Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

On the ordering of -modal cycles

The forcing relation on -modal cycles is studied. If is an -modal cycle then the -modal cycles with block structure that force form a -horseshoe above . If -modal forces , and does not have a block structure over , then forces a -horseshoe of simple extensions of .

     

On vanishing of characteristic numbers in Poincaré complexes

Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

     

On extension of cocycles to normalizer elements, outer conjugacy, and related problems

Let be an ergodic automorphism of a Lebesgue space and a cocycle of with values in an Abelian locally compact group . An automorphism from the normalizer of the full group is said to be compatible with if there is a measurable function such that at a.e. . The topology on the set of all automorphisms compatible with is introduced in such a way that becomes a Polish group. A complete system of invariants for the -outer conjugacy (i.e. the conjugacy in the quotient group is found. Structure of the cocycles compatible with every element of is described.

     

Powers in Finitely Generated Groups

In this paper we study the set of -powers in certain finitely generated groups . We show that, if is soluble or linear, and contains a finite index subgroup, then is nilpotent-by-finite. We also show that, if is linear and has finite index (i.e. may be covered by finitely many translations of ), then is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

     

A Concordance Extension Theorem

Let be a manifold approximate fibration between closed manifolds, where , and let be the mapping cylinder of . In this paper it is shown that if is any concordance on , then there exists a concordance such that and . As an application, if and are closed manifolds where is a locally flat submanifold of and and , then a concordance extends to a concordance on such that . This uses the fact that under these hypotheses there exists a manifold approximate fibration , where is a closed -manifold, such that the mapping cylinder is homeomorphic to a closed neighborhood of in by a homeomorphism which is the identity on .

     

Seifert manifolds with fiber spherical space forms

We study the Seifert fiber spaces modeled on the product space . Such spaces are ``fiber bundles' with singularities. The regular fibers are spherical space-forms of , while singular fibers are finite quotients of regular fibers. For each of possible space-form groups of , we obtain a criterion for a group extension of to act on as weakly -equivariant maps, which gives rise to a Seifert fiber space modeled on with weakly -equivariant maps as the universal group. In the course of proving our main results, we also obtain an explicit formula for for a cocompact crystallographic or Fuchsian group . Most of our methods for apply to compact Lie groups with discrete center, and we state some of our results in this general context.

     

Lévy group action and invariant measures on

For let be defined by . We investigate permutations of , which satisfy as with for (i.e. is in the Lévy group , or for in the subspace of Cesàro-summable sequences. Our main interest are -invariant means on or equivalently -invariant probability measures on . We show that the adjoint of maps measures supported in onto a weak*-dense subset of the space of -invariant measures. We investigate the dynamical system and show that the support set of invariant measures on is the closure of the set of almost periodic points and the set of non-topologically transitive points in . Finally we consider measures which are invariant under .

     

Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Let be the set of holomorphic functions on the unit disc with and Dirichlet integral not exceeding one, and let be the set of complex-valued harmonic functions on the unit disc with and Dirichlet integral not exceeding one. For a (semi)continuous function , define the nonlinear functional on or by . We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

     

Functorial structure of units in a tensor product

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let be an algebraically closed field. Let and be reduced rings containing , having connected spectra. Let be a unit. Then for some units and .

Here is a deeper consequence, stated for simplicity in the affine case only. Let be a field, and let be a homomorphism of finitely generated -algebras such that is dominant. Assume that every irreducible component of or is geometrically integral and has a rational point. Let be a faithfully flat homomorphism of reduced -algebras. For a -algebra, define to be . Then satisfies the following sheaf property: the sequence

is exact. This and another result are used to prove (5.2) of [7].

     

The Bergman kernel function of some Reinhardt domains

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points . Let be the Reinhardt domain

where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that

holds for every . Here

and is the defining function for . The constants and depend only on and , not on .

     

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Let be a real Banach space and a bounded, open and convex subset of The solvability of the fixed point problem in is considered, where is a possibly discontinuous -dissipative operator and is completely continuous. It is assumed that is uniformly convex, and A result of Browder, concerning single-valued operators that are either uniformly continuous or continuous with uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let The effect of a weak boundary condition of the type on the range of operators is studied for -accretive and maximal monotone operators Here, with sufficiently large norm and Various new eigenvalue results are given involving the solvability of with respect to Several results do not require the continuity of the operator Four open problems are also given, the solution of which would improve upon certain results of the paper.

     

Regularity theory and traces of -harmonic functions

In this paper we discuss two different topics concerning -
harmonic functions. These are weak solutions of the partial differential equation

where for some fixed , the function is bounded and for a.e. . First, we present a new approach to the regularity of -harmonic functions for . Secondly, we establish results on the existence of nontangential limits for -harmonic functions in the Sobolev space , for some , where is the unit ball in . Here is allowed to be different from .

     

Nonnegative Radix Representations for the Orthant

Let be a nonnegative real matrix which is expanding, i.e. with all eigenvalues , and suppose that is an integer. Let consist of exactly nonnegative vectors in . We classify all pairs such that every in the orthant has at least one radix expansion in base using digits in . The matrix must be a diagonal matrix times a permutation matrix. In addition must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set can be diagonally scaled to lie in . The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.

     

Uniform harmonic approximation of bounded functions

Let be an open set in and be a relatively closed subset of . We characterize those pairs which have the following property: every function which is bounded and continuous on and harmonic on can be uniformly approximated by functions harmonic on . Several related results concerning both harmonic and superharmonic approximation are also established.

     

Finite-dimensional lattice-subspaces of

Let be linearly independent positive functions in , let be the vector subspace generated by the and let denote the curve of determined by the function , where . We establish that is a vector lattice under the induced ordering from if and only if there exists a convex polygon of with vertices containing the curve and having its vertices in the closure of the range of . We also present an algorithm which determines whether or not is a vector lattice and in case is a vector lattice it constructs a positive basis of . The results are also shown to be valid for general normed vector lattices.

     

An -minimal Boolean algebra

For every linear order we define a notion of -minimal Boolean algebra and then give a consistent example of an -minimal algebra. The Stone space of our algebra contains a point such that is an example of a countably tight, initially -compact, non-compact space. This answers a question of Dow and van Douwen.