February 2011 - Banach - mathdept.okstate.edu

Abstract of a paper by A. Thiago L. Bernardino
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Remarks on cotype absolutely summing multilinear operators" by A. Thiago L. Bernardino. Abstract: In this short note we present some new results concerning cotype and absolutely summing multilinear operators. Archive classification: math.FA Remarks: 5 pages Submitted from: thiagodcea(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5119 or http://arXiv.org/abs/1101.5119

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Abstract of a paper by Boris Rubin
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Elementary inversion of Riesz potentials and Radon-John transforms" by Boris Rubin. Abstract: New simple proofs are given to some elementary approximate and explicit inversion formulas for Riesz potentials. The results are applied to reconstruction of functions from their integrals over Euclidean planes in integral geometry. Archive classification: math.FA Remarks: 9 pages Submitted from: borisr(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5105 or http://arXiv.org/abs/1101.5105

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Abstract of a paper by Daniel Fresen
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "A multivariate Gnedenko law of large numbers" by Daniel Fresen. Abstract: We show that the convex hull of a large i.i.d. sample from a non-vanishing log-concave distribution approximates a pre-determined body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For p-log-concave distributions with p>1 (such as the normal distribution where p=2) we also have approximation in the Hausdorff distance. These are multivariate versions of the Gnedenko law of large numbers which gaurantees concentration of the maximum and minimum in the one dimensional case. We give three different deterministic bodies that serve as approximants to the random body. The first is the floating body that serves as a multivariate quantile, the second body is given as a contour of the density function, and the third body is given in terms of the Radon transform. We end the paper by constructing a probability measure with an interesting universality property. Archive classification: math.PR math.FA Mathematics Subject Classification: 60D05, 60F99, 52A20, 52A22, 52B11 Remarks: 18 pages Submitted from: djfb6b(a)mail.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4887 or http://arXiv.org/abs/1101.4887

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Abstract of a paper by Guohui Song, Haizhang Zhang
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Reproducing kernel Banach spaces with the l1 norm II: Error analysis for regularized least square regression" by Guohui Song, Haizhang Zhang. Abstract: A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning. Archive classification: stat.ML cs.LG math.FA Submitted from: zhhaizh2(a)sysu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4439 or http://arXiv.org/abs/1101.4439

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Abstract of a paper by Guohui Song, Haizhang Zhang, Fred J. Hickernell
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Reproducing kernel Banach spaces with the l1 norm" by Guohui Song, Haizhang Zhang, Fred J. Hickernell. Abstract: Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given. Archive classification: stat.ML cs.LG math.FA Submitted from: zhhaizh2(a)sysu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4388 or http://arXiv.org/abs/1101.4388

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Abstract of a paper by Assaf Naor
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Sparse quadratic forms and their geometric applications (after Batson, Spielman and Srivastava)" by Assaf Naor. Abstract: We survey the work of Batson, Spielman and Srivastava on graph sparsification, and we describe some of its recently discovered geometric applications. Archive classification: math.FA Remarks: appeared as s\'eminaire Bourbaki expos\'e no. 1033, 2011 Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4324 or http://arXiv.org/abs/1101.4324

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Abstract of a paper by H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Multi-norms and the injectivity of $L^p(G)$" by H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden. Abstract: Let $G$ be a locally compact group, and take $p\in(1,\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces. Archive classification: math.FA Mathematics Subject Classification: 46H25, 43A20 Remarks: 27 pages Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4320 or http://arXiv.org/abs/1101.4320

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Grothendieck's Theorem, past and present" by Gilles Pisier. Abstract: Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as ``the fundamental theorem in the metric theory of tensor products''. That is now commonly referred to as ``Grothendieck's theorem'' (GT in short), or sometimes as ``Grothendieck's inequality''. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in $C^*$-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of ``operator spaces'' or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields:\ in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by ``semidefinite programming' and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes. Archive classification: math.FA math-ph math.MP math.OA Mathematics Subject Classification: 46B28, 46B07 Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4195 or http://arXiv.org/abs/1101.4195

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