Mathematical / Computational / Theoretical physics Books

Book SynopsisAn Introduction to Laplace Transforms and Fourier Series will be useful for second and third year undergraduate students in engineering, physics or mathematics, as well as for graduates in any discipline such as financial mathematics, econometrics and biological modelling requiring techniques for solving initial value problems.Table of ContentsThe Laplace Transform.- Further Properties of the Laplace Transform.- Convolution and the Solution of Ordinary Differential Equations.- Fourier Series.- Partial Differential Equations.- Fourier Transforms.- Wavelets and Signal Processing.- Complex Variables and Laplace Transforms.

Read more less

Book SynopsisIn this study we are concerned with Vibration Theory and the Problem of Dynamics during the half century that followed the publication of Newton's Principia. In fact, it was through problems posed by Vibration Theory that a general theory of Dynamics was motivated and discovered.Table of Contents1. Introduction.- 2. Newton (1687).- 2.1. Pressure Wave.- 2.2. Remarks.- 3. Taylor (1713).- 3.1. Vibrating String.- 3.2. Absolute Frequency.- 3.3. Remarks.- 4. Sauveur (1713).- 4.1. Vibrating String.- 4.2. Remarks.- 5. Hermann (1716).- 5.1. Pressure Wave.- 5.2. Vibrating String.- 5.3. Remarks.- 6. Cramer (1722).- 6.1. Sound.- 6.2. Remarks.- 7. Euler (1727).- 7.1. Vibrating Ring.- 7.2. Sound.- 8. Johann Bernoulli (1728).- 8.1. Vibrating String (Continuous and Discrete).- 8.2. Remark on the Energy Method.- 9. Daniel Bernoulli (1733; 1734); Euler (1736) …..- 9.1. Linked Pendulum and Hanging Chain.- 9.2. Laguerre Polynomials and J0.- 9.3. Double and Triple Pendula.- 9.4. Roots of Polynomials.- 9.5. Zeros of J0.- 9.6. Other Boundary Conditions.- 9.7. The Bessel Functions Jv.- 10. Euler (1735).- 10.1. Pendulum Condition.- 10.2. Vibrating Rod.- 10.3. Remarks.- 11. Johann II Bernoulli (1736).- 11.1. Pressure Wave.- 11.2. Remarks.- 12. Daniel Bernoulli (1739; 1740).- 12.1. Floating Body.- 12.2. Remarks.- 12.3. Dangling Rod.- 12.4. Remarks on Superposition.- 13. Daniel Bernoulli (1742).- 13.1. Vibrating Rod.- 13.2. Absolute Frequency and Experiments.- 13.3. Superposition.- 14. Euler (1742).- 14.1. Linked Compound Pendulum.- 14.2. Dangling Rod and Weighted Chain.- 15. Johann Bernoulli (1742) no.- 15.1. One Degree of Freedom.- 15.2. Dangling Rod.- 15.3. Linked Pendulum I.- 15.4. Linked Pendulum II.- Appendix: Daniel Bernoulli’s Papers on the Hanging Chain and the Linked Pendulum.- Theoremata de Oscillationibus Corporum.- De Oscillationibus Filo Flexili Connexorum.- Theorems on the Oscillations of Bodies.- On the Oscillations of Bodies Connected by a Flexible Thread.

