRODUCTION
Quantum mechanics provides some interesting ways for communicating
information securely between remote parties
<cit.>. However, in practice the
quantum channels such as optical fibers are noisy and lossy
<cit.>. The transmission loss and the decoherence of
photon systems increase exponentially with the distance, which
makes it extremely hard to perform a long-distance quantum
communication directly. To overcome this limitation, Briegel
et al. <cit.> proposed a noise-tolerant quantum
repeater protocol in 1998. The channel between the two remote
parties A and B is divided into smaller segments by
several nodes, the neighboring nodes can be entangled efficiently by
the indirect interaction through flying qubits, and the entanglement
between non-neighboring nodes is implemented by quantum entanglement
swapping, which can be cascaded to create the entanglement between
the terminate nodes A and B.
The implementation of quantum repeaters is compatible with different
physical setups assisted by cavity quantum electrodynamics, such as
nitrogen vacancy centers in diamonds <cit.>, spins in
quantum dots
<cit.>, single
trapped ions or atoms <cit.>. However, the
most widely known approach for quantum repeaters is based on
atomic ensembles <cit.> due to the collective enhancement
effect <cit.>. In a seminal paper by Duan et al.
<cit.>, the atomic ensemble is utilized to act as a local
memory node. The heralded collective spin-wave entanglement between
the neighboring nodes is established by the detection of a single
Stokes photon, emitted indistinguishably from either of the two
memory nodes via a Raman scattering process. However, due to the low
probability of Stokes photon emission required in the
Duan-Lukin-Cirac-Zoller (DLCZ) proposal <cit.>, the parties can
hardly establish the entanglement efficiently for quantum
entanglement swapping. In order to improve the success
probability, photon-pair sources and multimode memories are used to
construct a temporal multi-mode modification <cit.>,
and then the schemes based on the single-photon sources
<cit.> and spatial multiple modes
<cit.> are developed. Besides these protocols
based on Mach-Zehnder-type interference, Zhao et al.
<cit.> proposed a robust quantum repeater protocol
based on two-photon Hong-Ou-Mandel-type interference, which relaxes
the long-distance stability requirements and suppresses the vacuum
component to a constant item. Subsequently, the single-photon
sources are embedded to improve the performance of robust quantum
repeaters <cit.>. In addition,
Rydberg blockade effect <cit.> is used to perform
controlled-NOT gate between the two atomic ensembles in the middle
node <cit.>, which makes the quantum
entanglement swapping operation be performed deterministically.
Since the two-photon interference is performed with the polarization
degree of freedom (DOF) of the photons <cit.>, which
is incident to be influenced by the thermal fluctuation, vibration,
and the imperfection of the fiber <cit.>, the fidelity of
the entanglement created between the neighboring nodes will be
decreased when the photons are transmitted directly
<cit.>. In other words, the more the overlap
of the initial photon state used in the two-photon interference is,
the higher the fidelity of the entanglement created is. Following
the idea of Zhao's protocol <cit.>, quantum repeaters immune
to the rotational polarization noise are proposed with the time-bin
photonic state <cit.> and the antisymmetric Bell state
<cit.> $|\Psi^-\rangle=(|HV\rangle-|VH\rangle)/\sqrt{2}$,
respectively. When the noise on the two orthogonal polarized photon
states is independent, Zhang et al. <cit.> utilized
the faithful transmission of polarization photons <cit.> to
surmount the collective noise. In the ideal case, the two-fold
coincidence detection in the central node can successfully get the
stationary qubits entangled maximally in a heralded way. Apart from
this type of entanglement distribution, Kalamidas <cit.>
proposed an error-free entanglement distribution protocol in the
linear optical repeater. An entangled photon source is placed at the
center node, and the entangled photons transmitted to neighboring
nodes are encoded with their time-bin DOF. With two fast Pockels
cells (PCs), the entanglement distribution can be performed with a
high efficiency when the polarization-flip-error noise is relatively
In a recent work, Mei et al. <cit.> built a
controlled-phase-flip (CPF) gate between a flying photon and an
atomic ensemble embedded in an optical cavity, and constructed a
quantum repeater protocol, following some ideas in the original DLCZ
scheme <cit.>. In 2012, Brion et al. <cit.>
constituted a quantum repeater protocol with Rydberg blocked atomic
ensembles in fiber-coupled cavities via collective laser
manipulations of the ensembles and photon transmission. Besides,
Wang et al. <cit.> proposed a one-step
hyperentanglement distillation and amplification proposal, and Zhou
and Sheng <cit.> designed a recyclable protocol for the
single-photon entanglement amplification, which are quite useful to
the high dimensional or multiple DOFs optical quantum repeater.
In this paper, we give a general interface between a polarized
photon and an atomic ensemble trapped in a single-sided optical
cavity. Besides, we show that a deterministic faithful entanglement
distribution in a quantum repeater can be implemented with the
time-bin photonic state when two identical fibers act as the
channels of different spatial DOFs of the photons. Interestingly, it
does not require fast PCs and the time-slot discriminator
<cit.> is not needed
anymore. By using the input-output process of a single photon based
on our general interface, the entanglement between the neighboring
atom ensembles can be created in a heralded way, without any
classical communication after the clicks of the photon detectors,
and the quantum swapping can be implemented with almost unitary
success probability by a simplified parity-check gate (PCG) between
two ensembles, other than $1/2$ with linear optics. We analyze the
performance of our high-efficiency quantum repeater protocol with
current experimental parameters and show its robustness to the
imperfections involved in the reflection process. These good
features will make it a useful building block in long-distance
quantum communication in future.
§ RESULTS
§.§ A general interface between a polarized photon and an atomic ensemble.
The elementary node in our quantum repeater protocol includes an
ensemble with $N$ cold atoms trapped in a single-sided optical
cavity <cit.>. The atom has a four-level
internal structure and its relevant levels are shown in Fig. 1. The
two hyperfine ground states are denoted as $|g\rangle$ and
$|g_h\rangle$. The excited state $|e\rangle$ and the Rydberg state
$|r\rangle$ are two auxiliary states. The $|h\rangle$ polarized
cavity mode $a_h$ couples to the transition between $|g_h\rangle$
and $|e\rangle$. Initially, all of the atoms are pumped to the state
$|g\rangle$. With the help of the Rydberg state $|r\rangle$, one can
efficiently perform an arbitrary operation between the ground state
$|G\rangle=|g_1,\dots,g_j,\dots,g_{_N}\rangle$ and the single
collective spin-wave excitation state <cit.>
via collective laser manipulations of the ensembles
<cit.>. The single
collective excited state
When the Rydberg blockade shift is of the scale $ 2\pi\times
100MHz$, the transition between $|G\rangle$ and $|S\rangle$ can be
completed with an effective coupling strength $2\pi \times 1MHz$ and
the probability of nonexcited and doubly excited errors
<cit.> is about $10^{-3}$-$10^{-4}$. Recently, rotations
along axes $R_x$, $R_y$, and $R_z$ of a spin-wave excitation with
an average fidelity of $99\%$ are achieved in ${}^{87}Rb$ atomic
ensembles and they are implemented by making use of stimulated Raman
transition and controlled Larmor procession <cit.>. In
other words, the high-efficiency single qubit rotations of the
atomic ensemble can be implemented faithfully.
online) (a) Schematic diagram for a single-side cavity coupled to
an atomic ensemble system. (b) Atomic level
Let us consider an $|h\rangle$ polarized input photon with the
frequency $\omega$, which is nearly resonant to the cavity mode
$a_{_h}$ with the frequency $\omega_c$. The coupling rate between
the cavity and the input photon can be taken to be a real constant
$\sqrt{\frac{\kappa}{2\pi}}$ when the detuning $|\delta'|= |\omega
-\omega_c|$ is far less than the cavity decay rate $\kappa$
<cit.>. The Hamiltonian of
the whole system, in the frame rotating with respect to the cavity
frequency $\omega_c$, is ($\hbar=1$) <cit.>
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Equation 1
\hat{H}_s
\!\!&=&\!\!\sum_{j=1}^{N}\left[\left(\Delta-i\frac{\gamma_{e_{j}}}{2}\right)\!\hat{\sigma}_{e_{j}{e_{j}}}
+ig_j\!\left(\hat{a}_{_h}\hat{\sigma}_{e_{j}{s_{j}}}-\hat{a}_{_h}^{\dagger}\hat{\sigma}_{s_{j}{e_{j}}}\right)\right] \nonumber\\
\int\!
\label{hami}
\end{eqnarray}
where $\hat{a}$ and $\hat{b}$ are the operators of the cavity mode
and the input photon with the properties
$[\hat{a},\hat{a}^{\dagger}]=1$ and
respectively. $\Delta=\omega_0-\omega_c$ is the detuning between the
cavity mode frequency $\omega_c$ and the dipole transition
frequency $\omega_0$,
$\hat{\sigma}_{e_{j}{e_{j}}}=|e_{j}\rangle\langle e_{j}|$, and
$\hat{\sigma}_{e_{j}{s_{j}}}=|e_{j}\rangle\langle s_{j}|$.
$\gamma_{e_{j}}$ represents the spontaneous emission rate of the
excited state $|e_{j}\rangle$, while $g_{j}$ denotes the coupling
strength between the j-th atom transition and the cavity
mode $\hat{a}_{_h}$. Here and after, we assume $g_j = g$ and
$\gamma_{e_{j}} = \gamma$ for simplicity .
With the Hamiltonian $\hat{H}_s$ shown in Eq.(<ref>), the
Heisenberg-Langevin equations of motion for cavity $\hat{a}_h$ and
the atomic operator
$\hat{ \sigma}_{-}=|S\rangle{}\langle{}E|$ taking into account the atomic excited state decay $\gamma$ can be detailed as <cit.>
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Eq. 2
\begin{split}
\frac{d \hat{a}_h}{d t}\;=\;&-\left(
\frac{d \hat{\sigma}_{-}}{d t}\;=\;&-\left(
\right)\hat{\sigma}_{-}+ig\hat{\sigma}_{z}\hat{a}_h+\sqrt{\gamma}\,\hat{\sigma}_{z}\hat{N}.
\label{Heisenberg}
\end{split}
\end{eqnarray}
Here the Pauli operator
while $\hat{N}$ is corresponding to the vacuum noise field that
helps to preserve the desired commutation relations for the atomic
operator. Along with the standard cavity input-output relation
$\hat{a}_{out}=\hat{a}_{in}+\sqrt{\kappa}\,\hat{a}_h$, one can
obtain the reflection and noise coefficients $r(\delta')$ and
$n(\delta')$ in the weak excitation approximation where the
ensemble is hardly in the state $|E\rangle$ but predominantly in
$|S\rangle$, that is,
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 3
\label{r}
\begin{split}
\end{split}
\end{eqnarray}
where $\Delta'=\omega-\omega_0$ represents the frequency detuning
between the input photon and the dipole transition.
$|r{(\delta')}|^2+|n{(\delta')}|^2=1$ means that when the noise
field is considered, the energy is conserved during the input-output
process of the single-sided cavity.
(Color online) (a) $|r|$, $|n|$ and $|r_0|$ vs the scaled
detuning $\delta'/\kappa$, with the scaled coupling rate
$g/\kappa=4.0566$ and $\gamma/\kappa=0.0566$ <cit.>. (b)
$|r|$ vs the scaled coupling rate $g/\kappa$ with detuning
$\delta'/\gamma=0,0.5,$ and $1$.
If the atomic ensemble in the cavity is initialized to be the
state $|G\rangle$, it does not interact with the cavity mode (i.e.,
$g=0$). The input $|h\rangle$ polarized probe photon feels an empty
cavity and will be reflected by the cavity directly. Now, the
reflection coefficient can be simplified to be <cit.>
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 4
\label{r0}
\end{eqnarray}
Note that the detuning is small $|\delta'|\ll \kappa$, the pulse
bandwidth is much less than the cavity decay rate $\kappa$. If the
strong coupling condition $\gamma\kappa/4\ll{}g^2$ is achieved, one
can get the input probe photon totally reflected with
$r_{_0}{(\delta')}\simeq-1$ or $r{(\delta')}\simeq1$, shown in Fig.
2. The absolute phase shifts versus the scaled detuning are shown in
Fig. 3.
(Color online) The absolute phase shifts vs the scaled
detuning. The dashed and dashed-dot lines show the absolute phase
shifts $|\theta_0/\pi|$ and $|\theta/\pi|$ that the reflected photon
gets, with the ensemble in $|G\rangle$ and $|S\rangle$,
respectively. The solid line represents the absolute value of the
phase shifts difference $|\Delta\theta/\pi|$ =
$|\theta_0/\pi-\theta/\pi|$. The inset shows the phase shifts vs the
scaled detuning $\theta_0/\pi$ and $\theta/\pi$ that the reflected
photon gets, with the ensemble in $|G\rangle$ and $|S\rangle$,
§.§ Hybrid CPF gate on a photon-atomic-ensemble system and PCG on a two-atomic-ensemble system.
The principle of our CPF gate on a hybrid quantum system composed of
a photon $p$ and an atomic ensemble $E_A$ is shown in Fig. 4,
following some ideas in previous works
<cit.>. Suppose that the
photon $p$ is in the state
$|\varphi_p\rangle=\mu|h\rangle+\nu|v\rangle$ ($|\mu|^2+|\nu|^2=1$)
and the ensemble $E_{_A}$ is in the state
$|\phi_{_A}\rangle=\mu'|G\rangle+\nu'|S\rangle$ ($
|\mu'|^2+|\nu'|^2=1$). The $|h\rangle$ polarized component of the
photon $p$ transmits the polarization beam splitter (PBS) and then
be reflected by the cavity, while the $|v\rangle$ polarized
component is reflected by the mirror $M$. The optical pathes of the
$|h\rangle$ and $|v\rangle$ components are adjusted to be equal and
they will be combined again at the PBS with an extra $\pi$ phase
shift on the $|h\rangle$ component if the ensemble is in the state
$|G\rangle$. This process can be described as
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 5
\begin{split}
|\phi_{_A}\rangle \otimes |\varphi_p\rangle\rightarrow &\mu'|G\rangle\otimes(-\mu|h\rangle+\nu|v\rangle)\\
\end{split}
\end{eqnarray}
That is to say, the setup in Fig. 4(a) can be used to accomplish a
CPF gate on the atomic ensemble $E_{_A}$ and the photon $p$.
(Color online) Schematic setup for implementing a CPF gate
and a parity-check gate (PCG). $M$ stands for a mirror and the PBS
transmits the $|h\rangle$ polarized photon and reflects the
$|v\rangle$ component. HWP$_1$ and HWP$_2$ are half wave plates
performing the bit-flip operation while H represent a Hadamard
The schematic diagram of our PCG on two atomic ensembles $E_{_A}$
and $E_{_B}$ is shown in Fig. 4(b). Let us assume that $E_A$ and
$E_B$ are initially in the states
$|\phi_{i}\rangle=\mu_{i}|G\rangle_{i}+\nu_{i}|S\rangle_{i}$ (
$|\mu_{i}|^2+ |\nu_{i}|^2=1$ and $i=A$, $B$).
One can input a polarized photon p in the state
$|\varphi_p\rangle=\frac{1}{\sqrt{2}} (|h\rangle+|v\rangle)$ into
the import of the setup. HWP$_1$ (HWP$_2$) is used to perform the
bit-flip operation $|h\rangle \leftrightarrow|v\rangle$ on the
photon $p$ by using a half-wave plate (HWP) with its axis at $\pi/4$
with respect to the horizontal direction. After the two components
of $p$ are reflected by the two cavities, they combine with each
other at PBS$_2$. The state of the system composed of the two atom
ensembles and the photon evolves to be
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 6
|h\rangle\!\otimes\!(-\mu_{_{A}}|G\rangle_{_{A}}\!+\!\nu_{_{A}}|S\rangle_{_{A}})(\mu_{_{B}}|G\rangle_{_{B}}\!+\!\nu_{_{B}}|S\rangle_{_{B}}) \nonumber\\
\end{eqnarray}
And then, another HWP names $H$ whose axis is placed at $\pi/8$
is used to perform a Hadamard rotations
$|h\rangle\leftrightarrow1/\sqrt{2}(|h\rangle+|v\rangle)$ and
$|v\rangle\leftrightarrow1/\sqrt{2}(|h\rangle-|v\rangle)$ on the
photon. The state of the system becomes
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 7
\label{phipab}
- \mu_{_{A}}\mu_{_{B}}|G\rangle_{_{A}}|G\rangle_{_{B}}) \nonumber\\
|v\rangle\otimes(\nu_{_{A}}\mu_{_{B}}|S\rangle_{_{A}}|G\rangle_{_{B}} - \mu_{_{A}}\nu_{_{B}}|G\rangle_{_{A}}|S\rangle_{_{B}}).\nonumber\\
\end{eqnarray}
After the photon is measured with PBS$_3$ and two single-photon
detectors, the parity of $E_A$ and $E_B$ can be determined. In
detail, if the photon is in the state $|h\rangle$, the two
ensembles $E_A$ and $E_B$ have an even parity. If the photon is in
$|v\rangle$, $E_A$ and $E_B$ have an odd parity. With an effective
input-output process of a single photon, one can efficiently
complete the PCG on two atomic ensembles.
(Color online) Schematic setup for entanglement
distribution. p$_\pi$ is a $\pi$ phase shifter.
§.§ Entanglement distribution with faithful single-photon transmission.
Suppose that there is an entanglement source which is placed at a
central station between two neighboring nodes, say Alice and Bob.
The source produces a two-photon polarization-entangled Bell state
Here the subscripts $a$ and $b$ denote the photons sent to
Alice and Bob, respectively. As shown in Fig. 5 (a), the photons
a and b will pass through an encoder in each side
before they enter the noisy channels. The encoder is made up of a
PBS, an HWP, and a beam splitter (BS). Here BS is used for a
Hadamard rotation on the spatial DOF of the photon, i.e.,
$|u\rangle\leftrightarrow \frac{1}{\sqrt{2}} (|u\rangle+|d\rangle)$
and $|d\rangle\leftrightarrow \frac{1}{\sqrt{2}}
(|u\rangle-|d\rangle)$, where $|u\rangle$ and $|d\rangle$ represent
the upper and the down ports of the BS, respectively.
With our faithful single-photon transmission method (see Method),
Alice and Bob can share photon pairs in a maximally entangled
state, shown in Fig. 5. In detail, after a photon pair from the
source passes through the two encoders, its state becomes
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 8
\\&&
\end{eqnarray}
As the two photons a and b suffer from independent
collective noises from the two channels, the influence of the
channels on the two photons can be described with two unitary
rotations $U^a_{C}$ and $U^b_{C}$ as follows:
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 9--10
U^a_{C}|h\rangle &\xrightarrow[]{noise}& \delta_a|h\rangle+\eta_a|v\rangle,\\
U^b_{C}|h\rangle &\xrightarrow[]{noise}&
\delta_b|h\rangle+\eta_b|v\rangle,
\end{eqnarray}
where $|\delta_i|^2+|\eta_i|^2=1$ ($i=a,b$). The influence on the
polarization of the photons arising from the channel noises can be
totally converted into that on the spatial DOF. The state of the
photons a and b arriving at Alice and Bob becomes
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 11
|\varphi\rangle_{ab_2}&=&\frac{1}{\sqrt{2}}(|h\rangle_a|v\rangle_b+|v\rangle_a|h\rangle_b) \nonumber\\
&&\otimes(\delta_a|a_1\rangle+\eta_a|a_2\rangle)\otimes(\delta_b|b_1\rangle+\eta_b|b_2\rangle) \nonumber\\
&= &|\varphi\rangle_{ab}^p\otimes|\varphi\rangle_{ab}^s.
\end{eqnarray}
This is a two-photon Bell state $|\varphi\rangle_{ab}^p =
\frac{1}{\sqrt{2}}(|h\rangle_a|v\rangle_b+|v\rangle_a|h\rangle_b)$
in the polarization DOF of the photon pair $ab$. Simultaneously, it
is a separable superposition state $|\varphi\rangle_{ab}^s =
in the spatial DOF.
To entangle the stationary atomic ensembles $E_A$ and $E_B$, which
are initialized to be
only two CPF gates are required if Alice and Bob have shared some
photon pairs in the Bell state $|\varphi\rangle_{ab}^p$. Let us take
the case that the photons $a$ and $b$ come from the spatial modes
$a_2$ and $b_2$ as an example to detail the entanglement creation
process. As for the other cases, the same entanglement between
$E_A$ and $E_B$ can be obtained by a similar procedure with or
without some single-qubit operations.
First, the photon a suffers a Hadamard operation by passing
through a half-wave plate H. Second, it is reflected by the
cavity or the mirror $M$, which is used to complete the CPF gate
on the photon a and the ensemble $E_A$. Third, Alice
performs another Hadamard operation on the photon a. Now,
the state of the composite system composed of the photons a
and b and the ensembles $E_A$ and $E_B$ evolves into
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 12
\big[|h\rangle_{a}(|v\rangle_{b}|S\rangle_{_{A}}-|h\rangle_{b}|G\rangle_{_{A}}) \nonumber\\
\end{eqnarray}
Fourth, Alice measures the polarization state of the photon $a$ with
a setup composed of PBS and single-photon detectors $D_h$ and $D_v$.
If an $|h\rangle$ polarized photon is detected, the hybrid system
composed of b, $E_A$, and $E_B$ will be projected into
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 13
\end{eqnarray}
If a $|v\rangle$ polarized photon is detected, the remaining hybrid
system can also be transformed into the state $|\Phi\rangle_{PE_2}$
by a bit-flip operation
on the ensemble $E_A$.
Up to now, the original entanglement of the photon pair ab is
mapped to the hybrid entanglement between the photon b and
the ensemble $E_A$. In order to create the entanglement between
$E_A$ and $E_B$, Bob just performs the same operations as Alice
does. In brief, before and after the CPF operation on the photon
b and the ensemble $E_B$, Bob performs two local Hadamard
operations on the photon b with H. These operations
result in the entanglement between the photon b and the two
atomic ensembles. The state $|\Phi\rangle_{PE_2}$ is changed into
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 14
|\Phi\rangle_{PE_3}&=&\frac{1}{2}\big[|v\rangle_{b}\otimes(|S\rangle_{_{A}}|S\rangle_{_{B}}+|G\rangle_{_{A}}|G\rangle_{_{B}}) \nonumber\\
\end{eqnarray}
If the detector D$_{h}$ at Bob's node is clicked, the state of the
system composed of $E_A$ and $E_B$ will be collapsed into the
desired entangled state
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 15
\end{eqnarray}
As for the case that the photon b is in the state
$|v\rangle$, they can also obtain the desired entangled state
$|\Psi\rangle_{_{AB}}$ with an additional bit-flip operation
$\hat{\sigma}^B_{x}$ on $E_B$.
§.§ Entanglement swapping on atomic ensembles with a PCG.
After the parties produce successfully the entanglement between each
two atomic ensembles in the neighboring nodes, they can extend the
entanglement to a further distance by entanglement swapping. Let us
use the case with three nodes as an example to describe the
principle for connecting the two non-neighboring nodes.
(Color online) Schematic setup for entanglement swapping
with the simplified PCG.
Suppose the atomic ensembles $E_A$ and $E_C$ belong to the two
non-neighboring nodes Alice and Charlie, respectively, and the two
ensembles $E_{B_1}$ and $E_{B_2}$ belong to the middle node Bob,
shown in Fig. 6. The two ensembles $E_AE_{B_1}$ are in the state
and the two ensembles $E_{B_2}E_C$ are in the state
After a parity-check measurement performed on the two local
ensembles $E_{B_1}$ and $E_{B_2}$ with a PCG shown in Fig. 3 (b),
the state of the system composed of the four ensembles $E_A$, $E_C$,
$E_{B_1}$, and $E_{B_2}$ evolves into an entangled one. If the
outcome of the parity-check measurement on the ensembles $B_1B_2$ is
odd, the composite system composed of $E_{B_1}$, $E_{B_2}$,
$E_A$, and $E_{_C}$ will be projected into the state
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 16
\!+\!|S\rangle_{_{B_1}}|G\rangle_{_{B_2}}|G\rangle_{_{A}}|S\rangle_{_{C}}),\nonumber\\
\end{eqnarray}
which is a four-qubit Greenberger-Horne-Zeilinger state. The
decoherence of both $E_{B_1}$ and $E_{B_2}$ has an awful influence
on the system composed of $E_{A}$ and $E_{C}$ as it decreases the
fidelity of the entanglement of the system. In order to disentangle
the two ensembles $E_{B_1}$ and $E_{B_2}$ from the system, the party
at the middle node could first perform a Hadamard operation on the
two ensembles and then apply a parity-check measurement on them. If
the outcome of the second parity-check measurement is even, the
composite system composed of the four ensembles $E_{B_1}$,
$E_{B_2}$, $E_A$, and $E_C$ is projected into the state
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 17
\otimes(|S\rangle_{_{A}}|G\rangle_{_{C}}+|G\rangle_{_{A}}|S\rangle_{_{C}}),%\nonumber\\
\end{eqnarray}
where the ensembles $E_{B_1}$ and $ E_{B_2}$ are decoupled from
the system composed of the two nonlocal ensembles $E_{A}$ and
$E_{C}$ which are in the maximally entangled state
In the discussion above, we use the outcomes (odd, even) of the two
successive parity-check measurements as an example to describe the
principle of the entanglement swapping between the four atomic
ensembles. In fact, the other cases that the outcomes of each
parity-check measurement is either an odd one or an even one can
also be used for the entanglement swapping with only a single-qubit
operation on the ensemble $E_A$, shown in Table. 1.
The relation between the single-qbuit operation
on the ensemble $E_{A}$ for entanglement swapping and the outcomes
of the parity-check measurements on the two atomic ensembles at the
middle node. $P_1$ and $P_2$ denote the outcomes of the first and
the second parity-check measurements. Here $\hat{\sigma}_I=\vert
G\rangle_A\langle G\vert + \vert S\rangle_A\langle S\vert$,
$\hat{\sigma}_z=\vert G\rangle_A\langle G\vert - \vert
S\rangle_A\langle S\vert$, $\hat{\sigma}_y=\vert G\rangle_A\langle
S\vert - \vert S\rangle_A\langle G\vert$, and $\hat{\sigma}_x=\vert
G\rangle_A\langle S\vert +\vert S\rangle_A\langle G\vert$.
P$_1$ P$_2\quad\quad$ E$_{A}\quad\quad$
$v$ $h$ $\hat{\sigma}_I$
$v$ $v$ $\hat{\sigma}_z$
$h$ $v$ $\hat{\sigma}_y$
$h$ $h$ $\hat{\sigma}_x$
§ DISCUSSION
We would like to briefly discuss the imperfections of our quantum
repeater protocol. The photon loss is the main imperfection, which
is also of crucial importance for the previous quantum repeaters
with photon interference <cit.>.
The photon loss happens, due to the fiber absorbtion, diffraction,
the cavity imperfection, and the inefficiency of the single-photon
detectors. It will decrease the success probability and prolong the
time needed for establishing the quantum repeater. Since the memory
node in this protocol is implemented with the atomic ensemble, the
local operation between two collective quantum states $|G\rangle$
and $|S\rangle$ of the memory node, can be performed with collective
laser manipulations <cit.>, while excitations of
higher-order collective states can be suppressed efficiently with
the Rydberg blockade <cit.>. During the
entanglement swapping process, to detect the collective state of two
ensembles in the centering nodes, fluorescent detection
<cit.> can be used, since the detection efficiencies of
$99.99\%$ for trapped ions have been experimentally demonstrated
<cit.>. Moreover, with the current significant progress
on the source of entangled photon pairs, the repetition rate as high
as $10^6/10^7S^{-1}$ has been achieved <cit.>, so our
entanglement distribution process can be performed with a high
In summary, we have proposed a high-efficiency quantum repeater with
atomic ensembles embedded in optical cavities as the memory nodes,
assisted by single-photon faithful transmission. By encoding the
polarization qubit into the time-bin qubit, our faithful
single-photon transmission can be completed with only linear-optical
elements, and neither time-slot discriminator nor fast PCs is
required <cit.>. The
heralded entanglement creation between the neighboring nodes is
achieved with a CPF gate between the atomic ensemble and the photon
input in each node, which makes our scheme more convenient than the
one with post selection <cit.>, although both efficiencies
of our quantum repeaters are identical and maximal among all the
exciting quantum repeater schemes when multi-mode speed up is not
considered <cit.>. Besides, no
additional classical information is involved to determinate the
state of the entangled atomic ensembles, since the parties can
create a deterministic entanglement up to a feedback upon the
results of photon detection. The quantum swapping process is
deterministically completed with a simplified PCG involving only one
input-output process, which makes our scheme far more efficient
than the ones based on linear optical elements <cit.>.
§ METHODS
§.§ Faithful single-photon transmission
Our protocol for deterministic polarization-error-free
single-photon transmission can be details as follows. Assuming the
initial state of the single photon to be transmitted is
$|\varphi\rangle=\mu|h\rangle+\nu|v\rangle$ ($|\mu|^2+|\nu|^2=1$).
After passing through the encoder, the photon launched into the
noisy channel evolves into
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 18
\end{eqnarray}
where the subscripts l and s represent the photons
passing through the long path and short path of the encoder,
respectively. When the optical path difference between l and
s is small, the two time bins are so close that they suffer
from the same fluctuation from the optical fiber channels
The noise of the channel can be expressed with a unitary
transformation U$_C$ as follows:
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 19
U_C|h\rangle \xrightarrow[]{\text{ noise }}
\;\;\delta|h\rangle+\eta|v\rangle,
\end{eqnarray}
where $ |\delta|^2+|\eta|^2=1$. After the photon passes through the
channels, a $\pi$ phase shifter $P_{\pi}$ on the d channel
is applied, and the state of the photon becomes
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 20
|\varphi''\rangle= \frac{1}{\sqrt{2}}(\delta|h\rangle\!+\eta|v\rangle)\otimes(\nu|u_l\rangle\!-\!\nu|d_l\rangle+\mu|u_s\rangle+\!\mu|d_s\rangle).\nonumber\\
\end{eqnarray}
With a decoder composed of a BS, an HWP, and a PBS, shown in Fig. 5
(b), the evolution of the photon can be described as follows:
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 21
|\varphi''\rangle \!\!&\xrightarrow[]{\text{ BS }}&\!\! (\delta|h\rangle\!+\!\eta|v\rangle)\otimes(\nu|d_{ls}\rangle\!+\!\mu|u_{sl}\rangle) \nonumber\\
\!\! &\xrightarrow[]{\text{ HWP }}&\!\!
\nu|d_{ls}\rangle\otimes(\delta|h\rangle\!+\!\eta|v\rangle)
\!+\!\mu|u_{sl}\rangle\otimes(\delta|v\rangle\!+\!\eta|h\rangle) \nonumber\\
\!\!&\xrightarrow[]{\text{ PBS }}&\!\!
\delta|a_1\rangle(\nu|h\rangle\!+\!\mu|v\rangle)\!+\!\eta|a_2\rangle(\mu|h\rangle\!+\!\nu|v\rangle) \nonumber\\
\!\!& \xrightarrow[]{\text{ HWP }}&\!\!
\end{eqnarray}
Here the subscripts $ls$ ($sl$) represent the photon that passes
through the long (short) path of the encoder and the short (long)
path of the decoder, respectively. The difference between the long
path and the short one for the encoder is designed to be the same as
that for the decoder. Without any time-slot discriminator, one can
get the error-free photon in either the output $a_1$ or $a_2$ at
a deterministic time slot.
§.§ Performance of CPF and PCG with current experimental parameters.
Before we analyze the fidelity of the quantum entanglement
distribution and entanglement swapping in our quantum repeater
scheme, we first discuss the practical performance of the CPF gate
and the PCG based on the recent experiment advances
<cit.>. We define the fidelity of
a quantum process (or a quantum gate) as $F = |\langle \Psi_i
|\Psi_r \rangle|^2$, where $\vert \Psi_i\rangle$ and $\vert
\Psi_r\rangle$ are the output states of the quantum system in the
quantum process (or the quantum gate) in the ideal condition and
the realistic condition, respectively <cit.>.
By combining a fibre-based cavity with the atom-chip technology,
Colombe et al. <cit.> demonstrated the strong
atom-field coupling in a recent experiment in which each $^{87}$Rb
atom in Bose-Einstein condensates is identically and strongly
coupled to the cavity mode. In this experiment, all the atoms are
initialized to be the hyperfine zeeman state $|5S_{1/2}, F=2,
m_f=2\rangle$. The dipole transition of $^{87}$Rb $|5S_{1/2},
F=2\rangle$ $\mapsto$ $|5P_{3/2}, F'=3\rangle$ is resonantly coupled
to the cavity mode with the maximal single-atom coupling strength
$g_0=2\pi\times 215$ MHz. Meanwhile, the cavity photon decay rate is
$\kappa=2\pi\times53$ MHz and the atomic spontaneous emission rate
of $|5P_{3/2}, F'=3\rangle$ is $\gamma=2\pi\times3$ MHz. The
whispering-gallery microcavities (WGMC) <cit.> might be
another potential experimental realization of our scheme. The
parity-time-symmetry breaking is realized in a system of two
directly coupled WGMC <cit.> and the controlled loss is also
achieved with WGMC <cit.>, which enables the on-chip
manipulation and control of light propagation. In addition, the
routing of single photons has been demonstrated by the atom-WGMC
coupled unit controlled by a single photon <cit.>.
Under an ideal condition, the reflection coefficients of the
input-output processes are $r_{_0}{(\delta')}\simeq-1$ and
$r{(\delta')}\simeq1$. In this time, the input $|h\rangle$
polarized photon a will get a $\pi$ phase shift when the
embedded atomic ensemble $E_{_A}$ is in the state $|G\rangle$;
otherwise, there is no phase shift on the photon $a$. The fidelity
of both the CPF gate (shown in Fig.4 (a)) and the PCG (shown in Fig.
4 (b)) can reach unity. In a realistic atom-cavity system, the
relationship between the input and output field is outlined in Eqs.
(<ref>) and (<ref>). In this time, after the party operates the
photon a and the ensemble $E_{_A}$ with the CPF gate, the
output state of the composite system becomes
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 22
\begin{split}
&+\nu'|S\rangle\otimes(r \mu|h\rangle+\nu|v\rangle)].
\end{split}
\end{eqnarray}
Here the normalized coefficient
The fidelity of the CPF gate
$F_{cpf}=|_{_Ep}\langle\Phi|\Phi'\rangle{_{_Ep}}|^2$ depends on the
input state of the system composed of the photon and the atomic
ensemble. In the symmetric case with $\mu=\mu=\mu'=\nu'=1/\sqrt{2}$,
the fidelity $F_{cpf}$ can be simplified to be
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 23
\end{eqnarray}
Meanwhile, the efficiency $\eta_{cpf}$ of the CPF gate, which is
defined as the probability that the photon clicks either detectors
after being reflected by the CPF gate, can be detailed as
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 24
\eta_{cpf}= \frac{1}{2} + \frac{|r|^2 + |r_0|^2}{4}.
\end{eqnarray}
In a realistic condition, the output state of the composite system
composed of a, $E_A$, and $E_B$ in the PCG process before
the single photon is detected becomes
\begin{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 25
|\Phi''\rangle_{p_{AB}}&=&\frac{1}{\sqrt{C}}\{|v\rangle(r_0-r)(|G,S\rangle-|S,G\rangle) \nonumber\\
&&+|h\rangle[\sqrt{2}r_0|G,G\rangle+\sqrt{2}r_0|S,S\rangle \nonumber\\
&& +(r_0+r)(|G,S\rangle+|S,G\rangle)]\}.
\end{eqnarray}
Compared with the ideal output state described in Eq.(<ref>),
if an $|h\rangle$ polarized photon is detected, the fidelity of the
PCG gate $F_{pcg}$ can be expressed as
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 26
\end{eqnarray}
When the photon in the state $|v\rangle$ is detected, the fidelity
of the PCG is $F'_{pcg}=1$. The success of the PCG is heralded when
a single photon is detected after the parity-check process, no
matter what the state the photon evolves to be. The efficiency
$\eta_{pcg}$ of the PCG process can be defined as the probability
that the probe photon is detected after it is reflected by the two
cavities, that is,
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 27
\eta_{cpf}= \frac{|r|^2+|r_0|^2}{2}.
\end{eqnarray}
Since the absolute value of the relative phase shift during the
input-output process depends on the frequency of the input photon,
it decreases smoothly with the detuning $\delta'$ between the
input photon and the cavity mode, shown in Fig. 3.
(Color online) (a) Fidelities of CPF gate and PCG vs the
scaled detuning. $F'_{cpf}$ and $F'_{pcg}$ is performed with the
scaled coupling rate $g/\kappa=2.0283$ and $\gamma/\kappa=0.0566$,
$F_{cpf}$ and $F_{pcg}$ are performed with the scaled coupling rate
$g/\kappa=4.0566$ and $\gamma/\kappa=0.0566$ <cit.>. (b)
Fidelities of CPF gate and PCG gate VS the scaled coupling rate.
$F'_{cpf}$ and $F'_{pcg}$ are performed with the scaled detuning
$\delta'/\kappa=0.0283$ and $\gamma/\kappa=0.0566$, $F_{cpf}$ and
$F_{pcg}$ is performed with $\delta'/\kappa =
\gamma/\kappa=0.0566$.
(Color online) Efficiencies of CPF gate and PCG gate vs
the scaled coupling rate. $\eta'_{cpf}$ and $\eta'_{pcg}$ is
performed with the scaled detuning $\delta'/\kappa=0.0283$ and
$\gamma/\kappa=0.0566$, $\eta_{cpf}$ and $\eta_{pcg}$ are performed
with $\delta'/\kappa = \gamma/\kappa=0.0566$.
The fidelity of the CPF gate $F_{cpf}$ changes with the detuning
$\delta'$, shown in Fig. 7(a). Here the parameters are chosen as
$g/\kappa=2.0283$ or $4.0566$ and $\gamma/\kappa=0.0566$
<cit.>. When the linewidth of the input photon is
$\delta=2|\delta'|_{max}$ with the maximal detuning
$|\delta'|_{max}=0.5\gamma$ ($\gamma$), $F_{cpf}$ is larger than
$F_{cpf}(|\delta'|_{max})=0.9974$ ($0.9906$) for $g/\kappa=4.0566$.
The fidelity of the PCG depends on the coupling rate $g/\kappa$, as
shown in Fig. 7(b) with the detuning $|\delta'|_{max}=0.5\gamma$ or
$\gamma$. When the maximal detuning
of the input photon is $|\delta'|_{max}=0.5\gamma$,
the high-performance parity-check gate can be achieved with the
fidelity $F_{pcg}$ higher than $F_{pcg}(|\delta'|_{max})=0.9944$ and
$0.9938$ for $g/\kappa=2.0283$ and $g/\kappa=4.0566$, respectively.
The efficiencies of the CPF gate and the PCG process versus the
coupling rate $g/\kappa$ are shown in Fig. 8. When the bandwidth of
the probe photon is on the scale of $\gamma$, both efficiencies
$\eta_{cpf}$ and $\eta_{pcg}$ are robust to the variation of
$g/\kappa$ with the parameters above <cit.>.
In detail, when the maximal detuning $|\delta'|_{max}$ of the input
photon is less than $0.5\gamma$, $\eta_{cpf}$ and $\eta_{pcg}$ are
higher than $0.9966$ and $0.9932$, respectively. When
$|\delta'|_{max}=\gamma$, $\eta_{cpf}=0.9991$
and $\eta_{pcg}=0.9983$ are achievable.
(Color online) Fidelities of F$_{mh}$,
F$_{mv}$ and F$_s$ vs the detuing, with the scaled coupling
rate $g/\kappa=2.0283$ and $\gamma/\kappa=0.0566$.
(Color online) Efficiencies of entanglement distribution
and entanglement swapping processes vs the scaled coupling rate.
$\eta'_{m}$ and $\eta'_{s}$ is performed with the scaled detuning
$\delta'/\kappa=0.0283$ and $\gamma/\kappa=0.0566$, $\eta_{m}$ and
$\eta_{s}$ are performed with $\delta'/\kappa =
\gamma/\kappa=0.0566$.
§.§ Performance of entanglement distribution and entanglement swapping.
Now, let us discuss the fidelities and the efficiencies of the
entanglement distribution and entanglement swapping in our quantum
repeater scheme. After Alice performs the local operations on the
photon a and detects an $|h\rangle$ polarized photon, the
composite system composed of the photon b and the ensembles
$E_A$ and $E_B$ will be projected into the state
$|\Phi'_{PE_2}\rangle$, instead of $|\Phi\rangle_{PE_2}$. Here
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 28
[(r_0-1)|h\rangle\otimes|G\rangle_{_{A}} \nonumber\\
&&+(r_0+1)|v\rangle\otimes|G\rangle_{_{A}}+(r-1)|h\rangle\otimes|S\rangle_{_{A}} \nonumber\\
\end{eqnarray}
where the normalized coefficient
$C=2[|r_0-1|^2+|r_0+1|^2+|r-1|^2+|r+1|^2]$. And then, the same
operations, i.e., a CPF gate sandwiched by two Hadamard operations,
are performed by Bob on the photon b. After these
operations, the state of the composite system composed of the
photon b and the two ensembles $E_A$ and $E_B$ evolves into
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 29
|\varphi\rangle_{pE_2}&=&\frac{1}{\sqrt{C'}} \{|h\rangle\!\otimes\![(r_0^2-1)|G\rangle_{_{A}}\!\otimes\!|G\rangle_{_{B}} \nonumber\\
&& +(r_0\cdot{}r-1)(|G\rangle_{_{A}}\!\otimes\!|S\rangle_{_{B}}\!+\!|S\rangle_{_{A}}\!\otimes\!|G\rangle_{_{B}}) \nonumber\\
&& +(r^2-1)|S\rangle_{_{A}}\otimes|S\rangle_{_{B}}] \nonumber\\
&& +|v\rangle\!\otimes\![(r_0^2+1)|G\rangle_{_{A}}\!\otimes\!|G\rangle_{_{B}} \nonumber\\
&& +(r_0\cdot{}r+1)(|G\rangle_{_{A}}\!\otimes\!|S\rangle_{_{B}}\!+\!|S\rangle_{_{A}}\!\otimes\!|G\rangle_{_{B}}) \nonumber\\
&& +(r^2+1)|S\rangle_{_{A}}\otimes|S\rangle_{_{B}}]\},
\end{eqnarray}
where the normalized coefficient
$C'=2[|r_0^2|^2+|r^2|^2+2|r\cdot{}r_0|^2+4]$. One can obtain the
fidelity of the entanglement distribution process $F_{mh}$ and
$F_{mv}$ for the cases that $D_h'$ and $D_v'$ at the Bob's node are
clicked, respectively.
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 30--31
F_{mh}&=&\frac{2|r\cdot{}r_0-1|^2}{|r_0^2-1|^2+|r^2-1|^2+2|r\cdot{}r_0-1|^2}, \nonumber\\
\label{fmhfmv}
\end{eqnarray}
If one defines the efficiency $\eta^h_m$ as the probability that
Alice detects an $|h\rangle$ polarized photon while Bob detects a
photon in either $|h\rangle$ or $|v\rangle$ polarization, one has
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation 32
\eta^h_{m}=\frac{C}{32}\cdot\frac{C'}{C}=\frac{|r_0^2|^2+|r^2|^2+2|r\cdot{}r_0|^2+4}{16}.
\end{eqnarray}
In the above discussion, we detail the performance of our
entanglement distribution conditioned on the detection of an
$|h\rangle$ polarization photon at Alice's node. Considering the
symmetric property of the system, one can easily obtain the
performance of the entanglement distribution upon the detection of a
$|v\rangle$ polarization photon at Alice's node. Now, the
fidelities $F'_{mh}$ and $F'_{mv}$ for the cases that $D_h'$ and
$D_v'$ are clicked at Bob's node, have the following relations to
that for the cases that an $|h\rangle$ polarized photon is detected
by Alice, $F'_{mh}=F_{mv}$ and $F'_{mv}=F_{mh} $, see Eq.
(<ref>) for detail. Meanwhile, the efficiency $\eta^v_m$ of
the entanglement distribution process when Alice detects a photon in
$|v\rangle$ polarization is identical to $\eta^h_m$. The total
efficiency $\eta_m$ of the entanglement distribution can be written
\begin{eqnarray}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equation35-36
\eta_m=\eta^h_{m}+\eta^v_{m}=\frac{|r_0^2|^2+|r^2|^2+2|r\cdot{}r_0|^2+4}{8}.
\end{eqnarray}
In our entanglement swapping process, two PCGs are applied on the
two ensembles $E_{B_1}$ and $E_{B_2}$ at the middle node. In fact,
only one PCG is enough if a single-atomic-ensemble measurement on
each of the two ensembles $E_{B1}$ and $E_{B2}$ is utilized after
the local Hadamard operations. After these measurements, the system
composed of the two remote ensembles $E_A$ and $E_C$ is in the state
$|\Psi\rangle_{_{AC}}$ with or without a local unitary operation.
When the fluorescent measurement <cit.> or field-ionizing
the atoms <cit.> with the help of Rydberg excitation are
used, the state detection on atomic ensembles could be performed
with a near-unity efficiency. In other words, the fidelity of the
quantum entanglement swapping process can equal to that of the PCG
The fidelities of both the entanglement distribution and the
entanglement swapping in our repeater scheme are shown in Fig. 9.
One can see that all $F_{mh}$, $F_{mv}$, and $F_{s}=F_{pcg}$ are
larger than $0.9936$ with the parameters ($g$, $\kappa$,
$\gamma$)$\;=2\pi\times$($215$, $53$, $3$) MHz achieved in
experiment <cit.>. Meanwhile, all efficiencies involved in
our quantum repeater protocol, shown in Fig. 10, can be larger than
$0.9931$ when the effective coupling $g/\kappa>2.0283$ with
$\delta'/\kappa = \gamma/\kappa=0.0566$. In a recent experiment
with a fiber-based Fabry-Perot cavity constituted by CO$_2$
laser-machined mirrors <cit.>, the maximal coupling
strength as high as $g=2\pi\times2.8$ GHz is achieved for single Rb
atoms and the cavity decay rate is $\kappa=2\pi\times0.286$ GHz
$\simeq95\gamma$. In this time, $g/\kappa = 9.79$ is achieved, and a
better performance of our scheme is attainable.
§ ACKNOWLEDGEMENTS
TL was supported by the China Postdoctoral Science Foundation under
Grant No. 2015M571011.
FG was supported by the National Natural Science Foundation of
China under Grant Nos. 11174039 and 11474026, and the Open
Foundation of State key Laboratory of Networking and Switching
Technology (Beijing University of Posts and Telecommunications)
under Grant No. SKLNST-2013-1-13.
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|
1511.00535
|
Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
Institute of Physics, VAST, 10 Dao Tan, Ba Dinh Distr., 118011 Hanoi, Vietnam
Institute of Physics, VAST, 10 Dao Tan, Ba Dinh Distr., 118011 Hanoi, Vietnam
Institute for Bio-Medical Physics, 109A Pasteur, 1st Distr., 710115 Hochiminh City, Vietnam
We adapt the transfer matrix ($\T$-matrix) method originally designed for one-dimensional quantum mechanical problems to solve the circularly symmetric two-dimensional problem of graphene quantum dots. In similarity to one-dimensional problems, we show that the generalized $\T$-matrix contains rich information about the physical properties of these quantum dots. In particular, it is shown that the spectral equations for bound states as well as quasi-bound states of a circular graphene quantum dot and related quantities such as the local density of states and the scattering coefficients are all expressed exactly in terms of the $\T$-matrix for the radial confinement potential. As an example, we use the developed formalism to analyse physical aspects of a graphene quantum dot induced by a trapezoidal radial potential. Among the obtained results, it is in particular suggested that the thermal fluctuations and electrostatic disorders may appear as an obstacle to controlling the valley polarization of Dirac electrons.
§ INTRODUCTION
Transfer matrix ($\T$-matrix) is a classic quantum mechanics approach that is widely used to treat a variety of physical problems <cit.>. Linearly relating the parameters of the Schrödinger waves in the two sides of a potential barrier, the $\T$-matrix contains a rich information of quantum characteristics of the potential examined. The effectiveness of the $\T$-matrix approach relies on its analytic simplicity and on the fact that $\T$-matrices can be easily multiplied when treating relatively complicated potential barriers. Exact expressions for the energy structure as well as the transport characteristics of semiconductor super-lattices that were derived by Esaki and Tsu <cit.> could be seen as a typical example of elegant successes of this approach.
As for the graphene, when charge carriers behave like the two-dimensional (2D) Dirac relativistic fermions, the $\T$-matrix approach has also been shown to be an effective approach. For the graphene nanostructures induced by one-dimensional (1D) potentials, such as the multi-barrier structures or the $n$-$p$-$n$-junctions, the $\T$-matrix calculations have been developed to study the energy spectrum <cit.> as well as the dynamical characteristics <cit.>. In particular, the $\T$-matrix approach was successfully used to calculate the electronic band structure and the transport properties of various single-/bi-layer graphene superlattices induced by periodic electrostatic and/or magnetic potentials (see, for example, <cit.> and references therein). Note that, traditionally, the $\T$-matrix approach was just suggested for (quasi) 1D potential problems.
The present work is devoted to another class of graphene nano-structures that are induced by a cylindrically symmetric potentials, known as circular graphene quantum dots (GQDs) <cit.>. Experimentally, a circular GQD can be created using an appropriate circular top gate in the way as described in Ref. <cit.>. Thanks to the fact that the gate potential can be tuned externally, such a gate-induced GQD can be easily controlled as regards its carrier density and effective radius. Theoretically, circular GQDs were often modelled by confinement potentials of either rectangular <cit.> or power law forms <cit.>. Then, by solving the Dirac-like equation for the chosen potential one obtained the dot energy spectrum and the associated quantities. It was shown that for the gapless pristine graphene in the absence of a magnetic field, due to the Klein tunneling, in general, it is not truly bound states but just quasi-bound ones with a finite trapping time that can be induced by an electrostatic confinement potential (see below, the text following Eq. (<ref>), for exceptional cases). An energy gap <cit.> and/or a perpendicular magnetic field <cit.> can enhance the trapping time of quasi-bound states (QBSs) and induce even the bound states. A smoothness of confinement potential was also shown to enhance the trapping time of QBSs. However, solving the Dirac-like equation with a smooth potential is often rather problematic.
The purpose of this work is to extend the $\T$-matrix approach to study the electronic properties of circular GQDs induced by more general radial confinement potentials. For simplicity, our discussions are essentially limited to the case of zero magnetic field. Nevertheless, we briefly describe in an Appendix how to extend the approach to the case where a perpendicular homogeneous magnetic field is applied to the dot plane.
Note that, in reality, a GQD with well-defined discrete energy levels can be created by cutting a structure with the desired geometry from a flake of graphene <cit.>. However, so far there is a serious problem in fabricating such GQDs with atomic precision termination, while it was shown that the electronic properties of these GQDs are quantitatively sensitive to their precise terminations <cit.>. From the future electronics application point of view it is desirable to find the way of creating GQDs by the confinement potentials so that the trapping time of localized states is long enough to satisfy the application requirements and the electronic properties of the structure can be controlled externally.
The paper is organized as follows. Sec. <ref> presents the main results of the paper. It is there shown that for a very general class of circular GQDs, the bound and quasi-bound states spectral equations as well as the associated quantities, such as the local density of states and the resonance scattering characteristics, can all be expressed exactly in terms of the elements of the $\T$-matrix for the corresponding radial confinement potential. In Sec. <ref> we show, as an example, the numerical solutions of the presented equations for the case of trapezoidal radial potential. Among the obtained results, it is in particular suggested that thermal fluctuations and/or electrostatic disorders may appear as an obstacle to controlling the valley polarization of Dirac electrons. While the paper is closed with a brief summary in Sec. <ref>, the two Appendices are added to describe how the $\T$-matrix can be determined at some particular energies (A) and in the presence of a perpendicular magnetic field (B).
§ GENERAL CONSIDERATION
Let us consider a single-layer circular GQD defined by the radial confinement potential $U(r)$ that is assumed to be smooth on the scale of the graphene lattice spacing. Using the units such that $\hbar=1$ and the Fermi velocity $v_F = 1$ (quasi-relativistic quantum units), the low-energy electron dynamics in this structure can be described by the 2D Dirac-like Hamiltonian
\begin{equation}
H= \vec{\sigma} \vec{p} + \nu \Delta \sigma_z + U(r),
\label{eq: H U}
\end{equation}
where $\vec{\sigma}=(\sigma_x,\sigma_y)$ are the Pauli matrices, $\vec{p}=-i (\partial_x, \partial_y)$ is the 2D momentum operator, $\nu$ is the valley index ($\nu = \pm$ for the valleys $K$ and $K'$, respectively) and $\Delta \sigma_z$ is the constant mass term <cit.>.
We look for the eigen-functions of the Hamiltonian (<ref>) at energy $E$. Because of the cylindrical symmetry of $U(r)$, in the polar coordinates $(r, \phi )$ these eigen-functions can be found in the form
\begin{equation}
\Psi (r,\phi) = e^{i j \phi}
\left(
\begin{array}{cc}
e^{-i \phi/2} \chi_{A} (r) \\
e^{+i \phi/2} \chi_{B} (r)
\end{array}
\right),
\label{eq: eigen j}
\end{equation}
where the total angular momentum $j$ takes half-integer values and the radial spinor $\chi = (\chi_A,\chi_B)^{t}$ satisfies the following equation:
\begin{equation}
\left(
\begin{array}{cc}
U(r) - E + \nu \Delta & - i (\partial_r + \frac{j+\frac{1}{2}}{r}) \\
- i (\partial_r - \frac{j-\frac{1}{2}}{r}) & U(r)- E - \nu \Delta
\end{array}
\right)
\left(
\begin{array}{c}
\chi_A(r)\\
\chi_B(r)
\end{array}
\right)
= 0.
\label{eq: eigen chi U}
\end{equation}
This system of the two first order differential equations for the components $\chi_A$ and $\chi_B$ could be converted to a decoupled second order differential equation for either of these components. However, unless the potential $U(r)$ is simple enough, the resulting second order differential equations are often intractable. Nevertheless, we will show that the electronic characteristics of the circular GQDs described by the Hamiltonian (<ref>) can be exactly expressed in terms of the elements of a $(2 \times 2)$ $\T$-matrix defined below.
In order to define the expected $\T$-matrix, it should be noted that, in practice, we often have to deal with the confinement potentials $U(r)$ which are flat in the two limiting regions of small and large $r$, i.e.,
\begin{equation}
\left\{
\begin{array}{c}
\mbox{$U_i$, \ \ \ $r \le r_i$}, \\
\mbox{$U_f$, \ \ \ $r \ge r_f$}, \\
\mbox{arbitrary, \ otherwise}.
\end{array}
\right.
\label{eq: general potential}
\end{equation}
In these limiting regions, the eigenstates of Hamiltonian (<ref>) can be found exactly. Indeed, we consider some region $r_a < r < r_b$ where the potential $U(r)$ is constant, $U(r) = \bar{U}$. As is well-known <cit.>, for $E \neq \bar{U} \pm \nu \Delta$ the
general solution to Eq. (<ref>) in this region can be written in terms of two independent integral constants $C = ( C^{(1)}, C^{(2)} )^t$:
\begin{equation}
\chi (r) = \W (\bar{U} , r) C ,
\label{eq: chi W}
\end{equation}
where the columns of the $\W$-matrix are the two independent basic solutions of Eq. (<ref>),
\begin{equation}
\W (\bar{U} , r) =
\left(
\begin{array}{cc}
J_{j-\frac{1}{2}} (q r) & Y_{j-\frac{1}{2}} (q r) \\
i \tau J_{j+\frac{1}{2}} (q r) & i \tau Y_{j+\frac{1}{2}} (q r)
\end{array}
\right) .
\label{eq: W-matrix}
\end{equation}
Here $J_{j\pm\frac{1}{2}}$ is the Bessel function of the first kind and $Y_{j\pm\frac{1}{2}}$ is the Bessel function of the second kind <cit.>, $q = \sqrt{(E - \bar{U})^2 - \Delta^2}$ and $\tau = q / (E - \bar{U} + \nu \Delta )$. In the following, for definition, the integral constants $C = (C^{(1)}, C^{(2)})^t$ will be referred to as basic coefficients. In the 1D problems, these basic coefficients can be interpreted as the coefficients of the forward and backward waves <cit.>. A similar interpretation can be introduced when the Hankel functions <cit.> are used to present the basic solutions $\W$ <cit.>. In this paper, we however use the Bessel function representation for the sake of algebraic convenience.
A special care is needed in the case of energies $E \rightarrow \bar{U} \pm \nu \Delta$, when the basic solutions (<ref>) become divergent. To avoid such a divergence, maintaining the matrix $\W$ as independent basic solutions, one has to properly adjust the regularization coefficients for the matrix elements in getting the correct limiting form of $\W$. To keep our discussions continuous, in the following we always assume that $E \ne \bar{U} \pm \nu \Delta$ and the case $E \rightarrow \bar{U} \pm \nu \Delta$ will be discussed separately in Appendix <ref>.
We note that the basic coefficient $C$ can be considered as the spinor represented in a basis that depends on $r$ according to Eq. (<ref>). Then, Eq. (<ref>) actually describes a (non-unitary) basis transformation of the spinor. The advantage of using such a $r$-depending basis is that while $\chi(r)$ depends on $r$ explicitly, the wave coefficient $C$ is independent of $r$ in constant potential regions.
Now, the key feature of the differential Eq. (<ref>) is that it is linear and homogeneous. Consequently, the two radial spinors at $r = r_1$ and $r = r_2$ should be linearly related by some matrix $\G (r_2,r_1)$,
\begin{equation}
\chi (r_2) = \G (r_2,r_1) \chi (r_1).
\label{eq: G-matrix}
\end{equation}
This relation holds for any $r_2 \ge r_1$, including the case of $r_1 \le r_i$ and $r_2 \ge r_f$ [see Eq. (<ref>)]. Therefore, when we represent the spinors at $r_1 \le r_i$ and $r_2 \ge r_f$ by the basic coefficients $C_i$ and $C_f$ respectively, these basic coefficients should also be linearly related by some $\T$-matrix:
\begin{equation}
C_f = \T C_i .
\label{eq: T-matrix}
\end{equation}
Note that the variable $r$ is entirely dropped out of this equation. Thus, in the context of the studied problem, the $\T$-matrix is defined as the matrix that maps the basic coefficients in the limiting region of small $r$ to those in the limiting region of large $r$. In fact, Eq. (<ref>) is a just basis transformation of Eq. (<ref>). From Eqs. (<ref>), (<ref>), and (<ref>), we have the following elementary relation
\begin{equation}
\T = \W^{-1} (U_f,r_2) \G (r_2,r_1) \W (U_i,r_1).
\label{eq: T-G}
\end{equation}
This equation, like Eq. (<ref>), holds for any $r_1 \le r_i$ and $r_2 \ge r_f$, including $r_1=r_i$ and $r_2=r_f$.
Equation (<ref>) provides a practical way to compute the $\T$-matrix for any radial potential $U(r)$ of Eq. (<ref>) via computing $\G(r_f,r_i)$. By inserting (<ref>) into (<ref>), one finds an explicit differential equation for $\G(r_2,r_1)$, which resembles a dynamical equation in $r$-direction,
\begin{equation}
i\frac{\partial \G(r_2,r_1)}{\partial r_2}= \mathcal{H}(r_2) \G(r_2,r_1),
\end{equation}
with the formal Hamiltonian defined as
\begin{equation}
\mathcal{H} (r)=
\left(
\begin{array}{cc}
i\frac{j-\frac{1}{2}}{r} & U(r) - E - \nu \Delta \\
U(r)-E + \nu \Delta & -i \frac{j+\frac{1}{2}}{r}
\end{array}
\right).
\end{equation}
This dynamical equation is to be solved for $\G(r_2,r_1)$ with the initial condition such that $\G(r_1,r_1)$ is the $(2 \times 2)$ identity matrix. Note that the formal Hamiltonian $\mathcal{H}(r)$ is not hermitian, and thus the dynamics is non-unitary. Moreover, $\mathcal{H}(r)$ at different $r$ generally do not commute with each other, rendering the dynamics analytically intractable. However, for the purpose of numerically calculating the $\T$-matrix, a simple numerical method for ordinary differential equations (ODEs) such as the Runge–Kutta method is sufficient <cit.>.
Of particular importance is the case of one-step potential, $U(r)$ of Eq. (<ref>) with $r_i=r_f$. In this case, $\G(r_i,r_f)$ is simply the $(2 \times 2)$ identity matrix and we can easily write down the $\T$-matrix of Eq. (<ref>) explicitly,
\begin{eqnarray}
\T && = \left[ \tau_f J_{j-\frac{1}{2}} (q_f r_f) Y_{j+\frac{1}{2}} (q_f r_f) - \tau_f J_{j+\frac{1}{2}} (q_f r_f)Y_{j-\frac{1}{2}} (q_f r_f) \right]^{-1} \nonumber \\
&& \times \left(
\begin{array}{cc}
\tau_f Y_{j+\frac{1}{2}} (q_f r_f)J_{j-\frac{1}{2}} (q_i r_i)- \tau_i Y_{j-\frac{1}{2}} (q_f r_f)J_{j+\frac{1}{2}} (q_i r_i)& \tau_f Y_{j+\frac{1}{2}} (q_f r_f) Y_{j-\frac{1}{2}} (q_i r_i) - \tau_i Y_{j-\frac{1}{2}} (q_f r_f)Y_{j+\frac{1}{2}} (q_i r_i) \\
- \tau_f J_{j+\frac{1}{2}} (q_f r_f)J_{j-\frac{1}{2}} (q_i r_i) + \tau_i J_{j-\frac{1}{2}} (q_f r_f) J_{j+\frac{1}{2}} (q_i r_i) & - \tau_f J_{j+\frac{1}{2}} (q_f r_f) Y_{j-\frac{1}{2}} (q_i r_i)+ \tau_i J_{j-\frac{1}{2}} (q_f r_f) Y_{j+\frac{1}{2}} (q_i r_i)
\end{array}
\right), \nonumber \\
\label{eq: explicit T}
\end{eqnarray}
where $q_{i(f)}$ and $\tau_{i(f)}$ are defined as in Eq. (<ref>): $q_{i(f)} = \sqrt{(E - U_{i(f)} )^2 - \Delta^2}$ and $\tau_{i(f)} = q_{i(f)} / (E - U_{i(f)} + \nu \Delta )$.
Being a seemingly simple mathematical consequence of the linearity and the homogeneity of the wave equations, the $\T$-matrix of Eq. (<ref>), as can be seen below, holds rich information on the characteristics of the energy spectrum of the system. In order to derive these characteristics, we are going to impose appropriate boundary conditions for the basic coefficients $C_i$ and $C_f$, which in turn lead to corresponding constraints on the elements of the $\T$-matrix itself. It should be noted immediately that in the limiting region of small $r$, the Bessel function of the first kind $J_{j\pm\frac{1}{2}}(q_i r)$ is regular, while the Bessel function of the second kind $Y_{j\pm\frac{1}{2}}(q_i r)$ diverges. We should therefore set the condition $C_i \propto (1,0)^t $ for the basic coefficient in this region. We will first show that the localization behaviour of states is determined by the boundary condition for the basic coefficient $C_f$.
§.§ Bound states
For the bound states to emerge, the wave function should decay fast enough as $r$ increases. This happens only when the wave vector in the limiting region of large $r$, $q_f = \sqrt{( E - U_f )^2- \Delta^2}$, is imaginary, implying $-\Delta < E - U_f < \Delta$. Here, as mentioned above, we do not include the case of equalities, which may bring about a particular type of bound states (see also Appendix <ref> and references therein). Under this condition, although both Bessel functions $J_{{j\pm\frac{1}{2}}} (q_f r)$ and $Y_{{j\pm\frac{1}{2}}} (q_f r)$ diverge as $r$ increases, the Hankel function of the first kind, $H^{+}_{{j\pm\frac{1}{2}}} (q_f r)= J_{{j\pm\frac{1}{2}}} (q_f r) + i Y_{{j\pm\frac{1}{2}}}(q_f r)$, decays exponentially. Thus, for the bound states to emerge, the appropriate boundary condition for the basic coefficient $C_f$ should have the form $C_f \propto (1, i)^t $. With the boundary conditions for $C_i$ and $C_f$ just defined, Eq. (<ref>) leads to the following relation for the elements of the $\T$-matrix:
\begin{equation}
\T_{11} + i \T_{21} = 0.
\label{eq: bound states}
\end{equation}
This is the general equation to determine the energy spectrum of all the bound states in the considered energy regions for a GQD induced by the potential of Eq. (<ref>). To obtain this energy spectrum we first have to calculate the $\T$-matrix in the way described above and then to solve Eq. (<ref>). In the particular case of one-step potentials, using the explicit $\T$-matrix of Eq. (<ref>), we can easily recover the bound state spectral equation reported in Refs. <cit.> for the GQD induced by a rectangular potential.
§.§ Quasi-bound states
For $| E - U_f | > \Delta$, the wave vector in the limiting region of large $r$, $q_f$, is always real and the corresponding states cannot be truly bound. However, carriers may be temporally trapped at these states with some finite life-time. As mentioned above, such states are often referred to as the QBSs. Each QBS can be characterized by a complex energy $E = \Re(E) + i \Im(E)$ with $\Im(E)<0$. The real part of this energy, $\Re(E)$, defines the position of the QBS (i.e., the resonant level), while the imaginary part, $\Im(E)$, causes the probability density of the QBS to decay over time $t$ as $ \propto e^{2 \Im(E) t}$. This implies that $| \Im(E) |$ is a measure of the resonant level width and its inverse is a measure of the carrier life-time at the QBS, $\tau_0 \propto 1/ (2 | \Im(E) |)$.
Actually, the way we determined the spectral equation for bound states, Eq. (<ref>), can be easily extended to find the spectrum of QBSs. Indeed, as well-known <cit.>, the reasonable boundary condition for QBSs is that far from the origin the wave function should be an out-going wave. Letting $s = \sign(E - U_f )$, it is easy to see that the wave function $(H^{s}_{{j-\frac{1}{2}}} (q_f r), i \tau_f H^{s}_{{j+\frac{1}{2}}} (q_f r))^t $ with $H^{s}_{{j\pm\frac{1}{2}}} (q_f r) = J_{{j\pm\frac{1}{2}}} (q_f r) + is Y_{{j\pm\frac{1}{2}}} (q_f r)$ describes such an out-going wave. This can be confirmed by examining the current density of the radial wave function in the limiting region of large $r$ using the well-known asymptotic forms of the Hankel functions <cit.>. With the wave-function identified, in terms of the basic coefficients, it appears that the appropriate boundary condition for QBSs takes the simple from: $C_f \propto (1, is)^t$. Using this $C_f$ and the boundary condition for $C_i$ defined above, Eq. (<ref>) results in the general equation for determining the QBSs spectrum in circular GQDs:
\begin{equation}
\T_{11} + i s \T_{21} = 0.
\label{eq: quasi-bound states}
\end{equation}
Note that, to our best knowledge, the QBSs in circular GQDs were often identified by either numerically fitting asymptotic boundary conditions <cit.>, or intuitively analysing the behaviour of the local density of states <cit.>. Equation (<ref>) provides an alternative way to solve the problem, making it more definite and rather simple algebraically. In fact, this equation is in the same spirit as the equation suggested sometime ago for the QBSs in a 1D potential <cit.>.
§.§ Density of states
The local density of states (LDOS) for unbound states, as defined in <cit.>, can also be easily expressed in terms of the $\T$-matrix of the radial confinement potential. Note that for unbound states the wave functions are not normalizable and the usual definition of LDOS <cit.> should be used with care. Following <cit.>, we image that the considered GQD is entirely embedded in a large graphene disc of radius $R$, with the center of this disc coincides with that of the GQD. States are then bound within the large graphene disc, and the level spacing can be estimated to be $\Delta E= \pi/R$ <cit.>. The LDOS of the GQD is proportional to both the level density and the probability for the electron at that energy level to be inside the dot. For a wave function with basic coefficients $C_i = (F, 0)^t$ and $C_f = (P, Q)^t$, the latter is proportional to $\abs{F}^2 / \abs{N}^2$, where $N$ is the normalization factor of the wave function, which in turn can be estimated to be $\abs{N}^2 \propto (\abs{P}^2 + \abs{Q}^2) R/\abs{E}$ <cit.>. Overall, this gives the formula for the LDOS: $\rho^{(j)} (E) \propto \abs{E} \abs{F}^2 / ( \abs{P}^2+\abs{Q}^2 )$.
In order to get the LDOS in terms of the $\T$-matrix, we can use the relation (<ref>) to show that $\abs{F}^2/(\abs{P}^2+\abs{Q}^2) = 1 / ( \abs{T_{11}}^2+\abs{T_{21}}^2)$. Thus, for a given angular momentum $j$ and a given valley index $\nu$, the LDOS around the circular GQD can be calculated in terms of the $\T$-matrix as
\begin{equation}
\rho^{(j)} (E) \propto \frac{\abs{E}}{\abs{T_{11}^{(j)}}^2 + \abs{T_{21}^{(j)}}^2},
\label{eq: dos}
\end{equation}
where the superscript $(j)$ is added to explicitly indicate the $j$-dependence of the quantity calculated. Summing (<ref>) over all angular momenta, we obtain the total LDOS,
\begin{equation}
\rho(E)= \sum_{j=-\infty}^{+\infty} \rho^{(j)} (E).
\label{eq:tldos}
\end{equation}
It is easy to show that these general expressions, Eqs. (<ref>) and (<ref>), directly reduce to the corresponding ones given in Ref. <cit.> for circular GQDs with a rectangular confinement potential.
§.§ Scattering coefficients
The scattering states are those with the asymptotic wave functions far from the origin being a superposition of an in-coming plane wave and an out-going (scattering) circular wave <cit.>. Thus, for $r > r_f $, we write
\begin{equation}
\Psi_f (r,\phi) = \Psi_f^{(i)} (r, \phi) + \Psi_f^{(o)} (r, \phi),
\end{equation}
where the first and the second terms in the right-hand-side are the in-coming plane wave and out-going circular wave, respectively. The in-coming wave function $\Psi_f^{(i)}(r, \phi )$ is assumed to propagate along the $x$-direction with positive current density,
\begin{equation}
\Psi_f^{(i)} (r,\phi)= e^{isq_f r \cos \phi}
\left(
\begin{array}{c}
1 \\
\frac{s q_f}{E-U_f+ \nu \Delta}
\end{array}
\right),
\label{eq:plane-wave}
\end{equation}
where $q_f$ and $s$ have already been defined above. Note that for the electron to be propagated at large $r$, the energy should not be in the gap, $\abs{E-U}>\Delta$. Using the Jacobi–Anger identity <cit.>, the plane wave function of Eq. (<ref>) can be decomposed into the eigen-functions of the angular momentum as
\begin{equation}
\Psi_f^{(i)}(r,\phi)=\sum_{j=-\infty}^{+\infty} (is)^{j-\frac{1}{2}} e^{ij{\phi}}
\left(
\begin{array}{c}
e^{-\frac{i}{2}\phi} \ J_{j-\frac{1}{2}}(q_f r) \\
e^{+\frac{i}{2}\phi} \ i\tau_f J_{j+\frac{1}{2}}(q_f r)
\end{array}
\right),
\end{equation}
with $\tau_f$ also already defined.
The scattering wave can be also expanded in the out-going waves of different angular momenta,
\begin{equation}
\Psi_f^{(o)} (r,\phi)=\sum_{j= -\infty}^{+\infty} a^{(j)} (is)^{j-\frac{1}{2}} e^{ij{\phi}}
\left(
\begin{array}{c}
e^{-\frac{i}{2}\phi} \ H^{s}_{{j-\frac{1}{2}}}(q_f r) \\
e^{+\frac{i}{2}\phi} \ i\tau_f H^{s}_{{j+\frac{1}{2}}}(q_f r)
\end{array}
\right),
\label{eq: out-going}
\end{equation}
where $a^{(j)}$ are regarded as scattering coefficients <cit.>.
For $r<r_i$, similarly, the wave function can be decomposed into a linear combination of the wave functions of different angular momenta. Noting that to ensure the regularity of the wave function at the origin, the Bessel functions of the second kind are necessarily absent from this decomposition, one has
\begin{equation}
\Psi_i (r,\theta) =\sum_{j=-\infty}^{+\infty} c^{(j)} (is)^{j-\frac{1}{2}} e^{ij{\phi}}
\left(
\begin{array}{c}
e^{-\frac{i}{2}\phi} \ J_{{j-\frac{1}{2}}}(q_i r) \\
e^{+\frac{i}{2}\phi} \ i\tau_{i} J_{{j+\frac{1}{2}}}(q_i r)
\end{array}
\right),
\end{equation}
with $q_i$ and $\tau_{i}$ defined before and $c^{(j)}$ being some coefficients.
Further, since the basic coefficients in the two limiting regions, $r \le r_i$ and $r \ge r_f$, should be related to each other by the $\T$-matrix as in Eq. (<ref>), we find
\begin{equation}
\left(
\begin{array}{c}
a^{(j)}+1 \\
\end{array}
\right)
\left(
\begin{array}{c}
c^{(j)} \\
\end{array}
\right),
\label{eq: amplitude equation}
\end{equation}
where the superscript $(j)$ is again introduced to indicate the $j$-dependence of $\T$-matrix. Solving Eq. (<ref>) gives the scattering coefficients in terms of the $\T$-matrix elements:
\begin{equation}
\label{eq: scattering coefficients}
\end{equation}
Now it is important to note that for an unbound eigen-function of real energy, to ensure the probability current conservation, it requires that the coefficients for the total out-going waves and the total in-going waves should be equal in modulus <cit.>,
\begin{equation}
\abs{T_{11}^{(j)} - is T_{21}^{(j)}} = \abs{T_{11}^{(j)} + is T_{21}^{(j)}}.
\end{equation}
This implies that the scattering coefficients $a^{(j)}$ can be represented in terms of the so-called scattering phase-shifts <cit.>,
\begin{equation}
a^{(j)}= \frac{1}{2} \left(e^{-i 2 \delta^{(j)}} -1\right),
\end{equation}
\begin{equation}
\delta^{(j)} = \frac{1}{2} \arg \left( \frac{T_{11}^{(j)} + is T_{21}^{(j)}}{T_{11}^{(j)} - is T_{21}^{(j)}}\right).
\label{eq: phase-shift}
\end{equation}
The differential scattering cross section, defined as the ratio of the probability flux of the out-going wave per unit angle to the probability flux of the in-coming wave per unit length <cit.>, can be found as
\begin{equation}
\frac{\d \sigma }{\d \phi} = \frac{2}{\pi q_f} \abs{\sum_{j=-\infty}^{+\infty} a^{(j)} e^{j \phi}}^2.
\label{eq: scattering cross section}
\end{equation}
By integrating this expression over $\phi$, one finds the total scattering cross section,
\begin{equation}
\sigma = \frac{4}{q_f} \sum_{j=-\infty}^{+\infty} \sin^2 \delta_j.
\end{equation}
Thus, for circular GQDs with an arbitrary radial confinement potential of Eq. (<ref>), we have shown that the bound states as well as the QBSs spectra and the associated quantities such as the LDOS and the scattering coefficients can all be exactly expressed in terms of $\T$-matrix elements. Equations (<ref>), (<ref>), (<ref>), and (<ref>) are the key results of the present work. In particular cases, when the eigenstates of Hamiltonian (<ref>) can be found analytically (e.g., for a rectangular potential $U(r)$), these equations are exactly reduced to the corresponding expressions reported in various references. Generally, the $\T$-matrix can be calculated numerically. In the next section, as an example, we present numerical results obtained in the case of trapezoidal radial confinement potential.
§ EXAMPLE: TRAPEZOIDAL RADIAL POTENTIAL INDUCED GQDS
As a demonstration for the studies presented in the previous section, we consider a circular GQD induced by the radial potential of Eq. (<ref>) with: $U_i = U_0$, $r_i = (1-\alpha) L$, $U_f = 0$, $r_f = (1+\alpha) L$ and $U(r) = U_i + \frac{r - r_i }{ r_f - r_i }(U_f - U_i )$ for $r_i < r < r_f$. So, the considered confinement potential has a trapezoidal shape that is characterized by three parameters: the potential height $U_0$, the dot effective radius $L$, and the smoothness $\alpha$ that ranges from $0$ to $1$. In the limiting case of $\alpha = 0$, this potential is just the most studied rectangular one. The 1D trapezoidal potential are often used to describe the gate-induced graphene $n$-$p$-$n$-junctions <cit.>.
For given values of potential parameters as well as the angular momentum $j$, we first calculate the $\T$-matrix for the potential under study. In the case of $\alpha \ne 0$, the calculation of the $\T$-matrix requires solving the ODE (<ref>) numerically for the matrix $\G(r_i,r_f)$ with the Runge–Kutta method. Substituting the obtained $\T$-matrix elements into Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), and solving these equations, we respectively obtain the energy spectra, the associated LDOS, and the scattering coefficients [For the indicated parameters, the Runge–Kutta method with about $1024$ steps gave the typical accuracy of $10^{-5}$ for the elements of the $T$-matrix. The numerical solutions of Eqs. (<ref>) and (<ref>) presented in Fig. <ref> and Fig. <ref> (a) were obtained at the effective resolution of at least $4026$ grid-points in each dimension. Bessel functions were computed using the corresponding subroutines from Ref. <cit.>]. Such calculations can be carried out for various values of the potential parameters and the angular momenta. As an example, some of the obtained results are presented in Figs.1-4.
Note that we still use the quasi-relativistic quantum units ($\hbar=1$, $v_F=1$), so the dimension of energy is inverse of the length. For a comparison, to describe the usual experimental values of $L$ and $U_0$ ($L$ is of the order of $100$ nm and $U_0$ is of the order of $130$ meV), we choose $L$ to be about $1$ and $U_0$ to be about $20$.
(Colour online) Spectrum of bound states calculated from Eq. (<ref>) and QBSs
from Eq. (<ref>) for a GQD induced by the trapezoidal radial potential
of $U_0 = 15$ and $\alpha=0$. The lines represent the level
positions, plotted versus the dot effective radius L, while the
thickness of these lines represents the corresponding level widths.
Data are shown for $\nu=+$, $j = \frac{3}{2}$ and $\Delta=2$.
(Colour online) QBS spectra $(a)$ and LDOS $(b)$ of a GQD induced by the trapezoidal radial potential of $L=1$ and $U_0=20$ are presented for $\nu=+$, $j = \frac{3}{2}$ and various $\alpha$. In $(a)$: 5 curves correspond to 5 QBS levels, each describing how the QBS energy ($\Im(E)$ and $\Re(E)$) changes as $\alpha$ varying regularly from 0.3 (top) to 0.7 (bottom), correspondingly, from larger point-sizes to smaller point-sizes. In $(b)$: LDOS (in arbitrary unit) is shown for the three spectra with $\alpha$ given in the figure.
We first set $\alpha=0$ and study the spectra of bound states and QBSs as $L$ changing from $1$ to $3$. Obtained results are shown in Fig. <ref>. The limiting lines $E = \pm \Delta$ and $E = U_0 \pm \Delta$ define qualitatively different energy regions. The region $U_0 - \Delta \le E \le U_0 + \Delta $ appears as a gap, where there exists neither bound states nor QBSs. On the other hand, the states in the region of energies $- \Delta \le E \le +\Delta $ are truly bound, while those outside these regions are QBSs. For the QBSs presented, the thickness of the lines represents the corresponding level widths. When $L$ increases, starting from the low energy region ($E < -\Delta$), the QBS-levels gradually rise to approach the boundary at $E=-\Delta$, and, at the same time, their widths gradually narrow to vanish at this boundary. Throughout the region $-\Delta<E<+\Delta$, the states are truly bound with zero level widths. At the opposite boundary $E=+\Delta$ the states are again converted to QBSs. So, there may observe a continuous QBS - bound state - QBS transition in the energy spectra of circular GQDs as the dot radius $L$ varies. Note that, in the case of zero-gap, $\Delta = 0$, the bound states region actually collapses into the line $E = 0$ (That is why these states have been referred to as zero-energy ones <cit.>). At very large $L$, all levels converge to the two boundaries $E= U_0 \pm \Delta$ that describe the limiting case when a homogeneous potential of $U_0$ is applied on the entire graphene sheet.
In the gapless case, $\Delta = 0$, all the states other than zero-energy ones are just QBSs. In this case, the QBSs with energies in the region $0<E<U_0$ tend to have the level widths narrower than that for the QBSs with energies outside this region. It was suggested that the level widths of these QBSs can also be tuned by varying the smoothness of the confinement potential $\alpha$ <cit.>. Fig. <ref> $(a)$ shows how the complex energies of five different QBSs change as the smoothness $\alpha$ varies from $0.3$ to $0.7$ (correspondingly, point-sizes gradually decrease). Obviously, for any QBS under study, with increasing potential smoothness $\alpha$, while the real part of the energy $\Re(E)$ just changes slightly, the imaginary part $\Im(E)$ decreases substantially. This result is in a good agreement with those reported for 1D potentials <cit.> and 2D power law potentials <cit.>.
Next, we show in Fig. <ref> $(b)$ the LDOSs (in arbitrary unit) for the three spectra with the $\alpha$-values examined in Fig. <ref> $(a)$. Evidently, there is a good agreement between the positions of QBSs in $(a)$ and the corresponding resonant peaks of LDOS in $(b)$. Moreover, the imaginary parts of the QBS energies represent the widths of the corresponding LDOS peaks quite well. Thus, our results qualitatively demonstrate the correspondence between the QBSs and the LDOS peaks. In fact, the LDOS has already been used to determine QBSs indirectly <cit.>. Quantitatively, it should however be noted that for very broad LDOS peaks, such as those at $E \approx 1$ in Fig. <ref> $(b)$, the peak width may not correctly describe the life-time of the corresponding QBS.
To illustrate the $\T$-matrix-based scattering formalism developed in subsection <ref>, we calculate the low-energy differential scattering cross section $\d \sigma / \d \phi $ for the trapezoidal potential of $U_0=20$ and $L= 1$ ($\alpha$ is set to be zero for simplicity). In Fig. <ref>, obtained results of $\d\sigma / \d\phi $ are presented as a function of the scattering angle $\phi$ in three cases: $\Delta = 0$ (gapless), $0.5$, and $1$ (finite gap). In the gapless case (dash-dotted line), the differential scattering cross section vanishes at $\phi = \pm \pi$ (Fig. <ref>, inset), showing the undoubted effect of the Klein tunnelling. In the two cases of finite gap, on the contrary, $\d\sigma / \d\phi $ is always finite, implying an unavoidable presence of the back-scattering.
(Colour online) Low-energy differential scattering cross section is plotted as a function of scattering angle $\phi$ for the trapezoidal radial potential of $U_0=15$, $L=1$, and $\alpha=0$ in three cases of $\Delta$: $0$ (dash-dotted line), $0.5$ (dashed line), and $1$ (solid line). The inset zooms in the region of scattering angle around $\pi$. Data are shown for $E=2$ and $\nu=+1$.
Besides, the two curves of finite gap (solid and dashed) in Fig. <ref> clearly show an asymmetrical behaviour with respect to the sign of $\phi$. A similar asymmetry has been discussed in the context of scattering of Dirac electrons by the so-called mass-barriers in Ref. <cit.>. Note that by the reflection symmetry, $j \rightarrow -j$, $\nu \rightarrow -\nu$, electrons with opposite valley indices will scatter as if reflected along $\phi=0$, so no Hall-like voltage can be expected unless the injected current is valley-polarized. Nevertheless, with an unpolarised current, electrons of different valley indices are expected to accumulate on opposite edges of the graphene sample in the way similar to the spin Hall effect <cit.>. The valley-dependent asymmetric scattering was suggested to be used for the valley filtering purpose <cit.>.
Further, to learn if the examined electrostatic potential can support to control the valley polarisation of Dirac electrons like the mass potential does <cit.>, we calculate the transverse scattering cross section defined as
\begin{equation}
\eta= \int_{-\pi}^{+\pi} \d \sigma (\phi) \sin \phi.
\end{equation}
Calculations have been performed for potentials of $L = 1$, $\alpha = 0$, and different $U_0$. Obtained results for $\eta$ are plotted as a function of the incident energy $E$ in Fig. <ref>, where the three curves are different in $U_0$: $U_0 = 10$ (dash-dotted line), $20$ (dashed line), and $30$ (solid line). Remarkably, $\eta$ strongly fluctuates, changing its sign in a complicated way, depending on both $E$ and $U_0$. Consequently, the transverse scattering cross sections of valley-polarized electrons of slightly different energies (e.g., due to thermal fluctuations), or from slightly different potentials, might compensate each other, resulting in a vanishing net transverse scattering cross section. This is very different from the scattering of Dirac electrons by a mass-barrier studied in Ref. <cit.>, where it was shown that the transverse scattering cross section generally keeps its sign unchanged as the energy of electron varies. Given the fact that an energy gap in graphene is often induced by an underlying substrate <cit.>, a mass-barrier is likely to be accompanied by electrostatic disorders. Thus, although a more quantitative study is needed, we speculate that the electrostatic disorders and/or the thermal fluctuation may appear as an obstacle to controlling the valley polarization of Dirac electrons and, therefore, to observing the associated zero-field Hall and the valley filtering effects <cit.> .
Finally, to gain some insight into the discussed fluctuating behaviour of the transverse scattering cross section $\eta$ observed in Fig. <ref>, in the inset to this figure we compare three quantities, $\eta$, the total scattering cross section $\sigma$, and the total LDOS, all are plotted versus $E$. Obviously, there is a good correspondence between the peaks of the total LDOS resulted from QBSs of different angular momenta (labelled TLDOS) with those of the total (labelled $\sigma$) and transverse (labelled $\eta$) scattering cross sections. Note that the (rather shallow) peaks of the transverse scattering cross section come both as maxima and minima.
(Colour online) Transverse scattering cross section $\eta$ as a function of the incident energy $E$ for potentials of $L=1$, $\alpha=0$, and various $U_0$: $10$ (dash-dotted line), $20$ (dashed line), and $30$ (solid line). The inset zooms in a small region of energy (for $U_0=30$), where the total scattering cross section (labeled $\sigma$) and the (total) local density of states (labelled TLDOS) are also plotted for a comparison. [Note that the TLDOS (defined up to a constant factor) was rescaled to fit the figure.] Data are shown for $\Delta= 0.5$ and $\nu=+1$.
§ CONCLUSION
We have developed the $\T$-matrix formalism for studying electronic properties of the GQDs induced by a cylindrically symmetric confinement potential (circular GQDs). It was first shown that for circular GQDs with any radial confinement potential the equations for the bound states and QBSs spectra as well as the associated quantities such as the LDOS or scattering coefficients are all expressed explicitly in terms of the corresponding $\T$-matrix. In the case of simple confinement potentials (e.g., rectangular one), when the Dirac-like equation can be solved analytically, these equations give exactly the analytical results reported in various references. For any complicated potential, the $\T$-matrix can be determined numerically. As an example, we have in detail considered the case of trapezoidal radial confinement potentials, calculating the bound states and QBSs spectra, the LDOS, the differential scattering cross section, and the transverse scattering cross section for the potentials of different parameters. Apart from the role of a demonstration for the $\T$-matrix approach developed, obtained results in this example, in particular, suggest that controlling the valley polarization of Dirac electrons may turn out to be difficult in the presence of electrostatic disorders and/or thermal fluctuation. As an addition, we have shown how the developed $\T$-matrix formalism can be extended to study circular GQDs under a homogeneous perpendicular magnetic field (Appendix <ref>).
We thank Cong Huy Pham and Duy Quang To for useful discussions.
This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.02-2013.17.
§ $\W$-MATRIX FOR $E = \BAR{U} \PM \NU \DELTA$
As was mentioned above, the $\W (\bar{U},r)$-matrix as defined in (<ref>) diverges as $E \rightarrow \bar{U} \pm \nu\Delta$. Note that the basic solutions are always defined up to constant factors that do not depend on $r$. To cure this divergence, we introduce the regularization factors to the basic solutions so that they remain finite in the limit of $E \rightarrow \bar{U} \pm \nu \Delta$. For example, for $j > 0$, the regularized $\W$-matrix can be defined as
\begin{equation}
\tilde{\W} (\bar{U} ,r) =
\left(
\begin{array}{cc}
\tilde{J}_{j-\frac{1}{2}} (q r) & -i \epsilon^{(+)} \tilde{Y}_{j-\frac{1}{2}} (q r) \\
i \epsilon^{(-)} \tilde{J}_{j+\frac{1}{2}} (q r) & \tilde{Y}_{j+\frac{1}{2}} (q r)
\end{array}
\right),
\label{eq: W-matrix 1}
\end{equation}
where $\tilde{J}_{j\pm\frac{1}{2}} (q r) = q^{-\abs{j\pm\frac{1}{2}}}J_{j\pm\frac{1}{2}} (q r)$, $\tilde{Y}_{j\pm\frac{1}{2}} (q r) = q^{\abs{j\pm\frac{1}{2}}}Y_{j\pm\frac{1}{2}} (q r)$ and $\epsilon^{(\pm)} = E - \bar{U} \pm \nu \Delta$. Now, letting $E \rightarrow \bar{U} \pm \nu \Delta$, we find
\begin{equation}
\tilde{\W} (\bar{U} ,r) \rightarrow
\left(
\begin{array}{cc}
r^{\abs{j-\frac{1}{2}}} & - \frac{i\epsilon^{(+)}}{2j-1} r^{-\abs{j-\frac{1}{2}}} \\
\frac{i\epsilon^{(-)}}{2j+1} r^{\abs{j+\frac{1}{2}}} & r^{-\abs{j+\frac{1}{2}}}
\end{array}
\right),
\label{eq: W-matrix 1 degenerate 1}
\end{equation}
for $j>\frac{1}{2}$, where we have also removed common constant factors in taking the limit. For $j=\frac{1}{2}$, the limit is more tricky, where one also needs to linearly recombine the two solutions to find
\begin{equation}
\tilde{\W} (\bar{U} ,r) \rightarrow
\left(
\begin{array}{cc}
1 & i \epsilon^{(+)} \ln r \\
\frac{i \epsilon^{(-)}}{2} r & \frac{1}{r}
\end{array}
\right).
\label{eq: W-matrix 1 degenerate 2}
\end{equation}
It is easy to check that the wave functions of Eq. (<ref>) and Eq. (<ref>) are really the solutions to the Dirac equation (<ref>) at $E = \bar{U} \pm \nu \Delta$.
Similarly, for $j < 0$, we have
\begin{equation}
\tilde{\W} (\bar{U} , r) =
\left(
\begin{array}{cc}
-i \epsilon^{(+)} \tilde{J}_{j-\frac{1}{2}} (q r) & \tilde{Y}_{j-\frac{1}{2}} (q r) \\
\tilde{J}_{j+\frac{1}{2}} (q r) & i \epsilon^{(-)} \tilde{Y}_{j+\frac{1}{2}} (q r)
\end{array}
\right).
\label{eq: W-matrix 2}
\end{equation}
Using the same procedure of taking the limit $E \rightarrow \bar{U} \pm \nu \Delta$ as above, we finds
\begin{eqnarray}
\tilde{\W} (\bar{U} ,r) \rightarrow
\left(
\begin{array}{cc}
-\frac{i\epsilon^{(+)}}{2j-1} r^{\abs{j-\frac{1}{2}}} & r^{-\abs{j-\frac{1}{2}}} \\
r^{\abs{j+\frac{1}{2}}} & \frac{i \epsilon^{(-)}}{2j+1} r^{-\abs{j+\frac{1}{2}}}
\end{array}
\right),
\label{eq: W-matrix 2 degenerate 1}
\end{eqnarray}
for $j < - \frac{1}{2}$, and
\begin{eqnarray}
\tilde{\W} (\bar{U} ,r) \rightarrow
\left(
\begin{array}{cc}
\frac{i \epsilon^{(+)}}{2} r & \frac{1}{r} \\
1 & i \epsilon^{(-)} \ln r
\end{array}
\right),
\label{eq: W-matrix 2 degenerate 2}
\end{eqnarray}
for $j=-\frac{1}{2}$.
Note that, by definition, $\T$-matrix is basis-dependent. So, in the limiting case studied, when the $\W$-matrix of Eq. (<ref>) is replaced by $\tilde{\W}$ defined above, all the boundary conditions and the Eqs. (<ref>), (<ref>), (<ref>) and (<ref>) should be slightly modified accordingly.
Actually, the discussed degenerate solution is responsible for a special kind of bound states when the potential satisfies certain conditions. Since these special states only exist under very particular conditions, we do not examine them in any detail and interested readers are referred to Refs. <cit.>.
§ $\T$-MATRIX FOR CIRCULAR GQDS IN A MAGNETIC FIELD
In the presence of a uniform magnetic field $B$, the Hamiltonian of Eq. (<ref>) becomes
\begin{equation}
H_{\tau}= v_F \vec{\sigma} (\vec{p}+\frac{e}{c} \vec{A}) + \nu \Delta \sigma_z + U(r),
\label{eq: H B}
\end{equation}
where $\vec{A}$ is the vector potential <cit.>. Note that we explicitly reintroduce in this Hamiltonian the Fermi velocity $v_F$ and the Planck constant $\hbar$ to distinguish the scale of quasi-relativistic effects (defined by $v_F$) and the scale of electrodynamics (defined by $c$). The magnetic field is assumed to be perpendicular, $\vec{B} = (0, 0, B)$, and we choose the symmetric gauge, $\vec{A}=\frac{B}{2}(-y, x, 0)$. It is well-known that perpendicular magnetic field can induce localization of Dirac electrons even in the absence of the band gap <cit.>. In fact, for strong magnetic field, Dirac electrons are expected to exhibit the relativistic Landau levels <cit.>. The effects of weak and medium magnetic field on the electron localization in electrostatic GQDs have been also discussed early <cit.>. The spectral equation for a rectangular GQD with a perpendicular magnetic field can be written down explicitly <cit.>. We will show that for a general electrostatic potential of the form (<ref>), the spectral equation can also be written in terms of the $\T$-matrix with some modification.
Since the magnetic field preserves the cylindrical symmetry of the system, the Hamiltonian (<ref>) can be dealt with in terms of the $\T$-matrix in the same way as that described in Sec. <ref>. Indeed, using the ansatz (<ref>) for the eigenvalue problem of the Hamiltonian (<ref>), we obtain the equation for the radial spinor $\chi=(\chi_A,\chi_B)^t$ as
\begin{equation}
\left(
\begin{array}{cc}
U(r) - E + \nu \Delta & - i\hbar v_F \left(\partial_r + \frac{j+\frac{1}{2}}{r} + \frac{r}{2l_B^2} \right) \\
- i\hbar v_F \left(\partial_r - \frac{j-\frac{1}{2}}{r} - \frac{r}{2 l_B^2} \right) & U(r) -E - \nu \Delta
\end{array}
\right)
\left(
\begin{array}{c}
\chi_{A} (r) \\
\chi_{B} (r)
\end{array}
\right)
= 0,
\label{eq: chi A B}
\end{equation}
where $l_B$ is the magnetic characteristic length, $l_B=\sqrt{{\hbar c}/{eB}}$.
Again, we consider the Eq. (<ref>) in some region $r_a < r < r_b$ where the potential is constant, $U(r) = \bar{U}$. Following Ref. <cit.>, the general solution to this equation can be written in terms of the Kummer functions $U$ (not to be confused with the potential) and $M$ <cit.>,
\begin{eqnarray}
\chi (r)=&& e^{-br^2/2} r^{n_\sigma} \left[ C^{(1)} \alpha_\sigma M (q_\sigma,1+n_\sigma,br^2) \right. \nonumber \\
&&\qquad \left. + C^{(2)}\beta_\sigma U (q_\sigma,1+n_\sigma,br^2)\right],
\end{eqnarray}
where $q_{\sigma}= \frac{1}{4}\left[\frac{a_{\sigma }}{b}+2(1+n_\sigma) \right]$, $a_{\sigma } = 2 b \left(j+\frac{\sigma}{2}\right)- [(E - \bar{U})^2 - \Delta^2 ] /(\hbar v_F)^2$ and $n_{\sigma}=\left| j - \frac{\sigma}{2} \right|$ ($\sigma=A/B$ is identified with $\sigma=\pm 1$), $b=1/2l_B^2$. The coefficients $\alpha_\sigma$ and $\beta_\sigma$ are defined only up to their relative ratios, which are $\frac{\alpha_{-}}{\alpha_{+}}=2bi\frac{\hbar v_F}{E-\bar{U} +\nu \Delta}\left(1-\frac{q_{+}}{1+n_{+}}\right)$, $\frac{\beta_{-}}{\beta_{+}}=2bi\frac{\hbar v_F}{E - \bar{U} + \nu \Delta}$ for $j>0$; and $\frac{\alpha_{-}}{\alpha_{+}} = i\frac{E - \bar{U} -\nu \Delta}{\hbar v_F}\frac{1+n_{-}}{2bq_{-}}$, $\frac{\beta_{-}}{\beta_{+}}= - i\frac{E-\bar{U} - \nu \Delta}{\hbar v_F}\frac{1}{2bq_{-}}$ for $j<0$. This solution can be written in the form $\chi (r) = \W (\bar{U} , r) C$, with $C = (C^{(1)}, C^{(2)})^t$ and
\begin{equation}
\W(\bar{U}, r) = e^{-\frac{br^2}{2}} \left(
\begin{array}{cc}
\alpha_{+} r^{n_{+}} M (q_{+},1+n_{+},br^2) & \beta_{+} r^{n_{+}} U (q_{+},1+n_{+},br^2) \\
\alpha_{-} r^{n_{-}} M (q_{-},1+n_{-},br^2) & \beta_{-} r^{n_{-}} U (q_{-},1+n_{-},br^2)
\end{array}
\right).
\end{equation}
Further, viewing $C=(C^{(1)},C^{(2)})^t$ as the local basic coefficients we can consider an arbitrary radial potential of the form (<ref>) and follow the $\T$-matrix formalism just developed in this paper.
In particular, $\T$-matrix can be defined as the matrix that maps the basic coefficients $C_i$ in the limiting region of small $r$ to the basic coefficients $C_f$ in the limiting region of large $r$. The bound states in a circular GQD under a perpendicular magnetic field can be identified as follows. Since the Kummer function $U$ is singular at the origin <cit.>, the basic coefficients near the origin should have a vanishing component associated with $U$, $C_i \propto (1,0)^{t}$. On the other hand, in the limiting region of large $r$, the Kummer function $M$ is singular <cit.>, and should not be present in the basic coefficients in this region, implying $C_f \propto (0,1)^{t}$. As a result, the spectral equation for bound states of a circular GQD under a uniform magnetic field reads
\begin{equation}
\T_{11}= 0.
\end{equation}
For a rectangular potential this equation reduces to the spectral equations reported in Refs. <cit.>. Perpendicular magnetic fields may induce significant effects such as $(i)$ enhancing the localization of QBSs, $(ii)$ creating new bound states, and $(iii)$ lifting the valley degeneracy <cit.>. For a negative angular momentum a perpendicular magnetic field can even induce the localisation-delocalisation-localisation transition <cit.>. Particularly, the truly bound states as those in conventional semiconductor quantum dots can in principle be created by a spatially non-uniform magnetic field <cit.>.
|
1511.00317
|
Signal Processing in the MicroBooNE LArTPC
Jyoti Joshi, Xin Qian
(For THE MicroBooNE Collaboration)
The MicroBooNE experiment is designed to observe interactions of neutrinos with a
Liquid Argon Time Projection Chamber (LArTPC) detector from the
on-axis Booster Neutrino Beam (BNB) and off-axis Neutrinos at the Main
Injector (NuMI) beam at Fermi National
Accelerator Laboratory. The detector consists of a $2.5~m\times 2.3~m\times 10.4~m$ TPC including an array
of 32 PMTs used for triggering and timing purposes. The TPC is housed in an evacuable and foam insulated cryostat
vessel. It has a 2.5 m drift length in a uniform field up to 500 V/cm. There are 3 readout wire planes
(U, V and Y co-ordinates) with a 3-mm wire pitch for a total of 8,256 signal channels. The fiducial mass of the
detector is 60 metric tons of LAr.
In a LArTPC, ionization electrons from a charged particle track drift along the electric
field lines to the detection wire planes inducing bipolar signals on the U and V (induction) planes, and a unipolar
signal collected on the (collection) Y plane. The raw wire signals are
processed by specialized low-noise front-end readout electronics immersed in LAr which
shape and amplify the signal. Further signal processing and
digitization is carried out by warm electronics.
We present the techniques by which the observed final digitized waveforms,
which comprise the original ionization signal convoluted with detector
field response and electronics response as well as noise, are processed to recover
the original ionization signal in charge and time.
The correct modeling of these ingredients is critical for further event reconstruction in LArTPCs.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ LIQUID ARGON TIME PROJECTION CHAMBERS
Liquid Argon Time Projection Chambers (LArTPCs) <cit.>
provide a powerful, robust, and elegant solution for studying neutrino interactions and
probing the parameters that characterize neutrino oscillations. LArTPC technology offers a
unique combination of millimeter scale 3D precision particle tracking and calorimetry with
good dE/dx resolution. This combination results in high efficiencies for particle
identification and the background rejection. Due to its scalability and fine grained tracking
capability, LArTPC technology is a promising choice for the next generation massive neutrino
detectors. Liquid Argon is an ideal medium since it has high density, excellent properties
such as large ionization and scintillation yields, is intrinsically safe and cheap, and is readily
available anywhere as a standard by-product of the liquefaction of air. The operating principle
of large scale LArTPC detectors is based on the fact that in highly purified liquid argon, ionization tracks can
be transported by a uniform electric field over distances of the order of meters.
The signal properties of LArTPC
A single-phase LArTPC is basically a tracking wire chamber placed in
highly purified liquid argon with an electric field created within the
detector. Ionization electrons produced when charged particles go through
the detector volume would drift along the electric field until they reach the wire-planes and
hence produce signals that are utilized for imaging purposes. Several wire-planes
with different orientations using bias voltages chosen for optimal field shaping give several complimentary views of the same
interaction as a function of drift time, providing the necessary information for reconstructing a
three-dimensional image of the interaction <cit.>.
A schematic illustrating the LArTPC signal response is shown in Fig. <ref>. It shows the
planar illustration of electric field lines (i.e, electron trajectories) and the signals
induced by an ionizing track at $90\degree$ to the wire direction and at $0\degree$ to the
wire planes. In the simulation, the wires in the induction planes U and Y are inclined at
$\pm45\degree$ with respect to wires in the Y collection plane. Bipolar signals from two induction
planes, and the unipolar signal from the collection plane are
processed and readout by specialized low-noise front-end readout electronics immersed in LAr.
§ THE MICROBOONE LARTPC
MicroBooNE <cit.> is newly built LArTPC neutrino detector of 60 metric ton
fiducial mass (170 ton total) at Fermilab National Accelerator
Laboratory in Batavia, Illinois. MicroBooNE recently started its
operation and has been collecting neutrino data from the Booster
Neutrino Beamline (BNB) since October, 2015.
The experiment's primary motivation is to resolve the source of the
MiniBooNE low energy excess observed in $\nu_e$ candidates by taking advantage of the excellent electron-photon
particle identification capabilities of a LArTPC in addition to
carrying out a comprehensive suite of neutrino cross section measurements on Argon.
Schematic diagram of MicroBooNE detector
The TPC itself contains three wire planes, one collection plane at $0\degree$ from vertical and two
induction planes at $\pm60\degree$ with 3-mm wire pitch and 3-mm wire
plane separation. MicroBooNE also serves as a test bed for LArTPC
technologies for next generation very large scale detectors. There are many innovative technologies
implemented, such as 2.5 m long drift distance, cold front-end
low-noise readout
electronics <cit.> and a filling procedure
that does not include prior evacuation of the cryostat while still
maintaining ultra-high purity LAr. The use of cold electronics within
the LAr volume is critical for enabling the scaling up of the LArTPC
technology and to improve the signal-to-noise ratio. The light collection system <cit.> consists of 32 8-inch PMTs that are located just behind the wire planes to detect scintillation light from $\nu$-Ar interactions. The PMT information is used to trigger
on beam events and significantly reduce the data throughput. A
schematic diagram of the MicroBooNE detector is shown in Fig. <ref>.
§ SIGNAL PROCESSING CHAIN
The raw signal on the wires in the TPC consists of both the ionization
signal and the noise. The signal
is a convolution of the distribution of the electron cloud passing through the TPC wires, the field response
(i.e. the induced current on wires), and the electronics response. The background includes the noise
from various sources. The goal of signal processing is to extract both
the signal charge and time information
reliably and separate it from the noise. The following subsections will give details on each step.
