text:
\section{\label{sec:1} Motivation and Overview}

Among distinctive features of quantum mechanics is that, in some regards, quantum system
acts as being simultaneously in multiple states of a given type. Thus, Schr\"odinger
particle with sharp momentum is often said to reside at all positions, having an equal
chance of being detected anywhere. But how many of such ``position identities'' are
effectively present in a generic state which may give the available locations arbitrarily
varied relevance?
Variants of this counting problem appear throughout quantum mechanics. We will discuss
it in a prototypical setting where state $\ketpsi$ is the element of $\nrN$-dimensional
Hilbert space spanned by the orthonormal basis
$\{ \, \keti \} \equiv \{ \, \keti  \mid i=1,2,\ldots,\nrN \,\}$.\footnote{Albeit
discrete, this setting includes many-body and field-theoretic systems whose quantum
dynamics can be defined via lattice regularization.}
Informally, our aim is to inquire

\medskip
\noindent
     {\bf [Q]} {\em How many basis states from $\{ \, \keti \}$ is the system
     described by $\ketpsi$ effectively in?}
\medskip

\noindent
which will be referred to as the quantum identity problem here.
The answer would, among other things, provide a new angle on quantum
uncertainty~\cite{Hei27A, Hor18B}, offer a novel way to describe localization
phenomena~\cite{And58A, Hor18C}, or be used in the efficiency analysis of a quantum
algorithm. However, a well-founded resolution of [Q] is not readily
available\footnote{This is rooted in the fact that quantifiers of the desired type
cannot be expressed as quantum-mechanical expectation values in state $\ketpsi$.},
raising a deeper issue whether the effective number of quantum identities can be
elevated to a meaningful notion at all.

In this paper, we develop a theoretical framework (effective number theory)
which gives rationale to the following answer:

\medskip

   \noindent
   {\bf [A]} {\em Let $P \!=\! (p_1, p_2,\ldots,p_\nrN ) \;,\;
   p_i = \,\mid \!\! \langle \,i \!\mid \! \psi \,\rangle \!\! \mid^2$, be
   the probability
   vector assigned to quantum state $\ketpsi$ and basis $\{ \, \keti \}$, and
   let $\W \equiv \nrN P = (\w_1,\w_2,\ldots,\w_\nrN)$. The system described by
   $\ketpsi$ is effectively in $\efNm[\, \ketpsi , \{ \, \keti \} \,] = \efNm[\W]$
   states from $\{ \, \keti \}$, where}
   \begin{equation}
      \efNm[\W]  \,=\, \sum_{i=1}^\nrN \cfu(\w_i)   \quad,\quad
      \cfu(\w)  \; = \;   \min\, \{ \w, 1 \}    \quad
      \label{eq:015}
   \end{equation}

\noindent
To arrive at [A], we start with the axiomatic definition of {\em effective number
function} (ENF) $\efN[\, \ketpsi , \{ \, \keti \} \,] \!=\! \efN[\W]$, namely
a function consistently assigning effective totals. To solve [Q] then amounts to
finding such $\efN$ and using it to specify the effective number of quantum
identities in all situations.
The subsequent analysis shows, however, that there exists an entire continuum
of ENFs. Such multitude could render each fixed choice of $\efN$ too arbitrary
and its individual value uninformative on its own.\footnote{By construction,
each ENF conveys useful information on relations between effective totals in
different situations (for different arguments of $\efN$).}
Interestingly, this is not the case because it turns out that $\efNm$
is an ENF with absolute meaning. Indeed, we will prove that
$\efNm[\W] \le \efN[\W]$ for all $\W$ and
all~$\efN$, making $\efNm$ the unique minimum (least element) on the set of
all ENFs. Having revealed that the system in state $\ketpsi$ has to be
characterized as being simultaneously in at least
$\efNm[\, \ketpsi , \{ \, \keti \} \,]$ states from $\{ \, \keti \}$,
this result is used in [A] as a basis for the meaningful canonical
choice of ENF. It should be noted in this regard that the maximal ENF,
whose interpretation would otherwise be on equal footing with $\efNm$,
doesn't exist (see Theorem~2).\footnote{
The existence of multiple ENFs endows the constructed framework with
flexibility to accommodate quantum identity problems more structured
than [Q]. This may entail an additional problem-specific constraint(s)
on an ENF, possibly leading to a unique or privileged choice other than
$\efNm$. However, a generic extra requirement is that the effective total
determines the subset of $\{ \, \keti \}$ effectively present in $\ketpsi$.
For example, in the context of Anderson localization, it is of interest
to identify the spatial region effectively occupied by the electron.
In Ref.~\cite{Hor18D} we will show that $\efNm$
is the only ENF leading to a consistent selection of such
{\em effective support} of $\ketpsi$ on $\{ \, \keti \}$.}

A crucial novelty in our approach is the inclusion of additivity as a requirement
for ENFs. This step is necessary since the effective number of states is an additive
concept. However, a proper formulation requires some care. To that end, as well as
to start invoking parallels with localization, consider the simple setting
of a spinless Schr\"odinger particle on a finite lattice.
In the position basis, its state $\ketpsi$ is represented by $\nrN$-tuple
$( \psi(x_1),\ldots,\psi(x_\nrN) )$, with $p_i = \psi^\star \psi(x_i)$ being
the probability of detection at location $x_i$. Denoting by $\setW$ the set of
all {\em counting} weight vectors $\W=(\w_1,\ldots,\w_\nrN)$, $\w_i \!=\!\nrN p_i$,
namely\footnote{Working with counting vectors \eqref{eq:005} rather than probability
vectors $P \in \setP  =  \mathop{\cup}_{\nrN}  \, \setP_\nrN$ is simply a matter
of convenience. All results translate trivially.}
\begin{equation}
    \setW=\union_\nrN \setW_\nrN    \qquad , \qquad
    \setW_\nrN  \,=\,
    \bigl\{ \, (\w_1,\w_2, \ldots, \w_\nrN) \;\mid\;  \w_i \ge 0 \,,\,\,
    \sum_{i=1}^\nrN \w_i = \nrN \, \bigr\}  \quad
    \label{eq:005}
\end{equation}
the additivity property for $\efN$ arises as follows. Assume that the particle is
restricted to the lattice of $\nrN_1$ sites in a state generating the weight vector
$\W_1 \!\in\! \setW_{\nrN_1}$. Separately, let it be restricted to a non-overlapping
adjacent lattice of $\nrN_2$ sites and characterized by $\W_2 \!\in\! \setW_{\nrN_2}$.
With symbol $\boxplus$ representing the concatenation
operation\footnote{If $\W \!=\! (\w_1,\ldots, \w_\nrN) \in \setW_\nrN$ and
$\V \!=\! (\v_1,\ldots,\v_\nrM) \in \setW_\nrM$, then
$\W \boxplus \V \equiv (\w_1, \ldots, \w_\nrN, \v_1, \ldots, \v_\nrM)$.},
since
\begin{equation}
      \W \,=\, \W_1  \boxplus \W_2 \,\in\, \setW_{\nrN=\nrN_1+\nrN_2}
       \label{eq:025}
\end{equation}
there exists a state of the particle on the combined lattice, producing this composite
$\W$. Given the additivity of numbers, the sum rule for number of available states
($\nrN = \nrN_1 + \nrN_2$)
has to hold for its effective counterpart as well
(\,$\efN[\W] = \efN[\W_1] + \efN[\W_2]$\,).

Consequently, the additivity property we impose is
\begin{equation}
     \efN \bigl[ \W_1 \boxplus \W_2, \, \nrN_1 + \nrN_2 \bigr]  \;=\;
     \efN \bigl[ \W_1, \nrN_1 \bigr]  \,+\, \efN \bigl[ \W_2, \nrN_2 \bigr]
     \quad , \quad \forall \, \W_1, \W_2   \quad
     \tag{A}
     \label{eq:add}
\end{equation}
where the dimensions of vector arguments are indicated explicitly to emphasize that
$\efN[\W]$ represents $\nrN$ modified by distribution $\W$. Notice that $\efNm$ is
evidently additive, and that the above reasoning doesn't depend on the system, state
or basis in question.

Several decades ago, Bell and Dean~\cite{Bel70A} dealt with the problem analogous to [Q]
while analyzing localization properties of vibrations in glassy silica. In particular,
they asked how many atoms do these vibrations effectively spread over.
Their quantifier, the participation number $\pN$, is given by
\begin{equation}
    \frac{1}{\pN[\W]} \,=\, \frac{1}{\nrN^2} \sum_{i=1}^\nrN \w_i^2  \quad
    \label{eq:045}
\end{equation}
and is still widely used in the analysis of localization. In other areas, it is common
to exponentiate a suitable entropy, such as Shannon entropy~\cite{Sha48A}, and use it
for analogous purposes. However, none of these quantifiers is \eqref{eq:add}-additive.
Their interpretation as effective totals is thus vague and they tend to be too
arbitrary. In contrast, incorporating additivity into the definition of ENFs leads to
the resolution of the quantum identity problem and suggests new possibilities
both in physics (quantum uncertainty \cite{Hor18B}, localization \cite{Hor18C},
effective description of states \cite{Hor18D}) and measure-related aspects of
mathematics. The effective number theory, which we develop here as a tool to solve [Q],
provides a theoretical starting point for such developments.

In the rest of this section, we describe the construction of ENFs and discuss the key
results of the effective number theory. The goal here is to provide a concise but rigorous
overview, including the motivations for axiomatic properties, as well as the ramifications
of deduced features. The fully mathematical treatment in the technically convenient dual
form of effective complementary numbers (co-numbers)
is then given in Sec.~\ref{sec:eff_numbers}. Generalizations of the quantum identity
problem are discussed in Sec.~\ref{sec:count_idents}.
A broadly construed application of effective numbers in quantum theory, namely
as a tool to characterize quantum states, is described in Sec.~\ref{sec:str_states}.
Concluding remarks are given in Sec.~\ref{sec:conclude}.

\subsection{Effective Numbers}
\label{ssec:effnums_def}

We now develop the notion of ENF as a function $\efN \!=\! \efN[\W]$, assigning
effective total to each distribution of weights $\W \in \setW$ over the elements of a basis.
Such construction clearly doesn't depend on the fact that counted objects are quantum
states, and we will thus use generic terms in that regard from now on. The underlying goal
is to extend the ``counting measure'' for a collection of distinct but otherwise equivalent
objects (natural number $\nrN \in \N$) to the situation when these objects acquire varied
importance expressed by their counting weights (effective number $\efN[\W] \!\in \R$).
The additivity property \eqref{eq:add} is thus a basic consistency requirement for
acceptable ENFs.

