text:
\section{Introduction}
In mathematics, science and engineering many problems lead to sparse
linear algebraic systems. For example, in a discretization of a smooth
object the relations between variables defined in neighbouring points
typically involve only a small number of variables that is dependent
on the dimension of the object which is low.

For sparse linear systems there are computer algebra programs available,
for example, the code of Roman Pearce \cite{RoPe} that has been
incorporated into {\sc Maple} 14: {\tt SolveTools:-Linear}. It is
automatically applied if a sparse system is to be solved.

The typical concern of solvers for sparse systems is to avoid a choice
of pivot that increases the number of variables in equations and thus
leads to non-sparse equations during the solution process. For many
problems this can not be avoided, only delayed. For example, when
solving the sparse linear system resulting from the discretization of
the (partial differential) Laplace equation (i.e.\ heat equation in
the stationary limit) it is clear that the temperature at each single
point depends on the temperature at {\em all} boundary points and that
the temperature inside is typically non-zero.\footnote{Even if the
  temperature on the whole boundary is zero except on an
  $\varepsilon$-sized part where it is positive, the temperature at
  all inner points will be positive.}

The linear systems we studied are also sparse but different in nature.
The value of most of their variables in the solution is exactly zero.
Such systems play the role of selectors and are therefore called ``selection
systems'' in this paper, because from a large number of monomials, all
with a constant undetermined coefficient, some linear combinations of the
monomials are selected which satisfy some condition and the other monomials
are dropped by setting their coefficient to zero. This is very different from
linear sparse systems like those resulting from the discretization of the
Laplace equation. Table \ref{table0} compares both types of problems.

\begin{table*}[ht]

   \begin{center}
   \begin{tabular}{|l|c|c|} \hline
    type    & ``numerical'' systems    & ``selection'' systems          \\ \hline
    examples& systems resulting from a & systems resulting from a sym-  \\
            & discretization of PDEs   & metry investigation of PDEs    \\ \hline
    value of free parameters   & any floating     & 0 or 1              \\
    when applying the solution & point numbers    & (to isolate the individual  \\
    of the linear system       & (boundary values of PDE) & symmetries) \\ \hline
    number of zero-valued      & essentially none & most variables \\
    variables in solution      &                  &                \\ \hline
    sparsity                   &      yes         &     yes        \\ \hline
    overdetermination          &       no         &     yes        \\ \hline
    usability of iteration     &                  &                \\
    schemes for large problems &   useful         & not useful     \\
    of that type               &                  &                \\ \hline
   \end{tabular}
   \end{center}
   \caption{Characterization of two different types of sparse linear systems}
  \label{table0}
\end{table*}

Selection systems have a number of useful properties.
\begin{itemize} \itemsep=0pt
\item The existence of zero-valued variables allows an effective solution as
  performed by the program {\sc LSSS}, described in this paper. {\sc LSSS} is
  running under the computer algebra system {\sc Reduce} (\cite{RedSource}).
\item From the application it is clear whether a linear system belongs to this
  class, i.e.\ whether many variables take the value zero and thus
  a specialized computer program should be applied.
\item The efficiency of solving selection systems increases with the
  complexity of the problem. For example, the higher the degree of the
  polynomial ansatz in a symmetry investigation - the larger is the size of
  the linear system for the unknown coefficients but the higher is also the
  percentage of zero-valued coefficients and the higher is the efficiency of
  the solution technique that uses zero-valued variables (see section
  \ref{charac}).
\item Not only can such linear systems be solved efficiently, they can also be
  formulated more economically. More precisely, it is possible to formulate
  much smaller equivalent systems as shown in section \ref{iter}.
\end{itemize}

The strategy to have many different methods available to solve a system of
equations and to give those methods the highest priority which make most
progress and are least risky to lead to an explosion of size is not new.  In
the {\sc Maple} command {\tt solve} this is used since 1985 even with the
heuristic of substituting zero variables first (see \cite{GoMo}) and in the
{\sc Reduce} package {\sc Crack} such strategies including substitution of
variables by zero as highest priority are applied to the solution of
overdetermined algebraic and differential systems (\cite{Wo04}. What is new in
this paper is the realization that if variables take the value of zero in
linear systems, then usually many variables vanish and then
very fast routines for identifying 1-term equations, for dropping such
variables from equations and even for avoiding the construction of equations
pays of.

What justifies the creation of special purpose programs for selection
systems is the fact that these systems occur frequently, especially in
integrability investigations of differential equations when computing
infinitesimal symmetries, first integrals, conservation laws,
variational principles or other qualitative properties which may but
need not exist and for which an ansatz can be formulated.

The following section describes an application that leads to a series
of sparse systems with a high percentage of zero-valued variables.
This application was the starting point for the development of {\sc
  LSSS}. Readers interested essentially in the computational aspects
of formulating and solving the systems may proceed directly to section
\ref{charac} which describes methods for formulating and solving
selection systems efficiently.

The times for solving the linear systems resulting in the applications
are shown in section \ref{results}.  The subsequent section
\ref{reduce} discusses aspects of the computer algebra system {\sc
  Reduce} which become important in large computations. Sections
\ref{maple} and \ref{linbox} report on tests of other computer algebra
systems. The paper concludes with a list of available procedures in
section \ref{procedures} and a summary.

\section{The application} \label{appli}

\subsection{A non-abelian ODE-system}

In the programme of M.\ Kontsevich of creating a proper
non-commutative algebraic geometry inspired by modern quantum
field-theoretic requirements and challenges, he proposed a
non-commutative version of symplectic geometry in which integrable
non-abelian ODE-systems appear naturally (\cite{MKpriv}).

More specifically, he considered the discrete map
\begin{equation} \label{dsym}
  u \ \rightarrow \ u v u^{-1} \ , \ \ \ \
  v \ \rightarrow \ u^{-1} + v^{-1} u^{-1}
\end{equation}
(\cite{MKtalk}) and the following non-abelian ODE system
which is invariant under (\ref{dsym}):
\begin{equation}\label{inhom}
u_t=u v -u v^{-1} - v^{-1}, \ v_t=- v u + v u^{-1} + u^{-1}
\end{equation}
where $u,v$ are non-commutative variables (in particular, square
matrices of arbitrary size). In comparison to non-abelian ODE
systems with polynomial right hand sides investigated first in
\cite{olsok} and later in \cite{MikSok}, \cite{Ef} the system
(\ref{inhom}) involves Laurent polynomials, i.e.\ polynomials in $u,v$
and inverses $u^{-1}, v^{-1}$.

Based on numerical experiments Kontsevich conjectured that
(\ref{inhom}) is integrable itself. This was demonstrated recently in
\cite{WolEfi11} where a Lax pair has been found which also proves
integrability of (\ref{dsym}) (see section 2 of \cite{MKtalk}). In
addition a pre-Hamiltonian operator is given in \cite{WolEfi11} that
maps gradients of trace first integrals to infinitesimal symmetries.
The existence of infinitely many symmetries and even better of a Lax
pair is taken as an indicator of integrability of an ODE or PDE system
(\cite{olver}). To verify that the Lax pair produces all Laurent
polynomial first integrals our strategy is to compute by brute force
all Laurent polynomial infinitesimal symmetries up to some degree and
to compare them with the symmetries produced from the Lax pair. This
paper reports on the computer algebra program {\sc LSSS} that was
developed to solve the linear algebraic systems resulting in the
computation of all such symmetries up to degree 16.

\subsection{Symmetries} \label{sy}
For a given system of equations
\begin{equation}
\label{eq2}
u_{t}=P_{1}(u,v,u^{-1},v^{-1}), \ \ \ v_{t}=P_{2}(u,v,u^{-1},v^{-1})
\end{equation}
where $P_1, P_2$ are polynomials in $u,v,u^{-1},v^{-1}$ a Laurent
polynomial (infinitesimal) symmetry is defined through two polynomials
$Q_1, Q_2$ in $u,v,u^{-1},v^{-1}$ such that the flow
\begin{equation}
\label{symm}
u_{\tau}=Q_{1}(u,v,u^{-1},v^{-1}), \ \ v_{\tau}=Q_{2}(u,v,u^{-1},v^{-1})
\end{equation}
commutes with the system, i.e.\
\begin{subequations}  \label{symcon}
 \begin{equation}
   D_\tau D_t \ u = D_t D_\tau \ u,  \label{symcon1}
 \end{equation}
 \begin{equation}
   D_\tau D_t \ v = D_t D_\tau \ v . \label{symcon2}
 \end{equation}
\end{subequations}
The vector $(Q_{1},Q_{2})^{t}$ is called generator of the symmetry.

In this paper the system (\ref{eq2}) is given through (\ref{inhom}).
The polynomials $Q_1, Q_2$ are generated by a special purpose procedure that
creates the most general (inhomogeneous) polynomials up to some given
degree in the non-abelian variables $u,v,u^{-1},v^{-1}$ with symbolic
(abelian) coefficients $c_i$. The only condition satisfied by
$Q_1,Q_2$ is that $u,u^{-1}$ and $v,v^{-1}$ are not standing next to
each other anywhere in any term as they would cancel.

The symmetry conditions (\ref{symcon}) have to be satisfied identically for
any $u, v$. That means, after (\ref{symcon}) is formulated, the coefficients
of all different products of powers of $u,v,u^{-1},v^{-1}$ have to be set to
zero generating a linear system for the unknown coefficients in the
symmetry generators(\ref{symm}). This process is called 'splitting' in the
remainder of the paper.

The number $\hat{t}_n$ of terms of each $Q_i$ of degree $n$ (which is one half
of the number of unknown coefficients) satisfies the recursive relation
$\hat{t}_n = 3\hat{t}_{n-1}+2$ with $\hat{t}_0=1$ because apart from the
coefficients $c_i$ the polynomial of degree 0 has only the term 1 and the most
general polynomial of degree $n$ is obtained by multiplying each term of the
most general polynomial of degree $n-1$ from one side, say from the left, with
all possible three of the four factors $u,v,u^{-1},v^{-1}$ which do not give a
cancellation. One exception is the term 1 that is multiplied with each one of
the 4 factors i.e.\ from this term results one extra term and the extra term 1
is added giving $\hat{t}_n = 3\hat{t}_{n-1}+1+1 = 3\hat{t}_{n-1}+2$.  Thus the
total number of terms $t_n$ occuring in $Q_1$ and $Q_2$ of degree $n$
satisfies $t_n = 3t_{n-1}+4$.

