For well over a decade, the numerical approach to field computation has been gaining progressively greater importance. Analytical methods of field compu tation are, at best, unable to accommodate the very wide variety of configura tions in which fields must be computed. On the other hand, numerical methods can accommodate many practical configurations that analytical methods cannot. With the advent of high-speed digital computers, numerical field computations have finally become practical. However, in order to implement numerical methods of field computation, we need algorithms, numerical methods, and mathematical tools that are largely quite different from those that have been traditionally used with analytical methods. Many of these algorithms have, in fact, been presented in the large number of papers that have been published on this subject in the last two decades. And to some of those who are already experienced in the art of numerical field computations, these papers, in addition to their own original work, are enough to give them the knowledge that they need to perform practical numerical field computations."

The 1986 NATO Advanced Study Insti tute on Dynamics of Infini te Dimensional Systems was held at the Instituto Superior Tecnico. Lisbon. Portugal. In recent years. there have been several research workers who have been considering partial differential equations and functional differential equations as dynamical systems on function spaces. Such approaches have led to the formulation of more theoretical problems that need to be investigated. In the applications. the theoretical ideas have contributed significantly to a better understanding of phenomena that have been experimentally and computationally observed. The investigators of this development come wi th several different backgrounds - some from classical partial differential equations. some from classical ordinary differential equations and some interested in specific applications. Each group has special ideas and often these ideas have not been transmitted from one group to another. The purpose of this NATO Workshop was to bring together research workers from these various areas. It provided asoundboard for the impact of the ideas of each respective discipline. We believe that goal was accomplished. but time will be a better judge. We have included the list of participants at the workshop. with most of these giving a presentation. Although the proceedings do not include all of the presentations. it is a good representative sampie. We wish to express our gratitude to NATO. and.to Dr. M. di Lullo of NATO. who unfortunately did not live to see the completion of this project.

Infotext]((Kurztext))These are the proceedings of the 7th International Conference on Hyperbolic Problems, held in Zurich in February 1998. The speakers and contributors have been rigorously selected and present the state of the art in this field. 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.

((Volltext))These proceedings contain, in two volumes, approximately one hundred papers presented at the conference on hyperbolic problems, which has focused to a large extent on the laws of nonlinear hyperbolic conservation. Two-fifths of the papers are devoted to mathematical aspects such as global existence, uniqueness, asymptotic behavior such as large time stability, stability and instabilities of waves and structures, various limits of the solution, the Riemann problem and so on. Roughly the same number of articles are devoted to numerical analysis, for example stability and convergence of numerical schemes, as well as schemes with special desired properties such as shock capturing, interface fitting and high-order approximations to multidimensional systems. The results in these contributions, 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."

This IMA Volume in Mathematics and its Applications PARALLEL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS is based on the proceedings of a workshop with the same title. The work shop was an integral part of the 1996-97IMA program on "MATHEMAT ICS IN HIGH-PERFORMANCE COMPUTING." I would like to thank Petter Bj0rstad of the Institutt for Informatikk, University of Bergen and Mitchell Luskin of the School of Mathematics, University of Minnesota for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Founda tion (NSF), Department of Energy (DOE), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE The numerical solution of partial differential equations has been of major importance to the development of many technologies and has been the target of much of the development of parallel computer hardware and software. Parallel computers offer the promise of greatly increased perfor mance and the routine calculation of previously intractable problems. The papers in this volume were presented at the IMA workshop on the Paral lel Solution of PDE held during June 9-13, 1997. The workshop brought together leading numerical analysts, computer scientists, and engineers to assess the state-of-the-art and to consider future directions.

The subject of numerical methods in finance has recently emerged as a new discipline at the intersection of probability theory, finance, and numerical analysis. The methods employed bridge the gap between financial theory and computational practice, and provide solutions for complex problems that are difficult to solve by traditional analytical methods. Although numerical methods in finance have been studied intensively in recent years, many theoretical and practical financial aspects have yet to be explored. This volume presents current research and survey articles focusing on various numerical methods in finance. The book is designed for the academic community and will also serve professional investors.

One of the major concerns of theoretical computer science is the classifi cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function."

