text:
\section{Introduction.} \label{s.intro}
In this note we take up the investigation, initiated in \cite{K} and continued in \cite{IK}, concerning the \emph{discrepancy} of various geometrical shapes with respect to non-atomic measures (colorings). In order to discuss the problems we are interested in we need to introduce some notation. As in \cite{K,IK} we divide the Euclidean plane $\mathbb R^2$ into the unit cells
$$Q_p\eqdef[p_1,p_1+1)\times[p_2,p_2+1),\quad p=(p_1,p_2)\in\mathbb Z^2,$$
and color each one of the cells either black or white. Thus a \emph{checkerboard coloring} $f$ of the plane is a function
$$f:\mathbb R^2\to \{-1,+1\},$$
such that $f$ is constant on each unit cell $Q_p$. Now let $S$ be a simple curve lying in the checkerboard-plane and $f$ be a coloring as before. We define the \emph{discrepancy} of $S$ with respect to the given coloring $f$ to be the difference of the `white' length of $S$ against the `black' length of $S$, in absolute value. In \cite{K} it was proved that for any checkerboard coloring there exist arbitrarily long line segments $I$ with discrepancy at least $c\sqrt{|I|}$, for some numerical constant $c>0$. On the other hand in \cite{IK} the authors proved that for arbitrarily large $R>0$ there exists a circular arc of radius comparable to $R$ which has discrepancy at least $c\sqrt{R}$ for some numerical constant $c>0$. The authors in \cite{IK} also ask whether there is a \emph{full circle} $C$ with large discrepancy. We answer this question in a strong form by showing that for every radius $t\geq 1$ there exists a full circle of radius either $t$ or $2t$ with discrepancy at least $c\sqrt t$. Noting by $C(x,t)$ the circle of radius $x\in\mathbb R^2$ and radius $t>0$ we have:

\begin{theorem}\label{t.fullcircle}
	Let $f$ be a checkerboard coloring of the plane as before and let $t\geq 1$.There exists a $x\in\mathbb R^2$ such that
$$\mbox{either}\quad \ABs{\int_{C(x,t)} f} \geq c t^\frac{1}{2}\quad\mbox{or}\quad \ABs{\int_{C(x,2t)} f} \geq c (2t)^\frac{1}{2},$$
for some numerical constant $c>0$.
\end{theorem}

We also show that if we just care about finding \emph{arcs} with large discrepancy, then we can do so for any fixed radius $t\geq 1$.

\begin{theorem}\label{t.singleradius}
	Let $f$ be a checkerboard coloring of the plane as before and let $t\geq 1$.There exists a circular arc $K$ of radius $t$ such that
$$ \ABs{\int_K f} \geq c t^\frac{1}{2},$$
for some numerical constant $c>0$.
\end{theorem}

The results in \cite{K}, \cite{IK} as well as Theorem \ref{t.fullcircle} and Theorem \ref{t.singleradius}, are direct consequences of their finite counterparts. To make this precise, let $N$ be a positive integer and write $Q_N$ for the square $Q_N\eqdef[0,N)^2$. We now consider $Q_N$ as a union of congruent unit cells in the form
\begin{align}
Q(p)\eqdef p+[0,1)^2, \quad p=(p_1,p_2)\in G,
\end{align}
where $G$ is the part of the lattice $\mathbb Z^2$ that lies in $Q_N$, that is $G\eqdef\{(p_1,p_2):0\leq p_1,p_2\leq N-1\}$. A \emph{coloring} of $Q_N$ will be a function of the form
\begin{align}
	f_N:Q_N\to \{-1,+1\},\quad f_N\mbox{ constant in each cell }Q(p).
\end{align}
We extend $f_N$ to the whole plane $\mathbb R^2$ by setting $f_N\equiv 0$ outside $Q_N$. The discrepancy of a circle $C(x,t)$ with center $x\in\mathbb R^2$ and radius $t>0$ is defined as
$$ D_t(f_N,x)\eqdef \int_{C(x,t)} f_N=(f_N*\sigma_t)(x),$$
where $\sigma_t$ is the arc-length measure on a circle of center $0$ and radius $t$. A problem that arises is that discrepancy of circles with respect to a finite coloring in general only corresponds to discrepancy of arcs with respect to a coloring of the whole plane. The reason of course is that a circle $C(x,t)$ might intersect $Q_N$, and even have large discrepancy with respect to the finite coloring of $Q_N$, without necessarily lying entirely inside $Q_N$.

For example, Iosevich and the first author prove in \cite[ Theorem 1]{IK} that for any coloring $f_N$ of $Q_N$, there exists a circular arc $K$ of radius $R$, $N/5<R<N/4$, with
	$$\ABs{\int_{K} f_N}\geq c N^\frac{1}{2},$$
for some numerical constant $c>0$. The authors are not able to conclude that there is a full circle with large discrepancy since their main tool is to show that the $L^2$-type discrepancy
$$\frac{1}{N^3}\int_{N/5} ^{N/4} \int_{\mathbb R^2}|D_t(f_N,x)|^2 dx\, dt,$$
is large. However, the previous $L^2$ integral takes into account arcs as well as full circles. Furthermore, the averaging in the radial variable results to circles or circular arcs of radius \emph{comparable} to $N$ instead of radius exactly $N$.

In this note we partially fix the previous two problems by avoiding the radial averaging. We also show that circles that do not lie entirely inside $Q_N$ do not significantly contribute to the $L^2$ norm $\norm D_t(f,\cdot).L^2. ^2+\norm D_{2t}(f,\cdot).L^2. ^2$ when $N\gtrsim t^2$. This results to a full circle of radius either $t$ or $2t$ with large discrepancy if $t$ is small comparable to $N$.

Theorem \ref{t.fullcircle} is an immediate consequence of the following theorem:

\begin{theorem}\label{t.fullcircleN} Let $t\geq 1$ and for a positive integer $N\geq 100 t^2$ consider any finite coloring $f_N : Q_N\to\{−1, +1\}$ of $Q_N$. There exists $x\in\mathbb R$ such that the circle $C(x,2t)\subset Q_N$ and
$$\mbox{either}\quad\ABs { \int_{C(x,t)} f_N } \geq c t^\frac{1}{2} \quad\mbox{or}\quad \ABs { \int_{C(x,2t)} f_N } \geq c  (2t)^\frac{1}{2} ,$$
where $c>0$ is some numerical constant.
\end{theorem}

Similarly, Theorem \ref{t.singleradius} is a consequence of:
\begin{theorem}\label{t.singleradiusN} Let $f_N : Q_N\to\{−1, +1\}$ be a finite coloring of $Q_N$ and $N\simeq t$. There exists a circle $C$ of radius $t$ such that
	$$\ABs{\int_C f_N }\geq c\sqrt{t},$$
	where $c>0$ is some numerical constant.
\end{theorem}

\begin{remark}
	Note that in Theorem \ref{t.singleradiusN} we cannot guarantee that the circle $C$ is contained in $Q_N$. Thus, Theorem \ref{t.singleradiusN} only results to an \emph{arc} of radius $t$ in the infinite coloring of the plane with discrepancy $\sim \sqrt{t}$.
\end{remark}

We note that discrepancies with respect to non-atomic colorings have been considered by Rogers in \cite{RO1}, \cite{RO2} and \cite{RO3} where the author considers, among other things, the discrepancy of lines and half spaces with respect to finite colorings of the plane. Rogers proves lower bounds for the discrepancy of these families of sets with respect to generalized colorings. His results do not seem to be comparable to the results in this paper.

The rest of the paper is organized as follows. In Section \ref{s.fullcircles} we use the classical asymptotic estimates for the Fourier transform of the arc-length measure on the circle in order to prove Theorem \ref{t.fullcircleN}. In Section \ref{s.singleradius} we prove Theorem \ref{t.singleradiusN} by an appeal to the asymptotic estimates of the Fourier transform of the arc-length measure together with an appropriate Poincar\'e-type inequality. Finally in Section \ref{s.generalsets} we discuss the discrepancy of  more general families of sets with respect to a coloring of the plane. The corresponding lower bounds are contained in Theorem \ref{t.segment}. The main tool for these estimates are lower bounds for the averages of Fourier transforms of indicator functions. For the sake of completeness, we include these estimates and their proofs  in Section \ref{s.lowerfourier}.

\section{Notations} Throughout the paper $c$ denotes a numerical positive constant which might change even in the same line of text. We often suppress numerical constants by using the symbol $\lesssim$. Thus $A\lesssim B$ means that $A\leq c B$ for $c$ as described. Likewise the notation $A\simeq B$ means that $A\lesssim B$ and $A\gtrsim B$. We write $B(x,r)$ for the Euclidean disk of radius $r>0$ centered at $x\in\mathbb R^2$. We also write $C(x,r)=\partial B(x,r)$ for the circle of radius $r>0$, centered at $x\in\mathbb R^2$. For the unit circle of $\mathbb R^2$ we also use the symbol $S^{1}=C(0,1)$.

\section{Full circles of large discrepancy}\label{s.fullcircles} Recall that the discrepancy of a circle $C(x,t)$ with respect to the coloring $f_N$ of the square $Q_N=[0,N)^2$ is defined as
$$D_t(f_N)(x)\eqdef \int_{C(x,t)}f= (f*\sigma_t)(x),$$
where $\sigma_t$ is the arc-length measure on the circle $C(0,t)$. Observe that the function $(f_N *\sigma_t)(x)$ has support in $Q_N+B(0,t)$ in general. However in Theorems \ref{t.fullcircleN} and \ref{t.singleradiusN} we only need to consider values $t\lesssim N$ so the measure of the support is comparable to $N^2$. We thus study the $L^2$ discrepancy
$$D_t(f_N,2)\eqdef \bigg(\frac{1}{N^2}\int_{\mathbb R^2} |(f_N*\sigma_t)(x)|^2 dx \bigg)^\frac{1}{2},$$
since we obviously have the bound
$$\sup_{x\in\mathbb R^2}| D_t(f_N)(x) | \gtrsim D_t(f_N,2).$$
Furthermore, denoting by $\hat \sigma_1$ the Fourier transform of the measure $d\sigma_1$,
$$\hat \sigma_1 (\xi)=\int_{S^1}e^{-2\pi i x'\cdot\xi}d\sigma_1(x'),$$
we have that
$$D_t(f_N,2)^2=\frac{t^2}{N^2}\int_{\mathbb R^2} |\widehat{ f_N}(\xi)|^2 |\hat \sigma_1 (t \xi)|^2 d\xi.$$
The following Lemma is the most essential part of the proof of Theorem \ref{t.fullcircleN}.