Like in ordinary counting, no specific relation among individual objects is assumed.
Thus, in the same way the number of balls in a bag does not change upon their reshuffle,
the effective number will not change upon the permutation of counting weights. In other
words, ENFs are required to be totally symmetric in their
arguments, namely\footnote{We write $\efN(\w_1,\ldots,\w_\nrN)$ when weights need to be
distinguished, but use the functional notation $\efN[\W]$ otherwise.}
\begin{equation}
    \efN(\ldots \w_i  \ldots \w_j  \ldots)  \,=\,  \efN(\ldots \w_j  \ldots  \w_i  \ldots)
    \quad\; , \quad\; \forall \, i \ne j
     \quad
     \tag{S}
     \label{eq:sym}
\end{equation}

Extensions $\nrN \rightarrow \efN[\W]$ are by definition such that ordinary counting corresponds
to all objects being equally important, and thus to a uniform distribution. More precisely,
\begin{equation}
     \efN(1,1,\ldots,1) = \nrN      \quad\; , \quad\;
     (1,1,\ldots,1) \in \setW_\nrN  \quad\; , \quad\;
     \forall \, \nrN
     \quad
     \tag{B1}
     \label{eq:bou1}
\end{equation}
On the other hand, whenever all weight is given to a single object, all others being irrelevant,
the effective number is required to be one, namely
\begin{equation}
     \efN(\nrN,0,\ldots,0) \,=\, \efN(0,\nrN,0,\ldots,0) \,=\, \ldots \,=\,
     \efN(0,\ldots,0,\nrN) \,=\, 1
     \quad
     \tag{B2}
     \label{eq:bou2}
\end{equation}
within each $\setW_\nrN$.
Note that $(1,1,\ldots,1) \in \setW_\nrN$ and $(\ldots,0,\nrN,0,\ldots) \in \setW_\nrN$
are the opposite extremes in cumulation of weight. Hence, the effective number of objects with
arbitrary weights has to fall between the corresponding extremal values, namely
\begin{equation}
     1 \, \le \, \efN[\W] \, \le \, \nrN   \quad , \quad \forall \, \W \in \setW_\nrN
                                                      \quad , \quad   \forall \, \nrN   \quad
     \tag{B}
     \label{eq:bou}
\end{equation}

The degree of weight cumulation plays more detailed role in effective numbers than
just determining the boundary properties. Indeed, the concept has to respect that
increasing the cumulation in the distribution cannot increase the effective number.
To formulate such monotonicity, consider two objects
weighted by $\W \!=\! (\w_1,\w_2) \in \setW_2$ with $\w_1 \le \w_2$. The deformation
$\W \rightarrow \W_\epsilon = (\w_1-\epsilon, \w_2+\epsilon)$ leads to further cumulation
in favor of the second object, and thus $\efN[\W_\epsilon] \le \efN[\W]$ is imposed for all
$0 \le \epsilon \le \w_1$. In situation with arbitrary $\nrN$, we require the same for each
ordered pair $\w_i \le \w_j$ and deformation $0 \le \epsilon \le \w_i$,
namely\footnote{To visualize how the elementary deformation in \eqref{eq:mon-} increases
cumulation, one may picture each object as a cylindrical column of incompressible liquid in
the amount of its counting weight. Arranging the columns by increasing height from the left
to the right produces a half-peak profile with cumulation on the right. Consider the segment of
this profile delimited by columns $\w_i$ and $\w_j$ entering \eqref{eq:mon-}. The monotonicity
operation is represented by the transverse flow of liquid from the left to the right endpoint
through columns between them. It is understood that the columns are ordered at every moment of
the flow and thus, as the amount of liquid at the endpoints changes, the length of
the segment may increase. Since the liquid flows toward the center of cumulation at every
point of the process, the resulting distribution is more cumulated than the original one.}
\begin{equation}
      \efN(\ldots \w_i - \epsilon  \ldots  \w_j + \epsilon  \ldots)   \,\le\,
      \efN(\ldots  \w_i  \ldots  \w_j  \ldots)
      \quad
      \tag{M$^-$}
      \label{eq:mon-}
\end{equation}
It is easy to check that \eqref{eq:mon-}-monotonic $\efN$ attains its maximal value over
$\setW_\nrN$ at $(1,1,\ldots,1)$, while the minimum is at one or multiple fully cumulated
vectors $(\ldots,\nrN,\ldots)$. Conditions \eqref{eq:bou1}, \eqref{eq:bou2}, \eqref{eq:bou}
are thus compatible with~\eqref{eq:mon-}.\footnote{\label{note:schur} Monotonicity
\eqref{eq:mon-} is closely related to Schur concavity. The latter is equivalent
to imposing \eqref{eq:mon-} and symmetry \eqref{eq:sym} simultaneously
(see e.g.~\cite{Arn87A}).}
Note that, although not an ENF, the participation number \eqref{eq:045} satisfies
\eqref{eq:mon-} monotonicity.

\begin{figure}[t!]
\begin{center}
    \centerline{
    \hskip 0.00in
    \includegraphics[width=0.75\columnwidth,angle=0]{effectives_3.png}
     }
     \vskip -0.06in
     {\caption{Schematic representation of the extension from natural numbers (left) to effective numbers
     (right). See the discussion in the text.}
     \label{fig:numbers}}
     \vskip -0.45in
\end{center}
\end{figure}

The final requirement in definition of ENFs is continuity. The nature of problems with admitting
discontinuities can be illustrated by
\begin{equation}
     \mN[\W] = \sum_{i=1}^\nrN \mn(\w_i)    \quad , \quad
     \mn(\w)  \,=\,
     \begin{cases}
     \;0 \;,   &   \;  \w=0 \\[8pt]
     \;1 \;,   &   \;  \w > 0
     \end{cases}
     \quad
     \label{eq:075}
\end{equation}
which counts the number of non-zero weights in $\W$, and will be relevant later in our analysis.
Consider again two objects with $\W \!=\! (\w,2-\w)$. When $\w$ approaches zero, thus marginalizing
the first object to arbitrary degree, the effective number should approach one. However, this does
not materialize in $\mN$ due to its discontinuity. In general, we require that ENF cannot jump upon
arbitrarily small change of weights, namely
\begin{equation}
    \efN = \efN[\W]  \;\,
    \text{is continuous on} \;\, \setW_\nrN
    \quad , \quad \forall \,\nrN  \quad
    \tag{C}
    \label{eq:con}
\end{equation}

The properties discussed above define the set $\Nmaps$ of all effective number functions.
However, there are dependencies among these requirements. In particular, it can be easily
checked that the boundary condition \eqref{eq:bou1} is a consequence of \eqref{eq:bou2}
and additivity. Similarly,  \eqref{eq:bou} follows from \eqref{eq:bou1}, \eqref{eq:bou2},
symmetry and monotonicity. This leaves us with

\medskip

\noindent {\bf Definition~0.}   {\em A real-valued function $\efN \!=\! \efN[\W]$ on $\setW$ is
called an effective number function (belongs to set $\Nmaps$) if it is simultaneously additive
\eqref{eq:add}, symmetric \eqref{eq:sym}, continuous \eqref{eq:con}, monotonic \eqref{eq:mon-}
and satisfies the boundary condition \eqref{eq:bou2}.}

\medskip

Some of the features imprinted on the corresponding notion of effective numbers are
visualized in Fig.~\ref{fig:numbers}.\footnote{Once an ENF is fixed and used to assign totals,
the conventional ``number of objects'' is replaced by the ``effective number of objects''.
While we use the term effective number only in this restricted sense here, the underlying
algebraic structure makes the concept similar to standard types of numbers.}
On the left, natural numbers are shown as a theoretical model for expressing and
manipulating the quantities of like objects (bags of balls) or of varied objects treated
as equivalent. The bags containing differing amounts are assigned different discrete points
on the real axis (natural numbers), with the operation of ``merging the bags'' (\,$\sqcup$\,)
realized by ordinary addition. Extension to objects distinguished by counting weights is
shown on the right. Here the bags assigned equal amounts $\nrN$ by ordinary counting,
may be assigned different effective numbers $\efN$, depending on the cumulation of their
weight distributions. With maximal cumulation ($\delta$-function)
producing $\efN \!=\! 1$, the effective number continuously and monotonically
increases as cumulation decreases, reaching $\efN \!=\! \nrN$ when cumulation is absent
(uniform distribution). The operation of merging bags is represented by the additivity
property \eqref{eq:add}. Each element of $\Nmaps$, if any, implements a specific
version of this scheme.
Thus, to assess the conceptual value and practical impact of effective
numbers, it is necessary to decipher the structure of $\Nmaps$.

\subsection{Effective Counting}
\label{ssec:eff_counting}

It is not difficult to establish that ENFs do exist. For example, one can verify that
one parameter family of functions
\begin{equation}
     \efN_{(\alpha)}[\W] = \sum_{i=1}^\nrN \cf_{(\alpha)}(\w_i)    \quad , \quad
     \cf_{(\alpha)}(\w)  = \min\, \{ \w^\alpha, 1 \}               \quad , \quad
     0 < \alpha \le 1
     \quad
     \label{eq:085}
\end{equation}
belongs to $\Nmaps$, with $\efN_{(1)} \!=\! \efNm$. However, it is rather remarkable
that all $\efN \in \Nmaps$ have the additively separable structure of \eqref{eq:085}.
Indeed, Theorem 14 (Sec.~\ref{sec:eff_numbers}) implies
the following central result specifying $\Nmaps$ explicitly.

\bigskip

\noindent
{\bf Theorem 1.}   {\em Function $\efN$ on $\setW$ belongs to $\Nmaps$
if and only if there exists a real-valued function $\cf=\cf(\w)$ on $[0,\infty)$
that is concave, continuous, $\,\cf(0)=0\,$, $\,\cf(\w) = 1\,$
for $\,\w \ge 1$, and}
\begin{equation}
     \efN[\W]  \,=\,  \sum_{i=1}^\nrN \cf(\w_i)   \quad , \quad
     \forall \,\W \in \setW_\nrN    \quad , \quad
     \forall \,\nrN \quad
     \label{eq:095}
\end{equation}
{\em Such function $\cf$ associated with $\efN \in \Nmaps$ is unique.}

\bigskip

\noindent
Thus, there is one to one correspondence between ENFs and functions of single
variable specified by Theorem 1.\footnote{Note that it suffices to require
continuity at $\w=0$ since concavity guarantees it elsewhere.} Such $\cf$
associated with given $\efN$ will be referred to as its {\em counting function}.

