Mathematical / Computational / Theoretical physics Books

Book SynopsisThis volume contains the detailed text of the major lectures delivered during the I-CELMECH Training School 2020 held in Milan (Italy). The school aimed to present a contemporary review of recent results in the field of celestial mechanics, with special emphasis on theoretical aspects. The stability of the Solar System, the rotations of celestial bodies and orbit determination, as well as the novel scientific needs raised by the discovery of exoplanetary systems, the management of the space debris problem and the modern space mission design are some of the fundamental problems in the modern developments of celestial mechanics. This book covers different topics, such as Hamiltonian normal forms, the three-body problem, the Euler (or two-centre) problem, conservative and dissipative standard maps and spin-orbit problems, rotational dynamics of extended bodies, Arnold diffusion, orbit determination, space debris, Fast Lyapunov Indicators (FLI), transit orbits and answer to a crucial question, how did Kepler discover his celebrated laws? Thus, the book is a valuable resource for graduate students and researchers in the field of celestial mechanics and aerospace engineering.Table of Contents1) The contribution by Ugo Locatelli focuses on the explicit construction of invariant tori exploiting suitable Hamiltonian normal forms, with particular emphasis on applications to Celestial Mechanics. First, the algorithm constructing the Kolmogorov normal form is described in detail. Then the extension to lowerdimensional elliptic tori is provided. Both algorithms are then combined so as to accurately approximate the long-term dynamics of the HD 4732 extrasolar system. 2) The contribution by Gabriella Pinzari presents a review of some results of their research group, regarding the relation between some particular motions of the Three–Body problem (3BP) and the motions of the so–called Euler (or two–centre) problem, which is integrable. For the analysis of such relation, the authors make use of two novel results: on one hand, the two–centre problem (2CP) bears a remarkable property, here called renormalizable integrability, which states that the simple averaged potential of the 2CP and the Euler integral are one function of the other; on the other hand, the motions of the Euler integral are at least qualitatively explicit, and the averaged Newtonian potential is a prominent part of the 3BP Hamiltonian.3) The contribution by Alessandra Celletti deals with dissipative systems, a key topic in Celestial Mechanics. In particular the problem of the existence of invariant tori for conformally symplectic systems, which have the property to transform the symplectic form into a multiple of itself, is studied. Two different models are presented: a discrete system known as the standard map and a continuous system known as the spin–orbit problem. In both cases, both the conservative and dissipative versions are considered, in order to highlight the differences between the symplectic and conformally symplectic dynamics. 4) The contribution by Gwenael Boué provides basic tools to understand the rotational dynamics of extended bodies which could be either rigid or deformable by tides. The problem is described in a Lagrangian formalism as it was developed by H. Poincar´e in 1901. The case of rigid body is also presented in the corresponding Hamiltonian formalism. The mathematical description of the deformation of the extended body follows the approach used by C. Ragazzo and L. Ruiz in their two papers of 2015, 2017 due to the compactness and clarity of their formalism. In this Chapter, many applications to the rotation and the libration of celestial bodies are illustrated. 5) The contribution by Christos Efthymiopoulos concerns the phenomenon of Arnold diffusion. The authors begin with the famous example given by Arnold to describe the slow diffusion taking place in the action–space in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The text introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion and at the same time performed a qualitative investigation of the phenomenon of Arnold diffusion with many examples. The problem of the speed of diffusion is investigated using methods of perturbation theory, with particular emphasis on Nekhoroshevs theorem. 6) The contribution by Giovanni F. Gronchi deals with the problem of initial orbit determination of a solar system body, i.e. the determination of a preliminary orbit from observations collected for example by a telescope. The two methods that are presented, named Link2 and Link3, try to link together two and three, respectively, short arcs of optical observations of the same object which can possibly be quite far apart in time. The conservation laws of Kepler’s problem are used to derive a polynomial equation of degree 9 (Link2) and 8 (Link3) for the distance of the body from the observer. 7) The contribution by Catalin Gales provides an overview of some recent developments in the study of dynamics of space debris with focus on specific resonant interactions, in particular those related to the tesseral resonances. After an historical introduction to the topic, the authors provide a long–term picture of the dynamics that can help in the modeling and mitigation of the space debris problem, both in term of Cartesian coordinates and in the Hamiltonian framework. Some key terms in the perturbing functions are classified, while the effect of the dissipative force of the atmospheric drag is also formulated. 8) The contribution by Massimiliano Guzzo presents the use of the Fast Lyapunov Indicators (FLI) in the Three–Body problem, with the eventual aim of computing transit orbits. The FLI belong to the family of the finite time indicators, which are able to extract the information of the solutions of the variational equations on short time intervals. First, the FLI are applied to two model problems: the standard map and the double gyre. Then, it is described a modification of the FLI which was originally introduced to improve the computation of stable and unstable manifolds in model systems and the Three–Body problem. 9) The contribution by Antonio Giorgilli provides an answer to a simple question, how did Kepler discover his celebrated laws?. The answer however is not that simple and the present paper guides the reader by a short walk along the main works of Kepler, notably the Astronomia Nova, trying to follow his search of the perfection of the World till the discovery of his celebrated laws. At the end of the road, the consciousness that the finish line had not yet been reached.

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Book SynopsisThis book is devoted to the construction of space group representations, their tabulation, and illustration of their use. Representation theory of space groups has a wide range of applications in modern physics and chemistry, including studies of electron and phonon spectra, structural and magnetic phase transitions, spectroscopy, neutron scattering, and superconductivity. The book presents a clear and practical method of deducing the matrices of all irreducible representations, including double-valued, and tabulates the matrices of irreducible projective representations for all 32 crystallographic point groups. One obtains the irreducible representations of all 230 space groups by multiplying the matrices presented in these compact and convenient to use tables by easily computed factors. A number of applications to the electronic band structure calculations are illustrated through real-life examples of different crystal structures. The book's content is accessible to both graduate and advanced undergraduate students with elementary knowledge of group theory and is useful to a wide range of experimentalists and theorists in materials and solid-state physics.Table of ContentsScope and Overview.- Mathematical Preliminaries.- Induced Representations.- Projective Representations.- Representations of the Space Groups.- Tables.- Group Theory and Quantum Mechanics.

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Book SynopsisThis volume celebrates the 100th birthday of Professor Chen-Ning Frank Yang (Nobel 1957), one of the giants of modern science and a living legend. Starting with reminiscences of Yang's time at the research centre for theoretical physics at Stonybrook (now named C. N. Yang Institute) by his successor Peter van Nieuwenhuizen, the book is a collection of articles by world-renowned mathematicians and theoretical physicists. This emphasizes the Dialogue Between Physics and Mathematics that has been a central theme of Professor Yang’s contributions to contemporary science. Fittingly, the contributions to this volume range from experimental physics to pure mathematics, via mathematical physics. On the physics side, the contributions are from Sir Anthony Leggett (Nobel 2003), Jian-Wei Pan (Willis E. Lamb Award 2018), Alexander Polyakov (Breakthrough Prize 2013), Gerard 't Hooft (Nobel 1999), Frank Wilczek (Nobel 2004), Qikun Xue (Fritz London Prize 2020), and Zhongxian Zhao (Bernd T. Matthias Prize 2015), covering an array of topics from superconductivity to the foundations of quantum mechanics. In mathematical physics there are contributions by Sir Roger Penrose (Nobel 2022) and Edward Witten (Fields Medal 1990) on quantum twistors and quantum field theory, respectively. On the mathematics side, the contributions by Vladimir Drinfeld (Fields Medal 1990), Louis Kauffman (Wiener Gold Medal 2014), and Yuri Manin (Cantor Medal 2002) offer novel ideas from knot theory to arithmetic geometry.Inspired by the original ideas of C. N. Yang, this unique collection of papers b masters of physics and mathematics provides, at the highest level, contemporary research directions for graduate students and experts alike.Table of Contents1 Frank Yang at Stony Brook and the Beginning of Supergravity.- 2. A Stacky Approach to Crystals.- 3 The Potts Model, the Jones Polynomial and Link Homology.- 4 The Penrose–Onsager–Yang Approach to Superconductivity and Superfluidity.- 5 Quantum Operads.- 6 Quantum computational complexity withphotons and linear optics.- 7 Quantized Twistors, G2*, and the Split Octonions.- 8 Kronecker Anomalies and Gravitational Striction.- 9 Projecting Local and Global Symmetries to the Planck Scale.- 10 Gauge Symmetry in Shape Dynamics.- 11 Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?.- 12 Quantum Anomalous Hall Effect.- 13 Magic Superconducting States in Cuprates.

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Book SynopsisThis book presents the state-of-the-art in supercomputer simulation. It includes the latest findings from leading researchers using systems from the High Performance Computing Center Stuttgart (HLRS) in 2021. The reports cover all fields of computational science and engineering ranging from CFD to computational physics and from chemistry to computer science with a special emphasis on industrially relevant applications. Presenting findings of one of Europe’s leading systems, this volume covers a wide variety of applications that deliver a high level of sustained performance.The book covers the main methods in high-performance computing. Its outstanding results in achieving the best performance for production codes are of particular interest for both scientists and engineers. The book comes with a wealth of color illustrations and tables of results.Table of ContentsPart I Physics.- Part II Molecules, Interfaces, and Solids.- Part III Reactive Flows.- Part IV Computational Fluid Dynamics.- Part V Transport and Climate.- Part VI Computer Science.- Part VII Miscellaneous Topics.