\begin{lemma}\label{l.double} For all $|\xi|\geq \frac{1}{2\pi}$ we have that
	$$|\hat \sigma _1 (\xi)|^2+|\hat \sigma_1 (2\xi)|^2\gtrsim\frac{1}{|\xi|}.$$
\end{lemma}

\begin{proof} Setting $|\xi|=r$ we express the radial function $\hat \sigma_1$ by the well known formula
	 $$\hat \sigma_1 (r)=2\pi J_0(2\pi r) ,$$
where $J_0$ is the $0$-th order Bessel function. We use the asymptotic estimate
$$J_0(r)\simeq \frac{1}{\sqrt{ r}}\bigg(\cos(r-\frac{\pi}{4})+e(r) \bigg),$$
where the error term satisfies
$$|e(r)|\leq  \frac{1}{5r},$$
for $r\geq 1$. This is classical as $r\to+\infty$ but with a little more effort one can get the validity of the previous estimate for all $r\geq 1$. The previous asymptotic estimate easily implies that
\begin{align*}
	|\hat \sigma_1(r)|^2+|\hat \sigma_1(2r)|^2\gtrsim\frac{1}{r},
\end{align*}
for all $r\geq\frac{7}{2\pi}$.
For $\frac{1}{2\pi}\leq r \leq \frac{7}{2\pi}$ one can just directly check the zeros of $J_0$ to see that there is no $r$ so that $J_o(2\pi r)=J_0(4\pi r)=0$. We refer the interested reader to \cite[p. 113, \S 6.3]{M} where an identical argument is used for the derivation of a formula involving the $1$-st order Bessel function.
\end{proof}

\begin{corollary}\label{c.tor2t} For any $t\geq 1$ and any positive integer $N$ we have that
	$$D_t(f_N,2)^2+D_{2t}(f_N,2)^2\gtrsim t.$$
\end{corollary}

\begin{proof} We use Plancherel's theorem to write
	\begin{align*}
		D_t(f_N,2)^2+D_{2t}(f_N,2)^2 & =\frac{1}{N^2}\int_{\mathbb R^2} |\widehat{ f_N}(\xi)|^2 \big(|\hat \sigma_t(\xi)|^2+|\hat \sigma_{2t}(\xi)|^2 \big)\ d\xi \\
& \gtrsim \frac{1}{N^2}\int_{|\xi|\leq \frac{1}{2\pi}} |\widehat {f_N} (\xi/t)|^2 \big(|\hat \sigma_1 ( \xi)|^2+|\hat \sigma_ 1 (2 \xi)|^2 \big)\ d\xi \\ & \quad +\frac{1}{N^2}\int_{|\xi|> \frac{1}{2\pi}} |\widehat {f_N} (\xi/t)|^2 \big(|\hat \sigma_1 ( \xi)|^2+|\hat \sigma_ 1 (2 \xi)|^2 \big)\ d\xi\eqdef I+II.
	\end{align*}
For $I$ observe that $J_0 (2\pi \cdot)$  has no root in the range $|\xi|\leq \frac{1}{2\pi}$. We immediately get
$$ |I|\gtrsim \frac{1}{N^2}\int_{|\xi|\leq \frac{1}{2\pi}}|\widehat {f_N}(\xi/t)|^2 d\xi \geq \frac{t^2}{N^2}\int_{|\xi|\leq \frac{1}{2\pi t}}|\widehat{ f_N}(\xi)|^2 d\xi .$$
For $II$ we use Lemma \ref{l.double} to write
$$|II| \gtrsim \frac{1}{N^2}\int_{|\xi|>\frac{1}{2\pi}} |\widehat {f_N}(\xi/t)|^2 \frac{1}{|\xi|}d\xi \gtrsim\frac{t}{N^2}\int_{ [-\frac{1}{2},\frac{1}{2}]^2 \setminus \{\ |\xi|>\frac{1}{2\pi t } \ \}} |\widehat {f_N}(\xi)|^2 d\xi.$$
Combining the estimates and remembering that $t\geq 1$ we get
$$ 	D_t(f_N,2)^2+D_{2t}(f_N,2)^2 \gtrsim \frac{t}{N^2}\int_{[-\frac{1}{2},\frac{1}{2}]^2} |\widehat {f_N}(\xi)|^2d \xi \gtrsim t $$
where we have used that
\begin{align}\label{e.1period}
	|\widehat {f_N}(\xi)|^2=\bigg|\frac{\sin(\pi\xi_1)}{\pi\xi_1}\frac{\sin(\pi\xi_2)}{\pi\xi_2}\sum_{j,k=0} ^{N-1}z_{jk} e^{2\pi i (j\xi_1+k\xi_2)}\bigg|^2\gtrsim \bigg|\sum_{j,k=0} ^{N-1}z_{jk} e^{2\pi i (j\xi_1+k\xi_2)}\bigg|^2
	\end{align}
for $\xi\in[-\frac{1}{2},\frac{1}{2}]^2$. This in turn is a consequence of the elementary estimate $|\sin(\pi x)|\geq 2|x|$ for $|x|\leq \frac{1}{2}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t.singleradiusN}] Given $t\geq 1$ let $N\geq B t^2$ be a positive integer for some numerical constant $B>0$ to be determined later. By corollary \ref{c.tor2t} we have that
	$$D_s(f_N,2)\gtrsim \sqrt{s},$$
where $s$ is equal to either $t$ or $2t$. Consider the cube $Q_1\eqdef[s,N-s]^2$. We have
\begin{align*}
	\int_{[s,N-s]^2}|D_s(f_N)(x)|^2dx &=D_s(f_N,2)^2-\frac{1}{N^2}\int_{[-s,N+s]^2\setminus [s,N-s]^2 } |f_N*d\sigma_s(x)|^2dx \\
    &\gtrsim s(1-24 s^2/N )\gtrsim s,
\end{align*}
if $B$ is large enough, say $B\geq 100$. Since all the circles with centers in $[s,N-s]^2$ and radius $s$ are contained in $Q_N$ this proves Theorem \ref{t.fullcircleN}.
\end{proof}

\section{Single radius discrepancy for arcs} \label{s.singleradius}
Theorem \ref{t.fullcircleN} solves the problem of finding a full circle with large discrepancy. There is one element however that is not very satisfactory, namely the fact that we cannot guarantee that for \emph{every} radius $t\geq 1$ there corresponds a circle of radius \emph{exactly} $t$ with large discrepancy. The problem is caused by the roots of $\hat\sigma_1(\xi )$ which allow the expression
$$\int_{\mathbb R^2} |\hat f(\xi)|^2 |\hat \sigma_t (\xi)|^2d\xi$$
to become small. When $N\simeq t$ we can deal with this problem by essentially throwing away small neighborhoods of the roots of $\hat \sigma_1$ and showing that we don't loose much of the $L^2$ mass of the function $\hat f$.

We begin by analyzing the behavior of $\hat \sigma_1(|\xi|)$. By standard estimates we have the asymptotic expansion
\begin{equation}\label{e.asymptotic}
	\hat\sigma_1(\xi)=\hat\sigma_1(|\xi|)=2|\xi|^{-\frac{1}{2}}\cos\big(2\pi|\xi|-\frac{\pi}{4}\big)+O(|\xi|^{-\frac{3}{2}}),\quad |\xi|\to+\infty;
\end{equation}
see for example \cite{W}. Observe that the cosine term in the asymptotic formula above vanishes exactly when
$$|\xi|	=\beta_k\eqdef \big(\frac{k}{2}+\frac{3}{8}\big), \quad k=0,1,2,\ldots.$$
For a small parameter $0<w<\frac{1}{8}$ we define the neighborhoods
$$A_w(\beta_k)\eqdef \{\xi\in\mathbb R^2: | |\xi|-\beta_k| < w\}.$$
Observe that our choice of $w$ implies that the $A_w$'s do not overlap. The following lemma analyzes the behavior of $\hat \sigma_1$ away from the annuli $A_w$

\begin{lemma}\label{l.lowerestimate} For every sufficiently small $w>0$ there exists a constant $c(w)$ such that
	$$|\hat \sigma_1(|\xi|)|^2 \gtrsim_w \begin{cases} \frac{1}{|\xi|}, &\quad |\xi|>c(w), \quad \xi\notin \cup_k A_w(\beta_k) \\
		 1 ,&\quad |\xi|\leq c(w),\quad \xi\notin \cup_k A_w(\gamma_k),
	\end{cases}$$
where $\gamma_1<\gamma_2<\ldots<\gamma_M$ are the roots of $\hat \sigma_1$ in $\{|\xi|<c(w)\}$.
\end{lemma}

\begin{proof}
	By \eqref{e.asymptotic} there exist constants $c_1,c_2>0$ such that for $|\xi|>c_1$ we have

	\begin{align*}
		|\hat \sigma_1 (|\xi|)|^2&\gtrsim \frac{1}{|\xi|}\bigg( \Abs{\cos(2\pi |\xi|-\frac{\pi}{4})} ^2 -\frac{c_2}{|\xi|} \bigg).
	\end{align*}
Now the minimum of the cosine term in the region $\{|\xi|>c_1\}\setminus \cup_k A_w(\beta_k) $ is obviously achieved when $||\xi|-\beta_k|=w$ for some $k$.	If $w<\frac{1}{4}$ we have
$$\Abs{\cos(2\pi |\xi|-\frac{\pi}{4})}\geq 4 \Abs{|\xi|-\beta_k}=4w$$
We can thus estimate
	\begin{align*}
	|\hat \sigma_1(|\xi|)|^2 & \gtrsim \frac{1}{|\xi|} \bigg (16w^2 -\frac{c_2}{|\xi|}\bigg) \gtrsim_w \frac{1}{|\xi| },
	\end{align*}
	whenever $|\xi|>\frac{c_2}{8w^2}\eqdef c(w)$ and $\xi\notin \cup_k A_w(\beta_k)$.

Now there are finitely many roots of $\hat \sigma_1(\xi)$ in the ball $\{|\xi|\leq c(w)\}$ and let us denote them by $\gamma_1<\gamma_2<\cdots<\gamma_M$. By compactness we have that $|\hat \sigma_1(\xi)|^2 \gtrsim_w 1$ whenever $|\xi|\leq c(w)$ and $x\notin \cup_k A_w(\gamma_k)$. In order to make sure that all the annuli are non-overlapping we have to take $w<\min\{\frac{1}{8},\frac{1}{2}\min_k(\gamma_{k+1}-\gamma_k),\beta_0-\gamma_M\}\eqdef w_0$.
 \end{proof}

Lemma \ref{l.lowerestimate} can be used to obtain a favorable estimate for $D_t(f,2)$ as follows. Adopting the notations of Lemma \ref{l.lowerestimate} and invoking Plancherel's theorem we write for every $w<w_o$ small enough (remember $t\simeq N$)
	\begin{align*}
	D_t(f_N,2)^2  &\gtrsim  \frac{1}{N^2} \int_{ \{|\xi|<c(w) \}\setminus \cup_k A_w(\gamma_k) } |\widehat {f_N}(\xi/t)|^2 |\hat \sigma_1 (|\xi|)|^2d\xi + \frac{1}{N^2}\int_{\{c(w)<|\xi|<t)\}\setminus \cup_k A_w(\beta_k)} |\widehat {f_N}(\xi/t)|^2 |\hat\sigma_1(|\xi|)|^2d\xi \\
&\gtrsim_w \frac{1}{N^2} \int_{ \{|\xi|<c(w) \}\setminus \cup_k A_w(\gamma_k) }  |\widehat{f_N}(\xi/t)|^2 d\xi +  \frac{1}{N^2}\int_{\{c(w)<|\xi|<t)\}\setminus \cup_k A_w(\beta_k)}   |\widehat {f_N}(\xi/t)|^2 \frac{1}{|\xi|}d\xi.
\end{align*}
Setting $E_w\eqdef ( \cup_k A_w(\gamma_k)) \cup (  \cup_k A_w(\beta_k)  )$ and combining the previous estimates we have
\begin{align}\label{e.estimatewithholes}
D_t(f_N,2)^2&\gtrsim_w \frac{1}{tN^2} \int_{\{|\xi|<t \}\setminus E_w}  |\widehat {f_N}(\xi/t)|^2 d\xi =  \frac{t}{N^2}\int_{B(0,1)\setminus \frac{1}{t}E_w} |\widehat {f_N}(\xi)|^2 d\xi .
\end{align}

