Banach

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Abstract of a paper by Manor Mendel and Assaf Naor
by Dale Alspach 05 Dec '05

05 Dec '05

This is an announcement for the paper "Ramsey partitions and proximity data structures" by Manor Mendel and Assaf Naor. Abstract: This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in \cite{BFM86}). We then proceed to construct optimal Ramsey partitions, and use them to show that for every $\e\in (0,1)$, any $n$-point metric space has a subset of size $n^{1-\e}$ which embeds into Hilbert space with distortion $O(1/\e)$. This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor \cite{BLMN05}, in addition to considerably simplifying its proof. We use our new Ramsey partitions to design the best known approximate distance oracles when the distortion is large, closing a gap left open by Thorup and Zwick in \cite{TZ05}. Namely, we show that for any $n$ point metric space $X$, and $k\geq 1$, there exists an $O(k)$-approximate distance oracle whose storage requirement is $O(n^{1+1/k})$, and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking. Archive classification: Data Structures and Algorithms; Computational Geometry; Metric Geometry; Functional Analysis The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: anaor(a)microsoft.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/cs.DS/0511084 or http://arXiv.org/abs/cs.DS/0511084 or by email in unzipped form by transmitting an empty message with subject line uget 11084 or in gzipped form by using subject line get 11084 to: math(a)arXiv.org.

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Abstract of a paper by Jakub Duda
by Dale Alspach 05 Dec '05

05 Dec '05

This is an announcement for the paper "On Gateaux differentiability of pointwise Lipschitz mappings" by Jakub Duda. Abstract: We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f:X\to\R$ cone monotone, $g:X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Frechet differentiable. Archive classification: Functional Analysis Mathematics Subject Classification: 46G05; 46T20 Remarks: 11 pages; added name The source file(s), stronger_stepanoff.tex: 48652 bytes, is(are) stored in gzipped form as 0511565.gz with size 15kb. The corresponding postcript file has gzipped size 61kb. Submitted from: jakub.duda(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511565 or http://arXiv.org/abs/math.FA/0511565 or by email in unzipped form by transmitting an empty message with subject line uget 0511565 or in gzipped form by using subject line get 0511565 to: math(a)arXiv.org.

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Abstract of a paper by Julien Melleray
by Dale Alspach 19 Nov '05

19 Nov '05

This is an announcement for the paper "Computing the complexity of the relation of isometry between separable Banach spaces" by Julien Melleray. Abstract: We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris and Weaver. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 03E15 The source file(s), melleray_banachisometries.tex: 34260 bytes, is(are) stored in gzipped form as 0511456.gz with size 11kb. The corresponding postcript file has gzipped size 54kb. Submitted from: melleray(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511456 or http://arXiv.org/abs/math.FA/0511456 or by email in unzipped form by transmitting an empty message with subject line uget 0511456 or in gzipped form by using subject line get 0511456 to: math(a)arXiv.org.

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Activities In honor of Bill Johnson's 60th Birthday
by Dale Alspach 10 Nov '05

10 Nov '05

In honor of Bill Johnson's 60th birthday we have organized an /AMS Special Session on Extension of Functions/ <http://www.ams.org/amsmtgs/2095_program_ss5.html#title> at the 2006 Joint Mathematics Meeting in San Antonio, Texas. There will be a banquet on Thursday, January 12, at Biga's on the Bank <http://www.biga.com/> (*located on the River Walk, 203 South St. Mary's St, San Antonio, Texas 78205* ) starting at 7:00 pm. Biga's is a modern-American restaurant and its menu is inspired by the cuisines of Mexico and Asia. The price of the banquet is $20.00 and it is being supported in part by Texas A&M University. Please note that space is limited for the banquet and we are asking for attendees to register for it in advance. If you plan to attend, please send an email message to any of us Alvaro Arias aarias(a)math.du.edu <mailto:aarias@math.du.edu> Edward W. Odell odell(a)math.utexas.edu <mailto:odell@math.utexas.edu> Thomas Schlumprecht schlump(a)math.tamu.edu <mailto:schlump@math.tamu.edu> You can find more information at the website www.math.du.edu/~aarias/bill.htm

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Positions at Case
by Stanislaw J. Szarek 09 Nov '05