§.§ Field and Electronics Response Modeling
A detailed knowledge of the field and electronics response is necessary in order to characterize the
detector performance. Simulating the field response function is the first step in the chain of signal processing.
The drifting electrons are modeled as many small clouds of charge that diffuse as they travel toward
the collection wires. The response of the channels to the drifting electrons is parameterized as a function
of drift time, with separate response functions for collection and induction wires. The signals on the
induction-plane wires result from induced currents and are thus bipolar as a function of time as charge
drifts past the wires, while the signals on the collection plane wires are unipolar.
Fig. <ref> (left) shows the 2-D GARFIELD <cit.>
simulated response to a single electron blob generated in the MicroBooNE detector
geometry in terms of charge vs. time averaged for a single electron for both induction planes (U-Plane
in black and V-Plane in red) and collection plane (in blue).
[resp]Field Response Functions (left) and Electronics Shaping Functions (right)
The electronics response function for the MicroBooNE detector is shown in Fig. <ref> (right) in
terms of signal amplitude vs. time. Since the MicroBooNE front-end cold
electronics are designed to be programmable with 4 different
gain settings (4.7, 7.8, 14, and 25 mV/fC) and 4 shaping time settings
(0.5, 1, 2, and 3 us), the electronic
response function varies according to these settings. Different colored lines in Fig. <ref> (right)
show the electronics response for different shaping time settings. For a fixed gain setting, the peak
is always at the same height independent of the shaping time.
§.§ Raw and Convoluted Signal
In this section the true raw wire signal and the actual signal obtained
after processing by the readout electronics are described. The
digitized signal obtained after the ADC is formed when the ionization
signal is convoluted with the detector and the front-end cold
electronics response functions and then digitized at a
fixed frequency. The
top row of Fig. <ref>
shows the raw MIP signal in the U, V and Y-Planes and in the bottom
row, the raw signal convoluted with field and
electronics response for different shaping time settings are shown.
[raw_conv]Raw Signal (top row); Convoluted Signal (bottom row) for U, V and Y-Plane respectively
§.§ Noise Sources in Detector
The readout electronics and digitization circuits are the two main sources of noise in the detector.
In the case of the front-end readout electronics, the first transistor noise is the main component <cit.>.
The first transistor noise contribution to the measured signal charge is proportional to the
total capacitive load on the input channel, comprised of the sense wire capacitance, cable capacitance,
and input transistor capacitance. This total capacitance limits the signal-to-noise
ratio and it is the one dominant factor on which the feasibility and scalability of a
LArTPC design critically depends.
The other electronics noise sources are from thermal noise on the sense wires and
connection leads (signal cable). Usually, thermal noise is made negligible by
choosing appropriate resistors. Digitization noise arises during the
signal digitization by the ADC
which has a 12-bit resolution and a sampling rate of 2 mega-samples
per second (MS/s). The digitizer
has been chosen in such a way to ensure that digitization noise is
much smaller than the front-end electronics noise.
Other noise sources such as microphonics, and pick-up noise on the TPC
wires could also be present in the detector. These can be eliminated
using further signal processing steps that are not discussed in this
§.§ Deconvolution
The next stage of signal processing is termed “Deconvolution", which means reversing
the effects of convolution by unpacking and removing the readout electronics
and field response of the wire planes. The basic deconvolution process
is implemented in the standard LArSoft <cit.> software signal
processing procedure which was originally developed by the ArgoNeuT
experiment <cit.>,
and further developed by MicroBooNE. The process of deconvolution is
explained using the following equations.
\begin{equation}
\label{eq:decon_1}
M(t_0) = \int_{\{t\}} R(t-t_0) .\, S(t) \, dt
\end{equation}
\begin{equation}
\label{eq:decon_2}
M(w) = R(w) . S(w)
\end{equation}
\begin{equation}
\label{eq:decon_3}
S(w) = \frac {M(w)} {R(w)}
\end{equation}
If M is the measured signal i.e, the digitized signal convoluted with
the response functions R,
and S is the desired real signal, then the measured signal in the time domain is given by decon_1.
In order to remove the effects of the different response functions, a
fast fourier transformation (FFT) <cit.> is
performed on the measured signal in the time domain. The resulting
measured signal in the frequency domain is as shown in decon_2. By using simple factorization,
the real deconvoluted signal (number of electrons reaching wire
planes) is then extracted in the
frequency domain, decon_3. To obtain the real charge signal in time domain S(t),
an inverse fourier transformation is performed.
As discussed in the previous section, there are different types of noise sources present in
the detector and in order to extract the true charge signal from the
measured signal, the noise contribution to the measured signal needs
to be eliminated as much as possible. The filtering of the noise from
the measured signal will be discussed in next section.
§.§ Noise Filtering
To remove noise from the deconvoluted signal, a Wiener noise filter
<cit.> is constructed using
the expected signal and noise frequency response functions. The Wiener filter can be defined as
in filter, where S is the signal and N is the noise in the frequency domain.
\begin{equation}
\label{eq:filter}
F(w) = \frac {S^2(w)} {S^2(w)+N^2(w)}
\end{equation}
Wiener deconvolution is done in the frequency domain in order to minimize the impact of
deconvolved noise at frequencies which have a poor signal-to-noise
ratio. Using the noise filter,
$F(w)$, in the deconvolution method, decon_3 is modified as such:
\begin{equation}
\label{eq:decon_filt}
S(w) = \frac {M(w)} {R(w)} . F(w)
\end{equation}
The effect of noise and the process of noise filtering is described in Fig. <ref>
using toy Monte Carlo (MC) samples. Fig. <ref> shows (a) a very good agreement of the true
injected signal (blue) and the deconvoluted signal (red) without any noise in the time domain;
(b) when the signal digitization is introduced, noise can be clearly seen in deconvoluted signal (red);
(c) the deconvoluted signal (red) after adding random white noise as an example of electronics
noise where the deconvoluted signal peak can not be seen due to large amount of noise present;
(d) after introducing the Wiener noise filter in the deconvolution step using decon_filt,
the deconvoluted signal (red) is very close to the true injected signal (blue).
[noise_filt]Effect of Signal Processing and Noise Filtering
The Wiener Filter function is optimized for different gain and shaping time settings for both induction
and collection planes. The filter reduces the noise by a significant amount, however there is
a loss in signal amplitude too. In order to preserve the signal
strength and to obtain the true charge information from all the three
planes, the filter is normalized in such a way so that it
conserves the area. This step is very important in order to have a robust noise filter.
§ ADDITIONAL SIGNAL PROCESSING CHALLENGES
With the robust noise filtering and deconvolution, the full signal processing chain is complete
and the desired charge signal is obtained from all three planes. There are still some additional challenges
involved in the process. Due to the wire readout assembly of LArTPC, there is an effect of dynamic induced
charge as ionized electrons traveling through the TPC wires induce signal not only on the closest
wire but also on the adjacent wires. The field model described above does not take into account the
charge contributions from the adjacent wires and treats signal from each TPC wire independently.
To show the effect of induced current on the signal amplitude,
Fig. <ref> (left) shows the weighted
equipotential contours (green) for a U-Plane wire superimposed on the
electron drift lines (orange).
The induced charge on each wire is derived using dic along
the drift line of each electron.
\begin{equation}
\label{eq:dic}
i = - q_m E_w . v_d
\end{equation}
Fig. <ref> (right) (b) shows the induced current waveform for a
central U-Plane wire using the 2-D GARFIELD
simulation for a track $1.7\degree$ from the vertical (shown in
(a)). In comparison with the signal response
in Fig. <ref> (left), the induced charge signal is more complicated and strongly depends on the angle of the track.
[dic]2-D field simulations of the weighted potential distributions for a U-Plane wire.
The weighed equipotential contours (green) are shown superimposed on
the electron drift lines shown in orange (left); and (a) for a track $1.7\degree$ from the vertical, (b) the induced current waveform obtained from the Garfield simulation on the central U-Plane wire.
In order to account for the dynamic induced charge effect, the
traditional deconvolution scheme described above is revised using the
2-D fast fourier transformation method in both time and wire parameter
space. This implementation of a double deconvolution method is the
most recent development in the LArSoft signal processing procedure.
§ NEW 3D RECONSTRUCTION WITH CHARGE AND TIME
After the robust signal processing chain, a new 3D event reconstruction method using both
charge and time information is being developed. This new method is called “Wire-Cell
Reconstruction" <cit.>. Reconstruction with a LArTPC wire
plane readout is challenging
due to inherent ambiguities/degeneracies when using a projective wire geometry. In particular, the timing information alone is not enough
to remove various ambiguities in a complex electromagnetic shower consisting of many tracks.
An example of such a degeneracy is illustrated in Fig. <ref>
(left) using only two wire planes
for simplicity. The true hits are shown in red and the fake hits in
blue. For a given time slice, a
total of six possible hits on two U wires and three V wires generates ambiguation. This degeneracy
increases exponentially with the increase in number of hits. Additional information is required in order to
remove this degeneracy.
[wirecell]An example of hit degeneracy using two planes (left); and charge matrix equation to solve degeneracy (right)
Since the same charge is measured by all three wire planes, charge as
well as time information can be reliably used
to resolve this degeneracy. Fig. <ref> (right) shows the charge matrix equations, where $u_i$, $v_i$
are the measured charges on the wires, $H_i$ matrix is the true charge to be resolved. After solving these 2D equations, the
charge on the fake hits is expected to be close to zero and hence the degeneracy is greatly reduced. This technique
is used to obtain 3D hit maps by combining results from different time slices. This new algorithm is under rapid
development towards the goal of automated reconstruction for LArTPC.
§ CONCLUSIONS
The LArTPC is an excellent detector technology for precision neutrino
physics measurements. The MicroBooNE experiment, being the first
experiment in the future short
baseline program at Fermilab is an important step in the
development of LArTPC technologies
for future multi-kiloton detectors, in addition to enabling a wide range
of measurements of neutrino cross-sections and interactions. LArTPC
signal processing is the first and a critical
step in obtaining the correct charge and time information from all
three wire planes. After robust signal processing, we have
demonstrated that both the correct charge and
time information on all 3 planes can be obtained and is available
to be used in future improved 3-D tracking and calorimetry measurements.
C. Rubbia, “The Liquid Argon Time Projection Chamber: A New Concept for Neutrino Detectors," CERN-EP-INT-77-08.
W. Willis and V. Radeka, Nuclear Instruments and Methods 120, no. 2, 221-236 (1974).
O. Bunemann, T.E. Cranshaw, J.A. Harvey, Can. J. Res. 27, 191 (1949).
H. Chen et al., “Proposal for a New Experiment Using the Booster and NuMI Neutrino Beamlines: MicroBooNE," 2007. FERMILAB-PROPOSAL-0974.
MicroBooNE Technical Design Report,
V. Radeka et. al., “Cold Electronics for 'Giant' Liquid Argon Time Projection Chambers," J. Phys. Conf. Ser. 308 (2011) 012021.
J. Conrad et al., “The Photomultiplier Tube Calibration System of the MicroBooNE Experiment," JINST 10, T06001 (2015).
R. Veenhof,
“GARFIELD, recent developments,”
Nucl. Instrum. Meth. A 419, 726 (1998).
E. D. Church, “LArSoft: A Software Package for Liquid Argon Time Projection Drift Chambers," arXiv:1311.6774 [physics.ins-det].
B. Baller, “Liquid Argon TPC Signal Formation, Signal Processing and Reconstruction," (under preparation).
Fast Fourier Transform, http://mathworld.wolfram.com/FastFourierTransform.html
Wiener, Norbert, “Extrapolation, Interpolation, and Smoothing of
Stationary Time Series," New York: Wiley. ISBN 0-262-73005-7 (1949).
Wire-Cell Reconstruction Package, http://www.phy.bnl.gov/wire-cell/
|
1511.00526
|
Early-type galaxies (ETGs) host a hot ISM produced mainly by stellar
winds, and heated by Type Ia supernovae and the
thermalization of stellar motions. High resolution 2D
hydrodynamical simulations showed that
ordered rotation in the stellar component results in the formation of a centrifugally supported cold
equatorial disc. In a recent numerical investigation we
found that subsequent generations of stars are formed in this cold disc;
this process consumes most of the cold gas, leaving at the present epoch cold
masses comparable to those observed. Most of the new stellar mass
formed a few Gyrs ago, and resides in a disc.
§ INTRODUCTION
High resolution 2D hydrodynamical simulations with the ZEUS-MP2 code
showed that ordered rotation in the stellar
component affects significantly the evolution of the hot ISM in ETGs, and, among other effects,
results in the formation of a centrifugally supported cold
equatorial disc (Negri et al. 2014, hereafter N14). This disc can be extended ($\sim
0.5 -10$ kpc radius),
and as massive as $10^{10}$M$_{\odot}$ in the biggest ETGs.
Indeed there is evidence that $\sim 50$%
of massive ETGs host significant quantities of cold gas (Davis et al. 2011, Serra et al. 2014), often in settled
configurations, sharing the same kinematics of the stars, consistent
with an internal origin. Also, the cold gas seems to provide material for low level star formation (hereafter SF);
and, in the ATLAS$^{\rm 3D}$ sample,
molecular gas, SF and young stellar populations are detected only in fast rotators
(Sarzi et al. 2013, Davis et al. 2014).
We then added the possibility for the gas to form stars to the simulations of N14, to explore whether SF can
bring the amount of cold gas in the models
more in agreement with observed values, and whether it can explain
the low-level SF activity currently seen to be ongoing in rotating systems.
From top to bottom, final values for all rotating models of:
the cold gas mass $M_{\rm c}$ [without SF (left panel), and
with the two adopted $\eta_{\rm SF}$ values (other panels)];
the stellar mass in newly formed stars $M_*^{\rm new}$; the SF rate
normalized to the initial stellar mass of the galaxy $M_*$;
the mean formation time of the new stars $\langle t\rangle _*^{\rm new}$,
calculated from an initial time of 2 Gyr.
See Negri et al. (2015) for more details.
§ THE MODELS AND THE RESULTS
Hydrodynamical simulations were run for a representative subset of 12 rotating models from N14, including in the code
the removal of cold gas, and the injection of mass, momentum and
energy appropriate for the newly forming stellar population (Negri
et al. 2015). SF was implemented by
subtracting gas from the grid, at an adopted rate per unit volume of $\dot\rho_{\rm SF} = \eta_{\rm SF} \rho/{t_{\rm SF}}$,
where $\rho$ is the gas density, $\eta_{\rm SF}$ is the SF efficiency ($\eta_{\rm SF}=0.01$
and $0.1$ were adopted), and ${t_{\rm SF}}$ is the maximum between the cooling timescale
and the dynamical timescale.
In a typical (cyclical) evolution, gas injected by the stellar population
accumulates until radiative losses become catastrophic,
significant amounts of cold material are produced, and SF is enhanced. At the end
the cold gas mass $M_{\rm c}$ is much reduced in the models with SF, and the
mass in new stars $M^{\rm new}_*$ is close to the $M_{\rm c}$ values of the models without SF (Fig. 1).
The new stars reside mostly in a disc, and could be related to a younger, more metal rich
disky stellar component observed in fast rotators (Cappellari
et al. 2013). Most of $M^{\rm new}_*$ formed a few Gyrs ago;
the SF rate at the present epoch is low ($\le 0.1$M$_{\odot}$yr$^{-1}$), as observed,
at least for model ETGs of stellar mass $<10^{11}$M$_{\odot}$ (Fig. 1).
The adopted SF recipe reproduces the slope of the
Kennicutt-Schmidt relation, and even its normalization for $\eta_{\rm SF}=0.01$ (Negri et al. 2015).
Cappellari, M., et al. 2013, MNRAS, 432, 1862
Davis, T.A., et al. 2011, MNRAS, 417, 882
Davis, T. A., et al. 2014, MNRAS, 444, 3427
Negri, A., Posacki, S., Pellegrini, S., & Ciotti, L. 2014,
MNRAS, 445, 1351 (N14)
Negri, A., Pellegrini, S., & Ciotti, L. 2015, MNRAS, 451, 1212
Sarzi, M., et al. 2013, MNRAS, 432, 1845
Serra, P., et al. 2014, MNRAS, 444, 3388
|
1511.00587
|
Using kinetic Monte Carlo simulations, we show that molecular morphologies found
in non-equilibrium growth can be strongly different from those at equilibrium.
We study the prototypical hybrid inorganic-organic system 6P on ZnO(10-10) during thin
film adsorption, and find a wealth of phenomena including re-entrant growth, a critical
adsorption rate and observables that are non-monotonous with the adsorption rate. We identify
the transition from lying to standing molecules with a critical cluster size and discuss
the competition of time scales during growth in terms of a rate equation approach. Our results form a basis
for understanding and predicting collective orientational ordering during growth in hybrid
material systems.
(Color online) Growth schematic for submonolayer growth of 6P on ZnO(10-10). Part (a) depicts the substrate (alternating blue and red colors represent charge lines)
with molecules adsorbing, diffusing translationally and rotationally (see green arrows). Part (b) defines the three principles axes of a molecule, u, q and q$\prime$, as a function of the three Euler angles $\alpha$, $\vartheta$ and $\varphi$ (with q and q$\prime$ defining linear quadrupole moments).
Hybrid inorganic-organic systems (HIOS) have revolutionized opto-electronic semiconductor technologies
by combining the high charge carrier densities and high tunability of conjugated organic molecules with
stable, well controlled inorganic substrates <cit.>.
At the same time, the design of efficient devices poses fundamental physical questions on a multitude of length- and time-scales, from resonance energy transfer in complex
environments <cit.> and energy level alignment <cit.>
to classical statistical physics problems such as collective ordering of anisotropic molecules at interfaces, both in equilibrium and during
non-equilibrium growth, i.e., at finite flux <cit.>.
The final orientational ordering is indeed crucial fo HIOS functionality <cit.>. Experimentally,
a number of interesting structural phenomena in HIOS have been observed,
such as the increase of lying clusters with temperature <cit.>, the coexistence of domains with
different molecular tilt angles, and layer-dependent tilt angles <cit.>. However, so far there is no
consistent understanding from the theoretical side <cit.>.
A generic, yet unsettled phenomenon seems to be that single molecules lie
on substrates, while thin films generally form a standing configuration. Understanding this transition is further challenged
by the fact that in typical HIOS, the substrate generates electrostatic or topological
surface fields with significant impact on molecular ordering <cit.>.
In the present letter, we consider a prototypical HIOS system and ask the question: How does the molecular component transition from lying molecules to
standing clusters during non-equilibrium, sub-monolayer growth? We demonstrate, using theoretical calculations, that this transition is dominated by an interplay of anisotropic interactions and growth kinetics.
Specifically, we consider sexiphenyl (6P) molecules on a zinc-oxide (ZnO) 10-10 surface, see Fig. <ref>. This system combines two generic HIOS features: strongly anisotropic molecules with non-symmetric (here dominantly quadrupolar) charge distributions and an electrostatically patterned substrate, here " charge stripes" <cit.>. We have recently developed a coarse-grained model <cit.> based on DFT parametrization <cit.>,
which reproduces key equilibrium properties.
Here, we employ kinetic Monte Carlo (KMC) simulations to access the large time- and length-scales of collective ordering for different adsorption rates.
Currently, KMC simulations of anisotropic molecules are in their infancy. One conceptual problem is that the molecules'
mobility is influenced by its orientation <cit.>.
In earlier studies of molecular growth, the molecules' orientations are strongly restricted to discrete orientations <cit.>
or a 2D (stripe-patterned) plane <cit.>.
However, in reality most organic molecules explore the full, 3D space of orientations <cit.>. Here, we consider molecules that are
translationally confined to a 2D lattice, but have continuous, 3D rotational degrees of freedom.
As very few growth simulations of hybrid systems exist, and since 6P/ZnO(10-10) forms a prototypical case, our results provide first steps towards
a general understanding of growth phenomena in HIOS.
In KMC simulations, the surface evolution is described as a series of transitions between discrete states, where
each transition is characterized through a rate. Specifically, we use a `Composition and Rejection' algorithm to select events and determine the system time from event propensities,
a method previously used for biochemical models <cit.>.
This algorithm is advantageous for simulations with continuous rotational degrees of freedom, as it does not require the computationally expensive
calculation of a complete process rate catalogue after each process <cit.>.
The processes occuring during surface growth can be summarized in terms of three different types: rotational diffusion (r), translational diffusion (d)
and adsorption (a), described through the rates $r_i^{\textrm{r}}$, $r_i^{\textrm{d}}$ and $r_i^{\textrm{a}}$, respectively (see Fig. <ref>(a)).
Each process type is associated with an attempt frequency $\nu$, which we use as propensity in our algorithm.
The rotational rate allows continuous rotational degrees of freedom. Using an adiabatic approximation, it can be expressed as
\begin{equation}
r^{\textrm{r}}_i=\nu^{\textrm{r}} \cdot \min \left\{ \exp\left[ \beta\left(H^{\textrm{i}}(i)-H^{\textrm{f}}(i)\right)\right],1\right\}\textrm{,}\label{eq:rot_rate}
\end{equation}
where $H^{\textrm{i}}(i)$ and $H^{\textrm{f}}(i)$ are the effective coarse-grained interaction Hamiltonians for the initial and final configuration of molecule $i$, respectively,
and $\beta=(k_{\textrm{B}} T)^{-1}$ is defined through the Boltzmann constant $k_{\textrm{B}}$ and the temperature $T$. Here, we set $T=300$ K. Following <cit.>,
the effective interaction Hamiltonian is written as
\begin{equation}
\label{eq:Heff}
\end{equation}
where $H_{\textrm{m-m}}$ characterizes the intermolecular (6P-6P) interactions
consisting of an electrostatic quadrupolar contribution $V_{\textrm{QQ}}$ <cit.> between the
(linear) quadrupoles assigned by q in Fig. <ref>(b) and a non-electrostatic Gay-Berne interaction $V_{\textrm{GB}}$ <cit.>.
Correspondingly, the molecule-substrate Hamiltonian $H_{\textrm{m-s}}$ contains a quadrupole-field interaction for a quadrupole denoted by
q$\prime$ in Fig. <ref>(b). This electrostatic interaction
reflects the hybrid nature of the 6P/ZnO(10-10) system.
A single 6P molecule tends to align with the charge lines of the substrate <cit.>.
Further, $H_{\textrm{m-s}}$ includes a non-electrostatic interaction
$V_{\textrm{LJ}}$ <cit.> that involves attractive $z^{-3}$ interactions between molecule and substrate. The different contributions to $H_{\textrm{eff}}^{\textrm{pot}}$ are parametrized based on
DFT calculations <cit.>.
The rotational rate in Eq. (<ref>) lacks transitional information between the initial- and final configuration. This reflects the underlying "adiabatic"
approximation, i.e., a much faster relaxation of rotational versus translational motion.
The latter is modelled as an activated process between identical, energetically most favourable lattice sites of the ZnO(10-10) substrate <cit.>, as discussed in detail in
the SM1. As translational diffusion does not
include any rotation, the m-s interaction of initial site and destination are identical. Thus, $r_{i}^{\textrm{d}}$ involves
a constant contribution $E_{\textrm{d}}$ determined through the substrate, as well as the initial neighbour interaction energy $H_{\textrm{m-m}}^{\textrm{i}}$,
\begin{align}
\begin{cases}
&\min\left(\exp\left[\beta\left(H_{\textrm{m-m}}^{\textrm{i}}(i)-E_{\textrm{d}}\right) \right], 1 \right)\\
&\qquad \qquad \qquad \qquad \qquad \quad \textrm{if }H_{\textrm{m-m}}^{\textrm{f}} (i) \leq E_{\textrm{d}}\\
&\min\left(\exp\left[\beta\left(H_{\textrm{m-m}}^{\textrm{i}}(i)-H_{\textrm{m-m}}^{\textrm{f}}(i)\right)\right], 1 \right) \textrm{else}.
\end{cases}
\label{eq:diff_rate}
\end{align}
The second case comes into play if the molecule's destination is blocked through repulsive interaction with other molecules, i.e. if $H_{\textrm{m-m}}^{\textrm{f}}(i)>E_{\textrm{d}}$, as discussed in
In principle, information about $E_{\textrm{d}}$ can be obtained from atomistically resolved molecular dynamics (MD) simulations <cit.>. These have shown that $E_{\textrm{d}}$ is direction-dependent (as one might expect).
Nevertheless, we here set $E_{\textrm{d}} = 0$ eV for all directions of diffusion.
Test calculations for $E_{\textrm{d}} \leq 0.2$ eV (the free energy barrier found for diffusion along charge lines <cit.>)
indicate that this approximation has no substantial influence on our results.
The final process is the adsorption of molecules, expressed through the adsorption rate
\begin{equation}
r^{\textrm{a}}_i=\nu^{\textrm{a}}=f\cdot \nu^{\textrm{d}}\textrm{.}\label{eq:ads_rate}
\end{equation}
(Color online) Growth properties as a function of time for different adsorption rates $f$ . Part (a) depicts the overall density of molecules
while part (b) shows the densities of standing and lying molecules as a function of time. Each molecule with $\left|\vartheta_i \right| \leq 0.25\pi$ is classified as standing.
(Color online) Orientational order parameters as function of density. (a), (c) nematic order parameter $S$; (b), (d) biaxiality parameter $B$
as functions of the molecule density for different $f$. Parts (a) and (c) have underlying colourblocks that mark the different morphologies
arising for $f=0.0001$ and $f=0.001$, respectively. The vertical lines mark the transitions between these morphologies.
The transitions in equilibrium and at $f=0.01$ are not explicitely marked.
Note that we do not consider desorption and only count particles that adsorb on previously unoccupied substrate sites. The numerical results presented below have been obtained using a vertical adsorption orientation of molecules; the influence of different
adsorption configurations is discussed in SM2.
We use adsorption rates in the range of $10^{-6}\leq f \leq 1$, which we compare to the upper limit of the free diffusion rate $r^{\textrm{d}}_{\textrm{max}}\approx 2\cdot 10^{9}$ Hz determined on the basis
of MD simulations <cit.>. We find that for $f=10^{-6}$ maximally half a monolayer of molecules adsorbs within $\approx 0.4$ ms; thus we are far beyond typical time-scales of MD simulations. Simulation details are
discussed in SM1.
We characterize the orientational order using the order parameter tensor <cit.>,
$A_{\alpha\beta}(t)=N^{-1}\sum_{i=1}^N \left\langle u_{i,\, \alpha} u_{i,\, \beta}
- \frac{1}{3} \delta_{\alpha\beta}\textrm{Tr}(\textbf{u}_i\otimes \textbf{u}_j) \right\rangle$,
where Tr stands for the trace, $\otimes$ denotes a dyadic product and $\langle\dots\rangle$ is an average over all molecules and all runs in
the system at time $t$.
The nematic order parameter $S \in \left[-0.5,1\right]$ and the biaxiality parameter $B\in \left[0,1\right]$ are derived from the eigenvalues of $A_{\alpha\beta}(t)$ (see SM3).
For a perfectly uniaxal system $S = 1$, $B=0$, while $S = B=0$ represents a completely disordered system.
Positive values of both, $S$ and $B$, indicate nematic ordering with some degree of biaxiality. Finally, molecular ordering along two orthogonal axes within a plane <cit.> yields
$S=-1/2$, $B=0$, corresponding to maximal biaxiality (for further details, see SM3).
As a starting point, we consider the density of molecules, i.e. the number of molecules per surface unit cell of size $A$, as a function of time in Fig. <ref> (a).
The density increases monotonously for all adsorption rates.
This monotonic, yet non-linear behaviour is consistent with a rate-equation approach introduced below.
In the subsequent plots, we therefore replace the time axis by a density axis (adjusted to the adsorption rate considered).
Figure <ref> (b) gives first insight into the orientational ordering at small adsorption rates by depicting separately the density of standing and lying molecules.
Initially, the majority of molecules lies on the substrate (with preference of the $x$-direction as supported by the ZnO charge lines), until $t\approx3000\,(\nu^{\textrm{d}})^{-1}$.
Then, the fraction of standing molecules starts to dominate, and at large times (high density) all molecules stand.
The implications for cluster growth are discussed in SM4, where
we also estimate the critical cluster size for the lying-standing transition ($\approx 15$ molecules). Both, the lying and the standing configuration correspond to
equilibrium states (determined at $f=0$ and $r^{\textrm{d}}=0$ by scaling the lattice constants) at low and high densities, respectively <cit.>.
Moreover, the observation that molecules initially lie and later stand during thin film growth closely resembles experimental findings for self-assembled monolayers of decanthiol on silver
<cit.>: orientationally sensitive scattering measurements show that the first adsorbed molecules nearly immediately form a lying phase,
which, as the density increases, transforms to a standing phase <cit.>. This orientational reordering appears to be generic for a wide class of material
combinations <cit.> and
only very strongly attractive substrates do not support a transition to a standing molecular orientation <cit.>.
The dependency of the orientational ordering on the rate of adsorption is analyzed in Fig. <ref>, where we plot
$S$ and $B$ as functions of density for several values of $f$ (including $f=0$). An overview over different $f$ and densities is given in Fig. <ref>. Apart from statistical fluctuations, the data do not depend on the system size.
In equilibrium, the system displays two transitions (see Fig. <ref>(a)): First, from nematic (lying) to a biaxial
phase where the molecules orient either along the charge lines or the $z$-axis (at a density of $\approx 0.26\,A^{-1}$) or along the $z$-axis (standing). Second, from
biaxial to full nematic standing at $\approx 0.41\,A^{-1}$.
We now turn to non-equilibrium effects. For low, yet non-zero adsorption rates ($f=0.0001$), molecules have less time to form a nematical lying order.
As a consequence, both the transition from lying to biaxial and the transition from biaxial to
standing nematic move towards lower densities.
For very high adsorption rates ($f=0.01$), the standing order initiated through molecule adsorption dominates the
orientational ordering throughout the growth process, as depicted in Fig. <ref>(c). Here, the nematic order parameter $S$
never assumes negative values and the biaxiality parameter never is significantly larger than zero.
The intermediate range is dominated by competition of various morphologies, exemplified for $f=0.001$ in Fig. <ref> (c).
Initially, the standing orientation dominates. Then, the system transitions to biaxial
at $\approx 0.02\,A^{-1}$ and lying nematic at $\approx 0.078\,A^{-1}$, as the initially adsorbed molecules lie down. With increasing molecule density,
the molecular morphology observed in equilibrium becomes more significant. Correspondingly, the transition from lying to biaxial, and from biaxial to standing
resemble the transitions seen for $f=0.0001$ in Fig. <ref>(a) and (b), except that they are shifted to lower densities. We call this
phenomenon `re-entrant' growth, because the initial standing orientation vanishes, but reoccurs during the final stages of growth.
The rate-dependency of the collective orientation behaviour is summarized in Fig. <ref> for a range of adsorption rates and associated densities.
(Color online) `Growth state diagram' as a function of adsorption rate and density for
vertically adsorbed molecules. Regions of different orientation are separated through transition points that correspond to a change in
sign of $S$, as exemplified in Fig. <ref>. The dashed gray horizontal lines mark
the transitions expected in equilibrium (see Fig. <ref>(b)).
Note that the density is here expressed in units of nm$^{-2}$ (contrary to Fig. <ref>).
The snapshots correspond to the lying, biaxial and standing
configuration for $f=0.0001$, which correspond to molecule densities $0.39$ nm$^{-2}$, $1.59$ nm$^{-2}$ and $3.2$ nm$^{-2}$, respectively. The arrows mark
the adsorption rates shown in Fig. <ref>.
three interesting features become apparent. First, there is re-entrant growth for intermediate adsorption rates close to $f=0.001$, as discussed above. Second, the transitions of $S(f)$ (marked
in Fig. <ref>) are non-monotonous with $f$. This observation is verified by multiple runs for $0.005\leq f \leq 0.0005$ for
lattices with between $2500$ and $10^6$ lattice sites. As a consequence, the orientational ordering at a fixed density of 0.25 nm$^{-2}$ follows a sequence of states upon increase of $f$: lying,
biaxial, lying, biaxial and finally standing order. Third, there appears to be a `critical' adsorption rate $f_{\textrm{crit}}\approx0.005$ beyond which no transitions occur.
To rationalize the occurence of $f_{\textrm{crit}}$, we propose the following rate-equation model involving discretized rotational configurations (for details, see SM5).
Specifically, we assume that the number of vacant sites, $N_v$, sites occupied by standing particles, $N_s$, and sites occupied by lying particles, $N_l$, change
in time according to
$\dot{N}_v\equiv dN_v/dt=-N_v f$, $\dot{N}_s=N_v f- N_s r$,
$\dot{N}_l=N_s r$,
where $r$ is the rate of reorientations. The transition from standing to lying is identified with the condition $N_s(t)=N_l(t)$. Solving the rate equations accordingly we find the implicit equation
where $\Theta(t)=(N_s(t)+N_l(t))/N$ and $x=f/r$. This yields (see SM5) the critical adsorption rate $f_{\textrm{crit}}^,\approx 0.0055$,
in very good agreemnet with the corresponding simulation result. Physically, our model demonstrates that $f_{\textrm{crit}}$ results from
the competition between the time scales of adsorption ($f^{-1}$) and reorientation ($r^{-1}$).
In summary, our KMC simulations of the prototypical organic-inorganic system 6P on ZnO(10-10) clearly show that non-equilibrium morphologies can be strongly different from those found for $f=0$. The present calculations are based on spatially dependent pair potentials parametrized according to DFT results, and unlike earlier KMC studies <cit.>
we assume continuous 3D rotations. This allows us the explore the full complexity of non-equilibrium orientational ordering.
To which extent are the results generic for hybrid inorganic-organic systems? In fact, while our coarse-grained Hamiltonian (<ref>) is designed
for a specific material system (and surface temperature $T$), the observed competition between lying and standing orientations
as function of $f$ is a rather universal feature <cit.>. Regarding surface properties, one would clearly expect an impact
of $T$ and also of the surface structure: For higher $T$ the "orientational bias" exerted by charged stripes of ZnO(10-10) weakens, destabilizing the lying nematic phase already at $f=0$<cit.>. In non-equilibrium ($f>0$), we thus expect the critical rate $f_{\textrm{crit}}$ for vertical adsorption (and generally, the regime of rates related to lying phases)
to decrease. For substrate temperatures $T\lesssim 300~K$ <cit.>, the temperature difference between adsorbing molecules and the substrate may cause hot precursor states, which reduce the influence of $T$ <cit.>. Similarly, inorganic surfaces without a biasing field will lead to a destabilization of biaxial phases. Regarding the molecules, decreasing their length (e.g., from 6P to 2P)
reduces the anisotropy of sterical interactions even in the uncharged case <cit.>; thus, lying phases can exist
up to higher densities <cit.>. For quadrupolar molecules, shorter lengths weaken the electrostatic substrate interactions <cit.>, so we again expect $f_{\textrm{crit}}$ to decrease.
The understanding of these different, nonequilibrium molecular structures forms an important ingredient for later calculations of optical behaviour.
Hence, our understanding of the collective ordering during growth of hybrid structures contributes to the creation of greener, more sustainable and cost-efficient opto-electronic
devices <cit.>. We hope that our simulation results stimulate further experiments in this direction.
This work was supported by the Deutsche Forschungsgemeinschaft within CRC 951 (project A7).
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1511.00342
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Electronic address: [email protected]
Beijing Computational Science Research Center, Beijing 100084, China
CNR-SPIN, I-84084 Fisciano (Salerno), Italy and Dipartimento di Fisica “E. R. Caianiello", Universit$\grave{a}$ di Salerno, I-84084 Fisciano (Salerno), Italy
Beijing Computational Science Research Center, Beijing 100084, China
Electronic address: [email protected]
Center for Interdisciplinary Studies $\&$ Key Laboratory for Magnetism and Magnetic Materials of the MoE,
Lanzhou University, Lanzhou 730000, China
Beijing Computational Science Research Center, Beijing 100084, China
Electronic address: [email protected]
Beijing Computational Science Research Center, Beijing 100084, China
Beijing Computational Science Research Center, Beijing 100084, China
The Rabi model plays a fundamental role in understanding light-matter interaction. It reduces to the Jaynes-Cummings model via the rotating-wave approximation, which is applicable only to the cases of near resonance and weak coupling. However, recent experimental breakthroughs in upgrading light-matter coupling order require understanding the physics of the full quantum Rabi model (QRM). Despite the fact that its integrability and energy spectra have been exactly obtained, the challenge to formulate an exact wavefunction in a general case still hinders physical exploration of the QRM. Here we unveil a ground-state phase diagram of the QRM, consisting of a quadpolaron and a bipolaron as well as their changeover in the weak-, strong- and intermediate-coupling regimes, respectively. An unexpected overweighted antipolaron is revealed in the quadpolaron state, and a hidden scaling behavior relevant to symmetry breaking is found in the bipolaron state. An experimentally accessible parameter is proposed to test these states, which might provide novel insights into the nature of the light-matter interaction for all regimes of the coupling strengths.
42.50.Ct, 45.10.Db, 03.65.Ge, 42.50.Pq, 71.38.-k
Introduction.– In the past decade, it has been witnessed that the exploration of fundamental quantum physics in light-matter coupling systems has significantly evolved toward the (ultra-)strong coupling regime <cit.>. For example, in 2004, the strong coupling of a single microwave photon to a superconducting qubit was realized experimentally by using circuit quantum electrodynamics <cit.>. In 2010, this coupling rate was enhanced to reach a considerable fraction up to 12% of the cavity transition frequency<cit.>. Even with such small fractions the system has already entered into so-called ultrastrong-coupling limit<cit.>. In this situation, the well-known Jaynes-Cummings (JC) model<cit.> is no longer applicable because the JC model is valid only in the cases of near resonance and weak coupling <cit.>. Indeed, the experimentally observed anticrossing in the cavity transmission spectra <cit.> was due to counter-rotating terms, which are dropped in the JC model as a rotating-wave approximation. In addition, experimental observation of the Bloch-Siegert shift <cit.> also requires taking into account the counter-rotating terms in the description of the JC model. Thus the importance of the counter-rotating terms raises requests to comprehend the behavior of a full quantum Rabi model <cit.> (QRM) for all regimes of the coupling strengths <cit.>.
Remarkably, an important progress in the study of the QRM in the past years is the proof of its integrability <cit.>. As a result, its energy spectra have been exactly obtained <cit.>. However, to calculate the dynamics of the system, correlation
functions, and even other simpler physical observables, it is not enough to know only the exact eigenvalues, but the wavefunctions
(e.g., the exact eigenstates) are desirable. Based on series expansions of the eigenstates in terms of known basis sets, it was realized that a standard calculation with double precision, sufficient to compute the spectrum, fails for the eigenstates<cit.>. Therefore, the challenge to formulate an exact wave function in a general case still hampers access to a full understanding of the QRM.
In this work, by deforming the polaron and antipolaron <cit.> we propose a novel variational wavefunction ansatz to extract the ground state physics of the QRM. It is found that this ansatz is valid with high accuracy in all regimes of the coupling strengths. Thus a ground state phase diagram of the QRM is constructed. The nature of the system variation, by increasing the coupling strength from weak to strong, becomes transparent in the ground-state phase diagram with a quantum state changeover from quadpolaron to bipolaron, around a novel critical-like coupling scale analytically extracted. In particular, an unexpected overweighted antipolaron is revealed in the quadpolaron state and a hidden scaling behavior is found in the bipolaron state. Moreover, we propose an experimentally accessible parameter to test these states. For perspective, we also extend this ansatz to the multiple-mode case, which is expected to be useful to understand the physics of the spin-boson model <cit.>.
(Color) Schematic diagram for effective potentials induced by the tunneling between two levels.
a, In the absence of tunneling, i.e., $\Omega = 0$, the
original harmonic oscillator ($v_0$) is coupled with two
levels denoted by $\uparrow$ and $\downarrow$ to form two
polarons (associated with $v_\pm$) with the left($+,\uparrow$) and
right($-,\downarrow$) displacement $g'=\sqrt{2}g/ \omega$.
b, When the tunneling $\Omega$ is switched on, the left-
and right-polarons provide an effective
potential for each other $\delta v_\pm = \eta {\frac \Omega { \omega}} \frac
{\psi_\mp}{\psi_\pm}$ ($\eta = \pm$ represents the parity, here we
focus on the ground state with “$-$" parity), which induces an
antipolaron, as shown in c. c, The potential of the
left-polaron deforms from $v_+$ to $v_+ + \delta v_+$. The size of
$\uparrow$ indicates the weight of the polaron (blue) and
antipolaron (orange), respectively, in the same and opposite
directions of the potential displacement. The situation is
symmetric for the right-polaron. d-f, Typical
deformed potentials in the weak ($g < g_c$), intermediate ($g \sim g_c$), and strong
coupling ($g > g_c$) cases. There exist four tunneling channels between $\uparrow$ and
$\downarrow$ states, as shown in e, forming a quadpolaron state. In the strong coupling case, the tunneling
between left and right states decays until it is vanishingly small due to the large
potential barriers between them, yielding a bipolaron
state in f.
The model and effective potential.– The QRM <cit.> describes a quantum two-level
system coupled to a single bosonic mode or quantized harmonic
oscillator, which is a paradigm for interacting quantum systems
ranging from quantum optics <cit.> to quantum information
<cit.> to condensed matter <cit.>. The model
Hamiltonian reads
\begin{equation}
H= \omega a^{\dagger }a +\frac \Omega 2 \sigma_x +
g\sigma_z(a^{\dagger }+a), \label{ham}
\end{equation}
where $a^\dagger(a)$ is the bosonic creation (annihilation) operator
with frequency $\omega$ and $\sigma_{x,z}$ is the Pauli matrix
with level splitting $\Omega$. The last term describes the
interaction with coupling strength $g$.
In terms of the quantum harmonic oscillator with dimensionless
formalism<cit.> $a^{\dagger}=\left(\hat x-i\hat
p\right)/\sqrt{2}$, $a=\left( \hat x+i\hat p\right)/\sqrt{2}$,
where $\hat x=x$ and $\hat p=-i\frac \partial {\partial x}$ are
the position and momentum operators, respectively,
the model can be rewritten as
\begin{equation}
H = \sum_{\sigma_z=\pm } \left(h^{\sigma _z}|\sigma _z\rangle
\langle \sigma _z|+ {\frac \Omega 2} |\sigma _z\rangle \langle
\overline{\sigma}_z|\right) + {\cal E}_0, \label{Hx}
\end{equation}
where $\overline{\sigma }_z = -\sigma _z$ and $+$($-$) labels the
up $\uparrow$ (down $\downarrow$) spin in the $z$ direction,
respectively. $h^{\pm}=\frac12 \omega (\hat p^2+ v_\pm)$, with
$v_\pm = \left( \hat x \pm g'\right)^2$ and $g'=\sqrt{2}g/\omega$,
while ${\cal E}_0=-\frac12 \omega (g'^2 + 1)$ is a constant energy. Apparently, $h^{\pm}$ define two bare polarons <cit.> in the sense that the harmonic oscillator is bound by $\sigma_z$ due to the coupling $g'$, as shown in Fig. <ref>a. These two polarons form two bare potential wells but the existence of the level splitting $\Omega$ (resulting in the tunneling between these two wells<cit.>) makes the model difficult to solve analytically.
Let us begin with the wave-function $\Psi$ satisfying the Schrödinger equation $H\Psi = E\Psi$ with the eigenenergy $E$. Due to the fact that the model possesses the parity symmetry, namely, $[{\cal P},H]=0$ with ${\cal P}=\sigma_x(-1)^{a^{\dagger }a}$, $\Psi$ should take the form of
$\Psi = \frac{1}{\sqrt{2}}\left(\psi_{+}|\uparrow\rangle + \eta \psi_{-}|\downarrow\rangle\right)$,
where $\psi _{\pm} = \psi(\pm x)$ will be given below and $\eta = 1$ $(-1)$ for
positive (negative) parity. Without loss of generality, here we consider the ground state, with negative parity. The Schrödinger equation becomes
\begin{equation}
\frac12 \omega (\hat p^2 + v_\pm + \delta v_\pm )\psi_\pm = E\psi_\pm,
\end{equation}
where $\delta v_\pm = - \frac \Omega { \omega }\frac
{\psi_\mp}{\psi_\pm}$ is an additional effective potential originating from the tunneling, as shown in the lower
panel of Fig. <ref>b. The additional potential will deform the bare potential and as a result creates a subwell in the opposite direction of the bare potential $v_{\pm}$, as illustrated in Fig. <ref>c. The subwell induces an antipolaron as a
quantum effect. The above analysis from potential subwell verifies the existence of
antipolaron from wavefunction expansion<cit.>. Thus, the polaron and antipolaron constitute the basic ingredients of the ground-state wavefunction.
Deformed polaron and antipolaron.–
With the concept of polaron and antipolaron in hand, the competition between different energy scales $\omega, \Omega$ and $g'$ involved in the QRM will inevitably lead to deformations of the polaron and antipolaron. Physically, they can deform predominantly in two possible ways:
the position is shifted and the frequency is renormalized, which will introduce four independent variational parameters given below.
Explicit deformation depends on the coupling strength once the
tunneling is fixed, as shown in Fig. <ref> d-f from weak to
strong couplings according to a critical-like coupling strength
$g_c$. Thus a trial variational wave-function for $\psi(x)$
takes the superposition of the deformed polaron ($\varphi_\alpha$) and
antipolaron ($\varphi_\beta$),
\begin{equation}
\psi(x)=\alpha \varphi_\alpha(x) + \beta \varphi_\beta(x),
\label{wf-1}
\end{equation}
where $\varphi_\alpha (x)= \phi_0\left( \xi_\alpha \omega, x +
\zeta_\alpha g'\right)$ and $\varphi_\beta (x)= \phi_0\left(
\xi_\beta \omega, x - \zeta_\beta g'\right)$, with
$\phi_0(\omega, x)$ being the ground-state of standard harmonic oscillator with
frequency $\omega$. Here $\xi_i$ ($\zeta_i$), with $i = \alpha$
and $\beta$, describes the renormalization for frequency
(displacement) independently for the polaron and the antipoalron,
while the coefficients of $\alpha$ and $\beta$ denote their
weights, subject to the normalization condition
$\langle\psi|\psi\rangle = 1$. We stress that in contrast to the
direct expansion on basis series without frequency renormalization
<cit.>, we design our trial wavefunction based
on the dominant physics of deformation.
It turns out that our variational wavefunction is capable of
providing a reliable analysis on the QRM in the whole parameter regime,
ranging from weak to strong couplings, as shown for several
physical quantities for the ground-state including the energy, the mean
photon number, the coupling correlation and the tunneling
strength in Appendix <ref>. Obviously the
remarkable agreement between our results and the exact ones roots
in the fact that our trial wavefunction correctly captures the basic physics, as illustrated by the accurate wavefunction profiles
compared to the exact numerical ones for various couplings in
Fig.<ref>a. The variational wavefunction, with its
concise physical ingredients and its accuracy, in turn facilitates
unveiling more underlying physics.
0cm (Color) Mechanism for overweighted antipolaron in quadpolaron state. a, The calculated
(solid lines) spin-up groundstate wavefunctions, $\psi_+(x) =
\alpha\varphi_\alpha(x) + \beta \varphi_\beta(x)$, at $g/g_c = 0.5,
1., 1.25$ respectively, for weak (green), intermediate(navy),
and strong(red) couplings, with $\omega/\Omega = 0.1$. The symbols
denote the numerical exact results. The spin-down wavefunction is
given by $-\psi_+(-x)$(not shown). b and c,
$\alpha$- and $\beta$-components of $\psi_+(x)$, which correspond
to the polaron(blue) and antipolaron(orange), respectively, for
the intermediate $g \sim g_c$ and weak $g = 0.5g_c$ coupling
cases. d, The overlaps between different polarons and/or
antipolarons without the weights, $S_{i\bar{j}}=\langle
\varphi_i(x)|\varphi_j(-x)\rangle$ with $i,j = \alpha,\beta$. It
is clear that $S_{\beta\bar{\beta}}(\text{yellow}) >
S_{\alpha\bar{\alpha}}(\text{light magenta})$. e,
Schematic illustration of the physics for the overweighted
antipolaron. When decreasing the coupling strength $g'$,
the potential provided by the left-displaced oscillator for the
antipolaron gets lower, so the tunneling energy gain from large
$S_{\beta\bar{\beta}}$ in d overwhelms the potential
cost, which favors a larger weight of antipolaron. f, The
overweighted antipolaron with a larger weight than the polaron.
Quadpolaron/bipolaron quantum state changeover.– From
Fig.<ref>a-c one sees that when increasing the coupling, the wavepacket splits into visible polaron and
antipolaron (see animated plots in Supplementary Material for more vivid evolutions of
potentials and wavepackets). Before the full splitting, there are
significant tunnelings in all the four channels between the
polarons and antipolarons, as schematically shown in Fig.
<ref>e. Thus, in this sense we call this state a quadpolaron. After the splitting, only two same-side channels of
tunneling survives while the left-right channels are blocked gradually by
the increasing barrier, as sketched in Fig. <ref>f. This
state is termed here as a bipolaron. Despite the evolution from a transition-like feature in the low frequency limit to a crossover behavior in finite frequencies for the changover between quadpolaron and bipolaron states, the nature of the afore-mentioned splitting is essentially the same. This enables us to obtain an analytic coupling scale (see Appendix <ref>), $g_c = \sqrt{\omega^2 + \sqrt{\omega^4 + g^4_{c0}}}$, which generalizes the low frequency-limit result<cit.> $g_{c0}=\sqrt{\omega\Omega}/2$ and correctly captures the quantum state changeover between quadpolaron and bipolaron for the whole range of frequencies.
(Color) Renormalization factor and weight as a function of the coupling strength $g$. $\zeta _i$ ($i=\alpha,\beta$) is the displacement renormalization and $\xi _i$ is the frequency renormalization. $\alpha$ and $\beta$ denote the weights of the polaron and the antipolaron, respectively, in the variational groundstate wavefunction.
a and d, $\omega/\Omega = 0.5$.
b and e, $\omega/\Omega = 0.15$.
c and f, $\omega/\Omega = 0.005$.
Quadpolaron asymmetry and overweighted antipolaron in the regime of $g \lesssim g_c$.– We find that the polaron and antipolaron in the quadpolaron state have asymmetric displacements, which leads to a subtle competition depending on the frequency $\omega/\Omega$. Figure <ref> shows three types of distinct behaviors of the variational parameters in three different frequency regimes: high frequency ($\omega/\Omega \gtrsim 0.47$), intermediate frequency($\omega/\Omega \in [0.07, 0.47]$), and low frequency ($\omega/\Omega \lesssim 0.07$). The result is understandable due to the fact that the antipolaron always has a higher potential energy owing to its opposite direction to the displacement of $v_\pm$. Roughly speaking, at a high frequency, the antipolaron should have a lower weight than the polaron ($\beta < \alpha$) since the antipolaron is suppressed by the high potential. At a low frequency, the polaron benefits from both potential and tunneling energies. However, competition becomes subtle at an
intermediate frequency as each of these different energy scales may only favor either the polaron or the antipolaron respectively,
which may lead to overweighted antipolaron, as shown in Fig.<ref>e.
Below we give a more explicit analysis. Actually, the four channel
tunneling energies in the quadpolaron are proportional to the
overlaps of the polarons and antipolarons, $S_{\alpha
\bar{\alpha}}$, $S_{\beta \bar{\beta}}$, $S_{\alpha \bar{\beta}}$
and $S_{\beta \bar{\alpha}}$, respectively, as shown in
Fig.<ref>d. The mixture terms $S_{\alpha
\bar{\beta}}$ and $S_{\beta \bar{\alpha}}$ do not affect the
weight competition between the polaron and antipolaron, while
$S_{\alpha \bar{\alpha}}$ and $S_{\beta \bar{\beta}}$ yield
imbalances. Indeed, the antipolarons have larger overlap than the
polarons, i.e. $S_{\beta\bar{\beta}}
> S_{\alpha\bar{\alpha}}$ (see Fig. <ref>d). This is because the
antipolarons in up and down spins are closer to each other than
the polarons in order to reduce their higher potential energy, as indicated
in Fig. <ref>e and quantitatively shown by $ \zeta _\beta < \zeta
_\alpha$ in Fig. <ref>a and <ref>b.
Therefore, as far as the tunneling is concerned,
it would tend to have more weight of antipolarons to gain a maximum tunneling.
When the intermediate frequency reduces the cost of potential energy for such tendency,
a larger antipolaron weight might finally occur, as in Fig. <ref>f,
leading to an unexpected overweighted antipolaron. We find that this really
occurs as demonstrated in Fig. <ref>e where a
weight reversion appears at the crossing of $\alpha$ and $\beta$
for a weaker coupling.
At the low frequency, the harmonic potential becomes very flat,
the polarons may get closer than antipolarons, as indicated by
$\zeta _\alpha < \zeta _\beta$ in Fig.<ref>c in the
weak coupling regime. In this case, $S_{\alpha \bar{\alpha} }$ is
greater than $S_{\beta \bar{\beta} }$ so that polarons have
favorable energies in both potential and tunneling. Thus the
polaron regains its priority in weight.
-0.cm (Color) Scaling quantity $\gamma$ as a function of the coupling strength $g$. a, Our results
compared with exact numerics at $\omega/\Omega=0.01$ as an
example. b, $\gamma$ for different values of the ratios
$\omega/\Omega$. The scaling relation $\zeta_i \doteq \xi_i$ is
tested by $\gamma=1$ beyond $g_c$.
Bipolaron and hidden scaling behavior in the regime of $g\gtrsim g_c$.–
In the bipolaron regime, the remaining tunneling in channels $S_{\alpha\bar\beta}$ and $S_{\beta\bar\alpha}$ leads to intriguing physics,
showing a deeper nature of the interaction in the symmetry breaking aspect.
Indeed, Fig.<ref>a-c show that in this regime the frequency factor $\xi _i$ and the displacement factor $\zeta _i$ collapse into the same value,
i.e $\zeta_i \doteq \xi_i$. In fact, due to vanishing photon number below $g_c$ at low frequency limit, the parity ${\cal P}$ can be decomposed into
separate spin and spatial reversal sub-symmetries which are broken beyond $g_c$. However, further seeking the symmetry breaking character from these
sub-symmetries would fail at finite frequencies due to emergence of a finite number of photons below $g_c$. Nevertheless, the $\zeta_i$-$\xi_i$
symmetric aspect revealed here provides a compensation, from the beyond-$g_c$ side instead but valid also for finite frequencies.
To test this scaling behavior, we propose an experimentally accessible quantity, $\gamma
\equiv \frac \omega {gt} \sqrt{
\langle a^{+}a \rangle -\frac 1 4 (t+t^{-1})+\frac 12
where $t=-\langle \left( a^{+}-a\right) ^2\rangle $, which becomes the scaling ratio $\gamma \rightarrow \zeta /\xi$ (see Appendix <ref>) for their
average $\xi =\left( \xi _\alpha+\xi _\beta\right) /2$ and $\zeta =\left( \zeta _\alpha+\zeta _\beta\right) /2$
and thus equals to one above $g_c$, as shown in Fig. <ref> for various frequencies. The experimental measurement of $\gamma$ thus
provides a possible way to distinguish the states of bipolaron and quadpolaron as well as their changeover.
-0.0cm (Color) An overview of ground-state phase diagram for the QRM. Quadpolaron ($g \lesssim g_c$) and bipolaron ($g \gtrsim g_c$) as well as their crossover near the minimum of $\xi_i$ (red solid line) or the maximum of $\gamma$ around analytic $g_c$. The quadpolaron regime is further divided into the normal ($\alpha > \beta$) and overweighted antipolaron ($\alpha < \beta$) regimes. The dashed and dot-dashed lines have been obtained numerically from the cross points as shown in Fig. <ref> (e) and (f). The color density for $\gamma$ further distinguishes the characters of the different regimes and their changeovers.
Ground-state phase diagram.– The above discussions on
polaron-antipolaron competition can be summarized into a ground-state phase
diagram unveiled, as shown in Fig.
<ref>. The ground-state with different channels of tunneling is identified as a quadpolaron when $g \lesssim g_c$ and as a bipolaron when $g\gtrsim g_c$. An overweighted antipolaron is hidden in the quadpolaron
regime, while a scaling relation between the displacement and frequency renormalizations is revealed in the
bipolaron regime.
Note that the polaron and antipolaron structures might be detected by
optomechanics<cit.> and $\gamma$ is experimentally
measurable. The diagram may provide a renewed panorama for deeper
theoretical investigations and may raise more challenges for experiments.
Perspective in multiple modes.–
The basic physics in the QRM has a profound implication for the spin-boson model<cit.>, which is a multiple-mode version of the QRM.
The essential variational ingredients remain similar. The trial wavefunction can be written as
\psi [\{x_k\}]=\alpha \prod_{k=1}^M\varphi _\alpha^k+\beta \prod_{k=1}^M\varphi _\beta^k
with the extension $\{\omega,g,x,\xi_i,\zeta_i\}\rightarrow
\{\omega_k,g_k,x_k,\xi^k_i,\zeta^k_i\}$ for the $k$'th mode. We illustrate the same accuracy by a two-mode case in Appendix <ref>.
We thank Jun-Hong An for helpful discussions. Work at CSRC and Lanzhou University was supported by National Natural Science Foundation of China, PCSIRT (Grant No. IRT1251), National “973" projects of China. Z.-J.Y. also acknowledges partial financial support from the Future and Emerging Technologies (FET)
programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 618083 (CNTQC).
§ VARIATIONAL METHOD AND PHYSICAL PROPERTIES
Here we calculate the ground-state physical properties
from the variational method, including the energy $E$, the mean photon number $\langle a^{\dagger }a\rangle $,
the coupling correlation $\langle \sigma _z(a^{\dagger }+a)\rangle $ and
the spin flipping (tunneling) strength $\langle \sigma _x\rangle $.
§.§ The energy
As introduced in the main part of the paper, the wavefunction for the reformulated
Hamiltonian (<ref>) has the
following form
\begin{equation}
\Psi =\frac 1{\sqrt{2}}\left( \psi _{+}(x)\mid \uparrow \rangle
+\eta \psi _{-}(x)\mid \downarrow \rangle \right) ,
\end{equation}
where $\eta =\pm $ is the parity. We adopt the variational trial wavefunction as a superposition of
the polaron and the antipolaron
\begin{equation}
\psi _{+}(x)=\psi _{-}(-x)=\alpha \varphi _\alpha \left( x\right)
+\beta \varphi _\beta \left( x\right) , \label{wf-1}
\end{equation}
\begin{eqnarray}
\varphi _\alpha \left( x\right) =\phi _n(\xi _\alpha \omega,x+\zeta _\alpha g^{\prime }),\\
\varphi _\beta \left( x\right) =\phi _n(\xi _\beta \omega ,x-\zeta _\beta g^{\prime }),
\end{eqnarray}
with $\phi _n(\omega, x)$ being the $n$'th eigenstate of the
standard quantum harmonic oscillator with frequency $\omega $. In
this work we focus on the ground state so that $n=0$ and $\eta
=-$. The displacement of the bare potential $v_\pm = \left( \hat x \pm g'\right)^2$ in the single-well energy $ h^{\pm },$
\begin{equation}
g^{\prime }=\sqrt{2}g/\omega ,
\end{equation}
is driven by the interaction $g$ and for simplicity we have assumed
the unit $\hbar = m = 1$. Note that the polaron (antipolaron) has a displacement in the
same (opposite) direction as (to) that of the bare potential
$v^{\pm }$. The interplay of the interaction and the tunneling
leads to the deformation of the wavepacket: the frequency of the
polaron (antipolaron) is renormalized by $\xi _\alpha $ ($\xi
_\beta $) and the displacement by $\zeta _\alpha $ ($\zeta _\beta
$), respectively. The weights of the polaron and the antipolaron are
subject to the normalization condition $\langle \Psi \left| \Psi
\right\rangle =\langle \psi _{+}|\psi _{+}\rangle =1$. These
deformation parameters, independently $\{\xi _\alpha ,\xi _\beta
,\zeta _\alpha ,\zeta _\beta ,\alpha \}$, are optimized by
minimization of the total energy formulated in the following.
The energy can be directly obtained as
\begin{equation}
E\equiv \left\langle \Psi \right| H\left| \Psi \right\rangle
=h_{++}^{+}+\eta \frac{\hbar \Omega }2n_{+-}+{\cal E}_0,
\label{energy-1mode}
\end{equation}
\begin{eqnarray}
h_{++}^{+}&=&\left\langle \psi _{+}\right| h^{+}\left| \psi
_{+}\right\rangle \nonumber \\
&=&\alpha ^2h_{\alpha \alpha }^{+}+\beta
^2h_{\beta \beta }^{+}+2\alpha \beta h_{\alpha \beta }^{+}, \label{hpp}\\
n_{+-}&=&\langle \psi _{+}|\psi _{-}\rangle \nonumber \\
&=&\alpha ^2S_{\alpha
\bar \alpha }+\beta ^2S_{\beta \bar \beta }+2\alpha \beta
S_{\alpha \bar \beta }, \label{nab}
\end{eqnarray}
contribute to the single-well energy and the tunneling energy,
respectively. Here, we have defined
\begin{eqnarray}
h_{ij}^{+}&=&\langle \varphi _i\left( x\right) |h^{+}|\varphi
_j\left( x\right) \rangle , \nonumber \\
S_{ij}&=&\langle \varphi _i\left(
x\right) \mid \varphi _j\left( x\right) \rangle ,\label{defSij} \\
S_{i\,\bar j}&=&\langle \varphi _i\left( x\right) \mid \varphi _j\left(
-x\right) \rangle , \nonumber
\end{eqnarray}
for $i=\alpha ,\beta $, while ${\cal E}_0=-\frac 12 \omega
(1+g^{\prime 2})$ is a constant energy. Explicit formulas for
these quantities are readily available. In this Section we give the result
for the ground state
\begin{eqnarray}
h_{\alpha \alpha }^{+} &=&\frac 12\omega \left[ \frac 12(\xi
_\alpha +\xi
_\alpha ^{-1})+\left( 1-\zeta _\alpha \right) ^2g^{\prime 2}\right] , \label{hAaa}\\
h_{\beta \beta }^{+} &=&\frac 12\omega \left[ \frac 12(\xi _\beta
_\beta ^{-1})+\left( 1-\zeta _\beta \right) ^2g^{\prime 2}\right] , \label{hAbb}\\
h_{\alpha \beta }^{+} &=&\frac 12\omega
\left[ \left( 1-\xi _\alpha ^2\right) \left\langle \hat
x_\alpha ^2\right\rangle _{\alpha \beta } +\left( 1-\zeta _\alpha
\right) \left\langle \hat x_\alpha \right\rangle _{\alpha \beta
}2g^{\prime } \right. \nonumber\\
&& \left.
+\xi _\alpha S_{\alpha \beta }+\left( 1-\zeta _\alpha \right) ^2g^{\prime
2}S_{\alpha \beta }\right] ,\label{hAab}
\end{eqnarray}
\begin{eqnarray}
\left\langle \hat x_\alpha \right\rangle _{\alpha \beta }
&=&S_{\alpha\beta} \frac{\left( \zeta _\alpha +\zeta _\beta
\right) \xi _\beta }{\left( \xi
_\alpha +\xi _\beta \right) }g^{\prime }, \label{x1Aab} \\
\left\langle \hat x_\alpha ^2\right\rangle _{\alpha \beta }
&=&\frac {S_{\alpha\beta}} {(\xi _\alpha +\xi _\beta )}\left[ 1+\frac{\left( \zeta _\alpha +\zeta
_\beta \right) ^2\xi _\beta ^2}{(\xi _\alpha +\xi _\beta
)}g^{\prime 2}\right]
\label{x2Aab}
\end{eqnarray}
\begin{eqnarray}
S_{\alpha \beta }&=&S(\zeta _\alpha
,\zeta _\beta ,\xi _\alpha
,\xi _\beta ),
\nonumber \\
S_{\alpha \bar \beta }&=&S(\zeta _\alpha ,-\zeta
_\beta ,\xi _\alpha ,\xi _\beta ),
\nonumber \\
S_{\alpha \bar \alpha } &=& S(\zeta _\alpha ,\zeta _\alpha ,\xi
_\alpha ,\xi _\alpha ),
\nonumber \\
S_{\beta \bar \beta } &=& S(\zeta _\beta ,\zeta _\beta ,\xi _\beta
,\xi _\beta ),
\end{eqnarray}
are given by the function
\begin{eqnarray}
S(\zeta _1,\zeta _2,\xi _1,\xi _2) &=& \exp \left(
-\frac{\left( \zeta _1+\zeta _2\right) ^2g^{\prime 2}\xi _1\xi
_2}{2(\xi _1+\xi _2)}\right) \nonumber \\
&& \times \sqrt{2} \left[ \frac{\xi _1\xi _2}{(\xi
_1+\xi _2)^2}\right] ^{1/4}.
\end{eqnarray}
§.§ The mean photon number
From the relation
\begin{equation}
a^{\dagger }a=\frac{h^0}\omega -\frac 12,\quad h^0\equiv \frac
12\omega \left( \hat p^2+\hat x^2\right) ,
\end{equation}
and the symmetric relation $\psi _{-}\left( x\right) =\psi
_{+}\left( -x\right) $, the mean photon number simply reads as
\begin{equation}
\langle a^{\dagger }a\rangle \equiv \left\langle \Psi \right|
a^{\dagger }a\left| \Psi \right\rangle =\frac{h_{++}^0}\omega
-\frac 12 \label{PhotonN}
\end{equation}
\begin{equation}
h_{++}^0=\left\langle \psi _{+}\right| h^0\left| \psi
_{+}\right\rangle =\alpha ^2h_{\alpha \alpha }^0+\beta ^2h_{\beta
\beta }^0+2\alpha \beta h_{\alpha \beta }^0.
\end{equation}
For the ground state
\begin{eqnarray}
h_{\alpha \alpha }^0 &=&\frac 12\omega \left[ \frac 12(\xi _\alpha
_\alpha )+2\zeta _\alpha ^2g^{\prime 2}\right] , \\
h_{\beta \beta }^0 &=&\frac 12\omega \left[ \frac 12(\xi _\beta
_\beta )+2\zeta _\beta ^2g^{\prime 2}\right] , \\
h_{\alpha \beta }^0 &=&\frac 12\omega \left[ \left( 1-\xi _\alpha ^2\right) \left\langle \hat x_\alpha
^2\right\rangle _{\alpha \beta }-\zeta _\alpha \left\langle \hat
x_\alpha \right\rangle _{\alpha \beta }2g^{\prime } \right.\nonumber \\
&&\left.+\xi _\alpha S_{\alpha
\beta }+\zeta _\alpha
^2g^{\prime 2}S_{\alpha \beta }\right] ,
\end{eqnarray}
and $\left\langle \hat x_\alpha ^2\right\rangle _{\alpha \beta }$,
$ \left\langle \hat x_\alpha \right\rangle _{\alpha \beta }$ are
given by (<ref>) and (<ref>).
§.§ The coupling correlation $\langle \sigma _z(a^{\dagger
}+a)\rangle $ and the spin flipping (tunneling) strength $\langle
\sigma _x\rangle $
Now we calculate the coupling correlation $\langle \sigma
_z(a^{\dagger }+a)\rangle $. Since $(a^{\dagger }+a)=\sqrt{2}\hat
x$, we have
\begin{equation}
\langle \sigma _z(a^{\dagger }+a)\rangle \equiv \left\langle \Psi
\right| \sigma _z(a^{\dagger }+a)\left| \Psi \right\rangle
=\sqrt{2}\left\langle \hat x\right\rangle _{++}
\end{equation}
\begin{equation}
\left\langle \hat x\right\rangle _{++}=\left\langle \psi
_{+}\left( x\right) \right| \hat x\left| \psi _{+}\left( x\right)
\right\rangle =\alpha ^2x_{\alpha \alpha }+\beta ^2x_{\beta \beta
}+2\alpha \beta x_{\alpha \beta }
\end{equation}
\begin{equation}
x_{\alpha \alpha }=-\zeta _\alpha g^{\prime },\quad x_{\beta \beta
}=\zeta _\beta g^{\prime },\quad x_{\alpha \beta }=\left\langle
x_\alpha \right\rangle _{\alpha \beta }-\zeta _\alpha S_{\alpha
\beta }g^{\prime }.
\end{equation}
The strength of spin flipping or tunneling, $\sigma _x=\sigma
^{+}+\sigma ^{-}$, is simply
\begin{eqnarray}
\langle \sigma _x\rangle \equiv \left\langle \Psi \right| \sigma
_x\left| \Psi \right\rangle =\eta n_{+-}
\end{eqnarray}
which has been formulated in (<ref>).
§.§ Accuracy of our variational method
(Color online) Ground-state physical quantities as functions of the coupling strength $g/g_c$. $\omega/\Omega = 0.1$ is taken as an example. a, The ground state
energy. b, The mean photon number. c, The correlation
function $\langle \sigma_z (a^\dagger + a)\rangle$. d, The
tunneling strength $\langle \sigma_x\rangle$. The orange circles
denote the numerically exact results as a benchmark, the red
dashed-lines are calculated in adiabatic approximation(AA)
<cit.> or generalized rotating-wave approximation (GRWA)
<cit.>, and the blue lines are our results obtained by
the present variational method.
The most widely-used approximations in the literature are the
rotating-wave approximation (RWA)<cit.>, adiabatic
approximation(AA) <cit.>, generalized rotating-wave
approximation (GRWA) <cit.> and generalized variational
method (GVM) <cit.>, each working in some specific parameter
regime. The RWA neglects the counter-rotating terms in the
interaction, valid in regime $g\ll \omega ,\Omega $ under
near-resonance ($ \omega \sim \Omega $) condition. The AA and the
GRWA have the same groundstate, working for $g\gg \omega $ or
negative detuning ($\omega >\Omega $) regime. The GVM works for
$g\ll \omega $. Recently a mean-photon-number dependent
variational method was proposed to cover validity regimes
of both the GVM and the GRWA <cit.>. However, all the
approximations collapse when the ratio of $\omega /\Omega $ is
getting small, e.g. below around $0.5$ (see Ref. [LiuM]). An improved
variational method by including the antipolaron<cit.>
also finds breakdowns at $\omega /\Omega \sim 0.3$. It would be
favorable to have a variational method that always preserves a
high accuracy in varying all parameters which might facilitate and
even deepen the physical analysis.
Indeed, our variational wavefunction yields such accuracy
requirements. As an illustration, in Fig. <ref> we
compare with the exact numerics on the the ground state energy,
mean photon number, coupling correlation and tunneling strength,
at the example $\omega /\Omega =0.1$ (one can find other examples
for comparison at $\omega /\Omega =0.01$, $0.05$, $0.15$, $0.5$ for
another physical quantity $\gamma $ in Fig. <ref> and Fig. <ref>). As a comparison, the
results obtained by the AA or the GRWA are also shown. Clearly,
our results are completely consistent with the exact ones in the
whole parameter regime.
§.§ Physical necessity of the variational parameters
(Color online) Quantitative deviations and qualitative errors emerge in reducing variational parameters. Physical properties may deviate not only quantitatively but also qualitatively when the parameters are reduced, e.g., if imposing $\xi _\alpha=1,\ \xi _\beta=1$ (black line) or $\xi _\alpha=\xi _\beta, \ \zeta _\alpha=\zeta _\beta$ (red line), an incorrect cusp behavior appears in the energy $E$ and the spin flipping (tunneling) strength $\langle \sigma _x\rangle $ has a spurious jump around $g_c$ at $\omega/\Omega = 0.01$, in contradiction with the smooth crossover in the exact numerics (orange circles). The blue lines are our results in full minimal parameters which reproduce accurately the exact ones.
(Color online) An energy comparison of the excited states for the lowest
10 levels, at $\omega/\Omega = 0.1$ and $g/\Omega = 0.5$. The
orange dots, empty blue diamonds and empty red squares represent
the results of the exact numerics, our method and the GRWA,
It may be worthwhile to have further discussion on the physical necessity of the variational parameters. An unnecessary reduction of our parameters, on the one hand, will not lead to much of a reduction in the computational cost as the calculation in full parameters is actually quite easy to carry out, on the other hand, however, the price of physical loss would be too high. As discussed in the main part of the paper, our variational parameters physically correspond to the deformations of the polaron and the antipolaron with displacement and frequency renormalizations, which is justified by the behavior of the effective potential. In the subtle energy competitions of the potential of harmonic oscillator, the interaction and the tunneling, both the polaron and the antipolaron can adapt themselves via the variations of their displacements, frequencies and weights. Thus, corresponding to the key physical degree of freedom of the polaron and the antipolaron, the five variational parameters, $\xi _\alpha,\ \xi _\beta, \ \zeta _\alpha,\ \zeta _\beta, \alpha $, are the minimal necessary parameters to capture the true physics of the behavior of the polaron and the antipolaron, subject to the normalization of the wavefunction. Therefore, reducing the parameters would lead to mismatch of the physical degree of freedom and thus give rise to unreliable results, the physical properties may deviate not only quantitatively but also qualitatively. For an example, assuming $\xi _\alpha=\xi _\beta=1$ or imposing $\xi _\alpha=\xi _\beta, \ \zeta _\alpha=\zeta _\beta$ can reduce the parameter number by 2. However, as shown in Fig.<ref>, without mentioning the quantitative deviations, an incorrect cusp behavior appears in the energy $E$ at low frequencies as illustrated at $\omega/\Omega = 0.01$, and even worse, a spurious jump emerges in the tunneling (spin flipping) strength $\langle \sigma _x\rangle $ around $g_c$. The other physical quantities, such as the mean photon number $\langle a^{\dagger }a\rangle $, the coupling correlation $\langle \sigma _z(a^{\dagger }+a)\rangle $ also have a false discontinuity similar to $\langle \sigma _x\rangle $. Both the cusp and the discontinuity are qualitatively in contradiction with the smooth crossover in the exact numerics (orange circles). Nevertheless, our results using the full minimal variational parameters (blue line) reproduce accurately the exact results in the entire regime of the coupling strengths at different frequencies. Moreover, in the cases of reduced parameters, some important underlying physics would also be missing, such as the scaling relation of the displacement and frequency renormalizations as we revealed in the main text (see also Appendix <ref>).
§.§ Method extension to the excited states
Our method can also be useful for the excited states. As a first simple extension the variational energy of the excited
state can be obtained by replacing expressions (<ref>)-(<ref>) with
\begin{eqnarray}
h_{\alpha \alpha }^{+} &=&\frac \omega 2\left[ \left( n+\frac
12\right) \left( \xi _\alpha +\xi _\alpha ^{-1}\right) +\left(
1-\zeta _\alpha \right)
^2g^{\prime 2}\right] , \\
h_{\beta \beta }^{+} &=&\frac \omega 2\left[ \left( n+\frac
12\right) \left( \xi _\beta +\xi _\beta ^{-1}\right) +\left(
1-\zeta _\beta \right)
^2g^{\prime 2}\right] , \\
h_{\alpha \beta }^{+} &=&\frac \omega 2\ \left[
\left( 1-\xi _\alpha ^2\right) \left\langle \hat
x_\alpha ^2{}\right\rangle _{\alpha \beta }+
\left( 1-\zeta _\alpha
\right) \left\langle \hat x_\alpha \right\rangle _{\alpha \beta
}2g^{\prime }\right. \nonumber \\
&&+ \left. (2n+1)\xi _\alpha
S_{\alpha \beta }+
\left( 1-\zeta _\alpha \right) ^2g^{\prime
2}S_{\alpha \beta }\right],
\end{eqnarray}
where both $\langle \hat x_\alpha ^j{}\rangle _{\alpha \beta }$ and $%
S_{\alpha \beta }$ can be included by a unified function
\begin{equation}
\langle \hat x_\alpha ^j{}\rangle _{\alpha \beta }=X(n,j),\quad
S_{\alpha \beta }=X(n,0).
\end{equation}
Here the function $X(n,j)$ is defined by
\begin{eqnarray}
X(n,j) &=&n!j!\left[\frac{\left( \zeta _\alpha +\zeta _\beta \right) g^{\prime }}{%
2c}\right]^j\sum_{p=0}^{\min [n,j]}\sum_{q=0}^{\min [n,j-p]}\frac{\left(
-i\right) ^{j-p-q}a^pb^q}{p!q!\left( j-p-q\right)
!}\sqrt{\frac{2^{p+q}}{\left( n-p\right) !\left( n-q\right)
!}}H_{j-p-q}\left(\frac 12ab^2c\right)\tilde S_{n-p,n-q},
\\
\tilde S_{k,k^{\prime }} &=&\sum_{r=0}^{\min [k,k^{\prime
}r}H_{k-r}\left(\frac{ab^2c}{2\sqrt{1-a^2}}\right)H_{k^{\prime }-r}\left(-\frac{a^2bc}{2%
\sqrt{1-b^2}}\right), \\
C_{kk^{\prime }r} &=&\sqrt{\frac{ab}{2^{k+k^{\prime }}k!k^{\prime }!}}%
e^{-\left( abc\right) ^2/4}\frac{k!k^{\prime }!\left( 2ab\right)
^r\left( 1-a^2\right) ^{(k-r)/2}\left( 1-b^2\right) ^{(k^{\prime
}-r)/2}}{\left( k-r\right) !\left( k^{\prime }-r\right) !r!}.
\end{eqnarray}
and the factors $a,b,c$ depend on the variational parameters
\begin{equation}
a=\sqrt{\frac{2\xi _\alpha }{\xi _\alpha +\xi _\beta }},\quad b=\sqrt{\frac{%
2\xi _\beta }{\xi _\alpha +\xi _\beta }},\quad c=\left( \zeta
_\alpha +\zeta _\beta \right) g^{\prime }\sqrt{\frac{\left( \xi
_\alpha +\xi _{\beta}\right) }2}.
\end{equation}
For the other group of overlap in the tunneling term $n_{+-}$(<ref>),
one can also formulate using $S_{\mu \bar \mu ^{\prime }}=\left(
-1\right) ^nX(n,0)$ with
the corresponding replacement $\alpha ,\beta \rightarrow \mu ,\mu ^{\prime }$, but there is sign variation $\zeta _\beta \rightarrow -\zeta
_{\mu ^{\prime }}$. Here $n$ is the level number of the standard
quantum harmonic oscillator and $H_m\left( x\right) $ is the
standard Hermite polynomials. It is worthwhile to see that this
simple extension for the excited states has already yielded some
considerable improvements in strong couplings as illustrated in
Fig.<ref> for a number of lowest energy levels. With
the above expressions, one may further analytically construct an
improved extension of the variational energy for overall coupling
range by imposing the deformed polaron and antipolaron in the GRWA
form of wavefunction. On the other hand, the dynamics of the system also can be calculated in terms of $\tilde S_{k,k^{\prime}}$ which provides the intra-overlap and inter-overlap of the deformed polarons and antipolarons with different oscillator quantum number $k,k^{\prime }$.
Since here the focus is the ground state
which, as we show in the present work, already has rich underlying
physics to be uncovered, we shall present a more detailed method
description and systematical discussion for the excited state
properties in our future work.
§ QUADPOLARON/BIPOLARON CHANGEOVER AND SCALES OF COUPLING STRENGTH
§.§ Analytic approximation for $g_c$
In the variation of the coupling strength, the system undergoes a
phase-transition-like changeover around $g\sim g_c$. In the
super-strong tunneling or low-frequency limit, i.e. $\omega
/\Omega \rightarrow 0,$ this changeover is very sharp, it behaves
more like a phase transition, as discussed by Ashhab
<cit.>. In the other cases it behaves like a crossover.
We can get more insights into this transition-like behavior from
the profile deformation of the wavepacket. The increase of the
coupling strength is splitting the wavepacket into the polaron and the
antipolaron, while the tunneling is trying to keep them as close
as possible in the groundstate. Before a full splitting the system
remains in a quadpolaron state with four channels of tunneling,
$S_{\alpha \bar \alpha }$, $S_{\beta \bar \beta }$, $ S_{\alpha
\bar \beta }$, and $S_{\beta \bar \alpha }$, while after the
splitting the system enters a bipolaron state with only two
tunneling channels, $S_{\alpha \bar \beta }$ and $S_{\beta \bar
\alpha }$, surviving. Here we have labeled the tunneling channels
by the overlaps $S_{i\bar{j}}$, defined in (<ref>), to which the corresponding tunneling energies are proportional.
We show the tunneling channel difference for
these two regimes in Fig.<ref> a-c at various
frequencies. One can also see that the change in the tunneling
channel number is universal for different frequencies. Thus, the
two regimes distinguished by wavepacket splitting are essentially
different in the quantum states. Therefore, the coupling strength
at which the splitting really starts can be used to formulate
+0.0cm (Color online) Quadpolaron/bipolaron changeover and the behavior of variational parameters and related physical quantities. a-c, Weighted
groundstate tunneling of different channels, $\alpha ^2 S_{\alpha
\bar{\alpha}}$, $\beta ^2 S_{\beta \bar{\beta}}$, $\alpha \beta
S_{\alpha \bar{\beta}}$ and $\alpha \beta S_{\beta \bar{\alpha}}$
($S_{\beta \bar{\alpha}}= S_{\alpha \bar{\beta}}$) as functions of
the coupling strength $g/g_c$. The dashed lines roughly separate
the quadpolaron ($g\lesssim g_c$) regime and the bipolaron regime
($ g \gtrsim g_c$), the former has four channels of tunnelings
while the latter has two channels. d-f, The frequency
renormalization factors $ \xi_\alpha$ and $\xi_\beta$. g-i,
The scaling quantity $\gamma$. The results from our variational
method (solid lines) almost reproduce the ones from exact numerics
(orange circles) for all values of coupling at different
frequencies. The boundary of the quadpolaron and bipolaron regimes
is associated with the minimum of $\xi_i$ and the maximum of
$\gamma$. The black triangles mark the positions for $g_{c0}/g_c$
which becomes farther away from $1$ as $\omega$ increases. a,d $\omega/\Omega = 0.5$. b, e,
$\omega/\Omega =0.15$. c,f, $\omega/\Omega = 0.05$.
We adopt the value of $g_c$ at the point where the distance
between the polaron and the antipolaron is equal to their total radii,
\begin{equation}
\left( \zeta _\alpha +\zeta _\beta \right) g_c^{\prime }=r_\alpha
+r_\beta , \label{eqR12r12}
\end{equation}
where we take the radii by
\begin{equation}
r_\alpha =2\sqrt{\frac 1{\xi _\alpha }},\quad r_\beta
=2\sqrt{\frac 1{\xi _\alpha }},
\end{equation}
at which the value of the corresponding wavepacket is becoming
\begin{equation}
\frac{\varphi _i}{\varphi_i^{\max }}=\frac 1{e^2}
\end{equation}
for both $i=\alpha, \beta $.
Note that both sides of the above equation (<ref>) are
essentially averaging over the polaron and the antipolaron, thus
assuming symmetric polaron and antipolaron, i.e. $\zeta _\alpha
=\zeta _\beta $ and $\xi _\alpha =\xi _\beta $, would be a reasonable
approximation as far as $g_c$ is concerned. Under this constraint
the explicit results for the deformation parameters are available
for the well-separated polaron and antipolaron from the energy
minimization formulated in Appendix <ref>, reading as
\begin{equation}
\zeta _\alpha =\zeta _\beta =\sqrt{1-\frac{g_{c0}^4}{g^4}},\quad
\xi _\alpha =\xi _\beta =1, \label{Xi-noSqueez}
\end{equation}
where the critical point $g_{c0}=\sqrt{\omega \Omega }/2$ is
obtained in the semiclassical approximation at $\omega /\Omega
\rightarrow 0.$<cit.> We stress that we limit
the application of this approximation to the estimation of $g_c$,
while for other properties one should fall back upon asymmetric
polaron and antipolaron for higher accuracy. Actually, as mentioned in Appendix <ref>, imposing
symmetric polaron and antipolaron would lead to a spurious
discontinuous behavior of physical properties, such as the tunneling strength, around $g_c$ at
low frequencies, while in reality it
should be smooth as predicted by asymmetric polaron and
antipolaron in agreement with exact numerics.
Also, in the strong
coupling regime the displacement asymmetry of the polaron and the
antipolaron actually plays an important role in inducing the amplitude-squeezing effect ($\xi _\alpha <1$) which extends the wavepackets
of the polaron and the antipolaron to increase their overlap, thus enhancing the tunneling.
Without the asymmetry there would be no squeezing beyond $g_c$, as indicated by (<ref>), since the symmetric polaron and antipolaron in up and down spins would completely coincide, with an already-maximum overlap. In fact, as uncovered in the main text, there is a hidden relation between the squeezing and the displacement, which is also discussed in detail in Appendix <ref>.
Substitution of (<ref>) into (<ref>) leads us
to a simple analytic expression
\begin{equation}
g_c=\sqrt{\omega ^2+\sqrt{\omega ^4+g_{c0}^4}}. \label{eq-gC}
\end{equation}
It is easy to check $g_c\rightarrow g_{c0}$ in the slow oscillator
limit $ \omega /\Omega \rightarrow 0$. Besides the transition-like changeover in
this low frequency limit, our $g_c$ is also providing a valid
coupling scale for the quadpolaron/bipolaron changeover at finite
frequencies, which can be seen from Fig.<ref> where the
quadpolaron regime and bipolaron regime adjoin each other really
around $g_c$. A more quantitative way to identify the
transition-like point is, as shown by Fig.<ref> d-i, the
minimum point of the frequency renormalization factor or the
maximum point of the scaling quantity introduced in Appendix <ref>.
Still, one sees that it is well approximated by $g_c$ in
§.§ Novel scale for the coupling strength
At this point, it is worthwhile to further discuss
the scale of the coupling strength, the criterion for which is
actually a bit controversial in the literature<cit.>.
Although the terms for the coupling strengths were given in
relation to the validity of the RWA as well as the progress of
experimental accessibility, essentially the frequency $\omega $
has been conventionally taken as the evaluation scale: $ g/\omega \leq 0.01$ for
the weak coupling regime, $g/\omega \geq 0.01$ for the strong
coupling, $g/\omega \geq 0.1$ for the ultrastrong coupling
regime<cit.>, $g/\omega \geq 1$ for deep strong
coupling regime<cit.>. On the other hand, it should
be noticed that recently it has been proposed <cit.>
that the strength scale should be modified to be the semiclassical
critical point $g_{c0}$. Still, as afore-mentioned, $g_{c0}$ is
obtained in low-frequency semiclassical limit, while the situations at finite
frequencies would be different. The controversy essentially comes from the fact that a consensus on the nature of the interaction-induced variation in different frequencies is still lacking. Here, our expression of $g_c$ in
(<ref>) is obtained by the observation that it is the
wavepacket splitting that makes the essential change in increasing
the coupling strength, which controls the final effective coupling
tunneling strength and leads to transition (in low frequency
limit) or crossover (at finite frequencies) of the
quadpolaron/bipolaron states. We believe that $g_c$ is a more
universal scale valid for all frequencies, as indicated by Fig.
<ref>. Under these
considerations, we simply divide the coupling
strength into weak, intermediate and strong regimes under the conditions that $g$ is smaller than, comparable to and larger
than $g_c$, respectively. As a reference, we compare the different scales for the coupling strength used in the literature in Fig.<ref>.
+0.0cm (Color online) The conventional coupling regimes used in the literature. The conventional ultrastrong coupling regime (the green
shaded area) is enclosed by $g=0.1\omega$ and $g=\omega$, which
has been reached by experiments in rapid
progress<cit.>. The conventional deep strong coupling
regime (the light-cyan area) is surrounded by $g=\omega$ and
$g=10\omega$ (gray dash-dot lines), into which investigations have been
entering<cit.>. The black dotted line denotes the
semiclassical critical-like point in low frequency limit,
$g_{c0}$<cit.>, proposed as a different scale
of coupling strength<cit.>, while the blue solid line
schematically represents the quadpolaron/bipolaron boundary $g_c$ as a novel scale
generalized for the whole range of frequencies. Thus, the coupling strength is divided into weak, intermediate and strong regimes which correspond to that $g$ is smaller than, comparable to and larger than $g_c$, respectively. The orange-shaded
window edged by the dash lines opens for the overweighted
antipolaron discussed in our paper.
§ HIDDEN SCALING RELATION AND SYMMETRY-BREAKING-LIKE ASPECT
§.§ Scaling relation extracted from energy minimization
When the coupling strength grows beyond $g_c$, we find that
the squeezing factor $\xi _i$ and the displacement factor $\zeta
_i$ begin to collapse into the same values and scale with each other in
the further evolution, i.e.
\begin{equation}
\xi _\alpha \doteq \zeta _\alpha ,\quad \xi _\beta \doteq \zeta
_\beta . \label{scalingEq}
\end{equation}
This hidden scaling relation can be more explicitly formulated at
low frequencies. Note that the parameters can be extracted from
the energy minimization
\begin{equation}
\frac{\delta E}{\delta \alpha }=0,\quad \frac{\delta E}{\delta \xi
_i} =0,\quad \frac{\delta E}{\delta \zeta _i}=0.
\end{equation}
In the bipolaron regime, only the polaron-antipolaron tunneling
remains so the overlaps $S_{\alpha \bar{\alpha}}$ and
$S_{\beta \bar{\beta}}$ are vanishing, but $S_{\beta
\bar{\alpha}}$ is finite. In such a situation, controlling the
polaron-antipolaron center of mass, $ \zeta =\left( \zeta _\alpha
+\zeta _\beta \right) /2$, can be decoupled from the relative
motion in tunneling and squeezing, which enables us to extract
the weight of the polaron,
\begin{equation}
\alpha =\sqrt{\frac{1+\zeta _\beta }{2-\left( \zeta _\alpha -\zeta
_\beta \right) }}. \label{alpha}
\end{equation}
To obtain analytical results we assume a low frequency which enables a small-
$\omega $ expansion and leads us to
\begin{eqnarray}
\xi _{\alpha ,\beta }&=&\xi \left( 1\pm \frac{\omega
^2}{4g^2}\right) ,\nonumber \\
\zeta _{\alpha ,\beta }&=&\zeta \left( 1\pm
\frac{\omega ^2}{4g^2\left( 1-g_{c0}^4/g^4\right) }\right) ,
\end{eqnarray}
where $\xi _\alpha $ ($\xi _\beta $) takes the sign $+$ ($-$). In
the small-$ \omega $ limit, $\xi _i$ and $\zeta _i$ collapse into
their average $\xi =\left( \xi _\alpha +\xi _\beta \right) /2$ and
$\zeta =\left( \zeta _\alpha +\zeta _\beta \right) /2$ which are
\begin{equation}
\xi =\zeta =\sqrt{1-\frac{g_{c0}^4}{g^4}},
\end{equation}
up to $\omega ^2$ order. We can see the scaling relation from the
approximate analytic results: (i) in low-frequency limit, one sees
that $\xi _i\doteq \zeta _i$ holds, up to an $\omega ^2$
order correction which is negligible for small $\omega $. (ii) For
higher frequencies, the $\omega ^2$ terms in $\xi _{\alpha ,\beta
}$ and $\zeta _{\alpha ,\beta }$ become almost the same due to
$g_{c0}^4/g^4\ll 1$, since in bipolaron regime we have
$g>g_c>g_{c0}$ (e.g., for $\omega =0.5\Omega $,
$g_{c0}^4/g_c^4=0.056$ while $g_{c0}^4/g^4$ is negligible beyond
the crossover range.). These analytic considerations account for
the scaling relation as we showed in the main text for different
To test the scaling relation, we shall propose a physical
quantity that may be either measured experimentally or verified by
exact numerics. On one hand, applying the above expansion to the
photon number (<ref>) and neglecting the difference of
$\xi _\alpha ,\zeta _\alpha $ and $\xi _\beta ,\zeta _\beta $
leads us to an expression of $\zeta $ as a function of $
\langle a^{+}a\rangle $ and $\xi $
\begin{equation}
\zeta \doteq \frac \omega g\sqrt{\langle a^{+}a\rangle -\frac
14(\xi +\xi ^{-1})+\frac 12}.
\end{equation}
On the other hand, the same approximation yields
\begin{equation}
\xi \doteq -\langle \left( a^{+}-a\right) ^2\rangle .
\end{equation}
Thus, considering the ratio $\zeta /\xi $, we introduce the
following phyiscal quantity
\begin{equation}
\gamma \equiv \frac \omega {gt}\sqrt{\langle a^{+}a\rangle -\frac
14 (t+t^{-1})+\frac 12}, \label{ExactScaling}
\end{equation}
where we have defined $t=-\langle \left( a^{+}-a\right) ^2\rangle
$. In the bipolaron regime with strong couplings, this quantity
becomes the scaling ratio, $\gamma \rightarrow \zeta /\xi $. In
this regime, if the scaling relation (<ref>) holds, the
value of $\gamma $ will be equal to 1. Indeed, this scaling
relation is confirmed by the exact numerics, as shown in the main text.
In the quadpolaron regime with intermediate or weak couplings, not
only the scaling relation (<ref>) is violated but also
the relation between $\gamma $ and $\zeta /\xi $ is breaking
down, $\gamma \neq \zeta /\xi $. Nevertheless, we find that,
besides the bipolaron regime having the value $ \gamma =1$, the
quadpolaron with four strong channels of tunneling is located in
the range $\gamma <1$ and the state with decaying left-right
tunnelings ($S_{\alpha \bar \alpha }$, $S_{\beta \bar \beta }$)
falls in a range $\gamma >1$, as one can see in
Fig.<ref>a-c,g-i. In this sense, according to the values
and behavior of $\gamma $, one can distinguish the quantum
states of the bipolaron, quadpolaron and their changeover,
§.§ Scaling relation alternatively obtained from the lowest-order expansion of the effective
+0.0cm (Color online) Spin-up single-particle
effective potential, $v_+^{tot}=v_{+}+\delta v_{+}$, in the strong
coupling regime. As in this regime within the same spin component
there is no overlap between the polaron ($\alpha$, labelled by the
blue arrow) and antipolaron ($\beta$, labelled by the orange
arrow) in the two wells, to have both finite weights for the
polaron and the antipolaron the two sub-well energies have to be
degenerate, i.e., $v_{+}^{tot}(x^{min}_\alpha)+\varepsilon_\alpha=v_{+}^{tot}(x^{min}_\beta)+\varepsilon_\beta$.
Here $x^{min}_i$ is the position of local minimum potential and
$\varepsilon_i= \xi_i $ (scaled by $\omega/2$),
(Color online) Scaling relation alternatively obtained from the expansion of $v_+^{tot}$. $\omega / \Omega =0.001$ is taken. $\xi_i$ and $\zeta_i$
almost take the same values in the strong coupling regime above
the $g_c$. The inset shows their tiny differences by a zoom-in
Apart from variational method on energy minimization introduced in
Appendix <ref>, an alternative way to see the
scaling relation is to investigate the effective potential. As we
discussed in the main text, the eigenequation actually can be written in
a single particle form
\begin{equation}
\frac 12\omega\left( \hat p^2+v_{\pm }^{tot}\right) \psi _{\pm}=E\psi _{\pm },
\end{equation}
where we have assumed that the particle mass $m=1$ and the total
potential is composed of the bare potential $v_{\pm }$ and an
additional effective potential $\delta v_{\pm }$ induced by the
\begin{equation}
v_{\pm }^{tot}=v_{\pm }+\delta v_{\pm },
\end{equation}
\begin{equation}
v_{\pm }= \left( x\pm g^{\prime }\right) ^2,\quad \delta
v_{\pm }=\eta \frac \Omega {\omega}\frac{\psi _{\mp }}{\psi _{\pm
\end{equation}
and we have considered the ground state with $\eta =-1$. In
the strong coupling regime, the total potential exhibits an
obvious two-well structure, with a larger barrier separating the
wells, as shown in Fig. <ref>.
In the lowest order, the two wells can be considered as a local
harmonic potential. Actually, an expansion around the local
minimum point $x_i^{\min } $ of the potential leads to
\begin{equation}
v_{+}^{tot}\cong v_{+}^{tot}(x_i^{\min })+f_i^{(1)}\left(
x-x_i^{\min }\right) +f_i^{(2)}\left( x-x_i^{\min }\right) ^2,
\end{equation}
where $x_i^{\min }=\eta _i\zeta _ig^{\prime }$ with $i=\alpha
,\beta $, and $ \eta _\alpha =-1$, $\eta _\beta =1$ respectively
for the polaron and the antipolaron. The coefficients are defined
\begin{equation}
f_i^{(1)}=\frac{\partial v_{+}^{tot}}{\partial x}|_{x=x_i^{\min
}},\quad f_i^{(2)}=\frac 12\frac{\partial ^2v_{+}^{tot}}{\partial
x^2}|_{x=x_i^{\min }}.
\end{equation}
First, the approximation of local harmonic potential requires
\begin{eqnarray}
\text{condition-1:} &&\quad f_i^{(1)}=0, \\
\text{condition-2:} &&\quad f_i^{(2)}=\xi _i^2.
\end{eqnarray}
The condition-1 ensures the potential minimum point at $x_i^{min}$,
while the condition-2 indicates the same renormalized frequency
$\xi_i\omega$ of the local harmonic potential as that of the
wavefunction of the harmonic oscillator. Furthermore, since effectively there is no single-particle inter-site hopping (if regarding the wells as two sites) in the presence of the large barrier in the strong coupling regime, to have finite weights for both the polaron and the
antipolaron in the single-particle effective potential the local
energies of the two wells need to be degenerate
\begin{equation}
\text{condition-3:\quad }v_{+}^{tot}(x_\alpha ^{\min
})+\varepsilon _\alpha =v_{+}^{tot}(x_\beta ^{\min })+\varepsilon
_\beta ,
\end{equation}
\begin{equation}
\varepsilon _i= \xi _i
\end{equation}
is the energy of the local harmonic oscillator scaled by $\omega/2$
\begin{equation}
v_{+}^{tot}(x_i^{\min })=v_{+}(x_i^{\min })+\delta v_{+}(x_i^{\min
\end{equation}
corresponds to the reference energy. Taking the variational
wavefunction (<ref>), in the strong coupling regime we have
\begin{eqnarray}
\delta v_{+}(x_\alpha ^{\min })\doteq -\frac \Omega {\omega}
\frac{\beta \varphi
_\beta (-x_\alpha ^{\min })}{\alpha \varphi _\alpha (x_\alpha ^{\min })}%
,\nonumber \\
\delta v_{+}(x_\beta ^{\min })\doteq -\frac \Omega
\varphi _\alpha (-x_\beta ^{\min })}{\beta \varphi _\beta (x_\beta ^{\min })}.
\end{eqnarray}
while $\beta =\sqrt{1-\alpha ^2}$ in the strong coupling regime.
Now one can (i) control the displacement renormalization $\zeta
_i$ to satisfy condition-1 so that the linear term $f_i^{(1)}$ is
eliminated and the minimum is located at $x_i^{\min }$, (ii) tune
the frequency renormalization $\xi _i$ to fulfill condition-2 so
that both the local potential and the wavefunction
self-consistently shares the same frequency $ \xi _i\omega $,
(iii) balance the weight ratio of $\alpha /\beta $ to meet
condition-3 so that the two wells have degenerate local energies
to self-consistently guarantee the finiteness of the weights
$\alpha $ and $ \beta $. At this point, we see that the degree of
the basic deformation factors introduced for our variational
wavefunction is consistent with the minimum requirements of
self-consistence conditions.
From Conditions-1,2,3 we also refind the scaling relation as
illustrated by Fig.<ref>, which might provide
some alternative insights for the scaling relation that we
obtained from the energy minimization in last subsection. Still,
we should mention there is a small difference between the two ways,
since the above consideration from the effective potential is
based on the lowest order expansion which guarantees only the
local potential itself to be harmonic without taking care of
the energy, while the energy minimization ensures only the most
favorable energy but the effective potential $\delta v_{\pm }=\eta
\frac \Omega {\omega}\frac{\psi _{\mp }}{ \psi _{\pm }}$ includes
higher order terms beyond the harmonic potential approximation.
Despite the small difference, both approaches lead to the scaling
§.§ Symmetry-breaking-like aspect for the bipolaron-quadpolaron quantum state changeover
With the scaling relation at hand, it might provide some more
insight to discuss the quantum state changeover from the symmetry
point of view. Generally, for all eigenstates, the Hamiltonian has
the parity symmetry, ${\cal P}=\sigma_x(-1)^{a^{+}a}$ which involves
simultaneous reversion of the spin and the space. Specifically for
the ground state that we are focusing on in this work, one could find
extra symmetries. In fact, in the low frequency limit the photon
number vanishes for the ground state below $g_c$, as indicated
by Fig.<ref>b (this is more obvious for lower frequencies), so that additionally the total parity
symmetry can be decomposed into separate spin reversal symmetry
$\sigma_x$ and oscillator spatial reversal symmetry
$(-1)^{a^{+}a}$. These additional symmetries are broken beyond
$g_c$ due to the emergence of a number of photons, so that there
is a subsymmetry transition when the system goes across $g_c$.
Still, these spin and spatial subsymmetries are considered from
the weak coupling side and become less valid at finite frequencies
due to a nonvanishing photon number. Nevertheless, our finding of the scaling relation
provides compensation but from the strong coupling side.
Actually, as shown in Fig.4, the physical quantity we
proposed, $\gamma$, demonstrates an invariant behavior beyond
$g_c$, which confirms the scaling relation and thus the symmetric
aspect between the displacement and frequency renormalizations in
this regime. Note that, as shown in last subSection, in
this bipolaron regime the remaining left-left and right-right
tunnelings (i.e. polaron-antipolaron inter-tunnelings) render both the polaron and the antipolaron to have
finite weights, while to preserve finite weights as a quantum
effect in the absence of left-right tunneling channels (i.e. intra-polaron and intra-antipolaron tunnelings) the
polaron and the antipolaron have to maintain the
displacement-frequency scaling relation. In other words, this
displacement-frequency symmetry arises only in the absence of the
left-right channels, and conversely, there will be no left-right
channels if the symmetry is preserved there. Going from the
bipolaron regime to the quadpolaron regime, this symmetry will be
broken in the presence of the left-right tunneling channels. In
this sense, besides the afore-mentioned parity subsymmetry
breaking originating from the weak coupling side in the low frequency limit,
for both low frequency limit and finite frequencies there is
another hidden symmetry-breaking-like behavior in the changeover
of the two quantum states stemming from the strong coupling side.
Thus, it is interesting to see a deeper nature of the interaction that not only induces the bipolaron/quadpolaron quantum state changeover but also brings about the symmetry breaking.
§ PHYSICAL IMPLICATIONS AND METHOD EXTENSION TO THE MULTIPLE-MODE CASE
(Color online) Ground-state energy as a function of $g_1$ in the two-mode case. The parameters used are $\omega_1/\Omega=0.1, \omega_2/\Omega=0.01$ and $g_2/\Omega=0.025$. Our variational method (solid line) is in good agreement with exact numerics(dots).
§.§ Physical implications for the spin-boson model
Our ground-state phase diagram obtained for the Rabi model might also provide some insights or implications for the spin-boson model <cit.> which is a multiple-mode version of the Rabi model and has wide relevance to other fields, including the Kondo model <cit.> and the Ising spin chain <cit.> in condensed matter.
On the one hand, the bipolaron/quadpolaron changeover in the Rabi model can provide insights for localized/delocalized transition in the spin-boson model. In fact, the spin-boson model exhibits different behaviors in the Ohmic, super-Ohmic, and sub-Ohmic spectra, which actually have different weights of distributions for low and high frequency modes. Note that, as discussed in our work on the nature of the interaction-induced variation, the bipolaron and the quadpolaron states respectively have blockaded and enhanced left-right tunnelings, which is closely related to the situation of the localized and delocalized states involved in the spin-boson model. As indicated by our ground-state phase diagram and the obtained $g_c$ expression, the same coupling could be located in different regimes depending on whether frequency is low or high. Our ground-state phase diagram and $g_c$ expression might provide a primary reference and some insights into the different behaviours of the Ohmic, super-Ohmic, and sub-Ohmic spectra, since the distribution weights of low and high frequencies would make different contributions to blockaded and enhanced tunneling, thus affecting the competition in the quantum phase transition of the localized and delocalized states.
On the other hand, the overweighted antipolaron region might have some implication for the coherence-incoherence transition in the spin-boson model. It has been found that, within the delocalized phase of the spin-boson model, there is possibly another coherence-incoherence transition <cit.> for which the nature is still not very clear. Interestingly, in our ground-state phase diagram of the Rabi model, within the strong-tunneling regime in the quadpolaron state, there is also an underlying particular region characterized by an unexpected overweighted antipolaron, the possible implication and relation of the overweighted antipolaron regime in the Rabi model and the coherence-incoherence transition in the spin-boson model might be worthwhile exploring.
Since in the present work our focus is on the single-mode Rabi model, we would like to leave the investigations of these possible implications for the spin-boson model to some future works. Nevertheless, in the following we provide some indication of the method and variational wavefunction.
