text:
\section{Introduction}
Let $\Bbbk$ be an algebraically closed field. The {\it Cremona group}
$\operatorname{Cr}_{n}(\Bbbk)$ is the group of birational transformations of $\mathbb{P}^n_{\Bbbk}$, or
equivalently the group of $\Bbbk$-automorphisms of the field
$\Bbbk(x_1,\dots,x_n)$.
Finite subgroups of $\operatorname{Cr}_{2}(\mathbb{C})$ are completely classified (see \cite{Dolgachev-Iskovskikh}
and references therein).
In contrast, subgroups of $\operatorname{Cr}_{n}(\Bbbk)$ for $n\ge 3$
are not studied well (cf. \cite{Prokhorov2009b}).

In the present paper we study some kind of abelian subgroups of $\operatorname{Cr}_3(\mathbb{C})$.
Let $p$ be a prime number.
We say that a group $G$ is \textit{$p$-elementary} if
$G\simeq ({\boldsymbol{\mu}}_p)^s$ for some positive integer $s$.
In this case $s$ is called the \textit{rank} of $G$ and denoted by
$\operatorname{rk} G$.

\begin{theorem}[{\cite{Beauville2007}}]
\label{theorem-cr2}
Let $p$ be a prime $\neq \operatorname{char}(\Bbbk)$
and let $G\subset \operatorname{Cr}_2(\Bbbk)$ be a $p$-elementary subgroup.
Then:
\[
\operatorname{rk} G \le 2+\delta_{p,3}+2\delta_{p,2}
\]
where $\delta_{i,j}$ is Kronecker's delta.
Moreover, for any such $p$
this bound is attained for some $G$. These ``maximal'' groups $G$ are classified up
to conjugacy in $\operatorname{Cr}_2(\Bbbk)$.
\end{theorem}

More generally, instead of $\operatorname{Cr}_n(\Bbbk)$  we also can consider
the group $\operatorname{Bir}(X)$ of birational automorphisms
of an arbitrary rationally connected variety $X$.
Our main result is the following

\begin{theorem}
\label{theorem-cr3}
Let $X$ be a rationally connected threefold defined
over a field of characteristic $0$, let $p$ be a prime,
and let $G\subset \operatorname{Bir} (X)$ be a $p$-elementary subgroup.
Then
\begin{equation}
\label{eq-main}
\operatorname{rk} G\le
\begin{cases}
7&\text{if $p=2$,}
\\
5&\text{if $p=3$,}
\\
4&\text{if $p=5,\, 7,\,11$, or $13$,}
\\
3&\text{if $p\ge 17$.}
\end{cases}
\end{equation}
For any prime $p\ge 17$ this bound is attained for some subgroup $G\subset \operatorname{Cr}_3(\mathbb{C})$.
\textup(However we do not assert that the bound \eqref{eq-main}
is sharp for $p\le 13$\textup).
\end{theorem}

\begin{remark}
(i)
Note that $\operatorname{Cr}_1(\Bbbk)\simeq {\operatorname{PGL}}_2(\Bbbk)$.
Hence for any prime $p$ $\neq \operatorname{char}(\Bbbk)$
and any $p$-elementary subgroup $G\subset \operatorname{Cr}_1(\Bbbk)$,
we have $\operatorname{rk} G\le 1+ \delta_{p,2}$
(see, e.g., \cite[Lemma 2.1]{Beauville2007}).

(ii)
Since $\operatorname{Cr}_1(\Bbbk)\times \operatorname{Cr}_2(\Bbbk)$ admits (a lot of) embeddings to
$\operatorname{Cr}_3(\Bbbk)$, the group $\operatorname{Cr}_3(\Bbbk)$ contains a $p$-elementary
subgroup $G$ of rank $3+\delta_{p,3}+3\delta_{p,2}$.
This shows the last assertion of our theorem.
\end{remark}

The following consequence of
Theorem \ref{theorem-cr3}
was proposed by A. Beauville.
\begin{corollary}
\label{cor-cr}
The group $\operatorname{Cr}_3(\mathbb{C})$ is not isomorphic to $\operatorname{Cr}_n(\mathbb{C})$ for $n\neq 3$ as an abstract group.
\end{corollary}

\begin{proof}
Denote by $\xi(n,p)$ the maximal rank of a $p$-elementary group contained in $\operatorname{Cr}_n(\mathbb{C})$.
Then $\xi(2,17)=2<\xi(3,17)=3$ and $\xi(n,17)\ge n$ by Theorems \ref{theorem-cr2} and \ref{theorem-cr3}.
\end{proof}

Our method is a generalization of the method
used for study of finite subgroups of $\operatorname{Cr}_2(\Bbbk)$
\cite{Beauville2007}, \cite{Dolgachev-Iskovskikh}. Similarly to \cite{Prokhorov2009b}
we use the equivariant three-dimensional minimal model program.
This way easily allows us to reduce the problem to the
study of automorphism groups of some (not necessarily smooth) Fano threefolds.

\textbf{Acknowledgments.}
I would like  to thank J.-P. Serre for asking me the question considered here
and  for useful comments.
I am also grateful to A. Beauville for
proposing me Corollary \ref{cor-cr} and for his interest on my paper.
This paper was written
at the IHES (Bures-sur-Yvette) during my visit in 2009.
I am grateful to IHES for the support and hospitality.
Finally I would like to thank the referee for several suggestions that make
paper more readable and clear.

\section{Preliminaries}
Clearly, we may assume that $\Bbbk=\mathbb{C}$.
All the groups in this paper are multiplicative.
In particular, we denote a cyclic group of order $n$ by ${\boldsymbol{\mu}}_n$.

\begin{de}\label{Terminal-singularities}
\textbf{Terminal singularities.}
We need a few facts on the classification of
three-dimensional
terminal singularities (see  \cite{Mori-1985-cla}, \cite{Reid-YPG1987}).
Let $(X\ni P)$ be a germ of a three-dimensional terminal singularity.
Then $(X\ni P)$ is isolated, i.e., $\operatorname{Sing}(X)=\{P\}$.
The \textit{index} of $(X\ni P)$ is the minimal positive integer
$r$ such that $rK_X$ is Cartier.
If $r=1$, then $(X\ni P)$ is Gorenstein. In this case $(X\ni P)$ is analytically
isomorphic to a hypersurface
singularity in $\mathbb{C}^4$ of multiplicity $2$.
Moreover, any Weil $\mathbb{Q}$-Cartier divisor $D$ on $(X\ni P)$ is Cartier.
If $r>1$, then there is a cyclic, \'etale outside of $P$
cover $\pi : (X^{\sharp}\ni P^{\sharp})\to (X\ni P)$
of degree $r$ such that $(X^{\sharp}\ni P^{\sharp})$ is a Gorenstein
terminal singularity (or a smooth point). This $\pi$ is called
the \textit{index-one cover} of $(X\ni P)$.
\end{de}
\begin{theorem}[\cite{Mori-1985-cla}, \cite{Reid-YPG1987}]
\label{terminal-lemma}
In the above notation $(X^{\sharp}\ni P^{\sharp})$ is
analytically ${\boldsymbol{\mu}}_r$-isomorphic to a hypersurface
in $\mathbb{C}^4$ with ${\boldsymbol{\mu}}_r$-semi-invariant
\footnote{In invariant theory people often say ``relative invariant'' rather then
``semi-invariant''. We prefer to use the terminology of \cite{Mori-1985-cla}.} coordinates
$x_1,\dots,x_4$,
and the action is given by
\[
 (x_1,\dots,x_4) \longmapsto (\varepsilon^{a_1} x_1,\dots,\varepsilon^{a_4} x_4)
\]
for some primitive $r$-th root of unity $\varepsilon$, where
one of the following holds:
\begin{enumerate}
 \item
$(a_1,\dots,a_4)\equiv (1,-1,a_2,0)\mod r$, \quad
$\gcd(a_2,r)=1$,
 \item
$r=4$ and $(a_1,\dots,a_4)\equiv (1,-1,1,2)\mod 4$.
\end{enumerate}
\end{theorem}

\begin{definition}
A \emph{$G$-variety} is a variety $X$ provided with
a biregular faithful action of a finite group $G$.
We say that a normal $G$-variety
$X$ is \emph{$G\mathbb{Q}$-factorial} if any $G$-invariant Weil
divisor on $X$ is $\mathbb{Q}$-Cartier.
A projective normal $G$-variety $X$ is called \emph{$G\mathbb{Q}$-Fano}
if it is $G\mathbb{Q}$-factorial, has at worst terminal
singularities, $-K_X$ is ample, and $\operatorname{rk} \operatorname{Pic} (X)^G=1$.
\end{definition}

\begin{lemma}
\label{lemma-fixed-point}
Let $(X\ni P)$ be a germ of a threefold terminal singularity
and let $G\subset \operatorname{Aut}(X\ni P)$ be a $p$-elementary subgroup.
Then $\operatorname{rk} G\le 3+\delta_{2,p}$.
\end{lemma}
\begin{proof}
Assume that $\operatorname{rk} G\ge   4+\delta_{2,p}$.
First we consider the case where $(X\ni P)$ is Gorenstein.
The group $G$ acts faithfully on
the Zariski tangent space $T_{P,X}$, so
$G\subset {\operatorname{GL}} (T_{P,X})$, where $\dim T_{P,X}=3$ or $4$.
If $\dim T_{P,X}=3$, then
$G$ is contained in a maximal torus of ${\operatorname{GL}}_3(\mathbb{C})$, so
$\operatorname{rk} G\le 3$ and we are done.
Thus we may assume that $\dim T_{P,X}=4$.
Take semi-invariant coordinates
$x_1,\dots, x_4$ in $T_{P,X}$. There is a $G$-equivariant
analytic embedding $(X\ni P)\subset \mathbb{C}_{x_1,\dots, x_4}^4$.
As above, $\operatorname{rk} G\le 4$. Thus we may assume that $\operatorname{rk} G\le 4$ and $p>2$.
Let $\phi(x_1,\dots, x_4)=0$ be an equation of $X$, where $\phi$ is a $G$-semi-invariant function.
Regard $\phi$ as a power series and write $\phi=\sum_d \phi_d$,
where $\phi_d$ is the sum of all monomials of degree $d$.
Since the action of $G$ on $x_1,\dots, x_4$ is linear, all the $\phi_d$'s
are semi-invariants of the same $G$-weight $w=\operatorname{wt} \phi_d$.
Hence, for any $\phi_d$, $\phi_{d'}\neq 0$ we have $d-d'\equiv 0 \mod p$.
Since $(X\ni P)$ is a terminal singularity, $\phi_2\neq 0$ and so $\phi_3=0$.
Recall that $G\simeq ({\boldsymbol{\mu}}_p)^4$, $p\ge 3$.
In this case, $\phi_2$ must be a monomial.
Thus up to permutations of coordinates and scalar multiplication
we get either $\phi_2=x_1^2$ or $\phi_2=x_1x_2$.
In particular, we have $\operatorname{rk} \phi_2\le 2$ and $\phi_3=0$.
This contradicts
the classification of terminal singularities \cite{Mori-1985-cla}, \cite{Reid-YPG1987}.