The necessity of additively separable form \eqref{eq:095} for ENFs is interesting
conceptually. Indeed, it is common and familiar to represent the ordinary total
(natural number) by a sequential process of adding a unit amount for each object
in the collection. According to Theorem~1, this applies to every consistent extension
to the effective total (effective number), albeit with objects contributing
weight-dependent amounts specified by the counting function. It thus turns out
that the construction of ENFs generalizes the process of ordinary counting
to the process of {\em effective counting}.

\subsection{Minimal Effective Number}

A key insight into the nature of effective counting is provided by the following
results concerning the structure of set $\Nmaps$. They follow directly from
Theorem~18 in Sec.~\ref{sec:eff_numbers}.
\bigskip

\noindent {\bf Theorem 2.}   {\em Let $\efNm \!\in\Nmaps$ and $\mN \!\notin\! \Nmaps$ be
functions on $\setW$ defined by \eqref{eq:015} and \eqref{eq:075} respectively. Then}
\begin{displaymath}
     \begin{aligned}
         (a) \;\, & \;
               \efNm[\W]   \,\le\,  \efN[\W]  \,\le\,  \mN[\W]  \quad\; , \quad\;
               \forall\, \efN \in \Nmaps \;\;,\;\;  \forall\, \W \in \setW     \\[4pt]
         (b) \;\, & \;
               \bigl\{ \, \efN[\W]  \,\mid\, \efN \in \Nmaps \, \bigr\}  \;=\;
               [\, \alpha , \beta \,]   \quad\; , \quad\;
               \alpha=\efNm[\W] \;\;,\;\; \beta=\mN[\W]
               \;\;,\;\;  \forall\, \W \in \setW
      \end{aligned}
\end{displaymath}

To elaborate, first note that $(a)$ is the refinement of defining condition \eqref{eq:bou}.
While the upper bound is intuitive ($\,\mN[\W]$ counts the number of non-zero weights in $\W\,$),
the lower one is unexpected and consequential. In particular, the effective number of objects
weighted by $\W$ cannot be smaller than $\efNm[\W]$. Since $\efNm$ is an ENF, this feature
is inherent to the concept itself: there is a meaningful notion of minimal effective number.
In technical terms, $\efNm$ is the least element of function set $\Nmaps$ with respect
to partial order
$(\,\efN_1 \le \efN_2\,) \,\Leftrightarrow\, (\,\efN_1[\W] \le \efN_2[\W], \, \forall \,
\W\in \setW \,)$, and thus a unique ENF with this property.

Part $(b)$ conveys that, for any fixed $\W \!\in\! \setW$, effective counting can be
adjusted so that $\efN[\W]$ assumes any desired value from the allowed range specified
by $(a)$. While reflecting a certain degree of arbitrariness built into the concept of effective
numbers, the associated freedom of choice is in fact quite natural. To illustrate this,
consider $\nrN$ objects with non-zero weights of very disparate magnitudes so that the collection
is usefully characterized by an effective number. The insistence on ordinary count in this
situation constitutes a ``large extrapolation'' since it forces each object to contribute
equally despite the disparity. According to $(b)$, such extrapolation can be realized by
a sequence of ENFs that gradually bring the effective total to $\nrN$. This ability
to accommodate the needed continuum of consistent schemes can be considered a feature in
a framework designed to deal with generalized aspects of counting.

Note that $(b)$ also confirms an intuitive expectation that there is no maximal ENF since,
although specifying a supremal value for each $\W$, function $\mN$ doesn't belong to $\Nmaps$.
Taken together, the results of Theorem~2 form the basis for our canonical solution [A] of
quantum identity problem [Q]. The existence of minimal total $\efNm$ is particularly
consequential in applications of effective numbers. One notable example is that it facilitates
the notion of minimal (intrinsic) quantum uncertainty~\cite{Hor18B}.

\section{Effective Number Theory}
\label{sec:eff_numbers}

In this section, we will present the theory of effective numbers in a requisite mathematical
detail. The aim is to do this in a self-contained manner using elementary mathematics
so that it is readily accessible. Certain generalizations and abstract aspects
of the underlying structure will be elaborated upon in a dedicated mathematical account.

\subsection{Effective Complementary Numbers}

Due to its technical convenience, the discussion will be carried out in terms of
{\em effective complementary numbers} (effective co-numbers) realized by
functions\footnote{They could also be called {\em effective dual numbers} but
the duality is not a central property here.}
\begin{equation}
   \F[\P] = N-\G[\P] \quad,\quad  \P\in\Ps_N  \quad,\quad  \G \in \Nmaps
\end{equation}
where $\G=\G[\P]$ are the ENFs introduced in Sec.~\ref{ssec:effnums_def}. We start by
defining the effective co-number functions (co-ENFs), as implied by the above
relationship.

\begin{definition}\mylabel{def:fun}\setstretch{1.2}
  $\Fs$ is the set of effective co-number functions $\F$, where $\F :\, \Ps \rightarrow \R\,$ have
  the following properties.
  \phantom{}For all $N,M$, for all $1\le i,j \le N$, $i\ne j$, for all $\P = (c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_N) \in \Ps_N$, and for all ${Q} \in \Ps_M$

\medskip\noindent
   \/A additivity: $\F[\P \boxplus {Q}] = \F[\P] + \F[{Q}]$
  \\ \/{co-B2} boundary values: $\F(N,0,...,0) = N-1$, where  $(N,0,...,0)\in \Ps_N$
  \\ \/C continuity of $\F$ restricted to $ \Ps_N$
    whose topology is inherited from the standard topology on $\R^N$
  \\ \/{M$^+$} monotonicity: $0 < \varepsilon \le \min\{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,N-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\}$,\ \ $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \le c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j  \;\Rightarrow\;$
                $\F(...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j,...) \le \F(...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon,...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon,...)$

  \\ \/S symmetry: $\F(...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j,...) = \F(...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j,...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,...)$
\end{definition}

\medskip
The following examples will be useful in the context of our analysis.

\begin{example}\mylabel{ex:f_a}
The function $\F_{(\alpha)}[\P] = \sum_i \f_{(\alpha)}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) = \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i=0} 1 + \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \in (0,1)} (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i^\alpha)$, where
\dsskipp
\begin{align*}
  \f_{(\alpha)}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) =
  \begin{cases}
    1,  &  c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet = 0
    \\
    1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet^\alpha,  & 0 < c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet < 1
    \\
    0,  &      1 \le c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet
  \end{cases}
\end{align*}
belongs to $\Fs$ for $\alpha \in (0,1]$. This example is complementary to $\G_{(\alpha)}$ introduced in (6).
\end{example}

\begin{example}\mylabel{ex:fmin}
The function $\F_+[\P]=\F_{(0)}[\P]=\sum_{i=1}^N \f_+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$, where
\dsskipp
\begin{align*}
  \f_+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = \f_{(0)}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) =
  \begin{cases}
    1,  &  c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=0
    \\
    0,  &  0 < c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet
  \end{cases}
\end{align*}
satisfies Definition \ref{def:fun} except for continuity. Thus, $\F_+ \notin \Fs$
being complementary to $\G_+$ in (5).
\end{example}

\begin{example}\mylabel{ex:flin}
The function $\F_\star=\F_{(1)}[\P]=\sum_{i=1}^N \f_\star(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$, where
\dsskipp
\begin{align*}
  \f_\star(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = \f_{(1)}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) =
  \begin{cases}
    1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet,  &  0 \le c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet < 1
    \\
    0,  &  1 \le c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet
  \end{cases}
\end{align*}
satisfies all the properties from Definition \ref{def:fun}, so $\F_\star \in \Fs$.
This co-ENF is complementary to $\G_\star$ in (1).
\end{example}

Due to its repeated use in what follows, it is useful to formalize the following obvious Lemma.

\begin{lemma}\label{le:p1p}\setstretch{1.5
  If $\F$ satisfies (A) and (S) then  for all $N$

  \\ \/i $\F[\P] = \F(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{i-1}, 1, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{i+1},\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{N}) = \F(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{i-1}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{i+1},\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{N}) + \F(1)$,
  where $\P\in \Ps_N$,
  \\ \/{ii} $\F(1,\scalebox{0.8}{\ldots}, 1) = N\F(1)$, where  $(1,\scalebox{0.8}{\ldots}, 1)\in \Ps_N$.
\end{lemma}

\subsection{Separability}

\noindent We now focus on demonstrating the results that will ultimately clarify the structure of
effective co-numbers, and thus of effective numbers. The main conclusion will be that all co-ENFs
are of {\em additively separable} form, such as the one exhibited by the family of functions
$\F_{(\alpha)}$.
The property of additive separability is defined as follows.\footnote{Unless stated otherwise,
referencing ``function'' {\em in this section} applies to both real and complex-valued function, and
referencing ``number'' applies to both real and complex options. }

\begin{definition}\mylabel{df:separable}
    Additively separable function $\H$ on $\Ps$ is one that can be expressed as
    \dsskip
    \begin{align}\mylabel{eq:separable}
         \H[\P] = \sum_{i=1}^N \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)  \quad , \quad \P\in \Ps_N \quad ,\quad N=1,2,\ldots
    \end{align}
    where $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ is some function defined on $[0,\infty)$. Function $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ is called a generating function
    of $\H[\P]$.
\end{definition}

\medskip\noindent
Additively separable $\H$ is generated by infinitely many distinct functions.
However, for co-ENFs, a canonical representative can be singled out that is continuous and
bounded on $[0,\infty)$ (see Proposition \ref{prop:generating} and Corollary \ref{cor:bounded}).

\smallskip

The relevant insight into additive separability is provided by Lemma~\ref{le:separable} below.
Before formulating it, let us associate with every $\P \!\in\! \Ps_N$
the vector $\P^\uparrow \!\in\! \Ps_N$ obtained by permuting the components of $\P$ into ascending
order. Also, $\P^\uparrow_<$ will denote a vector obtained from $\P^\uparrow$ by keeping only
components less than one and removing the rest. Note that for any symmetric function $\F$ on $\Ps$,
we have $\F[\P]=\F[\P^\uparrow]$, and also that $\P^\uparrow_< \notin \Ps$.
In Proposition~\ref{prop:monoseparable} we will work with $\P^\uparrow_\le$ with an
analogous meaning.