\subsection{Necessary symmetry conditions}
Additional information available for our symmetry computation results
from a separate computation of full first integrals $I$ satisfying
$D_t I = 0$ where $D_t$ is the total time-derivative. The condition
that a Laurent polynomial in $u,v$ is a first integral is very
restrictive because it implies that each of the $m\times m$ components
of the matrix first integral is an integral of its own. Restrictive
conditions lead to very overdetermined systems (here linear) which are
easier to solve. Therefore it was possible to compute all first integrals
up to degree 14 with the result that
\begin{equation} \label{FI}
 I = u v u^{-1} v^{-1},\ \ \ I^{-1} = v u v^{-1} u^{-1}
\end{equation}
and their integer powers $I^k, \ \ k=-3,..,3$ are the only first
integrals up to degree 14.

 From the symmetry conditions (\ref{symcon})
and $D_t I = 0$ follows
\[ D_tD_\tau I=D_\tau D_t I = 0\]
and further from $I^k$ being the only first integrals (up to degree 14)
\begin{subequations} \label{nc}
\begin{equation} \label{nc1}
 D_\tau I = \sum_{k=-k_0}^{k_0} a_k I^k,
\end{equation}
and similarly
\begin{equation} \label{nc2}
 D_\tau I^{-1} = \sum_{k=-k_0}^{k_0} b_k I^k,
\end{equation}
\end{subequations}
for sufficiently high $k_0$, in our case $k_0=3$ because $I$ is of degree 4.
These conditions involve a few more extra unknown constants $a_k, b_k$
but adding (\ref{nc2}) is beneficial because
these are first order conditions involving fewer terms than the
full symmetry conditions (\ref{symcon}) which are of second order.

\section{The solution of selection systems} \label{charac}

The linear systems resulting from splitting the necessary and
sufficient conditions (\ref{symcon}) and the additional necessary
conditions (\ref{nc}) cover a wide range of sizes as shown in table
\ref{table1}. As discussed in section \ref{sy}
the numbers $k_n$ of unknowns for a symmetry ansatz of degree $n$
grow like $k_{n+1}=3k_n+4$ which is also the growth rate of the number
$e_1$ of equations in the necessary condition $D_\tau I=0$.\footnote{The
   vanishing of constants $a_k$ in (\ref{nc1}) and $b_k$ in (\ref{nc2})
   follows when formulating these conditions.}

\begin{table*}[ht]
  \tabcolsep=3pt
  \begin{center}
  \begin{tabular}{|r|r|r|r|r|r|r|} \hline
$n$ &  $k$       & $e_1$      & $t_1$      & $e_2$       & $t_2$        & $p$ \\ \hline
 3  & 106        & 142        & 192        & 448         & 1,034        &  1 \\
 4  & 322        & 430        & 616        & 1,412       & 3,706        &  2 \\
 5  & 970        & 1,294      & 1,904      & 4,448       & 12,914       &  4 \\
 6  & 2,914      & 3,886      & 5,784      & 13,878      & 44,098       &  5 \\
 7  & 8,746      & 11,662     & 17,440     & 43,052      & 148,346      &  7 \\
 8  & 26,242     & 34,990     & 52,424     & 132,954     & 493,162      &  8 \\
 9  & 78,730     & 104,974    & 157,392    & 409,470     & 1,623,842    & 12 \\
 10 & 236,194    & 314,926    & 472,312    & 1,258,526   & 5,304,562    & 13 \\
 11 & 708,586    & 944,782    & 1,417,088  & 3,862,086   & 17,212,778   & 17 \\
 12 & 2,125,762  & 2,834,350  & 4,251,432  & 11,835,758  & 55,535,578   & 18 \\
 13 & 6,377,290  & 8,503,054  & 12,754,480 & 36,228,892  & 178,298,450  & 24 \\
 14 & 19,131,874 & 25,509,166 & 38,263,640 & 110,777,292 & 569,970,466  & 25 \\
 15 & 57,395,626 & 76,527,502 & $>1.1\!\times\!10^8$ & $>3.3\times 10^8$ & $> 1.7\times 10^9$ & 31 \\
 16 & $>172\!\times\!10^6$ &
      $>229\!\times\!10^6$ &
      $>3.4\!\times\!10^8$ &
      $>1\!\times\!10^9$   &
      $>5.7\!\times\!10^9$ & 32 \\ \hline

  \end{tabular}
  \end{center}
  \caption{
 For each symmetry ansatz of degree $n=3..16$ are listed the numbers:
 $k$ of variables,
 $e_1$ of equations and $t_1$ of terms of system $D_\tau I=0$,
 $e_2$ of equations and $t_2$ of terms of system $[D_t,D_\tau](u,v)=0$ and
 $p$ of free parameters of the solution.}
 \label{table1}
\end{table*}

As the table indicates, the linear systems have a few characteristic
properties that need to be exploited in order to compute high degree
symmetries. These properties are:
\begin{itemize}
\item {\em Overdetermination:} There are 4.5 to 5.5 times as
      many equations as unknowns.
\item {\em Sparsity:} Equations have on average 4 to 5 terms.
\item {\em Zero-valued Variables:} Most of the variables take an exact value of
      zero in the solution. Even more importantly, the percentage of zero-valued
      variables is increasing as the degree of the ansatz increases, i.e.\ as
      the system to be solved increases in size. For systems resulting from symmetries
      of degree 4 this percentage is 93.2\% and for symmetries of degree 16 the
      percentage is 99.966\% . Nevertheless, the solution for degree 16 is not trivial.
      It has 58118 non-vanishing variables and 32 free parameters.
\end{itemize}

\subsection{Ideas for a solver of large sparse linear algebraic systems} \label{ideas}
The following ideas take advantage of the features of selection
systems as listed above and describe the stages of development
of the computer algebra program {\sc LSSS}. These stages have been to:
\begin{itemize}
\item start with a program for solving a stream of equations,

\item sort equations initially by size, shortest first,

\item apply 1-term equations very efficiently and apply them
      before any other equation,

\item be able to generate only 1-term equations before formulating the
      whole system,

\item iterate between the exclusive formulation and the application
      of 1-term equations,

\item be able to generate and take advantage of extra
      necessary conditions before working on the original system, and

\item choose from possibly different options to generate 1-term
      equations the most efficient one.
\end{itemize}
Any extra low level procedures that were written to implement
these concepts assume that systems are linear in the unknowns.
This allows them to be more efficient than universal routines.

In the following subsections more detailed comments are made to each
of the measures.

\subsection{A stream of equations} \label{stream}
In earlier work \cite{TsaWo} large and very overdetermined polynomial
systems were solved which had too many equations (in the order of
$10^{14}$) to be even formulated initially. A linear algebraic system
solver was developed at the time for related problems dealing with the
system of equations as an incoming stream of data not to be stored in
RAM memory, only to be processed equation by equation. A strength of
this algorithm is that any limitation on available RAM memory poses
only restrictions on the size of the preliminary solution of the
system and thus only on the number of variables, not on the number of
equations.

At the start of this program a preliminary solution is initialized as
an empty list. Then with each incoming equation the following steps are performed.
\begin{itemize}
\item The equation is simplified modulo a preliminary
  solution stored in the form of a substitution list $g_i = f_i(g_j)$
  where $g_k$ are the unknowns and $f_i$ are linear expressions in the unknowns
  except $g_k$ from any of the left hand sides.
\item If the resulting simplified equation  is not an identity then it
  is solved for one of the $g_m$,
\item substitutions $g_m=f_m(g_j)$ are performed in all $f_i$ of the
  preliminary solution, and
\item $g_m=f_m(g_j)$ is appended to the substitution list.
\end{itemize}

In the computation of high order symmetries this procedure is applied to the
remaining system after all 1-term equations have been determined and used.
The remaining system is small enough that it does not have to be stored
on hard disk and then read again from disk.

\subsection{Sorting equations} \label{sort}
A first speedup is obtained by sorting equations according to size,
shortest first, before processing them as described above leading to
the times shown in column (B) of table \ref{table2}.  For example,
for symmetries of degree 9 the speed up is a factor of 4.

Processing shorter equations earlier means that the substitution list
involves shorter right hand sides and reduces incoming equations to
shorter size earlier which adds shorter new substitution rules to the
preliminary solution, i.e.\ it is a self-amplifying increase of efficiency.

In symbolic mode of {\sc Reduce} the sorting of equations can be done very
effectively, for example, by establishing a list $L=\{l_1,l_2,l_3,...\}$ where $l_i$
is a list of (pointers to) all equations with $i$ terms and then
linking these lists.
Assigning an equation to a list $l_i$ is done efficiently by having 2 pointers, one
stepping from one term to the next in the equation and the other at
the same time stepping from $l_i$ to $l_{i+1}$.  When the first
pointer reached the end of the equation, the other pointer gives
the list $l_i$ under which the equation is then listed.

\subsection{Applying 1-term equations} \label{apply}
Sorting equations is beneficial if equations vary much in size. This
is especially the case for selection systems
with many equations having only one term.

But not all the variables that take zero value in the solution need to
appear in 1-term equations in the original system. Many 1-term
equations may result only after the first 1-term equations are
applied. Still, during the solution process many 1-term equations are
generated which justifies their special treatment.

What makes 1-term equations special is the fact that replacing a
variable by zero in a polynomial (or linear) expression can be done
very fast without re-writing the expression, at least in a
lisp-representation: the pointer from the previous term to the term
that vanishes is simply re-directed to the next term. Also, no
re-simplification is necessary. In contrast, already the
application of 2-term equations requires afterwards
simplifications as the term resulting from
substitution may combine with other terms.

