text:
\section{Introduction}\label{int}
\setcounter{equation}{0}
This work is a direct continuation of \cite{Sj1}, devoted to
semi-classical pseudo\-differential operators on ${\bf R}^n$ with small multiplicative
random perturbations, which was partly based on the work by M.~Hager
and the author \cite{HaSj}. The main goal in the present work is to
obtain the same results as in \cite{Sj1} but with ${\bf R}^n$ replaced
by a compact smooth $n$-dimensional manifold $X$. Hopefully this
extension will make it possible to obtain almost sure Weyl asymptotics
for the large eigenvalues of elliptic operators on compact
manifolds. Such results in the case of $X=S^1$ have recently been
obtained by W. Bordeaux-Montrieux \cite{Bo}.

On $X$ we consider an $h$-differential operator $P$ which in local
coordinates takes the form,
\ekv{int.1}
{
P=\sum_{|\alpha |\le m}a_\alpha (x;h)(hD)^\alpha ,
}
where we use standard multiindex notation and let
$D=D_x=\frac{1}{i}\frac{\partial }{\partial x}$. We assume that the
coefficients $a_\alpha  $ are uniformly bounded in $C^\infty $ for
$h\in ]0,h_0]$, $0<h_0\ll 1$. (We will also discuss the case when we
only have some Sobolev space control of $a_0(x)$.) Assume
\eekv{int.2}
{
&&a_\alpha (x;h)=a_\alpha ^0(x)+{\cal O}(h) \mbox{ in }C^\infty ,}
{&&a_\alpha (x;h)=a_\alpha (x)\hbox{ is independent of }h\hbox{ for
  }|\alpha |=m.
}
Notice that this assumption is invariant under changes of local
coordinates.

Also assume that $P$ is elliptic in the classical sense, uniformly
with respect to $h$:
\ekv{int.3}
{
|p_m(x,\xi )|\ge \frac{1}{C}|\xi |^m,
}
for some positive constant $C$, where
\ekv{int.4}
{
p_m(x,\xi )=\sum_{|\alpha |=m}a_\alpha (x)\xi ^\alpha
}
is invariantly defined as a function on $T^*X$.
It follows that $p_m(T^*X)$ is a closed cone in ${\bf C}$ and we
assume that
\ekv{int.5}
{
p_m(T^*X)\ne {\bf C}.
}
If $z_0\in {\bf C}\setminus p_m(T^*X)$, we see that $\lambda
z_0\not\in \Sigma (p)$ if $\lambda \ge 1$ is sufficiently large and
fixed, where $\Sigma (p):=p(T^*X)$ and $p$ is the semiclassical
principal symbol
\ekv{int.6}
{
p(x,\xi )=\sum_{|\alpha |\le m}a_\alpha ^0(x)\xi ^\alpha .
}
Actually, (\ref{int.5}) can be replaced by the weaker condition that
$\Sigma (p)\ne {\bf C}$.

\par Standard elliptic theory and analytic Fredholm theory now show
that if we consider $P$ as an unbounded operator: $L^2(X)\to L^2(X)$
with domain ${\cal D}(P)=H^m(X)$ (the Sobolev space of order $m$),
then $P$ has purely discrete spectrum.

\par We will need the symmetry assumption
\ekv{int'.7}{P^*=\Gamma P\Gamma ,}
where $P^*$ denotes the formal complex adjoint of $P$ in $L^2(X,dx)$,
with $dx$ denoting some fixed smooth postive density of integration
and $\Gamma $ is the antilinear operator of complex conjugation;
$\Gamma u=\overline{u}$. Notice that this assumption implies that
\ekv{int'.8}
{
p(x,-\xi )=p(x,\xi ),
}
and conversely, if $p$ fulfills (\ref{int'.8}), then we get
(\ref{int'.7}) if we replace $P$ by $\frac{1}{2}(P+\Gamma P^*\Gamma $),
which has the same semi-classical principal symbol $p$.

\par Let $V_z(t):=\mathrm{vol\,}(\{ \rho \in {\bf R}^{2n};\,|p(\rho
)-z|^2 \le t\} )$. For $\kappa \in ]0,1]$, $z\in \Omega $, we consider
the property that
\ekv{int.6.2}{V_z(t)={\cal O}(t^ \kappa ),\ 0\le t \ll 1.}
Since $r\mapsto p(x,r\xi )$ is a polynomial of degree $m$ in $r$ with
non-vanishing leading coefficient, we see that (\ref{int.6.2}) holds
with $\kappa =1/(2m)$.

The random potential will be of the form
\ekv{int.6.3}
{q_\omega (x)=\sum_{0<\mu _k\le L}\alpha _k(\omega )\epsilon
_k(x),\ |\alpha |_{{\bf C}^D}\le R,}
where $\epsilon _k$ is the orthonormal basis of eigenfunctions of
$h^2\widetilde{R}$, where $\widetilde{R}$ is an $h$-independent
positive elliptic 2nd order operator on $X$ with smooth
coefficients. Moreover, $h^2\widetilde{R}\epsilon _k=\mu
_k^2 \epsilon _k$, $\mu _k>0$.
We choose $L=L(h)$, $R=R(h)$ in the interval
\eekv{int.6.4}
{h^{\frac{\kappa -3n}{s-\frac{n}{2}-\epsilon }}\ll L\le Ch^{-M},&&
M\ge \frac{3n-\kappa }{s-\frac{n}{2}-\epsilon },}
{\frac{1}{C}h^{-(\frac{n}{2}+\epsilon )M+\kappa -\frac{3n}{2}}\le R\le
 C h^{-\widetilde{M}},&& \widetilde{M}\ge \frac{3n}{2}-\kappa
  +(\frac{n}{2}+\epsilon )M,}
for some $\epsilon \in ]0,s-\frac{n}{2}[$, $s>\frac{n}{2}$,
so by Weyl's law for the large eigenvalues of elliptic
self-adjoint operators, the dimension $D$ is of the order of magnitude
$(L/h)^n$. We introduce the  small parameter
$\delta =\tau _0 h^{N_1+n}$, $0<\tau _0\le \sqrt{h}$, where
\ekv{int.6.4.3}
{
N_1:=\widetilde{M}+sM+\frac{n}{2}.
}
The randomly perturbed operator is
\ekv{int.6.4.5}
{
P_\delta =P+\delta h^{N_1}q_\omega =:P+\delta Q_\omega .
}

\par The random variables $\alpha _j(\omega )$ will have a
joint probability distribution \ekv{int.6.5}{P(d\alpha )=C(h)e^{\Phi
(\alpha ;h)}L(d\alpha ),} where for some $N_4>0$,
\ekv{int.6.6}{ |\nabla _\alpha \Phi |={\cal
O}(h^{-N_4}),} and $L(d\alpha )$ is the
Lebesgue measure. ($C(h)$ is the normalizing constant,
assuring that the probability of
$B_{{\bf C}^D}(0,R)$ is equal to 1.)

\par We also need the parameter
\ekv{int.6.7.5}{\epsilon _0(h)=(h^{\kappa }+h^n\ln
\frac{1}{h})(\ln \frac{1}{\tau _0}+(\ln \frac{1}{h})^2)} and assume
that $\tau _0=\tau _0(h)$ is not too small, so that $\epsilon _0(h)$ is
small. Let $\Omega \Subset {\bf C}$ be open, simply connected not
entirely contained in $\Sigma (p)$. The main result of this work is:
\begin{theo}\label{int1} Under the assumptions above, let
$\Gamma \Subset \Omega $ have smooth boundary, let $\kappa \in
]0,1]$ be the parameter in \no{int.6.3}, \no{int.6.4}, \no{int.6.7.5} and assume that
\no{int.6.2} holds uniformly for $z$ in a
neighborhood of $\partial \Gamma $.  Then there
exists a constant $C>0$ such that for
$C^{-1}\ge r>0$,
$\widetilde{\epsilon }\ge C \epsilon _0(h)$
we have with probability
\ekv{int.6.8}{
\ge 1-\frac{C\epsilon _0(h)}
{rh^{n+\max (n(M+1), N_4+\widetilde{M})}}
e^{-\frac{\widetilde{\epsilon }}{C\epsilon _0(h)}} }
that:
\eekv{int.7}
{
&&|
\#(\sigma (P_\delta )\cap \Gamma )-\frac{1}{(2\pi h)^n
}\mathrm{vol\,}(p^{-1}(\Gamma ))
|\le
}
{&&
\frac{C}{h^n}\left( \frac{\widetilde{\epsilon }}{r}
+C(r+\ln (\frac{1}{r})\mathrm{vol\,}(p^{-1}(\partial
\Gamma +D(0,r))))
 \right).}
Here $\#(\sigma (P_\delta )\cap \Gamma )$ denotes the number of
eigenvalues of $P_\delta $ in $\Gamma $, counted with their algebraic multiplicity.
\end{theo}

Actually, we shall prove the theorem for the slightly more general
operators, obtained by replacing $P$ by $P_0=P+\delta
_0(h^{\frac{n}{2}}q_1^0+q_2^0)$, where $\Vert q_1^0\Vert_{H^s_h}\le
1$, $\Vert q_2\Vert_{H^s}\le 1$, $0\le \delta _0\le h$. Here, $H^s$ is
the standard Sobolev space and $H_h^s$ is the same space with the
natural semiclassical $h$-dependent norm. See Section \ref{hs}.

As in \cite{HaSj} we also have a result valid simultaneously for a
family ${\cal C}$ of domains $\Gamma \subset \Omega $ satisfying the
assumptions of Theorem \ref{int1} uniformly in the natural sense:
With a probability
\ekv{int.8}{
\ge 1-\frac{{\cal O}(1)\epsilon _0(h)}{r^2h^{n+\max (n(M+1), N_4+\widetilde{M})}}e^{-\frac{\widetilde{\epsilon }}{C\epsilon _0(h)}}, } the
estimate \no{int.7} holds simultaneously for all $\Gamma \in {\cal C}$.

\par In the introduction of \cite{Sj1} there is a discussion about the
choice of parameters and a corollary which carry over to the present
situation without any changes

\begin{remark}\label{int2}
{\rm When $\widetilde{R}$ has real coefficients, we may assume that the
eigenfunctions $\epsilon _j$ are real. Then (cf Remark 8.3 in \cite{Sj1}) we may
restrict $\alpha $ in (\ref{int.6.3}) to be in ${\bf R}^D$ so that
$q_\omega $ is real, still with $|\alpha |\le R$, and change
$C(h)$ in (\ref{int.6.5}) so that $P$ becomes a probability measure on
$B_{{\bf R}^D}(0,R)$. Then Theorem \ref{int1} remains valid.}
\end{remark}
\begin{remark}\label{int3}
{\rm The assumption (\ref{int.6}) cannot be
completely eliminated. Indeed, let $P=hD_x+g(x)$ on ${\bf T}={\bf
  R}/(2\pi {\bf Z})$ where $g$ is smooth and complex valued. Then (cf
Hager \cite{Ha2}) the spectrum of $P$ is contained in the line
$\Im z = \int_0^{2\pi }\Im g(x)dx/(2\pi )$. This line will vary only very
little under small multiplicative perturbations of $P$ so
Theorem \ref{int1} cannot hold in this case.}
\end{remark}

The proof follows the general scheme of \cite{Sj1}, we will recall the
intermediate steps but give proofs only when there is a difference
between the case of ${\bf R}^n$ and that of compact
manifolds. Actually, there will also be some simplifications since we
have no support condition on the random potential.

\medskip
\par\noindent {\bf Acknowledgement.}
A large part of this work was completed while attending the
special program ``Complex Analysis of Several Variables'' at the
Mittag-Leffler Institute in May--June 2008. We are grateful to the
organizers and the staff for very stimulating and pleasant working conditions.
\section{Semiclassical Sobolev spaces and multiplication}
\label{al}
\setcounter{equation}{0}
We let $H_h^s({\bf R}^n)\subset {\cal S}'({\bf R}^n)$, $s\in {\bf R}$,
denote the semiclassical Sobolev space of order
$s$ equipped with the norm $\Vert \langle hD\rangle^s u\Vert$ where
the norms are the ones in $L^2$, $\ell^2$ or the corresponding
operator norms if nothing else
is indicated. Here $\langle hD\rangle= (1+(hD)^2)^{1/2}$. Let
$\widehat{u}(\xi )=\int e^{-ix\cdot \xi }u(x)dx$ denote the Fourier
transform of the tempered distribution $u$ on ${\bf R}^n$. In
\cite{Sj1} we recalled the following result:
\begin{prop}\label{al1}
Let $s>n/2$. Then there exists a constant $C=C(s)$ such that for all
$u,v\in H_h^s({\bf R}^n)$, we have $u\in L^\infty ({\bf R}^n) $,
$uv\in H_h^s({\bf R}^n)$ and
\ekv{al.1}
{
\Vert u\Vert_{L^\infty }\le Ch^{-n/2}\Vert u\Vert_{H_h^s},
}
\ekv{al.2}
{
\Vert uv\Vert_{H_h^s} \le Ch^{-n/2} \Vert u\Vert_{H_h^s} \Vert v\Vert_{H_h^s}.
}
\end{prop}

