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1. CMB Online first
Path Decompositions of Kneser and Generalized Kneser Graphs Necessary and sufficient conditions are given for the existence
of a graph decomposition of the Kneser Graph $KG_{n,2}$ and of
the Generalized Kneser Graph $GKG_{n,3,1}$ into paths of length
three.
Keywords:Kneser graph, generalized Kneser graph, path decomposition, graph decomposition Categories:05C51, 05C70 
2. CMB 2013 (vol 57 pp. 870)
A Short Note on Short Pants It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.
Keywords:hyperbolic surfaces, geodesics, pants decompositions Categories:30F10, 32G15, 53C22 
3. CMB 2012 (vol 57 pp. 97)
Rationality and the JordanGattiViniberghi decomposition We verify
our earlier conjecture
and use it to prove that the
semisimple parts of the rational JordanKacVinberg decompositions of
a rational vector all lie in a single rational orbit.
Keywords:reductive group, $G$module, Jordan decomposition, orbit closure, rationality Categories:20G15, 14L24 
4. CMB 2012 (vol 56 pp. 606)
Characterization of Simple Highest Weight Modules We prove that for simple complex finite dimensional
Lie algebras, affine KacMoody Lie algebras, the
Virasoro algebra and the HeisenbergVirasoro algebra,
simple highest weight modules are characterized
by the property that all positive root elements
act on these modules locally nilpotently. We
also show that this is not the case for higher rank
Virasoro and for Heisenberg algebras.
Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action Categories:17B20, 17B65, 17B66, 17B68 
5. CMB 2011 (vol 56 pp. 442)
Closed Left Ideal Decompositions of $U(G)$ Let $G$ be an infinite discrete group and let $\beta G$ be the
StoneÄech compactification of $G$. We take the points of $Äta
G$ to be the ultrafilters on $G$, identifying the principal
ultrafilters with the points of $G$. The set $U(G)$ of uniform
ultrafilters on $G$ is a closed twosided ideal of $\beta G$. For
every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in
p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show
that if $G$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the
finest decomposition of $U(G)$ into closed left ideals of $\beta G$
such that the corresponding quotient space of $U(G)$ is Hausdorff.
Keywords:StoneÄech compactification, uniform ultrafilter, closed left ideal, decomposition Categories:22A15, 54H20, 22A30, 54D80 
6. CMB 2011 (vol 55 pp. 303)
Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$,
where the decomposition converges in $L^2_w$norm rather than in the distribution sense.
As applications of this decomposition, assuming that $T$ is a linear
operator bounded on $L^2_w$ and $0

7. CMB 2009 (vol 53 pp. 278)
CantorBernstein Sextuples for Banach Spaces Let $X$ and $Y$ be Banach spaces isomorphic
to complemented subspaces of each other with supplements $A$ and
$B$. In 1996, W. T. Gowers solved the SchroederBernstein (or
CantorBernstein) problem for Banach spaces by showing that $X$ is not
necessarily isomorphic to $Y$. In this paper, we obtain a necessary
and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in
$\mathbb N$
with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that
$X$ is isomorphic to $Y$, whenever these spaces satisfy the following
decomposition scheme
$$
A^u \sim X^p \oplus Y^q, \quad
B^v \sim X^r \oplus Y^s.
$$
Namely, $\Phi=(pu)(sv)(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called CantorBernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the wellknown PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the SchroederBernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, SchroederBernstein problem Categories:46B03, 46B20 
8. CMB 2007 (vol 50 pp. 504)
Asymptotic Existence of Resolvable Graph Designs Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block
design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$subsets of a
$v$set $X$ in which every unordered pair of elements from $X$ is
contained in exactly $\lambda$ elements of $\cA$. More generally, for a
fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a
collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such
that every unordered pair of elements from $X$ is an edge of exactly
$\lambda$ elements of $\cA$. A famous result of Wilson says that for a
fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all
sufficiently large $v$ satisfying certain necessary conditions. A
block (graph) design as above is \emph{resolvable} if $\cA$ can be
partitioned into partitions of (graphs whose vertex sets partition)
$X$. Lu has shown asymptotic existence in $v$ of resolvable
$\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for
resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle
asymptotic existence of resolvable graph designs.
Keywords:graph decomposition, resolvable designs Categories:05B05, 05C70, 05B10 
9. CMB 2003 (vol 46 pp. 356)
Branched Covers of Tangles in Threeballs We give an algorithm for a surgery description of a $p$fold cyclic branched
cover of $B^3$ branched along a tangle. We generalize constructions of
Montesinos and AkbulutKirby.
Keywords:tangle, branched cover, surgery, Heegaard decomposition Categories:57M25, 57M12 