Mathematical / Computational / Theoretical physics Books

Book SynopsisThis text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.Trade Review“This book has a large amount of new exercise problems that are uniformly distributed across the text. … this book is a very nice text which will serve well for the undergraduate as well as graduate students and will also become a ready reference for scholars.” (Murli M. Gupta, Mathematical Reviews, April, 2023)“Many of the SIAM Review readership will be interested in NMEPPDE from the standpoint of self-study or classroom education. … NMEPPDE offers the applied mathematics reader nearly a single point of entry to our broad and challenging area. … a bit of open space on the bookshelf could profitably be well filled with a copy of NMEPPDE.” (Robert C. Kirby, SIAM Review, Vol. 65 (1), March, 2023)Table of ContentsFor Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and A Posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- Beyond Coercivity, Consistency and Conformity.- Mixed and Nonconforming Discretization Methods.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Discretization Methods for Convection-Dominated Problems.- An Outlook to Nonlinear Partial Differential Equations.- Appendices.

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Book SynopsisThis book presents recent developments in nonlinear and complex systems. It provides recent theoretic developments and new techniques based on a nonlinear dynamical systems approach that can be used to model and understand complex behavior in nonlinear dynamical systems. It covers information theory, relativistic chaotic dynamics, data analysis, relativistic chaotic dynamics, solvability issues in integro-differential equations, and inverse problems for parabolic differential equations, synchronization and chaotic transient. Presents new concepts for understanding and modeling complex systems Table of Contents1. Preface by Dr. Dimitri Volchenkov2. Chapter 1 . J. A. Tenreiro Machado, Shannon Information Analysis of the Chromosome Code.3. Chapter 2. Dimitri Volchenkov, Veniamin Smirnov, An Unfair Coin of the Standard & Poor’s 5004. Chapter 3. Relativistic chaotic scattering by Juan D. Bernal, Jesus M. Seoane, Miguel A.F. Sanjuan5. Chapter 4. Artificial Intelligence for Studying Perception of Ambiguous Images and Decision-Marking Processes in the Human Brain by Alexander N. Pisarchik, Anastasija E. Runnova, Nikita S. Frolov, and Alexander E. Hramov6. Chapter 5. Fuhong Min, Chuang Li, Multistability Coexistence of Memristive Chaotic System, and the Application in Image Decryption7. Chapter 6: Veniamin Smirnov, Zhuanzhuan Ma, And Dimitri Volchenkov, Extreme Events and Emergency Scales 8. Chapter 7: M. Edelman, Evolution of Systems with Power-Law Memory: Do We Have to Die?9. Chapter 8: Dimitri Volchenkov, Probability Entanglement and Destructive Interference in Biased Coin Tossing10. Chapter 9: Messoud Efendiev, Vitali Vougalter, On the solvability of some systems of integro-differential equations with drift.11. Chapter 10: Vitali Vougalter, Vitaly Volpert, Solvability in The Sense of Sequences For Some Non Fredholm Operators With The Bi-Laplacian 12. Chapter 11: Vitali Vougalter, The Preservation of Nonnegativity of Solutions of A Parabolic System With The Bi-Laplacian

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Book SynopsisThis book introduces the traditional and novel techniques required to study the thermodynamic and transport properties of quark–gluon plasma. In particular, it reviews the construction of improved holographic models for QCD-like confining gauge theories and their applications in the physics of quark–gluon plasma. It also discusses the recent advances in the development of hydrodynamic techniques, especially those incorporating the effects of external magnetic fields on transport. The book is primarily intended for researchers and graduate students with a background in quantum field theory and particle physics but who may not be familiar with the theory of strong interactions and holographic and hydrodynamic techniques required to study said interactions.Table of ContentsIntroduction: AdS/CFT and heavy ion collisions.- Holographic QCD theories.- Improved holographic QCD - construction of the theory.- Thermodynamics and the confinement/deconfinement transition.- Flavor sector.- Hydrodynamics and transport coefficients.- Hard probes.- ihQCD at finite B.- Conclusion and a look ahead.

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Book SynopsisThis book is intended to serve as a basic introduction to scientific computing by treating problems from various areas of physics - mechanics, optics, acoustics, and statistical reasoning in the context of the evaluation of measurements. After working through these examples, students are able to independently work on physical problems that they encounter during their studies. For every exercise, the author introduces the physical problem together with a data structure that serves as an interface to programming in Excel and Python. When a solution is achieved in one application, it can easily be translated into the other one and presumably any other platform for scientific computing. This is possible because the basic techniques of vector and matrix calculation and array broadcasting are also achieved with spreadsheet techniques, and logical queries and for-loops operate on spreadsheets from simple Visual Basic macros. So, starting to learn scientific calculation with Excel, e.g., at High School, is a targeted road to scientific computing. The primary target groups of this book are students with a major or minor subject in physics, who have interest in computational techniques and at the same time want to deepen their knowledge of physics. Math, physics and computer science teachers and Teacher Education students will also find a companion in this book to help them integrate computer techniques into their lessons. Even professional physicists who want to venture into Scientific Computing may appreciate this book.Table of Contents

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Book SynopsisMany people, including physicists, are confused about what the Second Law of thermodynamics really means, about how it relates to the arrow of time, and about whether it can be derived from classical mechanics. They also wonder what entropy really is: Is it all about information? But, if so, then, what is its relation to fluxes of heat?One might ask similar questions about probabilities: Do they express subjective judgments by us, humans, or do they reflect facts about the world, i.e. frequencies. And what notion of probability is used in the natural sciences, in particular statistical mechanics?This book addresses all of these questions in the clear and pedagogical style for which the author is known. Although valuable as accompaniment to an undergraduate course on statistical mechanics or thermodynamics, it is not a standard course book. Instead it addresses both the essentials and the many subtle questions that are usually brushed under the carpet in such courses. As one of the most lucid accounts of the above questions, it provides enlightening reading for all those seeking answers, including students, lecturers, researchers and philosophers of science.Table of ContentsWhat We Need from Thermodynamics.- What Are Probabilities?.- Dynamical Systems.- Statistical Mechanics 1 : The Nature of Equilibrium.- Statistical Mechanics 2: Irreversibility.- Demystifying Entropy.- Comparison with Quantum Mechanics.

