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text:
Ghostnaps
Posted on April 9, 2018 by george
Richard Lathrop, better known by his stage name Ghostnaps, is a DJ and record producer based in Eugene, Oregon. Developing a passion for electronic music in 2011, Lathrop begun to DJ and produce immediately. Developing a progressive house sound, he started playing shows around his local area. In 2016, with five years of experience under his belt, Lathrop decided to rebrand himself as "Ghostnaps", and work on finding his own unique sound. On October 1 of that year, Lathrop released the single "No One Else", combining his old sound fans were used to with a modern future bass touch. In December 2017, Richard released the "Late Night" EP, which has seen praise from new and old fans alike. In March of 2018, Lathrop released his latest single "Sleepless", and announced he is working on an album, due to release in December of this year. In the meantime, he will be premiering his live set in May, which will push his own bounds musically, while keeping the energy of a Ghostnaps DJ set.
The Gooms
Sharlet Crooks

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

We study the question when the set of time-frequency shift s
\begin{equation}
  \label{eq:c1}
\mathcal{G}(g,\alpha , \beta ) = \{ e^{2\pi i \beta l x } g(x-\alpha k) : k,l \in \bZ \}
\end{equation}
is complete in $L^2(\bR ) $, where $g \in L^2(\bR ) $   and the
lattice parameters $\alpha , \beta >0$  are fixed.
The completeness  question arose
first in J.\ von Neumann's treatment of quantum mechanics~\cite{neumann} and
remains relevant in physics and in applied mathematics. The motivation
in signal analysis and time-frequency analysis\ comes from   Gabor's fundamental
paper~\cite{gabor46} on information theory. Gabor  tried to expand a
given function (signal) into a series of time-frequency shift s.  Correspondingly, in
mathematical terminology the set $\mathcal{G}(g,\alpha , \beta ) $ is called a Gabor
system with window function $g$.
In quantum mechanics the functions $e^{2\pi i \beta l x } g(x-\alpha
k)$ are called phase-space shift s of a (generalized) coherent state, and
$\mathcal{G}(g,\alpha , \beta ) $ can be
interpreted as a discrete set of coherent states  with respect to the
Heisenberg group   over a lattice $\alpha
\bZ \times \beta \bZ $~\cite{alanga00}.
In fact,
Perelomov's book~\cite{perelomov86} on coherent states contains several sections devoted
to the ``completeness of coherent state subsystems.''

In the applied mathematics literature of the last 20 years the
interest in Gabor systems has shifted to the frame property,
mainly for numerical reasons~\cite{feichtinger-strohmer98} and because Gabor frames can
be used to characterize function spaces~\cite{fg97jfa} and to describe pseudodifferential operator
s~\cite{gro06}. Here
$\mathcal{G}(g,\alpha , \beta ) $ is a (Gabor) frame for $L^2(\bR ) $, if there exist constants $A,B>0$ such
that
\begin{equation}
  \label{eq:c2}
A\norm{f}_2^2 \leq \sum _{k,l \in \bZ } | \langle f, e^{2\pi i \beta l
  \cdot } g(\cdot - \alpha k)\rangle |^2 \leq B \norm{f}_2^2 \qquad
\forall f\in L^2(\bR ) \, .
\end{equation}
Clearly \eqref{eq:c2} implies that $\mathcal{G}(g,\alpha , \beta ) $ is complete in $L^2(\bR ) $,
but in general the frame property of $\mathcal{G}(g,\alpha , \beta )$ is much  stronger than its
completeness.
This difference is already present in von Neumann's example $\mathcal{G} (\phi
, 1, 1)$ where $\phi (x) = e^{-\pi x^2}$ is the Gaussian, i.e.,  the
canonical  coherent state in quantum mechanics,  and $\alpha = \beta =1$. In this case it was
proved 40 years after von Neumann that $\mathcal{G} (\phi , 1,1)$ is
complete, but not a frame~\cite{bargmann71,perelomov71}.

The main  intuition for the results about the  completeness and the
frame property of $\mathcal{G}(g,\alpha , \beta ) $ is based on the uncertainty principle . According to the uncertainty principle\
every physical state $g$, i.e., $g\in L^2(\bR ), \|g\|_2^2 = \int _\bR
|g(x)|^2 \, dx =1$, occupies a cell in phase space (in the time-frequency\ plane)
of minimal area
one. The phase-space shift\ $e^{2\pi i \beta l x} g(x-\alpha k)$ is located roughly
at position $\alpha k$ and momentum $\beta l$ in phase space $\bR
^2$. Thus, in order to cover the entire phase space with a discrete
set of coherent states $\mathcal{G}(g,\alpha , \beta ) $, we must have necessarily $\alpha \beta
\leq 1$, otherwise there would be gaps in phase space that cannot be
reached by a phase-space shift\ of the form  $e^{2\pi i \beta l x} g(x-\alpha
k)$. The physical intuition has been made mathematically rigorous in
the form of numerous density theorems for Gabor systems: \emph{If
  $\mathcal{G}(g,\alpha , \beta ) $ is complete in $L^2(\bR ) $, then necessarily $\alpha \beta \leq
1$}~\cite{daubechies90,RS95,heil07}. The converse holds only for
special window functions. It was already noted in~\cite{bargmann71,perelomov71}
that for the Gaussian $\phi (x) = e^{-\pi x^2}$ the set $\mathcal{G} (\phi ,
\alpha , \beta )= \{ e^{2\pi i
  \beta l x} e^{- (x-k\alpha )^2} : k,l \in \bZ \}$ is
complete for $\alpha \beta \leq 1$ and incomplete for $\alpha \beta
>1$. In 1992  Lyubarskii\cite{lyub92} and Seip~\cite{seip92}
strengthened this statement and showed that $\mathcal{G} (\phi ,
\alpha , \beta )$ is frame, if and only if\ $\alpha \beta < 1$. (In fact, they
stated their results for arbitrary time-frequency shift s, not just lattice shifts.)
The  recent work ~\cite{GS13} shows that an analogous result also holds for
the class of  so-called totally positive
functions of finite type. Again, as in the case of the Gaussian,
these functions  possess good time-frequency\  localization, which in
mathematical terms amounts to additional analyticity properties.

In this paper we return to the  completeness problem for Gabor
systems.  In agreement with the physical description of quantum
states, we will assume
that the window functions  possess strong localization properties in
the time-frequency\ plane. Technically,
we will assume that $g$ and its Fourier transform have exponential decay.

Our main results provide a significant generalization of von Neumann
observation on the completeness of coherent states (Gabor systems)  and  may be summarized as follows.

\begin{tm} \label{tm:intro}
Assume that $g$ factors as  $g(x)= R(x) e^{-\gamma  x^2}$,   where
$\gamma >0$,  $R$ is
either a rational function with no real poles or a finite sum of complex
exponentials $R(x) = \sum _{j=1}^m  c_j  e^{\lambda _j x}$ with $c_j,
\lambda _j \in \bC $. If  $\alpha \beta $ is rational and  $\alpha
\beta \leq 1$,
then  the
Gabor system $\mathcal{G}(g, \alpha, \beta)$ is
complete in $L^2(\mathbb{R})$.
\end{tm}

One may say that  the physical intuition works far beyond the canonical
coherent states.
For rational lattices $\alpha \beta \in \bQ$ and windows as in Thm.~\ref{tm:intro}, the Gabor system
(coherent state subsystem) $\mathcal{G}(g,\alpha , \beta ) $ is complete, if and only if\ $\alpha \beta
\leq 1$.  Although it seems natural  that Theorem~\ref{tm:intro} can
be extended to
arbitrary rectangular lattices $\alpha \bZ \times \beta \bZ $ with
$\alpha \beta \leq 1$, we must leave this question  open because  there is no
useful  completeness characterization over
irrational lattices.

To  put this result in perspective, we note three special cases
(Theorems~\ref{t:pol}, \ref{t:rat}, and \ref{t:expsum} and their
corollaries  in
Section 3).

(a) Theorem~\ref{tm:intro} covers all Hermite functions $h_n = c_n
e^{\pi x^2} \tfrac{d^n}{dx^n } (e^{-2 \pi x^2})$ (with a suitable
  normalization constant $c_n$ and $n\in \bN $). So far it was known that $\mathcal{G} (h_n,
  \alpha  , \beta)$ is a frame, if $\alpha \beta <
  \tfrac{1}{n+1}$~\cite{GL09}. However, for  odd
  $n=2m+1$  and  $\alpha
  \beta = 1- \tfrac{1}{N} , N=2, 3, \dots , \infty $,  the Gabor system $\mathcal{G} (h_{2m+1}, \alpha , \beta
    )$   is not a frame~\cite{LN13}. The achievement of
    Theorem~\ref{tm:intro} is to assert the completeness of $\mathcal{G} (h_n
    , \alpha , \beta )$ for \emph{all} Hermite functions and
    \emph{all} rational lattices with $\alpha \beta \leq 1$.

The example of the Hermite functions also indicates the limits of the
physical intuition which does not explain the difference between
completeness (intuition is confirmed perfectly by
Theorem~\ref{tm:intro})  and the frame property (intuition is not
correct). It would be interesting to understand from physical
principles why the frame property can be destroyed by certain
symmetries, as is the case for the odd Hermite functions.

Let us mention that, after a suitable transformation and choice of
representation,  the Gabor systems $\mathcal{G} (h_n, \alpha , \beta )$ can
also be interpreted as a coherent state subsystem for the degenerate
Landau levels of a  particle in a constant magnetic
field. Thus our results extend to higher Landau levels what was previously known only for the ground state. The standard complex analytic techniques that can be used to deal with
the ground state
do not work with more general Landau levels.
See~\cite{MR3203099} for a detailed explanation of the
connection between Gabor systems and the Landau levels.

(b) If $g$ is given by its Fourier transform  $\hat{g}(\xi ) = \prod _{j=1}^m (1 +
i \delta _j \xi )^{-1} e^{-\gamma \xi ^2}$ for $\delta _j \in \bR $, then $g$ is a totally
positive function by Schoenberg's
characterization~\cite{schoenberg1947totally}. It was conjectured in
~\cite{G14} that the Gabor system $\mathcal{G}(g,\alpha , \beta ) $ for a totally positive
function $g$ generates a frame, if and only if\ $\alpha \beta  <1$, but so far
this statement is known to be true only for the class of totally
positive functions of finite type~\cite{GS13}. Theorem~\ref{tm:intro} offers a
similar result for the case of completeness for a complementary
class of totally positive functions and  supports the original
conjecture.

(c) Finally, Theorem~\ref{tm:intro} covers the case when $g$ is a
finite linear combination of time-frequency shift s of the Gaussian
$\phi (x) = e^{-\pi x^2}$. Again, this is in line with the physical
intuition.

The remainder of the paper will be organized as follows: In Section~2
we recall the abstract characterizations of complete Gabor systems over a
rational lattice due to Zeevi and Zibulski~\cite{zibulski1997analysis}
and derive a
specialized criterium for functions in the Gelfand-Shilov class. We
will juxtapose these characterizations with the corresponding
characterizations of the frame property.   The
comparison of these criteria  reveals the fundamental difference between
the completeness and the frame property and  is
particularly striking for window functions in the Gelfand-Shilov space.  The
completeness property  hinges on the
analyticity of $g$ and $\hat{g}$. It is easy to produce
counter-examples to Theorem~\ref{tm:intro} belonging to
the Schwartz class, where  $g$ and $\hat{g} $ are
$C^\infty $.

