Mathematical / Computational / Theoretical physics Books

Book SynopsisThe purpose of this book is twofold: first, it sets out to equip the reader with a sound understanding of the foundations of probability theory and stochastic processes, offering step-by-step guidance from basic probability theory to advanced topics, such as stochastic differential equations, which typically are presented in textbooks that require a very strong mathematical background. Second, while leading the reader on this journey, it aims to impart the knowledge needed in order to develop algorithms that simulate realistic physical systems. Connections with several fields of pure and applied physics, from quantum mechanics to econophysics, are provided. Furthermore, the inclusion of fully solved exercises will enable the reader to learn quickly and to explore topics not covered in the main text. The book will appeal especially to graduate students wishing to learn how to simulate physical systems and to deepen their knowledge of the mathematical framework, which has very deep connections with modern quantum field theory.Table of Contents1 Review of Probability Theory.- 2 Applications to Mathematical Statistics.- 3 Conditional Probability and Conditional Expectation.- 4 Markov Chains.- 5 Sampling of Random Variables and Simulation.- 6 Brownian Motion.- 7 Introduction to Stochastic Calculus and Ito Integral.- 8 Introduction to Stochastic Differential Equations and Applications.- Bibliography.- Solutions.

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Book SynopsisThe book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.Table of ContentsPreface xv Part I Mathematics 1 1 Functions of One Variable 3 1.1 Limits 3 1.2 Elementary Calculus 5 1.2.1 Differentiation Products and Quotients 6 1.2.2 Chain Rule 7 1.2.3 Inverse Functions 8 1.3 Integration 10 1.4 The Binomial Expansion 14 1.5 Taylor’s Series 15 1.6 Extrema 17 1.7 Power Series 17 1.8 Basic Functions 19 1.8.1 Exponential 19 1.8.2 Logarithm 22 1.9 First-Order Ordinary Differential Equations 24 1.10 Trigonometric Functions 25 1.10.1 L’Hôpital’s Rule 27 Problems 27 2 Complex Numbers 29 2.1 Exponential Function of a Complex Variable 30 2.2 Argand Diagrams and the Complex Plane 32 2.3 Complex Logarithm 34 2.4 Hyperbolic Functions 34 2.5 The Simple Harmonic Oscillator 36 2.5.1 Mechanics in One Dimension 38 2.5.2 Damped and Driven Oscillations 40 Problems 47 3 VectorsinR 3 51 3.1 Basic Operation 51 3.1.1 Scalar Triple Product 55 3.1.2 Vector Equations of Lines and Planes 56 3.2 Kinematics in Three Dimensions 57 3.2.1 Differentiation 57 3.2.2 Motion in a Uniform Magnetic Field 57 3.3 Coordinate Systems 59 3.3.1 Polar Coordinates 59 3.4 Central Forces 60 3.5 Rotating Frames 64 3.5.1 Larmor Effect 66 Problems 67 4 VectorSpaces 71 4.1 Formal Definition of a Vector Space 71 4.2 Fourier Series 75 4.3 Linear Operators 78 4.4 Change of Basis 89 Problems 91 5 Functions of Several Variables 95 5.1 Partial Derivatives 95 5.1.1 Definition of the Partial Derivative 95 5.1.2 Total Derivatives 98 5.1.3 Elementary Numerical Methods 104 5.1.4 Change of Variables 107 5.1.5 Mechanics Again 109 5.2 Extrema under Constraint 111 5.3 Multiple Integrals 113 5.3.1 Triple Integrals 116 5.3.2 Change of Variables 117 Problems 121 6 Vector Fields and Operators 125 6.1 The Gradient Operator 125 6.1.1 Coordinate Systems 127 6.2 Work and Energy in Vectorial Mechanics 130 6.2.1 Line Integrals 133 6.3 A Little Fluid Dynamics 135 6.3.1 Rotational Motion 138 6.3.2 Fields 141 6.4 Surface Integrals 142 6.5 The Divergence Theorem 146 6.6 Stokes’ Theorem 149 6.6.1 Conservative Forces 153 Problems 154 7 Generalized Functions 159 7.1 The Dirac Delta Function 159 7.2 Green’s Functions 163 7.3 Delta Function in Three Dimensions 165 Problems 169 8 Functions of a Complex Variable 173 8.1 Limits 174 8.2 Power Series 178 8.3 Fluids Again 179 8.4 Complex Integration 180 8.4.1 Application of the Residue Theorem 186 Problems 192 Part II Physics 195 9 Maxwell’s Equations: A Very Short Introduction 197 9.1 Electrostatics: Gauss’s Law 197 9.1.1 Conductors 203 9.2 The No Magnetic Monopole Rule 204 9.3 Current 205 9.4 Faraday’s Law 206 9.5 Ampère’s Law 208 9.6 The Wave Equation 210 9.7 Gauge Conditions 211 Problems 213 10 Special Relativity: Four-Vector Formalism 217 10.1 Lorentz Transformation 217 10.1.1 Inertial Frames 217 10.1.2 Properties and Consequences of the Lorentz Transformation 220 10.2 Minkowski Space 220 10.2.1 Four Vectors 220 10.2.2 Time Dilation 226 10.3 Four-Velocity 227 10.3.1 Four-Momentum 229 10.4 Electrodynamics 234 10.4.1 Maxwell’s Equations in Four-Vector Form 234 10.4.2 Field of a Moving Point Charge 237 10.5 Transformation of the Electromagnetic Fields 239 Problems 240 11 Quantum Theory 243 11.1 Bohr Atom 243 11.2 The de Broglie Hypothesis 246 11.3 The Schrödinger Wave Equation 246 11.4 Interpretation of the Wave function 249 11.5 Atom 251 11.5.1 The Delta Function Potential 252 11.5.2 Molecules 254 11.6 Formalism 257 11.6.1 Dirac Notation 257 11.7 Probabilistic Interpretation 258 11.7.1 Commutator Relations 259 11.7.2 Functions of Observables 261 11.7.3 Block’s Theorem 261 11.7.4 Band Structure 263 11.8 Time Evolution 266 11.9 The Stern–Gerlach Experiment 269 11.9.1 Successive Measurements 270 11.9.2 Spin Space 271 11.9.3 Explicit Matrix Representation 272 11.9.4 Larmor Precession 274 11.9.5 EPR Paradox 275 11.9.6 Bell’s Theorem 276 11.9.7 The Harmonic Oscillator 279 Problems 280 12 An Informal Treatment of Variational Principles and their History 287 12.1 Sin and Death 287 12.2 The Calculus of Variations 288 12.3 Constrained Variations 293 12.4 Hamilton’s Equations 293 12.5 Phase Space 296 12.6 Fixed Points 296 Problems 298 A Conic Sections 301 A.1 Polar Coordinates 303 A.2 Intersection of a Cone and a Plane 304 B Vector Relations 305 B.1 Products 305 B.2 Differential Operator Relations 305 B.3 Coordinates 306 Cylindrical Polar 306 Spherical Polar 307 Bibliography 309 Index 311

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Book SynopsisSince the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and its use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems. This volume collects the lecture notes of a C.I.M.E. course held in Summer 2006, when some of the most world recognized experts in the field reviewed the rigorous setting of mixed finite elements and revisited it after more than 30 years of practice. Applications, in this volume, range from traditional ones, like fluid-dynamics or elasticity, to more recent and active fields, like electromagnetism.Table of ContentsMixed Finite Element Methods.- Finite Elements for the Stokes Problem.- Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations.- Finite Element Methods for Linear Elasticity.- Finite Elements for the Reissner–Mindlin Plate.

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Book SynopsisThis book presents the state-of-the-art in simulation on supercomputers. Leading researchers present results achieved on systems of the High Performance Computing Center Stuttgart (HLRS) for the year 2010. The reports cover all fields of computational science and engineering, ranging from CFD to computational physics and chemistry to computer science, with a special emphasis on industrially relevant applications. Presenting results for both vector systems and microprocessor-based systems, the book makes it possible to compare the performance levels and usability of various architectures. As HLRS operates the largest NEC SX-8 vector system in the world, this book gives an excellent insight into the potential of vector systems, covering the main methods in high performance computing. Its outstanding results in achieving the highest performance for production codes are of particular interest for both scientists and engineers. The book includes a wealth of color illustrations and tables.

