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The main goal of this work is to revisit the proof of the global stability of Minkowski space by D. Christodoulou and S. Klainerman, [Ch-KI]. We provide a new self-contained proof of the main part of that result, which concerns the full solution of the radiation problem in vacuum, for arbitrary asymptotically flat initial data sets. This can also be interpreted as a proof of the global stability of the external region of Schwarzschild spacetime. The proof, which is a significant modification of the arguments in [Ch-Kl], is based on a double null foliation of spacetime instead of the mixed null-maximal foliation used in [Ch-Kl]. This approach is more naturally adapted to the radiation features of the Einstein equations and leads to important technical simplifications. In the first chapter we review some basic notions of differential geometry that are sys­ tematically used in all the remaining chapters. We then introduce the Einstein equations and the initial data sets and discuss some of the basic features of the initial value problem in general relativity. We shall review, without proofs, well-established results concerning local and global existence and uniqueness and formulate our main result. The second chapter provides the technical motivation for the proof of our main theorem.

Arnold Sommerfeld's'Mathematische Theorie der Diffraction'marks a milestone in optical theory, full of insights that are still relevant today. In a stunning tour de force, Sommerfeld derives the first mathematically rigorous solution of an optical diffraction problem. Indeed, his diffraction analysis is a surprisingly rich and complex mix of pure and applied mathematics, and his often-cited diffraction solution is presented only as an application of a much more general set of mathematical results. The body of Sommerfeld's work is devoted to the systematic development of a method for deriving solutions of the wave equation on Riemann surfaces, a fascinating but perhaps underappreciated topic in mathematical physics.

Broadly organized around the applications of Fourier analysis, Methods of Applied Mathematics with a MATLAB Overview covers both classical applications in partial differential equations and boundary value problems, as well as the concepts and methods associated to the Laplace, Fourier, and discrete transforms. Transform inversion problems are also examined, along with the necessary background in complex variables. A final chapter treats wavelets, short-time Fourier analysis, and geometrically-based transforms. The computer program MATLAB is emphasized throughout, and an introduction to MATLAB is provided in an appendix. Rich in examples, illustrations, and exercises of varying difficulty, this text can be used for a one- or two-semester course and is ideal for students in pure and applied mathematics, physics, and engineering.

This book is a snapshot of current research in multiscale modeling, computations and applications. It covers fundamental mathematical theory, numerical algorithms as well as practical computational advice for analysing single and multiphysics models containing a variety of scales in time and space. Complex fluids, porous media flow and oscillatory dynamical systems are treated in some extra depth, as well as tools like analytical and numerical homogenization, and fast multipole method.

The topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering. This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications. These proceedings present a selection of the latest exciting results by world leading researchers.

“Hypernumbers and Extrafunctions” presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics. This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students.

This volume showcases selected recent work presented at the 13th Regional Conference on Mathematical Physics held in Antalya, Turkey in 2010. The conference was dedicated to the memory of Faheem Hussain, one of the initiators of the Regional Conference series, and one of the organizers of the 12th Regional Conference. The “region”, originally comprised of Turkey, Iran and Pakistan, extends now to Bangladesh and Central Asia. However, the contributing researchers are not only limited to this region.Prominent contributors include B Ahmedov (Tashkent), F Ardalan (Tehran), N Dadhich (Pune), D A Demir (İzmir), R L Hall (Montreal), M Hortaçsu (İstanbul), M Koca (Oman), C S Lim (Kobe), F Mahomed (Johannesburg), A Qadir (Rawalpindi), M A Rashid (Rawalpindi), M Sakamoto (Kobe), M Sharif (Lahore), F Toppan (Rio), N Ünal (Antalya), amongst others. They sample a number of topics in the formal aspects of mathematical physics, general relativity and cosmology, quantum gravity, quantum field theory, and even applied physics.

This book summarizes and highlights progress in our understanding of Dy­ namical Systems during six years of the German Priority Research Program'Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems'. The program was funded by the Deutsche Forschungsgemeinschaft (DFG) and aimed at combining, focussing, and enhancing research efforts of active groups in the field by cooperation on a federal level. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications far into the neighboring disciplines of Science. Three fundamental topics in Dynamical Systems are at the core of our research effort: behavior for large time dimension measure, and chaos Each of these topics is, of course, a highly complex problem area in itself and does not fit naturally into the deplorably traditional confines of any of the disciplines of ergodic theory, analysis, or numerical analysis alone. The necessity of mathematical cooperation between these three disciplines is quite obvious when facing the formidahle task of establishing a bidirectional transfer which bridges the gap between deep, detailed theoretical insight and relevant, specific applications. Both analysis and numerical analysis playa key role when it comes to huilding that bridge. Some steps of our joint bridging efforts are collected in this volume. Neither our approach nor the presentations in this volume are monolithic.

