text:
\section{Introduction and main results}
Throughout let $k$ be a field.
All algebras in consideration are associative unital $k$-algebras and all automorphisms of algebras are $k$-automorphisms.
Let $D$ be an algebra,
$a$ be a central element of $D$ and
$\sigma$ be an automorphism of $D$.
The \emph{generalized Weyl algebra} (GWA, for short)
 $A=D(\sigma,a)$ over $D$ is defined as the algebra generated by $D$ and two indeterminates $x$ and $y$ subject to the relations:
\[
xd=\sigma(d)x,\ yd=\sigma^{-1}(d)y, \
yx=a,\ xy=\sigma(a), \
\text{ for all }d\in D.
\]
GWAs are natural generalizations of Weyl algebras and
form an important class of noncommutative algebras.
Many algebras of common interest fall into this class,
for instance, Weyl algebras,
certain quantum enveloping algebras, some iterated Ore extensions,
noetherian (generalized) down-up algebras, Heisenberg algebras,
 etc. (see \cite{Bavula1992Generalized,bavula2017quantum,Zhao2018Gelfand} for more examples).
GWAs were introduced by Bavula \cite{Bavula1992Generalized} in early 1990s.
Since then
their structures and representations have been studied intensively, for example,
their dimension and growth \cite{bavula1996global,Bavula1998Krull,bavula2001krull,Zhao2018Gelfand};
their derivations, isomorphisms and automorphisms
\cite{bavula2001isomorphism,Suarez-Alvarez_2015,Almulhem2018,Gaddis2019,Gutierrez2020};
and their homological and geometric properties \cite{Brzezinski2016,Won2018,Liu2018,Ferraro2020,liu2021batalin,klyuev2021twisted}.

The notion of growth,
introduced by Gelfand and Kirillov \cite{gelfand1966corps}
for algebras
and by Milnor \cite{milner1968curvature} for groups,
is a fundamental object of study in theory of algebras (not necessarily associative) and groups.
The Gelfand-Kirillov dimension (GK-dimension, for short) is a main tool to quantify growth of algebras and groups.
The GK-dimension of an algebra $A$ is defined as
\[
\operatorname{GKdim}(A):=\sup_V{\limsup_{n\to\infty}}\log_n \dim_k\left(\sum_{i=0}^nV^i\right)
\]
where the supremum is taken over all finite dimensional subspaces $V$ of $A$.
GK-dimension basically measures the growth of an algebra (rather than the growth of a ring, see \cite{Wu1991}).
For a finitely generated commutative algebra,
its Krull dimension and GK-dimension coincide.
Thus GK-dimension can be viewed as a noncommutative analogue of Krull dimension.
It turns out that the GK-dimension is a very useful and powerful tool for investigating noncommutative algebras,
for example, in recent studies of classification problems
(see e.g. \cite{Wang2015,GOODEARL2017,Andruskiewitsch2020})
and cancellation problems (see e.g. \cite{Bell2017Zariski,Lezama2018,tang2020cancellation,chan2022reflexive}).
For basic properties and applications of GK-dimension of algebras and groups we refer the reader to \cite{Krause-Lenagan_2000}.
For GK-dimension of other algebraic structures, see e.g.
\cite{Khoroshkin2015,Bao2020,Bai2021,2020Growth}.

There have been in the literature
a number of results concerning GK-dimension of skew polynomial extensions related to derivations and/or automorphisms,
for example, the GK-dimensions of Ore extensions \cite{lorenz1982gelfand, huh1996gelfand,zhang1997note},
 of PBW-extensions \cite{matczuk1988gelfand},
 and of differential difference\ algebras/modules \cite{ZhangZhao2013algebras}.
Recently, the author \cite{Zhao2018Gelfand} investigated general properties of GK-dimensions of GWAs.

In this paper, we mainly study the growth and GK-dimension of GWAs
over polynomial algebras $P_n:=k[z_1,z_2,\dots,z_n]$
(especially $P_2$)
and that over Laurent polynomial algebras $L_n:=k[z_1^{\pm1},z_2^{\pm1},\dots,z_n^{\pm1}]$.
We have two motivations for this paper.
The first motivation is the fact that
lots of GWAs studied in literature are ones over $P_n$ or $L_n$
and thus this subclass of GWAs is of fundamental importance in practice.
For instance, (quantum) Weyl algebras, quantum planes, and primitive quotients of the quantum enveloping algebra $\mathcal{U}_q(\mathfrak{sl}_2)$ of special linear Lie algebra
$\mathfrak{sl}_2$ are GWAs over $P_1$ or $L_1$;
noetherian generalized down-up algebras, quantum Heisenberg algebra,
and the universal enveloping algebra $U(\mathfrak{sl}_2)$
are GWAs over $P_2$;
the group algebra of the {(discrete) Heisenberg group}
(Example \ref{exam_Heisenberg}) and
the homogenized enveloping algebra of some Lie algebra
(\cite[Example 2.4 (ii)]{jordan2009reversible})
are GWAs over $L_2$.
Our second motivation is that,
since
the automorphism groups $\operatorname{Aut}(P_2)$ and $\operatorname{Aut}(L_n)$ are well-understood
(see Jung and van der Kulk \cite{Jung1942Uber,vanderKulk1953} for $\operatorname{Aut}(P_2)$ and Theorem \ref{thm_auto(L)} for $\operatorname{Aut}(L_n)$),
sharper theorems can be expected when one investigates the GK-dimension of GWAs over $P_2$ and $L_n$.

Let $A=D(\sigma,a)$ be a GWA.
It was proved in \cite{Zhao2018Gelfand} that the difference
$\operatorname{GKdim}(A)-\operatorname{GKdim}(D)$ can be any positive integer or infinity.
If $\sigma$ is \emph{locally algebraic} (Definition \ref{def-la}) meaning that
every finite dimensional\ subspace of $D$ is contained in a finite dimensional\ $\sigma$-stable subspace of $D$,
then $\operatorname{GKdim}(A)-\operatorname{GKdim}(D)=1$.
Although $\operatorname{GKdim}(A)-\operatorname{GKdim}(D)=1$ does not imply in general that $\sigma$ is locally algebraic (see Example \ref{exam-GK+1-NotLA}),
we prove that, for GWAs over $P_n$ and $L_n$, this implication does hold.
More generally,
we have the following theorem.

\begin{theorem}
  \label{thm-main-}
  Let $A=D(\sigma,a)$ be a GWA.
  Let $P_n=k[z_1,z_2,\dots,z_n]$ be the polynomial algebra in $n$ indeterminates and $Q_n$ be the fraction field of $P_n$.

  If $P_n\subseteq D\subseteq Q_n$,
  then the following statements are equivalent:
  \begin{enumerate}[(a)]
    \item\label{itemD+1}
    $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$.
    \item\label{itemD+LA}
    $\sigma$ is locally algebraic.
  \end{enumerate}

  If $k$ is algebraically closed, $D$ is a finitely generated\ field and $a\neq0$,
 then (\ref{itemD+1}) and (\ref{itemD+LA}) are equivalent to the following statements:
 \begin{enumerate}[(a)]
 \setcounter{enumi}{2}
    \item\label{itemD-FinOrd}
     $\sigma$ has a finite order.
     \item\label{itemD-NotSimple}
     $A$ is not simple.
 \end{enumerate}
\end{theorem}

Theorem \ref{thm-main-} is an analogue of \cite[Theorem 1.1]{zhang1997note} but note that we do not assume that
$k$ is algebraically closed to get $(a)\Leftrightarrow (b)$.
See Proposition \ref{prop-Zhang97} for more results for the algebraically closed case.

When considering GWAs over $P_2$,
we have more elaborate results.
Our main results on  the GK-dimension of GWAs over $P_2$ can be summarized as
\begin{theorem}
    \label{thm-mainP}
  Let $A=k[z_1,z_2](\sigma,a)$ be a GWA.
     Then the following statements are equivalent:
  \begin{enumerate}[(a)]
    \item\label{itemP+1}
    $\operatorname{GKdim}(A)=3$.
    \item\label{itemP-GKfinite}
    $\operatorname{GKdim}(A)<\infty$.
    \item\label{itemP-PolyGrowth}
    $A$ has polynomial growth.
    \item\label{itemP-NotExGrowth}
    $A$ does not have exponential growth.
    \item\label{itemP-LA}
    $\sigma$ is locally algebraic.
    \item\label{itemP-ConjTri}
    $\sigma$ is conjugate to a triangular automorphism.
    \item\label{itemP-PowerConjTri}
    $\sigma^m$ is conjugate to a triangular automorphism for some $m\geq 1$.
  \end{enumerate}
\end{theorem}

The growth of algebras (Definition \ref{def-growth}) can be prescribed within a rather wide range of functions.
Recently,
Bell and Zelmanov \cite{Bell2020} give a complete characterization of the functions that can occur as the growth functions of algebras.
Theorem \ref{thm-mainP} indicates that
GWAs over $P_2$ have \emph{alternative growth} (Definition \ref{def-growth}).
Note that not every algebra or group has alternative growth,
for example,
if a finitely generated\ Lie
algebra $\mathfrak{g}$ has polynomial growth then its universal enveloping algebra $U(\mathfrak{g})$ has intermediate
growth \cite{Smith1976}.
It is still a major open question in growth theory whether there exist finitely presented groups with intermediate growth \cite{Grigorchuk2008}.

The dichotomy of the GK-dimension of GWAs over $P_2$
(i.e., $\operatorname{GKdim}(P_2(\sigma,a))$ is either $3$ or $\infty$)
does not hold for that over $L_2$ (Example \ref{exam_Heisenberg}).
Note that some items in Theorem \ref{thm-mainP} are not equivalent for GWAs over $P_n$ with $n\geq 3$.
For example, the Nagata automorphism of $P_3$ is locally algebraic but
not conjugate to a triangular automorphism (Example \ref{exam-Nagata}).
The reason for this is that
the proof of Theorem \ref{thm-mainP} is
based on the famous theorem of Jung and van der Kulk \cite{Jung1942Uber,vanderKulk1953} asserting
that all automorphisms of $P_2$ are tame,
which is not true for $P_n$ with $n\geq3$ (\cite{Shestakov2003}).
See Remark \ref{remark-P2-P3} for more details.

We also study further the GK-dimension of GWAs over $L_n$.
An automorphism $\sigma\in\operatorname{Aut}(L_n)$ sends a unit to a unit and thus has the form
\begin{align*}
\sigma(z_i)=\alpha_iz_1^{a_{1i}}z_2^{a_{2i}}\cdots z_n^{a_{ni}},
 \ \alpha_i\in k^*, a_{1i},a_{2i},\dots, a_{ni}\in\mathbb{Z},\
i=1,2,\dots,n.
\end{align*}
The automorphism $\sigma$ is determined by the matrix
\(
M_{\sigma}=(a_{ij})\in \mathbb{Z}^{n\times n}
\)
and the scalars $\alpha_i\in k^*$, $i=1,2,\dots,n$.
The matrix $M_{\sigma}$ plays an important role in the
GK-dimension of the GWA $L_n(\sigma,a)$.
Our main results on the
GK-dimension of GWAs over $L_n$ are summarized in the following theorem.
\begin{theorem}
  \label{thm-mainL}
  Let $A=k[z_1^{\pm1},z_2^{\pm1},\dots,z_n^{\pm1}](\sigma,a)$ be a GWA.
  Then the following statements are equivalent.
  \begin{enumerate}[(a)]
    \item\label{itemL+1}
    $\operatorname{GKdim}(A)=n+1$
    \item\label{itemL-LA}
    $\sigma$ is locally algebraic.
    \item\label{itemL-FiniteOrd}
    $M$ has a finite order, i.e., $M_{\sigma}^m$ equals the identity matrix for some $m\geq1$.
    \item\label{itemL-PowerBound}
    $M_{\sigma}$ is power-bounded, i.e.,
    there exists $N>0$ such that the absolute values of all entries of $M_{\sigma}^m$ are less than $N$ for all $m\geq1$.
  \end{enumerate}
\end{theorem}

Theorem \ref{thm-mainL} enables us to determine the GK-dimension by computation of integer matrices under certain conditions.
Given $n\geq 1$,
Levitt \cite{Levitt1998} showed that the maximum order $G(n)$ of a matrix in $\operatorname{GL}(n,\mathbb{Z})$ satisfies a Landau-type estimate $\ln G(n)\sim \sqrt{n\ln n}$
and all possible orders of matrices in $\operatorname{GL}(n,\mathbb{Z})$ can be determined by factorization of natural numbers less than $G(n)$  (see Remark \ref{remark-MaxOrd}).

Our results can be applied to determine the GK-dimension, the simplicity (Corollary \ref{coro-Simple}),
and cancellation properties (Corollary \ref{coro-Cancellation}) of GWAs.

The rest of this paper is organized as follows.
In Section \ref{sec_preliminaries},
we recall basic definitions and list some results from the literature.
In Section \ref{sec_SMC},
we study the sensitive multiplicity condition and
prove Theorem \ref{thm-main-}.
In Section \ref{sce-Poly},
we focus on the growth and GK-dimension of GWAs over polynomial algebras,
where Theorem \ref{thm-mainP} is proved.
In Section \ref{sce-Laurent},
 we first investigate automorphisms of Laurent polynomial algebras,
 then study the GK-dimension of GWAs over Laurent polynomial algebras, and finally prove Theorem \ref{thm-mainL}.

\section{Preliminaries}\label{sec_preliminaries}
In this section,
we recall some terminology related to generalized Weyl algebras and Gelfand-Kirillov dimension.
Our main references for GK-dimension are \cite{Krause-Lenagan_2000} and \cite[Chapter 8]{McConnell-Robson2001}.