The following Poincar\'e-type inequality will allow us to show that the $L^2$ norm of $\widehat{f_N}$ on $B(0,1)\setminus \frac{1}{t} E_w$ is comparable to the the full $L^2$ norm of $\widehat {f_N}$.

\begin{proposition}\label{p.poincare}
	For any positive integer $d\geq 1$ let $B=B(0,R)\subset \mathbb R^d$ be a Euclidean ball in the $d$-dimensional Euclidean space, centered at the origin, and $g\in  C^1(B)$. Suppose that  $0<\beta_1<\beta_2<\cdots<\beta_N<R$. We set $\beta_0\eqdef 0$ and $\beta_{N+1}\eqdef R$ and
	$$\beta\eqdef \min_{1\leq n\leq N+1 }(\beta_{n}-\beta_{n-1}).$$
For $k=1,2,\ldots,N$,		 we set
$$A_w(\beta_k)\eqdef \{\xi\in\mathbb R^d: | |\xi|-\beta_k| < w\}.$$
Then for $0<w<\beta/3$ we have that
		$$ \int_{ B}|g(x)|^2 \lesssim \int_{ B\setminus (\cup_{n=1} ^N A_w(\beta_n))  }|g(x)|^2+ w^2\int_{ B} |\nabla g(x)|^2 .$$
\end{proposition}

\begin{proof}
	We first focus on a single annulus $A_s(\beta_n)$ for some $1\leq n \leq N$ and some real parameter $s$ in the interval $[w,2w)$. For $\beta_n-s<r\leq \beta_n+s$ and $u\in S^{d-1}$ we have that
	$$g(ru) = g((\beta_n-s)u)+ \int_{\beta_n-s} ^r \partial_t( g(tu)) dt.$$

	Using the simple inequality $\frac{1}{2}(a+b)^2\leq a^2+b^2$ for $a,b\in\mathbb R$, and the Cauchy-Schwarz inequality we conclude that
	$$|g(ru)|^2 \lesssim |g((\beta_n-s)u)|^2+ 2s \int_{\beta_n-s} ^{\beta_n+s} |\partial_t( g(tu))|^2 dt.$$

	Multiplying by $r^{d-1}$ and integrating for $r\in[\beta_n-s,\beta_n+s)$ and $u\in S^{d-1}$, we get
	\begin{align}
		\int_{A_s(\beta_n)} |g(x)|^2dx &\lesssim \int_{\beta_n-s} ^{\beta_n+s} r^{d-1}dr \int_{S^{d-1}}|g((\beta_n-s)u)|^2 d\sigma^{(d-1)} _1 (u)\\
		&+ 2s \int_{S^{d-1}}\bigg(\int_{\beta_n-s} ^{\beta_n+s}\Big(\int_{\beta_n-s} ^{\beta_n+s} |\partial_t( g(tu))|^2 dt\Big) r^{d-1}dr\bigg) d\sigma_1 ^{(d-1)}(u).
	\end{align}
	Now observe that for $w<s\leq 2w$ and $r,t\in[\beta_n-s,\beta_n+s)$, we have that $r\simeq t \simeq \beta_n$. Hence,
	$$\int_{A_s(\beta_n)} |g(x)|^2dx \lesssim s \beta_n ^{d-1}\int_{S^1}|g((\beta_n-s)u)|^2 du+ s^2 \int_{A_s (\beta_n)} |\nabla  g(x) |^2dx.$$
	Integrating the left hand side for $s\in[w,2w)$ we see that
	$$\int_w ^{2w} \int_{A_s(\beta_n)}|g(x)|^2dx ds \gtrsim w \int_{A_w(\beta_n)}|g(x)|^2dx.$$
	On the other hand
	\begin{align}  \beta_n ^{d-1} \int_w ^{2w}  \int_{S^1}|g((\beta_n-s)u)|^2 s du  ds & \lesssim w \int _{B(0,\beta_n-w) \setminus \ B(0,\beta_n-2w)}|g(x)|^2 dx
		\\ &\leq w\int_{B(0,\beta_n-w) \setminus \ B(0,\beta_{n-1}+w)}|g(x)|^2dx,
	\end{align}
	since $w<\beta/3$. Finally we readily see that
	$$\int_w ^{2w} s^2 \int_{A_s(\beta_n) } |\nabla g(x)|^2 dx ds \lesssim  w^3 \int_{A_{2w}(\beta_n)} |\nabla g(x)|^2 dx.$$
	Putting these estimates together get
	$$ w \int_{A_w(\beta_n)}|g(x)|^2dx \lesssim  w\int_{B(0,\beta_n-w) \setminus \ B(0,\beta_{n-1}+w)}|g(x)|^2dx +w^3 \int_{A_{2w}(\beta_n)} |\nabla g(x)|^2 dx.$$
Observe that we have
	 $$\cup_{n=1} ^N {B(0,\beta_n-w) \setminus \ B(0,\beta_{n-1}+w)} \subseteq   B\setminus \big(\cup_{n=1} ^N A_w (\beta_n) \big)$$
	and the unions on both sides of the inclusion above are disjoint. Summing in $n$ we thus get
		$$\int_{\cup_{n=1} ^N  A_w(\beta_n)}|g(x)|^2dx \lesssim  \int_{B\setminus \cup_{n=1} ^N A_w (\beta_n) }|g(x)|^2dx +w^2 \int_{\cup_{n=1} ^N A_{2w}(\beta_n)}|\nabla g(x)|^2 dx.$$
	Adding the term $\int_{B\setminus  \cup_{n=1} ^N A_w (\beta_n) }|g(x)|^2dx$ in both sides of the inequality completes the proof.\end{proof}

Now Proposition \ref{p.poincare} and estimate \eqref{e.estimatewithholes} will allow us to conclude the proof of Theorem \ref{t.singleradiusN}:

\begin{proof}[Proof of Theorem \ref{t.singleradiusN}] Estimate \eqref{e.estimatewithholes} and Proposition \ref{p.poincare} imply that
\begin{align}\label{e.break}
	D^2 _t(f_N,2) & =\frac{1}{t}\int_{B(0,1)\setminus \frac{1}{t}E_w} |\widehat {f_N}(\xi)|^2 d\xi   \gtrsim_w \frac{1}{t}\bigg( \int_{B(0,1)}|\widehat{f_N}(\xi)|^2d\xi -\frac{w^2}{t^2} \int_{B(0,1)}|\nabla{\widehat{f_N}}(\xi)|^2d\xi\bigg).
\end{align}
Using the bounds
$$\int_{B(0,1)}|\widehat f_N(\xi)|^2d\xi\gtrsim\int_{[-\frac{1}{2},\frac{1}{2}]^2}\bigg|\sum_{j,k=0} ^{N-1}z_{jk} e^{2\pi i (j\xi_1+k\xi_2)}\bigg|^2d\xi\geq N^2,$$
and
$$ \int_{B(0,1) }|\nabla{\widehat{f_N}}(\xi)|^2d\xi\leq \int_{\mathbb R^2} |x|^2|f_N(x)|^2dx\lesssim N^4$$
in estimate \eqref{e.break} we get
\begin{align*}
	D^2 _t(f_N,2)	\gtrsim\frac{1}{t} \big(N^2-\frac{w^2}{t^2} N^4\big)\simeq t (1-c w^2),
\end{align*}
for some constant $c>0$. If $w$ is sufficiently small we conclude that $D_t(f_N,2)\gtrsim_w \sqrt{t}$ as we wanted to prove.
\end{proof}

\begin{remark} The calculations in this section show that
		$$D_t(f_N,2)\gtrsim \sqrt{t},$$
for $N\simeq t$. This alone is not enough to conclude the existence of a \emph{full circle} with large discrepancy $\sim \sqrt{t}$. Indeed, the argument used in the proof of Theorem \ref{t.fullcircleN} requires the validity of the previous estimate for $N\gtrsim t^2$ while, here, we only have it for $N\simeq t$.
\end{remark}

\section{Discrepancy with respect to general sets}\label{s.generalsets}
In this section we study the discrepancy of a coloring $f$ of the plane with respect to more general families of sets. To keep the exposition relatively simple let us assume that $S$ is a simple, closed, piecewise $C^1$ curve in the Euclidean plane and let $K$ denote its interior. Let $d\sigma_S$ denote the arc-length measure on $S$. In the previous sections we have studied the discrepancy of $f$ with respect to the family of all dilations and translations of the unit circle. Here, the relevant families are
$$\{x+r\tau K:x\in\mathbb R^2,\ r>0,\ \tau\in SO(2)\},$$
and
$$\{x+r\tau S:x\in\mathbb R^2,\ r>0,\ \tau\in SO(2)\}.$$
Note that we introduce rotations which was superfluous in the case of the circle. Here however it is absolutely essential. Indeed, consider the standard chessboard-like alternating coloring (i.e. adjacent squares have different colors) and let $K$ be the unit square with its sides parallel to the coordinate axes. Obviously the discrepancy of this coloring with respect to the dilations and translations of $K$ (or $\partial K$) is $\sim 1$ so the problem is trivial. Another option would be to place certain assumptions on the curvature of $\partial K$ but we will not pursue this here.

For $x\in\mathbb R^2,r>0$ and $\tau \in SO(2)$ we define
$$D_K(f_N,x,\tau,r)\eqdef (f*\chi_{r\tau K})(x)$$
and
$$D_S(f_N,x,\tau,r)\eqdef (f*d\sigma_{r\tau S})(x).$$

\subsection{Average estimates for the Fourier transform.}\label{s.lowerfourier}
We will obtain lower bounds on the discrepancies described above by studying their $L^2 $ averages. The most important ingredient of this approach is the following lemma describing the average asymptotic behavior of the Fourier transform of $d\sigma_S$ and $\chi_K$. These estimates are essentially contained in the proof of \cite[Theorem 3, Chapter 6]{M}.
\begin{lemma}\label{l.fourierKS} Let $S$ be a simple, closed, piecewise $C^1$ curve in the Euclidean plane and denote by $K$ its interior so that $S=\partial K$. There exist numerical constants $A_o>1$ and $R_o>0$ such that, if $R>R_o$ and $A>A_o$, then
$$\int_{R\leq |\xi|\leq AR} |\widehat{\chi_K}(\xi)|^2d\xi \gtrsim_A \frac{|S|}{R},$$
and
$$\int_{R\leq |\xi|\leq AR} |\widehat{d\sigma_S}(\xi)|^2 d\xi \gtrsim_A |S| R.$$
Here $|S|$ denotes the arc-length of $S$.
\end{lemma}

\begin{proof}We follow Montgomery from \cite[Theorem 3, Chapter 6]{M}. For $r>0$ we set
	$$g(r)=\int_{S^1}|\widehat{\chi_K}(r\xi')|^2d\sigma_1(\xi').$$
Under our assumptions on $K$, Montgomery proves the asymptotic estimate
$$\int_0 ^R g(r)r^5 dr\simeq |S|R^3,$$
as $R\to +\infty$. 	This means that there exist numerical constants $R_o,c_1,c_2>0$ such that
$$c_1 |S| R^3 \leq \int_0 ^R g(r)r^5 dr \leq c_2|S|R^3,$$
whenever $R>R_o$. For $A>1$ and $R>R_o$ we thus have

\begin{align*}
	\int_{R} ^{AR} g(r) r^5 dr \geq |S| R^3 (A^3c_1 -c_2) \gtrsim |S| R^3
\end{align*}
if $A>A_0$ where $A_o>1$ is a numerical constant. We conclude that
\begin{align}
 \int_{R\leq |\xi|\leq AR}  |\widehat{\chi_K}(\xi)|^2 d\xi= \int_R ^{AR} g(r) r dr \gtrsim_A |S|/R,
\end{align}
whenever $R>R_o$ and $A>A_o$. This proves the first estimate of the lemma.