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Book SynopsisThis book presents the state-of-the-art in simulation on supercomputers. Leading researchers present results achieved on systems of the High Performance Computing Center Stuttgart (HLRS) for the year 2012. The reports cover all fields of computational science and engineering ranging from CFD via computational physics and chemistry to computer science with a special emphasis on industrially relevant applications. Presenting results for both vector-systems and micro-processor based systems the book allows to compare performance levels and usability of various architectures. As HLRS operates not only a large cluster system but also one of the largest NEC vector systems in the world this book gives an excellent insight also into the potential of vector systems. The book covers the main methods in high performance computing. Its outstanding results in achieving highest performance for production codes are of particular interest for both the scientist and the engineer. The book comes with a wealth of coloured illustrations and tables of results. ​Table of Contents1. Physics.- 2. Solid State Physics.- 3. Reacting Flows.- 4. Computational Fluid Dynamics.- 5. Transport and Climate.- 6. Miscellaneous Topics.

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Book SynopsisThis lecture note provides a tutorial review of non-Abelian discrete groups and presents applications to particle physics where discrete symmetries constitute an important principle for model building. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics—particularly model building beyond the standard model—non-Abelian discrete symmetries have been applied particularly to understand the three-generation flavor structure. The non-Abelian discrete symmetries are indeed considered to be the most attractive choice for a flavor sector: Model builders have tried to derive experimental values of quark and lepton masses, mixing angles and CP phases on the assumption of non-Abelian discrete flavor symmetries of quarks and leptons, yet lepton mixing has already been intensively discussed in this context as well. Possible origins of the non-Abelian discrete symmetry for flavors are another topic of interest, as they can arise from an underlying theory, e.g., the string theory or compactification via orbifolding as geometrical symmetries such as modular symmetries, thereby providing a possible bridge between the underlying theory and corresponding low-energy sector of particle physics. The book offers explicit introduction to the group theoretical aspects of many concrete groups, and readers learn how to derive conjugacy classes, characters, representations, tensor products, and automorphisms for these groups (with a finite number) when algebraic relations are given, thereby enabling readers to apply this to other groups of interest. Further, CP symmetry and modular symmetry are also presented.Table of ContentsIntroduction.- Basics of Finite Groups.- SN.- AN.- 5 T ′.- DN.- QN.- QD2N.- Σ(2N ).- Δ (3N2).- TN.- Σ(3N3).- Δ(6N2).- Subgroups and Decompositions of Multiplets.- Anomalies.- Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models.- Modular Group.- CP Symmetry.- Appendices.

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Book SynopsisThis book introduces the scattering theory of nonrelativistic systems, a standard tool for interpreting collision experiments with quantum particles at energies not too high. The goal is to explore the interaction between particles and their properties. The authors cover the basics of the theory through a detailed discussion of elastic scattering using the stationary Schrödinger equation and the Lippmann-Schwinger equation. These remarks are supplemented by a consideration of the time-dependent formulation of scattering theory. Selection rules for effective cross sections due to symmetries conditioned by the structure of the interparticle forces and the scattering of spin-polarized particles are discussed. The foundations for the treatment of inelastic processes are laid and explained by application to three-body and nucleotransfer processes.In all chapters, the more technical, mathematical aspect and the more physics-oriented explanations are separated as far as possible. The explanations are well comprehensible and suitable to introduce the reader to the physics of impact processes.This book is a translation of the original German 1st edition Streutheorie in der nichtrelativistischen Quantenmechanik by Reiner M. Dreizler, Tom Kirchner & Cora S. Lüdde, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2018. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). The present version has been revised extensively with respect to technical and linguistic aspects by the authors. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.Table of Contents1 Elastic scattering: stationary formulation - differential equations.- 2 Elastic scattering: stationary formulation - integral equations.- 3 Elastic scattering: time-dependent formulation.- 4 Conservation laws in scattering theory.- 5 Elastic scattering: the analytical structure of the S-matrix.- 6 Elastic scattering with spin-polarized particles.- 7 Remarks on multichannel problems.- Bibliography.

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Book SynopsisThe book provides a non-perturbative approach to the symmetry breaking in the standard model, in this way avoiding the critical issues which affect the standard presentations. The debated empirical meaning of global and local gauge symmetries is clarified. The absence of Goldstone bosons in the Higgs mechanism is non-perturbatively explained by the validity of Gauss laws obeyed by the currents which generate the relatedglobal gauge symmetry. The solution of the U(1) problem and the vacuum structure in quantum chromodynamics (QCD) are obtained without recourse to the problematic semiclassical instanton approximation, by rather exploiting the topology of the gauge group.Table of ContentsSpontaneous symmetry breaking.- Goldstone theorem. Breaking gauge symmetries.- Higgs mechanism.- U(1) problem in QCD; a solution without instantons.- Gauge group topology and $\theta$ vacuum structure.

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Book SynopsisThe book provides theoretical and phenomenological insights on the structure of matter, presenting concepts and features of elementary particle physics and fundamental aspects of nuclear physics. Starting with the basics (nomenclature, classification, acceleration techniques, detection of elementary particles), the properties of fundamental interactions (electromagnetic, weak and strong) are introduced with a mathematical formalism suited to undergraduate students. Some experimental results (the discovery of neutral currents and of the W± and Z0 bosons; the quark structure observed using deep inelastic scattering experiments) show the necessity of an evolution of the formalism. This motivates a more detailed description of the weak and strong interactions, of the Standard Model of the microcosm with its experimental tests, and of the Higgs mechanism. The open problems in the Standard Model of the microcosm and macrocosm are presented at the end of the book. Table of ContentsPreface.- 1. Historical Notes and Fundamental Concepts.- 2. Particle Interactions with Matter and Detectors.- 3. Particle Accelerators and Particle Detection.- 4. The Paradigm of Interactions: the Electromagnetic Case.- 5. First Discussion of the Other Fundamental Interactions.- 6 Invariance and Conservation Principles.- 7. Hadron Interactions at Low Energies and the Static Quark Model.- 8. Weak Interactions and Neutrinos.- 9. Discoveries in Electron-Positron Collisions.- 10. High Energy Interactions at the Dynamic Quark Model.- 11. The Standard Model of the Microcosm.- 12. CP-Violation and Particle Oscillations.- 13. Microcosm and Macrocosm.- 14. Fundamental aspects of Nucleon Interactions.- Appendix 1. Periodic Table.- Appendix 2. The natural units in subnuclear physics.- Appendix 3. Basic concepts of relativity and classical EM.- Appendix 4. Dirac’s equation and formalism.- Appendix 5. Physical and astrophysical constants.- References.- Index.

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Book SynopsisYoichiro Nambu was one of the giants in the physics of the last century. His profound ideas in fundamental physics are still playing an important role and are being rediscovered over and over again.He preferred to share some of his deepest insights in talks, rather than publications, but Nambu's papers and talks were not easy to understand. Like the Japanese gentleman he was, he did not want to humiliate the reader with obvious statements. Even if it is an interpretation, it fits very well with his character of a very polite and considerate human being with a self-reliance inside him that he did not show on the outside. It is no wonder that many of his breakthroughs were not immediately appreciated by his contemporaries, with a late award of a well-deserved Nobel Prize. He was probably the only one in the highest stratum of physicists who was respected by everybody.The purpose of this book, half-history, half-physics is to trace Nambu's progress formulating some of his greatest ideas. It is structured in seven chapters, describing a paper or a series of papers (or talks) where Nambu's originality and genius emerge. Each chapter begins with a historical background section that sets the physics climate of the time, followed by a somewhat detailed discussion of his papers/talks.This tribute to Nambu hopes that he will be remembered and admired for many years to come.

