text:
\section{Introduction}               \label{sec:1}

Let $\F$ be an arbitrary field and let $A,B,C$ be matrices of size
$n\times n$, $n\times m$ and $p\times n$, with entries in $\F$. These
matrices define a discrete time dynamical system through the
equations:
\begin{equation}                     \label{system}
  \begin{array}{ccl}
  x(t+1)&=&Ax(t)+Bu(t)\\
  y(t)&=&Cx(t).
  \end{array}
\end{equation}

An $m\times p$ matrix $K$ with entries in $\F$ defines the
feedback law:
\begin{equation}                   \label{feed-law}
u(t)=Ky(t).
\end{equation}
Applying\eqr{feed-law} to the system\eqr{system}, one gets the closed
loop system:
\begin{equation}
 x(t+1)=\left( A+BKC\right)x(t).
\end{equation}

The static output pole placement problem asks for conditions on
the matrices $A,B,C$ which guarantee that the characteristic
polynomial of the closed loop system, i.e., the characteristic
polynomial of the matrix $\left( A+BKC\right)$ can be made
arbitrary.

We can explain this problem also in terms of the so-called pole
placement map. For this, identify the set of monic polynomials of
degree $n$ of the form:
$$
s^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0\in \F[s]
$$
with  the vector space $\F^n$. Then we are seeking conditions
which guarantee that the pole placement map:
\begin{equation}                       \label{pole-map}
  \chi_{(A,B,C)}:\, \F^{m\times p}\longrightarrow\F^n,\hspace{3mm}
  K\longmapsto \det(sI-A-BKC)
\end{equation}
is surjective, or at least that the image contains a non-empty
Zariski-open set.

Many facets of this problem have been studied in the literature
and the reader is referred to~\cite{by89,ro98,wa92,wa96} where
also more references to the literature can be found.

If the base field is the complex numbers, then the major result
is due to Brockett and Byrnes~\cite{br81}:
\begin{theorem}                  \label{BrBy}
  If the base field $\F$ equals $\C$, the complex numbers, then
  $\chi$ is surjective for generic matrices $A,B,C$ if
  and only if $mp\geq n$.
Moreover if $mp=n$ and $\chi$ is surjective, then the general
fiber $\chi^{-1}(\phi)$ has  cardinality
\begin{equation}                       \label{cond-1}
d(m,p)= \frac{1!2!\cdots (p-1)!(mp)!}{m!(m+1)!\cdots(m+p-1)!}.
\end{equation}
\end{theorem}

In the next section we will go over the proof of
Theorem~\ref{BrBy} in the situation when the base field $\F$ is
algebraically closed and has characteristic zero. In
Section~\ref{sec:3} we will address the difficulties which
occur in positive characteristic. The main result of the paper is
a proof that Theorem~\ref{BrBy} holds over any algebraically closed
field in the case $n=mp$.

\section{Connection to Geometry and a proof
of Theorem~\ref{BrBy}  in characteristic zero} \label{sec:2}

Consider the transfer function $G(s):=C(sI-A)^{-1}B$ and
a left coprime factorization:
$$
G(s)=D^{-1}(s)N(s)=C(sI-A)^{-1}B.
$$
Over any field $\F$ we have the property that the $p\times (m+p)$
matrix $\left[ N(s)\ D(s)\right]$ has rank $p$ when evaluated at
an arbitrary element of the algebraic closure $\bar{\F}$ of $\F$.
In other words if $\lambda\in\bar{\F}$ then
$$
\mathrm{rank}\,\left[ N(\lambda)\ D(\lambda)\right]=p.
$$
It was the insight of Hermann and Martin~\cite{ma78} to realize
that every linear system $G(s)$ naturally defines a rational map
into the Grassmann variety Grass$(p,\F^{m+p})$:
$$
h:\ \mathbb{P}^1\longrightarrow \mathrm{Grass}(p,\F^{m+p}),\
s\longmapsto \mathrm{rowsp}\left[ N(s)\ D(s)\right].
$$
The map $h$ does not depend on the coprime factorization, and two
different linear systems $G_1(s)$ and $G_2(s)$ have different
associated rational maps. By the previous remark, the map is well
defined for every element $\lambda\in\bar{\F}$.  For this reason one
usually refers to $h$ as the Hermann-Martin map associated to the
linear system $G(s)$.

In order to arrive at an algebraic geometric formulation of the
pole placement problem, consider a left coprime factorization
$G(s)=D^{-1}(s)N(s)$ with the property that $\det (sI-A)=\det
D(s)$.  Then it is well known that the closed loop characteristic
polynomial can also be written as:
\begin{equation}\label{polypol}
  \det(sI-A-BKC)=\det\vier{I}{K}{N(s)}{D(s)}.
\end{equation}
Assume now that a desired closed loop characteristic polynomial
$\phi(s)$ factors over the algebraic closure as:
$$
\phi(s)=\prod_{i=1}^n(s-s_i),\ s_i\in\bar{\F},\ i=1,\ldots,n.
$$
The condition $\det(sI-A-BKC)=\phi(s)$ then translates into the
geometric condition:
$$
\mathrm{rowsp}\left[I\ K\right]\bigcap
\mathrm{rowsp}\left[N(s_i)\ D(s_i)\right]\neq \{ 0\}, \
i=1,\ldots,n.
$$

This formulation is closely connected to a theorem due to
Hermann Schubert:
\begin{theorem} \label{Schub} Given $n$ $p$-dimensional subspaces
  $U_i\subset \C^{m+p}$. If $n\leq mp$, then there is an
  $m$-dimensional subspace $V\subset \C^{m+p}$ such that
  \begin{equation} \label{cond-2} V \bigcap U_i\neq \{ 0\}, \
    i=1,\ldots,n.
  \end{equation}
  Moreover if $n=mp$ and the subspaces $U_i$ are in ``general
  position'', then there are exactly $d(m,p)$ (see
  Equation\eqr{cond-1}) different solutions $V\subset \C^{m+p}$
  satisfying Condition\eqr{cond-2}.
\end{theorem}

Theorem~\ref{Schub} was derived by Hermann Schubert towards the
end of the 19th century~\cite{sc1886,sc1891}. The mathematicians
at the time were not convinced with the proofs Schubert was
providing. The verification of the statements constituted
Hilbert's 15th problem, which he presented at the International
Congress of Mathematics in 1900 in Paris.  Theorem~\ref{Schub}
has been later verified rigorously and we refer to Kleiman's
survey article~\cite{kl76}.

It is not completely obvious how the geometric result of Schubert
implies Theorem~\ref{BrBy} of Brockett and Byrnes. The following
questions have to be addressed: \label{rem-1}
\begin{enumerate}
\item Given an $m$-dimensional subspace $\mathrm{rowsp}\left[K_1\
    K_2\right]\subset \C^{m+p}$, where $K_1$ is an $m\times m$
  matrix and $K_2$ is an $m\times p$ matrix. Assume
  $\mathrm{rowsp}\left[K_1\ K_2\right]\subset \C^{m+p}$ is a
  geometric solution, i.e.,
  \begin{equation} \label{cond-3}
    \det\vier{K_1}{K_2}{N(s_i)}{D(s_i)}=0, \ i=1,\ldots,n.
  \end{equation}
  Does it follow that $\left[K_1\ K_2\right]$ is row equivalent
  to $\left[I\ K\right]$ and $K$ represents a feedback law? For
  this to happen it is necessary and sufficient that $K_1$
  is invertible.
\item Assume $\mathrm{rowsp}\left[K_1\ K_2\right]\subset
  \C^{m+p}$ is a geometric solution in the sense of\eqr{cond-3}.
  Does it follow that $\det\vier{I}{K}{N(s)}{D(s)}$ is NOT the
  zero polynomial?
\item How is it possible to deal with multiple roots?
\end{enumerate}

These questions were all addressed in~\cite{br81}. A key
ingredient is the notion of non-degenerate system.
\begin{definition}
  An $m$-input, $p$-output linear system $G(s)=D^{-1}(s)N(s)$ is
  called degenerate, if there exist an $m\times m$ matrix $K_1$ and an
  $m\times p$ matrix $K_2$ such that $\left[K_1\ K_2\right]$ has full
  rank $m$ and
  \begin{equation} \label{cond-4}
    \det\vier{K_1}{K_2}{N(s)}{D(s)}=0.
  \end{equation}
  A system $G(s)$ which is not degenerate will be called
  non-degenerate.
\end{definition}
In more geometric terms, the Hermann Martin curve associated to a
non-degenerate system does not lie in any Schubert hyper-surface.

If $\left[N(s)\ D(s)\right]$ represents a non-degenerate system
of McMillan degree $n$, then $$\det\vier{K_1}{K_2}{N(s)}{D(s)}\neq
0$$ for any $\left[K_1\ K_2\right]$ of full rank.
If in addition $\left[K_1\ K_2\right]$ is a
geometric solution, then Condition\eqr{cond-3} is satisfied and
$\det\vier{K_1}{K_2}{N(s)}{D(s)}$ is a polynomial of degree at least
$n$. All the full size minors of $\left[N(s)\ D(s)\right]$ have degree
less than $n-1$, with the exception of the determinant of $D(s)$,
which has degree $n$. So the polynomial
$\det\vier{K_1}{K_2}{N(s)}{D(s)}$ cannot have degree $n$ unless $K_1$
is invertible. Hence it follows that a geometric solution for a
non-degenerate system results in a feedback solution $u=Ky$ on the
systems theory side.

Non-degenerate systems are therefore very desirable. The
following theorem was formulated in~\cite{br81} in the case when
the base field is the complex numbers.

\begin{theorem} \label{nondegenerate} Let $\F$ be an arbitrary
  field. If $n<mp$ then every system $(A,B,C)$ defined over $\F$ with
  $m$-inputs, $p$-outputs and McMillan degree $n$ is degenerate. If
  $\F$ is an algebraically closed field and $n\geq mp$, then a generic
  system $(A,B,C)\in\F^{n^2+n(m+p)}$ is non-degenerate.
\end{theorem}

The proof of the first part of the statement follows from basic
properties of
coprime factorizations of transfer functions. Indeed let the
$p\times (m+p)$ polynomial matrix $M(s)=\left[N(s)\ D(s)\right]$
represent an $m$-inputs, $p$-outputs system of McMillan degree
$n<mp$. Then, possibly after some row reductions, we find a
row of $M(s)$ whose degree is at most $m-1$. Using this row one
readily constructs a full rank $m\times (m+p)$ matrix such
that\eqr{cond-4} holds. This shows that $G(s)=D^{-1}(s)N(s)$ is
degenerate.

The second part of the statement, namely that a generic system defined
over $\F$ is non-degenerate, will be established through a series of
lemmas. Here $\F$ is an algebraically closed field of characteristic
zero.

Notice that it is enough to show that the set of degenerate systems
is contained in a proper algebraic set of $\F^{n^2+n(m+p)}$. In order
to prove this, we establish an algebraic relation between
the polynomial matrix $\left[N(s)\ D(s)\right]$ and the matrices
$(A,B,C)$. The following lemma is an ingredient of classical
realization theory. The proof and the concept of basis matrix is
found in~\cite{ro97a1}.

\begin{lemma} \label{lemma-1} Assume $G(s)=D^{-1}(s)N(s)$ is a
  left coprime factorization of a $p\times m$ transfer function
  of McMillan degree $n$. Then for every $p\times n$ basis matrix
  $X(s)$ there are matrices $A\in\F^{n\times n}, B\in\F^{n\times
    m}$ and $C\in\F^{p\times n}$ such that:
  \begin{equation} \label{cond-5} {\rm ker}_{\F(s)}\, [X(s) \;|\; N(s)\;|\;
    D(s)] \;=\; {\rm im}_{\F(s)}\, \left[ \begin{array}{cc} (sI_n-A) & B \\ 0
        & I_m\\ C & 0
      \end{array} \right].
  \end{equation}
  Furthermore $(A,B,C)$ is a minimal realization of $G(s)$,
  i.e.,
$$
G(s)=C(sI-A)^{-1}B,
$$
and for every minimal realization $(A,B,C)$ of $G(s)$ there
exists a basis matrix $X(s)$ such that\eqr{cond-5} is satisfied.
\end{lemma}

As pointed out in~\cite{ro97a1}, for certain basis
matrices $X(s)$ it is possible to compute $(A,B,C)$ just ``by
inspection''.