In Section~3 we will prove several versions of
Theorem~\ref{tm:intro}. The main idea is to use our knowledge that
the Gaussian window $\phi (x) = e^{-\pi x^2}$  generates a complete system
$\mathcal{G} (\phi , \alpha   , \beta ) $ for $\alpha \beta \leq 1$, and  to subsequently show that the
algebraic conditions that guarantee the completeness of $\mathcal{G}(g,\alpha , \beta ) $ for a
factorized function $g(x) = R(x) e^{-\pi x^2}$  are satisfied. This is
a new proof method in Gabor analysis that very likely can be sharpened
and extended. Ultimately we hope that our results will also lead to a
better understanding of several unsolved  problems about Gabor
\emph{frames} that were formulated in~\cite{G14}.

Notation: We write $f(x) \lesssim g(x)$ to say that there exists a
constant independent of $x$, such that $f(x) \leq C g(x)$ for all
$x$. Furthermore, $f(x)  \asymp g(x) $ means that $f(x) \lesssim g(x)$
and $g(x) \lesssim f(x)$.

\section{Gabor systems}

The translation and modulation operators act on a function $f: {\mathbb{R}} \to \bC$ by
\begin{align*}
&M_b f(x) := e^{2\pi i b x}f(x), \qquad b  \in {\mathbb{R}},
\\
&T_a f(x) := f(x-a), \qquad a \in {\mathbb{R}}.
\end{align*}
The composition $M_b T_a$ is called a time-frequency shift\ (or  phase-space shift\
in the language of quantum mechanics).
A Gabor system is a collection of time-frequency shift s  of a given function (a
window function in signal analysis, or a
quantum mechanical state).
Given $g\in L^2(\bR)$,  $\alpha , \beta >0$, we define
then
$$
\mathcal{G}(g,\alpha , \beta ) = \{ M_{\beta l } T_{\alpha k}g: k,l \in \bZ \}.
$$
The frame operator associated with this family is given by
\[
Sf = \sum_{k, l \in \mathbb{Z}} \langle f, M_{l \beta} T_{k \alpha} g \rangle M_{\beta l} T_{\alpha k } g.
\]
Under mild assumptions on $g$  the frame operator is
bounded~\cite{walnut92,MR1843717}. The   standard assumption is that $g$ belongs to the modulation space
\begin{align*}
M^1({\mathbb{R}}) := \set{f \in L^2({\mathbb{R}})}{\int_{{\mathbb{R}}^2} \abs{\ip{f}{M_\xi T_x
      f}} dx d\xi  < +\infty},
\end{align*}
also known as the \emph{Feichtinger algebra}. If  $g \in M^1({\mathbb{R}})$,
then $S$ is bounded
on $L^2({\mathbb{R}})$.
A Gabor system $\mathcal{G}(g,\alpha , \beta )$ is called a \emph{frame} if the frame operator
$S$ is invertible on $L^2({\mathbb{R}})$. This is equivalent to  the frame inequalities
\eqref{eq:c2}.

We say that $\mathcal{G}(g,\alpha , \beta )$ is complete (in $L^2$) if the linear span of $\mathcal{G}(g,\alpha , \beta )$ is a dense subspace of
$L^2({\mathbb{R}})$. Equivalently, $S$ is one-to-one on $L^2(\bR ) $.

While the completeness of $\mathcal{G}(g,\alpha , \beta )$ means that any function $f \in L^2({\mathbb{R}})$ can be approximated
by linear combinations of elements of $\mathcal{G}(g,\alpha , \beta )$, the frame property implies the existence of a convergent expansion
\begin{align*}
f = \sum_{k, l \in \mathbb{Z}} a_{k,l} M_{\beta l} T_{\alpha k } g,
\end{align*}
with $\norm{a}_2 \asymp \norm{f}_2$.  For  the Gaussian $\phi (t) :=
e^{-\pi t^2}$,
$\mathcal{G} (\phi , \alpha, \beta ) $ is complete if and only if $\alpha \beta \leq 1$
and $\mathcal{G} (\phi , \alpha, \beta ) $  is a frame if and only if $\alpha \beta < 1$
\cite{lyub92,perelomov71,seip92}.

\subsection{Completeness criteria for rational lattices}
Our starting point is the characterization of the completeness of a Gabor system
over a rational lattice by means of the Zak transform, due to Zeevi and
Zibulski \cite{zibulski1997analysis}.

We write $\alpha \beta = p/q$ for relatively prime positive integers
$p$ and $q$ and assume that $p\leq q$.  The Zak transform of $f \in L^2(\mathbb{R})$ with respect to a
parameter $ \alpha \in \mathbb{R}$ is
\[
Z_\alpha f(x, \xi) = \sum_{k \in \mathbb{Z}} f(x- \alpha k) e^{2 \pi i
  \alpha k \xi }.
\]
By quasi-periodicity  this
function is completely determined by its values on the rectangle
$I_\alpha = [0,\alpha ] \times [0, 1/\alpha ]$. Furthermore, the Zak transform is a unitary isomorphism
from
$L^2(\mathbb{R})$ to $L^2(I_\alpha)$.

A note on terminology:  In signal processing  $Z_\alpha $ is called  the Zak
transform in reference to one of its first applications by  J.\
Zak~\cite{zak67},  in  solid states physics $Z_\alpha $ is usually
called the
Bloch-Floquet transform ~\cite{kuch93}, and  in harmonic analysis it is
often called the Weil-Brezin transform~\cite{folland89}.

 We will also need the vector-valued version
\[
\overrightarrow{Z_{\alpha}}f(x, \xi) = \big( Z_\alpha f(x+ \tfrac{\alpha}{p} r, \xi) \big)_{r=0}^{p-1}, \quad x,\xi
\in
[0,
\alpha/p) \times [0, 1/\alpha),
\]
which is unitary  from $L^2(\mathbb{R})$ onto the  space $L^2 \big( [0, \alpha/p) \times [0,
1/\alpha), \mathbb{C}^p \big)$ of vector-valued functions.

For rational lattices $\alpha \bZ \times \beta \bZ $ with $\alpha
\beta \in \bQ $, the characterizations of the frame property and of the completeness of $\mathcal{G}(g,\alpha , \beta )$
are in terms of the spectrum of a certain matrix containing
Zak transforms of $g$.
Let $\mathcal{Q}_g(x,\xi )$  be the $p \times q   $-matrix with entries
\[
\mathcal{Q}_g(x,  \xi)_{jk}  = Z_\alpha g (x+ \frac{\alpha j}{p}, \xi - \beta k)
e^{2 \pi i j k /q}  \qquad   j=0, \dots , p-1, k= 0, \dots
,  q-1.
\]
and $\mathcal{A}_g(x,\xi)$ be the corresponding $p\times p$ square matrix
\[
\mathcal{A}_g(x,\xi) = \mathcal{Q}_g(x,\xi) \mathcal{Q}_g(x, \xi)^* \, .
\]

Then the  frame operator has the representation
\begin{equation} \label{frameopzak}
\overrightarrow{Z_{\alpha}}Sf(x, \xi)= \alpha^d \mathcal{A}_g(x,\xi) \overrightarrow{Z_{\alpha}}f(x, \xi), \quad x,\xi \in [0,
\alpha/p)
\times [0, 1/\alpha) ,
\end{equation}
which is well defined for  $g \in M^1({\mathbb{R}})$.
The following characterization of completeness is
 due to Zeevi and Zibulski~\cite[Theorem 2]{zibulski1997analysis}. (We use a slightly different setup that follows
\cite[Chapter 8]{MR1843717}.)
\begin{lemma}
  \label{l:comp1}
Let $g\in L^2({\mathbb{R}})$ and $\alpha \beta = p/q \in \bQ$. Then following
are equivalent.

(i) The Gabor system $\mathcal{G}(g,\alpha , \beta ) $ is complete in $L^2(\bR )$.

(ii) $\det \mathcal{A}_g(x,\xi)\neq 0$, for almost every $(x,\xi) \in {\mathbb{R}}^2$.

(iii) The matrix $\mathcal{Q}_g(x,\xi )$ has full rank $p$  for almost every $(x,\xi) \in {\mathbb{R}}^2$.
\end{lemma}

It is instructive to compare this characterization of completeness
with the corresponding characterization of the  frame property
in~\cite[Thm. 4]{zibulski1997analysis}:
$\mathcal{G}(g,\alpha , \beta )$ is a frame, if and only if\ there exist $0<\delta \leq \Delta $ such that
$\delta \leq \abs{\det
  \mathcal{A}_g(x,\xi)} \leq \Delta$, for almost every $(x,\xi) \in {\mathbb{R}}^2$.
When $g \in M^1({\mathbb{R}})$, the continuity and quasi-periodicity of $Z_g$
yield the following simple characterization
of the frame property.

\begin{lemma}
  \label{l:frame1}
Let $g\in M^1(\bR )$ and $\alpha \beta = p/q \in \bQ$. Then following
are equivalent:

(i) The Gabor system $\mathcal{G}(g,\alpha , \beta ) $ is a frame  in $L^2(\bR )$.

(ii)  $\det \mathcal{A}_g(x,\xi)\neq 0$, for all $(x,\xi) \in {\mathbb{R}}^2$.

(iii) The matrix $\mathcal{Q}_g(x,\xi )$ has full rank for all $(x,\xi) \in {\mathbb{R}}^2$.
\end{lemma}

Note the subtle difference in conditions (ii)! A single zero of $\det
\mathcal{A}_g $ may destroy the frame property of $\mathcal{G}(g,\alpha , \beta ) $.

We mention that in both cases one may write a formal reconstruction of
$f\in L^2(\bR ) $ from the correlations $\langle f, M_{\beta l } T_{\alpha
  k}g\rangle , k,l \in \bZ $ by inverting \eqref{frameopzak} as
follows:
\begin{equation}
  \label{eq:c7}
  f = \alpha ^{-d} \, \overrightarrow{Z_{\alpha}} ^{-1} \Big( \mathcal{A}_g (x,\xi )^{-1}
  \overrightarrow{Z_{\alpha}}S f(x, \xi) \Big) \, .
\end{equation}
If $\mathcal{G}(g,\alpha , \beta ) $ is a frame, then this reconstruction is stable, whereas for
a complete Gabor system this reconstruction may lead to instabilities
on certain subspaces of $L^2(\bR ) $, because $\mathcal{A}_g $ is not invertible
everywhere. For the classical coherent states $\phi (x) = e^{-\pi
  x^2}$ and $\alpha = \beta =1$ explicit reconstruction formulas are
known, see e.g.~\cite{Ner06}.

\subsection{Windows in the Gelfand-Shilov class}

For windows in the Gelfand-Shilov class $S^{1,1}$ the characterization
of completeness can be reformulated in a useful way, which  we now describe.

We say that a function $g:{\mathbb{R}} \to \bC$ is in the Gelfand-Shilov class $\saa({\mathbb{R}})$, if
$g$ and its Fourier transform $\hat{g}$ have exponential decay, i.e.,
$$
|g(x)| \lesssim   e^{-a|x|} \qquad  \forall x\in \mathbb{R}  $$
and
\begin{equation}
  \label{eq:c11}
|\hat{g}(\xi)| \lesssim  e^{-b|\xi|} \qquad  \forall \xi \in \mathbb{R} \,
\end{equation}
for some decay constants $a,b>0$.
This is not the standard definition but a simpler equivalent condition
due to ~\cite{MR1265470}.
In particular, every function of the form $g=R \phi $, where $R$ is a
rational function with no poles on the real axis and $\phi (x) =
e^{-\pi x^2}$, belongs to  the
Gelfand-Shilov class $\saa $.  Likewise,  $\saa$ includes functions
of the form
\[
g(x)= \sum_{k=1}^n c_k e^{2\pi i b_k x} \phi(x-a_k),
\]
where $a_1,\ldots,a_n,b_1,\ldots,b_n \in {\mathbb{R}}$.

