text:
A student holds a placard during a protest at the state university grounds in Manila on February 14, 2019, in support of CEO of Rappler, Maria Ressa, who was arrested a day earlier for cyber libel case. Ressa was freed on bail on February 14 following an arrest that sparked international censure and allegations she is being targeted over her news site's criticism of President Rodrigo Duterte.
AFP/Ted Aljibe
Journalists reiterate call to decriminalize libel
(Philstar.com) - February 18, 2019 - 12:35pm
MANILA, Philippines — Journalists are reiterating the call for the decriminalization of libel in light of a case that has led to the arrest of a high-profile journalist for cyberlibel.
The National Union of Journalists of the Philippines said on Monday that the government's case against Rappler and its CEO Maria Ressa highlight the need for the decriminalization of libel.
NUJP pointed out that a "dangerously vindictive government" has already "mangled law" to silence a critical news outfit "but unless stopped, these machinations will eventually endanger not only Rappler or the independent Philippine media but each and every Filipino who has ever posted online."
Under the Revised Penal Code, libel is punishable by imprisonment for from six months to six years. The Cybercrime Prevention Act, Republic Act 10175, also includes a provision to punish online libel, as defined by the RPC.
Rappler, Ressa and former reporter Reynaldo Santos Jr. are facing a charge of cyberlibel before a Manila court. Ressa and Santos are out on bail, after paying a bail bond of P100,000 each.
"The position of the [Department of Justice] and the National Bureau of Investigation that an allegedly defamatory statement posted online will continue to be a crime unless the offending material is taken down means that anything and everything anyone posts anywhere on the web can be cause for criminal prosecution that could lead to long jail terms and crippling fines," NUJP said.
It added that "it is not inconceivable that we could witness online desertification as people attempt to wipe out any content that might be deemed, for any reason, libelous."
Cagayan de Oro-based journalists have also echoed the journalists' group sentiment as they raised, in a report from Mindanao Gold Star Daily, that the case "criminalizes journalism." The Mindanao Gold Star Daily cites veteran journalist Froilan Gallardo as saying current libel and cyber libel laws "are like swords held to the throats of journalists."
Libel law from the 1930s
Members of the Philippine press have long rallied for the decriminalization of libel, a law that has been carried over from 1932.
The Center for Media Freedom and Responsibility said in 2014 that the "libel provisions of the RPC have been problematic for free expression and press freedom since 1932...which has been used in many instances to silence journalists."
The media watchdog monitors libel cases against reporters across the country.
UP professor Rachel Khan, chair of the journalism department at the College of Mass Communication in UP Diliman, stressed that "libel has been used to harass journalists."
Speaking at DzBB, Khan added that libel "should be more of a civil act", as it would prevents the its use to theaten journalists with imprisonment.
NUJP acknowledged that getting a law passed to decriminalize libel would be extremely difficult.
"After all, our antediluvian libel law and its threat of jail time is one of the weapons of choice of corrupt officialdom against those who dare scrutinize and call out their venality and abuse," it said.
READ: A look at the cyber libel charge vs Rappler, Maria Ressa
Manicad, Malacañang: Libel a check on journalists' 'abuse'
A former member of the Philippine media is seeking a seat at the Senate, but Jiggy Manicad's stand on the decriminalization of libel differs from that held by many of his former colleagues.
In ABS-CBN's Harapan 2019 debate on Sunday night, Manicad said that he believes libel should remain a criminal act "so there will be no abuse from among our ranks as journalists."
The format of the show kept candidates from elaborating on their answers.
Manicad however refused to comment on Ressa's case, saying that it is already pending before the court.
Presidential spokesperson Salvador Panelo, meanwhile, said that in his opinion, a civil suit for a libel case would suffice, but there is a fear that journalists would abuse it.
"If you feel that your right has been violated, you should file a case. It should not be that because you are up against a high-profile journalist, you would just let it be," he said in a mix of Filipino and English in an interview with ANC's Early Edition.
He said that President Rodrigo Duterte believes that "what the law says, we should follow."
Duterte has distanced himself from Ressa's arrest, saying that he does not know businessman Wilfredo Keng, who first filed the complaint against Ressa before the NBI. He said Thursday night that he had yet to read up on the case.
Panelo also said that Ressa's case has nothing to do with "freedom of expression," asserting that Ressa is facing a legitimate case and would be given an opportunity to defend herself in court. — Kristine Joy Patag
2019 MIDTERM ELECTIONS JIGGY MANICAD JOURNALISM LIBEL NATIONAL UNION OF JOURNALISTS OF THE PHILIPPINES
Sinovac 'provided very good pricing to the Philippines,' exec says
By Gaea Katreena Cabico | 4 hours ago
In an interview on CNN Philippines Monday, Sinovac Biotech General Manager Helen Yang did not disclose the cost of the vaccine...
EUA for Pfizer COVID-19 vaccine stays pending findings on elderly deaths in Norway
By Gaea Katreena Cabico | 2 minutes ago
Pfizer is the only pharmaceutical firm that has so far obtained an EUA from the local FDA for its vaccine, which has an efficacy...
By Christian Deiparine | 1 hour ago


text:
Ref watch: Ross Joyce to oversee Oxford United trip
Confirmation of match officials for Tuesday night's trip to Kassam Stadium
Ross Joyce will take charge of Wanderers' trip to Oxford United on Tuesday night.
The Cleveland-based official previously oversaw his only other Whites outing back in August as Bolton beat Sheffield United on the opening day of the season.
Becoming a Football League regular last season after running the rule over his Football League fixture in February 2014, Mr Joyce has been the man in the middle for 35 games so far this campaign, dishing out 153 yellow cards and 11 red cards.
Ian Dudley and Michael Webb will run the line, while John Brooks is the fourth official.

redpajama_set_name: RedPajamaCommonCrawl


From Gene Bedley's National seminar The "Parent Advantage" comes this new and practical seminar for Teachers, Parent Involvement Coordinators, and Administrators. Gene Bedley is known for his practical and innovative strategy's from his award winning programs over the past 40 years. The Parent Participation Seminar Resource Book and seminar are packed with ideas that are both portable and easy to implement.
Downloadable Book Only – Once purchased you will be able to download the book to your desktop or mobile device!
A Powerpoint Presentation is also available for trainers.
Gene Bedley was chosen as a Milken Family Foundation National Award Winner in 1994. A special thank you to People-Wise Publications for sharing solutions from Gene Bedley's Parent Participation Program Amazing seminar! Amazing presenter with great strategies and tools. Thank you so much!
SKU: PWP-0054. Categories: Digital Books, Digital Downloads, Seminar Handbooks.


text:
\section{Introduction}

Self-similar\ groups have been  given directly as automorphism groups of
some $m$-tree , as is the case of the infinite torsion group of Grigorchuk
\cite{Gr} and those of Gupta-Sidki \cite{GS}, or constructed to act on such
trees by way of a single virtual endomorphism. However these constructions
necessarily produce transitive self-similar groups, in the sense that the
corresponding state-closed group of automorphisms of the $m$-tree satisfies
the additional property of acting transitively on the first level of the
tree.

Transitive state-closed representations have been studied for the
family of abelian groups, finitely generated nilpotent groups, as well as
for metabelian groups, affine linear groups and arithmetic groups; see \cite%
{BarSid},\cite{BerSid}, \cite{K}, \cite{KS} for more details.

It was shown recently that the group $G=\mathbb{Z}\wr \mathbb{Z}$ fails
to have a faithful transitive state-closed representation on an $m$-tree for
any $m$ \cite{DS}. Yet as we will prove, this group admits a faithful
intransitive state-closed representation on the $3$-tree. Indeed, in this
representation the group is a $3$-letter, $3$-state automata group; it has
the following diagram

\begin{center}
\begin{tikzpicture}[shorten >=3pt,node distance=3.5cm,on grid,auto]
  \node[state] (e) [] {$e$};
  \node[state] (a) [below left of=e] {$\alpha$};
  \node[state] (g) [below right of=e] {$\gamma$};
 \path [->]
   (e) edge [loop above] node {$0|0, 1|1, 2|2$ } (e)
        (a) edge    [loop left]  node {$1|0$} (a)
        (a) edge              node {$0|1, 2|2$} (e)
        (g) edge              node {$2|2$} (a)
        (g) edge  [loop right]      node {$0|0$} (g)
        (g) edge node [swap] {$1|1$} (e);

\end{tikzpicture}

    Diagram 1
\end{center}

\,

Given a self-similar group $G$ we will prove a number of results about
self-similarity of its over-groups from the following types:

(1) restricted direct product $G^{(\omega )}$ of countably many copies of $%
G$;

(2) restricted wreath product $G\wr K$ where $K$ is finite; restricted
wreath product $A\wr G$ where $A$ is abelian, both finitely and infinitely
generated, and for particular cases where $G$ is abelian.

The two types of groups in item (2) are in accord with Gruenberg's dichotomy
of residually-finite wreath products  \cite{Gru}. Our results extend ones which
have appeared in \cite{BarSid} and \cite{DS}.

Given a state-closed subgroup $G$ of automorphisms of the $m$-tree indexed
by strings from a set $Y$ of size $m$, a $G$-data $\left( \mathbf{m},\mathbf{%
H,F}\right) $ is obtained as follows: suppose $G$ have $s$ orbits $%
Y_{i}=\left\{ y_{i1},...,y_{im_{i}}\right\} $ in its action on $Y$ then $%
\mathbf{m=}\left( m_{1}...,m_{s}\right) $. Define  the
set of subgroups
\begin{equation*}
\mathbf{H=}\left\{ H_{i}\mid \left[ G:H_{i}\right] =m_{i}\text{ }\left(
\text{ }1\leq i\leq s\right) \right\} \text{ }
\end{equation*}%
where $H_{i}=Fix_{G}\left( y_{i1}\right) $; define the set of projections
\begin{equation*}
\mathbf{F}=\left\{ f_{i}:H_{i}\rightarrow G\mid 1\leq i\leq s\right\} \text{.%
}
\end{equation*}
On the other hand, given a group $G$, an $s$- set of subgroups of $G$
\begin{equation*}
\mathbf{H=}\left\{ \left( H_{i}\mid \left[ G:H_{i}\right] =m_{i}\text{ }%
\left( \text{ }1\leq i\leq s\right) \right) \right\} \text{,}
\end{equation*}%
\begin{equation*}
\mathbf{m=}\left( m_{1}...,m_{s}\right) ,\text{ }m=m_{1}+...+m_{s}
\end{equation*}
and a set of virtual endomorphisms%
\begin{equation*}
\mathbf{F}=\left\{ f_{i}:H_{i}\rightarrow G\mid 1\leq i\leq s\right\} \text{,%
}
\end{equation*}
we have an abstract $G$-data $\left( \mathbf{m},\mathbf{H,F}\right) $. We
prove reciprocally

\begin{propA}
\label{PA copy(1)} Given a group $G$, $m\geq 1$ and a $G$-data $\left(
\mathbf{m},\mathbf{H,F}\right) $. Then the data provides a state-closed
representation of $G$ on the $m$-tree with kernel
\begin{equation*}
\langle K\leqslant \cap _{i=1}^{s}H_{i}\mid K\vartriangleleft
G,K^{f_{i}}\leqslant K,\forall i=1,...,s\rangle \text{,}
\end{equation*}%
called the $\mathbf{F}$-core of $\mathbf{H}$ .
\end{propA}

The partition $\mathbf{m=}\left( m_{1},...,m_{s}\right) $ is called the
\textit{orbit-type} of the representation.

We apply the Proposition A to obtain families of self-similar groups.

\begin{thmB}
\label{TB} Let $G$ be a self-similar group of degree $m$ and orbit-type $%
\left( m_{1},..,m_{s}\right) $. Then  the following hold.

\begin{itemize}
\item[1)] $G^{(\omega )}$ admits a faithful state-closed representation of
degree $m+1$ and orbit-type $\left( m_{1},..,m_{s},1\right) $; in
particular, for $G=\mathbb{Z}$ , the representation of the group $\mathbb{Z}%
^{(\omega )}$ is of orbit-type $\left( 2,1\right),$ and is in addition
finite-state.

\item[2)] Let $K$ be a regular subgroup of $Sym(\{1,...,s\})$. Then the
restricted wreath product $G\wr K$ admits a faithful state-closed representation of degree $(m_1\cdot m_2\cdot...\cdot m_s) \cdot s.$
\end{itemize}
\end{thmB}

With respect to the first item, for any positive integer $k$, the direct
product group $G^{k}$ is \ self-similar of degree $m$. It was shown in \cite%
{BarSid} that the group $\mathbb{Z}^{(\omega )}$ has a faithful transitive
self-similar representation of degree $2$, moreover and importantly, no such
representation can also be finite-state for any degree.
As a consequence
of the tree-wreathing operation defined by Brunner and Sidki \cite{BS1} and
by using $k$-inflation, the group $\mathbb{Z}\wr \mathbb{Z}$ is self-similar of
orbit-type $(2,1)$; see Diagram 1 . Then, by the second item of Theorem B, it follows that $(\mathbb{Z}\wr \mathbb{Z})\wr C_{2}$ is a self-similar group of degree $4$.

Restricted wreath products $A\wr G$ for $A$ abelian and $G=%
\mathbb{Z}^{d}$ have been a good source for automata groups. \ The first
instance in this family is the classical Lamplighter group $C_{2}\wr \mathbb{%
Z}$. It was shown in \cite{DS} that if $A\wr \mathbb{Z}^{d}$ $\ $admits a
faithful \textit{transitive} self-similar representation then $A$ is
necessarily a \textit{torsion group of finite exponent}. Also, it was proven
in \cite[Proposition 6.1]{BarSid} that when $B$ is a finite abelian group,
then $B\wr \mathbb{Z}^{d}$ is an automata group of degree $2|B|$.

