Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Joel A. Tropp
by alspach＠fourier.math.okstate.edu 01 Jul '08

01 Jul '08

This is an announcement for the paper "Column subset selection, matrix factorization, and eigenvalue optimization" by Joel A. Tropp. Abstract: Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing {\sf QR}, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the $(\infty, 1)$ norm of a matrix, which is generally {\sf NP}-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous {\sc maxcut} semidefinite program. Archive classification: math.NA math.FA Mathematics Subject Classification: 15A60; 15A23; 65F30; 90C25 Remarks: Conference version The source file(s), alg.sty: 7607 bytes macro-file.tex: 8456 bytes subset-selection-soda-v4.bbl: 4536 bytes subset-selection-soda-v4.tex: 88398 bytes, is(are) stored in gzipped %form as 0806.4404.tar.gz with size 30kb. The corresponding postcript file has gzipped size 109kb. Submitted from: jtropp(a)acm.caltech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.4404 or http://arXiv.org/abs/0806.4404 or by email in unzipped form by transmitting an empty message with subject line uget 0806.4404 or in gzipped form by using subject line get 0806.4404 to: math(a)arXiv.org.

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Abstract of a paper by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazzaa
by alspach＠fourier.math.okstate.edu 01 Jul '08

01 Jul '08

This is an announcement for the paper "A criterion of weak compactness for operators on subspaces of Orlicz spaces" by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazzaa. Abstract: To appear in J. Funct. Spaces and Appl. Archive classification: math.FA Mathematics Subject Classification: 46E30 Remarks: 18 pages The source file(s), critere.tex: 40456 bytes, is(are) stored in gzipped form as 0806.4204.gz with size 13kb. The corresponding postcript file has gzipped size 97kb. Submitted from: lefevre(a)euler.univ-artois.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.4204 or http://arXiv.org/abs/0806.4204 or by email in unzipped form by transmitting an empty message with subject line uget 0806.4204 or in gzipped form by using subject line get 0806.4204 to: math(a)arXiv.org.

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Abstract of a paper by Vladimir Kadets, Varvara Shepelska and Dirk Werner
by alspach＠fourier.math.okstate.edu 18 Jun '08

18 Jun '08

This is an announcement for the paper "Quotients of Banach spaces with the Daugavet property" by Vladimir Kadets, Varvara Shepelska and Dirk Werner. Abstract: We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46B25; 47B38 Remarks: 15 pages The source file(s), dpry_bullpol_june08.tex: 55217 bytes, is(are) stored in gzipped form as 0806.1815.gz with size 17kb. The corresponding postcript file has gzipped size 114kb. Submitted from: werner(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.1815 or http://arXiv.org/abs/0806.1815 or by email in unzipped form by transmitting an empty message with subject line uget 0806.1815 or in gzipped form by using subject line get 0806.1815 to: math(a)arXiv.org.

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SUMIRFAS 2008
by Bill Johnson 17 Jun '08

17 Jun '08

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Abstract of a paper by Jaegil Kim and Han Ju Lee
by alspach＠fourier.math.okstate.edu 04 Jun '08

04 Jun '08

This is an announcement for the paper "Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices" by Jaegil Kim and Han Ju Lee. Abstract: Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset. In the last part, using Reisner's graph theoretic-approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\ell_\infty^n$. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm. Archive classification: math.FA math.CO Mathematics Subject Classification: 46G25; 46B20; 46B22; 52A21; 46B20 The source file(s), graph-June3-08.tex: 54865 bytes, is(are) stored in gzipped form as 0806.0507.gz with size 15kb. The corresponding postcript file has gzipped size 117kb. Submitted from: hahnju(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0507 or http://arXiv.org/abs/0806.0507 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0507 or in gzipped form by using subject line get 0806.0507 to: math(a)arXiv.org.

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Abstract of a paper by G. Androulakis, S. J. Dilworth, and N. J. Kalton
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces" by G. Androulakis, S. J. Dilworth, and N. J. Kalton. Abstract: We give a new approach to the Ramsey-type results of Gowers on block bases in Banach spaces and apply our results to prove the Gowers dichotomy in F-spaces. Archive classification: math.FA Mathematics Subject Classification: 46A16, 91A05, 91A80 The source file(s), AndDilKal.tex: 70671 bytes, is(are) stored in gzipped form as 0806.0058.gz with size 20kb. The corresponding postcript file has gzipped size 132kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0058 or http://arXiv.org/abs/0806.0058 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0058 or in gzipped form by using subject line get 0806.0058 to: math(a)arXiv.org.