We cover $X$ by
finitely many coordinate neighborhoods $X_1,...,X_p$ and for
each $X_j$, we let $x_1,...,x_n$ denote the corresponding local
coordinates on $X_j$. Let $0\le \chi _j\in C_0^\infty (X_j)$ have the
property that $\sum_1^p\chi _j >0$ on $X$. Define $H_h^s(X)$ to be the
space of all $u\in {\cal D}'(X)$ such that
\ekv{al.4}
{
\Vert u\Vert_{H_h^s}^2:=\sum_1^p \Vert \chi _j\langle hD\rangle^s \chi
_j u\Vert ^2 <\infty .
}
It is standard to show that this definition does not depend on the
choice of the coordinate neighborhoods or on $\chi _j$. With different
choices of these quantities we get norms in \no{al.4} which are
uniformly equivalent when $h\to 0$. In fact, this follows from the
$h$-pseudodifferential calculus on manifolds with symbols in the
H\"ormander space $S^m_{1,0}$, that we quickly reviewed in the
appendix in \cite{Sj1}.
An equivalent definition of $H_h^s(X)$ is the following: Let
\ekv{al.5}
{
h^2\widetilde{R}=\sum (hD_{x_j})^*r_{j,k}(x)hD_{x_k}
}
be a non-negative elliptic operator with smooth coefficients on $X$,
where the star indicates that we take the adjoint with respect to some
fixed positive smooth density on $X$. Then $h^2\widetilde{R}$ is
essentially self-adjoint with domain $H^2(X)$, so
$(1+h^2\widetilde{R})^{s/2}:L^2\to L^2$ is a closed densely defined
operator for $s\in {\bf R}$, which is bounded precisely when $s\le
0$. Standard methods allow to show that $(1+h^2\widetilde{R})^{s/2}$
is an $h$-pseudodifferential operator with symbol in $S^s_{1,0}$ and
semiclassical principal symbol given by $(1+r(x,\xi ))^{s/2}$, where
$r(x,\xi )=\sum_{j,k}r_{j,k}(x)\xi _j\xi _k$ is the semiclassical
principal symbol of $h^2\widetilde{R}$.  See the appendix in
\cite{Sj1}.
The
$h$-pseudodifferential calculus gives for every $s\in {\bf R}$:
\begin{prop}\label{al2}
  $H_h^s(X)$ is the space of all $u\in {\cal D}'(X)$ such that
$(1+h^2\widetilde{R})^{s/2}u\in L^2$ and the norm $\Vert u\Vert_{H_h^s}$ is
equivalent to $\Vert (1+h^2\widetilde{R})^{s/2}u\Vert$, uniformly when $h\to 0$.
\end{prop}
\begin{remark}\label{al3}
\rm From the first definition we see that Proposition \ref{al1} remains
valid if we replace ${\bf R}^n$ by a compact $n$-dimensional
manifold $X$.
\end{remark}

\par Of course, $H_h^s(X)$ coincides with the standard Sobolev space
$H^s(X)$ and the norms are equivalent for each fixed value of $h$, but
not uniformly with respect to $h$. The following variant of
Proposition \ref{al1} will probably be useful when studying the high
energy limit (that we hope to treat in a future paper).

\begin{prop}\label{al4}
Let $s>n/2$. Then there exists a constant $C=C_s>0$ such that
\ekv{al.6}
{
\Vert uv\Vert_{H_h^s}\le C\Vert u\Vert_{H^s}\Vert v\Vert_{H_h^s},\
\forall u\in H^s({\bf R}^n),\, v\in H_h^s({\bf R}^n).
}
The result remains valid if we replace ${\bf R}^n$ by $X$.
\end{prop}
\begin{proof}
The adaptation to the case of a compact manifold is immediate by
working in local coordinates, so it is enough to prove (\ref{al.6}) in
the ${\bf R}^n$-case.

\par Let $\chi \in C_0^\infty ({\bf R}^n)$ be equal to one in a
neighborhood of $0$. Write $u=u_1+u_2$ with $u_1=\chi (hD)u$,
$u_2=(1-\chi (hD))u$. Then, with hats indicating Fourier transforms, we
have
$$
\langle h\xi \rangle^s\widehat{u_1v}(\xi )=\frac{1}{(2\pi )^n}
\int \frac{\langle h\xi \rangle^s}{\langle h\eta \rangle^s}
(\chi (h(\xi -\eta ))\widehat{u}(\xi -\eta ))\langle h\eta
\rangle^s\widehat{v}(\eta )d\eta .
$$
Here $\langle h\xi \rangle/\langle h\eta  \rangle={\cal O}(1)$ on the
support of $(\xi ,\eta )\mapsto \chi (h(\xi -\eta ))$, so
$$
\Vert u_1v\Vert_{H_h^s}\le {\cal O}(1)\Vert
\widehat{u}\Vert_{L^1}\Vert v\Vert_{H_h^s}\le {\cal O}(1)\Vert
u\Vert_{H^s}\Vert v\Vert_{H_h^s},
$$
where we also used that $s>n/2$ in the last estimate.

\par On the other hand, $\langle h\xi \rangle^s\le C h^s\langle
\xi \rangle ^s$ when $1-\chi (h\xi )\ne 0$, so $\Vert
u_2\Vert_{H_h^s}\le Ch^s\Vert u\Vert_{H^s}$. By Proposition \ref{al1},
we get
$$
\Vert u_2v\Vert_{H_h^s}\le Ch^{-\frac{n}{2}}\Vert
u_2\Vert_{H_h^s}\Vert v\Vert_{H_h^s}\le
\widetilde{C}h^{s-\frac{n}{2}}\Vert u\Vert_{H^s}\Vert
v\Vert_{H_h^s}\le \widetilde{C}\Vert u\Vert_{H^s}\Vert
v\Vert_{H_h^s},
$$
when $h\le 1$.
\end{proof}

\section{$H^s$-perturbations and eigenfunctions}\label{hs}
\setcounter{equation}{0}

\par This section gives a very straight forward adaptation of the
corresponding section in \cite{Sj1}. Let $S^m(T^*X)=S^m_{1,0}(T^*X)$,
$S^m(U\times {\bf R}^n)=S^m_{1,0}(U\times {\bf R}^n)$ denote the
classical H\"ormander symbol spaces, where $U\subset {\bf R}^n$ is
open. See for instance \cite{GrSj} and further references given
there. As in \cite{Ha, HaSj}, we can find $\widetilde{p}\in S^m(T^*X)$
which is equal to $p$ outside any given fixed neighborhood of
$p^{-1}(\overline{\Omega })$ such that $\widetilde{p}-z$ is
non-vanishing, for any $z\in \overline{\Omega }$. Let
$\widetilde{P}=P+\mathrm{Op}_h(\widetilde{p}-p)$, where
$\mathrm{Op}_h(\widetilde{p}-p)$ denotes any reasonable quantization
of $(\widetilde{p}-p)(x,h\xi )$. (See for instance the appendix in
\cite{Sj1}.) Then $\widetilde{P}-z:H^m_h(X)\to H^0_h(X)$ has a
uniformly bounded inverse for $z\in \overline{\Omega }$ and $h>0$
small enough. As in \cite{HaSj, Sj1}, we see that the eigenvalues of
$P$ in $\Omega $, counted with their algebraic multiplicity, coincide
with the zeros of the function $z\mapsto \det
((\widetilde{P}-z)^{-1}(P-z))=\det
(1-(\widetilde{P}-z)^{-1}(\widetilde{P}-P))$.

\par Fix $s>n/2$ and consider the perturbed operator
\ekv{hs.1}{
P_\delta =P+\delta (h^{\frac{n}{2}}q_1+q_2)=P+\delta
(Q_1+Q_2)=P+\delta Q,
}
where $q_j\in H^s(X)$,
\ekv{hs.2}{
\Vert q_1\Vert_{H^s_h}\le 1,\ \Vert q_2\Vert_{H^s}\le 1,\ 0\le \delta
\ll 1.
}
According to Propositions \ref{al1}, \ref{al4}, $Q={\cal
  O}(1):H_h^s \to H_h^s$ and hence by duality and interpolation,
\ekv{hs.3}
{
Q={\cal O}(1):H_h^\sigma \to H_h^\sigma ,\ -s\le \sigma \le s.
}

\par As in \cite{Sj1}, the spectrum of $P_\delta $ in $\Omega $
 is discrete and coincides with the set of zeros of
\ekv{hs.4}
{
\det ((\widetilde{P}_\delta -z)^{-1}(P_\delta -z))=\det (1-
(\widetilde{P}_\delta -z)^{-1}(\widetilde{P}-P)),}
where $\widetilde{P}_\delta :=P_\delta +\widetilde{P}-P$.
Here $(\widetilde{P}-z)^{-1}={\cal O}(1):H_h^\sigma \to H_h^\sigma $
for $\sigma $ in the same range and as in \cite{Sj1} we get the same
conclusion for $(\widetilde{P}_\delta -z)^{-1}$.

\par
Put
\ekv{hs.5}
{
P_{\delta ,z}:=(\widetilde{P}_\delta -z)^{-1}(P_\delta -z)=
1-(\widetilde{P}_\delta -z)^{-1}(\widetilde{P}-P)=:1-K_{\delta ,z},
}
\ekv{hs.6}
{
S_{\delta ,z}:=P_{\delta ,z}^*P_{\delta ,z}=1-(K_{\delta ,z}+K_{\delta ,z}^*-K_{\delta ,z}^*K_{\delta ,z})=:1-L_{\delta ,z}.
}
As in \cite{Sj1} we get
\ekv{hs.7}
{K_{\delta ,z},
L_{\delta ,z}={\cal O}(1):H_h^{-s}\to H_h^s.
}
For $0\le \alpha \le 1/2$, let $\pi _\alpha =1_{[0,\alpha ]}(S_{\delta
,z})$. Then as in \cite{Sj1}, we get
\ekv{hs.8}
{
\pi _\alpha ={\cal O}(1): H_h^{-s}\to H_h^s.
}

\par We also have the corresponding result for $P_\delta -z$. Let
\ekv{hs.9}
{
S_\delta =(P_\delta -z)^*(P_\delta -z)
}
be defined as the Friedrichs extension from $C^\infty (X)$ with
quadratic form domain $H_h^m(X)$. For $0\le \alpha \le {\cal O}(1)$,
we now put $\pi _\alpha =1_{[0,\alpha ](S_\delta )}$. Then as in
\cite{Sj1}, we see that this new spectral projection also fulfils
(\ref{hs.9}), for $0\le \alpha \ll 1$.

\section{Some functional and pseudodifferential cal\-culus}\label{fu}
\setcounter{equation}{0}

In this section we derive some results analogous to those of Section 4
in \cite{HaSj}. There we worked on ${\bf R}^n$ and by a simple
dilation and change of the semi-classical parameter from $h$ to
$h/\alpha $ we could reduce ourselves to a situation of more standard
$h/\alpha $-pseudodifferential calculus. On a manifold, this can
probably be done also, but appeared to us as quite heavy, so here we
take another route and develop directly a slightly exotic
pseudodifferential calculus, then use it to study resolvents and
functions of certain self-adjoint pseudodifferential operators.

\par Let $P$ be of the form (\ref{int.1}) and let $p$ in (\ref{int.6})
be the corresponding semi-classical principal symbol. Assume classical
ellipticity as in (\ref{int.3}) and let $z\in {\bf C}$ be fixed
throughout this section.
Let
\ekv{fu.1}
{
S=(P-z)^*(P-z),
}
that we realize as a self-adjoint operator in the sense of Friedrichs
extensions. Later on we will also consider a different choice of $S$,
namely
\ekv{fu.1.5}{
S=P_z^*P_z,\hbox{ where }P_z=(\widetilde{P}-z)^{-1}(P-z)} and
$\widetilde{P}$ is defined prior to (\ref{hs.1}). The main goal is to
make a trace class
study of $\chi (\frac{1}{\alpha }S)$ when $0<h\le \alpha \ll 1$, $\chi
\in C_0^\infty ({\bf R})$. With the second choice of $S$, we shall
also study $\ln \det (S+\alpha \chi (\frac{1}{\alpha }S))$, when $\chi
\ge 0$, $\chi (0)>0$. The main step will be to get enough information
about the resolvent $(w-\frac{1}{\alpha }S)^{-1}$ for $w={\cal O}(1)$,
$\Im w\ne 0$ and then apply the Cauchy-Riemann-Green-Stokes formula
\ekv{fu.2}
{
\chi (\frac{1}{\alpha }S)=-\frac{1}{\pi }\int \frac{\partial
  \widetilde{\chi }(w)}{\partial \overline{w}}(w-\frac{1}{\alpha
}S)^{-1}L(dw ),
}
where $\widetilde{\chi }\in C_0^\infty ({\bf C})$ is an almost
holomorphic extension of $\chi $, so that
\ekv{fu.3}
{
\frac{\partial \widetilde{\chi }}{\partial \overline{w}}={\cal O}(
|\Im w|^\infty ).
}
Thanks to (\ref{fu.3}) we can work in symbol classes with some
temparate but otherwise unspecified growth in $1/|\Im w|$.

\par Let
\ekv{fu.4}
{
s=|p-z|^2
}
be the semiclassical principal symbol of $S$ in (\ref{fu.1}). A basic
weight function in our calculus will be
\ekv{fu.5}
{
\Lambda :=\left(\frac{\alpha +s}{1+s} \right)^{\frac{1}{2}},
}
satisfying $\sqrt{\alpha }\le \Lambda \le 1$.

\par As a preparation and motivation for the calculus, we first
consider symbol properties of $1+\frac{s}{\alpha }$ and its powers.
\begin{prop}\label{fu1}
For every choice of local coordinates $x$ on $X$, let $(x,\xi )$
denote the corresponding canonical coordinates on $T^*X$. Then for all
$\ell\in {\bf R}$, $\widetilde{\alpha },\beta \in {\bf N}^n$, we have
uniformly in $\xi $ and locally uniformly in $x$:
\ekv{fu.6}
{
\partial _x^{\widetilde{\alpha }}\partial _\xi^\beta
(1+\frac{s}{\alpha })^\ell ={\cal O}(1) (1+\frac{s}{\alpha })^\ell
\Lambda ^{-|\widetilde{\alpha }|-|\beta |}\langle \xi \rangle
^{-|\beta |}.
}
\end{prop}
\begin{proof}
In the region $|\xi |\gg 1$ we see that $(1+\frac{s}{\alpha })^\ell$
is an elliptic element of the H\"ormander symbol class
$$
\alpha ^{-\ell}S^{2\ell m}_{1,0}=:\alpha ^{-\ell}S(\langle \xi \rangle
^{2\ell m}),
$$
and $\Lambda \asymp 1$ there, so (\ref{fu.6}) holds. In the region
$|\xi |={\cal O}(1)$, we start with the case $\ell =1$. Since $s\ge
0$, we have $\nabla s={\cal O}(s^{\frac{1}{2}})$, so
$$
|\nabla (1+\frac{s}{\alpha })|={\cal O}(\frac{s^{\frac{1}{2}}}{\alpha
})\le
{\cal O}(1)(1+\frac{s}{\alpha })(\alpha +s)^{-\frac{1}{2}}={\cal
  O}(1)(1+\frac{s}{\alpha })\Lambda ^{-1}.
$$
For $k\ge 2$, we have
$$
|\nabla ^k(1+\frac{s}{\alpha })|={\cal O}(\frac{1}{\alpha })={\cal
  O}(1)(1+\frac{s}{\alpha })\Lambda ^{-2}\le {\cal
  O}(1)(1+\frac{s}{\alpha })\Lambda ^{-k},
$$
and we get (\ref{fu.6}) when $\ell =1$.