Using the previous lemma, one readily establishes the following:

\begin{lemma} \label{lemma-2} Assume that $(A,B,C)$ is a minimal
  realization of $G(s)=D^{-1}(s)N(s)$ and $\det(sI-A)=\det D(s)$.
  Then
  \begin{equation}
    \det\vier{K_1}{K_2}{N(s)}{D(s)}=\det\vier{sI-A}{B}{K_2 C}{K_1}.
  \end{equation}
\end{lemma}

As before, identify an $m$-inputs, $p$-outputs system $(A,B,C)$ of
McMillan degree $n$ with a point of $\F^{n^2+n(m+p)}$. Let $S$ be the set:
\begin{equation}                   \label{coincidence}
\left\{ ((K_1,K_2),(A,B,C))\in
  \mathrm{Grass}(m,\F^{m+p})\times \F^{n^2+n(m+p)} :
  \det\vier{sI-A}{B}{K_2 C}{K_1}=0\right\}.
\end{equation}
Since Grass$(m,\F^{m+p})$ is a projective
variety, the projection of $S$ onto $\F^{n^2+n(m+p)}$ is an algebraic
set. This follows from the main theorem of elimination theory (see,
e.g.,~\cite{mu76}). We have therefore established that the set of
degenerate systems inside $\F^{n^2+n(m+p)}$ is an algebraic
set. We establish the genericity result as soon as we can
show the existence of one non-degenerate system, under the assumption
that $n\geq mp$.

\begin{remark}
  In the case of proper transfer functions, the dimension of the
  coincidence set $S$ was computed in~\cite[Theorem~5.5]{ro94}.
  With this result it was then shown in~\cite{ro94} that the set
  of non-degenerate systems inside the quasi-projective variety
  of proper transfer functions contains a dense Zariski-open
  set as soon as $n\geq mp$.
\end{remark}

\begin{definition}
  Let $\F$ be an algebraically closed field of characteristic
  $0$. The {\em osculating normal curve} $C_{p,m}$ is the closure
  of the image of the morphism
  \begin{equation}\label{ONC}
    \begin{array}{ccl}
      \F & \longrightarrow & \mathrm{Grass}(p,\F^{m+p}) \\
      s & \longmapsto &
      \mathrm{rowsp}\left[\frac{d}{d^i}s^j\right]_{i=0,\ldots,p-1;\,
        j=0,\ldots,m+p-1.}
    \end{array}
  \end{equation}
\end{definition}
We denote by $d/d^i$ the $i$-th derivative with respect to $s$,
i.e.,
$$
\frac{d}{d^i}s^j= \left\{\begin{array}{ll}
    \prod_{k=0}^{i-1}(j-k) s^{j-i} & \mbox{if $j\geq i$} \\
    0 & \mbox{if $j<i$.}
  \end{array}\right.
$$

The osculating normal curve is an example of a non-degenerate
curve in the Grassmannian $\mathrm{Grass}(p,\F^{m+p})$.  An
elementary matrix proof of this fact was first given
in~\cite{ro90}. We will say more about it in the next section.
If $n>mp$ one constructs a non-degenerate system by simply
multiplying the last column of the matrix representing the
osculating normal curve by $s^{n-mp}$.

In the case $p=1$, this is the rational normal curve of degree
$m$ in $\PP^m\cong \mathrm{Grass}(1,\F^{m+1})$. In the case
$m=1$, the osculating normal curve is isomorphic to the rational
normal curve of degree $p$ in $\PP^p\cong
\mathrm{Grass}(p,\F^{p+1})$.

So far we have shown that if $mp\geq n$, then a generic system
is non-degenerate. Moreover, if $n=mp$, the system is non-degenerate and the
desired closed loop polynomial has distinct roots, then pole
placement is possible with $d(m,p)$ different feedback
compensators.

It remains to be addressed the question of multiple roots in the
closed loop polynomial. This has been done in the literature by
lifting the pole placement map\eqr{pole-map} from $\F^{m\times
  p}$ to the Grassmann variety Grass$(m,\F^{m+p})$. We follow
the arguments in~\cite{ro98}.

We can expand the closed loop characteristic polynomial as:
\begin{equation}
  \det\vier{K_1}{K_2}{N(s)}{D(s)}=\sum_\alpha k_\alpha g_\alpha(s),
\end{equation}
where $k_\alpha$ are the Pl\"ucker coordinates of
$\mathrm{rowsp}\left[ K_1\
  K_2\right]\in\mathrm{Grass}(m,\F^{m+p})$ and where the
polynomials $g_\alpha(s)$ are (up to sign) the corresponding
Pl\"ucker coordinates of $[N(s)\ D(s)]$.  Let $\PP^N$ be the
projective space $\PP(\wedge^m\F^{m+p})$ and let
$$
E_{(A,B,C)}:=\left\{ k\in \PP^N \mid \sum_\alpha
  g_\alpha(s)k_\alpha =0 \right\} .
$$
As shown in~\cite{wa96}, one has an extended pole placement map
with the structure of a central projection:
\begin{equation}
  \label{central}
  L_{(A,B,C)} \; :\;
  \PP^N-E_{(A,B,C)} \  \longrightarrow \
  \PP^n,\hspace{9mm}
  k\  \longmapsto \ \sum_\alpha  k_\alpha g_\alpha(s).
\end{equation}
A system $[N(s)\ D(s)]$ is non-degenerate if and only if:
$$
E_{(A,B,C)}\cap \mathrm{Grass}(m,\F^{m+p})=\left\{ \right\}.
$$
For a non-degenerate system, the extended pole placement map
$L_{(A,B,C)}$ induces a finite morphism:
\begin{equation} \label{pole-map-ext} \hat{\chi}_{(A,B,C)}:\,
  \mathrm{Grass}(m,\F^{m+p})\longrightarrow\PP^n,\hspace{3mm}
  \mathrm{rowsp}\left[K_1\ K_2\right] \longmapsto
  \det\vier{K_1}{K_2}{N(s)}{D(s)}.
\end{equation}

The inverse image of a closed loop polynomial $\phi(s)\in \PP^n$
under the map $L_{(A,B,C)}$ is a linear space which intersects the
Grassmann variety $\mathrm{Grass}(m,\F^{m+p})$ in as many points
(counted with multiplicity) as the degree of the Grassmann variety.
This is equal to Schubert's number $d(m,p)$.

This completes the proof of Theorem~\ref{BrBy} of Brockett and
Byrnes in the case $n=mp$ not only for the field of complex numbers,
but also in the
case when the base field is algebraically closed and has
characteristic zero. In Remark~\ref{n<mp} we will discuss how to extend
the proof to the case when $n<mp$.

In the next section we will discuss how to extend
Theorem~\ref{nondegenerate} to the case of an algebraically closed
field of positive characteristic.
We will show that it is much more tricky to establish the existence of
non-degenerate systems in the case when the base field has positive
characteristic.

\section{A proof of Theorem~\ref{BrBy} in positive
  characteristic} \label{sec:3}

Let $\F$ be an algebraically closed field of characteristic
$q>0$. Lemma~\ref{lemma-1} and Lemma~\ref{lemma-2} as formulated
in the last section only depend on techniques from linear algebra
and are true over an arbitrary field, so in particular over an
algebraically closed field.

If $X$ is a projective variety, $Y$ is a quasi-projective
variety, and $S\subset X\times Y$ is an algebraic subset, then the
projection of $S$ onto $Y$ is a Zariski-closed subset of $Y$
(see, e.g.,~\cite[Chapter I, Section 5.2]{sh94b1}). This shows that
Theorem~\ref{nondegenerate} also holds over an algebraically
closed field. In order to establish Theorem~\ref{BrBy}, we have to
show that there exists at least one non-degenerate system for any
choice of the parameters $p,m,n\geq mp$.
We will also show that a generic fiber contains $d(m,p)$ elements
when $n=mp$.

The last statement is true as soon as the extended pole placement
map $\hat{\chi}_{(A,B,C)}$ is separable~\cite[Chapter II, Section
6.3]{sh94b1}. This is indeed the case: $\hat{\chi}_{(A,B,C)}$ can
be seen as the composition of the Pl\"ucker embedding (which just
involves the computation of minors) and the linear map
$L_{(A,B,C)}$. Both maps are separable, and we conclude therefore
that the composition map is separable.

So there remains the problem of establishing the existence of
non-degenerate systems in the case $n\geq mp$. As we will show next, the
osculating normal curve may be degenerate in
characteristic $q>0$. In Section~\ref{MDS} we will provide alternative
examples of non-degenerate systems in positive characteristic,
while in Section~\ref{ff} we will discuss the case of finite fields.

\subsection{The osculating normal curve}

Although the osculating normal curve is defined over a field of
characteristic zero, its reduction modulo $q$ defines a curve in
$\mathrm{Grass}(p,\F^{m+p})$, which can again be regarded as the
closure of the image of morphism (\ref{ONC}).  If $p=1$, the
curve is the rational normal curve of degree $m$ in
$\mathrm{Grass}(1,\F^{m+1})\cong\PP^m$. In particular it is
non-degenerate.  Notice however that the reduction of the
osculating normal curve is degenerate whenever $q\leq p+m$, provided
that $p\geq 2$.
This is easily checked if $q<p$, since in this case the $(q+1)$-st
row of the matrix defining the curve is identically zero. If $p\leq
q\leq p+m$, consider the minor of the sub-matrix consisting of
columns $1,\ldots,p-1,q$. This sub-matrix has the form:
$$
\left[
  \begin{array}{ccccc}
    1  &\ s & \ldots\ &  s^{p-2}    & s^q\\
    0  & 1\ &         & (p-2)s^{p-3}& 0\\
    \vdots &    &\ \ddots &   \vdots    & \vdots \\
    \vdots &    &         &  (p-2)!     & \vdots \\
    0  &\ldots&       &     0       &  0
  \end{array}
\right]
$$
It follows that the corresponding minor is zero. By choosing a
compensator $[K_1\ K_2]$ whose sub-matrix consisting of the
``complementary columns'' $p,p+1,\ldots,q-1,q+1,\ldots,p+m$ is the
identity matrix and where all other elements are zero, one
verifies that the osculating normal curve is also degenerate in
this situation.

\begin{remark} \label{zerominor} If at least one minor of the
  matrix $\left[N(s)\ D(s)\right]$ is 0, then the system
  $G(s)=D^{-1}(s)N(s)$ is degenerate.
\end{remark}

Notice that if $q\gg 0$, then the reduction modulo $q$ of the
osculating normal curve is
non-degenerate. This reflects the usual fact that ``fields with
large enough characteristic behave like fields of characteristic
zero''.

The appearance of many zero entries in the matrix over a field
$\F$ of ``small'' positive characteristic $q$ is due to the fact that
many derivatives vanish. More precisely, let $h\in\{0,\ldots,q-1\}$
s.t. $j=h$ mod. $q$. Then
$$
\frac{d}{d^i}s^j= \left\{\begin{array}{ll}
    \prod_{k=0}^{i-1}(j-k) s^{j-i} & \mbox{if $h\geq i$} \\
    0 & \mbox{if $h<i$}
  \end{array}\right.
$$
This was one of the reasons that motivated Hasse to introduce the
following concept.

\begin{definition}
  The {\em $i$-th Hasse derivative} of a polynomial
  $u(s)=\sum_{j=0}^d u_js^j$ is defined as:
$$
\frac{\partial}{\partial^i}u(s)=\sum_{j=i}^d {j\choose i}u_i
s^{j-i}.
$$
\end{definition}

Observe that in characteristic 0 one has
$$
\frac{\partial}{\partial^i}=\frac{1}{i!}\frac{d}{d^i}.
$$
Moreover, none of the Hasse derivatives vanishes identically for
all polynomials, regardless of the characteristic of the base
field, whereas in characteristic $q>0$, the $i$-th derivative of
any polynomial is identically zero for all $i\geq q$.

It is therefore natural that we define the osculating normal
curve in positive characteristic using the Hasse derivative
instead of the normal derivative.

\begin{definition}
  Let $\F$ be an algebraically closed field of characteristic
  $q>0$. The {\em osculating normal curve} $C_{p,m}$ is the
  closure of the image of the morphism
  \begin{equation}\label{ONCP}
    \begin{array}{ccl}
      \F & \longrightarrow & \mathrm{Grass}(p,\F^{m+p}) \\
      s & \longmapsto &
      \mathrm{rowsp}\left[\frac{\partial}{\partial^i}s^j\right]_{i=0,
        \ldots,p-1;\, j=0,\ldots,m+p-1.}
    \end{array}
  \end{equation}
  where $\partial$ denotes the Hasse derivative.
\end{definition}

For $p\leq 2$ the definition agrees with the one given at the
beginning of this section. In particular, for $p=1$ we have a
non-degenerate rational normal curve of degree $m$ in
$\mathrm{Grass}(1,\F^{m+1})\cong\PP^m$. Notice also that the curve is
well defined even if $p>q$, as we do not generate a zero row
in the defining matrix.

