text:
\section{Introduction}\label{sec1}
Let $H$ denote a complex Hilbert space, $A$ the generator of a $C_0$-semigroup $e^{tA}$ on $H$,
and $P$ a bounded projection. The convergence of the generalized Trotter's product formula
\begin{eqnarray}\label{pro}
\lim_{n \rightarrow \infty}(e^{\frac{t}{n}A}P)^n
\end{eqnarray}
was first studied (as a corollary of the main Theorem) in \cite{Ka2}. It was shown that \eqref{pro} converges strongly for all $t>0$
whenever $-A$ is a non-negative self-adjoint operator, and $P$ is an orthogonal projection. More
generally, the Addendum in \cite{Ka2} implies that the same result is true whenever $-A$ is
associated with a closed sectorial form (where the vertex of the sector is allowed to be any
real number $\omega$, i.e.
$S_{\phi ,\omega}:=\{z\in \mathbb{C} : -\phi <{\mathrm{arg}} \ (z-\omega)<
\phi\}$, $\omega\in\ensuremath{\mathbb{R}}$, $\phi\in (0,\frac{\pi}{2})$).

The convergence of \eqref{pro} was then studied in more general settings in \cite{AB} and
\cite{MS} where further convergence results, motivating examples, and some counterexamples were
given. The main result of this paper is to prove the converse of Kato's result, i.e. that the
strong convergence of \eqref{pro} for all orthogonal projections $P$, in fact, characterizes
generators $A$ such that $-A$ is associated with a closed sectorial form. To be more precise
we recall the following result (see \cite{Ka2} Addendum, and \cite{MS} Theorem 1 and 4.):
\begin{theorem}\label{thm1}
Let $A$ be the generator of a $C_0$-semigroup $e^{tA}$ on a Hilbert
space $H$. Consider the following statements:

(i) $A$ is bounded.

(ii) $-A$ is associated with a densely-defined, closed, sectorial form $a$ on $H$.

(iii) The formula $(e^{\frac{t}{n}A}P)^nx$ converges for all projections $P\in \mathcal{L} (H)$,
and all $x\in H$ and $t>0$.

(iv) The formula $(e^{\frac{t}{n}A}P)^nx$ converges for all {\it{orthogonal}} projections
 $P\in \mathcal{L} (H)$, and all $x\in H$ and $t>0$.

The following implications hold: (i) $\Rightarrow$ (iii)  and (ii) $\Rightarrow$ (iv).
\end{theorem}

We will show in Section \ref{sec2} that the converse implications also hold. In the course of
the proof we will need an auxillary result, given in Theorem \ref{thm2}, which can be regarded
as a complement of \cite{Ch}, and is of independent interest. Namely, we show that whenever the generator
$A$ of the semigroup $e^{tA}$ is unbounded, it is possible to introduce an equivalent scalar product
$( \ , \ )_0$ on $H$ such that $e^{tA}$ is non-quasi-contractive with respect to $( \ , \ )_0$.

\section{Main Result}\label{sec2}

In order to prove our main result [Theorem \ref{thm3}], first we need to characterize the class
of generators $A$ on $H$, such that the $C_0$-semigroup $e^{tA}$ is quasi-contractive for
{\it{every}} equivalent scalar product $( \ , \ )_0$ on $H$. The characterization is provided by
\begin{theorem}\label{thm2}
Let $A$ be the generator of a $C_0$-semigroup $e^{tA}$ on a Hilbert space $H$.
 The following are equivalent:

(i) A is bounded.

(ii) The semigorup $e^{tA}$  is quasi-contractive for every equivalent scalar product
$( \ , \ )_0$ on $H$.

(iii)  For every equivalent scalar product $( \ , \ )_0$ on $H$ there exists
$K_0\in \ensuremath{\mathbb{R}}$ such that for every vector $x\in D(A)$, $(x,x)_0=1$
implies ${\mathrm{Re}} \ (Ax,x)_0\le K_0$.

\begin{proof}
The implications (ii) $\Leftrightarrow$ (iii) are consequences of the Lumer-Phillips theorem
(see e.g. \cite{RN}, Proposition 3.23.). The implications (i) $\Rightarrow$ (ii) and (i)
$\Rightarrow$ (iii) are trivial. It remains to prove (iii) $\Rightarrow$ (i). We will need the
following
\begin{definition}\label{def1}
Let $T\in \mathcal{L} (H)$ be an injective operator, and $x\in H$, $\|x\|=1$, and $0<\delta \le 1$.
We say that $x$ is a $\delta$-quasi-eigenvector of $T$ if
\begin{eqnarray}
\delta \le \frac{|(x,Tx)|}{\|Tx\|}\le 1
\end{eqnarray}
\end{definition}
Note, that a 1-quasi-eigenvector is, in fact, an eigenvector of $T$.

Now, let $0<\delta <1$ be fixed. We prove the implication (iii) $\Rightarrow$ (i) by contradiction.
Assume, therefore, that $A\notin \mathcal{L} (H)$, and also, by rescaling, that $A^{-1}=:T\in\mathcal{L} (H)$.
Assume, furthermore, that a sequence $(h_n)\subset H$ is given with the following properties:

(a) $\|h_n\|=1$ for all $n\ge 1$.

(b) $\{h_k, \  Th_k\} \perp  \{h_j, \ Th_j\}$ for all $k \not= j$.

(c) $\lim_{n\to\infty}\|Th_n\|=0$

(d) For every $n\ge 1$ the vector $h_n$ is {\it{not}} a $\delta$-quasi-eginvector of $T$.

We construct an equivalent scalar product $( \ , \ )_0$ on $H$ with the help of the sequence
$h_n$.

Let $H_n={\rm span}\{h_n, \ Th_n\}$. Note, that $H_n$ is 2-dimensional because $h_n$ is not an
eigenvector of $T$.

Let $Th_n=c_{1,n}h_n+c_{2,n}h_n^{\perp}$, where $\|h_n^{\perp}\|=1$.
Note that
$$\frac{|c_{1,n}|^2}{|c_{1,n}|^2+|c_{2,n}|^2}<\delta^2  \ \  {\mathrm{and}} \  \
\frac{|c_{2,n}|^2}{|c_{1,n}|^2+|c_{2,n}|^2}>1-\delta^2$$
Hence,
$$\frac{|c_{1,n}|}{|c_{2,n}|}<\frac{\delta}{\sqrt{1-\delta^2}} \ \  {\mathrm{and}} \ \
\frac{|c_{2,n}|}{\|Th_n\|}>\sqrt{1-\delta^2}$$

Define $Q_n\in\mathcal{L} (H_n)$ by
\begin{eqnarray*}
Q_nh_n:=h_n+\overline{L}_nh_n^{\perp}\\
Q_nh_n^{\perp}:=L_nh_n+(|L_n|^2+1)h_n^{\perp}
\end{eqnarray*}
where $|L_n|=2\frac{\delta}{\sqrt{1-\delta^2}}$ and $\overline{{L}_n{c}_{2,n}}>0$ for
all $n\ge 1$. It is clear that $Q_n=Q_n^\ast \ge 0$, $Q_n^{-1}\in \mathcal{L}(H_n)$,
and $\|Q_n\|_{H_n}\le K$,  $\|Q_n^{-1}\|_{H_n}\le K$ for some universal constant $K$ (not
depending on $n$). Define $Q\in \mathcal{L} (H)$ by
\begin{eqnarray*}
Q:=Q_1\oplus Q_2\oplus\dots\bigoplus I_{(H_1\oplus H_2\oplus\dots)^{\perp}}
\end{eqnarray*}
It is easy to see that $Q$ is well-defined, $Q\in\mathcal{L} (H)$, $Q=Q^\ast\ge 0$, and
$Q^{-1}\in\mathcal{L} (H)$. This means that $Q$ defines an equivalent scalar product on $H$ by
$(x,y)_0:=(x,Qy)$.

Now, let $x_n:=\frac{Th_n}{\|Th_n\|}$. Then
\begin{eqnarray*}
{\mathrm{Re}} \ (Ax_n,x_n)_0=\frac{1}{\|Th_n\|^2}{\mathrm{Re}} \ (h_n,QTh_n)=
\frac{1}{\|Th_n\|^2}{\mathrm{Re}} \ (h_n,c_{1,n}h_n+c_{2,n}L_nh_n)=\\
=\frac{1}{\|Th_n\|^2}({\mathrm{Re}} \ c_{1,n}+\overline{c_{2,n}L_n})\ge
\frac{1}{\|Th_n\|^2}\frac{\delta}{\sqrt{1-\delta^2}}|c_{2,n}|\ge \frac{1}{\|Th_n\|}\delta\rightarrow +\infty
\end{eqnarray*}
Let $y_n:=\frac{x_n}{\|x_n\|}_0$. Then $ {\mathrm{Re}} \ (Ay_n,y_n)_0\rightarrow +\infty$ still holds
 due to the equivalence of the scalar products $( \ , \ )$ and $( \ , \ )_0$.

In order to complete the proof of the theorem it remains to construct the sequence $h_n$ with
the required properties. The construction is carried out in several steps.

{\it{Step 1.}} We construct an orthonormal sequence $(e_n)\subset H$, such that\\
$\lim_{n\to\infty}\|Te_n\|=0$.

Take the polar decomposition $T=UT_1$ of $T$, where $U$ is unitary and $T_1=T_1^\ast\ge 0$. It
 is clear from the spectral theorem that there exists an orthonormal sequence $(e_n)\subset H$
such that
$\lim_{n\to\infty}\|T_1e_n\|=0$ (otherwise $T_1$ and $T$ would be invertible, contrary
 to our assumption). Note, also, that $\|T_1e_n\|= \|Te_n\|$ for all $n\in\ensuremath{\mathbb{N}}$, therefore
$\lim_{n\to\infty}\|Te_n\|=0$ as required.

{\it{Step 2.}} We construct an orthonormal sequence $(f_n)\subset H$ such that \\
$\lim_{n\to\infty}\|Tf_n\|=0$ and $f_{n+1}\perp \{f_1,Tf_1,...f_n,Tf_n\}$.

We obtain the sequence $(f_n)$ by induction, with the help of the sequence $(e_n)$. Take an
index $i_1$ such that  $\|Te_{i_1}\|\le 1$, and let $f_1:=e_{i_1}$. Assume now that
$f_1, f_2,\dots ,f_n$ are already given such that
$$\|f_j\|=1, \ \ f_j\perp \{f_k,Tf_k\}, \  \ \|Tf_j\|\le\frac{1}{\sqrt{j}}$$
and $f_j\in{\rm span} \{e_1,e_2,\dots ,e_{l_n}\}$, for all $1\le j,k\le n$, $k<j$, and $l_n$ is an
index depending on $n$ only.

Let $H_n:={\rm span}\{Tf_1,Tf_2,\dots ,Tf_n\}$. Take indices $j_1,\dots j_{n+1}$ such that $j_k>l_n$
and  $\|Te_{j_k}\|\le\frac{1}{n+1}$ for all $1\le k\le n+1$. The subspace $H_n$ is at most
$n$-dimensional, therefore there exists a non-trivial linear combination
$$f_{n+1}:=\sum_{k=1}^{n+1}\lambda_ke_{j_k}$$
such that $\|f_{n+1}\|=1$ and $f_{n+1}\perp H_n$.

It is clear, by construction, that $f_{n+1}\perp \{f_1,Tf_1,...f_n,Tf_n\}$. Furthermore,
$$\|Tf_{n+1}\|\le\frac{1}{n+1}\sum_{k=1}^{n+1}|\lambda_k|\le
\sqrt{\frac{\sum_{k=1}^{n+1}|\lambda_k|^2}{n+1}}=\frac{1}{\sqrt{n+1}}$$

{\it{Step 3.}}  We construct an orthonormal sequence $(g_n)\subset H$ such that\\
$\lim_{n\to\infty}\|Tg_n\|=0$ and $\{g_j, Tg_j\}\perp \{g_k,Tg_k\}$ for all $j\not= k$.

We obtain the sequence $(g_n)$ by induction, with the help of the sequence $(f_n)$.

Let $g_1=f_1$.
Assume now that $g_1, g_2,\dots ,g_n$ are already given such that
$$\|g_j\|=1, \ \  \{g_j,Tg_j\}\perp \{g_k,Tg_k\}, \ \ \|Tg_j\|\le\frac{1}{\sqrt{2j-1}}$$
and $g_j\in{\rm span} \{f_1,f_2,\dots ,f_{b_n}\}$, for all $1\le j\not= k\le n$, and $b_n$ is an
index depending on $n$ only.

