text:
\section{Introduction}\label{s:intro}
Recent developments in ultracold atom and ion experiments have spurred renewed interest in dynamics of closed quantum many-body systems \cite{BlochDalibardZwerger08,Kinoshita_etal06}. Although it is expected that equilibration (in a suitable sense) takes place in a generic non-integrable quantum system \cite{vonNeumann}, a substantial body of recent research has shown that the interim evolution, which is becoming  accessible experimentally, is not yet fully understood \cite{Polkovnikov_etal11}. In particular, several qualitatively different scenarios of equilibration that depend on the initial state, including relaxation to a state that is different from the thermodynamic equilibrium, have been reported in a closed non-integrable spin chain \cite{banuls2011}. In this article a different spin model is studied, where the equilibrium state may not be independent of the initial conditions.

A model, where $N$ spin-$1/2$ particles on a lattice interact by a mean-field Curie-Weiss potential, has been discussed in \cite{paskauskas2012a}. From a previous study of a related model \cite{Kastner10}, it had been known that, with a restricted choice of coupling constants and initial conditions, certain observables approach their microcanonical expectation values with a rate $t_0$ that is system size-dependent and, in fact, diverges as $t_0\sim N^{1/2}$. With the aim of extending the range of parameters, and better understanding the course of equilibration of homogeneous states with arbitrary initial conditions, a study of the general Curie-Weiss quantum Heisenberg model was undertaken in \cite{paskauskas2012a}. For that purpose, reduced density operators $F_{1\dots n}$ for all $n=1\dots N$ spins were expanded in terms of a set of tensorial coefficients $\{f_n\}_{n=0}^N$, and the evolution equations for all $f_n$ were derived, starting from the Bogoliubov-Born-Green-Kirkwood-Yvon ({BBGKY}) hierarchy. Remarkably, these evolution equations, expressed in scaled units of time $\tau=t/t_0$ have a non-trivial $N=\infty$ limit. This thermodynamic \emph{coefficient {BBGKY} hierarchy} implies that in thermodynamic limit, $t_0$ is the slowest time scale of equilibration for arbitrary initial conditions and parameters of the model.

However, scaling is not the entire story, and additional information about equilibration is contained in the solutions of the equations which, unlike scaling, depend on a $3\times3$ matrix of spin-spin coupling constants $J$ and on the magnetic field vector $h$ in a non-trivial way. For example, it was shown that if the Curie-Weiss potential is of the form $V=-NJ_\perp(S^xS^x+S^yS^y)-NJ_zS^zS^z$ [ which corresponds to a diagonal matrix $J=\mathrm{diag}(J_\perp,J_\perp, J_z)$], and where $S^a$ is the $a\in\{x,y,z\}$ component of the average spin per particle, the coefficient BBGKY equations simplify considerably. Evolution of a certain subset of tensorial coefficients $f_n$ could be determined analytically to all orders $n=1,2,\dots$, showing a characteristic superexponential [$\sim\exp{(-\tau^2\Delta/2)}$, where $\Delta=(J_z-J_\perp)^2$]  approach to thermodynamic expectation values, which in this case are equal to zero. And yet, another subset of coefficients could be identified, which displays no relaxation at all. These analytic results are not expected to be qualitatively valid for more general couplings and, in fact, no such analytic results are known about as simple generalization, as $J=\mathrm{diag}(J_x,J_y,J_z)$, which differs from the one considered in \cite{paskauskas2012a} by allowing all three diagonal elements of $J$ to be different.

In this article, the thermodynamic coefficient {BBGKY} hierarchy equations are further investigated. The main goal is to classify, for a general $3\times3$ matrix $J$ of coupling constants and a general magnetic field vector $h=(h_x,h_y,h_z)$, all tensorial coefficients with different behavior. Although the number $D(N)$ of such coefficients depends on the system size $N$ and not on parameters $J$ or $h$, it will be shown that the matrix of effective thermodynamic limit coupling constants $\overline{J}(J,h)$, which is a function of $J$ and $h$, determines the partition of all the coefficients into several groups, decoupled by the {BBGKY} hierarchy. In this sense, $\overline{J}$ induces a topology in the vector space space of expansion coefficients.

This study provides an insight that the $N$-body quantum dynamics may impose symmetries on the Hilbert space. It is believed in this work that similar symmetries are present in a class of more realistic lattice spin models, where interactions among particles depend on the distance between them. Existence of invariant subspaces implies that equilibration and the equilibrium state may not be independent of the initial state, but not only that. It allows to divide a large computational problem into several smaller ones and moreover, should be taken in consideration when approximate dynamical models, such as kinetic theories, are constructed. The latter include a closure scheme as one step of approximation, whereby an ad-hoc relationship among correlators, involving different numbers of particles, is postulated. The existence of invariant subspaces with rigorously independent evolution implies causal relations among spin-spin correlators which should be respected by a good closure scheme.

The article is structured as follows. \Sref{s:summary} contains the preliminaries: The Curie-Weiss anisotropic quantum Heisenberg model is presented, a special expansion of reduced density operators is recalled, and the derivation of the equations for the expansion coefficients is outlined \cite{paskauskas2012a}.
In \sref{s:equations}, a useful representation of the coefficient {BBGKY} hierarchy equations is derived. Moreover, since the subsequent analysis of the equations can be succinctly expressed using concepts of graph theory, part of \sref{s:equations} is dedicated to the introduction to the relevant terminology.
In \sref{s:transformation}, equations are reformulated using the generalized Liouville operator. It is shown that this generalized Liouville operator is anti-Hermitian, provided that the scalar product in the vector space is appropriately defined.
In \sref{s:symmetries} the partition of the coefficient vector space is investigated. It is shown that all the coefficients can be grouped in either four, two or $O(N)$ disjoint sets, depending on whether the matrix of effective coupling constants is diagonal isotropic or anisotropic, or has a certain number of non-zero off-diagonal elements.
Finally, in \sref{s:conclusions} conclusions are drawn.

\section{Model and a summary of previous results}\label{s:summary}
This section briefly outlines some of the previous results which will be necessary for the subsequent discussion. The Reader is kindly referred to \cite{paskauskas2012a} for details of derivation.

\subsection{Curie-Weiss anisotropic quantum Heisenberg model}
Consider a Hilbert space $\mathcal{H}_N=\bigotimes_{i=1}^N \CC_i^2$ of $N$ identical spin-$1/2$ particles, where the Hilbert space of each spin-$1/2$ particle is $\CC_i^2$. The anisotropic Curie-Weiss Hamiltonian $H_{1\dots N}$, acting on $\mathcal{H}_N$, is defined by
\begin{equation}
  H_{1\dots N} = \sum_{i=1}^N H_i + \sum_{\scriptstyle i,j=1\atop \scriptstyle i<j}^N V_{ij}\,.\label{e:ham}
\end{equation}
The local potential $H_i$ and the interaction potential $V_{ij}$ are defined as
\begin{equation}
  H_i   = -\sum_{a\in\mathcal{I}} h_a \sigma_i^a\,,\qquad V_{ij} = -\lambda\sum_{a,b\in\mathcal{I}} J_{ab}\sigma_i^a \sigma_j^b\,. \label{e:V}
\end{equation}
where $\mathcal{I}=\{x,y,z\}$, and where
\begin{equation}\label{e:lambda}
  \lambda=1/N.
\end{equation}
The scaling factor $\lambda$ is necessary to render the energy per spin finite in the thermodynamic limit. The interaction potential $V_{ij}$ is determined by a symmetric $3\times3$ matrix of spin-spin coupling constants $J=(J_{ab})$. The local potential $H_i$ is determined by the three components of the magnetic field vector $h=(h_a)$.

\subsection{Operator {BBGKY} hierarchy}
The density operator of an $N$-spin system is a self-adjoint, positive, trace-class operator $\rho_N\in \mathcal{H}_N$. The normalization convention $\Tr_{1\dots N}\rho_N=1$ will be used, where $\Tr_{1\dots N}$ denotes the trace over $N$ degrees of freedom. The density operator satisfies the Von Neumann equation, associated to the Hamiltonian \eref{e:ham}
\begin{equation}\label{e:vn}
  \rmi\hbar\partial_t \rho_N=\pb{H_{1\ldots N},\rho_N}.
\end{equation}
Reduced $n$-particle density operators are defined by
\begin{equation}\label{e:F}
  F_{1\dots n}=\Tr_{n+1\dots N}\rho_N \qquad  1\le n\le N.
\end{equation}
where $\Tr_{n+1\dots N}$ denotes a partial trace over $N-n$ spin degrees of freedom. In this article, only homogeneous states will be considered. The $\rho_N$ for such states is invariant with respect to an arbitrary permutation of particle indices. As a result, the $n$-particle reduced density operators do not depend on \emph{which} $N-n$ particle complex is traced over in \eref{e:F}.

Reduced density operators satisfy \emph{trace properties} which are relations of the form
\begin{equation}\label{e:traceprop}
  \Tr_{n+1} F_{1\dots n+1} = F_{1\dots n}.
\end{equation}

The set of evolution equations for all the operators $\{F_{1\dots n}\}_{n=1}^N$ is called the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. It is derived from \eref{e:vn} using the standard formalism \cite{balescu1975}. For the Hamiltonian \eref{e:ham} it reads
\begin{equation}\label{e:spinkin}
  \rmi\hbar\partial_t F_{1\ldots n}=\sum_{\scriptstyle i,j=1\atop \scriptstyle i<j}^n\pb{V_{ij},F_{1\ldots n}}+ (N-n) \Tr_{n+1}\sum_{i=1}^{n}\pb{V_{i,n+1},F_{1\ldots n+1}}.
\end{equation}
The initial value problem for the reduced density operators $F_{1\dots n}$ consists of the set of equations \eref{e:spinkin} for each $1\le n\le N$, supplemented by the trace properties \eref{e:traceprop}.

It was shown \cite{paskauskas2012a} that there are two time scales involved in the dynamics: The slow time scale of long-range spin-spin interactions drives the equilibration and is the main focus of this article. This slow dynamics is coupled with the fast precession around the magnetic field vector $h=(h_x,h_y,h_z)$, appearing in the terms $H_i$. The latter terms can be eliminated by passing to an interaction picture, defined by
\begin{equation}\label{e:iPic}
  F_{1\ldots n}\rightarrow
  \tilde{F}_{1\dots n}= \exp\biggl[\frac{\rmi t}{\hbar}\sum_{i=1}^ n H_i\biggr] F_{1\ldots n} \exp\biggl[-\frac{\rmi t}{\hbar}\sum_{i=1}^ n H_i\biggr].
\end{equation}
The {BBGKY} hierarchy in the interaction picture is then given by
\begin{equation}\label{e:spinkin2}
  \fl \rmi\hbar\partial_t \tilde{F}_{1\ldots n} =
  \sum_{\scriptstyle i,j=1\atop \scriptstyle i<j}^n\pb{\tilde{V}_{ij}(t),\tilde{F}_{1\ldots n}}+ (N-n) \Tr_{n+1}\sum_{i=1}^{n}\pb{\tilde{V}_{i, n+1}(t),\tilde{F}_{1\ldots n+1}},
\end{equation}
where the two-body potential is now explicitly time-dependent,
\begin{equation}\label{e:V2}
  \tilde{V}_{ij}(t) =
  \exp\left[\frac{\rmi t}{\hbar} \left(H_i+H_j\right)\right]V_{ij}\exp\left[-\frac{\rmi t}{\hbar} \left(H_i+H_j\right)\right].
\end{equation}
The interaction picture potential $\tilde{V}_{ij}(t)$ is of the same form as \eref{e:V},
\begin{equation}
  \tilde{V}_{ij}(t)=-\lambda \sum_{a,b\in\mathcal{I}} \tilde{J}_{ab}(t)\sigma_i^a\sigma_i^b\,,
\end{equation}
where the matrix $J$ is replaced by a time-dependent $\tilde{J}(t)$, defined by
\begin{equation}\label{e:JPic}
  \tilde{J}(t)= B(th/\hbar)^\trsp J B(th/\hbar)\,.
\end{equation}
Define $b=th/\hbar$, $\hat{h}=h/\abs{h}$, and $c^0=\cos{\abs{2b}}$. Moreover, define $c^u=\hat{h}^u\sin{\abs{2b}}$, and $c^{uv}=2\hat{h}^u\hat{h}^v\sin^2{\abs{b}}$, where $u,v\in\{x,y,z\}$. Using these definitions, $B(b)$ is written as
\begin{equation}\label{e:B}
  B(b)=\left(\!\!\!\begin{array}{ccc}
  c^{xx}+c^0 & c^{xy}+c^z & c^{xz}-c^y \\
  c^{xy}-c^z & c^{yy}+c^0 & c^{yz}+c^x \\
  c^{xz}+c^y & c^{yz}-c^x & c^{zz}+c^0 \end{array}\!\!\!\right)\,.
\end{equation}

\subsection{Coefficient expansion and scaled variables}
Convinced that a ``good'' coefficient expansion should automatically satisfy the trace properties for reduced density operators \eref{e:traceprop}, and noting that the following definitions
\numparts\begin{eqnarray}
  F_1 &=& \frac12\big( \one_1 + \sum_{a\in \mathcal{I}} f_1^a\sigma_1^a\big) \,,\\
  F_{12} &=& \frac14\big( \one_{12} + \sum_{a\in\mathcal{I}} f_1^a( \sigma_1^a+\sigma_2^a) + \sum_{a,b\in\mathcal{I}}f_2\sigma_1^a\sigma_2^b\big)\,.
\end{eqnarray}\endnumparts
satisfy $\Tr_2 F_{12}=F_1$ and $\Tr_1F_1=1$, in \cite{paskauskas2012a} we attempted a generalization by
\begin{equation}\label{e:Fexp}
  F_{1\ldots n} = 2^{-n}\sum_{s=0}^{n}
  \sum_{a\in \mathcal{I}^s} f_s^{a}
  \sum_{p\in \mathfrak{P}_s(n)}{\bm\sigma}_{p}^{a}\,,
\end{equation}
where $\sigma_i^a$ is the $a$-th component of the Pauli operator on the $i$-th lattice site. Here $\{f_s\}_{s=0}^n$ is the set of coefficients of expansion of $F_{1\dots n}$, where $f_s\in\RR^{3\times\dots\times3}$ is a rank-$s$ symmetric tensor with components $f_s^{a}$, and where $a=(a_1a_2\dots a_s)\in \mathcal{I}^s$. By convention, $f_0=1$. Expansion \eref{e:Fexp} holds also for $\tilde{F}_{1\dots n}$ in terms of coefficients
$\{ \tilde{f}_s \}_{s=0}^n$.

