PRALAY – The Great Deluge by Vineet Bajpai is set to flood your brains with its gripping content and fierce style of writing. Read the full review here.
'For A Girl In A Star' by Ratna Chandu is an emotional tale that abounds in our ears as we close the book after reading it. Read the full review here.


text:
\section{\@startsection {section}{1}{\z@ }%
 {-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\sc }}

\def\thebibliography#1{%
 \section *{References\@mkboth {REFERENCES}{REFERENCES}}%
 \list {[\arabic {enumi}]}{\settowidth \labelwidth {[#1]}%
 \leftmargin \labelwidth \advance \leftmargin \labelsep %
 \usecounter {enumi}} \def \newblock %
 {\hskip .11em plus .33em minus -.07em} \sloppy \clubpenalty 4000%
 \widowpenalty 4000 \sfcode`\.=1000\relax}

\def\@maketitle{%
 \newpage \null \vskip 2em
 \begin{center}{\Large\sc \@title \par }
 \vskip 1.5em
 {\large \lineskip .5em
 \begin {tabular}[t]{c}\@author
 \end{tabular}\par }
 \end{center}\par \vskip 1.5em}

\def\abstract{%
\if@twocolumn \section *{Abstract}
 \else \small\quotation\noindent{\sc Abstract.}\fi}

\catcode`\@=13

\def\noindent{\sc Proof. }{\noindent{\sc Proof. }}
\def\hspace*{\fill{\hspace*{\fill}
\mbox{\hphantom{mm}{\sc q.e.d.}}\\
\par\vskip-5mm}

\def\hspace*{\fill{\hspace*{\fill}
\mbox{\hphantom{mm}{\bf --}}\\
\par\vskip-14mm\hphantom{m}}

\def\hskip.2mm\cdot\hskip.8mm{\hskip.2mm\cdot\hskip.8mm}
\def{\bf k}} \def\bk{{\bf k}{{\bf k}} \def\bk{{\bf k}}
\def\tp#1#2{{#1}^{\otimes{#2}}}
\def1\!\!1} \def\id{\ident{1\!\!1} \def\id{1\!\!1} \def\id{\ident}
\def\Delta{\Delta}
\def\otimes{\otimes}
\def{\overline V}{{\overline V}}
\def{\mbox{\rm Ker}}{{\mbox{\rm Ker}}}
\def\br#1{{\mbox{\rm Br}_{#1}}}
\def*{*}
\def\mbox{$\sum$\hskip1mm}{\mbox{$\sum$\hskip1mm}}
\def\mbox{$\bigodot$}{\mbox{$\bigodot$}}
\def\odot{\odot}
\def\bigodo{\mbox{$\bigodot$}}
\def{\cal B}_\b(V){{\cal B}_*(V)}
\defd_{\cal B}{d_{\cal B}}
\def{\cal B}{{\cal B}}
\def\bod(V,\bv){\bigodo(V,{\cal B}_\b(V))}
\def\uparrow\kern -.05em{\uparrow\kern -.05em}
\def\mbox{\rm Der}{\mbox{\rm Der}}
\def\der#1{\mbox{\rm Der}^{#1}_V(\bigodo(V,{\cal B}_\b(V)))}
\def\downarrow\kern -.05em{\downarrow\kern -.05em}
\def{\cal R}{{\cal R}}
\defd_{\cal R}{d_{\cal R}}
\def{\cal S}{{\cal S}}
\defd_{\cal S}{d_{\cal S}}
\def\mbox{\rm Im}} \def\bolds{{\bf s}{\mbox{\rm Im}} \def\bolds{{\bf s}}
\def{\cal L}} \def\boldt{{\bf t}{{\cal L}} \def\boldt{{\bf t}}
\def\cb#1{{\cal B}_{#1}(V)} \def{\bf u}{{\bf u}}
\def\cc#1#2{CC_{#1}(K_{#2})}
\def\Br#1#2{\mbox{\rm Br}_{#2}(#1)}
\def\hh#1#2#3{(\id^{\odot(#1)}\odot {#2}_N\odot \id^{\odot(#3)})}
\def{\bf e}{{\bf e}}
\def\mbox{\rm Aut}(A^*){\mbox{\rm Aut}(A^*)}

\begin{document}
\baselineskip18pt

\title{COHOMOLOGY OF DRINFEL'D ALGEBRAS:
\\ A GENERAL NONSENSE APPROACH}
\author{\rm Martin MARKL%
\thanks{Partially supported by the grant AV \v CR \#119 105 and
by the Gelbart Research Institute, Bar Ilan University, \hskip7mm
Israel}
\hskip1mm and Steve SHNIDER}

\maketitle

\section{Preliminaries}

Recall that a {\em Drinfel'd algebra\/} (or a
{\em quasi-bialgebra\/} in the
original terminology of~\cite{drinfeld:Alg.iAnaliz89}) is an object
$A=(V,\hskip.2mm\cdot\hskip.8mm,\Delta,\Phi)$, where
$(V,\hskip.2mm\cdot\hskip.8mm,\Delta)$ is an associative, not necessarily coassociative,
unital and counital ${\bf k}} \def\bk{{\bf k}$-bialgebra, $\Phi$ is an
invertible element of
$\tp V3$,
and the usual coassociativity property is replaced by
the condition which we shall refer to as quasi-coassociativity:
\begin{equation}
\label{14}
(1\!\!1} \def\id{\ident\otimes\Delta)\Delta \cdot \Phi
= \Phi\cdot(\Delta\otimes1\!\!1} \def\id{\ident)\Delta,
\end{equation}
where we use the dot $\hskip.2mm\cdot\hskip.8mm$ to indicate both the (associative)
multiplication on $V$ and the induced multiplication
on $V^{\otimes
3}$.
Moreover, the validity of the following
``pentagon identity'' is required:
\[
(1\!\!1} \def\id{\ident^2\otimes\Delta)(\Phi)\cdot(\Delta\otimes1\!\!1} \def\id{\ident^2)
(\Phi) =
(1\otimes\Phi)\cdot(1\!\!1} \def\id{\ident
\otimes\Delta\otimes1\!\!1} \def\id{\ident)(\Phi)\cdot(\Phi\otimes1),
\]
$1\in V$ being the unit element and $1\!\!1} \def\id{\ident$, the identity map
on $V$. If $\epsilon :V \to \bk$ ($\bk$ being the ground field) is the
counit of the coalgebra $(V,\Delta)$ then, by definition, $(\epsilon \otimes
\id)\Delta = (\id \otimes \epsilon)\Delta = \id$.
We have a natural
splitting $V = {\overline V} \oplus \bk$, ${\overline V} := {\mbox{\rm Ker}}(\epsilon)$,
given by the embedding $\bk \to V$, $\bk\ni c \mapsto c\cdot 1 \in V$.

For a $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule $N$, recall the following generalization of
the $M$-{\em construction\/}
of~\cite[par.~3]{markl-stasheff:Journ.ofAlgToApp} introduced
in~\cite{markl-shnider:preprint}.
Let $F^*=\bigoplus_{n\geq 0}F^n$ be
the free unitary nonassociative ${\bf k}} \def\bk{{\bf k}$-algebra generated by
$N$, graded by the length of words. The space $F^n$
is the direct sum of
copies of $N^{\otimes n}$ over the set $\br n$ of full bracketings of $n$
symbols, $F^n = \bigoplus_{u\in {\rm Br}_n}N^{\otimes n}_u$.
For example, $F^0=\bk$, $F^1 = N$, $F^2 =
N^{\otimes 2}$, $F^3 = N^{\otimes 3}_{(\bullet\bullet)\bullet} \oplus
N^{\otimes 3}_{\bullet(\bullet\bullet)}$,~etc.
The algebra $F^*$ admits a
natural left action,
$(a,f)\mapsto a\bullet f$, of the algebra $(V,\hskip.2mm\cdot\hskip.8mm)$ given by the
rules:
\begin{itemize}
\item[(i)]
on $F^0=\bk$, the action is given by the augmentation $\epsilon$,
\item[(ii)]
on $F^1=N$, the action is given by the left action of $V$ on $N$
and
\item[(iii)]
$a\bullet (f\star g)= \sum(\Delta'(a)\bullet
f)\star(\Delta''(a)\bullet g)$,
\end{itemize}
where $\star$ stands for the multiplication in $F^*$ and we use the
Sweedler
notation $\Delta(a)= \sum \Delta'(a)\otimes \Delta''(a)$. The right
action $(f,b)\mapsto f\bullet b$ is defined by similar rules. These
actions define on $F^*$ the structure of a $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule.

Let $\sim$ be the relation on $F^*$ $\star$-multiplicatively
generated
by the expressions of the form
\[
\sum\left((\Phi_1\bullet x)\star\left((\Phi_2\bullet
y)\rule{0mm}{4mm}\star(\Phi_3\bullet
z)\right)\right)
\sim\sum\left(\left((x\bullet\Phi_1)\star\rule{0mm}{4mm}
(y\bullet\Phi_2)\right)\star
(z\bullet\Phi_3)\right),
\]
where $\Phi = \sum\Phi_1\otimes \Phi_2\otimes\Phi_3$ and $x,y,z\in
F^*$.
Put $\mbox{$\bigodot$}(N):= F/\sim$.
Just as in~\cite[Proposition~3.2]{markl-stasheff:Journ.ofAlgToApp}
one proves that the $\bullet$-action induces on $\mbox{$\bigodot$}(N)$
the structure of
a $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule (denoted again by~$\bullet$) and that
$\star$ induces
on $\mbox{$\bigodot$}(N)$ a nonassociative multiplication denoted by
$\odot$. The operations are related by
\[
a\bullet (f\odot g)= \mbox{$\sum$\hskip1mm}(\Delta'(a)\bullet f)\odot(\Delta''(a)\bullet g)
\mbox{ and }
(f\odot g)\bullet b = \mbox{$\sum$\hskip1mm}(f\bullet \Delta'(b))\odot (g \bullet
\Delta''(b)),
\]
for $a,b\in V$ and $f,g\in \mbox{$\bigodot$}(N)$.
The multiplication $\odot$ is quasi-associative in the sense
\begin{equation}
\label{3}
\sum(\Phi_1\bullet x)\odot\left((\Phi_2\bullet
y)\rule{0mm}{4mm}\odot(\Phi_3\bullet
z)\right)
=\sum\left((x\bullet\Phi_1)\odot\rule{0mm}{4mm}
(y\bullet\Phi_2)\right)\odot
(z\bullet\Phi_3)
\end{equation}
The construction described above is functorial in the sense that any
$(V\hskip.2mm\cdot\hskip.8mm)$-bimodule map $f:N' \to N''$ induces a natural
$(V\hskip.2mm\cdot\hskip.8mm)$-linear algebra homomorphism $\bigodo(f):\bigodo(N')\to \bigodo(N'')$.
As it was shown in~\cite{markl-stasheff:Journ.ofAlgToApp},
for any $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule $N$
there exists a natural homomorphism of
$\bk$-modules $J= J(N): \bigodo(N)\to \bigotimes (N)$. If $f:N' \to N''$
is as above then $J(N'')\circ \bigodo(f)= \bigotimes(f)\circ J(N')$.

Since the defining relations~(\ref{3}) are homogeneous with
respect to length,
the grading of $F^*$ induces on $\bigodo(N)$ the grading $\bigodo^*(N)=
\bigoplus_{i\geq 0}\bigodo^i(N)$. If $N$ itself is a graded vector
space, we have also the obvious second grading, $\bigodo(N) =
\bigoplus_{j}\bigodo(N)^j$, which coincides with the first grading if
$N$ is concentrated in degree~1.