Let $G_n:={\rm span}\{g_1, Tg_1, g_2, Tg_2,\dots ,g_n, Tg_n\}$. Take indices $m_1,\dots m_{2n+1}$
such that $m_k>b_n$ and  $\|Tf_{m_k}\|\le\frac{1}{2n+1}$ for all $1\le k\le 2n+1$. The subspace
$G_n$ is at most $2n$-dimensional, therefore there exists a non-trivial linear combination
$$g_{n+1}:=\sum_{k=1}^{2n+1}\mu_kf_{m_k}$$
such that $\|g_{n+1}\|=1$ and $Tg_{n+1}\perp G_n$.

It is clear, by construction, that $\{g_{n+1},Tg_{n+1}\}\perp \{g_1,Tg_1,...g_n,Tg_n\}$.
Furthermore,
$$\|Tg_{n+1}\|\le\frac{1}{2n+1}\sum_{k=1}^{2n+1}|\mu_k|\le
\sqrt{\frac{\sum_{k=1}^{2n+1}|\mu_k|^2}{2n+1}}=\frac{1}{\sqrt{2(n+1)-1}}$$

{\it{Step 4.}} We construct the orthonormal sequence $(h_n)$ with the properties stated at the
beginning of the proof.

We obtain the sequence $(h_n)$ by induction, with the help of the sequence $(g_n)$.

Take an index $r_1$ such that $\|Tg_{r_1}\|\le \frac{\delta^2}{10}\|Tg_1\|$. Let
$$h_1:=\frac{\delta}{2}g_1+\sqrt{1-\frac{\delta^2}{4}}g_{r_1}$$
We need to prove that $h_1$ is not a $\delta$-quasi-eigenvector of $T$. It is clear that
$$1\ge \|Th_1\|\ge \left( \frac{\delta}{2}-\frac{\delta^2}{10}\sqrt{1-\frac{\delta^2}{4}}\right)\|Tg_1\|$$
Also,
$$|(h_1,Th_1)|= |(g_1,Tg_1)+(g_{r_1},Tg_{r_1})|\le \left(\frac{\delta^2}{4}+(1-\frac{\delta^2}{4})
\frac{\delta^2}{10}\right)\|Tg_1\|$$
Combining these two ineqalities a simple calculation shows that
$\frac{|(h_1,Th_1)|}{\|Th_1\|}< \delta$, as required.

Assume now that vectors $h_1, \dots ,h_n$ are already given, such that $h_j$ is not a
$\delta$-quasi-eigenvector of $T$,
$$\|h_j\|=1, \ \   \{h_j,Th_j\}\perp \{h_k,Th_k\}, \ \  \|Th_j\|\le\frac{1}{\sqrt{j}}$$
and $h_j\in{\rm span} \{g_1,g_2,\dots ,g_{a_n}\}$, for all $1\le j\not= k\le n$, and $a_n$ is an
index depending on $n$ only. Take indices $p_1$, $p_2$, such that $p_1,p_2>a_n$
and $\|Tg_{p_1}\|\le\frac{1}{\sqrt{n+1}}$, and  $\|Tg_{p_2}\|\le \frac{\delta^2}{10}\|Tg_{p_1}\|$.
Let
$$h_{n+1}:=\frac{\delta}{2}g_{p_1}+\sqrt{1-\frac{\delta^2}{4}}g_{p_2}$$
It is clear that $\|Th_{n+1}\|\le \frac{1}{\sqrt{n+1}}$, and it can be shown as above
that $h_{n+1}$ is not a $\delta$-quasi-eigenvector of $T$. Hence, the sequence $(h_n)$ satisfies
all requirements, and the proof is complete.
\end{proof}
\end{theorem}

The author conjectures that a result corresponding to Theorem \ref{thm2} holds
also in Banach spaces. Namely, whenever $A$ is not bounded it should be
possible to introduce an equivalent norm on the space such that $e^{tA}$ is
not quasi-contractive with respect to the new norm. This problem, however,
remains open.

Now we present the main result of the paper.

\begin{theorem}\label{thm3}
Let $A$ be the generator of a $C_0$-semigroup $e^{tA}$ on a Hilbert space $H$.
Consider the following statements.

(i) A is bounded.

(ii) $-A$ is associated with a densely-defined, closed, sectorial form $a$ on $H$.

(iii) The formula $(e^{\frac{t}{n}A}P)^nx$ converges for all projections $P\in \mathcal{L} (H)$, and
all $x\in H$ and $t>0$.

(iv) The formula $(e^{\frac{t}{n}A}P)^nx$ converges for all {\it{orthogonal}} projections
$P\in \mathcal{L} (H)$, and all $x\in H$ and $t>0$.

The following implications hold: (i) $\Leftrightarrow$ (iii), (ii) $\Leftrightarrow$ (iv).

\begin{proof}
The implication  (i) $\Rightarrow$ (iii) was proved in \cite{MS}, while the implication\\
(ii) $\Rightarrow$ (iv) is a consequence of \cite{Ka2}, Addendum (see also \cite{MS}, Theorem 4).

We prove the implication (iii) $\Rightarrow$ (i) by contradiction.

Assume first that the semigroup $e^{tA}$ is not quasi-contractive. By the Lumer-Phillips
theorem this is equivalent to the fact that the numerical range of $A$ is not contained in any
left half-plane.

We construct an element $g\in H$ such that $\|g\|=1$, and
$$\lim_{n\rightarrow\infty}(e^{\frac{1}{n}A}P_g)^ng$$
does not exist, where $P_g$ denotes the one-dimensional projection onto the subspace
spanned by $g$. The vector $g$ will be given as
$$g:=\frac{\lim_{k\rightarrow\infty}g_k}{\|\lim_{k\rightarrow\infty}g_k\|}$$
where $(g_k)$ denotes a convergent sequence in $H$ to be constructed in the sequel.

Let $g_1\in D(A)$, such that $\|g_1\|=1$. First, we show that
$$\lim_{n\rightarrow\infty}(e^{\frac{1}{n}A}P_{g_1})^ng_1=e^{(Ag_1,g_1)}g_1$$
Indeed,
$$(e^{\frac{1}{n}A}P_{g_1})^ng_1=e^{\frac{1}{n}A}(P_{g_1}e^{\frac{1}{n}A}P_{g_1})^{n-1}g_1=
e^{\frac{1}{n}A}(P_{g_1}e^{\frac{1}{n}A}P_{g_1}g_1, \ g_1)^{n-1}g_1$$
and
$$\lim_{n\rightarrow\infty}(P_{g_1}e^{\frac{1}{n}A}P_{g_1}g_1, \ g_1)^{n-1}=e^{(Ag_1, \ g_1)}$$
because
$$\lim_{n\rightarrow\infty}\frac{(P_{g_1}e^{\frac{1}{n}A}P_{g_1}g_1, \ g_1)-1}{1/n}=\lim_{n\rightarrow\infty}
\left( \frac{(e^{\frac{1}{n}A}-I)g_1}{1/n}, \ g_1\right)=(Ag_1, \ g_1)$$

Now, choose $g_1$ such that ${\mathrm{Re}} \ (Ag_1,g_1)\ge 1$ holds also.

Let $\varepsilon >0$ be fixed. Take an index $n_1$ so large that
$$\| \left(e^{\frac{1}{n_1}A}P_{g_1}\right)^{n_1}g_1-e^{(Ag_1,g_1)}g_1\|<\varepsilon$$
It is clear from standard continuity arguments that there exists a $\delta_1 >0$, such that for
all $h\in B(g_1,\delta_1)$ we have
$$\| \left(e^{\frac{1}{n_1}A}P_{\frac{h}{\|h\|}}\right)^{n_1}\frac{h}{\|h\|}-
e^{(Ag_1,g_1)}g_1\|<2\varepsilon$$
Without loss of generality we can assume that $\delta_1<\frac{1}{2}$.

Now assume, that vectors $g_1, g_2, \dots , g_k$, and positive numbers
$\delta_1,\delta_2,\dots , \delta_k$, and indices $n_1,n_2,\dots ,n_k$  are already given with
the properties that:
$$g_j\in D(A), \ \  {\mathrm{Re}} \ (Ag_j,g_j)\ge j$$
and
$$\| \left(e^{\frac{1}{n_j}A}P_{\frac{h}{\|h\|}}\right)^{n_j}\frac{h}{\|h\|}-
e^{(A\frac{g_j}{\|g_j\|},\frac{g_j}{\|g_j\|})}\frac{g_j}{\|g_j\|}\|<2\varepsilon$$
for all $1\le j\le k$ and all $h\in B(g_j,\delta_j)$. Assume, furthermore, that
$$\|g_{j+1}-g_j\|< {\mathrm{min}} \ \left\{\frac{\delta_1}{2^j}, \frac{\delta_2}{2^{j-1}}, \dots
\frac{\delta_j}{2}\right\}$$
for all $1\le j\le k-1$.

The numerical range of $A$ is not bounded from the right, hence there exists a vector
$f\in D(A)$ such that
$$\|f\|<  {\mathrm{min}} \ \left\{\frac{1}{\|Ag_k\|}, \frac{\delta_1}{2^k},
\frac{\delta_2}{2^{k-1}}, \dots  \frac{\delta_k}{2}\right\}$$
and
${\mathrm{Re}}\ \ (Af,f)\ge 2$.
Let $f_k:=e^{i\alpha}f$ with suitable $\alpha$ such that  ${\mathrm{Re}}\ \ (Af_k,g_k)\ge 0$. Let
$$g_{k+1}:=g_k+f_k$$
Then
\begin{eqnarray*}
{\mathrm{Re}}\ \ (Ag_{k+1},g_{k+1})=
{\mathrm{Re}}\ \ (Ag_{k},g_{k}) + {\mathrm{Re}}\ \ (Ag_{k},f_{k}) +\\
+{\mathrm{Re}}\ \ (Af_{k},g_{k}) + {\mathrm{Re}}\ \ (Af_{k},f_{k})
\ge k+(-1)+0+2=k+1
\end{eqnarray*}
Furthermore, we have
$$\lim_{n\to\infty} (e^{\frac{1}{n}A}P_{ \frac{g_{k+1}}{\|g_{k+1}\|}  })^{n}
\frac{g_{k+1}}{\|g_{k+1}\|}=e^{(A\frac{g_{k+1}}{\|g_{k+1}\|},\frac{g_{k+1}}{\|g_{k+1}\|})}
\frac{g_{k+1}}{\|g_{k+1}\|}$$
Take an index $n_{k+1}$ so large that $n_{k+1}>n_k$ and
$$\| \left(e^{\frac{1}{n_{k+1}}A}P_{ \frac{g_{k+1}}{\|g_{k+1}\|}   }\right)^{n_{k+1}}
\frac{g_{k+1}}{\|g_{k+1}\|}  -e^{(A \frac{g_{k+1}}{\|g_{k+1}\|}  ,
\frac{g_{k+1}}{\|g_{k+1}\|}  )} \frac{g_{k+1}}{\|g_{k+1}\|}  \|<\varepsilon$$
It is clear from standard continuity arguments that there exists a $\delta_{k+1} >0$, such that for
all $h\in B(g_{k+1},\delta_{k+1})$ we have
$$\|\left(e^{\frac{1}{n_{k+1}}A}P_{\frac{h}{\|h\|}}\right)^{n_{k+1}}\frac{h}{\|h\|}-
e^{(A \frac{g_{k+1}}{\|g_{k+1}\|}  , \frac{g_{k+1}}{\|g_{k+1}\|}  )}
\frac{g_{k+1}}{\|g_{k+1}\|}  \|<2\varepsilon$$

It is clear, by construction, that the sequence $g_k$ converges in $H$. Let
$$h:=\lim_{k\to\infty}g_k \ \ {\mathrm{and}} \ \  g:= \frac{h}{\|h\|}$$
Recall, that $\|g_1\|=1$ and $\delta_1<\frac{1}{2}$, therefore $\frac{1}{2}<\|g_k\|<\frac{3}{2}$
for all $k\ge 1$. It is also clear, by construction, that  $h\in B(g_k,\delta_k)$ for all $k\ge 1$.
Hence, for all $k\ge 1$ we have
$$\| \left(e^{\frac{1}{n_k}A}P_{g}\right)^{n_k}g-e^{(A\frac{g_k}{\|g_k\|},
\frac{g_k}{\|g_k\|})}\frac{g_k}{\|g_k\|}\|<2\varepsilon$$
Notice, that
$$\|e^{(A\frac{g_k}{\|g_k\|},\frac{g_k}{\|g_k\|})}\frac{g_k}{\|g_k\|}\|=
e^{\frac{1}{\|g_k\|^2}{\mathrm{Re}} \ (Ag_k,g_k)}> e^{\frac{1}{4}k}$$
This means that (the norm of) the sequence
$ (e^{\frac{1}{n}A}P_{g})^{n}g$ does not converge.

