Mathematical / Computational / Theoretical physics Books

Book SynopsisTrade Review“The book is an unexpectedly intimate look into a highly accomplished man, his colleagues and friends, the development of a new field of geometric analysis, and a glimpse into a truly uncommon mind.”—Nina MacLaughlin, Boston Globe"For decades, mathematician Shing-Tung Yau—a winner of the 1982 Fields Medal—has been central to the cross-fertilization between modern mathematics and physics. His work in geometry, for instance, underlies much of string theory. This volume, co-authored with science writer Steve Nadis, is an intimate account of Yau’s life”—Barbara Kiser, Nature“An eye-opening and insightful account. . . . Yau’s life story is an inspiring example of the power of education.”—Dan Eady, South China Morning Post“A real story of a remarkable mathematician and of contemporary mathematics, written with passion by one of the key players”—Peter Giblin, The Mathematical GazetteFinalist in the PROSE Awards mathematics category, sponsored by the Association of American Publishers“Yau and Nadis’s The Shape of a Life opens a window into the fascinating mind and world of today’s equivalent of Apollonius of Perga, ‘The Great Geometer’ of antiquity.”—Mario Livio, author of Brilliant Blunders"The interesting life of a remarkably influential modern mathematician."—Juan Maldacena, Institute for Advanced Study“This book tells a fascinating story of a life lived between multiple cultures—China and the West, and mathematics and physics. Yau's journey from poverty in Hong Kong to the top levels of the mathematics world was not a simple one.”—Edward Witten, Institute for Advanced Study"Candid, deep, and truly inspiring, The Shape of a Life is studded with unexpected insights into Yau's thinking. An extraordinary story about an extraordinary person."—Gish Jen, author of The Girl at the Baggage Claim: Explaining the East-West Culture Gap“The remarkable story of one of the world's most accomplished mathematicians, Shing-Tung Yau, who has made profound contributions in pure mathematics, general relativity, and string theory. Yau’s personal journey—from escaping China as a youngster, leading a gang outside Hong Kong, becoming captivated by mathematics, to making breakthroughs that thrust him on the world stage—inspires us all with humankind's irrepressible spirit of discovery.”—Brian Greene, author of The Elegant Universe

Read more less

Book SynopsisBased on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students.Trade Review`Shankar obviously enjoys his mathematics, and his attitude toward mathematics is simultaneously refreshing and contagious....Dirac notation is intriguingly introduced in the discussion of vector spaces. Finally, the book is richly endowed with well-chosen problems.' American Journal of Physics `Consistent with the needs of science students...a sound mathematical reference for anyone studying or practicing in the physical sciences.' Choice Table of ContentsDifferential Calculus of One Variable. Integral Calculus. Calculus of Many Variables. Infinite Series. Complex Numbers. Functions of a Complex Variable. Vector Calculus. Matrices and Determinants. Linear Vector Spaces. Differential Equations. Answers. Index.

Read more less

Book Synopsis1. Introductory Material.- 2. Green's Functions at Zero Temperature.- 3. Nonzero Temperatures.- 4. Exactly Solvable Models.- 5. Homogeneous Electron Gas.- 6. Strong Correlations.- 7. Electron-Phonon Interaction.- 8. dc Conductivities.- 9. Optical Properties of Solids.- 10. Superconductivity.- 11. Superfluids.- References.- Author Index.Trade ReviewAbout the First Edition: `This is a worthy newcomer and will make an excellent teaching text.' Physics Bulletin `The book should serve as a valuable contribution to the library of students and researchers in solid state and theoretical physics.' Philosophical Magazine About the Second Edition: `Since its first edition, this book has become one of the most popular textbooks in quantum many-body theory, thus guaranteeing the interest of the scientific community in this second edition.' Mathematical Reviews Table of Contents1: Introductory Material. 1.1. Harmonic Oscillators and Phonons. 1.2. Second Quantization for Particles. 1.3. Electron-Phonon Interactions. 1.4. Spin Hamiltonians. 1.5. Photons. 1.6. Pair Distribution Function. 2: Green's Functions at Zero Temperature. 2.1. Interaction Representation. 2.2. S Matrix. 2.3. Green's Functions. 2.4. Wick's Theorem. 2.5. Feynman Diagrams. 2.6. Vacuum Polarization Graphs. 2.7. Dyson's Equation. 2.8. Rules for Constructing Diagrams. 2.9. Time-Loop S Matrix. 2.10. Photon Green's Functions. 3: Nonzero Temperatures. 3.1. Introduction. 3.2. Matsubara Green's Functions. 3.3. Retarded and Advanced Green's Functions. 3.4. Dyson's Equation. 3.5. Frequency Summations. 3.6. Linked Cluster Expansions. 3.7. Real Time Green's Functions. 3.8. Kubo Formula for Electrical Conductivity. 3.9. Other Kubo Formulas. 4: Exactly Solvable Models. 4.1. Potential Scattering. 4.2. Localized State in the Continuum. 4.3. Independent Boson Models. 4.4. Bethe Lattice. 4.5. Tomonaga Model. 4.6. Polaritons. 5: Homogeneous Electron Gas. 5.1. Exchange and Correlation. 5.2. Wigner Lattice. 5.3. Metallic Hydrogen. 5.4. Linear Screening. 5.5. Model Dielectric Functions. 5.6. Properties of the Electron Gas. 5.7. Sum Rules. 5.8. One-Electron Properties. 6: Strong Correlations. 6.1. Kondo Model. 6.2. Single-Site Anderson Model. 6.3. Hubbard Model. 6.4. Hubbard Model: Magnetic Phases. 7: Electron-Phonon Interaction. 7.1. Fröhlich Hamiltonian. 7.2. Small Polaron Theory. 7.3. Heavily Doped Semiconductors. 7.4. Metals. 8: dc Conductivities. 8.1. Electron Scattering by Impurities. 8.2. Mobility of Frölich Polarons. 8.3. Electron-Phonon Relaxation Times. 8.4. Electron-Phonon Interactions in Metals. 8.5. Quantum Boltzmann Equation. 8.6. Quantum Dot Tunneling. 9: Optical Properties of Solids. 9.1. Nearly Free-Electron Systems. 9.2. Wannier Excitons. 9.3. X-Ray Spectra in Metals. 10: Superconductivity. 10.1. Cooper Instability. 10.2. Superconducting Tunneling. 10.3. Strong Coupling Theory. 10.4. Transition Temperature. 11: Superfluids. 11.1. Liquid 4He. 11.2. Liquid 3He. 11.3. Quantum Hall Effects.

Read more less

Book SynopsisTable of Contents1. Galilean relativity 2. Lorentz Boosts 3. Development of the Formalism 4. Electrodynamics 5. Gravity 6. Experiments and Applications Part II: Mathematics 7. Mathematics of Translations 8. The Rotation Group 9. The Lorentz Group

Read more less

Book SynopsisIntroduction: Newtonian Physics And Special Relativity.- Relativity Principles and Gravitation.- The Special Theory of Relativity.- The Mathematics Of The General Theory Of Relativity.- Vectors, Tensors, and Forms.- Basis Vector Fields and the Metric Tensor.- Non-inertial Reference Frames.- Differentiation, Connections, and Integration.- Curvature.- Einstein's Field Equations.- Einstein's Field Equations.- The Linear Field Approximation.- The Schwarzschild Solution and Black Holes.- Cosmology.- Homogeneous and Isotropic Universe Models.- Universe Models with Vacuum Energy.- Anisotropic and Inhomogeneous Universe Models.- Advanced Topics.- Covariant Decomposition, Singularities, and Canonical Cosmology.- Spatially Homogeneous Universe Models.- Israel's Formalism: The Metric Junction Method.- Brane-worlds.- Kaluza-Klein Theory.Trade ReviewFrom the reviews: "A sophisticated treatment of general relativity with a considerable number of applications to cosmology. … The book may be read in several different ways, depending on the interests of readers. A rich source of material; college libraries should have it on their shelves. Summing Up: Highly recommended. Upper-division undergraduates through professionals." (K. L. Schick, CHOICE, Vol. v4 (3), November, 2007) "This book is a carefully prepared overview about all essential aspects of relativity. Its 5 parts give the standard way to present the material … . Differently from other textbooks, the present authors emphasize much more concrete problems and examples … . The book contains a very large number of formulas, partially due to the fact, that many statements are given both in the old index-notation as well as in the modern index-free notation; I think this is helpful for the readers." (Hans-Jürgen Schmidt, Zentralblatt MATH, Vol. 1126 (3), 2008) "Even if you already have a sizable collection of books on general relativity, or if you are looking for a good modern book to teach a course from, the text by Grøn and Hervik is … a valuable addition to your collection." (David H. Delphenich, Mathematical Reviews, Issue 2008 i)Table of ContentsI. Introduction: Newtonian Physics and Special Relativity- 1. Relativity Principals and Gravitation 2. The Special Theory of Relativity II. The Mathematics of the General Theory of Relativity- 3. Vectors, Tensors, and Forms 4. Basis Vector Fields and Metric Tensor 5. Non-inertial Reference Frames 6. Differentiation, Connections and Integration 7. Curvature II. Einstein's Field Equations- 8. Einstein's Field Equations 9. The Linear Field Approximation 10. The Schwarzschild Solution and Black Holes IV. Cosmology- 11. Homogeneous and Isotropic Universe Models 12. Universe Models with Vacuum Energy 13. An Anisotropic Universe V. Advanced Topics- 14. Covariant decomposition, Singularities, and Canonical Cosmology 15. Homogeneous Spaces 16. Israel's Formalism: The metric junction method 17. Brane-worlds 18. Kaluza-Klein Theory VI. Appendices- A. Constrants of Nature B. Penrose diagrams C. Anti-de Sitter spacetime D. Suggested further reading

Read more less

Book SynopsisDifferentiable Dynamical Systems.- Notes.- References for Notes.- What Is Global Analysis?.- Stability and Genericity in Dynamical Systems.- Personal Perspectives on Mathematics and Mechanics.- Dynamics in General Equilibrium Theory.- Some Dynamical Questions in Mathematical Economics.- Review of Global Variational Analysis: Weier strass Integrals on a Riemannian Manifold.- Review of Catastrophe Theory: Selected Papers.- On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff.- Robert Edward Bowen (jointly with J. Feldman and M. Ratner).- On How I Got Started in Dynamical Systems.Table of ContentsDifferentiable Dynamical Systems.- Notes.- References for Notes.- What Is Global Analysis?.- Stability and Genericity in Dynamical Systems.- Personal Perspectives on Mathematics and Mechanics.- Dynamics in General Equilibrium Theory.- Some Dynamical Questions in Mathematical Economics.- Review of Global Variational Analysis: Weier strass Integrals on a Riemannian Manifold.- Review of Catastrophe Theory: Selected Papers.- On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff.- Robert Edward Bowen (jointly with J. Feldman and M. Ratner).- On How I Got Started in Dynamical Systems.