\medskip

\begin{lemma}{\bf (Separability)}\label{le:separable}
  Let $\H$ be a function on $\Ps$ satisfying (A), (S) and the property
  \dsskip
  \begin{align}\label{eq:(<)}
     \forall \,N  \;\;,\;\; \forall\, \P \,,{Q} \in \Ps_N \qquad:\qquad   \P^\uparrow_< \,=\, {Q}^\uparrow_<
         \quad\Longrightarrow\quad \H[\P] = \H[{Q}]
  \end{align}
  Then following statements hold:
  \br7
     \/a $\H$ is additively separable.
  \br7
     \/b  If, in addition, $\H$ is continuous\footnote{Continuity on $\Ps_2$ may appear weaker than $(C)$ continuity, but this lemma shows that they are equivalent.}
     on $\Ps_2$, then there exists a continuous function generating it.
\end{lemma}

\begin{proof}
  \setstretch{1.2}%
  \/{a} Assuming $\P \ne (1,1,\ldots,1)$, let $\P^\uparrow=(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, \scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_N) \in\Ps_N$ and
     $\P^\uparrow_< = (c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, \scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m)$. We will distinguish two cases, namely $2m \le N$ and $2m > N$,
     for which we respectively get by using \eqref{eq:(<)}
  \begin{eqnarray*}
     \H[\P] = \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1},\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_N)
      = \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, \scalebox{0.8}{\ldots}, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, 1,\scalebox{0.8}{\ldots},1),
     \\ \H[\P\boxplus (1,\scalebox{0.8}{\ldots},1)]  = \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1},\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_N, 1,\scalebox{0.8}{\ldots},1)
     = \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, \scalebox{0.8}{\ldots}, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m),
   \end{eqnarray*}
   with $2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell > 1$ for $\ell=1,...,m$. The vectors on the top line (case $2m \le N$) are from $\Ps_{N}$,
   while those on the bottom line (case $2m > N$) are from $\Ps_{2m}$.
   In both cases, Lemma \ref{le:p1p}, symmetry (S) and additivity (A) lead to
   \begin{eqnarray*}
       \H[\P] &=& \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots},c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m) \,+\, (N-2m) \, \H(1)
       \\   &=& \H\bigl[ (c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1) \boxplus \scalebox{0.8}{\ldots} \boxplus (c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m, 2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_m) \bigr]  \,+\, (N-2m) \, \H(1)
       \\   &=& \sum_{\ell=1}^m \bigl( \, \vphantom{F^f} \H(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell,2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell) - \H(1) \,\bigr) \,+\, (N-m) \, \H(1).
   \end{eqnarray*}
  Consequently, introducing the generating function
  \dsskipp
  \begin{align}\label{eq:f1}
    \h(x) =
    \begin{cases}
	\H(x,2-x) - \H(1),  & x\in [0,1]
	\\
        \H(1),  & x \in (1,\infty)
    \end{cases}
  \end{align}
  facilitates the claimed separability $\H[\P] = \sum_{i=1}^N \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$.
  Note that for $\P = (1,1,\ldots,1)$, which was initially excluded,
  the separability holds in the same form.
\br
\setstretch{1.1}%
  \/b Given the proof of (a), it is sufficient to show that continuity of $\H$ on $\Ps_2$ implies
  the continuity of $\h$ in \eqref{eq:f1}.  For that, one only needs to ascertain the continuity
  at the gluing point $x=1$, which holds since we have two continuous functions with the same value
  at the gluing point: $\H(1,2-1) - \H(1) = \H(1)$.
\end{proof}

\bigskip
We will now demonstrate that all co-ENFs satisfy \eqref{eq:(<)} and hence they are additively
separable.

\begin{proposition}\label{prop:monoseparable
  All functions $\F \in \Fs$ are additively separable.

\end{proposition}
\begin{proof}

  Because of symmetry (S), we will without loss of generality assume $\P=\P^\uparrow$, i.e. $\P$ is in ascending order.
  For $\P = (1,1,\ldots,1)$ the implication in \eqref{eq:(<)} is vacuously true\footnote{$\P^\uparrow_<$ is undefined here.},
  so $\P \ne (1,1,\ldots,1)$ is assumed in what follows.
  We will use index $\ell$ to label the elements of $\P^\uparrow_\le$ and index $j$ for the rest
  of entries in $\P$. Hence,
  $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \le 1 < c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j$ for $\ell=1,...,m$, $\,j=m+1,...,m+n=N$, and $n=\sum_j 1$.
    Then by monotonicity (M$^+$), Lemma \ref{le:p1p}, and $\F(1)\!=\!0$, which is the $N\!=\!1$ case of (co-B2), we have:
  \begin{eqnarray*}
     \F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j \scalebox{0.8}{\ldots})
     &\le \F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}-\varepsilon, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+2}, \scalebox{0.8}{\ldots},\hspace{8mm} &   c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n-1}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n}+\varepsilon)\;\;,\;\,
       \mathrm{\ where\ } \varepsilon=c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}-1
     \\ &=\F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; 1,\phantom{xxxxxi} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+2}, \scalebox{0.8}{\ldots},\hspace{8mm}&  c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n-1}, 1+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n}-1)+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}-1))
     \\ &\le \F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; 1,\phantom{xx} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+2}-\varepsilon, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+3}, \scalebox{0.8}{\ldots},&  c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n-1}, 1+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n}-1)+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}-1)+\varepsilon)
     \\ &\le \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; 1,\scalebox{0.8}{\ldots},1,\; 1 + \sum (c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j-1) \right)&
     \\ &= \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\;\phantom{xxxxxii} 1-n + \sum c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j \right).&
  \end{eqnarray*}
  The opposite inequality follows from additivity (A), (co-B2), and (M$^+$).
  We start with $\sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1)$ zeroes:
  \begin{eqnarray*}
     \F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j \scalebox{0.8}{\ldots})
     = \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j \scalebox{0.8}{\ldots},\; 0,\scalebox{0.8}{\ldots}, 0,\; 1+\sum_{j=m+1}^{m+n} (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1) \right)
       - \sum_{j=m+1}^{m+n} (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1)
     \\ \ge \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}+\varepsilon, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+2}, \scalebox{0.8}{\ldots}, c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n},\;  0,\scalebox{0.8}{\ldots}, 0,\; 1+\sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1) -\varepsilon \right)
       - \sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1)
     \\ \ge \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; \lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+1}\rceil, \scalebox{0.8}{\ldots}, \lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{m+n}\rceil,\; 0,\scalebox{0.8}{\ldots}, 0,\;
         1 + \sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1) - \sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j) \right)
       - \sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1)
     \\ = \F\left( \scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; 1 + \sum\!{\raisebox{-2mm}{\footnotesize\it j}} (-1) - \sum (-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j) \right)
       + \sum \F(0,\scalebox{0.8}{\ldots}, 0,\; \lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil) - \sum (\lceil c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j\rceil - 1)
     \\  = \F(\scalebox{0.8}{\ldots} c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \scalebox{0.8}{\ldots},\; 1-n + \sum c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j).
  \end{eqnarray*}
  The resulting equality implies\footnote{ Notice that $\sum c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j = m+n - \sum c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_{\ell}$.}
  the property~\eqref{eq:(<)} upon recalling
  that the value of $\F$ doesn't change when removing the entries $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_\ell \!=\! 1$ from $\P$,
  so that $\P^\uparrow_\le$ becomes $\P^\uparrow_<$.
  Moreover, since (A) and (S) are among the defining properties of co-ENFs, $\F$ is additively separable by Lemma~\ref{le:separable}~(a).

\end{proof}

\begin{corollary}\label{cor:(<)
   All functions $\F \in \Fs$ satisfy \eqref{eq:(<)}.
\end{corollary}

\bigskip

Next we investigate non-uniqueness of the generating function which requires some groundwork to begin with.

\begin{lemma}\label{le:mx+mz}  In real numbers, the following two statements are true
  \dsskip
  \begin{align*}
    \/{i}\quad  & (\forall a) (\forall b>a) (\forall x\ne 1)   \quad(\exists m,n\in\Z) (\exists z\in [a,b])   \quad(mx + nz = m+n)

 \\ \/{ii}\quad & (\forall a>1) (\forall b>a) \;\;(\exists B)\;\;  \bigl( \forall x\in \bigl[ 0,{\textstyle\frac{1}{2}} \bigr] \bigr)
             \;\;(\exists m,n\in\Z^+) (\exists z\in [a,b]) \;\;\Bigl( mx + nz = m+n, \,\text{ and }\, \frac{n}{m}\le B \Bigr)

   \end{align*}
 \end{lemma}

\begin{proof}
  \/{i} Fit $\frac{m}{n}$ between $\frac{a-1}{1-x}$ and $\frac{b-1}{1-x}$ using density of rationals in $\R$.
  Then choose $z=1+ (1-x)\frac{m}{n}$ to get the equality.

\medskip\noindent
  \/{ii}  Choose $B= \frac{1}{a-1}$ and
  $n = \lceil \frac{1}{b-a} \rceil$, then
  \begin{equation*}
    \frac{1}{n} \le b-a \le \frac{b-a}{1-x} = \frac{b-1}{1-x} - \frac{a-1}{1-x}
  \end{equation*}
  since  $0 \le x < 1$.
  Hence there exists $m$ to fit $\frac{m}{n}$ between $\frac{a-1}{1-x}$ and $\frac{b-1}{1-x}$.
  Then again choose  $z=1+ (1-x)\frac{m}{n}$ to get the equality.
  Moreover,
  \begin{equation*}
    (a-1) \le \frac{a-1}{1-x} \le \frac{m}{n} \quad\text{and so}\quad \frac{n}{m} \le \frac{1}{a-1} = B.
  \end{equation*}
  That concludes the proof.
\end{proof}

