text:
\section{Introduction}

In the representation theory of a given algebra or group, the usual
ultimate goal is a clear description of its irreducible modules to
begin with, indecomposable ones at the next step. Other goals are
sometimes also natural and reachable, cf. those we discuss here and
in review \cite{BF}.

\ssec{Discussion of open problems} The history of mathematics shows
that problems with a nice (at least in some sense, for example,
short) answer often turn out to be ``reasonable". I.~Gelfand used to
say that, the other way round, easy-to-formulate lists are answers
to ``reasonable" problems  and advised to try to formulate such
problems once we got a short answer.

\textbf{Hereafter, $\Kee$ is any field of characteristic $p>0$,
unless otherwise specified}.

Speaking of Lie algebras or superalgebras, and their representations
over $\Kee$, which of them is it natural to consider? In \cite{LL},
Deligne suggests that one should begin with restricted ones: unlike
nonrestricted ones, the restricted Lie (super)algebras correspond to
groups, in other words: to geometry. Certain problems concerning
nonrestricted algebras are also natural (although tough), and have a
short answer, e.g., classification of simple Lie algebras for $p>3$
(for its long proof, see \cite{S,BGP}).

Rudakov and Shafarevich \cite{RSh} were the first to describe ALL,
not only restricted, irreducible representations of  $\fsl(2)$ for
$p>2$. Dolotkazin solved the same problem for $p=2$, see \cite{Do};
more precisely, he described the irreducible representations of the
simple 3-dimensional Lie algebra $\fo^{(1)}(3)$. The difference of
Dolotkazin's problem from that considered in \cite{RSh} is that,
unlike $\fsl(2)$ for $p>2$, the Lie algebra $\fo^{(1)}(3)$ is not
restricted. These two results show that the description of all
irreducible representations looks feasible, to an extent, at least
for Lie (super)algebras with indecomposable Cartan matrix or their
simple ``relatives" (for the classification of both types of
(super)algebras over $\Kee$, see \cite{BGL}).

The problem we are solving here for $p>0$ resembles its super
version \textbf{over $\Cee$}; we recall the known results. Simple
Lie (super)algebras of vector fields with polynomial or formal
coefficients constitute another class of interesting Lie
(super)algebras. \textbf{Over $\Cee$}, they are classified and their
continuous (in the $x$-adic topology) irreducible representations
with a vacuum (highest or lowest) weight vector are classified as
well in the following cases:

$\bullet$ For the four series of simple Lie algebras, and their
direct super versions, the answer is as follows, see the review
\cite{GLS}: the module $T(V)$ of tensor fields coinduced from any
irreducible module $V$ with lowest weight vector over the subalgebra
of linear vector fields is irreducible, unless $T(V)\simeq\Omega^k$,
the module of exterior $k$-forms.

Rudakov proved that the exterior differential is the only
(nonscalar) operator between the modules of exterior differential
forms invariant with respect to all changes of indeterminates, or,
equivalently, for the general vectorial Lie algebra. Together with
the Poincar\'e lemma (stating that the sequence \eqref{ext} is exact
over any star-shaped domain) this yields a complete description of
continuous irreducible modules with lowest weight.

$\bullet$ For the remaining 3 series of simple vectorial Lie
algebras (and analogous simple Lie superalgebras \textbf{in their
standard grading}) the answer is essentially the same.

$\bullet$ For simple vectorial Lie superalgebras in their
\textbf{nonstandard} grading (listed in \cite{LS}), and for the
modules of tensor fields with \textbf{infinite-dimensional fibers}
(i.e.,  $\dim V=\infty$) even over finite-dimensional simple
vectorial Lie superalgebras, many never previously known
differential operators appear, see \cite{GLS,Lint}.

$\bullet$ In the purely odd case the Berezin integral has to be
added to the list of invariant differential operators, see
eq.~\eqref{intDR}.

$\bullet$ \textbf{Over $\Kee$}, it is tempting to conjecture that,
at least for $p$ ``sufficiently large", the description of
irreducible modules over vectorial Lie algebras (at least
$\Zee$-graded, and ``close to simple" ones) the description of
irreducible modules would resemble that given by Rudakov over
$\Cee$. For irreducible  restricted modules over restricted simple
vectorial Lie algebras of the 4 classical series and $p\geq 3$, this
is almost so, i.e., the spaces of differential $k$-forms areg the
only reducible spaces of tensor fields, as proven by Krylyuk, see
\cite{Kry}, although the complex of exterior differential forms is
not exact, see Theorem~\ref{Kryl}.

There are, however, natural filtered deforms of simple nonrestricted
vectorial Lie algebras; for $p\leq 5$, there are new types of simple
vectorial Lie algebras and no \textbf{description of their
irreducible modules is known; in particular, no description of
intertwining operators between modules of tensor fields --- the most
natural ones over these Lie algebras}. Description of this type has
a history.

\sssec{History: Veblen's problem} In 1928, at the IMC, O.~Veblen
formulated a problem which was later reformulated in more
comprehensible terms by A.~Kirillov and further reformulated as a
purely algebraic problem by J.~Bernstein who interpreted Rudakov's
solution of Veblen's problem for unary operators; for setting in
modern words and review, in particular, for a superization of
Veblen's problem, see \cite{GLS}.

\iffalse

In order to be able to speak about differential operators, recall
that A.~Gro\-then\-dieck defined \textit{differential operators
of~order} $\leq k$ for any commutative ring $R$ over a field $K$ or
$\Zee$ as the $K$- or $\Zee$-linear maps of~$R$\defis modules
$D:M\tto N$ such that
\begin{equation}
\label{2Oeq236} [l_{r_0}, [l_{r_1}, \ldots [l_{r_k}, D]\ldots]]=0
\quad \text{ for any }\; r_0, r_1, \ldots r_k\in R
\end{equation}
where $l_r$ denotes the operator of~left multiplication by $r$ in
$M$ and in $N$.

Denote by $\Diff_k(M, N)$\index{$\Diff_k(M, N)$} the $R$\defis
module of~differential operators of~degree $\leq k$; set
\begin{equation}
\label{2Oeq237} \Diff (M, N)=\mathop{\lim}\limits_{\tto} \Diff_k(M,
N).
\end{equation}

Define the $R$\defis module of~\textit{symbols of~differential
operators} to be the graded space associated with $\Diff(M, N)$
\begin{equation}
\label{2Oeq238} \Smbl(M, N)=\oplus \, \Smbl_k(M, N),
\end{equation}
where
\begin{equation}
\label{2Oeq239} \Smbl_k(M, N)=\Diff_k(M, N)/\Diff_{k-1}(M, N).
\end{equation}
If $M=N$, then, clearly, $\Diff~(M):= \Diff~(M, M)$ is
an~associative algebra with respect to the product and a~Lie algebra
--- denoted by $\fdiff(M)$ --- with respect to the commutator
of~operators. Clearly, $\Smbl (M)$ is a~commutative $R$\defis
algebra with respect to the product. The bracket in $\fdiff(M)$
induces a~Lie algebra structure in $\Smbl (M)$ (the Poisson
bracket); as is not difficult to see, this Lie algebra is isomorphic
to the Poisson Lie algebra in $2\rk_R M$ indeterminates over $R$. We
consider differential operators of fixed order, i.e., elements of
the spaces $\Diff^{\leq n}(M)/\Diff^{\leq n-1}(M)$, e.g.,
$\fvect_R(m)\simeq\Diff^{\leq 1}(R)/\Diff^{\leq 0}(R)$, the Lie
algebra of vector fields with polynomial coefficients over
$R=K[t_1,\dots, t_m]$.

\fi

Let $\fvect(m)$ be the Lie algebra of polynomial vector fields (over
a ground field of characteristic 0, say $\Cee$). Assuming the notion
of the $\fvect(m)$-module $T(V)$ of (formal) tensor fields of type
$V$, where $V$ is a $\fgl(m)$-module with lowest weight vector, is
known, see \cite{GLS}. Let us briefly recall the results concerning
the \textbf{nonscalar} unary and binary invariant differential
operators between spaces of tensor fields, although we only need the
simplest version of spaces $T(V)$, namely, the spaces of weighted
densities $\cF_a:=T(a\, \tr)$, where $\tr$ is the 1-dimensional
$\fgl(m)$-module given by the trace (supertrace for $\fgl(m|k)$) and
$a$ is in the ground field. We assume that modules $T(V)$ in what
follows are coinduced from \textbf{irreducible} modules $V$; this is
certainly so for modules $\cF_a$. (In Veblen's problem for simple
vectorial Lie algebras over $\Cee$, it suffices to consider only
irreducible modules $V$; for its superization and over $\Kee$, one
has to consider indecomposable modules $V$, but one has to begin
with irreducible ones, anyway.)

\paragraph{Unary operators} The only unary $\fvect(m)$-invariant
differential operators $T(V)\tto T(W)$ are the exterior differential
operators (so, practically, there is just one such operator) in the
de Rham complex, where $\Omega^i=T(E^i(\id))$, $E^i$ is the functor
of raising to the $i$th exterior power and $\id$ is the tautological
$m$-dimensional $\fgl(m)$-module:
\begin{equation}
\label{ext}
0\tto\Cee\tto\Omega^0\stackrel{d}{\tto}\Omega^1\stackrel{d}{\tto}\dots\stackrel{d}{\tto}
\Omega^{m-1} \stackrel{d}{\tto}\Omega^m\tto 0.
\end{equation}

In super setting, where $\sdim(\id)=m|k$, the analog of the complex
\eqref{ext} is infinite without the ``top term"
\begin{equation}
\label{extSup}
0\tto\Cee\tto\Omega^0\stackrel{d}{\tto}\Omega^1\stackrel{d}{\tto}\Omega^2
\stackrel{d}{\tto}\dots.
\end{equation}
its dualization brings the complex of \textit{integrable} forms
which \textbf{in the purely odd case of superdimension $0|k$}
terminates with an order-$k$ \textbf{differential} operator called
the \textit{Berezin integral}:
\begin{equation}
\label{intDR}
\renewcommand{\arraystretch}{1.4}
\begin{array}{ll}
\dots\stackrel{d}{\tto}\Sigma_{-1}(m|k)\stackrel{d}{\tto}
\Sigma_{0}(m|k)\stackrel{d}{\tto}0&\text{for $m\neq 0$},\\
\dots\stackrel{d}{\tto}\Sigma_{-1}
(0|k)\stackrel{d}{\tto}\Sigma_{0}(0|k)
 \stackrel{\int}{\tto}\Kee\tto 0&\text{for $m= 0$}.\\ \end{array}
\end{equation}
Over $\Cee$, the de Rham complex \eqref{extSup} is exact, while the
complex of integrable forms \eqref{intDR} has 1-dimensional
cohomology in the $(-m)$th  term; the cohomology is spanned, see
\cite{BL,MaGF,Del}, by (here the $u_i$ are the even indeterminates,
the $\theta_j$ are the odd ones for clarity; in Notation~\ref{Nota}
all indeterminates are called $u$)
\begin{equation}
\label{intCoh}\theta_1\dots\theta_k\partial_{u_1}\dots
\partial_{u_m}\vvol(u|\theta)
\end{equation}
and the super version of the volume element
$\vvol(u)=du_1\wedge\dots\wedge du_m$ is  the class
$\vvol(u|\theta)$ of the element
\begin{equation}
\label{vvol} du_1\wedge\dots\wedge du_m\otimes
\partial_{\theta_1}\wedge\dots\wedge\partial_{\theta_k}
\end{equation}
in the indecomposable $\fgl(V)=\fgl(m|k)$-module generated by the
element \eqref{vvol} modulo the codimension 1 submodule, see
\cite{Del,LSoS}.

\paragraph{Binary operators} These are classified by P.~Grozman, see
\cite{GInv}. Recall that the formal dual of $T(V)$ on the
$m$-dimensional space is not $T(V^*)$ but $T(V^*\otimes \tr)$.
Hence, with every unary operator $D\colon T(V)\tto T(W)$ there
exists its dual $D^*\colon T(W^*\otimes \tr)\tto T(V^*\otimes \tr)$.

In particular, on the line,  with every operator $D\colon \cF_a\tto
\cF_{a+\deg D}$ of order $\deg D$, there exists its dual $D^*\colon
\cF_{1-a-\deg D}\tto \cF_{1-a}$ of the same order.