Read more less

Book SynopsisThe Second Law of Thermodynamics has been called the most important law of nature: It is the law that gives a direction to processes that is not inherent in the laws of motion, that says the state of the universe is driven to thermal equilibrium.

Read more less

Book SynopsisIntended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics.Table of Contents1. Introduction; 2. Quantum Field Theory; 3. Statistical Mechanics; 4. Global Conformal Invariance; 5. Conformal Invariance in Two Dimensions; 6. The Operator Formalism I; 7. The Operator Formalism II; 8. Minimal Models; 9. The Coulomb Gas Formalism; 10. Modular Invariance; 11. Finite Size Scaling; 12. The Two-Dimensional Ising Model; 13. Simple Lie Algebras; 14. Affine Lie Algebras; 15. The WZNW Model; 16. Fusion Rules; 17. Modular Invariants; 18. The Coset Construction

Read more less

Book SynopsisRevised and updated, it includes expanded discussions of * the fundamentals of geostrophic turbulence * the theory of wave-mean flow interaction * thermocline theory * finite amplitude barocline instability.Trade ReviewFrom the reviews"The author has done a masterful job in presenting the theory with the necessary mathematical foundation, while keeping the physical aspects in clear view ... it is an outstanding introduction to a complex and important subject." (GEOPHYSICS)Table of ContentsPreliminaries * Fundamentals * Inviscid Shallow-Water Theory * Friction and Viscous Flow * Homogeneous Models of the Wind-Driven Oceanic Circulation * Quasigeostrophic Motion of a Stratified Fluid on a Sphere * Instability Theory * Ageostrophic Motion

Read more less

Book SynopsisI Newtonian Mechanics.- 1 Experimental facts.- 2 Investigation of the equations of motion.- II Lagrangian Mechanics.- 3 Variational principles.- 4 Lagrangian mechanics on manifolds.- 5 Oscillations.- 6 Rigid bodies.- III Hamiltonian Mechanics.- 7 Differential forms.- 8 Symplectic manifolds.- 9 Canonical formalism.- 10 Introduction to perturbation theory.- Appendix 1 Riemannian curvature.- Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids.- Appendix 3 Symplectic structures on algebraic manifolds.- Appendix 4 Contact structures.- Appendix 5 Dynamical systems with symmetries.- Appendix 6 Normal forms of quadratic hamiltonians.- Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories.- Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem.- Appendix 9 Poincaré's geometric theorem, its generalizations and applications.- Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters.- Appendix 11 Short wave asymptotics.- Appendix 12 Lagrangian singularities.- Appendix 13 The Korteweg-de Vries equation.- Appendix 14 Poisson structures.- Appendix 15 On elliptic coordinates.- Appendix 16 Singularities of ray systems.Trade ReviewSecond Edition V.I. Arnol’d Mathematical Methods of Classical Mechanics "The book's goal is to provide an overview, pointing out highlights and unsolved problems, and putting individual results into a coherent context. It is full of historical nuggets, many of them surprising . . . The examples are especially helpful; if a particular topic seems difficult, a later example frequently tames it. The writing is refreshingly direct, never degenerating into a vocabulary lesson for its own sake. The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview." —AMERICAN MATHEMATICAL MONTHLYTable of ContentsI Newtonian Mechanics.- 1 Experimental facts.- 2 Investigation of the equations of motion.- II Lagrangian Mechanics.- 3 Variational principles.- 4 Lagrangian mechanics on manifolds.- 5 Oscillations.- 6 Rigid bodies.- III Hamiltonian Mechanics.- 7 Differential forms.- 8 Symplectic manifolds.- 9 Canonical formalism.- 10 Introduction to perturbation theory.- Appendix 1 Riemannian curvature.- Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids.- Appendix 3 Symplectic structures on algebraic manifolds.- Appendix 4 Contact structures.- Appendix 5 Dynamical systems with symmetries.- Appendix 6 Normal forms of quadratic hamiltonians.- Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories.- Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov’s theorem.- Appendix 9 Poincaré’s geometric theorem, its generalizations and applications.- Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters.- Appendix 11 Short wave asymptotics.- Appendix 12 Lagrangian singularities.- Appendix 13 The Korteweg-de Vries equation.- Appendix 14 Poisson structures.- Appendix 15 On elliptic coordinates.- Appendix 16 Singularities of ray systems.

