Trace Formulas for Some Operators Related To Quadrature Domains in Riemann Surfaces

This note studies the trace formula of a class of pure operators A with finite rank self-commutators satisfying the condition that there is a finite dimensional subspace containing the image of the self-commutator and invariant with respect to A ^*. Besides, in this class, the spectrum of the operator A is covered by the projection of a union of quadrature domains in some Riemann surfaces.     

Divisible statistics

We consider divisible statistics with values in locally compact Abelian groups. We describe the class of weak limits (including contigual alternatives) of distributions of divisible statistics. We give a survey of results for special schemes of divisible statistics: spacings, higher order spacings, coverings of a circle by random arcs, spacing-frequencies.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostie, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 26, pp. 97–149, 1988.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

Divisible design graphs

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v,k,λ)-graphs, and like (v,k,λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v,k,λ)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.     

分次可除模

对于Ｇ—分次环Ｒ，我们证明如下结论：（１）若Ｒ是分次正则环，则Ｒ上的任一分次左Ｒ—模都是分次可除模；（２）若Ｒ分次非退化且Ｍ是分次可除左Ｒ—模，则Ｍｅ是可除左Ｒｅ—模；（３）若Ｇ是有序群，Ｍ是可除左Ｒ—模，则Ｍ～和Ｍ～是分次可除左Ｒ—模，其中Ｍ为分次左Ｒ—模Ｎ的子模     

Divisible rigid groups

A soluble group G is rigid if it contains a normal series of the form G = G₁ > G₂ > … > G_p > G_p+1 = 1, whose quotients G_i/G_i+1 are Abelian and are torsion-free as right ℤ[G/G_i]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients G_i/G_i+1 are divisible by any elements of respective groups rings Z[G/G_i]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 762–776, November–December, 2008.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Infinitely Divisible Point Processes

Journal of the Operational Research Society -     

Notes on Divisible and Torsionfree Modules

In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied.     

Constructions of Group Divisible Designs

In this paper we present constructions for group divisible designs from generalized partial difference matrices. We describe some classes of examples.     

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.     

Divisible and exponent actions of semigroups

In this paper we consider divisible actions of semigroups which have a completely simple minimal ideal and establish that these actions are determined by the divisible actions of the Schützenberger group of the minimal ideal. In the third section, we discuss the concept of an exponent action and present a result which relates the notion of divisible action to the notion of exponent action. The fourth section of this paper exhibits a characterization of exponent functions. To find such a characterization was the original objective of this project.     

An Introduction to Divisible Codes

This paper presents some basic facts about divisible codes, culminating in a divisible version of the Gleason-Pierce theorem on self-dual codes.     

Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.     

Constant Weight Codes and Group Divisible Designs

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.