text:
\section{Introduction}

In the limit
of large repulsion, when $U/t\rightarrow\infty$ Kusmartsev  \cite {Kus1}
found that the Aharonov-Bohm(AB) effect is fractional. In units of elementary
flux quantum  its period is equal to  $1/N$, where the $N$ is the
number of electrons on the ring.
This  effect is drastically different from conventional integer or
half-integer AB
effects\cite{Land},\cite{Kuli},\cite{Butt},\cite{Land1}\cite{Kus}.

This result has since been confirmed
in the investigation by Schofield, Wheatley and Xiang\cite {Scho}. The first
corrections were studied by Yu and Fowler
\cite{Yu}, where they first recognized the importance of
the magnon excitations for the Aharonov-Bohm effect.
 We extend their analysis and investigated the effects for the dilute system.
We find a new type of oscillations, related to a magnon excitation spectrum.
The new AB oscillations with  the period $M/N$ arising with the first
corrections
to the strong coupling limit or to the dilute density limit appears when
we take into account the polarized or partially polarized ring with $M$
down-spin electrons. That is
for the effect
it is very important  a role of the Zeeman interaction causing the polatization
effects. The importance of the Zeeman energy was noticed recently by
Choy \cite{Choy}.

We show that for the very dilute system, when the filling factor tends to zero,
as well as for strong-coupling limit, the fractional $1/N$
Aharonov-Bohm effect may coexist with  the integer conventional,
integer-fractional
or half-flux quantum periodic AB effect.  By the integer-fractional regime of
AB effect we mean the appearance of a new type of oscillations on the envelope
function of $1/N$ periodic oscillations, which display a new fractional
period. The period of these
oscillations is equal to $M/N$, where $M$ is the number of
down-spin particles. The $M/N$ oscillations
exist even when the $1/N$ oscillations are washed out.
 When the Zeeman interaction is neglected, i.e.
the Zeeman energy is zero and $M=N/2$,
there is a coexistence of the fractional $1/N$ and half-flux quantum periodic
oscillations. In contrast, when the system is almost polarized, i.e. $M=1$, the
fractional $1/N$ and integer periodic oscillations  coexist.

In previous work\cite{KWKT}  we found that finite scaling finite size effects
 were very important. There is a scaling behavior of the ground state energy
which  does not depend on the size $L$ or  on $U$, but depends only on
$N/UL=const=\alpha$,
where $U$ is measured in units of $t$ and $L$ is measured in units
of the lattice constant.  Such scaling occurs only at small values of $\alpha$.
Due to this scaling
symmetry  the fractional  Aharonov-Bohm effect may arise even for
small values of $U$, for the very dilute electron systems. In the present work,
we shall show that, such a symmetry is related to the structure
of the low energy excitations of the Hubbard Hamiltonian, namely,
with the spectrum of the spinon or magnon excitations.
In fact the parameter $\alpha$ defines the energy of the magnon excitations.
Thus, the shape of the new AB oscillations completely depends upon the
energy of the magnon excitations.

In fact, we get an amusing physical picture.
First, let us discuss the case $M=N/2$.
For an odd number of particles
 in the ground
state of the Hubbard ring, there already exists a spinon excitation.
The momentum of this excitation is associated with the top of the spin-wave
spectrum. This excitation is needed because of the parity effect,
discussed by Legget\cite{Legg}, Loss\cite{Loss} and Kusmartsev\cite{Kus},
for the case of interacting spinless fermions.

Thus, again with the first correction to the strong coupling limit
or to the dilute density limit we have shown that
on the Hubard ring there occurs the reconstruction of the parity effect,
reflecting
the structure of low-energy excitations of the Hubbard ring.
However, the parity effect obtained is distinct from the
parity effect for the mesoscopic ring with spinful electrons and disorder
\cite{Weis}
Inducing the flux the momentum of this spinon excitation, which exists
in the ground state, changes. This gives rise to the correspondence between
the spin excitation spectrum and the Aharonov-Bohm fine structure. With
the magnetic field the number of polarized electrons changes, i.e.
the spin excitation spectrum changes, this gives rise to the change
of the Aharonov-Bohm periodicity.

This occurs also for the general situation, with any number of particles on the
ring
and at different rational ratio $M/N$. In the latter case one may
see an additional fractional $M/N$ AB oscillations, which are due to
the structure of the spinon excitation spectrum;
the picture is as follows.
With the flux, the spinon changes its momentum, which is, in fact,
proportional to the flux. The change in the momentum does not
occur  continuously,
but via jumps related to the finite number $N$ of
the particles on the ring. Therefore these jumps are associated with the $1/N$
fraction of
elementary flux quantum. Thus, on the Hubbard ring with the flux
the spinons behave similar to spinless fermions, where the momentum
is also shifted with the flux, reconstructing the
 parity effects. The Fermi momentum of this fermions is related to the
number of polarized electrons $M$. This gives the new scale and the new
periodicity in Aharonov-Bohm effect, related to the ratio $M/N$.
However, the  fractional $M/N$ AB effects do not destroy
  the fractional $1/N$ effect associated with the bound state of $N$ particles.
Both effects may coexist, reflecting the separation of spin and charge
degrees of freedom.

\section{Main equations}

To demonstrate this picture we have studied the Hubbard Hamiltonian
\begin{equation}
H = - t \sum_{<i,j>, \sigma} a^\dagger_{i\sigma} a_{j\sigma}
     + U \sum^L_{i=1} n_{i+} n_{i-} \quad
\end{equation}
which is parametrized by the electron hopping integral $t$,
the on-site repulsive
Coulomb potential $U$ and the number of sites L.
The operator $a^\dagger_{i\sigma} (a_{i\sigma})$  creates (destroys)
an electron
with spin projection $\sigma$ ($\sigma = +\, \rm{or}\, {\bf - }$),
at the ring site $i$, and $n_{i\sigma}$
is the occupation number operator $a^\dagger_{i\sigma} a_{i\sigma}$. The
summation
in Eq. (1) extends over the ring sites $i$ or -- as indicated
by $<i,j>,\sigma$ -- over all distinct pairs of nearest-neighbor sites,
around the ring with the spin projection $\sigma$.
  For the case of the magnetic field we  use the same form
of the wave function as in  Refs \cite{LiebW},\cite{Suth} and
\cite {Kawa}, i.e.
\begin{equation}
\psi(x_1,\ldots,x_N)=\sum_P [Q,P]\exp[i \sum^N_{j=1} k_{Pj} x_{Qj}],
\end{equation}

where $P=(P_1,\ldots,P_N)$ and $Q=(Q_1,\ldots,Q_N)$
are two permutations of $(1,2,..., N)$, and N is the number of electrons.

     The   coefficients  $[Q,P]$,  as     well  as   $(k_1,...k_N)$,   are
determined  from the Bethe equations, which in a magnetic field  are
changed by the addition of the flux phase $2\pi f$ \cite{Suth},
\cite{Kawa},\cite {Staff}, \cite{Kus1},\cite{Fye}, \cite{Ferr}, and \cite{Sudb}
\begin{equation}
e^{i(k_j L-2\pi f)}=\prod^M_{\beta=1}
 \biggl({{it \sin k_j -i\lambda_{\beta}-U/4}
\over{{it \sin k_j -i\lambda_{\beta}+U/4}}} \biggr)
\end{equation}
and
\begin{equation}
-\prod^N_{j=1} \biggl({{it \sin k_j -i\lambda_{\alpha}-U/4}
\over{{it \sin k_j -i\lambda_{\alpha}+U/4}}} \biggr)=
\prod^M_{\beta=1}\biggl({{i\lambda_{\alpha} -i\lambda_{\beta}+U/2}
\over{i\lambda_{\alpha}} -i\lambda_{\beta}-U/2} \biggr)
\end{equation}
 $f$ being the flux in units of the elementary quantum flux $\phi_0$.
 The explicit form of the Bethe equations in a magnetic field is  \cite
{LiebW},\cite{Shas}, \cite{Kus1}, \cite{Fye},
\cite{Yu}
\begin{eqnarray}
Lk_j=2\pi I_j+2\pi f - \sum^M_{\beta=1} \theta(4(t\sin k_j -\lambda_{\beta})/U)
 \label{momenta}
\\
-\sum^N_{j=1} \theta(4(t\sin k_j -\lambda_{\beta})/U)=2\pi
J_{\beta}+\sum^M_{\lambda_{\alpha}=1}
\theta( 2(\lambda_\beta -\lambda_{\alpha})/U)\label{lambda}
\end{eqnarray}
where $\theta(x)=2 \arctan(x)$ and the quantum
numbers $I_j $ and $J_{\beta}$,
which are associated with the charge and spin degrees
of freedom, respectively, are either integers or hal- odd integers,
depending on the parities of the numbers of down and
 up-spin electrons, respectively, i.e.
\begin{equation}
I_j=\frac{M} {2} ~~~ {\rm (mod ~1) ~{and} }~~ J_{\beta}= \frac {N-M+1} {2}
{}~({\rm{mod} ~1).}
\end{equation}

\section{Expansion}

 In previous work\cite{KWKT} showned that
in the dilute density limit there exists a  small parameter $\alpha$, which
is
\begin{equation}
\alpha= \frac {N t} { L U} =\rho t/U
\label {parameter},
\end{equation}
where $\rho=N/L$ is
the filling factor.

{}From an expansion in $\alpha$ (
 $\alpha<<1$)
of the second Bethe equation (\ref {lambda}),  we get
the equations
\begin{eqnarray}
N\theta(t_{\beta}) -\frac{8} {U} \frac {1} {(1+t^2_\beta)} \sum^N_{j=1}
 \sin(k_j)=2\pi J_{\beta}+\sum^M_{\lambda_{\alpha}=1}
\theta( (t_\beta -t_{\alpha})/2),
\label{spin-lim-exp}
\end{eqnarray}
where $t_{\beta}=4 \lambda_{\beta}/U$.

Next step, we notice that the second term on
the left hand side  the eq.(\ref{spin-lim-exp}) is small and therefore
we may incorporate it into the argument $x$ of $\theta(x)-$, giving

\begin{eqnarray}
N\theta\biggl[t_{\beta} -\frac{4} {NU} \sum^N_{j=1}
 \sin(k_j)\biggr]=2\pi J_{\beta}+\sum^M_{\lambda_{\alpha}=1}
\theta( (t_\beta -t_{\alpha})/2),
\label{spin-lim-exp1}
\end{eqnarray}

With the substitution $x_\beta=t_\beta- \frac{4} {NU}\sum^N_{j=1}
 \sin(k_j)$, this equation is reduced to the Bethe equation
for an isotropic Heisenberg antiferromagnet on a ring having $N$ sites
and $M$ down spins:
\begin{eqnarray}
N\theta(x_{\beta})=2\pi J_{\beta}+\sum^M_{{\alpha}=1}
\theta( (x_\beta -x_{\alpha})/2),
\label{x-eq}
\end{eqnarray}
 One sees that the solution for $x_\alpha$ is independent
of both the flux $f$ and on its repulsion $U$.

