A characterization of strong measure zero sets

We show that a setXυR has strong measure zero iff for every closed measure zero setFυR,F+X has measure zero. Supported by KBN grant PB 2 1017 91 01.     

Diffusion approximation of branching migration processes

We study the behavior of a Galton-Watson process with homogeneous migration component stopped at zero (i.e., the state zero is absorbing). Assuming that the process is initiated at time zero by a large number of particles, we find a diffusion approximation for this process in the case where the average number of offspring per individual is close to one. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-00338 and 96-15-96092) and INTAS-RFBR (grant No. 95-0099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1044–1068, August, 1995.This research was partially supported by the International Science Foundation, grant No. UB2000.     

Zero sets of holomorphic functions in the bidisk

We characterize in geometric terms the zero sets of holomorphic functionsf in the bidisk such that log |f|∈L ^p (D ²) for 1<p<∞. Partially supported by the DGCYT grant PB95-0956-C02-02 and grant 1996-SGR-26.     

On Invertible and Dense Submodules

Abstract

In this paper, the authors give a partial characterization of invertible, dense and projective submodules. In the final section, they give the equivalent conditions to be invertible, dense and projective submodules for a given an R-module M. They also provide conditions under which a given ring R is a Dedekind domain if and only if every non zero submodule of an R-module is locally free.     

The Inflated Attractors of Non-autonomous Strongly Damped Wave Equations

We investigate the existence and the continuity of the inflated attractors for a class of nonautonomous strongly damped wave equations through differential inclusion.     

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix ?R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (?1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of ?R or ?R^{? 1} for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.     

The structure of bisimple left unipotent semigroups as ordered pairs

Call a semigroup S left unipotent if eachℒ-class of S contains exactly one idempotent. A structure theorem for bisimple left unipotent semigroups is given which reduces to that of N. R. Reilly [8] for bisimple inverse semigroups. A structure theorem, alternative to one given by R. J. Warne [13], is given for the case when the band E_S of idempotents of S is an ω-chain of right zero semigroups, and two applications of it are made. This research was partially supported by a grant from the National Science Foundation.     

An Adaptive Finite Element Method for the H- ψ Formulation of Time-dependent Eddy Current Problems

In this paper, we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the H − ψ formulation of eddy current problems with multiply connected conductors. Multiply connected domains are considered by making “cuts”. The competitive performance of the method is demonstrated by an engineering benchmark problem, Team Workshop Problem 7, and a singular problem with analytic solution.W. Zheng was supported in part by China NSF under the grant 10401040.Z. Chen was supported in part by China NSF under the grant 10025102 and 10428105, and by the National Basic Research Project under the grant 2005CB321701.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Choice Functions and Extensive Operators

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider classes of heritage choice functions satisfying conditions M, N, W, and C, or combinations of these conditions. In terms of extensive operators these classes can be considered as generalizations of symmetric, anti-symmetric and transitive binary relations. Among these classes we meet the well-known classes of matroids and convex geometries. Using a ‘topological’ language we discuss these classes of monotone extensive operators (or heritage choice functions) in terms of neighborhood systems. A remarkable inversion on the set of choice functions is introduced. Restricted to the class of heritage choice functions the inversion turns out to be an involution, and under this involution the axiom N is auto-inverse, whereas the axioms W and M change places. This research was supported in part by NWO–RFBR grant 047.011.2004.017, by RFBR grant 05-01-02805 CNRSL_a, and the grant NSh-929.2008.6, School Support. We want to thank B. Monjardet, the editor and a referee for discussions and useful suggestions.     

Strongly regular growth of entire functions of order zero

For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pflüger. Translated fromMatematicheskie Zameiki, Vol. 63, No. 2, pp. 196–208, February, 1998. This research was partially supported by the International Science Foundation under grant No. UCR000.     

ASPECTS OF BOUNDED PERTURBATION THEORY

1. Abstract

This paper is concerned with the stability of certain properties of linear operators in locally convex topological vector spaces under perturbations by operators which are small in some sense. Section 3 deals with the very useful concept of Banach balls which was introduced by Ra?kov [9]. Some properties are discussed. The following section investigates the invertibility of certain operators generalizing results of Robert [10] and de Bruyn [2],[3]. These results are used extensively in the sequel. We go on to discuss Riesz operators. We obtain results stronger than those of de Bruyn [1] with regard to asymptotically quasi-compact operators in locally convex spaces. The proofs are basically adaptations of those from [1]. In the final section we observe some results concerning the range ad null space of an operator perturbed by bounded operators. We obtain a result very similar to an unproved theorem of Vladimirski? [a] and point out their differences. MOS codes 4601, 4710, 4745, 4768, 4755.

This work was undertaken at Cambridge University and I would like to thank my research supervisor Dr. F. Smithies for his help and encouragement. I wish also to thank Dr. G.F.C. de Bruyn ad Dr. J.H. Webb for their interesting discussions on this subject. During my research I was financed by a Sir Henry Strakosch Memorial Scholarship and a grant from the South African Council for Scientific and Industrial Research.     

A nowhere-zero point in linear mappings

We state the following conjecture and prove it for the case whereq is a proper prime power:Let A be a nonsingular n by n matrix over the finite field GF_qq4, then there exists a vector x in (GF_q)ⁿ such that both x and Ax have no zero component.Research supported in part by Allon Fellowship and by a Bat Sheva de Rothschild grant.     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Convergence of Bieberbach polynomials in domains with interior cusps

We extend the results on the uniform convergence of Bieberbach polynomials for domains with certain interior zero angles (outward pointing cusps) and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss thecritical order of tangency at this interior zero angle, separating the convergent behavior of Bieberbach polynomials from the divergent one for sufficiently thin cusps. Research of both authors was supported in part by the National Science Foundation grant DMS-9707359. Research of the second author was also supported in part by the National Science Foundation grant DMS-9970659.     

On the rigidity of Hardy submodules

In contrast with the one-variable case, there is a large number of distinct submodules of the Hardy module over the polydisk algebra in the multi-variable use. We show that under hypotheses on the zero sets, two submodules which are equivalent in any reasonable sense must be equal. This is the rigidity referred to in the title.The research of the first author was partially supported by a grant from the National Science Foundation.