Similarly, with every binary operator $D\colon \cF_a\otimes
\cF_b\tto \cF_{a+b+\deg D}$, there exist the two its duals
$D^{*1}\colon \cF_{1-a-b-\deg D}\otimes \cF_b\tto \cF_{1-a}$ and
$D^{*2}\colon \cF_a\otimes \cF_{1-a-b-\deg D}\tto \cF_{1-b}$, as
well as the operator $r(D)\colon \cF_b\otimes \cF_a\tto
\cF_{a+b+\deg D}$, given by the interchange of the arguments
\[
r(D)(X,Y)\colon =D(Y,X)\text{~~for any $X\in \cF_b$ and
$Y\in\cF_a$.}
\]

\textbf{Summary}: Up to dualizations and interchange of the
arguments, there are 8 series of invariant operators of order 1, all
invariant differential operators of order 2 and 3 are compositions
of order-1 operators, except for the case $m=1$, where there is an
indecomposable operator $Gz$ of order $3$ discovered by Grozman; for
explicit expressions, see \cite{GInv} and Subsec.~\ref{BiCee}. There
are no operators of order $>3$.

\textbf{Off-topic}: 1) Among all 8 series of invariant operators of
order 1, there are 4 series that determine an associative algebra
and 3 series determine a Lie (super)algebra structure on their
domain of definition. These Lie (super)algebras are either simple or
simple modulo center or contain a simple algebra of codimension 1:
the corresponding products are the Poisson bracket, the Schouten
bracket (more popular now under the name anti-bracket) and a
deformed Schouten bracket, see \cite{GInv, LSh}.

2) Kirillov gave an example of the symbol of an invariant with
respect to all coordinate changes \textit{nonlocal} bilinear
operator on the circle but did not identify the operator itself; in
\cite{IoMo} this operator is described in terms of the
integral=residue; it is the only nonlocal invariant bilinear
operator on the circle. In characteristic $p>0$, same as on
superpoints, analogs of various operators nonlocal over $\Ree$
become local because if the space of functions is
finite-dimensional, then any operator in this space can be viewed as
differential, i.e., composition of derivations and multiplications
by some functions.

\paragraph{Invariant differential operators of higher arity}
The only $k$-ary invariant operators classified so far are
antisymmetric ones on $\Cee$ , see \cite{FeFu}, and ternary
operators between the spaces of weighted densities on $\Cee^n$, see
\cite{Bj}.

\ssec{Main result} The classification (Theorem~\ref{Th}) of binary
$\fvect(1;\underline{N})$-invariant operators between modules of weighted
densities; in particular, we discovered three series of
indecomposable operators of however high order: 1) related to the
analog of Berezin integral, 2) an analog of the Grozman operator,
and 3) a completely new series of operators, $Bj$.

\ssec{Open problems} 1) We conjecture that the Grozman operator has
something to do with the contact structure since $\fvect(1)=\fk(1)$.
Over $\Cee$, it seems feasible to classify binary
$\fk(2n+1)$-invariant differential operators for $n>0$ and complete
the partial result for binary $\fh(2n)$-invariant differential
operators, solved only for $n=1$, see \cite{Ghm}.

2) The only feasible super version of the problem we consider here
is, it seems, classification of the $\fk(1;\underline{N}|k)$-invariant
operators for $k=1$ or 2 for which the coinduced modules with lowest
weight vectors are of the form $\cF_a$. For the other pairs $m,k$
the analog of Veblen's problem does not seem to be feasible.

3) Prove Conjecture~\ref{Conj}.

\section{Veblen's problem over fields of
characteristic $p>0$}

\ssec{Notation}\label{Nota} For an $m+k$-tuple of nonnegative
integers $\underline{r}=(r_1, \ldots , r_a)$, where $r_i=0$ or 1 for
$i>m$, we introduce parity by setting $p(u_i)=\ev$ for $i\leq m$ and
$p(u_i)=\od$ for $i> m$. Set further
\begin{equation*}
\label{1.} u_i^{(r_{i})} := \frac{x_i^{r_{i}}}{r_i!}\quad
\text{and}\quad u^{(\underline{r})} := \prod\limits_{1\leq i\leq a}
u_i^{(r_{i})}.
\end{equation*}
The idea is to formally replace fractions with $r_i!$ in
denominators by inseparable symbols $u_i^{(r_{i})}$ which are
well-defined over fields of characteristic $p$ because the structure
constants are integers one can consider modulo $p$, in particular,
if $k=0$, the factor in the first parentheses below is equal to 1:
\begin{equation*}
\label{divp}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l} u^{(\underline{r})} \cdot u^{(\underline{s})} =
\left(\mathop{\prod}\limits_{m+1\leq i\leq n}
\min(1,2-r_i-s_i)\cdot(-1)^{\mathop{\sum}\limits_{m<i<j\leq a}
r_js_i}\right) \cdot \binom {\underline{r} + \underline{s}}
{\underline{r}}
u^{(\underline{r} + \underline{s})}, \\
\text{where}\quad \binom {\underline{r} + \underline{s}}
{\underline{r}}:=\mathop{\prod}\limits_{1\leq i\leq m}\binom {r_{i}
+ s_{i}} {r_{i}}.
\end{array}
\end{equation*}
For an $m+k$-tuple of positive integers $\underline{N} = (N_1,...,
N_m, 1, \dots, 1)$, set $\underline{N}_{ev} = (N_1,..., N_m)$. The
following supercommutative superalgebra is the analog of the algebra
of functions when $p>0$:
\begin{equation*}
\label{u;N}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
\cO(m;\underline{N}_{ev}|k):=\Kee[u;\underline{N}]:=
\Span_{\Kee}(u^{(\underline{r})}\mid r_i < p^{N_{i}}\;\text{ for
$i\leq m$, $0\leq r_i \leq 1$ for $i> m$}).\end{array}
\end{equation*}
The Lie superalgebra $\fvect(m;\underline{N}|k)$ of its \textit{distinguished}
derivations consists of the vector fields of the form $\sum
f_i(u)\partial_i$, where ${f_i\in\cO(m;\underline{N}_{ev}|k)}$ and $\partial_i$ are
\textit{distinguished} partial derivatives defined by the condition
${\partial_j(u_i^{(r_{i})})=\delta_{ji}u_i^{(r_{i}-1)}}$.

\ssec{Unary operators: De Rham complex for $k=0$ (no super)}
\sssbegin{Theorem}[\cite{Kry}]\label{Kryl} Set
\begin{equation*}\label{tau}
\tau(\underline{N})=(p^{N_1}-1, \dots, p^{N_m}-1).
\end{equation*}
Denote: $Z^i(m;\underline{N}):=\{\omega
\in\Omega^i(m;\underline{N})\mid d\omega=0\}$ and
$B^i(m;\underline{N}):=\{d\omega \mid \omega
\in\Omega^{i-1}(m;\underline{N})\}$.

On the $m$-dimensional space over $\Kee$, the sequence
\begin{equation*}
\label{ext-p}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
0\tto\Kee\tto\Omega^0(\underline{m;N})\stackrel{d}{\tto}
\Omega^1(m;\underline{N})\stackrel{d}{\tto}\Omega^2(m;\underline{N})
\stackrel{d}{\tto}\dots \stackrel{d}{\tto}\Omega^m(m;\underline{N})
\stackrel{d}{\tto}0 \end{array}
\end{equation*}
is not exact: the space
$H^a(m;\underline{N}):=Z^a(m;\underline{N})/B^a(m;\underline{N})$ is
spanned by the elements
\begin{equation*}
\label{coh} u_{i_1}^{(\tau(\underline{N})_{i_1})}\dots
u_{i_k}^{(\tau(\underline{N})_{i_k})}du_{i_1}\dots du_{i_k}\text{
where $a=i_1+\dots +i_k$}.
\end{equation*}
In other words, the supercommutative superalgebra
$H^{\bcdot}(m;\underline{N})$ is generated by
$H^1(m;\underline{N})$.
\end{Theorem}

\begin{proof}Induction on $m$. For $m=1$, this is obvious. \end{proof}

\sssbegin{Corollary}\label{Cor} There is an invariant differential
operator of order $|\tau(\underline{N})|:=(\sum p^{\underline{N}_i})-m$:
\begin{equation*}
\label{int-p}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
\Omega^m(m;\underline{N}) \stackrel{\int}{\tto}\Kee; \end{array}
\end{equation*}
this $\int$ is a, depending on $\underline{N}$, analog of the Berezin integral
defined as follows:
\[
\int  f(u)\vvol(u)=\text{the coefficient of
$u^{\tau(\underline{N})}=u_{1}^{(\tau(\underline{N})_{1})}\dots
u_{m}^{(\tau(\underline{N})_{m})}$, where
$\vvol(u)=du_1\wedge\dots\wedge du_m$}.
\]
\end{Corollary}

\parbegin{Remark} We did not find the statement of the above
Corollary in the literature; so formulation of the fact that there
exists an integral depending on $\underline{N}$ seems to be a new
result.\end{Remark}

\ssec{Unary operators: De Rham complex for $k\neq 0$, i.e., in super
setting} On the $m|k$-dimensional space over $\Kee$,  the de Rham
complex goes ad infinitum
\begin{equation*}
\label{superDR}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
0\tto\Kee\tto\Omega^0(m;\underline{N}|k)\stackrel{d}{\tto}
\Omega^1(m;\underline{N}|k)\stackrel{d}{\tto}\Omega^2(m;\underline{N}|k)
\stackrel{d}{\tto}\dots \end{array}
\end{equation*}

The dual to \eqref{superDR} complex of \textit{integrable} (Deligne
calls them \textit{integral}) forms is as follows, where
$\Sigma_{0}:=\Omega^0\vvol(u|\theta)$:
\begin{equation}
\label{intDRp>0}
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
\dots\stackrel{d}{\tto}\Sigma_{-1}
(m;\underline{N}|k)\stackrel{d}{\tto}\Sigma_{0}(m;\underline{N}|k)
 \stackrel{\int}{\tto}\Kee\tto 0. \end{array}
\end{equation}

Observe that, for $p>0$, the integral is defined for $m\neq0$ as
well, whereas for $p=0$, there is no such an invariant
\textit{differential} operator.

\sssbegin{Conjecture}\label{Conj} The only
$\fvect(m;\underline{N}|k)$-invariant unary operators between modules of
tensor fields coinduced from irreducible $\fgl(m|k)$-modules  are
the exterior operator $d$ and the Berezin integral $\int$.
\end{Conjecture}

\ssec{The binary differential operators over $\Cee$ for $m=1$ (recapitulation,
see \cite{GInv})}\label{BiCee} For $m=1$, we have
$\cF_a:=\{f(u)(du)^a\mid f(u)\in \Omega^0=\cF_0\}$ on which
the Lie derivative acts as follows:
\begin{equation}\label{F_a}
L_D(f(u)(du)^a)=(D(f)+afg')(du)^a)\text{~~for any
$D=g(u)\partial_u\in\fvect(1)$}.
\end{equation}

For any bilinear differential
operator
\[
D\colon \cF_a\otimes \cF_b\tto \cF_{a+b+\deg D},\ \ \
f(du)^a\otimes g(du)^b\longmapsto D(f,g)(du)^{a+b+\deg D},
\] it suffices to
indicate $(a,b)$ and $D(f,g)$. The operators are listed up to
proportionality.

