Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Turdebek N. Bekjan and Zeqian Chen
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Noncommutative integral inequalities for convex functions of maximal functions and applications" by Turdebek N. Bekjan and Zeqian Chen. Abstract: In this paper, we establish a Marcinkiewicz type interpolation theorem for convex functions of maximal functions in the noncommutative setting. As applications, we prove the noncommutative analogue of the Doob inequality for convex functions of maximal functions on martingales, the analogue of the classical Dunford-Schwartz maximal ergodic inequality for convex functions of positive contractions, and that of Stein's maximal inequality for convex functions of symmetric positive contractions. As a consequence, we obtain the moment Burkholder-Davis-Gundy inequality for noncommutative martingales. Archive classification: math.FA math.PR Remarks: 18 pages Submitted from: chenzeqian(a)hotmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.2795 or http://arXiv.org/abs/1108.2795

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Low-distortion embeddings of graphs with large girth" by Mikhail I. Ostrovskii. Abstract: The main purpose of the paper is to construct a sequence of graphs of constant degree with indefinitely growing girths admitting embeddings into $\ell_1$ with uniformly bounded distortions. This result answers the problem posed by N.~Linial, A.~Magen, and A.~Naor (2002). Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: Primary: 46B85, Secondary: 05C12, 54E35 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.2542 or http://arXiv.org/abs/1108.2542

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Abstract of a paper by Subhash Khot and Assaf Naor
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Grothendieck-type inequalities in combinatorial optimization" by Subhash Khot and Assaf Naor. Abstract: We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. Archive classification: cs.DS cs.CC math.CO math.FA Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.2464 or http://arXiv.org/abs/1108.2464

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Abstract of a paper by Grigory L. Litvinov
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Approximation properties of locally convex spaces and the problem of uniqueness of the trace of a linear operator" by Grigory L. Litvinov. Abstract: In the present article, it is proved that every nuclear operator in a locally convex space E has a well-defined trace if E possesses the approximation property. However, even if a space possesses the approximation property this still does not guarantee a positive solution of A. Grothendieck's uniqueness problem for this space. Below, we present an example of a quasi-complete space with the approximation property in which it is not possible to define the trace for all Fredholm operators (in the sense of A. Grothendieck). We prove that the uniqueness problem has a positive solution if E possesses the "bounded approximation property." Preliminary information and results are presented in Section 2. A number of approximation-type properties of locally convex spaces and relations between these properties are considered in Section 3. The principal results of the present study, along with certain corollaries from these results (for example, the existence of a matrix trace), may be found in Section 4. Archive classification: math.FA math.OA Mathematics Subject Classification: 46A32, 46A35 Citation: Selecta Mathematica Sovietica, vol. 11, No.1 (1992), p. 25-40 Remarks: 18 pages Submitted from: glitvinov(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.1721 or http://arXiv.org/abs/1108.1721

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Abstract of a paper by Diogo Diniz, G. A. Munoz-Fernandez, Daniel Pellegrino and J. B. Seoane-Sepulveda
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal" by Diogo Diniz, G. A. Munoz-Fernandez, Daniel Pellegrino and J. B. Seoane-Sepulveda. Abstract: In this note we provide a family of constants, $C_{n}$, enjoying the Bohnenblust--Hille inequality and such that $\lim_{n\rightarrow\infty}C_{n}/C_{n-1}=1$, i.e., their asymptotic growth is the best possible. As a consequence, we also show that the optimal constants, $K_n$, in the Bohnenblust--Hille inequality have the best possible asymptotic behavior. Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.1550 or http://arXiv.org/abs/1108.1550

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Abstract of a paper by Stephen Sanchez
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "On the supremal $p$-negative type of a finite metric space" by Stephen Sanchez. Abstract: We study the supremal $p$-negative type of finite metric spaces. An explicit expression for the supremal $p$-negative type $\wp (X,d)$ of a finite metric space $(X,d)$ is given in terms its associated distance matrix, from which the supremal $p$-negative type of the space may be calculated. The method is then used to give a straightforward calculation of the supremal $p$-negative type of the complete bipartite graphs $K_{n,m}$ endowed with the usual path metric. A gap in the spectrum of possible supremal $p$-negative type values of path metric graphs is also proven. Archive classification: math.FA math.MG Mathematics Subject Classification: 51F99 (Primary) 46B85, 54E35 (Secondary) Remarks: 11 pages, 6 figures Submitted from: stephen.sanchez(a)unsw.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.0451 or http://arXiv.org/abs/1108.0451