The assumption $g\in \saa $ implies that $g$ is real analytic. More precisely,
by the theorem of Paley-Wiener every $g\in \saa $ extends to a
function that is analytic on the strip $S_b = \{ z\in \bC : |\Im z| <
b\}$, where $b$ is the decay constant in \eqref{eq:c11}. Moreover, for every $b' \in (0,b)$ there exists a constant
$C_{b'}$ such that
\begin{align*}
\abs{g(x+iy)} \leq C_{b'} e^{-a|x|} \qquad  x,y\in \mathbb{R}
\mbox{ with } \abs{y} \leq b'.
\end{align*}
See for example \cite[Theorem 3.9]{MR1322917}

We now observe that the Zak transform of $g\in \saa $ can also be extended to an analytic function
of two variables.

\begin{lemma}
\label{lemma_zak_analytic}
  If $g\in \saa({\mathbb{R}})$, then the Zak transform $Z_\alpha g$ can be extended
  to an analytic function on $S_b \times S_b \subseteq \bC ^2$ for some $b>0$.
\end{lemma}
\begin{proof}
Since $g\in \saa$, there exists $b>0$ such that $g$ extends analytically to $S_b$ and satisfies the uniform
decay estimate:
\begin{align*}
\abs{g(x+iy)} \lesssim e^{-2 \pi a \abs{x}} \qquad x \in {\mathbb{R}}, \abs{y} < b,
\end{align*}
for some $a>0$.
Without loss of generality, we assume that $b < a$. Let us show that $Z_\alpha g$ extends analytically to $S_b \times
S_b$. It suffices to show that the series
\begin{align*}
Z_\alpha g(z, w) = \sum_{k \in \mathbb{Z}} g(z- \alpha k) e^{2 \pi i
  \alpha w k}
\end{align*}
converges uniformly and absolutely on compact subsets of $S_b \times S_b$. Fix
$C>0$ and let $(z,w) \in \bC^2$ with
$\abs{z}, \abs{w} \leq C$ and $\abs{\Im z}, \abs{\Im w} \leq b$. Then
\begin{align*}
&\sum_{k \in \mathbb{Z}} \abs{g(z- \alpha k) e^{2 \pi i \alpha w k}}
\lesssim
\sum_{k \in \mathbb{Z}} e^{-2 \pi a \abs{z- \alpha k}} e^{2 \pi \alpha
  \abs{\Im w} \abs{k}} \\
 & \qquad \leq
\sum_{k \in \mathbb{Z}}
e^{-2\pi (a \alpha \abs{k} - a\abs{z} - \alpha b \abs{k})}
\leq
e^{2\pi aC} \sum_{k \in \mathbb{Z}}
e^{-2\pi \alpha (a-b) \abs{k}}
<+\infty ,
 \end{align*}
since $a> b $. This completes the proof.
\end{proof}

Using the analyticity of the Zak transform, we now  obtain the following characterization for
the completeness of a Gabor system with windows $g \in \saa$.

\begin{prop} \label{propan}
  Let $g\in \saa({\mathbb{R}})$ and $\alpha \beta  =p/q \in \bQ $, $\alpha \beta  \leq 1$.
Then the following are equivalent.

(i) $\mathcal{G}(g,\alpha , \beta ) $ fails to be complete.

(ii) $\det \mathcal{A}_g(x,\xi) =0 $ for all $x  , \xi \in \bR $.

(iii) $\mathcal{Q}_g(x,\xi )$ has rank $ < p$ for all $x,\xi \in \bR $, in other
words,  all $p\times p$ submatrices of $\mathcal{Q}_g(x,\xi )$ have determinant
zero.
\end{prop}

\begin{proof}
The equivalence between $(ii)$ and $(iii)$ is clear. In addition, by Lemma
\ref{l:comp1}, $(iii)$ implies $(i)$. Let us show that $(i)$ implies
$(iii)$. Assume that $\mathcal{G}(g,\alpha , \beta )$ is incomplete,  and  let $l_0, \dots, l_{p-1} \in
\{ 0, \dots , q-1\}$ be distinct indices, and
denote by $\mathcal{Q}_g^{l_0, \dots, l_{p-1}}$ the  $p \times p$
submatrix of $\mathcal{Q}_g(x,\xi ) $ with columns  $l_0, \dots, l_{p-1}$. Let
\begin{align*}
E := \set{(x,\xi) \in {\mathbb{R}}^2}{\det
\mathcal{Q}_g^{l_0, \dots, l_{p-1}}(x,\xi)=0}.
\end{align*}
We will  show that $E={\mathbb{R}}^2$. By Lemma \ref{l:comp1}, we know that $E$ has positive Lebesgue measure,
since, otherwise, $\mathcal{Q}_g(x,\xi)$ would have full rank almost everywhere.
In addition, by Lemma \ref{lemma_zak_analytic}, there exists $b>0$ such that
$Z_\alpha$ extends analytically to $S_b \times S_b$ and, therefore, so does $\det \mathcal{Q}_g^{l_0, \dots, l_{p-1}}$.
Hence, $E$ is the zero set of a real analytic function on ${\mathbb{R}}^2$, and, having positive measure, it must be equal to
${\mathbb{R}}^2$. We provide a short argument for this (known)  fact.

By the continuity of $\det \mathcal{Q}_g^{l_0, \dots, l_{p-1}}$, we know that $E$ is a closed set.
Hence, in order to prove that $E={\mathbb{R}}^2$, it suffices to show that $|{\mathbb{R}}^2 \setminus E| = 0$.
Given $x \in {\mathbb{R}}$, the section
\begin{align*}
E_x := \set{\xi \in {\mathbb{R}}}{(x,\xi) \in E}
\end{align*}
is the zero set of the function $\det \mathcal{Q}_g^{l_0, \dots, l_{p-1}}(x,\cdot):{\mathbb{R}} \to \bC$, which admits an
analytic extension to $S_b$. Hence, $E_x$ is either ${\mathbb{R}}$ or has measure zero.
Therefore, $E=(X \times {\mathbb{R}}) \cup N_1$, for some measurable sets $X,N_1 \subseteq {\mathbb{R}}$ with $|N_1|=0$ and $|X|>0$.
A similar argument, with the roles of $x$ and $\xi $ interchanged,
shows  that $E=({\mathbb{R}}\times Y) \cup N_2$, with $|Y|>0$ and
$|N_2|=0$. The conditions on $E$ imply that
$1_E(x,y)=1_X(x)=1_Y(y)$  for almost all $(x,y) \in {\mathbb{R}}^2$. Since  $\abs{E}>0$,
this is only possible if $1_E \equiv 1$ a.e. Therefore, $|{\mathbb{R}}^2 \setminus E| = 0$, as desired.
\end{proof}

Motivated by Proposition \ref{propan}, we compute explicitly the determinant of $ \mathcal{Q}_g^{l_0,
  \dots, l_{p-1}} (x,\xi ) $ for a given selection of columns $L \equiv \{ l_0 , \dots , l_{p-1}\} \subseteq
\{ 0, \dots , q-1\}$.

Let $\mathrm{Perm} (l_0,
  \dots, l_{p-1})$ denote the group of all permutations of the chosen
  columns. Then
\begin{multline*}
\det \mathcal{Q}_g^{l_0, \dots , l_{p-1}}(x,\xi )  = \sum_{\sigma \in
  \mathrm{Perm}(l_0, \dots, l_{p-1})} (-1)^{\mathrm{sgn} (\sigma)}
\prod_{j=0}^{p-1} Z_{\alpha}g(x+ \frac{\alpha j}{p}, \xi -  \beta \sigma(l_j)) e^{2 \pi i j \sigma(l_j)/q} \\
= \sum_{\sigma \in \mathrm{Perm}(l_0, \dots, l_{p-1})}
(-1)^{\mathrm{sgn} (\sigma)}
\sum_{k_0, \dots, k_{p-1} \in \mathbb{Z}} \prod_{j=0}^{p-1} g(x +
\frac{\alpha j}{p} - \alpha k_j) \times \\
e^{2 \pi i  \sum_{j=0}^{p-1}\big[(\xi - \beta \sigma(l_j))\alpha k_j + j \sigma(l_j)/q \big]}=0.
\end{multline*}
We sum over the permutations $\sigma$ first and denote the resulting
coefficients by
\begin{align}
\label{eq_def_ck}
c(k_0, \dots, k_{p-1}):=
\sum_{\sigma \in
  \mathrm{Perm}(l_0, \dots, l_{p-1})} (-1)^{\mathrm{sgn} (\sigma)} e^{2 \pi i
  \sum_{j=0}^{p-1}\big[ - p \sigma(l_j)k_j/q + j \sigma(l_j)/q \big]}
.
\end{align}
Then
\begin{align*}
\det \mathcal{Q}_g^{l_0, \dots , l_{p-1}}(x,\xi )  &=  \sum_{k_0, \dots, k_{p-1} \in \mathbb{Z}} \prod_{j=0}^{p-1} g(x +
\frac{\alpha j}{p} - \alpha k_j)c(k_0, \dots , k_{p-1}) e^{2\pi i
  \alpha (k_0
  + \dots + k_{p-1} ) \xi } \\
&= \sum _{N\in \bZ } \Big( \sum_{k_0 +  \dots +  k_{p-1}  = N} \prod_{j=0}^{p-1} g(x +
\frac{\alpha j}{p} - \alpha k_j)c(k_0, \dots , k_{p-1}) \Big) e^{2\pi
  i N  \xi } \, .
\end{align*}
Thus for fixed $x \in [0, \alpha ]$ the determinant $\det
\mathcal{Q}_g^{l_0, \dots , l_{p-1}}(x,\xi ) $ is a Fourier series
with coefficients
\begin{equation} \label{sum2}
\Theta_g^{L}(x,N):=\sum_{k_0 + \dots+ k_{p-1} = N} \prod_{j=0}^{p-1} g\big(x+ \alpha j/p
- \alpha k_j\big)c(k_0, \dots, k_{p-1}).
\end{equation}
Note that the
selection of columns of $\mathcal{Q}_g ^{l_0, \dots , l_{p-1}}(x,\xi )$ enters
only in the coefficients $c(k_0, \dots, k_{p-1})$.

We now state one more reformulation of Lemma~\ref{l:comp1}.

\begin{lemma} \label{l:comsaa}
Let $g \in \saa({\mathbb{R}}) $ and $\alpha \beta = p/q \in \bQ $, $\alpha \beta \leq 1$.  Then the following
are equivalent.

(i) The Gabor system $\mathcal{G}(g,\alpha , \beta ) $ is incomplete in $L^2(\bR ) $.

(ii) For all choices of subsets $L\equiv \{l_0, \dots, l_{p-1}\} \subset \{0, \dots, q-1 \}$,
all $x\in \bR$ and all $N \in {\mathbb{Z}}$, we have $\Theta_g^{L}(x,N)=0$.
\end{lemma}

\subsection{Complete Gabor Systems versus Gabor Frames}
Although the formulations of Lemma~\ref{l:comp1} and \ref{l:frame1}
 are deceptively similar, the
difference between completeness and the frame property is dramatic. To
highlight this difference, we show that  the spanning properties of a
Gabor family over a lattice are extremely stable and cannot
be modified  by naive changes. In order to state the result precisely
(for arbitrary dimension), we recall
that the \emph{lower Beurling density} of a set $N \subseteq {\mathbb{R}}^d$
is given by
\begin{align*}
D^{-}(N) := \liminf_{R \longrightarrow +\infty} \inf_{x \in {\mathbb{R}}^d}
\frac{\#(N \cap B_R(x))}{\abs{B_R(x)}} \, ,
\end{align*}
where $B_R(x)$ denotes the ball of radius $R$ centered at $x\in \bR^d
$. Hence $D^{-}(N)=0$ if and only if $N$ contains arbitrarily large holes.

\begin{prop}
\label{prop_no_sup}
  Assume that $g\in M^1({\mathbb{R}^d})$, $\Lambda \subseteq {\mathbb{R}^{2d}}$ is a lattice
  and  $\mathcal{G} (g,\Lambda  ) $ is not a frame. If $N \subseteq {\bR^{2d}} $ is a set such that $D^-(N) = 0$, then $\mathcal{G} (g, \Lambda \cup
  N)$ fails to be a frame.
\end{prop}
\begin{proof}
Assume on the contrary that $\mathcal{G} (g, \Lambda \cup  N)$ is a frame. The proof is based on Theorem 5.1 from
\cite{grorro15}, which implies that
for every set $\Gamma$ that is a weak limit of translates of $\Lambda \cup  N$, $\mathcal{G} (g, \Gamma)$ is also a frame. (We
refer the reader to \cite{grorro15} for the precise definition of weak convergence of sets.)

