A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Cubature formulae which are exact on spacesP,intermediate betweenP k andQ k′

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of ⁿ , with weight function, and which are exact on the spaceR( _{k 1}, k₂, ..., k_n) of all polynomials of degree k _i respectively to each variablex _i , 1in.     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

Semi-groupes sous-Markoviens engendrés par des opérateurs matriciels

We characterize generators of sub-Markovian semigroups onL ^p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A _ij )_1i,jn on (L ^p ()) ⁿ is sub-Markovian if and only if the semigroup generated by the sum of each rowA _i 1+...+A _in (1in), is sub-Markovian. The corresponding result on (C ₀(X)) ⁿ characterizes dissipative operator matrices.

     

On bounds for the range of ordered variates II

Letx ₁, ,x _n be real numbers with ₁ ⁿ x _j =0, |x ₁ ||x ₂ ||x _n |, and ₁ ⁿ f(|x _i |)=A>0, wheref is a continuous, strictly increasing function on [0, ) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x _m | isf ^–1 (A/(n–m+1)). If (n–m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n–m+1) is odd provided either (i)f is strictly convex on [0, ), or (ii)f is strictly concave on [0, ). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x ^p forp>1 and 0<p<1 respectively.     

Nonoscillatory solutions of linear differential equations with deviating arguments

Summary The equation to be considered is of the form (1) x(ⁿ)(t)+p(t)x(g(t))=0 (t>a), where =±1, p(t) > 0 for ta and g(t) as t. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(ⁱ)(t)>0 (0il), (–1)^i–lx(t)x(ⁱ)(t)>0 (lin) for some integer l, 0ln, (–1)^n–l–1=1. In this paper, for a given l such that 0n–l–1=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every >0 the equation x(ⁿ)(t)+p(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

Asymptotic approximation of crossing probabilities of random sequences

Summary Let {X _i , i1} be a random sequence and {u _ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P _n =P{X _i u _ni ,1in} by . We give sufficient conditions on X _i such that P _n–P _n ^* 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace

Summary The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.The following principle governs the passage from results in the plane to results in space:Let B be a three-dimensional Brownian motion, and P ₁, ..., P _k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B ⁱ (1ik), defined respectively as the orthogonal projections of B on P _i (1ik), take place jointly, and the corresponding limit variables are independent.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Flat points of two-dimensional Brownian motion

Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt ₀ and >0 such thatZ(t ₀) is an extremal point of the convex hull of {Z(t)|t ₀–tt₀} and also an extremal point of the convex hull of {Z(t)|t ₀tt₀+} and, moreover, the tangent lines to the convex hulls atZ(t ₀) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t ₀ and >0 such that the intersection of the convex hulls of {Z(t)|t ₀–tt₀} and {Z(t)|t ₀tt₀+} contains onlyZ(t ₀)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture     

Error estimates of Hermite interpolation

This paper presents a procedure for obtaining error estimates for Hermite interpolation at the Chebyshev nodes {cos ((2j+1)/2n)} _j ^=0n–1 –1x1, for functionsf(x) of various orders of continuity. The procedure is applicable in many cases when the usual Lagrangian error bound is not, and is a better bound, in general, when both are applicable.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

Convex interpolating splines of arbitrary degree. III

For given data (x _i, f_i) _i=0 ⁿ (x ₀<x ₁<...<x _n) we consider the possibility of finding a spline functions of arbitrary degreek (k3) with preassigned smoothnessl, where 1l[(k-1)/2]. The splines should be such thats(x _i)=f _i (i=0, 1,...,n) ands is convex or nondecreasing and convex on [x ₀,x _n]. An explicit formula for this function as well as the conditions that guarantee the required properties are established. An algorithm for the determination of the splines and the error bounds is also included.     

A partial ordering of some of the elements of Algol

Them _Algol productions of Algol are of the formX _p0 ::=X _p1 X _p2 ...X _{pn p} , where 1pm _Algol, 1n _p,X _p0 is a defined type andX _pj, 1jn _p is either a defined type or a basic symbol or possibly, representing the empty string ifn _p=1 [1]. A partial ordering of that subset of Algol's basic symbols and defined types which for somep are eitherX _p0 orX _p1 is exhibited. This ordering is of interest in implementing the syntax-oriented translator described by Ingerman.     

On the stability of rational approximations to the cosine with only imaginary poles

Letr(z) be a rational approximation to cosz with only imaginary poles ±i ₁ ^–1/2 , ±i ₂ ^–1/2 , ..., ±i _m ^–1/2 such that |cozz –r(z)| C|z|^2m+2 as |z| 0. If the degree of the numerator ofr(z) is less than or equal to 2m and _i m/4,i=1, ...,m, then we show that |r(z)|1 for all realz.