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Book SynopsisThis book provides a contemporary treatment of the problems related to anomalous diffusion and anomalous relaxation. It collects and promotes unprecedented applications dealing with diffusion problems and surface effects, adsorption-desorption phenomena, memory effects, reaction-diffusion equations, and relaxation in constrained structures of classical and quantum processes. The topics covered by the book are of current interest and comprehensive range, including concepts in diffusion and stochastic physics, random walks, and elements of fractional calculus. They are accompanied by a detailed exposition of the mathematical techniques intended to serve the reader as a tool to handle modern boundary value problems. This self-contained text can be used as a reference source for graduates and researchers working in applied mathematics, physics of complex systems and fluids, condensed matter physics, statistical physics, chemistry, chemical and electrical engineering, biology, and many others.Table of ContentsPreface.- Integral Transforms and Special Functions.- Concepts in Diffusion and Stochastic Processes.- Random Walks.- Elements of Fractional Calculus.- Fractional Anomalous Diffusion.- Adsorption Phenomena and Anomalous Behavior.- Reaction-Diffusion Problems.- Relaxation under Geometric Constraints I: Classical Processes.- Relaxation under Geometric Constraints II: Quantum Processes.- Index.

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Book SynopsisThis third edition expands on the original material. Large portions of the text have been reviewed and clarified. More emphasis is devoted to machine learning including more modern concepts and examples. This book provides the reader with the main concepts and tools needed to perform statistical analyses of experimental data, in particular in the field of high-energy physics (HEP).It starts with an introduction to probability theory and basic statistics, mainly intended as a refresher from readers’ advanced undergraduate studies, but also to help them clearly distinguish between the Frequentist and Bayesian approaches and interpretations in subsequent applications. Following, the author discusses Monte Carlo methods with emphasis on techniques like Markov Chain Monte Carlo, and the combination of measurements, introducing the best linear unbiased estimator. More advanced concepts and applications are gradually presented, including unfolding and regularization procedures, culminating in the chapter devoted to discoveries and upper limits.The reader learns through many applications in HEP where the hypothesis testing plays a major role and calculations of look-elsewhere effect are also presented. Many worked-out examples help newcomers to the field and graduate students alike understand the pitfalls involved in applying theoretical concepts to actual data.Trade Review“The book is important because, as AI and data science continue to shape the future, much interdisciplinary work is being done in many different domains. It is a very good example of interdisciplinary physics research using AI and data science. ... Graduate students are often expected to apply theoretical knowledge. This book will be an invaluable resource for them, to jumpstart their research by getting equipped with the right statistical and data analysis toolsets.” (Gulustan Dogan, Computing Reviews, August 8, 2023)Table of ContentsPreface to the third edition Preface to previous edition/s 1 Probability Theory 1.1 Why Probability Matters to a Physicist 1.2 The Concept of Probability 1.3 Repeatable and Non-Repeatable Cases 1.4 Different Approaches to Probability 1.5 Classical Probability 1.6 Generalization to the Continuum 1.7 Axiomatic Probability Definition 1.8 Probability Distributions 1.9 Conditional Probability 1.10 Independent Events 1.11 Law of Total Probability 1.12 Statistical Indicators: Average, Variance and Covariance 1.13 Statistical Indicators for a Finite Sample 1.14 Transformations of Variables 1.15 The Law of Large Numbers 1.16 Frequentist Definition of Probability References 2 Discrete Probability Distributions 2.1 The Bernoulli Distribution 2.2 The Binomial Distribution 2.3 The Multinomial Distribution 2.4 The Poisson Distribution References 3 Probability Distribution Functions 3.1 Introduction 3.2 Definition of Probability Distribution Function 3.3 Average and Variance in the Continuous Case 3.4 Mode, Median, Quantiles 3.5 Cumulative Distribution 3.6 Continuous Transformations of Variables 3.7 Uniform Distribution 3.8 Gaussian Distribution 3.9 X^2 Distribution 3.10 Log Normal Distribution 3.11 Exponential Distribution3.12 Other Distributions Useful in Physics 3.13 Central Limit Theorem 3.14 Probability Distribution Functions in More than One Dimension 3.15 Gaussian Distributions in Two or More Dimensions References 4 Bayesian Approach to Probability 4.1 Introduction 4.2 Bayes’ Theorem 4.3 Bayesian Probability Definition 4.4 Bayesian Probability and Likelihood Functions 4.5 Bayesian Inference 4.6 Bayes Factors 4.7 Subjectiveness and Prior Choice 4.8 Jeffreys’ Prior 4.9 Reference priors 4.10 Improper Priors 4.11 Transformations of Variables and Error Propagation References 5 Random Numbers and Monte Carlo Methods 5.1 Pseudorandom Numbers 5.2 Pseudorandom Generators Properties 5.3 Uniform Random Number Generators 5.4 Discrete Random Number Generators 5.5 Nonuniform Random Number Generators 5.6 Monte Carlo Sampling 5.7 Numerical Integration with Monte Carlo Methods 5.8 Markov Chain Monte Carlo References 6 Parameter Estimate 6.1 Introduction 6.2 Inference 6.3 Parameters of Interest 6.4 Nuisance Parameters 6.5 Measurements and Their Uncertainties 6.6 Frequentist vs Bayesian Inference 6.7 Estimators 6.8 Properties of Estimators 6.9 Binomial Distribution for Efficiency Estimate 6.10 Maximum Likelihood Method 6.11 Errors with the Maximum Likelihood Method 6.12 Minimum X^2 and Least-Squares Methods 6.13 Binned Data Samples 6.14 Error Propagation 6.15 Treatment of Asymmetric Errors References7 Combining Measurements7.1 Introduction7.2 Simultaneous Fits and Control Regions7.3 Weighted Average7.4 X^2 in n Dimensions7.5 The Best Linear Unbiased EstimatorReferences 8 Confidence Intervals8.1 Introduction8.2 Neyman Confidence Intervals8.3 Binomial Intervals8.4 The Flip-Flopping Problem8.5 The Unified Feldman–Cousins ApproachReferences 9 Convolution and Unfolding9.1 Introduction9.2 Convolution9.3 Unfolding by Inversion of the Response Matrix9.4 Bin-by-Bin Correction Factors9.5 Regularized Unfolding9.6 Iterative Unfolding9.7 Other Unfolding Methods9.8 Software Implementations9.9 Unfolding in More DimensionsReferences10 Hypothesis Tests10.1 Introduction10.2 Test Statistic10.3 Type I and Type II Errors10.4 Fisher’s Linear Discriminant10.5 The Neyman–Pearson Lemma10.6 Projective Likelihood Ratio Discriminant10.7 Kolmogorov–Smirnov Test10.8 Wilks’ Theorem10.9 Likelihood Ratio in the Search for a New SignalReferences 11 Machine Learning11.1 Supervised and Unsupervised Learning11.2 Terminology11.3 Machine Learning Classification from a Statistical Point of View11.4 Bias-Variance tradeo11.5 Overtraining11.6 Artificial Neural Networks 11.7 Deep Learning11.8 Convolutional Neural Networks11.9 Boosted Decision Trees11.10 Multivariate Analysis ImplementationsReferences 12 Discoveries and Upper Limits12.1 Searches for New Phenomena: Discovery and Upper Limits12.2 Claiming a Discovery12.3 Excluding a Signal Hypothesis12.4 Combined Measurements and Likelihood Ratio12.5 Definitions of Upper Limit12.6 Bayesian Approach12.7 Frequentist Upper Limits12.8 Modified Frequentist Approach: the CLs Method12.9 Presenting Upper Limits: the Brazil Plot12.10 Nuisance Parameters and Systematic Uncertainties12.11 Upper Limits Using the Profile Likelihood12.12 Variations of the Profile-Likelihood Test Statistic12.13 The Look Elsewhere EffectReferences Index

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Book SynopsisGraduate students typically enter into courses on string theory having little to no familiarity with the mathematical background so crucial to the discipline. As such, this book, based on lecture notes, edited and expanded, from the graduate course taught by the author at SISSA and BIMSA, places particular emphasis on said mathematical background. The target audience for the book includes students of both theoretical physics and mathematics. This explains the book’s "strange" style: on the one hand, it is highly didactic and explicit, with a host of examples for the physicists, but, in addition, there are also almost 100 separate technical boxes, appendices, and starred sections, in which matters discussed in the main text are put into a broader mathematical perspective, while deeper and more rigorous points of view (particularly those from the modern era) are presented. The boxes also serve to further shore up the reader’s understanding of the underlying math. In writing this book, the author’s goal was not to achieve any sort of definitive conciseness, opting instead for clarity and "completeness". To this end, several arguments are presented more than once from different viewpoints and in varying contexts. Table of ContentsChapter 1. The Polyakov path integral. Chapter 2. Introduction to 2d conformal field theories. Chapter 3. Spectrum, vertices, and BRST quantization. Chapter 4. Tree and one-loop amplitudes in the bosonic string. Chapter 5. Consistent 10d superstring, modular invariance, and all that. Chapter 6. The Heterotic string: part I. Chapter 7. Toroidal compactifications and T-duality (bosonic string). Chapter 8. The Heterotic string: part II. Chapter 9. Superstring interactions and anomalies. Chapter 10. Superstring D-branes. Chapter 11. Strings at strong coupling. Chapter 12. Calabi-Yau compactifications. Appendix.