The set $\mathfrak{P}_s(n)$ consists of all $s$-element permutations $\left(p_1,\ldots,p_s\right)$ of particle labels $p_i\in\{1,\dots,n\}$ such that $p_i<p_j$ for all $1\le i<j\le n$ (i.e.\ all sequences are strictly increasing.) Considering for example $s=2$, we have $\mathfrak{P}_2(n)=\{(1,2),(1,3),\ldots,(1,n),(2,3),\ldots,(n-1,n)\}$. Lastly, a product of $s$ Pauli operators on different lattice sites is denoted by
\begin{equation}\label{e:pp}
  {\bm\sigma}_p^a=\prod_{i=1}^s \sigma_{p_i}^{a_i}\,.
\end{equation}

The expansion coefficients are related to the expectation values of many-particle spin observables by the identity
\begin{equation}
  f_s^a = \avg{{\bm\sigma}^a},
\end{equation}
where $\avg{{\bm\sigma}^a}$ is the expectation value of a product of $s$ spin operators, defined by
\begin{equation}
  \avg{{\bm\sigma}^a}=\Tr_{1\dots s} \big( F_{1\dots s}  \prod_{i=1}^s \sigma_i^{a_i} \big)\,.
\end{equation}

Mapping of the operator {BBGKY} hierarchy \eref{e:spinkin} and \eref{e:spinkin2} onto equations for the coefficients $\{f_s\}_{s=1}^N$ and $\{\tilde{f}_s\}_{s=1}^N$ is discussed in \cite{paskauskas2012a}. Here, only the final result for the interaction picture \eref{e:spinkin2} is recalled:
\begin{equation}\label{e:bbgky0}
 \frac{\hbar}{2}\partial_t \tilde{f}_n = \lambda \tilde{v}^-_{ n}[\tilde{f}_{n-1}] + (1-n\lambda)\tilde{v}^+_{n}[\tilde{f}_{n+1}]\,,
\end{equation}
where the components of $\tilde{v}_n^\pm$ are given by
\numparts\begin{eqnarray}
  (\tilde{v}^-_{n})^{(a_1\dots a_n)}[\tilde{f}_{n-1}]= -\sum_{b,c\in\mathcal{I}} \sum_{\scriptstyle i,j=1\atop \scriptstyle i\neq j}^ n \eps_{a_i bc} \tilde{J}_{ba_j}(t) \tilde{f}_{ n-1}^{a-a_i+c-a_j},\label{e:v-}\\
  (\tilde{v}^+_{n})^{(a_1\dots a_n)}[\tilde{f}_{n+1}]=-\sum_{b,c,d\in\mathcal{I}} \tilde{J}_{bd}(t) \sum_{i=1}^n \eps_{a_ibc}\tilde{f}_{ n+1}^{a-a_i+c+d}\,.\label{e:v+}
\end{eqnarray}
\endnumparts
Here, $\eps_{abc}$ is the Levi-Civita symbol defined according to the convention $\eps_{xyz}=1$, and
\begin{equation}
a-a_i+c-a_j=(a_1,\ldots,a_{i-1},c,a_{i+1},\ldots,a_{j-1},a_{j+1},\ldots,a_n)
\end{equation}
in \eref{e:v-} is derived from $a=(a_1\dots a_n)$ by replacing the $i$th element by $c$ and then deleting the $j$th entry of $a$, and
\begin{equation}
a-a_i+c+d=(a_1,\ldots,a_{i-1},c,a_{i+1},\ldots,a_n,d)
\end{equation}
\endnumparts
in \eref{e:v+} is obtained from $a$ by replacing the $i$th element by $c$ and then appending $d$.

Tensors $\tilde{v}_n^\pm[\tilde{f}_{n\pm1}]$ are linear in the tensors $\tilde{f}_{n\pm1}$ and in the time-dependent matrix $\tilde{J}(t)$. Moreover, $\tilde{v}_n^\pm$ do not depend on $\lambda$ explicitly. A change of  $t$ and $\tilde{f}_n$ variables to
\begin{equation}
  \tau=2t\lambda^{1/2}/\hbar, \qquad \tilde{f}'_n=\tilde{f}_n/\lambda^{n/2}\,\label{e:scaling}
\end{equation}
balances all the powers of $\lambda$ in the coefficients of $\tilde{v}_n^\pm$ in \eref{e:bbgky0}. These coefficients are then cancelled from the equation with the goal to arrive at a scale-free {BBGKY} hierarchy. The remaining dependence on the number of particles enters the scaled equations through the coefficient $1-n\lambda$ at order $n$, and in the matrix $\tilde{J}(t)$. By \eref{e:JPic}--\eref{e:B}, $\tilde{J}(t)$ is a sum of a constant term $\overline{J}$ and time-dependent terms which, by the form of $B(th/\hbar)$, are trigonometric functions such as $\cos{(2t\abs{h}/\hbar)}$. In $\tau$ units of time, such terms are a fast oscillation on top of the mean $\overline{J}$, and it was argued \cite{paskauskas2012a} that these fast oscillations can be taken into account by averaging (which becomes exact in thermodynamic limit.) With the additional assumption that $\tilde{f'}$ varies slowly in the period of oscillation $T_\lambda$, $\tilde{f'}$ is replaced by its average value $\overline{f}'(\tau)=(T_\lambda)^{-1}\int_\tau^{\tau+T}\tilde{f'}(\tau')d\tau'$. Application of this procedure on \eref{e:bbgky0} results in the scaled and averaged {BBGKY} hierarchy
\begin{equation}\label{e:bbgky1}
  \partial_\tau \overline{f}'_n\approx\overline{v}^-_{n}[\overline{f}'_{n-1}] + (1-\lambda n)\overline{v}^+_{n}[\overline{f}'_{n+1}]\,,
\end{equation}
where $\overline{v}_n^\pm$ are obtained from $\tilde{v}_n^\pm$ by replacing $\tilde{J}(t)$ by $\overline{J}$. The latter is defined by
\begin{equation}\label{e:Javg}
  \overline{J} =T^{-1} \int_0^{T} \tilde{J}(t) dt\qquad T=\pi \hbar/\abs{h}\,.
\end{equation}
 The approximation in \eref{e:bbgky1} is a result of a substitution $\tilde{f}\to \overline{f}'$. It turns into an equality in two cases. Firstly, if there is no magnetic field in \eref{e:ham} i.e. if $h=0$, then there is no need to perform averaging. In this case the non-averaged coefficient {BBGKY} hierarchy has the same form as \eref{e:bbgky1} where $\overline{J}$ is equal to the original coupling matrix $J$. Secondly, if separation of scales is a valid assumption\footnote{It is justified rigorously in a special case, treated in \cite{paskauskas2012a}}, the averaging is exact in the limit $\lambda=0$.

This limit describes thermodynamic evolution of the model by means of an infinite dimensional set of differential equations. The structure of these equations is the main focus of this article. Dropping the bars and the primes from $\overline{f}'$ and $\overline{v}_n^\pm$ for simplicity, the thermodynamic limit {BBGKY} hierarchy is defined as
\begin{equation}\label{e:bbgky2}
  \partial_\tau f_n=v^-_{n}[f_{n-1}] + v^+_{n}[f_{n+1}]\,.
\end{equation}
The distinction between the original matrix $J$ and the averaged one, $\overline{J}$, is maintained.

\section{Desymmetrized coefficient {BBGKY} hierarchy}\label{s:equations}
This section introduces the desymmetrized coefficient {BBGKY} hierarchy and presents the framework for the subsequent study of its properties in sections \ref{s:transformation}--\ref{s:symmetries}.
\subsection{Coefficient equations}\label{s:desym}
\Eref{e:Fexp} maps $F_{1\dots N}$ onto a set of tensor coefficients $\{f_n\}_{n=0}^N$, where $f_n$ is a rank-$n$ three dimensional symmetric tensor with components $f_n^a$, and where $a=(a_1\dots a_n)$ is a coordinate multiindex, and $a_i\in\{x,y,z\}$. A symmetric rank-$n$ tensor in three dimensions is specified completely by $d(n)$ numbers, where
\begin{equation}\label{e:dn}
  d(n) = \sum_{j=1}^{n+1}j = (n+1) (n+2) / 2.
\end{equation}
The number of degrees of freedom $D(N)$ is defined as the number of independent coefficients in the expansion of $F_{1\dots N}$. It is equal to a sum of the numbers of independent components from all the tensors $f_n$ excluding the trivial $f_0=1$,
\begin{equation}\label{e:Dn}
  D(N)=\sum_{n=1}^N d(n) = (N+1)(N+2)(N+3)/6-1.
\end{equation}
A sequence, formed from all the independent components of $f_n$, can be viewed as a vector $\mathrm{indep}(f_n)$ on a $d(n)$-dimensional vector space ${\mathcal S}_n$,
\begin{equation}
  \mathrm{indep}(f_n) \in {\mathcal S}_n\,.
\end{equation}
where ${\mathcal S}_n\subseteq\RR^{d(n)}$ is called the $n$-spin desymmetrized coefficient vector space, or simply the $n$-spin vector space. A sequence of all independent tensor components
\begin{equation}\label{e:f}
  \mathrm{indep}(f) = ( \mathrm{indep}(f_1), \dots, \mathrm{indep}(f_N) ) \in {\mathcal F}_N
\end{equation}
can be viewed as a vector on a direct sum of $n$-spin desymmetrized vector spaces,
\begin{equation}\label{e:hilbD}
  {\mathcal F}_N=\bigoplus_{n=1}^N{\mathcal S}_n\,.
\end{equation}

By the symmetry of $f_n$, any $f_n^{a'}$ is equal to $f_n^{a(m)}$, where
\begin{equation}\label{e:a(m)}
  a(m)=(\underbrace{x,\ldots,x}_{\hspace{-3mm}\displaystyle m^x\mbox{ elements}\hspace{-3mm}},y,\ldots,y,z,\ldots,z)
\end{equation}
is a permutation of $a'$ with the components $x$, $y$, $z$ arranged in contiguous blocks of $m^x$, $m^y$, and $m^z$ labels. Hence, all independent components of $f_n$ can be enumerated uniquely by a triple of nonnegative integers $m\in \mathfrak{M}_n$, where the set $\mathfrak{M}_n$ is defined by
\begin{equation}\label{e:setMn}
  \mathfrak{M}_n = \{ (m^x,m^y,m^z): m^a\ge0, m^x+m^y+m^z=n\}\,.
\end{equation}
In order to denote the allowed set of indices for an arbitrary $n$, define
\begin{equation}
  \mathfrak{M} = \{ (m^x,m^y,m^z): m^a\ge0, m^x+m^y+m^z\le N\}\,.
\end{equation}
Define the norm of $m\in\mathfrak{M}_n$ by
\begin{equation}\label{e:mnorm}
  \abs{m} = m^x+m^y+m^z\,.
\end{equation}
As a result, all $\mathrm{indep}(f)\in{\mathcal F}_N$ are uniquely labeled by elements of
\begin{equation}\label{e:setM}
  \mathfrak{M}^{(N)}=\bigcup_{n=1}^N \mathfrak{M}_n\,.
\end{equation}
In the subsequent, instead of a cumbersome $\mathrm{indep}(f_n)$, the sequence of all the independent components $f_n^{a(m)}$ will be denoted by $f_n(m)\in{\mathcal S}_n$ where $m\in\mathfrak{M}_n$.

The coefficient hierarchy \eref{e:bbgky2} is translated to a desymmetrized coefficient {BBGKY} hierarchy on ${\mathcal F}_N$. This is formally achieved by replacing $f_n^{a'}$ by $f_n(m)\equiv f_n^{a(m)}$, and by replacing $v_n^\pm[f_{n\pm1}]$ on the right-hand side of \eref{e:bbgky2} by $v_n^\pm(m, f_{n\pm1})$
\begin{equation}\label{e:bbgky3}
\partial_\tau f_n(m,\tau)= v_{n}^-(m,f_{n-1}) + (1-n\lambda) v_n^+(m,f_{n+1}).
\end{equation}
 The derivation of $v_n^\pm(m,f_{n\pm1})$ is given in \ref{s:derivation}. It is convenient to separate the diagonal and off-diagonal components of $\overline{J}$. Define vectors $\overline{K}=(\overline{K}_x,\overline{K}_y,\overline{K}_z)$ and $\overline{W}=(\overline{W}_x,\overline{W}_y,\overline{W}_z)$ as
\numparts\begin{eqnarray}
(\overline{K}_x,\overline{K}_y,\overline{K}_z) &=& (\overline{J}_{yy}-\overline{J}_{zz}, \overline{J}_{zz}-\overline{J}_{xx}, \overline{J}_{xx}-\overline{J}_{yy})\,, \label{e:K} \\
(\overline{W}_x,\overline{W}_y,\overline{W}_z) &=& (\overline{J}_{yz}, \overline{J}_{zx}, \overline{J}_{xy}) \label{e:W} \,.
\end{eqnarray}\endnumparts
Replacing $f_{n\pm1}$ by a generic $f$ in $v_n^{\pm}(m,f_{n\pm1})$ and using the notation \eref{e:K}--\eref{e:W}, terms $v_n^\pm(m,f)$ for general $\overline{K}$ and $\overline{W}$, and hence a general symmetric $\overline{J}$, are given by
\numparts
\begin{eqnarray}
  \fl v_{n}^+(m,f) &=& - \overline{K}_xm^x f(m^x-1,m^y+1,m^z+1) -\overline{K}_ym^y f(m^x+1,m^y-1,m^z+1)\nonumber\\
  \fl && - \overline{K}_zm^z f(m^x+1,m^y+1,m^z-1)\nonumber\\
  \fl  &-&\overline{W}_x(m^y-m^z) f(m^x+1,m^y,m^z) - \overline{W}_y(m^z-m^x)f(m^x,m^y+1,m^z) \nonumber\\
  \fl  && -\overline{W}_z(m^x-m^y)f(m^x,m^y,m^z+1) \nonumber \\
  \fl  &-& m^x\overline{W}_x\big[ f(m^x-1,m^y,m^z+2)-f(m^x-1,m^y+2,m^z) \big] \nonumber \\
  \fl  &-& m^y\overline{W}_y\big[ f(m^x+2,m^y-1,m^z)-f(m^x,m^y-1,m^z+2) \big] \nonumber \\
  \fl  &-& m^z\overline{W}_z\big[ f(m^x,m^y+2,m^z-1)-f(m^x+2,m^y,m^z-1) \big]\,. \label{e:vplus}\\
  \fl \tilde{v}_{n}^-(m;f) &=& \overline{K}_xm^ym^z f(m^x+1,m^y-1,m^z-1) + \overline{K}_ym^zm^x f(m^x-1,m^y+1,m^z-1) \nonumber \\
  \fl  && +  \overline{K}_zm^xm^y f(m^x-1,m^y-1,m^z+1)                                     \nonumber \\
  \fl  &+& \overline{W}_xm^x(m^y-m^z) f(m^x-1,m^y,m^z) +  \overline{W}_ym^y(m^z-m^x) f(m^x,m^y-1,m^z)  \nonumber \\
  \fl  && + \overline{W}_zm^z(m^x-m^y) f(m^x,m^y,m^z-1)\nonumber \\
  \fl  &+& m^x(m^x-1) \big[ \overline{W}_y f(m^x-2,m^y+1,m^z)-\overline{W}_zf(m^x-2,m^y,m^z+1)\big] \nonumber \\
  \fl  &+& m^y(m^y-1) \big[ \overline{W}_z f(m^x,m^y-2,m^z+1)-\overline{W}_xf(m^x+1,m^y-2,m^z)\big] \nonumber \\
  \fl  &+& m^z(m^z-1) \big[ \overline{W}_x f(m^x+1,m^y,m^z-2)-\overline{W}_yf(m^x,m^y+1,m^z-2)\big]\,.\label{e:vminus}
\end{eqnarray}\endnumparts
In \eref{e:vplus}--\eref{e:vminus}, $\mathrm{indep}(f)$ has been replaced by $f$, and in the subsequent the notation $f\equiv f(m)$ will imply an element of ${\mathcal F}_N$.

