Advances in mathematics and physics have often occurred together. The development of Newton's theory of mechanics and the simultaneous development of the techniques of calculus constitute a classic example of this phenomenon.Показать полностьюAdvances in mathematics and physics have often occurred together. The development of Newton's theory of mechanics and the simultaneous development of the techniques of calculus constitute a classic example of this phenomenon. However, as mathematics and physics have become increasingly specialized over the last several decades, a formidable language barrier has grown up between the two. It is thus remarkable that several recent developments in theoretical physics have made use of the ideas and results of modern mathematics and, in fact, have elicited the direct participation of a number of mathematicians. The time therefore seems ripe to attempt to break down the language barriers between physics and certain branches of mathematics and to re-establish interdisciplinary communication (see, for example, Robinson [1977]; Mayer [1977]). The purpose of this article is to outline various mathematical ideas, methods, and results, primarily from differential geometry and topology, and to show where they can be applied to Yang-Mills gauge theories and Einstein's theory of gravitation. We have several goals in mind. The first is to convey to physicists the bases for many mathematical concepts by using intuitive arguments while avoiding the detailed formality of most textbooks. Although a variety of mathematical theorems will be stated, we will generally give simple examples motivating the results instead of presenting abstract proofs. Another goal is to list a wide variety of mathematical terminology and results in a format which allows easy reference. The reader then has the option of supplementing the descriptions given here by consulting standard mathematical references and articles such as those listed in the bibliography. Finally, we intend this article to serve the dual purpose of acquainting mathematicians with some basic physical concepts which have mathematical ramifications; physical problems have often stimulated new directions in mathematical thought.

This text presents an introductory survey of the basic concepts and applied mathematical methods of nonlinear science as well as an introduction to some simple related nonlinear experimental activities.

This book is a self-contained guide to problem-solving and exploration in mathematical physics using the powerful Maple 9.5 computer algebra system (CAS). With a CAS one cannot only crunch numbers and plot results, but also carry out the symbolic manipulations which form the backbone of mathematical physics. The heart of this text consists of over 230 useful and stimulating “classic” computer algebra worksheets or recipes, which are systematically organized to cover the ma jor topics presented in the standard Mathematical Physics course offered to third or fourth year undergraduate physics and engineering students. The emphasis here is on applications, with only a brief summary of the underlying theoretical ideas being presented.Показать полностьюThis book is a self-contained guide to problem-solving and exploration in mathematical physics using the powerful Maple 9.5 computer algebra system (CAS). With a CAS one cannot only crunch numbers and plot results, but also carry out the symbolic manipulations which form the backbone of mathematical physics. The heart of this text consists of over 230 useful and stimulating “classic” computer algebra worksheets or recipes, which are systematically organized to cover the ma jor topics presented in the standard Mathematical Physics course offered to third or fourth year undergraduate physics and engineering students. The emphasis here is on applications, with only a brief summary of the underlying theoretical ideas being presented. The aim is to show how computer algebra can not only implement the methods of mathematical physics quickly, accurately, and efficiently, but can be used to explore more complex examples which are tedious or difficult or even impossible to implement by hand.

This course surveys various uses of “entropy” concepts in the study of PDE, both linear and nonlinear. We will begin in Chapters I–III with a recounting of entropy in physics, with particular emphasis on axiomatic approaches to entropy as (i) characterizing equilibrium states (Chapter I), (ii) characterizing irreversibility for processes (Chapter II), and (iii) characterizing continuum thermodynamics (Chapter III).Показать полностьюThis course surveys various uses of “entropy” concepts in the study of PDE, both linear and nonlinear. We will begin in Chapters I–III with a recounting of entropy in physics, with particular emphasis on axiomatic approaches to entropy as (i) characterizing equilibrium states (Chapter I), (ii) characterizing irreversibility for processes (Chapter II), and (iii) characterizing continuum thermodynamics (Chapter III). Later we will discuss probabilistic theories for entropy as (iv) characterizing uncertainty (Chapter VII). I will, especially in Chapters II and III, follow the mathematical derivation of entropy pro- vided by modern rational thermodynamics, thereby avoiding many customary physical ar- guments. The main references here will be Callen [C], Owen [O], and Coleman–Noll [C-N]. In Chapter IV I follow Day [D] by demonstrating for certain linear second-order elliptic and parabolic PDE that various estimates are analogues of entropy concepts (e.g. the Clausius inequality). I as well draw connections with Harnack inequalities. In Chapter V (conserva- tion laws) and Chapter VI (Hamilton–Jacobi equations) I review the proper notions of weak solutions, illustrating that the inequalities inherent in the definitions can be interpreted as irreversibility conditions. Chapter VII introduces the probabilistic interpretation of entropy and Chapter VIII concerns the related theory of large deviations. Following Varadhan [V] and Rezakhanlou [R], I will explain some connections with entropy, and demonstrate various PDE applications. B. Themes In spite of the longish time spent in Chapters I–III, VII reviewing physics, this is a mathematics course on partial differential equations. My main concern is PDE and how various notions involving entropy have influenced our understanding of PDE. As we will cover a lot of material from many sources, let me explicitly write out here some unifying themes: (I) the use of entropy in deriving various physical PDE, (II) the use of entropy to characterize irreversibility in PDE evolving in time (III) the use of entropy in providing variational principles. Another ongoing issue will be (IV) understanding the relationships between entropy and convexity.