Unfortunately, even with this adapted definition the osculating
normal curve is degenerate for many choices of the parameters,
as the following result points out:

\begin{proposition}\label{degen}
  Let $\F$ be an algebraically closed field of characteristic
  $q>0$. Assume that $q\leq m$. Then the osculating normal
  curve $C_{p,m}$ is degenerate.
\end{proposition}

\begin{proof} \smartqed
  By Remark~\ref{zerominor}, it suffices to show that one of the
  minors of the matrix:
  $$\left[{j\choose i}s^{j-i}\right]_{i=0,\ldots,p-1;\,
    j=0,\ldots,m+p-1}
  $$
  is zero. Consider the sub-matrix consisting of columns
  $0,\ldots,p-2,c$, where $c$  is a multiple of
  $q$, $c\in\{p+1,\ldots,p+m\}$. The corresponding minor is:
$$
\mathrm{det}\left[{j\choose i}\right]_{i=0,\ldots,p-1;\,
  j=0,\ldots,p-2,c}s={c\choose p-1}s=0.
$$
The first equality follows from the observation that the matrix
is upper triangular with ones on the diagonal, except for the
entry in the lower right corner which equals ${c\choose p-1}$.
\qed
\end{proof}

\begin{remark}
  If $q\mid p$ and $m\geq p$, the minor of the sub-matrix
  consisting of columns $p-1,p+1,\ldots,2p-1$ equals
$$
\mathrm{det}\left[{j\choose i}\right]_{i=0,\ldots,p-1;\,
  j=p-1,p+1,\ldots,2p-1}s^{p^2-1}=p s^{p^2-1}=0.
$$
The first equality follows from Lemma~9 in~\cite{ge85}.
\end{remark}

\begin{remark}
Degeneracy of the osculating normal curve over the field $\F_q$ with
$q\leq\max\{p,m\}$ also follows from Theorem~\ref{fieldsize}.
\end{remark}

In Proposition~\ref{degen} we saw that the osculating normal
curve may be degenerate over a field $\F$ of positive
characteristic $q$. Notice however that the curve may be
non-degenerate for certain choices of the parameters $p,m$. The
following example shows, e.g., that if the field $\F$ has
characteristic $2$, $m=1$ and $p$ is odd, then $C_{p,1}$ is
non-degenerate.

\begin{example}
  The curve $C_{p,1}$ is the closure of the image of the morphism
$$
\begin{array}{ccl}
  \F & \longrightarrow & \mathrm{Grass}(p,\F^{p+1}) \\
  s & \longmapsto &
  \mathrm{rowsp}\left[{j\choose i}s^{j-i}\right]_{i=0,\ldots,p-1;\,
    j=0,\ldots,p.}
\end{array}
$$
The minors of the matrix that defines the morphism are $${p\choose
  i}s^i\;\;\; \mbox{for $i=0,\ldots,p$.}$$
Hence the curve is non-degenerate if and
only if all the minors are non-zero, if and only if
$$q\nmid {p\choose i}\;\;\; \mbox{for any $i=0,\ldots,p.$}$$
\end{example}

Over a field of even characteristic, this is in fact the only
case when the osculating normal curve is non-degenerate.

\begin{corollary}
  Let $\F$ be an algebraically closed field of characteristic
  $2$. Then the osculating normal curve $C_{p,m}$ is degenerate,
  unless $m=1$ and $p$ is odd. In the latter case, $C_{p,1}$ is
  isomorphic to the rational normal curve of degree $p$ in
  $\PP^p$.
\end{corollary}

\subsection{Monomial systems and MDS matrices}\label{MDS}

\begin{definition}\label{monsyst}
A matrix $M(s)=\left[N(s)\ D(s)\right]$ is {\em monomial} if the
minors of all sizes of $M(s)$ are monomials.
A system $G(s)$ associated to a monomial matrix $M(s)$ is called
a {\em monomial system}.
\end{definition}

A monomial matrix $M(s)=[\alpha_{i,j}s^{d_{i,j}}]$ is determined by:
\begin{itemize}
\item the {\em coefficient matrix} $M=[\alpha_{i,j}]$,
\item the {\em degree matrix} $[d_{i,j}]$.
\end{itemize}
The degree matrix has the property that
$d_{i,j}+d_{k,l}=d_{i,l}+d_{k,j}$ for all $i,j,k,l$.

\begin{example}
The osculating normal curve defines a monomial system.
\end{example}

\begin{example}
  Let $\F$ be a field which contains at least three distinct
  elements $0,1,\alpha$. The matrix
$$
M(s)=\left[\begin{array}{cccc}
    1 & 0 & s^2 & \alpha s^3 \\
    0 & 1 & s   & s^2
  \end{array}\right]
$$
has minors
$$1, s, s^2, -s^2, \alpha s^3,
(1-\alpha)s^4.
$$
It therefore follows that $M(s)$ is a monomial matrix.  A direct
calculation shows that this system is non-degenerate.
\end{example}

\begin{definition}
A matrix $M$ with entries in $\F$ is {\em Maximum Distance Separable
  (MDS)} if all its maximal minors are non-zero.
\end{definition}

\begin{remark}
In coding theory a linear code $C\subset\F^n$ is called an MDS
code if all the maximal minors of the generator matrix of $C$ are
non-zero. This explains the choice of the name for these matrices.
\end{remark}

\begin{remark}
Let $M(s)$ be a monomial matrix. If the system associated to $M(s)$ is
non-degenerate, then $M$ is an MDS matrix. This follows from
Remark~\ref{zerominor}. It is not always the case that a monomial
matrix $M(s)$ with MDS coefficient matrix $M$ is non-degenerate.
\end{remark}

An example of degenerate $M(s)$ with MDS coefficient matrix is given
in the following example.

\begin{example}\label{MDSdeg}
Let $\F=\F_5$, the finite field of 5 elements.
 The following monomial system defined by the matrix
$$
M(s)=\left[\begin{array}{cccc}
    1 & s & s &  s^2 \\
    0 & 1 & 2 & 3s
  \end{array}\right]
$$
is left prime and
has an MDS matrix as coefficient matrix. Nonetheless the system is
degenerate as, e.g.,
$$
\left[ K_1\ K_2\right]:=
\left[\begin{array}{cccc}
    0 & 1 & 2 &  0 \\
    0 & 0 & 0 & 1
  \end{array}\right]
$$
results in the zero characteristic polynomial.
\end{example}

In the next theorem we show that an MDS matrix of given size
defined over a field $\F$ exists only if the ground field $\F$ has
enough elements.

\begin{theorem}\label{fieldsize}
Let $p,m\geq 2$ and let $M(s)$ be a monomial matrix of size
$p\times(m+p)$ defined over a field $\F$ with $q$ elements. If
$q\leq\max\{p,m\}$, then $M(s)$ is degenerate.
\end{theorem}

\begin{proof}
If $M(s)$ is non-degenerate, then its coefficient matrix $M$ is MDS.
Let $M^{\perp}$ be an $m\times (p+m)$-matrix defined over $\F$, such
that $\mathrm{rowsp}(M)=\mathrm{ker}(M^{\perp})$. Let $C^{\perp}$ be
the dual code of $C$. The generator matrix of $C^{\perp}$ is then
$M^{\perp}$. It is well known that $C$ is MDS if and only if
$C^{\perp}$ is MDS. Therefore, $M^{\perp}$ is an MDS matrix.

We want to show that if $M$ (resp. $M^{\perp}$) is MDS of size
$p\times (m+p)$ (resp. $m\times(m+p)$) defined over $\F_q$, then
$q\geq\max\{p+1,m+1\}$. The statement is symmetric in $p,m$, hence we
can assume without loss of generality that $2\leq p\leq m$. It
suffices to prove that $q\geq m+1$.

We first consider the case $p=2$. Since $M$ is MDS, every pair of
columns must be linearly independent. Over a field of $q$ elements,
there
are $q^2-1$ choices for the first column, $q^2-q$ for the second,
$q^2-2(q-1)-1$ choices for the third, and so forth. Since there are
$q^2-(m+1)(q-1)-1$ choices for the $m+2$-nd column, it must be
$q^2-(m+1)(q-1)-1=(q-1)(q-m)\geq 1$, hence $q\geq m+1$.

For an arbitrary $p$, we can assume that the matrix $M$ is of the form
$[I_p\ A]$, where $I_p$ is the $p\times p$ identity matrix and $A$ is
a matrix of size $p\times m$. The MDS property of $M$ translates into
the property that all the minors of all sizes of $A$ are
non-zero. Consider the submatrix $N$ obtained from $M$ by deleting the
last $p-2$ rows and the columns $3,\ldots,p$, $\ N=[I_2\ B]$ where $B$
consists of the first two rows of $A$. $N$ is a $2\times (m+2)$ MDS
matrix, since all the minors of all sizes of $B$ are non-zero. It
follows that $q\geq m+1$ for every $p\geq 1$.
\qed
\end{proof}

From the proposition it follows, e.g., that every monomial $M(s)$ of
size $2\times(m+2)$ defined over $\F_2$ is degenerate, unless
$m=1$. Clearly, there may be non-degenerate matrices which are not
monomial. E.g., the following is an example of a $2\times 4$
system
defined over $\F_2$ which is non-degenerate:

\begin{example}\label{2x4}
 Consider the system defined over $\F_2$  by the matrix
$$
M(s)=\left[\begin{array}{cccc}
    0 & s     & s+1 & s^2 \\
    1 & s^2+1 & 1   & s
  \end{array}\right].$$
The minors, listed in lexicographic order, are
$$
s, s+1, s^2, (s^3+s+1), s^4, s.
$$
A direct computation shows that the system is non-degenerate.
\end{example}

We conclude the paper with the main result.

\begin{theorem}
  Let $M(s)=[N(s)\ D(s)]$ be a monomial system having an MDS
  coefficient matrix $M$ of the form $M=[I_p\ R]$.  Let the
  degrees of the coefficient matrix be $d_{i,j}=j-i$ if $j\geq i$
  and zero else. Then $M(s)$ is non-degenerate of degree $mp$.
\end{theorem}

\begin{proof} \smartqed
Denote by $\alpha$ a multi-index
$\alpha=(\alpha_1,\ldots,\alpha_p)$ with the property that
$$
1\leq \alpha_1<\cdots \cdots < \alpha_p\leq m+p.
$$

Denote by $m_1(s),\ldots ,m_p(s)$ the $p$ row vectors of $M(s)$
and denote by $e_1,\ldots,e_{m+p}$ the canonical basis of
$\F^{m+p}$. One readily verifies that the Pl\"ucker expansion
of $M(s)$ has the form:
$$
m_1(s)\wedge \ldots \wedge m_p(s)=
\sum_{\alpha\in\left\{ n\atop p\right\}}
m_\alpha e_{\alpha_1}\wedge \ldots \wedge  e_{\alpha_p} s^{|\alpha|}
$$
where $|\alpha|:=\sum_{i=1}^p(\alpha_i-i)$ and $m_\alpha$ is the
minor of $M$ corresponding to the columns
$\alpha_1,\ldots,\alpha_p$.

The multi-indices $\alpha$ have a natural partial order, coming from
componentwise comparison of their entries.
If $\beta =(\beta_1,\ldots,\beta_p)$ is a multi-index, then
one defines:
$$
\alpha\leq \beta \ :\Longleftrightarrow \
\alpha_i\leq \beta_i\mbox{ for } i=1,\ldots,p.
$$

By contradiction assume now that $M(s)$ is degenerate. Let
$[K_1\ K_2]$ be a compensator which leads to the closed loop
characteristic polynomial zero:
\begin{equation}                    \label{zero-expan}
\det\vier{K_1}{K_2}{N(s)}{D(s)}=\sum_\alpha k_\alpha g_\alpha(s)=0.
\end{equation}
In the last expansion $k_\alpha$ denotes up to sign the $m\times
m$ minor of $[K_1\ K_2]$ corresponding to the columns $1\leq
\hat{\alpha}_1<\ldots<\hat{\alpha}_m\leq (m+p)$,
$\hat{\alpha}_i\not\in\{\alpha_1,\ldots,\alpha_p\}$.