Let us  substitute
the  variables $\lambda_\beta=U (x_\beta+ \frac{4} {NU}\sum^N_{j=1}
 \sin(k_j))/4$ into
  equation (\ref{momenta}) for the momenta $k_j$ and study the resulting
 equation
in the variables $x_{\beta}$.
Then, analogously to the expansion of  eq.(\ref{lambda}),
expanding the eq.(\ref{momenta}) in powers of $\alpha$ , we get
a new form of the first Bethe equation, namely
\begin{equation}
Lk_j=  2\pi I_j+2\pi f +\sum^M_{\beta=1} \theta(x_{\beta}) +
        \frac {8   \sum^N_{l=1}(1-N\delta_{lj}) \sin k_l } {NU}
\sum^M_{\beta=1} \frac  {1} {1+ x^2_{\beta}}
\label{momenta-1order}
\end{equation}
For parameter $\alpha$  small (dilute density),
this system of linear equations
may be solved, with the result for the momentum
\begin{equation}
k_j= \frac {2\pi} {\tilde{L}} (I_j +f +\frac {1 } {N} \sum^M_{\beta=1} J_\beta
 +\frac {8B } {UL} (\frac {1} {N}\sum^N_{l=1} I_l +f +\frac {1 } {N}
\sum^M_{\beta=1} J_\beta))
                               \label{2momenta-2order}
\end{equation}
and for the ground state energy
\begin{equation}
E_{ground}=- \tilde{D} \cos( \frac {2\pi} {\tilde{L}}
\left(f- \frac {p} {N} +\frac{I_{max}+I_{min}}{2}
+\frac {8B} {UL}  (\frac {1} {N}\sum^N_{l=1} I_l +f -\frac {p} {N}) \right))
 \label {en-corr}
\end{equation}
where $p=- \sum^M_{\beta=1} J_\beta$ , $I_{max}$ and $I_{min}$ are the
maximal and minimal charge quantum numbers,
$$\tilde {D}=2 \sin (\pi N/ \tilde {L})/ \sin (\pi/\tilde {L}),$$
 and
 $$\tilde{L}=L ( 1+ \frac {8B}{UL}).$$
 The magnitude
$$-B=-\sum^M_{\beta=1} \frac  {1}{1+ x^2_{\beta}},$$
 which is a real number,
has the physical meaning of the total energy( ground state + excitations)
of the antiferromagnetic Heisenberg Hamiltonian with
 the exchange constant equal to $1/4$.

The
variables $x_{\alpha}$ are satisfied by the Bethe ansatz equation
 for that Hamiltonian(\ref{x-eq}). Actually,
the value $B(k)$ is a function associated with the
excitations spectrum, where  $k$ is the momentum of the spin-wave
excitation of the Heisenberg Hamiltonian  $k=2 \pi \sum_{\alpha}
 J_{\alpha}/N$. For the antiferromagnetic state of  Heisenberg Hamiltonian
 the dependence $B(k)$ has been studied in detail in many works:
first being  estimated by Anderson\cite{Andsws},
within a spin wave approximation and then
  by  des Cloizeaux and Pearson\cite{Cloi}\cite{Cloi1}
 in the framework of the Bethe ansatz
obtaining the exact form of this spectrum.

At large number $N$ this dependence has the form:
\begin{equation}
B(k)=(N (\log 2-1/4) - \frac{\pi}{2} \mid\sin k \mid)/4
\end{equation}
 The ground state energy(first two terms) has been obtained by
Hulthen \cite{Hult} in the framework of the Bethe method\cite{Beth}.
It was seven years later after  Bethe  introduced his method
(the Bethe ansatz),
applying it to the ferromagnetic Heisenberg chain.

\section{ Single Spin}

Let us consider the case  with $M=1$, i.e. when the system
is strongly polarized.
Here, we find an analytical solution, permitting  eq.(\ref{x-eq})
to be immediately solved yielding
\begin{equation}
x_{\beta}=\tan(\frac{\pi J_{\beta}}{N})
\end{equation}
The function  $B(J_{\beta})$ can then be calculated from
 this solution, giving $B=\cos^2(\frac{\pi J_{\beta}}{N})$.

 Now we must consider two independent cases, when $N$ is odd
and when $N$ is even. Let $N$ be an even number, then $J_{\beta}$
is an integer,  $p$ say. Then,
for the ground state energy we get the explicit expression

\begin{equation}
E_{even-N}=-2 \frac{\sin\big(\frac{\pi N}{L+\frac{8} {U}\cos^2(p\pi/N)}
\big)}
{\sin\big(\frac{\pi}{L+\frac{8} {U}\cos^2(p\pi/N)}\big)}
 \cos \big[ \frac{2 \pi} {L} (f-\frac{p} {N}) \big].
\label{energy-even}
\end{equation}

The ground state energy-flux dependence is this a function
with two types of oscillation.
The period of the first  of oscillation
is equal to the single flux quantum, while the period of
the second type of oscillations is equal to the fraction $1/N$ of the
elementary flux quantum. Let us describe this dependence.
Each
period of the single flux quantum
then
 consists of $N$-parabolic like curves, where each of these curve is
labelled by  $p$: $0\leq p \leq N$.
In the limit of large $N$ we may put as $f=p/N$ to get an exact
single flux periodic function, which has  minima
and maxima  at integer flux values and at half-odd integer flux values
respectively.

For an illustration on Fig.1 is presented the energy-flux dependence
for the Hubbard ring having 1000 sites with 6 electrons and a single down spin
at the coupling $U/t=10$. On this Figure one sees 7 parabolic curves
having  minima at $p/6$ flux values, and 6 cusps associated with
 intersections between each two  neighbour parabolas. The envelope
of these parabolic curves is the single flux periodic
 function $\cos^2 \pi f$.

Next we turn  to the case of $N$ odd, when the numbers $J_{\beta}$ are
half-odd integer. So, let us take $J_{\beta}={ p+ N}/ {2}$, where
 $p$ is an arbitrary integer.
Given this,
then the ground state energy
may be expressed as

\begin{equation}
E_{odd-N}=-2 \frac{\sin\big(\frac{\pi N}{L+\frac{8} {U}\sin^2(p\pi/N)}
\big)}
{\sin\big(\frac{\pi}{L+\frac{8} {U}\sin^2(p\pi/N)}\big)}
 \cos \big[ \frac{2 \pi} {L} (f-\frac{p} {N}) \big],
\label{energy-odd2}
\end{equation}

Again, we find that the ground state energy dependence
 has both the single flux quantum and the fractional
flux quantum periodicity, described by the first and the second
factors, respectively. In the region of the one period
 of the single flux quantum
this dependence again consists of $N$-parabolic like curves,
labelled  by the integer  $p$: $0\leq p \leq N$.
In the limit of large $N$ we may put as $f=p/N$ to get an exact
single flux periodic function. However,
for an
odd number of particles, this function  now
 has maxima at integer flux values
and minima at half-odd integer flux values.

For an illustration on Fig.2 is presented the energy-flux dependence
for the ring having 1000 sites with 11 electrons and a single down spin
at the coupling $U/t=10$. On this Figure one sees 12 parabolic curves
having  minima at $p/11$ flux values, and 11 cusps associated with
 intersections between each two  neighbour parabolas. The envelope
of these parabolic curves is the single flux periodic
 function $\sim-\sin^2 \pi f$.

Several conclusions can be drawn immediately. Firstly,
the parity effect, which disappeared in
the $\alpha$ small limit, is  again recovered
when  the first correction is
taken into account. In other words, the ground state
energy-flux dependence is different for the  even and odd numbers of
fermions on the ring. To be more precise,
 the behavior of the flux-energy dependence for
an even number of particles is shifted by a half flux quantum in comparison
with the case of an odd number of particles.
 This is similar to the parity effect observed on a
 ring with interacting spinless fermions \cite {Legg}, \cite {Kus}
\cite{Loss}.  However, in  comparison
with  spinless fermions on the Hubbard ring the,
 cases of even and odd number of particles are reversed.
 The parity effect discussed might be related to
the number of down-spin electrons, which is $N-1$. In comparison
 with the case of spinless fermions, because of this shifting
of the particle number $N$ by $N-1$,  the
parity effect discussed might be reversed. In this case, again, one may connect
the parity effect with the creation of the statistical flux $\pi$,
due to Fermi statistics\cite{Kus}.

\section{Spin up-down equilibrium}

In this section we  will follow the important suggestion
 by Yu and Fowler \cite {Yu} about
the possible  role
of spin excitations in the Aharonov-Bohm effect on the Hubbard
 ring in the limit of large  $U/t$. Our results
 extend their analysis of the first
corrections in the strong coupling limit as well as
being applicable to systems with the low electron density  at arbitrary
coupling.
Now, let us consider the general case, when the number of  spin-up fermions
is approximately equal to the number of spin-down fermions, i.e. it may differ
by $\pm1$.

   To investigate the behavior of the flux-energy dependence
for such a general situation, we must
consider four independent cases, when the numbers $N$ and $M$ are even
or odd. As we will see below ( also,see, above the single spin case M=1)
for the ground state energy flux dependence
only  the parity of the number $N$ will be important.

 Let us consider the first of these cases, when
$N$  and $M$ are  even. The quantum
numbers (which are half-odd integer)
 of the Heisenberg Hamiltonian, ${J_{\alpha}}$,
for the ground state are distributed symmetrically with respect to
origin, so that $\sum_{\beta} J_{\beta}=0$. The quantum numbers of
the charge degrees of freedom $I_{\alpha}$ for the Hubbard Hamiltonian
are integer with $\sum I_j = -\frac{N}{2} = -M$.

However, at zero
flux the minimum energy of the Hubbard Hamiltonian will correspond to
the set of quantum number $\{J_{\alpha}\}$, but with nonvanishing sum
 $\sum J_{\alpha}=M$, which
 does not correspond to the ground state of the Heisenberg Hamiltonian.
Such a state is associated with the single magnon excitation of the Heisenberg
chain, having a momentum
$k_0 = \frac{2\pi P_0}{N} = \pi$ and  the excitation  energy
$B(k_0) = B_0$. The distribution of  quantum numbers
$\{J_{\alpha}\}$ associated with the ground state of the Hubbard
ring  at zero flux  has a hole near the origin, i.e. is of the form:
\begin{equation}
J_1,\ldots,J_M=-\frac{M-1}{2},\ldots,\frac{-1}{2}, {\bf hole} , \frac{3}{2},
\ldots,\frac{M-1}{2}
\end{equation}
With  increasing  flux    this hole in the distribution  is shifted
to  another place  so that $\sum_{\alpha} J_{\alpha}=M-1$ and, therefore,
 the magnon momentum decreases.  In other words with the flux
the hole in the distribution moves to its left
side. Therefore, with that motion of the hole, exactly
speaking, with the new position of the hole  the magnon
excitation will have the new momentum $k<k_0$ and the energy $B(k)<B_0$.
Exactly speaking, the obtained dependence $B(k)$ describes the low-energy
excitation of the discrete antiferromagnetic Heisenberg chain,
as first discussed  by des Cloiseaux and Pearson\cite{Cloi}\cite{Cloi1}.

With the dependence $B(k)$ taken into account
the appropriate ground
state energy dependence of the Hubbard model on the flux may be written in the
form:
\begin{equation}
E^{even-M}_{even-N}=-\tilde{D} \cos \big[ \frac{2 \pi} {L} (f-\frac{p} {N})
\big],
\label{even-en}
\end{equation}
where the flux  value $f$ is changed in the region $ (2p-1)/2N<f<(2p+1)/N$ and
$\tilde {D}(k)=2 \sin (\pi N/\tilde{L}(k))/ \sin (\pi/\tilde{L}(k))$,
with $k=2\pi p/N$ being the magnon momentum of the finite Heisenberg chain with
$N$ sites, where
$p$ takes integer values.

 The dependence consists of equidistant parabolic segments.
 The position of  each parabolic segment along
the vertical axis depends upon the value $-\tilde{D}(k)$.
As the integer $p$ increases,
the magnon momentum $k$ and
 the energy of this magnon excitation $B(k)$ also increases, so long as
 $p\leq\frac{N}{2}$. The magnon energy
then  has a maximal value at $p=\frac{N}{2}$.