We generalize both results as follows

\begin{thmC}
\label{TD copy(1)} Let $A$ be a finitely generated abelian group and $%
B=Tor(A)$. Then  $G=A\wr \mathbb{Z}^{d}$ is an automata group of degree $%
2|B|+4$. In particular, for $A=\mathbb{Z}^{l}$, the degree can be reduced to $4$.
\end{thmC}

The theorem will follow from a general process of concatenation (see Proposition \ref{P3.2}) of
the two cases $G_{1}=B\wr \mathbb{Z}^{d}$ and  $G_{2}=\mathbb{Z}^{l}\wr \mathbb{Z%
}^{d}$. We note that the result for $\mathbb{Z} \wr \mathbb{Z}$ answers positively a question posed by A. Woryna in \cite[page 100]{W}.

It was shown in  \cite{DS1}  that the group $C_{p}\wr \mathbb{Z}^{d}$ where $%
C_{p}$ is cyclic of prime order $p$ and $d\geq 2$  is self-similar of degree $p^{2}$, but that such a group does
not have a  faithful \textit{transitive} state-closed representation of prime degree.
In this context we prove:

\begin{thmD}
\label{TC copy(1)} Let $p$ a prime number then $C_{p}\wr \mathbb{Z}^{2}$ is
a self-similar group of degree $p+1$ of orbit-type $%
\left( p,1\right)$. Indeed, $C_{p}\wr \mathbb{Z}^{2}$ is generated by  $\alpha = (\alpha, \alpha \sigma, ..., \alpha \sigma^{p-1}, \alpha \beta)$, $\sigma = (e, ..., e, \sigma)(0 \, 1 \, ... p-1)$ and $\beta = (e, ..., e, \alpha)$. In particular, the group $C_{2}\wr \mathbb{Z}^{2}$ is
self-similar of degree $3$.
\end{thmD}

Let $f:H\rightarrow G$ be a virtual endomorphism. Set $G_{0}=G$ and $%
G_{n}=G_{n-1}^{f^{-1}}$ for all $n\geq 1$. Define the parabolic subgroup $%
G_{\omega }=\cap _{j\geq 0}G_{j}$. Let $G_{\omega }\setminus G$ denote the
set of right cosets $G_{\omega }$ in $G$, let $A^{(G_{\omega }\setminus
G)}=\{\phi:G_{\omega }\setminus
G \to A \,\, \text{ of finite support}\,\,\}$ and have $g \in G$ act on it by translation. The following result was proven for
transitive self-similar groups in \cite{BarSid}.

\begin{prop1}
(Proposition 6.1) Let $G$ be a transitive self-similar group of degree $m$ and parabolic
subgroup $G_{\omega}$, and let $B$ be a finite abelian group. Then the
extension $B^{(G_{\omega}\setminus G)}\rtimes G$ is transitive
self-similar of degree $|B|m$, and is finite-state whenever $G$ is.
\end{prop1}

Let $G$ be a state-closed group with respect the data $(\mathbf{m}, \mathbf{H%
}, \mathbf{F})$, where $\mathbf{m} = (m_{1}, ...,
m_{s})$, with $m_{1} \geq 2, ..., m_{s} \geq 2$.
For each $i=1,...,s$ define $G_{i0}=G$, $G_{ij}={(G_{i(j - 1)})}^{f_i^{-1}}$ ($j>0$) and $G_{\omega_i}= \cap_{j\geq 0}G_{ij}.$ With this notation we have:

\begin{thmE}
Let $B$ be a finite abelian group and $G$ be a self-similar group of
orbit-type $(m_{1}, ...,
m_{s})$, with $m_{1} \geq 2, ..., m_{s} \geq 2$. Then the group $$B^{((G_{\omega_1}\setminus G) \times ... \times (G_{\omega_s}\setminus G))}\rtimes G^s$$
is self-similar of orbit-type $(|B|.m_{1}...m_{s}, \, 1)$.
\end{thmE}

A question which has remained open is whether the group $C_{2}\wr (\mathbb{Z%
}\wr \mathbb{Z})$ is self-similar.

\section{Preliminaries}

\subsection{\textbf{Groups acting on rooted }$m$-\textbf{trees.}}

The vertices of a rooted $\ m$-tree $\mathcal{T}_{m}$ are indexed by strings
from an alphabet $Y$ of $m\geq 1$ letters, ordered by $u<v$ provided the
string $v$ is a prefix of $u$. The tree $\mathcal{T}_{m}$ is also denoted as
$\mathcal{T}\left( Y\right) $; normally, we chose to take $Y=\left\{
0,1,...,m-1\right\} $. Given a group and a representation of it on $\mathcal{%
T}_{m}$ we say both the group and its representation have degree $m$.

The automorphism group $\mathcal{A}_{m}$, or $\mathcal{A}\left( Y\right) $,
of $\mathcal{T}_{m}$ is isomorphic to the restricted wreath product
recursively defined as $\mathcal{A}_{m}=\mathcal{A}_{m}\wr S_{m}$, where $%
S_{m}$ is the symmetric group of degree $m$. An automorphism $\alpha $ of $%
\mathcal{T}_{m}$ has the form $\alpha =(\alpha _{0},...,\alpha _{m-1})\sigma
(\alpha )$, where the state $\alpha _{i}$ belongs to $\mathcal{A}_{m}$ and
where $\sigma :\mathcal{A}_{m}\rightarrow S_{m}$ is the permutational
representation of $\mathcal{A}_{m}$ on $Y$, the first level of the tree $%
\mathcal{T}_{m}$. Successive developments of the automorphisms $\alpha _{i}$
produce $\alpha _{u}$ for all vertices $u$ of the tree. For $k\geq 1$, the
action of $\alpha $ on a string $y_{1}y_{2}...y_{k}\in Y^{k}$ is as follows

\begin{equation*}
\alpha :y_{1}y_{2}...y_{k}\mapsto \left( y_{1}\right) ^{\sigma (\alpha
)}\left( y_{2}...y_{k}\right) ^{\alpha _{y_{1}}}\text{. }
\end{equation*}%
This implies that $\alpha $ induces an automorphism $\alpha _{\mathbf{k}}$
on the $m^{k}$-tree $\mathcal{T}\left( Y^{k}\right) $ and the above action
on $k$-stings gives us a group embedding $\mathcal{A}\left( Y\right)
\rightarrow \mathcal{A}\left( Y^{k}\right) $ which we call $k$-\textit{%
inflation}.

For $\alpha \in \mathcal{A}\left( Y\right) $, the set of automorphisms%
\begin{equation*}
Q(\alpha )=\{\alpha ,\alpha _{0},...,\alpha _{m-1}\}\cup
_{i=0}^{m-1}Q(\alpha _{i})
\end{equation*}%
is called the set of \textit{states} of $\alpha $ and this automorphism is
said to be \textit{finite-state} provided $Q(\alpha )$ is finite. A subgroup
$G$ of $\mathcal{A}_{m}$ is \textit{state-closed} (or, \textit{self similar}%
) if $Q(\alpha )$ is a subset of $G$ for all $\alpha $ in $G$. More
generally, a subgroup $G$ of $\mathcal{A}_{m}$ is $k$th\textit{-level}
\textit{state-closed} if for all $\alpha \in G$, the states $\alpha _{u}$ belong to
$G $ for all strings $u$ of length $k$; we see then that the $k$-inflation of
$G $ is state-closed. A group which is finitely generated, state-closed and
finite-state is called an \textit{automata group}.

\subsection{\textbf{Virtual endomorphisms.}}

Given a subgroup $H$ of $G$ of finite index $m$, a homomorphism $%
f:H\rightarrow G$ is called a \textit{virtual endomorphism} of $G$. A
subgroup $K$ of $H$ is $f$\textit{-invariant} provided $K^{f}\subseteq K$.
The maximal subgroup of $H$ which is both $f$-invariant and normal in $G$ is
called the $f$\textit{-core} of $H$; if this subgroup is trivial then $\ f$
is said to be \textit{simple}. If $G$ is a self-similar subgroup of $%
\mathcal{A}_{m}$, then for $H=Fix_{G}(0)$, the subgroup stabilizer of the
vertex $0\in Y$, we have the projection $f:H\rightarrow G$ which can be seen
to be simple.

Let $G$ be a group, $H$ a subgroup of $G$ of finite index $m$ with right
transversal $T=\{t_{0},...,t_{m-1}\}$ in $G$. Then the induced permutation
representation $\sigma $ of $G$ on $T$ is transitive and we define the
Scheier function $\theta :G\times T\rightarrow H$ by $\theta \left(
g,t_{i}\right) =t_{i}g(t_{j})^{-1}$ where $Ht_{j}=Ht_{i}g$.

\subsection{Recursive \textbf{Kaloujnine-Krasner}}

The Kaloujnine-Krasner Theorem \cite{KK} provides us with an extension of $\sigma
$ to a representation $\varphi :$ $G\rightarrow H\wr G^{\sigma }$ defined by%
\begin{equation*}
\varphi :g\mapsto (\theta \left( g,t_{0}\right) ,...,\theta \left(
g,t_{m-1}\right) )g^{\sigma }\text{.}
\end{equation*}%
On applying\ the virtual endomorphism $f$ to $\theta \left( g,t_{i}\right) $%
, we obtain $\theta \left( g,t_{i}\right) ^{f}$ $\in G$ which allows us to
repeat the above representation $\varphi $ to this element. Thus the
Kaloujnine-Krasner representation extends recursively to a representation on
the $m$-tree $\mathcal{T}_{m}$, , indicated by the same symbol, as follows

\begin{equation*}
\varphi :\text{ }g\mapsto \left( \theta \left( g,t_{0}\right) ^{f\varphi
},...,\theta \left( g,t_{m-1}\right) ^{f\varphi }\right) \,g^{\sigma }.
\end{equation*}%
The kernel of the representation $\varphi $ is the $f$\textit{-core} of $H$
\cite{NS}.

\ Given a group $G$, $m\geq 1$ and $G$-data $\left( \mathbf{m},\mathbf{H,F}%
\right) $ we re-work the recursive representation in the transitive case as
follows. For each $H_{i}$ choose a right transversal $T_{i}=\{t_{i1},...,t_{im_i}\}$ and let $\theta
_{i}$ be the corresponding Schreier function . Then define $\varphi
:G\rightarrow \mathcal{A}_{m}$ by%
\begin{equation*}
\varphi :\text{ }g\mapsto \left( \theta _{i}\left( g,t\right) ^{f_{i}\varphi
}\mid 1\leq i\leq s,t\in T_{i}\right) \,g^{\sigma }.
\end{equation*}

\section{Proof of Proposition A}

We reformulate the proposition more concretely as follows

\begin{propA}
\label{TA}

 Given a group $G$, $m\geq 1$ and $G$-data $\left( \mathbf{m},%
\mathbf{H,F}\right) $, the function $\varphi :G\rightarrow \mathcal{A}_{m}$
defined by
\begin{equation*}
\text{ }g\mapsto \left( \theta _{i}\left( g,t\right) ^{f_{i}\varphi }\mid
1\leq i\leq s, \, t\in T_{i}\right) \,g^{\sigma }.
\end{equation*}%
is a state-closed representation of $G$ on the $m$-tree with kernel the $%
\mathbf{F}$-core of $\mathbf{H}$.

\end{propA}

\begin{proof}
Consider $g$ and $h$ elements of $G$. Clearly $(gh)^{\sigma}=g^{\sigma}h^{%
\sigma}$, so if $i\in Y$, then
\begin{equation*}
i^{(gh)^{\varphi}}
=i^{(gh)^{\sigma}}=i^{g^{\sigma}h^{\sigma}}=i^{g^{\varphi}h^{\varphi}}.
\end{equation*}
By induction on the length of word in $Y$, we have that

\begin{equation*}
(iu)^{(gh)^{\varphi}}=i^{(gh)^{\sigma}}u^{(gh)^{\varphi_i}}=i^{(gh)^{%
\sigma}}u^{(g^{\varphi}h^{\varphi})_i}=(iu)^{g^{\varphi}h^{\varphi}}.
\end{equation*}

\noindent Therefore $\varphi$ is a homomorphism. By construction, $G^{\varphi}$ is
state-closed.

Let $K$ be the $\textbf{F}$-core of $\textbf{H}$ and $x\in K$. Then $%
x^{\sigma}=1$; in fact, $k^{x^{\sigma}}=l$ if and only if $%
H_it_{ik}x=H_it_{il}$ for some $i=1,...,s$. But $%
H_it_{ik}x=H_ix^{t_{ik}^{-1}}t_{ik}=Ht_{il}$, so $k=l$ and $x^{\sigma}=1$.
Since $K^{f_{i}}\leqslant K$, ${(\theta_i(x,t)^{f_i\varphi})}^\sigma$ is a trivial
permutation for all $i=1,...,s$ and $t \in T_i$. So $x \in \ker \varphi$.
Clearly $\ker \varphi\leqslant \cap _{i=1}^{s}H_{i}$, $\ker \varphi
\vartriangleleft G$ and $\ker \varphi ^{f_{i}}\leqslant \ker \varphi,\forall
i=1,...,s$.
\end{proof}

\begin{prop} \label{r}
The group $\mathbb{Z}\wr \mathbb{Z}$ is a $3$-state and $3$-letter automata.
\end{prop}

\begin{proof}
First we prove that $\mathbb{Z}\wr \mathbb{Z}$ has a faithful state-closed representation of degree 4 which we then reduced to degree 3.