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Book SynopsisThis book deals with mathematical modeling, namely, it describes the mathematical model of heat transfer in a silicon cathode of small (nano) dimensions with the possibility of partial melting taken into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type free boundary problems. The approach used is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the mathematical model including its parallel implementation. The results of numerical simulation concludes the book. The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.​Trade Review“The intended audience for this book includes researchers and specialists working in the field of electron emission processes and heat transfer, but this book also contains many details from the point of view of modeling and the corresponding mathematical architecture that may be useful for advanced students and postgraduates.” (Federico Zullo, Mathematical Reviews, March, 2021)Table of ContentsPreface Chapter 1. Introduction 1.1. Brief history of the electron emission discovery 1.2. Types of electron emission 1.3. Statement of the problem 1.4. Mathematical statement of the problem. Heat transfer model Chapter 2. Physical foundations of field emission 2.1. Band theory and Fermi levels 2.2. Specific conductance of semiconductors 2.2.1. Electron and hole concentration 2.2.2. Effective mass 2.2.3. Electron and hole mobility 2.2.4. Temperature dependence of specific conductance in silicon 2.3. Thermoelectricity 2.4. Heat conduction of solids 2.4.1. Electron heat conductivity 2.4.2. Heat conduction of crystal lattice 2.5. Emission current density and Nottingham effect 2.5.1. Support function in metals 2.5.2. Electron tunneling through potential barrier 2.5.3. Formula for the barrier transmission factor in the case of field emission cathode 2.5.4. Emission current density in metals 2.5.5. Specific characteristics of filed emission from semiconductor cathode 2.5.6. Approximation of the emission current density formula 2.5.7. Nottingham effect 2.5.8. Optimal values of approximation parameters 2.5.9. Inversion temperature dependence on the external electric field voltage Chapter 3. Mathematical model 3.1. Phase field system and its use in heat transfer modeling 3.2. Phase field system as regularization of limit problems with free boundary 3.3. Asymptotic solution of the phase field system and modified Stefan problem 3.3.1. Construction of asymptotic solution 3.3.2. Examples 3.4. Weak solution of the phase field system and the melting zone model 3.4.1. Weak solutions and Hugoniot-type conditions 3.4.2. ``Wavetrain''-type solutions and the corresponding limit problem 3.5. Derivation of the limit Stefan–Gibbs–Thomson problem solution from numerical solution of the phase field system 3.6. Generation and merging of dissipative waves 3.7. Cathode in the vacuum cube. Definition of a generalized solution to Poisson equation for electric field potential 3.8. Mathematical model of electron emission in a vacuum cube Chapter 4 Numerical modeling and its results 4.1. Nanocathode model 4.2. Computation of current density inside the cathode 4.3. Computation of emission current density and Nottingham effect modeling 4.4. Difference scheme 4.4.1. Difference scheme for the equation for the potential 4.4.2. Difference scheme for the equation for the function of order 4.4.3. Difference scheme for the heat conduction equation 4.4.4. Difference scheme stability 4.4.5. One more version of the difference scheme 4.4.6. Choice of the difference scheme step 4.5. Algorithm for solving difference equations and possible versions of its parallelization 4.6. Results of numerical experiments 4.6.1. Nonmonotone behavior of free boundaries 4.6.2. Results of modeling with physical parameters corresponding to experimental parameters 4.7. Formation of melting and crystallizing nuclei in the model 4.8. Conclusion

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Book SynopsisThis book is the first one that provides a solid bridge between algorithmic information theory and statistical mechanics. Algorithmic information theory (AIT) is a theory of program size and recently is also known as algorithmic randomness. AIT provides a framework for characterizing the notion of randomness for an individual object and for studying it closely and comprehensively. In this book, a statistical mechanical interpretation of AIT is introduced while explaining the basic notions and results of AIT to the reader who has an acquaintance with an elementary theory of computation.A simplification of the setting of AIT is the noiseless source coding in information theory. First, in the book, a statistical mechanical interpretation of the noiseless source coding scheme is introduced. It can be seen that the notions in statistical mechanics such as entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding in a natural manner. Then, the framework of AIT is introduced. On this basis, the introduction of a statistical mechanical interpretation of AIT is begun. Namely, the notion of thermodynamic quantities, such as free energy, energy, and entropy, is introduced into AIT. In the interpretation, the temperature is shown to be equal to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate measured by means of program-size complexity. Additionally, it is demonstrated that this situation holds for the temperature itself as a thermodynamic quantity. That is, for each of all the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T to be a fixed point on partial randomness.In this groundbreaking book, the current status of the interpretation from both mathematical and physical points of view is reported. For example, a total statistical mechanical interpretation of AIT that actualizes a perfect correspondence to normal statistical mechanics can be developed by identifying a microcanonical ensemble in the framework of AIT. As a result, the statistical mechanical meaning of the thermodynamic quantities of AIT is clarified. In the book, the close relationship of the interpretation to Landauer's principle is pointed out.Table of ContentsStatistical Mechanical Interpretation of Noiseless Source Coding.- Algorithmic Information Theory.- Partial Randomness.- Temperature Equals to Partial Randomness.- Fixed Point Theorems on Partial Randomness.- Statistical Mechanical Meaning of the Thermodynamic Quantities of AIT.- The Partial Randomness of Recursively Enumerable Reals.- Computation-Theoretic Clarification of the Phase Transition at Temperature T=1.- Other Related Results and Future Development.

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Book SynopsisThis book explores several key issues in beam phase space dynamics in plasma-based wakefield accelerators. It reveals the phase space dynamics of ionization-based injection methods by identifying two key phase mixing processes. Subsequently, the book proposes a two-color laser ionization injection scheme for generating high-quality beams, and assesses it using particle-in-cell (PIC) simulations. To eliminate emittance growth when the beam propagates between plasma accelerators and traditional accelerator components, a method using longitudinally tailored plasma structures as phase space matching components is proposed. Based on the aspects above, a preliminary design study on X-ray free-electron lasers driven by plasma accelerators is presented. Lastly, an important type of numerical noise—the numerical Cherenkov instabilities in particle-in-cell codes—is systematically studied.Table of Contents Chapter 1 Introduction 1.1 Introduction 1.2 Plasma Based Acceleration 1.3 Particle-in-Cell Simulations 1.4 Motivation and Outline Chapter 2 Phase Space Dynamics of Injected Electron Beams in Ionization Injection 2.1 Introduction 2.2 The Photoionization Process 2.3 The Residual Momentum 2.3.1 Initial Momentum from the Tunneling Ionization 2.3.2 The Momentum from the Lasers: Longitudinal Injection 2.3.3 The Momentum from the Lasers: Transverse Injection 2.3.4 The Thermal Emittance 2.4 Single Particle Motion in the Nonlinear Wake 2.5 Transverse Phase Mixing 2.5.1 Emittance Evolution: Growth and Oscillation in the Injection Stage 2.5.2 Emittance Evolution: Decrease and Regrowth in the Acceleration Stage 2.5.3 A Phenomenological Model 2.5.4 Comparisons with PIC Simulations 2.6 Longitudinal Phase Mixing 2.6.1 The Trapping Condition 2.6.2 Longitudinal Phase Mixing 2.7 Space Charge Effects 2.8 The Two-Color Ionization Injection 2.8.1 The Emittance in A Single Laser Case 2.8.2 The Two-Color Ionization Injection: Longitudinal Injection 2.8.3 The Two-Color Ionization Injection: Transverse Injection 2.9 Intrinsic Phase Space Discretization in Laser Triggered Ionization Injection 2.9.1 Single Laser Pulse Case 2.9.2 Beam Driver with a Laser Injector 2.10 Summary 3.1 Introduction 3.2 The Emittance Growth between Stages 3.2.1 Emittance Growth in Free Space Drifting 3.2.2 Emittance Growth in A Uniform Focusing Field 3.3 Theoretical Analysis of A Matching Plasma 3.3.1 How to Design the Matching Plasma? 3.3.2 The Effect of the Energy Spread 3.4 Verification by PIC Simulations 3.4.1 Matching Between Two-Stage LWFAs 3.4.2 Matching in External Injection 3.4.3 Matching between LWFAs and the Quadrupoles 3.5 Summary Chapter 4 X-FELs Driven by Plasma Based Accelerators 4.1 Introduction 4.1.1 The Basic Principles of FELs< 4.1.2 The Challenges and Opportunities of X-FELs Driven by plasma Based Accelerators 4.2 X-FEL Driven by A Two-Stage LWFA 4.2.1 Simulation of the Injector Stage 4.2.2 Simulation of the Accelerator Stage 4.2.3 Simulation of the Undulator Stage 4.3 Conclusions Chapter 5 Numerical Instability due to Relativistic Plasma Drift in EM-PIC Simulations 5.1 Introduction 5.1.1 The Boosted Frame Simulations of LWFA 5.1.2 Numerical Noise Induced by Relativistic Plasma Drift in PIC Codes 5.2 Numerical Dispersion Relation for Cold Plasma Drift 5.2.1 Derivation of Dispersion Relation 5.2.2 Elements of Dispersion Relation Tensor 5.2.3 EM Modes, and Wave-Particle Resonance 5.3 Numerical Instability Induced by Relativistic Plasma Drift for the Yee Solver 5.3.1 Theoretical Analysis of the 2D Dispersion Relation 5.3.2 Simulation Study of the Instability 5.4 Asymptotic Expression for Instability Growth Rate 5.4.1 Derivation of Asymptotic Expression 5.4 2 Parameter Scans for Minimal Instability Growth Rate 5.5 Elimination of the Numerical Cerenkov Instability for Spectral EM-PIC codes 5.5.1 The NCI Modes For the Spectral Solver 5.5.2 The Positions and the Growth Rates of the NCI Modes for the Spectral Solver 5.53 LWFA Simulation in the Lorentz Boosted Frame with Spectral Solver 5.6 Conclusions Chapter 6 Summary 6.1 Concluding Remarks 6.2 Future Work Reference Acknowledgement Appendix A A.1 Derivation of the Emittance Evolution in the Acceleration Stage A.2 Interpolation Tensor and Finite Difference Operator