\subsection{Permutation invariant form of equations}\label{s:permutation}
The {BBGKY} hierarchy \eref{e:bbgky1} can be written in an explicitly permutation invariant form, which is more concise and facilitates further analysis.

Define a cyclic permutation operator, acting on $\mathcal{I}= \{x,y,z\}$, by
\begin{equation}
  \pi (x,y,z) = (y,z,x)
\end{equation}
and three reflection operators $r_{x}$, $r_{y}$, $r_{z}$ which act by permuting pairs of elements. For example
\begin{equation}
  r_{x}(x,y,z) = (x,z,y)
\end{equation}
The group of permutations in three dimensions consists of six elements, ${\mathfrak g}=\{i, \pi, \pi^2, r_{1}, r_{2}, r_{3}\}$, where $i$ is the identity operator $i(x,y,z)=(x,y,z)$.

Define twelve vectors $t_i^a$, labelled by $i=1,2,3,4$ and $a\in\{x,y,z\}$, by
\begin{equation}\label{e:tai}
  \displaystyle
  \begin{array}{lll}
    t_1^x=(-1,1,1),\quad & t_1^y=(1,-1,1), \quad & t_1^z=(1,1,-1). \vspace{0.1cm}\\
    t_2^x=(1,0,0), &  t_2^y=(0,1,0), & t_2^z=(0,0,1). \vspace{0.1cm}\\
    t_3^x=(-1,0,2), & t_3^y=(2,-1,0), & t_3^z=(0,2,-1).\vspace{0.1cm}\\
    t_4^x=(-1,2,0), & t_4^y=(2,0,-1), & t_4^z=(0,-1,2).
  \end{array}
\end{equation}
Note that $t_i^y=\pi t_i^x$, $t_i^z=\pi t_i^y=\pi^2t_i^x$, and $t_4^a=r_at_3^a$. Using these definitions, the coefficient {BBGKY} hierarchy equations \eref{e:bbgky3} and \eref{e:vplus}--\eref{e:vminus} can be written as
\begin{eqnarray}\label{e:graph}
\fl \partial_\tau f_n(m) &=& \sum_{a\in \mathcal{I}}\overline{K}_a\Bigl\{ m^{\pi a}m^{\pi^{2}a} f_{n-1}(m-t_1^a) - m^a f_{n+1}(m+t_1^a)\Bigr\}\nonumber\\
\fl  &+& \sum_{a\in \mathcal{I}}\overline{W}_a(m^{\pi a}-m^{\pi^2a}) \Bigl\{ m^af_{n-1}(m-t_2^a) - f_{n+1}(m+t_2^a)\Bigr\}\nonumber\\
\fl  &+& \sum_{a\in \mathcal{I}}\overline{W}_a\Bigl\{m^{\pi^2a}(m^{\pi^2a}-1) f_{n-1}(m-t_3^a) - m^{\pi a}(m^{\pi a}-1)f_{n-1}(m-t_4^a) \nonumber \\
\fl  && - m^a\bigl[ f_{n+1}(m+t_3^a)-f_{n+1}(m+t_4^a)\bigr]\Bigr\}
\end{eqnarray}

\subsection{Representation by a graph}\label{s:graphs}
To study the properties of \eref{e:graph} it is convenient to use the language of the graph theory. A graph $\graph=(\mathfrak{M},\mathfrak{A})$ is an abstract representation of two sets: The set $\mathfrak{M}$ of objects called nodes $m\in\mathfrak{M}$ will be identified with the set of indices \eref{e:setM} of elements of ${\mathcal F}_N$. Certain pairs of nodes $m$ and $m_1$ of the graph will be connected by links, called edges and denoted by $\vv{m,m_1}$. The set of all the edges is denoted by $\mathfrak{A}$.

Couplings among tensor coefficients in \eref{e:graph} [or in \eref{e:bbgky2} or \eref{e:bbgky3} which are equivalent to \eref{e:graph}] will be allowed to determine $\mathfrak{A}$ by the following definition.
\begin{definition}\label{def:edge} An edge $\vv{m,m_1}$ is an ordered pair of elements $m, m_1\in\mathfrak{M}$. $\vv{m,m_1}\in\mathfrak{A}$ if and only if two conditions are satisfied: $\partial_\tau f_n(m)$ is coupled with $f_{n_1}(m_1)$ by \eref{e:graph} and the coupling coefficient is not equal to zero.
\end{definition}
Two more definitions will help establish the terminology.
\begin{definition}\label{def:adjoint}
  The node $m_1$ is said to be adjacent to $m$ if $\vv{m,m_1}\in\mathfrak{A}$. A set $\mathrm{adj} (m)\subset\mathfrak{M}$ is a set of all of the nodes, adjacent to $m$.
\end{definition}
\begin{definition}\label{def:value}
  A value function $\psi: \mathfrak{A}\to \RR$ is equal to the coupling coefficient between $\partial_\tau f_n(m)$ and $f_{n_1}(m_1)$ and is denoted by  $\psi[\vv{m,m_1}]$.
\end{definition}
The three-term recurrence structure of the {BBGKY} hierarchy implies that $\mathrm{adj} (\mathfrak{M}_n)\subseteq\{ \mathfrak{M}_{n-1},\mathfrak{M}_{n+1}\}$. The adjacency relations will be labelled by $\pm$ to distinguish this structure.
\begin{definition}\label{def:adj}
  A set $\mathrm{adj}^\pm(m)\subset\mathfrak{M}$ is a set of all the nodes $m_1$ such that $m_1$ is adjacent to $m$ and
$\abs{m_1} = \abs{m}\pm 1$.
\end{definition}
By this definition,
\begin{equation}\label{e:adj}
  \mathrm{adj}^\pm(\mathfrak{M}_n) \subseteq \mathfrak{M}_{n\pm1}.
\end{equation}

Having established the terminology of graph theory, let us now determine the implications of the structure of the {BBGKY} hierarchy \eref{e:graph} on the structure of the graph $\graph$. For this purpose the following two lemmas are provided.

Assume that $m=(m^x, m^y, m^z)\in\mathfrak{M}$, $m_1=(m_1^x,m_1^y, m_1^z)\in\mathfrak{M}$, $m_1\in\mathrm{adj}^+(m)$.
\begin{lemma}\label{l:upedge}
  Nodes $m$ and $m_1$ are connected by a edge $\vv{m,m_1}\in\mathfrak{A}$ iff for some $a\in\mathcal{I}$ both conditions in one of \eref{e:t1}--\eref{e:t4} are satisfied
   \numparts\begin{eqnarray}
   &m_1=m+t_1^a  &\psi[\vv{m,m_1}]=-\overline{K}_am^a\neq0.\label{e:t1}\\
   &m_1=m+t_2^a  &\psi[\vv{m,m_1}]=-\overline{W}_a(m^{\pi a}-m^{\pi^2a})\neq0.\label{e:t2}\\
   &m_1=m+t_3^a  &\psi[\vv{m,m_1}]=-\overline{W}_am^a\neq0.\label{e:t3}\\
   &m_1=m+t_4^a\qquad  &\psi[\vv{m,m_1}]=\overline{W}_am^a\neq0.\label{e:t4}
   \end{eqnarray}\endnumparts
\end{lemma}
\begin{proof}
  By \defref{def:edge} and by assumption $m_1\in\mathrm{adj}^+(m)$, an edge exists iff the term $\partial_\tau f_n(m)$ in \eref{e:graph} has a non-zero coupling with $f_{n+1}(m_1)$.  Since $\forall a,i:~\abs{m\pm t_i^a}=\abs{m}\pm1$, the first necessary condition for $\vv{m,m_1}\in\mathfrak{A}$ is that $m_1$ is equal to one of the terms listed in the first column of \eref{e:t1}--\eref{e:t4}. The second necessary condition is non-vanishing of the corresponding coupling coefficient, which are listed in the second column of \eref{e:t1}--\eref{e:t4}.
\end{proof}
\begin{lemma}\label{l:downedge}
  $\vv{m,m_1}\in\mathfrak{A}$ implies $\vv{m_1,m}\in\mathfrak{A}$.
\end{lemma}
\begin{proof}
  Assume that $\vv{m,m_1}\in\mathfrak{A}$. By \lemref{l:upedge}, $m_1=m+t_i^a$ for some $i=1,2,3,4$ and some fixed $a\in\mathcal{I}$. By \defref{def:adj}, the adjacent set $\mathrm{adj}^-{(m_1)}$ is determined by the non-zero couplings between $\partial_\tau f_{n+1}(m_1)$ and $f_n(m_2)$ where $m$ is substituted by $m_1$ in \eref{e:graph}. By \eref{e:graph} and \lemref{l:upedge}, $m_2=m+t_i^a-t_j^b$, therefore $m_2=m$ if $i=j$ and $a=b$. Examining the coupling coefficients for each $i$ and $a$ in \eref{e:graph} one finds
\numparts\begin{eqnarray}
&\psi[\vv{m_1,m}]=\overline{K}_a(m^{\pi a}+1)(m^{\pi^2a}+1)\qquad &i=1. \label{e:psi1} \\
&\psi[\vv{m_1,m}]=\overline{W}_a(m^a+1)(m^{\pi a}-m^{\pi^2 a})    &i=2. \label{e:psi2} \\
&\psi[\vv{m_1,m}]=\overline{W}_a(m^{\pi^2a}+1)(m^{\pi^2a}+2)      &i=3. \label{e:psi3} \\
&\psi[\vv{m_1,m}]=-\overline{W}_a(m^{\pi a}+1)(m^{\pi a}+2)      &i=4. \label{e:psi4}
\end{eqnarray}\endnumparts
Denote the absolute value of an integer coefficient of $\psi[\vv{m,m_1}]$, multiplying a coupling constant $\overline{K}_a$ or $\overline{W}_a$, by $\nu(m,m_1)$, and the one of $\psi[\vv{m_1,m}]$ by $\nu(m_1,m)$. By assumption that $\psi[\vv{m,m_1}]\neq0$, it follows that the relevant coupling constant $\overline{K}_a\neq0$ or $\overline{W}_a\neq0$, and $\nu(m,m_1)\neq0$. Comparing \eref{e:t1}--\eref{e:t4} with \eref{e:psi1}--\eref{e:psi4} and using $m^a+1>0$, it follows that under these conditions $\nu(m_1,m)\neq0$ as well.
\end{proof}

Comparing \eref{e:t1}--\eref{e:t4} with \eref{e:psi1}--\eref{e:psi4} we conclude that $\nu(m,m_1)\neq \nu(m_1,m)$ in general. In particular,
\begin{equation}\label{e:nu}
  -\frac{\psi[\vv{m,m_1}]}{\psi[\vv{m_1,m}]} = \frac{\nu(m,m_1)}{\nu(m_1,m)}
\end{equation}
is equal to a ratio of two positive integers.
For $m_1=m+t_i^a$, define $\gamma_i^a$ by
\begin{equation}\label{e:gai}
  \gamma_i^a =  -\psi[\vv{m,m+t_i^a}] \psi[\vv{m+t_i^a,m}].
\end{equation}
The following formulae for $\gamma_i^a$ are derived from lemmas \ref{l:upedge}--\ref{l:downedge}
\numparts\begin{eqnarray}
  \gamma_1^a = \overline{K}_a^2\max{(m^x,m_1^x)}\max{(m^y,m_1^y)}\max{(m^z,m_1^z)}\,,\label{e:gamma1}\\
  \gamma_2^a = \overline{W}_a^2(m^a+1)\left(m^{\pi a}-m^{\pi^2a}\right)^2\,,\label{e:gamma2}\\
  \gamma_{3}^a = \overline{W}_a^2m^a(m^{\pi^2 a}+1)(m^{\pi^2a}+2)\,,\label{e:gamma3}\\
  \gamma_{4}^a = \overline{W}_a^2m^a(m^{\pi a}+1)(m^{\pi a}+2)\,.\label{e:gamma4}
\end{eqnarray}\endnumparts

The graph $\graph$ is drawn as points and arrows on a plane. Since edges are directional, an edge $\vv{m,m_1}$, connecting the $m$-th and $m_1$-st nodes can be represented by an arrow, starting at the point $m$ and ending at $m_1$. However, by \lemref{l:downedge}, all arrows come in pairs. Therefore, instead  of connecting each adjacent pair of nodes by a pair of arrows, each pair of arrows can be substituted by a single line and an integer number on each side of the line to distinguish the direction of a link. The following convention will be adopted. For a pair of adjacent nodes $m$ and $m_1\in\mathrm{adj}^+(m)$, $\nu(m,m_1)$ is to the right of (or below), and $\nu(m_1,m)$ is to the left of (or above) the line, connecting $m$ and $m_1$.

