text:
\section{Introduction}

The Busemann-Petty problem was completely solved in the late 90's as a result of a series of papers of many mathematicians ([LR], [Ba], [Gi], [Bo], [Lu], [Pa], [Ga], [Zh1], [K1], [K2], [Zh2], [GKS]; see [K5, p.3] for the history of the solution).  The problem asks the following:

\noindent
Suppose $K$ and $L$ are two origin symmetric convex bodies in $\mathbb{R}^n$ such that for every $\xi \in S^{n-1},$

$$\mbox{\rm Vol}_{n-1}\bigl(K\cap \xi^{\perp}\bigr) \leq \mbox{\rm Vol}_{n-1}\bigl(L\cap \xi^{\perp}\bigr).$$
Does it follow that
$$\mbox{\rm Vol}_{n}\bigl(K\bigr) \leq \mbox{\rm Vol}_{n}\bigl(L \bigr) \ \ ?$$

The problem has an affirmative answer only if $n \leq 4.$ Since the answer is negative in most dimensions, it is natural to ask what conditions on the $(n-1)$-dimensional volumes of central sections do allow to compare the $n$-dimensional volumes.
Such conditions were found in [KYY]. The result is as
follows.

\noindent
For an origin symmetric convex body $K$ in $\mathbb{R}^n$ define the section function $$S_K(\xi)=\mbox{\rm Vol}_{n-1}(K\cap \xi^{\perp}), \ \xi \in S^{n-1}.$$
Suppose $K$ and $L$ are origin symmetric convex smooth bodies  in $\mathbb{R}^n$ and $\alpha \in \mathbb{R}$ with $ \alpha \geq n-4.$ Then, the inequality
$$\bigl(-\Delta \bigr)^{\alpha/2}S_{K}(\xi)\leq \bigl(-\Delta \bigr)^{\alpha/2}S_{L}(\xi), \ \xi \in S^{n-1}$$
implies that $\mbox{\rm Vol}_{n}(K)\leq \mbox{\rm Vol}_{n}(L).$ If $\alpha <n-4$ this is not necessarily true. Here, $\Delta$ is the Laplace operator on $\mathbb{R}^n.$

In this article we study the complex version of this problem.
 For $\xi \in \mathbb{C}^n, \ |\xi|=1$ we denote by
$$H_{\xi}=\{z\in \mathbb{C}^n \ : \ (z,\xi)=\sum\limits_{k=1}^{n}z_k \overline{\xi}_k=0\}$$

\noindent
the complex hyperplane perpendicular to $\xi.$

Origin symmetric convex bodies in $\mathbb{C}^n$ are the unit balls of norms on $\mathbb{C}^n.$ We denote by $\|\cdot\|_K$ the norm corresponding to the body $K$
$$K=\{z \in \mathbb{C}^n \ : \|z\|_{K}\leq 1\}.$$

\noindent
We identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ using the mapping
$$\xi=(\xi_1,\ldots ,\xi_n)=(\xi_{11}+i\xi_{12},\ldots ,\xi_{n1}+i\xi_{n2})\longmapsto (\xi_{11},\xi_{12},\ldots, \xi_{n1},\xi_{n2})$$
\noindent
and observe that under this mapping the complex hyperplane $H_{\xi}$ turns into a $(2n-2)$-dimensional subspace of $\mathbb{R}^{2n}$ orthogonal to the vectors

   \begin{center}
   $\xi=(\xi_{11},\xi_{12},\ldots, \xi_{n1},\xi_{n2})$   and   $\xi^{\perp}=(-\xi_{12},\xi_{11},\ldots, -\xi_{n2},\xi_{n1}).$
   \end{center}

\noindent
Since norms on $\mathbb{C}^n$ satisfy the equality
$$\|\lambda z\|=|\lambda|\|z\|, \ \ \forall z \in \mathbb{C}^n, \ \forall \lambda \in \mathbb{C}^n,$$
origin symmetric complex convex bodies correspond to those origin symmetric convex bodies $K$ in $\mathbb{R}^{2n}$ that are invariant with respect to any coordinate-wise two-dimensional rotation, namely for each $\theta \in [0,2\pi]$ and each $x=(x_{11},x_{12},\ldots, x_{n1},x_{n2}) \in \mathbb{R}^{2n}$
\begin{equation}
\|x\|_{K}=\|R_{\theta}(x_{11},x_{12}), \ldots, R_{\theta}(x_{n1},x_{n2})\|_{K},
\end{equation}

\noindent
where $R_{\theta}$ stands for the counterclockwise rotation of $\mathbb{R}^2$ by the angle $\theta$ with respect to the origin. If a convex body satisfies $(1)$ we will say that \emph{it is invariant with respect to all $R_{\theta}$}.

\medskip

The complex Busemann-Petty problem ([KKZ]) can now be formulated as follows:
Suppose $K$ and $L$ are origin symmetric invariant with respect to all $R_{\theta}$
 convex bodies in $\mathbb{R}^{2n}$ such that
 $$\mbox{\rm Vol}_{2n-2}(K\cap H_\xi)\leq \mbox{\rm Vol}_{2n-2}(L\cap H_\xi)$$
 for each $\xi$ from the unit sphere $S^{2n-1}$ of $\mathbb{R}^{2n}.$ Does it follow that
 $$\mbox{\rm Vol}_{2n}(K) \leq \mbox{\rm Vol}_{2n}(L) \  ?$$

As it is proved in [KKZ], the answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ In this article our aim is to extend the result from [KYY] to the complex case.

 \medskip

Let $D$ be an origin symmetric convex body in $\mathbb{C}^n.$ For every $\xi \in \mathbb{C}^n, \ |\xi |=1,$ we define the section function
\begin{equation}\label{eqt:defs}S_{CD}(\xi)=\mbox{\rm Vol}_{2n-2}(D\cap H_{\xi}), \ \forall \xi \in S^{2n-1}.\end{equation}

\noindent
Extending $S_{CD}$ to the whole $\mathbb{R}^{2n}$ as a homogeneous function of degree $-2$ we prove the following:

\begin{main}
Suppose $K$ and $L$ are two origin symmetric invariant with respect to all $R_{\theta}$ convex bodies in $\mathbb{R}^{2n}.$ Suppose that $\alpha \in [2n-6, 2n-2), \ n\geq 3.$ If
\begin{equation}\label{eqt:mSineq}\bigl(-\Delta \bigr)^{\alpha/2}S_{CK}(\xi)\leq \bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\xi),\end{equation}
for every $\xi \in S^{2n-1}.$ Then

$$\mbox{\rm Vol}_{2n}(K)\leq \mbox{\rm Vol}_{2n}(L).$$
If $\alpha \in (2n-7,2n-6)$ then one can construct two convex bodies $K$ and $L$ that satisfy (\ref{eqt:mSineq}), but $\mbox{\rm Vol}_{2n}(K) > \mbox{\rm Vol}_{2n}(L).$

\end{main}

This means that one needs to differentiate the section functions at least $2n-6$ times and compare the derivatives in order to obtain the same inequality for the volume of the original bodies. Note that if $\alpha=0$ the problem coincides with the original complex Busemann-Petty problem.

\medskip

\section{The Fourier analytic approach}

Throughout this paper we use the Fourier transform of distributions. The Schwartz class of the rapidly decreasing infinitely differentiable functions (test functions) in $\mathbb{R}^n$ is denoted by $\mathcal S(\mathbb{R}^n),$ and the space of distributions over $\mathcal S(\mathbb{R}^n)$ by $\mathcal S^{\prime}(\mathbb{R}^n).$ The Fourier transform $\hat{f}$ of a distribution $f \in \mathcal S^{\prime}(\mathbb{R}^n)$ is defined by $\langle\hat{f},\phi\rangle=\langle f,\hat{\phi}\rangle$ for every test function $\phi.$ A distribution is called even homogeneous of degree $p \in \mathbb{R}$ if $\langle f(x),\phi(x/\alpha)\rangle=|\alpha|^{n+p}\langle f,\phi\rangle$ for every test function $\phi$ and every $\alpha \in \mathbb{R}, \ \alpha\neq 0.$ The Fourier transform of an even homogeneous distribution of degree $p$ is an even homogeneous distribution of degree $-n-p.$ A distribution $f$ is called positive definite if, for every test function $\phi, \ \langle f, \phi\ast\overline{\phi(-x)}\rangle \geq 0.$ By Schwartz's generalization of Bochner's theorem, this is equivalent to $\hat{f}$ being a positive distribution in the sense that  $\langle\hat{f},\phi\rangle \geq 0$ for every non-negative test function $\phi.$

We denote by $\Delta$ the Laplace operator on $\mathbb{R}^n,$ and by $|\cdot|_2$ the Euclidean norm in the proper space. Then the fractional powers of the Laplacian are defined by
\begin{equation}\label{eqt:laplacef}
\Bigl(\bigl(-\Delta \bigr)^{\alpha/2}f\Bigr)^{\wedge}=\frac{1}{(2\pi)^n}|x|_2^{\alpha}\hat{f}(x),\end{equation}

\noindent
where the Fourier transform is considered in the sense of distributions.

 A compact set $K \subset\mathbb{R}^n$ is called a star body, if every straight line that passes through the origin crosses the boundary of the set at exactly two points and the boundary of $K$ is continuous in the sense that the \emph{Minkowski functional} of $K,$ defined by
$$\|x\|_K=\min \{\alpha \geq 0 : x \in \alpha K \}$$
is a continuous function on $\mathbb{R}^n.$
Using polar coordinates it is possible to obtain the following \emph{polar formula of the volume} of the body:

$$\mbox{\rm Vol}_n(K)=\int_{\mathbb{R}^n}\chi (\|x\|_K)dx=\frac{1}{n}\int_{S^{n-1}}\|\theta\|_K^{-n}d\theta.$$

A star body $K$ in $\mathbb{R}^n$ is called $k$-smooth (infinitely sooth) if the restriction of $\|x\|_{K}$ to the sphere $S^{n-1}$ belongs to the class of $C^k(S^{n-1} )\  (C^{\infty}(S^{n-1})).$ It is well-known that one can approximate any convex body in $\mathbb{R}^n$ in the radial metric,
$d(K, L)=\sup \{|\rho_{K}(\xi)-\rho_{L}(\xi)|,\ \xi \in S^{n-1} \},$
by a sequence of infinitely smooth convex bodies. The proof is based on a simple convolution argument (see for example [Sch, Theorem 3.3.1]). It is also easy to see that any convex body in $\mathbb{R}^{2n}$ invariant with respect to all $R_{\theta}$ rotations can be approximated in the radial metric by a sequence of infinitely smooth convex bodies invariant with respect to all $R_{\theta}.$ This follows from the same convolution argument, because invariance with respect to $R_{\theta}$ is preserved under convolutions.
This approximation argument allows us to consider only infinitely smooth origin symmetric convex bodies for the solution to the problem.

If $D$ is an infinitely smooth origin symmetric star body in $\mathbb{R}^n$ and $0<k <n,$ then the Fourier transform of the distribution $\|x\|_D^{-k}$ is a homogeneous function of degree $-n+k$ on $\mathbb{R}^n,$ whose restriction to the sphere is infinitely smooth (see [K5, Lemma 3.16]).

We use a spherical version of Parseval's identity, established in [K3]
(see also [K5, Lemma 3.22]).

 \begin{prop}\label{prop:parseval}
 Let $K$ and $L$ be two infinitely smooth origin symmetric convex bodies in $\mathbb{R}^n$ and $0<p<n.$ Then

 $$\int_{S^{n-1}}\bigl(\|x\|_K^{-p}\bigr)^{\wedge}(\xi)\bigl(\|x\|_L^{-n+p}\bigr)^{\wedge}(\xi)d\xi
 =(2\pi)^n\int_{S^{n-1}}\|x\|_K^{-p}\|x\|_L^{-n+p}dx.$$
\end{prop}

\bigskip

\medskip

Let $H$ be an $(2n-2)-$dimensional subspace of $\mathbb{R}^{2n}$ and $p\leq 2n-2.$  We fix an orthonormal basis $,\{e_1,e_2\},$ in the orthogonal subspace $H^{\perp}.$ For any convex body $D$ in $\mathbb{R}^{2n}$ we define the function $A_{D,H,p}$ as a function on $\mathbb{R}^2$ such that

\begin{equation}\label{def:A}A_{D,H,p}(u)=
\int_{D\cap H_u}|x|_2^{-p} dx, \ u\in \mathbb{R}^2, \end{equation}
where $H_u=\{x\in \mathbb{R}^{2n}:(x,e_1)=u_1,(x,e_2)=u_2\}.$

If the body $D$ is infinitely smooth and $0\leq p <2n-m-2,$ then $A_{D,H,p}$ is $m$-times continuously differentiable near the origin. This can be seen from an argument similar to [K5, Lemma 2.5].