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Book SynopsisThis book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was first proposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently.This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.Trade Review“The book is directed at readers with a solid foundation in algebraic geometry. … the main definitions and theorems are nicely illustrated by examples. … The book will serve as a guide to further reading for those wishing to learn more details about the theory.” (Matthew B. Young, Mathematical Reviews, March, 2023)Table of Contents1Donaldson–Thomas invariants on Calabi–Yau 3-folds.- 2Generalized Donaldson–Thomas invariants.- 3Donaldson–Thomas invariants for quivers with super-potentials.- 4Donaldson–Thomas invariants for Bridgeland semistable objects.- 5Wall-crossing formulas of Donaldson–Thomas invariants.- 6Cohomological Donaldson–Thomas invariants.- 7Gopakumar–Vafa invariants.- 8Some future directions.

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Book SynopsisThe book is intended for graduate students of theoretical physics (with a background in quantum mechanics) as well as researchers interested in applications of Lie group theory and Lie algebras in physics. The emphasis is on the inter-relations of representation theories of Lie groups and the corresponding Lie algebras.

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Book SynopsisThis open access book introduces the fundamentals of the space–time conservation element and solution element (CESE) method, which is a novel numerical approach for solving equations of physical conservation laws. It highlights the recent progress to establish various improved CESE schemes and its engineering applications. With attractive accuracy, efficiency, and robustness, the CESE method is particularly suitable for solving time-dependent nonlinear hyperbolic systems involving dynamical evolutions of waves and discontinuities. Therefore, it has been applied to a wide spectrum of problems, e.g., aerodynamics, aeroacoustics, magnetohydrodynamics, multi-material flows, and detonations. This book contains algorithm analysis, numerical examples, as well as demonstration codes. This book is intended for graduate students and researchers who are interested in the fields such as computational fluid dynamics (CFD), mechanical engineering, and numerical computation.Table of ContentsIntroduction.- Non-dissipative Core Scheme of CESE Method.- CESE Schemes with Numerical Dissipation.- Multi-Dimensional CESE Schemes on Cartesian Meshes.- CESE Schemes on Unstructured Meshes.- High-Order CESE Schemes.- Numerical Features of CESE Schemes.- Application: Hypersonic Aerodynamics.- Application: Compressible Multi-Fluid.- Other Applications.- Summary.

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Book SynopsisRiemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics.Trade Review"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry." -Publicationes Mathematicae "This is a very nice introduction to global Riemannian geometry, which leads the reader quickly to the heart of the topic. Nevertheless, classical results are also discussed on many occasions, and almost 60 pages are devoted to exercises." -Newsletter of the EMS "In the reviewer's opinion, this is a superb book which makes learning a real pleasure." -Revue Romaine de Mathematiques Pures et Appliquees "This mainstream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises." -Monatshefte F. MathematikTable of ContentsPreface to the 1st edition * Preface to the 2nd edition * Preface to the English edition * How to use this book * 0. Differentiable Manifolds * 1. Riemannian Metrics * 2. Affine Connections; Riemannian Connections * 3. Geodesics; Convex Neighborhoods * 4. Curvature * 5. Jacobi Fields * 6. Isometric Immersions * 7. Complete Manifolds; Hopf-Rinow and Hadamard Theorems * 8. Spaces of Constant Curvature * 9. Variations of Energy * 10. The Rauch Comparison Theorem * 11. The Morse Index Theorem * 12. The Fundamental Group of Manifolds of Negative Curvature * 13. The Sphere Theorem * References * Index

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

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Book SynopsisThis volume 6 of the Collected Works comprises 27 papers by V.I.Arnold, one of the most outstanding mathematicians of all times, written in 1991 to 1995. During this period Arnold's interests covered Vassiliev’s theory of invariants and knots, invariants and bifurcations of plane curves, combinatorics of Bernoulli, Euler and Springer numbers, geometry of wave fronts, the Berry phase and quantum Hall effect. The articles include a list of problems in dynamical systems, a discussion of the problem of (in)solvability of equations, papers on symplectic geometry of caustics and contact geometry of wave fronts, comments on problems of A.D.Sakharov, as well as a rather unusual paper on projective topology. The interested reader will certainly enjoy Arnold’s 1994 paper on mathematical problems in physics with the opening by-now famous phrase “Mathematics is the name for those domains of theoretical physics that are temporarily unfashionable.” The book will be of interest to the wide audience from college students to professionals in mathematics or physics and in the history of science. The volume also includes translations of two interviews given by Arnold to the French and Spanish media. One can see how worried he was about the fate of Russian and world mathematics and science in general.Table of Contents1 Bernoulli–Euler updown numbers associated with function singularities, their combinatorics and arithmetics.- 2 Congruences for Euler, Bernoulli and Springer numbers of Coxeter groups.- 3 The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups.- 4 Springer numbers and Morsification spaces.- 5 Polyintegrable flows.- 6 Bounds for Milnor numbers of intersections in holomorphic dynamical systems.- 7 Some remarks on symplectic monodromy of Milnor fibrations.- 8 Topological properties of Legendre projections in contact geometry of wave fronts [On topological properties of Legendre projections in contact geometry of wave fronts].- 9 Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques [On topological properties of Lagrangian projections in symplectic geometry of caustics].- 10 Plane curves, their invariants, perestroikas and classifications (with an appendix by F. Aicardi).- 11 Invariants and perestroikas of plane fronts.- 12 The Vassiliev theory of discriminants and knots.- 13 The geometry of spherical curves and the algebra of quaternions.- 14 Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect.- 15 Problems on singularities and dynamical systems.- 16 Sur quelques problèmes de la théorie des systèmes dynamiques [On some problems in the theory of dynamical systems].- 17 Mathematical problems in classical physics.- 18 Problèmes résolubles et problèmes irrésolubles analytiques et géométriques [Solvable and unsolvable analytic and geometric problems].- 19 Projective topology.- 20 Questions à V.I. Arnold (an interview with M. Audin and P. Iglésias) [Questions to V.I. Arnold].- 21 En todo matemático hay un ángel y un demonio (an interview with Marimar Jiménez) [In every mathematician, there is an angel and a devil].- 22 Will Russian mathematics survive?.- 23 Will mathematics survive? Report on the Zurich Congress.- 24 Why study mathematics? What mathematicians think about it.- 25 Preface to the Russian translation of the book by M.F. Atiyah “The Geometry and Physics of Knots”.- 26 A comment on one of A.D. Sakharov’s “Amateur Problems”.- 27 Comments on two of A.D. Sakharov’s “Amateur Problems”.- Acknowledgements.

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

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Book SynopsisThe Ising model provides a detailed mathematical description of ferromagnetism and is widely used in statistical physics and condensed matter physics. In this Student''s Guide, the author demystifies the mathematical framework of the Ising model and provides students with a clear understanding of both its physical significance, and how to apply it successfully in their calculations. Key topics related to the Ising model are covered, including exact solutions of both finite and infinite systems, series expansions about high and low temperatures, mean-field approximation methods, and renormalization-group calculations. The book also incorporates plots, figures, and tables to highlight the significance of the results. Designed as a supplementary resource for undergraduate and graduate students, each chapter includes a selection of exercises intended to reinforce and extend important concepts, and solutions are also available for all exercises.Table of Contents1. The Ising model; 2. Finite Ising systems; 3. Partial summations and effective interactions; 4. Infinite Ising systems in one dimension; 5. The Onsager solution and exact series expansions; 6. The mean-field approach; 7. Position-space renormalization-group techniques; Index.