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Book SynopsisIn recent years, cosmic rays have become the protagonists of a new scientific revolution. We are able today to film the Universe with telescopes of completely novel conception, recording information from many different messengers and accessing previously unknown cosmic regions.Written by a recognized authority in physics, this book takes readers on a captivating journey through the world of cosmic rays, their role in the revolutionary field of multi-messenger astronomy, their production from powerful accelerators close to the surfaces of black holes and compact objects, reaching the highest levels of energy observed in nature, and the implications this has for our understanding of the Universe. Through the stories of pioneering scientists, explorations of cutting-edge technologies, and simple explanations related to particle physics, quantum mechanics, and astrophysics, the book provides an illuminating state-of-the-art introduction to the current state of high-energy astrophysics. The book was written in straightforward yet rigorous language, so as to be accessible to the greater public. For those curious about the cosmos and cosmic gamma rays, nuclei, neutrinos, and gravitational waves, from casual observers to professional astronomers and physicists, the book is a must-read, offering a thrilling adventure into the future of astronomy and particle physics.Table of ContentsIntroduction 1 The Largest Energies in the Universe 1.1 The Universe around us 1.2 Particles and fields 1.3 Cosmic rays 2 The Mystery of Cosmic Rays 2.1 The discovery of natural radioactivity 2.2 Is natural radioactivity of extraterrestrial origin? 2.3 Father Wulf, a true experimental physicist 2.4 Pacini’s attenuation measures in water 2.5 Hess and balloon measurements 2.6 First developments and the tragedy of the first world war 3 Cosmic-Ray Research after the First World War 3.1 Research in Europe and the Pacini-Hess controversy 3.2 Research in the United States 3.3 Are cosmic rays predominantly charged or neutral? 3.4 Bruno Rossi and the discoveries after 1930 3.5 At the origins of elementary particle physics 3.6 The recognition of the scientific community 3.7 Hypothesis on the origin of cosmic rays: Tesla, Zwicky, Fermi 4 Cosmic Rays and the Physics of Elementary Particles 4.1 Leptons and mesons 4.2 The neutral pion 4.3 The discovery of strangeness 4.4 Laboratories on the mountains 4.5 Hunters become shepherds: particle accelerators 5 Fire Under the Ashes: the Discoveries at the End of the 20th Century and at the Beginning of the 21st Century 5.1 Cosmic rays of very-high-energy 5.2 Anomalous events 5.3 X-rays 5.4 Neutrinos from the Sun and the cosmos 6 Cosmic Ray Research Today: Multi-Messenger Astrophysics and the New Astronomy 6.1 Very-high-energy cosmic rays 6.2 Search for antimatter 6.3 Gamma rays 6.4 Cosmic neutrinos 6.5 Gravitational waves 6.6 Multi-messenger astronomy 7 Cosmic Rays and Climate 7.1 Cosmic rays and thunderstorms 7.2 Variations in the flux of cosmic rays 7.3 A correlation between cosmic rays and earthquakes? 8 Cosmic Rays and Life 8.1 Ionization and chemistry of the atmosphere 8.2 The Miller-Urey experiment 8.3 Biological effects of cosmic rays 8.4 Implications on evolution 9 Cosmic Rays and the Exploration of the Universe 10 Cosmic Rays and Archaeology 10.1 Dating techniques 10.2 Muon tomography 11 The Future

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Book SynopsisThis book presents the most common types of instabilities arising in classical field theories, namely tachyonic, Laplacian, ghost-like or strong coupling instabilities, also commenting on their quantum implications. The authors provide a detailed account on the Ostrogradski theorem and its implications for higher-order time-derivative field theories. After presenting the general concepts and formalism, they dive into its applications to particular field theories, using mainly modified gravity theories as examples. The book is intended for advanced undergraduate/graduate students, but can also be useful for researchers, for having a unified exposition of general results on instabilities in field theory and examples of their applications.Table of ContentsIntroduction to instabilities and some relevant examples.- Ostrogradski theorem and ghosts.- Examples of instabilities in gravity theories.- References.- Solutions.

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Book Synopsis- Introduction.- Existence of nonlinear FokkerPlanck flows.- Time dependent FokkerPlanck equations.- Convergence to equilibrium of nonlinear FokkerPlanck flows.- Markov processes associated with nonlinear FokkerPlanck equations.- Appendix.

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Book SynopsisGeometric structures, invariants and their uses in physics by A. Cardona and A.F. Reyes-Lega.- . Lectures on the Euler characteristic of affine manifolds by Camilo Arias-Abad and Sebastian Velez-Vasquez.- Elliptic Curves by Philip Candelas.-  The arithmetic of Calabi-Yau varieties, by Xenia de la Ossa.- Foliations and operator algebras by Georges Skandalis.- Pseudo-differential operators on groups and nonharmonic analysis, by Michael Ruzhansky.- Mathematical Foundations of Topological Matter, by Manuel Asorey.

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Book SynopsisAn energetic charged particle beam introduced to an rf cavity excites a wakefield therein. This wakefield can be decomposed into a series of higher order modes and multipoles, which for sufficiently small beam offsets are dominated by the dipole component. This work focuses on using these dipole modes to detect the beam position in third harmonic superconducting S-band cavities for light source applications. A rigorous examination of several means of analysing the beam position based on signals radiated to higher order modes ports is presented. Experimental results indicate a position resolution, based on this technique, of 20 microns over a complete module of 4 cavities. Methods are also indicated for improving the resolution and for applying this method to other cavity configurations. This work is distinguished by its clarity and potential for application to several other international facilities. The material is presented in a didactic style and is recommended both for students new to the field, and for scientists well-versed in the field of rf diagnostics.Table of ContentsIntroduction.- Electromagnetic Eigenmode Simulations of the Third Harmonic Cavity.- Measurements of HOM Spectra.- Analysis Methods for Beam Position Extraction from HOM.- Dependencies of HOM on Transverse Beam Offsets.- HOM-Based Beam Position Diagnostics.- Conclusions.- Bibliography.- Mathematics.- Eigenmodes of an Ideal Third Harmonic Cavity.- Technical Details of the HOM Measurements.

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Book SynopsisIn this "SpringerBrief" the author considers the underlying problems and questions that are common to numerical models of turbulence in different astrophysical systems. Turbulence has emerged as an important research topic in several areas of astrophysics. Understanding and modeling turbulence is particularly important for the dynamics of the interstellar medium, but also for the intergalactic medium, as well as in stars. The advancement of methods for numerical simulations of astrophysical turbulence, however, is still challenging because of gravity, strong compressibility, magnetic fields, and other effects.The book begins with a review of general aspects of numerical simulations of turbulence. In the main part the author presents findings from his numerical studies on astrophysical turbulence and discusses the astrophysical implications. He also explains in detail the numerical schemes utilized.Readers will find that this book offers a compact yet comprehensive introduction.Trade Review“This book provides an interesting and well-written account on an area witnessing encouraging progress.” (H. Muthsam, Monatshefte für Mathematik, 2016)“The primary focus of this book is on the numerical methods and their application in studying the fundamental statistical properties of compressible hydrodynamic turbulence. … The book can be a useful survey and reference source for researchers in the field. Also, it can be a good point of introduction for someone wanting to develop a research capability in this area.” (Stephen Wollman, zbMATH 1317.85004, 2015)Table of ContentsTurbulence theory.- Simulation techniques.- Phenomenology and statistics.- Complex processes.

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Book SynopsisThis textbook presents problems and exercises at various levels of difficulty in the following areas: Classical Methods in PDEs (diffusion, waves, transport, potential equations); Basic Functional Analysis and Distribution Theory; Variational Formulation of Elliptic Problems; and Weak Formulation for Parabolic Problems and for the Wave Equation. Thanks to the broad variety of exercises with complete solutions, it can be used in all basic and advanced PDE courses.Trade Review“The material, at an advanced undergraduate or first year graduate level, presents a very interesting mix of physical applications and abstract theory supporting rigorous existence and regularity results of PDEs. … Each chapter begins with a short formal summary of the basic concepts, which is followed by several dozen of problems covering all aspects of the materials in the corresponding chapters of the book.” (Peter Bernard Weichman, Mathematical Reviews, July, 2016)“Provides an excellent overview of the field and is highly recommended for modern PDEs courses designed for advanced students. … Each chapter first reviews the main theoretical concepts and tools and then provides solved problems and exercises. A broad range of problems and exercises at various levels of difficulty, as well as the extensive reference list, makes this book a valuable resource for advanced undergraduate or beginning graduate researchers … . Summing Up: Highly recommended. Upper-division undergraduates, graduate students, researchers/faculty.” (C. Park, Choice, Vol. 53 (9), May, 2016)“The authors’ aim is to give a systematic treatment of some of the classical methods in partial differential equations. … the book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study.” (Vicenţiu D. Rădulescu, zbMATH 1330.35003, 2016)Table of Contents1 Diffusion.- 2 The Laplace equation.- 3 First order equations.- 4 Waves.- 5 Functional analysis.- 6 Variational formulations.- 7 Appendix A Sturm-Liouville, Legendre and Bessel equations.- 8 Appendix B Identities.