This
means that the value $-\tilde{D}(k)$ associated with
 the position of bottom of the
$p$-th parabolic segment along the vertical axis
 decreases monotonically. As a result, we see
that the envelope curve which passes through the bottom of these
parabolic segments is similar to the  curve describing the
spectrum of the spin-wave excitations of the antiferromagnetic  Heisenberg
chain with $N$ sites. Note, that this curve is
described by the equation $E_{gr}{(k)} = -\tilde{D}(B(k))$.
The flux dependence of this envelope curve can be obtained
if we   substitute $k=2\pi f$.
It is a half
flux quantum periodic function.

    Now, let us consider the second case when {\it $N$ is even, but M is an
odd number}. In this case, at zero flux,
 the ground state energy
corresponds to half-integer numbers $I_j$ and integer numbers
$J_{\alpha}$ distributed  symmetrically  around the origin (see, for
example \cite{Kus}). That is, here,
\begin{equation}
\sum_{j} I_j=\sum_{\beta}J_{\beta}=0
\end{equation}

The analysis of this case is similar to that above.
The given  distribution of quantum numbers
${J_{\alpha}}$ corresponds  to the ground state energy,  both of the
antiferromagnetic Heisenberg chain and the Hubbard ring.
However, including the flux
the ground state energy of the Hubbard ring will now
correspond to the nonvanishing
 sum $\sum_{\beta} J_{\beta}$, which is negative.
This means that there will be
created a magnon excitation similar to that described above,
associated with the appearance of a hole in the distribution of
${J_{\alpha}}$. With the increasing flux the hole will move from
the left  to the right  of the distribution.

We, therefore, once again have the flux-energy dependence of
 the Hubbard ring determined via the low energy
 excitations of the Heisenberg chain.
This dependence again consists of $N$ parabolic segments
and is described by a  similar formula to  eq.(\ref{even-en}). The
envelope, which passes through the bottom of these parabolic segments is
described with the aid of the same equations $-\tilde{D(2\pi f)}$, i.e.
it has cusps at its minima, which are at integer and half-integer
 flux values and smooth
maxima at the values of the flux
exactly midway between the cusps.

    Let us now consider  the case when $N$ is  odd  and $M$ is
 even. The quantum numbers $I_j$ are integer and are
symmetrically distributed around the origin, i.e. $\sum I_j =0$. The
quantum numbers $J_{\alpha}$ are also integers. The ground state energy
of the Heisenberg Hamiltonian corresponds to the dense distribution
with the sum $\sum_{\alpha} J_{\alpha} = -\frac{M}{2}$. However, the
ground state energy of the Hubbard ring with zero flux values must
correspond to the state with $\sum_{\alpha}J_{\alpha}= 0$. This sum can
 equal  zero when the associated state of the Heisenberg
chain is not a ground state, at rather an excited state.
This means
that it is a state with the hole in the $\{J_{\alpha}\}$ distribution. One
sees that in this case the hole will be created exactly
in the origin:
\begin{equation}
J_1, \ldots, J_M :=-\frac{M}{2},\ldots,{-1},{\bf hole},{1},
\ldots,\frac{M}{2}
\end{equation}

This state corresponds to the top in the spin-wave excitations of the
Heisenberg chain, i.e.  to the spin-wave excitation with
momentum $k=\frac{\pi}{2}$. As the flux changes, the parabolic
segment with its minimum at zero flux is exchanged with another the
parabolic segment which has its  minimum at the  flux
$f=\frac{1}{N}$.

 This parabolic segment will correspond to the new set
of $J_{\alpha}$, where the hole is in the position
formerly occupied by
$J_{\alpha} = 1$, so now
 $\sum_{\alpha} J_{\alpha} = -1$.
The appropriate magnon excitation
associated with this parabolic segment will have   momentum
$k=\frac{\pi(N-2)}{2N}$  and   energy $B(\frac{\pi(N-2)}{2N})$,
which are lower
than the  momentum $k=\pi/2$ and the maximal energy
$B(\pi/2)$, respectively.

With the next increasing the flux, there will occur a transition to
a new parabolic segment, corresponding to a  new set of
${J_{\alpha}}$ with the hole at the position $J_{\alpha}= 2$.
    This is the parabolic segment corresponding to the magnon
excitations with  momentum $k=\frac{\pi(N-4)}{2N}$ which has a
lower energy than the magnon excitation in the previous case.
Therefore, at this
  flux the mimimum energy of the appropriate
parabolic segment decreases. The absolute minimum is reached
 at the quater and 3/4  of the flux
quantum, if the flux is changed within a single flux quantum.

 The envelope curve, which goes through the
minima of these parabolic segments is a half flux quantum periodic
function, which has
 smooth maxima at integer and half integer flux values. This curve
may be described with the aid of the formula $E_{ground} =-\tilde{D}(2\pi f)$.

    Let us consider now the last case, which is $N$  and $M$
odd. Here, both the  $J_{\alpha}$  and
$I_j$
are half-odd integer.
 The energy of the Hubbard Hamiltonian,
which corresponds to the dense distributions with $\sum I_j =
-\frac{N}{2}$ and with $\sum J_{\alpha} = \frac{M}{2}$
and assumed to be a ground state
is not a ground state of the
Hubbard ring at zero flux. But this dense distribution of $\{J_{\alpha}\}$ is
the true  ground
state of the Heisenberg Hamiltonian.

The ground state of the Hubbard ring will
correspond to the distribution of the quantum number $\{J_{\alpha}\}$
with the hole at the position $J_{\alpha} = -\frac{1}{2}$ and the sum
$\sum_{\alpha=1}^M J_{\alpha} = +\frac{2M+1}{2} = +\frac{N}{2}$. That
is, to say, in the ground state of the Hubbard ring
there already exists the magnon
excitation with  momentum $k= \frac{2M+1}{N} \pi =
{\pi}/{2}$, which is, in fact, associated with the top in low
energy spin wave excitation spectrum of the Heisenberg chain.
 The energy-flux dependence
 will again consist of parabolic segments
 as described above. The minima
 of these segments will be located at the flux
$f=\frac{p}{N}$. The energy behavior of the minima of these segments
will be exactly the same as described in the previous case.

The picture discussed is illustrated in Figs 3 and 4. In  Fig.3
one sees the coexistence of the half-integer and of the fractional
$1/N$
Aharonov-Bohm effects. The ring has 7500 sites filled with 25 electrons.
There are two types of oscillations present: half-flux quantum periodic,
which has the largest amplitude and the other fractional -
$1/25-$ flux quantum periodic, the amplitude of which
is much smaller.
  The envelope curve of the fractional oscillations is a half
flux quantum periodic function, with  smooth maxima
at integer and half-integer flux
quanta $ f=...,-1/2,0,1/2,1,...$ and cusps, which are minima,
 in between.
This is due to the parity effect, since the number of particles
on the ring is odd.

For  contrast, in Fig.4 we present a  case for an even
 number of electrons  on the ring with
$N=10$, and $L=1000$.
Again,
 as in  Fig.3  the coexistence of the half-integer and fractional
Aharonov-Bohm effects is observed and because there are
 10  particles
the fractional $1/10$ flux quantum oscillations arise.
  The envelope curve of these fractional oscillations
is as in  Fig.3
a half flux quantum periodic function, which, however, has minima at
integer and half-integer flux
quanta.

In the comparing these two Figures, one notes that
with the change of the number of particles from odd to even, the position of
the maxima is shifted by a quarter-flux quantum. This is precisly
due to the parity effect similar to  the parity
effect for spinless fermions\cite{Kus},\cite{Loss},\cite{Legg}.
That is on the Hubbard ring we have obtained the parity effect
which is similar to one for spinless
fermions. The effect  is due to  the
contribution of  the spinon excitation spectrum in the AB effect.
Note, that in the second case (Fig.4) the difference in amplitudes
between the half-integer and fractional $1/N$ oscillations is smaller,
  indicating
that the fractional effect is more important at small number of particles.
With  increasing $U$ or $L$, this difference may disappear.

\section{ Fractional $ {\bf\frac{M}{N}}$ Aharonov-Bohm effect.
         }

In previous sections we have shown that the AB periodicity for the cases
of $M=1$ and $M=N/2$ are different
( single and half flux quantum, recpectively). Therefore, there
 occurs the question: what kind of AB periodicity occurs at
arbitrary $M$: $1<M<N/2$
Let us now consider the case when $M$ is arbitrary, but $M<<N$.
For this general case, when the density of down-spin
  particles $M/N$ is much smaller, then
the density of up-spin  particles $1-M/N$ can again be found
 an analytic solution. In this case we  expand equation (\ref{x-eq})
with in the parameter $M/N$. We can assume that
the values of $x_{\beta}$ in eq.(\ref{x-eq}) are small. Using
these assumptions
the eq. (\ref{x-eq}) may be written in the form
\begin{equation}
x_{\beta}+\sum^M_{\alpha=1} \frac{x_{\alpha}-x_{\beta}}{2 N} =\frac{\pi
J_{\beta}} {N}
\end{equation}
i.e. we have obtained a system of linear equations, which can  be
immediately solved, giving
\begin{equation}
x_{\beta}=\frac{2 \pi J_{\beta}} {N-2 M}-\sum^M_{\alpha=1} \frac{\pi
J_{\alpha}}{(2 N -M) N}
\end{equation}
This solution looks like  the momentum spectrum of a free particle on
a chain of $2N-M$ sites ( the first term of the right hand side of the
equation), plus an additional fictitious flux associated with
the second term of the right hand side of
the equation. From the solution  obtained
 one sees that the values $x_{\beta}$
are small if $\pi M/(2N-M)<<1$, which is equivalent to our
small parameter $M/N<<1$, and therefore $x_\alpha<<1$
does, indeed, hold.

On the other hand this analysis has shown that
the second term of the right hand side of  eq.(\ref{x-eq}) is smaller
that the first one in the parameter $\frac{M}{N}\ll 1$.
Therefore, in
that case the solution may be obtained in the form:
\begin{equation}
x_{\alpha} = \tan \frac{\pi J_{\beta}}{N}
\end{equation}
The substitution of which into $B[x_{\alpha}]$ gives
\begin{equation}
B(k) = \frac{M}{2} + \frac{1}{2}  \sum_{\beta=1}^M
\cos \frac{2\pi J_{\beta}}{N}
\label{eq-B}
\end{equation}
The choice of the set ${J_{\alpha}}$ determines the dependence of the
low lying excitations $B(k)$. Now we must consider four independent
cases associated with the parity of numbers $N$ and $M$.

    Let, first us consider the case, when $N$ is even and $M$ is odd. Then
the ground state energy corresponds to the vanishing sums of the
integers $J_{\alpha}$,  $\sum_{\alpha} J_{\alpha}=0$ and of the
half-integers $I_j, \sum_j I_j=0$. Both of these sets are densly
distributed, symmetrically around the origin. The excitation will
correspond to the hole in the distribution of ${J_{\alpha}}$. For
example, when $p=1=-\sum J_{\alpha}$, then the hole is created
on the left hand side, from
the left edge
. For the value $p=2$, the hole is moved on
to the next position
. In general, if the hole in the position
$J_{\alpha}$ then  we can write

\begin{equation}
p= -\sum_{\beta=1}^{M+1} J_{\beta} + J_{\alpha}
\label{mom-1}
\end{equation}
where  the sum is taken over the dense distribution. Therefore, if
 $p\leq M$, this sum equals $\sum_{\beta=1}^{M+1} J_\beta = -(\frac{M+1}{2})$.
In
the similar way, by splitting into to terms,
 the sum over the dense distribution of
$M+1$ quantum numbers $J_\beta$ and the term associated with the
hole at the place of $J_\alpha$  we find for the energy of such excitations:
\begin{equation}
B(k) = \frac{1}{2} (\frac{M}{2} +
\frac{sin\frac{\pi(M+1)}{N}}{sin\frac{\pi}{N}} cos\frac{\pi}{N} - cos
\frac{2\pi J_{\alpha}}{N})
\end{equation}
Then, expressing $J_{\alpha}$ from eq.(\ref{mom-1})
and substituting it into eq.(\ref{eq-B}) we get the dependence
of $B(\frac{2\pi p}{N})=B(k)$ where $p\leq M$. Note, that the case of
$p=-\sum_{\beta=1}^{M} J_{\beta}=M$
 corresponds to the ground state distribution of $J_{\alpha}$ shifted
by 1 to its left side. When $p>M$, in this shifted distribution there
 will again
appear the hole, initially,  at the beginning on the left hand side,
 from the left edge.