Now assume that $(X\ni P)$ is non-Gorenstein of index $r>1$.
Consider the index-one cover $\pi\colon (X^\sharp\ni P^\sharp)\to
(X\ni P)$ (see \ref{Terminal-singularities}).
Here $(X^\sharp\ni P^\sharp)$ is a Gorenstein terminal
point and the map $X^\sharp\setminus \{P^\sharp\}\to
X\setminus \{P\}$ can be regarded as the topological universal cover.
Hence there exists a natural lifting
$G^\sharp\subset \operatorname{Aut} (X^\sharp\ni P^\sharp)$ fitting to the following exact sequence
\begin{equation}
\label{eq-seq-1}
1\longrightarrow {\boldsymbol{\mu}}_r \longrightarrow G^\sharp \longrightarrow G \longrightarrow 1.
\end{equation}
It is sufficient to show that there exists a
subgroup $G^{\bullet}\subset G^\sharp$ isomorphic to $G$
(but we do not assert that the sequence splits).
Indeed, in this case $G^{\bullet}\simeq G$ acts faithfully on the terminal Gorenstein
singularity $(X^\sharp\ni P^\sharp)$ and we can apply the above
considered case.
We may assume that $G^\sharp$ is not abelian
(otherwise a subgroup $G^{\bullet}\simeq G$ obviously exists).
The group $G^\sharp$ permutes
eigenspaces of ${\boldsymbol{\mu}}_{r}$.
By Theorem \ref{terminal-lemma} the subspace $T:=\{x_4=0\}\subset \mathbb{C}^4_{x_1,\dots,x_4}$
is $G^\sharp$-invariant and ${\boldsymbol{\mu}}_{r}$ acts on any eigenspace $T_1\subset T$ faithfully.
On the other hand, by \eqref{eq-seq-1} we see that the derived subgroup
$[G^\sharp, G^\sharp]$ is contained in ${\boldsymbol{\mu}}_{r}$.
In particular, $[G^\sharp, G^\sharp]$
is abelian and also acts on any eigenspace $T_1\subset  T$ faithfully.
Since $\dim T=3$, this implies that the representation of
$G^\sharp$ on $T$ is irreducible
(otherwise $T$ has a one-dimensional subrepresentation, say $T_1$,
and the kernel of the map $G\to {\operatorname{GL}}(T_1)\simeq \mathbb{C}^*$
must contain $[G^\sharp, G^\sharp]$).
Hence eigenspaces of ${\boldsymbol{\mu}}_{r}$ have the same dimension and so
${\boldsymbol{\mu}}_{r}$ acts on $T$ by scalar multiplication.
By Theorem \ref{terminal-lemma} this is possible only if $r=2$.

Let $G^\sharp_p\subset G^\sharp$ be a Sylow $p$-subgroup.
If ${\boldsymbol{\mu}}_r\cap G^\sharp_p=\{1\}$, then $G^\sharp_p\simeq G$ and we are done.
Thus we assume that ${\boldsymbol{\mu}}_r\subset G^\sharp_p$, so $p=r=2$ and $G^\sharp_p=G^\sharp$.
But then $G^\sharp$ is a $2$-group, so the dimension of its irreducible
representation must be a power of $2$. Hence
$T$ is reducible, a contradiction.
\end{proof}

\begin{lemma}
\label{lemma-fixed-points}
Let $X$ be a $G$-threefold with isolated singularities.
\begin{enumerate}
 \item
If there is a curve $C\subset X$ of $G$-fixed points,
then $\operatorname{rk} G\le 2$.
 \item
If there is a surface $S\subset X$ of $G$-fixed points,
then $\operatorname{rk} G\le 1$. If moreover $S$ is singular along a curve,
then $G=\{1\}$.
\end{enumerate}
\end{lemma}
\begin{proof}[Sketch of the Proof]
Consider the action of $G$ on
the tangent space to $X$ at a general point of $C$ (resp. $S$).
\end{proof}

\textbf{$G$-equivariant minimal model program.}
Let $X$ be a rationally connected three-dimensional algebraic variety
and let $G\subset \operatorname{Bir}(X)$ be a finite subgroup.
By shrinking $X$ we may assume that $G$ acts on $X$ biregularly.
The quotient $Y=X/G$ is quasiprojective, so
there exists a projective completion $\hat Y\supset Y$.
Let $\hat X$ be the normalization of $\hat Y$ in the function field
$\mathbb{C}(X)$. Then $\hat X$ is a projective variety birational to $X$
admitting a biregular action of $G$.
There is an equivariant resolution of singularities $\tilde X\to \hat X$,
see \cite{Abramovich-Wang}.
Run the $G$-equivariant minimal model program: $\tilde X\to \bar X$,
see \cite[0.3.14]{Mori-1988}.
Running this program we stay in the category of projective normal varieties
with at worst terminal $G\mathbb{Q}$-factorial singularities.
Since $X$ is rationally connected, on the final step we get
a Fano-Mori fibration $f: \bar X\to Z$. Here $\dim Z<\dim X$, $Z$ is normal,
$f$ has connected fibers, the anticanonical Weil
divisor $-K_{\bar X}$ is ample over $Z$, and the relative $G$-invariant Picard
number $\rho(\bar X)^G$ is one.
Obviously, we have the following possibilities:
\begin{enumerate}
\item
$Z$ is a rational surface and a general fiber $F=f^{-1}(y)$ is a conic;
\item
$Z\simeq \mathbb{P}^1$ and a general fiber $F=f^{-1}(y)$ is a smooth del Pezzo surface;
\item
$Z$ is a point and $\bar X$ is a $G\mathbb{Q}$-Fano threefold.
\end{enumerate}

\begin{proposition}
In the above notation assume that $Z$ is not a point.
Then $\operatorname{rk} G\le 3+\delta_{p,3}+3\delta_{p,2}$.
In particular, \eqref{eq-main} holds.
\end{proposition}

\begin{proof}
Let $G_0\subset G$ be the kernel of the homomorphism
$G\to \operatorname{Aut}(Z)$. The group $G_1:=G/G_0$ acts effectively on $Z$
and $G_0$ acts effectively on a
general fiber $F\subset X$ of $f$. Hence, $G_1\subset \operatorname{Aut}(Z)$ and $G_0\subset \operatorname{Aut}(F)$.
Clearly, $G_0$ and $G_1$ are $p$-elementary groups with
$\operatorname{rk} G_0 +\operatorname{rk} G_1=\operatorname{rk} G$.
Assume that $Z\simeq \mathbb{P}^1$. Then $\operatorname{rk} G_1\le 1+\delta_{p,2}$.
By Theorem \ref{theorem-cr2} we obtain
$\operatorname{rk} G_0\le 2+\delta_{p,3}+2\delta_{p,2}$.
This proves
our assertion in the case $Z\simeq \mathbb{P}^1$.
The case $\dim Z=2$ is treated similarly.
\end{proof}

\begin{de}
\label{Main-assumption}
\textbf{Main assumption.}
Thus from now on we assume that we are in the case (iii).
Replacing $X$ with $\bar X$ we may assume that our original
$X$ is a $G\mathbb{Q}$-Fano threefold.

The group $G$ acts naturally on $H^0(X,-K_X)$.
If $H^0(X,-K_X)\neq 0$, then there exists
a $G$-semi-invariant section $s\in H^0(X,-K_X)$
(because $G$ is an abelian group).
This section gives us an invariant member
$S\in |{-}K_X|$.
\end{de}

\begin{lemma}
\label{lemma-hyp-sect-0}
Let $X$ be a $G\mathbb{Q}$-Fano threefold, where $G$ is a $p$-elementary
group with $\operatorname{rk} G\ge \delta_{p,2}+4$.
Let $S$ be an invariant Weil divisor
such that $-(K_X+S)$ is nef.
Then the pair $(X,S)$ is log canonical \textup(LC\textup).
\end{lemma}
\begin{proof}
Assume that the pair $(X,S)$ is not LC.
Since $S$ is $G$-invariant and $\rho(X)^G=1$, we see that $S$ is
numerically proportional to $K_X$. Since  $-(K_X+S)$ is nef, $S$ is ample.
We apply quite standard
connectedness arguments of Shokurov \cite{Shokurov-1992-e-ba}
(see, e.g., \cite[Prop. 2.6]{Mori-Prokhorov-2008d}):
for a suitable $G$-invariant boundary $D$, the pair
$(X,D)$ is LC, the divisor $-(K_X+D)$ is ample,
and the minimal locus $V$ of log canonical singularities is also $G$-invariant.
Moreover, $V$ is either a point or a smooth rational curve.
By Lemma \ref{lemma-fixed-point} we may assume that $G$ has no fixed points.
Hence, $V\simeq \mathbb{P}^1$ and we have a map
$\varsigma: G\to \operatorname{Aut}(\mathbb{P}^1)$. If $p>2$, then $\varsigma(G)$ is a cyclic group,
so $G$ has a fixed point, a contradiction.
Let $p=2$ and let $G_0=\ker \varsigma$.
By Lemma \ref{lemma-fixed-points} $\operatorname{rk} G_0\le 2$.
Therefore $\operatorname{rk} \varsigma(G_0)\ge 3$.
Again we get a contradiction.
\end{proof}

\begin{lemma}
\label{lemma-hyp-sect}
Let $X$ be a $G\mathbb{Q}$-Fano threefold, where $G$ is a $p$-elementary
group with
\begin{equation}
\label{main-assumption-3}
\operatorname{rk} G\ge
\begin{cases}
7&\quad \text{if $p=2$,}
\\
5&\quad \text{if $p=3$,}
\\
4&\quad \text{if $p\ge 5$.}
\\
\end{cases}
\end{equation}
Let $S\in |{-}K_X|$ be a $G$-invariant member.
Then we have
\begin{enumerate}
\item
Any component $S_i\subset S$ is either rational or
birationally ruled over an elliptic curve.
\item
The group $G$  acts transitively on the components of $S$.
\item
For the stabilizer $G_{S_i}$ we have $\operatorname{rk} G_{S_i}\le \delta_{p,2}+4$.
\item
The surface $S$ is reducible \textup(and reduced\textup).
\end{enumerate}
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-hyp-sect-0} the pair $(X,S)$ is LC.
Assume that $S$ is normal (and irreducible).
By the adjunction formula $K_S\sim 0$.
We claim that $S$ has at worst Du Val singularities. Indeed, otherwise
by the Connectedness Principle \cite[Th. 6.9]{Shokurov-1992-e-ba}
$S$ has at most two non-Du Val points.
If $p>2$, these points must be $G$-fixed. This  contradicts Lemma \ref{lemma-fixed-point}.
Otherwise $p=2$ and these points are fixed for an index two subgroup
$G^{\bullet}\subset G$. Again we get a contradiction by Lemma \ref{lemma-fixed-point}.
Thus we may assume that $S$ has at worst Du Val singularities.
Let $\Gamma$ be the image of $G$ in $\operatorname{Aut}(S)$.
By Lemma \ref{lemma-fixed-points} $\operatorname{rk} G\le \operatorname{rk} \Gamma+1$.
Let $\tilde S\to S$ be the minimal resolution.
Here $\tilde S$ is a smooth K3 surface.
The natural representation of $\Gamma$ on $H^{2,0}(\tilde S)$
induces the exact sequence (see \cite{Nikulin-1980-aut})
\[
1 \longrightarrow \Gamma_0 \longrightarrow \Gamma \longrightarrow \Gamma_1 \longrightarrow 1,
\]
where $\Gamma_0$ (resp. $\Gamma_1$) is the kernel (resp. image)
of the representation of $\Gamma$ on $H^{2,0}(\tilde S)$.
The group $\Gamma_1$ is cyclic. Hence either $\Gamma_1=\{1\}$ or
$\Gamma_1\simeq {\boldsymbol{\mu}}_p$. In the second case by \cite[Cor. 3.2]{Nikulin-1980-aut}
$p\le 19$. Further, according to \cite[Th. 4.5]{Nikulin-1980-aut} we have
\[
\operatorname{rk} \Gamma_0\le
\begin{cases}
4 & \text{if $p=2$}
\\
2 & \text{if $p=3$}
\\
1 & \text{if $p=5$ or $7$}
\\
0& \text{if $p>7$.}
\end{cases}
\]
Combining this we obtain a contradiction with
\eqref{main-assumption-3}.