This book is motivated largely by a desire to solve shape optimization prob lems that arise in applications, particularly in structural mechanics and in the optimal control of distributed parameter systems. Many such problems can be formulated as the minimization of functionals defined over a class of admissible domains. Shape optimization is quite indispensable in the design and construction of industrial structures. For example, aircraft and spacecraft have to satisfy, at the same time, very strict criteria on mechanical performance while weighing as little as possible. The shape optimization problem for such a structure consists in finding a geometry of the structure which minimizes a given functional (e. g. such as the weight of the structure) and yet simultaneously satisfies specific constraints (like thickness, strain energy, or displacement bounds). The geometry of the structure can be considered as a given domain in the three-dimensional Euclidean space. The domain is an open, bounded set whose topology is given, e. g. it may be simply or doubly connected. The boundary is smooth or piecewise smooth, so boundary value problems that are defined in the domain and associated with the classical partial differential equations of mathematical physics are well posed. In general the cost functional takes the form of an integral over the domain or its boundary where the integrand depends smoothly on the solution of a boundary value problem."

This book constitutes the thoroughly refereed proceedings of the 38th International Workshop on Graph Theoretic Concepts in Computer Science (WG 2012) held in Jerusalem, Israel on June 26-28, 2012. The 29 revised full papers presented were carefully selected and reviewed from 78 submissions. The papers are solicited describing original results on all aspects of graph-theoretic concepts in Computer Science, e.g. structural graph theory, sequential, parallel, randomized, parameterized, and distributed graph and network algorithms and their complexity, graph grammars and graph rewriting systems, graph-based modeling, graph-drawing and layout, random graphs, diagram methods, and support of these concepts by suitable implementations. The scope of WG includes all applications of graph-theoretic concepts in Computer Science, including data structures, data bases, programming languages, computational geometry, tools for software construction, communications, computing on the web, models of the web and scale-free networks, mobile computing, concurrency, computer architectures, VLSI, artificial intelligence, graphics, CAD, operations research, and pattern recognition

This volume is a selection from the 281 published papers of Joseph Leonard Walsh, former US Naval Officer and professor at University of Maryland and Harvard University. The nine broad sections are ordered following the evolution of his work. Commentaries and discussions of subsequent development are appended to most of the sections. Also included is one of Walsh's most influential works, "A closed set of normal orthogonal function," which introduced what is now known as "Walsh Functions".

The course of lectures on numerical methods (part I) given by the author to students in the numerical third of the course of the mathematics mechanics department of Leningrad State University is set down in this volume. Only the topics which, in the opinion of the author, are of the greatest value for numerical methods are considered in this book. This permits making the book comparatively small in size, and, the author hopes, accessible to a sufficiently wide circle of readers. The book may be used not only by students in daily classes, but also by students taking correspondence courses and persons connected with practical computa tion who desire to improve their theoretical background. The author is deeply grateful to V. I. Krylov, the organizer ofthe course on numerical methods (part I) at Leningrad State University, for his considerable assistance and constant interest in the work on this book, and also for his attentive review of the manuscript. The author is very grateful to G. P. Akilov and I. K. Daugavet for a series of valuable suggestions and observations. The Author Chapter I NUMERICAL SOLUTION OF EQUATIONS In this chapter, methods for the numerical solution of equations of the form P(x) = 0, will be considered, where P(x) is in general a complex-valued function.

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.

Mathematical modelling of many physical processes involves rather complex dif- ferential, integral, and integro-differential equations which can be solved directly only in a number of cases. Therefore, as a first step, an original problem has to be considerably simplified in order to get a preliminary knowledge of the most important qualitative features of the process under investigation and to estimate the effect of various factors. Sometimes a solution of the simplified problem can be obtained in the analytical form convenient for further investigation. At this stage of the mathematical modelling it is useful to apply various special functions. Many model problems of atomic, molecular, and nuclear physics, electrody- namics, and acoustics may be reduced to equations of hypergeometric type, a(x)y" + r(x)y' + AY = 0 , (0.1) where a(x) and r(x) are polynomials of at most the second and first degree re- spectively and A is a constant [E7, AI, N18]. Some solutions of (0.1) are functions extensively used in mathematical physics such as classical orthogonal polyno- mials (the Jacobi, Laguerre, and Hermite polynomials) and hypergeometric and confluent hypergeometric functions.

