text:
\section{Introduction}
The Wigner distribution, introduced by Wigner in 1932 \cite{1} for the purpose of associating a phase space distribution to quantum systems with $\mathbb{R}^N$ as the configuration space, has, over the years, been extended and generalised in many directions
 \cite{2}-\cite{5}, differing
in the aspects of the orginal definition that one chooses to retain. Among these efforts, those aimed at setting up Wigner distributions for finite state systems have, of late, been of great interest because of their potential relevance to quantum information theory
\cite{7}. For the case of quantum systems with Hilbert space dimension $N$ equal to power of a prime, Gibbons et al \cite{5}  developed a formulation for Wigner distributions which elegantly ties up Wigner distributions or equivalently the phase point operators with mutually unbiased bases    \cite{8}, \cite{9} available in such dimensions. This approach yields not one but many possible definitions of phase point operators and hence of Wigner distributions and Gibbons et al propose a scheme for classifying them on the basis of the action of the Clifford group.  In the present work we examine
this classification,
taking the eigenvalue spectra of the phase point operators as the sole criterion. We highlight the role which the extended Clifford group plays in this context. In other words,
in line with the  observations in  \cite{10} in the context of SIC POVMS \cite{11}, we argue
that, in so far as the spectra of the phase point operators is concerned, what is relevant is not the Clifford Group, but the extended Clifford group. This has the consequence that the number of 'similarity' classes in the sense of spectra is
smaller than that given by the Clifford group action alone---for $N=7$ it is $210$ as opposed to $360$.

A brief summary of this work is as follows: In Section II we
recapitulate the ideas developed by Gibbons et al leading to the definition of generalised Wigner distributions.
We restrict ourselves to $N$ an odd prime, and set up the notation and terminology used later. In Section III,  we develop a convenient scheme for labelling the phase point operators by elements of $V_{N+1}(\mathbb{Z}_N)$, an $N+1$ dimensional vector space over the finite field $\mathbb{Z}_N$, and transcribe the actions of the Heisenberg-Weyl (H-W) group and the Clifford group in terms of linear transformations on $V_{N+1}(\mathbb{Z}_N)$. By a suitable choice of basis in $V_{N+1}(\mathbb{Z}_N)$, the scheme developed here is shown to yield a convenient way of labelling the `affine planes', sets of phase point operators related to each other by the H-W group action, as elements of $V_{N-1}(\mathbb{Z}_N)$. This machinery is put to use in Section IV to study the action of the Clifford group on the set of phase point operators and affine planes with the purpose of classifying them into orbits under the Clifford group. In Section V, we examine the spectra of the phase point operators belonging to distinct orbits and find that several orbits are degenerate in the spectral sense, suggesting the presence of an extra symmetry. In Section VI we
show that the extra symmetry is related to the operation of complex conjugation, and that the number of distinct spectra of phase point operators can be quantitatively understood in terms of the  Extended Clifford Group - the Clifford group augmented by the operation of complex conjugation. This section contains our principal results. We conclude in Section VI with a summary
and an outlook. The appendix contains a derivation of the conjugacy classes of the group
$\ESL(2,\mathbb{F}_N)$, where $\mathbb{F}_N$ is a finite field.

\section{Generalised Wigner distributions}

We begin by recalling the salient features of the framework laid out by Gibbons et al for setting up discrete phase spaces and defining Wigner distributions thereon though not necessarily in the same logical sequence. Their general framework applies to quantum systems of prime power dimensions, but we
confine ourselves to the odd prime case.
\newcounter{spacerule}
\begin{list}{\emph{\arabic{spacerule}.}}{\setlength{\leftmargin}{.5 in}\setlength{\rightmargin}{.5 in}\usecounter{spacerule}}
\item \emph{Hilbert Space}:
Consider the Hilbert space $\mathcal{H}_N$ associated with an $N$-level quantum system and introduce in it an orthonormal basis $\{|q\rangle \colon q\in\mathbb{Z}_N \}$, referred to as the coordinate or the standard basis, and a momentum basis $\{|p)\colon p\in\mathbb{Z}_N \}$ related to it by a finite Fourier transformation:
\begin{equation}
|p)=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}~\omega^{pq}|i\rangle  ;~~ \omega = e^{2\pi i/N}.
\end{equation}
\item{{\it Phase Space}:} With the $N$-level quantum system we associate a  phase space consisting of points $(q,p)$ with $q,p\in\mathbb{Z}_N$, for convenience arranged in the usual Cartesian fashion. The fact that $ \mathbb{Z}_N$ is a field has the consequence that the phase space can be decomposed into $N$ parallel lines, striations, in $N+1$ ways. For each striation, we can select the line through the origin as  its representative and call it a `ray'.
We thus have $N+1$ rays where each ray consists of points of the form $(sq,sp)$ with $q,p$ fixed, not both zero and $s$ taking all values in $\mathbb{Z}_N$. For obvious reasons, the two special rays $(s,0)$ and  $(0,s)$ are denoted as horizontal and  vertical rays respectively and the corresponding striations as horizontal and vertical striations. We arrange the striations with these two as the extremes:
\begin{equation}
{\rm vertical},\cdots \cdots\cdots \cdots,{\rm horizontal}.
\end{equation}
\item\emph{Transformations on Phase Space}:  Two kinds of linear operations on the phase space which map the phase space to itself naturally arise:
\begin{itemize}
\item Translations, $T(q_0,p_0)$, $N^2$ in number, have the effect of moving a phase point by an amount $(q_0,p_0)$. These operations constitute an abelian group and map a line in one striation to another in the same striation. Thus, for instance, starting from a ray, except the horizontal, all other members of the corresponding striation can be reached through $T(1,0)$. For the horizontal ray, one can achieve this through $T(0,1)$.
\item `Symplectic Rotations'
       \begin{equation} \left( \begin{array}{c} q^\prime  \\ p^\prime \end{array} \right) = \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right) \left( \begin{array}{c} q  \\ p \end{array} \right) ,
\end{equation}
with $\alpha,\beta,\gamma,\delta \in \mathbb{Z}_N$ and $\alpha\delta-\beta\gamma=1~~({\rm mod}~ N)$. Such $2\times 2$ matrices $S$ constitute the group $\SL(2,\mathbb{Z}_N)$, containing $N(N^2-1)$ elements.  Its generators are
\begin{equation}
g_1 = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right), \hspace{12mm}
g_2 = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)
\ .
\label{4}
\end{equation}
These transformations map rays onto rays, hence the figurative appellation `rotations'.
'Symplectic' means that when applied to $(q_1,p_1)$,$(q_2,p_2)$ they leave the symplectic product $(q_2p_1-q_1p_2)$ unchanged.
\end{itemize}
The semi-direct product $\SL(2,\mathbb{Z}_N)\ltimes T(q_0,p_0)$
will be of interest here.

\item\emph{Heisenberg-Weyl group}:
Returning to the Hilbert space we define the operators $X$ and $Z$ on $\mathcal{H}_N$ through their actions on the coordinate basis
$|q\rangle  $ (momentum basis $|p)$) as :
\begin{eqnarray}
X|q\rangle = |q+1\rangle && \hspace{5mm} (~~ X|p) = \omega^{-p}|p)~~),\\
Z|q\rangle = \omega^{q}|q\rangle   &&\hspace{5mm} (~~ Z|p) = |p+1)~~).
\end{eqnarray}
Consider the set of $N^2$ operators
\begin{equation} D(q,p) = \tau^{qp} X^qZ^p \ , \end{equation}
where $\tau=-e^{i\pi/N}$. These operators, called displacement operators, obey
\begin{eqnarray}
D(q,p)^\dagger&=&D(-q,-p),\\
D(q,p)D(q^\prime,p^\prime) &=& \tau^{(q^\prime p - qp^\prime)}D(q+q^\prime, p+p^\prime),\\
Tr[D(q,p)]&=&N\delta_{q,0}\delta_{p,0},\\
Tr[D^\dagger(q,p)D(q^\prime,p^\prime)]&=&N\delta_{q,q^\prime}\delta_{p,p^\prime}.
\end{eqnarray}

\noindent This set of unitary operators forms an orthonormal basis in the space
of all complex operators on $\mathcal{H}_N$ and furnishes a projective representation of the group of translations on the phase space. In the standard basis, they can be represented by
matrices as
\begin{equation} [D(q,p)]_{\ell k} = \tau^{qp}\omega^{kp}\delta_{\ell,k+q} \ . \end{equation}
In particular,
\begin{equation} [D(0,1)]_{\ell k}=Z = \omega^{kp}\delta_{\ell,k}; \ [D(1,0)]_{\ell k}=X = \delta_{\ell,k+1} \ . \end{equation}
Further $\{e^{i\phi}D(q,p)\}$ constitute the elements of the Weyl-Heisenberg group
(or the Pauli Group).

\item\emph{The Clifford group}:
\noindent The Clifford group consists of all unitary matrices
$U$ such that
\begin{equation} UD(q,p)U^\dagger = e^{i\theta_{q,p}}D(f(q,p), g(q,p)), \ . \end{equation}
for suitable phases $e^{\theta_{q,p}}$ and functions $f,g$ It is a semi-direct product of the Weyl-Heisenberg group (playing the role of translations) with the group
of $N$ dimensional unitary matrices $U(F)$ labelled by elements
\begin{equation}
F=\left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right);~~ {\rm det}F=1~~ {\rm mod}~ N,
\end{equation}
of $\SL(2,\mathbb{Z}_N)$. The unitaries $U(F)$, defined up to a phase, act on $D(q,p)$
as follows:
\begin{equation} U(F)D(q,p)U^\dagger(F) = D(\alpha q+\beta p, \gamma q+\delta p) ,
\label{16}
\end{equation}
and, up to a phase, are explicitly given by \cite{AF}
\begin{equation}
U(F)= \begin{cases}
\frac{1}{\sqrt{N}}\sum_{j,k}\tau^{\beta^{-1}(\alpha k^2-2jk+\delta j^2)}|j\rangle \langle k| \qquad & \beta\neq 0,\\
\sum_{j}\tau^{\alpha \gamma j^2}|\alpha j\rangle \langle j|\qquad & \beta= 0.
\end{cases}
\label{16a}
\end{equation}
We shall denote the unitaries corresponding to $g_1$ and $g_2$ by  $V$ and $W$. They generate all the $U$'s.
\item\emph{Mutually Unbiased Bases (MUB)}:
From the properties of the displacement operators listed above it is evident that two such operators
$D(q,p),D(q^\prime,p^\prime)$ commute with each other if and only if the `symplectic product'
$(q^\prime p-q p^\prime)$ of $(q,p)$ and $(q^\prime,p^\prime)$ vanishes. This in turn implies that the operators $D(q,p)$ with $(q,p)$ lying on a ray commute with each other and hence their simultaneous eigenvectors furnish an orthonormal basis. Moreover, the orthonormal bases corresponding to different rays can be shown to be mutually unbiased with respect to each other \cite{5}. Since there are $N+1$ rays we are naturally led to the full set of $N+1$ mutually unbiased bases. This association between rays and MUBs is a key ingredient in the construction of generalised Wigner distributions as we shall see later. As to explicit expressions for the MUBs there are many choices depending on the choice of phases for the vectors.
We find the following choice, with the MUB's arranged in a specific order, to be
convenient \cite{AF}:
$$M_0, M_1, \cdots, M_N,$$
with
 \begin{equation} M_m  =\begin{cases} \{|m,r\rangle  \colon r=0,\cdots,N-1\}\qquad & m=0,\cdots,N-1\\
  \{|\infty,r\rangle  \colon r=0,\cdots,N-1\} \qquad & m=N
  \end{cases}
\end{equation}
and
\begin{equation}
|m,r\rangle   = \begin{cases}
V^m|r\rangle  \qquad & m=0,\cdots,N-1\\  W|r\rangle  \qquad & m = \infty
\end{cases}
\end{equation}
where $V$ and $W$ denote unitaries corresponding to $g_1$ and $g_2$ respectively.
For the sets obtained by replacing each vector in the ordered sets  $ M_0, M_1, \cdots, M_N$ by its projector we will use the notation
$$
\mathcal{M}_0, \mathcal{M}_1, \cdots, \mathcal{M}_N.$$
Explicit expressions for the projectors are given below \cite{AF}:
\begin{equation}
|m,r\rangle \langle m,r|=
\begin{cases}\frac{1}{N}\sum_{j}\omega^{-rj} D(mj,j)\qquad &m\neq \infty,
\\
\frac{1}{N}\sum_{j}\omega^{-rj}D(j,0) \qquad &m=\infty.
\end{cases}
\label{21}
\end{equation}The MUB's at the extremes can clearly be identified as those associated with vertical and horizontal rays respectively. Again, as with striations, the vectors within each MUB, except for the last one, are related to each other (up to phases) by the action of $D(1,0)$. Those in the last one are related to each other (up to phases) by $D(0,1)$:
\begin{align}
D(1,0)\colon|m,r\rangle & \to  \begin{cases}|m,r+1\rangle \qquad & m\neq \infty
\\
|m,r\rangle \qquad & m =\infty,
\end{cases}
\label{22}
\\
D(0,1)\colon |m,r\rangle & \to
\begin{cases}|m,r-m\rangle,\qquad & m\neq \infty,
\\
 |m,r-1\rangle \qquad & m = \infty
\end{cases}
\label{23}
\end{align}