Let $k$ be a field and $k^*=k\setminus \{0\}$.
Unless otherwise stated,
all spaces in this article are $k$-spaces,
$\dim(V)$ denotes the dimension of a $k$-space $V$,
all algebras are associative unital $k$-algebras,
and
all automorphisms are $k$-algebra automorphisms.
Denote the space spanned by a set $S$ by $kS$,
particularly by $ka$ if
$S=\{a\}$.
Given subspaces $V$ and $W$ of an algebra $D$,
denote $VW=\sum_{v\in V, w\in W}k(vw)$,
$V^0=k$,
$V^n=V^{n-1}V$ for $n\geq 1$,
and write $VW=Va$ if $W=ka$.
We use $\Z(D)$ (respectively, $\operatorname{Aut}(D)$, $\operatorname{id}=\operatorname{id}_D$) to denote the center (respectively, automorphism group, identity automorphism) of $D$.

\begin{definition}[\cite{Bavula1992Generalized}]
  \label{def-GWA}
Let $D$ be an algebra,
$\sigma\in \operatorname{Aut}(D)$, and $a\in \Z(D)$.
The \emph{generalized Weyl algebra} (GWA, for short)
$A=D(\sigma,a)=D(x,y;\sigma,a)$ with the \emph{base algebra} $D$ is defined as the algebra generated by $D$ and indeterminates $x$ and $y$, subject to the defining relations:
\begin{align*}
  \label{eqn_def-GWA-n}
 xd=\sigma(d)x, \ yd=\sigma^{-1}(d)y,\
 yx=a, \ xy=\sigma(a), \text{ for all }d\in D.
 \end{align*}
The automorphism $\sigma$ and element $a$ are called \emph{defining automorphism} and \emph{defining element} of the GWA $A$ respectively.
\end{definition}

The following example indicates why the algebra $D(\sigma,a)$ is called a GWA.
\begin{example}
  \label{exam_WeylAlg}\upshape
 The \emph{first Weyl algebra} $A_1$ over $k$ is the ring of polynomials in indeterminates $x$ and $y$ with coefficients in $k$, subject to the relations
  $yx-xy=1$.
 Then
 \[
 A_1\cong D(\sigma,a), \ x\mapsto x, \ y\mapsto y,
 \]
  where $D=k[h]$, $a=h=yx$ and $\sigma(h)=h-1$.
 The $n$-th Weyl algebra can be presented as an iterated GWA. See \cite{Zhao2018Gelfand} for more detailed examples of GWAs.
\end{example}

A \emph{$\mathbb{Z}$-filtration} of algebra $A$ is a sequence of subspaces
  \[
  \cdots\subseteq F_{i-1}\subseteq  F_{i}\subseteq F_{i+1}\subseteq \cdots, i\in\mathbb{Z}
  \]
  such that $1\in F_0$,
  $F_iF_j\subseteq F_{i+j}$ for all $i,j\in\mathbb{Z}$,
  and $A=\bigcup_{i\in\mathbb{Z}}F_i$.
  An algebra with a $\mathbb{Z}$-filtration is called a $\mathbb{Z}$-filtered algebra.
  The vector space $\operatorname{gr}(A)=\oplus_{i\in\mathbb{Z}} F_i/F_{i-1}$
  equipped with the linear multiplication defined by the rule
  \[
  (a+A_{i-1})(b+F_{j-1})=ab+F_{i+j-1}
  \]
  is called the \emph{the associated graded algebra} of $A$.
  The linear mapping $\operatorname{gr}:A\to \operatorname{gr}(A)$ is defined by
  $\operatorname{gr}(a)=a+F_{i-1}$ for all $a\in F_i/F_{i-1}$.

  For a subspace $V\subseteq A$,
  denote $\operatorname{gr}(V)=\sum_{v\in V}k\operatorname{gr}(v)$.
  The following lemma lists some basic properties of the map $\operatorname{gr}$.
  For more properties related to filtered algebras and graded algebras, see \cite[Chapter 6]{Krause-Lenagan_2000}.

\begin{lemma}[\cite{zhang1997note}, Lemma 2.1]
  \label{lemma-LeadTermMap}
  Let $A$ be a $\mathbb{Z}$-filtered algebra, $W$ be a finite dimensional\ subspace of $A$, and
  $a\in A$.
  Then
\begin{enumerate}[(i)]
  \item
  $\operatorname{gr}(Wa)\supseteq \operatorname{gr}(W)\operatorname{gr}(a)$.
  \item
  $\dim(W)=\dim(\operatorname{gr}(W))$.
  \item
  $\dim(W^m)\geq\dim((\operatorname{gr}(W))^m)$.
\end{enumerate}
\end{lemma}

The GWA $A=D(x,y;\sigma,a)$ is a free left $D$-module with a basis
$\{x^i,y^j:i\geq 0, j>0\}$.
There is a natural
 $\mathbb{Z}$-graded structure for a GWA, i.e.,
$D(\sigma,a)=\bigoplus_{i\in \mathbb{Z}}A_i$, where $A_i=Dx^i$ if $i>0$,
$A_i=Dy^{-i}$ if $i<0$, and $A_i=D$ if $i=0$.
Let $F_i=\cup_{j\leq i}A_j$.
Then
$$
  \cdots\subseteq F_{i-1}\subseteq  F_{i}\subseteq F_{i+1}\subseteq \cdots, i\in\mathbb{Z}
$$
is a $\mathbb{Z}$-filtration for $A$.
Recall that, given $\sigma\in\operatorname{Aut}(D)$,
the Ore extension $D[x;\sigma]$ over an algebra $D$ is the algebra generated by $D$ and $x$ subject to the relation $xd=\sigma(d)x$ for all $d\in D$.
The following lemma follows immediately from the $\mathbb{Z}$-graded structure of GWAs.

\begin{lemma}
  \label{lemma_Ore-subalg}
The Ore extension $D[x;\sigma]$ is a subalgebra of the GWA $D(\sigma,a)$.
\end{lemma}

It is routine to check that GWAs have the following universal property (cf. \cite[1.2.5]{McConnell-Robson2001}).

\begin{proposition}
  \label{prop-universal}
  Let $D(x,y;\sigma,a)$ be a GWA and $B$ be an algebra.
  Suppose $\phi:D\to B$ is an algebra homomorphism,
  and $x',y'\in B$ satisfy
  \[
  x'\phi(d)=\phi(\sigma(d))x', y'\phi(d)=\phi(\sigma^{-1}(d))y', y'x'=\phi(a), x'y'=\phi(\sigma(a)), \text{ for all } d\in D.
  \]
  Then there exists a unique algebra homomorphism
  $\psi:D(x,y;\sigma,a)\to B$ such that $\psi(x)=x'$, $\psi(y)=y'$ and the following diagram
  \begin{align*}
  \xymatrix{
  D \ar[rr]^{} \ar[dr]_{\phi}
                &  &    D(x,y;\sigma,a) \ar[dl]^{\psi}    \\
                & B                 }
  \end{align*}
is commutative.
\end{proposition}

Let $\Phi$ denote the set of all eventually monotone increasing functions $f:\mathbb{N}\to \mathbb{R}^+$.
For $f,g\in\Phi$ set $f\leq^* g$ if and only if there exist $c,m,N\in\mathbb{N}$ such that
\[
f(n)\leq cg(mn) \text{ for all } n>N
\]
and set
\[
f\sim g\ \Leftrightarrow
f\leq^*g \text{ and }g\leq^*f.
\]
With a slight abuse of notation,
we often write $f(n)\sim g(n)$ to mean that $f\sim g$.
One can check that $\sim$ is an equivalent relation on $\Phi$.
For $f\in\Phi$ the equivalence class $\mathcal{G}(f)\in \Phi/\sim$ is called the \emph{growth of function $f$}.
At the risk of abusing the notation,
we also write the growth of $f$ as $\mathcal{G}(f(n))$.
The partial order on $\Phi/\sim$ induced by $\leq^*$ is denoted by $\leq $.

If an algebra $A$ is generated (as an algebra) by a subspace $V$,
then $V$ is called a \emph{generating subspace} of $A$.
If $V$ and $W$ are two finite dimensional\ generating subspaces of $A$,
then $\mathcal{G}(d_V)=\mathcal{G}(d_W)$ (\cite[Lemma 1.1]{Krause-Lenagan_2000}),
where $d_V(n)=\dim\left(\sum_{i=0}^nV^i\right)$ for all $n\in\mathbb{N}$.
Thus we can define the  notion of growth of algebras,
which is independent from the choice of generating subspaces.

\begin{definition}
  \label{def-growth}
  Let $A$ be a finitely generated\ algebra.
  \begin{enumerate}[(i)]
    \item
    Suppose $A$ is
  generated by a finite dimensional\ subspace $V$.
  Then $d_V$ is called the \emph{growth function} of $A$ with respected to $V$ and
  $\mathcal{G}(A):=\mathcal{G}(d_V)$ is called the \emph{growth of $A$}.
  \item
  A finitely generated\ algebra $A$ is said to have
  \begin{itemize}
    \item
    \emph{polynomial growth} if $\mathcal{G}(A)=\mathcal{G}(n^d)$ for some $d\in\mathbb{N}$;
    \item
    \emph{exponential growth} if
  $\mathcal{G}(A)=\mathcal{G}(e^n)$;
    \item
    \emph{intermediate growth} if
  $\mathcal{G}(A)<\mathcal{G}(e^n)$
  yet $\mathcal{G}(A)\not\leq\mathcal{G}(n^m)$ for all $m\in\mathbb{N}$.
  \item
  \emph{alternative growth} if $A$ has either polynomial growth or exponential growth.
  \end{itemize}
  \end{enumerate}
\end{definition}

Functions within a wide range can be realized as
growth functions of finitely generated\ algebras.
Particularly, exponential growth is the largest one that an algebra can have.
Recently,
Bell and Zelmanov \cite{Bell2020} give a complete characterization of the functions that can occur as the growth functions of algebras.

Gelfand-Kirillov dimension is a numerical characteristic of growth, which works for not only finitely generated\ algebras but also infinitely generated algebras.

\begin{definition}
  \label{def-GKdim}
  Let $A$ be an algebra.
The \emph{Gelfand-Kirillov dimension} (\emph{GK-dimension}, for short) of $A$ is defined as
\[
\operatorname{GKdim}(A):=\sup_V{\limsup_{n\to\infty}}\log_n d_V(n)
\]
where the supremum is taken over all finite dimensional subspaces $V$ of $A$.
\end{definition}

A finite dimensional\ subspace $V\subseteq A$ is called a \emph{subframe} of $A$ if $1\in V$.
If $A$ is generated by a subframe $V$ (and thus $A$ is finitely generated),
then
\[
\operatorname{GKdim}(A)={\limsup_{n\to\infty}}\log_n \dim(V^n).
 \]

The following well known fact is useful in the computation with growth functions and GK-dimension.

\begin{lemma}
  \label{lemma-GK-sum}
  If $f(n)\in\Phi$ is a polynomial of degree $d\in\mathbb{N}^*$,
  then $g(n):=f(0)+f(1)+\cdots+f(n)$ is a polynomial of degree $d+1$.
 \end{lemma}

The following lemma provides useful properties of GWAs,
which will be used later.
\begin{lemma}[Lemma 2.7 of \cite{bavula2001isomorphism}]
  \label{lemma_iso}
Let $A=D(\sigma,a)$ be a GWA and let $\tau\in\operatorname{Aut}(D)$.
Then
\begin{enumerate}[(i)]
\item
 $A\cong D(\tau^{-1}\sigma\tau,\tau^{-1}(a))$.
\item
$A\cong D(\sigma^{-1},\sigma(a))$.
\item
If $\lambda $ is a central unit in $D$, then $A\cong D(\sigma; a\lambda)$.
In particular, if $a$ is a unit then $A\cong D(x^{\pm1};\sigma,1)= D[x^{\pm1};\sigma]$, the twisted Laurent polynomial algebra over $D$.
\end{enumerate}
\end{lemma}
\begin{proof}
  The isomorphisms are given respectively by (i) $x\mapsto x$, $y\mapsto y$ and $d\mapsto \tau^{-1}(d)$ for all $d\in D$;
  (ii) $x\mapsto y$, $y\mapsto x$ and $d\mapsto d$ for all $d\in D$;
  (iii) $x\mapsto x\lambda^{-1}, y\mapsto y$ and $d\mapsto d$ for all $d\in D$.
\end{proof}

\begin{definition}\upshape
\label{def-la}
Suppose $D$ is an algebra and $\sigma,\tau\in\operatorname{Aut}(D)$.
\begin{enumerate}[(i)]
  \item
  $\tau$ is \emph{conjugate} to $\sigma$ if
  $\tau=\eta^{-1} \sigma \eta$ for some $\eta\in \operatorname{Aut}(D)$.
  \item
  A subspace $V$ of $D$ is said to be \emph{$\sigma$-stable} if $\sigma(V)=V$.
  \item
  $\sigma$ has a \emph{finite order} if $\sigma^m=\operatorname{id}$ for some $m\geq1$.
  \item
  $\sigma$ is \emph{inner} if there exists a unit $u\in D$ such that
  $\sigma(d)=udu^{-1}$ for all $d\in D$.
  \item
  $\sigma$ is \emph{locally algebraic}

  if every finite dimensional\ subspace of $D$  is contained in a finite dimensional\ $\sigma$-stable subspace of $D$.
\end{enumerate}
\end{definition}

\begin{lemma}
  \label{lemma-LocAlgIff}
  An automorphism $\sigma\in\operatorname{Aut}(D)$ is locally algebraic if and only if
  for every $a\in D$ the set $\{\sigma^i(a): i\in\mathbb{N}\}$
is contained in a finite dimensional\ subspace of $D$.
\end{lemma}
\begin{proof}
  ($\Rightarrow$) Suppose $\sigma$ is locally algebraic.
  Then, for every $a\in D$, the subspace $k\{a\}$ is finite dimensional\
  and thus contained in a finite dimensional\ $\sigma$-stable subspace $U$.
  As a result $\{\sigma^i(a): i\in\mathbb{N}\}\subseteq U$.

