Stable Algebraic Topology and Stable Topological Algebra

Algebraic topology is a young subject, and its foundations arenot yet firmly in place. I shall give some history, examplesand modern developments in that part of the subject called stablealgebraic topology, or stable homotopy theory. This is by farthe most calculationally accessible part of algebraic topology,although it is also the least intuitively grounded in visualizablegeometric objects. It has a great many applications to othersubjects such as algebraic geometry and geometric topology.Time will not allow me to say as much as I would like aboutthat. Rather, I shall emphasize some foundational issues thathave been central to this part of algebraic topology since theearly 1960s, but that have been satisfactorily resolved onlyin the last few years. 1991 Mathematics Subject Classification55P42, 55N20.     

Complex Stable Sums of Complex Stable Random Variables

A definition of complex stable random variables is presented which includes earlier definitions as special cases. The class of complex stable random variables is characterized and is shown to be a subclass of the operator stable random variables. The exact conditions under which a sum of independent complex stable random variables is again complex stable are also found.     

Stable theories

We studyK _T(λ)=sup {|S(A)| : |A|≦λ} and extend some results for totally transcendental theroies to the case of stable theories. We then investigate categoricity of elementary and pseudo-elementary classes. This paper is a part of the author’s doctoral dissertation written at the Hebrew University of Jerusalem, under the kind guidance of Profeossr M. Rabin.     

Stable multi-sets

In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets.  For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined.  The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks. Received: February 14, 2001/Revised version: September 7, 2001     

Stable husbands

Suppose n boys and n girls rank each other at random. We show that any particular girl has at least (1/2 ? ?) in n and at most (1 + ?) in n different husbands in the set of all Gale/Shapley stable matchings defined by these rankings, with probability approaching 1 as n → ∞, if ? is any positive constant. the proof emphasizes general methods that appear to be useful for the analysis of many other combinatorial algorithms.     

迹稳定秩1的C*-代数的稳定有限性与弱迹稳定秩1

主要给出了迹稳定秩1的C*-代数的稳定有限性,证明了如果A是有单位元迹稳定秩1的C*-代数,则A是稳定有限的,引入了弱迹稳定秩1的定义,并且证明了如果有单位元的C*-代数A是迹稳定秩1的,则A是弱迹稳定秩1的.对于单的具有SP性质的有单位元的C*-代数A,如果A是弱迹稳定秩1的,则A是迹稳定秩1的.同时给出了迹稳定秩1的C*-代数的一个等价条件,证明了一个有单位元的可分的C*-代数A是迹稳定秩1的,等价于A=(t4)limn→∞(An,Pn),其中tsr(AN)=1.     

迹稳定秩1的C~*-代数的稳定有限性与弱迹稳定秩1

主要给出了迹稳定秩1的C~*-代数的稳定有限性,证明了如果A是有单位元迹稳定秩1的C~*-代数,则A是稳定有限的,引入了弱迹稳定秩1的定义,并且证明了如果有单位元的C~*-代数A是迹稳定秩1的,则A是弱迹稳定秩1的．对于单的具有SP性质的有单位元的C~*-代数A,如果A是弱迹稳定秩1的,则A是迹稳定秩1的．同时给出了迹稳定秩1的C~*-代数的一个等价条件,证明了一个有单位元的可分的C~*-代数A是迹稳定秩1的,等价于A=(t_4)limn→∞(A_n,p_n),其中tsr(A_n)=1．     

Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

Stable Subnorms II

In this paper we continue our study of stability properties of subnorms on subsets of finite-dimensional, power-associative algebras over the real or the complex numbers.     

Stable Subnorms II

In this paper we continue our study of stability properties of subnorms on subsets of finite-dimensional, power-associative algebras over the real or the complex numbers.     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

Smooth Stable Planes

This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7]. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold $P\times {\cal L}$ (Theorem (1.14)), and the smooth structure on the set P of points and on the set ${\cal L}$ of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space T_pP carries the structure of a locally compact affine translation plane ${\cal A}_p$ , see Theorem (2.5). Dually, we prove in Section 3 that for any line $L \in {\cal L}$ the tangent space ${\rm T}_L{\cal L}$ together with the set ${\cal \rm S}_L=\lbrace {\rm T}_{L}{\cal L}_p\mid p \in L\rbrace$ gives rise to some shear plane. It turned out that the translation planes ${\cal A}_p$ are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.     

Stable local cohomology

For a module having a complete injective resolution, we define a stable version of local cohomology. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this functor behaves like the usual local cohomology functor. Our main result is that when there is only one nonzero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.     

Stable Moving Least-Squares

It is a common procedure for scattered data approximation to use local polynomial fitting in the least-squares sense. An important instance is the Moving Least-Squares where the corresponding weights of the data site vary smoothly, resulting in a smooth approximation. In this paper we build upon the techniques presented by Wendland and present a somewhat simpler error analysis of the MLS approximation. Then, we show by example that the factor, which appears in the bound on the Lebesgue constant in [Holger Wendland, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal. 21 (1) (2001) 285–300], where N is the number of points used in the approximation, can be realized. Hence, we devise a method for choosing the weights smoothly so that the corresponding Lebesgue constant can be bounded independently of N. This is done by employing Voronoi weights. We conclude with some numerical examples exhibiting the effectiveness of the suggested method for highly irregular data sites.