This book addressees a variety of techniques and applications in fractal geometry. It examines such topics as implicit methods and the theory of dimensions of measures, the thermodynamic formalism, the tangent of space method and the ergodic theorem.Показать полностьюThis book addressees a variety of techniques and applications in fractal geometry. It examines such topics as implicit methods and the theory of dimensions of measures, the thermodynamic formalism, the tangent of space method and the ergodic theorem. Each chapter ends with brief notes on the development and current state of the subject. Provides a clear guide to applications and recent trends in fractal geometry. There are numerous diagrams and illustrative examples.

The present monograph is devoted to the index theorem for deformation quantization generalizing the famous Atiyah-Singer index theorem for elliptic operators.Показать полностьюThe present monograph is devoted to the index theorem for deformation quantization generalizing the famous Atiyah-Singer index theorem for elliptic operators. The latter appeared some thirty years ago and seemed mysterious and incomprehensible at that time as the result was at the joint of many branches of mathematics: differential geometry and topology, functional analysis and partial differentia! equations. It took time and effort to master alt the ingredients necessary for its understanding. A vivid evidence of the fact is the well-known book [43] by Palais where in the form of seminar talks the necessary material and finally the proof itself are set for,th.

This book presents a functional approach to the construction, use and approximation of Green’s functions and their associated ordered exponentials. After a brief historical introduction, the author discusses new solutions to problems involving particle production in crossed laser fields and non-constant electric fields.Показать полностьюThis book presents a functional approach to the construction, use and approximation of Green’s functions and their associated ordered exponentials. After a brief historical introduction, the author discusses new solutions to problems involving particle production in crossed laser fields and non-constant electric fields. Applications to problems in potential theory and quantum field theory are covered, along with approximations for the treatment of color fluctuations in high-energy QCD scattering, and a model for summing classes of eikonal graphs in high-energy scattering problems. The book also presents a variant of the Fradkin representation which suggests a new non-perturbative approximation scheme, and provides a qualitative measure of the error involved in each such approximation. Covering the basics as well as more advanced applications, this book is suitable for graduate students and researchers in a wide range of fields, including quantum field theory, fluid dynamics and applied mathematics.

The goal of this book remains as before:To teach the powerful problem-solving methods of probability and statistics to students who have some background in either optics or linear systems theory.

Mathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space.Показать полностьюMathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space. Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics. Perhaps the most valuable feature of the book is the illuminating intuitive discussion of the “whys” of proofs and of axioms and definitions. This book, based on Geroch’s University of Chicago course, will be especially helpful to those working in theoretical physics, including such areas as relativity, particle physics, and astrophysics.

This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary.Показать полностьюThis book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.