Read more less

Book SynopsisI Asymptotic Solutions of Evolution Problems.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VIII Bifurcation of Periodic Solutions in the General Case.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case.- XII Stability and Bifurcation in Conservative Systems.Table of ContentsI Asymptotic Solutions of Evolution Problems.- I.1 One-Dimensional, Two-Dimensional n-Dimensional, and Infinite-Dimensional Interpretations of (I.1).- I.2 Forced Solutions; Steady Forcing and T-Periodic Forcing; Autonomous and Nonautonomous Problems.- I.3 Reduction to Local Form.- I.4 Asymptotic Solutions.- I.5 Asymptotic Solutions and Bifurcating Solutions.- I.6 Bifurcating Solutions and the Linear Theory of Stability.- I.7 Notation for the Functional Expansion of F(t µ,U).- Notes.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- II.1 The Implicit Function Theorem.- II.2 Classification of Points on Solution Curves.- 1I.3 The Characteristic Quadratic. Double Points, Cusp Points, and Conjugate Points.- II.4 Double-Point Bifurcation and the Implicit Function Theorem.- II.5 Cusp-Point Bifurcation.- II.6 Triple-Point Bifurcation.- II.7 Conditional Stability Theorem.- II.8 The Factorization Theorem in One Dimension.- II.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation.- II.10 Exchange of Stability at a Double Point.- II.1 1 Exchange of Stability at a Double Point for Problems Reduced to Local Form.- II.12 Exchange of Stability at a Cusp Point.- II.13 Exchange of Stability at a Triple Point.- II.14 Global Properties of Stability of Isolated Solutions.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- III.1 The Structure of Problems Which Break Double-Point Bifurcation.- III.2 The Implicit Function Theorem and the Saddle Surface Breaking Bifurcation.- III.3 Examples of Isolated Solutions Which Break Bifurcation.- III.4 Iterative Procedures for Finding Solutions.- III.5 Stability of Solutions Which Break Bifurcation.- III.6 Isolas.- Exercise.- Notes.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- IV.1 Eigenvalues and Eigenvectors of an n x n Matrix.- IV.2 Algebraic and Geometric Multiplicity—The Riesz Index.- IV.3 The Adjoint Eigenvalue Problem.- IV.4 Eigenvalues and Eigenvectors of a 2 x 2 Matrix.- 4.1 Eigenvalues.- 4.2 Eigenvectors.- 4.3 Algebraically Simple Eigenvalues.- 4.4 Algebraically Double Eigenvalues.- 4.4.1 Riesz Index 1.- 4.4.2 Riesz Index 2.- IV.5 The Spectral Problem and Stability of the Solution u = 0 in ?n.- IV.6 Nodes, Saddles, and Foci.- IV.7 Criticality and Strict Loss of Stability.- Appendix IV.I Biorthogonality for Generalized Eigenvectors.- Appendix IV.2 Projections.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- V.1 The Form of Steady Bifurcating Solutions and Their Stability.- V.2 Necessary Conditions for the Bifurcation of Steady Solutions.- V.3 Bifurcation at a Simple Eigenvalue.- V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue.- V.5 Bifurcation at a Double Eigenvalue of Index Two.- V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue of Index Two.- V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a Double Eigenvalue of Index One (Semi-Simple).- V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple Double Eigenvalue.- V.9 Examples of Stability Analysis at a Double Semi-Simple (Index-One) Eigenvalue.- V.10 Saddle-Node Bifurcation.- Appendix V.1 Implicit Function Theorem for a System of Two Equations in Two Unknown Functions of One Variable.- Exercises.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VI.1 The Evolution Equation and the Spectral Problem.- VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude.- VI.3 ?1 and ?1 in Projection.- VI.4 Stability of the Bifurcating Solution.- VI.5 The Extra Little Part for ?1 in Projection.- V1.6 Projections of Higher-Dimensional Problems.- VI.7 The Spectral Problem for the Stability of u = 0.- VI.8 The Spectral Problem and the Laplace Transform.- VI.9 Projections into ?1.- VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory).- VI.1 1 The Method of Projection at a Double Eigenvalue of Index Two.- VI.12 The Method of Projection at a Double Semi-Simple Eigenvalue.- VI.13 Examples of the Method of Projection.- VI.14 Symmetry and Pitchfork Bifurcation.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VII.1 The Structure of the Two-Dimensional Problem Governing Hopf Bifurcation.- VII.2 Amplitude Equation for Hopf Bifurcation.- VII.3 Series Solution.- VII.4 Equations Governing the Taylor Coefficients.- VII.5 Solvability Conditions (the Fredholm Alternative).- VII.6 Floquet Theory.- 6.1 Floquet Theory in ?1.- 6.2 Floquet Theory in ?2 and ?n.- VII.7 Equations Governing the Stability of the Periodic Solutions.- VII.8 The Factorization Theorem.- VII.9 Interpretation of the Stability Result.- Example.- VIII Bifurcation of Periodic Solutions in the General Case.- VIII.1 Eigenprojections of the Spectral Problem.- VIII.2 Equations Governing the Projection and the Complementary Projection.- VIII.3 The Series Solution Using the Fredholm Alternative.- VIII.4 Stability of the Hopf Bifurcation in the General Case.- VIII.5 Systems with Rotational Symmetry.- Examples.- Notes.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- Notation.- IX.1 Definition of the Problem of Subharmonic Bifurcation.- IX.2 Spectral Problems and the Eigenvalues ?( µ).- IX.3 Biorthogonality.- IX.4 Criticality.- IX.S The Fredholm Alternative for J( µ) —?( µ)and a Formula Expressing the Strict Crossing (IX.20).- IX.6 Spectral Assumptions.- IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality.- IX.8 The Operator $$\mathbb{J}$$ and its Eigenvectors.- IX.9 The Adjoint Operator $${{\mathbb{J}}^{*}}$$ Biorthogonality, Strict Crossing, and the Fredholm Alternative for $$\mathbb{J}$$.- IX.10 The Amplitude ?and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions.- IX.11 The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to ?at ? =0.- IX.12 Bifurcation and Stability of T-Periodic and 2 T-Periodic Solutions.- IX.13 Bifurcation and Stability of n T-Periodic Solutions with n >2.- IX.14 Bifurcation and Stability of 3T-Periodic Solutions.- IX.15 Bifurcation of 4 T-Periodic Solutions.- IX.16 Stability of 4 T-Periodic Solutions.- IX.17 Nonexistence of Higher-Order Subharmonic Solutions and Weak Resonance.- IX.18 Summary of Results About Subharmonic Bifurcation.- IX.19 Imperfection Theory with a Periodic Imperfection.- Exercises.- IX.20 Saddle-Node Bifurcation of T-Periodic Solutions.- IX.21 General Remarks About Subharmonic Bifurcations.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- X.1 Decomposition of the Solution and Amplitude Equation.- Exercise.- X.2 Derivation of the Amplitude Equation.- X.3 The Normal Equations in Polar Coordinates.- X.4 The Torus and Trajectories on the Torus in the Irrational Case.- X.5 The Torus and Trajectories on the Torus When ?0T/2? Is a Rational Point of Higher Order (n?5).- X.6 The Form of the Torus in the Case n =5.- X.7 Trajectories on the Torus When n =5.- X.8 The Form of the Torus When n >5.- X.9 Trajectories on the Torus When n?5.- X.10 Asymptotically Quasi-Periodic Solutions.- X.11 Stability of the Bifurcated Torus.- X.12 Subharmonic Solutions on the Torus.- X.13 Stability of Subharmonic Solutions on the Torus.- X.14 Frequency Locking.- Appendix X.1 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Irrational Points Using the Method of Two Times, Power Series, and the Fredholm Alternative.- Appendix X.2 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times.- Exercise.- Notes.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case.- Notation.- XI.1 Spectral Problems.- XI.2 Criticality and Rational Points.- XI.3 Spectral Assumptions About J0.- XI.4 Spectral Assumptions About $$\mathbb{J}$$ in the Rational Case.- XI.5 Strict Loss of Stability at a Simple Eigenvalue of J0.- XI.6 Strict Loss of Stability at a Double Semi-Simple Eigenvalue of J0.- XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two.- XI.8 Formulation of the Problem of Subharmonic Bifurcation of Periodic Solutions of Autonomous Problems.- XI.9 The Amplitude of the Bifurcating Solution.- XI.10 Power-Series Solutions of the Bifurcation Problem.- XI.11 Subharmonic Bifurcation When n =2.- XI.12 Subharmonic Bifurcation When n >2.- XI.13 Subharmonic Bifurcation When n = 1in the Semi-Simple Case.- XI.14 “Subharmonic” Bifurcation When n =1 in the Case When Zero is an Index-Two Double Eigenvalue of Jo.- XI.15 Stability of Subharmonic Solutions.- XI.16 Summary of Results About Subharmonic Bifurcation in the Autonomous Case.- XI.17 Amplitude Equations.- XI.18 Amplitude Equations for the Cases n?3 or ?0/?0Irrational.- XI.19 Bifurcating Tori. Asymptotically Quasi-Periodic Solutions.- XI.20 Period Doubling n =2.- XI.21 Pitchfork Bifurcation of Periodic Orbits in the Presence of Symmetry n = 1.- Exercises.- XI.22 Rotationally Symmetric Problems.- Exercise.- XII Stability and Bifurcation in Conservative Systems.- XII.1 The Rolling Ball.- XII.2 Euler Buckling.- Exercises.- XII.3 Some Remarks About Spectral Problems for Conservative Systems.- XII.4 Stability and Bifurcation of Rigid Rotation of Two Immiscible Liquids.- Steady Rigid Rotation of Two Fluids.

Read more less

Book SynopsisEquations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations.Trade ReviewJ.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations "This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples."—MATHEMATIKATable of ContentsI: Dimension One.- 1. Scalar Autonomous Equations.- 1.1. Existence and Uniqueness.- 1.2. Geometry of Flows.- 1.3. Stability of Equilibria.- 1.4. Equations on a Circle.- 2. Elementary Bifurcations.- 2.1. Dependence on Parameters - Examples.- 2.2. The Implicit Function Theorem.- 2.3. Local Perturbations Near Equilibria.- 2.4. An Example on a Circle.- 2.5. Computing Bifurcation Diagrams.- 2.6. Equivalence of Flows.- 3. Scalar Maps.- 3.1. Euler’s Algorithm and Maps.- 3.2. Geometry of Scalar Maps.- 3.3. Bifurcations of Monotone Maps.- 3.4. Period-doubling Bifurcation.- 3.5. An Example: The Logistic Map.- II: Dimension One and One Half.- 4. Scalar Nonautonomous Equations.- 4.1. General Properties of Solutions.- 4.2. Geometry of Periodic Equations.- 4.3. Periodic Equations on a Cylinder.- 4.4. Examples of Periodic Equations.- 4.5. Stability of Periodic Solutions.- 5. Bifurcation of Periodic Equations.- 5.1. Bifurcations of Poincaré Maps.- 5.2. Stability of Nonhyperbolic Periodic Solutions.- 5.3. Perturbations of Vector Fields.- 6. On Tori and Circles.- 6.1. Differential Equations on a Torus.- 6.2. Rotation Number.- 6.3. An Example: The Standard Circle Map.- III: Dimension Two.- 7. Planar Autonomous Systems.- 7.1. “Natural” Examples of Planar Systems.- 7.2. General Properties and Geometry.- 7.3. Product Systems.- 7.4. First Integrals and Conservative Systems.- 7.5. Examples of Elementary Bifurcations.- 8. Linear Systems.- 8.1. Properties of Solutions of Linear Systems.- 8.2. Reduction to Canonical Forms.- 8.3. Qualitative Equivalence in Linear Systems.- 8.4. Bifurcations in Linear Systems.- 8.5. Nonhomogeneous Linear Systems.- 8.6. Linear Systems with 1-periodic Coefficients.- 9. Near Equilibria.- 9.1. Asymptotic Stability from Linearization.- 9.2. Instability from Linearization.- 9.3. Liapunov Functions.- 9.4. An Invariance Principle.- 9.5. Preservation of a Saddle.- 9.6. Flow Equivalence Near Hyperbolic Equilibria.- 9.7. Saddle Connections.- 10. In the Presence of a Zero Eigenvalue.- 10.1. Stability.- 10.2. Bifurcations.- 10.3. Center Manifolds.- 11. In the Presence of Purely Imaginary Eigenvalues.- 11.1. Stability.- 11.2. Poincaré-Andronov-Hopf Bifurcation.- 11.3. Computing Bifurcation Curves.- 12. Periodic Orbits.- 12.1. Poincaré-Bendixson Theorem.- 12.2. Stability of Periodic Orbits.- 12.3. Local Bifurcations of Periodic Orbits.- 12.4. A Homoclinic Bifurcation.- 13. All Planar Things Considered.- 13.1. Structurally Stable Vector Fields.- 13.2. Dissipative Systems.- 13.3. One-parameter Generic Bifurcations.- 13.4. Bifurcations in the Presence of Symmetry.- 13.5. Local Two-parameter Bifurcations.- 14- Conservative and Gradient Systems.- 14.1. Second-order Conservative Systems.- 14.2. Bifurcations in Conservative Systems.- 14.3. Gradient Vector Fields.- 15. Planar Maps.- 15.1. Linear Maps.- 15.2. Near Fixed Points.- 15.3. Numerical Algorithms and Maps.- 15.4. Saddle Node and Period Doubling.- 15.5. Poincaré-Andronov-Hopf Bifurcation.- 15.6. Area-preserving Maps.- IV: Higher Dimensions.- 16. Dimension Two and One Half.- 16.1. Forced Van der Pol.- 16.2. Forced Duffing.- 16.3. Near a Transversal Homoclinic Point.- 16.4. Forced and Damped Duffing.- 17. Dimension Three.- 17.1. Period Doubling.- 17.2. Bifurcation to Invariant Torus.- 17.3. Silnikov Orbits.- 17.4. The Lorenz Equations.- 18. Dimension Four.- 18.1. Integrable Hamiltonians.- 18.2. A Nonintegrable Hamiltonian.- Farewell.- APPENDIX: A Catalogue of Fundamental Theorems.- References.