§.§ Method extension to the multiple-mode case
The basic variational physical ingredients introduced in the
single-mode case also should apply for the multiple-mode case. The
treatments are readily extendable from the single-mode case. The
Hamiltonian including $M$ modes of harmonic oscillators can be
written as
\begin{equation}
H=\sum_{k=1}^M\omega _ka_k^{\dagger }a_k+\sigma
_z\sum_{k=1}^Mg_k(a_k^{\dagger }+a_k)+\frac \Omega 2\sigma _x.
\end{equation}
We propose the variational trial wavefunction as
\begin{equation}
\psi [\{x_k\}]=\alpha \prod_{k=1}^M\varphi _\alpha ^k+\beta
\prod_{k=1}^M\varphi _\beta ^k,
\end{equation}
where $\varphi _{\alpha }^k$ ($\varphi _{\beta }^k$) is the $k$'th
mode polaron (antipolaron) under the direct extension $\{\omega
,g,x,\xi _i,\zeta _i\}\rightarrow \{\omega _k,g_k,x_k,\xi
_i^k,\zeta _i^k\}$. The energy is simply that of the single-mode
energy in (<ref>) with the overlap integrals replaced
by the product of all modes. We find the same accuracy as in the
single-mode case, as illustrated in Fig.<ref> by an
example of the two-mode case, for which exact numerics are available
for comparison. One can also include a bias term $\epsilon \sigma
_z$ with a broken parity for $\psi _\sigma \left( x\right) $ at
different spin $\sigma$. The multiple-mode case deserves special
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1511.00158
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Department of Mathematics, University of Michigan ([email protected])
Prediction of dynamical time series with additive noise using support
vector machines or kernel based regression has been proved to be consistent
for certain classes of discrete dynamical systems. Consistency implies
that these methods are effective at computing the expected value of
a point at a future time given the present coordinates. However, the
present coordinates themselves are noisy, and therefore, these methods
are not necessarily effective at removing noise. In this article,
we consider denoising and prediction as separate problems for flows,
as opposed to discrete time dynamical systems, and show that the use
of smooth splines is more effective at removing noise. Combination
of smooth splines and kernel based regression yields predictors that
are more accurate on benchmarks typically by a factor of 2 or more.
We prove that kernel based regression in combination with smooth splines
converges to the exact predictor for time series extracted from any
compact invariant set of any sufficiently smooth flow. As a consequence
of convergence, one can find examples where the combination of kernel
based regression with smooth splines is superior by even a factor
of $100$. The predictors that we compute operate on delay coordinate
data and not the full state vector, which is typically not observable.
§ INTRODUCTION
The problem of time series prediction is to use knowledge of a signal
$x(t)$ for $0\le t\leq T$ and infer its value at some future time
$t=T+t_{f}$. A time series is not predictable if it is entirely white
noise. Any prediction scheme has to make some assumption about how
the time series is generated. A common assumption is that the observation
$x(t)$ is a projection of the state of a dynamical system with noise
superposed <cit.>. Since the state of the dynamical
system can be of dimension much higher than $1$, delay coordinates
are used to reconstruct the state . Thus the state at time $t$ may
be captured as
\begin{equation}
\left(x(t),x(t-\tau),\ldots,x(t-(D-1)\tau)\right)\label{eq:s1-delay-coord}
\end{equation}
where $\tau$ is the delay parameter and $D$ is the embedding dimension.
Delay coordinates are (generically) effective in capturing the state
correctly provided $D\geq2d+1$, where $d$ is the dimension of the
underlying dynamics <cit.>.
Farmer and Sidorowich <cit.> used a linear framework
to compute predictors applicable to delay coordinates. It was soon
realized that the nonlinear and more general framework of support
vector machines would yield better predictors <cit.>.
Detailed computations demonstrating the advantages of kernel based
predictors were given by Muller et al <cit.>
and are also discussed in the textbook of Scholkopf and Smola <cit.>.
Kernel methods still appear to be the best, or among the best, for
the prediction of stationary time series <cit.>.
A central question in the study of noisy dynamical time series is
how well that noise can be removed to recover the underlying dynamics.
Lalley, and later Nobel, <cit.>
have examined hyperbolic maps of the form $x_{n+1}=F(x_{n})$, with
$F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$. It is assumed that observations
are of the form $y_{n}=x_{n}+\epsilon_{n}$, where $\epsilon_{n}$
is iid noise. They proved that it is impossible to recover $x_{n}$
from $y_{n}$, even if the available data $y_{n}$ is for $n=0,-1,-2,\ldots$
and infinitely long, if the noise is normally distributed. However,
if the noise satisfies $\abs{\epsilon_{n}}<\Delta$ for a suitably
small $\Delta$, the underlying signal $x_{n}$ can be recovered.
The recovery algorithm does not assume any knowledge of $F$. The
phenomenon of unrecoverability is related to homoclinic points. If
the noise does not have compact support, with some nonzero probability,
it is impossible to distinguish between homoclinic points.
Lalley <cit.> suggested that the case of flows could be
different from the case of maps. In discrete dynamical systems, there
is no notion of smoothness across iteration. In the case of flows,
the underlying signal will depend smoothly on time but the noise,
which is assumed to be iid at different points in time, will not.
Lalley's algorithm for denoising relies on dynamics and, in particular,
on recurrences. In the case of flows, we rely solely on smoothness
of the underlying signal for denoising. As predicted by Lalley, the
case of flows is different. Denoising based on smoothness of the underlying
signal alone can handle normally distributed noise or other noise
models. Thus, our algorithms are split into two parts: first the use
of smooth splines to denoise, and second the use of kernel based regression
to compute the predictor. Only the second part relies on recurrences.
Prediction of discrete dynamics, within the framework of Lalley <cit.>,
has been considered by Steinwart and Anghel <cit.>
(also see <cit.>). Suppose $x_{n}=F^{n}(x_{0})$
and $\tilde{x}_{n}=x_{n}+\epsilon_{n}$ is the noisy state vector.
The risk of a function $f$ is defined as
\[
\int\int\abs{F(x)+\epsilon_{1}-f(x+\epsilon_{2})}^{2}\,d\nu(\epsilon_{1})\,d\nu(\epsilon_{2})\,d\mu(x),
\]
where $\nu$ is the distribution of the noise and $\mu$ is a probability
measure invariant under $F$ and with compact support. Thus, the risk
is a measure of how well the noisy future state vector can be predicted
given the noisy current state vector. It is proved that kernel based
regression is consistent with respect to this notion of risk for a
class of rapidly mixing dynamical systems. Although the notion of
risk does not require denoising, consistency of empirical risk minimization
is proved for additive noise $\epsilon_{n}$ of compact support as
in <cit.>. In the case of empirical risk minimization,
compactness of added noise is not a requirement imposed by the underlying
dynamics but is assumed to make it easier to apply universality theorems.
Our results differ in the following ways. We consider flows and not
discrete time maps. In addition, we work with delay coordinate embedding
<cit.> and do not require the entire state
vector to be observable. Finally, we prove convergence to the exact
predictor, which goes beyond consistency. The convergence theorem
we prove is not uniform over any class of dynamical systems. However,
we do not assume any type of decay in correlations or rapid mixing.
Non-uniformity in convergence is an inevitable consequence of proving
a theorem that is applicable to any compact invariant set of a generic
finite dimensional dynamical system <cit.>.
This point is further discussed in section 2, which presents the main
algorithm as well as a statement of the convergence theorem. Section
3 presents a proof of the convergence theorem.
In section 4, we present numerical evidence of the effectiveness of
combining spline smoothing and kernel based regression. The algorithm
of section 2 is compared to computations reported in <cit.>
and the spline smoothing step is found to improve accuracy of the
predictor considerably. The numerical examples bring up two points
that go beyond either consistency or convergence. First, we explain
heuristically why it is not a good idea to iterate $1$-step predictor
$k$-times to predict the state $k$ steps ahead. Rather, it is a
much better idea to learn the $k$-step predictor directly. Second,
we point out that no currently known predictor splits the distance
vector between stable and unstable directions, a step which was argued
to be essential for an optimal predictor by Viswanath et al <cit.>.
In fact, the heuristic explanation for why iterating a $1$-step predictor
$k$ times is not a good idea relies on the same principle.
The concluding discussion in section 5 points out connections to related
lines of current research in parameter inference <cit.>,
prediction of nonstationary signals <cit.>,
and optimal consistency estimates for stationary data <cit.>.
§ PREDICTION ALGORITHM AND STATEMENT OF CONVERGENCE THEOREM
Let $\frac{dx}{dt}=\mathcal{F}(x)$, where $\mathcal{F}\in C^{r}(\mathbb{R}^{d},\mathbb{R}^{d})$,
$r\geq2$, be a flow that is limited to some compact subset of $\mathbb{R}^{d}$.
Let $\mathcal{F}_{t}(x_{0})$ be the time-$t$ map with initial data
$x_{0}$. It is assumed that $U(t):\mathbb{R}\rightarrow\mathbb{R}^{d}$
is a trajectory of the flow whose initial point $U(0)$ is picked
according to a probability measure that is invariant under the flow
and that has compact support.
Let $\pi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a generic nonlinear
projection. Let $u(t)=\pi U(t)$ be the projection of the random trajectory
$U(t)$. By the embedding theorem of Sauer et al <cit.>,
we assume that the delay coordinates give a $C^{r}$ diffeomorphism
into the state space implying that $U(t)$ can be recovered from the
delay vector, with delay $\tau>0$,
\[
\left(u(t),u(t-\tau),\ldots,u(t-(D-1)\tau)\right)
\]
for $D\geq2d+1$. This delay vector is denoted by $u(t;\tau)$. Because
of the embedding, we may assume that $u(t)$ is a random trajectory
with the initial point $u(0;\tau)$ chosen from some probability measure
$\mu$ that has compact support in $\mathbb{R}^{D}$ and that is invariant
under the flow lifted via the embedding.
Given such a random trajectory $u(t)$, it is assumed that the recorded
observations are $u_{\eta}(jh)=u(jh)+\epsilon_{j}$, where $\epsilon_{j}$
is iid noise. Following Eggermont and LaRiccia <cit.>,
we assume that $\mathbb{E}\epsilon_{j}=0$ and $\mathbb{E}\abs{\epsilon_{j}}^{\kappa}<\infty$
for some $\kappa>3$. To avoid inessential technicalities it is assumed
that $\tau/h\in\mathbb{Z}^{+}$ so that the delay is an integral multiple
of the time step $h$. In particular, we set $\tau=nh$. The delay
coordinates $u_{\eta}(jh;\tau)$ are assumed to be available for $j=0,\ldots,Nn$,
which implies that the observation interval is $t\in[-(D-1)\tau,N\tau]$.
The exact predictor $F:\mathbb{R}^{D}\rightarrow\mathbb{R}$ is a
$C^{r}$ function such that $F(u(t;\tau))=u(t+\tau)$, provided $u(t;\tau)$
lies in the support of the invariant measure $\mu$. The problem as
considered by Muller et al <cit.>
is to recover the exact predictor $F$ from the noisy observations
$u_{\eta}(jh)$. Let $\abs{\cdot}_{\epsilon}$ denote Vapnik's $\epsilon$-loss
function. The algorithm of Muller et al computes $f_{m}$ such that
the functional
\begin{equation}
\frac{1}{Nn+1}\sum_{j=0}^{Nn}\abs{f(u_{\eta}(jh;\tau))-u_{\eta}(jh+\tau)}_{\epsilon}+\Lambda\norm f_{K_{\gamma}}^{2}\label{eq:algo-muller}
\end{equation}
is minimized for $f=f_{m}$ in the reproducing kernel Hilbert space
$\mathcal{H}_{K_{\gamma}}$ corresponding to the kernel $K_{\gamma}$.
The kernel $K_{\gamma}$ is assumed to be given by $K_{\gamma}(x,y)=\exp\left(-\frac{\sum_{i=1}^{D}(x_{i}-y_{i})^{2}}{\gamma^{2}}\right)$.
The kernel bandwidth parameter $\gamma$ and the Lagrange multiplier
$\Lambda$ are both determined using cross-validation. We will compare
our predictor against that of Muller et al using some of the same
examples and the same framework as they do in section 4.
In our algorithm, the first step is to apply spline smoothing. In
particular, we apply cubic spline smoothing <cit.> to
compute a function $u_{s}(t)$, $t\in[-(D-1)\tau,N\tau]$ such that
the functional
\begin{equation}
\frac{1}{(N+D-1)n+1}\sum_{j=-(D-1)n}^{Nn}\left(u_{\eta}(jh)-u(jh)\right)^{2}+\lambda\int_{-(D-1)\tau}^{N\tau}u''(t)^{2}\,dt\label{eq:algo-1}
\end{equation}
is minimum for $u=u_{s}$ over $u\in W^{2,2}[-(D-1)\tau,N\tau]$.
The parameter $\lambda$ is determined using five-fold cross-validation.
The second step of our algorithm is similar to the method of Muller
et al. The predictor $f_{1}$ is computed as
\begin{equation}
f_{1}=\argmin_{f\in\mathcal{H}_{k}}\frac{1}{Nn+1}\sum_{j=0}^{Nn}\left(f(u_{s}(jh;\tau))-u_{s}(jh+\tau)_{\epsilon}\right)_{2}+\Lambda\norm f_{K_{\gamma}}^{2}.\label{eq:algo-2}
\end{equation}
Both the parameters $\gamma$ and $\Lambda$ are determined using
five-fold cross-validation.
The second step (<ref>) differs from the algorithm of Muller
et al in using the spline smoothed signal $u_{s}(t)$ in place of
the noisy signal $u_{\eta}(t)$. Our algorithm relies mainly on spline
smoothing to eliminate noise. The other difference is that we use
the least squares loss function in place of the $\epsilon$-loss function.
This difference is a consequence of relying on spline smoothing to
eliminate noise. As explained by Christmann and Steinwart <cit.>,
the $\epsilon$-loss function, Huber's loss, and the $L^{1}$ loss
function are used to handle outliers. However, spline smoothing eliminates
outliers, and we choose the $L^{2}$ loss function because of its
algorithmic advantages.
We now turn to a discussion of the convergence of the predictor $f_{1}$
to the exact predictor $F$. The first step is to assess the accuracy
of spline smoothing. We quote the following lemma, which is a convenient
restatement of the result of Eggermont and LaRiccia <cit.>
(see pages 132 and 133 of <cit.>).
Assume $2\leq m\leq r$. Suppose that $u(t)$ is a signal defined
for $t\in\mathbb{R}$ with $u(0;\tau)\in X$. For $j=-(D-1)n,\ldots,Nn$,
let $y_{j}=u(jh)+\epsilon_{j}$, where $h=\tau/n$ and $\epsilon_{j}$
are iid random variables. It is further assumed that $\mathbb{E}\epsilon_{j}=0$,
$\mathbb{E}\epsilon_{j}^{2}=\sigma^{2}$, and $\mathbb{E}\abs{\epsilon_{j}}^{\kappa}<\infty$
for some $\kappa>3$. Let $u_{s}(t)\in W^{m,2}[-(D-1)\tau,N\tau]$
be the spline that minimizes the functional
\[
\frac{1}{n(N+D-1)+1}\sum_{j=-(D-1)n}^{nN}(\tilde{u}(jh)-y_{j})^{2}+\lambda\int_{-(D-1)\tau}^{N\tau}\abs{\tilde{u}^{(m)}(t)}^{2}\,dt
\]
over $\tilde{u}\in W^{m,2}[-(D-1)\tau,N\tau]$. Assume
\[
\lambda=\left(\frac{\log(n(N+D-1))}{n(N+D-1)}\right)^{\frac{2m}{2m+1}}.
\]
Let $p=\pi\left(n,N,\Delta,u(0;\tau)\right)$ be the probability
\[
\norm{u_{s}-u}_{\infty}>\Delta>0.
\]
Then $\lim_{n\rightarrow\infty}\pi(n,N,\Delta,u(0;\tau)=0$.
Some remarks about the connection of this lemma to the algorithm given
by (<ref>) and (<ref>) follow. First, the lemma
assumes a fixed choice of $\lambda$ (the relevant theorem in <cit.>
in fact allows $\lambda$ to lie in an interval). In our algorithm,
$\lambda$ is determined using cross-validation because of its practical
effectiveness <cit.>. Second, the probability $\pi(n,N,\Delta,u(0;\tau))$
depends on $u(0;\tau)$ and therefore on the particular trajectory.
As discussed in <cit.>, uniform bounds on the
$\infty$-norm convergence of smooth splines are yet unavailable.
The achievability part of Stone's optimality result <cit.>
gives uniform bounds for the $\infty$-norms but the algorithm in
that proof does not appear practical. We also note that uniform bounds
are available for the $L^{2}$ error of smooth splines <cit.>.
The convergence analysis of the second half of the algorithm too alters
the algorithm slightly. In particular, the use of cross-validation
to choose parameters is not a part of the analysis. To state the convergence
theorem, we first fix $\epsilon>0$. By the universality theorem of
Steinwart <cit.>, we may choose $F_{\epsilon}\in\mathcal{H}_{K_{\gamma}}$
such that $\norm{F_{\epsilon}-F}_{\infty}<\epsilon$ in a compact
domain that has a non-empty interior and contains the invariant set
$X$. The convergence theorem also makes the technical assumption
$\epsilon^{2}/\norm{F_{\epsilon}}_{K_{\gamma}}^{2}<1$, which may
always be satisfied by taking $\epsilon$ small enough.
The choice of the kernel-width parameter $\gamma$ is important in
practice. In the convergence proof, the choice of $\gamma$ is not
explicitly considered. However, $\gamma$ still plays a role because
$\norm{F_{\epsilon}}_{K_{\gamma}}$ depends upon $\gamma$.
The parameter $\Lambda$ in (<ref>) is fixed as $\Lambda=\epsilon^{2}/\norm{F_{\epsilon}}_{K_{\gamma}}^{2}$for
the proof. Next we pick $\delta=\epsilon^{1/2}$ and $\ell\in\mathbb{Z}^{+}$
such that the covering of the invariant set $X$ using boxes of dimension
$2^{-\ell}$ ensures that the variation within each box is bounded
by $\delta/2$.
Suppose $A_{1},\ldots,A_{L}$ are boxes of dimension $2^{-\ell}$
that cover $X$ in the manner hinted above. We next choose $T^{\ast}$
such that the measure of the trajectories with respect to the ergodic
measure $\mu$ that sample each one of the boxes $A_{j}$ adequately
(in a sense that will be explained) is greater than $1-\epsilon$
if the time interval of the trajectory exceeds $T^{\ast}$.
The parameter $\Delta$ is a bound on the infinite norm accuracy of
the smooth spline as in Lemma <ref>. Choose
$\Delta>0$ small enough that
\[
\frac{B_{1}\Delta^{1/2}}{\Lambda}=\frac{B_{1}\Delta^{1/2}\norm{F_{\epsilon}}_{K}^{2}}{\epsilon^{2}}<\epsilon^{1/2},
\]
where $B_{1}$ is a constant specified later. The main purpose of
increasing $n$ is to make spline smoothing accurate. However, the
following condition requiring $n$ to be large enough is assumed in
the proof:
\[
\frac{B_{1}h^{1/2}}{\Lambda}=\frac{B_{1}\tau^{1/2}\norm{F_{\epsilon}}_{K}^{2}}{\epsilon^{2}n^{1/2}}<\epsilon^{1/2}.
\]
Within this set-up, we have the following convergence theorem.
For $\epsilon>0$, $T>T^{\ast}$, $N=T/\tau$, and $\Lambda$, $\Delta$
chosen as above, we have
\[
\mu\left\{ x\in X\biggl|\abs{f_{1}(x)-F(x)}>3\sqrt{\epsilon}\right\} <\frac{8\epsilon}{1-\epsilon}
\]
for $\left\{ u(0;\tau)\in X\right\} $ of measure greater than $1-\epsilon$
with denoising probability $\pi(n,N,\Delta,u(0;\tau))$ tending to
$1$ in the limit $n\rightarrow\infty$.
Nonuniform bounds implying a form of weak consistency are considered
by Steinwart, Hush, and Scovel <cit.>. However,
the algorithm of (<ref>) and (<ref>) does not
fit into the framework of <cit.>. The application
of spline smoothing to produce $u_{s}(t)$ means that $u_{s}(t)$
may not be stationary, and our method of analysis does not rely on
verifying a weak law of large numbers as in <cit.>.
The analysis summarized in this section and given in detail in section
4 relies on $\infty$-norm bounds.
§ PROOF OF CONVERGENCE
We begin the proof with a more complete account of how the embedding
theorem is applied. Let $\frac{dx}{dt}=\mathcal{F}(x)$, where $\mathcal{F}\in C^{r}(\mathbb{R}^{d},\mathbb{R}^{d})$,
$r\geq2$, be a flow. Let $\mathcal{F}_{t}(x_{0})$ be the time-$t$
map with initial data $x_{0}$. Let $\tilde{V}\subset\mathbb{R}^{d}$
be an open set with compact closure. It is assumed that $\mathcal{F}_{t}(x)$
is well-defined for $-\tau D\leq t\leq\tau$, where $D$ is the embedding
dimension (see below).
Assumption: For embedding dimension $D\geq2d+1$ and a suitably chosen
delay $\tau>0$, the map
\[
x\rightarrow(\pi x,\pi\mathcal{F}_{-\tau}x,\pi\mathcal{F}_{-2\tau}x,\ldots,\pi\mathcal{F}_{-(D-1)\tau}x)
\]
is a $C^{r}$ diffeomorphism between $\tilde{V}$ and its image in
$\mathbb{R}^{D}$ under the above map. This assumption is generically
true <cit.>. This map is called the delay
embedding. Denote the image of $\tilde{V}$ under the delay embedding
by $V$.
Let $\tilde{\mu}$ be an invariant measure with compact support and
with respect to which the flow is ergodic. Denote the compact support
of $\tilde{\mu}$ by $\tilde{X}\subset\tilde{V}$. Denote the image
of $\tilde{X}$ under the delay embedding by $X$, and let $\mu$
be the invariant measure of the induced flow on $X$.
Let $U(t)$, $t\in\mathbb{R}$, be a random trajectory of the flow.
The random trajectory is generated by picking $U(0)$ from the measure
$\tilde{\mu}$ over the compact set $\tilde{X}\subset\mathbb{R}^{d}$.
Let the signal $u(t)$ be equal to $\pi U(t)$. We denote
\[
\left(u(t),u(t-\tau),\ldots,u(t-(D-1)\tau)\right)
\]
by $u(t;\tau)$ as already noted. Evidently, $u(t;\tau)$ is the image
of $U(t)$ under the delay embedding. Thus, $u(t;\tau)$ is a point
in $X$, and the random embedded trajectory is generated by picking
$u(0;\tau)$ from the compact set $X\subset\mathbb{R}^{D}$ according
to the probability measure $\mu$.
Suppose $\xi(t)=\pi x(t)$ with $\frac{dx(t)}{dt}=\mathcal{F}(x(t))$
for $-\tau D\leq t\leq\tau$, $x(0)\in\tilde{V}$, and $\xi(0;\tau)\in V$.
There exists a unique and well-defined $C^{r}$ function $F:V\rightarrow\mathbb{R},$
called the exact predictor, such that
\[
\]
In particular, $F(u(t;\tau))=u(t+\tau)$ for all $t\in\mathbb{R}$
and delay trajectories $u(t;\tau)$ with $u(0;\tau)\in X$.
To map $\xi(0;\tau)$ to $\xi(\tau)$, first invert the delay map
to obtain a point in $\tilde{V}$, advance that point by $\tau$ by
applying $\mathcal{F}_{\tau}$, and finally project using $\pi$.
Each of the three maps in this composition is $C^{r}$ or better.
The predictor must be unique because $\mathcal{F}_{\tau}$ is uniquely
determined by the flow.
The following convexity lemma is an elementary result of convex analysis
<cit.>. It is stated and proved for completeness.
Let $\mathcal{L}_{1}(f)$ and $\mathcal{L}_{2}(f)$ be convex and
continuous in $f$, where $f\in\mathcal{H}$ and $\mathcal{H}$ is
a Hilbert space. If $w\in\nabla\mathcal{L}_{i}(f)$, the subgradient
at $f$, assume that
\[
\mathcal{L}_{i}(f+g)-\mathcal{L}_{i}(f)-\inner wg\geq\lambda\inner gg/2
\]
for $\lambda>0$, all $g\in\mathcal{H}$, and $i=1,2$. Let $f_{1}=\argmin\mathcal{L}_{1}(f)$
and $f_{2}=\argmin\mathcal{L}_{2}(f)$. Suppose that
\[
\abs{\mathcal{L}_{1}(f)-\mathcal{L}_{2}(f)}\leq\delta
\]
for $\norm f\leq r$, with $r$ large enough to ensure $\norm{f_{1}}<r$
and $\norm{f_{2}}<r$. Then,
\[
\norm{f_{1}-f_{2}}^{2}\leq\frac{2\delta}{\lambda}.
\]
Because $f_{1}$ minimizes $\mathcal{L}_{1}(f)$, we have $0\in\nabla\mathcal{L}_{1}(f_{1})$.
\[
\mathcal{L}_{1}(f_{2})-\mathcal{L}_{1}(f_{1})\geq\lambda\norm{f_{2}-f_{1}}^{2}/2.
\]
Similarly, $\mathcal{L}_{2}(f_{1})-\mathcal{L}_{2}(f_{2})\geq\lambda\norm{f_{2}-f_{1}}^{2}/2$.
By adding the two inequalities, we have
\[
\norm{f_{2}-f_{1}}^{2}\leq\frac{\abs{\mathcal{L}_{1}(f_{2})-\mathcal{L}_{1}(f_{1})+\mathcal{L}_{2}(f_{1})-\mathcal{L}_{2}(f_{2})}}{\lambda}\leq\frac{2\delta}{\lambda},
\]
proving the lemma.
If $u(t)$, $t\in\mathcal{T}$, is the noise-free signal, our arguments
are phrased under the assumption that $\abs{u(t)-u_{s}(t)}\leq\Delta$.
This assumption is realized with a high probability for small $h$
or large $n$ by Lemma <ref>. For now we assume
that the assumption is realized with a probability greater than $1-p$.
\[
Y=\left\{ y|\norm{y-x}_{\infty}\leq1\quad\text{for some}\quad x\in X\right\} ,
\]
where $X$ is the delay embedded compact invariant set in $\mathbb{R}^{d}$.
We will always assume $\Delta<1$ so that the spline-smoothed signal
maps to $Y$ under delay embedding with probability greater than $1-p$.
The exact predictor $F$ is only defined the set $V$ which contains
$X$, the support of the invariant measure $\mu$. We will assume
that $F$ is extended to the whole of $Y$ and that the extension
is also $C^{r}$ smooth. The extension can be carried out in many
ways. The convergence proof will assess the approximation to $F$
with respect to the measure $\mu$. Therefore, the manner in which
the extension is carried out is not highly relevant. The sole purpose
of the extension is to facilitate an application of the universality
theorem for Gaussian kernels.
\begin{equation}
B=\sup_{x\in X}\norm x_{\infty}+1\label{eq:defn-B-1}
\end{equation}
Thus, $B$ is a bound on the size of the embedded invariant set with
ample allowance for error in spline smoothing.
Let $u_{s}(t)$ denote the spline-smoothed signal and $u(t)$ the
noise-free signal with $u(t;0)\in X$. Define
\[
\mathcal{W}_{1}(f)=\frac{1}{Nn+1}\sum_{j=0}^{Nn}\left(f(u_{s}(jh;\tau))-u_{s}(jh+\tau)\right)^{2}+\Lambda\norm f_{K}^{2}.
\]
\[
\mathcal{W}_{2}(f)=\frac{1}{Nn+1}\sum_{j=0}^{Nn}\left(f(u(jh;\tau))-u(jh+\tau)\right)^{2}+\Lambda\norm f_{K}^{2}
\]
using the noise-free signal $u(t)$. Let $T=Nnh$ and define
\[
\mathcal{W}_{3}(f)=\frac{1}{T}\int_{0}^{T}\left(f(u(t;\tau))-u(t+\tau)\right)^{2}\,dt+\Lambda\norm f_{K}^{2}.
\]
For $\Lambda>0$, all three functionals are strictly convex and have
a unique minimizer. The unique minimizers of $\mathcal{W}_{1}$, $\mathcal{W}_{2}$,
and $\mathcal{W}_{3}$ are denoted by $f_{1}$, $f_{2}$, and $f_{3}$,
respectively. The functional $\mathcal{W}_{1}$ is the same as in
(<ref>), the second step of the algorithm. Thus, $f_{1}$
is the computed approximation to the exact predictor $F$.
The minimizer $f_{1}$ satisfies $\norm{f_{1}}_{K}\leq\frac{B}{\Lambda^{1/2}}$
with probability greater than $1-p$. The minimizers $f_{2}$ and
$f_{3}$ satisfy $\norm{f_{2}}_{K}\leq\frac{B}{\Lambda^{1/2}}$ and
Because $f_{1}$ minimizes $\mathcal{W}_{1}(f)$, we must have $\mathcal{W}_{1}(f_{1})\leq\mathcal{W}_{1}(0)$.
We have $\mathcal{W}_{1}(0)\leq B^{2}$ with probability greater than
$1-p$. Thus, $\Lambda\norm{f_{1}}_{K}^{2}\leq\mathcal{W}_{1}(f_{1})\leq\mathcal{W}_{1}(0)\leq B^{2}$
and the stated bound for $\norm{f_{1}}_{K}$ follows. The bounds for
$f_{2}$ and $f_{3}$ are proved similarly.
Assume $0<\Lambda\leq1$. For $f\in\mathcal{H}_{K}$ with $\norm f_{K}\leq\frac{B}{\Lambda^{1/2}}$,
we have $\abs{\mathcal{W}_{1}(f)-\mathcal{W}_{2}(f)}\leq\frac{B_{1}^{2}\Delta}{\Lambda}$
with probability greater than $1-p$. Here $B_{1}$ depends only on
$B$ and the kernel $K$.
First, we note that $\norm f_{\infty}\leq c_{0}\norm f_{K}$ and $\norm{\partial f}_{\infty}\leq c_{1}\norm f_{K}$,
where $\partial$ is the directional derivative of $f$ in any direction.
By a result of Zhou <cit.>, we may take $c_{0}=\norm{K(x,y)}_{\infty}$
and $c_{1}=2\sum_{j=1}^{d}\norm{\partial_{x_{j}}K(x,y)}_{\infty}$,
where $d$ is the embedding dimension and the $\infty$-norm is over
$x,y\in Y$. We define $B_{1}$ using
\begin{equation}
\end{equation}
and assume $\norm f_{2},\norm f_{\infty}$, $\norm{\partial f}_{\infty}$
(where $\partial$ is a directional derivative in any direction) as
well as $\norm f_{K}$ to all be bounded above by $B_{1}'/\Lambda^{1/2}$
with probability greater than $1-p$.
We may write
\begin{equation}
\abs{\mathcal{W}_{1}(f)-\mathcal{W}_{2}(f)}\leq\frac{1}{Nn+1}\sum_{j=0}^{Nn}\frac{4B_{1}'}{\Lambda^{1/2}}\left(\abs{f(u_{s}(jh;\tau))-f(u(jh;\tau))}+\abs{u_{s}(jh;\tau)-u(jh;\tau)}\right).\label{eq:W1minusW2}
\end{equation}
Here $\frac{4B_{1}'}{\Lambda^{1/2}}$ is an upper bound on $\abs{f(u_{s}(jh;\tau))}+\abs{f(u(jh;\tau))}+\abs{u_{s}(jh;\tau)}+\abs{u(jh;\tau)}$
obtained assuming $\Lambda\leq1$.
Now, $\abs{u_{s}(jh;\tau)-u(jh;\tau)}\leq\Delta$ with probability
greater than $1-p$, and
\[
\abs{f(u_{s}(jh;\tau))-f(u(jh;\tau))}\leq B_{1}'\Delta/\Lambda^{1/2}
\]
by the bound on $\norm{\partial f}_{\infty}$. Note that $B>1$ and
$\Lambda\leq1$ implying $B_{1}'/\Lambda^{1/2}\geq1$. The proof is
completed by utilizing these bounds in (<ref>) and redefining
$B_{1}$ as $B_{1}=8B_{1}'$.
Assume $0<\Lambda\leq1$. With probability greater than $1-p$, $\norm{f_{1}-f_{2}}_{K}\leq\frac{2B_{1}\Delta^{1/2}}{\Lambda}$.
Follows from Lemmas <ref> and <ref>.
Assume $0\leq\Lambda\leq1$. For $f\in\mathcal{H}_{K}$ and $\norm f_{K}\leq\frac{B}{\Lambda^{1/2}}$,
we have $\abs{\mathcal{W}_{2}(f)-\mathcal{W}_{3}(f)}\leq\frac{B_{1}^{2}h}{\Lambda}.$
We will argue as in Lemma <ref> and assume that $\norm f_{2}$,
$\norm f_{\infty}$, and $\norm{\partial f}_{\infty}$ are all bounded
by $B_{1}'/\Lambda^{1/2}$.
Suppose $\alpha\in[0,1]$. In the difference
\[
\begin{split}\frac{1}{h}\int_{kh}^{(k+1)h}\left(f(u(t;\tau))-u(t+\tau)\right)^{2}\,dt & -(1-\alpha)\left(f(u(kh;\tau))-u(kh+\tau)\right)^{2}\\
& -\alpha\left(f(u((k+1)h;\tau))-u((k+1)h+\tau)\right)^{2},
\end{split}
\]
we may apply the mean value theorem to the integral and argue as in
Lemma <ref> to upper bound the difference by $\left(B_{1}'\right)^{2}h/\Lambda$.
The proof is completed by summing the differences from $k=0$ to $k=Nn-1$
and dividing by $Nn$.
Assume $0\leq\Lambda<1$. Then $\norm{f_{2}-f_{3}}\leq\frac{2B_{1}h^{1/2}}{\Lambda}$.
Follows from Lemmas <ref> and <ref>.
Choose $\epsilon>0$. By the universality theorem of Steinwart, we
may find $F_{\epsilon}\in\mathcal{H}_{K}$ such that $\norm{F_{\epsilon}-F}_{\infty}\leq\epsilon$,
where the $\infty$-norm is over $Y$. In fact, we will need the difference
$\abs{F_{\epsilon}(x)-F(x)}$ to be bounded by $\epsilon$ only for
$x\in X$. The larger compact space $Y$ is needed to apply the universality
theorem and for other RKHS arguments.
Let $\Lambda=\epsilon^{2}/\norm{F_{\epsilon}}_{K}^{2}\leq1$. If $f_{3}$
minimizes $\mathcal{W}_{3}(f)$, we have
\[
\frac{1}{T}\int_{0}^{T}\left(f_{3}(u(t;\tau;\omega))-u(t+\tau)\right)^{2}\,dt\leq\Lambda\norm{F_{\epsilon}}_{K}^{2}+\epsilon^{2}=2\epsilon^{2}.
\]
In addition, $\norm{f_{3}}_{K}^{2}\leq2\norm{F_{\epsilon}}_{K}^{2}.$
We have
\[
\frac{1}{T}\int_{0}^{T}\left(f_{3}(u(t;\tau))-u(t+\tau)\right)^{2}\,dt\leq\mathcal{W}_{3}(f_{3}),
\]
$\mathcal{W}_{3}(f_{3})\leq\mathcal{W}_{3}(F_{\epsilon})$ because
$f_{3}$ is the minimizer, and
\[
\mathcal{W}_{3}(F_{\epsilon})\leq\epsilon^{2}+\Lambda\norm{F_{\epsilon}}_{K}^{2}.
\]
This last inequality uses $\int(F_{\epsilon}(u(t;\tau))-u(t+\tau))^{2}\,dt=\int(F_{\epsilon}(u(t;\tau))-F(u(t;\tau))^{2}\,dt$.
The proof of the first part of the lemma is completed by combining
the inequalities. To prove the second part, we argue similarly after
noting $\norm{f_{3}}_{K}^{2}\leq\mathcal{W}_{3}(F_{\epsilon})/\Lambda$.
Consider half-open boxes in $\mathbb{R}^{d}$ of the form
\[
\]
with $\ell\in\mathbb{Z}^{+}$and $j_{i}\in\mathbb{Z}$. The whole
of $\mathbb{R}^{d}$ is a disjoint union of such boxes. Because $X$
is compact, we can assume that $X\subset\cup_{j=1}^{L}A_{j}$, where
the union is disjoint, each $A_{j}$ is a half-open box of the form
above, and $A_{j}\cap X\neq\phi$ for $1\leq j\leq L$.
We will pick $\ell$ to be so large, that each box has a diameter
that is bounded as follows:
\[
\frac{\sqrt{d}}{2^{\ell}}<\frac{\delta}{2\sqrt{2}\norm{\partial^{2}K}_{2,\infty}\norm{F_{\epsilon}}_{K}}.
\]
Here $\delta>0$ is determined later, and $\norm{\partial^{2}K}_{2,\infty}$
is the $\infty$-norm in the function space $C^{2}(Y\times Y)$. Lemma
<ref> tells us that $\norm{f_{3}}_{K}\leq\sqrt{2}\norm{F_{\epsilon}}_{K}$,
and therefore $\norm{\partial f_{3}}_{\infty}\leq\sqrt{2}\norm{\partial^{2}K}_{2,\infty}\norm{F_{\epsilon}}_{K}$.
As a consequence of our choice of $\ell$, $x,y\in A_{j}$ implies
\begin{equation}
\abs{f_{3}(x)-f_{3}(y)}<\delta/2,\label{eq:f3-derv-bound}
\end{equation}
bounding the variation of $f_{3}$ within a single cell $A_{j}$.
The next lemma is about taking a trajectory that is long enough that
each of the sets $A_{j}$ is sampled accurately. By assumption $X$
is the support of $\mu$. However, we may still have $\mu(A_{j})=0$
for some $j$. In the following lemma and later, it is assumed that
all $A_{j}$ with $\mu(A_{j})=0$ are eliminated from the list of
boxes covering $X$.
Let $\chi_{A_{j}}$ denote the characteristic function of the set
$A_{j}$. There exist $T^{\ast}>0$ and a Borel measurable set
\[
S_{\epsilon,T^{\ast}}\subset X
\]
such that $u(0;\tau)\in S_{\epsilon,T^{\ast}}$ implies that for all
$T\geq T^{\ast}$ and $j=1,\ldots,L$
\[
\abs{\frac{1}{T}\int_{0}^{T}\chi_{A_{j}}\left(u(t;\tau;\omega)\right)\,dt-\mu(A_{j})}\leq\epsilon\mu(A_{j}).
\]
and with $\mu\left(S_{\epsilon,T^{\ast}}\right)>1-\epsilon$.
To begin with, consider the set $A_{1}$. By the ergodic theorem,
\[
\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\chi_{A_{1}}\left(u(t;\tau\right)\,dt=\mu(A_{j})
\]
for $u(0;\tau)\in S\subset X$ with $\mu(S)=1$. Let
\[
A_{s,\epsilon}=\left\{ u(0;\tau)\in X\biggl|\abs{\frac{1}{T}\int_{0}^{T}\chi_{A_{1}}(u(t;\tau)\,dt-\mu(A_{1})}>\epsilon\mu(A_{1})\:\:\text{for some}\:\:T\geq s\right\} .
\]
The sets $A_{s,\epsilon}$ shrink with increasing $s$. Then the measure
of $\cap_{s=1}^{\infty}A_{s,\epsilon}$ under $\mu$ is zero. Therefore,
there exists $s_{1}\in\mathbb{Z}^{+}$ such that $\mu(A_{s_{1},\epsilon})<\epsilon/L$.
We can find $s_{2},\ldots,s_{L}$ similarly by considering the sets
$A_{2},\ldots,A_{L}$. The lemma then holds with $T^{\ast}=\max(s_{1},\ldots,s_{L})$.
Suppose that $u(0;\tau)\in S_{\epsilon,T^{\ast}}$. Suppose that $f_{3}$
minimizes $\mathcal{W}_{3}(f)$ with $T\geq T^{\ast}$ and $\Lambda=\epsilon^{2}/\norm{F_{\epsilon}}_{K}^{2}\leq1$.
\[
\mu\left\{ x\in X\bigl|\abs{f_{3}(x)-F(x)}\geq\delta\right\} <\frac{8\epsilon^{2}}{\delta^{2}(1-\epsilon)}.
\]
Denote the set $\left\{ x\bigl|\abs{f_{3}(x)-F(x)}\geq\delta\right\} $
by $S_{\delta}$. Let $J$ be the set of all $j=1,\ldots,L$ such
that $\abs{f_{3}(x)-F(x)}\geq\delta$ for some $x\in A_{j}$. Evidently,
$S_{\delta}\subset\cup_{j\in J}A_{j},$ and it is sufficient to bound
the measure of $\cup_{j\in J}A_{j}$.
By (<ref>), for $y\in A_{j}$ we have $\abs{f_{3}(x)-F(x)}>\delta/2$.
By Lemma <ref>,
\[
\frac{1}{T}\int_{0}^{T}(f_{3}(u(t;\tau)-u(t+\tau))^{2}\,dt=\frac{1}{T}\int_{0}^{T}(f_{3}(u(t;\tau)-F(u(t;\tau))^{2}\,dt\geq\frac{\delta^{2}}{4}\mu\left(\cup_{j\in J}A_{j}\right)(1-\epsilon)
\]
provided $u(0;\tau)\in S_{\epsilon,T^{\ast}}$. Applying Lemma <ref>,
we get
\[
\frac{\delta^{2}}{4}\mu\left(\bigcup_{j\in J}A_{j}\right)(1-\epsilon)\leq2\epsilon^{2},
\]
completing the proof of the lemma.
Suppose $u(0;\tau)\in S_{\epsilon,T^{\ast}}$. Suppose that $f_{1}$,
$f_{2}$, and $f_{3}$ minimize $\mathcal{W}_{1}(f)$, $\mathcal{W}_{2}(f)$,
and $\mathcal{W}_{3}(f)$, respectively, with $T\geq T^{\ast}$ and
$\Lambda=\epsilon^{2}/\norm{F_{\epsilon}}_{K}^{2}\leq1$. Then
\[
\mu\left\{ x\in X\biggl|\abs{f_{1}(x)-F(x)}>\delta+\frac{B_{1}h^{^{1/2}}+B_{1}\Delta^{1/2}}{\Lambda}\right\} <\frac{8\epsilon^{2}}{\delta^{2}(1-\epsilon)}
\]
with probability greater than $1-\pi(n,N,\Delta,u(0;\tau))$.
Follows from Lemmas <ref>, <ref>,
and <ref>.
The above lemma implies Theorem <ref>
with the choice of $\delta$, $n$, and $\Delta$ specified above
§ NUMERICAL ILLUSTRATIONS
We compare three methods to compute an approximate predictor $f$.
The first method is that of Muller et al <cit.>
given in (<ref>). The second method is exactly the
same but with the least squares regression function. The third method
is the convergent algorithm given by (<ref>) and (<ref>).
In the convergence theorem, the point predicted is $u(t+\tau)$ using
delay coordinates $u(t;\tau)$. In the numerical examples, we apply
the method more generally so that the point predicted is $u(t+t_{f})$
using delay coordinates $u(t;\tau)$.
When comparing the methods, we always used the same noisy data for
all three methods. There can be some fluctuation due to the instance
of noise that is added to the exact signal $\tilde{x}(t)$ as well
as the segment of signal that is used (choice of $u(0;\tau)$). The
effect of this fluctuation on comparison is eliminated by using the
same noisy data in each case. For all three methods, the error in
the approximate predictor is estimated by applying it to a noise-free
stretch of the signal as in <cit.>,
which is standard because the object of each method is to approximate
the exact predictor.
The first signal we use is the same as in <cit.>,
except for inevitable differences in instantiation. The Mackey-Glass
\[
\frac{d\tilde{x}(t)}{dt}=-0.1\tilde{x}(t)+\frac{0.2\tilde{x}(t-D)}{1+\tilde{x}(t-D)^{10}},
\]
with $D=17$, is solved with time step $\Delta t=0.1$ and transients
are eliminated to produce the exact signal $\tilde{x}(t)$. This signal
will of course have rounding errors and discretization errors, but
those are negligible compared to prediction errors. The standard deviation
of the Mackey-Glass signal is about $0.23$. An independent normally
distributed quantity of mean zero is added at each point so that the
ratio of the variance of the noise to that of the signal ($0.23^{2}$)
is equal to the desired signal-to-noise ratio (SNR).
To confirm with <cit.>,
the Mackey-Glass signal was down-sampled so that $nh=1$ and $n=1$.
The spline smoothing method would fare even better if we chose $nh=.1$.
The delay and the embedding dimension used for delay coordinates were
$\tau=6$ and $D=6$, as in <cit.>.
The size of the training set was $N=1000$. For cross-validation,
the $\gamma/2D$ parameter was varied over $\left\{ 0.1,1.5,10.0,50.0,100.0\right\} $,
the $\Lambda$ parameter was varied over $\left\{ 10^{-10},10^{-6},10^{-2},10^{2}\right\} $,
and the $\epsilon$ was varied over $\left\{ 0.01,0.05,0.25\right\} $.
The phenomenon we will demonstrate is far more pronounced than the
slight gains obtained using more extensive cross-validation. For support
vector regression, we were able to reproduce the relevant results
reported in <cit.>.[The RMS error of $0.017$ reported for $t_{f}=1$ with SNR of 22.15%
in <cit.> appears
to be a consequence of an unusually favorable noise or signal. The
typical RMS error is around $0.03$.]
Root mean square errors in the prediction of the Mackey-Glass signal
with $t_{f}=1$ as a function of the signal to noise ratio. The superiority
of the method using smooth splines is evident.
Figure <ref> demonstrates that (<ref>)
produces predictors that are corrupted by errors in the inputs or
delay coordinates. The method with spline smoothing is more accurate
and deteriorates less with increasing SNR.
Comparison of the $1$-step least squares predictor (without spline
smoothing) iterated $t_{f}$ times with the $t_{f}$-step predictor
(without spline smoothing). The latter is seen to be superior.
A $t=t_{f}$ predictor can be obtained by iterating a $1$-step predictor
$t_{f}$ times, and this strategy is sometimes used to save cost <cit.>.
This is not a good idea as explained in <cit.>.
An optimal predictor would need to roughly split the distance to nearest
training sample such that the component of the distance along unstable
directions is small and with the component along stable directions
allowed to be much larger. The balance between the two components
depends upon how large $t_{f}$ is.
The plot on the left uses SNR of $0.2$ and the plot on the right
uses $0.4$. The method using smooth splines does better in all instances.
In Figure <ref>, we see that spline smoothing
becomes more and more advantageous as noise increases. The situation
in Figure <ref> is a little different. When $t_{f}$
is small, spline smoothing does help more for the noisier SNR of $0.4$
compared to $0.2$. However, for larger $t_{f}$, even though spline
smoothing helps, it does not help more when the noise is higher. This
could be because as $t_{f}$ increases capturing the correct geometry
of the predictor becomes more and more difficult, and this difficulty
may be constraining the accuracy of the predictor.
The McKey-Glass example is a delay-differential equation and does
not come under the purview of our convergence theorem. The Lorenz
example, $\dot{x}=10(y-x),\:\dot{y}=28x-y-xz,\:\dot{z}=-8z/3+xy$,
is a dynamical system with a compact invariant set and comes under
the purview of the convergence theorem. The Lorenz signal has a standard
deviation of $7.9$.
The advantage of spline smoothing for Lorenz is much less on the right
with $h=0.1$ than on the left with $h=0.01$.
Figure <ref> compares $h=.01$ and $h=.1$
for Lorenz. In both cases, the embedding dimension is $d=10$, the
delay parameter is $\tau=1$, and the lookahead is $t_{f}=h$. It
may be seen that spline smoothing is less effective when $h=0.1$
as compared to $h=0.01$. A typical Lorenz oscillation has a period
of about $0.75$, and when $h=0.1$ the resolution is too low causing
too much discretization error. Smooth splines are less effective in
reconstructing the noise-free signal if the grid on the time axis
does not have sufficient resolution. The left half of Figure <ref>
shows an example where prediction using spline smoothing improves
accuracy by a factor of $100$. In that case, $h=0.01$.
§ DISCUSSION
In many practical prediction problems, the signals are not necessarily
stationary. Algorithms for non-stationary prediction, such as the
Henderson predictor or the LOESS predictor, can be cast in an RKHS
framework <cit.>. However, recurrences
which are the basis of the convergence theorem in our setting do not
exist and cannot be assumed.
The setting of this paper is tied to prediction of dynamical time
series. Parameter estimation of dynamical time series is a related
problem <cit.>. As in the
case of prediction, much of the analysis in parameter estimation is
for maps as opposed to flow and with hyperbolicity assumptions that
imply rapid mixing. If the dynamical time series is noisy and generated
by a flow, spline smoothing may prove advantageous in the context
of parameter estimation as well.
The convergence theorem proved here is not uniform. It is possible
that a convergence theorem with uniform estimates can be proved with
hyperbolicity assumptions that ensure rapid mixing <cit.>.
If convergence is proved to be uniform, the question of optimality
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1511.00585
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The relativistic theory of the Dirac fermions moving on cylinders in external Aharonov-Bohm field is built starting with a suitably restricted Dirac equation whose spin degrees of freedom are not affected. The exact solutions of this equation on finite or infinite Aharonov-Bohm cylinders allow one to derive the relativistic circular and longitudinal currents pointing out their principal features. It is shown that all the circular currents are related to the energy in the same manner on cylinders or rings either in the relativistic approach or in the non-relativistic one. The specific relativistic effect is the saturation of the circular currents for high values of the total angular momentum. Based on this property some approximative closed formulas are deduced for the total persistent current at $T=0$ on finite Aharonov-Bohm cylinders. Moreover, it is shown that all the persistent currents on finite cylinders or rings have similar non-relativistic limits.
Keywords: Dirac equation; Aharonov-Bohm cylinder; saturation effect; persistent current.
§ INTRODUCTION
The non-relativistic quantum mechanics based on the Schrödinger equation is the principal framework for studying the electronic effects in mesoscopic systems <cit.>-<cit.> were the spin-orbit interaction is described by additional terms <cit.>-<cit.>.
However, there are nano-systems, as for example the graphenes, where several relativistic effects observed in the electronic transport can be satisfactory explained considering the electrons as massless Dirac particles moving on honeycomb lattices <cit.>-<cit.>.
For this reason, many studies <cit.>-<cit.> focus on the relativistic effects considering the electrons near the Fermi surface as being described by the $(1+2)$-dimensional Dirac equation corresponding to a restricted three-dimensional Clifford algebra. This method restricts simultaneously not only the orbital degrees of freedom but the spin ones too, reducing them to those of the $SO(1,2)$ symmetry and affecting thus the spin-orbit coupling. Starting with this remark, we have shown that there are situations when it is convenient to use the complete $(1+3)$-dimensional Dirac equation, restricting the orbital motion, according to the concrete geometry of the studied system, but without affecting the natural spin degrees of freedom given by the $SL(2,{\Bbb C})$ symmetry. Thus the spin-orbit coupling remains unharmed describing the orbital and spin effects with more accuracy.
A special problem that was treated only occasionally with the complete Dirac equation is that of the fermions in external Aharonov-Bohm (AB) field <cit.> - <cit.>. The persistent currents in AB rings were recently studied starting with a version of the Dirac equation in which the orbital restrictions induce an unwanted non-Hermitian term <cit.> that distorts the results, even though these may outline new realistic effects <cit.>. For this reason, we proposed a new approach based on the correctly restricted Dirac equation, involving only Hermitian terms, that can be obtained easily starting with a suitable restricted Lagrangian theory <cit.>. We derived thus the expression of the relativistic currents in AB rings pointing out the saturation effect and deducing the form of the corresponding persistent currents at $T=0$ <cit.>.
In the present paper we would like to extend this study to the relativistic ideal AB cylinders considered as geometric manifolds without internal structure. We derive the fundamental solutions of the Dirac equation on AB cylinders as common eigenspinors of a complete systems of commuting operators including the energy, total angular momentum and a specific operator analogous to the well-known Dirac spherical operator of the relativistic central problems <cit.>. Appropriate boundary conditions define different solutions, on infinite or finite AB cylinders, that can be normalized with respect to the relativistic scalar product. We obtain thus the systems of normalized fundamental solutions that allow us to write down the exact expressions of the relativistic circular and longitudinal currents and to derive the persistent circular currents on finite AB cylinders at $T=0$.
First we find that the circular currents on finite or infinite AB cylinders remain related to the derivative of the relativistic energies with respect to the flux parameter just as in the cases of the relativistic or even non-relativistic AB rings <cit.>. However, the principal relativistic effect is the saturation of the circular currents on finite or infinite AB cylinders in the limit of very high total angular momenta. It is remarkable that this effect is the same on AB cylinders or rings, the saturation value depending only on the radius of their transverse sections <cit.>.
The paper is organized as follows. In the second section we present the relativistic theory of the fermions in AB cylinders based on a suitable restriction of the complete Dirac equation. The fundamental solutions on infinite AB cylinders are derived in the third section pointing out the saturation effect. Ne next section is devoted to the solutions on finite AB cylinders determined by boundary conditions of MIT type that eliminate the longitudinal current but preserving the saturation properties of the circular one. In the fifth section we derive the persistent circular currents at $T=0$ on finite AB cylinders discussing the case of very short ones and the non-relativistic limit. Finally we briefly present our conclusions.
§ DIRAC FERMIONS ON AB CYLINDERS
We consider the motion of a Dirac fermion of mass $M$ on an ideal cylinder of radius $R$ whose axis is oriented along the homogeneous and static external magnetic field $\vec{B}$ given by the electromagnetic potentials $A_0=0$ and $\vec{A}=\frac{1}{2}\vec{B}\land \vec{x}$. This background is a two-dimensional manifold (without internal structure) embedded in the three-dimensional space obeying the simple equation $r=R$ in cylindrical coordinates $(t,\vec{x}) \to (t, r, \phi, z)$ with the $z$ axis oriented along $\vec{B}$.
Then, it is natural to assume that any field $\psi$ defined on this manifold depends only on the remaining coordinates $(t,\phi,z)$ such that we can put $\partial_r\psi=0$ in the kinetic term of the Lagrangian density. Thus we obtain the action of the Dirac fermion in the mentioned external magnetic field
\begin{equation}
{\cal S}={\cal S}_0-\beta\int dt\, d\phi\, dz\,\overline{\psi}\gamma^{\phi}\psi
\end{equation}
having the kinetic part
\begin{eqnarray}
{\cal S}_0&=&\int dt\, d\phi \,dz\,\left\{\frac{i}{2}\left[\overline{\psi}(\gamma^0\partial_t\psi +\gamma^{\phi}\partial_{\phi}\psi+\gamma^3\partial_z\psi)\right.\right.\nonumber\\ &&-\left.\left.(\partial_t\overline{\psi}\gamma^0+\partial_{\phi}\overline{\psi}\gamma^{\phi}+\partial_z\overline{\psi}\gamma^3)\psi \right]-M\overline{\psi}\psi\right\}\,,
\end{eqnarray}
where $\overline{\psi}=\psi^{\dagger}\gamma^0$ and
\begin{equation}\label{gamph}
\gamma^{\phi}=\frac{1}{R}(-\gamma^1\sin\phi+\gamma^2\cos\phi)\,.
\end{equation}
The notation $\beta=\frac{1}{2}eBR^2$ stands for the usual dimensionless flux parameter (in natural units).
From this action we obtain the correctly restricted Dirac equation, $E_D\psi=M\psi$, with the self-adjoint Dirac operator
\begin{equation}\label{D2}
E_D=i\gamma^0\partial_t +\gamma^{\phi}(i\partial_{\phi}-\beta)+\frac{i}{2}\,\partial_{\phi}(\gamma^{\phi})+i\gamma^3\partial_z\,,
\end{equation}
whose supplemental third term guarantees that $\overline{E}_D=E_D$.
This operator commutes with the energy operator $H=i\partial_t$, the momentum along to the third axis, $P_3=-i\partial_z$, and the similar component, $J_3=L_3+S_3$, of the total angular momentum, formed by the orbital part $L_3=-i\partial_{\phi}$ the spin one $S_3=\frac{1}{2}\,{\rm diag} (\sigma_3,\sigma_3)$.
Under such circumstances, we have the opportunity to look for particular solutions of the form
\begin{equation}\label{psi}
\psi_{E,\lambda}(t, \phi,z)=N\left(
\begin{array}{c}
\end{array}\right) e^{-iEt}
\end{equation}
which satisfy the eigenvalue problems,
\begin{equation}
H\psi_{E,\lambda}(t, \phi,z)=E\psi_{E,\lambda}(t, \phi,z)\,, \quad
J_3\psi_{E,\lambda}(t, \phi,z)=\lambda\psi_{E,\lambda}(t, \phi,z)\,,
\end{equation}
laying out the energy $E$ and the quantum number $\lambda=\pm\frac{1}{2},\pm \frac{3}{2},...$ of the total angular momentum.
The normalization constant $N$ has to be determined after we solve the functions of $z$ from the remaining reduced equation that in the standard representation of the gamma matrices (with diagonal $\gamma^0$) reads
\begin{equation}
\left(
\begin{array}{cccc}
E-M &0&i\partial_z &\frac{i}{R}(\lambda+\beta)\\
0&E-M &-\frac{i}{R}(\lambda+\beta)&-i\partial_z\\
\frac{i}{R}(\lambda+\beta)&i\partial_z&0&-E-M
\end{array}\right)\,\left(
\begin{array}{c}
\end{array}\right) =0\,.
\end{equation}
This is in fact a system of linear differential equations allowing us to solve the functions of $z$.
The general solutions of this system can be obtained reducing the number of functions with the help of the last two equations that yield
\begin{equation}\label{g12}
\left(
\begin{array}{c}
\end{array}\right)=\frac{1}{E+M}\left(
\begin{array}{cc}
-i\partial_z& -\frac{i}{R}(\lambda+\beta)\\
\frac{i}{R}(\lambda+\beta)&i\partial_z
\end{array}\right)\left(
\begin{array}{c}
\end{array}\right)
\end{equation}
leading to the second order equations
\begin{equation}\label{ene}
\left(E^2-M^2-\frac{1}{R^2}(\lambda+\beta)^2 +\partial_z^2\right)f_{1,2}(z)=0\,.
\end{equation}
Consequently, the solutions must be linear combinations of the form
\begin{equation}
f_{1,2}(z)=c_{1,2} e^{ikz}+c'_{1,2}e^{-ikz}
\end{equation}
where $k$ is the fermion momentum along the $z$ axis. The concrete form of these solutions depends on the boundary conditions we chose for determining the integration constants $c_{1,2}$ and $c'_{1,2}$ up to a normalization factor, $N$. This last constant has to be determined by imposing the desired normalization condition with respect to the relativistic scalar product
\begin{equation}
\langle \psi, \psi'\rangle=R\int_{0}^{2\pi}d\phi\int_{D_z}dz \psi^{\dagger}(t,\phi,z) \psi'(t,\phi,z)\,,
\end{equation}
calculated on the domain $D_z$ of the entire cylinder.
§ CURRENTS ON INFINITE AB CYLINDERS
The simplest case is of the infinite cylinder, with $D_z={\Bbb R}$, where the motion along its axis is a free one. We assume that the spin projections are measured just with respect to this axis such that we may chose
\begin{equation}
\left(
\begin{array}{c}
\end{array}\right)=Ne^{ikz} \xi_{\sigma}\,, \quad k\in {\Bbb R}\,,
\end{equation}
where $\xi_{\sigma}$ are the usual Pauli spinors,
\begin{equation}
\xi_{\frac{1}{2}}=\left(
\begin{array}{c}
\end{array}\right)\,,\quad \xi_{-\frac{1}{2}}=\left(
\begin{array}{c}
\end{array}\right)\,.
\end{equation}
of polarizations $\sigma=\pm\frac{1}{2}$. Then the functions $g_{1,2}(z)$ can be derived from Eq. (<ref>) as
\begin{equation}
\left(
\begin{array}{c}
\end{array}\right)=\frac{ Ne^{ikz}}{E+M}\left(
\begin{array}{cc}
k& -\frac{i}{R}(\lambda+\beta)\\
\frac{i}{R}(\lambda+\beta)&- k
\end{array}\right) \xi_{\sigma}\,,
\end{equation}
while the energy that depends on $k$ and $\lambda$ reads
\begin{equation}\label{Ekm}
E_{k,\lambda}=\left[ M^2+k^2+\frac{1}{R^2}\left(\lambda+\beta \right)^2 \right]^{\frac{1}{2}}
\end{equation}
as it results from Eq. (<ref>). We obtain thus a mixed energy spectrum whose ground level is given by $k=0$ and one of the values $\lambda=\pm\frac{1}{2}$ that minimizes the last term in Eq. (<ref>), e. g. $\lambda=-\frac{1}{2}$ if $\beta>0$. Consequently, the spinor components have to be tempered distributions that may be normalized in the momentum scale.
According to the above results we can write two types of fundamental solutions (<ref>) of the form
\begin{equation}
U_{k,\lambda}^{\pm}(t,\phi,z)=u^{\pm}_{k,\lambda}(\phi)\frac{1}{\sqrt{2\pi}}e^{-iE_{k,\lambda}t +ikz}\,,
\end{equation}
corresponding to the polarizations $\sigma =\pm\frac{1}{2}$. For $\sigma =\frac{1}{2}$ we obtain
\begin{equation}\label{psiUp}
\begin{array}{c}
\frac{k}{E_{k,\lambda}+M}e^{i\phi(\lambda-\frac{1}{2})}\\
\frac{i(\lambda+\beta)}{R(E_{k,\lambda}+M)}e^{i\phi(\lambda+\frac{1}{2})}
\end{array}\right)
\end{equation}
assuming that the normalization factor depend on $k$ and $\lambda$.
Similarly, for $\sigma=-\frac{1}{2}$ we deduce
\begin{equation}\label{psiUm}
u_{k,\lambda}^-( \phi)=N_{k,\lambda}^-\left(
\begin{array}{c}
\frac{-i(\lambda+\beta)}{R(E_{k,\lambda}+M)}e^{i\phi(\lambda-\frac{1}{2})}\\
\frac{-k}{E_{k,\lambda}+M}e^{i\phi(\lambda+\frac{1}{2})}
\end{array}\right)
\end{equation}
It remains to calculate the normalization in the momentum scale finding that by fixing the values
\begin{equation}
N_{k,\lambda}^{\pm}=\frac{1}{\sqrt{2\pi R}}\sqrt{\frac{E_{k,\lambda}+M}{2E_{k,\lambda}}}\,,
\end{equation}
we obtain the desired (generalized) orthogonality relations
\begin{equation}
\langle U^{\pm}_{k,\lambda}, U^{\pm}_{k',\lambda'}\rangle=\delta_{\lambda,\lambda'}\delta(k-k')\,,\quad \langle U^{\pm}_{k,\lambda}, U^{\mp}_{k',\lambda'}\rangle=0\,.
\end{equation}
These fundamental solutions are eigenspinors of the same set of commuting operators which seems to be incomplete as long as we cannot distinguish between $U^+$ and $U^-$. Therefore we need to introduce a new operator for completing this set. A short inspection suggests that this must be an analogous of the Dirac spheric operator <cit.> that reads now $K=\gamma^0(2 S_3 L_3+\frac{1}{2})$ giving the eigenvalues problems $K U_{k,\lambda}^{\pm} =\pm\lambda U_{k,\lambda}^{\pm}$. The conclusion is that the fundamental solutions we derived above are eigenspinors of the complete set of commuting operators $\{H,K,J_3, P_3\}$.
The spinors describing physical states are square integrable packets of a given total angular momentum $\lambda$ having the form
\begin{equation}\label{psilam}
\psi_{\lambda}=\int_{-\infty}^{\infty}dk \left[a_+(k) U^+_{k,\lambda}+a_-(k) U^-_{k,\lambda}\right]
\end{equation}
where the functions $a_{\pm}$ satisfy the condition
\begin{equation}\label{Calpha}
\int_{-\infty}^{\infty}dk \left[|a_+(k)|^2+|a_-(k)|^2\right]=1
\end{equation}
that assures the normalization condition $\langle \psi_{\lambda}, \psi_{\lambda}\rangle=1$. We say that the packets (<ref>) describe the states $(\lambda,a)$ in which the expectation value of the total angular momentum reads $\langle \psi_{\lambda}, J_3 \psi_{\lambda}\rangle= \lambda$ while the polarization degree can be defined as
\begin{equation}
{\cal P}=\langle \psi_{\lambda}, K\psi_{\lambda}\rangle=\lambda\int_{-\infty}^{\infty}dk \left[|a_+(k)|^2-|a_-(k)|^2\right]\,.
\end{equation}
Now we can derive the currents of the fermions in the states $(\lambda,a)$ by using the components of the current density $j_{\lambda}^{\mu}=\overline{\psi}_{\lambda}\gamma^{\mu}\psi_{\lambda}$. We consider first the total circular current
\begin{equation}
I^c_{\lambda}=R\int_{-\infty}^{\infty} dz \overline{\psi}_{\lambda}\gamma^{\phi}\psi_{\lambda}\,,
\end{equation}
that can be calculated according to Eqs. (<ref>), (<ref>) and (<ref>), that yield
\begin{equation}
\overline{u}^{\pm}_{\lambda}\gamma^{\phi}{u}^{\pm}_{\lambda}=\frac{\lambda+\beta}{2\pi R^3 E_{k,\lambda}}\,, \quad \overline{u}^{\pm}_{\lambda}\gamma^{\phi}{u}^{\mp}_{\lambda}=0\,.
\end{equation}
Then, observing that the integral over the $z$ axis generates a $\delta$-function, we obtain the definitive closed form
\begin{equation}
I_{\lambda}^c=\frac{\lambda+\beta}{2\pi R^2}\int_{-\infty}^{\infty}\frac{dk}{E_{k,\lambda}} \left[|a_+(k)|^2+|a_-(k)|^2\right]\,.
\end{equation}
Hereby we draw the conclusion that the circular current is stationary (i. e. independent on $t$)
depending only on the packet content as given by the arbitrary functions $a_{\pm}$.
Nevertheless, it is remarkable that there are two important properties of the circular currents that are independent on the form of these functions. The first one is the saturation effect for increasing total angular momenta,
\begin{equation}
\lim_{\lambda \to \pm \infty}I^c_{\lambda}=\pm \frac{1}{2\pi R}\int_{-\infty}^{\infty}dk \left[|a_+(k)|^2+|a_-(k)|^2\right] =\pm \frac{1}{2\pi R}\,.
\end{equation}
On the other hand, bearing in mind that the expectation value of the energy in the state $(\lambda,a)$ reads
\begin{equation}
E_{\lambda}=\int_{-\infty}^{\infty}{dk}{E_{k,\lambda}} \left[|a_+(k)|^2+|a_-(k)|^2\right]\,,
\end{equation}
we recover the familiar formula
\begin{equation}
I^c_{\lambda}=\frac{1}{2\pi}\frac{\partial E_{\lambda}}{\partial\beta}\,,
\end{equation}
that has the same form as in the case of the relativistic <cit.> or non-relativistic AB rings.
Note that, in contrast to the circular current, the longitudinal one,
\begin{equation}
I^3_{\lambda}=R\int_{0}^{2\pi} d\phi\, \overline{\psi}_{\lambda}\gamma^{3}\psi_{\lambda}\,,
\end{equation}
depends on time, reflecting thus the propagation and dispersion of the packet along the $z$ axis. The general expression of this current is presented in the Appendix A.
§ CURRENTS ON FINITE AB CYLINDERS
Another interesting problem is of a finite cylinder of length $L$ for which we must consider the boundary conditions $f_{1,2}(0)=f_{1,2}(L)=0$. Therefore, we may chose
\begin{equation}\label{fkn1}
\left(
\begin{array}{c}
\end{array}\right)= N \sin( k_nz)\,\xi_{\sigma} \,,\quad k_n=\frac{\pi n}{L}\,,n=1,2,...\,,
\end{equation}
denoting now $E_{n,\lambda}=E_{k_n,\lambda}$. Thus we obtain the countable discrete energy spectrum
\begin{equation}\label{Enm}
E_{n,\lambda}=\left[ M^2+\frac{\pi^2 n^2}{L^2}+\frac{1}{R^2}\left(\lambda+\beta\right)^2 \right]^{\frac{1}{2}}\,,
\end{equation}
corresponding to the square integrable spinors whose components are given by Eq. (<ref>) and Eq. (<ref>) that yields now
\begin{equation}
\left(
\begin{array}{c}
\end{array}\right)=\frac{i N}{E+M}\left(
\begin{array}{cc}
- k_n\cos(k_n z)& -\frac{1}{R}(\lambda+\beta)\sin(k_n z)\\
\frac{1}{R}(\lambda+\beta)\sin(k_n z)& k_n\cos(k_n z)
\end{array}\right) \xi_{\sigma}\,.
\end{equation}
Then, according to Eq. (<ref>) we can write down the form of two types of solutions corresponding to $\sigma =\pm\frac{1}{2}$. For $\sigma =\frac{1}{2}$ we obtain
\begin{equation}\label{psiUp}
U_{n,\lambda}^+(t, \phi,z)=N_{n,\lambda}^+\left(
\begin{array}{r}
\sin(k_n z)e^{i\phi(\lambda-\frac{1}{2})}\\
\frac{-ik_n}{E_{n,\lambda}+M}\cos(k_n z)e^{i\phi(\lambda-\frac{1}{2})}\\
\frac{i(\lambda+\beta)}{R(E_{n,\lambda}+M)}\sin(k_n z)e^{i\phi(\lambda+\frac{1}{2})}
\end{array}\right) e^{-iE_{n,\lambda}t}\,,
\end{equation}
and similarly for $\sigma=-\frac{1}{2}$,
\begin{equation}\label{psiUm}
U_{n,\lambda}^-(t, \phi,z)=N_{n,\lambda}^-\left(
\begin{array}{r}
\sin(k_n z)e^{i\phi(\lambda+\frac{1}{2})}\\
\frac{-i(\lambda+\beta)}{R(E_{n,\lambda}+M)}\sin(k_n z)e^{i\phi(\lambda-\frac{1}{2})}\\
\frac{ik_n}{E_{n,\lambda}+M}\cos(k_n z)e^{i\phi(\lambda+\frac{1}{2})}
\end{array}\right) e^{-iE_{n,\lambda}t}\,.
\end{equation}
After a little calculation we find that by fixing the value of the normalization constants as
\begin{equation}
N_{n,\lambda}^{\pm}=\frac{1}{\sqrt{\pi RL}}\sqrt{\frac{E_{n,\lambda}+M}{2E_{n,\lambda}}}\,,
\end{equation}
we obtain the desired orthogonality relations
\begin{equation}
\langle U^{\pm}_{n,\lambda}, U^{\pm}_{n',\lambda'}\rangle=\delta_{n,n'}\delta_{\lambda,\lambda'}\,,\quad \langle U^{\pm}_{n,\lambda}, U^{\mp}_{n',\lambda'}\rangle=0\,.
\end{equation}
The conclusion is that the above fundamental solutions are eigenspinors of the set of commuting operators $\{H,K,J_3, P_3^2\}$.
Let us consider the fermions in the states $(n,\lambda)$ given by the normalized linear combinations
\begin{equation}\label{psicc}
\psi_{n,\lambda}=c_+ U_{n,\lambda}^+ +c_-U_{n,\lambda}^-\,, \quad |c_+|^2+|c_-|^2=1\,,
\end{equation}
which satisfy $\langle \psi_{n,\lambda}, \psi_{n',\lambda'}\rangle=\delta_{n,n'}\delta_{\lambda,\lambda'} $. The constants $c_{\pm}$ give the polarization degree defined as in the previous case,
\begin{equation}\label{pol}
{\cal P}=\langle \psi_{n,\lambda}, K \psi_{n,\lambda}\rangle=\lambda (|c_+|^2-|c_-|^2)\,.
\end{equation}
Obviously, the fermions are unpolarized when $|c_+|=|c_-|=\frac{1}{\sqrt{2}}$.
With these ingredients we can calculate the quantities
\begin{eqnarray}
\overline{U}^{\pm}_{n,\lambda}\gamma^{\phi}{U}^{\pm}_{n,\lambda}&=&\frac{\lambda+\beta}{2\pi R^3 L E_{n,\lambda}}\sin^2 k_n z\,,\\ \overline{U}^{\pm}_{n,\lambda}\gamma^{\phi}{U}^{\mp}_{n,\lambda}&=&\frac{2n}{RL^2E_{n,\lambda}}\sin k_n z\cos k_n z\,.\label{usles}
\end{eqnarray}
that help us to derive the definitive form of the circular currents in the states $(n,\lambda)$ that read
\begin{equation}\label{Inm}
I^c_{n,\lambda}=R\int_{0}^{L} dz\, \overline{\psi}_{n,\lambda}\gamma^{\phi}\psi_{n,\lambda}=\frac{\lambda+\beta}{2\pi R^2 E_{n,\lambda}}=\frac{1}{2\pi}\frac{\partial E_{n,\lambda}}{\partial\beta}\,,
\end{equation}
since the integral over $z$ vanishes the mixed terms (<ref>) while the constants $c_{\pm}$ satisfy Eq. (<ref>). It is remarkable that this current is independent on polarization and has a similar form and relation with the energy as in the case of the AB rings <cit.>. The difference is that now the energy (<ref>) depends on two quantum numbers, $n$ and $\lambda$, as well as on the length $L$ of the AB cylinder.
The longitudinal current vanishes since we used boundary conditions that guarantee that
\begin{equation}
\overline{U}^{\pm}_{n,\lambda}\gamma^{3}{U}^{\pm}_{n,\lambda}=\overline{U}^{\pm}_{n,\lambda}\gamma^{3}{U}^{\mp}_{n,\lambda}=0\,.
\end{equation}
In other words, our boundary conditions are of the MIT type vanishing the currents but without canceling all the components of the Dirac spinors on boundaries.
§ PERSISTENT CURRENTS ON FINITE AB CYLINDERS
The properties of the currents (<ref>) can be better understood by introducing the appropriate dimensionless parameters
\begin{equation}
\mu=MR\,,\quad \nu=\frac{\pi R}{L}\,,
\end{equation}
that allow us to write
\begin{equation}\label{Ichi}
I^c_{n,\lambda}=\frac{1}{2\pi R}\chi_{\mu,\nu}(n,\lambda)\,, \quad \chi_{\mu,\nu}(n,\lambda)=\frac{\beta+\lambda}{\sqrt{\mu^2+\nu^2 n^2+(\beta+\lambda)^2}}\,,
\end{equation}
pointing out the function $\chi$ which gives the behavior of the circular currents.
This function is smooths with respect to all of its variables increasing monotonously with $\lambda$ from $-1$ to $1$ since
\begin{equation}
\lim_{\lambda\to \pm\infty}\chi_{\mu,\nu}(n,\lambda)=\pm 1\,,
\end{equation}
and vanishing for $n\to \infty$. This means that, as in previous case, for increasing total angular momenta, the circular current tends to the asymptotic saturation values $\pm (2\pi R)^{-1}$ just as it happens with the partial currents in AB rings <cit.>.
Now we can use these properties for estimating the persistent current at $T=0$ in semiconductor AB cylinders where the electron discrete energy levels $E_{n,\lambda}$ are given by Eq (<ref>). According to the Fermi-Dirac statistics, at $T=0$
the electrons occupy all the states $(n,\lambda)$ which satisfy the condition
\begin{equation}\label{Cond0}
E_{n,\lambda}\le E_F+M
\end{equation}
where $E_F\ll M$ is the (non-relativistic) energy of the Fermi level. Therefore, the total number of electrons $N_e$ and the persistent current $I$ can be calculated as
\begin{eqnarray}
N_e&=&\sum_{n,\lambda; E_{n,\lambda}\le E_F+M} 1=\sum_{n,\lambda>0; E_{n,\lambda}\le E_F+M} 2\,,\label{Ne}\\
I&=&\sum_{n,\lambda; E_{n,\lambda}\le E_F+M}I^c_{n,\lambda}=\sum_{n,\lambda>0; E_{n,\lambda}\le E_F+M} (I^c_{n,\lambda}+I^c_{n,-\lambda})\,.
\end{eqnarray}
In practice the flux parameter $\beta$ remains very small (less than $10^{-8}$) such that we can neglect the terms of the order $O(\beta^2)$ of the Taylor expansions of our functions (<ref>). Thus we can write
\begin{eqnarray}
2\pi R(I^c_{n,\lambda}+I^c_{n,-\lambda})&=&\chi_{\mu,\nu}(n,\lambda)+\chi_{\mu,\nu}(n,-\lambda)\nonumber\\
&=&2j_{\mu,\nu}(n,\lambda) \beta +O(\beta^3)\,,
\end{eqnarray}
\begin{equation}
j_{\mu,\nu}(n,\lambda)=\frac{\mu^2+\nu^2 n^2}{(\mu^2+\nu^2 n^2+\lambda^2)^{\frac{3}{2}}}\,,
\end{equation}
obtaining thus the expression of the relativistic persistent currents,
\begin{equation}\label{Ic}
I=\frac{\beta}{\pi R}\,c(\mu,\nu) \,, \quad c(\mu,\nu)=\sum_{n,\lambda>0;
E_{n,\lambda}\le E_F+M}j_{\mu,\nu}(n,\lambda)\,.
\end{equation}
The principal problem in evaluating such sums is the computation of the contributing states $(n,\lambda)$ (with $n=1,2,...$ and $\lambda=\pm\frac{1}{2},\pm\frac{3}{2},...$) which satisfy the condition (<ref>). We denote first by $n_F$ the greatest value of the quantum number $n$ and by $\lambda_n$ the greatest value of $|\lambda|$ for a given $n$, assuming that the states $(n_F,\pm\frac{1}{2})$ and respectively $(n,\pm\lambda_n)$ (with $n=1,2,...,n_F) $ are very close to the Fermi level, i. e. $E_{n_F,\pm\frac{1}{2}}\simeq E_{n,\pm\lambda_n}\simeq E_F+M$. In addition, we denote by $\lambda_F=\lambda_{n=1}$ the greatest value among the quantities $\lambda_n$. Then we can rewrite Eq. (<ref>) as
\begin{equation}\label{Cond}
\nu^2 n^2+\lambda^2\le \alpha^2\,, \quad \alpha=R\sqrt{E_F(E_F+2M)}\simeq R\sqrt{2M E_F}\,,
\end{equation}
obtaining the approximative identities
\begin{equation}\label{ident}
\nu^2 {n_F}^2+\frac{1}{4}\simeq \nu^2 n^2 + {\lambda_n}^2\simeq \nu^2+{\lambda_F}^2\simeq \alpha^2
\end{equation}
that help us to estimate the numbers $n_F$ and $\lambda_n$. Moreover, we observe that $\lambda_F$ must be much greater than $1$ since otherwise we cannot speak about statistics. Then we can use the approximative formula (<ref>) obtaining the compact results
\begin{eqnarray}
\sum_{n=1}^{n_F}(2\lambda_n+1)={n_F}+2\sum_{n=1}^{n_F} \lambda_n\,, \\ c(\mu,\nu)&=&\sum_{n=1}^{n_F}\sum_{\lambda=\frac{1}{2}}^{\lambda_n}j_{\mu,\nu}( n,\lambda)\simeq\frac{1}{\sqrt{\mu^2+\alpha^2}}\sum_{n=1}^{n_F} \lambda_n\,,
\end{eqnarray}
that represent a very good approximation for the systems with $\mu>200$ <cit.>. It remains to calculate on computer the sum over $n$ or to consider the estimation (<ref>) when $n_F\gg 1$.
An interesting case is of the very short cylinders with $1\ll\nu <\alpha<2\nu$ whose quantum number $n$ can take the unique value $n=n_F=1$ in order to satisfy Eq. (<ref>) that becomes now $\lambda^2\le \alpha^2-\nu^2=\lambda_F^2$. Consequently, the allowed states are $(1,\pm\frac{1}{2}), (1,\pm\frac{3}{2}),...(1,\pm\lambda_F)$ which means that $N_e=2 \lambda_F+1$ and
\begin{equation}\label{Ishort}
I_{short}\simeq \frac{\beta}{\pi R}\frac{\lambda_F}{\sqrt{\mu^2+\alpha^2}}= \frac{\beta}{\pi R}\sqrt{\frac{\alpha^2-\nu^2}{\alpha^2+\mu^2}}\,.
\end{equation}
However, for the very short cylinders with $\nu>\alpha$ the identities (<ref>) do not make sense such that we need to rebuild the entire theory without motion along the $z$ axis ($k=0$), retrieving thus the case of the ideal AB rings <cit.> for which we must substitute $\nu=0$ and $\alpha=\lambda_F$ in Eq. (<ref>). Finally, we note that for the non-relativistic short AB cylinders with $\alpha\ll \mu$ we recover the well-known result,
\begin{equation}
I_{nr}\simeq\frac{\beta}{\pi R}\frac{\lambda_F}{\mu}\simeq\frac{\beta}{\pi R}\frac{N_e}{2\mu}\,,
\end{equation}
of the non-relativistic persistent current in AB rings.
§ CONCLUDING REMARKS
We presented the relativistic theory of the Dirac fermions on AB cylinders based on the complete $(1+3)$-dimensional Dirac equation with restricted orbital degrees of freedom but without affecting the spin ones. This can be achieved by using the method we proposed recently
for the AB rings <cit.> according to which the orbital restrictions must be imposes on the Lagrangian density giving rise to a correct self-adjoint Dirac operator.
The results obtained here point out two principal features of the circular currents on the AB cylinders. The first one is the relation between the circular current and the derivative of the energy with respect to the flux parameter that is the same for AB rings or cylinders either in the relativistic approach or in the non-relativistic one. In other words this property is universal for all the AB systems with cylindric symmetry. The second feature is specific only for the relativistic circular currents that tend to saturation in the limit of very large total angular momenta, in contrast with the non-relativistic ones that are increasing linearly to infinity.
The saturation effect determines a specific form of the relativistic persistent current on finite AB cylinders that can be seen as a generalization of the persistent current in AB rings. Obviously, the dependence on parameters is more complicated in the case of the finite cylinders but these models are closer to the real devices involved in experiments. On the other hand, we must specify that the closed formulas derived in section 5 are only approximations that must be used prudently and completed by numerical calculations on computers.
We believe that only in this manner, by using combined analytical and numerical methods, we could step the threshold to the relativistic physics of the Aharonov-Bohm systems.
§.§ Appendix A: Longitudinal currents
On the infinite AB cylinders the longitudinal currents in the state $(\lambda,a)$ are determined by the structure of the wave packet (<ref>) as
\begin{eqnarray}
I^3_{\lambda}&=&R\int_{0}^{2\pi} d\phi\overline{\psi}_{\lambda}\gamma^{3}\psi_{\lambda}=\frac{1}{4\pi}\int_{-\infty}^{\infty}dkdk' \frac{e^{it(E_{k}-E_{k'})-iz(k-k')}}{\sqrt{E_{k}E_{k'}(E_{k}+M)(E_{k'}+M)}}\nonumber\\
&\times&\left\{\left[kE_{k'}+k'E_{k}+M(E_{k}+E_{k'})\right]\left[a_{+}^*(k)a_{+}(k')+a_{-}^*(k)a_{-}(k')\right]\right. \nonumber\\
&-&\left. \frac{i(\lambda+\beta)}{R}(E_{k}-E_{k'})\left[a_{+}^*(k)a_{-}(k')+a_{-}^*(k)a_{+}(k') \right] \right\}\,,
\end{eqnarray}
where the energies are given by Eq. (<ref>) and the arbitrary functions $a_{\pm}$ satisfy the condition (<ref>).
§.§ Appendix B: Approximating sums by integrals
Since $\lambda_F$ must be much greater than $1$ we can use the approximative formula
\begin{equation}\label{Bsum}
\sum_{\lambda=\frac{1}{2}}^{\lambda_n}j_{\mu,\nu}( n,\lambda)\simeq \int_{0}^{\lambda_n}d\lambda \,j_{\mu,\nu}( n,\lambda)=\frac{\lambda_n}{\sqrt{\mu^2+\nu^2 n^2+\lambda_n^2}}\simeq\frac{\lambda_n}{\sqrt{\mu^2+\alpha^2}}\,,
\end{equation}
that reproduces the numerical results with a satisfactory accuracy for $\mu>200$ <cit.>. Moreover, when $n_F >100$ we can evaluate
\begin{eqnarray}\label{Bsum1}
\sum_{n=1}^{n_F} \lambda_n&=& \sum_{n=1}^{n_F} \sqrt{\nu^2(n_F^2-n^2)+\frac{1}{4}}\nonumber\\
&\simeq& \int_{x=0}^{n_F}dx \sqrt{\nu^2(n_F^2-x^2)+\frac{1}{4}}\simeq \frac{1}{4}n_F \left(1+\frac{\pi n_F}{\nu}\right) \,.
\end{eqnarray}
§.§ Acknowledgments
I. I. Cotăescu is supported by a grant of the Romanian National Authority for Scientific Research, Programme for research-Space Technology and Advanced Research-STAR, project nr. 72/29.11.2013 between Romanian Space Agency and West University of Timisoara.
D.-M. Băltăţeanu is supported by the strategic grant POSDRU /159/1.5/S /137750, Project “Doctoral and Postdoctoral programs support for increased competitiveness in Exact Sciences research”, cofinanced by the European Social Fund within the Sectoral Operational Programme Human Resources Development 2007 - 2013.
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1511.00384
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This article provides a formalization of the W3C Draft Core SHACL Semantics specification using Z notation.
This formalization exercise has identified a number of quality issues in the draft.
It has also established that the recursive definitions in the draft are well-founded.
Further formal validation of the draft will require the use of an executable specification technology.
§ INTRODUCTION
The W3C RDF Data Shapes Working Group <cit.> is developing SHACL, a new language for describing constraints on RDF graphs.
A semantics for Core SHACL has been proposed <cit.>, hereafter referred to as the semantics draft.
The proposed semantics includes an abstract syntax, inference rules, and a definition of typing which allows for certain kinds of recursion.
The semantics draft uses precise mathematical language, but is informal in the sense that it is not written in a formal specification language
and therefore cannot benefit from tools such as type-checkers.
This document provides a formal translation of the semantics draft into Z Notation <cit.>.
The source for this article has been type-checked using the type-checker <cit.>
and is available in the GitHub repository <cit.>
Our motive for formalizing and type-checking the semantics draft is to help to improve its quality
and the ultimate design of SHACL.
§.§ Organization of this Article
The remainder of this article is organized as follows.
* Section <ref> formalizes some basic RDF concepts.
* Section <ref> translates the abstract syntax of SHACL into Z notation.
* Section <ref> formalizes the evaluation semantics of SHACL.
* Section <ref> formalizes the declarative semantics of shape expression schemas.
* Section <ref> summarizes the quality issues found in the draft.
* Section <ref> concludes with some remarks about the benefits of the formalization exercise and
possible next steps.
§ BASIC RDF CONCEPTS
This section formalizes some basic RDF concepts.
We reuse some formal definitions given in <cit.>, modifying the identifiers to match those used in the semantics draft.
§.§ $TERM$
Let $TERM$ be the set of all RDF terms.
§.§ $Iri$, $Blank$, and $Lit$
The set of all RDF terms is partitioned into IRIs, blank nodes, and literals.
Iri, Blank, Lit: TERM
⟨Iri, Blank, Lit ⟩TERM
§.§ $IRI$
The semantics draft introduces the term $Iri$, but it uses the term $IRI$ in the definitions of the abstract syntax.
We treat $IRI$ as a synonym for $Iri$.
IRI == Iri
§.§ $Triple$
An RDF triple is an ordered triple of RDF terms referred to as the subject, predicate, and object.
Triple == { s, p, o:TERM | s ∉Lit p ∈IRI }
* The subject is not a literal.
* The predicate is an IRI.
§.§ $subject$, $predicate$, and $object$
It is convenient to define generic functions that select the first, second, or third component of a Cartesian product of three sets.
fst[X,Y,Z] == (λx:X; y:Y; z:Z @ x )
snd[X,Y,Z] == (λx:X; y:Y; z:Z @ y )
trd[X,Y,Z] == (λx:X; y:Y; z:Z @ z )
The subject, predicate, and object of an RDF triple are the terms that appear in the corresponding positions.
subject == (λt:Triple @ fst(t) )
predicate == (λt:Triple @ snd(t) )
object == (λt:Triple @ trd(t) )
§.§ $Graph$
An RDF graph is a finite set of RDF triples.
Graph == Triple
§.§ $subjects$, $predicates$, and $objects$
The subjects, predicates, and objects of a graph are the sets of RDF terms that appear in the corresponding positions of its triples.
subjects == (λg: Graph @ { t:g @ subject(t) } )
predicates == (λg: Graph @ { t:g @ predicate(t) } )
objects == (λg: Graph @ { t:g @ object(t) } )
§.§ $nodes$
The nodes of an RDF are its subjects and objects.
nodes == (λg: Graph @ subjects(g) ∪objects(g) )
§.§ $PointedGraph$
A pointed graph is a graph and a distinguished node in the graph.
The distinguished node is variously referred to as the start, base, or focus node of the pointed graph, depending on the context.
PointedGraph == { g: Graph; n: TERM | n ∈nodes(g) }
§ ABSTRACT SYNTAX
This section contains a translation of the abstract syntax of SHACL into Z.
The semantics draft defines the abstract syntax using an informal Extended Backus-Naur Form (EBNF).
The approach used here is to interpret each term or expression that appears in the abstract syntax as
a mathematical set that is isomorphic to the set of abstract syntax tree fragments denoted by the corresponding term or expression.
Care has been taken to preserve the exact spelling and case of each abstract syntax term so that there is a direct correspondence
between the abstract syntax and Z.
For example, the term Schema is interpreted as the set $Schema$.
We give a Z definition for each abstract syntax term that appears on the left-hand side of the EBNF definition operator (::=).
The order in which these terms appear in the semantics draft has been preserved in this document.
If a Z term has a corresponding EBNF rule, we include it here for easy reference.
Refer to <cit.> for the complete definition of the abstract syntax.
A sequence of two or more abstract syntax terms is interpreted as the Cartesian product of the corresponding sets, i.e.
A B is interpreted as $A \cross B$.
The abstract syntax Kleene star (*) and plus (+) operators are interpreted as sequence ($\seq$)
and non-empty sequence ($\seq_1$) operators on the corresponding sets, i.e.
A+ is interpreted as $\seq_1 A$.
The abstract syntax optional operator (?) is interpreted as taking the union of the set of singletons and the
empty set of the corresponding set
using the generic function $OPTIONAL$ (defined below), i.e.
A? is interpreted as $OPTIONAL[A]$.
Abstract syntax terms that are defined as alternations (|) of two or more expressions are translated into either free types or unions
of sets.
A side effect of this process is that constructors may be required for each branch of the alternation.
In some cases the name of the constructors can be derived from a corresponding element of the abstract syntax.
For example, in ShapeDefinition, open and close are mapped to the constructors $open$ and $close$.
In the cases where there is no convenient element of the abstract syntax, we mint new constructor names.
We also introduce new Z identifiers when an element of the abstract syntax does not map to a valid alphanumeric Z identifier.
For example the the shape label negation operator (!) is mapped to $negate$.
§.§ $OPTIONAL$
An optional value is represented by a singleton set, if the value is present, or the empty set, if the value is absent.
OPTIONAL[X] == { v:X@{v} } ∪{ ∅}
§.§ $Schema$
A schema is a sequence of one or more rules.
Schema == _1 Rule
§.§ $Rule$
A rule consists of a shape label, a shape definition, and a sequence of zero or more extension conditions.
Rule == ShapeLabel ShapeDefinition ExtensionCondition
It is convenient to introduce functions that select the components of a rule.
shapeLabel == (λr:Rule @ fst(r) )
shapeDef == (λr:Rule @ snd(r) )
extConds == (λr:Rule @ trd(r) )
§.§ $ShapeLabel$
A shape label is an identifier drawn from some given set.
§.§ $ShapeDefinition$
A shape definition is either a closed shape or an open shape.
ShapeDefinition ::=
1 close ShapeExpr |
1 open OPTIONAL[InclPropSet] ShapeExpr
Note that abstract syntax terms that are defined using alternation are naturally represented as free types in Z Notation.
* $close$ is the constructor for closed shapes.A closed shape consists of a shape expression.
* $open$ is the constructor for open shapes. An open shape consists of an optional included properties set and a shape expression.
Given a shape definition $d$, let $shapeExpr(d)$ be its shape expression.
shapeExpr: ShapeDefinition ShapeExpr
∀x: ShapeExpr @
1 shapeExpr(close(x)) = x
∀o: OPTIONAL[InclPropSet]; x: ShapeExpr @
1 shapeExpr(open(o,x)) = x
§.§ $ClosedShape$
The set of closed shapes is the range of the $close$ shape definition constructor.
ClosedShape == close
§.§ $OpenShape$
The set of open shapes is the range of the $open$ shape definition constructor.
OpenShape == open
§.§ $InclPropSet$
An included properties set is a properties set.
InclPropSet == PropertiesSet
Note that there seems little motivation to introduce the term $InclPropSet$ since it is identical to $PropertiesSet$.
§.§ $PropertiesSet$
A properties set is a set of IRIs.
PropertiesSet == IRI
§.§ $ShapeExpr$
A shape expression defines constraints on RDF graphs.
ShapeExpr ::=
1 emptyshape |
1 triple DirectedTripleConstraint Cardinality |
1 someOf _1 ShapeExpr |
1 oneOf _1 ShapeExpr |
1 group _1 ShapeExpr |
1 repetition ShapeExpr Cardinality
* $emptyshape$ is the empty shape expression.
* $triple$ is the constructor for triple constraint shape expressions.
A triple constraint shape expression consists of a directed triple constraint and a cardinality.
* $someOf$ is the constructor for some-of shape expressions.
A some-of shape expression consists of a sequence of one or more shape expressions.
* $oneOf$ is the constructor for one-of shape expressions.
A one-of shape expression consists of a sequence of one or more shape expressions.
* $group$ is the constructor for grouping shape expressions.
A grouping shape expression consists of a sequence of one or more shape expressions.
* $repetition$ is the constructor for repetition shape expressions.
A repetition shape expression consists of a shape expression and a cardinality.
§.§ $EmptyShape$
The set of empty shape expressions is the singleton set that contains the empty shape.
EmptyShape == { emptyshape }
§.§ $DirectedPredicate$
A directed predicate is an IRI with a direction that indicates its usage in a triple.
$nop$ indicates the normal direction, namely the predicate relates the subject node to the object node.
$inv$ indicates the inverse direction, namely the predicate relates the object node to the subject node.
DirectedPredicate ::=
1 nop IRI |
1 inv IRI
The semantics draft uses the notation p for $inv(p)$.
Let $predDF(dp)$ denote the predicate of a directed predicate $dp$.
predDP : DirectedPredicate IRI
∀p: IRI @
1 predDP(nop(p)) = predDP(inv(p)) = p
§.§ $DirectedTripleConstraint$
A directed triple constraint consists of a directed predicate and a constraint.
The constraint is a value or shape constraint on the object of a triple if the direction is normal,
or a shape constraint on the subject of a triple if the direction is inverted.
DirectedTripleConstraint ==
1 { dp: DirectedPredicate; C: Constraint |
2 dp ∈inv C ∈ShapeConstr }
The semantics draft uses the notation p::C for $(nop(p),C)$ and p::C for $(inv(p),C)$.
Let $predDTC(dtc)$ denote the predicate of the directed triple constraint $dtc$.
predDTC: DirectedTripleConstraint IRI
∀dp: DirectedPredicate; C: Constraint |
1 (dp, C) ∈DirectedTripleConstraint @
2 predDTC(dp, C) = predDP(dp)
Let $constrDTC(dtc)$ denote the constraint of the directed triple constraint $dtc$.
constrDTC: DirectedTripleConstraint Constraint
∀dp: DirectedPredicate; C: Constraint |
1 (dp, C) ∈DirectedTripleConstraint @
2 constrDTC(dp, C) = C
§.§ $TripleConstraint$
A triple constraint places conditions on triples whose subject is a given focus node
and whose predicate is a given IRI.
TripleConstraint: DirectedTripleConstraint
TripleConstraint =
1 { p: IRI; C: Constraint @ (nop(p),C) }
§.§ $InverseTripleConstraint$
An inverse triple constraint places conditions on triples whose object is a given focus node
and whose predicate is a given IRI.
InverseTripleConstraint: DirectedTripleConstraint
InverseTripleConstraint =
1 { p: IRI; C: ShapeConstr @ (inv(p),C) }
§.§ $Constraint$
A constraint is a condition on the object node of a triple for normal predicates
or the subject node of a triple for inverse predicates.
Constraint ::=
1 valueSet (Lit ∪IRI) |
1 datatype LiteralDatatype OPTIONAL[XSFacet] |
1 kind NodeKind |
1 or _1 ShapeLabel |
1 and _1 ShapeLabel |
1 nor _1 ShapeLabel |
1 nand _1 ShapeLabel
* $valueSet$ is the constructor for value set value constraints.
A value set value constraint consists of a set of literals and IRIs.
* $datatype$ is the constructor for literal datatype value constraints.
A literal datatype value constraint consists of a literal datatype and an optional XML Schema facet.
* $kind$ is the constructor for node kind value constraints.
A node kind value constraint consists of a specification for a subset of RDF terms.
* $or$ is the constructor for disjunction shape constraints.
A node must satisfy at least one of the shapes.
* $and$ is the constructor for conjunction shape constraints.
A node must satisfy all of the shapes.
* $nor$ is the constructor for negated disjunction shape constraints.
A node must not satisfy any of the shapes.
* $nand$ is the constructor for negated conjunction shape constraints.
A node must not satisfy all of the shapes.
§.§ $Cardinality$
Cardinality defines a range for the number of elements in a set.
Cardinality == MinCardinality MaxCardinality
* A cardinality consists of a minimum cardinality and a maximum cardinality.
§.§ $MinCardinality$
Minimum cardinality is the minimum number of elements required to be in a set.
MinCardinality ==
§.§ $MaxCardinality$
Maximum cardinality is the maximum number of elements required to be in a set.
MaxCardinality ::= maxCard | unbound
* $maxCard$ is the constructor for finite maximum cardinalities.
A finite maximum cardinality is a natural number.
Note that a maximum cardinality of $0$ means that the set must be empty.
* $unbound$ indicates that the maximum number of elements in a set is unbounded.
§.§ $inBounds$
A natural number $k$ is said to be in bounds of a cardinality when $k$ is between the minimum and maximum
limits of the cardinality.
inBounds: Cardinality
∀k, n: @
1 k inBounds (n,unbound) n ≤k
∀k, n, m: @
1 k inBounds (n,maxCard(m)) n ≤k ≤m
§.§ Notation
Let $a$ be an IRI, let $C$ be a value or shape constraint, let $n$ and $m$ be non-negative integers.
The semantics draft uses the notation listed in Table <ref> for some shape expressions.
Notation Meaning
a::C[n;m] $triple(nop(a,C),(n,maxCard(m)))$
a::C[n;m] $triple(inv(a,C),(n,maxCard(m)))$
a::C a::C[1;1]
a::C a::C[1;1]
!a::C a::C[0;0]
!a:C a::C[0;0]
Meaning of shape expression notation
* If the cardinality is [1;1] it may be omitted.
* The negated shape expressions are semantically equivalent to the corresponding non-negated shape expressions with cardinality [0;0].
§.§ $none$, $one$
It is convenient to define some common cardinalities.
none == (0,maxCard(0))
one == (1,maxCard(1))
* A cardinality of $none =$ [0;0] is used to indicate a negated triple or inverse triple constraint.
* A cardinality of $one =$ [1;1] is the default cardinality of a triple or inverse triple constraint
when no cardinality is explicitly given in the notations a::C and a::C.
§.§ $NegatedTripleConstraint$
A negated triple constraint shape expression is a triple constraint shape expression that has a cardinality of $none$.
NegatedTripleConstraint ==
1 { tc: TripleConstraint @ triple(tc,none) }
§.§ $NegatedInverseTripleConstraint$
A negated inverse triple constraint shape expression is an inverse triple constraint shape expression that has a cardinality of $none$.
NegatedInverseTripleConstraint ==
1 { itc: InverseTripleConstraint @ triple(itc,none) }
§.§ $ValueConstr$
A value constraint places conditions on the object nodes of triples for normal predicates
and on the subject nodes of triples for inverse predicates.
ValueConstr == valueSet ∪datatype ∪kind
§.§ $ValueSet$
The set of value set value constraints is the range of the $valueSet$ constructor.
ValueSet == valueSet
§.§ $LiteralDatatype$
A literal datatype is an IRI that identifies a set of literal RDF terms.
We assume that this subset of IRIs is given.
LiteralDatatype: IRI
We also assume that we are given an interpretation of each literal datatype as a set of literals.
literalsOfDatatype: LiteralDatatype Lit
§.§ $NodeKind$
A node kind identifies a subset of RDF terms.
NodeKind ::= iri | blank | literal | nonliteral
* $iri$ identifies the set of IRIs.
* $blank$ identifies the set of blank nodes.
* $literal$ identifies the set of literals.
* $nonliteral$ identifies the complement of the set of literals, i.e. the union of IRIs and blank nodes.
Each node kind corresponds to a set of RDF terms.
termsOfKind: NodeKind TERM
termsOfKind(iri) = IRI
termsOfKind(blank) = Blank
termsOfKind(literal) = Lit
termsOfKind(nonliteral) = TERM ∖Lit
§.§ $XSFacet$
An XML Schema facet places restrictions on literals.
We assume this is a given set.
We also assume that we are given an interpretation of facets as sets of literals.
literalsOfFacet: LiteralDatatype XSFacet Lit
∀d: LiteralDatatype; f: XSFacet @
1 literalsOfFacet(d,f) ⊆literalsOfDatatype(d)
* The literals that correspond to a facet of a datatype are a subset of the literals that correspond to the datatype.
§.§ $ShapeConstr$
A shape constraint requires that a node satisfy logical combinations of one or more other shapes which are identified by their shape labels.
ShapeConstr == or ∪and ∪nor ∪nand
§.§ $DisjShapeConstr$
The set of all disjunctive shape constraints is the range of the $or$ constructor.
DisjShapeConstr == or
§.§ $ConjShapeConstraint$
The set of all conjunctive shape constraints is the range of the $and$ constructor.
ConjShapeConstraint == and
§.§ $SomeOfShape$
The set of some-of shape expressions is the range of $someOf$.
SomeOfShape == someOf
§.§ $OneOfShape$
The set of one-of shape expressions is the range of $oneOf$.
OneOfShape == oneOf
§.§ $GroupShape$
The set of grouping shape expressions is the range of $group$.
GroupShape == group
§.§ $RepetitionShape$
The set of repetition shape expressions is the range of $repetition$.
RepetitionShape == repetition
§.§ $ExtensionCondition$
An extension condition is the definition of a constraint written in an extension language
ExtensionCondition == ExtLangName ExtDefinition
§.§ $ExtLangName$
An extension language name is an identifier for an extension language, such as JavaScript.
We assume this is a given set.
§.§ $ExtDefinition$
An extension definition is a program written in some extension language that implements a constraint check.
We assume this is a given set.
An extension condition represents a function that takes as input a pointed graph, and returns
as output a boolean with the value true if the constraint is violated and false is satisfied.
We assume we are given a mapping that associates each extension condition with the set of pointed graphs that violate it.
violatedBy: ExtensionCondition PointedGraph
§.§ $ShapeLabel$ Definitions
Given a schema $S$, let $defs(S)$ be the set of all shape labels defined in $S$.
defs == (λS: Schema @
1 { r: S @ shapeLabel(r) } )
Each rule in a schema must be identified by a unique shape label.
SchemaUL == { S: Schema | # S = # (defs(S)) }
* In a schema with unique rule labels there are as many rules as labels.
§.§ $rule$
Given a schema $S$ with unique rule labels, and a label $T$ defined in $S$, let $rule(T,S)$ be the corresponding rule.
rule: ShapeLabel SchemaUL Rule
rule = { T: ShapeLabel; S: SchemaUL | T ∈defs(S) }
∀S: SchemaUL@
1 ∀r: (S) @
2 T == shapeLabel(r) @
3 rule(T,S) = r
§.§ $ShapeLabel$ References
Given a schema $S$, let $refs(S)$ be the set of shape labels referenced in $S$.
refs == (λS: Schema @ ⋃{ r: S @ refsRule(r) } )
* The set of references in a schema is the union of the sets of references in its rules.
Given a rule $r$, let $refsRule(r)$ be the set of shape labels referenced in $r$.
refsRule == (λr: Rule @ refsShapeDefinition(shapeDef(r)) )
* The set of references in a rule is the set of references in its shape definition.
