August 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by Vincent Lafforgue and Assaf Naor
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "A doubling subset of $L_p$ for $p>2$ that is inherently infinite dimensional" by Vincent Lafforgue and Assaf Naor. Abstract: It is shown that for every $p\in (2,\infty)$ there exists a doubling subset of $L_p$ that does not admit a bi-Lipschitz embedding into $\R^k$ for any $k\in \N$. Archive classification: math.MG 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/1308.4554 or http://arXiv.org/abs/1308.4554

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Abstract of a paper by Robert Bogucki
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "On explicit constructions of auerbach bases in separable Banach spaces" by Robert Bogucki. Abstract: This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pe{\l}czy{\'n}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite codimension and in the space $C(K)$ for $K$ compact countable metric space. Archive classification: math.FA Mathematics Subject Classification: 46B15, 46B20 Submitted from: r.bogucki(a)students.mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.4429 or http://arXiv.org/abs/1308.4429

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Abstract of a paper by Deping Ye
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "On the $L_p$ geominimal surface area and related inequalities" by Deping Ye. Abstract: In this paper, we introduce the $L_p$ Geominimal surface area for all $-n\neq p<1$, which extends the classical Geominimal surface area ($p=1$) by Petty and the $L_p$ Geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ Geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ Geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and the Santal\'{o} style inequality. Some cyclic inequalities are established to obtain the monotonicity of the $L_p$ Geominimal surface area. Comparison between the $L_p$ Geominimal surface area and the (formal) $p$-surface area is also provided. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: deping.ye(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.4196 or http://arXiv.org/abs/1308.4196

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Conference Announcement - 7th Conference on Function Spaces: May 2014
by Krzysztof Jarosz 27 Aug '13

27 Aug '13

7th Conference on Function Spaces will take place at the SIUE campus between May 20 and May 24, 2014. The Conference will follow the same format as the previous one: http://www.siue.edu/MATH/conference2010/ If you consider attending the Conference it would help our preparation if you could email us at kjarosz(a)siue.edu checking one of the following: I will participate, It is too early to decide, but I will likely come, Keep me on the mailing list but chances of me coming are rather low Comments: Could you also pass this information to your colleagues and graduate students? We received a small grant to cover some of the local expenses but at this point we are unable to offer any meaningful travel support. We are however applying for an NSF grant to defer travel and local cost for "graduate students, postdocs, young nontenured faculty, women and members of underrepresented groups" (NSF priority) as well as for invited speakers. Since the NSF founded the previous conferences in this series we are quite hopeful that they will provide participants' support again. Knowing well in advance the potential participants will increase chances for an adequate support. Sincerely yours, Krzysztof Jarosz Department of Mathematics and Statistics Southern Illinois University Edwardsville Edwardsville, IL 62026-1653, USA tel.: (618) 650-2354 fax: (618) 650-3771 e-mail: kjarosz(a)siue.edu http://www.siue.edu/~kjarosz/

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Abstract of a paper by Mirna Dzamonja
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Isomorphic universality and the number of pairwise non-isomorphic in the class of Banach spaces" by Mirna Dzamonja. Abstract: We study isomorphic universality of Banach spaces of a given density and a number of pairwise non-isomorphic models in the same class. We show that in the Cohen model the isomorphic universality number for Banach spaces of density $\aleph_1$ is $\aleph_2$, and analogous results are true for other cardinals (Theorem 1.2(1)) and that adding just one Cohen real to any model destroys the universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces and use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embeddings (very positive embeddings), is high (Theorem 4.8(1)), and similarly for the number of pairwise non-isomorphic models (Theorem 4.8(2)). Archive classification: math.LO math.FA Mathematics Subject Classification: 03E75, 46B26, 46B03, 03C45, 06E15 Submitted from: h020(a)uea.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.3640 or http://arXiv.org/abs/1308.3640

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Abstract of a paper by Petr Hajek and Thomas Schlumprecht
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "The Szlenk index of L_p(X)" by Petr Hajek and Thomas Schlumprecht. Abstract: We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=\omega^{\alpha}$, for some $\alpha<\omega_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le \begin{cases} \omega^{\alpha+1} &\text{ if $\alpha$ is a finite ordinal,} \omega^{\alpha} &\text{ if $\alpha$ is an infinite ordinal.} \end{cases}$$ Archive classification: math.FA Mathematics Subject Classification: 46B03 46B10 Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.3629 or http://arXiv.org/abs/1308.3629

