Mathematical / Computational / Theoretical physics Books

Book SynopsisFrom the reviews: "...I think the volume is a great success ... a welcome addition to the literature ..." The Mathematical Intelligencer, 1993 "... It is comparable in scope with the great Courant-Hilbert Methods of Mathematical Physics, but it is much shorter, more up to date of course, and contains more elaborate analytical machinery...." The Mathematical Gazette, 1993Trade ReviewFrom the reviews: "...I think the volume is a great success ... a welcome addition to the literature ..." The Mathematical Intelligencer, 1993 "... According to the authors ... the work was written for the nonspecialists and physicists but in my opinion almost every specialist will find something new for herself/himself in the text. ..." Acta Scientiarum Mathematicarum, 1993 "... On the whole, a thorough overview on the classical aspects of the topic may be gained from that volume." Monatshefte für Mathematik, 1993 "... It is comparable in scope with the great Courant-Hilbert Methods of Mathematical Physics, but it is much shorter, more up to date of course, and contains more elaborate analytical machinery...." The Mathematical Gazette, 1993Table of Contents1. Basic Concepts.- 1. Basic Definitions and Examples.- 1.1. The Definition of a Linear Partial Differential Equation.- 1.2. The Role of Partial Differential Equations in the Mathematical Modeling of Physical Processes.- 1.3. Derivation of the Equation for the Longitudinal Elastic Vibrations of a Rod.- 1.4. Derivation of the Equation of Heat Conduction.- 1.5. The Limits of Applicability of Mathematical Models.- 1.6. Initial and Boundary Conditions.- 1.7. Examples of Linear Partial Differential Equations.- 1.8. The Concept of Well-Posedness of a Boundary-value Problem. The Cauchy Problem.- 2. The Cauchy-Kovalevskaya Theorem and Its Generalizations.- 2.1. The Cauchy-Kovalevskaya Theorem.- 2.2. An Example of Nonexistence of an Analytic Solution.- 2.3. Some Generalizations of the Cauchy-Kovalevskaya Theorem. Characteristics.- 2.4. Ovsyannikov’s Theorem.- 2.5. Holmgren’s Theorem.- 3. Classification of Linear Differential Equations. Reduction to Canonical Form and Characteristics.- 3.1. Classification of Second-Order Equations and Their Reduction to Canonical Form at a Point.- 3.2. Characteristics of Second-Order Equations and Reduction to Canonical Form of Second-Order Equations with Two Independent Variables.- 3.3. Ellipticity, Hyperbolicity, and Parabolicity for General Linear Differential Equations and Systems.- 3.4. Characteristics as Solutions of the Hamilton-Jacobi Equation.- 2. The Classical Theory.- 1. Distributions and Equations with Constant Coefficients.- 1.1. The Concept of a Distribution.- 1.2. The Spaces of Test Functions and Distributions.- 1.3. The Topology in the Space of Distributions.- 1.4. The Support of a Distribution. The General Form of Distributions.- 1.5. Differentiation of Distributions.- 1.6. Multiplication of a Distribution by a Smooth Function. Linear Differential Operators in Spaces of Distributions.- 1.7. Change of Variables and Homogeneous Distributions.- 1.8. The Direct or Tensor Product of Distributions.- 1.9. The Convolution of Distributions.- 1.10. The Fourier Transform of Tempered Distributions.- 1.11. The Schwartz Kernel of a Linear Operator.- 1.12. Fundamental Solutions for Operators with Constant Coefficients.- 1.13. A Fundamental Solution for the Cauchy Problem.- 1.14. Fundamental Solutions and Solutions of Inhomogeneous Equations.- 1.15. Duhamel’s Principle for Equations with Constant Coefficients.- 1.16. The Fundamental Solution and the Behavior of Solutions at Infinity.- 1.17. Local Properties of Solutions of Homogeneous Equations with Constant Coefficients. Hypoellipticity and Ellipticity.- 1.18. Liouville’s Theorem for Equations with Constant Coefficients.- 1.19. Isolated Singularities of Solutions of Hypoelliptic Equations.- 2. Elliptic Equations and Boundary-Value Problems.- 2.1. The Definition of Ellipticity. The Laplace and Poisson Equations.- 2.2. A Fundamental Solution for the Laplacian Operator. Green’s Formula.- 2.3. Mean-Value Theorems for Harmonic Functions.- 2.4. The Maximum Principle for Harmonic Functions and the Normal Derivative Lemma.- 2.5. Uniqueness of the Classical Solutions of the Dirichlet and Neumann Problems for Laplace’s Equation.- 2.6. Internal A Priori Estimates for Harmonic Functions. Harnack’s Theorem.- 2.7. The Green’s Function of the Dirichlet Problem for Laplace’s Equation.- 2.8. The Green’s Function and the Solution of the Dirichlet Problem for a Ball and a Half-Space. The Reflection Principle.- 2.9. Harnack’s Inequality and Liouville’s Theorem.- 2.10. The Removable Singularities Theorem.- 2.11. The Kelvin Transform and the Statement of Exterior Boundary-Value Problems for Laplace’s Equation.- 2.12. Potentials.- 2.13. Application of Potentials to the Solution of Boundary-Value Problems.- 2.14. Boundary-Value Problems for Poisson’s Equation in Hölder Spaces. Schauder Estimates.- 2.15. Capacity.- 2.16. The Dirichlet Problem in the Case of Arbitrary Regions (The Method of Balayage). Regularity of a Boundary Point. The Wiener Regularity Criterion.- 2.17. General Second-Order Elliptic Equations. Eigenvalues and Eigenfunctions of Elliptic Operators.- 2.18. Higher-Order Elliptic Equations and General Elliptic Boundary-Value Problems. The Shapiro-Lopatinskij Condition.- 2.19. The Index of an Elliptic Boundary-Value Problem.- 2.20. Ellipticity with a Parameter and Unique Solvability of Elliptic Boundary-Value Problems.- 3. Sobolev Spaces and Generalized Solutions of Boundary-Value Problems.- 3.1. The Fundamental Spaces.- 3.2. Imbedding and Trace Theorems.- 3.3. Generalized Solutions of Elliptic Boundary-Value Problems and Eigenvalue Problems.- 3.4. Generalized Solutions of Parabolic Boundary-Value Problems.- 3.5. Generalized Solutions of Hyperbolic Boundary-Value Problems.- 4. Hyperbolic Equations.- 4.1. Definitions and Examples.- 4.2. Hyperbolicity and Well-Posedness of the Cauchy Problem.- 4.3. Energy Estimates.- 4.4. The Speed of Propagation of Disturbances.- 4.5. Solution of the Cauchy Problem for the Wave Equation.- 4.6. Huyghens’ Principle.- 4.7. The Plane Wave Method.- 4.8. The Solution of the Cauchy Problem in the Plane.- 4.9. Lacunae.- 4.10. The Cauchy Problem for a Strictly Hyperbolic System with Rapidly Oscillating Initial Data.- 4.11. Discontinuous Solutions of Hyperbolic Equations.- 4.12. Symmetric Hyperbolic Operators.- 4.13. The Mixed Boundary-Value Problem.- 4.14. The Method of Separation of Variables.- 5. Parabolic Equations.- 5.1. Definitions and Examples.- 5.2. The Maximum Principle and Its Consequences.- 5.3. Integral Estimates.- 5.4. Estimates in Hölder Spaces.- 5.5. The Regularity of Solutions of a Second-Order Parabolic Equation.- 5.6. Poisson’s Formula.- 5.7. A Fundamental Solution of the Cauchy Problem for a Second-Order Equation with Variable Coefficients.- 5.8. Shilov-Parabolic Systems.- 5.9. Systems with Variable Coefficients.- 5.10. The Mixed Boundary-Value Problem.- 5.11. Stabilization of the Solutions of the Mixed Boundary-Value Problem and the Cauchy Problem.- 6. General Evolution Equations.- 6.1. The Cauchy Problem. The Hadamard and Petrovskij Conditions.- 6.2. Application of the Laplace Transform.- 6.3. Application of the Theory of Semigroups.- 6.4. Some Examples.- 7. Exterior Boundary-Value Problems and Scattering Theory.- 7.1. Radiation Conditions.- 7.2. The Principle of Limiting Absorption and Limiting Amplitude.- 7.3. Radiation Conditions and the Principle of Limiting Absorption for Higher-Order Equations and Systems.- 7.4. Decay of the Local Energy.- 7.5. Scattering of Plane Waves.- 7.6. Spectral Analysis.- 7.7. The Scattering Operator and the Scattering Matrix.- 8. Spectral Theory of One-Dimensional Differential Operators.- 8.1. Outline of the Method of Separation of Variables.- 8.2. Regular Self-Adjoint Problems.- 8.3. Periodic and Antiperiodic Boundary Conditions.- 8.4. Asymptotics of the Eigenvalues and Eigenfunctions in the Regular Case.- 8.5. The Schrödinger Operator on a Half-Line.- 8.6. Essential Self-Adjointness and Self-Adjoint Extensions. The Weyl Circle and the Weyl Point.- 8.7. The Case of an Increasing Potential.- 8.8. The Case of a Rapidly Decaying Potential.- 8.9. The Schrödinger Operator on the Entire Line.- 8.10. The Hill Operator.- 9. Special Functions.- 9.1. Spherical Functions.- 9.2. The Legendre Polynomials.- 9.3. Cylindrical Functions.- 9.4. Properties of the Cylindrical Functions.- 9.5. Airy’s Equation.- 9.6. Some Other Classes of Functions.- References.- Author Index.

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Book SynopsisThis book has been long awaited in the "interacting particle systems" community. Begun by Claude Kipnis before his untimely death, it was completed by Claudio Landim, his most brilliant student and collaborator. It presents the techniques used in the proof of the hydrodynamic behavior of interacting particle systems.Trade Review"Das Buch ist nach Meinung des Rezensenten eine gelungene Einführung in ein interessantes Gebiet der modernen Stochastik und mathematischen Physik und stellt einen fest umrissenen Gegenstand umfassend dar, vor allem den analytisch-methodischen Aspekt. ... Die Beweise sind übersichtlich und gut gegliedert, was das Nachvollziehen der Argumente sehr erleichtert; die didaktische Leistung der Autoren in diesem Punkt ist beeindruckend. Ein sorgfältig zusammengestelltes Literaturverzeichnis von etwa 400 Titeln schließt das Buch ab. Ingesamt ein sehr gut geschriebener und nützlicher Band."DMV Jahresbericht, 103. Band, Heft 3, November 2001Table of Contents1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes.- 5. An Example of Reversible Gradient System: Symmetric Zero Range Processes.- 6. The Relative Entropy Method.- 7. Hydrodynamic Limit of Reversible Nongradient Systems.- 8. Hydrodynamic Limit of Asymmetric Attractive Processes.- 9. Conservation of Local Equilibrium for Attractive Systems.- 10. Large Deviations from the Hydrodynamic Limit.- 11. Equilibrium Fluctuations of Reversible Dynamics.- Appendices.- 1. Markov Chains on a Countable Space.- 1.1 Discrete Time Markov Chains.- 1.2 Continuous Time Markov Chains.- 1.3 Kolmogorov’s Equations, Generators.- 1.4 Invariant Measures, Reversibility and Adjoint Processes.- 1.5 Some Martingales in the Context of Markov Processes.- 1.6 Estimates on the Variance of Additive Functionals of Markov Processes.- 1.7 The Feynman-Kac Formula.- 1.8 Relative Entropy.- 1.9 Entropy and Markov Processes.- 1.10 Dirichlet Form.- 1.11 A Maximal Inequality for Reversible Markov Processes.- 2. The Equivalence of Ensembles, Large Deviation Tools and Weak Solutions of Quasi-Linear Differential Equations.- 2.1 Local Central Limit Theorem and Equivalence of Ensembles.- 2.2 On the Local Central Limit Theorem.- 2.3 Remarks on Large Deviations.- 2.4 Weak Solutions of Nonlinear Parabolic Equations.- 2.5 Entropy Solutions of Quasi-Linear Hyperbolic Equations.- 3. Nongradient Tools: Spectral Gap and Closed Forms.- 3.1 On the Spectrum of Reversible Markov Processes.- 3.2 Spectral Gap for Generalized Exclusion Processes.- 3.4 Closed and Exact Forms.- 3.5 Comments and References.- References.

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Book SynopsisThis is an approachable introduction to the important topics and recent developments in the field of condensed matter physics. First, the general language of quantum field theory is developed in a way appropriate for dealing with systems having a large number of degrees of freedom. This paves the way for a description of the basic processes in such systems. Applications include various aspects of superfluidity and superconductivity, as well as a detailed description of the fractional quantum Hall liquid.Table of ContentsQuantum Mechanics and Fundamentals of Quantum Field Theory.- Path Integral Quantization.- Broken Symmetry and Phase Transition.- Some Applications - Warming Ups.- Superconductivity.- Quantum Hall Liquids and Chern--Simons Gauge Field.- Appendix.

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Book SynopsisThe main characteristic of this classic exposition of the inverse scattering method and its applications to soliton theory is its consistent Hamiltonian approach to the theory. The nonlinear Schrödinger equation is considered as a main example, forming the first part of the book. The second part examines such fundamental models as the sine-Gordon equation and the Heisenberg equation, the classification of integrable models and methods for constructing their solutions.Trade Review Table of ContentsThe Nonlinear Schrödinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples and Their General Properties.- Fundamental Continuous Models.- Fundamental Models on the Lattice.- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models.- Conclusion.- Conclusion.