Now, assume only that $A\notin \mathcal{L} (H)$. Introduce, by Theorem \ref{thm2}, an equivalent scalar
product $(x,y)_0:=(x,Qy)$ on $H$, such that the semigroup $e^{tA}$ is not quasi-contractive
with respect to  $( \ ,  \ )_0$. Take an orthogonal projection $P_g$ (with respect to the
scalar product $( \ ,  \ )_0 \ $),  such that  $ (e^{\frac{1}{n}A}P_{g})^{n}g$ does not
converge. Then, $P_g$ is a bounded (possibly non-orthogonal) projection with respect to the
original scalar product $(  \ ,  \ )$, such that
$ (e^{\frac{1}{n}A}P_{g})^{n}g$ does not converge. This proves the implication
(iii) $\Rightarrow$ (i).

The implication (iv)  $\Rightarrow$ (ii) is also proved by contradiction.

Asume, that the numerical range of $A$ is not contained in any sector
$$\Sigma_{\phi ,\omega}:=\{z\in \mathbb{C} : \frac{\pi}{2}+\phi <{\mathrm{arg}} \ (z-\omega)<
\frac{3}{2}\pi-\phi\}$$
with $\omega\in\ensuremath{\mathbb{R}}$, $\phi\in (0,\frac{\pi}{2})$. There are two cases to consider.

If the semigroup $e^{tA}$ is not quasi-contractive , then, by the arguments above,
there exists a vector $g\in H$, such that $\|g\|=1$ and  $ (e^{\frac{1}{n}A}P_{g})^{n}g$
does not converge.

If the semigroup $e^{tA}$ is quasi-contractive then, by rescaling, we can assume that
${\mathrm{Re}} \ (Ax,x)\le -1$ for all $x\in D(A)$, $\|x\|=1$.

We construct an element $g\in H$ such that $\|g\|=1$, and
$\lim_{n\rightarrow\infty}(e^{\frac{1}{n}A}P_g)^ng$ does not exist, where $P_g$ denotes the
one-dimensional projection onto the subspace spanned by $g$. The vector $g$ will be given as
$$g:=\frac{\lim_{k\rightarrow\infty}g_k}{\|\lim_{k\rightarrow\infty}g_k\|}$$
where $(g_k)$ denotes a convergent sequence in $H$ to be constructed in the sequel.

Take an arbitrary vector $g_1\in D(A)$, $\|g_1\|=1$.
Let $(Ag_1,g_1)=:a_1+b_1i$. We know that
$$\lim_{n\rightarrow\infty}\left(e^{\frac{1}{n}A}P_{g_1}\right)^ng_1=e^{(Ag_1,g_1)}g_1$$
Let $\varepsilon>0$, and $\rho >0$ be fixed. Take an index $n_1$ so large that
$$\| \left(e^{\frac{1}{n_1}A}P_{g_1}\right)^{n_1}g_1-e^{(Ag_1,g_1)}g_1\|<\varepsilon$$
It is clear from standard continuity arguments that there exists a $\delta_1 >0$, such that for
all $h\in B(g_1,\delta_1)$ we have
$$\| \left(e^{\frac{1}{n_1}A}P_{\frac{h}{\|h\|}}\right)^{n_1}\frac{h}{\|h\|}-
e^{(Ag_1,g_1)}g_1\|<2\varepsilon$$
Without loss of generality we can assume that $\delta_1<\frac{1}{2}$.

Now assume, that vectors $g_1, g_2, \dots , g_k$, and positive numbers
$\delta_1,\delta_2,\dots , \delta_k$,\\
real numbers $\varepsilon_1 , \varepsilon_2 , \dots ,\varepsilon_k$, and indices $n_1,n_2,\dots ,n_k$  are already given
with the following properties: for all $1\le j\le k$ we have  $|\varepsilon_j|<\rho$,
$$g_j\in D(A), \ \ (A\frac{g_j}{\|g_j\|}, \frac{g_j}{\|g_j\|}  )= a_j+(\varepsilon_j+b_1+(j-1)\pi )i$$
(note that $\varepsilon_1=0$),
where $a_1-1<a_j\le -1$, and
$$\| \left(e^{\frac{1}{n_j}A}P_{\frac{h}{\|h\|}}\right)^{n_j}\frac{h}{\|h\|}-
e^{(A\frac{g_j}{\|g_j\|},\frac{g_j}{\|g_j\|})}\frac{g_j}{\|g_j\|}\|<2\varepsilon$$
for all $h\in B(g_j,\delta_j)$. Assume, furthermore, that
$$\|g_{j+1}-g_j\|< {\mathrm{min}} \ \left\{\frac{\delta_1}{2^j}, \frac{\delta_2}{2^{j-1}}, \dots
\frac{\delta_j}{2}\right\}$$ for all $1\le j\le k-1$.

Now, we construct the vector $g_{k+1}$. The numerical range of $A$ is not contained in any
sector, therefore there exists a sequence $(x_j)\subset D(A)$ such that,
$\lim_{j\to\infty}\|x_j\|=0$ and
$${\mathrm{Im}} \ \frac{(Ax_j,x_j)}{ \|g_k\|^2   }=
\pi \ \ {\mathrm{and}} \ \  \frac{{\mathrm{Re}} \ (Ax_j,x_j)}{\|g_k\|^2}<\frac{a_k-(a_1-1)}{2}$$

Take $y_j:=e^{i\alpha_j}x_j$ with suitable $\alpha_j$ such that $(Ay_j,g_k)\ge 0$ real. Then
\begin{eqnarray*}
\frac{(A(g_k+y_j), g_k+y_j)}{\|g_k\|^2}=\frac{(Ag_k,g_k)}{\|g_k\|^2}+
\frac{(Ag_k,y_j)}{\|g_k\|^2}+\\
+\frac{(Ay_j,g_k)}{\|g_k\|^2}+\frac{(Ay_j,y_j)}{\|g_k\|^2}=:c_j+d_ji
\end{eqnarray*}
The real part $c_j$ of this expression satisfies
$$c_j>(a_1-1)+(\frac{a_k-(a_1-1)}{2})-\frac{|(Ag_k,y_j)|}{\|g_k\|^2}$$
for all $j\ge 1$. For the imaginary part $d_j$, we have
$$\lim_{j\to\infty}d_j=\varepsilon_k+b_1+k\pi$$
This means that for large $j$ we have $\|y_j\|< {\mathrm{min}} \ \{\frac{\delta_1}{2^k},
\frac{\delta_2}{2^{k-1}}, \dots  \frac{\delta_k}{2}\}$, and
$$\frac{{\mathrm{Re}}(A(g_k+y_j),g_k+y_j)}{\|g_k+y_j\|^2}>a_1-1$$
and
$$\frac{{\mathrm{Im}}(A(g_k+y_j),g_k+y_j)}{\|g_k+y_j\|^2}=\varepsilon_{k+1}+b_1+k\pi$$
where $|\varepsilon_{k+1}|<\rho$. Take such an index $j$, and define
$$g_{k+1}:=g_k+y_j$$
Again, standard continuity arguments show that there exist a positive number $\delta_{k+1}$ and an
index $n_{k+1}$ such that
$$\| \left(e^{\frac{1}{n_{k+1}}A}P_{\frac{h}{\|h\|}}\right)^{n_{k+1}}\frac{h}{\|h\|}-
e^{(A\frac{g_{k+1}}{\|g_{k+1}\|},\frac{g_{k+1}}{\|g_{k+1}\|})}\frac{g_{k+1}}{\|g_{k+1}\|}\|<2\varepsilon$$
for all $h\in B(g_{k+1},\delta_{k+1})$.

It is clear, by construction, that the sequence $g_k$ converges.  Let
$$h:=\lim_{k\to\infty}g_k \ \ {\mathrm{ and}} \ \ g:= \frac{h}{\|h\|}$$
Recall, that $\|g_1\|=1$ and $\delta_1<\frac{1}{2}$, therefore $\frac{1}{2}<\|g_k\|<\frac{3}{2}$
for all $k\ge 1$. It is also clear, by construction, that  $h\in B(g_k,\delta_k)$ for all $k\ge 1$.
Hence, for all $k\ge 1$ we have
$$\| \left(e^{\frac{1}{n_k}A}P_{g}\right)^{n_k}g-e^{(A\frac{g_k}{\|g_k\|},
\frac{g_k}{\|g_k\|})}\frac{g_k}{\|g_k\|}\|<2\varepsilon$$

Notice, furthermore that
\begin{eqnarray*}
\|e^{(A\frac{g_{2k+1}}{\|g_{2k+1}\|},\frac{g_{2k+1}}{\|g_{2k+1}\|})}
\frac{g_{2k+1}}{\|g_{2k+1}\|}-e^{(A\frac{g_{2k}}{\|g_{2k}\|},\frac{g_{2k}}{\|g_{2k}\|})}
\frac{g_{2k}}{\|g_{2k}\|}\|=\\
\|e^{a_{2k+1}}e^{(\varepsilon_{2k+1}+b_1+2k\pi)i}\frac{g_{2k+1}}{\|g_{2k+1}\|}-
e^{a_{2k}}e^{(\varepsilon_{2k}+b_1+(2k-1)\pi)i}\frac{g_{2k}}{\|g_{2k}\|}\|\ge\\
\|e^{a_{2k+1}+b_1i}g_1-e^{a_{2k}+(b_1-\pi)i}g_1\|-
\|e^{a_{2k+1}+b_1i}(e^{\varepsilon_{2k+1}}\frac{g_{2k+1}}{\|g_{2k+1}\|}-g_1)\|-\\
-\|e^{a_{2k}+(b_1-\pi)i}(e^{\varepsilon_{2k}}\frac{g_{2k}}{\|g_{2k}\|}-g_1)\|\ge\\
 2e^{a_1-1}-\|e^{a_{2k+1}+b_1i}(e^{\varepsilon_{2k+1}}\frac{g_{2k+1}}{\|g_{2k+1}\|}-g_1)\|-
\|e^{a_{2k}+(b_1-\pi )i}(e^{\varepsilon_{2k}}\frac{g_{2k}}{\|g_{2k}\|}-g_1)\|
\end{eqnarray*}
We can now choose the values of $\varepsilon ,\delta_1, \rho$ so small that
\begin{eqnarray*}
\|e^{a_{2k+1}+b_1i}(e^{\varepsilon_{2k+1}}\frac{g_{2k+1}}{\|g_{2k+1}\|}-g_1)\|+
\|e^{a_{2k}+(b_1-\pi)i}(e^{\varepsilon_{2k}}\frac{g_{2k}}{\|g_{2k}\|}-g_1)\|\le e^{a_1-1}
\end{eqnarray*}
and $5\varepsilon \le e^{a_1-1}$

Then we have
$$\| (e^{\frac{1}{n_{2k+1}}A}P_{g})^{n_{2k+1}}g- (e^{\frac{1}{n_{2k}}A}P_{g})^{n_{2k}}g\|\ge \varepsilon$$
Therefore the sequence $(e^{\frac{1}{n}A}P_{g})^{n}g$ does not converge, and the proof is
complete.

