Bücher der Reihe International Series of Numerical Mathematics

Numerically oriented articles study finite difference, finite volume, and finite ele ment schemes, adaptive, multiresolution, and artificial dissipation methods.

Arguably, many industrial optimization problems are of the multiobjective type.

This monograph compiles, rearranges, and refines recent research results in the complex G-L theory with or without immediate applications to the theory of superconductivity.

Nineteen contributions cover recent topics in constructive approximation on varieties, approximation by solutions of partial differential equations, application of Riesz bases and frames, multiwavelets and subdivision.

These Proceedings form a record of the lectures presented at the interna tional Conference on Functional Analysis and Approximation held at the Ober wolfach Mathematical Research Institute, August 9-16, 1980.

"Optimal Control" reports on new theoretical and practical advances essential for analysing and synthesizing optimal controls of dynamical systems governed by partial and ordinary differential equations.

Symmetry is a property which occurs throughout nature and it is therefore natural that symmetry should be considered when attempting to model nature. In recent years, the tools provided by group theory and representation theory have proven to be highly effective in treating nonlinear problems involving symmetry.

The present Volume contains the contributions to the fourth meeting on Unilateral Problems in Structural Analysis, held at Capri on June 14 to 16,1989.

Multivariate Approximation Theory forms a rapidly evolving field in Applied Mathematics. The reason for its particular current interest lies in its impact on Computer Aided Geometric Design (CAGD), Image Processing, Pattern Recogni tion, and Mult idimensional Signal Processing. Mul ti var iate Bernstein polynomials and box splines, for example, play an important role in CAGD. Conversely, the highly important filter bank design problem of signal processing, for instance, gives rise to a new family of multivariate approximating functions, the Gabor wavelets, with interesting technological and biological applications. The conferences on Multivariate Approximation Theory held at the Mathematical Research Institute at Oberwolfach, Black Forest, in 1976, 1979, 1982, 1985 and 1989 ref lect the progress made in this area and related fie Ids. The present volume which is a continuation of the preceding volumes Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571 (1977) Multivariate Approximation Theory, ISNM 51 (1979) Multivariate Approximation Theory II, ISNM 61 (1982) Multivariate Approximation Theory III, ISNM 75 (1985) is based on the conference held on February 12-18, 1989. It includes most of the lectures presented at the Oberwolfach meeting and reveals the wide spectrum of activities in the field of multivariate approximation. The organizers are grateful to the Director of the Oberwolfach Mathematical Research Institute, Professor Dr. M. Barner, and his staff for providing the facili ties, and to Dr. G. Baszenski, Professor Dr. F. J. Delvos, Dr. H.

1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU¿s Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of the T-Operator.- 2.4. VEKUA¿s Theorems.- 2.5. Some Integral Operators on the Manifold.- 3. Orthogonal Decomposition of the Space L2,H(G).- 4. Some Boundary Value Problems of DIRICHLET¿s Type.- 4.1. LAPLACE Equation.- 4.2. HELMHOLTZ Equation.- 4.3. Equations of Linear Elasticity.- 4.4. Time-independent MAXWELL Equations.- 4.5. STOKES Equations.- 4.6. NAVIER-STOKES Equations.- 4.7. Stream Problems with Free Convection.- 4.8. Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity.- 5. H-regular Boundary Collocation Methods.- 5.1. Complete Systems of H-regular Functions.- 5.2. Numerical Properties of H-complete Systems of H-regular Functions.- 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems.- 5.4. Numerical Examples.- 6. Discrete Quaternionic Function Theory.- 6.1. Fundamental Solutions of the Discrete Laplacian.- 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator.- 6.3. Elements of a Discrete Quaternionic Function Theory.- 6.4. Main Properties of Discrete Operators.- 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations.- 6.6. Concluding Remarks.- References.- Notations.

It is probably more efficient to present such material after a reasonable competence in (at least) linear algebra and calculus has already been attained - but at this stage those not specializ ing in numerical mathematics are often interested in getting more deeply into their chosen field than in developing skills for later use.

The subjects covered treat bifurcation problems, ranging from theoretical investigations to numerical results, with emphasis placed upon applications. Different kinds of bifurcations are treated: Hopf bifurcation, bifurcation from continuous spec trum, complex bifurcation, and bifurcation near tori.

It is probably more efficient to present such material after a strong grasp of (at least) linear algebra and calculus has already been attained -but at this stage those not specializing in numerical mathematics are often interested in getting more deeply into their chosen field than in developing skills for later use.

The finite difference and finite element methods are powerful tools for the approximate solution of differential equations governing diverse physical phenomena, and there is extensive literature on these discre tization methods.