Read more less

Book SynopsisA. The Course.- Molecules Coupled to Their Environment.- Theories of Molecular Chirality: A Short Review.- Condensed Cooper Pairs and Macroscopic Quantum Phenomena.- Non-Equilibrium Statistical Mechanics: Dynamics of Macroscopic Observables.- Quantum Brownian Motion.- Localization Critical Exponents.- Stochastic Models of Population and Phase Relaxation.- Classical and Quantum, Lattice and Continuum Percolation.- Tunneling and Relaxation in Low Temperature Systems.- Dynamics of Quantum Particles: Coupled Coherent and Incoherent Motion.- Chaotic Motion of Molecular Chains.- Complex Surface Geometry in Nano-Structure Solids: Fractal versus Bernal-Type Models.- Aggregation Phenomena.- Electrodeposition: Phenomenology and Theory.- Screening in Electrolytes and in Polymer Solutions: The Charge Structure Function.- In Search of Scaling Laws in Porous Silica Gels.- Stochastic Aspects in Reaction Kinetics.- B. The Seminars.- I Quantum Theory of Large Systems.- Coherence and Quantum Mechanics.- The Dynamical Generation of Macroscopic Coherent Light.- Macroscopically Inhomogeneous Bose-Einstein Condensation.- Generalized Squeezing of Boson States.- Equilibrium States of Long-Range Interacting Quantum Lattice Systems.- Environment and Symmetry Breaking in Quantum Field Theory.- Gauge Chemistry.- II Localized and Extended States.- Coherence Effects in Excitation Transfer: Application to Hexagonal Photosynthetic Unit.- Random Matrix Theory and Anderson Localisation.- Local Moments and the Localization of Electrons in Liquids.- How Universal is the Scaling Theory of Localization?.- Dimers, Repulsions, and the Absence of Localisation.- III Transport and Reactions.- Electronic Excitations in Polysilanes: Frenkel Excitons of a Disordered Chain.- The Study of Energy Change in Radical Systems.- Decay From an Initial Unstable State in a Chemical Reaction Model.- The Effects of Static Disorder on Polaron Transport.- Diffusion in a Disordered Line.- Bimolecular Diffuion-Limited Reaction Kinetics at Steady State.- Dynamic Percolation Theory for Diffusion of Interacting Particles: Tracer Diffusion in a Multi-Component Lattice Gas.- Crossover from Dispersive to Diffusive Energy Transport.- The Coupling Scheme for Relaxations in Complex Correlated Systems.- Simulation of Excitation Transport in Disordered Media.- Dynamical Exponents for 1-D Random-Random Directed Walks.- IV Polymers.- Interactions between Poly(styrene-co-styrene Sulfonic Acid) and Poly(methylmethacrylate-co-4-vinyl Pyridine) in Dimethyl Sulfoxide Solution by Photon Correlation Spectroscopy.- Interactions of Sigma Conjugated Polymers With Strong Optical Fields.- Randomly Branched Polymers.- Multi-Particle Relaxation in Electronically Excited Polymers: Distribution of Transition Rates from Fluorescence Data - A Numerical Approach.- Understanding Heat Conduction in Oriented Polymers.- Static Correlations of Polymer Chains in Networks.- Theory of Polymers on Fractal Lattices.- V Disordered and Low-Dimensional Systems.- Self-Consistent Interpretation of Percolation in a Microemulsion.- Experimental Evidence of Fractal Aggregates in Dense Microemulsions.- Fractal Dynamics of the Catalytic CO-Oxydation - Application of Fractal Cellular Automata Models.- Electrodeposition: Fractal and Multifractal Measures.- High Resolution Spectroscopy of Langmuir-Blodgett Films.- Molecular Dynamics Simulation of the Transport of Small Molecules Across a Polymer Membrane.- Reactions in Microemulsions: Fractal Modeling.- Enhanced Membrane Rigidity in Charged Lamellar Phases.- VI Words of thanks.- Lecturers.- Participants.Table of ContentsA. The Course.- Molecules Coupled to Their Environment.- Theories of Molecular Chirality: A Short Review.- Condensed Cooper Pairs and Macroscopic Quantum Phenomena.- Non-Equilibrium Statistical Mechanics: Dynamics of Macroscopic Observables.- Quantum Brownian Motion.- Localization Critical Exponents.- Stochastic Models of Population and Phase Relaxation.- Classical and Quantum, Lattice and Continuum Percolation.- Tunneling and Relaxation in Low Temperature Systems.- Dynamics of Quantum Particles: Coupled Coherent and Incoherent Motion.- Chaotic Motion of Molecular Chains.- Complex Surface Geometry in Nano-Structure Solids: Fractal versus Bernal-Type Models.- Aggregation Phenomena.- Electrodeposition: Phenomenology and Theory.- Screening in Electrolytes and in Polymer Solutions: The Charge Structure Function.- In Search of Scaling Laws in Porous Silica Gels.- Stochastic Aspects in Reaction Kinetics.- B. The Seminars.- I Quantum Theory of Large Systems.- Coherence and Quantum Mechanics.- The Dynamical Generation of Macroscopic Coherent Light.- Macroscopically Inhomogeneous Bose-Einstein Condensation.- Generalized Squeezing of Boson States.- Equilibrium States of Long-Range Interacting Quantum Lattice Systems.- Environment and Symmetry Breaking in Quantum Field Theory.- Gauge Chemistry.- II Localized and Extended States.- Coherence Effects in Excitation Transfer: Application to Hexagonal Photosynthetic Unit.- Random Matrix Theory and Anderson Localisation.- Local Moments and the Localization of Electrons in Liquids.- How Universal is the Scaling Theory of Localization?.- Dimers, Repulsions, and the Absence of Localisation.- III Transport and Reactions.- Electronic Excitations in Polysilanes: Frenkel Excitons of a Disordered Chain.- The Study of Energy Change in Radical Systems.- Decay From an Initial Unstable State in a Chemical Reaction Model.- The Effects of Static Disorder on Polaron Transport.- Diffusion in a Disordered Line.- Bimolecular Diffuion-Limited Reaction Kinetics at Steady State.- Dynamic Percolation Theory for Diffusion of Interacting Particles: Tracer Diffusion in a Multi-Component Lattice Gas.- Crossover from Dispersive to Diffusive Energy Transport.- The Coupling Scheme for Relaxations in Complex Correlated Systems.- Simulation of Excitation Transport in Disordered Media.- Dynamical Exponents for 1-D Random-Random Directed Walks.- IV Polymers.- Interactions between Poly(styrene-co-styrene Sulfonic Acid) and Poly(methylmethacrylate-co-4-vinyl Pyridine) in Dimethyl Sulfoxide Solution by Photon Correlation Spectroscopy.- Interactions of Sigma Conjugated Polymers With Strong Optical Fields.- Randomly Branched Polymers.- Multi-Particle Relaxation in Electronically Excited Polymers: Distribution of Transition Rates from Fluorescence Data - A Numerical Approach.- Understanding Heat Conduction in Oriented Polymers.- Static Correlations of Polymer Chains in Networks.- Theory of Polymers on Fractal Lattices.- V Disordered and Low-Dimensional Systems.- Self-Consistent Interpretation of Percolation in a Microemulsion.- Experimental Evidence of Fractal Aggregates in Dense Microemulsions.- Fractal Dynamics of the Catalytic CO-Oxydation - Application of Fractal Cellular Automata Models.- Electrodeposition: Fractal and Multifractal Measures.- High Resolution Spectroscopy of Langmuir-Blodgett Films.- Molecular Dynamics Simulation of the Transport of Small Molecules Across a Polymer Membrane.- Reactions in Microemulsions: Fractal Modeling.- Enhanced Membrane Rigidity in Charged Lamellar Phases.- VI Words of thanks.- Lecturers.- Participants.

Read more less

Book SynopsisThis book explores the numerical algorithms underpinning modern finite element based computational mechanics software. It covers all the major numerical methods that are used in computational mechanics. It reviews the basic concepts in linear algebra and advanced matrix theory, before covering solution of systems of equations, symmetric eigenvalue solution methods, and direct integration of discrete dynamic equations of motion, illustrated with numerical examples. This book suits a graduate course in mechanics based disciplines, and will help software developers in computational mechanics. Increased understanding of the underlying numerical methods will also help practicing engineers to use the computational mechanics software more effectively.Trade Review"This book is a collection of the most relevant numerical methods used in computational mechanics. It is a clear and rigorous presentation of algorithms corresponding to numerical methods for solving systems of linear and nonlinear algebraic equations, for finding eigenvalues and eigenvectors of matrices, and for integration of dynamic equations of motion. This hands-on presentation will certainly be welcomed by users of computational mechanics software interested in gaining a better understanding of the implemented algorithms as well as developers of software. In addition, the book could be used as a textbook for a graduate level course in computational mechanics." — Corina S. Drapaca, Pennsylvania State University, USA"This book is a collection of the most relevant numerical methods used in computational mechanics. It is a clear and rigorous presentation of algorithms corresponding to numerical methods for solving systems of linear and nonlinear algebraic equations, for finding eigenvalues and eigenvectors of matrices, and for integration of dynamic equations of motion. This hands-on presentation will certainly be welcomed by users of computational mechanics software interested in gaining a better understanding of the implemented algorithms as well as developers of software. In addition, the book could be used as a textbook for a graduate level course in computational mechanics." — Corina S. Drapaca, Pennsylvania State University, USATable of ContentsReview of Matrix Analysis. Review of Methods of Analysis In Structural Mechanics. Solution of System of Linear Equations. Iterative Solution Methods for System of Linear Equations. Conjugate Gradient Methods. Solution Methods for System of Nonlinear Equations. Eigenvalue Solution Methods. Direct Integration of Dynamic Equation of Motion. The Generalized Difference Method.

Read more less

Book SynopsisThis volume records the proceedings of an international conference that explored recent developments and the interaction between mathematical theory and physical phenomena.Table of ContentsNON-COMMUTATIVE SPHERES and NUMERICAL QUANTUM MECHANICS; Matricial and Ultramatricial Topology; Remarks on the Three-Manifold Invariants ? p; The Crossed Product of the Irrational Rotation Algebra by the Flip; Quadratic and Exchange Algebras, and Modified Yang-Baxter Relations for the Selfdual Yang-Mills System and the WZNW Model; Regular Actions of Hopf Algebras on the C*-Algebra Generated by a Hilbert Space; Operator Algebras, Group Actions and Abstract Duals; A Classification of Certain Simple C*-Algebras; On Two Quantized Tensor Products 1; Geometry of Differential Equations and Projective Representations of the witt Algebra; Spin Model on Knot Projections; Towards Extracting Physical Predictions from Alain Connes' Version of the Standard Model (The New Grand Unification?); A Commutator Inequality; Duals of Compact Groups Realized by Semigroups of Non-Unital Endomorphisms of C *-Algebras; A New Index for Continuous Semigroups of *-Endomorphisms of B(H); Topological Orbit Equivalence; Toeplitz C *-Alegras on Pseudoconvex Domains with Transverse Symmetries; Normal Subgroups of the Automorphism Group of a Factor; Subfactors and Invariants of 3-Manifolds