For the second we modify the proof of \cite[Theorem 3, Chapter 6]{M}. With $h(x)=e^{-\pi R^2|x|^2}$ Montgomery shows that $ \|\chi_K*\nabla^2 h\|_2 ^2\simeq |S| / R  $ as $R\to +\infty$. On the other hand, by Green's theorem we have
$$\chi_K*\nabla^2 h=\int_S \frac{\partial h}{\partial\textbf n}(x-y) d\sigma_S(y).$$
Combining these two facts and using Plancherel's theorem we get
\begin{align*}
\frac{|S|}{R}&\simeq \int_{\mathbb R^2}\big| \widehat{\frac{\partial h}{\partial\textbf n}}(\xi)\big|^2	|\widehat{d\sigma_S}(\xi)|^2d\xi=\int_{\mathbb R^2}|\xi\cdot n|^2|\hat h(\xi)|^2 |\widehat{d\sigma_S}(\xi)|^2d\xi \\
&= \int_0 ^{+\infty}  |\hat h(r)|^2 \bigg( \int_{S^1} |\xi'  \cdot n|^2 |\widehat {d\sigma_S}( r \xi')|^2d\sigma(\xi') \bigg)\, r^3 dr,
\end{align*}
where $\hat h(r)=\hat h(|\xi|)=R^{-2}e^{-\pi R^{-2}r^2}$. Let us call
\begin{align}\label{e.y(r)}
	y(r)\eqdef  \int_{S^1} |\xi ' \cdot n|^2 |\widehat{d\sigma_S}( r \xi')|^2d\sigma(\xi') \leq \int_{S^1}  |\widehat{d\sigma_S}( r \xi')|^2d\sigma(\xi')  .
\end{align}
We have
$$\int_0 ^{+\infty} y(r) e^{-2\pi r^2/R^2} r^3 dr \simeq |S| R^3.$$
As in Montgomery \cite{M}, we use the Hardy-Littlewood Tauberian theorem \cite[Theorem 108]{HL} to conclude that
$$\int_0 ^R y(r) r^3 dr\simeq |S| R^3, $$
as $R\to+\infty$. Arguing as in the first part of the proof we conclude that there exist numerical constants $R_o$ and $A_o>1$ such that
$$\int_{R} ^{AR} y(r) r^3dr \gtrsim  |S| R^3,$$
whenever $R>R_o$ and $A>A_o$. By \eqref{e.y(r)} we conclude that
\begin{align*}
	\int_{R\leq |\xi|\leq AR} |\widehat{d\sigma_S}(\xi)|^2 d\xi  \geq \int_R ^{AR} y(r)r dr\gtrsim_A|S|R,
\end{align*}
as we wanted to prove.
\end{proof}

\subsection{Lower bounds for discrepancy with respect to general sets} Using the average estimates for the Fourier transform of $\chi_K$ and $d\sigma_S$ proved in the previous paragraph we can now show the desired lower bounds for the (average) discrepancy.

\begin{theorem}\label{t.segment} Let $S$ be a simple, closed, piecewise $C^1$ curve and denote by $K$ its interior.
	\begin{list}{}{}
		\item[(i)] For every positive integer $N$ there exists a $x\in Q_N$, a dilation $r\simeq N$ and a rotation $\tau\in SO(2)$ such that
	$$D_K (f_N,x,r,\tau)  \gtrsim_K \sqrt{N},$$
where the implied constant depends only on $K$.
\item [(ii)]  For every positive integer $N$ there exists a $x\in Q_N$, a dilation $r\simeq N$ and a rotation $\tau\in SO(2)$ such that
$$D_S (f_N,x,r,\tau)  \gtrsim_K \sqrt{N},$$
where the implied constant depends only on $S=\partial K$.
\end{list}
\end{theorem}

\begin{remark} As in Theorem \ref{t.singleradiusN} we cannot guarantee that the sets $x+r\tau K$, $x+r\tau S$ of the previous theorem are fully contained in $Q_N$. Thus, Theorem \ref{t.segment} only implies the existence of \emph{a segment} of $K$ or $S$ which has large discrepancy with respect the coloring of the whole plane $f$.
\end{remark}

In order to prove Theorem \ref{t.segment} we will consider the average discrepancy
$$D_K(f_N,2)^2\eqdef\frac{1}{N^3}\int_{SO(2)}\int_{a N} ^{\beta N}\bigg(  \int_{\mathbb R^2}D_{r\tau K}(f_N,x)^2 dx\bigg) \ dr d \tau ,$$
where $0<a<\beta$ will be appropriate numerical constants.
Similarly define
$$D_S(f_N,2)^2\eqdef\frac{1}{N^3}\int_{SO(2)}\int_{aN} ^{\beta N}\bigg(  \int_{\mathbb R^2}D_{r\tau S}(f_N,x)^2 dx\bigg) \ dr d  \tau .$$
The factor $1/N^3$ is there to almost normalize the measure while $d\tau$ is the normalized Haar measure on $SO(2)$.
\begin{proof} The proofs of \textit{(i)} and \textit{(ii)} are essentially identical so we will just prove \textit{(ii)}.
Using Plancherel's theorem we have
\begin{align*}
	D _S(f_N,2)^2 & =\frac{1}{N^3} \int_{\mathbb R^2} \Abs{\widehat{f_N}(\xi)}^2 \bigg( \int_{a N} ^{\beta N} r^2 \int_{SO(2)} \Abs{\widehat{d\sigma_S}(r \tau^{-1} \xi)}^2 d\tau dr \bigg) \ d\xi\\
	& = \frac{1}{N^3} \int_{\mathbb R^2} \Abs{\widehat{f_N}(\xi)}^2 \bigg(\int_{a |\xi| N} ^{\beta |\xi| N } \frac{r^2}{|\xi|^3} \int_{SO(2)} \Abs{\widehat{d\sigma_S}(r \tau e_\xi )}^2 d \tau dr\bigg)\  d\xi\\
	&\gtrsim \frac{1}{N^2}\int_{|\xi|<1}\Abs{\widehat{f_N}(\xi)}^2 \frac{1}{|\xi|^2} \bigg(\int_{a |\xi| N} ^{\beta |\xi| N }  \int_{SO(2)} \Abs{\widehat{d\sigma_S}(r \tau e_\xi )}^2 r d \tau  dr\bigg)\ d\xi \\
	&=\frac{1}{N^2}  \int_{|\xi|<1}\Abs{\widehat{f_N}(\xi)}^2  \frac{1}{|\xi|^2} \int_{\{ a|\xi|N<|y|<\beta |\xi|N \}}\Abs{\widehat{d\sigma_S}(y)}^2 dy\  d\xi\\
	&= \frac{1}{N^2} \int_{\{a|\xi|N< M\}} \Abs{\widehat{f_N}(\xi)}^2 \frac{1}{|\xi|^2}\bigg(  \int_{\{ a|\xi|N<|y|<\beta |\xi|N \}}\Abs{\widehat{d\sigma_S}(y)}^2 \ \bigg) d\xi \\
	& \quad +\frac{1}{N^2}\int_{\{M<a|\xi|N<a N\}} \Abs{\widehat{f_N}(\xi)}^2\frac{1}{|\xi|^2}   \int_{\{ a|\xi|N<|y|<\beta |\xi|N \}}\Abs{\widehat{d\sigma_S}(y)}^2 dy\  d\xi\eqdef I+II.
\end{align*}
Here we have set $e_\xi= \frac{\xi}{|\xi|}$. Using Lemma \ref{l.fourierKS} we get for $M>R_o$ and $\beta/a>A_o$, that
$$II\gtrsim \frac{1}{N^2} \int_{\{M<a|\xi|N<a N \}}  \Abs{\widehat{f_N}(\xi)}^2 \frac{N|S|}{|\xi|}d\xi\geq \frac{|S|}{N} \int_{\{\frac{M}{a N}< |\xi| <1 \}}  \Abs{\widehat{f_N}(\xi)}^2d\xi.$$

Now for small $\epsilon>0$ we write
\begin{align*}
	I& \geq \int_{ \{ \frac{\epsilon}{N}<|\xi|<\frac{M}{a N} \}}  \Abs{\widehat{f_N}(\xi)}^2  \int_{\{ a|\xi|N<|y|<\beta |\xi|N \}}\Abs{\widehat{d\sigma_S}(y)}^2 dy\  d\xi \\
	&\gtrsim_S \int_{ \{ \frac{\epsilon}{N}<|\xi|<\frac{M}{a N} \}}  \Abs{\widehat{f_N}(\xi)}^2  d\xi.
	\end{align*}
The last estimate is justified since the region $\{a|\xi|N<|y|<\beta|\xi|N\}$ is an annulus inside $B (0,M\beta/a )$, of width at least $(\beta-a)\epsilon$, and $\widehat{d\sigma_S}(y )$ does not vanish identically on any annulus. Adding the estimates we conclude

\begin{align*}
	D _S(f_N,2)^2 & \gtrsim_S \frac{1}{N }\int_{\{\frac{\epsilon}{N}<|\xi|<1\}} \Abs{\widehat{f_N}(\xi)}^2 d\xi
	\geq \frac{1}{N}\bigg(\int_{[-\frac{1}{2},\frac{1}{2}]^2}\Abs{\widehat{f_N}(\xi)}^2 d\xi - \int_{|\xi|<\frac{\epsilon}{N}}\Abs{\widehat{f_N}(\xi)}^2 d\xi  \bigg).
\end{align*}
Now using the trivial bound $\|\widehat{f_N}\|_{L^\infty(\mathbb R^2)} ^2 \leq \| f_N\|_{L^1(\mathbb R ^2)} ^2=N^4$ and \eqref{e.1period} we get
\begin{align*}
	D _S(f_N,2)^2  \gtrsim \frac{1}{N}(N^2-N^4\frac{\epsilon^2}{N^2})\gtrsim  N,
\end{align*}
if $\epsilon$ is small enough.
\end{proof}

\begin{remark}
	By using the same ideas as in the proof of Theorem \ref{t.fullcircleN} we can show a stronger result in the special case of the Euclidean ball. In particular, we have that for every checkerboard coloring $f$ of the whole plane and every $t\geq 1$, there is a $x\in\mathbb R^2$ such that
	$$ \mbox{either}\quad\bigg|\int_{B(x,t)} f(y)dy\bigg|\gtrsim \sqrt{t}\quad\mbox{or}\quad \bigg| \int_{B(x,2t)} f(y)dy\bigg|\gtrsim\sqrt{2t}.$$
\end{remark}
\begin{remark} The only limitation in the choice of the set $K$ and the curve $S$ come from Lemma \ref{l.fourierKS}. Going back to Montgomery's proof in \cite{M} one see that Lemma \ref{l.fourierKS} remains valid if $K$ is for example a multiply connected set and $S$ is replaced by $\partial K$. Furthermore, the $C^1$ condition of the boundary can be replaced by the weaker condition that the limit
	$$\lim_{\delta\to 0}\frac{|\{x\in\mathbb R^2: \textnormal{dist}(x,S)<\delta\}|}{\delta},$$
	exists and is finite.
\end{remark}

\begin{bibsection}
\begin{biblist}

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}

\bib{IK}{article}{
   author={Iosevich, Alex},
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\bib{K}{article}{
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}