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Book SynopsisVector calculus is the fundamental language of mathematical physics. It pro­ vides a way to describe physical quantities in three-dimensional space and the way in which these quantities vary. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. These top­ ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions. This book assumes no previous knowledge of vectors. However, it is assumed that the reader has a knowledge of basic calculus, including differentiation, integration and partial differentiation. Some knowledge of linear algebra is also required, particularly the concepts of matrices and determinants. The book is designed to be self-contained, so that it is suitable for a pro­ gramme of individual study. Each of the eight chapters introduces a new topic, and to facilitate understanding of the material, frequent reference is made to physical applications. The physical nature of the subject is clarified with over sixty diagrams, which provide an important aid to the comprehension of the new concepts. Following the introduction of each new topic, worked examples are provided. It is essential that these are studied carefully, so that a full un­ derstanding is developed before moving ahead. Like much of mathematics, each section of the book is built on the foundations laid in the earlier sections and chapters.Trade ReviewP.C. Matthews Vector Calculus "Written for undergraduate students in mathematics, the book covers the material in a comprehensive but concise manner, combining mathematical rigor with physical insight. There are many diagrams to illustrate the physical meaning of the mathematical concepts, which essential for a full understanding of the subject." — ZENTRALBLATT MATH Table of Contents1. Vector Algebra.- 1.1 Vectors and scalars.- 1.1.1 Definition of a vector and a scalar.- 1.1.2 Addition of vectors.- 1.1.3 Components of a vector.- 1.2 Dot product.- 1.2.1 Applications of the dot product.- 1.3 Cross product.- 1.3.1 Applications of the cross product.- 1.4 Scalar triple product.- 1.5 Vector triple product.- 1.6 Scalar fields and vector fields.- 2. Line, Surface and Volume Integrals.- 2.1 Applications and methods of integration.- 2.1.1 Examples of the use of integration.- 2.1.2 Integration by substitution.- 2.1.3 Integration by parts.- 2.2 Line integrals.- 2.2.1 Introductory example: work done against a force.- 2.2.2 Evaluation of line integrals.- 2.2.3 Conservative vector fields.- 2.2.4 Other forms of line integrals.- 2.3 Surface integrals.- 2.3.1 Introductory example: flow through a pipe.- 2.3.2 Evaluation of surface integrals.- 2.3.3 Other forms of surface integrals.- 2.4 Volume integrals.- 2.4.1 Introductory example: mass of an object with variable density.- 2.4.2 Evaluation of volume integrals.- 3. Gradient, Divergence and Curl.- 3.1 Partial differentiation and Taylor series.- 3.1.1 Partial differentiation.- 3.1.2 Taylor series in more than one variable.- 3.2 Gradient of a scalar field.- 3.2.1 Gradients, conservative fields and potentials.- 3.2.2 Physical applications of the gradient.- 3.3 Divergence of a vector field.- 3.3.1 Physical interpretation of divergence.- 3.3.2 Laplacian of a scalar field.- 3.4 Curl of a vector field.- 3.4.1 Physical interpretation of curl.- 3.4.2 Relation between curl and rotation.- 3.4.3 Curl and conservative vector fields.- 4. Suffix Notation and its Applications.- 4.1 Introduction to suffix notation.- 4.2 The Kronecker delta ?ij.- 4.3 The alternating tensor ?ijk.- 4.4 Relation between ?ijk and ?ij.- 4.5 Grad, div and curl in suffix notation.- 4.6 Combinations of grad, div and curl.- 4.7 Grad, div and curl applied to products of functions.- 5. Integral Theorems.- 5.1 Divergence theorem.- 5.1.1 Conservation of mass for a fluid.- 5.1.2 Applications of the divergence theorem.- 5.1.3 Related theorems linking surface and volume integrals.- 5.2 Stokes’s theorem.- 5.2.1 Applications of Stokes’s theorem.- 5.2.2 Related theorems linking line and surface integrals.- 6. Curvilinear Coordinates.- 6.1 Orthogonal curvilinear coordinates.- 6.2 Grad, div and curl in orthogonal curvilinear coordinate systems.- 6.2.1 Gradient.- 6.2.2 Divergence.- 6.2.3 Curl.- 6.3 Cylindrical polar coordinates.- 6.4 Spherical polar coordinates.- 7. Cartesian Tensors.- 7.1 Coordinate transformations.- 7.2 Vectors and scalars.- 7.3 Tensors.- 7.3.1 The quotient rule.- 7.3.2 Symmetric and anti-symmetric tensors.- 7.3.3 Isotropic tensors.- 7.4 Physical examples of tensors.- 7.4.1 Ohm’s law.- 7.4.2 The inertia tensor.- 8. Applications of Vector Calculus.- 8.1 Heat transfer.- 8.2 Electromagnetism.- 8.2.1 Electrostatics.- 8.2.2 Electromagnetic waves in a vacuum.- 8.3 Continuum mechanics and the stress tensor.- 8.4 Solid mechanics.- 8.5 Fluid mechanics.- 8.5.1 Equation of motion for a fluid.- 8.5.2 The vorticity equation.- 8.5.3 Bernoulli’s equation.- Solutions.

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Book SynopsisThis concise, class-tested book was refined over the authors’ 30 years as instructors at MIT and the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it for their own needs. Thus, the theoretical background is confined to introductory chapters. Subsequent chapters develop new theory alongside applications so that students can retain new concepts, build on concepts already learned, and see interrelations between topics. Essential problem sets between chapters aid retention of new material and consolidate material learned in previous chapters.Trade ReviewFrom the reviews:"It was developed for a graduate course taught mostly by Millie Dresselhaus at MIT for more than 30 years, with many revisions of lecture notes. Very much a graduate text or specialist monograph, the book covers a wealth of applications across solid-state physics. … The book can be warmly recommended to students and researchers in solid-state physics, either to serve as a text for an advanced lecture course or for individual study … ." (Volker Heine, Physics Today, November, 2008)"This textbook is based on the authors’ pedagogical experience during their 30 years at MIT. … the book develops all of the relevant mathematics (linear algebra) and the necessary physics (quantum mechanics), it is eminently suitable to a wide audience in physics, chemistry and materials science." (Barry R. Masters, Optics and Photonics News, July/August, 2009)“This is an excellent text … . originates from lectures by Charles Kittel and J. H. van Vleck in the 1950s and much of the material was presented in courses by the authors over the last three decades. The material is meant for Electrical Engineering and Physics students at the graduate level … . has exercises at the end of each chapter and an extensive set of appendices. The exposition is clear and detailed. This is a very good book for its target audience.” (W. Miller Jr., Zentralblatt MATH, Vol. 1175, 2010)“The goal of the book under review is to teach group theory in close connection to applications. … Every chapter of the book finishes with several selected problems. Specific to this book is the feature that every abstract theoretical group concept is introduced and applied in a concrete physical way. This is why the book is very useful for anyone interested in applications of group theory to the wide range of condensed matter phenomena.”­­­ (Oktay K. Pashaev, Mathematical Reviews, Issue 2010 i)“It is highly welcomed because of its well-thought structuring and the plenty of non-trivial examples. The authors develop those parts of the theory of groups which are interesting for physicists, from chapter to chapter offering nearly at any step one or more informative application.” (G. Kowol, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)Table of ContentsBasic Mathematics.- Basic Mathematical Background: Introduction.- Representation Theory and Basic Theorems.- Character of a Representation.- Basis Functions.- Introductory Application to Quantum Systems.- Splitting of Atomic Orbitals in a Crystal Potential.- Application to Selection Rules and Direct Products.- Molecular Systems.- Electronic States of Molecules and Directed Valence.- Molecular Vibrations, Infrared, and Raman Activity.- Application to Periodic Lattices.- Space Groups in Real Space.- Space Groups in Reciprocal Space and Representations.- Electron and Phonon Dispersion Relation.- Applications to Lattice Vibrations.- Electronic Energy Levels in a Cubic Crystals.- Energy Band Models Based on Symmetry.- Spin–Orbit Interaction in Solids and Double Groups.- Application of Double Groups to Energy Bands with Spin.- Other Symmetries.- Time Reversal Symmetry.- Permutation Groups and Many-Electron States.- Symmetry Properties of Tensors.

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Book SynopsisThis book is an often-requested reprint of two classic texts by H. Haken: "Synergetics. An Introduction" and "Advanced Synergetics". Synergetics, an interdisciplinary research program initiated by H. Haken in 1969, deals with the systematic and methodological approach to the rapidly growing field of complexity. Going well beyond qualitative analogies between complex systems in fields as diverse as physics, chemistry, biology, sociology and economics, Synergetics uses tools from theoretical physics and mathematics to construct an unifying framework within which quantitative descriptions of complex, self-organizing systems can be made. This may well explain the timelessness of H. Haken's original texts on this topic, which are now recognized as landmarks in the field of complex systems. They provide both the beginning graduate student and the seasoned researcher with solid knowledge of the basic concepts and mathematical tools. Moreover, they admirably convey the spirit of the pioneering work by the founder of Synergetics through the essential applications contained herein that have lost nothing of their paradigmatic character since they were conceived.Table of ContentsAn Introduction.- Advanced Topics.

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Book SynopsisA unique legacy, these lecture notes of Schwinger’s course held at the University of California at Los Angeles were carefully edited by his former collaborator Berthold-Georg Englert and constitute both a self-contained textbook on quantum mechanics and an indispensable source of reference on this fundamental subject by one of the foremost thinkers of twentieth century physics.Trade ReviewFrom the reviews: "Quantum Mechanics: Symbolism of Atomic Measurements is not just another textbook on quantum mechanics. Rather, it contains truly novel elements of both content and style. In particular, Schwinger begins his treatment not with de Broglie waves or the Schrödinger equation but rather with the measurement process. His idea is to derive, or at least make plausible, the formalism of state vectors, bras and kets, by reference to quantum measurements such as the Stern-Gerlach experiment. This [...] is simply the basis of a new way of teaching quantum mechanics. This opening chapter should be of interest to all scholars of quantum theory and might form a new topic of research for philosophers of quantum mechanics." (Contemporary Physics, 44/2, 2003) "There are dozen of excellent textbooks on the market. But this one really is different." (T. Kibble, The Times Higher Education Supplement, 2001) "The material covered is superficially similar to that of a typical graduate quantum mechanics course [...] However, each chapter has beautiful and unusual treatments of familiar topics. [...] This book would make an outstanding supplement and reference for a graduate quantum mechanics course. Theoretical physicists will delight in this wonderful book, which should be available in the library system of any institution with a research or graduate program in physics. Graduate students through professionals." (CHOICE, Dec. 2001) "The book is a tour-de-force. Once the groundwork is laid, he goes into subjects with the mathematical virtuosity for which he was famous – not advanced mathematics, but the incredible use of simple mathematics. … there are gems throughout the book. … it is a wonderful book for a professor to own, like Feyman’s lectures, because there is so much to learn from it. … The book was lovingly edited from some UCLA lecture notes, by Berthold-Georg Englert, a longtime student and assistant of Schwinger’s … ." (Daniel Greenberger, American Journal of Physics, Vol 71 (9), 2003) "Editor Englert has performed a service for physicists everywhere by making available this book, which is based on Schwinger’s unpublished UCLA lecture notes. … each chapter has beautiful and unusual treatments of familiar topics. … There are excellent problems at the end of each chapter. This book would make an outstanding supplement and reference for a graduate quantum mechanics course. Theoretical physicists will delight in this wonderful book, which should be available in the library system of any institution with a research or graduate program … ." (M. C. Ogilvie, CHOICE, December, 2001) "The book commences with an absorbing prologue in which Schwinger talks us through the development of quantum mechanic and quantum field theory in an easy conversational style. … The book is packed with exercises for the reader to attempt. … Anyone who works religiously through these exercises will acquire a thoroughly adequate command of quantum mechanics." (W. Cox, Mathematical Reviews, Issue 2002 h) "Quantum mechanics: Symbolism of Atomic Measurements is not just another textbook on quantum mechanics. Rather, it contains truly novel elements of both content and style. … This opening chapter should be of interest to all scholars of quantum theory and might form a new topic of research for philosophers of quantum mechanics. Throughout the text, new material is presented at a breathless pace. All the usual elements of the subject are there, but Schwinger’s presentation reveals surprises in even the most familiar of these." (S. M. Barnett, Contemporary Physics, Vol. 44 (2), 2003) "In the beginning, the editor has added an important material in the form of a prologue … . This is one of the best treatments of the philosophy of quantum mechanics, which I have come across. … One of the major features of the book is the incorporation of a large number of problems … . the contents of the problems are well integrated in the text and have become part of it. This has caused a rich and tight structure of the logical arguments." (S. S. Bhattacharyya, Indian Journal of Physics, Vol. 76B (3), 2002) "This unique textbook is based upon the lecture notes that Julian Schwinger wrote up for the students of the quantum mechanics course … . this book would probably make an ideal quantum mechanics reference … . There are a large number of problems included at the end of each chapter, which comprise an excellent resource for any lecturer … . this textbook is a unique resource, which provides an insight into the thoughts and deliberations of one of this century’s giants of quantum mechanics." (P. C. Dastoor, The Physicist, Vol. 38 (5), 2001) "There are dozens of excellent textbooks on the market. But this one really is different. … there is a carefully argued historical and philosophical prologue that sets the scene, centred on the two key features of quantum physics – atomicity and its probabilistic character; this alone would make the book worthwhile. The emphasis on discrete variables is a very modern approach… . To a theoretical physicist, this book is a delight and a wonderful resource. … This is a book I shall treasure." (Tom Kibble, Times Higher Education Supplement, September, 2001)Table of ContentsPrologue.- A. Fall Quarter: Quantum Kinematics.- 1 Measurement Algebra.- 2 Continuous q, p Degree of Freedom.- 3 Angular Momentum.- 4 Galilean Invariance.- B. Winter Quarter: Quantum Dynamics.- 5 Quantum Action Principle.- 6 Elementary Applications.- 7 Harmonic Oscillators.- 8 Hydrogenic Atoms.- C. Spring Quarter: Interacting Particles.- 9 Two-Particle Coulomb Problem.- 10 Identical Particles.- 11 Many-Electron Atoms.- 12 Electromagnetic Radiation.