\sssec{Order 1 operators} For generic values $(a,b)$, the invariant
operators form a 1-dimensional space. For $(a,b)=(0,0)$, and only in
this case, we have a 2-dimensional space of operators.

\begin{equation}
\label{.1} \renewcommand{\arraystretch}{1.4}
\begin{tabular}{|l|l|l|}
\hline
$(a,b)$&$D(f,g)$&\text{comment} \\
\hline
$(0,0)$&$P_{00}\colon \alpha f'g+\beta fg'$&\text{for any $\alpha,\beta\in\Cee$}\\
\hline $(a,b)$&$\{\cdot,\cdot\}_{P.B.}\colon =afg'-bf'g$
&\text{Poisson bracket in two
even indeterminates $u$ and $du$}\\
$(-1,-1)$&$\{\cdot,\cdot\}\colon =fg'-f'g$ &\text{contact bracket of
generating
functions $f\longmapsto K_f:=f\nfrac{d}{du}$}\\
\hline\end{tabular}
\end{equation}

\sssec{Order 2 operators}
\begin{equation}
\label{.2} \begin{tabular}{|l|l|l|}
\hline$(a,b)$&$D(f,g)$&\text{comment}\\
\hline $(0,b)$
&$f'g'-bf''g$&$\{df,g(du)^b\}_{P.B.}$\\
$(a,0)$&$afg''-f'g'$&$\{f(du)^a,dg\}_{P.B.}$ \\
$(a,-1-a)$&$afg''+(2a+1)f'g'+(a+1)f''g$&$(\{f(du)^a,g(du)^{-1-a}\}_{P.B.})'$ \\
\hline\end{tabular}
\end{equation}

\sssec{Order 3 operators}
\begin{equation}
\label{.3} \begin{tabular}{|l|l|}
\hline$(a,b)$&$D(f,g)$\\
\hline
$(0,0)$&$T_1(f,g)=\{f',g'\}$\\
$(-2,0)$&$T_1^{*1}(f,g)=f'g''+3f''g'+ 2f'''g$\\
$(0,-2)$&$T_1^{*2}(f,g)=f''g'+3f'g''+ 2fg'''$\\
\hline

$\left(-\nfrac23,-\nfrac23\right)$&$
Gz(f,g)=2\det\begin{pmatrix}f&g\\
f'''&g'''\end{pmatrix}+3\det\begin{pmatrix}f'&g'\\
f''&g''\end{pmatrix}$\\
\hline\end{tabular}
\end{equation}

\ssec{New binary differential operators: $p>0$ and $m=1$} For a
fixed $\underline{N}=N$, we list $\fvect(1;\underline{N})$-invariant operators up to
interchange of arguments.

Recall that $\int\colon  \Omega^1=\cF_1\tto\Kee\subset\cF_0$ is the
order $p^N-1$ operator, see Corollary~$\ref{Cor}$ and $1-p^N\equiv
1\pmod p$. Therefore, for any $p>0$ and $\underline{N}$, there are
$\fvect(1;\underline{N})$-invariant bilinear operators acting in the spaces
$\cF_a\otimes\cF_b$ in the following cases:
\begin{equation}\label{ind1}
\begin{tabular}{|l|c|c|}
\hline
operator$(a,b)$&order&case\\
\hline $\int\otimes \id(1,a)$,\ \ $(\int\otimes \id)^{*1}(1-a,a)$,\
\ $(\int\otimes
\id)^{*2}(1,1-a)$&$k=p^N-1$&1\\

$\int\otimes d=(\int\otimes d)^{*2}(1, 0)$,\ \ $(\int\otimes d)^{*1}(0,0)$,\ \
$r(\int\otimes d)(0,1)$ &$k=p^N$&2\\

$\int\otimes \int=(\int\otimes \int)^{*1}=(\int\otimes
\int)^{*2}(1,1)$&$k=2(p^N-1)$&3\\
\hline
\end{tabular}
\end{equation}
The operator $(\int\otimes \id)^{*1}(1-a,a)$ is of the form ($L$ for
``long" expression)
\begin{equation}\label{LongOp}
L_{p,N}(f,g)=\sum_{i=0}^{\lfloor (k-1)/2\rfloor} (-1)^{i}
\left ( f^{(i)}g^{(k-i)}+ f^{(k-i)}g^{(i)}\right ) +
\begin{cases} (-1)^{k/2}f^{(k/2)}g^{(k/2)}&\text{if $p>2$}\\
&\text{if $p=2$}.\end{cases}
\end{equation}

\sssbegin{Lemma}\label{leQ} For  $(a,b)=(1,1)$, there exists a
$\fvect(1;\underline{N})$-invariant bilinear operator $Bj_{p,m}$ of order
$k=p^m-1$ for any $m\leq N$ given by  the expression
\begin{equation}\label{ind2}
Bj_{p,m}(f,g)=\det
\begin{pmatrix}f &g\\
f^{(k)}&g^{(k)}\end{pmatrix}.
\end{equation}

Its duals for $(a,b)=(1,0)$ and $(0,1)$, respectively, are as
follows:

\begin{equation}\label{ind3}
Bj_{p,m}^{*2}(f,g)=f g^{(k)}+\sum_{i=1}^{\lfloor (k-1)/2\rfloor}
(-1)^{i} \left ( f^{(i)}g^{(k-i)}+ f^{(k-i)}g^{(i)}\right ) +
\begin{cases} (-1)^{k/2}f^{(k/2)}g^{(k/2)}&\text{if $p>2$}\\
&\text{if $p=2$ }\end{cases}
\end{equation}
and
\begin{equation}\label{ind4}
Bj_{p,m}^{*1}(f,g)=f^{(k)} g+\sum_{i=1}^{\lfloor (k-1)/2\rfloor}
(-1)^{i} \left ( f^{(i)}g^{(k-i)}+ f^{(k-i)}g^{(i)}\right ) +
\begin{cases} (-1)^{k/2}f^{(k/2)}g^{(k/2)}&\text{if $p>2$}\\
&\text{if $p=2$ }.\end{cases}
\end{equation}
\end{Lemma}

\begin{proof}Every order $k$ bilinear differential operator $D$ is of the form
\[
(f,g)\longmapsto\sum_{i+j=k} \alpha_{i,j} f^{(i)}
g^{(j)},\text{~~where $\alpha_{i,j}\in\Kee$}.
\]
For $N$ big enough (such that $p^N>\deg D+1$) and a given pair
$(a,b)$, the $\fvect(1;\underline{N})$-invariance of $D$ implies the following
system of equations
\begin{equation*}
\label{InvEq1}
\left ( a \binom{i}{r-1} + \binom{i}{r}  \right )\alpha_{i, k-i}+
\left ( b \binom{k-i+r-1}{r-1} + \binom{k-i+r-1}{r}  \right
)\alpha_{i-r+1, k-i+r-1}=0,
\end{equation*}
for $i=1,\ldots, k$ and $2\leq r \leq k+1$. In the case of Lemma,
$(a,b)=(1,1)$; besides $\alpha_{i,j}=0$ whenever $i+j\not=k$. The
above system becomes
\begin{equation}
\label{InvEq2} \binom{i+1}{r} \alpha_{i, k-i}+  \binom{k-i+r}{r}
\alpha_{i-r+1, k-i+r-1}=0\text{~~for $i=1,\ldots, k$ and $2\leq r
\leq k+1$}.
\end{equation}
If $r=k+1$, then $i$ must be equal to $k$, so we get the condition
$\alpha_{0,k}+\alpha_{k,0}=0.$ If $r<k+1$, then the above equations
are either identically zero (since $\alpha_{i,j}=0$ whenever
$i+j\not=k$) or of the form
\[
\binom{k+1}{r} \alpha_{k, 0}=0 \text{ and } \binom{k+1}{r}
\alpha_{0, k}=0.
\]
But $\binom{k+1}{r}=0\, (\text{mod } p)$, since $k+1=p^N$.

The claim on dual operators is subject to a direct verification.
\end{proof}

\sssec{The transvectants (also known as Cohen-Rankin brackets)} Recall that for any $p\neq 2$, there is an
embedding $\fsl(2)\hookrightarrow \fvect(1;\underline{N})$. (For $p=2$, the
analog of this embedding is the embedding
$\fo^{(1)}(3)\hookrightarrow \fvect(1;\underline{N})$ for $\underline{N}>1$.) The
\textit{transvectants} are $\fsl(2)$-invariant bilinear differential
operators discovered by Gordan. As a matter of fact, there is a
misprint in the explicit expressions of the transvectants
$J_{a,b}^k$ in \cite{BjOv}; the corrected expression is (up to a
numerical factor)
\begin{equation}\label{ind5} J_{a,b}^k(f,g)= \sum_{0\leq i\leq k}(-1)^i
\binom{2a +k-1}{k-i}\binom{2b+k-1}{i} f^{(i)} g^{(k-i)}.
\end{equation}
This expression is for generic $a$ and $b$, and $p\not =2$. For the
exceptional values of $a$ and $b$ over $\Cee$, see \cite{Bj2},
Prop.2. In this paper, we are mainly interested in the explicit
expressions
of the transvectants for $p=2$.

\sssbegin{Lemma}\label{CjGz} For  $(a,b)=(1,1)$, there exists a
$\fvect(1;\underline{N})$-invariant bilinear operator of order $k=p^m-2$ for
any $m\leq N$ if $p\not =2$ whereas for $p=2$ it should be any
$m\leq N+1$ (because of the term $f^{(k/2)}g^{(k/2)}$) given by the
expression
\begin{equation}\label{ind6}
Gz_{p,m}(f,g)=\sum_{i=0}^{\lfloor{(k-1)/2\rfloor}} (-1)^{i}\det
\begin{pmatrix}f^{(i)}&g^{(i)}\\
f^{(k-i)}&g^{(k-i)}\end{pmatrix}+
\begin{cases
f^{(k/2)}g^{(k/2)}&\text{if $p=2$}\\
&\text{otherwise}.\\
\end{cases}
\end{equation}
\end{Lemma}
This operator coincides with the Grozman operator if $p=5$ and
$m=1$. Observe that $Gz$ is selfdual over any field it is defined,
but $Gz\circ (d\otimes d)$ is not selfdual.

\begin{proof} Let, for simplicity, $p\not =2$ (for $p=2$ the proof is more or less the
same). Since $(a,b)=(1,1)$, the invariance with respect to
$\fvect(1;\underline{N})$ is equivalent to the system (\ref{InvEq2}). For
$i=k$ or $i=0$, we have $\alpha_{k,0}+\alpha_{0,k}=0$ which is
certainly satisfied. Now, for $i\not=0,k$, the system becomes
\[
\binom{i+1}{r} (-1)^i+  \binom{k-i+r}{r}  (-1)^{i-r+1}=0
\text{~~for $i=1,\ldots, k$ and $2\leq r \leq k+1$}.
\]

To show that the system is satisfied, we proceed by induction. If
$i=1$, then $r$ must be equal to 2. The result follows since
\[
\begin{array}{lcl}
\binom{k+1}{2} & =& \binom{p^m-1}{2} = 1 \text{ mod } (p).\\[3mm]
\end{array}
\]
Now suppose that the equality is
true at $i$ for every $r$ such that $r\leq i+1$. It follows that
\[
\begin{array}{lcl}
\binom{k-(i+1)+r}{r} & =&\binom{k-i+r}{r}-\binom{k-i+r-1}{r-1} \text{ `Pascal's triangle' }\\[3mm]
& =&(-1)^r\binom{i+1}{r}-(-1)^{r-1}\binom{i+1}{r-1} \text{ mod } (p) \text{ induction hypothesis}\\[3mm]
& =&(-1)^r \left (\binom{i+1}{r}+\binom{i+1}{r-1}\right ) \text{ mod } (p) \\[3mm]
& =&(-1)^r \binom{i+2}{r} \text{ mod } (p) \text{ `Pascal's triangle' } \hfill \qed\\
\end{array}
\]

\noqed\end{proof}

\ssbegin{Theorem}\label{Th} Up to the interchange of arguments, the
indecomposable $\fvect(1;\underline{N})$-invariant bilinear operators $D\colon
\cF_a\otimes\cF_b\tto\cF_c$ are only those of the form
$\eqref{ind1}$--$\eqref{ind6}$. For the complete list of
$\fvect(1;\underline{N})$-invariant bilinear operators of order $\leq 7$, see
tables $\eqref{t1}$--$\eqref{t7}$.
\end{Theorem}

Although we have found all invariant operators of order $\leq 20$,
the complete list is not that interesting, we think, unlike the
indecomposable operators. We illustrate the answer with a part of
the complete list.

\begin{proof} Computer-aided study with the aid of
Grozman's \textit{Mathematica}-based code \textit{SuperLie}, see
\cite{Gr}:  (1) We fix $\deg D\in\{1, 2, \dots, 20\}$ and
$p\in\{2,3,5,\dots, 19\}$. (2) For this $\deg D$ and $p$, we look
for which $\underline{N}$ and $(a,b)$ there is a $\fvect(1;\underline{N})$-invariant
operator.
\end{proof}

$\bullet$ \emph{Order 1 operators} are the same as for $p=0$, but
for $p=2$ we have
\begin{equation}
\label{t1} \renewcommand{\arraystretch}{1.4}
\begin{tabular}{|l|l|l|}
\hline
$(a,b)$&$D(f,g)$&\text{comment} \\
\hline
$(1,b)$ and $(a,1)$&$\int\otimes \id\colon  f'g$ and $fg'$&\text{for $\underline{N}=1$}\\
$(a,b)$&$P_{00}\colon \alpha f'g+\beta fg'$&\text{for any $\alpha,\beta\in\Kee$ and $\underline{N}=1$}\\
\hline
$(0,0)$&$P_{00}\colon \alpha f'g+\beta fg'$&\text{for any $\alpha,\beta\in\Kee$}\\
\hline
$(a,b)$&$\{\cdot,\cdot\}_{P.B.}\colon =afg'+bf'g$ &\text{Poisson bracket}\\
$(1,1)$&$\{\cdot,\cdot\}\colon =fg'+f'g$ &\text{contact bracket of generating functions}\\
\hline\end{tabular}
\end{equation}