Let $\mbox{\rm Der}^n_V(\bigodo(N))$ denote the set of
$(V,\hskip.2mm\cdot\hskip.8mm)$-linear derivations of degree $n$ (relative to the
second grading) of the (nonassociative) graded algebra
$\bigodo(N)^*$. One sees immediately that there is an one-to-one
correspondence between the elements $\theta \in\mbox{\rm
Der}^n_V(\bigodo N)$
and $(V,\hskip.2mm\cdot\hskip.8mm)$-linear homogeneous degree~$n$ maps $f:N^*\to
\bigodo(N)^*$.

If $N=X\oplus Y$, then $\bigodo(X\oplus Y)$ is naturally
bigraded, $\bigodo^{*,*}(X\oplus Y)= \bigoplus_{i,j\geq
0}\bigodo^{i,j}(X\oplus Y)$, the bigrading being defined by saying that
a monomial $w$ belongs to $\bigodo^{i,j}(X\oplus Y)$ if there are exactly
$i$
(resp.~$j$) occurrences of the elements of $X$ (resp.~$Y$) in $w$.
If $X,Y$ are graded vector spaces then there is a second bigrading
$\bigodo(X\oplus Y)^{*,*}= \bigoplus_{i,j}
\bigodo(X\oplus Y)^{i,j}$ just as above.

Let $({\cal B}_\b(V),d_{\cal B})$ be the (two-sided) normalized
bar resolution of the algebra
$(V,\hskip.2mm\cdot\hskip.8mm)$ (see~\cite[Chapter~X]{maclane:homology}),
but considered with the opposite grading.
This means that ${\cal B}_\b(V)$ is
the graded $(V\hskip.2mm\cdot\hskip.8mm)$-bimodule, ${\cal B}_\b(V) = \bigoplus_{n\leq
1}{\cal B}_n(V)$, where ${\cal B}_1(V):=V$ with the $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule
structure induced by the multiplication $\hskip.2mm\cdot\hskip.8mm$,
${\cal B}_0(V):=V\otimes V$ (the free $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule on $\bk$), and
for $n\leq- 1$, ${\cal B}_n(V)$ is the free $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule on
${\overline V}^{\otimes( -n)}$,
i.e.~the vector space $V\otimes {\overline V}^{\otimes(-n)}\otimes V$
with the action of ($V,\hskip.2mm\cdot\hskip.8mm)$ given by
\[
u\cdot (a_0\otimes\cdots\otimes a_{-n+1}):= (u\cdot
a_0\otimes\cdots\otimes a_{-n+1})\
\mbox{ and }\
(a_0\otimes\cdots\otimes a_{-n+1})\cdot w := (a_0\otimes\cdots\otimes
a_{-n+1}\cdot w)
\]
for $u,v,a_0,a_{-n+1}\in V$ and $a_1,\ldots,a_{-n}\in {\overline V}$.
If we use the more compact
notation (though a nonstandard one), writing $(a_0|\cdots|a_{-n+1})$
instead of $a_0\otimes\cdots\otimes a_{-n+1}$, the differential $d_{\cal B}
: {\cal B}_n(V)
\to {\cal B}_{n+1}(V)$ is, for $n\leq 0$, defined as
\[
d_{\cal B}(a_0|\cdots|a_{-n+1}):=
\sum_{0\leq i\leq -n}(-1)^{i}(a_0|\cdots|a_i\cdot a_{i+1}|
\cdots|a_{-n+1}).
\]
Here, as is usual in this context, we make no distinction between
the elements of $V/\bk \cdot 1$ and their representatives in ${\overline V}$. We
use the same convention throughout all the paper.
Notice that the differential $d_{\cal B}$ is a $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule map.

Put $\bod(V,\bv) := \bigodo(\uparrow\kern -.05em V \oplus \uparrow\kern -.05em {\cal B}_\b(V))$, where $\uparrow\kern -.05em$
denotes, as usual, the suspension of a graded vector space and $V$ is
interpreted as a graded vector space concentrated in degree zero.
Let $\der i$
denote, for each $i$, the space of degree $i$ derivations of the
algebra $\bod(V,\bv)$ which are also $(V,\hskip.2mm\cdot\hskip.8mm)$-linear maps.
Let us define the derivation $D_{-1}\in \der 1$ by
$D_{-1}|_{\uparrow\kern -.05em {\cal B}_\b(V)}:= \uparrow\kern -.05em d_{\cal B} \downarrow\kern -.05em$ and
$D_{-1}|_{\uparrow\kern -.05em V}:= 0$.
Clearly $D_{-1}(\bigodo(V,{\cal B}_*(V))^{i,j})
\subset \bigodo(V,{\cal B}_*(V))^{i,j+1}$ and
$D_{-1}(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset \bigodo^{i,j}(V,{\cal B}_\b(V))$
for any $i,j \geq 0$. We also see immediately that
$D_{-1}^2=0$.

Let us consider, for any $n\geq 1$,
the complex $(\bigodo^{n-1,1}(V,{\cal B}_*(V)),D_{-1})$,
i.e.~the complex
\[
0\longleftarrow \bigodo^{n-1,1}(V,V)
\stackrel{D_{-1}}{\longleftarrow} \bigodo^{n-1,1}(V,{\cal B}_0(V))
\stackrel{D_{-1}}{\longleftarrow}\bigodo^{n-1,1}(V,{\cal B}_{-1}(V))
\longleftarrow\cdots
\]
\begin{lemma}
The complex $(\bigodo^{n-1,1}(V,{\cal B}_*(V)),D_{-1})$ is acyclic, for
any $n\geq 1$.
\end{lemma}

\noindent{\sc Proof. } We have the decomposition \[
\bigodo^{n-1,1}(V,{\cal B}_*(V)) =
\bigoplus_{1\leq i\leq n}\bigodo^{n-1,1}_i(V,{\cal B}_*(V)),
\]
where $\bigodo^{n-1,1}_i(V,{\cal B}_*(V))$ denotes the subspace of
$\bigodo^{n-1,1}(V,{\cal B}_*(V))$ spanned by monomials having an
element of ${\cal B}_*(V)$ at the $i$-th place. The differential
$D_{-1}$
obviously respects this decomposition and the canonical isomorphism
$J$ of~\cite{markl-stasheff:Journ.ofAlgToApp} mentioned above
identifies $\bigodo^{n-1,1}_i(V,{\cal B}_*(V))$ to
$\tp V{(i-1)}\otimes {\cal B}_\b(V) \otimes \tp V{(n-1)}$. Under this identification the
differential $D_{-1}$ coincides with
$\tp \id{(i-1)}\otimes d_{\cal B}\otimes \tp\id{(n-i)}$ and the rest follows from the
K\"unneth formula and the acyclicity of $({\cal B}_\b(V),d_{\cal B})$.
\hspace*{\fill

\section{Properties of $\der *$}

Let $C = (C,\hskip.2mm\cdot\hskip.8mm,1_C)$ be a unital associative algebra and let $C
\stackrel{\epsilon}{\longleftarrow}({\cal R},d_{\cal R})$, $({\cal R},d_{\cal R})=
R_0 \stackrel{d_{\cal R}}{\longleftarrow}R_1
\stackrel{d_{\cal R}}{\longleftarrow}\cdots$, be a complex of free
$C$-bimodules (we consider $C$ as a $C$-bimodule with the bimodule
structure induced by the multiplication). Similarly, let $D
\stackrel{\eta}{\longleftarrow}({\cal S},d_{\cal S})$ with $({\cal S},d_{\cal S})=
S_0 \stackrel{d_{\cal S}}{\longleftarrow}S_1
\stackrel{d_{\cal S}}{\longleftarrow}\cdots$, be an acyclic complex of
$C$-bimodules. To simplify the notation, we write sometimes
$R_{-1}$ (resp.~$S_{-1}$, resp.~$d_{\cal R}$, resp.~$d_{\cal S}$) instead of $C$
(resp.~$D$, resp.~$\epsilon$, resp.~$\eta$).
Let
\[
Z := \{f=(f_i)_{i\geq -1};\ f_i:R_i\to S_i\ \mbox{a $C$-bimodule
map and}\ f_i\circ d_{\cal R} = d_{\cal S} \circ f_{i+1}\
\mbox{for any $i\geq -1$}\}.
\]
Let us define, for a sequence $\chi = (\chi_i)_{i\geq -1}$ of
$C$-bimodule maps $\chi_i: R_i \to S_{i+1}$,
$\nabla(\chi)= (\nabla(\chi)_i)_{i\geq -1}\in Z$ by
$\nabla(\chi)_i:= d_{\cal S} \circ \chi_i+\chi_{i-1}\circd_{\cal R}$. Let $B:=
\mbox{\rm Im}} \def\bolds{{\bf s}(\nabla)\subset Z$. For a $C$-bimodule $M$ let $M_I$ denote the set
of invariant elements of $M$, $M_I:= \{x\in M;\ cx = xc\ \mbox{for
any $c\in C$}\}$.

\begin{lemma}
\label{1}
Under the notation above, the correspondence
$Z\ni f=(f_i)_{i\geq -1}\mapsto f_{-1}(1_C)\in D_I$ induces an
isomorphism $\Omega : Z/B\cong D_I/\eta({S_0}_I)$. Moreover, if
$f_{-1}(1_C)=\eta(h)$ for some $h\in {S_0}_I$ then $f=\nabla(\chi)$
for some $\chi = (\chi_i)_{i\geq -1}$ with $\chi_{-1}(1)=h$.
\end{lemma}

\noindent{\sc Proof. } We show first that $\Omega$ is well-defined. If
$f=\nabla(\chi)$ then $f_{-1}=
\eta\circ \chi_{-1}$, therefore $f_{-1}(1_C)= \eta(h)$ with $h:=
\chi_{-1}(1_C)\in {S_0}_I$ and $\Omega(f)=0$.

Let us prove that $\Omega$ is an epimorphism. For $z\in D_I$
define a $C$-bimodule map $f_{-1}: C\to D$ by $f_{-1}(c):= cz$ (=
$zc$)
for $c\in C$. Because $({\cal R},d_{\cal R})$ is free and $({\cal S},d_{\cal S})$ is
acyclic,
$f_{-1}$ lifts to some $f=(f_i)_{i\geq -1}\in Z$ by
standard homological
arguments~\cite[Theorem~III.6.1]{maclane:homology}.