\end{proof}
\end{theorem}

redpajama_set_name: RedPajamaArXiv


Eton College is the principal fee paying school in Britain. Surprisingly, it's also a charity. As a result, those who attend Eton are subsidised by the taxpayer. Charities in Britain have a number of finanacial advantages reflecting their charitable work: their contribution to the well- being of society. At first sight it appears odd that Eton qualifies but successive governments have let this anomaly stand. Fees are reduced by these special privileges to the benefit of those who will, to coin a phrase, 'inherit the earth'.
Eton College is a very successful business. It has 1,300 pupils who pay fees amounting to circa £38.69 million p.a. ( 20% of their pupils pay less than the £33,270 annual fees.) VAT (UK transaction tax) exemption is worth circa £7.74 million to Eton. Additionally, Eton has property valued at £62 million and an investment portfolio of £213 million, both of which produce an income*. Each one of the the 1,010 pupils paying full fees are subsidised by £6,554 from VAT exemption. Full fees would be £39,924 if VAT exemption didn't apply. An interesting comparison is with Harvard University (USA), which charges $67,844 (£39, 908) for its students. This is virtually identical to Eton- £33,270 + £6,654 (VAT exemption) = £39,924. The key difference is that Harvard is a private university. Harvard is also a world- renowned research university and Eton came 37th in the GCSE** league tables (2013) beneath many state schools.
Charities in Britain have many other benefits. Business rates are one fifth of the full rate and Eton pays no corporation tax. (I can't calculate the impact of this for Eton.) Charitable Gift Aid legislation adds a further 28% to parental donations and fund raising activities creating a further income stream. Eton claims that a fifth of their pupils receive a discount. Pupils receiving this discount are an elite, selected to add lustre to Eton: they are cherry picked for Eton's 'charity'***. This appears to be 'window dressing' for the Charity Commissioners' perusal. Eton provides subsidies for poor pupils. The state then provides (invisible) subsidies for the vast majority of its (very rich) pupils. The deal for parents is quiet clear. Those paying full fees are subsidised by the state to total of £6,654 p.a.
Put that £6,654 benefit in other contexts. The state allocates £5,086 per pupil for schools in the outer London Borough of Havering, which is £1,568 less than the VAT subsidy to pupils at Eton. The basic state pension in Britain is £5,881 p.a. or £773 p.a. less than VAT subsidy provided for the children of the wealthiest people in Britain. Unemployed people, under 24 years old, can claim Job Seekers' Allowance (JSA) for six months amounting to £1,491, which is well below 50% of the VAT exemption given to pupils at Eton. JSA can only be claimed for six months, unlike five subsidised years for Eton pupils. Charitable status for Eton is a profitable anomaly to say the very least.
Invisible state aid to the independent education sector is unjust and inefficent. Parliament is dominated by a gigantic oligopoly of privately educated MPs lobbying for their old schools. In all 35% of MPs (229) were privately educated as opposed to just 7% of the general population. Astonishingly, 20 MPs were educated at Eton, a massive 3% of all MPs****. This is legislation by the rich for the rich. So the benefit culture is alive an well and living at Eton College.
**GCSE is Britain's national 16+ examination sat by all pupils.
The May 2015 British general election has seen 52% of the Conservative MPs, who were privately educated, elected to Parliament. The Conservative Party is the majority party and has formed a new government which will be in power until 2020. Doubtless they will continue looking after the charitable status of the private schools.
This entry was posted in Economics, Politics, School, statistics and tagged 2015 General Election, charity, Eton College, invisible subsidies, Mathletics 2015, MPs education, Privately educated Conservative MPs, VAT exemption, Warren Li. Bookmark the permalink.
Perhaps Eton College's financial statement doesn't appear to recognise the 'true impact of state subsidy' is because there's none received by them whatsoever. And I would entirely agree with them.
Politicians in particular, would like to have us believed that tax breaks and subsidies are one and the same thing whilst in reality they are two different animals. The definition of a subsidy is, "a direct pecuniary aid furnished by a government…"; "a grant or contribution of money"; "a sum paid…"; etc. Eton College received none of those – not a bean!
Hope the above explanation of the difference between a subsidy and tax break is useful. Thanks for responding.
An alteration in the tax relationship is nrmally seen as a (covert) subsidy.Additionally they do recognise it because they say that they don't 'profit' from it as they spend more on their charitible works theanthey get. (This is a point you also make.) Closet subsidies like all subsidies distort the market andthey don't cease to be subsidies merely because of tricky accounting. I sense I have no more to say on this matter but thank you for the opportunity; I try to write 600 words & it is a bit tight for a complex post.
Eton, for example, is on record to say they spend more each year on charity than what they receive in tax breaks. Thank you for your response.
Eton College is an elite school and a destination in the international education market unlike the majority of independent schools who are neighbourhood day schools. There are circa 512,000 children in independent schools and most are in the 'undistinguished' bracket they generate nothing beyond their own income. I've looked at the Eton College financial statement and it doesn't appear to me that they have 'recognised' the true impact of state subsidy on their accounts. I thought that the site you attached was full of artistic licence and double counting. The independent sector also has some companies that are For Profit, Cognita being a prime example, and they operate comfortably without state subsidy. Need I say any more? Thank you for your comments.
Oh really? If Eton and all the other fee-paying schools didn't exist, where do all these kids go for their education? And what about their teachers? They would ALL go to state schools and please tell us the figures that would cost the government seeing that you're so hot on statistics and figures.
Assuming the buildings still exist along with investment and property portfolios deleting the charity status does not imply deletion of the private school sector. However it does imply that the charity status is a state subsidy. So! Some parents will not be able to pay the real cost of their children's education and will, as you rightly say, revert to state education. And that will be an additional cost which will have to be met. The Parliamentary Committee estimates the direct cost of state subsidy to independent schools at £100 million p.a. And that is a clear under- estimate as it doesn't cost in subsequent savings. Do the two figures add up? I.e. Would the additional pupils decanted out of private schools cost more/less than circa £100M? I don't know but it does feel like unnecessary subsidy to me. So I am making both an economic and a political point. Thank you for your comment.
It's true. We need to start an alternative paper – not run by the Eton/Oxbridge elite! I wonder what the private/'public' school sector as a whole gain out of this system. It won't be just Eton..
I'm quite sure they all know because they too belong to same elite of 'Old Boys'. The consequence of a state supported elite is that it is self perpetuating. After 1997 I gave up all hope that the tax system would be fair: the demonisation of the poor since 2010 is just a continuation of the Blair policy but in trumps.
Staggering,Chris- I cannot take in at one sitting so will ask my butler to download this in its entirety.
It certainly seems to be in a 'Mess'. The whole 'Charity' system is in a mess. I expect there are much worse cases of 'Charity Status' abuse. ( another topic for you Chris ?). What you did not compare was the cost of the 'Uniforms' for Eton and Havering !!
I see that assets for the Cambridge colleges are worth £4 billion and Oxford £3.3 billion.
It is good that we have an elite school somewhere. Even the Nazi hierarchy wanted to send their kids to Eton after they had won.
I expect that most of the USA government top bods have had a privileged upbringing. It is not that they are the best for the posts but a case of jobs for the boys, which I am afraid happens in all walks of life; its just that it affects us most when rife in Parliament.
The following link expresses similar views to Chris's.
Well done Chris, concise and illuminating, like a laser beam.


text:
Debate Centre
Twitter's Toon Talk
Hartson gushes about Saint-Maximin
Date: 12th October 2019 at 8:00am
John Hartson has heaped praise on Newcastle United winger Allan Saint-Maximin for his role in the 1-0 victory over Manchester United last time out.
The pundit, regularly seen on BT Sport Score, singled out the Frenchman as the player who can cause real danger for the Magpies this season and insisted that Steve Bruce needs those sort of players to step up and be counted.
Hartson was particularly impressed with the 22-year-old's work on the ball, and his ability to beat a man in the final third.
"He's a box of tricks, he's powerful, he can cause problems," he told Football FanCast.
"He gave Man United all sorts of problems."
There is no doubting the fact that Saint-Maximin is Newcastle's standout player this season, and Bruce will rely on the Frenchman and his trickery at times throughout the campaign. The 22-year-old is one of few players who can get on the ball and beat his man to create an opportunity in the Magpies' squad, and Newcastle have desperately missed his creativity during his absence at the beginning of the season. The win over Manchester United was the first time that the Magpies' fans have got to see their summer signing for more than 45 minutes, and he didn't disappoint, completing four dribbles [WhoScored] as he terrorised the Red Devils all afternoon. Saint-Maximin also completed two key passes and had two shots on goal, proving his ability to be a major threat for Bruce going forward, who will be relying on the winger to consistently perform. Hartson is right to single out Saint-Maximin for his efforts, with the young winger surely relishing the opportunity to be Newcastle's main man this season.

redpajama_set_name: RedPajamaCommonCrawl


How can you understand if you are obese?
You can understand if you are obese or you border on obesity with the calculation of height and weight index. While to be slightly above the normal weight limit demonstrates that the person borders on obesity, to be much higher than the normal limit demonstrates that the person is obese. In this regard, the most important thing to know is that there is no need to weigh hundreds of kilos. Even a little bit more weight that your height can not lift shows that you are within the borders of obesity. Obesity treatment can be done with many different methods. In the selection of the method, the grade of obesity must be considered.
If the obesity level does not lead to death, obesity treatment can be realized with efficient diets. At the same time, the heart problems being caused by obesity are also regularly checked and it is tried to accelerate the metabolism in a natural way. In this kind of treatment, the most important point is patient's self control. If the patient's self control is strong, after weight loss, it is possible to maintain the healthy weight level. It is already the main aim of the treatment.
In obesity treatments, exercise method can be preferred. In obesity's lower and middle bound, it is very important that the patient watches himself, motivates himself when his self control disappears. In this type of treatment, psychological counselling has a great benefit on patients.
In obesity treatment, pharmacological treatment ways are preferred too. Drugs given in this treatment are only used in control of doctor. When obesity level reaches high dimensions, obesity drugs are in question. These drugs should not have side effects and cause addiction. Obesity patients must be careful about the products sold under the name of herbal slimming pills. Because many of these products are fake and their use can cause death.
If the patient cannot live and move independently, gastrostomy tube can be preferred as obesity treatment. This treatment which has high usefulness as gastric stapling is also known a gastric bypass and obesity surgery.


posts about sunn o))). | musikwiese.
felix, january 5, 2011 at 20:45. two comments.
sunn o))) is to music what david lynch is to film. if you have seen some of lynch's more obscure movies, such as eraserhead and inland empire, you know how strange, disturbing these films can be. these are films you either like or hate, and to see them you have to be in the right mood. the drone doom duo sunn o))) is very similar, only that they make music and not films.
the music is dark, or more like pitch black. or even blacker than that. it is disturbing. strange. creepy. more art than music. and definitely nothing you should listen to when being in the wrong mood. the following piece, orthodox caveman from their album black one, is a very good example of their music.


Turku, 27 May 2014 - Biotie Therapies today announced that patient enrollment into the Phase 2 study investigating nepicastat for cocaine dependence has been completed ahead of schedule. The 11-week, 179-patient study is being conducted at 10 US clinics under a Collaborative Research and Development Agreement (CRADA) with the National Institute on Drug Abuse (NIDA) at the US National Institutes of Health. Top-line results from the study are expected around the end of 2014.
"The extraordinary commitment that both NIDA and the trial investigators have shown for this program has helped the study exceed the already ambitious enrollment targets", commented Dr. Stephen Bandak, CMO of Biotie. "We look forward to completion of the clinical treatment phase and await the top-line results, scheduled to be available around year end, with great interest."
Nepicastat is an orally administered, potent and selective inhibitor of the enzyme dopamine beta-hydroxylase (DBH) which converts dopamine into norepinephrine. Like many other addictions, cocaine dependence is driven by dysregulation in the dopamine-reward system. Inhibition of DBH by nepicastat increases levels of dopamine, which may reduce craving for cocaine, and reduces the levels of norepinephrine, which may decrease the pleasurable responses to cocaine and the potential for stress-induced relapse following withdrawal.
Nepicastat is currently in Phase 2 development for cocaine dependence. A previously completed placebo-controlled Phase 2a study in non-treatment seeking cocaine addicts showed that nepicastat had a favourable safety profile and was well tolerated when administered with cocaine. An 11-week Phase 2 safety and efficacy trial in 179 treatment seeking cocaine addicted patients, funded by the U.S. National Institute on Drug Abuse under a Collaborative Research and Development Agreement, has completed patient enrollment, and top-line results are expected around the end of 2014.
Biotie retains full rights to nepicastat and will be able to use data from studies conducted with NIDA to support future potential regulatory submissions.


text:
Q: Prove $(E+A)^{n-1}$ is a nonnegative matrix with positive elements
Suppose $A$ is an $n\times n$ irreducible nonnegative matrix, prove that $(E+A)^{n-1}$ is a nonnegative matrix with positive elements.