To take into account the adjacency relation \eref{e:adj} and the composite structure \eref{e:setM}, all the nodes $m\in\mathfrak{M}_n$ with a fixed $n$ will be drawn at the same height. For example, a segment of $\graph^1$ \footnote{Some nodes and vertices of $\graph^1$ are not drawn in order to keep the image clear. } is shown in \eref{e:Gx}, where each node is denoted by $m^x.m^ym^z$.
\begin{equation}\label{e:Gx}
  \newcommand\av[2]{\ar@{-}@<0.15ex>[u]^{#1}_{#2}
  \newcommand\ad[2]{\ar@{-}@<0.15ex>[ur]^{#1}_{#2}
  \newcommand\add[2]{\ar@{-}@<0.15ex>[urr]^{#1}_{#2}
  \newcommand\lk{\ar@{-}@<0.15ex>[ul];
  \xymatrix@C=1pc@R=1pc{
    1.06 &1.24 &3.04&3.22&5.02 &7.00\\
    0.15\av{6}{1}\ad{2}{5}&2.13\av{8}{2}\ad{}{}\add{6}{3}&0.33\lk&4.11\av{4}{4}\ad{10}{1}&\\
    1.04\av{5}{1}\ad{2}{4}&1.22\av{6}{2}\ad{9}{1}&3.02\ad{4}{2}&5.00\av{1}{5}\\
    0.13\av{4}{1}\ad{2}{3}&2.11\av{4}{2}\ad{6}{1}&\\
    1.02\av{3}{1}\ad{2}{2}&3.00\av{1}{3}\\
    0.11\av{2}{1} &\\
    1.00\av{1}{1}
  }
\end{equation}

\section{Initial value problem for the coefficient {BBGKY} hierarchy}\label{s:transformation}

\subsection{Generalized Liouville operator}
Time evolution of coefficients $f$ is obtained by solving the initial value problem
\begin{equation}\label{e:iv}
  \partial_\tau f(\tau,f_{\mathrm{i}}) = \mathscr{L} f(\tau,f_{\mathrm{i}})\,,\qquad f(0,f_{\mathrm{i}})=f_{\mathrm{i}},
\end{equation}
where $\mathscr{L} f(\tau, f_{\mathrm{i}})$ denotes the right-hand side of the {BBGKY} hierarchy \eref{e:bbgky3}. The initial condition $f_{\mathrm{i}}$ is determined from the initial condition for the operator $F_{1\dots N}$.

The linear operator $\mathscr{L}: {\mathcal F}_N\mapsto{\mathcal F}_N$ is called the generalized Liouville operator \cite{balescu1975}. It is represented by a real-valued matrix, which multiplies vectors $f=(f_1,\dots,f_N)^\trsp$. The latter consist of $N$ blocks by the composition formula \eref{e:f}. The $n$-th block of $f$ consists of all the coefficients in the $n$-spin vector space arranged in a sequence. This block structure of the vector space ${\mathcal F}_N$ imparts a similar block structure on $\mathscr{L}$, comprising $N\times N$ rectangular matrices $L^{(ij)}\in\RR^{d(i)\times d(j)}$, where $1\le i,j\le N$. The three-term recurrence structure of \eref{e:bbgky3} implies that the only non-zero blocks $L^{(ij)}$ are such blocks that $\abs{i-j}=1$. Define
\begin{equation}\label{e:Ln}
\fl  L_n^+ = -L^{(n,n+1)}\in\RR^{d(n)\times d(n+1)}\,,\qquad L_n^-= L^{(n+1,n)}\in\RR^{d(n+1)\times d(n)}.
\end{equation}
The generalized Liouville operator $\mathscr{L}$ has the form of a block tridiagonal matrix with rectangular off-diagonal blocks $L_n^+$ and $L_n^-$
\begin{equation}\label{e:matrix}
  \mathscr{L} = \left(\!\!\!\begin{array}{cccccc}
  0 & -L^+_1 & & & & \\
  L_1^- & 0 & -L_2^+ & & & \\
  & L_2^- & 0 & & &\\
  &   &  & \ddots & & \\
  &   &  &        &0& -L_{N-1}^+ \\
  &   &  &        & L_{N-1}^- & 0
  \end{array}\!\!\!\right).
\end{equation}

\subsection{Self-adjoining transformation}
The {BBGKY} hierarchy \eref{e:bbgky3} can be rewritten using the graph notation as
\begin{equation}\label{e:graph2}
  \fl \partial_\tau f_n(m) = \sum_{m_1\in\mathrm{adj}^-(m)}\psi[\vv{m,m_1}] f_{n-1}(m_1) +  \sum_{m_1\in\mathrm{adj}^+(m)}\psi[\vv{m,m_1}]f_{n+1}(m_1)\,.
\end{equation}
Comparing this equation with \eref{e:matrix} we find that for any $m\in\mathfrak{M}_n$ and $m_1\in\mathfrak{M}_{n+1}$, one matrix element of $L_n^+$ is equal to $-\psi[\vv{m,m_1}]$ and similarly, one element of $L_n^-$ is equal to $\psi[\vv{m_1,m}]$. Relations between the indices $i$ and $j$ of a particular element $(L_n^\pm)_{ij}$ and value functions $\psi$ are determined by an arbitrary choice of a mapping of $m\in\mathfrak{M}_n$ onto a sequence of integers $\{1,\dots, d(n)\}$ for each $n$. It is not difficult to see that $(L_n^+)_{ij}=-\psi[\vv{m,m_1}]$ for some $i$ and $j$ implies $(L_n^-)_{ji}=\psi[\vv{m_1,m}]$ for the same pair $i$, $j$. By a corollary of \lemref{l:downedge} it follows that in general $(L_n^+)_{ij}\neq(L_n^-)_{ji}$, i.e. that $L_n^\pm$ are not mutually adjoint, and as a result that $\mathscr{L}$ is not anti-Hermitian
\begin{equation}\label{e:notaherm}
  L_n^-\neq L_n^{+\trsp} \quad \rightarrow \quad \mathscr{L} \neq - \mathscr{L}^\dagger\,.
\end{equation}
The evolution operator is formally obtained by exponentiating $\mathscr{L}$,
\begin{equation}\label{e:green}
  \mathscr{G}(\tau) = \rme^{\mathscr{L} \tau}.
\end{equation}
As a consequence of \eref{e:notaherm}, $\mathscr{G}(\tau)$ does not preserve the conventional inner product $(f,g)=\sum f_ig_i$ of $f, g\in{\mathcal F}_N$ for infinitesimal $\tau$ and therefore $\mathscr{G}(\tau)$ is not unitary with respect to this inner product. To get a positive result, one could try a change of basis and a similarity transformation of $\mathscr{L}$ by an invertible ${\sf D}$ such that $\mathscr{A}$, defined by
\begin{equation}\label{e:aherm}
  {\mathscr{A}} = {\sf D}^{-1} \mathscr{L} {\sf D}
\end{equation}
is anti-Hermitian,
\begin{equation}\label{e:aherm2}
  \mathscr{A}^\dagger=-\mathscr{A}\,.
\end{equation}
Since for a tridiagonal matrix (a matrix of the form \eref{e:matrix} where $L_n^\pm$ are scalars) this can be achieved by a scaling transformation, consider $\sf D$, defined by
\begin{equation}\label{e:diagD}
  {\sf D} = {\mathrm{diag}}( D_1, D_2, \dots, D_{N} )\,,
\end{equation}
where $D_i$ is a $d(i)\times d(i)$ square matrix which is yet to be found. The transformation \eref{e:aherm} by \eref{e:diagD} preserves the block tridiagonal structure of ${\mathscr{L}}$. In analogy with \eref{e:Ln}--\eref{e:matrix}, denote the off-diagonal blocks of ${\mathscr{A}}$ by $H_n^\pm$. By substituting into \eref{e:aherm}, the condition \eref{e:aherm2} is equivalent to
\begin{equation}\label{e:Hn}
  H_n^{+}= D_n^{-1} L_n^+ D_{n+1}\,, \qquad   H_n^{-}= D_{n+1}^{-1} L_n^- D_{n}\,.
\end{equation}
The condition that ${\mathscr{A}}$ is anti-Hermitian is
\begin{equation}\label{e:antihermitian}
  H_n^{-}=H_n^{+\trsp}
\end{equation}
and from \eref{e:Hn} and \eref{e:antihermitian} it follows that \eref{e:aherm2} is equivalent to existence of a solution of
\begin{equation}\label{e:symcond}
  (D_{n+1}D_{n+1}^\trsp) L_n^{+\trsp}  = L_n^-(D_nD_n^\trsp)\,, \qquad n=1,\dots, N-1,
\end{equation}
where $D_{n}D_{n}^\trsp$ must be determined by solving \eref{e:symcond} starting from an arbitrarily chosen $D_1$. For each $n$, \eref{e:symcond} is a set of $d(n)\times d(n+1)$ equations for the unknown elements of a square symmetric matrix, which has $d(n+1)[d(n+1)+1]/2$ free parameters. A necessary condition for solvability of \eref{e:symcond} is a greater number of unknowns than the number of equations, which amounts to satisfying $[ d(n+1)+1 ]/2 \ge d(n)$. Using \eref{e:dn} this inequality simplifies to $2n+6\ge0$ which is certainly satisfied for $n>0$. To construct an explicit form of $\mathscr{A}$, the following theorem is provided.
\begin{theorem}\label{the:transformation}
  Equations \eref{e:aherm}--\eref{e:aherm2} are solvable by a diagonal matrix ${\sf D}$. Each element of $H_n^\pm$ of $\mathscr{A}$ is equal to $\sqrt{\gamma_i^a}$, defined by \eref{e:gai}, for some $i$ and $a$.
\end{theorem}
\begin{proof}
  Assume $c\in{\mathcal F}_N$ such that $c(m)\neq0\forall m\in\mathfrak{M}$. Consider a transformation of $f$
\begin{equation}\label{e:f'}
  f'_n(m) = c(m) f_n(m)
\end{equation}
This transformation is of the form \eref{e:aherm} with a diagonal $\sf D$, where ${\sf D}_{ii}=1/c(m)$ $\forall i$ and some $m=m(i)\in\mathfrak{M}$.
It is convenient to express the {BBGKY} hierarchy equations for $f'_n(m)$ in the form \eref{e:graph2}. Substitution of \eref{e:f'} into \eref{e:graph2} results in the equation of the same form as \eref{e:graph2} with $f$s replaced by $f'$s, and $\psi$s replaced by $\psi'$s, and where the latter are defined by
\begin{equation}\label{e:psi'}
  \psi'[\vv{m,m_1}] = \psi[\vv{m,m_1}] c(m)/c(m_1)
\end{equation}
The proof will be established if $c(m)$ can be determined from \eref{e:psi'}, and the condition
\begin{equation}\label{e:psi'cond}
  \psi'[\vv{m,m_1}]=-\psi'[\vv{m_1,m}]
\end{equation}
is satisfied for each $m\in\mathfrak{M}$ and $m_1\in\mathrm{adj}(m)$, as this condition is equivalent to \eref{e:aherm2}.

Combining \eref{e:psi'} and \eref{e:psi'cond}, and using \eref{e:nu} one obtains
\begin{equation}\label{e:c2}
  c^2(m) = c^2(m_1)\frac{\nu(m_1,m)}{\nu(m,m_1)}.
\end{equation}
From \eref{e:c2} it follows that $c(m)$ is determined by its adjacent node $m_1$. \Eref{e:c2} can be applied recursively to determine $c(m)$ by $c(m_1)$, where $m_1$ is any node such that $m$ is reachable from $m_1$, multiplied by a ratio of products of numerical factors $\nu$ along the path, connecting $m_1$ with $m$.

\begin{definition}\label{d:path}A path $p$ of length $k>0$ is an ordered sequence of $k$ edges $\vv{m_j,m_{j+1}}$, $1\le j\le k$ such that $m_{j\pm1}$ is adjacent to $m_j$,
\begin{equation}
  p(m_1\rightarrow m_{k+1}) = ( \vv{m_1,m_2}, \vv{m_2,m_3}, \dots, \vv{m_{k}, m_{k+1}} ).
\end{equation}
A loop $p_0$ is a closed path $p$, i.e. a path where $m_{k+1}=m_1$
\begin{equation}
  p_0 = (\vv{m_1, m_2}, \vv{m_2,m_3}, \dots, \vv{m_{k},m_1}).
\end{equation}
\end{definition}
Using this definition
\begin{equation}\label{e:path}
  c^2(m_{k+1}) = c^2(m_1) \prod_{e\in p(m_1\rightarrow m_{k+1})} \frac{\nu(m'(e),m''(e))}{\nu(m''(e),m'(e))}\,,
\end{equation}
where $e$ denotes an element of $p$ such that $e=(m'(e),m''(e))$. Since most $m'$s have more than one adjacent $m''$, most pairs of connected nodes can be linked by several distinct paths. Therefore, solvability of \eref{e:aherm} by a diagonal $\sf D$ rests upon the condition that $c(m_{k+1})$ is uniquely determined by $m_1$ and is independent of $p$. Looking at a segment of $\graph^1$ in \eref{e:Gx} it is clear that any path $p(m_1\rightarrow m_{k+1})$ could be augmented by inserting loops at any of the intermediate nodes. Therefore, independence on the traversal is achieved if for any loop $p_0$
\begin{equation}\label{e:cond1}
  \prod_{e\in p_0} \frac{\nu(m'(e),m''(e))}{\nu(m''(e),m''(e))} =1\,.
\end{equation}
This condition could be turned into an additive condition for loops by taking a logarithm of \eref{e:cond1} and defining the current $j$ on each edge.
\begin{definition}
  For any $m'\in\mathfrak{M}$ and $m''\in\mathrm{adj}(m')$, and $e=(m',m'')\in\mathfrak{A}$, the current $j(e)\in\RR$ is defined by
  \begin{equation}\label{e:j}
    j(e) =     \log\big[\nu(m',m'')/\nu(m'',m')\big]\,.
  \end{equation}
\end{definition}
Solvability of \eref{e:aherm} by a scaling transformation can be formulated as a ``zero circulation'' condition: For any loop $p_0$ we require that
\begin{equation}\label{e:jcirc}
  \sum_{e\in p_0} j(e) = 0\,.
\end{equation}
Considering lines that make up any loop in \eref{e:Gx} as boundaries of that loop that separate the ``inside'' from the ``outside'', the zero circulation conditions is verified by multiplying all the numbers along the loop which are either outside or inside its boundary. As can be seen from \eref{e:Gx}, the most elementary non-trivial loop $p_0$ involves four nodes, and more complicated loops can be constructed from a sequence of elementary loops. It is therefore sufficient to verify the circulation condition \eref{e:jcirc} for each loop of the form
\begin{equation}\label{e:loop}
  \newcommand\av[2]{\ar@{-}@<0.15ex>[u]^{#1}_{#2}
  \newcommand\ak[2]{\ar@{-}@<0.15ex>[l]^{#1}_{#2}
  \newcommand\ad[2]{\ar@{-}@<0.15ex>[ur]^{#1}_{#2}
  \newcommand\add[2]{\ar@{-}@<0.15ex>[urr]^{#1}_{#2}
  \newcommand\lk{\ar@{-}@<0.15ex>[ul];
  \xymatrix@dr@C=2pc@R=2pc{
    m+t_i^a+t_j^b  & m+t_j^b \ak{\nu'_2}{\nu_2} \\
    m+t_i^a \av{\nu_3}{\nu'_3} & m\ak{\nu_4}{\nu'_4}\av{\nu'_1}{\nu_1} }
  \qquad \prod_i \nu_i = \prod_i \nu'_i\,.
\end{equation}
where $t_i^a\neq t_j^b$. As an example, consider in \eref{e:Gx} a loop, that visits $\{102,211,122,013\}$. We find
$\prod_i \nu_i =2\cdot2\cdot2\cdot3=24$, and $\prod_i \nu'_i =2\cdot4\cdot3\cdot1=24$. Therefore \eref{e:loop} is satisfied for this particular loop. A straightforward calculation (not detailed here) confirms that \eref{e:loop} is satisfied for arbitrary $t_i^a\neq t_j^b$.