Since $D^-(N) = 0$, $N$ contains arbitrarily large holes centered at points of $\Lambda$.
 For every  $n \in {\mathbb{N}}$
there exists $\lambda_n \in \Lambda$ such that $N \cap B_n(\lambda_n) = \emptyset$. This implies that the sequence
of translates $\sett{(\Lambda \cup N)-\lambda_n: n \in {\mathbb{N}}}$ converges weakly to $\Lambda$. Indeed
\begin{align*}
\left(
(\Lambda \cup N)-\lambda_n
\right) \cap B_n(0) =
\left(
\Lambda \cup (N-\lambda_n)
\right) \cap B_n(0) = \Lambda \cap B_n(0).
\end{align*}
 By \cite[Theorem 5.1]{grorro15}  $\mathcal{G} (g, \Lambda)$ is a frame, contradicting our assumptions.
\end{proof}

Proposition \ref{prop_no_sup} means that in order to extend a complete Gabor family (that is not already a frame)
into a (non-uniform) Gabor frame, we need to add a set of strictly positive density. This complements the
results of Balan, Casazza, Heil and Landau that also stress the strong rigidity of the frame property:
for a Gabor frame $\mathcal{G} (g,\Lambda  )$ with window $g \in M^1$, it is always possible to remove a certain infinite subset
$\Lambda' \subseteq \Lambda$ while preserving the frame property. Moreover,
it is possible to choose $\Lambda'$ so that the density of the remaining set $\Lambda \setminus \Lambda'$ is
arbitrarily close to 1 \cite{MR2837145}.

\section{Explicit Completeness Theorems}

In this section we investigate the completeness  of Gabor systems $\mathcal{G}(g,\alpha , \beta ) $
for several general,  explicit classes of window functions  in $\saa
$ on a rectangular rational lattice.
In all cases we will make use of the fact that the Gabor system $\mathcal{G}
(\phi , \alpha ,  \beta )$ for $\phi (x) = e^{-\pi x^2}$ is complete
when $\alpha \beta \leq 1$ and then apply the algebraic
characterization of Lemma~\ref{l:comsaa}.

 We start with the
class of  windows that factor into a polynomial and the Gaussian.
We first  prove that,  for an arbitrary polynomial $P$, the
function  $g(x)=P(x)e^{-\pi x^2}$ generates a complete
system when $\alpha \beta$ is rational and $  \alpha \beta \leq 1$.

\begin{tm} \label{t:pol}
Let $g(x)= P(x) e^{-\pi x^2}$ with a non-zero polynomial  $P$ and $\alpha \beta \in \bQ $, $\alpha \beta \leq 1$.
Then  the
Gabor system $\mathcal{G}(g, \alpha, \beta)$ is
complete in $L^2(\mathbb{R})$.
\end{tm}
\begin{proof}
We set $d:=\deg(P)$ and assume without loss of generality that the leading coefficient of $P$ is $1$. As before, we
write $\alpha \beta = p/q$ for relatively prime positive integers.

We recall the fundamental fact  that the Gaussian $\phi (x)=e^{-\pi
  x^2}$ generates a complete Gabor system $\mathcal{G} (\phi , \alpha   , \beta )$, if and only if\   $\alpha \beta \leq
1$~\cite{bargmann71,perelomov71}.  We will show that  this fact implies that $\mathcal{G} (g, \alpha, \beta )$
with $g = P \phi $ is also complete.
More precisely,  we will verify the conditions of
Proposition~\ref{propan}  and show that there
exist column indices
$L\equiv\{l_0, \dots, l_{p-1}\}$ such that the determinant of $\mathcal{Q}_{\phi} ^{l_0, \dots,
  l_{p-1}}$ does not vanish identically.

We argue by contradiction  and  assume that $\mathcal{G}(g,\alpha , \beta ) $ is not
complete. By Lemma \ref{l:comsaa},
\begin{equation}
\label{sum2b}
\Theta_g^{L}(x,N)=\sum_{k_0 + \dots+ k_{p-1} = N} \prod_{j=0}^{p-1} g\big(x+ \frac{\alpha j}{p} - \alpha k_j\big)c(k_0, \dots,
k_{p-1}) =0
\end{equation}
for all $x \in \mathbb{R}$ and all $N \in \mathbb{Z}$ where the
coefficients
$c(k_0, \dots, k_{p-1})$ are given by \eqref{eq_def_ck}.

\textbf{Step 1.} We first  compute the expression $\Theta_{\phi}^{L}(x,N)$ for the
Gaussian $\phi (x) = e^{-\pi x^2}$ in place of $g$.  The exponent of
each product is
\begin{align*}
\lefteqn{-\frac{1}{\pi} \log  \prod_{j=0}^{p-1} \phi \big(x+
  \frac{\alpha j}{p} -
\alpha k_j \big)} \\
&=  \sum_{j=0}^{p-2} \big(x+ \frac{\alpha j}{p}- \alpha k_j \big)^2 + \big( x+
\frac{\alpha(p-1)}{p}  - \alpha(N- k_0-\dots- k_{p-2} ) \big)^2 \\
&= px^2 + 2x \alpha \big[\sum_{j=0}^{p-2} (\frac{j}{p}-k_j ) + \frac{p-1}{p} - N + \sum_{j=0}^{p-2} k_j \big] \\
& \qquad \quad  + \alpha^2\Big[ \sum_{j=0}^{p-2} \Big(\frac{j}{p} - k_j \Big )^2 +
\Big(\frac{p-1}{p} -N+ \sum_{j=0}^{p-2} k_j \Big)^2 \Big] \\
& = px^2 +\frac{2 x \alpha}{p} \left(\sum_{j=0}^{p-1}j\right) - 2x \alpha N
+ \alpha^2\bigg[  \sum_{j=0}^{p-2} \Big(\frac{j}{p} - k_j \Big)^2 +
\Big(\frac{p-1}{p} -N+ \sum_{j=0}^{p-2} k_j \Big)^2  \bigg] \, .
\end{align*}
Since $\mathcal{G} (\phi, \alpha  , \beta )$ is complete for $\alpha \beta
\leq 1$,
there exists some
$N\in \bZ$, such that  the quantity
\begin{align}
   e^{-\pi (p x^2 + \alpha (p-1)x - 2N\alpha x)} &  \, \sum_{k_0, \dots, k_{p-2} \in \mathbb{Z}}
\exp \Big(- \pi \alpha^2 \bigg[ \sum_{j=0}^{p-2} \bigg(\frac{j}{p} - k_j
\bigg)^2 +  \bigg(\frac{p-1}{p} -N+ \sum_{j=0}^{p-2}k_j
\bigg)^2\bigg] \Big)  \times \notag \\
& \quad c(k_0, \dots, k_{p-2}, N-k_0 -\dots- k_{p-2}) \not \equiv
0\, .    \label{eq:c20}
\end{align}

Thus at least  one of the coefficients
\begin{align}
  s_0(N)& =   \sum_{k_0, \dots, k_{p-2} \in \mathbb{Z}}
\exp \Big(- \pi \alpha^2 \bigg[ \sum_{j=0}^{p-2} \bigg(j/p - k_j
\bigg)^2 +  \bigg(\frac{p-1}{p} -N+ \sum_{j=0}^{p-2}k_j
\bigg)^2\bigg] \Big)  \times  \notag
 \\
& \qquad c(k_0, \dots, k_{p-2}, N-k_0 -\dots- k_{p-2}) \neq  0 \,
\label{eq:c21}
\end{align}
must be non-zero.

\textbf{Step 2.}
We next evaluate $\Theta_g^{L}(x,N)$ for $g(x) = P(x) e^{-\pi x^2}$. Since
$P$ is a polynomial of degree $d$ with leading coefficient $1$, the
product
$\prod_{j=0}^{p-1} P \big(x+ \alpha j/p -
\alpha k_j \big)$ is a polynomial of degree $dp$. The coefficient of
$x^{dp} $ is one, whereas the coefficients of the lower order terms
depend on $(k_0, \dots , k_{p-1})$ in a complicated way. What matters
is that the leading coefficient does not depend on $(k_0, \dots ,
k_{p-1})$. After summing over $(k_0, \dots , k_{p-1})\in \bZ ^p$ with
$k_0 + \dots +  k_{p-1} = N$, we obtain  from \eqref{sum2b} and the
calculation in Step~1 that
\begin{multline} \label{sum3}
e^{-\pi \big[ px^2 +  x \alpha(p-1) - 2x \alpha N
  \big]} \bigg[ x^{pd} s_0(N) + x^{pd-1} s_1(N) + \dots+ x
s_{pd-1}(N) + s_{pd}(N) \bigg] = 0,
\end{multline}
for some coefficients $s_k(N) \in \bC$,
all $N \in \mathbb{Z}$ and $x \in {\mathbb{R}}$. The coefficient of the leading
term is exactly $s_0(N)$ from \eqref{eq:c21} obtained for the Gaussian.
Since we have assumed that $\mathcal{G}(g,\alpha , \beta ) $ is incomplete, \eqref{sum3}
vanishes identically for all $N\in \bZ $ by Lemma~\ref{l:comsaa}, and thus  all
coefficients $s_m(N), m=0, \dots, pd,$ must vanish. In particular,
$s_0(N) = 0$ for all $N\in \bZ $, contradicting \eqref{eq:c21}.

Altogether we have proved that $\mathcal{G}(g,\alpha , \beta ) $ is complete for rational
$\alpha \beta \leq 1$.
\end{proof}

We single out the special case of the Hermite functions $h_n$ defined
by $h_n(x) =
e^{\pi x^2} \tfrac{d^n}{dx^n } (e^{-2 \pi x^2}) $.

\begin{cor} \label{cor:herm}
If $\alpha \beta \in \bQ$ and
  $\alpha \beta \leq 1$,  then $\mathcal{G} (h_n, \alpha, \beta)$  is complete in $L^2(\bR )$.
\end{cor}

It is known that for $\alpha \beta = 1 - 1/N$ for $N=2,3, \dots ,
\infty $    the  Gabor system
$\mathcal{G} (h_{2n+1}, \alpha , \beta )$ cannot be a frame\cite{LN13}.
Furthermore, the frame property fails for $\mathcal{G} (h_{4m+2}, \alpha,
\beta )$ and $\mathcal{G} (h_{4m+3}, \alpha,
\beta )$ over certain rational lattices~\cite{lem15}.
 Corollary~\ref{cor:herm} shows that for all Hermite windows
and the lattice parameters where the frame property has been shown to fail, the weaker property of completeness still
holds.