Read more less

Book SynopsisThis is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory.Table of Contents1 Geometry in Regions of a Space. Basic Concepts.- §1. Co-ordinate systems.- 1.1. Cartesian co-ordinates in a space.- 1.2. Co-ordinate changes.- §2. Euclidean space.- 2.1. Curves in Euclidean space.- 2.2. Quadratic forms and vectors.- §3. Riemannian and pseudo-Riemannian spaces.- 3.1. Riemannian metrics.- 3.2. The Minkowski metric.- §4. The simplest groups of transformations of Euclidean space.- 4.1. Groups of transformations of a region.- 4.2. Transformations of the plane.- 4.3. The isometries of 3-dimensional Euclidean space.- 4.4. Further examples of transformation groups.- 4.5. Exercises.- §5. The Serret—Frenet formulae.- 5.1. Curvature of curves in the Euclidean plane.- 5.2. Curves in Euclidean 3-space. Curvature and torsion.- 5.3. Orthogonal transformations depending on a parameter.- 5.4. Exercises.- §6. Pseudo-Euclidean spaces.- 6.1. The simplest concepts of the special theory of relativity.- 6.2. Lorentz transformations.- 6.3. Exercises.- 2 The Theory of Surfaces.- §7. Geometry on a surface in space.- 7.1. Co-ordinates on a surface.- 7.2. Tangent planes.- 7.3. The metric on a surface in Euclidean space.- 7.4. Surface area.- 7.5. Exercises.- §8. The second fundamental form.- 8.1. Curvature of curves on a surface in Euclidean space.- 8.2. Invariants of a pair of quadratic forms.- 8.3. Properties of the second fundamental form.- 8.4. Exercises.- §9. The metric on the sphere.- §10. Space-like surfaces in pseudo-Euclidean space.- 10.1. The pseudo-sphere.- 10.2. Curvature of space-like curves in $$ \mathbb{R}_1^3 $$.- §11. The language of complex numbers in geometry.- 11.1. Complex and real co-ordinates.- 11.2. The Hermitian scalar product.- 11.3. Examples of complex transformation groups.- §12. Analytic functions.- 12.1. Complex notation for the element of length, and for the differential of a function.- 12.2. Complex co-ordinate changes.- 12.3. Surfaces in complex space.- §13. The conformal form of the metric on a surface.- 13.1. Isothermal co-ordinates. Gaussian curvature in terms of conformal co-ordinates.- 13.2. Conformal form of the metrics on the sphere and the Lobachevskian plane.- 13.3. Surfaces of constant curvature.- 13.4. Exercises.- §14. Transformation groups as surfaces in N-dimensional space.- 14.1. Co-ordinates in a neighbourhood of the identity.- 14.2. The exponential function with matrix argument.- 14.3. The quaternions.- 14.4. Exercises.- §15. Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions.- 3 Tensors: The Algebraic Theory.- §16. Examples of tensors.- §17. The general definition of a tensor.- 17.1. The transformation rule for the components of a tensor of arbitrary rank.- 17.2. Algebraic operations on tensors.- 17.3. Exercises.- §18. Tensors of type (0, k).- 18.1. Differential notation for tensors with lower indices only.- 18.2. Skew-symmetric tensors of type (0, k).- 18.3. The exterior product of differential forms. The exterior algebra.- 18.4. Skew-symmetric tensors of type (k, 0) (polyvectors). Integrals with respect to anti-commuting variables.- 18.5. Exercises.- §19. Tensors in Riemannian and pseudo-Riemannian spaces.- 19.1. Raising and lowering indices.- 19.2. The eigenvalues of a quadratic form.- 19.3. The operator ?.- 19.4. Tensors in Euclidean space.- 19.5. Exercises.- §20. The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space. Examples of invariant tensors.- §21. Rank 2 tensors in pseudo-Euclidean space, and their eigenvalues.- 21.1. Skew-symmetric tensors. The invariants of an electromagnetic field.- 21.2. Symmetric tensors and their eigenvalues. The energy-momentum tensor of an electromagnetic field.- §22. The behaviour of tensors under mappings.- 22.1. The general operation of restriction of tensors with lower indices.- 22.2. Mappings of tangent spaces.- §23. Vector fields.- 23.1. One-parameter groups of diffeomorphisms.- 23.2. The exponential function of a vector field.- 23.3. The Lie derivative.- 23.4. Exercises.- §24. Lie algebras.- 24.1. Lie algebras and vector fields.- 24.2. The fundamental matrix Lie algebras.- 24.3. Linear vector fields.- 24.4. Left-invariant fields defined on transformation groups.- 24.5. Invariant metrics on a transformation group.- 24.6. The classification of the 3-dimensional Lie algebras.- 24.7. The Lie algebras of the conformal groups.- 24.8. Exercises.- 4 The Differential Calculus of Tensors.- §25. The differential calculus of skew-symmetric tensors.- 25.1. The gradient of a skew-symmetric tensor.- 25.2. The exterior derivative of a form.- 25.3. Exercises.- §26. Skew-symmetric tensors and the theory of integration.- 26.1. Integration of differential forms.- 26.2. Examples of integrals of differential forms.- 26.3. The general Stokes formula. Examples.- 26.4. Proof of the general Stokes formula for the cube.- 26.5. Exercises.- §27. Differential forms on complex spaces.- 27.1. The operators d? and d?.- 27.2. Kählerian metrics. The curvature form.- §28. Covariant differentiation.- 28.1. Euclidean connexions.- 28.2. Covariant differentiation of tensors of arbitrary rank.- §29. Covariant differentiation and the metric.- 29.1. Parallel transport of vector fields.- 29.2. Geodesics.- 29.3. Connexions compatible with the metric.- 29.4. Connexions compatible with a complex structure (Hermitian metric).- 29.5. Exercises.- §30. The curvature tensor.- 30.1. The general curvature tensor.- 30.2. The symmetries of the curvature tensor. The curvature tensor defined by the metric.- 30.3. Examples: The curvature tensor in spaces of dimensions 2 and 3; the curvature tensor of transformation groups.- 30.4. The Peterson—Codazzi equations. Surfaces of constant negative curvature, and the “sine—Gordon” equation.- 30.5. Exercises.- 5 The Elements of the Calculus of Variations.- §31. One-dimensional variational problems.- 31.1. The Euler—Lagrange equations.- 31.2. Basic examples of functional.- §32. Conservation laws.- 32.1. Groups of transformations preserving a given variational problem.- 32.2. Examples. Applications of the conservation laws.- §33. Hamiltonian formalism.- 33.1. Legendre’s transformation.- 33.2. Moving co-ordinate frames.- 33.3. The principles of Maupertuis and Fermat.- 33.4. Exercises.- §34. The geometrical theory of phase space.- 34.1. Gradient systems.- 34.2. The Poisson bracket.- 34.3. Canonical transformations.- 34.4. Exercises.- §35. Lagrange surfaces.- 35.1. Bundles of trajectories and the Hamilton—Jacobi equation.- 35.2. Hamiltonians which are first-order homogeneous with respect to the momentum.- §36. The second variation for the equation of the geodesics.- 36.1. The formula for the second variation.- 36.2. Conjugate points and the minimality condition.- 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants.- §37. The simplest higher-dimensional variational problems.- 37.1. The Euler—Lagrange equations.- 37.2. The energy-momentum tensor.- 37.3. The equations of an electromagnetic field.- 37.4. The equations of a gravitational field.- 37.5. Soap films.- 37.6. Equilibrium equation for a thin plate.- 37.7. Exercises.- §38. Examples of Lagrangians.- §39. The simplest concepts of the general theory of relativity.- §40. The spinor representations of the groups SO(3) and O(3, 1). Dirac’s equation and its properties.- 40.1. Automorphisms of matrix algebras.- 40.2. The spinor representation of the group SO(3).- 40.3. The spinor representation of the Lorentz group.- 40.4. Dirac’s equation.- 40.5. Dirac’s equation in an electromagnetic field. The operation of charge conjugation.- §41. Covariant differentiation of fields with arbitrary symmetry.- 41.1. Gauge transformations. Gauge-invariant Lagrangians.- 41.2. The curvature form.- 41.3. Basic examples.- §42. Examples of gauge-invariant functionals. Maxwell’s equations and the Yang—Mills equation. Functionals with identically zero variational derivative (characteristic classes).

Read more less

Book SynopsisI Fundamentals.- 1 Ordinary Differential Equations.- 2 Difference Equations.- II Local Analysis.- 3 Approximate Solution of Linear Differential Equations.- 4 Approximate Solution of Nonlinear Differential Equations.- 5 Approximate Solution of Difference Equations.- 6 Asymptotic Expansion of Integrals.- III Perturbation Methods.- 7 Perturbation Series.- 8 Summation of Series.- IV Global Analysis.- 9 Boundary Layer Theory.- 10 WKB Theory.- 11 Multiple-Scale Analysis.Trade Review"This book is a reprint of the original published by McGraw-Hill \ref [MR0538168 (80d:00030)]. The only changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic methods and perturbation theory". This latter improvement is much needed, as the original title suggested that this was a teaching book for undergraduate scientists and engineers. It is not, but is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics. This is a book that has stood the test of time and I cannot but endorse the remarks of the original reviewer. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. I have owned a copy of the original for over twenty years, using it on a regular basis, and, after the original had gone out of print, lending it to my research students. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic." (A.D. Wood, Mathematical Reviews) Table of ContentsI Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index

Read more less

Book SynopsisEquations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case.Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.Table of ContentsSome Preliminaries from Analysis and the Theory of Wave Processes. Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium. Wave Fields in a Layer of Constant Thickness. Analytical Methods for Simply Connected Bounded Domains. Integral Equations in Diffraction by Linear Obstacles. Short-Wave Asymptotic Methods on the Basis of Multiple Integrals. Inverse Problems of the Short-Wave Diffraction. Ill-Posed Equations of Inverse Diffraction Problems for Arbitrary Boundary. Numerical Methods for Irregular Operator Equations.