The elements of $\mathscr{A}$ are expressed in terms of value functions $\psi'[\vv{m,m_1}]$. Substituting \eref{e:c2} into \eref{e:psi'}, with $k=1$, such that $m_2=m_1+t_i^a$, and using \eref{e:gai}
\begin{equation}
  \psi'[\vv{m,m+t_i^a}]=\sqrt{-\psi[\vv{m,m+t_i^a}]\psi[\vv{m+t_i^a,m}]}= \sqrt{\gamma_i^a}\,.
\end{equation}
It follows that $\psi'[\vv{m+t_i^a,m}]=\psi'[\vv{m,m+t_i^a}]$.
\end{proof}
Each coefficient $c(m)$ can be calculated by following the path to $m$ from some fixed reachable $m_0$, whose coefficient can be chosen arbitrarily. Setting $c(m_0)=1$,
\begin{equation}
  c(m) = \rme^{ j[p(m_0\rightarrow m)]/2}= \Bigl\{ \prod_{e\in p(m_0\rightarrow m)} \frac{\nu(m'(e),m{''}(e))}{\nu(m{''}(e),m'(e))} \Bigr\}^{1/2}\,,
\end{equation}
where $p(m_0\rightarrow m)$ is any path, connecting $m_0$ with $m$.

Since $\partial_\tau{\sf D}^{-1}f=\mathscr{A}{\sf D}^{-1}f$ where $\mathscr{A}$ is anti-Hermitian, and $({\sf D}^{-1}f)_i=c(m)f(m)$ for some $m=m(i)$ by \theref{the:transformation}, we conclude that the evolution operator \eref{e:green} is unitary, provided that the inner product on the vector space ${\mathcal F}_N$ is defined as follows
\begin{equation}\label{e:metric}
  \avg{f^1,f^2}_{{\mathcal F}_N} = \sum_{n=1}^N\sum_{m\in\mathfrak{M}_n} c^2(m) f^1_n(m) f^2_n(m) \quad f^1, f^2\in{\mathcal F}_N.
\end{equation}

\section{Invariant subspace decomposition}\label{s:symmetries}
The following definition of an induced graph will be useful for the discussion.
\begin{definition} A graph $\graph^1=(\mathfrak{M}^1,\mathfrak{A}^1)=\mathrm{ind}(\graph,m_1)$ is a subgraph of $\graph=(\mathfrak{M},\mathfrak{A})$ induced by $m_1\in\mathfrak{M}$, if $\mathfrak{M}^1$ and $\mathfrak{A}^1$ are defined as follows. The set $\mathfrak{M}^1\subset\mathfrak{M}$ consists of all the nodes $m\in\mathfrak{M}$ such that $m_1$ is reachable from $m$, and $\mathfrak{A}^1\subset\mathfrak{A}$ is the set of all the edges $\vv{m,m_1} \forall m$, by which $m_1$ can be reached from $m$.
\end{definition}
A graph $\graph$ is \emph{fully connected} if all of its nodes are reachable from any one of them. The same can be stated using induced graphs as $\mathrm{ind}(\graph,m)=\graph$ for any $m\in\mathfrak{M}$. Conversely, if for some $m_1\in\mathfrak{M}$, $\mathrm{ind}(\graph,m_1)\neq\graph$, then $\graph$ is a \emph{disconnected graph}. A disconnected graph is a union of several connected graphs $\graph^i$: $\graph=\graph^1\cup\graph^2 \cup \ldots$

If the {BBGKY} hierarchy \eref{e:bbgky3} is represented by a disconnected graph $\graph$, then all expansion coefficients in ${\mathcal F}_N$ can be grouped by the node set $\mathfrak{M}^i$ of its each connected subgraph $\graph^i=(\mathfrak{M}^i,\mathfrak{A}^i)$. Each group of coefficients evolves by a subset of {BBGKY} hierarchy equations for these coefficients. Since different subgraphs $\graph^i$ are disconnected, the {BBGKY} hierarchy equations do not couple coefficients in different $\mathfrak{M}^i$s and therefore each group of coefficients evolves independently from the remaining coefficients. Thus each set of nodes $\mathfrak{M}^i$ defines indices of a group of expansion coefficients that belong to an invariant subspace ${\mathcal F}_N^i$ of the full vector space ${\mathcal F}_N$. In this section it will be shown that the connectivity of $\graph$ depends on the structure of the effective couplings, and therefore $\overline{J}$ determines the topology of ${\mathcal F}_N$.

\subsection{Effective coupling constants}\label{s:oJ}
In \cite{paskauskas2012a} and in \sref{s:summary} it was shown that the effects of magnetic field $h$ can be accounted for, in thermodynamic limit, by replacing the matrix of coupling constants $J$ by a matrix of effective coupling constants $\overline{J}=(\overline{J}_{ab})$ which is defined by \eref{e:Javg}. If $h=0$, the interaction picture is not necessary since the equations with a non-averaged $J$ are of the same form \eref{e:bbgky2}. In this case assume $\overline{J}=J$. A common, but not the most general setting is when the couplings $J$ are diagonal,
\begin{equation}\label{e:Jdiag}
  J=\mathrm{diag}(J_x,J_y,J_z)
\end{equation}
and $h=(h_x,h_y,h_z)$ is an arbitrary vector with a condition $h\neq 0$. Define $\hat{h}=h/\abs{h}$. In this case, the matrix of effective coupling constants $\overline{J}$ can be written as
\numparts\begin{equation}\label{e:Jbar}
  \overline{J} = \frac{J_d}{2} - \frac{J_s}{2}\,,
\end{equation}
where
\begin{eqnarray}
  J_d =\left\{\hat{h}_x^2(J_y+J_z) + \hat{h}_y^2(J_x+J_z) + \hat{h}_z^2(J_x+J_y)\right\}\one_{3\times 3} \label{e:Jbard}\\
  J_s  = \left\{J_x+J_y+J_z-3(\hat{h}_x^2J_x+\hat{h}_y^2J_y+\hat{h}_z^2J_z)\right\} (\hat{h}_a\otimes\hat{h}_b)\label{e:Jbars}
\end{eqnarray}\endnumparts
and where $(\hat{h}_a\otimes\hat{h}_b)$ denotes the outer product of $\hat{h}=(\hat{h}_x,\hat{h}_y,\hat{h}_z)$.

Inspecting \eref{e:Jbar}--\eref{e:Jbars} we conclude that if $J$ is given by \eref{e:Jdiag} and either $h=0$ or $h$ has only one non-zero element [for example $h=(0,0,h_z)$], then $J_s=0_{3\times 3}$ and $\overline{J}$ is a diagonal matrix. In general, if two components of $h$ are non-zero, as in $h=(h_x,0,h_z)$, then $\overline{J}$ has one non-zero off-diagonal element; if all three components of $h$ are non-zero, then all off-diagonal elements of $\overline{J}$ are non-zero in general. The first two cases will be discussed in detail in the following, while it is assumed that $\graph$ is fully connected if all the diagonal elements of $\overline{J}$ are unequal, and all off-diagonal elements are non-zero.

\subsection{Diagonal couplings}\label{s:diagonal}
In this section, a special case will be considered where the coupling matrix $\overline{J}$ is diagonal
\begin{equation}\label{e:J1}
  \overline{J} = \mathrm{diag}( \overline{J}_x, \overline{J}_y, \overline{J}_z )\,.
\end{equation}
Define $\overline{J}$ to be \emph{anisotropic} if all three coupling constants $\overline{J}_a$ are distinct. Equivalently $\overline{J}$ is anisotropic if $\overline{K}_x=\overline{J}_y-\overline{J}_z\neq0$, $\overline{K}_y=\overline{J}_z-\overline{J}_x\neq0$, and $\overline{K}_z=\overline{J}_x-\overline{J}_y\neq 0$. A similar setup has been studied in \cite{paskauskas2012a} where the coupling matrix was diagonal but not anisotropic. This special case will be further studied in \sref{s:case}.

\Eref{e:J1} implies $\overline{W}_x=\overline{W}_y=\overline{W}_z=0$. As a result, $v_{n}^\pm(m,f)$ simplify considerably
\numparts
\begin{eqnarray}
  \fl v_{n}^-(m;f) &=& \overline{K}_xm_ym_z f(m_x+1,m_y-1,m_z-1) + \overline{K}_ym_zm_x f(m_x-1,m_y+1,m_z-1) \nonumber \\
  \fl  && +  \overline{K}_zm_xm_y f(m_x-1,m_y-1,m_z+1)\,.\label{e:v-2} \\
  \fl v_{n}^+(m;f) &=& - \overline{K}_xm_x f(m_x-1,m_y+1,m_z+1) -\overline{K}_ym_y f(m_x+1,m_y-1,m_z+1)\nonumber\\
  \fl && - \overline{K}_zm_z f(m_x+1,m_y+1,m_z-1)\,.\label{e:v+2}
\end{eqnarray}\endnumparts
It follows from \eref{e:v-2}--\eref{e:v+2} that all the edges of $\graph$ are generated by $t_1^a$, defined by \eref{e:tai}.

Assume that $\overline{J}$ is anisotropic. Define
\numparts
\begin{eqnarray}
  \mathfrak{N}_1^1=\{ (1,0,0)\}, \quad
  \mathfrak{N}_1^2=\{ (0,1,0)\}, \quad
  \mathfrak{N}_1^3=\{ (0,0,1)\}, \\
  \mathfrak{N}_2^4=\{ (2,0,0), (0,2,0), (0,0,2)\}.
\end{eqnarray}\endnumparts
Define four subgraphs of $\graph$, induced by these sets
\numparts\begin{eqnarray}
\graph^1=(\mathfrak{N}^1,\mathfrak{A}^1)=\mathrm{ind}(\graph,\mathfrak{N}_1^1),\\
\graph^2=(\mathfrak{N}^2,\mathfrak{A}^2)=\mathrm{ind}(\graph,\mathfrak{N}_1^2),\\
\graph^3=(\mathfrak{N}^3,\mathfrak{A}^3)=\mathrm{ind}(\graph,\mathfrak{N}_1^3),\\
\graph^4=(\mathfrak{N}^4,\mathfrak{A}^4)=\mathrm{ind}(\graph,\mathfrak{N}_2^4).
\end{eqnarray}\endnumparts
Here $\mathfrak{N}_n^i\subset\mathfrak{M}_n$. Similarly to the composition of $\mathfrak{M}^{(N)}$ \eref{e:setM}, we define
\begin{equation}
  \mathfrak{N}^{(N,i)} = \bigcup_{n=1}^N \mathfrak{N}_n^i\,.
\end{equation}
Set $\mathfrak{N}_1^4$ is empty. The following lemma describes the structure of the remaining $\mathfrak{N}_n^i$.
\begin{lemma}\label{l:2step}For any $1\le n<N-2$, and $i=1,2,3,4$,
  \begin{equation}\label{e:iter2}
    \mathfrak{N}^i_{n+2}= \bigcup_{m\in\mathfrak{N}^i_n} \{ m + (2,0,0), m + (0,2,0), m + (0,0,2) \}
  \end{equation}
\end{lemma}
\begin{proof}
  Assume that $n$ is fixed and that \eref{e:iter2} is valid for $n-2$ and $n$. Induction of nodes in one step from $\mathfrak{N}_n^i$ to $\mathfrak{N}_{n+1}^i$ is obtained by following all the edges which are defined by vectors $t_1^a$: $m\to m+t_1^a$. Induction in two steps results in
  \begin{equation}
    \mathfrak{N}^i_{n+2}=\bigcup_{\scriptstyle ab\in\mathcal{I}\atop \scriptstyle m\in\mathfrak{N}^i_n} \{ m + \{t_1^a+t_1^b\}\}\,.
  \end{equation}
where
\begin{equation}\label{e:iter2a}
  \fl\{ t_1^a+t_1^b\} =\{ (0,0,2), (0,2,0), (2,0,0), (-2,2,2), (2,-2,2), (2,2,-2) \}
  \end{equation}
  The first three terms in \eref{e:iter2a} satisfy \eref{e:iter2}. Each of the remaining three terms can be viewed as a three step path, e.g. $(-2,2,2)= (-2,0,0) + (0,2,0) + (0,0,2)$. By assumption $m'=m+(-2,0,0)\in \mathfrak{N}_{n-2}^i$ and $m''=m'+(0,2,0)\in\mathfrak{N}_n^i$. The last step $m'''=m''+(0,0,2)$ satisfies \eref{e:iter2} therefore $m'''\in\mathfrak{N}^i_{n+2}$.