\item\emph{Quantum Nets}: We have seen that each ray has a MUB associated with it. We have also seen that the passage from one to another member of the striation by translation, has a complete correspondence with the passage from one vector in the MUB to another through appropriate displacement operators. We now assign a specific vector (more precisely the corresponding projector) in the MUB to each ray and assign other vectors in the MUB to the other parallel lines parallel to the ray using the correspondence mentioned above. This assignment of vectors to lines in the phase space consistent with translations leads to a Quantum Net. Clearly, as there are $N+1$ rays and for each ray $N$ vectors to choose from the corresponding MUB's, there are  $N^{N+1}$ different ways of assigning vectors to the lines in phase space and hence $N^{N+1}$ distinct quantum nets. Now given a quantum net we can generate another unitarily equivalent by applying one of the $N^2$ displacement operators to all the vectors. This then means that we can divide $N^{N+1}$ quantum nets into $N^{N-1}$ equivalence classes containing   $N^2$ nets each.
\item\emph{Phase Point Operators}: Having assigned a vector to each line in the phase space we are now in a position to assign a Hermitian operator $\hat{A}(q,p)$ to each point $(q,p)$ in the phase space by adding up all the projectors $P_\lambda$ associated with the $N+1$ lines $\lambda$ passing through that point and subtracting the Identity operator from the sum to make its trace equal 1:
\begin{equation}
\widehat{A}(q,p) = \sum_{\lambda} P_\lambda - \mathbb{I}~~~~;~~~~ {\rm Tr}\widehat{A}(q,p)=1.
\end{equation}
One can invert this relation to obtain the projectors associated with a line $\lambda$ in terms of the phase point operators \begin{equation}
P_\lambda = \frac{1}{N}\sum_{(q,p)\in \lambda}\widehat{A}(q,p).
\end{equation}
Further using the mutually unbiasedness of the projectors one obtains:
\begin{equation}
{\rm Tr}[\widehat{A}(q,p)\widehat{A}(q^\prime,p^\prime)] = N\delta_{q,q^\prime}\delta_{p,p^\prime}.
\end{equation}
From the above discussion it is clear that the totality $\{\widehat{A}\}$ of $N^{N+1}$ phase point operators are obtained by picking one vector from each of the MUBs in all possible ways.

\item\emph{Affine Planes}: Given the phase point operator at say $\widehat{A}(0,0)$ we can generate $N^2$ operators $\widehat{A}(q,p)$ by applying all the $N^2$ displacement operators to it. Such a collection is taken to define an affine plane.  We thus have the following correspondences between the structures at the phase space level and those at the level of the Hilbert space:
\begin{eqnarray}
{\rm Points}~~ (q,p)&\longleftrightarrow& {\rm Phase ~point ~operators}~~\widehat{A}(q,p),\nonumber\\
{\rm Lines}~~ \lambda &\longleftrightarrow& {\rm Projectors},\\
{\rm Phase~ Space}~~  &\longleftrightarrow& {\rm Affine~ Plane}.  \nonumber
\end{eqnarray}
The full set of $N^{N-1}$ affine planes can thus be obtained by dividing $N^{N+1}$ $\widehat{A}$'s into subsets, related to each other by
the Heisenberg-Weyl group.

\item\emph{Wigner Distributions}:
 With the phase point operators at hand one can define Wigner distribution corresponding to a given density operator $\rho$ as
\begin{equation}
W_{\rho}(q,p)={\rm Tr}[\rho\widehat{A}(q,p)].
\end{equation}
which, among others, have the desired marginals property:
\begin{equation}
 \frac{1}{N}\sum_{(q,p)\in \lambda}W_{\rho}(q,p)={\rm Tr}[\rho P_\lambda].
\end{equation}
i.e. a Wigner distribution averaged over a line gives a probability.
\end{list}
\noindent
\section{A convenient labelling scheme for phase point operators and affine planes}
As discussed earlier, a selection of projectors, one from each $ \mathcal{M}_0, \mathcal{M}_1, \cdots, \mathcal{M}_N$, defines a phase point operator $\widehat{A}$. We can therefore label a phase point operator uniquely
by an array $(r_0,r_1,\cdots,r_N)$ :
$$\widehat{A}\leftrightarrow (r_0,r_1,\cdots,r_N).$$
This permits us to think of each phase point operator as a vector in $V_{N+1}(\mathbb{Z}_N)$, an $N+1$ dimensional vector space over $Z_N$. We now examine how they transform under the Clifford group. To this end we look at the action of the generators   $D(1,0)$ and $D(0,1)$ of the Heisenberg-Weyl group
as well as the action of the symplectic group.

From $(\ref{22})$ and $(\ref{23})$ it follows that :
\begin{eqnarray}
D(1,0) &:& (r_0,r_1,\cdots,r_N) \rightarrow (r_0,r_1,\cdots,r_N) + (1,1,\cdots,1,0), \\
 D(0,1) &:&(r_0,r_1,\cdots,r_N) \rightarrow (r_0,r_1,\cdots,r_N) - (0,1,\cdots,N-1,1).
\end{eqnarray}

Further, using $(\ref{16})$ and $(\ref{21})$ it is straightforward (though somewhat tedious) to confirm that an arbitrary $U(F)$ given by $(\ref{16a})$ acts on $|m,r\rangle  \langle m,r|$ as follows:
\begin{equation}
U(F) \colon |m,r\rangle\langle m,r|  \to
\begin{cases}
 \left|\frac{\alpha m+\beta}{\gamma m +\delta},\frac{r}{\gamma m +\delta}\right\rangle  \left\langle \frac{\alpha m+\beta}{\gamma m +\delta},\frac{r}{\gamma m +\delta}\right| \quad &\text{$m\neq \infty$, $\gamma m +\delta\neq 0$}
\\ \vphantom{\Bigl<}
|\infty,-\gamma r\rangle\langle \infty,-\gamma r| \quad & \text{$m\neq \infty$, $\gamma m+\delta = 0$}
\\
\left|\frac{\alpha}{\gamma},\frac{r}{\gamma}\right\rangle  \left\langle \frac{\alpha}{\gamma},\frac{r}{\gamma}\right| \quad & \text{$m=\infty$, $\gamma\neq 0$}
\\ \vphantom{\Bigl<}
|\infty,\delta r \rangle \langle \infty,\delta r| \quad &\text{$m=\infty$, $\gamma = 0$}.
\end{cases}
\label{e32}
\end{equation}
On $r_m$ these translate into
\begin{equation}
r_m \to
\begin{cases}
-(\gamma m-\alpha)r_{\frac{\beta-\delta m}{\gamma m-\alpha}} \qquad & \text{$m\neq N$, $\gamma m-\alpha \neq 0$}
\\
\frac{1}{\gamma}r_N \qquad & \text{$m\neq N$, $\gamma m-\alpha =0$}
\\
 -\gamma r_{\frac{-\delta}{\gamma}} \qquad &  \text{$m= N$, $\gamma\neq 0$}
 \\
\frac{1}{\alpha}r_N \qquad &  \text{$m= N$, $\gamma = 0$}
\end{cases}
\end{equation}
On the set of MUBs the action is that of a M\"obius transformation. In fact the set of MUBs
can be regarded as a projective line at infinity, added to the affine plane.

These actions can be written out as matrices acting on $\mathbf{r}\equiv (r_0,\cdots,r_N)$. Thus, for instance, for $N=3$ we find for the generators of $\SL(2, \mathbb{Z}_3)$
\begin{align}
V\colon \mathbf{r} \rightarrow \mathbf{r}^\prime &=\mathcal{U}(g_1)\mathbf{r},\\
W\colon \mathbf{r}\rightarrow \mathbf{r}^\prime &=\mathcal{U}(g_2)\mathbf{r},
\end{align}
where
\begin{equation}
\mathcal{U}(g_1)=\begin{pmatrix}0&0&1&0\\1&0&0&0\\0&1&0&0\\0&0&0&1\end{pmatrix}; \qquad
\mathcal{U}(g_2)=\begin{pmatrix}0&0&0&2\\0&0&1&0\\0&2&0&0\\1&0&0&0\end{pmatrix}.
\end{equation}

\noindent Note that these matrices have determinant equal to one and hence belong to
the group $\SL(N+1,\mathbb{Z}_N)$.

Next we develop a similar notation for the affine planes.
As discussed earlier, the set of phase point operators obtained by applying all elements of the Weyl-Heisenberg group to a fixed phase point operator defines an affine plane containing that phase point operator. This operation decomposes the set of $N^{N+1}$ phase point operator into $N^{N-1}$ disjoint subsets---the affine planes, containing $N^2$ points each. To
develop a useful representation for the affine planes as cosets in  $V_{N+1}(\mathbb{Z}_N)$ we
notice that the action of the Weyl-Heisenberg group on the phase point operators consists in adding vectors
which are linear combinations of $e_0$ and $e_1$,
where
\begin{eqnarray}
e_0&=& (1,1,\cdots,1,0), \\
e_{1}&=& (0,1,2,\cdots, N-1,1).
\end{eqnarray}
Further, under the action of $V$ and $W$ these transform into each other:
\begin{eqnarray}
V &:&  e_0\rightarrow e_0,~~ e_{1} \rightarrow e_1-e_0,  \\
W &:&  e_0\rightarrow e_1,~~ e_{1} \rightarrow -e_0.
\end{eqnarray}

It therefore proves convenient to choose a basis which contains these two. Any choice would do. We choose the remaining ones as:
\begin{eqnarray}
e_k&=& (0,1^k,2^k,\cdots,(N-1)^k,0);~~k=2,\cdots, N-1,  \nonumber
\\
e_{N}&=& (0,0,0,\cdots,0,1).
\end{eqnarray}
That this set of vectors is indeed a linearly independent set can easily be checked
using the properties of Van der Monde determinants.
We shall denote the components of a vector $(r_0,r_1,\cdots,r_N)$ in the e-basis as
 $[\alpha_0,\alpha_1,\cdots,\alpha_N]$ and the invertible matrix relating the two by $S$:
\begin{equation}
{\bf r}=S{ \boldsymbol{\alpha}}; \ S=(e_{0}^T, e_{1}^T,\cdots,e_{N}^T).
\label{S}
\end{equation}
Under the action of the Weyl-Heisenberg group, the components in the e-basis have rather simple properties
\begin{eqnarray} D(1,0) &:& [\alpha_0,\alpha_1,\cdots,\alpha_N] \rightarrow [\alpha_0+1,\alpha_1,\cdots,\cdots,\alpha_N], \label{65} \\
 D(0,1) &:& [\alpha_0,\alpha_1,\cdots,\alpha_N]\rightarrow [\alpha_0,\alpha_1-1,\cdots,\alpha_N].\label{66}
\end{eqnarray}

\noindent Thus the collection $[\alpha_0,\alpha_1,\cdots,\alpha_N]$, with $\alpha_0$
and $\alpha_1$ taking all values in $\mathbb{Z}_N$ with the rest fixed, defines an
affine plane. We label it by the $N-2$ coordinates
$[\alpha_2,\cdots,\alpha_{N-1},\alpha_N]$.