Given a shape definition $d$, let $refsShapeDefinition(d)$ be the set of shape labels referenced in $d$.
refsShapeDefinition: ShapeDefinition ShapeLabel
∀d: ShapeDefinition @
1 refsShapeDefinition(d) = refsShapeExpr(shapeExpr(d))
* The set of references in a shape definition is the set of references in its shape expression.
Given a shape expression $x$, let $refsShapeExpr(x)$ be the set of shape labels referenced in $x$.
refsShapeExpr: ShapeExpr ShapeLabel
refsShapeExpr(emptyshape) = ∅∀dtc: DirectedTripleConstraint; c: Cardinality @
1 refsShapeExpr(triple(dtc,c)) =
2 refsDirectedTripleConstraint(dtc)
∀xs: _1 ShapeExpr @
1 refsShapeExpr(someOf(xs)) =
1 refsShapeExpr(oneOf(xs)) =
1 refsShapeExpr(group(xs)) =
2 ⋃{ x: xs @ refsShapeExpr(x) }
∀x: ShapeExpr; c: Cardinality @
1 refsShapeExpr(repetition(x,c)) =
2 refsShapeExpr(x)
* The empty shape expression references no labels.
* A directed triple constraint shape expression references the labels referenced in the directed triple constraint.
* A some-of or one-of or group shape expression references the union of the labels referenced in each component shape expression.
* A repetition shape expression references the labels referenced in its unrepeated shape expression.
Given a directed triple constraint $dtc$, let $refsDirectedTripleConstraint(dtc)$ be the set of shape labels referenced in $dtc$.
1 DirectedTripleConstraint ShapeLabel
∀a: IRI; C: ValueConstr @
1 refsDirectedTripleConstraint((nop(a),C)) = ∅∀a: IRI; C: ShapeConstr @
1 refsDirectedTripleConstraint((nop(a),C)) =
1 refsDirectedTripleConstraint((inv(a),C)) =
2 refsShapeConstr(C)
* A value triple constraint references no labels.
* A shape triple constraint references the labels in its shape constraint.
Given a shape constraint $C$, let $refsShapeConstr(C)$ be the set of shape labels referenced in $C$.
refsShapeConstr: ShapeConstr ShapeLabel
∀ls: _1 ShapeLabel @
1 refsShapeConstr(or(ls)) =
1 refsShapeConstr(and(ls)) =
1 refsShapeConstr(nor(ls)) =
1 refsShapeConstr(nand(ls)) =
2 ls
* A shape constraint references the range of its sequence of shape labels.
Every shape label referenced in a schema must be defined in the schema.
SchemaRD == { s:Schema | refs(s) ⊆defs(s) }
A schema is well-formed if its rules have unique labels and all referenced shape labels are defined.
SchemaWF == SchemaUL ∩SchemaRD
§ EVALUATION
This section defines the interpretation of shapes as constraints on RDF graphs.
All functions that are defined in the semantics draft are given formal definitions here.
We assume that from this point on whenever the semantics draft refers to schemas they are well-formed.
§.§ $shapes$
Given a well-formed schema $S$, let $shapes(S)$ be the set of shape labels that appear in $S$.
shapes == (λS: SchemaWF @ defs(S) )
§.§ $expr$
Given a shape label $T$ and a well-formed schema $S$, let $expr(T,S)$ be the shape expression in the rule with label $T$ in $S$.
expr: ShapeLabel SchemaWF ShapeExpr
expr = { T: ShapeLabel; S: SchemaWF | T ∈shapes(S) }
∀T: ShapeLabel; S: SchemaWF | T ∈shapes(S) @
1 r == rule(T,S) @
2 expr(T,S) = shapeExpr(shapeDef(r))
* The shape expression for a shape label $T$ is the shape expression in the shape definition of the rule $r$ that has shape label $T$.
§.§ $incl$
Given a shape label $T$ defined in a well-formed schema $S$, let $incl(T,S)$ be the,
possibly empty, set of included properties.
incl: ShapeLabel SchemaWF InclPropSet
incl = { T: ShapeLabel; S: SchemaWF | T ∈shapes(S) }
∀T: ShapeLabel; S: SchemaWF | T ∈shapes(S) @
1 ∃_1 r: S | T = shapeLabel(r) @
2 incl(T,S) = inclShapeDefinition(shapeDef(r))
* The included properties set for a shape label $T$ is the included properties set in the shape definition of the rule $r$ that has shape label $T$.
Given a shape definition $d$, let $inclShapeDefinition(d)$ be its included properties set.
inclShapeDefinition: ShapeDefinition InclPropSet
∀x: ShapeExpr @
1 inclShapeDefinition(close(x)) =
1 inclShapeDefinition(open({ ∅},x))
2 = ∅∀ips: InclPropSet; x: ShapeExpr @
1 inclShapeDefinition(open({ ips },x)) = ips
* The included property set of a closed shape definition or an open definition with no included
property set is the empty set.
* The included property set of an open shape definition with an included property set is that
included property set.
§.§ $properties$
Given a shape expression $x$, let $properties(x)$ be the set of properties that appear in some
triple constraint in $x$.
properties: ShapeExpr PropertiesSet
properties(emptyshape) = ∅∀tc: TripleConstraint; c: Cardinality @
1 properties(triple(tc,c)) =
2 propertiesTripleConstraint(tc)
∀itc: InverseTripleConstraint; c: Cardinality @
1 properties(triple(itc,c)) =
2 ∅∀xs: _1 ShapeExpr @
1 properties(someOf(xs)) =
1 properties(oneOf(xs)) =
1 properties(group(xs)) =
2 ⋃{ x: xs @ properties(x) }
∀x: ShapeExpr; c: Cardinality @
1 properties(repetition(x,c)) = properties(x)
* An empty shape expression has no properties.
* The properties of a triple constraint shape expression are the properties of its triple constraint.
* Inverse triple constraint shape expressions have no properties.
* The properties of a some-of, one-of, or grouping shape expression are the union of the properties of their
component shape expressions.
* The properties of a repetition shape expression are the properties of the shape expression being repeated.
Given a triple constraint $tc$, let $propertiesTripleConstraint(tc)$ be its set of properties.
propertiesTripleConstraint: TripleConstraint PropertiesSet
∀a: IRI; C: Constraint @
1 propertiesTripleConstraint((nop(a),C)) = { a }
* The properties of a triple constraint is the singleton set that contains its IRI.
§.§ $invproperties$
Given a shape expression $x$, let $invproperties(x)$ be the set of properties that appear in some
inverse triple constraint in $x$.
invproperties: ShapeExpr PropertiesSet
invproperties(emptyshape) = ∅∀tc: TripleConstraint; c: Cardinality @
1 invproperties(triple(tc,c)) =
2 ∅∀itc: InverseTripleConstraint; c: Cardinality @
1 invproperties(triple(itc,c)) =
2 invpropertiesInverseTripleConstraint(itc)
∀xs: _1 ShapeExpr @
1 invproperties(someOf(xs)) =
1 invproperties(oneOf(xs)) =
1 invproperties(group(xs)) =
2 ⋃{ x: xs @ invproperties(x) }
∀x: ShapeExpr; c: Cardinality @
1 invproperties(repetition(x,c)) = invproperties(x)
* An empty shape expression has no inverse properties.
* A triple constraint shape expression has no inverse properties.
* The inverse properties of an inverse triple constraint shape expression are the inverse properties in its inverse triple constraint.
* The inverse properties of a some-of, one-of, or grouping shape expression is the union of the inverse properties of their
component shape expressions.
* The inverse properties of a repetition shape expression are the inverse properties of the shape expression being repeated.
Given an inverse triple constraint $itc$, let $invpropertiesInverseTripleConstraint(tc)$ be its set of inverse properties.
1 InverseTripleConstraint PropertiesSet
∀a: IRI; C: ShapeConstr @
1 invpropertiesInverseTripleConstraint((inv(a),C)) = { a }
* The inverse properties of an inverse triple constraint is the singleton set that contains its IRI.
§.§ $dep\_graph$
§.§.§ $DiGraph$
A directed graph consists of a set of nodes and a set of directed edges that connect the nodes.
nodes: X
edges : X X
edges ∈nodes nodes
* Each edge connects a pair of nodes in the graph.
§.§.§ $DepGraph$
Given a well-formed schema $S$, let the shapes dependency graph be the directed graph
whose nodes are the shape labels in $S$ and whose edges connect label $T1$ to label $T2$ when the shape expression
that defines $T1$ refers to $T2$.DepGraph
S: SchemaWF
nodes = shapes(S)
edges = { T1, T2 : nodes | T2 ∈refsShapeExpr(expr(T1,S)) }
* The nodes are the shapes of the schema.
* There is an edge from $T1$ to $T2$ when the definition of $T1$ refers to $T2$.
§.§.§ $dep\_graph$
Let $dep\_graph(S)$ be the dependency graph of $S$.
dep_graph: SchemaWF DiGraph[ShapeLabel]
dep_graph = { DepGraph @ S ↦θDiGraph }
§.§ $dep\_subgraph$
§.§.§ $reachable$
Given a directed graph $g$ and a node $T$ in $g$, a node $U$ is reachable from $T$ if there is a directed path of one or more edges that connects
$T$ to $U$.
reachable : DiGraph[X] X X
∀g: DiGraph[X]; T: X @
1 edges == g.edges @
2 reachable(g, T) = { U: X | T ↦U ∈edges }
§.§.§ $DepSubgraph$
Given a well-formed schema $S$ and a shape label $T$ in $S$, the shapes dependency
graph is the subgraph induced by the nodes that are reachable from $T$.
S: SchemaWF
T: ShapeLabel
T ∈shapes(S)
g == dep_graph(S) @
1 nodes = reachable(g,T)
1 edges = g.edges ∩(nodes nodes)
* The nodes of the subgraph consist of all the nodes reachable from $T$.
* The edges of the subgraph consist of all edges of the graph whose nodes are in the subgraph.
Note that the above formal definition of the dependency subgraph is a literal translation of the text in the semantics draft.
In particular, this literal translation does not explicitly include the label $T$ as a node.
Therefore $T$ will not be in the subgraph unless it is in a directed cycle of edges.
§.§.§ $dep\_subgraph$
Let $dep\_subgraph(T,S)$ be the dependency subgraph of $T$ in $S$.
dep_subgraph : ShapeLabel SchemaWF DiGraph[ShapeLabel]
dep_subgraph = { DepSubgraph @ (T,S) ↦θDiGraph }
§.§ $negshapes$
The definition of $negshapes$ makes use of several auxilliary definitions.
In the following we assume that $S$ is a well-formed schema and that $T$ is a shape label in $S$.
§.§.§ $inNeg$
Let $inNeg(S)$ be the set of labels that appear in some negated shape constraint.
inNeg: SchemaWF ShapeLabel
∀S: SchemaWF @
1 inNeg(S) = ⋃{ T: shapes(S) @ inNegExpr(expr(T,S)) }
Given a shape expression $x$, let $inNegExpr(x)$ be the set of labels that appear in some negated shape constraint in $x$.
inNegExpr: ShapeExpr ShapeLabel
inNegExpr(emptyshape) = ∅∀tc: TripleConstraint; c: Cardinality @
1 inNegExpr(triple(tc,c)) =
2 inNegTripleConstraint(tc)
∀itc: InverseTripleConstraint; c: Cardinality @
1 inNegExpr(triple(itc,c)) =
2 inNegInverseTripleConstraint(itc)
∀xs: _1 ShapeExpr @
1 inNegExpr(someOf(xs)) =
1 inNegExpr(oneOf(xs)) =
1 inNegExpr(group(xs)) =
2 ⋃{ x: xs @ inNegExpr(x) }
∀x: ShapeExpr; c: Cardinality @
1 inNegExpr(repetition(x,c)) = inNegExpr(x)
Given a triple constraint $tc$, let $inNegTripleConstraint(tc)$ be the set of labels that appear in some negated shape constraint in $tc$.
inNegTripleConstraint: TripleConstraint ShapeLabel
∀a: IRI; C: ValueConstr @
1 inNegTripleConstraint((nop(a),C)) = ∅∀a: IRI; C: ShapeConstr @
1 inNegTripleConstraint((nop(a),C)) = inNegShapeConstr(C)
Given an inverse triple constraint $itc$, let $inNegInverseTripleConstraint(tc)$ be the set of labels that appear in some negated shape constraint in $itc$.
1 InverseTripleConstraint ShapeLabel
∀a: IRI; C: ShapeConstr @
1 inNegInverseTripleConstraint((inv(a),C)) = inNegShapeConstr(C)
Given a shape constraint $C$, let $inNegShapeConstr(C)$ be the set of labels that appear in $C$ when it is negated, or the
empty set otherwise.
inNegShapeConstr: ShapeConstr ShapeLabel
∀Ts: _1 ShapeLabel @
1 inNegShapeConstr(or(Ts)) =
1 inNegShapeConstr(and(Ts)) =
2 ∅∀Ts: _1 ShapeLabel @
1 inNegShapeConstr(nor(Ts)) =
1 inNegShapeConstr(nand(Ts)) =
2 Ts
§.§.§ $underOneOf$
Let $underOneOf(S)$ be the set of labels that appear in some triple constraint or inverse triple constraint under a one-of constraint
in $S$.
underOneOf: SchemaWF ShapeLabel
∀S: SchemaWF @
1 underOneOf(S) =
2 ⋃{ T: shapes(S) @ underOneOfExpr(expr(T,S)) }
Given a shape expression $x$, let $underOneOfExpr(x)$ be the set of labels that appear in some triple constraint or inverse triple constraint under a one-of constraint in $x$.
underOneOfExpr: ShapeExpr ShapeLabel
∀x: ShapeExpr @
1 underOneOfExpr(x) =
2 x ∈someOf
3 refsShapeExpr(x)
3 ∅
§.§.§ $inTripleConstr$
Let $inTripleConstr(S)$ be the set of labels $T$ such that there is a shape label $T1$ and a triple constraint
p::C or an inverse shape triple constraint
p::C in $expr(T1, S)$, and $T$ appears in $C$.
Note that this definition looks wrong since it does not involve negation of shapes.
Nevertheless, a literal translation is given here.
The only difference between $inTripleConstr(S)$ and $refs(S)$ seems to be that the cardinality on the triple and inverse triple
constraints is [1,1] since it is not explicitly included in the notations p::C and p::C.
inTripleConstr: SchemaWF ShapeLabel
∀S: SchemaWF @
1 inTripleConstr(S) =
2 ⋃{ T1: shapes(S) @ inTripleConstrExpr(expr(T1,S)) }
Given a shape expression $x$, let $inTripleConstrExpr(x)$ be the set of labels $T$ such that $x$ contains a triple constraint
p::C or an inverse shape triple constraint p::C and $T$ appears in $x$.
inTripleConstrExpr: ShapeExpr ShapeLabel
inTripleConstrExpr(emptyshape) = ∅∀dtc: DirectedTripleConstraint; c: Cardinality @
1 inTripleConstrExpr(triple(dtc,c)) =
2 c = one
3 refsDirectedTripleConstraint(dtc)
3 ∅∀xs: _1 ShapeExpr @
1 inTripleConstrExpr(someOf(xs)) =
1 inTripleConstrExpr(oneOf(xs)) =
1 inTripleConstrExpr(group(xs)) =
2 ⋃{ x: xs @ inTripleConstrExpr(x) }
∀x: ShapeExpr; c: Cardinality @
1 inTripleConstrExpr(repetition(x,c)) = inTripleConstrExpr(x)
§.§.§ $negshapes$
The semantics draft makes the following statement.
Intuitively, negshapes(S) is the set of shapes labels for which one needs to check whether some nodes in a graph do not satisfy these shapes, in order to validate the graph against the schema S.
Let $negshapes(S)$ be the set of negated shape labels that appear in $S$.
negshapes: SchemaWF ShapeLabel
∀S: SchemaWF @
1 negshapes(S) = inNeg(S) ∪underOneOf(S) ∪inTripleConstr(S)
* A negated shape label is a shape label that appears in a negated shape constraint, or in a triple or inverse triple constraint under a one-of shape expression, or in a triple or inverse triple constraint that has cardinality [1,1].
Note that, as remarked above, the definition of $inTripleConstr$ seems wrong.
§.§ $ShapeVerdict$
The semantics draft defines the notation !T for shape labels $T$ to indicate that $T$ is negated.
The semantics of a schema involves assigning sets of shape labels and negated shape labels to the nodes of a graph,
which indicates which shapes must be satisfied or violated at each node.
A shape verdict indicates if a shape must be satisfied or violated.
An asserted label must be satisfied.
A negated label must be violated.
ShapeVerdict ::=
1 assertShapeLabel |
1 negateShapeLabel
The notation !T corresponds to $negate(T)$.
§.§ $allowed$
Given a value constraint $V$, let $allowed(V)$ be the set of all allowed values defined by $V$.
allowed: ValueConstr (Lit ∪IRI)
∀vs: (Lit ∪IRI) @
1 allowed(valueSet(vs)) = vs
∀dt: LiteralDatatype @
1 allowed(datatype(dt,∅)) = literalsOfDatatype(dt)
∀dt:LiteralDatatype; f: XSFacet @
1 allowed(datatype(dt,{f})) = literalsOfFacet(dt,f)
∀k: NodeKind @
1 allowed(kind(k)) = termsOfKind(k)
§.§.§ DAG
A directed, acyclic graph is a directed graph in which no node is reachable from itself.
g == θDiGraph @
1 ∀T: nodes @ T ∉reachable(g,T)
§.§ $ReplaceShape$
The semantics draft introduces the notation $S_{ri}$ for a reduced schema where $S$ is a schema,
$r$ is a rule-of-one node in a proof tree, and $i$ corresponds to a premise of $r$.
The reduced schema is constructed by replacing a shape with one in which the corresponding one-of component is eliminated.
This replacement operation is described here.
The full definition of $S_{ri}$ is given below following the definition of proof trees.
Given a schema $S$, a shape label $T$ defined in $S$, and a shape expression $Expr'$, the schema $replaceShape(S,T,Expr')$
is the schema $S'$ that is the same as $S$ except that $expr(T,S') = Expr'$.
S, S': SchemaWD
T: ShapeLabel
Expr': ShapeExpr
l: _1
d, d': ShapeDefinition
ecs: ExtensionCondition
l ∈S
S(l) = (T, d, ecs)
∀o: OPTIONAL[InclPropSet]; Expr: ShapeExpr |
1 d = open(o, Expr) @
2 d' = open(o, Expr')
∀Expr: ShapeExpr |
1 d = close(Expr) @
2 d' = close(Expr')
S' = S ⊕{ l ↦(T, d', ecs) }
1 SchemaWF ShapeLabel ShapeExpr SchemaWF
replaceShape = { ReplaceShape@ (S, T, Expr') ↦S' }
§.§ $SchemaWD$
Given a well-formed schema $S$, it is said to be well-defined if for each negated label $T$ in $negshapes(T)$, the dependency subgraph
$dep\_subgraph(T,S)$ is a directed, acyclic graph.
SchemaWD ==
1 { S: SchemaWF |
2 ∀T: negshapes(S) @
3 dep_subgraph(T,S) ∈DAG[ShapeLabel] }
The semantics of shape expression schemas is sound only for well-defined schemas.
Only well-defined schemas will be considered from this point forward.
§ DECLARATIVE SEMANTICS OF SHAPE EXPRESSION SCHEMAS
Recall that negated triple and inverse triple shape expressions are represented by the corresponding non-negated expressions with cardinality
$none =$ [0;0].
§.§ $LabelledTriple$
A labelled triple is either an incoming or outgoing edge in an RDF graph.
LabelledTriple ::=
1 out Triple |
1 inc Triple
Sometimes labelled triples are referred to simply as triples.
§.§ $matches$
A labelled triple matches a directed triple constraint when they have the same direction and predicate.
matches: LabelledTriple DirectedTripleConstraint
matches = matches_out ∪matches_inc
§.§.§ $matches\_out$
$matches\_out$ matches outgoing triples to triple constraints.
matches_out ==
1 { s, p, o: TERM; C: Constraint |
2 (s,p,o) ∈Triple @
3 out(s,p,o) ↦(nop(p),C) }
Note that this definition ignores any value constraints defined in $C$.
The absence of restrictions imposed by value constraints makes matching weaker than it could be.
This may be an error in the semantics draft.
The semantics drafts contains the following text.
The following definition introduces the notion of satisfiability of a shape constraint by a set of triples. Such satisfiability is going to be used for checking that the neighbourhood of a node satisfies locally the constraints defined by a shape expression, without taking into account whether the shapes required by the triple constraints and inverse triple constraints are satisfied.
Read literally, only shape constraints should be ignored, so unless value constraints are handled elsewhere,
the semantics draft has an error in the definition of $matches$.
§.§.§ $matches\_inc$
$matches\_inc$ matches incoming triples to inverse triple constraints.
matches_inc ==
1 { s, p, o: TERM; C: ShapeConstr |
2 (s,p,o) ∈Triple @
3 inc(s,p,o) ↦(inv(p),C) }
§.§ $satifies$
A set of labelled triples $Neigh$ is said to satisfy a shape expression $Expr$ if the constraints, other than shape constraints,
defined in $Expr$ are satisfied.
Note that the definition of $matches$ ignores both value and shape constraints.
satisfies: LabelledTriple ShapeExpr
This relation is defined recursively by inference rules for each type of shape expression.
satisfies =
1 rule_empty ∪
1 rule_triple_constraint ∪
1 rule_inverse_triple_constraint ∪
1 rule_some_of ∪
1 rule_one_of ∪
1 rule_group ∪
1 rule_repeat
§.§.§ $InfRule$
An inference rule defines a relation between a set of labelled triples and a shape expression.
It is convenient to define a base schema for the inference rules.
Neigh: LabelledTriple
Expr: ShapeExpr
§.§.§ $rule\_empty$
An empty set of triples satisfies the empty shape expression.
Expr = emptyshape
Neigh = ∅
rule_empty: LabelledTriple ShapeExpr
rule_empty =
1 { RuleEmpty @ Neigh ↦Expr }
§.§.§ $rule\_triple\_constraint$
A set of triples satisfies a triple constraint shape expression when each triple matches the constraint
and the total number of constraints is within the bounds of the cardinality.
p: IRI
C: Constraint
c: Cardinality
Expr = triple((nop(p),C),c)
k = # Neigh
k inBounds c
∀t: Neigh @ t matches (nop(p),C)
rule_triple_constraint: LabelledTriple ShapeExpr
rule_triple_constraint =
1 { RuleTripleConstraint @ Neigh ↦Expr }
§.§.§ $rule\_inverse\_triple\_constraint$
A set of triples satisfies an inverse triple constraint shape expression when each triple matches the constraint
and the total number of constraints is within the bounds of the cardinality.
p: IRI
C: Constraint
c: Cardinality
Expr = triple((inv(p),C),c)
k = # Neigh
k inBounds c
∀t: Neigh @ t matches (inv(p),C)
rule_inverse_triple_constraint: LabelledTriple ShapeExpr
rule_triple_constraint =
1 { RuleInverseTripleConstraint @ Neigh ↦Expr }
§.§.§ $rule\_some\_of$
A set of triples satisfies a some-of shape expression when the set of triples satisfies one of the component
shape expressions.
Exprs: _1 ShapeExpr
i: Expr = someOf(Exprs)
i ∈Exprs
Neigh satisfies Exprs(i)
rule_some_of: LabelledTriple ShapeExpr
rule_some_of =
1 { RuleSomeOf @ Neigh ↦Expr }
§.§.§ $rule\_one\_of$
A set of triples satisfies a one-of shape expression when the set of triples satisfies one of the component
shape expressions.
Exprs: _1 ShapeExpr
i: Expr = oneOf(Exprs)
i ∈Exprs
Neigh satisfies Exprs(i)
rule_one_of: LabelledTriple ShapeExpr
rule_one_of =
1 { RuleOneOf @ Neigh ↦Expr }
The semantics draft contains the following text.
Note that the conditions for some-of and one-of shapes are identical.
The distinction between both will be made by taking into account also the non-local, shape constraints.
§.§.§ $rule\_group$
A set of triples satisfies a group shape expression when the set of triples can be partitioned into a sequence of subsets whose length
is the same as the sequence of component shape expressions, and each subset satisfies the corresponding component shape expression.
Neighs: _1 (LabelledTriple)
Exprs: _1 ShapeExpr
Expr = group(Exprs)
Neighs Neigh
# Neighs = # Exprs
∀j: Neighs @
1 Neighs(j) satisfies Exprs(j)
rule_group: LabelledTriple ShapeExpr
rule_group =
1 { RuleGroup @ Neigh ↦Expr }
§.§.§ $rule\_repeat$
A set of triples satisfies a repetition shape expression when the set of triples can be partitioned into a sequence of subsets whose length is
in the bounds of the cardinality, and each subset satisfies the component shape expression of the repetition shape expression.
Expr1: ShapeExpr
Neighs: _1 (LabelledTriple)
c: Cardinality
Expr = repetition(Expr1,c)
k = # Neighs
k inBounds c
Neighs Neigh
∀j: Neighs @
1 Neighs(j) satisfies Expr1
rule_repeat: LabelledTriple ShapeExpr
rule_repeat =
1 { RuleRepeat @ Neigh ↦Expr }
§.§ Proof Trees
The preceding definition of $satisfies$ is based on the existence of certain characteristics of the set of triples.
For example, a set of triples satisfies one of a sequence of shape expressions when it satisfies exactly one of the them,
but the $satisfies$ relation forgets the actual shape expression that the set of triples satisfies.
We can remember this type of information in a proof tree.
§.§.§ $RuleTree$
A rule tree is a tree of inference rules and optional child rule trees.
Child rule trees occur in cases where the inference rule depends on other inference rules.
RuleTree ::=
1 ruleEmpty RuleEmpty |
1 ruleTripleConstraint RuleTripleConstraint |
1 ruleInverseTripleConstraint RuleInverseTripleConstraint |
1 ruleSomeOf RuleSomeOf RuleTree |
1 ruleOneOf RuleOneOf RuleTree |
1 ruleGroup RuleGroup _1 RuleTree |
1 ruleRepeat RuleRepeat _1 RuleTree
§.§.§ $baseRule$
Each node in a rule tree contains an inference rule and, therefore, a base inference rule.
baseRule: RuleTree InfRule
∀RuleEmpty @
1 rule == θRuleEmpty;
2 base == θInfRule @
3 baseRule(ruleEmpty(rule)) = base
∀RuleTripleConstraint @
1 rule == θRuleTripleConstraint;
2 base == θInfRule @
3 baseRule(ruleTripleConstraint(rule)) = base
∀RuleInverseTripleConstraint @
1 rule == θRuleInverseTripleConstraint;
2 base == θInfRule @
3 baseRule(ruleInverseTripleConstraint(rule)) = base
∀RuleSomeOf; tree: RuleTree @
1 rule == θRuleSomeOf;
2 base == θInfRule @
3 baseRule(ruleSomeOf(rule,tree)) = base
∀RuleOneOf; tree: RuleTree @
1 rule == θRuleOneOf;
2 base == θInfRule @
3 baseRule(ruleOneOf(rule,tree)) = base
∀RuleGroup; trees: _1 RuleTree @
1 rule == θRuleGroup;
2 base == θInfRule @
3 baseRule(ruleGroup(rule,trees)) = base
∀RuleRepeat; trees: _1 RuleTree @
1 rule == θRuleRepeat;
2 base == θInfRule @
3 baseRule(ruleRepeat(rule,trees)) = base
§.§.§ $baseNeigh$
Each node in a rule tree has a base set of labelled triples.
baseNeigh: RuleTree LabelledTriple
∀tree: RuleTree @
1 baseNeigh(tree) = (baseRule(tree)).Neigh
§.§.§ $baseExpr$
Each node in a rule tree has a base shape expression.
baseExpr: RuleTree ShapeExpr
∀tree: RuleTree @
1 baseExpr(tree) = (baseRule(tree)).Expr
§.§.§ $ProofTree$
A proof tree is a rule tree in which the child trees prove subgoals of their parent nodes.
ProofTree: RuleTree
The definition of proof tree is recursive so it is given by a set of constraints, one for each type of node.
Any rule tree whose root node contains an empty shape expression is a proof tree
since it has no subgoals.
ruleEmpty ⊂ProofTree
Any rule tree whose root node node contains a triple constraint shape expression is a proof tree
since it has no subgoals.
ruleTripleConstraint ⊂ProofTree
Any rule tree whose root node node contains an inverse triple constraint shape expression is a proof tree
since it has no subgoals.
ruleInverseTripleConstraint ⊂ProofTree
A rule tree whose root node contains a some-of shape expression is a proof tree if and only if
its child rule tree correspond to the distinguished shape expression at index $i$ and it is a proof tree.
∀RuleSomeOf; tree: RuleTree @
1 ruleSomeOf(θRuleSomeOf, tree) ∈ProofTree
2 baseNeigh(tree) = Neigh
2 baseExpr(tree) = Exprs(i)
2 tree ∈ProofTree
A rule tree whose root node contains a one-of shape expression is a proof tree if and only if
its child rule tree correspond to the distinguished shape expression at index $i$ and it is a proof tree.
∀RuleOneOf; tree: RuleTree @
1 ruleOneOf(θRuleOneOf, tree) ∈ProofTree
2 baseNeigh(tree) = Neigh
2 baseExpr(tree) = Exprs(i)
2 tree ∈ProofTree
A rule tree whose root node contains a group shape expression is a proof tree if and only if
its sequence of child rule trees correspond to its sequence of component neighbourhood and shape expressions
and each child rule tree is a proof tree.
∀RuleGroup; trees: _1 RuleTree @
1 ruleGroup(θRuleGroup, trees) ∈ProofTree
2 # Exprs = # trees
2 (∀i: trees @
3 baseNeigh(trees(i)) = Neighs(i)
3 baseExpr(trees(i)) = Exprs(i)
3 trees(i) ∈ProofTree)
A rule tree whose root node contains a repetition shape expression is a proof tree if and only if
its sequence of child rule trees correspond to its sequence of component neighbourhoods
and each child rule tree is a proof tree.
∀RuleRepeat; trees: _1 RuleTree @
1 ruleRepeat(θRuleRepeat, trees) ∈ProofTree
2 # Neighs = # trees
2 (∀i: trees @
3 baseNeigh(trees(i)) = Neighs(i)
3 baseExpr(trees(i)) = Expr1
3 trees(i) ∈ProofTree)
We have the following relation between proof trees and the $satisfies$ relation.
\[\vdash
satisfies = \\
\t1 \{~ tree: ProofTree @ baseNeigh(tree) \mapsto baseExpr(tree) ~\}
\]
§.§ Reduced Schema for rule-one-of
As mentioned above, inference rules and proof trees treat rule-one-of exactly the same as rule-some-of.
The difference between these rules appears when considering valid typings, which are described in detail later.
Let $t$ be a valid typing of graph $G$ under schema $S$.
Let $n$ be a node in $G$ and let $T$ be a shape label in $t(n)$.
Let $Expr = expr(T,S)$ be the shape expression for $T$.
Let $tree$ be a proof tree that the neighbourhood of $n$ satisfies $Expr$.
Let $r$ be a node of the proof tree that contains an application of rule-one-of
and let $i$ be the index of the component expression used in the application of the rule.
The intention of the one-of shape expression is that the triples match exactly one of the component expressions.
Therefore, if the matched shape expression is removed from the one-of expression then there must not be any valid typings
of $G$ under the reduced schema $S_{ri}$.
Note that a one-of shape expression may have one or more components.
The number of components is denoted by $k$ in the inference rule.
However, if it contains exactly one component then there no further semantic conditions that must hold and there
is no corresponding reduced schema.
Therefore, the definition of the reduced schema only applies to the case where the number of components is greater than one,
i.e. $k > 1$.
Rule trees are ordered trees.
A child tree can be specified by giving its index among all the children.
The maximum index of a child depends on the type of rule.
For leaf trees, the maximum child index is 0.
maxChild: RuleTree ∀tree: ruleEmpty @ maxChild(tree) = 0
∀tree: ruleTripleConstraint @ maxChild(tree) = 0
∀tree: ruleInverseTripleConstraint @ maxChild(tree) = 0
∀tree: ruleSomeOf @ maxChild(tree) = 1
∀tree: ruleOneOf @ maxChild(tree) = 1
∀r: RuleGroup; trees: _1 RuleTree @
1 maxChild(ruleGroup(r,trees)) = # trees
∀r: RuleRepeat; trees: _1 RuleTree @
1 maxChild(ruleRepeat(r,trees)) = # trees
Given a tree $tree$ and a valid child index $j$, the child tree at the index is $childAt(tree,j)$.
childAt: RuleTree _1 RuleTree
childAt =
1 { tree: RuleTree; ci: _1 | ci ≤maxChild(tree) }
∀r: RuleSomeOf; tree: RuleTree @
1 childAt(ruleSomeOf(r,tree),1) = tree
∀r: RuleOneOf; tree: RuleTree @
1 childAt(ruleOneOf(r,tree),1) = tree
∀r: RuleGroup; trees: _1 RuleTree @
1 tree == ruleGroup(r,trees) @
2 ∀ci: 1 maxChild(tree) @
3 childAt(tree,ci) = trees(ci)
∀r: RuleRepeat; trees: _1 RuleTree @
1 tree == ruleRepeat(r,trees) @
2 ∀ci: 1 maxChild(tree) @
3 childAt(tree,ci) = trees(ci)
The location of a node within a rule tree can be specified by giving a sequence of positive integers that specify the index of each child tree.
The root of the tree is specified by the empty sequence.
Such a sequence of integers is referred to as a rule path.
Given a rule tree $tree$, the set of all of its rule paths is $rulePaths(tree)$.
rulePaths: RuleTree (_1)
∀tree: RuleTree | maxChild(tree) = 0 @
1 rulePaths(tree) = { ⟨⟩}
∀tree: RuleTree | maxChild(tree) > 0 @
1 rulePaths(tree) =
2 ⋃{ ci : 1 maxChild(tree) @
3 { path: rulePaths(childAt(tree,ci)) @ ⟨ci ⟩path } }
Given a rule tree $tree$ and a rule path $path$, the tree node specified by the path is $treeAt(tree,path)$,
treeAt: RuleTree _1 RuleTree
treeAt =
1 { tree: RuleTree; path: _1 | path ∈rulePaths(tree) }
∀tree: RuleTree @ treeAt(tree, ⟨⟩) = tree
∀tree: RuleTree; ci: _1; path: _1 |
1 ⟨ci ⟩path ∈rulePaths(tree) @
2 treeAt(tree, ⟨ci ⟩path) = treeAt(childAt(tree,ci), path)
Given a one-of shape expression $Expr$ that has more than one component, and an index $i$ of one component,
$elimExpr(Expr,i)$ is the reduced expression in which component $i$ is eliminated.
Expr, Expr': ShapeExpr
Exprs, ExprsL, ExprsR: _1 ShapeExpr
i: Expr = oneOf(Exprs)
# Exprs > 1
i ∈Exprs
Exprs = ExprsL ⟨Exprs(i) ⟩ExprsR
Expr' = oneOf(ExprsL ExprsR)
elimExpr: ShapeExpr ShapeExpr
elimExpr = { ElimExpr @ (Expr,i) ↦Expr' }
Given a proof tree $tree$ with the shape expression $Expr$ as its base, and a path $path$ to some application $r$ of rule-one-of in $tree$
in which the rule-of expression has more than one component,
tree: ProofTree
path: _1
r, rChild: ProofTree
R: RuleOneOf
path ∈rulePaths(tree)
r = treeAt(tree,path) = ruleOneOf(R,rChild)
# R.Exprs > 1
* The path is a valid rule path in the proof tree.
* The tree at the path is an application of rule-one-of.
* There are more than one components in the one-of shape expression.
$reduceExpr(tree,path)$ is the reduced base shape expression with the corresponding one-of expression in $Expr$ replaced by the reduced one-of expression.
reduceExpr: ProofTree _1 ShapeExpr
reduceExpr = { RuleOneOfApplication @ (tree,path) }
∀RuleOneOfApplication |
1 path = ⟨⟩
1 tree = r @
2 reduceExpr(r,⟨⟩) = elimExpr(R.Expr, R.i)
* The domain of this function requires that the path be a valid rule path in the proof tree.
* In the case of an empty path, the tree must be a one-of tree and the branch taken is eliminated.
* When the path is not empty, this function is defined recursively by additional constraints which follow.
There are four possible cases in which the proof tree has children. These cases correspond to applications of
rule-some-of, rule-one-of, rule-group, and rule-repeat.
Each case is defined by a schema below.
child: ProofTree
tail: _1
ExprsL, ExprsR: ShapeExpr
Expr': ShapeExpr
tree = ruleSomeOf(θRuleSomeOf, child)
path = ⟨1 ⟩tail
Exprs = ExprsL ⟨Exprs(i) ⟩ExprsR
Expr' = someOf(ExprsL ⟨reduceExpr(child,tail) ⟩ExprsL)
∀ReduceSomeOf @
1 reduceExpr(tree,path) = Expr'
child: ProofTree
tail: _1
ExprsL, ExprsR: ShapeExpr
Expr': ShapeExpr
tree = ruleOneOf(θRuleOneOf, child)
path = ⟨1 ⟩tail
Exprs = ExprsL ⟨Exprs(i) ⟩ExprsR
Expr' = oneOf(ExprsL ⟨reduceExpr(child,tail) ⟩ExprsL)
∀ReduceOneOf @
1 reduceExpr(tree,path) = Expr'
children: _1 ProofTree
ci: _1
tail: _1
ExprsL, ExprsR: ShapeExpr
Expr': ShapeExpr
tree = ruleGroup(θRuleGroup, children)
path = ⟨ci ⟩tail
Exprs = ExprsL ⟨Exprs(ci) ⟩ExprsR
Expr' = group(ExprsL ⟨reduceExpr(children(ci),tail) ⟩ExprsL)
∀ReduceGroup @
1 reduceExpr(tree,path) = Expr'
children: _1 ProofTree
ci: _1
tail: _1
Expr': ShapeExpr
tree = ruleRepeat(θRuleRepeat, children)
path = ⟨ci ⟩tail
Expr' = repetition(reduceExpr(children(ci),tail),c)
∀ReduceRepeat @
1 reduceExpr(tree,path) = Expr'
* Something looks wrong here because if a repetition expression has a one-of expression as a child then there is no way
to associate the reduced one-of expression with just the path taken in the proof tree since all the children of a repetition expression
share the same shape expression. However, a rule-repeat node in the proof tree has many children and there is no requirement
that all children would use the same branch of the one-of expression.
To make progress, I'll assume that all children of the repeat will eliminate the same branch of the one-of.
I will report this to the mailing list later, along with the observation that the reduction should only one done when a one-of expression
has more than one component.
§.§ Witness Mappings
Given a set of labelled triples $Neigh$, a shape expression $Expr$ and a proof tree $tree$ that proves $Neigh$ satisfies $Expr$,
each labelled triple $triple$ appears in a unique leaf node of the proof tree whose rule matches $triple$ with a directed triple constraint $dtc$.
This association of $triple$ with $dtc$ is called a witness mapping, $wm(triple) = dtc$.
§.§ $WitnessMapping$
WitnessMapping == LabelledTriple DirectedTripleConstraint
§.§.§ $witness$
witness: ProofTree WitnessMapping
∀r: RuleEmpty @
1 tree == ruleEmpty(r) @
2 witness(tree) = ∅∀r: RuleTripleConstraint; dtc: DirectedTripleConstraint; c: Cardinality |
1 r.Expr = triple(dtc,c) @
2 tree == ruleTripleConstraint(r) @
3 witness(tree) = baseNeigh(tree) {dtc}
∀r: RuleInverseTripleConstraint; dtc: DirectedTripleConstraint; c: Cardinality |
1 r.Expr = triple(dtc,c) @
2 tree == ruleInverseTripleConstraint(r) @
3 witness(tree) = baseNeigh(tree) {dtc}
∀r: RuleSomeOf; subtree: ProofTree @
1 tree == ruleSomeOf(r, subtree) @
2 tree ∈ProofTree
3 witness(tree) = witness(subtree)
∀r: RuleOneOf; subtree: ProofTree @
1 tree == ruleOneOf(r, subtree) @
2 tree ∈ProofTree
3 witness(tree) = witness(subtree)
∀r: RuleGroup; subtrees: _1 ProofTree @
1 tree == ruleGroup(r, subtrees) @
2 tree ∈ProofTree
3 witness(tree) = ⋃{ subtree: subtrees @ witness(subtree) }
∀r: RuleRepeat; subtrees: _1 ProofTree @
1 tree == ruleRepeat(r, subtrees) @
2 tree ∈ProofTree
3 witness(tree) = ⋃{ subtree: subtrees @ witness(subtree) }
§.§ $outNeigh$
The outgoing neighbourhood of a node $n$ in an RDF graph $G$ is the set of outgoing labelled triples that correspond to triples
in $G$ with subject $n$.
outNeigh: Graph TERM LabelledTriple
∀G: Graph; n: TERM @
1 outNeigh(G,n) = { p, o: TERM | (n,p,o) ∈G @ out(n,p,o) }
§.§ $incNeigh$
The ingoing neighbourhood of a node $n$ in an RDF graph $G$ is the set of ingoing labelled triples that correspond to triples
in $G$ with object $n$.
incNeigh: Graph TERM LabelledTriple
∀G: Graph; n: TERM @
1 incNeigh(G,n) = { p, s: TERM | (s,p,n) ∈G @ inc(n,p,s) }
§.§ $Typing$
Given a schema $S$ and a graph $G$, a typing $t$ is a map that associates to each node $n$ of $G$ a, possibly empty, set $t(n)$ of shape labels and negated shape labels such that if T is a negated shape label then either T or !T is in $t(n)$.
Here I infer that T and !T are mutually exclusive.
A typing map associates a finite, possibly empty, set of shape verdicts to nodes.
Typing == TERM ShapeVerdict
G: Graph
S: SchemaWD
t: Typing
t = nodes(G)
∀n: nodes(G); T: ShapeLabel | assert(T) ∈t(n) @
1 T ∈shapes(S)
∀n: nodes(G); T: ShapeLabel | negate(T) ∈t(n) @
1 T ∈negshapes(S)
∀n: nodes(G); T: negshapes(S) @
1 assert(T) ∈t(n) negate(T) ∈t(n)
∀n: nodes(G); T: shapes(S) @
1 assert(T) ∉t(n) negate(T) ∉t(n)
* The typing associates a set of shape verdicts to each node in the graph.
* If a node is required to satisfy $T$ then $T$ must be a shape label of the schema.
* If a node is required to violate $T$ then $T$ must be a negated shape label of the schema.
* If $T$ is a negated shape label of the schema then each node must be required to either satisfy or violate it.
* No node must be required to both satisfy and violate the same shape.
typings: Graph SchemaWD Typing
∀G: Graph; S: SchemaWD @
1 typings(G,S) = { m: TypingMap | m.G = G m.S = S @ m.t }
§.§ $TypingSatisfies$
Given a typing $t$, a node $u$, and a shape constraint $C$, the typing satisfies the constraint at the node if the
boolean conditions implied by the shape constraint hold.
u: TERM
C: ShapeConstr
Ts: _1 ShapeLabel
u ∈nodes(G)
C = and(Ts)
1 (∀T: Ts @ assert(T) ∈t(u))
C = or(Ts)
1 (∃T: Ts @ assert(T) ∈t(u))
C = nand(Ts)
1 (∃T: Ts @ negate(T) ∈t(u))
C = nor(Ts)
1 (∀T: Ts @ negate(T) ∈t(u))
* The node is in the graph.
* The node is required to satisfy every shape in an and shape constraint.
* The node is required to satisfy some shape in an or shape constraint.
* The node is required to violate some shape in a nand shape constraint.
* The node is required to violate every shape in a nor shape constraint.
typingSatisfies: Typing TERM ShapeConstr
typingSatisfies =
1 { TypingSatisfies @ (t, u) ↦C }
§.§ $Matching$
Given a node $n$ in graph $G$, a typing $t$, and a directed triple constraint $X$, let $Matching(G,n,t,X)$ be the set of triples in the graph with focus node $n$ that match $X$ under $t$.
n, p: TERM
X: DirectedTripleConstraint
C: Constraint
triples: LabelledTriple
C ∈ValueConstr X = (nop(p),C)
1 triples = { u: TERM | (n,p,u) ∈G
2 u ∈allowed(C)@ out(n,p,u) }
C ∈ShapeConstr X = (nop(p),C)
1 triples = { u: TERM | (n,p,u) ∈G
2 (t,u) typingSatisfies C @ out(n,p,u) }
C ∈ShapeConstr X = (inv(p),C)
1 triples = { u: TERM | (u,p,n) ∈G
2 (t,u) typingSatisfies C @ inc(u,p,n) }
* An outgoing triple matches a value constraint if its object is an allowed value.
* An outgoing triple matches a shape constraint if the typing of its object satisfies the constraint.
* An incoming triple matches a shape constraint if the typing of its subject satisfies the constraint.
Matching: Graph TERM Typing DirectedTripleConstraint
1 LabelledTriple
Matching = { MatchingTriples @ (G, n, t, X) ↦triples }
§.§ $validTypings$
The definition of what it means for a graph to satisfy a shape schema is given in terms of the existence of a valid typing.
Given a graph $G$ and a schema $S$, a valid typing of $G$ by $S$ is a typing that satisfies certain additional conditions
at each node $n$ in $G$.
validTypings: Graph SchemaWD Typing
∀G: Graph; S: SchemaWD @
1 validTypings(G,S) ⊆typings(G,S)
§.§.§ $ValidTypingNodeLabel$
The definition of a valid typing is given in terms of a series of conditions that must hold at each node and for each shape verdict at that node.
It is convenient to introduce the following base schema for conditions.
n: TERM
T: ShapeLabel
ruleT: Rule
defT: ShapeDefinition
Expr: ShapeExpr
Xs: DirectedTripleConstraint
t ∈validTypings(G,S)
n ∈nodes(G)
assert(T) ∈t(n) negate(T) ∈t(n)
ruleT = rule(T,S)
defT = shapeDef(ruleT)
Expr = expr(T,S)
Xs = tripleConstraints(Expr)
§.§.§ $tripleConstraints$
Given a shape expression $Expr$ let $tripleConstraints(Expr)$ be the set of all triple or inverse triple constraints contained in it.
tripleConstraints: ShapeExpr DirectedTripleConstraint
tripleConstraints(emptyshape) = ∅∀dtc: DirectedTripleConstraint; c: Cardinality @
1 tripleConstraints(triple(dtc,c)) = {dtc}
∀Exprs: _1 ShapeExpr @
1 tripleConstraints(someOf(Exprs)) =
1 tripleConstraints(oneOf(Exprs)) =
1 tripleConstraints(group(Exprs)) =
2 ⋃{ Expr: Exprs @ tripleConstraints(Expr) }
∀Expr: ShapeExpr; c: Cardinality @
1 tripleConstraints(repetition(Expr,c)) = tripleConstraints(Expr)
§.§.§ $NegatedShapeLabel$
The semantics draft states:
for all negated shape label !T, if !T $\in$ t(n), then t1 is not a valid typing, where t1 is the typing that agrees with t everywhere, except for T $\in$ t1(n)
negate(T) ∈t(n)
* The shape $T$ is negated at node $n$.
§.§.§ $AssertShape$
t1: Typing
t1 = t ⊕{ n ↦(t(n) ∖{negate(T)} ∪{assert(T)}) }
* The typing $t1$ is the same as $t$ except that at node $n$ the shape label $T$ is asserted instead of negated.
In a valid typing if any node has a negated shape, then the related typing with this shape asserted is invalid.
∀AssertShape @
1 t1 ∉validTypings(G,S)
Although this condition on $t(n)$ is recursive in terms of the definition of $validTypings$,
it is well-founded since $t1(n)$ has one fewer negated shapes than $t(n)$.
Therefore it remains to define the meaning of $validTypings$ for typings that contain no negated shapes.
§.§.§ $assertShape$
Given a typing $t$, node $n$, and shape label $T$ such that $negate(T) \in t(n)$, define $assertShape(t,n,T)$ to be the typing $t1$
that is the same as $t$ except that $assert(T) \in t1(n)$.
assertShape: Typing TERM ShapeLabel Typing
assertShape =
1 { AssertShape @
2 (t, n, T) ↦t1 }
§.§.§ $AssertedShapeLabel$
The semantics draft defines the meaning of valid typings $t$ by imposing several conditions that must hold for all nodes $n$ and all
asserted shape labels $assert(T) \in t(n)$.
assert(T) ∈t(n)
* The shape label $T$ is asserted at node $n$.
The semantics draft states that the following conditions must hold for all valid typings $t$ and all nodes $n$ such that $T$ is asserted at $n$:
for all shape label $T$, if $T \in t(n)$, then there exist three mutually disjoint sets $Matching, OpenProp, Rest$ such that
* $out(G, n) \cup inc(G, n) = Matching \cup OpenProp \cup Rest$, and
* $Rest = Rest_{out} \cup Rest_{inc}$, where
* $Rest_{out} = \{(out, n, p, u) \in out(G, n) | p \notin properties(expr(T, S))\}$, and
* $Rest_{inc} = \{(inc, u, p, n) \in inc(G, n) | p \notin invproperties(expr(T, S))\}$, and
* $Matching$ is the union of the sets $Matching(n, t, X)$ for all triple constraint or inverse triple constraint $X$ that appears in $expr(T, S)$, and
* if $T$ is a closed shape, then $Rest_{out} = \emptyset$ and $OpenProp = \emptyset$
* if $T$ is an open shape, then $OpenProp \subseteq \{(out, n, p, u) \in out(G, n) | p \in incl(T, S)\}$
* there exists a proof tree with corresponding witness mapping $wm$ for the fact that $Matching$ satisfies $expr(T, S)$, and s.t.
* for all outgoing triple $(out, n, p, u)$, it holds $(out, n, p, u) \in Matching(n, t, wm((out, n, p, u)))$, and moreover if $wm((out, n, p, u))$ is a shape triple constraint, then there is no value triple constraint p::C in $expr(T, S) s.t. (out, n, p, u) \in Matching(n, t, p::C)$, and
* for all incoming triple $(inc, u, p, n) \in G$, it holds $(inc, u, p, n) \in Matching(n, t, wm((inc, u, p, n)))$, and
* for all node $r$ that corresponds to an application of rule-one-of in the proof tree, there does not exist a valid typing $t1$ of $G$ by $S_{ri}$ s.t. $T \in t1(n)$, and
* for all extension condition $(lang, cond)$, associated with the type $T$, $f_{lang}(G, n, cond)$ returns true or undefined.
§.§.§ $MatchingOpenRest$
for all shape label $T$, if $T \in t(n)$, then there exist three mutually disjoint sets $Matching, OpenProp, Rest$
MatchingNeigh, OpenProp, Rest: LabelledTriple
⟨MatchingNeigh, OpenProp, Rest ⟩
* There are three mutually disjoint sets of labelled triples.
* Note that the name $MatchingNeigh$ is used to avoid conflict with the previously defined $Matching$ function.
∀AssertedShapeLabel @
1 ∃MatchingNeigh, OpenProp, Rest: LabelledTriple @
2 MatchingOpenRest
§.§.§ $PartitionNeigh$
$out(G, n) \cup inc(G, n) = Matching \cup OpenProp \cup Rest$
⟨MatchingNeigh, OpenProp, Rest ⟩
1 outNeigh(G,n) ∪incNeigh(G,n)
∀AssertedShapeLabel @
1 ∃MatchingNeigh, OpenProp, Rest: LabelledTriple @
2 PartitionNeigh
§.§.§ $RestDef$
$Rest = Rest_{out} \cup Rest_{inc}$, where
* $Rest_{out} = \{(out, n, p, u) \in out(G, n) | p \notin properties(expr(T, S))\}$, and
* $Rest_{inc} = \{(inc, u, p, n) \in inc(G, n) | p \notin invproperties(expr(T, S))\}$, and
Rest_out, Rest_inc : LabelledTriple
Rest = Rest_out ∪Rest_inc
Rest_out =
1 { p, u: TERM |
2 out(n,p,u) ∈outNeigh(G,n)
2 p ∉properties(expr(T,S)) @
3 out(n,p,u) }
Rest_inc =
1 { p, u: TERM |
2 inc(u,p,n) ∈incNeigh(G,n)
2 p ∉invproperties(expr(T,S)) @
3 inc(u,p,n) }
∀MatchingOpenRest @
1 ∃_1 Rest_out, Rest_inc : LabelledTriple @
2 RestDef
§.§.§ $MatchingDef$
$Matching$ is the union of the sets $Matching(n, t, X)$ for all triple constraint or inverse triple constraint $X$ that appears in $expr(T, S)$
MatchingNeigh =
1 ⋃{ X: Xs @ Matching(G, n, t, X) }
∀MatchingOpenRest @
1 MatchingDef
§.§.§ $ClosedShapes$
if $T$ is a closed shape, then $Rest_{out} = \emptyset$ and $OpenProp = \emptyset$
defT ∈close
1 Rest_out = ∅
1 OpenProp = ∅
∀RestDef @
1 ClosedShapes
§.§.§ $OpenShapes$
if $T$ is an open shape, then
\[
OpenProp \subseteq \{(out, n, p, u) \in out(G, n) | p \in incl(T, S)\}
\]
defT ∈open
1 OpenProp ⊆
2 { p, u: TERM |
3 out(n,p,u) ∈outNeigh(G,n)
3 p ∈incl(T,S) @
4 out(n,p,u) }
∀MatchingOpenRest @
1 OpenShapes
§.§.§ $ProofWitness$
there exists a proof tree with corresponding witness mapping $wm$ for the fact that $Matching$ satisfies $expr(T, S)$, and s.t.
* for all outgoing triple $(out, n, p, u)$, it holds $(out, n, p, u) \in Matching(n, t, wm((out, n, p, u)))$, and moreover if $wm((out, n, p, u))$ is a shape triple constraint, then there is no value triple constraint p::C in $expr(T, S) s.t. (out, n, p, u) \in Matching(n, t, p::C)$, and
* for all incoming triple $(inc, u, p, n) \in G$, it holds $(inc, u, p, n) \in Matching(n, t, wm((inc, u, p, n)))$, and
* for all node $r$ that corresponds to an application of rule-one-of in the proof tree, there does not exist a valid typing $t1$ of $G$ by $S_{ri}$ s.t. $T \in t1(n)$, and
tree: ProofTree
wm: WitnessMapping
baseNeigh(tree) = MatchingNeigh
baseExpr(tree) = Expr
wm = witness(tree)
∀MatchingDef @
1 ∃tree: ProofTree; wm: WitnessMapping @
2 ProofWitness
§.§.§ $OutgoingTriples$
for all outgoing triple $(out, n, p, u)$, it holds
\[
(out, n, p, u) \in Matching(n, t, wm((out, n, p, u))),
\]
and moreover if $wm((out, n, p, u))$ is a shape triple constraint, then there is no value triple constraint p::C in $expr(T, S)$ s.t.
\[
(out, n, p, u) \in Matching(n, t, p::C)
\]
∀triple: outNeigh(G,n); p, u: TERM |
1 triple = out(n,p,u) @
2 X == wm(triple) @
3 triple ∈Matching(G, n, t, X)
3 (constrDTC(X) ∈ShapeConstr
4 ¬(∃C: ValueConstr | (nop(p),C) ∈Xs @
5 triple ∈Matching(G, n, t, (nop(p),C))))
∀ProofWitness @
1 OutgoingTriples
§.§.§ $IncomingTriples$
for all incoming triple $(inc, u, p, n) \in G$, it holds
\[
(inc, u, p, n) \in Matching(n, t, wm((inc, u, p, n)))
\]
∀triple: incNeigh(G,n) @
1 X == wm(triple) @
2 triple ∈Matching(G, n, t, X)
∀ProofWitness @
1 IncomingTriples
§.§.§ $OneOfNodes$
for all node $r$ that corresponds to an application of rule-one-of in the proof tree,
there does not exist a valid typing $t1$ of $G$ by $S_{ri}$ s.t. $T \in t1(n)$
Let $OneOfNodes$ describe the situation where we are
given a graph $G$, a schema $S$, a typing $t$ of $G$ under $S$, a node $n$ in $G$, a shape label $T$
in $t(n)$, a proof tree $tree$ for the triples $MatchNeigh$ and the expression $Expr= expr(T,S)$ and an application
of rule-one-of $r$ in the proof tree.
Expr_ri : ShapeExpr
S_ri : SchemaWD
Expr_ri = reduceExpr(tree, path)
S_ri = replaceShape(S, T, Expr_ri)
Whenever rule-one-of is applied in the proof tree, there must not be any valid typings $t1$
for the reduced schema $S\_ri$ in which the selected component of the one-of shape expression is eliminated.
∀OneOfNodes @
1 ¬(∃t1: validTypings(G,S_ri) @
2 assert(T) ∈t1(n))
§.§.§ $ExtensionConditions$
for all extension condition $(lang, cond)$, associated with the type $T$, $f_{lang}(G, n, cond)$ returns true or undefined
The semantics of an extension condition is given by a language oracle function
that evaluates the extension condition $cond$ on a pointed graph $(G,n)$
and returns a code indicating whether the pointed graph
satisfies the extension condition, or if an error condition holds, or if the extension condition is undefined.
f : ExtLangName Graph TERM ExtDefinition ReturnCode
∀G: Graph; n: TERM| (G, n) ∈PointedGraph @
1 ∀lang: ExtLangName; cond: ExtDefinition @
2 returnCode == f(lang, G, n, cond) @
3 returnCode = trueRC (G,n) ∉violatedBy(lang, cond)
3 returnCode = falseRC (G,n) ∈violatedBy(lang, cond)
* If the oracle returns true then the pointed graph satisfies the extension condition.
* If the oracle returns false then the pointed graph violates the extension condition.
Let the return codes for the language oracles be $ReturnCode$.
ReturnCode ::= trueRC | falseRC | errorRC | undefinedRC
* true means the extension condition is satisfied.
* false means the extension condition is violated.
* error means an error occurred.
* undefined means the extension condition is undefined.
lang: ExtLangName
cond: ExtDefinition
ecs == extConds(ruleT) @
1 (lang,cond) ∈ecs
* $(lang, cond)$ is an extension condition for $T$.
∀ExtensionConditions @
1 f(lang, G, n, cond) ∈{trueRC, undefinedRC }
§ ISSUES
Some areas of the semantics draft have multiple interpretations or appear to be wrong and therefore require further clarification.
These areas are discussed below.
§.§ dep-subgraph(T,S)
In the definition of dep-subgraph(T,S), is the shape T considered to be reachable from itself?
§.§ negshapes(S)
In the definition of negshapes(S), the third bullet states:
This statement looks wrong because it omits mention of negation.
If there is no negation involved, why would T be in negshapes(S)?
Does this definition only select directed triple constraints that have cardinality [1,1]
because that is the default?
If not then negshapes(S) is the set of all labels that are referenced in any shape definition ($refs(S)$), which seems wrong.
§.§ Triple matches constraint
The definition of matching p:C and p:C omits consideration of C.
The explanation is as follows.
This statement implies that only shape constraints should be ignored here.
However, the definition ignores the value set constraints too.
This looks wrong.
§.§ rule-triple-constraint
Add the condition that all the outgoing triples must be distinct.
§.§ rule-inverse-triple-constraint
Add the condition that all the incoming triples must be distinct.
§.§ rule-group
Add the condition that i and j must be different.
§.§ rule-repeat
Add the condition that i and j must be different.
§.§ Reduced Schema for rule-one-of
This is an edge case.
It only makes sense to reduce the schema if there are more than one components.
Applying rule-one-of to a sequence of one shape is equivalent to requiring that shape.
Add this condition to the definition.
§.§ Reduced Schema for rule-one-of under a repetition expression
Something looks wrong here because if a repetition expression has a one-of expression as a child then there is no way
to associate the reduced one-of expression with just the path taken in the proof tree since all the children of a repetition expression
share the same shape expression. However, a rule-repeat node in the proof tree has many children and there is no requirement
that all children would use the same branch of the one-of expression.
§ CONCLUSION
The exercise of formalizing the semantics draft has resulted in a considerable expansion in the size of the document.
The result has been the identification of a number of quality issues.
This exercise has also established that the recursive definitions in the semantics draft are well-founded.
However, it is not clear that these definitions produce results that agree with our intuition, or that they can be computed efficiently.
One possible way to further validate the semantics draft is to translate it into an executable formal specification system
such as Coq <cit.> and test it on a set of examples, including both typical documents and corner cases.
|
1511.00491
|
The Institute for Fundamental Study “The Tah Poe Academia Institute”, Naresuan University, Phitsanulok 65000, Thailand
The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11-34151, Trieste, Italy
The Institute for Fundamental Study “The Tah Poe Academia Institute”, Naresuan University, Phitsanulok 65000, Thailand
We give a brief review of the non-minimal derivative coupling (NMDC) scalar field theory in which there
is non-minimal coupling between the scalar field derivative term and the Einstein
tensor. We assume that the expansion is of power-law type or super-acceleration type for small redshift.
The Lagrangian includes the NMDC term, a free kinetic term, a cosmological constant term and a barotropic matter term.
For a value of the coupling constant
that is compatible with inflation,
we use the combined WMAP9 (WMAP9+eCMB+BAO+ $H_0$) dataset,
the PLANCK+WP dataset, and the PLANCK $TT,TE,EE$+lowP+Lensing+ext datasets
to find the value of the cosmological constant in the model.
Modeling the expansion with power-law gives a negative cosmological constants while
the phantom power-law (super-acceleration) expansion gives
positive cosmological constant with large error bar.
The value obtained is of the same order as in the $\Lambda$CDM model,
since at late times the NMDC effect is tiny due to small curvature.
§ INTRODUCTION
Recently, cosmic accelerating expansion has been confirmed by astrophysical observations.
Amongst these are supernova type Ia (SNIa) <cit.>, large-scale structure surveys <cit.>,
cosmic microwave background (CMB) anisotropies
and X-ray luminosity from galaxy clusters <cit.>.
The acceleration is responsible by an unknown energy form called dark energy <cit.> which is typically
in form of either cosmological constant or scalar field <cit.>.
There are many scalar field models proposed to explain the accelerating expansion of the universe, for example,
quintessence <cit.> and classes of k-essence type models <cit.>.
Modifications of gravity, for instance, braneworlds, $f(R)$ and others are as well
possible answers of present acceleration (see e.g. <cit.>). Acquiring the acceleration needs
the effective equation of state of matter species, especially a dynamical scalar field evolving under its potential, to be $p < - \rho c^2/3$.
It is possible to have a non-minimal coupling (NMC) between scalar field to Ricci scalar in GR in form of $\sqrt{-g}f(\phi)R$.
The NMC is motivated by scalar-tensor theories in the Jordan-Brans-Dicke models <cit.>,
re-normalizing term of quantum field in curved space <cit.> or supersymmetries, superstring and induced gravity theories
<cit.>. It was applied to extended inflations with first-order phase transition and other inflationary models
In context of quintessence field driving present acceleration,
non-minimal coupling to curvature has been studied as in <cit.>. In strong coupling regime, power-law and de-Sitter expansions
are found as late time attractor <cit.> and moreover the NMC term could also behave as effective cosmological constant <cit.>.
First cosmological consideration of the non-minimal curvature coupling to the derivative term of scalar field was proposed by Amendola in 1993 <cit.>.
Therein the coupling function is in form of $f(\phi, \phi_{,\mu}, \phi_{,\mu\nu}, \ldots)$.
This type of derivative coupling is required in scalar quantum electrodynamics to satisfy U(1) invariance of the theory and
is required in models of which the gravitational constant is function of the mass density of the gravitational source.
The non-minimal derivative coupling-NMDC terms are commonly found as lower energy limits of higher dimensional
theories which makes quantum gravity possible to be studied perturbatively. They are also found in Weyl anomaly in $\mathcal{N} = 4$ conformal supergravity
With simplest NMDC term, $R \phi_{,\mu}\phi^{,\mu}$,
class of inflationary attractors is enlarged from the previous NMC model of <cit.> and the NMDC renders
non-scale invariant spectrum without requirement of multiple scalar fields. Moreover it is possible to realize double inflation without
adding more fields to the theory <cit.>.
However conformation transformation can not transform the NMDC theory into the standard field equation in Einstein frame.
The conformal (metric) re-scaling transformation needs to be
generalized to Legendre transformation in order to recover the Einstein frame equations <cit.>.
There are various versions of the NMDC proposed in order to match plausible theory and to predict observation results as will be seen in the next section.
We give a brief review of the NMDC gravity models in this paper and
we consider a model in which the Einstein tensor couples to the kinetic scalar field term with a free kinetic term and a constant potential (considered as a cosmological constant).
In setups of power-law or phantom power-law (super) acceleration expansions and using inflation-estimated value of the coupling constant, we evaluate
value of the cosmological constant and show a parametric plots of the cosmological constant versus the power-law exponents.
Cosmological parameters given by WMAP9
(combined WMAP9+eCMB+BAO+$H_0$) dataset
<cit.> and PLANCK satellite dataset <cit.> are used here.
§ NON-MINIMAL DERIVATIVE COUPLING THEORY
§.§ Capozziello, Lambiase and Schmidt's result
Capozziello, Lambiase and Schmidt <cit.> found in 2000 that
all other possible coupling Lagrangian terms are not necessary in scalar-curvature coupling theory, leaving only
$R \phi_{,\mu}\phi^{,\mu}$ and $R^{\mu\nu} \phi_{,\mu} \phi_{,\nu}$ terms in the
Lagrangian without losing its generality, hence motivating cosmological study in the case of having
both terms. One character of the two new terms is to modulate gravitational strength with a free canonical kinetic term
without either scalar field potential $V(\phi)$ or $\Lambda$. This results in an
effective cosmological constant and hence effectively giving de-Sitter expansion <cit.>.
The conditions for which de-Sitter expansion is a late time attractor are given in <cit.>.
When considering only $R \phi_{,\mu}\phi^{,\mu}$ with free Ricci scalar, free kinetic term, potential and matter terms,
the equation of state, in absence of $V(\phi)$, goes to $-1$ at late time. When assuming slowly-rolling field and power-law expansion,
$V(\phi)$ is found directly <cit.>.
Another case is to consider only the $R^{\mu\nu} \phi_{,\mu} \phi_{,\nu}$ term as extra term to standard scalar field cosmology,
i.e. a free Ricci scalar with a free kinetic scalar term and a potential, the field equation contains third-order derivatives of $\phi$
and the continuity equation of the scalar field contains third-order derivative of $g_{\mu\nu}$.
This model is tightly constrained in weakly coupling regime, i.e. solar system constraint puts limit of the pressure,
$p_{\phi} < 10^{-6} \rho_{\rm c} c^2$, where $\rho_{\rm c}$ is critical density hence it can not play a role of quintessence.
If the coupling is strong with negative sign, the coupling term can flattens the slope of the inflationary potential
§.§ Granda's two coupling constant model
Another modification of the NMDC model is proposed by Granda in 2010 <cit.>.
The model contains the usual Einstein-Hilbert term, a scalar field kinetic term, a potential term and two separated dimensionless couplings, $\kappa$, $\eta$ re-scaled
by $1/\phi^2$ in form of $- (1/2) \kappa R \phi^{-2} g_{\mu\nu} \phi^{,\mu}\phi^{,\nu}$ and $- (1/2) \eta \phi^{-2} R_{\mu \nu} \phi^{\mu}\phi^{\nu}$.
In this model when there is no free kinetic scalar term (i.e. strictly NMDC) and no potential term, NMDC term takes a role of dark matter at early stage giving the power-law dust solution,
$a \sim t^{2/3}$ for $\eta = -2\kappa$ and accelerating solution for $\eta = -\kappa -1$ where $ 0 < \kappa < 1/3$.
Acceleration at present time is assured if including the potential into the Lagrangian. Motivation of such two separated couplings comes from an attempt to
approach quantum gravity perturbatively <cit.>. This gives ideas of the other versions of two coupling models without
the $1/\phi^2$ re-scaling factor <cit.> such as inclusion of Gauss-Bonnet invariance <cit.> or in context of Chaplygin gas
§.§ Sushkov's model
§.§.§ Constant or zero potential
Sushkov, in 2009, <cit.>
considered a special case $\kappa_1 R \phi_{,\mu}\phi^{,\mu}$ and $\kappa_2 R^{\mu\nu} \phi_{,\mu} \phi_{,\nu}$ with $\kappa \equiv \kappa_2 = -2 \kappa_1$.
This results in combination of the two NMDC terms into one Einstein tensor coupling to kinetic scalar field part, $\kappa G_{\mu \nu} \phi^{,\mu}\phi^{,\nu}$.
The chosen coupling constant $\kappa$ renders good dynamical theory, that is to say,
the field equations contain terms with second-order derivative of $g_{\mu\nu}$ and $\phi$ at most so that
the Lagrangian contains only divergence free tensors. Hence it consists of the $R$ term, free kinetic scalar $g_{\mu\nu} \phi^{,\mu}\phi^{,\nu}$
and $\kappa G_{\mu \nu} \phi^{,\mu}\phi^{,\nu}$ in absence of $V(\phi)$.
Cosmological study of the model for flat FLRW universe yields, for $\kappa > 0$, quasi-de-Sitter at very early stage but, for $\kappa < 0$, initial singularity at very early stage.
For any sign of the coupling, $a \propto t^{1/3}$ at very late time <cit.>.
A direct modification of this model is to have a constant potential with possibility of phantom behavior of the free kinetic term <cit.>.
In a range of coupling constant values, this modification enables the model to transit from de-Sitter phase to other types of
expansions giving various fates and various origins of the universe <cit.>. Alternative view point from de Rham, Gabadadze and Tolley massive gravity
<cit.>, is that in the decoupling limit, the massive gravity can give rise to a theory with $\kappa G_{\mu \nu} \phi^{,\mu}\phi^{,\nu}$ term as a subclass of Horndeski scalar-tensor theories
§.§.§ With potential but without free kinetic term
Inspired by Sushkov's model, in case of without free kinetic term, $ (1/2)g^{\mu \nu}\phi_{, \mu} \phi_{, \nu}$, but having Einstein tensor coupling kinetic term alone (strictly NMDC),
Gao in 2010 <cit.>, found that for $V(\phi) = 0$, the scalar field behaves like dust
in absence of other matters or in presence of pressureless matter. Its value of the equation of state parameter suggests that it could be a candidate of
dark energy and dark matter. However the model is not viable due to
superluminal sound speed. When adding more than one Einstein tensor coupling to the kinetic term <cit.>,
it was claimed not to be likely by <cit.>. Strictly NMDC term in curvaton model can also be seen in the work by <cit.>.
§.§.§ Purely kinetic coupling term and a matter term
The Sushkov's model, in absence of potential and absence of matter Lagrangians, is not able to explain phantom acceleration, i.e. no phantom crossing.
In order to fix the purely kinetic Lagrangian to allow phantom crossing, in 2011, Gubitosi and Linder proposed most general Lagragians
with purely kinetic term obeying shift symmetry. These are the $( a_1\phi_{, \mu}\phi^{,\mu} + a_2 \nabla^2 \phi )R$ term,
$\phi_{, \mu}\phi_{,\nu} R^{\mu\nu}$ term and
$R^{\alpha \beta \gamma \delta} f_{\alpha \beta \gamma \delta}(\phi_{,\mu}) $ term
where $f_{\alpha \beta \gamma \delta}$ is a function of $\phi_{,\mu}$ and a matter term
Absence of potential helps avoiding high energy quantum correction.
Their model is at lowest possible order of Planck mass and it verifies Sushkov's action <cit.>.
The model achieve wide range of $w$ values from stiff ($w=1$) to phantom crossing and is possible to result in loitering cosmological constant-like phase
before entering matter domination phase.
Sushkov's purely kinetic model with matter Lagrangian is found to be a special case of the Fab Four theory <cit.>.
Only positive coupling constant of the theory could result in phantom crossing
however it also gives non-causal scalar and tensor perturbation, hence making the purely-kinetic model discarded for inflation <cit.>.
Investigations of this model for $V(\phi) = 0 $ in blackhole spacetime are presented in <cit.>.
§.§.§ Adding potential term with matter term
As another way out of problem in purely kinetic model, potential is added into the theory (without matter term).
In order to have inflation, it is found that the potential needs to be less steep than quadratic potential <cit.>.
With constant potential and matter term in the model, it is able to describe transition from inflation to matter domination epoch
without reheating and later it describes the transit to late de-Sitter epoch.
The derivative coupling to curvature is strong at early time to drive inflation since the coupling constant acts as
another cosmological constant $\Lambda_{\rm NMDC}$. At late time the scalar field behaves like dark matter and the cosmological constant (or the constant potential)
together with the NMDC term (with little effect) drives the present acceleration <cit.>.
Dynamical analysis shows that for positive potential, the positive coupling gives unbound $\dot{\phi}$ value with restricted Hubble parameter <cit.>.
Indeed when considering constant potential and positive coupling, inflationary phase is always possible and the inflation depends solely on the value of coupling constant.
During inflation, gravitational heavy particles are less produced, if having stronger NMDC couplings to the inflaton field or to the particles <cit.>.
Perturbations analysis and inflationary analysis of the model with a constant potential considered as a cosmological constant was performed in
<cit.> to confront observational data.
§.§ Model with negative-sign NMDC
The model is related (by Germani and Kehagias in 2011 <cit.>)
to natural inflation of which pseudo-Nambu-Goldstone boson slowly rolling to create inflation
as well as related to three-form inflation <cit.>.
The model is related to Higgs inflation with $V(\phi) \sim \lambda \phi^4$ which is a NMDC coupling to gravity modification at tree-level of Higgs field <cit.>.
The Lagrangian looks similar to Sushkov's action but the free kinetic term and the NMDC term have opposite sign to each other, i.e. $g^{\mu \nu} - {G^{\mu\nu}}/M^2$.
The model gives a UV-protected inflation and enhances friction of the field dynamics gravitationally
<cit.>. The model is found to be a special case of the Fab Four theory (see, e.g. <cit.>).
Inflationary scenario of the model with quadratic potential and modifications of standard reheating by the NMDC term is found by Sadjadi and Goodarzi in 2013
<cit.>. Tsujikawa in 2012 showed that, due to gravitational friction produced by the NMDC, even with steep potentials,
a class of inflationary potentials is compatible with observation <cit.>.
Particle production of this action after inflation is reported in
<cit.> and one slow roll parameter is necessary for describing inflation <cit.>.
The NMDC coupling contributes to high-field friction making the energy scale reduce to sub-Planckian therefore more consistent to observation <cit.>.
The model is also investigated without free kinetic term for inflation <cit.>.
As dark energy, this model with matter term and a power-law potential is possible to give phantom crossing <cit.>.
Power-law quintessence potential $V_0 \phi^n$ gives rise to oscillatory dark energy. The oscillatory NMDC quintessence satisfies
EoS observational value for $n < 2$ <cit.> however inconsistencies are also reported in <cit.>.
Applying exponential and power-law potentials, perturbation analysis with combined SN Ia, BAO and CMB shows that NMDC coupling term
has very small effect on late acceleration if it is needed to satisfy instability avoidance. This suggests that the coupling needs to be small,
making $9 \kappa H^2$ term in the Friedmann equation small. Hence it behaves like quintessence at late time as it is driven by the potential.
However at early time the NMDC coupling plays major role in driving the acceleration due to large $H$ value at inflation
<cit.>. Phase space analysis for the case of exponential potential was performed in <cit.>. Static black hole scalar field solution of the model is found to be time dependent
<cit.>. Other investigation of the model on black holes and neutron stars can be seen in, e.g. <cit.>.
§ EQUATIONS OF MOTION
In this work, we consider the Sushkov's model which takes the action <cit.>,
S = ∫d^4x√(-g)ł[R8πG - ł(εg_μν + κG_μν)̊ϕ^,μϕ^,ν - 2V(ϕ)]̊ + S_m,
where $R$ is the Ricci scalar, $g$ is the determinant of metric tensor
$g_{\mu\nu}$, $G$ is the universal gravitational constant, $G_{\mu\nu}$ is the
Eintein tensor, $\phi$ is the scalar field, $V(\phi)$ is the scalar field
potential, $S_{\rm m}$ is ordinary matter action, $\varepsilon$ is a
constant with values $+1(-1)$ for canonical (and phantom)
scalar field, $\kappa > 0$ is the coupling constant as in <cit.>. Our universe is assumed to be a
spatially flat FLRW, with the metric
ṣ^2 = -c^2 ṭ^2 + a^2(t) x̣^2,
where $a(t)$ is the scale factor and $\d x^2$ is Euclidian metric.
Varying the action in Eq.(<ref>) with respect to metric tensor
$g_{\mu\nu}$ using line element in Eq. (<ref>) we obtain
3H^2 = 4πGϕ̇^2(ε- 9κH^2) + 8πGV(ϕ) + 8πGρ_m,
where $H$ is the Hubble parameter and $\rho_{\rm m}$ is the energy density of
matter. The Hubble parameter is a function of time $t$ and defined in a form
H = H(t) = {\dot{a}(t)}/{a(t)}
The acceleration equation takes the form,
2Ḣ + 3H^2 = -4πGł[ε+ κł(2Ḣ + 3H^2 +
4Hϕ̈ϕ̇^-1)̊]̊ + 8πGV(ϕ) - 8πGp_m,
where $p_{\rm m}$ is the pressure of matter. The scalar field equation is
ε(ϕ̈ + 3Hϕ̇) - 3κ(H^2ϕ̈ +
2HḢϕ̇ + 3H^3ϕ̇) = -V_,ϕ
where $V_{,\phi} \equiv \d V/ \d \phi$. The Eqs.
(<ref>), (<ref>) and (<ref>) are the dynamical system of the field equations.
We can write
ϕ̈ = - V_,ϕε- 3κH^2 - 3ε- 3κH^2ł(εH - 2κHḢ - 3κH^3)̊ϕ̇,
ϕ̈ = -3Hϕ̇ - V_,ϕε- 3κH^2 + 6κHḢϕ̇ε- 3κH^2.
Subtracting Eq. (<ref>) with (<ref>), we obtain
Ḣ = - 4πGł[ ϕ̇^2 ł( ε+ κḢ - 3κH^2 +
2κHϕ̈ϕ̇^-1 )̊ + p_m + ρ_m ]̊.
From above equations, energy density and pressure of
the scalar field is found to be
ρ_ϕ= 12ϕ̇^2(ε- 9κH^2) + V(ϕ),
p_ϕ= 12ϕ̇^2(ε- 9κH^2)ł[1 +
2κḢ(ε+ 9κH^2)(ε- 3κH^2)(ε-
9κH^2)]̊ - 2κHϕ̇V_,ϕε- 3κH^2 - V(ϕ).
Therefore we find the equation of state parameter as follow
w_ϕ= 12ϕ̇^2(ε- 9κH^2)ł(1 +
2κḢ(ε+ 9κH^2)(ε- 3κH^2)(ε-
9κH^2))̊ - 2κHϕ̇V_,ϕε- 3κH^2 -
V(ϕ)12ϕ̇^2(ε- 9κH^2) + V(ϕ).
Using the Friedmann equation, the potential is found as
V(ϕ) = 3H^28πG - 12(ε- 9κH^2)ϕ̇^2 -
One can check if this is correct by
substituting the scalar field potential in to
Eq.(<ref>) to obtain the usual Friedmann equation,
\rho_\phi + \rho_{\rm m} = {3H^2}/{8\pi G}
From Eq.(<ref>), we see that
ρ_ϕ+ p_ϕ= ϕ̇^2(ε+ κḢ - 3κH^2 + 2κH
Using Friedmann equation and Eq. (<ref>), hence Eq. (<ref>)
recovers its general kinematical form,
Ḣ = - 4πG ł[ ł(3H^2/8πG)̊ + p_m + p_ϕ]̊
and the equation of state parameter also recovers general kinematical form,
w_ϕ(H, Ḣ, ρ_m) = - 3H^2 + 2Ḣ + 8πGp_m3H^2 - 8πGρ_m.
Taking time derivative to the Friedmann equation (<ref>), hence
Ḣ = - 4πG3Hł[-ϕ̇ϕ̈(ε- 9κH^2) +
9κHḢϕ̇^2 - V_,ϕϕ̇ - ρ̇_m ]̊.
Using the continuity equation of matter, $\dot{\rho} = - 3H\rho$, with dust matter ($w_{\rm m} =
0$) to Eq.(<ref>), Eq.(<ref>) becomes
Ḣ = - 4πGł[ ł{ (ε- 9κH^2) - 2κḢ(ε-
9κH^2)(ε- 3κH^2) + 3κḢ}̊ ϕ̇^2 - 2κH V_,ϕϕ̇ε- 3κH^2 + ρ_m]̊.
Rearrange to obtain the kinetic term,
ϕ̇^2 = 2κHV_,ϕϕ̇ε- 3κH^2 - ρ_m -
Ḣ4πG(ε- 9κH^2) - 2κḢł(ε-
9κH^2ε- 3κH^2)̊ + 3κḢ.
Considering the case with constant potential, or equivalently a cosmological constant term, $V(\phi) = \Lambda/(8 \pi G)$ in
the system, with dust and scalar field term (both free kinetic term and the NMDC term),
the Friedmann equation can be written as
H^2 = H_0^2 ł[ Ω_Λ, 0 + Ω_m, 0a^3
+ Ω_ϕ, 0(ε- 9 κH^2)a^6 (ε- 3 κH^2)^2 ]̊
where $\Omega$ are density parameters of each component of cosmic fluids.
The system (<ref>), (<ref>) and (<ref>) with $\dot{\phi} = \psi(t) $ in absence of potential and barotropic fluid is a closed
autonomous dynamical system. An interesting particular solution of this system is when $\dot{\psi}_{\rm p} = 0 = \ddot{\phi}$
where $\psi \equiv \dot{\phi}$ hence $\psi_{\rm p} = \dot{\phi} = \text{constant}$.
As found in <cit.>, that the solution is a de-Sitter type. For the case of $\kappa \equiv \kappa_2 = -2 \kappa_1$, as of Sushkov's model, the solution gives,
H^2 = Λ_NMDC3 .
The effective cosmological constant is defined as
Λ_NMDC = ε κ
The solution is found as $
\psi_{\rm p} = \dot{\phi} = {1}/{\sqrt{\kappa}}
which is
ϕ_p = t√(κ) + ϕ_0
suggesting that the coupling constant should take a positive value and the effective cosmological
constant, $\Lambda_{\rm NMDC} $ should be positive. However general consideration
in <cit.> the NMDC term is strong at early time hence gives new inflation mechanism
that transition from a quasi-de-Sitter phase to power-law phase happens naturally.
Having constant $V = \Lambda/(8 \pi G)$, at late time, the transition from quasi-de-Sitter to de-Sitter phase is also possible.
The particular solution suggests that $\Lambda_{\rm NMDC} > 0$. Therefore, in presence of the usual
cosmological constant (or constant $V$), both $\Lambda$ and $\Lambda_{\rm NMDC}$
contribute both at late time. In order to have enough inflation, $\kappa$ is estimated to $10^{-74}\;{\sec}^{2}$.
Although $\Lambda_{\rm NMDC} \approx 10^{74}\;{\sec}^{-2}$ seems to be large, the NMDC term is suppressed
by its multiplication with curvature which is very small at late time.
§ RESULTS
We estimate that the present universe in very recent range of $z$ evolves as power-law $
a = a_0 \l({t}/{t_0}\r)^\alpha
$ for $\varepsilon = +1$.
Here $a_0$ is scale factor at a present time, $t_0$ is age of the universe and $\alpha$ is constant exponent. The power-law expansion has been considered widely in astrophysical observations, see e.g.
<cit.> (see also <cit.> for constraints).
It is realized as an attractor solution of a canonical scalar field evolving under exponential potential
<cit.> and solution of a barotropic fluid-dominant universe.
Space is under acceleration if $\alpha > 1$. We consider constant $\alpha$ in
a range $0 < \alpha < \infty$. Hence,
$ \dot{a} = {\alpha a}/{t},
and the acceleration is
\ddot{a} = {\alpha(\alpha-1)a}/{t^2}
The Hubble parameter and its time derivative are
H= {\dot{a}}/{a} = {\alpha}/{t},
$ and $
\dot{H} = - {\alpha}/{t^2}.
The value of $\alpha$ can be evaluated with data from gravitational lensing
statistics <cit.>, compact radio source <cit.>,
X-ray gas mass fraction measurements of galaxy cluster <cit.>.
Values of $\alpha$ from various observational data are listed in <cit.>.
To calculate $\alpha$ at the present we use $\alpha = H_0 t_0$ and dust
density is
\rho_{\rm m} = \rho_{\rm m,0}\l({t_0}/{t}\r)^{3\alpha},
where $\rho_{\rm m,0}$ is the dust density at present.
In the scenario of super-acceleration, i.e. the phantom power-law function for which $\varepsilon = -1$,
a = a_0\l[(t_{\rm s} - t)/(t_{\rm s} - t_0)\r]^\beta,
where $t_{\rm s}$ is the future singularity-the Big-Rip time defined as in <cit.>
t_{\rm s} \equiv t_0 \,+\, {|\beta|}/{H(t_0)},
and $\beta$ is a constant. In this case
\dot{a} = - a_0\beta {(t_{\rm s} - t)^{\beta - 1}}/{(t_{\rm s} - t_0)^\beta}
= - \beta {a}/{(t_{\rm s} - t)},
and cosmic acceleration is,
\ddot{a} = a_0\beta(\beta - 1) {(t_{\rm s} - t)^{\beta - 2}}/{(t_{\rm s} -
t_0)^\beta} = {\beta(\beta - 1)a}/{(t_{\rm s} - t)^2}.
Acceleration requires $\beta < 0$. The Hubble parameter is
H = - {\beta}/(t_{\rm s} - t),
and $
\dot{H} = - {\beta}/{(t_{\rm s} - t)^2}.
At present, $\beta = H_0(t_0 - t_{\rm s})$. Dust density in the
phantom power-law case is
\rho_{\rm m} = \rho_{\rm m,0}\l[(t_{\rm s} - t_0)/(t_{\rm s} - t)\r]^{3\beta}.
At present, $t = t_0$, the Big-Rip time $t_{\rm s}$ can be estimated from
t_s ≈t_0 - 23(1 + w_DE)1H_0√(1 - Ω_m,0)
Here, $w_{\rm DE}$ must be less than $-1$. To derive the above expression the
flat geometry and constant dark energy equation of state are assumed
<cit.>. This type of expansion function with phantom scalar field was considered in <cit.>.
We use
cosmological parameters are from WMAP9
(combined WMAP9+eCMB+BAO+$H_0$) dataset
<cit.>, PLANCK+WP dataset <cit.> and
PLANCK including polarization and other external parameters ($TT,TE,EE$+lowP+Lensing+ext.)
The value of $w_{\rm DE}$ is of the $w$CDM model obtained from observational data. The barotropic density contributes to power-law expansion shape
while the NMDC and $\Lambda$ contributes to de-Sitter expansion, in combination, the expansion function is a mixing
between these two. For the phantom case, the free kinetic part of the Lagrangian has negative kinetic energy, therefore
the combined effect to the expansion should be the phantom-power law (super acceleration) mixing with the de-Sitter expansion.
We will calculate the cosmological constant, $\Lambda$ of the model using observed value of
$w_{\rm DE}$ and using suggested value of $\kappa \approx 10^{-74} \;{\rm sec}^2$ as required by inflation <cit.>.
The coupling constant is regarded as a constant in data analysis.
The derived parameters from observations are shown in Table <ref> while Table <ref> shows
values of variables calculated from observations. Values of cosmological constant in this model using three datasets are shown in Table <ref>.
We show plots of $\Lambda$ versus varying value of the exponents $\alpha$ and $\beta$ in Figs. <ref> and <ref>.
Parameters WMAP9+eCMB+BAO+$H_0$ <cit.> PLANCK+WP <cit.> TT,TE,EE+lowP+Lensing+ext. <cit.>
2*$t_0$ $(4.346(4)\pm 0.018(6)) \times 10^{17}$ sec $(4.360(6)\pm 0.015(1)) \times 10^{17}$ sec $(4.354(9)\pm 0.006(6)) \times 10^{17}$ sec
$13.772 \pm 0.059$ Gyr $13.817 \pm 0.048$ Gyr $13.799 \pm 0.021$ Gyr
2*$H_0$ $(2.245(9) \pm 0.025(9)) \times 10^{-18} {\rm sec}^{-1}$ $(2.18(1)\pm 0.03(8)) \times 10^{-18} {\rm sec}^{-1}$ $(2.195(1)\pm 0.014(9)) \times 10^{-18} {\rm sec}^{-1}$
$69.32 \pm 0.80$ km/s/Mpc $67.3 \pm 1.2$ km/s/Mpc $67.74 \pm 0.46$ km/s/Mpc
$\Omega_{\rm m,0}$ $0.2865_{-0.0095}^{+0.0096}$ $0.315_{-0.018}^{+0.016}$ $0.3089 \pm 0.0062$
$\rho_{\rm c,0}$ $(9.019(6)\pm 0.208(8))\times 10^{-27} {\rm kg/m}^3$ $(8.50(6)\pm 0.14(8))\times 10^{-27} {\rm kg/m}^3$ $(8.618(6)\pm 0.117(0))\times 10^{-27} {\rm kg/m}^3$
$\rho_{\rm m,0}$ $(2.584(1)_{-0.145(5)}^{+0.146(4)})\times 10^{-27} {\rm kg/m}^3$ $(2.67(9)_{-0.19(9)}^{+0.18(3)})\times 10^{-27} {\rm kg/m}^3$ $(2.662(3) \pm 0.089(6))\times 10^{-27} {\rm kg/m}^3$
$w_{\rm DE}$ (of $w$CDM) $-1.073_{-0.089}^{+0.090}$ $-1.49_{-0.57}^{+0.65}$ $-1.019_{-0.080}^{+0.075}$
Derived parameters from the combined WMAP9 (WMAP9+eCMB+BAO+$H_0$), PLANCK+WP and TT,TE,EE+lowP+Lensing+external data.
Parameters WMAP9+eCMB+BAO+$H_0$ PLANCK+WP TT,TE,EE+lowP+Lensing+ext.
$\alpha$ $0.9761(6) \pm 0.0154(3)$ $0.951(0) \pm 0.019(9)$ $0.9559(4) \pm 0.0079(4)$
$q_{\rm power-law}$ $0.0244(2)\pm 0.0161(9)$ $0.0515(2)\pm 0.0220(0)$ $0.04613(4)\pm 0.00868(9)$
2*$t_{\rm s}$ $(5.248(1)_{-5.990(1)}^{+6.056(1)})\times 10^{18}$ sec $(1.19(0)_{-0.91(1)}^{+1.03(2)})\times 10^{18}$ sec $(1.96(6)_{-8.13(4)}^{+7.62(8)})\times 10^{19}$ sec
$166.2(9)_{189.8(0)}^{191.8(9)}$ Gyr $37.7(1)_{-28.8(7)}^{+32.7(0)}$ Gyr $622.9(4)_{-2577.3(1)}^{+2416.9(8)}$ Gyr
$\beta$ $-10.81(1)_{-13.58(1)}^{+13.73(1)}$ $-1.64(4)_{-2.01(8)}^{+2.28(2)}$ $-42.1(9)_{-178.9(9)}^{+167.8(8)}$
$q_{\rm phantom}$ $-1.0925(0)_{-0.1162(0)}^{+0.1174(8)}$ $-1.6082(7)_{-0.7440(6)}^{+0.8443(3)}$ $-1.0237(0)_{-0.1005(5)}^{+0.0943(1)}$
Expansion derived parameters from the three datasets
Parameters WMAP9+eCMB+BAO+$H_0$ PLANCK+WP TT,TE,EE+lowP+Lensing+ext.
$\Lambda_{\rm (\varepsilon = +1)}\; ({\rm sec}^{-2})$ $-8.5194(5)_{-4.5365(9)}^{+43.5168(8)}\times 10^{-35}$ $-1.3997(8)_{-0.6130(8)}^{+4.5685(7)}\times 10^{-35}$ $-2.9833(7)_{-2.3899(2)}^{+3.9590(8)}\times 10^{-34}$
$\Lambda_{\rm (\varepsilon = -1)} \; ({\rm sec}^{-2})$ $7.4792(3)_{-21.2910(5)}^{+38.2550(6)}\times 10^{-35}$ $2.6114(3)_{-32.4699(7)}^{+8.8452(5)}\times 10^{-35}$ $2.4939(1)_{-2.0660(4)}^{+3.3466(3)}\times 10^{-34}$
Value of the cosmological constant with power-law expansion (using $\varepsilon = +1$) and phantom power-law expansion (using $\varepsilon = -1$)
for each of observational data.
Parametric plots of $\Lambda$ versus $\alpha$ in a power-law expansion
Parametric plots of $\Lambda$ versus $\beta$ in a phantom power-law expansion
§ CONCLUSIONS
In this work we give a brief review of the canonical scalar field model with non-minimum derivative coupling to curvature in cosmology. Of our interest in
Sushkov's model <cit.>, we consider the case when the potential is constant, i.e. $V = \Lambda/ (8 \pi G)$ and the coupling constant is positive.
The NMDC coupling term behaves like an effective cosmological constant, $\Lambda_{\rm NMDC} = \varepsilon/\kappa$.
Hence the NMDC term together with the free kinetic term contributes to de-Sitter like acceleration to the dynamics in the slow-roll regime at early time, i.e. inflation.
At late time the NMDC contribution is very little due to small curvature.
At late time, in presence of barotropic matter term and cosmological constant, we use observational data from
WMAP9+eCMB+BAO+$H_0$, PLANCK+WP and TT,TE,EE+lowP+Lensing+external data to find cosmological constant of the theory, modeled with power-law
and super-acceleration (phantom power-law) expansion functions. We estimate that the universe kinematically expands with power-law or super acceleration
only from very recent redshifts.
For power-law expansion, the results are $\Lambda = -8.52 \times 10^{-35}\; {\rm sec}^{-2}$ (combined WMAP9), $-1.40\times 10^{-35}\; {\rm sec}^{-2}$ (PLANCK+WP) and
$\Lambda = -2.98 \times 10^{-34}\; {\rm sec}^{-2}$ (TT,TE,EE+lowP+Lensing+external data). These are of the same order as of $\Lambda$CDM model but negative.
Hence in this model, to have power-law expansion, the cosmological constant must be negative. Hence the power-law expansion is not suitable for modeling NMDC cosmology.
For the super-acceleration (phantom) expansion, the results are
$\Lambda = 7.48 \times 10^{-35}\; {\rm sec}^{-2}$ (combined WMAP9),
$\Lambda = 2.61 \times 10^{-35}\; {\rm sec}^{-2}$ (PLANCK+WP) and
$\Lambda = 2.49 \times 10^{-34}\; {\rm sec}^{-2}$ (TT,TE,EE+lowP+Lensing+external data). The value is very sensitive to $t_{\rm s}$ which has large error bar.
§ ACKNOWLEDGMENTS
We thank the reviewers for useful comments. We thank Eleftherios Papantonopoulos for initiating this project and for critical comments.
B. G. is supported by National Research Council of Thailand (grant no. R2557B072).
B.G. is also sponsored via the Abdus Salam ICTP Junior Associateship scheme which supports his visit to ICTP
where this work was partially completed.
P. R. is funded via the TRF's Royal Golden Jubilee Doctoral Scholarship (grant no. PHD/0040/2553).
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§ EQUATION OF STATE PARAMETER FOR POWER-LAW CASE
In this part, we apply the power-law expansion $
a = a_0 \l({t}/{t_0}\r)^\alpha
$ to the NMDC cosmology. The equation of state parameter in Eq. (<ref>) takes the form
w_ϕ=ϕ̇^2(t^2 - 9κα^2)ł[1 - 2κα(t^2 +
9κα^2)(t^2 - 3κα^2)(t^2 - 9κα^2)]̊ -
4καϕ̇V_,ϕt^3t^2 - 3κα^2 - 2V(ϕ)t^2ϕ̇^2(t^2 -
3κα^2) + 2V(ϕ)t^2.
Eq.(<ref>) takes the form,
ϕ̇^2 = F_1(t, ϕ, ϕ̇)(t^2 - 9κα^2).
Substituting Eq.(<ref>) into the equation of state parameter, Eq.(<ref>), we obtain
w_ϕ= F_1(t, ϕ, ϕ̇) ł[1 - 2κα(t^2 + 9κα^2)(t^2 - 3κα^2)(t^2 - 9κα^2)]̊ - 4καV_,ϕϕ̇t^3(t^2 - 3κα^2) - 2V(ϕ)t^2 F_1(t, ϕ, ϕ̇) + 2V(ϕ)t^2
F_1(t, ϕ, ϕ̇) = 2καV_,ϕϕ̇t^3(t^2 - 9κα^2) - ρ_m,0t_0^3α(t^3α- 2) + α(4πG)1 - κα(t^2+9κα^2)(t^2 - 3κα^2)(t^2 - 9κα^2)
§ EQUATION OF STATE PARAMETER FOR PHANTOM POWER-LAW CASE
Apply the phantom power-law expansion (super-acceleration),
a = a_0\l[(t_{\rm s} - t)/(t_{\rm s} - t_0)\r]^\beta,
The kinetic term can be written as
ϕ̇^2 = - F_2(t, ϕ, ϕ̇)ł[(t_s - t)^2 + 9κβ^2]̊
The equation of state parameter of a phantom power-law expansion is
w_ϕ= F_2(t, ϕ, ϕ̇) ł[1 + 2κβł[(t_s - t)^2 - 9κβ^2]̊ł[(t_s - t)^2
+ 3κβ^2]̊ł[(t_s - t)^2 + 9κβ^2]̊]̊ - 4κβV_,ϕϕ̇(t_s - t)^3ł[(t_s - t)^2 +
3κβ^2]̊ - 2V(ϕ)(t_s - t)^2 F_2(t, ϕ, ϕ̇) + 2V(ϕ)(t_s - t)^2
F_2(t, ϕ, ϕ̇) = 2κβV_,ϕϕ̇(t_s - t)^3(t_s - t)^2 + 3κβ^2
- ρ_m,0(t_s - t_0)^3β(t_s - t)^3β- 2 + β4πG1 + κβł[(t_s - t)^2
- 9κβ^2]̊ ł( ł[(t_s - t)^2 + 3κβ^2]̊ł[(t_s - t)^2 + 9κβ^2]̊ )̊
With constant potential in form of $
V(\phi) = {\Lambda}/{8\pi G}
hence $
V_{, \phi} = 0$ for both cases.
|
1511.00578
|
Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci
32, I-20133 Milano, Italy
Suppression of wave scattering and the realization of transparency effects in engineered optical media and surfaces have attracted great attention in the past recent years. In this work the problem of transparency is considered for optical wave propagation in a nonlinear dielectric medium with second-order $\chi^{(2)}$ susceptibility. Because of nonlinear interaction, a reference signal wave at carrier frequency $\omega_1$ can exchange power, thus being amplified or attenuated,
when phase matching conditions are satisfied and frequency conversion takes place. Therefore, rather generally the medium is not transparent to the signal wave because of 'scattering' in the frequency domain. Here we show that broadband transparency, corresponding to the full absence of frequency conversion in spite of phase matching, can be observed for the signal wave in the process of sum frequency generation whenever the effective susceptibility $\chi^{(2)}$ along the nonlinear medium is tailored following a suitable spatial apodization profile and the power level of the pump wave is properly tuned. While broadband transparency is observed under such conditions, the nonlinear medium is not invisible owing to an additional effective dispersion for the signal wave introduced by the nonlinear interaction.
42.65.Ky, 42.79.Nv, 42.25.Fx
§ INTRODUCTION
In the past decade considerable research efforts have
been devoted in developing synthetic materials appropriately
engineered to mold the flow of light in unprecedented ways, opening the way to
several important applications. A noteworthy example is provided by the possibility to suppress
wave reflection and scattering from inhomogeneities or surfaces in engineered optical media (see, for instance, <cit.> and reference therein).
Optical waves propagating in linear but inhomogeneous media generally experience reflection and scattering when the material properties rapidly change over a distance of the order of the optical wavelength <cit.>. In
one-dimensional purely dielectric systems, wave scattering suppression can be achieved by
tailoring the optical refractive index to realize reflectionless potentials. For
continuous media, the synthesis of reflectionless potentials was investigated in a pioneering work
by Kay and Moses in 1956 <cit.>, and then studied in great
detail in the context of the inverse scattering theory
<cit.> and supersymmetric quantum mechanics <cit.>, with applications to e.g. broadband omnidirectional antireflection coatings <cit.> and transparent optical intersections <cit.>. Exploiting the imaginary part of the dielectric permittivity $\epsilon$ (i.e. absorption) in addition to its real part, unidirectional antireflection can be also realized <cit.>.
In the electromagnetic domain, the full access to four quadrants of the
real $\epsilon-\mu$ plane by means of sub-wavelength structured metamaterials <cit.>, in connection with methods inspired by transformation optics <cit.>, has widely
extended the possibilities of controlling and suppressing wave scatting, with the demonstration of amazing phenomena like metamaterial cloaking and invisibility (for recent reviews in this broad research field see, for instance, <cit.>).In this work we consider the problem of transparency of optical waves that propagate in a nonlinear dielectric medium with second-order $\chi^{(2)}$ susceptibility. Because of nonlinear interaction, waves at different carrier frequencies can exchange power and, when phase matching conditions are satisfied, frequency conversion generally occurs <cit.>. In such a medium 'scattering' can be viewed in 'frequency' domain rather than in the spatial one. It is well-known that nonlinear interaction of light waves in a quadratic nonlinear crystal can be exploited to properly control the spectral transmission (both in amplitude and phase) of a given reference wave at carrier frequency $\omega_1$ (signal wave). For example, in a suitably-designed optical parametric amplifier it was shown <cit.> that a narrow transparency window for the signal wave can be opened, leading to superluminal group velocities. Such a narrow transparency effect, associated to superluminal propagation, basically reproduced the gain-assisted transparent pulse propagation experiment by Wang et al. <cit.> in atomic vapours and shares certain similarities with electromagnetically-induced transparency (EIT). The transparency windows that can be opened in a parametric down-conversion process as well as in EIT media, however, is rather narrow. An open question is whether broadband transparency can be realized in a nonlinear optical interaction process. Here we show that, while broadband transparency can not be observed in parametric amplification, it can arise (theoretically with an infinite bandwidth) in an up-conversion process, namely in sum frequency generation (SFG) <cit.> [Fig.1(a)]. To observe broadband transparency in SFG, the effective susceptibility $\chi^{(2)}$ along the nonlinear crystal has to be suitable apodized, which can be realized using well-established quasi-phase-matching (QPM) methods <cit.>. While broadband transparency is observed under such conditions, the nonlinear medium is not invisible, since the nonlinear interaction introduces an effective additional dispersion (phase delay) for the signal wave that can be detected by nonlinear-induced pulse distortion in a transmission experiment.
(Color online) (a) Schematic of sum frequency generation (SFG) in a nonlinear $\chi^{(2)}$ crystal with a periodic QPM grating. A weak signal field at frequency $\omega_1$ interacts with a strong pump field at frequency $\omega_2$ to generate a SFG wave at frequency $\omega_3=\omega_1+\omega_2$. $L$ is the crystal length, $\Lambda$ is the QPM grating period. For first-order QPM grating $\Lambda= 2 \pi / \Delta k$, where $\Delta k =| k_3-k_2-k_1|$ is the wave vector mismatch of the three interacting waves. (b) Coherently-driven two-level atom analogue of the SFG process. The Rabi frequency $q(z)$ of the exciting optical pulse, with carrier frequency detuned by $2 \delta$ from the atomic transition resonance, corresponds to the apodization profile of the QPM grating.
§ SUM FREQUENCY GENERATION: BASIC EQUATIONS AND THE DRIVEN TWO-LEVEL ATOM ANALOGUE
§.§ The model
We consider parametric interaction of three co-propagating waves with carrier frequencies $\omega_1$ (signal wave), $\omega_2$ (pump wave) and $\omega_3=\omega_1+\omega_2$ (SFG wave) in a nonlinear medium of length $L$ with a second-order $\chi^{(2)}$ nonlinearity, which are phase-matched via a QPM grating [Fig.1(a)]. The pump field at frequency $\omega_2$ is assumed to be a strong and continuous-wave field, whereas the signal at carrier frequency $\omega_1$ injected into the medium as well as the SFG wave are assumed weak but arbitrarily broadband. In the effective plane-wave
approximation and taking into account material dispersion, from Maxwell's equations the electric field $\mathcal{E}(z,t)$ is the medium is found to satisfy the nonlinear and dispersive wave equation (see, for instance, <cit.>)
\begin{equation}
\frac{\partial^2 {\cal E}}{\partial z^2}+ \int_{-\infty}^{\infty} d \omega k^2(\omega)
\tilde {\cal E}(z,\omega) \exp(-i \omega t) =\mu_0 \frac{\partial^2 {\cal
P}^{NL}}{\partial t^2} \; ,
\end{equation}
where $\tilde {\cal E}(z,\omega)=(2 \pi)^{-1} \int_{-\infty}^{\infty} d\omega {\cal
E}(z,t) \exp(i \omega t)$ is the Fourier transform of ${\cal E}(z,t)$,
$k(\omega)=(\omega / c_0) \sqrt {1 + \tilde \chi(\omega)}=(\omega/c_0)n(\omega)$ is the
dispersion relation defined by the complex linear susceptibility $\tilde {\chi}
(\omega)$ [or by the complex refractive index $n(\omega)= \sqrt{1+\tilde
{\chi}(\omega)}$], $c_0$ is the speed of light in vacuum, $\mu_0$ is the vacuum
magnetic permeability, and ${\cal P}^{NL}$ is the nonlinear driving polarization term.