\medskip

\begin{lemma}\label{le:generating}\setstretch{1.05}
  Let $\H$ be an additively separable function on $\Ps$
  and $\h,\h_1,\h_2$ its generating functions. Then
\br7
  \/i $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) + (1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)K$ is also a generating function of $\H$ for every  number $K$,
\br7
  \/{ii} if $\h_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)-\h_2(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ is bounded
  on some interval $[a,b],\; 0\le a < b$,
    then $\h_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)-\h_2(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = (1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)K_0$ for some  number $K_{0}\ $ and all $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet \in [0,\infty)$.
\end{lemma}
\begin{proof}
  \/i This follows from $\sum_{i=1}^N (1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) = 0$ for all $\P=(...c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i...)\in \Ps_N$.
\br
  \/{ii}
  Setting $\tilde{\h}=\h_1 - \h_2$ gives the following equation:
\dsskip
  \begin{equation}\label{eq:H=0}
    \sum \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) = 0 \mathrm{\quad for\ all\ } N  \mathrm{\ and\ all\ } \P=(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_1,\scalebox{0.8}{\ldots},c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_N).
  \end{equation}
  We will show that
  \begin{equation}\label{eq:h=linear}
    \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)\,\tilde{\h}(0)  \mathrm{\quad for\ all\ } c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet\in [0,\infty).
  \end{equation}
  Case $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=1$. We get $\tilde{\h}(1)=0$ from \eqref{eq:H=0} for $N=1$ and then \eqref{eq:h=linear} is satisfied for $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=1$.
\br
\setstretch{1.1}%
  Case $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet > 1$.
  Without loss of generality we can assume that $1\notin [a,b]$.
  Moreover, if $0\le a < b < 1$ then $\tilde{\h}$ is bounded also on $[2-b,\, 2-a]$
  because $\tilde{\h}(2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = - \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$, which follows from \eqref{eq:H=0} when $N=2$.
  As a result, we can assume without loss of generality even that $1<a<b$.
  Under this assumption, we will first show that $\tilde{\h}$, bounded on $[a,b]$, must be bounded also on $[0,\frac{1}{2}]$.
  According to Lemma \ref{le:mx+mz}(iii) we have
    \dsskip\[ (\forall a>1) (\forall b>a) \quad(\exists B)\quad \bigl( \forall x\in \bigl[ 0,{\textstyle\frac{1}{2}} \bigr] \bigr)
           \quad(\exists m,n\in\Z^+) (\exists z\in [a,b])  \quad\Bigl(mx + nz = m+n, \,\text{ and }\; \frac{n}{m}\le B \Bigr) \]
 Now we choose $\P=(x,\scalebox{0.8}{\ldots},x,z,\scalebox{0.8}{\ldots},z) \in \Ps_{m+n}$
 in equation \eqref{eq:H=0},
 where $x$ repeats $m$ times and $z$ repeats $n$ times.
 Then
   \[  m\tilde{\h}(x)+n\tilde{\h}(z)=0 \quad\text{and so}\quad |\tilde{\h}(x)| =  \left| \frac{-n}{m}\tilde{\h}(z) \right|  \le BA\quad , \quad\text{where} \]
 $A$ is a bound for $|\tilde{\h}|$ on $[a,b]$.
 Thus we have $\tilde{\h}$ bounded on $[0,\frac{1}{2}]$ by $BA$.

\medskip\noindent\setstretch{1.1}%
{\def\z{x}%
  Now suppose by contradiction that there is $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet > 1$ such that $\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) \ne (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)\,\tilde{\h}(0)$.
  Let $k$ be an integer large enough that
  \dsskip
  \begin{equation}\label{cond:k}
     N - k - 2 = \lceil kc}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet \rceil - k - 2 \ge 0 \quad\text{and}\quad
     \frac{4BA}{k} < \bigl| \tilde{\h}(c) - (1-c)\tilde{\h}(0) \bigr|
  \end{equation}
  and set $N=\lceil kc}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet \rceil $ , $\z = \frac{1}{2}(N - kc}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$.
  Then $kc}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet + 2 \z = N$ and $\z \in [0,\frac{1}{2}]$.

  Choosing $\P=(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet,\scalebox{0.8}{\ldots},c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet,0,\scalebox{0.8}{\ldots},0,\z,\z)\in \Ps_N$ in \eqref{eq:H=0},
  where $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet$ repeats $k$ times, will produce
  \dsskip\[ k\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) + (N - k - 2)\,\tilde{\h}(0) + 2 \tilde{\h}(\z) = 0 \]
  and the first condition in \eqref{cond:k} insures that $\P$ will not have a negative number of zeroes.
  Given that $N = kc}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet + 2 \z$, we get
  \dsskip
  \begin{align*}
     \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) + \left( c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet-1 + \frac{2 \z}{k} - \frac{2}{k} \right)\,\tilde{\h}(0) + \frac{2 \tilde{\h}(\z)}{k} = 0
  \end{align*}
  \begin{align}\label{eq:epsilon}
    \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)\,\tilde{\h}(0) + \varepsilon,
  \end{align}
  where we set $ \varepsilon = - \frac{2}{k} \left( (\z-1)\,\tilde{\h}(0) + \tilde{\h}(\z) \right) $.
  Since $k\varepsilon$ is bounded by $4BA$
  and from the second condition in \eqref{cond:k} we have
  \begin{align*}
    0 < |\varepsilon| \le \frac{4BA}{k}
      < \bigl| \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) - (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)\,\tilde{\h}(0) \bigr| ,
  \end{align*}
  which is a contradiction with \eqref{eq:epsilon}.
  Thus the original assumption $\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) \ne (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)\,\tilde{\h}(0)$ failed and \eqref{eq:h=linear} holds for $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet>1$.

\medskip\noindent
  Case $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet < 1$.
  We can use the previous case for $2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet > 1$ and get $\tilde{\h}(2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = (1 - (2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet))\,\tilde{\h}(0)$.
  This equation transforms into \eqref{eq:h=linear}
 since we already know that  $\tilde{\h}(2-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = -\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$
  and thus \eqref{eq:h=linear} holds for $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet < 1$ too.

\medskip\noindent
  We have shown that if $\tilde{\h}$ satisfies \eqref{eq:H=0} then it satisfies \eqref{eq:h=linear}.
  Finally, setting $K_0=\tilde{\h}(0)$ completes the proof.

  %
\end{proof}

\begin{proposition}\label{prop:generating}
\br7
   Let $\F \in \Fs$. Then for each number $t$, there is a unique generating function $\f$ of $\F$
   that is continuous and $\f(0)=t$.
\end{proposition}

\begin{proof}
 The existence of one continuous generating function, not necessarily satisfying $\f(0)=t$, follows from Corollary \ref{cor:(<)} and Lemma \ref{le:separable}(b).
 Part (i) of Lemma \ref{le:generating} implies that there is at least one continuous generating function for arbitrary
  value of $t=\f(0)$. To show the uniqueness of such function for every $t$, assume that there are two continuous generating functions $\f_1$ and $\f_2$, such that $\f_1(0)=\f_2(0)$.
 Since $\f_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) - \f_2(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ is then bounded on any finite interval due to continuity, we can use (ii) of Lemma \ref{le:generating}
 to infer that $0 = \f_1(0)-\f_2(0) = K_0$.  Using (ii) of Lemma \ref{le:generating} again, we finally conclude $\f_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)=\f_2(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$
 as claimed.

\end{proof}

\subsection{Description and Structure of Co-ENFs}

\noindent
Separability results of the previous section give us access to the content and the structure of set $\Fs$,
ultimately providing a key insight into the concept of effective (co-)numbers. We start with the following
proposition\footnote{The claim (i) of Proposition \ref{prop:convex} is likely to be known
in the context of majorization but we haven't found a suitable reference.}

\begin{proposition}\label{prop:convex}
\br7
  \/i Let $\H$ be a real additively separable function defined on $\Ps$.
    $\H$ is continuous (C) and monotone (M$^+$) if and only if it can be generated by a function $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ that is
    continuous at $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=0$ and convex.
\br7
  \/{ii} If $\F \in \Fs$
  then all its continuous generating functions $\f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ are convex.
\end{proposition}
\begin{proof}
  \/i \/{$\Leftarrow$} Convexity and continuity of $\h$ at $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=0$ imply its continuity on $[0,\infty)$, which guarantees
  continuity (C) of $\H$. In the presence of additive separability, conditions entailed by (M$^+$) take the form
\dsskip
  \begin{align}
    \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)+\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j) \le \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon)+\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon)\quad,\quad c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i\le c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j
  \end{align}
  To show that this also follows from stated properties of $\h$, consider function $\tilde{\h}$ that equals to $\h$
   everywhere except on interval $[c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j]$ where it is replaced by a linear segment with boundary values
   $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$ and $\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j)$. Such function $\tilde{\h}$ is still convex, which implies
  \begin{align*}
    \tilde{\h}\left( \frac{(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon)+(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon)}{2} \right)  \le  \frac{\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon) + \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon)}{2}.
  \end{align*}
    Then by linearity the left-hand side is
   \[  \tilde{\h}\left( \frac{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i+c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j}{2} \right)
       = \frac{\tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) + \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j)}{2}  =  \frac{\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) + \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j)}{2}. \]
    And the inequality turns into
\begin{samepage
\dsskip
  \begin{align*}
    \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)+\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j) \le \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon) + \tilde{\h}(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon) = \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon)+\h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon)
  \end{align*}
  as needed.
\end{samepage}
\br7
  \/{$\Rightarrow$} Consider the (M$^+$) condition  $\H(...c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i...c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j...) \le \H(...c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i-\varepsilon...c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j+\varepsilon...)$ for additively separable $\H$. Setting $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i = c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j = c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet$, and subsequently $a=c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet-\varepsilon,\; b=c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet+\varepsilon$, we obtain in turn
  \begin{eqnarray*}
      \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) + \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)  &\le&  \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet-\varepsilon) + \h(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet+\varepsilon)

     \\ \h\left( \frac{a+b}{2} \right) &\le& \displaystyle\frac{\h(a) + \h(b)}{2}.
  \end{eqnarray*}
  Hence any $\h$, a generating function of $\H$, is midpoint convex on $[0,N]$ for all $N$.
  It is well-known that every such function is convex if it is continuous.
  We thus select a continuous generating function $\h$, whose existence is guaranteed by Proposition 12. Such resulting $\h$ is then both continuous at $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet=0$ and convex.
   Note that $\h$ is an arbitrary continuous generating function,
   so all continuous generating functions $\h$ are convex.
   This is needed in the proof of (ii) that follows.
\br7
   \/{ii} Proposition \ref{prop:monoseparable} implies additive separability of $\F \in \Fs$ and the rest of
   the demonstration is contained in the proof of (i)($\Rightarrow$) above.
\end{proof}

\bigskip
We are now in a position to describe the set $\Fs$, specified by Definition \ref{def:fun}, explicitly.

\begin{theorem}{\bf (Set of co-ENFs)}\label{th:eff}
\br5
  $\F \in \Fs$ if and only if it is generated by a convex and continuous function $\f$, which is zero on $[1,\infty)$
  and $\f(0)=1$. Such a generating function $\f$ of $\F$ is unique.