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Book Synopsis1. Mathematical Introduction.- 1.1. Linear Vector Spaces: Basics.- 1.2. Inner Product Spaces.- 1.3. Dual Spaces and the Dirac Notation.- 1.4. Subspaces.- 1.5. Linear Operators.- 1.6. Matrix Elements of Linear Operators.- 1.7. Active and Passive Transformations.- 1.8. The Eigenvalue Problem.- 1.9. Functions of Operators and Related Concepts.- 1.10. Generalization to Infinite Dimensions.- 2. Review of Classical Mechanics.- 2.1. The Principle of Least Action and Lagrangian Mechanics.- 2.2. The Electromagnetic Lagrangian.- 2.3. The Two-Body Problem.- 2.4. How Smart Is a Particle?.- 2.5. The Hamiltonian Formalism.- 2.6. The Electromagnetic Force in the Hamiltonian Scheme.- 2.7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations.- 2.8. Symmetries and Their Consequences.- 3. All Is Not Well with Classical Mechanics.- 3.1. Particles and Waves in Classical Physics.- 3.2. An Experiment with Waves and Particles (Classical).- 3.3. The Double-Slit Experiment with Light.- 3.4. Matter Waves (de Broglie Waves).- 3.5. Conclusions.- 4. The Postulatesa General Discussion.- 4.1. The Postulates.- 4.2. Discussion of Postulates I -III.- 4.3. The Schrödinger Equation (Dotting Your i's and Crossing your ?'s).- 5. Simple Problems in One Dimension.- 5.1. The Free Particle.- 5.2. The Particle in a Box.- 5.3. The Continuity Equation for Probability.- 5.4. The Single-Step Potential: a Problem in Scattering.- 5.5. The Double-Slit Experiment.- 5.6. Some Theorems.- 6. The Classical Limit.- 7. The Harmonic Oscillator.- 7.1. Why Study the Harmonic Oscillator?.- 7.2. Review of the Classical Oscillator.- 7.3. Quantization of the Oscillator (Coordinate Basis).- 7.4. The Oscillator in the Energy Basis.- 7.5. Passage from the Energy Basis to the X Basis.- 8. The Path Integral Formulation of Quantum Theory.- 8.1. The Path Integral Recipe.- 8.2. Analysis of the Recipe.- 8.3. An Approximation to U(t) for the Free Particle.- 8.4. Path Integral Evaluation of the Free-Particle Propagator.- 8.5. Equivalence to the Schrödinger Equation.- 8.6. Potentials of the Form V=a + bx + cx2 + d? + ex?.- 9. The Heisenberg Uncertainty Relations.- 9.1. Introduction.- 9.2. Derivation of the Uncertainty Relations.- 9.3. The Minimum Uncertainty Packet.- 9.4. Applications of the Uncertainty Principle.- 9.5. The Energy-Time Uncertainty Relation.- 10. Systems with N Degrees of Freedom.- 10.1. N Particles in One Dimension.- 10.2. More Particles in More Dimensions.- 10.3. Identical Particles.- 11. Symmetries and Their Consequences.- 11.1. Overview.- 11.2. Translational Invariance in Quantum Theory.- 11.3. Time Translational Invariance.- 11.4. Parity Invariance.- 11.5. Time-Reversal Symmetry.- 12. Rotational Invariance and Angular Momentum.- 12.1. Translations in Two Dimensions.- 12.2. Rotations in Two Dimensions.- 12.3. The Eigenvalue Problem of Lz.- 12.4. Angular Momentum in Three Dimensions.- 12.5. The Eigenvalue Problem of L2 and Lz.- 12.6. Solution of Rotationally Invariant Problems.- 13. TheHydrogen Atom.- 13.1. The Eigenvalue Problem.- 13.2. The Degeneracy of the Hydrogen Spectrum.- 13.3. Numerical Estimates and Comparison with Experiment.- 13.4. Multielectron Atoms and the Periodic Table.- 14. Spin.- 14.1. Introduction.- 14.2. What is the Nature of Spin?.- 14.3. Kinematics of Spin.- 14.4. Spin Dynamics.- 14.5. Return of Orbital Degrees of Freedom.- 15. Addition of Angular Momenta.- 15.1. A Simple Example.- 15.2. The General Problem.- 15.3. Irreducible Tensor Operators.- 15.4. Explanation of Some Accidental Degeneracies.- 16. Variational and WKB Methods.- 16.1. The Variational Method.- 16.2. The Wentzel-Kramers-Brillouin Method.- 17. Time-Independent Perturbation Theory.- 17.1. The Formalism.- 17.2. Some Examples.- 17.3. Degenerate Perturbation Theory.- 18. Time-Dependent Perturbation Theory.- 18.1. The Problem.- 18.2. First-Order Perturbation Theory.- 18.3. Higher Orders in Perturbation Theory.- 18.4. A General Discussion of Electromagnetic Interactions.- 18.5. Interaction of Atoms with Electromagnetic Radiation.- 19. Scattering Theory.- 19.1. Introduction.- 19.2. Recapitulation of One-Dimensional Scattering and Overview.- 19.3. The Born Approximation (Time-Dependent Description).- 19.4. Born Again (The Time-Independent Approximation).- 19.5. The Partial Wave Expansion.- 19.6. Two-Particle Scattering.- 20. The Dirac Equation.- 20.1. The Free-Particle Dirac Equation.- 20.2. Electromagnetic Interaction of the Dirac Particle.- 20.3. More on Relativistic Quantum Mechanics.- 21. Path IntegralsII.- 21.1. Derivation of the Path Integral.- 21.2. Imaginary Time Formalism.- 21.3. Spin and Fermion Path Integrals.- 21.4. Summary.- A.l. Matrix Inversion.- A.2. Gaussian Integrals.- A.3. Complex Numbers.Trade Review`An excellent text....The postulates of quantum mechanics and the mathematical underpinnings are discussed in a clear, succint manner.' - American Scientist, from a review of the First EditionTable of ContentsMathematical Introduction.- Review of Classical Mechanics.- All Is Not Well With Classical Mechanics. The Postulates-A General Discussion.- Simple Problems in One Dimension.- The Classical Limit.- The Harmonic Oscillator.- The Path Integral Formulation of Quantum Theory.- The Heisenberg Uncertainty Relations.- Systems with N Degrees of Freedom.- Symmetries and Their Consequences.- Rotational Invariance and Angular Momentum.- The Hydrogen Atom.- Spin.- Addition of Angular Momenta.- Variational and WKB Methods.- Time-Independent Perturbation Theory.- Time-Dependent Perturbation Theory.- Scattering Theory.- The Dirac Equation.- Path Integrals-II.- Appendix.- Answers to Selected Exercises.- Table of Constants.- Index

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Book SynopsisThis introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles. While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. The goal of this book is to present important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other. Trade Review“He has written a book about principal bundles in the classical sense which is of great interest in and of itself … . a textbook which can be used in an advanced one-year course or for self-learning. … the book is also interesting for a physicist, because one can find the geometric basis of many mathematical tools used in physics. … reviewer has greatly enjoyed reading the book and acknowledges the author’s bravery in writing another text on differential geometry!” (Fernando Etayo Gordejuela, Mathematical Reviews, November, 2015)“The present book deals with principle bundles and their relevance in physics with a ground work on differential geometry. … The book will be helpful to the graduate and under graduate students of mathematics and physics. It can also be an informative hand book of the researchers in differential geometry and physics.” (Uday Chand De, zbMATH 1321.53004, 2015)Table of ContentsIntroduction.- Basics of Manifolds.- Vector Bundles.- Vectors and Covectors.- Differential Forms.- Lie Derivatives.- Lie Groups.- Frobenius Theorem.- Principle Bundles.- Connections on Principle Bundles.- Curvature of a Connection.- Classical Electromagnetism.- Yang-Mills Theory.- Gauge Theory.- The Dirac Monopole.- Instantons.- What Next?.- Discussion of the Exercises.