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Book SynopsisThe book is divided into two parts. The first part looks at the modeling of statistical systems before moving on to an analysis of these systems. This second edition contains new material on: estimators based on a probability distribution for the parameters; identification of stochastic models from observations; and statistical tests and classification methods.Table of Contents1 Statistical Physics: Is More than Statistical Mechanics.- I Modeling of Statistical Systems.- 2 Random Variables: Fundamentals of Probability Theory and Statistics.- 3 Random Variables in State Space: Classical Statistical Mechanics of Fluids.- 4 Random Fields: Textures and Classical Statistical Mechanics of Spin Systems.- 5 Time-Dependent Random Variables: Classical Stochastic Processes.- 6 Quantum Random Systems.- 7 Changes of External Conditions.- II Analysis of Statistical Systems.- 8 Estimation of Parameters.- 9 Signal Analysis: Estimation of Spectra.- 10 Estimators Based on a Probability Distribution for the Parameters.- 11 Identification of Stochastic Models from Observations.- 12 Estimating the Parameters of a Hidden Stochastic Model.- 13 Statistical Tests and Classification Methods.- Appendix: Random Number Generation for Simulating Realizations of Random Variables.- Problems.- Hints and Solutions.- References.

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Book SynopsisPercolation theory is the study of an idealized random medium in two or more dimensions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. Much new material appears in this second edition including dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.Table of Contents1 What is Percolation?.- 2 Some Basic Techniques.- 3 Critical Probabilities.- 4 The Number of Open Clusters per Vertex.- 5 Exponential Decay.- 6 The Subcritical Phase.- 7 Dynamic and Static Renormalization.- 8 The Supercritical Phase.- 9 Near the Critical Point: Scaling Theory.- 10 Near the Critical Point: Rigorous Results.- 11 Bond Percolation in Two Dimensions.- 12 Extensions of Percolation.- 13 Percolative Systems.- Appendix I. The Infinite-Volume Limit for Percolation.- Appendix II. The Subadditive Inequality.- List of Notation.- References.- Index of Names.

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Book SynopsisThis is an approachable introduction to the important topics and recent developments in the field of condensed matter physics. First, the general language of quantum field theory is developed in a way appropriate for dealing with systems having a large number of degrees of freedom. This paves the way for a description of the basic processes in such systems. Applications include various aspects of superfluidity and superconductivity, as well as a detailed description of the fractional quantum Hall liquid.Table of ContentsQuantum Mechanics and Fundamentals of Quantum Field Theory.- Path Integral Quantization.- Broken Symmetry and Phase Transition.- Some Applications - Warming Ups.- Superconductivity.- Quantum Hall Liquids and Chern--Simons Gauge Field.- Appendix.

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Book SynopsisThis book is aimed at theoretical as well as primarily physicists graduate students in field working quantum theory, quantum gravity, theories, gauge to sdme and and, it is not extent, general relativity cosmology. Although aimed at a I that it also be of level, hope in mathematically rigorous may terest to mathematical and mathematicians in physicists working spectral of differential mani geometry, spectral asymptotics on operators, analysis differential and mathematical methods in folds, geometry quantum theory. Thisbook will be considered too abstract some but certainly by physicists, not detailed and most mathematicians. This in completeenoughby means, thatthe material is at the level of particular, presented "physical" So, rigor. there theorems and areno and technicalcalculationsare lemmas, proofs long omitted. I tried detailed to a ofthe basic Instead, give presentation ideas, methodsandresults. Itried makethe to as andcom Also, exposition explicit as the lessabstractandhaveillustratedthe plete possible, methods language and results withsome As is well "onecannot examples. known, cover every in an text. The in this thing", especially introductory approach presented book the lines is a further of the so called along goes (and development) fieldmethod ofDe Witt. As a Ihavenot dealt at background consequence, allwithmanifoldswith boundary,non Laplacetype (ornonminimal) opera Riemann Cartan manifolds well with as as recent tors, developments many and advanced such Ashtekar's more as topics, approach,supergravity,strings, matrix etc. The membranes, interested reader is referred models, M theory tothe literature.Trade Review"This monograph rightly belongs to a series ‘Lecture notes in Physics’, as it represents a well-written review of main results by the author, who is a recognized expert on heat kernel techniques in quantum gravity. [...] The results exposed in this book reflect the major contributions of the author to differential geometry and the theory of differential operators. They have many applications in quantum field theory with background fields, and indeed, the book can be used as a text for a short graduate course in the heat kernel techniques and their quantum gravity." (Mathematical Reviews 2003a)Table of ContentsBackground Field Method in Quantum Field Theory.- Technique for Calculation of De Witt Coefficients.- Partial Summation of Schwinger-De Witt Expansion.- Higher-Derivative Quantum Gravity.- Conclusion.

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Book SynopsisThis book provides an original introduction to the geometry of Minkowski space-time. A hundred years after the space-time formulation of special relativity by Hermann Minkowski, it is shown that the kinematical consequences of special relativity are merely a manifestation of space-time geometry.The book is written with the intention of providing students (and teachers) of the first years of University courses with a tool which is easy to be applied and allows the solution of any problem of relativistic kinematics at the same time. The book treats in a rigorous way, but using a non-sophisticated mathematics, the Kinematics of Special Relativity. As an example, the famous "Twin Paradox" is completely solved for all kinds of motions.The novelty of the presentation in this book consists in the extensive use of hyperbolic numbers, the simplest extension of complex numbers, for a complete formalization of the kinematics in the Minkowski space-time.Moreover, from this formalization the understanding of gravity comes as a manifestation of curvature of space-time, suggesting new research fields.Table of ContentsIntroduction.- Hyperbolic Numbers.- Geometrical Representation of Hyperbolic Numbers.- Trigonometry in the Hyperbolic (Minkowski) Plane.- Equilateral Hyperbolas and Triangles in the Hyperbolic Plane.- The Motions in Minkowski Space-Time (Twin Paradox).- Some Final Considerations.

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Book SynopsisIn this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction. Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure. The book is arranged in four sections, devoted to realizing the universal principle force equals curvature: Part I: The Euclidean Manifold as a ParadigmPart II: Ariadne's Thread in Gauge TheoryPart III: Einstein's Theory of Special RelativityPart IV: Ariadne's Thread in Cohomology For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum. Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos). Trade ReviewFrom the reviews:“This book is the third volume of a complete exposition of the important mathematical methods used in modern quantum field theory. It presents the very basic formalism, important results, and the most recent advances emphasizing the applications to gauge theory. … the book’s greatest strength is Zeidler’s zeal to help students understand fundamental mathematics better. I thus find the book extremely useful since it signifies the role of mathematics for the road to reality … .” (Gert Roepstorff, Zentralblatt MATH, Vol. 1228, 2012)“The present book is a good companion to the literature on the subject of the volume title, especially for those already familiar with it. … the book touches upon a large number of subjects on the interface between mathematics and physics, providing a good overview of gauge theory in both fields. It contains lots of background material, many historical remarks, and an extensive bibliography that helps the interested reader to continue his or her more thorough studies elsewhere.” (Walter D. van Suijlekom, Mathematical Reviews, Issue 2012 m)Table of ContentsPrologue.- Part I. The Euclidean Manifold as a Paradigm: 1. The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure).- 2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Cli_ord Algebra, Lie Algebra).- 3. Representations of Symmetries in Mathematics and Physics.- 4. The Euclidean Manifold E3.- 5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry.- 6. Infinitesimal Rotations and Constraints in Physics.- 7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin.- 8. Changing Observers - A Glance at Invariant Theory Based on the Principle of the Correct Index Picture.- 9. Applications of Invariant Theory to the Rotation Group.- 10. Temperature Fields on the Euclidean Manifold E3.- 11. Velocity Vector Fields on the Euclidean Manifold E3.- 12. Covector Fields and Cartan's Exterior Differential - the Beauty of Differential Forms.- Part II. Ariadne's Thread in Gauge Theory: 13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field.- 14. Symmetry Breaking.- 15. The Noncommutative Yang{Mills SU(N)-Gauge Theory.- 16. Cocycles and Observers.- 17. The Axiomatic Geometric Approach to Bundles.- Part III. Einstein's Theory of Special Relativity: 18. Inertial Systems and Einstein's Principle of Special Relativity.- 19. The Relativistic Invariance of the Maxwell Equations.- 20. The Relativistic Invariance of the Dirac Equation and the Electron Spin.- Part IV. Ariadne's Thread in Cohomology: 21. The Language of Exact Sequences.- 22. Electrical Circuits as a Paradigm in Homology and Cohomology.- 23. The Electromagnetic Field and the de Rham Cohomology.- Appendix.- Epilogue.- References.- List of Symbols.- Index

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Book SynopsisThis graduate-level, course-based text is devoted to the 3+1 formalism of general relativity, which also constitutes the theoretical foundations of numerical relativity. The book starts by establishing the mathematical background (differential geometry, hypersurfaces embedded in space-time, foliation of space-time by a family of space-like hypersurfaces), and then turns to the 3+1 decomposition of the Einstein equations, giving rise to the Cauchy problem with constraints, which constitutes the core of 3+1 formalism. The ADM Hamiltonian formulation of general relativity is also introduced at this stage. Finally, the decomposition of the matter and electromagnetic field equations is presented, focusing on the astrophysically relevant cases of a perfect fluid and a perfect conductor (ideal magnetohydrodynamics). The second part of the book introduces more advanced topics: the conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations, the Isenberg-Wilson-Mathews approximation to general relativity, global quantities associated with asymptotic flatness (ADM mass, linear and angular momentum) and with symmetries (Komar mass and angular momentum). In the last part, the initial data problem is studied, the choice of spacetime coordinates within the 3+1 framework is discussed and various schemes for the time integration of the 3+1 Einstein equations are reviewed. The prerequisites are those of a basic general relativity course with calculations and derivations presented in detail, making this text complete and self-contained. Numerical techniques are not covered in this book.Trade ReviewFrom the reviews:“The monograph originating from lectures is devoted to the 3+1 formalism in general relativity. It starts with three chapters on basic differential geometry, the geometry of single hypersurfaces embedded in space-time, and the foliation of space-time by a family of spacelike hypersurfaces. … With the attempt to make the text self-consistent and complete, the calculations are … detailed such that the book is well suitable for undergraduate and graduate students.” (Horst-Heino von Borzeszkowski, Zentralblatt MATH, Vol. 1254, 2013)“This book is written for advanced students and researchers who wish to learn the mathematical foundations of various approaches that have been proposed to solve initial value problems (with constraints) for the Einstein equations numerically. … Even for experts it may be useful, as it includes an extensive bibliography up to 2011.” (Hans-Peter Künzle, Mathematical Reviews, January, 2013)Table of ContentsBasic Differential Geometry.- Geometry of Hypersurfaces.- Geometry of Foliations.- 3+1 decomposition of Einstein Equation.- 3+1 Equations for Matter and Electromagnetic Field.- Conformal Decompositon.- Asymptotic Flatness and Global Quantities.- The Initial Data Problem.- Choice of Foliation and Spatial Coordiinates.- Evolution Schemes.- Conformal Killing Operator and Conformal Vector Laplacian.- Sage Codes.

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Book SynopsisQuantum chromodynamics (QCD) is the fundamental quantum ?eld theory of quarks and gluons. In order to discuss it in a mathematically well-de?ned way, the theory has to be regularized. Replacing space-time by a Euclidean lattice has proven to be an e?cient approach which allows for both theor- ical understanding and computational analysis. Lattice QCD has become a standard tool in elementary particle physics. Asthetitlealreadysays:thisbookisintroductory!Thetextisintendedfor newcomerstothe?eld,servingasastartingpoint.Wesimplywantedtohavea bookwhichwecanputintothehandsofanadvancedstudentfora?rstreading on lattice QCD. This imaginary student brings as a prerequisite knowledge of higher quantum mechanics, some continuum quantum ?eld theory, and basic facts of elementary particle physics phenomenology. In view of the wealth of applications in current research the topics p- sented here are limited and we had to make some painful choices. We discuss QCD but omit most other lattice ?eld theory applications like scalar th- ries, gauge-Higgs models, or electroweak theory. Although we try to lead the reader up to present day understanding, we cannot possibly address all on- ing activities, in particular concerning the role of QCD in electroweak theory. Subjects like glueballs, topological excitations, and approaches like chiral p- turbation theory are mentioned only brie?y. This allows us to cover the other topics quite explicitly, including detailed derivations of key equations. The ?eld is rapidly developing. The proceedings of the annual lattice conferences provide information on newer directions and up-to-date results.Trade ReviewFrom the reviews:“This is a very nice and readable book on lattice gauge theories. It is conceived for non-specialists in the field and is quite self-contained. … It is a modern, updated introduction to lattice gauge theory, very easy to consult and conceived in a modern way. This is an excellent textbook for students or anyone wishing to be introduced to the subject.”­­­ (Giuseppe Nardelli, Mathematical Reviews, Issue 2010 k)Table of ContentsThe path integral on the lattice.- The path integral on the lattice.- QCD on the lattice — a first look.- QCD on the lattice — a first look.- Pure gauge theory on the lattice.- Pure gauge theory on the lattice.- Numerical simulation of pure gauge theory.- Numerical simulation of pure gauge theory.- Fermions on the lattice.- Fermions on the lattice.- Hadron spectroscopy.- Hadron spectroscopy.- Chiral symmetry on the lattice.- Chiral symmetry on the lattice.- Dynamical fermions.- Dynamical fermions.- Symanzik improvement and RG actions.- Symanzik improvement and RG actions.- More about lattice fermions.- More about lattice fermions.- Hadron structure.- Hadron structure.- Temperature and chemical potential.- Temperature and chemical potential.

