Banach August 2006

banach@mathdept.okstate.edu

1 participants
9 discussions

Abstract of a paper by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic
by Dale Alspach 29 Aug '06

29 Aug '06

This is an announcement for the paper "Sublinear Higson corona and Lipschitz extensions" by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic. Abstract: The purpose of the paper is to characterize the dimension of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{m+1}$. One should compare it to the result of Dranishnikov \cite{Dr1} who characterized the dimension of the Higson corona $\nu(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(\nu_L(X))\leq n$ if and only if $\R^{n+1}$ is an absolute extensor of $X$ in the category $\tilde\AAA$. \par As an application we reprove the following result of Dranishnikov and Smith \cite{DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension $\asdim_{AN}(X)$. If $X$ is cocompact and connected, then $\asdim_{AN}(X)$ equals the dimension of the sublinear Higson corona $\nu_L(X)$ of $X$. Archive classification: Metric Geometry; Functional Analysis; Geometric Topology Remarks: 13 pages The source file(s), SublinearHigson.tex: 51559 bytes, is(are) stored in gzipped form as 0608686.gz with size 15kb. The corresponding postcript file has gzipped size 76kb. Submitted from: dydak(a)math.utk.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0608686 or http://arXiv.org/abs/math.MG/0608686 or by email in unzipped form by transmitting an empty message with subject line uget 0608686 or in gzipped form by using subject line get 0608686 to: math(a)arXiv.org.

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Abstract of a paper by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann
by Dale Alspach 29 Aug '06

29 Aug '06

This is an announcement for the paper "Uniform uncertainty principle for Bernoulli and subgaussian ensembles" by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann. Abstract: We present a simple solution to a question posed by Candes, Romberg and Tao on the uniform uncertainty principle for Bernoulli random matrices. More precisely, we show that a rectangular k*n random subgaussian matrix (with k < n) has the property that by arbitrarily extracting any m (with m < k) columns, the resulting submatrices are arbitrarily close to (multiples of) isometries of a Euclidean space. We obtain the optimal estimate for m as a function of k,n and the degree of "closeness" to an isometry. We also give a short and self-contained solution of the reconstruction problem for sparse vectors. Archive classification: Statistics; Functional Analysis Mathematics Subject Classification: 46B07; 47B06; 41A05; 62G05; 94B75 Remarks: 15 pages; no figures; submitted The source file(s), uup-arx-21-08.tex: 48079 bytes, is(are) stored in gzipped form as 0608665.gz with size 16kb. The corresponding postcript file has gzipped size 71kb. Submitted from: alain.pajor(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.ST/0608665 or http://arXiv.org/abs/math.ST/0608665 or by email in unzipped form by transmitting an empty message with subject line uget 0608665 or in gzipped form by using subject line get 0608665 to: math(a)arXiv.org.

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Abstract of a paper by Christian Rosendal
by Dale Alspach 25 Aug '06

25 Aug '06

This is an announcement for the paper "Infinite asymptotic games" by Christian Rosendal. Abstract: We study infinite asymptotic games in Banach spaces with an F.D.D. and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into $\ell_p$ sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. These results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: Primary: 46B03, Secondary 03E15 The source file(s), AsymptoticGames18.tex: 61261 bytes, is(are) stored in gzipped form as 0608616.gz with size 19kb. The corresponding postcript file has gzipped size 83kb. Submitted from: rosendal(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0608616 or http://arXiv.org/abs/math.FA/0608616 or by email in unzipped form by transmitting an empty message with subject line uget 0608616 or in gzipped form by using subject line get 0608616 to: math(a)arXiv.org.

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Abstract of a paper by S. J. Dilworth, V. Ferenczi, Denka Kutzarova and E. Odell
by Dale Alspach 23 Aug '06

23 Aug '06

This is an announcement for the paper "On strongly asymptotic $\ell_p$ spaces and minimality" by S. J. Dilworth, V. Ferenczi, Denka Kutzarova and E. Odell. Abstract: We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20, 46B45 Remarks: 12 pages, AMSLaTeX The source file(s), dfko010206-archive.tex: 46987 bytes, is(are) stored in gzipped form as 0608550.gz with size 15kb. The corresponding postcript file has gzipped size 71kb. Submitted from: combs(a)mail.ma.utexas.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0608550 or http://arXiv.org/abs/math.FA/0608550 or by email in unzipped form by transmitting an empty message with subject line uget 0608550 or in gzipped form by using subject line get 0608550 to: math(a)arXiv.org.

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Abstract of a paper by Mohsen Alimohammady
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "Containment of $\c_{\bf 0}$ and $\ell_{\bf 1}$ in $\Pi_{\bf 1} \hbox{\bf (}\E\hbox{\bf ,}\ \F\hbox{\bf )}$" by Mohsen Alimohammady. Abstract: Suppose $\Pi_{1} (E, F)$ is the space of all absolutely 1-summing operators between two Banach spaces $E$ and $F$. We show that if $F$ has a copy of $c_{0}$, then $\Pi_{1} (E, F)$ will have a copy of $c_{0}$, and under some conditions if $E$ has a copy of $\ell_{1}$ then $\Pi_{1} (E, F)$ would have a complemented copy of $\ell_{1}$. Archive classification: Functional Analysis Mathematics Subject Classification: 47B10; 46B20 Remarks: 4 pages The source file(s), mat01.cls: 37258 bytes, mathtimy.sty: 20 bytes, pm2197new.tex: 11816 bytes, is(are) stored in gzipped form as 0607651.tar.gz with size 14kb. The corresponding postcript file has gzipped size 25kb. Submitted from: amohsen(a)umz.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0607651 or http://arXiv.org/abs/math.FA/0607651 or by email in unzipped form by transmitting an empty message with subject line uget 0607651 or in gzipped form by using subject line get 0607651 to: math(a)arXiv.org.

