The Measure of Noncompactness of Linear Operators between Spaces of Strongly C 1 Summable and Bounded Sequences

We give necessary and sufficient conditions for infinite matrices to map a sequence space X into a sequence space Y where X = l ₁ and Y = w ^p , w ^p , w ₀ ^p (1 p < ), or X = w ₀, w, w and Y = l _p (1 p ), or X = w ₀, w, w and Y = w ₀ ^p , w ^p , w ^p (1 p < ). Furthermore the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for a linear operator between these spaces to be compact.     

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.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

Summary In this paper, we derive error estimates inL _p-norm, 1p, for the ₂-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ₂-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.The results of this paper form a part of a Ph.D. thesis written at the University of Maryland under the direction of Professor J.E. Osborn     

О полноте систем функ ций и систем ортогона льных многочленов

The paper is devoted to the study of completeness problem of systems { ⁿ (x)} _n=0 inL ^p (a, b), where –a<b+,(x) is a weight function subject to mild assumptions, and(x) is a continuous function on (a,b), either bounded or unbounded in the neighbourhood of the end-points of (a,b). It turns out that this problem is connected with that of quasianalyticity of certain additive set of functions at a given point. As the most important application of the general results, the completeness problem is treated for systems of orthogonal polynomials.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

Completion of Countably Seminormed Spaces

Let E be a vector space with a topology generated by countably many increasing seminorms p1 p2 ...and let _{p i} be the completion of E/ker (pi) with respect to pi. If pi are norms compatible in a certain sense, it is known that E is a complete space if and only if E=_{i=1 p i} . In this paper we give a similar characterization of complete spaces in a general case when pi are seminorms, without any additional assumptions. Our characterization coincides with the known one if pi are compatible norms.     

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.

     

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.     

Sojourns and future infima of planar Brownian motion

Summary Let {W(t); 0t1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);rt1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

On optimal extreme-discrepancy point sets in the square

Summary The optimal extreme-discrepancyN-point sets in the unit square are found forN6. The point ordering is unique forN3, but there are 14, 29 and 22 distinct orderings up to reflection about the diagonal forN=4, 5 and 6, respectively. The minimum discrepancy is less than, equal to, and greater thanN ^–1 forN3, 4N6 andN7, respectively.This work was performed at the University of Wisconsin with support from the National Science Foundation.     

On the value of a logarithmic-trigonometric integral

A closed expression is derived for the integral ₀ ^/2 log ⁿ cosxlog ^p sinxdx, wheren andp are non-negative integers. As already remarked by Nielsen in a monograph on the generalized polylogarithms published early in this century, this integral is equal to times a homogeneous polynomial in (q) (the Riemann zeta function for integer arguments) and log 2, with rational coefficients. Explicit expressions for the integral are given for 0<n4, 0p4, most of which have been found from the general formula by means of a computer.Part of this work was done during the author's leave of absence at the Institute of Theoretical Physics at McGill University, Montreal, Quebec, Canada. This work was supported in part by the National Research Council of Canada.     

On a problem of Erdös and Alladi

We give an estimate for the quantity {f(n):nx, p(n)y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, ({(n):nx, p(n)y}) ({1:nx, p(n)y})^–1 tends to zero, asx, uniformly iny2, and thus settle a conjecture of Erdös.Supported by a grant from the Deutsche Forschungsgesellschaft.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Finding connected components of a semialgebraic set in subexponential time

Let a semialgebraic set be defined by a quantifier-free formula with atomic subformulas of the form f_i>0, 0,1 i where the polynomials f_i[X₁,..., X_n] of degree deg (f_i)<d have absolute value of the coefficients at most 2^M. An algorithm is constructed which finds the connected components of the semialgebraic set (i.e., giving formulas for them) in a time that is polynomial inM (kd) ^ro(1). Collins' previously known method has a bound which is polynomial inM (kd) ^ro(1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 3–46, 1991.     

Almost sure behaviour ofF-valued random fields

Summary LetX _n, n ^d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields _n, ^d of positive numbers, the almost sure behaviour of _knX_k/_n, n ^d is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.Supported in part by the Youth Science Foundation of China, No. 19001018Supported by the National Natural Science Foundation of China