The action of $V$ and $W$ is
\begin{align}
V\colon \boldsymbol{\alpha}\rightarrow \boldsymbol{\alpha}^\prime &=S^{-1}\mathcal{U}(g_1){S}\boldsymbol{\alpha},\\
W\colon \boldsymbol{\alpha}\rightarrow \boldsymbol{\alpha}^\prime &=S^{-1}\mathcal{U}(g_2)S\boldsymbol{\alpha}.
\end{align}
where the matrix $S$ is given in $(\ref{S})$.

If one is only interested in the actions of $V$ and $W$ on the affine planes then it suffices to look at the actions of the matrices $S^{-1}\mathcal{U}(g_1)S$ and $S^{-1}\mathcal{U}(g_2)S$ on the $N-1$ $\alpha$'s omitting $\alpha_0$ and $\alpha_1$.
These matrices have determinant equal to one and hence belong to the group $\SL(N-1,\mathbb{Z}_N)$.

To summarise, we have a representation of the $\SL(2,\mathbb{Z}_N)$ action on the phase space through matrices belonging to $\SL(N+1,\mathbb{Z}_N)$ ( $\SL(N-1,\mathbb{Z}_N)$ ) acting on  an $N+1$ ($N-1$) dimensional vector space over $\mathbb{Z}_N$ whose elements are in one to one correspondence with the phase point operators (affine planes). A natural question to ask is whether or not one has an analogue of the symplectic form left invariant by $\SL(2,\mathbb{Z}_N)$ action on the phase space.
To pursue this question we look for bilinear quadratic forms of the type $\Omega({\bf r}^{\prime},{\bf r})\equiv{\bf r}^{\prime T}~\Omega~{\bf r}$, left invariant by the $U(g_1), U(g_2)$ action on the ${\bf r}$'s, i.e we look for matrices $\Omega$ such that
\begin{equation}
\mathcal{U}^T (g_1)\Omega \mathcal{U} (g_1)=\Omega;~~ \mathcal{U}^T (g_2)\Omega \mathcal{U} (g_2)=\Omega.
\end{equation}
The antisymmetric matrix $\Omega$ turns out to be
\begin{equation}
\Omega=  \left( \begin{array}{cc} {\bf q} & \mathbb{I} \\ -\mathbb{I}^T & 0 \end{array} \right),
\end{equation}
where $\mathbb{I}$ denotes an $N$-dimensional column with all entries equal to $1$ and
${\bf q}$ is an $N\times N$ antisymmetric matrix with non-vanishing matrix elements
\begin{equation}
q_{ij}= \frac{1}{j-i}.
\end{equation}
In the e-basis, the symplectic form $\Omega$ has a very simple structure:
\begin{equation}
\Omega(e_i,e_j)=\begin{cases}
0 \qquad & \text{$i$ or $j=0$ or $1$},\\
             j\delta_{i+j,0} \qquad & 2\leq i,j\leq N-2,\\
           -1 \qquad & \text{ $i=N-1$ and  $j=N$},
\end{cases}
\end{equation}
which in turn suggest a basis in which $\Omega$ has the canonical form. Thus,
 for instance, for $N=5$ the basis in which $\Omega$ has the canonical form
\begin{equation}
  \left( \begin{array}{cc} {\bf 0} & \mathbb{I} \\ -\mathbb{I} & {\bf 0}\end{array} \right),
\end{equation}
is $$e_2,e_4,\frac{1}{3}e_3,-e_5 \ .$$
Similarly for $N=7$ we have  $$e_2,e_4,e_6,\frac{1}{5}e_5,\frac{1}{3}e_3,-e_7 \ .$$

\section{Orbits under the Clifford Group}

Having learnt how the generators of the Clifford group act on phase point operators and affine planes, our next task is to
investigate how they arrange themselves into orbits under the action of this group.   As is well known, to generate an orbit of a group $\mathcal{G}$ acting on a set  we need to pick an element of the set and apply all elements of the group to it. We then pick another element in the set not contained in the earlier set and generate its orbit and so on until all elements of the set are exhausted. The quantities of interest are then (a) the number of orbits (b) the size of each orbit. The total number of orbits can be calculated using the Burnside Lemma:
\begin{equation}
{\rm Number~ of~ orbits}=\frac{1}{|\mathcal{G}|}\sum_{g\in \mathcal{G}} \phi(g),
\end{equation}
where $|\mathcal{G}|$ denotes the order of the group and $\phi(g)$ the number of points of the set left fixed by the action of $g$. As $g$ and the elements related to it by conjugation have the same fixed points, to calculate the number of orbits we need to know the number of elements in each conjugacy class and the number of  points left fixed by a representative in each class.
As to the size of the orbits, it is given by the ratio of the order of the group to the order of the stability group of the starting element of the set from which the orbit is built up by group action. Knowing the order of the subgroups, one can then deduce the possible sizes of the orbits. In summary, to answer the questions pertaining to the orbits, we need
\begin{enumerate}
\item knowledge of the conjugacy classes and the number of elements therein
\item expressions for the class representatives in terms of the generators
\item orders of the cyclic subgroups generated by of elements of a conjugacy class.
\end{enumerate}
In the following we collect together some relevant facts from the literature
\cite{12} concerning these aspects for the group $\SL(2,\mathbb{Z}_N)$ (also see the Appendix).

 In our discussion we will need to distinguish the sets
\begin{align}
Q &= \{x \in \mathbb{Z}^{*}_N:  x = y^2 \text{ for some } y\in\mathbb{Z}^{*}_N\} \\
\bar{Q} & = \{x\in\mathbb{Z}^{*}_N: x \notin Q\}
\end{align}
where $\mathbb{Z}^{*}_N$ is the set of non-zero elements of $\mathbb{Z}_N$ (so $Q$ is the set of quadratic residues and $\bar{Q}$ is the set of non-quadratic residues~\cite{Hardy}).

It is immediate that elements of $\SL(2,\mathbb{Z}_N)$ having different trace must belong to different conjugacy classes.  It turns out~\cite{12} that there is in fact exactly one conjugacy class for each value of the trace, except when the trace $=\pm 2$ in which case there are three.  This gives us $N+4$ conjugacy classes:  the $N$ ``standard'' classes
\begin{equation}
C_{t} = \left[ \begin{pmatrix} 0 & -1 \\ 1 & t\end{pmatrix} \right]
\end{equation}
with $t=0,1,\dots (N-1)$ and the four additional classes
\begin{equation}
\bar{C}_{\pm 2}= \left[ \begin{pmatrix} 0 & -\frac{1}{\nu}  \\ \nu & \pm 2 \end{pmatrix} \right] \qquad \text{and} \qquad D_{\pm 2} = \left[ \begin{pmatrix} \pm 1 & 0  \\ 0 & \pm 1 \end{pmatrix} \right]
\end{equation}
where $\nu$ is any fixed element of $\bar{Q}$.  In these expressions $[F]$ denotes the conjugacy class containing $F$.  For more details see Theorem~\ref{thm:SLConjClass} in the Appendix.

For the standard classes we have
\begin{equation}
|C_t| =
\begin{cases}
N(N+1) \qquad & \text{if $t^2-4 \in Q$}\\
N(N-1) \qquad & \text{if $t^2-4 \in \bar{Q}$}\\
\frac{1}{2} (N^2-1)\qquad & \text{if $t^2-4 = 0$}
\end{cases}
\end{equation}
where the notation $|S|$ means ``number of elements in the set $S$''.   For the additional classes we have
 $|\bar{C}_{\pm 2}| = (N^2-1)/2$ and $|D_{\pm 2}| =1$.  It may be worth noting that the classes $C_t$ for which $t^2-4\in Q$ are precisely the ones whose elements are diagonalizable (apart from the two classes $D_{\pm 2}$).

For trace $= 0, \pm 1, \pm 2$  it is easy to calculate the orders of the cyclic subgroups generated by the elements in each conjugacy class for arbitrary $N$.   They are listed in Table~\ref{tb:genOrd}
\begin{table}[h]
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{| c | c | c| c | c |c |c | c | c | c  | }\hline
\te{2}{class} &
  \te{0.7}{$C_0$ }&
  \te{0.7}{$C_1$ }&
  \te{0.7}{$C_{-1}$ }&
  \te{0.7}{$C_2$}&
  \te{0.7}{$\bar{C}_2$ }&
  \te{0.7}{$D_2$ }&
  \te{0.7}{$C_{-2}$}  &
  \te{0.7}{$\bar{C}_{-2}$} &
  \te{0.7}{$D_{-2}$}
  \\ \hline
\te{2}{order }&
  \te{0.7}{$4$ }&
  \te{0.7}{$6$ }&
  \te{0.7}{$3$ }&
 \te{0.7}{$N$} &
 \te{0.7}{$N$} &
 \te{0.7}{$1$} &
 \te{0.7}{$2 N$ }&
 \te{0.7}{$2 N$ }&
 \te{0.7}{$2 $ } \\
\hline
\end{tabular}
\vspace{0.2 cm}
\caption{The order of the cyclic subgroups generated by elements of some special
conjugacy classes, for all $N$.}
\label{tb:genOrd}
\end{center}
\end{table}
(to derive the result for elements of $C_{\pm 2}$, $\bar{C}_{\pm 2}$ consider matrices of the form $\left(\begin{smallmatrix} \pm 1 & k \\0& \pm 1\end{smallmatrix}\right)$).
For the remaining conjugacy classes the orders of the cyclic subgroups depend
on number theoretical details. We have worked out the orders for $N \leq 19$.  They are listed in Table~\ref{tb:ordNLs19}.
\begin{table}[h]
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{| c| c| c| c| c| c| c| c| c| c| c| c| c| c| c|} \hline
&\te{   0.43}{$ \te{   0.43}{$C_3$} $}&\te{   0.43}{$ C_4 $}&\te{   0.43}{$ C_5$}&\te{   0.43}{$C_6$}&\te{   0.43}{$C_7$}&\te{   0.43}{$C_8$}&\te{   0.43}{$C_9$}&\te{   0.43}{$C_{10}$}&\te{   0.43}{$C_{11}$}&\te{   0.43}{$C_{12}$}&\te{   0.43}{$C_{13}$}&\te{   0.43}{$C_{14}$}&\te{   0.43}{$C_{15}$}&\te{   0.43}{$C_{16}$}
\\
\hline
\te{1.2}{$N=7 $}&\te{   0.43}{$ 8 $}&\te{   0.43}{$ 8 $}&
&& && && && && &
\\
\hline
\te{1.2}{$N=11 $}&\te{   0.43}{$ 5 $}&\te{   0.43}{$ 10 $}&\te{   0.43}{$12 $}&\te{   0.43}{$12 $}&\te{   0.43}{$5 $}&\te{   0.43}{$10 $}&
&& && && &
\\
\hline
\te{1.2}{$N=13 $}&\te{   0.43}{$ 14 $}&\te{   0.43}{$ 12 $}&\te{   0.43}{$14 $}&\te{   0.43}{$14 $}&\te{   0.43}{$7 $}&\te{   0.43}{$7 $}&\te{   0.43}{$12 $}&\te{   0.43}{$7 $}&
&& && &
\\
 \hline
\te{1.2}{$N=17 $}&\te{   0.43}{$ 18 $}&\te{   0.43}{$18 $}&\te{   0.43}{$ 16$}&\te{   0.43}{$ 8$}&\te{   0.43}{$ 9$}&\te{   0.43}{$ 16$}&\te{   0.43}{$16 $}&\te{   0.43}{$18 $}&\te{   0.43}{$8 $}&\te{   0.43}{$16 $}&\te{   0.43}{$9 $}&\te{   0.43}{$ 9$}&
&
\\
\hline
\te{1.2}{$N=19 $}&\te{   0.43}{$ 9 $}&\te{   0.43}{$ 5 $}&\te{   0.43}{$10 $}&\te{   0.43}{$20 $}&\te{   0.43}{$9 $}&\te{   0.43}{$20 $}&\te{   0.43}{$9 $}&\te{   0.43}{$18 $}&\te{   0.43}{$20 $}&\te{   0.43}{$18 $}&\te{   0.43}{$20 $}&\te{   0.43}{$ 5 $}&\te{   0.43}{$10 $}&\te{   0.43}{$18$}
\\
\hline
\end{tabular}
\vspace{0.2 cm}
\caption{The order of the cyclic subgroups generated by elements of all the conjugacy
classes, for $N \leq 19$.}
\label{tb:ordNLs19}
\end{center}
\end{table}
Since we know how many elements there are in each conjugacy class, the number of distinct cyclic subgroups can easily be worked out.
In particular we find that there is a unique subgroup of order $2$, generated
by the element $- I \in D_{-2}$;  $N+1$ (Sylow) subgroups of order $N$, generated by
elements of $C_{2}$, $\bar{C}_{2}$; and $N+1$ subgroups of order $2N$ generated by elements
of $C_{-2}$, $\bar{C}_{-2}$.