\par If $\ell\in {\bf R}$, then $\partial _x^{\widetilde{\alpha
  }}\partial _\xi ^\beta (1+\frac{s}{\alpha })^\ell$ is a finite
linear combination of terms
$$
(1+\frac{s}{\alpha })^{\ell -k}(\partial _x^{\widetilde{\alpha
  }_1}\partial _\xi^{\beta _1}(1+\frac{s}{\alpha }))\cdots
(\partial _x^{\widetilde{\alpha
  }_k}\partial _\xi^{\beta _k}(1+\frac{s}{\alpha })),
$$
with $\widetilde{\alpha }=\widetilde{\alpha }_1+...+\widetilde{\alpha
}_k$, $\beta =\beta _1+...+\beta _k$, and we get (\ref{fu.6}) in general.
\end{proof}

\par We next notice that when $w={\cal O}(1)$,
\ekv{fu.7}
{
\frac{|\Im w|}{C}(1+\frac{s}{\alpha })\le |w-\frac{s}{\alpha }|\le
C(1+\frac{s}{\alpha }).
}
In fact, the second inequality is obvious, and so is the first one,
when $\frac{s}{\alpha }\gg 1$. When $\frac{s}{\alpha }\le {\cal
  O}(1)$, it follows from the fact that
$$
1+\frac{s}{\alpha }={\cal O}(1),\quad |w-\frac{s}{\alpha }|\ge |\Im w|.
$$

\par From (\ref{fu.6}), (\ref{fu.7}), we get
\ekv{fu.8}
{
|\partial _x^{\widetilde{\alpha }}\partial _\xi ^\beta
(w-\frac{s}{\alpha })|\le {\cal O}(1)(w-\frac{s}{\alpha })\Lambda
^{-|\widetilde{\alpha }|-|\beta |}\langle \xi \rangle ^{-|\beta |}|\Im
w|^{-1}.
}
When passing to $(w-\frac{s}{\alpha })^\ell$ and applying the proof of
Proposition \ref{fu1}, we loose more powers of $|\Im w|$ that can
still be counted precisely, but we refrain from doing so and simply
state the following result:
\begin{prop}\label{fu2}
For all $\ell\in {\bf R}$, $\widetilde{\alpha },\beta \in {\bf N}^n$,
there exists $J\in {\bf N}$, such that
\ekv{fu.9}
{
\partial _x^{\widetilde{\alpha }}\partial _\xi ^\beta
(w-\frac{s}{\alpha })^\ell ={\cal O}(1)(1+\frac{s}{\alpha })^\ell
\Lambda ^{-|\widetilde{\alpha }|-|\beta |}\langle \xi \rangle^{-|\beta
  |}|\Im w|^{-J},
}
uniformly in $\xi $ and locally uniformly in $x$.
\end{prop}

\par We now define our new symbol spaces.
\begin{dref}\label{fu3}
Let $\widetilde{m}(x,\xi )$ be a weight function of the form
$\widetilde{m}(x,\xi )=\langle \xi \rangle ^k\Lambda^\ell $. We say
that the family $a=a_w\in C^\infty (T^*X)$, $w\in D(0,C)$, belongs to $S_\Lambda (\widetilde{m})$ if
for all $\widetilde{\alpha },\beta \in {\bf N}^n$ there exists $J\in
{\bf N}$ such that
\ekv{fu.10}
{
\partial _x^{\widetilde{\alpha }}\partial _\xi ^\beta a=
{\cal O}(1)\widetilde{m}(x,\xi )\Lambda ^{-|\widetilde{\alpha
  }|-|\beta |}\langle \xi \rangle^{-|\beta |}|\Im w|^{-J}.
}
\end{dref}

Here, as in Proposition \ref{fu2}, it is understood that that the
estimate is expressed in canonical coordinates and is locally uniform
in $x$ and uniform in $\xi $. Notice that the set of estimates
(\ref{fu.10}) is invariant under changes of local coordinates in $X$.

\par Let $U\subset X$ be a coordinate neighborhood that we shall view
as a subset of ${\bf R}^n$ in the natural way. Let $a\in S_\Lambda
(T^*U,\widetilde{m})$ be a symbol as in Definition \ref{fu3} so that
(\ref{fu.10}) holds uniformly in $\xi $ and locally uniformly in
$x$. For fixed values of $\alpha $, $w$ the symbol $a$ belongs
to $S^k_{1,0}(T^*U)$, so the classical $h$-quantization
\ekv{fu.11}
{
Au=\mathrm{Op}_h(a)u(x)=\frac{1}{(2\pi h)^n}\iint
e^{\frac{i}{h}(x-y)\cdot \eta }a(x,\eta ;h)u(y)dyd\eta
}
is a well-defined operator $C_0^\infty (U)\to C^\infty (U)$, ${\cal
  E}'(U)\to {\cal D}'(U)$. In order to develop our rudimentary
calculus on $X$ we first establish a pseudolocal property for the
distribution kernel $K_A(x,y)$:
\begin{prop}\label{fu4}
For all $\widetilde{\alpha },\beta \in {\bf N}^n$, $N\in {\bf N}$,
there exists $M\in {\bf N}$ such that
\ekv{fu.12}
{
\partial _x^{\widetilde{\alpha }}\partial _y^\beta K_A(x,y)={\cal
  O}(h^N|\Im w|^{-M}),
}
locally uniformly on $U\times U\setminus \mathrm{diag}(U\times U)$.
\end{prop}
\begin{proof}
If $\gamma \in {\bf N}^n$, then $(x-y)^\gamma K_A(x,y)$ is the
distribution kernel of $\mathrm{Op}_h((-hD_\xi )^\gamma a)$ and
$(-hD_\xi )^\gamma a\in S_\Lambda
\left(\widetilde{m}\left(\frac{h}{\Lambda \langle \xi \rangle}\right)^{|\gamma |}\right)
$ and we notice that $h/\Lambda \le h/\alpha ^{\frac{1}{2}}\le
h^{\frac{1}{2}}$. Thus for any $N\in {\bf N}$, we have
$$
(x-y)^\gamma K_A(x,y)={\cal O}(h^N|\Im w|^{-M})\hbox{ if }|\gamma |\ge
\gamma (N)
$$
is large enough. From this we get (\ref{fu.12}) when
$\widetilde{\alpha }=\beta =0$. Now, $\partial _x^{\widetilde{\alpha
  }}\partial _y^\beta K_A$ can be viewed as the distribution kernel of
a new pseudodifferential operator of the same kind, so we get
(\ref{fu.12}) for all $\widetilde{\alpha },\beta $.
\end{proof}

This means that if $\phi ,\psi \in C_0^\infty (U)$ have disjoint
supports, then for every $N\in {\bf N}$, there exists $M\in {\bf N}$
such that $\phi A\psi :H^{-N}({\bf R}^n)\to H^N({\bf R}^n)$ with norm
${\cal O}(h^N|\Im w|^{-M})$, and this leads to a simple way of
introducing pseudo\-differential operators on $X$: Let $U_1,...,U_s$ be
coordinate neighborhoods that cover $X$. Let $\chi _j\in C_0^\infty
(U_j)$ form a partition of unity and let $\widetilde{\chi }_j\in
C_0^\infty (U_j)$ satisfy $\chi _j\prec \widetilde{\chi }_j$ in the
sense that $\widetilde{\chi }_j$ is equal
to 1 near $\mathrm{supp\,}(\chi _j)$. Let $a=(a_1,...,a_s)$, where
$a_j\in S_\Lambda (\widetilde{m})$. Then we quantize $a$ by the
formula:
\ekv{fu.13}
{
A=\sum_1^s \widetilde{\chi }_j\circ \mathrm{Op}_h(a_j)\circ \chi _j.
}
This is not an invariant quantization procedure but it
will suffice for our purposes.

We next study the composition to the left with non-exotic
pseudodifferential operators. Let $U$ be a coordinate neighborhood,
viewed as an open set in ${\bf R}^n$, and take $A= \mathrm{Op}_h(a)$,
$a\in S_{1,0}(m_1)$, $m_1=\langle \xi \rangle^r$,
$B=\mathrm{Op}_h(b)$, $b\in S_\Lambda (m_2)$ with $m_2=\langle \xi
\rangle^k\Lambda ^\ell$ as in Definition \ref{fu3}. We will assume
that $\mathrm{supp\,}(b)\subset K\times {\bf R}^n$, where $K\subset U$
is compact. We are interested in $C=A\circ B$.

\par The symbol $c$ of this composition is given by
\eekv{fu.14}
{
&&c(x,\xi ;h)=e^{-\frac{i}{h}x\cdot \xi }A(b(\cdot ,\xi
)e^{\frac{i}{h}(\cdot )\cdot \xi })(x)}{&&=\frac{1}{(2\pi h)^n}\iint
a(x,\eta )b(y,\xi )e^{\frac{i}{h}(x-y)\cdot (\eta -\xi )}dyd\eta
}

\par In the region $|\eta -\xi |\ge \frac{1}{C}\langle \xi \rangle$ we can
make repeated integrations by parts in the $y$-variables and see that
the contribution from this region is a symbol $d(x,\xi ;h)$ satisfying
\eekv{fu.15}
{
\forall N\in {\bf N}, \widetilde{\alpha }, \beta \in {\bf N}^n,
\exists  M\in {\bf N}, \forall K\Subset U, \exists C>0;}
{|\partial
_x^{\widetilde{\alpha} }\partial _{\xi }^\beta d(x,\xi ;h)|\le
C\frac{h^N\langle \xi \rangle^{-N}}{|\Im w|^M},\ (x,\xi )\in K\times
{\bf R}^n.
}

\par Up to such a term $d$, we may assume that with $\chi \in
C_0^\infty (B(0,\frac{1}{2}))$ equal to 1 near 0,
\eekv{fu.16}
{c(x,\xi ;h)&\equiv&\frac{1}{(2\pi h)^n}\iint
a(x,\eta )b(y,\xi )\chi (\frac{\eta -\xi }{\langle \xi \rangle})e^{\frac{i}{h}(x-y)\cdot (\eta -\xi )}dyd\eta
}
{&=&\left(\frac{\langle \xi \rangle}{2\pi h}\right)^n
\iint
a(x,\langle \xi \rangle(\eta +\frac{\xi }{\langle \xi \rangle} ))
b(x+y,\xi )\chi (\eta )e^{\frac{-i\langle \xi
    \rangle}{h}y\cdot \eta }dyd\eta .
}
The method of stationary phase gives for every $N\in {\bf N}$:
\ekv{fu.17}
{
c(x,\xi ;h)=\sum_{|\beta |<N}\frac{h^{|\beta |}}{\beta !}\partial _\xi
^\beta a D_x^\beta b +R_N.
}
Here,
\eekv{fu.18}
{
R_N&=&\left(\frac{h}{\langle \xi \rangle}\right)^N\frac{1}{(N-1)!}\times }
{&&\int_0^1(1-t)^NJ\left( t\frac{h}{\langle \xi \rangle},
(\partial _\eta \cdot D_y)^N(a(x,\langle \xi
\rangle(\eta +\frac{\xi }{\langle \xi \rangle}))b(x+y,\xi )\chi (\eta
))
\right) dt,
}
where
$$
J(s,u)=\frac{1}{(2\pi s)^n}\iint u(y,\eta )e^{-\frac{i}{s}y\cdot \eta
}dyd\eta ,
$$
and we used the fact that $\partial _sJ(s,u)=J(s,\partial _\eta \cdot
D_y(u))$, $J(s,u)\to u(0)$ when $s\to 0$.
Noting that
$$
|J(s,u)|\le C\sum_{|\widetilde{\alpha }|+|\beta |\le
  2n+1}\Vert \partial _y^{\widetilde{\alpha }}\partial _\eta ^\beta u\Vert_{L^1},
$$
we see that there exist exponents $N_2,N_3$ independent of $N$, such
that
$$
|R_N|\le C\left(\frac{h}{\langle \xi \rangle}\right)^N m_1(\xi )
\langle \xi \rangle^{N_2}\alpha ^{N_3-\frac{N}{2}}|\Im w|^{-M(N)}.
$$
Similar estimates hold for the derivatives and we conclude:
\begin{prop}\label{fu5}
Let $A=\mathrm{Op}_h(a)$, $a\in S_{1,0}(m_1)$, $B=\mathrm{Op}_h(b)$,
$b\in S_\Lambda (m_2)$ and assume that $b$ has uniformly compact
support in $x$. Then $A\circ B=\mathrm{Op}_h(c)$, where $c$ belongs to
$S_\Lambda
(m_1m_2)$ and has the asymptotic expansion
$$
c\sim \sum \frac{h^{|\beta |}}{\beta !}\partial _\xi ^\beta a(x,\xi )D_x^\beta
b(x,\xi ),
$$
in the sense that for every $N\in {\bf N}$,
$$
c= \sum_{|\beta |<N} \frac{h^{|\beta |}}{\beta !}\partial _\xi ^\beta a(x,\xi )D_x^\beta
b(x,\xi ) + r_N(x,\xi ;h),
$$
where
$r_N\in S_\Lambda (\frac{m_1m_2}{(\Lambda \langle \xi \rangle)^N}h^N)$.
\end{prop}

