The K-theory and the invertibility of almost periodic Toeplitz operators

We use K-theory to study the invertibility of systems of almost periodic Toeplitz operators.Research supported in part by NSF under Grant No. DMS-8717185.     

Symmetries of planar growth functions

Supported in part by NSF Grant DMS 84-01919     

Hopf-algebraic structure of combinatorial objects and differential operators

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described, and we survey some applications of this structure to combinatorics and to differential operators. Some possible future directions for this work are indicated. Supported in part by NASA Grant NAG 2-513. Supported in part by NSF Grant DMS 870-1085.     

Local spectral properties of multipliers on Banach algebras

Supported in part by NSF Grant DMS 90-96108. This work was completed while the first author visited Mississippi State University. Financial support and generous hospitality are gratefully acknowledged.     

Weak stability of almost regular contact foliations

We prove that on a compact manifold, a contact foliation obtained by a smallC ¹ perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC ¹-close to almost regular contact forms.Supported in part by NSF Grant DMS 90-01861     

Transversal Lie group actions on abstract CR manifolds

Supported by NSF Grant DMS 8603176     

Linear complexity algorithm for semiseparable matrices

A new linear complexity algorithm for general nonsingular semiseparable matrices is presented. For symmetric matrices whose semiseparability rank equals to 1 this algorithm leads to an explicit formula for the inverse matrix.Supported in part by the NSF Grant DMS 9306357     

Existence and stability of large scale nonlinear oscillations in suspension bridges

A nonlinear model of a suspension bridge is considered in which large-scale, stable oscillatory motions can be produced by constant loading and a small-scale, external oscillatory force. Loud's implicit-function theoretic method for determining existence and stability of periodic solutions or nonlinear differential equations is extended to a case of a non-differentiable nonlinearity.Author partially supported by NSF under Grant DMS 8318204 and AFOSR Grant 85-0330.Author partially supported by NSF under Grant DMS 9519882.Author partially supported by NSF under Grant DMS 8519776.     

A minimal cutset of the boolean lattice with almost all members

Two almost explicit constructions are given satisfying the title.This research was done while the authors visited the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455.Research supported in part by Hungarian National Foundation for Scientific Research, grant no. 1812.Research supported in part by NSF Grant MDS 87-01475.Research supported in part by NSF grant DMS 86-06225 and Airforce Grant OSR-86-0076.     

Non-totally transcendental unidimensional theories

Supported by NSF Grant DMS 8601289 and a Humboldt Foundation Fellowship     

Hilbert's Tenth Problem for fields of rational functions over finite fields

Summary We prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 and hence, Hilbert's Tenth Problem for any such field is undecidable.Oblatum 1-XI-1989Supported in part by NSF Grant DMS 8605198.     

Diameter-extremal subsets of spheres

We investigate those spherical point sets which, relative to the Hausdorff metric, give local minima of the diameter function, and obtain estimates which, in principle, justify computer-generated configurations. We test the new sets against two classical conjectures: Borsuk's and McMullen's.Supported in part by NSF Grant DMS 8602645.     

Occupation times in systems of null recurrent Markov processes

Summary We study asymptotic properties of differences of occupation times for infinite systems of noninteracting Markovian particles. Under a suitable normalisation we prove convergence in law to a nondegerate Gaussian field. We also obtain large deviations properties. These results generalise previous results obtained separately by both authors.Supported in part by the Office of Graduate Studies and Research (1990), University of Maryland and by NSF Grant No. DMS 9207928Supported in part by the Fonds Institutionnel de Recherche, Université du Québec à Trois-Rivières and by the Natural Sciences and Engineering Research Council of Canada, Grant No. OGP0042137     

Weakly coupled semilinear parabolic evolution systems

Summary We obtain a priori bounds, global existence results, and a variation of parameters representation for classical solutions to weakly coupled semilinear parabolic systems which include spatial and temporal inhomogeneities.Supported in part by NSF Grant #DMS-8803151.Supported in part by NSF Grant #DMS-8813071.Supported in part by Texas Advanced Research Grant #1100.     

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.     

A quantitative maximum entropy theorem for the real line

A bound B is constructed for a large and natural class of entropy integrals on the real line. The construction involves Burg's maximum entropy theorem for the circle group and techniques from harmonic analysis. As the solution of an extension problem, the construction of B leads to a further construction problem which can be viewed as a means of refining Krein's theorem on positive definite extensions.Supported in part by the NSF under Grant DMS 86-01311.     

On analytic disks in the spectrum of a uniform algebra

Supported by NSF Grant DMS 85-03780. The author would also like to express Ms gratitude to the Alexander von Humboldt Foundation and to the Universität des Saarlandes for their support during the academic year 1986–87.     

Dominating points and large deviations for random vectors

Summary We establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a Banach space. These estimates are based on the existence of dominating points in this setting.Dedicated to Peter Ney on the occasion of his 65th birthday.Supported in part by NSF Grant DMS-9503665Supported in part by NSF Grant DMS-9400024     

The algebraic structure of linearly recursive sequences under hadamard product

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field. Supported in part by NSF Grant DMS 870-1085.     

A stochastic characterization of harmonic morphisms

Research supported in part by NSF Grant DMS 87-21347