$\bullet$ \emph{Order 2 operators} are the same as for $p=0$, but
the condition $a+b+1=0$ is to be replaced with $a+b+1=0 \pmod p$ and
for $p=2$ we have
\begin{equation}
\label{t2} \begin{tabular}{|l|l|l|}
\hline$(a,b)$&$D(f,g)$&\text{comment} \\
\hline
$(a,b)$ &$f'g'$&for $\underline{N}=1$; if $(a,b)=(1,1)$, then $D=\int\otimes \int$ \\
\hline
$(0,b)$ &$f'g'+bf''g$&$\{df,g(du)^b\}_{P.B.}$\\
$(a,1+a)$&$afg''+f'g'+(a+1)f''g$&$\{f(du)^a,dg(du)^{1+a}\}_{P.B.}$\\
$(a,b)$&$a g''f+f'g'+ b f'' g$&generalizes the above\\
\hline\end{tabular}
\end{equation}

whereas for $p=3$ we have
\begin{equation}
\label{t23} \begin{tabular}{|l|l|l|}
\hline$(a,b)$&$D(f,g)$&\text{comment} \\
\hline
$(a,b)$ for $a,b\not =0,1$&$J_{a,b}^2(f,g)
&for $\underline{N}=1$\\
$(1,b)$ &$\int\otimes \id(f,g)= f''g$ &for $\underline{N}=1$\\
$(0,b)$ &$f'g' -b fg''$ &for $\underline{N}=1$\\
\hline

$(a,-1-a)$&$\{f,g\}'=afg''+(1-a)f'g'+(a+1)f''g$& \\
$(1,1)$&$Bj_{3,1}(f,g)$&\\
$(a,0)$&$\{f,g'\}=-afg''+f'g'$& \\
\hline\end{tabular}
\end{equation}

$\bullet$ \emph{Order 3 operators} are the same as for $p=0$, but
the Grozman operator $Gz$ does not survive for $p=2$ and $3$. For
$p=2$, we additionally have (this might be unclear; in fact, for
$(a,b)=(1,1)$, we have $\int\otimes \id$ and $\id \otimes \int$ and
$Bj_{2,2}=\int\otimes \id-\id \otimes \int$):
\begin{equation}
\label{t32} \renewcommand{\arraystretch}{1.4}
\begin{tabular}{|l|l|l|}
\hline
$(a,b)$&$D(f,g)$&\text{comment} \\
\hline
$(1,b)$ for $b\not =0$&$\int\otimes \id(f,g)= f'''g$&\text{for $\underline{N}=2$}\\
$(1,0)$&$Bj^{*2}_{2,2}$, $\int\otimes \id$ &\text{for $\underline{N}=2$}\\
$(a,b)$ for $a,b\not=0,1$&$J_{a,b}^3(f,g)$
&\text{for $\underline{N}=2$}\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 4 operators}: no operators if $p>5$; for
$p\leq5$, we have

\begin{equation}
\label{t4} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comments\\
\hline

$(0,0)$&$Gz_{2,2}(df,dg)=f'''g'+f'g''' +f'' g''$&$2$& $\underline{N}=2$\\
$(1,0)$&$Gz_{2,2}^{*1}(df,dg)=fg^{(4)} +f''' g'$&$2$& $\underline{N}=2$\\
$(0,1)$&$Gz_{2,2}^{*2}(df,dg)=f^{(4)} g+ f' g'''$&$2$& $\underline{N}=2$\\
\hline \hline

$(1,1)$& $\int\otimes\int=f''g''$&$3$&for $\underline{N}=1$\\
\hline
$(0,0)$& $Bj_{3,1}\circ (d f,dg)$ &$3$&\\
\hline
\hline

$(1,1)$& $Bj_{5,1}(f,g)$&$5$&\\
$(1,0)$&$Bj_{5,1}^{*2}(f,g)=
fg^{(4)}-f'''g'-f'g'''+f''g''$&$5$&\\
$(0,1)$&$Bj_{5,1}^{*1}(f,g)= f^{(4)}g-f'''g'+f''g''-f'g'''$&$5$&\\
\hline

$(1, b)$, $b\not = 0$&$\int \otimes \id \colon  f^{(4)}g$&$5$& for $\underline{N}=1$\\
$(0, 1)$  &$\id \otimes \int, Bj_{5,1}^{*1}$&$5$& for $\underline{N}=1$\\
$(a, 1-a), a\not =0,1$&$L_{5,1}(f,g)=f^{(4)} g+f g^{(4)}  -f''' g'
-f'g'''+f''g''$&$5$&\\
\hline\end{tabular}
\end{equation}

$\bullet$ \emph{Order 5 operators}: no invariant operators if $p>7$;
for $p\leq7$,  we have
\begin{equation}
\label{t5} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comments\\
\hline
$(0,0)$& $Bj_{2,2}(df,dg)=f'g^{(4)}+f^{(4)}g'$ &$2$&\\
\hline

$(0,0)$& $Gz_{5,1}(df,dg)=f^{(4)} g'-g^{(4)}f'+f''g'''-f''' g''$ &$5$&\\
$(0,1)$& $Gz_{5,1}^{*2}(df,dg)=f^{(5)}g-f'g^{(4)}$&$5$&\\
$(1,0)$& $Gz_{5,1}^{*1}(df,dg)=fg^{(5)}-f^{(4)}g'$&$5$&\\
\hline

$(1,1)$&$Gz_{7,1}(f,g)$&$7$&\\
\hline\end{tabular}
\end{equation}

$\bullet$ \emph{Order 6 operators}: no invariant operators if $p>7$;
for $p\leq7$, we have
\begin{equation}
\label{t6} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(1,1)$& $\int\otimes\int$ & $2$&for $\underline{N}=2$\\
$(1,1)$& $Gz_{2,3}(f,g)$ & $2$&for $\underline{N}>2$\\
 \hline
$(0,0)$& $Bj_{5,1}(df,dg)=f^{(5)} g'-f' g^{(5)}$& $5$&\\
\hline
$(0,1)$& $Bj_{7,1}^{*1}(f,g)
& $7$&\\
$(1,1)$& $Bj_{7,1}(f,g) $& $7$&\\
$(1,0)$& $Bj_{7,1}^{*2}(f,g)
& $7$&\\
\hline
$(1,b)$ for $b\not =0$& $\int\otimes\id$& $7$&for $\underline{N}=1$\\
$(1,0)$& $\int\otimes\id$, $Bj_{7,1}^{*2}$& $7$&for $\underline{N}=1$\\
$(0,1)$& $Bj_{7,1}^{*1}$, $\id\otimes\int$& $7$&for $\underline{N}=1$\\
$(1,1)$ & $Bj_{7,1}(f,g)$ & $7$&for $\underline{N}=1$\\
$(a,1-a)$, $a\not =0,1$& $L_{7,1}(f,g)
& $7$&\\

\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 7 operators}: no invariant operators if $p>7$;
for $p\leq7$, we have
\begin{equation}
\label{t7} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline $(1,0)$ & $Bj_{2,3}^{*2}(f,g)$
&$2$&\\

$(1,1)$ &$Bj_{2,3}(f,g)$ &$2$&\\

$(1,b)$ for $b\not = 0$& $\int \otimes \id $& $2$& $N=3$\\
$(1,0)$& $\int \otimes \id$, $Bj_{2,3}^{*2}$& $2$& $N=3$\\
$(a,1-a), a\not =0,1$ &$L_{2,3}(f,g)$
& $2$& $N=3$\\

\hline $(1,1)$ & $Gz_{3,2}(f,g)$ &$3$&\\
\hline

$(0,0)$ &$Gz_{7,1}(df,dg)
& $7$ &\\
$(1,0)$ &$Gz_{7,1}^{*1}(df,dg)=fg^{(7)}-f^{(6)} g'$& $7$ &\\
$(0,1)$ &$Gz_{7,1}^{*2}(df,dg)=f^{(7)} g-f'g^{(6)}$& $7$ &\\
\hline
\end{tabular}
\end{equation}

\iffalse
From now and on, $N$ is ``big enough", i.e. $p^N\geq \deg
D+2$. Therefore the operators $L_{11,1}$  in Table \eqref{t10} and
$L_{13,1}$ in Table \eqref{t12}  are not missed because the operator
$L$ exists only when $\deg D=p^N-1$.

$\bullet$ \emph{Order 8 operators}: no invariant operators if $p>7$;
for $p\leq7$, we have
\begin{equation}
\label{t8} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0), (0,1), (1,0)$ &$Gz_{2,3}\circ (d\otimes d)$ \text{and its duals} & $2$&\\
\hline
$(1,0), (0,1), (1,1)$ & $Bj_{3,2}$ \text{and its duals}  &$3$&\\
\hline
$(0,0)$ & $Bj_{7,1}\circ (d\otimes d)$ &$7$&\\

\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 9 operators}: no invariant operators if
$p>11$; for $p\leq11$, we have
\begin{equation}
\label{t9} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0)$ & $Bj_{2,3}\circ (d\otimes d)$ \text{and its duals} &$2$&\\
\hline
$(1,0), (0,1), (0,0)$ & $Gz_{3,2}\circ (d\otimes d)$ \text{and its duals} &$3$&\\
\hline
$(1,1)$ & $Gz_{11,1}$ &$11$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 10 operators}: no invariant operators if
$p>11$; for $p\leq11$, we have
\begin{equation}
\label{t10} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0)$ & $Bj_{3,2}\circ (d\otimes d)$ \text{and its duals} &$3$&\\
\hline
$(1,1), (1,0), (0,1)$ &$Bj_{11,1}(f,g)$ &$11$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 11 operators}: no invariant operators if
$p>13$; for $p\leq13$, we have
\begin{equation}
\label{t11} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0), (0,1), (1,0)$ &$Gz_{11,1}\circ (d\otimes d)$ \text{and its duals}& $11$&\\
\hline
$(1,1)$ & $Gz_{13,1}(f,g)$ &$13$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 12 operators}: no invariant operators if
$p>13$; for $p\leq13$, we have
\begin{equation}
\label{t12} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0)$ & $Bj_{11,1}\circ (d\otimes d)$ \text{and its duals} &$11$&\\
\hline
$(1,1), (1,0), (0,1)$ & $Bj_{13,1}$ \text{and its duals}& $13$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 13 operators}: no invariant operators if
$p>13$; for $p\leq13$, we have
\begin{equation}
\label{t13} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0), (0,1), (1,0)$ &$Gz_{13,1}\circ (d\otimes d)$ \text{and its duals}& $13$&\\
\hline
\end{tabular}
\end{equation}

\BJ{I stopped here}

$\bullet$ \emph{Order 14 operators}:no invariant operators if
$p>13$; for $p\leq13$, we have
\begin{equation}
\label{.14} \begin{tabular}{|l|l|l|l|}
\hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(1,1)$ & $\int\otimes\int$ & $2$&for $\underline{N}=3$\\
\hline
$(0,0)$ & &$13$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 15 operators}: no invariant operators if
$p>17$; for $p\leq17$, we have
\begin{equation}
\label{.15} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(1,1), (0,1), (1,0)$ && $2$&\\
\hline
$(1,1)$ & &$17$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 16 operators}: no invariant operators if
$p>17$; for $p\leq17$, we have
\begin{equation}
\label{.16} \begin{tabular}{|l|l|l|l|} \hline
$(a,b)$&$D(f,g)$&$p$&comment\\
\hline
$(0,0), (0,1), (1,0)$ && $2$&\\
\hline
$(1,1), (1,0), (0,1)$ && $17$&\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 17 operators}:
\begin{equation}
\label{.17} \begin{tabular}{|l|l|} \hline
$(a,b)$&$p$\\
\hline
$(0,0)$ & $2$\\
\hline
$(0,0), (1,0), (0,1)$ & $17$\\
\hline
$(1,1)$ & $19$\\
\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 18 operators}:
\begin{equation}
\label{.18} \begin{tabular}{|l|l|} \hline
$(a,b)$&$p$\\
\hline
$(0,0)$ & $17$\\\hline
$(1,1), (0,1), (1,0)$ & $19$\\\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 19 operators}:
\begin{equation}
\label{.19} \begin{tabular}{|l|l|} \hline
$(a,b)$&$p$\\
\hline
$(0,0), (1,0), (0,1)$ & $19$\\\hline
\end{tabular}
\end{equation}

$\bullet$ \emph{Order 20 operators}:
\begin{equation}
\label{.20} \begin{tabular}{|l|l|} \hline
$(a,b)$&$p$\\
\hline
$(0,0)$ & $19$\\\hline
\end{tabular}
\end{equation}

\fi

\subsection*{Acknowledgements} We are thankful to Grozman for his wonderful
code \textit{SuperLie}, see \cite{Gr}. S.B. was partly supported by
the grant AD 065 NYUAD.