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Book SynopsisThis book provides an in-depth and accessible description of special relativity and quantum mechanics which together form the foundation of 21st century physics. A novel aspect is that symmetry is given its rightful prominence as an integral part of this foundation. The book offers not only a conceptual understanding of symmetry, but also the mathematical tools necessary for quantitative analysis. As such, it provides a valuable precursor to more focused, advanced books on special relativity or quantum mechanics.Students are introduced to several topics not typically covered until much later in their education.These include space-time diagrams, the action principle, a proof of Noether's theorem, Lorentz vectors and tensors, symmetry breaking and general relativity. The book also provides extensive descriptions on topics of current general interest such as gravitational waves, cosmology, Bell's theorem, entanglement and quantum computing.Throughout the text, every opportunity is taken to emphasize the intimate connection between physics, symmetry and mathematics.The style remains light despite the rigorous and intensive content. The book is intended as a stand-alone or supplementary physics text for a one or two semester course for students who have completed an introductory calculus course and a first-year physics course that includes Newtonian mechanics and some electrostatics. Basic knowledge of linear algebra is useful but not essential, as all requisite mathematical background is provided either in the body of the text or in the Appendices. Interspersed through the text are well over a hundred worked examples and unsolved exercises for the student.Table of Contents1 Introduction 91.1 The goal of physics . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The connection between physics and mathematics . . . . . . . 101.3 Paradigm shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . 162 Symmetry and Physics 172.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 172.2 What is Symmetry? . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Role of Symmetry in Physics . . . . . . . . . . . . . . . . . . . 182.3.1 Symmetry as a guiding principle . . . . . . . . . . . . . 182.3.2 Symmetry and Conserved Quantities: Noether's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 Symmetry as a tool for simplifying problems . . . . . . 192.4 Symmetries were made to be broken . . . . . . . . . . . . . . 202.4.1 Spacetime symmetries . . . . . . . . . . . . . . . . . . 202.4.2 Parity violation . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Spontaneously broken symmetries . . . . . . . . . . . . 242.4.4 Variational calculations: Lifeguards and light rays . . . 273 Formal Aspects of Symmetry 303.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Symmetries and Operations . . . . . . . . . . . . . . . . . . . 303.2.1 Denition of a symmetry operation . . . . . . . . . . . 303.2.2 Rules obeyed by symmetry operations . . . . . . . . . 323.2.3 Multiplication tables . . . . . . . . . . . . . . . . . . . 353.2.4 Symmetry and group theory . . . . . . . . . . . . . . . 363.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 The identity operation . . . . . . . . . . . . . . . . . . 373.3.2 Permutations of two identical objects . . . . . . . . . . 373.3.3 Permutations of three identical objects . . . . . . . . . 383.3.4 Rotations of regular polygons . . . . . . . . . . . . . . 393.4 Continuous vs discrete symmetries . . . . . . . . . . . . . . . 403.5 Symmetries and Conserved Quantities:Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Supplementary: Variational Mechanics and the Proof of Noether'sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6.1 Variational Mechanics: Principle of Least Action . . . . 423.6.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . 473.6.3 Proof of Noether's Theorem . . . . . . . . . . . . . . . 484 Symmetries and Linear Transformations 524.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Review of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.1 Coordinate free denitions . . . . . . . . . . . . . . . . 534.2.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . 584.2.3 Vector operations in component form . . . . . . . . . . 594.2.4 Position vector . . . . . . . . . . . . . . . . . . . . . . 604.2.5 Dierentiation of vectors: velocity and acceleration . . 624.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 644.3.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Reections . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Linear Transformations and matrices . . . . . . . . . . . . . . 684.4.1 Linear transformations as matrices . . . . . . . . . . . 684.4.2 Identity Transformation and Inverses . . . . . . . . . . 704.4.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 704.4.4 Reections . . . . . . . . . . . . . . . . . . . . . . . . . 724.4.5 Matrix Representation of Permutations of Three Objects 734.5 Pythagoras and Geometry . . . . . . . . . . . . . . . . . . . . 745 Special Relativity I: The Basics 775.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.1 Frames5.2.2 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . 785.2.3 Newtonian Relativity and Galilean Transformations . . 835.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.1 The Fundamental Postulate . . . . . . . . . . . . . . . 855.3.2 The problem with Galilean Relativity . . . . . . . . . . 855.3.3 Michelson-Morley Experiment . . . . . . . . . . . . . . 875.3.4 Maxwell's Equations . . . . . . . . . . . . . . . . . . . 905.4 Summary of Consequences . . . . . . . . . . . . . . . . . . . . 915.5 Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . 925.6 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.6.1 Derivation: . . . . . . . . . . . . . . . . . . . . . . . . 975.6.2 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . 995.6.3 Experimental Conrmation . . . . . . . . . . . . . . . 1015.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Lorentz Contraction . . . . . . . . . . . . . . . . . . . . . . . 1045.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7.3 Proper Length and Proper Distance. . . . . . . . . . . 1045.7.4 Examples: . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Special Relativity II: In Depth 1106.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 1106.2.1 Derivation of general form . . . . . . . . . . . . . . . . 1106.2.2 Properties of Lorentz Transformations . . . . . . . . . 1136.2.3 Lorentzian Geometry . . . . . . . . . . . . . . . . . . . 1166.3 The Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Proper time revisited . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Relativistic Addition of Velocities . . . . . . . . . . . . . . . . 1226.6 Relativistic Doppler Shift . . . . . . . . . . . . . . . . . . . . . 1246.6.1 Non-relativistic Doppler Shift Review . . . . . . . . . . 1246.6.2 Relativistic Doppler Shift . . . . . . . . . . . . . . . . 1246.7 Relativistic Energy and Momentum . . . . . . . . . . . . . . . 1276.7.1 Relativistic Energy Momentum Conservation . . . . . . 1276.7.2 Relativistic Inertia . . . . . . . . . . . . . . . . . . . . 1286.7.3 Relativistic Energy . . . . . . . . . . . . . . . . . . . . 1296.7.4 Relativistic Three-Momentum . . . . . . . . . . . . . . 1296.7.5 Relationship Between Relativistic Energy and Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.7.6 Kinetic energy: . . . . . . . . . . . . . . . . . . . . . . 1306.7.7 Massless particles . . . . . . . . . . . . . . . . . . . . 1316.8 Space-time Vectors . . . . . . . . . . . . . . . . . . . . . . . . 1336.8.1 Position Four-Vector: . . . . . . . . . . . . . . . . . . . 1346.8.2 Four-momentum: . . . . . . . . . . . . . . . . . . . . . 1356.8.3 Null four-vectors . . . . . . . . . . . . . . . . . . . . . 1376.8.4 Relativistic Scattering . . . . . . . . . . . . . . . . . . 1376.8.5 More Examples . . . . . . . . . . . . . . . . . . . . . . 1386.9 Relativistic Units . . . . . . . . . . . . . . . . . . . . . . . . . 1396.10 Symmetry Redux . . . . . . . . . . . . . . . . . . . . . . . . . 1406.10.1 Matrix form of Lorentz Transformations . . . . . . . . 1406.10.2 Lorentz Transformations as a Symmetry Group . . . . 1426.11 Supplementary: Four vectors and tensors in covariant form . . 1437 General Relativity 1497.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1497.2 Problems with Newtonian Gravity . . . . . . . . . . . . . . . . 1497.2.1 Review of Newtonian Gravity . . . . . . . . . . . . . . 1497.2.2 Perihelion Shift of Mercury . . . . . . . . . . . . . . . 1517.2.3 Action at a Distance . . . . . . . . . . . . . . . . . . . 1527.2.4 The Puzzle of Inertial vs Gravitational Mass . . . . . . 1537.3 Einstein's Thinking: the Strong Principle of Equivalence . . . 1537.4 Geometry of Spacetime . . . . . . . . . . . . . . . . . . . . . . 1557.5 Some Consequences of General Relativity: . . . . . . . . . . . 1587.6 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 1597.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1597.6.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . 1607.6.3 Recent Observations . . . . . . . . . . . . . . . . . . . 1617.7 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.3 Observational Evidence . . . . . . . . . . . . . . . . . . 1647.7.4 Further Information . . . . . . . . . . . . . . . . . . . 1667.8 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668 Introduction to the Quantum 1708.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1708.2 Light as particles . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2.1 Review: Light as Waves . . . . . . . . . . . . . . . . . 1718.2.2 Photoelectric Eect . . . . . . . . . . . . . . . . . . . . 1718.2.3 Compton Scattering . . . . . . . . . . . . . . . . . . . 1758.3 Blackbody Radiation and the Ultraviolet Catastrophe . . . . . 1798.3.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . 1798.3.2 Derivation of Rayleigh-Jeans Law . . . . . . . . . . . . 1818.3.3 The ultraviolet catastrophe . . . . . . . . . . . . . . . 1888.3.4 Quantum resolution: . . . . . . . . . . . . . . . . . . . 1898.3.5 The Early Universe: the ultimate blackbody . . . . . . 1918.4 Particles as waves . . . . . . . . . . . . . . . . . . . . . . . . . 1968.4.1 Electron waves . . . . . . . . . . . . . . . . . . . . . . 1968.4.2 de Broglie Wavelength . . . . . . . . . . . . . . . . . . 1978.4.3 Observational Evidence . . . . . . . . . . . . . . . . . . 1998.5 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . 2029 The Wave Function 2049.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 2049.2 Quantum vs Newtonian description of physical states . . . . . 2049.2.1 Newtonian description of the state of a particle . . . . 2059.2.2 Quantum description of the state of a particle . . . . . 2059.3 Physical Consequences and Interpretation . . . . . . . . . . . 2079.4 Measurements of position . . . . . . . . . . . . . . . . . . . . 2089.5 Example: Gaussian wavefunction . . . . . . . . . . . . . . . . 2099.6 \Spooky" Action at a Distance: Non-Locality in QuantumMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.6.1 The EPR \Paradox" . . . . . . . . . . . . . . . . . . . 2119.6.2 Bell's Theorem and the Experimental Repudiation ofEPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21410 The Schrodinger Equation 21710.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 21710.2 Momentum in Quantum Mechanics . . . . . . . . . . . . . . . 21810.2.1 Pure Waves . . . . . . . . . . . . . . . . . . . . . . . . 21810.2.2 The Momentum Operator . . . . . . . . . . . . . . . . 22010.3 Energy in Quantum Mechanics . . . . . . . . . . . . . . . . . 22310.4 The Time Independent Schrodinger Equation . . . . . . . . . 22410.4.1 Stationary States . . . . . . . . . . . . . . . . . . . . . 22410.4.2 The \Quantum" in Quantum Mechanics . . . . . . . . 22610.5 Examples of Stationary States . . . . . . . . . . . . . . . . . . 22610.5.1 Free particle in one dimension . . . . . . . . . . . . . . 22610.5.2 Example 2: Particle in a Box with Impenetrable Walls 22710.5.3 Example 3 : Simple Harmonic Oscillator . . . . . . . . 22910.6 Absorption and emission . . . . . . . . . . . . . . . . . . . . . 23110.7 Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.7.1 Tunnelling through a potential barrier of nite width . 23310.7.2 Particle in a Box with Penetrable Walls . . . . . . . . . 23510.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 23710.7.4 Applications of tunnelling . . . . . . . . . . . . . . . . 23810.8 The Quantum Correspondence Principle . . . . . . . . . . . . 24210.8.1 Recovering the everyday world . . . . . . . . . . . . . . 24210.8.2 The Bohr Correspondence Principle . . . . . . . . . . . 24310.9 The Time Dependent Schrodinger equation . . . . . . . . . . . 24410.9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 24611 The Hydrogen Atom 24911.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 24911.2 Newtonian (Classical) Dynamics . . . . . . . . . . . . . . . . . 24911.3 The Bohr Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 25111.4 Semi-classical spectrum from the Bohr correspondence principle25411.5 Emission and Absorption Spectra . . . . . . . . . . . . . . . . 25411.6 Three Dimensional Hydrogen Atom . . . . . . . . . . . . . . . 25611.6.1 Schrodinger Equation . . . . . . . . . . . . . . . . . . . 25611.6.2 Solutions and Quantum Numbers . . . . . . . . . . . . 25811.6.3 Fermions and the spin quantum number . . . . . . . . 26211.7 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.7.1 Hydrogen-like atoms . . . . . . . . . . . . . . . . . . . 26511.7.2 Chemical Properties and the Periodic Table . . . . . . 26612 Nuclear Physics 27012.1 Properties of the Nucleus . . . . . . . . . . . . . . . . . . . . . 27012.1.1 Mass of Nucleons . . . . . . . . . . . . . . . . . . . . . 27012.1.2 Structure of Nucleus . . . . . . . . . . . . . . . . . . . 27112.1.3 The Nuclear Force . . . . . . . . . . . . . . . . . . . . 27112.2 Binding Energy and Stability . . . . . . . . . . . . . . . . . . 27412.2.1 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 27412.2.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . 27512.2.3 Binding Energy per Nucleon . . . . . . . . . . . . . . . 27512.3 Formation of Elements: A Brief History of the Universe . . . . 27612.4 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27912.4.1 Unstable Isotopes . . . . . . . . . . . . . . . . . . . . . 27912.4.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . 28112.4.3 Beta decay . . . . . . . . . . . . . . . . . . . . . . . . . 28212.4.4 Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . 28312.4.5 Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . 28312.4.6 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 28513 Supplementary: Advanced Topics 28713.1 Quantum Information and Quantum Computation . . . . . . . 28713.2 Relativity and quantum mechanics . . . . . . . . . . . . . . . 28714 Conclusions 28815 Appendix: Mathematical Background 28915.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 28915.2 Probabilities and expectation values . . . . . . . . . . . . . . . 29115.2.1 Discrete Distributions . . . . . . . . . . . . . . . . . . 29115.2.2 Continuous probability distributions . . . . . . . . . . 29215.2.3 Dirac Delta Function . . . . . . . . . . . . . . . . . . . 29615.3 Supplementary: Fourier Series and Transforms . . . . . . . . . 29815.3.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . 29815.3.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . 30015.3.3 The mathematical uncertainty principle . . . . . . . . . 30215.3.4 Dirac Delta Function Revisited . . . . . . . . . . . . . 30315.3.5 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . 30315.4 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30415.4.1 Moving pure waves . . . . . . . . . . . . . . . . . . . . 30415.4.2 Complex Waves . . . . . . . . . . . . . . . . . . . . . . 30515.4.3 Group velocity and phase velocity . . . . . . . . . . . 30515.4.4 Wave packets . . . . . . . . . . . . . . . . . . . . . . . 30715.4.5 Wave number and momentum . . . . . . . . . . . . . . 30915.5 Derivation of Hydrogen Wave Functions . . . . . . . . . . . . 312

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Book SynopsisThis book deals with the interdisciplinary areas of nuclear physics, supernovae and neutron star physics. It addresses the physics and astrophysics of the spectacular supernova explosions, starting with the collapse of massive stars and ending with the birth of neutron stars or black holes. Recent progress in the understanding of core collapse supernova (CCSN) and observational aspects of future detections of neutrinos from CCSN explosions are discussed. The other main focus in this text is the novel phases of dense nuclear matter, its compositions and equation of state (EoS) from low to very high baryon density relevant to supernovae and neutron stars. The multi-messenger astrophysics of binary neutron star merger GW170817 and its relation to EoS through tidal deformability are also presented in detail. The synthesis of elements heavier than iron in the supernova and neutron star environment by the rapid (r)-process are treated here with special emphasis on the nucleosynthesis in the ejected material from GW170817. This monograph is written for graduate students and researchers in the field of nuclear astrophysics.Table of ContentsPREFACE1. INTRODUCTION 2. THEORY OF SUPERNOVA EXPLOSIONS 2.1 Overview- historical 2.2 Supernova Type Ia 2.3 Gravitational collapse and pre-supernova conditions 2.4 Production of neutrinos and their emission 2.5 Shock wave formation and its eventual stalling 2.6 The revival of the shock wave- the neutrino mechanism 2.7 Multi-dimensional hydrodynamic simulations and the present scenario 2.8 The supernova SN1987A 2.9 Detection of neutrinos from future supernova events 3. NEUTRON STARS 3.1 History and discovery of neutron stars 3.2 Observational Constraints on neutron stars 3.3 Compositions and novel phases of neutron stars - crust to core 3.4 Equation of State (EoS) models of neutron star matter 3.5 Relativistic field theoretical models for dense matter at zero and finite temperatures 3.6 Tolman-Oppenheimer-Volkoff Equation and Structures of neutron stars 3.7 A stable branch of compact stars beyond neutron star 3.8 Rotating neutron stars, moment of inertia (I) and quadrupole moment (Q) 3.9 Neutron star matter in strongly quantizing magnetic fields 3.10 EoS tables for supernova and binary neutron star merger simulations 4. BINARY NEUTRON STAR MERGERS 4.1 Gravitational waves as new window into neutron stars 4.2 First binary neutron star (BNS) merger GW170817 and multi-messenger astrophysics 4.3 Tidal deformability, LOVE number and EoS 4.4 I-Love-Q universal relations 4.5 Late inspiral phase of BNS merger, tidal deformability and cold EoS 4.6 Neutron Star radius determination from tidal deformability 4.7 Hot and neutrino trapped merger remnant and finite temperature EoS 5. SYNTHESIS OF HEAVY ELEMENTS IN THE UNIVERSE 5.1 s-, r- and p-processes 5.2 Conditions for production of elements by r- process and the sites 5.3 Electromagnetic counterpart of GW170817 and ejected matter in BNS merger 5.4 Decompression of ejected neutron rich matter in Lattimer and Schramm model 5.5 Kilonova model 5.6 Heavy element synthesis in neutron rich matter ejected in GW170817 INDEX BIBLIOGRAPHY (eventually at chapter-ends)

