Stochastic processes with sample paths in reproducing kernel Hilbert spaces

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either or . Driscoll also found a necessary and sufficient condition for that probability to be .

Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available.

Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.

     

A point set whose space of triangulations is disconnected

By the ``space of triangulations" of a finite point configuration we mean either of the following two objects: the graph of triangulations of , whose vertices are the triangulations of and whose edges are the geometric bistellar operations between them or the partially ordered set (poset) of all polyhedral subdivisions of ordered by coherent refinement. The latter is a modification of the more usual Baues poset of . It is explicitly introduced here for the first time and is of special interest in the theory of toric varieties.

We construct an integer point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the graph and the poset, which proves for the first time that these two objects cannot be connected.

     

An extremal property of Fekete polynomials

The Fekete polynomials are defined as

where is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known norm out of the polynomials with coefficients.

The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.

Theorem 0.1. Let with odd and . If

then must be an odd prime and is . Here

This result also gives a partial answer to a problem of Harvey Cohn on character sums.

     

Distribution of Hecke eigenvalues

We give two results concerning the distribution of Hecke eigenvalues of . The first result asserts that on certain average the Sato-Tate conjecture holds. The second result deals with the Gaussian central limit theorem for Hecke eigenvalues.

     

Extended admissible functions and Gaussian limiting distributions

We consider an extension of Hayman's notion of admissibility to bivariate generating functions that have the property that the coefficients satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a large class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is also presented. We apply this concept to various combinatorial examples.

     

On finite groups admitting a special noncoprime action

An important result of Turull (1984) is the following:

Let be a finite solvable group, and . Then , where denotes the Fitting height and denotes the composition length.

The purpose of this work is to give a treatment of the minimal configuration in this framework with additional conditions, yet without the coprimeness condition.

     

The power of standard information for multivariate approximation in the randomized setting

We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted -norm. We consider functions with an arbitrarily large number of variables, , and we focus on the randomized setting with algorithms using standard information consisting of function values at randomly chosen points.

We prove that standard information in the randomized setting is as powerful as linear information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the th minimal errors, i.e., of the minimal errors among all algorithms using function evaluations. Previously, it was only known that standard information in the randomized setting is no more powerful than the linear information in the worst case setting.

We also study (strong) tractability of multivariate approximation in the randomized setting. That is, we study when the minimal number of function evaluations needed to reduce the initial error by a factor is polynomial in  (strong tractability), and polynomial in and (tractability). We prove that these notions in the randomized setting for standard information are equivalent to the same notions in the worst case setting for linear information. This result is useful since for a number of important applications only standard information can be used and verifying (strong) tractability for standard information is in general difficult, whereas (strong) tractability in the worst case setting for linear information is known for many spaces and is relatively easy to check.

We illustrate the tractability results for weighted Korobov spaces. In particular, we present necessary and sufficient conditions for strong tractability and tractability. For product weights independent of , we prove that strong tractability is equivalent to tractability.

We stress that all proofs are constructive. That is, we provide randomized algorithms that enjoy the maximal speed of convergence. We also exhibit randomized algorithms which achieve strong tractability and tractability error bounds.

     

Braided symmetric and exterior algebras

The goal of the paper is to introduce and study symmetric and exterior algebras in certain braided monoidal categories such as the category for quantum groups. We relate our braided symmetric algebras and braided exterior algebras with their classical counterparts.

     

Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function defined in a domain and such that

0\}. \end{displaymath}">

We also assume that the interior boundary of the positivity set, \nobreak 0\}$">, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).

The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.

     

Napoleon in isolation

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic -manifolds, where hyperbolic Dehn surgeries on some collection of cusps leave the geometric structure at some other collection of cusps unchanged.

     

estimates for a class of oscillatory integrals

We obtain a simpler proof of Theorem 3.1 of The complete mapping properties of some oscillatory integrals in several dimensions, by G. Sampson and P. Szeptycki (Canad. Math. J. 53 (5) (2001), 1031-1056).

     

No embeddings of solenoids into surfaces

A quick proof of Bing's theorem indicated by the title is given. Indeed the inverse limit of a sequence of degree maps between closed oriented -manifolds can never be embedded into any closed orientable -manifold. The proof also concludes Gumerov's result on the covering of solenoids.

     

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets for which the following statement holds:

If is a properly immersed minimal surface in of finite topology that is eventually disjoint from then has finite total curvature.

The same question is also considered when the conclusion is finite type or parabolicity.

     

Random holomorphic iterations and degenerate subdomains of the unit disk

Given a random sequence of holomorphic maps of the unit disk to a subdomain , we consider the compositions

The sequence is called the iterated function system coming from the sequence We prove that a sufficient condition on the domain for all limit functions of any to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.

     

Covering maps that are not compositions of covering maps of lesser order

In 1995 J.W. Heath asked which exactly -to-one maps are compositions of exactly -to-one maps with . This paper deals with compositions of covering maps. Exactly -to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order are constructed for all 's, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly -to-one maps, open maps, or locally one-to-one maps.     

On finite minimal non-nilpotent groups

A critical group for a class of groups is a minimal non- -group. The critical groups are determined for various classes of finite groups. As a consequence, a classification of the minimal non-nilpotent groups (also called Schmidt groups) is given, together with a complete proof of Gol'fand's theorem on maximal Schmidt groups.

     

The Noetherian property in some quadratic algebras

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph . The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

A uqr mapping of an -manifold is a mapping which is rational with respect to a bounded measurable conformal structure on . Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.

     

Right-angled mock reflection and mock Artin groups

We define a right-angled mock reflection group to be a group acting combinatorially on a CAT(0) cubical complex such that the action is simply-transitive on the vertex set and all edge-stabilizers are . We give a combinatorial characterization of these groups in terms of graphs with local involutions. Any such graph not only determines a mock reflection group, but it also determines a right-angled mock Artin group. Both classes of groups generalize the corresponding classes of right-angled Coxeter and Artin groups. We conclude by showing that the standard construction of a finite space for right-angled Artin groups generalizes to these mock Artin groups.