\end{theorem}
\begin{proof}
  \/{$\Rightarrow$} $\F \in \Fs$ is additively separable by Proposition \ref{prop:monoseparable}. Then, as
  a consequence of additivity (A) and boundary conditions (co-B2), we have
  $\;\F(0,\scalebox{0.8}{\ldots},0,N) = (N-1)\cdot \f(0) + \f(N) = N-1$, so that
\dsskip
  \begin{align}\label{eq:B2c}
    \f(N) = (N-1)\cdot \bigl( 1 - \f(0) \bigr)
  \end{align}
  for all $N$. Given the continuity of $\F\in \Fs$, Proposition \ref{prop:generating} guaranties
  the existence of its unique continuous generating function with $\f(0)=1$.
  In conjunction with Eq.~\eqref{eq:B2c}, this implies that $\f(N)=0$ for all $N$.
  Furthermore, this continuous generating function is convex by (ii) of Proposition \ref{prop:convex}
    and, consequently, it is zero on the entire $[1,\infty)$.
  This demonstrates the existence of unique $\f$ with all required properties.
\br7
  \/{$\Leftarrow$} For the opposite direction, let $\F[\P]=\sum \f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$,
  where $\f$ is continuous, convex, $\f(0)=1$, and $\f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)=0$ on $[1,\infty)$.
  Then (A), (C), (S) and (co-B2) follow immediately, while (M$^+$) is a consequence of
  Proposition \ref{prop:convex}(i).
\end{proof}

\bigskip

Note that the unique choice of continuous generating function for co-ENF has been facilitated by a natural
choice $\f(0)=1$, expressing the fact that the object assigned zero probability should not contribute
to the effective number total ($\cf(0)=0$). However, it is worth pointing out that, as shown below, the same
unique choice of a generating function is selected by the requirement of boundedness on entire
$[0,\infty)$.

\begin{corollary}\label{cor:bounded}
  Let $\f$ be the generating function of $\F \in \Fs$, specified in Theorem \ref{th:eff}. Then
\br5
  \/{i} $0 \le \f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) \le 1$ for all $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet$
\br5
  \/{ii} $\f$ is the only generating function of $\F$ that is bounded on its whole domain $[0,\infty)$
\end{corollary}
\begin{proof}
   \/{i} This immediately follows from $\f(0) = 1$, $\f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)=0$ on $[1,\infty)$, and convexity.
\br7
  \/{ii} Boundedness of $\f$ follows from (i). To demonstrate uniqueness, assume there is another
  bounded generating function $\f_1$ of $\F$. Thus, $\f_1 - \f$ satisfies the assumptions of
  Lemma~\ref{le:generating}(ii), implying the existence of non-zero $K_0$ such that
  $\f_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) = \f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) + K_0 (1-c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$ for $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet \in [0,\infty)$. However, this contradicts the boundedness
  of $\f_1$ which demonstrates the claimed uniqueness.

\end{proof}

\bigskip
  Below we will make use of the following obvious lemma and a simple corollary.
\begin{lemma}\label{le:less}
   Let $\H_1$ and $\H_2$ be real additively separable functions on $\Ps$.
   If there exist respective generating functions such that $\h_1(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet) \le \h_2(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet)$, for all $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet$,
     then $\H_1(\P) \le \H_2(\P)$, for all $\P \in \Ps$.
\end{lemma}

\vfill\eject

\begin{corollary}\label{cor:Bounded}
  If $\F \in \Fs$, then $0 \le \F[\P] \le N-1$, for all $\P \in \Ps$.
\end{corollary}
\begin{proof}
  Let $\f$ be the generating function specified in Theorem \ref{th:eff}.
  From (i) of Corollary \ref{cor:bounded} we have $\sum_i 0 \le \sum \f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i) \le \sum_i 1$, which translates
  into  $0 \le \F[\P] \le N$ by Lemma \ref{le:less}. To put the second inequality into the claimed form, note that there is always at least one $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j \ge 1$.
  For this $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j$, we have $\f(c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_j)=0$ by Theorem \ref{th:eff}.
  This lowers the upper bound for
  $\F[\P]$ by unity and proves the second inequality.
\end{proof}

\bigskip
Using the above preparation, we will now demonstrate several structural properties of $\Fs$.

\begin{theorem}{\bf (Maximality)}\label{th:max}\setstretch{1.45}
\br1
  If $\F \in \Fs$ then the following holds for all $\P = (...,c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i,...) \in \Ps$
  \\ \/i $\F_{(0)}[\P] \,=\, \F_+[\P] \,\le\, \F[\P] \,\le\, \F_{\star}[\P] \,=\, \F_{(1)}[\P]$
  \\ \/{ii} $\F_+[\P] \,=\, \F[\P] \,=\, \F_\star[\P]  \quad\Leftrightarrow\quad  c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i\notin (0,1),\; i=1,2,...,N$
  \\ \/{iii} $\beta_0 \,=\, \F_+[\P] \;<\; \F_\star[\P] \,=\, \beta_1  \;\,\Rightarrow\;\,
   \{ \F[\P]: \F\in \Fs \}  =  [\beta_0,\beta_1]$
\end{theorem}
\begin{proof}\setstretch{1.05}
  \/i To show both inequalities, let $\f$ be the continuous generating function of $\F$ guaranteed by
   Theorem \ref{th:eff}. We will show that $\f$ satisfies $\f_+(x) \le \f(x) \le \f_{\star}(x)$ on $[0,\infty)$ and
   then Lemma \ref{le:less} will complete the proof of this part. The first inequality $\f_+(x) \le \f(x)$ follows
   directly from Theorem \ref{th:eff}, Corollary \ref{cor:bounded}(i), and the definition of $\f_+$.
   The second inequality holds as equality on $[1,\infty)$ by Theorem \ref{th:eff} and the definition
   of $\f_{\star}$. To show the second inequality on $[0,1)$, note that the graphs of both $\f$ and $\f_{\star}$
   pass through the points $(0,1)$ and $(1,0)$, $\f$ is convex by Proposition \ref{prop:convex}(ii), and
   $\f_{\star}$ is linear between those points. So $\f_+(x) \le \f(x) \le \f_{\star}(x)$ on $[0,\infty)$ as promised.
\br7
   \/{ii} \/{$\Leftarrow$} If $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i\notin (0,1)$ for $i=1,2,...,N$, then $\F_+[\P] = \F_\star[\P]$
     and the equality for $\F[\P]$ follows from (i).
\br7
  \/{$\Rightarrow$} Let $\F_+[\P] = \F_\star[\P]$. Then
  \[ \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i=0} 1  =  \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i=0} 1 + \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \in (0,1)} (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i). \]
   Hence $\; 0  =  \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \in (0,1)} (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i)$ and so $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i\notin (0,1)$ for $i=1,2,...,N$.
\br7
   \/{iii} From $\F_+[\P] < \F_\star[\P]$ we have the existence of at least one $c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \in (0,1)$ by (ii).
   Then for a fixed $\P$ define
  \[ g(\alpha) = \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i=0} 1 + \sum_{c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i \in (0,1)} (1 - c}\def\q{b}\def\P{C}\def\Q{B} %\def\q{\bullet_i^\alpha) = \F_{(\alpha)}[\P]. \]
   The function $g$ is continuous, increasing, and maps interval $[0,1]$ onto interval $[\beta_0,\beta_1]$.
   Then
  \[ \{ \F[\P]: \F\in \Fs \}  \supseteq  \bigl\{ \F_{(\alpha)}[\P]: \alpha \in [0,1] \bigr\}
         =  \bigl[ \F_+[\P],\; \F_\star[\P] \bigr] = [\beta_0,\beta_1]. \]
  The opposite inclusion follows from (i).
\end{proof}

\section{Counting The General Quantum Identities}
\label{sec:count_idents}

With the effective number theory in place, we now return to the topic that motivated its
construction, namely the quantum identity problem. In particular, we will make explicit
some of the straightforward but useful and relevant generalizations of [Q]. This serves,
in part, as a stepping stone toward the most generic application of effective numbers
in quantum theory, namely as a tool to characterize quantum states
(Sec.~\ref{sec:str_states}).

Conceptually important extension of [Q] and [A] is made possible by additive separability
of ENFs. Indeed, instead of an orthonormal basis, consider any collection $\{ \, \ketj \}$
of $n$ orthonormal states from $\nrN$-dimensional Hilbert space ($1 \!\le\! n \!\le\! \nrN$).
How many states from $\{ \, \ketj \}$ is a system described by $\ketpsi$ effectively in?
Let $\efN \in \Nmaps$ be an ENF and $\cf$ the counting function uniquely associated with
it by virtue of Theorem$\,$1. We define
\begin{equation}
     \efN\bigl[\, \ketpsi , \{ \, \ketj \} \,\bigr] \,\equiv\,
     \sum_{j=1}^n \cf(\w_j)  \quad\; , \quad\;
     \w_j = \nrN  \mid \!\! \langle \,j \!\mid \! \psi \,\rangle \!\! \mid^2 \quad
     \label{eq:4_10}
\end{equation}
for each $\ketpsi$ and $\{ \, \ketj \}$. This assignment is meaningful in the following
sense. Given a fixed $\{ \, \ketj \}$, let $\{ \, \keti \}$ be its arbitrary completion
into a basis of the Hilbert space. Then, owing to additive separability of ENFs,
\begin{equation}
     \efN \bigl[\, \ketpsi , \{ \, \keti \} \,\bigr]  \;=\;
     \efN \bigl[\, \ketpsi , \{ \, \ketj \} \,\bigr]  \;+\;
     \efN \bigl[\, \ketpsi , \{ \, \keti \} \setminus \{ \, \ketj \} \,\bigr]
     \label{eq:4_20}
\end{equation}
where ``$\setminus$'' denotes the set subtraction. In other words, the contribution of
$\{ \, \ketj \}$ to $\efN[\, \ketpsi , \{ \, \keti \} \,]$, defined by \eqref{eq:4_10}, is
independent of the completion $\{ \, \keti \}$, and is thus uniquely associated with
$\{ \, \ketj \}$ for fixed $\efN \in \Nmaps$. It is the effective number of states from
$\{ \, \ketj \}$ contained in $\ketpsi$ according to $\efN$. Moreover, it is
straightforward to check that $\efNm[\, \ketpsi , \{ \, \ketj \} \,]$ minimizes
the effective number so assigned, and we have

\medskip

\noindent
    {\bf [A$^\prime$]} {\em Let $\ketpsi$ be a state vector from $\nrN$-dimensional Hilbert
    space
    and $\{ \, \ketj \}  \equiv  \{ \, \ketj \mid\, j=1,2,\ldots,n \le \nrN \,\}$ the set of
    $n$ orthonormal states in this space. The physical system described by $\ketpsi$ is
    effectively in $\efNm [\, \ketpsi , \{ \, \ketj \} \,]$ states from
    $\{ \, \ketj \}$, specified by~\eqref{eq:4_10} with $\cf=\cfu$.}

\medskip

\noindent
Few simple points regarding the above are worth emphasizing.