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Book SynopsisThis book deals with the interdisciplinary areas of nuclear physics, supernovae and neutron star physics. It addresses the physics and astrophysics of the spectacular supernova explosions, starting with the collapse of massive stars and ending with the birth of neutron stars or black holes. Recent progress in the understanding of core collapse supernova (CCSN) and observational aspects of future detections of neutrinos from CCSN explosions are discussed. The other main focus in this text is the novel phases of dense nuclear matter, its compositions and equation of state (EoS) from low to very high baryon density relevant to supernovae and neutron stars. The multi-messenger astrophysics of binary neutron star merger GW170817 and its relation to EoS through tidal deformability are also presented in detail. The synthesis of elements heavier than iron in the supernova and neutron star environment by the rapid (r)-process are treated here with special emphasis on the nucleosynthesis in the ejected material from GW170817. This monograph is written for graduate students and researchers in the field of nuclear astrophysics.Table of ContentsPREFACE1. INTRODUCTION 2. THEORY OF SUPERNOVA EXPLOSIONS 2.1 Overview- historical 2.2 Supernova Type Ia 2.3 Gravitational collapse and pre-supernova conditions 2.4 Production of neutrinos and their emission 2.5 Shock wave formation and its eventual stalling 2.6 The revival of the shock wave- the neutrino mechanism 2.7 Multi-dimensional hydrodynamic simulations and the present scenario 2.8 The supernova SN1987A 2.9 Detection of neutrinos from future supernova events 3. NEUTRON STARS 3.1 History and discovery of neutron stars 3.2 Observational Constraints on neutron stars 3.3 Compositions and novel phases of neutron stars - crust to core 3.4 Equation of State (EoS) models of neutron star matter 3.5 Relativistic field theoretical models for dense matter at zero and finite temperatures 3.6 Tolman-Oppenheimer-Volkoff Equation and Structures of neutron stars 3.7 A stable branch of compact stars beyond neutron star 3.8 Rotating neutron stars, moment of inertia (I) and quadrupole moment (Q) 3.9 Neutron star matter in strongly quantizing magnetic fields 3.10 EoS tables for supernova and binary neutron star merger simulations 4. BINARY NEUTRON STAR MERGERS 4.1 Gravitational waves as new window into neutron stars 4.2 First binary neutron star (BNS) merger GW170817 and multi-messenger astrophysics 4.3 Tidal deformability, LOVE number and EoS 4.4 I-Love-Q universal relations 4.5 Late inspiral phase of BNS merger, tidal deformability and cold EoS 4.6 Neutron Star radius determination from tidal deformability 4.7 Hot and neutrino trapped merger remnant and finite temperature EoS 5. SYNTHESIS OF HEAVY ELEMENTS IN THE UNIVERSE 5.1 s-, r- and p-processes 5.2 Conditions for production of elements by r- process and the sites 5.3 Electromagnetic counterpart of GW170817 and ejected matter in BNS merger 5.4 Decompression of ejected neutron rich matter in Lattimer and Schramm model 5.5 Kilonova model 5.6 Heavy element synthesis in neutron rich matter ejected in GW170817 INDEX BIBLIOGRAPHY (eventually at chapter-ends)

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Book SynopsisThis textbook on Feynman integrals starts from the basics, requiring only knowledge of special relativity and undergraduate mathematics. Feynman integrals are indispensable for precision calculations in quantum field theory. At the same time, they are also fascinating from a mathematical point of view. Topics from quantum field theory and advanced mathematics are introduced as needed. The book covers modern developments in the field of Feynman integrals. Topics included are: representations of Feynman integrals, integration-by-parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, coactions and symbols, cluster algebras, elliptic Feynman integrals, and motives associated with Feynman integrals. This volume is aimed at a) students at the master's level in physics or mathematics, b) physicists who want to learn how to calculate Feynman integrals (for whom state-of-the-art techniques and computations are provided), and c) mathematicians who are interested in the mathematical aspects underlying Feynman integrals. It is, indeed, the interwoven nature of their physical and mathematical aspects that make Feynman integrals so enthralling.Trade Review“This book provides a detailed and up-to-date introduction to Feynman integrals … . The book is written in a very didactic way. … the book gives an excellent introduction to the field of Feynman integrals at the level of a master's/starting Ph.D. student … . The structure of the book and the fact that it contains many exercises make it a very useful resource for a course on this topic.” (Samuel Abreu, Mathematical Reviews, December, 2023)Table of ContentsThe file is attached

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Book SynopsisIntegrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products.Table of Contents1. Review of foundational concepts1.1. Sequences and Series 1.1.1. Sequences of Real Numbers and their Series – sequences, limits, series, convergence, harmonic numbers, summation by parts, change in the order of summation 1.1.2. Power Series and Generating Functions – definitions, radius of convergence, generating function representations of sequences, convolution 1.2. Integrals 1.2.1. Riemann Sums – definition, direct evaluation of certain sums 1.2.2. Fundamental Theorem - definition of definite integral, statement of theorem, verifications 1.2.3. Multiple Integrals – double integrals, conditions for reversal or order of integration 1.3. Evaluation Techniques 1.3.1. Integration by Parts - review 1.3.2. Conversion to Multiple Integrals – “Feynman’s Technique,” replacing a portion of an integrand with an integral representation and reversing the order of integration 1.3.3. Green’s Theorem – review, path integrals and parametrization, Stokes’ Theorem, applications 1.3.4. Partial Fractions review 1.4. Problems 2. Complex Integration 2.1. Analytic Functions 2.1.1. Cauchy-Riemann Conditions – complex functions and their derivatives, defining analytic functions as a direction-independent derivative, harmonic functions 2.1.2. Evaluating Complex Integrals – numerical examples of parametrizations 2.1.3. Path Independence – demonstrate for analytic functions and demonstrate invalidity for nonanalytic integrands 2.2. Cauchy’s Theorems 2.2.1. Winding Numbers – definition in terms of a complex integral 2.2.2. Cauchy’s Integral Theorem – derivation and illustration for a wide variety of integrands and contours 2.2.3. Cauchy’s Theorem – statement, examples, Liouville’s Theorem, Morera’s Theorem 2.3. Useful Results 2.3.1. Taylor Series – review, error analysis in complex plane, convergence 2.3.2. Laurent Series – regions of validity (e.g., annuli), analytic continuation 2.3.3. Argument Principle – derivation for zeroes and poles 2.3.4. Rouche’s Theorem – derivation, illustration for determining poles within integration contours 2.4. Multivalued Functions – branch points, branch cuts, Riemann surfaces 2.5. Problems 3. Evaluation of Real Integrals and Sums 3.1. Preliminary Matters 3.1.1. Poles and Residue Theory – residue definition, residue computation 3.1.2. Essential Singularities – computation of residues of essential singularities 3.1.3. Branch Points – illustration of a unified approach to expressing an integral of a function in terms of its singularities 3.2. Definite Integrals 3.2.1. Integrands Having Both Poles and Branch Points – e.g., integrands featuring logs and exponents less than -1 3.2.2. Integrands Defined Over - insertion of one higher power of log(z) in the integrand, residue backpropagation 3.2.3. Integrands Having Rational Functions of Polynomials and Trigonometric Functions – integration over the unit circle, modifying the unit circle in the presence of singularities, replacing monomial with a branch point in constructing a contour integral 3.2.4. Alternative Contours: Wedges, Rectangles, and Others – reducing the number of singularities in a contour to simplify calculation 3.2.5. Integrands Having Algebraic Functions and the Residue At Infinity – whole new paradigm in evaluating definite integrals with finite limits of an integrand having branch points at the finite limits, defining the residue at infinity, branch point at infinity 3.3. Sums 3.3.1. Complex Integral Representations – selection of integrand and contour to produce sums, demonstration of convergence of complex integral as contour expands to infinity 3.3.2. Examples – rational summands, summands with trigonometric functions 3.4. Problems 4. Cauchy Principal Value 4.1. Integrands Having Poles On the Contour 4.1.1. Definition of a Cauchy Principal Value – definition as a limit, illustration with simple examples 4.1.2. Managing Divergent Terms of a Contour Integral – detailed illustrations of evaluating definite integrals via complex integrals having contributions with divergent terms that cancel 4.2. Analytic Signals and Hilbert Transforms – equivalence of Cauchy-Riemann equations and Hilbert transforms of real and imaginary parts of an analytic function, illustrations of analytic signals having harmonic real and imaginary parts, examples of deriving imaginary parts of analytic function from real part 4.3. Problems 5. Integral Transforms 5.1. Preliminary Matters 5.1.1. The Dirac Delta Function – derivation via self-transform in Hilbert transform integrals, review of properties 5.1.2. A General Discussion of Integral Transforms - integral transforms require a computable inverse to be of any use, conditions under which inverses exist, general format of integral transforms 5.2. The Fourier Transform 5.2.1. Definition and Plancherel’s Theorem – mean square error, and inner product spaces, the Fourier Transform as a Principal Value 5.2.2. Jordan’s Lemma – evaluating Fourier integrals using complex integration, convergence conditions 5.2.3. Parseval’s Theorem – statement, examples of integral evaluations, Fourier series and application of theorem to sums 5.2.4. Convolution Theorem – statement and derivation, applications 5.2.5. Analyticity of the Fourier Transform In the Complex Plane - theorem relating rates of convergence of Fourier transforms and their inverses in the complex plane, strips of convergence, causality 5.2.6. Poisson Sum Formula – derivation, application to computation of error function to machine precision anywhere in the complex plane 5.3. The Laplace Transform 5.3.1. Definition – extending the discussion of analyticity of the Fourier transform with an exponentially decaying kernel rather than an oscillatory kernel, derivation of inverse as an integral in the complex plane 5.3.2. Convolution Theorem – derivation, examples, application to computing certain classes of definite integrals 5.3.3. Inversion Via Complex Integration 5.3.3.1. Solutions to Ordinary Differential Equations and Rational Transforms – initial conditions, homogeneous and inhomogeneous equations, inversion via the residue theorem 5.3.3.2. Solutions to Partial Differential Equations and Multivalued Transforms – heat equation produces multivalued transforms, evaluation of inverse Laplace transforms to derive solutions 5.4. The Mellin Transform 5.4.1. Definition discussion of strip of convergence, inverse Mellin transform 5.4.2. Convolution Theorem – derivation; NB this will be used in the next chapter 5.4.3. Scaling – expression of scaled integrals in terms of residues 5.5. Problems 6. Asymptotic Analysis 6.1. Definitions 6.1.1. Big-O, Little-O, and The Squiggle – i.e., definitions of asymptotic equivalence in specific limits 6.1.2. Asymptotic Series – definition, properties, numerical calculations, summation acceleration techniques 6.2. Integration by Parts – development of asymptotic series; limitations 6.2.1. Euler-Maclurin Formula – derivation of asymptotic series using integration by parts, application to evaluation of sums 6.3. Watson’s Lemma and h-Transforms – asymptotic behavior of monotonic integrands, application of Mellin transforms in derivation 6.3.1. Application to Complex Integration – evaluation of integrals with branch points at infinity using h-transforms 6.4. Laplace’s Method – asymptotic behavior of nonmonotomic, nonoscillatory integrals 6.5. The Method of Steepest Descents – deriving asymptotic behavior of complex integrals, derive behavior of real integrals by using Cauchy’s Theorem 6.6. Problems

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Book SynopsisDa Esiodo ad Orazio.- 2 Sull'unità della conoscenza.- 3 Archelao.- 4 Sulle tracce di Pitagora.- 5 Dio salvi Ippaso!.- 6 Gli atomi geometrici non esistono.- 7 Su filosofia e scienza.- 8 Filosofia e condizione femminile.- 9 La terza parte.- 10 Eudosso di Cnido.- 11 Archimede e la scienza.- 12 Arte e scienza.- 13 Sul ruolo del De Rerum Natura.- 14 Roghi di persone, di libri e di idee.- 15 Dante, i matematici ed i filosofi.- 16 Ma perché si chiama "seno"?.- 17 Un gesuita euclideo.- 18 Contare gli atomi?.- 19 Le radici del metodo scientifico.- 20 Ancora sul metodo scientifico.- 21 Scienza greca (o no?).- 22 Dai suoni all'antimateria.- A I poliedri regolari.- B Scala pitagorica.- C Medie pitagoriche.- D La grandezza degli atomi.- E I blocchi della piramide.- F Sulla nozione di numero reale.- G Sulla congettura di Archimede.- H Alcune illusioni ottiche.- Ringraziamenti.- Riferimenti bibliografici.- Indice delle persone.