Read more less

Book SynopsisThe theory of holomorphic functions of several complex variables emerged from the attempt to generalize the theory in one variable to the multidimensional situation. Research in this area has led to the discovery of many sophisticated facts, structures, ideas, relations, and applications. This deepening of knowledge, however, has also revealed more and more paradoxical differences between the structures of the two theories. The authors of this Research Note were driven by the quest to construct a theory in several complex variables that has the same structure as the one-variable theory. That is, they sought a reproducing kernel for the whole class that is universal and from same class. Integral Theorems for Functions and Differential Forms in Cm documents their success. Their highly original approach allowed them to obtain new results and refine some well-known results from the classical theory of several complex variables. The 'hyperholomorphic" theory they developed proved to be a kind of direct sum of function theories for two Dirac-type operators of Clifford analysis considered in the same domain.In addition to new results and methods, this work presents a first-look at a brand new setting, based upon the natural language of differential forms, for complex analysis. Integral Theorems for Functions and Differential Forms in Cm reveals a deep link between the fields of several complex variables theory and Clifford analysis. It will have a strong influence on researchers in both areas, and undoubtedly will change the general viewpoint on the methods and ideas of several complex variables theory.Trade Review"…the book will be interesting to specialists in complex analysis and its applications".- Mathematical Reviews, 2003a"This well-written book is a valuable contribution to the broad field of interactions between complex analysis and partial differential equations...Moreover, the book can be used for individual studies, because fundamental concepts and important theorems are explained in detail."-Mathematical Reviews, Issue 94aTable of ContentsIntroduction. Differential Forms. Differential Forms with Co-Efficients in 2x2 Matrices. Hyperholomorphic Functions and Differential Forms in Cm. Cauchy's Theorem. Morera's Theorem. Cauchy's Integral Representation. Hyperholomorphic D-problem. Complex Hodge-Dolbeault System. Relations with Clifford Analysis.

Read more less

Book SynopsisThis book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models.The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial manifolds.Table of ContentsDynamical Processes. Attractors of Semiflows. Attractors for Semilinear Evolution Equations. Exponential Attractors. Inertial Manifolds. Examples. A Non-Existence Result for Inertial Manifolds. Appendix: Selected Results from Analysis. Bibliography. Index. Nomenclature

Read more less

Book SynopsisDue to the increase in computational power and new discoveries in propagation phenomena for linear and nonlinear waves, the area of computational wave propagation has become more significant in recent years. Exploring the latest developments in the field, Effective Computational Methods for Wave Propagation presents several modern, valuable computational methods used to describe wave propagation phenomena in selected areas of physics and technology.Featuring contributions from internationally known experts, the book is divided into four parts. It begins with the simulation of nonlinear dispersive waves from nonlinear optics and the theory and numerical analysis of Boussinesq systems. The next section focuses on computational approaches, including a finite element method and parabolic equation techniques, for mathematical models of underwater sound propagation and scattering. The book then offers a comprehensive introduction to modern numerical methods for time-dependent elastic wave propagation. The final part supplies an overview of high-order, low diffusion numerical methods for complex, compressible flows of aerodynamics. Concentrating on physics and technology, this volume provides the necessary computational methods to effectively tackle the sources of problems that involve some type of wave motion.Table of ContentsPreface. Nonlinear Dispersive Waves: Numerical Simulations of Singular Solutions of Nonlinear Schrödinger Equations. Numerical Solution of the Nonlinear Helmholtz Equation. Theory and Numerical Analysis of Boussinesq Systems: A Review. The Helmholtz Equation and its Paraxial Approximations in Underwater Acoustics: Finite Element Discretization of the Helmholtz Equation in an Underwater Acoustic Waveguide. Parabolic Equation Techniques in Underwater Acoustics. Numerical Solution of the Parabolic Equation in Range-Dependent Waveguides. Exact Boundary Conditions for Acoustic PE Modeling over an N2-Linear Half-Space. Numerical Methods for Elastic Wave Propagation. Introduction and Orientation. The Mathematical Model for Elastic Wave Propagation. Finite Element Methods with Continuous Displacement. Finite Element Methods with Discontinuous Displacement. Fictitious Domains Methods for Wave Diffraction. Space Time Mesh Refinement Methods. Numerical Methods for Treating Unbounded Media. Waves in Compressible Flows. High-Order Accurate Space Discretization Methods for Computational Fluid Dynamics. Governing Equations. High-Order Finite-Difference Schemes. ENO and WENO Schemes. The Discontinuous Galerkin (DG) Method. Index.

Read more less

Book SynopsisExtending and generalizing the results of rational equations,Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures focuses on the boundedness nature of solutions, the global stability of equilibrium points, the periodic character of solutions, and the convergence to periodic solutions, including their periodic trichotomies. The book also provides numerous thought-provoking open problems and conjectures on the boundedness character, global stability, and periodic behavior of solutions of rational difference equations. After introducing several basic definitions and general results, the authors examine 135 special cases of rational difference equations that have only bounded solutions and the equations that have unbounded solutions in some range of their parameters. They then explore the seven known nonlinear periodic trichotomies of third order rational difference equations. The main part of the book presents the known results of each of the 225 special cases of third order rational difference equations. In addition, the appendices supply tables that feature important information on these cases as well as on the boundedness character of all fourth order rational difference equations. A Framework for Future Research The theory and techniques developed in this book to understand the dynamics of rational difference equations will be useful in analyzing the equations in any mathematical model that involves difference equations. Moreover, the stimulating conjectures will promote future investigations in this fascinating, yet surprisingly little known area of research.Table of ContentsPreface. Introduction. Preliminaries. Equations with Bounded Solutions. Existence of Unbounded Solutions. Period Trichotomies. Known Results for Each of the 225 Special Cases. Appendices. Bibliography. Index.

Read more less

Book SynopsisDesigned for the 21st century classroom, this textbook poses, refines, and analyzes questions of sustainability in a quantitative environment. Building mathematical knowledge in the context of issues relevant to every global citizen today, this text takes an approach that empowers students of all disciplines to understand and reason with quantitative information. Whatever conclusions may be reached on a given topic, this book will prepare the reader to think critically about their own and other people’s arguments and to support them with careful, mathematical reasoning. Topics are grouped in themes of measurement, flow, connectivity, change, risk, and decision-making. Mathematical thinking is at the fore throughout, as students learn to model sustainability on local, regional, and global scales. Exercises emphasize concepts, while projects build and challenge communication skills. With no prerequisites beyond high school algebra, instructors will find this book a rich resource for engaging all majors in the mathematics classroom. From the Foreword No longer will you be just a spectator when people give you quantitative information—you will become an active participant who can engage and contribute new insights to any discussion.[…] There are many math books that will feed you knowledge, but it is rare to see a book like this one that will help you cultivate wisdom.[…] As the authors illustrate, mathematics that pays attention to human considerations can help you look at the world with a new lens, help you frame important questions, and help you make wise decisions. Francis Edward Su, Harvey Mudd CollegeTrade Review“This is a most revolutionary book, and I would love to teach a course using it! There is much interest in what is variously termed quantitative literacy (QL), quantitative reasoning (QR), and critical thinking (CT). This book is the first in a series of Texts for Quantitative Critical Thinking … . I do not want to fail to mention that the book’s layout is simply beautiful.” (Paul J. Campbell, Mathematics Magazine, Vol. 92 (1), November, 2018)Table of Contents1. Measuring.- 2. Flowing.- 3. Networking.- 4. Changing.- 5. Risking.- 6. Deciding.- 7. Case Studies.- 8. Resources.- List of Figures.- List of Tables.- Bibliography.- Index.