In addition, if we consider the action of the distribution $|u|_2^{-q-2}/ \Gamma(-q/2)$ on $A_{D,H,p}$ we may apply a standard regularization argument (see [GS, p.71-74]) and define the function
\begin{equation}\label{eqt:q-A}
q \longmapsto \left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})},A_{D,H,p}(u)\right>.
\end{equation}

For $q \in \mathbb{C}$ with $ \mbox{\rm Re} q \leq 2n-p-3,$ the function is an analytic function of $q.$
If $q<0$
\begin{equation}\label{eqt:q<0}\left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})},A_{D,H,p}(u)\right>=
\frac{1}{\Gamma(-q/2)}\int_{\mathbb{R}^2}|u|_2^{-q-2} A_{D,H,p}(u)du.\end{equation}

If $q=2d, \ d \in \mathbb{N}\cup \{0\},$ then
$$\left<\frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}\Big|_{q=2d}, A_{D,H,p}(u)\right>$$
\begin{equation}\label{eqt:q=2d}=\frac{(-1)^d \pi}{2^{2d}d!}\Delta^d A_{f,D,H}(0),
\end{equation}
where $\Delta=\sum_{i=1}^2 \partial^2/\partial u_i^2$ is the $2$-dimensional Laplace operator (see [GS, p.71-74]).
Note that the function (\ref{eqt:q-A}) is equal, up to a constant, to the fractional power of $\Delta^{q/2}A_{D,H,p}.$ (see [KKZ, p.6-7] or [K4, p.6-7] for complete definition).

\noindent
If the body $D$ is origin symmetric the function $A_{D,H,p}$ is even and for $0< q <2$ we have  (see also [K5, p.39])
$$\left<\frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right>$$
\begin{equation}\label{eqt:0q2}
=\frac{1}{\Gamma(-\frac{q}{2})}\int_0^{2\pi}
\Bigl(\int_0^{\infty}\frac{A_{D,H,p}(t\theta)-A_{D,H,p}(0)}{t^{1+q}}dt\Bigr)d\theta.
\end{equation}

\medskip

The following proposition is a generalization of [K4], (see also [KKZ, Proposition 4]) with $k=2$. We prove it using a well-known
formula (see for example [GS, p.76]): for any $v\in \mathbb{R}^2$
and $q< -1,$
\begin{equation} \label{sph}
(v_1^2+v_2^2)^{\frac{-q-2}{2}}= {{\Gamma(-q/2)}\over {2\Gamma((-q-1)/2) \pi^{1/2}}}
\int_{0}^{2\pi} |(v,u)|^{-q-2}\ du.
\end{equation}

\begin{prop}\label{prop:A}
Let $D$ be an infinitely smooth origin symmetric convex body in $\mathbb{R}^{2n}.$ If $-2<q<2n-2, \ 0\leq p \leq 2n-q-3.$ Then for every $(2n-2)$-dimensional subspace $H$ of $\mathbb{R}^{2n}$

$$\left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right>$$
\begin{equation}\label{eqt:A^{q}}=\frac{2^{-q-2}}{\pi\Gamma\bigl(\frac{q+2}{2}\bigr)(2n-q-p-2)}
\int\limits_{S^{2n-1}\cap H^{\perp}}
\Bigl(\|x\|_D^{-2n+q+p+2}|x|_2^{-p}
\Bigr)^{\wedge}(\theta)d\theta.
\end{equation}
Also, for every $ d \in \mathbb{N} \cup {0}, \ d <n-1$
\begin{equation}\label{eqt:d-A}
\Delta^d A_{D,H,p}(0)=\frac{(-1)^d}{8\pi^2(n-d-1)}\int\limits_{S^{2n-1}\cap H^{\perp}}
\Bigl(\|x\|_D^{-2n+2d+p+2}|x|_2^{-p}
\Bigr)^{\wedge}(\eta)d\eta.
\end{equation}

\end{prop}

\medskip

\textbf{Proof.}
First we assume that $q\in (-2,-1).$ Then
$$\left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right>
=\frac{1}{\Gamma(-q/2)}\int_{\mathbb{R}^2}|u|_2^{-q-2} A_{D,H,p}(u)du$$
Using the expression (\ref{def:A}) for the function $A_{D,H,p}$, writing
the integral in polar coordinates and then using (\ref{sph}), we
see that the right-hand side of the latter equation is equal to
$${1\over {\Gamma(\frac{-q}{2})}} \int_{\mathbb{R}^n} \big((x,e_1)^2+(x,e_2)^2)^{\frac{-q-2}{2}}  |x|_2^{-p}\chi(\|x\|_D)\ dx$$
$$={1\over {\Gamma(\frac{-q}{2})(n-q-p-2)}} \int_{S^{n-1}} \left((\theta,e_1)^2+(\theta,e_2)^2 \right)^{\frac{-q-2}{2}} \|\theta\|_D^{-n+q+2}\ d\theta$$
$$={1\over {2\Gamma(\frac{-q-1}{2})\pi^{\frac{1}{2}}(n-q-p-2)}}\ \times$$
$$\int_{S^{n-1}} \|\theta\|_D^{-n+q+p+2} \left(
\int_{0}^{2\pi} \big| ( u_1e_1+u_1e_2, \theta) \big|^{-q-2}\ du \right)\ d\theta$$
$$={1\over {2\Gamma(\frac{-q-1}{2})\pi^{\frac{1}{2}}(n-q-p-2)}}\times$$
\begin{equation} \label{comput}
\int_0^{2\pi} \left(
\int_{S^{n-1}} \|\theta\|_D^{-n+q+p+2} \big| ( u_1e_1+u_2e_2, \theta)
\big|^{-q-2}\ d\theta\right) du.
\end{equation}

Let us show that the function under the integral over $[0,2\pi]$
is the Fourier transform of $\|x\|_D^{-n+q+p+2}|x|_2^{-p}$ at the point
$ u_1e_1+u_2e_2$. For any even test function $\phi\in {\mathcal S}(\mathbb{R}^n),$
using the well-known connection between the Fourier and Radon transforms
(see [K5, p.27]) and the
expression for the Fourier transform of the distribution
$|z|_2^{q-1}$ (see [K5, p.38]), we get

$$\langle (\|x\|_D^{-n+q+p+2}|x|_2^{-p})^\wedge, \phi \rangle =
\int_{\mathbb{R}^n} \|x\|_D^{-n+q+p+2}|x|_2^{-p} \hat\phi(x)\ dx $$
$$=\int_{S^{n-1}} \|\theta\|_D^{-n+q+p+2}\left(
\int_0^\infty r^{q+1} \hat\phi(r\theta)\ dr\right) d\theta$$
$$={1\over 2} \int_{S^{n-1}} \|\theta\|_D^{-n+q+p+2}
\Big\langle |r|^{q+1}, \hat\phi(r\theta) \Big\rangle \ d\theta$$
$$={{2^{q+2}\sqrt{\pi}\ \Gamma((q+2)/2)}\over
{2\Gamma((-q-1)/2)}}
\int_{S^{n-1}} \|\theta\|_D^{-n+q+p+2}
\Big\langle |t|^{-q-2},
\int_{(y,\theta)=t} \phi(y)\ dy \Big\rangle \ d\theta$$
$$= {{2^{q+1}\sqrt{\pi} \Gamma((q+2)/2)}\over
{2\Gamma((-q-1)/2)}} \int_{\mathbb{R}^n} \Big(\int_{S^{n-1}}
|(\theta, y)|^{-q-2} \|\theta\|_D^{-n+q+p+2}\ d\theta \Big) \phi(y)\ dy.$$
Since $\phi$ is an arbitrary test function, this
proves that, for every $y\in \mathbb{R}^n\setminus \{0\},$
$$\big(\|x\|_D^{-n+q+p+2}|x|_2^{-p}\big)^\wedge(y)$$
$$ =
{{2^{q+2}\sqrt{\pi} \Gamma((q+2)/2)}\over
{2\Gamma((-q-1)/2)}} \int_{S^{n-1}}
|(\theta, y)|^{-q-2} \|\theta\|_D^{-n+q+p+2}\ d\theta.$$
Together with (\ref{comput}), the latter equality shows that
$$\Big\langle {{|u|_2^{-q-2}}\over {\Gamma(-q/2)}},
A_{D,H,p}(u) \Big\rangle$$
\begin{equation} \label{fracform}
 = {{2^{-q-2}\pi^{-1}}\over {\Gamma((q+2)/2)(n-q-p-2)}}
\int_{S^{n-1}\cap H^\bot} \big(\|x\|_D^{-n+q+p+2}|x|_2^{-p}\big)^\wedge(\theta)
\ d\theta,
\end{equation}
because in our notation $S^{n-1}\cap H^\bot=[0,2\pi].$

We have proved (\ref{fracform}) under the assumption that $q\in (-2,-1).$
However, both sides of (\ref{fracform})
are analytic functions of $q\in \mathbb{C}$ in the domain where
$-2<\mbox{\rm Re} q<2n-2.$
This implies that the equality (\ref{fracform}) holds for every $q$ from this domain
(see [K5, p.61] for the details of a similar argument).

Putting $q=2m,\ m\in \mathbb{N}\cup \{0\},\ m< n-1$ in (\ref{fracform}) and applying
(\ref{eqt:q=2d}) and the fact that $\Gamma(x+1)=x\Gamma(x)$, we get the second
formula.
\qed

\medskip

\bigskip

Brunn's theorem (see for example [K5, Theorem 2.3]) states that for an origin symmetric convex body and a fixed direction, the central hyperplane section has the maximal volume among all the hyperplane sections perpendicular to the given direction. As a consequence we have the following generalization proved in [KKZ, Lemma 1] for $p=0.$

\begin{prop}\label{prop:brunn}
Suppose $D$ is a $2$-smooth origin symmetric convex body in $\mathbb{R}^{2n},$ then the function $ A_{D,H,p}$ is twice differentiable at the origin and $$\Delta A_{D,H,p}(0) \leq 0.$$
Moreover, for any $q \in (0, 2)$,
$$\left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right> \geq 0.$$
\end{prop}

\medskip

\textbf{Proof.} Differentiability follows from the same argument as in [K5, Lemma 2.4].

The body $D$ is origin symmetric and convex, so, to prove the first inequality we need to observe that the function $u\longmapsto A_{D,H,p}(u), \ u\in \mathbb{R}^2,$ attains its maximum at the origin:

If $p=0$ then it follows immediately from Brunn's theorem (see [K5, Theorem 2.3] and [KKZ, Lemma 1].)

\noindent
Let $p>0.$ Since $|x|_2^{-p}=p\int_{0}^{\infty}\chi(z|x|_2)z^{q-1}dz,$ we have that for any $u\in \mathbb{R}^2$
\begin{eqnarray*}A_{D,H,p}(u)&=&\int_{D\cap H_u}|x|_2^{-p} dx=p\int_{D\cap H_u}\int_{0}^{\infty}\chi(z|x|_2)z^{q-1}dzdx \\
&=&p\int_{0}^{\infty}z^{q-1}\int_{D\cap H_u}\chi(z|x|_2)dxdz \\
&=&p\int_{0}^{\infty}z^{q-1}\int_{B(1/z)\cap H_u}\chi(\|x\|_D)dx dz,  \end{eqnarray*}

\noindent
where $B(1/z)$ is the unit ball of radius $1/z,.$ Applying Brunn's theorem to the body $B(1/z)\cap D, $ we have that the latter integral is
$$\leq p\int_{0}^{\infty}z^{q-1}\int_{ H}\chi(\|x\|_{B(1/z)\cap D})dx dz=A_{D,H,p}(0).$$

If $q\in (0,2)$ then $ \Gamma(-q/2)<0.$ Hence, for the second inequality we use (\ref{eqt:0q2}) to get that
$$\left<\frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right>$$
$$
=\frac{1}{\Gamma(-\frac{q}{2})}\int_0^{2\pi}
\Bigl(\int_0^{\infty}\frac{A_{D,H,p}(t\theta)-A_{D,H,p}(0)}{t^{1+q}}dt\Bigr)d\theta \geq 0,$$
since $A_{D,H,p}(u)\leq A_{D,H,p}(0),$ for every $u\in \mathbb{R}^2.$

 \section{Distributions of the form $|x|_2^{-\beta}\|x\|^{-\gamma}$ }\label{distr}

As in the modified real Busemann-Petty problem the solution is closely related to distributions of the form $|x|_2^{-\beta}\|x\|^{-\gamma}.$

First, we need a simple observation. The following lemma is crucial for the solution of the problem.
\medskip

\begin{lemma}\label{modconst}
 For every infinitely smooth origin symmetric invariant with respect to all $R_{\theta}$ convex body $D$ in $\mathbb{R}^{2n}$ and every $\xi \in S^{2n-1},$ the Fourier transform of the distribution $|x|_2^{-\beta}\|x\|_{D}^{-\gamma}, \ 0< \beta , \gamma<2n$ is a constant function on $S^{2n-1}\cap H_{\xi}^{\perp}.$
\end{lemma}

\noindent
\textbf{Proof.}
The proof (see [KKZ, Theorem 1], when $\beta=0$) is based on the following observation:

 \noindent
 The body $D$ is invariant with respect to all $R_{\theta}.$ So, because of the connection between the Fourier transform and linear transformations, the Fourier transform of $|x|_2^{-\beta}\|x\|_{D}^{-\gamma}$ is also invariant with respect to all $R_{\theta}.$ This implies that it is a constant function on $S^{2n-1}\cap H^{\perp}_{\xi}$ because this circle can be represented as the set of all the rotations $R_{\theta},\ \theta \in [0, 2\pi],$ of the vector $\xi \in S^{2n-1}.$