To set many variables in an expression to zero in a most efficient way, the
value cell of the variable was set to nil. Testing a variable for a zero value
can then be done by just testing whether the variable is bound.  After this is
done for all known vanishing variables, the (large linear) expression in these
variables is pruned only once by the special low level routine {\tt
  PruneZeros}. The pruning is done 'in place' avoiding copying of the whole
expression and also reducing future garbage collections.

All efficiency improving measures introduced in the sections
\ref{stream}, \ref{sort}, \ref{apply} are applied in the procedure
{\sc LSSS} \cite{liso} but can also be performed alone on given
expressions.

In the following subsections techniques are described that make the
formulation of the linear system more efficient, or avoid the
formulation of large selection systems.

\subsection{Selective splitting of equations} \label{selective}
The following steps are all computationally expensive:
\begin{itemize} \itemsep=0pt
\item the formulation of two large symmetry conditions (\ref{symcon}),
\item their separation (called 'splitting' in the following) into two large
  linear systems with many redundant equations by setting coefficients of
  different products of powers of $u, v$ separately to zero, and
\item millions of simplifications of the large linear systems due to
millions of vanishing variables.
\end{itemize}
The idea is to avoid most of these computations by
\begin{itemize} \itemsep=0pt
\item extracting selectively only 1-term equations from the
symmetry conditions (\ref{symcon}), and
\item using them to simplify the symmetry conditions themselves, and
\item to repeat this procedure as long as 1-term equations can be found (see section \ref{iter}).
\end{itemize}
Finally, the much smaller symmetry conditions are completely split and
the resulting linear system is solved using {\sc LSSS} as
described in the previous three subsections.

The only remaining large step is the initial formulation of the symmetry
conditions. One can save half of that computation by formulating only
one symmetry condition, extracting from it and applying to it
repeatedly 1-term equations, then pruning the ansatz for the symmetry
(\ref{symm}) before computing the second symmetry condition.

Furthermore, even the formulation of the first symmetry condition can
be postponed if short additional necessary conditions are known as
described in the next subsection. From them 1-term equations, i.e.\
vanishing variables can be extracted and the ansatz for the symmetry
(\ref{symm}) be pruned before any new symmetry condition is formulated.

The interplay between formulating and solving equations may be the
only way to approach a large problem which otherwise could even not be
formulated.  For example, this iterative approach was necessary to
compute symmetries of degree 14 on the available computer hardware
with 128 GB memory under PSL {\sc Reduce}. The same approach was also
applied in \cite{TsaWo} to solve a non-linear system which was too
large to be formulated at once.

The key to be able to selectively split up 1-term equations is a low level
procedure written in {\sc Symbolic Mode of Reduce} that takes only two lines.
It recursively steps through an expression and identifies 1-term coefficients
of any monomial in $u,v,u^{-1},v^{-1}$. If such a coefficient is found, it
directly sets the value cell of the variable to nil (see section \ref{apply}
for the case that a system of linear equations is given from which 1-term
equations are picked).

\subsection{Utilizing additional necessary conditions}
Apart from a system of equations to be solved sometimes additional
necessary conditions are available. The question is whether these
conditions can be utilized to speed up the solution of the original
system. If a program is performing its computation with the whole
system that is to be solved at once then solving in addition extra
conditions results in an increase of the total computation time. On
the other hand, if the program is able to extract information from the
system selectively then having extra necessary equations available
means that the program has more options and may be able to solve the
combined system faster than the original system alone.

If a program is able to generate repeatedly 1-term equations and to
apply them then it is beneficial for the program to have extra
necessary conditions (\ref{nc1}). Even if such opportunities do not
exist, extra necessary conditions can already be useful if the length
of equations varies much and if all equations are sorted by size and
if system plus extra necessary conditions are very overdetermined.

\subsection{Optimizing iterations} \label{iter}
An optimal strategy to speed up computation is not to find and use as many as
possible vanishing variables but to find and use as many as possible {\em per
time}. This means one wants to find an optimum between formulating new
conditions (\ref{nc}), (\ref{symcon})
which each provide many vanishing variables but which take
considerable time to formulate, or to utilize already formulated
conditions by extracting again vanishing variables, utilizing them, extracting
more, and so on. The second process is faster than formulating new
conditions but the number of vanishing variables that are found is gradually
decreasing as shown in table \ref{table10}.
The following comments refer to the computation of symmetries of
degree 13.\footnote{Given times refer to the computation of degree 13
  symmetries on a single CPU of a 8 core node (2 sockets x 4 cores per
  socket) Xeon @ 2.93 GHz using 48 GB memory of the node.}

After extracting vanishing variables from the necessary condition (\ref{nc1})
one has the options to
  \begin{itemize} \itemsep=0pt
  \item prune $D_\tau (u,v)$ (8 sec) and re-formulate (\ref{nc1}) (39 sec), or

  \item prune $D_\tau (u,v)$ (8 sec) and formulate the other necessary condition
  (\ref{nc2}) (39 sec)

  \item prune the already formulated condition (\ref{nc1}) (37.3 sec) \vspace{-3pt}

  \end{itemize}
  before extracting vanishing variables from that equation which was
  just formulated or pruned. Because in all 3 cases about 1,535,270
  new vanishing variables are found, the third option is the fastest
  and most efficient one which is therefore repeated several times.
\begin{table*}[ht]
  \tabcolsep=3pt
  \begin{center}
  \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline
runs             &   1   &   2   &  3   &  4  &  5 & 6  & 7  & 8   \\ \hline
$t_1$            &  116  &  37.3 & 14.6 & 12.2&11.7&11.7&11.7&11.7 \\
$t_2$            &  1.8  &  .49  &  .32 & .26 & .26&.25 &.25 &.25  \\
new              &5314423&1535271&288684&32076&3564&396 &44  & 0   \\ \hline
  \end{tabular}
  \end{center}
  \caption{Times and results for extracting 1-term equations repeatedly
in the computation of symmetries of degree 13.
$t_1$: time to formulate (\ref{nc1}) in run 1 and to prune it in later runs,
$t_2$: time to 1-term-split (\ref{nc1}),
new: number of new vanishing variables found in each run.}
 \label{table10}
\end{table*}
As shown on table \ref{table10} the number of vanishing variables that
are found, shrinks and the time to scan the condition stays constant,
thus it pays off after three runs to invest in the formulation of a
new condition
\begin{equation}
u_{\tau t} = u_{t \tau}.  \label{symcon3}
\end{equation}
After a first selective split of (\ref{symcon3}) it is most effective
to prune and selectively split (\ref{nc1}) again, even twice before
continuing to prune and split selectively (\ref{symcon3}). It turns
out that formulating in addition the second symmetry condition
\begin{equation}
v_{\tau t} = v_{t \tau}  \label{symcon4}
\end{equation}
only for the purpose of selective splitting does not result in
additional 1-term equations and is therefore not beneficial.

To summarize, denoting the pruning and selective splitting of
necessary condition (\ref{nc1}) by $n$ and the pruning and selective
splitting of symmetry condition (\ref{symcon3}) by $s$, the sequence
$n^3(snn)^4(sn)^4$ followed by a formulation of the symmetry condition
(\ref{symcon3}), a complete splitting of (\ref{symcon4}) and complete
splitting of what is left of (\ref{symcon3}) and a call of {\sc
  LSSS} to solve the linear system brings down the total time for
formulating and solving symmetry conditions for degree 13 to 467 sec,
compared to 3615 sec when straightforwardly formulating and splitting
(\ref{symcon3}), (\ref{symcon4}) and solving them with {\sc LSSS}.

An additional benefit of utilizing 1-term equations consists in a
drastic reduction of memory needs to only 13 GB for PSL {\sc Reduce}
or 7 GB for CSL {\sc Reduce} (see section \ref{reduce} about the
differences between PSL and CSL {\sc Reduce}) whereas the ad hoc formulation
of the large symmetry conditions (\ref{symcon3}), (\ref{symcon4})
requires 120 GB in PSL {\sc Reduce} or 60 GB in CSL {\sc Reduce} for
degree 13 symmetries.  Without selective splitting of 1-term
equations, i.e.\ determination of vanishing variables, it would not
have been possible to determine symmetries of degree 15 on the
computer with 128 GB and of degree 16 on a computer with
256 GB.

\subsection{Applying Solutions of large linear Systems}
In applications the solution of a system of equations has usually to be
substituted in expressions that are the main interest of the application. But
millions of variables to be substituted in expressions with millions of terms
is expensive. In such situations the labelling of zero-valued variables and
the pruning of large expressions containing them becomes useful again.

In table \ref{table2}, column (D) subst.\ shows drastically reduced
substitution times of the solution into the symmetry ansatz (\ref{symm})
compared to column (C) subst. After the computation in column (D) no explicit
substitutions are necessary. Any expression containing the unknowns needs
only a) to be pruned to drop terms involving zero-valued variables and b) be
simplified where non-zero variables get replaced by their computed value.

