Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4–)-expansion for the scaling function of the pair correlator of the O(n)-symmetric ⁴-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the ³ order of the expansion of the scaling function in the small- domain.     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Convergence Analysis of Perturbed Feasible Descent Methods

We develop a general approach to convergence analysis of feasible descent methods in the presence of perturbations. The important novel feature of our analysis is that perturbations need not tend to zero in the limit. In that case, standard convergence analysis techniques are not applicable. Therefore, a new approach is needed. We show that, in the presence of perturbations, a certain -approximate solution can be obtained, where depends linearly on the level of perturbations. Applications to the gradient projection, proximal minimization, extragradient and incremental gradient algorithms are described.     

Distance functional and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A - d(x, A) is continuous for each x X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) - {d(ta, b): a A, b B} is continuous on CL(X) × CL(X).Visiting the University of Minnesota.Visiting California State University, Los Angeles.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Convergence of the Gradient Projection Method for Generalized Convex Minimization

This paper develops convergence theory of the gradient projection method by Calamai and Moré (Math. Programming, vol. 39, 93–116, 1987) which, for minimizing a continuously differentiable optimization problem min{f(x) : x } where is a nonempty closed convex set, generates a sequence x_k+1 = P(x_k – _k f(x_k)) where the stepsize _k > 0 is chosen suitably. It is shown that, when f(x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either x_k x^* and x^* is a minimizer (stationary point); or x_k arg min{f(x) : x } = , and f(x_k) inf{f(x) : x }.     

Inverse power law potentials in rectangular configurations

Summary Calculations based on a (distance)^– intermolecular potential (>3) enable study of the effects on adsorption of the geometry of the solid. This paper gives the closed form solution for the adsorptive potential about a homogeneous solid rectangular corner; and, through systematic superposition, closed form solutions for the following configurations also: the rectangular corner of a cavity; laminae and rectangular cracks occupying a quarter plane; semi-infinite rectangular prisms and prismatic cavities; rectangular parallelepipeds and brick-shaped cavities. These various results are developed in detail for the cases =6 and =4. The paradox that potentials for >3 seem to be obtainable more readily than Newtonian potentials (=1) is explained by the existence only for >3 of simple fundamental solutions for infinite homogeneous solid configurations.

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Zusammenfassung Berechnungen, denen ein intermolekulares Potential der Form (Abstand)^– (>3) zugrunde gelegt ist, ermöglichen eine Untersuchung von Effekten der Adsorption auf die Geometrie des Festkörpers. Die vorliegende Arbeit gibt die Lösung in geschlossener Form für das Adsorptionspotential um eine feste, homogene, rechtwinklige Ecke an. Ausserdem werden durch systematische Superposition Lösungen in geschlossener Form für die folgenden Konfigurationen angegeben: die rechtwinklige Innenecke einer Mulde; viertelunendliche, ebene Platten und rechteckige Spalten; halbunendliche, reckteckige Prismen und prismatische Mulden; Quader und quaderförmige Höhlen. Diese Ergebnisse sind ausführlich dargestellt für die Fälle =4. Das Paradoxon. dass Potentiale mit >3 scheinbar leichter zugänglich sind als das Gravitationspotential (=1), wird dadurch erklärt, dass nur für >3 einfache Grundlösungen für unendliche, homogene Festköperkonfigurationen existieren.

     

Stochastic search in a convex region

Summary A stochastic search strategy is proposed for locating a possibility mobile target in a bounded, convex region of the plane. The strategy is asymptotically minimax as 0 with respect to the time required to get within of the target. The proof involves the study of first passages to time-dependent boundaries by a certain semi-Markov process.Supported by NSF grant DMS 82-01723     

Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [–^–, ^–], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0⁺, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as the center of the instanton is far from the endpoints of the interval. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit 0⁺ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Exchangeable random variables and the subsequence principle

Summary Call a sequence {X _n } of r.v.'s -exchangeable if on the same probability space there exists an exchangeable sequence {Y _n } such thatP(|X _n –Y _n |) for alln. We prove that any tight sequence {X _n } defined on a rich enough probability space contains -exchangeable subsequences for every >0. The distribution of the approximating exchangeable sequences is also described in terms of {X _n }. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

Continuity of martingales in the Brownian excursion filtration

Summary Given a Brownian motionB, we consider the filtration ( _{x xR} ), where _x is defined as the -field generated by the excursions ofB belowx. In this paper we prove a conjecture of Walsh which says that all -martingales are continuous.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Pointwise displacement errors in linear shell theory resulting from errors in the stress-strain relations

It is rigorously proved that relative errors of order in the stress-strain relations of linear shell theory result in relative pointwise errors in the solution displacement field of order .

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Zusammenfassung Für die Theorie dünner Schalen wird bewiesen, daß ein relativer Fehler der Größe in den Spannungs-Dehnungs-Beziehungen einen relativen lokalen Fehler der Größe in der Lösung für das Verschiebungsfeld erzeugt.

     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.