December 2010 - Banach - mathdept.okstate.edu

Abstract of a paper by Rafal Latala
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Weak and strong moments of random vectors" by Rafal Latala. Abstract: We discuss a conjecture about comparability of weak and strong moments of log-concave random vectors and show the conjectured inequality for unconditional vectors in normed spaces with a bounded cotype constant. Archive classification: math.PR math.FA Mathematics Subject Classification: Primary 60E15, Secondary 52A40, 60B11 Remarks: 8 pages Submitted from: rlatala(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1012.2703 or http://arXiv.org/abs/1012.2703

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Abstract of a paper by Taras Banakh and Arkady Leiderman
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Uniform Eberlein compactifications of metrizable spaces" by Taras Banakh and Arkady Leiderman. Abstract: We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class. Archive classification: math.GN math.FA Mathematics Subject Classification: 54D35, 54G12, 54D30, 54D20 Remarks: 6 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/1012.0920 or http://arXiv.org/abs/1012.0920

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Abstract of a paper by Olivier Guedon and Emanuel Milman
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Interpolating thin-shell and sharp large-deviation estimates For isotropic log-concave measures" by Olivier Guedon and Emanuel Milman. Abstract: Given an isotropic random vector $X$ with log-concave density in Euclidean space $\Real^n$, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \[ \P(\abs{|X| -\sqrt{n}} \geq t \sqrt{n}) \leq C \exp(-c n^{\frac{1}{2}} \min(t^3,t)) \;\;\; \forall t \geq 0 ~, \] for some universal constants $c,C>0$. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when $X$ is $\psi_\alpha$ ($\alpha \in (1,2]$), in precise agreement with both estimates of Paouris. The upper bound on the thin-shell width $\sqrt{\Var(|X|)}$ we obtain is of the order of $n^{1/3}$, and improves down to $n^{1/4}$ when $X$ is $\psi_2$. Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. Archive classification: math.FA Remarks: 27 pages - also resolved the negative moment and deviation estimates, interpolating now between the thin-shell and the Paouris small-ball estimate Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.0943 or http://arXiv.org/abs/1011.0943

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Abstract of a paper by Radoslaw Adamczak, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Sharp bounds on the rate of convergence of the empirical covariance matrix" by Radoslaw Adamczak, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in S^{n-1}} \Big|\frac{1/N}\sum_{i=1}^N (|<X_i, x>|^2 - \E|<X_i, x>|^2\r)\Big| \leq C \sqrt{\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\leq N, x\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\sqrt n)_{i\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $\lambda_{\rm min}$ and $\lambda_{\rm max}$ of $AA^\top$ satisfy the inequalities $ 1 - C\sqrt{{n/N}} \le {\lambda_{\rm min}/N} \le \frac{\lambda_{\rm max}/N} \le 1 + C\sqrt{{n/N}} $ which is a quantitative version of Bai-Yin theorem \cite{BY} known for random matrices with i.i.d. entries. Archive classification: math.PR math.FA Mathematics Subject Classification: 52A20, 46B09, 52A21 (Primary) 15A52, 60E15 (Secondary) 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/1012.0294 or http://arXiv.org/abs/1012.0294

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Abstract of a paper by Rafal Latala
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Order statistics and concentration of l_r norms for log-concave vectors" by Rafal Latala. Abstract: We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15 (52A38, 60B11) Remarks: 17 pages Submitted from: rlatala(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.6610 or http://arXiv.org/abs/1011.6610

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Abstract of a paper by O.F.K. Kalenda, H. Pfitzner and J. Spurny
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "On quantification of weak sequential completeness" by O.F.K. Kalenda, H. Pfitzner and J. Spurny. Abstract: We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G.~Godefroy, N.~Kalton and D.~Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G.~Godefroy. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 9 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.6553 or http://arXiv.org/abs/1011.6553

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Abstract of a paper by Gustavo Corach and Alejandra Maestripieri
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Products of orthogonal projections and polar decompositions" by Gustavo Corach and Alejandra Maestripieri. Abstract: We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all products $PQP$, where $P,Q$ run over all orthogonal projections and we solve the problems \newline $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or $\YY.$ We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of $\XX.$ Archive classification: math.FA Mathematics Subject Classification: 47A05 Submitted from: gcorach(a)fi.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.5237 or http://arXiv.org/abs/1011.5237

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Announcement of Meeting
by Dale Alspach 10 Dec '10

10 Dec '10

Announcement of Meeting: SET THEORETIC TECHNIQUES IN FUNCTIONAL ANALYSIS To be held in Castro Urdiales (Cantabria) Spain, from February 21 to February 24, 2011. Organized by Jesús M. F. Castillo (Univ. of Extremadura) and Manuel González (Univ. of Cantabria), in collaboration with the CIEM (International Center for Matematical Encounters). The meeting includes six mini-courses lectured by: Antonio Avilés (Univ. Murcia, Spain) (2 hours) Push-out constructions in Banach spaces and Boolean algebras Valentin Ferenczi (Univ. Sao Paulo, Brasil) (2 hours) Groups of isometries on Banach spaces Piotr Koszmider (Univ. Lodz, Poland) (2 hours) Some applications of set-theoretic topological methods in C(K) spaces Wieslaw Kubis (Univ. Praga, Czech Republic) (2 hours) Category-theoretic methods for constructing universal Banach spaces Jordi López Abad (ICMAT-CSIC, Madrid, Spain) (2 hours) Banach Spaces and Ramsey Theory: some open problems Stevo Todorcevic (Univ. Toronto, Canada) (3 hours) Combinatorial dichotomies in set theory and their applications to analysis Participants will have the possibility of delivering short lectures of 20 or 30 minutes. Registration for the meeting can be done through the web-site: http://www.ciem.unican.es/encuentros/banach2011. There is no registration fee. Additional information can be found in that web-site. The meeting is supported by CIEM, Universidad de Cantabria, Ayuntamiento de Castro Urdiales and Ingenio Mathematica. Antonio MArtínez-Abejón (University of Oviedo, Spain)

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