We can now compute
quantities of interest concerning the orbits generated by Clifford group action on phase point operators and affine planes.   Using the results in the previous section we have done this for   $N=3,5, 7$.  The results are listed in Table~\ref{tb:fixedEtc}.
\begin{table}
\begin{center}
\begin{tabular}{| c| c| c| c| c| c| c|} \hline
\te{1.4}{N} & \te{1.4}{Class} & \te{1.4}{number of elements} & \te{1.4}{order of cyclic subgroup} & \te{1.4}{fixed points in plane} & \te{1.4}{fixed $A$s}& \te{1.4}{fixed planes}
\\ \hline
$3$ & $C_0$ & 6 & 4 & 1 & 1 & 1 \\
& $C_{1}$ & 4 & 6 & 1 & 1 & 1 \\
& $\bar{C}_{1}$ & 4 & 6 & 1 & 1 & 1 \\
& $D_{1}$ & 1 & 2 & 1 & 1 & 1 \\
& $C_2$ & 4 & 3 & 3 & 9 & 3 \\
& $\bar{C}_2$ & 4 & 3 & 3 & 9 & 3 \\
 & $D_2$ & $1$ & $1$ & $9$ & $81$ & $9$ \\
\hline
5 & $C_{0}$ & 30 & 4 & 1 & 1 & 1 \\
& $C_{1}$ & 20 & 6 & 1 & 1 & 1 \\
 & $C_2$ & 12 & 5 & 5 & 25 & 5 \\
& $\bar{C}_2$ & 12 & 5 & 5 & 25 & 5 \\
 & $D_2$ & 1 & 1 & 25 & 15625 & 625 \\
& $C_{3}$ & 12 & 10 & 1 & 1 & 1 \\
& $\bar{C}_{3}$ & 12 & 10 & 1 & 1 & 1 \\
& $D_{3}$ & 1 & 2 & 1 & 1 & 1 \\
& $C_{4}$ & 20 & 3 & 1 & 25 & 25 \\
\hline
7 & $C_{0}$ & 42 & 4 & 1 & 1 & 1 \\
& $C_{1}$ & 56 & 6 & 1 & 1 & 1 \\
& $C_2$ & 24 & 7 & 7 & $7^2$ & 7 \\
& $\bar{C}_2$ & 24 & 7 & 7 & $7^2$ & 7 \\
 & $D_2$ & 1 & 1 & $7^2$ & $7^8$ & $7^6$ \\
& $C_{3}$ & 42 & 8 & 1 & 1 & 1 \\
& $C_4$ & 42 & 8 & 1 & 1 & 1 \\
& $C_{5}$ & 24 & 14 & 1 & 1 & 1 \\
& $\bar{C}_{5}$ & 24 & 14 & 1 & 1 & 1 \\
& $D_{5}$ & 1 & 2 & 1 & 1 & 1 \\
& $C_6$ & 56 & 3 & 1 & $7^2$ & $7^2$ \\ \hline
\end{tabular}
\vspace{0.2 cm}
\caption{Fixed points under the action of the elements in $SL(2, Z_N)$, $N \leq 7$.
The action can be on an affine plane, on the set of phase point operators, and on
the set of affine planes.}
\label{tb:fixedEtc}
\end{center}
\end{table}
Some things can be said in general. Notably cyclic subgroups of even order have only one
fixed point. It appears that cyclic subgroups of an odd order $m$ always leave exactly
$N^k$ affine planes fixed, where $N + 1 = km + m'$, $m' <  m$. (We proved this for subgroups
of order 3, and checked it in all cases for $N \leq 19$.)

On the affine plane the number of orbits is always $2$. On the set of affine planes the number of orbits can be found using Burnside's lemma.  The results for $N\le 11$ are listed in Table~\ref{tb:noOrbits}.
\begin{table}[h]
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|c|} \hline
\te{2}{$N$} & \te{1.5}{$3$} & \te{1.5}{$5$} &\te{1.5}{$7$}&\te{1.5}{$11$}\\ \hline
No.\ of orbits & $2$ & $11$ &$360$& $19650810$\\ \hline
\end{tabular}
\vspace{0.2 cm}
\caption{Number of orbits of $SL(2, Z_N)$ when acting on the set of affine planes.}
\label{tb:noOrbits}
\end{center}
\end{table}
They should be compared with the estimate $N^{N-1}/N^3$ which for $N=5,7,11$ gives
$5,343,19487171$ respectively.
The estimate is rather good.
For $N=3$ the sizes of the orbits are $1+8=3^2$, and for $N = 5$ they are $1+24+40+40+40+40+40+40+120+120+120=5^4$. For $N=7$ the results are too numerous to be given here. As expected, each term in the sum is a divisor of the order $N(N^2-1)$ of the group.
We observe that for all $N$ there is a unique singlet, and a unique $(N^2-1)$-plet consisting
of affine planes left invariant by some Sylow subgroup..

\section{Spectra of Phase Point Operators, Complex Conjugation and the Extended Clifford Group}
We now turn to the spectra of the phase point operators. Obviously those in the same affine plane have the same spectra and hence we need to focus only on the affine planes. For
$N=3$ one finds two distinct spectra: $(1,1,-1)$ and $(\Phi,1-\Phi,0)$ where $\Phi=
(1+\sqrt{5})/2=1.61803..$, the Golden Ratio. No surprises here.

For $N=5$ one finds only 9 distinct spectra as against the naive expectation of 11, as listed in Table~\ref{tb:spectra}.
\begin{table}[h]
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|c|c|}
\hline
spectrum & no. of occurrences \\
\hline
$ \{-1.,-1.,1.,1.,1.\} $ & 1  \\ \hline
$ \{-1.,-0.61803,0,1.,1.61803\}$ & 24  \\ \hline
 $\{-0.94658,-0.5169,-0.18438,0.93842,1.70944\}$ & 120  \\ \hline
 $\{-0.90932,-0.48701,0,0.46853,1.9278\}$ & 120  \\ \hline
 $\{-0.90039,-0.64018,-0.14531,1.06785,1.61803\}$ & 40  \\ \hline
 $\{-0.83726,-0.58152,-0.09576,0.6287,1.88584\}$ & 120  \\ \hline
 $\{-0.83607,-0.81,0,1.05469,1.59139\}$ & 80  \\ \hline
 $\{-0.79859,-0.36221,0,0.10661,2.05419\}$ & 80  \\ \hline
 $\{-0.70281,-0.61803,-0.13294,0.48666,1.96712\}$ & 40 \\ \hline
\end{tabular}
\vspace{0.2 cm}
\caption{Possible spectra of the phase point operators in an affine plane, $N = 5$.}
\label{tb:spectra}
\end{center}
\end{table}
For $N=7$ one gets 210 distinct spectra (too numerous to reproduce here). In both cases one finds that sizes of the spectral orbits do not divide the order of the group
but rather twice the order.
This leads one to suspect that as far as the spectra are concerned, the relevant group is not the Clifford group but rather an augmented Clifford group.
The extra symmetry is provided by the operation of complex conjugation which also maps the MUB's into themselves. The Extended Clifford Group~\cite{10}
includes the complex conjugation operation and has twice as many elements as the Clifford group. This is consistent with the sizes of the spectral orbits found. With this in mind we investigate and encode the action of complex conjugation on the MUB's, so as to have explicit actions of all the operations on the MUB's relevant for discussing the spectra of the phase point operators which in turn enable us to classify
distinct Wigner distributions.

Complex conjugation, denoted by $C$, is represented by an anti-unitary operator. Under $C$
the MUB's transform into each other
as follows:
\begin{align}
C\colon |m,r\rangle & \to
\begin{cases}
|-m, r\rangle \qquad & m \neq \infty \\
|\infty, -r\rangle \qquad & m = \infty
\end{cases}
\\
\intertext{which implies}
C\colon r_m & \to
\begin{cases}
r_{-m} \qquad & m = 0, 1, \dots, N-1 \\
-r_N \qquad & m = N
\end{cases}
\\
\intertext{and}
C\colon e_k & \to (-1)^k e_k \qquad k = 0, 1, \dots, N
\end{align}
These actions, when translated into actions on the $\alpha$'s, have
the pleasant feature that under complex conjugation the $\alpha$'s alternately change signs:
\begin{equation}
{\rm C}:[\alpha_0,\alpha_1,\cdots,\alpha_N]\rightarrow
[\alpha_0,-\alpha_1,\alpha_2,\cdots,-\alpha_N].
\end{equation}
Thus complex conjugation acts on the phase point operators $[\alpha_0,\alpha_1,\cdots,\alpha_N]$  by a diagonal matrix ${\rm diag}(1,-1,1\cdots,-1)$,
with determinant $-1$ if $N=4k+1$ and $1$ if $N=4k+3$ .
On the affine planes $[\alpha_2,\alpha_{3},\cdots,\alpha_N]$, the action is again
by a diagonal matrix but with determinant  $1$ if $N=4k+1$ and $-1$ if $N=4k+3$.

As described in ref.~\cite{10} the extended Clifford group is obtained by admitting matrices in $\ESL(2,\mathbb{Z}_N)$ (the group consisting of $2\times 2$ matrices with entries in $\mathbb{Z}_N$ and   determinant $=\pm 1$).  The conjugacy classes of $\ESL(2,\mathbb{Z}_N)$ are described in the Appendix to this paper.  Unlike  $\SL(2,\mathbb{Z}_N)$ there is a distinction between the cases $N = 1 \text{ (mod $4$)}$ and $N = 3 \text{ (mod $4$)}$.    It turns out that for all values of $N$ we have the $2 N$ ``standard'' classes
\begin{equation}
C_{\Delta, t} = \left[\begin{pmatrix} 0 & -\Delta \\ 1 & t\end{pmatrix} \right]
\end{equation}
where $\Delta = \pm 1$ and $t = 0, 1, \dots , (N-1)$, and the $2$ additional classes
\begin{equation}
D_{\pm 2} = \left[ \begin{pmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{pmatrix} \right]
\end{equation}
If $N= 3 \text{ (mod $4$)}$ these are the only conjugacy classes (see Theorem~~\ref{thm:ESLconj3mod4} in the Appendix).  If, however, $N = 1 \text{ (mod $4$)}$ there are another $6$ conjugacy classes, which we denote $\bar{C}_{\pm 2}$, $\bar{C}_{\pm 2 i}$, $D_{\pm 2 i}$ (see Theorem~\ref{thm:ESLconj1mod4} in the Appendix).