As a consequence of the above we have that
\begin{equation}\label{eqt:invariance}
\int_{S^{2n-1}\cap H_{\xi}^{\perp}}\Bigl(|x|_2^{-\beta}\|x\|_{D}^{-\gamma}\Bigr)^{\wedge}(\theta) d\theta=2\pi \Bigl(|x|_2^{-\beta}\|x\|_{D}^{-\alpha}\Bigr)^{\wedge}(\xi).
\end{equation}
\qed

\bigskip

\begin{lemma}\label{lm:posdef}
Let $D$ be an origin symmetric invariant with respect to all $R_{\theta}$ convex body in $\mathbb{R}^{2n}, n\geq 3.$ If $q \in (-2,2]$ and $0\leq p< 2n-q-3$ then $|x|_2^{-p}\|x\|_D^{-2n+p+q+2}$ is a positive definite distribution.
\end{lemma}

\medskip

\noindent
\textbf{Proof.}

\medskip

If $p=0$ then by [KKZ, Theorem 3],   $\bigl(\|x\|_D^{-2n+q+2}\bigr)^{\wedge} \geq 0,$ since $2n-q-2 \in [2n-4, 2n).$

\medskip

Let $p>0.$ If $q \in (-2,0)$ then by equation (\ref{eqt:q<0}) and Proposition \ref{prop:A} (formula (\ref{eqt:A^{q}})) we have that
$$\frac{2^{-q-2}}{\pi\Gamma\bigl(\frac{q+2}{2}\bigr)(2n-q-p-2)}
\int\limits_{S^{2n-1}\cap H^{\perp}}
\Bigl(\|x\|_D^{-2n+q+p+2}|x|_2^{-p}
\Bigr)^{\wedge}(\theta)d\theta$$
$$=\frac{1}{\Gamma(-q/2)}\int_{\mathbb{R}^2}|u|_2^{-q-2} A_{D,H,p}(u)du\geq 0.$$
By Lemma \ref{modconst}, the Fourier transform of the distribution
$|x|_2^{-p}\|x\|_{D}^{-2n+p+q+2}$ is a constant function on $S^{2n-1}\cap H^{\perp}_{\xi}.$ So,

$$\Bigl(|x|_2^{-p}\|x\|_{D}^{-2n+p+q+2}\Bigr)^{\wedge} \geq 0,$$
since $\Gamma(\frac{q+2}{2})>0, \ \Gamma(-\frac{q}{2})>0$ and $q<2n-p-2.$

\medskip

Now, if $q=0,$ (\ref{eqt:d-A}) and (\ref{eqt:invariance}) give that
$$A_{D,H,p}(0)=\frac{1}{4\pi (n-1)}\Bigl(|x|_2^{-p}\|x\|_{D}^{-2n+p+q+2}\Bigr)^{\wedge}(\xi) \geq 0.$$

\medskip

For the case where $q \in (0,2)$ we use Proposition \ref{prop:A} and the Remark to get that
$$\left< \frac{|u|_2^{-q-2}}{\Gamma(-\frac{q}{2})}, A_{D,H,p}(u)\right>$$
$$=\frac{2^{-q-1}}{\Gamma\bigl(\frac{q+2}{2}\bigr)(2n-q-p-2)} \Bigl(\|x\|_D^{-2n+q+p+2}|x|_2^{-p}
\Bigr)^{\wedge}(\xi).$$
Then, by the generalization of Brunn's theorem, Proposition \ref{prop:brunn}, the desired follows.

 Lastly, if $q=2,$ (\ref{eqt:d-A}) and (\ref{eqt:invariance}) imply that
 $$\Delta A_{D,H,p}(0)=\frac{-1}{4\pi(n-2)}
\Bigl(\|x\|_D^{-2n+p+4}|x|_2^{-p}
\Bigr)^{\wedge}(\xi).$$
Combining this with Brunn's generalization, since the Laplacian of the function $A_{D,H,p}$ at $0$ is non-positive, we have that
$$\Bigl(|x|_2^{-p}\|x\|_{D}^{-2n+p+4}\Bigr)^{\wedge}(\xi) \geq 0.$$

 \qed

 Before we prove the main result of this article we need the following:

\medskip
\begin{lemma}
 Let $D$ be an infinitely smooth origin symmetric invariant with respect to all $R_{\theta}$ convex body in $\mathbb{R}^{2n}$ and $\alpha \in \mathbb{R}.$ Then
 \begin{equation}\label{eqt:laplaces}\bigl(-\Delta \bigr)^{\alpha/2}S_{CD}(\xi)=\frac{1}{4\pi (n-1)}\bigl(|x|_2^{\alpha}\|x\|_{D}^{-2n+2}\bigr)^{\wedge}(\xi)
\end{equation}

\end{lemma}

\noindent
\textbf{Proof.} Let $\xi \in S^{2n-1}.$ As proved in [KKZ, Theorem 1], using the same idea as in Lemma \ref{modconst} (with $r=0$)
\begin{equation}\label{eqt:vol}
\mbox{\rm Vol}_{2n-2}(D\cap H_{\xi})=\frac{1}{4\pi (n-1)}\Bigl(\|x\|_{D}^{-2n+2}\Bigr)^{\wedge}(\xi).
\end{equation}

By the definition of the section function of $D,$ and equation (\ref{eqt:vol}) we obtain the following formula:
\begin{equation}S_{CD}(\xi)=\frac{1}{4\pi (n-1)}\Bigl(\|x\|_{D}^{-2n+2}\Bigr)^{\wedge}(\xi).
\end{equation}

We extend $S_{CD}$ to the whole $\mathbb{R}^{2n}$ as a homogeneous function of degree $-2$ and apply the definition of the fractional powers of the Laplacian. Then, since $\|x\|_{D}^{-2n+2}$ is an even distribution, equation (\ref{eqt:laplaces}) immediately follows.
\qed

\bigskip

                    \section{The solution of the problem.}\label{solMCBP}

\noindent
We consider the affirmative and negative part of the main result separately. The proof follows by the next two theorems.

\begin{thm}\label{thm:pos}(AFFIRMATIVE PART) Let $K$ and $L$ be two infinitely smooth origin symmetric invariant with respect to all $R_{\theta}$ convex bodies in $\mathbb{R}^{2n}.$ Suppose that $\alpha \in [2n-6, 2n-2), n\geq 3.$ Then
for every $\xi \in S^{2n-1}$
\begin{equation}\label{eqt:Sineq}\bigl(-\Delta \bigr)^{\alpha/2}S_{CK}(\xi)\leq \bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\xi)\end{equation}
implies that

$$\mbox{\rm Vol}_{n}(K)\leq \mbox{\rm Vol}_{n}(L).$$
\end{thm}

\noindent
\textbf{Proof.} The bodies $K$ and $L$ are infinitely smooth and invariant with respect to all $R_{\theta}$ convex bodies. So by equation (\ref{eqt:laplaces}) the condition in (\ref{eqt:Sineq}) can be written as
\begin{equation}\label{eqt:normineq}\Bigl(|x|_2^{\alpha}\|x\|_{K}^{-2n+2}\Bigr)^{\wedge}\leq \Bigl(|x|_2^{\alpha}\|x\|_{L}^{-2n+2}\Bigr)^{\wedge}.
\end{equation}

 We apply Lemma \ref{lm:posdef} with $p=\alpha$ and $q=2n-\alpha -4$  so that the distribution $|x|_2^{\alpha}\|x\|_{K}^{-2}$ is positive definite. By Bochner's theorem this implies that its Fourier transform is a non-negative function on $\mathbb{R}^{2n}\setminus \{0\}.$ By [K5, Lemma 3.16], it is also continuous, since $K$ is infinitely smooth. Multiply both sides in (\ref{eqt:normineq}) by$\bigl(|x|_2^{-\alpha}\|x\|_{K}^{-2}\bigr)^{\wedge}$ and integrate over the unit sphere $S^{2n-1}.$ Then we can apply Parseval's spherical version, Proposition \ref{prop:parseval}, to get that
 \begin{equation}\label{eqt:volK}
\int_{S^{2n-1}}\|x\|_K^{-2n}dx\leq \int_{S^{2n-1}}\|x\|_K^{-2}\|x\|_L^{-2n+2}.\end{equation}

 \noindent
 Then, by a simple application of H\"older's inequality on formula (\ref{eqt:volK}) and the polar formula of the bodies (see Section 2.) we obtain the affirmative answer to the problem, since
 $$2n\ \mbox{\rm Vol}_{2n}(K)\leq \Bigl(2n\ \mbox{\rm Vol}_{2n}(K)\Bigr)^{1/n}\Bigl(2n\ \mbox{\rm Vol}_{2n}(L)\Bigr)^{(n-1)/n}.$$
\qed

 \medskip

 \bigskip
To prove the negative part we need the following lemma.
 \begin{lemma}\label{lm:nonposdef}
 Let $\alpha \in (2n-7, 2n-6).$ There exists an infinitely smooth origin symmetric convex body $L$ with positive curvature, so that
 $$|x|_2^{-\alpha}\|x\|_{L}^{-2}$$
 is not a positive definite distribution.
 \end{lemma}

We postpone the proof of Lemma \ref{lm:nonposdef} until the end of this section to show that the existence of such a body gives a negative answer to the problem.

\medskip

\begin{thm}\label{thm:neg}(NEGATIVE PART)
Suppose there exists an infinitely smooth, origin symmetric convex body $L$ for which $|x|_2^{-\alpha}\|x\|_{L}^{-2}$ is not a positive definite distribution. Then one can construct an origin symmetric convex body $K$ in $\mathbb{R}^{2n}, n\geq 3,$ so that together with $L$ they satisfy (\ref{eqt:Sineq}), for every $\xi \in S^{2n-1}$ but
$$\mbox{\rm Vol}_{2n}(K)> \mbox{\rm Vol}_{2n}(L).$$
\end{thm}

\noindent
\textbf{Proof.}
The body $L$ is infinitely smooth, so by [K5, Lemma 3.16] the Fourier transform of the distribution $|x|_2^{-\alpha}\|x\|_{L}^{-2}$ is a continuous function on the unit sphere $S^{2n-1}.$ Moreover there exists an open subset $\Omega$ of $S^{2n-1}$ in which $\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}<0 .$
Since $L$ is invariant with respect to all $R_{\theta}$ we may assume that $\Omega$ is also invariant we respect to rotations $R_{\theta}$.

We use a standard perturbation procedure for convex bodies, see for example [K5, p.96] (similar argument was used in [KKZ, Lemma 5]).
 Consider a non-negative infinitely differentiable even function $g$ supported on $\Omega$ that is also invariant with respect to rotations $R_{\theta}.$ We extend it to a homogeneous function of degree $-\alpha-2$ on $\mathbb{R}^{2n}.$ By [K5, Lemma 3.16] its Fourier transform is an even homogeneous function of degree $-2n+\alpha+2$ on $\mathbb{R}^{2n},$ whose restriction to the sphere is infinitely smooth: $\bigl(g(x/|x|_2)|x|_2^{-\alpha-2}\bigr)^{\wedge}(y)=h(y/|y|_2)|y|_2^{-2n+\alpha+2},$ where $h\in C^{\infty}(S^{2n-1}).$

We define a body $K$ so that
$$\|x\|_{K}^{-2n+2}=\|x\|_{L}^{-2n+2}+\varepsilon |x|_2^{-2n+2}h\bigl(\frac{x}{|x|_2}\bigr),$$
for small enough $\varepsilon >0$ so that the body $K$ is strictly convex. Note that $K$ is also invariant with respect to all $R_{\theta}.$ We multiply both sides by $\frac{1}{4\pi(n-1)}|x|_2^{\alpha}$ and apply Fourier transform. Then
\begin{eqnarray}\label{eqn:Sc}\bigl(-\Delta \bigr)^{\alpha/2}S_{CK}(\xi)&=&\bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\xi)+
\frac{\varepsilon(2\pi)^{2n}}{4\pi(n-1)}|x|_2^{-\alpha-2}g\bigl(\frac{x}{|x|_2}\bigr) \\
&\leq & \bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\xi), \nonumber \end{eqnarray}
since $g$ is non-negative.

On the other hand, we multiply both sides of (\ref{eqn:Sc}) by $\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}$ and integrate over the sphere,

$$\int_{S^{2n-1}}\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}(\theta)
\bigl(-\Delta \bigr)^{\alpha/2}S_{CK}(\theta) d\theta$$

\begin{eqnarray*}&=&\int_{S^{2n-1}}\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}(\theta)
\bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\theta) d\theta \\
&+& \varepsilon\frac{(2\pi)^{2n}}{4\pi(n-1)}\int_{S^{2n-1}}
\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}(\theta)g(\theta)d\theta \\
&>& \int_{S^{2n-1}}\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}(\theta)
\bigl(-\Delta \bigr)^{\alpha/2}S_{CL}(\theta) d\theta,\end{eqnarray*}
since $\Bigl(|x|_2^{-\alpha}\|x\|_{L}^{-2}\Bigr)^{\wedge}<0$ on the support of $g.$
\noindent
Using equation (\ref{eqt:laplaces}) and the spherical version of Parseval's identity, the latter becomes
$$\int_{S^{2n-1}}\|x\|_L^{-2}\|x\|_K^{-2n+2} > \int_{S^{2n-1}}\|x\|_L^{-2n}dx.$$As in Theorem \ref{thm:pos} we apply H\"older's inequality and the polar representation of the volume to obtain the desired inequality for the volumes of the bodies.