This IMA Volume in Mathematics and its Applications MULTIPARTICLE QUANTUM SCATTERING WITH APPLICATIONS TO NUCLEAR, ATOMIC AND MOLECULAR PHYSICS is based on the proceedings of a workshop with the same title, which was an integral part of the 1994-1995 IMA program on'Waves and Scattering.'We would like to thank Donald G. Truhlar and Barry Simon for their ex­ cellent work as organizers of this meeting and as editors of the proceedings. We also take this opportunity to thank the National Science Foundation (NSF), the Army Research Office (ARO), and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE The workshop on Multiparticle Quantum Scattering with Applications to Nuclear, Atomic, and Molecular Physics was held June 12-16, 1995 at the Institute for Mathematics and Its Applications in the University of Min­ nesota Twin Cities campus as part of the 1994-95 Program on Waves and Scattering. There were about seventy participants including the plenary lecturers whose contributions are included in this volume. The workshop was preceded by a two-day tutorial featuring lectures by Donald J. Kouri and Gian Michele Graf, and we are pleased that both Professors Graf and Kouri were able to write up their tutorials as opening chapters of this volume.

Special relativity is one of the high points of the undergraduate mathematical physics syllabus. Nick Woodhouse writes for those approaching the subject with a background in mathematics: he aims to build on their familiarity with the foundational material and the way of thinking taught in first-year mathematics courses, but not to assume an unreasonable degree of prior knowledge of traditional areas of physical applied mathematics, particularly electromagnetic theory. His book provides mathematics students with the tools they need to understand the physical basis of special relativity and leaves them with a confident mathematical understanding of Minkowski's picture of space-time. Special Relativity is loosely based on the tried and tested course at Oxford, where extensive tutorials and problem classes support the lecture course. This is reflected in the book in the large number of examples and exercises, ranging from the rather simple through to the more involved and challenging. Theauthor has included material on acceleration and tensors, and has written the book with an emphasis on space-time diagrams. Written with the second year undergraduate in mind, the book will appeal to those studying the'Special Relativity'option in their Mathematics or Mathematics and Physics course. However, a graduate or lecturer wanting a rapid introduction to special relativity would benefit from the concise and precise nature of the book.

This collection of essays spans pure and applied mathematics. Readers interested in mathematical research and historical aspects of mathematics will appreciate the enlightening content of the material. Highlighting the pervasive nature of mathematics today in a host of different areas, the book also covers the spread of mathematical ideas and techn

Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange's reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigate existence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.

The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.

Fractals and disordered systems have recently become the focus of intense interest in research. This book discusses in great detail the effects of disorder on mesoscopic scales (fractures, aggregates, colloids, surfaces and interfaces, glasses, and polymers) and presents tools to describe them in mathematical language. A substantial part is devoted to the development of scaling theories based on fractal concepts. In 10 chapters written by leading experts in the field, including E. Stanley and B. Mandelbrot, the reader is introduced to basic concepts and techniques in disordered systems and is lead to the forefront of current research. In each chapter the connection between theory and experiment is emphasized, and a special chapter entitled'Fractals and Experiments'presents experimental studies of fractal systems in the laboratory. The book is written pedagogically. It can be used as a textbook for graduate students, by university teachers to prepare courses and seminars, and by active scientists who want to become familiar with a fascinating new field.

Proceedings of the Conference on Differential Geometry, Budapest, Hungary, July 27-30, 1996

Proceedings of the Workshop, Bharathidasan University, Tiruchirapalli, India, November 29 - December 2, 1989

This study began as an attempt to understand mechanics in the nineteenth century. The terms mechanics and mechanical world view were being used as general descriptions of nineteenth-century physicists'assumptions and interpretations of nature. However, there were no studies of the particulars of these assumptions or the range and content of these interpretations. Rene Dugas'work on classical mechanics focused on France. The search for the particulars of these forms of'mechanics'led me to explore precisely what mechanics meant to physicists of a century and more ago. However, none of Lagrange's, Hamilton's, or Jacobi's'mechanics,'while ele­ gant, fits easily within the history of physics. Lagrange reduced mechanics to an exercise in analysis; Hamilton and Jacobi used mechanics to explore solutions to partial differential equations. They were mathematicians doing mathematics. As I went deeper into the matter it became obvious that, in the nineteenth century, there were two kinds of mechanics, each containing a variety of forms, one physical, the other mathematical. There were a group of men using mechanics to understand nature and another group using the equations of mechanics to explore the calcu­ lus. However, when tracing these two traditions back into the eighteenth century, physics disappeared altogether.

This is a textbook written for use in a graduate-level course for students of mechanics and engineering science. It is designed to cover the essential features of modern variational methods and to demonstrate how a number of basic mathematical concepts can be used to produce a unified theory of variational mechanics. As prerequisite to using this text, we assume that the student is equipped with an introductory course in functional analysis at a level roughly equal to that covered, for example, in Kolmogorov and Fomin (Functional Analysis, Vol. I, Graylock, Rochester, 1957) and possibly a graduate-level course in continuum mechanics. Numerous references to supplementary material are listed throughout the book. We are indebted to Professor Jim Douglas of the University of Chicago, who read an earlier version of the manuscript and whose detailed suggestions were extremely helpful in preparing the final draft. He also gratefully acknowledge that much of our own research work on variational theory was supported by the U.S. Air Force Office of Scientific Research. He are indebted to Mr. Ming-Goei Sheu for help in proofreading. Finally, we wish to express thanks to Mrs. Marilyn Gude for her excellent and pains­ taking job of typing the manuscript. J. T. ODEN J. N. REDDY Table of Contents PREFACE 1. INTRODUCTION 1.1 The Role of Variational Theory in Mechanics. 1 1.2 Some Historical Comments.......... 2 1.3 Plan of Study................ 5 7 2. MATHEMATICAL FOUNDATIONS OF CLASSICAL VARIATIONAL THEORY 7 2.1 Introduction........

Astronomers do not do experiments. They observe the universe primarily through detect­ ing light emitted by stars and other luminous objects. Since this light must travel through space to reach us, variations in the metric of space affects the appearance of astronomical objects. These variations lead to dramatic changes in the shape and brightness of astronom­ ical sources. Because these variations are sensitive to mass rather than to light, observations of gravitational lensing enable astronomers to probe the mass distribution of the universe. With gravitational lensing observations, astronomers are addressing many of the most important scientific questions in astronomy and physics: • What is the universe made of? Most of the energy and mass in the universe is not in the form of luminous objects. Stars account for less than 1% of the energy density of the universe. Perhaps, as much as another 3% of the energy density of the universe is in the form of warm gas that fills the space between galaxies. The remaining 96% of the energy density is in some yet unidentified form. Roughly one third of this energy density of the universe is'dark matter,'matter that clusters gravitationally but does not emit light. Most cosmologists suspect that this dark matter is composed of weakly interacting subatomic particles. However, most of the energy density of the universe appears to be in an even stranger form: energy associated with empty space.

More mathematicians have been taking part in the development of digital image processing as a science and the contributions are reflected in the increasingly important role modeling has played solving complex problems. This book is mostly concerned with energy-based models. Through concrete image analysis problems, the author develops consistent modeling, a know-how generally hidden in the proposed solutions. The book is divided into three main parts. The first two parts describe the materials necessary to the models expressed in the third part. These materials include splines (variational approach, regression spline, spline in high dimension), and random fields (Markovian field, parametric estimation, stochastic and deterministic optimization, continuous Gaussian field). Most of these models come from industrial projects in which the author was involved in robot vision and radiography: tracking 3D lines, radiographic image processing, 3D reconstruction and tomography, matching, deformation learning. Numerous graphical illustrations accompany the text showing the performance of the proposed models. This book will be useful to researchers and graduate students in applied mathematics, computer vision, and physics.

The topic of this book is the theory of state spaces of operator algebras and their geometry. The states are of interest because they determine representations of the algebra, and its algebraic structure is in an intriguing and fascinating fashion encoded in the geometry of the state space. From the beginning the theory of operator algebras was motivated by applications to physics, but recently it has found unexpected new applica­ tions to various fields of pure mathematics, like foliations and knot theory, and (in the Jordan algebra case) also to Banach manifolds and infinite di­ mensional holomorphy. This makes it a relevant field of study for readers with diverse backgrounds and interests. Therefore this book is not intended solely for specialists in operator algebras, but also for graduate students and mathematicians in other fields who want to learn the subject. We assume that the reader starts out with only the basic knowledge taught in standard graduate courses in real and complex variables, measure theory and functional analysis. We have given complete proofs of basic results on operator algebras, so that no previous knowledge in this field is needed. For discussion of some topics, more advanced prerequisites are needed. Here we have included all necessary definitions and statements of results, but in some cases proofs are referred to standard texts. In those cases we have tried to give references to material that can be read and understood easily in the context of our book.