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History Of New England
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Visit State Of Maine
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Places to Visit In Vermont
Halloween In New England
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Halloween Resources-Salem Witchcraft Trials 1692
The Multitude of Salem Witch Trials Descendants
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Hunted Location In New England
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TouringNewEngland.org Visits Salem MA
My Research of Salem Witch Trials
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*Our New Video Series 8/1/20
WELCOME TO TOURING NEW ENGLAND.ORG
New England – the Birthplace of America – is filled with outdoor adventures, rich history, cultural attractions, fascinating cities, scenic villages — along with delightful food and superb shopping at every turn. Discover brilliant fall foliage, expansive lakes, white sand beaches, lighthouses, panoramic mountain views, and restaurants with delicious seafood chowder, lobster, and blueberry pie. In New England, you'll find a wealth of diverse travel experiences within just a few hour's drive from Boston's Logan International Airport.
During the 4 seasons
What is so special about New England?
New England begins in late September and ends in late December, it marks the transition from summer to winter and know for it's vibrant colors and picturesque beauty – September and October are the best months to visit, but depending on what exact area you're visiting the trees can quickly change from fiery red to the barren in a matter of days. By keep in mind these leaf- peeping tips for your New England excursion, you're sure to experience the magic of fall in the Northeast
The New England Great State Fair (Sept 17 – Oct 3, 2021)
In the Fall the New England Great State Fair comes to West Springfield Massachusetts which is called The Eastern States Exposition (aka The Big E) in September. For 17 Days of Food, Fun, and all type of Attractions. All six New England States have a Replica of its state building on the Avenue of States and Carnival rides, The Big E Horse Show, The Big E Circus, all types of farm animals, all type of different Foods from around the World, The Big E Arena Concerts Some of the biggest names from the past and present, as well as those on their way to the top, grace The Big E Arena stage, sponsored by TRULY. Sing, dance and sway to your favorite acts underneath the stars in this open-air 7,000-seat outdoor arena. such as Reba McEntire, Clint Black, George Jone, The Judds, Willie Nelson, Destiny's Child, Jessica Simpson, Chubby Checker and many others
New England Great State Fair Website Link
Yes … Springtime in New England is just so Awesome, this is the time of the year when we shake off our winters blue ans wen the sun shines brighter and the warm sun starts to help the grass to grow and flowers to poke the heads up out of the cold winter ground
Is spring a good time to visit New England?
The best time to visit New England depends on what you want to do while you're there. There are four distinct seasons, each one bringing its own touch to the region. We particularly recommend visiting during late spring. The thawing spring temperatures bring wildflowers, and baseball when Boston's Fenway Park opens.
Coming late and staying only briefly, spring in New England is often a disappointment.
The week or two of spring days in April, May and June are a delight, with cool temperatures in the evening and just the perfect degree of warmth during the day, in bright, clear sun.
But the delight of spring may be so brief that New Englanders joke "Oh, spring…yeah, last year it was on a Tuesday."
In the countryside the thaw brings "mud time," the period between frost and spring planting.
Mud time is the slowest season for tourist facilities, coming after ski season and before summer warmth and school vacations bring out the city folk.
Thus many country inns, resorts, and amusements close for a few weeks in April.
The Summer is filled with outdoor adventures, rich history, cultural attractions, fascinating cities, scenic villages — along with delightful food and superb shopping at every turn. Discover brilliant fall foliage, expansive lakes, white sand beaches, lighthouses, panoramic mountain views, and restaurants with delicious seafood chowder, lobster, and blueberry pie. There are so many places to enjoy in New England in all six states
The Winter months in New England December, January, February are very cold and is if we get Snow is great for many different winter sports such as Snowmobiling, Snow skiing (Downhill, Cross Country skiing) and Ice Hockey, Downhill sledding, Ice Skating, Snow Tubing, Snow Shoeing, but after a great snow storm here it nice to see the trees covered with the snow, it can be hard to drive in the snow if you are not used to it and it's important to have snow tires on your car during this winter months
Discover why visitors travel back to New England again and again...
TouringNewEngland.Org History
The Founder of our video production group Steven Provost started when he was born until his father passed away in 1996. He was just a young kid who did a lot of traveling with his family and did a lot camping here in the New England States of Maine, New Hampshire, Massachusetts, Connecticut and Rhode Island. In the 1993, Steven started working at Local Collage Radio station 90.7 Fm Wtcc in Springfield Massachusetts on the campus of Springfield Technical Community Collage this show was called called Ziggie Corner and ran during the early Thursday mornings from 3am to 6am weekly and later he took a course in video production at Continental Cablevision and passed the class and started a video Disc jockey show called Ytv Video Time, which turned out be a christian music video show which lasted 1 year In 2017, he joined up with a close friend Mike Austin and started Touringnewengland.net which in time changed to touringnewengland.org due to internet issues. Steven has been tried hard to make each year better then the last and reaches out to find events and other interesting things to film, we have even done some food business reviews and the videos have done well like our Malden Train Store-Charles Ro Supply Company which as 1/13/22 has 585 views and our trip to visit the MBTA North Maintenance Facility has 1,929 views as of 1/13/2022. its been so awesome that in 2022 we have started our 6th year in video Production and we are gaining more new subscribers all the time.
Our Future Plans before 2023
We hope to expand our coverage area before 2023 to allow more coverage of the states of Connecticut, Rhode Island, Vermont,
New Hampshire, Maine by
Purchasing a motor vehicle (2016 or later van in awesome shape?)
Purchasing Better Video Equipment (Camera with XLR connection)
Better video Editing software for PC
Shure sm58 microphone
We Would like to Personally Send a Thank You To each of our Subscribers for joining our Adventures, and those who visit our website, and those who like our videos, it really means so much to the both of us and you are helping our dreams come true, making these videos is so much fun and meeting people. We believe that 2022 will be another great year and I hope to get a lot of videos and exploring some new locations. If you are not a subscriber to our YouTube.com Channel, Please hit the Subscribe button and hit the bell so you don't miss when we upload a new video.
if you have any questions or ideas for videos please send us a email at stevenprovost570@gmail.com

redpajama_set_name: RedPajamaCommonCrawl


text:
Opening Friday July 29th: Michael Kenna - A Retrospective
photo-eye Gallery Opening Friday July 29th: Michael Kenna - A Retrospective photo-eye Gallery is excited to announce Michael Kenna: A Retrospective, an exhibition of black-and-white silver gelatin prints by seminal photographer Michael Kenna opening Friday, July 29th and continuing through September 10th, 2016.
Michael Kenna - Huangshan Mountains, Study 31, Anhui, China. 2009
MICHAEL KENNA: A Retrospective
Selected Works from 30 Years
Artist Reception & Book Signing
Friday Aug. 5, 5–7pm PHOTO-EYE GALLERY
541 SOUTH GUADALUPE, SANTA FE, NM 87501
FOR MORE INFORMATION CONTACT ANNE KELLY
ANNE@PHOTOEYE.COM 505-988-5152 X121
Michael Kenna - Hillside Fence, Study 4,
Teshikaga, Hokkaido, Japan. 2002
photo-eye Gallery is excited to announce Michael Kenna: A Retrospective, an exhibition of black-and-white silver gelatin prints by seminal photographer Michael Kenna opening Friday, July 29th and continuing through September 10th, 2016. An artist reception and book signing will take place on Friday, August 5th from 5 – 7 PM.
ABOUT THE ARTWORK
Michael Kenna is master of contemporary photography. Known for clean compositions, long exposures and minimalist aesthetics, Kenna's signature style remains highly influential among a generation of photographers. Kenna views his process as a conversation with the landscape, each individual communion lasting anywhere between a few minutes to a number of hours. The images are transformative, and photographed in black-and-white because, as the artist describes, "color is what we see all the time." He furthers this approach in the darkroom, spending hours on his exquisite silver prints, shaping the photograph to match his uncompromising vision. "I want photographs to be a mystery, an interpretation, a catalyst for one's imagination… we fill all our time up with noise and chaos, so I hope a viewer can enter into a photograph and feel the beauty of this place, the calmness, the tranquility." Michael Kenna: A Retrospective showcases more than 50 works produced by the artist between 1987–2015 and features 15 countries including Japan, China, France, Egypt, South Korea and the United States.
Michael Kenna - Single Tree, Mita, Hokkaido,
Japan. 2007
Born in Widnes, Lancashire, England in 1953, Michael Kenna studied at the Banbury School of Art and the London College of Printing before relocating to the US in the 1980s to work with acclaimed photographer Ruth Bernhard. Since that time Kenna's photographs have been exhibited worldwide and have appeared in more than 100 major collections including: Musée National d'Art Moderne, Paris, France; Art Institute of Chicago, Chicago, Illinois, USA; National Gallery of Art, Washington, D.C., USA; Victoria and Albert Museum, London, UK. During his expansive career, Kenna has authored over 40 photographic books covering journeys to more than 30 countries across the globe, including 2015's Forms of Japan. The artist currently lives and works in Seattle, Washington.
View books by Michael Kenna. including limited editions
Labels: Events, Gallery, photo-eye Gallery


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Margaret from TN Asked Note: This answer was provided prior to the change to planDisney and may still contain references to Disney Parks Moms Panel.
When will we be able to reserve rooms for the Star Wars hotel?"
Hello Margaret,
The force is strong in bringing you to the Disney Parks Moms Panel with your question. Last summer, plans were unveiled for a Star Wars themed Resort hotel that will be built at the Walt Disney World Resort very soon. I checked with some of my Rebel Spies and they could only give me an update on the construction of Star Wars: Galaxy's Edge. The transmission could only confirm that park expansion will open at Disney's Hollywood Studios in 2019. I tried to reach someone who was a big deal in the Resistance for more information. Currently, this one-of-a-kind totally immersive vacation experience is still in the planning phase. My powers with the force are not developed enough for me to see into the future to know how many Republic credits will be needed for a night's stay. Then I remembered the teachings of Jedi Master Yoda and Master Obi-Wan, "we must learn patience." For now, I would suggest keeping up to date with transmissions received from the Disney Parks Blog for all things Star Wars.
May the force be with you,
Meet the Panelist: Andrea, Ohio
Hi there, I am Andrea from the Buckeye State. A very proud mom to my princess who is a former Miss Ohio. Friends and family describe me as a "Disney Nerd" and Star Wars fangirl. I have made 30 trips and counting to the Walt Disney World Resort in all stages of life since 1972 as a small child, a young mom with a toddler and now as an empty nester. The love of vacation planning runs strong in my family. My uncle had it, I have it and with my help, you will have it too. Learn More About Andrea
Walt Disney World Disney's Hollywood Studios
Walt Disney World Resort Hotel Reservations
Star Wars Activities throughout Walt Disney World