The result concerning Hermite functions has also  an interpretation in
terms of complex analysis. We define the
\emph{Bargmann-Fock spaces of polyanalytic functions} as
\[
A^2_n := \set{f:\bC\to\bC}{\int_{\bC} |f(z)|^2 e^{- \pi |z|^2} \mathrm{d} A(z)
<\infty, \bar \partial^n f =0},
\]
where $\mathrm{d} A(z)$ is the area measure on the complex plane and $\bar \partial = \frac12(\partial_x+ i
\partial_y)$. The corresponding orthogonal difference spaces
\[
\delta A^2_n := A^2_n \ominus A^2_{n-1}
\]
are called \emph{true polyanalytic Bargmann-Fock spaces}. These appear naturally in quantum mechanics
\cite{MR3203099, haimi2013polyanalytic}, where they are sometimes
called the Landau levels.
As in the case of the Gaussian $n=0$, one verifies that
$\mathcal{G}(h_n, \alpha, \beta)$ is a complete system if and only if $\alpha \mathbb{Z} \times \beta
\mathbb{Z}$ is a uniqueness set for $\delta A^2_n$ (see \cite{MR2672228,MR3203099}).

According to the previous corollary, any rectangular lattice of rational density larger or equal to $1$ is a
uniqueness set for $\delta A^2_q$. It would be interesting to obtain an alternative proof of this fact using complex
analysis. The standard techniques for analytic functions do not seem to work in the polyanalytic setting.

Next we consider windows of the form $g(x) = R(x)  \, e^{-\pi  x^2}$, where $R= P/Q$ is  a rational function with two
polynomials $P$ and $Q$. We may assume without loss of generality that the
leading coefficients of $P$ and $Q$ are $1$.  For well-posedness  we must assume
that $R$ does not have any poles  on the real axis. Then, in fact, the
poles of $g$ are outside a strip $S_a = \{ z\in \bC : |\Im z| < a\}$
for some $a>0$. It is then  easy to see that $g\in \saa $.

Before stating a completeness theorem for windows of this type, we
formulate a fact about rational functions required later.
\begin{lemma}
\label{lemma_poly}
Let  $P,Q
\in \mathbb{C}[X]$ be two monic polynomials  of degree $n$, and let
\begin{align*}
M := \max\sett{\abs{z}: z \in \bC, P(z)=0\mbox{ or }Q(z)=0}.
\end{align*}
Then   there exists a constant $C_n >0$ depending only on the degree
$n$, but not on $M$,  such that
\begin{align*}
\abs{\frac{P(x)}{Q(x)} - 1} \leq
\frac{C_n M^n}{\abs{x}},
\qquad
\text{ for all } \abs{x} \geq \max\{2 M,1\}.
\end{align*}
\end{lemma}
\begin{proof}
Let $\abs{x} \geq 2 M$ and let $Q(z) := \prod_{k=1}^n (z-z_k)$ with $z_k \in \bC$. Then
\begin{align*}
\abs{x-z_k} \geq \abs{x}-\abs{z_k} \geq \abs{x}-M \geq \frac{\abs{x}}{2},
\end{align*}
and therefore $\abs{Q(x)} \geq 2^{-n} \abs{x}^n$.

Writing  $P(x) = x^n + \sum_{k=0}^{n-1} a_k x^k$ and
$Q(x) = x^n + \sum_{k=0}^{n-1} b_k x^k$, the coefficients obey the estimates
$\abs{a_k}, \abs{b_k} \leq c_n M^n$. (This follows, for example,
by expressing $a_k$ and $b_k$ as sums of products of the corresponding roots.)

For  $\abs{x} \geq \max\{2M,1\}$, we now simply estimate
\begin{align*}
\abs{\frac{P(x)}{Q(x)} - 1} &=
\abs{\frac{\sum_{k=0}^{n-1} (a_k-b_k) x^k}{Q(x)}}
\leq
\frac{2^n  \sum_{k=0}^{n-1} \abs{a_k-b_k}\abs{x}^k}
{\abs{x}^n}
\leq \frac{2^{n+1} c_n M^n}{\abs{x}},
\end{align*}
and we may take $C_n = 2^{n+1} c_n$.
\end{proof}

We then have the following completeness theorem. Note that it contains
Theorem~\ref{t:pol} as a special case, but the proof is more involved, so we presented the simpler case first.

\begin{tm} \label{t:rat}
  Let $g(x)= R(x)  e^{-\pi x^2}$ with a non-zero rational function  $R$
  that does not have any poles on  $\mathbb{R} $, and let  $\alpha \beta = p/q \in \bQ $, $\alpha \beta \leq 1$.   Then  the
Gabor system $\mathcal{G}(g, \alpha, \beta)$ is
complete in $L^2(\mathbb{R})$.
\end{tm}

\begin{proof}
  The proof follows a similar outline as before for
  Theorem~\ref{t:pol}. In the proof of Theorem~\ref{t:pol} we
  evaluated $\Theta_g^{L}(x,N)$ in ~\eqref{sum2} for both $g(x)=P(x) e^{-\pi x^2}$
  and for the Gaussian $\phi (x) = e^{-\pi x^2}$ and then showed that
  the expression for the coefficient of the highest power in $x$
  equalled, up to an exponential factor, precisely the expression
  $\Theta_{\phi}^{L}(x,N)$ for the Gaussian. Since $\Theta_{\phi}^{L}(x,N)$ cannot vanish identically,
neither can $\Theta_g^{L}(x,N)$ for $g = P  \phi $, whence the completeness of
  $\mathcal{G}(g,\alpha , \beta ) $.

In the case of a function $g(x) = R(x) e^{-\pi x^2}$ with rational
$R$, we investigate the behavior of $\Theta_g^{L}(x,N)$ for $x \rightarrow \infty$.

Assume that $\mathcal{G}(g,\alpha , \beta ) $ is not complete. Then, by Lemma \ref{l:comsaa}
and the calculation in Step 1 of the proof of Theorem~\ref{t:pol}, we conclude that,
for all $x \in {\mathbb{R}}$ and $N \in {\mathbb{Z}}$,
\begin{align}
\nonumber
0&= \Theta_g^{L}(x,N)
\\
\nonumber
&= \sum_{k_0 + \dots+ k_{p-1} = N} \prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big) \prod_{j=0}^{p-1} \phi \big(x+
\frac{\alpha j}{p} - \alpha k_j\big) c(k_0, \dots, k_{p-1})
\notag \\
&= e^{-\pi (px^2 + \alpha (p-1) x  - 2x \alpha N
  )} \sum_{k_0 + \dots+ k_{p-1} = N} \prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big) \times  \notag \\
& \quad  \exp \Big(- \pi \alpha^2 \bigg[ \sum_{j=0}^{p-2} \bigg(j/p - k_j
\bigg)^2 +  \bigg(\frac{p-1}{p} -N+ \sum_{j=0}^{p-2}k_j
\bigg)^2\bigg] \Big) \, c(k_0, \dots, k_{p-1})\, . \label{eq:c24}
\end{align}
For $(k_0, \dots, k_{p-1}) \in {\mathbb{Z}}^p$, let
\begin{align}
\label{eq_def_dk}
d(k_0, \dots, k_{p-1}) &:=
\exp \Big(- \pi \alpha^2 \sum_{j=0}^{p-1}
(j/p - k_j)^2\Big) c(k_0, \dots, k_{p-1}),
\end{align}
and drop one term in exponential in \eqref{eq:c24} to obtain
\begin{equation}
\label{eq:c24b}
\sum_{k_0 + \dots+ k_{p-1} = N} d(k_0, \dots, k_{p-1})
\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)=0, \qquad x\in{\mathbb{R}}, N \in {\mathbb{Z}}.
\end{equation}
Now let $R=P/Q$, with $P,Q \in \mathbb{C}[X]$ monic,
$\Delta = \mathrm{deg}\, (P) -   \mathrm{deg}\, (Q) \in {\mathbb{Z}}$ and note
that
\begin{equation}
  \label{eq:c23}
  \lim _{x\to \infty } \frac{P(x)}{x^\Delta Q(x)} = 1.
\end{equation}
After multiplying \eqref{eq:c24b} by $x^{-p\Delta }$, we obtain
\begin{align}
  \label{eq:c25}
0=   &\sum_{k_0 + \dots+ k_{p-1} = N} x^{-p \Delta } \prod_{j=0}^{p-1}  R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big) d(k_0, \dots, k_{p-1}) \, .
\end{align}
Now we let $x$ tend to infinity, formally interchange the sum and the limit
- this is carefully justified below - and use \eqref{eq:c23} obtaining
\begin{align*}
  0 &=   \lim _{x\to \infty }  \sum_{k_0 + \dots+ k_{p-1} = N}
  x^{-p \Delta } \prod_{j=0}^{p-1}  R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big) d(k_0, \dots, k_{p-1})  \\
&=    \sum_{k_0 + \dots+ k_{p-1} = N}  \lim _{x\to \infty }  x^{-p \Delta } \prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big) d(k_0, \dots, k_{p-1}) \\
&=   \sum_{k_0 + \dots+ k_{p-1} = N}   d(k_0, \dots, k_{p-1})
= s_0(N),
\end{align*}
where the last expression is exactly $\Theta_{\phi}^{L}(x,N)$ for the Gaussian $\phi$
(apart for an exponential term). Since $s_0(N)$ must be non-zero
for some $N$, we have arrived at a contradiction and conclude that
$\mathcal{G}(g,\alpha , \beta ) $ must be complete.

For a rigorous proof we need to justify the interchange of the limit
and the sum. Let $M>1$ and $x \in {\mathbb{R}}$ be arbitrary.
Using
\eqref{eq:c24b}, we write the partial sums of  $s_0(N)$
as:
\begin{align*}
s^M_0(N)&:=
\sum_{
\stackrel{k_0 + \dots+ k_{p-1} = N}{\abs{(k_0, \dots, k_{p-1})}_2 \leq M}}
d(k_0, \dots, k_{p-1})
\\
&=
\sum_{
\stackrel{k_0 + \dots+ k_{p-1} = N}{\abs{(k_0, \dots, k_{p-1})}_2 \leq M}}
d(k_0, \dots, k_{p-1})
\left(1-x^{-p\Delta}\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)\right)+
\\
&\qquad
\sum_{
\stackrel{k_0 + \dots+ k_{p-1} = N}{\abs{(k_0, \dots, k_{p-1})}_2 \leq M}}
d(k_0, \dots, k_{p-1})
x^{-p\Delta}\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)
\\
&=
\sum_{
\stackrel{k_0 + \dots+ k_{p-1} = N}{\abs{(k_0, \dots, k_{p-1})}_2 \leq M}}
d(k_0, \dots, k_{p-1})
\left(1-x^{-p\Delta}\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)\right)-
\\
&\qquad -
\sum_{
\stackrel{k_0 + \dots+ k_{p-1} = N}{\abs{(k_0, \dots, k_{p-1})}_2 > M}}
d(k_0, \dots, k_{p-1})
x^{-p\Delta}\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)
\\
&=: t_N^{M}(x)-u_N^{M}(x).
\end{align*}
The  numbers $t^{M}_N(x)$ and  $u^{M}_N(x)$ depend on $x$, whereas the
partial sum $s_0^M(N)$ is independent of $x$. To show that $\lim
_{M\to \infty }
s_0^M(N) = 0$, we will choose  $x= x_M$ judiciously  and
obtain suitable bounds. First we note that the coefficients $d(k_0,\ldots,k_{p-1})$ satisfy the decay condition:
\begin{align}
\label{eq_dec_dk}
\abs{d(k_0,\ldots,k_{p-1})} \lesssim e^{-\gamma \abs{(k_0, \dots, k_{p-1})}_2^2}
\end{align}
for some $\gamma>0$. Indeed, for each $j \in \{0,\ldots,p-1\}$ and $k_j \in {\mathbb{Z}}$,
since $\abs{j/p} \leq \tfrac{p-1}{p} <1$, it follows that
$\abs{k_j - j/p} \gtrsim \abs{k_j}$.