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Abstract of a paper by Assaf Naor
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Comparison of metric spectral gaps" by Assaf Naor. Abstract: Let $A=(a_{ij})\in M_n(\R)$ be an $n$ by $n$ symmetric stochastic matrix. For $p\in [1,\infty)$ and a metric space $(X,d_X)$, let $\gamma(A,d_X^p)$ be the infimum over those $\gamma\in (0,\infty]$ for which every $x_1,\ldots,x_n\in X$ satisfy $$ \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n d_X(x_i,x_j)^p\le \frac{\gamma}{n}\sum_{i=1}^n\sum_{j=1}^n a_{ij} d_X(x_i,x_j)^p. $$ Thus $\gamma(A,d_X^p)$ measures the magnitude of the {\em nonlinear spectral gap} of the matrix $A$ with respect to the kernel $d_X^p:X\times X\to [0,\infty)$. We study pairs of metric spaces $(X,d_X)$ and $(Y,d_Y)$ for which there exists $\Psi:(0,\infty)\to (0,\infty)$ such that $\gamma(A,d_X^p)\le \Psi\left(\gamma(A,d_Y^p)\right)$ for every symmetric stochastic $A\in M_n(\R)$ with $\gamma(A,d_Y^p)<\infty$. When $\Psi$ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if $n\in \N$ and $p\in (2,\infty)$ then for every $f_1,\ldots,f_n\in L_p$ there exist $x_1,\ldots,x_n\in L_2$ such that \begin{equation}\label{eq:p factor} \forall\, i,j\in \{1,\ldots,n\},\quad \|x_i-x_j\|_2\lesssim p\|f_i-f_j\|_p, \end{equation} and $$ \sum_{i=1}^n\sum_{j=1}^n \|x_i-x_j\|_2^2=\sum_{i=1}^n\sum_{j=1}^n \|f_i-f_j\|_p^2. $$ This statement is impossible for $p\in [1,2)$, and the asymptotic dependence on $p$ in~\eqref{eq:p factor} is sharp. We also obtain the best known lower bound on the $L_p$ distortion of Ramanujan graphs, improving over the work of Matou\v{s}ek. Links to Bourgain--Milman--Wolfson type and a conjectural nonlinear Maurey--Pisier theorem are studied. Archive classification: math.MG 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/1308.2851 or http://arXiv.org/abs/1308.2851

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Abstract of a paper by Alon Dmitriyuk and Yehoram Gordon
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Large distortion dimension reduction using random variable" by Alon Dmitriyuk and Yehoram Gordon. Abstract: Consider a random matrix $H:\mathbb{R}^n\longrightarrow\mathbb{R}^m$. Let $D\geq2$ and let $\{W_l\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\mathbb{R}^n$. We ask what is the probability that for all $1\leq l\leq p$ and $x,y\in W_l$, \[ \|x-y\|_2\leq\|Hx-Hy\|_2\leq D\|x-y\|_2. \] We show that for $m=O\big(k+\frac{\ln{p}}{\ln{D}}\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$. Archive classification: math.FA Remarks: 18 pages Submitted from: gordon(a)techunix.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.2768 or http://arXiv.org/abs/1308.2768

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Abstract of a paper by Ludek Zajicek
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces" by Ludek Zajicek. Abstract: The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces. Archive classification: math.FA Mathematics Subject Classification: Primary: 46G05, Secondary: 26B05 Submitted from: zajicek(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.2415 or http://arXiv.org/abs/1308.2415

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Abstract of a paper by Enrico Boasso
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "On the Moore-Penrose inverse, EP Banach space operators and EP Banach algebra elements" by Enrico Boasso. Abstract: The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and $C^*$-algebra elements. Archive classification: math.FA Mathematics Subject Classification: Primary 15A09, Secondary 47A05 Citation: J. Math. Anal. Appl. 339(2) (2008), 1003-1014 Remarks: 20 pages, original research article Submitted from: eboasso(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1897 or http://arXiv.org/abs/1308.1897

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