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Abstract of a paper by Yanqi Qiu
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "On the operator space $OUMD$ property for the column Hilbert space $C$" by Yanqi Qiu. Abstract: The operator space $OUMD$ property was introduced by Pisier in the context of verctor-valued noncommutative $L_p$-spaces. It is still unknown whether the property is independent of $p$ in this setting. In this paper, we prove that the column Hilbert space $C$ is $OUMD_p$ for all $1 < p < \infty$, this answers positively a question asked by Ruan. Archive classification: math.FA math.OA Submitted from: yqi.qiu(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1107.4941 or http://arXiv.org/abs/1107.4941

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Abstract of a paper by G. A. Munoz-Fernandez, D. Pellegrino and J. B. Seoane-Sepulveda
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "Estimates for the asymptotic behavior of the constants in the Bohnenblust--Hille inequality" by G. A. Munoz-Fernandez, D. Pellegrino and J. B. Seoane-Sepulveda. Abstract: A classical inequality due to H.F. Bohnenblust and E. Hille states that for every positive integer $n$ there is a constant $C_{n}>0$ so that $$\left(\sum\limits_{i_{1},\dots,i_{n}=1}^{N}\left\vert U(e_{i_{^{1}}}, \dots ,e_{i_{n}})\right\vert^{\frac{2n}{n+1}}\right)^{\frac{n+1}{2n}}\leq C_{n}\left\Vert U\right\Vert$$ for every positive integer $N$ and every $n$-linear mapping $U:\ell_{\infty}^{N}\times\cdots\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$. The original estimates for those constants from Bohnenblust and Hille are $$C_{n}=n^{\frac{n+1}{2n}}2^{\frac{n-1}{2}}.$$ In this note we present explicit formulae for quite better constants, and calculate the asymptotic behavior of these estimates, completing recent results of the second and third authors. For example, we show that, if $C_{\mathbb{R},n}$ and $C_{\mathbb{C},n}$ denote (respectively) these estimates for the real and complex Bohnenblust--Hille inequality then, for every even positive integer $n$, $$\frac{C_{\mathbb{R},n}}{\sqrt{\pi}} = \frac{C_{\mathbb{C},n}}{\sqrt{2}} = 2^{\frac{n+2}{8}}\cdot r_n$$ for a certain sequence $\{r_n\}$ which we estimate numerically to belong to the interval $(1,3/2)$ (the case $n$ odd is similar). Simultaneously, assuming that $\{r_n\}$ is in fact convergent, we also conclude that $$\displaystyle \lim_{n \rightarrow \infty} \frac{C_{\mathbb{R},n}}{C_{\mathbb{R},n-1}} = \displaystyle \lim_{n \rightarrow \infty} \frac{C_{\mathbb{C},n}}{C_{\mathbb{C},n-1}}= 2^{\frac{1}{8}}.$$ Archive classification: math.FA Remarks: 7 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/1107.4814 or http://arXiv.org/abs/1107.4814

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Abstract of a paper by Apostolos Giannopoulos, Grigoris Paouris, and Beatrice-Helen Vritsiou
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "A remark on the slicing problem" by Apostolos Giannopoulos, Grigoris Paouris, and Beatrice-Helen Vritsiou. Abstract: The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I_1(K,Z_q^o(K))=\int_K ||< : ,x> ||_{L_q(K)}dx. We show that an upper bound of the form I_1(K,Z_q^o(K))\leq C_1q^s\sqrt{n}L_K^2, with 1/2\leq s\leq 1, leads to the estimate L_n\leq \frac{C_2\sqrt[4]{n}log(n)} {q^{(1-s)/2}}, where L_n:= max {L_K : K is an isotropic convex body in R^n}. Archive classification: math.FA math.MG Mathematics Subject Classification: 52A23, 46B06, 52A40 Remarks: 24 pages Submitted from: bevritsi(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1107.4527 or http://arXiv.org/abs/1107.4527

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Abstract of a paper by Radoslaw Adamczak, Rafal Latala, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠math.okstate.edu 26 Aug '11

26 Aug '11

This is an announcement for the paper "Chevet type inequality and norms of submatrices" by Radoslaw Adamczak, Rafal Latala, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of ``symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $\Gamma_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with independent log-concave unconditional rows. We show also that our Chevet type inequality does not extend to general isotropic log-concave random matrices. Archive classification: math.PR math.FA math.MG Mathematics Subject Classification: Primary 52A23, 46B06, 46B09, 60E15 Secondary 15B52, 94B75 Submitted from: radamcz(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1107.4066 or http://arXiv.org/abs/1107.4066

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