    With  increasing $p$ this hole will move from the
left to the right side, once $M<p\leq 2 M$. At the value $p=M$, the
distribution of $J_{\alpha}$ will be dense, but shifted by 2 to the left
in  comparison with the distribution associated with the value
$p=0$. Thus, for an arbitrary value of $p$, we can write

\begin{equation}
p= + l (M+1) + (\frac{M+1}{2}) - J_{\alpha}
\label {mom-gen}
\end{equation}

The energy of the associated excitations is equal to:

\begin{equation}
B(p) = \frac{1}{2} [M+\frac{sin\frac{\pi(M+1)}{N}}{sin\frac{\pi}{N}}
cos(\frac{2\pi l + \pi}{N}) - cos \frac{2\pi}{N} (p - \frac{M+1}{2}  -l(M+1))]
\label{gse-even,odd}
\end{equation}

where $l=0$, if $0\leq p \leq M $,
 $l=1$, if $M\leq p\leq 2M$,   $l=2$, if
$2M\leq p\leq 3M$ and so on.

    Thus, the ground state energy has cuspoidal minima (cusps) at $p=nM$,
where $n$ is an integer. So, if we substitute $p=lM$, we have for
the envelope functions of these cusps:
\begin{equation}
B(p) = \frac{1}{2} (M + \frac{sin(\frac{\pi M}{N})}{sin\frac{\pi}{N}}
cos\frac{2\pi l}{N})
\end{equation}
 Since $p\leq\frac{N}{2}$, the function $B(l)$ is monotonically
 decreasing with $l$, having a maximal value when
$l\sim[\frac{N}{2M}]$, where the brackets indicate that we take the integer
part of the value $\frac{N}{2M}$. The spectrum of the excitations
of the Heisenberg chain reflects the flux quantum periodicity of the
Hubbard ring, as described above. Therefore, we come to the conclusion,
 that in addition to fractional ${1}/{N}$, and integer  on the Hubbard
ring there also occurs fractional $\frac{M}{N}$ flux quantum periodicity.

For an illustration we give several examples for the ring with
 $U/t=10$ and $L=10^4$,
presented in Fig.5. In the first example (see, Fig.5a) the energy-flux
dependence is given for
$N=20$ and $M=3$ in the region of the half flux quantum. Since
the number of particles is not large one sees, the very pronounced
$1/N$ oscillations associated with the single parabolic like segment.
However, one sees already $M/N$ oscillations with much smaller
amplitude. So, the first, fourth and seventh parabolic like segments
have minima lower than nighbouring parabolas. If the number
of particles increases, for example, $N=30$ (see, Fig.5b, where we
 taken $M=5$) the amplitude
of $M/N$ oscillations increases. In this case the amplitudes of $1/N$
and are $M/N$ oscillations are, already, compared.
With the next increasing number of particles $N$, for example,
when $N=40$ and $M=7$ presented in Fig.5c, the amplitude
of $M/N$ oscillation becomes larger than the amplitude of the
$1/N$ oscillation.

    Let us consider the second case, when $N$ is odd and $M$ is even,
which corresponds to two types of integer quantum numbers
${J_{\alpha}}$ and ${I_j}$, with $\sum I_j=0$ and
$\sum_{\alpha}J_{\alpha}= -\frac{M}{2}$. However, the ground state of
the Hubbard ring will correspond to an excited state of the Heisenberg
chain with the hole in the origin, i.e. the ground state
 corresponds to the sum:
$\sum_{\alpha} J_{\alpha}=0$.

    The excitation associated with the value $p=1$ corresponds to the
shift of this hole to the position $J_{\alpha}=1$. So with the
increasing $p$ the hole moves to right.
Therefore, we can
write $p= -\sum_{\alpha=1}^{M+1}J_{\alpha} + J_{\beta}=J_{\beta}$,
where $p\leq {M}/{2}$ and the sum is taken over the dense
distribution.

 As $p$ is  increased  from
${M}/{2}$ the hole will be created on the left edge and moved to
the right edge, where $p\leq(M+{M}/{2})$. With the next increase
of $p$ this process will be repeated and so on. Thus, we may write in
general form that

\begin{equation}
p= +l(M+1) + J_{\beta}  ,
\end{equation}
where the first term related to the sum $\sum^{M+1}_{\alpha=1} J_{\alpha}$.

    The envelope function
of parabolic like curves
associated with the ground state energy-flux dependence
 is calculated in a similar way and
equals:
\begin{equation}
B(p) = \frac{1}{2} [M+ \frac{sin\frac{\pi(M+1)}{N}}{sin\frac{\pi}{N}}
cos(\frac{2\pi l}{N}) - cos\frac{2\pi}{N}(p-l(M+1))]
\label{gse-odd,even}.
\end{equation}
where  $l=0$, if $-(M)/2\leq p \leq (M)/2$; $l=1$, if
 $(M)/2<p\leq  (M)/2+M $,   $l=2$, if
$ (M)/2+M\leq p\leq  (M)/2+2M$ and so on.

 This dependence is shifted by $\frac{M}{2N}$ is comparison with the
previous case. It consists of cuspoidal minima and smooth maxima in
midway between.
 In comparison with the previous case
here the cuspoidal minima, one of which
existed at the  value $p=0$, are shifted by the same value of the flux,
too, and are exchanged in the positions with
smooth maxima. Thus,
 the properties of the envelope function are the same as
in the previous case.
For an illustration in Fig.6 we present the energy-flux
dependence for the ring with $U/t=10$. So, it is clearly
coexist $6/19 \sim 1/3$ and $1/19$
flux quantum periodicities for the ring with 19 electrons having 6 down-spins
for number of sites $L=1000$
(see, Fig.6a). The parabolic like curves are associated
with $1/N$ type oscillation.
 The envelope function of these curves is $6/19 \sim 1/3$ flux quantum
periodic function. The cusps of this function are located at $3/19$ and
$9/19$ flux quantum. With the increasing number of particles
the amplitudes of both type of fractional oscillations decrease.
This can be seen in Fig.6b, where the particles number was changed to $49$
and the number of sites was taken as $10^4$.
Although with the next increase
of the number of particles  at the fixed value of $M$
the amplitudes of
both type of oscillation decrease,
the fractional $1/N$ oscillations begin
to wash out, however the fractional $M/N$
oscillations are still exist. This
is shown in Fig.6c where the number of particles
was taken as $N=99$ on the ring with the same the number of sites,
as in Fig.6b. On the other hand when both numbers, the number of
particles $N$ and the number of dowm-spins $M$ increase,
the amplitude of $M/N$ oscillation increases and it is compared
with the amplitude of the main single flux quantum
periodic AB oscillation. In the latter the fractional $1/N$ oscillation
is completely washed out, although it is an origin for the fractional
$M/N$ oscillation. The fractional $23/99$ AB
oscillation is presented
in Fig.6d, where we consisder the ring with $M=23$ and $N=99$.
One sees on this Figure
that the fractional $1/99$ oscillation is practically
washed out.

The other two cases only differ from these already considered
ones only
by an appropriate shift.
In both of them the ground state of the Hubbard ring corresponds to the
excited state of the Heisenberg chain with the value
$$p_0=-\sum_{\alpha=1}^{M}J_{\alpha} = -\frac{N}{2} .$$

Because of the
$M$-periodicity, this state may be mapped into another one,
 where the sum
$ \sum^M_{\alpha=1} \tilde{J_{\alpha}}=0$ vanishes, i.e.
 this can be obtained with the aid of the transformation:
$$
\tilde{J_{\alpha}}=J_{\alpha}-l_1,
$$
where $l_1$ is the integer part of
the ratio $\frac{N}{2M}$. Therefore,
 we can write
$$
\sum_\alpha\tilde{J_{\alpha}}= \sum_{\alpha}J_{\alpha}-l_1 M=q ,
\label{eq-q}
$$
 where $q<[\frac{M}{2}]$.

Thus,
 the problem has been reduced to the one, discussed above, but with the
difference, that the ground state of the Hubbard ring will correspond
to the excited state of the Heisenberg chain where the hole in the
dense distribution of the set ${J_{\alpha}}$ is located at the $-q$-th
position, which is determined
by eq.(\ref {eq-q}). The other excited state
of the Heisenberg chain (or the ground state energy-flux dependence
of the Hubbard ring)
 is associated with the shift of this hole
to the right edge. Then all excitations may be described exactly by the
same manner as above. That is we may write
\begin{equation}
\tilde{p}= -\sum_{\alpha=1, dense}^{M+1}\tilde{J_{\alpha}} + \tilde{J_{\beta}}
\label{m30}
 \end{equation}

  Now we must consider the two particular cases ( 1) $N$ and $M$ are
even and 2) $N$ and $M$ are odd), independently.

 In the first case, the
values $J_{\alpha}$ or $\tilde{J_{\alpha}}$ are half-odd integer and from
eq.(\ref{m30} ) we get
\begin{equation}
\tilde{p}= \frac{M+1}{2} + \tilde{J_{\beta}}
\label{mom-3}.
\end{equation}
    The excitation energy is estimated as above yielding:
\begin{equation}
B(p) = \frac{1}{2} [ M+ \frac{sin\frac{\pi(M+1)}{N}}{sin\frac{\pi}{N}} cos
(\frac{2\pi l + \pi}{N}) - cos\frac{2\pi\tilde{J_{\beta}}}{N}]
\label{energy-3} .
\end{equation}

{}From
 eq.(\ref{mom-3}) and making the inverse
transformation from
$\tilde{J_{\alpha}}$ to $J_{\alpha}$ we finally get
\begin{equation}
B(p) = \frac{1}{2}( M+ \frac{sin\frac{\pi(M+1)}{N}}{
\sin\frac{\pi}{N}}
\cos(\frac{2\pi l+\pi}{N}) -
\cos\frac{2\pi}{N} [p+ \frac{N}{2} - \frac{M+1}{2} -l(M+1)])
\label{gse-even,even}
\end{equation}
where the value $l$ is choosen thus
$l=0$, if $-N/2\leq p \leq -N/2+M$;
 $l=1$, if $-N/2+M\leq p\leq  -N/2+2M$,
 $l=2$, if $-N/2+2M\leq p\leq -N/2+3M$ and so on.

For an illustration we present in Fig.7 the dependence of the
ground state energy for the ring with 1000 sites and $U/t=10$.
The number of electrons is taken as $N=20$ amd $M=6$. One sees  on this
Figure that the dependence of the envelope function of the parabolic curves
is monotonically decreased with the flux increases from zero.
This dependence has cusps, which are repeated with the periodicity
$6/20$. So we have here a quasi $\sim 1/3$ flux quantum periodicity.

