Microstructure evolution model in micromagnetics

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopical quantity of energy required to change one pole of a magnet to another. The microstructure is described on a mesoscopical level in terms of Young measures. Such mesoscopical, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteresis macroscopical response is demonstrated together with the influence of material properties.     

Preconditioning nonconforming finite element methods for treating Dirichlet boundary conditions. I

Summary This work deals with theL ² condition numbers and the distribution of theL ² singular values of the preconditioned operators {B _h ^–1 A_h}_{0, whereA h andB h are finite element discretizations of second order elliptic operators,A andB respectively. For conforming finite elements, it was shown in the work of Goldstein, Manteuffel and Parter that if the leading part ofB is a scalar multiple (1/) of the leading part ofA, then the singular values ofB h ^–1 A h cluster and fill-in the interval [ min, max], where 0< min max are the minimum and maximum of the factor . As a generalization of these results, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, theL ² condition numbers of {B h ^–1 A h } are uniformly bounded. Moreover, the singular values also cluster and fill-in the same interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuka and the method of nearly zero boundary conditions of Nitsche.This research was supported by the National Science Foundation under grant number DMS-8913091.}     

Mesoscopic model for ferromagnets with isotropic hardening

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopic quantity of energy required to change one pole of a magnet to another. This energy can increase within the evolution which is the effect like hardening in plasticity. The microstructure is described on a mesoscopic level in terms of Young measures. Such a mesoscopic, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteretic macroscopic response, including effects as virgin curves and minor loops, is demonstrated.     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

Computations in choice groupoids

Defining achoice as a mapping of the subsets of a setX into their respective subsets, a one-to-one (and naturally) corresponding binary operation,sequential choice, is identified under which the power set ofX is closed as achoice groupoid. A complete logical diagram is given, exhibiting all the implications between conjunctions of the seven conditions: (1) idempotence, (2) consistency, (3) absorbence, and (4) homomorphism of a choice, and (5) commutativity, (6) associativity, and (7) path-independence of the corresponding sequential choice.     

Partly convex programming and zermelo's navigation problems

Mathematical programs, that become convex programs after freezing some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require cooperation of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a sandwich condition.The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.Research partly supported by NSERC of Canada.     

The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with non-constant coefficients and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with constant coefficients and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birthThis work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme Boundary Element Methods within the guest-programme We-659/19-2.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

On the integrability of “finite energy” solutions for <Emphasis Type="Italic">p</Emphasis>-harmonic equations

We prove a higher integrability result for the gradient of solutions to some degenerate elliptic PDEs, whose model arises in the study of mappings with finite distortion.The nonnegative function which measures the degree of degeneracy of ellipticity bounds lies in the exponential class, i.e. is integrable for some > 0.Our result states that if is sufficiently large, then the gradient of a finite energy solution actually belongs to the Zygmund space L^plog^{L, 1}.     

Maximum monoreflections

It is shown that, in a category with a specified class of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in . Thus, for example, any variety, and most SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each completion lies a maximum monoreflection, so that, for example, any category of rings has amaximum functorial ring of quotients.     

When do Toeplitz and Hankel operators commute?

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolz, where is the symbol of the Toeplitz operator, and is an affine function of the characteristic function of certain anti-symmetric sets of the unit circle.     

Chapter 1 Introduction: Extensions and new developments in DEA

The extensions, new developments and new interpretations for DEA covered in this paper include: (1) new measures of efficiency, (2) new models and (3) new ways of implementing established models with new results and interpretations presented that include treatments of congestion, returns-to-scale and mix and technical inefficiencies and measures of efficiency that can be used to reflect all pertinent properties. Previously used models, such as those used to identify allocative inefficiencies, are extended by means of assurance region approaches which are less demanding in their information requirements and underlying assumptions. New opportunities for research are identified in each section of this chapter. Sources of further developments and possible sources for further help are also suggested with references supplied to other papers that appear in this volume and which are summarily described in this introductory chapter.     

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known comb inequalities, the path inequalities and the 3-star constraints as special cases. Its relation to the clique tree inequalities is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

A Nonlinear Problem Associated to the Motion of a Membrane

A nonlinear model associated to the motion of a membrane is considered as limit of a sequence of approximate models, for which a global existence and uniqueness theorem can be proved. The paper investigates the relationship between the solutions of the real and approximate models.     

SharpL ∞-error estimates for semilinear elliptic problems with free boundaries

Summary A numerical scheme to approximate a semilinear PDE involving a (singular) maximal monotone graph is analyzed inL . A preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; the discrete maximum principle is thus avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization procedures. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems, provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

Types in a mixed language system

As a support for writing software, a comprehensive set of problem oriented languages appears preferable to any so-called universal language, as soon as static checking is sufficient to ensure type correctness of the mixed language program. We lay the basis for a mixed language system where this requirement is fulfilled. The general outline of the system is first sketched. Detailed consideration is then given to our basic constructs for establishing communication between languages, namely standard types and foreign types. Abstract types, such as defined in CLU, are finally shown to be a particular class of foreign types.     

Product integration of singular integrands based on cubic spline interpolation at equally spaced nodes

Summary In this paper the convergence of product integration rules, based on cubic spline interpolation at equally spaced nodes, with not-a-knot end condition, is investigated for integrand functions with a interior or endpoint singularity in the integration interval.Work sponsored by the Ministero della Pubblica Istruzione of Italy