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Book SynopsisThe book aims to present current knowledge concerning the propagation of electro­ magnetic waves in a homogeneous magnetoplasma for which temperature effects are unimportant. It places roughly equal emphasis on the radio and the . hydromagnetic parts of the electromagnetic spectrum. The dispersion properties of a magnetoplasma are treated as a function both of wave frequency (assumed real) and of ionization density. However, there is little discussion of propagation in a stratified medium, for of collisions is included only which reference may be made to Budden [1] . The effect in so far as this can be done with simplicity. The book describes how pulses are radiated from both small and large antennas embedded in a homogeneous magneto­ plasma. The power density radiated from a type of dipole antenna is studied as a function of direction of radiation in all bands of wave frequency. Input reactance is not treated, but the dependence of radiation resistance on wave frequency is described for the entire electromagnetic spectrum. Also described is the relation between beaming and guidance for Alfven waves.Table of Contents1. Elementary properties of a plasma.- Plasma.- Equations of drift motion.- Isothermal atmosphere in equilibrium.- Types of wave.- Effect of collisions.- The continuity equations.- 2. Maxwell’s equations.- Equations in terms of current and charge densities.- Equations in terms of electric moment per unit volume.- The exponential wave function.- The concept of a dispersion relation.- Calculation of the dispersion relation (electric current method).- Calculation of the dispersion relation (electric moment method).- 3. Isotropic plasma.- Mobility and conductivity of an isotropic plasma.- Susceptibility and dielectric constant of a collisionless isotropic plasma.- The plasma frequency.- Refractive index of a collisionless isotropic plasma.- Wave dispersion in a collisionless isotropic plasma.- Effect of collisions in an isotropic plasma.- Importance of ordered kinetic energy in a plasma.- Poynting’s theorem in a plasma.- The energy significance of the complex dielectric constant of an isotropic plasma.- 4. Alternating current in a magnetoplasma.- Mobility tensor for a magnetoplasma.- Conductivity tensor for a magnetoplasma.- Low-frequency conduction properties of an infinite homogenous magnetoplasma.- Low-frequency conduction properties of a slab of magnetoplasma.- Effect of plasma scale on wave propagation.- 5. General properties of phase propagation in a magnetoplasma.- Susceptibility tensor for a magnetoplasma.- Alternative expressions for the susceptibility tensor components in the absence of multiple ion species.- Dispersion relation for a magnetoplasma.- Elliptic polarization.- Alternative derivation of the dispersion relation for a magnetoplasma.- The radio and hydromagnetic approximations.- Effect of collisions in a magnetoplasma.- 6. General properties of group propagation in a magnetoplasma.- Frequency and angular spectra.- Velocity of a wave packet.- Relation between phase and group propagation.- Method for calculating group velocity in a magnetoplasma.- Formulae for group velocity in a magnetoplasma.- Beam radiation in a magnetoplasma.- 7. Propagation of phase along the imposed magnetic field.- Circular polarization.- The dispersion relation for longitudinal propagation.- Longitudinal Alfvén waves.- The violin-string approach to longitudinal Alfven waves.- The hydromagnetic approximation for longitudinal propagation.- The radio approximation for longitudinal propagation.- The Eckersley approximation for longitudinal propagation.- Comparison of approximations.- Pass and stop bands of frequency for longitudinal propagation.- Particle vibration for longitudinal propagation.- Plasma motion in a longitudinal Alfvén wave.- Longitudinal propagation in low-density and high-density magnetoplasmas.- Effect of collisions on longitudinal propagation.- Effect of an additional ion species on longitudinal propagation.- Pass and stop bands of ionization density for longitudinal propagation.- 8. Energy flow and group velocity for longitudinal propagation.- Electromagnetic energy density for longitudinal propagation.- Kinetic energy density for longitudinal propagation.- Energy flow and group velocity for longitudinal propagation.- Energy in a longitudinal Alfvén wave.- Faraday rotation for longitudinal Alfvén waves.- A resonator for longitudinal Alfvén waves.- The mode of operation of a hydromagnetic violin-string.- Freezing of the magnetic field in the plasma (longitudinal Alfven waves).- Energy in a longitudinal whistler wave in the band ?Mi ? ? ? ?Me.- A resonator for longitudinal whistler waves in the band ?Mi ? ? ? ?Me.- Freezing of the magnetic field in the electron gas (longitudinal whistler wave).- Solid-state plasmas.- 9. Propagation of phase transverse to the imposed magnetic field.- The O wave.- The X wave.- Superposition of the O and X waves.- Pass and stop bands of frequency for transverse propagation.- The hybrid resonant frequencies.- Transverse propagation in a low-density magnetoplasma.- Pass and stop bands of ionization density for transverse propagation.- Effect of collisions on transverse propagation.- 10. Elliptic polarization of the X wave for transverse propagation.- The electric ellipse for transverse propagation of the X wave.- Frequency dependence of the electric ellipse.- Particle vibration for transverse propagation of the X wave.- Plasma compressions and dilations for transverse propagation of the X wave.- Non-reciprocity.- 11. Energy behaviour of the X wave for transverse propagation.- Electromagnetic energy density for transverse propagation of the X wave.- Kinetic energy density for transverse propagation of the X wave.- Energy flow and group velocity for transverse propagation of the X wave.- A resonator for transverse Alfvén waves.- The mode of operation of a hydromagnetic organ-pipe.- Freezing of the magnetic field in the plasma (transverse Alfvén waves).- 12. Propagation at any angle to the imposed magnetic field..- The zeros in the frequency dispersion curves.- Nomenclature for the characteristic waves.- The cross-connection phenomenon for frequency dispersion curves.- Frequency dispersion curves for nearly transverse propagation.- Frequency dispersion curves for nearly longitudinal propagation.- The elliptic polarizations of the O and X waves at the plasma frequency.- Effect of an additional ion species on cross-connection phenomena.- The infinities in the frequency dispersion curves.- Permitted regions for the frequency dispersion curves.- The cross-connection phenomenon for ionization dispersion curves.- Permitted regions for the ionization dispersion curves.- Propagation into a magnetoplasma from free space.- 13. The radio approximation.- The radio approximation to the dispersion relation.- Frequency dispersion curves in the radio band.- Frequency dependence of elliptic polarization in the radio band.- Frequency dependence of the direction of group propagation in the radio band.- Variation in the angle of squint of a rotating broadside antenna in the radio band.- Dependence of refractive index on ionization density in the radio band.- Dependence of elliptic polarization on ionization density in the radio band.- Dependence of the direction of group propagation on ionization density in the radio band.- 14. The hydromagnetic approximation.- The hydromagnetic approximation to the dispersion relation.- Frequency dispersion curves in the hydromagnetic band.- Effect of ionic collisions in the hydromagnetic band.- The fit between the hydromagnetic and radio approximations.- Frequency dependence of elliptic polarization in the hydromagnetic band.- Frequency dependence of the tilts of the electronic and current ellipses in the hydromagnetic band.- Frequency dependence of the direction of group propagation in the hydromagnetic band.- Polar diagrams for group velocity in the hydromagnetic band.- Dependence of refractive index on ionization density in the hydromagnetic band.- Dependence of elliptic polarization on ionization density in the hydromagnetic band.- Dependence of the direction of group propagation on ionization density in the hydromagnetic band.- 15. The quasi-longitudinal and quasi-transverse approximations.- The transition angle between the quasi-longitudinal and quasi-transverse approximations.- The regions of validity for the first- order angular approximations.- Importance of avoiding angular approximations that upset an infinity of a refractive index.- The regions of validity for angular approximations of practical value.- Accuracy of ?2n/??p2 using angular approximations.- The quasi-transverse approximation when ? ? ?Mi.- The quasi-longitudinal approximation when ? ? Max(?N,?Me).- The quasi-longitudinal approximation when ?Mi ? ? < ?Me ? ?N.- Group propagation for the whistler wave when ??1 ? ? < ?Me ? ?N.- Comparison of the zero-order quasi-longitudinal approximation in the whistler band with the unapproximated formulae.- 16. Directional behaviour of group velocity in a magnetoplasma.- Group propagation of the X wave in the pass band ? > ?C2.- Group propagation of the O wave in the pass band ? > ?N.- Group propagation in the upper part of the whistler band [??1 < ? < Min(?N,?Me)].- Group propagation in the lower part of the whistler band (? ? ??1).- Group propagation of the O wave in the pass band ? < ?Mi.- Group propagation of the X wave in the pass band ?C1 ?Me.- Group propagation of the X wave in the pass band ?C1 < ? < ??2 when ?N < ?Me.- 17. The field of an antenna in a magnetoplasma.- Axes of coordinates.- Angular spectra of O and X waves.- The predominant directions of group and phase propagation in the far field.- The method of steepest descent.- Simplification of the notation.- The power density in the far field.- Use of the angle of phase propagation as an independent variable.- Radiation from a gaussian dipole in a homogeneous magnetoplasma.- A reference isotropic medium.- Radiation ?C2.- Radiation in the frequency band ?C2 > ? > ??2.- Radiation in the frequency band ?? > ? > ?N.- Radiation in the frequency band ?N > ? > ?C1.- Radiation in the frequency band ?Me > ? > ??1.- Radiation in the frequency band ??1 > ? > ?Mi.- Radiation in the frequency band ? < ?Mi.- Frequency dependence of radiation resistance.- The relation between beaming and guidance in a homogeneous magnetoplasma.- The relation between beaming and guidance for the whistler wave when ??1 < ? ? Min (?N, ?Me).- The relation between beaming and guidance for the O wave when ? ? ?Mi.- The relation between beaming and guidance for the combined O and X waves when ? ? ?Mi.- Effect of energy absorption on Alfvén guidance.- Symbols.- Index of subjects.