    Similarly way we can obtain the excitation energy for the second
case, when the parity of  particles number is odd, i.e.
 $N$ and $M$ are odd:
\begin{equation}
B(p) = \frac{1}{2} [ M+ \frac{sin\frac{\pi(M+1)}{N}}{sin\frac{\pi}{N}}
cos\frac{2\pi l}{N} - cos\frac{2\pi}{N} [ p+\frac{N}{2} - l(M+1)]]
\label{gse-odd,odd}
\end{equation}
where the value $l$ is chosen
thus
 $l=0$, if $-N/2-(M-1)/2 \leq p \leq -N/2+(M-1)/2$;
 $l=1$, if $-N/2+(M-1)/2 \leq p\leq  -N/2+(3M-1)/2$,
 $l=2$, if $-N/2+(3M-1)/2\leq p\leq -N/2+(5M-1)/2$ and so on.

For an illustration we present in Fig.8 the dependence of the
ground state energy for the ring with 1000 sites and $U/t=10$.
The number of electrons is taken as $N=19$ and $M=5$. One sees  on this
Figure that the dependence of the envelope function of the parabolic curves
is monotonically decreased with the flux increases from zero.
This dependence has cusps, which are repeated with the periodicity
$5/19$. So, here again as in the previous case
 we have  a quasi-$\sim 1/3$- flux quantum periodicity.

    Thus, for all four cases the excitation energy is as ${M}/{N}$
periodic function, having cusps repeated with the
same periodicity. To study the flux dependence we must
 substitute the flux value $f$ instead of
the value $p/N$ of the excitations of Heisenberg ring.
 Generally speaking, in the
last two cases  the ground state  energy-flux dependence
of the Hubbard ring related to these excitations of the Heisenberg
chain is shifted by half-flux quantum in comparison with two
previous cases, respectively. But, the qualitative
 shape of these dependencies are not changed.

\section{Fine Structure}

    One general conclusion that can be drawn here that
the complete picture of the Aharonov-Bohm effect must include
a fine structure. The latter appears when we take into
account the electron-electron interactions and is characterized
by oscillations with an amplitude and a period smaller than
the conventional one. In fact, there is a hierarchy of  flux periodicities
which depends implicitly on the Zeeman interaction.

The Fourier spectrum
of the energy-flux
dependence for the Hubbard ring mainly consists of two or three harmonics.
The number and type of these harmonics depend on the contribution
from the Zeeman interaction.
 One of these harmonics is conventional and
has a single flux quantum period; the second one has the fractional
${1}/{N}$ flux quantum period; the third one has half-flux
quantum period or fractional $M/N$ period. The latter appears
 when the number of
down spin particles $M$  and
the number of up-spin particles
$N-M$ are different.

Let us analyse the fine structire.
The amplitude of the single-flux or the fractional $M/N$
periodic oscillations is proportional to the parameter $ n^2 \alpha$,
where $n={\pi N}/{L}$ and we use dimensionless units.
The amplitude of the fractional $M/N$ oscillation
is smaller, by factor $M/N$, than the amplitude of
oscillations with the integer period. Here, we assume that $M<<N$.
When $M \sim N/2$ the situation is reversed and the $M/N$ oscillation has
a dominant amplitude.
 The amplitude
of the fractional ${1}/{N}$ flux quantum periodic oscillations
 is proportional to
$\frac{n^2}{ N^3}$ and does not depend on the value $U$.
One sees also that both types of oscillations only depend
upon $U$ and $L$ through
  their product $UL$ or in other words
they only depend upon $\alpha$. With  increasing  $N$ at
 constant $n$ and $\alpha$,
the amplitude of the  fractional $1/N$ oscillations vanishes.

    The  single flux quantum harmonic coexisted with fractional $M/N$
 periodic harmonics show the parity
dependence. That is there is a difference in the energy-flux
dependencies for the cases of an  even and an odd number of fermions. This is
in  contrast with the  fractional ${1}/{N}$ periodicity, which has no the
parity
dependence. In the present case the parity effect
for an even and an odd number of fermions is transposed
  with respect to
 the case of
 spinless
fermions.

    It is useful explicitly  to write  an envelope of the
flux-dependence. For example in the case, where there is up spin- down spin
equilibrium (i.e. M=N/2),
 for the two cases of an even or an odd number of particles on the ring
the flux energy dependence has the form:

\begin{equation}
E_{erv} = \frac{n^3}{3} \frac{t^2}{U}
\left\{
\begin{array}{ll}
        \mid \sin 2\pi f \mid &\mbox { $N$ is even}\\
       \mid \cos 2\pi f\mid &\mbox{ $N$ is odd}
       \end{array}
\right\} .
\end{equation}

For the other values of $M$ the prefactor will be the same,
but the oscillation factor will be changed to reflect the fractional
$M/N$ flux quantum periodicity.

 On the
other hand the amplitude of the fractional ${1}/{N}$ oscillations
is estimated to be $\sim {tn^2}/{N^3}$, which is rapidly
 decreases  with
increasing $N$. Comparing this amplitude
 with the amplitude of $E_{erv}$ we get the
criterion for when the single flux quantum or $M/N$ flux
quantum  periodicity becomes dominant over the fractional $1/N$ ones;
 this
happens when
\begin{equation}
N\gg \sqrt[4]{\frac{UL}{t}}
\end{equation}
    However, to investigate this problem we have used the parameter
$\alpha = \frac{Nt}{UL} \ll 1$. The region where
the single or $M/N$ flux quantum periodicity is dominant is
\begin{equation}
\sqrt[4]{\frac{UL}{t}} \ll N\ll \frac{UL}{t}
\end{equation}
One sees that at large $L$ the limits when such behavior exists may be
easily recovered.

    In conclusion, then in the dilute density limit or for
 strong coupling
 we have discovered two
different regimes of the Aharonov-Bohm effect, namely, fractional $1/N$ and
integer coexisting with fractional $M/N$ and $1/N$. Integer and fractional
$M/N$ periodic oscillations appear simultaneously and may have
comparable amplitudes, which are distinct by numerical factor.
As a rule, in high magnetic fields
 the amplitude of $M/N$ oscillations is always amaller
than the amplitude of integer ones.

Thus, a very surprising implicit role of the Zeeman interaction has been
found, i.e.  with  increasing  magnetic field, as the
contribution of the Zeeman energy increases, in addition to
the fractional $1/N$ flux quantum periodicity, there appears the
integer-fractional $M/N$ flux quantum periodicity. With  increasing
 the magnetic field, i.e. with  decreasing $M$
the half flux  quantum periodicity at $M=N/2$ transforms
 into fractional $M/N$ ones and integer one.
Finally, in strong field, when the system is almost
polarized, only the integer flux quantum periodicity remains.

This transformation is not continuous but singular. The period
of this fine structure of the AB effect changes
with each  spin flip.
 For almost polarized system, we have
the coexistence of integer  and fractional $1/N$ flux quantum periodic
oscillations, only.
 On the ring with a small number of electrons
$N\leq\sqrt[4]{\frac{UL}{t}}$, there occurs
the fractional ${1}/{N}$ Aharonov-Bohm effect.
 However with  increasing  $N$, when
$N\gg \sqrt[4]{\frac{UL}{t}}$,
the single-fractional $M/N$ flux quantum
periodicity becomes dominant over the fractional $1/N$ effect,
although they  may both coexist.

\section{               CHANGE OF THE FINE STRUCTURE PERIOD
                         WITH MAGNETIC FIELD}

    The period of fractional oscillations
$\frac{M}{N}$ related to the number of down-spin particles. However,
 with the increase of magnetic field on the ring there may
occur spin-flip processes. As the result the number of the down-spin
particles changes (decreases) and the magnetization of the ring
related to the number $M$ decreases. This Zeeman magnetization may be
described with the aid of the formula:

\begin{equation}
(N-2M) = [\frac{2\chi H}{\mu_B}]
\end{equation}
where brackets means an integer part of the value.
The value $\chi$ is a
susceptibility of the Hubbard chain . First, in thermodynamic limit the
value of $\chi$ has been calculated by Takahashi \cite{Taka} for the case
of the half-filling, then for an arbitrary filling the value of $\chi$
has been calculated by Shiba \cite{Shib}. For the low density limit
$\alpha<<1$ we discuss the value of $\chi$ may be estimated by the similar
way. For the large values of $L$ the thermodynamic limit is
acceptable. Therefore we  use Shiba's result

\begin{equation}
\chi = \frac {3 U  L} { 2 \pi^4 t^2 n^2}
\end{equation}
whence
\begin{equation}
M = \frac{N}{2} - [\frac{\chi H}{\mu_B}]
\end{equation}
    One sees that with the field the value of $M$, and, therefore,
 the period of the
fine structure decreases
\begin{equation}
f_T = \frac{1}{2} - [\frac{\chi H}{\mu_B}]
\frac{1}{N}.
\end{equation}

    Thus we came to the conclusion that in the Aharonov-Bohm effect of
strongly correlated system such as the Hubbard ring may occur the
oscillation of the smallest amplitude, which period decreases, when
the field increases. In such an effect the spin orbit interaction
might be important (see, for example, Ref.\cite {Fuji})

\section { towards reality from exact solutions}

In the present work we have presented the exact solution
for the Hubbard ring, showing
a fine structure in the Aharonov-Bohm effect.
The mesoscopic rings in existing experiments have many channels,
large sizes and many electrons ( see, also, discussion in Ref.\cite{Avis}).
They also have disorder or impurities. The measurements are
carried at finite temperatures. Of course, all these factors make
a large difference between the presented model calculations and
the realistic situations.

However  on such  rings the electron density is much smaller
than in a 3D metal, where  screening exists.
Therefore  the electron-electron
interaction is very important.
The effects of the strong
electron-electron correlations, which give rise
the finite structure of AB effect, of course, will appear there.
 But at present time there is no
tool to take into account  correctly the  many-body effects related
to this interaction. The
Bethe ansatz method is only one single tool, where the effects of strong
correlations may be studied for large number of electrons
and  a system of a large size, but for some models only.
So, using the Bethe ansatz  we found, that
the interaction between particles also may create the finite
structure, which is characterized by several types of
periodic oscillations with the amplitude and period smaller than
one unit.

    In the integer-fractional regime the effects of
 disorder and  temperature will be to make
the flux-energy dependence smoother leading to
the disappearence of the fine structure. In experimental situations, where
the disorder or the temperature always exist, but have a low level
for the ring with large number of
electrons one may see three types of oscillations
in the flux energy dependence. With the next increase of the disorder
or the temperature the oscillations having the smallest
amplitude will disappear. So, for example, instead of integer-fractional $M/N$
periodic oscillations
on the single ring, one may see only the integer flux quantum
periodic oscillations.

 The discovered single flux quantum periodicity, when the finite structure
has disappeared, may be relevant to
two recent experimental studies of the Aharonov-Bohm effect on
single metallic and semiconductor rings\cite{21},\cite{22}.
Recent experiments\cite{21} have described only the full quantum period in
the case of individual metallic rings. On a semiconductor single
 loop in the $GaAs/GaAlAs$ system \cite {33}
such single flux periodicity has been detected.
To explain this integer flux  quantum periodicity we assume  that
 the individual rings are sufficiently disordered. Recently,
 similar  results\cite{Weis} \cite{Gogo}
were obtained that free electrons on
the single ring with disorder or with the temperature may show
 only the full single flux quantum periodicity.
Both these findings are consistent with   existing experiments.

The interaction between particles  may also support the full
single quantum periodicity, and, therefore, it may  also be
responsible for the observed single flux quantum period.
The prediction of fractional $1/N$ and fractional $M/N$
oscillations could be made for the ring
having lowest level of disorder. Once the resolution of the experimental
studies has been improved such periodicities  may be observed.