By direct calculation one verifies that $\mathfrak{N}^1_2=\{(0,1,1)\}$, $\mathfrak{N}^1_3=\{(1,0,2), (1,2,0)\}$, and $\mathfrak{N}^1_4=\{(2,1,1),(0,1,3), (0,3,1)\}$ satisfy the assumption of the lemma for $n=1,3$, and $n=2,4$. The corresponding sets with $i=2,3$ are obtained by permutations of $\mathfrak{N}_n^1$.
The case of $i=4$ also satisfies the assumption with $n=2,4$ and $n=3,5$ as $\mathfrak{N}^4_3=\{(1,1,1)\}$, $\mathfrak{N}^4_4=\{(0,2,2),(2,0,2),(2,2,0)\}$, and $\mathfrak{N}_5^4=\{(3,1,1), (1,3,1), (1,1,3)\}$.
\end{proof}
The main result of this section is contained in the following theorem.
\begin{theorem}\label{the:graph}
Assume that $\overline{J}$ is a diagonal anisotropic matrix. Then $\graph^i$ are disjoint subgraphs of $\graph$, and $\graph=\graph^1\cup\graph^2\cup\graph^3\cup\graph^4$.
\end{theorem}
\begin{proof}
For the first part of the theorem it is sufficient to show that for each $1\le n\le N$,
\begin{equation}\label{e:cap}
  \mathfrak{N}_n^i\cap\mathfrak{N}_n^j=0\qquad i\neq j, \forall n.
\end{equation}
For the second part one must show that
\begin{equation}\label{e:cup}
  \mathfrak{M}_n = \bigcup_{i=1}^4\mathfrak{N}_n^i  \qquad \forall n.
\end{equation}
In analogy with \eref{e:dn}, denote the number of elements in $\mathfrak{N}_n^i$ by $d_i(n)$. The connectivity structure of $\graph^1$, $\graph^2$ and $\graph^3$ is the same because their node sets $\mathfrak{N}^i$ are related by cyclic permutations, in addition to $\overline{K}_x\neq0$, $\overline{K}_y\neq0$, $\overline{K}_z\neq0$, implied by the anisotropy of $\overline{J}$. Therefore the corresponding node sets $\mathfrak{N}_n^i$ have the same number of elements for each $n$, and the condition \eref{e:cup} can be replaced by \eref{e:cap} together with
\begin{equation}\label{e:proofdn}
  d(n) = 3d_1(n) + d_4(n) \qquad  \forall n.
\end{equation}
For $n=1,2$ conditions \eref{e:cap} and \eref{e:cup} are verified directly. In particular
\numparts\begin{eqnarray}
  \mathfrak{M}_1=\{(1,0,0),(0,1,0),(0,0,1)\}=\cup_{i=1}^4\mathfrak{N}_1^i\,,\\
  \mathfrak{M}_2=\{(2,0,0),(0,2,0),(1,1,0),(1,0,1),(0,1,1)\}=\cup_{i=1}^4\mathfrak{N}_2^i\,.
\end{eqnarray}\endnumparts
Consider the parity of  $m^x$, $m^y$, $m^z$ of the elements $m=(m^x,m^y,m^z)\in\mathfrak{N}_n^i$. Induction of nodes in one step by $m\to m+t_1^a$, adds or subtracts unity to each element $m^a$ of $m$. This operation inverts the parity of each $m^a$ in one step. In two steps, induction does not alter parity, as can be seen from \lemref{l:2step}. By examining $\mathfrak{N}_1^i$, $\mathfrak{N}_2^i$ and applying this simple parity alternation rule, the parity composition of $\mathfrak{N}_n^i$ can be determined. Define $i_a$ by $i_x=1$, $i_y=2$, $i_z=3$. For all $k=0,1,\dots, N/2$ and $a=x,y,z$
\numparts
\begin{eqnarray}
  m\in\mathfrak{N}_{2k}^{i_a}   &:& m^a\, \mathrm{is\,\, even,}\quad m^{\pi a}, m^{\pi^2a}\, \mathrm{are\,\, both\,\, odd},\label{e:parity1}\\
  m\in\mathfrak{N}_{2k+1}^{i_a} &:& m^a\, \mathrm{is\,\, odd,} \quad m^{\pi a}, m^{\pi^2a}\, \mathrm{are\,\, both\,\, even}.\label{e:parity2}
  \end{eqnarray}
Similarly, if $i=4$,
\begin{eqnarray}
  m\in\mathfrak{N}_{2k}^4 &:& m^x, m^y, m^z\, \mathrm{are\,\, all\,\, even},\label{e:parity3}\\
  m\in\mathfrak{N}_{2k+1}^4  &:& m^x,m^y,m^z\, \mathrm{are\,\, all\,\, odd}.\label{e:parity4}
\end{eqnarray}\endnumparts
It follows from \eref{e:parity1}--\eref{e:parity4} that the patterns of parity are incompatible among all $\mathfrak{N}_n^i$ for each $n$, and therefore these sets are disjoint. This proves \eref{e:cap}.

Counting of the elements in $\mathfrak{N}_n^1$ and $\mathfrak{N}_n^4$ can be achieved by parameterizing all $m=(m^x,m^y,m^z)$ with the necessary properties\eref{e:parity1}--\eref{e:parity4}. Define
\begin{equation}\label{e:mij}
  m_{kij}=(2k-2i,2i-2j,2j)\qquad k\ge i\ge j\ge 0.
\end{equation}
Using this definition all the cases in \eref{e:parity1}--\eref{e:parity4} can be parameterized as follows,
\numparts\begin{eqnarray}
  \mathfrak{N}_{2k+1}^1 &= \{ m= m_{kij} + (1,0,0) \} \qquad k\ge0\,, \label{e:d1}\\
  \mathfrak{N}_{2k+2}^1 &= \{ m=m_{kij} + (0,1,1) \}  \qquad k\ge0\,, \\
  \mathfrak{N}_{2k}^4 &= \{ m=m_{kij} \} \qquad k\ge1\,, \\
  \mathfrak{N}_{2k+3}^4 &= \{ m=m_{kij}+(1,1,1) \} \qquad k\ge0\,.\label{e:d4}
\end{eqnarray}\endnumparts
Parametrizations of $\mathfrak{N}^2_n, \mathfrak{N}^3_n$ are derived similarly from $\mathfrak{N}_n^1$.  From \eref{e:d1}--\eref{e:d4} it follows that the number elements in various sets $\mathfrak{N}^i_n$ is related to the number of elements $m_{ijk}$ with a fixed $k$ by
\begin{eqnarray}
  d_1(2k+1)=d_1(2k+2)=d_4(2k)=d_4(2k+3) \nonumber \\
  =\sum_{i,j}\big(1_{k\ge i\ge j}\big)\big(1_{i \ge j\ge 0}\big)=\sum_{i=0}^k (i+1) = d(k)\,.
\end{eqnarray}
where $1_X=1$ if $X$ is true and $1_X=0$ if $X$ is false. Counting of terms by \eref{e:d1}--\eref{e:d4}, done separately for the even $n=2k$ and odd $n=2k+1$ results in, respectively
\numparts\begin{eqnarray}
  3d_1(2k)+d_4(2k)= 3d(k-1)+d(k) = d(2k)\,,\\
  3d_1(2k+1)+d_4(2k+1)=3d(k)+d(k-1) = d(2k+1)\,.
\end{eqnarray}\endnumparts
Since the argument on the right hand side is $n$ in both cases, this proves \eref{e:proofdn}.
\end{proof}

\subsection{Non-diagonal couplings}\label{s:offdiag}
Assume that one off-diagonal element of the coupling matrix is non-zero, i.e.  $\overline{W}_a\neq0$ for some $a\in\{x,y,z\}$, and $\overline{W}_{\pi a}=\overline{W}_{\pi^2a}=0$. For example if $a=x$, $\overline{J}$ has the form
\begin{equation}\label{e:Joffdiagonal}
  \overline{J} = \left(\begin{array}{ccc} \overline{J}_x & \overline{W}_x & 0\\ \overline{W}_x & \overline{J}_y & 0\\ 0 & 0 & \overline{J}_z\end{array}\right)
\end{equation}
Define
\begin{equation}
  \mathfrak{N}^A= \mathfrak{N}^{i_a} \cup\, \mathfrak{N}^4\,, \qquad
  \mathfrak{N}^B= \mathfrak{N}^{i_{\pi a}}\cup \mathfrak{N}^{i_{\pi^2 a}}\,.
\end{equation}
Define graphs $\graph^A\subset\graph$ and $\graph^B\subset\graph$ as induced from these sets
\numparts\begin{eqnarray}
  \graph^A=(\mathfrak{N}^A, \mathfrak{A}^A)=\mathrm{ind}(\graph,\mathfrak{N}^A)\,,\\
  \graph^B=(\mathfrak{N}^A, \mathfrak{A}^A)=\mathrm{ind}(\graph,\mathfrak{N}^B)\,.
\end{eqnarray}\endnumparts
\begin{theorem}\label{the:graph2} Assume that $\overline{K}_x\neq 0$, $\overline{K}_y\neq 0$, $\overline{K}_z\neq0$, and $\overline{W}_a\neq0$ for some $a\in\{x,y,z\}$ but $\overline{W}_{\pi a}=\overline{W}_{\pi^2a}=0$. Then graphs $\graph^A$ and $\graph^B$ are disjoint subgraphs of $\graph$, and $\graph=\graph^A\cup\graph^B$.
\end{theorem}
\begin{proof}
  Here, the complete node sets $\mathfrak{N}^A$ and $\mathfrak{N}^B$ are provided and it is sufficient to show that induction of nodes, generated by vectors $t_1^x$, $t_1^y$, $t_1^z$, and $t_2^a$, $t_3^a$, $t_4^a$ which correspond to non-zero effective coupling coefficients, leaves the node sets invariant. The first three vectors generate the sets $\mathfrak{N}^{1/2/3/4}$, discussed extensively in \sref{s:diagonal}. From the definition \eref{e:tai} it follows that induction in one step by the remaining three vectors inverts the parity of $m^a$ only, and does not change the parity of $m^{\pi a}$ and $m^{\pi^2a}$. By inspecting the parity structure of $\mathfrak{N}^i_n$, discussed in \theref{the:graph}, it follows that the addition of these vectors modifies the adjacency relations to $\mathrm{adj}^\pm(\mathfrak{N}^{i_a}_n)=\{ \mathfrak{N}^{i_a}_{n\pm1},\mathfrak{N}^4_{n\pm1}\}$,  $\mathrm{adj}^\pm(\mathfrak{N}^4_n)=\{\mathfrak{N}^4_{n\pm1}, \mathfrak{N}^{i_a}_{n\pm1}\}$, and similarly, $\mathrm{adj}^\pm(\{\mathfrak{N}^{i_{\pi a}}_n,\mathfrak{N}^{i_{\pi^2a}}_n\})=\{\mathfrak{N}^{i_{\pi a}}_{n\pm1},\mathfrak{N}^{i_{\pi^2}}_{n\pm1}\}$, from which the invariance requirement of $\mathfrak{N}^A$ and $\mathfrak{N}^B$ follows. Furthermore, disjointness and completeness of the graphs $\graph^A$ and $\graph^B$ follows from from the corresponding properties of $\graph^i$ and $\oplus_i\graph^i$, proved in \theref{the:graph}.
\end{proof}

\subsection{Isotropic diagonal couplings}\label{s:case}
A rather different situation is encountered when $\overline{J}$ is diagonal and isotropic, as in
\begin{equation}\label{e:Jiso}
  \overline{J}=\mathrm{diag}( \overline{J}_\perp,\overline{J}_\perp,\overline{J}_z )\,.
\end{equation}
Now we have $\overline{W}_x=\overline{W}_y=\overline{W}_z=\overline{K}_z=0$. Since $\sum_i\overline{K}_i=0$, $\overline{K}$ must be of the form
\begin{equation}
  \overline{K} = (-\kappa,\kappa,0)\,,
\end{equation}
where $\kappa=\overline{J}_z-\overline{J}_\perp$. Since $\kappa$ is the only coupling coefficient appearing in each term on the right-hand side of \eref{e:graph}, it can be scaled to $\pm1$ by an additional transformation of the time variable. Since the evolution is unitary, we may set $\kappa=+1$, and thus
\begin{eqnarray}\label{e:gspec}
  \fl \partial_\tau f_n(m) = m^xf_{n+1}(m^x-1,m^y+1,m^z+1)- m^yf_{n+1}(m^x+1,m^y-1,m^z+1) \nonumber\\
 \fl  + m^z\big[m^x f_{n-1}(m^x-1,m^y+1,m^z-1) - m^yf_{n-1}(m^x+1,m^y-1,m^z-1)\big]\,.
\end{eqnarray}
Since each term on the right-hand side of \eref{e:gspec} is multiplied by $m^x$ or $m^y$, the left-hand size of \eref{e:gspec} is zero whenever $m$ is of the form $m_n=(0,0,n)$. As a result, $f_n(m_n)$ is a constant of motion for each $n=1,2,\dots, N$.

To determine the remaining invariant subspaces, it is instructive to study a graph that corresponds to $\partial_\tau^2 f = \mathscr{L}^2 f$. A straightforward differentiation of \eref{e:gspec} results in
\begin{eqnarray}
  \fl\partial_\tau^2 f_n(m) =&-(2m^z+1)\bigl\{C_0f_n(m)-C_xf_n(m+t_{\perp})-C_yf_n(m-t_{\perp})\bigr\} \nonumber\\
  \fl &+C_x\bigl\{f_{n+2}(m+t_{\perp}+t_{\parallel})+C_zf_{n-2}(m+t_{\perp}-t_{\parallel})\bigr\}\nonumber\\
  \fl &+C_y\bigl\{f_{n+2}(m-t_{\perp}+t_{\parallel})+C_zf_{n-2}(m-t_{\perp}-t_{\parallel}) \bigr\} \nonumber\\
  \fl &-C_0\bigl\{f_{n+2}(m+t_{\parallel})+C_zf_{n-2}(m-t_{\parallel})\bigr\}\,. \label{e:gspec2}
\end{eqnarray}
where
\numparts\begin{equation}
  \fl C_a(m)=m^a(m^a-1)\quad a\in\{x,y,z\},\qquad C_0(m)=2m^xm^y+m^x+m^y\,,
\end{equation}
and where
\begin{equation}\label{e:tpie}
  t_{\perp}=(-2,2,0) \qquad t_{\parallel}=(0,0,2).
\end{equation}\endnumparts
As in previous sections, we associate a graph $\graph=(\mathfrak{M},\mathfrak{A})$ to the operator $\mathscr{L}^2$ on the right-hand side of \eref{e:gspec2}, and determine the disconnected subgraphs of $\graph$ by following all possible paths, induced by \eref{e:tpie} from various initial elements $m$. For example, all changes in $m^z$ are provided by $\pmt_{\perp}$, which implies that the node set of any disconnected subgraph $\graph^i\subset\graph$ is such that $m^z$ is either an even or an odd integer from an interval $[m^z_{\mathrm{min}},N]$, where $0\le m^z_{\mathrm{min}}<N$ is the smallest possible value of $m^z$ of a particular subgraph. Moreover, if $\graph^i$ is a disconnected subgraph with $m^z=0,2,\dots$, then another graph $\graph'^{i}$, which differs from $\graph^i$ only by replacing $m^z$ by $m^z+1$, is also a disconnected subgraph of $\graph$: $\graph'^{i}\subset\graph$.