  ($\Leftarrow$)
  Suppose $V=k\{a_1,a_2,\dots,a_m\}$ is a finite dimensional\ subspace of $D$.
  By the assumption, each space $k\{a_i\}$ is contained in a finite dimensional\
  $\sigma$-stable subspace $U_i\subseteq D$, $1\leq i\leq m$.
  Then $V\subseteq U:=U_1+U_2+\dots+U_m$.
  Clearly $U$ is finite dimensional\ and $\sigma$-stable.
\end{proof}

Locally algebraicity has a close relation with the GK-dimension of algebras extended by automorphisms.

\begin{lemma}[\cite{Zhao2018Gelfand}, Lemma 3.1 and Theorem 3.4]
  \label{lem_GK-CA}
  Let $A=D(\sigma,a)$ be a GWA.
  Then
  \begin{enumerate}[(i)]
    \item
    $\operatorname{GKdim}(A)\geq \operatorname{GKdim}(D)+1$.
    \item
    $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$ if moreover $\sigma$ is locally algebraic.
  \end{enumerate}

\end{lemma}

\begin{lemma}[\cite{zhang1997note},Theorem 1.1]
  \label{lemma-zhang97}
  Let $D$ be a commutative domain over an algebraically closed field $k$ such that the fraction field of $D$ is finitely generated as a field,
  and let $\sigma\in\operatorname{Aut}(D)$.
  Then the following statements are equivalent:
  \begin{enumerate}[(i)]
    \item $\operatorname{GKdim}(D[x,\sigma])=\operatorname{GKdim}(D)+1$.
    \item
    $\sigma$ is locally algebraic.
  \end{enumerate}
 If moreover $D$ is a field,
 then (i) and (ii) are equivalent to the following statement:
 \begin{enumerate}
    \item[(iii)]
     $\sigma$ has a finite order.
 \end{enumerate}
\end{lemma}

We conclude this section by two lemmas
of GWAs due to Bavula.
\begin{lemma}[\cite{Bavula1992Generalized}]
  \label{lemma-Noeth}
Let $A=D(\sigma,a)$ be a GWA.
  Suppose $a\neq0$.
  If $D$ is Noetherian (respectively, without zero divisors),
then $A$ is Noetherian (respectively, without zero divisors).
\end{lemma}

\begin{lemma}[\cite{Bavula1996}, Theorem 4.2]
  \label{lemma-Simplicity}
  Let $A=D(\sigma,a)$ be a GWA.
  Then $A$ is simple if and only if the following conditions hold:
  \begin{enumerate}[(i)]
    \item
    $D$ has no proper ideals that are $\sigma$-stable;
    \item
    $\sigma^m$ is not inner for all $m\geq 1$;
    \item
    $Da+D\sigma^i(a)=D$ for all $i\in\mathbb{N}$;
    \item
    $a$ is regular in $D$, i.e., $ad\neq 0$ for all $0\neq d\in D$.
  \end{enumerate}
\end{lemma}

\section{Sensitive multiplicity condition}
\label{sec_SMC}
Let
$P_n=k[z_1,z_2,\cdots,z_n]$ and
$Q_n$ be the fraction field of $P_n$.
In this section, we first prove that all algebras $D$ such that
$P_n\subseteq D\subseteq Q_n$ satisfy the sensitive multiplicity condition (Definition \ref{def-SMC}) without assuming that
$k$ is algebraically closed and then we prove Theorem \ref{thm-main-}.

The notion of {sensitive multiplicity condition} was introduced by Zhang \cite{zhang1997note},
and it plays an important role in GK-dimension of skew polynomial extensions.
Recall that $a\in A$ is called a \emph{regular element} in algebra $A$ if
$ab\neq 0$ and $ba\neq 0$ for all $0\neq b\in A$.

\begin{definition}
\label{def-SMC}
  Let $A$ be an algebra of $\operatorname{GKdim}(A)=d$.
We say $A$ satisfies the \emph{sensitive multiplicity condition}
(SMC for short) $\operatorname{SM}(V_0,c,d)$ if there exist a finite dimensional\ subspace $V_0$ of $A$ and a constant $c>0$ such that
if $W$ is a finite dimensional\ subspace of $A$ containing $V_0a$ for some regular element $a\in A$, then
$\dim(W^m)\geq c\dim(W) m^d$ for all $m\in\mathbb{N}$.
\end{definition}

\begin{lemma}[\cite{zhang1997note}, Lemma 3.1(2)]
  \label{lemma-SMCQuotient}
  Let $A$ be a commutative domain and $Q(A)$ be the fraction field of $A$.
  Then $Q(A)$ satisfies $\operatorname{SM}(V_0,c,d)$ if and only if $A$ satisfies
  $\operatorname{SM}(V_0a,c,d)$ for some $0\neq a\in A$ such that $V_0a\subseteq A$.
\end{lemma}
\begin{proof}
  Suppose $Q(A)$ satisfies $\operatorname{SM}(V_0,c,d)$.
  Since $V_0$ is finite dimensional,
  we can write $V_0=Va^{-1}$ for some finite dimensional\ subspace $V\subseteq A$ and $0\neq a\in A$.
  Suppose $W$ is a finite dimensional\ subspace of $A$ such that $W\supseteq Vb$ for some regular element $b\in A$.
  Then $W\supseteq V_0(ab)$ and $ab\in Q(A)$ is regular.
  By SMC for $Q(A)$,
  \begin{align*}
    \dim(W^m)\geq c\dim(W)m^d, \ \ m\in\mathbb{N}.
  \end{align*}
  Thus $A$ satisfies $\operatorname{SM}(V_0a,c,d)$ for $a\in A$ regular and $V_0a=V\subseteq A$.

  Suppose $A$ satisfies
  $\operatorname{SM}(V_0a,c,d)$ for some $V_0\subseteq Q(A)$ and $0\neq a\in A$ such that $V_0a\subseteq A$.
  Let $W$ be a finite dimensional\ subspace of $Q(A)$ and
  $V_0q\subseteq W$ for some $0\neq q\in Q(A)$.
  We can write $W=W'b^{-1}$ for some finite dimensional\ subspace $W'\subseteq A$ and $0\neq b\in A$
  such that $a^{-1}qb\in A$.
  Then $(V_0a)(a^{-1}qb)=V_0qb\subseteq Wb=W'$ and thus, by SMC for $A$,
  \begin{align*}
    \dim(W^m)=\dim(W'^m)\geq c\dim(W')m^d=c\dim(W)m^d, \ \ m\in\mathbb{N}.
  \end{align*}
  Thus $Q(A)$ satisfies $\operatorname{SM}(V_0,c,d)$.
\end{proof}

\begin{lemma}
  \label{lemma-SMC-GK+2}
  If $D$ is a commutative algebra satisfying SMC and
  $\sigma\in\operatorname{Aut}(D)$ is not locally algebraic,
  then $\operatorname{GKdim}(D(\sigma,a))\geq \operatorname{GKdim}(D)+2$.
\end{lemma}
\begin{proof}
  By Lemma \ref{lemma_Ore-subalg}, $D[x,\sigma]$ is a subalgebra of $D(\sigma,a)$.
  Thus, by \cite[Lemma 3.1]{Krause-Lenagan_2000} and \cite[Proposition 3.3 (1)]{zhang1997note}, we have
  \[
  \operatorname{GKdim}(D(\sigma,a))\geq \operatorname{GKdim}(D[x,\sigma])\geq \operatorname{GKdim}(D)+2,
  \]
  which completes the proof.
\end{proof}

Following the proof of \cite[Theorem 3.2]{zhang1997note}, we have the following lemma without the assumption that $k$ is algebraically closed.

\begin{lemma}
  \label{lemma-PnSMC}
   For $n\geq 0$, the polynomial algebra $P_n$ satisfies $\operatorname{SM}(V_n,c_n,n)$
  where
  \[
  V_n=k+kz_1+\cdots+kz_n,\
  c_n=\frac{1}{2^n\sqrt5^{n(n+1)}n!}.
  \]
\end{lemma}
\begin{proof}
The proof of \cite[Theorem 3.2]{zhang1997note} with small modification still works for our lemma.
For the reader's convenience, we include the slightly modified proof in the Appendix.
\end{proof}

\begin{corollary}
  \label{coro-SMCPoly}
  Suppose $D$ is an algebra such that $P_n\subseteq D\subseteq Q_n$.
  Then $D$ satisfies SMC.
\end{corollary}
\begin{proof}
  Let $V=k+kz_1+\cdots+kz_n\subseteq D$.
  Suppose that $W$ is a finite dimensional\ subspace of $D$ such that
  $Va\subseteq W$ for some $0\neq a\in D$.
  Note that $\operatorname{GKdim}(D)=n$ since $\operatorname{GKdim}(P_n)=\operatorname{GKdim}(Q_n)=n$.
  By Lemmas  \ref{lemma-PnSMC} and \ref{lemma-SMCQuotient}, $Q_n$ satisfies $\operatorname{SM}(V_n,c_n,n)$, where $V_n, c_n$ are as in Lemma \ref{lemma-PnSMC}.
  Hence by the definition of SMC
  \[
  \dim(W^m)\geq c\dim(W) m^n \text{ for some $c>0$ and all}\ m\in\mathbb{N},
  \]
  which means that $D$ satisfies SMC.
\end{proof}

Now we have the following result without the assumption that
$k$ is algebraically closed.

\begin{proposition
  \label{prop-Zhang97PL}
  Let $A=D(\sigma,a)$ be a GWA,
   where $P_n\subseteq D\subseteq Q_n$.
  Then the following statements are equivalent:
  \begin{enumerate}[(i)]
    \item $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$.
    \item
    $\sigma$ is locally algebraic.
  \end{enumerate}
\end{proposition}

\begin{proof}
  By Corollary \ref{coro-SMCPoly}, $D$ satisfies SMC.
  Then the statement follows from Lemmas \ref{lem_GK-CA} and \ref{lemma-SMC-GK+2}.
\end{proof}

Proposition \ref{prop-Zhang97PL} is a partial converse of Lemma \ref{lem_GK-CA} (ii).
If $k$ is algebraically closed,
we have the following proposition,
which is an analogue of Lemma \ref{lemma-zhang97}.

\begin{proposition}
  \label{prop-Zhang97}
  Let $A=D(\sigma,a)$ be a GWA,
   where $D$ is a commutative domain over an algebraically closed field $k$ such that the fraction field of $D$ is finitely generated as a field.
  Then the following statements are equivalent:
  \begin{enumerate}[(i)]
    \item $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$.
    \item
    $\sigma$ is locally algebraic.
  \end{enumerate}
 If moreover $D$ is a field,
 then (i) and (ii) are equivalent to the following statement:
 \begin{enumerate}
    \item[(iii)]
     $\sigma$ has a finite order.
    \end{enumerate}
\end{proposition}

\begin{proof}
  (ii)$\Rightarrow$(i) follows from Lemma \ref{lem_GK-CA}.
  (i)$\Rightarrow$(ii) follows from Lemma \ref{lemma-SMC-GK+2}.

  (ii)$\Leftrightarrow$(iii).
  The proof of Lemma \ref{lemma-zhang97} (see \cite[p.369]{zhang1997note}) still works for our statements.
\end{proof}

Note that (i)$\Rightarrow$(ii) in Proposition \ref{prop-Zhang97} fails if $D$ is not commutative,
see Example \ref{exam-GK+1-NotLA}.

\begin{example}\upshape
  \label{exam-GK+1-NotLA}
  Let $D=k[z^{\pm1},s][d;\sigma]$ be an Ore extension over polynomial ring $k[z^{\pm1},s]:=k[z^{\pm1}][s]$,
  where the automorphism $\sigma$ is defined by
  $\sigma(z)=z$, $\sigma(s)=zs$.
  Let $A=D(x,y;\tau,1)$ be a GWA, where $\tau\in\operatorname{Aut}(D)$ satisfies
  $\tau(z)=z$, $\tau(s)=s$, $\tau(d)=z^{-1}d$.
  Then $\tau$ is not locally algebraic but
  $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1=5$.
\end{example}

\begin{proof}
  First note that $D$ is the differential difference algebra defined in \cite[Example 3.12]{ZhangZhao2013algebras} and
  $\operatorname{GKdim}(D)=4$ (see \cite[p.494]{ZhangZhao2013algebras}).
  By Lemma \ref{lem_GK-CA},
  $\operatorname{GKdim}(A)\geq \operatorname{GKdim}(D)+1=5$.
  It remains to show that
  $\operatorname{GKdim}(A)\leq 5$.
  Rewrite $A$ as an Ore extension over $C:=k[x^{\pm1},z^{\pm1},s]$,
  that is,
  $A=C[d;\delta]$,
  where $\delta\in\operatorname{Aut}(C)$ is defined by
  $\delta(z)=z$, $\delta(x)=zx$, $\delta(s)=zs$.
  Let $V=k\{x^{\pm1},z^{\pm1},s,1\}$.
  Then $W=V+kd$ is a finite dimensional\ generating subspace of $A$.
  By induction, we can prove that, for $m\geq 1$,
  \begin{align*}
    W^m=\sum_{i=0}^m W_i^{m-i}d^i,\ \
    W_i:=V+\delta(V)+\cdots+\delta^i(V)
  \end{align*}
  (see also \cite[Eq. (3.3.1) and (3.3.2)]{zhang1997note})
  and that,
  for $0\leq i\leq m$,
  \begin{align*}
   W_i=k\{1,
   z^{\pm1},z^jx,z^{-j}x^{-1},
   z^js: 0\leq j\leq i\}.
  \end{align*}
  Thus
  \begin{align*}
    W_i^{m-i}\subseteq k\{z^{p_1}x^{p_2}s^{p_3}:-i(m-i)\leq p_1\leq i(m-i),
    -(m-i)\leq p_2\leq m-i,
    0\leq p_3\leq m-i\}.
  \end{align*}
  As a result,
  \begin{align*}
    \dim(W^m)=
    \sum_{i=0}^m \dim(W_i^{m-i})
    \leq \sum_{i=0}^m(2i(m-i)+1)(2(m-i)+1)(m-i+1)\sim m^5
  \end{align*}
  where $\sim$, meaning the equivalence of two functions in $\Phi$ (see Section \ref{sec_preliminaries}),
  follows from Lemma \ref{lemma-GK-sum}.
  Therefore, $\operatorname{GKdim}(A)={\limsup_{m\to\infty}}\log_m\dim (W^m)\leq 5$
  as desired.
\end{proof}

Proposition \ref{prop-Zhang97} together with Lemma \ref{lemma-Simplicity} give a criterion of the simplicity for certain GWAs.
\begin{corollary}
  \label{coro-Simple}
  Suppose that $k$ is algebraically closed and $A=D(\sigma,a)$ is a GWA,
  where $D$ is a finitely generated\ field and $a\neq 0$.
  Then the following statements are equivalent:
  \begin{enumerate}[(i)]
    \item
    $A$ is simple.
    \item
    $\operatorname{GKdim}(A)\neq\operatorname{GKdim}(D)+1$.
  \end{enumerate}
\end{corollary}
\begin{proof}
Note that if $D$ is a field and $a\neq 0$ then
conditions (i)--(iii) in Lemma \ref{lemma-Simplicity} are satisfied automatically.
As a result,
under the assumption,
$A$ is simple if and only if $\sigma^m$ is not inner for all $m\geq 1$,
if and only if $\sigma^m\neq \operatorname{id}$ for all $m\geq1$,
if and only if $\operatorname{GKdim}(A)\neq\operatorname{GKdim}(D)+1$
(by Proposition \ref{prop-Zhang97}).
\end{proof}

\

Now we are ready to prove Theorem \ref{thm-main-}.

