Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

Summary The Lyapunov exponents _{1 2}... _d for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting exponential growth rates of tangent vectors under the flow. This paper shows how the Lyapunov exponents are related to measure preserving properties of the stochastic flow on M and of the induced stochastic flow on the projective bundle PM. Relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow. The results include the following. If M is compact and if the one-point motion on M is a non-degenerate diffusion with stationary probability measure then ₁+...+ _d 0 with equality if and only if the flow preserves almost surely; if in addition the induced one-point motion on PM satisfies a weak non-degeneracy condition then ₁=...= _d if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely.     

Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

Surface order large deviations for Ising,Potts and percolation models

Summary We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq1 andd3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.     

Characterization of allA-stable methods of order 2m-4

The characterization ofA-stable methods is often considered as a very difficult task (see e.g. [1]). In recent years, simple proofs have been found for methods of orderp2m-2 (see [2], [3], [7]). In this paper, we characterize theA-acceptable approximations of orderp 2m-4 and apply the result to 12-parameter families of implicit Runge-Kutta methods.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Summary A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (-valued) random variables {X _j , j1} and a kernel h: ^m H, m2, which is degenerate for the common distribution function of X _j , j1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=.     

Pointwise lower and upper bounds for eigenfunctions of ordinary differential operators

Summary For a differential operatorL as defined in (2.1) we consider the eigenvalue problemL u=u and describe a method to obtain a pointwise bound |u–v| wherev denotes the Rayleigh-Ritz approximation to an exact eigenfunctionu. The upper bound is continuous, but only piecewise differentiable and calculated by solving certain (inverse-positive!) auxiliary problems.Our method uses well-known estimations for (t _i) at a finite number of pointst _i[a, b] to calculate an upper, bound in the whole interval [a, b].The author would like to, thank the Battelle Institute Geneve for their assistance     

Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

Summary All rational approximations to exp(z) of order 2m– (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.     

Excursions of a Markov process induced by continuous additive functionals

Summary Let be a continuous additive functional with supportC of a Hunt processX={X _t;t0}. LetS={S _t;t0} be the inverse of and put . For each time of discontinuityu ofS, letZ _u be the corresponding excursion ofX outside ofC. The conditional structure of the excursion process {Z _u;u0} given the paths ofY={Y _t;t0} is studied. It is shown that conditionally, givenY, the excursion process is a Poisson random measure.Support from the Office of Naval Research (Contract Number N-00014-67-A-0112-0011) and the National Science Foundation (Grant Number ENG 75-02026) is gratefully acknowledged     

On the diophantine equation tan (kπ/m)=k tan π/m

It is proved that the equation tan (k/m)=k tan /m has no solution in integersk andm withk2,m3. This answers a question concerning the problem of approximating a convex disc by polygons.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

A convergent splitting of matrices

LetA, M, N ben ×n real matrices, letA = M– N, letA andM be nonsingular, letM y 0 implyN y 0, and letA y 0 implyN y 0 (where the prime denotes the transpose). Then the spectral radius(M ^–1 N) ofM ^–1 N is less than one, and the iterative processx ⁱ⁺¹ =M ^–1 N x ⁱ +M ^–1 b converges to the solution ofA x = b starting from anyx ⁰.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No. DA-31-124-ARO-D-462, and in part by the National Science Foundation under Grant NSF GP-6070.     

Estimates from below for Lebesgue constants of Fourier series on compact Lie groups

In this paper the Lebesque constants (L _R ^K (G))_R>0 of Fourier series on compact Lie groups G corresponding to general one-dimensional groupings on the dual object G^{^} are estimated from below by the associated (abelian) Lebesgue constants (L _R ^K (T))_R>0 on a maximal torus T in G. For spherical groupings this leads to the estimate L _R (G)const.R^(l-1)/2, l=dimT2.     

Absolut stetige Maße auf lokalkompakten Halbgruppen

LetS be a locally compact semigroup. It is shown that if a measure is absolutely continuous and ifS is cancellative, then the measure concentrated on a Borel subsetB ofS (i. e. =(B.)) is also absolutely continuous. Other properties of absolutely continuous measures will be obtained. Moreover we will answer the question when absolutely continuous probability measures exist. This is the case ifS admits an invariant integral on the space of all continuous functions onS with compact support. Another result is the following: If the compact semigroupS has a connected kernel then there exist absolutely continuous probability measures if and only ifS is amenable.     

Remark on the Helly number for strongly convex sets on Riemannian manifolds

Replacing convex by strongly convex we show that Helly's famous intersection theorem holds on every Riemannian n-manifold in the following form: The intersection of k relatively compact, strongly convex subsets of M (kn+i2) is nonvoid as soon as any n+i of these sets have a nonvoid intersection, where i=2 if M is homeomorphic to the standard n-sphere and i=1 otherwise.     

Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems

Summary This paper deals with polynomial approximations ø(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument, denoted by –R(ø), at which ø is absolutely monotonic. For given integersp1,m1 we determine the maximum ofR(ø) when ø varies over the class of all polynomials of a degree m with (forx0).     

Gauge theorems for resolvents with application to Markov processes

Summary LetV=(V )0 be a (not necessarily sub-Markovian) resolvent such that the kernelV for some 0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV ₀ is bounded.This result will be applied to resolventsU ^M arising from perturbation of sub-Markovian right resolventsU by multiplicative functionalsM (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao.     

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

Zur Lototsky — Transformation über kompakten Räumen von Wahrscheinlichkeitsmassen

As is well known, the theory of the classical Bernstein polynomials is connected with the theory of probability on the one hand and with the theory of matrix transformations and summability on the other hand. It is the purpose of the present paper to define and to investigate the Lototsky method of summability on the space of Radon probability measures on a compact topological space T. By the aid of an extended version of the Bohman-Korovkin approximation theorem we shall prove a convergence theorem for the sequence (L_n,,P)_n1 of so-called Lototsky-Schnabl operators, having as its sequence of ray functions. By specializing in an appropriate manner the underlying space T as well as the matrix P of weights, we shall deduce from this general theorem a result concerning the approximation properties of the sequence (L_n,)_n1 of Lototsky-Bernstein operators acting on the space of real-valued functions which are continuous on a compact N-simplex.     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.