\bib{M}{book}{
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   harmonic analysis},
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   publisher={Published for the Conference Board of the Mathematical
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\bib{RO1}{article}{
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\bib{RO2}{article}{
   author={Rogers, Allen D.},
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}

\bib{RO3}{article}{
   author={Rogers, A. D.},
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}

\bib{W}{book}{
   author={Wolff, Thomas H.},
   title={Lectures on harmonic analysis},
   series={University Lecture Series},
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   Edited by \L aba and Carol Shubin},
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}

\end{biblist}
\end{bibsection}

\end{document}

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Sixty-two years ago, in September of 1956, I arrived at the Uberfall—a total neophyte; an Appie beginner.
That first climb! All the way to the top of the cliff on a rope led by Bob Larsen.
It was The Easy Overhang.
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The next spring he offered my first "leg on a climb" toward becoming an Appie leader.
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There I smoked my first joint.
Eventually, inspired by Bob's account of his own Teton ascent, I bought a rope and went west to climb the Grand.


Leonardo Jardim kills Victor Valdes deal?
According to our colleagues at L'Equipe, AS Monaco have decided not to sign Barcelona goalkeeper Victor Valdes, despite having agreed a pre-contract with the club at the start of the calendar year.
It is possible that new boss Leonardo Jardim is behind this, after having recently killed a deal which would have seen Ashley Cole join the Ligue 1 side.
L'Equipe however claim that the combination of Valdes' injury and Subasic's impressive form are the reasons behind the deal being pulled, if that is at all the case at the time of writing.


Greetings, and welcome to the WRX Property Group website and blog! Today we continue on our journey through the various retirement communities in Kitchener-Waterloo and beyond. Our present location is found in the Kingsdale neighbourhood of Kitchener: the Trinity Village.
The Trinity Village is a Christian-affiliated, not-for-profit retirement community and registered charity; it's owned by Lutheran Homes Kitchener-Waterloo. The whole Trinity Village is spread out over 11.5 acres of space, and it consists of a large Care Centre, bungalow town homes, and a complex of residential suites.
The Trinity Village has an excellent location. For one thing, it's right near the convergence of Highway 7 and Highway 8 on the Conestoga Parkway, meaning it's got quick, easy access to the rest of Kitchener-Waterloo, Cambridge, and the rest of southern Ontario via Highway 401 just a few minutes south (great for out-of-town visitors, or travelling to other parts of the Waterloo Region).
Four major grocery stores are quite nearby (especially Freshco, which is roughly a block away), and Fairview Park Mall (the largest mall in the Waterloo Region) is a stone's throw away (it's best not to throw a stone toward it, though).
Fairway Road South and King Street East/Weber Street East both have a wide variety of shops, restaurants, cafés and more, and the nearby Cineplex Cinemas almost always has something great to watch. All in all, there's a lot to do in this part of Kitchener, and its access to both the Conestoga Parkway and numerous shops and restaurants makes life quite convenient.
The Trinity Village consists of four primary areas, three of which are permanent residences. The three residential areas are: the Care Centre (Long Term Care); the Studios (Retirement Residence); and the Terraces (Independent Living).
The Trinity Village Terrace is an exciting opportunity for adults aged 55 and over. The area feels like a village unto itself (as the name suggests) and it consists of 60 individual town homes. What makes these town homes special is that they are Life Lease properties.
What is a Life Lease, you might ask? Well, you can read all about them in this WRX article, but essentially, it's a property or residence that you purchase the right to live in for your lifetime, paying only monthly fees similar to those you'd pay for a condo.
There are many benefits, two of which are: investing your money in the Kitchener-Waterloo real-estate market, as the initial purchase price can potentially appreciate in value upon resale; and living in a community of fellow Life Lease residents.
Properties in the Trinity Village Terrace range in size from 762 to 1407 square feet, and they include a basement on top of this (below this, I suppose?) for additional recreational and storage space. In terms of layouts, there are four basic models: a one bedroom; a one bedroom with sunroom; a two bedroom; and a two bedroom with sunroom.
Homes include such things as air conditioning, decks, and wide corridors, doorways, and bathrooms.
The Trinity Village Studios is a large Assisted Living residence. This residential area provides both a welcoming community and a dedicated staff capable of providing whatever level of care you need. Nurses are on duty 24 hours a day, all year round, and care packages include medication administration, dressing and bathing assistance, nursing treatments, mobility assistance, and more – packages are designed to accommodate your specific needs.
Within the Studios are a therapeutic spa, a recreation room, a community kitchen, a hair salon and beauty parlour, a woodworking room, and much, much more. Delicious meals are part of the package, as well as weekly housekeeping and linen service, weekly bus outings, and recreation opportunities.
Rooms here come in a variety of sizes and styles, including one or two room suites, and semi-private suites. Prices vary based on room configuration and care packages, but semi private rooms start under $1700 per month.
The Care Centre is Trinity Village's facility for long term care. It's got three separate floors, divided into five Home Areas, each of which has accommodations for up to 30 residents (for a total of 150).
90 of the residential rooms are private, and 60 are basic rooms. Professional, compassionate staff that operate out of the Care Centre ensure that residents' needs are all attended to. Residents have their own, separate entrance into the Care Centre.
Each of the 5 Home Areas include a recreation room, a living room with a crackling fireplace (great for sitting and chatting during the winter months), a dining room with a fully operational kitchenette, and a balcony or patio area (great for sitting and chatting during the non-winter months).
The meals offered here are nutritious and homemade. The Trinity Village Care Centre has received licensing from the Ministry of Health and accreditation by CARF (Commission on Accreditation of Rehabilitation Facilities), and it was awarded as the Workplace of the Year in 2016 by the Ontario Long Term Care Association.
Caring professionalism is their mandate, and they live up to it.
The fourth area is the Trinity Village Adult Program (unlike the other three, this is not a permanent residence). This takes place in 'the Gathering Place' in the main Care Centre. It has a separate entrance than the Long Term Care area; participants in the program enter through an entrance near the 'Day Program' garden.
The Adult Program is run by a dedicated, caring, experienced staff, including specialists such as 4 recreation therapists, and community volunteers. It's designed for adults over the age of 55 with health (physical or mental) struggles, who are looking for a supportive community in which they can develop life skills and foster healthy self-esteem.
There is a $10 cost per day of participation, and each day can accommodate up to 30 participants. Participants will be able to enjoy therapeutic games and activities, music, nutritious food, and more.
Being so close to the bustling Fairway Road area of Kitchener (and of course Fairview Park Mall) means there's always plenty to see and do nearby. Trinity Village also offers a wide range of events and activities every month – everything from coffee with the chaplain to choir practice, and from shopping excursions to trivia extravaganzas.
To get an idea of what sorts of things you can get involved with here, take a look at the January 2018 calendar of events here.
The Trinity Village has three major residential configurations, and each of these has specific options designed to fit your specific needs. It's a great place to live your golden years. If you're looking to sell your current property before moving into a retirement community, WRX Property Group would be happy to help you out – so please don't hesitate to contact us!


Mathematics is an important subject without which we cannot imagine the modern world of science and technology. In mathematics, arithmetic is the back bone of mathematical sciences. Here we deal with some basic concepts of arithmetic and their applications which you will also feel enchanting and make more interesting to read a lot. The approach could be different than normal. This book is written in a simple language with examples, and definitions for easy understanding of the readers from various backgrounds. We hope this book will provide a better understanding of concepts in arithmetic.


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text:
Q: Why did the Seanchan hang the watchers? As the Seanchan invaded Falme, they hang the watchers of Watchers over the wave because "They watched for the wrong thing". But what is the right thing to watch? It seems to they were watching for the right thing and the army the Artur Hawkwing sent across the ocean did come back.
From WoT Wikipedia:

The Watchers Over the Waves - A group who believe that the armies
  Artur Hawkwing sent across the Aryth Ocean will one day return, and so
  keep watch from the town of Falme on Toman Head.

From The Great Hunt, Ch 29

Falme stood on a spit of land at the very tip of Toman Head, with
  nothing further west of it except the Aryth Ocean. High cliffs ran to
  the harbor mouth on both sides, and atop one of those, where every
  ship running into the harbor had to pass under them, stood the towers
  of the Watchers Over the Waves. A cage hung over the side of one of
  the towers, with a man sitting in it despondently, legs dangling
  through the bars.
"Who is that?" Domon asked.
Caban had finally given over sharpening his sword, after Domon had
  begun to wonder if he meant to shave with it. The Seanchan glanced up
  to where Domon pointed. "Oh. That is the First Watcher. Not the one
  who sat in the chair when we first came, of course. Every time he
  dies, they choose another, and we put him in the cage."
"But why?" Domon demanded.
Caban's grin showed too many teeth. "They watched for the wrong thing,
  and forgot when they should have been remembering."

A: First Watcher is the title of the head citizen of Falme.  The title originates from the Watchers over the Waves, and the job of watching for the Return, but over time the people have forgotten about that and just use First Watcher as a title.  That's what they're being punished for.

A: I have my own theory on this. I believe that they were left there as original followers of Artur Hawkwing and were intended to watch for The Forerunners' return so that they could be welcomed and the way made clear for them. However, unlike their kin across the ocean, they forgot (an easy thing to do over 1000 years) their purpose and were therefore punished for their dereliction of duty.
I believe there is more than sufficient evidence for this position. You must remember that the Seanchan consider everyone on the East side of the ocean to be in violation of their oaths and make them "reswear." This implies that not only The Watchers, but everyone in the East has broken their oaths, but it was the one job The Watchers had, hence the higher price of failure.
I think most readers, as well as the people East of the ocean, assume that they were set there to guard against AH's return, but the context leads me to believe the opposite. That is why the Seanchan punished them.

redpajama_set_name: RedPajamaStackExchange


Employees take part in a summer cookout.
Oftentimes, when we are able to provide bonuses., Our president will cash these checks in person for the employees. This gives him an opportunity to personally say thank you for the outstanding service our employees provide.
Operation Feed. We have a group of employees that leads our Operation Feed fundraising initiative every year. They accept donations from the employees and the organization to sell via an online auction.