The other experiment\cite{22}, which is worth to mention
is that the half-flux quantum periodicity simultaneously of $10^7$ rings
 has been measured. It is
 also consistent with our results.
It is reasonable to assume that  there is
 a disorder. So the single ring must show
the single flux periodicity. For such large ensemble of
the rings we must take also into account the parity effect. Then
it is also reasonable that half of these rings has an even number
 of particles and the other half of these rings has an odd number of particles.
Because of the discovered  and described parity effect,
averaging over the two cases of odd and even number of particles indicated
above, immediately gives the $\Phi_0/2$
period. This interpretation is then natural for the experimental results
concerning many rings\cite{22}.
Thus in the framework of our approach we have obtained the unified
pictures why the
$\Phi_0$ period is seen in individual rings and why $\Phi_0/2$ period
is seen an ensemble of  $10^7$ rings.

In addition to these explanations, we predict the existence of
oscillations with  periods smaller than $\Phi_0$
or  $\Phi_0/2$.
To check the predictions of our theory,
 it is important
to perform the Fourier analysis of the persistent current oscillations
 in order
to detect the fractional Aharonov-Bohm effect, which
coexists with integer one, i.e.
whether there are  fractional oscillations on a background of
integer ones.

 Recently the fractional $1/4-$ AB effect
has been observed
\cite{Liu} in $AuIn$-rings prepeared
by $e-$beam lithography .
Since this effect has been observed in the region of a
superconducting phase transition,
 where the phase separation occurs. It seems that
there the effective system
reminding the quantum dot chain ring is created. Therefore,
the discussed Hubbard model for the description of
unpaired electrons in these droplets may be applicable.

In this case  the fractional $1/4-$ effect  may be created
as a result of averaging over rings
 having different parity of electron's number.
This occurs only at a small magnetic field
when Zeeman energy may be neglected. With the increasing magnetic field
the electrons become partially polarized and
such periodicity is destroyed. But here the fractional $M/N$ may appear
instead.

 Thus, our
finding of a coexistence of the fractional $1/N$ effect with fractional
$1/4$ or $1/2$  integer AB effects may not only to explain the observed
$1/4$ flux quantum periodicity but also predicts smallest, namely
$1/N$ and $M/N$ fractional AB effects, once the experimental
resolution has been improved.

\vspace{3mm}
\begin{flushleft}
{\bf Acknowledgement}
\end{flushleft}
\vspace{2mm}

We thank  T. Ando, S. Katsumoto, M. Kohmoto,
M. Ueda, K.  Kawarabayashi, F. Assaad
and M. Kohno for useful discussions of obtained results
and especially S.M. Manning for careful reading of the manuscript
and corrections.
 The work by FVK  has been supported by
 Ministery of Education, Science and Culture of Japan, he also appreciates
the hospitality of ISSP.

permanent address:

{\parskip=0pt
$*)$L.D. Landau Institute for Theoretical Physics \par
Moscow,117940, GSP-1, Kosygina 2, V-334, Russia \par }

{\bf REFERENCES}

redpajama_set_name: RedPajamaArXiv


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EPA announced the release of the final report,Biofuels and the Environment: The First Triennial Report to Congress (EPA/600/R-10/183F), prepared by the National Center for Environmental Assessment (NCEA) within EPA's Office of Research and Development, as the first EPA report published on this issue. The 2007 Energy Independence and Security Act (EISA) mandates increased production of biofuels (fuels derived from organic materials) from 9 billion gallons per year in 2008 to 36 billion gallons per year by 2022. Additionally, EISA (Section 204) also requires that the U.S. Environmental Protection Agency (EPA) assess and report to Congress every three years on the current and potential future environmental and resource conservation impacts associated with increased biofuel production and use.
This is the first triennial Report to Congress required under Section 204 of the 2007 Energy Independence and Security Act (EISA). EISA increases the renewable fuel standards (RFS) to 36 billion gallons per year by 2022. Section 204 requires an assessment of environmental and resource conservation impacts of the RFS program. Air and water quality, soil quality and conservation, water availability, ecosystem health and biodiversity, invasive species, and international impacts are assessed, as well as opportunities to mitigate these impacts.
The feedstocks compared in this qualitative assessment of peer-reviewed literature include corn starch, soybeans, corn stover, perennial grasses, woody biomass, algae, and waste. Biofuels compared include conventional and cellulosic ethanol and biodiesel. This report concludes that (1) the extent of negative impacts to date are limited in magnitude and are primarily associated with the intensification of corn production; (2) whether future impacts are positive or negative will be determined by the choice of feedstock, land use change, cultivation and conservation practices; and (3) realizing potential benefits will require implementation and monitoring of conservation and best management practices, improvements in production efficiency, and implementation of innovative technologies at commercial scales. This report provides a foundation for comprehensive environmental assessments of biofuel production.
U.S. EPA. Biofuels and the Environment: the First Triennial Report to Congress (2011 Final Report). U.S. Environmental Protection Agency, Washington, DC, EPA/600/R-10/183F, 2011.
Dec 2009 EPA conducted an interagency review for comments on the draft report.
Nov 2010 EPA finalizes the 2011 Renewable Fuel Standards (PDF, 3pp, 505Kb, About PDF).
Jan 2011 EPA released the biofuels report to Congress external review draft for public review and comment.
Jan 2012 EPA released the final report.


text:
Jackass is an American reality series, originally shown on MTV from 2000 to 2002, featuring people performing various dangerous, crude self-injuring stunts and pranks. The show served as a launchpad for the television and acting careers of Bam Margera, Steve-O, and also Johnny Knoxville, who previously had only a few minor acting roles.
Since 2002, three Jackass films have been produced and released by MTV corporate sibling Paramount Pictures, continuing the franchise after its run on television. The show sparked several spin-offs including Viva La Bam, Wildboyz, Homewrecker, Dr. Steve-O and Blastazoid.
The show placed #68 on Entertainment Weekly's "New TV Classics" list.
Genre: Comedy, Documentary, Reality
Director: Jeff Tremaine, Johnny Knoxville, Spike Jonze
Studio: Dickhouse Productions
Duration: 30,25 min
Networks: MTV, MTV2
The adventures of a late-20th-century New York City pizza delivery boy, Philip J. Fry, who, after being unwittingly cryogenically frozen for one thousand years, finds employment at Planet Express, an…
Genre: Animation, Comedy, Drama, Sci-Fi & Fantasy
An American reality competition show in which chefs compete against each other in culinary challenges and are judged by a panel of professional chefs and other notables from the food…
Two worlds collide when Josh, a wealthy young tech entrepreneur, meets Gabi, a feisty young food blogger, looking to be his personal chef. Gabi is desperate for the job and…
CSI: NY is an American police procedural television series that ran on CBS from September 22, 2004 to February 22, 2013 for a total of nine seasons and 197 original…
Bleak House is a fifteen-part BBC television drama serial adaptation of Charles Dickens' novel Bleak House, which was originally published in 1852–53. Produced with an all-star cast, the serial was…
An anthology series centered around people who believe themselves to be the modern-day descendants of the Romanov family.
Mackenzie "Mickey" Murphy is a hard-living, foul-mouthed, cigarette-smoking woman who moves to affluent Greenwich, CT to raise the spoiled kids of her wealthy sister who fled the country to avoid…
A comedy-drama following a chaste young woman who is accidentally impregnated via artificial insemination as she struggles to inform her devoutly religious family and make the right choices concerning the…
After 250 years on ice, a prisoner returns to life in a new body with one chance to win his freedom: by solving a mind-bending murder.
Genre: Crime, Drama, Mystery, Sci-Fi & Fantasy
Braceface
Braceface is a Canadian animated series that aired on Teletoon in Canada, and on Disney Channel and ABC Family in the United States, as well as Pop Girl in Australia….
Genre: Animation, Comedy, Drama
Law & Order is an American police procedural and legal drama television series, created by Dick Wolf and part of the Law & Order franchise. It originally aired on NBC…
Profiles of some of the men who choose to live off the grid in the unspoiled wilderness, where dangers like mudslides, falling trees and bears are all part of everyday…

redpajama_set_name: RedPajamaCommonCrawl


June 15-17—KILLINGTON— Mountain Bike Vermont (MTBVT) and Killington Resort, home of the largest downhill bike park in eastern North America, are thrilled to welcome the second annual Vermont Bike & Brew Festival to Killington Bike Park, June 15-17. This weekend-long event will include entertainment for the entire family, including three days of organized mountain biking for all skill levels, the downhill critical mass ride, onsite camping, vendor garden, bike swap, craft beer garden, whip-off competition, pond cross skinny challenge and live music from mountain biking's house band, The CopOuts.
The event will kick off on Friday, June 15 with a pre-registration party and afternoon lift laps. Dusk brings a locals' night cross-country ride tour of Green Mountain Trails followed by an evening around the campfire at the Vale Lot camping area.
Saturday morning the "Kegs n' Eggs" breakfast returns before the lifts open at 9 a.m. The 10 a.m. Critical Mass ride will line up the largest downhill train of riders Vermont has ever seen. Help them beat last year's 150 rider total and join for a peak to creek romp with a few hundred friends.
Join the skinny bridge contest or spectate on Saturday afternoon while riders test their balance on a floating skinny bridge strung across the Snowshed Pond.
Following the pond-cross an open entry whip-off competition will be held adjacent the beer garden. Speaking of beer garden, MTBVT and Killington are curating a tap list to please the most discerning beer nerd – and beer tickets are included with registration.
Sunday morning brings an additional day of riding kicked off with the MTBVT Recovery Ride for those that didn't (or did) get their fill. For those with an cross-country bike in tow and brave enough to take on the mountain, climb to the top with MTBVT (yes, climb) before kicking off a day of downhill laps.
For more information, visit mtbvt.com.


Unified Program is the consolidation of six state environmental programs into one program under the authority of a Certified Unified Program Agency.
Our organization represents local agencies certified by the California Environmental Protection Agency (Cal-EPA) as Certified Unified Program Agencies (CUPAs). Our member agencies regulate facilities with hazardous materials, hazardous waste, above-ground storage tanks and underground storage tanks throughout the State. Every year, the California CUPA Forum, with our partner State agencies, host the annual CUPA Training Conference.
The CUPA Conference Student Committee would like to extend an invitation to all students interested in a career path in the field of hazardous materials to attend the conference's Free Student Day, Wednesday, February 27, 2019, beginning with complementary breakfast at 7am and concluding with a drawing for scholarship money during the student debriefing from 5:15 – 5:45pm. The application enclosed (Form CCSD-2) and can be distributed to interested students. This opportunity is open to full-time undergraduate and graduate candidates.
The Unified Program is the consolidation of six state environmental programs into one program under the authority of a Certified Unified Program Agency. These can be a county, city or JPA (Joint Powers Authority). The mission of the Unified Program is to protect public health and safety, restore and environmental quality, and sustain economic vitality through effective and efficient implementation of the program.