\noindent
\emph{\textbf{Proof of Theorem \ref{thm-main-}.}}
It follows from Propositions \ref{prop-Zhang97PL} and \ref{prop-Zhang97}, and Corollary \ref{coro-Simple}.
\hfill\qed
\section{GK-dimension of GWAs over polynomial algebras}
\label{sce-Poly}
In this section,
we study GK-dimension of GWAs over $P_n:=k[z_1,z_2,\dots,z_n]$,
in particular over $P_2$.
We first prove a lemma, which will be used later.
\begin{lemma}
  \label{lemma_Power}
  Let $A=D(x,y;\sigma,a)$ be a GWA, $m\in\mathbb{N}^*$
  and $B$ be the subalgebra of $A$ generated by $D\cup \{x^m,y^m\}$.
  Then $B=D(x^m,y^m;\sigma^m,b)$ is a GWA and $\operatorname{GKdim}(A)=\operatorname{GKdim}(B)$, where
  \[
  b=b_m=\sigma^{-(m-1)}(a) \sigma^{-(m-2)}(a)\cdots \sigma^{-1}(a)a.
  \]
\end{lemma}
\begin{proof}
  First note that $x^md=\sigma^m(d)x^m$ and
   $y^md=\sigma^{-m}(d)y^m$ for all $d\in D$.
   Let \[
  b=\sigma^{-(m-1)}(a) \sigma^{-(m-2)}(a)\cdots \sigma^{-1}(a)a\in D.
  \]
  By using the defining relations of $A$,
  it is straightforward to check that $b=y^mx^m$.
  We claim that $b$ is central in $D$.
  For each $d\in D$ and $i\in\mathbb{Z}$,
  \[
  \sigma^i(a)d

  =\sigma^i(a\sigma^{-i}(d))
  =\sigma^i(\sigma^{-i}(d)a)

  =d\sigma^i(a),
  \]
  which implies $bd=db$ and the claim holds.

  Similarly, $\sigma^i(b)$ is central in $D$ for all $i\in \mathbb{N}$.
  Thus
  \begin{align*}
    \sigma^m(b)

    =\sigma(a)\sigma^2(a)\cdots\sigma^m(a)
    =\sigma^m(a)\sigma^{m-1}(a)\cdots\sigma(a)
    =x^my^m.
  \end{align*}
  Therefore, $B=D(x^m,y^m;\sigma^m,b)$ is a GWA.

  Since $A$ is finitely generated\ as a right $B$-module,
  we obtain by \cite[Proposition 5.5]{Krause-Lenagan_2000} that
  $\operatorname{GKdim}(A)=\operatorname{GKdim}(B)$.
\end{proof}

Now we can prove the following proposition.

\begin{proposition}\label{prop_GK_FiniteOrd}
  Suppose $A=D(\sigma,a)$ is a GWA.
  \begin{enumerate}[(i)]
    \item
    If $\sigma^m$ is an inner automorphism for some $m\geq 1$,
  then $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$.
  \item
  If $\sigma$ has a finite order then $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$.
  \end{enumerate}
\end{proposition}
\begin{proof}
(i) Let $A=D(x,y;\sigma,a)$ and $B$ be the subalgebra of $A$ generated by $D\cup \{x^m,y^m\}$.
Suppose $u\in D$ is an invertible element such that
  $\sigma^m(d)=udu^{-1}$ for all $d\in D$.
By Lemma \ref{lemma_Power}, $B=D(x^m,y^m;\sigma^m,b)$
where $b$ is a central element in $D$.
We claim further that $B=D(u^{-1}x^m,uy^m;\operatorname{id},b)$.
For each $d\in D$,
\[
u^{-1}x^md=u^{-1}\sigma^m(d)x^m=u^{-1}udu^{-1}x^m=du^{-1}x^m
\]
and
$uy^md=duy^m$ similarly.
Note that $\sigma^{\pm m}(u)=u$ and $\sigma^{\pm m}(u^{-1})=u^{-1}$.
Thus \[
uy^mu^{-1}x^m=u\sigma^{-m}(u^{-1})y^mx^m=y^mx^m=b
\]
and
\[
u^{-1}x^m uy^m=x^m y^m=\sigma^m(b)=ubu^{-1}=b,
\]
where the last equality holds since
$b$ is central in $D$.
Thus,
$B=D(u^{-1}x^m,uy^m;\operatorname{id},b)$.

Now it follows from Lemmas \ref{lemma_Power} and \ref{lem_GK-CA} that
$\operatorname{GKdim}(A)=\operatorname{GKdim}(B)=\operatorname{GKdim}(D)+1$.

(ii) follows from (i).
\end{proof}

Proposition \ref{prop-Zhang97} implies that the converse of Proposition \ref{prop_GK_FiniteOrd} (ii) also holds under certain conditions, while it does not hold in general,
see Example \ref{exam_FiniteOrd}.

\begin{corollary}
  Let $D$ be an algebra and $D[x,y]$ be the polynomial algebra in two variables over $D$.
  Suppose that
  $I$ is the ideal of $D[x,y]$ generated by $xy-a$ for some $a\in \Z(D)$.
  Then $\operatorname{GKdim}(D[x,y]/I)=\operatorname{GKdim}(D)+1$.
\end{corollary}
\begin{proof}
  By Proposition \ref{prop-universal},
  there exists a homomorphism $\psi:D(x,y;\operatorname{id},a)\to D[x,y]/I$ such that $\psi(x)=x$, $\psi(y)=y$ and $\psi(d)=d$ for all $d\in D$.
  Note that the homomorphism
  \[
  \psi':D[x,y]\to D(x,y;\operatorname{id},a), x\mapsto x,y\mapsto y, d\mapsto d, \text{ for all } d\in D
  \]
  maps $I$ to $0$ since
  $\psi'(xy-a)=xy-a=0$.
  Hence $\psi'$ induces a homomorphism $\psi'':D[x,y]/I\to D(x,y;\operatorname{id},a)$.
  It is easy to check that $\psi''\psi=\operatorname{id}_{D(\operatorname{id},a)}$ and $\psi\psi''=\operatorname{id}_{D[x,y]}$.
  Thus $D[x,y]/I\cong D(x,y;\operatorname{id},a)$.
  It  follows from
  Proposition \ref{prop_GK_FiniteOrd} that
  $\operatorname{GKdim}(D[x,y])=\operatorname{GKdim}(D(x,y;\operatorname{id},a))=\operatorname{GKdim}(D)+1$.
\end{proof}

Now we consider GWAs $P_n(\sigma,a)$ over $P_n$.
Denote by $\deg(f)$ the total degree of $f\in P_n$.
If $n=0$, then $\sigma=\operatorname{id}_k$ and $\operatorname{GKdim}(k(\sigma,a))=1$.
If $n=1$, then $\sigma\in\operatorname{Aut}(k[z])$ is determined by $\sigma(z)=bz+c$ for some $b\in k^*$ and $c\in k$.
Hence every $\sigma\in\operatorname{Aut}(k[z])$ is locally algebraic and thus $\operatorname{GKdim}(k[z](\sigma,a))=\operatorname{GKdim}(k[z])+1=2$.
For $n\geq2$, we have the following

\begin{lemma}
  \label{lem_GK-infty}
  Let $D=P_n$, $n\geq 2$, and $A=D(x,y;\sigma,a)$ be a GWA over $D$.
  Suppose that there exist $z\in D$ and real number $r\geq2$ such that
  \(
  \deg(\sigma^{m+1}(z))\geq r \deg (\sigma^{m}(z))
  \)
  for all $m\in\mathbb{N}$.
  Then $A$ has exponential growth.
  In particular, $\operatorname{GKdim}(A)=\infty$.
\end{lemma}
\begin{proof}
   Suppose there exist $z\in D$ and $r\geq2$ such that \(
  \deg( \sigma^{m+1}(z))\geq r \deg (\sigma^{m}(z))
  \)
  for all $m\in\mathbb{N}$.
  Consider elements of $A$ of the form
  \[
  z^{\epsilon_0}xz^{\epsilon_1}x\cdots z^{\epsilon_{p-1}}xz^{\epsilon_p},
  \]
  where $p\in\mathbb{N}$ and each $\epsilon_i$ is either $0$ or $1$.
  There are $2^{p+1}$ such elements (denoted by $a_1,\ldots,a_{2^{p+1}})$ for each $p\in\mathbb{N}$.
  It is easy to see that $a_i\in V^{2p+1}$ where $V=k+kz_1+\cdots+ kz_n+kx$.

  \emph{Claim.} $a_1,\ldots,a_{2^{p+1}}$ are linearly independent for all $p\in \mathbb{N}$.

  \emph{Proof of the claim.}
  Suppose there exists
  $q\in\mathbb{N}$ such that $a_1,\ldots,a_{2^{q+1}}$ are linearly dependent.
  Note that
  \begin{align*}
    z^{\epsilon_0}xz^{\epsilon_1}x\cdots z^{\epsilon_{p-1}}xz^{\epsilon_q}
    &=z^{\epsilon_0}\sigma(z^{\epsilon_1})\sigma^2(z^{\epsilon_2})\cdots \sigma^q(z^{\epsilon_q}) x^q\\
    &=z^{\epsilon_0}(\sigma(z))^{\epsilon_1}(\sigma^2(z))^{\epsilon_2}\cdots (\sigma^q(z))^{\epsilon_q} x^q
\end{align*}
  and thus the degree with respect to $x$ of each $a_i$ is $q$.
  Write $a_i=t_ix^q$, $i=1,2,\ldots,2^{q+1}$, $t_i\in D$.
  Since $a_1,\ldots,a_{2^{q+1}}$ are linearly dependent,
  so are $t_1,\ldots,t_{2^{q+1}}$.
  Thus there exist $t_i$ and $t_j$ such that $\deg (t_i)=\deg (t_j)$.
  Namely,
  there are two sequences $(\epsilon_0,\ldots, \epsilon_q)\neq (\epsilon'_0,\ldots, \epsilon'_q)$, each $\epsilon_i,\epsilon'_i\in\{0,1\}$,
  such that
  \begin{align}\label{eqn-deg}
  \epsilon_0d_0+\dots+\epsilon_qd_q
  =\epsilon'_0d_0+\dots+\epsilon'_qd_q
  \end{align}
  where each $d_i:=\deg( \sigma^i(z))$.
  Let $l$ be the maximal index such that $\epsilon_l\neq \epsilon'_l$.
  Clearly $l\geq1$.
  Without loss of generality, assume $\epsilon_l=1$ and $\epsilon'_l=0$.
  Then Eq. (\ref{eqn-deg}) becomes
  \[
  \epsilon_0d_0+\dots+\epsilon_{l-1}d_{l-1}+d_l

  =\epsilon'_0d_0+\dots+\epsilon'_{l-1}d_{l-1}.
  \]
  By hypothesis,
  $d_{m+1}\geq rd_m$ for all $m\in\mathbb{N}$,
  which implies that $d_i\leq (1/r)^{l-i}d_l$.
  Thus
  \begin{align*}
    d_l&=(\epsilon'_0-\epsilon_0)d_0+\dots+
  (\epsilon'_{l-1}-\epsilon_{l-1})d_{l-1}\\
  &\leq   d_0+\dots+d_{l-1}
  \leq \sum_{i=0}^{l-1} \left(\frac1r\right)^{l-i}d_l
  =\frac {1-\left(\frac1r\right)^{l}}{r-1}d_l
  <d_l,
 \end{align*}
 a contradiction.
 Hence the claim holds.

 It follows from the claim that
 $\dim(V^{2p+1})\geq 2^{p+1}$ for all $p\in \mathbb{N}$ and thus $A$ has exponential growth.
\end{proof}

\begin{convention}\upshape
  \label{conv-auto}
For an automorphism $\sigma:P_n\to P_n$ defined by $\sigma(z_i)=f_i\in P_n$, $1\leq i\leq n$,
we denote
$\sigma=(f_1, f_2, \dots,f_n)$.
\end{convention}
Recall that
an automorphism $\sigma=(f_1, f_2, \dots,f_n)\in\operatorname{Aut}(P_n)$ is said to be \emph{triangular}
if
$
f_i=\lambda_i z_i+g_i,
$
where $\lambda_i\in k^*$ for $1\leq i\leq n$, $g_i\in k[z_{i+1},\ldots,z_n]$ for $1\leq i<n$ and $g_n\in k$.

\begin{lemma}
  \label{lemma_ConLAlgebraic}
  Suppose that $\sigma\in\operatorname{Aut}(P_n)$ is conjugate to a triangular automorphism.
  Then $\sigma$ is locally algebraic.
\end{lemma}
\begin{proof}
  Let $\delta, \tau\in \operatorname{Aut}(P_n)$ and $\tau=\delta^{-1}\sigma\delta$ be triangular.
Note that for every finite dimensional\ subspace $U\subseteq P_n$,
the subset $\delta^{-1}(U)\subseteq P_n$ is also  finite dimensional.
Since $\tau$ is triangular and
a triangular automorphism of $P_n$ is locally algebraic (\cite[Proposition 2]{Leroy1988On}),
we have that $\tau$ is locally algebraic.
Hence, there exists a finite dimensional\ subspace $V_1\subseteq P_n$ such that
$\delta^{-1}(U)\subseteq V_1$ and $\tau(V_1)=V_1$.
Thus $\delta(V_1)$ is a finite dimensional\ subspace of $P_n$ and $U\subseteq \delta(V_1)$.
Note that $\sigma(\delta(V_1))=\delta\tau\delta^{-1}(\delta(V_1))=\delta(V_1)$,
i.e.,
$\delta(V_1)$ is $\sigma$-stable.
Therefore $\sigma$ is locally algebraic.
\end{proof}

The converse of Lemma \ref{lemma_ConLAlgebraic} is true in the case $n=2$ (see Theorem \ref{thm_GKdichotomy}).
However it does not hold in general, see the following example.

\begin{example}
  \label{exam-Nagata}\upshape
  The Nagata automorphism (\cite{nagata1972automorphism})
  \[
  \left(
  x-2y(xz+y^2)-z(xz+y^2)^2, y+z(xz+y^2), z
  \right)
   \in \operatorname{Aut}(k[x,y,z])\]
is locally algebraic but not conjugate to a triangular automorphism (see \cite{bass1984non,furter2009quasi}).
\end{example}

\begin{corollary}
  \label{coro_PowerConjTri}
  Let $A=P_n(\sigma,a)$ be a GWA.
  If $\sigma^m$ is conjugate to a triangular automorphism for some $m\geq 1$,
  then $\operatorname{GKdim}(A)=n+1$.
\end{corollary}
\begin{proof}
  Let $D=P_n$, and $B$ and $b$ be as in Lemma \ref{lemma_Power}.
  Then $B=D(x^m,y^m;\sigma^m,b)$ and $\operatorname{GKdim}(A)=\operatorname{GKdim}(B)$ by Lemma \ref{lemma_Power}.
  By the assumption and Lemma \ref{lemma_ConLAlgebraic}, $\sigma^m$ is locally algebraic and thus $\operatorname{GKdim}(B)=n+1$
  by Lemma \ref{lem_GK-CA}.
  Hence $\operatorname{GKdim}(A)=n+1$.
\end{proof}

Now we consider the case $n=2$.
\begin{theorem}
  \label{thm_GKdichotomy}
  Let $A=k[z_1,z_2](\sigma,a)$ be a GWA.
  Then $\operatorname{GKdim}(A)=3$ if $\sigma$ is conjugate to a triangular automorphism, and $\operatorname{GKdim}(A)=\infty$ otherwise.
\end{theorem}
\begin{proof}
Let $D=k[z_1,z_2]$.
Suppose that $\sigma$ is conjugate to a triangular automorphism.
By Corollary \ref{coro_PowerConjTri},  $\operatorname{GKdim}(A)=3$.

