March 2005 - Banach - mathdept.okstate.edu

Abstract of a paper by Madjid Mirzavaziri and Mohammad Sal Moslehian
by Dale Alspach 29 Mar '05

29 Mar '05

This is an announcement for the paper "Parallelogram norm" by Madjid Mirzavaziri and Mohammad Sal Moslehian. Abstract: Replacing the triangle inequality by \|x+y\|^2\leq 2(\|x\|^2 + \|y\|^2) in the definition of norm we obtain the notion of parallelogram norm. We establish that every parallelogram norm is a norm in the usual sense. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 46C05 Remarks: 3 pages The source file(s), Paral1.tex: 4582 bytes, is(are) stored in gzipped form as 0503616.gz with size 2kb. The corresponding postcript file has gzipped size 27kb. Submitted from: msalm(a)math.um.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503616 or http://arXiv.org/abs/math.FA/0503616 or by email in unzipped form by transmitting an empty message with subject line uget 0503616 or in gzipped form by using subject line get 0503616 to: math(a)arXiv.org.

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Abstract of a paper by Mark Rudelson and Roman Vershynin
by Dale Alspach 23 Mar '05

23 Mar '05

This is an announcement for the paper "Sampling from large matrices: an approach through geometric functional analysis" by Mark Rudelson and Roman Vershynin. Abstract: We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix. This improves known algorithms for computing low-rank approximations of large matrices. We also estimate norms of random submatrices of A. This yields an improved approximation algorithm for all MAX-2CSP problems (which includes MAX-CUT and other graph problems). Our results are essentially dimension-free; the picture is only controlled by the norms of the matrix and not by its size or rank. We use methods of Probability in Banach spaces, in particular the law of large numbers for random operators. Archive classification: Functional Analysis; Numerical Analysis Mathematics Subject Classification: 15A60, 68W20, 15A18 The source file(s), rv-random-submatrices.tex: 50699 bytes, is(are) stored in gzipped form as 0503442.gz with size 16kb. The corresponding postcript file has gzipped size 82kb. Submitted from: vershynin(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503442 or http://arXiv.org/abs/math.FA/0503442 or by email in unzipped form by transmitting an empty message with subject line uget 0503442 or in gzipped form by using subject line get 0503442 to: math(a)arXiv.org.

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Abstract of a paper by V.Yaskin and M.Yaskina
by Dale Alspach 22 Mar '05

22 Mar '05

This is an announcement for the paper "Centroids and comparison of volumes" by V.Yaskin and M.Yaskina. Abstract: For $-1<p<1$ we introduce the concept of a polar $p$-centroid body ${\Gamma^*_p K}$ of a star body $K$. We consider the question of whether ${\Gamma^*_p K}\subset {\Gamma^*_p L}$ implies $\mathrm{vol}(L)\le \mathrm{vol}(K).$ Our results extend the studies by Lutwak in the case $p=1$ and Grinberg, Zhang in the case $p> 1$. Archive classification: Functional Analysis Mathematics Subject Classification: 52Axx Remarks: 18 pages The source file(s), centr.tex: 51970 bytes, is(are) stored in gzipped form as 0503290.gz with size 13kb. The corresponding postcript file has gzipped size 71kb. Submitted from: yaskinv(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503290 or http://arXiv.org/abs/math.FA/0503290 or by email in unzipped form by transmitting an empty message with subject line uget 0503290 or in gzipped form by using subject line get 0503290 to: math(a)arXiv.org.

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Abstract of a paper by V.Yaskin
by Dale Alspach 22 Mar '05

22 Mar '05

This is an announcement for the paper "A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space" by V.Yaskin. Abstract: The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if $k>3$. The problem is still open for $k=2,3$. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space $\mathbb{H}^n$. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A55, 52A20, 46B20 Remarks: 12 pages, 2 figures The source file(s), LDHBP.tex: 70816 bytes, pic04.eps: 9457 bytes, pic06.eps: 9542 bytes, is(are) stored in gzipped form as 0503289.tar.gz with size 25kb. The corresponding postcript file has gzipped size 59kb. Submitted from: yaskinv(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503289 or http://arXiv.org/abs/math.FA/0503289 or by email in unzipped form by transmitting an empty message with subject line uget 0503289 or in gzipped form by using subject line get 0503289 to: math(a)arXiv.org.

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Abstract of a paper by Miguel Martin, Javier Meri and Rafael Paya
by Dale Alspach 09 Mar '05

09 Mar '05

This is an announcement for the paper "On the intrinsic and the spatial numerical range" by Miguel Martin, Javier Meri and Rafael Paya. Abstract: For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a bounded linear operator $T:X\longrightarrow Y$ such that $\ecc W(T)\neq V(T)$. We also show that, up to renormig, for every non-reflexive Banach space $Y$, one can find a closed subspace $X$ and a bounded linear operator $T\in L(X,Y)$ such that $\ecc W(T)\neq V(T)$. Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobas property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 47A12 Remarks: 12 pages The source file(s), MartinMeriPaya.tex: 40725 bytes, is(are) stored in gzipped form as 0503076.gz with size 13kb. The corresponding postcript file has gzipped size 70kb. Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503076 or http://arXiv.org/abs/math.FA/0503076 or by email in unzipped form by transmitting an empty message with subject line uget 0503076 or in gzipped form by using subject line get 0503076 to: math(a)arXiv.org.

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