It remains to prove that $\Omega$ is a monomorphism. For
$f=(f_i)_{i\geq -1} \in Z$, $\Omega(f)=0$ means that $f_{-1}(1_C)=
\eta(h)$ for some $h\in {S_0}_I$. The $C$-bimodule map $\chi_{-1}:
C\to S_0$ defined by $\chi_{-1}(c):= ch$ (= $hc$) for $c\in C$ clearly
satisfies $f_{-1}= \eta \circ \chi_{-1}$. A standard homological
argument (see again \cite[Theorem~III.6.1]{maclane:homology}
then enables one to extend $\chi_{-1}$ to a `contracting
homotopy' $\chi = (\chi_i)_{i\geq -1}$ with $f= \nabla(\chi)$.
\hspace*{\fill

\begin{definition}
\label{0}
For $n\geq 2$ and $k\geq 0$ let $J_k(n)$ be the subspace of
$\der {n-1-k}$ consisting of derivations $\theta$ satisfying
\begin{itemize}
\item[(i)]
$\theta(\bigodo(V,{\cal B}_\b(V))^{i,j})\subset \bigodo(V,{\cal B}_\b(V))^{i+n-1,j-k}$,
\item[(ii)]
$\theta(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset \bigodo^{i+n-1,j}(V,{\cal B}_\b(V))$,
\item[(iii)]
$[D_{-1},\theta]=0$ if $k=0$ and $\theta|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$ if
$k\geq 1$.
\end{itemize}
\end{definition}

Let us observe that, for $\theta \in J_{\geq 1}(n)$, $\theta|_{\uparrow\kern -.05em
V}=0$.
This follows from item~(i) of the definition above.
Observe also that $J_*(n)$ is
invariant under the differential $\nabla$ defined by
$\nabla(\theta):= [D_{-1},\theta]$,
$\nabla(J_k(n))\subset J_{k-1}(n)$ for $k\geq 1$ and
$\nabla(J_0(n))=0$.

\begin{proposition}
\label{12}
$H_{\geq 1}(J_*(n),\nabla)=0$ while
\begin{equation}
\label{2}
H_0(J_*(n),\nabla) =
\bigodo^{n-1,1}(V,{\cal B}_1(V))_I
\oplus \bigodo^n(V)_I.
\end{equation}
\end{proposition}

\noindent{\sc Proof. }
Let $k>0$ and let $\theta \in J_k(n)$. As $\theta|_{\uparrow\kern -.05em V}=0$,
$\theta$ is
given by its restriction to ${\cal B}_\b(V)$, namely by a sequence of
$(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule maps
$\theta_i:{\cal B}_i(V)\to \bigodo^{n-1,1}(V,{\cal B}_{i-k}(V))$, $i\leq 1$.
Suppose that $\theta$ is a $\nabla$-cocycle, i.e.~that
$\nabla(\theta)=0$. This means that the diagram
\[
\unitlength=1.45mm
\begin{picture}(100.00,25.00)(25,10)
\put(40.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,{\cal B}_{1-k}(V))$}}
\put(70.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,{\cal B}_{-k}(V))$}}
\put(100.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,%
{\cal B}_{-1-k}(V))$}}
\put(40.00,30.00){\makebox(0,0)[cc]{${\cal B}_1(V)=V$}}
\put(70.00,30.00){\makebox(0,0)[cc]{${\cal B}_0(V)$}}
\put(100.00,30.00){\makebox(0,0)[cc]{${\cal B}_{-1}(V)$}}
\put(40.00,26.00){\vector(0,-1){11.00}}
\put(70.00,26.00){\vector(0,-1){11.00}}
\put(55.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(85.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(55.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(85.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(118.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(45.00,21.00){\makebox(0,0)[cc]{$\theta_1$}}
\put(76.00,21.00){\makebox(0,0)[cc]{$\theta_0$}}
\put(105.00,21.00){\makebox(0,0)[cc]{$\theta_{-1}$}}
\put(20.00,30.00){\makebox(0,0)[cc]{0}}
\put(20.00,10.00){\makebox(0,0)[cc]{0}}
\put(100.00,26.00){\vector(0,-1){10.89}}
\put(115.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(128.00,30.00){\makebox(0,0)[cc]{$\cdots$}}
\put(128.00,10.00){\makebox(0,0)[cc]{$\cdots$}}
\put(30.00,30.00){\vector(-1,0){8.00}}
\put(64.00,30.00){\vector(-1,0){15.93}}
\put(94.00,30.00){\vector(-1,0){17.93}}
\put(124.00,30.00){\vector(-1,0){17.93}}
\put(26.00,10.00){\vector(-1,0){4.00}}
\put(86.00,10.00){\vector(-1,0){4.00}}
\put(57.00,10.00){\vector(-1,0){4.1}}
\put(124.00,10.00){\vector(-1,0){9}}
\end{picture}
\]
is commutative. Since $\theta_1=0$ by item~(iii) of
Definition~\ref{0}, Lemma~\ref{1} (with $C = (V,\hskip.2mm\cdot\hskip.8mm,1_V)$ and
$D=\bigodo^{n-1,1}(V,{\cal B}_{1-k}(V))$) gives a sequence
$\chi_i:{\cal B}_i(V)\to \bigodo^{n-1,1}(V,\cb {i-k-1})$ of
$(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule maps, $i\leq 1$, such that $\theta_i=
D_{-1}\circ\chi_i + \chi_{i+1}\circ d_{\cal B}$. We can, moreover, suppose
that $\chi_1=0$, thus
the sequence $(\chi_i)_{i\leq 1}$ determines a derivation
$\chi \in J_{k+1}(n)$ with $\nabla(\chi)=\theta$. This proves
$H_k(J_*(n),\nabla)=0$ for $k\geq 1$.

A derivation $\theta \in J_0(n)$ is given by two independent data: by
the restriction $\theta_V := \theta|_{\uparrow\kern -.05em V}: V\to\bigodo^n(V)$ and by
the restriction
$\theta_{{\cal B}_\b(V)}:= \theta|_{\uparrow\kern -.05em {\cal B}_\b(V)}: {\cal B}_\b(V) \to \bigodo^{n-1,1}(V,{\cal B}_\b(V))$.
As $D_{-1}|_{\uparrow\kern -.05em V}=0$, the
condition $\nabla(\theta)= [D_{-1},\theta]=0$
imposes no restrictions
on $\theta_V$ and, because $\nabla(\chi)|_{\uparrow\kern -.05em V}=0$ for any
$\chi \in J_1(n)$, the contribution of $\theta_V$ to
$H_0(J_*(n),\nabla)$ is parametrized by $\theta_V(1_V)$, i.e.~by an
element of $\bigodo^n(V)_I$. This explains the second summand
in~(\ref{2}).

The restriction $\theta_{{\cal B}_\b(V)}$ is in fact a sequence $\theta_i:\cb i
\to \bigodo^{n-1,1}(V,\cb i)$, $i\leq 1$, of $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule maps and
the condition $\nabla(\theta)=0$ means that the diagram
\[
\unitlength=1.45mm
\begin{picture}(100.00,25.00)(25,10)
\put(40.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,{\cal B}_{1}(V))$}}
\put(70.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,{\cal B}_{0}(V))$}}
\put(100.00,10.00){\makebox(0,0)[cc]{$\bigodo^{n-1,1}(V,{\cal B}_{-1}(V))$}}
\put(40.00,30.00){\makebox(0,0)[cc]{${\cal B}_1(V)=V$}}
\put(70.00,30.00){\makebox(0,0)[cc]{${\cal B}_0(V)$}}
\put(100.00,30.00){\makebox(0,0)[cc]{${\cal B}_{-1}(V)$}}
\put(40.00,26.00){\vector(0,-1){11.00}}
\put(70.00,26.00){\vector(0,-1){11.00}}
\put(55.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(85.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(55.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(85.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(118.00,15.00){\makebox(0,0)[cc]{$D_{-1}$}}
\put(45.00,21.00){\makebox(0,0)[cc]{$\theta_1$}}
\put(76.00,21.00){\makebox(0,0)[cc]{$\theta_0$}}
\put(105.00,21.00){\makebox(0,0)[cc]{$\theta_{-1}$}}
\put(20.00,30.00){\makebox(0,0)[cc]{0}}
\put(20.00,10.00){\makebox(0,0)[cc]{0}}
\put(100.00,26.00){\vector(0,-1){10.89}}
\put(115.00,35.00){\makebox(0,0)[cc]{$d_{\cal B}$}}
\put(128.00,30.00){\makebox(0,0)[cc]{$\cdots$}}
\put(128.00,10.00){\makebox(0,0)[cc]{$\cdots$}}
\put(30.00,30.00){\vector(-1,0){8.00}}
\put(64.00,30.00){\vector(-1,0){15.93}}
\put(94.00,30.00){\vector(-1,0){17.93}}
\put(124.00,30.00){\vector(-1,0){17.93}}
\put(26.00,10.00){\vector(-1,0){4.00}}
\put(86.00,10.00){\vector(-1,0){4.00}}
\put(57.00,10.00){\vector(-1,0){4.1}}
\put(124.00,10.00){\vector(-1,0){9}}
\end{picture}
\]
is commutative. Similarly as above, a derivation
$\chi \in J_1(n)$ is given by a sequence
$\chi_i: \cb i\to \bigodo^{n-1,1}(V,\cb{i-1})$, $i\leq 1$, of
$(V,\hskip.2mm\cdot\hskip.8mm)$-linear maps. The condition $\nabla(\chi)=\theta$ then means
that $\theta_i = D_{-1}\circ \chi_i + \chi_{i+1}\circ d_{\cal B}$,
especially, $\theta_1=D_{-1}\circ \chi_1$. This last equation
implies, since $\chi_1=0$ by~(iii) of Definition~\ref{0}, that
$\nabla(\chi)=\theta$ forces $\theta_1(1_V)=0$. On the other hand, if
$\theta_1(1_V)=0$ then Lemma~\ref{1} gives a $\chi \in J_1(n)$ with
$\theta = \nabla(\chi)$ and we conclude that the contribution of
$\theta_{{\cal B}_\b(V)}$ to $H_0(J_*(n),\nabla)$ is parametrized by
$\theta_{{\cal B}_\b(V)}(1_V)\in \bigodo^{n-1,1}(V,\cb 1)_I$ which is the first
summand of~(\ref{2}).
\hspace*{\fill

Let us recall that a (right) {\em differential graded (dg) comp
algebra\/}
(or a {\em nonunital operad\/} in the terminology of~\cite{zebrulka})
is a
bigraded differential space $L =(L_*(*),\nabla)$, $L_*(*)=
\bigoplus_{k\geq 0,n\geq 2} L_k(n)$,
$\nabla(L_k(n))\subset L_{k-1}(n)$,
together with a system of bilinear operations
\[
\circ_i : L_p(a) \otimes L_q(b) \to L_{p+q}(a+b-1)
\]
given for any $1\leq i \leq b$ such that, for $\phi \in L_p(a)$,
$\psi \in L_q(b)$ and $\nu \in L_r(c)$,
\begin{equation}
\label{4}
\phi\circ_i(\psi\circ_j \nu) =
\left\{
\begin{array}{ll}
(-1)^{p\cdot q}\cdot\psi\circ_{j+a-1}(\phi\circ_i \nu),&
\mbox{ for }1\leq i\leq j-1,
\\
(\phi \circ_{i-j+1} \psi)\circ_j \nu, &
\mbox{ for }j\leq i \leq b+j-1, \mbox{ and}
\\
(-1)^{p\cdot q}\cdot \psi\circ_j(\phi\circ_{i-b+1} \nu),&
\mbox{ for }i\geq j+b.
\end{array}
\right.
\end{equation}
We suppose, moreover, that for any $\phi \in L_p(a)$,
and $\psi \in L_q(b)$,
$1\leq i \leq b$,
\[
\nabla(\phi\circ_t \psi) = \nabla(\phi)\circ_t \psi +
(-1)^p\cdot \phi\circ_t \nabla(\psi).
\]

Any dg comp algebra determines a
nonsymmetric (unital) operad in the monoidal category of differential
graded spaces (see~\cite{may:1972} for the terminology).
To be more precise, let $L = (L_*(*),\circ_i,\nabla)$
be a dg comp
algebra as above and let us define the bigraded vector space $
{\cal L}} \def\boldt{{\bf t}_*(*) = \bigoplus_{k\geq 0, n\geq 1}$ by ${\cal L}} \def\boldt{{\bf t}_*(n):=
L_*(n)$ for $n\geq 2$ and
${\cal L}} \def\boldt{{\bf t}_*(1)={\cal L}} \def\boldt{{\bf t}_0(1):= \mbox{Span}(1_{\cal L}} \def\boldt{{\bf t})$, where $1_{\cal L}} \def\boldt{{\bf t}$ is a degree zero
generator. Let us extend the definition of structure maps $\circ_i$
to ${\cal L}} \def\boldt{{\bf t}$ by putting $f\circ_1 1_{\cal L}} \def\boldt{{\bf t}:= f$ and $1_{\cal L}} \def\boldt{{\bf t}\circ _i g:= g$, for
$f\in {\cal L}} \def\boldt{{\bf t}_*(m)$, $g\in {\cal L}} \def\boldt{{\bf t}_*(n)$ and $1\leq i\leq n$. Let us extend
the differential $\nabla$ by $\nabla(1_{\cal L}} \def\boldt{{\bf t}):=0$. In~\cite{zebrulka} we
proved the following proposition.