Suppose that $\sigma$ is not conjugate to a triangular automorphism.
Let $\pi=(z_2,z_1)\in\operatorname{Aut}(D)$ (see Convention \ref{conv-auto}).
 By \cite[Lemma 3]{Lane1975Fixed},
 $\sigma$ is conjugate to an automorphism $\sigma'$ of the form
 \begin{align*}
   \label{eqn-square}
 \sigma'=\tau_1\pi\tau_2\cdots\pi\tau_s\pi,\
 s\geq 1,
 \end{align*}
 where,
 for each positive integer $i\leq s$,
 \[
 \tau_i=(\alpha_i z_1+\beta_i(z_2),u_iz_2+v_i),\
 \alpha_i,u_i\in k^*, v_i\in k,\
 \beta_i(z_2)\in k[z_2],
 d_i:=\deg(\beta_i)>1.
  \]
 We claim that
 \[
 \sigma'(z_2)=\lambda z_2^{d_1\cdots d_s}+\text{(lower total degree terms)}, \ \lambda\in k^*.
 \]
 We prove the claim by induction on $s$.
 If $s=1$,
 then $\sigma'(z_2)=\tau_1(z_2)=\alpha_1z_1+\beta_1(z_2)
 =\lambda z_2^{d_1}+\alpha_1z_1$.
 Thus the claim holds for $s=1$.
 Suppose the claim holds for automorphisms with $s\geq 1$.
 For $\sigma'=\tau_1\pi\tau_2\cdots\pi\tau_{s+1}\pi$,
 we have that
 \begin{align*}
   \sigma'(z_2)&=\tau_1\pi
   \left(\lambda z_2^{d_2d_3\cdots d_{s+1}}+T \right)\\
   &=\tau_1
   \left(\lambda z_1^{d_2d_3\cdots d_{s+1}} \right)+\tau_1\pi(T)\\
   &=\lambda (\alpha_1z_1+\beta_1(z_2))^{d_2d_3\cdots d_{s+1}}+\tau_1\pi(T)\\
   &=\lambda' z_2^{d_1d_2\cdots d_{s+1}}+T' +\tau_1\pi(T)
 \end{align*}
 where $\lambda,\lambda'\in k^*$,
$\deg (T)<d_2d_3\cdots d_{s+1}$
  and $\deg (T')<d_1d_2\cdots d_{s+1}$.
 Since $\deg (\tau_1\pi(T))=\deg (\tau_1(T))<d_1d_2\cdots d_{s+1}$,
 we have that
 \[
 \sigma'(z_2)=\lambda' z_2^{d_1d_2\cdots d_{s+1}}+\text{ (lower total degree terms)}
  \]
  as desired.

 It follows from the claim that
\begin{align*}
 \deg (\sigma'^{m+1}(z_2))&= \deg((\tau_1\pi\tau_2\cdots\pi\tau_s\pi)^{m+1}(z_2))\\
 &=(d_1\cdots d_s)^{m+1}\\
 &\geq (d_1\cdots d_s)^{m}=2\deg (\sigma'^{m}(z_2)).
\end{align*}
Now, by Lemma \ref{lem_GK-infty},
$D(\sigma',b)$ has exponential growth and
$\operatorname{GKdim}(D(\sigma',b))=\infty$ for all $b\in k[z_1,z_2]$.
Since $\sigma$ is conjugate to $\sigma'$,
we may suppose $\sigma=\delta^{-1}\sigma'\delta$ for some $\delta\in \operatorname{Aut}(k[z_1,z_2])$.
Then, by Lemma \ref{lemma_iso}, we have $k[z_1,z_2](\sigma,a)\cong k[z_1,z_2](\sigma',\delta^{-1}(a))$.
Hence $\operatorname{GKdim}(D(\sigma,a))=\operatorname{GKdim}(D(\sigma',\delta^{-1}(a)))=\infty$.
\end{proof}

\begin{remark}\upshape
  \label{remark-P2-P3}
Note that a key element in
the proof of Theorem \ref{thm_GKdichotomy} is \cite[Lemma 3]{Lane1975Fixed},
which is based on the famous theorem of Jung and van der Kulk \cite{Jung1942Uber,vanderKulk1953} that asserts
that $\operatorname{Aut}(P_2)$ is generated by triangular automorphisms and
the permutation $\pi=(z_2,z_1)$,
that is,
all $\sigma\in\operatorname{Aut}(P_2)$ can be written as a product of triangular automorphisms (and their inverses) and $\pi$.
Shestakov and Umirbaev \cite{Shestakov2003}
proved that there exist automorphisms $\tau\in\operatorname{Aut}(P_3)$ that are not tame.
As a result,
such $\tau$ cannot be presented as a product of triangular automorphisms (and their inverses) and affine automorphisms (i.e., automorphisms $\delta=(f_1,f_2, f_3)\in\operatorname{Aut}(P_3)$ with $\deg(f_i)=1$ for $i=1,2,3$) of $P_3$.
\end{remark}

\

We can now put everything together to prove Theorem \ref{thm-mainP}.

\noindent
\emph{\textbf{Proof of Theorem \ref{thm-mainP}.}}
(\ref{itemP-PolyGrowth}) $\Rightarrow$
(\ref{itemP-NotExGrowth})
and (\ref{itemP-ConjTri}) $\Rightarrow$
(\ref{itemP-PowerConjTri})
are trivial.
(\ref{itemP+1}) $\Leftrightarrow$
(\ref{itemP-GKfinite}) $\Leftrightarrow$ (\ref{itemP-LA})
follow from Theorem \ref{thm_GKdichotomy} and
Proposition \ref{prop-Zhang97PL} respectively.
(\ref{itemP-NotExGrowth}) $\Rightarrow$ (\ref{itemP-ConjTri})
follows from the proof of Theorem \ref{thm_GKdichotomy}.
(\ref{itemP-PowerConjTri}) $\Rightarrow$ (\ref{itemP+1})
follows from Corollary \ref{coro_PowerConjTri}.

(\ref{itemP-LA}) $\Rightarrow$ (\ref{itemP-PolyGrowth}).
Suppose $\sigma$ is locally algebraic.
Let $V$ be a $\sigma$-stable generating subframe of $P_2$ such that $a\in V$ (and thus $\sigma(a)\in V$).
Then $W:=V+kx+ky$ is a generating subframe of $A$.
Since
  \[
  W^{2m}=(V+kx+ky)^{2m}\supseteq V^m+V^mx+\cdots+V^mx^m, m\geq1,
  \]
we have that
\begin{align}
  \label{eqn-Lower}
  \dim(W^{2m})> (m+1)\dim(V^m), m\geq1.
\end{align}
On the other hand,
we claim that
  \begin{align}\label{claim_W^n}
    W^m\subseteq\sum_{i=1}^mV^{m}x^i+\sum_{i=1}^mV^{m}y^i+V^{m}, \ m\geq 1.
  \end{align}
The claim is obvious when $m=1$.
Suppose that the inclusion (\ref{claim_W^n}) holds for some $m\geq 1$. Then
 \begin{align*}
   W^{m+1}&\subseteq W\left(\sum_{i=1}^mV^{m}x^i+\sum_{i=1}^mV^{m}y^i+V^{m}\right)\ \ (\text{induction hypothesis})\\
   &=kx\sum_{i=1}^mV^{m}x^i+kx\sum_{i=1}^mV^{m}y^i+kxV^{m}\\
   &\ \ \ +ky\sum_{i=1}^mV^{m}x^i+ky\sum_{i=1}^mV^{m}y^i+kyV^{m}\\
 &\ \ \ \ \ \ +V\sum_{i=1}^mV^{m}x^i+V\sum_{i=1}^mV^{m}y^i+VV^{m}\\
 &=\sum_{i=1}^mV^{m}x^{i+1}+ \sum_{i=1}^mV^{m}\sigma(a)y^{i-1}+V^{m}x\ \ [\text{since } xV^{m}=V^{m}x]\\
   &\ \ \ +\sum_{i=1}^mV^{m}ax^{i-1}+\sum_{i=1}^mV^{m}y^{i+1}+V^{m}y\ \ [\text{since } yV^{m}=V^{m}y]\\
 &\ \ \ \ \ \ +\sum_{i=1}^mV^{m+1}x^i+\sum_{i=1}^mV^{m+1}y^i+V^{m+1}\\
 &\subseteq\sum_{i=1}^{m+1}V^{m+1}x^i+\sum_{i=1}^{m+1}V^{m+1 }y^i+V^{m+1 }\ \ [\text{since } a,\sigma(a)\in V].
  \end{align*}
Hence the inclusion (\ref{claim_W^n}) holds for  $m+1$ and thus
for all $m\geq 1$.

By the $\mathbb{Z}$-graded structure of GWAs, the sum in Eq. (\ref{claim_W^n}) is direct and thus
\begin{align}
  \label{eqn-Upper}
  \dim(W^m)\leq (2m+1)\dim(V^{m}), m\geq1.
\end{align}
Combining Eq. (\ref{eqn-Lower}) and (\ref{eqn-Upper}) gives
\[
(m+1)\dim(V^m)\leq \dim(W^{2m})\leq (4m+1)\dim(V^{2m}), m\geq 1.
\]
Since the growth of $P_2$ is independent from the choice of generating subspaces,
$\dim(V^m)\sim \dim(kz_1+kz_2)^m\sim m^2$.
We obtain that $\dim(W^m)\sim m^3$ and thus $A$ has polynomial growth.
\begin{figure}[h!]
\begin{align*}
  \xymatrix{
  (a) \ar@2{<-}[d]_{\ref{coro_PowerConjTri}} \ar@2{<->}[r]^{\ref{thm_GKdichotomy}}
                & (b) \ar@2{<->}[r]^{\ref{prop-Zhang97PL}}
                &  (e)\ar@2{->}[r]
                & (c)  \ar@2{->}[d] \\
  (g)  \ar@2{<-}[r]
                & (f)
                &
                &(d)  \ar@2{->}[ll]_{\text{Proof of } \ref{thm_GKdichotomy}}   }
\end{align*}
\caption{Implication relations among the items in Theorem \ref{thm-mainP}}
\label{fig}
\end{figure}
Figure \ref{fig} summarizes the relations we have proved among the items in Theorem \ref{thm-mainP},
which completes the proof.
\hfill \qed

\

We conclude this section with an example,
which shows that the dichotomy of GK-dimension appears in Theorem \ref{thm_GKdichotomy} (i.e., $\operatorname{GKdim}(P_2(\sigma,a))$ is either $3$ or $\infty$)
cannot be extended to that of GWAs over a general base algebra of GK-dimension two.
Note that
it is proved by Rogalski \cite{rogalski2009gk} that
if $k$ is algebraically closed and $K/k$ is a finitely generated field extension with
$\operatorname{GKdim}(K)=2$,
then every ``big subalgebra" \cite[Definition 6.1]{rogalski2009gk} of the GWA $K(x,y;\sigma,1)=K[x,x^{-1};\sigma]$
has the same GK-dimension $d\in\{3,4,5,\infty\}$.
\begin{example}
  \label{exam_Heisenberg}\upshape
  Let
  \[
  H_m=H_m(\alpha_1,\alpha_2):=k[z_1^{\pm1},z_2^{\pm1}](x,y;\sigma_m,a)
  \]
  be a GWA, where
  \[
  \sigma_m(z_1)=\alpha_1z_1, \sigma_m(z_2)=\alpha_2z_1^mz_2, \
  \alpha_1,\alpha_2\in k^*, m\in\mathbb{N}.
  \]
  Then $\operatorname{GKdim}(H_m)=4$ for all $m\neq 0$ and $\operatorname{GKdim}(H_0)=3$.
\end{example}
\begin{proof}
Without loss of generality,
we assume $\alpha_1=\alpha_2=1$.
In fact,  since $V=k\{z_1^{\pm1},z_2^{\pm1}, x,y\}$ is a generating subspace of both $H_m(\alpha_1,\alpha_2)$ and $H_m(1,1)$,
we have that $\operatorname{GKdim}(H_m(\alpha_1,\alpha_2))=\operatorname{GKdim}(H_m(1,1))$.
We may also suppose $m\geq 0$, since
$H_m\cong H_{-m}$ via the isomorphism determined by
$z_1\mapsto z_1^{-1}$, $z_2\mapsto z_2$, $x\mapsto x$, $y\mapsto y$.

It is clear that $\sigma_0$ is locally algebraic and thus
$\operatorname{GKdim}(H_0)=3$. Now assume $m\geq1$.
Note that $H_1$ is the group algebra of the \emph{first (discrete) Heisenberg group}
(also known as the \emph{generic quantum torus} \cite[Example 2.5(ii)]{jordan2009reversible}).
It is well-known that $\operatorname{GKdim}(H_1)=4$ (see e.g. \cite[Example 11.10]{Krause-Lenagan_2000}).
 By Lemma \ref{lemma_Power},
  the subalgebra $B=k[z_1^{\pm1},z_2^{\pm1}](x^m,y^m;\sigma_1^m,1)\subseteq H_1$ is a GWA and $\operatorname{GKdim}(B)=\operatorname{GKdim}(H_1)=4$.
  By the universal property of GWAs (Proposition \ref{prop-universal}), the mappings
  \[
  \psi:H_m\to B, z_1\mapsto z_1, z_2\mapsto z_2, x\mapsto x^m, y\mapsto y^m
  \]
  and
  \[
  \psi':B\to H_m, z_1\mapsto z_1, z_2\mapsto z_2, x^m\mapsto x, y^m\mapsto y
  \]
  determine two algebra homomorphisms such that
  $\psi\psi'=\operatorname{id}_{B}$ and $\psi'\psi=\operatorname{id}_{H_m}$.
  Hence $H_m\cong B$ and thus $\operatorname{GKdim}(H_m)=\operatorname{GKdim}(B)=4$.
\end{proof}

\section{GK-dimension of GWAs over Laurent polynomial algebras}
\label{sce-Laurent}
In this section, we first characterize the automorphisms of Laurent polynomial algebra
$$
L_n=k[z_1^{\pm1},z_2^{\pm1},\dots,z_n^{\pm1}]
$$
 and then investigate the GK-dimension of GWAs over $L_n$.
Finally we prove Theorem \ref{thm-mainL} and give an application on cancellation problem.