Now assume that $S$ is not normal.
Let $S_i\subset S$ be an irreducible component (the case $S_i=S$
is not excluded).
Let $\nu\colon S'\to S_i$
be the normalization.
Write $0\sim \nu^*(K_X+S_i)=K_{S'}+D'$,
where $D'$ is the different, see \cite[\S 3]{Shokurov-1992-e-ba}.
Here $D'$ is an effective reduced divisor
and the pair is LC \cite[3.2]{Shokurov-1992-e-ba}.
Since $S$ is not normal, $D'\neq 0$.
Consider the minimal resolution
$\mu\colon \tilde S\to S'$ and let $\tilde D$ be the crepant
pull-back of $D'$, that is,
$\mu_*\tilde D=D'$ and
\[
K_{\tilde S}+\tilde D=\mu^*(K_{S'}+D')\sim 0.
\]
Here $\tilde D$ is again an effective reduced divisor.
Hence $\tilde S$ is a ruled surface. If it is not rational,
consider the Albanese map
$\alpha: \tilde S\to C$. Clearly $\alpha$ is
$\Gamma$-equivariant and the action of $\Gamma$ on $C$ is not trivial.
Let $\tilde D_1\subset \tilde D$ be an $\alpha$-horizontal
component. By Adjunction $\tilde D_1$ is an elliptic curve.
So is $C$. This proves (i).

If the action on components $S_i\subset S$
is not transitive,
we have an invariant divisor $S'<S$.
Since $X$ is $G\mathbb{Q}$-factorial and $\rho(X)^G=1$,
we can take $S'$ so that $-(K_X+2S')$ is nef.
This contradicts Lemma \ref{lemma-hyp-sect-0}.
So, (ii) is proved.

Now we prove (iii).
Let $\Gamma$ be the image of $G_{S_i}$ in $\operatorname{Aut}(S_{i})$.
By Lemma \ref{lemma-fixed-points} $\operatorname{rk} G_{S_i}\le \operatorname{rk} \Gamma+1$.
If $S_i$ is rational, then we get the assertion by Theorem \ref{theorem-cr2}.
Assume that $S_i$ is birationally ruled surface over an elliptic curve.
As above, let $\tilde S_i\to S_i$ be the composition of the normalization and
the minimal resolution, and let $\alpha: \tilde S_i\to C$ be the Albanese map.
Then $\Gamma$ acts faithfully on $\tilde S_i$ and $\alpha$ is $\Gamma$-equivariant.
Thus we have a homomorphism $\alpha_* : \Gamma \to \operatorname{Aut} (C)$.
Here $\operatorname{rk} \Gamma \le\operatorname{rk} \alpha_*(\Gamma)+1+\delta_{p,2}$.
Note that $\alpha_*(\Gamma)$ is a $p$-elementary subgroup
of the automorphism group of an elliptic curve.
Hence, $\operatorname{rk} \alpha_*(\Gamma)\le 2$. This implies (iii).

It remains to prove (iv).
Assume that $S$ is irreducible. By (iii)
the surface $S$ is not rational.
So, $S$ is birational to a ruled surface over an elliptic curve.
By Lemma \ref{lemma-fixed-points} the group $G$ acts
on $S$  faithfully.
Hence, in the above notation,
$\operatorname{rk} G=\operatorname{rk} \Gamma \le \operatorname{rk} \alpha_*(\Gamma)+1+\delta_{p,2}\le 3+\delta_{p,2}$,
a contradiction.
\end{proof}

\section{Proof of Theorem \ref{theorem-cr3}}

\begin{pusto}
\label{notation-3}
In this section we prove Theorem \ref{theorem-cr3}.
As in \xref{Main-assumption} we assume that
$X$ is a $G\mathbb{Q}$-Fano threefold, where $G$ be a $p$-elementary subgroup
of $\operatorname{Aut}(X)$.
\end{pusto}

First we consider the case where $X$ non-Gorenstein, i.e., it has at least one
point of index $>1$.
\begin{proposition}
\label{proposition-non-Gorenctein}
Let $G$ be a $p$-elementary group and
let $X$ be a non-Gorenstein $G\mathbb{Q}$-Fano threefold. Then
\[
\operatorname{rk} G\le
\begin{cases}
7& \text{if $p=2$,}
\\
5& \text{if $p=3$,}
\\
4& \text{if $p=5$, $7$, $11$, $13$,}
\\
3 & \text{if $p\ge 17$.}
\end{cases}
\]
\end{proposition}
\begin{proof}
Let $P_1$ be a point of index $r>1$ and let $P_1,\dots,P_l$
be its $G$-orbit.
Here $l=p^{t}$ for some $t$ with $t\ge s-\delta_{2,p}-3$, where $s=\operatorname{rk} G$ (see Lemma \ref{lemma-fixed-point}).
By the orbifold Riemann-Roch formula \cite{Reid-YPG1987} and
a form of Bogomolov-Miyaoka
inequality \cite{Kawamata-1992bF}, \cite{KMMT-2000} we have
\begin{equation*}
\label{eq-Kawamata-Bogomolov}
\sum \left( r_{P_i}-\frac1{r_{P_i}} \right)<24.
\end{equation*}
Since $r_i-1/r_i\ge 3/2$, we have $3l/2<24$ and so
\[
p^{s-\delta_{2,p}-3}\le l<16.
\]
This gives us the desired inequality.
\end{proof}

From now on we assume that our $G\mathbb{Q}$-Fano threefold $X$ is
Gorenstein, i.e., $K_X$ is a Cartier divisor.
Recall (see, e.g., \cite{Iskovskikh-Prokhorov-1999})
that the Picard group of a Fano variety $X$ with at worst
(log) terminal singularities is a torsion free finitely generated
abelian group ($\simeq H^2(X,\mathbb{Z})$). Then we can define the \textit{Fano index} of
$X$ as the maximal positive integer that divides $-K_X$ in $\operatorname{Pic}(X)$.

\begin{proposition-definition}[see, e.g., \cite{Iskovskikh-Prokhorov-1999}]
Let $X$ be a Fano threefold with at worst terminal Gorenstein
singularities.
The positive integer $-K_X^3$ is called the \emph{degree} of $X$.
We can write $-K_X^3=2g-2$, where
$g$ is an integer $\ge 2$ called the \emph{genus} of $X$.
Then $\dim |{-}K_X|=g+1\ge 3$.
\end{proposition-definition}

\begin{corollary-notation}
\label{corollary-notation}
In notation \xref{notation-3} the linear system
$|{-}K_X|$ is not empty, so there exists a $G$-invariant
member $S\in |{-}K_X|$. Write $S=\sum _{i=1}^N S_i$,
where $S_i$ are irreducible components.
\end{corollary-notation}

\begin{theorem}[{\cite{Namikawa-1997}}]
\label{theorem-Namikawa}
Let $X$ be a Fano threefold with terminal Gorenstein
singularities. Then $X$ is smoothable, that is,
there is a flat family $X_t$
such that $X_0\simeq X$ and a general member $X_t$ is
a smooth Fano threefold of the same
degree, Fano index and Picard number.
Furthermore, the number of singular points is bounded as follows:
\begin{equation}
\label{eq-number-singular-points}
|\operatorname{Sing}(X)|\le 20-\rho(X_t)+h^{1,2}(X_t).
\end{equation}
where $h^{1,2}(X_t)$ is the Hodge number.
\end{theorem}

Combining the above theorem with the classification of
smooth Fano threefolds
\cite{Iskovskikh-1980-Anticanonical}, \cite{Mori1981-82}
(see also \cite{Iskovskikh-Prokhorov-1999}) we get the following

\begin{theorem}
\label{classification-Fano}
Let $X$ be a Fano threefold with at worst terminal Gorenstein
singularities and let $X_t$ be its smoothing. Let $g$ and $q$ be the genus
and Fano index of $X$, respectively.
\begin{enumerate}
\item
$q\le 4$.
\item
If $q=4$, then $X\simeq \mathbb{P}^3$.
\item
If $q=3$, then $X$ is a quadric in $\mathbb{P}^4$ \textup(with $\dim \operatorname{Sing}(X)\le 0$\textup).
\item
If $q=2$, then $\rho(X)\le 3$
and $-K_X^3=8d$, where $1\le d\le 7$. Moreover $\rho(X)=1$ if and only if
$d\le 5$.
\item
If $q=1$ and $\rho(X)=1$, then there are the following possibilities:
{\rm \begin{center}
\begin{tabular}{|c|cccccccccc|}
\hline
&&&&&&&&&&
\\[-5pt]
$g$&2&3&4&5&6&7&8&9&10&12
\\[5pt]
\hline
&&&&&&&&&&
\\[-5pt]
$h^{1,2}(X_t)$&52&30&20&14&10&7&5&3&2&0
\\[5pt]
\hline
\end{tabular}
\end{center}}
\end{enumerate}
\end{theorem}

\begin{lemma}
\label{lemma-base-point-free}
Let $G$ be a $p$-elementary group and
let $X$ be a Gorenstein $G\mathbb{Q}$-Fano threefold. If
the linear system $|{-}K_X|$ is not base point free, then $\operatorname{rk} G\le 3+\delta_{p,2}$.
\end{lemma}
\begin{proof}
Assume that $\operatorname{Bs} |{-}K_X|\neq \emptyset$.
Clearly, $\operatorname{Bs} |{-}K_X|$ is $G$-invariant.
By
\cite{Iskovskikh-1980-Anticanonical}, \cite{Shin1989}
$\operatorname{Bs} |{-}K_X|$ is either a single point or a smooth rational curve.
In the first case the assertion
immediately follows by Lemma \ref{lemma-fixed-point}.
In the second case $G$ acts on the curve $C=\operatorname{Bs} |{-}K_X|$.
Since $C\simeq \mathbb{P}^1$,
the assertion
follows by Lemma \ref{lemma-fixed-points}.
\end{proof}

\begin{proposition}
\label{proposition-Gorenctein-p}
Let $G$ be a $p$-elementary group, where $p\ge 5$, and
let $X$ be a Gorenstein $G\mathbb{Q}$-Fano threefold. Then
\[
\operatorname{rk} G\le
\begin{cases}
4&\text{if $p=5,\, 7,\,11,\,13$,}
\\
3&\text{if $p\ge 17$.}
\end{cases}
\]
\end{proposition}

\begin{proof}
Assume that the above inequality does not hold.
We use the notation of \ref{corollary-notation}.
In particular, $N$ denotes the number of components of $S=\sum S_i\in |-K_X|$.
By Lemma \ref{lemma-hyp-sect} $N=p^l$, where $l\ge 1$.
Hence $p$ divides $-K_X^3=2g-2=\bigl((-K_X)^2\cdot S_i\bigr)N$.
First we claim that $\rho(X)=1$.
Indeed, if $\rho(X)>1$, then the natural representation of $G$ on $\operatorname{Pic}_\mathbb{Q}(X):=\operatorname{Pic}(X)\otimes \mathbb{Q}$
is decomposed as $\operatorname{Pic}_\mathbb{Q}(X)=V_1\oplus V$, where
$V_1$ is a trivial subrepresentation generated by the class of $-K_X$
and $V$ is a subrepresentation such that $V^G=0$.
Since $G$ is a $p$-elementary group,
$\dim V\ge p-1$.
Hence, $\rho(X)\ge p\ge 5$ and by the classification \cite{Mori1981-82} we have two possibilities:
\begin{itemize}
\item
$-K_X^3=6(11-\rho(X))$,\quad $5\le \rho(X)\le 10$, or
\item
$-K_X^3=28$,\quad $\rho(X)=5$.
\end{itemize}
In the last case $p=5$, so $-K_X^3\not\equiv 0\mod p$, a contradiction.
In the first case $p$ divides $-K_X^3$ only if $p=5$.
Then $\rho(X)=6$. So, $\dim V=5$ and $V^G\neq 0$. Again we get a contradiction.