\begin{proposition}
The composition maps
$\gamma :{\cal L}} \def\boldt{{\bf t}_*(a)\otimes {\cal L}} \def\boldt{{\bf t}_*(n_1)\otimes \cdots
\otimes {\cal L}} \def\boldt{{\bf t}_*(n_a)\to {\cal L}} \def\boldt{{\bf t}_* (n_1+\cdots+ n_a)$ given by
\[
\gamma(\phi;\nu_1,\ldots,\nu_a):= \nu_1 \circ_1(\nu_2
\circ_2(\cdots \circ_{a-1}(\nu_a \circ_a \nu)))
\]
for $\phi \in {\cal L}} \def\boldt{{\bf t}_*(a)$ and $\nu_i \in {\cal L}} \def\boldt{{\bf t}_*(n_i)$, $1\leq i\leq a$,
define on ${\cal L}} \def\boldt{{\bf t}_*(*)$ a structure of a nonsymmetric differential
graded
operad in the monoidal category of differential graded vector spaces.
\end{proposition}

Let $N$ be a (graded) $(V,\hskip.2mm\cdot\hskip.8mm)$-bimodule, let $d: N\to N$ be a
$(V,\hskip.2mm\cdot\hskip.8mm)$-linear differential and define
$X_k(n):=\{\theta \in \mbox{\rm Der}^k_V(\bigodo(N));\
\theta(N)\subset \bigodo^n(N)\}$. For $\omega \in X_*(m)$,
$\theta \in X_*(n)$ and $1\leq i\leq n$ let $\omega_N
:= \omega|_N : N\to \bigodo^m(N)$ and $\theta_N := \theta|_N
: N\to \bigodo^n(N)$ be the restrictions. Let then $\omega \circ_i
\theta \in X_*(m+n-1)$ be a derivation defined by $(\omega \circ_i
\theta)|_N := (\id^{\odot(i-1)}\odot \omega_N
\odot \id^{\odot (n-i)})\circ \theta_N$. Let us extend the
differential
$d$ to a derivation $D$ of $\bigodo(N)$ and define $\nabla(\theta):=
[D,\theta]$.

\begin{lemma}
The object $X_*(*)=(X_*(*), \circ_i,\nabla)$ constructed above
is a differential graded comp algebra.
\end{lemma}

\noindent{\sc Proof. }
Let $\phi\in X_p(a)$, $\psi \in X_q(b)$ and $\nu \in X_r(c)$. The
composition $\phi\circ_i(\psi\circ_j\nu)$ is, by definition, given by
its restriction $[\phi\circ_i(\psi\circ_j\nu)]_N$ to $N$ as
\[
[\phi\circ_i(\psi\circ_j\nu)]_N =
\hh{i-1}{\phi}{b+c-i-1}\circ (\psi\circ_j \nu)_N,
\]
with $(\psi\circ_j \nu)_N = \hh{j-1}{\psi}{c-j}\circ \nu_N$.
This implies that
\[
[\phi\circ_i(\psi\circ_j\nu)]_N= \hh{i-1}{\phi}{b+c-i-1}\circ
\hh{j-1}{\psi}{c-j}\circ \nu_N.
\]
For $i\leq j-1$ we have (taking into the account that
$\mbox{\rm Im}(\phi_N)\subset \bigodo^a(N)$ and
$\mbox{\rm Im}(\psi_N)\subset \bigodo^b(N)$)
\begin{eqnarray*}
\lefteqn{
\hh{i-1}{\phi}{b+c-i-1}\circ \hh{j-1}{\psi}{c-j}=\hskip.5cm}
\\
&&\hskip.5cm=(-1)^{pq}\cdot \hh{j+a-2}{\psi}{c-j}\circ
\hh{i-1}{\phi}{c-i},
\end{eqnarray*}
which means that $[\phi\circ_i(\psi\circ_j\nu)]_N=
(-1)^{pq}\cdot [\psi\circ_{j+a-1}(\phi\circ_i \nu)]_N$.
This is the axiom~(\ref{4}) for $i\leq j-1$.

Similarly, for $j\leq i\leq b+j-1$ we have
\begin{eqnarray*}
\lefteqn{
\hh{i-1}{\phi}{b+c-i-1}\circ \hh{j-1}{\psi}{c-j}=\hskip.5cm}
\\
&&\hskip.5cm=\id^{\odot (j-1)}\odot
[\hh {i-j}{\phi}{b-i+j-1}\circ \psi_N]\odot
\id^{\odot(c-j)}
\end{eqnarray*}
which means that $[\phi\circ_i(\psi\circ_j\nu)]_N=
[(\phi\circ_{i-j+1}\psi)\circ_j \nu]_N$.
This is the axiom~(\ref{4}) for $j\leq i\leq b+j-1$.
The discussion of the remaining case $i\geq j+b$ is similar.
\hspace*{\fill

Let us consider the special case of the construction above with $N :=
\uparrow\kern -.05em V \oplus \uparrow\kern -.05em {\cal B}_\b(V)$ and $d := 0 \oplus \suspd_{\cal B}\downarrow\kern -.05em$.

\begin{lemma}
The bigraded subspace $J_*(*)$ of $X_*(*)$ introduced in
Definition~\ref{0} is closed
under the operations $\circ_i$ and the differential $\nabla$.
\end{lemma}

The proof of the lemma is a straightforward verification.
The lemma says that the dg comp algebra structure
on $X_*(*)$ restricts to a dg comp algebra structure
$(J_*(*),\circ_i,\nabla)$ on $J_*(*)$.

\section{More about $\der *$}

Let $J_*(*)= (J_*(*),\circ_i,\nabla)$ and $D_{-1}\in \der 1$
be as in the previous section.

\begin{definition}
\label{8}
An infinitesimal deformation of $D_{-1}$ is an element
$D_0\in J_0(2)$ such that $\nabla(D_0) =0$.
An integration of an infinitesimal deformation $D_0$ is a
sequence $\tilde D = \{D_i\in J_i(i+2);\ i\geq 1\}$ such that $D:=
D_{-1}+ D_0 + D_1+ \cdots$ satisfies $[D,D]=0$.
\end{definition}

Let $K_n$ be, for $n\geq 2$, the Stasheff
associahedron~\cite{stasheff:TAMS63}.
It is an $(n-2)$-dimensional cellular complex whose $i$-dimensional
cells are indexed by the set $\Br in$ of all (meaningful) insertions
of $(n-i-2)$ pairs of brackets between $n$ symbols, with suitably
defined incidence maps. There is, for any $a,b \geq 2$, $0\leq i \leq
a-2$, $0\leq j \leq b-2$ and $1\leq t \leq b$, a map
\[
(-,-)_t : \Br ia \times \Br jb \to \Br{i+j}{a+b-1},\ u\times v \mapsto
(u,v)_t,
\]
where $(u,v)_t$ is given by the insertion of $(u)$ at the $t$-th
place in $v$.
This map defines, for $a,b \geq 2$ and $1\leq t\leq b$, the inclusions
$\iota_t :K_a\times K_b \hookrightarrow \partial K_{a+b-1}$.
It is well-known that the sequence $\{K_n\}_{n\geq 1}$ form a
topological operad, see again~\cite{stasheff:TAMS63}.

Let $\cc in$ denote the set of $i$-dimensional oriented
cellular chains with coefficients in ${\bf k}} \def\bk{{\bf k}$ and let $d_C
:\cc in\to \cc {i-1}n$ be the cellular differential.
For $\bolds\in \cc pa$ and $\boldt \in \cc qb$, $p,q \geq 0$,
$a,b\geq 2$ and $1\leq i\leq b$,
let $\bolds \times\boldt \in C_{p+q}(K_a\times K_b)$
denote the cellular cross product and put
\[
\bolds\circ_i\boldt:=
({\iota_i})_*
(\bolds\times\boldt)\in \cc {p+q}{a+b-1}.
\]

\begin{proposition}
\label{5}
The cellular chain complex $(\cc *\b,d_C)$ together with
operations $\circ_i$ introduced above forms a
differential graded comp algebra.
\end{proposition}

The simplicial version of this proposition was proved
in~\cite{markl-shnider:preprint},
the proof of the cellular version is similar. The dg comp algebra
structure of Proposition~\ref{5} reflects the topological operad
structure of $\{K_n\}_{n\geq 2}$ mentioned above.

Let $c_n$ be, for $n\geq 0$, the unique top dimensional cell of
$K_{n+2}$, i.e.~the unique element of $\Br{n+2}n$ corresponding to
the insertion of no pairs of brackets between $(n+2)$ symbols. Let us
define ${\bf e}_n \in \cc n{n+2}$ as ${\bf e}_n:= 1\cdot c_n$.

The following proposition was proved in~\cite{zebrulka}, see also the
comments below.

\begin{proposition}
\label{6}
The graded comp algebra $\cc {*}{*}= (\cc*\b,\circ_i)$ is a free
graded
comp algebra on the set $\{{\bf e}_0,{\bf e}_1,\ldots\}$.
\end{proposition}

Let us recall that the freeness in the proposition above means that
for any graded comp algebra $L_*(*)= (L_*(*),\circ_i)$ and
for any sequence $\alpha_n\in L_n(n+2)$, $n\geq 0$, there exists a
unique graded comp algebra map $f : \cc {*}{*}\to L_*(*)$ such
that $f({\bf e}_n)= \alpha_n$, $i\geq 0$.

The proof of Proposition~\ref{6}
is based on the following observation. There is a description of the
free graded comp algebra (= free nonsymmetric
nonunital operad) on a given set in terms of oriented planar trees.
The free comp algebra ${\cal F}({\bf e}_0,{\bf e}_1,\ldots)$ on
the set $\{{\bf e}_0,{\bf e}_1,\ldots\}$ has
${\cal F}({\bf e}_0,{\bf e}_1,\ldots)(n) =$ the vector space spanned by oriented
connected planar trees with $n$ input edges. Each such a tree $T$ then
determines an element of $\Br ni$ where $i =$ the number of
vertices of $T$, i.e.~a cell of $\cc{n-i-2}n$. This correspondence
defines a map ${\cal F}({\bf e}_0,{\bf e}_1,\ldots)(n) \to \cc *n$ which
induces the requisite isomorphism of operads.