Read more less

Book SynopsisUnlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity.The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects.The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.Trade ReviewFrom the book reviews:“The aim of the textbook is twofold. First, it is a concise and self-contained quick introduction to the basics of differential geometry, including differential forms, followed by the main ideas of Riemannian geometry. Second, the last two chapters are devoted to some interesting applications to geometric mechanics and relativity. … the book is well written and also very readable. I warmly recommend it to specialists in mathematics, physics and engineering, especially to Ph.D. students.” (Miroslaw Doupovec, zbMATH 1306.53001, 2015)Table of ContentsDifferentiable Manifolds.- Differential Forms.- Riemannian Manifolds.- Curvature.- Geometric Mechanics.- Relativity.

Read more less

Book SynopsisThis textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used for the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems.With over 220 figures, numerous examples and more than one hundred exercise on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on numerics, and as a reference for CFD programmers and researchers. Trade Review“Directed towards future practitioners such as engineers the authors first provide an introduction to fluid dynamics presupposing but a modicum of mathematical and physical knowledge. … . A number of exercises plus special chapters on modelling incompressible and compressible flow make the book very useful for its purpose.” (H. Muthsam, Monatshefte für Mathematik, Vol. 187 (1), September, 2018)“The book is very attractive, carefully written and easy to read by those interested in learning about finite volume methods for fluid dynamics. The authors have made an important effort to bridge the gap between classroom material and actual model development questions. The text is well illustrated by means of quality figures helping to understand the described concepts. Furthermore, the book contains pieces of academic codes in MATLAB … . It is certainly a useful, practical and valuable book.” (Pilar Garcia-Navarro, Mathematical Reviews, May, 2016)Table of ContentsFoundation1 Introduction2 Review of Vector Calculus3 Mathematical Description of Physical Phenomena4 The Discretization Process5 The Finite Volume Method6 The Finite Volume Mesh7 The Finite Volume Mesh in OpenFOAM® and uFVMDiscretization8 Spatial Discretization: The Diffusion Term9 Gradient Computation10 Solving the System of Algebraic Equations11 Discretization of the Convection Term12 High Resolution Schemes13 Temporal Discretization: The Transient Term14 Discretization of the Source Term, Relaxation, and Other DetailsAlgorithms15 Fluid Flow Computation: Incompressible Flows16 Fluid Flow Computation: Compressible FlowsApplications17 Turbulence Modeling18 Boundary Conditions in OpenFOAM® and uFVM19 An OpenFOAM® Turbulent Flow Application 20 Closing RemarksAppendices<20 Closing RemarksAppendices20 Closing RemarksAppendices

Read more less

Book SynopsisThis volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental role in the current development of probability theory and statistical mechanics. The lectures were: Random Polymers by E. Bolthausen, Spontaneous Replica Symmetry Breaking and Interpolation Methods by F. Guerra, Derrida's Random Energy Models by N. Kistler, Isomorphism Theorems by J. Rosen and Spectral Properties of Wigner Matrices by B. Schlein.This book is the first in a co-edition between the Jean-Morlet Chair at CIRM and the Springer Lecture Notes in Mathematics which aims to collect together courses and lectures on cutting-edge subjects given during the term of the Jean-Morlet Chair, as well as new material produced in its wake. It is targeted at researchers, in particular PhD students and postdocs, working in probability theory and statistical physics.Table of Contents1 Random Polymers.- 2 Spontaneous replica symmetry breaking and interpolation methods for complex statistical mechanics systems.- 3 Derrida’s random energy models: from spin glasses to the extremes of correlated random fields.- 4 Isomorphism Theorems: Markov processes, Gaussian processes and beyond.- 5 Spectral properties of Wigner matrices.

Read more less

Book SynopsisThis textbook is for the standard, one-semester, junior-senior course that often goes by the title "Elementary Partial Differential Equations" or "Boundary Value Problems". The audience consists of students in mathematics, engineering, and the sciences. The topics include derivations of some of the standard models of mathematical physics and methods for solving those equations on unbounded and bounded domains, and applications of PDE's to biology. The text differs from other texts in its brevity; yet it provides coverage of the main topics usually studied in the standard course, as well as an introduction to using computer algebra packages to solve and understand partial differential equations.For the 3rd edition the section on numerical methods has been considerably expanded to reflect their central role in PDE's. A treatment of the finite element method has been included and the code for numerical calculations is now written for MATLAB. Nonetheless the brevity of the text has been maintained. To further aid the reader in mastering the material and using the book, the clarity of the exercises has been improved, more routine exercises have been included, and the entire text has been visually reformatted to improve readability.Trade Review“The aim of this book is to provide the reader with basic ideas encountered in partial differential equations. … The mathematical content is highly motivated by physical problems and the emphasis is on motivation, methods, concepts and interpretation rather than formal theory. The textbook is a valuable material for undergraduate science and engineering students.” (Marius Ghergu, zbMATH 1310.35001, 2015)Table of ContentsPreface to the Third Edition.- To the Students.- 1: The Physical Origins of Partial Differential Equations.- 1.1 PDE Models.- 1.2 Conservation Laws.- 1.3 Diffusion.- 1.4 Diffusion and Randomness.- 1.5 Vibrations and Acoustics.- 1.6 Quantum Mechanics*.- 1.7 Heat Conduction in Higher Dimensions.- 1.8 Laplace’s Equation.- 1.9 Classification of PDEs.- 2. Partial Differential Equations on Unbounded Domains.- 2.1 Cauchy Problem for the Heat Equation.- 2.2 Cauchy Problem for the Wave Equation.- 2.3 Well-Posed Problems.- 2.4 Semi-Infinite Domains.- 2.5 Sources and Duhamel’s Principle.- 2.6 Laplace Transforms.- 2.7 Fourier Transforms.- 3. Orthogonal Expansions.- 3.1 The Fourier Method.- 3.2 Orthogonal Expansions.- 3.3 Classical Fourier Series.-4. Partial Differential Equations on Bounded Domains.- 4.1 Overview of Separation of Variables.- 4.2 Sturm–Liouville Problems - 4.3 Generalization and Singular Problems.- 4.4 Laplace's Equation.- 4.5 Cooling of a Sphere.- 4.6 Diffusion inb a Disk.- 4.7 Sources on Bounded Domains.- 4.8 Poisson's Equation*.-5. Applications in the Life Sciences.-5.1 Age-Structured Models.- 5.2 Traveling Waves Fronts.- 5.3 Equilibria and Stability.- References.- Appendix A. Ordinary Differential Equations.- Index.

