TheC M -embedded problem and optimality conditions

In this paper theC _M -embedded problem which is also called the design centering problem in other papers will be described, and new optimality conditions and some results associated with optimality conditions will be presented. These results hold for general non-convex regions. To a certain extent they provide the possibility to develop search techniques. It should be pointed out that, in this paper, the only case where the Minkowski norm is just the Euclidean norm is treated.     

An interpolation theorem for holomorphic automorphisms ofC n

We construct automorphisms of C ⁿ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.     

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x ⁰ of the convex SIP problem by such conditions for x ⁰ in some nonlinear programming (NLP) problem denoted by NLP(I(x ⁰)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x ⁰)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.     

A simple proof of an approximation theorem of H. Minkowski

It is proved that every convex bodyC inR ⁿ can be approximated by a sequenceC _k of convex bodies, whose boundary is the intersection of a level set of a homogeneous polynomial of degree 2k and a hyperplane. The Minkowski functional ofC _k is given explictly. Some further nice properties of the approximantsC _k are proved.Supported in part by BSF and Erwin Schrödinger Auslandsstipendium J0630.     

On a modification of the Koenig theorem

We present a modified Koenig theorem for the simultaneous determination of all distinct poles ofG(z)/F(z), whereG(z) is an analytic function inside a simple smooth closed contourC, F(z) is an analytic function inside and onC, with a known numbern of simple zeros insideC, andF(z), G(z) have no common zeros insideC. It turns out that complex and interval iterations of higher order can be constructed, and several algorithms are available for doing this. Some of them are well known and discussed in many papers.The author is grateful to the referees for their valuable comments and suggestions. Also, she would like to thank Andrey Andreev and Nikolay Kjurkchiev for their helpful discussions.     

Variational approach and optimal control of a PEM fuel cell

The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions.     

Optimal control of a heat transfer problem with convective boundary condition

We consider the problem of controlling the solution of the heat equation with the convective boundary condition taking the heat transfer coefficient as the control. We take as our cost functional the sum of theL ²-norms of the control and the difference between the temperature attained and the desired temperature. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We show how the solution depends in a differentiable way on the control using appropriate a priori estimates. We establish existence and uniqueness of the solution of the optimality system, and thus determine the unique optimal control in terms of the solution of the optimality system.This research was sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under Contract DE-AC05-84OR21400 with the Martin Marietta Energy Systems. The authors thank David R. Adams for his assistance in clarifying the proof of Proposition 2.1 and appreciate the comments of the referees for needed revisions.     

An Optimal Control Problem for a Singular System of Solid–Liquid Phase Transition

In this article, we deal with a control problem for a singular system regarding a phase-field model which describes a solid–liquid transition by the Ginzburg–Landau theory. The purpose is to control the system by the means of the heat supply r able to guide it into a certain state with a solid (or liquid) part in a prescribed subset Ω₀ of the space domain Ω, and maintain it in this state during a period of time. The transition is described by a nonlinear differential system of two equations for the phase field and temperature. The control problem is set for some expressions of the cost functional which might reveal cases of physical interest. An approximating control problem is introduced and the existence of at least an optimal pair is proved. The first-order optimality conditions for the approximating problem are determined and a convergence result is given.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

Regularized parametric Kuhn-Tucker theorem in a Hilbert space

For a parametric convex programming problem in a Hilbert space with a strongly convex objective functional, a regularized Kuhn-Tucker theorem in nondifferential form is proved by the dual regularization method. The theorem states (in terms of minimizing sequences) that the solution to the convex programming problem can be approximated by minimizers of its regular Lagrangian (which means that the Lagrange multiplier for the objective functional is unity) with no assumptions made about the regularity of the optimization problem. Points approximating the solution are constructively specified. They are stable with respect to the errors in the initial data, which makes it possible to effectively use the regularized Kuhn-Tucker theorem for solving a broad class of inverse, optimization, and optimal control problems. The relation between this assertion and the differential properties of the value function (S-function) is established. The classical Kuhn-Tucker theorem in nondifferential form is contained in the above theorem as a particular case. A version of the regularized Kuhn-Tucker theorem for convex objective functionals is also considered.     

Constraint qualification in a general class of Lipschitzian mathematical programming problems

The Kuhn–Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a locally Lipschitzian function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the condition of a generalized Kuhn–Tucker constraint qualification or a generalized Arrow–Hurwicz–Uzawa constraint qualification. The case when the set C is open is shown to be a special one of our results, which helps us to improve some of the existing results in the literature. To finish we consider several test problems.     

Remarks on a theorem of Taskinen on spaces of continuous functions

We clarify and prove in a simpler way a result of Taskinen about symmetric operators on C(Kⁿ), K an uncountable metrizable compact space. To do this we prove that, for any compact space K and any n ∈ ?, the symmetric injective n–tensor product of C(K), , is complemented in C(B_C(K)*), a result of independent interest. The techniques we develop allow us to extend the result in several directions. We also show that the hypothesis of metrizability and uncountability cannot be removed.     

An exact rate of convergence in the functional central limit theorem for special martingale difference arrays

Summary Since the topology of weak convergence of probability distributions on the Borel -field of the space C= C([0, 1]) is metrizable, it is natural to describe the speed of convergence in weak functional limit theorems by means of an appropriate metric. Using the metric proposed by Prokhorov it is shown that under suitable conditions the rate of convergence in the functional central limit theorem for C-valued partial sum processes based on martingale difference arrays is the same as in the special case of row-wise independent random variables where this rate is known to be an optimal one.     

Numerical solutions of the Dirichlet problem via a density theorem

Under mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for L^p boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.     

Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data

We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class C ^1,1. Second-order optimality conditions for local Pareto solutions are derived as a special case.     

The second order optimality conditions for nonlinear mathematical programming with C 1,1 data

To find nonlinear minimization problems are considered and standard C ²-regularity assumptions on the criterion function and constrained functions are reduced to C ^1,1-regularity. With the aid of the generalized second order directional derivative for C ^1,1 real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.     

A note on a theorem of Yamamuro

Conditions are presented which are necessary for the existence of a regular fixed point of aC ¹ map.Work supported in part by NSF Grant No. MCS 77-15509.Work supported in part by ONR Grant No. N0014-75-C-0495 and NSF Grant No. Eng. 76-81058.     

On a generalization of a theorem of B. tyler

LetB be an arbitrary normal matrix, satisfying some conditions. AbsoluteB-summability factors in a sequence for Cesàro methodC ^α if α≧1 or α=0 and absolute convergence factors in a sequence forC ^α if 0<α<1 are obtained.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.