The Fifth International Conference on General Inequalities was held from May 4 to May 10, 1986, at the Mathematisches Forschungsinstitut Oberwolfach (Black Forest, Germany). The organizing committee consisted of W.N. Everitt (Birmingham), L. Losonczi (Debrecen) and W. Walter (Karlsruhe). Dr. A. Kovacec served efficiently an'd enthusiastically as secretary to the con­ ference. The meeting was attended by 50 participants from 16 countries. In his opening address, W. Walter had to report on the death of five colleagues who had been active in the area of inequali­ ties and who had served the mathematical community: P.R. Beesack, G. Polya, D.K. Ross, R. Bellman, G. Szegö. He made special mention of G. Polya, who had been the last surviving author of the book InequaZities (Cambridge University Press, 1934), who died at the age of 97 years and whose many and manifold contributions to mathematics will be recorded elsewhere, in due course. Inequalities continue to play an important and significant role in nearly all areas of mathematics. The interests of the participants to this conference reflected the many different fields in which both classical and modern inequalities continue to influence developments in mathematics. In addition to the established fields, the lectures clearly indicated the importance of inequalities in functional analysis, eigenvalue theory, con­ vexi ty., number theory, approximation theory, probability theory, mathematical prograrnrning and economics.

The Third International Symposium on Hultivariate Approximation Theory was held at the Oberwolfach !1athematical Research Insti­ tute, Black Forest, February 8-12, 1982. The preceding conferen­ ces on this topic were held in 1976* and 1979**. The conference brought together 50 mathematicians from 14 coun­ tries. These Proceedings form arecord of most of the papers pre­ sented at the Symposium. The topics treated cover different problems on multivariate approximation theory such as new results concerning approxima­ tion by polynomials in Sobolev spaces, biorthogonal systems and orthogonal series of functions in several variables, multivariate spline functions, group theoretic and functional analytic methods, positive linear operators, error estimates for approximation procedures and cubature formulae, Boolean methods in multivari­ ate interpolation and the numerical application of summation procedures. Special emphasis was posed on the application of multivariate approximation in various fields of science. One mathematician was sorely missed at the Symposium. Professor Arthur Sard who had actively taken part in the earlier conferen­ ces passed away in August of 1980. Since he was a friend of many of the participants, the editors wish to dedicate these Procee­ dings to the memory of this distinguished mathematician. Abrief appreciation of his life and mathematical work appears as well *"Constructive Theory of Functions of Several Variables". Edited by w. Schempp and Karl Zeller. Lecture Notes in 1-1athematics, Vol.

Brannigan, M. A Multivariate Adaptive Data Fitting Algorithm 30 Brass, H. Zur numerischen Berechnung konjugierter Funktionen 43 Bultheel, A.

This monograph contains a collection of 16 papers that were presented at the confer ence "Free Boundary Problems: Numerical 7reatment and Optimal Control", held at the Mathematisches Forschungsinstitut Oberwolfach, West Germany, July 9-15, 1989.

The publication of Oberwolfach conference books was initiated by Birkhauser Publishers in 1964 with the proceedings of the conference 'On Approximation Theory', conducted by P.

The articles, both theoretical and numerical, encompass a wide range of applications, such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phenomena and geometrical optics.

Particular examples of problems on which substantial progress has been made are the control and stabilization of mechatronic structures, the control of growth of thin films and crystals, the control of Laser and semi-conductor devices, and shape optimization problems for turbomachine blades, shells, smart materials and microdiffractive optics.

The international symposium held in October 1984 at the Uni versity of Mannheim was the first with the special aim to expose the connection of the Theory of Delay Eauations and Approximation Theory with the emphasis on constructive methods and applications.

This volume contains the proceedings of a conference held in Wiirzburg, August 20-24, 1990. How trends vary may be seen by comparing these proceedings with previous ones, in particular with the conference held in Dortmund 1986 (proceedings published in ISNM 79).

This book presents different formulations of the equations governing incompressible viscous flows, in the form needed for developing numerical solution procedures.

Jim Douglas, Jr.' These proceedings reflect some of the thoughts expressed at the Oberwolfach Con ference on Porous Media held June 21-27, 1992, organized by Jim Douglas, Jr., Ulrich Hornung, and Cornelius J, van Duijn.

Herbert Hornlein, Klaus Schittkowski The finite element method (FEM) has been used successfully for many years to simulate and analyse mechanical structural problems. They proceed from the design model as defined for structural analysis, to perform an internal adaption of design pa rameters based on formal mathematical methods.

22 papers on control of nonlinear partial differential equations highlight the area from a broad variety of viewpoints. A significant part of the volume is devoted to applications in engineering, continuum mechanics and population biology.

Many phenomena of interest for applications are represented by differential equations which are defined in a domain whose boundary is a priori unknown, and is accordingly named a "free boundary".

A collection of articles summarizing the state of knowledge in a large portion of modern homotopy theory. This welcome reference for many new results and recent methods is addressed to all mathematicians interested in homotopy theory and in geometric aspects of group theory.

This book introduces general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators.