June 2011 - Banach - mathdept.okstate.edu

Abstract of a paper by Guillaume Aubrun, Stanislaw J. Szarek and Deping Ye
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Entanglement thresholds for random induced states" by Guillaume Aubrun, Stanislaw J. Szarek and Deping Ye. Abstract: For a random quantum state on $H=C^d \otimes C^d$ obtained by partial tracing a random pure state on $H \otimes C^s$, we consider the whether it is typically separable or typically entangled. We show that a threshold occurs when the environment dimension $s$ is of order roughly $d^3$. More precisely, when $s \leq cd^3$, such a random state is entangled with very large probability, while when $s \geq Cd^3 \log^2 d$, it is separable with very large probability (here $C,c>0$ are appropriate effectively computable universal constants). Our proofs rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces. Our methods work also for multipartite systems and for "unbalanced" systems such as $C^{d} \otimes C^{d'}$, $d \neq d' $. Archive classification: quant-ph math.FA math.PR Report Number: Mittag-Leffler-2010fall Remarks: 29 pages Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2264 or http://arXiv.org/abs/1106.2264

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Abstract of a paper by Taras Banakh and Ivan Hetman
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "A hidden characterization of polyhedral convex sets" by Taras Banakh and Ivan Hetman. Abstract: We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$. Archive classification: math.FA math.CO Mathematics Subject Classification: 46A55, 52B05, 52A07, 52A37 Remarks: 8 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2227 or http://arXiv.org/abs/1106.2227

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Abstract of a paper by Jarno Talponen
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Convex-transitivity of Banach algebras via ideals" by Jarno Talponen. Abstract: We investigate a method for producing concrete convex-transitive Banach spaces. The gist of the method is in getting rid of dissymmetries of a given space by taking a carefully chosen quotient. The spaces of interest here are typically Banach algebras and their ideals. We also investigate the convex-transitivity of ultraproducts and tensor products of Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04, 47L20, 46Mxx, 47L10 Remarks: 18 pages Submitted from: talponen(a)cc.hut.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.1292 or http://arXiv.org/abs/1106.1292

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Abstract of a paper by Mark Rudelson and Shuheng Zhou
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Reconstruction from anisotropic random measurements" by Mark Rudelson and Shuheng Zhou. Abstract: Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. The current paper discusses the recently introduced Restricted Eigenvalue (RE) condition, which is among the most general assumptions on the matrix, guaranteeing recovery. We prove a reduction principle showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces. This principle allows us to establish the RE condition for several broad classes of random matrices with dependent entries, including random matrices with subgaussian rows and non-trivial covariance structure, as well as matrices with independent rows, and uniformly bounded entries. Archive classification: math.ST cs.IT math.FA math.IT stat.TH Report Number: Technical Report 522, University of Michigan, Department of Statistics Remarks: 30 Pages Submitted from: szhou(a)cs.cmu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.1151 or http://arXiv.org/abs/1106.1151

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Abstract of a paper by Lasse Leskela and Matti Vihola
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Stochastic order characterization of uniform integrability and tightness" by Lasse Leskela and Matti Vihola. Abstract: We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. Especially, we show that whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15, 60B10, 60F25 Remarks: 14 pages, 1 figure Submitted from: lasse.leskela(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.0607 or http://arXiv.org/abs/1106.0607

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Abstract of a paper by Christian Rosendal
by alspach＠math.okstate.edu 03 Jun '11

03 Jun '11

This is an announcement for the paper "Characterising subspaces of Banach spaces with a Schauder basis having the shift property" by Christian Rosendal. Abstract: We give an intrinsic characterisation of the separable reﬂexive Banach spaces that embed into separable reﬂexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht. Archive classification: math.FA Mathematics Subject Classification: 46B03 Submitted from: rosendal(a)math.uic.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.0472 or http://arXiv.org/abs/1106.0472

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Abstract of a paper by Philipp Hoffmann and Michael Mackey
by alspach＠math.okstate.edu 03 Jun '11

03 Jun '11

This is an announcement for the paper "On the second parameter of an $(m, p)$-Isometry" by Philipp Hoffmann and Michael Mackey. Abstract: A bounded linear operator $T$ on a Banach space $X$ is called an $(m, p)$-isometry if it satisfies the equation $\sum_{k=0}^{m}(-1)^{k} {m \choose k}\|T^{k}x\|^{p} = 0$, for all $x \in X$. In the first part of this paper we study the structure which underlies the second parameter of $(m, p)$-isometric operators. More precisely, we concentrate on the question of determining conditions on $q \neq p$ for which an $(m, p)$-isometry can be a $(\mu, q)$-isometry for some $\mu$. In the second part we extend the definition of $(m, p)$-isometry, to include $p=\infty$. We then study basic properties of these $(m, \infty)$-isometries. Archive classification: math.FA Submitted from: philipp.hoffmann(a)ucdconnect.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.0339 or http://arXiv.org/abs/1106.0339

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Abstract of a paper by G. Botelho, V. V. Favaro, D. Pellegrino and J. B. Seoane-Sepulveda
by alspach＠math.okstate.edu 03 Jun '11

03 Jun '11

This is an announcement for the paper "$L_{p}[0,1] \setminus \bigcup\limits_{q>p} L_{q}[0,1]$ is spaceable for every $p>0$" by G. Botelho, V. V. Favaro, D. Pellegrino and J. B. Seoane-Sepulveda. Abstract: In this short note we prove the result stated in the title; that is, for every $p>0$ there exists an infinite dimensional closed linear subspace of $L_{p}[0,1]$ every nonzero element of which does not belong to $\bigcup\limits_{q>p} L_{q}[0,1]$. This answers in the positive a question raised in 2010 by R. M. Aron on the spaceability of the above sets (for both, the Banach and quasi-Banach cases). We also complete some recent results from \cite{BDFP} for subsets of sequence spaces. Archive classification: math.FA Remarks: 3 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.0309 or http://arXiv.org/abs/1106.0309

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Abstract of a paper by Petr Hajek and Jarno Talponen
by alspach＠math.okstate.edu 03 Jun '11

03 Jun '11

This is an announcement for the paper "Smooth approximations of norms in separable Banach spaces" by Petr Hajek and Jarno Talponen. Abstract: Let X be a separable real Banach space having a k-times continuously Fr\'{e}chet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth norms. Archive classification: math.FA math.AG Mathematics Subject Classification: Primary 46B03, 46T20, Secondary 47J07, 14P20 Remarks: 10 pages Submitted from: talponen(a)cc.hut.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.6046 or http://arXiv.org/abs/1105.6046

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Abstract of a paper by Piotr Niemiec
by alspach＠math.okstate.edu 03 Jun '11

03 Jun '11

This is an announcement for the paper "Central points and measures and dense subsets of compact metric spaces" by Piotr Niemiec. Abstract: For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the metric space $K$. With use of the generalized Chebyshev centers, the central measure $\mu_X$ of an arbitrary compact metric space $X$ is defined. For a large class of compact metric spaces, including the interval $[0,1]$ and all compact metric groups, another `central' measure is distinguished, which turns out to coincide with the Lebesgue measure and the Haar one for the interval and a compact metric group, respectively. An idea of distinguishing infinitely many points forming a dense subset of an arbitrary compact metric space is also presented. Archive classification: math.FA Mathematics Subject Classification: Primary 46S30, 47H10, Secondary 46A55, 46B50 Remarks: 13 pages Submitted from: piotr.niemiec(a)uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.5706 or http://arXiv.org/abs/1105.5706

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