I think that this excise can be proved by definition. So I tried to prove by contradiction. I suppose there exists an element of $(E+A)^{n-1}$ equals zero and then deduce to a contradiction, but I didn't know how, since I needed to consider so many situations.

redpajama_set_name: RedPajamaStackExchange


Let us take care of all your tree care needs.
Serving Ithaca, NY and the surrounding area since 2000.
Specializing in crane assisted hazard tree removal.
We are committed to providing our customers with professional, reliable, high quality tree services.
We have the equipment and expertise to safely and efficiently remove even the most challenging trees.


Welcome to Q36.5 Absolutely Equipment!
Q36.5 is an extreme vision of the future of competitive cycling apparel, producing some of the lightest pieces of equipment in the world. Weight is never an objective of the design, but the result of the integrity of the process: proprietary fabrics developed by the Q36.5 studio combined with the radically essentialist pure pattern construction, creates ergogenic garments that maintain a stable body temperature during physical activity.


text:
The badminton women's singles tournament at the 2014 Asian Games in Incheon took place 24–28 September 2014 at Gyeyang Gymnasium.

Schedule
All times are Korea Standard Time (UTC+09:00)

Results
Legend
WO — Won by walkover

Final

Top half

Bottom half

References

External links
Official website

Badminton at the 2014 Asian Games

redpajama_set_name: RedPajamaWikipedia


Boxing Day delivered a special gift to BrewDog and Crowdcube. The Scottish craft brewer closed the largest ever UK crowdfunding round at £10 million. The funding was listed on Crowdcube and stands as the largest to date. Crowdcube also points out that their record was topped three times in 2016 helping to push the total raised since platform launch to £210 million. It is also indicative of the fact that later stage companies are recognizing the power and potential of crowdfunding as an effective method to sell securities.
BrewDog, one of the UK's fastest growing food brands, raised its £10 million target in just three weeks. More than 2,700 people invested in the BrewDog Bond, which will pay 7.5% interest per annum. The average investment was £3,600 and the largest £200,000.
Luke Lang, co-founder of Crowdcube, labeled 2016 "another standout year" for his crowdfunding company. He also said it was an important year for the entire industry. Beset with economic challenges, this sector of finance continued to evolve and growth.
Recapping the year, Crowdcube said that over 51,000 investments were made through Crowdcube in 2016. During the year more than £80 million was successfully raised for startup, early and growth stage businesses from the UK and Europe.
With more established businesses raising finance on Crowdcube, such as Biscuiteers, Innis & Gunn and Vivobarefoot, Crowdcube has seen the average funding round increase to £642,000, up from £550,000 in 2015.
goHenry, a fintech trailblazer, was the first businesses in 2016 to set a new record for the largest crowdfunding round after raising £4 million in April. The record was then broken by Crowdcube's own funding round, which saw £7.7 million raised from over 3,500 people who invested alongside Balderton Capital, one of Europe's largest investment firms. This was an important event as Crowdcube filed a prospectus to hurdle a legal barrier to raising the sizeable sum – thus paving the path for others to follow.
Focusing on delivering capital gains for investors and potential exits, Lang said that over the next year he expects other companies to follow prior exiting firms to drive solid investor returns.
This entry was posted in Global, Investment Platforms and Marketplaces, Offerings and tagged brewdog, crowdcube, luke lang, uk, united kingdom. Bookmark the permalink.


Abby Wambach, Remarks as Delivered: Stunning commencement address to Barnard College.
What to Read when It's Time for Sports:: "Here is a list of books that tackle the complexities of sports in America, from nonfiction to poetry to fiction. ... And just a "heads up" (as they say in the sports world): this isn't your average sports list."
With words and a strong grip, an MLB umpire pulls a woman back from the brink in Pittsburgh : As a Major League Baseball umpire, John Tumpane often has to diffuse tense situations at the ballpark. None compared to the scene he came upon Wednesday as he walked across the Roberto Clemente Bridge around 3 p.m. on his way back from a run and lunch: A few hundred yards from PNC Park, he saw a woman climb over a railing and look toward the Allegheny River below.
Mud and Glory: "Of a wet and thunderous sodden Saturday they couldn�t put on any lights fifty-two years ago, there weren�t any to put on. It grew much darker and blacker in vilest intensity with coagulated punch-ups going oddly unnoticed due to the annoying and most profound darkness that at least cost nothing. The dark has always been on the house I think you�ll find. Dark and mud sponsored the game of life back then before the new artificial grass authorities put the kybosh on free thinking."
NCAA pulls championship events from North Carolina over HB2: Having warned the state of North Carolina at the Final Four that it risked losing NCAA events over House Bill 2, the NCAA showed Monday night just how serious it was. Bonus opinion piece by local sports writer, Luke DeCock.
The Great Sports Myth: the widespread assumption that sport is, inherently, a force of good�despite the fact that it can both empower and humiliate, build bonds and destroy them, blur boundaries and marginalize.
The Ringer: Bill Simmons' spiritual successor to Grantland, is open for business.
Baylor fires Art Briles over criminal coverups: Details are still a bit thin but apparently Coach Briles protected multiple players on his Baylor football team from sexual assault cases. Baylor President and Chancellor Ken Starr is now just Chancellor, athletic director Ian McCaw has been sanctioned and placed on probation but Briles' assistant coaches all remain in place, at least for now, including Kendall Briles. The details are nauseating so I'll leave them out of the post.
Grosse Ile H.S. lacrosse team accused of killing animal before game: Members of the Grosse Ile High School lacrosse team are under investigation by police and the school district after allegations surfaced Monday that some of the teens killed an animal prior to a game.
Former U.S. House Speaker Dennis Hastert's career was propelled by wrestling: which also allowed him to abuse, and get away with abusing, boys for many years. "When calls for his removal from office reverberated through the capital a decade ago, Speaker J. Dennis Hastert struggled to explain why he had not aggressively investigated allegations that a Florida lawmaker had sent flirtatious messages to a teenage boy who had served as a House page. At that perilous moment, an impassioned group of supporters stepped forward to speak up on Mr. Hastert's behalf: wrestling coaches."
It's Who We Are?: It�s really not surprising that, on December 10, 2015, the school board of McLoud, Oklahoma voted to maintain their racist mascot. All of the usual arguments were made, in support of keeping the R word: �It�s an honor,� tearful pleas of �We�ve been the R** for generations�, and, my favorite, �It�s who we are.� The latter is by far the most accurate and telling. Put simply, the events during the McLoud meeting was not only a by-product of American history, but an indictment of it.
Athletics doping: Wada commission wants Russia ban: The report says London 2012 was "sabotaged" by "widespread inaction" against athletes with suspicious doping profiles. Russia was also accused of running a "state-supported" doping program.
"Shuttering of Grantland Hints at Corroding ESPN Culture, Competency, and Gumption": Awful Announcing's finance director on why he thinks ESPN's cancelling of Grantland is, "the network�s dumbest decision in the five years I have professionally been involved in tracking the company". Linked from there, ESPN.com's former ombudsman on the situation.
Grantland is dead: "Grantland distinguished itself with quality writing, smart ideas, original thinking and fun. So we're junking it."
Craig Finn's guide to the best ever sports music: Hate to have to break it to him about Ragnar.
The Cleveland Cavaliers catch heat for in-game promotional video.: The video features a couple re-enacting the final performance scene from Dirty Dancing; however after the male catches his girlfriend he proceeds to slam her into the ground when it's revealed she's wearing a Bulls t-shirt.
Yes, But Is It A Sport?: A landmark court case in Britain to judge whether Bridge is a sport or not. Significant lottery funding is at stake.
Can popularity ruin a sport?: Rafa Honigstein on how success has changed the Budesliga.
"That rise to all-conquering prominence has attracted the country�s brightest minds and developed into a highly professional, productive industry. But football�s heightened social relevance is also reflected in the sort of insufferably grim, po-faced seriousness that used to be confined to political struggles or actual tribal conflicts.
People Who Like Sport Remember Their Lives Better Than People Who Don't.: A long rambling look through 2005. Life, death, divorce. And cricket.
Served up on a platter for SportsFilter: the question of which is the superior sport, cricket or baseball?
Requiem for Phillip Hughes: "So rest in peace my little brother. I'll see you out in the middle."
MLB umpire Dale Scott comes out as gay in quietest way possible.: (Scott) has worked three World Series, three All-Star Games, two no-hitters and numerous playoff games. He is gay and married to his partner of 28 years. That last fact is just part of who Dale Scott is and has had no impact on his abilities as an umpire for the past 29 seasons. Yet it is understandably the one that most people will notice, because Scott is the first Major League Baseball umpire to publicly say he is gay while active (and the first out active male official in the NBA, NHL, NFL or MLB).
ESPN Gets Ready to Launch Its First Web Video Subscription Service � For Cricket�s World Cup: ESPN's efforts to sell streaming video subscriptions for some sports programming without requiring viewers to pay for cable TV could start early next year.
People familiar with ESPN's plans say that Russell Wolff, who heads up the company's international group, is spearheading the plan to sell online access in the U.S. Earlier this year, Wolff pegged cricket's U.S. fan base at 30 million.
The Rise and Fall of The Great Kamala: how James Harris from Mississippi became a Ugandan giant.
Lisa Saxon, the woman who helped change sports writing forever.: A profile well worth a read from Vice Sports.
The End of Daisy Buchanan's: a Boston institution "where anything can happen".
All while benefiting from the same situation with a grad student from FSU who's competing for the starting QB job.
The NBPA Makes History With The Hiring Of A Female Executive Director: Receiving 32 of 36 possible votes, Michele Roberts takes over the position left vacant after Billy Hunter was fired amid accusations of shady business practices. Roberts becomes the first woman elected to lead a union representing male professional athletes.
"That makes two of us, doesn�t it Steve?": Mike Vaccaro on New York sports teams, Steve Philllips and Lupica.
"The unofficial king of sports paperbacks", Zander Hollander,: died at 91.
Before televised sports were pervasive and the Internet a nonstop gusher of sports trivia, Mr. Hollander found a niche in the market by annually providing statistics, team rosters, records, schedules and predictions for the coming season in the form of brick-size tomes he titled "Complete Handbooks."
In Fake Classes Scandal, UNC Fails Its Athletes�and Whistle-Blower: Beginning in the 1990s and continuing at least through 2011, UNC�s Department of African, African American, and Diaspora Studies offered more than 200 lecture courses that never met. The department also sponsored hundreds of independent study classes of equally dubious value. Internal reviews have identified forged faculty signatures and more than 500 grades changed without authorization. The students affected were disproportionately football and basketball players. Behind the cover of this week's edition.
A Forbes click-through-the-pictures list of the 40 most valuable brands in sports. It's divided into 4 categories (top 10 in each): businesses, teams, events, and athletes.


text:
Q: Avoiding duplicate record insertion into SQL table I have a windows service which basically watches a folder for any CSV file. Each record in the CSV file is inserted into a SQL table. If the same CSV file is put in that folder, it can lead to duplicate record entries in the table. How can I avoid duplicate insertions into the SQL table?

A: Try INSERT WHERE NOT EXISTS, where a, b and c are relevant columns, @a, @b and @c are relevant values.
INSERT INTO table
(
    a,
    b,
    c
)
VALUES
(
    @a,
    @b,
    @c
)
WHERE NOT EXISTS
(
    SELECT 0 FROM table WHERE a = @a, b = @b, c = @c
)

A: The accepted answer has a syntax error and is not compatible with relational databases like MySQL.
Specifically, the following is not compatible with most databases:
values(...) where not exists

While the following is generic SQL, and is compatible with all databases:
select ... where not exists

Given that, if you want to insert a single record into a table after checking if it already exists, you can do a simple select with a where not exists clause as part of your insert statement, like this:
INSERT
INTO      table_name (
            primay_col,
            col_1,
            col_2
          )
SELECT    1234,
          'val_1',
          'val_2'
WHERE     NOT EXISTS (
  SELECT  1
  FROM    table_name
  WHERE   primary_col=1234
);

Simply pass all values with the select keyword, and put the primary or unique key condition in the where clause.