In this book we display the fundamental structure underlying classical electro­ dynamics, i. e., the phenomenological theory of electric and magnetic effects. The book can be used as a textbook for an advanced course in theoretical electrodynamics for physics and mathematics students and, perhaps, for some highly motivated electrical engineering students. We expect from our readers that they know elementary electrodynamics in the conventional (1 + 3)-dimensional form including Maxwell's equations. More­ over, they should be familiar with linear algebra and elementary analysis, in­ cluding vector analysis. Some knowledge of differential geometry would help. Our approach rests on the metric-free integral formulation of the conservation laws of electrodynamics in the tradition of F. Kottler (1922), E. Cartan (1923), and D. van Dantzig (1934), and we stress, in particular, the axiomatic point of view. In this manner we are led to an understanding of why the Maxwell equa­ tions have their specific form. We hope that our book can be seen in the classical tradition of the book by E. J. Post (1962) on the Formal Structure of Electro­ magnetics and of the chapter'Charge and Magnetic Flux'of the encyclopedia article on classical field theories by C. Truesdell and R. A. Toupin (1960), in­ cluding R. A. Toupin's Bressanone lectures (1965); for the exact references see the end of the introduction on page 11..

The study of spatial patterns in extended systems, and their evolution with time, poses challenging questions for physicists and mathematicians alike. Waves on water, pulses in optical fibers, periodic structures in alloys, folds in rock formations, and cloud patterns in the sky: patterns are omnipresent in the world around us. Their variety and complexity make them a rich area of study. In the study of these phenomena an important role is played by well-chosen model equations, which are often simpler than the full equations describing the physical or biological system, but still capture its essential features. Through a thorough analysis of these model equations one hopes to glean a better under­ standing of the underlying mechanisms that are responsible for the formation and evolution of complex patterns. Classical model equations have typically been second-order partial differential equations. As an example we mention the widely studied Fisher-Kolmogorov or Allen-Cahn equation, originally proposed in 1937 as a model for the interaction of dispersal and fitness in biological populations. As another example we mention the Burgers equation, proposed in 1939 to study the interaction of diffusion and nonlinear convection in an attempt to understand the phenomenon of turbulence. Both of these are nonlinear second-order diffusion equations.

This invaluable book offers engineers and physicists working knowledge of a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, but nevertheless extremely useful when applied to typical problems in many different fields. It deals principally with linear algebraic equations, quadratic and Hermitian forms, operations with vectors and matrices, the calculus of variations, and the formulations and theory of linear integral equations. Annotated problems and exercises accompany each chapter.

Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with stochastic coefficients, etc. ​

The conceptofspontaneous symmetry breaking plays a fundamental role in contemporary physics. It is essential for the description of degenerate ground states, massless modes, and topological defects. Examples are abundant in condensed matter physics, atomic and particle physics, as well as in astro­ physics and cosmology. In fact, spontaneous symmetry breaking can be re­ garded as a cornerstone ofa whole branch ofphysics which intersects the above mentioned traditionally distinct fields. In the year 2000 the European Science Foundation (ESF) started the Pro­ gramme'Cosmology in the Laboratory'(COSLAB), with the goal to search for and to develop analogies betweencondensed matterphysics, particle physics, and cosmology. Not surprisingly, spontaneous symmetry breaking is among the most useful notions in that endeavour. It has been decided that in the sec­ ond year of the Programme a School should be held in order to work out and deliver to a wide audience of students synthetic overviews of achievements and of current research topics of COSLAB. This idea has been supported by the Scientific and Environmental Affairs Division of NATO by including the School in the renowned series of its Advanced Study Institutes. The School, entitled'Patterns of Symmetry Breaking', was held in Cracow during 16-28 September 2002. It gathered 17 lecturers and about 60 students. The present volume contains notes ofmost of the lectures from that School. We hope that of the physics of spon­ it will convey to the reader the breadth and the beauty taneous symmetry breaking.