Denote by $\mathbb{Z}^{m\times n}$ (respectively $\operatorname{GL}(n,\mathbb{Z})$) the set of all $m\times n$ integer matrices (respectively the general linear group of degree $n$ over $\mathbb{Z}$).
For $M=(a_{ij})\in\mathbb{Z}^{n\times n}$, let $M[i,j]:=a_{ij}$ be the $(i,j)$-th element of $M$,
$M[i,-]:=(a_{i1},a_{i2},\dots,a_{in})$ be the $i$-th row of $M$,
and $M[-,j]:=(a_{1j},a_{2j},\dots,a_{nj})^T$ be the $j$-th {column} of $M$,
 where $T$ stands for the transpose operation.

{It is well-known} that the unit group of $L_n$ is
\[
U(L_n)=\{
\alpha z_1^{i_1}z_2^{i_2}\cdots z_n^{i_n}:\alpha\in k^*, i_1,i_2,\dots,i_n\in \mathbb{Z}\}.
\]
Since an automorphism $\sigma\in\operatorname{Aut}(L_n)$ sends a unit to a unit,
we have
\begin{align} \label{eqn_auto}
\sigma(z_i)=\alpha_iz_1^{a_{1i}}z_2^{a_{2i}}\cdots z_n^{a_{ni}},
 \ \alpha_i\in k^*, a_{1i},a_{2i},\dots, a_{ni}\in\mathbb{Z},\
i=1,2,\dots,n.
\end{align}
The matrix
$M=M_{\sigma}=(a_{ij})\in \mathbb{Z}^{n\times n}$
(respectively, the scalar $\alpha=\alpha_{\sigma}:=(\alpha_1,\alpha_2,\dots,\alpha_n)\in (k^*)^n$)
is uniquely determined by the automorphism $\sigma$ and we write $\sigma=(M,\alpha)$.
We introduce the following convention.
\begin{convention}\upshape
  \label{conv-a^b}
Denote $z:=(z_1,z_2,\dots,z_n)$.
Given rows $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)\in L_n^n$ and
$\beta=(\beta_1,\beta_2,\dots,\beta_n)\in L_n^n$,  a column $b=(b_1,b_2,\dots,b_n)^T\in \mathbb{Z}^{n\times 1}$,
and a matrix $M\in \mathbb{Z}^{n\times n}$,
we use the following notation:
\begin{align*}
  \alpha^b&:=
\alpha_1^{b_1}\alpha_2^{b_2}\dots \alpha_n^{b_n}\in L_n,\\
\alpha^M&:=
(\alpha^{M[-,1]},\alpha^{M[-,2]},\dots,\alpha^{M[-,n]})\in L_n^n,\\
\alpha \beta&:=(\alpha_1\beta_1,\alpha_2\beta_2,\dots,\alpha_n\beta_n)\in L_n^n.
\end{align*}
\end{convention}
In particular, the element-wise multiplication
\[
\alpha\beta=(\alpha_1\beta_1,\alpha_2\beta_2,\dots,\alpha_n\beta_n),\
\forall \alpha,\beta\in (k^*)^n
\]
gives a group structure on $(k^*)^n$.

With the Convention \ref{conv-a^b}, we can write the automorphism $\sigma=(M,\alpha)$ given in Eq. (\ref{eqn_auto}) as
\begin{align}\label{eqn_auto2}
  \sigma(z_i)&=\alpha_iz_1^{M[1,i]}z_2^{ M[2,i]}\cdots z_n^{ M[n,i]}=
 \alpha_i z^{M[-,i]},\ \ i=1,2,\dots,n
\end{align}
or
\begin{align}
\label{eqn_sigma(z)}
  \sigma(z):=(\sigma(z_1),\sigma(z_2),\dots,\sigma(z_n))
  =(\alpha_1z^{M[-,1]},\alpha_2z^{M[-,2]},\dots,\alpha_nz^{M[-,n]})
  =\alpha z^M.
\end{align}

\begin{lemma}\label{lemma_exponent}
Suppose $\alpha,\beta,\gamma\in {L_n^n}$ and
$M,N\in \mathbb{Z}^{n\times n}$.
Then
\begin{enumerate}[(i)]
  \item
  $(\alpha\beta)^M=\alpha^M\beta^M$.
  \item
  $(\alpha^M)^N=\alpha^{MN}$.
  \item
  $\alpha^M\alpha^N=\alpha^{M+N}$.
\end{enumerate}
\end{lemma}
\begin{proof}
  (i) and (iii) are obvious.
  (ii) can be checked directly.
\end{proof}

\begin{lemma}
  \label{lemma_composition}
  Suppose $\sigma=(M,\alpha), \tau=(N,\beta)\in \operatorname{Aut}(L_n)$.
  Then
  \begin{enumerate}[(i)]
   \item
    $\tau\sigma=(N,\beta)(M,\alpha)=(NM,\beta^M\alpha).$
    \item
    $M$ is invertible and $\sigma^{-1}= (M^{-1},\alpha^{-M^{-1}})$.
  \end{enumerate}
\end{lemma}
\begin{proof}
(i) Suppose  $C=NM$.
By Eq. (\ref{eqn_auto2}), it is routine to check that, for $1\leq i\leq n$,
$(\tau\sigma)(z_i)=\beta^{M[-,i]}\alpha_iz^{C[-,i]}.$
Thus
$\tau\sigma(z)=\beta^M\alpha z^{NM}$ and
$\tau\sigma=(NM,\beta^M\alpha)$.

(ii) follows from (i) and Lemma \ref{lemma_exponent}.
\end{proof}

The following theorem is well known,
see for instance \cite[Lemma 3.6(3)]{CPWZ15} or \cite[Exercise 5.19 (b)]{BG09}.

\begin{theorem}
  \label{thm_auto(L)}
  Let $L_n=k[z_1^{\pm1},z_2^{\pm1},\dots,z_n^{\pm1}]$.
  Then \(
  \operatorname{Aut}(L_n)\cong \operatorname{GL}(n,\mathbb{Z})\ltimes (k^*)^n
  \) as groups,
  where the group operation of the semiproduct is given by Lemma \ref{lemma_composition} (i).
\end{theorem}
\begin{proof}
It follows from Lemma \ref{lemma_composition} (ii) that the mapping
  \[
  f:\operatorname{Aut}(L_n)\to \operatorname{GL}(n,\mathbb{Z})\times (k^*)^n,\ \sigma\mapsto(M_{\alpha},\alpha_{\sigma})
  \]
  is well-defined and bijective.
It is routine to check that
 $\operatorname{GL}(n,\mathbb{Z})\times (k^*)^n$ has a semidirect structure with
the product given by Lemma \ref{lemma_composition} (i).
\end{proof}

\begin{definition}
Suppose $M\in\mathbb{Z}^{n\times n}$ and
\[
\|M\|:=\max_{1\leq i,j\leq n} |M[i,j]|.
\]
\begin{enumerate}[(i)]
  \item
  $M$ is called \emph{power-bounded} (by $N$)
if there exists $N\in\mathbb{N}^*$ such that $\|M^m\|<N$
for all $m\in \mathbb{N}$.
  \item
  $M$ has a \emph{finite order} if $M^m=I$ for some $m\in\mathbb{N}^*$, and the smallest such $m$ is called the \emph{order} of $M$, denoted by ${\operatorname{o}}(M)$.
\end{enumerate}
\end{definition}

Note that if $\sigma=(\alpha,M)\in\operatorname{Aut}(L_n)$ has a finite order,
then so does $M$.
But, as demonstrated in Example \ref{exam_FiniteOrd},
the other way around is not true in general.

\begin{lemma}
  \label{lemma_pb=la}
  Let $\sigma=(\alpha,M)\in \operatorname{Aut}(L_n)$, $a\in L_n$ and $A=L_n(x,y;\sigma,a)$.
  Then the following statements are equivalent:
  \begin{enumerate}[(i)]
    \item
    $M$ is power-bounded.
    \item
    $M$ has a finite order.
    \item
    $\sigma$ is locally algebraic.
  \end{enumerate}
\end{lemma}
\begin{proof}
(i)$\Rightarrow$(ii).
Suppose that $\|M^m\|<N$ for some $N\geq 1$ and all $m\in\mathbb{N}$.
Then the entries of matrix $M^m$ must be integers between $-N$ and $N$ for all $m\in \mathbb{N}$.
Hence $\{M^m:m\in \mathbb{N}\}$ is a finite subgroup of $\operatorname{GL}(n,\mathbb{Z})$.
Thus $M$ is of finite order.

(ii)$\Rightarrow$(iii).
Suppose $M$ has a finite order.
Then $\{M^i:i\geq 0\}$ is a finite set.
Let $N$ be the maximal number that appears as an entry of the matrices $\{M^i:i\geq 0\}$.

\emph{Claim.}  $\sigma^m(z)=\beta_m z^{M^m}$ for $m\geq 1$, where
\[
\beta_m=\alpha^{I+M+M^2+\dots+M^{m-1}}, m\geq 1.
\]

\emph{Proof of the claim.} The claim holds for $m=1$ by Eq. (\ref{eqn_sigma(z)}).
Assume that the claim holds for some $m\geq 1$.
Then, by Eq. (\ref{eqn_sigma(z)}) and Lemma \ref{lemma_exponent},
\begin{align*}
  \sigma^{m+1}(z)
  =\sigma(\beta_m z^{M^m})
=\beta_m\sigma( z)^{M^m}
=\beta_m( \alpha z^M)^{M^m}
=\beta_m \alpha^{M^m} z^{M^{m+1}}
=\beta_{m+1} z^{M^{m+1}}.
\end{align*}
Thus the claim holds for all $m\geq 1$.

By the claim, we have that
 \[
 \sigma^m(z_i)\in V:= k\{z_1^{i_1}z_2^{i_2}\dots z_n^{i_n}:-N\leq i_j\leq N, 1\leq j\leq n\},  m\geq 1,
 \]
where $V$ is a finite dimensional\ subspace with $\dim(V)\leq (2N+1)^n$.
Suppose $T=z_1^{a_1}z_2^{a_2}\dots z_n^{a_n}\in L_n$, each $a_i\in \mathbb{Z}$.
Then
 $\sigma^m(T)=(\sigma(z_1))^{a_1}(\sigma(z_2))^{a_2}\dots (\sigma(z_n))^{a_n}$,
 which is contained in the finite dimensional\ subspace
  \[
 V_T:=k\{z_1^{i_1}z_2^{i_2}\dots z_n^{i_n}:-N'\leq i_j\leq N', 1\leq j\leq n\}
 \]
 for all $m\geq 1$, where $N'=N(|a_1|+|a_2|+\dots+|a_n|)$.
 As a result,
 for each Laurent polynomial $p\in L_n$, say
 \[
  p=c_1T_1+c_2T_2+\dots+c_sT_s,
  c_i\in k^*, T_i \text{ Laurent monomial}, i=1,2,\dots,n,
  \]
 we have that $\sigma^m(p)$ is contained in the finite dimensional\ space
 $V_{T_1}+V_{T_2}+\dots+V_{T_s}$
 for all $m\geq 1$.
 Therefore, by Lemma \ref{lemma-LocAlgIff},
 $\sigma$ is locally algebraic.

 (iii)$\Rightarrow$(i).
 Suppose that $M$ is not power-bounded,
 i.e., for every $l>1$, there exists $m=m(l)>1$ such that
 $\|M^m\|>l$.
 Without loss of generality,
 assume $\|M^m\|=|M^m[1,1]|$ for infinitely many $m\geq 1$.
 Let $V=kz_1+kz_2+\dots+kz_n+kz_1^{-1}+kz_2^{-1}+\dots+kz_n^{-1}$.
 It follows from the above claim that
 \[
 \sigma^m(z_1)=cz_1^{M^m[1,1]}z_2^{M^m[2,1]}\dots z_n^{M^m[n,1]}\not\in
 V^l.
 \]
Thus the subspace $k\{\sigma^m(z_1):m\geq 1\}$ is not finite dimensional.
Therefore $\sigma$ is not locally algebraic.

\end{proof}

\noindent
\emph{\textbf{Proof of Theorem \ref{thm-mainL}.}}
  It follows from Proposition \ref{prop-Zhang97PL} and Lemma \ref{lemma_pb=la}.
  \hfill\qed

\begin{remark}\upshape
  \label{remark-MaxOrd}
It follows from Lemma \ref{lemma_pb=la} that
the elements of finite order in $\operatorname{GL}(n,\mathbb{Z})$ play an important role.
Given $n\geq 1$, there are only finitely many integers that can be the orders of matrices in $\operatorname{GL}(n,\mathbb{Z})$.
Levitt \cite{Levitt1998} shows that the maximum order $G(n)$ of elements of finite order in $\operatorname{GL}(n,\mathbb{Z})$ satisfies a Landau-type estimate $\ln G(n)\sim \sqrt{n\ln n}$ and
there is a method of dynamical programming to compute $G(n)$.
Following \cite{Levitt1998}, Table \ref{tab-Gn} gives the values of $G(n)$ for $n\leq 20$ where $G(n)$ is omitted for odd $n=2p+1$ since $G(2p+1)=G(2p)$.
\begin{table}[h]
  \centering
  \begin{spacing}{1.15}
  \caption{Maximum orders of matrices in $\operatorname{GL}(n,\mathbb{Z})$}\label{tab-Gn}
  \begin{tabular}{c|ccccccccccc}
  \hline
  $n$&$1$&$2$&$4$&$6$&$8$&$10$
  &$12$&$14$&$16$&$18$&$20$\\
  \hline
  $G(n)$&$2$&$6$&$12$&$30$&$60$&$120$
  &$210$&$420$&$840$&$1260$&$2520$\\
  \hline
\end{tabular}
\end{spacing}
\end{table}
By using \cite[Proposition 1.1]{Levitt1998},
we can determine whether an integer is the order  of an element of $\operatorname{GL}(n,\mathbb{Z})$.
Note that for a given order $o$ there may be many matrices of order $o$ (see Example \ref{exam_FiniteOrd}).
\end{remark}

\begin{example}
\label{exam_FiniteOrd}\upshape
  Suppose that $c\in k^*$ and $\sigma=\sigma_q \in \operatorname{Aut}(L_2)$ is defined as
  \[
  \sigma(z_1)=cz_1^{1-q}z_2^{2-q},
  \sigma(z_2)=z_1^{q}z_2^{q-1},\ q\in\mathbb{Z}.
  \]
  Then $\sigma^m\neq \operatorname{id}$ for all $m\geq 1$.
But $M_{\sigma}=\begin{pmatrix}
    1-q&q\\
    2-q&q-1
  \end{pmatrix}$ has a finite order ${\operatorname{o}}(M)=2$ for all $q\in\mathbb{Z}$
and thus $\operatorname{GKdim}(L_2(\sigma,a))=\operatorname{GKdim}(L_2)+1=3$
by Theorem \ref{thm-mainL}.
\end{example}