$[K_1\ K_2]$ has a well defined row reduced echelon form with
Pivot indices
$\hat{\beta}=(\hat{\beta_1},\ldots,\hat{\beta_m})$. It follows
that $k_\alpha=0$ for $\alpha\not\leq \beta$. But this means that
the term $m_\beta s^{|\beta|}$ cannot cancel in the
expansion\eqr{zero-expan} and this is a contradiction. $M(s)$ is
therefore non-degenerate.
\qed
\end{proof}

\begin{remark}
If $n>mp$ choose $d_{i,m+p}=n-mp+m+p-i$ in order to obtain once
more a non-degenerate system of degree $n$.
\end{remark}

By establishing the existence of a non-degenerate
system, we have shown that Theorem~\ref{BrBy} holds true for
any algebraically closed field for $n=mp$.

\begin{remark}\label{n<mp}
  In order to prove Theorem~\ref{BrBy} in the situation when
  $n<mp$, one can show that for a generic system $(A,B,C)$
  the set of dependent compensators, i.e., the set of compensators
  which results in a zero closed loop characteristic polynomial,
  has minimum possible dimension, namely $mp-n-1$. This is clearly sufficient to
  establish the result. In order to prove this statement, one
  can proceed in two ways. Either one shows that the condition is
  algebraic and constructs an example of a system
  of degree $n$ satisfying the condition. Alternativeley one shows
  that the coincidence set $S$ introduced in\eqr{coincidence} has
  dimension $n^2+n(m+p)+mp-n-1$. The generic fiber of the
  projection onto the second factor has then dimension $mp-n-1$.
  This last argument was developed for the dynamic pole placement
  problem in~\cite{ro94}.
\end{remark}

\subsection{Non-degenerate systems over finite fields}\label{ff}

In this last subsection, we show that in general non-degeneracy
does not guarantee that the pole placement map is surjective over
a finite field.

\begin{theorem}
  Let $\F_2$ be the binary field. Then
  no non-degenerate system defined over $\F_2$
  induces an onto pole placement map:
  $$
  \mathrm{Grass}(2,\F_2^4)\longrightarrow\PP^4(\F_2).
  $$
\end{theorem}

\begin{proof}
  Let $M(s)$ be a non-degenerate matrix with entries in
  $\F_2[s]$. Let $\F$ denote the algebraic closure of
  $\F_2$ and let
  $$
  \chi:\mathrm{Grass}(2,\F^4)\longrightarrow\PP^4(\F)
  $$
  be the pole placement map associated to $M(s)$ over $\F$.
  $\chi$ is a morphism, since $M(s)$ is
  non-degenerate.  We will now show that the restriction of
  $\chi$ to $\F_2$-rational points
  $$
  \mathrm{Grass}(2,\F_2^4)\longrightarrow\PP^4(\F_2)
  $$
  is never surjective.

  Let $\mathrm{rowsp}(A)\in \mathrm{Grass}(2,\F_2^4)$. Denote by
  $A_{i,j}$ the determinant of the sub-matrix of $A$ consisting of
  columns $i$ and $j$. Then:
  $$
  \chi(A)=\left[\sum_{i<j}\chi_{ijk}A_{i,j}\right]_{k=0,\ldots,4}$$
  where $$\det\left[\begin{array}{c} M(s)\\

      A\end{array}\right]=\sum_{k=0}^4\sum_{i<j}\chi_{ijk}A_{i,j}s^k.
  $$
  Since the system is non-degenerate, the $5\times 6$ matrix:
  $$
  C=\left[\begin{array}{ccc}
      \chi_{120} & \ldots & \chi_{340} \\
      \vdots & & \vdots \\
      \chi_{124} & \ldots & \chi_{344}
  \end{array}
\right]$$ has full rank, hence its kernel is 1-dimensional and
generated by a unique element of $\F_2^6$. By non-degeneracy, the
generator of the kernel corresponds to a point in $\PP^6(\F_2)$
which does not belong to $\mathrm{Grass}(2,\F^4)$.
Hence we have the following
possibilities for the generator of $\ker C$:
$$(1,0,0,0,0,1),(0,1,0,0,1,0),(0,0,1,1,0,0),(1,1,1,1,1,1),$$
$$(1,1,0,0,0,1),(1,0,1,0,0,1),(1,0,0,1,0,1),(1,0,0,0,1,1),$$
$$(1,1,0,0,1,0),(0,1,1,0,1,0),(0,1,0,1,1,0),(0,1,0,0,1,1),$$
$$(1,0,1,1,0,0),(0,1,1,1,0,0),(0,0,1,1,1,0),(0,0,1,1,0,1),$$
$$(0,0,1,1,1,1),(0,1,0,1,1,1),(0,1,1,0,1,1),(0,1,1,1,0,1),$$
$$(1,0,0,1,1,1),(1,0,1,0,1,1),(1,0,1,1,1,0),(1,1,0,1,0,1),$$
$$(1,1,0,1,1,0),(1,1,1,0,0,1),(1,1,1,0,1,0),(1,1,1,1,0,0).$$
Observe that the problem is symmetric with respect to the
following changes of basis of $\F_2^6=\langle
e_{12},\ldots,e_{34}\rangle$ (which correspond to automorphisms
of $\mathrm{Grass}(2,\F^4)$) and composition thereof:\begin{itemize}
\item exchange $e_{12}$ and $e_{34}$ and leave the rest
  unaltered,
\item exchange $e_{13}$ and $e_{24}$ and leave the rest
  unaltered,
\item exchange $e_{14}$ and $e_{23}$ and leave the rest
  unaltered,
\item exchange $e_{12}$ and $e_{13}$, exchange $e_{34}$ and
  $e_{24}$,
\item exchange $e_{12}$ and $e_{14}$, exchange $e_{34}$ and
  $e_{23}$,
\item exchange $e_{13}$ and $e_{14}$, exchange $e_{24}$ and
  $e_{23}$.
\end{itemize}
Hence, reducing to the analysis of the following possibilities is
non-restrictive:
$$(1,0,0,0,0,1),(1,1,0,0,0,1),(0,0,1,1,1,1),(1,1,1,1,1,1).$$
Up
to a change of coordinates in $\PP^4$, we may assume that the
corresponding matrix $C$ is respectively:
$$\left[\begin{array}{cccccc}
    1 & 0 & 0 & 0 & 0 & 1 \\
    0 & 1 & 0 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 0 & 1 & 0 \\
  \end{array}\right],
\left[\begin{array}{cccccc}
    1 & 0 & 0 & 0 & 0 & 1 \\
    0 & 1 & 0 & 0 & 0 & 1 \\
    0 & 0 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 0 & 1 & 0 \\
  \end{array}\right],
\left[\begin{array}{cccccc}
    1 & 0 & 0 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 & 1 \\
    0 & 0 & 0 & 1 & 0 & 1 \\
    0 & 0 & 0 & 0 & 1 & 1 \\
  \end{array}\right],
\left[\begin{array}{cccccc}
    1 & 0 & 0 & 0 & 0 & 1 \\
    0 & 1 & 0 & 0 & 0 & 1 \\
    0 & 0 & 1 & 0 & 0 & 1 \\
    0 & 0 & 0 & 1 & 0 & 1 \\
    0 & 0 & 0 & 0 & 1 & 1 \\
  \end{array}\right].$$
Analyzing each case, it is now easy to prove that the
corresponding $\chi$ is not onto. E.g., in the first case we have:
$$
\chi(A)=\left[A_{1,2}+A_{3,4},A_{1,3},A_{1,4},A_{2,3},A_{2,4}\right]
$$
which is surjective if and only if the equations
$$
A_{1,2}+A_{3,4}=\alpha_0, A_{1,3}=\alpha_1,
A_{1,4}=\alpha_2,A_{2,3}=\alpha_3, A_{2,4}=\alpha_4,
$$
$$
A_{1,2}A_{3,4}+A_{1,3}A_{2,4}+A_{1,4}A_{2,3}=0
$$
have a solution in $\F_2^6$ for any choice of
$[\alpha_0:\ldots:\alpha_4]\in\PP^4(\F_2)$.  Letting $x=A_{1,2}$,
the equations reduce to:
$$
x^2+\alpha_0 x+\alpha_1 \alpha_4+\alpha_2 \alpha_3=0,
$$
which has no solution over $\F_2$ for
$\alpha_0=\alpha_1=\alpha_2=\alpha_4=1, \alpha_3=0$.

In the second case we have:
$$
\chi(A)=\left[A_{1,2}+A_{3,4},A_{1,3}+A_{3,4},A_{1,4},A_{2,3},A_{2,4}\right]
$$
which is surjective if and only if the equations
$$
A_{1,2}+A_{3,4}=\alpha_0, A_{1,3}+A_{3,4}=\alpha_1,
A_{1,4}=\alpha_2,A_{2,3}=\alpha_3, A_{2,4}=\alpha_4,
$$
$$A_{1,2}A_{3,4}+A_{1,3}A_{2,4}+A_{1,4}A_{2,3}=0$$
have a
solution in $\F_2^6$ for any choice of
$[\alpha_0:\ldots:\alpha_4]\in\PP^4(\F_2)$.  Letting $x=A_{3,4}$,
the equations reduce to:
$$
x^2+(\alpha_0+\alpha_4)x+\alpha_1 \alpha_4+\alpha_2
\alpha_3=0,
$$
which has no solution over $\F_2$ for
$\alpha_0=\alpha_1=\alpha_2=\alpha_3=1, \alpha_4=0$.

In the third case we have:
$$
\chi(A)=\left[A_{1,2},A_{1,3},A_{1,4}+A_{3,4},
  A_{2,3}+A_{3,4},A_{2,4}+A_{3,4}\right]
$$
which is surjective if and only if the equations
$$
A_{1,2}=\alpha_0, A_{1,3}=\alpha_1,
A_{1,4}+A_{3,4}=\alpha_2,A_{2,3}+A_{3,4}=\alpha_3,
A_{2,4}+A_{3,4}=\alpha_4,
$$
$$
A_{1,2}A_{3,4}+A_{1,3}A_{2,4}+A_{1,4}A_{2,3}=0
$$
have a solution in $\F_2^6$ for any choice of
$[\alpha_0:\ldots:\alpha_4]\in\PP^4(\F_2)$.  Letting $x=A_{3,4}$,
the equations reduce to:
$$x^2+(\alpha_0+\alpha_1+\alpha_2+\alpha_3)x+\alpha_1
\alpha_4+\alpha_2 \alpha_3=0,$$
which has no solution over $\F_2$
for $\alpha_0=\alpha_2=\alpha_3=0, \alpha_1=\alpha_4=1$.