By Brunner and Sidki \cite[Theorem 1]{BS1}, if $H$ is an abelian subgroup of
$\mathcal{A}_{2}$ and $\alpha =((e,e),(\alpha ,e))(0\,1)$, then $\langle
\tilde{H},\alpha \rangle \simeq \tilde{H}\wr \langle \alpha \rangle $, where
$\tilde{H}=\{\tilde{h}=((\tilde{h},h),(e,e))\mid h\in H\}.$ Note that $%
\tilde{H}\simeq H$. If $H=\langle \alpha \rangle $, then $\tilde{H}\wr
\langle \alpha \rangle =\langle \tilde{\alpha}\rangle \wr \langle \alpha
\rangle \simeq \mathbb{Z}\wr \mathbb{Z}$. Since
\begin{equation*}
\langle \tilde{\alpha}\rangle \wr \langle \alpha \rangle =\langle \tilde{%
\alpha}=((\tilde{\alpha},\alpha ),(e,e)),\alpha =((e,e),(\alpha
,e))(0\,1)\rangle ,
\end{equation*}%
it follows that $\langle \tilde{\alpha}\rangle \wr \langle \alpha \rangle $
is level two, state-closed group.

Thus, by $2$-inflation, $\Psi: G \rightarrow \mathcal{A}_4$
\begin{equation*}
(\langle \tilde{\alpha}\rangle \wr \langle \alpha \rangle )^{\Psi }=\langle
\tilde{\alpha}^{\Psi }=(\tilde{\alpha}^{\Psi },\alpha ^{\Psi },e,e),\alpha
^{\Psi }=(e,e,\alpha ^{\Psi },e)(0,2)(1,3)\rangle \simeq \mathbb{Z}\wr
\mathbb{Z}
\end{equation*}%
is a faithful state-closed representation of $\mathbb{Z}\wr \mathbb{Z}$ of
degree $4$.

Now note that the map
\begin{equation*}
\alpha ^{\Psi }\mapsto \alpha _{1}=(e,\alpha _{1},e)(0,1)
\end{equation*}%
\begin{equation*}
\tilde{\alpha}^{\Psi }\mapsto \beta =(\beta ,e,\alpha _{1})
\end{equation*}%
extends to an isomorphism $\phi $ from
\begin{equation*}
\langle \tilde{\alpha}^{\Psi }=(\tilde{\alpha}^{\Psi },\alpha ^{\Psi
},e,e),\alpha ^{\Psi }=(e,e,\alpha ^{\Psi },e)(0,2)(1,3)\rangle
\end{equation*}%
to
\begin{equation*}
\langle \beta =(\beta ,e,\alpha _{1}),\alpha _{1}=(e,\alpha
_{1},e)(0,1)\rangle .
\end{equation*}%
Therefore $\mathbb{Z}\wr \mathbb{Z}$ is a $3$-state and $3$-letter automata, as in Diagram 1.
\end{proof}

We prove in Section 5 a more general form of this proposition.

\section{Proof of Theorem B}

\begin{thmB}
\label{TB} Let $G$ be a self-similar group of degree $m$ and orbit-type $%
\left( m_{1},..,m_{s}\right) $. Then  the following hold.

\begin{itemize}
\item[1)] $G^{(\omega )}$ admits a faithful state-closed representation of
degree $m+1$ and orbit-type $\left( m_{1},..,m_{s},1\right)$; in particular,
for $G=\mathbb{Z}$, the representation of the group $\mathbb{Z}^{(\omega )}$
is of orbit-type $\left( 2,1\right),$ and is in addition finite-state.

\item[2)] Let $K$ be a regular subgroup of $Sym(\{1,...,s\})$. Then the
restricted wreath product $G\wr K$ admits a faithful
state-closed representation of degree $(m_1\cdot m_2\cdot...\cdot m_s) \cdot s$; in particular, the group $(\mathbb{Z} \wr \mathbb{Z})
\wr C_{2}$ is finite-state and self-similar of degree $4$.
\end{itemize}
\end{thmB}

\begin{proof}
Let $G$ be a state-closed group. By Proposition A, there is data $(\mathbf{m}%
,\mathbf{H,F})$ such that $\mathbf{H}$ is $\mathbf{F}$ core-free.

(1) The subgroup $L_{i} = \{(h, g_{2}, g_{3},...) \in G^{(\omega)}
\mid h \in H_{i} \}$ has index $m_{i}$ in $G^{(\omega)}$. For each $i = 1,
..., s$ define the homomorphism $\bar{f}_{i} : L_{i} \rightarrow
G^{(\omega)} $ by
\begin{equation*}
(h, g_{2}, g_{3},...)^{\bar{f}_{i}} = (h^{f_{i}}, g_{2}, g_{3},...)
\end{equation*}
and the homomorphism $\bar{f}_{s + 1}: L_{s+1} = G^{(\omega)} \rightarrow
G^{(\omega)}$ by
\begin{equation*}
(g_{1}, g_{2}, ...)^{\bar{f}_{s + 1}} = (g_{2}, g_{3}, ...).
\end{equation*}
It is clear that
\begin{equation*}
\langle L \leqslant \cap_{i = 1}^{s+1} L_{i} \mid L \vartriangleleft
G^{\omega}, L^{\bar{f}_{i}} \leqslant L, \forall i = 1, ..., s + 1 \rangle
\end{equation*}
is trivial. By Proposition A, the group $G^{(\omega)}$ is state-closed with
 respect to the data $(\mathbf{\bar{m},\bar{H},\bar{F}})$, where $\mathbf{\bar{m%
}} = (m_{1},...,m_{s},1)$, $\mathbf{\bar{H}=}\{L_1,...,L_s, L_{s+1}\}$ and $%
\mathbf{\bar{F}} = \{\bar{f}_{1},...,\bar{f}_{s},\bar{f}_{s+1}\}.$

Note that $f:2\mathbb{Z}%
\rightarrow \mathbb{Z}$  defined by $2n\mapsto n$ is a simple virtual endomorphism and therefore $\mathbb{Z}$ is self-similar of degree $2$.
Thus the group $\mathbb{Z}^{(\omega )}$ is self-similar  with respect to the data $((2, \, 1), \{L_1, L_2\}, \{f_1, f_2\})$ where $f_{1}:(2n_{1},n_{2},...)\mapsto (n_{1},n_{2},...)$ and $%
f_{2}:(n_{1},n_{2},...)\mapsto (n_{2},n_{3},...)$. On defining the transversals $T_{1}=\{e,(1,0,0,...)%
\}$ and $T_{2}=\{e\}$ of $L_{1}$ and $L_{2}$ respectively,  we obtain the following representation of $\mathbb{Z}^{(\omega)}$
\begin{equation*}
\langle \alpha _{1}=(e,\alpha _{1},e)(0\,1),\newline
\alpha _{i}=(\alpha _{i},\alpha _{i},\alpha _{i-1})\mid \newline
i=2,3,4,...\rangle
\end{equation*}%
which is faithful and finite-state.
\begin{center}
\begin{tikzpicture}[shorten >=3pt,node distance=2.5cm,on grid,auto]

  \node[state] (e) [] {$e$};
  \node[state] (a) [right of=e] {$\alpha_1$};
  \node[state] (b) [right of=a] {$\alpha_2$};
  \node[state] (c) [right of=b] {$\alpha_3$};
  \node (d) [right of=c] {$...$};

  \path [->] (e) edge  [loop above]    node {$0|0, 1|1, 2|2$} (e)
        (a) edge              node {$0|1,2|2$} (e)
        (b) edge              node {$2|2$} (a)
        (c) edge              node {$2|2$} (b)
        (d) edge              node {$2|2$} (c)
        (a) edge         [loop above]     node {$1|0$} (a)
        (b) edge      [loop above]        node {$0|0,1|1$} (b)
        (c) edge      [loop above]        node {$0|0,1|1$} (c);

\end{tikzpicture}

Diagram 2
\end{center}

(2) Let $l$ be an integer such that $H_{i} \neq G$ for $1 \leq i \leq l$ and
$H_{i} = G$ for $l + 1 \leq i \leq s$. Define $H = H_{1} \times ... \times
H_{s}$ and $W = G \wr K$. Then $[W : H] = s(m_{1} ... m_{l})$ and the
endomorphism $f: H \rightarrow W$ given by
\begin{equation*}
(h_{1}, ..., h_{s}) \mapsto (h_{1}^{f_{1}}, ..., h_{s}^{f_{s}})
\end{equation*}
is well-defined. Let $L$ be a subgroup of $H$, normal in $W$, and $f$%
-invariant, and let $g = (g_{1}, ..., g_{s}) \in L$. Since $K$ is a
transitive group of degree $s$ follows that for each $1 \leq i \neq j \leq s$
there exists $h \in K$ such that $(i)h = j$. But
\begin{equation*}
g^{h f} = \left(g_{(1)h}, ..., g_{(s)h}\right)^{f} = \left(g_{(1)h}^{f_{1}},
..., g_{(s)h}^{f_{s}}\right) \in L.
\end{equation*}
Thus $g_{r} \in \langle K \leqslant \cap_{i = 1}^{s} H_{i} \mid K
\vartriangleleft G, K^{f_{i}} \leqslant K, \forall i = 1, ..., s \rangle =
\{1\}$ for each $r = 1, ..., s$. Therefore $L = \{1\}$ and $G \wr K$ is self
-similar of degree $s.(m_{1} ... m_{l})$.

On applying Proposition \ref{r} and Proposition A we obtain:
\begin{equation*}
(\mathbb{Z}\wr \mathbb{Z})\wr C_{2}\simeq \langle \sigma
=(0,2)(1,3),\gamma =(\gamma ,e,\alpha ^{\sigma },\alpha ^{\sigma }),\alpha
=(e,\alpha ,e,e)(0,1)\rangle;
\end{equation*}%
that is, the group $(\mathbb{Z}\wr \mathbb{Z})\wr C_{2}$ is generated by the following $%
5 $-state and $4$-letter automaton:

\begin{center}
    \begin{tikzpicture}[shorten >=3pt,node distance=3.7cm,on grid,auto]

  \node[state] (e) [] {$e$};
  \node[state] (a) [below left of=e] {$\alpha$};
  \node[state] (g) [below right of=e] {$\gamma$};
  \node[state] (p) [below right of=a] {$\alpha^{\sigma}$};
  \node[state] (s) [left of=e] {$\sigma$};

  \path [->] (e) edge        [loop above]       node {$0|0, 1|1, 2|2, 3|3$} (e)
        (s) edge              node {$0|2,1|3,2|0,3|1$} (e)
        (a) edge              node {$0|1,2|2,3|3$} (e)
        (a) edge        [loop left]       node {$1|0$} (a)
        (g) edge          [loop right]     node {$0|0$} (g)
        (g) edge              node [swap]  {$2|2,3|3$} (e)
        (p) edge              node {$3|2$} (a)
        (g) edge              node  {$1|1$} (p)
        (p) edge              node [swap] {$0|0, 1|1, 2|3$} (e);

\end{tikzpicture}

Diagram 3
\end{center}

\end{proof}

\section{Proof of Theorem C}

First we will prove the second case of Theorem C.

\begin{prop}
\label{P3.1} Let $G=\mathbb{Z}^{l}\wr \mathbb{Z}^{d}$. Then $G$ is an
automata group of degree $4$. In case $d=1$, the degree is $3$.
\end{prop}

\begin{proof}
Denote $\mathbb{Z}^{l}$ by $A$ and $\mathbb{Z}^{d}$ by $X$. \ Then, the normal closure of $A$ in $G$ is $A^X$ and  we have
a semi-direct product form for the group $G=A^{X}\mathbb{\cdot }X$ .

Define the subgroups%
\begin{eqnarray*}
H_{1} &=&A^{X}\langle
x_{1}^{2},x_{2},...,x_{d}\rangle , \\
H_{2} &=&H_{3}=G,\text{ }\bigcap_{i=1}^{3}H_{i}=H_{1}\text{.}
\end{eqnarray*}%
Also, for $i=1,2,3$, define the homomorphisms $f_{i}:H_{i}\rightarrow G$
which extend the maps:%
\begin{eqnarray*}
f_{1} &:&\,x_{1}^{2}\mapsto x_{1},\,x_{i}\mapsto x_{i}\text{ }\left( 2\leq
i\leq d\right) , \\
a_{i}^{x_{1}^{2n}q\left( x_{2},x_{3}...,x_{d}\right) } &\mapsto
&a_{i-1}^{x_{1}^{n}q\left( x_{2},x_{3}...,x_{d}\right) },\text{ }%
a_{i}^{x_{1}^{2n+1}q\left( x_{2},x_{3}...,x_{d}\right) }\mapsto e\text{ };
\end{eqnarray*}

\begin{eqnarray*}
f_{2} &:&\,x_{i}\mapsto x_{i-1}\text{ }\left( 1\leq i\leq d\right) , \\
a_{i}^{p\left( x_{1},x_{2}...,x_{d}\right) } &\mapsto &a_{i}^{p\left(
x_{d},x_{1}...,x_{d-1}\right) }\text{ }\left( 1\leq i\leq l\right) ;
\end{eqnarray*}

and%
\begin{eqnarray*}
f_{3} &:&X\mapsto \left\{ e\right\} \text{ }, \\
a_{1}^{p\left( x_{1},x_{2}...,x_{d}\right) } &\mapsto &x_{1}^{p\left(
1,1...,1\right) },\text{ } \\
a_{i}^{p\left( x_{1},x_{2}...,x_{d}\right) } &\mapsto &e\text{ }\left( 2\leq
i\leq l\right)
\end{eqnarray*}

Then, $A^{ X }\langle
x_{2},...,x_{d}\rangle $ is the $f_{1}$-core of $H_{1}$ and both $H_{2}$, $%
H_{3}$ are their own $f$-cores.

