- Banach - mathdept.okstate.edu

Abstract of a paper by Piotr Indyk and Stanislaw Szarek
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "A simple construction of almost-Euclidean subspaces of $\ell_1^N$ via tensor products" by Piotr Indyk and Stanislaw Szarek. Abstract: It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\Omega(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding arbitrarily small distortions, or "almost Euclidean" subspaces. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B25, 52A21, 68P30 Remarks: 10 pages The source file(s), tensoring3e.tex: 37038 bytes, is(are) stored in gzipped form as 1001.0041.gz with size 13kb. The corresponding postcript file has gzipped size 99kb. Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1001.0041 or http://arXiv.org/abs/1001.0041 or by email in unzipped form by transmitting an empty message with subject line uget 1001.0041 or in gzipped form by using subject line get 1001.0041 to: math(a)arXiv.org.

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Abstract of a paper by F. Baudier, N. J. Kalton, and G. Lancien
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "A new metric invariant for Banach spaces" by F. Baudier, N. J. Kalton, and G. Lancien. Abstract: We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B20; 46T99 Remarks: 22 pages The source file(s), new_invariant_BKL.tex: 63462 bytes, is(are) stored in gzipped form as 0912.5113.gz with size 19kb. The corresponding postcript file has gzipped size 132kb. Submitted from: florent.baudier(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.5113 or http://arXiv.org/abs/0912.5113 or by email in unzipped form by transmitting an empty message with subject line uget 0912.5113 or in gzipped form by using subject line get 0912.5113 to: math(a)arXiv.org.

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Abstract of a paper by Mohammed Yahdi
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "A coanalytic rank on super-ergodic operators" by Mohammed Yahdi. Abstract: Techniques from Descriptive Set Theory are applied in order to study the Topological Complexity of families of operators naturally connected to ergodic operators in infinite dimensional Banach Spaces. The families of ergodic, uniform-ergodic,Cesaro-bounded and power-bounded operators are shown to be Borel sets, while the family of super-ergodic operators is shown to be either coanalytic or Borel according to specific structures of the Space. Moreover, trees and coanalytic ranks are introduced to characterize super-ergodic operators as well as spaces where the above classes of operators do not coincide. Archive classification: math.GN math.FA Mathematics Subject Classification: 47A35; 54H05 Remarks: 9 pages The source file(s), YahdiCoanalyticRankOnSuperErgodicOperators.tex: 28531 bytes, is(are) stored in gzipped form as 0912.5389.gz with size 9kb. The corresponding postcript file has gzipped size 80kb. Submitted from: myahdi(a)ursinus.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.5389 or http://arXiv.org/abs/0912.5389 or by email in unzipped form by transmitting an empty message with subject line uget 0912.5389 or in gzipped form by using subject line get 0912.5389 to: math(a)arXiv.org.

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Abstract of a paper by Mark Kozdoba
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "On the smallest L_2 projection of a curve in R^n" by Mark Kozdoba. Abstract: For a curve T:[0,1] -> R^n, we consider the directions theta in R^n which T "misses" the most and quantify this, as a function of the L_2 norm of T's differential. Archive classification: math.FA The source file(s), curvL2arch.tex: 21640 bytes, is(are) stored in gzipped form as 0912.5323.gz with size 8kb. The corresponding postcript file has gzipped size 79kb. Submitted from: marikk(a)tx.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.5323 or http://arXiv.org/abs/0912.5323 or by email in unzipped form by transmitting an empty message with subject line uget 0912.5323 or in gzipped form by using subject line get 0912.5323 to: math(a)arXiv.org.

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Abstract of a paper by Daniel Li, Herve Queffelec, Luis Rodriguez-Piazza
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "Some new thin sets of integers in Harmonic Analysis" by Daniel Li, Herve Queffelec, Luis Rodriguez-Piazza. Abstract: We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in $\Lambda$ have uniformly convergent series, and their Fourier coefficients are in $\ell_p$ for all $p>1$; moreover, all the Lebesgue spaces $L^q_\Lambda$ are equal for $q<+\infty$. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in $\Lambda$ is non separable. So these sets are very different from the thin sets of integers previously known. Archive classification: math.FA Mathematics Subject Classification: MSC: Primary: 42A36 ; 42A44 ; 42A55 ; 42A61 ; 43A46; Secondary: The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.4214 or http://arXiv.org/abs/0912.4214 or by email in unzipped form by transmitting an empty message with subject line uget 0912.4214 or in gzipped form by using subject line get 0912.4214 to: math(a)arXiv.org.