A feeling for "inverse problems" can be obtained, for example, by noting that not every object in nature is accessible to direct study, and consequently, its properties must be judged indirectly.Показать полностьюA feeling for "inverse problems" can be obtained, for example, by noting that not every object in nature is accessible to direct study, and consequently, its properties must be judged indirectly. The bowels of the Earth may serve as an example and problems of this type have been known for some time in geophys- geophysics. At the same time the posing of inverse problems is characteristic of scientific investigation and interest in them has been on the increase in many fields of science, physics in particular. What distinguishes mathematical physics is the fact that in it the study of nature proceeds within the framework of precise mathematical models, formulated on the basis of known regularities. Such models serve as the basis for the solution of inverse problems. However, due to the specificity of inverse problems—their mathematical "wrongness"—substantial progress has been achieved in the posing and solving of inverse problems only in the last 10-20 years. Responsible for this was the mathematical "regularization theory" developed by Soviet scientists,71'85148 as well as the intensive development of computational tech- techniques. Familiarity with elements of regularization theory and its applications to the solution of inverse problems, as well as the description of possible fields of application, constitute an essential part in the training of scientists and are useful to workers in many branches of knowledge. This monograph is based on a special topics course taught by the author in the Physics Department of Moscow State University and constitutes a short version of the course "Inverse problems of mathematical physics." It is not meant as a substitute for the excellent monographs on the theory of regularization and some of its applications19'86132139148 for readers interested in the development of the mathematical theory. Neither does it pretend to provide a full review of known results in the field. However, for the reader not familiar with the subject under discussion, or interested in practical applications of regularization theory, this book should serve as a kind of introduction. The monograph is addressed to a reader familiar with elements of mathematical analysis, with a course in mathematical physics similar to Ref. 162, with some elements of functional analysis, and with some computational methods. Unfortunately the size of this monograph precludes the replacing of references to various results by their explanation. The content is based on the fundamental papers [Refs. 71, 85, and 148] including a number of results obtained with the author's participation. The material is distributed among four chapters. In Chap. 1, we give a broad characterization of the class of inverse problems. In contrast to existing publications and in accordance with the essence of the subject, we consider questions of uniqueness of solutions of inverse problems (Chap. 3), which might be useful from both an instructional and pedagogical point of view. In a slight departure from tradition, we emphasize in Chap. 2 questions relating to the mathematical posing of inverse problems. Regularizing algorithms for their solution are considered in Chap. 4. Concrete examples of inverse problems of mathematical physics are used to illustrate the main assumptions.

Percolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations.Показать полностьюPercolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. Percolation plays a pivotal role in studying more complex systems exhibiting phase transition. The mathematical theory is mature, but continues to give rise to problems of special beauty and difficulty. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed beyond undergraduate analysis and probability. This new volume differs substantially from the first edition through the inclusion of much new material, including: the rigorous theory of dynamic and static renormalization; a sketch of the lace expansion and mean field theory; the uniqueness of the infinite cluster; strict inequalities between critical probabilities; several essays on related fields and applications; numerous other results of significant. There is a summary of the hypotheses of conformal invariance. A principal feature of the process is the phase transition. The subcritical and supercritical phases are studied in detail. There is a guide for mathematicians to the physical theory of scaling and critical exponents, together with selected material describing the current state of the rigorous theory. To derive a rigorous theory of the phase transition remains an outstanding and beautiful problem of mathematics.

On 21 March 1990 John Hammersley celebrates his seventieth birthday. A number of his colleagues and friends wish to pay tribute on this occasion to a mathematician whose exceptional inventiveness has greatly enriched mathematical science.Показать полностьюOn 21 March 1990 John Hammersley celebrates his seventieth birthday. A number of his colleagues and friends wish to pay tribute on this occasion to a mathematician whose exceptional inventiveness has greatly enriched mathematical science. The breadth and versatility of Hammersley’s interests are remarkable, doubly so in an age of increased specialisation. In a range of highly individual papers on a variety of topics, he has theorised, and posed (and solved) problems, thereby laying the foundations for many subjects currently under study. By his evident love for mathematics and an affinity for the hard problem, he has been an inspiration to many. If one must single out one particular area where Hammersley’s contribution has proved especially vital, it would probably be the study of random processes in space. He was a pioneer in this field of recognised importance, a field abounding in apparently simple questions whose resolutions usually require new ideas and methods. This area is not just a mathematician’s playground, but is of fundamental importance for the understanding of physical phenomena. The principal theme of this volume reflects various aspects of Hammersley’s work in the area, including disordered media, subadditivity, numerical methods, and the like. The authors of these papers join with those unable to contribute in wishing John Hammersley many further years of fruitful mathematical activity.

A slim little text designed to acquaint advanced undergraduate and beginning graduate students of physics (knowledge of classical mechanics, statistical mechanics and computer programming is assumed) with the art and excitement of the topic to which the title refers.Показать полностьюA slim little text designed to acquaint advanced undergraduate and beginning graduate students of physics (knowledge of classical mechanics, statistical mechanics and computer programming is assumed) with the art and excitement of the topic to which the title refers. Four chapters (Introductory examples, Computer-simulation methods, Deterministic methods, Stochastic methods) provide the student with what is in effect a computer-lab workbook, an invitation to learn "hands-on" some exciting things both about applied computation and about physics. An appendix provides information about random number generators, and there is a useful bibliography. (NW) Annotation copyright Book News, Inc. Portland, Or.