Olivier Award winner Harriet Walter will make her first ever London stage appearance alongside her husband Guy Paul next February when the couple stars in Boa at the Trafalgar Studios.
Clara Brennan's two-hander, which tells an acerbic, warm and honest account of a lifetime in love, will play in the intimate Trafalgar Studio 2 from 5 February to 7 March.
Walter has just completed the run of Phyllida Lloyd's all-female Henry IV at the Donmar Warehouse, a production that followed the success of 2013's Julius Caesar, in which the actress also starred.
One of Britain's best-loved stage actors, Walter's other numerous stage credits include Women Beware Women at the National Theatre and extensive credits at the Royal Shakespeare Company, while on screen she has appeared in films including Sense And Sensibility, The Young Victoria and Atonement.
Her husband, American actor Paul, is a well-known presence on Broadway, having appeared in productions of Twelve Angry Men, King Lear and The King And I. The actor's film credits include Hyde Park On Hudson, George Washington II: The Forging Of A Nation and The Fifth Estate.
Reuniting on stage following their performances together in Broadway's Mary Stuart, where they met for the first time, Walter and Paul tell the tender and powerful story of Louis, Belinda and their 30 years of overwhelming affection, wine, words and dance.
Directed by Hannah Price, the production adds to an eclectic winter season for the Trafalgar Studio 2. World War-I drama Not About Heroes ends its run this weekend and will be followed by two Charles Dickens-inspired plays, Sikes & Nancy and Miss Havisham's Expectations, and Moses Raine's Moscow-set familial drama Donkey Heart.


text:
Ю́рій Микола́йович Щерба́к (* 12 жовтня 1934, Київ, Українська Соціалістична Радянська Республіка, СРСР) — український письменник, сценарист, публіцист, політик, діяч екологічного руху, дипломат, лікар-епідеміолог. Доктор медичних наук (1983 рік). Почесний доктор медицини Національного медичного університету імені О. О. Богомольця (2014 рік), професор. Лауреат Літературної премії ім. Ю. Яновського (1984).

Голова Комітету з Національної премії імені Тараса Шевченка (23 грудня 2016 — 19 грудня 2019).

Біографія
Народився в Києві, під час Німецько-радянської війни евакуйований до Росії, звідки на початку березня 1944-го повернувся в рідне місто, а в 1958 році закінчив Київський медичний інститут за фахом «санітарний лікар».

У 1958–1987 роках працював у Київському науково-дослідному інституті епідеміології та інфекційних хвороб імені Л. В. Громашевського молодшим, а потім і старшим науковим співробітником. Займався профілактикою сказу, брав також участь у боротьбі з епідеміями холери та іншими захворюваннями на території України та Узбекистану, за що нагороджений орденом Трудового Червоного Прапора.

Кандидатська (1965) і докторська (1983) дисертації — з епідеміології особливо небезпечних інфекційних хвороб.

З 28 березня 2014 року Ю. М. Щербак — голова Наглядової ради Національного медичного університету імені О. О. Богомольця.

Був головою Комітету з Національної премії України імені Тараса Шевченка (з березня 2016 по грудень 2019).

Володіє польською та англійською мовами. Доктор медичних наук. Академік УЕАН. З січня 1999 р. — державний службовець 3-го рангу.

Дружина Марія Йосипівна та син Ярослав, лікар-кардіолог, померли 2020 року. Має доньку. Його старший брат — Микола Щербак — учений-зоолог, завідувач Зоологічного музею НАН України — ще в молодості був звинувачений КДБ у націоналізмі й засуджений.

Літературна діяльність
Літературну діяльність почав у середині 1950-х років як активний член літературного об'єднання свого інституту. У студентській газеті «За медичні кадри» він пише перші оповідання, публікує сатиричні «мальовані рецензії» в дусі літературних прямувань народжуваного тоді руху «шестидесятників». Перша повість «Як на війні» (1966) розповідає про будні радянських медиків, а перші свої оповідання, опубліковані в журналі «Юність», навіть оформляє власними малюнками.

З 1966 — член Спілки письменників України (у 1987—1991 рр. — Секретар правління), з 1971 року — Спілки кінематографістів України.

У 1975 році в Харківському академічному театрі ім. О. Пушкіна п'єсою «Відкриття» він дебютував як драматург. Прекрасно володіючи польською мовою, перекладав поезію польських авторів, неодноразово читав студентам Варшавського університету лекції.

Більшу частину його творів можна віднести до умовного жанру «міської прози». Він автор роману про проблеми трансплантації серця «Бар'єр несумісності», документального роману «Причини і наслідки» про боротьбу зі сказом, повістей, збірок новел, віршів і п'єс, а також ряду кіносценаріїв художніх, науково-публіцистичних, і документальних фільмів. Твори Юрія Щербака перекладали мовами народів колишнього СРСР і видавали на території соціалістичних країн. За збірку новел «Світлі танці минулого» удостоєний літературної премії Юрія Яновського (1984), за кіносценарій фільму «Державне відношення» — премії імені Олександра Довженка. Як публіцист Ю. Щербак здобув популярність своїми статтями про Чорнобильську трагедію, що публікувалися в українській і московській пресі.

У повісті «Хроніка міста Ярополя», опублікованій в 1968 році й переробленій у 1983-му, описана гротескно-фантастична історія невеликого містечка Ярополя, в якій у літописній манері розповідається про реальні й неймовірні події, що відбуваються тут протягом кількох століть. У повісті поєднуються елементи наукової фантастики, легенди й казки. Крім того, перу Щербака належать також декілька фантастичних оповідань: «Допит», «Синтез», «Одіссея-2482» та ін.

Політична діяльність
Після розпаду СРСР і здобуття Україною незалежності він відійшов від літературної творчості й зайнявся політикою. У 1991 році оприлюднив заяву, в якій сказав, що відтепер не займатиметься так званим «красним письменством», тобто не буде писати романів чи оповідань «Про Васі і Асі, Вані та Ані». У 1998 році Український науковий Інститут Гарвардського університету видав політико-публіцистичну книгу Щербака «Стратегічна роль України» (The Strategic Role of Ukraine) англійською мовою, а в 2003 році вийшла його публіцистична книга «Україна: виклик і вибір (Перспективи України в глобалізованому світі XXI століття)».

Один із засновників і голова Української екологічної асоціації "Зелений світ", перший лідер Партії зелених України.

1989–1991 — народний депутат СРСР, член опозиційної Міжрегіональної депутатської групи, очолюваної академіком Андрієм Сахаровим, член Верховної Ради СРСР, голова підкомітету з питань ядерної енергетики і екології.

1990–1992 — Голова Партії зелених України.

19 червня 1991 — 16 жовтня 1992 — перший міністр охорони навколишнього середовища незалежної України, член Ради Національної Безпеки України.

1992–1994 — надзвичайний і Повноважний Посол України в Ізраїлі

1994–1998 — надзвичайний і Повноважний Посол України в США (з 1997 р. — також посол України в Мексиці — за сумісництвом).

1998–2000 — радник Президента України.

2000–2003 — надзвичайний і Повноважний Посол України в Канаді (одночасно — представник України при Міжнародній організації цивільної авіації (ICAO, Оттава).

З грудня 2009 — один із засновників та член Ради із зовнішньої та безпекової політики.

Академік АЕНУ.

Громадська позиція
У липні 2018 року підтримав відкритий лист українських діячів культури до ув'язненого у Росії українського режисера Олега Сенцова.

Нагороди та премії
 Орден Свободи (25 червня 2016) — за значний особистий внесок у державне будівництво, соціально-економічний, науково-технічний, культурно-освітній розвиток України, вагомі трудові здобутки та високий професіоналізм
 Орден «За заслуги» II ст. (16 січня 2009) — за вагомий особистий внесок у справу консолідації українського суспільства, розбудову демократичної, соціальної і правової держави та з нагоди Дня Соборності України
 Почесна відзнака Президента України (22 серпня 1996) — за визначні досягнення у праці, що сприяють економічному, науково-технічному і соціально-культурному розвиткові України, зміцненню її державності і міжнародного авторитету, та з нагоди п'ятої річниці незалежності України
 Лауреат премії ім. Ю. Яновського (1984)
 Премія фундації Антоновичів (2018)

Літературні твори
 1966 Як на війні (збірка оповідань і повістей)
 1971 Бар'єр несумісності (роман)
 1973 Маленька футбольна команда (збірка оповідань і повістей)
 1986 Причини і наслідки (документальний роман про боротьбу зі сказом)
 1987 Чорнобиль (документальна повість)
 2003 Україна: виклик і вибір. Перспективи України в глобалізованому світі XXI століття
 2010 Україна в зоні турбулентності
 2011 Час смертохристів: Міражі 2077 року (гостросюжетний політичний трилер)
 2012 Час Великої Гри. Фантоми 2079 року
 2014 Час тирана. Прозріння 2084 року
 п'єси
 кіносценарії

Кінематографічні роботи
Автор дикторського тексту до стрічки «Ранкові острови» (1966), сценаріїв фільмів:

 «Ми студенти різних континентів»,
 «Вони відстояли весну» (1965),
 «Ранкові острови» (1966),
 «Сини Баштанської республіки» (1967),
 «Роки і секунди»,
 «Карантин» (1968),
 «Хортиця» (1969),
 «Шлях до серця» (1970),
 «В. М. Глушков, кібернетик» (1980, у співавт.),
 «Острів співучих пісків» (1981, т/ф),
 «Відкриття» (1982, т/ф),
 «Державне ставлення» (1983),
 «Наближення до майбутнього» (1986, у співавт.) та ін.

За його оповіданням «Гармонійне натхнення» знято телефільм «Якось у грудні» (1988).