\qed

 \noindent
 \textbf{Proof of Lemma \ref{lm:nonposdef}.}
 The construction of the body follows similar steps as in [KYY]. We put $q=2n-\alpha-4,$ so $q \in (2,3).$ From the definition of the fractional derivatives, Proposition \ref{prop:A} and the Remark we see that for a $\xi \in S^{2n-1}$ we need to construct a convex body $D$ so that

$$\int_0^{2\pi}\int_0^{\infty}
t^{-q-1}\Bigl(A_{D,H_{\xi},\alpha}(t\theta)-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt d\theta<0$$
since $\Gamma (-\frac{q}{2})>0$ for $q \in (2,3).$

We define the function $$f(|u|)=(1-|u|_2^2-N|u|_2^4)^{\frac{1}{2n-\alpha-2}}, \ u \in \mathbb{R}^2 $$
and consider the body $D$ in $\mathbb{R}^{2n}$ as
$$D=\Big\{(x_{11},x_{12},\ldots ,x_{n1},x_{n2})  \in \mathbb{R}^{2n} : |\bar{x}|_2=|(x_{n1},x_{n2})|_2 \in [-\alpha _{N}, \alpha _{N}],$$
$$\bigl(\sum\limits_{i=1 \atop j=1,2}^{n-1}x^2_{ij}\bigr)^{1/2}\leq f(|\bar{x}|_2)\Big\},
$$
where $a_N$ is the first positive root of the equation $f(t)=0.$
From its definition, the body $D$ is strictly convex with an infinitely smooth boundary. We choose $\xi \in S^{2n-1}$ in the direction of $\bar{x}.$ For $u \in \mathbb{R}^2$ with $|u|_2 \in [0, a_N],$ we write equation (\ref{def:A}) in polar coordinates and get that
\begin{eqnarray*}A_{D,H_{\xi},\alpha}(u)&=&\int_{S^{2n-3}_{u}}
\int_0^{f(|u|_2)}(r^2+|u|_2^2)^{-\frac{\alpha}{2}}r^{2n-3}drd\theta \\
&=&|S^{2n-3}_{u}|\int_0^{f(|u|_2)}(r^2+|u|_2^2)^{-\frac{\alpha}{2}}r^{2n-3}dr,
\end{eqnarray*}
where $|S^{2n-3}_{u}|$ is the volume of the $(2n-3)$-dimensional unit sphere. Note that if $|u|_2 > \alpha_N$ then $A_{D,H_{\xi},\alpha}(u)=0.$ Moreover, if $u=t\theta, \ t\in [0, \infty)$ and $\theta \in S^1,$ the parallel section function $A_{D,H_{\xi},\alpha}(t\theta)$ is independent of $\theta$ since
\begin{equation}\label{eqt:A_t}
A_{D,H_{\xi},\alpha}(t\theta)=|S^{2n-3}_{t}|\int_0^{f(t)}(r^2+t^2)^{-\frac{\alpha}{2}}r^{2n-3}dr.
\end{equation}
Hence, we need to prove that the above construction of the body $D$ gives that
\begin{equation}\label{eqt:countrxample}
\int_0^{\infty}
t^{-q-1}\Bigl(A_{D,H_{\xi},\alpha}(t\theta)-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt <0.
\end{equation}
Note that the condition $|u|_2 \in [0, \alpha_N]$ is now equivalent to $t \in [0, \alpha_N].$
In order to prove the above we first compute
$$A_{D,H_{\xi},\alpha}(0)=\frac{|S^{2n-3}|}{2n-\alpha-2}$$
and
$$\Delta A_{D,H_{\xi},\alpha}(0)=-|S^{2n-3}|\Big[\frac{1}{2n-\alpha-2}+\frac{\alpha}{2n-\alpha-4}\Big].$$

Let $\beta_N$ be the positive root of the equation $1-t^2-Nt^4=t^{q+1}.$ We split the integral in
(\ref{eqt:countrxample}) in three parts: $[0, \beta_N], \ [\beta_N,\alpha_N]$ and $[\alpha_N, \infty)$ and work separately. It is not difficult to see that for large $N, \ \alpha_N , \beta_N \simeq N^{-\frac{1}{4}}.$ Also, for every $t \in [0, \alpha_N], \ f(t)>0$ and $f(t)\geq t $ if and only if $t \in [0, \beta_N].$

\medskip

 For the first part, the interval $[0,\beta_N],$ since $f(t)\geq t $, the 2-dimensional parallel section function $A_{D,H_{\xi},\alpha}$ can be easily estimated if we split it into two integrals. For the second we use the inequality $(1+x)^{-\gamma}\leq 1-\gamma x +\frac{\gamma(\gamma+1)}{2}x^2,$ for $\gamma >0$ and $0<x<1.$ Then

 $$\int_0^t(r^2+t^2)^{-\frac{\alpha}{2}}r^{2n-3}dr
 \leq \int_0^tr^{-\alpha+2n-3}dr=\frac{t}{2n-\alpha-2}$$
 and
 $$\int_t^{f(t)}(r^2+t^2)^{-\frac{\alpha}{2}}r^{2n-3}dr\leq
 \int_t^{f(t)} \Bigl[1-\frac{\alpha}{2}\frac{t^2}{r^2}+\frac{\alpha(\alpha+2)}{4}
 \frac{t^4}{r^4}\Bigr]r^{2n-\alpha-3}dr$$

$$=\frac{r^{2n-\alpha-2}}{2n-\alpha-2}-\frac{\alpha}{2}t^2\frac{r^{2n-\alpha-4}}{2n-\alpha-4}
+\frac{\alpha(\alpha+2)}{4}t^4\frac{r^{2n-\alpha-6}}{2n-\alpha-6}\Bigg|_t^{f(t)}$$

$$=\frac{f^{2n-\alpha-2}(t)}{2n-\alpha-2}-\frac{\alpha}{2}t^2\frac{f^{2n-\alpha-4}(t)}{2n-\alpha-4}
+\frac{\alpha(\alpha+2)}{4}t^4\frac{f^{2n-\alpha-6}(t)}{2n-\alpha-6}
-Ct^{2n-\alpha-2},$$

\noindent
where $C=\frac{1}{2n-\alpha-2}-\frac{p}{2(2n-\alpha-4)}+\frac{\alpha(\alpha+2)}{4(2n-\alpha-6)}>0,$ since $n\geq 3$ and $\alpha \in (2n-7,2n-6).$

\noindent
We now use the definition of the function $f$ and the inequality $(1-x)^{\gamma}\geq 1-\gamma x(1-x)^{\gamma-1},$ for $0<\gamma<1$ and $0<x<1.$ We then write

$$=\frac{1-t^2-Nt^4}{2n-\alpha-2}-\frac{\alpha}{2}
\frac{t^2(1-t^2-Nt^4)^{\frac{2n-\alpha-4}{2n-\alpha-2}}}{2n-\alpha-4}$$
$$+\frac{\alpha(\alpha+2)}{4}\frac{t^4(1-t^2-Nt^4)^{\frac{2n-\alpha-6}{2n-\alpha-2}}}
{2n-\alpha-6}
-Ct^{2n-\alpha-2}$$
$$\leq \frac{1-t^2-Nt^4}{2n-\alpha-2}-\frac{\alpha t^2}{2(2n-\alpha-4)}+
\frac{\alpha t^2(t^2+Nt^4)}{2(2n-\alpha-2)(1-t^2-Nt^4)^{\frac{2}{2n-\alpha-2}}}$$
$$+
\frac{\alpha(\alpha+2)t^4}{4(2n-\alpha-6)}-
\frac{\alpha(\alpha+2)}{4(2n-\alpha-2)}\frac{t^4(t^2+Nt^4)}{(1-t^2-Nt^4)^{\frac{4}{2n-\alpha-2}}}-Ct^{2n-\alpha-2}.$$

Hence, we have that
$$\int_0^{\beta_N}
t^{-q-1}\Bigl(A_{D,H_{\xi},\alpha}(t\theta)-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt$$

$$=\int_0^{\beta_N}
t^{-q-1}\Bigl(Ct^{2n-\alpha-2}-Dt^4+E\frac{t^2(t^2+Nt^4)}{(1-t^2-Nt^4)^{\frac{2}{2n-\alpha-2}}}$$
\begin{equation}\label{betaN}
-F\frac{t^4(t^2+Nt^4)}{(1-t^2-Nt^4)^{\frac{4}{2n-\alpha-2}}}\Bigr)dt, \end{equation}

where  $E=\frac{\alpha}{2(2n-\alpha-2)}>0,$
$F=\frac{\alpha(\alpha+2)}{2n-\alpha-2}>0$ and $D=\frac{N}{2n-\alpha-2}-\frac{\alpha(\alpha+2)}{4(2n-\alpha-6)}>0, $ for $N$ large enough.

Now, in order to obtain an upper bound for (\ref{betaN}) we need to estimate four different integrals. The first one simply gives $\frac{C}{2}\beta_N^2\simeq C_1 N^{-\frac{1}{2}},$ and the second  $\frac{D}{4-q}\beta_N^{-q+4}\simeq D_1 N^{\frac{q-4}{4}},$ for large $N.$ For the third one, we make a change of variables, $u=N^{\frac{1}{4}}t$ and get
$$E\int_0^{\beta_N}\frac{t^{-q+1}(t^2+Nt^4)}{(1-t^2-Nt^4)^{\frac{2}{2n-\alpha-2}}}dt=
EN^{\frac{q-2}{4}}\int\limits_0^{\beta_N N^{\frac{1}{4}}}
\frac{u^{-q+3}(N^{-\frac{1}{2}}+u^2)}{(1-u^2N^{-\frac{1}{2}}-u^4)^{\frac{2}{2n-\alpha-2}}}du$$
$$\leq E_1N^{\frac{q-2}{4}},$$
since $\beta_N N^{\frac{1}{4}}\longrightarrow 1$ as $N \rightarrow \infty$ and the integral $\int_0^1
\frac{u^{-q+5}}{(1-u^4)^{\frac{2}{2n-\alpha-2}}}du$ converges.

We apply the same change of variables, $u=N^{\frac{1}{4}}t,$ for the last integral and find that it is comparable to $ N^{\frac{q}{4}-1}.$
\begin{equation}\label{F}F\int_0^{\beta_N}\frac{t^{-q+3}(t^2+Nt^4)}{(1-t^2-Nt^4)^{\frac{4}{2n-\alpha-2}}}dt=
FN^{\frac{q}{4}-1}\int\limits_0^{\beta_N N^{\frac{1}{4}}}
\frac{u^{-q+3}(u^2N^{-\frac{1}{2}}+u^4)}{(1-u^2N^{-\frac{1}{2}}-u^4)^{\frac{4}{2n-\alpha-2}}}du.
\end{equation}
The integrand in the latter is a positive increasing function of $u$ and $\beta_N N^{\frac{1}{4}}\longrightarrow 1$ as $N \rightarrow \infty.$ So, we can roughly bound the integral from below by a positive constant and have that equation (\ref{F}) is greater than $F_1N^{\frac{q}{4}-1},$ where $F_1>0.$

\medskip

 In the second interval, we use the fact that $A_{D,H,\alpha}(t\theta) \leq A_{D,H,\alpha}(0)$ since central sections have maximum volume. Then, since $t<<1,$ we have that
$$\int_{\beta_N}^{\alpha_N}
t^{-q-1}\Bigl(A_{D,H_{\xi},\alpha}(t\theta)-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt$$
$$\leq \int_{\beta_N}^{\alpha_N}
t^{-q-1}\Bigl(\frac{1}{2n-\alpha-2}+\frac{\alpha}{2(2n-\alpha-4)}\Bigr)t^2dt
<A\int_{\beta_N}^{\alpha_N}t^{-q-1}dt.$$

\noindent
Recall that $\alpha_N$ and $\beta_N$ are the positive solutions of the equations $f(t)=0$ and
$1-t^2-Nt^4=t^{q+1}$ respectively, and that for large $N, \ \alpha_N\simeq N^{-\frac{1}{4}}.$ Then, it is
not difficult to see that
$$A\int_{\beta_N}^{\alpha_N}t^{-q-1}dt \leq \frac{A}{(\alpha_N+\beta_N)(1+N(\alpha_N^2+\beta_N^2))}
\simeq AN^{-\frac{1}{4}},$$

\noindent
see [KYY, p.204] for details.