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Book SynopsisThis text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.Trade Review“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)Table of ContentsPreface.- 1 Introduction and Overview.- 2 The Group U(1) and its Representations.- 3 Two-state Systems and SU(2).- 4 Linear Algebra Review, Unitary and Orthogonal Groups.- 5 Lie Algebras and Lie Algebra Representations.- 6 The Rotation and Spin Groups in 3 and 4 Dimensions.- 7 Rotations and the Spin 1/2 Particle in a Magnetic Field.- 8 Representations of SU(2) and SO(3).- 9 Tensor Products, Entanglement, and Addition of Spin.- 10 Momentum and the Free Particle.- 11 Fourier Analysis and the Free Particle.- 12 Position and the Free Particle.- 13 The Heisenberg group and the Schrödinger Representation.- 14 The Poisson Bracket and Symplectic Geometry.- 15 Hamiltonian Vector Fields and the Moment Map.- 16 Quadratic Polynomials and the Symplectic Group.- 17 Quantization.- 18 Semi-direct Products.- 19 The Quantum Free Particle as a Representation of the Euclidean Group.- 20 Representations of Semi-direct Products.- 21 Central Potentials and the Hydrogen Atom.- 22 The Harmonic Oscillator.- 23 Coherent States and the Propagator for the Harmonic Oscillator.- 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1.- 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d.- 26 Complex Structures and Quantization.- 27 The Fermionic Oscillator.- 28 Weyl and Clifford Algebras.- 29 Clifford Algebras and Geometry.- 30 Anticommuting Variables and Pseudo-classical Mechanics.- 31 Fermionic Quantization and Spinors.- 32 A Summary: Parallels Between Bosonic and Fermionic Quantization.- 33 Supersymmetry, Some Simple Examples.- 34 The Pauli Equation and the Dirac Operator.- 35 Lagrangian Methods and the Path Integral.- 36 Multi-particle Systems: Momentum Space Description.- 37 Multi-particle Systems and Field Quantization.- 38 Symmetries and Non-relativistic Quantum Fields.- 39 Quantization of Infinite dimensional Phase Spaces.- 40 Minkowski Space and the Lorentz Group.- 41 Representations of the Lorentz Group.- 42 The Poincaré Group and its Representations.- 43 The Klein-Gordon Equation and Scalar Quantum Fields.- 44 Symmetries and Relativistic Scalar Quantum Fields.- 45 U(1) Gauge Symmetry and Electromagnetic Field.- 46 Quantization of the Electromagnetic Field: the Photon.- 47 The Dirac Equation and Spin-1/2 Fields.- 48 An Introduction to the Standard Model.- 49 Further Topics.- A Conventions.- B Exercises.- Index.

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Book SynopsisThis new edition is a concise introduction to the basic methods of computational physics. Readers will discover the benefits of numerical methods for solving complex mathematical problems and for the direct simulation of physical processes. The book is divided into two main parts: Deterministic methods and stochastic methods in computational physics. Based on concrete problems, the first part discusses numerical differentiation and integration, as well as the treatment of ordinary differential equations. This is extended by a brief introduction to the numerics of partial differential equations. The second part deals with the generation of random numbers, summarizes the basics of stochastics, and subsequently introduces Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. The final two chapters discuss data analysis and stochastic optimization. All this is again motivated and augmented by applications from physics. In addition, the book offers a number of appendices to provide the reader with information on topics not discussed in the main text. Numerous problems with worked-out solutions, chapter introductions and summaries, together with a clear and application-oriented style support the reader. Ready to use C++ codes are provided online.Table of ContentsSome Basic Remarks.- Part I Deterministic Methods.- Numerical Differentiation.- Numerical Integration.- The KEPLER Problem.- Ordinary Differential Equations – Initial Value Problems.- The Double Pendulum.- Molecular Dynamics.- Numerics of Ordinary Differential Equations - Boundary Value Problems.- The One-Dimensional Stationary Heat Equation.- The One-Dimensional Stationary SCHRÖDINGER Equation.- Partial Differential Equations.- Part II Stochastic Methods.- Pseudo Random Number Generators.- Random Sampling Methods.- A Brief Introduction to Monte-Carlo Methods.- The ISING Model.- Some Basics of Stochastic Processes.- The Random Walk and Diffusion Theory.- MARKOV-Chain Monte Carlo and the POTTS Model.- Data Analysis.- Stochastic Optimization.- Appendix: The Two-Body Problem.- Solving Non-Linear Equations. The NEWTON Method.- Numerical Solution of Systems of Equations.- Fast Fourier Transform.- Basics of Probability Theory.- Phase Transitions.- Fractional Integrals and Derivatives in 1D.- Least Squares Fit.- Deterministic Optimization.

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Book SynopsisThis book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory.This book is Open Access under a CC BY licence. Trade Review“Quantum theory has frequent applications in the subjects of quantum information theory and quantum optics. The purpose of this book is to present the foundations of quantum theory in connection with classical physics, from the point of view of classical-quantum duality. … This good book is recommended for mathematicians, physicists, philosophers of physics, researchers and advanced students in this field.” (Michael M. Dediu, Mathematical Reviews, Decemeber, 2017)Table of ContentsIntroduction.- Part I Co(X) and B(H): Classical physics on a finite phase space.- Quantum mechanics on a finite-dimensional Hilbert space.- Classical physics on a general phase space.- Quantum physics on a general Hilbert space.- Symmetry in quantum mechanics.- Part II Between Co(X) and B(H): Classical models of quantum mechanics.- Limits: Small hbar.- Limits: large N.- Symmetry in algebraic quantum theory.- Spontaneous Symmetry Breaking.- The Measurement Problem.- Topos theory and quantum logic.- Appendix A: Finite-dimensional Hilbert spaces.- Appendix B: Basic functional analysis.- Appendix C: Operator algebras.- Appendix D: Lattices and logic.- Appendix E: Category theory and topos theory.- References.

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Book SynopsisThis text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.Trade Review“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)Table of ContentsPreface.- 1 Introduction and Overview.- 2 The Group U(1) and its Representations.- 3 Two-state Systems and SU(2).- 4 Linear Algebra Review, Unitary and Orthogonal Groups.- 5 Lie Algebras and Lie Algebra Representations.- 6 The Rotation and Spin Groups in 3 and 4 Dimensions.- 7 Rotations and the Spin 1/2 Particle in a Magnetic Field.- 8 Representations of SU(2) and SO(3).- 9 Tensor Products, Entanglement, and Addition of Spin.- 10 Momentum and the Free Particle.- 11 Fourier Analysis and the Free Particle.- 12 Position and the Free Particle.- 13 The Heisenberg group and the Schrödinger Representation.- 14 The Poisson Bracket and Symplectic Geometry.- 15 Hamiltonian Vector Fields and the Moment Map.- 16 Quadratic Polynomials and the Symplectic Group.- 17 Quantization.- 18 Semi-direct Products.- 19 The Quantum Free Particle as a Representation of the Euclidean Group.- 20 Representations of Semi-direct Products.- 21 Central Potentials and the Hydrogen Atom.- 22 The Harmonic Oscillator.- 23 Coherent States and the Propagator for the Harmonic Oscillator.- 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1.- 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d.- 26 Complex Structures and Quantization.- 27 The Fermionic Oscillator.- 28 Weyl and Clifford Algebras.- 29 Clifford Algebras and Geometry.- 30 Anticommuting Variables and Pseudo-classical Mechanics.- 31 Fermionic Quantization and Spinors.- 32 A Summary: Parallels Between Bosonic and Fermionic Quantization.- 33 Supersymmetry, Some Simple Examples.- 34 The Pauli Equation and the Dirac Operator.- 35 Lagrangian Methods and the Path Integral.- 36 Multi-particle Systems: Momentum Space Description.- 37 Multi-particle Systems and Field Quantization.- 38 Symmetries and Non-relativistic Quantum Fields.- 39 Quantization of Infinite dimensional Phase Spaces.- 40 Minkowski Space and the Lorentz Group.- 41 Representations of the Lorentz Group.- 42 The Poincaré Group and its Representations.- 43 The Klein-Gordon Equation and Scalar Quantum Fields.- 44 Symmetries and Relativistic Scalar Quantum Fields.- 45 U(1) Gauge Symmetry and Electromagnetic Field.- 46 Quantization of the Electromagnetic Field: the Photon.- 47 The Dirac Equation and Spin-1/2 Fields.- 48 An Introduction to the Standard Model.- 49 Further Topics.- A Conventions.- B Exercises.- Index.