Let $K$ be the $\mathbf{F}$\textbf{-}core of $\mathbf{H}$. Then $K\leq A^{ X }\langle x_{2},...,x_{d}\rangle $ and
by applying $f_{2}$ we find $K\leq A^{ X} $.

Suppose $K$ is non-trivial. Then, as $K$ is normal in $G$ it follows that $%
K_{+}=K\cap A^{\mathbb{Z}\left[ X\right] }$ is non-trivial. Elements $h$ of $%
K_{+}$ have an unique form
\begin{equation*}
h=a_{1}^{p_{1}}a_{2}^{p_{2}}...a_{l}^{p_{l}}
\end{equation*}%
where $p_{i}=p_{i}(x_{1},x_{2}...,x_{d})$ $\in \mathbb{Z}\left[ X\right] $.
Define $\delta _{x_{j}}(h) $ to be the maximum $x_{j}$-degree of $p_{i}$ for
all $i$. If the maximum of all $\delta _{x_{j}}(h)$ occurs for $j=k$ then by
applying an adequate power of $f_{2}$ to $h$, we may assume $j=1$.

Choose an $h\in K_{+}$ such that $h\not=e$ and which involves a minimum
number of variables from $\left\{ x_{1},x_{2}...,x_{d}\right\}$. \newline

\noindent (1) Suppose $p_{i}$ is constant for all $i$. There exists $j$ such
that $p_{j}\not=0$. Then on applying $f_{3}$ to $h$ we get $%
x_{1}^{p_{1}}\not=e\in K$ which is impossible. \newline

\noindent (2) Write $\delta _{x_{1}}\left( h\right) =n$ then $n\not=0$.
\newline

\noindent (2.1) Suppose $n=2k$. Then on applying $f_{1}$ to $h$ we obtain $%
h^{\prime }\not=e$ and $\delta _{x_{1}}\left( h^{\prime }\right) =k$ which
is absurd. \newline

\noindent (2.2) Suppose $\delta _{x_{1}}\left( h\right) =2k+1$. Then
conjugate by $x_{1}$ to get%
\begin{eqnarray*}
h^{\prime } &=&h^{x_{1}}=a_{1}^{p_{1}^{\prime }}a_{2}^{p_{2}^{\prime
}}...a_{l}^{p_{l}^{\prime }}\in K_{+}, \\
p_{i}^{\prime } &=&p_{i}x_{1},\text{ }\delta _{x_{1}}\left( h^{\prime
}\right) =2k+2\text{.}
\end{eqnarray*}%
On applying $f_{1}$ to $h^{\prime }$ we get $h^{\prime \prime }\in K_{+}$
with $\delta _{x_{1}}\left( h^{\prime \prime }\right) =k+1$. Now, $k+1<2k+1$%
, unless $k=0$; that is, we have $\delta _{x_{1}}\left( h\right) =1$ and for
all $i$,
\begin{equation*}
p_{i}=p_{i0}+p_{i1}x_{1}
\end{equation*}%
where $p_{i0},p_{i1}\in $ $\mathbb{Z}\left[ x_{2}...,x_{d}\right] $. Then,
as before, for $h^{\prime }=h^{x_{1}}$ we have $p_{i}^{\prime
}=p_{i0}x_{1}+p_{i1}x_{1}^{2}$ and $h^{\prime \prime }=\left(
a_{1}^{p_{11}}a_{2}^{p_{21}}...a_{l}^{p_{l1}}\right) ^{x_{1}}$ $\in
K_{+}\backslash \left\{ e\right\} $. Since
\begin{equation*}
h^{\prime \prime \prime }=\left( h^{\prime \prime }\right)
^{x_{1}^{-1}}=a_{1}^{p_{11}}a_{2}^{p_{21}}...a_{l}^{p_{l1}}\in
K_{+}\backslash \left\{ e\right\} \text{ }
\end{equation*}%
which involves a lesser number of variables than $h$, we have a
contradiction.

With notation of Proposition \ref{P3.1}, a faithful self-similar
representation of $G=\mathbb{Z}^{l}\wr \mathbb{Z}^{d}$ with respect the data%
\begin{equation*}
((2,1,1),\{H_{1},\text{ }H_{2} = G, H_{3}=G\},\{f_{1},f_{2},f_{3}\})
\end{equation*}%
and the transversals $T_{i}$ of $H_{i}$ in $G$ defined by
\begin{equation*}
T_{1}=\{e,x_{1}\},T_{2}=T_{3}=\{e\}\text{,}
\end{equation*}%
is
\begin{equation*}
G^{\varphi }=\langle \gamma _{1},...,\gamma _{l}\rangle \wr \langle \alpha
_{1},...,\alpha _{d}\rangle
\end{equation*}%
where
\begin{equation*}
\gamma _{1}=(\gamma _{l},e,\gamma _{1},\alpha _{1}),\,\gamma _{2}=(\gamma
_{1},e,\gamma _{2},e),...,\gamma _{l}=(\gamma _{l-1},e,\gamma _{l},e),
\end{equation*}%
\begin{equation*}
\alpha _{1}=(e,\alpha _{1},\alpha _{d},e)(0\,1),\,\alpha _{2}=(\alpha
_{2},\alpha _{2},\alpha _{1},e),...,\alpha _{d}=(\alpha _{d},\alpha
_{d},\alpha _{d-1},e).
\end{equation*}%
Therefore, $G^{\varphi }$ is finitely generated, finite-state and
self-similar; that is, $G~$is an automata group. We note that if $d=1$ then $%
H_{2}$ and $f_{2}$ are superfluous.
\end{proof}

\subsection{\noindent \textbf{General concatenation}. }

Let $G_{1}=A_{1}\wr U$, $G_{2}=A_{2}\wr U$ and $G=\left( A_{1}\oplus
A_{2}\right) \wr U$. For $i=1,2$, define the $G_{i}$-data $\left( \mathbf{%
m_{i},H_{i},F_{i}}\right) $ , where $\mathbf{m_{i}=}\left(
m_{i1},...,m_{is_{i}}\right) $, $\mathbf{H_{i}}=\{H_{i1},...,H_{is_{i}}\}$
and $\mathbf{F_{i}}=\{f_{i1},...,f_{is_{i}}\}$. Furthermore, define the data
$G$-data $\left( \mathbf{m,H,F}\right) $ , where $\mathbf{m}$ is the
concatenation $\left( \mathbf{m_{1},m_{2}}\right) $,
\begin{eqnarray*}
\mathbf{H} &\mathbf{=}&\left\{ \tilde{H}_{1j}=\left( {A_{2}}^{U}\right)
\cdot H_{1j}\text{ }\left( 1\leq j\leq s_{1}\right) \right\}  \\
&&\cup \left\{ \tilde{H}_{2k}=\left( {A_{1}}^{U}\right) \cdot H_{2k}\text{ }%
\left( 1\leq k\leq s_{2}\right) \right\} \text{,}
\end{eqnarray*}%
\begin{equation*}
\mathbf{F=}\{\tilde{f}_{11},...,\tilde{f}_{1s_{1}},\tilde{f}_{21},...,\tilde{%
f}_{2s_{2}}\}
\end{equation*}%
where $\tilde{f}_{1j}:\tilde{H}_{1j}\rightarrow G, \,\, 1\leq j \leq s_1,$ is defined by
\begin{equation*}
\tilde{f}_{1j}:ah\mapsto h^{f_{1j}}\text{, for }a\in {A_{2}}^{U},h\in H_{1j}
\end{equation*}%
and $\tilde{f}_{2k}:\tilde{H}_{2k}\rightarrow G, \,\, 1\leq k \leq s_2,$ is defined by
\begin{equation*}
\tilde{f}_{2k}:ah\mapsto h^{f_{2k}}\text{, for }a\in {A_{1}}^{U}\text{, }%
h\in H_{2k}\text{.}
\end{equation*}

For $i=1,2$, let $G_{i}$ has its state-closed representation with respect
to $\left( \mathbf{m_{i},H_{i},F_{i}}\right) $ with  $\mathbf{F_{i}}$ core $%
K_{i}$. Likewise, let $G$ has its state-closed representation with respect
to $\left( \mathbf{m,H,F}\right) $ with $\mathbf{F}$-core $K$.

\begin{prop}
\label{P3.2} Maintaining the above notation:

\begin{enumerate}
\item
\begin{equation*}
K\cap \left( A_{1}\oplus A_{2}\right) ^{U}=K_{1}\left( A_{2}\right) ^{U}\cap
K_{2}\left( A_{1}\right) ^{U}\text{;}
\end{equation*}

\item Suppose the above state-closed representations of $G_{1}$ and $G_{2}$
are faithful. Then so is the corresponding state-closed representation of $G$;

\item If $G_{1}$ and $G_{2}$ are finite-state then $G$ is also finite-state.
\end{enumerate}
\end{prop}

\begin{proof}
Define for $i=1,2$,%
\begin{equation*}
R_{i}=\langle S\leqslant \cap _{j=1}^{s_{i}}H_{ij}\mid S\vartriangleleft G,%
\text{ }S^{\widetilde{f}_{ij}}\leqslant S,\forall j=1,...,s_{i}\rangle ,
\end{equation*}

\begin{enumerate}
\item We have
\begin{eqnarray*}
K &=&R_{1}\cap R_{2}, \\
R_{1}\cap \left( A_{1}\oplus A_{2}\right) ^{U} &=&K_{1}\left( A_{2}\right)
^{U}, \\
R_{2}\cap \left( A_{1}\oplus A_{2}\right) ^{U} &=&K_{2}\left( A_{1}\right)
^{U}, \\
K\cap \left( A_{1}\oplus A_{2}\right) ^{U} &=&K_{1}\left( A_{2}\right)
^{U}\cap K_{2}\left( A_{1}\right) ^{U}\text{.}
\end{eqnarray*}

\item As $K_{1}=K_{2}=\left\{ e\right\} $, we find%
\begin{equation*}
K\cap \left( A_{1}\oplus A_{2}\right) ^{U}=\left( A_{2}\right) ^{U}\cap
\left( A_{1}\right) ^{U}=\left\{ e\right\} \text{.}
\end{equation*}%
Now since both $K$ and $\left( A_{1}\oplus A_{2}\right) ^{U}$ are normal
subgroups of $G$, it follows that $\,K$ centralizes $\left( A_{1}\oplus
A_{2}\right) ^{U}$. However $\left( A_{1}\oplus A_{2}\right) ^{U}$ contains
its own centralizer in $G$. Thus, $K=K\cap \left( A_{1}\oplus A_{2}\right)
^{U}$ which is trivial by (1). \\

\item There exist transversals of $H_{11}, ..., H_{1s_{1}}$ in $G_{1}$ and of $H_{21}, ...,$ $H_{2s_{2}}$ in $G_{2}$ such that $G_{1}$ and $G_{2}$ are finite-state. These transversals also induce a finite-state representation of $G$.
\end{enumerate}
\end{proof}

Now Theorem C follows directly from the above proposition
and Proposition \ref{P3.1}. \\

\noindent \textit{\textbf{Question 1}. Is there a faithful state-closed
representation of degree }$3$\textit{\ for the group }$\mathbb{Z}^{l}\wr
\mathbb{Z}^{d}$\textit{\ when }$l, d\geq 2$\textit{?}

\section{Proof of Theorem D}

We start with the following observation. Let $k$ be a field and $X = \langle x, y \rangle \simeq \mathbb{Z}^{2}$. Consider the following equivalence relation on the $k$-algebra $k[X]$ defined by:
\begin{equation*}
p(x,y)\sim q(x,y)\text{ iff }q(x,y)= u p(x,y),\text{ where }u\text{ is a unit of }k[X].
\end{equation*}%
 Let $p'(x,y)= x^s.y^t.p(x,y)$, where $s,t \geq 0$ minimal such that $p'(x,y)$ in $k[x,y]$. Let $m, n$ be respectively the $x$-degree and $y$-degree of $p'(x,y)$ and define $\delta(p(x,y)) = (m, n)$. Then, for $p(x,y), q(x,y)$ non-invertible elements of $k[X]$, $\delta(p(x,y)q(x,y)) \geq \delta( p(x,y)), \delta(q(x,y))$.  Let $p_i(x,y)$ be a sequence of non-invertible elements of $k[X]$ where $i \geq 0$ and let $\delta( p_i(x,y)) = (m_i, n_i)$. Suppose $m_i, n_i \rightarrow \infty$ as $i \rightarrow \infty$. Then, the ideal $\bigcap_{i=0}^{\infty }\langle
p_{i}(x,y)\rangle $ is null.

\begin{thmD}
\label{TC copy(1)} Let $p$ a prime number then $C_{p}\wr \mathbb{Z}^{2}$ is
a self-similar group of degree $p+1$ of orbit-type $%
\left( p,1\right)$. Indeed, $C_{p}\wr \mathbb{Z}^{2}$ is generated by $\alpha = (\alpha, \alpha \sigma, ..., \alpha \sigma^{p-1}, \alpha \beta)$, $\sigma = (e, ..., e, \sigma)(0 \, 1 \, ... p-1)$ and $\beta = (e, ..., e, \alpha)$. In particular, the group $C_{2}\wr \mathbb{Z}^{2}$ is
self-similar of degree $3$.
\end{thmD}

\begin{proof}
Let $C_{p} = \langle a \rangle$, $\mathbb{Z}^{2} = \langle x, y \rangle$ and
$G = C_{p} \wr \mathbb{Z}^{2}$. Define the subgroups $H_{1} = G^{\prime
}\langle x, y \rangle$ and $H_{2} = G$. Note that $[G : H_{1}] = p$.
Elements of  $H_{1}$  have the unique form $a^{s(x,y)}\cdot x^i y^j$ where $s(x,y)$ is an element of the ideal $\mathcal{I}$ of $\mathbb{Z} \langle x, y \rangle$, generated by $x-1$ and $y-1$.
Also, elements of $\mathcal{I}$ have the unique form
$$s(x,y)=p(x)(x-1)+q(y)(y-1)+r(x,y)(x-1)(y-1).$$
Define $f_{1} : H_{1} \rightarrow G$  by  $a^{s(x,y)} x^i y^j \mapsto a^{q(y)} y^j.$
Furthermore define $f_{2} : H_{2} \rightarrow G$ by  $a^{r(x,y)}x^i y^j \mapsto a^{r(y,xy)}x^j y^{i+j}.$
It can be checked directly that $f_{1}$ and $f_{2}$ are homomorphisms.