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Book SynopsisDieses Buch ist bis heute eine der populärsten Darstellungen der Relativitätstheorie geblieben. In der vorliegenden Version haben J. Ehlers und M. Pössel vom Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) in Golm/Potsdam den Bornschen Text kommentiert und einen den anschaulichen, aber präzisen Stil Borns wahrendes, umfangreiches Ergänzungskapitel hinzugefügt, das die stürmische Entwicklung der Relativiatätstheorie bis hin zu unseren Tagen nachzeichnet. Eingegangen wird auf Gravitationswellen und Schwarze Löcher, auf neuere Entwicklungen der Kosmologie, auf Ansätze zu einer Theorie der Quantengravitation und auf die zahlreichen raffinierten Experimente, welche die Gültigkeit der Einsteinschen Theorie mit immer größerer Genauigkeit bestätigt haben. Damit bleibt dieses Buch nach wie vor einer der unmittelbarsten Zugänge zur Relativitätstheorie für alle die sich für eine über das rein populärwissenschaftliche hinausgehende Einführung interessieren.Trade Review"Allen interessierten Laien sehr zu empfehlen, die ohne höhere Mathematik tiefer in diese Materie eindringen möchten." (Weltraum-Facts mit Space-Informer, 2001) "Alle, die sich über eine rein populärwissenschaftliche Einführung hinaus für die Relativitätstheorie interessieren – besonders Physikstudenten – werden an dem eingehenden Werk Freude haben." (Der Sternenbote, 2001) "Die Erweiterung bereichert den Text von Born aber nicht nur um neuere Entwicklungen der Physik. Rückverweise verzahnen die neuen Kapitel inhaltlich mit den vorstehenden Überlegungen, und auch das Bornsche Projekt, für den mit Schulmathemaitk und einem "gesunden Menschenverstand" ausgestatteten Leser verständlich zu sein, wird erfolgreich fortgeführt." (Wissenschaftlicher Literaturanzeiger, 2001) "Das Buch zeichnet sich in besonderm Maße gegenüber fast allen anderen Abhandlungen über dieses Thema dadurch aus, daß es in einer zuweilen geradezu brillianten Ausdrucksweise die Problematik der klassischen Physik und deren Hintergrund aufzeigt, die dann durch Einsteins Theorien eine Auflösung fand. [...] Alles in allem ein sehr lesenswertes Buch, jedoch kein populärwissenschaftliches und für den Laien sicherlich auch kein leichtes. Denn trotz der außerordentlich gut verständlichen Darstellungsweise Borns erfordert das Buch außer etwas Grundlagen-Mathematik ein hohes Maß an Aufmerksamkeit und die bereitschaft, physikalsiche Sachverhalte gedanklich zu durchdringen. Es ist mit sicherheit ein Verdienst von Herausgebern und Verlag, Borns Buch durch die Neuauflage mit den aktuellen Ergänzungen wieder einer interessierten Leserschaft verfügbar zu machen." (Nachrichten der Olbers-Gesellschaft, 2001) "Wer eine fundierte, gründliche Einführung in die Welt der Relativitätstheorie sucht, kommt an Borns Klassiker nicht vorbei. Freilich muß man sich mitunter bemühen und auch einiges an Mathematik bewältigen, läuft dafür aber nicht Gefahr, durch aus dem Alltag entlehnte Analogien verwirrt zu werden." (Sirius – Zeitschrift der Vereinigten Amateur-Astronomen, 2002) "Wer die populärwissenschaftlichen Darstellungen der Relativitätstheorie als nicht ausreichend betrachtet und gern etwas tiefer schürfen möchte, dem sei dieses Buch als Lektüre ans Herz gelegt." (Astrokurier, 2002) "Ehlers und Pössel ist damit ein gelungenes Remake von Borns wegweisendem Werk gelungen. Sie haben bewiesen, dass der 80 Jahre alte Zugang zu Einsteins Theorie auch heute noch gangbar ist." (Physik in unserer Zeit, 2002) "[...] weiterhin einer der unmittelbarsten Zugänge zur Relativitätstheorie für Schüler-, Lehrer- und Studentenschaft sowie für alle, die sich nicht beruflich mit relativistischer Physik beschäftigen möchten, insbesondere für jene Leserschaft aus der Amateur-Astronomie, die an einem tieferen Verständnis dieses spannenden Themas interessiert ist. Gerade diesem Kreis ist das vorliegende Buch als Einführung in die moderne Kosmologie sehr zu empfehlen." (ORION 61/314, 2003)Table of ContentsGeometrie und Kosmologie.- Die Grundgesetze der klassischen Mechanik.- Das Newtonsche Weltsystem.- Die Grundgesetze der Optik.- Die Grundgesetze der Elektrodynamik.- Das spezielle Einsteinsche Relativitätsprinzip.- Die allgemeine Relativitätstheorie Einsteins.- Neuere Entwicklungen der relativistischen Physik.

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Book SynopsisShafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles.Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.Trade Review“This is a very good book and very good introduction to algebraic geometry and it serves an entry door to this enormous subject in mathematics. … This book is classical and I strongly recommend it as the first book on algebraic geometry. … It is an excellent book and every mathematician should have a copy.” (Philosophy, Religion and Science Book Reviews, Bookinspections.wordpress.com, July, 2016)“I find the book wonderfully put together, and I am sure the reader will learn a lot, either from systematic study or from browsing particular topics. … In each chapter, the theorems, propositions, corollaries, examples, remarks, etc., each have their own independent numbering system, running consecutively throughout the chapter. This makes it a real chore to track any internal reference in the book.” (Robin Hartshorne, SIAM Review, Vol. 56 (4), December, 2014)“The author’s two-volume textbook ‘Basic Algebraic Geometry’ is one of the most popular standard primers in the field. … the author’s unique classic is a perfect first introduction to the geometry of algebraic varieties for students and nonspecialists, and the current, improve third edition will maintain this outstanding role of the textbook in the relevant literature without any doubt.” (Werner Kleinert, zbMATH, Vol. 1273, 2013)Table of ContentsPreface.- Book 1. Varieties in Projective Space: Chapter 1. Basic Notions.- Chapter II. Local Properties.- Chapter III. Divisors and Differential Forms.- Chapter IV. Intersection Numbers.- Algebraic Appendix.- References.- Index

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Book SynopsisThe application of statistical methods to physics is essential. This unique book on statistical physics offers an advanced approach with numerous applications to the modern problems students are confronted with. Therefore the text contains more concepts and methods in statistics than the student would need for statistical mechanics alone. Methods from mathematical statistics and stochastics for the analysis of data are discussed as well. The book is divided into two parts, focusing first on the modeling of statistical systems and then on the analysis of these systems. Problems with hints for solution help the students to deepen their knowledge. The third edition has been updated and enlarged with new sections deepening the knowledge about data analysis. Moreover, a customized set of problems with solutions is accessible on the Web at extras.springer.com.Trade ReviewFrom the book reviews:“The book is carefully divided into two parts. The first part deals with modeling of statistical systems. The second part is devoted to the analysis of the respective systems. … followed by a section that offers helpful hints and solutions to problems throughout the text, making it easier for students to deepen their understanding and confidence in their newfound knowledge. … topics in each chapter are carefully selected and are well presented, making it a reliable reference for ‘statistical physics.’” (Technometrics, Vol. 55 (2), May, 2013)Table of ContentsStatistical Physics is more than Statistical Mechanics.- Part I: Modeling of Statistical Systems.- Random Variables: Fundamentals of Probability Theory and Statistics.- Random Variables in State Space: Classical Statistical Mechanics of Fluids.- Random Fields: Textures and Classical Statistical Mechanics of Spin Systems.- Time-Dependent Random Variables: Classical Stochastic Processes.- Quantum Random Systems.- Changes of External Conditions.- Part II: Analysis of Statistical Systems.- Estimation of Parameters.- Signal Analysis: Estimation of Spectra.- Estimators Based on a Probability Distribution for the Parameters.- Identification of Stochastic Models from Observations.- Estimating the Parameters of a Hidden Stochastic Model.- Statistical Tests and Classification Methods.- Appendix: Random Number Generation for Simulating Realizations of Random Variables.- Problems.- Hints and Solutions.

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Book SynopsisOne of the driving forces behind much of modern science and technology is the desire to foresee and thereby control the future. In recent years, however, it has become clear that, even in a deterministic world, there is alimit to the accuracy with which we can predict the future. This book details, in a largely nontechnical style, the extent to which we can predict the future development of various physical, biological and socio-economic processes.Table of Contents1. Introduction.- References.- 2. Forecasting Weather and Climate.- 2.1 Weather and Climate.- 2.2 Dynamical Systems and Their Properties.- 2.3 Weather Predictability.- 2.4 Elements of Stationary Random Process Prediction Theory.- 2.5 Predictability of Climatic Processes.- 2.6 Ways to Improve Statistical Forecasting.- 2.7 Utilization of Forecasting Results.- 2.8 Conclusion.- References.- 3. How an Active Autowave Medium Can Be Used to Predict the Future.- 3.1 Prediction.- 3.2 Active Autowave Media.- 3.3 Autowave Propagation in Energy-Restoring Active Media.- 3.4 Dynamics of Autowave Interaction.- 3.5 The External Medium Model and Its Fourier Image.- 3.6 Non-isochronism of Cyclic Processes.- 3.7 Harmonious Modulation and Modulation of Harmonics.- 3.8 The Fourier Image Cleared by the Active Autowave Medium.- References.- 4. Synergetics, Predictability and Deterministic Chaos.- 4.1 Dynamical Chaos.- 4.2 Nonlinearity and Open Systems Behavior.- 4.3 Synergetics and Order Parameters.- 4.4 Strangeness of the Strange Attractors.- 4.5 Dynamical Chaos and Reality.- 4.6 Dynamical Chaos. Gates of Fairyland.- References.- 5. The Information-Theoretic Approach to Assessing Reliability of Forecasts.- 5.1 Assessing Forecasts.- 5.2 Forecasting as the Subject Matter of Information Theory.- 5.3 An Example.- 5.4 Optimization of Forecasting Methods.- 5.5 Properties Shared by Prediction Methods.- 5.6 The Connection Between Discounting and Non-stationarity.- 5.7 Conclusion.- References.- 6. Prediction of Time Series.- 6.1 The Problem.- 6.2 Genesis of Random Phenomena.- 6.3 Time Series Prediction Based on Dynamical Chaos Theory.- 6.4 Prediction of Point Processes.- 6.5 The Nature of Errors Hindering Prediction.- 6.6 Prediction of Strong Earthquakes.- References.- 7. Fundamental and Practical Limits of Predictability.- 7.1 Predictability.- 7.2 Real, Observed, and Model Processes.- 7.3 Degree of Predictability. The Predictability Horizon.- 7.4 Searching for Prediction Models.- 7.5 Limits to Predictability.- 7.6 Dynamical Analogs to Social and Economic Phenomena.- 7.7 Conclusion.- References.- 8. The Future is Foreseeable but not Predictable: The ‘Oedipus Effect’ in Social Forecasting.- 8.1 Historical Background.- 8.2 The ‘Oedipus Effect’ in Social Forecasting.- 8.3 The Problem of Foresight and Prediction in Globalistics.- 8.4 The Problem of Foreseeing and Predicting the Development of the Former Soviet Society.- References.- Appendix A: Looking Back on the August 1991 Coup.- Appendix B: Looking Ahead.- 9. The Self-Organization of American Society in Presidential and Senatorial Elections.- 9.1 Historical Background.- 9.2 The American Presidential Election: Formal Analysis.- 9.3 Midterm Senatorial Elections: Formal Analysis.- 9.4 Discussion.- References.- 10. Problems of Predictability in Ethnogenic Studies.- References.