Therefore, $\rho(X)=1$. Let $q$ be the Fano index of $X$.
We claim that $X$ is singular.
Indeed, otherwise all the $S_i$ are Cartier divisors. Then
$-K_X=NS_1$, where $N\ge p$, and so $q\ge 5$.
This contradicts (i) of Theorem \ref{classification-Fano}.
Hence $X$ is singular. By Lemma \ref{lemma-fixed-point} and our assumption we have
$|\operatorname{Sing}(X)|\ge p$. In particular, $q\le 2$ (see Theorem \ref{classification-Fano}).
If $q=1$, then by Theorem \ref{classification-Fano} either $2\le g\le 10$ or
$g=12$.
Thus $N=p$ and we get the following possibilities:
$(p,g)=$ $(5,6)$, $(7,8)$, or $(11,12)$.
Moreover, $(-K_X)^2\cdot S_i= (2g-2)/N=2$.
Therefore, the restriction $|{-}K_X||_{S_i}$ of the
(base point free) anticanonical linear system defines
either an isomorphism to a quadric $S_i\to Q\subset \mathbb{P}^3$ or
a double cover $S_i\to \mathbb{P}^2$.
In both cases the image is rational, so
we get a map $G_i \to \operatorname{Cr}_2(\mathbb{C})$ whose kernel is of rank $\le 1$
by Lemma \ref{lemma-fixed-points} and because $p>2$.
Then by Theorem
\ref{theorem-cr2} \
$\operatorname{rk} G_{S_i}\le 3$.
Hence, $\operatorname{rk} G\le 4$ which contradicts our assumption.

Finally, consider the case $q=2$. Then $-K_X=2H$
for some ample Cartier divisor $H$ and $d:=H^3\le 7$.
Therefore, $N S_i\cdot H^2=S\cdot H^2=2d$.
Since $\rho(X)=1$, by Theorem \ref{classification-Fano}
we get $p=d=5$. Then we apply \eqref{eq-number-singular-points}.
In this case, $h^{1,2}(X_t)=0$  (see \cite{Iskovskikh-Prokhorov-1999}). So,
$|\operatorname{Sing}(X)|\le 19$. On the other hand,
$|\operatorname{Sing}(X)|\ge 25$ by Lemma \ref{lemma-fixed-point} and our assumption.
The contradiction proves the proposition.
\end{proof}

We need the following result which is a very weak form of much more general Shokurov's
toric conjecture \cite{McKernan-2001}, \cite{Prokhorov-2003e}.
\begin{lemma}
\label{lemma-toric-conjecture}
Let $V$ be a smooth Fano threefold and let
$D\in |{-}K_V|$ be a divisor such that the pair
$(V,D)$ is LC.
Then $D$ has at most $3+\rho(V)$ irreducible components.
\end{lemma}

\begin{proof}
Write $D=\sum_{i=1}^n D_i$.
If $\rho(V)=1$, then all the $D_i$ are linearly proportional:
$D_i\sim n_i H$, where $H$ is an ample generator of $\operatorname{Pic}(V)$.
Then $-K_V\sim \sum n_i H$ and
by Theorem \ref{classification-Fano} we have $\sum n_i=q\le 4$.

If $V$ is a blowup of a curve on another smooth Fano threefold $W$,
then we can proceed by induction replacing $V$ with $W$.
Thus we assume that $V$ cannot be obtained by blowing up
of a curve on another smooth Fano threefold. In this situation $V$  is
called \textit{primitive} (\cite{Mori1983}).
According to \cite[Th. 1.6]{Mori1983} we have $\rho(V)\le 3$
and $V$ has a conic bundle structure $f: V\to Z$, where $Z\simeq \mathbb{P}^2$
(resp. $Z\simeq \mathbb{P}^1\times \mathbb{P}^1$)
if $\rho(V)=2$ (resp. $\rho(V)=3$). Let $\ell$ be a general fiber.
Then $2=-K_V\cdot \ell=\sum D_i\cdot \ell$.
Hence $D$ has at most two $f$-horizontal components
and at least $n-2$ vertical ones.
Now let $h: V\to W$ be an extremal contraction other than $f$
and let $\ell'$
be any curve in a non-trivial fiber of $h$.
For any $f$-vertical component
$D_i\subset D$ we have $D_i=f^{-1}(\Gamma_i)$, where  $\Gamma_i\subset Z$ is a curve,
so $D_i\cdot \ell'=\Gamma_i \cdot f_*\ell'\ge 0$.
If $\rho(V)=2$, then $D_i\cdot l'\ge 1$. Hence, $-K_V\cdot \ell'\ge n-2$.
On the other hand, $-K_V\cdot \ell'\le 3$ (see \cite[\S 3]{Mori1983}).
This immediately gives us $n\le 5$ as claimed.
Finally consider the case $\rho(V)=3$.
Assume that $n\ge 7$. Then we can take $h$ so that
$\ell'$ meets at least three $f$-vertical components,
say $D_1$, $D_2$, $D_3$. As above, $-K_V\cdot \ell'\ge 3$
and by the classification of extremal rays (see \cite[\S 3]{Mori1983})
$h$ is a del Pezzo fibration.
This contradicts our assumption $\rho(V)=3$.
\end{proof}

\begin{proposition}
\label{proposition-Gorenctein-2}
 Let $G$ be a $2$-elementary group and
let $X$ be a Gorenstein $G\mathbb{Q}$-Fano threefold. Then $\operatorname{rk} G\le 7$.
\end{proposition}
\begin{proof}
Assume that $\operatorname{rk} G\ge 8$.
By Lemma \ref{lemma-hyp-sect} we have
$\operatorname{rk} G_{S_i}\le 5$. Hence, $N\ge 8$.
If $X$ is smooth, then
by Lemma \ref{lemma-toric-conjecture} we have $\rho(X)\ge 5$.
If furthermore $X\simeq Y\times \mathbb{P}^1$, where $Y$ is a del Pezzo surface, then
the projection $X\to Y$ must be $G$-equivariant.
This contradicts $\rho(X)^G=1$.
Therefore, $\rho(X)=5$ and $-K_X^3=28$ or $36$ (see \cite{Mori1981-82}).
On the other hand, $-K_X^3$ is divisible by $N$, a contradiction.

Thus $X$ is singular.
Assume that $|\operatorname{Sing}(X)|\ge 32$.
Then for a smoothing $X_t$ of $X$
by \eqref{eq-number-singular-points} we have
$h^{1,2}(X_t)\ge 13$. Since $N$ divides $-K_X^3=-K_{X_t}^3$,
using the classification of Fano threefolds
\cite{Iskovskikh-1980-Anticanonical}, \cite{Mori1981-82}
(see also \cite{Iskovskikh-Prokhorov-1999}) we get:
\[
\rho(X)=1,\quad -K_X^3=8,\quad  N=8,\quad  |\operatorname{Sing}(X)|=32.
\]
Consider the representation of $G$ on $H^0(X,-K_X)$.
Since
\[
 7=\dim H^0(X,-K_X)< \operatorname{rk} G,
\]
this representation is not faithful (otherwise $G$ is contained in
a maximal torus of ${\operatorname{GL}}(H^0(X,-K_X))={\operatorname{GL}}_7(\mathbb{C})$).
Therefore, the linear system $|{-}K_X|$ is not very ample.
On the other hand, $|{-}K_X|$ is base point free (see
Lemma \ref{lemma-base-point-free}). Hence $|{-}K_X|$ defines a double cover
$X\to Y\subset \mathbb{P}^{6}$ \cite{Iskovskikh-1980-Anticanonical}.
Here $Y$ is a variety of degree $4$ in $\mathbb{P}^6$,
a variety of minimal degree.
If $Y$ is smooth, then according to the Enriques theorem
(see, e.g., \cite[Th. 3.11]{Iskovskikh-1980-Anticanonical})
$Y$ is a rational scroll
$\mathbb{P}_{\mathbb{P}^1}({\mathscr{E}})$, where
${\mathscr{E}}$ is a rank $3$ vector bundle on $\mathbb{P}^1$.
Then $X$ has a $G$-equivariant projection to a curve.
This contradicts $\rho(X)^G=1$.
Hence $Y$ is singular. In this case, $Y$ is a cone (again by the the Enriques theorem
\cite[Th. 3.11]{Iskovskikh-1980-Anticanonical}).
If its vertex $O\in Y$ is zero-dimensional, then  $\dim T_{O,Y}=6$.
On the other hand, $X$ has only hypersurface singularities (see \ref{Terminal-singularities}).
Therefore the double cover $X\to Y$ is not \'etale over $O$ and so
$G$ has a fixed point
on $X$. This contradicts Lemma \ref{lemma-fixed-point}.
Thus $Y$ is a cone over a rational normal curve of degree $4$
with vertex along a line. Then $X$ cannot have isolated singularities, a contradiction.

Therefore, $|\operatorname{Sing}(X)|< 32$.
Then for any point $P\in\operatorname{Sing}(X)$ by Lemma \ref{lemma-fixed-point}
we have $\operatorname{rk} G_P \ge 4$. Hence the orbit of $P$ contains $16$ elements
and coincides with $\operatorname{Sing}(X)$, i.e. the action of $G$ on $\operatorname{Sing}(X)$ is transitive.
Since $S\cap \operatorname{Sing}(X)\neq \emptyset$, we have $\operatorname{Sing}(X)\subset S$.
On the other hand, our choice  of $S$ in \ref {Main-assumption} is not unique:
there is a basis $s^{(1)}$, \dots, $s^{(g+2)}\in H^0(X,-K_X)$
consisting of eigensections. This basis gives us $G$-invariant divisors
$S^{(1)}$, \dots, $S^{(g+2)}$ generating $|-K_X|$.
By the above $\operatorname{Sing}(X)\subset S^{(i)}$ for all $i$.
Thus $\operatorname{Sing}(X)\subset \cap S^{(i)}=\operatorname{Bs} |-K_X|$.
This contradicts  Lemma \ref{lemma-base-point-free}.
Proposition \ref{proposition-Gorenctein-2} is proved.
\end{proof}

\begin{proposition}
\label{proposition-Gorenctein-3}
Let $G$ be an $3$-elementary group and
let $X$ be a Gorenstein $G\mathbb{Q}$-Fano threefold. Then $\operatorname{rk} G\le 5$.
\end{proposition}
\begin{proof}
Assume that $\operatorname{rk} G\ge 6$.
By Lemma \ref{lemma-hyp-sect} we have
$\operatorname{rk} G_{S_i}\le 5$.
Hence, $N\ge 9$.
If $X$ is smooth, then by Lemma \ref{lemma-toric-conjecture} we have $\rho(X)\ge 6$ and so
$X\simeq Y\times \mathbb{P}^1$, where $Y$ is a del Pezzo surface \cite{Mori1981-82}.
Then the projection $X\to Y$ must be $G$-equivariant.
This contradicts $\rho(X)^G=1$.
Therefore, $X$ is singular.
By Lemma \ref{lemma-fixed-point} $|\operatorname{Sing}(X)|\ge 3^{6-3}=27$.
Hence, for a smoothing $X_t$ of $X$
by \eqref{eq-number-singular-points} we have $h^{1,2}(X_t)\ge 7+\rho(X)$.
Recall that $N$ divides $-K_X^3=-K_{X_t}^3$.
Then we use the classification of smooth Fano threefolds
\cite{Iskovskikh-1980-Anticanonical}, \cite{Mori1981-82}
and get a contradiction.
\end{proof}

Now Theorem \ref{theorem-cr3} is a consequence of Propositions
\ref{proposition-non-Gorenctein},
\ref{proposition-Gorenctein-p},
\ref{proposition-Gorenctein-2}, and
\ref{proposition-Gorenctein-3}.