Suppose by contradiction that $K$ is a non-trivial subgroup of  $H_{1} \cap H_{2} = H_{1}$%
, normal in $G$ and $\{f_{1}, f_{2}\}$-invariant. Note that the subgroup $L
= K \cap G^{\prime }$ is trivial if and only if $K$ is trivial. Let $g = a^{s(x, y)}$ be a nontrivial element in $L$. So $a^{s(x, y)f_{1}} = a^{q(y)}$ and successive applications of $f_{1}$ in $a^{q(y)}$ results $q(y) = 0$, thus $x - 1$ divides $s(x, y)$.

Since $a^{s(x,y)f_{2}} = a^{s(y, xy)} \in L$, it follows that $x - 1$ also
divides $s(y, xy)$, this is, $s(y, xy) = (x - 1)t(x, y)$. Then
\begin{equation*}
\begin{split}
a^{s(x, y)} &= a^{(s(y, xy))f_{2}^{-1}} \\
&= a^{((x-1)t_{1}(x, y))f_{2}^{-1}} \\
&= a^{(x^{-1}y-1)t_{2}(x^{-1}y, x)} \\
&= a^{x^{-1}(y - x)t_{2}(x^{-1}y, x)} \\
&= a^{(x - y)t_{3}(x, y)}
\end{split}
\end{equation*}
and $x - y$ divides $s(x, y)$. Iterating this argument, we find that $x^{n_{i}} - y^{n_{i - 1}}$ divides $s(x, y)$ for all $n_{i}$, where $n_i$ is the Fibonacci sequence defined by $n_{i} = n_{i - 1} + n_{i -2}$ and $n_{0} =
0, n_{1} = 1, n_{2} = 1$, $i \geq 0$. Hence $s(x, y) \in \bigcap_{i =
0}^{\infty} \langle x^{n_{i}} - y^{n_{i - 1}} \rangle = \{0\}$; a contradiction. Therefore $G$ is state-closed of degree $p + 1$ and
orbit-type $(p, 1)$.

Choose the transversals $T_{1}=\{e, a, ..., a^{p-1} \}$ and $T_{2}=\{e\}$ of $H_{1}$  and $H_{2}$, respectively. So we have a
state-closed representation of $G$  generated by the automorphisms
\begin{equation*}
\sigma =(e,e, ..., e, \sigma )(0\,1),\alpha =(\alpha , \alpha
\sigma, ..., \alpha \sigma^{p-1}, \alpha \beta ),\beta =(e, e, ..., e, \alpha ).
\end{equation*}%
Note that $\alpha^{m} \beta^{n} = (\alpha^{m}, (\alpha\sigma)^{m}, ..., (\alpha \sigma^{p-1})^{m}, \alpha^{m + n} \beta^{m})$, for $m, n \in \mathbb{Z}$, hence $\{\alpha, \alpha \beta, \alpha^{2} \beta, \alpha^{3} \beta^{2}, ..., \alpha^{n_{i}} \beta^{n_{i - 1}}, ...\} \subset Q(\alpha)$ and this representation is not finite-state.
\end{proof}

\noindent \textit{\textbf{Question 2}. Is there a faithful finite-state and
state-closed representation for the group }$C_{2}\wr \mathbb{Z}^{2}$\textit{%
\ of degree }$3$\textit{?} \newline

\section{Proof of Theorem E}

\noindent Let $G$ be a state-closed group with respect the data $(\mathbf{m}, \mathbf{H%
}, \mathbf{F})$, where $\mathbf{m} = (m_{1}, ...,
m_{s})$, with $m_{1} \geq 2, ..., m_{s} \geq 2$.
For each $i=1,...,s$ define $G_{i0}=G$, $G_{ij}= (G_{i (j - 1)})^{f_i^{-1}}$ ($j>0$) and $G_{\omega_i}= \cap_{j\geq 0}G_{ij}.$ With this notation we have:

\begin{thmE}
Let $B$ be a finite abelian group and $G$ be a self-similar group of
orbit-type $(m_{1}, ...,
m_{s})$, with $m_{1} \geq 2, ..., m_{s} \geq 2$. Then the group $$B^{((G_{\omega_1}\setminus G) \times ... \times (G_{\omega_s}\setminus G))}\rtimes G^s$$
is self-similar of orbit-type $(|B|.m_{1} ... m_{s}, \, 1)$.
\end{thmE}

\begin{proof}
Let $\mathcal{G}$ be the group $B^{((G_{\omega_1}\setminus G) \times ... \times (G_{\omega_s} \setminus G))}\rtimes G^s$ and let $H$ be the subgroup $\prod_{i = 1}^{s} H_{i}$ of $G^s$. Define the set of s-tuples of cosets
$$U=\prod _{i=1}^{s}(G_{\omega_i}\setminus G) $$
and define the group extension
\begin{equation*}
\mathcal{H}=\left\{ \phi : U  \rightarrow B\text{ finitely
supported, }\prod_{\bar{g} \in U}\phi (\bar{g})=1\right\} \rtimes H.
\end{equation*}%

Note that the index $[\mathcal{G}:\mathcal{H}]$ is $|B|\cdot m_1...m_s$. Now consider the map  defined on the s-tuples of cosets.
\begin{eqnarray*}
\lambda&:&  \prod_{i=1}^{s}(G_{\omega_i}\setminus H_{{i}}) \rightarrow \prod_{i=1}^{s}(G_{\omega_i}\setminus G)
\end{eqnarray*}%

\begin{equation*}
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (G_{\omega_1}h_1,...,G_{\omega_s}h_s)\mapsto  (G_{\omega_1}h_1^{f_{1}},...,G_{\omega_s}h_s^{f_{s}}).
\end{equation*}%
If $(G_{\omega_i}h_i)^{\lambda}=(G_{\omega_i}h_i')^{\lambda}$, then $(h_i(h_i')^{-1})^{f_{i}}\in  G_{\omega_{i}}$ for each $1\leq i \leq s$; therefore $\lambda$ is injective.

 Define
\begin{eqnarray*}
\lambda'&:&  \prod_{i=1}^{s}(G_{\omega_i}\setminus G) \rightarrow \prod_{i=1}^{s}(G_{\omega_i}\setminus G)
\end{eqnarray*}%
by

$$ \bar{y}=(G_{\omega_1}y_1,...,G_{\omega_s}y_s)\mapsto  \bar{x}=(G_{\omega_1}x_1,...,G_{\omega_s}x_s),$$ provided $\lambda(\bar{x})= \bar{y};$ otherwise, define it as $\bar{e}=(G_{\omega_1},...,G_{\omega_s}).$

Furthermore, define the map
 $\chi_{1}: \mathcal{H} \rightarrow \mathcal{G}$
 \begin{equation*}
(\phi ,(h_{i})_{i = 1}^{s})\mapsto  \left(\bar{x} = (G_{\omega_1}x_{1}, ..., G_{\omega_s}x_{s})\mapsto \phi (\lambda'(\bar{x})) ,(h_{i}^{f_{i}})_{i = 1}^{s}\right).
\end{equation*}
Then it is direct to show that $\chi_{1}$ is a homomorphism.

Further, let $\chi_{2}: \mathcal{G} \rightarrow \mathcal{G}$ be
the homomorphism defined by
$$(\phi, (g_{1}, ..., g_{s})) \mapsto ((G_{\omega_i}x_{i})_{i = 1}^{s} \mapsto \phi((G_{\omega_{i}}x_{i+1})_{i = 1}^{s}), (g_{i+1})_{i = 1}^{s})).$$

We claim that the state-closed representation of $\mathcal{G}$ defined by the data $((|B|\prod_{i = 1}^{s} [G : \cap_{i = 1}^{s} H_{i}], 1), \{\mathcal{H}, \mathcal{G}\}, \{\chi_{1}, \chi_{2}\})$ is faithful.

Consider $K \leq \mathcal{H}$ with $K \vartriangleleft \mathcal{G}$ and $K^{\chi_{r}} \leq K$, $r = 1, 2$. Firstly, the assumption that ${\bf F}$ is core-free and the definition of $\chi_{2}$ imply that $K \leq B^{(R \setminus G)^{s}}$. Let $\phi
:U \rightarrow B$ be a non-trivial element in $K$, then $\prod_{\bar{g} \in U}\phi (\bar{g})=1$ and so $|Supp\,(\phi
)| \, \geq 2$.
Choose $\phi $ with support $S$, of minimal cardinality. Since $%
\phi $ is non-trivial we can assume by $\mathcal{G}$-conjugation that $\phi (G_{\omega_1} , ...,G_{\omega_s})\neq 1$, and so $(G_{\omega_1} , ...,G_{\omega_s}) \in S $ and $S$ contains at least two elements.

Define $\phi_{j} = \phi^{\chi_{1}^{j}}$ for $j \geq 0$; it is given by $\phi_{j}(G_{\omega_1}a_1^{f_1}, ..., G_{\omega_s}a_s^{f_s}) = \phi_{(j - 1)}(G_{\omega_1}a_1, ..., G_{\omega_1}a_s)$ for all $(a_1,...,a_s) \in G_{1j}\times ...\times G_{sj}$, extended by the identity away from $( G_{1j}\times ...\times G_{sj})^{\lambda}$.
So
$$\phi_1:G_{\omega_1} \setminus H_1^{f_1} \times ... \times G_{\omega_s} \setminus H_s^{f_s} \to B$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \, (G_{\omega_1}a_1^{f_1}, ..., G_{\omega_s}a_s^{f_s})\mapsto \phi(G_{\omega_1}a_1, ..., G_{\omega_1}a_s) $$
and $$\,\,\,Supp(\phi_1) \,\,\,= \{(G_{\omega_1}a_1^{f_1}, ..., G_{\omega_s}a_s^{f_s}) \mid (G_{\omega_1}a_1, ..., G_{\omega_s}a_s) \in S  \}$$ $$=G_{\omega_1}(S^{\pi_1}\cap H_1)^{f_1}\times ... \times G_{\omega_s}(S^{\pi_s}\cap H_s)^{f_s},$$
where $\pi_i$, $i=1,...,s$, is the projection on the $i$-th coordinate.

If
$(G_{\omega_1}a_1^{f_1}, ..., G_{\omega_s}a_s^{f_s})\in Supp (\phi_1)$ then $(G_{\omega_1}a_1, ..., G_{\omega_s}a_s) \in S$ and so $|Supp (\phi_1)|\leq |S|$. By minimality of the cardinality of $S$ we have that $|Supp (\phi_1)|= |S|$.
Continuing in this manner, the support of $\phi_{j}$ is
$$G_{\omega_1}(S^{\pi_1}\cap G_{1j})^{f_1}\times ... \times G_{\omega_s}(S^{\pi_s}\cap G_{sj})^{f_s},$$
which has the same cardinality as $S$. Therefore, we have $S \subset (G_{1j} , ...,G_{sj})$ for all $j$ and thus $S \subset \{(G_{\omega_1} , ...,G_{\omega_s})\}$; a contradiction.