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Book SynopsisChapter 1. Introduction.- Chapter 2. Fibred manifolds.- Chapter 3. Vector fields and differential forms.- Chapter 4. Calculus of variations.- Chapter 5. Dynamical forms and the inverse problem.- Chapter 6. Hamiltonian systems.- Chapter 7. Elements of the variational sequences.- Chapter 8. Extension: Geometrical structures for field theories.- Chapter 9. Variational physics.- Chapter 10. Appendix.

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Book SynopsisIntroduction.- Capitolo 1 La teoria della Relatività di Einstein: una nuova visione del mondo.- Capitolo 2 La Meccanica Quantistica: il bizzarro mondo atomico e subatomico.- Capitolo 3 Materia condensata e l'impatto tecnologico della prima rivoluzione quantistica.- Capitolo 4 La seconda rivoluzione quantistica e le tecnologie quantistiche.

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Book SynopsisIntroduction.- Fluctuations Relations and Large Deviations.- Examples.- Two Fluctuation Theorems for Chaotic Maps.- Examples.

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Book SynopsisThe primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers’ convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.Trade ReviewFrom the reviews:“The book is well written and not unnecessarily wordy. There is an up-to-date bibliography and a nice index. … a mathematician who wishes to know what the important issues concerning eq. (1) are and what has been achieved, would find this an excellent source. Equally, a mathematically-minded student, with a good grounding in analysis and who has decided to work in this area, or the teacher who wants to teach a course on this material would find this a valuable text.” (P. N. Shankar,Current Science, Vol. 85 (2), July, 2003)Table of ContentsPreliminary Results.- The Stationary Navier-Stokes Equations.- The Linearized Nonstationary Theory.- The Full Nonlinear Navier-Stokes Equations.

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Book SynopsisMany nonlinear systems around us can generate a very complex and counter-intuitive dynamics that contrasts with their simplicity, but their understanding requires concepts that are outside the basic training of most science students. This textbook, which is the fruit of graduate courses that the authors have taught at their respective universities, provides a richly illustrated introduction to nonlinear dynamical systems and chaos and a solid foundation for this fascinating subject. It will satisfy those who want discover this field, including at the undergraduate level, but also those who need a compact and consistent overview, gathering the concepts essential to nonlinear scientists.The first and second chapters describe the essential concepts needed to describe nonlinear dynamical systems as well as their stability. The third chapter introduces the concept of bifurcation, where the qualitative dynamical behavior of a system changes. The fourth chapter deals with oscillations, from their birth to their destabilization, and how they respond to external driving. The fifth and sixth chapters discuss complex behaviors that only occur in state spaces of dimension three and higher: quasi-periodicity and chaos, from their general properties to quantitative methods of characterization. All chapters are supplemented by exercises ranging from direct applications of the notions introduced in the corresponding chapter to elaborate problems involving concepts from different chapters, as well as numerical explorations.

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Book SynopsisHow does a quantum computer work and how can photons be used to transmit messages securely? Intended for engineering and computer science students, this introduction to quantum technologies presents the fundamentals of quantum computing, quantum communication, and quantum sensing without requiring extensive previous knowledge of physics.

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Book SynopsisData Management for Natural Scientists offers a practical guide for scientific processing of data. It covers the way from “getting hands on” experimental results to ensuring their use for addressing various scientific questions. Code snippets are provided in order to introduce the proposed workstream and to demonstrate the adjustability to specific challenges.

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Book SynopsisThis book offers an essential bridge between college-level introductions and advanced graduate-level books on special relativity. It begins at an elementary level, presenting and discussing the basic concepts normally covered in college-level works, including the Lorentz transformation. Subsequent chapters introduce the four-dimensional worldview implied by the Lorentz transformations, mixing time and space coordinates, before continuing on to the formalism of tensors, a topic usually avoided in lower-level courses. The book’s second half addresses a number of essential points, including the concept of causality; the equivalence between mass and energy, including applications; relativistic optics; and measurements and matter in Minkowski space-time. The closing chapters focus on the energy-momentum tensor of a continuous distribution of mass-energy and its co-variant conservation; angular momentum; a discussion of the scalar field of perfect fluids and the Maxwell field; and general coordinates.Every chapter is supplemented by a section with numerous exercises, allowing readers to practice the theory. These exercises constitute an essential part of the textbook, and the solutions to approximately half of them are provided in the appendix.Trade ReviewFrom the reviews:“The book is one of the best texts in special relativity designed for readers between the college-level and advanced level. … A number of useful and new examples is added at the end of every chapter of the book. … A very useful table of constants is added at the end of the book. … The book represents one of the best conspects in special relativity and is useful for professors of special relativity. It is good for students and every other reader.” (Alex Gaina, zbMATH, Vol. 1277, 2014)Table of ContentsFundamentals of Special Relativity.- Introduction.- The Principle of Relativity.- Groups—the Galilei group.- Galileian law of addition of velocities.- The lesson from electromagnetism.- The postulates of Special Relativity.- Consequences of the postulates.- Conclusion.- Problems.- The Lorentz transformation.- Introduction.- The Lorentz transformation.- Derivation of the Lorentz transformation.- Mathematical properties of the Lorentz transformation.- Absolute speed limit and causality.- Length contraction from the Lorentz transformation.- Time dilation from the Lorentz transformation.- Transformation of velocities and accelerations in Special Relativity.- Matrix representation of the Lorentz transformation.- The Lorentz group.- The Lorentz transformation as a rotation by an imaginary angle with imaginary time.- The GPS system.- Conclusion.- Problems.- The 4-dimensional world view.- Introduction.- The 4-dimensional world.- Spacetime diagrams.- Conclusion.- Problems.- The formalism of tensors.- Introduction.- Vectors and tensors.- Contravariant and covariant vectors.- Contravariant and covariant tensors.- Tensor algebra.- Tensor fields.- Index-free description of tensors.- The metric tensor.- The Levi-Civita symbol and tensor densities.- Conclusion.- Problems.- Tensors in Minkowski spacetime.- Introduction.- Vectors and tensors in Minkowski spacetime.- The Minkowski metric.- Scalar product and length of a vector in Minkowski spacetime.- Raising and lowering tensor indices.- Causal nature of 4-vectors.- Hypersurfaces.- Gauss’ theorem.- Conclusion.- Problems.- Relativistic mechanics.- Introduction.- Relativistic dynamics of massive particles.- The relativistic force.- Angular momentum of a particle.- Particle systems.- Conservation of mass-energy.- Conclusion.- Problems.- Relativistic optics.- Introduction.- Relativistic optics: null rays.- The drag effect.- The Doppler effect.- Aberration.- Relativistic beaming.- Visual appearance of extended objects.- Conclusion.- Problems.- Measurements in Minkowski spacetime.- Introduction.- Energy of a particle measured by an observer.- Frequency measured by an observer.- A more systematic treatment of measurement.- The 3+1 splitting.- Conclusion.- Problems.- Matter in Minkowski spacetime.- Introduction.- The energy-momentum tensor.- Covariant conservation.- Energy conditions.- Angular momentum.- Perfect fluids.- The scalar field.- The electromagnetic field.- Conclusion.- Problems.- Special Relativity in arbitrary coordinates.- Introduction.- The covariant derivative.- Spacetime curves and covariant derivative.- Physics in Minkowski spacetime revisited.- Conclusions.- Problems.- Solutions to selected problems.- References.- Index.

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Book SynopsisA certain curious feature of random objects, introduced by the author as “super concentration,” and two related topics, “chaos” and “multiple valleys,” are highlighted in this book. Although super concentration has established itself as a recognized feature in a number of areas of probability theory in the last twenty years (under a variety of names), the author was the first to discover and explore its connections with chaos and multiple valleys. He achieves a substantial degree of simplification and clarity in the presentation of these findings by using the spectral approach.Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012.The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.Table of ContentsPreface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand’s method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.

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Book SynopsisThis small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future.At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas. These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible.There is another aspect to Geometric Algebra, which is less tangible, and goes beyond questions of mathematical power and range. This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics. Examples of this are peppered throughout ‘Space-Time Algebra’, despite its short length, and some of them are effectively still research topics for the future.From the Foreward by Anthony LasenbyTable of ContentsPreface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure of Clifford Algebra.- 5.Reversion, Scalar Product.- 6.The Algebra of Space.- 7.The Algebra of Space-Time.- Part II:Electrodynamics.- 8.Maxwell's Equation.- 9.Stress-Energy Vectors.- 10.Invariants .- 11. Free Fields.- Part III:Dirac Fields.- 12.Spinors.- 13.Dirac's Equation.- 14.Conserved Currents.- 15.C, P, T.- Part IV:Lorentz Transformations.- 16.Reflections and Rotations.- 17.Coordinate Transformations.- 18.Timelike Rotations.- 19.Scalar Product.- Part V:Geometric Calculus.- 20.Differentiation.- 21.Coordinate Transformations.- 22.Integration.- 23.Global and Local Relativity.- 24.Gauge Transformation and Spinor Derivatives.- Conclusion.- Appendices.- A.Bases and Pseudoscalars.- B.Some Theorems.- C.Composition of Spacial Rotations.- D.Matrix Representation of the Pauli Algebra.

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Book SynopsisThis book – specifically developed as a novel textbook on elementary classical mechanics – shows how analytical and numerical methods can be seamlessly integrated to solve physics problems. This approach allows students to solve more advanced and applied problems at an earlier stage and equips them to deal with real-world examples well beyond the typical special cases treated in standard textbooks.Another advantage of this approach is that students are brought closer to the way physics is actually discovered and applied, as they are introduced right from the start to a more exploratory way of understanding phenomena and of developing their physical concepts.While not a requirement, it is advantageous for the reader to have some prior knowledge of scientific programming with a scripting-type language. This edition of the book uses Python, and a chapter devoted to the basics of scientific programming with Python is included. A parallel edition using Matlab instead of Python is also available.Last but not least, each chapter is accompanied by an extensive set of course-tested exercises and solutions.Table of ContentsIntroduction.- Getting started with programming.- Units and measurement.- Motion in one dimension.- Forces in one dimension.- Motion in two and three dimensions.- Forces in two and three dimensions.- Constrained motion.- Forces and constrained motion.- Work.- Energy.- Momentum, impulse, and collisions.- Multiparticle systems.- Rotational motion.- Rotation of rigid bodies.- Dynamics of rigid bodies.- Proofs.- Solutions.- Index.

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Book SynopsisThis monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets.The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.Trade Review“This monograph gives a complete and detailed account of the most recent techniques developed to obtain a precise mathematical description of the phenomenon of metastability. … The book is well organized and well written, it contains a large amount of fundamental ideas and techniques, and it is an important reference for any researcher interested in the study of long-time behavior of Markov processes and applications to statistical mechanics.” (Jean-Baptiste Bardet, Mathematical Reviews, April, 2017)“No doubt, this is a fundamental book written by well established scientists whose contribution to this area is widely recognized. The book is addressed to readers with serious mathematical background and interests in metastability of stochastic dynamical systems. The books is also an excellent references source.” (Jordan M. Stoyanov, zbMATH 1339.60002, 2016)Table of ContentsPart I Introduction.- 1.Background and motivation.- 2.Aims and scopes.- Part II Markov processes 3.Some basic notions from probability theory.- 4.Markov processes in discrete time.- 5.Markov processes in continuous time.- 6.Large deviations.- 7.Potential theory.- Part III Metastability.- 8.Key definitions and basic properties.- 9.Basic techniques.- Part IV Applications: Diffusions with small noise.- 10.Discrete reversible diffusions.- 11.Diffusion processes with gradient drift.- 12.Stochastic partial differential equations.- Part V Applications: Coarse-graining at positive temperatures.- 13.The Curie-Weiss model.- 14.The Curie-Weiss model with a random magnetic field: discrete distributions.- 15.The Curie-Weiss model with random magnetic field: continuous distributions.- Part VI Applications: Lattice systems in small volumes at low temperatures.- 16.Abstract set-up and metastability in the zero-temperature limit.- 17.Glauber dynamics.- 18.Kawasaki dynamics.- Part VII Applications: Lattice systems in large volumes at low temperatures.- 19.Glauber dynamics.- 20.Kawasaki dynamics.- Part VIII Applications: Lattice systems in small volumes at high densities.- 21.The zero-range process.- Part IX Challenges.- 22.Challenges within metastability.- 23.Challenges beyond metastability.- References.-Glossary.- Index.

