A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers

Given a Banach spaceX, letc ₀(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc ₀(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc ₀(X) admits a center iffX is quasi uniformly rotund. We construct a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the condition (Q) andX is not quasi uniformly rotund. It follows that the Banach spaceE=c ₀(X) has the property from the title. Eine überarbeitete Fassung ging am 4. 7. 2001 ein     

Towards Operations Research for all,CLAIO 2018

Annals of Operations Research -     

Cayley-like representations are for all algebras, not merely groups

Cayley 's Theorem represents an arbitrary group as a set of permutations with the group operation captured by the composition of permutations. A few other examples with related representations are monoids, Boolean algebras and Menger algebras, permutations now being replaced by functions with one or more arguments. Although Cayley-like representations appear to be rare, this article shows that they are not. The idea is to represent the elements of an arbitrary algebra by multivariable functions, and its operations by particular compositions of these functions. Any finite algebra can be so represented,and so can any variety generated by one finite subdirectly irreducible algebra. It will follow that these varieties are Cayley-like: semilattices, distributive lattices, median algebras, elementary Abelian p -groups (for fixed p), and those generated by a primal algebra. If the definition of Cayley-like is stretched to allow the representing functions to have an infinite number of arguments, then all algebras are Cayley-like.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

There are no infinite order polynomially complete lattices, after all

If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power has an antichain of size . Hence there are no infinite opc lattices. However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices. Received November 2, 1998; accepted in final form March 19, 1999.     

Closed classes of k-valued logic,all of whose congruences are trivial

In this article a criterion for the existence of only trivial congruences on a closed class of k-valued logic containing selectors is formulated and proved. All homomorphic, but nonisomorphic, images of a given closed class of k-valued logic containing selectors are described.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 499–509, October, 1977.The author thanks S. V. Yablonskii for guidance and assistance with the article.     

Primitive rispetto all’ oscillazione

We study the real valued functions having a primitive with respect to the oscillation or a primitive with respect to the oscillation up to anegligible set.

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Supported by MURST of Italy.     

A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres

Summary. The algorithm proved here solves the problem of orthogonal distance regression for the maximum norm with hyperplanes and hyperspheres. For each finite set of points in a Euclidean space of any dimension, the algorithm determines – through finitely many arithmetic operations – all the hyperplanes and hyperspheres that minimize the maximum Euclidean distance measured perpendicularly from the data. The algorithm finds all the slabs (bounded by parallel hyperplanes) and all the spherical shells (bounded by concentric hyperspheres) that contain all the data and are “rigidly supported” by the data (for which there does not exist any other pair of parallel hypersurfaces of the same type that intersect the data at the same points.) The computational complexity of the algorithm increases as the number of data points raised to the dimension of the ambient space. The solutions are then the midrange hyperplanes in the thinnest slabs, and the midrange hyperspheres in the thinnest shells. Their sensitivity to perturbations of the data is of the order of a power of the reciprocal of the smallest angle between two median hyperplanes separating two pairs of data points. The methods of proof consist in showing that if a pair of parallel hyperplanes or hyperspheres is not rigidly supported but encompasses all the data, then there exists a projective shift of their common projective center producing a thinner slab or shell that still contains all the data. Received December 14, 1999 / Revised version received August 30, 2000 / Published online September 19, 2001     

Dreamland with classic higgs field, dimensional reduction and all that

This text, on the one hand, is related to the talk delivered at the conference “Gauge Fields. Yesterday, Today, Tomorrow” dedicated to the 70th anniversary of Andrei Slavnov; on the other—in the form of a fairy tale—it summarizes some results of the research performed after this Fest, mainly due to discussion around the talk.     

All for one and one for all: Theoretical models,sociological theory,and mathematical sociology

I make the argument that mathematical sociology uniquely contributes to sociological theory through the theoretical models it develops to bridge the gap between the ideas of sociological theorists and data relevant to their empirical evaluation. My work on intergroup association and social integration, the fruit of a long-time collaboration with Fararo and more recently with Karpiński, is used throughout to illustrate my points.     

Non-intersection bodies, all of whose central sections are intersection bodies

We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of .

     

On solvable groups,all proper factor groups of which have finite ranks

This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1274–1281, September, 1993.     

A Class of Spaces Containing all Connected and all Locally Connected Spaces

A simultaneous generalization of connectedness and local connectedness, called sum connectedness, is introduced. The category of sum connected spaces forms the smallest co-reflective subcategory of TOP containing all connected spaces. A product theorem, analogous to that for locally connected spaces, is proved for sum connected spaces and a necessary and sufficient condition for the STONE -?ECH compactification of a TYCHONOFF space to be sum connected is obtained.