\def$'${$'$}
  \def\mathbb#1{\mathbf#1}\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
  \lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}

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I finally get to say it, "My first book is now available!" I got the email today saying that all of the pre-publication work is complete, and people can now start buying it.
If you are local to South Bend (and you're interested), I will have some in a week or so. If you don't want to wait, or if you're not local, you can order it on Amazon.com today.
I want to give a big thanks to everyone who has helped me with this process - proofing, editing, questioning my conclusions, etc. I am deeply grateful.
Congrats. . . . Now you're going to have to work those hand muscles for all the book signings you'll be doing. . . . maybe you should come up with a generic saying for people when you sign yearbooks in highschool and people say. . don't ever change. . . stay the same. . . . keep in touch. . . and i know you'll write on mine BFF. (best friends forever) , just a thought. . . . anyway congrats.
A famous author! I am so pleased for you!! I think I still have the rough draft...HA! Well, anyway, the class you held helped me understand this very controversial topic and I know God will bless others with the message you give.


Prime Minister Justin Trudeau is apologizing on Friday in Iqaluit for the federal government's treatment of Inuit with tuberculosis in the mid-20th century, an appearance postponed the day before because of bad weather.
The apology was expected to be delivered on Thursday in the Nunavut capital, but the weather diverted Trudeau's flight and the ceremony was cancelled.
Friday's event started at 9:30 a.m. ET at the Frobisher Inn.
As the ceremony got underway, Aluki Kotierk, president of Nunavut Tunngavik Incorporated, said it will be an emotional day, and counsellors were being made available in Iqaluit.
Representatives from the four Inuit regions of Canada, which stretch across the Arctic to Labrador, have been in Iqaluit for much of the week for the apology.
Inuit were separated from their family members and taken to Southern Canada for tuberculosis treatment from the 1940s through the '60s.
Many patients lived in sanatoriums for years, and their families often weren't told about their whereabouts or conditions, even when they died.
Trudeau was expected to apologize specifically to Inuit.
Former patients and immediate family members of people who died in Southern Canada will be among those hearing the prime minister deliver the apology.
It's believed Trudeau's apology will also mark the launch of the Nanilavut initiative — a database that will make records available to Inuit to facilitate finding family members' gravesites.


Pastor's email: Use the email above while in transition.
For problems or questions regarding this Web site contact the [WebMaster].
Note: This website is a work in progress. Every attempt is being made to provide accurate and complete information. Please forward any website corrections/questions/comments to the webmaster.


text:
fou un futbolista català de les dècades de 1940 i 1950.

Començà la seva carrera a la UA Horta, essent fitxat pel RCD Espanyol el 1947, amb qui debutà a primera divisió. La seva carrera continuà dues temporades al Girona FC i tres al CE Sabadell, totes cinc a segona divisió. La temporada 1953-54 retornà a primera divisió de la mà del CA Osasuna. Els darrers anys de la seva carrera els visqué al CE Manresa, CD Fabra i Coats i CF Gavà.

Referències

Enllaços externs

Futbolistes del Baix Llobregat
Esportistes viladecanencs
Futbolistes de la UA Horta
Futbolistes del RCD Espanyol de la dècada de 1940
Futbolistes del Girona FC
Futbolistes del CE Sabadell de la dècada de 1940
Futbolistes del CE Sabadell de la dècada de 1950
Futbolistes del CA Osasuna
Futbolistes del CE Manresa
Atlètic Catalunya Club de Futbol
Futbolistes del CF Gavà
Morts a Barcelona

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text:
Home All Difference between Term Sheet of Equity, CCPS and CCD under different clauses
Difference between Term Sheet of Equity, CCPS and CCD under different clauses
Image source - https://bit.ly/39Zy3kt
This article is written by Abhishek Dubey, a law student. The article compares the term sheet of an angel investor in all three cases such as issue of equity, CCPS and CCD share and also the difference in any of the particular clause of the term sheet.
Instrument clause
Interest clause
Valuation clause
Advisory clause
Board composition and incidental matter clause
Pre-emptive rights clause
Conversion clause
Promoters and non-selling investors right of the first offer
Employee stock option plan clause
Exit mechanism clause
Holders rights
Under the equity term sheet, equity share has a face value of INR at the price of per-share equity share. And in CCPS(Compulsary Convertible preference Share) term sheet, companies have a face value of INR at each price of share along with certain rights as agreed by the parties in the documentation of preference and subscription share. And in CCD(Compulsory Convertible Debenture) term sheet, the total value of INR and certain rights are attached in the debenture documentation.
In terms of equity, there is no interest clause. There is a dividend clause which says that holders of preference share shall be entitled to the payment of % non-cumulative per annum on each of preference share by way of dividend from the company in preference to holders of equity share or any other securities of the company.
In the term sheet of CCD, the debenture shall bear an interest of % on a non-cumulative basis per annum.
In term sheet of equity, there is pre-money valuation is required in INR, in term sheet of CCPS the pre-money valuation is required but it is not required in CCD term sheet.
In the term sheet of equity, the company agrees to issue the equity share to an advisor on an ongoing basis in value and consideration of advice.
In the term sheet of CCPS, the company agrees to issue advisory equity equivalent to the investment per cent in value in consideration of the advisory services on an ongoing basis.
In the term sheet of CCD, the company at the time of closing agrees is to issue equity shares.
The difference in these term sheets for advisory equity is optional and the investor will only issue when he has taken advisory services but it is compulsory in case of term sheet of CCD where the investor has to definitely issue advisory equity.
In all 3 term sheets, investors shall be entitled to nominate at least one director, others will be nominated by the promoters. But in the term sheet of CCPS, if the investor does not have representation for the board of directors then the board observer position may be granted to the investor.
Click Above
The investor shall have pro-rata right to participate in case of issue of share to the third party in future and retain their shareholding on same terms and condition but it is only applicable in equity and CCPS term sheet but not in CCD term sheet.
There is no conversion clause in the term sheet of equity. In terms of the CCPS sheet after liquidation and after 20 years from the date of allotment of share, the preference share shall convert into equity share in the ratio of 1:1. The conversion shall be adjusted in future according to bonus, employee stock option plan and equity splits etc.
But in the term sheet of CCD, there is a conversion clause which says that if the company raises the qualified financing then debenture will be automatically converted into equity share. Qualified financing refers to the investment lower than the INR.
In the event selling, investors sell whole or any part of their shareholding to the third party. The selling investor has to first offer it to the promoters. This is applicable only in CCPS and equity term sheet but not in CCD term sheet.
The company shall implement an employee stock option plan not exceeding than the post-issue share capital of the company on a fully diluted basis with the approval of investor and director. This is applicable in case of term sheet of equity and CCD but in CCPS term sheet, the employee stock option plan shall be at least 10 per cent on the post-issue share of capital. However, there is no minimum condition in the term sheet of equity and CCD.
The exit mechanism provides that the company and promoters shall make all ways to provide the exit opportunity to the investor.
In terms of equity, there are various ways to provide the exit opportunity such as initial public offering at a time mutually agreed upon, within 5 years from the date of closing.
At the strategic sale of equity share means at the price acceptable to the investor within 6 years from the date of closing.
Drag along with rights: In the event, if the company and promoters fail to provide the exit opportunity then the investor can sell its share to the third party.
In terms of CCD also, there are various ways to give the exit opportunity to the investor such as IPO, strategic sale of debenture and drag along with options etc.
But in term sheet of CCPS, there is qualified financing that the company shall endeavour to ensure that any future issuance by it of securities of a cumulative value equal to or higher than 125 per cent. It shall be done in such a way that 25 percent of the amount proposed to be raised.
The investor shall be granted all rights that are granted to all the investors subscribing to the securities of the company. This is given only in the term sheet of CCD but not in the term sheet of equity and CCPS.
This clause is given only in the term sheet of CCPS which says that the investor shall be the power of attorney to other investors defined in the documentation.
There is governing law which says that it shall be governed by the law of India and dispute will be settled in the court but there is arbitration in CCPS where the party can appoint the arbitrator but in equity it and CCD the dispute will be settled in the court.
S. NO
Basis of difference in clauses
Under the equity term sheet, equity share has a face value of INR at the price of equity share.
Under the CCD term sheet, certain rights are agreed by the parties in the documents called debenture definitive documentation.
In the CCPS, the rights of the parties are governed in the documentation called preference share and equity and preference share have face value collectively known as subscription share.
In equity term sheet there is no interest clause.
In CCD term sheet, the debenture shall bear interest on non-cumulative basis per annum.
In CCPS term sheet, the holders of the preference share shall be entitled to the payment at certain percentage on non-cumulative coupon per annum on each of preference share by way of dividend from the company.
The pre-money valuation is required in case of the equity term sheet.
But no such valuation is required in case of CCD term sheet.
The pre-money valuation is required in case of CCPS term sheet.
In the equity term sheet, the company agrees to issue the equity share to an advisor on an ongoing basis and in consideration of advice.
In CCD, the company agrees to issue equity shares at the time of closing to an advisor.
In CCPS term sheet, the company agrees to issue the advisory equity on an ongoing basis equivalent to the investment per cent in value.
Board composition and incidental matter
Investors shall be entitled to nominate at least one director, others shall be nominated by the promoter.
Investors shall be entitled to nominate only one director, others shall be nominated by the promoters.
Investors shall be entitled to nominate at least one director and others shall be nominated by the promoter and in case of the investor does not have representation, the board observer position may be granted to the investor.
Pre emptive rights clause
The investor shall have pro-rata right to participate in case of equity share to the third party and retain their shareholding.
Pre emptive rights clauses are not specified in case of CCD term sheet.
There is no conversion in equity term sheets.
In the case of CCD term sheet, if the company raises qualified financing, the debenture will be automatically converted into equity.
In the case of CCPS term sheet, after 20 years and after liquidation, the preference share shall convert into equity share in the ratio1:1.
Promoters and non-selling rights of the first offer
The investor if selling the share to any third party, has to first offer it to the promoter.
The investor without giving offer to the promoter can sell the share to the third party.
The investor before selling the share to the third party has to offer it to the promoters.
The company shall implement the employee stock option plan on the fully diluted basis with the consent of the director and investor.
The company shall implement the employee stock option plan not exceeding than the post-issue share of the capital on fully diluted basis along with the consent of the director and promoters.
The company shall have the stock option plan for at least 10 per cent of the post-issue share capital with the fully diluted basis with the consent of directors and promoters.
There are various ways to provide the exit opportunity to the investor such as Initial public offering, strategic sale of equity share and drag along option.
There are various ways to provide the exit opportunity to the investor such as initial public offering, strategic sale of debenture and drag along option etc.
There are various ways to provide the exit opportunity such as initial public offering, sale of subscription share and drag along with options.
The investor shall not be granted all the rights subscribing to the securities of the company.
The investor shall be granted all the rights subscribing to the securities of the company.
The investor shall not be the power of attorney to other investors defined in the documentation.
The investor shall be the power of attorney to other investors defined in the documentation.
Governing law jurisdiction and arbitration
This term sheet shall be governed according to the law of India. And any disputes will be settled in the court.
The term sheet shall be governed according to the law applicable to India. And any dispute will be settled in the court.
The term sheet shall be governed according to the law applicable in India and any dispute shall be settled by the arbitrator.
Complusary Convertible Preference Share
Compulsory Convertible Debenture
Term Sheet
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Hello, Thank you so much for sharing this useful information with all of the readers. Keep sharing such useful information.

