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1. CMB 2014 (vol 58 pp. 7)

Boulabiar, Karim
 Characters on $C(X)$ The precise condition on a completely regular space $X$ for every character on $C(X)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience. Keywords:characters, realcompact, evaluation, real-valued continuous functionsCategories:54C30, 46E25

2. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
 2-dimensional Convexity Numbers and $P_4$-free Graphs For $S\subseteq\mathbb R^n$ a set $C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\subseteq\mathbb R^2$ without an uncountable $3$-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in $\mathbb R^2$ without uncountable 3-cliques in terms of clopen, $P_4$-free graphs on Polish spaces. Keywords:convex cover, convexity number, continuous coloring, perfect graph, cographCategories:52A10, 03E17, 03E75

3. CMB 2011 (vol 55 pp. 449)

Bahreini, Manijeh; Bator, Elizabeth; Ghenciu, Ioana
 Complemented Subspaces of Linear Bounded Operators We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$. Feder proved that if $X$ is infinite-dimensional and $c_0 \hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Bator and Lewis showed that if $X$ is not a Grothendieck space and $c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in $L(X,Y)$. In this paper, classical results of Kalton and separably determined operator ideals with property $(*)$ are used to obtain complementation results that yield these theorems as corollaries. Keywords:spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operatorsCategories:46B20, 46B28

4. CMB 2011 (vol 55 pp. 623)

Pan, Jiaqing
 The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations In this paper, we consider the Cauchy problem $$\begin{cases} u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3, \\ % ^^----- here u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}. \end{cases}$$ We will prove that: (i) for $m_{c} Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearityCategories:35K05, 35K10, 35K15 5. CMB 2010 (vol 53 pp. 466) Dubarbie, Luis  Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied. Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuityCategories:47B38, 46E15, 46E40, 46H40, 47B33 6. CMB 2010 (vol 53 pp. 360) Porter, Jack; Tikoo, Mohan  Separating H-sets by Open Sets In an H-closed, Urysohn space, disjoint H-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the H-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint H-sets can be separated by disjoint open sets. Keywords:H-set, H-closed, θ-continuousCategories:54C08, 54D10, 54D15 7. CMB 2008 (vol 51 pp. 217) Filippakis, Michael E.; Papageorgiou, Nikolaos S.  A Multivalued Nonlinear System with the Vector$p$-Laplacian on the Semi-Infinity Interval We study a second order nonlinear system driven by the vector$p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis$\mathbb{R}_+.$Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval$[0,n]$and then via a diagonalization method we produce a solution for the original infinite time-horizon system. Keywords:semi-infinity interval, vector$p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous mapCategory:34A60 8. CMB 2008 (vol 51 pp. 15) Aqzzouz, Belmesnaoui; Nouira, Redouane; Zraoula, Larbi  The Duality Problem for the Class of AM-Compact Operators on Banach Lattices We prove the converse of a theorem of Zaanen about the duality problem of positive AM-compact operators. Keywords:AM-compact operator, order continuous norm, discrete vector latticeCategories:46A40, 46B40, 46B42 9. CMB 2007 (vol 50 pp. 632) Zelenyuk, Yevhen; Zelenyuk, Yuliya  Transformations and Colorings of Groups Let$G$be a compact topological group and let$f\colon G\to G$be a continuous transformation of$G$. Define$f^*\colon G\to G$by$f^*(x)=f(x^{-1})x$and let$\mu=\mu_G$be Haar measure on$G$. Assume that$H=\Imag f^*$is a subgroup of$G$and for every measurable$C\subseteq H$,$\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable$C\subseteq G$, there exist$S\subseteq C$and$g\in G$such that$f(Sg^{-1})\subseteq Cg^{-1}$and$\mu(S)\ge(\mu(C))^2$. Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 10. CMB 2005 (vol 48 pp. 267) Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.  Continuous Adjacency Preserving Maps on Real Matrices It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form$ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix$R$. The result extends previously known theorems where the map was assumed to be also injective. Keywords:adjacency of matrices, continuous preservers, affine transformationsCategories:15A03, 15A04. 11. CMB 2003 (vol 46 pp. 268) Puls, Michael J.  Group Cohomology and$L^p$-Cohomology of Finitely Generated Groups Let$G$be a finitely generated, infinite group, let$p>1$, and let$L^p(G)$denote the Banach space$\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of$G$with coefficients in$L^p(G)$, and the first reduced$L^p$-cohomology space of$G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups. Keywords:group cohomology,$L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functionalCategories:43A15, 20F65, 20F18 12. CMB 1999 (vol 42 pp. 118) Rao, T. S. S. R. K.  Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space$\WC(K,X)^\ast$For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions,$M$-idealsCategories:46B20, 46E40 13. CMB 1998 (vol 41 pp. 261) Barthwal, S.; Jhingan, S.; Kanwar, P.  A simple ring over which proper cyclics are continuous is a$\PCI$-ring It is shown that simple rings over which proper cyclic right modules are continuous coincide with simple right$\PCI$-rings, introduced by Faith. Keywords:Simple rings,$\PCI$-rings,$\PCQI\$-rings, continuous modules,, quasi-continuous modulesCategories:16D50, 16D70
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