\medskip

 Lastly, for the interval $[\alpha_N, \infty),$ we use the fact that $A_{D,H_{\xi},\alpha}(t\theta)=0.$ Then, we have that
$$\int_{\alpha_N}^{\infty}t^{-q-1}\Bigl(-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt$$
$$=\int_{\alpha_N}^{\infty}\Bigl[-\frac{t^{-q-1}}{2n-\alpha-2}+\Bigl(\frac{2}{2n-\alpha-2}+\frac{\alpha}{2n-\alpha-4}\Bigr)\frac{t^{-q+1}}{2}\Bigr]dt$$
$$=-A_1\alpha_N^{-q}+A_2\alpha^{-q+2}\simeq -A_1N^{\frac{q}{4}}+A_2N^{\frac{q-2}{4}},$$

\noindent
where $A_1,A_2 >0.$

Combining all the above estimations, for $N$ large enough, we obtain the following upper bound for the integral in (\ref{eqt:countrxample}),

$$\int_0^{\infty}
t^{-q-1}\Bigl(A_{D,H_{\xi},\alpha}(t\theta)-A_{D,H_{\xi},\alpha}(0)-\Delta A_{D,H_{\xi},\alpha}(0)\frac{t^2}{2}\Bigr)dt d\theta$$
$$< C_1 N^{-\frac{1}{2}}+D_1 N^{\frac{q-4}{4}}+E_1N^{\frac{q-2}{4}}-F_1 N^{\frac{q}{4}-1}+AN^{-\frac{1}{4}}
-A_1N^{\frac{q}{4}}+A_2N^{\frac{q-2}{4}},$$
which clearly shows that it is negative since all the constants are positive and $q\in (2,3).$

\qed

\bigbreak

{\bf Acknowledgments:}
The author was partially supported by the
NSF grant DMS-0652571. Also, part of this work was carried out when the author was visiting the Department of Mathematics of University of Crete, which the author thanks
for its hospitality.

redpajama_set_name: RedPajamaArXiv


Processed from hardwood and softwood woody substrates. Dyed using natural iron oxide pigment for long lasting colour.
Dyed black mulch with uniform, fibrous texture.
Black mulch provides contrast to planting bed, highlighting plants and flowers.
Fibrous texture allows mulch to knit and create a weed barrier against airborne weed seeds.


text:
Declerck Sibo (born 19 May 1994) is an Angolan handball player for Primeiro de Agosto and the Angolan national team.

He represented Angola at the 2019 World Men's Handball Championship.

References

1994 births
Living people
Angolan male handball players
Competitors at the 2019 African Games
African Games competitors for Angola
African Games medalists in handball
African Games gold medalists for Angola

redpajama_set_name: RedPajamaWikipedia


text:
Posted on May 14, 2017 by Caroline Chiasson // 0 Comments
On this day in 1936, Waldon Robert Cassatto – a.k.a., Bobby Darin – was born in the Bronx borough of New York City.
Darin was raised to believe his grandparents were his actual parents. He extremely close with his "mother" Vivian Fern, a fun-loving vaudeville singer, and was devastated when, at 32 years old, he was told the truth about his birth. He grew up with an older sister, Giovannina "Gina" Cassotto, with whom he had a close bond, but he never would he imagine that it was Gina who was his biological mother. He only learned of this relationship in 1968, and it was a hard lie to grow up with. But, he did eventually realize that, in 1936, a young, unwed girl could not have a child on her own. He eventually acknowledged the maternal relationship with Gina.
Darin was born weak and frail. At age eight, he struggled through recurring bouts of rheumatic fever, leaving him with a very weak heart. He wouldn't have heart surgery until January 1971, when doctors made an attempt to mend some of the heart damage he had lived with since childhood. Darin spent that year recovering from the surgery. By the end of his career, he would have an oxygen tank just off the stage when needed.
With Vivian's encouragement, the young Darin was playing several instruments by the time he hit his teens, including piano, guitar, and drums. Later he would pick up the xylophone and harmonica. Darin was able to graduate from the esteemed Bronx High School of Science.
Darin started out his musical career by writing songs for Connie Francis (the two would have a romantic relationship during this period, and, later in her life, she wrote that her biggest regret was that she did not marry Darin).
He went on to record his own major hit, 'Splish Splash', in 1958, selling over one million copies. To him, the song was a joke – it took Darin literally minutes to write. It always bothered him that a song so silly could be so respected. Three of the greatest songs ever written were to follow: 'Make The Knife', 'Dream Lover', and 'Beyond the Sea'. In 1962 Darin started acting in films, earning a Golden Globe award for his first role in Come September, which co-starred his first wife, actress Sandra Dee (the two would have one son together, Dodd Darin).
With the enormous amount of success in his career, Darin would be able to fulfil a dream of playing the Copacabana, where he ended up setting the all-time attendance record at the Manhattan nightclub. He was headlining in Las Vegas casinos at the time, as well.
In the 1960s, Darin opened, owned, and operated a music publishing and production company called TM Music/Trio. He was the man who signed the young Wayne Newton, even giving him the song 'Danke Schoen'. The 1960s brought many changes to Darin's life: he made the switch over to country music, writing and recording the lovely track 'Things' (which was later turned into a swing song by Dean Martin). Many more of his songs also had reasonable success.
As the 1960s continued, Darin was having a different outlook on life, and, therefore, his music styles were changing. He dove into the folk scene and had a hit with Tim Hardin's 'If I Were a Carpenter'. He had also become active in politics, and traveled with Robert Kennedy, working on the 1968 presidential campaign. Darin was with Kennedy on June 4, 1968, for the California primary at the Ambassador Hotel, where, later that evening, Kennedy would be assassinated while walking through the kitchen of the hotel. This death, combined with learning of his true parentage, had a deep impact on Darin, and he spent the next year living in seclusion in a trailer near Big Sur in Calfironia.
Darin returned to Los Angeles in 1969 and opened Direction Records, a place for folk and protest music. Darin wrote the ballad 'Simple Song of Freedom', which was later recorded by Tim Hardin. By 1972, Darin was starring in his own television show on NBC, The Bobby Darin Amusement Company, which ran until his death.
Darin re-married in June 1973, and continued to make television guest appearances. Though he was still selling out nightclubs in Las Vegas, his health was beginning to fail.
On December 20, 1973, Darin died at age 37 in the recovery room at Cedars-Sinai Medical Center, after an operation to repair two artificial heart valves.
In 1990, Bobby Darin was inducted into the Rock and Roll Music Hall of Fame. 'Mack The Knife' has since been honoured with a Grammy Hall of Fame Award.
caroline chiasson
today in music
Scoop Trumbull & the Wrong Notes :: Pull the Cork

redpajama_set_name: RedPajamaCommonCrawl


Home » Senator Rockefeller to Cruise Lines: You Can't Operate for Free in the U.S.
Senator Rockefeller to Cruise Lines: You Can't Operate for Free in the U.S.
Over the last year, Senator Rockefeller has raised concerns that the cruise industry has used the infrastructure of U.S. ports, the resources of the U.S. Navy and Coast Guard, and more than 20 U.S. agencies, but has paid virtually no U.S. taxes. At the same time, the U.S. Coast Guard is substantially under-funded and is increasingly called upon to assist cruise lines which are experiencing fires and engine failures.
Efforts to repeal the exception have been a long time coming. Earlier this year, as the disabled Carnival cruise ship Triumph was being towed back to the U.S. at U.S. taxpayer expense, Forbes published an article: Ship Isn't The Only Thing That Stinks At Carnival: Low Tax Rate Stirs Ire.
The cruise industry enjoys a substantial tax advantage over shore-side hotels, restaurants and amusement parks by incorporating their businesses and registering their cruise ships in foreign countries. This permits the cruise lines to sell their cruises at artificially low prices. Although Carnival and Royal Caribbean are based here in the U.S., they are registered in Panama and Liberia respectively to avoid U.S. taxes.
Rockefeller's proposed legislation would eliminate the tax exemption for cruise lines and impose a 5 percent excise tax on gross income if passengers get on or off a ship in the U.S. The tax would be targeted to improve the transportation infrastructure.
Avoiding taxes is a cornerstone of the cruise industry's business model. Expect CLIA and the cruise lines to mount a heavy PR campaign to try and kill the new bill.
Your Tax Dollars At Sea – Who Pays When Things Go Wrong on Cruises?
Have a thought about this issue? Leave a comment below or join the discussion on our Facebook page.


text:
Q: FCEUX emulator doesn't show the tools menu I have the FCEUX emulator 2.2.1 on Ubuntu 14.04, the problem is it doesn't show the tools menu:

As it should have:

I need the tools menu to get the RAM values for a bot I am creating.
Could someone help me with this please?

A: No. There is no tools menu in the linux version of FCEUX. There are no tools. Learn to use wine and use the windows version.

A: There is no tools menu in the linux version HOWEVER, you can still access the cheats interface by pressing "F1" while playing.
If you want other features beyond cheat interface then you might have to use the windows version :(
If you want the cheat interface, keep reading
The Linux cheat interface is command line only (no UI).
Here is how it works:

*

*Make sure you start FCEUX from a terminal

*While playing the game press F1

*In the terminal you should see something like that

     1) List Cheats
     2) New Cheats...
     D) Display Menu
     X) Return to Previous

     Command >

Your game will be frozen while this menu is shown. You can press X to return back to your game.
Enter 2 to access the cheat functions, you should see output like that

1) Add Cheat
2) Reset Search
3) Do Search
4) Set Original to Current
5) Unhide Excluded
6) Show Results
7) Add Game Genie Cheat
8) Add PAR Cheat
D) Display Menu
X) Return to Previous

First you select 2 to reset the search function then you can select 3 to search for the value and you can enter 6 to view search result.
Of course first search would return lot of result so after you do that you press X twice to return back to the game, play a little bit then press F1 again, choose 2 again (for new cheats) and perform anther search (without resetting this time!!).
repeat unit you find the value you are looking for

redpajama_set_name: RedPajamaStackExchange


text:
Rare Bird Alert: January 13, 2023
The ABA Area hosts an impressive set of rarities continuing into this week, with the highlight remaining the ABA Area's 3rd record of Eurasian Blackbird (ABA Code 5) in Labrador. California continues to host a Red-flanked Bluetail (4) and a Little Stint (4), and the Common Shelduck (5) is still being seen and discussed in Pennsylvania.
The influx of winter wanderers southward into the interior of the continent was not predicted to include many Bohemian Waxwings, but one was seen in Apache, Arizona. The mountain ash crops in the western half of the continent were reported as good to bumper, but that doesn't always stop birds from traveling to new places. EDIT: Previous versions of this post stated this was a 1st record for Arizona.
From New Hampshire comes the still unconfirmed but compelling report of a Garganey (4) in Cheshire, which would represent a 1st for that state and potentially a 3rd for Vermont if it crosses the dozen meters between where it was spotted and the state line.
Nova Scotia's 8th record of Common Ringed Plover in Yarmouth is one of very few, if any, winter records on the continent.
In Maine, a Harris's Sparrow was visiting a feeder in Androscoggin.
Good for New York was a young Swainson's Hawk in Richmond.
Good numbers of Common Murres were seen this week off of Virginia Beach, Virginia, where there are only a handful of records for the state.
In Florida, a Bahama Mockingbird (4) and the state's 3rd record of Mountain Plover were both seen in Brevard.
Ohio's 3rd Slaty-backed Gull was found among a large gull flock in Cuyahoga, where both Common and Glaucous-winged Gulls have been seen in recent weeks.
In Indiana, a Mountain Bluebird was seen in Newton, the state's 8th.
Illinois also had a Bohemian Waxwing this week, in Stephenson.
And in British Columbia, the province's 3rd Fieldfare (4) was seen in Kelowna.
Omissions and errors are not intended, but if you find any please message blog AT aba.org and I will try to fix them as soon as possible. This post is meant to be an account of the most recently reported birds. Continuing birds not mentioned are likely included in previous editions listed here. Place names written in italics refer to counties/parishes.
Readers should note that none of these reports has yet been vetted by a records committee. All birders are urged to submit documentation of rare sightings to the appropriate state or provincial committees. For full analysis of these and other bird observations, subscribe to North American Birds, the richly illustrated journal of ornithological record published by the ABA.
Nate Swick2023-01-17T14:17:16-05:00
Nate Peterson
The Bohemian Waxwing in Arizona isn't a first state record, but rather the first in the state since the last huge irruption 1980-something?
Reply to Nate Peterson
Nate Peterson is basically correct. There are published records in Arizona from 1919 and 1936(7?)
https://academic.oup.com/condor/article-abstract/41/3/123/5263641?redirectedFrom=fulltext .
Not addressed by the Arizona Committee.
A record in Arizona in 1977.
http://abc.azfo.org/reports/AZ_1974_1996_part2.pdf .
A large flock from Flagstaff in 1984.
http://abc.azfo.org/reports/AZ_1996_1999.pdf .
And a single bird from 2004.
A great find. This bird is still on Arizona Review List.
http://abc.azfo.org/lists/review_list.html .
Nate Swick
Sorry about that. I had made an edit to his post last Friday but it didn't save for whatever reason.
Flight Calls Email Newsletter Signup
Confirm that you are not a bot.
Frank Izaguirre2023-01-26T12:38:25-05:00
Belted Kingfisher, the 2023 ABA Bird of the Year!
In 2023, the ABA honors the Belted Kingfisher, Megaceryle alcyon, as its Bird of the Year! Bold, resolute, and recognizable along much of the ABA Area's shorelines, look for Belted Kingfisher content throughout the year in all the ABA's media.
Ted Floyd2023-01-16T21:34:09-05:00
How to Know the Birds: No. 72, Layover Birding
Even when everything goes smoothly, airport layovers are an "exercise" in sedentariness. Which isn't to say birdlessness. On the contrary, airport layovers sometimes result in life birds; that's happened to me more than once. For those of us on eBird consecutive days streaks, airport layovers . . .
Snowy Owl Viewing Ethics
American Birding Association2023-01-27T17:17:53-05:00January 27th, 2023|0 Comments
Closer to Owls, but in the Right Ways.
Snowy Owls are one of the most recognized and beloved birds in the world. To protect them, it's essential that birders enjoy these birds ethically.
Amazing Milestone for Guy McCaskie
Ted Floyd2023-01-27T15:26:58-05:00May 29th, 2019|0 Comments
May 29, 2019 How many bird species are on the official list of your state or province? If you're an ABA member, there's a decent chance that total is somewhat less than read more >>
Birding is a force for good in our society. Learning and sharing about birds translates into concern for birds and the environment, and the American Birding Association provides resources and community for all people interested in birds!
Please consider joining or donating to the American Birding Association.
Also, logged-in members don't see this message.


