August 2012 - Banach - mathdept.okstate.edu

Abstract of a paper by Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Volume inequalities and additive maps of convex bodies" by Franz E. Schuster. Abstract: Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A40, 52A39 Citation: Mathematika 53 (2006), 211–234 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7290 or http://arXiv.org/abs/1207.7290

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Abstract of a paper by Rolf Schneider and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Rotation invariant Minkowski classes of convex bodies" by Rolf Schneider and Franz E. Schuster. Abstract: A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 33C55 Citation: Mathematika 54 (2007), 1–13 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7286 or http://arXiv.org/abs/1207.7286

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Abstract of a paper by Cleon S. Barroso, Michel P. Reboucas and Marcus A. M. Marrocos
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces" by Cleon S. Barroso, Michel P. Reboucas and Marcus A. M. Marrocos. Abstract: In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fr\'echet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe{\l}czy\'nski on Banach spaces containing $L_1(\mu)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence of $\ell_1$-isomorphs inside the underline space. We also study a kind of algebraic genericity for the Cauchy-Peano problem in spaces $E$ having complemented subspaces with unconditional Schauder basis. It is proved that if $\mathscr{K}(E)$ denotes the family of all continuous vector fields $f\colon E\to E$ for which $u'=f(u)$ has no solutions at any time, then $\mathscr{K}(E)\bigcup \{0\}$ is spaceable in sense that it contains a closed infinite dimensional subspace of $C(E)$, the locally convex space of all continuous vector fields on $E$ with the linear topology of uniform convergence on bounded sets. Archive classification: math.FA Remarks: 13 pages Submitted from: cleonbar(a)mat.ufc.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6777 or http://arXiv.org/abs/1207.6777

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Abstract of a paper by Jamilson Ramos Campos
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Multiple Cohen strongly $p$-summing operators, ideals, coherence and compatibility" by Jamilson Ramos Campos. Abstract: Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly $p$-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals is discussed in detail. Archive classification: math.FA Submitted from: jamilson(a)dce.ufpb.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6664 or http://arXiv.org/abs/1207.6664

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Abstract of a paper by Bjorn Kjos-Hanssen
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Effective Banach spaces" by Bjorn Kjos-Hanssen. Abstract: This thesis addresses Pour-El and Richards' fourth question from their book "Computability in analysis and physics", concerning the relation between higher order recursion theory and computability in analysis. Among other things it is shown that there is a computability structure that is uncountable. The example given is a structure on the Banach space of bounded linear operators on the set of almost periodic functions. Archive classification: math.LO math.FA Mathematics Subject Classification: 03D Remarks: Master's thesis, University of Oslo, 1997. Adviser: Dag Normann. The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6622 or http://arXiv.org/abs/1207.6622

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nsc
by Dale Alspach 13 Aug '12

13 Aug '12

Dear Colleagues, The Institute of Mathematics of the Hebrew University is planning to organize a memorial conference for Joram Lindenstrauss. The conference, titled "Banach spaces: geometry and analysis" will be held at the Institute of Advanced Studies of the Hebrew University in Jerusalem , May 26-31, 2013. More details will follow. Organizers Gideon Schechtman, Tomek Szankowski, Benjamin Weiss.

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