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Kenny Pickett announces he will opt out of Peach Bowl, prepare for 2022 NFL Draft
Pittsburgh quarterback Kenny Pickett has played his final game of college football.
Pickett announced on Friday that he will forego the opportunity to play for the Panthers in the Peach Bowl in order to begin preparing for the 2022 NFL Draft.
Forever a Pitt Man #H2P pic.twitter.com/fq0Sw06dIa
— Kenny Pickett (@kennypickett10) December 17, 2021
Simply put, there was very little left for Pickett to accomplish at Pitt. He already holds every feasible single-season and career passing record at Pitt, surpassing marks set by players like Dan Marino. Pickett also now holds the ACC's single-season touchdown record with 42 after passing Deshaun Watson in the ACC Championship Game.
Pickett is second in ACC history with 12,303 passing yards, trailing only Phillip Rivers (13,484). Pickett and Rivers are the only quarterbacks in ACC history to reach 12,000 yards in a career.
He also led Pitt to its first ACC Championship in program history. The Panthers won 11 games for the first time since 1981.
After entering the season off the NFL Draft radar, Pickett now has a chance to be the first quarterback off the board in next year's draft. With highly-touted passers like Spencer Rattler and Sam Howell struggling for much of 2021, there's room for Pickett to make a statement during the pre-draft process.
Pitt faces Michigan State in the Peach Bowl on Dec. 30.
Damar Hamlin shares lengthy video message thanking numerous people for love and support
Pitt getting NCAA Tournament buzz after knocking off No. 20 Miami
Pittsburgh trolls Wake Forest after record-breaking performance from 3 in win
Pittsburgh reportedly lands commitment from transfer WR with 3 years of eligibility remaining


STRATIS is the only IoT Platform built for the complexities of MultiFamily and Student Housing.
STRATIS is installed in 300,000 units in the U.S. and Japan.
STRATIS provides seamless "sidewalk to sofa" credentialing, for staff, maintenance, vendors, residents, and their guest and service providers.
STRATIS offers automation beyond "Smart Apartments." STRATIS creates Intelligent Buildings.
STRATIS is the only IoT Platform built specifically for MultiFamily and Student Housing, with focus on return on investment for Property Owners & Managers and Residents as natural beneficiaries.
As "Technology as Amenity" gains traction, it will exist, for a short time as a competitive differentiator for early adopters. As the return on investment is continuously realized, "Technology as Amenity" becomes a competitive advantage. And finally, as industries adopt market-wide, this technology becomes the new industry standard.
Our Mission: To be YOUR standard.
STRATIS provides seamless, "sidewalk to sofa" mobile access, for staff, maintenance, vendors, residents, and their guests and service providers.
STRATIS was created by a team with more than 72 years of combined IoT experience.
The STRATIS mobile app has been tested and retested, and is approaching two years in development.
We created STRATIS with Property Managers in mind. The mobile app saves both time and money for properties.
STRATIS prioritizes technology security & safety for Residents & Property Managers.
We focus on ease-of-use for Property Owners & Residents.
STRATIS is award-winning technology, disrupting the MultiFamily & Student Housing industries.


Global Volcanism Program, 2009. Report on Sangay (Ecuador). In: Sennert, S K (ed.), Weekly Volcanic Activity Report, 10 June-16 June 2009. Smithsonian Institution and US Geological Survey.
Based on analysis of satellite imagery, the Washington VAAC reported that on 15 June possible small ash plumes from Sangay drifted WNW. A thermal anomaly was detected.


For over 20 years, Rebecca has designed customized diet plans and experimented with healthy ways to to make meals simple, quick, and tasty. Many of her clients have various food sensitivities and preferences, which has pushed Rebecca to discover perfect options for any palette.
Eating clean and healthy doesn't have to be a chore and shouldn't taste boring. Rebecca is a fan of meals that can be made in under 10 minutes that still taste amazing! Rebecca will teach you how to prep and cook meals that you will love!
Rebecca designs easy to follow and easy to make meal plans which takes the guesswork out of eating healthy! Reach your weight loss goals with customized nutrition plans and let Rebecca keep you on track as she also identifies any sabotaging habits that may be blocking you from reaching your goals.
Check out what Rebecca's client Michelle G. said about working with Rebecca to lose 10 pounds.
"As diet coach, Rebecca helped me change my diet habits in no time. We had one in-person consultation that involved her asking me a series of questions about my goals, my meals and my workout routine.
She attentively listened to what I wanted to accomplish but lacked the knowledge to do on my own. Previously, I was on a low-carb, high-protein diet, which left me feeling hungry, and did not provide me with enough energy to do high-intensity interval training. I felt stuck.
Rebecca was a great source of inspiration and motivation. Because of the individualized attention she gave me, she was able to customize and tailor a meal plan specifically designed for me. And I loved her recommendations! For example, instead of buying pizza, I now make my own Ezekiel pizza through a recipe she gave me.
Another favorite of mine is the protein shake she taught me how to make. By sticking to her recommendations, I feel so much better about my body because I have enough energy to work out and I see myself slimming down.
Let Rebecca share her meal planning tips for you and your entire family!
Stop the endless cycle of feeling lethargic and gaining weight when you ' r e traveling. Rebecca is passionate about speaking to corporations about how to be a Nutrition Ninja While On The Road!


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River North Gallery District, Near North Side, Chicago
The River North Gallery District, in Chicago, is in the Near North Side, Chicago. It hosts the largest concentration of art galleries in the United States, outside of Manhattan. River North has experienced vast changes in the years 1990 - 2012 including the development of vast highrise buildings, nightclubs and restaurants. River North has become one of Chicago's top neighborhoods for nightlife especially along Franklin Avenue and Hubbard Street.
A common definition puts the District in the area north of the Chicago River and the Merchandise Mart, south of Oak Street, east of the Chicago River and west of Michigan Avenue. As it has grown and the area has gentrified, galleries can also now be found west of Orleans and east of LaSalle, though the core area still contains the highest concentration of galleries. Along with hundreds of art galleries, the area holds many bars, dance clubs, popular restaurants and entertainment venues. Subsections of River North include:
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/River_North_Gallery_District,_Near_North_Side,_Chicago
Latest News for: river north
Lake Mead's decline may slow, thanks to wet winter
The North Platte Telegraph 30 Jan 2023
... Courthouse squirrel loves peanut butter crackers Sutherland man arraigned on three burglary charges Lip Sync Battle at the Fox rocks for charity Judge tells parties seeking funds from Marshbanks estate to 'work it out' Flood advisory continues on ice-jammed North Platte River.
This week's Elite 11 rankings for swimming, wrestling and basketball
Toms River North (2);13-2 ... North Wildwood man appears on Saturday Night LIve Old Brigantine gas station an eyesore to residents, tourists Local radio broadcaster, journalist David Spatz dies Earth's inner core may have stopped turning and could go into ... Toms River North (11);14-4.
How will Georgia QB Stetson Bennett's NFL draft stock be affected by Dallas arrest?
Mikaela Shiffrin must wait for record win 86 but takes slalom title
Eagles rout 49ers for NFC title, soar into Super Bowl
David Murphy: Three thoughts: Nick Sirianni's fourth-down guts set the tone for the Super Bowl-bound Eagles
From hockey brawls to debt limit: Emmer wrangles House GOP votes
Gregory Allen Howard, who wrote 'Remember the Titans,' dies
'Again!?': Aucklanders warned of more heavy rain possible tonight ahead of second 'atmospheric river'
NewstalkZB 29 Jan 2023
... for a second "atmospheric river" sweeping into the upper north later in the week ... Niwa Weather forecaster Ben Noll said details of that later-week event were still coming through, "but it looks like we could be in for another intense atmospheric river event for the North Island".
Proposal looks for pavement preservation on roads in Great Falls
The Montana Department of Transportation (MDT) is inviting the public to submit comments on the proposal for work on River Drive North, Park Drive North, 2nd Street North/South, 6th Street North and 3rd Avenue South ... Proposed work on a portion of River Drive North and ...


LAST week I attended the relaunch of Common on Northern Quarter's Edge Street with three blokes I didn't know wearing socks with sandals and two doctors I did; the smoking gambler and the boozy cyclist (weirdly I've often found medical types to be amongst the heaviest drinkers and smokers... who can blame 'em eh?).
There were plenty of plates floating around the place, but trying to keep anything down with those two around is bloody impossible. Working in hospitals, conversations usually turn to some poor, curious sod having something awkward removed from their Khyber Pass.
So we necked a few pints of a marvellous Indy Man Brew House pale, stayed until the free gin ran out and left.
The menu stuck with me though; it looked highly considered, pared down but far-reaching (with splashes of India, the Middle East and 'pig bits' from your local boozer). So myself and Meatless returned one Tuesday lunchtime to get stuck in.
Common - now over a decade old - is one of those places you can't imagine the Northern Quarter without; like Affleck's, Koffee Pot, Fred Aldous or one of those adult book stores that somehow, in the age of Pornhub and Chaturbate in your pocket, still manage to hang in there. Good on 'em. Two fingers to gentrification.
The new place feels more grown-up. And that's not a bad thing. Gone is the Berlin hostel snot-green paint job, the chip-board, the goofy graffiti art and the beer mats plastered across the walls. They've been replaced by pricey retractable window walls, a restaurant extension and all four volumes of How To: Relaunch A Bar... Scandi-Style. You can pick up a copy from Magma on Oldham Street, probably.
This has, inevitably, miffed a good few regular punters, who now presumedly sit in a corner of Common's sister venue, Port Street Beer House, and weep into their hellishly strong Brewmaster Armageddon beer. No matter. The new fit-out looks great, even if the furniture has been thumbed from an A1 plywood press out board. You can pick up your plywood press out furniture from Fred Aldous, probably.
We were quickly sat down, then moved to a sunny table, then away from the sunny table, and finally settled on a table near the one from The Inbetweeners with the naff quiff and the wardrobe malfunction.
We ordered a small bowl of hot, crunchy oven-roasted chickpeas spiced with a masala salt (a little steep at £3). Meatless loved 'em, but she would; she's a veggie and they're chickpeas. I found them similar to popcorn - a good idea until the fourth handful.
A small plate of salt and pepper squid with soy and lime aoili (£4.5) was a delight. Girthier than usual; golden and light. The best I've had since Red Chilli. Perhaps better. A wonderful little dish.
A chopped kale and avocado dish (steep at £4.50) was less successful; fresh and served at a suitably cool temperature, but a bit of a nothing dish. Nutrient-packed kale might be, but it's also terribly dull.
And then something wonderful happened. A revelation. Enough to make me reconsider our Top 10 Dishes In Manchester. Out came a quad of brillianty red, gooey, sticky Korean fried chicken (£4.50).
Now currently we have to tread carefully with Korean food. Jay Rayner has not only put the cat amongst the pigeons with his latest Guardian review but slipped amphetamine in the Whiskers. Still, you don't have to 'understand Korean food' to know these are special.
Chunky, slightly crispy, boneless chicken bites glazed in a syrupy sweet Gangjeong sauce (a Korean confectionary of rice, honey, malt and cereal) and sesame seeds. Every now and then you discover a dish that makes your day, this made my week.
Next came a perfectly cooked hake with lentil dahl, charred cauliflower and curry sauce (£9). Another winner and another surprisingly elegant dish from this indie stronghold.
I'll admit I hadn't expected a great deal from the new Common menu.
Northern Quarter bar restaurant screams burgers (of course there are burgers), chilli fries (yes those too) and skateboards on the walls (none of those, but I think there used to be). Steer away from that, sit, order a clean, crisp Flensburger lager (£4) or a bottle of Veranza Tinto 2013 (£18.50), and five plates of those gorgeous, little, red dollops.
Common, 39 Edge Street, Northern Quarter, M4 1HW. 0161 832 9245.


The main causes of noise pollution -- machines, transportation, poor urban planning, people -- vary considerably in terms of location and intensity throughout the year. Consequently, measuring these noise levels on a large scale is often a time consuming and sometimes inaccurate procedure.
However, Rajib Rana, Chun Tung Chou, Nirupama Bulusu, Salil Kanhere and Wen Hu at the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia's national science agency, have devised a novel way of combating many of the problems that have hindered previous methods of monitoring noise in the past. Rather than spending thousands on equipment, Rana and his team have crowdsourced the data from smartphones -- they call it the "Ear-Phone".
The average smartphone has enough sophisticated technology (on-board microphones, GPS, time stamping) to make it an extraordinary mobile monitoring device. However, the implementation of the Ear-Phone came with its own set of challenges -- there are many ambient noises picked up by a smartphone that would be of no use to the researchers, namely intimate conversations that sound loud to the phone, but wouldn't be disruptive to passersby, as well as the sound of rustling clothes, or keys or money jingling in pockets.
The team at CSIRO isolated these problems and fixed them accordingly, teaching the phone how to recognise a conversation, wait until it was over, and then start recording again. They also made sure the GPS was only collecting data when outdoors and whilst being held in the user's hand.
This was made possible by taking advantage of the automatic proximity sensors and accelerometers that are standard in most smartphones. These sensors generate specific feedback when the phone is handled, which the software developed by Rana and his team can then use to asses whether or not it's an appropriate time to take a reading. Using this method, handheld usage can be detected with an accuracy of 84 percent.
Once all of the criteria have been met, the phone will take a sound recording, complete with exact location and time. The data is then sent to a central server the moment the phone is connected to Wi-Fi.
The only major downside with the crowdsourced method is battery life -- always-on GPS and Wi-Fi can drain a phone's power in a matter of hours, but the team claims it is working on a solution.
Noise pollution is a particularly destructive environmental hazard. Not only can it cause hearing-loss, stress and tinnitus in humans, but it's devastating to local Wildlife, reducing livable habitats, disrupting predator or prey detection and avoidance, and even causing infidelity in finches.
The Ear-Phone has been tested on various Android and Nokia phones in Australia.