\par We next make a parametrix construction for $w-\frac{1}{\alpha
}S$, still with $S$ as in (\ref{fu.1}), and most of the work will take
place in a coordinate neighborhood $U$, viewed as an open set in ${\bf
  R}^n$. The symbol of $w-\frac{1}{\alpha }S$ is of the form
\ekv{fu.19}
{
F=F_0+F_{-1},\quad F_0=w-\frac{1}{\alpha }s,\ F_{-1}=\frac{h}{\alpha }s_{-1}\in
S(\frac{h}{\alpha }\langle \xi \rangle^{2m-1}).
}
Put
\ekv{fu.20}
{
E_0=\frac{1}{w-\frac{1}{\alpha }s}\in S_\Lambda (\frac{\alpha
}{\Lambda ^2\langle \xi \rangle^{2m}}).
}
With Proposition \ref{fu5} in mind, we first consider the formal
composition
\eekv{fu.21}
{F\# E_0&\sim& \sum \frac{h^{|\beta |}}{\beta !}(\partial _\xi ^\beta F)
  (D_x^{\beta }E_0)}
{&\sim& 1+\sum_{|\beta |\ge 1}\frac{h^{|\beta |}}{\beta !}(\partial
  _\xi ^\beta F_0) (D_x^\beta E_0)+F_{-1}\# E_0.
}
Here,
$$
F_{-1}\# E_0\in S_\Lambda (\frac{h}{\alpha }\langle \xi
\rangle^{2m-1}\frac{\alpha }{\Lambda ^2\langle \xi
  \rangle^{2m}})=S_\Lambda (\frac{h}{\Lambda ^2\langle \xi \rangle }).
$$
Since $F_0$ also belongs to $S_\Lambda (\frac{1}{\alpha }\Lambda
^2\langle \xi \rangle^{2m})$, we see that for $|\beta |\ge 1$,
$$
h^{|\beta |}(\partial _\xi ^\beta F_0)(D_x^\beta E_0)\in S_\Lambda
(\frac{h^{|\beta |}}{\Lambda ^{2|\beta |}\langle \xi \rangle^{|\beta
  |}})\subset
S_\Lambda (\frac{h}{\Lambda ^2\langle \xi \rangle}),
$$
and this can be improved for $|\beta |\ge 2$, using that
$F\in S_{1,0}(\frac{1}{\alpha }\langle \xi \rangle^{2m})$. Hence,
$$
F\# E_0=1+r_1,\ r_1\in S_\Lambda (\frac{h}{\Lambda ^2\langle \xi \rangle}).
$$

\par Now put $E_1=E_0-r_1/(w-s/\alpha )$. Then by the same estimates
with an extra power of $h\Lambda ^{-2}\langle \xi \rangle^{-1}$, we
get
$$
F\# E_1=1+r_2,\ r_2\in S_{\Lambda }((\frac{h}{\Lambda ^2\langle \xi \rangle})^2),
$$
and iterating the procedure we get
\ekv{fu.22}
{
E_N\equiv \frac{1}{w-\frac{s}{\alpha }}\ \mathrm{mod}\ S_\Lambda
(\frac{\alpha }{\Lambda ^2\langle \xi \rangle ^{2m}}\frac{h}{\Lambda
  ^2\langle \xi \rangle}),
}
such that
\ekv{fu.23}
{
F\# E_N=1+r_N,\ r_N\in S_\Lambda ((\frac{h}{\Lambda ^2\langle \xi
\rangle})^{N+1}).
}
Actually, in this construction we can work with finite sums instead
of asymptotic ones and then
\ekv{fu.24}
{
E_N\hbox{ is a holomorphic function of }w,\hbox{ for }|\xi |\ge C,
}
where $C$ is independent of $N$.

\par Now we return to the manifold situation and denote by
$E_N^{(j)}$, $r_N^{(j)}$ the corresponding symbols on $T^*U_j$,
constructed above. Denote the operators by the same symbols, and put
on the operator level:
\ekv{fu.25}
{
E_N=\sum_{j=1}^s \widetilde{\chi }_jE_N^{(j)}\chi _j,
}
with $\chi _j$, $\widetilde{\chi _j}$ as in (\ref{fu.13}). Then
\eeekv{fu.26}
{
(w-\frac{1}{\alpha }S)E_{N-1}&=&1-\sum_{j=1}^s \frac{1}{\alpha
}[S,\widetilde{\chi }_j]E_{N-1}^{(j)}\chi _j+\sum_{j=1}^s\widetilde{\chi
}_j
r_N^{(j)}\chi _j
}{&=:&1+R_N^{(1)}+R_N^{(2)}
}
{&=:&1+R_N.}
Proposition \ref{fu4} implies that for every $\widetilde{N}$, there
exists an $\widetilde{M}$ such that the trace class norm of
$R_N^{(1)}$ satisfies
\ekv{fu.27}
{
\Vert R_N^{(1)}\Vert_{\mathrm{tr}}\le {\cal O}(h^{\widetilde{N}}|\Im
w| ^{-\widetilde{M}}).
}

\par As for the trace class norm of $R_N^{(2)}$, we review some easy
facts about such norms for pseudodifferential operators:

\par If $A=a(x,D)$ is a pseudodifferential operator on ${\bf R}^n$,
either in the
Weyl or in the classical quantization, then $A$ is of trace class and
we have
$$
\Vert A\Vert_{\mathrm{tr}}\le C\iint \sum_{_{|\beta |\le
    2n+1}}|\partial _{x,\xi }^\beta a | dxd\xi ,
$$
provided that the integral is finite. In that case we also know that
$$
\mathrm{tr\,}(A)=\frac{1}{(2\pi )^n}\iint a(x,\xi )dxd\xi .
$$
 See Robert \cite{Ro}, and also
$\cite{DiSj}$ for a sharper statement. If instead we consider an
$h$-pseudodifferential operator $A=a(x,hD)$, then it is unitarily
equivalent to $\widetilde{A}=a(h^{\frac{1}{2}}x,h^{\frac{1}{2}}D_x)$,
so
$$
\Vert A\Vert_{\mathrm{tr}}\le \frac{C}{h^n}\iint \sum_{_{|\beta |\le
    2n+1}}|(h^{\frac{1}{2}}\partial _{x,\xi })^\beta a | dxd\xi ,
$$
where the factor $h^{-n}$ is the Jacobian, when passing from
$h^{1/2}x, h^{1/2}\xi $ to $x,\xi $.

\par Now, let $a\in S_\Lambda (m)$ be a symbol on $T^*U$ with
uniformly compact support in $x$. Then for $|\beta |\le 2n+1$, we
have
$$
h^{\frac{|\beta |}{2}}\partial _{x,\xi }^\beta a={\cal
  O}(1)m\left(\frac{h}{\alpha }
\right)^{\frac{|\beta |}{2}}|\Im w|^{-M(\beta )}.
$$
Thus there exists $M\ge 0$ such that $a(x,hD_x)$ is of trace class and
\ekv{fu.28}
{
\Vert a(x,hD)\Vert_{\mathrm{tr}}\le Ch^{-n}\iint_{U\times {\bf R}^n}
m(x,\xi )dxd\xi\, |\Im w|^{-M},
}
provided that the integral converges.

\par From (\ref{fu.26}), (\ref{fu.23}), we now get
$$
\Vert R_N^{(2)}\Vert_{\mathrm{tr}}\le Ch^{-n}|\Im w|^{-M(N)}\iint
\left(\frac{h}{\Lambda ^2\langle \xi \rangle} \right)^N dxd\xi ,
$$
and (\ref{fu.27}) then shows that we have the same estimate for $R_N$:
\ekv{fu.29}
{
\Vert R_N\Vert_{\mathrm{tr}}\le Ch^{-n}|\Im w|^{-M(N)}\iint
\left(\frac{h}{\Lambda ^2\langle \xi \rangle} \right)^N dxd\xi .
}
The contribution to this expression from the region where $\Lambda \ge
1/C$ is ${\cal O}(h^{N-n})|\Im w|^{-M(N)}$.

\par The volume growth assumption (\ref{int.6.2}), that we now assume
for our fixed $z$, says that
\ekv{fu.30}
{
V(t):=\mathrm{vol\,}(\{ \rho \in T^*X;\, s\le t \})={\cal O}(t^\kappa
),\ 0\le t\ll 1,
}
for $0<\kappa \le 1$. The contribution to the integral in
(\ref{fu.29}) from the region $0\le s\le t_0$, $0<t_0\ll 1$, is equal
to some negative power of $|\Im w|$ times
\begin{eqnarray*}
&&{\cal O}(1)\int_0^{t_0}\left(\frac{h}{\alpha +t} \right)^N dV(t)\\
&=&
{\cal O}(1)\left[ \left(\frac{h}{\alpha +t}
  \right)^NV(t) \right]_{t=0}^{t_0}+{\cal
  O}(1)\int_0^{t_0}\frac{h^N}{(\alpha +t) ^{N+1}}V(t)dt\\
&=& {\cal O}(1)h^N+{\cal O}(1)h^N\int_0^{t_0}\frac{t^\kappa }{(\alpha +t)^{N+1}}dt.
\end{eqnarray*}
The last integral is equal to
$$
\int _0^{t_0/\alpha }\frac{(\alpha s)^\kappa }{\alpha ^{N+1}(1+s)^{N+1}}\alpha
ds\le \alpha ^{\kappa -N}\int_0^\infty \frac{s^\kappa }{(1+s)^{N+1}}ds.
$$
Thus,
\ekv{fu.31}
{
\Vert R_N\Vert_{\mathrm{tr}}\le {\cal O}(1) h^{-n}\alpha ^{\kappa
}\left(\frac{h}{\alpha } \right)^N|\Im w|^{-M(N)}.
}

\par From (\ref{fu.26}), we get
$$
(w-\frac{1}{\alpha }S)^{-1}=E_{N-1}-(w-\frac{1}{\alpha }S)^{-1}R_N.
$$
Write
$$
E_{N-1}=\frac{1}{w-\frac{s}{\alpha }}+F_{N-1},\quad F_{N-1}\in S_\Lambda
(\frac{\alpha h}{\Lambda ^4\langle \xi \rangle^{2m+1}}).
$$
More precisely we do this for each $E_{N-1}^{(j)}$ in (\ref{fu.25}). Then
quantize and plug this into (\ref{fu.2}):
\eekv{fu.32}
{
\chi (\frac{1}{\alpha }S)&=&-\frac{1}{\pi} \int \frac{\partial
    \widetilde{\chi }}{\partial
    \overline{w}}\mathrm{Op}_h(\frac{1}{w-\frac{s}{\alpha }})L(dw)
-\frac{1}{\pi }\int \frac{\partial
    \widetilde{\chi }}{\partial
    \overline{w}}F_{N-1} L(dw)
}
{&&-\frac{1}{\pi} \int \frac{\partial
    \widetilde{\chi }}{\partial
    \overline{w}}(w-\frac{1}{\alpha }S)^{-1}R_N L(dw)=:\mathrm{I}+\mathrm{II}+\mathrm{III}.
}
Here by definition,
$$
\mathrm{Op}_h\left( \frac{1}{w-\frac{s}{\alpha
  }}\right)=\sum_{j=1}^s\widetilde{\chi }_j\mathrm{Op}_h\left(\frac{1}{w-\frac{s}{\alpha
  }} \right)\chi _j
$$
with the coordinate dependent quantization appearing to the right.
$$
\mathrm{tr\,}(-\frac{1}{\pi }\int \frac{\partial \widetilde{\chi
  }}{\partial \overline{w}}(w)\widetilde{\chi }_j \mathrm{Op}_h\left( \frac{1}{w-\frac{s}{\alpha
  }}\right) \chi _j L(dw))
$$
is equal to
\begin{eqnarray*}
\frac{1}{(2\pi h)^n}\iint -\frac{1}{\pi }\int
\frac{\partial \widetilde{\chi }}
{\partial \overline{w}}(w) \frac{1}{w-\frac{s}{\alpha
  }}L(dw)\chi _j(x)dxd\xi \\ =\frac{1}{(2\pi h)^n}\iint \chi (\frac{s(x,\xi
  )}{\alpha })\chi _j(x)dxd\xi ,
\end{eqnarray*}
so
\ekv{fu.33}{
\mathrm{tr\,}(\mathrm{I})=\frac{1}{(2\pi h)^n}\iint \chi (\frac{s(x,\xi
  )}{\alpha })dxd\xi .
}
As at the last estimate in the proof of Proposition 4.4 in
\cite{HaSj} we see that this quantity is ${\cal O}(\alpha
^\kappa h^{-n})$ and more generally,
$$
\Vert \mathrm{I}\Vert_{\mathrm{tr}}={\cal O}(\alpha ^\kappa h^{-n}).
$$

\par For II, we get, using the fact that the symbol is holomorphic in
$w$ for large $\xi $,
\begin{eqnarray*}
\Vert \mathrm{II}\Vert_{\mathrm{tr}}&=&{\cal O}(h^{-n})\iint_{|\xi |\le
  C}\frac{h\alpha }{(\alpha +s)^2}dxd\xi \\
&=& {\cal O}(1)h^{-n}\frac{h}{\alpha
}\int_0^{t_0}\frac{1}{(1+\frac{t}{\alpha })^2}dV(t)\\
&=&{\cal O}(1)h^{-n}\frac{h}{\alpha}\left( \left[ (1+\frac{t}{\alpha
  })^{-2}V(t)
\right]_0^{t_0}+\int_0^{t_0}(1+\frac{t}{\alpha })^{-3}V(t) \frac{dt}{\alpha
}
\right)\\
&=&{\cal O}(1)h^{-n}\frac{h}{\alpha }(\alpha ^2+\alpha ^\kappa
\int_0^{t_0}
(1+\frac{t}{\alpha })^{-3}\left(\frac{t}{\alpha } \right)^\kappa
\frac{dt}{\alpha })\\
&=&{\cal O}(1)\frac{\alpha ^\kappa }{h^n}\frac{h}{\alpha }.
\end{eqnarray*}

\par It is also clear that
$$\Vert \mathrm{III}\Vert_{tr}={\cal O}(1)\frac{\alpha ^\kappa
}{h^n}\left(\frac{h}{\alpha } \right)^N.$$

Summing up our estimates, we get under the assumption (\ref{fu.30})
(equivalent to (\ref{int.6.2})) the following result:
\begin{prop}\label{fu6}
Let $\chi \in C_0^\infty ({\bf R})$. For $0<h\le \alpha <1$, we have
\ekv{fu.34}
{
\Vert \chi (\frac{1}{\alpha }S)\Vert_{\mathrm{tr}}={\cal
  O}(1)\frac{\alpha ^\kappa }{h^n},
}
\ekv{fu.35}
{
\mathrm{tr\,}\chi (\frac{1}{\alpha }S)=\frac{1}{(2\pi h)^n}
\iint \chi (\frac{s(x,\xi )}{\alpha })dxd\xi +{\cal O}(\frac{\alpha
  ^\kappa }{h^n}\frac{h}{\alpha }).
}
\end{prop}

\begin{remark}\label{fu7}
Using simple $h$-pseudodifferential calculus (for instance as in
the appendix of \cite{Sj1}, we see that if we redefine $S$ as in
(\ref{fu.1.5}), then in each local coordinate chart,
$S=\mathrm{Op}_h(S)$, where $S\equiv s\,\mathrm{mod\,}S_{1,0}(h\langle
\xi \rangle^{-1})$ and $s$ is now redefined as
\ekv{fu.36}
{
s(x,\xi )=\left(\frac{|p(x,\xi )-z|}{|\widetilde{p}(x,\xi )-z|}
\right)^2.
}
The discussion goes through without any changes (now with $m=0$) and
we still have Proposition \ref{fu6} with the new choice of $S$, $s$.
\end{remark}

In the remainder of this section, we choose $S$, $s$ as in
(\ref{fu.1.5}), (\ref{fu.36}). In this case we notice that $S$ is a
trace class perturbation of the identity, whose symbol is $1+{\cal
  O}(h^\infty /\langle \xi \rangle^\infty )$ and similarly for all its
derivatives, in a region $|\xi |\ge \mathrm{Const}$.