Let $L= (L_*(*),\circ_i,\nabla)$ be a dg comp
algebra. Let us define, for $\phi \in L_p(a)$ and $\psi \in L_q(b)$,
\[
\phi\diamond \psi :=
\sum_{1\leq i\leq b}(-1)^{(a+1)(i+q+1)}\cdot \phi\circ_i \psi
\mbox{\hskip2mm and \hskip2mm}
[\phi,\psi]:= \phi\diamond \psi -
(-1)^{(a+p+1)(b+q+1?)} \cdot\psi\diamond \phi.
\]
In~\cite{markl-shnider:preprint} we proved the following proposition.

\begin{proposition}
The operation $[-,-]$ introduced above endows $L^*:=
\bigoplus_{a-p-1=*}L_p(a)$ with a structure of a differential graded
(dg) Lie algebra, $L = (L^*,[-,-],\nabla)$.
\end{proposition}

The construction above thus defines a functor from the category of
dg comp algebras to the category of dg Lie algebras.
Let us observe that for the comp algebras $X_*(*)$
and $J_*(*)$ this structure coincides with the Lie algebra
structure induced by the graded commutator of derivations.
We can also easily prove that the elements $\{{\bf e}_n\}_{n\geq 0}$
satisfy, for each $m\geq 0$,
\begin{equation}
\label{7}
\mbox{$
d_C({\bf e}_m)+\frac12 \sum_{i+j=n-1}[{\bf e}_i,{\bf e}_j]=0
$}
\end{equation}
in the dg Lie algebra $CC(K)^*=(CC(K)^*,[-,-],d_C)$.

\begin{proposition}
There exists an one-to-one correspondence between integrations of an
infinitesimal deformation $D_0$ in the sense of
Definition~\ref{8} and dg comp algebra homomorphisms
$m: \cc *\b \to J_*(*)$ with $m({\bf e}_0)=D_0$.
\end{proposition}

\noindent{\sc Proof. }
Let us suppose we have a map $m: \cc *\b
\to J_*(*)$ with $m({\bf e}_0)=D_0$
and define, for $n\geq 1$, $D_n:= m({\bf e}_n)$.
We must prove that the derivation $D:= D_{-1}+ D_0 + D_1 +\cdots$
satisfies $[D,D]=0$. This condition means that
\begin{equation}
\label{9}
\mbox{$
\nabla(D_m)+\frac12 \sum_{i+j=n-1}[D_i,D_j]=0
$}
\end{equation}
for any $m\geq 0$, which is exactly what we get
applying on~(\ref{7}) the Lie algebra homomorphism $m$.

On the other hand, suppose we have an integration
$\{D_n\}_{n\geq 1}$. The freeness of the graded comp algebra $\cc
*\b$
(Proposition~\ref{6}) ensures the existence of a graded comp algebra
map $m: \cc *\b \to J_*(*)$ with $m({\bf e}_n)= D_n$ for $n\geq 0$. We
must verify that this unique map commutes with the differentials,
i.e.~that $m(d_C(\bolds))= \nabla(m(\bolds))$ for any $\bolds\in
\cc**$.
Because of the freeness, it is enough to verify the last condition for
$\bolds\in \{{\bf e}_n\}_{n\geq 0}$, i.e.~to verify that $m(d_C({\bf e}_n))=
\nabla(m({\bf e}_n))$ for $n\geq 0$. Expanding $d_C({\bf e}_n)$
using~(\ref{7}) we see that this follows from~(\ref{9}) and from the
fact that the map $m$ is a homomorphism of graded Lie algebras.
\hspace*{\fill

\begin{proposition}
\label{13}
An infinitesimal deformation $D_0\in J_0(2)$ can be
integrated if and only if
$[D_0,D_0]|_{\uparrow\kern -.05em V} =[D_0,D_0]|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$.
\end{proposition}

\noindent{\sc Proof. }
Standard obstruction theory.
Let us suppose that we have an integration $\tilde D =
\{D_i\}_{i\geq 1}$. Condition~(\ref{9}) with $m=1$ means that
\begin{equation}
\label{11}
\mbox{$\nabla(D_1)+\frac12[D_0,D_0]=0.$}
\end{equation}
Because $D_1\in J_1(3)$,
$D_1|_{\uparrow\kern -.05em V} = D_1|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$ by (iii) of
Definition~\ref{0},
therefore $\nabla(D_1)|_{\uparrow\kern -.05em V} =\nabla(D_1)|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$
and~(\ref{11}) implies that $[D_0,D_0]|_{\uparrow\kern -.05em V} =
[D_0,D_0]|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$.

On the other hand, let us suppose that $[D_0,D_0]|_{\uparrow\kern -.05em V}
=[D_0,D_0]|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$. By the description of
$H_0(J_*(3),\nabla)$ as it is given in Proposition~\ref{12} we see
that the homology class of $[D_0,D_0] \in J_0(3)$ is zero, therefore
there exists some $D_1\in J_1(3)$ with
$\nabla(D_1)+\frac12[D_0,D_0]=0$.

Let us suppose that we have already constructed a sequence
$D_i\in J_i(i+2)$, $1\leq i\leq N$, such that basic equation~(\ref{9})
holds for any $m\leq N$. The element $\frac12
\sum_{i+j=N}[D_i,D_j]\in J_N(N+3)$ is a $\nabla$-cycle (this follows
from the definition of $\nabla(-)$ as $[D_{-1},-]$ and the Jacobi
identity) and the triviality of $H_N(J_*(N+3))$ (again
Proposition~\ref{12}) gives some $D_{N+1}\in J_{N+1}(N+3)$ which
satisfies~(\ref{9}) for $m=N+1$. The induction may go on.
\hspace*{\fill

Let $G$ be the subgroup of $\mbox{\rm Aut}(\bigodo(V,{\cal B}_\b(V))^*)$ consisting
of automorphisms of the form $g= \id+\phi_{\geq 2}$, where
$\phi_{\geq 2}$ is a $(V,\hskip.2mm\cdot\hskip.8mm)$-linear map satisfying
$\phi_{\geq 2}(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset\bigodo^{\geq i+2,j}(V,{\cal B}_\b(V))$ for
any $i,j$.

Let us observe that $G$ naturally acts on the set of integrations of
a fixed infinitesimal deformation $D_0$. To see this, let $\tilde D =
\{D_i\}_{i\geq 1}$ be such an integration and let us denote, as
usual, $D := D_{-1}+D_0+D_1 +\cdots$. Then $g^{-1}Dg$ is, for
$g\in G$, clearly also a $(V,\hskip.2mm\cdot\hskip.8mm)$-linear derivation from $\der 1$
and $(g^{-1}Dg)(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset (\bigodo^{\geq i,j}(V,{\cal B}_\b(V)))$.
We may thus decompose $g^{-1}Dg$ as $g^{-1}Dg = \sum_{k\geq -1}D'_k$
with $D'_k(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset \bigodo^{i+k+1,j}(V,{\cal B}_\b(V))$. We
observe that $D'_{-1}=D_{-1}$, $D'_0=D_0$ and that, from degree
reasons, $D'_k(\bigodo(V,{\cal B}_\b(V))^{i,j})\subset \bigodo(V,{\cal B}_\b(V))^{i+k+1,j-k}$.
This means that $D'_k \in J_k(k+2)$ for $k\geq 1$. The equation
$[g^{-1}Dg,g^{-1}Dg]=0$ is immediate, therefore the correspondence
$(g,\{D_i\}_{i\geq 1}) \mapsto \{D'_i\}_{i\geq 1}$ defines the
requisite action. The following proposition shows that this action is
transitive.

\begin{proposition}
\label{16}
Let $\tilde D' = \{D'_i\}_{i\geq 1}$ and $\tilde D'' =
\{D''_i\}_{i\geq 1}$ be two integrations of an infinitesimal
deformation $D_0$. If we denote $D' := D_{-1}+D_0+\sum_{i\geq 1}D'_i$
and $D'' := D_{-1}+D_0+\sum_{i\geq 1}D''_i$, then $D' = g^{-1}D''g$
for some $g\in G$.
\end{proposition}

\noindent{\sc Proof. }
Again standard obstruction theory.
As we already observed, for any $g\in G$, $g^{-1}D''g$ decomposes
as $g^{-1}D''g = D_{-1}+D_0+\sum_{i\geq 1}\{g^{-1}D''g\}_i$ with some
$\{g^{-1}D''g\}_i \in J_i(i+2)$. Let us suppose that we have already
constructed some $g_N \in G$, $N\geq 1$, such that
$\{g_N^{-1}D''g_N\}_i= D'_i$ for $1\leq i \leq N$. We have
\[
\mbox{$\nabla(D'_{N+1})-\frac12 \sum_{i+j=N}[D'_i,D'_j]=0$}
\]
and, similarly,
\[
\mbox{$\nabla(\{g_N^{-1}D''g_N\}_{N+1})-\frac12
\sum_{i+j=N}[\{g_N^{-1}D''g_N\}_i,\{g_N^{-1}D''g_N\}_j]=0$.}
\]
By the induction, the second terms of the above equations are the
same, therefore $\nabla(D'_{N+1}) = \nabla(\{g_N^{-1}D''g_N\}_{N+1})$
which means that $D'_{N+1} -
\{g_N^{-1}D''g_N\}_{N+1}\in J_{N+1}(N+3)$ is a cycle. The triviality
of $H_{N+1}(J_*(N+3))$ (Proposition~\ref{12}) gives some
$\phi \in J_{N+2}(N+3)$ such that $D'_{N+1} -
\{g_N^{-1}D''g_N\}_{N+1}= \nabla(\phi)$.
The element $\mbox{\rm exp}(\phi)\in G$ is of the form $\id +
\phi +\phi_{\geq N+3}$ with
\[
\phi_{\geq N+3}(\bigodo^{i,j}(V,{\cal B}_\b(V)))\subset
\bigodo^{i+N+3,j}(V,{\cal B}_\b(V)),
\]
therefore $g_{N+1}:= \mbox{\rm exp}(\phi)g_N$ satisfies
$\{g_{N+1}^{-1}D''g_{N+1}\}_i= \{g_{N}^{-1}D''g_{N}\}_i
=D'_i$ for $1\leq i \leq N$ and $\{g_{N+1}^{-1}D''g_{N+1}\}_{N+1}=
\{g_{N}^{-1}D''g_{N}\}_{N+1}+\nabla(\phi)= D'_{N+1}$ and the
induction goes on. The prounipotency of the group $G$ assures that
the sequence $\{g_N\}_{N\geq 1}$ converges to some $g\in G$ as
required.
\hspace*{\fill

\section{Applications to Drinfel'd algebras}

In~\cite{markl-shnider:preprint} we
introduced two $(V,\hskip.2mm\cdot\hskip.8mm)$-linear `coactions' $\lambda
:{\cal B}_\b(V) \to V\odot {\cal B}_\b(V)$ and $\rho : {\cal B}_\b(V) \to {\cal B}_\b(V)\odot V$ as
\begin{eqnarray*}
\lambda(a_0|\cdots|a_{-n+1})&:=&
\sum\Delta'(a_0)\cdots\Delta'(a_{-n+1})\odot
(\Delta''(a_0)|\cdots|\Delta''(a_{-n+1})),\mbox{ and}
\\
\rho(a_0|\cdots|a_{-n+1})&:=&\sum
(\Delta'(a_0)|\cdots|\Delta'(a_{-n+1}))\odot\Delta''(a_0)\cdots
\Delta''(a_{-n+1}).
\end{eqnarray*}
Let us define a derivation $D_0\in J_0(2)$ by
\[
D_0|_{\uparrow\kern -.05em {\cal B}_\b(V)}:= (\uparrow\kern -.05em\bigodo\uparrow\kern -.05em)(\lambda+\rho)(\downarrow\kern -.05em)
\mbox{ and }
D_0|_{\uparrow\kern -.05em V}:= (\uparrow\kern -.05em\bigodo\uparrow\kern -.05em)(\Delta)(\downarrow\kern -.05em).
\]

\begin{proposition}
The derivation $D_0$ defined above is an integrable infinitesimal
deformation of $D_{-1}$.
\end{proposition}

\noindent{\sc Proof. }
To prove that $D_0$ is an infinitesimal deformation of $D_{-1}$ means
to show that $\nabla(D_0)= [D_{-1},D_0]= 0$.
This was done in~\cite{markl-shnider:preprint}.

