Stability of localized operators

Let ?p, 1?p?∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ?p-stability of the convolution operator Ca for different 1?p?∞ are equivalent to each other, i.e., if Ca has ?p-stability for some 1?p?∞ then Ca has ?q-stability for all 1?q?∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ?p-stability (or Lp-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.     

ψ-Stability of a cooperative game without side payments

In the present paper we defineψ-stability for theAumann-Peleg theory of cooperative games without side payments, and we prove some theorems which are analogous to the core theorem byAumann andBurger. These theorems provide foundations of the theory ofψ-stability for cooperative games without side payments in addition to being of interest for their own sake. We also consider the composition of two admissible functionsψ ₁ andψ ₂.     

Stability analysis of methods employing reducible rules for Volterra integral equations

The concept of (A ₀,S)-stability, for numerical methods approximating solutions of Volterra integral equations, is formally defined. New stability polynomials for the recent multi-lag type methods are obtained. (A ₀, 1)-stability of these and other methods employing reducible quadrature rules are also investigated.     

The Elliott conjecture for Villadsen algebras of the first type

We study the class of simple C^∗-algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C^∗-algebraic (Z-stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z-stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C^∗-algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C^∗-algebras.     

Non-hyperelliptic modular curves of genus 3

A curve C defined over Q is modular of level N if there exists a non-constant morphism from X₁(N) onto C defined over Q for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is Q-isogenous to a given three dimensional Q-quotient of J₁(N). Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J₁(N) and Jac C factors through the new part of J₁(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves and we present some examples that show the ambiguity of the non-new modular case.     

Extension of Turán's Theorem to the 2-Stability Number

 Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α₂(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α₂(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turán's Theorem to the q-stability number, for q>2. Received: July 21, 1999 Final version received: August 22, 2000 Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca     

Improved K1-stability for the embedding D5 into E6

Abstract

This article is dedicated to the surjective stability of the K₁-functor for Chevalley groups for the embedding D₅ into E₆. This case was already studied by Plotkin. In this article, we improve his result by showing that surjective stability holds under a weaker assumption on a ring. Another result of this article shows how the K₁-stability can help to study overgroups of subsystem subgroups.     

A purely infinite AH-algebra and an application to AF-embeddability

We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit ofC*-algebras of the formC ₀([0, 1),M _k ) and it absorbs the Cuntz algebra ctive limit of the finite and elementaryC*-algebrasC ₀([0, 1),M _k ). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exactC*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class ofC*-algebras can be embedded.     

An Enumeration of Binary Self-Dual Codes of Length 32

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C ₁ and C ₂, are equivalent if and only if there is a permutation of the coordinates of C ₁ that takes C ₁ into C ₂. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself.The main topic of this paper is the enumeration of inequivalent binary self-dual codes. We have developed algorithms that will take lists of inequivalent small codes and produce lists of larger codes where each inequivalent code occurs only a few times. We have defined a canonical form for codes that allowed us to eliminate the overenumeration. So we have lists of inequivalent binary self-dual codes of length up to 32. The enumeration of the length 32 codes is new. Our algorithm also finds the size of the automorphism group so that we can compute the number of distinct binary self-dual codes for a specific length. This number can also be found by counting and matches our total.     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.