All elements of $ESL$ with determinant $-1$ are represented on the symplectic vector
space of affine planes, $V_{N-1}(\mathbb{Z}_N)$, by matrices $\mathcal{A}$ that obey

\begin{equation} \mathcal{A}^T\Omega \mathcal{A} = - \Omega \ . \end{equation}

\noindent They are therefore anti-canonical transformations. In Table~\ref{tb:fixedESL}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{| c| c| c| c| c| c| c|}  \hline
\te{1.4}{N} & \te{1.4}{Class} & \te{1.4}{number of elements} & \te{1.4}{order of cyclic subgroup} & \te{1.4}{fixed points in plane} & \te{1.4}{fixed $A$s}& \te{1.4}{fixed planes}
\\ \hline
3 & $C_{-1,0}$ & 12 & 2 & 3 & $3^2$ & 3 \\
& $C_{-1,1}$ & 6 & 8 &  1 & 1 & 1 \\
& $C_{-1,2}$ & 6 & 8 &  1 & 1 & 1 \\
\hline
5 & $C_{-1,0}$ & 30 & 2 & 5 & $5^3$ & $5^2$ \\
& $C_{-1,1}$ & 12 & 20 & 1 & 1 & 1 \\
& $\bar{C}_{1}$ & 12 & 20 & 1 & 1 & 1 \\
& $D_{1}$ & 1 & 4 & 1 & 1 & 1 \\
& $C_{-1,2}$ & 20 & 20 & 1 & 1 & 1 \\
& $C_{-1,3}$ & 20 & 12 & 1 & 1 & 1 \\
& $C_{-1,4}$ & 12 & 20 & 1 & 1 & 1 \\
& $\bar{C}_{4}$ & 12 & 20 & 1 & 1 & 1 \\
& $D_{4}$ & 1 & 4 & 1 & 1 & 1 \\
\hline
7 & $C_{-1,0}$ & 56 & 2 & 7 & $7^4$ & $7^3$ \\
& $C_{-1,1}$ & 42 & 16 & 1 & 1 & 1 \\
& $C_{-1,2}$ & 56 & 6 & 1 & 7 & 7 \\
& $C_{-1,3}$ & 42 & 16 & 1 & 1 & 1 \\
& $C_{-1,4}$ & 42 & 16 & 1 & 1 & 1 \\
& $C_{-1,5}$ & 56 & 6 & 1 & 7 & 7 \\
& $C_{-1,6}$ & 42 & 16 & 1 & 1 & 1 \\
\hline
\end{tabular}
\vspace{0.2 cm}
\caption{Fixed points under the action of elements in $ESL(2, Z_N)$ with
determinant $-1$, $N \leq 7$.}
\label{tb:fixedESL}
\end{center}
\end{table}
we list the number of fixed points for the conjugacy classes of $\ESL(2,\mathbb{Z}_N)$
with determinant $-1$. For all $N$ one can show that elements of order 2 leave
$N^{(N-1)/2}$ affine planes fixed. Because the transformations are anti-canonical it
follows that these fixed planes form Lagrangian subspaces of $V_{N-1}(\mathbb{Z}_N)$.
Furthermore we believe
that if an element with determinant $-1$ squares to an element leaving $N^{2k}$ $A$s
fixed, then that element itself leaves $N^k$ $A$s fixed; we checked this for $N \leq 11$.

Again we can use Burnside's lemma to compute the number of orbits under $\ESL(2,\mathbb{Z}_N)$.  The results for $N\le 11$ are listed in Table~\ref{tb:noOrbitsE}.
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|c|} \hline
\te{2}{$N$} & \te{1.5}{$3$} & \te{1.5}{$5$} &\te{1.5}{$7$}&\te{1.5}{$11$}
\\ \hline
No.\ of orbits & 2 & 9 &210& 9833460\\ \hline
\end{tabular}
\vspace{0.2 cm}
\caption{Number of orbits of $ESL(2, Z_N)$ when acting on the set of affine planes.}
\label{tb:noOrbitsE}
\end{center}
\end{table}
For large $N$ the number approaches the estimate $N^{N-1}/2N^3$. For $N \leq 7$ we know the number
of distinct spectra, and we find complete agreement with the number of orbits.

\section{Summary}
\bigskip
\noindent

To summarise:
\begin{itemize}
\item For $N$ an odd prime, a compact, mathematica friendly way of representing phase point operators and affine planes as vectors and cosets in an $N+1$ dimensional vector space over $Z_N$ is given.
\item In this representation the $\SL(2,Z_N)$ actions on phase point operators
actions by matrices in $\SL(N+1,Z_N)$.
\item A convenient basis, factoring out the Heisenberg-Weyl group actions, is introduced in which $\SL(2,Z_N)$ actions on the affine planes can be described in terms of matrices in $\SL(N-1,Z_N)$.
\item The action of complex conjugation on the phase point operators
is by matrices in   $\GL(N+1,Z_N)$ with determinant $-1$ if $N=4k+1$ and $1$ if $N=4k+3$.
\item The action of complex conjugation on the affine planes
is by  matrices in   $\GL(N-1,Z_N)$ with determinant= $1$ if $N=4k+1$ and $-1$ if $N=4k+3$.
\item The $(N-1)$-dimensional vector space of affine planes is symplectic. Elements of
$ESL(2, Z_N)$ with determinant $1$ preserve the symplectic form, elements with
determinant $-1$ are anti-canonical.
\item For $N \leq 7$ the number of distinct spectra in the phase point operators
equals the number of orbits under the group $\ESL(2, \mathbb{Z}_N)$.

\end{itemize}

The work reported here suggests a number of avenues for future research.   In the first place one would like to know if the degeneracies of the phase point operators are completely determined by $\ESL(2, \mathbb{Z}_N)$ for every odd prime dimension $N$.  Beyond that one would like to know if the result generalizes to the case of arbitrary odd prime \emph{power} dimension.    Finally, there is the question of even prime power dimensions.  In that connection let us observe that Gibbons et al \cite{5} find 4
similarity classes when $N = 4$.  However, when we computed the spectra we found only the $3$ distinct spectra tabulated in  Table~\ref{tb:spectra2}.
\begin{table}[h]
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|c|c|}
\hline
spectrum & no. of occurrences \\
\hline
 $\{-0.896802,-0.14204,0.278768,1.76007\}$ & 384  \\ \hline
$ \{-0.866025,-0.5,0.866025,1.5\}$ & 320  \\ \hline
 $ \{-0.5,-0.5,0.133975,1.86602\} $ & 320  \\ \hline
\end{tabular}
\vspace{0.2 cm}
\caption{Possible spectra of the phase point operators in an affine plane, $N = 4$.}
\label{tb:spectra2}
\end{center}
\end{table}
 The cause of this additional degeneracy cannot be the same as the cause in odd prime dimension since when $N$ is an even prime power the fact that $-1 = 1 \text{ (mod $2$)}$ means that $\SL(2, \mathbb{F}_N)= \ESL(2, \mathbb{F}_N)$.  There must therefore be some other explanation, which it would be interesting to investigate.

\vspace{15mm}

{\bf Acknowledgements:}

\

\noindent We thank Ernesto Galv\~ao for sharing some preliminary results in this direction.
Our work was financially supported by the Wenner-Gren Foundations, as well as by the
Swedish Research Council. One of us (SC) also wishes to thank the Leverhulme
Trust for a Visiting Professorship at the Imperial College London where a
part of this work was done.

redpajama_set_name: RedPajamaArXiv


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The shower and bathtub require special attention and maintenance. Check the tiles and replace if cracked. Make sure that there is no missing grout between the tiles. Inspect and replace cracked caulking at joints where the walls meet the floor or the bathtub. Clogged drains and pipes problems will prevent the bathtub from drying and may indicate serious problems beneath the bathtub. Consult with a professional immediately to prevent structural damage. Pay attention to discolorations or soft areas around the bathtub walls as they may indicate an internal leak.
Signs for water damage are hard to detect since most pipes are installed inside the walls. Pay special attention to flooring and walls moisture and stains as they may indicate an invisible plumbing problem. Check moisture levels in adjoining rooms as well.
Sinks and cabinets are exposed to moisture and humidity daily and are often overlooked. Inspect regularly under the sink and on the countertop above it. Repair any drip in the trap as it may suggest drain problems. Look around the sink, slow draining pipes may indicate a blocked drain. Replace sink seals if they are cracked or loose.
The toilet is a vulnerable water junction. Check the water lines and search for leaks around the toilet seat, in the hose, and under the water tank. If you detect any signs of moisture on the floor around the toilet, check for leaks in the toilet rim and tank seals. Be aware that hanging toilet bowl deodorants increases the chances for clogs.


Fellow at the interdisciplinary research center called Aesthetic Experience and the Dissolution of Artistic Limits (the Free University, Berlin). Currently she is working at the creation of the Retroformalism research group and plans to work out poetonomy of modern art.
Anke is a fellow at the interdisciplinary research center Aesthetic Experience and the Dissolution of Artistic Limits (the Free University, Berlin). She studied modern German literature and Russian philology at the universities of Berlin, Bochum and Moscow. In 2005 she took a PhD representing a paper on Soviet Cinema Art of the 1930s. She is an active participant of the Russian Cinema Research Group (UCL, London) and is a Fulbright Program scholarship holder at the University of New York (2012). She shared the results of her research in Moscow, Cambridge, London, Florence, Vienna, New-York, Washington, Bochum, and Leipzig. Currently she is working at the creation of the Retroformalism research group and plans to work out poetonomy of modern art.
Metanoia, co-authored with Armen Avanessian. Berlin: Merve 2014.


text:
Vendor Expo
Le Grand Marketing, Inc.
Ignacio (Nacho) Gomez
Field Merchandiser/Trainer - Southern California
Ignacio (Nacho) Gomez grew up Guadalajara, Mexico and came to the United States on a work visa at age 18. He landed a job at a restaurant and bakery now located in Huntington Beach, California where for seventeen years he learned how to formulate and bake scratch recipes for breads, cakes and cookies.
Nacho was then recruited by Boudin San Francisco Sourdough Company and received intensive training in San Francisco on the same mother dough cultivated from a gold miner's sourdough starter originally formulated in 1849. After his training he went to work at the Boudin located in Costa Mesa.
After three years there, Nacho took his scratch baking skills to Gelson's Market in Newport Beach. At the same time he worked at Chapman University which serves 1,700 college students and 5,000 plates a day. Nacho also worked under Chapman Chef Jim Douglas' catering business doing the baking for catering events for VIP Guests such as former Governor Arnold Schwarzenegger, Supreme Court Justice Anton Scalia, Spanish President Jose Marie Aznar and Nobel Peace Prize Recipient Eli Wiesel.
Now as a merchandiser and trainer for Le Grand, Nacho has used his skills at store level to provided training at Ralphs, Gelsons, Smart & Final, Vons, Albertsons, Super King and Pavilions. His bilingual skills and Hispanic heritage have allowed Ignacio to effectively train and roll out new items at predominantly Hispanic chains like Superior Grocers and Northgate Markets. He also does a great job of presentation baking and has attended several Hispanic sales presentations where his impute has aided in a number of successful calls.
In 2014 Nacho was sworn in as a US Citizen and wants to continue to improve his skills as a baker and trainer. Nacho's passion for baking is fueled by the satisfaction that his end product will bring a smile to someone's face.
Maria Serna
Maria has been in the grocery industry for the last twelve years, eight of which have been in the bakery. There she learned to bake and soon after to cake decorate.
Maria is currently assigned to Ralph's Grocery Company.
Valerie Beard
Valerie Beard was seven years old when she began training to bake and cake decorate in the family's Corona California bakery. Val's earliest memories were of going to the bakery in the morning with her father before he would drop her off at kindergarten. Dad taught her how to make Danish and, for fun, he would let her ice cupcakes. Spending time with her father really paid off as she developed the skills that set her on the path to become a successful baker.
By the time Valerie was sixteen years old the bakery had become a second home. The whole family worked and played together as the bakery was the focal point. Val drew knowledge from everyone that worked in the bakery to hone her skills to be able to reach her potential.
In 2002 Valerie's baking career took a very different turn when she joined Vons Grocery Company as a full time baker and cake decorator. Soon she was promoted to bakery manager. This allowed her to continued her hands- on education by allowing her to see baking on a much higher production level, learn about the grocery business and understanding the role in-store bakeries play. What Val loved the most about Vons/Safeway was they were still a mostly scratch bakery.
Fast forward to August of 2015 and Valerie's career moved on to Le Grand Marketing. She was introduced to the world of trainer/merchandiser and has been using her extensive experience and skills effectively ever since. If anyone asks, Val will be quick to say that she loves what she does and strives to share the passion and art of good baking with others.
Diana Keller
Field Merchandiser/Trainer - Northern California
Diana started in the grocery industry at the age of sixteen. She worked as a deli clerk before moving into the bakery as a baker. Diana has been cake decorating for the last fifteen years. In her spare time, she enjoys creating three-dimensional cakes. She appreciates the level of creativity this provides. Additionally, she makes wedding cakes.
Diana joined Le Grand in the capacity of store level training in June 2007. She provides primary technical support throughout Northern California.
Jessica Soo
Jessica Soo started at Ralphs Grocery Company in 2004 and was quickly promoted into the bakery department. Within two years, after running the bread aisle and becoming the main cake decorator and full time baker, she was promoted to bakery manager and ran four successful bakeries over the next eight years.
Jessica was dedicated to her job as manager and focused on building a strong team. While at a fast growing Fresh Fare in Irvine, she became the district trainer and gained a passion for teaching others.
In 2015 Jess joined the Le Grand team. She is eager to learn more herself and train others to be successful bakers. When she's not working, Jessica dedicates all her free time to her two amazing kids and husband.
Ryan Davenport
Field Merchandiser/Trainer - Pacific Northwest
Ryan was born and grew up in Longview Washington where he graduated from R.A. Long High School. He always excelled in Home Economics and baking at home for his family.
Shortly after graduation, he started in the bakery business going to work briefly for Top Food and Drug in Kelso as a donut fryer and back up baker for the day crew. Soon after he started, he moved on to work for an Oregon independent grocer, Lamb's Thriftway in Wilsonville, Oregon. Beginning as a donut fryer and baker for the evening shift, over the years he moved up the ranks of the Lamb's crew adding bread making , baking and cake decorating, eventually becoming Production Manager. In 2014, after 19 years, Ryan left Lamb's and purchased a small bakery/cake shop in Portland. He enjoyed building relationships with the local customers with his baked goods and decorated cakes. After 5 years serving the greater Portland area, he decided to move on, working briefly for Bales as a cake decorator before it went out of business in 2019.
In October of 2019 Ryan joined the LeGrand Marketing team as a Field Merchandiser for the Oregon and Washington Region. He frequently works with Safeway/Albertsons as a bake tech, training and teaching handling of our vendor's products. He also focuses on helping to grow Le Grand's reach in the Pacific Northwest Independent Markets. Ryan works hard at being a friendly, hard working team player.
Brian Trujillo
Field Merchandiser/Trainer - Arizona
Field Merchandisers