Read more less

Book Synopsis

Read more less

Book SynopsisFourier transform theory is of central importance in a vast range of applications in physical science, engineering and applied mathematics. Providing a concise introduction to the theory and practice of Fourier transforms, this book is invaluable to students of physics, electrical and electronic engineering, and computer science. After a brief description of the basic ideas and theorems, the power of the technique is illustrated through applications in optics, spectroscopy, electronics and telecommunications. The rarely discussed but important field of multi-dimensional Fourier theory is covered, including a description of Computer Axial Tomography (CAT scanning). The book concludes by discussing digital methods, with particular attention to the Fast Fourier Transform and its implementation. This new edition has been revised to include new and interesting material, such as convolution with a sinusoid, coherence, the Michelson stellar interferometer and the van CittertâZernike theorem, Trade ReviewFrom previous editions: 'It is the wide range of topics that makes this book so appealing … I highly recommend this book for the advanced student … Even the expert who wants a deeper appreciation of the Fourier transform will find the book useful.' Computers in Physics'… this is an excellent book to initiate students who possess a reasonable mathematical background to the use of Fourier transforms …' Microscopy and AnalysisTable of Contents1. Physics and Fourier transforms; 2. Useful properties and theorems; 3. Applications 1: Fraunhofer diffraction; 4. Applications 2: signal analysis and communication theory; 5. Applications 3: spectroscopy and spectral line shapes; 6. Two-dimensional Fourier transforms; 7. Multi-dimensional Fourier transforms; 8. The formal complex Fourier transform; 9. Discrete and digital Fourier transforms; 10. Appendix; 11. Bibliography; 12. Index.

Read more less

Book SynopsisThis textbook is an accessible introduction to stochastic processes and their applications, as well as methods for numerical simulation, for graduate students and researchers in physics. It includes coverage of the more exotic Levy processes, and a concise account of numerical methods for simulating stochastic systems driven by Gaussian noise.Trade Review'Jacobs is an enthusiastic, clear, and concise writer. He presents each theory by means of heuristic arguments and calculations.' Cosma Shalizi, Physics Today'I think this book is a very nice introduction to the subject of stochastic processes.' Zentralblatt MATHTable of Contents1. A review of probability theory; 2. Differential equations; 3. Stochastic equations with Gaussian noise; 4. Further properties of stochastic processes; 5. Some applications of Gaussian noise; 6. Numerical methods for Gaussian noise; 7. Fokker–Planck equations and reaction-diffusion systems; 8. Jump processes; 9. Levy processes; 10. Modern probability theory; Appendix; References; Index.

Read more less

Book SynopsisSelf-contained, comprehensive, and consistent, this definitive new edition is a unique resource for graduate students and researchers in theoretical physics. This second edition contains over 300 pages of new material, covers an extensive array of topics, and is accompanied by an exhaustive index and bibliography. An exceptional reference work.Table of Contents1. Differential geometry; 2. Symmetries and Noether's theorems; 3. A perturbative introduction to general relativity; 4. Action principles for gravity; 5. Pure N=1,2,d=4 supergravities; 6. Matter-coupled N=1,d=4 supergravity; 7. Matter-coupled N=2,d=4 supergravity; 8. A generic description of all the N>2,d=4 SUEGRAS; 9. Matter-coupled N=1,d=5 supergravity; 10. Conserved charges in general relativity; 11. The Schwarzschild black hole; 12. The Reissner–Nordström black hole; 13. The Taub–NUT solution; 14. Gravitational pp-waves; 15. The Kaluza–Klein black hole; 16. Dilaton and dilaton/axion black holes; 17. Unbroken supersymmetry I: supersymmetric vacua; 18. Unbroken supersymmetry II: partially supersymmetric solutions; 19. Supersymmetric black holes from supergravity; 20. String theory; 21. The string effective action and T duality; 22. From eleven to four dimensions; 23. The type-IIB superstring and type-II T duality; 24. Extended objects; 25. The extended objects of string theory; 26. String black holes in four and five dimensions; 27. The FGK formalism for (single, static) black holes and branes; Appendices: A.1 Lie groups, symmetric spaces, and Yang–Mills fields; A.2 The irreducible, non-symmetric Riemannian spaces of special holonomy; A.3 Miscellanea on the symplectic group; A.4 Gamma matrices and spinors; A.5 Kähler geometry; A.6 Special Kähler geometry; A.7 Quaternionic-Kähler geometry.

Read more less

Book SynopsisThis textbook provides a comprehensive and pedagogical introduction to supersymmetry and other aspects of particle physics at the high-energy frontier. Aimed at graduate students and researchers, it also discusses concepts of physics beyond the Standard Model, including extended Higgs sectors, grand unification, and the origin of neutrino masses.Table of ContentsPreface; Acknowledgements; Acronyms and abbreviations; Part I. Spin-1/2 Fermions in Quantum Field Theory, the Standard Model, and Beyond: 1. Two-component formalism for spin-1/2 fermions; 2. Feynman rules for spin-1/2 fermions; 3. From two-component to four-component spinors; 4. Gauge theories and the standard model; 5. Anomalies; 6. Extending the standard model; Part II. Constructing Supersymmetric Theories: 7. Introduction to supersymmetry; 8. Supersymmetric Lagrangians; 9. The supersymmetric algebra; 10. Superfields; 11. Radiative corrections in supersymmetry; 12. Spontaneous supersymmetry breaking; Part III. Realistic Supersymmetric Models: 13. The Minimal Supersymmetric Standard Model; 14. Realizations of supersymmetry breaking; 15. Supersymmetric phenomenology; 16. Beyond the MSSM; Part IV. Sample Calculations in the Standard Model and Its Supersymmetric Extension: 17. Practical calculations involving two-component fermions; 18. Tree-level supersymmetric processes; 19. One-loop calculations; Part V. The Appendices: Appendix A. Notations and conventions; Appendix B. Compendium of sigma matrix and Fierz identities; Appendix C. Behavior of fermion bilinears under C, P, T; Appendix D. Kinematics and phase space; Appendix E. The spin-1/2 and spin-1 wave functions; Appendix F. The spinor helicity method; Appendix G. Matrix decompositions for fermion mass diagonalization; Appendix H. Lie group and algebra techniques for gauge theories; Appendix I. Interaction vertices of the SM and its seesaw extension; Appendix J. MSSM and RPV fermion interaction vertices; Appendix K. Integrals arising in one-loop calculations; Bibliography; References; Index.

Read more less

Book SynopsisSuitable for graduate students, this book presents detailed derivations of cosmological and black hole processes in which curved spacetime plays a key role. It explains how such processes in the early universe leave observable consequences today, and how these processes uncover deep connections between gravitation and elementary particles.Trade Review"While readers of Birrel and Davies will certainly like this new book, newcomers and students will appreciate the breadth and the style of a treatise written by two well known scientists who have dedicated their lives to the understanding of the treatment of quantum fields in a fixed gravitational background." Massimo Giovannini, CERN Courier"This is an interesting book which contains a lot of material about an important topic of theoretical physics." Michael Keyl, Mathematical ReviewsTable of ContentsPreface; Conventions and notation; 1. Quantum fields in Minkowski spacetime; 2. Basics of quantum fields in curved spacetimes; 3. Expectation values quadratic in fields; 4. Particle creation by black holes; 5. The one-loop effective action; 6. The effective action: non-gauge theories; 7. The effective action: gauge theories; Appendixes; References; Index.

Read more less

Book SynopsisIntended for graduates and researchers, this textbook, first published in 2007, provides a simple introduction to supersymmetry. This elementary, practical treatment places emphasis on physical understanding, and detailed derivations. Many short exercises are included, making for a valuable and accessible self-study tool.Trade Review'Any student or practising physicist who wants to learn about the phenomenological implications of TeV-scale supersymmetry without spending the time to master the more mathematical approaches to the subject would do well to acquire a copy of Supersymmetry in Particle Physics. It is a unique text that has value both on its own accord and as a supplement to larger and more comprehensive texts. This is indeed entry-level supersymmetry in its best and most practical sense.' Physics TodayTable of Contents1. Introduction and motivation; 2. Spinors: Weyl, Dirac and Majorana; 3. A simple supersymmetric Lagrangian, and a first glance at the MSSM; 4. The supersymmetry algebra and supermultiplets; 5. The Wess-Zumino model; 6. Superfields; 7. Vector (or gauge) supermultiplets; 8. The MSSM; 9. SUSY breaking; 10. The Higgs sector and electroweak symmetry breaking in the MSSM; 11. Sparticle masses in the MSSM; 12. Some simple tree-level calculations in the MSSM; References; Index.

