Phase transitions in Mandelbrot's percolation process in three dimensions

Summary We study the phase structure and transitions in three-dimensional Mandelbrot percolation—a process which generates random fractal sets. We establish the existence of three distinct phase transitions, and we show that two of these transitions, corresponding to percolation across the initial set by paths and sheets, are discontinuous.Work partially supported by NSF Grant: DMS 8806552Work partially supported by NSF Grant: DMS 8902152     

Rigidity of measures—The high entropy case and non-commuting foliations

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures. To Hillel Furstenberg with friendship and admiration Manfred Einsiedler is partially supported by the NSF Grant DMS 0400587. Anatole Katok is partially supported by the NSF Grant DMS 0071339.     

Some remarks on symplectic actions of compact groups

The Research was partially supported by NSF Grant DMS 8505550     

Classification of certain commutator ideals and the tensor product closure property

Research partially supported by NSF Grant DMS 8503390.     

On degenerate elliptic operators of infinite type

The author was partially supported by NSF Grant DMS 91-01161     

Torsion-free nilpotent groups andE-modules

Research partially supported by NSF Grant no. DMS 8701074, and Baylor University's Summer Sabbatical Program.     

Multi-linear measure theory and multiple stochastic integration

Summary Stochastic integration in a Lebesgue Stieltjes sense is developed in a general context of multi-linear measure theory based on the Fundamental Grothendieck Inequality and Factorization Theorem.Research partially supported by NSF Grant DMS 8601485     

Optimal nonlinear approximation

We introduce a definition of nonlinear n-widths and then determine the n-widths of the unit ball of the Sobolev spaceW _p ^r inL _q. We prove that in the sense of these widths the manifold of splines of fixed degree with n free knots is optimal for approximating functions in these Sobolev spaces.This author was supported by NSF Grant DMS 8620108This author was supported by NSF Grant DMS 8803585     

A very thin thick superatomic Boolean algebra

The existence of a very thin thick superatomic Boolean algebra is consistent with the axioms of set theory.Author partially supported by NSF Grant # MCS-8102599.Presented by Jan Mycielski.     

A solution to a problem of Dubins and savage

A problem left open in Dubins and Savages’ “How to Gamble if You Must” is solved. Research partially supported by NSF Grant DMS 91-57461.     

Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry

We study the class ofn-Riemannian manifolds in the title such that the torsion elements in the fundamental group have a definite bound on their orders. Our main result asserts the existence of a kind of generalized Seifert fiber structure onM ⁿ , for which the fundamental group of fibers injects into that ofM ⁿ . This provides a necessary and sufficient topological condition for a manifold to admit a sufficiently collapsed metric in our class. Among other consequences we obtain a strengthened version of the gap conjecture in this context.The work of the first author is partially supported by NSF Grant DMS 9303999. The work of the second author is supported by MSRI through NSF grant DMS 9022140 and partially supported by NSF Grant DMS 9204095.     

Null boundary controllability for semilinear heat equations

We consider null boundary controllability for one-dimensional semilinear heat equations. We obtain null boundary controllability results for semilinear equations when the initial data is bounded continuous and sufficiently small. In this work we also prove a version of the nonlinear Cauchy-Kowalevski theorem.W. Littman was partially supported by NSF Grant DMS 90-02919. The results of this paper were presented by Yung-Jen Lin Guo at the P.D.E. seminar at the University of Minnesota on January 27, 1993 and by W. Littman at the First International Conference on Dynamics Systems and Applications held in Atlanta in May 1993.     

On the connected components of the space of line transversals to a family of convex sets

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ³. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220.     

The existence of smooth densities for the prediction filtering and smoothing problems

The qualitative behavior of buckled states of two different models of elastic beams is studied. It is assumed that random imperfections affect the governing nonlinear equations. It is shown that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation whose coeffficients are Gaussian random variables. The corresponding asymptotic expansion for the displacement is to lowest order a Gaussian stochastic process.Work supported by NSF Grant No. DCR81-14726.Work supported by NSF Grant No. DMS87-01895.     

Approximation and extension of random functions

Stochastic versions of the extension theorems of Tietze and Dugundji are obtained, as well as an existence theorem for partitions of unity by random continuous functions. A form of the classical approximation theorem of Mergelyan valid for random holomorphic functions on random compact sets is presented. A similar approach yields versions of the approximation theorems of Runge, Arakelyan, and Vitushkin.Research of both authors was partially supported by the NSF under Grant No. DMS 85-02308     

Finite groups of uniform logarithmic diameter

We demonstrate the existence of an infinite family of finite groups with 2 generators and logarithmic diameter with respect to any set of generators. This answers a question of A. Lubotzky. Moreover, in our groups, all minimal sets of generators have at most 3 elements. Research partially supported by NSF Grant DMS 0401006     

Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients

Uniqueness is proved for the Dirichlet problem for second order nondivergence form elliptic operators with coefficients continuous except at a countable set of points having at most one accumulation point. Moreover, gradient estimates are proved.The authors are partially supported by the National Science Foundation Grant no. NSF/DMS 8421377-04.     

Uniform stabilization of a nonlinearly perturbed Kirchoff plate

This paper considers the problem of local uniform stability of certain plate equations where unstructured nonlinearities appear in the boundary conditions. Using appropriate feedback laws, we prove that the solutions to these equations possess the necessary regularity to achieve the desired stability.This research was partially supported by NSF Grant DMS 8903747 for Ph.D. students, and is part of the author's Ph.D. dissertation written under the supervision of Professor I. Lasiecka at the University of Virginia.     

A note on subgroups generated by reflections in Coxeter groups

Partially supported by NSF Grant DMS 8502310.     

Tangential interpolation with symmetries and two-point interpolation problem for matrix-valued H2-functions

We describe all solutions of the two-sided tangential interpolation problem in the class of matrix-valued Hardy functions when symmetries are added: these symmetries are defined in terms of involutions ofH ₂. The obtained results are applied to a one-sided two-points tangential interpolation for matrix functions.The research of this author is partially supported by the NSF Grant DMS 9500924 and by the Binational United States-Israel Foundation Grant 9400271.