\par Let $0\le \chi \in C_0^\infty ([0,\infty [)$ with $\chi (0)>0$ and
let $\alpha _0>0$ be small and fixed. Using standard
pseudodifferential calculus in the spirit of \cite{MeSj}, we get
\ekv{fu.37}
{
\ln\det (S+\alpha _0\chi (\frac{1}{\alpha _0}S))=
\frac{1}{(2\pi h)^n}(\iint \ln (s+\alpha _0\chi (\frac{1}{\alpha
  _0}s))dxd\xi +{\cal O}(h)).
}

\par Extend $\chi $ to be an element of $C_0^\infty ({\bf R};{\bf C})$
in such a way that $t+\chi (t)\ne 0$ for all $t\in {\bf R}$. As in
\cite{HaSj}, we use that
\ekv{fu.39}
{
\frac{d}{dt}\ln (E+t\chi (\frac{E}{t}))=\frac{1}{t}\psi (\frac{E}{t}),
}
where
\ekv{fu.39.5}
{
\psi (E)=\frac{\chi (E)-E\chi '(E)}{E+\chi (E)},
}
so that $\psi \in C_0^\infty ({\bf R})$. By standard functional
calculus for self-adjoint operators, we have
\ekv{fu.40}
{
\frac{d}{dt}\ln \det (S+t\chi (\frac{S}{t}))=\mathrm{tr\,}
\frac{1}{t}\psi (\frac{S}{t}).
}
Using (\ref{fu.35}), we then get for $t\ge \alpha \ge h>0$:
$$
\frac{d}{dt}\ln\det (S+t\chi (\frac{1}{t}S))
=\frac{1}{(2\pi h)^n}(\iint \frac{1}{t}\psi (\frac{s}{t})dxd\xi +{\cal
  O}(ht^{\kappa -2})).
$$
Integrating this from $t=\alpha _0$ to $t=\alpha $ and using
(\ref{fu.37}), (\ref{fu.39}), we get
\ekv{fu.41}
{
\ln \det (S+\alpha \chi (\frac{1}{\alpha }S))=
\frac{1}{(2\pi h)^n}(\iint \ln (s+\alpha \chi (\frac{s}{\alpha
}))dxd\xi +{\cal O}(\frac{h}{\alpha })r_\kappa (\alpha )),
}
where $r_\kappa (\alpha )=\alpha ^\kappa $ when $\kappa <1$, and
$r_1(\alpha )=\alpha \ln \alpha $.

\par Improving the calculation prior to (4.22) in \cite{HaSj}, we get
\begin{eqnarray*}
\iint \ln (s+\alpha \chi (\frac{s}{\alpha }))dxd\xi &=&\iint \ln
(s)dxd\xi +\int_0^\alpha \iint \frac{1}{t}\psi (\frac{s}{t})dxd\xi \,
dt\\
&=&\iint \ln
(s)dxd\xi +\int_0^\alpha t^{\kappa -1} dt\\
&=&\iint \ln (s)dxd\xi +{\cal O}(\alpha ^\kappa ).
\end{eqnarray*}
and together with (\ref{fu.41}) this leads to
\begin{prop}\label{fu8}
If $0\le \chi \in C_0^\infty ([0,\infty [)$, $\chi (0)>0$, we have
uniformly for $0<h\le \alpha \ll 1$
\ekv{fu.42}
{
\ln\det (S+\alpha \chi (\frac{1}{\alpha }S))=\frac{1}{(2\pi h)^n}
(\iint \ln s(x,\xi )dxd\xi +{\cal O}(\alpha ^\kappa \ln \alpha )).
}
Here the remainder term can be replaced by ${\cal O}(\alpha ^\kappa )$
when $\kappa <1$ and by ${\cal O}(\alpha +h\ln \alpha )$ when $\kappa =1$.
\end{prop}

\section{End of the proof}\label{en}
\setcounter{equation}{0}

Having established in Section \ref{fu} the analogues for manifolds of
the results in Section 4 of \cite{HaSj}, the remainder of the proof of
Theorem \ref{int1} is basically identical to the proof  in \cite{Sj1}
for the ${\bf R}^n$ case. For that reason, we will only give a brief
outline.

\par Let $P$ be as in (\ref{int.1}), (\ref{int.2}), classically
elliptic as in (\ref{int.3}) and let $p$ be the semiclassical
principal symbol. Assume (\ref{int.5}). To start with, we let $z\in \Omega $ be fixed and
assume (\ref{fu.30}), where $s=|p-z|^2$ is the semiclassical principal
symbol of $S=(P-z)^*(P-z)$. Also, put
\ekv{en.1}
{
S_z =P_z^*P_z,\ P_z=(\widetilde{P}-z)^{-1}(P-z).
}

\par We fix the choice of an operator $P_0=P_{\delta _0}$ as in
(\ref{hs.1}), (\ref{hs.2}) (with $\delta =\delta _0$ still depending
on $h$) and define $\widetilde{P}_{\delta _0}$, $P_{\delta _0,z}$,
$S_{\delta_0 ,z}$, $S_{\delta _0}$ as in that section. As in
\cite{Sj1}, ${\cal D}(S_{\delta _0})=\{ u\in H^m(X);\, P_{\delta
  _0}u\in H^m(X)\}$. As there, we also introduce the self-adjoint
operator $T_{\delta _0}=(P_{\delta _0}-z)(P_{\delta _0}-z)^*$ with
domain ${\cal D}(T_{\delta _0})=\{ u\in H^m(X);\, P^*_{\delta
  _0}u\in H^m(X)\}$. In some fixed ($h$-independent) neighborhood of
$0$ the spectra of
$S_{\delta _0}$, $T_{\delta _0}$ are discrete and coincide. If $0<\alpha \ll 1$, denote the (common)
eigenvalues in $[0,\alpha [$ by $t_1^2,t_2^2,...,t_N^2$, where $0\le
t_1\le t_2\le ...\le t_N$. Then, there are orthonormal families of
eigenfunctions, $e_1,...,e_N\in {\cal D}(S_{\delta _0})$,
$f_1,...,f_N\in {\cal D}(T_{\delta _0})$ such that
\ekv{en.2}{(P_{\delta _0}-z)e_j=t_jf_j,\ (P_{\delta
    _0}-z)^*f_j=t_je_j.}

\par Define $R_+:L^2(X)\to {\bf C}^N$, $R_-:{\bf C}^N\to L^2(X)$ by
\ekv{en.3}
{
R_+u(j)=(u|e_j),\ R_-u_-=\sum_{1}^N u_-(j)f_j.
}
The Grushin problem
\ekv{en.4}
{(P_{\delta _0}-z)u+R_-u_-=v,\ R_+u=v_+,}
has a unique solution $u=E^0v+E_+^0v_+\in H^m(X)$, $u_-=E^0_-v+E^0_{-+}v_+\in
{\bf C}^N$ for every $(v,v_+)\in L^2(X)\times {\bf C}^N$, and
$E^0_\pm$ and $E^0_{-+}$ can be given explicitly. In particular,
$E^0_{-+}=-\mathrm{diag\,}(t_j)$.

\par Let now $P_\delta =P_{\delta _0}+\delta Q$ be a small perturbation
(in a suitable sense) of $P_{\delta _0}$. Then we still have a
wellposed problem after replacing $P_{\delta _0}$ by $P_\delta $ in
(\ref{en.4}) with the solution $u=E^{\delta}v+E_+^{\delta}v_+\in H^m$, $u_-=E^{\delta}_-v+
E^{\delta}_{-+}v_+\in {\bf C}^N$ and the new solution operators have Neumann
series expansions. In particular,
\ekv{en.5}
{
E_{-+}^\delta = E_{-+}^0-\delta E_-^\delta QE_+^0+\delta ^2E_-^0QE^0QE_+^0
-...,}
where we can write the leading perturbation
$-\delta E_-^0Qe_+^0=-\delta M$, where $M=(M_{j,k})_{1\le j,k\le N}$,
$M_{j,k}=(Qe_k|f_j)$. When $Q$ is a multiplication operator,
$Qu(x)=q(x)u(x)$, then
\ekv{en.6}
{
M_{j,k}=\int q(x)e_k(x)\overline{f}_j(x)dx.
}

\par Now, adopt the symmetry assumption (\ref{int'.7}). Then we can
replace the orthonormal family $f_j$ by the new orthonormal family of
eigenfunctions $\widetilde{f}_j=\overline{e}_j$ without changing the
singular values of $E_{-+}^0$, $E_{-+}^\delta $ and we get
\ekv{en.7}
{
M_{j,k}=\int q(x)e_k(x)e_j(x) dx.
}
In \cite{Sj1} we showed how to find admissible potentials $q$ as in
(\ref{int.6.3}), (\ref{int.6.4}), such
that $M$ gets at least $N/2$ ``large'' singular values and this was
used in an iteration procedure in order to find perturbations of the
form $P_\delta $ where $q$ is an admissible potential, for which the
small singular values are not ``too small''.

\par Strengthen the assumption on $\delta _0$ to
\ekv{en.8}
{
\delta _0\le h.
}
Then combining Proposition \ref{fu6} with the perturbative functional
calculus in Section 4 of \cite{Sj1}, we obtain that for $0<h\le \alpha
\ll 1$, the number of eigenvalues of $S_{\delta _0}$ in $[0,\alpha ]$
satisfies $N={\cal O}(\alpha ^\kappa h^{-n})$. The iteration scheme in
Sections 5 to 7 in \cite{Sj1} now works without any changes and we
get the following analogue of Proposition 7.3 there:
\begin{prop}\label{en1}
We make the assumptions above (with $z$ fixed). Let
$s>\frac{n}{2}$, $0<\epsilon <s-\frac{n}{2}$, $N_1=\widetilde{M}+sM+
\frac{n}{2}$, $N_2=
2(N_1+n)+\epsilon _0$, where $\epsilon _0>0$ and $M,\widetilde{M}$ are
as in (\ref{int.6.4}). Let $L$, $R$ be
$h$-dependent parameters as in (\ref{int.6.4}).
Let $0<\tau_0\le \sqrt{h}$ and let $N^{(0)}={\cal O}(h^{\kappa -n})$
be the number of singular
values of $P_{\delta _0}-z$ in $[0,\tau_0[$. Let $0<\theta <\frac{1}{4}$ and let
$N(\theta )\gg 1$ be sufficiently large. Define $N^{(k)}$, $1\le k\le
k_1$ iteratively in the following way. As long as $N^{(k)}\ge
N(\theta )$, we put $N^{(k+1)}=[(1-\theta )N^{(k)}]$ (the integer part
of $(1-\theta )N^{(k)}$). Let $k_0\ge 0$
be the last $k$ value we get in this way. For $k>k_0$ put $N^{(k+1)}=N^{(k)}-1$ until
we reach the value $k_1$ for which $N^{(k_1)}=1$.

\par Put $\tau_0^{(k)}=\tau_0h^{kN_2}$, $1\le k\le k_1+1$. Then there
exists $q=q_h(x)$ of the
form \no{int.6.3}, satisfying \no{int.6.4}, so that by the
choice of $L$,
$$
\Vert q\Vert_{H^s}\le {\cal O}(1)h^{-N_1+\frac{n}{2}}, \
\Vert q\Vert_{L^\infty }\le {\cal O}(1)h^{-N_1},
$$
such that if $P_{\delta}=P_{\delta _0}+\frac{1}{C}\tau
_0h^{2N_1+n}q=P+\delta Q$, $\delta =\frac{1}{C}h^{N_1+n}\tau_0$, $Q=h^{N_1}q$,
we have the following estimates on the singular values of
$P_{\delta }-z$:
\begin{itemize}
\item If $\nu >N^{(0)}$, we have
$t_\nu (P_{\delta }-z)\ge (1-\frac{h^{N_1+n}}{C})t_\nu (P-z)$.
\item If $N^{(k)}<\nu \le N^{(k-1)},$ $1\le k\le k_1$, then $
t_\nu (P_\delta -z)\ge (1-{\cal O}(h^{N_1+n}))\tau_0^{(k)}$.