The relation between teaching and research has been a perennial theme in academia as well as the Oersted Lectures, with no apparent progress on resolving the issues. Physics Education Research (PER) puts the whole matter into new light, for PER makes teaching itself a subject of research. This shifts attention to the relation of education research to scientific research as the central issue.

This book provides a synopsis of spacetime calculus with applications to classical electrodynamics, quantum theory and gravitation. The calculus is a coordinate-free mathematical language enabling a unified treatment of all these topics and bringing new insights and methods to each of them.

This book is based on lectures given by the authors at an instructional conference on integrable systems held at the Mathematical Institute in Oxford in September 1997.Показать полностьюThis book is based on lectures given by the authors at an instructional conference on integrable systems held at the Mathematical Institute in Oxford in September 1997. Most of the participants were graduate students from the United Kingdom and other European countries. The lectures emphasized geometric aspects of the theory of integrable systems, particularly connections with algebraic geometry, twistor theory, loop groups, and the Grassmannian picture. We are grateful for support for the conference from the London Mathematical Society, the Engineering and Physical Sciences Research Council (contract No. 00985SCI96), the University of Oxford Mathematical Prizes Fund, the Mathematical Institute, Wadham College, and Oxford University Press.

Spaces of particles have long been studied in homotopy theory, partly for their intrinsic interest but also for their role in describing the structure of loop spaces.Показать полностьюSpaces of particles have long been studied in homotopy theory, partly for their intrinsic interest but also for their role in describing the structure of loop spaces. Recently the structure of these spaces has been put to good use in understanding several moduli spaces of solutions to variational problems, such as the moduli of holomorphic maps of surfaces into certain complex manifolds, the moduli of instantons, and the Chow varieties. In these notes, we give a detailed description of the particle structures involved in the first two cases, and then explain how well-established results on the topology of particle spaces can be exploited to prove stability theorems for the topology of the moduli spaces, theorems which state that the moduli space approximates in a suitable homotopic sense the topology of the function spaces in which they sit, provided one stabilises with respect to a charge or degree.

This book is based on lecture notes for the introductory course on modern, coordinate-free differential geometry. The course is concerned entirely with the mathematics itself, although the emphasis and detailed topics have been chosen with an eye on the way in which differential geometry is applied to theoretical physics these days.Показать полностьюThis book is based on lecture notes for the introductory course on modern, coordinate-free differential geometry. The course is concerned entirely with the mathematics itself, although the emphasis and detailed topics have been chosen with an eye on the way in which differential geometry is applied to theoretical physics these days. Such applications include not only the traditional area of general relativity, but also the theory of Yang-Mills fields, non-linear sigma models, superstring theory, and other types of non-linear field systems that feature in modern elementary particle theory and quantum gravity. The course is in four parts dealing with, respectively, (i) an introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; and (iv) the basic ideas of fibre bundle theory.

