Banach December 2014

banach@mathdept.okstate.edu

1 participants
19 discussions

Abstract of a paper by N. Albuquerque, G. Araujo, and D. Pellegrino
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "H\"{o}lder's inequality: some recent and unexpected applications" by N. Albuquerque, G. Araujo, and D. Pellegrino. Abstract: H\"{o}lder's inequality, since its appearance in 1888, has played a fundamental role in Mathematical Analysis and it is, without any doubt, one of the milestones in Mathematics. It may seem strange that, nowadays, it keeps resurfacing and bringing new insights to the mathematical community. In this expository article we show how a variant of H\"{o}lder's inequality (although well-known in PDEs) was essentially overlooked in Functional Analysis and has had a crucial (and in some sense unexpected) influence in very recent and major breakthroughs in Mathematics. Some of these recent advances appeared in 2012-2014 and include the theory of Dirichlet series, the famous Bohr radius problem, certain classical inequalities (such as Bohnenblust--Hille or Hardy--Littlewood), or even Mathematical Physics. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.2017 or http://arXiv.org/abs/1412.2017

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Abstract of a paper by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Networks for the weak topology of Banach and Frechet spaces" by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski. Abstract: We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr\'echet lcs, then $E$ endowed with its weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr\'echet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fr\'echet lcs $E$ in the weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space. Archive classification: math.FA Mathematics Subject Classification: Primary 46A03, 54H11, Secondary 22A05, 54C35 Remarks: 18 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1748 or http://arXiv.org/abs/1412.1748

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Abstract of a paper by Michael Megrelishvili
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "A note on dependence of families having bounded variation" by Michael Megrelishvili. Abstract: We show that for arbitrary linearly ordered set $X$ any bounded family of real valued functions on $X$ with bounded total variation does not contain independent subsequences. As a corollary we generalize Helly's sequential compactness theorem. Archive classification: math.GN math.FA Mathematics Subject Classification: 54F15, 54D30, 06A05 Remarks: 7 pages Submitted from: megereli(a)math.biu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1515 or http://arXiv.org/abs/1412.1515

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Abstract of a paper by Jarno Talponen
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers" by Jarno Talponen. Abstract: In this note various geometric properties of a Banach space $X$ are characterized by means of weaker corresponding geometric properties involving an ultrapower $X^\mathcal{U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal{U}$. For example, a point $x\in S_X$ is an MLUR point if and only if $j(x)$ (given by the canonical inclusion $j\colon X \to X^\mathcal{U}$) in $\B_{X^\mathcal{U}}$ is an extreme point; a point $x\in S_X$ is LUR if and only if $j(x)$ is not contained in any non-degenerate line segment of $S_{X^\mathcal{U}}$; a Banach space $X$ is URED if and only if there are no $x,y \in S_{X^\mathcal{U}}$, $x\neq y$, with $x-y \in j(X)$. Archive classification: math.FA math.LO Mathematics Subject Classification: 03H05, 46B20, 46M07, 46B10 Submitted from: talponen(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1279 or http://arXiv.org/abs/1412.1279

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Abstract of a paper by Gilles Pisier and Eric Ricard
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "The non-commutative Khintchine inequalities for $0<p<1$" by Gilles Pisier and Eric Ricard. Abstract: We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are H\"older on semifinite von Neumann algebras. Archive classification: math.OA math.FA Mathematics Subject Classification: 2000 MSC 46L51, 46L07, 47L25, 47L20 Submitted from: pisier(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.0222 or http://arXiv.org/abs/1412.0222

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Abstract of a paper by Astrid Berg, Lukas Parapatits, Franz E. Schuster, and Manuel Weberndorfer
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Log-concavity properties of Minkowski valuations" by Astrid Berg, Lukas Parapatits, Franz E. Schuster, and Manuel Weberndorfer. Abstract: New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new classification of generalized translation invariant valuations. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A38, 52B45 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.7891 or http://arXiv.org/abs/1411.7891

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Abstract of a paper by Aleksandar Nikolov
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Randomized rounding for the largest $j$-simplex problem" by Aleksandar Nikolov. Abstract: The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{R}^d$. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem is known to be $\mathsf{NP}$-hard to approximate within a factor of $2^{cj}$ for some constant $c$. Our algorithm also approximates the problem of finding the largest determinant principal $j\times j$ submatrix of a rank $d$ positive semidefinite matrix, with approximation ratio $e^{j + o(j)}$. We achieve our approximation by rounding solutions to a generlization of the $D$-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid probelm. Our arguments give a short and simple proof of a restricted invertibility principle for determinants. Archive classification: cs.CG cs.DS math.FA Submitted from: anikolov(a)cs.rutgers.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.0036 or http://arXiv.org/abs/1412.0036

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Banach J. Math Anal and Ann. Funct. Anal.
by Dale Alspach 16 Dec '14

16 Dec '14

Duke University Press partners with the Tusi Mathematical Research Group to publish the Annals of Functional Analysis (AFA) and the Banach Journal of Mathematical Analysis (BJMA). In 2015, Duke University Press will begin publishing both journals. AFA, started in 2010, and BJMA, started in 2007, are online-only journals included in the prestigious "Reference List Journals" covered by MathSciNet and indexed by Zentralblatt Math, Scopus and Thomson Reuters (ISI). With the start of their 2015 volumes under the guidance of strong editorial boards, the journals will increase in frequency from two to four issues per year. The journals publish research papers and critical survey articles that focus on, but are not limited to, functional analysis, abstract harmonic analysis and operator theory. AFA and BJMA have rapidly established themselves as providing high-level scholarship that addresses important questions in the study of mathematical analysis. The journals are no longer open access but papers will be freely available in Project Euclid 5 years after publication. As before, they will be available on Project Euclid at http://projecteuclid.org/euclid.bjma [1] and http://projecteuclid.org/euclid.afa [2] Editor-in-chief M. S. Moslehian ===========================================================

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Address Changes
by Dale Alspach 15 Dec '14

15 Dec '14

The Banach list is changing its address. Messages should be sent to banach(a)mathdept.okstate.edu. The list location for subscribing or unsubscribing is now https://www.mathdept.okstate.edu/cgi-bin/mailman/listinfo/banach The URL for lists of past postings and other information is https://math.okstate.edu/people/alspach/banach/index.html My current address alspach(a)math.okstate.edu should continue to work but banach(a)math.okstate.edu will stop working sometime in the next few weeks. Best Wishes for the New Year, Dale Alspach

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Banach December 2014