The integrability of $D_0$ means, by Proposition~\ref{13}, that
$[D_0,D_0]|_{\uparrow\kern -.05em V}= [D_0,D_0]|_{\uparrow\kern -.05em {\cal B}_1(V)}=0$. For $v\in V$
we have
\begin{eqnarray*}
[D_0,D_0](\uparrow\kern -.05em v)&=& (D_0 \circ D_0)(\uparrow\kern -.05em v)= D_0 (\uparrow\kern -.05em
\odot \uparrow\kern -.05em)(\Delta)(\uparrow\kern -.05em v)=
\\
&=&[((\uparrow\kern -.05em \odot \uparrow\kern -.05em)(\Delta)\odot \uparrow\kern -.05em)(\Delta)-
(\uparrow\kern -.05em \odot(\uparrow\kern -.05em \odot \uparrow\kern -.05em)(\Delta))(\Delta)](\uparrow\kern -.05em v) =
\\
&=&(\uparrow\kern -.05em\odot\uparrow\kern -.05em\odot\uparrow\kern -.05em)\circ [(\Delta\odot \id)(\Delta)-
(\id \odot \Delta)(\Delta)(\uparrow\kern -.05em v)]
\end{eqnarray*}
which is zero by the quasi-coassociativity~(\ref{14}) and
by~(\ref{3}).

Similarly, for $(v)\in {\cal B}_1(V)$ we have
\[
D_0^2 = (\uparrow\kern -.05em \odot \uparrow\kern -.05em \odot \uparrow\kern -.05em)[ (\Delta\odot \id)\lambda -
(\id \odot \lambda)\lambda -(\id \odot\rho)\lambda
+ (\lambda \odot \id)\rho + (\rho \odot \id)\rho
-(\id \odot \Delta)\rho ](\downarrow\kern -.05em (v))
\]
and~(\ref{14}),~(\ref{3}) again imply that this is zero.
\hspace*{\fill

\begin{definition}
\label{15}
An integration of the infinitesimal deformation $D_0$ above is called
a homotopy comodule structure.
\end{definition}

Let $\bigodo'(V,{\cal B}_\b(V))= \bigodo^{*,1}(V,{\cal B}_\b(V))$ denote the submodule of
$\bigodo^{*,*}(V,{\cal B}_\b(V))$ with precisely one factor of $\uparrow\kern -.05em {\cal B}_\b(V)$.
Let $C^n(A)$ be the set of all degree $n$ homogeneous maps $f:
\bigodo'(V,{\cal B}_\b(V)) \to \bigodo^*(\uparrow\kern -.05em V)$ which are both $\bigodo^*(\uparrow\kern -.05em V)$
and $(V,\hskip.2mm\cdot\hskip.8mm)$-linear. Let us define also a degree one derivation
$d_C$ on $\bigodo^*(\uparrow\kern -.05em V)$ by $d_C|_{\uparrow\kern -.05em V}:=
(\uparrow\kern -.05em \odot \uparrow\kern -.05em)(\Delta)(\uparrow\kern -.05em)$.

Let $\{D_i\}_{i\geq 1}$ be a homotopy comodule structure in the sense
of Definition~\ref{15} and let $D:= D_{-1}+D_0+D_1+\cdots$. Define a
degree one endomorphism $d$ of $C^*(A)$ by $d(f):= f\circ D+ (-1)^n
d_C\circ f$. It is easy to show that $d$ is a differential and,
following~\cite{markl-shnider:preprint}, we define the
{\em cohomology of our Drinfel'd algebra
$A$\/} by $H^*(A):= H^*(C^*(A),d)$.

\begin{proposition}
The definition of the cohomology of a Drinfel'd algebra does not
depend on the particular choice of a homotopy comodule structure.
\end{proposition}

\noindent{\sc Proof. }
Let $\{D'_i\}_{i\geq 1}$ and $\{D''_i\}_{i\geq 1}$ be two homotopy
comodule structures, $D':= D_{-1}+D_0+D'_1+\cdots$ and
$D'':= D_{-1}+D_0+D''_1+\cdots$. Let $d'(f):= f\circ D'+ (-1)^n
d_C\circ f$ and $d'(f):= f\circ D''+ (-1)^n
d_C\circ f$. Proposition~\ref{16} then gives
some $g\in G$ such that $D' \circ g = g \circ D''$. We see
immediately that the map $\Psi :(C^*(A),d') \to (C^*(A),d'')$ defined
by $\Psi(f):= f\circ g$ is an isomorphism of complexes.
\hspace*{\fill

redpajama_set_name: RedPajamaArXiv


text:
Stevie Wonder has had a unique career as a singer, composer, instrumentalist, performer and humanitarian. After his early years with Motown, who signed him up as 'Little Stevie Wonder' aged eleven, he gradually released a striking array of innovative music that would influence the world of contemporary music as we know it today.
Born Stevland Hardaway Judkins (later Morris) on May 13, 1950, two months premature, Stevie was placed in an incubator for life support. However an excess of oxygen resulted in the scarring and detachment of his retina leading to a condition known as Retinopathy of Prematurity.
In 1963, he had his first US number 1 hit single and album with the first live recording to ever top the charts. He is still the youngest person to have reached those number 1 spots.
FINGERTIPS, AGED 12
During the 60's, under the control of the Motown formula, Stevie evolved from the status of child prodigy to music super stardom. With a string of hits that progressively involved more and more of his song writing, musicianship and production skills he entered the 70's on a solid foundation to create some of the most enduring music of the century.
UPTIGHT, 1965 (GRAMMY AWARD NOMINATION, 1967)
FOR ONCE IN MY LIFE, 1965 (GRAMMY AWARD, 2007 ON RE-RELEASE)
At age 21, Stevie broke away from the Motown hit machine and demanded full artistc control for creating and producing his own music. This resulted in an impressive array of albums that are still considered ground breaking achievements.
I WAS MADE TO LOVE HER, 1969
UNTIL YOU COME BACK TO ME
With this new found freedom in the early 1970's Stevie pioneered the use of synthesizers, incorporating electronic sounds seamlessly with acoutic sounds. This venture hugely influenced the face of popular music and the development of electronic musical genres.
SIGNED, SEALED, DELIVERED, I'M YOURS, 1970 (GRAMMY AWARD NOMINATION, 1971)
In 1973 he won a Grammy for the best album of the year, a feat that was repeated for 3 consecutive albums. He has since gone on to win 24 Grammys and be awarded the Grammy Lifetime Achieviement Award in 1996. He has also picked up Academy and Golden Globe awards for his soundtracks to movies.
SUPERSTITION, 1972 (GRAMMY AWARD, 1974)
LIVING FOR THE CITY, 1973 (GRAMMY AWARD, 1975)
INNERVISIONS 1976, ALBUM OF THE YEAR GRAMMY AWARD
SONGS IN THE KEY OF LIFE ALBUM, 1976 (TWO GRAMMY AWARDS, 1977)
I WISH, 1976, BEST RnB MALE VOCAL PERFORMANCE GRAMMY AWARD, 1977
MASTER BLASTER (JAMMIN) 1980, GRAMMY AWARD NOMINATION 1981
IN SQUARE CIRCLE, 1985, BEST RnB MALE VOCAL PERFORMANCE, 1986
He was inducted in the Rock and Roll Hall of fame in 1989, the youngest solo artist to achieve that feat and is also the youngest individual to have been awarded with the Kennedy Center Honour in 1999.
FOR YOUR LOVE, 1995 (GRAMMY AWARD, 1996)
The results of his amalgamation of electronic and acoustic music with unique rhythms and melodies overlaid with poignant lyrics spanning every conceivable life situation has made him an absolutely pivotal figure in modern music.
The longevity of his music is evident in so many facets of our life's, where his melodies and lyrics are frequently sampled/covered on records and movies and has gained him the respect of his peers in the entertainment industry.
However behind the music scene, he is committed to many humanitarian activities such as Dr King Jnr.'s National Holiday, fight against Apartheid in South Africa, fundraising for AIDS research, numerous Children's and Cancer charities and always available for Disaster Relief fundraising. Stevie has also held his House Full of Toys annual charity concert since 1995 to raise funds for underprivileged children during the festive Christmas season.
He recently played a pivotal role in the election of Barack Obama to the presidency of the United States, performing and speaking at many rallies and offering his song, Signed Sealed Delivered, as the theme of the campaign.
In 2009 Stevie was awarded the Gershwin Prize for Popular Music by the US Library of Congress, again the youngest person to achieve this award, which was presented to him at the White House by President Obama. In addtion Stevie was commissioned by the Library to write a classical piece, an honour not normally bestowed on popular music artists.
Currently a United Nations Messenger of Peace, Stevie has been able to win the rights to publish copyrighted works into formats accessible to the visually impaired throughout the world.
Stevie Wonder is without question one of the great personalities in music history. His powerfully expressive vocal presence, coupled with his musical elegance, forms part of the broad, personal spectrum of irresistible music and compelling emotion which is his personal hallmark.
Stevie's back catalogue and his musical legacy is unparalleled and like Michael Jackson, Marvin Gaye and Bob Marley, it is highly improbable that we will see his kind again. 26 albums, 25 Grammy Awards and countless collaborations coupled with timeless, sublime musical talent make Stevie a King amongst musicians.
Source: steviewonder.org.uk
Blazingseollc March 31, 2022
I absolutely love your blog and find many of your post's to be just what I'm looking for. Would you offer guest writers to write content to suit your needs? I wouldn't mind creating a post or elaborating on many of the subjects you write regarding here. Again, awesome site!
Previous post AL GREEN
Next post SAM COOKE

redpajama_set_name: RedPajamaCommonCrawl


I've searched quite a bit regarding the error I'm seeing in OpenOCD, when connecting to the Particle Programmer Shield and the Photon. I haven't been able to get past this for weeks.
Are there any tips or hints on how to troubleshoot this error? Even narrowing it down between firmware build, hardware error, configuration, etc. would help.
I attach an ARM GDB debugger to port 3333 on OpenOCD for remote debugging. In the simplest case, whenever I try to examine memory, I receive a JTAG-DP STICKY ERROR. I can examine registers fine, continue execution, etc.
openocd -f particle-ftdi.cfg -f stm32f2x.cfg -c "gdb_port 3333"
I'll see the following error on the OpenOCD console.
And I'll get back zero's for memory content in the debugger.
Any idea? Where could I look? I've tried different adapter speeds (ex: 500, 1000, 10000).


