Sensitivity analysis of distributed-parameter optimal control problems for nonlinear parabolic equations

A sequence of optimal control problems for systems described by nonlinear parabolic equations is considered. It is proved that, under the -convergence of objective functionals, the parabolicG-convergence of operators in the state equations, and the Kuratowski convergence of control constraint sets, a convergent sequence of optimal pairs has a limit which is an optimal pair for the limit control problem. The convergence of minimal values is also obtained.This research was supported in part by the Istituto Nazionale di Alta Matematica F. Severi, Rome, Italy. Part of this research was carried out while the author was visiting the Scuola Normale Superiore, Pisa, Italy.     

On the variational convergence of non-coercive quadratic integral functionals and semicontinuity problems

In this paper we give some results about convergence of non coercive quadratic integral functionals by examining the behaviour of coefficients. We apply our results to semicontinuity problems and we illustrate them by some examples.AMS Subject Classification: 40A10, 49J45.     

SOLUTIONS CONTAINING DELTA－WAVES OF CAUCHY PROBLEMS FOR A NONSTRICTLY HYPERBOLIC SYSTEM

ＳＯＬＵＴＩＯＮＳＣＯＮＴＡＩＮＩＮＧＤＥＬＴＡ－ＷＡＶＥＳＯＦＣＡＵＣＨＹＰＲＯＢＬＥＭＳＦＯＲＡＮＯＮＳＴＲＩＣＴＬＹＨＹＰＥＲＢＯＬＩＣＳＹＳＴＥＭ￥ＨＵＡＮＧＦＥＩＭＩＮ（黄飞敏）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ,Ｓｈａｎｔｏｕ...     

A successive quadratic programming method for a class of constrained nonsmooth optimization problems

In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.     

A Note on Base Change,Identities Involving τ(n), and a Congruence of Ramanujan

We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) ₁₁(n) (mod 691) follows.     

Large Deviations for Products of Empirical Probability Measures in the τ-Topology

We prove a large deviation principle (LDP) for products of empirical measures, where the state space S of the underlying sequence of i.i.d. random variables is Polish and the set of probability measures on S respectively S×S is endowed with the -topology. An improved form of a LDP for U-statistics and some conclusions from that are obtained as a particular application.     

Large deviations for empirical measures of Markov chains

In this paper we obtain large-deviation upper and lower bounds for the empirical measure of a Markov chain with general state space, as well as for the associated multivariate empirical measure and empirical process. In each of these instances we improve in various ways the results in the literature.     

Bankruptcy games

Bankruptcy problems are considered from a game theoretic point of view. Solution concepts from cooperative game theory are studied for bankruptcy games. A necessary and sufficient condition for a division rule for bankruptcy problems to be a game theoretic rule is given. A new division rule which is an adjustment of the proportional rule is given. This rule coincides with theT-value for bankruptcy games. Properties of the new rule are treated and a set of characterizing properties is given.

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Zusammenfassung In dieser Arbeit werden Bankrottprobleme von spieltheoretischer Warte aus behandelt; insbesondere werden Lösungskonzepte der kooperativen Spieltheorie für Bankrottspiele untersucht. Eine notwendige und hinreichende Bedingung wird angegeben dafür, Daß eine Aufteilungsregel für Bankrottprobleme spieltheoretischer Natur ist. Ferner wird eine neue Aufteilungsregel angegeben, welche eine passende Modellierung der Proportionalitätsregel ist. Diese Regel fällt mit dem-Wert für Bankrottspiele zusammen. Schließlich werden Eigenschaften dieser neuen Regel untersucht und eine Axiomatisierung angegeben.

     

Descent methods for composite nondifferentiable optimization problems

We present a framework for the development of globally defined descent algorithms for the minimization of non-differentiable objective functionsF := h º f withh convex. Within our structure the global convergence properties of the Cauchy, Modified Newton, Gauss—Newton, and Variable-Metric methods are easily established along with that of several new approaches. Examples illustrating the calculational techniques are provided.     

Topological existence and stability for stackelberg problems

The aim of this paper is to study, in a topological framework, existence and stability for the solutions to a parametrized Stackelberg problem. To this end, approximate solutions are used, more precisely, -solutions and strict -solutions. The results given are of minimal character and the standard types of constraints are considered, that is, constant constraints, constraints defined by a finite number of inequalities, and more generally constraints defined by an arbitrary multifunction.     

Perturbation Theory and the Analytic Approach in the Context of the Inclusive τ-Lepton Decay

We perform a comparative analysis of different forms of pertubative expansions in spacelike and timelike regions. In the context of the inclusive -lepton decay, we compare the results obtained using the standard perturbation theory and the Shirkov–Solovtsov analytic approach, which modifies the perturbative expansions such that the new approximations reflect basic principles of the theory, such as renormalization invariance, spectrality, and causality. We show the advantages and self-consistency of the analytic approach in describing the -lepton decay.     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Given a family of real-valued functions defined in a normed vector space X, we study a class of -convex functions having a simpler representation for the --subdifferential. The case =X^* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X^*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.This work is based on research material supported in part by CONICYT-Chile through FONDECYT 101-0116 and FONDAP-Matemáticas Aplicadas II.     

Some statistical problems in random translations of stochastic point processes

A random translation of a marked point process is considered. The distribution of random translation is assumed to be dependent upon the mark through a certain functionh(·). The main concern is to make inferences about the functionh(·) for different types of data. Complete identification and estimation are not possible in general, but some interesting particular solutions are presented.     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Maximum likelihood estimation in exponential orthogeodesic models

An orthogeodesic statistical model is defined in terms of five conditions of differential geometric nature. These conditions are reviewed together with a characterization theorem for exponential orthogeodesic models. Orthogonal projections, relevant for maximum likelihood estimation in exponential orthogeodesic models, are described in a simple way in terms of some of the quantities in the characterization theorem. A unified procedure for performing maximum likelihood estimation in exponential orthogenodesic models is given and the use of this procedure is illustrated for some of the most important models of this kind such as -parallel models, -parallel models and certain transformation models.     

ε-solutions in vector minimization problems

This paper presents some properties of -solutions for vector minimization problems where the function to be optimized takes its values in the Euclidean space ^p . The results obtained generalize the classical ones for exact Pareto solutions.     

Sato-Tate Conjectures and Chebyshev Polynomials

Results of Shahidi on the analytic continuation of certain L-functions have been used by Serre to obtain partial information on the distribution of Sato-Tate angles of the Ramanujan function. Employing the Christoffel numbers associated to the orthogonal polynomials used by Serre we derive upper bounds on the lower density of primes with small angles. We also make effective his lower bounds on the upper density of such primes. Assuming that more analytic information is available, we sharpen similar estimates of Ram Murty, and, using the Christoffel numbers attached to the second order Chebyshev polynomials, derive general bounds on the discrepancy between the empirical angles distribution and the Sato-Tate distribution.