Preface In 1999, when we started teaching this course at the Department of Physics in Oslo, Compu- tational Physics and Computational Science in general were still perceived by the majority of physicists and scientists as topics dealing with just mere tools and number crunching, and not as subjects of their own.Показать полностьюPreface In 1999, when we started teaching this course at the Department of Physics in Oslo, Compu- tational Physics and Computational Science in general were still perceived by the majority of physicists and scientists as topics dealing with just mere tools and number crunching, and not as subjects of their own. The computational background of most students enlisting for the course on computational physics could span from dedicated hackers and computer freaks to people who basically had never used a PC. The majority of graduate students had a very rudimentary knowl- edge of computational techniques and methods. Four years later most students have had a fairly uniform introduction to computers, basic programming skills and use of numerical exercises in undergraduate courses. Practically every undergraduate student in physics has now made a Mat- lab or Maple simulation of e.g., the pendulum, with or without chaotic motion. These exercises underscore the importance of simulations as a means to gain novel insights into physical sys- tems, especially for those cases where no analytical solutions can be found or an experiment is to complicated or expensive to carry out. Thus, computer simulations are nowadays an inte- gral part of contemporary basic and applied research in the physical sciences. Computation is becoming as important as theory and experiment. We could even strengthen this statement by saying that computational physics, theoretical physics and experimental are all equally important in our daily research and studies of physical systems. Physics is nowadays the unity of theory, experiment and computation. The ability "to compute" is now part of the essential repertoire of research scientists. Several new fields have emerged and strengthened their positions in the last years, such as computational materials science, bioinformatics, computational mathematics and mechanics, computational chemistry and physics and so forth, just to mention a few. To be able to e.g., simulate quantal systems will be of great importance for future directions in fields like materials science and nanotechonology. This ability combines knowledge from many different subjects, in our case essentially from the physical sciences, numerical analysis, computing languages and some knowledge of comput- ers. These topics are, almost as a rule of thumb, taught in different, and we would like to add, disconnected courses. Only at the level of thesis work is the student confronted with the synthesis of all these subjects, and then in a bewildering and disparate manner, trying to e.g., understand old Fortran 77 codes inherited from his/her supervisor back in the good old ages, or even more archaic, programs. Hours may have elapsed in front of a screen which just says ’Underflow’, or ’Bus error’, etc etc, without fully understanding what goes on. Porting the program to another machine could even result in totally different results! The first aim of this course is therefore to bridge the gap between undergraduate courses in the physical sciences and the applications of the aquired knowledge to a given project, be it either a thesis work or an industrial project. We expect you to have some basic knowledge in the physical sciences, especially within mathematics and physics through e.g., sophomore courses in basic calculus, linear algebraand general physics. Furthermore, having taken an introductory course on programming is something we recommend. As such, an optimal timing for taking this course, would be when you are close to embark on a thesis work, or if you’ve just started with a thesis. But obviously, you should feel free to choose your own timing. We have several other aims as well in addition to prepare you for a thesis work, namely 􏰇 We would like to give you an opportunity to gain a deeper understanding of the physics you have learned in other courses. In most courses one is normally confronted with simple systems which provide exact solutions and mimic to a certain extent the realistic cases. Many are however the comments like ’why can’t we do something else than the box po- tential?’. In several of the projects we hope to present some more ’realistic’ cases to solve by various numerical methods. This also means that we wish to give examples of how physics can be applied in a much broader context than it is discussed in the traditional physics undergraduate curriculum. 􏰇 To encourage you to "discover" physics in a way similar to how researchers learn in the context of research. Hopefully also to introduce numerical methods and new areas of physics that can be stud- ied with the methods discussed. To teach structured programming in the context of doing science. The projects we propose are meant to mimic to a certain extent the situation encountered during a thesis or project work. You will tipically have at your disposal 1-2 weeks to solve numerically a given project. In so doing you may need to do a literature study as well. Finally, we would like you to write a report for every project. 􏰇 The exam reflects this project-like philosophy. The exam itself is a project which lasts one month. You have to hand in a report on a specific problem, and your report forms the basis for an oral examination with a final grading. Our overall goal is to encourage you to learn about science through experience and by asking questions. Our objective is always understanding, not the generation of numbers. The purpose of computing is further insight, not mere numbers! Moreover, and this is our personal bias, to device an algorithm and thereafter write a code for solving physics problems is a marvelous way of gaining insight into complicated physical systems. The algorithm you end up writing reflects in essentially all cases your own understanding of the physics of the problem. Most of you are by now familiar, through various undergraduate courses in physics and math- ematics, with interpreted languages such as Maple, Mathlab and Mathematica. In addition, the interest in scripting languages such as Python or Perl has increased considerably in recent years. The modern programmer would typically combine several tools, computing environments and programming languages. A typical example is the following. Suppose you are working on a project which demands extensive visualizations of the results. To obtain these results you need however a programme which is fairly fast when computational speed matters. In this case you would most likely write a high-performance computing programme in languages which are tay- lored for that. These are represented by programming languages like Fortran 90/95 and C/C++. However, to visualize the results you would find interpreted languages like e.g., Matlab or script- ing languages like Python extremely suitable for your tasks. You will therefore end up writing e.g., a script in Matlab which calls a Fortran 90/95 ot C/C++ programme where the number crunching is done and then visualize the results of say a wave equation solver via Matlab’s large library of visualization tools. Alternatively, you could organize everything into a Python or Perl script which does everything for you, calls the Fortran 90/95 or C/C++ programs and performs the visualization in Matlab as well. Being multilingual is thus a feature which not only applies to modern society but to comput- ing environments as well. However, there is more to the picture than meets the eye. This course emphasizes th