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Abstract of a paper by Shuheng Zhou
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "Restricted eigenvalue conditions on subgaussian random matrices" by Shuheng Zhou. Abstract: It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in $\R^p$, where $p$ is the dimensionality of the data, and show that a class of random matrices with independent rows, but not necessarily independent columns, satisfy the RE condition, when the sample size is above a certain lower bound. Here we explicitly introduce an additional covariance structure to the class of random matrices that we have known by now that satisfy the Restricted Isometry Property as defined in Candes and Tao 05 (and hence the RE condition), in order to compose a broader class of random matrices for which the RE condition holds. In this case, tools from geometric functional analysis in characterizing the intrinsic low-dimensional structures associated with the RE condition has been crucial in analyzing the sample complexity and understanding its statistical implications for high dimensional data. Archive classification: math.ST math.FA stat.ML stat.TH Remarks: 23 Pages The source file(s), graphs.tex: 71862 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.4045 or http://arXiv.org/abs/0912.4045 or by email in unzipped form by transmitting an empty message with subject line uget 0912.4045 or in gzipped form by using subject line get 0912.4045 to: math(a)arXiv.org.

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Abstract of a paper by Romain Demazeux
by alspach＠fourier.math.okstate.edu 07 Jan '10

07 Jan '10

This is an announcement for the paper "Weighted composition operators as Daugavet centers" by Romain Demazeux. Abstract: We investigate the norm identity $\|uC_\varphi + T\| = \|u\|_\infty + \|T\|$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T : C(S)\to C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D).$ Archive classification: math.FA Mathematics Subject Classification: 47B33, 47B38,46E15 Remarks: 18 pages The source file(s), Weighted_composition_operators_as_Daugavet_centers.tex: 57655 bytes, is(are) stored in gzipped form as 0912.4032.gz with size 15kb. The corresponding postcript file has gzipped size 112kb. Submitted from: romain.demazeux(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.4032 or http://arXiv.org/abs/0912.4032 or by email in unzipped form by transmitting an empty message with subject line uget 0912.4032 or in gzipped form by using subject line get 0912.4032 to: math(a)arXiv.org.

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Abstract of a paper by Mikko Kemppainen
by alspach＠fourier.math.okstate.edu 18 Dec '09

18 Dec '09

This is an announcement for the paper "On the Rademacher maximal function" by Mikko Kemppainen. Abstract: This paper studies a new maximal operator introduced by Hyt\"onen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The L^p-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to sigma-finite measure spaces with filtrations and the L^p-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for L^p-boundedness and also to provide a characterization by concave functions. Archive classification: math.FA Mathematics Subject Classification: 46E40 (Primary); 42B25 (Secondary) Remarks: 22 pages, 4 figures The source file(s), RMF.bbl: 4575 bytes RMF.tex: 148459 bytes averages.pdf: 1054 bytes filtrations.pdf: 1394 bytes mart11.pdf: 1111 bytes mart33.pdf: 1082 bytes, is(are) stored in gzipped form as 0912.3358.tar.gz with size 39kb. The corresponding postcript file has gzipped size . Submitted from: mikko.k.kemppainen(a)helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.3358 or http://arXiv.org/abs/0912.3358 or by email in unzipped form by transmitting an empty message with subject line uget 0912.3358 or in gzipped form by using subject line get 0912.3358 to: math(a)arXiv.org.

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Abstract of a paper by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazza
by alspach＠fourier.math.okstate.edu 18 Dec '09

18 Dec '09

This is an announcement for the paper "Some translation-invariant Banach function spaces which contain $c_0$" by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazza. Abstract: We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space $c_0$. Archive classification: math.FA Mathematics Subject Classification: MSC: Primary: 43A46, 46B20; Secondary: 42A55, 42B35, 43A07, 46E30 Citation: Studia Mathematica 163, 2 (2004) 137 - 155 The source file(s), LLQR3D.TEX: 56689 bytes, is(are) stored in gzipped form as 0912.3133.gz with size 18kb. The corresponding postcript file has gzipped size 109kb. Submitted from: daniel.li(a)euler.univ-artois.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.3133 or http://arXiv.org/abs/0912.3133 or by email in unzipped form by transmitting an empty message with subject line uget 0912.3133 or in gzipped form by using subject line get 0912.3133 to: math(a)arXiv.org.

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Abstract of a paper by Longyun Ding
by alspach＠fourier.math.okstate.edu 18 Dec '09

18 Dec '09

This is an announcement for the paper "Borel reducibility and Holder($\alpha$) embeddability between Banach spaces" by Longyun Ding. Abstract: We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called H\"older$(\alpha)$ embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in[1,+\infty)$. We also answer a problem presented by Kanovei in the affirmative by showing that $C({\Bbb R}^+)/C_0({\Bbb R}^+)$ is Borel bireducible to ${\Bbb R}^{\Bbb N}/c_0$. Archive classification: math.LO math.FA Mathematics Subject Classification: 03E15, 46B20, 47H99 Remarks: 29 pages The source file(s), Banach.tex: 57984 bytes, is(are) stored in gzipped form as 0912.1912.gz with size 16kb. The corresponding postcript file has gzipped size 128kb. Submitted from: dingly(a)nankai.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0912.1912 or http://arXiv.org/abs/0912.1912 or by email in unzipped form by transmitting an empty message with subject line uget 0912.1912 or in gzipped form by using subject line get 0912.1912 to: math(a)arXiv.org.

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