Read more less

Book SynopsisThese course-tested lectures provide a technical introduction to Supersymmetric Grand Unified Theories (SUSY GUTs), as well as a personal view on the topic by one of the pioneers in the field. While the Standard Model of Particle Physics is incredibly successful in describing the known universe it is, nevertheless, an incomplete theory with many free parameters and open issues. An elegant solution to all of these quandaries is the proposed theory of SUSY GUTs. In a GUT, quarks and leptons are related in a simple way by the unifying symmetry and their electric charges are quantized, further the relative strength of the strong, weak and electromagnetic forces are predicted. SUSY GUTs additionally provide a framework for understanding particle masses and offer candidates for dark matter. Finally, with the extension of SUSY GUTs to string theory, a quantum-mechanically consistent unification of the four known forces (including gravity) is obtained. The book is organized in three sections: the first section contains a brief introduction to the Standard Model, supersymmetry and the Minimal Supersymmetric Standard Model. Then SUSY GUTs in four space-time dimensions are introduced and reviewed. In addition, the cosmological issues concerning SUSY GUTs are discussed. Then the requirements for embedding a 4D SUSY GUT into higher-dimensional theories including gravity (i.e. String Theory) are investigated. Accordingly, section two of the course is devoted to discussing the so-called Orbifold GUTs and how in turn they solve some of the technical problems of 4D SUSY GUTs. Orbifold GUTs introduce a new set of open issues, which are then resolved in the third section in which it is shown how to embed Orbifold GUTs into the E(8) x E(8) Heterotic String in 10 space-time dimensions.Trade Review“I enjoyed this book very much and found it useful for refreshing my views and learning something new about SUSY, namely about the GUT state of affairs. I recommend it to individual researchers and to libraries in research universities, physics departments, and HEP laboratories.” (Paulo Moniz, Mathematical Reviews, January, 2018)Table of ContentsThe Standard Model:background.- The Minimal Supersymmetric Standard Model (MSSM).- Supersymmetric GUTs in 4 space-time dimensions.- SUSY GUTs meets data: LHC, fermion masses and mixing angles, dark matter.- Problems of 4 D SUSY GUTs.- SUSY GUTs in 5 or 6 dimensions : Orbifold GUTs.- SUSY breaking in extra dimensions.- Orbifold GUTs meet data.- SUSY GUTs in string theory : background.- Heterotic orbifold constructions.- Guaranteeing the MSSM, proton decay and precise gauge coupling unification.- Smooth heterotic constructions.- Type II string models and F theory – lectures.- Stabilizing moduli and SUSY breaking.- Cosmology.- Conclusions and Outlook.

Read more less

Table of ContentsMorse-smale fields of geodesics.- Periodic points and topological entropy of one dimensional maps.- Ergodicity of linked twist maps.- Infinitesimal hyperbolicity implies hyperbolicity.- A qualitative singular perturbation theorem.- On a theorem of conley and smoller.- Positively expansive maps of compact manifolds.- An algorithm for finding closed orbits.- Linked twist mappings are almost anosov.- Symbolic dynamics, homology, and knots.- Anomalous anosov flows.- Efficiency vs. hyperbolicity on tori.- Dynamical behavior of geodesic fields.- The growth of topological entropy for one dimensional maps.- Separatrices, non-isolated invariant sets and the seifert conjecture.- Construction of invariant measures absolutely continuous with respect to dx for some maps of the interval.- The estimation from above for the topological entropy of a diffeomorphism.- Ergodicity in (G,?)-extensions.- A probabilistic version of bowen — Ruelle's volume lemma.- Periodically forced relaxation oscillations.- Moduli of stability for diffeomorphisms.- Uncountably many distinct topologically hyperbolic equilibria in ?4.- Dynamical properties of certain non-commutative skew-products.- A note on explosive flows.- Intertwining invariant manifolds and the lorenz attractor.- Counting compatible boundary conditions.- Stable manifolds for maps.- Singular points of planar vector fields.- Gradient vectorfields near degenerate singularities.- Invariant curves near parabolic points and regions of stability.- Motion under the influence of a strong constraining force.- Conjugacies of topologically hyperbolic fixed points: A necessary condition on foliations.- Coleman's conjecture on topological hyperbolicity.- Population dynamics from game theory.

Read more less

Book SynopsisAstronomie mit dem PC vermittelt dem Leser eine fundierte Einführung in die Welt der himmelsmechanischen Berechnungen, die für die astronomische Beobachtungspraxis von besonderer Bedeutung sind.Von den theoretischen Grundlagen der Astronomie und Numerik bis zur Erstellung schneller und präziser Anwendungsprogramme vermittelt das Buch die notwendigen Kenntnisse und Softwarelösungen für die Bestimmung und Vorhersage von:- Positionen der Sonne, des Mondes und der Planeten- Auf- und Untergangszeiten- physischen Ephemeriden der Sonne und der großen Planeten- Kometen- und Kleinplanetenpositionen (mit Störungen)- Mondphasen- Zentrallinie und lokalen Umständen von Sonnenfinsternissen - Sternbedeckungen durch den Mond- Bahnelementen aus drei Beobachtungen (auch mehrere Lösungen)- Koordinaten aus Himmelsaufnahmen. Die Verwendung der weitverbreiteten objektorientierten Programmiersprache CC++ ermöglicht die effiziente Realisierung eigener Anwendungen auf der Basis einer leistungsfähigen Modul-Bibliothek. Die Begleit-CD enthält neben den vollständigen, ausgiebig dokumentierten und kommentierten Quelltexten auch die ausführbaren Programme - damit können Leser ohne Programmierkenntnisse alle im Buch beschriebenen Programme ebenfalls nutzen. Zusätzlich befinden sich zwei Sternkataloge (Position und Proper Motion Katalog und Zodialkatalog) sowie die Lowell-Datenbank aktueller Kleinplaneten-Bahnelemente auf der CD, die den Nutzwert der entsprechenden Programme weiter erhöhen. Die vorliegende 4. Auflage stellt, neben einigen Überarbeitungen der Texte und Bilder, die ausführbaren Programme für die Betriebssysteme Windows 98/2000/XP und LINUX sowie die akualisierten Kataloge und Datenbanken zur Verfügung.Table of ContentsEinführung.- Koordinatensysteme.- Auf- und Untergangsrechnung.- Kometenbahnen.- Störungsrechnung.- Planetenbahnen.- Physische Planetenephemeriden.- Die Mondbahn.- Sonnenfinsternisse.- Sternbedeckungen.- Bahnbestimmung.- Astrometrie.