In the last case we have:
$$
\chi(A)=\left[A_{1,2}+A_{3,4},A_{1,3}+A_{3,4},A_{1,4}+A_{3,4},
  A_{2,3}+A_{3,4},A_{2,4}+A_{3,4}\right]
$$
which is surjective if and only if the equations
$$
A_{1,2}+A_{3,4}=\alpha_0, A_{1,3}+A_{3,4}=\alpha_1,
A_{1,4}+A_{3,4}=\alpha_2,A_{2,3}+A_{3,4}=\alpha_3,
A_{2,4}+A_{3,4}=\alpha_4,
$$
$$
A_{1,2}A_{3,4}+A_{1,3}A_{2,4}+A_{1,4}A_{2,3}=0
$$
have a solution in $\F_2^6$ for any choice of
$[\alpha_0:\ldots:\alpha_4]\in\PP^4(\F_2)$.  Letting $x=A_{3,4}$,
the equations reduce to:
$$
x^2+(\alpha_0+\alpha_1+\alpha_2+\alpha_3+\alpha_4)x+\alpha_1
\alpha_4+\alpha_2 \alpha_3=0,
$$
which has no solution over $\F_2$ for
$\alpha_0=\alpha_1=\alpha_4=1, \alpha_2=\alpha_3=0$.
\qed
\end{proof}

\begin{acknowledgement}
  The work of both authors is supported by the Swiss National
  Science Foundation through grants \#123393, \#113251, and
  \#107887.
\end{acknowledgement}

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text:
When hosts go without: how to aid refugee camp hosts
Anna Jefferys
(Anna Jefferys/IRIN)
MBERA/BASSIKOUNOU
Since April, when the UN Refugee Agency (UNHCR) started biometric registration of the Malian refugees in Mauritania's Mbera camp, 8,000 of the 68,000 registered refugees have been de-activated from the system - almost all of them Mauritanians from nearby towns and villages hoping to access aid handouts. In September, a small group of camp residents tried to destroy the registration centre in protest of the new system, raising a key question: when and how should aid agencies helping refugees also try to assist vulnerable host populations?
The situation in Mauritania is by no means unique; many refugee camp populations are swollen with locals trying to access aid. Locals have even outstripped refugee populations in some crises, said a UNHCR source.
UNHCR and other aid agencies often assist host populations to allay potential tensions between refugees and locals, but doing so is typically not a first priority. Some agencies do it from the get-go, while UNHCR usually does so after the emergency phase of protracted refugee situations.
Loubna Benhayoune, head of the UN Office for the Coordination of Humanitarian Affairs (OCHA) in Mauritania, urges early aid, telling IRIN: "The locals are just as vulnerable [as the refugees], if not worse." OCHA is currently mapping out who is doing what, and where, for local populations.
UNHCR said its initial priority is to respond to the immediate refugee crisis. Just getting the basics - food, water, shelter and healthcare - in place for tens of thousands of refugees, in a situation where the agency had no staff, and no office within 1,000km "was quite a challenge," said Hovig Etyemezian, who runs the camp at Mbera for UNHCR. "Now we're focusing more on quality issues, including how to better assist host populations."
Locals impoverished
Mbera camp is situated in Hodh el Chargui, Mauritania's most food-insecure region, where 37 percent of inhabitants are at risk of hunger, according to the World Food Programme (WFP). The region received very little rain in 2011, and this year fared little better. In October, when pasture should appear in even the Sahel's northern desert zone, there was little in Hodh el Chargui.
A Joint Assessment Mission by WFP and UNHCR earlier this year revealed while 80 percent of refugee households were eating three meals a day, just 14 percent of nearby families were doing the same.
Mauritanians near the camp whom IRIN spoke to were gracious about the refugee presence. Bassikounou resident Fatouma Mint Lhessen told IRIN, "These people are our people. They are like our brothers and sisters, and we are here to help them."
But her family is struggling, too. She lives in a one-room house with her three children as her husband searches for work in the bush. Had she lived nearer the camp, she would have tried to access food aid, she said. "Prices are too high. I cannot afford to make sauce many days. In some ways, we are more refugees than the refugees!"
She continued: "We live. We survive. Sometimes we eat well, sometimes we eat nothing - rice, no sauce. I'm used to living on the edge. It's my routine."
"This area is impoverished," said UNHCR's Etyemezian. "Locals were experiencing malnutrition and hunger before the camp." He noted, however, that some locals had been receiving government food assistance for several years.
According to Mohamed Ould Zeidane, head of the planning department at the Interior Ministry, which manages the refugee response, vulnerable families in Hodh el Chargui were receiving government food aid, backed up by WFP. Following the protests in the camp, the Department of Hodh el Chargui was instructed to set aside US$2 million to help local populations. "Since then, tensions have eased," said Ould Zeidane. IRIN was unable to track the money flow.
Impact of refugees
Tensions often arise between refugees and host communities, who find themselves competing for scarce resources such as water, food or firewood, and sometimes for access to education, health services and infrastructure support, say numerous studies. Common pressures caused by refugee camps in desert environments are deforestation, depletion of the water table and degradation of water points, said a background note to the 2011 World Bank Development Report.
The area around Mbera refugee camp in eastern Mauritania has been denuded of plants or desert scrub as refugees have cut it down to cook their food. October 2013
Anna Jefferys/IRIN
http://www.irinnews.org/photo/
Photo: Anna Jefferys/IRIN
The area around Mbera camp has been denuded of trees
According to a UNHCR study on the impact of refugees on a host population in western Tanzania, refugees often use more firewood than their hosts, as it takes longer to cook the cereal rations they receive, and they keep their fires burning because they do not have enough matches. UNHCR and the NGO Lutheran World Federation have distributed energy-conserving stoves to many refugees in Mbera, but more are needed.
The population pressure at Mbera camp has put stress on an already fragile desert environment. The 68,000 or so Malians at Mbera tripled the local population, turning the camp into Mauritania's third-largest city. Wasteland surrounds the camp, with most of the bushes and trees denuded by refugees who collected firewood to cook, for example, though no impact assessment has as yet been undertaken.
Smart programming can help abate some of these challenges, said Ahmed Ibrahim, head of the NGO SOS Desert. The organization tried to slow deforestation by collecting dead wood from uninhabited areas to distribute to 20,000 refugees. Uncollected, this wood is usually eaten by termites and creates a bush fire risk, but local environmental ministry officials stopped the project, insisting the wood belonged to Mauritanians. Distributing it to both host and refugee populations would help relieve tensions, said Ibrahim, who is still discussing options with the government.
"In an ideal world, we would provide each household with a solar panel and with an energy-efficient cooking stove. but that would require a new generation of donors," said Etyemezian.
There is also increased pressure on water for livestock (the majority of refugees at Mbera are herders). UNHCR, SOS Desert, the German NGO SPANA and others are now trying to address how to improve water points along traditional grazing routes, among other potential efforts.
Often unrecognized are the benefits that refugees can bring to vulnerable populations.
201311181211590039.jpg
Malian refugees have started setting up boutiques selling basics - food, soap, condiments, even clothes - around Mbera camp in eastern Mauritania. October 2013
Interruption de l'aide alimentaire dans un camp de réfugiés maliens en Mauritanie
Malians are opening up small shops all over the camp
Numerous studies have shown that refugees create economic opportunities for themselves and their hosts, but there are winners and losers on both sides: Qualified Mauritanians benefit from job opportunities at the dozen or so aid agencies that have set up shop in Bassikounou, 18km from the camp, but the cheap labour offered by refugees can push locals out of already low-paid jobs.
Refugees may also expand commercial markets for local goods, but this, too, has a flip-side: in Tanzania, local farmers sold so much of their produce to the Congolese and Rwandan refugees that they did not leave enough for themselves, causing food prices to skyrocket. In Bassikounou, one wealthy landowner has raised his rental fee from 10,000 to 180,000 ouguiyas (from US$33 to $603) per month, making a handsome profit from the crisis, but local labourers cannot afford to deal with such rent hikes.
The only way to understand these dynamics is through an impact assessment and the only way to ascertain locals' needs is through detailed needs assessments, says Ann Maymann, head of UNHCR in Mauritania. "Targeting host populations will require needs-assessments and evaluations, all of which must be resourced," she said.
Arriving aid groups often bring assistance that benefits locals, as well. Mauritanians near Mbera can now water their animals at two reservoirs built just outside by the camp by Oxfam. UNICEF has helped build latrines in surrounding villages, and is working with a local partner to rebuild two primary schools in Mbera Village and a pre-school in Bassikounou. The NGO Solidarité extended electricity to people living in surrounding villages. Oxfam and the International Organization for Migration are building or rehabilitating dozens of wells outside of the camp, and SOS Desert has rehabilitated 12 wells used by pastoralists within 2km of the camp.
Perhaps most noticeable is the improvement in health services: Mauritanians can access any of the camp's four well-equipped, well-staffed clinics, three of them run by Médecins sans Frontières and one by the UN Population Fund. The clinics have sonogram machines, and can transport patients with complications to MSF's emergency surgery centre in Bassikounou. Ministry of Health nurse Fatimata Wahalatte Ahmed, who works in one of the camp clinics, estimated about 10 percent of the patients come from surrounding villages. She added that the camp had provided work opportunities for Ministry of Health staff.
Fabien Kabongo, head of MSF in Bassikounou, estimates 70 percent of the emergency surgery patients are Malians and the rest Mauritanians. Before, women had to travel to Nema, 240km away, when they needed a Caesarean section. "Two out of five died en route," said Cheride Kasongo, the surgeon at the centre.
Links to development
The best way to aid host populations is to link assistance with ongoing development efforts, say refugee aid experts, including UNHCR.
MSF runs an emergency surgery block in Bassikounou, 18km from Mbera camp. Most of the operations are C-sections. They say 70% of their clients are refugees from Mbera
Quand les hôtes n'ont plus rien : comment aider les populations qui accueillent le camp de réfugiés
An MSF-run emergency surgery block in Bassikounou, 18km from Mbera
But coordinating refugee aid and ongoing development aid projects is usually not done well, said the 2011 World Bank report. Creative development projects that might benefit both refugees and hosts are often the "exception rather than the rule" because of lack of donor support and lack of awareness of the opportunities that refugee populations can bring, according to the report.
While refugees can attract further development aid to a region, bringing benefits, one expert in the capital, Nouakchott, told IRIN she feared emergency aid for refugees could also cause development actors to diminish their activities. She said more research is needed in this area.
UNHCR head Maymann told IRIN: "Extending more help to the hosts now would be a nice bridge to the process of deactivating 'refugees'." She added that the government has not given any directives regarding targeting local populations, but has "welcomed it" when UNHCR has mentioned trying to increase their aid.
Since the protests at Mbera in September, there has been no visible tension between host communities and refugees, said Etyemezian. Individuals who feel they have been wrongly de-activated can bring their cases to UNHCR; so far, 5 percent of cases have been re-activated following an inquiry.
aj/rz
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redpajama_set_name: RedPajamaCommonCrawl


End of a short Era!
Walked away with just a few scratches from the airbag deployment and some small mussel tears which have just about healed.
A very good example of the safety features of the Exy's, as I still can't believe I walked away from it!
This is what I hit. The guys where all right as well. Only damage to their car was I moved tray and put a slight crease into the cabin and busted a spring on the rear left and of course the tail lights as well.
Will have a new car by the end of next week. Upgrading to the series 5, ST-L with options as well. Will keep you guys in the loop once I get it. Looking at the Tempest Blue as well.
South of the River - Where the Real Enthusiasts Live!!!
Good to hear of the only minor damage to you and looks like the ute got off better than the exy did.
Good to hear everyone is OK.
That nudge bar didn't help much at all, I suppose most of your Xy went under the ute.
Nudge bars are only there to stop minor scratches (wall tapping, car tapping) and hold the driving lights in a better location.
If you want more then get a bull bar.
But neither is of any use when you go under a ute.
Well I did find out that the nudge bar is actually chrome plated ally! I went to pick it up have a look and expected some weight, but it was like picking up a feather!
Going shopping today and hoping to get a good deal.
Good thing I planned to have today off at the start of the year! Was planning to go kayak fishing today.
Re: End of a short Era!
I definitely don't recommend your method as an excuse to upgrade my exy in the future.
It looks as if the ute tray managed to miss all of the usual crumple zones so most of the impact absorption was left to the bonnet and the airbags.
2011/12 Australia:- 2.5 litre petrol 4WD, Series IV ST, CVT with Nissan bonnet & headlight protectors, mats + after-market HD towbar with added step for old dogs. ;) Garmin GPS, K&N air filter, DashMate, Black Pearl Tint , Dirt. X-2429(c).
Oooh... Noooooo So bad .