Примітки

Джерела
 Сайт державного управління справами

Література
 В. М. Матвієнко. Щербак Юрій Миколайович //
 І. С. Стрикун. Щербак Юрій Миколайович  //

 Довідник «Хто є хто в Україні», видавництво «К. І.С»

Посилання
 Щербак Юрій Миколайович // Україна в міжнародних відносинах. Енциклопедичний словник-довідник. Випуск 6. Біографічна частина: Н–Я / Відп. ред. М. М. Варварцев. — К.: Ін-т історії України НАН України, 2016. — с.343-344
 ЮРІЙ ЩЕРБАК | Програма «Велич особистості» | 2014 (ВІДЕО)
 Універсум
 Український тиждень, № 19 (80), 15-21 травня 2009. С. 16-19.
 Український тиждень, № 12 (125), 26 березня — 1 квітня 2010. С. 30-32.
 Юрій ЩЕРБАК: Колись Путіна прокляне не лише український, але й російський народ
 Юрій ЩЕРБАК. Восьма окупація України.  Опубліковано у Літературній Україні 01 лютого 2020, № 3-4,

Відповідальні працівники структури Державного управління справами
Голови Комітету з Національної премії України імені Тараса Шевченка
Наукові фантасти України
Українські письменники-фантасти
Народні депутати СРСР від України
Посли України в США
Посли України в Ізраїлі
Посли України в Канаді
Посли України в Мексиці
Уродженці Києва
Члени Національної спілки кінематографістів України
Члени НСПУ
Лауреати літературної премії імені Юрія Яновського
Письменники-лікарі
Почесні доктори
Доктори медичних наук СРСР
Професори
Науковці Інституту епідеміології та інфекційних хвороб

Енциклопедисти України

redpajama_set_name: RedPajamaWikipedia


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redpajama_set_name: RedPajamaGithub


text:
Q: realmtasks not able to sync data after logout , re-login and kill app in realm After Logout and re-login the lists are not syncing , and also kill app and relaunch app but not sync data in realm database on realm object server with realm studio and data update with android studio app.
How to solve this problem, please help me . Thanks in advance.


text:
Dad: Killers must die in prison
A father's fight to change the law so life means life for convicted murderers is gathering momentum after two families whose children were killed joined him.
Martin Kane, whose daughter, Laura, nine, was murdered by a family friend, says strong public support has made him confident of getting 30,000 petition signatures.
He has support from Liz Neailey, whose son, Wesley, 11, was murdered in Newcastle in 1998 by sex offender Dominic McKilligan, and June Richardson, whose son, Martin Brown, was one of the two victims of child killer Mary Bell in 1968.
The three will step up the fight with a special display at Newcastle's Grey's Monument, on August 25.
Mr Kane, of Houghton-le-Spring, who already has 2,000 signatures on his petition, said: "We're going to set up a stall by the monument where people can sign the petition." Signing days will also be held in Sunderland, Durham City and Stanley.
He said: "I've been pleased by the support and I believe the campaign has to be recognised now.
"It's ridiculous when killers are released and they just do it again. The Government is not learning the lessons. The more that parents get involved in this the more the Government will recognise it."
Mr Kane plans to take the petition to the Home Office and to 10 Downing Street.
In August, 1999, Laura was lured from her home in Penshaw, near Washington, by Colin Bainbridge, who raped her, suffocated her and hid her body under the floorboards of his home in Murton, County Durham.
Bainbridge, nicknamed Billy Liar, was jailed for life, with the recommendation that he serve 22 years.
Mr Kane, 44, says Bainbridge will only be in his 50s when he is eligible for release.


As far as the splinter goes, it's been my experience that they hurt more coming out than going in. Right?
Hell of a learning experience.
Been there done that and it will defiantly get your attention!!!
I built the sled that Jerry Bennett talked about and he was even kind enough to us vets he sent us 4 of his templetes. I am told by someone I know that they built a sled like his to work on his Bandsaw also.
I think it turned out cool looking.
All part of woodworking unfortunately I get them all the time (not as bits as the Titanic one of yours) and they are so painful I have to stop work and remove them beacause they are so distracting.
I managed to brad nail my finger one day, dont ask how as to this day even I dont know, but it happened.
Almost went right through, look closely and you can see the skin bulging opposite the entry point.
Didnt go to the hospital, too much woodwork to do and didnt want to waste half a day or more.
Nice work on the bowl. That must of been a hands on experience.
Wow! That's a bad splinter for sure. However, the bowl sure did turn out beautifully. Nice work!
Still a doozy though! Good thing it was shallow.
Let me guess, you were feeling the lip of the bowl while it was spinning?
As a token bonus, it looks like you saved yourself crawling on the floor looking for that piece. Pull it out an glue it back on!
BTW, I really like the pattern you created with those segments. Did you create the circles in one tall cylinder then cut them to thickness or did you make them individually?
Beautiful bowl … worth the "pain"!


Rev. Frank E. Ray Sr.
Rev. Frank E. Ray, Sr. is one of the most profound African American preachers in the land. He pastors a large church in Memphis,Tennessee and he preaches in most of the major Houses of Worship within the National Baptist Convention. Enjoy this clip of his preaching!


This entry was posted on Thursday, May 7th, 2009 at 9:15 am	and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
Aww Edel and David….sorry to hear that you both got sick! I hope you are both feeling better and that will be the only time. I'm glad to see that you are writing a journal. I'm going to be able to stalk you now for the next year! David, you take such beautiful photographs – you are truly talented. You should try to sell them to travel magazines! You could make a fortune! I look forward to hearing about your travels so keep posting! Hope all is well!


Choice A: now of course the increase in the Mexican American population would cause a greater demand in building more Muchos Tacos restaurants. I mean what about me.. I like muchos tacos!
Beyonce is quite an entrepreneur I suppose.
LOL now... hopefully this girl does not find this and hunt me down.. but I had to post this! As a citizen with both and savings and a checking account. haha shouldn't you be worried about Identity theft. Secondly, since when did "30" become "3o" with a lower case "o". Girl.


text:
Sony Classical Announces Upcoming Box Set Releases in April 2021 - Szigeti, Rodzinski, and Bernstein
Joseph Szigeti – The Complete Columbia Album Collection
April 23: Artur Rodziński and the New York Philharmonic – The Complete Album Collection
Bernstein Conducts Stravinsky
Sony Classical releases a 17-CD box set collecting the recordings made between 1940 and 1956 for American Columbia by the renowned Hungarian violinist Joseph Szigeti. This first-ever complete recording collection totals 31 works, remastered from the original analogue discs and tapes using 24 bit / 192 kHz high resolution.
Szigeti had a remarkable career. Born in 1892 in Budapest, where he studied with Jenő Hubay, one of most celebrated virtuosos and teachers of that golden era of violin playing, he was praised by the iconic German violinist Joseph Joachim at his Berlin debut in 1905; lived in London for several years following his acclaimed 1907 debut and played chamber music with, among others, Myra Hess and Ferruccio Busoni; was a frequent visitor after the war to the Soviet Union, where he introduced Prokofiev's First Violin Concerto; made his triumphant American debut at Carnegie Hall under Stokowski in 1925; toured the world during the 1930s before finally settling in the US in 1940.
It was in that year that Szigeti renewed his friendship with fellow Hungarian émigré Béla Bartók, and in April the two gave a now-legendary recital in Washington which featured Bartók's First Violin Rhapsody of 1928 – a work dedicated to and premiered by Szigeti in Europe. In May 1940, Columbia recorded their interpretation of this "vehicle for Szigeti's biting and wholly magnificent fiddling" (MusicWeb International) in New York. That performance appears here for the first time on CD, along with another important work by Bartók, the classic first recording of his Contrasts for clarinet, violin and piano, written for and performed with Szigeti and Benny Goodman.
The rest of the new collection displays many more treasures of Szigeti's passionate dedication to chamber music: in Bach, Handel, Tartini, Mozart, Beethoven, Schubert, Brahms, Dvořák, Debussy, Ravel, Bloch, Busoni, Prokofiev, Stravinsky and Henry Cowell, collaborating with such artists as Andor Foldes – another Hungarian émigré – as well as with Mieczyslaw Horzowski, Myra Hess, Pablo Casals and Igor Stravinsky.
There are, of course, major orchestral works represented in the new Szigeti edition, including two towering concertos in D major – the Brahms, recorded in 1945 with Eugene Ormandy conducting the Philadelphia Orchestra, and the Beethoven, recorded in 1947 with Bruno Walter and the New York Philharmonic ("an account of impassioned grandeur" – MusicWeb International) – along with Busoni's early Violin Concerto in D major, recorded in 1954 with Thomas Sherman conducting the Little Orchestra Society. Szigeti's numerous Bach concerto recordings for Columbia are here as well, conducted by Casals, Fritz Stiedry and George Szell.
As Nathan Milstein, one of his great colleagues, said in a touching tribute to Szigeti, who died in 1973: "He was an incredibly cultured musician. Actually, his talent grew out of his culture. … I always admired him, and he was respected by musicians." In his late years, Joseph Szigeti finally got the appreciation he deserved from the general public as well. Sony Classical's new collection his Columbia recordings, many never before released on CD at Sony Classical, can only further enhance that appreciation.
Conductor Artur Rodziński and the New York Philharmonic – The Complete Album Collection
Artur Rodziński's legacy for Columbia Masterworks with the New York Philharmonic is presented in a 16-CD set. The collection features 15 works recorded with the New York Philharmonic for the first time on CD at Sony Classical, transferred and remastered from the original analogue discs and tapes using 24 bit / 192 high resolution.
He has been called the most exciting younger conductor in the USA during the 1930s. Artur Rodziński (1892–1958) galvanized audiences wherever he appeared as a guest, including Philadelphia, where he worked with Stokowski, and New York, where his 1937 concert performance of Strauss's Elektra is still considered a landmark in the Philharmonic's history. Earning a reputation as a builder of great orchestras, the Polish conductor headed up and developed the Los Angeles Philharmonic (1929–33) and the Cleveland Orchestra (1933–43), with which he gave the American premiere of Shostakovich's opera Lady Macbeth of the Mtsensk District in 1935. Meanwhile, he was also active in Europe, becoming the first naturalized American to conduct the Vienna Philharmonic at the Salzburg Festival. There Toscanini admired his work and in 1938 picked Rodziński to train his new NBC Symphony Orchestra.
Rodziński amassed a fairly sizable discography in a relatively short recording career. During his Cleveland tenure, he made a number of important recordings for Columbia Masterworks, and he continued his association with the label after taking up his post in New York. Sony Classical's new 16-disc box set of Rodziński's New York recordings, largely recorded in Carnegie Hall between 1944 and 1946, will fill an important gap in the classical catalogue.
Rodziński was perhaps most closely associated with the Russian repertoire – Tchaikovsky, Rachmaninoff, Prokofiev, Mussorgsky – as well as Wagner. Both are prominent in Sony's new collection. Artur Rodziński's range is further reflected by generous samplings of French music (Saint-Saëns' Fourth Piano Concerto with Robert Casadesus and various works by Bizet, Ibert, Franck, Satie, Offenbach and Debussy), American repertoire (Gershwin, Copland and Morton Gould) and symphonies by Brahms (Nos. 1 and 2) and Sibelius (No. 4). This meticulously transferred and remastered collection of Rodziński's New York Philharmonic recordings represents a generous reward for the conductor's many patient admirers.
-Leonard Bernstein, conductor
-New York Philharmonic
-Boston Symphony Orchestra
-London Philharmonic Orchestra
-London Symphony Orchestra
"The last great father-figure of Western music" was how Leonard Bernstein eulogized Igor Stravinsky in 1972 before his homage concert in London a year after the composer's death. Stravinsky's works, he asserted, "sum up and embrace all of music itself – from primitive folk art to highly sophisticated serialism, from rarefied church music to outspoken jazz."
In fact, Bernstein performed surprisingly few selections from Stravinsky's vast output, recording most of those for Columbia Masterworks and RCA Victor, yet they include some of his most indelible interpretations. He also delivered eloquent and revelatory lectures on the composer. Sony Classical is now pleased to collect these performances and talks in a new 8-CD box set.
One of Bernstein's earliest passions was Histoire du Soldat (The Soldier's Tale). He studied the droll score at Tanglewood with his mentor, Serge Koussevitzky, first performing it in 1940 at a Koussevitzky family picnic. Then in 1947, after conducting it again at Tanglewood with Boston Symphony players, he recorded it, along with the enchanting Octet, on 78s for RCA.
His most celebrated Stravinsky recording came a decade later, Le Sacre du printemps with the New York Philharmonic in 1958 and from around the same time comes Bernstein's New York Philharmonic recording of the Firebird Suite (1919 version). In 1962 there followed recordings of the seldom-heard Concerto for Piano and Wind Instruments (with soloist Seymour Lipkin) and the suite from Pulcinella.
After that he didn't return to the studio with Stravinsky's music until 1970, with the New York Philharmonic's recording of Petrushka. Two years later, this time with the London Symphony Orchestra, Bernstein revisited Le Sacre du printemps and in the same London Symphony sessions, joined by the English Bach Festival Chorus, he recorded the Symphony of Psalms.
That autumn, Bernstein took up residence at his alma mater, Harvard University, as Charles Eliot Norton Professor of Poetry for the academic year 1972–73. The heart of his memorable last Harvard appearance, entitled "The Poetry of Earth," was an "outstanding" (ClassicsToday) performance of the opera-oratorio Oedipus Rex, recorded earlier at Symphony Hall with Bernstein conducting the Boston Symphony Orchestra and Harvard Glee Club, and featuring René Kollo as Oedipus, Tatiana Troyanos as Jocasta and Tom Krause as Creon.
This shattering performance of Stravinsky's massive work is aptly presented in Sony's new edition within the original context of his "Poetry of Earth" lecture. Like every item in this indispensable new edition of Bernstein's Stravinsky – also containing the conductor's spoken introduction to Petrushka from 1970 – it is offered in the newest remastering from analogue tape.