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Table of Contents18 Elementary Group Theory.- 18.1 The group axioms; examples.- 18.2 Elementary consequences of the axioms; further definitions.- 18.3 Isomorphism.- 18.4 Permutation groups.- 18.5 Homomorphisms; normal subgroups.- 18.6 Cosets.- 18.7 Factor groups.- 18.8 The Law of Homomorphism.- 18.9 The structure of cyclic groups.- 18.10 Translations, inner automorphisms.- 18.11 The subgroups of ?4.- 18.12 Generators and relations; free groups.- 18.13 Multiply periodic functions and crystals.- 18.14 The space and point groups.- 18.15 Direct and semidirect products of groups; symmorphic space groups.- 19 Continuous Groups.- 19.1 Orthogonal and rotation groups.- 19.2 The rotation group SO(3); Euler’s theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics; Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3); the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General; Rigid Motions; Bessel Functions.- 21.1 Equivalence; unitary representations.- 21.2 The reduction of representations.- 21.3 Schur’s Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration; Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds; method of identification.- 23.2 Coordinate systems or charts; compatibility; smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold; Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness; components of a manifold.- 23.7 Global topology; homotopic curves; fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(·, ·) and l(·, ·).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping; logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms; the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts; compatibility; G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25—Two nonlinear Lie groups.- 26 Metric and Geodesics on a Manifold.- 26.1 Scalar and vector fields on a manifold.- 26.2 Tensor fields.- 26.3 Metric in Euclidean space.- 26.4 Riemannian and pseudo-Riemannian manifolds.- 26.5 Raising and lowering of indices.- 26.6 Geodesies in a Riemannian manifold.- 26.7 Geodesies in a pseudo-Riamannian manifold M.- 26.8 Geodesies; the initial-value problem; the Lipschitz condition.- 26.9 The integral equation; Picard iterations.- 26.10 Geodesies; the two-point problem.- 26.11 Continuation of geodesies.- 26.12 Affmely connected manifolds.- 26.13 Riemannian and pseudo-Riemannian covering manifolds.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 27.1 Topology and metric.- 27.2 Geodesic or Riemannian coordinates.- 27.3 Normal coordinates in Riemannian and pseudo-Riemannian manifolds.- 27.4 Geometric concepts; principle of equivalence.- 27.5 Covariant differentiation.- 27.6 Absolute differentiation along a curve.- 27.7 Parallel transport.- 27.8 Orientability.- 27.9 The Riemann tensor, general; Laplacian and d’Alembertian.- 27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian manifold.- 27.11 The Riemann tensor and the intrinsic curvature of a manifold.- 27.12 Flatness and the vanishing of the Riemann tensor.- 27.13 Eisenhart’s analysis of the Stäckel systems.- 28 The Extension of Einstein Manifolds.- 28.1 Special relativity.- 28.2 The Einstein gravitational field equations.- 28.3 The Schwarzschild charts.- 28.4 The Finkelstein extensions of the Schwarzschild charts.- 28.5 The Kruskal extension.- 28.6 Maximal extensions; geodesic completeness.- 28.7 Other extensions of the Schwarzschild manifolds.- 28.8 The Kerr manifolds.- 28.9 The Cauchy problem.- 28.10 Concluding remarks.- 29 Bifurcations in Hydrodynamic Stability Problems.- 29.1 The classical problems of hydrodynamic stability.- 29.2 Examples of bifurcations in hydrodynamics.- 29.3 The Navier-Stokes equations.- 29.4 Hilbert space formulation.- 29.5 The initial-value problem; the semiflow in ?.- 29.6 The normal modes.- 29.7 Reduction to a finite-dimensional dynamical system.- 29.8 Bifurcation to a new steady state.- 29.9 Bifurcation to a periodic orbit.- 29.10 Bifurcation from a periodic orbit to an invariant torus.- 29.11 Subharmonic bifurcation.- Appendix to Chapter 29—Computational details for the invariant torus.- 30 Invariant Manifolds in the Taylor Problem.- 30.1 Survey of the Taylor problem to 1968.- 30.2 Calculation of invariant manifolds.- 30.3 Cylindrical coordinates.- 30.4 The Hilbert space.- 30.5 Separation of variables in cylindrical coordinates.- 30.6 Results to date for the Taylor problem.- Appendix to Chapter 30—The matrices in Eagles’ formulation.- 31 The Early Onset of Turbulence.- 31.1 The Landau-Hopf model.- 31.2 The Hopf example.- 31.3 The Ruelle-Takens model.- 31.4 The co-limit set of a motion.- 31.5 Attractors.- 31.6 The power spectrum for motions in ?n.- 31.7 Almost periodic and aperiodic motions.- 31.8 Lyapounov stability.- 31.9 The Lorenz system; the bifurcations.- 31.10 The Lorenz attractor; general description.- 31.11 The Lorenz attractor; aperiodic motions.- 31.12 Statistics of the mapping f and g.- 31.13 The Lorenz attractor; detailed structure I.- 31.14 The symbols [i, j] of Williams.- 31.15 Prehistories.- 31.16 The Lorenz attractor; detailed structure II.- 31.17 Existence of 1-cells in F.- 31.18 Bifurcation to a strange attractor.- 31.19 The Feigenbaum model.- Appendix to Chapter 31 (Parts A-H)—Generic properties of systems:.- 31.A Spaces of systems.- 31.B Absence of Lebesgue measure in a Hilbert space.- 31.C Generic properties of systems.- 31.D Strongly generic; physical interpretation.- 31.E Peixoto’s theorem.- 31.F Other examples of generic and nongeneric properties.- 31.G Lack of correspondence between genericity and Lebesgue measure 308 31.H Probability and physics.- References.

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Book SynopsisAt the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has been made in our understanding of and computational abilities with regard to scattering amplitudes in gauge theories, going beyond the traditional textbook approach. These advances build upon on-shell methods that focus on the analytic structure of the amplitudes, as well as on their recently discovered hidden symmetries. In fact, when expressed in suitable variables the amplitudes are much simpler than anticipated and hidden patterns emerge.These modern methods are of increasing importance in phenomenological applications arising from the need for high-precision predictions for the experiments carried out at the Large Hadron Collider, as well as in foundational mathematical physics studies on the S-matrix in quantum field theory.Bridging the gap between introductory courses on quantum field theory and state-of-the-art research, these concise yet self-contained and course-tested lecture notes are well-suited for a one-semester graduate level course or as a self-study guide for anyone interested in fundamental aspects of quantum field theory and its applications.The numerous exercises and solutions included will help readers to embrace and apply the material presented in the main text.Trade Review“Aimed at the advanced graduate student or a practitioner of high energy theory interested in the subject, the book begins with a review of non-abelian gauge theory and its conventional Feynman methods before immediately delving into on-shell recursion relations of BCFW (Britto-Cachazo-Feng-Witten) and factorization properties. … Of particular usefulness to the student are the exercises and an entire appendix dedicated to their detailed solutions.” (Yang-Hui He, zbMATH 1315.81005, 2015)Table of ContentsIntroduction and Basics.- Tree-Level Techniques.- Loop-Level Structure.- Advanced Topics.- Renormalization Properties of Wilson Loops.- Conventions and Useful Formulae.- Solutions to the Exercises.- References.

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Book SynopsisThis book reviews the basic ideas of the Law of Large Numbers with its consequences to the deterministic world and the issue of ergodicity. Applications of Large Deviations and their outcomes to Physics are surveyed. The book covers topics encompassing ergodicity and its breaking and the modern applications of Large deviations to equilibrium and non-equilibrium statistical physics, disordered and chaotic systems, and turbulence.Table of ContentsErgodicity – A Basic Concept.- Large Deviations in Statistical Mechanics: Rigorous and Non-Rigorous.- Large Deviation Techniques for Long-Range Interactions.- Fluctuation-Dissipation and Fluctuation Relations: From Equilibrium to Nonequilibrium Phenomena and Back.- Stochastic Fluctuations in Deterministic Systems.- Large Deviation and Disordered Systems.- Large Deviations in Turbulence.- Ergodicity Breaking Challenges Monte Carlo Methods.- Anomalous Diffusion: Deterministic and Stochastic Perspectives.- The Use of Fluctuation Relations for the Analysis of Free-Energy Landscapes.

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Book SynopsisThis monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem.This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.Table of ContentsPreface.- Conventions.- List of Symbols.- 1 Symmetric bilinear forms.- 2 Clifford algebras.- 3 The spin representation.- 4 Covariant and contravariant spinors.- 5 Enveloping algebras.- 6 Weil algebras.- 7 Quantum Weil algebras.- 8 Applications to reductive Lie algebras.- 9 D(g; k) as a geometric Dirac operator.- 10 The Hopf–Koszul–Samelson Theorem.- 11 The Clifford algebra of a reductive Lie algebra.- A Graded and filtered super spaces.- B Reductive Lie algebras.- C Background on Lie groups.- References.- Index.

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Book SynopsisThe German edition of this book appeared in 1932 under the title "Die gruppentheoretische Methode in der Quantenmechanik". Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory to the Quantum Mechanics of Atoms and Molecules. The book was mainly written for the benefit of physicists who were supposed to be familiar with Quantum Mechanics. However, it turned out that it was also used by. mathematicians who wanted to learn Quantum Mechanics from it. Naturally, the physical parts were too difficult for mathematicians, whereas the mathematical parts were sometimes too difficult for physicists. The German language created an additional difficulty for many readers. In order to make the book more readable for physicists and mathe­ maticians alike, I have rewritten the whole volume. The changes are most notable in Chapters 1 and 6. In Chapter t, I have tried to give a mathematically rigorous exposition of the principles of Quantum Mechanics. This was possible because recent investigations in the theory of self-adjoint linear operators have made the mathematical foundation of Quantum Mechanics much clearer than it was in t 932. Chapter 6, on Molecule Spectra, was too much condensed in the German edition. I hope it is now easier to understand. In Chapter 2-5 too, numerous changes were made in order to make the book more readable and more useful.Table of Contents1. Fundamental Notions of Quantum Mechanics.- § 1. Wave Functions.- § 2. Hilbert Spaces.- § 3. Linear Operators.- § 4. Hypermaximal Operators.- § 5. Separation of Variables.- § 6. One Electron in a Central Field.- § 7. Perturbation Theory.- § 8. Angular Momentum and Infinitesimal Rotations.- 2. Groups and Their Representations.- § 9. Linear Transformations.- § 10. Groups.- § 11. Equivalence and Reducibility of Representations.- § 12. Representations of Abelian Groups. Examples.- § 13. Uniqueness Theorems.- § 14. Kronecker’s Product Transformation.- § 15. The Operators Commuting with all Operators of a Given Representation.- § 16. Representations of Finite Groups.- § 17. Group Characters.- 3. Translations, Rotations and Lorentz Transformations.- § 18. Lie Groups and their Infinitesimal Transformations.- A. Lie Groups.- B. One-dimensional Lie Groups and Semi-Groups.- C. Causality and Translations in Time.- D. The Lie Algebra of a Lie Group.- E. Representations of Lie Groups.- § 19. The Unitary Groups SU(2) and the Rotation Group O3.- § 20. Representations of the Rotation Group O3.- § 21. Examples and Applications.- A. The Product Representation ?j × ?j’.- B. The Clebsch-Gordan Series.- C. Applications of (21.1).- D. The Reflection Character.- § 22. Selection and Intensity Rules.- § 23. The Representations of the Lorentz Group.- A. The Group SL(2) and the Restricted Lorentz Group.- B. Infinitesimal Transformations.- C. The Relation between World Vectors and Spinors.- IV. The Spinning Electron.- § 24. The Spin.- § 25. The Wave Function of the Spinning Electron.- A. Pauli’s Pair of Functions (?1, ?2).- B. Transformation of the Pair (?1, ?2).- C. Infinitesimal Rotations.- D. The Angular Momenta.- E. The Doublet Splitting of the Alkali Terms.- G. The Inversion s.- § 26. Dirac’s Wave Equation.- § 27. Two-Component Spinors.- A. Dirac’s Equation Rewritten.- B. Weyl’s Equation.- § 28. The Several Electron Problem. Multiplet Structure. Zeeman Effect.- V. The Group of Permutations and the Exclusion Principle.- § 29. The Resonance of Equal Particles.- § 30. The Exclusion Principle and the Periodical System.- § 31. The Eigenfunctions of the Atom.- § 32. The Calculation of the Energy Values.- § 33. Pure Spin Functions and their Transformation under Rotations and Permutations.- § 34. Representations of the Symmetric Group Sn.- VI. Molecule Spectra.- § 35. The Quantum Numbers of the Molecule.- § 36. The Rotation Levels.- § 37. The Case of Two Equal Nuclei.- Author and Subject Index.

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Book SynopsisIn structure mechanics analysis, finite element methods are now well estab­ lished and well documented techniques; their advantage lies in a higher flexibility, in particular for: (i) The representation of arbitrary complicated boundaries; (ii) Systematic rules for the developments of stable numerical schemes ap­ proximating mathematically wellposed problems, with various types of boundary conditions. On the other hand, compared to finite difference methods, this flexibility is paid by: an increased programming complexity; additional storage require­ ment. The application of finite element methods to fluid mechanics has been lagging behind and is relatively recent for several types of reasons: (i) Historical reasons: the early methods were invented by engineers for the analysis of torsion, flexion deformation of bearns, plates, shells, etc ... (see the historics in Strang and Fix (1972) or Zienckiewicz (1977». (ii) Technical reasons: fluid flow problems present specific difficulties: strong gradients,l of the velocity or temperature for instance, may occur which a finite mesh is unable to properly represent; a remedy lies in the various upwind finite element schemes which recently turned up, and which are reviewed in chapter 2 (yet their effect is just as controversial as in finite differences). Next, waves can propagate (e.g. in ocean dynamics with shallowwaters equations) which will be falsely distorted by a finite non regular mesh, as Kreiss (1979) pointed out. We are concerned in this course with the approximation of incompressible, viscous, Newtonian fluids, i.e. governed by N avier Stokes equations.Table of ContentsNotations.- 1. Elliptic Equations of Order 2: Some Standard Finite Element Methods.- 1.1. A 1-Dimensional Model Problem: The Basic Notions.- 1.2. A 2-Dimensional Problem.- 1.3. The Finite Element Equations.- 1.4. Standard Examples of Finite Element Methods.- 1.4.1. Example 1: The P1-Triangle (Courant’s Triangle).- 1.4.2. Example 2: The P2-Triangle.- 1.4.3. Example 3: The Q1-Quadrangle.- 1.4.4. Example 4: The Q2-Quadrangle.- 1.4.5. A Variational Crime: The P1 Nonconforming Element.- 1.5. Mixed Formulation and Mixed Finite Element Methods for Elliptic Equations.- 1.5.1. The One Dimensional Problem.- 1.5.2. A Two Dimensional Problem.- 2. Upwind Finite Element Schemes.- 2.1. Upwind Finite Differences.- 2.2. Modified Weighted Residual (MWR).- 2.3. Reduced Integration of the Advection Term.- 2.4. Computation of Directional Derivatives at the Nodes.- 2.5. Discontinuous Finite Elements and Mixed Interpolation.- 2.6. The Method of Characteristics in Finite Elements.- 2.7. Peturbation of the Advective Term: Bredif (1980).- 2.8. Some Numerical Tests and Further Comments.- 2.8.1. One Dimensional Stationary Advection Equation (56).- 2.8.2. Two Dimensional Stationary Advection Equation.- 2.8.3. Time Dependent Advection.- 3. Numerical Solution of Stokes Equations.- 3.1. Introduction.- 3.2. Velocity—Pressure Formulations: Discontinuous Approximations of the Pressure.- 3.2.1. uh: P1 Nonconforming Triangle (§1-4-5); ph: Piecewise Constant.- 3.2.2. uh: P2 Triangle ph: P0 (Piecewise Constant).- 3.2.3. uh: “P2+bubble” Triangle (or Modified P2); ph: Discontinuous P1.- 3.2.4. uh: Q2 Quadrangle; ph: Q1 Discontinuous.- 3.2.5. Numerical Solution by Penalty Methods.- 3.2.6. Numerical Results and Further Comments.- 3.3. Velocity—Pressure Formulations: Continuous Approximation of the Pressure and Velocity.- 3.3.1. Introduction.- 3.3.2. Examples and Error Estimates.- 3.3.3. Decomposition of the Stokes Problem.- 3.4. Vorticity—Pressure—Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 3.5. Vorticity Stream-Function Formulation: Decompositions of the Biharmonic Problem.- 4. Navier-Stokes Equations: Accuracy Assessments and Numerical Results.- 4.1. Remarks on the Formulation.- 4.2. A review of the Different Methods.- 4.2.1 Velocity—Pressure Formulations: Discontinuous Approximations of the Pressure.- 4.2.2. Velocity—Pressure Formulations: Continuous Approximations of the Pressure.- 4.2.3. Vorticity—Pressure—Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 4.2.4. Vorticity Stream-Function Formulation.- 4.3. Some Numerical Tests.- 4.3.1. The Square Wall Driven Cavity Flow.- 4.3.2. An Engineering Problem: Unsteady 2-D Flow Around and In an Air-Intake.- 5. Computational Problems and Bookkeeping.- 5.1. Mesh Generation.- 5.2. Solution of the Nonlinear Problems.- 5.2.1. Successive Approximations (or Linearization) with Under Relaxation.- 5.2.2. Newton-Raphson Algorithm.- 5.2.3. Conjugate Gradient Method (with Scaling) for Nonlinear Problems.- 5.2.4. A Splitting Technique for the Transient Problem.- 5.3. Iterative and Direct Solvers of Linear Equations.- 5.3.1. Successive Over Relaxation.- 5.3.2. Cholesky Factorizations.- 5.3.3. Out of Core Factorizations.- 5.3.4. Preconditioned Conjugate Gradient.- Appendix 2. Numerical Illustration.- Three Dimensional Case.- References.