\section{Results} \label{results}

\begin{table*}[ht]
  \begin{center}
  \tabcolsep=3pt
  \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline
  &\multicolumn{2}{c|}{(A):}
  &\multicolumn{2}{c|}{(B):}
  &\multicolumn{2}{c|}{(C): 1-term equ.,}
  &\multicolumn{2}{c|}{(D):} \\
n &\multicolumn{2}{c|}{}
  &\multicolumn{2}{c|}{sorting by size,}
  &\multicolumn{2}{c|}{sorting by size,}
  &\multicolumn{2}{c|}{all techniques} \\
  &\multicolumn{2}{c|}{stream solve}&\multicolumn{2}{c|}{stream solve}
  &\multicolumn{2}{c|}{stream solve}&\multicolumn{2}{c|}{of section \ref{charac}} \\
   \cline {2-9}
  &    solve&  subst.&     solve&  subst. &    solve&  subst. &  solve & subst.  \\ \hline
 3&       .02      &      .01      &       .01      &      .00      &       .00      &       .00      &      .01     &    0  \\
 4&       .14      &      .01      &       .15      &      .02      &       .02      &       .02
                                                                                .03      &    0  \\
 5&      1.6 \ \   &      .17      &      1.3 \ \   &      .23      &       .20      &       .16
                                                                                .08      &    0  \\
 6&     20.8 \ \   &     1.46      &     15.3 \ \   &     2.4 \ \   &      1.7 \ \   &      1.3 \ \
                                                                                .26      &    0  \\
 7&    206 \ \ \ \ &     9.0 \ \   &     85 \ \ \ \ &     9.0 \ \   &     16.3 \ \   &     15.4 \ \
                                                                                .77      &    0  \\
 8&  2,740 \ \ \ \ &    82.5 \ \   &    808 \ \ \ \ &    92.8 \ \   &    241 \ \ \ \ &    223 \ \ \ \
                                                                                2.7 \ \   &   .01 \\
 9& 53,940 \ \ \ \ & 1,190 \ \ \ \ & 13,335 \ \ \ \ & 1,482 \ \ \ \ &  4,645 \ \ \ \ &  2,675 \ \ \ \
                                                                                9.1 \ \   &   .01 \\
10&                &               &                &               & 60,140 \ \ \ \ & 20,360 \ \ \ \
                                                                                30 \ \ \ \ &   .03 \\
11&                &               &                &               &                &
                                                                                96 \ \ \ \ &   .07 \\
12&                &               &                &               &                &
                                                                                302 \ \ \ \ &   .13 \\
13&                &               &                &               &                &
                                                                                927 \ \ \ \ &   .34 \\
14&                &               &                &               &                &                &  2284 \ \ \ \ &   .42 \\
15&                &               &                &               &                &                &  7587 \ \ \ \ &  1.58 \\
16&                &               &                &               &                &                & 27970 \ \ \ \ &  3.16 \\ \hline
  \end{tabular}
  \end{center}
  \caption{Times in sec for solving the symmetry conditions
           (\ref{symcon}) for each degree $n$ by different methods
           and for substituting the computed values into the symmetry
           ansatz (\ref{symm}).}
  \label{table2}
\end{table*}

The impact of measures described in the previous section is
shown in table \ref{table2}.
The entries in columns with the header 'solve' give the time
in sec to solve the linear algebraic system resulting from splitting
the symmetry condition (\ref{symcon}) (denoted in the following by
$D_{[t,\tau]}(u,v)=0$) wrt.\ monomials in
$u,v,u^{-1},v^{-1}$. Columns with the header 'subst.' show
the time to substitute the solution into the symmetry ansatz
(\ref{symm}).\footnote{Columns (A)-(C) have been run on one
Opteron 2.2 GHz core of a 32 core node (8 sockets x 4 cores per socket) with 128 GB memory.
In column (D)
$n=3..15$ were run on one Xeon 2.4 GHz core of a 16 core node with 128 GB memory and
$n=16$ was run on a single core of a machine equipped with 4 AMD Opteron
processors each having 12 cores running at 2.2.GHz and with 256 GB of main
memory.  Computation times turned out to be strongly dependent on the load on
the remaining cores of the node and the other nodes of the cluster.  The times in
column (D) are conservative times which are always reproducible. Sometimes
computations have been up to 30\% faster than shown in column (D).}

As described in the previous section, column (A) was produced with the
original stream solver \cite{streamsol}. Sorting equations by size
results in a speedup of a factor up to 4 shown in column (B) and column
(C) shows the times when 1-term equations are applied first as described
in section \ref{apply}. Computations in columns (A)-(C) use standard
substitutions and in column (C) the explicit formulation of 1-term equations
and complete splittings of equations.
The drastic improvement shown in column (D)
became possible through direct detection of vanishing variables (specialized
partial splitting), repeated partial splitting,
assigning nil to the value cell of a variable and thus avoiding substitutions
and enabling fast pruning of expressions from zero variables.

\begin{table*}[ht]
  \tabcolsep=3pt
  \begin{center}
  \begin{tabular}{|l|c|c|c|c|} \hline
                                   &      (E)           &       (F)           &    (G)     &   (H)     \\
                                  &$D_{[t,\tau]}(u,v)=0$&$D_{[t,\tau]}(u,v)=0$&$D_\tau I=0$& Iteration \\
                                   & at once            & in two stages       & first      & first     \\ \hline
formulation of ansatz for $D_\tau$ &    47.7            &    47.7             &  47.7      &   47.7    \\ \hline
multiple runs to find vanishing    &     $-$            &     $-$             &   $-$      &   777     \\
variables (see section \ref{iter}) &                    &                     &            &           \\ \hline
formulation and extraction of      &                    &                     &            &           \\
vanishing variables once from      &     $-$            &     $-$             &  238.7     &   $-$     \\
necessary condition  (\ref{nc1})   &                    &                     &            &           \\ \hline
formulation and complete           &                    &                     &            &           \\
splitting of $D_{[t,\tau]}u=0$     &
                                         1624           &
                                                             1624             &  621.3     &   4.5     \\ \hline
extraction of vanishing            &                    &                     &            &           \\
variables from complete set of     &     $-$            &    101.7            &   64.5     &  0.28     \\
conditions $D_{[t,\tau]}u=0$ once  &                    &                     &            &           \\ \hline
formulation and complete           &                    &                     &            &           \\
splitting of $D_{[t,\tau]}v=0$     &
                                        1274            &    46.9             &   46.7     &  1.77     \\ \hline
complete solution of all           &                    &                     &            &           \\
remaining equations                &     365            &    125              &   142      &  92.7     \\ \hline
substitution of solution           &                    &                     &            &           \\
in $D_\tau(u,v)$                   &     2.95           &     0.39            &   0.35     &  0.18     \\ \hline
total time (rounded)               &    3314            &   1946              &  1161      &  924      \\ \hline

  \end{tabular}
  \end{center}
  \caption{Times in sec of the whole symmetry computation for $n=13$}
  \label{table3}
\end{table*}

Table \ref{table3} compares four runs of the $n=13$ case.\footnote{All
  times in this table include CPU time and garbage collection time of CSL
  {\sc Reduce} running on one of 16 cores (4 sockets x 4 cores per socket)
  Xeon \@ 2.4 GHz node with 128 GB memory.}
In all these computations the program {\sc LSSS} is used to
solve the linear algebraic systems (the row 'complete solution of all
remaining equations'). {\sc LSSS} repeatedly applies 1-term equations
and sorts the remaining equations by size before starting the stream-solver.
What table \ref{table3} shows is the benefit of
generating and solving the linear algebraic system in stages.
With the availability of {\sc LSSS} as an effective solver of large
selection systems, the main cost shifted to the formulation of the linear
system.

In the first column the whole system is
formulated and solved at once. In the second column at first $D_{[t,\tau]}u=0$
is formulated, then 1-term equations are extracted and utilized to simplify
the ansatz for $D_\tau u$ and $D_\tau v$ before $D_{[t,\tau]}v=0$ is formulated and
the complete system is solved. This provides a speedup of nearly 1.7 for
$n=13$.

The third column gives the times when the whole computation is done in
3 stages.  In the first step the necessary condition
(\ref{nc1}): $D_\tau I = \sum_{k=-k_0}^{k_0} a_k I^k$ is
formulated, and once vanishing variables are extracted
which are used to simplify the symmetry ansatz (\ref{symm}) and speed up
the formulation of the system.
Although that first step is an
additional computation not performed in the first two runs costing an
extra 238 sec, this is more than compensated afterwards by the speedup
of computing and solving $D_{[t,\tau]}(u,v)=0$ (also in two stages)
leading to another overall speedup factor of about 1.7.

The fourth column reports the times when at first
a sequence of runs is performed where vanishing variables are extracted
and set to nil as explained in section \ref{iter}.
This leads to another modest overall speedup of 1.25 but the main advantage of
the computation in the fourth column is to lower the maximum memory
requirements that occur when the first half of symmetry conditions is formulated.

The next section describes how differences between two versions of the
computer algebra system {\sc Reduce} become important for large computations.
\begin{table*}[ht]
  \begin{center}
  \tabcolsep=3pt
  \begin{tabular}{|r|r|r|r|r|} \hline
 n &  (I)  &  (J)  &   (K)   &   (L)   \\
   &s solve&s tools&ns solve &ns tools \\ \hline
 3 & .0036 & .024  &  .035   &    .027 \\
 4 & .11   & .089  &  .11    &    .085 \\
 5 & .38   & .20   &  .3     &    .23  \\
 6 & 1.1   & .86   &  1.2    &    .80  \\
 7 & 4.3   & 2.97  &  4.9    &   3.75  \\
 8 & 20.4  & 13    &  24     &  16.9   \\
 9 & 128   & 73    &  162    &  81.3   \\
10 & 792   & 262   &  929    &   294   \\
11 & 7560  & 871   & 8615    &   972   \\
12 & DNF   & 3527  &   DNF   &  3232   \\
13 &       & 11786 &         & 11840   \\ \hline
  \end{tabular}
  \end{center}
  \caption{{\sc Maple} 14 times}
  \label{table2m1}
\vspace*{-20pt}
\begin{tabbing}
 \ \ \ \  ns tools: \= solving the union of both systems with the {\tt SolveTools:-Linear} command  \kill \\
 \ \ \ \  n         \> degree of the symmetry \\
 \ \ \ \  s solve:  \> solving the system (\ref{symcon}) with the
                       {\tt solve} command \\
 \ \ \ \  s tools:  \> solving the system (\ref{symcon}) with the
                       {\tt SolveTools:-Linear} command \\
 \ \ \ \  ns solve: \> solving both systems (\ref{nc}) and (\ref{symcon})
                       with the {\tt solve} command \\
 \ \ \ \  ns tools: \> solving both systems (\ref{nc}) and (\ref{symcon})
                        with the {\tt SolveTools:-Linear} command \\
 \ \ \ \  DNF:      \> does not finish in one week
\end{tabbing}
\end{table*}

\section{{\sc Reduce} issues} \label{reduce}
The open-source computer algebra system {\sc Reduce} can be run on two
different LISP implementations.  They are PSL (Portable Standard Lisp) and CSL
(Codemist Standard Lisp).