redpajama_set_name: RedPajamaCommonCrawl


"1949 Not Out" - The movement of the DD to the "Hill"
"So Kev this is it - D DAY" Bill said to me as he picked me up on a cold June Friday.
"You look how I Feel old boy " I replied to Bill - knowing that both of us had had a rather restless night worrying about what we were gonna do, what could happen and the certain humiliation we faced if it all went very wrong. Jamie - the younger Nile looked on giving a cheerful smile - he didn't care he just finished his school sentence and exams and was a now a man of leisure - (he was as worried as his dad was!). I suspect mother Nile ordering to him to keep on eye on a beleaguered dad, his function this day.
Anyway that's the scene the day we moved the Double Decker would prove to be a day of fun, hard work and general running about.
At 08.15 Bill, Jamie & myself appeared at NLR to prepare the Coach for movement to the Hill by low-loader, which as we saw was already there, Nick, Andris & Ivor duly materialised along the rest of Nick's cohorts who all had a vested interest in the move.
The loading was done in quick time and before we knew it we were speeding ahead to Hunsbury to prepare the ground for the arrival of the beast.
Anyway the unloading of the Beast was something that will live in my memory for many, many years. First of all the lorry had trouble traversing the crossing, which had been removed the day before by the combined efforts of Bill, Jamie myself, Nick and Young Eddie. We had hoped that the preparation we had done would be enough to enable a smooth reversal in straight in to the yard. How wrong we were. We were first faced with the problem of severe lack of traction that the lorry had, but with the rapid laying of some ballast and stone we quickly overcame this. We had one other small problem - the gate. We knew we could get through with six inches spare on each side, but with the constraints of the lorry and other factors Nick made a decision with the Railway committee - aka Chris Osbourne to remove a panel of fencing, this done the shunt went smoothly.
We had one other small problem - we had the sleepers laid and ready for rail installation, unfortunately Alleys had forgotten to bring it with them. All though was not lost with some fast thinking from Grant & Eddie we soon had found 120 foot of rail which was laid as the and gauged on the sleepers as the DD slowly moved down the ramp from the trailer.
Anybody there on the day will remember the amount of people who were running about with rail on rail dogs, people with crowbars and the general looks of panic on the people as we frantically tried to make a bad idea look like a good one.
It got to the point when the all of us were pinch barring the DD along the panel of track and the tractor unit desperately tried to give us a helping hand, how the tractor did help I will not divulge as I never saw what happened. All I know is that one minute we had 20 foot of space and the next I was I was shouting stop as I called to Nick to slowly draw up to the end of the panel of track.
That done we made our piece with the world and christened the DD in the only was possible and maybe for the first time she got her first proper christening - of champagne.
In the meantime Nick, Simon & co were slowly re-erecting the panel of fencing as the people who appeared suddenly disappeared, before I knew it - Nick & myself were all alone with our thoughts as we talked rubbish.
Anyway the Double Decker is here at Hunsbury Hill - we as a group are very pleased to be here on this site and look forward to years ahead. When we will move 4002 around at least half the running line, the future for the site is good, the future of the DD is secure and in October we celebrate the 50th birthday of the design.
Finally we owe the following people thanks - the Hunsbury Hill Management Team for their assistance, Northants Police, Northants County Council, Mother Nile & family for keeping dad sane - one hell of a job, one bottle of Vodka (in joke I heard all about young lady!) Nicks Missus for keeping him sane, Simon, Brendan and the others who broke their backs we thank you all.
Finally thanks to Oliver Vernon Snell Bulleid for creating the DD.


Wanna make your 'do that lil' extra this party season. Done. Here's eight hair hacks to take your hair from whatever to woah.
Want curls that actually hold? Get your hair done ahead of your night out and let it set for an hour before hitting the dancefloor. Or if you wanna make sure your work will last the night, get your kirby grip friends and pin up your 'do 'til you hit your destination and when you do, carefully unpin and voila, hair that held.
Want effortless worn in waves? Plan it girl. The day before get your wand and spend 10 minutes curling your hair. Separate into 4-6 sections depending on the thickness of your hair, and curling away from your face, wrap around your wand just above your jawline for tight Blair Waldorf inspired curls. Hairspray and let set. Then, the day after or at night, gently run your fingers through to let the curls drop to a natural wave. Grab your hairspray and give one last spritz.
Sounds obvious but like any tool you use, they need looking after. So once a month, set aside 10 minutes to give your brushes the cleanse they need. Grab some baby shampoo or whatever you have in and scrub away. Let air dry overnight for healthier looking hair.
Want a bigger ponytail? Don't we all. If you're limited to equipment, double it up. Grab half of your hair and tie into a mid-low pony before tying the top section into a high pony that gets a lil' lift from the one below.
Kirby grips are a girl's best friend: fact. So if you really wanna make sure they stay put, take two, and cross them over for extra strength.
Got extra time to kill before going out? Ditch the heat and let your hair air dry once in a while as it'll get your hair looking way healthier, saving you a trip to the salon.
Get your party prep in girls and treat yo'self to a luxe hair mask that you can pop on and leave to work its magic whilst you binge on a Netflix boxset.


text:
Asturias Foundation
admin February 20, 2014 General books
The program manages franchise corresponding to February 8 interviewed live Jose Maria Garcia-Caballero, responsible for Etam franchising, a chain store of lingerie and underwear from the opening, in 1984, on his first operational unit in Spain, has not had any closure to date, which gives an idea of the strength and balance of the network. In his speech, he highlighted the role played Etam in the modernization of a sector which was represented in our country by haberdashery and small neighborhood shops, far removed from the current boutiques in lingerie and underwear that are, often, locomotives in streets and shopping malls. And it also counted with the presence of the head of the Nails 4 Us in Spain, Juan Jose Munoz masterfranquicia. During the interview, this businessman pointed out that the Spanish woman is revealing as a customer very interested in this type of manicure and pedicure in gel. And although it is true that other countries take them certain advantage, because they started earlier, while the Spaniards say they are still educating, Nails 4 Us we see it as an opportunity, rather than as a handicap: much ground remains and much business ahead for our present and future franchisees. The section of the program, led by Jose Ramon Inguanzo, corresponding to the theoretical aspect of this formula of business collaboration tried raising awareness to the public about the little credibility that deserve the statistical figures on the world of franchising in Spain, contributing each year between four and five different sources, none of them official.
In the words of Jaime Bosqued, journalist specialized, left it quite clear: as trading system, the franchise carries seated less than a quarter of a century in our country. Are we going to be on par with countries where a lot takes more time working, as France or United States?. The space of Gestiona Radio dedicated to this business model also He did a review of today, with news of the start of construction of the new Logistics Park of Mango, the arrival in Spain of the firm of young fashion Forever 21, the first contest of skewers of taverns Lizarran, the Ensign partnership network the thimble of the Tata with the Prince of Asturias Foundation, which Almeida travel has signed with the health insurer Asisa or the imminent opening of the third shopping centre and leisure larger Europe, in La Coruna, on April 14. Listen to program (can download the audio file by clicking the link with the right mouse button and choosing the option Save target). The next program, don't miss Tuesday 15 February, at 11.30 pm at Gestiona Radio. More information: manages Radio is a radio chain attached to the Group COPE, which emits a programming specialized in economic information since 2009. Currently, manages Radio have dial in Madrid (94.8 FM and FM 108.0), Valencia (100.7 FM), Seville (90.8 FM), Zaragoza (98.3 FM), Malaga (98.5 FM), Bilbao (107.9 FM), Valladolid (88.9 FM) in brief points of the dial in other provincial capitals are added. The program manages franchise is the first space of radio, specializing in franchise, issued by a large national chain in Spain. Led by two journalists with more than 15 years of experience in the sector, such as Jaime Bosqued and Jaime Parrondo, it can also be followed live by Internet (www.gestionaradio.com/radio.php).