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Book SynopsisThis book presents a self-contained derivation of van der Waals and Casimir type dispersion forces, covering the interactions between two atoms but also between microscopic, mesoscopic, and macroscopic objects of various shapes and materials. It also presents detailed and general prescriptions for finding the normal modes and the interactions in layered systems of planar, spherical and cylindrical types, with two-dimensional sheets, such as graphene incorporated in the formalism. A detailed derivation of the van der Waals force and Casimir-Polder force between two polarizable atoms serves as the starting point for the discussion of forces: Dispersion forces, of van der Waals and Casimir type, act on bodies of all size, from atoms up to macroscopic objects. The smaller the object the more these forces dominate and as a result they play a key role in modern nanotechnology through effects such as stiction. They show up in almost all fields of science, including physics, chemistry, biology, medicine, and even cosmology. Written by a condensed matter physicist in the language of condensed matter physics, the book shows readers how to obtain the electromagnetic normal modes, which for metallic systems, is especially useful in the field of plasmonics.Table of ContentsIntroduction.- Part I - Background Material.- Electromagnetic.- Complex Analysis.- Statistical Physics.- Electromagnetic Normal Modes.- Different Approaches.- General Method to find the Normal Modes in Layered Structures.- Part II - Non-retarded Formalism: van der Waals.- Van der Waals Force.- Van der Waals Interaction in Planar Structures.- Van der Waals Interaction in Spherical Structures.- Van der Waals Interaction in Cylindrical Structures.- Part III - Fully Retarded Formalism: Casimir.- Casimir Interaction.- Dispersion Interaction in Planar Structures.- Dispersion Interaction in Spherical Structures.- Dispersion Interaction in Cylindrical Structures.- Summary and Outlook.

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Book SynopsisDieses Buch ist eine besonders geeignete Einführung zum Thema Chaos und Simulation dynamischer Systeme für Studenten der Ingenieur- und der Naturwissenschaften. Die Untersuchung von einfachen Modellen aus der Populationsdynamik dient als Vehikel, einen schnellen und zielgerichteten Einstieg zu erzielen. Das nötige mathematische und simulationstechnische Werkzeug wird nach Bedarf eingeführt und gut verständlich erklärt. Unterstützt wird die Darstellung durch Programme auf beiliegender Diskette, die die Zusammenhänge verdeutlichen und zum Experimentieren anregen.Table of ContentsPopulationsdynamik - Räuber-Beute-Systeme - Simulation nichtlinearer Differentialgleichungen - Prozeßorientierte Simulation - Stabilität - Bifurkation - seltsame Attraktoren - Chaos - dissipative Systeme - fraktale Ddimension - Lorenzmodelle - Dynamik der Krankheitsepidemien

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Book SynopsisMathematische Grundlagen - Kinematik - Dynamik - Statik starrer Systeme - Statik deformierbarer Systeme - Kinetik starrer Systeme - Kinetik deformierbarer Systeme - Prinzipien der Mechanik.Trade Review,,(...) Vor allem auch aufgrund der systematisch gegliederten Darstellung sowie der klar formulierten Aussagen kann dieses Buch allen Studenten, Naturwissenschaftlern und Ingenieuren sehr empfohlen werden, die sich in die Grundlagenwissenschaft 'Mechanik' einarbeiten wollen und/oder die diese als ein wertvolles Instrument zum Lösen technischer Probleme benötigen."VDI-Z 18/1986Table of ContentsMathematische Grundlagen - Kinematik - Dynamik - Statik starrer Systeme - Statik deformierbarer Systeme - Kinetik starrer Systeme - Kinetik deformierbarer Systeme - Prinzipien der Mechanik.

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Book SynopsisAus moderner Sicht werden in diesem Teubner-Taschenbuch die Grundlagen und wichtige Anwendungen der statistischen Physik dargestellt. Auf eine gründliche Darstellung der Begriffsbildungen der statistischen Physik, auf die korrekte Herleitung grundlegender Gleichungen und auf die Durchführung wichtiger Beweise wird besonderer Wert gelegt. Das Buch eignet sich als Begleittext für Kurs- und Spezialvorlesungen, als Repetitorium zur Prüfungsvorbereitung und als Nachschlagewerk zur raschen Information für breite Leserkreise aus Mathematik, Naturwissenschaften und technischen Disziplinen, insbesondere für Studenten dieser Fachrichtungen.Table of ContentsKombinatorik - Wahrscheinlichkeitstheorie - Quantenmechanik und Wahrscheinlichkeit - Thermodynamik - Statistische Physik der Gleichgewichtssysteme - Statistische Physik der Systeme im Nichtgleichgewicht - Statistische Physik und Informationstheorie - Phasenraummethoden der Quantenstatistik - Fraktaltheorie und Perkolationstheorie - Theorie dynamischer Systeme, Chaostheorie, Ergodentheorie - Statistische Thermodynamik chemischer Systeme - Statistische Theorie biologischer Systeme - Synergetik, weitere Anwendungen der statistischen Physik

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Book SynopsisZusammen mit einer kurzen Einführung in das System der Maxwellschen Gleichungen und einer Definition der Feldgrößen lehrt das Buch mit charakteristischen Beispielen die Lösungsmethodik der Feldtheorie. Schwerpunkte sind dabei statistische und stationäre elektrische und magnetische Felder, quasistationäre elektromagnetische Felder und elektromagnetische Wellen. Für das Verständnis besonders hilfreich ist die Darstellung von Feldlinienbildern. Dieses Lehrbuch bietet eine Sammlung ausgewählter anspruchsvoller Übungsaufgaben mit Lösungen, die es ermöglichen, die elektromagnetische Feldtheorie zu verstehen und sachgerecht anzuwenden.Trade Review"Das Buch enthält in untadeliger Darstellung etliche Aufgaben mit Ausarbeitung zu den klassischen Teilgebieten der Elektrodynamik, wobei die ausgezeichneten Feldbilder besonders hervorgehoben werden müssen. Den zitierten Wunsch des Autors hat sich dieser mit seinem Buch ohne Frage erfüllt." Impulse, 01/2003Table of ContentsDie Maxwellschen Gleichungen - Elektrostatische Felder - Das stationäre Strömungsfeld - Das magnetische Feld stationärer Ströme - Das quasistationäre elektromagnetische Feld: der Skineffekt - Elektromagnetische Wellen

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Book SynopsisDas vorliegende Buch stellt eine Einführung in die Theorie der Distributionen (verallge­ meinerte Funktionen) und ihrer Anwendungen in der Physik dar. Der zum Verständnis der Theorie notwendige topologische Apparat wurde auf ein Minimum reduziert. Lediglich das erste Kapitel gibt eine Einführung in die Theorie der abzählbar normierten Räume. Es wird angenommen, daß der Leser vertraut mit den elementaren Begriffen der Funktionalanalysis (Hilbert- und Banachraum) ist. Das Buch enthält die bereits klassisch gewordenen Kapitel der Theorie der Distributionen, wie: Lokale Eigenschaften von Distributionen, Distributionen mit kompaktem Träger, temperierte Distributionen, Regularisierung divergenter Integrale, Fourier- und Fourier­ Laplace-Transformation, den Satz von Paley-Wiener-Schwartz, Distributionen als Rand­ werte analytischer Funktionen usw. In Kapitel 11 werden Distributionen untersucht, die auf Flächen konzentriert sind; insbesondere auf dem Lichtkegel konzentrierte Distri­ butionen. In den Kapiteln 8, 9, 10 werden verschiedene Anwendungen der Theorie der Distributionen in der relativistischen Physik (Feldtheorie) entwickelt. Kapitel 12 schließlich enthält Probleme der Theorie der Distributionen im Hilbertraum und ihre Anwendungen in der Quantenphysik (Vertauschungsrelationen, Fock-Raum, Quanten­ feldtheorie usw.). Das Buch wendet sich sowohl an Mathematiker, die auch die Anwendungen der Theorie der Distributionen in der Physik kennenlernen wollen; als auch an Physiker, die sich für die Theorie der Distributionen als Teilgebiet der mathematischen und theoretischen Physik interessieren. Das vorliegende Buch entstand aus Vorlesungen, die ich im Jahre 1970 als Humboldt­ Stipendiat an der Universität München gehalten habe. Mein besonderer Dank gilt daher an dieser Stelle Herrn Prof. Dr. W. Güttinger für die Unterstützung in meinen ersten Arbeitsjahren in Deutschland.Table of Contents1. Normierte und abzählbar normierte Räume.- 2. Die Testfunktionenräume.- 3. Die Distributionenräume.- 4. Lokale Eigenschaften von Distributionen.- 5. Einfache Beispiele von Distributionen.- 6. Das Rechnen mit Distributionen.- 7. Distributionen mit kompaktem Träger und die allgemeine Form der temperierten Distributionen.- 8. Funktionen mit algebraischen nichtintegrierbaren Singularitäten.- 9. Das Tensorprodukt und die Faltung von Distributionen.- 10. Die Fouriertransformation.- 11. Mit dem Lichtkegel verknüpfte Distributionen.- 12. Hilbertraum und Distributionen. Anwendungen in der Physik.- Literatur.