In understanding of nature we may co-create with nature, our technology, science, art and lifestyle, in tune with our natural habitat. When our natural intelligence is Universally Connected there is the wisdom of holographic holism - as above, so below, and within. This is a natural law of healing and evolution. Blog http://johannkotze.com/blog/a-law-of-healing-holographic-nature-holism/ Recorded at Johann Kotze Music & Yoga Studio, Muizenberg, Cape Town.


Now you can have the elegance and beauty of a nixie tube clock in your own pocket. Regular nixie tube clocks sell for hundreds, and are not easily re-configurable. This one is inexpensive and you can adjust the configuration to suit your preference; just touch the screen to flip it to the configuration side. Gorgeous nixie tube imagery. Large or Small nixie tubes. Portrait and Landscape displays (simply rotate the device). 12 or 24 hour time modes. Simulated tube flicker.


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imgutil
=======
Image operation utility

Installation
------------

Installation using pip

.. code-block:: bash

    $ pip install imgutil

Prerequesite
------------

* jpegtran
* pngcrush

Command line utilities
----------------------

imgoptimize
~~~~~~~~~~~

.. code-block:: bash

    $ imgoptimize [-h] [-o OUTPUT] src_img

* Optimize image to reduce file size.
* Should install ``jpegtran`` and ``pngcrush``.

Usage
`````
* ``-o``: (Optional) Specifiy the output image.
* ``src_img``: Specify the input image.

imgresize
~~~~~~~~~

.. code-block:: bash

    $ imgresize [-h] [-o OUTPUT] [-W WIDTH] [-H HEIGHT] src_img

* Resize image by specifying width or height.

Usage
`````

* ``-o``: (Optional) Specifiy the output image.
* ``-W`` or ``--width``: (Optional) Specify the desired image width. If only width is specified, height will be determined by the aspect ratio of image.
* ``-H`` or ``--height``: (Optional) Specify the desired image height. If only height is specified, width will be determined by the aspect ratio of image.
* ``src_img``: Specify the input image.

Usage
-----

imgoptimize
~~~~~~~~~~~

.. code-block:: python

    imgoptimize(input_filename, output_filename=None)

* Should install ``jpegtran`` and ``pngcrush``.
* It reduces jpeg/png file size.
* It converts CMYK to RGB.

**quick example**

.. code-block:: python

    import os
    from imgutil import imgoptimize

    test_img = '/tmp/test.jpg'
    os.stat(test_img).st_size              # 81073
    imgoptimize(test_img, '/tmp/opt.jpg')  # output to /tmp/opt.jpg
    os.stat('/tmp/opt.jpg').st_size        # 81026
    imgoptimize(test_img)                  # optimize the original file
    os.stat(test_img).st_size              # 81026

imgresize
~~~~~~~~~

.. code-block:: python

    imgresize(input_filename, width=None, height=None, output_filename=None)

* It can resize image and preserve aspect ratio.
* It can resize image to specified dimension.

**quick example**

.. code-block:: python

    from imgutil import imgresize

    test_img = '/tmp/test.jpg'    # It is a 1024x768 image
    imgresize(test_img, width=600, output_filename='/tmp/resize.jpg') # /tmp/resize.jpg is 600x450
    imgresize(test_img, height=600, output_filename='/tmp/resize.jpg') # /tmp/resize.jpg is 800x600
    imgresize(test_img, width=100, height=100, output_filename='/tmp/resize.jpg') # /tmp/resize.jpg is 100x100
    imgresize(test_img, width=200, height=400) # /tmp/test.jpg is 200x400 now

redpajama_set_name: RedPajamaGithub


I am pleased to report that I kept up on my January book group reading and am excited for my neighborhood book group's discussion of Wonder this coming week. I actually picked this novel and rereading it reminded me how much I love the characters and it's sweet and simple message.
This book is not my usual choice of genre. I don't read a lot of middle grade fiction nor books about disabilities. But I just love it.
The different points of view in this novel are well done. We get inside many characters' heads–I almost wished for a bit more.
This is a quick read. Only 300 pages and short chapters.
Wonder by R. J. Palacio follows the coming of age story of August Pullman, a 5th grader with a facial deformity who is attending normal school for the first time. Although his face is horribly disfigured, August is blessed with a loving family, several new friends, and a witty and engaging personality. Regardless, he must navigate his new school and classes in addition to surviving lunch period, the stares and whispers of his peers, and a particularly important class trip to a nature camp. Can August really fit in at a new school? And can his peers move past his surprising face?
I love the different perspectives in this novel. We get inside August's head, his sister Via, his best friend Jack, and several others. Palacio cleverly gives us the same scenes several times from different perspectives. We feel sorry for August because of his face. We connect with Via who must give up a lot as August's sister. We appreciate Jack's courage to finally stand up for August. And we get more background on each character and their feelings about August. I think this adds depth and greater emotion to the story because we get a full picture of how people react to and interact with August.
As the novel progresses, we get more details about August's face. He has a genetic mutation that caused his face to be complete deformed at birth–at one point his features are described as looking like they have been burned in a fire. Regardless of the details, his face certainly gives most people–students, teachers, and parents alike a shock when they first see it. But I love when Jack says he got used to how August looked and doesn't even notice it after a while. Or when August's dad admits to throwing away the astronaut helmet August wore everyday for months (to hide his face) because he loves August's face and doesn't want him to hide it. It's amazing that although his facial deformity is the center of the novel, it also loses importance as the novel progresses.
My absolute favorite part of this book is the ending. I love when the 5th grade goes to the nature camp and several boys (some surprisingly) stand up for August against some bullies from another school. While the experience is scary for August, it also is a big turning point for how his peers treat him. These boys protect him and befriend him. They are able to look past his face and get to know him. That is just awesome. Then I love the graduation scene, Mr Tushman's speech, and August's standing ovation. It seriously makes me cry.
The messages this novel forwards are truly inspiring. I think everyone should read this book–from my growing children, to my friends, to my sister, to everyone I know. We all need to experience the great lessons taught in this book. This book teaches us the importance and the value of kindness, courage, friendship, and love.
Although August Pullman has a physical disability, he is blessed in so many other ways. His family is loving and kind. He has parents that trust and support him. He makes good friends. And he learns to stand up for himself. Life is not just made up of trials or difficulties; there is joy and happiness to be found all around us. Truly, a great novel.
Have you read Wonder? What did you think?
Book Traveling Thursdays: Book to Movie!
I hope you can read it soon! It really is fantastic!
I have read this book SO many times, and I cry every time. It is such a great book, with such wonderful (no pun intended) lessons about friendship and acceptance. Great review!
Thank you so much! I'm glad you loved it as much as me. And thanks for your observation about August and the staring. That is so true. And makes me think twice before I stare at others or make judgments. There are so many great quotes in this book! Hard to pick only a few 🙂 Thanks again for stopping by and reading!
I loved this book. Nice review. Did you know that the author has added some short stories of their characters connected to this story? I think there's 3 more. I've read one and hope to read the others soon.
Thank you! Isn't it just great? 🙂 I haven't read any of those but I really want to read the Julian chapter. That sounds super interesting.
Julian is the one I read and I did end up liking it. It was a good addition.
Cool. I'm really intrigued by what's happening inside his head. And also a bit nervous too!


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Be Quick Or Be Dead - 2015 Remaster
From Here to Eternity - 2015 Remaster
Afraid to Shoot Strangers - 2015 Remaster
Turbo Lover - Remastered
Seek & Destroy - Remastered
Hangar 18 - Remastered 2004
Chemical Wedding - 2001 Remastered Version
Sabbath Bloody Sabbath - 2009 Remastered Version
Black Sabbath - 2009 Remastered Version
In My Darkest Hour - Remastered 2004
Man of Sorrows - 2001 Remastered Version
Fade To Black - Remastered
There are multiple artists with this name: 1) Iron Maiden are an English heavy metal band formed in Leyton, east London, in 1975 by bassist and primary songwriter Steve Harris. The band's discography has grown to thirty-seven albums, including fifteen studio albums, eleven live albums, four EPs, and seven compilations. Pioneers of the New Wave of British Heavy Metal, Iron Maiden achieved initial success during the early 1980s. After several line-up changes, the band went on to release a series of US and UK platinum and gold albums, including 1982's The Number of the Beast,… read more
There are multiple artists with this name: 1) Iron Maiden are an English heavy metal band formed in Leyton, east London, in 1975 by bassist and primary songwriter Steve Harris. The band… read more
There are multiple artists with this name: 1) Iron Maiden are an English heavy metal band formed in Leyton, east London, in 1975 by bassist and primary songwriter Steve Harris. The band's discography has grown to thirty-seven albu… read more


Now that you understand what an ERP is and how it can transform your business, it's time to evaluate vendors and select the right software for your business needs. It may seem like a daunting task but it's vital you select the right vendor and implementation partner or it could be a costly mistake for years to come.
We keep it simple with our free downloadable tool kit to help you evaluate and select the right vendor. Inside the tool set you'll find sample business requirements, business process maps, vendor scorecards, request for information (RFI) templates and top 10 questions to ask your ERP vendor.
It's important to have an evaluation team, because each department requires different things. You'll need to collaborate to make sure your ERP vendor is a fit.
The Executive Team / Steering Team is responsible to make sure there is a buy-in and needs to assign a project lead.
Your company's subject matter experts (SMEs) need to fully define your current state and understand what the future business requirements are. Business process maps are a vital part of this activity.
Business process maps will allow you and an ERP vendor to better understand the business workflows. If you have your own already that's great! If you don't, feel free to download our business process maps as an example template that you can build from.
Each department has different requirements, you'll want to document each of them across all departments. A good vendor will help you identify key business requirements and ensure your new ERP system will support industry standards and leading practices. If you're not sure or need help identifying requirements and leading practices across each department give us a call and we'll set a time to assist you.
Use them as a template or guide to get started mapping out your current business process and to understand your primary ERP requirements.
This will help you with evaluating a vendor's ability to meet your unique business requirements. You'll want to take your defined business requirements you identified in the previous step, assign a weight of importance to each requirement (ideally 1-5), and track a vendor's response (yes, no, custom, or 3rd party add-on).
If you aren't sure how to create a scorecard, below is a template you can utilize.
This will empower vendors to submit a proper proposal for evaluation.
Identify 5 to 10 possible vendors, call them, share goals, timeframes and expectations along with your Request for Information (RFI).
Refer to the "Top 10 Questions To Ask Your ERP Vendor" (PDF Download) when evaluating vendors.
Each vendor should be able to give you a tailored demonstration using your own sample data, so you can see and understand how a new ERP system will function with your data.
Your Customers, Vendors, Projects, Items, BOMs, etc..
Sample Quotes, Sales Orders, Contracts, Purchase Orders, etc..
You'll want to provide a demo script or outline to save you and the vendor time so that the focus is on what's important to see, in order to properly evaluate. Make sure all key decision makers are users are present for the demo.
You'll want to use the internal/external score card you previously created and assign a score for each requirement during the live demo. Take on a don't believe it until you see it attitude and be true to your evaluation score.
Match the vendor and software product with the requirement defined in the scorecard.
Call references to make sure you'll enjoy partnering together and have a successful experience.
ERP Software, Implementation and 3rd Party Add-Ons should be considered with total cost.
Understand their ability to meet requested start and go live dates.
Is custom development needed? If so, consider the costs and timeframes accordingly.
Review all contracts and agreements in detail.
Is the vendor a strong implementation partner? This is often overlooked and can lead to much higher costs than initially budgeted for. You'll want to be sure your implementation partner has the required experience rolling out ERP systems, so things go as smoothly as planned.
Executive review – everyone has to be on the same page to ensure you're making the right decision with the right partner.
If you follow these 8 vital steps you're organization will be on the surefire path to breakthrough the barriers of growth.
Includes business process maps, vendor scorecard, sample RFI and top 10 questions to ask your ERP vendor.