We conclude this section by an application of Theorem \ref{thm-mainL} to cancellation problem.
\begin{definition}[Definitions 0.1 and 0.8 of \cite{tang2020cancellation}]
Let $A$ be an algebra.
\begin{enumerate}[(i)]
\item
We say $A$ is \emph{cancellative} (respectively, \emph{strongly cancellative})
if the isomorphism
\begin{align*}
 A[x]\cong B[x] \ \ (\text{respectively }
 A[x_{1}, \cdots, x_{m}] \cong B[x_{1}, \cdots, x_{m}])
\end{align*}
implies $A\cong B$
for all algebra $B$ (respectively for all algebra $B$ and all integer $m\geq 1$).
\item
We say $A$ is \emph{$\sigma$-algebraically cancellative} (respectively, \emph{strongly $\sigma$-algebraically cancellative})
if the isomorphism between Ore extensions (respectively iterated Ore extensions)
\begin{align*}
 A[x;\sigma']\cong B[x;\sigma''] \ \ (\text{respectively }
 A[x_{1};\sigma_1']\cdots[x_{m};\sigma_m'] \cong B[x_{1};\sigma_1'']\cdots[x_{m};\sigma_m''])
\end{align*}
implies $A\cong B$
for all algebra $B$ and locally algebraic automorphisms $\sigma'$ and $\sigma''$
(respectively for all algebra $B$,
 locally algebraic automorphisms $\sigma'_i$ and $\sigma''_i$, $1\leq i\leq m$, and all integer $m\geq 1$).
\end{enumerate}
\end{definition}
The following lemma follows directly from the definitions and its proof is omitted.

\begin{lemma}
  \label{lemma-SCancel}
  Let $A$ be an algebra.
  \begin{enumerate}[(i)]
    \item
    If $A$ is strongly cancellative (respectively strongly $\sigma$-algebraically cancellative),
  then $A$ is cancellative (respectively $\sigma$-algebraically cancellative).
    \item
    If $A$ is $\sigma$-algebraically cancellative (respectively strongly $\sigma$-algebraically cancellative),
  then $A$ is cancellative (respectively strongly cancellative).
  \end{enumerate}

\end{lemma}

Theorem \ref{thm-mainL} together with results from \cite{tang2020cancellation} can be applied to get some cancellation properties of GWAs.

\begin{corollary}
  \label{coro-Cancellation}
  Let $A=L_n(\tau,a)$ be a GWA with $a\neq 0$ such that one of the equivalent conditions in Theorem \ref{thm-mainL} is satisfied.
  Then $A$ is strongly $\sigma$-algebraically cancellative.
  As a consequence,
  $A$ is strongly cancellative and cancellative.
\end{corollary}
\begin{proof}
  Since $L_n$ is a finitely generated\ domain and $a\neq 0$,
  it follows from Lemma \ref{lemma-Noeth} that $A$ is a finitely generated\ domain.
  It is easy to check that the identity element $1$ is a \emph{controlling element} (see \cite[Definition 4.2]{tang2020cancellation}) of $A$, i.e., ${\mathbb D}(1)=A$.
  By the assumption and Theorem \ref{thm-mainL},
  $\operatorname{GKdim}(A)=n+1<\infty$ and thus
  $A$ is strongly $\sigma$-algebraically cancellative by \cite[Theorem 0.9]{tang2020cancellation}.
\end{proof}

\

\textbf{Acknowledgements.}
The author would like to thank the anonymous referees for their
careful reading, valuable suggestions and useful references.
Part of this work was done during a visit to James J. Zhang at the University of Washington.
The author would like to express deep gratitude to James J. Zhang
for many stimulating conversations on the subject,
and to the department of Mathematics at the University of Washington
for the hospitality during his visit.
The author wishes to thank Yu Li, Wenchao Zhang and Xiaotong Sun for valuable discussions.
This work was supported in part by Huizhou University (hzu202001, 2021JB022) and the Guangdong Provincial Department of Education (2020KTSCX145, 2021ZX088).

redpajama_set_name: RedPajamaArXiv


I visited Launch Fishers today. Nice venue. Ample parking. Integrated coffee shop. Simple program. $500 / year membership fee. Gains 24 hour access. Common first come first serve space. For $250 / year get a mailing address and mailbox. For $150 / month get your own space. Common space is door access controlled. Permanent space has an additional level of door access control.
Great bandwidth. Several large screen TVs for group projection. Printer available for a per copy fee. Variety of meeting / conference rooms from 4 people to 100 people. Online signup for any member. Wide variety of seating options: open tables, chairs and private phone booths.
Kitchen facility. Starting a bike share program for members. Probably 40 people in the common areas and didn't seem crowded. Have a weekly lunch for members and a monthly after hours meet and greet.
Typed while waiting on LISC presentation to start. www.liscindianapolis.org.


The cost for a one-month course of treatment will be between $4,720 and $6,785, said Janssen, and experts said it will give the company a foothold in the $12 billion global antidepressant market, where most drugs now are generic.
An earlier version of this article misstated the cost of treatment with esketamine. The correct cost is between $4,720 and $6,785, not between $2,360 and $3,540.


text:
Q: Installing Rcmdr/car/quantreg packages in OS X Trying to install the Rcmdr (R version 3.2.2) package in my Mac (Yosemite 10.10.4), I ran into a few problems:

*

*After I installed Rcmdr with all its dependencies I get the following when I try to load the package:

Loading required package: RcmdrMisc
          Loading required package: car
          Error in loadNamespace(j <- i[[1L]], c(lib.loc, .libPaths()), versionCheck = vI[[j]]) : there is no package called 'quantreg'
          Error: package 'car' could not be loaded

*So I tried to install the car package along with its dependency, the quantreg package, and this is what happens:

** package 'quantreg' successfully unpacked and MD5 sums checked
          ** libs
          gfortran-4.8   -fPIC  -g -O2  -c akj.f -o akj.o
          make: gfortran-4.8: No such file or directory
          make: *** [akj.o] Error 1
          ERROR: compilation failed for package 'quantreg'
          * removing '/Library/Frameworks/R.framework/Versions/3.2/Resources/library/quantreg'
          Warning in install.packages :
            installation of package 'quantreg' had non-zero exit status

*Then I realized the problem might be the fortran compiler. I don't have the 4.8 version, but I did a symlink for gfortran-4.8 to my gfortran-5.0 and yet I couldn't solve the problem:

ld: warning: directory not found for option '-L/usr/local/lib/gcc/x86_64-apple-darwin13.0.0/4.8.2'
          ld: warning: directory not found for option '-L/usr/local/lib/gcc/x86_64-apple-darwin13.0.0/4.8.2'
          ld: library not found for -lquadmath
          clang: error: linker command failed with exit code 1 (use -v to see invocation)
          make: *** [quantreg.so] Error 1
          ERROR: compilation failed for package 'quantreg'
          * removing '/Library/Frameworks/R.framework/Versions/3.2/Resources/library/quantreg'

I would be very grateful if someone could help me get to the bottom of this problem.

A: Here is how I solved it. [OSX Mojave, R 4.0.2]
I hope it helps, especially for those who could not solve it with the answers provided here: OS X package installation depends on gfortran-4.8

*

*Make sure R is updated to version >= 4.0
(for previous versions this might help: https://cran.ism.ac.jp/bin/macosx/tools/)

*Download and install the GNU fortran compiler found here: https://mac.r-project.org/tools/

*Make sure you add /usr/local/gfortran/bin to your PATH in order to use this compiler. This can be done, e.g. by

export PATH=$PATH:/usr/local/gfortran/bin

redpajama_set_name: RedPajamaStackExchange


Part old-school lounge, part sports bar, part patio hangout, and part outdoor music venue, Shady Park has become a go-to spot for Tempe locals and those seeking some fun on Mill Avenue toward the end of the week. Shady Park's dog-friendly, tree-laden back patio includes a full bar, plenty of seating, a weatherproof sound system, and a small stage backed with a large projector screen. Scott Price, also the owner of C.A.S.A. SunBa a little farther north on Mill, and the Shady Park crew have hosted live bands, DJs, pizza parties, the monthly Shady AF Comedy Night, and, we're sure, more to come.
The venue is labeled as "TreeHouse" on Sunday's for its vibe outside.
There is a huge variety of food options available, from sushi to pizza to ramen.
The venue is well-loved by locals for intimate house/techno shows.
Where is Shady Park located?
Shady Park is located in Tempe.
How much is cover charge at Shady Park?
Shady Park does not generally charge cover.
How long will I have to wait at Shady Park?
When is Shady Park open?
Shady Park is open every day.
What are the best nights to go to Shady Park?
Nights with the most well-known artists are most popular.
What is the dress code at Shady Park?
How much do drinks cost at Shady Park?
What kind of music do they play at Shady Park?
Shady Park features mostly techno and house music.
Where can I find an list of upcoming events / DJs at Shady Park?
You can download the Discotech app to see upcoming events, and book tables for Shady Park.


Low Grow Wildflower Mix is great for many diverse urban landscapes. Great for golf courses, driveway approaches, hiding low retaining walls, low growing meadows, ditches, and sunny beds or borders.
African Daisy, California Bluebell, California Poppy, Candytuft, Cottage Pinks, Dwarf Cornflower, Dwarf Evening Primrose, Dwarf Godetia, Dwarf Lance-Leaved Coreopsis, Dwarf Plains Coreopsis, Iceland Poppy, Johnny Jump-Up, Moss Verbena, Siberian Wallflower, Spurred Snapdragon, Sweet Alyssum and Tussock Bellflower.
Planting Rate: 10-15 lb per acre, 1 lb per 2500 sq. ft.
Grows Well In: Midwest, Northeast, Northwest, Southeast, Southwest, Texas/Oklahoma, southwestern Canada and Western states.


Today's Daily Deals is sponsored by Stories for Boys Who Dare to Be Different and Running Press Kids.
We're Going to Need More Wine: Stories That Are Funny, Complicated, and True by Gabrielle Union for $2.99. Get it here, or just click on the cover image below.
Fly Girls: How Five Daring Women Defied All Odds and Made Aviation History by Keith O'Brien for $2.99. Get it here, or just click on the cover image below.
A River in Darkness: One Man's Escape from North Korea by Masaji Ishikawa, translated by Risa Kobayashi and Martin Brown, for $1.99. Get it here, or just click on the cover image below.
The Once and Future King by T. H. White for $1.99. Get it here, or just click on the cover image below.
The Indifferent Stars Above: The Harrowing Saga of the Donner Party by Daniel James Brown for $1.99. Get it here, or just click on the cover image below.


text:
var React = require('react');
var LocationItem = require('./LocationItem.react');

var LocationList = React.createClass({

  render: function() {
    var self = this;

    var locations = this.props.locations.map(function(l) {
      var active = self.props.activeLocationAddress == l.address;

      return <LocationItem address={l.address} timestamp={l.timestamp}
          active={active} onClick={self.props.onClick} />
    });

    if (!locations.length) {
      return null;
    }

    return (

        Saved Locations

        {locations}

    )
  }

});

module.exports = LocationList;

redpajama_set_name: RedPajamaGithub


text:
The Mississippi Delta was covered with forests and heavy cane and most of the region lay undeveloped, "as if the foot of man had never trod." It was the building of feeder rail lines into the rich and potentially productive cotton region that brought vitality and color to the life of the Delta -Green Diamond, IC Historical Society, December 2004.
The goal of the Railroad Heritage Museum is to preserve and promote the history and the culture of the railroad and its impact of the establishment, growth, and development of Cleveland, Bolivar County, and the Delta Region.
The museum houses and displays many railroad artifacts that capture the history of railroading - from tools used by crews known as "Gandy Dancers" to the timetables and schedules used in the depots and railroad offices.
The Model Train
The largest O-gauge model train layout in Mississippi is housed inside the museum. Measuring 71 feet by 17 feet, the layout's attention to detail and history is amazing and ever-changing. Representative of "Anytown," Mississippi, the layout depicts an era between the 1940s and the 1960s, including just about every facet of small town life and the countryside built around the railroad.
The 1941 Caboose
Our 1941 caboose was built in Centralia, Illinois for the Illinois Central Railroad. Original features include the offset cupola, sliding wooden doors, cast iron stove, water fountain, gauges, and brake controls. Many items that were once inside the caboose are on display inside the City of Cleveland's Martin and Sue King Railroad Heritage Museum located in Historic Downtown Cleveland. The caboose is located by the museum and is open to the public during museum hours. For more information call us at 662-843-3377.
Anna Sledge
Email Anna Sledge
115 S Bayou Road
115 S Bayou Road Cleveland MS 38732
No admission fee and groups are welcome.
Railroad Museum Website
Finance & Purchasing
Municipal Airport
Waste Water Department
Mayor Welcome
Residential Utilities