\item Finally, for $\nu =N^{(k_1)}=1$, we have  $
t_1(P_\delta -z)\ge (1-{\cal O}(h^{N_1+n}))\tau_0^{(k_1+1)}$.
\end{itemize}
\end{prop}

\par As shown in \cite{Sj1} we have an equivalence between lower
bounds for the small singular values of $P_\delta -z$ in the above
proposition and for the singular values of $E_{-+}^\delta $ appearing
in the solution of the Grushin problem for $P_\delta -z$ (and that
is used in the proof of the proposition). We also have an equivalence
between lower bounds for the small singular values of $P_\delta -z$
and those of $P_{\delta ,z}$. For the latter operator we have a well
posed Grushin problem analogous to (\ref{en.4}) and an equivalence
between lower bounds for the small singular values of $P_{\delta ,z}$
and for the singular values of $E_{-+}^{\delta ,z}$, appearing in the
solution of the new Grushin problem. Using perturbative functional
calculus we also have an asymptotic formula for $\ln \det {\cal
  P}_{\delta ,z}$, where
$${\cal P}_{\delta ,z}=\left(\begin{array}{ccc}P_{\delta ,z}
    &R_-^z\\R_+^z &0 \end{array}\right)$$ is the matrix associated to
the new Grushin problem. As showed in \cite{HaSj} by means of
calculations from \cite{SjZw}, we have
\ekv{en.10}
{
\det P_{\delta ,z}=\det {\cal P}_{\delta ,z}\det E_{-+}^{\delta ,z}.
}

\par The perturbative functional calculus gives a general upper bound
on $\ln \det P_{\delta ,z}$, and for the special admissible
perturbation in Proposition \ref{en1}, we have a lower bound on $\ln
|\det E_{-+}^{\delta ,z}|$ (using the lower bound on the singular
values of $E_{-+}^{\delta ,z}$ and the fact the modulus of the
determinant is equal to the product of the singular values). We get as
in \cite{Sj1}:
\begin{prop}\label{en2}
For the special admissible perturbation $P_\delta $ in
Proposition \ref{en1}, we have
\eekv{en.11}
{
&&\ln |\det P_{\delta ,z}|\ge }
{&&\frac{1}{(2\pi h)^n}
\left( \iint \ln |p_z|dxd\xi -{\cal O}\left(
(h^{\kappa }+h^n\ln \frac{1}{h})(\ln \frac{1}{\tau_0}+
(\ln \frac{1}{h})^2)\right)
\right) .
}
\end{prop}

On the other hand, for more general operators of the form $P_\delta
=P_{\delta _0}+\tau_0h^{2N_1+n}q$ with $q$ admissible as in
(\ref{int.6.3}), (\ref{int.6.4}) we get as in \cite{Sj1} the upper
bound:
\ekv{en.12}
{
\ln |\det P_{\delta ,z}|\le \frac{1}{(2\pi h)^n}
\left( \iint \ln |p_z|dxd\xi +{\cal O}\left(h^{\kappa }\ln \frac{1}{h}\right)
\right) .
}

Section 8 of \cite{Sj1} (based on Jensen type arguments in the $\alpha
$-variables) now applies and shows that with probability close to 1,
we have
\ekv{en.13}
{
\ln |\det P_{\delta ,z}|\approx \frac{1}{(2\pi h)^n}\iint \ln
|p_z|dxd\xi .
}

So, far $z$ was fixed and we now let it vary in a neighborhood of
$ \Gamma $, recalling that the eigenvalues $P_\delta $ in this region
coincide with the zeros of the holomorphic function $F_\delta (z)=\det
P_{\delta ,z}$. Assuming now that (\ref{int.6.2}) holds uniformly for
$z$ in a neighborhood of $\partial \Gamma $, we can then conlude the
proof as in Section 9 of \cite{Sj1}, by applying a general result
about the number of zeros of holomorphic functions with exponential
growth, from \cite{HaSj}. Recall that this result (applied to
$F_\delta (z)$) requires an upper bound on $\ln |F_\delta (z)|$ in a
fixed neighborhood of $\partial \Gamma $, in our case provided by
(\ref{en.12}), as well as a corresponding lower bound at finitely many
points along $\partial \Gamma $. The latter is provided by the lower
bound part of (\ref{en.13}) and holds with probability close to 1.

redpajama_set_name: RedPajamaArXiv


text:
Home | Business | Russia: US pursuing aggravation of crisis for Iran
Russia: US pursuing aggravation of crisis for Iran
Moscow, Russia has reiterated its opposition to the US oil sanctions against Iran, saying that the United States is trying to limit all measures by the Iranian government to complicate the internal situation in the country and thus influence its foreign actions in the world.
Foreign Ministry spokeswoman Maria Zakharova, at a weekly press conference on Friday said, 'US actions cover all aspects of Iran's domestic politics and its activities internationally.
We consider the US decision not to extend the exemption of the eight countries from purchasing Iran's oil as part of the US global strategy against Iran and condemn it., the Russian official expressed.
The spokesman for the Russian Foreign Ministry went on to say that 'We have voiced opposition to the principle of unilateral sanctions, which has not been approved by international organizations, including the Security Council, which legitimizes sanctions.
The Russian official added that Moscow has repeatedly announced what is related to Iran regarding it is opposed to American antiIranian policies.
'Objectives announced by the United States, even if it looks good with American efforts, has nothing to do with the specific policy of unilateral sanctions. In fact, this is a public and very hostile pressure on an independent state., Zakharova noted.
A few days ago, in an exclusive interview with IRNA, Zakharova condemned US sanctions against Iran and confirmed that this policy has long been on the agenda of the United States.
ON 22 April Trump administration took its last step to zero Iran's oil exports. The White House has announced that since May 2, there has been no extension of any exemption for countries willing to buy Iranian oil.
Although this move was, in fact, a demand that Trump was repeating from the very first days of taking office at the White House, but the global and regional conditions were such that he was not allowed to do so.
In his meeting with members of the National Security Council of the White House less than a month ago, the US president insisted on imposing maximum pressure on Iran by terminating the waivers.
In response to the White House, Iran's foreign ministry spokesman said that refusal to issue waivers for purchase of Iran's oil was basically unlawful. Seyyed Abbas Mousavi said, 'Given the illegality of the sanctions, the Islamic Republic of Iran has not, in principle, gave no credibility for granting exemptions to sanctions.'
However, given the negative effects of these sanctions and the possibility of increasing these negative impacts without extending the exemptions, the FOREIGN Ministry has been continuously in contact with all relevant domestic institutions and has had intensive consultations with many foreign partners such as Europeans, internationals and neighbors.
He added that the outcome of these internal and external contacts and consultations would report to the decisionmaking authorities and that a prompt decision would be made and announced.
Source: Islamic Republic News Agency IRNA
RA Holding Corp. Announcement To U.S. Shareholders – Tax Information
كافلان تفوز بجوائز "القمة" في أولى مسابقات المشروبات الروحية 2019 في سان فرانسيسكو

redpajama_set_name: RedPajamaCommonCrawl


Creates a timestamp for the date and time represented by a specified number of seconds since 12:00 a.m., January 1, 1904, Universal Time.
Number of seconds elapsed since the epoch time of 12:00 a.m., January 1, 1904, Universal Time [01-01-1904 00:00:00].
This input can also be any data type that contains only numerics, such as an array or cluster of numerics.
Date and time that results from adding number to 12:00 a.m., January 1, 1904, Universal Time [01-01-1904 00:00:00].


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text:
Isle of Ghosts
Sun Sep 15, 2013 11:03 am by admin
Every year thousands of tourists visit the Isle of Wight, unaware that they are stepping into the land of the supernatural
ISLE OF GHOSTS BY EVAN LEGHTFOOT
FOR A place of such small size, the Isle of Wight must be one of the most haunted spots in Britain. Countless tales of the supernatural and the unknown have their roots in her green and mysterious countryside. Tales of horror and violence, sudden death and bizarre intrigue. From the spine-tingling legend or a ghostly queen, to the baffling mystery of a murdered Frenchman From the horrifying story of a headless pilot, to the ancient tale of a phantom coach.
No sooner has the visitor stepped off Ryde Pier and set foot for the first time on Island soil, than he is plunged into a world of spine-chilling terror and eerie legend. For the sea-front of Ryde, so down-to-earth in daylight, is a place of awe and dread during the hours of darkness. Barely two hundred yards to the west of the pier lies the muddy expanse of Players Beach, where according to the locals, the restless spirit of a drowned child is supposed to wander by the sea-wall.
Behind the beach lie the rambling grounds of Buckland Grange, reputably Walked by it's own shadowy phantom, and a mile away, in the opposite direction, the forbidding shape of Appley Tower, complete with goblin. Beyond Appley lies Puckpool, and the mysterious story of The Grey Lady.
The Golden Coffin
Indeed there must be almost as many ghosts in Ryde as pubs. From a poltergeist with a dislike for mortals playing darts, to a Victorian house full of screams, groans, and "things that go bump in the night." Just outside Ryde lies the crumbling ruins of old Quarr Abbey, and the tale of a long-dead monk in ghostly cowl walking an overgrown
tunnel beneath the Ryde-Newport road (a supernatural bid to avoid the traffic no doubt). Immediately north of this road is the leafy tangle of Eleanour's copse, so named because the mortal remains of Eleanour of Provence, unpopular wife of Henry III are supposed to be buried in a golden
coffin nearby. But beware, all ye treasure seekers. At night the Queen's uneasy spirit can be seen roaming the copse with "much wringing of hands," and the coffin itself is said to be protected by various magic spells from all who might be tempted to chip a lump off.
Most commercialised of all the local "spooks " is poor old "Lonely Louis," the unfortunate wraith of a French traveller foully murdered in Brading's oldest house. The luckless apparition is doomed to walk the creaking building — which now houses the famous Osborne-Smith Wax Museum — until such time as his poor, blanched bones are buried in his homeland.
Witch's spell
To the west of Brading lies the tiny hamlet of Knighton, probably named after the Black Knight who thunders down the shute every Halloween, October rain or moonshine. This sorry gentleman is possibly in an even greater plight than Monsieur Lonely Louis, as he must make this annual journey strapped to his horse's saddle for ever more as a result of a witch's curse. Another knight, Sir Tristram Dillington, was once supposed to haunt the grounds and terrace of Knighton Manor nearby. A priest from Brading was called in to exorcise him. however, and he seems to have put paid to Sir Tristram's nocturnal wanderings.
Overlooking Knighton on the wind-swept heights of Ashey Down stands the solitary phantom of Michael Morey, the hermit hanged in 1735 for the murder of his grandson. Further south is Godshill, and its celebrated legend of church stones moved by the Devil.
Almost every Island town has its own ghost or ghosts. Workmen demolishing a Ventnor hospital recently complained repeatedly of a strong smell of chloroform coming from the old operating theatre, and one poor chap had the fright of his life when a ghostly surgeon, complete with mask and scalpel appeared before his eyes.
Strange noises have been heard coming from a sealed-off room in the old Shanklin Rectory, and a phantom horse has been seen galloping through the streets of Bonchurch at night.
An irate ghost in Newport's Bugle Hotel objects to alterations being made in the kitchen, and spellbound members of the staff have watched in awe as plugs and plates have lifted themselves up off the tables, and floated unaided in mid air.
As the season of peace and good cheer is not too far behind us, it seems only right to end this some-what ghoulish article on a festive note. I take you back, therefore, some three hundred years or so to a certain New Year's Ball held. somewhere in Southern Wight.
Slippery shute
It was a freezing January morning as the gentleman and his lady left the happy gathering for home. The family coach was ready and waiting but the couple were unaware of the fact that the coachman, not to be outdone by his master and mistress, had been having a private celebration of his own. With no police to fear in those pre-breathalyser days, the man was more than a little drunk.
Everything ran smoothly for the first part of the journey however, the coach and team rattled through the narrow Island lanes without incident.
It was not until they reached Newchurch that Phantoms walk while an Island sleeps things began to happen ! The team descended the steep shute much too fast for safety, and the coachman, only half sober, was unable to control them! The road was cold and slippery, and the horses slipped on the treacherous surface. The coach overturned, and the coachman. the gentleman, and his lady, were thrown to their death at the bottom of the slope.
The story would have ended there, had it not been for certain inexplicable events that happened afterwards. A few years later, a local farmhand, returning home late at night, watched in horror as a phantom vehicle, drawn by four ghostly horses, crashed down the hill in exactly the same fashion. Dozens of people since have sworn to have seen various similar re-enactments of the ghastly tragedy.
To this day, the phantom coach is supposed to return to the scene of the disaster. Rattling down Newchurch shute every New Year's Eve. The grim tale is as old as the warning. IF YOU DRINK—DON'T DRIVE !


text:
Jules Richard (Blet, 12 de agosto de 1862 - Châteauroux, 14 de outubro de 1956) foi um matemático francês.

Vida e obras
Jules Richard ensinou no ensino médio em Tours, Dijon e Châteauroux. Ele obteve seu doutorado aos 39 anos pela Faculté des Sciences em Paris. Sua tese de 126 páginas discute a superfície-onda de Fresnel. Richard trabalhou primariamente nas fundações da matemática e geometria, relacionando-se com os trabalhos de David Hilbert, von Staudt e Charles Mérav.

Em um tratado mais filosófico sobre a natureza de axiomas na geometria, Richard discute e rejeita os seguintes princípios básicos:
 (1) A geometria é construída baseada em axiomas arbitrários - existem infinitas geometrias igualmente corretas.
 (2) Da experiência provêm os axiomas da geometria, a base é experimental e o desenvolvimento é dedutivo.
 (3) Os axiomas da geometria são definições (em contraste com (1)).
 (4) Axiomas não são nem experimentais, nem arbitrários, eles se impõem sobre nós já que experiência é uma impossibilidade sem eles.
A última proposição foi essencialmente proposta por Immanuel Kant.

Richard chegou ao resultado de que a noção de intensidade de dois objetos e a invariância de um objeto são demasiadamente vagas e precisam ser especificadas mais precisamente. Isto dever ser feito através de axiomas.
 Axiomas são proposições, cuja tarefa é tornar precisa a noção de identidade de dois objetos pré-existentes em nossas mentes.
Mais adiante, de acordo com Richard, é a principal meta da ciência explicar o universo material. E (apesar de geometria não-euclidiana não ter encontrado aplicações até Albert Einstein publicar sua Teoria da Relatividade em 1915) Richard já havia dito:

Pode-se perceber, que tendo admitido a noção de ângulo, estamos livres para escolher a noção de linha reta de forma que uma ou outra de três geometrias são possíveis.

Richard correspondeu-se com Giuseppe Peano e Henri Poincaré. Ele se tornou conhecido além de grupos de especialistas ao formular o paradoxo usado extensivamente por Poincaré para atacar a atual Teoria dos Conjuntos.

Paradoxo de Richard

O paradoxo de Richard foi primeiro descrito em 1905 em uma carta para Louis Olivier, diretor do Revue générale des sciences pures et appliquées. Foi publicado em 1905 no artigo intitulado Les Principes des mathématiques et le problème des ensembles. A Principia Mathematica de Alfred North Whitehead e Bertrand Russel o referencia junto a seis outros paradoxos ligados à auto-referência. Em um dos mais importantes compêndios de lógica matemática, compilado por Jean van Heijenoort, o artigo é traduzido para inglês. O paradoxo pode ser interpretado como uma aplicação do argumento da diagonal de Cantor. Ele inspirou Kurt Gödel e Alan Turing em suas famosas obras. Kurt Gödel considerou seu teorema da incompletude como um análogo ao paradoxo de Richard, que, em sua versão original, diz:

Seja E o conjunto de números reais que pode ser definido por um conjunto finito de palavras. Este conjunto é enumerável. Seja p o n-ésimo número do conjunto E; nós formamos um número N tendo zero para a parte integral e p+ 1 para o n-ésimo decimal, se p não é igual a 8 ou 9, e união no caso contrário. Este número N não pertence ao conjunto E porque ele difere de qualquer número neste conjunto, por exemplo do n-ésimo número vizinho do n-ésimo digito. Porém N foi definido por um número finito de palavras. Ele, portanto, deve pertencer ao conjunto E. Isto é uma contradição.