The computations of this article require a very large number of identifiers.
CSL has no restriction on the number of identifiers that can be used, whereas
PSL is usually limited to 65,000 identifiers. To run the symmetry computations
reported in this paper in PSL, an extended version of PSL {\sc Reduce} had
been developed by Winfried Neun that allows for 20 Mio identifiers. This
extended version of PSL can be downloaded for free \cite{reduce20m}.

When applied straightforwardly, PSL {\sc Reduce} typically runs somewhat
faster than CSL {\sc Reduce}.\footnote{The timings reflect the state of CSL/PSL
  at the time of writing the paper.  The performance of both LISP systems is
  under active development.}
However, CSL allows for compilation of critical routines
into C. When this is done, CSL runs in most cases as fast as PSL or even
faster. Compilation into C can be achieved by creating a package and making a
profiling run followed by a rebuild of the CSL-Reduce system.

A big advantage of CSL is its garbage collector. CSL uses both a copying and a
mark-sweep collector. Copying garbage collectors have little overhead but can
give only half of the memory to the application as the other half is needed
for copying. A mark-sweep garbage collector allows the use of all of the
memory by the application.  In CSL the copying collector is used as long as
the memory pressure is low, but switches to mark-sweep when the requirement
for more memory rises. This way, CSL leverages performance and available
memory. This feature of CSL made it possible to ultimately compute symmetries
of degree 15 with 115 GB of memory and symmetries of degree 16 on a different
machine with 256 GB.
\begin{table*}[ht]
  \begin{center}
  \tabcolsep=3pt
  \begin{tabular}{|r|c|c|c|c|c|c|} \hline
n   &   (M)   &   (N)   &  (O)  &  (P)   &  (Q): read       & (M/(O+P+Q)) \\
    & unsimp. & unsorted& sorted&simplify&simplified sys.&factor of speedup \\ \hline
3   & .024    & .0012   & .0014 &  0     &  .010      & 2.1 \\
4   & .089    & .0084   & .0084 &  .01   &  .012      & 2.9 \\
5   & .20     & .011    & .0092 &  .01   &  .012      & 6.4 \\
6   & .86     & .036    & .037  &  .03   &  .016      & 10  \\
7   & 3.0     & .047    & .047  &  .09   &  .021      & 19  \\
8   & 13      & .19     & .22   &  .31   &  .037      & 23  \\
9   & 73      & .41     & .38   & 1.04   &  .060      & 49  \\
10  & 262     & 2.6     & 2.7   & 3.44   &  .118      & 42  \\
11  & 871     & 4.7     & 4.9   & 11.2   &  .228      & 53  \\
12  & 3527    & 32      & 31    & 31.7   &  .443      & 56  \\
13  & 11786   & 78      & 71    & 87.5   &  .940      & 74  \\
14  &         & 402     & 387   &        & 1.574      &     \\ \hline
  \end{tabular}
  \end{center}
  \caption{{\sc Maple} 14 times for pre-simplified systems}
  \label{table3c}
\end{table*}

\begin{table*}[ht]
  \begin{center}
  \tabcolsep=3pt
  \begin{tabular}{|r|r|r|r|r|r|} \hline
   & {\sc Maple} 14
   & \multicolumn{3}{c|}{{\sc LinBox}}
   & {\sc Reduce} \\ \cline{2-6}
 n & SolveTools
   & default & \multicolumn{2}{c|}{sparse}
   & LSSS \\ \cline{2-6}
   & eqn.\ (\ref{symcon})
   & eqn.\ (\ref{symcon}) & eqn.\ (\ref{symcon}),(\ref{nc})  & eqn.\ (\ref{symcon})
   & eqn.\ (\ref{symcon}),(\ref{nc}) \\ \hline
 3 &     .024 &             &              &              &  .01 \\
 4 &     .09  &   .02       &     .02      &     .02      &  .03 \\
 5 &     .20  &             &     .13      &     .12      &  .08 \\
 6 &     .86  & 30.6\, \    &     .90      &    1.1\, \   &  .26 \\
 7 &    3     &             &   12.9\, \   &   14.9\, \   &  .77 \\
 8 &   13     &3080 \ \ \ \ &  210 \ \ \ \ &  283.5\, \   &  2.7 \\
 9 &   73     &             & 1812 \ \ \ \ & 2318 \ \ \ \ &  9.1 \\
10 &  262     &             &21210 \ \ \ \ &21610 \ \ \ \ &   30 \\
11 &  871     &             &              &              &   96 \\
12 &  3527    &             &              &              &  302 \\
13 &  11786   &             &              &              &  927 \\
14 &          &             &              &              & 2284 \\
15 &          &             &              &              & 7587 \\
16 &          &             &              &              &27970 \\  \hline
  \end{tabular}
  \end{center}
  \caption{A comparison of times of {\sc Maple, LinBox} and {\sc Reduce}}
\label{tablexx}
\end{table*}

\section{Comparison with {\sc Maple}} \label{maple}
In this section we report on applying the computer algebra system {\sc Maple} 14 on
solving our linear systems. Table \ref{table2m1} shows
times\footnote{Computations were run on one Xeon 2.4 GHz core of a 16 core
node with 128 GB memory} for solving the
symmetry conditions (\ref{symcon}) with the {\sc Maple} standard {\tt solve} command
in column (I) and with the {\tt SolveTools:-Linear} command in column (J) and
the times for solving the combined systems of additional conditions (\ref{nc})
and symmetry conditions (\ref{symcon}) in columns (K) and (L).

We see that for small systems, {\tt solve} and {\tt SolveTools:-Linear} take
similar times. For larger systems ($N=11$) {\tt SolveTools:-Linear} is 9 times
as fast and even bigger systems could not be solve with the command
{\tt solve} within a week.

Another observation is that neither {\tt solve} nor {\tt SolveTools:-Linear}
can take advantage of additional necessary equations except for $n=12$ in
column (L) to a small extent.
The best times for {\sc Maple} 14 in column (J) of table \ref{table2m1} and {\sc LSSS} ({\sc Reduce})
in column (D) in table \ref{table2} have been measured on the same nodes of
the same cluster and can be compared. To explain the 11786/927=12.7 times faster
performance of {\sc LSSS} for symmetries of degree 13 the following test has been made.
Table \ref{table3c} shows the times in sec for {\sc Maple} 14 procedure {\tt SolveTools:-Linear} to solve
\begin{itemize}
  \item the unsimplified system (\ref{symcon}) in column (M) (=column (J) of
        table \ref{table2m1}),
  \item the same system simplified after 1-term equations have been solved
        repeatedly in column (N),
  \item and then in addition all equations being sorted by size, shortest first in column (O).
\end{itemize}
The cost of these simplifications using the {\sc Reduce} procedures
{\sc FindZeros} to find and utilize all 1-term equations repeatedly and {\sc LengthSort}
to sort the remaining system by size is shown in column (P). The time for
{\sc Maple 14} to read the simplified system is given in column (Q). The rightmost
column gives the factor of speedup of {\sc Maple} if it would use at first the
{\sc Reduce} procedures to simplify the system before solving it.
The potential speedup of {\sc Maple} is impressive and increasing with
increasing size of the system.

\section{Comparison with {\sc LinBox}}  \label{linbox}
As described on its web page \cite{libo}
{\sc LinBox} is a C++ template library for exact, high-performance
linear algebra computations with dense, sparse, and structured matrices
over the integers and over finite fields.

Because the solution of our linear system (\ref{symcon}) contains free
parameters, {\sc LinBox} can not be applied straightforwardly to solve our
system.  However, what can be computed easily is the rank of these
systems. Table \ref{tablexx} compares the best times of {\sc Maple} solving
the system (\ref{symcon}) (column (J) in table \ref{table2m1}) with the best
times of {\sc Reduce} achieved when solving (\ref{symcon}) and using
additional necessary conditions (\ref{nc}) and with {\sc LinBox} at least
determining the dimension of the nullspace of (\ref{symcon})
and of (\ref{symcon}) together with (\ref{nc}). The timings of
{\sc LinBox} include reading of the sparse coefficient matrix of the linear
system which is the only way to enter data into LinBox.

\section{The procedures} \label{procedures}
The following four procedures are available for the formulation and solution of
linear algebraic system with an emphasis on selection systems which have many
zero variables in their solution.

The procedure {\sc LSSS} can solve linear systems with an arbitrary number
of equations and unknowns, i.e. the linear system can be under- or
overdetermined. In case the linear system is inconsistent a message will be
printed. The unknowns to be solved for must be ordered ahead of the rest of
the variables. The coefficients of the linear variables can be rational
functions of parameters.

The procedure {\sc FindZeros} repeatedly picks all 1-term equations from
an input list of equations and sets the value cell of the vanishing variables
to nil. It returns a list of vanishing variables and list of remaining
equations.

The procedure {\sc LengthSort} effectively sorts a list of equations
by their size.

The procedure {\sc PruneZeros} drops all terms of the input expression
which should be zero due to an earlier run of {\sc LSSS} or {\sc FindZeros}.

The procedures {\sc LSSS, FindZero} and {\sc LengthSort} are freely available from
\cite{liso} where more details about syntax are given.  These procedures
are also included as module in the open source computer algebra system {\sc
  Reduce}.  The server at \cite{liso} contains also files for all linear
systems for symmetries of degree 3 to 15 in {\sc Reduce} format and in
{\sc LinBox} format which can also be read by {\sc Maple}.

\section{Summary} \label{summary}
A class of sparse linear systems that allows efficient solution
strategies and that occurs frequently, for example, in integrability
investigations, is characterized by the vanishing of many of the
unknowns in its solution. In association with the purpose of such
systems we call them selection systems. A series of such systems that
we investigated results from symmetry investigations of a non-abelian
ODE system of Kontsevich. It is demonstrated how the vanishing of many
variables of a selection system can not only be used to speed up the
solution of the system but also to avoid formulating most of the
system and to apply the solution of the system to large
expressions. It is demonstrated how a pre-processor step in which
repeatedly 1-term equations are identified and applied and afterwards
the remaining equations are sorted by size could speed up {\sc Maple}
routines that are specialized in sparse systems by a considerable
factor.