09 Nov '05

(This announcement, with additional active hyperlinks, is also accessible via http://www.case.edu/artsci/dean/searches/ ) The Department of Mathematics in the College of Arts & Sciences at Case Western Reserve University invites applications for one or more faculty positions. Although rank is open and commensurate with qualifications, we prefer to appoint at the rank of assistant professor. We especially emphasize coordination with Department, College and University goals, including undergraduate teaching in the University's SAGES (Seminar Approach to General Education and Scholarship) program. Areas of preference identified to complement existing department activities include: (1) Functional analysis, convexity theory and related high-dimensional phenomena, the area that recently has been often referred to as "asymptotic geometric analysis" and of which members of the Department are internationally recognized leaders. See http://www.cwru.edu/artsci/math/szarek/ and http://www.cwru.edu/artsci/math/werner/ for examples of recent research directions. Besides hires that would directly augment this research, the Department envisions expanding into related areas of non-commutative geometry/functional analysis or even theoretical computer science or complexity theory. (2) Probability theory. Areas currently represented in the Department include large deviation theory and stochastic differential equations. As indicated on our website, the Department has researchers in a number of mathematical fields, and a candidate's potential ability to interact and collaborate with colleagues in other areas of mathematics will be regarded as a valuable asset. Candidates are invited to address this point. Case has strong presences in biomedical, engineering and other scientific areas and, in addition to strong theoretical credentials, a candidate's potential ability to interact and collaborate with colleagues in other disciplines will also be regarded as a valuable asset. Notwithstanding the above, (3) exceptionally strong candidates in other areas will be considered. Depending on needs, (4) visiting positions/instructorships/lectureships may also be open. Indicate in which area you wish to be considered. The successful candidate will hold the Ph.D. or equivalent (Masters for lectureship) and have, relative to career stage, a distinguished record of publication, research, service, and teaching. Compensation commensurate with qualifications. Case is an integral part of one of the major research medical complexes in the country. It also has a major presence in various science and engineering disciplines. Geographically, it is located on the eastern edge of Cleveland, in northeast Ohio, adjacent to University Circle, home to the Cleveland Symphony Orchestra, the Cleveland Museum of Art, and many other cultural institutions. There is a wide variety of housing, schooling, and other amenities nearby. The Department has approximately 20 faculty, with several focused research areas. The Department is responsible for service (beginning with calculus), majors, and graduate instruction. Nominal teaching load is 2/2. The Department has a dedicated 8 CPU computational server with an SGI 3D graphics front end. Facilities of the Ohio Supercomputer Center are also available. Electronic applications only, to: James Alexander, math-faculty-position(a)case.edu, consisting of a letter of application, which indicates in which area of preference you wish to be considered, AMS cover sheet, a c.v., and the names and contact information for four referees to whom we may write. Evaluation of applications will begin December 15, 2005. Case is a recipient of a National Science Foundation ADVANCE institutional transformation grant to increase the participation of women in science and engineering. Case Western Reserve University is committed to diversity and is an affirmative action, equal opportunity employer. Applications from women or minorities are especially encouraged.

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Abstract of a paper by Valentin Ferenczi
by Dale Alspach 08 Nov '05

08 Nov '05

This is an announcement for the paper "On the number of non permutatively equivalent sequences in a Banach space" by Valentin Ferenczi. Abstract: This paper contains results concerning the Borel reduction of the relation $E_0$ of eventual agreement between sequences of $0$'s and $1$'s, to the relation of permutative equivalence between basic sequences in a Banach space. For more clarity in this abstract, we state these results in terms of classification by real numbers. If $R$ is some (analytic) equivalence relation on a Polish space $X$, it is said that $R$ is classifiable (by real numbers) if there exists a Borel map $g$ from $X$ into the real line such that $x R x'$ if and only if $g(x)=g(x')$. If $R$ is not classifiable, there must be $2^{\omega}$ $R$-classes. It is conjectured that any separable Banach space such that isomorphism between its subspaces is classifiable must be isomorphic to $l_2$. We prove the following results: - the relation $\sim^{perm}$ of permutative equivalence between normalized basic sequences is analytic non Borel, - if $X$ is a Banach space with a Schauder basis $(e_n)$, such that $\sim^{perm}$ between normalized block-sequences of $X$ is classifiable, then $X$ is $c_0$ or $\ell_p$ saturated for some $1 \leq p <+\infty$, - if $(e_n)$ is shrinking unconditional, and $\sim^{perm}$ between normalized disjointly supported sequences in $X$, resp. in $X^*$, are classifiable, then $(e_n)$ is equivalent to the unit vector basis of $c_0$ or $\ell_p$, - if $(e_n)$ is unconditional, then either $X$ is isomorphic to $l_2$, or $X$ contains $2^{\omega}$ subspaces or $2^{\omega}$ quotients which are spanned by pairwise non permutatively equivalent normalized unconditional bases. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 03E15 Remarks: 28 pages The source file(s), permutative_ferenczi.tex: 72505 bytes, is(are) stored in gzipped form as 0511170.gz with size 21kb. The corresponding postcript file has gzipped size 81kb. Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511170 or http://arXiv.org/abs/math.FA/0511170 or by email in unzipped form by transmitting an empty message with subject line uget 0511170 or in gzipped form by using subject line get 0511170 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi
by Dale Alspach 08 Nov '05