text:
19-year-old Paterson man charged in Summer Street shooting
April 29, 2020 by Jonathan Greene
A city man has been arrested and charged for his alleged role in a shooting on Summer Street that took place on January 11, according to the Passaic County Prosecutor's Office.
Leighmeir Singletary, 19-year-old, was arrested outside of 51 Godwin Street at around 6:30 p.m. on Tuesday. He allegedly led police on a brief foot chase before being captured.
Police determined Singletary was the alleged gunman in the January 11 shooting on Summer Street and Godwin Avenue. At around 2:45 a.m., police responded to report of a shooting in the area and located ballistics evidence and a blood trail. A 26-year-old man was injured and suffered non-life-threatening injuries.
During the arrest Singletary had 42 envelopes of heroin in his possession, police said. He was charged with several drug offenses. He was also charged with eluding related to a March 16 incident that happened on Summer Street at around 4:51 p.m.
Authorities said Singletary faces 20 years on the first-degree attempted murder, 20 years for first-degree armed robbery, 10 years for possession of handgun, and 10 years for aggravated assault charges.
Prosecutors have filed a motion for pretrial detention, authorities said.
Upto 20 years? Give these low lifes that. Don't feel sorry for them. Don't give into them and don't give into the racism tactic always used. These are.awful, evil people killing all people.and poisoning ALL people no matter what race they are. They are savages and.treat them as such.
Paterson Resident
Throw the book at them. Period
this place is a dump
Thank god for taking this trash off the streets. But this wont solve the problem. Police need to attack all the suppliers. You take one of these fools down, 10 more will take his place. FIND THE SUPPLIER!


text:
NPR Creates: Best Stereotypical Major Media Trade Paperback List
by Rob McMonigal December 19, 2008
Okay, so I love a lot of things about NPR. But certainly not their list of best superhero graphic novels for 2008. First off, that name--good God! Just call them Superhero Comics, okay? Don't worry, reading the word "comics" won't shave points off the IQs of the $6.00 coffee set as they read in Starbucks, honest.
Second of all, this list reads like an article titled, "Comics just aren't for kids anymore" with selections that are so hopelessly tied to mainsteam press preconceived notions about comics it's not even funny.
Let's see....Joker? A tie-in to The Dark Knight Defends Bush's Torture Policy. Death of Captain America? Probably read about it from the articles in the New York Times when the story first broke. Joss Wheadon? Seriously? Did the author of this piece even bother to go past the cover? Or read the Brian K. Vaughn version? Give onto me a break.
I'm not sure how Warren Ellis ended up on there, maybe an accidental Google search. And last but not least--some random Superman story that echoes 1970s comic books.
Please, NPR--if you're going to try this, do it right. Take more than five minutes cobbling the list together, and we'll all be happier.
Full disclosure--I've not read any of these titles. But from reading about them and knowing the authors involved, I can just about guarantee they would not get any person who doesn't normally read comics to start browsing the graphic novel section of Border's. So in addition to looking sloppy, I can't even see the use to NPR's yuppy set, either.
This is whyI don't really like "Best of..." lists. They don't really help, other than to make a publication look like they're "hip" and can rank things about which they'd probably be better off leaving alone.
On that note...next on Panel Patter: My list of the best 2008 Brain Surgeries!


Baltimore, MD – After an extensive nationwide search, Deborah Graves has been selected as the new president and chief operating officer of Levindale Hebrew Geriatric Center and Hospital.
Graves brings a wide breath of experience in health care, strategic planning, business proficiency, and in creating successful and collaborative work environments to engage residents, patients and employees.
Graves joins Levindale from Benedictine Health System, headquartered in Duluth, Minnesota, where she was the senior vice president of Client Services. Benedictine provides a complete spectrum of post-acute services that includes independent living, assisted living, memory care, inpatient rehabilitation, and transitional, home health and long-term care to seniors.
In 2008, she was selected as one of Minnesota's 100 Influential Health Care Leaders, which recognizes leaders in the industry who contribute to the high quality of health care in Minnesota.
"I am very much looking forward to furthering the work that is already underway by a dedicated hard-working staff at Levindale which encompasses a unique blend of long-term care, behavioral health, chronic hospital services and skilled nursing home care, as well outpatient day treatment programs and adult day services," says Deborah Graves.
In addition to the usual hiring process, members of Levindale's Employee Action Committee were instrumental in the interview and selection of Graves. She begins her tenure at Levindale on November 27.
Graves earned her Bachelor of Nursing and her bachelor of Sociology from Louisiana State University and an MBA from the University of New Orleans. She is a member of the American College of Healthcare Executives, the Association of periOperative Registered Nurses, the League of Nursing, the American Association of Critical Care Nurses and the American Organization of Nurse Executives.
Levindale is part of LifeBridge Health, one of the largest and most comprehensive providers of health care in Maryland. In addition to Levindale, LifeBridge Health includes Sinai Hospital of Baltimore, Northwest Hospital, Carroll Hospital and related subsidiaries and affiliates.


Fresno (AAA Pacific Coast): After a rainout on Tuesday night, the Fresno Grizzlies (5-1) and Salt Lake Bees (1-5) attempted to play two on Wednesday at Smith's Ballpark, before the second game was suspended by rain after three innings with the Grizzlies leading 8-1. The early onslaught inlcuded Carter Kieboom's first Triple-A homer (2-run) in the second inning. That and the rest of the individual stats will not be made official until the second game is completed during Fresno's next trip to Salt Lake City, June 20-24 (exact resumption date is TBA).
In the opener on Wednesday, Jake Noll's solo home run, an opposite field blast to right in the first inning turned out to be the game-winning hit in a 2-0 Fresno victory. Alec Keller added the second run with a bases loaded sacrifice fly in the fifth. Noll's homer was his first of the season and of his Triple-A career.
Right-hander Kyle McGowin made everything stand up by tossing 5.2 scoreless innings, scattering only four hits without a walk, striking out six. The Grizzlies are now 2-0 this season with McGowin on the mound, as he picked up his first winning decision of the 2019 campaign. Lefty Vidal Nuno converted a four-out save to secure Fresno's fourth consecutive victory, keeping the Grizzlies in first place in the Pacific Northern Division as the PCL's opening week comes to a close.
ON THAT FRES-NOTE: Fresno's Game 1 win was the first time Salt Lake was shutout in a home opener. Under nicknames "Buzz," "Stingers," and "Bees" since Salt Lake joined the PCL in 1994, Wednesday was their 26th home opener in the Beehive State.
Thursday's starting pitching matchup will be an all-University of Washington alumni affair. Fresno's Austin Voth pitched three seasons for the Huskies before turning pro; as UW's Friday night starter during his junior season (2013), he posted a 2.99 ERA in 17 games (15 starts) over 105.0 IP. Voth's 98 strikeouts in 2013 ranked second in the Pac-12 Conference, behind only future/former Grizz Mark Appel (2015). Probable Salt Lake starter Forrest Snow was not a teammate of Voth's at Washington; he pitched three seasons at the school as well from 2008-2010 before being drafted by and signing with the Seattle Mariners following his junior year.
The Senators completed their season opening homestand with a 5-1 win over the Hartford Yard Goats and with the win, swept the Goats in the three-game series. Wil Crowe retired the first 17 batters he faced, and Tres Barrera continued his scorching start with two home runs to lead the Sens. The Senators moved to 6-1 on the season while Hartford fell to 1-6.
With one out in the third inning, Chuck Taylor drew a walk and Tres Barrera followed with a two-run home run into the Senators bullpen. The home run gave the Senators a 2-0 lead, a lead they held through the remainder of the game.
Wil Crowe faced 19 batters in six innings of work, striking out five and allowing just an infield single. Crowe went to a three-ball count just twice in the outing. Crowe earned his second win against no losses. Taylor Guilbeau pitched two innings and allowed four hits and a run and struck out three. Jacob Condra-Bogan pitched the ninth and after allowing a leadoff single, he struck out the next two hitters before retiring the final batter on a flyball to center field.
Tres Barrera went 4-for-4 with two home runs and drove in five runs as he continued scorching the baseball. Austin Davidson went 2-for-3 with a double, his third double in his past two games. Spencer Kieboom had the only other hit for the Senators.
Neither Drew Ward or Rhett Wiseman batted so their hitting streaks are in tact at six games each. Barrera is now 11-for-19 on the season. The Senators lead the E.L in HR's with 11. Wil Crowe retired 17 straight batters to begin the game after retiring the final seven in his first start for a total of 24 consecutive batters.
The Senators and Bowie Baysox open a four-game series Friday night in Bowie. The game is the home opener for the Baysox. The Sens took three of four from Bowie last weekend at FNB Field. RH Erick Fedde is on the mound for the Sens opposed by RH Marcos Molina for Bowie.
Potomac (High-A Carolina League): For the second straight game, the Potomac Nationals (3-4) failed to plate a run, as the Carolina Mudcats (5-2) won 2-0 on Wednesday night and earned a series sweep. In a bounce back pitching performance, RHPKyle Johnston (L, 0-2) allowed a run in the opening frame, but kept Carolina off the scoreboard over the next five innings, as he tallied Potomac's first quality start of the season. The P-Nats' offense had plenty of opportunities, though finished 0-10 with RISP and left 12 runners on base.
Johnston issued a one-out walk to 1B Ryan Aguilar in the opening frame, and he crossed the plate on a two-out, two-strike double from C Payton Henry. Henry's double gave Carolina a 1-0 lead, which proved to be enough. Johnston allowed just one more hit and struck out five over six innings.
Potomac's first two batters of the night, 2B Cole Freeman and DH Telmito Agustin , walked off of RHP Bowden Francis(W, 1-0), but were both left in scoring position. Potomac left two men in scoring position in the second inning, a runner on second base in the fourth frame, two men in scoring position in the fifth inning, and the bases full in their eighth trip to the plate. Bowden allowed three hits over five innings in the victory.
After six strong innings from Johnston, Potomac went to the bullpen. RHP Frankie Bartow worked a scoreless seventh inning, but allowed a solo home run to CF Joantgel Segovia, which began the eighth inning. Segovia's second home run in as many games made it 2-0 in favor of Carolina, the eventual final.
RHP J.T. Hintzen tossed a clean seventh inning out of the Carolina bullpen, but ran into trouble in the eighth. 1B Aldrem Corredor drew a leadoff walk, while 3B Anderson Franco singled with one out. After LF Jack Sundberg flied out, Carolina went to RHP Rodrigo Benoit (SV, 2) in search of a four-out save. Benoit walked C Alex Dunlap on five pitches, which loaded the bases, but SS David Masters flied out to centerfield and ended the inning. Potomac got a baserunner in the ninth against Benoit but failed to do anything with it.
The P-Nats will head on the road for the first time in 2019 in an effort to snap a three-game losing streak. Thursday night, the P-Nats will send LHP Grant Borne (0-0, 4.50) to the hill against Winston-Salem. Borne allowed two runs over four innings in his 2019 debut, a no-decision in the front half of a doubleheader on 4/6. For the Dash, LHP John Parke (1-0, 1.80) will take the series-opening start. Parke allowed just one run over five innings in a victory over Frederick on 4/6.
First pitch in Winston-Salem on Thursday night is set for 7:00pm.
Hagerstown (Low-A South Atlantic League): Behind a KJ Harrison three-for-five outing, Hagerstown outlasted the BlueClaws 5-3 Wednesday night at FirstEnergy Park.
Harrison drove in the tie-breaking run in the seventh after both teams were tied 2-2 since the third inning. After Harrison's single to score Kyle Marinconz, Jamori Blash lined his own base knock to right to score Harrison and put Hagerstown (2-5) up 4-2.
In the eighth inning, Gilbert Lara added icing to the cake in the form of an RBI base hit to push Gage Canning home. The RBI was Lara's team-leading eighth of the season and gave Hagerstown a 5-2 lead.
Those runs proved to be critical as Lakewood (3-4) scored one run in both the eighth and ninth innings. Jake Holmeshit a sacrifice fly to plate Luis Garcia in the eighth and inch the BlueClaws to 5-3. Then, in the ninth, Alec Bohm hit a sacrifice fly to score Carlos De La Cruz before Aaron Fletcher (S, 1) slammed the door shut.
Fletcher spun 1.2 innings, allowing just one run and two hits to lock down his first South Atlantic League save. The lefty came in for Ryan Tapani (W, 1-0), who worked 2.2 innings. The Creighton product fanned four batters before getting lifted.
Tim Cate breezed through his second Suns start of 2019. The lefty offered 4.2 innings of work and fanned five on his way to a no-decision. Cate now has a 1.93 ERA on the year.
The offense kicked-off the game in the third inning from the bat of Nic Perkins. The backstop brought Blash home from first, lacing a double to right. Two batters later, Canning added an RBI to his statsheet with a single to score Perkins.
The Suns return home tomorrow to face the Asheville Tourists at Municipal Stadium at 6:05 p.m. The contest begins a seven-game homestand in which the Suns will welcome the Tourists and the defending South Atlantic League Champion Lexington Legends to the Hub City. It all kicks off tomorrow with a magnet schedule giveaway for the first 1,000 fans in attendance. Hagerstown starts RHP Jake Irvin (0-0, 7.20 ERA). while the Tourists counter with Shelby Lackey (0-0, 6.75 ERA).