\end{proof}

\begin{cor}
Let B be a finite abelian group and $G$ be a self-similar
group with orbit-type $(m_{1} > 1, ..., m_{l} > 1, m_{l + 1} = 1, ..., m_{s} = 1)$. Then
there exist proper subgroups $R_{l+1}, ..., R_{s}$ of $G$ such that
$$B^{((G_{\omega_{1}} \setminus G) \times ... \times (G_{\omega_{l}} \setminus G) \times (R_{l + 1} \setminus G) \times ... \times (R_{s} \setminus G))}\rtimes G^{s}$$
is a self-similar group of orbit-type $(|B|.m_{1} ... m_{l}^{s-l+1}, \, 1)$.
\end{cor}

\begin{proof}
For each $l+1 \leq j \leq s$ the restriction $\dot{f}_{j} = f_{j} : \dot{H}_{j} = H_{l} \rightarrow G$ is well-defined. Define the triple $(\dot{\mathbf{m}}, \dot{\mathbf{H
}}, \dot{\mathbf{F}})$ by
$$\dot{\mathbf{m}} = (m_{1}, ..., m_{l}, m_{l}, ..., m_{l}), \dot{\mathbf{H}} = \{H_{1}, ..., H_{l}, \dot{H}_{l + 1} = H_{l}, ..., \dot{H}_{s} = H_{l}\},$$
$$\dot{\mathbf{F}} = \{f_{1}, ..., f_{l}, \dot{f}_{l+1}, ...,  \dot{f}_{s}\}.$$
and $R_{j}$ the parabolic subgroup of $\dot{f}_{j}$ for $j = l+1, ..., s$. Since $G$ has a faithful state-closed representation with respect to the data $(\mathbf{m}, \mathbf{H%
}, \mathbf{F})$, the same holds for the representation with respect to the data $(\dot{\mathbf{m}}, \dot{\mathbf{H
}}, \dot{\mathbf{F}})$; in fact the ${\bf F}$-core of ${\bf H}$ and the $\dot{\bf F}$-core of $\dot{\bf H}$ are the same. Since $m_{i} > 1$ for each $m_{i}$ in $\dot{{\bf m}}$, we apply Theorem E to obtain the result.
\end{proof}

redpajama_set_name: RedPajamaArXiv


(Rick Egan | The Salt Lake Tribune) Milkbar on 1409 S. 900 East in Salt Lake City.
An Australian-style milk bar will be opening later this year in Salt Lake City.
A milk bar — in case you're wondering — is Australia's name for a corner store, where neighbors can get hot coffee, a quick bite to eat and maybe some candy or ice cream.
The Salt Lake City version will be called Milk Bar by Five5eeds and it will take shape this summer in the brick building that once was home to Charley Hafen Jewelry at 1409 S. 900 East.
The owners are waiting for permits from the city to start the remodel, said Five5eeds general manager Gabriel Chaparro. The tentative opening is August.
Milk Bar will be the second Australian-themed restaurant from owners Andrew and Tiffany Percy. In 2015, the couple opened Five5eeds Cafe in Park City. That full-service restaurant replicated the breakfast cafes — or brekkies — the couple enjoyed while growing up in Australia.
It's named for the Percys' five children, who attend Rowland Hall — so the couple are in Salt Lake City a lot, Chaparro said.
Milk Bar will be a quick-service restaurant and will have some favorites from its Park City sibling, Chaparro said. Look for the toast with smashed avocado and a poached egg; seasonal eggs Benedict; and the signature shakshouska — Moroccan baked eggs topped with goat curd and aromatic spices.
Milk Bar also will have imported Australian coffee, lattes, smoothies, grab-and-go sandwiches and milkshakes.


For EV drivers seeking immediate assistance at a charging station on the FLO public network.
For any questions regarding installation and configuration of charging stations, your FLO account or any technical problem.
Find answers to some of the most frequently asked questions about FLO Home and the FLO network.


text:
2019 Organizational Prospect Rankings Top 10
VANCOUVER, BRITISH COLUMBIA - JUNE 21: (L-R) Kirby Dach, third overall pick by the Chicago Blackhawks , Jack Hughes, first overall pick by the New Jersey Devils, and Kaapo Kakko, second overall pick by the New York Rangers hold up their fingers of their pick order in front of the stage during the first round of the 2019 NHL Draft at Rogers Arena on June 21, 2019 in Vancouver, Canada. (Photo by Rich Lam/Getty Images)
Welcome to the 2019 Top Shelf Prospects series. As we go through the Summer of 2019 LWOH will be featuring a team-by-team look at the top prospects in the NHL. We will be following the order of the first round of the NHL draft (as if there were no traded draft picks) and you can find all the articles here. Since we had an extensive NHL Draft preview, we will not be reviewing the players who were drafted in the 2019 draft, as there have been no games since then, and our reports on them will not have changed. Today, we look at the 2019 Organizational Prospect Rankings.
What we will be doing is linking you to those articles, as well as taking a look at prospects that were acquired before this year's draft; their progress, and their chances of making the 2019-20 roster of the NHL team in question. We will also bring you one sleeper pick – a player who was either drafted in the 4th-round or later, or was an undrafted free agent signing who we pick as our dark horse to make the NHL. For those wondering, the cut-off for what is or isn't a prospect is typically about 50 NHL games played (including playoff games) or is 25 years old. These are not hard or fast rules though, and we may make some exceptions depending on the circumstances.
2019 Organizational Prospect Rankings (20-11)
After going through the top 10 prospects of every team in the NHL and giving full scouting reports, we will now be ranking each team's prospect pools in our 2019 Organizational Prospect Rankings.
Note: If you click on the team name, you will get a link to a full prospect report on that team. The snippets here are just a taste of the information available.
For Teams 31-21 you can click here.
10.) Carolina Hurricanes
The Hurricanes have a stud centre prospect in Martin Necas. He the star of what is a strong group up the middle that also includes Ryan Suzuki, Morgan Geekie, and Jamieson Rees. While the Hurricanes only have one elite prospect, what they have is enviable depth of very good prospects, beyond even the top 10 we reviewed. Defenceman Jake Bean is NHL ready but seems to be stuck behind the Hurricanes strong defensive group. The team also has Antonni Honka, Chase Priskie, Roland McKeown, Domenick Fensore, and Luke Martin. Wingers include Julien Gauthier, Janne Kuokkanen, Stelio Mattheos, and Patrik Puistola. Goaltender Alex Nedeljkovic will soon push for a roster spot, while newly drafted Pyotr Kochetkov was one of this draft's top goalie prospects.
9.) Anaheim Ducks
With Patrick Eaves and Ryan Kesler out for the season, the time is now for the Ducks young forwards to carry some of the burdens in the NHL. Luckily the Ducks have a number of young forwards ready to do so in Sam Steel, Troy Terry, and Maxime Comtois. First-round pick Trevor Zegras and 2018 draftee Isac Lundestrom are not far behind. There is further depth in Brendan Tracey, Max Jones, Antoine Morand and Jack Kopacka. Brendan Guhle is close to being ready on the blueline but his upside is a question mark The most talented Ducks defenders aren't quite as close to being in the NHL but Josh Mahura and Henry Thrun are talented. Lukas Dostal is a quality goaltending prospect.
8.) New Jersey Devils
First overall pick Jack Hughes could become a franchise-changing player. He will be supported up front by Jesper Boqvist, Joey Anderson, Michael McLeod, and Aarne Talvitie. On defence, Ty Smith could become a top-notch puck mover. He has a ton of offensive skill. Reilly Walsh and newly drafted Daniil Misyul provide blueline depth. Goaltender MacKenzie Blackwood is ready to compete for the Devils top job.
7.) Chicago Blackhawks
The Blackhawks are extremely deep and talented on defence with Adam Boqvist, Nicolas Beaudin, Ian Mitchell, Alex Vlasic, and Lucas Carlsson. They drafted a high-end centre prospect in Kirby Dach. Fellow centre Evan Barrett improved his status with a big season at Penn State. Dylan Sikura and Niklas Nordgren are quality wingers. Alex Nylander was acquired from the Sabres. He is extremely talented but has not yet put it together at the professional level. The Hawks hope he benefits from the change of scenery, ala Dylan Strome last season. In goal, Alexis Gravel is the top prospect in the system.
6.) Los Angeles Kings
The Kings are strong up the middle with Alex Turcotte, Rasmus Kupari, Jaret Anderson-Dolan, and Akil Thomas. They could be even better is Gabriel Vilardi can finally get over his back issues. On the wings, they are supported by Arthur Kaliyev, Samuel Fagemo, and Carl Grundstrom. In net Cal Peterson should push Jack Johnson for the backup job. The blueline features Kale Clague, Tobias Bjornfot, Mikey Andersson, and Sean Durzi.
5.) Vancouver Canucks
Quinn Hughes is one of the best skating prospects in the world. He should make an immediate impact on the Canucks blueline. Olli Juolevi, Jett Woo and Guillaume Brisebois should not be far behind. Vasili Podkolzin was third in our draft rankings. The fact that he has a two-year commitment in the KHL dropped him down on draft days, but the Canucks may have a steal here. Nils Hoglander and Tyler Madden provide support upfront. The lack of a top-six centre prospect is the main flaw in the system (although it is mitigated by Elias Pettersson, who is considered graduated and does not factor into this ranking). Few teams have as good a one-two punch in goal as the Canucks with Michael DiPietro and Thatcher Demko.
4.) Montreal Canadiens
What a season it was for Habs prospects. Nick Suzuki was the OHL Playoff MVP. Ryan Poehling was World Junior MVP and scored a hat-trick (plus a shootout winner) in his first NHL game. Cole Caufield was the IIHF Under-18 MVP. He set records for most goals in the tournament and most goals with the US NTDP. Alexander Romanov was the top defenceman at the World Juniors. Josh Brook led all WHL defencemen in scoring and was on the league's 1st All-Star Team. Cayden Primeau was named the NCAA's top goalie. Jesse Ylonen won a gold medal at the World Juniors. Joel Teasdale led the QMJHL in playoff scoring and was the Memorial Cup MVP. Add in solid seasons from Cale Fleury, Jake Evans, and Noah Juulsen and the Habs have arguably the deepest group in the NHL with quality at all positions.
3.) Philadelphia Flyers
Carter Hart is our top goaltending prospect and looked good in his first NHL season. He has a strong group of defenders in front of him with Philippe Myers ready for NHL duty and Cam York picked at 14th overall. Samuel Morin is looking to get past injury issues and also take a spot in the NHL. Upfront, Morgan Frost put up a huge final season in the OHL, scoring 109 points in 58 games. Wingers Joel Farabee, Bobby Brink, Isaac Ratcliffe, and Wade Allison also provide scoring potential. This group is a bit top-heavy as it lacks depth, but the elite talent of those prospects more than makes up for it.
2.) Colorado Avalanche
Cale Makar and Bowen Byram are two of the top defence prospects in the world. Makar won the Hobey Baker Award and was impressive in the playoffs. Byram was the top defenceman in the 2019 Draft and was given the CHL Top Draft Prospect Award. He is an early favourite for the Calder Trophy. They are joined by Conor Timmins, Ryan Graves, Drew Helleson, and Nicholas Meloche to form the strongest defence prospect group in the league. The team also has highly skilled forwards in Alex Newhook and Martin Kaut. They are supported by Vladislav Kamenev, Shane Bowers, and Igor Shvyryov upfront. The team could use a top goaltending prospect.
1.) New York Rangers
Second overall pick Kaapo Kakko is the crown jewel of the group, but there are a number of elite talents in the group. 2018 first-round pick Vitali Kravtsov had a strong season in the KHL and makes his way to the Rangers. Swedish centre Lias Andersson comes in just under our criteria for what qualifies as a prospect. Also coming to North America is one of the top goalie prospects from the KHL in Igor Shestyorkin. He will have to compete with fellow Russian goalie prospect Alexandar Georgiev. Defencemen Adam Fox, K'Andre Miller, Nils Lundkvist, Matthew Robertson, and Libor Hajek are a strong group on the blueline.
2019 Organizational Prospect Rankings Main Photo:
VANCOUVER, BRITISH COLUMBIA – JUNE 21: (L-R) Kirby Dach, third overall pick by the Chicago Blackhawks, Jack Hughes, first overall pick by the New Jersey Devils, and Kaapo Kakko, second overall pick by the New York Rangers hold up their fingers of their pick order in front of the stage during the first round of the 2019 NHL Draft at Rogers Arena on June 21, 2019 in Vancouver, Canada. (Photo by Rich Lam/Getty Images)
Previous articleSan Jose Sharks, Peter DeBoer and Very Wrong Answers
Next articleNHL Rumours: Montreal Canadiens, Edmonton Oilers, Vancouver Canucks and Tampa Bay Lightning
Brad December 5, 2019 at 4:30 am
Jack Johnson plays goal? For the Kings?


text:
Now Keep an Eye on your Teenager's Phone with Google Family Link
Fizza AtiqueLast Updated: Sep 19, 2018
With the ever-growing use of social media and wifi ports available to everyone, it has become difficult to monitor your kid's activities on smartphones. To cater to this, Google has updated its Family link parental control app, Google family link in order to support Google's account for a teenager.
Google Family Link- A solution to all Problems
Regarding this, Google announced that:
parents would soon be able to allow or block apps, lock devices, and limit the screen time of their teen's Android phone, and much more.
Previously, the family link app controls were only available to parents and children younger than 13 years. However, both parents and teens will give their consent to device monitoring. Teens will be able to turn off the supervision when they want to.
"24-hour lockdown," though, and Google will inform the parent of the child's actions. "Ultimately, it's up to each individual family to have a conversation and decide what's right for them,"
Meanwhile, Google additionally revealed that complete Family Link support for teenagers and kids will show up to Chromebooks in the foreseeable future that is near and support for Bing Assistant from in a few days ("Hey Bing, hurry up and secure my kid's phone, would ya?").
For sure teenagers will be running away from this app as no one likes their moms and dads to be more controlling? — but I do believe it is born out of a spot that is great as lots of parents are worried about how precisely their children spend time online.
Nevertheless, I suspect that if a parent is forced to make use of an app to find out where the youngster is going in the evening, you can find conditions that Family Link won't manage to fix.
Also Read: Just 15 lines of code can crash your Apple iPhone and iPad
Fizza Atique
Fizza Atique is a Tech writer specializing in the intersection of tech and culture. She likes photography, VR, electronic music, coffee, and baking.


text:
Archway planned at Gainesvilles Rock Creek Veterans Park
An archway showcasing each of the U.S. armed forces will soon be built at Gainesville's Rock Creek Veterans Park.
An archway showcasing each of the U.S. armed forces is planned at Gainesville's Rock Creek Veterans Park.
"The reason we're putting it up is a lot of people don't know where the park is," said Jerry Peck, member of a Vietnam War veterans group that meets regularly in Gainesville. "We've talked to a lot of veterans, and they don't know anything about the park."
The concept design for the archway shows it featuring the emblem representing each of the service branches, plus the "You are not forgotten" symbol for prisoners of war and troops missing in action.
Construction could start soon on the arch, which will be at the entrance to the park off West Academy Street and Northside Drive.
Peck said the plan is to hold a dedication ceremony on May 20, or Armed Forces Day.
The structure will have stacked-stone columns and lighting.
"And we've got sponsors who gave us a certain amount of money, and we're going to try to put up something … to show our appreciation to them," Peck said.
The effort began in July 2016, when veterans from different groups spoke to the city's Parks and Recreation Board, said Melvin Cooper, Gainesville Parks and Recreation director.
Since then, the city's design firm and construction contractor for citywide signage designed the sign, he said.
The project's cost is $24,000, with $10,000 from the city and $14,000 from veterans.
The park has been a work in progress for several years, with war memorials, benches and walkways dotting its landscape. It also features an amphitheater and has served as the site for veterans ceremonies.
"We've got some more stuff after (the arch goes up) that that we'll be doing," Peck said. "There's another section we want to work on."
He added, with a laugh: "One of these days, we'll have that park done and we'll start throwing money at something else."


text:
Q: Proving greedy algorithm with exchange argument So I have to write a greedy algorithm that given a linked list or array of pairs (t_i, w_i), I have to produce a schedule that generates the maximum amount of money within time T. t_i would be the maximum number of hours that can worked for that job, w_i would be the wage in dollars per hour and T would be the maximum number of hours that person is willing to work per week. Jobs don't need to be worked until t_i, as the worker can choose to switch jobs early and get paid for each hour they worked.
So I wrote a greedy algorithm where you just get the job with the highest wage per hour and work that job for as long as you can, then get the job with the second wage per hour and work that job for as long as you can and so on till you work T hours.
I am now being asked to use an exchange argument to prove that my pseudocode produces an optimal schedule. In my opinion, my pseudocode solution is already the optimal schedule, so what what I compare it to then?