A: Problems with the answers using WHERE NOT EXISTS are:

*

*performance -- row-by-row processing requires, potentially, a very large number of table scans against table

*NULL handling -- for every column where there might be NULLs you will have to write the matching condition in a more complicated way, like
(a = @a OR (a IS NULL AND @a IS NULL)).
Repeat that for 10 columns and viola - you hate SQL :)

A better answer would take into account the great SET processing capabilities that relational databases provide (in short -- never use row-by-row processing in SQL if you can avoid it. If you can't -- think again and avoid it anyway).
So for the answer:

*

*load (all) data into a temporary table (or a staging table that can be safely truncated before load)

*run the insert in a "set"-way:

INSERT INTO table (<columns>)
    select <columns> from #temptab
    EXCEPT
    select <columns> from table

Keep in mind that the EXCEPT is safely dealing with NULLs for every kind of column ;) as well as choosing a high-performance join type for matching (hash, loop, merge join) depending on the available indexes and table statistics.


This gold Nazana Ashrafi which weighs around 9.6g was issued by Sadiq Muhammad Khan V. The obverse of this depicts the portrait of Nawab Sadiq Muhammad Khan in military uniform facing towards left in the centre and the legend Ba-Faziah Taalah Sadiq Mohammad Abbasi Khamis Farman Rawai Mamlakat Khudadad Bahawalpur around. The reverse of this coin depicts 'Coats Of Arms in the centre, mint name Bahawalpur written above and 'Yek Ashrafi' at the bottom.
This gold Nazarana Ashrafi is listed for INR 1,20,000 in the upcoming auction of Classical Numismatic Gallery, which will be held on 11th August 2018.


FHI 360 seeks applications from South African local non-governmental organizations (NGOs) and Faith Based Organizations (FBOs) based in Kwa Zulu Natal for grants to implement a one-year HIV Service Delivery and Support for Families Caring for Orphans and Vulnerable Children (OVC) Project. The project will be implemented in Harry Gwala district, KZN province. This initiative is funded by the United States Agency for International Development (USAID) and the President's Emergency Plan for AIDS Relief (PEPFAR) and managed by FHI 360.
This opportunity is open to local South African NGOs and FBOs with a solid track record of facilitating access to HIV testing, care and treatment for orphans and vulnerable children, including adolescents, and providing a package of services for OVC, their caregivers and their families in the identified districts. The NGOs must be able to use creative and evidence-based approaches (Let's Talk; YOLO & Vhutshilo curricula series).
A detailed Annual Programme Statement (APS) entitled "ReACH APS III in the KwaZulu-Natal Province" with application guidelines is found at http://www.capacitydev.org. The APS provides guidelines on the grant application process and details on expected key activities.
Questions on the APS should be addressed to cds@fhi360.org and ensure the subject line states "ReACH in KwaZulu-Natal province".


To celebrate this holiday season, Perlinski Design is offering you 20% off of your next graphic or web design purchase* until the end of 2011.
Don't forget to get in last minute projects for your 2011 taxes!
Mention this offer when calling or contacting us online to receive the discount.
We appreciate your business, and hope you will shop with us again!
*Printing not included in this offer.


text:
Reviews and thoughts on movies, music and tv shows
The Godfather III (1990)
October 4, 2020 Posted in Movies
Directed by Francis Ford Coppola
Starring Al Pacino, Andy García, Diane Keaton, Eli Wallach, Joe Mantegna, Bridget Fonda
It has been confirmed that on December 8, 2020, a new version of this film will be released. Titled Mario Puzo's The Godfather Coda: The death of Michael Corleone I can't wait to see this version since it will be the true and original vision both Mario and Francis had for the film. it appears Paramount Pictures wanted the movie to go other ways and now Francis has been allowed to revisit the film and make this new cut.
The importance of The Godfather trilogy in films can't be denied. The first film launched the careers of so many great actors like Pacino, Keaton, Duvall, and Cane, and of course, it featured a brilliant Marlon Brando. The main success of the film was the story and how Mario was able to take a mafia story and make it about the family, in that way making us like the family despite the fact that they killed sometimes, as Francis once said in an interview. The Godfather II was a brilliant sequel, the way they told the story of the rise of Vito together with Michael's was brilliant and with the amazing performances of Al Pacino and Robert De Niro, the film could not go wrong. The Godfather III was released 16 years later and it was successful but not at the level the first two films were.
In this installment, the main focus of the film is Michael trying to finally make the Corleone family legit, he wanted out and fulfill what he so naively promised in the first family, to make the family business legit only to become more powerful and maybe evil than his father and brother were. But as he tries this, Vincent Corleone, Sonny's son, comes and wants to work for his uncle and he and his aunt don't want out causing trouble for Michael. We see Michael trying to reconnect with his children and Kay.
What went wrong with the film, I can't say much because I always liked the film. There are a few details that prevent this installment to be at the same level as the first two but there were not that many that I can say. What I loved about the film, the first thing has to be Al Pacino's performance, he was at his best in this one. in terms of performances, Andy Garcia and Diane Keaton were both great also. One thing missing was the absence of Tom, reading on Wikipedia it says that Duvall declined to participate in the film because of the salary he was going to get, a shame because it would have made the film better. Sofía Coppola turned out to be a great director, but her performance was not at par with the rest and it affects the film a little bit. Also, the romance between Mary and Vincent did not feel right for me.
I don't know how the cut will be or what things might change, but I do know that I can't wait to see it and I'll let you know what I think about it when I see it.
#alpacino#dianekeaton#francisfordcoppola#godfather#mariopuzzo
Published by luisreviewer
View all posts by luisreviewer
Previous: Previous post: Elton John: Peachtree Road (2005)
Next: Next post: Bon Jovi: 2020 (2020)
One thought on "The Godfather III (1990)"
Pingback: The Godfather Coda: The death of Michael Corleone (1993) re-edited 2020 – World of Entertainment
TV series/Documentaries
Movie reviews and talk about entertainment.
Blog at WordPress.com. World of Entertainment


Start growing coffee beans under the perfect conditions provided by this mini greenhouse. The set comes with everything needed to start growing coffee beans indoors. Once the plants start to sprout, transplant them outside or into a larger pot to allow them to grow to their full size. You'll be growing coffee beans in no time!


Hodo kwaja is a popular snack food sold by street vendors throughout Korea. Walking through the markets in Songtan-si, Kunsan, Suwon, or Seoul, the aroma of fresh cooked walnut cakes would stand out.
My young sons would drag me to the vendor's cart and demand that I purchase "wad ja" for them. These treats are baked in a walnut shaped mold and were usually filled with red bean paste and chopped walnuts, but could contain various fillings.
Specialty mold pans are now available for home cooks to re-create these treats.
Crack the eggs into a medium mixing bowl and whisk together.
Add salt, sugar flour, baking soda, and milk, and gently whisk into a smooth batter.
Add the melted butter and whisk into the batter.
Lightly oil the cooking pan on top and bottom.
Pre-heat both sides over very low flame.
Leave the pan on the stove over very low flame and add batter to each depression, filling it to just over half.
Place a few walnut pieces into the batter, then add a small portion of the bean paste, and top with more pieces of walnut.
Dribble some batter on top, close the pan and turn it over.
Cook on very low heat for about one to one and a half minutes and turn the pan. Attempt to lift the lid – the walnut cakes should release and stay in the bottom half.
Cakes are done when both sides are golden brown.


text:
package debop4k.timeperiod.samples;

import debop4k.timeperiod.TimeBlock;
import org.joda.time.DateTime;

public class Lesson extends TimeBlock {

  public Lesson(DateTime start) {
    super(start, start.plus(SchoolDay.LessonDuration));
  }

  private static final long serialVersionUID = -5322914051240174741L;
}

redpajama_set_name: RedPajamaGithub


VietNamNet Bridge - Northern folk singing will be introduced as an official bachelor subject at the Faculty of Heritage Music, set up in 2007 and part of the Hue Music Academy.
Course subjects will include xam (ballads by blind musicians), ca tru (ceremonial singing) and quan ho (love duets).
Up to 20 students will sit entrance exams next month in order to access the course, to be taught by experts from the Ha Noi-based Viet Nam Traditional Music Development Centre, according to Prof Pham Minh Khang, the centre's director.
"In the near future, the faculty plans to offer additional subjects based on folk music from various regions of the country. Students enrolled in the new courses will be exempt from 70 per cent of tuition fees," said Nguyen Viet Duc, deputy director of the academy.
Khang added that the course on northern folk singing would be a first, even though faculties of traditional music already existed at various music institutes throughout the country.
"Although we initially focused on older audiences, we have realised that many young people love traditional music," he said.
According to Khang, the Hue Music Academy, mainly focused on teaching music heritage and traditional music studies, will also offer courses on Tay Nguyen (Central Highlands) gongs and nha nhac (Hue royal court music), which form the foundation for students to further study traditional music.
The academy, in co-ordination with the centre, will also teach students how to play folk instruments and sing folk songs in order to help them understand northern folk music better.
In the case of hat van (spiritual singing), students will be taught how to play the dan nguyet (moon-shaped two-string lute) and, in the case of ca tru (ceremonial singing), students will be taught how to play the phach (bamboo bars used as instruments in ca tru).
According to composer Thao Giang, deputy director of the centre, various instrumentalists, singers and famed researchers such as Prof Ngo Duc Thinh and Prof Phan Dang Nhat, will be teaching on the course.
The centre has trained students in traditional folk singing for six years. It has also helped organise frequent performances at the Dong Xuan Market in Ha Noi's Old Quarter, the Dong Lac Temple (38 Hang Dao Street), the Kim Ngan Temple (42 Hang Bac Street) and the Quan De Temple (28 Hang Buom Street), in order to assist in turning each venue into a tourist hub.


text:
SportsColumnistsMark La Monica
By Mark La Monica @LaMonicaMark
Au revoir, Georges St-Pierre, you were one of the greatest ever to do it
Georges St-Pierre announces his retirement from mixed martial arts during a press conference in Montreal, Thursday, Feb. 21, 2019. Credit: AP/Ryan Remiorz
The cloud over Georges St-Pierre's legacy in mixed martial arts does not hang. It floats.
No rain looms, only sunshine. No doom. Just smiles and applause for one of the greatest MMA fighters the world ever encountered.
St-Pierre officially retired on Thursday at a news conference inside the Bell Centre in Montreal, a venue he once ruled in the city he called home, in the country he hoisted onto his shoulders to help deliver his sport to his people, and his people to his sport.
"It takes a lot of discipline to retire on top," St-Pierre said Thursday. "I always said I wanted to retire on my own, not be told to retire.
"I don't always know what I want, but I do know what I don't want. What I don't want is to retire too late."
St-Pierre, 37, hasn't fought since November 2017, and that was his first fight in more than four years. So, "on top" is subjective, as is his place in the pantheon of the sport (it should be very high on everyone's lists, however; if not, find another list).
What is not subjective, though, is this: a 26-2 career record, the most title fight victories in UFC history with 13, nine straight defenses of his welterweight title (the most in the division's history and third longest streak in UFC history), most takedowns landed with 90, and becoming the fourth fighter in UFC history to win titles in two different weight classes when he beat Michael Bisping at UFC 217 at Madison Square Garden.
Those are some facts. Not all of them. With St-Pierre, a list of stat facts may challenge the generally accepted principle that the internet has unlimited space.
Subscribe to Sports Now newsletter
Yet with all those victories, massive career-defining moments, dollars and championships, do you know what St-Pierre called his proudest moment of his career? That time he got blasted in the head by Carlos Condit. It was a head kick in the third round of their title fight at UFC 154 on Nov. 17, 2012.
"I fell down and I was able to stand up. It sounds weird, people think it will be the wins, but for me it was when I was dropped and got back up," St-Pierre said. "It shows I had the guts to come back from an obstacle and was able to overcome it."
St-Pierre won that fight, as he did 93 percent of the time he made that walk to the cage.
One of the two times he didn't win was against Long Island's Matt Serra at UFC 69 in 2007. St-Pierre was the champion heading into that night, with Serra a 10-1 underdog. The first-round knockout of St-Pierre remains the biggest upset in UFC history. It also remains the most important moment in St-Pierre's fighting career. He never lost a fight again. Never took another opponent lightly again.
"When I lost to Serra and got humiliated, it took a huge hit to my ego, my pride," St-Pierre said.
When St-Pierre returned in 2017 to challenge Bisping for the middleweight title after stepping away from the sport for four years, he did so with his legacy in mind. He had nothing left to prove, and very few avenues to pursue improvement to his resume. The challenge of moving up 15 pounds in weight, and doing so after four years away from competitive fighting, was enough to motivate St-Pierre. He won the title from Bisping that night via third-round submission, only to vacate it 33 days later when diagnosed with ulcerative colitis.
Before making his retirement official on Thursday, St-Pierre had been vying for a fight with UFC lightweight champion Khabib Nurmagomedov. When it didn't materialize, St-Pierre decided to bring an official end to his fighting career.
"For a fighter where I am right now in my career, the most exciting thing and also the scariest thing to do is to fight an opponent who seems invincible," he said. "And that's where Khabib is now. It would lift my legacy."
St-Pierre said he was more interested in one fight at a time rather than committing to a number of fights. He said the UFC had other plans.
"I don't have the same motivation I used to have," St-Pierre said. "I'm not the same person I used to be ... I don't have the hunger anymore."
He doesn't need any of that. He walks away from MMA on his own terms. Most fighters don't get that opportunity, let alone hold news conferences in a 21,000-seat arena that are well-attended by the media and well-chronicled via those watching on live streams. No, most fighters retire via a tweet that you may or not see, or via an interview you may or may not watch, or they simply aren't signed to new contracts and decide to move on.
St-Pierre never was "most fighters," though. Not in the cage, not out of it. He wore suits to media events well before anyone else. He actively spoke out for drug testing in his sport. He conducted himself with class and professionalism and brought a level of credibility to the sport, helping carry it from fringe to mainstream. Before Jon Jones had his Nike deal, and before UFC signed on with Reebok, there were commercials, magazine ads and in-store cardboard cutouts of St-Pierre in UnderArmour.
"Georges has cemented his legacy as one of the pound-for-pound greatest fighters ever," UFC President Dana White said in a statement. "He beat all the top guys during his welterweight title reign and even went up a weight class to win the middleweight championship. He spent years as one of the biggest names in MMA and remains one of the best ambassadors for the sport. He put Canada on the MMA map."
Mark La Monica is the deputy sports editor for cross-media at Newsday. He also covers mixed martial arts.
UFC Hall of FameSee which MMA fighters have received the ultimate honor in sports, being recognized as the best of the best. UFC champions in multiple weight divisionsThough many have had the opportunity, only five fighters in UFC history have won UFC ... UFC champions history: WelterweightOriginally known as lightweight, the UFC's welterweight class is one of the oldest and most ... UFC champions history: LightweightsThe UFC's lightweight division began in 2001 and has had its ups and downs early ...
WATCH NOW Highlights: Knicks vs. Cavaliers
Mets' Marcus Stroman chides Houston Astros for stealing signs
WATCH NOW HIghlights: Nets vs. 76ers
Will Jeter be unanimous pick in Hall of Fame voting?
Islanders get another chance to beat Rangers
Quinn: Rangers need to forget loss, focus on Isles