\medskip

\noindent
(i) The extension \eqref{eq:4_10} and the ensuing generalization of [A] to [A$^\prime$]
arises because the abundance of quantum identities is determined ``locally'', namely
without reference to basis elements orthogonal to the subspace in question. Apart from
being natural for a measure-like characteristic, this feature has practical consequences
in many-body applications where the dimension of Hilbert space grows exponentially with
the size of the system. Indeed, the above avoids such complexity in certain calculations,
thus providing a computational benefit.

\medskip

\noindent
(ii) None of the above applies to the abundance of quantum identities determined by
the participation number $\pN$ of Eq.~\eqref{eq:045}, since this value doesn't split
into contributions from orthogonal subspaces generated by the partitioned basis.
This is of course due to the lack of additivity and hence of additive separability.

\medskip

\noindent
(iii) The above considerations are clearly not limited to counting quantum identities.
In a generic situation, the inquiry is concerned with the contribution to effective
number from a subset of weighted objects. To formalize such assignments directly in
the effective number theory, one simply extends $\efN \in \Nmaps$ from domain $\setW$
of counting vectors to the domain of general weights
\begin{equation}
    \setWl=\union_n \setWl_n    \qquad , \qquad
    \setWl_n  \,\equiv\,
    \bigl\{ \, \Wl=(\wl_1,\wl_2, \ldots, \wl_n) \;\mid\;  \wl_j \ge 0 \; \bigr\} \quad
    \label{eq:4_30}
\end{equation}
by virtue of its counting function $\cf$, namely $\efN[\Wl]=\sum_j \cf(\wl_j)$.

\smallskip

While the effective number $\efN[\, \ketpsi , \{ \, \keti \} \,]$ specifies how many
identities from basis $\{ \, \mid \! i \,\rangle \}$ can the state
$\mid \! \psi \,\rangle$ effectively take, it is frequently useful to inquire about
a coarse-grained version of such effective count. To formalize the corresponding
generalization, consider
the decomposition of Hilbert space $\Hsp$ into $M$ mutually orthogonal subspaces
$\{ \Hsp_m \}$
\begin{equation}
     \Hsp = \Hsp_1  \oplus \Hsp_2 \oplus \ldots \oplus \Hsp_M    \qquad , \qquad
     \{ \Hsp_m \}  \equiv  \{ \, \Hsp_m \mid\, m=1,2,\ldots,M \,\}
     \label{eq:4_40}
\end{equation}
Subspaces $\Hsp_m$ are implicitly treated as equivalent entities.\footnote{What is
considered ``equivalent'' is dictated by the physics involved, rather than the concept
of effective number itself. For example, it is possible to encounter a situation
where dimensions of ``usefully equivalent'' subspaces are not the same.}
How many subspaces from $\{ \Hsp_m \}$ is the system described by $\ketpsi$
effectively in?

Given a state of the system, the rules of quantum mechanics assign probability to each
subspace of the associated Hilbert space. The effective number theory therefore
provides an immediate answer to the above question. More specifically, let $\ketchim$ be
the (unnormalized) projection of $\ketpsi$ onto $\Hsp_m$. Then the probability vector
is $P=(p_1,p_2,\ldots,p_M)$, where $p_m = {\langle \,\chi_m \!\mid\! \chi_m \,\rangle}$,
and the corresponding counting vector is $\W=MP$. Hence, given an ENF, we assign
$\efN[\, \ketpsi , \{ \, \Hsp_m \} \,] \equiv \efN[\W]$, which leads to the following
generalization [A$_g$] of [A].

\medskip

   \noindent
   {\bf [A$_g$]} {$\,$ \em Let $\W$ be the counting vector assigned by quantum mechanics
   to state $\ketpsi$ and the orthogonal decomposition $\{ \Hsp_m \}$ of the Hilbert
   space. Then the system described by $\ketpsi$ is effectively contained in
   $\efNm[\, \ketpsi , \{ \, \Hsp_m \} \,] = \efNm[\W]$ subspaces from $\{ \, \Hsp_m \}$.}

\medskip

Applying the logic identical to one producing [A$^\prime$], it is straightforward
to generalize [A$_g$] into [A$^\prime_g$] for counting the identities from arbitrary sets
(not necessarily full decompositions) of mutually orthogonal subspaces.

\vfill\eject

\section{Application: The Structure of Quantum States}
\label{sec:str_states}

While our discussion was carried out in the context of quantum identity problem,
the constructed effective number framework clearly offers a much larger scope of
uses. Apart from physics and mathematics, these also appear in the areas of
applied science simply due to the very basic role of measure and probability in
quantitative analysis. Examples of such applications will be discussed in
the follow-up works. Here we describe a broader outlook on the utility of
effective numbers in quantum theory.

With the quantum state encoding all ``options'' for the system, the physical
content of $\ketpsi$ closely relates to all probability vectors $P$ induced
in this manner.
Retracing the steps leading to [A], it is then meaningful to associate
the effective number $\efNm$ with any complete set of mutually exclusive
possibilities. We propose the collection of all such reductions
\begin{equation}
    \ketpsi  \quad \longrightarrow \quad  P
    \quad \longrightarrow \quad   \efNm
    \label{eq:4_45}
\end{equation}
as a systematic and physically relevant characterization of a quantum state.
The general quantum identities of Sec.~\ref{sec:count_idents}, which can
also be thought of as measurement outcomes, are prime examples
of objects/possibilities in question. However, any meaningful
$\ketpsi \rightarrow P$ can be considered. The resulting variety offers a wide
range of options for targeted insight into the structure of $\ketpsi$.
We wish to highlight few points in this regard.

\smallskip

\noindent (i)
Each effective number characteristic can be {\em refined} by totals assigned
to subsets of the associated sample space or {\em coarse-grained} by counting
its partitions. Note that for quantum identities in [Q] this corresponds
to refining by totals involving parts of the basis and coarse-graining
by totals involving orthogonal decompositions.

\smallskip

\noindent (ii)
The proposed approach is universal with respect to the nature of
the quantum system being described. Indeed, $\ketpsi$ may be as simple
as a state of harmonic oscillator, but as complex as a many-body state of
quantum spins or the vacuum of non-Abelian gauge theory.

\smallskip

\noindent (iii)
Our present focus on the finite discrete case is not restrictive either. Even
in situations involving the space-time continuum, the intermediate steps of
ultraviolet (lattice) and infrared (volume) regularizations yield such
descriptions. Removing the cutoffs then involves a suitable
$\nrN \!\to\! \infty$ limit.
For this purpose, it is more convenient to work with {\em effective fraction}
rather than effective number. In terms of probabilities, it is defined as
 \begin{equation}
    \efF[P] \,\equiv\, \frac{1}{\nrN} \; \efN[ \nrN P ]
    \quad , \quad   \efN \in \Nmaps
    \quad,\quad P \in \setP_\nrN    \quad
    \label{eq:105}
 \end{equation}
Function $\efNm$ produces the minimal effective fraction, namely
\begin{equation}
     \efFm[P] = \sum_{i=1}^\nrN \ffu(p_i,\nrN)    \quad,\quad
     \ffu(p,\nrN)  \equiv  \min\, \{\, p, 1/\nrN \,\}    \;
     \label{eq:115}
\end{equation}
Restricting ourselves to quantum identities of [Q], the cutoff removal schematically
proceeds as follows. Assume that $\ketpsi$ is the target state (from infinite-dimensional
Hilbert space) to be assigned an effective fraction in basis $\{ \,\keti \}$.
Let ${\mid \! \psi^{\scrk} \,\rangle}$ represents $\ketpsi$ at $k$-th step
of the regularization process, involving the state space of increasing dimension
$\nrN_k$. If the latter is spanned by $\{ \,\mid \! i^{\scrk} \,\rangle \}$
targeting $\{ \,\keti \}$ then~\footnote{The meaning of
$\{ \,\mid \! i^{\scrk} \,\rangle \} \equiv
\{ \,\mid \! i^{\scrk} \,\rangle  \mid i=1,2,\ldots,\nrN_k \}$
targeting $\{ \,\mid \! i \,\rangle \}$ depends on the context but is usually
clear on physics grounds.}
\begin{equation}
     \efFm \bigl[ \,\mid \! \psi \,\rangle, \, \{ \,\mid \! i \,\rangle \} \, \bigr]
     \;\equiv\;
     \lim_{k \to \infty} \efFm
     \bigl[ \,\mid \! \psi^{\scrk} \,\rangle, \, \{ \,\mid \! i^{\scrk} \,\rangle \} \, \bigr]
     \;=\;
     \lim_{k \to \infty} \efFm [P_k]
     \label{eq:117}
\end{equation}
where $P_k \!=\! (p_1^\scrk,\ldots,p_{\nrN_k}^\scrk)$ ,
$p_i^\scrk \!=\! \,{\mid \! \langle \,i^{\scrk} \!\mid \! \psi^{\scrk} \,\rangle \! \mid^2}$.
Note that  $\efFm$ inversely reflects ``localization'' of $\ketpsi$ in $\{ \,\keti \}$.

\smallskip

\noindent (iv)
Example of an application where the assignment \eqref{eq:4_45} is carried out without
direct reference to the underlying Hilbert space is provided by the problem of vacuum
structure in Quantum Chromodynamics (QCD). This is often studied in the Euclidean path
integral formalism, with regularized vacuum represented by the statistical ensemble of
lattice gauge configurations $U\!=\! \{U_{x,\mu} \}$.
Given a composite field $O\!=\!O(x,U)$ and the induced space-time probability
distribution $P(x,U) \propto |O(x,U)|$, effective number framework can be used
to determine the effective fraction of space-time occupied by $O$ for each $U$.
The corresponding quantum averages are of vital interest in this context.
Yet more indirect vacuum characteristics of such type reflect the space-time properties
of Dirac eigenmodes. Their main utility is in probing the features of quark dynamics.

\section{Concluding Remarks}
\label{sec:conclude}

Emergence of a quantum system in one of many possible ``identities'' upon
(strong) probing is among the key features of quantum behavior. Indeed, it underlies
the notion of quantum uncertainty and is closely connected to a fruitful concept
of localization. A well-founded prescription for the corresponding abundances is thus
desirable. As a contemporary example, one may use it in the analysis of a quantum algorithm
which produces the state $\ketpsi_o$ as an output of quantum computation, and follows up
with a measurement involving the basis $\{ \,\keti \}$. The effective abundance
of distinct collapsed states $\keti$ obtained upon repetition of these steps is relevant
for the assessment of algorithm efficiency.