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Abstract of a paper by G. Androulakis and K. Beanland
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators" by G. Androulakis and K. Beanland. Abstract: The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $\omega_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied. Archive classification: math.FA Mathematics Subject Classification: 47B07, 47A15 The source file(s), AlmostSC.tex: 41247 bytes, is(are) stored in gzipped form as 0806.0056.gz with size 12kb. The corresponding postcript file has gzipped size 95kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0056 or http://arXiv.org/abs/0806.0056 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0056 or in gzipped form by using subject line get 0806.0056 to: math(a)arXiv.org.

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Abstract of a paper by Hamed Hatami
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Graph norms and Sidorenko's conjecture" by Hamed Hatami. Abstract: Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for $A_G$, the adjacency matrix of a graph $G$, we have $h_H(A_G)=h_H(G)$. Let $m$ be the number of edges of $H$. It is easy to see that when $H$ is the cycle of length $2n$, then $h_H(\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann norm. We investigate a question of Lov\'{a}sz that asks for a characterization of graphs $H$ for which the function $h_H(\cdot)^{1/m}$ is a norm. We prove that $h_H(\cdot)^{1/m}$ is a norm if and only if a H\"{o}lder type inequality holds for $H$. We use this inequality to prove both positive and negative results, showing that $h_H(\cdot)^{1/m}$ is a norm for certain classes of graphs, and giving some necessary conditions on the structure of $H$ when $h_H(\cdot)^{1/m}$ is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the $h_H(\cdot)^{1/m}$ norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the $2n$-th Schatten-von Neumann norms. Archive classification: math.FA math.CO Mathematics Subject Classification: 46E30; 05C35 Remarks: to appear in Israel Journal of Mathematics The source file(s), arxiv/normFinal.bbl: 6949 bytes arxiv/normFinal.tex: 57888 bytes arxiv/normFinal.toc: 1082 bytes, is(are) stored in gzipped form as 0806.0047.tar.gz with size 20kb. The corresponding postcript file has gzipped size 125kb. Submitted from: hamed(a)cs.toronto.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0047 or http://arXiv.org/abs/0806.0047 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0047 or in gzipped form by using subject line get 0806.0047 to: math(a)arXiv.org.

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Abstract of a paper by Joaquim Martin and Mario Milman
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Isometry and symmetrization for logarithmic Sobolev inequalities" by Joaquim Martin and Mario Milman. Abstract: Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts. Archive classification: math.FA math.AP The source file(s), Gauss-final-rev.tex: 69485 bytes, is(are) stored in gzipped form as 0806.0021.gz with size 19kb. The corresponding postcript file has gzipped size 129kb. Submitted from: mario.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0021 or http://arXiv.org/abs/0806.0021 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0021 or in gzipped form by using subject line get 0806.0021 to: math(a)arXiv.org.

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Abstract of a paper by Markus Biegert
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Lattice homomorphisms between Sobolev spaces" by Markus Biegert. Abstract: We show that every vector lattice homomorphism $T$ from $W^{1,p}_0(\Omega_1)$ into $W^{1,q}(\Omega_2)$ for $p,q\in (1,\infty)$ and open sets \Omega_1,\Omega_2\subset\IR^N$ has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\Omega_1$ and $g:\Omega_2\to[0,\infty)$. This representation follows as an application of an abstract and more general representation theorem, which can be applied in many other situations. We prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $W^{1,q}(\Omega_2)$ admits a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\overline\Omega_1$ and $g:\Omega_2\to[0,\infty)$. Here $\tsW^{1,p}(\Omega_1)$ denotes the closure of $W^{1,p}(\Omega_1)\cap C_c(\overline\Omega_1)$ in $W^{1,p}(\Omega_1)$ and every $u\in\tsW^{1,p}(\Omega_1)$ admits a trace on the boundary $\partial\Omega_1$ of $\Omega_1$. Finally we prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $\tsW^{1,q}(\Omega_2)$ where $\Omega_1$ is bounded has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_{q,\Omega_2}$-quasi everywhere on }\overline\Omega_2$ with mappings $\xi:\overline\Omega_2\to\overline\Omega_1$ and $g:\overline\Omega_2\to[0,\infty)$. At the end of this article we consider also lattice isomorphisms between Sobolev spaces and the representation of their inverses. Archive classification: math.AP math.FA The source file(s), orderhomomorphism.bbl: 4468 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0805.4740 or http://arXiv.org/abs/0805.4740 or by email in unzipped form by transmitting an empty message with subject line uget 0805.4740 or in gzipped form by using subject line get 0805.4740 to: math(a)arXiv.org.

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