Geez, I tried. But the office where my packages were delivered isn't open during normal working hours. 1/3 Heavy packages from chewy. Com were accepted by the office in error. Representatives from office said they'd be dropped off by maintenance. Not here 1.5 weeks after original delivery that was marked for delivery to my apt, and several phone calls to the office. I am an employed old lady with heart disease and a bad back. Guess the only choice I have is to show up on the weekend with my wheel barrow and open these boxes in the office lobby and wheel the stuff home. What a crummy way to treat the residents.
I am so sorry that your package was not delivered to your home. As a courtesy, we do accept resident packages at the office. We are open from 8:30am to 5:30pm as well as on the weekends for resident's convenience. We are happy to offer the help of our maintenance staff as a courtesy if they are not on an onsite emergency call. If the delivery service does not deliver your packages to you door, you will need to contact them directly about the issue. Please let us know if you need any thing else. We are happy to help!
First, the property overall is clean and well maintained. All of the Fogelman staff, including maintenance staff, are amazing. They are always friendly and responsive, and I feel that they genuinely care about their residents. Another positive is safety and security. Thus far, I have not experienced any issues while living here, and I feel generally safe. Cons: I gave Maintenance 4 stars instead of 5 because of their pest control contractor. My understanding is that the exterior of each unit is supposed to be sprayed every few months. I work from home, and in a year and a half, I have never once seen Terminix spray the exterior of the apt. units. I've asked the service tech multiple times to spray the entire apt. interior, and he has not done it. I've had to resort to purchasing my own pest control products, and in the last month, I have found 6 dead and 4 live roaches in my apt. There have been other insects in the apt., which I consider common trespassers. Roaches on the other hand are completely unacceptable and avoidable if Terminix were doing what they're being paid to do. I urge mgmt. to address this with their contractor. I gave Amenities 4 instead of 5 stars, because the apartment interior (appliances, fixtures, etc.) is pretty dated. Last, I'm disappointed by how much and how often I can hear my neighbors above me. I don't blame them, as it's nothing more than their normal, everyday routine. I don't know if anything can be done about this, as it seems to be due to the age of the property. However if anything can be done to improve this, such as better quality carpet padding, I would suggest investing in it.
The staff is phenomenal... Go out of their way To make sure you are ok. I love the quiet gym at night. It really does feel like home. I've lived in 5 different units.
We love the staff, they take great care of you. Very close to everything, convenient location.
The staff is very friendly and very helpful. Maintenance is usually pretty fast about getting things fixed. I only wish there was less to be fixed.
This is our first home with our new baby. Overall, we really enjoy living here. We did have some maintenance issues when we first moved in, but it was all fixed within a couple weeks. Compared to the last place I stayed, that's pretty quick. The staff has all been really nice and I appreciate the fact that they get to know their residents. As long as everything continues to go well, we plan on staying for the next few years.
Pest control issues, lizards make there way into my home and I am a VERY clean person not to mention the large cockroaches. There is poor and inconsistent communication from office personal and they often take a long time to answer your questions. Okay quality apartment but not worth the fees and prices that come with it. Lived here last year and was very satisfied, if I known how it was going to be this year I would have never resigned.
The layouts are spacious and the grounds are kept neat and clean. The people that are in the neighborhood seem to be really nice and quiet. It is a community, especially tailored to dogs. The staff and maintenance individuals are very nice. Sometimes, the communication is lacking in the office depending on how busy they are. I think this is just being human, so it does not bother me much.
Pathetic customer service. I ve been trying to talk to them about the availability of apartments, but they never pick the phone. The call always goes into voice messages, and they never reply to those messages. Frustrating.
In terms of community, it is great. I feel welcome and safe here and the apartment itself is spacious. The apartments are in a prime location as well I mostly just have complaints with the layout of the one bedroom apartments. I couldn't opt out of getting a second story one bedroom without a fireplace. I would have done without it as I would rather have more space. I think the counter space in the kitchen of the one bedroom layout leaves a lot to be desired I had to buy a kitchen island in order to store all of our kitchen appliances. The tiny cabinets above the fridge are also a complete waste of space and exceedingly difficult to reach. I don't like that the washer and dryer room is right next to the kitchen. Maintenance also seems to take a little longer than I would prefer. I have, however, toured the two bedroom and fell in love with the layout, so my only complaints are for the one bedroom. Overall, I only listed my complaints as I have countless more compliments--these apartments are worth considering.
Nicely kept and clean gated apartment complex!
I m sad to be moving from here (out of state). The front staff and maintenance are all superb. They are always smiling, listen when you report a problem, and take care of it quickly. The neighbors are friendly and (from my experience) quiet. The complex is super dog friendly, will help adopt out any stray animals that are found even! There is tons of natural light with huge windows and nice balconies. The apartments are quiet, or at least I only hear my neighbors very rarely. The apartments are super clean when you move in, and the grounds are well maintained. The pool is always clean and pleasant. The neighborhood is great too, super close to target, restaurants, and a movie theater since you re right behind crossroads plaza. My ONLY complaint is that you don t get a notice if you receive a package at the front office, but they re so nice at the front office that if you go up there and aren t sure, they re always happy to double and triple check. Honestly, this is the best apartment complex ever, and if I could, I would stay. Can t recommend it highly enough!
The staff is very friendly and very helpful and make it there business to learn who you are. They try to do whatever they can to address and solve issues. It is quiet and clean. I enjoy living here and appreciate all that they do.
I have been here for almost 2 years and it has been wonderful! Great staff!
Nice, small gated community. Gate doesn t always work and the communities amenities could be better. Great maintenance, but overall a very overpriced typical apartment community where you re mostly paying for location.
I lived here for a year and it was quite pleasant. The neighborhood feels very safe to walk around at night and the crossroads shopping area is walk-able distance. The employees are very friendly and wonderful people. They even quickly learned my name as well as my room-mates. I had a water leak from my water heater unit a while back and they quickly replaced it the same day with a new one. They even cleaned and dried the carpet for us! I really appreciated everything they did. They really went out of their way to make us comfortable with hardly any compromising. I was sad when I left Enclave but it will definitely be one of my first choices if I ever had to move back into the area again.
Best apt community I have ever lived in.
Good Place to live. Staff is wonderful. Notice some delays in maintenance issues. Old apartments but maintained very well.
It's a great location and quite nice overall. I like the place, but there are a couple of annoyances I've had to deal with since moving in. For example, the apt wasn't clean when I moved in, along with a lot of maintenance issues upon moving in e.g. light didn't work, door painted completely shut, leaky faucets, etc. When I later reported the priority issues they were fixed, and maintenance folks have seemed to be very timely otherwise , neighbors slamming doors and others who don't know how to keep their dogs from an ungodly amount of yapping annoying but I guess is common with apartment living and not the fault of the organization . The recycling area and mailboxes could definitely be better maintained not enough bins so people put trash on the ground, some people's mailboxes seem like they are always open, not enough lighting at night .
Property is great, maintenance not so great.
Great place to live, quite neighborhood. My only issue is that maintenance is slow and also the gym is small, but other than that I would really recommend it to a friend.
I enjoy the property very much.


text:
Kennesaw Knights
Team: Kennesaw Knights [ID #12]
Location: Kennesaw, GA (Southeast)
League: V.15 - East
Owner: None
Manager: Loren Burton
Ballpark: Kennesaw Stadium
Rank: #345 - Rating: 94.33
Fan Mood: 94.84
Color: Navy & Gold
• May 29 2058: Jason Montgomery was signed as a free agent.
• Apr 07 2058: Matías Alvarez was signed as a free agent.
• Mar 14 2058: Gerardo Fernández was released.
• Mar 14 2058: Sergio Aguilera was released.
• Mar 14 2058: Timmy Sales was signed as a free agent.
• Jan 13 2058: Byung Ho Kim was selected to the national team of South Korea.
• Jan 13 2058: Sergio Aguilera was selected to the national team of Panama.
• Jan 13 2058: Gerardo Fernández was selected to the national team of Costa Rica.
• Jan 13 2058: Albert Smits was selected to the national team of Aruba.
• Nov 02 2057: Rodger Culver was hired as the new coach.
Ave: 4.1 - 5.0 RS: 186 RA: 227 DIFF: -41 Power Index: 104.2
Jan 21 2038: second year - by Bostonboy2113 on October 9th, 2018
we gonna win watch out
Jun 15 2023: North Port Stuns Austin; Takes 4 of 5 - by cqk328 on July 15th, 2015
With this being North Port's 10th straight season in League VI.29 (their 5th under the leadership of cqk328), there is one thing the team knows that has always been constant in this league: the West is better than the East. This definitely showed last year, as the West contained 5 teams over .500, while Independence, the mediocre East's division winner, was 80-80. However, North Port's hard work in player development and focus on a balance between pitching, hitting, and fielding may finally be paying off. In the opening series of inter-divisional play, the Knights dropped only one game to the then 31-19 Ravens. The manager takes full responsibility for the Game 5 loss. "I made a late starting pitcher change hours before gametime," he said. "Mad Dog Martin has been our setup guy recently, so he probably wasn't ready to start on such short notice." North Port is looking very good this year, as they are 28-27 and are the only team besides Summerville with a positive run differential. Today the Knights are in Coeur d Alene to take on the Indians. Vince Caputo (6-1, 2.09 ERA) will be on the hill, while the Indians will start lefty Javier Lozano (2-6, 4.25 ERA).
Jul 14 2021: Halfway Point - by cqk328 on February 17th, 2015
80 games down, 80 to go. Man, these seasons go by so fast...Anyway, can't believe I'm saying this, but the Knights are 2 games over .500 headed into the All-Star break. I knew the 23-9 start was only a fluke, but this is still much better than I expected this team to do this year. Why are we better? Let's see.
This year's starting rotation, particularly Thomas and Aguilar, has matured much quicker than expected. Unfortunately, elbow soreness prevents Aguilar from making an All-Star appearance that he would most likely get, but Thomas should definitely get the nod. His 8-7, 3.50 ERA does not tell the whole story. Although this year's offense is much better than anything I have gotten in three years as GM, he has had to endure nights where the bats were simply extinct. With 10 years in the system now, he deserves a second nomination to the team. Hopefully the rotation will continue to improve as we attempt to bring playoff baseball to North Port.
Generally, this team's offense had been lethargic for years. However, last year's team showed signs of life in the last 20 games, and this has continued into 2021. Obviously, you cannot forget Dennis White, who leads the league in average at .345. Ottenello and Rojas have also been influential, with 12 and 10 HR's respectively. I believe that although our offense is impressive, it could still be much better. To make the playoffs, our offense still needs to mature in a hurry.
Jul 14 2021: Double Post...Sorry - by cqk328 on February 17th, 2015
Mar 02 2021: Okay so I'm the only non not left in my division... - by cqk328 on January 9th, 2015
Is this good or bad??


text:
March 20, 2016 Vincent Gragnani
More than 100 miles from home, in an antique shop in the Hudson River Valley, I discovered this postcard with an image of a street just a few blocks from my office in St. George, Staten Island. Mailed from Port Richmond on July 18, 1908, the postcard depicts a scene from Richmond Terrace, in New Brighton, Staten Island.
"We are out on a trolley trip to Port Richmond, Staten Island. Passed through this place," wrote N.A. Chase to a Miss Elsie McKay in Fishkill-on-Hudson, New York.
I stared at the image, looking for clues as to which block this postcard depicted. It is clearly not the Richmond Terrace of today, which is largely lined with industry on the North side, facing the Kill van Kull, and residential developments on the South Side. The postcard looks more like a scene out of Brooklyn, lined with two- and three-story buildings—businesses on the lower floor and, presumably, residences above. Power lines, trolley tracks and horse-drawn carriages line the middle of the street.
I shared the postcard first with colleagues who are longtime New Brighton residents. They were stumped. I then shared it with a local author and preservationist. He, too, could not identify the block, but he promised to do some research and, sure enough, he got back to me with a letter from the chief curator of Historic Richmondtown/Staten Island Historical Society.
If you look carefully at the left side of the street, you can see the inscription "Wanty/Harness" on the front of one of the buildings. Joseph Wanty was a harnessmaker who was located on Richmond Terrace in New Brighton. Here's some of the information we have about him.
The Richmond County Fair program for 1895 has an ad: "All Grades of HARNESS / Every Article necessary for / HORSE, / CARRIAGE and / STABLE. / Joseph W. Wanty, / 371 Richmond Terrace, New Brighton, S.I. / ALL GOODS / CARED FOR AND DELIVERED / FREE."
He had a full page ad in the 1905 Richmond County Agricultural Society program: "J.W. WANTY, / Hand Made Harness for all Purposes / A GOOD STOCK OF BLANKETS, WHIPS, BITS, / COLLARS, SADDLES AND STABLE UTENSILS, ETC. / 371 RICHMOND TERRACE, NEW BRIGHTON, N.Y."
The Richmond Borough Directory for 1912 lists Joseph W. Wanty under the category "Harness Makers," with an address of 509 Richmond Terrace, New Brighton.
Both of his addresses, 371 Richmond Terrace and later 509 Richmond Terrace, are close to the foot of Jersey Street.
Mystery solved, thanks to some sharp eyes and diligent research from Historic Richmondtown!
Now, my only question is, why was N.A. Chase going to Port Richmond?
Today, you would have to look closely for clues to know Port Richmond's rich history as a commercial and industrial hub. Port Richmond was settled by Dutch and French colonists in the 1690s and early 1700s. Aaron Burr, our nation's third vice president, died there in 1836—around the same time that the neighborhood's street grid was laid out.
When N.A. Chase visited the neighborhood more than 70 years later, trolley lines connected it to St. George to the east, and Bulls Head to the South. Port Richmond Avenue, known then as Richmond Avenue, was on its way to becoming known as the "Fifth Avenue" of Staten Island, a designation that slipped away as the shopping hub shifted to the Staten Island Mall in the later half of the 20th Century.
Though it may not seem like a destination today, chances are, N.A. Chase was en route to do some shopping in what was then a bustling neighborhood of Staten Island.
Postcards of Old New York, Staten Island
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2 thoughts on "Passing through New Brighton on a trolley ride to Port Richmond"
Liz September 2, 2016 / 12:45 pm
Hi there. I enjoyed this post very much and really appreciate the effort you went through to identify the location in the postcard. I'm in the process of writing a blog post for my family history blog, Rooted in Elizabeth, about an ancestor who was born in West New Brighton in 1909. I'm coming up short on illustrations for this post – in other words, I have none! I would love to use the old postcard image to illustrate a bit of what the area was like at the time he was born. Any chances I can use your image, with all appropriate links back and credit, etc.?
Vincent Gragnani September 2, 2016 / 12:50 pm
Yes, please do! I'm eager to read your post!


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Contoocook
Contoocook, New Hampshire, United States
February 4, 2021 @ 04:38
March 12, 2021 @ 21:23
April 3, 2021 @ 23:04
April 11, 2021 @ 15:32
May 11, 2021 @ 08:01
June 1, 2021 @ 20:26
June 17, 2021 @ 16:54
July 1, 2021 @ 10:12
July 16, 2021 @ 23:11
August 8, 2021 @ 02:50
August 15, 2021 @ 04:21


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Q: R: Impute missing values from other categorical values in same vector I have a large dataset and am trying to determine how to impute an existing categorical value (Seminar #) for a missing values in the same vector for the same Student #.  An example is below where the first observation of student "813" is missing his Seminar number, which should be "Sem5" according to the second observation.
Stu#  Sem#
812   Sem1
812   Sem1
813
813   Sem5
814   Sem9
814   Sem9

I've been at this a few hours and have note had any success.  Thanks in advance.