Richard nunca apresentou este paradoxo em outra forma, porém existem várias versões diferentes, algumas apenas vagamente parecidas com a original. Algumas listadas a seguir.

Outras versões do paradoxo de Richard

(A) A versão exposta em Principia Mathematica por Whitehead e Russell é similar à versão original de Richard, porém não precisamente igual. Aqui o digito 9 é trocado pelo digito 0, para que entidades como 1.000... = 0.999... podem quebrar o resultado.

(B) Paradoxo de Berry, primeiramente mencionado em Principia Mathematica como o quinto de sete outros paradoxos, é creditado ao Sr. G. G. Berry da Biblioteca Bodleian. Ela usa "O primeiro número inominável em menos de dez palavras"; na realidade, em inglês ela denota 111,777. Porém "O primeiro número inominável em menos de dez palavras" é por si só um nome consistindo de nove palavras; portanto pelo menos um inteiro não nomeável em menos de dez palavras pode ser nomeado em nove palavras, o que é uma contradição.

(C) Paradoxo de Berry com letras em vez de silabas é comummente relacionado ao conjunto de todos os números naturais que podem ser definidas em menos de 100(ou qualquer outro número grande) letras. Já que os números naturais são um conjunto bem-ordenado, deve existir o menor número que não pode ser definido com menos de 100 letras. Porem este número acaba de ser definido com 64 letras incluindo espaços.

(D) Paradoxo de König também foi publicado em 1905 por Julius König. Todos os números reais que podem ser definidos por um número finito de palavras formam um subconjunto dos números reais. Se os números reais podem ser bem-ordenados, então deve haver um primeiro número real (de acordo com esta ordem) que não pode ser definido por um número finito de palavras. Mas o primeiro número real que não pode ser definido em um número finito de palavras acaba de ser definido em um número finito de palavras.

(E) O menor número natural sem propriedades interessante adquire uma propriedade interessante devido à sua falta de propriedades interessante.

(F) um empréstimo do Paradoxo de Grelling-Nelson. O número de todas as definições finitas é contável. Em ordem lexical, nós obtemos uma sequência de definição D1, D2, D3, ... Agora, uma definição pode definir seu próprio número. Isto seria o caso se D1 lesse "o menor número natural". Pode ocorrer, que uma definição não descreve seu próprio número. Isto seria verdade se D2 lesse "o menor número natural". Além disso, a sentença "esta definição não define seu número" é uma definição finita. Seja Dn. n é descrita por Dn? Se sim, então não, se não, então sim. O dilema é insatisfazível.

Reações ao Paradoxo de Richard

Georg Cantor escreveu em uma carta para David Hilbert:
 (Ex., definições que não podem ser feitas em tempo finito) são um absurdo. Se a declaração de König estiver "correta", de acordo com a qual todos os números reais "finitamente definidos" formam uma coleção de número cardial , isto implicaria que a inumerabilidade do continuum inteiro; mas isto é obviamente errôneo. A questão agora é qual erro a prova deste teorema errado se baseia. O erro (que também aparece na nota de um Sr. Richard na última edição de Acta Mathematica, que Sr. Poincaré enfatiza na última edição de Revue de Métaphysique et de Morale) é, em minha opinião, a seguinte: É assumido que o sistema {B} de noções B, que precisa ser usado para a definição de números individuais, é no máximo inumeravelmente infinito. Esta suposição "deve ser um erro" por que caso contrário nós teríamos um o teorema errado: "o continuum de números tem a cardinalidade ".

Aqui, Cantor está errado. Hoje nós sabemos que existem incontáveis números reais sem a possibilidade de definição finita.

Ernst Zermelo comenta o argumento de Richard:
 A noção de "finitamente definível" não é absoluta, mas uma definição relativa e sempre relacionada à "linguagem" escolhida. A conclusão de acordo com a qual todos os finitamente definíveis objetos são contáveis é apenas válida caso um e apenas um sistema de símbolos é usado; a questão de se um único indivíduo pode ser atribuído uma definição finita é nula pois para cada coisa um nome arbitrário pode ser atribuído.

Zermelo aponta para a razão pela qual o paradoxo de Richard falha. Sua última proposição, porém, é impossível de se satisfazer. Um número real com infinitos dígitos, que não são determinados por alguma "regra", tem um conteúdo de informação infinitamente grande. Tal número só pode ser identificado por um nome curto se existe pelo menos um ou poucos deles existentes. Se houverem incontáveis números, como é o caso, identificação é impossível.

Artigos e livros por Jules Richard

Thèses présentées à la Faculté des sciences de Paris par M. Jules Richard, 1re thèse: Sur la surface des ondes de Fresnel..., Chateauroux 1901 (126 pages).
Sur la philosophie des mathématiques, Gauthier-Villars, Paris 1903 (248 pages).
Sur une manière d'exposer la géométrie projective, L'Enseignement mathématique 7 (1905) 366-374.
Les principes des mathématiques et le problème des ensembles, Revue générale des sciences pures et appliquées 16 (1905) 541-543.
The principles of mathematics and the problem of sets (1905), English translation in Jean van Heijenoort, "From Frege to Gödel - A Source Book in Mathematical Logic", 1879-1931. Harvard Univ. Press, 1967, p. 142-144.
Lettre à Monsieur le rédacteur de la Revue Générale des Sciences, Acta Math. 30 (1906) 295-296.
Sur les principes de la mécanique, L'Enseignement mathématique 8 (1906) 137-143.
Considérations sur l'astronomie, sa place insuffisante dans les divers degrés de l'enseignement, L'Enseignement mathématique 8 (1906) 208-216.
Sur la logique et la notion de nombre entier, L'Enseignement mathématique 9 (1907 ) 39-44.
Sur un paradoxe de la théorie des ensembles et sur l'axiome Zermelo, L'Enseignement mathématique 9 (1907) 94-98.
Sur la nature des axiomes de la géométrie, L'Enseignement mathématique 10 (1908 ) 60-65.
Sur les translations,  L'Enseignement mathématique 11 (1909) 98-101.
Contre la géométrie expérimentale Revue de l'Enseignement des Sciences (1910) 150.

Referências

Bibliografia
J. Itard: Richard, Jules Antoine, Dictionary of Scientific Biography, 11, Charles Scribner's Sons, New York (1980) 413-414.
[This seems to be the only original source, used by all other biographers.]
S. Gottwald: Richard, Jules Antoine in: Lexikon bedeutender Mathematiker, Harri Deutsch, Thun und Frankfurt (M) 1990.
J. J. O'Connor, E. F. Robertson: The MacTutor History of Mathematics archive

Bibliografia sobre Paradoxo de Richard
H. Meschkowski, W. Nilson: Georg Cantor - Briefe, Sphinhubyringer, Berlin 1991, p. 446.
W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.
A. N. Whitehead, B. Russell: Principia Mathematica I, Cambridge Univ. Press, Cambridge 1910, p. 64.
E. Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908) p. 107-128.
Proof of impossibility

Links

Matemáticos da França

redpajama_set_name: RedPajamaWikipedia


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Tickets For Troops is delighted to announce that it has been allocated funds from the LIBOR Fund. As a small charity which was only launched three years ago and relies on individual donations, this extra support will be a fantastic boost to the charity and allow it to continue providing the wide range of free tickets to major events to thousands of members of the Armed Forces in recognition of the enormous sacrifice they make on our behalf. To find out more about the LIBOR Fund and allocation, please click here.


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The Metalheadz x Concrete Junglist colab now in Black!
Inspired by the recent Mikal album on Metalheadz of the same name, Concrete Junglist delivers one of the finest Headz T-Shirts to date.
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The Minister of Defence, Brigadier-General Mansur Dan-Ali (retd), has denied an online media report credited to him that Boko Haram insurgents still controlled two local government areas in the North-East.
The minister stated this in a statement issued in Abuja, on Wednesday, by Colonel Tukur Gusau, public relations officer in his office.
He also denied reports in the same medium credited to him that the abducted female students of Government Secondary School, Chibok, were being used as suicide bombers.
According to the statement, the minister was quoted out of context, as he did not imply what was reported by the online medium.
'The minister was a guest in the studio of Voice of America, where he spoke in Hausa language on the appraisal of the successes recorded in the ongoing war against terrorists in the North- East.
'He said that before now, over 60 local government areas were under the occupation of Boko Haram in the North-East, but now only two were having some remnants of Boko Haram activities and not under their complete control.
'On the issue of the Chibok girls, the minister was not referring to the girls abducted on April 14, 2014 from Government Secondary School, Chibok.
'Rather, he was referring to other girls abducted by the terrorists from the local government areas and environs,'' the statement quoted Gusau as saying.


text:
<?php

// *** Make sure the file isn't accessed directly
defined('APPHP_EXEC') or die('Restricted Access');
//--------------------------------------------------------------------------

if(Modules::IsModuleInstalled('booking') &&
  (ModulesSettings::Get('booking', 'is_active') != 'no') &&
  ($objLogin->IsLoggedInAs('owner','mainadmin') || ($objLogin->IsLoggedInAs('hotelowner') && $objLogin->HasPrivileges('view_hotel_reports')))
){

	$task = isset($_POST['task']) ? prepare_input($_POST['task']) : '';
	$sel_type = isset($_POST['sel_type']) ? prepare_input($_POST['sel_type']) : '';
	$sel_hotel = isset($_POST['sel_hotel']) ? prepare_input($_POST['sel_hotel']) : '';
	$date_from = isset($_POST['date_from']) ? prepare_input($_POST['date_from']) : '';
	$date_to = isset($_POST['date_to']) ? prepare_input($_POST['date_to']) : '';
	$msg = '';
	$where_clause = '';
	$date_from_sql = '';
	$date_to_sql = '';
	$nl = "\n";

	$arr_hotels = array();
	$hotels_list = '';
	$hotels_count = Hotels::HotelsCount();
	if($hotels_count > 1){
		if($objLogin->IsLoggedInAs('hotelowner')) $hotels_list = implode(',', $objLogin->AssignedToHotels());
		$total_hotels = Hotels::GetAllActive((!empty($hotels_list) ? TABLE_HOTELS.'.id IN ('.$hotels_list.')' : ''));
		foreach($total_hotels[0] as $key => $val){
			$arr_hotels[$val['id']] = $val['name'];
		}
	}

	if($objSettings->GetParameter('date_format') == 'mm/dd/yyyy'){
		$calendar_date_format = '%m-%d-%Y';
		$date_from_sql = date('Y-m-d', mktime(0, 0, 0, substr($date_from, 0, 2), substr($date_from, 3, 2), substr($date_from, 6, 4)));
		$date_to_sql   = date('Y-m-d', mktime(0, 0, 0, substr($date_to, 0, 2), substr($date_to, 3, 2), substr($date_to, 6, 4)));
	}else{
		$calendar_date_format = '%d-%m-%Y';
		$date_from_sql = date('Y-m-d', mktime(0, 0, 0, substr($date_from, 3, 2), substr($date_from, 0, 2), substr($date_from, 6, 4)));
		$date_to_sql   = date('Y-m-d', mktime(0, 0, 0, substr($date_to, 3, 2), substr($date_to, 0, 2), substr($date_to, 6, 4)));
	}

	if($task == 'prepare_report'){
		if($sel_type == ''){
			$msg = draw_important_message(_SELECT_REPORT_ALERT, false);
		}else if($sel_type == 'arriving' && $date_from == ''){
			$msg = draw_important_message(str_replace('_FIELD_', '\''._FROM.'\'', _FIELD_CANNOT_BE_EMPTY), false);
		}else if($sel_type == 'departing' && $date_to == ''){
			$msg = draw_important_message(str_replace('_FIELD_', '\''._TO.'\'', _FIELD_CANNOT_BE_EMPTY), false);
		}else if($sel_type == 'staying' && ($date_from == '' || $date_to == '')){
			$msg = draw_important_message(_DATE_EMPTY_ALERT, false);
		}

		if($msg == ''){
			if($sel_type == 'arriving'){
				$where_clause = ' AND br.checkin >= \''.$date_from_sql.'\''.(($date_to != '') ? ' AND br.checkout <= \''.$date_to_sql.'\'' : '');
			}else if($sel_type == 'departing'){
				$where_clause = ' AND br.checkout <= \''.$date_to_sql.'\''.(($date_from != '') ? ' AND br.checkin >= \''.$date_from_sql.'\'' : '');
			}else if($sel_type == 'staying'){
				$where_clause = ' AND br.checkin <= \''.$date_from_sql.'\' AND br.checkout >= \''.$date_to_sql.'\'';
			}

			if($sel_hotel != ''){
				$where_clause = ' AND r.hotel_id = '.(int)$sel_hotel;
			}

			$sql = 'SELECT
						b.booking_number,
						IF(
							b.is_admin_reservation = 1,
							"Admin",
							CONCAT(c.first_name, " ", c.last_name)
						) as full_name,
						br.checkin,
						br.checkout,
						br.room_numbers,
						r.room_type
					FROM '.TABLE_BOOKINGS.' b
						INNER JOIN '.TABLE_BOOKINGS_ROOMS.' br ON b.booking_number = br.booking_number
						INNER JOIN '.TABLE_ROOMS.' r ON br.room_id = r.id
						LEFT OUTER JOIN '.TABLE_CUSTOMERS.' c ON b.customer_id = c.id
					WHERE 1=1
						'.$where_clause;
			$result = database_query($sql, DATA_AND_ROWS, ALL_ROWS);
			if($result[1] <= 0) $msg = draw_important_message(_NO_RECORDS_FOUND, false);
		}
	}

	// Start main content
	draw_title_bar(
		prepare_breadcrumbs(array(_BOOKINGS=>'',_INFO_AND_STATISTICS=>'',_REPORTS=>'')),
		(($task == 'prepare_report' && $msg == '') ? ' '._PRINT.'' : '')
	);

	//if($objSession->IsMessage('notice')) echo $objSession->GetMessage('notice');
	echo $msg;

	draw_content_start();

	echo '<link type="text/css" rel="stylesheet" href="modules/jscalendar/skins/aqua/theme.css" />

    ';

    echo '

		<form name="frmReport" action="index.php?admin=mod_booking_reports" method="post">
			'.draw_hidden_field('task', 'prepare_report', false).'
			'.draw_token_field(false).'