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The songs of Kobra And The Lotus deserve to be heard by a large audience and on the strength of this performance, that is bound to happen at some point in time.
But on Sunday evening in Camden there was a growing concern that The Barfly would be only half-full, or half-empty, depending on your outlook as Metalworks were playing five doors down in The Monarch, Eye Hate God were fifteen minutes walk away in The Koko and the hottest ticket in town, Alice Cooper at The 100 Club (click here for Johnny Churchill's review...), would surely account for some of the inevitable lack of audience at The Barfly?
However, by the time Kobra hit the stage the attendance was much more than respectable for a band who could take this show straight into the arenas and make it work. It's all in place; it just needs a little fine tuning.
More on Kobra And The Lotus shortly. Triaxis opened proceedings in The Barfly with their own unique and impressive brand of gothically tinted, avant-garde mysticism and they have the songs to carry them further forward in this ever adapting industry.
I'm afraid I didn't get a set-list from the band so I can't quote any song titles but be assured, Triaxis' material has true quality and if they can secure enough bookings to really fine tune their craft, they could go far.
That was a prompt to my friend Martin Jarvis who made a cameo appearance at last night's event. Go on son, get on the phone and get them some more gigs! They're worth it.
The Praying Mantis. Synchronicity with the Kobra and/or the Lotus? A totem to some perhaps and most certainly a British Metal institution and they were on fine form tonight and played one of the best sets they have played all year.
Metalworks lost a guitarist recently when Richie Faulkner got the call to join Judas Priest, one of the top Metal bands in the world today. Before he became an overnight sensation two months ago, Richie played here at The Monarch with Metalworks and he left behind some mighty shoes to fill.
Tonight, Judas Priester from Germany were here and I am sure it meant a lot to this rather good Priest tribute band to be treading the boards where the current Priest lead guitarist was in residence before his sudden and dramatic rise to world fame.
It was great to hear 'Ripper', 'Victim Of Changes', 'Turbo Lover', 'The Green Manalishi With The Two Pronged Crown' and 'Painkiller' and the band executed them well.
And then back to The Barfly for the main attraction of the night and I have to say, they were simply fantabulous!
'Kobra And The Lotus' could easily be the title of a book and if the words in the pages were as rich as the band's songs, it would be a book that would be difficult to put down.
Brittany Paige has the spirit of a modern day Jutta Weinhold (Zed Jago) and she carries these songs off with her presence that resonates from both her voice and the way she holds an audience. And the songs are epics as well; carefully crafted pieces of fine art, presented in a bejewelled casket.
Kobras' set list will make your hair stand on end at several points throughout and you will be left wanting to hear more. They are in England for another three weeks and if you decide to make an effort to catch them, you will be pleased that you did.
My message to Kobra And The Lotus is if you see a circle of light on a shadowy path, step into it because you are meant to shine and your music will bring joy to the masses. Hail!


Once again the electoral politics affecting Newark is heaving change. A new crop of legislative officials is now slated to replace a crop of seasoned incumbents. Teresa Ruiz, Grace Spencer and Albert Coutino are the new Senator and Assemblywoman and Assemblyman representing the 29th Legislative District in New Jersey, which includes a substantial portion of Newark. Even before last Tuesday's election, gone were Senator Sharpe James who chose not to run and Assemblywoman Odaline Truitt who was defeated in the June primary. It took a sound thumping in Tuesday's general election to vanquish former Assemblyman William Payne who sought the Senate seat as an independent.
The future has not been so up in the air in a long time. What's next? Who will emerge as the dominant political forces going forward? How long will Black politics be relevant in Newark? All these questions and more have wafted quietly in the background of Newark's political scene for several years but are now raging naked on "front street."
In a matter of a few months the political fortunes of the Payne dynasty have been cut nearly in half. In May there were four family members holding five elective offices. Now there are just two members holding three offices. So what's next for the Paynes? Will they regroup and try to rebuild or will they reach an accommodation with reality and forge an alliance to protect their remaining holdings? And if so, with whom?
And what of the next generation political insurgents who were left languishing when Sharpe James departed Newark's political arena? Many rallied around Senator Rice in his primary victory, no doubt hoping it would provide sufficient momentum to jump-start a political movement destined to help them regain their status. Will Tuesday's hide tanning tame their enthusiasm? Will they finally stop dabbling with low risk political skirmishes and step up to the real battle? There are a lot of questions to which answers will no doubt reveal themselves in the coming days and weeks.
But here's what we know now. There is a major political transformation underway in Newark. Its face will be largely ethnic as Blacks yield leadership to Latinos, but its root will be economic as the business community seeks the best front-line political partner with which to execute their aggressive agenda. And so long as the community places its focus on the competition for power between Blacks and Latinos, both will be the losers in the economic sweepstakes that is poised to define a new Newark.
Because African American politicians in Newark have primarily viewed and played politics as an enterprise of individual empowerment, we are fast becoming defenseless in the emerging political climate. All the old horses are going or being sent out to pasture. They did not bother to train and ready a new breed. The competition is well heeled, well disciplined and well financed. Newark's next generation of African American political leaders have been held back too long by the old horses and just might be passed over all together.
But will it really matter? We believe it will not so long as Black politicians are unable and unwilling to translate their leadership into opportunity that breeds sustainable political power and creates institutions that transcend politics and strengthen community. Anything less is worthless-except to some individual politicians. By now we should have had our fill of their type of political selfishness.
We see a significant opportunity for Black political maturation on the horizon. True, Black politics is old. But is it mature? We think not. And the obvious very public dissolution that is now underway gives us all a chance to recognize and openly acknowledge that peril is at hand. We can act all grown up. We can accept collective blame. Then we can choose to work together. Yes. Right now we can move to close breaches, heal rifts, span gaps and rebuild badly damaged important relationships. If we refuse and adopt recalcitrance as methodology, we can chalk it all up.
A partnership with Latinos and others is the appropriate course of action. It does not mean giving up aspirations of the Black community. It can mean assuring them. A partnership with the business community is also essential. And a truly equitable relationship will not be possible so long as Blacks and Latinos are engaged in an intractable political struggle over little or nothing. But none of these critical arrangements can emerge if the Black community continues to prove incapable of hammering out the required internal agreements between its vital sectors.
Cory Booker has become a lightening rod of false focus. Truth is, he is a transitional figure. His role is significant and must be acknowledged. Pretending to discount and ignore his significance is folly. He is in place and a transition is taking place. What that transition means for Newark's Black community is murky because we have few collective goals that are driven by clear thinking. Yet our community can still define our future. It will not wait forever though. It is now being defined by events and circumstances not within our reach. This can change only if our behavior changes.
Black politicians have dominated the local scene for too long. And in many ways and for many people they have become irrelevant. Major players in many sectors view them as a nuisance factor. They tend to be engaged only when absolutely necessary or for purely ceremonial purposes. Serious political interaction regularly takes place around them and outside their realm of competency. Members of the African American clergy have even stepped into the void left by ineffective political organization in the Black community. And if you think carefully about it you will be hard pressed to find another community of people where the religious leaders are the political spokespersons. It is an outrage run amuck. But it is our community and if we want to be taken seriously and make real progress we must change it without further delay.
Sharpe James left office at 70 years old, Don Bradley at about 70, Bill Payne around 70. This is amazing. Young political aspirants in the African American community should be livid and determined to end the practice of generational stifling in Black local politics. The election of Grace Spencer as 29th District Assemblywoman is an encouraging sign. But just as soon as one can absorb the positive flavor of the Spencer victory, political talk fast forwards to the notion of a Newark mayoral bid by our good friend Cliff Minor in 2010. At that time Cliff will be around 70. Hard to believe young Newarkers will swallow our man Cliff for the top spot at 70 years old. We are having a hard enough time being taken seriously without behaving stupidly.
Things are changing-moving forward fast. It is no time to entrench ourselves in backwardness or turn aside progress. Forward is where we must go if we don't just plain want to disappear from the political spectrum. History does record people becoming extinct. Let's just grow up before we're gone.
Tuesday was but one more wakeup call. It signals the closing of an era. Perhaps there is still a chapter or two to be played out but by and large it is time to change the subject. We need a new script. It is important that it lean heavily on history but it must portray the present and envision the future. There is no reason that we should not all contribute to its authorship. No good reason. A perfect moment for unity has arrived. Will we take it?
Imagine this: You were born and raised in Newark and you have been working there since you were 14. Then one day, you find out that you have to move out your house you were raised in and move somewhere else because you couldn't afford rent.
This is the harsh result of the unmistakable makeover Newark is going through currently.
In a newspaper article called, "Downtown Newark," by Carol Sharif, it clearly states the possible results of the Newark makeover. I have never thought that so many disadvantages would actually come because of the makeover. In the makeover, tax dollars will increase and Newarkers will not be able to afford anything or even live in houses. Also, businesses in Newark are going to migrate because they won't fit the standards of the upcoming clientele.
My classmates and I really adore your article. We attend North Star Academy in Newark, New Jersey. We read your article and analyzed it well. We connected it back to our homes in Newark. We started realizing that our homes, Newark were being changed by the government. We discussed this in class thoroughly and believe that you should come visit us. We would love for you to come visit us anytime becuase we know you will change history. So please Mr. Shariff take this into consideration and think about coming to North Star Academy.
I really enjoyed reading your new paper article "Down Town Newark." It was very interesting and informative. There were many things in this article that I have never known about or have heard. I believe that New York residents moving into Newark is a good idea. I understand that them moving to Newark will bring big businesses and make profit but what about those who already were here an down small businesses and stores in Down Town? If those big businesses come in and take over, how will those small businesses make money if they have been pushed out of the picture? New businesses will provide more revenue for the city but what about those that have been living in the city? This new unmistakable make over in Down Town Newark will create much chaos between those indigenous people of Newark and those new comers of Newark that are used to the Big Apple life style and business.
You stated in your article that, "Change is inevitable." There is no point in change if about over half of the current residents living in Newark are not supporting the alteration. Sharif, you even mentioned yourself, "How Newark residents will benefit is far less clear." This obviously means that Newark residents have fewer advantages to this "unmistakable makeover" but the New Yorkers will. It is less important that those New Yorkers who move into Newark benefit form this change because the Newark current residents are the ones that have to go through the pain. Vendors and merchants will have to move to other areas because the new businesses will make all of the money that they once have been making, natives will have to dislocate, and a large gap between both the small already established businesses and new big businesses. Newark citizens will have more advantages if their city that they have lived in and stood by for so many years remain the same.You should really come to my school, North Star Academy Charter School of Newark, to speak to the students adn my peers of this issue. I would really appreciate it. Thanks.
Int eh "DownTown Newark" article, you bring up the topic of how in Newark, New Jersey has been having many changes. The fact is that I agree with the changes, however i believe that there will believe that there will also be many disadvantages that come through the way. The reasons that still happens is that many New Yorkers are coming to live here to New Jersey is less expensive both economically and physically. The fact is that many New Yorkers are investing money to build new things. This is a preposterous idea because at the same time if the New Yorkers are coming here to live the rest of their lives, they have to know that we always lived here and they can't kick us out. Change will impact us in so many good ways and bad ways. This is extremely inconcievable because many people here who are indigenous to this town will have to leave because other people will want to move in. If all of these investments come along through out here in Newark, wouldn't Newark become another New York? New York is overflowing with so many people. If Newark is really wide spread, Newark wouldn't be the same. New York is filled with investments, property, and money. If Newark also becomes filled up with those things, there will be no exact difference. Newark would become so different, that the people that live here already will definetly move out. But in my school, North Star Academy in Newark, New Jersey, my class has read your article and felt that it is a very important issue. We would really like to have you over as a guest speaker to speak to us about this situation.
In the "DownTown Newark" article, you bring up the topic of how in Newark, New Jersey has been having many changes. The fact is that I agree with the changes, however I believe that there will believe that there will also be many disadvantages that come through the way. The reasons that still happens is that many New Yorkers are coming to live here to New Jersey is less expensive both economically and physically. The fact is that many New Yorkers are investing money to build new things. This is a preposterous idea because at the same time if the New Yorkers are coming here to live the rest of their lives, they have to know that we always lived here and they can't kick us out. Change will impact us in so many good ways and bad ways. This is extremely inconcievable because many people here who are indigenous to this town will have to leave because other people will want to move in. If all of these investments come along through out here in Newark, wouldn't Newark become another New York? New York is overflowing with so many people. If Newark is really wide spread, Newark wouldn't be the same. New York is filled with investments, property, and money. If Newark also becomes filled up with those things, there will be no exact difference. Newark would become so different, that the people that live here already will definetly move out. But in my school, North Star Academy in Newark, New Jersey, my class has read your article and felt that it is a very important issue. We would really like to have you over as a guest speaker to speak to us about this situation.