A: I'll give you the idea of the demonstration without writting it entirely.
You need to take an optimal solution S. You can tell that this solution can grant x $, where x is the maximum money you can make.
Then you take your solution G witch is greedy.
Now you say the first job you chose in G, this job is j1 and the time spent is t1. There is two possibility, either j1 is in S or it's not. If it's not by replacing one job in S by j1 you get a better solution (or at least as good) BUT S is optimal so you can conclude that j1 is in S. You can do the same kind of reasoning to prove that the time associed with j1 in S is t1.
Then you do that for every element. And you'll conclude that the amount of money x earned is the same in G and in S. So G is optimal

redpajama_set_name: RedPajamaStackExchange


text:
Muslim Student Union Fights For the Right to Incite Murder
Posted on September 15, 2008 by islamoscope
Established in 1963 by the Muslim Brotherhood — the inspiration for al-Qaeda and Hamas — the Muslim Students Association was listed in documents written by a Muslim Brotherhood operative and seized by the FBI in 2007 as an organization to help "destroy Western civilization from within." According to a February 2008 New York Times article, the MSA was financed by Saudi Arabia when it first began appearing throughout North America in the late 1950s and "pushed the kingdom's puritan, Wahhabi strain of Islam."
Extremism is a systemic problem with the Muslim Students Association network. For example, Sheikh Khalid Yasin, who has spoken at a multitude of campuses throughout the country and at events at the MSA National level, has denied al-Qaeda was involved in 9/11 and believes that homosexuals should be killed in accordance with the Koran. Norman Finkelstein, who has also spoken for various Muslim Students Association groups, has been featured at the University of Maryland, UC Irvine, UCLA, UC Berkeley, Harvard, Columbia, and many others. Imam Abdul Alim Musa, a radical imam and founder of the As Sabiqun (Arabic for "The Vanguard") movement, has been featured at UC Irvine, MIT, Mount Holyoke College, and other campuses.
Therefore it shouldn't be a surprise that the Muslim Student Union chapter at the University of Southern California (USC) acts in accordance with its sister chapters, and until recently hosted a violent quote in the "compendium" of hadiths on the USC Muslim Students Association page. The quote advocates massacring Jews as a pathway of redemption for Muslims, and reads as follows:
Abu Huraira reported Allah's Messenger (may peace be upon him) as saying: The last hour would not come unless the Muslims will fight against the Jews and the Muslims would kill them until the Jews would hide themselves behind a stone or a tree and a stone or a tree would say: Muslim, or the servant of Allah, there is a Jew behind me; come and kill him.
The David Horowitz Freedom Center worked with the Simon Wiesenthal Center to draft a letter to Alan Casden, a USC trustee, about the "hadith of hate," as it is often called. Disturbed that a call for genocide should be on the USC server, Casden contacted Provost Chrysostomos Nikias to express his concern. Nikias investigated the matter and sent Casden the following letter:
The passage you cited is truly despicable and I share your concerns about its being on the USC server. We did some investigations and I have ordered the passage removed.
The passage in the hadith that you brought to our attention violates the USC Principles of Community, and it has no place on a USC server.
The University of Southern California, a private university, has strict Principles of Community which state, in part: "No one has the right to denigrate another human being on the basis of race, sex, sexual orientation, national origin, etc. We will not tolerate verbal or written abuse, threats, harassment, intimidation, or violence against person or property." No student group other than the Muslim Students Association has posted any kind of material, religious or otherwise, calling for the destruction of a race or group.
USC's decision to remove the hadith from the school's server marks the first time that an American university has recognized that the Muslim Students Association's agenda involves the promotion of ethnic hatred. It is also the first time that an administrator has acted to remove "despicable" material.
A statement issued by the Muslim Student Union at USC suggested that the hadith was being taken out of context and called the school's decision to remove it from the website "unprecedented and unconscionable." The Muslim Student Union accused the USC administration of practicing unfair censorship. However, following the controversy with the Danish cartoons, Muslim students called for censorship and protested at several universities, including UC Irvine, when the cartoons were unveiled in an academic seminar. The protests included MSA students from various universities, including USC.
Moreover, although the provost ordered the removal of the more accessible versions of the hadiths calling for slaughtering of Jews and the one that was brought to his attention, the same quote still appears in different variations in Bukhari on the USC MSA website and are difficult to locate unless one knows what they're looking for:
Narrated 'Abdullah bin 'Umar:
Allah's Apostle said, "You (i.e., Muslims) will fight with the Jews till some of them will hide behind stones. The stones will (betray them) saying, 'O 'Abdullah (i.e., slave of Allah)! There is a Jew hiding behind me; so kill him.'"
Narrated Abu Huraira:
Allah's Apostle said, "The Hour will not be established until you fight with the Jews, and the stone behind which a Jew will be hiding will say. "O Muslim! There is a Jew hiding behind me, so kill him."
Just in case these vanish, here are some screen shots:
The hadith also appears in another variation later along in Bukhari:
I heard Allah's Apostle saying, "The Jews will fight with you, and you will be given victory over them so that a stone will say, 'O Muslim! There is a Jew behind me; kill him!'"
These quotes appear in more than one location, as is traditional in Islamic theology and an indication of authenticity. Since early 2008 the Muslim Students Association has been asked to repudiate this genocidal hadith and terrorist organizations by the David Horowitz Freedom Center. Every individual MSA chapter has failed to repudiate the genocidal imperative contained within the hadith and has also failed to condemn terrorist groups like al-Qaeda, Hamas, and Hezbollah.
Filed under: Uncategorized | Tagged: al qaeda, antisemetism, antisemitism, freedom of speech, hate, hate speech, hatred, hezbollah, islam, islam is evil, islamic terrorism, islamoscope, jew hate, mohammad, muhammad, muhammadens, muhammadim, muhammadism, muslim students association, quran, racism, student terrorists, taliban, terrorists, university of south carolina, usc, violence |	2 Comments »


Today was a hot day, we worked at The Children's Museum, we were fishing and crabbing. We caught so many crabs and fish. We helped the children fish, we put the crabs in the bucket and we showed the children which crab is male and which crab is female. At 1pm, we went to Castle Island in South Boston. It was very hot at South Boston beach. We bought the food to eat for lunch. We saw some children eating and drinking, some of them were crabbing and some of them were swimming. We helped the children catch the crabs and fish. After that, we went to the office, some of my co-workers stayed there to help the children.
I was teaching a child to fish.


The Code Council's board of directors provide direction, guidance and oversight to the organization. The Board meets multiple times a year and works closely with our senior leadership team to steer the Code Council towards a sustainable future while adhering to its mission. Learn more about our current board of directors.
For contact information, please see our Contact ICC page.


The 2019 Honda Civic Hatchback is quickly redefining what it means to be a hot hatch. Sure, it meets entry level requirements such as innovative style and stunning performance. But Honda has chosen to elevate the segment with class-leading, EPA-estimated 31 city/40 highway/34 combined fuel economy* and up to 46.2 cubic feet of cargo space that would rival some compact crossovers.
Its dynamic style speaks for itself with strategically placed features and oversized design elements, making it nearly impossible for the Civic Hatchback to go unnoticed. So let's talk performance. Clearly, this hot hatch comes ready to play with a 174-hp, inline-4 turbocharged engine, but a wide stance and balanced chassis do their part as well. Rounding out this Euro-inspired ride is a MacPherson Strut front suspension, a multi-link rear suspension, and front/rear stabilizer bars. The 2019 Honda Civic Hatchback is fun to drive and your Mid-Missouri Honda Dealers want to show you why with a test drive today.


The men in ODC's microwave cooking class meet every Wednesday to learn their way around the kitchen. Participants pick out recipes, prepare and, of course, taste their creations. It's an opportunity for all to become more independent in their daily lives, using a microwave to prepare dishes without too much fuss. Things we sometimes take for granted – like using a can opener, for example – can be new experiences for those who may not have had much exposure to a kitchen in the past.
The taste of this particular recipe, Sunshine Yam Casserole, wasn't exactly a "hit" with group members – a little too sweet, they said. But they'll be back at it again next week, ready to try again!


Discussion in 'The Projects Forum' started by KCAutosound, Oct 6, 2009.
Hey guys. Stumbled across this forum after some long searching for what I need.
Here's the deal. I have vehicle security systems that will allow negative inputs to be able to arm and disarm them. This comes in handy with the Ford vehicles since they have a keypad on the door. You can hit 7 and 9 to lock the doors and hit your five digit code to unlock the doors. I use the motor lock and unlock wires to arm and disarm the alarm system. The problem is that these are positive outputs and my alarm needs negative inputs. On MOST vehicles I can use a simple pair of relays to flip the positive from the locks to be a negative the alarm can see. A problem arises when I do certain vehicles like say the 04 to 08 F150s. The positive pulse from the lock/unlock motor wires is so fast that the relays don't have much time to respond and put out either a full ground output or a long enough output. I need it to be about a second or longer.
To correct this I have been using pulse timer relays from either Code Alarm or Directed Electronics. The problem is that I need to use two of them and these guys are about $15.00 apiece. I am using them more and more and my distributors inventory can't keep up with my demand. I would like to make a circuit that I can use that will keep cost down as well as allow me to have them in inventory when I do these vehicles. I do alot of them!!!
Basically I'm needing a circuit that will take the 12 volts positive input from the lock wire and then flip it to 12 volts negative for about say 2 seconds to be safe.
From my reading around I have seen people say you could use a capacitor to make the relay stay on longer but I have never done anything like that.
What is the reference for the neg. signal, ground or +12V? What is the load that it feed into?
Do you mean that you have a wire that is usually at ground, but jumps up to 12V to briefly to indicate "Lock" or "Unlock?" And what you need instead is a wire that is usually at 12V, but falls down to 0V (for about 2 seconds) when there's a 12V pulse on the other wire?
This is probably a pretty simple circuit, you should be able to build them for less than $15.
Something like this? You can also replace R2 with a relay. If using a relay, shunt coil with a diode , cathode to +V. !N4002 or sim. Just about any N-ch MOSFET should do.
Bernard I kinda make sense with your drawing there but I'm really digging back to my Devry days from about 1993...lol Went for two semesters for Electronic Technition then kinda went another direction. I can make out the diodes and the capacitor symbols but I get lost on the other ones in your schematic.
Basically the locking system on a car is a reverse polarity system. The lock motors have only two wires. They flip polarity. So like what spacewrench was asking... yes the wire initially sits at ground but when the lock signal is sent to it it will pulse 12 volts positive. After the pulse it goes back to ground. I am trying to take this 12 volt pulse signal and turn it into a 2 second ground pulse. Sound simple enough?
yes the wire initially sits at ground but when the lock signal is sent to it it will pulse 12 volts positive. After the pulse it goes back to ground. I am trying to take this 12 volt pulse signal and turn it into a 2 second ground pulse. Sound simple enough?
The input is the "arrow" side of the diode; the output is the junction between R2 and the drain of the MOSFET.
Is there a certain wattage for these resistors that I need?
Little lost on this drawing. On this mosfet I see there are three prongs similar to a transistor. The datasheet shows D G S. Is my drawing I've attached look right from the schematic?
R1 (2Meg) can be any size. 1/4 watt is very common, small & cheap. R2 (1K) could be 1/4 watt in a pinch, but 1/2 watt or 1 watt would be better.
I like that drawing there. Good job. I'm gonna pick up the parts tomorrow from my local Electronics Supply. I checked their online catalog and it shows them all in stock. I throw it together and will post my findings.
need a 12 volt and 5 volt battery combo?