A: Assuming your data is in a data frame where Sem# is character (if not, convert to character) and that the only possible values for Sem# are either blank or the correct seminar name, try:
df.ord <- df[order(nchar(df$Sem), decreasing=T), ]
merge(df, df.ord[!duplicated(df.ord$Stu), ], by="Stu", all.x=T)[c(1, 3)]
#   Stu Sem.y
# 1 812  Sem1
# 2 812  Sem1
# 3 813  Sem5
# 4 813  Sem5
# 5 814  Sem9
# 6 814  Sem9

Note I changed your column headers to avoid the #.
Basically, what I'm doing here is first re-ordering the data so that values with blank Sem# end up last, and then using duplicated to subset the data-frame just to non-duplicated values that have non-blank Sem# values.  Once we have that, we can just merge back to the original data to attach the Sem#.

redpajama_set_name: RedPajamaStackExchange


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Oakland A's vs St. Louis Cardinals-6/26/19- MLB Over Under Picks.
Oakland A's versus St. Louis Cardinals (June 26)
The Oakland A's, currently 2nd in the AL West, take on the NL Central's 3rd place Cardinals in this interleague battle.
Tanner Anderson is scheduled to pitch for the A's and Adam Wainwright is set to take the mound for the Cardinals.
The A's have an offense, like many in the league, that rely on the long ball.
They are hitting a modest .248 but have 121 home runs and average five runs per game.
Matt Chapman leads the team in home runs with 18.
Khris Davis adds to a power duo with 16 home runs and 41 RBI.
Matt Olson and Ramon Laureano have chipped in with 12 home runs each.
The Cardinals have a more average offense, averaging 4.6 runs with a .243 average and 95 home runs.
Marcell Ozuna heads the attack with 20 home runs and 62 RBI.
Paul Goldschmidt had 14 home runs but a modest 31 RBI, while Paula DeJong has contributed 13 long balls to the offense.
Matt Carpenter has struggled and needs to pick up the pace, hitting just .218 with 10 home runs.
On the mounds for the A's, Tanner Anderson has only thrown 15 innings, posting an ERA of 4.20, striking out 12 with two home runs allowed.
Close Blake Treinen has 16 saves but has been far from automatic, posting a mediocre 4.08 ERA.
On the other hand, Liam Hendriks as been extremely effective, posting an ERA of 1.49 in 42 1/3 innings.
Yusmeiro Petit is another frequently used member of the bullpen and have been solid with an ERA of 2.55 over 42 innings.
As a team, the A's have allowed slightly above the league average at 4.8 per game.
Adam Wainwright enters with a record of 5-6 and ERA of 4.50. He' struck out 67 in 76 innings and has allowed a respectable nine home runs.
The biggest problem with the bullpen now is who will close after Jordan Hicks suffered a torn UCL in his right elbow.
John Gant, who has an ERA of 2.40 in 41 1/3 innings, could enter that role.
Other frequently used relievers include John Brebbia (3.52 ERA, 38 1/3 innings) and Giovanny Gallegos (2.27 ERA, 39 2/3 innings).
As a team, the Cardinals have given up 4.4 runs per game.
The is a very evenly match game with the A's having a slight edge offensive and the Cardinals slight edge on the mound.
The key could be who gets to which starter first because both bullpens are very good.
Balls should stay in the park with neither pitcher surrendering a huge number of home run balls.
Look for a tight, relatively low scoring game. Go with the "under" and the Cardinals to squeak out a 4-3 victory.
PICK- UNDER
Tampa Bay Rays vs Minnesota Twins-6/25/19- MLB Over Under Picks.
Seattle Mariners vs Milwaukee Brewers-6/27/19-MLB Over Under Picks.


The going advice for whether or not you should be dating a co-worker is plainly, 'don't'. You are not destined for a Jim and Pam style romance. Television has tricked you into thinking that everybody in the office wants to see you in a dating relationship and happy – but they don't.
So if, by chance or stupidity, you become one half of an office couple, the best advice we can give is to keep it on the down-est of the lows. Eventually everyone will find out you're dating but they will do their damnedest not to acknowledge it.
Keeping your feelings on the inside and covertly being a couple in the office is a mission and a half, though not impossible. Here are five tips for dating someone in the workplace on the sly.
When you leave the house (if you come to work together) leave the dating conversations at home. Try your best to revert to how you were before the fateful Christmas Party/Summer Party/Friday night lock-in. You'll be more productive and it'll make speaking at the end of the day all the more special.
We're asking you to dig deep into yourself people. Resurface that amazing skill of texting under the table that you used to do in school! It's easier than ever now, you don't even have to press a button four times to get the letter 's'. Emailing is too risky, so keep things casual and on the phone.
If you're dating seriously, you probably have breakfast together. Personally, I don't think eating a soggy bowl of cereal together is the most intimate a date a couple can have, but since the two of you will most likely be having dinner together too, use lunch as a chance to get some alone time or socialise with other co-workers.
Breaking up is something you never hope to have to think of but, being honest, you need to. The post-breakup awkwardness is the reason most shy away from office dating, but if you prepare in advance, you can make the whole transition smoother. If it's not working out, destroy the relationship cleanly and without collateral, then move on with your careers and lives.


All the way home from work, riding the bus, I dug deeper and deeper into negativity. Everything seemed stupid and useless and overwhelming, especially the endless masses and masses of cars going back and forth on four different busy roads. All going nowhere.. I thought of everything that was wrong with my life, that was hopeless and irreparable, all the insurmountable obstacles to overcome…. so bad I started swearing silently to myself, a whole nasty cauldron of inner negativity.
I glanced out the window and saw a car moving alongside the bus. A small compact car. The driver could barely see out the back it was so packed with various envelopes, boxes, stuffed shopping bags, and papers, and what looked like masses of crumpled plastic Jewel bags.. on top of all that an air filter, 20x20x1. Funny to notice that… I always seem to have to change my furnace filter.
As the car whizzed by I glanced inside and couldn't see the driver as the front passenger seat was also completely stuffed with junk, especially masses of those crumpled plastic bags. Like a homeless stash under Wacker Drive. I wondered if he was homeless and lived in his car. I assumed it was a he, maybe not.
That carload of garbage was like my mind.
The bus was full of sad and depressing looking people, tired, worn out, unattractive and badly dressed. Shame on me!!!! But that is the fact. I am sure I looked the same.
Then someone got on the next stop, and I realized it was the Jamaican woman I befriended a few months ago. She already noticed me as she was getting on struggling with the baby and carriage, but flashed me a big beautiful smile that lit up the whole sad bus interior. I went and helped her and we chatted happily the rest of the way home. I felt cleansed by her very presence.
Home, pulled out various remnants from Thanksgiving: pate, chutney, white bread, cashews, peanuts– too tired to pull out the turkey and veggies and make a decent meal. So much for the uplift. I am so lazy!
Stack of bills and papers piled up on desk to review. Again.
Made tea, took out pumpkin pie from Thanksgiving, took it into the dining room. I brought in the Cyclamen from kitchen and put it on the dining room table where it caught the light. A mass of fiery magenta birds.
The tea hot and cinnamony, the cup deliciously warm in my hands, the tasty pie my darling little sister made with her own little hands. Those precious hands that were almost stilled forever last March. It was a little uncooked in the center but was suave and luxurious in my mouth, the spicing just right. We all like plain pumpkin pie, so she acquiesced, not making the Elizabeth David version with macerated prunes and cognac. Too fussy. She spent four hours making it and rushing to get it done in time. I almost wept eating it.
Then some movement at the window– and I saw some tail feathers, but first I heard some piping like flutes or piccolos, and thought it was the radio. It was a group of birds singing sweetly in the shrubs outside the windows. I went to look and they flew away. Just a touch of their magic but it was enough to make me bright again. Their incredible winter presence. My little ballerinas in the evergreen shrubs.
Sitting here cup in hand, the harsh winter seemed to ease up a bit (actually it has just begun!), and I felt great peace, and realized that it started with the beautiful Jamaican Cindie, and her smile, an angel sent from God.


The welcoming aroma of the food says everything about the place. Saravana Bhavan is truly one of the legendary South Indian outlets in Delhi, at least according to the people who love eating South Indian delicacies. Heavenly food which is not even expensive makes up for a perfect combo. It also provides breakfast, home delivery. Plus, they serve Jain food.
Nothing, as this spot is simply awesome.
I would call you lucky if you did not have to wait to get your seat.
The Coconut Rava Masala Dosa, Filter Coffee, Crispy Dosa, Pomegranate Juice, Paper Roast, Paneer Dosa, Mini Idli comes highly recommended.


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Villa Eleni is a delightful, secluded villa, surrounded by cypress trees, olive trees and colourful aromatic flowers, offering absolute privacy in a quiet, ancient cretan village. The villa is complimented by a beautiful, infinity pool (7x4m., depth: 1.1-2.1m.)and enjoys views to the countryside, the White mountains and the sea beyond. The location is perfect, as it is within a short walk from Douliana village center, the minimarket and lovely tavernas.
The area is perfect for walking and biking as there are several country paths and quiet small roads.
including several local bakeries, butchers, fruit and vegetable shops as well as small supermarkets.
for the delicious fresh cooked local dishes and the reasonable prices.
The villa itself, is very well appointed, decorated in traditional style, with a fully equipped kitchen, two spacious bedrooms and two bathrooms. It is a perfect choice for sophisticated couples or small families!
Villa Eleni is licensed for holiday rentals with the Greek Tourist Authorities (EOT).
Douliana is a traditional and ancient Cretan village, it is reliant on sheep and olive farming for its main sources of income. A few beautifully restored and some new villas surrounded by beautiful countryside, a couple of family-run tavernas with a cool, tree shaded courtyard a mini market and a couple of cafeneions, make the village a perfect escape from the busy resorts.
There are amazing views over up to the White Mountains and beautiful country paths perfect for walking or biking. Gavalocori and Vamos, larger villages with a selection of great tavernas,traditional bakeries, small shops and artifact museums are less than 5 minutes drive and the sandy beach of Almyrida is just 3 kms away. distance of the villa. In Kalives at 3km there are small supermarkets, several tavernas and a choice of beaches.
The area is a nature lovers paradise, especially in Spring when the fields and lanes are a riot of wild flowers, and the nearby beaches are only minutes away by car. Archaeological sites abound in Crete, with one of the most interesting – Ancient Aptera – a short distance away. This site is still being slowly excavated, the latest discovery being a well-preserved Roman amphitheatre. More famous sites and Heraklion museum are feasible in a day. For other activities, there is a water park, Botanical Garden, and two lakes within easy reach, as is the city of Chania with its Venetian harbour, elegant Neo-classical houses, and fascinating alleys. Water sports such as sailing, windsurfing, and snorkelling can be arranged, as can day trips on a variety of boats or yachts from Chania.
Beautiful villa with fabulous pool and garden.
Perfect location, beautiful villa, lovely holiday!


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Photo: Casey B. Gibson
HECHT FLIES TO SECOND AT U23 TIME TRIAL CHAMPIONSHIPS
With just five days of racing under his belt, Gage Hecht launched his 2019 comeback tour into orbit with an incredible second place at the U23 time trial championships.
"With the way that the season played out with injuries, I came into the race feeling underprepared," Hecht said, who crashed badly during a training ride in late March. "I felt pretty good out on course and found a good rhythm. I felt smoother and smoother as the course rolled on. I did start to lose it a bit rolling up the final straightaway. I'm pretty thankful for the healing I've been blessed with. It has granted me the ability to compete with some of the best U23s in the country. I'm super excited to have finished on the podium this year and I look forward to the rest of the weekend here in Maryland."
The Parker, Colo. native completed the 30 kilometer rolling out and back course in a time of 37:59, 1'51" back from eventual winner, Ian Garrison.
"A great ride from Gage, especially considering it's his first TT of the year and he's just back from his injury," team director Mike Creed said.
Riley Sheehan and Andrew Vollmer, first and second in the 2018 junior TT, also competed in their first U23 national championships. Vollmer took 12thwith a time of 39:12 and Sheehan clocked 39:33 to claim 15thplace.


Serve your individual bite sized treats with Make'n Mold's® Foil Candy Cups. If you're candy making for a seasonal holiday or hosting a party with specific colors, our Foil Candy Cups come in just the colors you are looking for. From red & green to solid gold, use these foil cups for your homemade desserts.