We let $n:=\max\{\deg(P),\deg(Q)\}$. For $M \gg 1$ and
$\abs{(k_0, \dots, k_{p-1})}_2 \leq M$, the function
\begin{align*}
x^{-p\Delta}\prod_{j=0}^{p-1} R\big(x+
\frac{\alpha j}{p} - \alpha k_j\big)
\end{align*}
is a quotient of two monic polynomials of degree $pn$ whose complex roots
lie inside a ball of radius $\lesssim M$. Using Lemma \ref{lemma_poly},
we now choose $x_M \in {\mathbb{R}}$ with $\abs{x_M} \asymp M^{n+1}$ such that
\begin{align*}
\abs{1-x_M^{-p\Delta}\prod_{j=0}^{p-1} R\big(x_M+
\frac{\alpha j}{p} - \alpha k_j\big)} \lesssim \frac{1}{M}.
\end{align*}
Combining this estimate with \eqref{eq_dec_dk} we obtain
\begin{align}
\label{eq_tm}
\abs{t^{M}_N(x_M)} \lesssim \frac{1}{M}
\sum_{(k_0, \dots, k_{p-1})} \abs{d_{(k_0, \dots, k_{p-1})}}
\lesssim \frac{1}{M}.
\end{align}
Second, since $R$ is a rational function without real poles, it satisfies the estimate
\begin{align*}
\abs{R(x)} \lesssim (1+\abs{x})^{l}, \qquad x \in {\mathbb{R}},
\end{align*}
for $l:=\max\{\Delta,0\}$. This allows us to bound,
for $(k_0, \dots, k_{p-1}) \not= 0$,
\begin{align*}
&\abs{x_M^{-p\Delta}\prod_{j=0}^{p-1}R\big(x_M+
\frac{\alpha j}{p} - \alpha k_j\big)}
\lesssim M^{(n+1)p\abs{\Delta}}
\prod_{j=0}^{p-1}
(1+|x_M+\frac{\alpha j}{p} - \alpha k_j|)^{l}
\\
&\qquad
\lesssim M^{(n+1)p\abs{\Delta}}
\left(
M^{(n+1)lp}+\abs{(k_0, \dots, k_{p-1})}_2^{lp}\right)
\lesssim
M^s \abs{(k_0, \dots, k_{p-1})}_2^s,
\end{align*}
for some $s>0$. Therefore
\begin{align*}
\abs{u^{M}_N(x_M)} \lesssim
M^s \sum_{\abs{(k_0, \dots, k_{p-1})}_2>M}
\abs{(k_0, \dots, k_{p-1})}_2^s e^{-\gamma\abs{(k_0, \dots, k_{p-1})}^2_2}.
\end{align*}
The last bound shows that $\abs{u^{M}_N(x_M)} \longrightarrow 0$
as $M \longrightarrow +\infty$. Combining this with \eqref{eq_tm} we conclude that
\begin{align*}
s_0(N)=\lim_{M \longrightarrow +\infty} s^M_0(N)=0,
\end{align*}
and  the proof is complete.
\end{proof}

As a consequence of Theorem \ref{t:rat}, we obtain the completeness of Gabor systems for a
class of  totally positive functions. Let $\delta _j \in \bR $, $j=1,
\dots , M$, $\gamma >0$ and set
\begin{equation}
  \label{eq:c26}
\hat{g}(\xi ) = e^{-\gamma \xi ^2} \prod _{j=1}^M (1+2\pi  i \delta _j
\xi )^{-1} \, .
\end{equation}
Schoenberg's factorization theorem \cite{schoenberg1947totally} asserts that $g$ is a
totally positive function. Since $\hat{g}$ satisfies the
assumptions of Theorem~\ref{t:rat} and the completeness of a Gabor
system is invariant under the Fourier transform, we obtain the following corollary.

 \begin{cor} \label{cortp}
Let $g$ be a totally positive function whose Fourier transform factors
as in \eqref{eq:c26} and assume that $\alpha \beta \leq 1$ is
rational. Then $\mathcal{G}(g,\alpha , \beta ) $ is complete in $L^2(\bR ) $.
\end{cor}
To put the corollary into perspective, we note that it is conjectured
that for an  arbitrary  totally positive function $g$  the Gabor system
$\mathcal{G}(g,\alpha , \beta ) $ is a frame, if and only if\ $\alpha \beta < 1$. At this time, this
conjecture is known to be true only for totally positive functions
whose  Fourier transform factors as  $\prod _{j=1}^M (1+2\pi  i \delta _j
\xi )^{-1} $~\cite{GS13}. Corollary \ref{cortp} shows at least
completeness for another class of totally positive functions.

As a final application of the method of proof used in Theorem
\ref{t:pol} we treat  windows of the type
$E(x)\phi(x)$, where $E$ is an exponential polynomial
\begin{equation} \label{expsum}
E(x)= \sum_{\lambda \in \Lambda} a_\lambda e^{\lambda x},
\end{equation}
for some  finite set $\Lambda \subset \bC $ and non-zero coefficients  $a_\lambda \in \bC$.
\begin{tm} \label{t:expsum}
Let $g(x)=E(x)\phi(x)$ where $E(x)$ is of the form \eqref{expsum},
$\alpha \beta \leq 1$, and $\alpha \beta \in \bQ$.  Then $\mathcal{G}(g,\alpha , \beta )$ is
complete in $L^2(\bR )$.
\end{tm}
\begin{proof}
We proceed  as in the proofs of Theorems \ref{t:pol}
and \ref{t:rat}. Writing $\alpha \beta =p/q$ and with the notation from \eqref{eq_def_dk},
the incompleteness condition of Lemma \ref{l:comsaa} becomes
\begin{multline} \label{horriblesum}
0=
\sum_{\lambda_1, \dots, \lambda_p \in \Lambda} a_{\lambda_0} \cdots a_{\lambda_{p-1}}
e^{\sum_{j=0}^{p-1} \lambda_j x} \sum_{k_0+\dots+k_{p-1}=N} e^{-\alpha \sum_{j=0}^{p-1} \lambda_j k_j}
d(k_0, \dots, k_{p-1}),
\end{multline}
for all $N \in {\mathbb{Z}}$. Next, let $\lambda ^*\in \Lambda $ be the
minimal element of $\Lambda $  in the lexicographic ordering of $\bR
^2$. In other words, let $\tilde \Lambda$ be the subset
of $\Lambda$ which contains the elements of $\Lambda$ with  minimal
real part and  let $\lambda^*$ be the
element of $\tilde{\Lambda}$ with minimal  imaginary part.
Then $\sum_{j=0}^{p-1} \lambda_j = p \lambda^*$ if and only if
$\lambda_1=\dots=\lambda_p=\lambda^*$.
Since the complex exponentials $x \to e^{wx}, w\in \bC, x\in \bR$, are
linearly independent, the coefficient of $e^{p\lambda ^*x} $ must
vanish. Since $a_\lambda \neq 0$ for all $\lambda \in \Lambda $,
it follows from \eqref{horriblesum} that
\begin{align*}
0= e^{-\alpha Np \lambda^*} \sum_{k_0+\dots+k_{p-1}=N}
d(k_0, \dots, k_{p-1}) = e^{-\alpha Np \lambda^*} s_0(N),
\end{align*}
for all $N \in {\mathbb{Z}}$. As in the proofs of Theorems \ref{t:pol} and Theorem \ref{t:rat},
this contradicts the completeness of $\mathcal{G}(\phi,\alpha,\beta)$, with $\phi$ the Gaussian.
\end{proof}

\begin{cor}
  Let $g = \sum _{j=1}^n d_j M_{b _j} T_{a_j} \phi $ be a non-zero
  finite linear combination of time-frequency shifts of the Gaussian
$\phi$, $\alpha \beta \leq 1$, and $\alpha\beta \in \bQ$. Then
$\mathcal{G}(g,\alpha , \beta ) $
is complete in $L^2(\bR )$.
\end{cor}

\begin{proof}
  This is clear since sums of time-frequency shift s can be written in the
  form~\eqref{expsum}, as $M_b T_a \phi (x) = e^{2\pi \lambda x}
  \phi (x) e^{-\pi b ^2}$ for $a,b \in \bR $ and $\lambda = a
  +ib $.
\end{proof}

\textbf{Concluding remarks:}
1. In view of Theorem~\ref{tm:intro} it is tempting to conjecture that
$\mathcal{G}(g,\alpha , \beta ) $ with $\alpha \beta \leq 1$ is complete for \emph{every}
function
$g\in \saa $. If true, the proof of this statement must heavily depend
on the analyticity of $g$ and $\hat{g}$ and its Zak transform
$Z_\alpha g$. The following simple counter-example shows what may
happen without analyticity. Let  $g $ be  a Schwartz function  with support in $\bigcup _{j\in \bZ } [2j, 2j+1]$, then $\mathcal{G}(g,\alpha , \beta )
$ is incomplete  whenever $\alpha \in [0,1)+2\bZ $ and  $\beta >0$,
although $g$ and $\hat{g}$ are $C^\infty $.

2.  Furthermore we remark that the Gabor systems $\mathcal{G}(g,\alpha , \beta ) $ with $\alpha
\beta <1$ in
Theorem~\ref{tm:intro}, although being complete, cannot be   Schauder
bases.  A  Gabor
Schauder basis $\mathcal{G}(g,\alpha , \beta ) $ must satisfy  $\alpha \beta = 1$ and the
corresponding  window $g$ is  poorly
localized either in the time or the frequency domain \cite{dehe00,
  MR2278667}.

3. The completeness results depend heavily on the lattice structure of
the phase space shifts. Although the examples seem counter-intuitive
from the point of view of quantum mechanics,
 one may construct
complete Gabor systems  without lattice structure that are  of density zero~\cite{ALS09}.

\vspace{3 mm}

\textbf{Acknowledgements:} The second author would like to thank
Kristian Seip and Yurii Lyubarskii for useful discussions.

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redpajama_set_name: RedPajamaArXiv


text:
package org.nem.monitor.config;

import org.nem.core.utils.ExceptionUtils;

import java.io.InputStream;
import java.util.Properties;

/**
 * Class responsible for loading all monitor configuration settings.
 */
public class MonitorConfiguration {
	private final String nemFolder;

	/**
	 * Creates a new configuration object from the default properties.
	 */
	public MonitorConfiguration() {
		this(loadDefaultProperties());
	}

	protected static Properties loadDefaultProperties() {
		return ExceptionUtils.propagate(() -> {
			try (final InputStream inputStream = MonitorConfiguration.class.getClassLoader().getResourceAsStream("config.properties")) {
				final Properties properties = new Properties();
				properties.load(inputStream);
				return properties;
			}
		}, IllegalStateException::new);
	}

	/**
	 * Creates a new configuration object around the specified properties.
	 *
	 * @param properties The specified properties.
	 */
	public MonitorConfiguration(final Properties properties) {
		this.nemFolder = getOptionalString(properties, "nem.folder", this.getDefaultFolder()).replace("%h", this.getDefaultFolder());
	}

	protected static String getOptionalString(final Properties properties, final String name, final String defaultValue) {
		final String value = properties.getProperty(name);
		return null == value ? defaultValue : value;
	}

	/**
	 * Get the default folder for database and log files.
	 *
	 * @return path to the folder location.
	 */
	private String getDefaultFolder() {
		return System.getProperty("user.home");
	}

