Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

The maximum size of a convex polygon in a restricted set of points in the plane

Assume we havek points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at mostk, for some positive constant. We show that there exist at leastk ^1/4 of these points which are the vertices of a convex polygon, for some positive constant=(). On the other hand, we show that for every fixed>0, ifk>k(), then there is a set ofk points in the plane for which the above ratio is at most 4k, which does not contain a convex polygon of more thank ^1/3+ vertices.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

Generalized kolmogorov inequalities for martingales

The classical ebyev inequality leads to an inequality for martingales which is often called the Kolmogorov inequality. It is shown here that many generalized ebyev inequalities for random variables lead in a similar way to martingale inequalities, and that the corresponding martingale inequality is sharp when the ebyev inequality is.On leave from Tel-Aviv University. Presently at the University of California, Berkeley.Research supported by National Science Foundation Grant MPS75-06173     

Vertex-reinforced jump processes on trees and finite graphs

We study the continuous time process on the vertices of the b-ary tree which jumps to each nearest neighbor vertex at the rate of the time already spent at that vertex times , plus 1, where is a positive constant. We show that for fixed b>1, if is large enough the process is transient, and if is close enough to zero it is recurrent. Related results for some other graphs and trees are also proved.The research is partly supported by The Nuffield Foundation.     

Nonlinear programming and stationary equilibria in stochastic games

Stationary equilibria in discounted and limiting average finite state/action space stochastic games are shown to be equivalent to global optima of certain nonlinear programs. For zero sum limiting average games, this formulation reduces to a linear objective, nonlinear constraints program, which finds the best stationary strategies, even when-optimal stationary strategies do not exist, for arbitrarily small. The work of the first author was supported in part by the Air Force Office of Scientific Research, and by the National Science Foundation under Grant No ECS-8704954.The work of the third author was supported by The Netherlands Organization for Scientific Research NWO, project 10-64-10.     

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.     

Existence of strong solutions for Itô's stochastic equations via approximations

Summary Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on any probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ^d and in domains in ^d are considered.Research supported by the Hungarian National Foundation of Scientific Research No. 2990.Supported in part by NSF Grant DMS-9302516     

Probability and expectation inequalities

Summary This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZA) P(ZA) for some class of setsA, (b)k(Z)k(Z) for some class of functionsk, and (c)l(Z)k(Z) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ, that, in several cases, (a) (b) (c); included here are cases of normal and elliptically contoured distributions. A case where (a) (b) (c) is studied and is expressed in terms ofn monotone functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748     

Self-intersections of 1-dimensional random walks

Summary Consider a random walk S _n on the integers, where the steps _i have mean 0 and variance ². Let T be the time of first self-intersection of the random walk. It is shown that, as , T grows at rate ^2/3. More precisely, T^2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.Research supported by National Science Foundation Grant MCS80-02698.     

On the fractional matching polytope of a hypergraph

For a hypergraph andb:⁺ define Conjecture. There is a matching of such that For uniform andb constant this is the main theorem of [4]. Here we prove the conjecture if is uniform or intersecting, orb is constant.The research was done while the author visited the Department of Mathematics at Rutgers University. Research supported in part by the Hungarian National Science Foundation under grant No. 1812Supported in party by NSF and AFOSR grants and by a Sloan Research Fellowship     

Sharp regularity theory for second order hyperbolic equations of Neumann type

Summary We consider the mixed problem for a general, time independent, second order hyperbolic equation in the unknown u, with datum g L₂() in the Neumann B.C., with datum f L₂(Q) in the right hand side of the equation and, say, initial conditions u₀=u₁=0. We obtain sharp regularity results for u in Q and ù| in , by a pseudo-differential approach on the half-space.Research partially supported by the National Science Foundation under Grants DMS-83-016668 and DMS-87-96320.     

Point homogeneous flat affine planes

All flat projective planes (P,) whose automorphism group contains a 2-dimensional, connected closed subgroup fixing at most one line 1, are classified, except the following 2 classes: (1) – L₂, and not transitive on P1, and (2) S₁xR.Research supported in part by National Research Council of Canada (NRC)Herrn Professor Dr. Werner Burau zum 70. Geburtstag     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.     

Some inequalities of Hardy-Littlewood type

[3] , >0 n ^– a _n , . , . . , .

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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant #234.     

A note on small ball probability of a Gaussian process with stationary increments

Let {X(t), 0t1} be a Gaussian process with mean zero and stationary increments. Let ²(h) =EX ²(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.Research supported by Charles Phelps Taft Post-doctoral Fellowship of the University of Cincinnati and by the Fok Yingtung Education Foundation of China.     

Colouring a graph frugally

We prove that any graph with maximum degree sufficiently large, has a proper vertex colouring using +1 colours such that each colour class appears at most log⁸ times in the neighbourhood of any vertex. We also show that for 1, the minimum number of colours required to colour any such graph so that each vertex appears at most times in the neighbourhood of any vertex is (+^1+1//), showing in particular that when =log/loglog, such a colouring cannot always be achieved with O() colours. We also provide a polynomial time algorithm to find such a colouring. This has applications to the total chromatic number of a graph.The second two authors were supported by NATO Collaborative Research Grant #CRG950235.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

On the spectra of finite-dimensional perturbations of matrix multiplication operators

Let be a closed rectifiable curve and a region in the complex plane. Suppose for each , R() represents multiplication by an nxn-matrix of rational functions and F() is a finite rank operator, both acting on the Hilbert space L ₂ ⁿ (). Sufficient conditions are given for the integer valued function dim ker (R()+F()) to be continuous at all but finitely many points in . This result is applied to singular integral operators.This work was partially supported by the National Science Foundation.