I want a popup to show every time it gets trigger, I don't any setting to block it after x triggers, so I don't know why it only is showing once. can anyone help me with this?
I checked your site and PopUp from PopUp Pro plugin shows when I try to leave window - it works on different pages during one session. Is it not opening for you? What browser do you use?
I'm using Chrome, Thanks I'll keep checking it.


text:
Match pack: Hull (H)
Get all the news ahead of Nottingham Forest's Sky Bet Championship match against Hull City at The City Ground on Saturday (kick-off 3pm).
Nottingham Forest take on Hull City tomorrow knowing that victory will see them move level on points with Steve Bruce's side in the Sky Bet Championship table ahead heading into the October international break.
Dougie Freedman's men have been in good recent form with two wins and a draw from their last four matches but will be presented with a tough test in the form of a Hull side packed with high-quality players.
With both teams enjoying solid starts to the season, tomorrow's game is set to be an entertaining affair as Forest aim to claim three points against Hull at The City Ground for the first time in 38 years.
Nottingham Forest defender Matt Mills is set to return to the matchday squad after missing the trip to Huddersfield through injury but central-midfield duo Ben Osborn and Robert Tesche remain sidelined.
Jonny Williams played 90 minutes for Forest's Under 21s on Monday and could be included in the squad to face Hull but Chris Cohen, Andy Reid, Matty Fryatt and Britt Assombalonga are still long-term absentees.
Hull midfielder Jake Livermore is in contention to make his first start of the season tomorrow while Shaun Maloney and David Meyler are also pushing for a starting berth after strong recent performances.
Last time out
Nottingham Forest earned a 1-1 draw at Huddersfield Town in their last outing as Ryan Mendes's first-half opener was cancelled out late on by a deflected striker from Emyr Huws.
Forest: De Vries, Lichaj, Wilson, Hobbs, Pinillos, Mancienne, Vaughan, Ward (Burke 76'), Lansbury (c), Mendes (Walker 80'), O'Grady
Hull City suffered last-gasp disappointment on Saturday as Jordan Rhodes equalised in the final minute for Blackburn Rovers at the KC Stadium after Abel Hernandez had opened the scoring with 17 minutes to play.
Hull: McGregor, Odubajo, Dawson, Davies, Robertson (Livermore 65'), Elmohamady, Huddlestone, Diame (Maloney 61'), Clucas, Akpom, Hernandez (Meyler 88')
Tickets for Nottingham Forest's Sky Bet Championship game against Hull City are still available.
For details and a full list of prices, please click here.
Hospitality for Hull City
Nottingham Forest supporters can see the game against Hull City in supreme comfort with a matchday hospitality package, starting from just £62 + VAT pp.
To book one of the hospitality packages below, call the commercial team on 0115 982 4450 (Mon - Fri, 9am-5pm).
Castle Club
- Padded seat in the Trent End viewing gallery
- Access to the Pitch Sports Diner
- Pre-match interview with Larry Lloyd
Adults - £62 + VAT pp
Under 16s - £31 + VAT pp
- Padded seat by the half-way line
- Access to private members bar
£82 + VAT pp
What else is happening?
Prostate Cancer UK will be holding a bucket collection before tomorrow's game against Hull City and supporters are asked to give generously to such a worthy cause.
The charity have selected tomorrow's game as their awareness match and the club will be launching a new half-time game at The City Ground, sponsored by Prostate Cancer UK, to mark this event.
Live audio commentary will be available on Forest Player HD, as well as live text commentary on www.nottinghamforest.co.uk and via @Official_NFFC on Twitter.
Forest Player HD will also have extended match highlights from 12pm on Sunday, as well as all the reaction from the Forest camp.
The official matchday programme will be available to buy and download for Saturday's meeting with Hull.
The 76-page edition, priced at £3, includes columns from chairman and owner Fawaz Al Hasawi, manager Dougie Freedman and cover to make the passing of the club's former assistant manager Peter Taylor.
The life of Peter is remembered through a feature interview with his daughter Wendy and son Phil, a special feature with his former captain John McGovern at the thoughts of many of the players who played under him.
Forest fans are reminded that they can download the official matchday magazine to their iPad, iPhone or Android device. Perfect for supporters unable to attend the game, the download can be accessed via this app now for Apple users or via this link for Android users.
Fancy a flutter?
Nottingham Forest are the 17/10 slight outsiders to taste victory against Hull City at The City Ground tomorrow with the draw priced at 11/5 and a win for the visitors priced as the current favourite at 13/8.
Supporters can back The Reds to be winning at half-time and at full-time at 100/30 or there are odds of 33/1 on Forest coming from behind at half-time to win the game against the recently-relegated Tigers.
Chris O'Grady has already put in some superb performances since joining on loan from Brighton and the striker can be backed at 6/1 to be the first goalscorer of the match or at 9/4 to open his account in the 90 minutes.
For all the latest odds on Saturday's game, please visit www.skybet.com/forest and make a prediction for Forest v Hull.
All games to be played at 3pm on Saturday unless stated
Rotherham v Burnley (Friday, kick-off 7.45pm)
Wolves v Huddersfield (Saturday, kick-off 12.30pm)
Blackburn v Ipswich
Brighton v Cardiff
Bristol City v MK Dons
Derby v Brentford
Leeds v Birmingham
QPR v Bolton
Reading v Middlesbrough
Sheffield Wednesday v Preston
Charlton v Fulham (Sunday, kick-off 12pm)


Hairstyles For Thin Hair Over 60 – Every people worries concerning hair care, is not it? Stylists always urge for a easy hairstyle if your program doesn't allow you to maintain sophisticated hairstyles. Hairstyles For Thin Hair Over 60 offer you a good deal of designing options. As a consequence, you have the opportunity of showcasing your own hair. These hairstyles match women of any border. Simply speaking, there is something for everyone on this list. Take a rough idea about what you need before going through any list.
Hairstyles For Thin Hair Over 60 can be attained from your comfort zone, when you have the skills. Consequently, it is also easy to maintain when compared to many different hairstyles within this listing. Offer your whole hair a blond look throughout your mind. Divide both the mid head and fringe hair into two sections through a line. Each section needs to conjoin to your sideways hair, depending on its position. Form both a layered along with a curly look across the head prior to combing your sideways hair and on the back towards their various directions.
Side parted voluminous of Hairstyles For Thin Hair Over 60 is attributed by some uniqueness. This cut conjoins a huge percentage of your fringe and mid head hair along with your right sideways hair. Hair over the left side of the mind should rest behind your left ear. Hair on the right and on the back ought to be taken straight. Side parted volume curls hairstyle largely suits casual events.
Short Hairstyles Fine Hair Over 60 Short Layered Hairstyles for Over 60 Short Haircut for Thick.
Short Hairstyles for Thin Hair Over 50 with Glasses with Short Layered Hairstyles for Over 60.
Bob Hairstyles and Colours Inspirational Bobs Hairstyle New Bob Hairstyles Gorgeous I Pinimg 1200x 0d 60.
hairstyles for short medium hair luxury i pinimg 1200x 0d 60 8a.
Older Women Hairstyles Easy Very Short Hairstyles For Women Over 60 With Round Faces And Thin.
Hairstyles for Thin Hair Over 60 Short Medium Hairstyles 2015 Graph 50 New Hairstyles for Fine.
Short Hairstyles For Fine Hair Over 60.
Side swept Hairstyles For Thin Hair Over 60 is credited by a mix of black and blonde look throughout the mind. A section of your head and fringe hair ought to conjoin with hair on both sides of the mind. Hair on the right side should break on your ear, whereas on the back and left side of their mind ought to be combed straight. Pick how big curls to form on both sides of the head. Hairstyles For Thin Hair Over 60 is a causal hairstyle. Consider this hairstyle since it blends with almost any outfit you're wearing.
You are free to pick any dull color when designing this simple curls hairstyle. Here, curls should be visible on both sides and on the rear of the mind. That is not all, form a layered look throughout the mind. The layered appearance on the head helps distinguish poker confront fringe hairstyle from other comparable hairstyles. Rest your pubic hair on your forehead and ought to conjoin with your hair. Similarly, your mid head hair ought to conjoin with hair on both sides of the head. Offer your hair look up to its borders.
Brushed Hairstyles For Thin Hair Over 60 is among the simplest and simple to achieve hairstyles within this list. It is attributed with a blonde appearance on the mind. But a black appearance ought to be visible at some point on the middle mind. Decide which part of the front view hair rests on either the left or right side of the mind. Form a Hairstyles For Thin Hair Over 60 look on the mind along with a part of hair on the back of the head. Hairstyles For Thin Hair Over 60 provides you the opportunity to showcase your face.


iWorldGeography is used by schools, libraries, and families in over 60 countries!!!!
"Your iWorld apps were the first we found that we could endorse whole-heartedly … Thank you for your quality, real, Montessori apps."
iWorldGeography United States includes a Montessori 3-part-lesson [introduce, recall, reconstruct], plus labeling. Student-centered, self-correcting, independent work blends seamlessly with existing curriculum or stands alone for home use. Free worksheets can be printed for follow-up, physical-world activities at home or in the classroom. Available to all at TenSunInteractive.com - Extras.
• Active Intro - Touch any state to hear its name.
• Passive Intro - Movie introduces each state name.
• Quiz - Name is played. Touch any state to reveal answer.
• Puzzle - Freely drag pieces to reconstruct map.
• Label - Pins drag freely to label states and capitals.
• "Grid" - scatters pieces to screen edges.
• "Toss" - scatters pieces randomly across screen.
• "Lines/No Lines" - toggle to work with or without outline.
• "Pass" - skips current label.
• "Lightbulb" - toggles to show or hide control map.
TenSun puts children in charge of their learning by allowing them to control the challenge level and depth of their lesson. There is no penalty for a wrong answer. The correct answer is simply reinforced. Activities are self-correcting. Children are free to work independently, in a small group, or with the help of a supervising adult at home or in the classroom.
Touch Classroom Printouts button on app's title screen to access printouts for additional color, trace, and label activities. Or, visit TenSunInteractive.com's Extras page.
Follow us on Twitter @tensuninteract for App, Update, and Online Extra alerts.
Follow TenSun Interactive on FaceBook.
iWorldGeography colors match American Montessori Society classroom materials.
iWorldGeography Series blends self-correcting, child-directed-activity design with Geography curriculum used in Montessori Children's House and Lower Elementary classrooms. Concept and visual design is thoughtfully integrated to supplement existing curriculum and classroom materials. The series was inspired by 3-year's observation in Lexington (MA) Montessori School Children's House classrooms where MyWorld, TenSun's desktop-based version of this series, has been used since 2004.
Now runs on iOS 11.