	/**
	 * Gets the path to the folder where database and log files should be located.
	 *
	 * @return The path to the folder.
	 */
	public String getNemFolder() {
		return this.nemFolder;
	}
}

redpajama_set_name: RedPajamaGithub


text:
Gaming History: Just how Far Back Perform Gaming Games Go? Potentially With the particular Egyptian Pharaohs
Home / Others /
mrebvies Others
At what period of time in historical past did individuals start to gamble? The precise period of time is unidentified but imagine it or not, pairs of dice have been identified in Egyptian tombs over 4,000 many years aged! Also, gambling video games had been played in historic China, exactly where Poker is considered to have originated. In 1492 for the duration of the Columbus landing, Native People in america were sporting activities betting on the end result of a recreation resembling LaCross. So when did it all start off in The usa? Go through on.
Early The us
Gambling in The united states began with the first English settlers in the 1600's. Their traditions incorporated card game titles that ended up element of the aristocratic lifestyle. Even so, when Puritans colonized in Massachusetts Bay they experienced the independence to produce their possess culture which provided hostility in direction of gambling. They outlawed the possession of dice, playing cards, and gambling desk game titles in their communities. Nevertheless, gambling prevailed in other localities. Several English colonists considered gambling to be a appropriate type of entertainment.
The colony of Virginia was the initial to comprehend that lotteries could elevate capital for regional governments. Sooner or later all thirteen colonies have been boosting lottery earnings. Proceeds assisted build Universities like Harvard, Yale, and Princeton. Lotteries also funded churches and libraries. Founding Fathers George Washington, Ben Franklin, and John Hancock have been promoters of certain lotteries for public performs tasks. When the Revolutionary War commenced, the Continental Congress voted for a $10 million lottery to finance the war.
Westward Ho
In the course of the early 1800's the taverns and street homes permitted dice and card games, creating the first variation of casinos. As America's population started to boost, casinos turned more lavish. https://citysoups.com/ was a major trade route where merchants and entrepreneurs introduced their money. Gambling on riverboats became a favourite pastime and New Orleans turned the gambling capitol of America. In 1849 gambling followed the pioneers to California during the gold hurry. Gambling institutions commenced to flourish there and west of the Mississippi, like Nevada. In the late 1800's Roulette was adopted from France and the Slot Device was invented.
A lot of the general public seen gambling as a social ill since it was connected to alcoholism and prostitution. Reformers confident jurisdictions to shut down the Dens of Iniquity. Most states discontinued lotteries as well. Riverboat gambling dried up with the introduction of the railroad. By the end of the century only Nevada allowed gambling.
In 1910 Nevada ultimately shut the door on gambling, which remaining horse race wagering the only lawful entity in America. In 1912 Arizona and New Mexico were granted statehood beneath the problem that gambling remain outlawed. In the course of the 1920's prohibition era, the public's thirst for gambling matched that of alcohol. Casinos went underground along with the speakeasys. In 1931 Nevada legalized gambling yet again and remained the only condition to do so until finally the latter 50 percent of the century. Gambling flourished underground as structured crime produced weighty investments in Nevada, and prospered by managing off observe betting and the quantities lottery.
For the duration of the 1950's the U.S. Senate investigated organized crime's website link to unlawful gambling. Ultimately the mob departed Las Vegas. States place bookies out of company by legalizing off keep track of betting and quantities video games. Atlantic Metropolis approved gambling in 1976, the Indian Gaming Act was accepted by congress in the late 1980's. Dockside riverboat gambling manufactured a comeback, racetracks set up slot equipment while Las Vegas reinvented itself by constructing mega resorts in the course of the 1990's.
The American Gaming Affiliation documented that there are 832,988 slot equipment unfold out over 1,151 casinos and racetracks throughout forty four states with a lot more on the way. It seems that the American culture's thirst for gambling matches that of the Egyptian Pharaohs! The usa has embraced gambling as an suitable type of enjoyment.
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From torchlight parades to weekly Singo Music Trivia downtown, there are plenty of events and winter activities happening all season long!
Now that seasons have changed and the hustle and bustle of summer has slowed down, it's a great time to list your home! Here are 4 reasons to list your home this fall.
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text:
Memorandum of Understanding concerning the Conservation of the Manatee and Small Cetaceans of Western Africa and Macaronesia
Anas querquedula
Information d'évaluation
Instruments de la CMS
Date d'inscription à l'Annexe II
Répartition géographique
Afghanistan, Albania, Algeria, Andorra, Armenia, Austria, Azerbaijan, Bahrain, Bangladesh, Belarus, Belgium, Benin, Bosnia and Herzegovina, Bulgaria, Burkina Faso, Burundi, Cambodia, Cameroon, Central African Republic, Chad, China, Croatia, Cyprus, Czech Republic, Democratic People's Republic of Korea, Democratic Republic of the Congo (Kinshasa), Denmark, Djibouti, Egypt, Eritrea, Estonia, Ethiopia, European Union, Finland, France, Gambia, Georgia, Germany, Ghana, Greece, Guinea, Guinea-Bissau, Hungary, India, Iran, Iraq, Ireland, Israel, Italy, Japan, Jordan, Kazakhstan, Kenya, Kuwait, Kyrgyzstan, Lao People's Democratic Republic, Latvia, Lebanon, Libya, Liechtenstein, Lithuania, Luxembourg, Mali, Malta, Mauritania, Monaco, Mongolia, Montenegro, Morocco, Myanmar, Netherlands, Niger, Nigeria, North Macedonia, Norway, Oman, Pakistan, Poland, Portugal, Qatar, Republic of Korea, Republic of Moldova, Romania, Russian Federation, San Marino, Saudi Arabia, Senegal, Serbia, Slovakia, Slovenia, Somalia, Spain, Sudan, Sweden, Switzerland, Syrian Arab Republic, Tajikistan, Thailand, Togo, Tunisia, Turkey, Turkmenistan, Uganda, Ukraine, United Arab Emirates, United Kingdom, United Republic of Tanzania, Uzbekistan, Viet Nam, Yemen, Zambia
Pas de photo pour Anas querquedula
Noms communs
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Knäckente
COVID-19 ET ESPÈCES MIGRATRICES
Faits et informations sur la maladie du coronavirus (COVID-19) et la faune migratrice. En savoir plus
Coup d'oeil sur les média


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The lion fiercely attacks Hercules. The poet stands in the shop and recites a verse.


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Heard that sophomore to be and baseball/basketball stand out Isaac Kelly has now been out at practice for the Lions. They've been working him at QB, WR, and safety. He will play receiver this season for the Lions and likely take over under center after Parks graduates this year. Kelly is a very nice athlete that will help Lion football out a lot.
That's good news for the Lions. Good to hear they getting more players out.
I like seeing athletes from other sports going out for football.
Hopefully this young man can get a friend from his round ball team to join him on the field.
There was a hotel close to I 64 in Bath County. It was a chain hotel and then changed to something else. I don't know if it is still open.
1 week until we all get to see this event.
Someone want to explain to the new guy how this works.
Can't wait to see old friends and the same goes for making new ones.
1 week until we all get to see this event. Someone want to explain to the new guy how this works. Can't wait to see old friends and the same goes for making new ones.
EKC grid is usually three normal quarters for varsity and the last quarter is a running clock for JV and freshmen.
Looking forward to the game. I won't be able to attend but I will have it on hudl as soon as it is posted. I do know that senior RB/LB Kirk Everman will most likely rest in this as he's trying to nurse a hamstring injury sustained in summer practice. Hope the Rams are healthy and ready to go. Good luck to both teams!
Reserved my room for Friday.
I will be doing game reports on some if not all five games.
Some might not be up until Monday after work and the gym, but they will be up and pics will be included.
Stupid, why a running clock for JV and Freshman. There is a reason some of the programs in this part of the state never amount to much.
Time constraints is why. The 2nd night of the grid has 3 games, and most years you can't start play early enough due to the heat index. For most teams, the quarter of running clock JV is more than enough, as JV regular season games are about growth and experience, which is similar to a scrimmage.
Scrimmages can be up to 3 hours...Ive never liked the EKC grid for that reason...it limits the teams on their time...in which in this case the JV and frosh suffer.
See you guys all tomorrow.
I will be the only one there with a green Newark Catholic hat on.


text:
Israel lifts Muslim age limit at Al-Aqsa as 90 wounded in Gaza, West Bank
JERUSALEM World
Israel lifted age restrictions for the main weekly prayers at Jerusalem's flashpoint Al-Aqsa mosque compound, on October 23, 2015. (AA Photo)
Israel lifted age limits for prayers at Jerusalem's Al-Aqsa mosque compound Friday in an apparent bid to ease tensions as nearly 90 people were injured in clashes in Gaza and the West Bank.
While thousands of Muslims prayed without incident at the ultra-sensitive site, young Palestinians clashed with Israeli police in the occupied West Bank and Gaza Strip, leaving scores injured by Israeli gunfire, according to emergency services.
The international community has scrambled to defuse a wave of violence that many fear heralds a Palestinian uprising, and UN Secretary-General Ban Ki-moon said it was time for Israeli and Palestinian leaders to meet face to face.
"There is no substitute to direct talks," he said after meeting Israeli Prime Minister Benjamin Netanyahu and Palestinian president Mahmud Abbas.
US Secretary of State John Kerry held long talks with Netanyahu in Berlin on Thursday, voicing hope that "we can seize this moment and pull back from the precipice".
The top American diplomat is due to hold talks with Abbas and Jordan's King Abdullah II at a meeting in Amman on Saturday.
Kerry, his Russian counterpart Sergei Lavrov, EU foreign policy chief Federica Mogherini and Ban -- members of the Middle East peacemaking Quartet -- met on Friday in Vienna and issued a call for Israel to work with Jordan, historic steward of the Al-Aqsa compound.
A Quartet statement called for "maximum restraint and avoidance of provocative rhetoric and actions."
According to emergency services, 65 people, including three journalists, were hit by Israeli fire in the blockaded Gaza Strip.
The Israeli army said an Israeli woman and her two young daughters were injured in the West Bank when her car was hit with a Molotov cocktail.
The latest wave of violence erupted over the status of the Al-Aqsa mosque compound, a site sacred to both Jews and Muslims that has long been a crucible for tensions fuelling the decades-old Israeli-Palestinian conflict.
The Jordanian-run Islamic Waqf, which administers Al-Aqsa, said 25,000 Muslims attended Friday prayers after age restrictions were lifted for the first time since mid-September, when Palestinians and Israeli police clashed.
Israeli police estimated the crowd Friday at 30,000.
"Of course it is better but there are still checkpoints and searches. There is still no respect," said worshipper Wissam Abu Madi, 20, who said he believed a wave of attacks on Israelis would continue.
Clashes erupted during Jewish religious holidays last month as an increase in visits by Jews to the compound raised fears among Muslims that Israel was planning to change longstanding rules governing the site.
To avoid tensions Jews are allowed to visit but not pray at the site located in east Jerusalem, which was annexed by Israel in 1967.
It is managed by the Jordanian-run Waqf but Israel controls access.
Since October 1, at least 50 Palestinians and one Israeli Arab have died in clashes with Israeli police or while carrying out attacks.
One Israeli Jew and one Eritrean have also been killed after being mistaken for attackers.
Last Update: Oct 23, 2015 11:48 pm


text:
using pb = global::Google.Protobuf;
using pbc = global::Google.Protobuf.Collections;
using pbr = global::Google.Protobuf.Reflection;
using scg = global::System.Collections.Generic;
namespace Google.Ads.GoogleAds.V10.Enums {

  /// Holder for reflection information generated from google/ads/googleads/v10/enums/value_rule_set_dimension.proto
  public static partial class ValueRuleSetDimensionReflection {