text:
Late-Race Tire Issue Results in Disappointing Finish for Busch
#51 TruckBy Ryan Kingsbury March 5, 2017
Date: March 4, 2017 Event: Active Pest Control 200 (Race 2 of 23) Series: NASCAR Camping World Truck Series Location: Atlanta (Ga.) Motor Speedway (1.54-mile oval) Format: Three Stages; 40 laps/40 laps/50 laps Start/Finish: 2/26 (running, completed 127 of 130 laps)…
Gragson Rebounds for 14th-Place Finish at Atlanta
Date: March 4, 2017 Event: Active Pest Control 200 (Race 2 of 23) Series: NASCAR Camping World Truck Series Location: Atlanta (Ga.) Motor Speedway (1.54-mile oval) Format: Three Segments; 40 laps/40 laps/50 laps Start/Finish: 10th / 14th (running, completed 130 of 130 laps) Winner: Christopher Bell of Kyle Busch…
Ready to Seal the Deal
#4 TruckBy Ryan Kingsbury February 28, 2017
MOORESVILLE, N.C. (Feb. 28, 2017) – Christopher Bell could nearly taste victory in last year's NASCAR Camping World Truck Series race at Atlanta Motor Speedway in Hampton, Ga., but instead ended up with a torn up JBL Tundra and a disappointing 26th-place finish after blowing a tire late in the race. He returns to the 1.54-mile…
Gragson Puts Daytona in the Rear-View Mirror Headed to Atlanta
#18 TruckBy Ryan Kingsbury February 28, 2017
MOORESVILLE, N.C. (Feb. 28, 2017) – Noah Gragson has been looking forward to racing at Atlanta (Ga.) Motor Speedway ever since last Friday night when he was taken out of contention after only one lap at Daytona (Fla.) International Speedway. By the time he changed out of his Switch firesuit, he was already looking ahead…
Busch Looks to Score First Atlanta Win for KBM
MOORESVILLE, N.C. (Feb. 28, 2017) – Kyle Busch will take the green flag for his first 2017 NASCAR Camping World Truck Series (NCWTS) start of the season Saturday afternoon at Atlanta (Ga.) Motor Speedway behind the wheel of the No. 51 Cessna Toyota Tundra in the Active Pest Control 200. The 1.5-mile oval in Hampton, Ga.,…
234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374


dEAR friends, summer sales still going strong! But I am blocked from sharing to groups far more then the time I was allowed to. Therefore would be great for those who see this post / offer, can share it to those whom you think they will be interested in. As usual, much appreciated!
we are making a great offer! all herbal titles (http://herbalinternational.blogspot.com) are now available as pay as you wish (no minimum, very zen, we know)! do let us know if you are interested with any of the herbal titles! the offer will run until 31st July 2018!
4cd onward depend on the packaging.


text:
Holoubek ist der Familienname folgender Personen:

 Gustaw Holoubek (1923–2008), polnischer Schauspieler, Film- und Theaterregisseur und Pädagoge
 Jakub Holoubek (* 1992), tschechischer Skeletonsportler
 Jan Holoubek (* 1978), polnischer Kameramann, Drehbuchautor und Filmregisseur
 Karl Holoubek (1899–1974), österreichischer Politiker (SPÖ)
 Michael Holoubek (* 1962), österreichischer Jurist, Universitätsprofessor und Verfassungsrichter

Siehe auch:
 Holubec
 Holubek
 Holubeck


text:
Makerere University Non-teaching Staff Strike
August 2, 2015 Staff Writer
Makerere University non-teaching and support staff have joined other public universities resolving to withdraw their labor starting August 02, 2015 until government enhances their salaries.
All non-teaching staff raised their hands uninanimously in support of the industrial action at Makerere University,mainhall
Under their Umbrella, Public Universities' Non-Teaching Staff Executive Forum (PUNTSEF), they reached at the resolution on Friday evening at Makerere University main hall, led by PUNTSEF chairman Jackson Betihamah.
"We Makerere university non-teaching staff have resolved to lay down our tools with effect from 3rd August until government enhances our salaries. All staff have been advised not to report on duty until further notice. Whoever will be found working will be disciplined," Said Bruce Twesigye, the Forum's General Secretary.
Uganda has got six public universities namely; Busitema University, Gulu University, Kyambogo University, Makerere University, Mbarara University of Science and Technology and Muni University, who have over 4000 employees as non-teaching staff.
They have resolved that all custodians of various buildings including halls of residence should not open, Makerere security to handover to the police station, Registrars should not give out provisional admission forms to joining students, and as well the Directorate of ICT has resolved to shut down all internet facilities among others.
With all public universities set to re-open for new academic year beginning August 8, the strike announcement appears to have been intended to coincide with the opening of the academic year.
Last year, president Museveni promised a pay rise for all academic staff in public universities with a professor earning 15M pro-rata, a pledge that has been fulfilled with government allocating sh50b in the 2015/2016 national budget.
Whereas the presidential pay rise pledge emphasized only academic staff, the non-teaching staff looks at it as unfair, saying instead the president would have pledged to all public universities staff without segregating.
The non teaching staff revealed that they have had enough meetings with all relevant stake holders including the minister of education, the prime minister, the public service parliamentary committee among others but they have all turned a deaf ear.
They are demanding Shs39bn for this financial year 2015/2016 compared to 50bn that was allocated to the academic staff.
In a closed-door meeting chaired by Education minister Jessica Alupo, on Thursday the workers had vowed that nothing will deter them from laying down their tools come Monday August 03, after the minister said government did not have any money to meet their demands.
"We are aware that they are to lay down their tools but we are not able to enhance their salaries this financial year as requested because government has financial constraints and has to keep other public sectors running," said the minister .
Mr. Jackson Betihamah, the chairperson of the Public Universities Non-Teaching Staff Executive Forum (PUNTSEF), said he was disappointed with the meeting.
"I think they just called us here to make us drink their tea which was not even enough due to the large numbers from PUNTSEF," Betihamah told Red Pepper.
"Where is it written that when academic staff gets pay rise, non-academic staff gets satisfied? Let the government learn to respect her employees. It is every laborers right to get a better pay without discrimination, and the only tool we have is to withdraw the labor which is our right, and the only tool to fight this discrimination," said PUNTSEF chairman Jackson Betihamah.
They have noted that the segregation of enhancing salary for one category of staff and leaving out others is discriminatory, unfair, demotivating, destroys team work and will negatively affect the achievement of the institutions' goals and objectives.
Guild President Supports PUNTSEF
Makerere University guild president, Bala David Bwiruka told the staff, that as students' leaders they support the industrial action, but also warned that with two weeks remaining for the semester to begin, the students might also take action.
"The struggle you are in is a struggle against injustice and for us as students' guild we have sat and agreed that we will support you. When things become abnormal you also become abnormal. As guild presidents in all public universities, we are meeting next week Tuesday where we shall arrange to meet the minister for Education and the IGP to chat a way forward,"Bala said.
Makerere university convocation also approved the industrial action and they said they support anything to do with the staff welfare.
"Makerere convocation represents all staff students and the alumni therefore we shall support anything to do with the staff welfare .We want to assure you that we will not depart from that mandate and hence we are supporting you," said Deus Kamunyu Muhwezi the Publicity secretary of the convocation.
Makerere has since set up the industrial action committee that will oversee everything to ensure that no worker reports on duty.
The committee is comprised of Bruce Twesigye, Yunus Musisi, Denis Mbabazi, Francis Angida, Wilberforce Bubolu, Joseph Musoke, Isaac Egesa, Nelson Akampulira, John K Muhumuza and Justine Namuddu.
In a petition dated April 19th that was presented to the speaker of parliament through the clerk, the staff observed that, the proposed enhancement will be in gross violation of the principle of equal opportunity and a clear act of open discrimination among the three categories of staff of public universities of Uganda.
"We request that the proposed gradual enhancement of the salaries should cover all the categories of the workers in all public universities on pro-rata to enable us to avoid the pain staking and laborious industrial and court action by all public universities including the upcoming new ones Muni and Soroti," reads the petition.
"We don't like strikes neither, but we have been stretched to do so. We gave 21 days as required by law to give room for negotiations. There are supplementary budgets and budgetary omissions where they can get this money from. For us we need only Shs39bn, for this financial year," Bruce Twesigye said.
Twesigye revealed that, they have already written to the minister of education, and also copied the Speaker of Parliament, Prime minister, Minister of Public Service, Presidency, IGP, Councils to all Public Universities and other relevant Parliamentary Committees.
Previous Bloody Fight For Woman Leaves One Dead
Next China Donates To Disabled Soldiers


What is a woodworking hand plane?
What is a woodworking hand plane used for?
What are the parts of a standard metal hand plane?
What are the parts of a wooden hand plane?
How does a plane work?
What are the different types of woodworking hand plane?
Which type of plane should you choose?
What are metal hand planes made of?
What are wooden hand planes made of?
What are plane kits and carpenter's kits?
What are bevel-up and bevel-down planes?
What is the ideal shape for plane irons?
What is the perfect pitch for plane irons?
What is a scrub plane?
What are the parts of a metal scrub plane?
What are the parts of a wooden scrub plane?
What is a bench plane?
What are the parts of a metal bench plane?
What are the parts of a low-angle bench plane?
What are the parts of a wooden bench plane?
What is a Norris-style adjuster?
What is a jack plane?
What is a fore plane?
What is a jointer plane?
What is a smoothing plane?
What is a block plane?
What are the parts of a metal block plane?
What are the parts of a wooden block plane?
What are Japanese woodworking hand planes?
What are specialised woodworking planes?
What is a rebate, or rabbet, plane?
What is a shoulder plane?
What is a bullnose plane?
What is a moulding plane?
What is a combination plane?
What is a router plane?
What is a chisel plane?
What is a compass, or circular, plane?
What are finger and palm planes?
What is a shooting board plane?
Can you get spare parts for woodworking hand planes?
What are the different types of woodworking hand plane spare parts?
How did you do it, Stanley?
The firm that has had most influence on the design and development of the hand plane over the past 150 years or so is the American manufacturer, Stanley.
Its influence has been so great that many modern plane manufacturers – including Stanley themselves – still use the Stanley model numbering system invented as part of the description of their products.
For instance, one type of block plane is known as the No. 102.
Experienced woodworkers will know that a No. 1 plane is a small smoothing plane and the No. 2 is a slightly larger smoothing plane.
By the time we get to No. 5, we have moved on to jack planes.
No. 6 is a fore plane and Nos. 7 and 8 are jointer planes, with the first block plane coming in at No. 9.
The numbering system doesn't have a lot a rhyme or reason to it except that, generally, the higher a number is for a particular type of plane, the bigger the plane is.
To complicate things, ½ and ¼ sizes were also introduced. For instance, a No.5½ jack plane is a slightly wider, heavier version of the No.5, while the No. 5¼ is a narrower, lighter version of the No. 5.
Some planes have their model numbers on the toe, or front end of the plane. Sometimes the number is on the heel or another part of the plane's body, and some types don't bear the number at all!
The list was continually added to as new and improved planes were introduced. It is very extensive and includes all types of plane: metal, wooden, transitional and infill.
Letters were also used to signify certain features. For instance, the number 608-C signifies a 24" long Stanley jointer plane with a corrugated sole.
As another example, the letter "A" before a plane's number indicates that it was made from aluminium – for instance, the Stanley No. A45 aluminium combination plane, which has interchangeable irons for a variety of specialist work.
It's useful to have some knowledge of the Stanley model numbering system. Apart from Stanley Black & Decker, most modern plane manufacturers also advertise their planes as, for instance, 'No .1' (the smallest smoothing plane), 'No. 5½' (a jack plane) and 'No. 62' (a low-angle block plane), because they are based on the same original designs.
It's also handy to know if a plane has a Stanley number when ordering spare parts.
The numbers are meaningful to collectors of woodworking hand planes, too. Examples of some planes identified by Stanley numbers can be exceedingly rare – and consequently very valuable.
Some manufacturers do not use Stanley model numbers, preferring to describe their planes by type and length. For instance, a plane might be described as "A 250mm smoothing plane", which is actually the equivalent of a No. 4½ plane.