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Abstract of a paper by Jakub Duda
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "On Gateaux differentiability of pointwise Lipschitz mappings" by Jakub Duda. Abstract: We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f:X\to\R$ cone monotone, $g:X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Frechet differentiable. Archive classification: Functional Analysis Mathematics Subject Classification: 46G05; 46T20 Remarks: 11 pages; updated version The source file(s), ongatdif.tex: 43273 bytes, is(are) stored in gzipped form as 0511565.gz with size 13kb. The corresponding postcript file has gzipped size 61kb. Submitted from: jakub.duda(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511565 or http://arXiv.org/abs/math.FA/0511565 or by email in unzipped form by transmitting an empty message with subject line uget 0511565 or in gzipped form by using subject line get 0511565 to: math(a)arXiv.org.

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Abstract of a paper by Yunan Cui, Henryk Hudzik, Narin Petrot, Suthep Suantai and Alicja Szymaszkiewicz
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "Basic topological and geometric properties of Cesaro--Orlicz spaces" by Yunan Cui, Henryk Hudzik, Narin Petrot, Suthep Suantai and Alicja Szymaszkiewicz. Abstract: Necessary and sufficient conditions under which the Cesaro--Orlicz sequence space $\cfi$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesaro--Orlicz spaces $\cfi$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\cfi$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\cfi$ are given. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20, 46B45, 46E30 Remarks: 16 pages The source file(s), mat01.cls: 37258 bytes, mathtimy.sty: 20 bytes, pm2563new.tex: 46836 bytes, is(are) stored in gzipped form as 0607730.tar.gz with size 23kb. The corresponding postcript file has gzipped size 55kb. Submitted from: Yunan Cui, Henryk Hudzik, Narin Petrot, Suthep Suantai and Alicja Szymasz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0607730 or http://arXiv.org/abs/math.FA/0607730 or by email in unzipped form by transmitting an empty message with subject line uget 0607730 or in gzipped form by using subject line get 0607730 to: math(a)arXiv.org.

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Abstract of a paper by Erwin Lutwak, Deane Yang, and Gaoyong Zhang
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "Volume inequalities for isotropic measures" by Erwin Lutwak, Deane Yang, and Gaoyong Zhang. Abstract: A direct approach to Ball's simplex inequality is presented. This approach, which does not use the Brascamp-Lieb inequality, also gives Barthe's characterization of the simplex for Ball's inequality and extends it from discrete to arbitrary measures. It also yields the dual inequality, along with equality conditions, and it does both for arbitrary measures. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52A40 Remarks: 10 pages, to appear in American Journal of Mathematics The source file(s), bb2_copy7.tex: 32473 bytes, is(are) stored in gzipped form as 0607753.gz with size 10kb. The corresponding postcript file has gzipped size 45kb. Submitted from: dyang(a)poly.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0607753 or http://arXiv.org/abs/math.MG/0607753 or by email in unzipped form by transmitting an empty message with subject line uget 0607753 or in gzipped form by using subject line get 0607753 to: math(a)arXiv.org.

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Abstract of a paper by Oscar Valero
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "Quotient normed cones" by Oscar Valero. Abstract: Given a normed cone $(X,p)$ and a subcone $Y,$ we construct and study the quotient normed cone $(X/Y,\tilde{p})$ generated by $Y$. In particular we characterize the bicompleteness of $(X/Y,\tilde{p})$ in terms of the bicompleteness of $(X,p),$ and prove that the dual quotient cone $((X/Y)^{*},\|\cdot \|_{\tilde{p},u})$ can be identified as a distinguished subcone of the dual cone $(X^{*},\|\cdot \|_{p,u})$. Furthermore, some parts of the theory are presented in the general setting of the space $CL(X,Y)$ of all continuous linear mappings from a normed cone $(X,p)$ to a normed cone $(Y,q),$ extending several well-known results related to open continuous linear mappings between normed linear spaces. Archive classification: Functional Analysis; General Topology Mathematics Subject Classification: 54E35; 54E50; 54E99; 54H11 Remarks: 17 pages The source file(s), mat01.cls: 37258 bytes, mathtimy.sty: 20 bytes, pm2745new.tex: 58553 bytes, is(are) stored in gzipped form as 0607619.tar.gz with size 26kb. The corresponding postcript file has gzipped size 61kb. Submitted from: o.valero(a)uib.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0607619 or http://arXiv.org/abs/math.FA/0607619 or by email in unzipped form by transmitting an empty message with subject line uget 0607619 or in gzipped form by using subject line get 0607619 to: math(a)arXiv.org.

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Banach August 2006