For a quadratic medium and neglecting dispersion and absorptive effects of second-order polarization,
one can take ${\cal P}^{NL}(z,t)=\epsilon_0 \chi^{(2)}(z) {\cal E}^2(z,t)$, where
$\chi^{(2)}$ is the spatially-modulated nonlinear susceptibility that accounts for the
QPM grating. To study the process of SFG, the electric field ${\cal E}(z,t)$
is assumed to be given by the superposition of three wave trains with carrier frequencies
$\omega_1$ (signal field), $\omega_2$ (pump field) and $\omega_3=\omega_1+\omega_2$ (SFG field), co-propagating along the longitudinal $z$ direction.
Phase matching is accomplished by a QPM grating, i.e. the susceptibility $\chi^{(2)}$ is a quasi-periodic
function of $z$ with period $\Lambda$
\begin{equation}
\chi^{(2)}(z)= \sum_{n=-\infty}^{\infty} \chi^{(2)}_{n}(z) \exp(-2 i n \pi z / \Lambda)
\; ,
\end{equation}
where the Fourier coefficients ${\chi^{(2)}_{n}(z)}$ are slowly varying functions of
$z$ over one period $\Lambda$. In practice, the slow dependence of coefficients on $z$
can be achieved by a +/- reversal of domains in the ferroelectric crystal with a local
period and local duty cycle that are slowly varying along the $z$ axis; methods to apodize the QPM grating are described and demonstrated, for
instance, in <cit.>. For first-order QPM, the grating period $\Lambda$ satisfies the condition
$\Lambda=2 \pi / \Delta k$, where $\Delta k \equiv |k_3-k_2-k_1|$ is the wave vector mismatch of interacting waves and $k_l=k(\omega_l)$ ($l=1,2,3$).
After setting
\begin{eqnarray}
{\cal E}(z,t) & = & \frac{1}{2}\left\{ A_1(z,t) \exp(-i \omega_1 t+i k_1z) \nonumber \right. \\
& + & A_2 (z,t)
\exp(-i\omega_2 t+i k_2 z) \\
&+ & A_3(z,t) \exp(-i\omega_3 t+ik_3z) +c.c. \left. \right\} \; ,
\end{eqnarray}
the evolution equations of the slowly-varying envelopes
$A_l(z,t)$ ($l=1,2,3$) can be derived, in the limit of weak nonlinearity and
quasi-monochromatic wave trains, by a multiple-scales asymptotic expansion analysis (see, for
instance, <cit.>). The resulting equations read <cit.>:
\begin{eqnarray}
2ik_1 \frac{\partial A_1}{\partial z}=\left[ k_{1}^{2}-k^2(\omega_1+i \partial_t)
\right] A_1-\frac{2k_{1}^{2}}{n_{1}^{2}}d_{eff}A_{2}^{*}A_3 \;\;\;\;\;\; \\
2ik_2 \frac{\partial A_2}{\partial z}=\left[ k_{2}^{2}-k^2(\omega_2+i \partial_t)
\right] A_2-\frac{2k_{2}^{2}}{n_{2}^{2}}d_{eff}A_{1}^{*}A_3 \;\;\;\;\;\; \\
2ik_3 \frac{\partial A_3}{\partial z}=\left[ k_{3}^{2}-k^2(\omega_3+i \partial_t)
\right] A_3-\frac{2k_{3}^{2}}{n_{3}^{2}}d_{eff}^{*} A_1A_2 \;\;\;\;\;\;
\end{eqnarray}
where $n_l=n(\omega_l)$ ($l=1,2,3$) are the refractive indices at the three carrier wavelengths,
\begin{equation}
d_{eff}(z) \equiv \frac{1}{2} \overline {\chi^{(2)}(z) \exp(i \Delta k z) }=
\frac{1}{2} \chi^{(2)}_{1}(z),
\end{equation}
is the effective nonlinear interaction coefficient for first-order QPM, and the overline denotes a spatial average over a few modulation periods of the QPM
grating. For a square-wave (+/-) modulation of the ferroelectric domains with 50$\%$ duty cycle and uniform period, one has
\begin{equation}
d_{eff}(z)=\frac{2} {\pi} d_0 W(z),
\end{equation}
where $d_0$ is the nonlinear coefficient in the absence of the grating and the real envelope $W$, with $0 \leq W(z) \leq 1$, can be tailored rather arbitrarily with the methods demonstrated in Ref.<cit.>, for example by means of the domain cancellation technique.
The linear operators on the right hand side of Eqs.(5) describe the
linear dispersive and absorptive properties of the medium at any order of approximation. In the following, we will consider spectral
regions of transparency for the medium, so that we will neglect the imaginary part of
$k(\omega)$. In addition, we will assume a strong and continuous-wave pump field, so that $A_2$ can be taken to be constant (independent of space and time) in Eqs.(5a) and (5c). Under the no-pump-depletion approximation, one can thus write
\begin{eqnarray}
2ik_1 \frac{\partial A_1}{\partial z}=\left[ k_{1}^{2}-k^2(\omega_1+i \partial_t)
\right] A_1-\frac{2k_{1}^{2}}{n_{1}^{2}}d_{eff}A_{2}^{*}A_3 \;\;\;\;\;\; \\
2ik_3 \frac{\partial A_3}{\partial z}=\left[ k_{3}^{2}-k^2(\omega_3+i \partial_t)
\right] A_3-\frac{2k_{3}^{2}}{n_{3}^{2}}d_{eff}^{*} A_1A_2 \;\;\;\;\;\;
\end{eqnarray}
§.§ Driven two-level atom analogy
The coupled equations (8a) and (8b) describing the SFG process in the no-pump-depletion approximation, when written for monochromatic waves, bear a close analogy to the optical Bloch equations describing the dynamics of a two-level atomic system driven by a near-resonant optical pulse. Such an analogy, which is fruitful for the prediction of the transparency effect presented in the next section, has been previously discussed in the monochromatic case in Ref.<cit.> and applied to efficient broadband SFG based on the analogue of rapid adiabatic passage using chirped QPM gratings <cit.>. Other analogies between multi-step frequency conversion processes in nonlinear second-order optical media and coherent population transfer in coherently-driven multi-level atomic systems, including stimulated Raman adiabatic passage, have been highlighted in the recent literature as well <cit.>.
To show the equivalence of Eqs.(8a) and (8b) with the optical Bloch equations of a driven two-level atom <cit.>, let us consider a monochromatic signal wave with frequency offset $\Omega$ from the reference frequency $\omega_1$. Since Eqs.(8a) and (8b) are linear ones, the general case of an incident pulsed wave is obtained by standard Fourier analysis starting from the solution of the monochromatic case. After setting in Eqs.(8a) and (8b)
\begin{eqnarray}
A_1(z,t) & = & u(z) \exp[-i \Omega t+i \beta(\Omega) z] \\
A_2(z,t) & = & \frac{n_1}{n_3} \sqrt{\frac{k_3}{k_1}} v(z) \exp[-i \Omega t+i \beta(\Omega) z]
\end{eqnarray}
\begin{equation}
\beta(\Omega) \equiv -\frac{k_1^2-k^2(\omega_1+\Omega)}{4 k_1}-\frac{k_3^2-k^2(\omega_3+\Omega)}{4 k_3}
\end{equation}
one obtains the following coupled equations for the amplitudes $u(z)$ and $v(z)$
\begin{eqnarray}
i \frac{du}{dz} & = - \delta u -q(z) v \\
i \frac{dv}{dz} & = \delta v -q^*(z) u
\end{eqnarray}
where we have set
\begin{eqnarray}
q(z) & = & \frac{\sqrt{k_1 k_3}A_2^* d_{eff}(z)}{n_1n_3} \\
\delta & = & \delta(\Omega)=\frac{k_3^2-k^2(\omega_3+\Omega)}{4 k_3}-\frac{k_1^2-k^2(\omega_1+\Omega)}{4 k_1}. \;\;\;\;\;
\end{eqnarray}
Equations (11) are analogous to the optical Bloch equations for a driven two-level atom describing the transition between the two atomic levels induced by a nearly resonant optical pulse with Rabi frequency $q(z)$ and frequency detuning $2 \delta$ [Fig.1(b)] <cit.>. Note that the detuning $\delta$, as given by Eq.(13), accounts for material dispersion at any order. A simple expression of $\delta(\Omega)$ is obtained when group velocity dispersion (and higher-order dispersion effects) are negligible, and $k^2(\omega_l + \Omega)$ can be expanded in power series up to first order in $\Omega$. After setting $k^2(\omega_l+\Omega) \simeq k_l^2+2 k_l (\omega_l) \Omega / v_{gl}$, where $v_{gl}=(d k / d \omega)_{\omega_l}^{-1}$ is the group velocity at the carrier frequency $\omega_{l}$, one simply obtains
\begin{equation}
\delta(\Omega) \simeq \frac{\Omega}{2} \left( \frac{1}{v_{g1}}-\frac{1}{v_{g3}} \right)
\end{equation}
where $v_{g1}$ and $v_{g3}$ are the group velocities of signal and SFG waves, respectively. In the following analysis, we will assume that the QPM grating is not chirped, so that the Rabi frequency $q(z)$ entering in Eqs.(11) can be assumed to be real.
As shown in the next section, the broadband transparency effect predicted in this work is based on the two-level atom analogy and existence of off-resonance Rabi pulses $q(z)$, which do not transfer population between the two atomic levels. It should be noted that the two-level atom analogy can be established for the SFG process, but not for other second-order nonlinear interactions like parametric amplification involving a down-conversion process. In the latter case, which was considered in <cit.>, the underlying equations for idler and signal waves differ from Eqs.(11) because of the replacement $ - q^*(z)u \rightarrow q^*(z)u$ on the right hand side of Eq.(11b). The resulting coupled equations describe an exponential (rather than oscillatory) behavior of interacting waves, and are similar to coupled-mode equations found in Bragg scattering theory of counter-propagating waves <cit.>. As a result, broadband transparency effects are prevented in parametric amplification, where only narrow transparency windows can be opened in the spectral gain curve and associated to superluminal group velocities <cit.>.
§ TRANSPARENCY IN SUM FREQUENCY GENERATION
§.§ Theoretical analysis
The solution to Eqs.(11), from the input $z=-L/2$ to the output $z=L/2$ planes of the nonlinear medium, can be written in the general form
\begin{equation}
\left(
\begin{array}{c}
u(L/2) \\
\end{array}
\right)=
\left(
\begin{array}{cc}
\mathcal{M}_{11}(\delta) & \mathcal{M}_{12}(\delta) \\
\mathcal{M}_{21}(\delta) & \mathcal{M}_{22}(\delta)
\end{array}
\right) \times \left(
\begin{array}{c}
u(-L/2) \\
\end{array}
\right)
\end{equation}
where the transfer matrix $\mathcal{M}$ is unimodular with $\mathcal{M}_{11}=\mathcal{M}_{22}^*$, $\mathcal{M}_{21}=-\mathcal{M}_{12}^*$ and $|\mathcal{M}_{11}|^2+|\mathcal{M}_{12}|^2=1$. For signal excitation at the input plane $z=-L/2$, i.e. for $v(-L/2)=0$, the spectral transmission of the signal wave is simply given by
\begin{equation}
t(\Omega)=\left( \frac{u(L/2)}{u(-L/2)}\right)_{v(-L/2)=0} =\mathcal{M}_{11}.
\end{equation}
Note that $t(\Omega)$ can be factorized as
\begin{equation}
t(\Omega)=t_0(\Omega) \exp[i \delta(\Omega)L],
\end{equation}
where $\exp[i \delta(\Omega)L]$ is the spectral transmission (phase delay) introduced by the medium in the absence of the nonlinearity, i.e. for $q(z) \equiv 0$, and $t_0(\Omega)$ accounts for the nonlinear interaction. Broadband transparency is realized provided that
\begin{equation}
\mathcal{M}_{12}(\delta)=\mathcal{M}_{21}(\delta)=0 \; ,\;\; |\mathcal{M}_{11}(\delta)|=|\mathcal{M}_{22}(\delta)|=1
\end{equation}
for any detuning $\delta$, i.e. $|t_0(\Omega)|=1$ for any frequency offset $\Omega$ of the carrier wave from the reference frequency $\omega_1$. This means that for an arbitrary optical signal pulse propagating into the nonlinear medium no SFG field is produced at the output of the crystal in spite of phase matching. We note that invisibility is a more stringent condition than transparency, since it requires $t_0(\Omega)=1$ for any frequency $\Omega$. If the nonlinear medium is transparent but the phase of $t_0(\Omega)$ is not flat, a propagating signal pulse in the medium would suffer for an additional phase delay arising from the nonlinear interaction, resulting in pulse distortion as compared to the invisible regime $\chi^{(2)}=0$ of linear propagation.
A necessary condition for the observation of transparency can be readily established as follows. Exact solution to the Bloch equations (11) is available at exact resonance $\delta=0$ for an arbitrary shape of the Rabi frequency $q(z)$. In fact, for $\delta=0$ one simply has $\mathcal{M}_{11}= \cos \mathcal{A} $, where
\begin{equation}
\mathcal{A}=\int_{-L/2}^{L/2} q(z) dz
\end{equation}
is the 'area' of the driving pulse in the quantum mechanical analogy. Hence transparency at $\delta=0$ requires
\begin{equation}
\mathcal{A}= N \pi
\end{equation}
with $N$ integer. Equation (20) provides a necessary condition for broadband transparency, since it ensures transparency at resonance $\delta=\Omega=0$. However, for a general profile $q(z)$ transparency is not found far from resonance, i.e. for $\delta \neq 0$, especially if there is a non-neglibile group velocity mismatch between signal and sum-frequency waves [see Eq.(14)]. For example, let us consider the simplest case $W(z)=1$, corresponding to a non-apodized (uniform) QPM grating, so that $q(z)=q_0$ constant in the range $-L/2<z<L/2$. According to Eq.(20), transparency at resonance $\delta=0$ requires $q_0=N \pi/L$. However, for $\delta \neq 0$ the transmittance in not unity. In fact, after a simple calculation one finds
\begin{equation}
|t_0(\delta)|^2=\cos^2 \left( \sqrt{q_0^2+\delta^2} L \right) +\frac{\delta^2}{\delta^2+q_0^2} \sin^2 \left( \sqrt{q_0^2+\delta^2} L \right).
\end{equation}
The question thus arises whether there exist special profiles $q(z)$ such that $|t_0(\delta)|^2=1$ for any vaue of the detuning $\delta$. In the theory of driven two-level atoms, it is known <cit.> that for $q(z)$ of the form
\begin{equation}
q(z)=\frac{q_0}{\cosh( \alpha z)},
\end{equation}
transparency at any value of detuning $\delta$ can be realized whenever the condition (20) on the area is satisfied. The parameter $\alpha$ entering in Eq.(22) can be taken arbitrarily, and its inverse $ 1/ \alpha$ basically determines the characteristic length of nonlinear interaction.
For such a profile of $q(z)$, exact solution for the optical Bloch equations (11) can be obtained in terms of hypergeometric functions <cit.>, and the transparency at special values of the amplitudes $q_0$ satisfying the area condition (20) can be explained in term of supersymmetric quantum mechanics <cit.>. Assuming a medium length $L$ such that $\cosh(\alpha L/2) \gg 1$, the nonlinear correction $t_0(\Omega)$ to the transmission coefficient $t$ can be obtained in a closed form and reads explicitly
\begin{equation}
t_0(\Omega)=\frac{\Gamma(1/2+i \Delta) \Gamma(1/2+i \Delta) }{\Gamma(1/2+i \Delta- \mathcal{A} / \pi) \Gamma(1/2+i \Delta+ \mathcal{A} / \pi)}
\end{equation}
where we have set
\begin{equation}
\Delta \equiv \frac{\delta( \Omega)}{\alpha} \simeq \frac{\Omega}{2 \alpha} \left( \frac{1}{v_{g1}}-\frac{1}{v_{g3}} \right),
\end{equation}
$\mathcal{A}= \pi q_0/ \alpha$ is the area (defined by Eq.(19) with $L \rightarrow \infty$), and $\Gamma(.)$ is the Gamma function. From Eq.(23) the spectral transmittance $T(\Omega)=|t(\Omega)|^2=|t_0(\Omega)|^2$ for the signal wave can be calculated, which reads
\begin{equation}
T(\Omega)=1- \frac{\sin^2 \mathcal{A}}{\cosh^2 ( \pi \Delta)}
\end{equation}
where $\Delta=\Delta(\Omega)$ is given by Eq.(24).
Note that broadband transparency $T=1$ is obtained provided that $\mathcal{A}= N \pi$ with $N$ integer, according to Eq.(20). Once the normalized spatial profile $W(z)$ of the QPM grating is designed according to $W(z)=1/ \cosh( \alpha z)$, from Eqs.(7) and (12) it follows that the transparency condition $\mathcal{A}= N \pi$ is met for special values of the pump amplitude $A_2$. In terms of the intensity $I_2=(1/2) \epsilon_0 c_0 n_2 |A_2|^2$ of the strong pump wave, the transparency condition is satisfied provided that $I_2= N I_{tr}$, where $N$ is an integer number and $I_{tr}$ is the transparency pump intensity given by
\begin{equation}
I_{tr}=\frac{1}{32} \frac{\epsilon_0 c_0 n_1 n_2 n_3 \lambda_1 \lambda_3 \alpha^2}{d_0^2}
\end{equation}
For a pump intensity $I_2=NI_{tr}$, the nonlinear medium is broadband transparent, i.e. no SFG wave is generated at the output of the crystal for any arbitrarily broadband incident signal pulse. In fact, once the area condition (20) is satisfied the transparency bandwidth is in principle infinite according to Eq.(25). In practice, however, deviations of the profile of the effective susceptibility from the ideal sech shape or pump intensity deviations from the transparency value result in the appearance of a spectral region around the phase matching condition where the transmittance is not unitary. For example, if the pump intensity $I_{2}$ is close to but slightly detuned from the transparency value $I_{tr}$, according to Eq.(25)
transparency is not observed in a spectral region with a bandwidth $\Delta \Omega$ determined by the condition $\pi | \Delta| \sim 1$, i.e. by the group velocity mismatch and interaction length $\Delta \Omega \sim (2 \alpha / \pi) |1/v_{g1}-1/v_{g3}|^{-1}$ [see Eq.(24)].
It should be noted that, even thought the transparency condition is met, the nonlinearity of the medium is not invisible since the phase of $t_0(\Omega)$, as given by Eq.(23), is not flat. For example, in the simplest case $N=1$, i.e. for
$\mathcal{A}= \pi$, one has
\begin{equation}
t_0(\Omega)=\frac{\delta+i \alpha/2}{\delta-i \alpha/2}=\exp[i \phi(\Omega)]
\end{equation}
\begin{equation}
\phi(\Omega)=2 {\rm atan} \left( \frac{\alpha}{ 2 \delta( \Omega) }\right) \simeq 2 {\rm atan} \left[ \frac{\alpha v_{g1} v_{g3} }{ \Omega(v_{g3}-v_{g1})} \right].
\end{equation}
The additional phase delay $\phi(\Omega)$ leads to an effective non-linear induced contribution to the linear material dispersion, and can be detected in pulse transmission experiments, as discussed in the next subsection.
(Color online) Numerically-computed (a) spectral transmittance, and (b) phase delay for the signal wave in a 2.5-cm-long PPLN crystal with $\cosh^{-1}(\alpha z)$-apodized profile for $\alpha=5$ cm$^{-1}$ and for increasing values of the pump intensity $I_2$. Curve 1: $I_2=0.5 I_{tr}$; curve 2: $I_2= I_{tr}$; curve 3: $I_2=1.3 I_{tr}$; curve 4: $I_2=2 I_{tr}$; curve 5: $I_2=2.6 I_{tr}$. The pump intensity at transparency is $I_{tr} \simeq 24.37$ MW/cm$^2$.
(Color online) Numerically-computed propagation of a Gaussian signal pulse at carrier wavelength $\lambda_1=1.55 \; \mu$m in a 2.5-cm-long PPLN crystal with $\cosh^{-1}(\alpha z)$-apodized profile for $\alpha=5$ cm$^{-1}$ and for a FWHM pulse width (a) $\Delta \tau_p=23.5$ ps, and (b) $\Delta \tau_p=589$ fs. Curve 1 shows the transmitted pulse intensity distribution of the signal field for a continuous-wave pump intensity $I_2=I_{tr} \simeq 24.37$ MW/cm$^2$, whereas curve 2 is the transmitted pulse distribution of the signal waveform when the pump field is switched off ($I_2=0$, linear propagation regime). The thin dotted curve [almost overlapped with curve 2 in (a)] is the pulse intensity distribution of the weak Gaussian signal pulse at the input plane of the crystal.
§.§ Numerical results
To illustrate the phenomenon of SFG transparency and to provide some design parameters, let us consider as an example nonlinear frequency conversion in a periodically-poled lithium niobate (PPLN) crystal pumped at the wavelength $\lambda_2=810$ nm and probed with a weak signal field
at $\lambda_1=1.55 \; \mu$m. The SFG wave corresponds to the wavelength $\lambda_3=532$ nm. We assume extraordinary wave propagation, corresponding to a nonlinear coefficient $d_0=d_{33} \simeq 27$ pm/V. The temperature-dependent dispersion relation $k=k(\omega)=n(\omega)
\omega / c_0$ for extraordinary waves in lithium niobate is determined using Sellmeier
equations from Ref.<cit.>. At 25$ ^o$C, one can estimate $n_1=2.1381$, $n_2=2.1748$, $n_3=2.2343$, the group velocities
$v_{g1} \simeq 0.4581 c_0$, $v_{g3} \simeq 0.4069 c_0$, and a first-order QPM grating with period $\Lambda=2 \pi / \Delta k=7.38 \; \mu$m, which is accessible with current poling technology.
As an example, Fig.2 shows the numerically-computed transmittance (modulus square of $t_0$) and phase delay (phase of $t_0$) versus wavelength in a $L=2.4$ cm long PPLN crystal for $\alpha=5$ cm$^{-1}$ and for increasing values of the pump intensity $I_2$. The intensity at transparency is given by $I_{tr} \simeq 24.37$ MW/cm$^2$ according to Eq.(26). Note that, for a non-integer value of the normalized pump intensity $I_2/I_{tr}$, SFG is observed in a wavelength range of the signal wave corresponding to phase matching of the nonlinear interaction. This is shown by curves 1,3, and 5 in Fig.2(a), where the spectral transmittance shows a dip near the wavelength of perfect phase matching. The central wavelength of the dip and its width are determined by the phase matching condition in the nonlinear interaction, i.e. by the QPM grating period, the decay length $1/ \alpha$ of the QPM grating, and the material dispersion.
As the ratio $I_2 / I_{tr}$ is an integer number (curves 2 and 4 in Fig.2), there is no SFG wave, i.e. the medium in transparent for the signal wave according to the theoretical analysis. Nevertheless, a wavelength-dependent phase delay is accumulated in the nonlinear interaction [see Fig.2(b)], corresponding to an additional non-linear-induced dispersion term for the signal field. In an experiment, the effect of the nonlinear-induced dispersion at the transparency regime can be detected by comparing the propagation of a short optical pulse along the medium with the pump field switched off and on. This is illustrated in Fig.3, which shows the numerically-computed propagation of a Gaussian input signal pulse $A_2(-L/2,t) \propto \exp[-(t/ \tau_p)^2]$ along the 2.4 cm-long PPLN crystal when the pump intensity is tuned at the transparency value $I_2=I_{tr}$ (curve 1) and when it is switched off $I_2=0$ (curve 2). The pulse duration $\Delta \tau_p$, defined as the full-width at half maximum of the field intensity, is related to $\tau_p$ by the relation $\Delta \tau_p=\sqrt{2 {\rm ln} 2} \tau_p$. For a relatively long input pulse [Fig.3(a), $\Delta \tau_p \simeq 23.5$ ps], the linear dispersion of the medium is negligible, and the nonlinear-induced dispersion is responsible for a time delay of the transmitted pulse, given by the group delay $\tau_d=d (\phi / d \Omega)=2(v_{g1}-v_{g3})/(\alpha v_{g1}v_{g3}) \simeq 3.6$ ps. For shorter pulses [Fig.3(b), $\Delta \tau_p \simeq 589$ fs], the linear dispersion of the medium is non-negligible [curve 2 in Fig.3(b)], and the additional dispersion introduced by the non-linear interaction at $I_2=I_{tr}$ is responsible for strong pulse reshaping. In particular, one can observe pulse splitting with a long pulse tail [curve 1 in Fig.3(b)].
§ CONCLUSIONS AND OUTLOOK
Optical waves propagating in a linear but inhomogeneous medium generally show reflection and scattering when the material properties rapidly change over a distance of the order of the optical wavelength <cit.>. However, proper tailoring of the dielectric permittivity can suppress scattering and the medium thus appears to be transparent <cit.>. A different kind of 'scattering' can occur in the frequency domain when the optical waves propagate in a nonlinear $\chi^{(2)}$ medium. When phase matching conditions are met, efficient frequency conversion can occur, and an optical wave at a reference frequency (signal field) can be amplified or attenuated owing to frequency conversion. Here we have investigated the possibility to realize optical transparency in the process of sum frequency generation in a second-order nonlinear crystal. By exploiting the quantum-optical analogy between the process of SFG in the undepleted pump approximation and the coherent excitation of a two-level atom by a near-resonant pulse with tailored shape and pulse area, we have shown that broadband transparency can be realized in the nonlinear crystal with an engineered QPM grating. Such a result could be of interest in the nonlinear control of material transparency and is expected to motivate further theoretical and experimental studies in the field of transparency and invisibility in nonlinear media. A few natural extensions to the present study can be envisaged. For example, is it possible to engineer the nonlinear interaction to realize one-way transparency? Also, can one tailor the nonlinearity of the medium to make it invisible? One possibility might be to engineer the material properties to allow for an imaginary part of the nonlinear susceptibility <cit.>, i.e. to explore the full domain of complex non-linear susceptibility. In this case, transferring the recent proposal by Horsley and coworkers <cit.> of spatial Kramers-Kronig relations for linear susceptibilities to the nonlinear ones, it would be possible to realize one-way transparency and non-linear invisibility. Other extensions of the present study might be the analysis of transparency and invisibility in two-dimensional QPM gratings, in nonlinear interactions with phase-matched counter-propagating waves <cit.>, and in frequency wave mixing based on third-order nonlinear media.
RESPONSE TO YOUR QUERIES
Q1. There are no funding institutions/organization to acknowledge for this article.
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|
1511.00403
|
1]Ken Kikuchi
1,2]Tadakatsu Sakai
[1]Department of Physics, Nagoya University
[2]KMI, Nagoya University
We revisit a study of local renormalization group (RG)
with background gauge fields incorporated using
the AdS/CFT correspondence.
Starting with a $(d+1)$-dimensional bulk gravity coupled to
scalars and gauge fields, we derive a local RG equation
from a flow equation by working in the Hamilton-Jacobi
formulation of the bulk theory.
The Gauss's law constraint associated with gauge symmetry plays an
important role.
RG flows of the background gauge fields are governed by
vector $\beta$-functions, and some interesting properties
of them are known to follow.
We give a systematic rederivation of them on the basis of
the flow equation.
Fixing an ambiguity of local counterterms in such a manner
that is natural from the viewpoint of the flow equation,
we determine all the coefficients uniquely appearing
in the trace of the stress tensor for $d=4$.
A relation between a choice of schemes and a Virial current
is discussed.
As a consistency check, these are found to satisfy
the integrability conditions of local RG transformations.
From these results, we are led to a proof of a
holographic $c$-theorem by finding out a full family
of schemes where a trace anomaly coefficient is related
with a holographic $c$-function.
§ INTRODUCTION
Coupling `constants' are literally regarded as constants
in ordinary quantum field
theories (QFTs). However, it is an interesting question
to ask what happens when
they have spacetime dependence.
A method called local renormalization group
(RG) puts this idea in practice.
That is, we lift spacetime independent coupling
constants $g$ to spacetime dependent coupling functions
$g(x)$ <cit.>.
The coupling functions can be regarded as external fields.
Correlation functions are thus obtained by functional derivatives
of the generating functional $\Gamma[g(x)]$ of
connected graphs (a.k.a. the Schwinger functional) with respect to $g(x)$.
This rule is called the Schwinger's action principle <cit.>.
In the AdS/CFT correspondence <cit.>,
$\Gamma[g(x)]$ is identified with an on-shell action of a bulk
gravity dual with the external fields corresponding to
boundary values of bulk fields <cit.>.
Studies of RG flows using the AdS/CFT correspondence have been
done extensively so far. In particular, an analysis in
this line using a flow equation was
first made by de Boer, Verlinde and Verlinde <cit.>,
and then generalized in <cit.>. For a review, see
In these papers,
bulk gravity theories coupled to scalar fields with
a generic metric were investigated.
It was revealed that the flow equation of a bulk gravity
yields a local RG equation of $\Gamma[g(x)]$ with
the Weyl anomalies of the boundary QFT reproduced
correctly in this framework.
For recent developments in local RG, see also
One of the purposes of this paper is to generalize these results
by introducing gauge fields in the bulk side.
This is partially motivated by a desire to bring a somewhat
mysterious quantity called a vector $\beta$-function
<cit.> to light.
This characterizes how background gauge fields coupled to
currents flow under RG transformations.
Some of the properties mentioned above were derived from
the Wess-Zumino consistency conditions concerning
local RG transformations. A paper <cit.> found that
the AdS/CFT correspondence explains these properties nicely.
In this paper, we make a systematic rederivation of these
results on the basis of the flow equation.
One of the advantages of our analysis throughout this paper
is to clarify scheme dependence of the results, that is,
how they are affected by a choice of local counterterms.
In particular, we point out a close relationship
between a choice of schemes and a Virial current.
The organization of the paper is as follows.
In section 2, we formulate a $(d+1)$-dimensional gravity
dual model in the Hamilton-Jacobi formalism.
We derive the flow equation and give some
comments on its general aspects. Some suggested properties of
the vector $\beta$-functions such as (i) gradient property,
(ii) orthogonality, (iii) Higgs-like relation between anomalous
dimensions and (iv) the relation between a vanishing vector
$\beta$-function and non-renormalization of current operators can be readily
confirmed. In section 3, we perform explicit calculations in $d=4$.
We obtain closed expressions of anomaly coefficients including
central charges
with an emphasis on a scheme choice we made.
We can see the coefficient functions satisfy integrability conditions
that come from the Wess-Zumino consistency condition associated
with the local RG transformation <cit.>.
For an earlier work in this line, a paper <cit.>
discusses five-dimensional bulk gravity coupled to scalar fields
without gauge fields.
It is shown there that
the anomaly coefficients computed in that model
satisfy the Wess-Zumino consistency conditions.
As an application of our results, a holographic $c$-theorem in
$d=4$ dimensions is proven
by finding out a full family of schemes
where a monotonically decreasing function
under RG flows can be constructed from an anomaly coefficient.
This is an extension of a paper <cit.>.
We collect our notations, some useful formulae and lengthy equations
in appendices.
§ FORMALISM
We start with a bulk action in $(d+1)$-dimensions:
\begin{align}
\mathbf S&\left[\hat\gamma_{\mh\nh}(x,\tau),\hat\phi^I(x,\tau),\Ah_{\mh}(x,\tau)
\right]\nonumber\\
=&\int_{M_{d+1}}d^{d+1}X\sqrt{\hat\gamma}\,\Big\{V(\hat\phi)-\hat R_{(d+1)}
\hat\nabb_{\nh}\,\hat\phi^J
+\frac14J(\hat\phi)\hat F^a_{\mh\nh}\hat F^{a\mh\nh}\Big\}\nonumber\\
&-2\int_{\Sigma_d}d^dx\sqrt{\hat h}\,\hat K \ .
\label{bulkS}
\end{align}
Here $\hat\gamma_{\mh\nh}$ denotes a bulk metric.
Using ADM decomposition, it becomes
\begin{align}
=\hat N^2(x,\tau)d\tau^2+\hat h_{\mu\nu}
d\tau) \ ,
\end{align}
where $\hat h_{\mu\nu}$ is an induced metric on a $d$-dimensional hypersurface
$\displaystyle{\Sigma_d:=\{X\in M_{d+1}|\tau=\text{const.}\}}$.
We have
$\hat\gamma=-\det(\hat\gamma_{\mh\nh}),\hat h=-\det(\hat h_{\mu\nu})$.
On each slice,
an extrinsic curvature is defined as
\begin{align}
\hat K_{\mu\nu}:=\frac1{2\hat N}(\partial_\tau\hat h_{\mu\nu}
-\hat\nabla_\mu\hat \lambda_\nu-\hat\nabla_\nu\hat \lambda_\mu)
\ ,
\end{align}
$\hat K:=\hat h^{\mu\nu}\hat K_{\mu\nu}$.
The hatted quantities mean off-shell without the equations
of motion imposed.
The covariant derivatives are defined as
\begin{align}
\hat \nabb_{\mh}\,\hat\phi^I:=
\hat\nabla_{\mh}\,\hat\phi^I-i\hat A^a_{\mh}\,(T^a\hat\phi)^I
\ .
\end{align}
Here $\hat\nabla_{\mh}$ denotes a covariant derivative
associated with the Levi-Civita connection, $\hat\Gamma^{\mh}_{\nh\rhoh}$ ,
constructed from
$T^a$ is a generator of the gauge group $G$.
Since we want to recognize
$\phi$ as real coupling functions, we restrict the symmetry $G$ to a group which
has real representations such as $SO(N)$.
For details, see Appendix <ref>.
Before proceeding, some comments on earlier works on the
AdS/CFT correspondence with bulk gauge fields are in order.
This was first investigated by a paper <cit.>
on the basis of holographic renormalization, and since then
has been discussed extensively from many perspectives.
A systematic algorithm for solving the HJ equations in the
cases with Abelian gauge fields and neutral scalar fields coupled
is obtained in <cit.>.
Using this algorithm,
a paper <cit.> studies a class of AdS/CMP models that are
parametrized with a Lifshitz exponent $z$.
In this section, we aim to give a formalism
for analyzing more general systems of bulk gravity coupled with
nonAbelian gauge fields and charged scalar fields
by extending a flow equation that was first developed in
Many of the results we give below coincide with those
found already in the papers quoted above.
Working in a Hamilton formalism with $\tau$ regarded as
a time direction rewrites the action (<ref>)
in a first-order form:
\begin{align}
\mathbf S&
=\int d^dxd\tau
\sqrt{\hat h}\Bigg\{\hat\pi^{\mu\nu}\partial_\tau\hat h_{\mu\nu}
+\hat\pi_I\partial_\tau\hat\phi^I+\hat\pi^{a\mu}\partial_\tau\hat A^a_\mu
\nonumber\\
&\hspace{20pt}+\hat N\Big[\frac1{d-1}({\hat\pi^\mu}_\mu)^2-\hat\pi_{\mu\nu}^2
-\frac12L^{IJ}(\hat\phi)\hat\pi_I\hat\pi_J-\frac1{2J(\hat\phi)}\hat h^{\mu\nu}
\hat\pi^a_\mu\hat\pi^a_\nu\nonumber\\
&\hspace{60pt}+V(\hat\phi)-\hat R_{(d)}+\frac12L_{IJ}(\hat\phi)\hat h^{\mu\nu}
\hat \nabb_\mu\hat\phi^I\hat \nabb_\nu\hat\phi^J+\frac14J(\hat\phi)
\hat F^a_{\mu\nu}\hat F^{a\mu\nu}\Big]\nonumber\\
-\hat\pi_I\hat \nabb_\mu\hat\phi^I-\hat F^a_{\mu\nu}\hat\pi^{a\nu}\right]
\nonumber\\
&\hspace{20pt}+\hat A^a_\tau\Big[{\hat \nabb^a}_{b\nu}\hat\pi^{b\nu}-i(T^a\hat\phi)^I
\hat\pi_I\Big]\Bigg\} \ .
\label{bulkS:1st}
\end{align}
Here the canonical momenta conjugate to
$\hat h_{\mu\nu},\ \hat\phi^I$ and
$\hat A^a_\mu$ are respectively computed to be
\begin{align}
\hat\pi^{\mu\nu}:=&\pdif{\mathcal L_{d+1}}{(\partial_\tau\hat h_{\mu\nu})}
=\hat K^{\mu\nu}-\hat h^{\mu\nu}\hat K,
\ ,\\
\hat\pi_I:=&\pdif{\mathcal L_{d+1}}{(\partial_\tau\hat\phi^I)}=\frac1{\hat N}
\ ,\\
\hat\pi^{a\mu}:=&\pdif{\mathcal L_{d+1}}{(\partial_\tau\hat A^a_\mu)}=\frac1
{\hat N^3}J(\hat\phi)\Big[\hat N^2\hat h^{\mu\nu}\hat F^a_{\tau\nu}
-\hat\lambda^\nu(\hat N^2\hat h^{\rho\mu}
+\hat\lambda^\rho\hat\lambda^\mu)\hat F^a_{\nu\rho}\Big]\ ,
\end{align}
with $L^{IJ}=L_{IJ}^{-1}$.
As evident, $\hat N$, $\hat\lambda^\mu$ and $\hat A_\tau$ are
the auxiliary fields, and their equations of motion
yield the first-class constraints
\begin{align}
\hat H:=\frac1{\sqrt{\hat h}}\ddif{\mathbf S}{\hat N}
-\frac1{2J(\hat\phi)}\hat h^{\mu\nu}\hat\pi^a_\mu\hat\pi^a_\nu
\nonumber\\
&+V(\hat\phi)-\hat R_{(d)}+\frac12L_{IJ}(\hat\phi)\hat h^{\mu\nu}
\hat \nabb_\mu\,\hat\phi^I\hat \nabb_\nu\,\hat\phi^J+\frac14J(\hat\phi)
\hat F^a_{\mu\nu}\hat F^{a\mu\nu}\approx0 \ ,
\label{Hcon}
\\
\hat P_\mu:=\frac1{\sqrt{\hat h}}\ddif{\mathbf S}{\hat\lambda^\mu}
=&2\hat\nabla^\nu\hat\pi_{\mu\nu}-\hat\pi_I\hat \nabb_\mu\,\hat\phi^I
-\hat F^a_{\mu\nu}\hat\pi^{a\nu}\approx0 \ ,
\label{Momcon}
\\
\hat G^a:=\frac1{\sqrt{\hat h}}\ddif{\mathbf S}{\hat A^a_\tau}
=&\hat \nabb^a_{b\mu}
\hat\pi^{b\mu}-i(T^a\hat\phi)^I\hat\pi_I\approx0 \ .
\label{Gcon}
\end{align}
(<ref>) and (<ref>) are the Hamiltonian and momentum
constraints respectively that result from diffeomorphism in
the $(d+1)$-dimensional bulk spacetime. (<ref>) is the Gauss's
law constraint due to the gauge symmetry $G$.
Suppose that we find a solution to the equations of motion of
$\hat h_{\mu\nu}, \hat A_{\mu}$ and $\hat\phi^I$
with the constraints (<ref>),
(<ref>) and (<ref>) under a Dirichlet boundary
condition at $\tau=\tau_0$:
\begin{align}
\bar h_{\mu\nu}(x,\tau=\tau_0)=h_{\mu\nu}(x) \ ,~~
\bar A_{\mu}(x,\tau=\tau_0)=A_\mu(x) \ ,~~
\bar\phi^I(x,\tau=\tau_0)=\phi^I(x) \ .\nn
\end{align}
Here the bulk fields with a bar means on-shell.
Substituting the classical solutions into (<ref>),
we obtain the on-shell action as a functional of the
boundary values
\begin{align}
:=\int d^dx\int_{\tau_0}^\infty d\tau\sqrt{\bar h}\,\Big\{\bar\pi
^{\mu\nu}\partial_\tau\bar h_{\mu\nu}+\bar\pi_I\partial_\tau\bar\phi^I
+\bar\pi^{a\mu}\partial_\tau\bar A^a_\mu\Big\}
\ .
\label{onshellS}
\end{align}
Following the standard procedure in the Hamilton-Jacobi formalism,
it is verified that the variation of the on-shell action
under the boundary values and the location of $\Sigma_d$
is given by
\begin{align}
\delta S[h(x),\phi(x),A(x);\tau_0]=-\int d^dx\sqrt{h}\,
\Big\{\bar\pi^{\mu\nu}
(x,\tau_0)\delta h_{\mu\nu}(x)+\bar\pi_I(x,\tau_0)\delta\phi^I(x)+\bar\pi^{a\mu}
(x,\tau_0)\delta A^a_\mu(x)\Big\} \ .
\end{align}
We then obtain the Hamilton-Jacobi equations
\begin{align}
\bar\pi^{\mu\nu}(x,\tau_0)
=-\frac1{\sqrt{h}}\ddif S{h_{\mu\nu}(x)} \ ,~~
\bar\pi_I(x,\tau_0)=-\frac1{\sqrt{h}}\ddif S{\phi^I(x)}\ ,~~
\bar\pi^{a\mu}(x,\tau_0)&=-\frac1{\sqrt{h}}\ddif S{A^a_{\mu}(x)}\ ,~~
\frac{\partial S}{\partial\tau_0}=0 \ .
\end{align}
Inserting these into the Hamilton constraint (<ref>)
gives the flow equation
\begin{equation}
\{S,S\}=\mathcal L_d \ ,
\label{floweq}
\end{equation}
\begin{equation}
\{S,S\}:=\left(\frac1{\sqrt{h}}\right)^2\left[-\frac1{d-1}\left(h_{\mu\nu}\ddif S
{h_{\mu\nu}}\right)^2+\left(\ddif S{h_{\mu\nu}}\right)^2+\frac12L^{IJ}(\phi)\ddif
S{\phi^I}\ddif S{\phi^J}+\frac1{2J(\phi)}h_{\mu\nu}\ddif S{A^a_\mu}\ddif S
\ ,
\label{SS}
\end{equation}
\begin{equation}
\mathcal L_d:=V(\phi)-R_{(d)}+\frac12L_{IJ}(\phi)\nabb^\mu\phi^I
\nabb_\mu\phi^J
+\frac14J(\phi)F^a_{\mu\nu}F^{a\mu\nu}\ .
\label{Ld}
\end{equation}
\begin{align}
\nabb_{\mu}\,\phi^I:=
\nabla_{\mu}\phi^I-i A^a_{\mu}\,(T^a\phi)^I
\ .
\end{align}
$\nabla_{\mu}$ denotes a covariant derivative
associated with the Levi-Civita connection, $\Gamma^{\mu}_{\nu\rho}$ ,
constructed from the boundary metric
We show that the momentum constraint and the
Gauss's law constraint ensures $d$-dimensional diffeomorphism invariance
and gauge invariance of the on-shell action, respectively.
First, we note that the Gauss's law constraint (<ref>) and
the Hamilton-Jacobi equations give
\begin{align}
0&=\int d^dx\sqrt{h}\,\alpha^a\left({\nabb^a}_{b\mu}\pi^{b\mu}
\nn\\
&=\int d^dx\left\{\nabb_\mu\alpha^a\ddif S{A^a_\mu}+i\alpha^a(T^a\phi)^I\ddif S{\phi^I}
\right\}\label{Gaussopid}\\
&=\int d^dx\left(\delta_\alpha^{\rm gauge} A^a_\mu\ddif S{A^a_\mu}
+\delta_\alpha^{\rm gauge}\phi^I\ddif S{\phi^I}\right)
=\delta^\mathrm{gauge}_\alpha S\ .\label{Gausslaw}
\end{align}
\begin{align}
\delta^{\rm gauge}_\alpha A^a_\mu
A^b_\mu\alpha^c \ ,~~
\delta^{\rm gauge}_\alpha \phi^I:=
i\alpha^a(T^a\phi)^I \ ,
\end{align}
denote an infinitesimal gauge transformation.
Further, the momentum constraint (<ref>) and
the Hamilton-Jacobi equations lead to
\begin{align}
0=&\int d^dx\sqrt{h}\,\epsilon^\mu\left(2\nabla^{\nu}\pi_{\mu\nu}
\nn\\
=&\int d^dx\left\{(\nabla_\mu\epsilon_\nu+\nabla_\nu\epsilon_\mu)
\ddif S{h_{\mu\nu}}+\epsilon^\mu \nabb_\mu\,\phi^I\ddif S{\phi^I}
+\epsilon^\mu F^a_{\mu\nu}\ddif S{A^a_\nu}\right\}
\nn\\
%=\int d^dx&\Bigg\{\delta_\epsilon h_{\mu\nu}\ddif S{h_{\mu\nu}}+\delta_\epsilon
%\phi^I\ddif S{\phi^I}
%+\delta_\epsilon A^a_\mu\ddif S{A^a_\mu}
%&-i\epsilon^\mu A_\mu\phi^I\ddif S{\phi^I}-[\partial_\mu
% (\epsilon^\nu A^a_\nu)+{f^a}_{bc}A^b_\mu(\epsilon^\nu A^c_\nu)]
% \ddif S{A^a_\mu}\Bigg\}
=&\delta_\epsilon S
-\int d^dx\sqrt{h}\,\epsilon^\mu A^a_\mu\left\{{\nabb^a}_{b\nu}\pi^{b\nu}
-i(T^a\phi)^I\pi_I\right\} \ .
\label{eAgauss}
\end{align}
\begin{align}
\delta_\epsilon\phi^I:=\cL_\epsilon\phi^I
\equiv\epsilon^\mu\partial_\mu\phi^I\ ,
\quad
\delta_\epsilon A^a_\mu:=\cL_\epsilon A^a_\mu\equiv\epsilon^\nu\partial_\nu A^a_\mu
+\partial_\mu\epsilon^\nu A^a_\nu \ ,
\quad
\delta_\epsilon h_{\mu\nu}:=\cL_\epsilon h_{\mu\nu}
\equiv\nabla_\mu\epsilon_\nu+\nabla_\nu\epsilon_\mu \ ,
\end{align}
are Lie derivatives with respect to $d$-dimensional diffeomorphism.
Noting that the second term in (<ref>) vanishes
because of (<ref>) implies invariance of the on-shell action
under $d$-dimensional diffeomorphism.
For the purpose of solving the flow equation (<ref>) using
systematic derivative expansions,
we divide the on-shell action into local and non-local parts:
\begin{align}
\frac1{2\kappa_{d+1}^2}S[h(x),\phi(x),A(x)]
\equiv\frac1{2\kappa_{d+1}^2}S_\mathrm{loc}
[h(x),\phi(x),A(x)]-\Gamma[h(x),\phi(x),A(x)]\ .
\label{S:divided}
\end{align}
We next assign an additive
number called weight to each ingredient of the action as in a table below.
elements weight
$h_{\mu\nu}(x),\phi^I(x),\Gamma[h,\phi,A]$ 0
$\partial_\mu,A^a_\mu(x)$ 1
$R,R_{\mu\nu},R_{\mu\nu\rho\sigma},\partial^2,\dots$ 2
$\ddif{}{A^a_\mu(x)}$ $d-1$
$\ddif{}{h_{\mu\nu}(x)},\ddif{}{\phi^I(x)}$ $d$
assignment of weights
The weight of the gauge field is assigned to be $w=1$
because of gauge invariance.
We parametrize the local Lagrangian as below
\begin{align}
\cL_{\rm loc}=\sum_{w=0,2,4,\cdots}[\cL_{\rm loc}]_w \ ,
\end{align}
\begin{align}
[\cL_{\rm loc}]_0=&W(\phi) \ ,
\\
[\cL_{\rm loc}]_2=&-\Phi(\phi)R_{(d)}+\frac12M_{IJ}(\phi)
\nabb^\mu\,\phi^I\nabb_\mu\,\phi^J\ .
\end{align}
It is important here that all the local terms are taken to
be gauge invariant.
We also define
\begin{align}
S_{{\rm loc};w-d}:=\int d^dx\sqrt{h}\,[\cL_{\rm loc}]_w \ .
\end{align}
Note that $d^dx$ has a weight $-d$.
Inserting (<ref>) into the flow equation and then
decomposing it depending on weights, we find
for $w=0$
\begin{equation}
V(\phi)=-\frac d{4(d-1)}W^2(\phi)+\frac12L^{IJ}(\phi)\partial_IW(\phi)
\partial_JW(\phi) \ .
\label{feq0}
\end{equation}
For $w=2$,
\begin{align}
\partial_J\Phi(\phi) \ ,
\label{feq2-1}\\
\frac12L_{IJ}(\phi)=&-\frac{d-2}{4(d-1)}W(\phi)M_{IJ}(\phi)-L^{KL}(\phi)
\partial_KW(\phi)\Gamma_{L;IJ}(\phi)
\nonumber\\
M_{JL}(\phi)(T^a\phi)^K(T^a\phi)^L\ ,
\label{feq2-2}\\
0=&W(\phi)\partial_K\Phi(\phi)+L^{IJ}(\phi)\partial_IW(\phi)M_{JK}(\phi)\ . \label{feq2-3}
\end{align}
For $w=4$,
\begin{equation}
\frac14J(\phi)F^a_{\mu\nu}F^{a\mu\nu}=[\{S,S\}]_4\ .
\label{feq4}
\end{equation}
For $w=d$ with $d\neq 4$,
\begin{align}
[\mathcal L_d]_d=&\frac{2\kappa_{d+1}^2W(\phi)}{2(d-1)}
\frac2{\sqrt{h}}h_{\mu\nu}\ddif\Gamma{h_{\mu\nu}}-\frac{2\kappa_{d+1}^2}
\ddif{S_{\mathrm{loc};2-d}}{A^a_\mu}\ddif\Gamma{A^a_\nu}+[\{S_\mathrm{loc},
S_\mathrm{loc}\}]_d\ .
\label{feqd}
\end{align}
In the AdS/CFT correspondence,
$h_{\mu\nu}(x), \phi^I(x)$ and $A_\mu^a(x)$ are identified with
a background metric, a coupling function associated with an
gauge invariant operator $O_I$, and a background gauge potential
of $G$, respectively, in the boundary QFT.
Then, we see that
(<ref>) is equivalent to
the local RG equation,
specifies how the coupling functions $\phi^I(x)$ and $A_\mu^a(x)$ flow under
local Weyl transformation <cit.>.
In particular, by rewriting (<ref>) as
\begin{align}
2h_{\mu\nu}\frac{\delta\Gamma}{\,\,\delta h_{\mu\nu}}
\frac{\delta\Gamma}{\delta\phi^J}
\frac{1}{\sqrt{h}}\,\frac{\delta S_{{\rm loc};2-d}}{\delta A_\mu^a}
\frac{\delta \Gamma}{\delta A_\nu^a}
\nn\\
\sqrt{h}\,\bigg(
[\cL_d]_d-\left[\{S_{\rm loc},S_{\rm loc}\}\right]_d
\bigg)
\ ,
\label{localRG}
\end{align}
the scalar $\beta$-function
associated with a coupling function $\phi^I$
\begin{equation}
\beta^I(\phi):=-\frac{2(d-1)}{W(\phi)}L^{IJ}(\phi)\partial_JW(\phi) \ ,
\label{beta:grad}
\end{equation}
where $\partial_I=\partial/\partial\phi^I$.
Using (<ref>), this can be recast as
\begin{equation}
\beta^I(\phi)=+2(d-1)M^{IJ}(\phi)\partial_J\Phi(\phi) \ ,
\end{equation}
with $M^{IJ}=M_{IJ}^{-1}$.
The coefficient of
$\delta\Gamma/\delta A^a_\mu$ defines a vector $\beta$-function as
\begin{equation}
\beta^a_\mu(\phi,A):=
-\frac1{\sqrt h}\frac{2(d-1)}{W(\phi)J(\phi)}h_{\mu\nu}
\ddif{S_{\text{loc;}2-d}}{A^a_\nu}
M_{IJ}(\phi)(T^a\phi)^I\nabb_\mu\phi^J\ .
\label{vecbeta}
\end{equation}
Following <cit.>, we define
$\beta_\mu^a\equiv\rho_I^a\,\nabb_\mu\phi^I$ so that
\begin{align}
\rho_I^a=+i\frac{2(d-1)}{W(\phi)J(\phi)}
M_{IJ}(\phi)(T^a\phi)^J \ .
\end{align}
We now show that $W, L_{IJ}$ and $V$ can be expressed in
terms of $\Phi$ and $M_{IJ}$.
From (<ref>) and (<ref>), we obtain
\begin{align}
+\frac{d-2}{2(d-1)}\,\Phi \ .
\label{solW}
\end{align}
Next, using (<ref>), (<ref>) becomes
\begin{align}
\left[-\frac{d-2}{4(d-1)}\,M_{IJ}
\right] \ .
\label{solL}
\end{align}
Here, $\Gamma^I_{JK}$ is the Levi-Civita connection
with respect to $M_{IJ}$, and $D_I$ is a covariant derivative defined with the
connection. For a consistency check of (<ref>),
we rewrite (<ref>) as
\begin{align}
W^{-1}\,L_{IJ}\,M^{JK}\,\partial_K\Phi \ .
\end{align}
Using (<ref>) and (<ref>), the RHS takes the form
\begin{align}
\nn\\
\partial_I\left(
\right)
\ .
\end{align}
Because $\Phi(\phi)$ is gauge invariant by definition, we have
\begin{align}
(T^a\phi)^I\,\partial_I\Phi=0 \ .
\label{delPhi=0}
\end{align}
This ensures
that (<ref>) is consistent indeed.
Finally, (<ref>) and (<ref>)
\begin{align}
\partial^I\Phi\,\partial^J\Phi\left(
\right)
\right] \ .
\end{align}
Vacuum expectation value of the stress tensor
in the
presence of the background fields $h,\phi$ and $A$ is defined by
\begin{align}
%\langle T^{\mu\nu}(x)\rangle_{h,\phi,A}:=
\langle T^{\mu\nu}(x)\rangle:=
\frac2{\sqrt{h}}\ddif{\Gamma
[h,\phi,A]}{h_{\mu\nu}(x)} \ .
\end{align}
It follows from (<ref>) that the trace of the
stress tensor becomes
\begin{align}
\left\langle{T^\mu}_\mu\right\rangle
\bigg([\mathcal L_d]_d
\bigg)
\ddif\Gamma{A^a_\mu} \ .
\label{traceT}
\end{align}
As explained in <cit.>, the flow equation cannot
determine $S_{{\rm loc};0}$ uniquely, reflecting
an ambiguity of adding local counterterms to $\Gamma$.
In a computation of the Weyl anomaly, $S_{{\rm loc};0}$
is manifested as
a degree of freedom of adding a total derivative <cit.>.
To see this, we note that under an infinitesimal Weyl transformation,
$S_{{\rm loc};0}$ transforms as
\begin{align}
% \delta_\sigma\,S_{{\rm loc};0}=
\int d^dx\, 2\,\sigma(x)\,h_{\mu\nu}(x)
\frac{\delta}{\delta h_{\mu\nu}(x)}\,S_{{\rm loc};0}
=\int d^dx\sqrt{h}\,
\partial_\mu\sigma\cJ_d^\mu \ ,
\end{align}
because $S_{{\rm loc};0}$ is invariant under global
scale transformations. Thus,
\begin{align}
\frac{\delta}{\delta h_{\mu\nu}(x)}\,S_{{\rm loc};0}
=-\sqrt{h}\,\nabla_\mu\cJ_d^\mu \ .
\end{align}
From this relation, we obtain
\begin{align}
\left\langle{T^\mu}_\mu\right\rangle
\bigg([\mathcal L_d]_d
\bigg)
\nn\\
\frac{\delta}{\delta\phi^I}
\left(
\Gamma-\frac{1}{2\kappa_{d+1}^2}S_{{\rm loc};0}
\right)
\frac{\delta}{\delta A_\mu^a}
\left(
\Gamma-\frac{1}{2\kappa_{d+1}^2}S_{{\rm loc};0}
\right) \ .
\end{align}
Here, $\{S_\mathrm{loc},S_\mathrm{loc}\}'$
denotes the bracket $\{S_\mathrm{loc},S_\mathrm{loc}\}$ with
$[\mathcal L_\mathrm{loc}]_d$ removed from $S_{\rm loc}$.
the Weyl anomaly, which is defined as
\begin{align}
\mathcal W_d(x):=\left\langle{T^\mu}_\mu(x)\right\rangle\Big|_{\beta=0}
\ ,
\end{align}
contains a total derivative that comes from
$S_{{\rm loc};0}$.
It is important to note that in the AdS/CFT
correspondence, $\cJ_d^\mu$ is the only origin of the Virial
current, which might spoil conformal symmetry of scale invariant
field theories. For an excellent review on relations
between scale and conformal symmetry, see <cit.>.
As an operator, $\cJ_d^\mu$ is proportional to an identity
operator, and therefore gives no obstacle to having CFTs.
This is natural because we are working on QFTs with gravity duals.
Effects of the Virial current here are only manifested as
ambiguities of local counterterms addded to $\Gamma$.
For the moment, we work in the scheme
$S_{\rm{loc};0}=0$ because this is simple and natural
in the flow equation.
For a discussion on how $S_{{\rm loc};0}$ affects
the coefficients in $\langle T^\mu_\mu\rangle$, see <cit.>.
Another choice of the scheme
will be discussed below
for the purpose of studying a holographic $c$-theorem.
An analysis of the Gauss's law constraint is straightforward.
With the HJ equations, (<ref>) can be regarded as a constraint
on the on-shell action:
\begin{align}
{\nabb^a}_{b\mu}\frac{\delta S}{\delta A^b_\mu}
-i(T^a\phi)^I\,\frac{\delta S}{\delta \phi^I}=0 \ .
\label{gauss;S}
\end{align}
Inserting (<ref>) into this,
we see that the local terms give no contribution
because they are gauge invariant by definition.
Then, (<ref>) reduces to
\begin{align}
{\nabb^a}_{b\mu}\frac{\delta \Gamma}{\delta A^b_\mu}
-i(T^a\phi)^I\,\frac{\delta \Gamma}{\delta \phi^I}=0 \ .
\end{align}
Because the vev's of the gauge invariant
operators and currents in the presence of the background
fields are given by
\begin{equation}
\langle O_I(x)\rangle:=\frac1{\sqrt h}\ddif\Gamma{\phi^I(x)}\ ,
\langle J^{a\mu}(x)\rangle:=\frac1{\sqrt h}\ddif\Gamma{A^a_\mu(x)}
\ ,
\end{equation}
we obtain an
operator identity
\begin{equation}
\nabb_\mu J^{a\mu}=i(T^a\phi)^IO_I\ .\label{opid}
\end{equation}
We now give some comments on properties of the vector
$\beta$-function that hold for $d$-dimensional QFTs with gravity duals.
These were first obtained in <cit.>, and
the rest of this section may be regarded as a review
of part of that paper.
First, we have already observed that (<ref>)
exhibits the gradient property.
Second, an orthogonal relation between scalar and
vector $\beta$-functions is easy to verify in the AdS/CFT correspondence
thanks to the gauge invariance (<ref>):
\begin{align}
\rho^a_I\beta^I=
i\frac{4(d-1)^2}{JW}\,(T^a\phi)^I\partial_I\Phi=0\ .\label{ortho}
\end{align}
In addition, anomalous dimensions receive non-trivial contributions
from operator mixing: differentiating the local RG equation
\begin{align}
(\text{local terms})
=\int d^dx\Bigg\{2h_{\mu\nu}(x)\ddif{}{h_{\mu\nu}(x)}+\beta^I
\Bigg\}\Gamma[\phi,h,A] \ ,
\end{align}
with $\phi^I$ and $A_\mu^a$,
we obtain the RG equations of correlation functions of
$O_I$ and $J^{a\mu}$, of which
the anomalous dimensions of $O_I$ and $J^{a\mu}$ are
read off as
\begin{align}
\ ,~~
{\gamma^a}_b=i\rho^c_I\delta_{bc}(T^a\phi)^I \ ,
%(T^c\phi)^J\ ,
\end{align}
respectively. Here we employ the operator identity (<ref>).
It then follows
\begin{align}
M_{JK}(\phi)(T^a\phi)^K(T^a\phi)^I\ ,
\\
(T^c\phi)^J\ .
\end{align}
These expressions exhibit the suggested Higgs-like relation
Finally, the equivalence
\begin{equation}
\beta^a_\mu=0\iff\nabb_\mu J^{a\mu}=0\label{nonreno}
\end{equation}
can also be shown.
Recalling (<ref>), $\Leftarrow$ is obviously true because the
conservation of the current implies $(T^a\phi)^I=0$. On the other hand, if
$\beta^a_\mu=0$, we have two possibilities: (i) $(T^a\phi)^I=0$ and (ii)
$\nabb_\mu\phi^I=0$ because we are assuming $M_{IJ}$ to be invertible (and
$d\neq1$). In case of (i), we have the current conservation via the
operator identity (<ref>).
In case of (ii), since $\phi^I$ does not belong to a singlet, we must have
$\phi^I=\text{const.}=0$, and this again results in the current conservation.
§ EXPLICIT CALCULATIONS IN FOUR DIMENSIONS
For $d=4$, the equation (<ref>) should not be imposed, and
for $w=4$ the flow equation (<ref>) yields
the local RG equation (<ref>) instead.
Using the formulae given in appendix <ref>
together with (<ref>),
we arrive at
the explicit expression of the trace of the stress tensor:
\begin{align}
\langle{T^\mu}_\mu\rangle=&
\frac6{2\kappa_5^2W(\phi)}\frac14J(\phi)F^a_{\mu\nu}F^{a\mu\nu}
\nn\\
\nn\\
&+\frac{12\Phi}{2\kappa_5^2W}\partial_I\Phi\cdot E^{\mu\nu}
\nabb_\mu \nabb_\nu\,\phi^I
-\frac6{2\kappa_5^2W}L^{JK}\partial_{(J}\Phi M_{K)I}\cdot
\nn\\
+M_{IJ})E^{\mu\nu}\nabb_\mu\,\phi^I \nabb_\nu\,\phi^J
\nn\\
&+\frac1{2\kappa_5^2W}\left(\Phi M_{IJ}
\nn\\
\nn\\
+\frac3{2\kappa_5^2W}\partial_I\Phi M_{JK}
\nabb^2\phi^I\nabb^\mu\,\phi^J\nabb_\mu\,\phi^K
\nn\\
\nabb^\mu \nabb^\nu\,\phi^I\nabb_\mu \nabb_\nu\,\phi^J
\nn\\
+M_{JK}\right)\cdot \nabb^\mu \nabb^\nu\,\phi^I
\nabb_\mu\,\phi^J\nabb_\nu\,\phi^K
\nn\\
\partial_I\partial_J\Phi)
\nn\\
\partial_K\Phi)+\frac6{2\kappa_5^2W}(\partial_I\partial_J\Phi\partial_K\partial_L\Phi-\partial_I\partial_K\Phi\partial_J\partial_L\Phi)
\nn\\
\nabb^\nu\,\phi^K\nabb_\nu\,\phi^L
\ .
\label{explicitT}
\end{align}
Now we compare these results with those
in <cit.>, where a generic form of
$\langle T_\mu^\mu\rangle$ is given in accord with
symmetry constraints.
For details, see Appendix <ref>.
(<ref>), (<ref>), (<ref>) and (<ref>)
are easily solved as
\begin{equation}
A=-\frac{12\Phi^2}{2\kappa_5^2W}\ ,
\quad
C=-\frac{3\Phi^2}{2\kappa_5^2W}\ ,
\quad
\end{equation}
\begin{equation}
\label{WI}
\end{equation}
Using (<ref>), (<ref>) yields
\begin{equation}
{2\kappa_5^2W}\partial_I\Phi\partial_J\Phi-\frac6{2\kappa_5^2W}\Phi M_{IJ}.
\label{GIJ}
\end{equation}
From (<ref>), (<ref>) and (<ref>), we find
\begin{align}
H_I&=0\ ,
\\
E_I&=\frac{36}{2\kappa_5^2W}L^{JK}\partial_{(J}\Phi M_{K)I}\ ,
\\
-\frac6{2\kappa_5^2W}\Phi M_{IJ}
+\frac{36}{2\kappa_5^2W}L^{KL}\partial_{(K}\Phi\Gamma_{L);IJ}\ .
\end{align}
(<ref>) and (<ref>) give
\begin{equation}
\ ,
\end{equation}
from which (<ref>) leads to
\begin{equation}
A_{IJ}=-\frac6{2\kappa_5^2W}L^{KL}M_{IK}M_{JL}\ .
\label{AIJ}
\end{equation}
The rest of the equations given in Appendix <ref>
requires hard work to solve.
However, since there are six
equations left and six coefficient functions to be determined, viz.
$B_{IJK},T_{IJK}$, $C_{IJKL},\beta_f,Q_I$ and $P_{IJ}$, thus it is expected that there is a
It is argued in <cit.> that the quantities appearing in
$\langle T^\mu_\mu\rangle$ must satisfy integrability conditions,
that is, Wess-Zumino consistency conditions associated
with local RG transformations:
\begin{align}
\left[ \Delta_\sigma,\Delta_{\sigma'}\right]\Gamma=0 \ .
\end{align}
\begin{align}
\Delta_\sigma:=\int d^dx\,\sigma(x)\left(
2h_{\mu\nu}\frac{\delta}{\,\,\delta h_{\mu\nu}}
\frac{\delta}{\delta\phi^J}
\frac{\delta}{\delta A_\mu^a}
\right) \ .
\end{align}
As discussed before, the AdS/CFT correspondence always
gives us trivial Virial currents.
Then, no nontrivial modification of the scalar
and vector $\beta$-functions arises that is necessary
for gauge invariance of the $\beta$ functions.
Therefore, we can show that a number of integrability conditions derived in
<cit.> still hold on their own. We also obtain additional integrability
conditions concerned with the external gauge field.
To list some of the integrability conditions that
play an important role in this paper, we have
\begin{align}
\partial_IA&=G_{IJ}\beta^J-\cL_\beta W_I \ ,
\label{WZ;delA}
\\
\rho^a_I\beta^I&=0 \ .\label{WZ;ortho}
\end{align}
Here $\cL_\beta$ denotes a Lie derivative associated with
the vector $\beta^I$, which acts on $W_I$ as
\begin{align}
\cL_\beta W_I\equiv\beta^K\partial_K W_I+\partial_I\beta^KW_K \ .
\end{align}
The coefficients given in
(<ref>)-(<ref>) should satisfy all the integrability
conditions, because
the flow equation is formulated on the basis that
the effective action $\Gamma$ does exist once a bulk
gravity model is given.
In fact, a straightforward computation shows that
(<ref>) holds indeed.
(<ref>) is nothing but the orthogonality
condition (<ref>),
which is already verified from the gauge invariance
of $\Phi$.
We end this paper by making some comments on
the $c$-theorem of RG flows in higher dimensions and a holographic
As shown by Jack and Osborn in <cit.>,
(<ref>) can be rewritten as
\begin{align}
\beta^I\partial_I\tilde{A}=G_{IJ}\beta^I\beta^J \ ,
\label{delAtilde}
\end{align}
\begin{align}
\tilde{A}:=A+W_I\beta^I \ .
\end{align}
Proving positive-definiteness of $G_{IJ}$, if possible, implies
that $\tilde{A}$ decreases monotonically under
RG flows.
$A$ and $G_{IJ}$ in (<ref>) and (<ref>)
are obtained in the scheme $S_{\rm loc;0}=0$.
As evident, this $G_{IJ}$ is not positive definite even if
$L_{IJ}$ is assumed to be so.
However, $G_{IJ}$ takes a different
expression by working in
a different scheme with $S_{\rm loc;0}\ne 0$.
In fact, it is argued in <cit.> that adding local counterterms
to $\Gamma$ gives rise to a shift
\begin{align}
\tilde A\to \tilde A':=\tilde A+g_{IJ}\beta^I\beta^J \ ,~~
G_{IJ}\to G'_{IJ}:=G_{IJ}+\cL_\beta g_{IJ} \ ,
\label{shift}
\end{align}
which leaves (<ref>) unchanged.
Here, we show that an appropriate choice of $g_{IJ}$
maps $\tilde A$ to a holographic $c$-function.
Relations between a holographic $c$-function
and $\tilde A$ were first studied in <cit.>.
Our aim in this paper is to make a full identification
of schemes
where a holographic $c$-function is related directly
with an trace anomaly coefficient.
As discussed in <cit.>,
a holographic $c$-function for $d=4$ is defined as
\begin{align}
c_h:=-\frac{27}{2\kappa_5^2}\,\frac{1}{W^3} \ .
\label{ch}
\end{align}
Here the overall factor is chosen so that the value of
$c_h$ at a fixed point equals that of $C$ given
in (<ref>).
It follows from (<ref>), (<ref>) and (<ref>)
\begin{align}
\beta^I\partial_Ic_h=\frac{1}{2}\,c_h\,L_{IJ}\,\beta^I
\,\beta^J\ .
\label{del;ch}
\end{align}
This relation was first derived in <cit.>, although
we prove it when $\phi^I$ is promoted
to space-time dependent couplings.
The gradient flow nature becomes more manifest by rewriting
the scalar $\beta$-function (<ref>) as
\begin{align}
\beta^I=\frac{2}{c_h}L^{IJ}\partial_J c_h \ .
\end{align}
The positivity of
$c_h$ together with positive definiteness of $L_{IJ}$
guarantees that $c_h$ is indeed a monotonically decreasing
function. $c_h L_{IJ}$ is to be identified with a Zamolodchikov
For the purpose of relating $\tilde A'$ to $c_h$,
we take $g_{IJ}$ to the most general form:
\begin{align}
\ ,
\label{gIJ}
\end{align}
\begin{align}
+x_2(\Phi) \ ,
\nn\\
+\Phi^2 \ ,
\end{align}
with $x_1,x_2$ being arbitrary functions of $\Phi=\Phi(\phi^I)$
and $(\partial\Phi\cdot\partial\Phi)=M^{IJ}\partial_I\Phi\partial_J\Phi$.
From this mapping, it turns out that $\tilde A'=4c_h$.
Furthermore, we can easily show that
\begin{align}
=2c_h\,L_{IJ}\beta^I\beta^J \ .
\end{align}
This implies that
(<ref>) after a shift
(<ref>) with (<ref>)
is identical with (<ref>).
§ ACKNOWLEDGMENTS
We would like to thank Yu Nakayama for many useful discussions.
§ NOTATIONS
Let $\phi^I$ be a charged scalar.
We divide the index $I$ into two parts: $I=(i,\alpha_i)$.
For each $i$, the charged field transforms
as a representation $R_i$ under the
gauge group $G$.
$\alpha_i=1,2,\cdots,{\rm dim}R_i$ is an index of $R_i$.
The generator of $G$, $(T^a)^I_{~J}$, defined in this paper
refers to
\begin{align}
(T^a)^I_{~J}=\delta^i_j\,(t^a_{(i)})^{\alpha_i\beta_i} \ .
\end{align}
with $I=(i,\alpha_i),\,J=(j,\beta_j)$.
$t^a_{(i)}$ is the generators of $G$ that belong to
the representation $R_i$.
The covariant derivative $\nabb$ acts on $\phi^I$ as
\begin{align}
\nabb_\mu\phi^I=
\nabb_\mu\phi^{i,\alpha_i}
\nn\\
\phi^{i,\beta_i} \ .
\end{align}
In addition, we define a symbol $(~)$ to denote symmetric parts of tensors:
\begin{equation}
\end{equation}
and denote the Levi-Civita connection in the theory space constructed from
$M_{IJ}$ $\Gamma_{I;JK}$, i.e.
\[ \Gamma_{I;JK}:=\frac12\left(\partial_JM_{IK}+\partial_KM_{IJ}-\partial_IM_{JK}\right).
\]
§ SOME USEFUL FORMULAE
The following formulae are useful and valid for any $d$:
\begin{align}
\ddif{}{h_{\mu\nu}}S_{\mathrm{loc};-d}&=
\frac12\sqrt{h}\,h^{\mu\nu}W(\phi) \ ,
\\
\ddif{}{\phi^I}S_{\mathrm{loc};-d}=&\sqrt{h}\,\partial_IW(\phi) \ ,
\end{align}
\begin{align}
\ddif{}{h_{\mu\nu}}S_{\mathrm{loc};2-d}&=
\sqrt{h}\,\Bigg\{\Phi(\phi)(R^{\mu\nu}
\nn\\
\phi^I\nabb_\rho\phi^J
-\nabb^\mu\phi^I\nabb^\nu\phi^J\right]\Bigg\} \ ,\\
\ddif{}{\phi^I}S_{\mathrm{loc};2-d}=&
\sqrt{h}\,\Big\{-\partial_I\Phi(\phi)R_{(d)}
\nabb^{2}\phi^J\Big\} \ .
\end{align}
From these results, we find
\begin{align*}
=&R^{\mu\nu}R_{\mu\nu}\cdot\Phi^2+R_{(d)}^2\left(-\frac d{4(d-1)}\Phi^2
&+E^{\mu\nu}\nabb_\mu \nabb_\nu\phi^I(-2\Phi\partial_I\Phi)\\
&+R_{(d)}\nabb^2\phi^I\Big[L^{JK}\partial_{(J}\Phi M_{K)I}\Big]\\
&+R_{(d)}\nabb^\mu\phi^I\nabb_\mu\phi^J\left[-\frac{d-2}{4(d-1)}\Phi M_{IJ}+L^{KL}
\partial_{(K}\Phi\Gamma_{L);IJ}\right]\\
-\frac12\partial_I\Phi M_{JK}+L^{LM}M_{I(L}\Gamma_{M);JK}\right]\\
&+\nabb^\mu \nabb^\nu\phi^I\nabb_\mu \nabb_\nu\phi^J(\partial_I\Phi\partial_J\Phi)\\
&+\nabb^\mu \nabb^\nu\phi^I\nabb_\mu \phi^J\nabb_\nu\phi^K\Big[\partial_I\Phi(2\partial_J
\partial_K\Phi+M_{JK})\Big]\\
&+\nabb^\mu\phi^I\nabb_\mu\phi^J\nabb^\nu\phi^K\nabb_\nu\phi^L\Bigg[-\frac d{16(d-1)}
\partial_I\partial_J\Phi)\\
\Phi)-(\partial_I\partial_J\Phi\partial_K\partial_L\Phi-\partial_I\partial_K\Phi
\partial_J\partial_L\Phi)\\
&\hspace{150pt}+\frac12L^{MN}\Gamma_{M;IJ}\Gamma_{N;KL}\Bigg]\ .
\end{align*}
§ TRACE OF STRESS TENSOR DEFINED IN <CIT.>
AND ITS RELATION TO THAT OBTAINED FROM BULK GRAVITY
In <cit.>, Jack and Osborn wrote down the explicit form
of the trace of the stress tensor as
\begin{align}
\hspace{-20pt}\langle{T^\mu}_\mu\rangle
\nabb_\mu\phi^I\nabb_\nu\phi^J
\nn\\
\nn\\
\nabb_\mu\phi^J\nabb^\nu\phi^J\nabb_\nu\phi^K
\nn\\
&+\frac14(F^{\mu\nu}F_{\mu\nu})\beta_f+F^{\mu\nu}\cdot P_{IJ}\nabb_\mu\phi^I\nabb_\nu\phi^J
\nn\\
\nabb^\mu\phi^I+S_{IJ}\nabb^\mu\phi^I\nabb^2\phi^J+T_{IJK}
\nabb^\mu\phi^I\nabb^\nu\phi^J\nabb_\nu\phi^K
\right.
\nn\\
&\left.\qquad\qquad+F^{\mu\nu}\cdot Q_I\nabb_\nu\phi^I\right)
\nn\\
\nabb_\mu\phi^J\right)+(\text{terms proportional to $\beta$-functions}) \ .
\end{align}
Here the most general total derivative terms are added.
For the purpose of matching with those results computed
from the bulk gravity, however,
it is sufficient to set $D=0=U_I$
because there is no term proportional to $\nabb^2R_{(4)}$ or
$\nabb^4\phi^I$ there with $S_{{\rm loc};0}=0$.
It is then straightforward to verify
\begin{align}
\langle{T^\mu}_\mu\rangle
+(-2C+A)R_{\mu\nu}^2+\left(\frac C3-\frac A4-\frac B{72}\right)R_{(4)}^2
\nn\\
\left[-G_{IJ}+2\partial_{(I}W_{J)}-2S_{(IJ)}\right]\nn\\
\phi^J\left(-\frac16F_{IJ}+\frac13\partial_{(I}H_{J)}\right)\nn\\
&+\nabb^\mu R_{(4)}\nabb_\mu\phi^I\cdot\frac13H_I+\nabb^2\phi^I\nabb^2\phi^J
\left(\frac12A_{IJ}+2S_{(IJ)}\right)+\nabb^\mu\nabb^\nu\phi^I\nabb_\mu
\nabb_\nu\phi^J(-2V_{IJ})
\nn\\
\nn\\
\nn\\
&+\nabb_\mu\phi^{i,\alpha_i}{(\nabb_\nu F^{\nu\mu})^{\beta_j}}_{\gamma_j}\left[2\delta_i^j\delta^{\gamma_j}_{\alpha_i}Q_{j,\beta_j}-2(S_{ij})_{\alpha_i\beta_j}
\phi^{j,\gamma_j}\right]
\nn\\
&+(\text{terms proportional to $\beta$-functions}).
\label{traceT:JO}
\end{align}
Comparing the coefficients of operators appearing in (<ref>)
and those in (<ref>) gives
\begin{align}
C-\frac14A&=0\ ,
\label{1st}\\
-2C+A&=-\frac{6\Phi^2}{2\kappa_5^2W}\ ,
\label{2nd}
\\
\frac C3-\frac A4-\frac B{72}&=\frac{2\Phi^2}{2\kappa_5^2W}
-\frac3{2\kappa_5^2W}L^{IJ}\partial_I\Phi\partial_J\Phi\ ,
\label{3rd}
\\
2W_I&=\frac{12}{2\kappa_5^2W}\Phi\partial_I\Phi\ ,
\label{4th}
\\
\partial_J\Phi+\frac6{2\kappa_5^2W}\Phi M_{IJ},
\label{5th}\\
&=-\frac6{2\kappa_5^2W}L^{JK}\partial_{(J}\Phi M_{K)I}\ ,
\label{6th}\\
&=\frac1{2\kappa_5^2W}\Phi M_{IJ}
-\frac6{2\kappa_5^2W}L^{KL}\partial_{(K}\Phi\Gamma_{L);IJ}\ ,
\label{7th}\\
H_I&=0\ ,
\label{8th}\\
\frac12A_{IJ}+2S_{(IJ)}
{2\kappa_5^2W}L^{KL}M_{IK}M_{JL}\ ,
\label{9th}\\
-2V_{IJ}&=-\frac6{2\kappa_5^2W}\partial_I\Phi\partial_J\Phi\ ,
\label{10th}\\
2(S_{ij})_{\alpha_i\beta_j}-2(V_{ij})_{\alpha_i\beta_j}&=0\ ,
\label{11th}\\
\partial_I\Phi\partial_J\partial_K\Phi+\frac3{2\kappa_5^2W}\partial_I\Phi M_{JK}
\partial_J\partial_K\Phi-\frac6{2\kappa_5^2W}\partial_I\Phi M_{JK},\\
\partial_I\partial_K\Phi)+\frac6{2\kappa_5^2W}(\partial_I\partial_J\Phi
\partial_K\partial_L\Phi-\partial_I\partial_K\Phi\partial_J\partial_L\Phi)\nn\\
\frac14{(\beta_f)^{\beta_i}}_{\alpha_i}+\phi^{i,\beta_i}Q_{i,\alpha_i}&=\frac6{2\kappa_5^2W}\frac14B(\phi)
\end{align}
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|
1511.00488
|
[2010]Primary: 43A85; secondary: 58J50, 22E30
§ INTRODUCTION
The study of resonances has started in quantum mechanics, where they are linked to the metastable states of a system. Mathematically, the resonances appear as poles of the meromorphic continuation of the resolvent $(H-z)^{-1}$ of a Hamiltonian $H$ acting on a space of functions $\mathcal{F}$ on which $H$ is not selfadjoint. In the last thirty years, several articles have considered the case where $H$ is the Laplacian of a Riemannian symmetric space of the noncompact type $\X$ and $\mathcal{F}$ is the space $C_c^\infty(\X)$ of smooth compactly supported functions on $\X$. The basic problems are the existence, location, counting estimates and geometric interpretation of the resonances. All these problems are nowadays well understood when $\X$ is of real rank one, such as the real hyperbolic spaces.
The situation is completely different for Riemannian symmetric spaces of higher rank. The
pioneering articles proving the analytic continuation of the resolvent of the Laplacian operator across its continuous spectrum are <cit.> and <cit.>. However, in these articles, the domains where the continuation was obtained is not sufficiently large to cover the region where the resonances could possibly be found. Indeed, the existence of resonances is linked to the singularities of the Plancherel measure on $\X$. The basic question, whether resonances exist or not for general Riemannian symmetric spaces for which the Plancherel measure is singular, is still open. If the general picture is still unknown, some complete examples in rank 2 have been treated recently: $\SL(3,\R)/\SO(3)$ in <cit.> and
the direct products $\X_1\times \X_2$ of two rank-one Riemannian symmetric spaces of the noncompact type in <cit.>.
The present paper is a natural continuation of <cit.> and deals with the cases of
Riemannian symmetric spaces $\X=\G/\K$ of real rank two and restricted root system $BC_2$ or $C_2$ except the case when $\G=\SO_0(p,2)$ with $p>2$ odd. The reason is that for all the spaces $\X$ considered here the analysis of the meromorphic continuation of the resolvent of the Laplacian can be deduced from the same problem on a direct product $\X_1\times \X_1$ of a Riemannian symmetric space of rank one not isomorphic to the real hyperbolic space.
We prove that for all the spaces $\X$ we consider, the resolvent of the Laplacian of $\X$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb C$. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. If $z_0$ is a resonance of the Laplacian, then the (resolvent) residue operator at $z_0$ is the linear operator
\begin{equation}
\label{residueop1}
{\Res}_{z_0} \wt R: C^\infty_c(\X)\to C^\infty(\X)
\end{equation}
defined by
\begin{equation}
\label{residueop2}
\big({\Res}_{z_0} \wt R f\big)(y)={\Res}_{z=z_0} [R(z)f](y) \qquad (f\in C_c^\infty(\X), \, y\in \X)\,.
\end{equation}
Since the meromorphic extension takes place on a Riemann surface, the right-hand side of
(<ref>) is computed with respect to some coordinate charts and hence determined up to constant multiples. However, the image ${\Res}_{z_0}\wt R\big(C^\infty_c(\X)\big)$ is a well-defined subspace of $C^\infty(\X)$.
Its dimension is the rank of the residue operator at $z_0$. We prove that
${\Res}_{z_0}\wt R$ acts on $C^\infty_c(\X)$ as a convolution by a finite linear combination of
spherical functions of $\X$ and is of finite rank. More precisely, write $\X=\G/\K$ for a
connected noncompact real semisimple Lie group with finite center $\G$ with maximal compact subgroup $\K$. Then the space ${\Res}_{z_0}\wt R\big(C^\infty_c(\X)\big)$ is a $\G$-module which is a finite direct sum of finite-dimensional irreducible spherical representations of $\G$. The trivial representation of $\G$
occurs for the residue operator at the first singularity, associated with the bottom of the spectrum of the Laplacian.
§.§.§ Acknowledgements
The second author would like to thank the University of Oklahoma as well as the organizers of the XXXV Workshop on Geometric Methods in Physics, Bialowieza, for their hospitality and financial support.
The third author gratefully acknowledges partial support from the NSA grant H98230-13-1-0205.
§ PRELIMINARIES
§.§ General notation
We use the standard notation $\mathbb Z$, $\R$, $\R^+$, $\C$ and $\C^\times$ for the integers, the reals, the positive reals, the complex numbers and the non-zero complex numbers, respectively. For $a\in \Zb$, the symbol $\mathbb Z_{\geq a}$ denotes the set of integers $\geq a$. We write $\leftFpar a,b \rightFpar=[a,b]\cap \Zb$ for the discrete interval of integers in $[a,b]$.
The interior of an interval $I\subseteq \R$ (with respect to the usual topology on the real line) will be indicated by $I^\circ$.
The upper half-plane in $\C$ is $\C^+=\{z \in \C:\Im z>0\}$; the lower half-plane $-\C^+$ is denoted $\C^-$.
If $\X$ is a manifold, then $C^\infty(\X)$ and $C^\infty_c(\X)$ respectively denote the space of smooth functions and the space of smooth compactly supported functions on $\X$.
§.§ Noncompact irreducible Riemannian symmetric spaces of type $BC_2$ or $C_2$
Let $\X=\G/\K$ be an irreducible Riemannian symmetric space of the noncompact type and (real) rank $2$. Hence $\G$ is a connected noncompact semisimple real Lie group with finite center and $\K$ is a maximal compact subgroup of $\G$. We can suppose that $\G$ is simple and admits a faithful linear representation. Let $\g$ and $\k$ be respectively the Lie algebras of $$ and $$, and let $=⊕̨$ be the corresponding Cartan decomposition.
Let us fix a maximal abelian subspace $$ of $$. The (real) rank $2$ condition means that
$$ is a 2-dimensional real vector space. We denote by $^*$ the dual space of $$ and by
$_^*$ the complexification of $^*$. The Killing form of $$ restricts to an inner product on $$. We extend it to $^*$ by duality. The $$-bilinear extension of $··$ to $_^*$ will be indicated by the same symbol.
Let $Σ$ be the root systems of $(,)$.
In the following, we suppose that $Σ$ is either of type $BC_2$ or of type $C_2=B_2$.
The set $Σ^+$ of positive restricted roots is the form
$Σ^+=Σ^+_⊔Σ^+_⊔Σ^+_$, where
\begin{eqnarray}
\label{eq:roots}
\Sigma^+_\rl=\{\beta_1, \beta_2\}\,, \notag\, \quad
\Sigma^+_\mrm=\big\{\frac{\beta_2\pm \beta_1}{2}\}\,,\quad
\Sigma^+_\rs=\big\{\frac{\beta_1}{2}, \frac{\beta_2}{2}\big\}
\end{eqnarray}
with $Σ^+_=∅$ in the case $C_2=B_2$.
The two elements of $Σ^+_$ form an orthogonal basis of $^*$ and have same norm $b$. The elements of $Σ^+_$ and $Σ^+_$ have therefore norm $√(2)/2 b$ and $b/2$, respectively. We define
$𝔞^*_+={λ∈𝔞^*: $\inner{\lambda}{\beta}>0$ for all $\beta\in\Sigma^+$}$.
The system of positive unmultipliable roots is $Σ_*^+=Σ^+_⊔Σ^+_$.
The set $Σ_*$ of unmultipliable roots is a root system. A basis of positive simple roots
for $Σ_*$ is ${β_1 ,β_2- β_1/2}$.
The Weyl group $$ of $Σ$ acts on the roots by permutations and sign changes. For $a∈{,,}$ set $Σ_a=Σ^+_a ⊔(-Σ^+_a)$. Then each $Σ_a$ is a Weyl group orbit in $Σ$.
The root multiplicities are therefore triples $m=(m_,m_,m_)$ so that $m_a$ is the (constant) value of $m$ on $Σ_a$ for $a∈{,,}$. By classification, if $=/$ is Hermitian, then $m_=1$. We adopt the convention that $m_=0$ means that $Σ^+_=∅$, i.e. $Σ$ is of type $C_2$. In this case, if $$ is Hermitian, then $$ is said to be of tube type.
The half sum of positive roots, counted with their multiplicities, is indicated by $ρ$. Hence
\begin{equation}
\label{eq:rho}
2\rho=\sum_{\alpha \in \Sigma^+} m_\alpha \alpha= \Big(m_\rl+ \frac{m_\rs}{2}\Big) \beta_1 + \Big(m_\rl+m_\mrm+\frac{m_\rs}{2}\Big) \beta_2\,.
\end{equation}
Table 1 contains the rank-two irreducible Riemannian symmetric spaces $/$ with root systems of type $BC_2$, their root systems, the multiplicities $m=(m_,m_,m_)$, and the value of $ρ$.
\smallskip
\begin{table}[!ht]
\begin{adjustwidth}{-1.5cm}{}
\setlength{\extrarowheight}{.2em}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
Type &AIII & BDI & CII & DIII & EIII\\[.2em]
\hline
$\G$ & $\SU(p,2)$ $(p>2)$ &$\SO_0(p,2)\; (p>2)$ & $\Sp(p,2)\; (p\geq 2)$
& $\SO^*(10)$ & $E_{6(-14)}$
\\[.2em]
\hline
$\K$ & $\Sg(\Ug(p)\times \Ug(2))$ &$\SO(p)\times\SO(2)$ & $\Sp(p)\times\Sp(2)$ &
$\Ug(5)$ & $\Spin(10)\times\Ug(1)$
\\[.2em]
\hline
\scriptsize{Hermitian} & yes & yes & no & yes & yes
\\[.2em]
\hline
$\Sigma$ & $BC_2$ & $C_2$ & \begin{tabular}{l}
$p=2$: $C_2$ \\
$p>2$: $BC_2$
\end{tabular} & $BC_2$ & $BC_2$
\\[.2em]
\hline
{\footnotesize $m=(m_\rl,m_\mrm,m_\rs)$} & {\footnotesize $(1,2,2(p-2))$} & {\footnotesize $(1,p-2,0)$} &
& {\footnotesize $(1,4,4)$} & {\footnotesize $(1,6,8)$} \\[.2em]
\hline
$2\rho$ & {\footnotesize $(p-1)\beta_1+(p+1)\beta_2$} & {\footnotesize $\beta_1+(p-1)\beta_2$} & {\footnotesize $5\beta_1+(5+2(p-2))\beta_2$} & {\footnotesize $3\beta_1+7\beta_2$} & {\footnotesize $5\beta_1+8\beta_2$}
\\[.2em]
\hline
\end{tabular}
\medskip
\caption{Rank-two irreducible symmetric spaces with root systems $BC_2$ or $C_2$}
\end{adjustwidth}
\end{table}
Notice that we are using special low rank isomorphisms (see e.g. \cite[Ch. X, \S 6, no.4]{He1}), which allow us to omit some cases:
\begin{eqnarray}
\SU(2,2)/\Sg(\Ug(2)\times \Ug(2)) &\cong & \SO_0(4,2)/ (\SO(4)\times \SO(2))\,,\\
\Sp(2,\R)/\Ug(2) &\cong &\SO_0(3,2)/ (\SO(3)\times \SO(2))\,,\\
\SO^*(8)/\Ug(4) &\cong &\SO_0(6,2)/ (\SO(6)\times \SO(2))\,.
\end{eqnarray}
Observe also that $_0(2,2)/((2)×(2)) ≅(2,)×(2,)$ is not in the list because not irreducible.
\begin{rem}
Up to isomorphisms, there are four additional irreducible Riemannian symmetric spaces of rank two:
\begin{enumerate}
\item
$\SL(3,\R)/\SO(3)$ (type AI, with root system of type $A_2$ and one root multiplicity $m=1$; see \cite{HPP14}),
\item
$\SU^*(6)/\Sp(3)$ (type AII, with root system of type $A_2$ and one even root multiplicity $m=4$; see \cite{Str05}),
\item
$E_{6(-26)}/F_4$ (type EIV, with root system of type $A_2$ and one even root multiplicity $m=8$; see \cite{Str05}),
\item
$G_{2(-14)}/(\SU(2)\times \SU(2))$ (type G, with root system of type $G_2$ and one root multiplicity
\end{enumerate}
\end{rem}
\subsection{The Plancherel density of $\G/\K$}
For $λ∈𝔞_^*$ and $β∈Σ$ we shall employ the notation
\begin{equation}\label{eq:la}
\lambda_\beta=\frac{\inner{\lambda}{\beta}}{\inner{\beta}{\beta}}\,.
\end{equation}
Observe that
\begin{eqnarray}
\label{eq:lambdabetahalf}
\lambda_{\beta/2}&=&2\lambda_{\beta}\,,\\
\label{eq:lambdabeta12}
\lambda_{(\beta_2\pm \beta_1)/2}&=&
2 \frac{\inner{\lambda}{\beta_2}\pm \inner{\lambda}{\beta_1}}
{\inner{\beta_2}{\beta_2}+ \inner{\beta_1}{\beta_1}}=\lambda_{\beta_2}\pm \lambda_{\beta_1}\,.
\end{eqnarray}
For $β∈Σ_*$, we set
\begin{equation}
\label{eq:wtrhob}
\wt\rho_\beta=\frac{1}{2} \left( m_{\beta}+ \frac{m_{\beta/2}}{2}\right)\,,
\end{equation}
where $m_β$ denotes the multiplicity of the root $β$, and
\begin{equation}
\label{eq:cbeta}
c_\beta(\lambda)= \frac{2^{-2\lambda_\beta}\Gamma(2\lambda_\beta)}{\Gamma\big(\lambda_\beta+\frac{m_{\beta/2}}{4}+\frac{1}{2}\big) \Gamma\big(\lambda_\beta+\wt\rho_\beta\big)}\,.
\end{equation}
Observe that $ρ_β=ρ_β=⟨ρ,β⟩/⟨β,β⟩$ if $β$ is a simple root in $Σ_*$ (but not in general).
In particular, $ρ_β_1=ρ_β_1$.
Harish-Chandra's $c$-function $$ (written in terms of unmultipliable
instead of indivisible roots) is defined by
\begin{equation}
\label{eq:c}
\cHC(\lambda)=c_0 \; \prod_{\beta\in\Sigma_*^+} c_\beta(\lambda)\,.
\end{equation}
where $c_0$ is a normalizing constants so that $(ρ)=1$.
In the following we always adopt the convention that empty products are equal to $1$.
As a consequence of the properties of the gamma function, we have the following explicit expression.
\begin{lem}
\label{lemma:Planch-density}
The Plancherel density is given by the formula
\begin{equation}
\label{eq:Planch-density}
[\cHC(\lambda)\cHC(-\lambda)]^{-1}=C \Pi(\lambda) P(\lambda) Q(\lambda)\,,
\end{equation}
\begin{eqnarray}
\label{eq:Pi}
\Pi(\lambda)&=&\prod_{\beta \in \Sigma_*^+} \lambda_\beta\,,\\
\label{eq:P}
\prod_{\beta \in \Sigma_*^+} \Big( \prod_{k=0}^{(m_{\beta/2})/2-1} \big[ \lambda_\beta -\big( \tfrac{m_{\beta/2}}{4}-\tfrac{1}{2} \big)+k\big]
\prod_{k=0}^{2 \wt \rho_\beta-2} [ \lambda_\beta -(\wt \rho_\beta-1)+k]\Big)\,, \\
\label{eq:Q}
Q(\lambda)&=&\prod_{\stackrel{\beta\in \Sigma_*^+}{\textup{$m_\beta$ odd}}}
\cot(\pi(\lambda_\beta - \wt\rho_\beta))\,.
\end{eqnarray}
and $C$ is a constant.
Consequently, the singularities of the Plancherel density $[\cHC(\lambda)\cHC(-\lambda)]^{-1}$ are at most simple poles located along the hyperplanes of the equation
\begin{equation*}
\pm \lambda_\beta=\wt\rho_\beta +k
\end{equation*}
where $\beta \in \Sigma_*^+$ has odd multiplicity $m_\beta$, and $k \in \Zb_{\geq 0}$.
\end{lem}
\begin{proof}
The singularities of $[\cHC(\lambda)\cHC(-\lambda)]^{-1}$ are those of $\cot(\pi(\lambda_\beta-\wt\rho_\beta))$,
for $\beta \in \Sigma_*^+$ with $m_ \beta$ odd, which are not killed by zeros of the polynomial
$\Pi(\lambda) P(\lambda)$.
\end{proof}
The following corollary will allow us to establish a region of holomorphic extension of the
\begin{cor}
\begin{equation}
\label{eq:L}
L=\min\{\wt\rho_\beta |\beta|: \text{$\beta \in \Sigma_*^+$, $m_\beta$ odd} \}\,.
\end{equation}
Then, for every fixed $\omega \in \mathfrak{a}^*$ with $|\omega|=1$, the function
$r \mapsto [\cHC(r\omega)\cHC(-r\omega)]^{-1}$ is holomorphic on $\C \setminus \big(]-\infty,-L] \cup [L,+\infty[ \big)$.
\end{cor}
The values of $ρ_β$ for the roots in $Σ_*^+$, as well as the value of
$L$, are given in Table 2.
Recall that
\smallskip
\begin{table}[!ht]
\setlength{\extrarowheight}{.2em}
\begin{adjustwidth}{-1.5cm}{}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
$\G$ & $\SU(p,2)$ $(p>2)$ &$\SO_0(p,2)\; (p>2)$ & $\Sp(p,2)\; (p\geq 2)$
& $\SO^*(10)$ & $E_{6(-14)}$
\\[.2em]
\hline\hline
$\wt \rho_{\beta_j}=\frac{1}{2}\big(m_\rl+\frac{m_\rs}{2}\big)$ &
$(p-1)/2$ & $1/2$ & $p-1/2$ & $3/2$ & $5/2$ \\[.2em]
\hline
$\wt \rho_{(\beta_2\pm \beta_1)/2}=\frac{m_\mrm}{2}$ &
1 & $(p-2)/2$ & $2$ & $2$ & $3$ \\[.2em]
\hline
\footnotesize{$\Sigma^+_{*,{\rm odd}}=\{\beta\in \Sigma_*^+:\text{$m_\beta$ odd}\}$ } &
$\{\beta_1,\beta_2\}$ &
\begin{tabular}{l}
\footnotesize{$p$ even: $\{\beta_1,\beta_2\}$} \\
\footnotesize{$p$ odd: $\{\beta_1,\beta_2, \frac{\beta_2\pm \beta_1}{2}\}$}
\end{tabular}
& $\{\beta_1,\beta_2\}$ & $\{\beta_1,\beta_2\}$ & $\{\beta_1,\beta_2\}$ \\[.2em]
\hline
{\footnotesize $L=\min\{\wt\rho_\beta |\beta|:
\text{$\beta \in \Sigma^+_{*,{\rm odd}}$}\} $} & $\frac{\sqrt{p-1}}{2}b$ &
\begin{tabular}{l}
\footnotesize{$p=3$: $\frac{\sqrt{2}}{4}b$} \\
\footnotesize{$p> 3$: $\frac{b}{2}$}
\end{tabular} & $\big(\frac{3}{2}+2(p-2)\big) b$ & $\frac{3}{2}b$ & $\frac{5}{2}b$ \\[.2em]
\hline
\end{tabular}
\medskip
\caption{The values of $\wt \rho_\beta$ for $\beta \in \Sigma_*^+$ and of $L$}\label{table tilde rho}
\end{adjustwidth}
\end{table}
A computation using the values in the tables together with \cite[\S 2]{HPP15} yields the following corollary.
\begin{cor}\label{Cor:Plancherel density}
If $\G\neq \SO_0(p,2)$ with $p$ odd, then $\{\beta\in \Sigma_*^+: \text{$m_\beta$ is odd}\}$
is equal to $\{\beta_1,\beta_2\}$. Hence
\begin{equation}
\label{eq:cHCandproducts}
\Pi_0(\lambda)P_0(\lambda)[\cHC^\times(\lambda)\cHC^\times(-\lambda)]^{-1}\,,
\end{equation}
\begin{eqnarray}
\label{eq:Pi0}
\Pi_0(\lambda)&=&\lambda_{(\beta_2-\beta_1)/2} \lambda_{(\beta_2+\beta_1)/2}
\label{eq:P0}
\prod_{\beta=(\beta_2\pm \beta_1)/2}
\prod_{k=0}^{2 \wt \rho_\beta-2} [ \lambda_\beta -(\wt \rho_\beta-1)+k]\,,
\end{eqnarray}
$[\cHC^\times(\lambda)\cHC^\times(-\lambda)]^{-1}$ is the Plancherel density of the product
$\X_1\times \X_1$ of two isomorphic rank-one Riemannian symmetric spaces with root systems of type $BC_1$ (or $A_1$) and multiplicities $(m_{\beta_j},m_{\beta_j/2})=(m_\rl,m_\rs)$.
If $\G=\SO_0(p,2)$ with $p\geq 3$ odd, then $\Sigma^+_*=\Sigma^+$ and
\begin{eqnarray*}
\Pi(\lambda)&=&\lambda_{\beta_1} \lambda_{\beta_2} (\lambda_{\beta_2}^2-\lambda_{\beta_1}^2)\\
P(\lambda)&=&\prod_{k=0}^{p-2} \Big( \lambda_{(\beta_2-\beta_1)/2}-\big(\frac{p-2}{2}-1\big)+k\Big)\Big( \lambda_{(\beta_2+\beta_1)/2}-\big(\frac{p-2}{2}-1\big)+k\Big)\\
&=&\prod_{k=0}^{p-2} \Big( \lambda_{\beta_2}-\lambda_{\beta_1}-\big(\frac{p-2}{2}-1\big)+k\Big)\Big( \lambda_{\beta_2}+\lambda_{\beta_1}-\big(\frac{p-2}{2}-1\big)+k\Big)\\
Q(\lambda)&=&\cot\big(\pi \big( \lambda_{\beta_1}-\tfrac{1}{2}\big)\big)
\cot\big(\pi \big( \lambda_{\beta_2}-\tfrac{1}{2}\big)\big)
\cot\big(\pi \big( \lambda_{\beta_2}-\lambda_{\beta_1}-\tfrac{p}{2}+1\big)\big)\\
&&\hskip 7truecm \times \cot\big(\pi \big( \lambda_{\beta_2}+\lambda_{\beta_1}-\tfrac{p}{2}+1\big)\big)
\end{eqnarray*}
\end{cor}
\subsection{The resolvent of $\Delta$}
Endow the Euclidean space $𝔞^*$ with the Lebesgue measure normalized so that the unit cube has volume $1$. On the Furstenberg boundary $=/$ of $$, where $$ is the centralizer of $𝔞$ in $$, we consider the $$-invariant measure $db$ normalized so that the volume of $$ is equal to $1$.
Let $$ be equipped with its (suitably normalized) natural $$-invariant Riemannian measure
and let $Δ$ denote the corresponding (positive) Laplacian.
As in the cases treated in \cite{HPP14} and \cite{HPP15}, it will be convenient to identify $𝔞^*$ with $$ as vector spaces over $$. More precisely, we want to view $𝔞_1^*$ and $𝔞_2^*$ as the real and the purely imaginary axes, respectively. To distinguish the resulting complex structure in $𝔞^*$ from the natural complex structure of $𝔞^*_$, we shall indicate the complex units in $𝔞^*≡$ and $𝔞_^*$ by $i$ and $$, respectively.
So $𝔞^*≡=+i $, whereas $𝔞_^*=𝔞^*+𝔞^*$. For $r,s∈$ and $λ,ν∈𝔞^*$ we have $(r+is)(λ+ν)=(rλ-sν)+(rν+sλ)∈𝔞^*_$.
By the Plancherel Theorem \cite[Ch. III, \S 1, no. 2]{He3}, the Helgason-Fourier transform
$ℱ$ is a unitary equivalence of $Δ$ acting on $L^2()$ with the
multiplication operator $M$ on $L^2(𝔞^*_+×B,[(λ)(-λ)]^-1 dλ db)$ given by
\begin{equation} \label{eq:Mmult}
MF(\lambda,b)=\Gamma(\Delta)(\bi\lambda)F(\lambda,b)=(\inner{\rho}{\rho}+\inner{\lambda}{\lambda})F(\lambda,b) \qquad ((\lambda,b)\in \mathfrak{a}^*\times B)\,.
\end{equation}
It follows, in particular, that the spectrum of $Δ$ is the half-line
$[ρ_^2, +∞[$, where $ρ_^2=ρρ$.
By the Paley-Wiener theorem for $ℱ$, see e.g. \cite[Ch. III, \S 5]{He3}, for every
$u ∈∖[ρ_^2, +∞[$ the resolvent of $Δ$ at $u$ maps $C^∞_c()$ into $C^∞()$.