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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 Matlab, and a chapter devoted to the basics of scientific programming with Matlab is included. A parallel edition using Python instead of Matlab is also available.Last but not least, each chapter is accompanied by an extensive set of course-tested exercises and solutions.Table of Contents Introduction.- 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 SynopsisSpecial relativity is the basis of many fields in modern physics: particle physics, quantum field theory, high-energy astrophysics, etc. This theory is presented here by adopting a four-dimensional point of view from the start. An outstanding feature of the book is that it doesn’t restrict itself to inertial frames but considers accelerated and rotating observers. It is thus possible to treat physical effects such as the Thomas precession or the Sagnac effect in a simple yet precise manner. In the final chapters, more advanced topics like tensorial fields in spacetime, exterior calculus and relativistic hydrodynamics are addressed. In the last, brief chapter the author gives a preview of gravity and shows where it becomes incompatible with Minkowsky spacetime. Well illustrated and enriched by many historical notes, this book also presents many applications of special relativity, ranging from particle physics (accelerators, particle collisions, quark-gluon plasma) to astrophysics (relativistic jets, active galactic nuclei), and including practical applications (Sagnac gyrometers, synchrotron radiation, GPS). In addition, the book provides some mathematical developments, such as the detailed analysis of the Lorentz group and its Lie algebra. The book is suitable for students in the third year of a physics degree or on a masters course, as well as researchers and any reader interested in relativity. Thanks to the geometric approach adopted, this book should also be beneficial for the study of general relativity. “A modern presentation of special relativity must put forward its essential structures, before illustrating them using concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Éric Gourgoulhon.” (excerpt from the Foreword by Thibault Damour)Table of ContentsMinkowski Spacetime.- Worldlines and Proper Time.- Observers.- Kinematics 1: Motion with Respect to an Observer.- Kinematics 2: Change of Observer.- Lorentz Group.- Lorentz Group as a Lie Group.- Inertial Observers and Poincaré Group.- Energy and Momentum.- Angular Momentum.- Principle of Least Action.- Accelerated Observers.- Rotating Observers.- Tensors and Alternate Forms.- Fields on Spacetime.- Integration in Spacetime.- Electromagnetic Field.- Maxwell Equations.- Energy-Momentum Tensor.- Energy-Momentum of the Electromagnetic Field.- Relativistic Hydrodynamics.- What about Relativistic Gravitation?.- A Basic Algebra.- B Web Pages.- C Special Relativity Books.

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Book SynopsisNoncommutative geometry combines themes from algebra, analysis and geometry and has significant applications to physics. This book focuses on cyclic theory, and is based upon the lecture courses by Daniel G. Quillen at the University of Oxford from 198892, which developed his own approach to the subject. The basic definitions, examples and exercises provided here allow non-specialists and students with a background in elementary functional analysis, commutative algebra and differential geometry to get to grips with the subject. Quillen''s development of cyclic theory emphasizes analogies between commutative and noncommutative theories, in which he reinterpreted classical results of Hamiltonian mechanics, operator algebras and differential graded algebras into a new formalism. In this book, cyclic theory is developed from motivating examples and background towards general results. Themes covered are relevant to current research, including homomorphisms modulo powers of ideals, traces onTrade Review'The monograph is an excellent introduction to cyclic theory and an absolute must to any academic library, let alone a superb first-hand account and a selfless tribute to the late Daniel G. Quillen.' Igor V. Nikolaev, zbMATHTable of ContentsIntroduction; 1. Background results; 2. Cyclic cocycles and basic operators; 3. Algebras of operators; 4. GNS algebra; 5. Geometrical examples; 6. The algebra of noncommutative differential forms; 7. Hodge decomposition and the Karoubi operator; 8. Connections; 9. Cocycles for a commutative algebra over a manifold; 10. Cyclic cochains; 11. Cyclic cohomology; 12. Periodic cyclic homology; References; List of symbols; Index of notation; Subject index.

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Book SynopsisThis book and CD-ROM package has been developed in a lab-oriented course taught at Cal Tech in 1995 and 1996, and simultaneously augmented by artificial life research conducted there.Table of Contents1 Flavors of Artificial Life.- 1.1 Whither a Theory of the Living State?.- 1.2 Emulation and Simulation.- 1.3 Carbon-Based Artificial Life.- 1.4 Turing and von Neumann Automata.- 1.5 Cellular Automata.- 1.6 Overview.- 2 Artificial Chemistry and Self-Replicating Code.- 2.1 Virtual Machines and Self-Reproducing CA.- 2.2 Viruses and Core Worlds.- 2.3 The tierra System.- 2.4 avida, amoeba, and the Origin of Life.- 2.5 Overview.- 3 Introduction to Information Theory.- 3.1 Information Theory and Life.- 3.2 Channels and Coding.- 3.3 Uncertainty and Shannon Entropy.- 3.4 Joint and Conditional Uncertainty.- 3.5 Information.- 3.6 Noiseless Coding.- 3.7 Channel Capacity and Fundamental Theorem.- 3.8 Information Transmission Capacity for Genomes.- 3.9 Overview.- 4 Statistical Mechanics and Thermodynamics.- 4.1 Phase Space and Statistical Distribution Function.- 4.2 Averages, Ergodicity, and the Ergodic Theorem.- 4.3 Thermodynamical Equilibrium, Relaxation.- 4.4 Energy.- 4.5 Entropy.- 4.6 Second Law of Thermodynamics.- 4.7 Tèmperature.- 4.8 The Gibbs Distribution.- 4.9 Nonequilibrium Thermodynamics.- 4.10 First-Order Phase Transitions.- 4.11 Overview.- 5 Complexity of Simple Living Systems.- 5.1 Complexity and Information.- 5.2 The Maxwell Demon.- 5.3 Kolmogorov Complexity.- 5.4 Physical Complexity and the Natural Maxwell Demon.- 5.5 Complexity of tRNA.- 5.6 Complexity in Artificial Life.- 5.7 Overview.- 6 Self-Organization to Criticality.- 6.1 Self-Organization and Sandpiles.- 6.2 SOC in Forest Fires.- 6.3 SOC in the Living State.- 6.4 Theories of SOC.- 6.5 Overview.- 7 Percolation.- 7.1 Site Percolation.- 7.2 Cluster Size Distribution.- 7.3 Percolation in 1D.- 7.4 Higher-Dimensional Euclidean Lattices.- 7.5 Percolation on the Bethe Lattice.- 7.6 Scaling Theory.- 7.7 Percolation and Evolution.- 7.8 Overview.- 8 Fitness Landscapes.- 8.1 Theoretical Formulation.- 8.2 Example Landscapes.- 8.3 Fractal Landscapes.- 8.4 Diffusive and Nondiffusive Processes.- 8.5 RNA Landscapes.- 8.6 Fitness Landscape in avida.- 8.7 Overview.- 9 Experiments with avida.- 9.1 Choice of Chemistry.- 9.2 A Simple Experiment.- 9.3 Experiments in Adaptation.- 9.4 Experiments with Species and Genetic Distance.- 9.5 Overview.- 10 Propagation of Information.- 10.1 Information Transport and Equilibrium.- 10.2 The Artificial Life System sanda.- 10.3 Diffusion and Waves.- 10.4 Comparison: Theory and Experiment.- 10.5 Overview.- 11 Adaptive Learning at the Error Threshold.- 11.1 Information Processing at the Edge of Chaos.- 11.2 Adaptation to Computation in avida.- 11.3 Eigen’s Error Threshold.- 11.4 Molecular Evolution as an Ising Model.- 11.5 The Race to the Error Threshold.- 11.6 Approach to Error Threshold in avida.- 11.7 Overview.- A The avida User’s Manual.- A.1 Introduction.- A.2 A Beginner’s Guide to avida.- A.3 Time Slicing and the Fitness Landscape.- A.4 Reproduction.- A.5 The Virtual Computer.- A.6 Mutations.- A.7 Installing avida.- A.8 The Text Interface.- A.9 Configuring avida Runs.- A.10 Guide to Output Files.- A.11 Summary of Variables.- A.12 Glossary.- References.