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Book SynopsisTable of Contents1. Mathematische Grundlagen.- 1.1. Der Begriff des Feldes und des Gradienten.- 1.1.1. Definition der Feldgröße.- 1.1.2. Änderung (Differentiation) der Feldgrößen.- 1.2. Integration der Feldgrößen.- 1.2.1. Kurvenintegrale.- 1.2.2. Flächenintegrale.- 1.3. Tensoren.- 1.3.1. Der Begriff des Tensorfeldes.- 1.3.2. Rechenregeln für Tensoren in kartesischen Koordinatensystemen.- 1.3.3. Der 5-Tensor und e-Tensor.- 1.4. Koordinatentransformationen.- 1.5. Einfachste Differentialoperatoren.- 1.5.1. Die Divergenz und der Satz von Gauß.- 1.5.2. Die Rotation und der Satz von Stokes.- 1.5.3. Sprungflächenoperatoren.- 1.5.4. Divergenz und Rotor in krummlinigen Koordinaten.- 1.6. Übungsbeispiele zu Kap. 1.- 2. Partielle Differentialgleichungen der Physik.- 2.1. Die Poissonsche Differentialgleichung.- 2.1.1. Beschreibung eines Feldes durch Quellen und Wirbel.- 2.1.2. Eindeutigkeit der Lösung. Randbedingungen.- 2.2. Die partielle Differentialgleichung von Schwingungsvorgängen.- 2.2.1. Die schwingende Saite.- 2.2.2. Die schwingende Membran und räumliche Schwingungen.- 2.3. Die Differentialgleichungen der Diffusion und Wärmeleitung.- 2.4. Einfachste Differentialgleichungen der Quantenmechanik.- 2.5. Übungsbeispiele zu Kap. 2.- 3. Lösungsansätze für partielle Differentialgleichungen.- 3.1. Trennung der Variablen.- 3.2. Die Laplacegleichung.- 3.2.1. Die Laplacegleichung für ein Rechteck.- 3.2.2. Die Laplacegleichung in Polarkoordinaten.- 3.3. Die schwingende Saite.- 3.3.1. Die beidseitig eingespannte schwingende Saite.- 3.3.2. Die d’Alembertsche Lösung der schwingenden Saite.- 3.4. Übungsbeispiele zu Kap. 3.- 4. Rand und Eigenwertaufgaben.- 4.1. Problemstellung.- 4.2. Sturm-Liouville-Differentialoperatoren.- 4.2.1. Selbstadjungierte Differentialoperatoren.- 4.2.2. Sturm-Liouville-Randwertaufgaben.- 4.2.3. Sturm-Liouville-Eigenwertaufgaben.- 4.2.4. Die Sturm-Liouville -Transformation.- 4.3. Der Entwicklungssatz.- 4.3.1. Eigenwerte und Eigenfunktionen.- 4.3.2. Der Entwicklungssatz für beschränkte Intervalle.- 4.4. Die Lösung der Anfangsrandwertaufgabe.- 4.5. Die inhomogene Randwertaufgabe.- 4.6. Nadelartige Funktionen.- 4.7. Ergänzungen und Bemerkungen.- 4.8. Übungsbeispiele zu Kap. 4.- 5. Singuläre Differentialgleichungen.- 5.1. Der Begriff der singulären Differentialgleichung. Differentialgleichungen der Fuchsschen Klasse.- 5.2. Die hypergeometrische Differentialgleichung.- 5.3. Die konfluente hypergeometrische Differentialgleichung.- 5.4. Übungsbeispiele zu Kap. 5.- 6. Spezielle Funktionen.- 6.1. Kugelfunktionen.- 6.1.1. Die Laplacegleichung in Kugelkoordinaten.- 6.1.2. Die Legendreschen Polynome und ihre erzeugende Funktion.- 6.1.3. Die Formel vom Rodrigues.- 6.1.4. Die Integraldarstellung von Laplace.- 6.1.5. Die zugeordneten Legendreschen Funktionen.- 6.1.6. Kugelflächenfunktionen als Eigenfunktionen.- 6.1.7. Das Additionstheorem der Kugelflächenfunktionen.- 6.1.8. Der Entwicklungssatz nach Kugelflächenfunktionen.- 6.1.9. Die Randwertaufgaben der Potentialtheorie.- 6.2. Zylinderfunktionen.- 6.2.1. Die Laplacegleichung in Zylinderkoordinaten.- 6.2.2. Besselfunktionen.- 6.2.3. Besselfunktionen als Eigenfunktionen.- 6.2.4. Integraldarstellung und erzeugende Funktion der Besselfunktion Jn (?).- 6.2.5. Das Additionstheorem der Besselfunktionen mit ganzzahligem Zeiger.- 6.2.6. Die Wellengleichung. Sphärische Besselfunktionen.- 6.2.7. Entwicklung einer ebenen Welle nach Kugelwellen.- 6.2.8. Asymptotische Darstellungen für sphärische Besselfunktionen.- 6.3. Hermitesche und Laguerresche Polynome.- 6.3.1. Der harmonische Oszillator (Hermitesche Polynome).- 6.3.2. Die erzeugende Funktion der Hermiteschen Polynome.- 6.3.3. Die Schrödingergleichung für das Wasserstoffatom (Laguerresche Polynome).- 6.4. Übungsbeispiele zu Kap. 6.- 7. Verallgemeinerte Funktionen.- 7.1. Problemstellung.- 7.2. Testfunktionen.- 7.3. Verallgemeinerte Funktionen.- 7.4. Die Diracsche Deltafunktion.- 7.5. Die Derivierte einer verallgemeinerten Funktion.- 7.6. Produkte von verallgemeinerten Funktionen. Das Funktional ?(g(x)).- 7.7. Die uneigentliche Funktion ?(1/r).- 7.8. Ergänzungen und Bemerkungen.- 7.9. Übungsbeispiele zu Kap. 7.- 8. Die Methode der Greenschen Funktionen für partielle Differentialgleichungen.- 8.1. Die klassische Lösung der Poissongleichung.- 8.2. Greensche Funktionen und die Deltafunktion.- 8.3. Die Greensche Funktion der Poissongleichung.- 8.3.1. Der eindimensionale Fall.- 8.3.2. Der dreidimensionale Fall mit natürlichen Randbedingungen.- 8.4. Die Greensche Funktion der Wärmeleitung (Diffusion).- 8.4.1. Die Wärmeleitung im unendlich langen Stab.- 8.4.2. Anfangs- und Randbedingungen der homogenen Wärmeleitungsgleichung.- 8.4.3. Die Wärmeleitung im Raum.- 8.5. Die Greenschen Funktionen der Wellengleichung und ihrer Verallgemeinerungen.- 8.5.1. Allgemeine Randbedingungen.- 8.5.2. Greensche Funktionen im unendlichen Raum.- 8.6. Übungsbeispiele zu Kap. 8.- A. Funktionentheorie.- B. Die Gammafunktion.- Literatur.- Sachwortverzeichnis.

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

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Book SynopsisIt is unanimously accepted that the quantum and the classical descriptions of the physical reality are very different, although any quantum process is "mysteriously" transformed through measurement into an observable classical event. Beyond the conceptual differences, quantum and classical physics have a lot in common. And, more important, there are classical and quantum phenomena that are similar although they occur in completely different contexts. For example, the Schrödinger equation has the same mathematical form as the Helmholtz equation, there is an uncertainty relation in optics very similar to that in quantum mechanics, and so on; the list of examples is very long. Quantum-classical analogies have been used in recent years to study many quantum laws or phenomena at the macroscopic scale, to design and simulate mesoscopic devices at the macroscopic scale, to implement quantum computer algorithms with classical means, etc. On the other hand, the new forms of light – localized light, frozen light – seem to have more in common with solid state physics than with classical optics. So these analogies are a valuable tool in the quest to understand quantum phenomena and in the search for new (quantum or classical) applications, especially in the area of quantum devices and computing.Trade ReviewFrom the reviews: "The main role of quantum classical analogies presented in ten distinct chapters is to shed some light on the genuine significance of the quantum and classical worlds. … The book addresses a large category of readers, especially graduates and PhD students … . The book is also useful for researchers working in advanced topics … . It can be used as an additional source for a course on quantum mechanics … . The hard cover book is nicely edited … ." (Roland Carchon, Physicalia, Vol. 57 (3), 2005) "The authors … devote their new book to the striking analogies between classical and quantum physics. … the authors wish to show that the classical and quantum worlds share many common concepts despite striking differences. … The wealth of analogies … discovered and presented in ten distinct chapters sheds some light on the genuine significance of both the quantum world and its classical counterpart. The book addresses students and researchers alike specialising in the study of quantum devices, atom optics or quantum optics." (Gert Roepstorff, Zentralblatt MATH, Vol. 1093 (19), 2006) "Analogies are a powerful cognitive tool that allow us to make inferences and learn new aspects from the comparison of two things by highlighting their similarities. … It is important to mention that the book is intended to be a catalogue of phenomena shared between classical and quantum physics … . the references given are an invaluable asset. … This book is therefore a very good choice for those interested in bridging ideas from classical physics into the quantum world or vice versa." (Dr. J. Rogel-Salazar, Contemporary Physics, Vol. 46 (6), 2005) "This book develops and explores in a systematic manner a large number of analogs between quantum and classical theories. … It follows closely the recent experimental developments, and for each chapter there is a large number of current references. … It will be very valuable for a large category of readers ranging from graduate and Ph. D. students to researchers working in these areas, and on to teachers looking for nontrivial modern applications and developments in both quantum and classical physics." (Vitor R. Vieira, Mathematical Reviews, Issue 2007 c)Table of Contents1 Introduction.- 2 Analogies Between Ballistic Electrons and Electromagnetic Waves.- 3 Electron/Electromagnetic Multiple Scattering and Localization.- 4 Acoustic Analogies for Quantum Mechanics.- 5 Optical Analogs for Multilevel Quantum Systems.- 6 Particle Optics.- 7 Quantum/Classical Nonlinear Phenomena.- 8 Quantum/Classical Phase Space Analogies.- 9 Analogies Between Quantum and Classical Computing.- 10 Other Quantum/Classical Analogies.- References.