Recall that for sufficiently regular functions $f_1, f_2:→$, the convolution $f_1×f_2$ is the function on $$ defined by $(f_1 ×f_2) ∘π= (f_1 ∘π) * (f_2 ∘π)$. Here $π:→=/$ is the natural projection and $*$ denotes the convolution product of functions on $$.
The Plancherel formula yields the following explicit expression for the image of $f ∈C^∞_c()$ under the resolvent operator $R(z)=(Δ-ρ_^2-z^2)^-1$ of the shifted Laplacian $Δ- ρ_^2$:
\begin{equation}\label{eq:resolvent}
[R(z)f](y)=\int_{\mathfrak{a}^*} \frac{1}{\inner{\lambda}{\lambda}-z^2}\; (f \times \varphi_{\bi\lambda})(y) \; \frac{d\lambda}{\cHC(\bi\lambda)\cHC(-\bi\lambda)}
\qquad (z \in \C^+\,, y\in \X)\,.
\end{equation}
See \cite[formula (14)]{HP09}. Here and in the following, resolvent equalities as \eqref{eq:resolvent} are given up to non-zero constant multiples.
\section{Meromorphic extension in the case $\G\neq \SO_0(2,p)$, $p>2$ odd}
\subsection{The resolvent kernel in polar coordinates}
We write $𝔞_1^*=β_1$ and $𝔞_2^*=β_2$, so that $𝔞^*=𝔞_1^*⊕𝔞^*_2$ and
Introduce the coordinates
\begin{equation}\label{1.1}
\R^2\ni (x_1,x_2)\to x_1\beta_1+x_2\beta_2\in \a_1^*\oplus\a_2^*=\a^*.
\end{equation}
Hence $x_j=λ_β_j$ if $λ=x_1β_1+x_2β_2$.
In view of Table~\ref{table tilde rho}, the functions $Π_0$ and $P_0$ from \eqref{eq:Pi0} and \eqref{eq:P0} can be rewritten in these coordinates, as
\begin{eqnarray}
\label{eq:Pi0'}
\Pi_0(\lambda)&=&\Pi_0(x_1\beta_1+x_2\beta_2)\ =\ x_2^2-x_1^2\,,\\
\label{eq:P0'}
P_0(\lambda)&=&\nonumber P_0(x_1\beta_1+x_2\beta_2)
\ =\ \prod_{\beta=(\beta_2\pm \beta_1)/2} \prod_{k=0}^{m_\mrm-2} \big[ \lambda_\beta -\big(\frac{m_\mrm}{2}-1\big)+k\big]\\
&=&\nonumber \prod_{k=1}^{m_\mrm-1} \big[ (x_2+x_1)-\frac{m_\mrm}{2}+k\big]\,\big[ (x_2-x_1)-\frac{m_\mrm}{2}+k\big]\\
%&=&\nonumber \prod_{k=1-m_\mrm}^{m_\mrm-1} [ (x_1+x_2)-k]\,[ (x_1-x_2)-k]\\
&=&\prod_{k=1}^{m_\mrm-1} \big[ (x_2-\frac{m_\mrm}{2}+k)^2-x_1^2\big]
\end{eqnarray}
(\beta_j)_{(\beta_2\pm\beta_1)/2}=(\beta_j)_{\beta_2}\pm (\beta_j)_{\beta_1}=
\begin{cases}
\pm 1 &\text{for } j=1,\\
1 &\text{for } j=2.\\
\end{cases}
We write
\begin{equation}\label{eq:vartheta0}
\vartheta_0(x_1,x_2)= \Pi_0(\lambda)P_0(\lambda) = (x_2^2-x_1^2)\prod_{k=1}^{m_\mrm-1} \big[ (x_2-\frac{m_\mrm}{2}+k)^2-x_1^2\big]
\end{equation}
Further we write
\begin{equation}\label{eq:Rang 1 c-function}
[\cHC^{\X_1}(\bi \lambda)\cHC^{\X_1}(-\bi \lambda)]^{-1} = C_1 \Pi_1(\bi \lambda) P_1(\bi \lambda) Q_1(\bi \lambda)
\end{equation}
for the Plancherel density of the space $_1$ in Corollary~\ref{Cor:Plancherel density}, so that
\begin{equation}\label{eq:Rang 1 c-function}
[\cHC^\times(\bi\lambda)\cHC^\times(-\bi\lambda)]^{-1} = C_1^2 \Pi_1(i x_1)\Pi_1(i x_2) P_1(i x_1)P_1(i x_2) Q_1(i x_1)Q_1(i x_2).
\end{equation}
See \cite[\S 1 and \S 2]{HPP15}.
Using \eqref{eq:cHCandproducts} and omitting non-zero constant multiples, we can therefore rewrite \eqref{eq:resolvent} as
\begin{multline*}
=\int_{\a^*}\frac{1}{\langle \lambda,\lambda\rangle-z^2}(f\times \varphi_{\bi\lambda})(y)\frac{1}{\cHC(\bi\lambda)\cHC(-\bi\lambda)}\,d\lambda\\
=\int_{\R^2}\frac{(f\times \varphi_{\bi x_1\beta_1+\bi x_2\beta_2})(y)}{x_1^2b^2+x_2^2b^2-z^2} \vartheta_0(ix_1,ix_2)x_1x_2P_1(ix_1)P_1(ix_2)Q_1(ix_1)Q_1(ix_2)\,dx_1\,dx_2\,.
\end{multline*}
Introduce the polar coordinates
\[
x_1=\frac{r}{b}\cos \theta,\ \ \ x_2=\frac{r}{b}\sin \theta \qquad (0<r\,,\ 0\leq \theta<2\pi)
\]
on $^2$ and set
\begin{equation}
\label{eq:pq}
p_1(x)=P_1\big(i\tfrac{x}{b}\big)\qquad \text{and} \qquad q_1(x)=Q_1\big(i\tfrac{x}{b}\big).
\end{equation}
In these terms (up to a non-zero constant multiple)
\[
\]
\begin{eqnarray}\label{2.1}
F(r)&=&\int_0^{2\pi}(f\times \varphi_{\bi\frac{r}{b}\cos \theta\,\beta_1+\bi\frac{r}{b}\sin \theta\,\beta_2})(y)\,\vartheta_{0,{\rm pol}}(r,\theta)r^2\cos\theta \sin\theta\\
&&\times \; p_1(r\cos \theta) q_1(r\cos \theta) p_1(r\sin \theta) q_1(r\sin \theta)\,d\theta\,,\nn
\end{eqnarray}
$$\vartheta_{0,{\rm pol}}(r,\theta)= \vartheta_0(i\,x_1,i\,x_2) = -\frac{r^2}{b^2}(\sin^2 \theta -\cos^2 \theta) \prod_{k=1}^{m_\mrm-1} \big[ \big(\frac{r}{b}i\sin \theta-\frac{m_\mrm}{2}+k\big)^2+\frac{r^2}{b^2}\cos^2 \theta\big].$$
Here and in the following, we omit from the notation the dependence of $F$ on the function $f∈C^∞_c()$ and on $y∈$.
Recall the functions
\begin{equation}
\label{eq:c-s}
\cz(w)=\frac{w+w^{-1}}{2}\,,\qquad \sz(w)=\frac{w-w^{-1}}{2}=i\cz(-iw) \qquad (w\in\C^\times)
\end{equation}
from \cite[(20)]{HPP15} and notice that
\[
\cos\theta=\cz(e^{i\theta})\,, \qquad \sin\theta=\frac{\sz(e^{i\theta})}{i}=\cz(-ie^{i\theta})\,,\qquad d\theta=\frac{d e^{i\theta}}{ie^{i\theta}}.
\]
For $z ∈$ and $w ∈^×$ define
\begin{eqnarray}
\label{eq:psiz}
\psi_z(w)&=&(f\times \varphi_{\bi\frac{z}{b}\cz(w)\,\beta_1+\bi\frac{z}{b}\cz(-iw)\,\beta_2})(y)\\
\label{eq:phiz}
\phi_z(w)&=&-z^2\cz(w) \frac{\sz(w)}{w} p_1\big(z\cz(w)\big) q_1\big(z\cz(w)\big) p_1\big(z\cz(-iw)\big) q_1\big(z \cz(-iw)\big).
\end{eqnarray}
as in \cite[(32) and (33)]{HPP15} together with
\begin{equation}\label{eq:vathetaz}
\vartheta_z(w)=\frac{z^2}{b^2}\big(\cz(w)^2-\cz(-iw)^2\big)\prod_{k=1}^{m_\mrm-1} \big[ \big(\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k\big)^2+\frac{z^2}{b^2}\cz(w)^2\big]\,,
\end{equation}
which is a polynomial function of $z$.
\begin{equation}\label{F(r)}
\end{equation}
\begin{lem}
\label{lemma:functions-z}
Let $z\in \C$ and $w\in \C^\times$. Then:
\medskip
\begin{center}
\begin{tabular}{llll}
\setlength{\extrarowheight}{.3em}
&$\psi_{-z}(w)=\psi_{z}(w)$, \qquad\qquad &$\psi_{z}(-w)=\psi_{z}(w)$, \qquad\qquad
&$\phi_{-z}(w)=\phi_{z}(w)$, \qquad &$\phi_{z}(-w)=-\phi_{z}(w)$, \qquad
&$\vartheta_{-z}(w)=\vartheta_{z}(w)$, \qquad &$\vartheta_{z}(-w)=\vartheta_{z}(w)$, \qquad
\end{tabular}
\end{center}
\smallskip
\end{lem}
\begin{proof}
Set $\mu(z,w)=\bi \frac{z}{b}\cz(w) \beta_1 +\bi \frac{z}{b}\cz(-iw) \beta_2$, so that
$\psi_z(w)=f\times \varphi_{\mu(z,w)}(y)$. Then $\mu(-z,w)$, $\mu(z,-w)$ and $\mu(z,iw)$
are transformed into $\mu(z,w)$ by sign changes and transposition of $\beta_1$ and of $\beta_2$.
The equalities for $\psi_z(w)$ then follow because the spherical function $\varphi_\lambda$ is
$W$-invariant in the parameter $\lambda$.
The equalities for $\phi_{z}(w)$ are an immediate consequence of \eqref{eq:c-s} and the fact that the functions $\cz$, $\sz$ and $p_1q_1$ are odd.
To prove the relations for $\vartheta_z(w)$, notice that
$-\frac{m_\mrm}{2}+k = \frac{m_\mrm}{2}-h$
where $h=m_\mrm-k \in \{1,\dots,m_\mrm-1\}$ when $k \in \{1,\dots,m_\mrm-1\}$.
\prod_{k=1}^{m_\mrm-1} \Big[\big(-\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k\big)^2+\frac{z^2}{b^2}\cz(w)^2 \Big] =
\prod_{h=1}^{m_\mrm-1}
\Big[ \big(\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+h\big)^2+\frac{z^2}{b^2}\cz(w)^2 \Big] \,.
This proves the first two equalities for $\vartheta_z(w)$ since $\sz(-w)=-\sz(w)$. For the last equality, notice that
\begin{eqnarray*}
&&\qquad =\big[\frac{z}{b}i\cz(w)-\frac{m_\mrm}{2}+k+i\frac{z}{b}i\sz(w)\big]
\big[\frac{z}{b}i\cz(w)-\frac{m_\mrm}{2}+k-i\frac{z}{b}i\sz(w)\big]\\
&&\qquad =\big[-\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k+i\frac{z}{b}\cz(w)\big]
\big[\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k-i\frac{z}{b}\cz(w)\big]\,.
\end{eqnarray*}
Hence, since $m_\mrm$ is even,
\begin{eqnarray*}
\Big[ \big(\frac{z}{b}\sz(iw)-\frac{m_\mrm}{2}+k\big)^2+\frac{z^2}{b^2}\cz(iw)^2 \Big] \\
&&\qquad =
\prod_{k=1}^{m_\mrm-1}
\big[\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k+i\frac{z}{b}\cz(w)\big] \cdot
(-1)^{m_\mrm-1} \prod_{h=1}^{m_\mrm-1}
\big[\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+h-i\frac{z}{b}\cz(w)\big]\\
&&\qquad =
-\prod_{k=1}^{m_\mrm-1} \Big[ \big(\frac{z}{b}\sz(w)-\frac{m_\mrm}{2}+k\big)^2+\frac{z^2}{b^2}\cz(w)^2 \Big]\,.
\end{eqnarray*}
This proves the claim because $\cz(w)^2-\cz(-iw)^2$ changes sign under the transformation
$w \to iw$.
\end{proof}
Thus \cite[Lemma 3]{HPP15} generalizes as follows.
\begin{lem}
\label{lemma:holoextF}
The function $F(r)$, \eqref{F(r)}, extends holomorphically to
\begin{eqnarray}\label{2.2}
\end{eqnarray}
\[
z\in \C\setminus i((-\infty,- L]\cup[L, +\infty))
\]
and $L$ is the constant defined in \eqref{eq:L}.
The function $F(z)$ is even and $F(z)z^{-2}$ is bounded near $z=0$.
\end{lem}
The following proposition, giving an initial holomorphic extension of the resolvent across the spectrum of the Laplacian, has been independently proven by Mazzeo and Vasy \cite[Theorem 1.3]{MV05} and by Strohmaier \cite[Proposition 4.3]{Str05} for general Riemannian symmetric spaces of the noncompact type and even rank. It shows that all possible resonances of the resolvent are located along the half-line $i(-∞, -L]$. According to our conventions, we will omit $f$ and $y$ from the notation and write $R(z)$ instead of $[R(z)f](y)$.
\begin{pro}
The resolvent $R(z)=[R(z)f](y)$ extends holomorphically from $\C \setminus \big((-\infty,0] \cup i(-\infty, -L]\big)$ to a logarithmic Riemann
surface branched along $(-\infty,0]$, with the preimages of $i\big((-\infty, -L]\cup [L, +\infty)\big)$ removed and, in terms of monodromy, it satisfies the following equation
\begin{equation*}
R(z e^{2i\pi})=R(z)+2 i\pi\, F(z)\\ \qquad (z \in \C \setminus \big((-\infty,0]\cup i(-\infty, -L]\cup i[L,+\infty) \big)).
\end{equation*}
\end{pro}
The starting point for studying the meromorphic extension of $R$ across $i(-∞, -L]$ is the Proposition \ref{pro:holoextRF} below. It says that this meromorphic extension is equivalent to that of function $F$. This proposition is analogous to \cite[Proposition 4]{HPP15} and its proof is omitted.
\begin{pro}
\label{pro:holoextRF}
Fix $x_0>0$ and $y_0>0$. Let
\begin{eqnarray*}
Q&=&\{z\in \C; \Re z>x_0,\ y_0> \Im z \geq 0\}\\
U&=&Q\cup \{z\in \C; \Im z <0\}
\end{eqnarray*}
Then there is a holomorphic function $H:U\to \C$ (depending on $f\in C^\infty_c(\X)$ and $y\in \X$, which are omitted from the notation) such that
\begin{equation}
\label{eq:holoextRF}
R(z)=H(z)+\pi i\, F(z) \qquad (z\in Q).
\end{equation}
As a consequence, the resolvent $R(z)=[R(z)f](y)$ extends holomorphically from $\C^+$ to
$\C \setminus \big((-\infty,0] \cup i(-\infty, -L]\big)$.
\end{pro}
\subsection{Meromorphic extension and residue computations}
This section is devoted to the meromorphic extension of the function $F$ (and hence of the resolvent) across the half-line $i(-∞, -L]$. We set
\begin{equation}
\label{eq:psi-theta}
\psi^\vartheta_z(w)=\vartheta_z(w)\psi_z(w)
\end{equation}
and follow the stepwise extension procedure for $F$ from \cite[\S2 and \S 3]{HPP15} with $ψ_z(w)$ replaced by $ψ^ϑ_z(w)$. Some formulas are simplified by the fact that we are only dealing with the special case of $_1=_2$ with $β_1$ and $β_2$ of equal norms $b_1=b_2=b$ and equal odd multiplicities $m_β_1=m_β_2$. Notice also that in this paper, studying the singularities of the Plancherel density, we are replacing the elements $ρ_β_1$ and $ρ_β_2$ used in \cite{HPP15} with $ρ_β_1$ and $ρ_β_2$, which are equal and have value $1/2( m_+m_/2)$. Indeed, in the case of direct product of rank one symmetric spaces treated in \cite{HPP15}, there was no need of introducing multiple notation by distinguishing between $ρ_β$ and $ρ_β$ for $β∈Σ_*$. The distinction is now necessary since $ρ_β_1=ρ_β_1=ρ_β_2≠ρ_β_2$.
Furthermore, we omit the index $j$ from the notation used in \cite{HPP15} when it only refers to which of the two factors one considers. So, for instance \cite[(38)]{HPP15} yields, for the set of singularities of the product $p_1 q_1$ from \eqref{eq:pq}, the set
\begin{equation}
\label{eq:Sj}
S=S_+ \cup (-S_+)\,,
\end{equation}
\begin{equation}
\label{eq:Splus}
S_+=ib(\wt\rho_{\beta_1}+\Bbb Z_{\geq 0})=ib\Big(\frac{1}{2}\big(m_\rl+\frac{m_\rs}{2}\big)+\Bbb Z_{\geq 0}\Big).
\end{equation}
For $r>0$ and $c,d ∈∖{0}$ recall the sets
\begin{eqnarray*}
\Dg_r&=&\{z\in \C;\ |z|<r\}\,,\\
\Eg_{c,d}&=&\big\{\xi+i\eta\in \C;\ \left(\tfrac{\xi}{c}\right)^2+ \left(\tfrac{\eta}{d}\right)^2<1\big\}\,.
\end{eqnarray*}
and the role they play in \cite[\S 1.4]{HPP15} for the functions $$ and $$ introduced in \eqref{eq:c-s}. Then \cite[Prop. 6]{HPP15} translates in the following proposition.
\begin{pro}\label{2.5}
Suppose $z\in \C\setminus i((-\infty,- L]\cup[L, \infty))$ and $r>0$ are such that
\begin{equation}\label{2.5.1}
S\cap z\partial E_{\cz(r),\sz(r)}=\emptyset.
\end{equation}
\begin{equation}
\label{eq:F-contour}
F(z)=F_r(z)+2\pi i \, F_{r,{\rm res}}(z),
\end{equation}
\begin{eqnarray*}
F_r(z)&=&\int_{\partial D_r}\psi^\vartheta_z(w)\phi_z(w)\,dw,\\
F_{r,{\rm res}}(z)&=&{\sum}_{w_0}'\psi^\vartheta_z(w_0)\Res_{w=w_0}\phi_z(w),
\end{eqnarray*}
and $\sum_{w_0}'$ denotes the sum over all the $w_0$ such that
\begin{equation}\label{2.5.2}
z\cz(w_0)\in S\cap z(\Eg_{\cz(r),\sz(r)}\setminus [-1,1])
\end{equation}
\begin{equation}\label{2.5.3}
z\cz(-iw_0)\in S\cap z(\Eg_{\cz(r),\sz(r)}\setminus [-1,1]).
\end{equation}
Both $F_r$ and $ F_{r,{\rm res}}$ are holomorphic functions on the open subset of $\C\setminus i((-\infty,- L]\cup[L, \infty))$ where the condition \eqref{2.5.1} holds. Furthermore, $F_r$ extends to a holomorphic function on the open subset of $\C$ where the condition \eqref{2.5.1} holds.
\end{pro}
To make the function $ F_r,res(z)$ explicit, we proceed as in \cite[\S 3.1]{HPP15}. The present situation is in fact simpler, because only the case
$L_1,ℓ=L_2,ℓ$ occurs. We denote this common value by $L_ℓ$, i.e.
we define for $ℓ∈_≥0$
\begin{equation}
\label{eq:Lell}
L_\ell=b(\wt \rho_{\beta_1}+\ell)=b\big(\tfrac{m_\rl}{2}+\tfrac{m_\rs}{4}+\ell\big)\,.
\end{equation}
So $S_+={iL_ℓ; ℓ∈_≥0}$.
If $0≠z∈∖i ((-∞,-L_ℓ]∪[L_ℓ,+∞))$, then $
i L_ℓ/z ∈∖[-1,1]$ and we can
uniquely define $w_1^±∈_1∖{0}$ satisfying
\begin{eqnarray}
\label{eq:cw1pm-eps-k1}
z\cz(w_1^\pm)&=&\pm i L_{\ell}\,.
\end{eqnarray}
Since $c(-w)=-c(w)$, we obtain that $w_1^-=-w_1^+$. Moreover,
$w_1$ satisfies \eqref{2.5.2} if and only if $w_2=iw_1$ satisfies \eqref{2.5.3}
because $z(_(r),(r)∖[-1,1])$ is symmetric with respect to the origin $0∈$.
\begin{multline}
\label{eq:Gr-1}
F_{r,{\rm res}}(z)={\sum}_{w_1^+}'
\Big[\psi^\vartheta_z(w_1^+)\Res_{w=w_1^+}\phi_z(w)+
\psi^\vartheta_z(w_1^-)\Res_{w=w_1^-}\phi_z(w)+\\
\psi^\vartheta_z(iw_1^+)\Res_{w=iw_1^+}\phi_z(w)+
\psi^\vartheta_z(iw_1^-)\Res_{w=iw_1^-}\phi_z(w)\Big]\,,
\end{multline}
where $∑_w_1^+'$ denotes the sum over all the $w_1^+$ such that
and $w_1^-=-w_1^+$.
Then, using Lemma \ref{lemma:functions-z}, we obtain the following analogue of \cite[Lemma 9]{HPP15}.
\begin{lem}
\label{lemma:Gjkj}
For $\ell\in \Zb_{\geq 0}$ and
$0\neq z\in \C\setminus i \big((-\infty,-L_{\ell}]\cup [L_{\ell},+\infty)\big)$,
let $w_1^\pm$ be defined by \eqref{eq:cw1pm-eps-k1}. Then
\begin{eqnarray}
\label{eq:psithetazw1}
\psi^\vartheta_z(iw_1^-)\,,\\
\label{eq:Res-phiw1}
\Res_{w=w_1^+} \phi_z(w)=\Res_{w=w_1^-} \phi_z(w)=
\Res_{w=iw_1^+} \phi_z(w)=\Res_{w=-iw_1^-} \phi_z(w)\,.\\
\end{eqnarray}
\begin{eqnarray*}
\label{eq:psithetazw1-1}
\psi^\vartheta_z(w_1^+)&=&\psi^\vartheta_z\Big(\cz^{-1}\big(\tfrac{iL_{\ell}}{z}\big)\Big)\,,\\
\label{eq:Res-phiw1}
\Res_{w=w_1^+} \phi_z(w)&=&-C_{\ell}\,
p_1\Big(i z (\sz\circ \cz^{-1})\big(\tfrac{iL_{\ell}}{z}\big)\Big) \,
q_1\Big(i z (\sz\circ \cz^{-1})\big(\tfrac{iL_{\ell}}{z}\big)\Big)\,,
\end{eqnarray*}
\begin{equation}\label{constants}
C_{\ell}= \frac{b}{\pi}L_{\ell}\, p_1(iL_{\ell}) \neq 0\,.
\end{equation}
\end{lem}
\begin{cor}
\label{cor:Gell}
Let $\ell\in \Zb_{\geq 0}$ and $0\neq z\in \C\setminus i \big((-\infty,-L_{\ell}]\cup [L_{\ell},+\infty)\big)$. Set
\begin{eqnarray}
\label{eq:Gell}
G_{\ell}(z)&=&-C_\ell \psi^\vartheta_z\Big(\cz^{-1}\big(\tfrac{iL_{\ell}}{z}\big)\Big)
p_1\Big(i z (\sz\circ \cz^{-1})\big(\tfrac{iL_{\ell}}{z}\big)\Big) \,
q_1\Big(i z (\sz\circ \cz^{-1})\big(\tfrac{iL_{\ell}}{z}\big)\Big)\,,\\
\label{eq:Sjrz}
\Sg_{r,z,+}&=&\{\ell\in \Zb_{\geq 0}: i L_{\ell}\in z\big(\Eg_{\cz(r),\sz(r)}\setminus [-1,1]\big)\}\,.
\end{eqnarray}
Then the function $ F_{r,{\rm res}}(z)$ on the right-hand side of \eqref{eq:F-contour} is given by
\begin{equation}
\label{eq:Gr-Splus}
F_{r,{\rm res}}(z)=4 \sum_{\ell \in \Sg_{r,z,+}} G_\ell(z)\,.
\end{equation}
\end{cor}
The following proposition is analogous to \cite[Proposition 10]{HPP15}.
\begin{pro}
\label{prop:SjrzW}
For $0<r<1$ and $0\neq z\in \C\setminus i((-\infty,- L]\cup[L, +\infty))$,
let $\Sg_{r,z,+}$ be as in \eqref{eq:Sjrz}. Moreover,
let $W \subseteq \C$ be a connected open set such that
\begin{equation}
\label{eq:intersection-W}
S \cap W \partial \Eg_{\cz(r),\sz(r)}=\emptyset
\end{equation}
and set
\begin{equation}
\label{eq:SjrzW}
S_{r,W,+}=\{\ell\in \Zb_{\geq 0}: i L_{\ell}\in W\Eg_{\cz(r),\sz(r)}\}
\qquad (z \in W\setminus i\R)\,.
\end{equation}
Then $S_{r,z,+}=S_{r,W,+}$. In particular, $S_{r,z,+}$ does not depend on
$z \in W\setminus i\R$.
\end{pro}
Proceeding now as in \cite[Corollaries 11, 13 and Lemma 12]{HPP15}, we obtain the following
result for points on $i$.
\begin{cor}
\label{cor:SjrvW}
For every $iv \in i\R$ and for every $r$ with $0<r<1$ and $vc(r)\notin iS$
there is a connected open neighborhood $W_v$ of $iv$ in $\C$ satisfying the following conditions.
\begin{enumerate}
\item
$S \cap W_v \partial \Eg_{\cz(r),\sz(r)}=\emptyset$\,,
\item
$S_{r,W_v,+}=\{\ell\in \Zb_{\geq 0}:i L_{\ell}\in iv\Eg_{\cz(r),\sz(r)}\}=
\leftFpar 0, N_{v}\rightFpar$ for some $N_{v}\in \Zb_{\geq 0}$.
\item
For $n\in \Zb_{\geq 0}$, set
\begin{equation}
\label{eq:Ijm}
\end{equation}
If $v \in I_n$ then $N_{v}=n$. Hence
\begin{equation}
\label{eq:Gr-Gell}
F_{r,{\rm res}}(z)=4 \sum_{\ell=1}^n G_{\ell}(z) \qquad (z \in W_v\setminus i\R)\,.
\end{equation}
\end{enumerate}
\end{cor}
We recall the relevant Riemann surfaces from \cite[(76)]{HPP15}. Fix $ℓ∈Z_≥0$. Then
\begin{equation}
\label{eq:Mjk}
\M_{\ell}=\Big\{(z,\zeta)\in\C^\times\times(\C\setminus\{i, -i\})\ :\ \zeta^2=\Big(\frac{iL_{\ell}}{z}\Big)^2-1\Big\}
\end{equation}
is a Riemann surface above $^×$, with projection map $π_ℓ:_ℓ ∋(z,ζ)→z∈^×$\,.
The fiber of $π_ℓ$ above $z∈^×$ is ${(z,ζ), (z,-ζ)}$.
In particular, the restriction of $π_ℓ$ to $_ℓ∖{(±i L_ℓ,0)}$ is a double cover
of $^×∖{±i L_ℓ}$.
Now \cite[Lemma 15]{HPP15} has the following analogue.
The difference is that we have replaced $ψ_z(w)$ by $ψ^ϑ_z(w)$.
So we have to look for possible cancellations of singularities arising from the additional polynomial factor $ϑ_z$.
\begin{lem}\label{2.8}
In the above notation,
\begin{eqnarray}\label{eq:tGell}
\wt G_{\ell}:\M_{\ell}\ni (z,\zeta)&\to&\frac{b}{\pi} \, L_{\ell}\, p_1(i L_{\ell})\psi^\vartheta_z\big(\frac{i L_{\ell}}{z}-\zeta\big)
p_1(iz\zeta) q_1(iz\zeta)\in\C
\end{eqnarray}
is the meromorphic extension to $\M_{\ell}$ of a lift of $G_{\ell}$.
The function $\wt G_{\ell}$ has simple poles at all points $(z,\zeta)\in \M_{\ell}$ such that
\begin{equation}\label{2.9.1}
z=\pm i \sqrt{ L_{\ell}^2+L_{m}^2},
\end{equation}
where $m\in\Zb_{\geq 0} \setminus \bigleftFpar \ell -\big(\frac{m_\mrm}{2}-1\big),
\ell +\big(\frac{m_\mrm}{2}-1\big)\bigrightFpar$.
\end{lem}
\begin{proof}
Formula \eqref{eq:tGell} is obtained using the lifts of $\cz^{-1}$ and $\sz\circ \cz^{-1}$,
as in \cite[Lemma 15]{HPP15}.
The poles of $\wt G_{\ell}$ are the points $(z,\zeta)\in \M_\ell$
for which the function $\vartheta_z\big(\frac{i L_{\ell}}{z}-\zeta\big)
p_1(iz\zeta) q_1(iz\zeta)$ is singular, i.e. the points for which $p_1(iz\zeta) q_1(iz\zeta)$ is singular and $\vartheta_z\big(\frac{i L_{\ell}}{z}-\zeta\big)\neq 0$. By construction,
$p_1(iz\zeta) q_1(iz\zeta)$ is singular if and only if $iz\zeta\in S$, see \eqref{eq:Sj}. In this case,
there exist $\epsilon\in \{\pm 1\}$ and $m\in \Zb$ so that $\zeta=\frac{\epsilon L_m}{z}$. Hence $\zeta^2=\frac{L_m^2}{z^2}$. Since $(z,\zeta)\in \M_\ell$, we also have
$\zeta^2=-\frac{L_\ell^2}{z^2}-1$. Thus $z=\pm i \sqrt{ L_{\ell}^2+L_{m}^2}$.
We now compute $\vartheta_z\big(\frac{i L_{\ell}}{z}-\zeta\big)$ for such $(z,\zeta)$.
w=\frac{i L_{\ell}}{z}-\zeta=\frac{i L_{\ell}}{z}-\frac{\epsilon L_m}{z}=
\frac{i L_{\ell}-\epsilon L_m}{\pm i \sqrt{ L_\ell^2+ L_m^2}}\,.
w^{-1}=\frac{\pm i \sqrt{L_\ell^2+ L_m^2}}{i L_{\ell}-\epsilon L_m}=
\frac{i L_{\ell}+\epsilon L_m}{\pm i \sqrt{L_\ell^2+ L_m^2}}\,.
\begin{eqnarray*}
\cz(w)&=&\frac{w+w^{-1}}{2}=\frac{L_{\ell}}{\pm \sqrt{ L_\ell^2+ L_m^2}}\,,\\
\cz(-iw)&=&\frac{w-w^{-1}}{2i}=\frac{\epsilon L_m}{\pm \sqrt{ L_\ell^2+ L_m^2}}\,.
\end{eqnarray*}
z\cz(w)=iL_\ell\,, \qquad z\cz(-iw)=i\epsilon L_m\,, \qquad z\sz(w)=-\epsilon L_m\,.
Substituting in \eqref{eq:vathetaz}, we obtain
\begin{equation}
\label{eq:vartheta-lm-1}
\vartheta_z(w)
=\frac{\big(L_m^2-L_\ell^2\big)}{b^2}\prod_{k=1}^{m_\mrm-1} \Big[ \big(\frac{-\epsilon L_m}{b}-\frac{m_\mrm}{2}+k\big)^2-\frac{L_\ell^2}{b^2}\Big]\,.
\end{equation}
The same argument used in the proof of Lemma \ref{lemma:functions-z} shows that the right-hand side of this equation is independent of $\epsilon\in \{\pm 1\}$. Using \eqref{eq:Lell}, we therefore obtain
\begin{eqnarray*}
\vartheta_z(w)&=&
\big((\wt\rho_{\beta_1}+m)^2-(\wt \rho_{\beta_1}+\ell)^2\big)\prod_{k=1}^{m_\mrm-1} \big[ \big(\wt\rho_{\beta_1}+m-\frac{m_\mrm}{2}+k\big)^2-(\wt\rho_{\beta_1}+\ell)^2\big]\\
\prod_{k=1}^{m_\mrm-1}
\Big(m-\ell-\frac{m_\mrm}{2}+k\Big) \Big(m+\ell+2\wt\rho_{\beta_1}-\frac{m_\mrm}{2}+k\Big)\\
\prod_{h=-(\frac{m_\mrm}{2}-1)}^{\frac{m_\mrm}{2}-1}
\Big(m-\ell+h\Big) \Big(m+\ell+2\wt\rho_{\beta_1}+h\Big)\,.
\end{eqnarray*}
The values of $m\in \Zb_{\geq 0}$ making this polynomial vanish are:
\begin{eqnarray}
\label{eq:m=l}
&&m=\ell\,, \\
\label{eq:otherm}
&&m\in \Zb_{\geq 0} \cap \bigleftFpar \ell -\big(\tfrac{m_\mrm}{2}-1\big),
\ell +\big(\tfrac{m_\mrm}{2}-1\big)\bigrightFpar\,,\\
&&m\in \Zb_{\geq 0} \cap \bigleftFpar -\ell-2\wt\rho_{\beta_1} -\big(\tfrac{m_\mrm}{2}-1\big), -\ell -2\wt\rho_{\beta_1} + \big(\tfrac{m_\mrm}{2}-1\big)\bigrightFpar\,,
\end{eqnarray}
Observe that $-\ell -2\wt\rho_{\beta_1} + \big(\frac{m_\mrm}{2}-1\big)\geq 0$ if and only if
$(0\leq) \ell \leq -2\wt\rho_{\beta_1} + \big(\frac{m_\mrm}{2}-1\big)$. Looking at the first two rows of Table \ref{table tilde rho}, we see that
this can happen if and only if $\G=\SO_0(p,2)$ with even $p\geq 6$. In this case,
\begin{eqnarray*}
\bigleftFpar\ell -\big(\tfrac{m_\mrm}{2}-1\big),
\ell +\big(\tfrac{m_\mrm}{2}-1\big)\bigrightFpar&=&\bigleftFpar\ell+2-\tfrac{p}{2}, \ell -2+\tfrac{p}{2}\bigrightFpar\\
\bigleftFpar -\ell-2\wt\rho_{\beta_1} -\big(\tfrac{m_\mrm}{2}-1\big),
-\ell -2\wt\rho_{\beta_1} + \big(\tfrac{m_\mrm}{2}-1\big)\bigrightFpar
&=&\bigleftFpar -\ell+1-\tfrac{p}{2},-\ell-3+\tfrac{p}{2}\bigrightFpar.
\end{eqnarray*}
$$\Zb_{\geq 0} \cap \bigleftFpar -\ell+1-\tfrac{p}{2}, -\ell-3+\tfrac{p}{2}\bigrightFpar =
\bigleftFpar 0,-\ell-3+\tfrac{p}{2}\bigrightFpar$$
does not add zeros to those in \eqref{eq:otherm}.
In fact, $-\ell-3+\tfrac{p}{2}\leq \ell-2+\tfrac{p}{2}$ and, if $-\ell-3+\tfrac{p}{2}\geq 0$, i.e.
$\ell\leq \tfrac{p}{2}-3$, then $\ell+2-\tfrac{p}{2}\leq 0$.
\end{proof}
For $ℓ, m∈_≥0$, set
\begin{equation}
\label{eq:z-lm}
z_{\ell,m}=i \sqrt{L_\ell^2+L_m^2}
\end{equation}
\begin{equation}\label{2.9.5'}
\zeta_{\ell,m}= i \sqrt{\frac{L_m^2}{L_\ell^2+L_m^2}}\,.
\end{equation}
Let $ϵ∈{±1}$.
Then all points $(±z_ℓ,m,ϵζ_ℓ,m)$ are in $_ℓ$.
Open neighborhoods in $_ℓ$ of these points
are the sets
\begin{equation}
\label{eq:U1pm}
\Ug_{\ell,\pm}=\{(z,\zeta)\in \M_{\ell}\ ;\ \pm\Im z >0\}\,.
\end{equation}
and local charts on them are
\begin{equation}\label{charts}
\kappa_{\ell,\pm}:\Ug_{\ell,\pm}\ni (z,\zeta)\to \zeta\in \C\setminus i\big((-\infty, -1] \cup [1,+\infty)\big)\,,
\end{equation}
inverted by setting $z=±i L_ℓ/√(ζ^2+1)$.
\begin{lem}\label{lem:chart-expressions}
The local expressions for $\wt G_\ell$ in terms of the charts (\ref{charts}) are
\begin{eqnarray}
\label{expression in terms of charts 1}
\big(\wt G_\ell\circ\kappa_{\ell,\pm}^{-1}\big)(\zeta) %&=&
=\pm \frac{b}{\pi} \, L_{\ell}\, p_1(i L_\ell)
p_2\Big(\tfrac{L_\ell\zeta}{\sqrt{\zeta^2+1}}\Big) q_1\Big(\tfrac{L_\ell\zeta}{\sqrt{\zeta^2+1}}\Big)
\psi^\vartheta_{i\,\frac{L_\ell}{\sqrt{\zeta^2+1}}}\big(\sqrt{\zeta^2+1}\mp \zeta\big)\,.
\end{eqnarray}
Suppose $m\in\Zb_{\geq 0} \setminus \bigleftFpar \ell -\big(\frac{m_\mrm}{2}-1\big),
\ell +\big(\frac{m_\mrm}{2}-1\big)\bigrightFpar$. Then
the residue of the local expression of $\wt G_\ell$ at a point $(z,\zeta)\in \M_\ell$ with
$z=\pm z_{\ell,m}$ is
\begin{equation}\label{2.11.1}
\Res_{\zeta=\pm \zeta_{\ell,m}} (\wt G_\ell\circ \kappa_{\ell,\pm}^{-1})(\zeta)
=\pm \frac{1}{i\pi^2} C_{\ell,m} (f\times \varphi_{\frac{L_\ell \beta_1 +L_m\beta_2}{b}})(y)\,.
\end{equation}
In \eqref{2.11.1},
\begin{equation}
\label{eq:const-lm}
C_{\ell,m}=b L_\ell\, p_1(i L_\ell) p_2(i L_m) \vartheta_0\Big(\frac{L_\ell}{b},\frac{L_m}{b}\Big)\,,
\end{equation}
where $\vartheta_0$ is as in \eqref{eq:vartheta0}, is a positive constant.
\end{lem}
\begin{proof}
The computation of the residues is as in \cite[Lemma 16]{HPP15}.
The constant $\vartheta_0\big(\frac{L_\ell}{b},\frac{L_m}{b}\big)$ agrees with \eqref{eq:vartheta-lm-1} with $(z,\zeta)=(z_{\ell,m},\epsilon \zeta_{\ell,m})$,
and we only need to prove that it is positive. Recall that \eqref{eq:vartheta-lm-1} is independent of $\epsilon$. Hence
\begin{equation}
\label{eq:vartheta-lm-2}
\vartheta_0\Big(\frac{L_\ell}{b},\frac{L_m}{b}\Big)
\Big(\frac{L_m}{b}-\big(\frac{m_\mrm}{2}-k\big)-\frac{L_\ell}{b}\Big)
\Big(\frac{L_m}{b}-\big(\frac{m_\mrm}{2}-k\big)+\frac{L_\ell}{b}\Big)\,.
\end{equation}
If $m> \ell+(\frac{m_\mrm}{2}-1\big)\geq \ell$, then all factors appearing in the above product are positive. If $m< \ell-(\frac{m_\mrm}{2}-1\big)\leq \ell$, then all factors $\frac{L_m}{b}-\big(\frac{m_\mrm}{2}-k\big)+\frac{L_\ell}{b}$ are positive, whereas $L_m^2-L_\ell^2$ as well as the $m_\mrm-1$ factors $\frac{L_m}{b}-\big(\frac{m_\mrm}{2}-k\big)-\frac{L_\ell}{b}$
are negative. Since $m_\mrm$ is even, we conclude that $\vartheta_0\big(\frac{L_\ell}{b},\frac{L_m}{b}\big)>0$ in all cases.
\end{proof}
A different parametrization of the singularities of $G_ℓ$ will turn out to be more convenient.
Observe first that, by \eqref{eq:rho} and \eqref{eq:wtrhob},
$$\wt \rho_{\beta_1}=\rho_{\beta_1}=\rho_{\beta_2}-\frac{m_\mrm}{2}\,.$$
We will use the following notation for $(ℓ_1,ℓ_2)∈_≥0^2$:
\begin{equation}
\label{eq:lambdal1l2}
\lambda(\ell_1,\ell_2)=(\rho_{\beta_1}+\ell_1)\beta_1+(\rho_{\beta_2}+\ell_2)\beta_2=
\frac{1}{b}\big( L_{\ell_1} \beta_1+L_{\ell_2+\frac{m_\mrm}{2}}\beta_2\big).
\end{equation}
\begin{cor}
\label{cor:singGell}
Keep the notation of Lemma \ref{lem:chart-expressions}.
If $\ell\in\leftFpar 0,\frac{m_\mrm}{2}-1\rightFpar$, then $\wt G_\ell$ has simple poles at the points $(z,\zeta)\in \M_\ell$ with $z=\pm i|z|$ and
\begin{equation}
\label{eq:singular1}
b^{-2}|z|^2=(\rho_{\beta_1}+\ell)^2 +(\rho_{\beta_2}+\ell+k)^2 \qquad (k\in \Zb_{\geq 0})\,.
\end{equation}
If $\ell\in\frac{m_\mrm}{2}+\Zb_{\geq 0}$, then $\wt G_\ell$ has simple poles at the points $(z,\zeta)\in \M_\ell$ with $z=\pm i|z|$ and satisfying either \eqref{eq:singular1} or
\begin{equation}
\label{eq:singular2}
b^{-2}|z|^2=(\rho_{\beta_1}+m)^2 +(\rho_{\beta_2}+\ell_0)^2 \qquad
(m\in\leftFpar 0,\ell_0\rightFpar )\,,
\end{equation}
where $\ell_0=\ell-\frac{m_\mrm}{2}$.
The residue of the local expression of $\wt G_\ell$ at a point $(z,\zeta)\in \M_\ell$ with $z=\pm i |z|$ satisfying \eqref{eq:singular1} is
\begin{equation}
\label{eq:resGell1}
\Res_{\zeta=\pm \zeta_{\ell,\ell+\frac{m_\mrm}{2}+k}} (\wt G_\ell\circ \kappa_{\ell,\pm}^{-1})(\zeta)
=\pm \frac{1}{i\pi^2} C_{\ell,\ell+\frac{m_\mrm}{2}+k} \big(f\times \varphi_{\lambda(\ell,\ell+k)}\big)(y)\,.
\end{equation}
The residue of the local expression of $\wt G_\ell$ at a point $(z,\zeta)\in \M_\ell$ with $z=\pm i |z|$ satisfying \eqref{eq:singular2} is
\begin{equation}
\label{eq:resGell2}
\Res_{\zeta=\pm \zeta_{\ell,m}} (\wt G_\ell\circ \kappa_{\ell,\pm}^{-1})(\zeta)
=\pm \frac{1}{i\pi^2} C_{\ell,m} \big(f\times \varphi_{\lambda(m,\ell_0)}\big)(y)\,.
\end{equation}
\end{cor}
\begin{proof}
We have $\ell\in\leftFpar 0,\frac{m_\mrm}{2}-1\rightFpar$ if and only if $0\in
\leftFpar \ell-\big(\frac{m_\mrm}{2}-1\big),\ell+\big(\frac{m_\mrm}{2}-1\big)\rightFpar$.
In this case, $m\in \Zb_{\geq 0}\setminus \leftFpar \ell-\big(\frac{m_\mrm}{2}-1\big),\ell+\big(\frac{m_\mrm}{2}-1\big)\rightFpar=\ell + \frac{m_\mrm}{2}+ \Zb_{\geq 0}$
is of the form $m= \ell + \frac{m_\mrm}{2}+k$ with $k\in \Zb_{\geq 0}$. Hence
$\frac{L_\ell}{b}=\rho_{\beta_1}+\ell$ and $\frac{L_m}{b}=\wt \rho_{\beta_1}+\frac{m_\mrm}{2}+ \ell+k=\rho_{\beta_2}+\ell+k$.
On the other hand, if $\ell\in\frac{m_\mrm}{2}+\Zb_{\geq 0}$ and $m\in \Zb_{\geq 0}\setminus \leftFpar \ell-\big(\frac{m_\mrm}{2}-1\big),\ell+\big(\frac{m_\mrm}{2}-1\big)\rightFpar$, then
either $m\in \ell + \frac{m_\mrm}{2}+ \Zb_{\geq 0}$ (and the above applies), or $m \in
\leftFpar 0,\ell_0\rightFpar$. In the latter case, $\frac{L_\ell}{b}= \wt\rho_{\beta_1}+\frac{m_\mrm}{2}+\ell_0=\rho_{\beta_2}+\ell_0$ and
$\frac{L_m}{b}=\rho_{\beta_1}+m$. Observe also that
by $W$-invariance.
\end{proof}
%%%%%%%%%%%%%% Figures here
\begin{figure}
\includegraphics[trim = 10mm 100mm 10mm 100mm, clip, width=16cm]{figures-residue-final.pdf}
\caption{On the left: $\lambda(\ell,\ell+k)$ for $\ell\in \leftFpar 0,\frac{m_{\rm m}}{2}\rightFpar$.
On the right: $\lambda(\ell,\ell+k)$ and $\lambda(m,\ell_0)$ for $\ell\geq \frac{m_{\rm m}}{2}$}
%\caption{$\lambda(\ell,\ell+k)$, $(k\in \Zb_{\geq 0})$ and $\lambda(m,\ell_0)$, $(m\in \leftFpar 0,\ell_0\rightFpar)$ when $\ell\geq \frac{m_{\mrm}{2}}$}
\end{figure}
%\vskip -6truecm
We now proceed with the piecewise extension of $F$ along the negative imaginary half-line
$-i[L,+∞)$. Recall from Corollary \ref{cor:SjrvW} that for $v∈I_n=[L_n,L_n+1)$ with $n∈_≥0$ there exists
$0< r_v <1$ and an open neighborhood $W_v$ of $-iv$ in $$ such that
\begin{equation}
\label{eq:Fresidues1}
F(z)=F_{r_v}(z)+ 4\sum_{\ell=0}^{n} G_{\ell}(z)
\qquad (z\in W_v\setminus i\R)\, ,
\end{equation}
where the function $F_r_v$ is holomorphic in $W_v$. This equality extends then to $I_-1=(0,L)$ by allowing empty sums. By possibly shrinking $W_v$, we may also assume that $W_v$ is an open disk around $-iv$ such that
W_v\cap i\R \subseteq
\begin{cases}
-iI_{n} &\text{for $v\in I_n^\circ$},\\
-i(I_{n}-\frac{b}{2}) &\text{for $v=L_n$}\,.
\end{cases}
In addition, for $0<v < L$ we define $W_v$ to be an open ball around $-iv$ in $$ such that $W_v∩i⊂(0,L)$.
If $v∈I_n$, $v'≥L$ and $W_v∩W_v'≠∅$, then we obtain for $z∈W_v∩W_v'$
\begin{cases}
0& \text{if $v'\in I_n$},\\
4\, G_n(z) &\text{if $v'\in I_{n-1}$}.
\end{cases}
Now we set
\begin{eqnarray*}
W_{(-1)}=\bigcup_{v\in I_{-1}} W_v \quad &\text{and} & \quad
W_{(n)}=\bigcup_{v\in I_n} W_v \qquad (n\in \Zb_{\geq 0})\,.
\end{eqnarray*}
For $n∈_≥1$ we define a holomorphic function $F_(n): W_(n)→$ by
\[
F_{(n)}(z)= \begin{cases}
F_{r_v}(z) &\text{if $n\in \Zb_{\geq 0}$, $v\in I_n$ and $z\in W_v$}\\
F(z) &\text{if $n=-1$ and $z\in W_{(-1)}$}\,.
\end{cases}
\]
We therefore obtain the following analogue of \cite[Proposition 18]{HPP15}.
\begin{pro}
\label{cor:FextendedMainSection}
For every integer $n\in \Zb_{\geq -1}$ we have
\begin{equation}
\label{eq:Fresidues2}
F(z)=F_{(n)}(z)+4\sum_{\ell=0}^{n} G_{\ell}(z)
\qquad (z\in W_{(n)}\setminus i\R)\,,
\end{equation}
where $F_{(n)}$ is holomorphic in $W_{(n)}$, the $G_\ell$ are as in \eqref{eq:Gell}, and empty sums are defined to be equal to $0$.
\end{pro}
We can continue $F$ across $-i(0,+∞)$ inductively, as in the case of the direct product of two rank one symmetric spaces in \cite{HPP15}. Our specific case $_1=_2$ is slightly easier, as for instance one gets just one regularly spaced sequence of branching points $L_ℓ$. Since the procedure does not involve new steps, we will limit ourself to overview the different parts and state the final result, referring the reader to \cite{HPP15} for the details.
For a fixed positive integer $N$, we construct a Riemann surface $_(N)$ by ``pasting together'' the Riemann surfaces $_ℓ$ to which all functions $G_ℓ$, with $ℓ=0,1,…,N$, admit meromorphic extension. Namely, we set
\begin{equation}
\label{eq:MN}
\M_{(N)}
=\left\{(z,\zeta)\in \C^- \times \C^{N+1} ; \zeta=(\zeta_0,\dots, \zeta_N),\
(z,\zeta_\ell)\in \M_\ell, \ \ell\in\Zb_{\geq 0}, \ 0\leq \ell\leq N\right\}.
\end{equation}
Then $_(N)$ is a Riemann surface, and the map
\begin{eqnarray}\label{finite cover 1}
&&\pi_{(N)}: \M_{(N)} \ni(z,\zeta)\to z\in \C^-
\end{eqnarray}
is a holomorphic $2^N+1$-to-$1$ cover, except when
$z=-i L_ℓ$ for some $ℓ∈_≥0$ with $0≤ℓ≤N$.
The fiber above each of these elements $-i L_ℓ$ consists of $2^N$ branching points of $_(N)$. A choice of square root function $ζ_ℓ^+(z)$, see \cite[(81)]{HPP15},
for every coordinate function $ζ_ℓ$ on $_(N)$ yields a section
\[
\sigma_{(N)}^+: z\to (z,\zeta_0^+(z),\dots,\zeta_N^+(z))
\]
of the projection $π_(N)$. All possible sections of $π_(N)$ are obtained by choosing a sign $±ζ_ℓ^+$ for each coordinate function. We obtain in this way a parametrization of all
sections of $π_(N)$ by means of elements $ε=(ε_0,…,ε_N)∈{±1}^N+1$.
For $0≤ℓ≤N$ consider the holomorphic projection
\begin{eqnarray}\label{finite cover pi}
&&\pi_{(N,\ell)}: \M_{(N)} \ni(z,\zeta)\to (z,\zeta_\ell)\in \M_\ell.
\end{eqnarray}
Then the meromorphic function
\begin{equation}
\label{eq:wtGNell}
\wt G_{(N,\ell)}=\wt G_{\ell}\circ \pi_{(N,\ell)}: \M_{(N)}\to \C
\end{equation}
is holomorphic on $(π_(N))^-1(^-∖i)$. Moreover, on $^-∖i$,
\[
\wt G_{(N,\ell)}\circ \sigma_{(N)}^+=G_{\ell}\,.
\]
So, $G_(N,ℓ)$ is the meromorphic extension of a lift of $G_ℓ$ to $_(N)$.
Using the right-hand side of \eqref{eq:Fresidues2} with $F_(n)$ constant on the $z$-fibers, we obtain a lift of $F$ to $π_(N)^-1(W_(n)∖i)$.
The next step is to ``glue together'' all these local meromorphic extensions of $F$, moving from branching point to branching point, to get a meromorphic extension of $F$ along the branched curve $γ_N$ in $_(N)$ covering the interval $-i(0,L_N+1)$.
Define, as in \cite[section 4.3]{HPP15}, the open sets $U_n,$, $U_(n^∨)$ (with $n∈_≥0$, $∈{±1}^N+1$) and the open neighborhood $_γ_N$ of $γ_N$ in $_(N)$. Every open set $U_(n^∨)∪U_n,$ is a homeomorphic lift to $_(N)$ of the neighborhood $W_(n)$ of $-[L_n,L_n+1)$.
Then we have the following analogue of \cite[Theorem 19]{HPP15}.
\begin{thm}
\label{thm:meroliftF}
For $n\in \{-1,0,\dots,N\}$, $\eps \in \{\pm 1\}^{N+1}$ and $(z,\zeta) \in U_{\eps(n^\vee)}\cup U_{n,\eps}$ define
\begin{equation}
\label{eq:widetildeF}
\wt F(z,\zeta)=
\displaystyle{F_{(n)}(z)+4 \sum_{\ell=0}^n\wt G_{(N,\ell)}(z,\zeta)+4
\sum_{\stackrel{n<\ell\leq N}{\text{with $\eps_\ell=-1$}}} \big[\wt G_{(N,\ell)}(z,\zeta)-
\wt G_{(N,\ell)}(z,-\zeta) \big]}\,,
\end{equation}
where the first sum is equal to $0$ if $\ell=-1$ and the second sum is $0$ if $\eps_\ell=1$ for all $\ell>n$.
Then $\wt F$ is the meromorphic extension of a lift of $F$ to the open neighborhood
$\M_{\gamma_N}$ of the branched curve $\gamma_{N}$
lifting $-i(0,L_{N+1})$ in $\M_{(N)}$.
\end{thm}
Order the singularities according to their distance from the origin $0∈$, and let
${z_(h)}_h∈_≥0$ be the resulting ordered sequence. For a fixed $h∈_≥0$ set
\begin{equation}
\label{eq:Sh}
S_{h}=\{\ell\in \Zb_{\geq 0}; \ \text{$\exists k \in \Zb_{\geq 0}$ so that
$b^{-2}|z_{(h)}|^2=(\rho_{\beta_1}+\ell)^2 +(\rho_{\beta_2}+\ell+k)^2$}\}\,.
\end{equation}
Notice that if $ℓ∈S_h$, then the corresponding element $k$ is uniquely determined.
Let $N∈_≥0$ be such that $|z_(h)|<L_N+1$ and $n∈0, N$ such that $|z_(h)|∈[L_n,L_n+1)$.
Then the possible singularities of $F$ at points of $_(N)$ above $z_(h)$ are those of
\[
\sum_{\ell=0}^n \wt G_{(N,\ell)}(z,\zeta)=\sum_{\ell=0}^n \wt G_\ell(z,\zeta_\ell)\,.
\]
Indeed, the singularities of $G_(N,ℓ)(z,ζ)=G_ℓ(z,ζ_ℓ)$ occur at
points $(z,ζ)∈_(N)$ with $|z|^2=L_ℓ^2+L_m^2>L_ℓ^2$. Hence the second sum on the right-hand side of \eqref{eq:widetildeF} is holomorphic on $U_(n^∨)∪U_n,$.
The singular points of $F$ above $z_(h)$ are parametrized by $ε∈{±1}^N+1$. We denote by $(z_(h), ζ^(h,ε))$ the one in $U_(n^∨)∪U_n,$. The local expression of $F$ on $U_(n^∨)∪U_n,$ is computed in terms of the chart $κ_n,$ defined for $(z,ζ) ∈U_(n^∨)∪U_n,$ by $κ_n,(z,ζ)=ζ_n$.
Suppose $G_(N,ℓ)(z,ζ)$ is singular at $(z_(h), ζ^(h,ε))$. Then, by \cite[Proposition 21]{HPP15},
\begin{equation}
\label{eq:compres1}
\Res_{\zeta_n=\zeta^{(h,\eps)}_n} \big( \wt G_{(N,\ell)}
\circ \kappa_{n,\eps}^{-1}\big)(\zeta_n)
\eps_\ell\eps_n \;\frac{L_n^2}{L_\ell^2} \; \frac{\sqrt{|z_{(h)}|^2-L_\ell^2}}{\sqrt{|z_{(k)}|^2-L_n^2}}\; \Res_{\zeta_\ell=\zeta^{(h,\eps)}_\ell} \big( \wt G_{\ell}
\circ \kappa_{\ell,-}^{-1}\big)(\zeta_\ell)\,.
\end{equation}
If $ℓ$ satisfies \eqref{eq:singular1} with $z=z_(h)$ for some $k ∈_≥0$, then
If $ℓ≥m_/2$ satisfies \eqref{eq:singular2} with $z=z_(h)$ for some $m ∈0,ℓ_0$ and $ℓ_0=ℓ-m_/2$, then
In the first case, by \eqref{eq:resGell1}, the right-hand side of \eqref{eq:compres1} is equal to
\begin{multline*}
\frac{\eps_n L_n^2}{\sqrt{|z_{(h)}|^2-L_n^2}} \; \frac{L_{\ell+\frac{m_\mrm}{2}+k}}{L_\ell^2}\; \Res_{\zeta_\ell=-\zeta_{\ell,\ell+\frac{m_\mrm}{2}+k}} \big( \wt G_{\ell}
\circ \kappa_{\ell,-}^{-1}\big)(\zeta_\ell)\\
=\frac{i}{\pi^2}\; \frac{\eps_n L_n^2}{\sqrt{|z_{(h)}|^2-L_n^2}} \; \frac{L_{\ell+\frac{m_\mrm}{2}+k}}{L_\ell^2}\; C_{\ell,\ell+\frac{m_\mrm}{2}+k} \big( f\times \varphi_{\lambda(\ell,\ell+k)}\big)(y)\,.
\end{multline*}
In the second case, by \eqref{eq:resGell2}, the right-hand side of \eqref{eq:compres1} is equal to
\[
\frac{\eps_n L_n^2}{\sqrt{|z_{(h)}|^2-L_n^2}} \; \frac{L_{m}}{L_\ell^2}\; \Res_{\zeta_\ell=-\zeta_{\ell,m}} \big( \wt G_{\ell}
\circ \kappa_{\ell,-}^{-1}\big)(\zeta_\ell)
=\frac{i}{\pi^2}\;\frac{\eps_n L_n^2}{\sqrt{|z_{(h)}|^2-L_n^2}} \; \frac{L_{m}}{L_\ell^2}\; C_{\ell,m} \big( f\times \varphi_{\lambda(m,\ell_0)}\big)(y)\,.
\]
Observe that in both cases, the constants appearing are $i$ times a positive constant.
Observe also that if $ℓ≥m_/2$ and $G_(N,ℓ)$ is singular at
$(z_(h), ζ^(h,ε))$ with $ℓ$ satisfying \eqref{eq:singular2} with $z=z_(h)$, some $m ∈0,ℓ_0$ and $ℓ_0=ℓ-m_/2$, then
$(z_(h), ζ^(h,ε))$ is also a singularity of $G_(N,m)$ and
$m$ satisfies \eqref{eq:singular1} with $z=z_(h)$ and $k=ℓ_0-m∈_≥0$.
Of course, $φ_λ(m,ℓ_0)=φ_λ(m,m+k)$ in this case. It follows that
the set $S_h$ is sufficient to parametrize the residues of $F$ at $(z_(h), ζ^(h,ε))$.
It follows that
\begin{equation}
\label{eq:compresF}
\Res_{\zeta_n=\zeta^{(h,\eps)}_n} \big( \wt F
\circ \kappa_{n,\eps}^{-1}\big)(\zeta_n)=
\frac{i\eps_n L_n^2}{\sqrt{|z_{(h)}|^2-L_n^2}} \sum_{\ell\in S_h} c_\ell \big(f \times \varphi_{\lambda(\ell,\ell+k)}\big)(y)\,,
\end{equation}
where $k∈_≥0$ is associated with $ℓ$ as in the definition of $S_h$ and $c_ℓ$ is a positive constant depending only on $ℓ$.
By Proposition \ref{pro:holoextRF}, the meromorphic extensions on the half-line $i(-∞, -L]$ of $F$ and of the resolvent $R$ of the Laplacian are equivalent. Thus
the resolvent $R$ can be lifted and meromorphically extended along the curve $γ_N$ in $_γ_N$. Its singularities
(i.e. the resonances of the Laplacian) are those of the meromorphic extension $F$ of $F$ and are located at the points
of $_γ_N$ above the elements $z_(h)$. They are simple poles.
The precise description is given by the following theorem.
\begin{thm}
\label{thm:meroextshiftedLaplacian}
Let $f \in C^\infty_c(\X)$ and $y \in \X$ be fixed. Let $N\in \mathbb N$ and let $\gamma_{N}$ be the curve lifting the inteval $-i(0,N+1)$ in $\M_{(N)}$.
Then the resolvent $R(z)=[R(z)f](y)$ lifts as a meromorphic function to the neighborhood $\M_{\gamma_{N}}$ of the curve $\gamma_{N}$ in $\M_{(N)}$. We denote the lifted meromorphic function by
$\wt R_{(N)}(z,\zeta)=\big[\wt R_{(N)}(z,\zeta)f\big](y)$.
The singularities of $\wt R_{(N)}$ are at most simple poles at the points $(z_{(h)},\zeta^{(h,\eps)})\in \M_{(N)}$ with $h\in \Zb_{\geq 0}$
so that $|z_{(h)}|<L_{N+1}$ and $\eps \in\{\pm 1\}^{N+1}$. Explicitly, for $(n,\eps)\in \leftFpar 0,N\rightFpar\times \{\pm 1\}^{N+1}$,
\begin{eqnarray}
\label{eq:RNnearwn}
\wt R_{(N)}(z,\zeta)=\wt H_{(N,m,\eps)}(z,\zeta)+2\pi i \sum_{\ell=0}^m \wt G_{(N,\ell)}(z,\zeta) \qquad
((z,\zeta)\in U_{\eps(n^\vee)} \cup U_{n,\eps})
\,,
\end{eqnarray}
where $\wt H_{(N,m,\eps)}$ is holomorphic and $\wt G_{(N,\ell)}(z,\zeta)$
is in fact independent of $N$ and $\eps$ (but dependent on $f$ and $y$, which are omitted from the notation).
The singularities of $\wt R_{(N)}(z,\zeta)$ in $U_{\eps(n^\vee)} \cup U_{n,\eps}$ are simple poles at the points $(z_{(h)},\zeta^{(h,\eps)})$ belonging to $U_{\eps(n^\vee)} \cup U_{n,\eps}$. The residue of the local expression of $\wt R_{(N)}$ at one such point is $i\pi$ times the right-hand side of \eqref{eq:compresF}.
\end{thm}
\section{The residue operators}
Recall the notation $λ(ℓ_1,ℓ_2)= (ρ_β_1+ℓ_1)β_1+(ρ_β_2+ℓ_2)β_2$ introduced in \eqref{eq:lambdal1l2}.
For a fixed $h∈_≥0$, the sum over $S_h$ appearing on the right-hand side of \eqref{eq:compresF} is independent either of $N$ or $n$. It can be used to define
the residue operator $_z_(h)R$ of the meromorphically extended resolvent at $z_(h)$.
\begin{equation}\label{a1}
{\Res}_{z_{(h)}}\wt R=\sum_{\ell\in S_{h}} c_\ell R_{\lambda(\ell,\ell+k_\ell)}
\end{equation}
where, $c_ℓ$ are non-zero constants and, as in \cite[(57)]{HPP14}, $R_λ: C_c^∞()→C^∞()$ is defined by $R_λf=f×φ_λ$.
We know from \cite[chapter IV, Theorem 4.5]{He2} that $R_λ(C_c^∞())$ is an irreducible representation of $$. Furthermore, two such representations are equivalent if and only if the spectral parameters $λ$ are in the same Weyl group orbit. Since, in our case, the Weyl group acts by transposition and sign changes, the element $λ(ℓ_1,ℓ_2)$
is dominant with respect to the fixed choice of positive roots if and only if
\rho_{\beta_2}+\ell_2\geq \rho_{\beta_1}+\ell_1\geq 0\,, \quad \text{i.e.}\quad
\ell_2+ \frac{m_\mrm}{2}\geq \ell_1\,.
In particular, all $λ(ℓ,ℓ+k_ℓ)$ are distinct and dominant.
Hence, as a $$-module,
\begin{equation}\label{a2}
{\Res}_{z_{(h)}}\wt R(C_c^\infty(\X))=\bigoplus_{\ell\in S_{(h)}} R_{\lambda(\ell,\ell+k_\ell)}(C_c^\infty(\X))\,.
\end{equation}
\begin{thm}\label{a3ell}
If $(\ell, k)\in \Zb_{\geq 0}^2$, then
\dim R_{\lambda(\ell,\ell+k)}(C_c^\infty(\X))<\infty\,.
Thus ${\Res}_{z_{(h)}}\wt R(C_c^\infty(\X))$ is a finite dimensional $\G$-module.
The $\G$-module ${\Res}_{z_{(h)}}\wt R(C_c^\infty(\X))$ has $\K$-finite matrix coefficients (and is unitary) if and only
it is the trivial representation, which occurs for $h=0$, i.e. when $z_{(0)}=-i\sqrt{\inner{\rho}{\rho}}$.
\end{thm}
\begin{proof}
By \cite[Ch II, \S 4, Theorem 4.16]{He3}, $\dim R_{\lambda(\ell_1,\ell_2)}(C_c^\infty(\X))<\infty$
if and only if there is $w\in W$ such that
\begin{equation}\label{a4}
(w\lambda(\ell_1,\ell_2)-\rho)_\beta\in \Bbb Z_{\geq 0} \qquad (\beta\in \Sigma_*^+, \text{$\beta$ simple})\,.
\end{equation}
Recall that the simple roots in $\Sigma_*^+$ are $\beta_1$ and $\frac{\beta_2-\beta_1}{2}$.
Moreover, $\mu_{\frac{\beta_2-\beta_1}{2}}=\mu_{\beta_2} -\mu_{\beta_1}$ for $\mu\in \a_\C^*$.
\lambda(\ell,\ell+k)-\rho=(\rho_{\beta_1}+\ell)\beta_1+(\rho_{\beta_2}+\ell+k)\beta_2-
(\rho_{\beta_1}\beta_1+\rho_{\beta_2}\beta_2)=\ell \beta_1+(\ell+k)\beta_2\,,
we conclude that
\begin{eqnarray*}
&&(\lambda(\ell,\ell+k)-\rho)_{\beta_1}=\ell\in \Zb_{\geq 0}\,,\\
&&(\lambda(\ell,\ell+k)-\rho)_{(\beta_2-\beta_1)/2}=k\in \Zb_{\geq 0}\,,
\end{eqnarray*}
which satisfies \eqref{a4} with $w=\id$.
\end{proof}
\begin{thebibliography}{222}
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Helgason, S:.
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Orlando, FL, 1978.
\bibitem{He2}%HelgasonGeometric
\bysame: \textit{Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions.} Pure and Applied Mathematics, 113. Academic Press, Inc., Orlando, FL, 1984.
\bibitem{He3}
\bysame: \textit{Geometric Analysis on Symmetric Spaces}. Second edition,
American Mathematical Society, Providence, 2008.
\bibitem{HP09}
Hilgert, J. and A. Pasquale:
Resonances and residue operators for symmetric spaces of rank one,
\textit{J. Math. Pures Appl. (9)} \textbf{91} (2009), no. 5, 495--507.
\bibitem{HPP14}
Hilgert, J., Pasquale, A. and T. Przebinda: Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3,R)/SO(3), preprint arxiv:1411.6527, 43 pages, 2014.
\bibitem{HPP15}
\bysame: Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces, preprint arxiv:1508.07032, 40 pages, 2015.
\bibitem{MV05}
Mazzeo, R. and A. Vasy:
Analytic continuation of the resolvent of the Laplacian
on symmetric spaces of noncompact type. \emph{J. Funct. Anal.} \textbf{228} (2005), no. 2, 311--368.
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Strohmaier, A.:
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\emph{Math. Z.} \textbf{250} (2005), no. 2, 411--425.
\end{thebibliography}
\end{document}
|
1511.00264
|
The CMS Beam Halo Monitor Detector System
Kelly Stifter
On behalf of the CMS collaboration
A new Beam Halo Monitor (BHM) detector system has been installed in the CMS cavern to measure the machine-induced background (MIB) from the LHC. This background originates from interactions of the LHC beam halo with the final set of collimators before the CMS experiment and from beam gas interactions. The BHM detector uses the directional nature of Cherenkov radiation and event timing to select particles coming from the direction of the beam and to suppress those originating from the interaction point. It consists of 40 quartz rods, placed on each side of the CMS detector, coupled to UV sensitive PMTs. For each bunch crossing the PMT signal is digitized by a charge integrating ASIC and the arrival time of the signal is recorded. The data are processed in real time to yield a precise measurement of per-bunch-crossing background rate. This measurement is made available to CMS and the LHC, to provide real-time feedback on the beam quality and to improve the efficiency of data taking. Here, I present the detector system and first results obtained in Run II.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The new Beam Halo Monitor (BHM) is a novel detector that provides an online, bunch-by-bunch, Machine Induced Background (MIB) rate at large radii in the CMS experiment at the LHC.
Machine Induced Background is produced through several interactions, but mainly through protons scattering off LHC collimators. MIB particles often arrive at a high radius, and can increase the fake trigger rate of muon chambers or leave energy deposits in calorimeters. The rate of MIB particles also gives information about the condition of the condition of the beam and can alert the LHC to any instabilities in the beam.
The detector units, comprised of quartz bars read out by photomultiplier tubes, take advantage of precision timing and the directional nature of Cherenkov radiation in order to distinguish the low rate of MIB particles from the large flux of collision products.
§ DETECTOR SYSTEM
§.§ Concept
In order to separate the order of [1]$\textrm{Hz}/\textrm{cm}^{2}$ MIB signal from the [$10^4$]$\textrm{Hz}/\textrm{cm}^{2}$ collision (PP) product background, BHM uses a direction-sensitive detector and precision timing.
The base of a BHM detector unit is a fused silica cylinder optically coupled to a photomultiplier tube (PMT). The detector units are oriented in such a way that when a MIB particle arrives with the incoming beam and produces Cherenkov radiation in the quartz, the light propagates forward and is collected by the PMT. On the other hand, collision products arrive in the opposite direction. While they still produce Cherenkov radiation, the resulting photons propagate in the opposite direction and are absorbed by black paint that is applied to the other end of the cylinder. Consequently, signals from backward particles are much smaller than those from forward particles. Both of these processes are shown in Fig. <ref>.
This concept was tested with particle beams at CERN in 2012 <cit.> and DESY in 2014 <cit.>, and it was shown that by implementing a charge amplitude cut on all signals, it is possible to suppress signals from backward particles by a factor of $10^{4}$ while maintaining an efficiency close to 100$\%$ for MIB signals.
Top: A MIB muon arrives with the incoming beam and produces Cherenkov radiation, which is detected by the PMT. Bottom: A PP product arrives in the opposite direction, and the Cherenkov radiation that is produced is absorbed by black paint at the end of the cylinder.
In addition to using the directional nature of the MIB and PP particles, the BHM detector uses precise timing to separate these two types of events. Since the bunches from the LHC collide in CMS at the interaction point every [25]ns, and all products are traveling with the beams at the velocity of light, it is possible to select a location along the beam line where the time difference between the arrival of MIB and PP particles is maximized. The BHM detectors were placed at one of these locations on each side of CMS in order to take advantage of this property.
§.§ Detector units
The Cherenkov medium used in the detector units is a UV-transmissive, radiation hard, [10]cm cylinder of SQ0 synthetic fused silica, which is [5.2]cm in diameter. The cylinder is optically coupled to a Hamamatsu R2059 photomultiplier tube of the same diameter using a silicon disk. This PMT was chosen for its sensitivity to UV light. The front of the cylinder is painted black in order to absorb the Cherenkov radiation that propagates in the opposite direction.
In order to protect the sensitive PMT from any residual magnetic field from the CMS magnet, three layers of magnetic shielding were used. The first is a layer of Permalloy. In addition to shielding, it serves as the mechanical support for the PMT and quartz bar. The second is a layer of mu-metal. A [1]cm thick cylinder of iron is used as the third and final layer, as shown in Fig. <ref>. This layer also absorbs much of the large flux of low-energy particles present in the cavern.
[Detector Units]
[Unit Orientation]
Left: The base of a detector unit, and the final layer of [1]cm iron shielding. Right: Azimuthal distribution of BHM detector units around CMS magnetic shielding. The units point away from the bulk of CMS, in order to detect incoming particles.
Twenty of these units are installed on each side of CMS, [20.65]m from the interaction point, which is outside the bulk of the detector. This location was selected for a number of different reasons, including a high MIB flux compared to collision products, a small residual magnetic field from the CMS solenoid, a low overall radiation dose, and easy access for installation, cabling, and services.
The units are oriented parallel to the beam line, and are distributed azimuthally, at a radius of [1.8]m from the beam line. The detectors located on either end measure the background for the incoming beam, with a total acceptance of [424]cm$^2$.
§.§ Read-out system
The backbone of BHM electronics comes from the HCAL Phase 1 Upgrade <cit.>. The analog signal from each PMT is read out by front-end digitizer cards, called HF Readout Modules <cit.>, and then sent, via [5]Gbps asynchronous optical link, to the back-end histogramming unit, called a MicroTCA HCAL Trigger and Readout unit ($\mu$HTR) <cit.>. From there, the data is sent out to software, which is a part of xDAQ-based architecture called BRIL-DAQ <cit.>.
The data comes as eight-bit charge information and six-bit timing information for each detector unit during each bunch crossing. A per-channel amplitude cut is performed, and then events passing the threshold are binned in an occupancy histogram by their bunch crossing and the timing information, which is mapped into four TDC bins. The current mapping sets one bin to the time a MIB signal is expected, another to the time a collision product is expected, and the other two for early and late hits. The histograms are then read out by the software, where a flux is calculated and published to CMS and the LHC every [23]s.
This system serves as a first demonstration of full functionality of all electronics components, including the pre-production front-end cards and the high-speed [5]Gbps asynchronous link used.
§.§ Calibration system
A calibration system <cit.> was installed in order to assess the detector performance over time as radiation damage and aging effects start to become significant. The calibration system is designed to distribute light pulses of known amplitude to all detector units in order to measure variation in PMT response.
§ COMMISSIONING RESULTS
The first BHM commissioning results were recorded during the LHC “beam splash” events in April 2015. During a splash event, one bunch of protons is directed into the tertiary collimators, which are located immediately before the experimental cavern. This produces a large flux of secondary particles, which travel through CMS in the same direction, at the same time.
Fig. <ref> shows the distribution of the charge recorded in a single detector unit that measures the halo for Beam 2. An increased fraction of high-amplitude events (green) was seen during the Beam 2 splashes, whereas the response to the Beam 1 splashes (red) is similar to the one with no beam (black). This shows that while the unit saw many high-amplitude forward particles, no high-amplitude backward particles were observed. This served as the first indication of the directional sensitivity of the BHM detectors.
Amplitude data from a beam 2 detector during beam 1 splashes, beam 2 splashes, and no beam.
Narrow collimator aperture is expected to be a large source of MIB particles. The sensitivity of BHM to this movement is demonstrated in Fig. <ref>. The dashed lines show the collimator aperture for the two opposing beams, while the solid lines show the average background rate in the BHM detectors that look in opposing directions. A sharp increase in BHM rate occurs for each end when the collimator aperture of the respective beam is reduced, while the rate in the opposite detectors remains the same.
MIB rate shows distinct correlation to the collimator gap.
As the commissioning process continued, data from stable collisions were collected with low signal amplitude thresholds. The details of the fill structure can be seen as changes in the Machine Induced Background (MIB) rate for a selected detector unit, as shown in fig. <ref>. In this fill, there were six bunch trains, two single bunches, and two probe bunches. Each bunch train contained six colliding bunches spaced at [50]ns, and is followed by a tail of hits, the timing of which is consistent with the expected albedo in the CMS cavern. The two single bunches have the same amplitude as the bunch trains, and are similarly followed by an albedo tail. The two probe bunches were of much lower intensity, which account for their smaller amplitude and lack of albedo tail.
A closer look at a single bunch train reveals more details, as can be seen in Fig. <ref>. The first three filled bins are purely Machine Induced Background (MIB) particles. The first collision products are seen approximately 5.5 bunch crossings later, as expected. This pattern then continues for six subsequent bunches. Note that all MIB events appear in TDC bin 1, which represents the same [6.25]ns window within each bunch. Alternatively, the particles from collisions come mainly in different TDC bins - either 0 or 3. This indicates that BHM will be able to successfully distinguish MIB and PP particles based on the sub-bunch crossing timing information.
[Occupancy Histogram]
[Single Bunch Train]
Left: Summed occupancy histograms for fill 3858 from a single detector unit. Right: A closer look at a bunch train in fill 3858 from a single detector unit. Please note that the given bunch crossing number was not yet aligned with the global bunch crossing number at this stage of commissioning.
§ SUMMARY
The new BHM detector was successfully installed in CMS during LS1. Preliminary results indicate that the detector is sensitive to MIB particles produced through interactions with collimators, the directional aspect of the detector units can be used to discriminate MIB particles and PP products, and these signals can be further differentiated through the use of sub-bunch crossing timing.
These activities are the collaborative work of a number of physicists, engineers, and technicians from several institutions, including CERN, University of Minnesota, INFN Bologna, and National Technical
University of Athens. Worthy of special note are the contributions of A. E. Dabrowski, J. Mans, S. Orfaneli, R. Rusack, and N. Tosi.
S. Orfanelli, et al. Design of a novel Cherenkov detector system for Machine Induced Background monitoring in the CMS cavern. IBIC2013, (Oxford, UK, 2013) 33–36.
N Tosi. The new Beam Halo Monitor for the CMS experiment at the LHC. PhD thesis, (Alma Mater Studiorum Università di Bologna, 2015).
A. Baumbaugh et al. QIE10: a new front-end custom integrated circuit for high rate experiments. JINST, 9(01), (2014).
J Mans. HCAL uHTR Specifications and Operational Documentation. CMS-DOC-12306-v12. (2014).
J. Mans, et al. CMS Technical Design Report for the Phase 1 Upgrade of the Hadron Calorimeter. Technical Report CERN-LHCC-2012-015, CMS-TDR-10 CERN (Geneva, Switzerland, 2012).
V. Brigljevic et al. Using XDAQ in application scenarios of the CMS experiment. FERMILAB-CONF-03-293, CHEP-2003-MOGT008, (2003).
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1511.00034
|
Univerzita Mateja Bela, FPV, Tajovského 40,
97401 Banská Bystrica, Slovakia
České vysoké učení technické v Praze, FJFI, Břehová 7,
11519 Prague 1, Czech Republic
Žilinská univerzita, Elektrotechnická fakulta, Akademická 1, 01026 Žilina, Slovakia
A rapidly expanding fireball which undergoes first-order phase transition
will supercool and proceed via spinodal decomposition. Hadrons are produced
from the individual fragments as well as the left-over matter filling the space between them.
Emission from fragments should be visible in rapidity correlations, particularly
of protons. In addition to that, even within narrow centrality classes, rapidity distributions will
be fluctuating from one event to another in case of fragmentation. This can be
identified with the help of Kolmogorov-Smirnov test. Finally, we present a method
which allows to sort events with varying rapidity distributions
in such a way, that events with similar rapidity histograms are grouped together.
§ INTRODUCTION
Experiments at NICA aim to explore the region of the phase diagram where highly compressed
and excited matter may undergo a first-order phase transition. It is argued elsewhere
in this volume that such a phase transition in a rapidly expanding system
may bring it out of equilibrium and end up in its spinodal decomposition.
Such a process then generates enhanced fluctuations in spatial distributions of
the baryon density and the energy density.
In this paper we focus on observables which could help to identify such processes.
Before we explain various possible observables, we introduce DRAGON: the Monte Carlo
tool suited for generation of hadron distributions coming from a fragmented fireball
Then, we report on an idea proposed in <cit.>
and further elaborated in <cit.>:
clustering of baryons can be visible in rapidity correlations of protons. Further, we turn
our attention to the whole rapidity distributions of produced hadrons and present an
idea to search for nonstatistical differences between them
with the help of Kolmogorov-Smirnov test <cit.>.
Finally, we propose a novel treatment now being developed which also
compares momentum distributions from individual events and sorts events according
to their similarity with each other <cit.>.
§ MONTE CARLO HADRON PRODUCTION FROM FRAGMENTS
In order to test the effects of fireball fragmentation into droplets it is
useful to have Monte Carlo tool for the generation of artificial events with such features
included. One possibility is to construct hydrodynamic models
which include such a behaviour in the evolution
They allow to link the resulting
effects in fireball evolution with the underlying properties of the hot matter.
On the other hand, they offer less freedom for systematic investigation
of how the fragmentation is indeed seen in data. Interesting questions
of this kind are: what is the minimum size and abundance of fragments that can
be seen? What exactly is their influence on spectra, correlations, anisotropies,
and femtoscopy? How are these observables influenced by the combination
of droplet production and collective expansion?
Such questions can be conveniently explored with the help of Monte Carlo generator
that uses a parametrization of the phase-space distribution of hadron
production. Such a tool has been developed in <cit.> under the
title DRAGON (DRoplet and hAdron Generator fOr Nuclear collisions).
All studies presented here have been performed on events generated
with its help.
The bedding of the generator is the blast-wave model. The probability to
emit a hadron in phase-space is described by the emission function
\begin{multline}
S(x,p)\, d^4x = \frac{g}{(2\pi)^3} \, m_t \cosh(y-\eta)\,
\exp\left ( - \frac{p_\mu u^\mu}{T} \right )\\
\times \Theta (R - r)\,
\exp\left ( -\frac{(\eta - \eta_0)^2}{2 \Delta\eta^2} \right )
\delta(\tau - \tau_0)\\
\times \tau\,d\tau\, d\eta\, r\,dr\, d\phi\, .
\label{e:S}
\end{multline}
It is formulated in Milne coordinates $\tau = \sqrt{t^2 - z^2}$,
$\eta = ({1}/{2}) \ln ((t+z)/(t-z))$ and polar coordinates $r$, $\phi$ in
the transverse plane.
Emission points are distributed uniformly in transverse direction within
the radius $R$ and freeze-out occurs along the hypersurface given by
constant $\tau = \tau_0$. Azimuthal anisotropy has not been used in
studies presented here although the model includes such a possibility.
There is collective longitudinal and transverse expansion parametrized
by the velocity field
\begin{eqnarray}
\nonumber
u^\mu &=& ( \cosh\eta\, \cosh\eta_t, \cos\phi\, \sinh\eta_t, \\
&& \phantom{\cosh\eta\, \cosh\eta_t} \sin\phi \sinh\eta_t,\,
\sinh\eta \cosh\eta_t)\\
\eta_t & = & \eta_t(r) = \sqrt{2}\rho_0 \frac{r}{R}\, .
\end{eqnarray}
The fireball is locally thermalized with the temperature $T$.
A part of the hadrons, which can be specified in the model, is emitted from
the droplets. The droplets stem from the fragmentation of the
same hypersurface as assumed in eq. (<ref>). The actual picture is that
when the fireball fragments, some free hadrons are born between the produced
droplets. The volume of droplets is distributed according to
\begin{equation}
{\cal P}_V(V) = \frac{V}{b^2} e^{-V/b}\, .
\end{equation}
The average volume of droplets is then $2b$. The minimal mass is practically set
by the lightest hadron in simulation: usually the pion. The probability to emit hadron
from a droplet drops exponentially in droplet proper time $\tau_d$
\begin{equation}
{\cal P}_\tau(\tau_d) = \frac{1}{R_d} e^{-\tau_d/R_d}\, ,
\end{equation}
where $R_d$ is the radius of the droplet. Momenta of hadrons from droplets
are chosen from the Boltzmann distribution with the same temperature as bulk production.
Currently, neither momentum nor charge conservation is taken into account in droplet decays, but an upgrade of the model including these effects is envisaged.
DRAGON also includes production of hadrons from resonance decays. Baryons
up to 2 GeV and mesons up to 1.5 GeV of mass are included. Chemical
composition is specified by chemical freeze-out temperature and chemical potentials
for baryon number and strangeness. (Chemical potential for $I_3$ should also be
introduced but is practically very small and thus neglected in the simulations.)
§ PROTON CORRELATIONS
Hadrons emitted from the same droplet will have similar velocities.
This should be seen in their correlations <cit.>.
Protons appear best suited for
such a study. Their mass is higher than that of most mesons, so their deflection
from the velocity of the droplet due to thermal smearing
will be less severe. Pions would have better
statistics thanks to their high abundance, but their smearing due to thermal
motion and resonance decays is too big.
Correlation function can be measured as a function of rapidity difference
$\Delta y = y_1 - y_2$ or (better) of the relative rapidity
\begin{equation}
y_{12} = \ln \left [ \gamma_{12} + \sqrt{\gamma_{12}^2 -1} \right ]
\end{equation}
with $\gamma_{12} = p_1\cdot p_2 / m_1 m_2$.
The correlation function is conveniently sampled as
\begin{equation}
C_{12}(y_{12}) = \frac{P_2(y_{12})}{P_{2,\mathrm{mixed}}(y_{12})}
\end{equation}
where $P_2(y_{12})$ is the probability to observe a pair of protons with relative
rapidity $y_{12}$. The reference distribution $P_{2,\mathrm{mixed}}(y_{12})$
in the denominator is obtained via the mixed events technique.
It is instructive to first consider a simple model where the rapidities of
droplets follow Gaussian distribution
\begin{equation}
\zeta(y_d) = \frac{1}{\sqrt{2\pi \xi^2}} \exp\left ( - \frac{(y_d - y_0)^2}{2\xi^2}\right )
\, .
\end{equation}
Within the droplet $i$ which has rapidity $y_i$, rapidities of protons are also
distributed according to Gaussian
\begin{equation}
\rho_{1,i}(y) = \frac{\nu_i}{\sqrt{2\pi \sigma^2}}
\exp\left ( - \frac{(y-y_i)^2}{2\sigma^2} \right )\, .
\end{equation}
This distribution is normalized to the number of protons from droplet $i$,
which is denoted as $\nu_i$.
The resulting correlation function in this simple model is <cit.>
\begin{multline}
C(\Delta y)-1 = \frac{\xi \langle N_d \rangle \langle \nu (\nu -1) \rangle_M}{%
\langle N_d (N_d - 1) \rangle\langle \nu \rangle_M^2}
\sqrt{1 + \frac{\sigma^2}{\xi^2}}
\\
\frac{1}{\sigma}
\exp\left ( - \frac{\Delta y^2}{4\sigma^2 \left ( 1 + \frac{\sigma^2}{\xi^2}\right ) }
\right )
\end{multline}
where $\langle N_d \rangle$ is the average number of droplets in one event
and $\langle \cdots \rangle_M$ denotes averaging over various droplets.
Naturally, the width of the correlation function depends on $\sigma^2$,
as might have been expected. However, it also depends on the width of the
rapidity distribution of droplets: through the factor $(1 +\sigma^2/\xi^2)$,
growing $\xi^2$ leads to narrower proton correlation function.
As an illustration relevant for NICA we generated sets of events with the
help of DRAGON. On these samples we studied the influence of droplet size
and the share of particles from droplets on the resulting correlation functions.
It turns out that the relative rapidity $y_{12}$ yields better results, so we have mainly
used this observable in our analyses. A more detailed study, though not with
specific NICA fireball settings, can be found in <cit.>.
DRAGON was set with Gaussian rapidity distribution with the width of 1.
Within the rapidity acceptance window $-1<y<1$ there were about 1200
hadrons; this number includes all neutral stable hadrons. Momentum
distribution has been set by the temperature of 120 MeV and the transverse
velocity gradient $\eta_f = 0.4$. Chemical composition was according to
$T_{ch} = 140$ MeV and $\mu_B = 413$ MeV. Recall that resonance
decays are included in the model. The same kinetic temperature and
chemical composition was assumed for the droplets. Total mass of
each droplet is given by its size and the energy density 0.7 GeV/fm$^3$.
Transverse size of the fireball was set to 10 fm and the lifetime $\tau=9$ fm/$c$,
but these parameters have no influence on the presented results. Note that
we have imposed acceptance cut in rapidity $-1<y<1$, so that we do not show
results that would not be measurable due to limited acceptance.
In order to see the effect of droplet formation on the correlation function we
simulated one data set with no droplets and three sets which differ in droplet
settings. We have sets with:
$b = 50$ fm$^3$ and the fraction of 25% of hadrons from droplets,
$b = 20$ fm$^3$ and 50%,
$b = 20$ fm$^3$ and 75%. Recall that the mean droplet volume is
The resulting proton correlation functions in $y_{12}$ are plotted in
Fig. <ref>.
Proton correlation functions for four different settings of hadron
production from droplets.
As expected, without fragmentation the correlation function is flat. The widths of
the correlation functions are given by the smearing of the momenta of protons
within one droplet, mainly due to temperature.
The level of correlation is expressed in the height of the
peak at $y_{12}=0$. Naturally, this is expected to grow if a larger number of
protons is correlated. This can be achieved in two ways: by increasing
droplet sizes so that more protons come from each droplet,
or by increasing the number of droplets by enhancing the share of particles
produced by droplets. By coincidence we thus obtained very similar results
for the cases with droplet fractions 25% and 75%, since the latter one assumes smaller
Note the width scale of the correlation function which is larger than the
typical scale of strong interactions. Thus any modification
due to final state interactions which have not been included here
is expected to be concentrated around the peak of our correlation
§ COMPARISON OF RAPIDITY DISTRIBUTIONS
The fragmentation of the fireball actually leads to event-by-event
fluctuations of rapidity distributions. In each event hadrons are
produced from a different underlying rapidity distribution. In <cit.>
it was proposed to use a standard statistical tool for the comparison of hadron
rapidity distributions from individual events: the Kolmogorov-Smirnov (KS)
test. The KS test has been designed to answer the question, to what extent
two empirical distributions seem to correspond to the same underlying probability
To apply the test on empirical distributions one first has to define a measure
of how much they differ. For the sake of clarity and brevity we shall call
empirical distributions events and the measure of difference will
be their scaled distance, to be defined later. A distance is defined in Fig. <ref>.
Definition of the distance between two events. The measured values of
variable $x$ are indicated on horizontal axis. Lines of different
thickness represent
two different events.
Consider measuring the quantity $x$ (this may be e.g. the rapidity) for
all particles in two different events. We mark the values of $x$ on the horizontal axis.
Then, in the same plot we draw for each event its empirical cumulative distribution
function. It is actually a staircase: we start at 0 and in each position where there is
measured $x$ we make a step with the height $1/n_i$, where $n_i$ is the
multiplicity of the event. The maximum vertical distance $D$ between the two
obtained staircases is taken as the measure of difference between the two events.
For further work one takes the scaled distance
\begin{equation}
d = \sqrt{\frac{n_1 n_2}{n_1+n_2}} D
\end{equation}
where $n_1$, $n_2$ are the multiplicities of the two events.
Next one defines
\begin{equation}
Q(d) = P(d' > d)
\end{equation}
i.e. the probability that the scaled distance $d'$ determined for a pair of random
events generated from the same underlying distribution will be bigger than $d$.
The formulas for obtaining $Q(d)$ for any $d$ are given in the Appendix of
<cit.>. Thus defined, for large $d$, the value of $Q$ will be small because there
is little chance that two events will be so much different. If all events come from
the same underlying distribution, then the $Q$'s determined on a
large sample of event pairs will be distributed uniformly.
In a sample of events where the shape and dynamical state of the fireballs
fluctuate, e.g. due to fragmentation, large scaled distance $d$ will be more
frequent. This is then translated into higher abundance of low $Q$ values. Thus
non-statistical differences between events will show up as a peak at low $Q$
in the histogram of $Q$ values for large number of event pairs. In order to quantify the
significance of the peak above the usual statistical fluctuations
we introduce
\begin{equation}
R = \frac{N_0 - \frac{N_{\mathrm{tot}}}{B}}{\sigma_0}
= \frac{N_0 - \frac{N_{\mathrm{tot}}}{B}}{\frac{N_{\mathrm{tot}}}{B}}
\end{equation}
where $N_0$ is the number of event pairs in the first $Q$-bin,
$N_{\mathrm{tot}}$ is the number of all event pairs and $B$ is the number
of $Q$-bins.
To illustrate the application at NICA, we have used event samples with the same
settings as in the previous Section and show in Fig. <ref> the
$Q$-histograms for samples of $10^4$ simulated events. Rapidities
of charged pions (top) and all charged hadrons (bottom) are taken into account.
$Q$-histograms for pion rapidity distributions as well as rapidity distributions
of all charged hadrons.
The signal is very strong and the one for charged hadrons is generally
more pronounced than the one for pions. The comparison of different data sets
is consistent with results for correlation functions from the previous section.
Note that there is basically very weak signal for the case without droplets, which
shows that clustering
effect due to resonance decays cannot mask the investigated mechanism.
§ EVENT SHAPE SORTING
In presence of fireball fragmentation, rapidity distributions of different events show
large variety. This motivates the quest to select among them groups of events
which will be similar. Such groups allow to appreciate the range of fluctuations
of the momentum distribution. They also may be useful for the construction
of mixed events histograms used in correlation functions.
A method for sorting events according to their similarity with each other has been
proposed <cit.>. The application in <cit.> was on azimuthal
angle distributions. Here we use it for rapidity distributions. Details can be
found in <cit.>; here we only shortly explain the sorting algorithm.
An event is characterized when all its bin entries $n_i$ are given; $i$ numbers the
bins in rapidity. Full bin record will be denoted $\{ n_i \}$.
* Events are initially sorted in a chosen way and divided into $N$ quantiles
of the distribution. We use deciles, numbered by Greek letters.
* For each event, characterized by record $\{ n_i \}$, calculate the
probability that it belongs
to the event bin $\mu$, $P(\mu|\{n_i\})$, using the Bayes' theorem
\begin{equation}
\label{e:Bay}
P(\mu|\{ n_i\} ) = \frac{P(\{ n_i\}| \mu ) P(\mu)}{P(\{n_i\})}\, .
\end{equation}
The probability $P(\{ n_i\}| \mu )$ that the event with bin record $\{n_i\}$ belongs
to the event bin $\mu$ can be expressed as
\begin{equation}
P(\{ n_i\}| \mu ) = M! \prod_i \frac{P(i|\mu)^{n_i}}{n_i!}
\end{equation}
where $M$ is the event multiplicity, the product goes over all (rapidity) bins, and
$P(i|\mu)$ is the probability that a particle falls into bin $i$ in an event from
event bin $\mu$
\begin{equation}
P(i|\mu) = \frac{n_{\mu,i}}{M_\mu}\, .
\end{equation}
($M_\mu$ is the total multiplicity of all events in event bin $\mu$ and
$n_{i,\mu}$ is the total number of particles in bin $i$.)
Coming back to eq. (<ref>): $P(\mu)=1/N$ is the prior and
\begin{equation}
P(\{n_i\}) = \sum_{\mu = 1}^N P(\{n_i\}|\mu) P(\mu)\, .
\end{equation}
* For each event determine
\begin{equation}
\bar \mu = \sum_{\mu = 1}^{N} P(\mu|\{n_i\}) \mu
\end{equation}
and re-sort all events according to $\bar\mu$. Then divide again into quantiles.
* If the ordering of events changed, re-iterate from point 2. In a less strict version
of the algorithm, the ordering is re-iterated only if the assignment to quantiles has changed.
This iterative algorithm organizes events in such a way, that those which are
similar to each other by the shapes of their histograms end up close
to each other. It is not specified a priori, however, whether there is any specific
observable according to which the sorting proceeds. The algorithm itself picks
the best ordering automatically.
The method actually provides a more sophisticated version of the Event
Shape Engineering.
We have tested the algorithm on a set of events generated by DRAGON with the same
parameters as in previous two Sections. For illustration, we show in Fig. <ref>
Average rapidity histograms of the 10 event bins after the sorting algorithm
with 5000 events
(with rapidity flip - see text) converged. Droplet fraction 25% and $b = 50$ fm$^3$.
the average histograms in different event bins after the sorting algorithm.
We have chosen the data set with droplet fraction 25% and $b=50$ fm$^3$
and the algorithm works with rapidity distributions of pions.
As a result of the fluctuations in rapidity distributions, the differences
between event bins are large. On one end there are events with almost
symmetric distributions, whereas on the other end there are events with strong
emphasis on one side.
It should be noted that the simulation setting assumes symmetric Gaussian rapidity
distribution and corresponded to symmetric nuclear collisions. Consequently, there
is no reason to favour one rapidity direction over the other. The resulting
sorting in Fig. <ref> is obtained when in the middle of the iteration
process one half of the events is flipped over the mid-rapidity.
The difference between event bins is much bigger here than in a sample of events
where no droplets are present.
§ CONCLUSIONS
We have sketched and explained two kinds of observables that can be used
for identification of the fragmentation process: proton correlations in rapidity
<cit.> and the
Kolmogorov-Smirnov test comparing the event-by-event
rapidity distributions <cit.>. The motivation to look for the
fragmentation comes
from the fact that a first order phase transition actually should proceed this way.
It should be mentioned that in <cit.> it has been argued
that potentially there is a mechanism which may lead to fireball fragmentation even in
absence of the first order phase transition. A sharp peak of the bulk viscosity
as a function of temperature may suddenly cause resistance of the bulk matter against
expansion. Driven by the inertia, the fireball could choose to fragment. This possibility
puts the uniqueness of the fragmentation process as the signature for the first order
phase transition under question. Nevertheless, it is still certainly worthwhile to investigate
the consequences of such a process.
A process that could mask the signals of fragmentation is
rescattering of hadrons emitted from droplets. It would be interesting to combine
the presented methods with models including such a possibility.
Finally, we presented a method which is still being developed and which allows
to sort the measured events automatically according to the most pronounced
features in their histograms and build groups of similar events <cit.>.
This would
allow to study such groups, where event-by-event fluctuations are suppressed, in more
We acknowledge partial support
by grants APVV-0050-11,
VEGA 1/0469/15 (Slovakia). BT was also supported by
grants RVO68407700, LG15001 (Czech Republic).
RK acknowledges support from SGS15/093/OHK4/1T/14.
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|
1511.00531
|
Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6418, USA.
Quantum nonlocality is a counterintuitive phenomenon that lies beyond the purview of causal influences. Recently, Bell inequalities have been generalized to the case of quantum inputs, leading to a powerful family of semi-quantum Bell inequalities that are capable of detecting any entangled state. Here, we focus on a different problem and investigate how the local-indistinguishability of quantum inputs and postselection may affect the requirements to detect semi-quantum nonlocality. To this end, we consider a semi-quantum nonlocal game based on locally-indistinguishable qubit inputs and derive its postselected local and quantum bounds by using a novel connection to the local-distinguishability of quantum states. Interestingly, we find that the postselected local bound is independent of the measurement efficiency and that the Bell violation increases with lower measurement efficiencies.
It is known that in quantum physics, there exist experiments in which correlations from measurements on entangled systems are at odds with our causal world views. These correlations may be verified by using a family of statistical tests called Bell inequalities <cit.>, which are linear constraints on the set of correlations that are compatible with the principle of local causes <cit.>. In other words, if the correlations violate a Bell inequality, then the underlying physics must be nonlocal in nature. Remarkably, apart from their foundational significance, Bell inequalities have also found practical applications in quantum cryptography and quantum state estimation <cit.>. For these reasons, quantum nonlocality is one of the most widely studied topics in quantum information science.
Recently, a new paradigm called semi-quantum nonlocality has emerged <cit.>, where observers use quantum inputs—instead of classical inputs—to specify their desired measurement settings. Interestingly, by doing so, all entangled states are “nonlocal”, in that for any entangled state there is always a semi-quantum Bell inequality with which violation is achieved. This feature suggests that certain semi-quantum Bell inequalities are strong entanglement witnesses and thus could provide an unprecedented level of confidence in detecting entanglement using untrusted measurement devices. For instance, see Ref. <cit.> for a generic procedure that converts entanglement witnesses into measurement-device-independent entanglement witnesses, and Ref. <cit.> for the corresponding proof-of-principle experiment. See also Refs. <cit.> for the connection to quantum steering <cit.>.
On a more general level, semi-quantum nonlocality admits the possibility of working with locally-indistinguishable quantum inputs, a notion that is central to local quantum state discrimination <cit.> and quantum data hiding <cit.>. For our purposes, we define such quantum inputs as quantum states that are indistinguishable at the level of local operations and shared randomness (LOSR) <cit.>, but distinguishable at the level of local quantum measurements assisted with shared entanglement (henceforth referred to as quantum strategies). In particular, our theoretical contribution recognizes that semi-quantum Bell inequalities using locally-indistinguishable quantum inputs can acquire the following two interesting properties: (1) the ability to safely perform postselection and (2) the ability to achieve higher Bell violations with decreasing measurement efficiencies <cit.>.
The first property is based on the fact that postselection strategies due to the detection loophole <cit.> are local filtering processes assisted with shared randomness. Thus by the above definition, it is impossible for LOSR models to produce postselected correlations that are semi-quantum nonlocal—even if arbitrarily low measurement efficiencies are allowed. The second property is due to the fact that the violation of a semi-quantum Bell inequality is directly related to the local-distinguishability of the quantum input states. This connection implies that with a suitable choice of quantum inputs, it is possible to devise a semi-quantum Bell inequality whose optimal violation is achieved only if the measurement efficiencies fall below a certain threshold; this is analogous to the optimal discrimination of non-orthogonal quantum states whereby inconclusive measurement elements are necessary <cit.>.
To illustrate the above properties, we analyze a semi-quantum Bell experiment inspired by the Clauser-Horne-Shimony-Holt (CHSH) Bell experiment <cit.>, and derive its postselected local bound and postselected maximum quantum bound for a given measurement efficiency. To start with, let us first clarify the meaning of using quantum states to choose the measurements. While it is clear what it means by classically choosing a measurement setting (e.g., turning a knob), in the case of quantum inputs, the notion of choosing a measurement is somewhat less obvious. To sharpen this notion, we propose to think in terms of programmable quantum measurement (PQM) devices <cit.>. More specifically, a PQM device is a measurement device that accepts two quantum inputs, namely a quantum target system and a quantum program system, and then performs a measurement (determined by the state of the program system) on the target system. In other words, one uses the state of the quantum program system to choose the desired measurement. Therefore, we may view measurements in the semi-quantum nonlocality framework as untrusted PQM devices whose measurements are purportedly determined by trusted quantum input systems, i.e., see Fig. (<ref>).
Semi-quantum CHSH inequality. We consider a semi-quantum Bell experiment involving two distant observers, called Alice and Bob, who each have a trusted local source of randomness, a trusted qubit preparation device, and an untrusted PQM device. Note that the measurement-independence condition <cit.> is thus implicitly assumed. In each run of the experiment, Alice generates two random bits $\bar{x}=x_1x_2$ and prepares a program qubit using the following encoding scheme: $\proj{\bar{x}}=H^{x_1}\proj{x_2}H^{x_1}$, where $\{\ket{x_2}\}_{x_2=0,1}$ is the computational basis and $H$ is the Hadamard matrix. Then, she sends the prepared qubit to her PQM device for measurement and receives an outcome $a\in\{0,1,\varnothing\}$, where all inconclusive outcomes are assigned to $\varnothing$. Likewise for Bob, we write $\bar{y}=y_1y_2$ and $b$ to denote his measurement choice and measurement outcome, respectively. Furthermore, in what follows, we will refer to Alice's and Bob's qubit input systems as $\mathsf{A}$ and $\mathsf{B}$, respectively, and their corresponding quantum target systems as $\mathsf{A}'$ and $\mathsf{B}'$.
In the LOSR framework, untrusted measurements are modeled by a classical distribution $\{\Pr[\lambda]\}_\lambda$ and a corresponding set of conditional local positive-operator valued measure (POVM) operators, $\{Q^\lambda_a\}_{a}$, $\{R^\lambda_b\}_{b}$, acting on systems $\mathsf{A}$ and $\mathsf{B}$, respectively. Here, the classical variable $\lambda$ is a diagonal quantum state living in the Hilbert space of $\mathsf{A}'\otimes \mathsf{B}'$, and thus captures all the classical randomness that is pre-shared between the two measurement devices. For a given pair of measurement choices, $\omega_{\bar{x}}:=\proj{\bar{x}}$ and $\tau_{\bar{y}}:=\proj{\bar{y}}$, the conditional probability of observing outcomes $a$ and $b$ is given as
[ a,b|x̅,y̅]=∑_λ[λ][Q_a^λω_x̅] [R_b^λτ_y̅ ],
which is synonymous to the locality condition assumed in standard Bell inequalities. Also, we write $\{M_{a,b}\}_{a,b}$ to denote the effective two-qubit measurement acting on the qubit inputs, i.e., $M_{a,b} = \sum_{\lambda} \Pr[\lambda] {Q}_a^\lambda \otimes {R}_b^\lambda$. Note that if $M_{a,b}$ is not separable for some $a,b$, then by definition the joint target state must be entangled, i.e., see Fig. (<ref>). Accordingly, a violation of Eq. (<ref>) implies that the local PQM devices must share entanglement.
