Measures with Bounded Variation with Respect to a Normed Ideal of Operators and Applications

We introduce in a natural way the notion of measure with bounded variation with respect to a normed ideal of operators and prove that for each maximal normed ideal of operators (, ), is true the following result: If U ∈ L(C(T,X), Y) with G the representing measure of U and G : Σ → ((X, Y),) has bounded variation, then U ∈ (C(T,X), Y). As an application of this result we prove that an injective tensor product of an integral operator with an operator belonging to a maximal normed ideal of operators (,) belongs also to (, ).     

Universality of critical behaviour in a class of recurrent random walks

Let X₀=0, X₁, X₂,.. be an aperiodic random walk generated by a sequence ₁, ₂,... of i.i.d. integer-valued random variables with common distribution p(·) having zero mean and finite variance. For anN-step trajectory and a monotone convex functionV: withV(0)=0, define Further, let be the set of all non-negative paths compatible with the boundary conditionsX₀=a, X_N=b. We discuss asymptotic properties of under the probability distribution N and 0, Z^a,b_N,+, being the corresponding normalization. If V(·) grows not faster than polynomially at infinity, define H() to be the unique solution to the equation Our main result reads that as 0, the typical height of X_[, ^N] scales as H() and the correlations along decay exponentially on the scale H()². Using a suitable blocking argument, we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3/2. In the particular case of linear potential V(·), the characteristic length H() is proportional to ^-1/3 as 0.Mathematics Subject Classification (2000):60G50, 60K35; 82B27, 82B41     

The Medvedev lattice of computably closed sets

Simpson introduced the lattice of Π⁰₁ classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π⁰₁ classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context. Supported by the Austrian Research Fund (Lise Meitner grant M699-N05).     

Prediction for discrete time series

Let {X_n} be a stationary and ergodic time series taking values from a finite or countably infinite set Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times _n along which we will be able to estimate the conditional probability P(=x|X₀,...,) from data segment (X₀,...,) in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then _n is upperbounded by a polynomial, eventually almost surely.Mathematics Subject Classification (2000): 62G05, 60G25, 60G10     

Lyapunov exponent for the parabolic anderson model with lévy noise

We consider the asymptotic almost sure behavior of the solution of the equationwhere {Y_x:x Z^d} is a field of independent Lévy processes and is the discrete Laplacian.Research of the first two authors supported in part by a grant from NSF, of the third author by JSPS Grant-in-Aid for Scientific Research, Kiban(C) 13640103     

On Cyclic Packing of a Tree

We prove that there exists a packing of copies of a tree of size into K_n. Moreover, the proof provides an easy algorithm.Acknowledgments. The research of the second author was partially supported by Deutscher Akademischer Austauschdienst.     

Reconstructing under Group Actions

We give a bound on the reconstructibility of an action GX in terms of the reconstructibility of a the action NX, where N is a normal subgroup of G, and the reconstructibility of the quotient G/N. We also show that if the action GX is locally finite, in the sense that every point is either in an orbit by itself or has finite stabilizer, then the reconstructibility of GX is at most the reconstructibility of G. Finally, we give some applications to geometric reconstruction problems.     

Translation invariant Gibbs states for the Ising model

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for . Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's coarse graining [Pi] up to the critical temperature. We would like to thank C. Pfister and Y. Velenik for very useful comments.     

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C₀(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝ^D of order m≥D.     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Let a sequence of iid. random variables ξ ₁, . . . ,ξ _n be given on a space with distribution μ together with a nice class of functions f(x ₁, . . . ,x _k ) of k variables on the product space For all f ∈ we consider the random integral J _n,k (f) of the function f with respect to the k-fold product of the normalized signed measure where μ _n denotes the empirical measure defined by the random variables ξ ₁, . . . ,ξ _n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too. Supported by the OTKA foundation Nr. 037886     

Canonical height functions for affine plane automorphisms

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.     

Maximality of Positive Operator-valued Measures

A positive operator-valued measure is a (weak-star) countably additive set function from a σ-field Σ to the space of nonnegative bounded operators on a separable complex Hilbert space . Such functions can be written as M = V*E(·)V in which E is a spectral measure acting on a complex Hilbert space and V is a bounded operator from to such that the only closed linear subspace of , containing the range of V and reducing E (Σ), is itself. Attention is paid to an existing notion of maximality for positive operator-valued measures. The purpose of this paper is to show that M is maximal if and only if E, in the above representation of M, generates a maximal commutative von Neumann algebra.     

On the exact Hausdorff dimension of the set of Liouville numbers

Let denote the set of Liouville numbers. For a dimension function h, we write () for the h-dimensional Hausdorff measure of . In this paper we locate the exact cut-point at which the Hausdorff measure of drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then ()=, and if h is a dimension function that increases slower than some power function near 0, then ()=0. This answers a question asked by R. D. Mauldin.Mathematics Subject Classification (2000): 28A80     

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

Density results for the <Emphasis Type="Italic">W</Emphasis><Superscript>1/2</Superscript> energy of maps into a manifold

We show that maps from Bⁿ to a smooth compact boundaryless manifold which are smooth out of a singular set of dimension n−2 are dense for the strong topology in W^1/2(Bⁿ,). We also prove that for n≥2 smooth maps from Bⁿ to are dense in W^1/2(Bⁿ,) if and only if π₁()=0, i.e. the first homotopy group of is trivial.     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

On certain categories of modules for affine Lie algebras

In this paper, we exploit basic formal variable techniques to study certain categories of modules for an (untwisted) affine Lie algebra , motivated by Chari-Pressleys work on certain integrable modules. We define and study two categories and of -modules using generating functions, where is proved to contain the well known evaluation modules and to unify highest weight modules, evaluation modules and their tensor product modules. We classify integrable irreducible -modules in categories and and we determine the isomorphism classes of those irreducible modules. Finally we prove a result that relates fusion rules in the context of vertex operator algebras with integrable irreducible modules of Chari-Pressley.in final form: 12 November 2003Partially supported by a NSA grant and a grant from Rutgers Research Council.