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

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Book SynopsisDas Ziel des nun auch in deutscher Übersetzung erhältlichen Buches ist es, angewandte Mathematik und Ingenieurmathematik so darzustellen, wie sie heutzutage Anwendung findet. Das Buch basiert auf dem Kurs „Wissenschaftliches Rechnen" des Massachusetts Institute of Technology und versucht, Konzepte und Algorithmen zusammenzuführen. Beginnend mit der angewandten linearen Algebra entwickeln die Autoren die Methoden der finiten Differenzen und finiten Elemente – stets in Verbindung mit Anwendungen in zahlreichen Wissensgebieten.Trade Review“... Der in jeder Hinsicht hervorragende Text von Strang wird für lange Zeit eine Standardreferenz in der Literatur zur Ausbildung im wissenschaftlichen Rechnen sein. Es gibt viele gute Gründe, ihn zu besitzen ...” (H.R. Schneebeli, in: Elemente der Mathematik, jg. 67, S. 200 f. 2012)“... Das Buch enthält zahlreiche Aufgaben in unterschiedlichen Schwierigkeitsgraden und drei Anhänge und ist, wenn man es nicht als Begleitbuch zu einer Vorlesung gebraucht, hervorragend zum Stöbern geeignet. ... ist das Buch äußerst empfehlenswert nicht nur für Ingenieure. Teile des Buches sind sicher auch schon für begabte und interessierte Schülerinnen und Schüler geeignet und von Pseudoanwendungen in Schulbüchern genervte Lehrkräfte an unseren Gymnasien können hier realistische Anwendungen finden, die man mit Mathematik bearbeiten kann.“ (in: Mathematische Semesterberichte, 2012, Issue 1)“... in diesem umfangreichen, ursprünglich in englisch erschienenem Werk mit mathematischer Exaktheit und anwendungsorientierter Zielrichtung ... Angesprochen sind vorwiegend Studierende der Mathematik und der Ingenieurwissenschaften. ... Der kurze Anhang ist allenfalls geeignet, Erinnerungslücken zu schließen.“ (Wolfgang Grölz, in: ekz-Informationsdienst, 2010, Vol. 2010/34)Table of ContentsAngewandte lineare Algebra.- Ein Grundmuster der angewandten Mathematik.- Randwertprobleme.- Fourier-Reihen und Fourier-Integrale.- Analytische Funktionen.- Anfangswertprobleme.- Große Systeme.- Optimierung und Minimumprinzip.

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Book SynopsisDer Grundkurs Theoretische Physik deckt in 7 Bänden die im Diplom- und Bachelor/Master-Studium maßgeblichen Gebiete ab und vermittelt das im jeweiligen Semester benötigte theoretisch-physikalische Rüstzeug. Der erste Teil von Band 5 beginnt mit einer Begründung der Quantenmechanik und der Zusammenstellung ihrer formalen Grundlagen, um dann Konzepte und Begriffsbildungen an Modellsystemen zu illustrieren. Der Band enthält Übungsaufgaben und Kontrollfragen zur Vertiefung des Stoffs. Die überarbeitete und ergänzte Neuauflage ist zweifarbig gestaltet.Table of ContentsInduktive Begründung der Wellenmechanik.- Schrödinger-Gleichung.- Grundlagen der Quantenmechanik (Dirac-Formalismus).- Einfache Modellsysteme.- Lösungen der Übungsaufgaben.

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012. Volume II includes the papers on PDE, SDE, diffusions, and random media.​​Table of ContentsVol. II: Diffusion processes with continuous coefficients - I (with D. W. Stroock).- Diffusion processes with continuous coefficients - II (with D. W. Stroock).- Diffusion processes with boundary conditions (with D. W. Stroock).- On degenerate elliptic-parabolic operators of second order and their associated diffusions (with D. W. Stroock).- On the support of diffusion processes with applications to the strong maximum principle (with D. W. Stroock).- Diffusion processes (with D. W. Stroock).- A probabilistic approach to Hp(Rd) (with D. W. Stroock).- Kac functional and Schrodinger equation (with K. L. Chung).- Brownian motion in a wedge with oblique reection (with R. J. Williams).- A multidimensional process involving local time (with A.S. Sznitman).- Etat fondamental et principe du maximum pour les operateurs elliptiques du second ordre dans des domaines generaux. [The ground state and maximum principle for second-order elliptic operators in general domains] (with H. Berestycki and L. Nirenberg).- The principal eigenvalue and maximum principle for second-order elliptic operators in general domains (with H. Berestycki and L. Nirenberg).- Diffusion semigroups and di_usion processes corresponding to degenerate divergence form operators (with J. Quastel).- Random Media.- Diffusion in regions with many small holes (with G. Papanicolaou).- Boundary value problems with rapidly oscillating random coefficients (with G. Papanicolaou).- Diffusions with random coefficients (with G. Papanicolaou).- Ohrnstein-Uhlenbeck process in a random potential (with G. Papanicolaou).- Large deviations for random walks in a random environment.- Random walks in a random environment.- Stochastic homogenization of Hamilton-Jacobi-Bellman equations (with E. Kosygina and F. Rezakhanlou).- Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium (with E. Kosygina).- Behavior of the solution of a random semilinear heat equation (with N. Zygouras).​

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012.Volume III includes the papers on large deviations. ​​Table of ContentsLarge Deviations.- Asymptotic probabilities and differential equations.- On the behavior of the fundamental solution of the heat equation with variable coefficients .- Diffusion processes in a small time interval .- On a variational formula for the principal eigenvalue for operators with maximum principle.- Asymptotic evaluation of certain Markov process expectations for large time I.- Asymptotic evaluation of certain Markov process expectations for large time II.- Asymptotic evaluation of certain Wiener integrals for large time.- Asymptotics for the Wiener sausage.- Erratum: Asymptotics for the Wiener sausage.- Asymptotic evaluation of certain Markov process expectations for large time III.- On the principal eigenvalue of second-order elliptic differential operators.- On laws of the iterated logarithm for local times.- Some problems of large deviations.- On the number of distinct sites visited by a random walk.- A law of the iterated logarithm for total occupation times of transient Brownian motion.- Some problems of large deviations .- The polaron problem and large deviations.- Asymptotic evaluation of certain Markov process expectations for large time IV.- Asymptotics for the polaron.- Large deviations for stationary Gaussian processes.- Large deviations and applications.- Large deviations for non-interacting infinite-particle systems.- Some familiar examples for which the large deviation principle does not hold.- The large deviation principle for the Erdös-Rényi random graph.- Large deviations for random matrices. ​

