Banach July 2004

banach@mathdept.okstate.edu

2 participants
14 discussions

Abstract of a paper by S. Artstein, V. Milman, and S. J. Szarek
by Dale Alspach 15 Jul '04

15 Jul '04

This is an announcement for the paper "Duality of metric entropy" by S. Artstein, V. Milman, and S. J. Szarek. Abstract: For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in $R^n$. We prove that the two functions of t, N(K,tD) and N(D, tK^o), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K in $R^n$ and over positive integers n. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B10; 47A05; 52C17; 51F99 Remarks: 17 p., LATEX The source file(s), ArtMilSzaAoM.tex: 40692 bytes, is(are) stored in gzipped form as 0407236.gz with size 14kb. The corresponding postcript file has gzipped size 68kb. Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407236 or http://arXiv.org/abs/math.FA/0407236 or by email in unzipped form by transmitting an empty message with subject line uget 0407236 or in gzipped form by using subject line get 0407236 to: math(a)arXiv.org.

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Abstract of a paper by Stanislaw Szarek
by Dale Alspach 14 Jul '04

14 Jul '04

This is an announcement for the paper "The volume of separable states is super-doubly-exponentially small" by Stanislaw Szarek. Abstract: In this note we give sharp estimates on the volume of the set of separable states on N qubits. In particular, the magnitude of the "effective radius" of that set in the sense of volume is determined up to a factor which is a (small) power of N, and thus precisely on the scale of powers of its dimension. Additionally, one of the appendices contains sharp estimates (by known methods) for the expected values of norms of the GUE random matrices. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces. Archive classification: Quantum Physics; Functional Analysis Remarks: 20 p., LATEX; an expanded version of the original submission: more background material from convexity and geometry of Banach spaces, more exhaustive bibliography and improved quality of references to the bibliography The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/quant-ph/0310061 or http://arXiv.org/abs/quant-ph/0310061 or by email in unzipped form by transmitting an empty message with subject line uget /0310061 or in gzipped form by using subject line get /0310061 to: math(a)arXiv.org.

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Abstract of a paper by Jeremy J. Becnel
by Dale Alspach 13 Jul '04

13 Jul '04

This is an announcement for the paper "About countably-normed spaces" by Jeremy J. Becnel. Abstract: Here we present an overview of countably normed spaces. In particular, we discuss the main topologies---weak, strong, inductive, and Mackey---placed on the dual of a countably normed spaces and discuss the sigma fields generated by these topologies. In particlar, we show that the strong, inductive, and Mackey topologies are equivalent under reasonable conditions. Also we show that all four topologies induce the same Borel field under certain conditions. The purpose in mind is to provide the background material for many of the results used in White Noise Analysis. Archive classification: Functional Analysis Mathematics Subject Classification: 46A11 Remarks: 23 pages, 0 figures, Background material for White Noise Analysis The source file(s), NuclearSpace.bbl: 1198 bytes, NuclearSpace.tex: 1472 bytes, borel.tex: 5271 bytes, cns.tex: 16479 bytes, compare.tex: 6600 bytes, conclusion.tex: 4430 bytes, inductive.tex: 6567 bytes, nuclear.sty: 4578 bytes, strong.tex: 17400 bytes, tvs.tex: 14418 bytes, weak.tex: 3536 bytes, is(are) stored in gzipped form as 0407200.tar.gz with size 23kb. The corresponding postcript file has gzipped size 103kb. Submitted from: beck(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407200 or http://arXiv.org/abs/math.FA/0407200 or by email in unzipped form by transmitting an empty message with subject line uget 0407200 or in gzipped form by using subject line get 0407200 to: math(a)arXiv.org.

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Abstract of a paper by Christian Rosendal
by Dale Alspach 08 Jul '04

08 Jul '04

This is an announcement for the paper "Incomparable, non isomorphic and minimal Banach spaces" by Christian Rosendal. Abstract: A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence. Archive classification: Functional Analysis; Logic The source file(s), ArchiveIncomparable.tex: 57150 bytes, is(are) stored in gzipped form as 0407111.gz with size 19kb. The corresponding postcript file has gzipped size 81kb. Submitted from: rosendal(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407111 or http://arXiv.org/abs/math.FA/0407111 or by email in unzipped form by transmitting an empty message with subject line uget 0407111 or in gzipped form by using subject line get 0407111 to: math(a)arXiv.org.

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Banach July 2004