Training/Jens: Recruitment training means to increase the percentage of the muscle fibers in use. This is what happens when you get stronger by the number of attempts on a crux move or at the campus board. For some it takes 45 minutes to get fully loaded meanwhile others are fully recruited directly after the warming up.
The most extreme way for recruiting the fibers in your muscles is to do downward moves at a campus board and this is how Wolfgang Güllich prepared himself for Action Directe.
Recruitment training is also perfect for optimizing the crimping angle of your grip or even adjusting different finger angels. Beginners should start by standing on the feet while doing moves on a campus board in order to avoid injuries. Stop the training or take longer rest as soon as you are stagnating.


text:
Digital Connectivity plays key role in London Infrastructure Plan 2050
The Mayor of London's office published a consultation on the London Infrastructure Plan 2050 today.
The core focus of the proposals is how ensure that London's infrastructure continues to underpin and improve its position as a global city – whilst dealing with a growing population.
Digital connectivity features prominently and is the topic of an accompanying paper. Ideas and actions outlined in this paper include:
The creation of a Connectivity Advisory Group, which will work with the Mayor and the wider Infrastructure Delivery Board
A city-wide mapping exercise to create a 'connectivity map' to help identify barriers to digital access
Improvements to the regulatory regime, including the permit scheme and the possibility of bringing planning applications for communications infrastructure under the Mayor's strategic remit
Consideration of an open communications network, a domestic property version of the SME voucher scheme, or other proposals to 'fill gaps in high speed connectivity'
There was also the eye-catching ambition for London to be the first capital city to deploy 5G.
The deadline for the consultation is 31 October and the BSG will be working with stakeholders over the coming weeks on a response.
Audience protection standards on Video-on-Demand Services
The government is calling for views and evidence on potential changes to the regulation of video-on-demand services. At present traditional... read more
New social housing in Wales must be ready for gigabit connections
The Welsh Development Quality Requirements 2021 - Creating Beautiful Homes and Places, is a document that sets out the minimum... read more
Project Gigabit: Open Market Review request for information – first release of Phase 2
Building Digital UK has launched an Open Market Review (OMR) to identify potential areas for government intervention to support... read more
Proposed modification to the Universal Service Conditions
Ofcom has launched a consultation to modify the universal broadband service rules, to clarify how BT should provide quotes to... read more


This weekend I visited Mum and stayed at Madge's place. Now the keen reader (yes, you again) will recall that it was Madge who was instrumental in my getting Minou back in November. I'd been staying with her and mentioned that Idgy and I wanted to get a new cat after the untimely death of darling Hank. Well good old Madge had hit Gumtree right away and that's how we found the wonderful Minou.
Anyway Madge has never met the lady herself and has always wanted to. "I'll bring her with me next time I visit Mum" said I in the casual throw away manner of one who hasn't really thought this through.
So when I rang Madge and said I was coming through she reminded me of my promise to bring The Lady Of The House with me. "No problem".
Saturday morning dawned and I brought out the cat box. This was Minou's cue to leave the house at something akin the the speed of an Olympic finalist.
On her return I grabbed her and encouraged her into the basket. She immediately adopted the starfish position so beloved of toddlers when you are trying to strap them into a car seat. However unlike a toddler who can have a tummy tickle and drop their arms, this cat was maintaining position at all costs. I prised her craws one at a time from the edge of the basket and tried to ease her in. She gripped harder and held firm.
It became a two man (woman) opperation to get her in. The other member of the party was the fourteen year old who was keeling over laughing and making all sorts of comments about how this journey could pan out and how glad she was Not to be coming with us.
Cat in basket and basket in car, the mood is not carnival. Minou starts to sing before I have turned the key in the ignition. Now when I use the word Sing I am using it in it's loosest form. Minou's singing compares not unfavourably with the noise made when you inadvertently stand on a squeaky toy.
Fourteen year old hanging over gate laughing as I drive away.
There is only one thing for it. Up the volume of Garth Brooks. She sings louder, Garth sings louder. I sing along with Garth in vain hope that the principal of 2 against 1 will work. It doesn't.
By the time we reach our destination after an hour and a quarter my ears are ringing with all the noise of our joint singing.
Madge is thrilled to meet Minou. The visit is a huge success.
When I brought out the basket for the return journey Monou braced herself and took a deep breath. I just knew she was preparing to sing.


In this final video in the series, Amy D. explains the difference in marketing and branding and how you can use that information when becoming the neighborhood expert in your farm. Learn how to distinguish yourself from others and learn the importance of effectively branding yourself.
0 Responses on Becoming a Neighborhood Expert Series Part Five: Branding vs. Marketing"
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text:
Archives|Link to Lobbyist Brings Scrutiny to G.O.P. Figure
Link to Lobbyist Brings Scrutiny to G.O.P. Figure
By KATE ZERNIKE and ANNE E. KORNBLUT MAY 23, 2005
WASHINGTON, May 18 - In Republican Washington, Jack Abramoff and Grover Norquist worked all the angles.
One was a $750-an-hour lobbyist, the other an antitax activist, and they helped drive the Republican takeover of the capital and cement the party's power. Both had a close ally in the House majority leader, Tom DeLay. And they shared a conservative ideology and a friendship going back to their days in college.
Now, with widening Congressional and criminal inquiries in the capital into Mr. Abramoff's dealings, they are sharing trouble, too.
While Mr. Abramoff has been under scrutiny for more than a year, Mr. Norquist has attracted unwelcome attention in recent weeks. A Congressional committee investigating whether Mr. Abramoff defrauded Indian tribes has subpoenaed records from Mr. Norquist's group, Americans for Tax Reform, after he refused for six months to turn them over voluntarily.
The Justice Department is reviewing records of an advocacy group Mr. Norquist started with Gale A. Norton, now secretary of the interior, after reports that Mr. Abramoff instructed Indian tribes to give it $250,000. And Mr. Norquist's name appears over and over in newly disclosed documents outlining Mr. Abramoff's work in the Northern Mariana Islands, an American protectorate in the Pacific, which Democrats are agitating to investigate.
In interviews, Mr. Norquist dismissed any suggestion of wrongdoing on his part and said that the only reason he is "getting dragged into this" is because Senator John McCain, the head of the Senate Indian Affairs Committee, which is investigating Mr. Abramoff, holds a grudge against Mr. Norquist for campaigning for President Bush in the 2000 Republican primaries.
"McCain hates me," he said.
Mr. McCain, Republican of Arizona, said in a statement that he subpoenaed the records only after six months of negotiations with Mr. Norquist failed.
Temperamental and sound-bite-ready, Mr. Norquist, 48, has had a hand in nearly every conservative cause in Washington for nearly two decades -- from naming institutions after Ronald Reagan to the current battle over judicial nominees -- largely through his Wednesday meetings, weekly beat-the-drum sessions attended by lobbyists, think tank employees and politicians that command such influence that the Bush administration routinely sends an envoy.
With Mr. DeLay's eager backing and Mr. Abramoff's help, Mr. Norquist engineered a shift in the city's lobbying corps by making it clear to K Street firms and trade associations that if they wanted access on Capitol Hill, they had to hire and donate to more Republicans.
As Mr. Abramoff's problems touched Mr. DeLay -- the majority leader may face an ethics inquiry over trips arranged by Mr. Abramoff -- Mr. Norquist was their most vocal defender. But in recent weeks he has distanced himself from the two men whose success has been so intertwined with his own.
At a gala dinner this month to support Mr. DeLay, Mr. Norquist declined a seat on the dais, despite being listed as a host. He slipped out after a predinner reception, he said later, for a dinner party at his home.
Mr. Abramoff attended his wedding on April 2, yet Mr. Norquist described him as simply a "friend," someone he has lunch or dinner with a few times a year. "I knew him when we were in college," he said. "But there's no business or financial relationship."
Mr. Norquist later elaborated: "I haven't seen that much of him recently, and I don't -- and it's mostly because -- I'm not shunning him or anything. It's just, I'm busy, he's busy, he's in a different world. He took the path I didn't take, which was to go make money as a consultant, and I decided to build A.T.R."
But Mr. Abramoff's lobbying billing records for one client, recently released under a Freedom of Information Act request, show that Mr. Abramoff billed for consulting with Mr. Norquist.
The two men's paths have intersected frequently since their days as young Republicans, organizing Massachusetts campuses for Ronald Reagan in 1980; Mr. Norquist was at Harvard, Mr. Abramoff at Brandeis.
The next year, Mr. Norquist managed Mr. Abramoff's successful campaign to become chairman of College Republicans, a national campus organization; the two men left amid complaints of financial mismanagement. Similar criticism followed them to Citizens for America, a conservative group that shut down soon after they left.
They seemed an odd pair, Mr. Norquist as rumpled as Mr. Abramoff is hand-tailored. But they shared a fierce ambition: when they talked about Democrats, they tended to use fiery language and were unapologetic about being strongly partisan.
Mr. Norquist went on to be president of Americans for Tax Reform, founded in 1985 to mobilize support for tax cuts, and he and the group became prominent in the conservative movement. From the group's offices near K Street, Mr. Norquist runs the Wednesday meetings, which started as strategy sessions to defeat the Clinton health care plan in 1993, then evolved to try to develop grass-roots support for every item on the Republican agenda.
"He has for years successfully worn the hat of being an ideologue rather than a typical business lobbyist, and in a way in this town that gets you more respect," said Hilary Rosen, a Democrat and the former president of the Recording Industry Association of America.
Mr. Norquist and Mr. Abramoff resumed working in tandem in 1994, when Mr. Abramoff returned to Washington from a stint in Hollywood as a producer of B movies. The Republicans were newly triumphant, and Mr. Norquist had become close to Newt Gingrich, the House speaker.
"Abramoff would have had no value without Norquist," said J. Michael Waller, a scholar at the Institute of World Politics who followed the two men at College Republicans. "Norquist was the pivot, he had the speaker of the House as his friend, all the new leadership, all the visionaries for more than a decade."
In what became known as the K Street Project, the two men pressed lobbying firms to do more business with Republicans, helping to enrich Mr. Abramoff, strengthen Mr. Norquist's influence and fortify Mr. DeLay's power.
Mr. Norquist and Mr. Abramoff worked together to help Mr. Abramoff's first big client, the Northern Mariana Islands. The garment industry there relies on cheap labor from China, and the local government was trying to avoid imposition of American minimum wage and immigration laws. Mr. Norquist promoted the Marianas in his meetings and speeches as a model of free enterprise, while Mr. Abramoff collected nearly $9 million in lobbying fees and sent members of Congress on fact-finding and golf trips to the islands.
Mr. Abramoff also billed the islands for thousands of dollars for "discussions" with Mr. Norquist and to send a staff member to the Wednesday meetings, as well as for airfare to the Marianas for staff members of Americans for Tax Reform.
Mr. Norquist sounded surprised to learn that he showed up so frequently in Mr. Abramoff's billing records. Mr. Abramoff, he said, must have been name-dropping.
Indian tribes say Mr. Abramoff dropped Mr. Norquist's name when he began trying to win their business. Mr. Norquist used his platform to argue against taxing Indian gambling. Mr. Abramoff billed the tribes tens of millions of dollars to try to fend off antigambling groups and regulators and to send members of Congress on lavish overseas trips. The tribes say Mr. Abramoff also instructed them to give money to Mr. Norquist's groups as way of getting an audience with the Bush administration. The tribes gave $1.5 million to Americans for Tax Reform and $250,000 to the Council of Republicans for Environmental Advocacy, the group founded by Mr. Norquist and Ms. Norton.
Mr. Norquist has tried to portray as a misunderstanding any perception that he was involved in buying or selling influence. Earlier this month, Americans for Tax Reform sent letters to the Coushatta tribe of Louisiana and the Saginaw Chippewas of Michigan, saying there "may be some confusion" about the reason some were invited into White House meetings in earlier years.
"Recent press reports appeared to suggest that some staff of your tribe's previous leadership thought that they were making a contribution to A.T.R. in order to be invited to a White House event," the letter said.
Once a year, Mr. Norquist said,he brings members of Indian tribal councils who have worked to pass local tax initiatives to the White House for a "thank you" from the administration. Some may have donated to Americans for Tax Reform, he said, but others did not.
We are continually improving the quality of our text archives. Please send feedback, error reports, and suggestions to archive_feedback@nytimes.com.
A version of this article appears in print on May 23, 2005, on Page A00001 of the National edition with the headline: Link to Lobbyist Brings Scrutiny to G.O.P. Figure. Order Reprints| Today's Paper|Subscribe


Welcome to the Elementary Education Resources Page. This page contains handbooks and resources for students, supervisors, cohort leaders and cooperating teachers.
All students in all Elementary Education cohorts should download and read our student guide. It contains contact information, course sequence guides, practicum information, student teaching rules and responsibilities, and important information on testing and performance standards.


