Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

Minimax risk overl p -balls forl p -error

Summary Consider estimating the mean vector from dataN _n (, ² I) withl _q norm loss,q1, when is known to lie in ann-dimensionall _p ball,p(0, ). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).     

Energy functional depending on elastic strain and chemical composition

In this paper we deal with energy functionals depending on elastic strain and chemical composition and we obtain lower semicontinuity results, existence theorems and relaxation in the spacesH ^1,p(; ⁿ)×L ^q (; ^d) with respect to weak convergence. Our proofs use parametrized measures associated with weakly converging sequences.The research was partially supported by the National Science Foundation under Grants No. DMS-9000133 and DMS-9201215 and also by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis.The research was partially supported by the National Science Foundation uncer Grants No. DMs 911572, the AFOSR 91 0301, the ARO DAAL03 92 G 003 and also by the ARO and the NSF through the Center for Nonlinear Analysis.The research was supported by DGICYT (Spain) through Programa de Perfeccionamiento y Movilidad del Personal Investigador and through grant PB90-0245, by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis and also by the project EurHomogenization SC1-CT91-0732 of the European Comunity.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

A weak approach to finite elasticity

On the ground of four axioms we define thekinematics of perfectly elastic bodies and in particular the notion ofweak deformations of a perfectly elastic body. Weak deformations turn out to agree withweak diffeomorphisms introduced in [10], a class of rectifiable currents which enjoys good closure and compactness properties. Defining thedynamics of perfectly elastic bodies in terms of twoconstitutive conditions on the stored energy function, we can therefore prove existence of stable equilibrium weak deformations for mixed boundary value problems, which moreover satisfy equilibrium and conservation equations.This work has been partially supported by the Ministero dell'Universitá e della Ricerca Scientifica, by C.N.R. contract n. 91.01343.CT01, by the European Research Project GADGET, and by Grant n.11957 of the zech Acad. of Sciences.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

Join-continuous frames,Priestley's duality and biframes

The primary purpose of this paper is to study join-continuous frames. We present two representation theorems for them: one in terms of -subframes of complete Boolean algebras and the other in terms of certain Priestley spaces. This second representation is used to prove that the topological spaces whose frame of open sets is join-continuous are characterized by a condition which says that certain intersections of open sets are open. Finally, we show that Priestley's duality can be viewed as a partialization of the dual adjunction between the categories of, respectively, bitopological spaces and biframes, stated by B. Banaschewski, G. C. L. Brümmer and K. A. Hardie in [5].This work was partially supported by Centro de Matemáíica da Universidade de Coimbra.