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Book SynopsisA thorough introduction to the computation of celestial mechanics, covering everything from astronomical and computational theory to the construction of rapid and accurate applications programs. The book supplies the necessary knowledge and software solutions for determining and predicting positions of the Sun, Moon, planets, minor planets and comets, solar eclipses, stellar occultations by the Moon, phases of the Moon and much more. This completely revised edition takes advantage of C++, and individual applications may be efficiently realized through the use of a powerful module library. The accompanying CD-ROM contains the complete, fully documented and commented source codes as well as executable programs for Windows 98/2000/XP and LINUX.Table of Contents1 Introduction.- 2 Coordinate Systems.- 3 Calculation of Rising and Setting Times.- 4 Cometary Orbits.- 5 Special Perturbations.- 6 Planetary Orbits.- 7 Physical Ephemerides of the Planets.- 8 The Orbit of the Moon.- 9 Solar Eclipses.- 10 Stellar Occultations.- 11 Orbit Determination.- 12 Astrometry.- A.1 The Accompanying CD-ROM.- A.1.1 Contents.- A.1.2 System Requirements.- A.1.3 Executing the Programs.- A.2 Compiling and Linking the Programs.- A.2.1 General Advice on Computer-Specific Modifications.- A.2.2 Microsoft Visual C++ for Windows 98/2000/XP.- A.2.3 GNU C++ for Linux.- A.3 List of the Library Functions.- Symbols.

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Book SynopsisThis second English edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis.The main difference between the second and first English editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. This second volume presents classical analysis in its current form as part of a unified mathematics. It shows how analysis interacts with other modern fields of mathematics such as algebra, differential geometry, differential equations, complex analysis, and functional analysis. This book provides a firm foundation for advanced work in any of these directions.Table of Contents9 Continuous Mappings (General Theory).- 10 Differential Calculus from a General Viewpoint.- 11 Multiple Integrals.- 12 Surfaces and Differential Forms in Rn.- 13 Line and Surface Integrals.- 14 Elements of Vector Analysis and Field Theory.- 15 Integration of Differential Forms on Manifolds.- 16 Uniform Convergence and Basic Operations of Analysis.- 17 Integrals Depending on a Parameter.- 18 Fourier Series and the Fourier Transform.- 19 Asymptotic Expansions.- Topics and Questions for Midterm Examinations.- Examination Topics.- Examination Problems (Series and Integrals Depending on a Parameter).- Intermediate Problems (Integral Calculus of Several Variables).- Appendices: A Series as a Tool (Introductory Lecture).- B Change of Variables in Multiple Integrals.- C Multidimensional Geometry and Functions of a Very Large Number of Variables.- D Operators of Field Theory in Curvilinear Coordinates.- E Modern Formula of Newton–Leibniz.- References.- Index of Basic Notation.- Subject Index.- Name Index.

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Book SynopsisThis new edition of the near-legendary textbook by Schlichting and revised by Gersten presents a comprehensive overview of boundary-layer theory and its application to all areas of fluid mechanics, with particular emphasis on the flow past bodies (e.g. aircraft aerodynamics). The new edition features an updated reference list and over 100 additional changes throughout the book, reflecting the latest advances on the subject.Trade ReviewFrom the reviews: "We find here a book where the theory is developed with rigours in parallel with a strong physical intuition. Comparison with experiments and simulations are always proposed and carefully analysed. The book contains at the end a very rich and complete bibliography ... I warmly encourage everyone interested in boundary-layer theory to have this book in his bookcase." Physicalia "... I do recommend the book highly, especially for its long historical perspective, including all the diagrams comparing theory and experiment that remind us that engineering is practical ..." SIAM ReviewsTable of ContentsPart I. Fundamentals of Viscous Flows.- 1. Some Features of Viscous Flows.- 2. Fundamentals of Boundary–Layer Theory.- 3. Field Equations for Flows of Newtonian Fluids.- 4. General Properties of the Equations of Motion.- 5. Exact Solutions of the Navier–Stokes Equations.- Part II. Laminar Boundary Layers.- 6 Boundary–Layer Equations in Plane Flow; Plate Boundary Layer.- 7 General Properties and Exact Solutions of the Boundary–Layer Equations for Plane Flows.- 8 Approximate Methods for Solving the Boundary–Layer Equations for Steady Plane Flows.- 9 Thermal Boundary Layers Without Coupling of the Velocity Field to the Temperature Field.- 10 Thermal Boundary Layers with Coupling of the Velocity Field to the Temperature Field.- 11. Boundary–Layer Control (Suction/Blowing).- 12. Axisymmetric and Three–Dimensional Boundary Layers.- 13. Unsteady Boundary Layers.- 14. Extensions to the Prandtl Boundary–Layer Theory.- Part III. Laminar–Turbulent Transition.- 15. Onset of Turbulence (Stability Theory).- Part IV. Turbulent Boundary Layers.- 16. Fundamentals of Turbulent Flows.- 17. Internal Flows.- 18. Turbulent Boundary Layers Without Coupling of the Velocity Field to the Temperature Field.- 19. Turbulent Boundary Layers with Coupling of the Velocity Field to the Temperature Field.- 20. Axisymmetric and Three–Dimensional Turbulent Boundary Layers.- 21. Unsteady Turbulent Boundary Layers.- 22. Turbulent Free Shear Flows.- Part V. Numerical Methods in Boundary–Layer Theory.- 23. Numerical Integration of the Boundary–Layer Equations.

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Book SynopsisAs a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics.The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.Table of ContentsRemarks on Mathematial Physics.- 1 Introduction.- 2 Dynamical Systems.- 3 Ordinary Differential Equations.- 4 Linear Dynamics.- 5 Classification of Linear Flows.- 6 Hamiltonian Equations and Symplectic Group.- 7 Stability Theory.- 8 Variational Principles.- 9 Ergodic Theory.- 10 Symplectic Geometry.- 11 Motion in a Potential.- 12 Scattering Theory.- 13 Integrable Systems and Symmetries.- 14 Rigid and Non-Rigid Bodies.- 15 Perturbation Theory.- 16 Relativistic Mechanics.- 17 Symplectic Topology.- A Topological Spaces and Manifolds.- B Differential Forms.- C Convexity and Legendre Transform.- D Fixed Point Theorems, and Results about Inverse Images.- E Group Theory.- F Bundles, Connection, Curvature.- G Morse Theory.- H Solutions of the Exercises.- Bibiography.- Index of Proper Names.- Table of Symbols.- Image Credits.- Index.

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Book SynopsisThis second edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.Trade Review“This is a thorough and easy-to-follow text for a beginning course in real analysis … . In coverage the book is slanted towards physics (mostly mechanics), and in particular there is a lot about line and surface integrals. … Will be popular with students because of the detailed explanations and the worked examples.” (Allen Stenger, MAA Reviews, maa.org, May, 2016)Table of Contents1 Some General Mathematical Concepts and Notation: 1.1 Logical Symbolism.- 1.2 Sets and Elementary Operations on them.- 1.3 Functions.- 1.4 Supplementary Material.- 2 The Real Numbers: 2.1 Axioms and Properties of Real Numbers.- 2.2 Classes of Real Numbers and Computations.- 2.3 Basic Lemmas on Completeness.- 2.4 Countable and Uncountable Sets.- 3 Limits: 3.1 The Limit of a Sequence.- 3.2 The Limit of a Function.- 4 Continuous Functions: 4.1 Basic Definitions and Examples.- 4.2 Properties of Continuous Functions.- 5 Differential Calculus: 5.1 Differentiable Functions.- 5.2 The Basic Rules of Differentiation.- 5.3 The Basic Theorems of Differential Calculus.- 5.4 Differential Calculus Used to Study Functions.- 5.5 Complex Numbers and Elementary Functions.- 5.6 Examples of Differential Calculus in Natural Science.- 5.7 Primitives.- 6 Integration: 6.1 Definition of the Integral.- 6.2 Linearity, Additivity and Monotonicity of the Integral.- 6.3 The Integral and the Derivative.- 6.4 Some Applications of Integration.- 6.5 Improper Integrals.- 7 Functions of Several Variables: 7.1 The Space Rm and its Subsets.- 7.2 Limits and Continuity of Functions of Several Variables.- 8 Differential Calculus in Several Variables: 8.1 The Linear Structure on Rm.- 8.2 The Differential of a Function of Several Variables.- 8.3 The Basic Laws of Differentiation.- 8.4 Real-valued Functions of Several Variables.- 8.5 The Implicit Function Theorem.- 8.6 Some Corollaries of the Implicit Function Theorem.- 8.7 Surfaces in Rn and Constrained Extrema.- Some Problems from the Midterm Examinations: 1. Introduction to Analysis (Numbers, Functions, Limits).- 2. One-variable Differential Calculus.- 3. Integration. Introduction to Several Variables.- 4. Differential Calculus of Several Variables.- Examination Topics: 1. First Semester: 1.1. Introduction and One-variable Differential Calculus.- 2. Second Semester: 2.1. Integration. Multivariable Differential Calculus.- Appendices: A Mathematical Analysis (Introductory Lecture).- B Numerical Methods for Solving Equations (An Introduction).- C The Legendre Transform (First Discussion).- D The Euler–Maclaurin Formula.- E Riemann–Stieltjes Integral, Delta Function, and Generalized Functions.- F The Implicit Function Theorem (An Alternative Presentation).- References.- Subject Index.- Name Index.

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Book SynopsisThis book is about supergravity, which combines the principles of general relativity and local gauge invariance with the idea of supersymmetries between bosonic and fermionic degrees of freedom. The authors give a thorough and pedagogical introduction to the subject suitable for beginning graduate or advanced undergraduate students in theoretical high energy physics or mathematical physics. Interested researchers working in these or related areas are also addressed. The level of the presentation assumes a working knowledge of general relativity and basic notions of differential geometry as well as some familiarity with global supersymmetry in relativistic field theories. Bypassing curved superspace and other more technical approaches, the book starts from the simple idea of supersymmetry as a local gauge symmetry and derives the mathematical and physical properties of supergravity in a direct and “minimalistic” way, using a combination of explicit computations and geometrical reasoning. Key topics include spinors in curved spacetime, pure supergravity with and without a cosmological constant, matter couplings in global and local supersymmetry, phenomenological and cosmological implications, extended supergravity, gauged supergravity and supergravity in higher spacetime dimensions.Table of ContentsIntroduction.- From Global to Local SUSY.- Gravity and spinors.- D=4 N=1 SUGRA.- Matter couplings in global SUSY.- Matter couplings in SUGRA.- SUGRA phenomenology.- Extended supergravities.- Gauged supergravity.- SUGRA in any dimension.

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Book SynopsisThis book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area of ongoing research for both graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.Trade Review"The presentation provides significant insight into a large part of the current issues of interest in the physics of Bose systems and especially into the "kitchen" of several relevant mathematical techniques. As such, it is highly recommended to both advanced researchers and students preparing to work in this field." (Mathematical Reviews)Table of ContentsThe Dilute Bose Gas in 3D.- The Dilute Bose Gas in 2D.- Generalized Poincaré Inequalities.- Bose-Einstein Condensation and Superfluidity for Homogeneous Gases.- Gross-Pitaevskii Equation for Trapped Bosons.- Bose-Einstein Condensation and Superfluidity for Dilute Trapped Gases.- One-Dimensional Behavior of Dilute Bose Gases in Traps.- Two-Dimensional Behavior in Disc-Shaped Traps.- The Charged Bose Gas, the One- and Two-Component Cases.- Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.

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Book Synopsis"José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced, and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach in mathematics from the early nineteenth century." --Bulletin of Symbolic Logic (Review of first edition)Trade ReviewFrom the book reviews:“The book is a thorough, deep, fascinating work. It is not only recommended, it is compulsory for anyone interested in the history of mathematical ideas.” (László I. Szabó, Acta Scientiarum Mathematicarum (Szeged), Vol. 75 (1-2), 2009)Table of ContentsThe Emergence of Sets within Mathematics.- Institutional and Intellectual Contexts in German Mathematics, 1800–1870.- A New Fundamental Notion: Riemann’s Manifolds.- Dedekind and the Set-theoretical Approach to Algebra.- The Real Number System.- Origins of the Theory of Point-Sets.- Entering the Labyrinth-Toward Abstract Set Theory.- The Notion of Cardinality and the Continuum Hypothesis.- Sets and Maps as a Foundation for Mathematics.- The Transfinite Ordinals and Cantor’s Mature Theory.- In Search of an Axiom System.- Diffusion, Crisis, and Bifurcation: 1890 to 1914.- Logic and Type Theory in the Interwar Period.- Consolidation of Axiomatic Set Theory.