There is no doubt nowadays that numerical mathematics is an essential component of any educational program. 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. An alternative approach is to incorporate the numerical aspects of linear algebra and cal culus as these subjects are being developed. Long experience has persuaded us that a third attack on this problem is the best and this is developed in the present two volumes, which are, however, easily adaptable to other circum stances. The approach we prefer is to treat the numerical aspects separately, but after some theoretical background. This is often desirable because of the shortage of persons qualified to present the combined approach and also because the numerical approach provides an often welcome change which, however, in addition, can lead to better appreciation of the fundamental con cepts. For instance, in a 6-quarter course in Calculus and Linear Algebra, the material in Volume 1 can be handled in the third quarter and that in Volume 2 in the fifth or sixth quarter."

This text is an introduction to methods of grid generation technology in scientific computing. Special attention is given to methods developed by the author for the treatment of singularly-perturbed equations, e.g. in modeling high Reynolds number flows. Functionals of conformality, orthogonality, energy and alignment are discussed.

Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz 1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis."

The book is a revised and updated version of the lectures given by the author at the University of Timi oara during the academic year 1990-1991. Its goal is to present in detail someold and new aspects ofthe geometry ofsymplectic and Poisson manifolds and to point out some of their applications in Hamiltonian mechanics and geometric quantization. The material is organized as follows. In Chapter 1 we collect some general facts about symplectic vector spaces, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study ofHamiltonian mechanics. We present here the gen- eral theory ofHamiltonian mechanicalsystems, the theory ofthe corresponding Pois- son bracket and also some examples ofinfinite-dimensional Hamiltonian mechanical systems. Chapter 3 starts with some standard facts concerning the theory of Lie groups and Lie algebras and then continues with the theory ofmomentum mappings and the Marsden-Weinstein reduction. The theory of Hamilton-Poisson mechan- ical systems makes the object of Chapter 4. Chapter 5 js dedicated to the study of the stability of the equilibrium solutions of the Hamiltonian and the Hamilton- Poisson mechanical systems. We present here some of the remarcable results due to Holm, Marsden, Ra~iu and Weinstein. Next, Chapter 6 and 7 are devoted to the theory of geometric quantization where we try to solve, in a geometrical way, the so called Dirac problem from quantum mechanics. We follow here the construc- tion given by Kostant and Souriau around 1964.

This book constitutes the refereed proceedings of the 7th International Workshop on Algorithms and Computation, WALCOM 2013, held in Kharagpur, India, in February 2013. The 29 full papers presented were carefully reviewed and selected from 86 submissions. The papers are organized in topical sections on computational geometry, approximation and randomized algorithms, parallel and distributed computing, graph algorithms, complexity and bounds, and graph drawing.

The classical theories of Linear Elasticity and Newtonian Fluids, though trium phantly elegant as mathematical structures, do not adequately describe the defor mation and flow of most real materials. Attempts to characterize the behaviour of real materials under the action of external forces gave rise to the science of Rheology. Early rheological studies isolated the phenomena now labelled as viscoelastic. Weber (1835, 1841), researching the behaviour of silk threats under load, noted an instantaneous extension, followed by a further extension over a long period of time. On removal of the load, the original length was eventually recovered. He also deduced that the phenomena of stress relaxation and damping of vibrations should occur. Later investigators showed that similar effects may be observed in other materials. The German school referred to these as "Elastische Nachwirkung" or "the elastic aftereffect" while the British school, including Lord Kelvin, spoke ofthe "viscosityofsolids." The universal adoption of the term "Viscoelasticity," intended to convey behaviour combining proper ties both of a viscous liquid and an elastic solid, is of recent origin, not being used for example by Love (1934), though Alfrey (1948) uses it in the context of polymers. The earliest attempts at mathematically modelling viscoelastic behaviour were those of Maxwell (1867) (actually in the context of his work on gases; he used this model for calculating the viscosity of a gas) and Meyer (1874)."