Operational interpretation. Alice's and Bob's measurement choices are encoded into trusted qubit systems and then sent to their respective untrusted PQM devices. The PQM devices share a bipartite state (denoted by $\phi_{\mathsf{A'B'}}$) which may or may not be entangled. To test for entanglement, Alice and Bob compute Eq. (<ref>): if the inequality is violated, they conclude $\phi_{\mathsf{A'B'}}$ is entangled, otherwise, the experiment is not conclusive. It is useful to mention that like standard Bell experiments, the PQM devices and the source device are all part of the test.
Following standard arguments <cit.>, we suppose $\sum_{a \not = \varnothing}\Pr\left[a |\bar{x} \right] = \gamma$, $\sum_{b \not = \varnothing}\Pr\left[b|\bar{y} \right]= \gamma$, and $\sum_{a,b \not = \varnothing} \linebreak \Pr\left[a,b |\bar{x},\bar{y} \right] = \gamma^2$ for all measurements choices, where $\gamma \in (0,1]$ is the measurement efficiency. With that, our postselected inequality reads
S(γ)=1/4∑_x̅,y̅(-1)^f(x̅,y̅)C(x̅,y̅)/γ^2 ≤β(γ| LOSR) ,
where $f(\bar{x},\bar{y}):=x_1 \wedge y_1 \oplus x_2 \oplus y_2$ is a balanced boolean function, and $C(\bar{x},\bar{y}):=\Pr[a=b|\bar{x},\bar{y}]-\Pr[a\not=b|\bar{x},\bar{y}]$ for $a,b\not= \varnothing$ is the conditional correlation function. Our goal is to derive the postselected local bound, $\beta(\gamma| \tn{LOSR})$, and to see how it scales with the measurement efficiency, $\gamma$.
For pedagogical reasons, we first discuss what happens when the inputs are classical. In this picture, our inequality can be seen as a symmetric extension of the CHSH inequality. To see this connection, we note that the first bit of each party, $x_1,y_1$, determines his or her measurement setting, and the second bit, $x_2,y_2$, determines if he or she should flip the measurement outcome. Indeed, it can be easily verified that Eq. (<ref>) is an average of four CHSH inequalities conditioned on $x_2$ and $y_2$, therefore the local bound of our inequality assuming classical inputs is 2. However, despite these similarities, there is a subtle difference between the CHSH inequality and Eq. (<ref>) with respect to classical local models. That is, a classical local model that outputs fixed correlated outcomes independently of the inputs would give a CHSH value of 2, whereas with Eq. (<ref>) the Bell value is zero. This example illustrates that the additional randomness injected via $x_2$ and $y_2$ plays an interesting role in constraining the efficacy of certain classical local models.
Moving on to quantum inputs, the measurement basis is now determined by the basis in which the program qubit is prepared, and the bit flip value is given by the eigenvector of that basis. Notice that this encoding scheme is inspired by the celebrated quantum conjugate coding scheme used in quantum cryptography <cit.>. The main advantage of this scheme is that the probability of learning each bit is upper bounded by $(1+1/\sqrt{2})/2 \approx 0.853$ <cit.>. Based on these observations, we thus expect correlations generated by LOSR models to be weakly correlated with Alice's and Bob's measurement choices.
Recall that we want to derive the postselected local bound and the postselected maximum quantum bound for Eq. (<ref>). As mentioned earlier, the former is denoted by $\beta(\gamma|\tn{LOSR})$ and is defined as the maximization of $S(\gamma)$ over all LOSR measurements for a fixed measurement efficiency $\gamma$. At this point, it is useful to mention that all postselection strategies conceivable by LOSR models are automatically accounted for in the maximization. That is, any postselection strategy employed by the underlying LOSR model must be captured by the local filtering POVMs ${Q}_0^\lambda+{Q}_1^\lambda$ and ${R}_0^\lambda+{R}_1^\lambda$, which are also optimized as part of the maximization together with the distribution $\{\Pr[\lambda]\}_\lambda$. Moving on, the postselected maximum quantum bound is denoted by $\beta(\gamma)$ and is defined as the maximization of $S(\gamma)$ over the set of quantum strategies, $\{\phi_{\mathsf{A'B'}},\{Q_a\}_a,\{R_b\}_b\}$.
Connection to quantum state discrimination. The above maximization problems can be solved by using a connection to the local-distinguishability of quantum inputs. To illustrate this connection, we first note that the proposed semi-quantum CHSH experiment is equivalent to a guessing game in which the untrusted local measurement devices have to guess the bit value $f(\bar{x},\bar{y})$ when given quantum inputs $\omega_{\bar{x}}\otimes \tau_{\bar{y}}$. More precisely, the devices win the game if they output $a\oplus b=f(\bar{x},\bar{y})$ whenever $a,b\not= \varnothing$, i.e., the game is counted only for jointly conclusive events. The conditional guessing probability can be written in terms of Eq. (<ref>),
G(γ):=[a⊕b = f(x̅,y̅) ]/γ^2=1/2+S(γ)/8, where $S(\gamma)/8$ can be seen as the distinguishing advantage. Then, it can be easily verified that
[a⊕b = f(x̅,y̅) ]=[ρ_0Π_a ⊕b =0+ρ_1Π_a ⊕b =1 ]/2,
where we used
\[
\rho_{0}=\frac{1}{8}\!\!\!\!\!\!\!\sum_{\substack{\bar{x},\bar{y}\\\tn{s.t.}f(\bar{x},\bar{y})=0}}\!\!\!\!\!\!\!\omega_{\bar{x}}\otimes \tau_{\bar{y}},\quad \, \rho_{1}=\frac{1}{8}\!\!\!\!\!\!\!\sum_{\substack{\bar{x},\bar{y}\\\tn{s.t.}f(\bar{x},\bar{y})=1}}\!\!\!\!\!\!\!\omega_{\bar{x}}\otimes\tau_{\bar{y}},\]
and the measurement assignments $\Pi_{a\oplus b=0}=M_{0,0}+M_{1,1}$, $\Pi_{a\oplus b=1}=M_{0,1}+M_{0,1}$ and $\Pi_{\varnothing}=\mathds{1}-\Pi_{a\oplus b=0}-\Pi_{a\oplus b=1}$.
We may interpret Eq. (<ref>) as follows. In each run of the experiment, the measurement devices are given a product quantum state $\omega_{\bar{x}}\otimes \tau_{\bar{y}}$ randomly chosen from one of the two sets of states, $\{\omega_{\bar{x}}\otimes \tau_{\bar{y}}: f(\bar{x},\bar{y})=0\}$ and $\{\omega_{\bar{x}}\otimes \tau_{\bar{y}}: f(\bar{x},\bar{y})=1\}$, and the devices have to guess which set the given state is drawn from. In other words, the local devices have to collectively guess the global identity $f(\bar{x},\bar{y})$ of $\omega_{\bar{x}}\otimes \tau_{\bar{y}}$ using whatever resources they are given with. Indeed, the figure of merit in this case is exactly given by Eq. (<ref>), which is the conditional guessing probability uniformly averaged over all product states. Using the Born's rule and the linearity of the trace operator, this guessing game can be simplified to the local-distinguishability of two non-orthogonal mixed states $\rho_0$ and $\rho_1$ assuming a fixed conclusive rate of $\gamma^2$. Therefore, the maximization Eq. (<ref>) is equivalent to the maximization of Eq. (<ref>) (up to the constant normalization factor of $1/\gamma^2$).
The advantage of local-distinguishability games is that they can be analytically solved through semidefinite programming <cit.>, a form of convex optimization that maximizes a linear function over the intersection of a semidefinite cone and an affine plane <cit.>. For brevity, we present only the primal programs and defer the corresponding dual programs and optimal solutions to the Supplementary Material. The primal program for computing the maximum quantum guessing probability assuming a fixed $\gamma^2 \in (0,1]$ is given by
\begin{eqnarray*}\nonumber
\texttt{maximize}&:& \frac{1}{2}\Tr \left[\rho_0 \Pi_{a\oplus b=0} + \rho_1 \Pi_{a\oplus b=1} \right] \\ \nonumber
\texttt{subject to}&:& \Pi_{a\oplus b=0}+ \Pi_{a\oplus b=1}+ \Pi_{\varnothing} = \mathds{1}_{\mathsf{A} \otimes \mathsf{B}},\\ \nonumber
&& \Tr\left[ (\omega_{\bar{x}}\otimes \tau_{\bar{y}}\Pi_{\varnothing}\right] =1-\gamma^2,\quad \forall~\bar{x},\bar{y} \\
&& \Pi_i \succeq 0,\quad i=0,1,\varnothing,
\end{eqnarray*}
and the optimal values are found to be
max G(γ) ={
1/2(1+1/γ^22√(2)) if γ>1/√(2)
1/2+1/2√(2) if γ≤1/√(2)
Here, an important remark is in order. These optimal values are obtained over the whole set of two-qubit POVMs acting on $\mathsf{A}\otimes\mathsf{B}$, which is larger than the set of quantum strategies, i.e., $M_{a,b}=\Tr_{\mathsf{A}'\mathsf{B}'}\left[ \phi_{\mathsf{A'B'}}(Q_a \otimes R_b)\right]$. Thus strictly speaking, Eq. (<ref>) is an upper bound on the maximum quantum bound, i.e., $\beta(\gamma)\leq 2\sqrt{2}$ for $0<\gamma \leq 1/\sqrt{2}$ and $\beta(\gamma)\leq \sqrt{2}/\gamma^2$ for $1/\sqrt{2}<\gamma \leq 1$. However, as we will see later, this upper bound is tight for $\gamma \in (0,1/2]$.
Similarly, the maximization for LOSR measurements is based on a circuitous method, which nevertheless also leads to a tight upper bound on $G(\gamma|\tn{LOSR})$. More precisely, we optimize over all measurements compatible with the positive partial transpose (PPT) condition <cit.> instead of LOSR measurements. The reason is that PPT measurements admit a much simpler characterization and can be formulated as linear constraints in the semidefinite programs, i.e., we only need to add $ \Pi_i^{T_\mathsf{B}} \succeq 0$, for $i=0,1,\varnothing$, where $T_\mathsf{B}$ means the partial transpose with respect to Bob's measurements. Moreover, we use the fact that PPT and separable measurements are equivalent at the level of two-qubit positive operators <cit.>. Therefore, the optimal bound for PPT measurements is an upper bound on that of LOSR measurements, i.e., $\beta(\gamma|\tn{LOSR}) \leq \beta(\gamma|\tn{Sep}) = \beta(\gamma|\tn{PPT})$. The optimal value for PPT models is found to be independent of $\gamma$,
max G( ·|PPT) =1/2+1/4√(2).
Interestingly, it turns out that the optimal measurements are given by LOSR measurements. To show this, suppose the qubit inputs are given by $\omega_{0x_2}= (\mathds{1}_{\mathsf{A}}+(-1)^{x_2}\mathbbm{X})/2$, $\omega_{1x_2}= (\mathds{1}_{\mathsf{A}}+(-1)^{x_2}\mathbbm{Y})/2$, and $\tau_{y_1y_2}= (\mathds{1}_{\mathsf{B}}+(-1)^{y_2}(\mathbbm{X}+(-1)^{y_1}\mathbbm{Y})/\sqrt{2})/2$, where $\mathbbm{X}$ and $\mathbbm{Y}$ are Pauli matrices <cit.>. Then, it can be verified that the joint input states are jointly diagonal in the standard Bell basis:
α^+ 0 0 0
0 α^- 0 0
0 0 1/4 0
0 0 0 1/4
, ρ_1=
α^- 0 0 0
0 α^+ 0 0
0 0 1/4 0
0 0 0 1/4
where the eigenvalues are $\alpha^\pm:=(1\pm1/\sqrt{2})/4$, and the corresponding eigenvectors are ordered as: $\ket{\Psi^+}$, $\ket{\Psi^-}$, $\ket{\Phi^+}$ and $\ket{\Phi^-}$ <cit.>. For example, we have $\bra{\Psi^+} \rho_0 \ket{\Psi^+}=\bra{\Psi^-} \rho_1 \ket{\Psi^-}=\alpha^+$. A simple LOSR measurement that achieves Eq. (<ref>) is one that uses only local measurements, i.e., no shared randomness is needed. More specifically, the strategy is $Q_a=\gamma(\mathds{1}_{\mathsf{A}}+(-1)^{a}\mathbbm{X})/2$, $R_b=\gamma(\mathds{1}_{\mathsf{B}}+(-1)^{b}\mathbbm{X})/2$ for $a,b=0,1$, and $Q_\varnothing=(1-\gamma)\mathds{1}_{\mathsf{A}}$, $R_\varnothing=(1-\gamma)\mathds{1}_{\mathsf{B}}$ for the inconclusive outcomes. That is, each measurement device with probability $\gamma$ measures in the $\mathbbm{X}$ basis, and with probability $1-\gamma$ outputs $\varnothing$ without measurement. Another strategy is to measure in the $\mathbbm{Y}$ basis instead of $\mathbbm{X}$, or to use a combination of these two strategies assisted with shared randomness.
Quantum violation vs efficiency. The vertical axis is the postselected Bell value $S(\gamma)$ and the horizontal axis is the measurement efficiency, $\gamma\in (0,1]$. The (black) dashed line is given by the maximum quantum bound, Eq. (<ref>), which is obtained using general two-qubit measurements. The (red) solid line is the postselected local bound, Eq. (<ref>). The (blue) shaded area is due to the pretty good quantum strategy.
A pretty good quantum strategy. As mentioned above, the optimal solutions to Eq. (<ref>) are given in terms of two-qubit POVMs and thus do not provide a clear exposition on the optimal quantum strategy (i.e., the optimal entangled bipartite state and local PQMs) needed to achieve the maximum quantum bound. To this end, we provide an explicit quantum strategy that reaches the upper bound in the region of $0<\gamma<1/2$, i.e., see the shaded area in Fig. (<ref>). Again, we refer to the aforementioned encoding scheme, i.e., Eq. (<ref>). The optimal joint target system is a two-qubit maximally entangled state, $\phi_{\mathsf{A'B'}}=\proj{\Psi^+}$, and the optimal PQMs are
\begin{eqnarray*}
Q_\varnothing&=&\mathds{1}_{\mathsf{A} \otimes \mathsf{B}} - Q_0-Q_1,
\end{eqnarray*}
and likewise $R_i=Q_i$ for $i=0,1,\varnothing$, where $\gamma_1=\min\{2\gamma,1\}$ and $\gamma_2=\max\{\gamma-1/2,0\}$. Note that the PQMs are inefficient Bell-state measurements (BSMs), i.e., they can only discriminate between $\ket{\Psi^+}$ and $\ket{\Psi^-}$. The Bell values using these states and measurements are $S(\gamma)=2\sqrt{2}$ for $0<\gamma \leq 1/2$ and $S(\gamma)=1/(\gamma^2 \sqrt{2})$ for $1/2< \gamma <1/\sqrt{2}$. We remark that this quantum strategy is however sub-optimal when it comes to detecting weakly entangled states. For instance, in the case of two-qubit Werner states <cit.>, $\phi_{\mathsf{A'B'}}=F\proj{\Psi^-}+(1-F)\mathds{1}_{\mathsf{A'}\otimes \mathsf{B'}}/4$, it can be shown that violation is obtained only for $F>1/2$; note that these Werner states are separable for $F \leq 1/3$. On the other hand, we have upper bounds on the achievable Bell violations for $F>1/3$, which suggest that Eq. (<ref>) might be able to detect all entangled two-qubit Werner states; see Supplementary Material.
Discussion. A way to interpret our result is to examine the optimality conditions for discriminating $\rho_0$ and $\rho_1$. To begin with, we remind that these mixed states share the same support and can be simultaneously diagonalized in the Bell basis. The first point implies that unambiguous state discrimination <cit.> is not possible, thus the best measurement scheme, for our purpose, is probabilistic minimum-error state discrimination <cit.>. From the optimality conditions of this scheme, it can be easily verified that the maximum success probability for which $\rho_0$ and $\rho_1$ are optimally discriminated is 1/2, which is indeed the value given in Eq. (<ref>). This also explains the trend seen in Fig. (<ref>) wherein higher Bell violations are achieved with higher inconclusive rates/lower measurement efficiencies.
From the second point, it is clear that the optimal measurement that discriminates between $\rho_0$ and $\rho_1$ must consists of entangled POVMs: the positive and negative eigenspaces of $\rho_0-\rho_1$ are maximally entangled subspaces. This means that no LOSR measurement (or more generally, separable measurement) can coherently access these entangled eigenspaces. Crucially, this limitation also holds in the presence of inconclusive outcomes, i.e., entanglement cannot be created using local operations and classical communication (local filtering with shared randomness in our case).
Conclusion. In the above, we have provided a semi-quantum Bell experiment that safely allows for postselection and is defined by a loss-independent local bound that is violated only in the region of imperfect measurement efficiencies. On the conceptual level, our result suggests that semi-quantum nonlocality is much more powerful than previously recognized. For instance, Eq. (<ref>) does not require the so-called fair-sampling condition <cit.>, which is typically assumed in standard Bell experiments involving postselection to ensure that the conclusive/detected events are representative of the underlying quantum system. Most interestingly, Fig. (<ref>) shows that in order to (optimally) violate Eq. (<ref>), it is necessary to use highly inefficient measurements, which up to the best of our knowledge, is the first time that such a trend has been found. Furthermore, the maximal quantum violation $2\sqrt{2}$ can be achieved for a continuum of measurement efficiencies, i.e., $\gamma \in (0, 1/2]$, unlike standard Bell inequalities which can only reach their maximum violations in the limit of perfect measurement efficiency.
Finally, we remark that on the practical side, our inequality provides a semi-device-independent method for testing entanglement in detected quantum systems. That is, as mentioned above, the inequality allows one to restrict the analysis to detected events without assuming the fair-sampling condition. For example, this application could be useful for entanglement-based experiments suffering from high detection losses, e.g., those based on practical entangled photon-pair sources <cit.>.
Acknowledgements. We thank J.-D. Bancal, A. Martin, V. Scarani, D. Rosset, N. Gisin, H.-K. Lo, R. Thew, B. Qi, W. Grice, N. Johnston and A. Cosentino for helpful discussions. This work was performed at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. The author acknowledges support from the laboratory directed research and development program.
§ TECHNICAL RESULTS
In order for us to provide a more precise description of our semidefinite programs, we would need to introduce a few mathematical notations; some of which may be different from those used in the main text. We let Alice's and Bob's complex Hilbert spaces be denoted by $\mathcal{A}$ and $\mathcal{B}$, respectively. The set of linear operators, Hermitian operators and positive semidefinite operators acting on the composite Hilbert space are written as $\tn{L}(\mathcal{A}\otimes \mathcal{B})$, $\tn{Herm}(\mathcal{A}\otimes \mathcal{B})$ and $\tn{Pos}(\mathcal{A}\otimes \mathcal{B})$, respectively. Furthermore, we write $Q \succeq 0$ to indicate that $Q$ is positive semidefinite. The set of density operators acting on Alice's and Bob's systems is defined as $\tn{D}(\mathcal{A} \otimes \mathcal{B}):=\{\rho \in \tn{Pos}(\mathcal{A}\otimes \mathcal{B}) : \Tr[ \rho ]=1\}$. The set of separable operators, which is a closed convex cone, is denoted by $\tn{Sep}(\mathcal{A}:\mathcal{B})$. Additionally, we would require the partial transpose operation, $T_\mathcal{B}=\mathbb{I}_{\tn{L}(\mathcal{A})} \otimes T$, which performs the transpose operation, $T$, on Bob's Hilbert space. Accordingly, the set of positive partial transpose (PPT) operators is defined as $\tn{PPT}(\mathcal{A} : \mathcal{B}):=\{Q:T_\mathcal{B}(Q) \succeq 0, Q \in \tn{Pos}(\mathcal{A}\otimes \mathcal{B}) \}$. Also, we denote a diagonal matrix by $Q=\tn{diag}[\lambda_1, \lambda_2, \lambda_3,\lambda_4]$.
§.§ Accessible entanglement in separable states
Let us first point out a key observation that explains why global measurements are more predictive than separable measurements for our choice of quantum input states. Recall that the goal is to (locally) discriminate between two mixed separable states, namely,
\[\rho_{0}=\frac{1}{8}\!\!\!\!\!\!\sum_{\substack{\bar{x},\bar{y}\\\tn{s.t.}f(\bar{x},\bar{y})=0}}\!\!\!\!\!\!\omega_{\bar{x}}\otimes \tau_{\bar{y}},\quad \rho_{1}=\frac{1}{8}\!\!\!\!\!\!\sum_{\substack{\bar{x},\bar{y}\\\tn{s.t.}f(\bar{x},\bar{y})=1}}\!\!\!\!\!\!\omega_{\bar{x}}\otimes\tau_{\bar{y}}. \]
where $\omega_{\bar{x}}$ and $\tau_{\bar{y}}$ are defined as $\omega_{\bar{x}}=H^{x_1}\proj{x_2}H^{x_1}$ for $\bar{x}=x_1x_2 \in \{0,1\}^2$ and $\tau_{\bar{y}}=H^{y_1}\proj{y_2}H^{y_1}$ for $\bar{y}=y_1y_2 \in \{0,1\}^2$, respectively. An important feature of these mixed states is that they can be simultaneously diagonalized in an entangled eigenbasis, whose eigenvectors are given by entangled states. That is, using the standard Bell state definitions, i.e., $\ket{\Phi^\pm}=(\ket{00}\pm\ket{11})/\sqrt{2}$ and $\ket{\Psi^\pm}=(\ket{01}\pm\ket{10})/\sqrt{2}$, we have
λ^+ 0 0 0
0 λ^- 0 0
0 0 1/4 0
0 0 0 1/4
, ρ_1=
λ^- 0 0 0
0 λ^+ 0 0
0 0 1/4 0
0 0 0 1/4
where the eigenvalues are $\lambda^\pm=(1\pm1/\sqrt{2})/4$, and the corresponding eigenvectors given as $\ket{\phi_1}=\sqrt{2\lambda^+}\ket{\Phi^-}+\sqrt{2\lambda^-}\ket{\Psi^+}$, $\ket{\phi_2}=\sqrt{2\lambda^-}\ket{\Phi^-}-\sqrt{2\lambda^+}\ket{\Psi^+}$, $\ket{\phi_3}=\ket{\Phi^+}$ and $\ket{\phi_4}=\ket{\Psi^-}$. For example, we have $\lambda^+=\bra{\phi_1} \rho_0 \ket{\phi_1}=\bra{\phi_2} \rho_1 \ket{\phi_2}$. From equation (<ref>), we immediately see that a good guess for the optimal global measurement strategy is to first project the unknown state onto the subspace $\proj{\phi_1}+\proj{\phi_2}$, and then discriminate between the two orthogonal maximally entangled states $\ket{\phi_1}$ and $\ket{\phi_2}$ (i.e., to pick the maximum eigenvalue). Indeed, this gives us a conclusive rate of $\lambda^+ + \lambda^-=1/2$, which agrees with our optimal solution found using semidefinite programming. If we were to use separable measurements, then it is clear that some amount of mixing between the eigenvalues would occur, thus leading to a guessing value that is less than the maximum eigenvalue.
§.§ Optimal guessing probabilities
As mentioned in the main text, the bounds for general and PPT measurements can be analytically solved using convex optimization techniques, namely, semidefinite programming <cit.>. More specifically, the idea is to find feasible solutions for the primal and dual programs, which provide lower and upper bounds on the optimal value, i.e., by virtue of the weak duality principle. If the feasible solutions lead to values that coincide, then we say that the optimal solution for the semidefinite program is found. That is, by the strong duality principle, the duality gap is zero. In fact, the considered semidefinite programs have zero duality gaps.
Recall that the generic guessing probability defined in the state discrimination game for a fixed conclusive rate $\gamma^2$ is given as
\begin{eqnarray} \nonumber
G(\gamma)&:=&\frac{\Tr\left[\frac{1}{2}\rho_0\Pi_{a \oplus b =0}+\frac{1}{2}\rho_1\Pi_{a \oplus b =1} \right]}{\gamma^2}\\&=&\frac{\Pr\left[a\oplus b = f(\bar{x},\bar{y}) \right]}{\gamma^2}=\frac{1}{2}+\frac{S(\gamma)}{8}, \end{eqnarray}
where we used the measurement assignments $\Pi_{a\oplus b=0}=M_{0,0}+M_{1,1}$, $\Pi_{a\oplus b=1}=M_{0,1}+M_{0,1}$ and $\Pi_{\varnothing}=\mathds{1}-\Pi_{a\oplus b=0}-\Pi_{a\oplus b=1}$. Here, we remind that the measurement $\{M_{a,b}\}_{a,b}$ for all $a,b=0,1,\varnothing$ can be either a PPT measurement or a general global measurement, depending on which bound we want to solve. In the following, we will first show the computation for general global measurements.
(Optimal guessing probability for general measurements). The maximum probability of discriminating $\rho_0$ and $\rho_1$ using measurements $\{\Pi_0,\Pi_1,\Pi_\varnothing \} \in \tn{Pos}(\mathcal{A} \otimes \mathcal{B}) $ with a fixed conclusive rate of $\gamma^2 \in (0,1]$ is
max G(γ) = {
1/2(1+1/γ^22√(2)) if γ>1/√(2)
1/2+1/2√(2) if γ≤1/√(2)
The optimal solution is obtained if the feasible solutions for the primal and dual programs lead to a common optimization value. To this end, the primal program for general measurements under the constraint that the conclusive rate is fixed to $\gamma^2 \in (0,1]$ is
Primal program (general)
\begin{eqnarray*}\nonumber
\texttt{maximize}&:& \frac{1}{2}\Tr \left[\rho_0 \Pi_{a\oplus b=0} + \rho_1 \Pi_{a\oplus b=1} \right], \\ \nonumber
\texttt{subject to}&:& \Pi_{a\oplus b=0}+ \Pi_{a\oplus b=1}+ \Pi_{\varnothing} = \mathds{1}_{\mathcal{A} \otimes \mathcal{B}}\\ \nonumber
&& \Tr\left[ (\omega_{\bar{x}}\otimes \tau_{\bar{y}}\Pi_{\varnothing}\right] =1-\gamma^2,\quad \forall~\bar{x},\bar{y}\\
&& \Pi_i \in \tn{Pos}(\mathcal{A} \otimes \mathcal{B}),\quad i=0,1,\varnothing,
\end{eqnarray*}
and the corresponding dual program is found to be
Dual program (general)
\begin{eqnarray*}\nonumber
\texttt{minimize}&:& \Tr\left[Y \right] - (1-\gamma^2)\gamma\\ \nonumber
\texttt{subject to}&:& 2Y - \rho_i \succeq 0,\quad i=0,1\\ \nonumber
&& 4Y - \gamma\mathds{1}_{\tn{L}(\mathcal{A}\otimes \mathcal{B})} \succeq 0 \\ \nonumber
&& Y \in \tn{Herm}(\mathcal{A} \otimes \mathcal{B})\\
&& \gamma \in \mathbb{R}.
\end{eqnarray*}
For the region $0<\gamma \leq 1/\sqrt{2}$, we use the observations from the preceding section to construct a feasible solution for the primal program that is diagonal in the basis of equation (<ref>), i.e.,
\begin{eqnarray*}
\tilde{\Pi}_{a\oplus b=0}&=&\tn{diag}\left[2\gamma^2, 0, 0 ,0 \right], \\ \tilde{\Pi}_{a\oplus b=1}&=&\tn{diag}\left[0, 2\gamma^2,0,0\right],\\ \tilde{\Pi}_{\varnothing}&=&\tn{diag}\left[1-2\gamma^2, 1-2\gamma^2,1,1 \right].
\end{eqnarray*}
A direct computation of the primal objective function using this solution gives $\tn{max}\, \gamma^2{G}(\gamma) \geq \gamma^2\tilde{G}(\gamma) =\gamma^2\left(1+1/\sqrt{2}\right)/2$. A feasible solution for the dual in the same region is
\begin{eqnarray*}
\tilde{Y}=\frac{1}{8}\left(1+\frac{1}{\sqrt{2}}\right)\mathbb{I}_{\tn{L}(\mathcal{A}\otimes \mathcal{B})},\quad \tilde{\gamma}=\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right),
\end{eqnarray*}
which gives $\tn{max}\, \gamma^2{G}(\gamma) \leq \gamma^2\tilde{G}(\gamma) =\gamma^2\left(1+1/\sqrt{2}\right)/2$. Therefore, we arrive at the optimal solution (i.e., equation (<ref>)) for the region $0<\gamma\leq 1/\sqrt{2}$.
For the other half of the region, $1/\sqrt{2} < \gamma \leq 1$, a feasible solution for the primal program is
\begin{eqnarray*}
\tilde{\Pi}_{a\oplus b=0}&=&\tn{diag}\left[1,\, \gamma^2-\frac{1}{2},\, \gamma^2-\frac{1}{2},0 \right], \\ \tilde{\Pi}_{a\oplus b=1}&=&\tn{diag}\left[0, 1, \,\,\gamma^2-\frac{1}{2},\gamma^2-\frac{1}{2} \right],\\ \tilde{\Pi}_{\varnothing}&=&\tn{diag}\left[0, 0,2(1-\gamma^2),2(1-\gamma^2) \right],
\end{eqnarray*}
which leads to a lower bound of $\tn{max}\, \gamma^2{G}(\gamma) \geq \gamma^2\tilde{G}(\gamma) =(2\gamma^2+1/\sqrt{2})/4$. A feasible solution for the dual program is
\begin{eqnarray*}
\tilde{Y}=\begin{bmatrix}
\mu_1 & 0 & 0 & -\mu_2 \\
0 & \mu_1 & \mu_2 & 0 \\
0 & \mu_2 & \mu_1 & 0 \\
-\mu_2 & 0 & 0 & \mu_1
\end{bmatrix},\quad \tilde{\gamma}=\frac{1}{2}, \end{eqnarray*}
where $\mu_1=(2+1/\sqrt{2})/16$ and $\mu_2=(1+1/\sqrt{2})/8-\lambda_1$. Plugging these into the dual objective gives $\tn{max}\, \gamma^2{G}(\gamma) \leq \gamma^2\tilde{G}(\gamma) =\left(2\gamma^2+1/\sqrt{2}\right)/4$. Combining the lower and upper bounds, we thus get the other half of equation (<ref>), that is, $\tn{max}\, {G}(\gamma) =\left(1+1/(\gamma^2 2\sqrt{2})\right)/2$ for $\gamma > 1/\sqrt{2}$.
The upper bound for the local operations and shared randomness (LOSR) bound is computed using PPT measurements, which admit a concise mathematical characterization. Furthermore, under the assumption of two-qubit measurements, PPT measurements are necessarily separable measurements, since for any linear operator $Q\in \tn{L}(\mathcal{A}\otimes \mathcal{B})$, it is separable if and only if it is PPT <cit.>. That is, the set of separable operators and the set of PPT operators are equivalent up to some constant factor for two-qubit positive operators. Note that this is generally not the case if we consider higher dimension Hilbert spaces where PPT does not imply separability. In the below, we first show the optimal guessing probability assuming PPT measurements.
(Optimal guessing probability for PPT measurements). The maximum probability of discriminating $\rho_0$ and $\rho_1$ using measurements $\{\Pi_0,\Pi_1,\Pi_\varnothing \} \in \tn{PPT}(\mathcal{A} \otimes \mathcal{B}) $ for any conclusive rate $\gamma^2 \in (0,1]$ is
max G( ·|PPT) =1/2+1/4√(2).
The primal program for separable/PPT measurements is given as
Primal program (separable/PPT)
\begin{eqnarray*}\nonumber
\texttt{maximize}&:& \frac{1}{2}\Tr \left[\rho_0 \Pi_{a\oplus b=0} + \rho_1 \Pi_{a\oplus b=1} \right] \\ \nonumber
\texttt{subject to}&:& \Pi_{a\oplus b=0}+ \Pi_{a\oplus b=1}+ \Pi_{\varnothing} = \mathds{1}_{\mathcal{A} \otimes \mathcal{B}}\\ \nonumber
&& \Tr\left[ (\omega_{\bar{x}}\otimes \tau_{\bar{y}}\Pi_{\varnothing}\right] =1-\gamma^2,\quad \forall~\bar{x},\bar{y}\\
&& \Pi_i \in \tn{PPT}(\mathcal{A} :\mathcal{B}),\quad i=0,1,\varnothing,
\end{eqnarray*}
and the corresponding dual program is
Dual program (separable/PPT)
\begin{eqnarray*}\nonumber
\texttt{minimize}&:& \Tr\left[Y \right] - (1-\gamma^2)\gamma\\ \nonumber
\texttt{subject to}&:& 2\left(Y -T_{\mathcal{B}}(Q_i)\right)- \rho_i \succeq 0,\quad i=0,1\\ \nonumber
&& 4\left(Y-T_{\mathcal{B}}(Q_2)\right) - \gamma\mathds{1}_{\tn{L}(\mathcal{A}\otimes \mathcal{B})} \succeq 0 \\ \nonumber
&& Y \in \tn{Herm}(\mathcal{A} \otimes \mathcal{B})\\
&& Q_i \in \tn{Pos}(\mathcal{A} \otimes \mathcal{B}),\quad i=0,1,2 \\
&& \gamma \in \mathbb{R}.
\end{eqnarray*}
Similarly, we construct feasible solutions for the primal and dual programs and show that their optimization values are identical. For the primal program, a feasible solution is
\begin{eqnarray*}
\tilde{\Pi}_{a\oplus b=0}&=&\tn{diag}\left[\gamma^2, 0, \frac{\gamma^2}{2}, \frac{\gamma^2}{2}\right], \\ \tilde{\Pi}_{a\oplus b=1}&=&\tn{diag}\left[0,\gamma^2, \frac{\gamma^2}{2}, \frac{\gamma^2}{2}\right],\\ \tilde{\Pi}_{\varnothing}&=&\tn{diag}\left[1-\gamma^2, 1-\gamma^2,1-\gamma^2,1-\gamma^2 \right],
\end{eqnarray*}
where each element is diagonal in the basis of equation (<ref>). Using this solution, we get $\tn{max}\, \gamma^2{G}(\gamma|\tn{Sep}) \geq \gamma^2 \tilde{G}(\cdot|\tn{Sep}) =\gamma^2\left(2+1/\sqrt{2}\right)/4$. For the dual program, a feasible solution is
\begin{eqnarray*}
\tilde{Y}=\frac{1}{8}\left(1+\frac{1}{2\sqrt{2}}\right)\mathds{1}_{\tn{L}(\mathcal{A}\otimes \mathcal{B})},\quad \tilde{\gamma}=\frac{1}{2}\left(1+\frac{1}{2\sqrt{2}}\right), \\
Q_0=\frac{1}{8\sqrt{2}}\proj{\phi_2}, \quad Q_1=\frac{1}{8\sqrt{2}}\proj{\phi_1},\quad Q_2=0_{\tn{L}(\mathcal{A}\otimes \mathcal{B})},
\end{eqnarray*}
which gives $\tn{max}\, \gamma^2{G}(\gamma|\tn{Sep}) \leq \gamma^2\tilde{G}(\cdot|\tn{Sep}) =\gamma^2\left(2+1/\sqrt{2}\right)/4$. Therefore, after normalization, the obtained upper and lower bounds give equation (<ref>).
§.§ Possible detection of all entangled two-qubit Werner states
The detection of entangled two-qubit Werner states <cit.> can be shown by explicitly modeling the measurement operators in terms of two local measurements and a two-qubit Werner state $\phi_\xi:=\xi \proj{\Psi^-}+(1-\xi)\mathds{1}/4$ defined in two auxiliary systems $\mathcal{A}'$ and $\mathcal{B}'$. More specifically, the resulting measurements on systems $\mathcal{A}$ and $\mathcal{B}$ are given as
M_a,b=_𝒜'ℬ'[ϕ_ξM_a,b^+ ],
where $M_{a,b}^+ \in \tn{Sep}(\mathcal{A}\otimes \mathcal{A}':\mathcal{B}\otimes \mathcal{B}')$. Indeed, the resulting measurements $\{M_{a,b}\}_{a,b}$ can only be entangled only if the underlying Werner state is entangled. We compute (upper) quantum bounds for this choice of modeling using the following semidefinite program assuming $\gamma^2=1/4$.
Primal program (Werner states with fixed $\xi$)
\begin{eqnarray*}\nonumber
\texttt{maximize}&:& \frac{1}{2}\Tr \left[(\rho_0 \otimes \phi_\xi) \Pi_{a\oplus b=0} + (\rho_1\otimes \phi_\xi) \Pi_{a\oplus b=1} \right] \\ \nonumber
\texttt{subject to}&:& \Pi_{a\oplus b=0}+ \Pi_{a\oplus b=1}+ \Pi_{\varnothing} = \mathds{1}_{\mathcal{A} \otimes \mathcal{A}' \otimes \mathcal{B} \otimes \mathcal{B}'}\\ \nonumber
&&\frac{1}{2}\tr\left[ (\rho_0+\rho_1)\otimes \phi_\xi \Pi_{\varnothing}\right] =3/4,\quad i=0,1\\
&& \Pi_i \in \tn{PPT}(\mathcal{A}\otimes \mathcal{A}':\mathcal{B}\otimes \mathcal{B}'),\quad i=0,1,\varnothing.
\end{eqnarray*}
Quantum violation vs maximally entangled fraction. Here, we see that quantum violations are obtained only for $\xi > 1/3$, which means that the proposed semi-quantum CHSH inequality could be capable of detecting all entangled two-qubit Werner states.
Here, the optimization of the semidefinite program is taken over measurements separable with respect to the bi-partition $\mathcal{A}\otimes \mathcal{A}':\mathcal{B}\otimes \mathcal{B}'$.
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1511.00453
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$^1$ Max Planck Institut für Wissenschaftsgeschichte, Boltzmannstraße 22, 14195 Berlin, Germany
$^2$ INFN Frascati National Laboratories, Via E. Fermi 40, I00044 Frascati, Italy
$^1$ [email protected]
$^2$ [email protected]
This article recalls the birth of the first electron-positron storage ring AdA, and the construction of the higher energy collider ADONE, where early photon-photon collisions were observed. The events which led the Austrian physicist Bruno Touschek to propose and construct AdA will be recalled, starting with early work on the Widerøe's betatron during World War II, up to the construction of ADONE, and the theoretical contribution to radiative corrections to electron-positron collisions.
§ INTRODUCTION
Photon-photon physics started its long way towards measurement and observation long time ago. Theoretical calculations of $\gamma \gamma $ processes were developed early before the advent of QED, and then refined in mid 1950, as described in other contributions to this session. The first phenomenological studies started with electron-positron collisions, at VEPP-2 and ADONE. The construction of ADONE has been proposed by Bruno Touschek in 1960 soon after AdA, the first storage ring ever to be built and function, where electron-positron collision were first observed <cit.>.
AdA opened the way to higher luminosity, higher energy, more modern machines and in this contribution we shall recall the birth of AdA, and the construction of ADONE, and the events which led to the first observation of electron-positron collisions. Since much has been written on this subject, to avoid repetition we shall mostly discuss here less known parts of the story of AdA, Touschek and ADONE, in particular focusing on the extraordinary combination of events which allowed Touschek to propose AdA in February 1960. The main reference work for Touschek's life and his contribution to the construction of both AdA and ADONE is the biography written by E. Amaldi after Touschek's death in 1978 <cit.>. Further details about Touschek's life and science can be found in <cit.>.[ Volume <cit.> and the LNF internal Notes since 1953, mentioned later in this article, are available at http://www.lnf.infn.it/sis/ .]
Touschek's life spans Europe in space and time, from before to after the Second World War, from the native Austria to Italy, where he gave his greatest contribution to modern day science by proposing the construction of the first electron-positron storage ring. Fig. <ref> summarizes the major stages of his life.
A cartoon showing the main stages of Bruno Touschek's life.
The milestones in Touschek's scientific life can be identified as having taken place in the following major periods:
* 1943-45: during these years, Touschek was in Germany, and worked with Rolf Widerøe to build a betatron;
* 1947-1952: he was awarded his Ph.D, published several papers on quantum field theory, double $\beta$-decay and meson physics, and was involved in the design and construction of an electron-synchrotron;
* 1953-59: these are the years during which Bruno Touschek, who had been given a position in Rome with the newly established Istituto Italiano di Fisica Nucleare (INFN, ), deepened his knowledge in theoretical physics, studying the new symmetries and the breaking of the old ones;
* 1960-64: this is the period for which his contribution to modern science is best known, the proposal and construction of AdA, the discovery of the Touschek effect in Orsay, and the first ever observation of $e^+e^-$ collisions;
* November 1960 onwards: Touschek drafted a proposal for a large collider, ADONE, and then followed its construction and developments.
§ HOW TOUSCHEK LEARNT TO BUILD ACCELERATORS
Bruno Touschek learnt the art of building accelerators in Germany, during World War II, while working on a secret betatron project, led by the Norwegian engineer Rolf Widerøe <cit.>, financed by the Aviation Ministry of the Reich, the Reichsluftfahrtministerium (RLM). The encounter and collaboration of Touschek and Widerøe, which ultimately led to the AdA proposal in 1960, follows from a series of rather extraordinary coincidences, which originated on the one side in Norway, in Trondheim and Oslo, and on the other in Vienna and Munich. We know for sure that in September 1943 Touschek and Widerøe were already working together on the betatron project, and in fact it is at that time that Widerøe mentioned to Touschek the idea of oppositely charged colliding particles. But how did they meet? One was a 22 year old physics student, born in Vienna from a Jewish mother, who died young, while his father, who later remarried, had been an officer in the Austrian army. The other was an experienced engineer from Oslo, who had done early work on the theory and construction of accelerators for the PhD he had obtained in Karslruhe. Not irrelevant to the story is that Widerøe's brother had been working for the Norwegian underground and was kept prisoner by the Germans. The desire to help him is one of the reasons Widerøe ultimately accepted to come to Germany to build a betatron. Touschek had left Vienna to continue his studies in physics in the relatively anonymity of Hamburg and Berlin, after he had to stop attending classes at the University, where he had enrolled to study physics, and where he had already shown excellence. In Fig. <ref>, we show two photographs of Touschek and Widerøe.
At left, a photograph of the 18 year old from <cit.> . At right, a photograph of Bruno Touschek from his passport. It was probably prepared in 1939, for his last travel to Italy to visit his maternal aunt Adele, nicknamed Ada.
We shall try to separately outline the two stories which came together in Berlin in 1943, and will start describing the path which brought Touschek to learn of Widerøe's work.
Bruno had left Vienna, where he was born in 1921. His Jewish origin from mother's side had brought many difficulties after the annexation of Austria to Germany in 1938. Enrolled at the University of Vienna to study physics in 1939, at the end of the academic year he had to stop attending classes and could only study at home with books borrowed by his young teacher Paul Urban. Thus he moved to Germany, under the patronage of Arnold Sommerfeld, who had been contacted by Bruno about some small errors in the fundamental treaty Atombaum und Spektrallinien, which Bruno had started studying. In February 1942 Bruno was in Munich, visiting Sommerfeld, and then went to Hamburg, where he started attending courses at the University, studying and doing odd jobs to make a difficult living. In Figs. <ref> and <ref>, we show two drawings from this period, which Touschek had included in letters to his father and step mother.[In these letters Touschek addresses them as Dear parents.]
A drawing by Bruno Touschek, from a 23 August 1942 letter from Hamburg.
A 1942 portrait of Bruno Touschek, included in a letter to his parents.
Under Sommerfeld's recommendation, Bruno was unofficially attending classes at both University of Hamburg and University of Berlin, and frequently moved between the two cities. Sometime, probably in fall 1942, he met a girl, also half-jewish, who worked in Berlin, at Loewe-Opta, a radio and television manufacturing company, and suggested he could also obtain a job there. And so it happened that Bruno ended up working with Karl A. Egerer, who was at the time the director of a special department within the company, that was now also producing electronic devices of war interest. Egerer was the editor of the scientific journal , too. In this journal, in 1928, had published his article on the theory of betatrons and to this journal, as we shall presently see, he would send on September 15, 1942 an article entitled Der Strahlentransformator, where he presented the proposal to construct a 15 MeV betatron, and even gave hints about a more powerful 100-MeV machine. This is the article, accepted but never published, which was to put in motion the RLM betatron project. We have in fact some evidence from a February 1943 letter
to his parents, that Touschek read this article, in his capacity as assistant to Egerer, and commented upon it to him. In this letter, Touschek tells his parents that Egerer became excited and started making crazy plans for some war related project to present to the RLM, or even to Heisenberg.[ This letter and a translation of it were published in <cit.>.] Egerer was in contact with other scientists and engineers gravitating around various high ranking officers at the RLM, and it is suggestive to think that this is how the project must have reached them and started its way to realization. Once accepted, the article was classified and could not be published.[A copy of the article in its proofs was kindly provided to us by Aashild Sørheim, of the University Radium Museum in Oslo. A more detailed account of the events involving the unpublished article can be found in <cit.>.]
At this point, we shall now step back to see the train of events which made submit the article and then come to Hamburg in September 1943 to build the 15 MeV German betatron.
In the United States, the 1928 article by Widerøe had interested accelerator scientist. His first unsuccessful attempt to build a betatron inspired Ernest Lawrence to build the first cyclotron <cit.> and was later central to the construction of the betatron by Donald Kerst, who reported it in his Physical Review articles on the betatron <cit.>. It is worth noticing here that the same articles were the subject of the 1941 thesis work by Giorgio Salvini, the physicist who would be called in 1953 to build the Frascati electro-synchrotron and who, in 1960, as Director of the Frascati National Laboratories, approved the construction of AdA.
The issue of the Physical Review with Kerst's article was one of the last to reach Nazi-occupied Norway
and it was read, in the fall of 1941, by the physicist Roald Tangen, from the
Physical Institute of Trondheim University. In Tangen's words [6]: I can well
remember the events of 1941 [….] In the autumn of 1941 the Physics Association
invited me to give a lecture on modern accelerators in Oslo. We had
been denied access to American magazines by then, and we were completely
ignorant of the betatron. A few days before my trip to Oslo a single copy of
the Physical Review arrived in Trondheim by ordinary mail. Mysteriously, it
had found its way to us. It contained an article by Donald Kerst on the first
working betatron. This fitted well in my lecture in which I went on to explain
that Kerst mentioned a German doctorate thesis by a R. Widerøe in which a
fundamental equation for the betatron was developed. I did not know anyone
by the name of Widerøe at the time, but I told my audience that the name
indicated that he could be a Norwegian. As we were to discover soon enough,
Rolf Widerøe was sitting in the auditorium.
Widerøe had left working on accelerators after his thesis work, and had joined the Norwegian branch of Brown-Boveri. But his interest was rekindled by Kerst's article and he immediately went back to work to propose a similar machine to be built in Europe. Thus, in September 1942 he submitted the article to the . And here is where the two stories of Touschek and Widerøe meet. In his autobiography, Widerøe writes :
A very strange thing happened when my first article appeared. One day, it
must have been in March or April 1943, several German Air Force officers
came to NEBB [Norsk Elektrisk og Brown Boveri] wanting to speak with me.
Norway had been under occupation since April 1940. I cannot remember
exactly whether there were two or three of them.
They asked whether we could go
to the Grand Hotel together to talk about something.
They said that it could be a matter of some
importance to my brother …The German officers hinted that it may be
possible to release my brother if I helped them. This decided things for me,
and I agreed to go to Berlin. Two days later I was flown there for a short
visit, and they told me about their plans to build betatrons.
The coincidence of dates, namely Touschek's reading the submitted article, in mid February, and the visit to Widerøe by the German officers in Spring 1945, suggests that, following Touschek's comments on the article, Egerer may have contacted the RLM. The interest of the German war authorities in betatron research and possible war applications is documented in <cit.> and this may have prompted the contacts with Widerøe in Oslo. We also notice that Widerøe had previously tried in vain to obtain a more lenient treatment of his brother's imprisonment conditions <cit.>. Several groups at the time were quite interested in the possibility of having sources of artificially accelerated particles. And even if it was already clear that the betatron could be employed only as a source of X-rays used mainly for medical purposes, building such a machine was of course an exciting scientific project per se.
Bruno worked alongside with Widerøe from 1943 until March 1945. During this period Touschek contributed with theoretical work, the great part of which was later used for his thesis work at Göttingen, at the end of the war. In particular he tackled the problem of the energy losses due to radiation damping, which would define an upper limit for the energy obtainable with the betatron. As the war approached its end, the betatron, which was almost completed, was moved from Hamburg to Kellinghusen, a supposedly safer location, 40 Km North of Hamburg, in the property of one of the scientists of the betatron group. As recounted many times, around March 15th, Touschek was arrested and kept in the infamous jail of Fuhlsbüttel. He received some comfort from Widerøe's
visits, which brought him books and cigarettes, and promises of release, but the future destination of prisoners in this jail, mostly Jews, was the Kiel concentration camp, some 30 Km North of Hamburg. And it was to this camp, as the allied troops were approaching Hamburg in mid April, that Touschek was directed, together with other 200 prisoners, guards in front and guards at the back. Luckily for him and for science, Touschek did not reach Kiel: he fell to the ground, being sick and burdened by a load of books and few belongings, was shot and left for dead on the side of the road. Further developments of Touschek's story during the war are outlined in <cit.>, where the two letters written by Touschek to his family in June and October 1945 are presented.
§ THE BIRTH OF $E^+E^-$ COLLIDERS
The construction of AdA and its final success is the work of many scientists. Three, among them, had a pivotal role: Bruno Touschek, who proposed it and contributed to its commissioning, Giorgio Salvini who made the Frascati synchrotron to work in 1959 and created, from nothing, a pool of scientists, technicians and engineers of world capacities, and Edoardo Amaldi, Fermi's youngest collaborator in Rome before the war.
Edoardo Amaldi, who was one of the leading actors in the resurgence of physics in Italy and in Europe after the war, called to Rome both Touschek and Salvini, two scientists who had, in their training, the knowledge and the mindset of constructing accelerators.
In Fig. <ref> we show Touschek with Amaldi and other scientist friends, during a excursion to the Tuscolo hills, above the town of Frascati.
Bruno Touschek (center), in Italy in 1953, at Tuscolo hills with Edoardo and Ginestra Amaldi to his right.
During his earlier period in Rome, where he joined as INFN researcher in 1952, Touschek does not seem to have been interested to the synchrotron work. He was keen in expanding his knowledge of theoretical physics and engaged in the challenges posed by the renewed post-war activities and quantum field theory formulations, as testified, notably, by his intense correspondence with Wolfgang Pauli during the 1950s. But as the synchrotron approached its completion, his interest grew, and in 1959, he was coming regularly to the Laboratory, attending seminars and meetings. It was one such seminar, by the Director of the High Energy Physics Laboratory of Stanford University, Wolfgang Panofsky, which seems to have ignited the spark which started AdA. A first reconstruction of this event can be found in <cit.>, where the sources of different recollections are discussed. A possible date for this seminar is October 26th, 1959, since the list of 1959-60 seminars at Frascati Laboratories shows a seminar by Panofsky on that day.
Nicola Cabibbo, who had graduated with Touschek a few years before, recalled [Bruno Touschek's life during the period spanning from his arrival in Italy in 1952 to the building of AdA and ADONE is outlined in the docu-film Bruno Touschek and the art of physics, by E. Agapito and L. Bonolis, ©INFN 2003] that after this seminar and a discussion on the electron-electron tangential ring collider built in the US,
Touschek asked: Why not using electrons against positrons? According to Cabibbo, it was thus during this seminar that the idea of having electrons and positrons circulate and collide within the same ring, was put forward by Bruno. In the months to follow he pursued the idea and started making some calculations for a draft proposal, which he presented on February 17th, 1960 at a meeting called to discuss the future programs of the Laboratory.
He had envisaged to modify the Frascati electro synchrotron to make it into a storage ring, but this was unthinkable given the expectations of the Frascati physicists to start experimentation with the synchrotron set-up. On February 18th, the day after the meeting, he then started work on a realistic proposal, whose first page is shown in Fig. <ref>.
First page of the notebook where Touschek started the actual proposal for the construction of AdA.
As a comment to the intense work, which started taking place in those months, such as the comprehensive calculations of expected processes <cit.>,
we show two (later) drawings by Bruno Touschek, in Fig. <ref>.
Two drawings by Bruno Touschek.
The proposal was approved on March 7th and in less than one year AdA started functioning <cit.>.
As Pierre Marin would later recall <cit.>, a team of first class scientists designed and built a small ring in which they made electrons and positrons circulate, in opposite directions, for hours.
They were Carlo Bernardini, Gianfranco Corazza, Giorgio Ghigo, Mario Puglisi, Ruggero Querzoli, Giancarlo Sacerdoti, Peppino di Giugno and, of course Bruno Touschek. AdA started working in February 1961 but the road to prove the viability of this type of accelerator was long. It took two yeas and the transfer of AdA to Orsay before obtaining proof that annihilation had taken place.[The transfer of AdA to Orsay is described in the docu-film Touschek with AdA in Orsay, by E. Agapito, L. Bonolis and G. Pancheri, ©INFN 2013.]
§ PHOTONS WITH ADA AND TWO PHOTON OBSERVATIONS WITH ADONE
AdA's luminosity had been too low to observe annihilation into final particles. This had been true in Frascati, but remained true even after AdA had been transported to Orsay to make use of the higher intensity of the electron beam from the linear accelerator, jokingly called by students and researcher's alike OLGA, Orsay Linear Great Accelerator. Instead, the proof of collisions had come from the observation of events consistent with single bremsstrahung <cit.>, namely
\begin{equation}
e^+e^-\rightarrow e^+e^- \gamma \label{eq:brem}
\end{equation}
Touschek however had been firmly convinced that such type of machine would work, and, as early as November 1960, less than a year after he had proposed to build AdA, a proposal to build a much bigger and more powerful machine was put in writing and presented to the Frascati Laboratory management. In Fig. <ref> we show the typewritten draft, which was transformed in a joint internal laboratory note <cit.> a few months later.
\frame{\includegraphics[width=20pc]{AdoneprogettoTouschek-LB}}}}$
At left we show the draft of the proposal for the construction of ADONE prepared by Bruno Touschek as soon as he was certain that the storage ring principle of AdA was working. Notice the value of the c.m. energy he proposed, namely 3 GeV, chosen so as to observe pairs of all the particles known at the time. At right Bruno Touschek in a photograph in the second half of 1950s and one of his drawings, most probably from the early AdA period, elaborated by C. Federici.
AdA remained in Orsay, at the Laboratoire de l'Accélérateur Linéaire, two years, 1962-1964. During this time, important
effects, such as the Touschek effect <cit.>, were discovered.
After AdA's final measurements in Orsay in 1964 and the confirmation of collisions through Eq. (<ref>) <cit.>, the Italian team returned to Rome and Frascati, and the construction of ADONE started in earnest. Touschek followed the construction of ADONE, often contributing to discussions about beam instability problems, as described in <cit.>. He was also particularly worried about radiation problems, whose calculation became the central focus of the theory group gathered around him in Frascati.
Calculations of radiative effects at ADONE's energies had initiated with the single bremsstrahlung,
and continued, under Touschek's leadership and guidance, with double bremstrahlung <cit.> and photon resummation to all orders <cit.>. ADONE started functioning at the end of 1968 and began taking data in 1969.
Concerning two photon processes, early in its operation ADONE observed the process which had been proposed by Touschek as monitor for the machine luminosity, i.e. annihilation into two photons:
\begin{equation}
e^+e^-\rightarrow \gamma \gamma
\end{equation}
This process, which was observed in Novosibirsk with VEPP-2 <cit.> and in Frascati, with ADONE <cit.> in the range with $E_{beam}=0.7-1.2\ GeV$,
was aimed
at verification of QED. Other channels of annihilation into pairs of muons, pions, kaons followed, as Touschek had envisaged in his draft proposals.
Soon, both ADONE and VEPP-2 reported the observation of the more complicate final state $e^+e^-\rightarrow e^+e^- + X$, such as production of an electron-positron pair accompanied by hadrons, or by a second electron-positron pair, or a muon pair. As recalled in the contribution to this conference by Elena Pakhtusova, the first observations of the process
\begin{equation}
e^+e^-\rightarrow e^+e^- \gamma \gamma \rightarrow e^+e^- e^+e^-
\end{equation}
were made with VEPP-2 <cit.>
and with ADONE <cit.>. The ADONE results had also been earlier reported at the 1971 Bologna Conference in April,[ International Conference on Meson Resonances and Related Electromagnetic Phenomena, Bologna 1971.] and at Cornell,[The International Conference on Electrons and Photons Interactions, Cornell 1971.] in August 1971 <cit.>.
Soon, there followed the first observation of the process
\begin{equation}
e^+e^-\rightarrow e^+e^- \gamma \gamma \rightarrow e^+e^- \mu^+\mu^-
\end{equation}
which was reported in <cit.>.
Later, in 1979, the results of various other measurements of the two photon collision channels at ADONE, namely
\begin{equation}
e^+e^-\rightarrow e^+e^- \ X \ \ \ \ \ \ \ \ \ \ \ X=\gamma \gamma \rightarrow e^+e^-,\ \mu^+\mu^-,\ \pi^+\pi^-,\ \eta'(958)
\end{equation}
which had been taken in the energy range $\sqrt{s}=750=1500\ MeV$, were reported in <cit.>.
The complete history, as well as the theoretical developments which led to this field of photon-photon physics, can be found in other talks in the historical session of this conference. In particular, the theoretical developments, to which the group in Frascati and Rome contributed as well, are illustrated in the talks by I. Ginzburg and F. Kapusta.
As a final comment to this brief historical contribution, we show in the right hand panel of Fig. <ref> a photo of Bruno Touschek, with his well known drawing entitled Magnetic discussion.
§ ACKNOWLEDGEMENTS
G.P. thanks Valery Telnov and the other organizers, who have invited her to present this talk and allowed it to be done remotely. The collaboration of the Communication Service of Frascati National Laboratories is gratefully acknowledged. We are also very grateful to the Touschek family who holds the copyright of photos and drawings by Bruno Touschek and provided us with unpublished material from Touschek's letters home during WWII.
§ REFERENCES
|
1511.00579
|
Department of Theoretical Physics, Ho Chi Minh City
University of Science, Vietnam
Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
Institute of Physics, Vietnamese Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam
Theoretical Particle Physics and Cosmology Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Faculty of Applied Sciences,
Ton Duc Thang University, Ho Chi Minh City, Vietnam
We consider the baryogenesis picture in the Zee-Babu model. Our analysis shows that electroweak phase transition (EWPT) in the model is a first-order phase transition at the $100$ GeV scale, its strength is about $1 - 4.15$ and the masses of charged Higgs are smaller than $300$ GeV. The EWPT is strengthened by only the new bosons and this strength is enhanced by arbitrary $\xi$ gauge. However, the $\xi$ gauge does not make lose the first-order electroweak phase transition. The process of the EWPT kinetics corresponding to B violation is received through the sphaleron probability. By a thin-wall approximation, we assume that sphaleron rate is larger than the cosmological expansion rate at the temperature being higher than the critical temperature; and after the phase transition, the sphaleron process is decoupled. This also suggests that the phase transition is a transition depending at each point in space. This may provide baryon-number violation (B-violation) necessary for baryogenesis in the relationship with non-equilibrium physics in the early universe.
11.15.Ex, 12.60.Fr, 98.80.Cq
Keywords: Spontaneous breaking of gauge symmetries,
Extensions of electroweak Higgs sector, Particle-theory models (Early Universe)
§ INTRODUCTION
Physics, at present, has entered into a new period, understanding of the early Universe. In that context, almost Cosmology and Particle Physics are on the same way. Being as a central issue of cosmology and particle physics, at present the baryon asymmetry is an interesting problem. If we could explain this problem, we can understand the true nature of the smallest elements and reveal a lot about an imbalances matter-antimatter from the early Universe.
The electroweak baryogenesis (EWBG) is a way to explaining the Baryon Asymmetry of Universe (BAU) in the early universe, has been known by Sakharov conditions, which are B, C, CP violations, and deviation from thermal equilibrium <cit.>. These conditions can be satisfied when the EWPT must be a strongly first-order phase transition. Because that not only leads to thermal imbalance <cit.>, but also makes a connection between B and CP violation via nonequilibrium physics <cit.>.
The EWPT has been investigated in the Standard Model (SM) <cit.> as well as its various extended versions <cit.>. For the SM, although the EWPT strength is larger than unity at the electroweak scale, but the mass of the Higgs boson must be less than $125$ GeV <cit.>; so the EWBG requires new physics beyond the SM at the weak scale <cit.>.
Many extensions such as the Two-Higgs-Doublet model, the reduced minimal 3-3-1 model, the economical 3-3-1 model or the Minimal Supersymmetric Standard Model, have a strongly first-order EWPT and the new sources of CP violation, which are necessary to account for BAU; triggers for the first-order EWPT in these models are heavy bosons or dark matter candidates <cit.>. However, the most researches of the EWPT are the Landau gauge. Recently gauge invariant also made important contributions in the electroweak phase transition as researching in Ref. <cit.>.
The quantity of sphaleron rate which is B violation rate, has been calculated in the SM <cit.> and the reduced minimal 3-3-1 model <cit.>. In addition, by using non-perturbative lattice simulations, a powerful framework and set of analytic and numerical tools have been developed in Refs.<cit.>.
The Zee-Babu (ZB) model is one of the simplest extensions of the SM which has some interesting features <cit.>. We have considered the EWPT and sphaleron rate in the ZB model due to its simplicity.
The ZB model has more two charged scalars $h^{\pm}$ and $k^{\pm\pm}$ in the Higgs potential. The kind of new scalars can play an important role in the early universe. They can be a reason for tiny mass of neutrinos through two loops or three loop corrections <cit.>. One important property of these particles which will be shown in this paper, is that they can be triggers for the first-order phase transition.
This paper is organized as follows. In Sect. <ref> we give a short review of the ZB model and we drive an effective potential which has a contribution from heavy scalars and the $\xi$ gauge at one-loop level. In Sect. <ref>, we find the range mass of charged scalar particles by a first-order phase transition condition. In Sect. <ref>, we offer the
solutions of VEV and estimation sphaleron rate by our approximations, and show that this rate can satisfy the decoupling condition. Finally, Sect. <ref> is devoted to constraints on the mass of the charged Higgs boson. In Sect. <ref> we summarize and describe outlooks.
§ EFFECTIVE POTENTIAL IN THE ZEE-BABU MODEL
In the ZB model, by adding two charged scalar fields $h^{\pm}$ and $k^{\pm\pm}$ <cit.>, the Lagrangian becomes
ℒ = L_SM+f_abψ_aL^cψ_bLh^+ +h_ab^'l_aR^cl_bRk^++ + V(ϕ,h,k)
+ (D_μh^+)^†(D^μh^+) +
In this model, the Higgs potential has four couplings between $h^{\pm}$ or $k^{\pm\pm}$ and neutral Higgs <cit.>:
V(ϕ,h,k) = μ^2ϕ^†ϕ+ u^2_1|h|^2 + u_2^2|k|^2 +λ_H(ϕ^†ϕ)^2 +
λ_h |h|^4 + λ_k |k|^4
+λ_hk |h|^2 |k|^2 + 2p^2|h|^2 ϕ^†ϕ+ 2q^2|k|^2 ϕ^†ϕ+ (μ_hk h^2 k^++ + H.c) ,
and $\rho^0$ has a Vacuum Expectation Value (VEV)
ρ^0=1/√(2)( v_0+σ+iζ) .
The masses of $h^\pm$ and $k^{{\pm\pm}}$ are given by
m^2_h^± = p^2v^2_0+u^2_1,m^2_±± = q^2v^2_0+u^2_2.
Diagonalizing matrices in the kinetic component of the Higgs potential and retain Goldstone bosons, we obtain
\begin{gather}\label{eq:SMfieldDepmasses}
\begin{aligned}
m_H^2(v_0)&=-\mu^2+3\lambda v^2_0\,,\\
m_G^2(v_0)&=-\mu^2+\lambda v^2_0\,,\\
\end{aligned}
\begin{aligned}
m_Z^2(v_0)&=\textstyle\frac{1}{4}(g^2+g'^2) v^2_0=a^2v^2_0\,,\\
m_W^2(v_0)&=\textstyle\frac{1}{4}g^2 v^2_0=b^2v^2_0\,.\\
\end{aligned}
\end{gather}
§.§ EFFECTIVE POTENTIAL WITH LANDAU GAUGE
From Eq. (<ref>), ignoring Goldstone bosons, we obtain an effective potential with contributions of $h^{\pm}$ and $k^{{\pm\pm}}$ in the Landau gauge:
V _eff(v) = V _0(v)+3/64π^2( m _Z^4(v)lnm _Z^2(v)/Q^2
+2m _W^4(v)lnm _W^2(v)Q^2 - 4m _t^4(v)lnm _t^2(v)Q^2) + 1/64π^2(2m_h^±^4(v)lnm_h^±^2(v)Q^2 + 2m_k^±±^4(v)lnm_k^±±^2(v)Q^2+ m _ H^4(v)lnm_H^2(v)Q^2 ) + 3 T^4/4π^2 {F_- (m_Z(v)/T)+F_-(m_W(v)/T) + 4F_+(m_t(v)/T)} + T^4/4π^2{2F_-(m_h^±(v)/T)+2F_-(m_k^±±(v)/T) + F_-(m_H(v)/T)} ,where $v_\rho$ is a variable which changes with temperature, and at $0^o K$, $v_\rho \equiv v_0=246$ GeV. Here
F_±(m_ϕT) = ∫_0^m_ϕTαJ_∓^1 (α,0)dα,J_∓^1 (α,0) = 2∫_α ^∞(x^2-α^ 2)^1/2e^x∓1dx.
§.§ EFFECTIVE POTENTIAL WITH $\xi$ GAUGE
However, we know that in high levels, the contribution of Goldstone boson cannot be ignored. Therefore, we must consider an effective potential in arbitrary $\xi$ gauge given by
𝒱_1^T=0(v) = 1/4(4π)^2(m_H^2)^2[ln(m_H^2/Q^2)-3/2]+1/4(4π)^2(m_h^±^2)^2[ln(m_h^±^2/Q^2)-3/2]
𝒱^T≠0_1(v,T) = T^4/2π^2[J_B(m_H^2/T^2)+J_B(m_h^±^2/T^2)+2J_B(m_k^±±^2/T^2)]
where “free” represents a free-field subtraction.
§ ELECTROWEAK PHASE TRANSITION IN THE ZEE-BABU MODEL
§.§ EWPT in Landau gauge
Ignoring $u_1$ and $u_2$ (i.e., $u_1$ and $u_2$ are assumed to be very small), we can write the high-temperature expansion of the potential (<ref>) as a quartic equation in $v$:
in which
D = 1/24 v_0^2 [
6 m_W^2(v_0) +3m_Z_1^2(v_0)
+m_H^2(v_0)+2m_h^±^2(v_0) +2m_k^±±^2(v_0) +6 m_t^2 (v_0)
],T_0^2 = 1/D{m_H^2(v_0)/4
.. .. +2m_h^±^4(v_0) +2m_k^±±^4(v_0) -12 m_t^4 (v_0)
)},E = 1/12 πv_0^3 (
6 m_W^3(v_0) +3m_Z_1^3(v_0)
+m_H^3(v_0) +2m_h^±^3(v_0) +2m_k^±±^3(v_0)
λ_T =
m_H^2(v_0)/2 v_0^2{
1- 1/8π^2 v_0^2 (m_H^2(v_0))[
6 m_W^4(v_0) lnm_W^2(v_0)/a_b T^2
.. ..
+3m_Z_1^4(v_0) lnm_Z_1^2(v_0)/a_b T^2+m_H^0^4(v_0) lnm_H^0^2(v_0)/a_b T^2
.. ..
+2m_k^±±^4(v_0)lnm_k^±±^2(v_0)/a_bT^2+2m_h^±^4(v_0)lnm_h^±^2(v_0)/a_bT^2-12 m_t^4(v_0)lnm_t^2(v_0)/a_F T^2
] },where $v_0$ is the value where the zero-temperature effective potential $V^{0^o K}_{eff}(v)$ gets the minimum. Here, we acquire $V^{0^o K}_{eff}$ from $V_{eff}$ in Eq.(<ref>) by neglecting all terms in the form $F_{\mp}\left(\frac{m}{T}\right)$.
The minimum conditions for $V^{0^o K}_{eff}(v)$ are
V^0^o K_eff(v_0)=0,∂V^0^o K_eff(v)/∂v|_v=v_0=0,∂^2 V^0^o K_eff(v)/∂v^2|_v=v_0=
[m^2_H(v)] |_v=v_0=125^2 GeV^2.
We also have the minima of the effective potential (<ref>)
v=0, v ≡v_c=2ET_c/λ_T_c,
where $v_c$ is the critical VEV of $\phi$ at the broken state, and $T_c$ is the critical temperature of phase transition given by
Now let us investigate the phase transition strength
of this EWPT. In the limit $E \rightarrow 0$, the transition strength $S\rightarrow 0$ and the phase transition is a second-order. To have a first-order phase transition, we requires $S \geq 1$. We plot $S$ as a function of $m_{h^{\pm}}$ and $m_{k^{\pm\pm}}$ in Fig. <ref>.
According to Ref. <cit.>, the accuracy of a high-temperature expansion for the effective potential such as that in Eq. (<ref>) will be better than $5\%$ if $\frac{m_{boson}}{T}< 2.2$, where $m_{boson}$ is the relevant boson mass. Therefore, as shown in Fig. <ref>, for $m_{h^{\pm}}$ and $m_{k^{\pm\pm}}$ which are respectively in the ranges $0-350\, \mathrm{GeV}$, the transition strength is in the range $1 \leq S <2.4$.
We see that the contribution of $h^{\pm}$ and $k^{\pm\pm}$ are the same. The larger mass of $h^{\pm}$ and $k^{\pm\pm}$, the larger cubic term ($E$) in the effective potential but the strength of phase transition cannot be strong. Because the value of $\lambda$ also increases, so there is a tension between $E$ and $\lambda$ to make the first order phase transition. In addition when the masses of charged Higgses are too large, $T_0, \lambda$ will be unknown or $S\longrightarrow \infty$.
When the solid contour of $s=2E/\lambda_{T_c}=1$, the
dashed contour: $2E/\lambda_{T_c}=1.5$, the dotted contour: $2E/\lambda_{T_c}=2$,
the dotted-dashed contour: $2E/\lambda_{T_c}=2.4$, even and no-smooth contours: $s\longrightarrow \infty$
§.§ EWPT in $\xi$ gauge
We can rewrite the high-temperature expansion of the potential (<ref>) and (<ref>) as a like-quartic equation in $v$
𝒱 = (𝒟_1+𝒟_2+𝒟_3+𝒟_4+ℬ_2)v^2+ℬ_1v^3+Λv^4+f(T,u_1,u_2,μ,ξ),
\[
f(T,u_1,u_2,\mu,\xi, v)=\mathcal{C}_1+\mathcal{C}_2,\]
𝒟_1 = T^2/24 v^2_0 (3m^2_Z(v_0)+6 m^2_W(v_0)+6 m^2_t(v_0)+2(m^2_h^±(v_0)-u_1^2)+2(m^2_k^±±(v_0)-u_2^2)+6λv^2_0) ,𝒟_2 = 1/32 v^2_0π^2{3m^4_Z(v_0)+6m^4_W(v_0)-12m^4_t(v_0)+2(m^2_h^±(v_0)-u_1^2)^2-8π^2 v^2_0 m^2_H_0. .
+2(m^2_k^±±(v_0)-u_2^2)^2+12v_0^4λ^2+2m^2_Z(v_0)v^2_0λξ+4m^2_W(v_0)v^2_0λξ},𝒟_3 = 1/32π^2{2p^2 u_1^2 ln[a_b T^2/p^2 v_0^2+u_1^2]+2 q^2 u_2^2 ln[a_b T^2/q^2 v_0^2+u_2^2]. .
-3λμ^2 ln[a_b T^2/3v_0^2λ-μ^2]-λμ^2 ln[a_b T^2/v_0^2 (λ+a^2ξ)-μ^2]. .
-2λμ^2 ln[a_bT^2/v_0^2 (λ+b^2ξ)-μ^2]-a^2 ξμ^2 ln[a_b T^2/v_0^2 (λ+a^2 ξ)-μ^2]. .
-2b^2 ξμ^2ln[a_bT^2/v_0^2(λ+b^2 ξ)-μ^2]} ,𝒟_4 = 1/32π^2(2p^2u_1^2+2q^2 u_2^2-6λμ^2-a^2ξμ^2-2b^2ξμ^2) ,Λ = 1/64π^2{2p^4 ln[a_bT^2/u_1^2+p^2v_0^2]+2q^4 ln[a_bT^2/u_2^2+q^2v_0^2]+3a^4ln[a_bT^2/a^2v_0^2]+6b^4ln[a_bT^2/b^2v_0^2]. .
-12k^4 ln[a_F T^2/k^2v_0^2]+9λ^2ln[a_bT^2/3λv_0^2-μ^2]+8π^2 m^2_H_0/v_0^2. .
-a^4ξ^2ln[a_bT^2/a^2ξv_0^2]-2b^4ξ^2ln[a_bT^2/b^2ξv_0^2]. .
+a^4ξ^2ln[a_b T^2/v_0^2 (λ+a^2 ξ)-μ^2]+2b^4ξ^2ln[a_b T^2/v_0^2(λ+b^2ξ)-μ^2]. .
+2a^2λξln[a_b T^2/v_0^2(λ+a^2ξ)-μ^2]+4b^2λξln[a_b T^2/v_0^2 (λ+b^2ξ)-μ^2]. .
+λ^2ln[a_b T^2/v_0^2(λ+a^2ξ)-μ^2]+2λ^2ln[a_b T^2/v_0^2(λ+b^2ξ)-μ^2]} ,ℬ_1 = T/12 πv^3_0(-3m^3_Z(v_0)-6m^3_W(v_0)+m^3_Z(v_0)ξ^3/2+2m^3_W(v_0)ξ^3/2
) ,ℬ_2 = T(-p^2 √(u_1^2+p^2 v^2)/6 π-q^2√(u_2^2+q^2 v^2)/6 π-λ√(3λv^2-μ^2)/4π-λ√(λv^2+a^2 ξv^2-μ^2)/12π. .
-a^2ξ√(λv^2+a^2ξv^2-μ^2)/12π-λ√(λv^2+b^2ξv^2-μ^2)/6π-b^2ξ√(λv^2+b^2 ξv^2-μ^2)/6π) ,
𝒞_1 = -Tu_1^2√(u_1^2+p^2 v^2)/6π-T u_2^2√(u_2^2+q^2 v^2)/6 π-T^2μ^2/6+3μ^4/32π^2 + T μ^2√(3λv^2-μ^2)/12π+T μ^2√(λv^2+a^2ξv^2-μ^2)/12π+Tμ^2 √(λv^2+b^2 ξv^2-μ^2)/6π + u_1^4ln[a_b T^2/p^2v_0^2]/32π^2+u_2^4 ln[a_b T^2/q^2 v_0^2]/32π^2
+μ^4 ln[a_bT^2/3v_0^2λ]/64π^2+μ^4 ln[a_bT^2/v_0^2(λ+a^2 ξ)]/64π^2 + μ^4 ln[a_b T^2/v_0^2 (λ+b^2ξ)]/32π^2 ,𝒞_2 = T^2u_1^2/12+3u_1^4/64π^2+T^2 u_2^2/12+3u_2^4/64π^2+δΩ ,
δΩ = -1/128π^2(-4p^2 u_1^2 v_0^2-4q^2u_2^2 v_0^2+3a^4v_0^4+6b^4v_0^4-12k^4v_0^4+2p^4v_0^4+2q^4 v_0^4. .
+12v_0^4λ^2+2a^2v_0^4λξ+4b^2v_0^4λξ+12v_0^2λμ^2+2a^2v_0^2ξμ^2+4b^2v_0^2ξμ^2. .
+4 u_1^4ln[u_1^2+p^2v_0^2/p^2v_0^2]+4u_2^4 ln[u_2^2+q^2v_0^2/q^2 v_0^2]+2μ^4 ln[3 v_0^2λ-μ^2/3v_0^2λ]. .
+2μ^4 ln[v_0^2λ+a^2v_0^2ξ-μ^2/v_0^2(λ+a^2ξ)]+4μ^4ln[v_0^2λ+b^2 v_0^2 ξ-μ^2/v_0^2 (λ+b^2 ξ)]-16π^2v_0^2m^2_H_0).
The potential (<ref>) is not a quartic equation because $\mathcal{B}_2, \mathcal{D}_3, \mathcal{D}_4$ and $f(T,u_1,u_2,\mu,\xi, v)$ depend on $v$, $\xi$ and $T$. The above effective potential has seven variables ($u_1, u_2, p, q, \mu, \lambda, \xi$). If Goldstone bosons are neglected and $\xi=0$, the effective potential will be reduced to those in the Landau gauge. So, the efficiency is distorted by $u_1, u_2, p,q, \xi$ but not so much and the electroweak phase transition can be slightly altered in the case of $\xi=0$. Because (<ref>) is, in form, similar to (<ref>), the minimum conditions for (<ref>) are still like (<ref>) with just a condition: $m_{H_0}=-\mu^2+3\lambda v^2_0=125 \text{ GeV}$.
Our problem is that there are many variables and some of them, for example, $u_1, u_2, p, q, \mu$ play the same role. However, $\xi$ and $ \lambda$ are two important variables and have different roles. Therefore, in order to reduce a number of variables, we simplify the latter, meanwhile, do not lose the generality of the problem.
We can see that $\mu$ reduces the strength of phase transitions. Because components containing $\mu$ in the effective potential (<ref>) (i.e., the term ($-\mu^2$) in $\Lambda, \mathcal{B}_2$) are always associated with the negative sign, so $\mu$ gives negative contributions. As a consequence, we will ignore $\mu$ or fix $\mu \ll \lambda v^2$.
In $\mathcal{D}_2, \mathcal{B}_2$ factors, $u_1, u_2$ are the components of the Goldstone and gauge bosons. Therefore they make the contribution of these particles to be depended on $v$, $u_1$ and $u_2$, so $u_1$ and $u_2$ alway follow $v$. However the effective potential is a function of $v$, thus these two values ($u_1, u_2$) also distort the shape of phase transitions. Hence, we ignore $u_1$, $u_2$.
More importantly we want to find the main contribution of the $\xi$ gauge to the phase transition. However $u_1, u_2, p, q,\mu$ are in the mass composition of the particles. As we pointed out in the previous section, when $\xi=0$, heavy particles which were sufficiently stocked, make the strength of phase transition greater than one in this model. So we look for the simplest approximation to investigate the role of $\xi$, i.e., we retain $\xi$ and simplify the role of $u_1, u_2, \mu$.
Furthermore $\xi$ and $u_1, u_2,\mu$ are not mutually exclusive, except for the $\mathcal{D}_3$ factor which is a factor of $v^2$, containing $T$. Thus the strength of phase transition depends directly on $\mathcal{B}$ and $\Lambda$. As a result we want to estimate a contribution of the $\xi$ gauge and we simplify $\mathcal{D}_3$ being a compound function of $T, \mu, u_1, u_2$ by ignoring the action of $u_1, u_2, \mu$.
Let us summarize our arguments. Firstly, ignoring $u_1$, $u_2$, $\mu$, we obtain $f(T,u_1,u_2,\mu,\xi,v)=0$, $\mathcal{D}_3=\mathcal{D}_4=0$ and $\delta\Omega\approx const$.
Secondly, we can approximate $\mu \ll \lambda v^2$, $u^2_1 \ll p^2v^2$ and $u^2_2\ll q^2v^2$ in $\mathcal{B}_2$ actually, this case is quite similar to ignoring $u_1$, $u_2$, $\mu$) and rewrite $\mathcal{B}_2$ as follows:
ℬ_2≈ℬ_3 = T v(-p^3/6π-q^3/6π-λ√(3λ)/4π-λ√(λ+a^2ξ)/12π. .
-a^2ξ√(λ+a^2ξ)/12π-λ√(λ+b^2ξ)/6π-b^2ξ√(λ+b^2ξ)/6π) = T/12v^3_0πv(-2(m^2_h^±(v_0)-u^2_1)^3/2-2(m^2_k^±±(v_0)-u^2_2)^3/2. .
-3v^3_0λ√(3λ)-v^2_0λ√(v^2_0λ+m^2_Z(v_0)ξ)-2v^2_0λ√(v^2_0λ+m^2_W(v_0)ξ). .
-m^2_Z(v_0)ξ√(v^2_0λ+m^2_Z(v_0)ξ)-2m^2_W(v_0)ξ√(v^2_0λ+m^2_W(v_0)ξ)).Finally, we can rewrite (<ref>) as follows:
where $\mathcal{B}=\mathcal{B}_1+\mathcal{B}_3$. If we ignore Goldstone boson and fix $\xi=0$, Eq. (<ref>) will return to Eq.(<ref>). We do not write $\delta\Omega$ in (<ref>) because it is a constant.
The potential (<ref>) is a quartic equation that likes (<ref>). So the form of the critical temperature and strength EWPT also are like (<ref>) and (<ref>). As well as the comments in the previous section, two charged Higgs bosons have played the same role in the effective potential. In order to ease calculating, we accept that $m_{h^\pm}=m_{k^{\pm\pm}}$. Therefore, the effective potential only depends on $\lambda$, $\xi$ and $m_{h^\pm}$.
In figures from Fig.<ref> to Fig.<ref>, we have plotted $S$ as a function of $m_{h^\pm}, \xi$ with $\lambda=0.1, 02, 035, 0.5$.
$\lambda=0.1$ when the solid contour of $S=1$, the
dashed contour: $S=1.5$, the dotted contour: $S=2$,
the dotted-dashed contour: $S=4.15$, even and no-smooth contours: $S\longrightarrow \infty$
$\lambda=0.2$ when the solid contour of $S=1$, the
dashed contour: $S=1.5$, the dotted contour: $S=2$,
the dotted-dashed contour: $S=4.15$, even and no-smooth contours: $S\longrightarrow \infty$
$\lambda=0.35$ when the solid contour of $S=1$, the
dashed contour: $S=1.5$, the dotted contour: $S=2$,
the dotted-dashed contour: $S=4.15$, even and no-smooth contours: $S\longrightarrow \infty$
$\lambda=0.5$ when the solid contour of $S=1$, the
dashed contour: $S=1.5$, the dotted contour: $S=2$,
the dotted-dashed contour: $S=4.1$, even and no-smooth contours: $S\longrightarrow \infty$
According to Figures <ref>-<ref>, if $m_{h^\pm}=m_{k^{\pm\pm}}$ is smaller than $300$ GeV, we have the first order phase transition. We also found that the larger $\Lambda$ be, the smaller maximum strength of EWPT will be. In this case, the strength of the EWPT is in range, $1<S<4.15$.
The dependence of $\mathcal{B}$ on $\xi$
The dependence of $\Lambda$ on $\xi$
From Fig. <ref> and <ref>, if $\xi$ increases, $\mathcal{E}$ increases and $\lambda$ decreases, this means that the larger $\xi$ be, the stronger strength of EWPT will be. The contributions from new particles ($h^{\pm}$, $k^{\pm\pm}$) make of the first order phase transition that the Standard Model cannot. In addition, we found that $\xi$ will not lose the first order phase transition as mentioned in Ref. <cit.>. From Fig. <ref>, we do not find the value of $\xi$ that makes $\mathcal{B}=0$. Moreover, ignoring $\xi$ we still search a first order EWPT in the previous section.
§ SPHALERON RATE IN THE ZEE-BABU MODEL
When temperature drops below $T_1$, the effective potential appears a non-zero VEV. The transition from a zero vacuum to a non-zero vacuum through two ways. The first way which cross-over a barrier, was called sphaleron. The second way is quantum tunneling, was called instanton.
The $B$ violation can be seen through lepton number violation. Lepton violation can be read through Chern-Simon number or Winding number <cit.>. But we obtain the kinetic processes of Higgs field in EWPT which can be described by the transition rate between two VEVs. This rate is sphaleron rate. So sphaleron rate or Chern-Simon number is different from zero, driving $\Delta B$ will not be zero <cit.>.
The sphaleron rate is also important for determining whether the baryon asymmetry produced at the bubble exterior, then diffusing into the bubble interior or not.
The criteria $S>1$ is a very approximate condition, subject to considerable theoretical uncertainties. We see that $S >1$ is required for a first order phase transition but one can certainly have a first order phase transition for $S<1$. However the sphaleron rate may be too large to preserve the baryon asymmetry in this case. Therefore we state that $S >1$ is required for a first order phase transition and the sphaleron rate must be satisfied the sphaleron decoupling.
In addition, the sphaleron decoupling condition must be imposed at the temperature $T_E$ lower than $T_C$ where the EWPT terminates. In practice, since it is difficult to determine this temperature, we can substitute $T_N$, a nucleation temperature of the critical bubble, in place of $T_E$. The critical bubble is defined as the bubble whose surface energy and volume energy becomes balanced. Only such bubbles can nucleate and expand in the symmetric phase.
From above analysis we see that the heavy particles (with masses larger or equal mass of the $W^{\pm}$ boson) give main contribution to the sphaleron rate. Therefore, to study the sphaleron processes in the ZB model, we begin from the Lagrangian of the gauge-Higgs system
ℒ_gauge-Higgs =-1/4F^a_μνF^aμν
+( 𝒟_μϕ)
^†( 𝒟^μϕ)-V(ϕ).
From Eq.(<ref>), the energy functional being the sum of the kinetic and potential constituents in the temporal gauge, will be the following form
ℰ = ∫d^3x
[( 𝒟_μϕ)
^†( 𝒟^μϕ)+V(ϕ)],
here we assume that the least energy has the pure-gauge configurations. Thus $F^a_{ij}=0$.
Assuming that the EWPT processes occur in the form of nucleation bubbles, and using the temperature expansion of the effective potential at one loop given in the section <ref>, we can rewrite the energy functional in the spherical coordinate system as follows
ℰ = 4π∫^∞_0d^3x[1/2(∇^2 v)^2
Using the static field approximation, i.e., VEV variable do not change in time, as follows
we obtain
ℰ=∫d^3x [1/2(∂_i v)^2+
From the Lagrangian (<ref>), it follows the equation of motion for the VEV $v_\rho$:
v̈+∇^2 v-∂V_eff(v,T)/∂v=0.
When VEV does not change in time according to the condition (<ref>), we can rewrite Eq. (<ref>) in spherical coordinates
Lastly, from Eq. (<ref>) and (<ref>), the sphaleron energy in the EWPT process, $\mathcal{E}_{sph.su(2)}$, has the following form
Let us consider the electroweak phase transition in the form of the formation of bubble nucleation. In order to calculate the energy, we must solve the equation of motion (<ref>) for the VEV of the Higgs field and obtain $v(r)$. Eq. (<ref>) does not have an exact solution because the factor $\frac{2}{r}\frac{dv}{dr}$ oscillates very strongly at $r\longrightarrow 0$. Therefore, we should use approximations presented in the next subsections. The sphaleron rate per unit time, per unit volume, $\Gamma/V$, is characterized by a Boltzmann factor, $\exp\left(-\mathcal{E}/T\right)$, as follows <cit.>:
Γ/V = κ_brokα^4 T^4 exp(-ℰ/T),
where $V$ is the volume of universe, $T$ is the temperature, $\mathcal{E}$ is the sphaleron energy, and $\alpha=1/30$. $\kappa_{brok}$ specifies the strength of EWPT. In this model, the strength is about 1, so $\kappa_{brok} \simeq 1$. We will compare the sphaleron rate with the Hubble constant describing the cosmological expansion rate at the temperature $T$ <cit.>
H^2=π^2 g T^4/90 M_pl^2,
where $g=106.75$, $M_{pl}=2.43 \times 10^{18}$ GeV.
In order to have B violation, the sphaleron rate must be larger than the Hubble rate at the temperatures above the critical temperature (otherwise, B violation will become negligible during the Universe's expansion); however, the sphaleron process must be decoupled after the EWPT to ensure the generated BAU is not washed out <cit.>.
§.§ An upper bound of the sphaleron rates
From the equation of motion (<ref>), it can be seen that the VEV of the Higgs field cannot be equal at every point in space. So we suppose that the VEV of the Higgs fields dose not change from point to point. Due to this supposition, we have $\frac{dv}{dr}=0$. Hence, from Eq. (<ref>) we obtain
Eq. (<ref>) shows that $v_\rho$ is the extremes of the effective potential. Therefore, the sphaleron energy (<ref>) can be rewritten as
ℰ_sph=4π∫V_eff(v,T) r^2dr=4πr^3/3V_eff(v_ρ,T) |_v_m,
where $v_{m}$ is the VEV at the maximum of the effective potentials. From Eq. (<ref>), the sphaleron energy is equal to the maximum heights of the potential barriers.
The Universe's volume at a temperature $T$, after the inflation and re-heating epoch, is given by $V=\frac{4\pi r^3}{3}\sim \frac{1}{T^3}$. Because the whole Universe is an identically thermal bath, the sphaleron energies are approximately
ℰ_sph∼E^4 T/4λ^3_T.
From the definition (<ref>), the sphaleron rates take the form
Γ∼α_w^4 Texp(-E^4T/4λ_T^3T).
For the heavy particles, $E, \lambda$ are constants. Hence, the sphaleron rate in this approximation is the linear function of temperature. From Eq. (<ref>), we estimate the value of the sphaleron rates as follows
Γ∼10^-4 ≫H ∼10^-13.
This value is very large, so we assume it as an upper bound of the sphaleron rate. Therefore, sphaleron decoupling condition cannot be satisfied. For instance, as the temperature drops below the phase-transition temperature $T_c$ and the Universe switches to the symmetry-breaking phase, the sphaleron rate is still much larger than the Hubble constant, and this makes the B violation washed out. By this consequence, the sphaleron process cannot occur identically in large regions of space; it can only take place in the microscopic regions or at each point in space.
§.§ Sphaleron rate in a thin-wall approximation
At every point in the early Universe, the effective potential varies as a function of VEV of the Higgs field and temperature, as illustrated in Fig. <ref>. If the temperature at a spatial location is higher than $T_1$, then $V_{eff}(v)$ at this location has only one the zero minimum, and this location is in a symmetric phase region.
As the temperature goes below $T_1$, the second minimum of $V_{eff}(v)$ gradually forms, and a potential barrier which separates two minima gradually appears. The VEV can be transformed by thermal fluctuations. The phase transition occurs microscopically, resulting in a tiny bubble of broken phase where the Higgs field $v$ gets a nonzero vacuum expectation value.
As the temperature goes to $T_c$, $V_{eff}(v)$ at two minimums are equal
each other. But when the temperature goes below $T_c$, the second minimum becomes the lower one corresponding to a true vacuum, while the first minimum becomes the false vacuum. Such tiny true-vacuum bubbles at various locations in the Universe can occur randomly and expand in the midst of false vacuum. If the sphaleron rate is larger than the Universe's expansion rate, the bubbles can collide and merge until the true vacuum fills all space.
However, if the sphaleron decoupling condition is satisfied after the transition, the sphaleron rate must be smaller than the cosmological expansion rate when the temperature goes from $T_N$ ($T_N$ lower than $T_C$, at which the EWPT terminates, $T_C<T_N<T_0$).
The dependence of the effective potential on temperature, $m_{h^\pm}=m_{k^{\pm\pm}}=220 \textrm{GeV}$, $\lambda=0.35$, $\xi=2$
The broken phase of the EWPT can be started at $T_1\approx 120.5$ GeV. At $T_1$, the sphaleron rate can be larger than the Hubble rate ($H=2.1231 \times 10^{-14}$ GeV). As the temperature drops below $T_1$, the EWPT really starts at $T_C$, the sphaleron rate is still larger than the Hubble rate and this lasts until the temperature reaches the nucleation temperature $T_N$ when the transition ends. As the temperature goes down $T_N$, the sphaleron rate is smaller than the Hubble rate.
In a bubble of the EWPT, we have the following approximation
∂V_eff(v)/∂v≈ΔV_eff(v)/Δv=const ≡M,
here $\Delta v=v_{c}$, $\Delta V_{eff}(v)= V_{eff}(v_{c})-V_{eff}(0)$, and $v_{c}$ is a second minimum of the effective potential for the phase transition.
Now, we solve the equations of motion (<ref>) for the VEV $v$ by the approximation (<ref>). Rewriting Eq. (<ref>) in this approximation, we have
d^2v/dr^2+2/rdv/dr= M.
In the cases that $r \to \infty$ (the spatial locations are in the symmetric phase) or $r \to 0$ (the spatial locations are in the broken phase), the VEVs must satisfy the boundary conditions
lim_r→∞v(r)=0; dv(r)/dr|_r=0=0.
In the bubble walls, the solutions of Eq. (<ref>) take the form
where $A, B$ are the parameters to be specified.
The continuity of the scalar fields in the bubble results in the following system of equations:
where $R_{b}$ and $\Delta l$ are, respectively, the radius and the wall thickness of a bubble nucleated.
Solving the systems of Eq. (<ref>), we obtain the solutions $v$, which are of the forms
v_c; when r ≤R_b,
M/6r^2-A/r+B; when R_b<r ≤R_b.su(2)+Δl ,
0; when R_b+Δl<r.
The system (<ref>) has four equations, so we have to specify four unknown parameters [$A$, $B$, $\Delta l$, and $R_{b}$]. Therefore the decoupling condition in which the sphaleron rate is equal to the Hubble rate at $T_N$, has been used. This supposition relies on the requirement for avoiding the washout of the generated BAU after a phase transition, by which the sphaleron rate must be larger than the Hubble rate at temperatures above $T_N$, but the sphaleron rate must be smaller than the Hubble rate at temperatures below $T_N$.
The masses of heavy particles $(h^{\pm}, k^{\pm\pm})$ are unknown so far. However, we can estimate their mass regions which satisfy the first-order phase transition conditions, and we choice any values in these regions for calculating the sphaleron energy. Although the strengths of the first-order EWPT are sufficiently strong ($>1$), they are not so strong ($<4.15$), hence the coefficients $(\Lambda, \mathcal{B}, \mathcal{D})$ in the effective potential are not meaningfully different for the different values in these regions. Here, as an example, we choice $m_{h^\pm}=m_{k^{\pm\pm}}=220\,\textrm{GeV}$ (our choices are random), to illustrate the determination of the radius of nucleation.
In Fig. <ref>, our respective solution $v(r)$ is not as smooth as those in Refs. <cit.>. Because in this work, the bubble walls were been considered very thin, ie., $\Delta l \ll 1/T$ (while in Ref. <cit.>, for instance, $\Delta l \gg 1/T$). Inside the thin walls of bubbles, $\frac{dv_\rho}{dr}$ is very large; this allows the Higgs field $\phi$ to change their values over potential barriers. Therefore, the thinner the bubble walls, the larger the sphaleron rates.
The solutions $v(r)$ with $m_{h^\pm}=m_{k^{\pm\pm}}=220 \, \textrm{GeV}$, $\lambda=0.35$, $\xi=2$. The regions in grey portray the thin walls of vacuum bubbles nucleated in each phase transition.
In order to estimate $R_{b}$ at $T_N$ (the radius of nucleation), we accept that $\Delta l=\frac{R_{b}}{10}$. In addition, it is difficult to determine $T_N$, we only know that $T_0<T_N<T_C$. Therefore, we can only estimate the upper and lower bounds of the radius of nucleation. They are estimated in Table <ref>.
Sphaleron rate with $m_{h^\pm}=m_{k^{\pm\pm}}=220\, \textrm{GeV}, \lambda=0.35, \xi=2$
T[GeV] $R[10^{-4}\times\, \textrm{GeV}^{-1}]$ $R/\Delta {l}$ $\mathcal{\varepsilon} [\textrm{GeV]}$ $\Gamma [10^{-14}\times\, \textrm{GeV}]$ $H [10^{-14}\times \, \textrm{GeV}]$ $T_N$ Bound
$116.964$ $(T_C)$ $8.82948$ $10$ $2659.49$ $2$ $2$ $T_N=T_C$ Upper
$96.25$ $(T_0)$ $4.77064$ $10$ $2207.27$ $1.3$ $1.3$ $T_N=T_0$ Lower
From Table <ref>, in this case, $m_{h^\pm}=m_{k^{\pm\pm}}=220 \textrm{GeV}$, $\lambda=0.35$, $\xi=2$, the largest and smallest radius are $8.82948 \times 10^{-4}\times\, \textrm{GeV}^{-1}$ and $4.77064 \times 10^{-4}\times\, \textrm{GeV}^{-1}$ respectively.
§ CONSTRAINTS ON COUPLING CONSTANTS IN THE HIGGS POTENTIAL
In order to have the first order phase transition, $m_{h^{\pm}}$ and $m_{k^{\pm\pm}}$ must be smaller than $350\, \mathrm{GeV}$. Therefore, we obtain
p^2v^2_0<(300 GeV)^2,
q^2v^2_0<(300 GeV)^2 .
From the above equations, we obtain $0<p<1.22$ and $0<q<1.22$. However, we need to have other considerations in order to find these accurate values of $m_{h^\pm}$ and $m_{k^{\pm\pm}}$.
In the ZB model, the tiny masses of neutrino are generated at two loops, so $m_{h^\pm}$ and $m_{k^{\pm\pm}}$ cannot be very heavy <cit.>. From the experimental point of view it is interesting to consider new scalars light enough to be produced at the LHC, theoretical arguments introduce that the scalar masses should be a few TeVs, to avoid unnaturally large one-loop corrections to the Higgs mass which would introduce a hierarchy problem. Therefore, these upper bounds of new scalar masses can be 2 TeVs <cit.>. Contacting to neutrino oscillation data, in the decay $k^{\pm\pm}\longrightarrow ll$, the branching ratio to $\tau\tau$ is very small in the ZB model, less than about $1\%$. Then, a conservative limit is $m_{k{\pm\pm}}>200$ GeV. In the ZB model, we can have the decay $k^{\pm\pm}\longrightarrow h^{\pm}h^{\pm}$, so $2m_{h^\pm}<m_{k^{\pm\pm}}$. Therefore, our results in Eqs. (<ref>), (<ref>) are consistent with the above estimates.
Recently, the experimental groups at LHC (ATLAS and CMS Collaborations) <cit.> have reported an experimental anomaly in diboson production with apparent excess in boosted jets of the $W^+ W^-, W^\pm Z$ and $ZZ$ channels at around 2 TeV invariant mass of the boson pair.
In addition the calculation the Higgs coupling to photons (due to charged particles in the loop diagram) can be related to neutrino mass and CP violation which are the key of matter and antimatter asymmetry. This study will be investigated in a future publication.
§ CONCLUSION AND OUTLOOKS
In this paper we have investigated the EWPT and sphaleron rate in the ZB model using the high-temperature effective potential. The EWPT is strengthened by the new scalars to be the strongly first-order, the phase transition strength is in the range $1 - 4.15$. By using the sphaleron decoupling condition, we also propose a way to estimate the radius of the nucleation. $h^{\pm}, k^{\pm\pm}$ are triggers for the first-order EWPT. Our results may be further than the results in Ref. <cit.>.
In addition, the EWPT can be calculated in a different way as in <cit.>. In order to determine $T_N$ or $T_E$, we will examine this problem in conjunction with the CP-violation.
In the ZB model, the tiny mass of neutrino which can be explain in two loops interactions of charged Higgs with neutrino, can be a reason of the matter-antimatter asymmetry and CP-violation. The behavior of charged Higgs is also very interested. Therefore, in the next works, we can investigate the ratio $m'_{h^{\pm}/k^{\pm\pm}}$ by using neutrino data. We will investigate the CP- violation and beyond issues of the baryon asymmetry problem through neutrino physics.
With this region of self couplings in the Higgs potential, we can serve as basis for the calculation of cross section of the decay Higgs to photons and evaluate and extend the Zee-Babu model when connected to the data of LHC.
§ ACKNOWLEDGMENTS
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2017.22.
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1511.00258
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Raman Research Institute, C. V. Raman Avenue, Sadashivanagar,
Bangalore 560065, India
Many powerful radio quasars are associated with large scale jets, exhibiting bright knots as shown by high resolution images from the Hubble Space Telescope and Chandra X-ray Observatory. The radio–optical flux component from these jets can be attributed to synchrotron radiation by accelerated relativistic electrons while the IC/CMB model, by far has been the most popular explanation for the observed X-ray emission from these jets.
Recently, the IC/CMB X-ray mechanism has been strongly disfavoured for 3C 273 and PKS 0637-752 since the anomalously hard and steady gamma-ray emission predicted by such models violates the observational results from Fermi-LAT.
Here we propose the proton synchrotron origin of the X ray–gamma ray flux from the knots of PKS 0637-752 with a reasonable budget in luminosity, by considering synchrotron radiation from an accelerated proton population. Moreover, for the source 3C 273, the optical data points near 10$^{15}$ Hz could not be fitted using electron synchrotron (Meyer et al. 2015). We propose an updated proton synchrotron model, including the optical data from Hubble Space Telescope, to explain the common origin of optical–X-ray–gamma-ray emission from the knots of quasar 3C 273 as an extension of the work done by Kundu $\&$ Gupta (2014).
We also show that TeV emission from large scale quasar jets in principle, can arise from proton synchrotron, which we discuss in the context of knot wk8.9 of PKS 0637-752.
§ INTRODUCTION
Quasars are a class of active galaxies, which are highly energetic and powered by accretion of mass around central supermassive
black-holes. Jets of FRII radio galaxies and quasars often exhibit regions of extreme brightness or knots, as have been
observed by radio telescopes
since a long time back. But more recently, images
from Hubble Space Telescope (HST) and Chandra X-Ray Observatory have shown that significant amount of high energy
radiation is also produced from the bright knots present in these jets. The first discovery of high-energy X-ray emission from
the kpc-scale relativistic jet of quasar PKS 0637-752 by the Chandra Observatory (Chartas et al. (2000)) has led to many similar
significant discoveries later (Harris & Krawczynski (2006)).
PKS 0637-752, located at redshift z = 0.651 (Savage,Browne and Bolton (1976)) was the first X-ray target of the Chandra Observatory
(Weisskopf et al. (2000); Schwartz et al. (2000); Chartas et al. (2000)) which accidentally discovered a 100 kpc one-sided jet from
the source coincident with the radio jet reported by Tingay et al. (1998). The strong X-ray flux observed from this jet is
difficult to explain via standard mechanisms such as Synchrotron Self Compton (SSC) or thermal Bremsstrahlung (Schwartz et al.
2000). Since the launch of Chandra in 1999, many tens of such quasar jets luminous in X-rays have been detected. The periodic structure in knots in the megaparsec scale jet of PKS0637-752 has been observed by the Australian
Telescope (Godfrey et al. (2012)).
3C 273, located at redshift z=0.158 (Strauss et al.(1992)) is the brightest and
most studied AGN and is accompanied by a large scale jet of projected length 57 kpc (Harris $\&$ Krawczynski (2006)).
Analysis has been performed on its broadband energy spectrum by various authors (Jester et al. (2001), (2005); Uchiyama et al. (2006); Meyer $\&$ Georganopulos (2014)). Soldi et al. (2008) have analyzed long-term multiwavelength data to study its temporal variability properties.
Sambruna et al. (2001) and Uchiyama et al. (2006) have found two distinct components in the jet emission of 3C 273.
The radio emission from the kpc scale jet has been explained with synchrotron radiation from shock-accelerated relativistic electrons by Marscher $\&$ Gear (1985) and Türler et al. (2000).
The radio to optical spectral energy distribution from jets of both 3C 273 and PKS 0637-752, can be well explained by synchrotron radiation of
relativistic electrons present in the jets (Samburna et al. (2001)). However in several jets, the X-ray emission
from the knots is much higher and/or harder than expected from the radio-optical synchrotron spectrum as explained by Schwartz et al. (2000) for quasar PKS 0637-752. Synchrotron self-Compton (SSC) mechanism could not explain the observed
X-radiation in both the sources, as explained by Chartas et al. (2000) for PKS 0637-752 and Samburna et al. (2001) for 3C 273.
Hence it was suggested by Tavecchio et al. (2000) and Celotti et al. (2001),
that the X-ray emission from PKS 0637-752 can arise due to inverse Compton scattering of the cosmic microwave background photons by shock-accelerated relativistic electrons in the jet (IC/CMB). The IC/CMB model has also been applied to explain X-ray radiation from Knot A of 3C 273 (Samburna et al. (2004)).
IC/CMB mechanism by far has been the most popular explanation for X-radiation in quasar jets. Samburna et al. (2004) explained
the X-ray emission from most of the radio jets included in their survey, by IC/CMB. IC/CMB model predictions was verified for
the source PKS 1150+49 by follow-up observations by Samburna et al. (2006).
However, a number of problems have been noticed with the IC/CMB X-ray model.
It requires the jet to remain highly relativistic (i.e. high Lorentz factor$\sim$10-20) even upto kpc scales and pointed at a very small angle to our line
of sight. But it was predicted by Arshkian $\&$ Longair (2004) that the jets decelerate and can only remain mildly relativistic when they reach kpc scales, although this lacks direct experimental support.
Moreover, the small angle subtended by the jet to our line of sight sometimes gives rise to deprojected jet length of Mpc size. Also IC/CMB models require huge jet kinetic power, sometimes exceeding the Eddington limit for the source (Uchiyama et al. (2006)). As an alternative to IC/CMB, it was proposed that X-radiation
from large scale quasar jets can also be explained by synchrotron radiation from a second shock accelerated electron population, different from the one giving rise to the radio-optical spectra (Jester et al. (2006), Uchiyama et al. (2006)).
Although this second synchrotron model overcomes the problem of super-Eddington power requirements, high Lotentz factor and Mpc scale jet length of the IC/CMB to explain the observed X-Ray data, the problem lies in its unexplained co-spatial existence with the first high energy electron population (Schwartz et al. (2000)).
Recently it has been shown by Meyer et al. (2014 and 2015) using observational results from Fermi-LAT, that IC/CMB incorrectly predicts
the gamma-ray flux at GeV energies. They have shown using long term Fermi monitoring data that the hard and steady gamma-ray
emission implied by the IC/CMB X-ray models, overproduces the GeV flux thus violating observational results from Fermi
for both quasar jets PKS 0637-752 and 3C 273. Thus IC/CMB is ruled out as possible X-ray emission mechanism in both of our target
The implication of explaining X-ray emission from the knots with electron synchrotron also has been discussed by Meyer et al. (2015). The shock accelerated electrons emitting X-rays in synchrotron emission would also give GeV-TeV gamma rays by inverse Compton scattering off the CMB photons. The luminosity expected in TeV gamma rays is very high in this case.
Aharonian (2002) proposed synchrotron radiation by a shock-accelerated proton population which explained the radio to X-ray spectrum from knot A of 3C 273. Also the jet emission from PKS 0637-752 in optical to X ray spectrum was well explained by the proton synchrotron model in this paper. A broken power law spectrum of accelerated protons having energy upto $10^{20}$ eV was used, where the spectral indices were determined by three important time scales the synchrotron loss and escape time scales of the protons and the age of the jet. The protons lose energy very slowly in the magnetic field of order milli Gauss in the jet as a result they can diffuse through the length of the kpc scale jet.
We have used this proton synchrotron model (Aharonian 2002) to propose the possible common origin of the high energy photons in kpc scale jets of quasars PKS 0637-752 and 3C 273. For the source PKS 0637-752 we have modeled the X-ray–gamma-ray flux by proton synchrotron. Kundu $\&$ Gupta (2014) demonstrated the possible proton synchrotron origin of X-ray and gamma-ray emission from the large scale jet of 3C 273. In the recent work of Meyer et al. (2015), the optical HST data near 10$^{15}$ Hz for the source 3C 273 has not been included in the radio-optical synchrotron fit, which we have included in our updated proton
synchrotron model proposing a common origin of optical, X-ray and gamma-ray photons. We have also discussed about the possibility of TeV photon emission from large scale quasar jets within the proton synchrotron model.
For the knot wk8.9 of PKS 0637-752, we have shown that proton synchrotron mechanism can in principle, give rise to a TeV flux within a reasonable budget in luminosity if protons are accelerated to energy close to 10$^{21}$ eV.
§ THE PROTON SYNCHROTRON MODEL
§.§ High Energy Spectral Energy Distribution
We describe the formalism used in our work (earlier discussed in Aharonian 2002, Kundu & Gupta 2014). The shock accelerated protons are diffusing through the large scale jets of the quasars and losing energy due to synchrotron emission and diffusion. We have calculated the Doppler factors ($\delta_D$) of the jets assuming their Lorentz factor to be $\Gamma=3$ to fit the observational data.
Within a spherical blob of size $R$ and magnetic field $B$ the relativistic protons
are trapped. Their escape time scale is
\begin{equation}
t_{\rm esc}\simeq 4.2\times 10^5 \eta^{-1} B_{\rm mG} R^{2}_{\rm kpc} (E/10^{19}\rm eV)^{-1} {\rm yr}.
\label{bohm_esc}
\end{equation}
In Bohm diffusion limit the gyrofactor $\eta=1$.
Another expression for the escape time which is energy dependent reduces the energy budget (Aharonian 2002)
\begin{equation}
t_{\rm{esc}}=\frac{1.4\times 10^7}{(E/10^{14}\rm {eV})^{0.5}} \rm {yr}.
\label{time_esc}
\end{equation}
The synchrotron energy loss time scale of the relativistic protons in the
jet is
\begin{equation}
t_{\rm{synch}}\simeq 1.4\times 10^{7} B^{-2}_{\rm{mG}} (E/10^{19} \rm eV)^{-1} \rm{yr}.
\label{synch_time}
\end{equation}
We have considered the broken power law spectrum of the shock accelerated
relativistic protons
\begin{equation}
\frac{dN_{p}(E_p)}{dE_p}
=A \left\{ \begin{array}{l@{\quad \quad}l}
{E_p}^{-p_1} &
{E_p}^{-p_2} & E_p>E_{p,br}.
\end{array}\right.
\end{equation}
We compare the synchrotron loss and escape time scales of the protons in the jet whose age is assumed to be 3$\times 10^{8}$ years. When the synchrotron loss time scale is shorter than the escape or diffusion time scale given in eqn (<ref>) and the age of the jet, synchrotron loss becomes important. As a result the spectrum of high energy protons steepens by $E_p^{-1}$ above the break energy $E_{p,br}$ (our model 1).
For knot wk8.9 of PKS 0637-752 at $E_{p,br}$=10$^{16}$ eV, the synchrotron loss time scale is $t_{sync}$ = 1.4$\times10^{8}$
yrs, which is smaller than the age of the jet and the escape time ($t_{esc}$=7.56$\times10^{9}$ yrs); similar is the case
in the combined knot scenerio, thus increasing the spectral index by 1. For knot A and knots A+B1 combined of 3C 273, at $E_{p,br}$=10$^{16}$ eV (and 5.62$\times10^{15}$ eV),
$t_{sync}$ = 1.4$\times10^{8}$ yrs (and 2.49$\times10^{8}$ yrs) whereas $t_{esc}$ = 1.52$\times10^{10}$ yrs and 8.14$\times10^{10}$ yrs
respectively. We have also considered another scenario where the escape time scale given in eqn(<ref>) becomes shorter than the synchrotron time scale and the age of the jet for very high energy protons. This results in a steeper spectrum by a factor of $E_p^{-0.5}$ above the break energy $E_{p,br}$ (our model 2).
For PKS 0637-752 both for the single knot and combined knot scenarios, at $E_{p,br}$=10$^{12}$ eV, the escape time is
1.4$\times10^{8}$ yrs which thus becomes dominant over the synchrotron loss ($t_{sync}\sim10^{12}$ yrs) and age of the jet.
Similar is the case for knot A and the combined knots of 3C 273, where also according to our model 2, the escape loss becomes
more important compared to other time scales.
The high energy photon spectrum from knots of PKS 063-752 and 3C 273 are compared with the theoretical predictions
of our model 1 and model 2 (see Fig.1. to Fig.4.). The values of the parameters used in our flux calculations are given
in Table 1 and 2.
§.§ Proton Synchrotron Origin of TeV Gamma-rays
In this section we show that proton synchrotron radiation can in principle, give rise to TeV gamma-rays from extended quasar jets within a reasonable budget in photon luminosity. TeV blazars can exhibit luminosity beyond $10^{42}$ erg/sec (Abramowski et al. (2014)).
We discuss the implications of our proton synchrotron model in the context of TeV emission, for knot wk8.9 of PKS 0637-752. The expressions for escape and synchrotron time scales used in our models are given in eqn (<ref>) and eqn (<ref>). For the range of parameters considered, the synchrotron loss becomes shorter than escape loss and jet age resulting in a steeper proton spectrum by $E_{p}^{-1}$ above the break energy. We propose that proton synchrotron radiation can give rise to TeV emission from knot wk8.9, if proton acceleration to energies near $10^{21}$ eV is possible.
Ebisuzaki $\&$ Tajima (2013) have discussed that protons/nuclei in AGN jets can be accelerated to beyond $10^{21}$ eV by plasma wakefield field formed by intense electromagnetic field.
Our parameters estimates are listed in Table 3 and the model fits can be found in Fig.5. The models have been constructed under the assumption of equipartition in energy density of the particles and magnetic field.
In our calculations we have shown the intrinsic source spectrum, not the observed spectrum. However, due to severe absorption of the TeV gamma-rays by the extragalactic background light (EBL), direct observation of the TeV spectra would be difficult. Our photon luminosity budget near 1 TeV, for the four models considered $\sim10^{41}-10^{43}$ erg/sec, which is reasonable for TeV blazars, thus implying the validity of our proposition.
§ RESULTS AND CONCLUSION
The radio to optical data from the single knot wk8.9 and the knots wk7.8, wk8.9,wk9.7,wk10.6 of PKS 0637-752, the single knot A and the combined knots A, B1 of 3C 273 are fitted by the synchrotron emission of shock accelerated electrons by Meyer et al. 2015. We have fitted the higher energy photon data (X-ray data and Fermi LAT upper limits) from these knots with proton synchrotron mechanism. Moreover,
in the case of 3C 273 the optical data at $10^{15}$ Hz which cannot be fitted by electron synchrotron emission (Meyer et al. 2015) has been included within our updated proton synchrotron models.
The existence of very high energy protons ($\sim 10^{20}$ eV) in the kpc-scale knots, is the basic assumption of this model.
In the work of Aharonian (2002), it was proposed that proton-synchrotron can give rise to radio to X-ray flux from extended quasar jets. In this work it was assumed that during the jet lifetime, protons with a time independent energy spectrum are injected (quasi) continuously into a spherically symmetric blob. Aharonian considered three models to explain the observed spectral energy distribution, each of which reduces the energy budget compared to the previous one.
For Knot A of 3C 273 with jet lifetime 3$\times10^7$ yrs, the first model uses a broken power-law spectrum of protons with spectral indices 2.4 and 3.4 below and above the break and a magnetic field B = 5 mG. This model fits the radio to X-ray data with luminosity in magnetic field $L_B$= 1.33$\times10^{44}$ erg/s and that in protons $L_p$= 1.2$\times10^{47}$ erg/s, implying a large deviation from equipartition. To reduce the energy budget Aharonian's second model considers magnetic field B = 10mG, initial proton spectral index p1 = 2, an energy-dependent escape time scale which becomes dominant over synchrotron losses, thus resulting in spectral index p2 = 2.5 after break. This model also explains the radio to X-ray spectrum of 3C 273, but for reduced energy requirements ($L_B=1.1\times10^{45}$ erg/s; $L_p=1.1\times10^{44}$ erg/s).To further reduce the luminosity, in his third model Aharonian adopted a power-law spectrum with exponential cut-off at E = 10$^{18}$ eV, which fits only the X-ray data for a magnetic field value of 3 mG and spectral index 2. The luminosities in cosmic ray protons and magnetic field are $L_p$=10$^{45}$ erg/s and $L_B$=3.7$\times10^{44}$ erg/s respectively, which is less compared to the other two models.
For PKS 0637-752 instead of taking the individual knots, Aharonian considered that the overall X-ray emission is coming from a single source. The first model considers a broken power law with exponential cut-off at E = 10$^{20}$ eV and spectral indices 1.75 and 2.75 below and above break. It was assumed that particles propagate in relaxed-Bohm diffusion limit with magnetic field 1.5 mG and size of emitting region 5 kpc, where escape losses dominate over synchrotron losses. This model fits the optical to X-ray spectrum of PKS 0637-752 with a large proton acceleration power $L_p$=3$\times10^{46}$ erg/s. The second model adopted a higher value of magnetic field (B=3 mG), size was reduced to 3kpc and it was assumed that particle propagation takes place in the Bohm regime which results in dominance of the synchrotron loss time scale over escape time scale. In order to reduce the proton acceleration power by another order of magnitude, in his third model Aharonian uses an early exponential cut-off at E=2$\times10^{18}$ eV, which requires a proton power of 2.9$\times10^{45}$ erg/s.
In our work, we revisit the proton synchrotron models to explain the higher energy observations from knots of PKS 0637-752 and 3C 273, under the assumption that particles diffuse in the Bohm limit ($\eta$=1). We fit the X-ray to gamma-ray observational data from Knot wk8.9 and combined knots wk7.8, wk8.9, wk9.7, wk10.6 of PKS 0637-752 with parameter estimates according to Table 1.
The spectral index of protons has been varied in the range of 1.35-1.9 below the break in model 1 and 2 and luminosities required to explain the observed X-ray to gamma ray spectral energy distribution from the knots of PKS 0637-752 are $\sim 6\times10^{43}$ erg/s which is about 0.6$\%$ of Eddington luminosity of the source (see Table 1).
Our models also explain the optical to gamma-ray energy spectrum from Knots A and A+B1 of 3C 273 with luminosity $\sim5\times10^{43}$ erg/s in model 1 and 2 (0.5$\%$ of Eddington luminosity)(see Table 2). For this source in order to match the experimental observations, we consider an initial proton spectra in the range p1 = 1.57-2.03 which changes by 1 or 0.5 (according to model 1 or model 2) after break. In all cases our model 2 fits the observed photon data with equipartition of energy between the magnetic field and the relativistic cosmic-ray protons. Thus our model 2 remains more favorable.
Also we discuss the possible proton synchrotron origin of TeV component from extended quasar jets, in the context of knot wk8.9 of PKS 0637-752 and show that TeV emission is in principle possible with photon luminosity $10^{41}-10^{43}$ erg/s at the peak near 1 TeV in the energy spectrum, if protons in the kpc-scale jets are accelerated upto 5.6$\times10^{21}$ eV. However direct observation of such TeV spectrum would be difficult due to severe EBL absorption.
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TABLE 1 : Model parameters for PKS 0637-752
Knot Parameter Notation Model 1 Model 2
wk8.9 Size of knot (m) R 2.2$\times$10$^{19}$ 3.6$\times$10$^{19}$
Lorentz factor $\Gamma$ 3 3
Viewing angle $\theta$ 35$^\circ$ 30$^\circ$
Doppler factor $\delta_D$ 1.46 1.79
Magnetic field (mG) B 8 5
Minimum proton energy(eV) E$_{p,min}$ 10$^{14}$ 10$^{10}$
Maximum proton energy(eV) E$_{p,max}$ 7.2$\times10^{19}$ 5.2$\times10^{19}$
Break proton energy(eV) E$_{p,br}$ 10$^{16}$ 10$^{12}$
Low energy proton spectral index p1 1.35 1.63
High energy proton spectral index p2 2.35 2.13
Luminosity in magnetic field (erg/sec) L$_{B}$ 1.19$\times$10$^{43}$ 2.06$\times$10$^{43}$
Luminosity in proton (erg/sec) L$_{p}$ 7.07$\times$10$^{42}$ 2.06$\times$10$^{43}$
wk7.8+ Size of knot (m) R 2.1$\times$10$^{19}$ 4.6$\times$10$^{19}$
wk8.9+ Lorentz factor $\Gamma$ 3 3
wk9.7+ Viewing angle $\theta$ 35$^\circ$ 23$^\circ$
wk10.6 Doppler factor $\delta_D$ 1.46 2.47
Magnetic field (mG) B 9 7
Minimum proton energy(eV) E$_{p,min}$ 10$^{14}$ 10$^{10}$
Maximum proton energy(eV) E$_{p,max}$ 5.62$\times10^{19}$ 5.2$\times10^{19}$
Break proton energy(eV) E$_{p,br}$ 1.58$\times$10$^{16}$ 10$^{12}$
Low energy proton spectral index p1 1.35 1.9
High energy proton spectral index p2 2.35 2.4
Luminosity in magnetic field (erg/sec) L$_{B}$ 1.3$\times$10$^{43}$ 8.4$\times$10$^{43}$
Luminosity in protons (erg/sec) L$_{p}$ 1.3$\times$10$^{43}$ 8.4$\times$10$^{43}$
The SED of knot wk8.9 of PKS 0637-752. The radio to X-ray data (Mehta et al. (2009),Meyer et al. (2015))
are shown by blue points and the Fermi-LAT upper
limit (Meyer et al. (2015)) shown by magenta arrows. Black dotted line: Electron synchrotron fit to radio-optical data (Meyer et al. (2015));
Red solid line: Proton synchrotron fit according to our Model 1 as in Table 1; Black dot-dashed line: Our Model 2. The code used
in our work is from Krawczynski et al. (2004)
The SED of combined knots wk7.8+wk8.9+wk9.7+wk10.6 of PKS 0637-752. Fermi-LAT upper limits, radio to X-ray
data and model fits follow the same notation as Fig. 1 (references as described in figure 1)
TABLE 2 : Model parameters for 3C 273
Knot Parameter Notation Model 1 Model 2
A Size of knot (m) R 1.9$\times$10$^{19}$ 3.15$\times$10$^{19}$
Lorentz factor $\Gamma$ 3 3
Viewing angle $\theta$ $45^\circ$ 23$^\circ$
Doppler factor $\delta_D$ 1 2.47
Magnetic field (mG) B 10 9
Minimum proton energy(eV) E$_{p,min}$ 10$^{14}$ 10$^{10}$
Maximum proton energy(eV) E$_{p,max}$ 1.9$\times10^{20}$ 8.9$\times10^{19}$
Break proton energy(eV) E$_{p,br}$ 10$^{16}$ 10$^{12}$
Low energy proton spectral index p1 1.62 2.02
High energy proton spectral index p2 2.62 2.52
Luminosity in magnetic field (erg/sec) L$_{B}$ 1.2$\times$10$^{43}$ 4.45$\times$10$^{43}$
Luminosity in protons (erg/sec) L$_{p}$ 7.59$\times$10$^{42}$ 4.45$\times$10$^{43}$
A+ Size of knot (m) R 3.3$\times$10$^{19}$ 3.6$\times$10$^{19}$
B1 Lorentz factor $\Gamma$ 3 3
Viewing angle $\theta$ $45^\circ$ 22$^\circ$
Doppler factor $\delta_D$ 1 2.59
Magnetic field (mG) B 10 9
Minimum proton energy(eV) E$_{min}$ 10$^{14}$ 10$^{10}$
Maximum proton energy(eV) E$_{max}$ 1.25$\times10^{20}$ 7.24$\times10^{19}$
Break proton energy(eV) E$_b$ 5.62$\times$10$^{15}$ 10$^{12}$
Low energy proton spectral index p1 1.57 2.03
High energy proton spectral index p2 2.57 2.53
Luminosity in magnetic field (erg/sec) L$_{B}$ 6.3$\times$10$^{43}$ 6.65$\times$10$^{43}$
Luminosity in protons (erg/sec) L$_{p}$ 1.38$\times$10$^{43}$ 6.65$\times$10$^{43}$
The SED of knot A of 3C 273. Fermi-LAT upper limits (Meyer et al. (2015)) and radio to X-ray
data (Jester et al. (2005), Uchiyama et al. (2006)) are shown with magenta arrows and blue points. Black dotted line: Electron synchrotron
spectrum (Meyer et al. 2015); Solid red line:
Our Model 1 as in Table 2; Black dot-dashed line: Our Model 2
The SED of combined knot A+B1 of 3C 273. Data references as in Fig. 3. Symbols and model fits follow same notation as in Fig.3.
TABLE 3: TeV emission from Knot wk8.9 of PKS 0637-752 at a distance of 3.8Gpc
Parameter Notation Model 1 Model 2 Model 3 Model 4
Size of knot (m) R 1.53$\times$10$^{19}$ 1.53$\times$10$^{19}$ 1.75$\times$10$^{19}$ 3.2$\times$10$^{19}$
Lorentz factor $\Gamma$ 3 3 3 3
Viewing angle $\theta$ 28$^\circ$ 23$^\circ$ 23$^\circ$ 23$^\circ$
Doppler factor $\delta_D$ 1.9 2.47 2.47 2.47
Magnetic field (mG) B 10 8 7 5
Min proton energy(eV) E$_{min}$ 10$^{14}$ 10$^{14}$ 10$^{14}$ 10$^{14}$
Max proton energy(eV) E$_{max}$ 3.16$\times10^{21}$ 3.16$\times10^{21}$ 5.6$\times10^{21}$ 5.6$\times10^{21}$
Break proton energy(eV) E$_b$ 7.08$\times10^{15}$ 10$^{16}$ 1.99$\times10^{16}$ 3.98$\times10^{16}$
Low energy proton spec. index p1 1.8 1.9 2 2.25
High energy proton spec. index p2 2.8 2.9 3 3.25
Luminosity in magnetic field (erg/sec) L$_{B}$ 6.29$\times$10$^{42}$ 4.06$\times$10$^{42}$ 4.63$\times$10$^{42}$ $1.44\times$10$^{43}$
Luminosity in protons (erg/sec) L$_{p}$ 6.29$\times$10$^{42}$ 4.06$\times$10$^{42}$ 4.63$\times$10$^{42}$ 1.44$\times$10$^{43}$
Photon luminosity in jet (erg/sec) L$_{j,ph}$ 1.81$\times$10$^{43}$ 6.74$\times$10$^{42}$ 2.75$\times$10$^{42}$ 2.28$\times$10$^{41}$
Proton-synchroton modelling of TeV emission from Knot wk8.9 of PKS 0637-752. Fermi-LAT and lower energy photon data references and notation as in Fig. 1;.
Red dotted line: electron synchrotron spectrum; Black dotted line: Model 1 according to Table 3;
Dot-dashed line: Model 2; Dot-dot-dashed line: Model 3; Solid line: Model 4
|
1511.00589
| |
1511.00232
|
Peter Grünberg Institut & Institute for Advanced Simulation,
Forschungszentrum Jülich & JARA, D-52425 Jülich,
The magnetic anisotropy energy defines the energy barrier that stabilizes a magnetic moment.
Utilizing density functional theory based simulations and analytical formulations, we establish that
this barrier is strongly modified by long-range contributions very similar to Friedel oscillations
and Rudermann-Kittel-Kasuya-Yosida interactions. Thus, oscillations are expected and observed, with different decaying
factors and highly anisotropic in realistic materials, which can switch non-trivially the sign of the magnetic anisotropy energy.
This behavior is general and for illustration we address transition metals adatoms, Cr, Mn, Fe and Co deposited on Pt(111) surface.
We explain in particular the
mechanisms leading to the strong site-dependence of the magnetic anisotropy energy observed for Fe adatoms on Pt(111) surface as revealed previously via first-principles
based simulations
and inelastic scanning tunneling spectroscopy (A. A. Khajetoorians et al. Phys. Rev. Lett. 111, 157204 (2013)). The same mechanisms are
probably active for the site-dependence of the magnetic anisotropy energy obtained for Fe adatoms on Pd or Rh(111) surfaces and for Co adatoms on Rh(111) surface
(P. Blonski et al. Phys. Rev. B 81, 104426 (2010)).
§ INTRODUCTION
As magnetic devices shrink toward atomic dimensions with
the ultimate goal of encoding information in the smallest possible magnetic entity,
the understanding of magnetic stability down to the single atomic limit becomes crucial. Here,
a critical ingredient is the magnetic anisotropy energy (MAE) that provides directionality and stability to magnetization.
The larger the MAE, the more protected is the magnetic bit against, for example, thermal fluctuations.
Thus the search for nanosystems with enhanced MAE is a very active field giving the perspective of stabilizing and simultaneously reducing the size of magnetic bits.
Recently, it was demonstrated that nanostructures with only a few atomic spins, ranging from single atoms, clusters on metal
surfaces (see for example Refs. <cit.>
to molecular magnets (e.g. Refs. <cit.>) can exhibit MAEs that are large enough
to maintain in principle a stable spin orientation at low temperatures.
A celebrated example is the giant MAE ($\sim$ 9 meV) discovered by Gambardella et al. <cit.> for a single Co adatom
on Pt(111) surface. There the right ingredients for a large MAE are satisfied: a large magnetic moment carried by the 3$d$ transition element, Co, being
at the vicinity of heavy substrate atoms characterized by a large spin-orbit interaction (SOI). Naturally, here details of the electronic structure and
hybridization effects are decisive. Thus exchanging the Co adatom by an Fe adatom leads to an extremely small MAE as demonstrated recently by
inelastic scanning tunneling spectroscopy and ab-initio simulations based on density functional theory (DFT) <cit.>. Most intriguing in the latter work is the dramatic change
of the MAE magnitude and sign once the Fe adatom was moved from an fcc–stacking site, where the moment points out-of-plane, to an hcp–stacking site,
where the moment lies in-plane. This was assigned to the proximity effect leading to a large spin polarization cloud induced by Fe in the Pt substrate, which is notorious
for its high magnetic polarizability <cit.> as seen also for Pd <cit.>.
A similar site-dependent MAE for Fe adatoms on the (111) surfaces
of Pd and Rh and for Co on Rh(111) was noticed with ab-initio simulations <cit.>. The physical
mechanism behind such a behavior has not been, to our knowledge, identified convincingly.
Even on surfaces with low polarizability, such as gold,
the MAE follows an oscillating behavior depending on the distance to the surface of buried magnetic nanostructures <cit.>.
Thus the polarizability is probably not the only ingredient modifying the strength of the MAE since Au is much less polarizable than Pt.
One has to keep in mind that the polarizability of the substrate atoms is determined by the Stoner
product $I \cdot N_F$ with the exchange integral $I$ and the number of states at the Fermi level $N_F$
($I \cdot N_F = 0.29$ for Ir, 0.59 for Pt and 0.05 for Au) <cit.>.
The goal of our work is to demonstrate with a formal proof that a strong contribution to the MAE can be highly non-local and long-ranged and may endow up to
$\pm 50 \%$ of the total MAE. Strong similarities can be foreseen with respect to Friedel <cit.> and
Rudermann-Kittel-Kasuya-Yosida (RKKY) <cit.> oscillations in terms of the impact of the
nature of the mediating electronic states, their localization in real-space and their shape in reciprocal-space (e.g. Fermi surface) on the decay of the oscillations
and their focusing ( see e.g. Refs. <cit.>). A particularity of this long-range
contribution to the MAE is, as expected, its dependence on the strength of SOI. Taking as an illustration 3$d$ adatoms (Cr, Mn, Fe and Co) deposited on Pt(111) surface,
we demonstrate that the contribution of the substrate Pt
atoms to the total MAE oscillates and decays with their distance to the adatom.
§ METHOD
The MAE can be determined from the magnetic force theorem <cit.> taking the energy difference,
$\epsilon_{\perp} - \epsilon_{||}$, between
the band energies of the two configurations: out-of-plane ($\perp$) and in-plane ($||$) orientations of
the magnetic moment. A reference magnetic configuration is chosen, here the out-of-plane orientation, where the self-consistent calculations
are performed and the related band energy is obtained. Then the magnetic moment is rotated in-plane and one iteration is done in order to extract the band energy.
With such a traditional technique, one reduces the error made by taking differences between the total energies, which are large numbers.
A positive sign of the MAE indicates an in-plane preferable orientation of the adatom's magnetic moment.
We utilize the full potential relativistic Korringa-Kohn-Rostoker Green function
method (FP-KKR-GF) <cit.>. The local Spin Density Approximation as parametrized by Vosko, Wilk and Nusair was used <cit.>.
First, the electronic structure of a 22 layers Pt slab with two additional vacuum regions (8 layers)
is calculated. The experimental lattice parameter (3.92 Å) was considered without surface relaxations
which are negligible <cit.>.
Then, each adatom is embedded on the surface of
this slab, in real space, together with its neighboring sites, defining a cluster of atoms, where the charge is allowed to be updated during
self-consistency. We note that the cluster still interacts with the rest of the host surface via the Coulomb interaction. The adatoms
are allowed to relax towards the surface, and we found qualitatively a similar behavior for the magnetic moments and the MAE in the range of relaxation
from 15 to 25% towards the surface. As indicated in Ref.<cit.>, Fe was found to relax by 20% towards the surface. The same relaxed geometry was found for
Co adatoms<cit.>. Thus for the sake of comparison, the four investigated adatoms were assumed at the same relaxed position 20% towards the surface.
The MAE is extracted for clusters of different sizes, for which the Green functions of the impurity-free surface are generated with
$200\times200$ k-points in the two-dimensional Brillouin zone and a maximum angular quantum number $l = 3$.
To provide an idea of the convergence of the MAE versus the number of k-points we address the case of the
Fe adatom in contact with the Pt substrate where 221 Pt atoms are allowed to be perturbed by the impurity. The MAE is found to
change by about 0.002% with respect to one obtained for $200\times200$ k-points when the number of k-points is decreased to $180\times180$ or $150\times150$ k-points.
§ RESULTS AND DISCUSSIONS
§.§ Fe adatoms, fcc versus hcp stacking sites
MAE of Fe impurity adsorbed on an fcc (square) or an hcp (circle) site on top of Pt(111) surface versus the number of Pt atoms in the cluster. A
positive MAE corresponds to an in-plane orientation of the magnetic moment and a negative MAE corresponds to an out-of-plane magnetic moment. Readapted from
Fig.3 of the Supplement of Ref. <cit.>.
Fig. <ref> displays the MAE obtained with the band energy differences of an Fe adatom sitting on an fcc- or an hcp-site on Pt(111) surface
versus the number of Pt atoms included in the real-space calculations. This figure is part of the Supplement of Ref. <cit.>. If only the nearest
neighbors (NN) Pt atoms to the Fe impurity are considered, in this case 3 Pt atoms, the MAE yields an out-of-plane
easy axis with the same value of -2.8 meV for both binding sites. However, considering more Pt atoms, the neighborhoods of
the two stacking sites differ, and therefore the MAE becomes
strongly dependent on the binding site and even changes sign for the hcp-site. The latter occurs when including in the surrounding
cluster 16 Pt atoms in addition to the NN atoms. The MAE first decreases from -2.8 meV to -3.3 meV by
adding the 10 closest Pt neighboring atoms and then surprisingly jumps to +0.1 meV by adding the further distant 6 Pt atoms (colored in blue in Fig. <ref>a-b).
The latter means that these 6 Pt atoms, with their positive contribution (+3.4 meV) to the MAE, play a key-role in
switching the preferable orientation of the adatom's magnetic moment.
These switcher atoms are equivalent, belong to the subsurface-layer and are equidistant ($\sim 0.5$ nm)
from the adatom. Interestingly, the switcher atoms occur also for the fcc binding site, and are located similarly to the hcp binding site
at the subsurface-layer equidistantly from the adatom. However, their number is lower than in the hcp stacking site: 3 instead of 6 (see Fig. <ref>c-d).
Therefore, their contribution to the MAE (+1.5 meV) is about half their contribution for the hcp binding site. This is not sufficient to compete
against the preferable orientation of the adatom and its NN (MAE = -2.8 meV). Indeed, once the switching atoms included
the MAE jumps from -2.2 meV, obtained
with a cluster containing 15 Pt atoms, to -0.7 meV. After adding more substrate atoms, reaching a cluster
of $\sim$ 221 atoms, the MAE tends to +0.5 meV and -2.9. meV for the hcp and fcc binding sites, respectively.
The latter values are rather converged since smaller clusters with a number of atoms (not shown in Fig. <ref>) close to the largest one show a
stable MAE.
Atomic structures of Fe impurity adsorbed on an hcp a (side view) and b (top view) or an fcc site c (side view) and d (top view) of Pt(111) surface.
The Pt atoms with blue color are the switching atoms that have a large contribution to the MAE.
We have also examined the effect of the Pt polarization cloud on the total spin and orbital magnetic moments. Interestingly, the impact on the total moment is less
impressive than on the MAE as summarized in the Table 1 for the case of the Fe adatom with a magnetic moment pointing out-of-plane.
When only the NN Pt atoms are included the total spin moment reaches a value of
$\sim4\mu_B$ while the total orbital moment is around 0.2$\mu_B$.
Inclusion of a larger number of neighboring Pt atoms increases the total spin moment by a
maximum of $\sim 0.4\mu_B$ while the total orbital moment reached saturation already with the NN atoms. This observation can be extracted from Fig.<ref> where the induced Pt total z-components of the spin and orbital moments are plotted for the case the impurity sits at the hcp stacking-site. The z-direction is perpendicular to the substrate.
1c| Fe hcp Fe fcc
Number of Pt atoms $m_s$ $m_{orb}$ $m_s$ $m_{orb}$
3 4.11 0.22 4.03 0.23
53 4.57 0.216 4.427 0.227
221 4.59 0.21 4.42 0.212
Total magnetic spin and orbital moments of the Fe adatom including different sets of neighboring Pt atoms. The total spin moment converges after considering 53 Pt atoms, while the total orbital moment
is already saturated with the NN Pt atoms.
Convergence of the total z-component of the spin moment and orbital moment induced in the Pt atoms of different cluster sizes. The case of an Fe adatom sitting on the
hcp binding site is considered and the z-direction is perpendicular.
As a summary, one realizes that the contributions of the different Pt shells to the total MAE is not uniform and
oscillates with the distance and is certainly not correlating perfectly with
the change of the total spin moment or total orbital moment. The latter quantities describe the polarization of the Pt cloud.
At first sight, one could ask whether Fig. <ref> is the result of numerical artifacts related to the KKR embedding scheme. In principle,
whenever a cluster is considered, the atoms sitting at the edge of the cluster would feel the boundary conditions more strongly than the atoms close to the Fe impurity. As illustrated
in Fig. <ref>, the edge atoms are not that affected by the boundary conditions.
The spread of the plotted values gives an idea on the impact of the cluster size on the individual Pt magnetic moments.
As an example, the spin moment of the edge atom, located at $\sim0.65$ nm, in the cluster
containing 34 atoms is on top of the spin moment of the same Pt atom when the boundary conditions have been improved by extending the size of the cluster to 220 Pt atom. The same
conclusion can be drawn for the orbital moment, although here the values are much smaller than the spin moments. In general, the boundary conditions will affect slightly the values
obtained for the magnetic properties including the MAE. However, the general oscillatory behavior observed in Fig. <ref> seems to go beyond the numerical conditions needed to
extract it. The main reason is that the Pt spin moment, for instance, has two contributions: either induced by the magnetic adatom or by the surrounding magnetic Pt atoms.
The former has in general a much stronger contribution than the latter. Also, within the KKR embedding scheme the atoms at the edge feel
the Coulomb interaction of the neighboring atoms beyond the cluster.
The individual Pt atomic spin moment (left) and orbital moment (right) as function of distance with respect to the Fe adatom sitting
on the hcp binding site. The spread of the magnetic moments for the different cluster sizes is rather small,
highlighting the low impact of
the boundary conditions of the KKR simulations on these magnetic properties.
In the following the
origin of the oscillatory behavior of the MAE will be discussed by realizing that the band energies, $\epsilon$, can be evaluated from $-\int\limits_{-\infty}^{E_{F}}dE\, N(E)$, i.e. an integration up to the Fermi energy, $E_F$,
of the integrated density of states (IDOS), $N(E)$, which in turn can be extracted from the celebrated Lloyd's formula <cit.>. Indeed, if a system described by
a Green function, $G$, is perturbed by a potential $V$, the change in the IDOS, $\delta N(E)$, is given simply by $-\frac{1}{\pi}\Im \mathrm{Tr}\,\mathrm{ln}(1 - V G(E))$, where the trace is taken over the site index,
orbital and spin angular momentum quantum numbers. This permits the aforementioned decomposition of the MAE into local and non-local contributions by evaluating wisely the
change in the IDOS.
§.§ Long-range contributions to the MAE: Formalism and results
First, we note that once the adatom is deposited on the substrate, it
perturbs strongly the potentials of the NN Pt atoms, which are 3 Pt atoms in total.
If we consider solely the adatom and its NN Pt atoms, the corresponding Green function, $G_1$, can be obtained from the Dyson equation
\begin{equation}
G_{1}(E)= G_{0}(E)+G_{0}(E)V_{1}G_{1}(E),\
\end{equation}
where $G_0$ is the Green function of the ideal surface of Pt without SOI while $V_1$ is the perturbing potential limited to the region of the adatom and its NN and
is induced by the presence of the impurity and the SOI. Instead of the potential
$V_1$, one can use the scattering matrix $T_1$:
\begin{equation}
\label{Dyson1}
\end{equation}
Out of the previous Dyson equation, the local electronic and magnetic properties of the adatom can be reasonably described. For instance, it leads to a MAE
of -2.8 meV for the Fe adatom. To grasp the
effect of the rest of Pt atoms, i.e. the hundreds outer Pt atoms, on the MAE, we solve a second Dyson equation to obtain the new Green function, $G_2$:
\begin{equation}
G_{2}(E) = G_{1}(E)+G_{1}(E)V_{2}G_{2}(E).
\label{Dyson2}
\end{equation}
where the perturbing potential, $V_2$, describes simultaneously the change induced by the adatom on the additionally incorporated 217 Pt outer
atoms ($V_2^{'}$) and their SOI ($V_2^{so}$). In fact, $V_2^{so} = \xi (E) \mathbf{L.S}$, with $\xi(E)$ being the
strength of SOI. Thus, $V_{2}=\displaystyle\sum_{j}(V_{2j}^{'}+V_{2j}^{so})$ where the sum runs over all outer Pt atoms. In contrast to $T_1$, $V_2$ is
limited to the rest of Pt atoms and
is expected to be relatively small since the perturbation decays with the distance from the adatom, which permits the use of Taylor expansions when solving Eq. <ref>.
The change in the IDOS, $\delta N(E)$, due to the coupling of the adatom and its NN to the
rest of the Pt substrate atoms is then given as: $-\frac{1}{\pi}\Im \mathrm{Tr}\, ln(1 - V_2 G_1(E))$, which for small $V_2$ can be expanded up to second order:
\begin{equation}
\delta N(E) = \frac{1}{{2\pi}}\Im \mathrm{Tr} [2V_{2} G_{1}(E)+V_{2} G_{1}(E)V_{2} G_{1}(E)].
\end{equation}
We express $G_1$ in terms of $G_0$ as given in Eq. <ref>, drop terms leading to third and fourth-order processes (these are expected to be much smaller than the second order-processes)
and find:
\begin{eqnarray}
\delta N &=& \frac{1}{{2\pi}}\Im \mathrm{Tr} [2V_{2} G_{0}+2V_{2} G_{0}T_{1}G_{0} +V_{2} G_{0}V_{2} G_{0}],
\end{eqnarray}
where the energy argument, $E$, was taken out for the sake of simplicity. Since $V_2$ is written in terms of non-SOI- and SOI-dependent terms, this allows to disentangle the previous expression:
\begin{eqnarray}
\delta N &=&\frac{1}{\pi}\Im \mathrm{Tr}\displaystyle\sum_{j} \Large{\{} V_{2j}^{'}G_{0}+ V_{2j}^{so}G_{0}\\ \nonumber
& &+ T_{1} G_{0}V_{2j}^{'} G_{0} + T_{1} G_{0}V_{2j}^{so} G_{0}\\ \nonumber
% & &+ T_{1} G_{0}V_{2j}^{so} G_{0}]\\ \nonumber
& &+ \frac{1}{2}\sum_{j'}(V_{2j}^{'}+V_{2j}^{so}) G_{0}(V_{2j'}^{'}+V_{2j'}^{so})G_{0}\Large{\}}.
\end{eqnarray}
In view of our interest in the band energies that depend on the rotation of the magnetic moment, i.e. contributing to the MAE, not all terms in Eq. 6 are relevant.
For instance the term of first order in $V_2$ or $G_0$ contain either no spin orbit coupling or only the linear SOI term.
Therefore they vanish when one evaluates the MAE. From the last term, only the contribution from the scattering at $V_2^{'}$ and at $V_2^{so}$ is finite.
Since these atoms are only weakly spin-polarized, the latter term is negligible as verified numerically and therefore it is not considered in the following. The contribution to the band energy relevant for the MAE is then given by
\begin{eqnarray}
-\frac{1}{\pi} \Im \mathrm{Tr}\int\limits_{-\infty}^{E_{F}}dE \displaystyle\sum_{j} \Large\{ T_{1} G_{0}V_{2j}^{'} G_{0}+ T_{1} G_{0}V_{2j}^{so} G_{0}\LARGE\},
\label{Final}
\end{eqnarray}
which has to be evaluated at the different configurations $\perp$ and $||$ orientations of the magnetic moment in order to extract the MAE.
The first term is the most simple one. It is independent of the SOI of the outer Pt atoms and just describes a renormalization of
the MAE of the small cluster consisting of the Fe atoms and the 3Pt atoms due to the scattering at the potentials $V_{2j}^{'}$ of
the outer Pt atoms, which does not include the SOI of these atoms. Therefore we name this contribution the no-so-term.
The second term, called so-term, is also important and describes the double scattering at the SOI-term of $T_1$
and the SOI potential $V_{2j}^{so}$ of the outer atoms. These two terms might therefore be described as non-local, since they
connect the scattering at the SOI of the inner cluster with the scattering at the potentials of the outer atoms.
The analogy of these non-local terms with the celebrated formula from Lichtenstein et al. <cit.>
for the evaluation of the magnetic exchange interactions is appealing, and as
for the magnetic interactions, we expect these two terms to oscillate and decay with the distance between the two regions.
Instead of the magnetic part of the potential, the scattering occurs at the SOI term but the mediation is made in both cases via the Green functions.
In order to clarify the importance of the non-local terms in the MAE, we have therefore
recalculated the anisotropy by switching on and off the SOIs of individual outer Pt atoms, based on
Eq. <ref>. In this way, we demonstrate how the relatively small so and no-so contributions of an outer Pt atom changes the
MAE of the complex system containing the Fe atom, its NN and that preselected outer Pt atom, and shows Friedel-like oscillations.
For this analysis, the cluster thus contains an Fe adatom, its 3 NN Pt atoms and one additional single Pt atom. That Pt atom probes the non-locality of
the MAE following Eq. <ref> by considering it along different directions and distances away from the magnetic adatom. In this investigation and
to simplify the discussion, we do not include
the nearest neighboring atoms of that particular additional Pt atom in our cluster. Of course these boundary conditions will affect the final values of the non-local contributions but the general conclusions of this work are not affected. We perform two steps: (step 1) SOI is switched on within the additional Pt atom. After removing the MAE of the Fe adatom and its NN 3 Pt atoms, we obtained the sum
of the two terms given in Eq. <ref>. Then we proceed with (step 2) and switch off SOI, getting thereby the no-so-term, with which one extracts the so-term to the
sum in Eq. <ref>.
Contributions to the MAE from different shells of Pt atoms versus their distance with respect to the adatom sitting either at
the fcc or at the hcp stacking sites. In (a), (b) and (c), the plotted values correspond to the Pt atoms sitting along the direction
connecting the adatom with one of the switching Pt atoms, i.e. ($\theta=125^{\circ}, \phi=-60^{\circ}$) for the hcp stacking site and ($\theta=128^{\circ}, \phi=30^{\circ}$)
for the fcc stacking site. The inset enhances the oscillations observed in the non-local terms. While in (a) the sum of the non-local contributions to the MAE is plotted,
in (b-c) the no-so and so-terms are plotted separately for respectively the hcp and fcc sites.
(d) Anisotropy of the non-local contribution to the MAE obtained for two different set of angles:
($\theta=125^{\circ}, \phi=-60^{\circ}$) compared to ($\theta=150^{\circ}, \phi=80^{\circ}$). The magnitude of MAE is clearly more
enhanced along the direction passing by the switching Pt atom, i.e. the red curve.
Fig. <ref> shows the non-local contributions from a single Pt atom as
function of the distance, $d$, from the adatom for hcp-
and fcc-sites along two directions connecting the adatom to one of the Pt switching atoms.
While in Fig. <ref>(a) we plot the sum of the non-local contributions, in Fig. <ref>(b) and
(c) these contributions are resolved into the so and no-so-terms for respectively the hcp and fcc sites.
In Figs.5(a, b, and c), the chosen polar and azimuthal angles ($\theta, \phi$) are ($125^{\circ}$, $-60^{\circ}$) (hcp stacking sites) and
($128^{\circ}$, $30^{\circ}$) (fcc stacking sites).
Naturally, here we allow for an error bar for the angles ($\delta\theta=\pm3^{\circ}$
and $\delta\phi=\pm8^{\circ}$) since a straight line will not cross a sufficient
number of Pt atoms at reasonable distances.
One clearly sees, that the sum of non-local terms are important outside
the small inner region with the largest contribution emanating from the switcher atom,
which reaches a value of $0.37$ meV for the Fe fcc-site and $0.73$ meV for the hcp-site.
As explained earlier, since there are only three switching atoms for the fcc-site instead of six for
the hcp-site, the barrier given by the MAE of the adatom and its NN is not overcome.
By increasing the distance from the adatom,
the induced term oscillates and changes even sign. Its magnitude, however, is not sufficient to
overcome the aforementioned barrier. These oscillations as function of distance have a Friedel-like character
and are similar to those obtained for long-ranged magnetic exchange interactions <cit.>.
From Figs. <ref>(b) and <ref>(c), we notice that the so-term is not behaving
similarly to the no-so-term. These two terms can counteract each other as for the contribution from the switcher atom.
Thus, for this particular atom the so-term is dominant and favors an in-plane orientation of the moment in contrast to the no-so-term.
For large distances both terms oscillate non-trivially. Although the values plotted in Fig. <ref> can look small at first sight, one should not forget that these
are contributions from a single Pt atom. At the end, one has to sum up contributions from all the surrounding Pt atoms to get the full-non local
part of the MAE.
These oscillating non-local parts of the MAE can be highly anisotropic as demonstrated in Fig. <ref> (d), where two directions are probed. First, along the
direction already shown in Fig. <ref>(a) that connects the Fe-adatom with one switcher atom leading to a very large peak at $0.5$nm. The second probed direction does not cross
such switcher atoms and interestingly the calculated values are considerably smaller at short distances but show similar Friedel-like oscillations at large distances.
Thus, the non-local MAE contribution from the outer Pt-atoms show Friedel-like oscillations, but are highly anisotropic which is expected when looking at the Fermi surface
of Pt presented in Fig. <ref>. Indeed the Fermi surface, extracted utilizing the scheme described in Ref. <cit.>,
is extremely anisotropic such that isotropic oscillations resulting from
a simple spherical Fermi surface cannot be expected in our particular system.
Bulk Fermi surface of Pt, with directions of probed atoms as indicated by red and blue arrows and the [111] direction by a black arrow. The color code on the Fermi surface corresponds to the magnitude of the Fermi velocity (red and blue corresponding to high and low velocity, respectively).
§.§ Case of Cr, Mn and Co adatoms on fcc and hcp stacking sites
For completeness, we examined the impact of the Pt spin-polarization cloud on the MAE of Cr, Mn and Co adatoms.
Like Fe adatom, Co adatom and its NN prefer an out-of-plane orientation of the magnetic moment independently from
the binding site (Fig. <ref>). The MAE found in this case ($-8.2$ meV) is however larger
than the one of Fe adatom, making the barrier higher for an in-plane reorientation of the magnetic moment when including a large number of Pt substrate atoms (up to 221 atoms).
Besides that, here the non-local contribution of the switching atoms to the MAE is even smaller than for Fe-adatom. The total MAE for the largest studied system decreases to
$-6.9$ meV and $-5.5$ meV for respectively the hcp- and fcc-sites. We point out that the experimental
value of Gambardella et al. <cit.> is around $-9$ meV. This large value has generated a lot of theoretical investigations based on density functional theory.
Usual simple exchange and correlation functionals, such as the local spin density approximation (LDA) or the
generalized gradient approximation (GGA) lead to rather small MAE. Therefore, correlation effects beyond LDA or GGA were considered, e.g. by including a correlation
$U$ as a parameter or the orbital polarization scheme to tune the MAE and understand the origin of its large magnitude. Our work demonstrates that even without the invoked
correlation effects, the non-local contribution to
the MAE, not considered up to now, can be crucial in the case of Co as well. We predict that in the case of the hcp-stacking site the MAE reaches $\sim-7$ meV.
The case of Mn is interesting since contrary to what has been observed for Fe and Co, both the local and non-local contributions to the MAE from the switching Pt atom favor
an in-plane orientation of the magnetic moment. However, the rest of Pt atoms are decisive. By increasing their number,
the adatom on the fcc binding-site switches first to an out-of-plane magnetic orientation before
converging to an in-plane orientation. Cr adatom behaves similarly to Mn, i.e. both the local and non-local contributions to the MAE favor an in-plane orientation
of the magnetic moment but unlike Mn, the local term is large: $+5.6$ meV and $+4.5$ meV for respectively the hcp and fcc stacking sites.
Furthermore, when compared to Mn, Fe and Co adatoms, the switching atoms at the vicinity of Cr adatom contribute to the MAE differently and favor an out-of-plane
orientation of the moment. This contribution is, however, not large enough to overcome
the barrier created by the adatom and its NN. When the rest of Pt atoms are included, Cr-adatoms on both binding sites prefer an in-plane magnetic orientation.
By changing the chemical nature of the adatom, the non-local behavior of the MAE is modified. As it can be realized from Eq. <ref>, the scattering properties
at the adatom site, described by $T_1$, can renormalize strongly the total MAE. $T_1$ depends obviously on the electronic properties of the adatom and its nearest surrounding.
It is not a single number but a matrix and therefore after taking the trace in
Eq.<ref> besides the impact on the magnitude of the MAE, non-trivial interference effects can occur, which affect the oscillating behavior of the MAE.
MAE of Co, Mn, Cr impurities adsorbed on an fcc (square) or on an hcp (circles) site on top of Pt(111) surface versus the number of Pt atoms in the cluster. The convention of the sign of
the MAE is identical to the one used in Fig. <ref>.
§ DISCUSSIONS AND CONCLUSIONS
To summarize, for 3d adatoms on Pt(111) we demonstrated the existence of long-range, RKKY-like,
contributions to the MAE mediated by the electronic states of the substrate.
Since they oscillate as a function of the distance with different kind of decaying factors, they affect the
magnitude of the total MAE and can even switch its sign.
This depends on the details of the electronic structure, and as for Friedel oscillations or RKKY interactions,
they can be highly anisotropic with a possibility of observing a focusing effect induced by
the shape of the constant energy contours (e.g. the Fermi surface) <cit.>. Our results go beyond the approximations
assumed along our theoretical investigations. We expect non-negligible non-local contributions to the MAE independently from
the assumptions related to the exchange and correlation functionals, geometrical relaxations, and inclusion of a $U$ as done in traditional LDA + $U$.
established effect is expected to occur in other substrates with high polarizability (e.g. Rh, W, Ir, Pd substrates), but also when confined
electronic states are present in
low dimensional systems (e.g. surface states of Ag and Au(111) surfaces) since the latter
favor a lower decay of the usual Friedel oscillations. We believe that such an effect is active in the recently investigated surfaces of
CuN/Cu(001) <cit.> and Graphene/Rh(111) <cit.> were unusual behavior of the MAE of different types of adsorbates
has been observed. To verify experimentally the theoretical facts described in our work, one would have for example to switch off/on the spin-orbit interaction of a remote substrate
Pt atom at will. This is certainly impossible, however, we believe that the signature of the non-locality of the MAE could be detectable for two magnetic adatoms on a surface,
for example two Fe adatoms on a Pt(111) surface. We expect the MAE to be dependent on the inter-adatom distances, which is expect to be related to the non-local effect discussed
in the main text. Thus, we expect and oscillatory behavior of the MAE measurable with state-of-the-art inelastic scanning tunneling spectroscopy, wherein the MAE leads to
a gap in the excitation spectra.
We acknowledge fruitful discussions with Stefan Blügel and the teams of Jens Wiebe and Alex Khajetoorians.
This work is supported by the HGF-YIG Programme VH-NG-717 (Functional Nanoscale Structure and Probe Simulation Laboratory–Funsilab)
and the DFG project LO 1659/5-1.
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Recognition of objects in still images has traditionally been regarded as a difficult computational problem. Although modern automated methods for visual object recognition have achieved steadily increasing recognition accuracy, even the most advanced computational vision approaches are unable to obtain performance equal to that of humans. This has led to the creation of many biologically-inspired models of visual object recognition, among them the HMAX model. HMAX is traditionally known to achieve high accuracy in visual object recognition tasks at the expense of significant computational complexity. Increasing complexity, in turn, increases computation time, reducing the number of images that can be processed per unit time. In this paper we describe how the computationally intensive, biologically inspired HMAX model for visual object recognition can be modified for implementation on a commercial Field Programmable Gate Array, specifically the Xilinx Virtex 6 ML605 evaluation board with XC6VLX240T FPGA. We show that with minor modifications to the traditional HMAX model we can perform recognition on images of size 128$\times$128 pixels at a rate of 190 images per second with a less than 1% loss in recognition accuracy in both binary and multi-class visual object recognition tasks.
§ INTRODUCTION
Object recognition has received a lot of attention in recent years and is an important step towards building machines which can understand and interact meaningfully with their environment. In this context, both a high recognition accuracy and a short recognition time are desirable. By shortening recognition time even further, we foresee applications that include rapidly searching and categorizing images on the internet based on features extracted from their pixel content on the fly. Many currently available image search and characterization platforms rely on image metadata and watermarks rather than the images' actual pixel values, while those platforms which do make use of actual pixel values typically rely on previously extracted image features rather than creating and extracting new features on the fly.
The challenge of consistently recognizing an object is complicated by the fact that the appearance of the object can vary significantly depending on its location, orientation, and scale within an image. Reliable object recognition must therefore be invariant to translation, scale, and orientation. Some methods of object recognition incorporate these invariances, such as the Scale Invariant Feature Transformation (SIFT) <cit.> or Speeded Up Robust Features (SURF) <cit.>. These models achieve good recognition rates, but still fall far short of the recognition rates achieved by humans. There is evidence suggesting that after viewing an object for the first time, a biological system is capable of recognizing that object again at a novel position and scale <cit.>. The object can also be recognized if it is slightly rotated, but the recognition accuracy decreases when the object is rotated too far from a familiar view <cit.>. A biologically inspired model which shares this property of scale and translation invariance, but also achieves only limited rotation invariance is the Hierarchical Model and X (HMAX) <cit.> in which the `X' represents a non linearity.
Jarrett et al. <cit.> investigated which architecture is best for object recognition. They found that non-linearities are the most important feature in such models. Their results show that rectification and local normalization significantly improve recognition accuracy. Their results also indicate that a multistage method of feature extraction outperforms single stage feature extraction. The HMAX model is a multistage model which mixes Gabor filters in the first stage with learned filters in the second. HMAX is intended to model the first 100-200ms of object recognition due to purely feed-forward mechanisms in the ventral visual pathway <cit.>. HMAX is biologically inspired and incorporates rectification and local-normalization non-linearities, both of which were later recommended by Jarret et al. <cit.> as important properties for object recognition models.
In this paper, we focus specifically on the version of HMAX described in <cit.>. The recognition accuracy of HMAX is well below that of the biological counterparts it attempts to mimic for real world tasks because it only mimics the first stages of the feed-forward pathways. However, HMAX performs comparably to its biological counterparts on rapid characterization tasks in which a stimulus is presented long enough for feed forward recognition to take place, but short enough to prevent top down feedback from having an effect <cit.>. HMAX provides a valuable step towards achieving higher recognition accuracy and better understanding the operation of the ventral stream in visual cortex. Biological processing systems (networks of neurons) are inherently distributed and massively parallel. If we intend to achieve comparable recognition rates by mimicking biological processing, then we too should use distributed and massively parallel hardware which is suited to the task.
Originally, object recognition models were typically run on sequential processors (CPUs), for which Mutch and Lowe developed the Feature Hierarchy Library (FHLib) tool in 2006 <cit.> for implementing hierarchical models such as HMAX. CPUs require little effort to program and offer great flexibility, allowing them to be used for a large variety of tasks, but the sequential nature of their processing makes them ill suited to an application such as HMAX. Modern CPUs are capable of impressive performance and allow some parallel processing, but depending on the nature of the algorithm to be implemented, it can be very difficult, if not impossible, to fully utilize the theoretical computational capacity of such devices. In 2008 Chikkerur <cit.> reported a multithreaded CPU implementation of HMAX, showing that the increased parallelism outperformed previous CPU implementations.
GPUs allow even more parallel processing paths, but writing code for GPUs requires a larger effort than for CPUs. GPUs also offer greater control of data flow and storage during computation, which allows programmers to make greater use of the theoretical computational capacity. In the same paper as his multithreaded CPU implementation <cit.>, Chikkerur presented a GPU implementation of HMAX with even more parallel processing paths, which outperformed the multithreaded CPU implementation by 3$\times$-10$\times$ depending on input image size. Soon GPU technologies were being used extensively for HMAX and in 2010 Mutch and Lowe released the Cortical Network Simulator (CNS) <cit.> which uses a GPU for processing and can speedup the HMAX model by 97$\times$ compared to the FHLib software it was intended to replace. Later in 2010, Sedding et al. <cit.> presented another GPU implementation of HMAX which is claimed to outperform the CNS implementation in both accuracy and speed. There are also many other examples in the literature of the application of GPU processing to object recognition <cit.>.
Application Specific Integrated Circuits (ASICs) offer an even greater level of control than GPUs through intentional design of the hardware to suit the task at hand, but once fabricated, an ASIC is typically ill suited to other applications. Furthermore, ASICs require a large design effort, a long time to implement (while waiting for fabrication), and come at high cost, which excludes them from use in many cases. Nevertheless, high performance still makes ASICs an attractive option for some tasks. An example of such work is the object recognition processor developed by Kim et al. <cit.> which can recognize up to 10 objects at a rate of 60fps at an image size of 640$\times$480 pixels.
Field Programmable Gate Arrays (FPGAs) fall in the space between GPUs and ASICs in terms of time to implementation and level of control. FPGA hardware (fabric) is designed to be highly reconfigurable, thereby giving more control than with GPUs, but the hardware is already fabricated, thereby eliminating the time for fabrication which plagues ASICs. FPGAs also offer an advantage over GPUs in that they can operate in a standalone manner and interface directly with external sensors. A disadvantage of FPGAs is that their use often requires knowledge of a hardware descriptor language (such as Verilog or VHDL) which can be difficult to learn.
In an attempt to make FPGAs more accessible and user friendly, Impulse Accelerated Technologies Inc. <cit.> has developed a C-to-FPGA compiler to make FPGA acceleration more accessible to those not familiar with hardware design languages. A review of this and other C-to-FPGA approaches can be found in <cit.>. The E-lab at Yale is also working on easing the transition to FPGA with the development of “NeuFlow" <cit.>, an FPGA based system which can be programmed using the easier to learn Lua <cit.> scripting language. This approach significantly reduces time to implementation, but does not necessarily allow the user to fully exploit the performance capabilities of the FPGA. Despite being a valuable tool, the NeuFlow architecture is not well suited to implementing large filters (the original HMAX model requires filters up to 37$\times$37 pixels in size). Other architectures for implementing HMAX on FPGA, developed in parallel with the work in this paper, have been recently published <cit.>. These implementations also show considerable speedup over GPU and CPU implementations. Most interesting of these works is a paper from Kestur et al. <cit.> which operates on higher resolution images (2352$\times$1724 pixels), but uses a saliency algorithm to identify regions of interest, thereby obtaining further speedup by circumventing the need for an exhaustive search. Further discussion and comparison with these works can be found in the discussions (Section <ref>).
Despite the difficulties of learning hardware design languages, many other vision algorithms have also been implemented in FPGA, including the Lucas-Kanade <cit.> optical flow algorithm <cit.>, SIFT <cit.>, SURF <cit.> spatiotemporal energy models for tracking <cit.> and segmentation <cit.> as well as bioinspired models of gaze and vergence control <cit.>. There are also many examples of Neural Networks (NNs) implemented in FPGA, including multilayer perceptrons <cit.>, Boltzmann machines <cit.>, and spiking NNs <cit.>.
In work on multilayer perceptrons, Savich et al. <cit.> compared the use of fixed point and floating point representations for FPGA implementation and found that fixed point representation used less physical resources, fewer clock cycles, and allowed a higher clock speed than floating point representation while achieving similar precision and functionality. In this work fixed point representation is used throughout.
Himavathi et al. <cit.> described a Neural Network implementation in FPGA which multiplexed resources for computation in different layers, to reduce the total resources required at the expense of computation time. The ultimate aim was to use resources more effectively. In HMAX cells differ by layer, so instead resources are multiplexed for different cells within the same layer. The ultimate aim is similar, to use resources as effectively as possible, thereby achieving maximum throughput with the available resources.
The computation performed by the first four layers of HMAX is task independent, allowing us to easily estimate required computation and allocate resources accordingly. The classifier, which follows the fourth HMAX layer, differs depending on the task (binary or multi-class), and in the case of multi-class, the required computation is further dependent on the number of classes (see Section <ref>). To simplify implementation and maintain flexibility of the system, we implement the classification stage in the loop on a host PC. We show through testing in Section <ref> that implementing the classifier in the loop on a host PC does not affect the system throughput. Implementing a classifier in FPGA is nevertheless possible, as is evidenced by numerous examples of FPGA classifier implementations in the literature, including Gaussian Mixture Models (GMMs) <cit.>, NNs <cit.>, Naive Bayes <cit.>, K-Nearest Neighbour (KNN) <cit.>, Support Vector Machines (SVMs) <cit.>, and even a core-generator for generating classifiers in FPGA <cit.>.
To remain consistent with previous work <cit.> and provide a fair comparison, a boosting classifier was used when performing binary classification, and a linear (SVM) classifier was used when performing multi-class classification. The use of linear SVM is further supported by Misaki et al. <cit.>, who did a comparison of multivariate classifiers in a visual object discrimination task using FMRI data from early stages of human visual and inferior temporal cortex. Linear classifiers were found to perform better than non-linear classifiers, which they note is consistent with previous similar investigations <cit.>. Misaki et al. also note that non-linear classifiers may perform better if larger datasets are used for training, or if fewer features are used. Non-linear classifiers can better fit the training data, but this comes with the risk of overfitting the classifier to the data, which is particularly problematic when only a few training samples are used.
The rest of this paper describes how the original model <cit.> was adapted for implementation on an FPGA to increase throughput and how these adaptations affect recognition accuracy. To test the FPGA implementation we performed a binary classification task on popular categories from the commonly referenced, publicly available Caltech 101 <cit.> dataset as well as a tougher minaret dataset comprised of images downloaded from Flickr. We also investigated multi-class classification accuracy using Caltech 101. Results are compared to previously-published test results on the same dataset using a software implementation of the HMAX model <cit.>. An analysis of how the image throughput rate and required hardware would change with input image size is also presented. The aim of this paper is not to beat the state of the art in terms of recognition accuracy, but rather to show how a given model can be adapted for implementation on an FPGA to drastically increase throughput while maintaining the same level of recognition accuracy.
§ ORIGINAL MODEL DESCRIPTION
The version of the HMAX model used <cit.> has two main stages, each consisting of a simple and complex substage. We will call these Simple-1 (S1), Complex-1 (C1), Simple-2 (S2) and Complex-2 (C2) as is done in the original paper.
§.§ S1
In S1 the image is filtered at each location with Gabor filters applied at 16 different scales with the side length of a filter ranging from 7 to 37 pixels in increments of 2 pixels as shown in Table <ref>.
For each filter size the filter is applied at four different orientations (0$^o$, 45$^o$, 90$^o$, and 135$^o$). For each filter position the underlying image region is normalized before filtering to increase illumination invariance. The output of S1 consists of 64 filtered versions of the original image (16 scales $\times$ 4 orientations). The sign of the result is dropped and only the magnitude is passed to C1.
Parameters used in FPGA implementation of HMAX. This table is adapted from the parameter table shown in <cit.>.
2*Size Band# Subsampling Filter Sizes 2*$\sigma$ 2*$\lambda$
period $\Delta$ $\diameter \times \diameter$
2*Band 1 2*4 7$\times$7 1.3 3.9
9$\times$9 1.7 5.0
2*Band 2 2*5 11$\times$11 2.1 6.2
13$\times$13 2.5 7.4
2*Band 3 2*6 15$\times$15 2.9 8.7
17$\times$17 3.3 10.0
2*Band 4 2*7 19$\times$19 3.8 11.3
21$\times$21 4.2 12.7
2*Band 5 2*8 23$\times$23 4.7 14.1
25$\times$25 5.2 15.5
2*Band 6 2*9 27$\times$27 5.7 17.0
29$\times$29 6.2 18.5
2*Band 7 2*10 31$\times$31 6.7 20.1
33$\times$33 7.2 21.7
2*Band 8 2*11 35$\times$35 7.8 23.3
37$\times$37 8.3 25.0
§.§ C1
Filter responses are grouped by filter sizes into 8 size-bands as shown in Table <ref>. Within each size-band the response of a C1 unit is the maximum of the S1 units in that size-band over a small local spatial region ($2\Delta\times2\Delta$ from Table <ref>). The result is then subsampled (every $\Delta$ pixels) and output to S2. The output is therefore 32 sets of C1 units (8 size-bands $\times$ 4 orientations).
§.§ S2
S2 units have as their inputs C1 units from all four orientations. They compute the Euclidean distance between a predefined patch and the C1 units at every location. The patch sizes are $4\times4\times4$, $8\times8\times4$, $12\times12\times4$ and $16\times16\times4$ ($x \times y \times orientation$). For every S2 unit the patch distance is computed at every (x,y) location within every size-band and passed to C2.
§.§ C2
The C2 layer computes the minimum of the S2 distance for each patch across all locations in all size-bands. The number of C2 outputs is therefore equal to the number of S2 patches used.
§.§ Classification
Classification is performed directly on the C2 outputs. The choice of classifier can vary based on the required task. Previous work <cit.> presented results using a boosting classifier for binary classification, and a linear SVM one-vs-all classifier for multi-class classification.
§ FPGA IMPLEMENTATION
§.§ Hardware Description
The large number of Multiply ACcumulate (MAC) operations required to implement the 64 filters in S1 and the 1000 patches in S2 make the number of multipliers available on an FPGA one of the limiting constraints for throughput. The second limiting constraint is the amount of internal memory available. We need to ensure we have enough memory to store all intermediate results, S2 patches, and S1 filters since we can save time by not loading S1 filters and S2 patches from external memory, as will be shown in Section <ref>. Multiple block RAMs are used in parallel whenever data wider than 16 bits needs to be stored. We chose to use the Xilinx XC6VLX240T from the Virtex 6 family for its large number of multipliers (768) combined with its reasonable price of $1800 for a development board (Xilinx `EK-V6-ML605-G' board). The S1, C1, S2, and C2 stages were each implemented as separate modules in VHDL using a pipelined architecture.
§.§ Edge Effects
The most obvious way to speed up the model is to not waste resources on unnecessary computation. For this reason we chose to only compute filter responses and patch distances when the filter (S1) or patch (S2) has full support. We effectively ignored any computation which involves regions beyond the image edges.
§.§ S1 Filters
Block diagram of our hardware implementation of HMAX. Red rectangles indicate the usage of block RAM. Input to the 1 dimensional S1 filters can come from either the input image or from RAM holding intermediate results. S1 results are sent to C1 where the maximum is computed over a local region (see Table <ref>) and stored. Each size band has its own dedicated RAM. A demultiplixer controls reading of C1 results for the S2 stage. C2 computes the global maximum of S2 outputs and stores the results in RAM before transferring them to the Ethernet transmit buffer.
The S1 layer consists of directionally selective Gabor receptive fields, similar to the selectivity of simple cells found by Hubel and Weisel <cit.> in V1. We implement cells at four different orientations (0$^o$, 45$^o$, 90$^o$ and 135$^o$) as was done in the original model <cit.>. Due to symmetry, we need not compute cells at orientations at or above 180$^o$. Each orientation is implemented at sixteen different scales and at every location in the image where full support is available. The equations defining the filters used in the original HMAX model <cit.> are repeated in (<ref>) for convenience. The equations for the filters are the product of a cosine function and a Gaussian weighted envelope:
\begin{equation}
\label{eq:Original Filters}
\begin{array}{l l}
F_{\theta}(x,y) &= e^{(-\frac{x_0^2+\gamma^2y_0^2}{2\sigma^2})} \times \cos{(\frac{2\pi}{\lambda}x_0)}\\[8pt]
x_0 &= x\cos{\theta} + y\sin{\theta}\\[8pt]
y_0 &= -x\sin{\theta} + y\cos{\theta}.\\
\end{array}
\end{equation}
Here $\lambda$ determines the spatial frequency at the filter's peak response, $\sigma$ specifies the radius of the Gaussian window and $\gamma$ squeezes or stretches the Gaussian window in the $y_0$ direction to create an elliptical window. For the 0$^o$, and 90$^o$ cases we can easily rewrite this equation as product of two separate functions as shown in (<ref>). The 45$^o$, and 135$^o$ terms are not separable unless we change the Gaussian weighting function to an isotropic function by specifying $\gamma~=~1$. By doing this we arrive at the equations for the 45$^o$ and 135$^o$ filters shown below:
\begin{equation}
\label{eq:Separable Filters}
\begin{array}{l l}
F_{0}(x,y) = E(x,y)*G(x,y)^T\\[8pt]
F_{90}(x,y) = E(x,y)^T*G(x,y)\\[8pt]
F_{45}(x,y) = E(x,y)*E(x,y)^T + O(x,y)*O(x,y)^T\\[8pt]
F_{135}(x,y) = E(x,y)*E(x,y)^T - O(x,y)*O(x,y)^T\\[8pt]
E(x,y) = e^{(\frac{-x^2}{2\sigma^2})}\cos{(\frac{2\pi x}{\lambda})}\\[8pt]
G(x,y) = e^{(\frac{-\gamma^2x^2}{2\sigma^2})}\\[8pt]
O(x,y) = e^{(\frac{-x^2}{2\sigma^2})}\sin{(\frac{2\pi x}{\lambda})}\\[8pt]
%N(x,y) = \displaystyle\sqrt{\sum_{(x,y)} I(x,y)^2}.\\
\end{array}
\end{equation}
Here $(x,y)$ is the location of the kernel value within the filter, $O(x,y)$ is an odd Gabor filter, $E(x,y)$ is an even Gabor filter, and $G(x,y)$ is a pure Gaussian filter. $A*B$ designates the convolution of $A$ and $B$, while $A^T$ designates the transpose of $A$. By writing the filters in a separable manner, we can implement them using two passes of a one dimensional filter rather than one pass of a two dimensional filter <cit.>. The number of MAC operations required to implement a separable filter grows linearly with the side length of the filter rather than as the square of the side length and therefore results in a significant speed up, or in the case of FPGA implementation, a significant saving of resources. If we consider the specific case of implementing the 64 S1 filters at a single image location, we can compute the number of multiply accumulates required using
\begin{equation}
\label{eq:separable cost}
\begin{array}{l l l}
MAC_{original} &= 4 \times \sum_{i=1}^{16} [\diameter(j)^2] &= 36416\\
\\
MAC_{separable} &= 4 \times \sum_{i=1}^{16} [2\times \diameter(j)] &= 2816\\
\end{array}
\end{equation}
where $\diameter(j)$ is the side length of filter $j$ as indicated in Table <ref> and in (<ref>).
Using separable filters reduces the number of required multiply accumulates from 36416 down to 2816, a reduction to less than 8% of the originally required computation. Furthermore, each one-dimensional filter used has either even or odd symmetry about the origin, allowing us to sum values in the filter support either side of the origin before performing multiplication. By exploiting the symmetry of the filter the required multiplications are reduced by a further 50%, freeing up more dedicated hardware multipliers for use in the more computationally intensive S2 stage of processing. Using separable instead of non-separable filters reduces the time taken to compute the S1 filter responses from 2.3 seconds to 0.3 seconds per 128$\times$128 image in Matlab.
To increase illumination invariance, the filter response at each location is normalized by the $l^2$ norm of its support, as is done in the original model. This normalization ensures that filters capture information about the local contrast and are unaffected by the absolute brightness of a pixel region. The $l^2$ norm is computed by first summing the squares in the x-direction, then summing the result in the y-direction and taking the square root. We timed this result to be available simultaneously with the filter results so that we can immediately perform division without the need to store intermediate results. Responses for filters at all four orientations are computed in parallel, eliminating the need to recompute or store the $l^2$ norm of the filter support for each orientation.
The filter kernels are all pre-computed and stored in a look up table (see Fig. <ref>). Each filter is modified to have zero mean and an $l^2$ norm of $(2^{16}-1)$ to ensure that results are always less than 16 bits wide. The parameters used for these separable filters is shown in Table <ref>. These parameters can be written into equations as shown in (<ref>) below.
\begin{equation}
\label{eq:Parameterise}
\begin{array}{l l}
\diameter(j) &= 5 + 2\times j\\[8pt]
\Delta(b) &= 3+b\\[8pt]
\kappa(k) &= (4\times k)^2\\
\end{array}
\end{equation}
where $j$ is an index for filter sizes arranged from the smallest to largest (1 to 16). The diameter of filter $j$ is $\diameter(j)$. The filter is actually square with side length $\diameter(j)$ to avoid the complexity of implementing a round filter. The subsampling period of size band $b$ is written $\Delta(b)$. $k$ is an index for the size of patches (1 to 4 for the four different patch sizes). At each orientation a patch of size index $k$ will have size $\kappa(k)$.
§.§ C1
The C1 layer requires finding the maximum S1 response over a region of $2\Delta\times2\Delta$ and subsampling every $\Delta$ pixels in both $x$ and $y$ (for values of $\Delta$ see Table <ref>). We computed the maximum of a $2\Delta\times2\Delta$ region by first computing the maximum over adjacent non-overlapping regions of size $\Delta\times\Delta$. By taking the maximum across every 4 adjacent $\Delta\times\Delta$ regions we obtained the maximum over a $2\Delta\times2\Delta$ region, subsampled every $\Delta$ pixels in both $x$ and $y$.
Computing on data as it streams from S1 eliminates the need to store non-maximal S1 results (see Fig. <ref>). As with the S1 layer, computation in C1 is performed on all four orientations in parallel. Each time C1 finishes computing the results for a size band, a flag is set which indicates to S2 that it can begin computation on that size band.
§.§ S2
Even though the data coming into S2 has already been reduced by taking the maximum across a local pool and subsampling in C1, the S2 layer is where most of the computation takes place. The number of MAC operations required to compute all patch responses at a single location in the original model is:
\begin{equation}
\label{eq:patchMAC}
\begin{array}{l l}
250\times4\times\sum_{k=1}^4\kappa(k) &= 480 000\\
\end{array}
\end{equation}
where there are 250 patches per size and 4 orientations per patch, each of size $\kappa(k)$, which was defined in (<ref>). The computation of these patch responses must be repeated at all locations within all size-bands.
We decided to use 1280 patches (320 per size) which was a compromise between speed of implementation and the number of patches. As in the original model, S2 patches are obtained from previously computed C1 results on images from both the positive and negative classes. Since S2 patches are simply portions of previously computed C1 outputs, the number of bits required to store each patch coefficient is 16. The closeness of a patch to a C1 region is computed as the Euclidean distance between the patch and that region.
We computed patch responses starting with the smallest sized patches ($x\times y\times orientation \rightarrow 4\times4\times4$) and computing their response at a single location. We then repeat this computation for all locations in the current size band, before moving onto the next patch size. Once all patch sizes have been computed for all locations in the current image size-band we move onto the next size-band as soon as it is available from C1. All patches of the size currently being considered are computed in parallel. Furthermore, the response at two different orientations is considered in parallel. This results in $320\times2 = 640$ parallel multiply-accumulate operations every clock cycle. This uses 640 multipliers and requires that 640 patch coefficients be read every clock cycle. Patch coefficients are stored in the FPGA's internal block RAM since the bandwidth to external RAM would not allow such high datarates. Using external RAM would require a data rate of $640\times16 bits\times100MHz~=~1Tb/s$ for a 100MHz clock.
§.§ C2
C2 simply consists of a running minimum for each S2 patch, computed by comparing new S2 results with the previously stored S2 minimum. This is performed for all 320 S2 patches of the current size simultaneously (see Fig. <ref>).
§.§ Classifier
Results from <cit.> suggest that a boosting classifier is better than SVM for the binary classification problem. We used the gentleboosting algorithm <cit.> with weak learners consisting of tree classifiers each with a maximum of three decision branches before reaching a result as shown in Fig. <ref>. We used 1280 weak learners in the classifier, each computed in series.
A weak learner used in the gentle boosting algorithm. Each weak learner is a tree consisting of 7 nodes. $F(x,y)$ represents the feature used at node $y$ in weak learner $x$. $O_{(x,1)}$ through $O_{(x,8)}$ are the binary outputs of classifier $x$. Each output is a binary value 1 or -1.
For multi-class classification a linear one-vs-all SVM classifier was chosen <cit.>. This is a simple linear classifier, but is memory intensive in its requirement for storing coefficients, as is discussed in Section <ref>.
In order to not restrict the FPGA implementation to only binary problems or only multi-class problems, the classifier was implemented separately on a host PC.
§.§ Scheduling
The FPGA implementation has an input FIFO buffer capable of holding up to four complete 128$\times$128 pixel images. As soon as at least one full image has been loaded into the buffer S1 will read the image. S1 then computes responses at all four orientations for the smallest filter simultaneously and outputs the results in a streaming fashion to C1. After computing the responses from the smallest filter, S1 filters will read in coefficients for the next filter size and compute the new filter responses. S1 will continue in this manner until responses for all filter sizes have been computed. S1 will read a new image from the input buffer as soon as it has completed the first pass with the largest separable filter, or as soon as an image becomes available if none are available at the time.
The C1 and C2 layers operate on the results of S1 and S2 as they are output in a streaming fashion during computation, thereby reducing the internal memory required to store intermediate results. This approach also ensures that C1 and C2 only add a negligible amount of processing time to the algorithm (less than 100 $\mu$seconds for an entire image).
Each stage (S1, C1, S2, C2) uses its own dedicated FPGA resources, thereby allowing all stages to run simultaneously. Sharing of memory occurs between C1 and S2, where access is managed by setting and clearing flags. There is a separate memory unit and flag for each image band. When a flag is low, C1 has exclusive read/write access to the corresponding memory unit. Once C1 has finished storing results in the memory unit, it will set the corresponding flag high. When a flag is high, the S2 stage has exclusive read/write access to the corresponding memory block and will clear the flag once it has finished processing all data from that memory block, thereby transferring control back to C1.
If waiting for access to a particular memory block, a stage (C1 or S2) will begin processing as soon as access is granted (the very next clock cycle). Since results for each image band are stored separately, the S1 and C1 stages can process the next image band (and loop around) without having to wait. This allows S1 and C1 to be almost an entire image ahead in computation than the S2 stage, which is important because although the S1 and C1 stages take the same length of time to process each image band, the time taken by S2 varies. The S2 stage takes longer to compute on smaller image bands because their higher frequency of subsampling produces more C1 results on which computation must be performed (see Table <ref>).
Buffering of C1 outputs in the manner described allows us to focus on matching the throughput of the S1 and C1 stage with the average throughput (across image bands) of the S2 stage, without being troubled by how computation time in S2 varies with each image band.
S1 will not compute new results for an image band if the current results for that image band (from the previous image) have not yet been processed by S2 (i.e. if the relevant memory flag is still high). S1 will however still perform the first pass with a separable filter in the meanwhile to ensure it can start outputting results as soon as the flag is cleared.
Results from S2 stream to C2, which writes the final results to an output buffer for communication back to the host PC.
§ SCALABILITY OF FPGA IMPLEMENTATION
In this section we show how the input image size affects the hardware resources and time required for computation using the FPGA implementation described in Section <ref>. The described FPGA implementation was specifically designed to operate on images of size 128$\times$128 pixels and is therefore not necessarily recommended as the best implementation for larger or smaller images. Nevertheless, if implementing a new design to operate on larger (or smaller) images, extrapolating the current design to different sizes provides a good starting point.
§.§ Hardware Resources
The number of bits in the counters used to track the progress of computation on the input image and intermediate results in stages S1, C1, and S2 will need to increase to handle larger images. This increase scales as:
\begin{equation}
\label{eq:counter size}
\begin{array}{l l}
Counter Bits \propto \log_2{\sqrt{N}}\\
\end{array}
\end{equation}
where $N$ is the number of pixels in the input image and the image is assumed to be square, having side length $\sqrt{N}$. This increase in required hardware is negligible, especially in comparison to the increase in internal RAM required to store the input image and intermediate results in the S1 and C1 stages.
The internal RAM requirement scales proportionally to $N$ for large images. Due to the nature of computation in S2 and C2, no additional RAM is required in those stages when the image size increases. The number of elements required to compute multiplication, addition, division, and square roots remains unchanged in all stages. The total required internal RAM is the sum of the RAM required by all stages.
Internal RAM is required for three purposes in S1: storing the input image, storing intermediate results between the first and second passing of the separable filter and finally, to store the S1 filter coefficients. The required RAM can be explicitly calculated using (<ref>) below.
\begin{equation}
\label{eq:S1bits}
\begin{array}{l l}
S1_{bits} &= S1_{input} + S1_{intermediate}+ S1_{filters}\\[8pt]
S1_{input} &= 4\times N \times 8\\[8pt]
S1_{intermediate} &= 5\times N \times 23\\[8pt]
S1_{filters} &= \sum_{j = 1}^{16}{(2 \times(3+j) \times 16)}\\[8pt]
\end{array}
\end{equation}
$N$ represents the number of pixels in the input image. The $input$ buffer has to hold four images (a FIFO buffer) with 8 bits per pixel. The $intermediate$ results require 5 buffers (one for each orientation and one for calculating the $l^2$ norm of the filter support). Each result consists of 23 bits. For storage of the $filters$, the $j^{th}$ filter (ordered smallest to largest) consists of 2 separable filters, each with $(3+j)$ coefficients and 16 bits per coefficient.
The output of the S1 stage does not require RAM for storage since each result is processed by C1 as soon as it becomes available, but C1 does require RAM for intermediate and final results. The RAM required by C1 can be explicitly calculated using (<ref>) below.
\begin{equation}
\label{eq:C1bits}
\begin{array}{l l}
C1_{bits} &= \sum_{b = 1}^{8}C1_{size}(b)\times 16\\
\\
C1_{size}(b) &= \frac{S1_{size}(b)}{\Delta(b)^2}\\
\\
S1_{size}(b) &= 4\times(\sqrt{N}-\diameter(2b)+1)^2\\
\end{array}
\end{equation}
The number of valid S1 results in image band $b$ is then given by $S1_{size}(b)$, where $\diameter(2b)$ was previously defined in (<ref>) and there are 4 orientations. The number of C1 results can then be calculated knowing the number of S1 results and the subsampling period $\Delta(b)$, which was also previously defined in (<ref>). Each C1 result occupies 16 bits.
The RAM required for S2 is constant across image sizes and can be written explicitly as:
\begin{equation}
\label{eq:S2bits}
\begin{array}{l l}
S2_{bits} &= \sum_{k = 1}^{4}320\times 4\times \kappa(k) \times 16 \\
\end{array}
\end{equation}
where $k$ is an index of patch size. There are 320 patches per size and 4 orientations per patch, each with $\kappa(k)$ coefficients as previously defined in (<ref>). Each coefficient occupies 16 bits.
C2 requires only enough RAM to hold the final C2 results.
\begin{equation}
\label{eq:C2bits}
\begin{array}{l l}
C2_{bits} &= 1280\times42\\
\end{array}
\end{equation}
where there are 1280 C2 features each consisting of 42 bits.
Although we implement the classifier on the host PC, it is possible to determine the resources required by the classifier. The most memory intensive classifier used in this paper is the 102 class one-vs-all linear SVM classifier, for which the memory requirements are:
\begin{equation}
\label{eq:ClassifierBits}
\begin{array}{l l}
Classifier_{bits} &= 102\times1280\times32 + 84\\
&= 4178004 bits\\
\end{array}
\end{equation}
where there are 102 possible classes, 1280 C2 features, 32 bits per coefficient, and up to 84 bits required to hold the result. The current FPGA implementation does not have enough remaining internal memory to hold all these coefficients, but the coefficients could easily fit into external RAM, or the classifier could be run on a second FPGA. If running at 190 images per second, an external memory bandwidth of $102\times1280\times32\times190 = 794Mbps$ per second would be required, which is only about 6% of the available 12.8Gbps bandwidth on the targeted FPGA platform. In our implementation, running the classifier on a host PC did not affect the system throughput.
§.§ Time
The time taken to process an image is dominated by the S1 and S2 stages. The C1 and C2 stages perform simple maximum operations on each valid data point as it becomes available and therefore do not contribute significantly to the time taken to process an image. The time computed in the equations below is in units of clock cycles and the actual time taken for computation therefore depends on the FPGA clock frequency.
The time taken to compute S1 can be accurately approximated as the time required to do 2 passes of the image for each of the 16 separable filter sizes (<ref>). All four orientations are simultaneously computed in parallel and therefore the multiple orientations do not add to computation time.
\begin{equation}
\label{eq:S1Time}
\begin{array}{l l}
S1_{time} &= 2\times N \times 16\\
\end{array}
\end{equation}
where $S1_{time}$ is in units of clock cycles, $N$ is the number of pixels per image and 16 filter sizes are implemented.
In S2, all 320 patches of the same size are considered simultaneously and within each patch, computation is performed at two orientations simultaneously.
\begin{equation}
\label{eq:S2Time}
\begin{array}{l l}
S2_{time} = \sum_{b = 1}^{8}\sum_{k = 1}^{4}S2_{size}(b,k) \times \kappa(k) \times2\\
\\
S2_{size}(b,k) = (\sqrt{C1_{size}(b)}-\sqrt{\kappa(k)} +1)^2\\ %iff C1_{size}(b)>= \kappa(k) \\
%S2_{size}(b,k) = \\
%0 & if \: C1_{size}(b)< \kappa(k) \\
%(\sqrt{C1_{size}(b)}-\sqrt{\kappa(k)} +1)^2 & otherwise
\\
\end{array}
\end{equation}
where $S2_{size}(b,k)$ is the number of valid S2 results for size band $b$ and patch size index $k$. $S2_{size}(b,k)$ is zero whenever the size of the C1 results is smaller than the patch size, that is when $C1_{size}(b)< \kappa(k)$. $\kappa(k)$ is the patch size and was previously defined in (<ref>). $S2_{time}$ is the total time (in clock cycles) taken to compute all patch responses of all sizes in every size band.
If the multi-class one-vs-all linear SVM classifier were to be implemented on the FPGA with 102 classes and only a single hardware multiplier, the time taken could be computed as
\begin{equation}
\label{eq:classTime}
\begin{array}{l l}
Classifier_{time} &= 1280\times102\\
\end{array}
\end{equation}
for 1280 C2 features and 102 classes. The time taken for classification would not be dependent on the input image size. Using a single multiplier would enable a throughput of up to 765 images per second when using a 100MHz clock.
§ SIMULATION
Four different sets of code were used in simulation. The first is a Matlab implementation of the HMAX model which was retrieved from HMAX website <cit.>. This was used as a benchmark against which to compare our modified implementation of HMAX for FPGA to verify that the modifications made did not severely compromise recognition accuracy. We refer to this original HMAX implementation as `HMAX CPU'.
The second, third, and fourth sets of code are Matlab, C++, and VHDL implementations respectively of our modified version of HMAX for FPGA. These implementations are functionally equivalent and we refer to them as `HMAX FPGA'. The Matlab code was used to make initial changes to the model and test accuracy on small datasets. Once satisfied with the changes made, a faster C++ implementation was written and used to verify the modified model on larger datasets. Finally, the actual VHDL code required to implement the proposed model in FPGA was written. This VHDL code was used to determine possible clock speeds and image throughput as well as to verify that the proposed FPGA model could be implemented using the resources available on the targeted FPGA platform (Xilinx Virtex 6 XC6VLX240T). Both final and intermediate results from the modified Matlab, C++, and VHDL codes were compared to verify that all three were performing the same computation.
§ HARDWARE VALIDATION
The results of simulation were verified through implementation on the Xilinx Virtex 6 ML605 development board. A C++ interface was written for the host PC which handles Ethernet communications with the ML605 board and performs classification. The C++ code transmits four images to the ML605 board to fill the input buffer (described in Section <ref>), then waits for all 1280 C2 values from an image to be returned before transmitting the next image. Reading of images from the hard drive and classification are both performed while waiting for the next set of C2 values from the FPGA, thereby adding negligibly to the overall computation time. Classification results are written to an output file as they are computed. For further verification C2 results from FPGA could be optionally written to disk for direct comparison against simulated C2 results.
Theoretical analysis of the internal RAM required by each stage as well as the total RAM required and total RAM available on the selected FPGA. The vertical line shows the number of pixels in a 128$\times$128 image, for which this implementation was designed.
§ RESULTS
§.§ FPGA code analysis
Using the Xilinx ISE, the VHDL code for implementing HMAX on FPGA was analyzed. For simplicity we use a single clock for all stages within the model. All lookup tables, S1 filters, and S2 patches as well as all intermediate results are stored in internal block RAM, as shown in Fig. <ref>. The system has a latency of 600k clock cycles when processing a single image, but can maintain a throughput of an image every 526k clock cycles. Implementation of the full model indicates that the design can run at a clock frequency of 100MHz (10ns period). A 100MHz clock results in a latency of 6ms for processing a single image and a maximum throughput of 190 images per second when processing multiple images. These figures are achieved assuming that the input figure is a 128$\times$128 pixel 8-bit per pixel grayscale image.
The throughput of the design is determined by the throughput of the slowest stage in the pipeline. Computational resources should therefore be allocated in such a way that all stages have roughly the same throughput. This has been done as is evident in the distribution of multipliers between the S1 and S2 stages. S1 is the slowest stage, limiting the throughput to 190 images per second using 77 multipliers at 100MHz clock frequency, while S2 is capable of a throughput of 193 images per second, but uses 640 multipliers.
If we were to create an optimal implementation of S1 using non-separable filters with a 100MHz clock, then S1 alone would require over 1600 multipliers to achieve the same throughput of 190 images per second (unless a scale space approach was adopted). This is over double the number of hardware multipliers available on the chosen FPGA.
Table <ref> shows the total resources used by the HMAX implementation.
FPGA resources used by HMAX
Resource Used Available % used
Multipliers 2*717 2*768 2*93
Internal RAM 2*373 2*416 2*89
2*Slice Registers 2*66 196 2*301 440 2*21
2*Slice LUTs 2*60 872 2*150 720 2*40
§.§ Scalability
Fig. <ref> shows the internal RAM requirements computed using the equations presented in Section <ref>, as well as the total block RAM available on the selected Virtex 6 FPGA (14976kb, dashed line) and the image size for which the algorithm was designed (128$\times$128 pixels, vertical line).
Since all S2 patches of the same size are computed in parallel, the number of patches does not affect computation time, but will be limited by the number of available multipliers and amount of RAM available (see Table <ref>).
The time taken to compute the S1 and S2 stages is shown in Fig. <ref> along with the number of pixels for which the current implementation was designed (vertical line). The throughput of the complete system is limited to the throughput of the slowest stage.
A theoretical analysis of the time taken to compute each stage of HMAX in the current architecture. Due to the pipelined nature of the computation, the rate at which images can be processed is limited by the stage which takes the longest time. The vertical line shows the number of pixels in a 128$\times$128 image, for which this implementation was designed. The time required to compute the two longest stages is equal at this point as a result of the effort to allocate resources in such a way as to maximize throughput.
The time taken to compute S2 can be seen as the time which would be taken to compute all results (even partial results on edges) minus the time which is saved by not computing edge results. The time saved by not computing at edges is significant at an image size of 128$\times$128. The time saved grows proportionally to the side length of the image $\sqrt{N}$, which is much slower than the time to compute all results (which grows linearly with $N$). This is why the time for S2 grows linearly with $N$ only for large $N$. S1 always grows linearly with $N$.
The design of the current framework ensures that the time taken for S1 and S2 is roughly equal (within 2%) for images of size 128$\times$128, thereby ensuring that computational resources in each stage are not sitting idle waiting for the other stage to finish computing. If working with images of a different size, resources would ideally be reallocated to ensure that S1 and S2 still take equal time.
§.§ Caltech 101 binary classification
Two datasets were used to test the recognition accuracy of our modified HMAX model. The first is the often referenced Caltech 101 dataset <cit.>. Recognition accuracy of popular categories in this dataset were presented for the HMAX model in <cit.>. We ran our own binary classification simulations on these categories using both the downloaded and modified versions of HMAX. The binary task constituted discriminating the class in question (airplanes, cars, faces, leaves, or motorbikes) from the background class. In each case, half the images from the class in question and half images from the background class were used for training. The remaining images from both the class in question and the background class were used for testing. In each case 10 trials were run. The accuracy reported in Table <ref> is the percentage of correct classifications at the point on the ROC curve (Fig. <ref>) where the false positive and false negative rates are equal. Looking at the mean accuracy for this metric, the FPGA implementation achieves 0.24% higher accuracy than the original CPU implementation. This shows that the modifications made for the FPGA implementation have not adversely affected recognition accuracy.
Comparison of recognition accuracies obtained from original HMAX code and FPGA implementation on popular categories in Caltech 101
Category HMAX <cit.> HMAX CPU HMAX FPGA
Airplanes 96.7 97.1 98.2
Cars 99.7 99.3 99.2
Faces 98.2 95.8 96.4
Leaves 97.0 94.6 93.7
Motorbikes 98.0 98.3 98.8
Receiver Operating Characteristics for the binary classification task on Caltech 101 popular image categories and Minaret datasets. Each curve is the result of a mean over 10 trials. Note that the True Positive Rate axis is different for the Minaret classification task.
A sample of images from the minaret (top row) and background (bottom row) classes used in the minaret binary classification task.
§.§ Binary classification on Flickr dataset
The binary minaret classification task was performed on a dataset containing 662 images of minarets and 1332 background images. The minaret (positive) images were obtained from Flicker by searching for “Minaret" while negative images were obtained by periodically downloading the most recently uploaded Flicker image. Examples of these images are shown in Fig. <ref>. Ten random splits were used for classification and testing, with the test set consisting of 1000 negative and 500 positive images. The remaining images constitute the training set. This test was performed with both the downloaded HMAX code and the modified HMAX code for FPGA. The results are shown in Table <ref>. The metric used is the percentage of correct classifications at the point where false positive and false negative rates are equal. As expected, using 2000 features instead of 1280 improves the accuracy for both the CPU and FPGA implementations. The accuracy of the FPGA implementation is within 1% of that of the original model.
Comparison of results obtained from original HMAX code and FPGA implementation on minaret classification task
Model HMAX CPU HMAX CPU HMAX FPGA HMAX FPGA
Features 2000 1280 2000 1280
Accuracy 82.9 82.2 82.2 81.3
§.§ Caltech 101 multi-class one-vs-all
A second test using the Caltech 101 database is the multi-class one-vs-all test. For this we used 15 training examples per category, as was done in <cit.>. Testing was performed using 50 examples per category or as many images as remained if fewer than 50 were available. Each of the categories was weighted such that it contributes equally to the result as was done in <cit.>. This is a 102 category problem including the background category. Using the one-vs-all linear SVM multi-class classifier from <cit.> we achieved a mean accuracy of 47.2 $\pm$ 1.0% over 10 trials, which is in agreement with the result of 44 $\pm$ 1.14% reported in <cit.> for the same task. The slight increase in accuracy can be attributed to the fact that our FPGA implementation uses 1280 features compared to 1000 features used in <cit.>. The confusion matrix for the 101 multi-class one-vs-all problem is shown in Fig. <ref>.
Confusion matrix averaged over 10 trials for the 102 category multi-class one-vs-all test performed on the Caltech 101 database. The low accuracy on the extreme bottom left of the diagonal is the background category. The largest confusion is between the `schooner' (81) and `ketch' (57) categories, which are similar cases of sailboats.
§.§ Hardware Validation
Maximum Throughput for Each Stage in Images/sec
Stage Input Buffer S1 C1 S2 C2
Throughput 6100 190 552 193 10 000
Results from analyzing VHDL code were verified by implementing the code on the ML605 board and processing the Caltech 101 database. The entire dataset consisting of 9144 images was processed ten times in different trials. The time taken to complete processing was measured from when the first image is read from disk until the last classification result is written to disk. The time taken to process the entire Caltech 101 database was measured as 48.12s $\pm 57\mu$s, which is a throughput of 190 images/sec and agrees with VHDL simulation predictions (shown in Table <ref>) to within 0.01%. Accuracy of the VHDL implementation was also verified against simulations. Both classification results and C2 outputs from testing were verified against simulation and found to exactly match.
§.§ Comparison to other approaches
To the best of our knowledge 190 images/sec is the fastest reported implementation of this version HMAX. Direct comparisons with other versions are not always straightforward because both the number of patches and their sizes can vary, as well as the size of the input image or even the model itself.
In 2010 Sedding et al. <cit.> presented a time of 86.4ms for 4075 patches using custom code on an NVIDIA GeForce285 GTX. They used sparse features as proposed by Mutch and Lowe <cit.> and claimed a shorter runtime than both the Feature Hierarchy Library (FHLib) <cit.> and the GPU based Cortical Network Simulator (CNS) <cit.>.
In our aim to recreate the original model we chose not to use sparse features, but using sparse features would allow us either a 4$\times$ speedup or it would allow us to implement 4$\times$ as many patches at the same speed (resulting in 5120 patches) on the ML605 board. Their implementation also operates on larger images, with shortest side measuring 140 pixels. If our 1280 dense patch implementation was to run on an image measuring 140$\times$186 pixels (assuming a 3$\times$4 aspect ratio), it would still take under 12ms to complete.
On Caltech 101 with 15 training and 50 test samples per category, our 1280 patch 128$\times$128 pixel model achieves an accuracy of 47.2% (see Section <ref>) whereas Sedding <cit.> achieves 37%, most likely a result of using sparse features. In terms of speed our implementation takes 5.3ms whereas theirs takes 86.4ms. They can reduce their processing time to 8.9ms if they only compute 240 patches, but this will come at the expense of even lower accuracy (less than 30% on the same task).
§ DISCUSSION
The previous section shows that a massive increase in throughput can be achieved with almost no change in recognition accuracy. In this paper the aim has been to achieve a very high throughput as an argument for the use of FPGA in hierarchical models, but one could just as easily trade speed for accuracy. Interestingly our FPGA implementation of HMAX uses more S2 patches (1280) than the 1000 used in <cit.>. This increase in the number of patches was implemented simply because the additional resources required for the patches were available and the parallel processing of patches means that as long as resources are available, adding more patches does not affect throughput.
The issues of image acquisition, rescaling and conversion to grayscale are not tackled by the current model since these will be application specific. The model requires that images are prescaled to 128$\times$128 pixels and converted to 8 bit grayscale before they are processed. The FPGA model requires an input image in the form of raw pixel values. For 190 images per second this translates to just over 3MB of data per second, which is well within the capabilities of the evaluation board's PCI express or gigabit Ethernet interfaces, as has been verified through testing in Section <ref>. If using a laptop, the system can run over gigabit Ethernet allowing it to be portable as shown in Fig. <ref>.
An illustration of the portable hardware setup for the binary classification system showing a laptop communicating pixel values over gigabit Ethernet to a Xilinx ML605 evaluation board containing the Xilinx Virtex 6 XC6VLX240T FPGA on which the HMAX model runs. C2 features are returned to the laptop via the same gigabit Ethernet interface.
The HMAX model used in this paper is one which was freely available in easy to follow Matlab code. It does not represent the least computationally intensive, or most accurate version of the HMAX model. The creators of the model are continuously working on improvements and a number of newer iterations have been presented <cit.>. One of the most significant changes is the use of a scale-space approach such that the image is rescaled and reprocessed multiple times by filters of a single fixed size rather than keeping the image the same size and using multiple filters of varying size. Many recent implementations <cit.> make use of 12 orientations instead of 4, which increases accuracy although it comes at the expense of extra computation time.
We achieved a key speedup in the S1 layer by exploiting the known structure of filters, which allowed us to implement the Gabor filters as separable. The unsupervised learning in S2 means that its structure is not known a priori. If the model were changed to S2 patches of a known structure which could be similarly exploited then further significant speedups could be achieved, but the effect on recognition accuracy would have to be further investigated.
Another change which greatly reduces computational complexity is the use of sparse S2 patches as proposed by Mutch and Lowe <cit.>. In their model only the S1 orientation with maximal response is considered at each image location, thereby reducing the number of orientations in S2 from 4 to 1, which reduces the number of required multiply accumulates to only a quarter of the original. These sparse S2 features are used in most recent works <cit.>. The effect on throughput of using sparse versus dense features, and of changing the number of orientations from 4 to 12, can be found in a recent paper by Park et al. <cit.>. Despite running on four FPGAs, each of which is more than twice as large as our FPGA (Virtex 6 SX475T versus LX240T), their dense implementation of HMAX using four orientations runs at roughly 45 images per second. However there are certain differences, they operate on larger images (256$\times$256 versus 128$\times$128), and use more patches (4075 versus 1280). Using four FPGAs, we could run four copies of our model in parallel, each with different patches, thereby giving us $1280\times4 = 5120$ patches while maintaining throughput of 190 images per second. We also use an equal number of patches of each size, whereas more recent approaches typically use more small (4$\times$4) and less large (16$\times$16) patches to reduce computation. To summarize in comparison with Park et al., we could implement more patches (5120 versus 4075), with a higher percentage of large patches, and a 4$\times$ higher throughput if 4 FPGAs were used. Their implementation uses significantly larger FPGAs than ours (containing 2016 versus 768 multipliers), but also operates on 4$\times$ larger images, making a direct comparison difficult.
A common bottleneck for parallel architectures lies in the available bandwidth to memory and structuring how memory is accessed. For example, if two cores simultaneously request data from memory, one will have to wait for the other before it can access memory. In the presented FPGA implementation this was overcome by using the internal block RAM of the FPGA which resulted in a bandwidth of over 1 Terabit per second, which could be difficult to maintain on other platforms. Other implementations of HMAX which have recently been published also make use of internal block RAM to overcome this memory access bottleneck <cit.>.
The size of the current filters and patches are designed to operate on small images. Even if higher resolution images are available, they should be rescaled to 128$\times$128 if they are to be processed with the current filters and patches. Nevertheless, extension to larger images is possible. Scalability of the current implementation has been presented and shows that larger images can be processed on the current FPGA with minor adjustments, but will ultimately be limited by the amount of internal memory available for buffering images and storing intermediate results. To overcome this one could use a larger FPGA, use multiple FPGA's operating in parallel, reduce the number of S2 patches to free up memory, or change the model to use sparse features.
To provide a fair comparison with the original HMAX model we used the same classifiers (boosting for binary and linear one-vs-all SVM for multi-class). Linear SVM classifiers remain the top choice for most HMAX implementations due to their computational simplicity and speed. The choice of linear SVM classifiers is also supported by other work on discriminating between visual objects based on fMRI recordings of early stages of visual cortex <cit.>. In our implementation we were able to run the classifier in the loop on a host PC without affecting the system throughput because classification was performed in parallel with feature extraction for the next image. Nevertheless, various classifiers can and have been implemented in FPGA <cit.>, including SVM <cit.>, and even a core generator for parameterized generation of your own classifier in FPGA <cit.>.
Comparison with other approaches shows that this is currently the fastest complete HMAX implementation and outperforms reported CNS <cit.> and custom <cit.> GPU implementations, as well as many FPGA implementations, although direct comparison with other FPGA implementations is not always possible. As more powerful GPU platforms become available these GPU implementations will achieve even better results, however the same can be said for FPGAs. The platform we have used (Xilinx Virtex 6 XC6VLX240T) is only in the middle of the range of the Virtex 6 family and is an entire technology generation behind the currently available Virtex 7 family.
§ CONCLUSION
We have shown how a neuromorphic bio-inspired hierarchical model of object recognition can be adapted for high speed implementation on a mid-range COTS FPGA platform. This implementation has a throughput of 190 images per second which is the fastest reported for a complete HMAX model. We have performed binary classification tests on popular Caltech 101 categories as well as on a more difficult Flickr dataset to show that adaption for FPGA does not have a significant effect on recognition accuracy. We have also shown that accuracy is not compromised on a multi-class classification task using Caltech 101.
§ ACKNOWLEDGMENT
This work was partially supported by the Defense Advanced Research Projects Agency NeoVision2 program (government contract no. HR0011-10-C-0033) and the Research Program in Applied Neuroscience.
[]Garrick Orchard
[]Jacob G. Martin
[]R. Jacob Vogelstein
[]Ralph Etienne-Cummings
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