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Book SynopsisThis book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos).Table of ContentsIntroduction.- Linear Systems with Constant Coefficients.- Nonlinear Systems: Local Theory.- Nonlinear Systems: Global Theory.- Nondimensionalization and Scaling.- Trajectories Near Equilibria.- Oscillations in ODEs.- Bifurcation from Equilibria.- Examples of Global Bifurcation.- Epilogue.- Appendices.

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Book SynopsisAn integrated guide to cellular biophysics and nonlinear dynamics, introducing students to the mathematical modeling of excitable cells. It combines empirical physiology and mathematical theory to present key interdisciplinary tools, highlighting how quantitative approaches can complement and advance bench research.Trade Review'In this text, Conradi Smith does an excellent job of teaching students with no mathematical training beyond calculus how to use differential equations to understand the basic principles of cell physiology and excitability. He skilfully walks students through the steps of modeling and analysis, all the while working to develop intuition and insight into how things work. His emphasis on computational methods for solution as well as graphical and geometrical means for interpretation enables him to communicate complex ideas in understandable ways. Furthermore, his patience and attention to detail will be appreciated by those students who have not had extensive exposure to the art of mathematical modeling. This text is a wonderful addition to the mathematical biology textbook literature.' James P. Keener, University of UtahTable of Contents1. Introduction; Part I. Models and Odes: 2. Compartmental modeling; 3. Phase diagrams; 4. Ligands, receptors and rate laws; 5. Function families and characteristic times; 6. Bifurcation diagrams of scalar ODEs; Part II. Passive Membranes: 7. The Nernst equilibrium potential; 8. The current balance equation; 9. GHK theory of membrane permeation; Part III. Voltage-Gated Currents: 10. Voltage-gated ionic currents; 11. Regenerative ionic currents and bistability; 12. Voltage-clamp recording; 13. Hodgkin-Huxley model of the action potential; Part IV. Excitability and Phase Planes: 14. The Morris-Lecar model; 15. Phase plane analysis; 16. Linear stability analysis; Part V. Oscillations and Bursting: 17. Type II excitability and oscillations; 18. Type I excitability and oscillations; 19. The low-threshold calcium spike; 20. Synaptic currents.

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Book SynopsisThis text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject ''elliptic cohomology'' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten''s genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.Trade Review'It has a down-to-earth and inviting style (no small achievement in a book about functorial algebraic geometry). It is elegant, precise, and incisive, and it is strong on both theory and calculation.' Michael Berg, MAA Reviews'This book is likely to be quite useful to graduate students in algebraic topology. For years it has been an informal tradition for students of algebraic topology to teach themselves enough of the foundations of algebraic geometry to be able to translate between theorems about Hopf algebroids and theorems about algebraic stacks, and then to proceed to translate, as much as possible, calculations and theorems in algebraic topology into equivalent formulations in terms of moduli stacks of formal groups and related objects. This book does a great service to such students (and their advisors!), as it gives good answers to many of the questions such students inevitably ask.' Andrew Salch, MatSciNet'The presentation is lucid, pedagogical, and also offers a fresh point of view on classical topics. It draws from several mostly unpublished sources, for instance Strickland's manuscripts or various sets of notes by Goerss, Hopkins, and Lurie, and combines them in a single uniform treatment. Moreover, it contains a wealth of references to the published and unpublished literature that guides the interested reader to further topics that are only discussed in passing.' Tobias Barthel, zbMATH OpenTable of ContentsForeword Matthew Ando; Preface; Introduction; 1. Unoriented bordism; 2. Complex bordism; 3. Finite spectra; 4. Unstable cooperations; 5. The σ-orientation; Appendix A. Power operations; Appendix B. Loose ends; References; Index.

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Book SynopsisGroup theory helps readers in understanding the energy spectrum and the degeneracy of systems possessing discrete symmetry and continuous symmetry. The fundamental concepts of group theory and its applications are presented with the help of solved problems and exercises. The text covers two essential aspects of group theory, namely discrete groups and Lie groups. Important concepts including permutation groups, point groups and irreducible representation related to discrete groups are discussed with the aid of solved problems. Topics such as the matrix exponential, the circle group, tensor products, angular momentum algebra and the Lorentz group are explained to help readers in understanding the quark model and theory composites. Real-life applications including molecular vibration, level splitting perturbation, crystal field splitting and the orthogonal group are also covered. Application-oriented solved problems and exercises are interspersed throughout the text to reinforce understaTable of ContentsPreface; Acknowledgement; Dedication; 1. Introduction; 2. Molecular symmetry; 3. Representations of finite groups; 4. Elementary applications; 5. Lie groups and lie algebras; 6. Further applications; Reference; Index.

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Book SynopsisFrom Bäcklund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Bäcklund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed compTable of ContentsIntroduction; 1. A bundle approach to conformal surfaces in space-forms; 2. The mean curvature sphere congruence; 3. Surfaces under change of flat metric connection; 4. Willmore surfaces; 5. The Euler–Lagrange constrained Willmore surface equation; 6. Transformations of generalized harmonic bundles and constrained Willmore surfaces; 7. Constrained Willmore surfaces with a conserved quantity; 8. Constrained Willmore surfaces and the isothermic surface condition; 9. The special case of surfaces in 4-space; Appendix A. Hopf differential and umbilics; Appendix B. Twisted vs. untwisted Bäcklund transformation parameters; References; Index.

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Book SynopsisPerturbation Theory: Advances in Research and Applications begins with a deliberation on the development of a formalism of the Exchange perturbation theory (EPT) that accounts for the general identity principle of electrons that belong to different atomic centres. The possible applications of the theory concerning scattering and collision problems are discussed, and the authors apply the TDEPT to the description of the positron scattering on a Lithium atom as an example. Next, spin fluctuations in metallic multiband systems are discussed, including how to calculate the effect of itinerant spin excitations on the electronic properties and formulate a theory of spin fluctuation-induced superconductivity. The function of spin-orbit coupling is emphasized. Following this, the authors review how, governed by chiral symmetry, the long- and intermediate-range parts of the $NN$ potential unfold order by order, proceeding up to sixth order where convergence is achieved. Perturbative and nonperturbative approaches to nuclear amplitude are discussed, including the implications for renormalization. Continuing, this book presents proof of the good convergence properties of the new expansions on mathematical models that simulate the physical polarization function for light quarks and its derivative (the Adler function), in various prescriptions of renormalization-group summation. An overview of the calculation of one-loop corrections to the baryon axial vector current in the large-Nc heavy baryon chiral perturbation theory is offered, where Nc is the number of color charges. The matrix elements of the space components of the renormalization of the baryon axial vector current between SU(6) symmetric states yield the values of the axial vector couplings.

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Book SynopsisIt is well-known that the fundamental problem in contemporary theoretical physics is the "pacific coexistence" between General Relativity and Quantum Mechanics. The scenarios of the explorable relationships between classical space-time and quantum land are various: the geometrodynamic one (by a proper extension of geometry), the stochastic fractal one (defining a middle land mediated by QFT-like hypotheses), the emergent one (from a physical viewpoint, by the collective behaviours of discrete entities, which mathematically means that the geometry derives from an algebraic structure of events).This anthology includes some of the most significant voices on the problem of the possible relations between the space-time dynamics and the quantum networks of events.

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