\section*{Acknowledgement}
The first author would like to thank Winfried Neun for providing a PSL
version for large numbers of identifiers.  The second author would
like to thank Arthur C. Norman for very helpful discussions concerning
CSL.  Computations were run on computer hardware of the Sharcnet
consortium ({\tt www.sharcnet.ca}).

redpajama_set_name: RedPajamaArXiv


text:
SAM MERRILL
Without Giannis, Bucks beat Hawks 123-112 for 3-2 lead
Brook Lopez scored a playoff career-high 33 points and the Milwaukee Bucks withstood Giannis Antetokounmpo's absence and beat the Atlanta Hawks 123-112 on Thursday to regain the lead in the Eastern Conference finals.
Young out, Capela available for Hawks in Game 5 vs. Bucks
Atlanta Hawks guard Trae Young will join Milwaukee's Giannis Antetokounmpo in sitting out Game 5 of the Eastern Conference finals on Thursday night due to injuries.
Hawks dealing with injuries to Young, Capela in East final
The Atlanta Hawks are dealing with another injury that could affect their preparations for Game 5 of the Eastern Conference final.
Holiday, Bucks ink 4-year extension worth at least $134M
Milwaukee Bucks guard Jrue Holiday has agreed to a four-year contract extension that will be worth at least $134 million and possibly much more with incentives.
Bucks set NBA record for 3's, roll past Heat 144-97
On Tuesday, they set the NBA record for 3's in a game. The Bucks made a record 29 3-pointers in their 144-97 romp past the reigning Eastern Conference champion Miami Heat on Tuesday night. The previous record for 3's in a game was 27, set by the Houston Rockets against the Phoenix Suns on April 7, 2019. It was the biggest road win in Bucks history, topping the 39-point margin at New York on April 10, 1987. It was the first time since Feb. 13, 2017 that he played, scored less than 10 points and the Bucks won anyway.

redpajama_set_name: RedPajamaCommonCrawl


text:
Setback for new Aldi store in Flitwick after legal challenge from Tesco
Housebuilder relocates to Bedford business park
Housebuilder strikes £31m deal to build 80 luxury eco homes close to Bedford railway station
Bromham village scheme will play 'crucial' part in meeting company's delivery forecasts
By Reporter
Wednesday, 10th February 2021, 9:51 am
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Mark Booth, managing director of Hayfield, said: "Our live schemes in Silsoe and Clifton are among the best-selling developments in the county. But more of the same is never enough, so Hayfield Park will deliver something very different, featuring our brand new 'Green Revolution' homes.
"These 80 highly efficient homes will demonstrate how conserving energy and the environment doesn't need to make any compromises to our signature luxury style.
"This new development will play a crucial part in meeting our 2021/2 delivery forecasts and the continued support from OakNorth ensures we can keep growing the business at a rapid pace."
The site was marketed by Savills' Cambridge office, on behalf of land promoter The Fairfield Partnership. Thirty per cent of the homes have been designated for affordable housing.
Ben Barbanel, head of debt finance at OakNorth Bank, said: "We're in constant admiration when looking at Hayfield's growth story to date, as they continue to defy the challenges encountered by much of the property industry since the pandemic struck.
"This prime site in Bedfordshire sits well within our current sweet spot of funding high quality and spacious out of city centre residential projects.
"We have no doubt that Hayfield will be going above and beyond to create a new eco-friendly community and are once again delighted to be supporting Hayfield's development activity and enabling the rapid expansion of this innovative young housebuilder."
Flourish of awards take Bedford businesses flying into 2022
Plans for Aldi supermarket in Bedfordshire town have been resubmitted following legal challenge from Tesco
New larger Co-op store opening in Cranfield after £1.2m investment
Anything is paw-sible as purpose-built doggie day care centre near Bedford is unveiled
Bathstore goes into administration - but Bedford branch is staying open for now
Discount home store to open in Bedford - creating 75 jobs
50 new jobs at 'energy recovery' plant near Bedford
AudioMob founder and business graduate from Bedford Christian Facey makes Forbes 30 Under 30
Bedford's Swan Hotel owners submit last planning application as part of £5million refurbishment
35 new homes to be built in Bedfordshire


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Vojnik (in tedesco Hoheneck) è un comune di 8.368 abitanti della Slovenia centrale. È un piccolo borgo a Nord di Celje, lungo le rive del fiume Hudinja. Il luogo fu abitato dai Romani e di quell'epoca conserva un tratto di strada e molte pietre tombali incorporate nei muri delle chiese.

