Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Bin Han and Zhiqiang Xu
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Robustness properties of dimensionality reduction with gaussian random matrices" by Bin Han and Zhiqiang Xu. Abstract: In this paper we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal RIP constants, which plays a central role in the study of phaseless compressed sensing. Archive classification: cs.IT math.FA math.IT math.NA math.PR Remarks: 22 pages Submitted from: xuzq(a)lsec.cc.ac.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.01695 or http://arXiv.org/abs/1501.01695

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Abstract of a paper by Vladimir G. Troitsky and Foivos Xanthos
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Spaces of regular abstract martingales" by Vladimir G. Troitsky and Foivos Xanthos. Abstract: In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm. Archive classification: math.FA Submitted from: foivos(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.01685 or http://arXiv.org/abs/1501.01685

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Abstract of a paper by Sergo A. Episkoposian
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On the divergence of greedy algorithms with respect to Walsh subsystems in $L$" by Sergo A. Episkoposian. Abstract: In this paper we prove that there exists a function which $f(x)$ belongs to $L^1[0,1]$ such that a greedy algorithm with regard to the Walsh subsystem does not converge to $f(x)$ in $L^1[0,1]$ norm, i.e. the Walsh subsystem $\{W_{n_k}\}$ is not a quasi-greedy basis in its linear span in $L^1$ Archive classification: math.FA Citation: Journal of Nonlinear Analysis Series A: Theory, Methods & The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00832 or http://arXiv.org/abs/1501.00832

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Abstract of a paper by Erhan Caliskan and Pilar Rueda
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "s-Numbers sequences for homogeneous polynomials" by Erhan Caliskan and Pilar Rueda. Abstract: We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials. Archive classification: math.FA Submitted from: pilar.rueda(a)uv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00785 or http://arXiv.org/abs/1501.00785

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Abstract of a paper by Daniel Pellegrino
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On the optimal constants of the Bohnenblust--Hille and inequalities" by Daniel Pellegrino. Abstract: We find the optimal constants of the generalized Bohnenblust--Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $ \left( 1,2,...,2\right)$, sometimes called mixed $\left( \ell _{1},\ell _{2}\right) $-Littlewood inequality. We show that these optimal constants are precisely $\left( \sqrt{2}\right) ^{m-1}$ and this is somewhat surprising since a series of recent papers have shown that the constants of the Bohnenblust--Hille inequality have a sublinear growth, and in several cases the same growth was obtained for the constants of the generalized Bohnenblust--Hille inequality. This result answers a question raised by Albuquerque et al. (2013) in a paper published in 2014 in the Journal of Functional Analysis. We also improve the best known constants of the generalized Hardy--Littlewood inequality in such a way that an unnatural behavior of the old estimates (that will be clear along the paper) does not happen anymore. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00965 or http://arXiv.org/abs/1501.00965

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Abstract of a paper by Anthony Weston
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "An application of virtual degeneracy to two-valued subsets of $L_{p}$-spaces" by Anthony Weston. Abstract: Suppose $0 < p < 2$ and that $(\Omega, \mu)$ is a measure space for which $L_{p}(\Omega, \mu)$ is at least two-dimensional. Kelleher, Miller, Osborn and Weston have shown that if a subset $B$ of $L_{p}(\Omega, \mu)$ does not have strict $p$-negative type, then $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). Examples show that the converse of this statement is not true in general. In this note we describe a class of subsets of $L_{p}(\Omega, \mu)$ for which the converse statement holds. We prove that if a two-valued set $B \subset L_{p}(\Omega, \mu)$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space), then $B$ does not have strict $p$-negative type. This result is peculiar to two-valued subsets of $L_{p}(\Omega, \mu)$ and generalizes an elegant theorem of Murugan. It follows, moreover, that of certain types of isometry with range in $L_{p}(\Omega, \mu)$ cannot exist. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B85 Remarks: 3 page note Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.8481 or http://arXiv.org/abs/1412.8481

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Abstract of a paper by Alexander Koldobsky
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Slicing inequalities for measures of convex bodies" by Alexander Koldobsky. Abstract: We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $\lambda n$ for some $\lambda\in (0,1).$ The constant depends only on $\lambda.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20 Submitted from: koldobskiya(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.8550 or http://arXiv.org/abs/1412.8550

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Informal Analysis Seminar, March 14-15, 2015
by Artem Zvavitch 14 Jan '15

14 Jan '15

Dear Colleague, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, March 14-15, 2015. The plenary lecture series will be given by: Alexandre Eremenko (Purdue University) and Grigoris Paouris (Texas A&M University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please, register before February 15, 2014. Please feel free to contact us at informal(a)math.kent.edu for any further information. Attached is a poster that you are welcome to forward to any colleagues you think may be interested. Sincerely, Analysis Group at Kent State University

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Abstract of a paper by Brendan Pass and Susanna Spektor
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "On Khinchine type inequalities for pairwise independent Rademacher random variables" by Brendan Pass and Susanna Spektor. Abstract: We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is independent of $N$; in fact, we prove that the best constant one can hope for is at least $N^{1/2-1/p}$. Furthermore, we show that this estimate is sharp for exchangeable vectors when $p = 4$. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors. Archive classification: math.FA math.PR Submitted from: sanaspek(a)yandex.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.7859 or http://arXiv.org/abs/1412.7859

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Abstract of a paper by Mikhail I. Ostrovskii and Beata Randrianantoanina
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces" by Mikhail I. Ostrovskii and Beata Randrianantoanina. Abstract: For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G.~Schechtman. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary: 46B85, Secondary: 05C12, 30L05, 46B15, 52A21 Remarks: 35 pages, 4 figures Submitted from: randrib(a)miamioh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.7670 or http://arXiv.org/abs/1412.7670

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