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Book SynopsisThis book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.Trade ReviewFrom the reviews: “It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. … This book should be interesting to anybody who is interested in applications of hypercomplex numbers … . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers.” (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)Table of ContentsThe Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries.- Commutative Hypercomplex Number Systems.- The General Two-Dimensional System.- Linear Transformations and Geometries.- The Geometries Associated with Hypercomplex Numbers.- Conclusions.- 2 Trigonometry in the Minkowski Plane.- Geometrical Representation of Hyperbolic Numbers.- Basics of Hyperbolic Trigonometry.- Geometry in Pseudo-Euclidean Cartesian Plane.- Trigonometry in the Pseudo-Euclidean Plane.- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane.- Some Examples of Triangle Solutions in the Minkowski Plane.- Conclusions.- 3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox).- Inertial Motions.- Inertial and Uniformly Accelerated Motions.- Non-uniformly Accelerated Motions.- Conclusions.- 4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation.- Geometry and Trigonometry in Two-Dimensional Algebras.- Some Properties of Fundamental Conic Section.- Numerical Examples.- 5 Functions of a Hyperbolic Variable.- Some Remarks on Functions of a Complex Variable.- Functions of Hypercomplex Variables.- The Functions of a Hyperbolic Variable.- The Elementary Functions of a Canonical Hyperbolic Variable.- H-Conformal Mappings.- Commutative Hypercomplex Systems with Three Unities.- 6 Hyperbolic Variables on Lorentz Surfaces.- Introduction.- Gauss: Conformal Mapping of Surfaces.- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces.- Beltrami: Complex Variables on a Surface.- Beltrami’s Integration of Geodesic Equations.- Extension of Beltrami’s Equation to Non-Definite Differential Forms.- 7 Constant Curvature Lorentz Surfaces.- Introduction.- Constant Curvature RiemannSurfaces.- Constant Curvature Lorentz Surfaces.- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces.- Conclusions.- 8 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle).- Physical Meaning of Transformations by Hyperbolic Functions.- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature.- Einstein’s Way to General Relativity.- Conclusions.- II An Introduction to Commutative Hypercomplex Numbers.- 9 Commutative Segre’s Quaternions.- Introduction.- Hypercomplex Systems with Four Units.- Historical Introduction of Segre’s Quaternion.- Algebraic Properties of Commutative Quaternions.- Functions of a Quaternion Variable.- Mapping by Means of Quaternion Functions.- Elementary Functions of the Quaternions.- Elliptic-Hyperbolic Quaternions.- Elliptic-Parabolic Generalized Segre’s Quaternions.- 10 Constant Curvature Segre’s Quaternion Spaces.- Introduction.- Quaternion differential geometry and geodesic equations.- Orthogonality in Segre’s Quaternion Space.- Constant Curvature Quaternion Spaces.- Geodesic Equations in Quaternion Space.- Beltrami’s Integration Method for Quaternion Spaces.- Beltrami’s Integration Method for Quaternion Spaces.- Conclusions.- 11 A Matrix Formalization for Commutative Hypercomplex Systems.- Mathematical Operations.- Properties of the Characteristic Matrix M.- Functions of Hypercomplex Variable.- Functions of a Two-Dimensional Hypercomplex Variable.- Derivatives of a Hypercomplex Function.- Characteristic Differential Equation.- A Equivalence Between the Formalizations of Hypercomplex Numbers.

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Book SynopsisOne studying the motion of fluids relative to particulate systems is soon impressed by the dichotomy which exists between books covering theoretical and practical aspects. Classical hydrodynamics is largely concerned with perfect fluids which unfortunately exert no forces on the particles past which they move. Practical approaches to subjects like fluidization, sedimentation, and flow through porous media abound in much useful but uncorrelated empirical information. The present book represents an attempt to bridge this gap by providing at least the beginnings of a rational approach to fluid­ particle dynamics, based on first principles. From the pedagogic viewpoint it seems worthwhile to show that the Navier-Stokes equations, which form the basis of all systematic texts, can be employed for useful practical applications beyond the elementary problems of laminar flow in pipes and Stokes law for the motion of a single particle. Although a suspension may often be viewed as a continuum for practical purposes, it really consists of a discrete collection of particles immersed in an essentially continuous fluid. Consideration of the actual detailed boundary­ value problems posed by this viewpoint may serve to call attention to the limitation of idealizations which apply to the overall transport properties of a mixture of fluid and solid particles.Table of Contents1. Introduction.- 1–1 Definition and purpose, 1. 1–2 Historical review, 8. 1–3 Application in science and technology, 13..- 2. The Behavior of Fluids in Slow Motion.- 2–1 The equations of change for a viscous fluid, 23. 2–2 Mechanical energy dissipation in a viscous fluid, 29. 2–3 Force and couple acting on a body moving in a viscous fluid, 30. 2–4 Exact solutions of the equations of motion for a viscous fluid, 31. 2–5 Laminar flow in ducts, 33. 2–6 Simplifications of the Navier-Stokes equations, especially for slow motion, 40. 2–7 Paradoxes in the solution of the creeping motion equations, 47. 2–8 Molecular effects in fluid dynamics, 49. 2–9 Non-newtonian flow, 51. 2–10 Unsteady creeping flows, 52..- 3. Some General Solutions and Theorems Pertaining to the Creeping Motion Equations.- 3–1 Introduction, 58. 3–2 Spherical coordinates, 62. 3–3 Cylindrical coordinates, 71. 3–4 Integral representations, 79. 3–5 Generalized reciprocal theorem, 85. 3–6 Energy dissipation, 88..- 4. Axisymmetrical Flow.- 4–1 Introduction, 96. 4–2 Stream function, 96. 4–3 Relation between stream function and local velocity, 98. 4–4 Stream function in various coordinate systems, 99. 4–5 Intrinsic coordinates, 100. 4–6 Properties of the stream function, 102. 4–7 Dynamic equation satisfied by the stream function, 103. 4–8 Uniform flow, 106. 4–9 Point source or sink, 106. 4–10 Source and sink of equal strength, 107. 4–11 Finite line source, 108. 4–12 Point force, 110. 4–13 Boundary conditions satisfied by the stream function, 111. 4–14 Drag on a body, 113. 4–15 Pressure, 116. 4–16 Separable coordinate systems, 117. 4–17 Translation of a sphere, 119. 4–18 Flow past a sphere, 123. 4–19 Terminal settling velocity, 124. 4–20 Slip at the surface of a sphere, 125. 4–21 Fluid sphere, 127. 4–22 Concentric spheres, 130. 4–23 General solution in spherical coordinates, 133. 4–24 Flow through a conical diffuser, 138. 4–25 Flow past an approximate sphere, 141. 4–26 Oblate spheroid, 145. 4–27 Circular disk, 149. 4–28 Flow in a venturi tube, 150. 4–29 Flow through a circular aperture, 153. 4–30 Prolate spheroid, 154. 4–31 Elongated rod, 156. 4–32 Axisymmetric flow past a spherical cap, 157..- 5. The Motion of a Rigid Particle of Arbitrary Shape in an Unbounded Fluid.- 5–1. Introduction, 159. 5–2 Translational motions, 163. 5–3 Rotational motions, 169. 5–4 Combined translation and rotation, 173. 5–5 Symmetrical particles, 183. 5–6 Nonskew bodies, 192. 5–7 Terminal settling velocity of an arbitrary particle, 197. 5–8 Average resistance to translation, 205. 5–9 The resistance of a slightly deformed sphere, 207. 5–10 The settling of spherically isotropic bodies, 219. 5–11 The settling of orthotopic bodies, 220..- 6. Interaction between Two or More Particles.- 6–1 Introduction, 235. 6–2 Two widely spaced spherically isotropic particles, 240: 6–3 Two spheres by the method of reflections and similar techniques, 249. 6–4 Exact solution for two spheres falling along their line of centers, 270. 6–5 Comparison of theories with experimental data for two spheres, 273. 6–6 More than two spheres, 276. 6–7 Two spheroids in a viscous liquid, 278. 6–8 Limitations of creeping motion equations, 281..- 7. Wall Effects on the Motion of a Single Particle.- 7–1 Introduction, 286. 7–2 The translation of a particle in proximity to container walls, 288. 7–3 Sphere moving in an axial direction in a circular cylindrical tube, 298. 7–4 Sphere moving relative to plane walls, 322. 7–5 Spheroid moving relative to cylindrical and plane walls, 331. 7–6 k-coefficients for typical boundaries, 340. 7–7 One- and two-dimensional problems, 341. 7–8 Solid of revolution rotating symmetrically in a bounded fluid, 346. 7–9 Unsteady motion of a sphere in the presence of a plane wall, 354..- 8. Flow Relative to Assemblages of Particles.- 8–1 Introduction, 358. 8–2 Dilute systems—no interaction effects, 360. 8–3 Dilute systems—first-order interaction effects, 371. 8–4 Concentrated systems, 387. 8–5 Systems with complex geometry, 400. 8–6 Particulate suspensions, 410. 8–7 Packed beds, 417. 8–8 Fluidization, 422..- 9. The Viscosity of Particulate Systems.- 9–1 Introduction, 431. 9–2 Dilute systems of spheres—no interaction effects, 438. 9–3 Dilute systems—first-order interaction effects, 443. 9–4 Concentrated systems, 448. 9–5 Nonspherical and nonrigid particles, 456. 9–6 Comparison with data, 462. 9–7 Non-newtonian behavior, 469..- Appendix A. Orthogonal Curvilinear Coordinate Systems.- A-l Curvilinear coordinates, 474. A-2 Orthogonal curvilinear coordinates, 477. A-3 Geometrical properties, 480. A-4 Differentiation of unit vectors, 481. A-5 Vector differential invariants, 483. A-6 Relations between cartesian and orthogonal curvilinear coordinates, 486. A-7 Dyadics in orthogonal curvilinear coordinates, 488. A-8 Cylindrical coordinate systems, 490. A-9 Circular cylindrical coordinates, 490. A-10 Conjugate cylindrical coordinate systems, 494. A-ll Elliptic cylinder coordinates, 495. A-12 Bipolar cylinder coordinates, 497. A-l3 Parabolic cylinder coordinates, 500. A-14 Coordinate systems of revolution, 501. A-l5 Spherical Coordinates, 504. A-l6 Conjugate coordinate systems of revolution, 508. A-17 Prolate spheroidal coordinates, 509. A-18 Oblate spheroidal coordinates, 512. A-19 Bipolar coordinates, 516. A-20 Toroidal coordinates, 519. A-21 Paraboloidal Coordinates, 521..- Appendix B. Summary of Notation and Brief Review of Polyadic Algebra.- Name Index.