The pier, built in 1890, is the perfect place to stop and gaze out to sea, or to enjoy watching the constant comings and goings in the adjacent marina.
Arsenal Soccer School now have a permanent base at the Riviera International Centre.
The Soccer School work with 75 Primary and Secondary Schools and 22 Sports Coaches throughout Devon.
The Torquay Land Train is a fun way to get around Torquay for all ages! Further information can be found on www.torquaylandtrain.co.uk.
Agatha Christie was born in Torquay in 1890 and the English Riviera celebrated her centenary in 1990 with the creation of a hand-crafted bronze bust in her likeness.
If you're looking for a pleasant, idyllic spot for somewhere to relax then Watcombe is the beach for you.
There are so many ways to bowl with us here at TLH. Whether you're a serious competitive bowler, or simply enjoy it as a sociable sport, TLH has a range of flexible bowling break options so you can enjoy bowling throughout the year.
The Golden Palms Family Entertainment Centre, Torquay is the families favourite venue for rainy day fun!
The Grand Hotel offers an excellent standard of accommodation with breath-taking sea views. Adjacent to the train Station and Cricket and Rugby Clubs of Torquay.
Coming on holiday and want to save money on Car Parking order a Torbay Council weekly or 3 day car park permit. Spend your time not having to worry about finding change or having to rush back to your car.
Torquay has been one of the UK's top holiday destinations since Victorian Times when the gentry promenaded, bathing huts were wheeled down the beaches and anyone who was anyone had an attractive villa overlooking the sea.
Do not touch! is the warning for all visitors to the Potent Plants Garden at Torre Abbey. Discover a world of poisons, antidotes and remedies in the serene setting of the Abbey grounds.


text:
Eclectic, Bohemian, Beach, Country Club, Cruise, Loft, Backyard Wedding Photos
A Bohemian, Coastal Wedding at The Lookout in Brooklin, Maine
Alexandra (Alle) Weil (31 and a holistic nutritionist) and Tim Peper (35 and an actor) combined their love of Maine's coastal charm and Georgi...
A Bohemian, Industrial Tie-Dye Wedding at The Guild in Kansas City, Missouri
Nicole Deister (32 and a landscape architect) and Josh Campbell (39 and a youth construction training manager) balanced The Guild's industrial...
A Colorful, Late-Summer Backyard Wedding at a Private Home in Perrysburg, Ohio
Because the bride always envisioned a wedding on the Maumee River, Maude Kasperzak (creative director at Charm City Cakes) and Zachary Wimberl...
A Glam, Rustic Chic Wedding at the Riverhouse at Rumson Country Club in Rumson, New Jersey
Lindsay Messina (29 and a senior research analyst at the NYC Department of Education) and Blake Abbot (33 and a physical education teacher at ...
A Casual, Playful Wedding at Discovery Cove Orlando in Orlando, Florida
Sascha Voss (41 and a catering sales manager) and Tom Keller (49 and a director of human resources) used light blue, sand tones and oranges to...
A Provencale Glam Wedding at a Private Residence in Niantic, Connecticut
Instead of more traditional nautical hues like navy and white, Alexanderson (Alex) Dowling (23 and a marketing coordinator for a real estate c...
A Rustic-Glam Barn Wedding at Rancho Dos Pueblos in Goleta, California
Personalized details, gold accents and beachy tones of apricot, white and coral added flair to Megan Rollins (28 and principal at Precision St...
A Whimsical Garden Wedding at a Private Home in Boulder, Colorado
Set in the backyard if the bride's childhood home, Jessica Von Eschen (27 and co-owner of The Urban Garden) and Eric Beck (35 and co-owner of ...
A Romantic Beach Wedding at the Four Seasons Resort Maui in Wailea, Hawaii
Soon after getting engaged to Leonard Gordon Jr. (28 and a physical therapist), Bethany Christian (29 and an occupational therapist) started b...
A Romantic Bohemian Beach Wedding at the Jellyfish in Punta Cana, Dominican Republic
With white sandy beaches in Punta Cana, Dominican Republic, swaying palm trees and crystal blue seas providing the backdrop, Victoria Chae (27...
A Boho Beach Wedding at Sunset Restaurant in Malibu, California
Gwynne Crowley (35 and a freelance producer) and David Lowrie-Reed (40 and in marketing) exchanged vows under a dream catcher wedding arch on ...
A Barefoot Beachside Wedding at La Siesta Resort & Marina in Islamorada, Florida
Florida natives McKenzie Coburn (27 and a national accounts manager) and Chris Cherry (29 and a general contractor) threw their beachy boho-ch...
A Bohemian Backyard Wedding at a Private Residence in Franklin, New Hampshire
Nicole Holguin (35 and a physician) and Dor Aharony (35 and an electrical engineer) whisked their family and friends off to New Hampshire's sc...
A Lakeside Cabin Wedding at a Private Residence in Fifty Lakes, Minnesota
Danielle Voigt (25 and a designer) and Kyle Randall (27 and an education engineer) met when Danielle, who was a 16-year-old server at the time...
A Vintage Bohemian Backyard Wedding at a Private Residence in Shaker Heights, Ohio
Tonya Brown (39 and a pastry chef) and Michael Bradford (34 and a business development manager) met at a common friend's house, when Tonya was...
A Romantic Seaside Wedding at a Private Residence in Weekapaug, Rhode Island
Tryn Collins (30 and an artist and teacher) and Matthew Kaelin (34 and a photographer) planned a romantic seaside wedding with rustic, romanti...
A Low-Key Backyard Wedding at a Private Residence in Gladwyne, Pennsylvania
Caroline Fenkel (28 and a clinical social worker) and Michael Blanche (43 and a clinical social worker) planned a fun-filled backyard fete for...
A Bohemian Forest Wedding at a Private Residence in Krasnoyarsk, Russia
For Tatyana Tereshkova (31 and an interior designer) and Vadim Zagidullin (31 and an engineer), art lit their spark. "My drawings inspired hi...
A Bohemian Destination Wedding at the El Blok Hotel in Vieques, Puerto Rico
Diana Dozier (30 and a digital strategist) and Stephan Hein (36 and an innovation consultant) were friends and coworkers in New York City for ...
A DIY Bohemian Wedding at a Private Residence in Okanagan, British Columbia
Rebecca Nickel (28 and in marketing) and Judson Luciak (27 and a student) crossed paths more than a few times before sparks flew. "We bumped i...
A Romantic, Bohemian Backyard Wedding at a Private Residence in Gloucester, Massachusetts
Inspired by their love of nature and a desire to highlight the organic beauty of their woodland home, Jenna Craig (28 and a financial manager)...


text:
PAUL WURTH TO SUPPLY KEY TECHNOLOGIES FOR EVRAZ NEW BLAST FURNACE AT NIZHNIJ TAGIL
For the development and balancing of their hot metal capacities at NTMK, Nizhnij Tagil, Russia, the EVRAZ Group has recently decided in favour of building a new blast furnace which will become the No.7 unit at this site. It will be of similar design as the existing No.5 and No.6 furnaces. At a hearth diameter of 9.8 meters and an inner volume of 2,200 cubic meters, with 22 tuyeres and 2 tapholes, each unit can produce 2.5 million tons of hot metal per year. Thus, the EVRAZ-NTMK blast furnaces ar
INAUGURATION OF PAUL WURTH INCUB HOUSE
On October 26, 2016, the premises of the Paul Wurth InCub House have been officially inaugurated in presence of the Secretary of State of the Economy, Francine Closener.Having undergone several months of renovation works, the building located at 5, rue de l'Industrie in Luxembourg-Hollerich, close to the headquarters of Paul Wurth S.A., can welcome up to 38 start-up representatives. Holding 20 dedicated and 18 flexible desks as well as some meeting rooms, the Paul Wurth InCub House is equipped
PAUL WURTH TO MODERNISE COKING PLANT FACILITIES OF ARCELORMITTAL ASTURIAS
In order to ensure the steelmaking future of the Asturias province in Spain, ArcelorMittal decided to invest into the reconstruction and modernisation of the coking plant at its Gijon works, which has been out of operation since 2013.In the context of this extensive project, Paul Wurth has been awarded with the engineering, supply of key equipment as well as erection and commissioning supervision services for the rebuild of two coke oven batteries and the modernisation of the adjacent by-product
NEW CSP COKING PLANT NOW FULLY OPERATIONAL
After the first of two new coke oven batteries was put in operation in May 2016, one month later first coke was successfully pushed also from the second new battery of Companhia Siderúrgica de Pecèm (CSP) in Brazil.This operation marks an important milestone in CSP's new coking plant project, being part of their greenfield integrated steel plant construction in the state of Cearà in Brazil's Northeast region. The steel plant will have the capacity to produce 3 million tons of steel slabs
PAUL WURTH AND RINL CONDUCTED SUCCESSFUL PERFORMANCE GUARANTEE TESTS FOR BF3 AT VIZAG STEEL PLANT
Between the 16th of March and 21st of April 2016, all Performance Guarantee tests for No.3 Blast Furnace have been completed successfully at Rashtriya Ispat Nigam's Vizag Steel Plant, Visakhapatnam, Andhra Pradesh, India.The Consortium (Paul Wurth Italia S.p.A., Paul Wurth India Pvt. Ltd and Larsen & Toubro Ltd) responsible for the construction of the furnace, Customer's consultant DASTUR Co. and all concerned Vizag Steel departments signed protocols certifying the successful completion
PRESS RELEASE: ALGERIAN QATARI STEEL TO BUILD 2.5 MTPY MIDREX® PLANT
Midrex and Paul Wurth Awarded Project for HDRI/CDRI Combination FacilityLuxembourg & Charlotte, NC, 14 June 2016. Algerian Qatari Steel (AQS) has awarded Midrex Technologies, Inc., and its consortium partner Paul Wurth, a contract to supply Equipment, Engineering and Technical Services for one of the world's largest direct reduced ironmaking (DRI) plants. The new natural gas-based MIDREX NG™ DRI plant will be located in Bellara, Algeria, 375 km east of Algiers. AQS is a joint venture bet
RESULTS 2015: FOLLOWING NEW PATHS WITH PROVEN EXPERTISE
Under the chairmanship of Michel Wurth, the Ordinary General Meeting of shareholders of Société Anonyme Paul WURTH was held on 10 May 2016 to approve amongst others the financial statements of the year 2015.Despite very gloomy business conditions, 2015 was quite a satisfactory year for the Paul Wurth Group, in particular with regard to new orders. At the same time, the company stepped up a number of growth initiatives, giving thereby important impetus for the group's future business developm
JINDAL SOUTH WEST STEEL AND PAUL WURTH RESTART BLAST FURNACE AFTER MAJOR RELINE PROJECT
After a complete rebuild with modernisation, size increase and capacity expansion, No.1 Blast Furnace at JSW Steel's Vijayanagar Works, Toranagallu, Karnataka, India has been blown in on 8 February 2016 at 06:15 a.m.This was achieved after a very short phase of cold and hot commissioning which started 12 days before with a first furnace leak test followed by pressure test, furnace drying, heating and first filling. A smooth and safe blow-in and ramp-up allowed achieving 83% of nominal producti
POSCO ORDERS NEW PAUL WURTH BLT CHARGING SYSTEM FOR POHANG WORKS
In February 2017, POSCO will execute the third reline of their No. 3 Blast Furnace at Pohang works in Korea. Since 1989, this furnace has been operating a Central Feed® type Bell Less Top® which has been supplied by Paul Wurth's licensee IHI of Japan.Now, enlarging the furnace and increasing the hot metal production to 14,000 tons per day, the customer decided to switch to a classical type Parallel Hopper top with a hopper volume of 108 m3 each. All components of this charging system will be
NLMK RELIES ON PAUL WURTH FOR MAJOR BLAST FURNACE RELINE
In preparation of a complete reline of No. 6 Blast Furnace at Lipetsk/Russia, NLMK recently signed a contract awarding to Paul Wurth engineering, supply of equipment and technology as well as site supervision related to the rebuild of the BF's central unit, i.e. the blast furnace proper and directly attached systems and equipment.Paul Wurth will adapt the Basic Engineering (delivered at an earlier stage) to the final technical solutions chosen with the customer, and will be further in charge o
PAUL WURTH TO SUPPLY COKE MAKING TECHNOLOGY TO SHANDONG
Luxembourg, 18 February 2016. The contract signed between Shandong Province Metallurgical Engineering Co., Ltd (SDM) – the engineering company associated to the Chinese Shangdong Iron & Steel Group - and Paul Wurth officially came into force on 22 January 2016. It includes the supply of engineering and supervision services for the construction of four Paul Wurth jumbo oven batteries of top charging type, with 58 ovens each and an oven height of 7.3 m, in the new coke making plant of Shando
SHANDONG RIZHAO IRON & STEEL TO EQUIP NEW IRONMAKING PLANT WITH PAUL WURTH TOP CHARGING SYSTEMS
Luxembourg, 12 February 2016. For the construction of a brand-new, 8.5 mtpy ironmaking plant, Shandong Rizhao Iron & Steel relies on the latest technology of the international market leader in Blast Furnace top charging.Shandong Rizhao Iron & Steel and Paul Wurth signed at the end of 2015 a contract for the delivery of two Bell Less Top® charging systems for the two blast furnaces of customer's new ironmaking plant. Each blast furnace will have an inner volume of 5,100 m3, designed fo
FAC RECEIVED FOR NEW BLT INSTALLED AT SALZGITTER
In the frame of a general reline of "B" blast furnace at Salzgitter Flachstahl GmbH (SZFG), Germany, Paul Wurth installed and commissioned a modernised Bell Less Top charging system. Salzgitter's BF "B" reline has been finished within 80 days, ahead of the 85-days schedule. First hot metal has been tapped on Saturday, the 14th of November. Paul Wurth received an order from SZFG in August 2014 which foresaw a complex job of modifications and replacement of components for the existing BL


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and Bathurst (1989), the expression was referred to as the stress–force–fabric (SFF) relationship.
normal stress, the deviatoric stress ratio as well as the principal stress directions.
multiplication, an attractive feature to conduct computer program aided analyses.