Proceedings of the NATO Advanced Research Workshop, Chicago, USA, July 22-26, 2000

This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this second volume, special emphasis is placed on functional analytic methods and applications to differential geometry. The following topics are treated: solvability of operator equations in Banach spaces linear operators in Hilbert spaces and spectral theory Schauder's theory of linear elliptic differential equations weak solutions of differential equations nonlinear partial differential equations and characteristicsnonlinear elliptic systemsboundary value problems from differential geometryThis new second edition of this volume has been thoroughly revised and a new chapter on boundary value problems from differential geometry has been added.In the first volume, partial differential equations by integral representations are treated in a classical way.This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.

In the last decade there has been an extraordinary confluence of ideas in mathematics and theoretical physics brought about by pioneering discoveries in geometry and analysis. The various chapters in this volume, treating the interface of geometric analysis and mathematical physics, represent current research interests. No suitable succinct account of the material is available elsewhere. Key topics include: • A self-contained derivation of the partition function of Chern- Simons gauge theory in the semiclassical approximation (D.H. Adams) • Algebraic and geometric aspects of the Knizhnik-Zamolodchikov equations in conformal field theory (P. Bouwknegt) • Application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems (A.L. Carey and E. Langmann) • A study of variational methods in Hermitian geometry from the viewpoint of the critical points of action functionals together with physical backgrounds (A. Harris) • A review of monopoles in nonabelian gauge theories (M.K. Murray) • Exciting developments in quantum cohomology (Y. Ruan) • The physics origin of Seiberg-Witten equations in 4-manifold theory (S. Wu) Graduate students, mathematicians and mathematical physicists in the above-mentioned areas will benefit from the user-friendly introductory style of each chapter as well as the comprehensive bibliographies provided for each topic. Prerequisite knowledge is minimal since sufficient background material motivates each chapter.

The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems.'Fractal-Based Methods in Analysis'draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters. For each topic the theoretical framework is carefully explained using examples and applications. The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises. This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences. Herb Kunze is a professor of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo inOntario. The major focus of their research is on fractals and the applications of fractals.

Falling Liquid Films gives a detailed review of state-of-the-art theoretical, analytical and numerical methodologies, for the analysis of dissipative wave dynamics and pattern formation on the surface of a film falling down a planar inclined substrate. This prototype is an open-flow hydrodynamic instability, that represents an excellent paradigm for the study of complexity in active nonlinear media with energy supply, dissipation and dispersion. It will also be of use for a more general understanding of specific events characterizing the transition to spatio-temporal chaos and weak/dissipative turbulence. Particular emphasis is given to low-dimensional approximations for such flows through a hierarchy of modeling approaches, including equations of the boundary-layer type, averaged formulations based on weighted residuals approaches and long-wave expansions. Whenever possible the link between theory and experiment is illustrated, and, as a further bridge between the two, the development of order-of-magnitude estimates and scaling arguments is used to facilitate the understanding of basic, underlying physics. This monograph will appeal to advanced graduate students in applied mathematics, science or engineering undertaking research on interfacial fluid mechanics or studying fluid mechanics as part of their program. It will also be of use to researchers working on both applied, fundamental theoretical and experimental aspects of thin film flows, as well as engineers and technologists dealing with processes involving isothermal or heated films. This monograph is largely self-contained and no background on interfacial fluid mechanics is assumed.

The book comprises an assembly of benchmarks and examples for porous media mechanics collected over the last twenty years. Analysis of thermo-hydro-mechanical-chemical (THMC) processes is essential to many applications in environmental engineering, such as geological waste deposition, geothermal energy utilisation, carbon capture and storage, water resources management, hydrology, even climate chance. In order to assess the feasibility as well as the safety of geotechnical applications, process-based modelling is the only tool to put numbers, i.e. to quantify future scenarios. This charges a huge responsibility concerning the reliability of computational tools. Benchmarking is an appropriate methodology to verify the quality of modelling tools based on best practices. Moreover, benchmarking and code comparison foster community efforts. The benchmark book is part of the OpenGeoSys initiative - an open source project to share knowledge and experience in environmental analysis and scientific computation.

Mathematics, taught and learned appropriately, improves the mind and implants good habits of thought. This tenet underlies all of Professor Pólya's works on teaching and problem-solving. The distinctive feature of Mathematical Methods in Science is the stress on the history of certain elementary chapters of science; these can be a source of enjoyment and deeper understanding of mathematics even for beginners who have little, or perhaps no, knowledge of physics.

Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed. Features and Topics: • The mathematical foundations of geometric algebra are explored • Applications in computational geometry include models of reflection and ray-tracing and a new and concise characterization of the crystallographic groups • Applications in engineering include robotics, image geometry, control-pose estimation, inverse kinematics and dynamics, control and visual navigation • Applications in physics include rigid-body dynamics, elasticity, and electromagnetism • Chapters dedicated to quantum information theory dealing with multi- particle entanglement, MRI, and relativistic generalizations Practitioners, professionals, and researchers working in computer science, engineering, physics, and mathematics will find a wide range of useful applications in this state-of-the-art survey and reference book. Additionally, advanced graduate students interested in geometric algebra will find the most current applications and methods discussed.

This book is one of the finest I have ever read. To write a foreword for· it is an honor, difficult to accept. Everyone knows that architects and master masons, long before there were mathematical theories, erected structures of astonishing originality, strength, and beauty. Many of these still stand. Were it not for our now acid atmosphere, we could expect them to stand for centuries more. We admire early architects'visible success in the distribution and balance of thrusts, and we presume that master masons had rules, perhaps held secret, that enabled them to turn architects'bold designs into reality. Everyone knows that rational theories of strength and elasticity, created centuries later, were influenced by the wondrous buildings that men of the sixteenth, seventeenth, and eighteenth centuries saw daily. Theorists know that when, at last, theories began to appear, architects distrusted them, partly because they often disregarded details of importance in actual construction, partly because nobody but a mathematician could understand the aim and func­ tion of a mathematical theory designed to represent an aspect of nature. This book is the first to show how statics, strength of materials, and elasticity grew alongside existing architecture with its millenial traditions, its host of successes, its ever-renewing styles, and its numerous problems of maintenance and repair. In connection with studies toward repair of the dome of St. Peter's by Poleni in 1743, on p.

This book is an outgrowth of the activities of the Center for Geometry and Mathematical Physics (CGMP) at Penn State from 1996 to 1998. The Center was created in the Mathematics Department at Penn State in the fall of 1996 for the purpose of promoting and supporting the activities of researchers and students in and around geometry and physics at the university. The CGMP brings many visitors to Penn State and has ties with other research groups; it organizes weekly seminars as well as annual workshops The book contains 17 contributed articles on current research topics in a variety of fields: symplectic geometry, quantization, quantum groups, algebraic geometry, algebraic groups and invariant theory, and character­ istic classes. Most of the 20 authors have talked at Penn State about their research. Their articles present new results or discuss interesting perspec­ tives on recent work. All the articles have been refereed in the regular fashion of excellent scientific journals. Symplectic geometry, quantization and quantum groups is one main theme of the book. Several authors study deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting two transverse polarizations, and studies the moment map in the case of semisimple coadjoint orbits. Bieliavsky constructs an explicit star-product on holonomy reducible sym­ metric coadjoint orbits of a simple Lie group, and he shows how to con­ struct a star-representation which has interesting holomorphic properties.

This book considers methods of approximate analysis of mechanical, elec­ tromechanical, and other systems described by ordinary differential equa­ tions. Modern mathematical modeling of sophisticated mechanical systems consists of several stages: first, construction of a mechanical model, and then writing appropriate equations and their analytical or numerical ex­ amination. Usually, this procedure is repeated several times. Even if an initial model correctly reflects the main properties of a phenomenon, it de­ scribes, as a rule, many unnecessary details that make equations of motion too complicated. As experience and experimental data are accumulated, the researcher considers simpler models and simplifies the equations. Thus some terms are discarded, the order of the equations is lowered, and so on. This process requires time, experimentation, and the researcher's intu­ ition. A good example of such a semi-experimental way of simplifying is a gyroscopic precession equation. Formal mathematical proofs of its admis­ sibility appeared some several decades after its successful introduction in engineering calculations. Applied mathematics now has at its disposal many methods of approxi­ mate analysis of differential equations. Application of these methods could shorten and formalize the procedure of simplifying the equations and, thus, of constructing approximate motion models. Wide application of the methods into practice is hindered by the fol­ lowing. 1. Descriptions of various approximate methods are scattered over the mathematical literature. The researcher, as a rule, does not know what method is most suitable for a specific case. 2.