08 Nov '05

This is an announcement for the paper "Real hereditarily indecomposable Banach spaces and uniqueness of complex structure" by Valentin Ferenczi. Abstract: There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the example with complex quotient space has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable; this extends results of J. Bourgain and S. Szarek from 1986. The quaternionic example, on the other hand, has unique complex structure up to isomorphism; there also exists a space with an unconditional basis, non isomorphic to $l_2$, which admits a unique complex structure. These examples answer a question of Szarek. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 46B04; 47B99 Remarks: 29 pages The source file(s), cplexstructure_ferenczi.tex: 70811 bytes, is(are) stored in gzipped form as 0511166.gz with size 22kb. The corresponding postcript file has gzipped size 87kb. Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511166 or http://arXiv.org/abs/math.FA/0511166 or by email in unzipped form by transmitting an empty message with subject line uget 0511166 or in gzipped form by using subject line get 0511166 to: math(a)arXiv.org.

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Abstract of a paper by Aaron Zerhusen
by Dale Alspach 07 Nov '05

07 Nov '05

This is an announcement for the paper "An embedding theorem for pseudoconvex domains in Banach spaces" by Aaron Zerhusen. Abstract: We show that a pseudoconvex open subset of a Banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a Banach space with an unconditional basis. Archive classification: Complex Variables; Functional Analysis Mathematics Subject Classification: 32T; 46G20 Remarks: 14 pages The source file(s), zerhusen.tex: 30421 bytes, is(are) stored in gzipped form as 0511095.gz with size 10kb. The corresponding postcript file has gzipped size 60kb. Submitted from: azerhus(a)math.purdue.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.CV/0511095 or http://arXiv.org/abs/math.CV/0511095 or by email in unzipped form by transmitting an empty message with subject line uget 0511095 or in gzipped form by using subject line get 0511095 to: math(a)arXiv.org.

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Abstract of a paper by Bas Lemmens, Beata Randrianantoanina, and Onno van Gaans
by Dale Alspach 03 Nov '05

03 Nov '05

This is an announcement for the paper "Second derivatives of norms and contractive complementation in vector-valued spaces" by Bas Lemmens, Beata Randrianantoanina, and Onno van Gaans. Abstract: We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell_p(\ell_q)$ with $p,q\in (1,2)\cup (2,\infty)$ and obtain a complete characterization of its 1-complemented subspaces. Archive classification: Functional Analysis Mathematics Subject Classification: 46B45, 46B04 (Primary) 47B37 (Secondary) Remarks: 22 pages, LaTeX The source file(s), lplqsub.tex: 52714 bytes, is(are) stored in gzipped form as 0511044.gz with size 15kb. The corresponding postcript file has gzipped size 80kb. Submitted from: lemmens(a)maths.warwick.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511044 or http://arXiv.org/abs/math.FA/0511044 or by email in unzipped form by transmitting an empty message with subject line uget 0511044 or in gzipped form by using subject line get 0511044 to: math(a)arXiv.org.

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Abstract of a paper by Jakub Duda
by Dale Alspach 01 Nov '05

01 Nov '05

This is an announcement for the paper "Cone monotone mappings: continuity and differentiability" by Jakub Duda. Abstract: We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. First we show some relationships between these classes. Then, we study continuity and differentiability (also in the metric and $w^*$ senses) of mappings in these classes. Archive classification: Functional Analysis Mathematics Subject Classification: 46T20; 26B25 Remarks: 13 page; better abstract The source file(s), domdif_prep.tex: 55009 bytes, is(are) stored in gzipped form as 0510678.gz with size 16kb. The corresponding postcript file has gzipped size 56kb. Submitted from: jakub.duda(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510678 or http://arXiv.org/abs/math.FA/0510678 or by email in unzipped form by transmitting an empty message with subject line uget 0510678 or in gzipped form by using subject line get 0510678 to: math(a)arXiv.org.

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