Pták function and symmetry

We investigate subadditivity of Pták function as a sufficient and necessary condition for symmetry in the class of involutive spectrally bounded algebras. Among others it is proved an analogue of Raikov's criterion for complete lmc^*-algebras in which the hermitian elements have finite spectral radius.     

Pták function and symmetry

We investigate subadditivity of Pták function as a sufficient and necessary condition for symmetry in the class of involutive spectrally bounded algebras. Among others it is proved an analogue of Raikov's criterion for complete lmc^*-algebras in which the hermitian elements have finite spectral radius. This paper is based on the author's Doctoral Thesis, University of Athens, written under the supervision of Professor M. Fragoulopoulou.     

Pták Homomorphism Theorem Revisited

Rodrigues' extension (1989) of the classical Pták's homomoiphism theorem to a non-necessarily locally convex setting stated that a nearly semi-open mapping between a semi-B-complete space and an arbitrary topoiogical vector space is semi-open. In this paper we study this extension and, as a consequence of the results obtained, provide an improvement of Pták's homomorphism theorem.     

On the projective Finsler metrizability and the integrability of Rapcsák equation

A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.     

Pták’s characterization of reflexivity in tensor products

We characterize the reflexivity of the completed projective tensor products of Banach spaces in terms of certain approximative biorthogonal systems.     

The majorant method in the theory of newton-kantorovich approximations and the pták error estimates

For a certain modified Newton-Kontorovich method, sharp error estimates are obtained by menas of the majorant method. In particular, these error estimates generalize Pták's estimates for the usual Newton-Kontorovich method.     

On the k—smoothness and k—strong smoothness

5l.IntroductionFrompal,er[11,L2],to[3j,tI1efol1owingresultsareobtained:1)LetXbeaBanachspace,thenxes(X)={xeX;llxIl=1}isG-differetiableifandonIyif,foreachyeX,letXI=span(x,y).2)LetXbeaBanachspace,thenxeS(X)isF-differetiableifandonlyifIn[4j,NanchaoxunandWangjianhuaintroducedk-smoothnessandk-strongsmooth-ness.lnthisPaPer,wegive:Theorem1LetXbeaBanachspace,thcnthefollowingstatementsareequivalent.(i)xo6S(X)isk-smooth.(ii)IfX*cyXisk-dimensionalsubspaceandx,eX*,thenwhereU(x,,r)={xeX;l1x-xo…     

The Riemann Hypothesis, the Generalized Riemann Hypothesis, and the Cesáro Operator

A result due to Nyman establishes the equivalence of the Riemann hypothesis with the density of a set of functions in L ²[0, 1]. Here a large class of analytic functions is considered, which includes the Riemann zeta function and the Dirichlet L-functions as well as functions not given by a Dirichlet series. For each such function there is an associated integral operator T on L ²[0, 1] such that has no zeros in Re(s) > 1/2 iff the operator T has dense range iff a specified set of functions is dense in L ²[0, 1].      

Randomized graph products, chromatic numbers, and the Lovász ϑ-function

For a graphG, let (G) denote the size of the largest independent set inG, and let (G) denote the Lovász -function onG. We prove that for somec>0, there exists an infinite family of graphs such that \alpha (G)n/2^{c\sqrt {\log n} }$$ " align="middle" border="0"> , wheren denotes the number of vertices in a graph. this disproves a known conjecture regarding the function.As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest.Incumbent of the Joseph and Celia Reskin Career Development Chair. Yigal Alon Fellow     

The life and work of Zbyněk Šidák (1933–1999)

Zbynk idák, the chief editor of the Applications of Mathematics, an outstanding Czech statistician and probabilist, died on November 12, 1999, aged 66 years. This article is devoted to memory of him and outlines his life and scientific work.     

Combinatorial Groupoids,Cubical Complexes,and the Lovász Conjecture

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature”, etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial “Theorema Egregium” for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend Babson–Kozlov–Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs. The author was supported by grants 144014 and 144026 of the Serbian Ministry of Science and Technology.     

Ball polytopes and the Vázsonyi problem

Let V be a finite set of points in the Euclidean d-space (d ≧ 2). The intersection of all unit balls B(υ, 1) centered at υ, where υ ranges over V, henceforth denoted by $ \mathcal{B} $ (V) is the ball polytope associated with V. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. (a) Define the boundary complex of $ \mathcal{B} $ (V), i.e., define its vertices, edges and facets in dimension 3, and investigate its basic properties. (b) Apply results of this investigation to characterize finite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950’s by means of ball polytopes, that the diameter of V is attained at most 2|V| ? 2 times. Call V extremal if its diameter is attained this maximal number (2|V| ? 2) of times. We extend the aforementioned result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope $ \mathcal{B} $ (V) and show that in this case the boundary complex of $ \mathcal{B} $ (V) is self-dual in some strong sense. The problem of constructing new types of extremal configurations is not addressed in this paper, but we do present here some such new types.