Do you all too often get an OM (out of memory) error? Are you tired of waiting for a pro- gram to load or to get a few dozen data files into memory? On the other hand, maybe your Level II, 16K system is now very, very reliable and you are afraid of the problems associated with the Radio Shack expansion in- terface. There is another way.
LNW Research offers a circuit board and a manual for $69.95. It claims to have all of the features of the Radio Shack expansion chassis and then some. Also mentioned is software compati- bility. They further claim a "quiet bus" design that elimi- nates some of the memory prob- lems.
After receiving additional data from LNW, I carefully com- pared their specs with those of Radio Shack. It appeared that LNW's board would do every- thing that Radio Shack's does, plus the serial interface was not an accessory that connects via- a poor connector, as is the Radio Shack version. The board had heavy power buses and lots of bypass capacitors.
I ordered the kit last spring and received it only 30 hours later. I spent some time compar-ing LNW's circuitry with that of Radio Shack. Anyone building the LNW board should purchase the expansion interface hard-ware from Radio Shack. The manual has a good parts list. Actually, the parts are listed in several ways. The system al-lows partial construction for those who don't need all the fea-tures. For instance, if you only want additional memory, and don't need a floppy disk, the manual explicitly tells you what parts to buy. On the other hand, parts are sorted alphanumeri-cally to make it easy to order and inventory. The hardest item to find was the cable. Radio Shack wanted $25 (love/hate... hate this time). One supplier had both ribbon connectors with one end termi-nated, and also40-pin soldertail connectors, so out with the sol-dering iron! This is the part of the job I really hated. I detest symmetrical conneators that can be wired and plugged in up-side down. These dumb un-keyed connectors are the worst design feature of all.
Anyway, all the parts finally, were found from three suppliers, not including Radio Shack. Many of the parts (such as bridge rectifiers, SCRs) were called out as Radio Shack "276-" type part numbers.
At a local parts store I found a 16" x 13" x 3" chassis. I wanted lots of space for a good power supply, a modem, and an inter-face for my ham radio. While having these mounted on a sin-gle chassis makes a neater as-sembly, my real objective was to keep radio interference to a minimum.
keyboard. Check at your local Radio Shack to see if they have a bad one laying around. Chances are they do and will sell it to you for less than $5.00. You can go home, pry it apart and replace the fuse. A sche-matic is included in the con-struction manual.
Fig. 1. This circuit will take the place of the Radio Shack power sup-ply. The power line filter generate transients of sufficient magnitude to activate the crowbar circuits if the GE-750 is not installed. Even larger transformers may be purchased if you anticipate using floppy disks and a common supply.
pins on the sockets were ta- pered so that they barely ex- -tended through the board. This made inspection of soldering hard, but I managed by going slower than I normally would. There was nothing else tricky or unusual about assembling the board.
While the manual directs you to plug in the chips prior to pre- liminary power checks, certain jumpers are not installed. With these removed, no power is ap- plied to the chips. Follow the di- rections exactly. This section performs voltage checks to find errors or bad parts that could cause damage.
When power was first applied to my unit, both fuses blew. This was caused by the crowbar cir- cuit, which is designed to short the power supply to ground if the regulator fails. Testing de- termined that turn-on transients generated by the line filter were to blame.
The problem was corrected by installation of a metal oxide varistor (MOV), General Electric Part No. GE-750. This compo- nent draws almost no currerit at 120 V ac but at higher voltages the resistance decreases dra- matically so that spikes are shorted out. The MOV totally coriects the problem. The short probably would not have hap- pened if I had used the Radio Shack power supply, but a MOV is good to use across the ac line of any sensitive equipment. After this was fixed, all power supply voltages checked. I in- stalled the jumpers, powered up the chassis, and rechecked the power supply voltages. I then re- moved power and connected to my keyboard.
since the TRS-80 has ROM resi-dent routines that automatically check and .size the memory. With the additional 32K, it takes longest. I then eritered PRINT MEM and the computer returned 48340, the correct value for a 48K machine. The next thing I tried was to PEEK and POKE in-to the new memory locations. Attempts to POKE or PEEK above 32767 (top of memory for a 16K machine) returned an OV erior (overflow error).
While I thought it interesting that the system recognized the new memory, the fact that I couldn't PEEK or POKE into it convinced me that there was a hardware problem. Out came the old oscilloscope. I pulled aut all the chips except those to support the new memory. One came out; socket and all; I had completely missed soldering it to the board. While this would not have caused my problem, I pulled the board and carefully inspected for missing parts and integrity of soldering. Finding no further problems, I then pow-ered up and continued trouble-shooting. After hours of exten-sive testing, I tried loading some machine language programs in-to the new memory. .Radio Shack's RENUM con-tains modules for 16, 32 and 48K machines. When I tried the 32 and 48K versions, they loaded and worked. I then loaded T-BUG and found I could wiite and read to high memory ad-dresses with no problem. RSM has a memory test option, and this ran faultlessly. I began to suspect cockpit errors. After more troubleshooting, I got out my TRS-80 owner's manual, and found that to POKE and PEEK above 32767, you have to sub-tract 32767 from the desired memory address. Thus to POKE X into 32768, you POKE(32767 – 32768),X or POKE – 1,X. I tried it, and of course it worked.
ly off the expansion port, such as a page printer. The bus termi-nation resistors are installed near one of these connectors. The manual suggests that these resistors may not be necessary if you are using a device on the expansion connector that has its own termination networks. l. installed the termination resis-tors and have experienced no problems.
The real-time clock is only of limited use without a disk oper-ating system. Various programs such as the one on page 21 of Radio Shack's Expansion Inter-face Hardware Manual can be used to test the clock. The pro-gram seemed to work well but the clock gained about three minutes in ten hours. Using my frequency counter, I checked the frequency of the internal 4 MHz xtal. oscillator. and found it running high.
right on frequency and made the clock accurate. The exact value varies with the tolerance of the crystal. If the frequency is high, the value of C15 should be in-creased, if low, vice-versa. Ac-tually this whole effort is unnec-essary since the clock loses time anyway during disk read and write operations. The clock has proven useful in disk opera-tion for timing program events in BASIC programs.
'A', etc. the system will energize the last recorder selected from previous cassette operations.
hase the additional parts, but the cost is less than $15.
The line printer port, like every other LNW feature, works like Radio Shack's and is designed to work with a parallel printer, ' using the Centronics interface. For those using a serial printer, the serial interface can drive it; With no modifications to the printer or the LNW board, you can set up the serial interface .for the right baud rate and punch in a software routine that will link the LLISTs and LPRINTs to the serial interface. The driver software program is included in the. manual.
A better way is to follow the .instructions of Chapter 6 which route signals that would normal- ly go out .the normal parallel printer port through the serial port. This allows use of the LPRINT and LLIST commands without any special software. A switch is provided to allow the serial interface to be configured back to work with something like a modem.
A still better approach would be to build a separate parallel to serial converter which would plug into the printer port and convert to the type of serial level (usually RS-232) required by the printer. While either of the above techniques work well, I wanted to leave the serial port alone for . a modem in the future. I con-structed a small wirewrap card that is mounted inside and de-rives power from my Heath H-14 line printer. Since it plugs into the normal printer port, LLISTs and LPRINTs work the same as they do using a Radio Shack or Centronics printer.
Those of you who are handy with a wirewrap tool can receive a schematic of what I am using by sending me five dollars. The schematic also has optional cir-cuitry to operate a Heath H-14 directly off the expansion port of an unmodified 16K Level II ma-chine.
running for almost a month be-fore my Pertec drive arrived. Dur-ing this time I ran every test I could to ensure that it was work-ing.
The Pertec box was ripped open and the four-drive cable quickly connected to the drive. Without pausing to read any di-rections, I plugged it into power and to the chassis interface. I then inserted the TRSDOS 2.3 system disk that I had pur-chased and turned everything on. The drive came on, the screen went blank, and up came DOS READY.
You may not care about this if you are not a radio amateur or shortwave listener. The TRS-80 generates a lot of energy in the radio frequency spectrum. It seems to be the worst in the 40 meter region. Someday I am go-ing to hook my antenna tuner up to it and I probably will be able to talk to. Japan if I can figure out some way to key it.
It seems that every accessory added causes the noise level to come up. I built my LNW board in a large metal box with the in-tent of building in my M80 (ham radio interface). This would eliminate one cable as a radiat-ing source. The line filter also helps. I have done nothing to correct the radiation caused by the TRS-80 but I do not observe any increase in noise when the expansion interface is connect-ed or disconnected.
I have tried NEWDOS+, NEWDOS 80, Electric Pencil, Percom's DOS, etc., etc. All run exactly as on a standard TRS-80. The LNW system does not expe-rience the mysterious system crashes that some of my friends occasionally have.
The LNW board performs. It is "totally hardware and software compatible with Radio Shack products designed for the TRS-80. While the manual is good, the project is such that I would recommend it only for the advanced kit builder.


text:
build:
	docker build -t jgeiger/vault .

redpajama_set_name: RedPajamaGithub


Oct. 19, 2012 • Landsat satellites have witnessed over four decades of changes on Earth. In advance of the next Landsat spacecraft launch, the Landsat Data Continuity Mission, the U.S. Geological Survey (USGS), in cooperation with the National Aeronautics and Space Administration (NASA), announces the selection of the Landsat Science Team. This expert team of scientists and engineers will serve a five-year term, from 2012-2017, and provide technical and scientific input to USGS and NASA on issues critical to the success of the Landsat program.
Since 1972, the United States has acquired and maintained a unique, continuous record of the global land surface. This impartial record has become indispensable for detecting and monitoring natural and human-induced changes to the Earth's landscape.
The Landsat Data Continuity Mission (LDCM), which will become Landsat 8 following launch in February 2013, is designed to extend Landsat's comprehensive global record for at least five years.
As recognized national and international leaders in land remote sensing, Landsat Science Team members will evaluate operational and data management strategies to meet the requirements of all Landsat users, including the needs of policy makers at all levels of government. They will play a key role in ensuring that the LDCM mission is successfully integrated with past, present, and future remotely sensed data for the purpose of observing national and global environmental systems.


Minoru Sasazuka has got a handsome physique that should make him popular with girls, but his overdone romanticism drives them away. He's now age 19, at university, and still a virgin, so he hopes to find the right girl soon.
He and his friends decide to rent a cabin in the mountains and have a winter ski trip. For his friends it's a dream vacation, but for Minoru it's like going back home, since it's a place he frequented often as a kid. Traveling with him is Kyoka, a childhood friend, her stylish sister, Keika, and his sister, Honami.
At the lodge, they enjoy the hospitality of the owner, Kasumi, a beautiful but soft-spoken woman, and the lovely yet mysterious helper, named Shizuka. As the snow begins to fall, love starts to develop and hearts begin to melt. Who will fall in love with Minoru, and will she be "the one" for him?
In Snow Drop you experience love as pure as the winter snow, that heats up the cabin where you and your friends are staying. But, there is more than just love in these mountains, and it's up to you to uncover the secrets. A beautiful story with memorable characters, in this classic romantic eroge.
She's the childhood friend of Minoru, and is very energetic and tomboyish. Although they're childhood friends, there's no romantic involvement between them...yet. Still, she constantly teases him and puts down his hopelessly romantic tendencies, promoting her own "super-realist" point of view. Age 19.
The newly hired maid at the ski lodge where Minoru and the others are staying for vacation. She is very helpful and polite, and seems strangely taken with Minoru. Unlike Kyoka, she seems to carry an aura of romantic tranquility with her. Minoru is beset with a puzzle as to why she seems so familiar... Age ???
The owner of the ski resort. She has a great deal of knowledge about the local area because it's her hometown. Her 'hobby' is creating Ghost Stories to tell to her guests to scare them out of their wits before bedtime. Her husband has always run the resort with her, but he seems to be missing... Age 28.
Minoru hopes a ski trip to the mountains will be the opportunity for love to blossom in this memorable winter tale.


