A generalized Laplace transform of generalized functions

- . . . , .     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

A note on generating generalized two-dimensional plate and shell theories

Summary Ordinary two dimensional linear shell theory may be formulated by using six differential equations for stress resultants and couples and for middle surface force and moment load intensities, in conjunction with the principle of virtual work, for the derivation of strain displacement relations. The present paper deals with a more general formulation, involving additional two-dimensional equilibrium equations, as a consequence of three-dimensional equations for force and moment stresses, in conjunction with a stipulation of surface force and moment load intensities for two face surfaces in place of the one middle surface. The main intent of the analysis is an illumination of the concept of a mechanical Cosserat-surface theory, in comparison with ordinary two-dimensional shell theory.

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Zusammenfassung Die übliche zweidimensionale Schalentheorie kann formuliert werden, indem man sechs Differentialgleichungen für Spannungsresultierende, Momente, Mittel flächenkraft und Momentlastintensität zusammen mit dem Prinzip der virtuellen Verschiebungen dazu benutzt, um ein System von Dehnungs-Verschiebungsgleichungen abzuleiten. Die vorliegende Arbeit beschäftigt sich mit einer verallgemeinerten Formulierung, mit zusätzlichen zweidimensionalen Gleichgewichtsbedingungen, die von einer dreidimensionalen Theorie für Kraft- und Momentenspannungen abgeleitet werden, zusammen mit einer Vorschrift über zwei Systeme von Ober flächenkräften und Momentlastintensitäten anstelle des einen Mittelfllächenlastsystems. Der Hauptzweck der Analyse ist die Beleuchtung des Begriffs der mechanischen Cosserat-Flächentheorie im Vergleich zur üblichen zweidimensionalen linearen Schalentheorie.

     

Equations de convolution et formes quadratiques

Résumé On étudie, sans hypothèse de convexité, les équations f=g, f=g et f=g.

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Summary We study, without any convexity hypothesis, equations f=g, f=g and f=g where and respectively denote infimal convolution and deconvolution. We give an explicit formulation of these results in the quadratic hilbertian frame, and we interpret them in terms of parallel addition and subtraction of non necessarily semi-definite positive operators.

     

Hardy spaces of predictable martingales

, P _p Q _p .

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This research was partially supported by the Hungarian Scientific Foundation.     

Three theorems on ρ* random fields

This paper generalizes results by Bradley.⁽³⁾ Suppose that for 1=1,2,...X _k ¹ :k ^d is a centered, weakly stationary ^*-mixing random field, and suppose lim_l Cov(X ₀ ¹ ,x _k ¹ ) exists, anyk ^d . Then the successive spectral densities converge uniformly to a continuous function. For a sequence of strictly stationary random fields that are uniformly ^*-mixing and satisfy a indeberg condition, a CLT is proved for sequences of sums from the fields. This result is then applied: given a centered strictly stationary ^*-mixing random field whose probability density and joint densities are continuous, then a kernel estimator for the probability density obeys the CLT.     

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Inequalities of Hardy-Littlewood type

( ). - .     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Collocation methods for parabolic partial differential equations in one space dimension

Summary Collocation at Gaussian points for a scalarm-th order ordinary differential equation has been studied by C. de Boor and B. Swartz. J. Douglas, Jr. and T. Dupont, using collocation at Gaussian points, and a combination of energy estimates and approximation theory have given a comprehensive theory for parabolic problems in a single space variable. While the results of this report parallel those of Douglas and Dupont, the approach is basically different. The Laplace transform is used to lift the results of de Boor and Swartz to linear parabolic problems. This indicates a general procedure that may be used to lift schemes for elliptic problems to schemes for parabolic problems. Additionally there is a section on longtime integration and A-stability.Supported by the Office of Naval Research under contract N-00014-67-A-0128-0004     

Theory of the classical gravitational field and Mach's principle

Equations for a massive gravitational field are found in the framework of the special theory of relativity on the geometrization principle. The existence of a graviton mass has fundamental significance for the construction of the theory. In accordance with this theory of gravitation, a homogeneous and isotropic universe evolves cyclically from a high density to a minimum density, etc., and can only be flat. The theory predicts the existence of an appreciable amount of dark matter in the universe. The existence of black holes in the universe is completely ruled out. The theory explains all known observational facts in the solar system.Institute of High Energy Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 3–27, October, 1994.     

On the approximation of discrete functions by fourier-hahn sums

— f: {0,1, ..., N–1}R. .     

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 }.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

On aU-set for multiple Walsh series

m- (m2) , - , .     

A study of the rate of rational approximation of a class of analytic functions

This paper is devoted to results belonging to the theory of rational approximation of analytic functions. We study the rate of decrease of the best approximations _n of a functionf holomorphic in a disk ¦z ¦z¦< , >1> by rational functions of order at most n in the uniform metric on the unit disk with center at z=0. We prove theorems that connect the rate of decrease of the quantities _n with the order 0 and type 0 of the functionf. The proofs of these results are based on the methods of the theory of Hankel operators.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 32–39.The author is grateful to Prof. V. A. Prokhorov for posing this problem and assistance with the preparation of this article.     

Economic and biological risk analysis of the Norwegian spring‐spawning herring fishery

The purpose of this paper is to study the biological and economic risks involved in the management of the Norwegian springspawning herring fishery. We use a discrete time and agestructured model based on historical data. The current paper investigates, under different levels of fishing mortalities, the risk probabilities related to the time behaviour of the spawning stock and profit. We show that the exploitation of the herring stock is vulnerable to small changes in harvesting and price level.     

Inverspositive Operatoren und Taubersätze in der Erneuerungstheorie

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ₊ the aim is (a,a+) with 0a<, 0< instead of (a, ), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a0(a), with fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchröder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBene.

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Herrn Professor Collatz zum 65. Geburtstag