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1. CMB 2016 (vol 60 pp. 510)

Haase, Christian; Hofmann, Jan
 Convex-normal (Pairs of) Polytopes In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between $k$- and $(k+1)$-convex-normality (for $k\geq 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement of the normal fan of $Q$. If every edge $e_P$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap \mathbb{Z}^d = (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$. Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopesCategories:52B20, 14M25, 90C10

2. CMB 2015 (vol 58 pp. 877)

Zaatra, Mohamed
 Generating Some Symmetric Semi-classical Orthogonal Polynomials We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda$ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example. Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relationCategories:33C45, 42C05

3. CMB 2015 (vol 58 pp. 610)

Rodger, C. A.; Whitt, Thomas Richard
 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 decompositionCategories:05C51, 05C70

4. CMB 2013 (vol 57 pp. 870)

Parlier, Hugo
 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 decompositionsCategories:30F10, 32G15, 53C22

5. CMB 2012 (vol 57 pp. 97)

Levy, Jason
 Rationality and the Jordan-Gatti-Viniberghi decomposition We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit. Keywords:reductive group, $G$-module, Jordan decomposition, orbit closure, rationalityCategories:20G15, 14L24

6. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
 Characterization of Simple Highest Weight Modules We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro 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 actionCategories:17B20, 17B65, 17B66, 17B68

7. CMB 2011 (vol 56 pp. 442)

Zelenyuk, Yevhen
 Closed Left Ideal Decompositions of $U(G)$ Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Äech compactification of $G$. We take the points of $Äta G$ 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 two-sided 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:Stone--Äech compactification, uniform ultrafilter, closed left ideal, decompositionCategories:22A15, 54H20, 22A30, 54D80

8. CMB 2011 (vol 55 pp. 303)

Han, Yongsheng; Lee, Ming-Yi; Lin, Chin-Cheng
 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 $0 Keywords:$A_p$weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 9. CMB 2009 (vol 53 pp. 278) Galego, Elói M.  Cantor-Bernstein Sextuples for Banach Spaces Let$X$and$Y$be Banach spaces isomorphic to complemented subspaces of each other with supplements$A$and$B$. In 1996, W. T. Gowers solved the Schroeder--Bernstein (or Cantor--Bernstein) problem for Banach spaces by showing that$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,$\Phi=(p-u)(s-v)-(q+u)(r+v)$is different from zero and$\Phi$divides$p+q$and$r+s$. These sextuples are called Cantor--Bernstein sextuples for Banach spaces. The simplest case$(1, 0, 0, 1, 1, 1)$indicates the well-known PeÅczyÅski's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder--Bernstein problem become evident. Keywords:Pel czyÅski's decomposition method, Schroeder-Bernstein problemCategories:46B03, 46B20 10. CMB 2007 (vol 50 pp. 504) Dukes, Peter; Ling, Alan C. H.  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 designsCategories:05B05, 05C70, 05B10 11. CMB 2003 (vol 46 pp. 356) Ishiwata, Makiko; Przytycki, Józef H.; Yasuhara, Akira  Branched Covers of Tangles in Three-balls 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 Akbulut-Kirby. Keywords:tangle, branched cover, surgery, Heegaard decompositionCategories:57M25, 57M12
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