Altri progetti

Collegamenti esterni

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OpinionCase details
Young v. Weirich
Lee v. Bonner
In a prior case before this Court, the Court summarized the alleged facts involving the statutory rape…
1 Citing Cases
Full title:DELISHA YOUNG, ET AL., Plaintiffs, v. AMY WEIRICH, ET AL., Defendants.
Court:UNITED STATES DISTRICT COURT WESTERN DISTRICT OF TENNESSEE WESTERN DIVISION
Date published: Nov 26, 2018
No. 18-2157-JDT-tmp (W.D. Tenn. Nov. 26, 2018)
UNITED STATES DISTRICT COURT WESTERN DISTRICT OF TENNESSEE WESTERN DIVISION
No. 18-2157-JDT-tmp
DELISHA YOUNG, ET AL., Plaintiffs, v. AMY WEIRICH, ET AL., Defendants.
JAMES D. TODD UNITED STATES DISTRICT JUDGE
ORDER GRANTING MOTIONS TO PROCEED IN FORMA PAUPERIS, ASSESSING PLAINTIFF LEE'S PRO RATA FILING FEE UNDER THE PLRA, DENYING MOTION FOR SUMMARY JUDGMENT, DISMISSING COMPLAINT, CERTIFYING AN APPEAL WOULD NOT BE TAKEN IN GOOD FAITH AND DENYING LEAVE TO APPEAL IN FORMA PAUPERIS
On March 7, 2018, Plaintiffs Delisha Young, Tatyana McGee (T. McGee), Emma McGee (E. McGee) and Wendolyn Lee, booking number 18103194, filed a pro se civil complaint. (ECF No. 1.) Plaintiff Lee is incarcerated at the Shelby County Criminal Justice Center (Jail) in Memphis, Tennessee, and it appears he also was incarcerated when the complaint was filed. Each Plaintiff also filed a motion to proceed in forma pauperis. (ECF Nos. 1 & 2.) The Clerk shall record the Defendants as Amy Weirich, District Attorney General, and Chris Craft, Criminal Court Judge, both in the Thirtieth Judicial District of Tennessee.
Federal law provides that the "clerk of each district court shall require parties instituting any civil action, suit or proceeding in such court, whether by original process, removal or otherwise," to pay a filing fee. 28 U.S.C. §§ 1914(a)-(b). To ensure access to the courts, however, 28 U.S.C. § 1915(a) permits an indigent plaintiff to avoid payment of filing fees by filing an in forma pauperis affidavit. In this case, Plaintiffs Young, T. McGee and E. McGee have submitted properly completed and executed in forma pauperis affidavits. The information set forth in those affidavits satisfies Plaintiffs' burden of demonstrating that they are unable to pay the civil filing fee. Accordingly, the motions to proceed in forma pauperis filed by the three non-prisoner Plaintiffs are GRANTED.
The civil filing fee is $350. See 28 U.S.C. § 1914(a). The schedule of fees set out following the statute also requires the Clerk to collect an administrative fee of $50 for filing any civil case. However, the additional $50 fee does not apply if a plaintiff is granted leave to proceed in forma pauperis.
As stated, it appears that Plaintiff Lee was incarcerated when this case was filed. Under the Prison Litigation Reform Act (PLRA), 28 U.S.C. §§ 1915(a)-(b), a prisoner bringing a civil action must pay the filing fee required by 28 U.S.C. § 1914(a). Although the obligation to pay the fee accrues at the moment the case is filed, see McGore v. Wrigglesworth, 114 F.3d 601, 605 (6th Cir. 1997), partially overruled on other grounds by LaFountain v. Harry, 716 F.3d 944, 951 (6th Cir. 2013), the PLRA provides the prisoner the opportunity to make a "down payment" of a partial filing fee and pay the remainder in installments. Id. at 604. Therefore, the Court GRANTS Plaintiff Lee's motion to proceed in forma pauperis, and he is assessed his $87.50 share of the $350 filing fee in accordance with the terms of the PLRA.
Accordingly, pursuant to 28 U.S.C. § 1915(b)(1), Plaintiff Lee is ORDERED to cooperate fully with prison officials in carrying out this order. It is further ORDERED that the trust account officer at Plaintiff Lee's prison shall calculate a partial initial filing fee equal to twenty percent (20%) of the greater of the average balance in or deposits to Plaintiff Lee's trust account for the six months immediately preceding the completion of the affidavit. When the account contains any funds, the trust account officer shall collect them and pay them directly to the Clerk of the Court. If the funds in Plaintiff Lee's account are insufficient to pay the full amount of the initial partial filing fee, the trust account officer is instructed to withdraw all of the funds in the Plaintiff Lee's account and forward them to the Clerk of the Court.
On each occasion that funds are subsequently credited to Plaintiff Lee's account the trust account officer shall immediately withdraw those funds and forward them to the Clerk of the Court, until the initial partial filing fee is paid in full.
It is further ORDERED that after the initial partial filing fee is fully paid, the trust account officer shall withdraw from Plaintiff Lee's account and pay to the Clerk of this Court monthly payments equal to twenty percent (20%) of all deposits credited to Plaintiff's account during the preceding month, but only when the amount in the account exceeds $10, until his $87.50 share of the filing fee is paid.
Each time the trust account officer makes a payment to the Court as required by this order, he shall print a copy of the prisoner's account statement showing all activity in the account since the last payment under this order and submit it to the Clerk along with the payment. All payments and account statements should be sent to:
Clerk, United States District Court, Western District of Tennessee, 167 North Main St., Rm. 242, Memphis, TN 38103
and shall clearly identify Plaintiff Lee's name and the case number as included on the first page of this order.
If Plaintiff Lee is transferred to a different prison or released, he is ORDERED to notify the Court immediately, in writing, of his change of address. If still confined, he shall provide the officials at the new facility with a copy of this order.
The Clerk shall mail a copy of this order to the official in charge of trust fund accounts at Plaintiff Lee's prison. The Clerk is further directed to forward a copy of this order to the Director of the Jail to ensure that the custodian of Plaintiff's inmate trust account complies with that portion of the PLRA pertaining to the payment of filing fees.
The allegations of the complaint are somewhat convoluted, but Plaintiffs allege that Plaintiff Lee was arrested on rape charges after his step-daughter, Plaintiff T. McGee, accused him of raping her when she was fourteen years old. (ECF No. 1-1 at PageID 4.) The rape charges remain pending. Several months after Lee's arrest, T. McGee recanted her accusation of rape and stated instead that she artificially inseminated herself with Lee's sperm. (Id. at PageID 4-5; see also ECF No. 1-2 at PageID 6.) Plaintiff E. McGee, who is T. McGee's mother, asserts that she gave T. McGee a condom with Lee's sperm in it and told her to throw it away, but T. McGee instead took the sperm and inseminated herself with a syringe. (ECF No. 1-3 at PageID 8.) Plaintiff Young, who is T. McGee's sister, maintains that she helped T. McGee inseminate herself. (ECF No. 1-4 at PageID 9.) As a result, T. McGee became pregnant and gave birth to a child. (Id.; see also ECF No. 1-1 at PageID 5.)
Plaintiff T. McGee now alleges that Defendant Weirich, the Shelby County District Attorney, threatened to put her in jail if she did not say she was raped by Lee. (ECF No. 1-1 at PageID 5.) When T. McGee recanted and submitted her statement and corroborating statements by her mother and sister, Weirich allegedly became angry and refused to drop the rape charges against Lee. (Id.)
Plaintiff Lee has submitted various amendments to the complaint, the first of which was filed on September 7, 2018. (ECF No. 12.) He alleges that T. McGee accused him of rape after she became angry with her family. (Id. at PageID 48.) Plaintiff T. McGee and Weirich allegedly were in a gay relationship, and Weirich convinced T. McGee to come to Memphis to press charges against Lee because he knew of that relationship and had made derogatory comments about Weirich on Facebook. (Id. at PageID 48-49.) However, Lee alleges that even when T. McGee first made her false accusation, she stated the events occurred in West Memphis, Arkansas, not in Memphis, Tennessee; therefore, he contends Tennessee has no jurisdiction in the criminal proceeding. (Id. at PageID 49-50.) Weirich also allegedly told T. McGee that she could get Defendant Craft, a Criminal Court Judge, to help her convict Lee, so T. McGee should keep quiet about it. (Id. at PageID 50.) Lee alleges that Weirich was arrested by the FBI for false imprisonment, tampering with evidence, conspiracy, and sexual battery against Plaintiff Young. (Id.) He further alleges that it is a conflict of interest for Weirich to prosecute him when it was Young who had her arrested. (Id.) Lee also alleges that Defendant Craft issued an order for him to undergo a mental evaluation. (Id.)
The allegation that Weirich was arrested by the FBI or any other law enforcement agency appears to be demonstrably false. There are no charges pending against Weirich in any federal or Tennessee state court, and such a high-profile arrest would have been widely reported by the media.
Plaintiff Lee's second amendment was filed September 25, 2018. (ECF No. 13.) Lee repeats his allegation that Weirich was arrested, stating that she appeared in federal court for issuing false indictments. (Id. at PageID 62.) A pro se motion which Lee filed in his criminal case is attached, in which he alleges that Weirich forged the grand jury foreperson's name on the indictment; therefore, he contended he was never properly indicted and the criminal court was without jurisdiction. (ECF No. 13-2 at PageID 64-65.)
Lee also appears to be attempting to add several defendants, including the Shelby County Sheriff, the Shelby County Board of Supervisors, the Shelby County Mayor, and an Assistant District Attorney. (Id.) However, none of the individuals are identified, and there are no factual allegations made against them. Therefore, the Court DENIES leave to amend to add those defendants. --------
On October 11, 2018, Lee filed a motion for summary judgment. (ECF No. 14.) This case is being dismissed; therefore that motion is DENIED.
Lee sent a letter to the Clerk of this Court, which was docketed on November 1, 2018, in which he again repeats some of his allegations and also claims that Weirich was found guilty of withholding evidence but did not go to jail. (ECF No. 17.)
The Court is required to screen prisoner complaints and to dismiss any complaint, or any portion thereof, if the complaint—
(1) is frivolous, malicious, or fails to state a claim upon which relief may be granted; or
(2) seeks monetary relief from a defendant who is immune from such relief.
28 U.S.C. § 1915A(b); see also 28 U.S.C. § 1915(e)(2)(B).
In assessing whether the complaint in this case states a claim on which relief may be granted, the standards under Fed. R. Civ. P. 12(b)(6), as stated in Ashcroft v. Iqbal, 556 U.S. 662, 677-79 (2009), and in Bell Atlantic Corp. v. Twombly, 550 U.S. 544, 555-57 (2007), are applied. Hill v. Lappin, 630 F.3d 468, 470-71 (6th Cir. 2010). "Accepting all well-pleaded allegations in the complaint as true, the Court 'consider[s] the factual allegations in [the] complaint to determine if they plausibly suggest an entitlement to relief.'" Williams v. Curtin, 631 F.3d 380, 383 (6th Cir. 2011) (quoting Iqbal, 556 U.S. at 681). "[P]leadings that . . . are no more than conclusions, are not entitled to the assumption of truth. While legal conclusions can provide the framework of a complaint, they must be supported by factual allegations." Iqbal, 556 U.S. at 679; see also Twombly, 550 U.S. at 555 n.3 ("Rule 8(a)(2) still requires a 'showing,' rather than a blanket assertion, of entitlement to relief. Without some factual allegation in the complaint, it is hard to see how a claimant could satisfy the requirement of providing not only 'fair notice' of the nature of the claim, but also 'grounds' on which the claim rests.").
"A complaint can be frivolous either factually or legally." Hill, 630 F.3d at 470 (citing Neitzke v. Williams, 490 U.S. 319, 325 (1989)). "Any complaint that is legally frivolous would ipso facto fail to state a claim upon which relief can be granted." Id. (citing Neitzke, 490 U.S. at 328-29).
Whether a complaint is factually frivolous under §§ 1915A(b)(1) and 1915(e)(2)(B)(i) is a separate issue from whether it fails to state a claim for relief. Statutes allowing a complaint to be dismissed as frivolous give judges not only the authority to dismiss a claim based on an indisputably meritless legal theory, but also the unusual power to pierce the veil of the complaint's factual allegations and dismiss those claims whose factual contentions are clearly baseless. Unlike a dismissal for failure to state a claim, where a judge must accept all factual allegations as true, a judge does not have to accept "fantastic or delusional" factual allegations as true in prisoner complaints that are reviewed for frivolousness.
Id. at 471 (citations and internal quotation marks omitted).
"Pro se complaints are to be held 'to less stringent standards than formal pleadings drafted by lawyers,' and should therefore be liberally construed." Williams, 631 F.3d at 383 (quoting Martin v. Overton, 391 F.3d 710, 712 (6th Cir. 2004)). Pro se litigants, however, are not exempt from the requirements of the Federal Rules of Civil Procedure. Wells v. Brown, 891 F.2d 591, 594 (6th Cir. 1989); see also Brown v. Matauszak, 415 F. App'x 608, 612, 613 (6th Cir. Jan. 31, 2011) (affirming dismissal of pro se complaint for failure to comply with "unique pleading requirements" and stating "a court cannot 'create a claim which [a plaintiff] has not spelled out in his pleading'" (quoting Clark v. Nat'l Travelers Life Ins. Co., 518 F.2d 1167, 1169 (6th Cir. 1975))); Payne v. Sec'y of Treas., 73 F. App'x 836, 837 (6th Cir. 2003) (affirming sua sponte dismissal of complaint pursuant to Fed. R. Civ. P. 8(a)(2) and stating, "[n]either this court nor the district court is required to create Payne's claim for her"); cf. Pliler v. Ford, 542 U.S. 225, 231 (2004) ("District judges have no obligation to act as counsel or paralegal to pro se litigants."); Young Bok Song v. Gipson, 423 F. App'x 506, 510 (6th Cir. 2011) ("[W]e decline to affirmatively require courts to ferret out the strongest cause of action on behalf of pro se litigants. Not only would that duty be overly burdensome, it would transform the courts from neutral arbiters of disputes into advocates for a particular party. While courts are properly charged with protecting the rights of all who come before it, that responsibility does not encompass advising litigants as to what legal theories they should pursue.").
The complaint is filed on the form used for commencing actions pursuant to 42 U.S.C. § 1983, which provides:
Every person who, under color of any statute, ordinance, regulation, custom, or usage, of any State or Territory or the District of Columbia, subjects, or causes to be subjected, any citizen of the United States or other person within the jurisdiction thereof to the deprivation of any rights, privileges, or immunities secured by the Constitution and laws, shall be liable to the party injured in an action at law, suit in equity, or other proper proceeding for redress, except that in any action brought against a judicial officer for an act or omission taken in such officer's judicial capacity, injunctive relief shall not be granted unless a declaratory decree was violated or declaratory relief was unavailable. For the purposes of this section, any Act of Congress applicable exclusively to the District of Columbia shall be considered to be a statute of the District of Columbia.
To state a claim under § 1983, a plaintiff must allege two elements: (1) a deprivation of rights secured by the "Constitution and laws" of the United States (2) committed by a defendant acting under color of state law. Adickes v. S.H. Kress & Co., 398 U.S. 144, 150 (1970).
Any claims that Plaintiffs Young, T. McGee and E. McGee are attempting to assert on their own behalf fail to state a claim under § 1983, as the complaint does not explain how Plaintiff Lee's arrest and prosecution violates their own personal constitutional rights. The fact that Defendant Weirich did not drop the charges against Lee even after T. McGee's recantation does not amount to a constitutional violation against McGee herself. In addition, general allegations that Weirich is harassing their family do not state a constitutional claim.
Plaintiff Lee is asserting a claim of malicious prosecution. In order to succeed on such a claim, he must show the following:
First, . . . that a criminal prosecution was initiated against the plaintiff and that the defendant made, influenced, or participated in the decision to prosecute. Second, . . . that there was a lack of probable cause for the criminal prosecution. Third, . . . that, as a consequence of the legal proceeding, the plaintiff suffered a deprivation of liberty . . . . Fourth, the criminal proceeding must have been resolved in the plaintiff's favor.
Sykes v. Anderson, 625 F.3d 294, 308-09 (6th Cir. 2010) (citations, internal quotation marks and brackets omitted). In this case, Lee has failed to state a claim for malicious prosecution because the criminal proceeding is ongoing; therefore, he cannot show that the prosecution was resolved in his favor.
Furthermore, both Defendant Weirich and Defendant Craft are entitled to absolute immunity for the actions taken in the course of their duties as prosecutor and judge. Prosecutors are absolutely immune from suit for actions taken in initiating and pursuing criminal prosecutions because that conduct is "intimately associated with the judicial phase of the criminal process." Imbler v. Pachtman, 424 U.S. 409, 430-31 (1976). "A prosecutor's decision to initiate a prosecution, including the decision to file a criminal complaint or seek an arrest warrant, is protected by absolute immunity." Howell v. Sanders, 668 F.3d 344, 351 (6th Cir. 2012).
Likewise, judges are absolutely immune from civil liability in the performance of their judicial functions. Mireles v. Waco, 502 U.S. 9, 9-10 (1991); Stump v. Sparkman, 435 U.S. 349, 363 (1978); Bright v. Gallia Cnty., Ohio, 753 F.3d 639, 648-49 (6th Cir. 2014); Leech v. DeWeese, 689 F.3d 538, 542 (6th Cir. 2012). The "touchstone" for applicability of absolute judicial immunity is "performance of the function of resolving disputes between parties, or of authoritatively adjudicating private rights." Antoine v. Byers & Anderson, Inc., 508 U.S. 429, 435-36 (1993). In this case, Lee specifically alleges only that Defendant Craft ordered him to undergo a mental evaluation, which is within the scope of his judicial function. Therefore, any claim against Defendant Craft is barred by judicial immunity.
To the extent that Lee may be asking this Court to intervene in his criminal proceeding and order the charges dismissed, the Court cannot do so. Under the Anti-Injunction Act, 28 U.S.C. § 2283, "[a] court of the United States may not grant an injunction to stay proceedings in a State court except as expressly authorized by Act of Congress, or where necessary in aid of its jurisdiction, or to protect or effectuate its judgments." The Sixth Circuit has explained that "[t]he Act thereby creates 'an absolute prohibition against enjoining state court proceedings, unless the injunction falls within one of three specifically defined exceptions,' which are set forth in the statutory language." Andreano v. City of Westlake, 136 F. App'x 865, 879-80 (6th Cir. 2005) (quoting Atl. Coast Line R.R. Co. v. Bhd. of Locomotive Eng'rs, 398 U.S. 281, 286 (1970)). Federal injunctions against state criminal proceedings can be issued only "under extraordinary circumstances where the danger of irreparable loss is both great and immediate." Younger v. Harris, 401 U.S. 37, 45 (1971) (internal quotation marks and citation omitted). The Supreme Court has emphasized that
[c]ertain types of injury, in particular, the cost, anxiety, and inconvenience of having to defend against a single criminal prosecution, could not by themselves be considered "irreparable" in the special legal sense of that term. Instead, the threat to the plaintiff's federally protected rights must be one that cannot be eliminated by his defense against a single criminal prosecution.
Id. at 46. In this case, Lee does not allege the type of extraordinary circumstances that would permit the Court to become involved in his state-court criminal matter.
For all of the foregoing reasons, the complaint is subject to dismissal in its entirety for failure to state a claim.
The Sixth Circuit has held that a district court may allow a prisoner to amend his complaint to avoid a sua sponte dismissal under the PLRA. LaFountain v. Harry, 716 F.3d 944, 951 (6th Cir. 2013); see also Brown v. R.I., 511 F. App'x 4, 5 (1st Cir. 2013) (per curiam) ("Ordinarily, before dismissal for failure to state a claim is ordered, some form of notice and an opportunity to cure the deficiencies in the complaint must be afforded."). Leave to amend is not required where a deficiency cannot be cured. Curley v. Perry, 246 F.3d 1278, 1284 (10th Cir. 2001) ("We agree with the majority view that sua sponte dismissal of a meritless complaint that cannot be salvaged by amendment comports with due process and does not infringe the right of access to the courts."). In this case, the Court concludes that leave to amend is not warranted.
In conclusion, the Court DISMISSES the complaint for failure to state a claim on which relief can be granted, pursuant to 28 U.S.C. §§ 1915(e)(2)(B)(ii) and 1915A(b)(1). Leave to amend is DENIED.
Pursuant to 28 U.S.C. §1915(a)(3), the Court must also consider whether an appeal by any Plaintiff in this case would be taken in good faith. The good faith standard is an objective one. Coppedge v. United States, 369 U.S. 438, 445 (1962). The same considerations that lead the Court to dismiss this case for failure to state a claim also compel the conclusion that an appeal would not be taken in good faith. Therefore, it is CERTIFIED, pursuant to 28 U.S.C. §1915(a)(3) and Federal Rule of Appellate Procedure 24(a), that an appeal in this matter by any Plaintiff would not be taken in good faith. Accordingly, leave to appeal in forma pauperis is DENIED.
The Clerk is directed to prepare a judgment. IT IS SO ORDERED.
s/ James D. Todd
JAMES D. TODD
UNITED STATES DISTRICT JUDGE