PITTSBURGH -- Minnesota Wild forward Matt Cooke and coach Mike Yeo have fond memories of their time in Pittsburgh, but they are not looking to the past.
Cooke and Yeo, members of the 2008-09 Stanley Cup champion Pittsburgh Penguins, will return to Consol Energy Center for the first time since exiting the Penguins organization. Each said he will take a moment to reflect on his tenure with Pittsburgh, but that won't linger after the puck drop on Thursday.
"It's always emotional," Cooke said. "Especially having success, winning a Cup here, the fans really make you feel like you're a Pittsburgher forever when you win here. They supported me through tough times too. So, I have a special place for Pittsburgh.
"But unfortunately, this is a business and I play for the Minnesota Wild now and as emotional as it is to see everybody for the first time, today I have to go help my team win a hockey game."
Cooke, who spent the past five seasons with the Penguins before signing with Minnesota during the offseason, was one of Pittsburgh's more physical forwards, but became a fan favorite after adjusting to a more disciplined style of play.
He has carried that discipline to Minnesota and is sixth on the team with 12 points (five goals). Forward Mike Rupp, Cooke's teammate for two seasons in Pittsburgh, said there is a noticeable difference between the way Cooke played then and the way he performs now.
"He provides a lot on the ice," Rupp said. "He's one of those guys who obviously kills penalties and adds experience. So, you can't have enough of those guys and at the beginning of the year, he was adding to the offense too. Those are all good things that you want to add to any team.
"I wasn't surprised [that Cooke changed his game] from knowing him. I've talked to him about the steps he's done and I think everyone knows the steps he's done. He's a guy that when he decides to do something, he's a student of the game and wants to learn how to be better."
Yeo, a Penguins assistant coach from 2005-2010, said Cooke brought the same positive traits he remembers him having in Pittsburgh to Minnesota.
"He's a guy we kind of looked and said we identified a need and he would fit it," Yeo said. "We were looking for a guy who would play a real important role in our penalty kill, a guy who could play against the top players, a guy who's going to bring you momentum and obviously the speed and physicality that he brings.
"The fact that he's playing as disciplined as he is a bonus for us."
Cooke credited Yeo with making his transition to the Wild seamless.
"Having Mike's trust, knowing him and being able to come into a situation where the system was fairly familiar, I think allowed me to be successful right away," Cooke said. "It wasn't a big transition and that was important for me in my decision making in July."
After coaching Penguins captain Sidney Crosby for five seasons, Yeo called the thought of coaching against Crosby, who leads the League with 49 points, "scary," but thinks he is prepared due to a coaching style similar to Penguins coach Dan Bylsma.
"It's hard to put it on one or two things [learned from Bylsma]. Obviously, I can point to Xs and Os and other things strategically," Yeo said. "The way [Bylsma] conducts himself, the way he communicates, the way that he prepares his teams, those are things that stick out most."
The Wild have earned consecutive shootout victories and are in eighth place in the Western Conference. Yeo said he is more focused on winning the game Thursday than reminiscing about his time with the Penguins.
"For me personally, I think enough time has passed," Yeo said. "Obviously, it's exciting to come in and see the faces, but it's an important game for our group, for our team. We want to go into Christmas break feeling good about ourselves and we've won a couple games in a row, we'd like to keep that going."
Despite playing the majority of his career with the Vancouver Canucks, Cooke enjoyed his greatest success while with the Penguins. He had more than 30 points in four of his five seasons with Pittsburgh (he had 21 during last season's 48-game regular season). He surpassed that mark in three of his nine seasons before joining the Penguins.
But his view on the game Thursday is similar to Yeo's.
Cooke said he would like to have a tribute video to be played, similar to one played for his former Pittsburgh linemate, San Jose Sharks forward Tyler Kennedy, earlier this season. But he said his primary motivation is beating the Penguins.
"I need to go out and prepare myself to play a game. That's what's most important to me," Cooke said. "Obviously, I hope I don't get booed. But it wouldn't be the first time, probably won't be the last."


text:
Discover Resources Electronic periodicals
Journal of Contemporary Central & Eastern Europe
A scientific journal running in Great Britain and publishing analytical articles, reviews, studies, discussions and book reviews related to social matters in Central and Eastern Europe: politics, economics and culture. A printed version is also available in the library.
The oldest news journal in Germany, publishing political and social news and analytical articles. A printed version is also available in the library.
Published since 1925 in New York, the United States, The New Yorker is a weekly magazine offering coverage of both American and global news, commentary on political, social and cultural topics, modern literary texts and reviews. The magazine has won a Peabody Award (2012), a GLAAD Award (2014) and a Pulitzer Prize (2018). A printed version is also available at the library.
You can read the publications below using any desktop computer in the National Library's reading rooms. To sign in as a registered user, please approach the library counter staff for assistance.
More foreign electronic periodicals are available in the subscribed databases.


text:
Working at KEMP Technologies
Kemp Technologies: where technology and enthusiasm collide
The five minute CIO: Simon Roach, KEMP Technologies
SDN will bring cohesion to the networking world – KEMP co-founder (video)
SDN Focus: Interview with KEMP Technologies' CMO Atchison Frazer (video)
Limerick life and working at KEMP Technologies (video)
Limerick's got talent, KEMP Technologies wants more of it
KEMP Technologies to create 50 new jobs in Limerick
22 Aug 2014532 Views
Simon Roach, CTO at KEMP Technologies
"We are growing incredibly rapidly and scaling our development teams is the biggest challenge," says Simon Roach, chief technology officer at KEMP Technologies.
US load balancing company KEMP Technologies is to create 50 new jobs in Limerick over the next three years, bringing its current Irish workforce to 80.
The company, which has seven offices worldwide, with Limerick heading up the EMEA region, is a developer of load balancers that enable all types of SMEs and global groups to optimise their e-commerce and online traffic capabilities, such as online banking, email, web and cloud-based applications.
The company also intends to invest even more resources to accommodate its rapid expansion in the Limerick's National Technology Park.
What are the big trends and challenges in your area?
I'm based in Munich. We obviously have the R&D facility in Limerick and I spend a fair bit of time there and in an office in Santa Clara and we have a core dev team here in Munich.
There would be 35 people in the core R&D team and the worldwide support team report to me additionally. We have an Irish guy who runs the worldwide support team who is based in Long Island, New York, and we located him from Limerick to Manhattan, which must have been a culture shock.
We are spread across locations in Long Island, Limerick and Singapore – we follow the sun.
We are growing incredibly rapidly and obviously scaling the development teams is one of the big challenges we've been working on for the last 24 months and Limerick has been very, very good for us so far.
What factors led to KEMP deciding to make Limerick the company's EMEA headquarters?
The founder of KEMP had chosen Limerick back in 2009 and obviously we wanted to have a European dev base and an EMEA headquarters and Ireland was the lowest-hanging fruit in the sense that the language was the same. That was a big factor from a location point of view. The UK would have been the other option, but the close vicinity of the universities, such as Limerick Institute of Technology (LIT) and University of Limerick (UL), whom we work with, was a big deciding point.
The relationship and the way that LIT, in particular, has engaged with us has just been unbelievable. We've taken on a large number of graduates – five or six students in their final year doing projects with us – this is the fourth year's intake we've had in and most of them have joined KEMP once they have graduated. They work in software engineering, software development and our customer support teams.
We have been getting first pick of the next batch of students from the network and computer science courses based on good experiences that people had.
One of the things we do quite well is we are very respectful to undergraduates when they come into the company and we treat their bachelor's projects as if they were KEMP internal projects so they treat it as a professional project. We track it through all of our processes and give them the time and the focus that it deserves and that certainly has persuaded a number of guys to come back.
A couple have got first-class honours degrees based on their good projects. That has worked very well.
It is working less well on Long Island and we are only now starting to engage more with some of the colleges and universities there and using the model that we developed in Limerick as a kind of a guiding light.
We want to expand this to Cork, as well, and engage with Cork Institute of Technology (CIT) and University College Cork (UCC).
In terms of acquiring fresh talent, it has been great. Obviously, acquiring more experience in the short term is starting to become quite tricky. The past 18 months has become very difficult. It is a worldwide phenomenon, it just feels particularly difficult in Ireland because of the size of the population.
How complex is the infrastructure, are you taking steps to simplify it?
It's complex but it comes down to skills ultimately. First of all, getting the kind of people we need fresh and trained up is very, very difficult. If I can give our QA processes and the way we go about quality assurance at KEMP you will find there are not many methodologies out there that suit us and not many companies do it probably as well as we strive to. There aren't many people with finished skills.
If you look at it from a development point of view, we are looking at people with Linux skills, people who have worked on the Linux kernel, people who worked with protocol stacks, TCP/IP, people who've worked on the support side with the kind of enterprise workloads we support, such as Lync, Oracle, SAP, Exchange, so depending on where we bring people into the organisation there's a varying skill set that is required.
What are the main points of your company's development strategy?
We have a strategy in the R&D department which is built on three pillars and we call them our technology core values, like companies have core values on behaviour and so forth.
Technology core values sounds corny but it works for us, it gets everyone on the same page.
The three areas are: platform ubiquity, interoperability and API economy, and the third area is platform richness.
The idea of that within the area of application delivery is that it is not just load balancing, but the broader area of application delivery.
On the ubiquity value we are looking at porting software because primarily we are a software company. To the outside world we are a company that provides gold boxes, appliances and virtualised software, but internally we've always been a software company.
We build an operating system that is standardised, we put that out on standard x86 hardware, which in some cases is accelerating, but fundamentally the way that the company ticks is the way that most software companies would tick. What we are trying to do with our operating system KEMP OS, which is a Linux-based operating system, is to make that available on as many platforms and as many creatures as possible.
How diverse is the range of platforms you are addressing?
Today we provide our ADCs on gold appliances, which was the traditional business. We are also starting to provide them as bare metal – you can buy a bare metal OS from KEMP that will run on C platforms, HP, Oracle, or a Solaris-based system, it will run on a number of diverse systems and then of course we have running the OS on the cloud-based operating systems and hypervisors, so there is a very broad-based operating system component here that we need to get in place.
Then we have the whole API economy and interoperability stuff where we starting to find ways of better embedding our technology to other environments, be it orchestrators with OpenStack and System Centre, SDN and NFV, as well.
SDN is an area we are looking at for people at the moment. We hired a young guy from LIT and he's been with us for six months on the support side and he's a very good programmer with some great networking skills and we are in the process of building an SDN integration test lab in Limerick and that is one of the things he will be working on.
But to complete the three values – platform expansion – we want to expand the platform into other areas, tick more security boxes. We launched a security pack last year and we are in the process now of bringing a WAF application firewall and tailoring our WAF approach and making it application centric.
That is one of the strategies we always followed at KEMP as opposed to competitors – putting big pieces of IT infrastructure in place that can be used by everybody for all sorts of processes.
It has been said that if software is eating the world, APIs are eating software. What challenge does this create for KEMP?
You've hit it on the head – our ubiquity is about supporting as many platforms as possible and we support something like 14 or 15 platforms at the moment so whenever we make a release on an eight-week cycle we try to maintain this religiously – this includes 13 or 14 OS platforms every time we release something, two cloud platforms, a large number of hypervisors and bare metal platforms for Dell, Cisco, HP and Oracle.
Every time we go through a release cycle one of the big challenges is making sure our software operates on all those platforms. It was clear from the outset that we had to create a highly automated test lab so from day one we put a lot of effort into process automation and test automation, and we've got literally hundreds of thousands of test cases that have been coded and programmed.
It is a continuous challenge to support more platforms and broaden and deepen the test cases, in particular, adding functionality and the APIs is the big challenge.
We are a load balancer, in the middle – above us are orchestrators and below us platforms like SDN controllers, and just coming to terms with the number of these APIs is a challenge. So supporting the OpenStack, being involved in the committees, being a member of OpenStack, the orchestration side is challenging. We have our own Restful API and try to adapt that – but just tracking all of this a challenge. A challenge we relish all the same.
KEMP Technologies is a Silicon Republic Featured Employer, comprised of top tech companies that are hiring now
In-Depth: Life at KEMP Technologies
Related: software development, employers, Microsoft, IT, Linux, SDN, KEMP Technologies, Oracle, Limerick IT, software
How an internship can prepare you for a tech career
9 May 2019623 Views
Kemp Technologies
Workplace wellbeing should be more than just a box-ticking exercise
'It's important to embrace the concept that HR is everybody's responsibility'
5 Dec 2018622 Views
WANT TO KNOW MORE ABOUT EMPLOYER BRAND?
Photo via Gajus/Shutterstock
Silicon Republic's Employer Brand Programme highlights top companies that are hiring right now in Ireland.
To find out more about the programme and how to become part of it, please submit an enquiry
Skies are Azure and the cloud is still golden in Microsoft Q3 results
Singlepoint to create 50 new jobs in Dublin's North Dock
Private equity firm and management acquire Kemp Technologies