    #region Descriptor
    /// File descriptor for google/ads/googleads/v10/enums/value_rule_set_dimension.proto
    public static pbr::FileDescriptor Descriptor {
      get { return descriptor; }
    }
    private static pbr::FileDescriptor descriptor;

    static ValueRuleSetDimensionReflection() {
      byte[] descriptorData = global::System.Convert.FromBase64String(
          string.Concat(
            "Cj1nb29nbGUvYWRzL2dvb2dsZWFkcy92MTAvZW51bXMvdmFsdWVfcnVsZV9z",
            "ZXRfZGltZW5zaW9uLnByb3RvEh5nb29nbGUuYWRzLmdvb2dsZWFkcy52MTAu",
            "ZW51bXMikAEKGVZhbHVlUnVsZVNldERpbWVuc2lvbkVudW0icwoVVmFsdWVS",
            "dWxlU2V0RGltZW5zaW9uEg8KC1VOU1BFQ0lGSUVEEAASCwoHVU5LTk9XThAB",
            "EhAKDEdFT19MT0NBVElPThACEgoKBkRFVklDRRADEgwKCEFVRElFTkNFEAQS",
            "EAoMTk9fQ09ORElUSU9OEAVC9AEKImNvbS5nb29nbGUuYWRzLmdvb2dsZWFk",
            "cy52MTAuZW51bXNCGlZhbHVlUnVsZVNldERpbWVuc2lvblByb3RvUAFaQ2dv",
            "b2dsZS5nb2xhbmcub3JnL2dlbnByb3RvL2dvb2dsZWFwaXMvYWRzL2dvb2ds",
            "ZWFkcy92MTAvZW51bXM7ZW51bXOiAgNHQUGqAh5Hb29nbGUuQWRzLkdvb2ds",
            "ZUFkcy5WMTAuRW51bXPKAh5Hb29nbGVcQWRzXEdvb2dsZUFkc1xWMTBcRW51",
            "bXPqAiJHb29nbGU6OkFkczo6R29vZ2xlQWRzOjpWMTA6OkVudW1zYgZwcm90",
            "bzM="));
      descriptor = pbr::FileDescriptor.FromGeneratedCode(descriptorData,
          new pbr::FileDescriptor[] { },
          new pbr::GeneratedClrTypeInfo(null, null, new pbr::GeneratedClrTypeInfo[] {
            new pbr::GeneratedClrTypeInfo(typeof(global::Google.Ads.GoogleAds.V10.Enums.ValueRuleSetDimensionEnum), global::Google.Ads.GoogleAds.V10.Enums.ValueRuleSetDimensionEnum.Parser, null, null, new[]{ typeof(global::Google.Ads.GoogleAds.V10.Enums.ValueRuleSetDimensionEnum.Types.ValueRuleSetDimension) }, null, null)
          }));
    }
    #endregion

  }
  #region Messages
  ///
  /// Container for enum describing possible dimensions of a conversion value rule
  /// set.
  ///
  public sealed partial class ValueRuleSetDimensionEnum : pb::IMessage<ValueRuleSetDimensionEnum>
  #if !GOOGLE_PROTOBUF_REFSTRUCT_COMPATIBILITY_MODE
      , pb::IBufferMessage
  #endif
  {
    private static readonly pb::MessageParser<ValueRuleSetDimensionEnum> _parser = new pb::MessageParser<ValueRuleSetDimensionEnum>(() => new ValueRuleSetDimensionEnum());
    private pb::UnknownFieldSet _unknownFields;
    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public static pb::MessageParser<ValueRuleSetDimensionEnum> Parser { get { return _parser; } }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public static pbr::MessageDescriptor Descriptor {
      get { return global::Google.Ads.GoogleAds.V10.Enums.ValueRuleSetDimensionReflection.Descriptor.MessageTypes[0]; }
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    pbr::MessageDescriptor pb::IMessage.Descriptor {
      get { return Descriptor; }
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public ValueRuleSetDimensionEnum() {
      OnConstruction();
    }

    partial void OnConstruction();

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public ValueRuleSetDimensionEnum(ValueRuleSetDimensionEnum other) : this() {
      _unknownFields = pb::UnknownFieldSet.Clone(other._unknownFields);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public ValueRuleSetDimensionEnum Clone() {
      return new ValueRuleSetDimensionEnum(this);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public override bool Equals(object other) {
      return Equals(other as ValueRuleSetDimensionEnum);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public bool Equals(ValueRuleSetDimensionEnum other) {
      if (ReferenceEquals(other, null)) {
        return false;
      }
      if (ReferenceEquals(other, this)) {
        return true;
      }
      return Equals(_unknownFields, other._unknownFields);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public override int GetHashCode() {
      int hash = 1;
      if (_unknownFields != null) {
        hash ^= _unknownFields.GetHashCode();
      }
      return hash;
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public override string ToString() {
      return pb::JsonFormatter.ToDiagnosticString(this);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public void WriteTo(pb::CodedOutputStream output) {
    #if !GOOGLE_PROTOBUF_REFSTRUCT_COMPATIBILITY_MODE
      output.WriteRawMessage(this);
    #else
      if (_unknownFields != null) {
        _unknownFields.WriteTo(output);
      }
    #endif
    }

    #if !GOOGLE_PROTOBUF_REFSTRUCT_COMPATIBILITY_MODE
    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    void pb::IBufferMessage.InternalWriteTo(ref pb::WriteContext output) {
      if (_unknownFields != null) {
        _unknownFields.WriteTo(ref output);
      }
    }
    #endif

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public int CalculateSize() {
      int size = 0;
      if (_unknownFields != null) {
        size += _unknownFields.CalculateSize();
      }
      return size;
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public void MergeFrom(ValueRuleSetDimensionEnum other) {
      if (other == null) {
        return;
      }
      _unknownFields = pb::UnknownFieldSet.MergeFrom(_unknownFields, other._unknownFields);
    }

    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public void MergeFrom(pb::CodedInputStream input) {
    #if !GOOGLE_PROTOBUF_REFSTRUCT_COMPATIBILITY_MODE
      input.ReadRawMessage(this);
    #else
      uint tag;
      while ((tag = input.ReadTag()) != 0) {
        switch(tag) {
          default:
            _unknownFields = pb::UnknownFieldSet.MergeFieldFrom(_unknownFields, input);
            break;
        }
      }
    #endif
    }

    #if !GOOGLE_PROTOBUF_REFSTRUCT_COMPATIBILITY_MODE
    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    void pb::IBufferMessage.InternalMergeFrom(ref pb::ParseContext input) {
      uint tag;
      while ((tag = input.ReadTag()) != 0) {
        switch(tag) {
          default:
            _unknownFields = pb::UnknownFieldSet.MergeFieldFrom(_unknownFields, ref input);
            break;
        }
      }
    }
    #endif

    #region Nested types
    /// Container for nested types declared in the ValueRuleSetDimensionEnum message type.
    [global::System.Diagnostics.DebuggerNonUserCodeAttribute]
    [global::System.CodeDom.Compiler.GeneratedCode("protoc", null)]
    public static partial class Types {
      ///
      /// Possible dimensions of a conversion value rule set.
      ///
      public enum ValueRuleSetDimension {
        ///
        /// Not specified.
        ///
        [pbr::OriginalName("UNSPECIFIED")] Unspecified = 0,
        ///
        /// Used for return value only. Represents value unknown in this version.
        ///
        [pbr::OriginalName("UNKNOWN")] Unknown = 1,
        ///
        /// Dimension for geo location.
        ///
        [pbr::OriginalName("GEO_LOCATION")] GeoLocation = 2,
        ///
        /// Dimension for device type.
        ///
        [pbr::OriginalName("DEVICE")] Device = 3,
        ///
        /// Dimension for audience.
        ///
        [pbr::OriginalName("AUDIENCE")] Audience = 4,
        ///
        /// This dimension implies the rule will always apply.
        ///
        [pbr::OriginalName("NO_CONDITION")] NoCondition = 5,
      }

    }
    #endregion

  }

  #endregion

}

#endregion Designer generated code


Click here to download the full report. Overall, the Fort Lauderdale area market is showing continual improvement.
For more information on any Fort Lauderdale home or condo, contact me directly at 954-632-5100.


I have no real content yet, so for the moment you'll just have to goto my old Geocities page. You can also view it without frames.
I'm also putting up a Xoom page.
Silly CDs you can buy.
[Main][More][More to come] Finally if you want free email but aren't sure where to go, check out my Reviews.


The GGP/CAD converter is provided without any warranty. Prices might differ from those given by financial institutions as banks, brokers or money transfer companies.
The best day to change Guernsey Pound in Canadian dollars was the Friday, 15 March 2019 . At that time the currency had reached its highest value.
The worst day to change Guernsey Pound in Canadian dollars was the Tuesday, 14 August 2018 . The exchange rate had fallen to its lowest value.
History of daily rates GGP /CAD since Monday, 23 June 2014.


text:
Tsitsi Dangarembga and Maaza Mengiste Make 2020 Booker Prize Longlist
NOBANTU SHABANGU
Zimbabwean literary giant Tsitsi Dangarembga and Ethiopian-born Maaza Mengiste appear in the latest Booker Prize longlist.
Zimbabe's Tsitsi Dangarembga has once again gained international prominence through her third novel This Mournable Body. The announcement of the annually anticipated Booker Prize Longlist recognises the third book from Dangarembga as an auspicious body of work worthy to be on the list. Ethopian-American Maaza Mengiste's The Shadow King is another historic African selection for the reputable prize.
Originally only for commonwealth countries, The Booker Prize is an international literary award institution that financially rewards orignal English literary submissions from around the world.
In 51 years of The Booker Prize, African authors have been nominated over 40 times but these nominations have been controversially dominated by European African authors. Nigerian authors, Ben Okri and Bernardine Evaristo are recorded recipients of the prize.
This Mournable Body picks up from Dangarembga's seminal work, Nervous Conditions. This Mournable Body is a somber outlook on life in Harare for Black female protagonist Tambu (the protagonist of Nervous Conditions) 30 years later. The latest narrative was well-received as it reflected the volatile political landscape in Zimbabwe and maintained Tambu's aspirational resilience.
Maaza Mengiste joins the literary trail of Black African women with The Shadow King, a novel that revisits the Italian-Ethopian war under Mussolini's reign. Much like Dangarembga, Mengiste highlights the wars unseen, the wars that women have to face in political upheaval and the unfair resilience required of them.
Tsitsi Dangarembga and Maaza Mengiste are the only two writers from Africa in the list of 13 authors. Winning authors from this year will receive 50,000 pounds.
The Booker Prize longlist 2020:
Diane Cook (US) for The New Wilderness (Oneworld)
Tsitsi Dangarembga (Zimbabwe) for This Mournable Body (Faber & Faber)
Avni Doshi (US) for Burnt Sugar (Hamish Hamilton)
Gabriel Krauze (UK) for Who They Was (4th Estate)
Hilary Mantel (UK) for The Mirror & the Light (4th Estate)
Colum McCann (Ireland/US) for Apeirogon (Bloomsbury)
Maaza Mengiste (Ethiopia/US) for The Shadow King (Canongate)
Kiley Reid (US) for Such a Fun Age (Bloomsbury Circus)
Brandon Taylor (US) for Real Life (Originals, Daunt Books)
Anne Tyler (US) for Redhead by the Side of the Road (Chatto & Windus)
Douglas Stuart (Scotland/US) for Shuggie Bain (Picador)
Sophie Ward (UK) for Love and Other Thought Experiments (Corsair)
C Pam Zhang (US) for How Much of These Hills Is Gold (Virago)
Tags Tsitsi Dangarembga
Retired Springbok Legend Tendai Mtawarira Speaks on Black Lives Matter
Hyde Park couple start AYO, a West African frozen food line
Tsitsi Dangarembga Receives 2021 PEN Freedom of Expression Award


Home» Account Dev» Community» VideoScribe ideas and feature requests» Keyboard shortcuts PLEASE!
I have just started using VideoScribe. I really, badly, miss having proper keyboard shortcuts. I agree that you want make VideoScribe usable to everyone and I don't see keyboard shortcuts being a hindrance to that.
Shortcuts will cut down our development time significantly.
(5) Delete a component directly by pressing DELETE.
(6) Pressing SHIFT to move object faster.