text:
Miltiades' Disgrace
Posted by Ellopos Blog on October 31, 2017
The victory of Marathon was a great triumph for the Athenians; and Miltiades, who had so successfully led them, was loaded with honors. His portrait was painted by the best artist of the day, and it was placed in one of the porticos of Athens, where every one could see it.
At his request, the main part of the booty was given to the gods, for the Greeks believed that it was owing to divine favor that they had conquered their enemies. The brazen arms and shields which they had taken from the ten thousand Persians killed were therefore melted, and formed into an immense statue of Athene, which was placed on the Acropolis, on a pedestal so high that the glittering lance which the goddess held could be seen far out at sea when the sunbeams struck its point.
The Athenians vented their triumph and delight in song and dance, in plays and works of art of all kinds; for they wished to commemorate the glorious victory which had cost them only a hundred and ninety men, while the enemy had lost ten thousand.
One of their choicest art treasures was made by Phidias, the greatest sculptor the world has ever known, out of a beautiful block of marble which Darius had brought from Persia. The Great King had intended to set it up in Athens as a monument of his victory over the Greeks. It was used instead to record his defeat; and when finished, the statue represented Nemesis, the goddess of retribution, whose place it was to punish the proud and insolent, and to make them repent of their sins.
Miltiades was, as we have seen, the idol of the Athenian people after his victory at Marathon. Unfortunately, however, they were inclined to be fickle; and when they saw that Miltiades occupied such a high rank, many began to envy him.
Themistocles was particularly jealous of the great honors that his friend had won. His friends soon noticed his gloomy, discontented looks; and when they inquired what caused them, Themistocles said it was because the thought of the trophies of Miltiades would not let him sleep. Some time after, when he saw that Miltiades was beginning to misuse his power, he openly showed his dislike.
Not very far from Athens, out in the Aegean Sea, was the Island of Paros. The people living there were enemies of Miltiades; and he, being sole head of the fleet, led it thither to avenge his personal wrongs.
The expedition failed, however; and Miltiades came back to Athens, where Themistocles and the indignant citizens accused him of betraying his trust, tried him, and convicted him of treason.
Had they not remembered the service that he had rendered his country in defeating the Persians at Marathon, they would surely have condemned him to death. As it was, the jury merely sentenced him to pay a heavy fine, saying that he should remain in prison until it was paid.
Miltiades was not rich enough to raise this large sum of money, so he died in prison. His son Cimon went to claim his body, so that he might bury it properly; but the hardhearted judges refused to let him have it until he had paid his father's debt.
Alcibiades in Disgrace
Acropolisart treasuresatheneathenian peopleAtheniansAthensCimonDariusdivine favorglorious victorygoddess of retributionimmense statueMiltiadesNemesisPersiansPhidiasThemistoclestreason


text:
A Look Ahead at our Winter 2017 Programming
By Film at Lincoln Center on December 7, 2016 in Dance on Camera, Film Comment Selects, New Directors/New Films, NYJFF, Rendez-Vous, Retrospective
Jean-Luc Godard's La Chinoise. Image courtesy of The Kobal Collection / Gueville/Parc/Simar/Anouchka/Athos.
Get your calendars out! We've announced dates for our winter season of repertory programming and festivals…
Illuminating Moonlight
With the ravishing, unforgettable Moonlight, Barry Jenkins has established himself as one of today's major voices in independent American filmmaking. This series brings together Jenkins's two features (including his ripe-for-rediscovery debut, Medicine for Melancholy) with a selection of films that informed the making of his latest, handpicked by the director himself. These major works of queer, black, and international art cinema are, like Moonlight, rich, stylistically sensual, and compassionate portraits of outsiders. Taken together, the films in this series serve to contextualize Jenkins's work and offer insight into the making of a modern masterpiece. Barry Jenkins in person!
New York Jewish Film Festival
The year 2017 marks the 26th edition of the New York Jewish Film Festival, a partnership between the Jewish Museum and the Film Society of Lincoln Center. JFF brings you the finest narrative and documentary films from around the world that explore the diversity of Jewish experience. This year's festival features a wide-ranging and exciting lineup of films and shorts from the iconic to the iconoclastic. Including global, U.S., and New York premieres, a tribute to actress and cabaret artist Valeska Gert, a 50th anniversary screening of Mel Brooks's The Producers, and a related poster exhibition honoring the life and work of Zero Mostel and Gene Wilder, this year's NYJFF will entertain film lovers from all backgrounds.
Neighboring Scenes
Now in its second year, Neighboring Scenes is the Film Society's showcase of contemporary Latin American cinema. Highlighting impressive recent productions from across the region, this selective slate of premieres exhibits the breadth of styles, techniques, and approaches employed by Latin American filmmakers today. Neighboring Scenes spans a wide geographic range, featuring established auteurs as well as fresh talent from the international festival scene. Presented by the Film Society of Lincoln Center and Cinema Tropical.
Andrzej Wajda's Ashes and Diamonds
Dance on Camera
Dance on Camera Festival remains the longest-running dance film festival in the world, providing a platform for choreographic storytelling and creative expression, and intimate access to innovative media artists and their cinematic works. Each February, the Festival presents feature-length documentary and narrative films, inventive short films, filmmaker panels and special events, cutting edge media and art installations, as well as engaging community and student programs. Presented by the Film Society of Lincoln Center and Dance Films Association.
Tribute to Andrzej Wajda
Andrzej Wajda passed away on October 9, 2016 in Warsaw at the age of 90, nearly a month after the world premiere of his last film, Afterimage. For more than half a century, Wajda crafted a filmography that encapsulated the essence of postwar Poland and constitutes, quite simply, one of the great legacies of world cinema. No single visual style or strategy characterizes his films. His work often employed intricately illuminated deep spaces as well as looser, more vérité methods; many served as counter-narratives to the officially sanctioned records kept by Stalinized Poland; others were more oblique and meditative as they reckoned with concepts including individualism, one's duty toward others, and the meaning of freedom. This February, the Film Society is honored to present a selection of the Polish master's previous films in celebration of his monumental lifework, as well as the New York premiere of Afterimage, an impassioned memorial to the great avant-garde artist Władysław Strzemiński. Presented in partnership with the Polish Cultural Institute.
Film Comment Selects
Of all the annual film festivals in New York, there is no other quite like this one. Film Comment's festival of movies returns in its 17th edition with a selection of titles curated by the magazine's editors. It's an offering of strikingly bold visions, mixing New York premieres of new films and long-unseen older titles that deserve the big-screen treatment. As evidenced by such past selections as Claire Denis's Trouble Every Day, Olivier Assayas's demonlover, Park Chan-wook's Oldboy, Kathryn Bigelow's The Hurt Locker, Werner Herzog's Cave of Forgotten Dreams, Christian Petzold's Phoenix, and Terence Davies's Sunset Song, these are films that play by their own rules, works of considered artistry that reflect the philosophy of a magazine that has been essential for film lovers for more than 50 years.
William Wyler's Wuthering Heights (1939). Image courtesy of United Artists / The Kobal Collection.
In the 170 years since its publication, Emily Brontë's only novel, Wuthering Heights, has been one of the most frequently adapted works of literature, fascinating, inspiring, and provoking some of cinema's greatest directors to try to render its dark, romantic, politically charged majesty. Its timeless story—the impossible love of Catherine Earnshaw and Heathcliff, and the far-reaching scars it leaves on their families—has been transposed to various historical periods and countries, has had the races and genders of its characters changed to striking effect, and has been incarnated through a diverse assortment of visual aesthetics and performance styles. Which is the definitive adaptation? Wyler's classical Hollywood rendition? Buñuel's surrealist reimagining? Rivette's materialist ghost story? Yoshida's stark expressionist take? Arnold's kitchen-sink realist interpretation? Decide for yourself by joining the Film Society in revisiting five of the greatest attempts to put Wuthering Heights on the screen.
Rendez-Vous with French Cinema
Rendez-Vous with French Cinema returns in March with another edition that exemplifies the variety and vitality of contemporary French filmmaking. The films on display, by emerging talents and established masters, raise ideas both topical and eternal, and many take audiences to entirely unexpected places. Highlights from recent Rendez-Vous with French Cinema editions include Claire Denis's 35 Shots of Rum, François Ozon's In the House, Robin Campillo's Eastern Boys, Michel Gondry's Mood Indigo, and Benoît Jacquot's 3 Hearts. Co-presented with UniFrance Films, the 22nd edition of Rendez-Vous will demonstrate that the landscape of French cinema is as fertile, inspiring, and distinct as ever.
New Directors/New Films
Celebrating its 46th edition in 2017, the New Directors/New Films festival introduces New York audiences to the work of emerging filmmakers from around the world. Throughout its rich, nearly half-century history, New Directors has brought previously little-known talents like Pedro Almodóvar, Chantal Akerman, Hou Hsiao-hsien, Christopher Nolan, Laura Poitras, Spike Lee, and Kelly Reichardt to wider audiences. We hope you'll join us in celebrating a group of filmmakers who represent the present and anticipate the future of cinema: daring artists whose work pushes the envelope and is never what you'd expect. Presented by the Film Society of Lincoln Center and the Museum of Modern Art.
March 29–April 6
Jean-Pierre Léaud is to the French New Wave what Anna Magnani was to Italian Neorealism and what John Wayne was to American westerns: its spirit, its emblem, its avatar. The actor, who last year received the Cannes Film Festival's Honorary Palme d'Or in recognition of a career spanning nearly 60 years, first broke through as François Truffaut's on-screen surrogate Antoine Doinel in 1959's The 400 Blows, and he won Best Actor at the 1966 Berlin Film Festival for Jean-Luc Godard's Masculin féminin. Since then he has worked with French masters Jacques Rivette, Jean Eustache, Philippe Garrel, Bertrand Bonello, and Olivier Assayas, and such key international filmmakers as Tsai Ming-liang, Bernardo Bertolucci, Pier Paolo Pasolini, Aki Kaurismäki, and Raúl Ruiz. On the occasion of the release of Albert Serra's The Death of Louis XIV (NYFF54)—in which he delivers a magisterial, career-capping performance as the longest-reigning French monarch during his final days—the Film Society is proud to pay tribute to the prolific actor's irresistible presence and undeniable legacy.
Director Adam Leon on the Making of Italian Studies and Working with Vanessa Kirby


The Smithers Art Gallery is located in the historic Central Park Building, on the corner of Main and Highway 16. We feature local and regional artwork that changes monthly. Artist cards and prints available for puchase. Admission is by donation. Open Monday to Saturday 12 to 4 PM. Summer Art Camp now taking enrollment.
Experience the finest art of B.C.'s Northwest. Located on vibrant Main St. in downtown Smithers, the Four Seasons Gallery showcases the work of regional artists-- from sculptures and paintings to native carvings. Look for us beside CIBC at 1230 Main Street. Private appointments are welcome by calling in advance.


A brilliant but naive maths student is reluctantly drawn into the dark and treacherous world of industrial espionage when he is recruited by an old school spy with a hidden agenda.
Cultures collide in the treacherous world of industrial espionage. A burnt-out old school spy attempts to turn a bunch of reluctant young cyber truants into an effective counter intelligence unit, while thwarting sinister forces determined to settle old scores.


"I came to NKU with only knowing one person. I registered for Freshstart to get to know campus and some other people. Once Freshstart ended I came out with a whole friend group that I still hang out with to this day. Three were in Freshstart with me and one was a roommate with my friends from Freshstart. Later on me and a guy from my Freshstart group built a relationship. On October 24, 2017 we started our relationship together and have been together ever since."


text:
Koreyoshi Kurahara (jap. , Kurahara Koreyoshi; * 31. Mai 1927 in Kuching, Sarawak, Malaysien; † 28. Dezember 2002 in Yokohama, Japan) war ein japanischer Filmregisseur und Drehbuchautor.

Kurahare studierte an der Fakultät für Künste der Nihon-Universität. Ab 1952 war er Direktionsassistent des Shochiku Kyoto Studio. 1957 debütierte er als Regisseur mit dem Film Ore wa matteru ze (I am waiting). Mit Aru kyōhaku (Intimidation) schuf er 1960 den ersten japanischen Film Noir. Es folgten Filme wie Kyōnetsu no kisetsu (The Warped Ones, 1960), Nikui an-chikushō (I hate but love, 1962), Kuroi taiyō (Black Sun, 1964) und Ai no kawaki (Thirst for Love, 1966). Spätere Erfolge waren Nankyoku monogatari (Antarctic, 1983) Haru no kane (1985) und – in Zusammenarbeit mit Roger Spottiswoode – Hiroshima (1995).

Außerdem drehte Kurahara auch zwei Dokumentarfilme (Der Gletscherfuchs, 1978 und Zou monogatari, 1980) und wirkte an mehreren Fernsehserien mit. Zu mehreren seiner Filme (u. a. Antarctica) schrieb er auch das Drehbuch. Sein Onkel war der Literaturkritiker Korehito Kurahara, sein Bruder Koretsugu Kurahara wurde gleich ihm als Regisseur bekannt.

Filmografie (Auswahl)
 1957: Ore wa matteruze ()
 1960: Kyonetsu no kisetsu
 1964: Die Liebe des Mädchens Kiyono (, Shuen)
 1969: Safari 5000 ()
 1978: Der Gletscherfuchs (, Kita-kitsune monogatari)
 1981: Seishun no mon ()
 1983: Taro und Jiro in der Antarktis (, Nankyoku monogatari)
 1988: Umi e, See You ( See You)
 1991: Strawberry Road (, Sutoroberī rōdo)
 1995: Hiroshima

Einzelnachweise
Asia-Team: Koreyoshi Kurahara (1927–2002)

Filmregisseur
Drehbuchautor
Japaner
Geboren 1927
Gestorben 2002
Mann