One of the most well-known of all network optimization problems is the shortest path problem, where a shortest connection between two locations in a road network is to be found. This problem is the basis of route planners in vehicles and on the Internet. Networks are very common structures; they consist primarily of a ?nite number of locations (points, nodes), together with a number of links (edges, arcs, connections) between the locations. Very often a certain number is attached to the links, expressing the distance or the cost between the end points of that connection. Networks occur in an extremely wide range of applications, among them are: road networks; cable networks; human relations networks; project scheduling networks; production networks; distribution networks; neural networks; networks of atoms in molecules. In all these cases there are "objects" and "relations" between the objects. A n- work optimization problem is actually nothing else than the problem of ?nding a subset of the objects and the relations, such that a certain optimization objective is satis?ed.

Parallel computers have started to completely revolutionize scientific computation. Articles in this volume represent applied mathematics, computer science, and application aspects of parallel scientific computing. Major advances are discussed dealing with multiprocessor architectures, parallel algorithm development and analysis, parallel systems and programming languages. The optimization of the application of massively parallel architectures to real world problems will provide the impetus for the development of entirely new approaches to these technical situations.

The chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important topics at the forefront of nonlinear analysis and optimization.

This book contains the main results of the German project POPINDA. It surveys the state of the art of industrial aerodynamic design simulations on parallel systems. POPINDA is an acronym for Portable Parallelization of Industrial Aerodynamic Applications. This project started in late 1993. The research and development work invested in POPINDA corresponds to about 12 scientists working full-time for the three and a half years of the project. POPINDA was funded by the German Federal Ministry for Education, Science, Research and Technology (BMBF). The central goals of POPINDA were to unify and parallelize the block-structured aerodynamic flow codes of the German aircraft industry and to develop new algorithmic approaches to improve the efficiency and robustness of these programs. The philosophy behind these goals is that challenging and important numerical appli cations such as the prediction of the 3D viscous flow around full aircraft in aerodynamic design can only be carried out successfully if the benefits of modern fast numerical solvers and parallel high performance computers are combined. This combination is a "conditio sine qua non" if more complex applications such as aerodynamic design optimization or fluid structure interaction problems have to be solved. When being solved in a standard industrial aerodynamic design process, such more complex applications even require a substantial further reduction of computing times. Parallel and vector computers on the one side and innovative numerical algorithms such as multigrid on the other have enabled impressive improvements in scientific computing in the last 15 years."

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schroedinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and the self-dual Yang-Mills equations, and describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.

There is no doubt nowadays that numerical mathematics is an essential component of any educational program. 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. An alternative approach is to incorporate the numerical aspects of linear algebra and calculus as these subjects are being developed. Long experience has persuaded us that a third attack on this problem is the best and this is developed in the present two volumes, which are, however, easily adaptable to other circumstances. The approach we prefer is to treat the numerical aspects separately, but after some theoretical background. This is often desirable because of the shortage of persons qualified to present the combined approach and also because the numerical approach provides an often welcome change which, however, in addition, can lead to better appreciation of the fundamental concepts. For instance, in a 6-quarter course in Calculus and Linear Algebra, the material in Volume 1 can be handled in the third quarter and that in Volume 2 in the fifth or sixth quarter.

This volume contains the contributions to the 10th DGLR I AG ST AB-Symposium held at the German Aerospace Research Establishment (DLR) -Braunschweig Research Centre - November, 11 to 13, 1996. AG STAB is the German Aerospace AerodynapIics Association, founded at the end of the 70', while DGLR is the German Society for Aeronautics and Astronautics (Deutsche Gesellschaft fur Luft- und Raumfahrt, LilienthaI Oberth Gesellschaft). In the AG STAB German scientists and engineers from universities, research-establishments and industries are involved, who are doing research and project work in numerical and experimental fluidmechanics and aerodynamics for aerospace and other applications. About 20 years ago it became obvious for this community that a joint effort of members of universities, the DLR and industry was necessary to counter-act declining budgets in the field. It was decided to approach high-level persons in industry, ministries and the parliament for help to shift the trend with its negative effects for research and industry. From the begin it was clear that an effort should be built around a central theme. ''Flow with Separation" became the topic of the AG STAB (Arbeitsgemeinschaft Stromung mit Ablosung), which developed fast into a lively association, with, however, a larger scope than just flow with separation."