text:
Home National News British Parliament Opposition Leader applauds role of PM Khan in peace process
British Parliament Opposition Leader applauds role of PM Khan in peace process
Lahore: Governor Punjab Chaudhry Mohammad Sarwar during his visit to United Kingdom, he met with Labor Party Leader and Opposition Leader Keir Starmer and Shadow Secretary of State for Foreign Affairs MP David Lemmy at the British Parliament. Issues faced by people of Afghanistan, Afghan peace process, current situation in the region and Pak-UK relations were also discussed.
According to the details issued from Governor House Lahore. During his visit to the British Parliament, Governor Punjab Chaudhry Mohammad Sarwar met with opposition leaders Kier Starmer and David Lemmy. British MPs Yasmin Qureshi, Mohammad Afzal, MP Sheeran and others were also present in the meeting. During the meeting, Governor Punjab spoke about the steps being taken by Pakistan for peace and stability in Afghanistan, the current situation in the region and other issues including Kashmir issue and urged the UK government to not only increase aid to Afghanistan but also to play a role in providing basic services there.
During the meeting, Carey Starmer and David Lemmy praised Pakistan's role under the leadership of Prime Minister Imran Khan in the Afghan peace process and eradicating terrorism and said we will fully support Pakistan for the establishment of peace and eradication of terrorism as the establishment of peace and eradication of terrorism is in the interest of the whole world. Pakistan is fighting on the front line against terrorism.
Governor Punjab while talking during the meeting said that Pakistan Army and the Pakistani nation made exemplary sacrifices in the war against terrorism. The success of any country, region or world requires the elimination of terrorism. Pakistan has no personal agenda regarding Afghanistan. We want only peace and stability in Afghanistan and the provision of basic amenities to the Afghan people.
Governor Punjab Chaudhry Mohammad Sarwar said that whenever the world has ignored Afghanistan in the past, the whole world has seen the consequences, so now the whole world, including the United States and the United Kingdom, should not make the mistake of leaving Afghanistan as it is because if things go wrong there, then the world must understand that its effects will be felt not only in Afghanistan, Pakistan or the region but the whole world. Therefore, the world must help Afghanistan as much as possible and as soon as possible and Pakistan is also playing its role to help the Afghan people.
During the meeting Governor Punjab asked Labor Party Leader and Opposition Leader Keir Starmer and Shadow Secretary of State for Foreign Affairs MP David Lemmy to play their part in ending the ongoing atrocities and human rights abuses in Kashmir.
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30 is one of those milestone birthdays that many people (especially girls) dread. I honestly don't dread it, as my grandma says turning 30 "sure beats the alternative!" But I have always thought of 30 as adult.Like a real adult. Don't laugh. I know that I'm a married mom of two and that I should consider myself an adult at this point but that's not what I'm talking about. I'm talking about the I've-got-this-all-figured-out adult.
Back when I was 17 and just graduating high school, I was certain there would be a point (surely by the age of 30!) that I would be an adult and have everything figured out. You know, I would somehow have managed the discover how to maintain a spotless house, cook amazing meals every day,and keep up with all the errands.I would have a beautiful flower garden and maybe even grow some veggies or herbs. I just knew I would have 3 or 4 perfectly behaved, Jesus lovin', funny, well-rounded and absolutely adorable children. We would craft, and sing and read and they would NEVER bicker or throw temper tantrums. I would always be wonderful,even-tempered, attentive, supporting and loving wife. I would have a deep and consistently growing relationship with the Lord. I would be active in my church. And of course I would do all of this while managing to be a size four in an adorable dress and the perfect shoes.
So, here I am a week away from 30 and certainly don't have it all figured out. So many of my dreams have come true. Some haven't yet. I haven't figured out how to perfectly balance my wonderful husband, 2 kids, housework, exercise, errands, and time with the Lord. But I've learned to give myself grace when I fail to meet what I think I have to get done, as long as I'm focusing on my priorities.
I certainly don't have everything figured out. But I'm ok with that too. I've learned that seasons seem to change at the blink of an eye. It's so quick that it sends my head swirling in a million directions but in the middle of my chaos stands One who is steady and faithful. One who is never taken off guard or surprised. One who does have it all figured out — I love that I don't have to have it all figured out because I know who to run to with my cares and concerns, my dreams and desires — and there is so much peace in that.
This summer I am a part of an amazing small group. It is hands down my favorite summer small group ever. It is filled with mostly single girls – there are a couple of other married girls & one engaged. Oh, how I love Monday nights. Once you move out of that season, you forget what it's like to be so excited about a first date, the anxious feelings before you get married, or the deep longing to have your 1st baby. Our tendency is to want to rush through whatever season we are in right now instead of sitting back and savoring the moment. I love watching these girls enjoy their seasons.
I remember being single and praying to be in a relationship. Then praying for a spouse. After getting married praying for kids. My twenties have been amazing. I've gone through several different seasons but I've savored few of them.
I wonder sometimes what blessings we miss in our current season, because we are so focus on what's next.
So, I may not have it all figured out. In fact, I probably had more "figured out" at 17 than I do now but I will enjoy where I am now. I'm married to an amazing man and blessed with two crazy, Jesus lovin' babies. I am blessed indeed.


Earlier this month the LifeKeepers team brought their new e-learning programme to the Counties Manukau Steelers' for their personal development day.
This August twelve nominees from across Aotearoa were recognised in Christchurch for their contributions to preventing suicide in their communities - ordinary people doing extraordinary things to create communities of C.A.R.E.
Waitangi, Aotearoa hosted delegates from 32 countries at the IASP Asia-Pacific Regional Conference in May. We were joined at the conference by the winner of the 2017 LifeKeepers Supreme Award, Zack Makoare, who has written a blog about his experience.
Due to popular demand, the national LifeKeepers Awards for suicide prevention are returning for 2018. Plans are underway to once again celebrate the efforts, dedication and commitment of the individuals and organisations out there championing this cause in our communities across the motu.


Going on Northwest to China?
Any one get ill upon returning from India?
Ideas for rest & relaxation in Ho Chi Minh City?
KL Airport - Pan Pacific Hotel / others?
Bangkok to Phuket or Ko Samui?
Bali - Sanur or Candidasa?
Up North - Chaing Mai, Rai, Pai or Mae Hon Song?


text:
Q: Firefox for Android and HTML5 canvas transparency bug Consider this simple webpage:
<!DOCTYPE html><html>
<head>
    <title>test</title>
    <style type="text/css">
        canvas { border: 1px solid black; }
    </style>
</head>
<body>
    <canvas width="100" height="100" id="the-canvas"></canvas>
    <script type="text/javascript">
        var canvas = document.getElementById('the-canvas');
        var context = canvas.getContext('2d');
        context.fillStyle = 'rgba(0, 0, 255, 0.5)';
        context.fillRect(0, 0, 100, 100);
    </script>
</body></html>

Basically, it contains only one <canvas/> element that has semi-transparent blue background. The expected behavior is to have it rendered in light blue color, since the background of the web page is white.
It renders fine on the desktop machine.
On Android it renders fine as well, except for Firefox, where it acts like the page underneath canvas was actually.. black. The strangest thing is that it acts like it only if there is any part of Firefox's interface visible on the screen!
Examples below.
Good behavior - color is light blue. No parts of Firefox's interface are visible.

Wrong behavior - the resulting color suddenly gets way too dark. Note that Firefox's interface comes into view as I sweep my screen to see the address bar.

Here's another example; this time I try to render corny snow animation to celebrate incoming Christmas. To do that, I render small opaque circles on transparent canvas. Version on the left - Firefox interface visible, circles have ugly anti-aliasing problems; version on the right - the same page after scrolling a bit down, circles are rendered correctly.

I tried to use various blending features to get around this, but without success.

A: Just wondering is this is a "web safe" colour issue!

redpajama_set_name: RedPajamaStackExchange


Hiking event open to all members and non-members!
For people who will be driving and have extra space, please contact Debbie (deb_chou@hotmail.com) so we can car pool...much appreciated!
Food will not be provided, so please bring snack/lunch for this event. Please bring plenty of water to stay hydrated, given the perfect sunny day!
CISC will be providing one sports drink to each participant.
Fullerton Loop is an easy hike that provides excellent views of the Elbow Valley and nearby foothills. The ample south-facing exposure of this loop makes a great relax easy hike.
When you get to Alan Bill Pond, park in the lot there, and then look for your trailhead at the east end, toward the river. The trail goes through the fence once or twice before heading up into the woods. When you get to the fork (stairs on left, bridge on right), stay right to do the loop in a counter-clockwise direction.
Event Directions: Follow the Trans-Canada Highway (Hwy 1) to the Highway 22 overpass, follow Highway 22 through Bragg Creek to the Highway 66 Junction, turn right and follow Highway 66 to the Allen Bll Pond day use area (on the left side). The east side of the parking lot is blocked off - just park as close as you can to that.
Carpool Departure Time: 11 am MST (sharp! We will leave without you) Please arrive at least 15mins early to figure out carpooling arrangements.
If you choose to drive out on your own, please meet at the Allen Bill Pond Parking by noon.


text:
This Proposal For The Brand New Los Angeles NFL Stadium Is Incredible (Video)
By Julian Sonny
By now, we know an NFL franchise coming back to Los Angeles isn't just a possibility: It's inevitable.
And with the proposal for a new stadium in Carson that could house both the Raiders and the Chargers, we may have found the centerpiece for Southern California sports for the next 50 years.
This dope video narrated by Kiefer Sutherland reveals a state-of-the-art entertainment and sports complex that will surely cost a sh*tload of money, but it will be too f*cking lit when it's done.
This is the place to be seen watching a game -- whether it's because of the farmers market or the VIP valet for you rich bastards. Football in LA is already a match made in heaven, but you know the city needs the stadium to match.
But these are the real questions: Will there be a marijuana dispensary on site, and can we blow loud during the games?