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Book SynopsisOn a geometric quantization scheme generalizing those of Kostant-Souriau and Czyz.- Further applications of geometric quantization.- General vector field representations of local Heisenberg systems.- Aspects of relativistic quantum mechanics on phase space.- On the confinement of magnetic poles.- SU(3) and SU(4) as spectrum-generating groups.- The phase space for the Yang-Mills equations.- Instantons in nonlinear ?-models, gauge theories and general relativity.- Gauge-theoretical foundation of color geometrodynamics.- Non-associative algebras and exceptional gauge groups.- Atiyah-Singer index theorem and quantum field theory.- Topological concepts in phase transition theory.- Life without T2.- Affine model of internal degrees of freedom in a non-euclidean space.- Jet bundles and weyl geometry.- Line fields and Lorentz manifolds.- The manifold of embeddings of a closed manifold.- The manifold of embeddings of a non-compact manifold.Table of ContentsOn a geometric quantization scheme generalizing those of Kostant-Souriau and Czyz.- Further applications of geometric quantization.- General vector field representations of local Heisenberg systems.- Aspects of relativistic quantum mechanics on phase space.- On the confinement of magnetic poles.- SU(3) and SU(4) as spectrum-generating groups.- The phase space for the Yang-Mills equations.- Instantons in nonlinear ?-models, gauge theories and general relativity.- Gauge-theoretical foundation of color geometrodynamics.- Non-associative algebras and exceptional gauge groups.- Atiyah-Singer index theorem and quantum field theory.- Topological concepts in phase transition theory.- Life without T2.- Affine model of internal degrees of freedom in a non-euclidean space.- Jet bundles and weyl geometry.- Line fields and Lorentz manifolds.- The manifold of embeddings of a closed manifold.- The manifold of embeddings of a non-compact manifold.

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Table of ContentsContinuum mechanics and geometric integration theory.- Structure of continuum physics.- On differentiable spaces.- Cartesian closed categories and analysis of smooth maps.- to synthetic differential geometry, and a synthetic theory of dislocations.- Synthetic reasoning and variable sets.- Recent research on the foundations of thermodynamics.- Global and local versions of the second law of thermodynamics.- Thermodynamics and the hahn-banach theorem.- What is the length of a potato?.

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Book SynopsisIn this book we describe the elementary theory of operator algebras and parts of the advanced theory which are of relevance, or potentially of relevance, to mathematical physics. Subsequently we describe various applications to quantum statistical mechanics. At the outset of this project we intended to cover this material in one volume but in the course of develop­ ment it was realized that this would entail the omission ofvarious interesting topics or details. Consequently the book was split into two volumes, the first devoted to the general theory of operator algebras and the second to the applications. This splitting into theory and applications is conventional but somewhat arbitrary. In the last 15-20 years mathematical physicists have realized the importance of operator algebras and their states and automorphisms for problems of field theory and statistical mechanics. But the theory of 20 years aga was largely developed for the analysis of group representations and it was inadequate for many physical applications. Thus after a short honey­ moon period in which the new found tools of the extant theory were applied to the most amenable problems a longer and more interesting period ensued in which mathematical physicists were forced to redevelop the theory in relevant directions. New concepts were introduced, e. g. asymptotic abelian­ ness and KMS states, new techniques applied, e. g. the Choquet theory of barycentric decomposition for states, and new structural results obtained, e. g. the existence of a continuum of nonisomorphic type-three factors.Table of Contents(Volume 1).- C*-Algebras and von Neumann Algebras.- Groups, Semigroups, and Generators.- Decomposition Theory.- References.- Books and Monographs.- Articles.- List of Symbols.

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Book SynopsisSoliton theory, the theory of nonlinear waves in lattices composed of particles interacting by nonlinear forces, is treated rigorously in this book. The presentation is coherent and self-contained, starting with pioneering work and extending to the most recent advances in the field. Special attention is focused on exact methods of solution of nonlinear problems and on the exact mathematical treatment of nonlinear lattice vibrations. This new edition updates the material to take account of important new advances.Table of Contents1. Introduction.- 1.1 The Fermi-Pasta-Ulam Problem.- 1.2 Hénon-Heiles Calculation.- 1.3 Discovery of Solitons.- 1.4 Dual Systems.- 2. The Lattice with Exponential Interaction.- 2.1 Finding of an Integrable Lattice.- 2.2 The Lattice with Exponential Interaction.- 2.3 Periodic Solutions.- 2.4 Solitary Waves.- 2.5 Two-Soliton Solutions.- 2.6 Hard-Sphere Limit.- 2.7 Continuum Approximation and Recurrence Time.- 2.8 Applications and Extensions.- 2.9 Poincaré Mapping.- 2.10 Conserved Quantities.- 3. The Spectrum and Construction of Solutions.- 3.1 Matrix Formalism.- 3.2 Infinite Lattice.- 3.3 Scattering and Bound States.- 3.4 The Gel’fand-Levitan Equation.- 3.5 The Initial Value Problem.- 3.6 Soliton Solutions.- 3.7 The Relationship Between the Conserved Quantities and the Transmission Coefficient.- 3.8 Extensions of the Equations of Motion and the Kac-Moerbeke System.- 3.9 The Bäcklund Transformation.- 3.10 A Finite Lattice.- 3.11 Continuum Approximation.- 4. Periodic Systems.- 4.1 Discrete Hill’s Equation.- 4.2 Auxiliary Spectrum.- 4.3 Relation Between ?j (k) and ?j (0).- 4.4 Related Integrals on the Riemann Surface.- 4.5 Solution to the Inverse Problem.- 4.6 Time Evolution.- 4.7 A Simple Example (A Cnoidal Wave).- 4.8 Periodic System of Three-Particles.- 5. Application of the Hamilton-Jacobi Theory.- 5.1 Canonically Conjugate Variables.- 5.2 Action Variables.- 6. Recent Advances in the Theory of Nonlinear Lattices.- 6.1 The KdV Equation as a Limit of the TL Equation.- 6.2 Interacting Soliton Equations.- 6.3 Integrability.- 6.4 Generalization of the TL Equation.- 6.5 Two-Dimensional TL.- 6.6 Bethe Ansatz.- 6.7 The Thermodynamic Limit.- 6.8 Hierarchy of Nonlinear Equations.- 6.9 Some Numerical Results.- Appendices.- Simplified Answers to Main Problems.- References.- List of Authors Cited in Text.

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Book SynopsisLet us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne­ tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.Table of ContentsI. Eleven coordinate systems.- II. Transformations in the complex plane.- III. Cylindrical systems.- IV. Rotational systems.- V. The vector Helmholtz equation.- VI. Differential equations.- VII. Functions.- Appendix. Symbols.- Author Index.

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Book SynopsisHigh resolution upwind and centered methods are a mature generation of computational techniques. They are applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. For its third edition the book has been thoroughly revised to contain new material.Table of ContentsThe Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014;linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014;State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013;Order and TVD Methods for Scalar Equations.- High#x2013;Order and TVD Schemes for Non#x2013;Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013;Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.

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Book SynopsisUnderstanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. Significant progress in the understanding of such systems has been made recently. These books present the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications.Table of Contentsto the Theory of Linear Operators.- to Quantum Statistical Mechanics.- Elements of Operator Algebras and Modular Theory.- Quantum Dynamical Systems.- The Ideal Quantum Gas.- Topics in Spectral Theory.

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Book SynopsisFew books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms. From the reviews: "Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation." --SIAM REVIEWTrade ReviewFrom the reviews: "Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation … . The new edition is highly recommended as a general reference for the essential theory of ordinary differential equations and as a textbook for an introductory course for serious undergraduate or graduate students. … In the US system, it is an excellent text for an introductory graduate course." (Carmen Chicone, SIAM Review, Vol. 49 (2), 2007) "Vladimir Arnold’s is a master, not just of the technical realm of differential equations but of pedagogy and exposition as well. … The writing throughout is crisp and clear. … Arnold’s says that the book is based on a year-long sequence of lectures for second-year mathematics majors in Moscow. In the U.S., this material is probably most appropriate for advanced undergraduates or first-year graduate students." (William J. Satzer, MathDL, August, 2007)Table of ContentsBasic Concepts.- Basic Theorems.- Linear Systems.- Proofs of the Main Theorems.- Differential Equations on Manifolds.

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Book SynopsisBryce DeWitt, a student of Nobel Laureate Julian Schwinger, was himself one of the towering figures in 20th century physics, particularly renowned for his seminal contributions to quantum field theory, numerical relativity and quantum gravity. In late 1971 DeWitt gave a course on gravitation at Stanford University, leaving almost 400 pages of detailed handwritten notes. Written with clarity and authority, and edited by his former student Steven Christensen, these timeless lecture notes, containing material or expositions not found in any other textbooks, are a gem to be discovered or re-discovered by anyone seriously interested in the study of gravitational physics.Trade ReviewFrom the reviews:“DeWitt’s lectures cover interesting and detailed material which is rarely found in other text books. It is a book for the advanced reader.” (Norbert Dragon, General Relativity and Gravitation, Vol. 44, 2012)Table of ContentsReview of the Uses of Invariants in Special Relativity.- Accelerated Motion in Special Relativity.- Realization of Continuous Groups.- Riemannian Manifolds.- The Free Particle Geodesics.- Weak Field Approximation. Newton`s Theory.- Ensembles of Particles.- Production of Gravitational Fields by Matter.- Conservation Laws.- Phenomenological Description of a Conservative Continuous Medium.- Solubility of the Einstein and Matter Equations.- Energy, Momentum and Stress in the Gravitational Field.- Measurement of Asymptotic Field.- The Electromagnetic Field.- Gravitational Waves.- Spinning Bodies.- Weak Field Gravitational Wave.- Stationary Spherically (or Rotationally) Symmetric Metric.- Kerr Metric Subcalculations.- Friedmann Cosmology.- Dynamical Equations and Diffeomorphisms.