Proceedings of the NATO Advanced Research Workshop, held in Cadiz, Spain, from 12 to 16 June 2002

Lawrence Sirovich will turn seventy on March 1, 2003. Larry's academic life of over 45 years at the Courant Institute, Brown University, Rockefeller University and the Mount Sinai School of Medicine has touched many peo­ ple and several disciplines, from fluid dynamics to brain theory. His con­ tributions to the kinetic theory of gases, methods of applied mathematics, theoretical fluid dynamics, hydrodynamic turbulence, the biophysics of vi­ sion and the dynamics of neuronal populations, represent the creative work of an outstanding scholar who was stimulated mostly by insatiable curios­ ity. As a scientist, Larry has consistently offered fresh outlooks on classical and difficult subjects, and moved into new fields effortlessly. He delights in what he knows and does, and sets no artificial boundaries to the range of his inquiry. Among the more than fifty or so Ph. D. students and post­ docs that he has mentored, many continue to make first-rate contributions themselves and hold academic positions in the US and elsewhere. Larry's scientific collaborators are numerous and distinguished. Those of us who have known him well will agree that Larry's charm, above all, is his taste, wit, and grace under fire. Larry has contributed immensely to mathematics publishing. He be­ gan his career with Springer by founding the Applied Mathematical Sci­ ences series together with Fritz John and Joe LaSalle some 30 years ago. Later he co-founded the Texts in Applied Mathematics series and more re­ cently the Interdisciplinary Applied Mathematics series.

Recent results from high-energy scattering and theoretical developments of string theory require a change in our understanding of the basic structure of space-time. This book is about the advancement of ideas on the stochastic nature of space-time from the 1930s onward. In particular, the author promotes the concept of space as a set of hazy lumps, first introduced by Karl Menger, and constructs a novel framework for statistical behaviour at the microlevel. The various chapters address topics such as space-time fluctuation and random potential, non-local fields, and the origin of stochasticity. Implications in astro-particle physics and cosmology are also explored. Audience: This volume will be of interest to physicists, chemists and mathematicians involved in particle physics, astrophysics and cosmology.

1. Introduction. 1 2. Areas and Angles.. 6 3. Tessellations and Symmetry 14 4. The Postulate of Closest Approach 28 5. The Coexistence of Rotocenters 36 6. A Diophantine Equation and its Solutions 46 7. Enantiomorphy........ 57 8. Symmetry Elements in the Plane 77 9. Pentagonal Tessellations. 89 10. Hexagonal Tessellations 101 11. Dirichlet Domain 106 12. Points and Regions 116 13. A Look at Infinity. 122 14. An Irrational Number 128 15. The Notation of Calculus 137 16. Integrals and Logarithms 142 17. Growth Functions... 149 18. Sigmoids and the Seventh-year Trifurcation, a Metaphor 159 19. Dynamic Symmetry and Fibonacci Numbers 167 20. The Golden Triangle 179 21. Quasi Symmetry 193 Appendix I: Exercise in Glide Symmetry. 205 Appendix II: Construction of Logarithmic Spiral. 207 Bibliography. 210 Index.................... 225 Concepts and Images is the result of twenty years of teaching at Harvard's Department of Visual and Environmental Studies in the Carpenter Center for the Visual Arts, a department devoted to turning out students articulate in images much as a language department teaches reading and expressing one­ self in words. It is a response to our students'requests for a'handout'and to l our colleagues'inquiries about the courses : Visual and Environmental Studies 175 (Introduction to Design Science), YES 176 (Synergetics, the Structure of Ordered Space), Studio Arts 125a (Design Science Workshop, Two-Dimension­ al), Studio Arts 125b (Design Science Workshop, Three-Dimensional),2 as well as my freshman seminars on Structure in Science and Art.

Physics has long been regarded as a wellspring of mathematical problems. Mathematical Methods in Physics is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines.

The most important thing is to write equations in a beautiful form and their success in applications is ensured. Paul Dirac The uniqueness and existence theorems for the solutions of boundary and initial value problems for systems of high-order partial differential equations (PDE) are sufficiently well known. In this book, the problems considered are those whose solutions can be represented in quadratures, i.e., in an effective form. Such problems have remarkable applications in mathematical physics, the mechanics of deformable bodies, electro­ magnetism, relativistic quantum mechanics, and some of their natural generalizations. Almost all such problems can be set in the context of Clifford analysis. Moreover, they can be obtained without applying any physical laws, a circumstance that gives rise to the idea that Clifford analysis itself can suggest generalizations of classical equations or new equations altogether that may have some physical content. For that reason, Clifford analysis represents one of the most remarkable fields in modem mathematics as well as in modem physics.

This volume represents the outgrowth of an ongoing workshop on stochastic analysis held in Lisbon. The nine survey articles in the volume extend concepts from classical probability and stochastic processes to a number of areas of mathematical physics. It is a good reference text for researchers and advanced students in the fields of probability, stochastic processes, analysis, geometry, mathematical physics, and physics. Key topics covered include: nonlinear stochastic wave equations, completely positive maps, Mehler-type semigroups on Hilbert spaces, entropic projections, and many others.