Read more less

Book SynopsisThe first book containing a rigorous construction and uniqueness proof for the largest and most famous sporadic simple group, the Monster. The author provides a systematic exposition of the theory of the Monster group, which so far remains largely unpublished, and explores the theory of groups generated by Majorana involutions.Trade Review'This book contains the basic knowledge on the Monster group in a very accessible way. some results are published in this book for the first time. Many are not even easily found in literature. Hence the book is a very good source for any group theorist who is interested in sporadic simple groups.' Zentralblatt MATHTable of ContentsPreface; 1. M24 and all that; 2. The Monster amalgam M; 3. 196 883-representation of M; 4. 2-local geometries; 5. Griess algebra; 6. Automorphisms of Griess algebra; 7. Important subgroups; 8. Majorana involutions; 9. The Monster graph; 10. Fischer's story; References; Index.

Read more less

Book SynopsisThis book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson''s work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson'Table of ContentsThe infinitesimal rule of threeNonstandard methods in the precalculus curruculumDifference quotients and smoothnessContinuous maps with special propertiesSome nonstandard methods in geometric topologyDelayed bifurcations in perturbed systems analysis of slow passage of Suhl-thresholdFunctional analysis and NSANear-standard compact internal linear operatorsDiscrete Fredholm's equationsNonstandard theory of generalized functionsRepresenting distributions by nonstandard polynomialsContributions of nonstandard analysis to partial differential equationsLoeb measure theoryUnions of Loeb nullsets: the contextGredient lines and distributions of functionals in infinite dimensional Euclidean spacesNonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interactionNonstandard analysis in selective uniersesLattices and monadsA neometric surveyLong sequences and neocompact sets

Read more less

Book SynopsisScientists are using simulations to recreate the universe, revealing the hidden nature of reality.Cosmology is a tricky science—no one can make their own stars, planets, or galaxies to test its theories. But over the last few decades a new kind of physics has emerged to fill the gap between theory and experimentation. Harnessing the power of modern supercomputers, cosmologists have built simulations that offer profound insights into the deep history of our universe, allowing centuries-old ideas to be tested for the first time. Today, physicists are translating their ideas and equations into code, finding that there is just as much to be learned from computers as experiments in laboratories.    In The Universe in a Box, cosmologist Andrew Pontzen explains how physicists model the universe’s most exotic phenomena, from black holes and colliding galaxies to dark matter and quantum entanglement, enabling them to study the

Read more less

Book SynopsisTable of Contents*Frontmatter, pg. i*Acknowledgements, pg. v*Abstract, pg. vii*tables of contents, pg. ix*Introductions, pg. 1*CHAPTER I. Renormalization in Lagrangian Field Theory, pg. 5*CHAPTER II. Definition of Generalized Amplitudes, pg. 43*CHAPTER III. Analytic Renormalization, pg. 61*CHAPTER IV. Summation of Feynman Amplitudes, pg. 81*CONCLUSION, pg. 95*APPENDIX A. Graphs, pg. 97*APPENDIX B. Distributions, pg. 103*APPENDIX C. The Free Field, pg. 109*BIBLIOGRAPHY, pg. 119

Read more less

Book SynopsisFocuses on heat equations associated with non self-adjoint uniformly elliptic operators. This book provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. It then treats Lp properties of solutions to a wide class of heat equations.Trade Review"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"Table of ContentsPreface ix Notation xiii Chapter 1. SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS 1 1.1 Bounded sesquilinear forms 1 1.2 Unbounded sesquilinear forms and their associated operators 3 1.3 Semigroups and unbounded operators 18 1.4 Semigroups associated with sesquilinear forms 29 1.5 Correspondence between forms, operators, and semigroups 38 Chapter 2. CONTRACTIVITY PROPERTIES 43 2.1 Invariance of closed convex sets 44 2.2 Positive and Lp-contractive semigroups 49 2.3 Domination of semigroups 58 2.4 Operations on the form-domain 64 2.5 Semigroups acting on vector-valued functions 68 2.6 Sesquilinear forms with nondense domains 74 Chapter 3. INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS 79 3.1 Sub-Markovian semigroups and Kato type inequalities 79 3.2 Further inequalities and the corresponding domain in Lp 88 3.3 Lp-holomorphy of sub-Markovian semigroups 95 Chapter 4. UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS 99 4.1 Examples of boundary conditions 99 4.2 Positivity and irreducibility 103 4.3 L1-contractivity 107 4.4 The conservation property 120 4.5 Domination 125 4.6 Lp-contractivity for 1 134 4.7 Operators with unbounded coefficients 137 Chapter 5. DEGENERATE-ELLIPTIC OPERATORS 143 5.1 Symmetric degenerate-elliptic operators 144 5.2 Operators with terms of order 1 145 Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS 155 6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities 155 6.2 Holder-continuity estimates of the heat kernel 160 6.3 Gaussian upper bounds 163 6.4 Sharper Gaussian upper bounds 174 6.5 Gaussian bounds for complex time and Lp-analyticity 180 6.6 Weighted gradient estimates 185 Chapter 7. GAUSSIAN UPPER BOUNDS AND Lp-SPECTRAL THEORY 193 7.1 Lp-bounds and holomorphy 196 7.2 Lp-spectral independence 204 7.3 Riesz means and regularization of the Schrodinger group 208 7.4 Lp-estimates for wave equations 214 7.5 Singular integral operators on irregular domains 228 7.6 Spectral multipliers 235 7.7 Riesz transforms associated with uniformly elliptic operators 240 7.8 Gaussian lower bounds 245 Chapter 8. A REVIEW OF THE KATO SQUARE ROOT PROBLEM 253 8.1 The problem in the abstract setting 253 8.2 The Kato square root problem for elliptic operators 257 8.3 Some consequences 261 Bibliography 265 Index 283

Read more less

Book SynopsisPresents an overview of the developments in the area of localization for quasi-periodic lattice Schrodinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. This book emphasises on so-called 'non-perturbative' methods and the role of subharmonic function theory and semi-algebraic set methods.Trade Review"This text is an up to date introduction to localization problems for lattice Schrodinger operations with deterministic ergodic potentials by one of the leading experts... I can recommend it to any graduate student or researcher in the field."--G. Teschl, Monatschefte fur MathematikTable of ContentsAcknowledgment v CHAPTER 1: Introduction 1 CHAPTER 2: Transfer Matrix and Lyapounov Exponent 11 CHAPTER 3: Herman's Subharmonicity Method 15 CHAPTER 4: Estimates on Subharmonic Functions 19 CHAPTER 5: LDT for Shift Model 25 CHAPTER 6: Avalanche Principle in SL2( R ) 29 CHAPTER 7: Consequences for Lyapounov Exponent, IDS, and Green's Function 31 CHAPTER 8: Refinements 39 CHAPTER 9: Some Facts about Semialgebraic Sets 49 CHAPTER 10: Localization 55 CHAPTER 11: Generalization to Certain Long-Range Models 65 CHAPTER 12: Lyapounov Exponent and Spectrum 75 CHAPTER 13: Point Spectrum in Multifrequency Models at Small Disorder 87 CHAPTER 14: A Matrix-Valued Cartan-Type Theorem 97 CHAPTER 15: Application to Jacobi Matrices Associated with Skew Shifts 105 CHAPTER 16: Application to the Kicked Rotor Problem 117 CHAPTER 17: Quasi-Periodic Localization on the Z d -lattice ( d > 1) 123 CHAPTER 18: An Approach to Melnikov's Theorem on Persistency of Non-resonant Lower Dimension Tori 133 CHAPTER 19: Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrodinger Equations 143 CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159 Appendix 169

