An infinitesimal characterization of the complexity of homogeneous spaces

A characterization of the complexity of a homogeneous space of a reductive groupG is given in terms of the mutual position of the tangent Lie algebra of the stabilizer of a generic point of and the (–1)-eigenspace of a Weyl involution of .Supported by Russian Foundation for Basic Research, Grant No. 98-01-00598.Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, and Russian Foundation for Basic Research, Grant No. 98-01-00598.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

（Z2）^k-Actions with Fixed Point Set of Constant Codimension 2^k＋ 2

Let J.^r ,k denote the ideal in MO. of cobordism classes containing a representative that admits (Z2)^k-actions with a fixed point set of constant codimension r. In this paper we determine J,k^2k 2 and Ja,3^2^3 1.     

Deforming metrics with negative curvature by a fully nonlinear flow

By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for , any metric g with its modified Schouten tensor always can be deformed in a natural way to a conformal metric with constant -scalar curvature at exponential rate.Received: 2 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Jiayu Li: Supported in part by a grant from the National Science Foundation of China. Weimin Sheng: Supported by the Zhejiang Provincial Natural Science Foundation of China (No.102033).     

Conditions of topological conjugacy of gradient-like diffeomorphisms on irreducible 3-manifolds

In this paper, we obtain the topological classification of gradient-like diffeomorphisms and the conditions of topological conjugacy of Morse-Smale diffeomorphisms with finite sets of heteroclinic trajectories on three-dimensional manifolds.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 73–80, January, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-01-407, by the International Science Foundation under grant R99000, and by the Foundation Cultural Initiative.     

On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1

Summary We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr1 andd=3r+1 for almost all triangulations.Dedicated to R. S. Varga on the occasion of his sixtieth birthdaySupported in part by National Science Foundation Grant DMS-8701121Supported in part by National Science Foundation Grant DMS-8602337     

The Existence of Maximal w^＊-closed Submodules in Nest Algebra Modules

Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.     

A Result On the Probability Measures Dominated by g-Expectation

It is proved that a probability measure is dominated by g-expectation ε_μ[·] if and only if it can begenerated by Girsanov transformation via a process which is uniformly bounded by μ.     

On arrangements of Jordan arcs with three intersections per pair

Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union ofn regions bounded by Jordan curves in the plane. We show that, under some fairly weak conditions, a simply connected surface can be constructed that exactly covers this union and whose boundary has combinatorial complexity that is nearly linear, even though the covered region can have quadratic complexity. In the case where our regions are delimited by Jordan acrs in the upper halfplane starting and ending on thex-axis such that any pair of arcs intersect in at most three points, we prove that the total number of subarcs that appear on the boundary of the union is only (n(n)), where(n) is the extremely slowly growing functional inverse of Ackermann's function.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fourth and seventh authors has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. The seventh author in addition wishes to acknowledge support by a research grant from the NCRD—the Israeli National Council for Research and Development. The fifth author would like to acknowledge support in part by NSF grant DMS-8501947. Finally, the eighth author was supported in part by a National Science Foundation Graduate Fellowship.     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.*supported in part by Tian Yuan Foundation of NNSF (A0324612)**Supported by 973 Chinese NSF and Foundation of Chinese Academy of Sciences.***Supported in part by NNSF of China.Received: September 23, 2002; revised: November 30, 2003     

Error Reduction by Parallel Repetition—A Negative Result

We show that no xed number of parallel repetitions suces in order to reduce the error in two-prover one-round proof systems from one constant to another. Our results imply that the recent bounds proven by Ran Raz, showing that the number of rounds that suffice is inversely proportional to the answer length, are nearly best possible.* Incumbent of the Joseph and Celia Reskin Career Development Chair. Supported in part by grant No. MQT 300 from the International Science Foundation and by an AMS-FSU grant.     

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.     

∑-flat manifolds and Riemannian submersions

In this paper, we show that a certain rigidity condition (-flatness) for open nonnegatively curved manifoldsM is preserved by Riemannian submersions. The result can be applied to quotients ofM by groups of isometries. -flat metrics are also used to derive a splitting theorem for distance tubes of maximal volume growth.Research supported by the Heinrich Hertz Foundation (first author), and by grant DMS88-01999 from the National Science Foundation (second author).     

The closure of a similarity orbit is always arcwise connected

If T, A are Hilbert space operators such that A is the norm-limit of operators similar to T, then there exists a continuous function from [0,1] into the algebra of operators such that (t) is a limit of similarities of T for all t in the interval [0,1], (0)=T and (1)=A.This research was partially supported by a Grant of the National Science Foundation.     

On certain quotients of Hardy spaces

The quotient space of a Hardy space on a half-plane Imz> modulo the subspace of elements containing a factore ^iz is in some sense independent of . A formula is derived which exhibits a correspondence between any two such quotient spaces.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

On the reduction of Holt's problem to a finite interval

Summary The reduction of Holt's problem to a finite interval is studied. We give the error estimate for the reduction. The solution of the boundary value problem is computed by a variant of the chasing method.This work was partly supported by the Hungarian Academic Research Foundation under project N⁰1-3-86-221     

Parallel and semi-parallel hypersurfaces of \mathbb{S}^n × ℝ

We give a complete classification of totally umbilical, parallel and semi-parallel hypersurfaces of the Riemannian product space × ℝ. The author is a postdoctoral researcher supported by the Research Foundation — Flanders (F.W.O.).     

The overlay of lower envelopes and its applications

Let and be two collections of a total ofn (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of are the planar maps obtained by the xy-projections of the lower envelopes of , respectively. We show that the combinatiorial complexity of the overlay of the minimization diagrams of and of is O(n^2+ɛ), for any ɛ>0. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopes in three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply shaped convex sets in three dimensions. Work on this paper by the first author was supported by National Science Foundation Grant CCR-93-01259 and an NYI award. Work on this paper by the second author was supported by the Netherlands’ Organization for Scientific Research (NWO) and partially supported by ESPRIT Basic Research Action No. 6546 (project PROMotion). His current address is Department of Computer Science, Postech, San 31, Hyoja-Dong, Pohang, Kyungbuk 790–784, South Korea. Email: otfried@vision.postech.ac.kr. Work on this paper by the third author was supported by NSF Grant CCR-91-22103, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

On the complexity of a PL homotopy algorithm for zeros of polynomials

A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity. We prove that the cost of locating all zeros of a polynomial of degreen to an accuracy of (measured by the number of evaluations of the polynomial) grows no faster thanO(max{n ⁴,n ³log₂(n/)}).This work is in response to a question raised in a paper by S. Smale as to the efficiency of piecewise linear methods in solving equations. In comparison with a few results reported, the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.This work is supported in part by the Foundation of Zhongshan University Advanced Research Centre, and in part by the National Natural Science Foundation of China.