Read more less

Book SynopsisAn application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.Trade ReviewM.P. Do Carmo Differential Forms and Applications "This book treats differential forms and uses them to study some local and global aspects of differential geometry of surfaces. Each chapter is followed by interesting exercises. Thus, this is an ideal book for a one-semester course."—ACTA SCIENTIARUM MATHEMATICARUMTable of Contents1. Differential Forms in Rn.- 2. Line Integrals.- 3. Differentiable Manifolds.- 4. Integration on Manifolds; Stokes Theorem and Poincaré’s Lemma.- 1. Integration of Differential Forms.- 2. Stokes Theorem.- 3. Poincaré’s Lemma.- 5. Differential Geometry of Surfaces.- 1. The Structure Equations of Rn.- 2. Surfaces in R3.- 3. Intrinsic Geometry of Surfaces.- 6. The Theorem of Gauss-Bonnet and the Theorem of Morse.- 1. The Theorem of Gauss-Bonnet.- 2. The Theorem of Morse.- References.

Read more less

Book Synopsis"Number Theory in Science and Communication" is a well-known introduction for non-mathematicians to this fascinating and useful branch of applied mathematics . It stresses intuitive understanding rather than abstract theory and highlights important concepts such as continued fractions, the golden ratio, quadratic residues and Chinese remainders, trapdoor functions, pseudo primes and primitive elements. Their applications to problems in the real world are one of the main themes of the book. This revised fifth edition is augmented by recent advances in coding theory, permutations and derangements and a chapter in quantum cryptography. From reviews of earlier editions – "I continue to find [Schroeder’s] Number Theory a goldmine of valuable information. It is a marvelous book, in touch with the most recent applications of number theory and written with great clarity and humor.’ Philip Morrison (Scientific American) "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin GardnerTrade ReviewFrom the reviews of the fifth edition:“Number theory has been a very active field in the last twenty-seven years, and Schroeder’s text has a palimpsest quality, with later mathematical advances layered on earlier ones. … Number Theory in Science and Communication is rewarding to browse, or as a jumping-off point for further research … . It would be a good source of student projects in an undergraduate discrete mathematics or number theory course.” (Ursula Whitcher, The Mathematical Association of America, March, 2011)Table of ContentsA Few Fundamentals.- The Natural Numbers.- Primes.- The Prime Distribution.- Some Simple Applications.- Fractions: Continued, Egyptian and Farey.- Congruences and the Like.- Linear Congruences.- Diophantine Equations.- The Theorems of Fermat Wilson and Euler.- Permutations Cycles and Derangements.- Cryptography and Divisors.- Euler Trap Doors and Public-Key Encryption.- The Divisor Functions.- The Prime Divisor Functions.- Certified Signatures.- Primitive Roots.- Knapsack Encryption.- Residues and Diffraction.- Quadratic Residues.- Chinese and Other Fast Algorithms.- The Chinese Remainder Theorem and Simultaneous Congruences.- Fast Transformation and Kronecker Products.- Quadratic Congruences.- Pseudoprimes, Möbius Transform, and Partitions.- Pseudoprimes Poker and Remote Coin Tossing.- The Möbius Function and the Möbius Transform.- Generating Functions and Partitions.- From Error Correcting Codes to Covering Sets.- Cyclotomy and Polynomials.- Cyclotomic Polynomials.- Linear Systems and Polynomials.- Polynomial Theory.- Galois Fields and More Applications.- Galois Fields.- Spectral Properties of Galois Sequences.- Random Number Generators.- Waveforms and Radiation Patterns.- Number Theory Randomness and “Art”.- Self-Similarity, Fractals and Art.- Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter.

Read more less

Book SynopsisDer Grundkurs Theoretische Physik deckt in 7 Bänden alle für das Diplom und für Bachelor/Master-Studiengänge maßgeblichen Gebiete ab. Jeder Band vermittelt das im jeweiligen Semester notwendige theoretisch-physikalische Rüstzeug. Übungsaufgaben mit ausführlichen Lösungen dienen der Vertiefung des Stoffs. Der 6. Band zur Statistischen Physik wurde für die Neuauflage grundlegend überarbeitet und um aktuelle Entwicklungen ergänzt. Durch die zweifarbige Gestaltung ist der Stoff jetzt noch übersichtlicher gegliedert.Table of ContentsKlassische statistische Physik.- Quantenstatistik.- Quantengase.- Phasenübergänge.- Lösungen der Übungsaufgaben.

Read more less

Book SynopsisDer Band bietet eine Schritt-für-Schritt-Einführung in das numerische Rechnen mit den Programmen MATLAB, Scilab und Octave. Anhand zahlreicher Beispiele zeigen die Autoren, wie die mathematischen Tools zur Lösung mathematischer, physikalischer und insbesondere ingenieurwissenschaftlicher Aufgaben eingesetzt werden können. Dazu gehören die Lösung von linearen Gleichungssystemen, nichtlinearen Gleichungen und Differentialgleichungen, die Fourier- und Wavelet-Transformation, Kurvenanpassung und Interpolation sowie die numerische Integration.Trade Review“... Zum Nachschlagen enthält jedes Kapitel eine Zusammenfassung, und ganz am Ende stehen einige Testfragen sowie eine kleine Literaturliste, so dass sich der neugierige Leser weiteres Futter verschaffen kann. ... erkunden die Autoren einige Teilbereiche der numerischen Mathematik, die für Anfänger geeignet sind. ... Die vielen Beispiele und Aufgaben unterstützen den Leser enorm ... Wer sich für das Lösen numerischer Probleme mit zeitgemäßen Werkzeugen interessiert, der kann zunächst bedenkenlos zu diesem Werk greifen.” (Harald Löwe, in: Mathematische Semesterberichte, Jg. 62, 2015, S. 114 f.) Table of Contents

Read more less

Book SynopsisThe study of dynamical systems is a well established field. This book provides a panorama of several aspects of interest to mathematicians and physicists. It collects the material of several courses at the graduate level given by the authors, avoiding detailed proofs in exchange for numerous illustrations and examples. Apart from common subjects in this field, a lot of attention is given to questions of physical measurement and stochastic properties of chaotic dynamical systems.Trade ReviewFrom the reviews: "The book is a good introduction to the field of dynamical systems with a particular emphasis on statistical properties and applications. In particular, the relations both with real experiments with numerical simulations are discussed. … The book contains many figures that really help the understanding of the text. The book can be used as a text for an introductory course in dynamical systems (at the master’s or Ph.D. level). It is particularly suited for students with interests in applications (either physics, economy or biology)." (Carlangelo Liverani, Mathematical Reviews, Issue 2007 m) "Two thoughts crossed my mind when I picked up this book. The first was: ‘what a physically attractive book.’ The second was: ‘what a short book to have on such a wide ranging topic.’ … It is a perfect size to carry in a knapsack, the print is clear and the layout of text, equations, and figures is marvelously done. … images are multi-colored stereo images, and allow the reader to ‘see’ a three dimensional effect that helps illustrate the phenomena." (David S. Mazel, MathDL, December, 2007)Table of ContentsA Basic Problem.- Dynamical Systems.- Topological Properties.- Hyperbolicity.- Invariant Measures.- Entropy.- Statistics and Statistical Mechanics.- Other Probabilistic Results.- Experimental Aspects.