Thermal energy, provided by heating water Let's look at how all these elements work together.
Assume we have a great, big, oily, greasy stain on one of our favorite shirts. Water alone is not enough to remove the stain and get our shirt clean.
Now let's add some laundry detergent.
The laundry detergent provides chemical energy. This energy is created because of the way the ingredients in the detergent interact with the stain on our shirt. You cannot see it with your eyes, but the molecules in the detergent are attracted to the stain and help pull the dirt from the shirt to the wash water.
Next, let's see what happens when we add mechanical energy.
Inside a washing machine, clothes move back and forth rubbing together with other items. This is mechanical energy. The rubbing actions helps to loosen the stain and free it from the surface of our shirt. When you look in your washing machine and see laundry moving, you are watching mechanical energy in action.
Mechanical energy can even be created by hand.
The final type of energy is thermal energy.
Thermal energy means temperature. Warm or hot water can help the stain dissolve quicker. While warm water can help speed things along, most laundry detergents today can work just as well at colder temperatures.
All three types of energy need the right amount of time to work best. The more one type of energy is used, the less others are needed.
For example, delicate clothes would be damaged by the mechanical energy of the washing machine and hotter water. So instead, we can use chemical energy—detergents—and increase time to get them clean.


text:
Вадим Феопентович Ладонин (09.07.1930—16.09.2007) — российский учёный в области агрохимии, физиологии и защиты растений, академик РАСХН (1997).

Родился 9 июля 1930 г. в г. Тутаев Ярославской области. Сын юриста.

Окончил Пошехонский сельскохозяйственный техникум (1949) и Ярославский СХИ (1954, с отличием).
 1954—1977 — научный сотрудник, заведующий лабораторией, заместитель директора (с 1958) ВНИИ удобрений и агропочвоведения им. Д. Н. Прянишникова (ВИУА),
 1977—1982 — эксперт Международного агентства по атомной энергии,
 1982—1984 — заведующий лабораторией Центрального НИИ агрохимического обслуживания сельского хозяйства,
 1984—2007 — заведующий лабораторией, одновременно заместитель директора Всероссийского научно-исследовательского института агрохимии им. Д. Н. Прянишникова.

Учёный в области агрохимии, физиологии и защиты растений.
Разработал научные основы комплексного применения удобрений при возделывании с.-х. культур. Установил физиолого-биохимические закономерности действия гербицидов.

Доктор сельскохозяйственных наук (1974), профессор (1984), академик РАСХН (1997).

Заслуженный деятель науки РСФСР (1989). Награждён орденом «Знак Почёта» (1971), медалями СССР и ВДНХ.

Автор более 500 научных трудов.
Книги:
 Интенсивные технологии производства кормов: справ. / соавт. Д. А. Алтунин и др. — М.: Росагропромиздат, 1991. — 352 с.
 Плодородие черноземов России / соавт.: В. Д. Панников и др.; ВИУА. — М., 1998. — 686 с.
 Экотоксикологические аспекты загрязнения почв тяжелыми металлами / соавт.: Н. А. Черных, Н. З. Милащенко. — М.: Агроконсалт, 1999. — 176 с.
 Агроэкология: учеб. пособие для студентов вузов агрон. спец. / соавт.: В. А. Черников и др. — М.: Колос, 2000. — 535 с. — (Учеб. и учеб. пособия для студентов вузов).
 Стратегия использования осадков сточных вод и компостов на их основе в агрикультуре / соавт.: Г. Е. Мерзлая и др.; ВИУА. — М.: Агроконсалт, 2002. — 138 с.
 Агроэкология. Методология, технология, экономика: учеб. для студентов вузов…/ соавт.: В. А. Черников и др. — М.: КолосС, 2004. — 399 с.
 Научные основы оптимизации питания и фитосанитарного состояния посевов в ландшафтном земледелии / Всерос. НИИ агрохимии им. Д. Н. Прянишникова. — М., 2005. — 94 с.
 Агрохимические средства в адаптивно-ландшафтном земледелии Центрального района Нечернозёмной зоны России / соавт.: Л. М. Державин и др.; Всерос. НИИ агрохимии им. Д. Н. Прянишникова. — М., 2006. — 267 с.

Источники
 http://www.cnshb.ru/AKDiL/akad/base/RL/000530.shtm
 Памяти Вадима Феопентовича Ладонина // Агрохимия, М., 2008

redpajama_set_name: RedPajamaWikipedia


Katrina Frlan recently launched a weekly YouTube program that has left curling fans astonished by how much she knows about the game. The reason they're so surprised? Katrina is just 11 years old.
There's a new Canadian curling expert winning over fans with her in-depth knowledge of the sport. Katrina Frlan recently launched a weekly YouTube program called the Lazy Handle Show that has left people astonished by how much she knows about the game.
The reason they're so surprised? Katrina is just 11 years old.
"I think I was very nervous at first about what people might think," says Katrina, who's from Kanata, Ont., in Ottawa's west end. "I thought people might like it because they see a cute little girl, but then I was nervous. Once it got famous and got a lot of views, then I was really happy and I want to keep on going."
Katrina posted her first episode at the end of August and didn't really know what to expect. In the 10-minute video, she takes an inside look at the newest Canadian curling teams, breaking down the positions and players and giving her analysis on how they might perform.
"Kevin Koe, you obviously know him because he's a three-time Brier champion. He's amazing," Katrina proclaims in the video. "His new third is B.J. Neufeld, the former third for Mike McEwen. The new second is Colton Flasch. I think this team is the top team right now. Maybe [Brad] Gushue is better, but Koe is right there."
At first, Katrina was worried about putting her opinions out there. In the first episode, she chooses her words carefully while breaking down the teams. But now she's embracing the feedback.
"I was a little worried because of course people are going to disagree and I like that because it means we have different opinions. I don't want everyone to agree with me all the time," she says.
Katrina has been a fan of curling since discovering the game five years ago. It was her father, Lou Frlan, who urged her to try it when she was six years old. She wasn't too keen on the idea to begin with.
"I said I didn't want to go. Then my dad said I just had to try it once," Katrina says. "I had a lot of fun. And I just kept watching curling. And I just joined a competitive curling team for the first time."
Curling is her life now. When Katrina comes home from school she watches hours' worth of games. Over the summer she watched every game from the 2018 Olympics, learning the names of all the international players.
"I don't really care who's playing," she says. "My two favourite teams to watch are [John] Epping and [Rachel] Homan. Homan is from Ottawa. That's where I'm from and they're really good. I cheer for Epping because he's so nice and friendly all the time. I want to support a team like that."
After watching those old games, Katrina would go to her bedroom and record herself on her iPad, pretending she was already hosting her own show. One day her father overheard her and suggested she start her own YouTube channel.
The name of the show comes from the term curlers use to describe a rock thrown with poor rotation that stops curling in the intended direction.
"We didn't want to pick a name already used," Katrina says. "Lazy Handle Show didn't seem like it already existed. I really liked it because I see a lot of lazy handles. My dad has some lazy handles."
Katrina says that, because of all of the feedback she's received, she's going to continue posting episodes throughout the season, analyzing bonspiels and making predictions about upcoming events.
Early in the year she's keeping a close eye on three men's teams she believes are going to have successful seasons. Katrina thinks the rinks skipped by Koe, Gushue and Epping are the ones to beat on the men's side.
"I like Gushue. They're a good team. They're going to be the top team. They might have an advantage because they didn't make any changes. I think Epping is coming up too," she says. "Koe's team really gelled fast as we saw at the Curling World Cup. I think they're also going to be one of the top teams."
On the women's side, Katrina has her eyes on Homan, Casey Scheidegger and Kerri Einarson.
"I'm really excited for the season because there are so many new teams," she says.
And you can be sure Katrina will be watching every rock very closely all season and talking about it on her show.
"It's definitely going to go on for a while," she says. "It's pretty fun how this all works."


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The search committee invites applications, nominations and expressions of interest for this new position. This is an internal search: candidates must currently hold, or qualify for, a full-time academic appointment at Memorial.
Reporting to the provost and vice-president (academic), the associate vice-president (academic) faculty affairs will have various major areas of functional responsibility that will include the following: strategic involvement in faculty recruitment and retention, and faculty complement analysis and planning at the university level; strategic involvement in collective bargaining and in resolving complaints, grievances and other similar matters; operational oversight of the Office of Faculty Relations; leadership in establishing and maintaining a comprehensive career and professional development program for faculty, and a development program for academic leaders; oversight of processes for academic administrative appointments and review of recommendations arising; review of all recommendations for full-time academic appointment, promotion, tenure, and all types of leave; and involvement in the development, revision and administration of university policies that relate to academic staff. In addition, the incumbent will be an active participant in a team-based approach to senior academic leadership.
The successful candidate for associate vice-president (academic) faculty affairs will have substantial successful experience in administration, preferably in a university setting, and will have a deep understanding of the environment of a research university and what faculty members do. He or she will bring a vision to the portfolio, and have strong analytical skills and a high level of comfort of working with data; excellent interpersonal skills and strategic leadership qualities; capacity to inspire trust and confidence, and to build a consensus and then make a decision. The initial appointment will be for a five-year, renewable term.
Consideration of applications, nominations and expressions of interest will begin the week of Nov. 26, 2012, and will continue until the position is filled. Requests for a copy of the complete position profile should be sent to the address below. Memorial University is actively committed to diversity and to the principles of employment equity, and invites applications from all qualified candidates. Women, Aboriginal Peoples members of visible minorities and people with disabilities are especially encouraged to apply.
Applications, nominations and expressions of interest should be submitted in confidence to Dr. Sam Scully, sam.scully@jwasearch.com, of Janet Wright & Associates, 174 Bedford Rd., Toronto, Ont., M5R 2K9.


On the road and out of Metropolis—and carrying a secret that could disrupt Superman's life—Lois Lane embarks on a harrowing journey to uncover a threat to her husband and a plot that reaches the highest levels of international power brokers and world leaders. Critically acclaimed and best-selling author Greg Rucka and master storyteller Mike Perkins team up for a tale of conspiracy, intrigue and murder that pushes even Lois to her limits.
Jimmy Olsen lives! Superman's best friend and Daily Planet photographer Jimmy Olsen tours the bizarre underbelly of the DC Universe in this new miniseries featuring death, destruction, giant turtles and more! It's a centuries-spanning whirlwind of weird that starts in Metropolis and ends in Gotham City. And then we kill Jimmy.