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Book SynopsisFrom the Preface: Srinivasa Varadhan began his research career at the Indian Statistical Institute (ISI), Calcutta, where he started as a graduate student in 1959. His first paper appeared in Sankhyá, the Indian Journal of Statistics in 1962. Together with his fellow students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy, Varadhan began the study of probability on topological groups and on Hilbert spaces, and quickly gained an international reputation. At this time Varadhan realised that there are strong connections between Markov processes and differential equations, and in 1963 he came to the Courant Institute in New York, where he has stayed ever since. Here he began working with the probabilists Monroe Donsker and Marc Kac, and a graduate student named Daniel Stroock. He wrote a series of papers on the Martingale Problem and Diffusions together with Stroock, and another series of papers on Large Deviations together with Donsker. With this work Varadhan's reputation as one of the leading mathematicians of the time was firmly established. Since then he has contributed to several other areas of probability, analysis and physics, and collaborated with numerous distinguished mathematicians. Varadhan was awarded the Abel Prize in 2007. These Collected Works contain all his research papers over the half-century spanning 1962 to early 2012. Volume IV includes the papers on particle systems.Table of ContentsVolume 4: Particle Systems and Their Large Deviations.- Nonlinear diffusion limit for a system with nearest neighbor interaction.- Hydrodynamics and large deviation for simple exclusion processes.- Large deviations from a hydrodynamic scaling limit.- On the derivation of conservation laws for stochastic dynamics.- Scaling limits for interacting diffusions.- Scaling limit for interacting Ornstein-Uhlenbeck processes.- Entropy methods in hydrodynamical scaling.- Hydrodynamical limit for a Hamiltonian system with weak noise.- Nonlinear diffusion limit for a system with nearest neighbor interactions II.- Regularity of self-diffusion coefficient.- Entropy methods in hydrodynamic scaling.- Spectral gap for zero-range dynamics.- The complex story of simple exclusion.- Non-gradient models in hydrodynamic scaling.- Relative entropy and mixing properties of interacting particle systems.- Diffusive limit of lattice gas with mixing conditions.- Large deviations for the symmetric simple exclusion process in dimensions d > 3.- Large deviations for interacting particle systems.- Infinite particle systems and their scaling limits.- Lectures on hydrodynamic scaling.- Scaling limits of large interacting systems .- Asymptotic behavior of a tagged particle in simple exclusion processes.- Large deviation and hydrodynamic scaling.- Symmetric simple exclusion process: regularity of the self-diffusion coefficient.- Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process.- Large deviations for the asymmetric simple exclusion process.- Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes.- On viscosity and fluctuation-dissipation in exclusion processes.- Large deviations for the current and tagged particle in 1d nearest neighbor.- Symmetric simple exclusion.- List of Publications of S.R.S. Varadhan.- Acknowledgements. ​

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Book SynopsisThe Springer Handbook of Spacetime is dedicated to the ground-breaking paradigm shifts embodied in the two relativity theories, and describes in detail the profound reshaping of physical sciences they ushered in. It includes in a single volume chapters on foundations, on the underlying mathematics, on physical and astrophysical implications, experimental evidence and cosmological predictions, as well as chapters on efforts to unify general relativity and quantum physics. The Handbook can be used as a desk reference by researchers in a wide variety of fields, not only by specialists in relativity but also by researchers in related areas that either grew out of, or are deeply influenced by, the two relativity theories: cosmology, astronomy and astrophysics, high energy physics, quantum field theory, mathematics, and philosophy of science. It should also serve as a valuable resource for graduate students and young researchers entering these areas, and for instructors who teach courses on these subjects.The Handbook is divided into six parts. Part A: Introduction to Spacetime Structure. Part B: Foundational Issues. Part C: Spacetime Structure and Mathematics. Part D: Confronting Relativity theories with observations. Part E: General relativity and the universe. Part F: Spacetime beyond Einstein.Trade Review“This is a complete comprehensive textbook of all areas of classical and relativistic Physics including mechanics, E & M, quantum theory, perturbation, solid state, and particle physics. … It is good enough to be read cover to cover and will not disappoint the reader reviewer. I highly recommend this book for physics students, and investigators in physics theories.” (Joseph J. Grenier, Amazon.com, January, 2016)“This is a splendid and very comprehensive review of the special and general theories of relativity and their applications, in a collection of about 40 articles by experts in the field. … the book will appeal to a wide variety of readers, from advanced undergraduates to experts in the field. … I doubt that there is any physicist who would not find something new and interesting here.” (Alan Heavens, The Observatory, Vol. 135 (1245), April, 2015)Table of ContentsPreface (A. Ashtekar, V. Petkov).- Part A – Introduction to Spacetime Structure.- Chap. 1 From Aether Theory to Special Relativity.- Chap. 2 The Historical Origins of Spacetime.- Chap. 3 Relativity Today.- Chap. 4 Acceleration and Gravity: Einstein's Principle.- Chap. 5 The Geometry of Newton's and Einstein's Theories.- Part B – Foundational Issues.- Chap. 6 Time in Special Relativity.- Chap. 7 Rigid Motion and Adapted Frames.- Chap. 8 Physics as Spacetime Geometry.- Chap. 9 Electrodynamics of Radiating Charges.- Chap. 10 The Nature and Origin of Time-Asymmetric Spacetime Structures.- Chap. 11 Teleparallelism: A new Insight into Gravity.- Chap. 12 Gravity and the Spacetime: An Emergent Perspective.- Chap. 13 Spacetime and the Passage of Time.- Part C – Spacetime Structure and Mathematics.- Chap. 14 Unitary Representations of the Inhomogeneous Lorentz Group and Their Significance in Quantum Physics.- Chap. 15 Spinors.- Chap. 16 The Initial Value Problem in General Relativity.- Chap. 17 Dynamical and Hamiltonian Formulation of General Relativity.- Chap. 18 Positive Energy Theorems in General Relativity.- Chap. 19 Conserved Charges in Asymptotically (Locally) AdS Spacetimes.- Chap. 20 Spacetime Singularities.- Chap. 21 Singularities in Cosmological Spacetimes.- Part D – Confronting Relativity theories with observations.- Chap. 22 The experimental status of Special and General Relativity Chap. 23. Observational Constraints on Local Lorentz Invariance.- Chap. 24 Relativity in GNSS.- Chap. 25 Quasi Local Black Hole Horizons.- Chap. 26 Gravitational Astronomy.- Chap. 27 Probing Dynamical Spacetimes with Gravitational Waves.- Part E – General Relativity and the Universe.- Chap. 28 Einstein's Equation, Cosmology and Astrophysics.- Chap. 29 Viscous Universe Models.- Chap. 30 Friedmann-Lemaitre-Robertson-Walker Cosmology.- Chap. 31 Exact Approach to Inflationary Universe Models.- Chap. 32 Cosmology with the Cosmic Microwave Background.- Part F – Spacetime Beyond Einstein.- Chap. 33 Quantum Gravity.- Chap. 34 Quantum Gravity via Causal Dynamical Triangulations.- Chap. 35 String Theory and Primordial Cosmology.- Chap. 36 Quantum Spacetime.- Chap. 37 Gravity, Geometry and the Quantum.- Chap. 38 Spin Foams.- Chap. 39 Loop Quantum Cosmology.- Acknowledgements.- About the Authors.- Subject Index.

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Book Synopsis

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Book Synopsis

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Book SynopsisDer beliebte Grundkurs Theoretische Physik deckt in sieben Bänden alle für das Bachelor-/Master- oder Diplomstudium maßgeblichen Gebiete ab. Jeder Band vermittelt gut durchdacht das im jeweiligen Semester nötige theoretisch-physikalische Rüstzeug. Zahlreiche Übungsaufgaben mit ausführlichen Lösungen dienen der Vertiefung des Stoffes. Der zweite Teil des fünften Bandes befasst sich mit Anwendungen und mit dem Ausbau der im ersten Teil entwickelten Konzepte der Quantenmechanik.Die vorliegende neue Auflage enthält einige neue Aufgaben, wurde grundlegend überarbeitet und durch einige Zusatzkapitel zur Streutheorie ergänzt. Sie ermöglicht durch die zweifarbige Darstellung einen sehr übersichtlichen und schnellen Zugriff auf den Lehrstoff.Table of ContentsQuantentheorie des Drehimpulses.- Zentralpotential.- Näherungsmethoden.- Mehr-Teilchen-Systeme.- Streutheorie.- Lösungen der Übungsaufgaben.

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Book SynopsisThis text presents the mathematical concepts of Grassmann variables and the method of supersymmetry to a broad audience of physicists interested in applying these tools to disordered and critical systems, as well as related topics in statistical physics. Based on many courses and seminars held by the author, one of the pioneers in this field, the reader is given a systematic and tutorial introduction to the subject matter. The algebra and analysis of Grassmann variables is presented in part I. The mathematics of these variables is applied to a random matrix model, path integrals for fermions, dimer models and the Ising model in two dimensions. Supermathematics - the use of commuting and anticommuting variables on an equal footing - is the subject of part II. The properties of supervectors and supermatrices, which contain both commuting and Grassmann components, are treated in great detail, including the derivation of integral theorems. In part III, supersymmetric physical models are considered. While supersymmetry was first introduced in elementary particle physics as exact symmetry between bosons and fermions, the formal introduction of anticommuting spacetime components, can be extended to problems of statistical physics, and, since it connects states with equal energies, has also found its way into quantum mechanics. Several models are considered in the applications, after which the representation of the random matrix model by the nonlinear sigma-model, the determination of the density of states and the level correlation are derived. Eventually, the mobility edge behavior is discussed and a short account of the ten symmetry classes of disorder, two-dimensional disordered models, and superbosonization is given.Trade Review“This volume of Lecture Notes in Physics presents in three parts the topics of Grassmann algebra and its applications and the subject of supermathematics, where commuting and anticommuting variables are treated on equal footing, and its applications. … it may be of interest to those already in the field who want to expand their knowledge in both the underlying mathematics and its applications in physics.” (Moorad Alexanian, Mathematical Reviews, January, 2017)“Each chapter contains a set of illustrating problems supplied with answers in the book's end. Thus, this monograph can be used for practical teaching as well.” (Eugene Postnikov, zbMATH 1345.81003, 2016)Table of ContentsPart I Grassmann Variables and Applications.- Part II Supermathematics.- Part III Supersymmetry in Statistical Physics.- Summary and Additional Remarks.- References.- Solutions.- Index.

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