In this work, we showed that requiring the desired characteristics to be measure-like
(additive) produces meaningful answers.
It results in the theoretical structure (effective number theory) revealing that
a consistent assignment of totals to collections of objects with probability weights
requires the existence of an inherent (minimal) amount $\efNm$.
The appearance of such qualitative feature in basic measure considerations suggests
the utility of the constructed framework already in contexts much less abstract than
quantum mechanics. For example, the effective number theory can be viewed as an extension
of the ordinary counting measure to what can be referred to as a {\em diversity measure}
$\efNm$, with its wide range of contemporary applications
(social sciences, ecosystems; see e.g.~\cite{Lei12A}).
Other viewpoints can cast it as a {\em choice measure}, facilitating a probabilistic
notion of effective choices, or as a {\em support measure}, conveying the effective size
of a function support (effective domain)~\cite{Hor18D}. Given this universality, $\efNm$
may find uses in multiple areas of quantitative science.

Finally, we wish to point out that the existence of minimal amount is rooted in the simultaneous
requirement of monotonicity (Schur concavity) and additivity for ENFs. This combination is
rather unusual from the mathematics standpoint. Indeed, while monotonicity is important for
the theory of majorization~\cite{Mar11A}, it has no role in the standard formalization
of measure. Conversely, additivity is crucial for the latter but not native to the
former. In fact, relaxing either \eqref{eq:mon-} or \eqref{eq:add} in Definition~0 leaves
the respective effective pseudo-number assignments too arbitrary. But their combination,
which is necessary on conceptual grounds, leads to $\efNm$ and the associated insight.

\begin{acknowledgments}
{\bf Acknowledgments. }
I.H. acknowledges the support by  the Department of Anesthesiology at
the University of Kentucky.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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text:
The Same Yet Inn
The Same Yet was handy for the farmers and labourers who work and live locally to have a well earned pint at the end of the day.
Mrs Jowett is the landlady having taken over from Mrs Booth.
The legend is it was originally The Star Inn until it was refurbished when the sign writers asked what the name of it was, the answer was, "the same yet', meaning it was still The Star Inn, but, to everyone's surprise,
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HAUNTED: The Same Yet exhibits ghostly manifestations. these include the ghost of a man called James Heyward who was robbed and murdered nearby in 1843. His body was carried to the pub for the police to investigate. Every now and again, there is no time limit, his bearded ghost appears in the bars and quite startle all
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Here is an episode from Hauntings of England paranormal TV show.

redpajama_set_name: RedPajamaCommonCrawl


text:
Gators reveal throwback uniforms for Auburn game
gatorsports
[Courtesy of UAA]
Special to Gatorsports.com
Florida will be wearing retro uniforms – blue jerseys, white pants, black shoes and white helmets with an "F" on them – for Saturday's soldout game against Auburn.
It's all part of the Gators honoring their storied tradition on homecoming weekend with a throwback uniform to Steve Spurrier's playing days in the 1960s.
This piece combines a multitude of years and traditions all within one helmet.
The white helmet, grey facemask and solid orange strip across the middle is from the 1950s and early 1960s.
Meanwhile, the block "F" with a circle decal that was used throughout the early 1960s.
Rather than a patch on the jersey as it has been all season, fans will notice the CFB 150 logo on the bumper of the helmet.
The blue jersey is from the 1960s, when the Head Ball Coach played quarterback for the Gators and won the Heisman Trophy in 1966. The jersey has an over the shoulder orange & white insert stripe.
This is the first time that the Gators have worn this jersey in a game since Sept. 30, 2006.
Also from the 1960s, the Gators white pants for Saturday has an Orange & Blue insert stripe down each side of the pants.
This is the first time that UF has have worn these pants in a game since the aforementioned game against Alabama in 2006.
Florida will wear white crew socks, that include a blue stripe around the top, which was a very common look back in the 1950s and 1960s.
For the first time in recent memory, the Gators will wear black cleats with white shoelaces. Unlike nowadays, cleats did not have all the exotic colors and styles in the 1950s and 60s
Previous articleCollege GameDay returns to Gainesville
Next articleSunday Blog: A campus spot for College GameDay
http://gatorsports.wpengine.com
The Back Nine: Rest in peace, Aubrey
Gators' 2020 SEC football schedule announced
Dooley: The SEC East race may be decided early
Gator-6 September 29, 2019 At 2:14 pm
Brother. What genius dreamed this one up? Hey, I gotta better idea: If you really want to honor SOS, why not play like him instead of just dressing up like him?
Joe Shiver September 30, 2019 At 9:01 am
Hey, how 'bout Trask waves off McPherson and kicks the game winning field goal himself?
Gator-6 September 30, 2019 At 12:27 pm
With a drop kick at that! Which was perfected by SOS himself.
Ocalarob September 30, 2019 At 6:50 am
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Whilst finding sales is obviously an issue, financing sales seems to be much higher up the agenda of SME businesses in the UK.


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The Grand Cherokee is no slouch in the standard equipment department. Although the Overland features the best bits, such as HID headlamps, heated seats in the first and second row, the ParkSense parking aid, a navigation system, a roof rack, chrome 18-inch wheels and Sirius satellite radio. A wealth of options and packages are available, including a rear-seat entertainment system, the MyGig music storage system, and Sirius Backseat TV.
For 2009, Jeep's Grand Cherokee SUV gets a boost in the engine output. The 5.7L Hemi is improved to deliver 357 hp and 389 ft-lb of torque. Also new is variable valve timing (VVT), which improves fuel economy, and Multi-Displacement System, which shuts down half the engine's cylinders during certain periods further improve fuel efficiency.
A new Leather Appearance Group is standard on Limited and Overland models and includes leather trim on the doors, armrests, center console, shift knob and instrument binnacle. The Laredo model now includes a soft-touch center armrest cover, leather-wrapped parking brake, and chromed interior door handles and audio control buttons.
The high-performance Cherokee SRT8 receives upgraded leather and a bigger DVD screen.
The 2009 Jeep Grand Cherokee pairs a comfortable, luxurious interior with impressive off-road ability, and in top trims gets enough high-tech conveniences to compete with the most luxurious sport utility vehicles. The top-of-the-range Overland model is a true rival to some of the top luxury brand SUVs, while the 420-hp 6.1L Hemi V8-powered SRT8 delivers world-class performance.


Interview with Zero Waste Nerd | The Humble Co.
At The Humble Co., we're always on the lookout for people and organizations who are making positive changes to protect our planet. One of these amazing individuals is Megean Weldon, also known as Zero Waste Nerd. Megean's zero waste blog not only chronicles her and her family's journey but also helps others limit their trash. We interviewed Megean to learn more about zero waste practices and what we can do to help reduce our environmental footprint.
How would you describe the zero waste lifestyle movement?
I know Zero Waste can have various meanings to everyone so I'll describe what it means to me. The term in itself can come across very exclusive. That in order to be apart of the zero waste party, you have to make absolutely zero trash which, news flash, isn't possible. As long as we live in modern society, we will have some sort of impact on this planet. Zero Waste to me is less about the number and more about the perspective. What kinds of decisions do I make on a daily basis to reduce my waste and to tread a little lighter on this Earth? Do I make trash? Occasionally, yes! That's just part of living. But, I can reduce my waste to nearly a jar a month with just making better decisions and rethinking my perspective on waste and consumption. Should everyone strive for a jar of waste a month? No. All of our stories, access, and journeys are very uniquely different. So, do your best. That's the most important thing.
What prompted you to go zero waste?
Three years ago, my husband and I decided to pick up trash around our neighborhood in commemoration of Earth Day. Like every Earth Day, we always tried to do something to show our love for the planet. One year we started composting, another year we planted fruit trees, and another year we installed a rain barrel. It was always a fun activity. However, after what normally took us a half an hour to walk around our block, we spent nearly 2 whole hours filling bag after bag of filthy waste that had been carelessly thrown out and mismanaged. I was angry. So angry. Until I got home and realized that I couldn't fit any of that garbage into my own trash can because our 30 gallon can was already overflowing with our own waste. A definite reality check for us. How could we call ourselves environmentalists? I felt like a huge hypocrite. It was then that we knew we needed to change our habits and look for ways to reduce the amount of waste we produced.
What has been the biggest challenge of a zero waste lifestyle?
The biggest challenge for me has been pet waste. We have 7 pets including 4 cats, 2 dogs, and a sassy cockatiel. Why we have so many pets is another long story. Pet care comes with a lot of packaged products. Pet food comes in plastic, dog poop has to be bagged and thrown into the garbage, and cat litter is not compostable. We do our best to make the best decisions we can like buying the biggest bag of food, reusing trash we find on the road to pick up our dog's waste, and we source out wood chip waste for our cat boxes.
What has been the most rewarding part thus far?
The amount of people I've inspired has truly been the most rewarding part of this journey. I truly believe living by example is the best thing anyone can do and to see people such as my family make small changes fills my heart.
In addition to using bamboo toothbrushes, what other alternatives have you found for your day-to-day personal care products?
In addition to bamboo toothbrushes, I use many locally, handmade products that come in sustainable packaging. I also look for beauty products in bulk when I can and use my own containers to fill up, use products that focus on a circular business model, and have replaced all of my disposable objects such as razors and menstrual products with reusable alternatives.
What is some advice for someone wanting to reduce the amount of waste they produce?
My advice for someone wanting to reduce their waste is to first assess what they are already throwing away. What fills up your bin? Once you know what you are throwing away, then you can start developing a plan of attack. I have a great beginner's guide on my blog that will walk you through the steps. But ultimately, to set yourself up for success, go at your own pace. Go slow, develop those habits, then tackle the next area of opportunity. Overall, waste reduction can be quite overwhelming so be gentle with yourself and strive for progress, not perfection.
To learn more about reducing needless waste, visit www.zerowastenerd.com and follow Zero Waste Nerd on Instagram and Facebook!


text:
Q: API call limit in Salesforce Can I get to know or check how many calls have been made to Salesforce in C#?
This will help me in checking Call count limitation to user and update same to him.

A: Yes, you can use the LimitInfoHeader from the Partner API to monitor the current API usage.
<soapenv:Header>
    <LimitInfoHeader>
        <limitInfo>
            <current>5</current>
            <limit>5000</limit>
            <type>API REQUESTS</type>
        </limitInfo>
    </LimitInfoHeader>
</soapenv:Header>

It is also available in the REST API - Limit Info Header.
HTTP/1.1 200 OK
Date: Mon, 20 May 2013 22:21:46 GMT
Sforce-Limit-Info: api-usage=18/5000

redpajama_set_name: RedPajamaStackExchange


Click the Sugar Skull coloring pages to view printable version or color it online (compatible with iPad and Android tablets).
You might also be interested in coloring pages from Sugar Skulls category and Tattoo tag.
This Coloring page was posted on Thursday, September 1, 2016 - 14:23 by painter.