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Showing articles about gears 5
12/29/2020 at 11:10 AM by Nick Puleo 0
It's Gearsmas!
Gears 5 Co-Op Stream Recap
It's been over a year since Gears 5 launched, and the Co-Opticrew is finally getting back to it. Nick just wrapped up the campaign and will be heading into the new Hivebusters DLC that just launched, and the whole team decided to dive into the Escape mode, the new 3 player co-op scenario in the game.
1/1/2020 at 4:00 PM by Matthew Squaire 0
Updates, Patches And Presents!
ICYMI: Gears 5 Received New Content and Last Day for "Gearsmas" Goodies
Gears 5 kicked off its second "Operation" (i.e., season) last month, called "Operation: Free For All." The Title Update 3 brought a slew of bug fixes, maps, and the new Free For All PvP mode to the game, but that's not all we got for December. Much like in real life, Christmas came upon us and the COGs faster than we had anticipated and with it came some holiday-themed content for Gears 5 in the form of the annual Gearsmas event.
9/11/2019 at 2:30 PM by Enoch Starnes 0
Get those chainsaws a-revvin'.
Gears 5 Gets A Full Release And A New Trailer
Time to bust out your favorite bandana and crush the nearest creatine shake, cause Gears 5 has slammed its way into stores. While the latest entry into the Gears franchise has technically been out since its "Early Access" release on September 5th, it wasn't officially available to the general public until the 10th.
9/9/2019 at 4:21 PM by Nick Puleo 0
Woooo, the co-op train is back.
Gears 5 Co-Op FAQ
Gears 5 is here and with it comes a brand new co-op mode called Escape. Don't fret - returning is Horde and of course campaign co-op. Below we dive into all the technical details of what makes co-op, co-op in Gears 5. Let us know if you have any additional questions and we'll update the FAQ.
8/22/2019 at 3:01 PM by Jason Love 0
The Coalition further evolves the mode they created over 10 years ago
Gears 5 Horde Focuses on the Heroes You Play
A few days ago, The Coalition lifted up the curtain on the new Horde mode for the forthcoming Gears 5. While many of the same systems from Gears of War 4 are returning (i.e., using a Fabricator to build fortifications/guns, gathering energy to power the Fabricator, equipping cards on your character), there are a few tweaks, too.
7/2/2019 at 4:00 PM by Jason Love 0
Pretty sure all these videos should just have Neil Cicierega's "Bustin" playing in the background
Meet The Hive Bustin' Crew From Gears 5's Escape Mode
Last month, The Coalition showed off the new three player co-op mode that, Escape, will be a part of Gears 5 when it arrives in September. Rather than staying in one spot and fighting off wave after wave of enemies, Escape puts you in the boots of one of three members of a "Hivebuster" crew and tasks you with being able to get out from the middle of a Hive before the angry Swarm, or the deadly bio weapon you deployed, takes you out. In three new videos, you can now get a chance to "meet" the members of the crew and go ahead and call "dibs" on your fave.
A lady, an Irishman, and the original Hulu guy walk into a Hive...
[E3 2019] - Gears 5 Arrives This September With a New Co-Op Mode
So much news, so little time! Rod Fergusson, studio head for The Coalition, took the stage during Microsoft's E3 press conference this year to talk about the newest entry in the Gears franchise, Gears 5. The last time we heard anything much about this game was during last year's E3, and it was a little surprising that not only did they have a new mode to show off and talk about, but a release date: September 10, 2019.
Some cool games are on the way, and hopefully we'll learn more this week
[E3 2018] - Microsoft Press Conference Recap
Microsoft's E3 2018 press conference wrapped up just a few hours ago, and there were some exciting things announced, like a new Halo title, Dying Light 2, and Gears 5. While Microsoft Studios is making the right moves towards "hey, come check out our excellent console," it was a bit disappointing that they didn't have more to say about what they were showing.
6/10/2018 at 7:22 PM by Nick Puleo 0
Them gears keep on churning...
[E3 2018] - Gears 5 Screenshots and Video
As predicted, Microsoft have announced two new Gears of War games coming out in the next few years. While the other one, Gears of War Tactics is a PC exclusive strategy game similar to XCOM, Gears 5 is the PC and Xbox One title that continues the series 3rd person action.
9 stories found


In an increasingly global and technical economy, employers are looking for graduates who have critical thinking skills, an understanding of the importance of technology, and an international worldview. Liberal studies provide you with a global perspective and a robust critical thinking skillset for students. These skills will be beneficial for a myriad of professions.
The cost of classes at a college or university is called tuition. A college or university typically defines tuition by credit hour, and tuition is, generally, significantly cheaper at a community college than a university. For example, the cost of a credit hour at GTCC is $76 for in-state students and $268 for out-of-state students; the cost of a credit hour at a regional university is $552 for in-state students and $1814 for out-of-state students. For a full-time student taking 15 hours at GTCC, the cost for tuition for an in-state student will be $1140* for a semester; the cost for tuition for this same student at a regional university will be $8280 for a semester. For most students, attending a community college is a cheaper alternative to higher education than a state university.
Which GTCC campuses offer Liberal Studies/Humanities courses?
Humanities courses such as Technology & Society and Critical Thinking are available on every GTCC campus and online.


I have more or less stage or prepared for my trip next time to the Everest Base Camp and have arranged to have a lot of products made in Nepal. I will complete my whole backpack system when I return.
That has got to be bad English, "Steal Proof." I wanted to prepare for Russia and Europe Trains where the backpacks have a good chance of getting nabbed. Jeff said something about a Dutch girl in Pokhala. I will wait to hear the results.


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Pack a Trowel, It's Masonry Camp for Architects
By Violet Law
Summer camp is usually for kids, but imagine trading your drafting pencil and computer mouse for a hammer and chisel—or laying the bricks that form a building you design. Some 40 early-career architects did just that at a weeklong masonry camp, which ends today, hosted by the International Masonry Institute (IMI) in Bowie, Maryland.
Photo: by Hazel Bradford, Courtesy the International Masonry Institute
Some 40 early-career architects spent the week learning six different masonry techniques from journeymen at a camp sponsored by the International Masonry Institute in Bowie, Maryland. For their final project, which is being critiqued today, the architects teamed with craftsmen to build sections of a school facility they designed.
The goal of this summer camp, says Maria Viteri, AIA, IMI's director of program development, "is to foster a level of collaboration between the people who are designing and the people who are building." Young architects from 39 firms nationwide teamed up with masonry journeymen who taught them skills with six different materials or project types: brick/block, stone, terrazzo, plaster, tiles and restoration. Armed with this knowledge, the architects then designed a neighborhood school and worked with the craftsmen to build small sections of their plans. Guest architects for a critique included Jeanne Gang, Monica Ponce de Leon, and Stanley Tigerman.
Some participants compare the camp to a condensed version of studio—only more practical. "Building it with your own hand is an experience I never had in school or on the job," says Valerie Towe, an architect with Einhorn Yaffee Prescott in Boston. "Architects are very focused on the design process and may lose sight of the reality. But to design along with craftsmen, it's a way of stepping back."
Seth Wentz, a first-year architect with LSC Design in York, Pennsylvania, says that he has learned how to design a project more efficiently from the builder's perspective. "When you work side-by-side with the actual craftsman, you learn what you do in [the office] may not always be the most logical thing to do. It may actually make the job harder for the craftsman."
The IMI has offered its summer camp, which takes place in two weeklong-sessions during August, for 12 years as a way to revive the tradition of architects as master builders. Although previous participants labored in an outdoor tent, this year's campers enjoyed the indoor convenience of a new 60,000-square training center. Designed by Tigerman, the building officially opens next month and will be home to the International Union of Bricklayers and Allied Craftworkers, which supports the IMI.
Top 300 Firms of 2018: Gensler Leads the Pack for Seventh Consecutive Year
Brad Pitt Selects More Architects for "Make It Right" Project
Stone Creek Camp by Andersson-Wise Architects
Modern Sustainable Residential Design: A Guide for Design Professionals
Building Great Schools for a Great City
CARBON 101: What It Is and How To Talk About It
NOW ON DEMANDCredits: 1 AIA LU/HSW; 1 AIBD P-CE; 0.1 IACET CEUThis course defines and contextualizes carbon, explains the impact of human activity on the natural carbon cycle, and provides insight into how companies can measure and reduce their climate impact.


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The Gustav Adolf Church (, ) is a church building in Hamburg-Neustadt, Germany. It belongs to the Church of Sweden, and was built between 1906-1907.

References

External links

Church of Sweden in Hamburg

Church of Sweden churches
Lutheran churches in Hamburg
Buildings and structures in Hamburg-Mitte
Hamburg GustafAdolf
1907 establishments in Germany
20th-century churches in Germany

redpajama_set_name: RedPajamaWikipedia


Our practice is focused on helping clients pursue personal injury claims for accident injuries, including but not limited to injuries suffered as a result of car accidents, accidents caused by drunk drivers, accidents caused by texting and driving, other motor vehicle accidents, 18 wheeler accidents, and more areas.
We have been in practice since 1998. "Service with Integrity" is our Motto. Practice areas: Personal injuries (auto accident cases), immigration, divorce, DWI, criminal, and civil law.
Schorr Law firm, P.C. is located in Garland, Texas and represents clients from throughout the DFW Metroplex in family law, personal injury, medical malpractice and bankruptcy.
Though our main office is located in Garland, Texas, our clients come from all over the Dallas/Fort Worth and Houston areas. Case types handled: Car accidents, motorcycle accidents, truck accidents, pedestrian accidents, slip and fall -premises liability, hit by drunk driver, work site injury, wrongful death, and pharmaceutical.


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Xiaomi confirms it has hired Google's Hugo Barra, who will take up VP Global role from October
by Josh Ong — in Asia
Hugo Barra, Google's Vice President of product management for Android, is reportedly leaving the company and headed to Chinese smartphone maker Xiaomi, according to AllThingsD.
Update: Xiaomi president Bin Lin has confirmed the Barra's impending arrival on Sina Weibo. Bin's fellow co-founder Lei Jun says Barra will take up the position of Xiaomi vice president of global and begin work at the company from October — hat tip Tech In Asia. Barra has penned a farewell note on Google+, we've included the text at the bottom of this post.
Coming just months after Andy Rubin stepped down from the Android team, Barra's alleged departure represents yet another shakeup at Google, as well as a huge win for Xiaomi.
Google, for its part, has confirmed Barra's departure. The company noted that it is "excited that he is staying within the Android ecosystem". It's not a given that he's headed to Xiaomi, but Google's statement does seem to hint at it.
Xiaomi has been growing at a breath-taking pace as of late, recently closing a funding round that valued the company at $10 billion. The company has dabbled briefly in international markets, but it has yet to firmly establish its reputation overseas. Hiring away a high-profile executive from a US tech giant is a crucial step toward being perceived as a global company, rather than a domestic Chinese player.
Considering how vital Android is to Xiaomi's long-term strategy, Barra's direct experience at Google would also make him an extremely valuable addition to the team. Assuming that Barra is indeed headed to Xiaomi, this is an important turning point for the company.
Chinese hardware makers like Huawei, Lenovo and ZTE have enjoyed massive success worldwide lately, but local Internet giants such as Tencent, Baidu and Alibaba have yet to succeed in gaining widespread consumer adoption in the US and Europe. Xiaomi bounces among hardware, software and e-commerce, so it doesn't fit squarely into either camp, though its smartphones and software already have a niche appeal to Android fans around the world. The right international team would set up Xiaomi to quickly hit its stride in the global mid-range smartphone market just as it has at home.
Here's Barra's own confirmation, via Google+:
A new Android chapter
After nearly 5½ years at Google and almost 3 years as a member of the Android team — the most amazing group of people I've ever worked with in my life — I have decided to start a new career chapter.
In a few weeks, I'll be joining the Xiaomi team in China to help them expand their incredible product portfolio and business globally — as Vice President, Xiaomi Global. I'm really looking forward to this new challenge, and am particularly excited about the opportunity to continue to help drive the Android ecosystem.
It has been an amazing ride and true honor to be part of the Android team at Google, and I especially want to thank Andy Rubin , Hiroshi Lockheimer , Sundar Pichai and Vic Gundotra for their phenomenal support and mentorship over the years.
Android team, I will truly miss you all!
Read next: Apple pursues trademark for 'startup' in Australia, again
GoogleHugo Barra