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Book SynopsisChoice Outstanding Title! (January 2006) This richly illustrated text covers the Cauchy and Neumann problems for the classical linear equations of mathematical physics. A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging! What makes the book unique is Arnold's particular talent at holding a topic up for examination from a new and fresh perspective. He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject. No other mathematical writer does this quite so well as Arnold.Trade ReviewFrom the reviews of the German edition: "This book provides an introductory text (in German) to basic partial differential equations, based on the author's lectures at Moscow University. […] Most of the standard themes are treated (see list below), but some unusual topics are covered as well. For instance, in chapter 10 double layer potentials are considered, and chapters 11 and 13 deal (among others) with Maxwell's theorem on the multipole expansion of spherical functions. The style of the book is quite non-technical (it contains almost no estimates), taking a mainly geometric viewpoint. [...]" Markus Kunze, Zentralblatt für Mathematik 1076.35001 From the reviews: "[...] This excellent and stimulating textbook gives a beautiful first view on some basic aspects of the theory of partial differential equations and can be warmly recommended to any graduate student in mathematics and physics." M.Günther, Zeitschrift für Angewandte Analysis und Ihre Anwendungen, Vol. 24, Issue 4, 2005 "…..Arnold .. has long held a reputation as one of the world's leaders in dynamics and geometry. His Lectures survey big ideas; accordingly, he largely suppresses both the functional analytic formalism and the delicate estimates so characteristic of the subject. He takes the viewpoint that the most important PDEs arise in physics and the most important mathematical ideas contributing to their solution derive from physical principles. Amold concentrates on the simplest equations of a given type and shows how the key ideas play out. For example, he attacks the general theory of one first-order equation, first via wave-particle duality, then via Hamiltonian dynamics. .... The author's stature and the book's lucidity make this an essential acquisition for all College libraries. …." D.V.Feldman, CHOICE, January 2005 Vol. 42 No. 05 "... Professor Arnold's Lectures on Partial Differential Equations is an ambitious, intensely personal effort to reconnect the subject with some of its roots in modeling physical processes. He does so in a lively lecture-style format, resulting in a book that would complement almost any course in PDEs. ... As can be gleaned from the previous paragraph, we bouth found the book by V.I.Arnold most stimulating and thought provoking, leading to statements such as, "It has been years since I enjoyed a book so much" by RBG and "I cannot point to any other book in mathematics written with the same intensity" by EAT. ... ... what follows [...] is a beautiful book on PDEs, interwoven with the exposition of deep physical, geometrical, and topological insights that contribute to both the understanding and history of PDEs. Prof. Arnold's book ... connects with the roots of the field and brings in concepts from geometry, continuum mechanics, and analysis. It can be used together with any book on PDEs and students will welcome its directness and freshness. We know of no other book like it on the market and highly recommend it for individual reading and as an accompaniment to any course in PDEs. ..." R.B. Guenther, E.A.Thomann, SIAM Review, Vol. 47, No. 1, 2005 "This book contains the transcripts of twelve lectures on partial differential equations … . The presentation gives a vivid sense of what was actually said and discussed in the lecture course, and in this fashion the book differs markedly from many text books with similar titles. … The author uses physical intuition to derive the various mathematical theories, and is thus able to explain the ideas … in a fashion which is clear and helpful to both novice and expert." M. Groves, ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 85 (4), 2005 "[...] In brief, this book contains beautifully structured lectures on classical theory of linear partial differential equations of mathematical physics. Professor Arnold stresses the importance of physical intuitions and offers in his lecture a deep geometric insight into these equations. The book is highly recommended to anybody interested in partial differential equations as well as those involved in lecturing on these topics. I encourage readers of this book to take note of the Preface which contains very interesting comments on the role of Bourbaki's group in mathematics, a theme which resurfaces many times in these lectures." J.Chabrowski, Gazette, Australian Mathematical Society, Vol. 31, Issue 5, 2004 "... As a result the author has aimed to impart to students with pre-knowledge of only a basic kind (linear algebra, basic analysis, ordinary differential equations, ...) the essence of the theory and applications of the subject of partial differential equations. Of course the subject is fundamental in mathematics and in physics and the author is an evangelist for keeping the subject mainstream for mathematicians and for physicists. He has attempted, he writes, to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation! This is successfully done, so that this book should prove attractive in length and in scope to its target readership. ... In this new excellent text are included a large number of interesting problems; at the end of the book there is a full set of problems from examinations given in Moscow. ..." F.H.Berkshire, Imperial College London, Contemporary Physics 2004, Vol. 45, Issue 6 "Like all Vladimir Arnold’s books, this one is full of geometric insight. Arnold illustrates every principle with a figure. This book aims to cover the most basic parts of the subject … . A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging!" L’Enseignement Mathematique, Vol. 50 (1-2), 2004 "Dieses Buch betont geometrische Einsicht und physikalische Intuition. Die Prinzipien werden an Bildern erläutert, und das Buch enthält mehr Text als Formeln und Sätze. […]. Neben einer großen Anzahl von Übungsaufgaben, die im Buch verstreut sind, finden sich interessante Prüfungsbeispiele der Moskauer Universität." J. Hertling, Internationale Mathematische Nachrichten, 2004, Issue 197, p. 47-48 "The book is based on a short course of lectures delivered to the third year mathematics students of the Independent University of Moscow … . The book can serve as a nonstandard, geometrically motivated introduction to PDEs for students … . It is, probably, worth mentioning that the introduction contains some general philosophical views of the author on the subject of PDEs and modern mathematics as a whole and will be of interest to a broad mathematical audience." (Victor Shubov, MathDL, January, 2001) "Like other books of Arnold, this is a very original introduction to the subject. It is … based on a course delivered to third-year students of mathematics. The aim of this book is to teach the fundamental ideas of partial differential equations and mathematical physics. … Not only students but also professional mathematicians from other fields of mathematics can learn the basic and simple ideas of partial differential equations from this unique book." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 74, 2008)Table of Contents1. The General Theory for One First-Order Equation.- 2. The General Theory for One First-Order Equation (Continued).- 3. Huygens’ Principle in the Theory of Wave Propagation.- 4. The Vibrating String (d’Alembert’s Method).- 5. The Fourier Method (for the Vibrating String).- 6. The Theory of Oscillations. The Variational Principle.- 7. The Theory of Oscillations. The Variational Principle (Continued).- 8. Properties of Harmonic Functions.- 9. The Fundamental Solution for the Laplacian. Potentials.- 10. The Double-Layer Potential.- 11. Spherical Functions. Maxwell’s Theorem. The Removable Singularities Theorem.- 12. Boundary-Value Problems for Laplace’s Equation. Theory of Linear Equations and Systems.- A. The Topological Content of Maxwell’s Theorem on the Multifield Representation of Spherical Functions.- A.1. The Basic Spaces and Groups.- A.2. Some Theorems of Real Algebraic Geometry.- A.3. From Algebraic Geometry to Spherical Functions.- A.4. Explicit Formulas.- A.6. The History of Maxwell’s Theorem.- Literature.- B. Problems.- B.1. Material from the Seminars.- B.2. Written Examination Problems.

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

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Book SynopsisSince its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.Trade Review"... These three bulky volumes [EMS 124, 125, 127], written by one of the most prominent researchers of the area, provide an introduction to this repidly developing theory. ... These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of matematics. Furthermore, they should be on the bookshelf of every researcher of the area." (László Kérchy, Acta Scientiarum Mathematicarum, Vol. 69, 2003)Table of ContentsFundaments of Banach Algebras and C*-Algebras.- Topologies and Density Theorems in Operator Algebras.- Conjugate Spaces.- Tensor Products of Operator Algebras and Direct Integrals.- Types of von Neumann Algebras and Traces.- Appendix: Polish Spaces and Standard Borel Spaces.

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Book SynopsisThis book is a new edition of Volumes 3 and 4 of Walter Thirring’s famous textbook on mathematical physics. The first part is devoted to quantum mechanics and especially to its applications to scattering theory, atoms and molecules. The second part deals with quantum statistical mechanics examining fundamental concepts like entropy, ergodicity and thermodynamic functions.Trade ReviewFrom the reviews of the second edition: "Just as the general theory of relativity leads to many new mathematical advances and applications, the same is true of quantum mechanics. It is these mathematical advances that are the topic of this extensive volume, a volume which also delineates how these advances made possible the difficult transition from understanding hydrogen to understanding complex atoms, molecules, and ‘large systems’. As such this volume will serve as an excellent source book for the mathematical basis of the many recent advances in quantum mechanics. It will also serve as an excellent text book for an advanced course in either quantum physics or applied mathematics." (Physicalia, 25/3, 2003) "This work is written uncompromisingly for the mathematical physicist … . Thirring writes concisely but with a clarity that makes the book easy to read. … There are extensive bibliographies, with references mostly to articles in journals … . There are copious problems and–even better-all the solutions. … the volume would make a valuable addition to the library of … a mathematical physicist." (Prof. A.I. Solomon, Contemporary Physics, Vol. 46 (4), 2005) "This volume will serve as an excellent source book for the mathematical basis of the many recent advances in quantum mechanics. It will also serve as an excellent textbook … . Each chapter is chock full of mathematical derivations and proofs but perhaps the most interesting part of each proof is the following section entitled ‘Remarks’ sections which are full of interesting details, ideas, drawbacks, comments, and references. … As is usually the case with Springer-Verlag, this book has been beautifully produced … ." (Fernande Grandjean and Gary J. Long, Physicalia, Vol. 25 (3), 2003)Table of ContentsI Quantum Mechanics of Atoms and Molecules.- 1 Introduction.- 2 The Mathematical Formulation of Quantum Mechanics.- 3 Quantum Dynamics.- 4 Atomic Systems.- II Quantum Mechanics of Large Systems.- 1 Systems with Many Particles.- 2 Thermostatics.- 3 Thermodynamics.- 4 Physical Systems.- Bibliography to Part I.- Bibliography to Part II.