The ``horizontal'' induction of nodes (by paths that do not change $m^z$, nor $\abs{m}$) is obtained by applying $m\mapsto m\pmt_{\perp}$ iteratively with the usual constraint $m\in\mathfrak{M}$. Four cases are possible, depending on the parity of $m^x$ and $m^y$. Define $\mathfrak{o}_{1,kn}, \mathfrak{o}_{2,kn}\subseteq\mathfrak{M}_{n+2k+1}$, $\mathfrak{o}_{3,kn}\subseteq\mathfrak{M}_{n+2k}$, and $\mathfrak{o}_{4,kn}\subseteq\mathfrak{M}_{n+2k+3}$ by
\numparts\begin{eqnarray}
\fl\mathfrak{o}_{1kn}=\bigcup_{j=0}^k (1+2j,2k-2j,n)\,, \quad & \mathfrak{o}_{2kn}=\bigcup_{j=0}^k (2k-2j,1+2j,n) \,, \label{e:kodes-1}\\
\fl\mathfrak{o}_{3kn}=\bigcup_{j=0}^k (2k-2j,2j,n) \,,        & \mathfrak{o}_{4kn}=\bigcup_{j=0}^k (1+2k-2j,1+2j,n) \,.\label{e:kodes-2}
\end{eqnarray}\endnumparts
Each of the sets $\mathfrak{o}_{ikn}\subseteq\mathfrak{M}_{n'}$ is invariant with respect to the horizontal induction.

Define even and odd node sets by combining the corresponding $\mathfrak{o}_{ikn}$ as follows
\begin{equation}
  \mathfrak{O}^{e}_{ik} = \bigcup_{n=0}^{N/2} \mathfrak{o}_{ik,2n} \qquad \mathfrak{O}^{o}_{ik} = \bigcup_{n=0}^{N/2} \mathfrak{o}^3_{k,2n+1}\,.
\end{equation}
By construction, $\mathfrak{O}^{e/o}_{ik}$ are invariant with respect to node induction by \eref{e:tpie}, therefore a family of induced graphs, parametrized by $e/o$, $i=1,2,3,4$ and $k=0,1,\dots, N/2$ and defined by
\begin{equation}\label{e:ON}
  \fl\graph^{e/o}_{ik} = (\mathfrak{O}^{e/o}_{ik}, \mathfrak{A}^{e/o}_{ik}) = \mathrm{ind}(\graph, \mathfrak{O}^{e/o}_{ik}) \quad k=0,1,\dots, N/2, \quad i=1,2,3,4
\end{equation}
are disconnected subgraphs of $\graph$.

The range of the parameter $k$ in \eref{e:ON} implies that the number of disconnected subgraphs of $\graph$ scales as $O(N)$ and consequently, the same scaling holds for the number of invariant vector spaces ${\mathcal F}^{e/o}_{ik,N}$, associated to each disconnected subgraph $\graph^{e/o}_{ik}$.

Associated to each $\mathfrak{o}_{ikn}$ there is an $n'$-spin vector space ${\mathcal S}_{n'}$, where $n'=n+2k+1$ if $i=1,2$, $n'=n+2k$ if $i=3$, and $n'=n+2k+3$ if $i=4$. By definition \eref{e:kodes-1}--\eref{e:kodes-2}, $\mathfrak{o}_{ikn}$ has $k+1$ elements, thus each ${\mathcal S}_{n'}$ subspace is $k+1$ dimensional. As a result, the operator $\mathscr{L}^2$ associated to a disconnected subgraph $\graph^{e/o}_{ik}$ has a block structure where all the blocks are square $(k+1)\times (k+1)$ size matrices, and where $k$ is fixed.

\subsection{Dependence on initial conditions}
In the preceding sections it was demonstrated that, depending on the structure of $\overline{J}$, connectivity properties of $\graph$ vary. They may be summarized as follows.
\begin{equation}
  \graph = \bigcup_{\alpha_i\in\alpha} \graph^{\alpha_i}\,,
\end{equation}
where $\alpha=\{\alpha_1,\alpha_2,\dots \}$ is a set of partition indices and where $\graph^{\alpha_i}=(\mathfrak{M}^{\alpha_i},\mathfrak{A}^{\alpha_i})$ is a connected graph for each $\alpha_i\in\alpha$. If $\overline{J}$ is a diagonal anisotropic matrix as defined in \sref{s:diagonal}, $\alpha=\{1,2,3,4\}$; if $\overline{J}$ is diagonal anisotropic and, in addition, one off-diagonal element of $\overline{J}$ is non-zero, as defined in \sref{s:offdiag}, $\alpha=\{A,B\}$. Lastly, if $\overline{J}$ is a diagonal isotropic matrix then, as discussed in \sref{s:case}, $\alpha=\cup_k\cup_j (e/o,j,k)$.

A collection of components $f(m)$, $m\in\mathfrak{M}^{\alpha_i}$ was viewed as a vector in a vector space ${\mathcal F}^{\alpha_i}_N\subseteq{\mathcal F}_N$. Connectendess of a graph $\graph^{\alpha_i}$ implies that the subset of {BBGKY} equations for the evolution of vectors in ${\mathcal F}^{\alpha_i}_N$ involves only vectors in ${\mathcal F}^{\alpha_i}_N$ and therefore ${\mathcal F}^{\alpha_i}_N$ is an invariant subspace of ${\mathcal F}_N$. It follows that
\begin{equation}\label{e:liouvdecomp}
  \mathscr{L} = \bigoplus_{\alpha_i\in\alpha}\mathscr{L}^{\alpha_i}\,.
\end{equation}
where $\mathscr{L}$ is the generalized Liouville operator such that $\mathscr{L}^{\alpha_i}: {\mathcal F}^{\alpha_i}_N\mapsto{\mathcal F}^{\alpha_i}_N$ acts on each invariant subspace.

Almost every $\graph^{\alpha_i}$ (the only exception is $\alpha=(e/o,3,k)$ in \sref{s:case}) slices across the entire graph $\graph$, meaning that $\graph^{\alpha_i}$ describes tensor coefficients from $n$-spin vector subspace with all $n_{\mathrm{min}}\le n\le N$ (possibly $n$ is constrained to either even or odd values) where $n_{\mathrm{min}}$ is some smallest order spin vector space ${\mathcal S}_{n_{\mathrm{min}}}$ of that subspace ${\mathcal F}_N^{\alpha_i}$. As a result, the initial value problem \eref{e:iv} in each ${\mathcal F}^{\alpha_i}_N$ has to be solved to all orders in the expansion coefficients $f_n$ in general.

From \eref{e:liouvdecomp} it follows that dynamics a-priori depends on the initial conditions. For example, if the initial condition is such that the only non-zero components of $f$ are in one of the invariant subspaces ${\mathcal F}^{\alpha_i}_N$, then the properties of equilibration and the equilibrium state will be determined by, respectively, the spectrum and the null-space of $\mathscr{L}^{\alpha_i}$. Both these properties may depend on $\alpha_i$ and thus on the initial state.

\section{Conclusions}\label{s:conclusions}
Aspects of the dynamics of the anisotropic Curie-Weiss quantum Heisenberg model are addressed by studying the structure of its equations of motion. This study is based on the {BBGKY} theory and a special expansion of reduced density operators in a basis of Pauli operators, and it builds on a recent work where the coefficient {BBGKY} hierarchy \eref{e:bbgky2} was derived \cite{paskauskas2012a}.

This work was motivated by a special case of coupling constants \cite{paskauskas2012a}, where time dependence of a certain subset of spin-spin correlators could be determined analytically to all orders, allowing to determine the evolution of a class of observables. It was hoped that the method of solution could be extended to all observables, initial conditions, and more general coupling constants. As in became apparent that a special symmetry of the coefficient vector space was behind the analytic solution, a comprehensive study of symmetries of a general case was called for.

To examine the structure of the coefficient vector space, the {BBGKY} hierarchy was represented by a graph, whose vertices are the expansion coefficients, and edges are couplings in the {BBGKY} hierarchy. It was shown that, under certain conditions, this graph is disconnected. A disconnected graph implies that certain groups of expansion coefficients are causally unrelated. To each such group one can associate an invariant subspace of the underlying vector space. It therefore follows that the vector space is partitioned into invariant subspaces, and that this partition depends on the structure of the matrix of effective coupling constants. Three scenarios of partition were identified in addition to the most general case: If $\overline{J}$ is a diagonal anisotropic matrix, there are four invariant subspaces. If in addition $\overline{J}$ has one non-zero off-diagonal element, there are two invariant subspaces. In the most general case, the vector space is irreducible, which corresponds to $\overline{J}$ that is diagonally anisotropic and has two or three non-zero off-diagonal elements. Lastly, it was found that, if $\overline{J}$ is diagonal and isotropic, the number of invariant subspaces scales as $O(N)$ with the size of the system $N$.

In addition it was shown that, to make the evolution in the coefficient vector space unitary, it is necessary to scale the components of vectors by certain weight factors, which is equivalent to redefining the conventional inner product, where the products of vector components are multiplied by the squares of these factors.

This work provides several insights about symmetries of the Hilbert space of a simple quantum model. It is believed in this work that similar symmetries are present in a class of more realistic lattice spin models, where interactions among particles depend on the distance between them. It is also believed that these symmetries should be taken into consideration when low-dimensional approximations, such as kinetic theories, are constructed.

redpajama_set_name: RedPajamaArXiv


text:
Economic Information Daily says it enabled players to bet through online casinos in the Philippines and Cambodia
Macau's Suncity singled out by Chinese state media in online gambling report
Junket operators are middlemen who bring high rollers to play extending them credit and collecting on their debts. Suncity's operations in Macau include the Studio City (photo).
Macau | 07/10/2019
The state-backed report says the annual amount bet through online gambling in the mainland is more than one trillion yuan (USD 145 billion), equivalent to nearly twice the annual income of China's lottery. Macau's biggest junket operator stated it didn't operate any online gaming business and all its operations were permitted under local government regulations.
acau's biggest junket operator Suncity, which operates VIP gambling rooms around Asia, has raked in billions of dollars in online gaming and proxy betting, causing great harm to China's social-economic order, a state-backed report said this week.
The report by Economic Information Daily was the first time a Macau junket has been singled out for online gaming activity and comes as China faces slowing economic growth and seeks to keep a lid on capital outflows.
Gambling is illegal in mainland China apart from in the special administrative region of Macau where Chinese nationals are able to bet in casinos. Suncity enabled Chinese players to bet through online casinos in the Philippines and Cambodia and utilized underground banks to move capital out of the country. "The annual amount bet through online gambling in the mainland is more than one trillion yuan (USD 145 billion), equivalent to nearly twice the annual income of China's lottery," the report said.
Suncity said in a statement that it didn't operate any online gaming business and all its operations were permitted under local government regulations. Macau's gaming regulator said in an email to Reuters that online gambling in Macau is illegal and if there were violations of local laws and regulations, outside of the former Portuguese colony, it would affect the qualifications of the junket operators and the bureau would "take serious action."
Suncity's publicly listed entity in Hong Kong, Suncity Group Holdings, does not include its junket operations. Junket operators are middlemen who bring high rollers to play, extending them credit and collecting on their debts.
The group's other ventures include an equity stake in a $4 billion casino project in Vietnam together with VinaCapital, a Vietnamese investment management and real estate firm, and Hong Kong-based VMS Investment Group. The project is due to open this year.
Alvin Chau, the 45-year-old founder of Suncity, has grown the company from operating a high roller table in Wynn Macau's casino in 2007 to a sprawling conglomerate with businesses ranging from property to autos and thousands of employees. Chau in June became chairman of Summit Ascent Holdings, which controls a casino in Russia's Vladivostok, after Suncity increased its stake in the company to 30 percent.
Eager to shed the industry's shady image and avert scrutiny from regulators, particularly in the United States which alleges that many junkets have ties to organized crime and facilitate illicit money flows, companies like Suncity have tried to diversify into industries such as dining and film.
Suncity - Macau
Macau: Beijing-backed candidate could lead the award of new casino licenses

redpajama_set_name: RedPajamaCommonCrawl


text:
Review: Numark Electrowave Heaphones
Hollin Jones on Sep 18, 2013 in Review 0 comments
Aside from monitors, a good, comfy pair of cans can make all the difference when mixing and DJing. Hollin Jones tries a pair of Electrowaves by Numark on for size.
Numark's Electrowave headphones ($149.95) are designed primarily for DJs, but in truth, anyone making or playing music should invest in a decent pair of headphones. Even if you mix on great speakers you still need to know what your tracks sound like on "cans". That said, these cans do have some features aimed directly at DJs. For a start, they are double hinged so that they fold up really quite small, making them much easier to get into a bag or backpack alongside your laptop and audio interface. They feel solid and chunky but they're not heavy, and the extending arms that let you fit them to your head are firm, meaning they shouldn't slip off as you nod rhythmically to the tunes you're spinning.
Softly does it
The uppermost part of the headband is cushioned, and when wearing the headphones they certainly feel anchored on, but without pressing too much on your head. The sides do stand away from your head quite a lot when you wear them making a sort of arc shape, though this is more of an observation than a problem. The ear cups are soft too and finished in furred material, making them comfortable to wear for long periods. They are billed as isolating headphones and they certainly do block out a lot of external noise though not all at low volumes, partly because the cups are a little smaller than, say, a heftier pair of studio headphones. With the volume up however you won't notice much if any sound seeping in from outside.
Cable, not wireless
One interesting touch is that you get two cables : a mini jack and full size jack, rather than a single cable and an adaptor. Whichever you choose to use, it connects to one of the cups using a clever locking plug so that it can't be accidentally pulled out. I'd venture that the reason for the removable cable is that when you pack the headphones away, removing the cable makes them much more portable. They apparently have glow-in-the-dark sides too, though I couldn't actually see this working! There's a carry case included in the box to help prevent scuffs and scratches.
An easy drive
The headphones have a frequency response of 10 Hz to 22 kHz, a sensitivity of 100 +/- 3 dB and a maximum input power of 3000 mW, if all that stuff tips the scales for you. In practice, they sound great. Playing back some heavy dance and hip hop from Traktor DJ on an iPad, the sound was crisp and punchy with bags of weight and clarity. They seemed quite a lot louder than my studio BeyerDynamic DT550s used with the same iPad, probably because they are easier to drive using the iPad's circuitry. Connected to a pro audio interface, they offered similarly great results. They're maybe a little excitable for accurate studio mixing, but then in fairness that's not what they're really designed for. As DJ headphones they are comfortable, lightweight and deliver the required levels of volume and immediacy. If you're in the market for some affordable but serious new cans for performing, the Electrowaves are well worth checking out.
Web: http://www.numark.com/product/electrowave
Hollin Jones was classically trained as a piano player but found the lure of blues and jazz too much to resist. Graduating from bands to composition then production, he relishes the chance to play anything with keys. A sometime lecturer in videographics, music production and photography post production, Hollin has been a freelance w... Read More
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Kullu (Himachal Pradesh): A landslide in Kullu blocked the Manali-Leh highway for 14 hours, leaving people stranded. The Border Road Organization (BRO) started road clearance work late on Saturday. Speaking to media, a BRO worker, Veer Singh Bahadur, said they are trying to restore the problem at the earliest. "The road is blocked since yesterday. We worked yesterday till 9 pm and have been working here since 5 am," Bahadur said.