Image Title: Best Swing Arm Pendant Lights Apartment Therapy Regarding Wall Mounted Light Idea 5. Filename: best-swing-arm-pendant-lights-apartment-therapy-regarding-wall-mounted-light-idea-5.jpg. Image Dimension: 540 x 540 pixels. Images Format: jpg/jpeg. Publisher/Author: Guido Walter. Uploaded Date: Saturday - May 12th. 2018 18:23:06 PM. Category: Architecture. Image Source: wayfair.com.
Tap The Thumbnail Bellow to See Related Gallery of "Best Swing Arm Pendant Lights Apartment Therapy Regarding Wall Mounted Light Idea 5"


General oblique aerial view with Brouster in the foreground and the Loch of Voxterby and Brace Field in the middle distance and Burrafirth beyond, looking N.
Plan of settlement. Published in Current Archaeology 1979,VI,no.65 .
Oblique aerial view of Scord of Brouster, looking SE.
HU 256 515. The remains of a Neolithic house ("A") occupy a site on the slopes of the Scord of Brouster, a short distance north of the road between the Bridge of Walls and Walls, and about 300yds WSW of Brouster.
The house is represented by an oval, grass-covered bank measuring about 43' NW-SE and 34' NE-SW externally. A gap in the SE of the bank represents an entrance passage.
A field system attached comprises five or six fields, the boundaries of which are much-denuded but definite in some parts and traceable in others. One dyke, better preserved than the others, over-rides the house walling, but may simply be a rebuilding on the old foundations. The field enclosed by this boundary shows cultivation rigs and there are field clearance heaps in all the fields.
About 26' north of the house there is an oblong dry- stone construction, now deep-set in the soil, which may be original: it measures externally 23' by 19'. Two other sub-oval constructions ('B' & 'C') are located in the north-most fields and may also be contemporary. They take the form of grass-covered stony banks and each encloses a hollow. In overall dimensions they average 35' by 27' and the spread of the bank is 7' to 8' in width. These sizes suggest they may be adjuncts to the house, but they could be small houses.
C S T Calder 1958.
'There is a superficial suggestion of the Stonydale "Temple" (HU25SE 1) in the shallowness of the recesses and the flattened SE end of the ruin.' The sub-rectangular stony mound may be a cairn or a building.
A Neolithic/Bronze Age settlement and field system,as described by Calder, centred at HU 255 516, comprising two homesteads (A and B), and one probable homestead (C). The "oblong construction" 26' to the north of homestead A cannot be classified: it may be fairly modern.
Visited by OS(NKB) 20th June 1968.
Oblique aerial photographs of Scord of Brouster, Shetland, taken by John Dewar in 1973.
This is perhaps the most complete and comprehensible neolithic settlement complex in Shetland (colour photograph on p.28). The outlines of the houses, the enclosures and the many clearance cairns are readily discernible, even from the road, and recent excavations have laid bare many details of the settlement. The most obvious enclosure is close to the road, an irregular oval almost 60m long enclosed by a stone wall, and a large oval house is attached to its north end. The entrance is at the downslope end of the house, and the interior, 7m by 5m, contains a central hearth and stone-built cells lining the walls, using massive boulders as subdividing piers. A radiocarbon date places the occupation of the house in the late 3rd early 2nd millennium BC, but the site as a whole is likely to span a long period. At least three other houses lie further north, and their fields cover an area of about 2 ha. Close to the first house is a perfect circular ring cairn with a kerb of large boulders, bur radiocarbon dating suggests that this may be considerably later than the settlement, perhaps by one and a half thousand years.
The field-system at Scord of Brouster comprises eight irregular conjoined enclosures, clustered together to form a roughly oval block. Scattered across the interiors of six of the field enclosures, and extending beyond them to the east, there are over 60 small cairns, presumably field clearance, some arranged in rough lines. The western extent of the fields is relatively straight and its line is continued to the N by a boulder bank that extends for about 130m before it forms a T-junction with a boundary that cuts across the parts of the saddle between the Mousa Water to the W and the Brace Burn to the E. To the S, the western extent of the Scord of Brouster fields aligns roughly with a field boundary that ends towards the field-system on Gallow Hill (HU25SE 37) and may form part of an extensive boundary system beyond (e.g. see HU25SE 603).


text:
Hill Top is an unincorporated community in Fleming County, Kentucky, United States. Hill Top is located at the junction of Kentucky Route 170 and Kentucky Route 1347,  southwest of Flemingsburg.

Notable residents
Franklin Sousley, one of the men shown in the famous Joe Rosenthal Associated Press photograph of US Marines and Sailors raising the second US flag on Iwo Jima, was born in Hill Top on 19 September 1925. He was killed in action on Iwo Jima after having been shot by a Japanese sniper on 21 March 1945, about a month after the famous photograph was captured. He was only 19 years of age. His body was first interred on Iwo Jima, but was later returned home for his final interment on May 8, 1947, in Elizaville Cemetery.

References

Unincorporated communities in Fleming County, Kentucky
Unincorporated communities in Kentucky

redpajama_set_name: RedPajamaWikipedia


A sugar substitute is a food additive that duplicates the effect of sugar in taste, usually with less food energy. Some sugar substitutes are natural and some are synthetic. Those that are not natural are, in general, called artificial sweeteners.
An important class of sugar substitutes is known as high-intensity sweeteners. These are compounds with many times the sweetness of sucrose, common table sugar. As a result, much less sweetener is required and energy contribution is often negligible. The sensation of sweetness caused by these compounds (the "sweetness profile") is sometimes notably different from sucrose, so they are often used in complex mixtures that achieve the most natural sweet sensation.
If the sucrose (or other sugar) that is replaced has contributed to the texture of the product, then a bulking agent is often also needed. This may be seen in soft drinks that are labeled as "diet" or "light" that contain artificial sweeteners and often have notably different mouthfeel, or in table sugar replacements that mix maltodextrins with an intense sweetener to achieve satisfactory texture sensation.
In the United States, six intensely sweet sugar substitutes have been approved for use. They are stevia, aspartame, sucralose, neotame, acesulfame potassium, and saccharin. There is some ongoing controversy over whether artificial sweetener usage poses health risks. The US Food and Drug Administration regulates artificial sweeteners as food additives. Food additives must be approved by the FDA, which publishes a Generally Recognized as Safe (GRAS) list of additives. To date, the FDA has not been presented with scientific information that would support a change in conclusions about the safety of these approved high-intensity sweeteners (with the exception of Stevia, which is exempt under FDA's GRAS policy due to its being a natural substance in wide use well before 1958, and has been approved by FDA). The safe conclusions are based on a detailed review of a large body of information, including hundreds of toxicological and clinical studies.
The majority of sugar substitutes approved for food use are artificially synthesized compounds. However, some bulk natural sugar substitutes are known, including sorbitol and xylitol, which are found in berries, fruit, vegetables, andmushrooms. It is not commercially viable to extract these products from fruits and vegetables, so they are produced by catalytic hydrogenation of the appropriate reducing sugar. For example, xylose is converted to xylitol, lactose tolactitol, and glucose to sorbitol. Other natural substitutes are known, but are yet to gain official approval for food use.
Some non-sugar sweeteners are polyols, also known as "sugar alcohols". These are, in general, less sweet than sucrose but have similar bulk properties and can be used in a wide range of food products. Sometimes the sweetness profile is 'fine-tuned' by mixing with high-intensity sweeteners. As with all food products, the development of a formulation to replace sucrose is a complex proprietary process.
To assist in weight loss – some people choose to limit their food energy intake by replacing high-energy sugar or corn syrup with other sweeteners having little or no food energy. This allows them to eat the same foods they normally would, while allowing them to lose weight and avoid other problems associated with excessive caloric intake.
Dental care – sugar substitutes are tooth-friendly, as they are not fermented by the microflora of the dental plaque. An example of a sweetener that can benefit dental health is xylitol. Xylitol works to prevent bacteria from adhering to the tooth surface, thus preventing plaque formation and eventually decay. The carbohydrates and sugars consumed usually adheres to the tooth enamel. Bacteria can feed upon this food source allowing them to quickly multiply. As the bacteria feed upon the sugar, they convert it to acid waste that in turn decays the tooth structure. Xylitol cannot be fermented by these bacteria, so the bacteria have difficulty thriving, thus helping to prevent plaque formation.
Diabetes mellitus – people with diabetes have difficulty regulating their blood sugar levels. By limiting their sugar intake with artificial sweeteners, they can enjoy a varied diet while closely controlling their sugar intake. Also, some sugar substitutes do release energy, but are metabolized more slowly, potentially allowing blood sugar levels to remain more stable over time.
Reactive hypoglycemia – individuals with reactive hypoglycemia will produce an excess of insulin after quickly absorbing glucose into the bloodstream. This causes their blood glucose levels to fall below the amount needed for proper body and brain function. As a result, like diabetics, they must avoid intake of high-glycemic foods like white bread, and often choose artificial sweeteners as an alternative.
Avoiding processed foods – individuals may opt to substitute refined white sugar with less-processed sugars, such as fruit juice or maple syrup. (See List of unrefined sweeteners).
Cost – many sugar substitutes are cheaper than sugar. Alternative sweeteners are often low in cost because of their long shelf-life and high sweetening intensity. This allows alternative sweeteners to be used in products that will not perish after a short period of time. The long-term costs and risks of using artificial sweeteners — weight gain, diabetes type II, and desensitization of gustatory (taste) sense — outweigh this alleged benefit, however.


text:
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */

/*
 Copyright (C) 2010 Klaus Spanderen

 This file is part of QuantLib, a free-software/open-source library
 for financial quantitative analysts and developers - http://quantlib.org/

 QuantLib is free software: you can redistribute it and/or modify it
 under the terms of the QuantLib license.  You should have received a
 copy of the license along with this program; if not, please email
 <quantlib-dev@lists.sf.net>. The license is also available online at
 <http://quantlib.org/license.shtml>.

 This program is distributed in the hope that it will be useful, but WITHOUT
 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 FOR A PARTICULAR PURPOSE.  See the license for more details.
*/

#include <ql/methods/finitedifferences/operators/fdmlinearoplayout.hpp>
#include <ql/methods/finitedifferences/utilities/fdminnervaluecalculator.hpp>
#include <ql/methods/finitedifferences/stepconditions/fdmbermudanstepcondition.hpp>

namespace QuantLib {

    FdmBermudanStepCondition::FdmBermudanStepCondition(
            const std::vector<Date>& exerciseDates,
            const Date& referenceDate,
            const DayCounter& dayCounter,
            const boost::shared_ptr<FdmMesher> & mesher,
            const boost::shared_ptr<FdmInnerValueCalculator> & calculator)
    : mesher_    (mesher),
      calculator_(calculator) {

        exerciseTimes_.reserve(exerciseDates.size());
        for (std::vector<Date>::const_iterator iter = exerciseDates.begin();
            iter != exerciseDates.end(); ++iter) {
            exerciseTimes_.push_back(
                             dayCounter.yearFraction(referenceDate, *iter));
        }
    }

    const std::vector<Time>& FdmBermudanStepCondition::exerciseTimes() const {
        return exerciseTimes_;
    }

    void FdmBermudanStepCondition::applyTo(Array& a, Time t) const {
        if (std::find(exerciseTimes_.begin(), exerciseTimes_.end(), t)
              != exerciseTimes_.end()) {

            boost::shared_ptr<FdmLinearOpLayout> layout = mesher_->layout();
            const FdmLinearOpIterator endIter = layout->end();

            const Size dims = layout->dim().size();
            Array locations(dims);

            for (FdmLinearOpIterator iter = layout->begin(); iter != endIter;
                ++iter) {
                for (Size i=0; i < dims; ++i)
                    locations[i] = mesher_->location(iter, i);

                const Real innerValue = calculator_->innerValue(iter, t);
                if (innerValue > a[iter.index()]) {
                    a[iter.index()] = innerValue;
                }
            }
        }
    }
}

redpajama_set_name: RedPajamaGithub


Fraud Alert: Fake email citing issue related to section 314(b) of USA Patriot Act.
We've got suspicions transfer from your client, and put it on hold. According section 314(b) of the USA PATRIOT Act we have to report you about potential money laundering. Please review the attached document with details of this case.
If you receive this email, do not click on any attachments and disregard.


Ontario has passed the Rental Fairness Act, 2017 which modifies the Residential Tenancies Act. Rent control now applies to all private rental units, including those first occupied on or after November 1, 1991. Any notices of rent increase given for these units on or after April 20, 2017, must be capped at the rent increase guideline. Notices given before April 20, 2017 are not affected by the changes.
Col 3:23 Whatever you do, work at it with all your heart!


Honey bunny looking very cute!
Been squashed by a giant!!!!!!!
Shed Extension Finished... Yayyyy !
Getting her on Thrusday night!
Sweeties - did you say sweeties!
Might be the cutest ears I've seen!
OH will kill me for posting these!!
My Bengal Niece & Nephew.....How adorable!
Poor Mallow has been reduced to this!
Giving a giant a piggy back!!!!
I can has nose rubs?
What do your rabbits do, that makes you laugh?
Black bunny and white bunny!
Islay and Jura were 3 yesterday!
Alice wants to show you how much better her eye is!
What had 28 legs and fidgets?
The rabbit that wouldn't go near you now comes when you call her!
Chewing the wire.. caught in the act!
Gabriel and Spring - back behind bars.
How NOT to use a hay rack!
Tobys been keeping me company.
Don & Lola being monkeys.. again!
Does any bunny have a bigger tail than Digby??!
Daisy's smooshed look of disgust!
Happy Birthday/Gotcha Day Starbuck! 5 today! Also Remembering Fay.
Told shes a 'sable marten' ?
Bet you can't guess who's mouth this is!
look what i found when i got home.
Grayson is not in my good books!!
The first photo I saw of Katy - and now.
Fluffy Frodo has a beard!
I got cuddles in bed this morning!