Read more less

Book SynopsisQuantum effects may be modelled by means of stochastic perturbation of non-linear partial differential (field) equations. Contributions to this field of research are collected in this volume. Finite dimensional stochastically perturbed Hamiltonian systems and infinite dimensional white noise analysis are treated. The main part concerns problems encountered in deterministic equations. Papers treat the existence of solutions for given initial data, the existence of non-linear bound states or solitary waves including a thorough discussion of various approaches to stability, and global properties (e.g. time decay properties) for non-linear wave equations. This volume provides a good survey of present-day research in non-linear problems of quantum theory for researchers and graduate students.Table of ContentsSome remarks on stochastically perturbed (Hamiltonian) systems.- Stability of ground states for nonlinear classical field theories.- A note on solutions of two-dimensional semilinear elliptic vector-field equations with strong nonlinearity.- Some remarks on the nonlinear Schrödinger equation in the subcritical case.- The Cauchy problem for the Dirac equation with cubic nonlinearity in three space dimensions.- The Cauchy problem for the non-linear Klein-Cordon equation.- Conformal invariance and time decay for nonlinear wave equations.- Energy forms and white noise analysis.- Principles of solitary wave stability.

Read more less

Book SynopsisThis guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Easy to understand, and convenient to use, this guide book gives concisely the information necessary to evaluate most problems which occur in concrete applications. In the newer editions emphasis was laid on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes, namely Numerical Mathematics, Probability Theory and Statistics, as well as Information Processing. Besides many enhancements and new paragraphs, new sections on Geometric and Coordinate Transformations, Quaternions and Applications, and Lie Groups and Lie Algebras were added for the sixth edition.Trade Review“Russian scholars Bronshtein and Semendyayev created a math classic over seven decades ago. … This new Springer edition details over 1,500 entries in its table of contents, including new entries for analytical geometry, Lie groups and Lie algebra, nonlinear optimization, and computer algebra systems. … Summing Up: Recommended. All mathematics library collections.” (K. L. Swetland, Choice, Vol. 53 (11), July, 2016)Table of ContentsArithmetics.- Functions.- Geometry.- Linear Algebra.- Algebra and Discrete Mathematics.- Differentiation.- Infinite Series.- Integral Calculus.- Differential Equations.- Calculus of Variations.- Linear Integral Equations.- Functional Analysis.- Vector Analysis and Vector Fields.- Function Theory.- Integral Transformations.- Probability Theory and Mathematical Statistics.- Dynamical Systems and Chaos.- Optimization.- Numerical Analysis.- Computer Algebra Systems-Example Mathematica.

Read more less

Book SynopsisOur esteemed colleague C. V. Vishveshwara, popularly known as Vishu, turned sixty on 6th March 1998. His colleagues and well wishers felt that it would be appropriate to celebrate the occasion by bringing out a volume in his honour. Those of us who have had the good fortune to know Vishu, know that he is unique, in a class by himself. Having been given the privilege to be the volume's editors, we felt that we should attempt something different in this endeavour. Vishu is one of the well known relativists from India whose pioneer­ ing contributions to the studies of black holes is universally recognised. He was a student of Charles Misner. His Ph. D. thesis on the stability of the Schwarzschild black hole, coordinate invariant characterisation of the sta­ tionary limit and event horizon for Kerr black holes and subsequent seminal work on quasi-normal modes of black holes have passed on to become the starting points for detailed mathematical investigations on the nature of black holes. He later worked on other aspects related to black holes and compact objects. Many of these topics have matured over the last thirty years. New facets have also developed and become current areas of vigorous research interest. No longer are black holes, ultracompact objects or event horizons mere idealisations of mathematical physicists but concrete entities that astrophysicists detect, measure and look for. Astrophysical evidence is mounting up steadily for black holes.Table of ContentsPreface. 1. The Black Hole Equilibrium Problem; B. Carter.2. Stability of Black Holes; B.F. Whiting. 3. Separability of Wave Equations; E.G. Kalnins, et al. 4. Energy-Conservation Laws for Perturbed Stars and Black Holes; V.Ferrari. 5. Gravitational Collapse and Cosmic Censorship; R.M.Wald. 6. Disturbing the Black Hole; J.D. Bekenstein.7. Notes on Black Hole Fluctuations and Back-Reaction; B.L. Hu, et al. 8. Black Holes in Higher Curvature Gravity; R.C. Myers. 9. Micro-Structure of Black Holes and String Theory; S. Wadia. 10. Quantum Geometry and Black Holes; A. Ashtekar, K. Krasnov. 11. Black Holes, Global Monopole Charge and Quasi-Local Energy; N. Dadhich. 12. Kinematical Consequences of Inertial Forces in General Relativity; A.R. Prasanna, S. Iyer. 13. Gyroscopic Precession and Inertial Forces in General Relativity; R.Nayak. 14. Analysis of the Equilibrium of a Charged Test Particlein the Kerr - Newman Black Hole; J.M. Aguirregabiria, et al. 15. Neutron Stars and Relativistic Gravity; M. Vivekanand. 16. Accretion Disks around Black Holes; P.J. Wiita. 17. Astrophysical Evidence for Black Hole Event Horizons; K. Menou, et al. 18. Black Holes in Active Galactic Nuclei; A.K. Kembhavi. 19. Energetic Photon Spectra as Probes of the Process of Particle Acceleration in Accretion Flows around Black Holes; R. Cowsik. 20. Black Hole Perturbation Approach to Gravitational Radiation: Post-Newtonian Expansion for Inspiralling Binaries; M.Sasaki. 21. More Quasi Than Normal! N. Andersson. 22. The Two Black Hole Problem: Beyond Linear Perturbations; R.H. Price. 23. The Synergy between Numerical and Perturbative Approaches to Black Holes; E. Seidel. 24. Cauchy-Characteristic Matching; N.T. Bishop, et al. 25. Astrophysical Sources of Gravitational Waves; B.S. Sathyaprakash.26. Gravitational Radiation from Inspiraling Compact Binaries: Motion, Generation and Radiation Reaction; B.R. Iyer. 27. Ground-Based Interferometric Detectors of Gravitational Waves; B. Bhawal. 28. Detection of Gravitational Waves from Inspiraling Compact Binaries; S.V. Dhurandhar. 29. Perturbations of Cosmological Backgrounds; P.K.S. Dunsby, G.F.R. Ellis. 30.Mach's Principle in Electrodynamics and Inertia; J.V. Narlikar.31. The Early History of Quantum Gravity (1916&endash;1940); J. Stachel. 32. Geometry in Color Perception; A. Ashtekar, etal. 33. C.V. Vishveshwara &endash; A Profile; N. Panchapakesan. Publications of C.V. Vishveshwara.