Read more less

Book SynopsisProvides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. This book describes how to create codes that simulate the internal dynamics of planets and stars.Trade Review"This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves and magnetic field generation in the interiors and atmospheres of rotating planets and stars. It is very useful for readers having a basic understanding of classical physics, vector calculus, partial differential equations, and simple computer programming."--Claudia-Veronika Meister, Zentralblatt MATHTable of ContentsPreface xi PART I. THE FUNDAMENTALS 1 Chapter 1 A Model of Rayleigh-Benard Convection 3 1.1 Basic Theory 3 1.2 Boussinesq Equations 10 1.3 Model Description 13 Supplemental Reading 15 Exercises 15 Chapter 2 Numerical Method 17 2.1 Vorticity-Streamfunction Formulation 17 2.2 Horizontal Spectral Decomposition 19 2.3 Vertical Finite-Difference Method 21 2.4 Time Integration Scheme 22 2.5 Poisson Solver 24 Supplemental Reading 25 Exercises 25 Chapter 3 Linear Stability Analysis 27 3.1 Linear Equations 27 3.2 Linear Code 29 3.3 Critical Rayleigh Number 30 3.4 Analytic Solutions 31 Supplemental Reading 34 Exercises 34 Computational Projects 34 Chapter 4 Nonlinear Finite-Amplitude Dynamics 35 4.1 Modifications to the Linear Model 35 4.2 A Galerkin Method 36 4.3 Nonlinear Code 38 4.4 Nonlinear Simulations 43 Supplemental Reading 48 Exercises 49 Computational Projects 49 Chapter 5 Postprocessing 51 5.1 Computing and Storing Results 51 5.2 Displaying Results 51 5.3 Analyzing Results 54 Supplemental Reading 57 Exercises 57 Computational Projects 57 Chapter 6 Internal Gravity Waves 59 6.1 Linear Dispersion Relation 59 6.2 Code Modifications and Simulations 62 6.3 Wave Energy Analysis 66 Supplemental Reading 66 Exercises 67 Computational Projects 67 Chapter 7 Double-Diffusive Convection 68 7.1 Salt-Fingering Instability 69 7.2 Semiconvection Instability 72 7.3 Oscillating Instabilities 74 7.4 Staircase Profiles 76 7.5 Double-Diffusive Nonlinear Simulations 79 Supplemental Reading 80 Exercises 80 Computational Projects 80 PART II. ADDITIONAL NUMERICAL METHODS 83 Chapter 8 Time Integration Schemes 85 8.1 Fourth-Order Runge-Kutta Scheme 85 8.2 Semi-Implicit Scheme 87 8.3 Predictor-Corrector Schemes 89 8.4 Infinite Prandtl Number: Mantle Convection 91 Supplemental Reading 92 Exercises 93 Computational Projects 93 Chapter 9 Spatial Discretizations 95 9.1 Nonuniform Grid 95 9.2 Coordinate Mapping 97 9.3 Fully Finite Difference 98 9.4 Fully Spectral: Chebyshev-Fourier 102 9.5 Parallel Processing 108 Supplemental Reading 112 Exercises 112 Computational Projects 112 Chapter 10 Boundaries and Geometries 115 10.1 Absorbing Top and Bottom Boundaries 115 10.2 Permeable Periodic Side Boundaries 117 10.3 2D Annulus Geometry 122 10.4 Spectral-Transform Method 130 10.5 3D and 2.5D Cartesian Box Geometry 133 10.6 3D and 2.5D Spherical-Shell Geometry 135 Supplemental Reading 162 Exercises 162 Computational Projects 164 PART III. ADDITIONAL PHYSICS 167 Chapter 11 Magnetic Field 169 11.1 Magnetohydrodynamics 170 11.2 Magnetoconvection with a Vertical Background Field 173 11.3 Linear Analyses: Magnetic 179 11.4 Nonlinear Simulations: Magnetic 182 11.5 Magnetoconvection with a Horizontal Background Field 184 11.6 Magnetoconvection with an Arbitrary Background Field 187 Supplemental Reading 189 Exercises 190 Computational Projects 191 Chapter 12 Density Stratification 193 12.1 Anelastic Approximation 194 12.2 Reference State: Polytropes 207 12.3 Numerical Method: Anelastic 214 12.4 Linear Analyses: Anelastic 219 12.5 Nonlinear Simulations: Anelastic 222 Supplemental Reading 227 Exercises 227 Computational Projects 228 Chapter 13 Rotation 229 13.1 Coriolis, Centrifugal, and Poincare Forces 229 13.2 2D Rotating Equatorial Box 233 13.3 2D Rotating Equatorial Annulus: Differential Rotation 241 13.4 2.5D Rotating Spherical Shell: Inertial Oscillations 247 13.5 3D Rotating Spherical Shell: Dynamo Benchmarks 259 13.6 3D Rotating Spherical Shell: Dynamo Simulations 264 13.7 Concluding Remarks 275 Supplemental Reading 277 Exercises 278 Computational Projects 279 Appendix A A Tridiagonal Matrix Solver 283 Appendix B Making Computer-Graphical Movies 284 Appendix C Legendre Functions and Gaussian Quadrature 288 Appendix D Parallel Processing: OpenMP 291 Appendix E Parallel Processing: MPI 292 Bibliography 295 Index 307

Read more less

Book SynopsisProvides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. This book describes how to create codes that simulate the internal dynamics of planets and stars.Trade Review"This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves and magnetic field generation in the interiors and atmospheres of rotating planets and stars. It is very useful for readers having a basic understanding of classical physics, vector calculus, partial differential equations, and simple computer programming."--Claudia-Veronika Meister, Zentralblatt MATHTable of ContentsPreface xi PART I. THE FUNDAMENTALS 1 Chapter 1 A Model of Rayleigh-Benard Convection 3 1.1 Basic Theory 3 1.2 Boussinesq Equations 10 1.3 Model Description 13 Supplemental Reading 15 Exercises 15 Chapter 2 Numerical Method 17 2.1 Vorticity-Streamfunction Formulation 17 2.2 Horizontal Spectral Decomposition 19 2.3 Vertical Finite-Difference Method 21 2.4 Time Integration Scheme 22 2.5 Poisson Solver 24 Supplemental Reading 25 Exercises 25 Chapter 3 Linear Stability Analysis 27 3.1 Linear Equations 27 3.2 Linear Code 29 3.3 Critical Rayleigh Number 30 3.4 Analytic Solutions 31 Supplemental Reading 34 Exercises 34 Computational Projects 34 Chapter 4 Nonlinear Finite-Amplitude Dynamics 35 4.1 Modifications to the Linear Model 35 4.2 A Galerkin Method 36 4.3 Nonlinear Code 38 4.4 Nonlinear Simulations 43 Supplemental Reading 48 Exercises 49 Computational Projects 49 Chapter 5 Postprocessing 51 5.1 Computing and Storing Results 51 5.2 Displaying Results 51 5.3 Analyzing Results 54 Supplemental Reading 57 Exercises 57 Computational Projects 57 Chapter 6 Internal Gravity Waves 59 6.1 Linear Dispersion Relation 59 6.2 Code Modifications and Simulations 62 6.3 Wave Energy Analysis 66 Supplemental Reading 66 Exercises 67 Computational Projects 67 Chapter 7 Double-Diffusive Convection 68 7.1 Salt-Fingering Instability 69 7.2 Semiconvection Instability 72 7.3 Oscillating Instabilities 74 7.4 Staircase Profiles 76 7.5 Double-Diffusive Nonlinear Simulations 79 Supplemental Reading 80 Exercises 80 Computational Projects 80 PART II. ADDITIONAL NUMERICAL METHODS 83 Chapter 8 Time Integration Schemes 85 8.1 Fourth-Order Runge-Kutta Scheme 85 8.2 Semi-Implicit Scheme 87 8.3 Predictor-Corrector Schemes 89 8.4 Infinite Prandtl Number: Mantle Convection 91 Supplemental Reading 92 Exercises 93 Computational Projects 93 Chapter 9 Spatial Discretizations 95 9.1 Nonuniform Grid 95 9.2 Coordinate Mapping 97 9.3 Fully Finite Difference 98 9.4 Fully Spectral: Chebyshev-Fourier 102 9.5 Parallel Processing 108 Supplemental Reading 112 Exercises 112 Computational Projects 112 Chapter 10 Boundaries and Geometries 115 10.1 Absorbing Top and Bottom Boundaries 115 10.2 Permeable Periodic Side Boundaries 117 10.3 2D Annulus Geometry 122 10.4 Spectral-Transform Method 130 10.5 3D and 2.5D Cartesian Box Geometry 133 10.6 3D and 2.5D Spherical-Shell Geometry 135 Supplemental Reading 162 Exercises 162 Computational Projects 164 PART III. ADDITIONAL PHYSICS 167 Chapter 11 Magnetic Field 169 11.1 Magnetohydrodynamics 170 11.2 Magnetoconvection with a Vertical Background Field 173 11.3 Linear Analyses: Magnetic 179 11.4 Nonlinear Simulations: Magnetic 182 11.5 Magnetoconvection with a Horizontal Background Field 184 11.6 Magnetoconvection with an Arbitrary Background Field 187 Supplemental Reading 189 Exercises 190 Computational Projects 191 Chapter 12 Density Stratification 193 12.1 Anelastic Approximation 194 12.2 Reference State: Polytropes 207 12.3 Numerical Method: Anelastic 214 12.4 Linear Analyses: Anelastic 219 12.5 Nonlinear Simulations: Anelastic 222 Supplemental Reading 227 Exercises 227 Computational Projects 228 Chapter 13 Rotation 229 13.1 Coriolis, Centrifugal, and Poincare Forces 229 13.2 2D Rotating Equatorial Box 233 13.3 2D Rotating Equatorial Annulus: Differential Rotation 241 13.4 2.5D Rotating Spherical Shell: Inertial Oscillations 247 13.5 3D Rotating Spherical Shell: Dynamo Benchmarks 259 13.6 3D Rotating Spherical Shell: Dynamo Simulations 264 13.7 Concluding Remarks 275 Supplemental Reading 277 Exercises 278 Computational Projects 279 Appendix A A Tridiagonal Matrix Solver 283 Appendix B Making Computer-Graphical Movies 284 Appendix C Legendre Functions and Gaussian Quadrature 288 Appendix D Parallel Processing: OpenMP 291 Appendix E Parallel Processing: MPI 292 Bibliography 295 Index 307

Read more less

Book SynopsisTrade Review"A tour de force of the mathematical description of waves. . . . I sincerely wish I had encountered such a book early in my teaching career. The material presented in it would have provided a very useful enhancement to a number of courses I have taught to undergraduate physics majors over the years."---James A. Lock, American Journal of Physics

Read more less

Book Synopsis

Read more less

Book SynopsisThis book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann''s conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation lawsPDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlin

Read more less

Book SynopsisThis book provides an in-depth and self-contained treatment of the latest advances achieved in quantitative spectroscopic analyses of the observable outer layers of stars and similar objects. Written by two leading researchers in the field, it presents a comprehensive account of both the physical foundations and numerical methods of such analyses.Trade Review"It is an excellent guide for anyone interested in radiation transport and spectral analyses in astrophysics."--Claudia-Veronika Meister, Zentralblatt MATH "A magisterial work that will surely be the definitive reference for many years to come."--Ian D. Howarth, The ObservatoryTable of ContentsPreface xi Chapter 1. Why Study Stellar Atmospheres? 1 1.1 A Historical Precis 1 1.2 The Bottom Line 15 Chapter 2. Observational Foundations 20 2.1 What Is a Stellar Atmosphere? 20 2.2 Spectroscopy 23 2.3 Spectrophotometry 29 2.4 Photometry 32 2.5 Mass, Luminosity, and Radius 46 2.6 Interpretation of Color-Magnitude Diagrams 53 Chapter 3. Radiation 61 3.1 Specific Intensity 61 3.2 Mean Intensity and Energy Density 65 3.3 Radiation Flux 72 3.4 Radiation Pressure Tensor 75 3.5 * Transformation Properties of I, E, F, P 78 3.6 Quantum Theory of Radiation in Vacuum 80 Chapter 4. Statistical Mechanics of Matter and Radiation 86 4.1 Thermodynamic Equilibrium 86 4.2 Boltzmann Statistics 88 4.3 Thermal Radiation 98 4.4 Quantum Statistics 103 4.5 Local Thermodynamic Equilibrium 111 Chapter 5. Absorption and Emission of Radiation 113 5.1 Absorption and Thermal Emission 114 5.2 Detailed Balance 116 5.3 Bound-Bound Absorption Probability 121 5.4 Bound-Bound Emission Probability 130 5.5 Photoionization 136 5.6 Free-Free Transitions 137 Chapter 6. Continuum Scattering 144 6.1 Thomson Scattering: Classical Analysis 145 6.2 Thomson Scattering: Quantum Mechanical Analysis 150 6.3 * Rayleigh and Raman Scattering 153 6.4 Compton Scattering 159 6.5 Compton Scattering in the Early Universe 165 Chapter 7. Atomic and Molecular Absorption Cross Sections 170 7.1 Hydrogen and Hydrogenic Ions 171 7.2 Multi-Electron Atoms 192 7.3 Molecules 208 Chapter 8. Spectral Line Broadening 228 8.1 Natural Damping Profile 228 8.2 Doppler Broadening: Voigt Function 231 8.3 Semiclassical Impact Theory 233 8.4 Statistical Theory: Quasi-Static Approximation 241 8.5 * Quantum Theory of Line Broadening 248 8.6 Applications 258 Chapter 9. Kinetic Equilibrium Equations 262 9.1 LTE versus Non-LTE 262 9.2 General Formulation 264 9.3 Transition Rates 267 9.4 Level Dissolution and Occupation Probabilities 278 9.5 Complete Rate Equations 282 Chapter 10. Scattering of Radiation in Spectral Lines 290 10.1 Semiclassical (Weisskopf-Woolley) Picture 291 10.2 * Quantum Mechanical Derivation of Redistribution Functions 301 10.3 Basic Redistribution Functions 308 10.4 More Complex Redistribution Functions 321 10.5 Emission Coefficient 327 Chapter 11. Radiative Transfer Equation 334 11.1 Absorption, Emission, and Scattering Coefficients 334 11.2 Formulation 339 11.3 Moments of the Transfer Equation 347 11.4 Time-Independent, Static, Planar Atmospheres 352 11.5 Schwarzschild-Milne Equations 361 11.6 Second-Order Form of the Transfer Equation 367 11.7 Discretization 370 11.8 Probabilistic Interpretation 373 11.9 Diffusion Limit 374 Chapter 12. Direct Solution of the Transfer Equation 378 12.1 The Problem of Scattering 379 12.2 Feautrier's Method 387 12.3 Rybicki's Method 397 12.4 Formal Solution 400 12.5 Variable Eddington Factors 418 Chapter 13. Iterative Solution of the Transfer Equation 421 13.1 Accelerated Lambda Iteration: A Heuristic View 421 13.2 Iteration Methods and Convergence Properties 425 13.3 Accelerated Lambda Iteration (ALI) 434 13.4 Acceleration of Convergence 440 13.5 Astrophysical Implementation 443 Chapter 14. NLTE Two-Level and Multi-Level Atoms 448 14.1 Formulation 448 14.2 Two-Level Atom 457 14.3 Approximate Solutions 471 14.4 Equivalent-Two-Level-Atom Approach 482 14.5 Numerical Solution of the Multi-level Atom Problem 488 14.6 Physical Interpretation 505 Chapter 15. Radiative Transfer with Partial Redistribution 511 15.1 Formulation 511 15.2 Simple Heuristic Model 515 15.3 Approximate Solutions 519 15.4 Exact Solutions 524 15.5 Multi-level Atoms 533 15.6 Applications 539 Chapter 16. Structural Equations 546 16.1 Equations of Hydrodynamics 546 16.2 1D Flow 554 16.3 1D Steady Flow 555 16.4 StaticAtmospheres 557 16.5 Convection 558 16.6 Stellar Interiors 565 Chapter 17. LTE Model Atmospheres 569 17.1 Gray Atmosphere 569 17.2 Equation of State 588 17.3 Non-Gray LTE Radiative-Equilibrium Models 593 17.4 Models with Convection 604 17.5 LTE Spectral Line Formation 606 17.6 Line Blanketing 620 17.7 Models with External Irradiation 627 17.8 Available Modeling Codes and Grids 631 Chapter 18. Non-LTE Model Atmospheres 633 18.1 Overview of Basic Equations 633 18.2 Complete Linearization 645 18.3 Overview of Possible Iterative Methods 660 18.4 Application of ALI and Related Methods 667 18.5 NLTE Metal Line Blanketing 676 18.6 Applications: Modeling Codes and Grids 684 Chapter 19. Extended and Expanding Atmospheres 691 19.1 Extended Atmospheres 691 19.2 Moving Atmospheres: Observer's-Frame Formulation 705 19.3 Moving Atmospheres: Comoving-Frame Formulation 713 19.4 Moving Atmospheres: Mixed-Frame Formulation 736 19.5 Sobolev Approximation 743 19.6 NLTE Line Formation 754 Chapter 20. Stellar Winds 764 20.1 Qualitative Picture 765 20.2 Thermally DrivenWinds 766 20.3 Radiation-Driven Winds 772 20.4 Global Model Atmospheres 800 Appendix A. Relativistic Particles 815 A.1 Kinematics and Dynamics of Point Particles 815 A.2 Relativistic Kinetic Theory 822 Appendix B. Photons 829 B.1 Lorentz Transformation of the Photon Four-Momentum 829 B.2 Photon Distribution Function 830 B.3 Thomas Transformations 831 Glossary of Symbols 833 Bibliography 849 Index 915

Read more less

Book SynopsisThis book presents an authoritative and in-depth treatment of potential energy landscape theory, a powerful analytical approach to describing the atomic and molecular interactions in condensed-matter phenomena. Drawing on the latest developments in the computational modeling of many-body systems, Frank Stillinger applies this approach to a diverseTrade Review"Remarkably comprehensive, clearly presented, and rich with examples. The scope of topics is encyclopedic, taking readers from broad classes to paradigmatic specifics such as helium and water. The book is quite different from and complementary to David Wales's book, Energy Landscapes."—R. Stephen Berry, University of Chicago"This is an extraordinary book, remarkable for its breadth of coverage, depth of physical insight, clarity, and technical rigor. Invoking the energy landscape viewpoint as an overarching and unifying theme, Stillinger takes the reader on a fascinating journey whose ports of call include crystals, liquids, glasses, clusters, helium, polymers, and that most ubiquitous and intriguing of substances, water. A masterful panorama of condensed-matter physics and chemistry as viewed through the lens of the inherent structure formalism, presented by one of the subject's acknowledged masters."—Pablo G. Debenedetti, Princeton University"Stillinger has produced the most readable of all books on the subject, equally suitable for a course in chemistry or physics. It is also ideal for self-study by practitioners who want to better understand some of the more complex ideas that characterize current work in this area."—H. Eugene Stanley, Boston University"Stillinger provides a formalism for describing energy landscapes along with many examples of how this formalism may be used to understand the energetics of condensed-matter phenomena. His presentation will be of significant value to junior scientists, including beginning graduate students, as well as senior researchers. The book is a pleasure to read, with many interesting insights and discussions."—Paul Whitford, Northeastern UniversityTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. ix*I. Potential Energy Functions, pg. 1*II. Statistical Mechanical Basics, pg. 28*III. Basins, Saddles, and Configuration-Space Mapping, pg. 55*IV. Crystal Phases, pg. 79*V. Liquids at Thermal Equilibrium, pg. 134*VI. Supercooled Liquids and Glasses, pg. 195*VII. Low-Density Matter, pg. 240*VIII. The Helium Isotopes, pg. 278*IX. Water, pg. 313*X. Polymeric Substances, pg. 375*XI. Protein Folding Phenomena, pg. 428*References, pg. 463*Index, pg. 489

Read more less

Book SynopsisTrade Review"Nahin's writing style, as in previous books, is clear, conversational, humorous and chatty... [A]nd the discussions in the book are careful and appropriately rigorous."--MAA Reviews "Fun, accessible physics/math problems along with some humor."--Antonio Cangiano, Math-Blog "[Nahin] knows how to catch the attention of his reader. You will not regret buying any of his books, and I am sure after reading it, you will pick up this one to check again on one of his models and his solution methods."--European Mathematical Society "A superb book... [D]emonstrates clever ways to solve simply physics problems."--ChoiceTable of ContentsForeword by T. M. Helliwell ix Preface with Challenge Problems xi 1 How's Your Math? 1 2 The Traffic-Light Dilemma 20 3 Energy from Moving Air 25 4 Dragsters and Space Station Physics 32 5 Merry-Go-Round Physics and the Tides 42 6 Energy from Moving Water 51 7 Vectors and Bad Hair Days 63 8 An Illuminating Problem 67 9 How to Measure Depth with a Stopwatch 74 10 Doing the Preface Problems 79 11 The Physics of Stacking Books 92 12 Communication Satellite Physics 103 13 Walking a Ladder Upright 110 14 Why Is the Sky Dark at Night? 115 15 How Some Things Float (or Don't) 126 16 A Reciprocating Problem 141 17 How to Catch a Baseball (or Not) 146 18 Tossing Balls and Shooting Bullets Uphill 153 19 Rapid Travel in a Great Circle Transit Tube 163 20 Hurtling Your Body through Space 177 21 The Path of a Punt 194 22 Easy Ways to Measure Gravity in Your Garage 200 23 Epilogue Newton's Gravity Calculation Mistake 218 Postscript 227 Acknowledgments

Read more less