I don't think I need to say more about this project, than the post title. I adore it! Let me show you why.
Look how cozy it is! I am so glad I decided to add a zip instead of a the knitted strap at the top of the garment. I've been wearing this as a jacket in our cold Spring weather and it has worked perfectly when zipped right up. If the jacket was only fixed at the top, like the pattern, I think it wouldn't be warm enough when it is cold enough outside to wear this. The jacket is immensely warm because of the dense and fuzzy fabric. The camel content in the yarn is definitely going to pill, in fact it already is, but, it makes for a wonderful warm fabric, so I can't and won't complain.
In order to have full flexibility with the jacket, I decided to install a two way zip. I love that I can wear it fully, or part, closed. In fact, you can zip it so it looks like it would if it were held together at the top the way it was designed to.
I cannot say how much I love the look of this cable pattern. I particularly like the nice flare on the sleeves. The sleeves are a little long, but I love that because I suffer from cold hands. I often have to wear mitts indoors because of poor circulation in my extremities. I've not had to wear mitts indoors when I've worn this garment.
The detail in this pattern is gorgeous. From the beautifully turned hem, to the cable pattern around the pockets. The pockets are just deep enough to keep your hands warm, but I probably wouldn't use them for anything else other than possibly a handkerchief!
When I blocked the garment, after knitting, I used wires, and I angled the bottom half of the jacket outwards. This has, unfortunately, led to a little flaring of the hem. I think this will disappear after wearing it, as it is already less noticeable. When blocking the jacket, I used my polystyrene head form (that I use for blocking hats) and placed it in the hood. This pulled the fabric at the top of the hood into a nice soft shape that hid the short row shaping beautifully. The hood keeps my upper back and shoulders really warm, when it's not being worn up.
I love this jacket and I have been wearing it as much as possible since I finished it. It is a wonderful cozy cardigan for Winter and it's a great light and warm jacket for Spring and Autumn. I could not be happier with how it turned out.
I'm so glad I was finally well enough to take these photos to share, as I have been wearing it for a while now, but I didn't feel like the project was complete until I showed you guys! I can officially mark this off my list for Knit The Queue 2013 and Challenge To Myself. I've already pulled out my next KTQ2013 project, it's a kit for some stranded mitts designed by Bombella. The next CTM is the Dew Drops Shawl, which is being knit in Artisan Yarns Soft Silk Camel Lace.
Hopefully I'll be able to take some photos of my progress soon.


text:
Penelope Niven on Thornton Wilder, reading
Guest blog post by Penelope Niven, author of the forthcoming biography, Thornton Wilder: A Life (HarperCollins)
I have the instinctive habit-formed impulse, at any unoccupied moment during the day, to reach for a book to read . . . .
—Thornton Wilder, Journal Entry 24, May 21, 1940
"Dear Papa," Thornton Wilder wrote when he was nine. "Books I have read this month." He listed Shakespeare's Othello, John White's translation of Plutarch's Lives, Washington Irving's Sketch Book, Wilbur Fist Crafts's Successful Men of Today, and Sarah Knowles Bolton's Poor Boys Who Became Famous. Thornton concluded with his opinion: "None of the books were unsatisfactory."
Two of the books would have particularly satisfied his father, Dr. Amos Parker Wilder, journalist, orator, diplomat, and temperance advocate. American authors Crafts and Bolton were leaders in the temperance movement and wrote prolifically for children, hoping to instill in them such virtues as obedience, punctuality, hard work, and, of course, abstinence.
Thornton, his brother, Amos, and their three sisters grew up being read to—the Bible, and works by Bunyan, Scott, Dickens, Thackeray, Thoreau, James Fenimore Cooper, and John Greenleaf Whittier. On Sunday afternoons their father read to his precocious children from "edifying" books on religion, philosophy, history, and civics. On other days, their mother, a poet and lover of the arts, read mythology to them, or poetry, fiction, and drama. Thornton eagerly read every genre. His father encouraged him to read biography, not only the didactic children's books about poor boys growing up to be famous, but Boswell's life of Samuel Johnson and the four-volume biography of abolitionist William Lloyd Garrison. When Thornton was twenty-eight and working on The Cabala, his first novel, he wrote, "I never cry for fiction (save when I'm composing it) but I weep myself ill over biography."
During their high school and college years, Dr. Wilder dispatched his sons to do summer work on farms, believing the experience provided a wider understanding of grassroots America. In the summer of 1916, afflicted with mosquito bites, bedbugs, and sunburn, Thornton worked six ten-hour days a week, at fifteen cents per hour, on a farm in Massachusetts. After the day's work, he escaped into books. He was so captivated by J. M. Barrie's plays that he read "to the cows in the stanchions the judge's speech from Barrie's The Legend of Leonora." Thornton's grounding in biblical literature came in handy another summer when he worked as entertainment director in a Connecticut camp for boys. On Sunday nights he told the boys Bible stories, and because many of them had "never heard of Goliath or Esau or Belshazzar," he said, they gave him "credit for a great deal of talent."
From boyhood Thornton was writing plays and stories, fed by his eclectic reading. He noted in his letters and journals the books he read and the writers who absorbed his attention. As an adult, he often read a book once for sheer pleasure, and then re-read it analytically, dissecting it and retrieving the techniques or themes he wanted to try with his own hand. He was schooling himself in the demanding arts of drama and fiction.
He read globally, exploring cultures through their books and working on his Latin, French, German, Italian, and Spanish proficiency so that he could read a text in its original language. He didn't want another person to stand between him and the author. Books could be bridges to friendship with other American authors—Hemingway, Fitzgerald, Glenway Wescott, Alexander Woollcott, Mabel Dodge Luhan, and Edmund Wilson, among others. In 1934, Thornton met Gertrude Stein and embarked on one of the most important literary friendships of his life. Before they met, he had read Stein's work closely enough to write a wicked parody of her style, but from 1935 until Stein's death in 1946, he was the American champion and interpreter of her work.
In his fifties Wilder concentrated on books by nineteenth-century American authors. He was appointed Charles Eliot Norton Professor of Poetry at Harvard for 1950–51, to give a series of lectures he called "The American Characteristics in Classic American Literature." He turned his prodigious intellectual energy to the mission he set for himself: "To interpret the American experience and the American character as revealed in nineteenth century American literature; to define what it meant to be an American in the twentieth century; and to examine the role of America and Americans in the global community."
He filled his journals with research and reflections about nineteenth-century American authors and their significance in American life, enough material for the nonfiction book he dreamed about writing but never finished. He was immersed in the work of Whitman, Poe, Melville, Henry James, Thoreau, and Dickinson. He excluded Emerson. "Isn't he awful?" Wilder wrote to Malcolm Cowley.
Yet how that colossus bestrode the world for so long! His ideas basely, soothingly, flattering all that is facile and evasive in the young republic . . . Melville's copies of the Essays are in the Harvard Library and it's a joy to see how Melville dug his pencil into the page in scornful annotation.
Wilder much preferred Thoreau. "Walden," he wrote, "is a manual of self-reliance so much more profound than Emerson's famous essay that the latter seems to be merely on the level of that advice to melancholics which directs them to take walks and drink a lot of milk." Three of Wilder's lectures were published in the Atlantic Monthly in 1952: "Toward an American Language," "The American Loneliness," and "Emily Dickinson," a biographical portrait articulating a fundamental theme in Wilder's own work—the importance of "loving the particular while living in the universal."
By the time he was seventy-eight his vision was failing, and Wilder could not read as copiously as he had always loved to do. Nevertheless, in the months before his death in December 1975, he "devoured with joy" Lewis Thomas's The Lives of a Cell and James D. Watson's The Double Helix. "What sublime reaches," he told his brother, adding that he was "glad to have lived long enough to peek into these processes."
Back in 1926, when he was a young man, Wilder had reflected that "the whole purport of literature" is "the notation of the heart"—a line he wrote in a letter to his former schoolmate Henry Luce, and then embedded in a passage in The Bridge of San Luis Rey. That maxim held true for Thornton Wilder for a lifetime of writing—and a lifetime of reading.
Read more about Thornton Wilder at the official website of the Thornton Wilder Family
The Library of America has interviewed volume editor J. D. McClatchy twice about Wilder: about the works in The Bridge of San Luis Rey volume and about the works in The Eighth Day volume
Wilder's autobiographical writings about China were the subject of both a previous Reader's Almanac post and a Story of the Week
Related LOA works: Thornton Wilder: Collected Plays and Writings on Theater; The Bridge of San Luis Rey and Other Novels 1926–1948 (includes Wilder's essay, "Gertrude Stein's Four in America"; The Eighth Day, Theophilus North, Autobiographical Writings
Labels: Thornton Wilder
Elmore Leonard to be honored by the National Book ...
September 17, 1862: The Battle of Antietam (Sharps...
Dawn McGuire on Christian Wiman, Ambition and Surv...
An interview with Caroline Fraser: why the Little ...
Lightness v. pungency: Michael Gorra on Henry Jame...


Add a fun and safe roost for wrens or chickadees to your garden using the BestNest Classic Hanging Wren House. The traditional lines and diminutive size make this unit a perfect roosting spot for small songbirds. A 1.25" diameter entry hole on the front panel allows wrens or chickadees to enter, while preventing larger birds from accessing the home. The pitched, wooden roof overhangs the entry, protecting it from rain and other precipitation. Its lapped detailing lends added depth and dimension to the piece, furthered by the flat base. This bottom section of wood removes for seasonal cleanouts. The all-wood construction insulates the piece, while its light color keeps it from overheating. Gaps at the top of both the front and back panel ventilate the home with ease for avian comfort. Place this house on a bracket or hook using its attached, plastic coated cable, so the birds may access it easily. Add unique shape and timeless housing to your bird garden at once with this Classic Hanging Wren House.


Other than cleaning with a damp cloth, vinyl interior shutters are virtually maintenance free. Vinyl�s ability to withstand moisture makes these interior shutters a great choice in your kitchen , bathrooms or kids� rooms, or if you live in a locale subject to high humidity.... Vinyl window shutters are decorative exterior accents and mimic the look of functional wooden shutters for visual interest. Available styles include louver, raised panel and board-and-batten. Shutters may be flat-topped or arched to complement the window.
Vinyl window shutters are decorative exterior accents and mimic the look of functional wooden shutters for visual interest. Available styles include louver, raised panel and board-and-batten. Shutters may be flat-topped or arched to complement the window.... Clean your shutters before painting. Before painting anything, you will want to prepare the surface of your plantation shutters. In most cases, the shutters will need to be cleaned.
Other than cleaning with a damp cloth, vinyl interior shutters are virtually maintenance free. Vinyl�s ability to withstand moisture makes these interior shutters a great choice in your kitchen , bathrooms or kids� rooms, or if you live in a locale subject to high humidity.... Just as with interior shutters, avoid using anything that is excessively abrasive to clean them. Abrasive items could potentially damage the finish or sealant that is on the shutter, allowing them to become weathered much more quickly.
Vinyl window shutters are decorative exterior accents and mimic the look of functional wooden shutters for visual interest. Available styles include louver, raised panel and board-and-batten. Shutters may be flat-topped or arched to complement the window.
Just as with interior shutters, avoid using anything that is excessively abrasive to clean them. Abrasive items could potentially damage the finish or sealant that is on the shutter, allowing them to become weathered much more quickly.
Other than cleaning with a damp cloth, vinyl interior shutters are virtually maintenance free. Vinyl�s ability to withstand moisture makes these interior shutters a great choice in your kitchen , bathrooms or kids� rooms, or if you live in a locale subject to high humidity.