Read more less

Book SynopsisA course in angular momentum techniques is essential for quantitative study of problems in atomic physics, molecular physics, nuclear physics and solid state physics. This book has grown out of such a course given to the students of the M. Sc. and M. Phil. degree courses at the University of Madras. An elementary knowledge of quantum mechanics is an essential pre-requisite to undertake this course but no knowledge of group theory is assumed on the part of the readers. Although the subject matter has group-theoretic origin, special efforts have been made to avoid the gro- theoretical language but place emphasis on the algebraic formalism dev- oped by Racah (1942a, 1942b, 1943, 1951). How far I am successful in this project is left to the discerning reader to judge. After the publication of the two classic books, one by Rose and the other by Edmonds on this subject in the year 1957, the application of angular momentum techniques to solve physical problems has become so common that it is found desirable to organize a separate course on this subject to the students of physics. It is to cater to the needs of such students and research workers that this book is written. A large number of questions and problems given at the end of each chapter will enable the reader to have a clearer understanding of the subject.Table of ContentsPreface. 1. Angular Momentum Operators and Their Matrix Elements. 2. Coupling of Two Angular Momenta. 3. Vectors and Tensors in Spherical Basis. 4. Rotation Matrices - I. 5. Rotation Matrices - II. 6. Tensor Operators and Reduced Matrix Elements. 7. Coupling of Three Angular Momenta. 8. Coupling of Four Angular Momenta. 9. Partial Waves and the Gradient Formula. 10. Identical Particles. 11. Density Matrix and Statistical Tensors. 12. Products of Angular Momentum Matrices and Their Traces. 13. The Helicity Formalism. 14. The Spin States of Dirac Particles. Appendices. References. Subject Index.

Read more less

Book SynopsisWhat is deep learning for those who study physics? Is it completely different from physics? Or is it similar? In recent years, machine learning, including deep learning, has begun to be used in various physics studies. Why is that? Is knowing physics useful in machine learning? Conversely, is knowing machine learning useful in physics? This book is devoted to answers of these questions. Starting with basic ideas of physics, neural networks are derived naturally. And you can learn the concepts of deep learning through the words of physics.In fact, the foundation of machine learning can be attributed to physical concepts. Hamiltonians that determine physical systems characterize various machine learning structures. Statistical physics given by Hamiltonians defines machine learning by neural networks. Furthermore, solving inverse problems in physics through machine learning and generalization essentially provides progress and even revolutions in physics. For these reasons, in recent years interdisciplinary research in machine learning and physics has been expanding dramatically. This book is written for anyone who wants to learn, understand, and apply the relationship between deep learning/machine learning and physics. All that is needed to read this book are the basic concepts in physics: energy and Hamiltonians. The concepts of statistical mechanics and the bracket notation of quantum mechanics, which are explained in columns, are used to explain deep learning frameworks.We encourage you to explore this new active field of machine learning and physics, with this book as a map of the continent to be explored.Trade Review“The book has the feel of a graduate thesis. It could be quite useful to a researcher investigating the relationship between ANNs and dynamical physical systems.” (Anoop Malaviya, Computing Reviews, February 16, 2023)Table of ContentsChapter 1: Forewords: Machine learning and physics.- Part I Physical view of deep learning.- Chapter 2: Introduction to machine learning.- Chapter 3: Basics of neural networks.- Chapter 4: Advanced neural networks.- Chapter 5: Sampling.- Chapter 6: Unsupervised deep learning.- Part II Applications to physics.- Chapter 7: Inverse problems in physics.- Chapter 8: Detection of phase transition by machines.- Chapter 9: Dynamical systems and neural networks.- Chapter 10: Spinglass and neural networks.- Chapter 11: Quantum manybody systems, tensor networks and neural networks.- Chapter 12: Application to superstring theory.- Chapter 13: Epilogue.- Bibliography.- Index.

Read more less

Book SynopsisComputation is the process of applying a procedure or algorithm to the solution of a mathematical problem. This book covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum mechanics in computation.Trade Review"A beautiful little book.... It succeeds so well because Geroch believes that 'physics is a human activity' and wants to share some of its joy with others." - Physics Today"

Read more less

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 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 SynopsisIntroduces quaternions for scientists and engineers, and shows how they can be used in a variety of practical situations. This book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. It also presents the conventional and familiar 3 x 3 (9-element) matrix rotation operator.Trade Review"This book will appeal to anyone with an interest in three-dimensional geometry. It is a competent and comprehensive survey... This book is unique in that it is probably the only modern book to treat quaternions seriously... A valuable asset."--Aeronautical Journal "[A] splendid book ... everything one could wish for in a primer. It is also beautifully set out with an attractive layout, clear diagrams, and wide margins with explanatory notes where appropriate. It must be strongly recommended to all students of physics, engineering or computer science."--Peter Rowlands, Contemporary PhysicsTable of ContentsList of FiguresAbout This BookAcknowledgements1Historical Matters32Algebraic Preliminaries133Rotations in 3-space454Rotation Sequences in R[superscript 3]775Quaternion Algebra1036Quaternion Geometry1417Algorithm Summary1558Quaternion Factors1779More Quaternion Applications20510Spherical Trigonometry23511Quaternion Calculus for Kinematics and Dynamics25712Rotations in Phase Space27713A Quaternion Process30314Computer Graphics333Further Reading and References365Index367

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 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

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

Book Synopsis

Read more less

Book Synopsis

Read more less

Book SynopsisSystems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picturTrade Review"[This book] should be read by any mathematician who wants to see something of the exciting connections between geometry and the nonlinear systems of mathematical physics."--Bulletin of the American Mathematical SocietyTable of Contents*FrontMatter, pg. i*CONTENTS, pg. v*PREFACE, pg. vii*INTRODUCTION, pg. 1*CHAPTER 1. The Monopole Equations, pg. 9*CHAPTER 2. Geometry of the Monopole Spaces, pg. 14*CHAPTER 3. Metric of Monopole Spaces, pg. 21*CHAPTER 4. Hyperkahler Property of the Metric, pg. 28*CHAPTER 5. The Twistor Description, pg. 38*CHAPTER 6. Particles and Symmetric Products, pg. 51*CHAPTER 7. The 2-monopole Space, pg. 58*CHAPTER 8. Spectral Radii and the Conformal Structure, pg. 64*CHAPTER 9. The Anti-self-dual Einstein Equations, pg. 70*CHAPTER 10. Some Inequalities, pg. 79*CHAPTER 11. The Metric on M02, pg. 90*CHAPTER 12. Detailed Properties of the Metric, pg. 96*CHAPTER 13. Geodesics on M02, pg. 102*CHAPTER 14. Particle Scattering, pg. 109*CHAPTER 15. Comparisons with KdV Solitons, pg. 116*CHAPTER 16. Background Material, pg. 119*BIBLIOGRAPHY, pg. 129*INDEX, pg. 132

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisFocuses on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices.Table of Contents Introduction Integrability and symmetries of nonlinear differential and difference equations in two independent variables Symmetries as integrability criteria Construction of lattice equations and their Lax pair Transformation groups for quad lattice equations Algebraic entropy of the nonautonomous Boll equations Translation from Russian of R. I. Yamilov, ''On the classification of discrete eqautions'', reference [841] No quad-graph equation can have a generalized symmetry given by the narita-Itoh-Bogoyavlensky equation Bibliography Subject Index

Read more less