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NYC Startup Funding, Month-end Wrap-up: March 2018
March was another terrific month for NYC-based startup funding. Below are five of the largest rounds of funding closed by NYC-based startups in March of 2018.
CommonBond – $50M
Round: Series D
Industry: Education, Finance, Fintech
Investors: 8 – August Capital, Columbia Seligman Investments, Fifth Third Capital, First Republic Bank, Neuberger Berman Group, Nyca Partners, Thomas Glocer, Vikram Pandit
Company: Founded in 2011, CommonBond is a financial services lender specializing in the refinancing of student loans. The company aims to make student loan debt more affordable, while also allowing investors to see a healthy ROI. To date, CommonBond has raised $803.6 million after eight rounds of funding.
Prevail Therapeutics – $75M
Round: Series A
Industry: Biotechnology
Investors: 9 – Adage Capital Management, Alexandria Venture Investments, Biotechnology Value Fund, Boxer Capital, EcoR1 Capital, Omega Funds, OrbiMed, Pontifax, RA Capital Management
Company: Founded in 2017, Prevail Therapeutics is a biotechnology company specializing in the development of new gene therapies for the treatment of Parkinson's, and other neurodegenerative diseases. To date, Prevail Therapeutics has raised $75 million after one round of funding.
Warby Parker – $75M
Industry: E-commerce, Fashion, Retail
Investors: 1 – T. Rowe Price
Company: Founded in 2010, Warby Parker is a fashion and lifestyle brand offering eyewear at a fraction of the cost of traditional brands. As a pioneer in socially conscious business practices, Warby Parker has donated more than 3 million pairs of eye glasses to those who would otherwise not be able to afford them. To date, Warby Parker has raised $290.5 million from seven rounds of funding.
UiPath – $153M
Round: Series B
Industry: Robotics, Automation
Investors: 6 – Accel Partners, CapitalG, Credo Ventures, Earlybird Venture Capital, Kleiner Perkins Caufield & Byers, Seedcamp
Company: Founded in 2012, UiPath is a software and automation robotics company. UiPath aims to streamline tedious tasks such as document management and data extraction, and is currently used by thousands of companies. To date, UiPath has raised $183 million from three rounds of funding.
Oscar Health – $165M
Round: Venture Round
Industry: Healthcare, Information Technology
Investors: 8 – 8VC, CapitalG, Fidelity Investments, Founders Fund, General Catalyst, Khosla Ventures, Thrive Capital, Verily
Company: Founded in 2013, Oscar Health is a medical insurance company aimed at easing the process for its patients. The company offers individual, couple, and family plans, and is currently available in New York, New Jersey, California, and Texas. To date, Oscar Health has raised $892.5 million from seven rounds of funding.
Spotify Files for IPO
Spotify, the world's largest music-streaming service has filed to go public. In a rather unorthodox move, the…
Dropbox Files for IPO
Dropbox announced that it had filed paperwork for its long-awaited IPO. After six rounds…
5 of the Largest Rounds of Funding for NYC-based Startups in 2018
2018 was a banner year for NYC-based startup funding. Below are five of the…
How Tiger Woods Made $1 Billion


Leather sectional with chaise. Should you be in the market for new living room, bonus room or den furniture then you should consider getting one of the numerous leather sectionals available on the market. Leather sectionals are becoming more and more favored by consumers. Modular furniture allows the consumer to organise their sectional in many different configurations. Some have optionally available left or right side chaise lounges and recliners. This allows the consumer to position the leather sectional to fit the room they are designing.
You might not want the chaise lounge placed near your doorway as it would impede the flow of foot traffic into and from your room. By choosing to have the chaise lounge at the opposite end of the leather sectional you are able to still have a comfortable sitting arrangement as well as idealized traffic design in your room.
Leather sectionals come in many different sizes. There are lots of available on the market so you can have one to suit your room no matter how small or how big it might be. Because they are modular in design you can add armless pieces to extend the length in a direction. You would not want to possess a small leather sectional within a large living or bonus room. It would throw from the scale of your room and leave holes in your design flow. You might, however , want small scale leather sectional for the den as these rooms are usually much smaller in size. For a smaller room you also might choose to leave off the optional recliners and chaise lounge pieces. Leather sectional with chaise and recliner, leather sectional with chaise lounge, leather sectional with chaise and sleeper, leather sectional with chaise moving furniture.


I have been working on computers for over 30 years and was not able to gain many certifications because of the cost. The MTA certification path has provided practical certifications that I can gain and also implement in my classroom. I wanted to experience the exams before encouraging my students to take these. We have now implemented these certifications as a part of our classes and program. The changes for my students is incredible.
I have implemented five certifications into our programs as a final exam for my students. They now leave with not only a degree but certifications that help them get jobs. The program was started this year so most of our students will not graduate till next year. I expect the number who find jobs immediately will increase. I am so excited about giving these students a great start on a new career.


Click to show race results for IMSA Continental Tire SportsCar Challenge - GS 2018.
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Click to show race results for SCCA Pro Racing Trans-Am Series - TA2 2011.
The following graph shows the change in the DriverDB Rating of Cameron Lawrence over time since January 1st 2013. The rolling average of the graph is 10 days.


Type II Cast host Steve Olker and panelists Scott Bernstein, Tanya Sperry and Justin Wendt each had a tough time waiting for two weeks to discuss Phish's three-night stand at Madison Square Garden, so much so that you can hear the energy in each participant's voice as they detail the run's highlights.
Tanya starts us off by recapping the December 30th performance including discussions and listening sessions devoted to the energy at the venue, the unique version of Tweezer and the highly melodic Bathtub Gin thrown down that evening. Scotty explains how the run, and specifically the New Year's Eve concert, channeled the best of the band's previous year-ending shows. Clips from the 31st include a machine gun-laden Rock and Roll, a buttery Ghost jam that the panel flipped over and the grand Meatstick - The Musical segment that put an exclamation point on 2010.
Justin led a discussion about the improv-heavy run closer from New Year's Day. The panelists gave their thoughts on the Walls of a Cave jam, the Crosseyed jam and the gorgeous full-band interplay that developed out of Simple. All four Type IIers summed up their thoughts on the MSG Run before one of the most fun sequence of the three-night stand played us out.
2001, Crosseyed, Ghost, Gin, Manteca, Meatstick, Rock and Roll, Simple, Tweezer, Walls.


text:
Hedia Trabelsi – (2 de enero de 1992) es una deportista tunecina que compite en lucha libre. Ganó dos medallas en el Campeonato Africano entre los años 2011 y 2013. Obtuvo una medalla de bronce en los Juegos Mediterráneos de 2009.

Palmarés internacional

Referencias

Luchadores de Túnez


text:
film December 2001
1. Bully (Larry Clark) My favorite movie of the year: a dirty true-crime sexploitation picture that dares to be art. Larry Clark invents the "crotch-cam" shot and inspires the most outraged New York Times review of the season.
2. Faithless (Liv Ullmann) Liv Ullmann channels Ingmar Bergman. See it on acid.
3. L.I.E. (Michael Cuesta) A feel-good child molester with a hard-on of gold befriends a confused Long Island teen and his Gacy-bait sidekick.
4. Mulholland Drive (David Lynch) Lipstick lesbians never had this much celluloid fun.
5. Jay and Silent Bob Strike Back (Kevin Smith) GLAAD was
1. Dancer in the Dark (Lars von Trier) The most hilariously moving, "feel-insane" movie of the year.
2. O Brother, Where Art Thou? (Joel Coen) The jaw-dropping all-singing, all-dancing Ku Klux Klan—Busby Berkeley number is a real beaut.
3. L'Humanité (Bruno Dumont) The endless saga of a simpleton cop so desperate to feel emotion that he spies on the sex life of his lusty neighbors and smells and kisses his crime suspects during interrogations.
4. American Psycho (Mary Harron) A chain-saw movie for the elite; the funniest American comedy of the year.
5. The Idiots (Lars von Trier) A
print December 1999
BEST OF THE '90s: FILM
CINDY SHERMAN, artist: Thomas Vinterberg's brilliant The Celebration (1998) is especially important because it signals the future of the medium, away from Hollywood's excesses.
JOHN WATERS, filmmaker: During the 1994 Cannes Film Festival I was sick in bed with the flu on the night Pulp Fiction premiered. Suddenly, from blocks away I heard the most stupendous roar of approval from the opening-night audience. I was so pissed to have missed the night Quentin Tarantino became an instant cinematic icon. But once I saw the movie I knew he deserved it. I guess you could call me a Quentin-hag.
DAVE HICKEY
Bruce Nauman retrospective, Museum of Contemporary Art, Los Angeles: Bad news from the studio, but real news nevertheless. Twenty-five years of pieces, each of which seems to have arisen out of a condition of sudden panic—out of the terror of not knowing, of having forgotten, willfully, day after day, what art is and what an artist might do—of having forgotten, even, what an artist is. A fountain? A source of mystic truths? A cruel instructor? A tortured clown? We get one brutal, last-ditch guess after another, and the whole practice of "artmaking" is reinvented, again
GORE GIRL!
JOHN WATERS' NEW MOVIE, Serial Mom, is not what you would call an especially plot-driven narrative. In the gentle language of literary criticism, the diegesis is subsumed by the rabid, frothy-mouthed semiosis. From the very first scene, which shows Beverly Sutphin (Kathleen Turner), the serial mom of the title, stalking a pesky fly through her exquisitely hygienic kitchen, it's abundantly clear that this woman is totally wacko. Establishing her, uh, dark side early on, the film provides not the usual pleasures of fear and suspense but a series of increasingly ornate vignettes of mayhem. And