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Book SynopsisVery few people have contributed as much to twentieth-century physics as Julian Schwinger. It is therefore appropriate to offer a retrospective of his work on the occasion of his sixtieth birthday (February 12, 1978). We hope, in offering this selection of his papers, to bring to light ideas and results that may have been partly overlooked at the time of the original publication. Schwinger has published prodigiously on a great variety of subjects, as is evident from the comprehensive list of publications arranged in chronological order which appears on p. xiii. Needless to say, only a small subset could be included in the present modest volume. In the selection, great weight was assigned to papers that seem to be less widely known or appreciated than they deserve. Many important papers are therefore omitted. (Examples: Paper [64] 'On Gauge Invariance and Vacuum Polarization' and Paper [69] 'On Angular Momentum', both of which have been reprinted elsewhere. ) The collection is a personal one, having been chosen by Schwinger himself, and is therefore of particular interest. It would probably not be interesting to offer an analysis, by the editors, of Schwinger's contributions to physics. However, we are very pleased to be able to include Schwinger's own informal and very personal comments about each article that appears in this volume. These comments indicate his reasons for choosing these particular articles and, in many cases, provide a capsule synopsis of what he considers most valuable.Table of Contents[8] ‘The Scattering of Neutrons by Ortho- and Parahydrogen’ (with E. Teller), Phys. Rev.52, 286 (1937)..- [11] ‘The Neutron-Proton Scattering Cross Section’ (with V. Cohen and H. Goldsmith), Phys. Rev.55, 106 (1939)..- [15] ‘On Pair Emission in the Proton Bombardment of Fluorine’ (with J. R. Oppenheimer), Phys. Rev.56, 1066 (1939)..- [25] ‘On a Theory of Particles with Half-Integral Spin’ (with W. Rarita), Phys. Rev.60, 61 (1941)..- [26] ‘On the Interaction of Mesotrons and Nuclei’ (with J. R. Oppenheimer), Phys. Rev.60, 150 (1941)..- [31] ‘On a Field Theory of Nuclear Forces’, Phys. Rev.61, 387 (1942)..- [34] ‘Polarization of Neutrons by Resonance Scattering in Helium’, Phys. Rev.69, 681 (1946)..- [42] ‘On the Polarization of Fast Neutrons’, Phys. Rev.73, 407 (1948)..- [43] ‘On Quantum-Electrodynamics and the Magnetic Moment of the Electron’, Phys. Rev.73, 416 (1948)..- [44] ‘A Note on Saturation in Microwave Spectroscopy’ (with R. Karplus), Phys. Rev.73, 1020 (1948)..- [58] ‘On the Charge Independence of Nuclear Forces’, Phys. Rev.78, 135(1950)..- [66] ‘On the Green’s Functions of Quantized Fields I, II’, Proc. Natl. Acad. Sci., U.S.A.37, 452, 455 (1951)..- [74] ‘The Theory of Quantized Fields. III’, Phys. Rev.91, 728 (1953)..- [76] ‘The Theory of Quantized Fields. IV’, Phys. Rev.92, 1283 (1953)..- [78] ‘The Quantum Correction in the Radiation by Energetic Accelerated Electrons’, Proc. Natl. Acad. Sci., U.S.A.40, 132 (1954)..- [82] ‘A Theory of the Fundamental Interactions’, Ann. Phys. (N. Y.)2, 407 (1957)..- [86] ‘On the Euclidean Structure of Relativistic Field Theory’, Proc. Natl. Acad. Sci., U.S.A.44, 956 (1958)..- [88] ‘Euclidean Quantum Electrodynamics’, Phys. Rev.115, 721 (1959)..- [91] ‘The Algebra of Microscopic Measurement’, Proc. Natl. Acad. Sci., U.S.A.45, 1542 (1959)..- [98] ‘The Special Canonical Group’, Proc. Natl. Acad. Sci., U.S.A.46, 1401 (1960)..- [100] ‘On the Bound States of a Given Potential’, Proc. Natl. Acad. Sci., U.S.A.47, 122 (1961)..- [101] ‘Brownian Motion of a Quantum Oscillator’, J. Math. Phys.2, 407(1961)..- [104] ‘Gauge Invariance and Mass’, Phys. Rev.125, 397(1962). 188.- [105] ‘Non-Abelian Gauge Fields. Commutation Relations’, Phys. Rev.125, 1043 (1962)..- [106] ‘Exterior Algebra and the Action Principle. I’, Proc. Natl. Acad. Sci., U.S.A.48, 603 (1962)..- [107] ‘Non-Abelian Gauge Fields. Relativistic Invariance’, Phys. Rev.127, 324(1962)..- [108] ‘Gauge Invariance and Mass. II’, Phys. Rev.128, 2425 (1962)..- [109] ‘Quantum Variables and Group Parameters’, IlNuovo Cimento30, 278(1963)..- [111] ‘Commutation Relations and Conservation Laws’, Phys. Rev.130, 406 (1963)..- [112] ‘Energy and Momentum Density in Field Theory’, Phys. Rev.130, 800(1963)..- [114] ‘Quantized Gravitational Field.II’ Phys. Rev.132, 1317(1963)..- [116] ‘Coulomb Green’s Function’, J. Math. Phys.5, 1606 (1964)..- [117] ‘Non-Abelian Vector Gauge Fields and the Electromagnetic Field’, Rev. Mod. Phys.26, 609 (1964)..- [118] ‘Field Theory of Matter’, Phys. Rev.135, B816 (1964)..- [124] ‘Field Theory of Matter. II’, Phys. Rev.136, B1821 (1964)..- [128] ‘Field Theory of Matter.IV’, Phys. Rev.140, B158 (1965)..- [132] ‘Relativistic Quantum Field Theory’, Nobel Lecture, in Nobel Lectures — Physics, 1963–1970, Elsevier, Amsterdam, 1972..- [135] ‘Particles and Sources’, Phys. Rev.152, 1219 (1966)..- [137] ‘Chiral Dynamics’, Phys. Letters24B, 473 (1967)..- [139] ‘Partial Symmetry’, Phys. Rev. Letters18, 923 (1967)..- [144] ‘Gauge Fields, Sources and Electromagnetic Masses’, Phys. Rev.165, 1714 (1968); Phys. Rev.167, 1546 (1968)..- [147] ‘Sources and Magnetic Charge’, Phys. Rev.173, 1536 (1968)..- [150] ‘A Magnetic Model of Matter’, Science, 165, 757(1969)..- [151] ‘Theory of Sources’, Contemporary Physics (Trieste Symposium 1968), IAEA Vienna, 1969, Vol. II, p. 59..- [155] ‘How Massive is the W Particle?’, Phys. Rev.D7, 908 (1973)..- [156] ‘Classical Radiation of Accelerated Electrons. II. A Quantum Viewpoint’, Phys. Rev.D7, 1696 (1973)..- [157] ‘How to Avoid ?Y=1Neutral Currents’, Phys. Rev.D8, 960 (1973)..- [160] ‘A Report on Quantum Electrodynamics’, in The Physicist’s Conception of Nature, edited by J. Mehra, Reidel, Dordrecht, 1973, p. 413..- [164] ‘Photon Propagation Function: Spectral Analysis of Its Asymptotic Form’, Proc. Natl. Acad. Sci., U.S.A.71, 3024 (1974)..- [167] ‘Source Theory Viewpoints in Deep Inelastic Scattering’, Proc. Natl. Acad. Sci., U.S.A.72, 1 (1975)..- [172] ‘Magnetic Charge and the Charge Quantization Condition’, Phys. Rev.D12, 3105 (1975)..- [174] ‘Casimir Effect in Source Theory’, Lett. Math. Phys.1, 43 (1975)..- [178] ‘Deep Inelastic Scattering of Leptons’, Proc. Natl. Acad. Sci., U.S.A.73, 3351 (1976)..

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Table of ContentsMathematical and Quantum-Mechanical Background.- General Concepts of the Theory of Resonance States and Processes.- Theory of Resonance States Based on the Hilbert-Schmidt Expansion.- Projection Methods.- Theory of Resonance States and Processes Based on Analytical Continuation in the Coupling Constant.- S-matrix Parametrization of Scattering Data. Extraction of Resonance Parameters from Experimental Data.- Resonances in Atomic Physics.- Conclusion, Open Problems.

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Book SynopsisThe Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization or perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. This volume deals with the application of this method to many problems of physics, including some frontier problems which have previously required much more computationally-intensive approaches. The opening chapters deal with various fundamental aspects of the decomposition method. Subsequent chapters deal with the application of the method to nonlinear oscillatory systems in physics, the Duffing equation, boundary-value problems with closed irregular contours or surfaces, and other frontier areas. The potential application of this method to a wide range of problems in diverse disciplines such as biology, hydrology, semiconductor physics, wave propagation, etc., is highlighted. For researchers and graduate students of physics, applied mathematics and engineering, whose work involves mathematical modelling and the quantitative solution of systems of equations. Trade Review`I recommend Adomian's new book to all researchers in the area of mathematical modeling and solving complex dynamical systems.' Foundations of Physics, 1994 Table of ContentsPreface. Foreword. 1. On Modelling Physical Phenomena. 2. The Decomposition Method for Ordinary Differential Equations. 3. The Decomposition Method in Several Dimensions. 4. Double Decomposition. 5. Modified Decomposition. 6. Applications of Modified Decomposition. 7. Decomposition Solutions for Neumann Boundary Conditions. 8. Integral Boundary Conditions. 9. Boundary Conditions at Infinity. 10. Integral Equations. 11. Nonlinear Oscillations in Physical Systems. 12. Solution of the Duffing Equation. 13. Boundary-Value Problems with Closed Irregular Contours or Surfaces. 14. Applications in Physics. Appendix I: Padé and Shanks Transform. Appendix II: On Staggered Summation of Double Decomposition Series. Appendix III: Cauchy Products of Infinite Series. Index.

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Book SynopsisFor many physicists quantum theory contains strong conceptual difficulties, while for others the apparent conclusions about the reality of our physical world and the ways in which we discover that reality remain philosophically unacceptable. This book focuses on recent theoretical and experimental developments in the foundations of quantum physics, including topics such as the puzzles and paradoxes which appear when general relativity and quantum mechanics are combined; the emergence of classical properties from quantum mechanics; stochastic electrodynamics; EPR experiments and Bell's Theorem; the consistent histories approach and the problem of datum uniqueness in quantum mechanics; non-local measurements and teleportation of quantum states; quantum non-demolition measurements in optics and matter wave properties observed by neutron, electron and atomic interferometry. Audience: This volume is intended for graduate students of physics and those interested in the foundations of quantum theory.Table of Contents1. The subject of our discussions; E. Santos. 2. Measurement of the Schrödinger wave of a single particle; Y. Aharonov, L. Vaidman. 3. The emergence of classical properties from quantum mechanics: New problems from old; L.E. Ballentine. 4. Deformations of space-time symmetries and fundamental scales; A. Ballesteros, et al. 5. Aspects of quantum reality; S. Bergia. 6. Kochen-Specker diagram of the Peres--Mermin example; A. Cabello. 7. Zeropoint waves and quantum particles; A.M. Cetto, L. de la Peña. 8. Results of atom interferometry experiments with potassium; J.F. Clauser. 9. On the uncertainty relations; J.R. Croca. 10. Continuously diagonalized density operator of open systems; L. Diósi. 11. The hazy spacetime of the Károlyházy model of quantum mechanics; A. Frenkel. 12. Can the experiments based on parametric-down conversion disprove Einstein locality? A. Garuccio. 13. Quantum-mechanical histories and the uncertainty principle; J.J. Halliwell. 14. Experiments with coherent electron wave packets; F. Hasselbach. 15. The ontological interpretation of quantum field theory applied in a cosmological context; B.J. Hiley, A.H. Aziz Muft. 16. State vector reduction via spacetime imprecision; F. Károlyházy. 17. Analyses of classical and thermodynamic limits of quantum mechanics and quantum measurements on the basis of nonstandard analysis; T. Kobayashi. 18. A realistic interpretation of lattice gauge theories; M. Lorente. 19. Is there abridge connecting stochastic and quantum electrodynamics? T.W. Marshall. 20. Action-angle variables inherent in quantum dynamics; J. Martínez-Linares. 21. A philosopher struggles to understand quantum theory: Particle creations and wavepacket reduction; N. Maxwell. 22. Consistent histories and the interpretation of quantum mechanics; R. Omnès. 23. Is quantum mechanics a limit cycle theory? L. de la Peña, A.M. Cetto. 24. Realization and characterization of quantum nondemolition measurements in optics; J.Ph. Poizat, et al. 25. Fuzzy sets and infinite-valued Łukasiewicz logic in foundations of quantum mechanics; J. Pykacz. 26. A model of topological quantization of the electromagnetic field; A.F. Rañada. 27. Postselection and squeezing in neutron interferometry and EPR-experiments; H. Rauch. 28. Macroscopic decoherence and classical stochastic gravity; J.L. Sanchez-Gomez. 29. Dynamics and measurement of the absolute phase in macroscopic quantum systems; F. Sols, R.A. Hegstrom. 30. Realistic quantum theory and relativity; E.J. Squires. 31. On the empirical law of epistemology: Physics as an artifact of mathematics; N.A. Tambakis. 32. Search of a first principle for quantum physics; A.C. de la Torre. 33. Decoherence in an isolated macroscopic quantum system: A parameter-free model involving gravity; J. Unturbe. 34. Nonlocal measurements and teleportation of quantum states; L. Vaidman. 35. Quantum noise in optical photon detectors; A. Vidiella-Barranco, E. Santos.

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Book SynopsisThere exist essentially two levels of investigation in theoretical physics. One is primarily descriptive, concentrating as it does on useful phenomenological approaches toward the most economical classifications of large classes of experimental data on particular phenomena. The other, whose thrust is explanatory, has as its aim the formulation of those underlying hypotheses and their mathematical representations that are capable of furnishing, via deductive analysis, predictions - constituting the particulars of universals (the asserted laws)- about the phenomena under consideration. The two principal disciplines of contemporary theoretical physics - quantum theory and the theory of relativity - fall basically into these respective categories. General Relativity and Matter represents a bold attempt by its author to formulate, in as transparent and complete a way as possible, a fundamental theory of matter rooted in the theory of relativity - where the latter is viewed as providing an explanatory level of understanding for probing the fundamental nature ofmatter indomainsranging all the way fromfermis and lessto light years and more. We hasten to add that this assertion is not meant to imply that the author pretends with his theory to encompass all ofphysics or even a tiny part of the complete objective understanding of our accessible universe. But he does adopt the philosophy that underlying all natural phenomena there is a common conceptualbasis,and then proceeds to investigate how far such a unified viewcan take us at its present stage of development.Trade Review`...to read it should be a rewarding experience for anyone who is concerned with understanding the most fundamental features of the physicist's world view. ...well written and contains some very useful material on both the conceptual and technical aspects of relativity.' Foundations of Physics, 15 (1985) Table of ContentsA.- 1 / Concepts.- B : Mathematical Preliminaries.- 2 / Vector-Tensor Analysis in Relativity Theory.- 3 / Spinor-Quaternion Analysis in Relativity Theory.- C: The Field Equations.- 4 / The Matter Field Equations.- 5 / The Electromagnetic Field Equations.- 6 / The Gravitational Field Equations and Unification with Inertia and Electromagnetism.- 7 / Astrophysics and Cosmology.- Selections from the Author’s Bibliography.

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Book SynopsisAddressing students and researchers as well as Computational Fluid Dynamics practitioners, this book is the most comprehensive review of high-resolution schemes based on the principle of Flux-Corrected Transport (FCT). The foreword by J.P. Boris and historical note by D.L. Book describe the development of the classical FCT methodology for convection-dominated transport problems, while the design philosophy behind modern FCT schemes is explained by S.T. Zalesak. The subsequent chapters present various improvements and generalizations proposed over the past three decades. In this new edition, recent results are integrated into existing chapters in order to describe significant advances since the publication of the first edition. Also, 3 new chapters were added in order to cover the following topics: algebraic flux correction for finite elements, iterative and linearized FCT schemes, TVD-like flux limiters, acceleration of explicit and implicit solvers, mesh adaptation, failsafe limiting for systems of conservation laws, flux-corrected interpolation (remapping), positivity preservation in RANS turbulence models, and the use of FCT as an implicit subgrid scale model for large eddy simulations.Table of ContentsThe conception, gestation, birth and infancy of FCT.- The design of flux-corrected transport (FCT) algorithms for structured grids.- On monotonically integrated large eddy simulation of tubulent flows based on FCT algorithms.- Large scale urban simulations with FCT.- 30 years of FCT.- Algebraic flux corretion I.- Algebraic flux correction II.- Algebraic flux correction III.- Algebraic flux correction IV.- An evaluation of the FCT method for high-speed flows.- Flux-corrected and optimization-based remap.

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