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Book SynopsisThis is a book about spectral methods for partial differential equations: when to use them, how to implement them, and what can be learned from their of spectral methods has evolved rigorous theory. The computational side vigorously since the early 1970s, especially in computationally intensive of the more spectacular applications are applications in fluid dynamics. Some of the power of these discussed here, first in general terms as examples of the methods have been methods and later in great detail after the specifics covered. This book pays special attention to those algorithmic details which are essential to successful implementation of spectral methods. The focus is on algorithms for fluid dynamical problems in transition, turbulence, and aero­ dynamics. This book does not address specific applications in meteorology, partly because of the lack of experience of the authors in this field and partly because of the coverage provided by Haltiner and Williams (1980). The success of spectral methods in practical computations has led to an increasing interest in their theoretical aspects, especially since the mid-1970s. Although the theory does not yet cover the complete spectrum of applications, the analytical techniques which have been developed in recent years have facilitated the examination of an increasing number of problems of practical interest. In this book we present a unified theory of the mathematical analysis of spectral methods and apply it to many of the algorithms in current use.Table of Contents1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat Equation.- 1.2.3. A Legendre Tau Method for the Poisson Equation.- 1.2.4. Basic Aspects of Galerkin, Tau and Collocation Methods.- 1.3. The Equations of Fluid Dynamics.- 1.3.1. Compressible Navier-Stokes.- 1.3.2. Compressible Euler.- 1.3.3. Compressible Potential.- 1.3.4. Incompressible Flow.- 1.3.5. Boundary Layer.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.1.1. The Continuous Fourier Expansion.- 2.1.2. The Discrete Fourier Expansion.- 2.1.3. Differentiation.- 2.1.4. The Gibbs Phenomenon.- 2.2. Orthogonal Polynomials in ( — 1, 1).- 2.2.1. Sturm—Liouville Problems.- 2.2.2. Orthogonal Systems of Polynomials.- 2.2.3. Gauss-Type Quadratures and Discrete Polynomial Transforms.- 2.3. Legendre Polynomials.- 2.3.1. Basic Formulas.- 2.3.2. Differentiation.- 2.4. Chebyshev Polynomials.- 2.4.1. Basic Formulas.- 2.4.2. Differentiation.- 2.5. Generalizations.- 2.5.1. Jacobi Polynomials.- 2.5.2. Mapping.- 2.5.3. Semi-Infinite Intervals.- 2.5.4. Infinite Intervals.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.1.1. Fourier Galerkin.- 3.1.2. Fourier Collocation.- 3.1.3. Chebyshev Tau.- 3.1.4. Chebyshev Collocation.- 3.2. Convolution Sums.- 3.2.1. Pseudospectral Transform Methods.- 3 2 2 Aliasing Removal by Padding or Truncation.- 3.2.3. Aliasing Removal by Phase Shifts.- 3.2.4. Convolution Sums in Chebyshev Methods.- 3.2.5. Relation Between Collocation and Pseudospectral Methods.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.4.1. Polar Coordinates.- 3.4.2. Spherical Polar Coordinates.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.2.1. The First-Derivative Operator.- 4.2.2. The Second-Derivative Operator.- 4.3. Some Standard Schemes.- 4.3.1. Multistep Schemes.- 4.3.2. Runge—Kutta Methods.- 4.4. Special Purpose Schemes.- 4.4.1. High Resolution Temporal Schemes.- 4.4.2. Special Integration Techniques.- 4.4.3. Lerat Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.1.1. Fourier Approximations.- 5.1.2. Chebyshev Tau Approximations.- 5.1.3. Schur-Decomposition and Matrix-Diagonalization.- 5.2. Fundamentals of Iterative Methods.- 5.2.1. Richardson Iteration.- 5.2.2. Preconditioning.- 5.2.3. Non-Periodic Problems.- 5.2.4. Finite-Element Preconditioning.- 5.3. Conventional Iterative Methods.- 5.3.1. Descent Methods for Symmetric, Positive-Definite Systems.- 5.3.2. Descent Methods for Non-Symmetric Problems.- 5.3.3. Chebyshev Acceleration.- 5.4. Multidimensional Preconditioning.- 5.4.1. Finite-Difference Solvers.- 5.4.2. Modified Finite-Difference Preconditioners.- 5.5. Spectral Multigrid Methods.- 5.5.1. Model Problem Discussion.- 5.5.2. Two-Dimensional Problems.- 5.5.3. Interpolation Operators.- 5.5.4. Coarse-Grid Operators.- 5.5.5. Relaxation Schemes.- 5.6. A Semi-Implicit Method for the Navier—Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier—Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.2.1. A Spectral Galerkin Solution Technique.- 7.2.2. Treatment of the Nonlinear Terms.- 7.2.3. Refinements.- 7.2.4. Pseudospectral and Collocation Methods.- 7.3. Inhomogeneous Flows.- 7.3.1. Coupled Methods.- 7.3.2. Splitting Methods.- 7.3.3. Galerkin Methods.- 7.3.4. Other Confined Flows.- 7.3.5. Unbounded Flows.- 7.3.6. Aliasing in Transition Calculations.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.4.1. Choice of Mesh.- 7.4.2. Coupled Methods.- 7.4.3. Splitting Methods.- 7.4.4. Other Methods.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.5.1. Potential Flow.- 8.5.2. Ringleb Flow.- 8.5.3. Astrophysical Nozzle.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. Fourier Approximation.- 9.1.1. Inverse Inequalities for Trigonometric Polynomials.- 9.1.2. Estimates for the Truncation and Best Approximation Errors.- 9.1.3. Estimates for the Interpolation Error.- 9.2. Sturm—Liouville Expansions.- 9.2.1. Regular Sturm—Liouville Problems.- 9.2.2. Singular Sturm—Liouville Problems.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.4.1. Inverse Inequalities for Algebraic Polynomials.- 9.4.2. Estimates for the Truncation and Best Approximation Errors.- 9.4.3. Estimates for the Interpolation Error.- 9.5. Chebyshev Approximations.- 9.5.1. Inverse Inequalities for Polynomials.- 9.5.2. Estimates for the Truncation and Best Approximation Errors.- 9.5.3. Estimates for the Interpolation Error.- 9.5.4. Proofs of Some Approximation Results.- 9.6. Other Polynomial Approximations.- 9.6.1. Jacobi Polynomials.- 9.6.2. Laguerre and Hermite Polynomials.- 9.7. Approximation Results in Several Dimensions.- 9.7.1. Fourier Approximations.- 9.7.2. Legendre Approximations.- 9.7.3. Chebyshev Approximations.- 9.7.4. Blended Fourier and Chebyshev Approximations.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.1.1. A Fourier Galerkin Method for the Wave Equation.- 10.1.2. A Chebyshev Collocation Method for the Heat Equation.- 10.1.3. A Legendre Tau Method for the Poisson Equation.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.4.1. Galerkin Methods.- 10.4.2. Tau Methods.- 10.4.3. Collocation Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.5.1. Conditions for Stability and Convergence: The Parabolic Case.- 10.5.2. Conditions for Stability and Convergence: The Hyperbolic Case.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.1.1. Legendre Methods.- 11.1.2. Chebyshev Methods.- 11.1.3. Other Boundary Value Problems.- 11.2. Advection-Diffusion Equation.- 11.2.1. Linear Advection-Diffusion Equation.- 11.2.2. Steady Burgers Equation.- 11.3. Navier—Stokes Equations.- 11.3.1. Compatibility Conditions Between Velocity and Pressure.- 11.3.2. Direct Discretization of the Continuity Equation: The “inf-sup” Condition.- 11.3.3. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser—Schumann Method.- 11.3.4. Navier—Stokes Equations in Streamfunction Formulation.- 11.4. The Eigenvalues of Some Spectral Operators.- 11.4.1. The Discrete Eigenvalues for Lu = ? uxx.- 11.4.2. The Discrete Eigenvalues for Lu = ? vuxx + bux.- 11.4.3. The Discrete Eigenvalues for Lu = ux.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.1.1. Periodic Boundary Conditions.- 12.1.2. Non-Periodic Boundary Conditions.- 12.1.3. Hyperbolic Systems.- 12.1.4. Spectral Accuracy for Non-Smooth Solutions.- 12.2. Heat Equation.- 12.2.1. Semi-Discrete Approximation.- 12.2.2. Fully Discrete Approximation.- 12.3. Advection-Diffusion Equation.- 12.3.1. Semi-Discrete Approximation.- 12.3.2. Fully Discrete Approximation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.2.1. Notation.- 13.2.2. Discretization.- 13.2.3. Solution Techniques.- 13.2.4. Examples.- 13.3. Variational Methods.- 13.3.1. Formulation.- 13.3.2. The Spectral-Element Method.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.5.1. Patching Methods.- 13.5.2. Equivalence Between Patching and Variational Methods.- 13.6. Some Stability and Convergence Results.- 13.6.1. Patching Methods.- 13.6.2. Variational Methods.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi—Gauss—Lobatto Roots.- References.

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Book SynopsisSymmetries in Physics presents the fundamental theories of symmetry, together with many examples of applications taken from several different branches of physics. Emphasis is placed on the theory of group representations and on the powerful method of projection operators. The excercises are intended to stimulate readers to apply the techniques demonstrated in the text.Table of Contents1. Introduction.- 2. Elements of the Theory of Finite Groups.- 2.1 Symmetry and Group Concepts: A Basic Example.- 2.2 General Theorems on Group Theory.- 2.3 Conjugacy Classes.- 3. Discrete Symmetry Groups.- 3.1 Point Groups.- 3.1.1 Symmetry Elements.- 3.1.2 Proper Point Groups.- 3.1.3 Improper Point Groups.- 3.2 Colour Groups and Magnetic Groups.- 3.3 Double Groups.- 3.4 Lattices, the Translation Group and Space Group.- 3.4.1 Normal Space Groups.- 3.4.2 Colour and Magnetic Space Groups.- 3.4.3 Double Space Groups.- 3.5 Permutation Groups.- 3.6 Other Finite Groups.- 4. Representations of Finite Groups.- 4.1 Linear Spaces and Operators.- 4.1.1 Linear and Unitary Spaces.- 4.1.2 Linear Operators.- 4.1.3 Special Operators and Eigenvalues.- 4.2 Introduction to the Theory of Representations.- 4.2.1 Operator Representations by Matrices.- 4.2.2 Equivalent Representations and Characters.- 4.2.3 Reducible and Irreducible Representations.- 4.2.4 Orthogonality Theorems.- 4.2.5 Subduction. Reality of Representations.- 4.3 Group Algebra.- 4.3.1 The Regular Representation.- 4.3.2 Projection Operators.- 4.4 Direct Products.- 4.4.1 Representations of Direct Products of Groups.- 4.4.2 The Inner Direct Product of Representations of a Group. Clebsch-Gordan Expansion.- 4.4.3 Simply Reducible Groups.- 5. Irreducible Representations of Special Groups.- 5.1 Point and Double Point Groups.- 5.2 Magnetic Point Groups. Time Reversal.- 5.3 Translation Groups.- 5.4 Permutation Groups.- 5.5 Tensor Representations.- 5.5.1 Tensor Transformations. Irreducible Tensors.- 5.5.2 Induced Representations.- 5.5.3 Irreducible Tensor Spaces.- 5.5.4 Direct Products and Their Reduction.- 6. Tensor Operators and Expectation Values.- 6.1 Tensors and Spinors.- 6.2 The Wigner-Eckart Theorem.- 6.3 Eigenvalue Problems.- 6.4 Perturbation Calculus.- 7. Molecular Spectra.- 7.1 Molecular Vibrations.- 7.1.1 Equation of Motion and Symmetry.- 7.1.2 Determination of Eigenvalues and Eigenvectors.- 7.1.3 Selection Rules.- 7.2 Electron Functions and Spectra.- 7.2.1 Symmetry in Many-Particle Systems.- 7.2.2 Symmetry-Adapted Atomic and Molecular Orbitals.- 7.2.3 The Hückel Method and Ligand Field Theory.- 7.3 Many-Electron Problems.- 7.3.1 Permutation Symmetry.- 7.3.2 Point and Permutation Symmetry. Molecular States.- 7.3.3 The H2 Molecule.- 8. Selection Rules and Matrix Elements.- 8.1 Selection Rules of Tensor Operators.- 8.2 The Jahn-Teller Theorem.- 8.2.1 Spinless States.- 8.2.2 Time Reversal Symmetry.- 8.3 Radiative Transitions.- 8.4 Crystal Field Theory.- 8.4.1 Crystal Field Splitting of Energy Levels.- 8.4.2 Calculation of Splitting.- 8.5 Independent Components of Material Tensors.- 9. Representations of Space Groups.- 9.1 Representations of Normal Space Groups.- 9.1.1 Decompositions into Cosets.- 9.1.2 Induction of the Representations of R.- 9.2 Allowable Irreducible Representations of the Little Group Gk.- 9.2.1 Projective Representations. Representations with a Factor System for $${\mathcal{G}_{{0k}}} = {\mathcal{G}_k}/\mathbb{T}$$.- 9.2.2 Vector Representations of the Groupe $${\mathcal{J}_k} = {\mathcal{G}_k}/{\mathbb{T}_k}$$.- 9.2.3 Representations of Double Space Groups. Spinor Representations.- 9.3 Projection Operators and Basis Functions.- 9.4 Representations of Magnetic Space Groups.- 9.4.1 Corepresentations of Magnetic Space Groups.- 9.4.2 Time Reversal Symmetry in MII Groups.- 10. Excitation Spectra and Selection Rules in Crystals.- 10.1 Spectra — Some General Statements.- 10.1.1 Bands and Branches.- 10.1.2 Compatibility Relations.- 10.2 Lattice Vibrations.- 10.2.1 Equation of Motion and Symmetry Properties.- 10.2.2 Vibrations of the Diamond Lattice.- 10.3 Electron Energy Bands.- 10.3.1 Symmetrization of Plane Waves.- 10.3.2 Energy Bands and Atomic Levels.- 10.4 Selection Rules for Interactions in Crystals.- 10.4.1 Determination of Reduction Coefficients.- 10.4.2 General Selection Rules.- 10.4.3 Electron-Phonon Interaction.- 10.4.4 Electron-Photon Interaction: Optical Transitions.- 10.4.5 Phonon-Photon Interaction.- 11. Lie Groups and Lie Algebras.- 11.1 General Foundations.- 11.1.1 Infinitesimal Generators and Defining Relations.- 11.1.2 Algebra and Parameter Space.- 11.1.3 Casimir Operators.- 11.2 Unitary Representations of Lie Groups.- 11.3 Clebsch-Gordan Coefficients and the Wigner-Eckart.- Theorem.- 11.4 The Cartan-Weyl Basis for Semisimple Lie Algebras.- 11.4.1 The Lie Group LU (n, ?) and the Lie Algebra An-1..- 11.4.2 The Cartan-Weyl Basis.- 12. Representations by Young Diagrams. The Method of Irreducible Tensors.- 13. Applications of the Theory of Continuous Groups.- 13.1 Elementary Particle Spectra.- 13.1.1 General Remarks.- 13.1.2 Hadronic States.- 13.1.3 Colour States of Quarks.- 13.1.4 A Possible LU (4) Classification.- 13.2 Atomic Spectra.- 13.2.1 Russell-Saunders (LS) Coupling.- 13.2.2 jj Coupling.- 13.3 Nuclear Spectra.- 13.3.1 jj-JI Coupling.- 13.3.2 LSI Coupling.- 13.4 Dynamical Symmetries of Classical Systems.- 14. Internal Symmetries and Gauge Theories.- 14.1 Internal Symmetries of Fields.- 14.2 Gauge Transformations of the First Kind.- 14.2.1 U (1) Gauge Transformations.- 14.2.2 LU (n) Gauge Transformations.- 14.3 Gauge Transformations of the Second Kind.- 14.3.1 U (1) Gauge Transformations of the Second Kind.- 14.3.2 LU (n) Gauge Transformations of the Second Kind.- 14.3.3 A Differential Geometric Discussion of the Yang-Mills Fields.- 14.4 Gauge Theories with Spontaneously Broken Symmetry.- 14.4.1 General Remarks.- 14.4.2 Spontaneous Breaking of a Gauge Symmetry of the First Kind: Goldstone Model.- 14.4.3 Spontaneous Breaking of an Abelian Gauge Symmetry of the Second Kind: Higgs-Kibble Model.- 14.5 Non-Abelian Gauge Theories and Symmetry Breaking.- 14.5.1 The Glashow-Salam-Weinberg Model of the Electro-Weak Interaction.- 14.5.2 Symmetry Breaking in the Glashow-Salam-Weinberg Model.- 14.5.3 Grand Unified Theories: General Remarks.- 14.5.4 LU (5) Group and Georgi-Glashow Model.- 14.5.5 Some Consequences of LU (5) Theory.- Appendices.- A. Character Tables.- B. Representations of Generators.- C. Standard Young-Yamanouchi Representations of the Permutation Groups P3 - P5.- D. Continuous Groups.- E. Stars of k and Symmetry of Special k-Vectors.- F. Noether’s Theorem.- G. Space-Time Symmetry.- G.1 Canonical Transformations and Algebra.- G.2 The Galilei Group and Classical Mechanics.- G.3 Lorentz and Poincaré Groups.- G.4 The Physical Quantities.- H. Goldstone’s Theorem.- I. Remarks on 5-fold Symmetry.- J. Supersymmetry.- K. List of Symbols and Abbreviations.- References.- Additional Reading.