Many physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods. Such problems abound in applied mathematics, theoretical mechanics, and mathematical physics. This uncorrected soft cover reprint of the second edition places the emphasis on applications and presents a variety of techniques with extensive examples.Originally published in 1971, Linear Integral Equations is ideal as a text for a beginning graduate level course. Its treatment of boundary value problems also makes the book useful to researchers in many applied fields.

This graduate textbook presents the basics of representation theory for finite groups from the point of view of semisimple algebras and modules over them. The presentation interweaves insights from specific examples with development of general and powerful tools based on the notion of semisimplicity. The elegant ideas of commutant duality are introduced, along with an introduction to representations of unitary groups. The text progresses systematically and the presentation is friendly and inviting. Central concepts are revisited and explored from multiple viewpoints. Exercises at the end of the chapter help reinforce the material.Representing Finite Groups: A Semisimple Introduction would serve as a textbook for graduate and some advanced undergraduate courses in mathematics. Prerequisites include acquaintance with elementary group theory and some familiarity with rings and modules. A final chapter presents a self-contained account of notions and results in algebra that are used. Researchers in mathematics and mathematical physics will also find this book useful.A separate solutions manual is available for instructors.

The book'TransMath - Innovative Solutions from Mathematical Technology'has been conceived as a tool for the dissemination of scientific knowledge. This publication is addressed to those companies with innovation needs that could be met through mathematical technology.The book maps both existing and possible interactions and connections that enable technology transfer between Spanish mathematical research and industrial and business sectors. Businesses can determine the level of implementation and demand for such technology within their sector and understand the benefits and innovations achieved in other companies and industries with the application of mathematical techniques.The information is classified into eleven sectors of economic activity: Biomedicine & Health; Construction; Economics & Finance; Energy & Environment; Food; ICT; Logistics & Transport; Management & Tourism; Metal & Machinery; Public Administration; and Technical Services.

The Classical Theory of Integral Equations is a thorough, concise, and rigorous treatment of the essential aspects of the theory of integral equations. The book provides the background and insight necessary to facilitate a complete understanding of the fundamental results in the field. With a firm foundation for the theory in their grasp, students will be well prepared and motivated for further study. Included in the presentation are: A section entitled Tools of the Trade at the beginning of each chapter, providing necessary background information for comprehension of the results presented in that chapter;Thorough discussions of the analytical methods used to solve many types of integral equations;An introduction to the numerical methods that are commonly used to produce approximate solutions to integral equations;Over 80 illustrative examples that are explained in meticulous detail; Nearly 300 exercises specifically constructed to enhance the understanding of both routine and challenging concepts; Guides to Computation to assist the student with particularly complicated algorithmic procedures. This unique textbook offers a comprehensive and balanced treatment of material needed for a general understanding of the theory of integral equations by using only the mathematical background that a typical undergraduate senior should have. The self-contained book will serve as a valuable resource for advanced undergraduate and beginning graduate-level students as well as for independent study. Scientists and engineers who are working in the field will also find this text to be user friendly and informative.

This book gives a systematic investigation of convection in systems comprised of liquid layers with deformatable interfaces.This new edition includes completely updated and new material on flows in ultra thin films and brings up to date progress made in the technology on micro and nano scales. Also, this revised edition will reflect progress in the dynamics of complex fluids.

Mathematical Modeling for Complex Fluids and Flows provides researchers and engineering practitioners encountering fluid flows with state-of-the-art knowledge in continuum concepts and associated fluid dynamics. In doing so it supplies the means to design mathematical models of these flows that adequately express the engineering physics involved. It exploits the implicit link between the turbulent flow of classical Newtonian fluids and the laminar and turbulent flow of non-Newtonian fluids such as those required in food processing and polymeric flows.The book develops a descriptive mathematical model articulated through continuum mechanics concepts for these non-Newtonian, viscoelastic fluids and turbulent flows. Each complex fluid and flow is examined in this continuum context as well as in combination with the turbulent flow of viscoelastic fluids. Some details are also explored via kinetic theory, especially viscoelastic fluids and their treatment with the Boltzmann equation. Both solution and modeling strategies for turbulent flows are laid out using continuum concepts, including a description of constructing polynomial representations and accounting for non-inertial and curvature effects.Ranging from fundamental concepts to practical methodology, and including discussion of emerging technologies, this book is ideal for those requiring a single-source assessment of current practice in this intricate yet vital field.