text:
User Ratings: 4
Our Review: Scroll Down
Samurai Shodown Review (Neo Geo)
VICTOLY!
Version Reviewed: North American
review by Damien McFerran Fri, 30 May 2008 09:00:00 +0100
SNK became something of a fighting game factory in the early '90s, releasing titles like Art of Fighting and Fatal Fury to capitalize on public interest in the genre that had been kick started by a certain 2D fighting title from Cacpom. However, Samurai Showdown (or Samurai Spirits, as it was known in Japan) marked a real change of pace for the company: instead of the typical modern-day setting with white t-shirt wearing thugs throwing fireballs at each other, SNK set their latest epic in the past. The decision was an inspired one.
Widely regarded as the first weapon-based game of its type, Samurai Shodown is set in 18th century Japan. Although it features special moves like any other fighting game, the tone is extremely authentic.
The music is based on traditional Japanese themes and gives the game an almost Zen-like quality at times. The levels are also somewhat restrained, but in a good way – the falling cherry blossom and crashing waves found in some of the stages are the complete opposite of the brash effects of other games of the era.
All this would be enough to make Samurai Showdown stand out against the competition, but SNK wisely decided to back it all up with one of the finest fighting game engines of the day. Weapons clash and blows are deflected, special moves are balanced perfectly and the challenge is considerable. It's also a more methodical game than its rival, rewarding players who take their time and look for openings before attacking; button bashing will not win fights here.
When it was first released, Samurai Showdown caused concern in America due to the amount of blood it contained. It was subsequently censored when released in the US, sadly removing much of its appeal (not to mention realism – it's hard to hit someone with a katana and not draw a little blood, after all). Astonishingly, the Virtual Console release has all the gore intact - this should come as quite a shock to anyone who remembers the toned-down version of Mortal Kombat that SNES fans had to endure all those years ago! Nintendo has obviously relaxed its stance!
With Fatal Fury, SNK proved it could compete with Capcom, but with Samurai Showdown it showed it had the talent to surpass its rival. Fans still argue over which game is better – this or Street Fighter 2 – but one thing is for sure, any self-respecting Wii owner should really own both.
Great 8/10
Please Note: This review was originally published on vc-reviews.com
About Damien McFerran
Damien has over a decade of professional writing experience under his belt, as well as a repulsively hairy belly. Rumours that he turned down a role in The Hobbit to work on Nintendo Life are, to the best of our knowledge, completely and utterly unfounded.
michaelmax
Great review. Samurai Showdown rocks! A bit difficult to beat the AI though...but welcome to SNK.
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Game Profile
Wii Shop Release Date:
16th Jun 2008 (USA), 900 points
30th May 2008 (UK/EU), 900 points
16th Oct 2008 (JPN), 900 points
7th Jul 1993 (USA)
Switch eShop Release Date:
20th Jul 2017 (UK/EU), £6.29
20th Jul 2017 (USA), $7.99
Review 198X - A Fabulous, If Fleeting, Celebration Of Arcade Gaming's Golden Era
Switch eShop
Review Hypercharge: Unboxed - Lighthearted FPS Action That Mixes Toy Story With Small Soldiers
Review To The Moon - A Powerful Interactive Story You Really Need To Experience
Review AO Tennis 2 - A Decent Tennis Sim Ruined By Woeful Performance On Switch
Review Super Crush KO - The Only Issue With This Superb Side-Scrolling Brawler Is Longevity
Here's Your First Look At The Third Season Of Castlevania...


She was selected to be a representative of OHV interests for the Travel Management Strategy Working Group on the Deschutes and Ochoco National Forests and the Crooked River National Grassland.
She helped create the Oregon Recreation Coalition which is a group representing multiple recreation uses in the land use and political arena. As a member of the group's steering committee, she is able to draw from her extensive background to provide the group direction and guidance to successfully navigate through complicated arena of land use, politics, the media, and government agencies.
She represented motorized recreation for the BLM's Upper Deschutes Resource Management Plan which made critical decisions on the use of motorized vehicles for the next 20 years.
She was invited to attend the formation meeting of the Blue Ribbon Coalition (BRC) and has been a charter member since 1986.
She was president of the Blue Ribbon Coalition for three years during the period when BRC was instrumental in writing and passing the National Recreation Trails Fund Act (RTP) which provides millions of dollars annually to recreation.
Served on the BRC executive board and was the Oregon representative.
She served as secretary/treasurer for the Western Chapter of the International Snowmobile Council.
As a certified Master Gardener, she likes to care for the land as well as play on it.
She represented motorized recreation for the Trail User Group on the Deschutes National Forest. This is a multiple use group which was formed to help reduce user conflicts by building trust, communication, and education among the various interest groups.
She has an extensive background in working with land managers and the Congressional delegation.
She is a charter member of the Central Oregon Motorcycle and ATV Club (COMAC) and lifetime board member for land use.


text: Collector - September 2016 - (Page 2) COLLECTOR ® UPFRONT CHIEF EXECUTIVE OFFICER Patrick J. Morris You've Got a Friend in ACA ACA International's Industry Advancement Program continues to be a strong voice for the credit and collection industry throughout the judicial branch. DESIGN FIRM The YGS Group COMMUNICATIONS SPECIALIST Katy Zillmer By Patrick J. Morris Y our association is constantly thinking about what's next for its members and for the credit and collection industry as a whole. ACA International has taken on a growing role in advocating for legitimate collection businesses at the federal, state and local levels. From contacting consumers on their cell phones to navigating the collection of federal student loan debt, not a day goes by that ACA is not fighting to advance our industry's interests with lawmakers and regulators. But what about the other branch of government? Our nation's judicial system is designed to interpret the law in a way that is fair, balanced and-frankly-low-profile. And your association has taken an active role in working with the courts on cases critical to the credit and collection industry. ACA's Industry Advancement Program (IAP) has done outstanding advocacy work in the judicial branch on behalf of the credit and collection industry in the nearly three years since the inception of the ACA Industry Advancement Fund. In that time, 39 court cases have received some sort of support through IAP. This is a precipitous rise in our interaction since the ACA Board of Directors took the important step of beginning the IAP, instead of relying only on voluntary contributions. The data, and our effectiveness, tells the tale. Specifically, ACA has filed amicus, or friend of the court, briefs in 24 cases impacting the credit and collection industry. These amicus briefs are often the only opportunity for nonparties like ACA to participate in appeals that present significant legal issues or may lead to harmful precedents for the industry. Think of it as ACA's opportunity to bring its expertise to court: Amicus briefs can supplement or bolster another party's brief, provide historical perspective or technical assistance, or mitigate or expand the effects of previous opinions from other courts. Learn more about the importance of ACA's amicus briefs on pg. 30 of this issue. Also, be sure to check out the IAP page at acainternational.org for the most up-to-date information about ACA's advocacy work in the judicial branch. Patrick J. Morris is ACA International's chief executive officer. He can be reached at morris@acainternational.org. 02 EDITOR Anne Rosso May Collector is published monthly by ACA International. 4040 W. 70th St., Minneapolis, MN 55435 Telephone: (952) 926-6547 Web: www.acainternational.org Email: comm@acainternational.org Twitter: @acacollector Annual subscription rates: $35 for ACA members, $70 for nonmembers. Single copies: $3.25 for ACA members, $6.50 for nonmembers. Collector welcomes article submissions. However, we cannot guarantee publication of the articles we receive. Collector editors reserve the right to edit submissions. For advertising information, please contact Leonard Media Group at (215) 675-9208, ext. 201; info@leonardmedia.com; or www.acainternational.org/advertising. Advertisement of products or services in Collector does not constitute an endorsement by ACA. Information contained in Collector is not intended to be legal advice and may not be used as legal advice. Every effort has been made to ensure this information is up-to-date at the time of publication. It is not intended to be a full and exhaustive explanation of the law, nor should it be used to replace the advice of your counsel. ©2016 ACA International. All rights reserved. Materials ACAINTERNATIONAL.ORG may not be reproduced without written permission. http://www.acainternational.org http://www.acainternational.org http://www.acainternational.org/advertising Table of Contents for the Digital Edition of Collector - September 2016 Upfront


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Country / Region: United States


text:
Q: How to derive an identity between summations of totient and Möbius functions I have the following identities

*

*$$\sum_{n \le x} \varphi(n) = \frac{1}{2} \sum_{n \le x} \mu(n) \left[\frac{x}{n}\right]^2 + \frac{1}{2}$$

*$$\sum_{n \le x} \frac{\varphi(n)}{n} = \sum_{n \le x} \frac{\mu(n)}{n} \left[\frac{x}{n}\right]$$

how can I discover a similar one for $$\sum_{n \le x} \frac{\varphi(n)}{n^2}?$$

The reason for changing them from $\varphi$ to $\mu$ is that they are easier to asymptotic analysis. Proving both required the identity $\varphi(n) = \sum_{d|n} \mu(d) \frac{n}{d}$ as the first step so I am guessing that is the first step for this too, but when I try that I get $\sum_{n \le x} \sum_{d|n} \frac{\mu(d)}{nd}$ which doesn't look like a convolution or anything simple like that.

A: You are on the correct path, it is very similar to the previous one.  As you noted, we have $$\sum_{n\leq x}\frac{\phi(n)}{n^{2}}=\sum_{n\leq x}\frac{1}{n}\sum_{d|n}\frac{\mu(d)}{d}.$$  Switching the order of summation yields $$\sum_{d\leq x}\frac{\mu(d)}{d}\sum_{r\leq\frac{x}{d}}\frac{1}{rd}=\sum_{d\leq x}\frac{\mu(d)}{d^{2}}\sum_{r\leq \frac{x}{d}}\frac{1}{r}=\sum_{d\leq x}\frac{\mu(d)}{d^{2}}H_{\left[\frac{x}{d}\right]}.$$
Where $H_m$ is the $m^{th}$ harmonic number.  Now this is possible to evaluate since $$\sum_{n\leq x}\frac{1}{n}=\log x +\gamma+O\left(\frac{1}{x}\right).$$ We then get $$\sum_{d\leq x}\frac{\phi(d)}{d^2}=\sum_{d\leq x}\frac{\mu(d)}{d^{2}}\left(\log\frac{x}{d}+\gamma+O\left(\frac{d}{x}\right)\right).$$
We may rewrite this sum as
$$\log x\sum_{d\leq x}\frac{\mu(d)}{d^{2}}-\sum_{d\leq x}\frac{\mu(d)\log d}{d^{2}}+\gamma\sum_{d\leq x}\frac{\mu(d)}{d^{2}}+O\left(\frac{1}{x}\sum_{d\leq x}\frac{|\mu(d)|}{d}\right)$$
Since $\sum_{d> x}\frac{\mu(d)\log d}{d^2}=O(\log x/x)$, and $\sum_{d\leq x}\frac{\mu (d)}{d}=O(\log x)$, we see that
$$\sum_{n\leq x}\frac{\phi(n)}{n^2}=\log x \sum_{d=1}^\infty \frac{\mu(d)}{d^2}+\gamma \sum_{d=1}^\infty \frac{\mu(d)}{d^2}-\sum_{d=1}^\infty \frac{\mu(d)\log d}{d^2}+O\left(\frac{\log x}{x}\right).$$
Looking at these series, $\sum_{d=1}^\infty \frac{\mu(d)}{d^2}=\frac{1}{\zeta(2)}$ and $\sum_{d=1}^\infty \frac{\mu(d)\log d}{d^2}=\frac{\zeta^{'}(2)}{\zeta(2)^2}$, and so
$$\sum_{n\leq x}\frac{\phi(n)}{n^2}=\frac{\log x}{\zeta(2)}+\frac{\gamma}{\zeta(2)} -\frac{\zeta^{'}(2)}{\zeta(2)^2}+O\left(\frac{\log x}{x}\right).$$
Hope that helps,
Edit: Fixed error with the constant term.  I originally missed the $-\log d$ term.
Edit 2: Changed the bound $\sum_{d> x}\frac{\mu(d)\log d}{d^2}=O(1/x)$ to $\sum_{d> x}\frac{\mu(d)\log d}{d^2}=O(\log x/x).$ While the previous one is also true, it requires the prime number theorem. For this result the $\log x$ does not change anything, and there is a much simpler proof. See this answer.

redpajama_set_name: RedPajamaStackExchange


July 7 was the day lovers of classical music in the St. Louis area could hear it no more at 99.1 on their FM radio dial, as they had for 35 years.
KFUO-FM Classic 99 stopped broadcasting classical and instead, under new ownership, began playing Christian contemporary pop music.
The station was bought from the Lutheran Church – Missouri Synod by Gateway Creative Broadcasting and the station is now called Joy-FM. The $18 million deal was approved by the Federal Communications Commission in May, despite complaints from enraged listeners of the classical music.
Joy-FM is listener supported and two prominent givers – the Cardinals' Albert Pujols and ex-Cardinal pitcher Andy Benes – made large donations to help finance the sale. (Their dollar amounts were not reported).
The impending sale was in the news steadily since it became known last October. Letters-to-the-editor by dismayed listeners were countered by supporters of the Christian programs. "Just change the station if you don't want to hear the word of God," wrote one woman.
Some suggested that the classical lovers find their music elsewhere, like on some other station. KWMU-FM said it would play classical, but on an HD channel that few listeners could get.
Sarah Bryan Miller, classical music critic of the St. Louis Post-Dispatch wrote on May 21 that the Radio Arts Foundation had given up its hunt for a replacement station, at least for now. "Currently there is no appropriate station available to buy," said a statement from the Foundation. It said it was placing the hunt for another station on hold until one becomes available.
The Foundation had originally tried to buy KFUO-FM from the Lutheran Church – Missouri Synod but was told the church preferred selling to a Christian enterprise, Miller reported.
(Originally published in the St. Louis Journalism Review 7/2010).