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Author by Rue McClanahan
Genre eBook Biography & Autobiography
More than 30 years after it first aired, the Golden Girlsretains a steadily-growing cult following. The show ran for seven seasons, and over the years, these four fierce and sassy broads charmed millions of viewers with wit and verve, and most of all, the power of their friendship. Thank You For Being a Friend brings you the ultimate wisdom of Dorothy, Rose, Sophia and Blanche, with recipes, quotes sex tips, quizzes, style advice and the secrets to maintaining the lifelong bonds of friends who become family.
Author by Emma Lewis
Genre eBook Golden girls (Television program)
The Essential Fan Guide To The Golden Girls
Everything you need to know about the enduring hit TV comedy The Golden Girls is in the pages of this book. This is the definitive guide to The Golden Girls. More than thirty years after it first aired, perhaps more than any other TV show, this show's cult fan base continues to grow. It ran for seven seasons (collecting a staggering fifty-eight Emmy nominations and eleven wins along the way) and over the years, this hit comedy about four fierce and sassy roommates in Miami charmed millions of viewers with its incomparable wit. Above all, The Golden Girls celebrated the strength and depth of the friendship between its four iconic characters. The Essential Fan Guide to The Golden Girls brings you everything you need to know about the enduring hit TV comedy. Inside you'll find: overviews of its seven seasons; top episode guides; behind-the-scenes trivia; cast and writer profiles; guides to stand out guest stars; and plenty of deep dives into the unforgettable aesthetic of costumes and set design. Whether you're a new or old fan, this book will satisfy your craving for all things related to The Golden Girls.
Genre eBook Humor
All I Need To Know I Learned From The Golden Girls
If you're looking for the ultimate Golden Girls trivia book then this is not the book for you and you should probably look somewhere else. Same if you're looking for a summary of each episode, interviews with the cast, gossip from backstage, or anything factual or related to any sort of biographical or informative type of work. But, if you're looking for a fun and silly memoir that lovingly chronicles how the wit, wisdom, and storylines of the Golden Girls television show can touch our lives THEN this IS the book for YOU. We've been told that everything we need to know about life can be found in the lessons of kindergarten, Little Golden Books, or from the Twilight Zone. In truth, while those sources make great points, the real lessons of life can be found by hanging out with The Golden Girls. For over thirty years the wit and wisdom of those four sassy broads has withstood the test of time and brought to us not only a blueprint for living well, but also the knowledge we need to do good (in bed if Blanche has anything to do it) and be good (also in bed) as well as vivacious, strong, and fierce (those are both in bed and out of bed). This collection of anecdotes is replete with musings on life, death, love, and the many Italian dishes of Sophia Petrillo. Through the lens of one of the best shows ever to be on television (well, it's no T.J. Hooker, but c'mon, what is?) we go on a journey with the author that is both personal and amusing. The author engages us by relating the virtues and moral ideas gleaned from the show to incidents and characters from her own life. Personal narratives that include: the required endurance needed to be married to an Englishman (they're crazy, if you don't believe me, ask the Queen... I bet she has tales to tell about ol' Phillip), how to date ex-convicts when you don't know you are (seriously, this happened), and even life with a family that includes murderers and drug dealers (same person, but I have to make this exciting). All of this along with a back drop that includes tales from Miami in the 80s.So, grab your shoulder pads, a box of condoms, your St. Olaf yearbook, and a bucket of marinara and relax the one book guaranteed to change your life - even if the only change is having lost a few hours of it to the nonsense in these pages. Or, as the girls would teach us - grab life by the ziti, love each other to the fullest, and never give up on ourselves or our sex lives.
Author by T. A. Pryor


text:
Pojem rychlodráha může mít více významů:

 Rychlodrážní tramvaj – lehká (městská) železnice, spojující výhody (lehkých) tramvají a (rychlého) metra

konkrétní železniční spojení
 Rychlodráha Praha – letiště Ruzyně – Kladno – mnohaletý dopravní středočeský projekt z přelomu 20. a 21. století
 Rychlodráha z Prahy do Drážďan – českoněmecký dopravní projekt z 21. století
 Rychlodráha Praha–Brno – plánovaná vysokorychlostní trať z Prahy do Brna
 Rychlodráha z Říma do Milána – italská železniční trať
 a mnoho dalších

Externí odkazy

 Idnes.cz: GLOSA: Rychlovlak není důležitý pro Česko, chce ho Evropa (5/2016)

Železnice