text:
Nov. 2014 Minutes
Official Minutes from the November 2014 Regents Meeting
New Mexico Institute of Mining and Technology
1:30 p.m. Friday, November 21, 2014
Embassy Suites Hotel, Sandia Ballroom 7
1000 Woodward Place NE
The New Mexico Tech Board of Regents convened at 1:40 p.m. with Board of Regents Chairman Richard N. Carpenter presiding. Other Board members present were Regent Jerry Armijo, Regent Debra Hicks, Regent Deborah Peacock, and Student Regent Israel Rodriguez-Rios. University administrators, staff, and guests who were also present included:
Daniel H. López
Warren Ostergren
Melissa Jaramillo-Fleming
Van Romero
Lonnie Marquez
Lala Garcia
Greer Price
John Meason
Richard Cervantes
Leyla Sedillo
Bill Stone
JoAnn Salome
Colleen Guengerich
Mona Torres
Tony Ortiz
Theresa Kappel
W.D. Peterson
Lorie Liebrock
Thomas Guengerich
Valerie Kimble
Proof of Meeting Notice. The meeting was called to order at 1:40 p.m. by Chairman Carpenter after the President affirmed that proper legal notice had been given.
Approval of the Agenda. The Chairman announced the addition of Item 16 and an Executive Session (Item 20), should the Board deem it appropriate. The Agenda then was approved unanimously on a motion by Regent Armijo and a second from Regent Peacock.
Minutes. The Minutes from the October 16, 2014 meeting were unanimously approved following a motion by Regent Peacock and a second by Regent Hicks.
Announcements.
Enrollment Updates. Mr. Ortiz reported that applications have increased significantly and a high percentage of these applications have included a tuition payment. When an application includes a tuition payment, it is very likely that the student will actually enroll at Tech. He went on to point out that he is impressed with the quality of many of these applicants. Recruiting efforts in the El Paso and Rio Grande Valley areas of Texas also have boosted minority applications by more than 400 over last year. Mr. Ortiz and his staff received praise for their successful efforts.
Update Funding Formula. The President informed Regents that after three months of focused effort, a steering committee of the N.M. Council of Presidents has come to a consensus regarding a revised formula for funding higher education in New Mexico, and has been able to convince all parties, including the Governor's Office and the Department of Finance and Administration (DFA), of its viability. Unfortunately, he added, there is no way to predict the amount of funding available to the state with the recent plunge in oil prices. President López pronounced it a very good formula, and predicted that as the economy improves, "New Mexico Tech stands to gain."
Report on Exit Conference of Annual Audit. Mr. Marquez reported that the University submitted results of the audit to the State Auditor's Office on Nov. 14 with no significant findings. An issue described as a misunderstanding with the auditors eventually was worked out. Chairman Carpenter cited findings repeated from last year, and urged officials to eliminate them. Mr. Marquez responded that the University is close to completing the migration to a new accounting system, which should resolve most of these repeat findings. The Chairman forwarded comments from the auditors on the talent and attentiveness of University staff who assisted them. Once approved by the state, the audit will be brought to the Board for review and acceptance. This will probably happen at the February meeting.
Sabbatical Leave. Dr. Ostergren informed Regents that sabbatical has been granted for the 2015 fall semester for Dr. Jun Zheng, Associate Professor of Computer Science and Engineering, who will spend the time on a computer security project. Chairman Carpenter expressed his appreciation in receiving biographical data on Dr. Zheng in advance.
Faculty Senate Report. Dr. Stone, new Dean of Arts and Sciences, reported on behalf of Dr. Raymond, citing the formation of several Faculty Senate (FS) committees that now have evolved into University committees. The FS may reactivate its faculty mentoring program with the hiring of new, tenure-track faculty; and the Faculty Development committee will try to host a workshop on research funding to include senior graduate students.
Other. The President reported that filming at Playas will continue until Dec. 19 on a Warner Brothers film featuring Batman versus Superman. In addition, Aerojet will be leaving the University's Research Park in March or April 2015, a move the University was not in a position to announce until recently. None of the 35 workers affected belong to New Mexico Tech; however, the University will try to absorb at least some and perhaps the balance of employees not moving with the company, if possible. Aerojet is consolidating its operations as contract work has decreased. President López expressed confidence the University would replace the million-dollar loss in revenue over time.
Dr. Romero announced good news regarding the MRO: An exclusive team agreement with the NRAO puts the University in a good position to win a five-year, $5 million USAF project. One requirement is a study fusing optical interferometry with radio interferometry. The contract will be announced in March 2016. Winning the contract would get the MRO to first imaging. President López cited Dr. Romero's "dogged determination to make this happen." Meanwhile, the University was notified on Nov. 5 of its selection to conduct an independent study of a recent leak at the WIPP site in Carlsbad under an $800,000 contract with the Department of Energy or the State Energy and Minerals Department. The University's task will be to process review – no health aspects are involved. In addition to chemists and other scientists, the project team includes a technical communicator, Dr. Julie Ford.
Dr. Romero updated Regents on a report given them at last summer's retreat regarding developing a bioengineering Ph.D. program by acquiring the National Center for Genome Resources (NCGR). One sticking point was the real property it occupies. Now that the NCGR has been able to secure a loan, the University has had discussions with numerous entities on bringing the company under our Research Park. As such, overhead would come to New Mexico Tech, with no liability ties to the property. While the outlook is favorable, Dr. Romero said the University would "proceed with caution." Dr. Stone added that, in general, faculty response to adding a new Ph.D. has been favorable.
The President outlined the history of the University's Investment Policy as one based on "the prudent man rule" which addresses both potential opportunities and risk in handling public monies. Both the President and Foundation member W.D. Peterson provided details. Regents raised several issues during discussion, including comparable returns and periodic reporting to the Board, both of which Mr. Marquez will research. Mr. Peterson noted that a policy-writing committee is working to clean up and clarify Foundation procedures, and is attempting to work with the academic community regarding intellectual property. President López assured Regents he would bring to them a listing of board members for the Research Park Corp. and the Research Foundation.
Public Comment. None
Degree Conferrals for October 2014. Two Master of Science degrees were conferred during the month of October.
Report on Fundraising Activities. Ms. Guengerich reported the results of the third Annual Campaign, noting that donations are well ahead of prior years. Proceeds from the annual President's Golf Tournament were 8 percent above those from 2013, and attendance at 49ers also was up over previous years, bringing longtime alumni to campus for a series of Tech Talks and other activities. The Office for Advancement will offer major-gifts training to faculty with a workshop on Feb. 4, 2015; and is considering a campaign for faculty donations, an initiative Chairman Carpenter suggested should come from the Faculty Senate, to which the President agreed. Ms. Guengerich addressed other measures and potential donors, including a proposal to Intel to fund mini-courses as a recruitment tool. Regent Peacock pronounced Advancement efforts "something to celebrate," and noted that year-to-date data would be helpful.
Financial Analysis for October 2014. Mr. Marquez reported revenue 3 percent ahead of last year. Overhead and tuition/fees are up; expenses are behind those of last year. All research divisions are within budget, and the Playas deficit is receding. Regent Peacock moved to approve the report, with a second from Regent Armijo. The motion passed unanimously.
Quarterly Financial Ending September 2014. Mr. Marquez asked for Board approval for the quarterly report, noting that it reflected a reconciliation of budget numbers from fund accounting to GASB. The request was unanimously approved on a motion by Regent Armijo and a second by Regent Peacock.
Establish a Bachelor of Science Degree in Biomedical Science. Dr. Ostergren submitted for Board approval a new degree offering in biomedical science described as an interdisciplinary study involving 10 academic departments. He noted that Dr. Stone provided the primary leadership in program development. Dr. Stone said students could choose among three emphases in Biology, Chemistry and Psychology, for 12 possible degrees. During discussion, it was noted that new Living/Learning Communities could be developed in support of the new offering; and developing a full biomedical program could be daunting under ABET requirements. Regent Rodriguez-Rios moved for program approval. Regent Peacock seconded the motion, and it passed unanimously.
Emeritus Status Request. The President requested Board approval to grant emeritus status to Professor Andrew Campbell, who retired in 2013 after 30 years with the E&ES Department, and continues as a student advisor. Regent Armijo moved for approval. Regent Rodriguez-Rios seconded the motion, and it passed unanimously.
Unrestricted Fund Purchase. The President brought forward a request for $380,654 on behalf of the N.M. Bureau of Geology. Director Price told Regents that professional services are required to move the contents of the Bureau and its 60 employees, which includes five laboratories and millions of dollars in equipment. The award to A1 Freeman Moving Group includes transferring the 50,000 pounds of cuttings bound for the Research Office Building near EMRTC. A crane will provide access to the Bureau's map collection. Mr. Price noted that the move will take 55 days or longer, and costs will be paid out of a reserve fund started years ago. Regent Armijo moved to approve the award. Regent Rodriguez-Rios seconded the motion, and it passed unanimously.
Adoption of Strategic Planning Mission Statement. President López, stating he preferred the previous version, but noting his respect for the Strategic Planning Committee he put into place, as brought forward for Board approval the revised Mission Statement. Regent Rodriguez-Rios moved for approval. Regent Armijo seconded the motion and then opened discussion. The Socorro Regent wanted the word "diversified" added to the third line in referring to the University's student body. "We need to make that clear in our Mission," he said. Regent Armijo then asked that Regent Rodriguez-Rios amend his motion to reflect the change. It was agreed. When brought forward for a vote, the motion as amended passed unanimously.
Property Resolutions. Mr. Marquez presented for Board approval the following:
Resolution 141121-01, to approve removal of property items under $5,000. Regent Peacock moved for approval. Regent Armijo seconded the motion and it passed unanimously.
Resolution 141121-02, to approve removal of property items over $5,000. Regent Armijo moved for approval. Regent Peacock seconded the motion and it passed unanimously.
Naming of the Bureau of Geology Building. President López requested Board approval to name the new Bureau of Geology complex The Charles "Chuck" F. Headen Center. Regent Armijo noted that Mr. Headen's late wife Jessie should be included in the name. To accommodate the additional letters, Regent Rodriguez-Rios suggested that "Chuck" be omitted. Regent Armijo then moved to name the facility The Charles and Jessie Headen Center. Regent Peacock seconded the motion, and it passed unanimously.
NM State Mine Inspector Nomination/Recommendation. The President said it is a practice to nominate an individual to serve as State Mine Inspector, and the University must abide by what the State has decreed. He noted that the nominee is a miner with background and training in the field. Regent Peacock recommended the nomination of Terrence Foreback. Regent Armijo seconded the motion, and it passed unanimously.
Individual Board Comments. Regent Rodriguez-Rios, whose term ends on Dec. 31, 2014, said his experience on the Board was fantastic, and wished happy holidays to all. (The terms of Regents Armijo and Carpenter also expire on Dec. 31, and all three were recognized at a luncheon preceding the meeting.) Regent Armijo reflected over his 12 years on the Board, citing the strength of the University to continually adapt to changes and challenges. He noted the many personnel changes over the past dozen years, including the loss of faculty; and felt comfortable leaving the University "in as good a position as it was 12 years ago." Regent Armijo will remain as a member of the Research Foundation Board of Directors. Chairman Carpenter, who joined the Board with Armijo, said he enjoyed his tenure, appreciated the assistance given him and thanked his Board colleagues. Regent Peacock thanked the Chairman for his "incredible leadership" and called working with "an amazing Board" a fantastic experience. She led a standing ovation for the departing Regents.
New Business. The President recognized Dr. Liebrock, who reported on Graduate Studies. Fall 2015 enrollment is up 15 percent over 2014, and the office is negotiating details of an MOU to provide funding for graduate research assistants as part of a two-year program. She noted that the program is recruiting Central American students seeking graduate degrees in STEM fields. In other business, Ms. Salome reported that the Employee Manual has been revised with some policy modifications and additions. The document will be posted for internal comments in January 2015.
Employee Benefit Trust. The Board recessed at 3:35 p.m. to convene as the Benefit Trust. Mr. Marquez thanked Regents for their guidance over the year. He reported a deficit due to several large claims. The recent Benefits Fair was successful, he added. New premium rates will take effect Jan. 1, 2015. The Trust recessed at 3:36 p.m. to reconvene as the Board of Regents.Possible Executive Session, NMSA 1978 Section 10-15-1 (H) (2); (If Required to Consider Legal and/or Personnel Matters). Chairman Carpenter announced a request to discuss potential pending personnel claims and lawsuits within the purview of the Open Meetings Act (OMA). Regent Armijo moved to so meet, followed by a second by Regent Peacock. The Board recessed at 3:37 p.m. following a roll-call vote.
Reconvene in Open Session and Report on Final Action, if any, on Such Limited Personnel Matters in Open Session Following Conclusion of the Closed Session. Chairman Carpenter reconvened the regular meeting at 4:15 p.m. and announced that discussion related only to matters within the purview of the OMA, and no action was taken.
Adjournment. The meeting was adjourned by acclamation at 4:16 p.m.