The Hamaz Mission is to present the pharmaceutical industry operating in Pakistan & worldwide in the way that: Assures all the patients access to the best available medicines. Creates favorable natural & human friendly environment. Encourges innovative research & products development. Believes in professional & standardized services.


8vo., rebound in blue cloth with gilt titling to spine, pp. , 220, with engraved coats of arms and 16 folding genealogical tables. Ex library (National Gallery, London), pages browned and foxed, table II a little crinkled and offset; otherwise very good, pages uncut. Text in Italian.


text:
Home»Posts tagged with»Judge Amy Reedy
Tennessee Tenth Judicial District Prosecutor: Government Influence of Grand Jury "Unethical"
By Sharon Rondeau on Monday, January 12, 2015 National
"THE GRAND JURY IS OPEN TO ANY CITIZEN WHO WANTS TO COME AND APPEAR" by Sharon Rondeau (Jan. 12, 2015) — The chief prosecutor of Tennessee's Tenth Judicial District, which includes the counties of McMinn, Monroe, Polk and Bradley, told a local radio show host last month that "the grand jury is open to any […]
Monroe County, TN: Sentencing for George Raudenbush Monday Without Counsel?
By Sharon Rondeau on Monday, December 15, 2014 States
"THE PEOPLE…HAVE TURNED THEIR BACK ON GOD" by Sharon Rondeau (Dec. 15, 2014) — Christian missionary George Joseph Raudenbush, III is scheduled to be sentenced on seven convictions from a November 25 trial on Monday, December 15, at noon, but for the second time, he has no attorney. In August 2011, Raudenbush was convicted on […]
Eyewitness to Raudenbush Trial Believes Jury was Rigged
By Sharon Rondeau on Friday, December 12, 2014 National
"IT WAS JUST A PRETENSE'" by Sharon Rondeau (Dec. 12, 2014) — On November 25, Christian missionary George Joseph Raudenbush III was convicted on seven counts of vehicular infractions stemming from an incident on December 30, 2010 in the town of Tellico Plains, TN. Last December, Raudenbush's initial convictions from 2011 were reversed by an […]
Lawrence County, TN Citizen Addresses the Grand Jury
By Sharon Rondeau on Wednesday, September 10, 2014 National
"BE JUST" by Sharon Rondeau (Sep. 10, 2014) — Over the last several weeks, The Post & Email has reported on the case of Arthur Hirsch, who was arraigned in Lawrence County, TN in January without an indictment or arrest warrant, subjected to a subsequent humiliating hearing in April by Judge Patricia McGuire, and then […]
How Did the Fitzpatrick Jury Reach Its Verdicts?
By Sharon Rondeau on Tuesday, August 26, 2014 National
JURORS REFUSE TO COMMENT ABOUT ALLEGATION OF BRIBERY, COERCION by Sharon Rondeau (Aug. 26, 2014) — On June 24, CDR Walter Francis Fitzpatrick, III (Ret.) was convicted of "aggravated perjury" and "extortion" as a result of bringing petitions to the McMinn County grand jury which claimed that former grand jury foreman Jeffrey Cunningham, criminal court […]
Breaking: Report: McMinn County Courthouse "Loaded"
BUT HOW MANY ARE "LAW ENFORCEMENT?" by Sharon Rondeau (Aug. 19, 2014) — The Post & Email received a report from an individual on the way to the McMinn County courthouse on Tuesday morning for the sentencing hearing for CDR Walter Francis Fitzpatrick, III (Ret.) stating that the courthouse is "loaded with people." While not […]
The Post & Email and Abel Danger Announce Professional Media Company to Cover Fitzpatrick Case
HAS JUDICIAL CORRUPTION MET ITS MATCH? by Sharon Rondeau (Aug. 12, 2014) — The Abel Danger radio show held a special episode on Tuesday to announce that William Fain of William Fain Productions has agreed to provide video coverage of the August 18 prayer breakfast, barbecue, and August 19 sentencing hearing for CDR Walter Francis […]
Bombshell: State of Tennessee Admits No Victim Exists for Fitzpatrick's "Crimes"
By Sharon Rondeau on Monday, August 11, 2014 National
INDICTED, CHARGED, CONVICTED AND SOON TO BE SENTENCED, WITHOUT A VICTIM by Sharon Rondeau (Aug. 11, 2014) — The State of Tennessee Board of Probation and Parole has admitted that in the case of Walter Francis Fitzpatrick, III, 14-CR-69, in which Fitzpatrick was found guilty of "aggravated perjury" and "extortion," there was no victim. In […]
Fitzpatrick Trial Observer: "I Just Don't Understand How it Happened So Quickly"
By Sharon Rondeau on Saturday, August 9, 2014 National
WHAT WAS THE RUSH? by Sharon Rondeau (Aug. 9, 2014) — An eyewitness to the trial of CDR Walter Francis Fitzpatrick, III (Ret.) on Monday, June 23 and Tuesday, June 24 in McMinn County, TN told The Post & Email that she does not believe the 12 jurors could have properly reviewed the evidence provided […]
Will Field McConnell Be Suing Obama on August 19th?
By Sharon Rondeau on Sunday, August 3, 2014 National
RETIRED MARINE OPTIMISTIC JUDGE WILL VACATE FITZPATRICK VERDICTS by Sharon Rondeau (Aug. 3, 2014) — On Thursday, The Post & Email reported (as pictured at right) that a former U.S. Marine aviator, instructor, private commercial pilot and current radio show host and intelligence analyst plans to file a lawsuit against Barack Hussein Obama on August […]
Fitzpatrick Case on Display at Arpaio Rally in Southern California
By Sharon Rondeau on Sunday, July 27, 2014 National
"COMMANDER FITZPATRICK " MEETS SHERIFF JOE ARPAIO by Sharon Rondeau (Jul. 27, 2014) — Neil Turner and Jeff Schwilk were present at an address given by Maricopa County, AZ Sheriff Joe Arpaio in southern California on Saturday. In addition to hearing what Arpaio had to say about the tens of thousands of illegal aliens allowed […]
Fitzpatrick Conviction To Be Publicized in Overpasses Protest in California on Saturday
By Sharon Rondeau on Saturday, July 26, 2014 National
"ON THE OFFENSIVE" by Sharon Rondeau (Jul. 26, 2014) — Capt. Neil Turner and Dr. James David Manning have been working with President and Founder of Overpasses for America James Neighbors to further publicize the conviction of CDR Walter Francis Fitzpatrick, III (Ret.) on two false charges by a McMinn County, TN jury which should […]
Tennessee Judicial Corruption Goes National
By Sharon Rondeau on Wednesday, July 23, 2014 National
SECOND "BATTLE OF ATHENS" INVOLVES CROOKED JUDGES, PROSECUTORS, JURORS by Sharon Rondeau (Jul. 23, 2014) — On Wednesday, this writer was a guest on the Abel Danger radio show to discuss the impending sentencing and imprisonment of CDR Walter Francis Fitzpatrick, III (Ret.), who was convicted of "aggravated perjury" and "extortion" on June 24 by […]
Multiple Interviews on Fitzpatrick Verdict This Week Expose Tennessee Corruption pb
By Sharon Rondeau on Tuesday, July 22, 2014 National
ATTORNEY: "I'M NOW SCARED TO GO TO THE GRAND JURY" by Sharon Rondeau (Jul. 22, 2014) — On Monday evening, radio host Dr. Laurie Roth interviewed Atty. Van Irion, who represented CDR Walter Francis Fitzpatrick, III (Ret.) on charges filed after Fitzpatrick attempted to bring evidence of criminality on the part of judges and other […]
The Post & Email to Appear on The Roth Show Thursday Evening
By Sharon Rondeau on Thursday, July 17, 2014 National
JUDICIAL CORRUPTION IN ONE STATE AFFECTS ALL AMERICANS by Sharon Rondeau (Jul. 17, 2014) — The Post & Email's editor will be a guest on Thursday evening with Dr. Laurie Roth, host of The Roth Show. Dr. Roth is an accomplished musician, radio and former television show host, and former presidential candidate. She has a […]
Is the FBI Doing Its Job?
"BRING YOUR HANDCUFFS" by Sharon Rondeau (Jul. 13, 2014) — On June 24, CDR Walter Francis Fitzpatrick, III (Ret.) was convicted of "aggravated perjury" and "extortion" for attempting to submit a packet of evidence incriminating public officials to the McMinn County, TN grand jury. McMinn is one of four counties in the Tenth Judicial District […]
U.S. Senate Judiciary Media Spokeswoman on Judicial Corruption: "Feel Free to Send Me What You Have" pb
By Sharon Rondeau on Monday, July 7, 2014 National
"I'LL BE GLAD TO FLAG IT FOR REVIEW" by Sharon Rondeau (Jul. 7, 2014) — The Post & Email has made several phone calls to the offices of U.S. representatives and senators to request a congressional investigation into the Tennessee state courts and specifically, the Tenth Judicial District, where Judges Carroll Lee Ross, Amy Reedy, […]
Why is Judge Blackwood Taking the Fitzpatrick Case to Trial?
By Sharon Rondeau on Sunday, June 22, 2014 National
AND WHY IS HE HEARING IT? by Sharon Rondeau (Jun. 22, 2014) — On Monday, June 23, a trial is scheduled in the case of State of Tennessee v. Walter Francis Fitzpatrick, III (Ret.) which accuses Fitzpatrick of three felonies and one misdemeanor charge. The allegations arose on March 18 of this year when Fitzpatrick […]
Will "The Grand Jury" Go on Trial on Monday?
By Sharon Rondeau on Saturday, June 21, 2014 National
"THE GRAND JURY IS WHAT MAKES US A CONSTITUTIONAL REPUBLIC" by Sharon Rondeau (Jun. 21, 2014) — On Monday, a trial will take place beginning at 10:00 a.m. at the McMinn County, TN courthouse of CDR Walter Francis Fitzpatrick, III (Ret.) on three felonies and one misdemeanor charge stemming from an arrest on March 18 […]
Tennessee Judges and Grand Jury Foremen: Violators of Law for More Than Three Decades
By Sharon Rondeau on Monday, June 16, 2014 States
HOW MANY GENERATIONS HAVE BEEN AFFECTED BY AN OUT-OF-CONTROL JUDICIARY? by Sharon Rondeau (Jun. 16, 2014) — Appointing orders obtained from various Tennessee courthouses show that criminal court judges have engaged in the practice of hand-selecting grand jury foremen for at least 34 years. The Fifth Amendment to the Bill of Rights included the grand […]


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Q: css id in custom html helper I have my custom html helper and want to add css id tag there.
this is my custom html helper:
 public static class CustomHtmlHelpers
{
    public static IHtmlString Image(this HtmlHelper helper, string src)
    {
        TagBuilder tb = new TagBuilder("img");
        tb.Attributes.Add("src", VirtualPathUtility.ToAbsolute(src));
        return new MvcHtmlString(tb.ToString(TagRenderMode.SelfClosing));
    }
}

and here's my code in razor view:

   @Html.DisplayNameFor(model => model.Photo)

   @Html.Image(@Model.Photo)

I have tried
@Html.Image(@Model.Photo, new { @id = "myId" })

but it doesn't works

A: Can you try change your HTML helper to this:
public static IHtmlString Image(this HtmlHelper helper, object htmlAttributes)

and add code:
tb.MergeAttributes(new RouteValueDictionary(htmlAttributes));

Link:
http://www.asp.net/mvc/tutorials/older-versions/views/using-the-tagbuilder-class-to-build-html-helpers-cs

A: public static class CustomHtmlHelpers
{
    public static IHtmlString Image(this HtmlHelper helper,
        string src,
        object htmlAttributes)
    {
        TagBuilder tb = new TagBuilder("img");
        tb.Attributes.Add("src", VirtualPathUtility.ToAbsolute(src));
        tb.MergeAttributes(new RouteValueDictionary(htmlAttributes));
        return new MvcHtmlString(tb.ToString(TagRenderMode.SelfClosing));
    }
}

and html
@Html.Image(@Model.Photo, new { @id = "myId" })