June 2004 - Banach - mathdept.okstate.edu

Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 25 Jun '04

25 Jun '04

This is an announcement for the paper "Some results about the Schroeder-Bernstein Property for separable Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of V. Ferenczi and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03, 46B20 Remarks: 25 pages The source file(s), ferenczigalegoSB.tex: 74499 bytes, is(are) stored in gzipped form as 0406479.gz with size 22kb. The corresponding postcript file has gzipped size 87kb. Submitted from: eloi(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0406479 or http://arXiv.org/abs/math.FA/0406479 or by email in unzipped form by transmitting an empty message with subject line uget 0406479 or in gzipped form by using subject line get 0406479 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 25 Jun '04

25 Jun '04

This is an announcement for the paper "Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_{\sigma}}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_{\sigma}} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 03E15; 46B03 Remarks: 22 pages; 2 figures The source file(s), sjm16.tex: 74499 bytes, is(are) stored in gzipped form as 0406477.gz with size 22kb. The corresponding postcript file has gzipped size 86kb. Submitted from: eloi(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0406477 or http://arXiv.org/abs/math.FA/0406477 or by email in unzipped form by transmitting an empty message with subject line uget 0406477 or in gzipped form by using subject line get 0406477 to: math(a)arXiv.org.

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Abstract of a paper by Artem Zvavitch
by Dale Alspach 22 Jun '04

22 Jun '04

This is an announcement for the paper "The Busemann-Petty problem for arbitrary measures" by Artem Zvavitch. Abstract: The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in $\R^n$ and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52A15, 52A21, 52A38 The source file(s), GBP_Zvavitch.tex: 44254 bytes, is(are) stored in gzipped form as 0406406.gz with size 12kb. The corresponding postcript file has gzipped size 65kb. Submitted from: zvavitch(a)math.kent.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0406406 or http://arXiv.org/abs/math.MG/0406406 or by email in unzipped form by transmitting an empty message with subject line uget 0406406 or in gzipped form by using subject line get 0406406 to: math(a)arXiv.org.

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Abstract of a paper by Peter A. Loeb and Erik Talvila
by Dale Alspach 21 Jun '04

21 Jun '04

This is an announcement for the paper "Lusin's Theorem and Bochner integration" by Peter A. Loeb and Erik Talvila. Abstract: It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $\varepsilon$ of the integral, with the sum for the local errors also less than $\varepsilon$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion. Archive classification: Classical Analysis and ODEs; Functional Analysis Mathematics Subject Classification: 28A20, 28B05; 26A39 Remarks: To appear in Scientiae Mathematicae Japonicae The source file(s), bochnerbox.tex: 34366 bytes, is(are) stored in gzipped form as 0406370.gz with size 11kb. The corresponding postcript file has gzipped size 52kb. Submitted from: etalvila(a)math.ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.CA/0406370 or http://arXiv.org/abs/math.CA/0406370 or by email in unzipped form by transmitting an empty message with subject line uget 0406370 or in gzipped form by using subject line get 0406370 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 16 Jun '04

16 Jun '04

This is an announcement for the paper "Equilateral sets in finite-dimensional normed spaces" by Konrad J. Swanepoel. Abstract: This is an expository paper on the largest size of equilateral sets in finite-dimensional normed spaces. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52A21 (Primary) 46B20, 52C17 (Secondary) Remarks: 30 pages The source file(s), equilateral.tex: 94432 bytes, is(are) stored in gzipped form as 0406264.gz with size 29kb. The corresponding postcript file has gzipped size 128kb. Submitted from: swanekj(a)unisa.ac.za The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0406264 or http://arXiv.org/abs/math.MG/0406264 or by email in unzipped form by transmitting an empty message with subject line uget 0406264 or in gzipped form by using subject line get 0406264 to: math(a)arXiv.org.

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Abstract of a paper by V. Farmaki and S. Negrepontis
by Dale Alspach 10 Jun '04

10 Jun '04

This is an announcement for the paper "Block combinatorics" by V. Farmaki and S. Negrepontis. Abstract: In this paper we extend the block combinatorics partition theorems of Hindman and Milliken in the setting of the recursive system of the block Schreier families (B^xi) consisting of families defined for every countable ordinal xi. Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology. Archive classification: Combinatorics; Functional Analysis Mathematics Subject Classification: 05D10; 46B20 Remarks: 26 pages, AMS-LaTeX The source file(s), fn04.tex: 83752 bytes, is(are) stored in gzipped form as 0406188.gz with size 20kb. The corresponding postcript file has gzipped size 98kb. 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.CO/0406188 or http://arXiv.org/abs/math.CO/0406188 or by email in unzipped form by transmitting an empty message with subject line uget 0406188 or in gzipped form by using subject line get 0406188 to: math(a)arXiv.org.

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