';
				if($hotels_count > 1 && count($arr_hotels) > 0){
					echo '';
					echo ''._HOTEL.':';
					echo '<select style="font-size:12px" name="sel_hotel">';
					echo '<option value="">-- '._ALL.' --</option>';
					foreach($arr_hotels as $key => $val){
						echo '<option value="'.$key.'" '.(($sel_hotel == $key) ? 'selected="selected"' : '').'>'.$val.'</option>';
					}
					echo '</select>';
					echo '	';
				}

			echo '

					'._TYPE.':
					<select style="font-size:12px" name="sel_type">
						<option value="">-- '._SELECT.' --</option>
						<option value="arriving" '.(($sel_type == 'arriving') ? 'selected="selected"' : '').'>'._PEOPLE_ARRIVING.'</option>
						<option value="departing" '.(($sel_type == 'departing') ? 'selected="selected"' : '').'>'._PEOPLE_DEPARTING.'</option>
						<option value="staying" '.(($sel_type == 'staying') ? 'selected="selected"' : '').'>'._PEOPLE_STAYING.'</option>
					</select>

					'._FROM.':
					<input type="textbox" size="9" readonly="readonly" name="date_from" id="date_from" value="'.$date_from.'" />

					'._TO.':
					<input type="textbox" size="9" readonly="readonly" name="date_to" id="date_to" value="'.$date_to.'" />

					'._ROWS.':
					<select style="font-size:12px" onchange="setNumberOfPages(this.value)">
						<option value="">'._ALL.'</option>
						<option value="3">3</option>
						<option value="5">5</option>
						<option value="10">10</option>
						<option value="15">15</option>
						<option selected="selected" value="20">20</option>
						<option value="50">50</option>
						<option value="100">100</option>
					</select>

					'.(($task == 'prepare_report') ? '<input class="mgrid_button" type="button" name="btnReset" value="'._RESET.'" onclick="javascript:appGoTo(\'admin=mod_booking_reports\')" />' : '').'
					<input class="mgrid_button" type="submit" name="btnSubmit" value="'._SEARCH.'" />

	    </form>

	';

	draw_content_end();
}else{
	draw_title_bar(_ADMIN);
	draw_important_message(_NOT_AUTHORIZED);
}

?>

redpajama_set_name: RedPajamaGithub


Carrageenan is a common food additive that is extracted from a red seaweed, Chondrus crispus, which is popularly known as Irish moss. Carrageenan, which has no nutritional value, has been used as a thickener and emulsifier to improve the texture of ice cream, yogurt, cottage cheese, soy milk, and other processed foods.
All told, as far as carrageenan safety goes, I recommend avoiding regular consumption of foods containing carrageenan. This is especially valuable advice for persons with inflammatory bowel disease. (Image from: https://thetruthaboutcancer.com/almond-milk-cancer-causing-ingredient/).
Post by, Andrew Weil, M.D.


text:
Chernobyl villages are the most secret and remote places of the Chernobyl zone. They are difficult for tourists. In this category we will show you the most unique photos from the villages of Chernobyl where the things of the Soviet Union!
The well saved abandoned house in the Chernobyl zone
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What is the abandoned Korogod village ? Korogod village was founded in the 17th century. It is located 15 km. from the Chernobyl NPP. During the 1986 accident,...
Abandoned villages in Chernobyl – beautiful photo
Abandoned villages in Chernobyl - beautiful photo Abandoned villages in Chernobyl - beautiful photo. These are the most beautiful places in the Chernobyl zone....
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Zimovishche – an abandoned village in the Chernobyl zone
Zimovishche - an abandoned village in the Chernobyl zone on the Left Bank of the Pripyat River Zimovishche - an abandoned village. It is located in the Chernob...
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The terrible buried village of Chernobyl – Kopachi
The terrible buried village of Chernobyl - Kopachi is the legendary The terrible buried village of Chernobyl very famous and has a sad fate. Kopachi is an a...
Lisove – an unknown village in the Red Forest of Chernobyl
As we found this Lisove this summer - an unknown village in the Red Forest of Chernobyl A new discovery for us was a village in the Red Forest of Chernobyl. Th...
Chernobyl abandoned village Rassoha
Rassokha in Chernobyl is not only a cemetery of technology, but a village Everybody heard about the cemetery of radioactive technology in the Chernobyl zone: "...
Chernobyl Church in the Krasno
Church in Krasno - one of the most enchanting places in Chernobyl One of the most beautiful buildings in the Chernobyl zone. Church of the Archangel Michael....
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The most preserved village in Chernobyl – Mashevo
Fantastic village Mashevo - a living Soviet Union in Chernobyl Villages located on the left bank of the Pripyat River in the Exclusion Zone suffered at the h...
Evacuation of Pripyat. How it was
26.04.1986 8.00-9.00 Request of the director of the Chernobyl NPP to evacuate the population from Pripyat from the Chairman of the Government Commission. The...


Bring your family and friends for an exclusive behind the scenes tour of this stunning new, state-of-the-art Stadium.
You'll be treated to breathtaking views of the playing surface, the Swan River and Perth city from the Stadium's highest vantage point, and also experience the inner sanctum. Along the way you will discover sports memorabilia detailing WA's rich sporting history, and don't forget your camera because there will be plenty of photo opportunities around the Stadium.
Check in at the rebel Fan Gear Store (next to Gate D) 10 minutes prior to your tour time.
Wear comfortable walking shoes, bring your own water, selfie stick and a fun attitude.
Enjoy a coffee or café favourite at City View, before or after your Stadium Tour. Open 7.00am-3.00pm, Monday to Sunday.
Parking: Parking is available within the Marlee Loop car park, accessed via Victoria Park Drive, and the Bus Station and Camfield car parks, accessed via Roger Mackay Drive. Please note, parking is only available on non major event days.
ACROD Parking: ACROD parking is available within the Marlee Loop car park, accessed via Victoria Park Drive.
Walk or Cycle: There are a range of public access paths to Optus Stadium and over 600 bike racks are located around the Stadium Park.
Taxi: A dedicated taxi rank is located adjacent to Victoria Park Drive.
Private Buses & Coaches: For large groups travelling by bus there is a dedicated bus drop off area in P1 Marlee Loop. Bus parking is on Roger Mackay Drive.
Ferry: The Little Ferry Co. stops daily at the Burswood Jetty.
Bus: The Hop On and Hop Off bus stops daily in Marlee Loop.


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This gorgeous doll is the compliment to Capucine. She has dark brown hair and was an FAO Exclusive. This doll is very hard to find. I bought her from someone who had taken the earrings out of her ears so they didn't discolor the face of the doll; and they are in a plastic baggie.
I never displayed this doll, but I listed her as "used" because her stole is not secured by threads in the box and I cannot be sure if the previous owner removed her from the box and replaced her. So to be safe, the honest thing is to do is list her as Mint in Box. She displays beautifully in her box. There is no tissue on the inside of the box. Here's your chance to get an extremely rare doll for your collection! The item "Chataine Silkstone Barbie MIB -WithSHIP- B4425 FAO Excl EXTREMELY RARE HTF" is in sale since Tuesday, January 1, 2019.
This item is in the category "Dolls & Bears\Dolls\Barbie Contemporary (1973-Now)\Barbie Dolls\Silkstone, Fashion Models". The seller is "crazy4barbie" and is located in Grand Forks, North Dakota. This item can be shipped to United States, Canada, United Kingdom, Denmark, Romania, Slovakia, Bulgaria, Czech republic, Finland, Hungary, Latvia, Lithuania, Malta, Estonia, Australia, Greece, Portugal, Cyprus, Slovenia, Japan, China, Sweden, South Korea, Indonesia, Taiwan, Thailand, Belgium, France, Hong Kong, Ireland, Netherlands, Poland, Spain, Italy, Germany, Austria, Bahamas, Israel, Mexico, New Zealand, Philippines, Singapore, Switzerland, Norway, Saudi arabia, Ukraine, United arab emirates, Qatar, Kuwait, Bahrain, Croatia, Malaysia, Chile, Colombia, Costa rica, Panama, Trinidad and tobago, Guatemala, Honduras, Jamaica, Antigua and barbuda, Aruba, Belize, Dominica, Grenada, Saint kitts and nevis, Saint lucia, Montserrat, Turks and caicos islands, Barbados, Bangladesh, Bermuda, Brunei darussalam, Bolivia, Egypt, French guiana, Guernsey, Gibraltar, Guadeloupe, Iceland, Jersey, Jordan, Cambodia, Cayman islands, Liechtenstein, Sri lanka, Luxembourg, Monaco, Macao, Martinique, Maldives, Nicaragua, Oman, Pakistan, Paraguay, Reunion, Uruguay.


Recently our cousin, Chris, in New York, kindly sent me a copy of the December 1916 edition of the National Geographic magazine. It is devoted entirely to Australia and provides a unique view of us from the perspective an American writing from a century ago.
If you have wondered what 200,000 bags of wheat look like, then here you are!
Australia is the most isolated of all inhabited continents and is remote from the centre of all the world's activities.
The aboriginal population is the lowest in the scale of beings having human form.
Curiously enough, the establishment of the first colony on the new-found continent is an episode in the history of the US. It was proposed by the British Government to utilize the land as a home for the "Loyalists" (Tories) who found life in the American Colonies uncomfortable at the close of the Revolutionary War. They were to be supplied with land and money, Malay slaves or English convicts were to be provided as laborers.
Fear of the French fleet and the removal of many Tories to Canada led to the abandonment of this scheme. ..Place must be found for undesirable citizens, who, before the Revolution, had been sent to America ….
Australia's White Australia policy was introduced to exclude Chinese, Japanese, and Polynesian labour.
On the basis of doing a moderate amount of work amidst agreeable surroundings, most Australians are workers. Short hours are the rule, and there is a tendency to ward off competition by legislative enactments rather than to meet and overcome it. The desire for money is in most cases a desire to secure a competence, not to secure power and prestige by amassing a huge bank account.
(Then he describes our Primary producers of sheep, cattle, and cotton. Our rabbit proof fences are also highlighted.
Even in 1916 a US Company was quick to recognise a marketing opportunity!


Is exposing the default plesk page dangerous?
Is exposing the default plesk page on an IP dangerous? I can imagine it is somehow usable for fingerprinting a version maybe. But is it dangerous anyhow?
Well, it tells people you don't really know what you're doing, in that it reveals that you use Plesk (or any web hosting panel in general).
At the same time, I suspect it would be almost trivial to suss out that information anyway, with or without the presence of that parking page. So, I'd say that the dangerous thing here is using a web control panel at all, as opposed to understanding and properly administrating your web server.
Based on the image you provide, it doesn't show any version information. It merely tells the user that there is no website at that address.
Note that it is possible to change the default website templates in Plesk. Consult its manual for more details.
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text:
THE TURING TEST FALLS
Robots can now officially imitate humans
Image: AP
Kabir Chibber
A computer that has convinced humans it is a 13-year-old Ukrainian boy has potentially passed a benchmark for artificial intelligence for the first time.
The programme, named "Eugene Goostman" and created by Russian developers, managed to convince 33% of the judges that it was human at an event at the Royal Society in London, The Independent reported. The so-called Turing Test states that a computer should be judged as "thinking" if it can convince more than 30% of those interacting that it is human. "Eugene" was not that excited by its success. "I feel about beating the turing test in quite convenient way," it replied. "Nothing original."
The test was derived from Alan Turing, the brilliant British mathematician and computer pioneer who died 60 years ago yesterday. Turing didn't actually name the test after himself, but after a Victorian parlor trick called the imitation game. He said in 1950:
I believe that in about fifty years' time it will be possible to programme computers… to make them play the imitation game so well that an average interrogator will not have more than 70 percent chance of making the right identification after five minutes of questioning.
While this didn't happen by the year 2000, it seems Turing was off by only 14 years. His test brings to mind the Voight-Kampff Test from the seminal film Blade Runner, used to distinguish "replicants" (humanoid robots) from real people by asking them a serious of awkward questions designed to provoke an emotional response. The scene in the Philip K. Dick-penned short story that inspired the film was reportedly inspired by Turing's thoughts.
Separately, the Chicago Police Department made its first arrest using facial recognition software. After committing a robbery at gunpoint, Pierre D. Martin was identified by computer software scanning the police's 4.5 million criminal booking shots, the Chicago Sun-Times reported. Chicago police used NeoFace, software developed by Australia's NEC, which was acquired by the police department using a $5.4 million federal grant.
As humanity hurtles towards its robot-controlled future, try speaking to "Eugene," the creepy Turing test-passing bot, here.


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Dear Blanca have released their new song "Out of View" (listen to it below). The track is taken from the band's upcoming sophomore album To Tell A Half-Truth, being released on September 16th.
2016 has been a busy year for the band. Between opening gigs for Bernie Sanders and Delta Spirit's Matthew Logan Vasquez, Dear Blanca has embarked on two national tours, traded Instagram shout-outs with Killer Mike, filmed a pair of live sessions with BalconyTV Atlanta and recently appeared at the Secret Stages Festival in Birmingham, Alabama. This all follows the release of March's widely-praised EP, I Don't Mean to Dwell – recorded with NC producer Scott Solter (St. Vincent, the Mountain Goats).


Battersea Dogs Home is devoted to finding dogs loving and permanent new homes.
A dedicated team interviews everyone who wants to offer a home to a dog and their unique computer matching system helps them find the perfect match.
The average stay for a dog at Battersea Dogs Home is 23 days, but some dogs end up staying a bit longer through no fault of their own. Sometimes a dog needs an extra-special home, with experience of their breed, temperament, or they may be unsuitable with children or other animals.
Whatever the reason, the dedicated staff here at the Dogs Home take care of them until the right owner comes along.