redpajama_set_name: RedPajamaCommonCrawl


text:
Movies|Movies, Mountains and High Hopes
Movies | Film
Movies, Mountains and High Hopes
By A. O. SCOTT SEPT. 6, 2010
Natalie Portman in "Black Swan," the new film from Darren Aronofsky. Credit Niko Tavernise/Fox Searchlight Pictures
TELLURIDE, Colo. — "If what we hear from down on the plains is true, and film culture is dying," said the Australian director Peter Weir, up in the mountains here to present his latest movie, "The Way Back," "then perhaps this place can become a kind of Machu Picchu of cinema." Not a ruin, of course, but a fortress and a shrine, a sacred spot where embattled traditions can survive in troubled times.
Telluride is already a pilgrimage spot for film lovers, who mingled over Labor Day weekend with fly-fishing enthusiasts, mountain bikers and other worshipers in the church of high-end leisure, turning a remote former mining town into a holy city of movies. The atmosphere is hardly ascetic — the Telluride Film Festival calls itself "the show" and conducts its business with low-key Western flair — but the exalted sense of seriousness that Mr. Weir evoked is an important part of the show's self-image.
The program, which is announced at the beginning of the festival rather than trumpeted via news release months in advance, includes rarities and restorations, and also seminars and discussions with visiting filmmakers, writers and scholars. This year, for example, pass holders lined up for the original, silent "Chicago" and the Italian silent film "Rotaie." They also flocked to documentaries about cinematic giants of yore, including Elia Kazan, the cinematographer Jack Cardiff and Jacques Tati. One of Tati's unproduced screenplays inspired "The Illusionist," a lovely and melancholy animated feature by Sylvain Chomet that was a high point of the festival.
Whatever Mr. Weir, in his remarks at the first screening of "The Way Back" on Friday night, meant by "film culture" — the term has a long and controversial history — it is certainly preserved and celebrated here at the end of each summer. Telluride, now in its 37th year, is at once more relaxed and more intense than other festivals. Lasting only 4 days rather than the usual 10, the show features elaborate line-waiting protocols, open-air question-and-answer sessions with stars and directors, and bustling public picnics on the opening and closing days. This festival guards its sense of specialness jealously, and it guards traditions that are intended to keep hype and scene making at bay.
But the exalted, rarefied atmosphere is to some extent a carefully maintained illusion, an aspect of the show. (Not for nothing is the official festival headquarters known as Brigadoon.) Telluride, which takes place right before Toronto and in the middle of Venice, is hardly isolated from the ordinary business of movies. It is a transit point for movies from Sundance and Cannes and also a showcase for high-profile, Oscar-aspiring autumn releases.
In this season some of the anxieties that afflict the movie world — and the world beyond it — have filtered up into the highlands. Like other nonprofit institutions Telluride has lost corporate support in the wake of the financial crisis, a situation that was mentioned repeatedly at screenings in the introductory remarks, when patrons and sponsors were thanked and audience members were reminded to silence their phones.
That is easily done. It can be harder, though, to tune out the festival's natural setting, which frequently upstages the movies on its screens, at least as far as sheer visual sublimity is concerned. Equally dramatic landscapes figured in a number of movies here this year. Danny Boyle's latest, "127 Hours," which made one of the unannounced "sneak" appearances that are a festival trademark, was shot in neighboring Utah. It stars James Franco as Aron Ralston, a hiker whose real-life ordeal was something of a sensation in 2003. You may remember the story: Trapped in a narrow crevasse in a deep canyon, Mr. Ralston escaped by cutting off part of his right arm, which was pinned by a boulder.
His experience is disconcerting enough just to think about, and to see it recreated in Mr. Boyle's characteristically fast-moving, immersive style is jarring, thrilling and weirdly funny. At a question-and-answer session after the first screening on Saturday afternoon, Mr. Boyle — the director of "Trainspotting," "28 Days Later" and of course "Slumdog Millionaire," which sneaked into Telluride two years ago — described himself as a thoroughly urban type with no great love for or interest in nature. And the jangly, jumpy energy he brings to a story of silence, solitude and confinement gives the film an irreverent kick that deepens and sharpens its emotional and spiritual insights.
Mr. Weir, unlike Mr. Boyle, has frequently shown a reverence for the beauties and terrors of nature, going back to early films like "The Last Wave" and "Picnic at Hanging Rock," in which the topography of his native country assumes a haunting, even demonic presence. The drama of human beings confronting the elemental power of nature figures in later work like "The Mosquito Coast" and "Master and Commander: The Far Side of the World," and also in "The Way Back."
Compared with Mr. Boyle's, Mr. Weir's style is stately, almost classical, and the astonishing story he has to tell — about a group of men who escaped from a Soviet labor camp in 1941 and walked to India from Siberia — has an old-fashioned gravity and grandeur. There are fine performances from Ed Harris, Saoirse Ronan and Jim Sturgess as Janusz, the Polish prisoner who leads the trek toward freedom, and breathtaking images of tundra, desert forest and grassland.
"Le Quattro Volte," directed by Michelangelo Frammartino, never ventures beyond a small rural village in southern Italy, and yet its view of nature is among the most profound, expansive and unsettling I have ever encountered on film. There is virtually no dialogue, but the film is far from silent: the rustling of trees, the sounds of agricultural labor, the barking of a dog and in particular the cries of goats supply a meaning that transcends words, while Mr. Frammartino's eye for both comedy and mystery produces compositions that are so strange and memorable that they seem to reinvent the very act of perception.
"Le Quattro Volte" will soon make its way into the urban setting of the New York Film Festival, along with some other travelers from Telluride, including "Another Year" by Mike Leigh, "Carlos" by Olivier Assayas and "Inside Job," a documentary about the roots of the current economic crisis by Charles Ferguson, whose "No End in Sight" is one of the definitive documentaries about the war in Iraq. "Inside Job" was part of a strong documentary field that also included the queasy, marvelously bizarre "Tabloid" from Errol Morris and "Precious Life," a moving film about Palestinian family members from Gaza trying to save one of their children from a fatal genetic disorder with the help of an Israeli doctor and the filmmaker, the television journalist Shlomi Eldar.
The second sneak of the festival was "Black Swan," a backstage melodrama by Darren Aronofsky that often plays like a horror movie. It popped up on Sunday along with Mr. Aronofsky himself, making his first visit to this festival en route from Venice, where "Black Swan" opened last week and where his previous film, "The Wrestler," won the top prize two years ago.
"Black Swan" is, in many ways, a companion to the earlier film in that it is the story of a performer driven to sometimes gruesome extremes of physical and psychological self-punishment in the pursuit of professional glory. In this case the protagonist is an up-and-coming ballerina (played by Natalie Portman) rather than a down-and-out gladiator, but if anything, the violence is more intense.
And intensity — a swirl of dread, suspense and almost tactile beauty, brought to fever pitch by Clint Mansell's Tchaikovsky-on-steroids score — is certainly what "Black Swan" delivers. It also builds a gaudy, gothic frame around a disturbing and virtuosic central performance, as Ms. Portman travels the spectrum from meek to monstrous and discovers a terrifying new variation on the old idea of suffering for art.
The visceral impact of "Black Swan" is hard to dispute — at least my own pounding heart and clammy palms could not dispute it — but a calmer discussion of its merits and faults will wait for another occasion. The same is true of "127 Hours," "Another Year," "The King's Speech" (starring Colin Firth as George VI) and the other films that have now begun their climb down the mountain, hoping to arrive on Feb. 27 on the stage of the Kodak Theater. Film culture may or may not be ailing, but the business of show goes on.
A version of this article appears in print on September 7, 2010, on Page C1 of the New York edition with the headline: Movies, Mountains and High Hopes. Order Reprints| Today's Paper|Subscribe


text:
by Steve St-Amand · Published April 21, 2016
The world of music lost an icon today as Prince died at the age of 57.
by Steve St-Amand · Published March 22, 2016
Sara Diamond releases her first single "Just Give In"
The Montreal singer-songwriter Sara Diamond has released her very first single "Just Give In".
by Steve St-Amand · Published February 16, 2016
Paul McCartney denied access at Grammy Party
You think being a music legend will get you anywhere you want??? Guess again!!!
Here's the complete list of the 2016 Grammy Awards Winners List:
by Marco Paradis · Published February 8, 2016
A breathtaking halftime
With a spectacular entry into the scene, Coldplay move up the audience of Levi's Stadium, Bruno Mars and Beyonce have made an outstanding appearance…the trio nail it for the 50th Super Bowl.
The national anthem, a dream come true
The 29 year old singer Lady Gaga fulfill his dream by singing the national anthem at the most important game of 2016 in the NFL, the 50 th Super Bowl.
by David Peloquin · Published August 22, 2015 · Last modified August 28, 2015
Children of Bodom will release a new album!
The Finnish metal band, Children of Bodom will release a new album called I Worship Chaos on October 2nd. If you want to have a sample of the new album follow this link to the...


Tennis is a game that a lot of people enjoy playing and is a fun sport. Whether you are young or old, playing tennis can be a lot of fun. If you are always playing tennis with your friends, you can show off how much joy tennis brings you with this Take Joy in Tennis decal.


The Department of Industrial and Manufacturing Systems Engineering of The University of Hong Kong has developed a virtual aircraft loading bridge operations training (VALBOT) system for the Airport Authority Hong Kong. The system is developed based on the imseCAVE system that uses the state-of-the-art mixed reality (MR) technology to recreate an authentic virtual environment for the training of engineers and operators working in the airfield. VALBOT is installed in the International Aviation Academy at the Hong Kong International Airport.
Following the ceremony, the Chief Executive, accompanied by AA Chairman Jack So and AA CEO Fred Lam, visited our brand-new training facilities, including the Virtual Aircraft Loading Bridge Operation and the Flight Simulator at the Virtual Reality (VR) Room of the Hong Kong International Aviation Association (HKIAA). As pioneering aviation VR training facilities in Hong Kong, they provide students with an immersive training environment to enhance their practical skills in flight or ramp operations. They come equipped with the innovative imseCAVE, a low-cost, high-performance Cave Automatic Virtual Environment (CAVE) system designed and deployed by the Department of Industrial and Manufacturing Systems Engineering of the University of Hong Kong.
The Chief Executive Carrie Lam (left of front row) viewed the facility which demonstrates ramp operations at HKIA with the assistance of the VR technology. Also present are AA Chairman Jack So (right of back row), AA CEO Fred Lam (middle of back row) and Secretary for Transport and Housing Frank Chan (left of back row).
Dr. Henry Lau (middle) and his team.


'Parade' is an interactive cinema installation shown at the town of Hudson in New York on December 4th 2004. Between 4pm and 8pm, 10,000 people took part in a 'Winterwalk', during which time we projected on two windows of 330 U Gallery. 'Parade' is made from an exaggerated physiognomy, reacting to the movements of the passers by. Parade is part of a site-specific new media project called Third Person.
Example of collage with silhouette of pedestrian and video clip.
Another of the exaggerated physiognomies.
Opening night of the Winterwalk.
Opening night of the Winterwalk. Installation reacts to and collages with the movement of pedestrians.
Left, camera, computer and projector installation.
Right, windows with sound transducers for sound.
Parade is a part of Third Person, an interactive video installation that displays a site-specific cinematic narrative.


Is Cazenovia safe? Find crime statistics for the college campus and the surrounding area.
Cazenovia College reported 338 safety-related incidents involving students on or near campus or other Cazenovia affiliated properties in 2017. Of the 2,795 colleges and universities that reported crime and safety data, 2,518 of them reported fewer incidents than this.
Based on a student body of 1,072 that works out to about 315.30 reports per thousand students. In 2017, 2,774 colleges and universities reported fewer incidents per thousand students than did Cazenovia.
Crime and safety incidents fall into four broad categories. Disciplinary actions represented 92.9% of all incidents. Arrests related to possession represented 2.4% of all incidents. Violence against women represented 1.5% of all incidents. Arrests for major crimes represented 3.3% of all incidents.
94.7% of all incidents occurred on-campus. 0.3% of all incidents occurred in the Cazenovia neighborhood surrounding the campus. 5.0% of all incidents occurred off campus but while on school affiliated property.
Cazenovia College reported 320 safety-related incidents involving students while on campus in 2017. Of the 2,795 colleges and universities that reported crime and safety data, 2,520 of them reported fewer incidents than this.
Based on a student body of 1,072 that works out to about 298.51 reports per thousand students. In 2017, 2,775 colleges and universities reported fewer incidents per thousand students than did Cazenovia.
Disciplinary actions represented 93.4% of all on-campus incidents. Arrests related to possession represented 2.2% of all on-campus incidents. Violence against women represented 1.6% of all on-campus incidents. Arrests for major crimes represented 2.8% of all on-campus incidents.
Cazenovia College reported 1 safety-related incidents involving students while on Cazenovia public property near campus in 2017. Of the 2,795 colleges and universities that reported crime and safety data, 1,661 of them reported fewer incidents than this.
Based on a student body of 1,072 that works out to about 0.93 reports per thousand students. In 2017, 2,081 colleges and universities reported fewer incidents per thousand students than did Cazenovia.
Cazenovia College reported 17 safety-related incidents involving students while off campus but still on property owned or controlled by the school in 2017. Of the 1,470 colleges and universities that reported crime and safety data, 1,323 of them reported fewer incidents than this.
Based on a student body of 1,072 that works out to about 15.86 reports per thousand students. In 2017, 1,430 colleges and universities reported fewer incidents per thousand students than did Cazenovia.
Disciplinary actions represented 88.2% of all off-campus incidents. Arrests related to possession represented none of all off-campus incidents. Violence against women represented none of all off-campus incidents. Arrests for major crimes represented 11.8% of all off-campus incidents.
Even in low-crime areas, students still run the risk of encountering violence and unsafe situations. What kind of policies does Cazenovia have in place to protect students or help students who are victims of crimes?
One way a student may feel more comfortable might be to take a self-defense or safety class. Are there classes like this offered at Cazenovia?


text:
O núcleo externo é a camada terrestre que se situa entre o núcleo interno (sólido) e o manto terrestre. Ele é formado por ferro que se apresenta em estado líquido, enquanto o núcleo interno se encontra no estado sólido. Essa descoberta se deve em grande parte ao estudo das ondas sísmicas e da sismologia.

É essa região que forma o campo magnético da Terra. O campo é causado devido a movimentação do fluido condutor de eletricidade, em um fenômeno parecido com o movimento das bobinas em um gerador elétrico.

Atualmente, cientistas acreditam que o núcleo externo está ligado à inversão da polaridade magnética do planeta, ocorrida no passado.

Ver também
 Núcleo (geologia)
 Núcleo interno

Ligações externas

Estrutura da Terra

redpajama_set_name: RedPajamaWikipedia


text:
Published on August 27th, 2016 | by Justin G.
Q5: New World Order
Cult classic heavy metal band Q5 shocked their fans when they returned after a lengthy absence in 2014 to start playing live. When they announced that they were working on their first new studio album in 30 years, that was an even bigger surprise. The eagerly anticipated New World Order is only the band's third album, and expectations were understandably high.
We now have three albums from Q5, and they each have very different approaches. Their classic 1984 debut Steel the Light is a pure heavy metal album, and their controversial 1985 follow-up When the Mirror Cracks was almost an AOR album. New World Order is somewhere between the two, rocking hard but never quite hitting metal territory. Actually, if you took the more pure hard rock songs off of the last few Saxon albums and put them in one place, you'd be pretty close to the New World Order sound. Jonathan Scott even sounds quite a lot like Biff Byford on a lot of these songs.
So we've established that New World Order is essentially a rock album and it sounds like Saxon. The next big question is: is it any good? For the most part, yes. At 14 songs this album runs way too long, and not all of the songs are home runs. It sure starts strong though, with the hard charging "We Came to Rock," "One Night in Hellas" and "New World Order." "Just One Kiss" is a surprisingly good love song, and "A Warrior's Song" is another highlight.
Overall, New World Order is a solid, enjoyable hard rock album. If it was Uriah Heep putting this out as their 39th album, it wouldn't raise any eyebrows. As a comeback album after a 30-year absence though, "solid and enjoyable" aren't good enough. Q5 needed to make a bold statement here, and this just doesn't get the job done. If you're a Q5 fan, you pretty much have to give it a try though. Fans of Saxon, Uriah Heep and UFO may dig this as well.
Q5: New World Order Justin G.
Album Rating
Summary: It's not bad at all, but it's not great either.
Tags: 2016 Reviews, Frontiers, Q5