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New Injustice 2 Trailer Reveals Enchantress's Mystic Abilities
Posted on January 6, 2018 by Graphic Policy Team
Warner Bros. Interactive Entertainment and DC Entertainment have unveiled a new gameplay trailer for the newest addition to the Injustice 2 character roster, Enchantress. The trailer gives players their first glimpse at June Moone's magic-wielding alter-ego Enchantress, in-action as she struggles to contain the mystical powers that have taken hold of her. Enchantress will arrive on Jan. 9 for players who have purchased the Injustice 2 Ultimate Edition, Injustice 2 Ultimate Pack or Fighter Pack #3, and will be available as a standalone purchase beginning Jan. 16 from the PlayStation® Store, Xbox Store, Steam, and Microsoft Store.
Developed by the award-winning NetherRealm Studios, Injustice 2 features a massive roster of DC Super Heroes and Super-Villains and allows players to build and power up the ultimate version of their favorite DC characters. The game is currently available for PlayStation 4, Xbox One, and PC via Steam and the Microsoft Store, with DLC characters Red Hood, Sub-Zero, Starfire, Black Manta, Raiden, Hellboy and The Atom available now on console and PC. Injustice 2 mobile is available now for free on iOS and Android devices.
tagged with dc entertainment, enchantress, injustice 2, netherrealm studios, video, warner bros. interactive entertainment
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Artists in Residence: Joli Vyann's 'Imbalance'
"Just how much does my life revolve around technology?"
That is the question we were left asking ourselves after Joli Vyann's prototype performance at PDSW last week. Following the highly successful 'Stateless', Olivia Quayle and Jan Patzke return with an exciting fusion that explores our obsession with technology. In a world where smartphones, tablets and laptops have become necessities in our day-to-day lives, it is easy to forget to turn off your devices and reconnect with the 'real world'. Delving into the virtual chaos of the modern age, this thought-provoking piece leaves audiences questioning their own reliance on technology and how it impacts on their face-to-face relationships.
Speaking about the influence behind 'Imbalance' choreographer and Olivier award nominee, Jonathan Lunn, explained how the performance had strong roots in the ideas explored in Godfrey Reggio's 'Koyaanisqatsi' (a Hopi word meaning 'life out of balance'), a renowned documentary that explores how humanity has grown apart from nature. Through dance, acrobatics and circus, the piece eloquently conveys this theme of balance (or lack thereof), with Olivia and Jan telling the story of a couple obsessively dependent on the digital world.
In preparation for the 'Imbalance' tour, Joli Vyann spent two weeks as our artists-in-residence, building and tweaking the piece. We caught up with Jan who talked through their time at PDSW and described what they had been working on during their time with us, "The entire cyberbullying section was developed here, and we also worked a lot on the beginning section. It's been great to have access to a theatre environment". We also took it back to basics and spoke about how they got into dance and acrobatics. Giving us a brief glance at the sheer breadth of their skills, Jan told us "Olivia did artistic gymnastics from a young age and went on to dance school, doing some circus jobs on the side. I have a background in martial arts. I also worked as a stunt man and went to circus school. When we first met we were able to teach each other new skills. I knew more about the hand-to-hand side of acrobatics, whereas Olivia had dance skills". The piece clearly strikes a personal note with the performers too. Speaking about the creative process, Jan mentioned that it is a reflection of his and Olivia's 'real life' relationship off-stage, which started out long-distance and heavily reliant on technology.
'Imbalance' is remarkably relatable and the audience are likely to see hints of their own relationships in the couple portrayed by Jan and Olivia. It is a pertinent piece, accessible to audiences of all literacies. It has elements of humour, plenty of breath-taking movement and a take-home message for the audience.
This performance was commissioned by Pavilion Dance South West as part of the Shift and Share small-scale touring project with commissioning partners Activate, Exeter Northcott Theatre, The Performance Centre – Falmouth University, ICIA – University of Bath, Tacchi Morris Arts Centre, Theatre Bristol and Tobacco Factory. Shift and Share has been designed in collaboration with partner venues in the South West committed to bringing dance to more audiences in the South West and is supported by Arts Council England.
'Imbalance' will be performed at PDSW on Thursday 24th March at 7:30pm.


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Miss Sharon Jones!
Synopsis & Film Details
Fresno Filmworks presents the American independent documentary "Miss Sharon Jones!" a powerful crowd-pleaser that captures iconic soul diva Sharon Jones during the most intense and courageous year of her life. Often compared to the legendary James Brown for her energetic performances, Jones finds her career taking off after years of being kept down by a music industry that branded her "too short, too Black, too fat." After decades of working odd jobs, from a corrections officer to a wedding singer, she enjoys a breakthrough in middle age after joining forces with the Brooklyn R&B outfit The Dap-Kings. Then in 2013, on the eve of the release of her album "Give the People What They Want," Jones is diagnosed with cancer, but she battles back with a magnetizing spirit of heart, wit, and humor to triumphantly reclaim center stage. Directed by two-time Academy Award winner Barbara Kopple.
Director: Barbara Kopple
Distributor: Starz Digital
USA • 94 minutes • Not Rated
Screening Sponsors
Alpha Kappa Alpha Sorority, Inc. — Iota Omicron Omega Chapter
Judie and Alan Pierrot
Filmworks thanks Jewel FM 99.3, The Fresno Bee, Vida en el Valle, and Stella Artois for their ongoing support.
Categories: Monthly Film Series Genres: documentary, music Tags: barbara kopple, feature, USA


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Last edited by Mazuran
Friday, August 7, 2020 | History
5 edition of Noise and the Worker (Health & Safety at Work, New) found in the catalog.
Noise and the Worker (Health & Safety at Work, New)
Dept.of Employment
by Dept.of Employment
Published November 1971 by Stationery Office Books .
Occupational / industrial health & safety,
OCLC/WorldCa 516699
However, in the Ministry of Labour published guidance in the form of their booklet 'Noise and the Worker' that, in effect, put the 'safe' limit at about 90dB(A) for workers who are exposed to a continuous steady noise for eight hours a day, five days a week. publication called Noise and the Worker was published in that year. The second stems from the landmark legal judgement made in the case of Thompson v Smith Ship Repairers. In that case, because of Noise and the Worker's publication and other information available at the time, the Judge decided that by employers ought to have been awareFile Size: KB.
Exposure to industrial noise may increase the heart rate of workers. Although rises in heart rate, systolic, and diastolic blood pressure of workers in the case group were observed after exposure to noise, the values of heart rate, systolic, and diastolic blood pressure were Cited by: Noise Exposure In evaluating worker exposure to noise, the industrial hygienist should answer two main questions: 1. Is the OSHA PEL met or exceeded? 2. Does the worker need to be in the hearing conservation program? Modern dosimeters calculate dose both ways 46File Size: KB.
Noise levels should be regularly monitored and work practices continuously improved to preserve the hearing of workers. In accordance with the Occupational Health and Safety Regulations (regulation ), employers that are required to provide employees with hearing protection to use in the workplace must provide workers with hearing tests. This short book is for owners and managers of business enterprises, especially in the developing world, to help you identify, reduce, and eliminate hazards in the workplace. This guide also serves as a tool to promote a health and safety-oriented culture of cooperation among workers, managers,File Size: 2MB.
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Noise and the Worker (Health & Safety at Work, New) by Dept.of Employment Download PDF EPUB FB2
COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this 's WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
Noise and the worker. by Great Britain. Factory Inspectorate. Published by H.M.S.O. Written in EnglishPages: His ambitious new book, The Signal and the Noise, is a practical handbook and a philosophical manifesto in one, following the theme of prediction through a series of case studies ranging from hurricane tracking to professional poker to counterterrorism.
It will be a supremely valuable resource for anyone who wants to make good guesses about the Cited by: Noise-induced hearing loss (NIHL; see Glossary) affects workers' quality of life and increases the risk of injury – for instance, when a worker cannot hear approaching vehicles or warning signals.
1 The U.S. Occupational Safety and Health Administration (OSHA) set the permissible exposure limit (PEL; see Glossary) for construction noise to. Among the many factors that contribute to the deterioration of the working environment, noise and vibration have an important place.
Noise and the Worker book code of practice, adopted at a meeting of experts convened by the ILO, sets out the principles that should be followed for the control of workplace noise and vibration, and contains the information required for the establishment of control programmes for.
Buy Noise and the Worker (Health & Safety at Work, New) 3rd Revised edition by Employment, (ISBN: ) from Amazon's Book Store. Everyday low prices and free delivery on Author: Employment.
Noise is usually defined as unwanted sound: loud music from a neighbor, the honk of a taxicab, the roar of a supersonic jet. But as Garret Keizer illustrates in this probing examination, noise is as much about what we want as about what we seek to by: Noise Noise means unwanted sound or loud discordant or disagreeable sound or sounds.
The effect of noise on hearing can be temporary or permanent. Temporary deafness is often experienced after leaving a noisy place. Noise is present in many workplaces, but being exposed to too much can cause permanent and irreversible damage to workers.
Noise levels and the duration of the worker's exposure to noise should be reduced. Loud machines and equipment are among the most common causes of harm to workers' hearing.
As a guideline, if you are within 1 metre of. The resulting book, Noise and Military Service: Implications for Hearing Loss and Tinnitus, presents findings on the presence of hazardous noise in military settings, levels of noise exposure necessary to cause hearing loss or tinnitus, risk factors for noise-induced hearing loss and tinnitus, the timing of the effects of noise exposure on.
The law in In'Noise and the Worker' was the first major publication produced which highlighted the dangers of excessive noise and gave recommendations of what should be done to protect employees from exposure. In light of the publication, the case of Thompson v Smith Ship Repairs [] held that fromemployers exposing employees to noise levels exceeding 90dB were in.
Inthe Minister of Labour published "Noise and the Worker", the first major publication on the subject. It made employers aware of the dangers of excessive noise and recommended what they needed to do to protect their workers from exposure to excessive noise.
Noise-induced hearing loss is the most common preventable occupational health condition in the world. Noise is defined as 'unwanted sounds', while sound is a term used for sensation that the brain receives when pressure variations in the air are detected by the ear.
Noise at work: A brief guide to controlling the risks Page 4 of 10 other workplaces similar to yours; and data from suppliers of machinery. Do not make any allowance for the wearing of personal hearing protection when you estimate workers' noise exposure levels.
Personal noise exposure may also be calculated over a week rather than a day, ifFile Size: KB. Free download or read online The Sound and the Fury pdf (ePUB) book. The first edition of the novel was published in October 7thand was written by William Faulkner.
The book was published in multiple languages including English, consists of pages and is available in Paperback format/5. KEYWORDS: Wheat processing, noise, machines, workers, audiometric examination ABSTRACT: This study assessed the intensity and likely effects of noise on workers during wheat processing.
Noise measurements were taken using HD sound level meter. Subjective assessment of the effects of noise was undertaken using semi structured questionnaire.
Also. Search the world's most comprehensive index of full-text books. My library. Books shelved as noise: Noisy Night by Mac Barnett, Swish and Squeak's Noisy Day by Birgitta Sif, The Rooster Who Would Not Be Quiet. by Carmen Agra Deed. Part of a Noise Management Program is to identify noise hazards in the workplace and those workers most likely to be exposed to noise exceeding the OELs.
This is done by conducting a survey in which sound levels are measured and the duration of worker exposures in the various work areas are determined. These results establish the need forFile Size: KB.
Posted By Dave Daniel Noise and the Worker was a Department of Employment publication which set out for the first time in a widely available Government publication the fact that noise caused deafness and could and should be controlled. Measures recommended included quieting noisy machinery and wearing earplugs, including at the time I think wax.
Even now, careful analysis of the worker's audiological evaluation can lead to rehabilitation through individual actions and group therapies. As noise is an occupational hazard, prevention efforts must be implemented in the work by: The Signal and the Noise is a very interesting book with mixed success: 3 1/2 stars, were this permitted.
I found it somewhat difficult to review; however, my entire book group without exception had similar opinions. I would encourage you to view this as a group opinion. At its best, TSANTN is interesting, illustrative, educational, and 4/5(K).Any noise problem may be described in terms of a source, a transmission path and a receiver (in this context, a worker) and noise control may take the form of altering any one or all of these elements.
The noise source is where the vibratory mechanical energy originates, as.
johnsonout.com - Noise and the Worker book © 2020


Ron Hardy is a Winnebago County Board Supervisor in Oshkosh, and former Chair of the City of Oshkosh Sustainability Advisory Board. He is a librarian at UW Oshkosh, and a father of two.
Hardy faces four other candidates for Wisconsin State Treasurer. "After the primaries, I'm the only progressive candidate in the treasurer's race and the only candidate supporting the common-sense solution of a public bank. With support from progressives, fiscal conservatives, independents, and anyone who's fed up with politics as usual, I can win this race."
The General Election is November 4, 2014.


Bridges connect, enabling transport and communication. They also have a symbolic function. Their constructions can be quite different, ranging from the clapper bridges in Dartmoor to the industrial beauty of the bridge over the Firth of Forth. Of course forces are important - a topic you discussed in physics class. We're going to think about and experiment with different materials and constructions, closing with a contest: who will design the best and most beautiful light-weight bridge?
This is the Golden Gate Bridge in San Francisco. The other 9 most beautiful or well known bridges are found here.
Download Bridge designer 2016 and design bridges at home.
These three apps are great tools for unstanding the characteristics of a proper bridge and they contain a lot of levels to keep you challenged. Bridge Architect is a free app, but both Link! as Bridge Constructor have free versions available. At least Bridge Architect is also available for Android. Play them and learn how you make a stable and strong bridge and use your findings to make a great bridge yourself to win the contest!