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Book SynopsisThe new edition provided the opportunity of adding a new chapter entitled "Principles and Lessons of Quantum Physics". It was a tempting challenge to try to sharpen the points at issue in the long lasting debate on the Copenhagen Spirit, to assess the significance of various arguments from our present vantage point, seventy years after the advent of quantum theory, where, after ali, some problems appear in a different light. It includes a section on the assumptions leading to the specific mathematical formalism of quantum theory and a section entitled "The evolutionary picture" describing my personal conclusions. Alto­ gether the discussion suggests that the conventional language is too narrow and that neither the mathematical nor the conceptual structure are built for eter­ nity. Future theories will demand radical changes though not in the direction of a return to determinism. Essential lessons taught by Bohr will persist. This chapter is essentially self-contained. Some new material has been added in the last chapter. It concerns the char­ acterization of specific theories within the general frame and recent progress in quantum field theory on curved space-time manifolds. A few pages on renor­ malization have been added in Chapter II and some effort has been invested in the search for mistakes and unclear passages in the first edition. The central objective of the book, expressed in the title "Local Quantum Physics", is the synthesis between special relativity and quantum theory to­ gether with a few other principles of general nature.Trade Review"Indeed, both the expert in the field and the novice will enjoy Haags insightful exposition... This (superb) book is bound to occupy a place on a par with other classics in the mathematical physics literature." Physics Today "...enjoyable reading to anybody interested in the development of fundamental physical theories." Zentralblatt f. MathematikTable of ContentsI. Background.- 1. Quantum Mechanics.- Basic concepts, mathematical structure, physical interpretation..- 2. The Principle of Locality in Classical Physics and the Relativity Theories.- Faraday’s vision. Fields..- 2.1 Special relativity. Poincaré group. Lorentz group. Spinors. Conformal group..- 2.2 Maxwell theory..- 2.3 General relativity..- 3. Poincaré Invariant Quantum Theory.- 3.1 Geometric symmetries in quantum physics. Projective representations and the covering group..- 3.2 Wigner’s analysis of irreducible, unitary representations of the Poincare group. 3.3 Single particle states. Spin..- 3.4 Many particle states: Bose-Fermi alternative, Fock space, creation operators. Separation of CM-motion..- 4. Action Principle.- Lagrangean. Double rôle of physical quantities. Peierls’ direct definition of Poisson brackets. Relation between local conservation laws and symmetries..- 5. Basic Quantum Field Theory.- 5.1 Canonical quantization..- 5.2 Fields and particles..- 5.3 Free fields..- 5.4 The Maxwell-Dirac system. Gauge invariance..- 5.5 Processes..- II. General Quantum Field Theory.- 1. Mathematical Considerations and General Postulates.- 1.1 The representation problem..- 1.2 Wightman axioms..- 2. Hierarchies of Functions.- 2.1 Wightman functions, reconstruction theorem, analyticity in x-space..- 2.2 Truncated functions, clustering. Generating functionals and linked cluster theorem..- 2.3 Time ordered functions..- 2.4 Covariant perturbation theory, Feynman diagrams. Renormalization..- 2.5 Vertex functions and structure analysis..- 2.6 Retarded functions and analyticity in p-space..- 2.7 Schwinger functions and Osterwalder-Schrader theorem..- 3. Physical Interpretation in Terms of Particles.- 3.1 The particle picture: Asymptotic particle configurations and collision theory..- 3.2 Asymptotic fields. S-matrix..- 3.3 LSZ-formalism..- 4. General Collision Theory.- 4.1 Polynomial algebras of fields. Almost local operators..- 4.2 Construction of asymptotic particle states..- 4.3. Coincidence arrangements of detectors..- 4.4 Generalized LSZ-formalism..- 5. Some Consequences of the Postulates.- 5.1 CPT-operator. Spin-statistics theorem. CPT-theorem..- 5.2 Analyticity of the S-matrix..- 5.3 Reeh-Schlieder theorem..- 5.4 Additivity of the energy-momentum-spectrum..- 5.5 Borchers classes..- III. Algebras of Local Observables and Fields.- 1. Review of the Perspective.- Characterization of the theory by a net of local algebras. Bounded operators. Unobservable fields, superselection rules and the net of abstract algebras of observables. Transcription of the basic postulates..- 2. Von Neumann Algebras. C*-Algebras. W*-Algebras.- 2.1 Algebras of bounded operators. Concrete C*-algebras and von Neumann algebras. Isomorphisms. Reduction. Factors. Classification of factors..- 2.2 Abstract algebras and their representations. Abstract C*-algebras. Relation between the C*-norm and the spectrum. Positive linear forms and states. The GNS-construction. Folia of states. Intertwiners. Primary states and cluster property. Purification. W*-algebras..- 3. The Net of Algebras of Local Observables.- 3.1 Smoothness and integration. Local definiteness and local normality..- 3.2 Symmetries and symmetry breaking. Vacuum states. The spectral ideals..- 3.3 Summary of the structure..- 4. The Vacuum Sector.- 4.1 The orthocomplemented lattice of causally complete regions..- 4.2 The net of von Neumann algebras in the vacuum representation..- IV. Charges, Global Gauge Groups and Exchange Symmetry.- 1. Charge Superselection Sectors.- “Strange statistics”. Charges. Selection criteria for relevant sectors. The program and survey of results..- 2. The DHR-Analysis.- 2.1 Localized morphisms..- 2.2 Intertwiners and exchange symmetry (“Statistics”)..- 2.3 Charge conjugation, statistics parameter..- 2.4 Covariant sectors and energy-momentum spectrum..- 2.5 Fields and collision theory..- 3. The Buchholz-Fredenhagen-Analysis.- 3.1 Localized 1-particle states..- 3.2 BF-topological charges..- 3.3 Composition of sectors and exchange symmetry..- 3.4 Charge conjugation and the absence of “infinite statistics”..- 4. Global Gauge Group and Charge Carrying Fields.- Implementation of endomorphisms. Charges with d = 1. Endomorphisms and non Abelian gauge group. DR categories and the embedding theorem..- 5. Low Dimensional Space-Time and Braid Group Statistics.- Statistics operator and braid group representations. The 2+1-dimensional case with BF-charges. Statistics parameter and Jones index..- V. Thermal States and Modular Automorphisms.- 1. Gibbs Ensembles, Thermodynamic Limit, KMS-Condition.- 1.1 Introduction..- 1.2 Equivalence of KMS-condition to Gibbs ensembles for finite volume..- 1.3 The arguments for Gibbs ensembles..- 1.4 The representation induced by a KMS-state..- 1.5 Phases, symmetry breaking and the decomposition of KMS-states..- 1.6 Variational principles and autocorrelation inequalities..- 2. Modular Automorphisms and Modular Conjugation.- 2.1 The Tomita-Takesaki theorem..- 2.2 Vector representatives of states. Convex cones in H..- 2.3 Relative modular operators and Radon-Nikodym derivatives..- 2.4 Classification of factors..- 3. Direct Characterization of Equilibrium States.- 3.1 Introduction..- 3.2 Stability..- 3.3 Passivity..- 3.4 Chemical potential..- 4. Modular Automorphisms of Local Algebras.- 4.1 The Bisognano-Wichmann theorem..- 4.2 Conformal invariance and the theorem of Hislop and Longo..- 5. Phase Space, Nuclearity, Split Property, Local Equilibrium.- 5.1 Introduction..- 5.2 Nuclearity and split property..- 5.3 Open subsystems..- 5.4 Modular nuclearity..- 6. The Universal Type of Local Algebras.- VI. Particles. Completeness of the Particle Picture.- 1. Detectors, Coincidence Arrangements, Cross Sections.- 1.1 Generalities..- 1.2 Asymptotic particle configurations. Buchholz’s strategy..- 2. The Particle Content.- 2.1 Particles and infraparticles..- 2.2 Single particle weights and their decomposition..- 2.3 Remarks on the particle picture and its completeness..- 3. The Physical State Space of Quantum Electrodynamics.- VII. Principles and Lessons of Quantum Physics. A Review of Interpretations, Mathematical Formalism and Perspectives.- 1. The Copenhagen Spirit. Criticisms, Elaborations.- Niels Bohr’s epistemological considerations. Realism. Physical systems and the division problem. Persistent non-classical correlations. Collective coordinates, decoherence and the classical approximation. Measurements. Correspondence and quantization. Time reflection asymmetry of statistical conclusions..- 2. The Mathematical Formalism.- Operational assumptions. “Quantum Logic”. Convex cones..- 3. The Evolutionary Picture.- Events, causal links and their attributes. Irreversibility. The EPR-effect. Ensembles vs. individuals. Decisions. Comparison with standard procedure..- VIII. Retrospective and Outlook.- 1. Algebraic Approach vs. Euclidean Quantum Field Theory.- 2. Supersymmetry.- 3. The Challenge from General Relativity.- 3.1 Introduction..- 3.2 Quantum field theory in curved space-time..- 3.3 Hawking temperature and Hawking radiation..- 3.4 A few remarks on quantum gravity..- Author Index and References.

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Book SynopsisFor almost two decades, this has been the classical textbook on applications of operator algebra theory to quantum statistical physics. Major changes in the new edition relate to Bose-Einstein condensation, the dynamics of the X-Y model and questions on phase transitions. Table of ContentsVolume 2.- States in Quantum Statistical Mechanics.- 5.1. Introduction.- 5.2. Continuous Quantum Systems. I.- 5.3. KMS-States.- 5.4. Stability and Equilibrium.- Models of Quantum Statistical Mechanics.- 6.1. Introduction.- 6.2 Quantum Spin Systems.- 6.3. Continuous Quantum Systems. II.- 6.4. Conclusion.- References.- Books and Monographs.- Articles.- List of Symbols.

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Book SynopsisIn this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.Table of ContentsPreface.- Contents.- Chap. I: Ordinary differential equations.- Chap. II: Scalar first order equations with one space variable.- Chap. III: Scalar first order equations with several variables.- Chap. IV: First order systems of conservation laws with one space.- Chap. V: Compensated compactness.- Chap. VI: Nonlinear perturbations of the wave equation.- Chap. VII: Nonlinear perturbations of the Klein-Gordon equation.- Chap. VIII: Microlocal analysis.- Chap. IX: Pseudo-differential operators of type 1,1.- Chap. X: Paradifferential calculus.- Chap. XI: Propagation of singularities.- Appendix on pseudo-Riemannian geometry.- References.- Index of notations.- Index.

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