Mathematical / Computational / Theoretical physics Books

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

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Book SynopsisBased on Jost function theory this book presents an approach useful for different types of quantum mechanical problems. These include the description of scattering, bound, and resonant states, in a unified way. The reader finds here all that is known about Jost functions as well as what is needed to fill the gap between the pure mathematical theory and numerical calculations. Some of the topics covered are: quantum resonances, Regge poles, multichannel scattering, Coulomb interaction, Riemann surfaces, multichannel analog of the effective range theory, one- and two-dimensional problems, many-body problems within the hyperspherical approach, just to mention few of them. These topics are relevant in the fields of quantum few-body theory, nuclear reactions, atomic collisions, and low-dimensional semiconductor nanostructures. In light of this, the book is meant for students, who study quantum mechanics, scattering theory, or nuclear reactions at the advanced level as well as for post-graduate students and researchers in the fields of nuclear and atomic physics. Many of the arguments that are traditional for textbooks on quantum mechanics and scattering theory, are covered here in a different way, using the Jost functions. This gives the reader a new insight into the subject, revealing new features of various mathematical objects and quantum phenomena.Trade Review“This book has to be recommended to graduate students and to young researchers as well who want to enter the difficult field of modern scattering theory.” (Giorgio Cattapan, Mathematical Reviews, July, 2023)Table of ContentsChapter 1: The Basic Concepts.- Part I: Single-Channel Problems.- Chapter 2: Schr¨Odinger Equation and its Solutions.- Chapter 3: Riemann Surface and the Spectral Points.- Chapter 4: Scattering States and the S-Matrix.- Chapter 5: Complex Angular Momentum.- Chapter 6: Green’s Functions.- Chapter 7: Short-Range Potential Extending to Infinity.- Chapter 8: Single-Channel Potential with Coulombic Tail.- Part II: Multi-Channel Problems.- Chapter 9: Non-Central Potential.- Chapter 10: Systems with Non-Zero Spin.- Chapter 11: Multi-Channel Schr Odinger Equation.- Chapter 12: Multi-Channel Jost Matrix.- Chapter 13: Riemann Surfaces for Multi-Channel Systems.- Chapter 14: Multi-Channel Problems of Charged Particles.- Chapter 15: Effective-Range Expansion and its Generalizations.- Part III: Special Issues.- Chapter 16: Singular and Low-Dimensional Potentials.- Chapter 17: Miscellaneous Extensions of the Jost Function Approach.- Chapter 18: Some Exactly Solvable Potential Models.- Appendices.- References and Index.

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Book SynopsisThis book introduces and critically appraises the main proposals for how to understand quantum mechanics, namely the Copenhagen interpretation, spontaneous collapse, Bohmian mechanics, many-worlds, and others. The author makes clear what are the crucial problems, such as the measurement problem, related to the foundations of quantum mechanics and explains the key arguments like the Einstein-Podolsky-Rosen argument and Bell’s proof of nonlocality. He discusses and clarifies numerous topics that have puzzled the founding fathers of quantum mechanics and present-day students alike, such as the possibility of hidden variables, the collapse of the wave function, time-of-arrival measurements, explanations of the symmetrization postulate for identical particles, or the nature of spin. Several chapters are devoted to extending the different approaches to relativistic space-time and quantum field theory. The book is self-contained and is intended for graduate students and researchers who want to step into the fundamental aspects of quantum physics. Given its clarity, it is accessible also to advanced undergraduates and contains many exercises and examples to master the subject.Table of ContentsPreface1. Waves and Particles 1.1 Overview 1.2 The Schrodinger Equation 1.3 Unitary Operators in Hilbert Space 1.3.1 Existence and Uniqueness of Solutions of the Schrodinger Equation 1.3.2 The Time Evolution Operators 1.3.3 Unitary Matrices and Rotations 1.3.4 Inner Product 1.3.5 Abstract Hilbert Space 1.4 Classical Mechanics 1.4.1 Definition of Newtonian Mechanics 1.4.2 Properties of Newtonian Mechanics 1.4.3 Hamiltonian Systems 1.5 The Double Slit Experiment 1.5.1 Classical Predictions for Particles and Waves 1.5.2 Actual Outcome of the Experiment 1.5.3 Feynman's Discussion 1.6 Bohmian Mechanics 1.6.1 Definition of Bohmian Mechanics 1.6.2 Historical Overview 1.6.3 Equivariance 1.6.4 The Double Slit Experiment in Bohmian Mechanics 1.6.5 Delayed Choice Experiments Summary Exercises References 2. Some Observables 2.1 Fourier Transform and Momentum 2.1.1 Fourier Transform 2.1.2 Momentum 2.1.3 Momentum Operator 2.1.4 Tunnel Effect 2.2 Operators and Observables 2.2.1 Heisenberg's Uncertainty Relation 2.2.2 Self-Adjoint Operators 2.2.3 The Spectral Theorem 2.2.4 Conservation Laws in Quantum Mechanics 2.3 Spin 2.3.1 Spinors and Pauli Matrices 2.3.2 The Pauli Equation 2.3.3 The Stern-Gerlach Experiment 2.3.4 Bohmian Mechanics with Spin 2.3.5 Is an Electron a Spinning Ball? 2.3.6 Is There an Actual Spin Vector? 2.3.7 Many-Particle Systems 2.3.8 Representations of SO(3) 2.3.9 Inverted Stern-Gerlach Magnet and Contextuality Summary Exercises References 3. Collapse and Measurement 3.1 The Projection Postulate 3.1.1 Notation 3.1.2 The Projection Postulate 3.1.3 Projection and Eigenspace 3.1.4 Remarks 3.2 The Measurement Problem 3.2.1 What the Problem Is 3.2.2 How Bohmian Mechanics Solves the Measurement Problem 3.2.3 Decoherence 3.2.4 Schrodinger's Cat 3.2.5 Positivism and Realism 3.3 The GRW Theory 3.3.1 The Poisson Process 3.3.2 Definition of the GRW Process 3.3.3 Definition of the GRW Process in Formulas 3.3.4 Primitive Ontology 3.3.5 How GRW Theory Solves the Measurement Problem 3.3.6 Empirical Tests 3.3.7 The Need for a Primitive Ontology 3.4 The Copenhagen Interpretation 3.4.1 Two Realms 3.4.2 Positivism 3.4.3 Purported Impossibility of Non-Paradoxical Theories 3.4.4 Completeness of the Wave Function 3.4.5 Language of Measurement 3.4.6 Complementarity 3.4.7 Complementarity and Non-Commuting Operators 3.4.8 Reactions to the Measurement Problem 3.5 Many Worlds 3.5.1 Schrodinger's Many-Worlds Theory 3.5.2 Everett's Many-Worlds Theory 3.5.3 Bell's First Many-Worlds Theory 3.5.4 Bell's Second Many-Worlds Theory 3.5.5 Probabilities in Many-World Theories 3.6 Special Topics 3.6.1 The Mach-Zehnder Interferometer 3.6.2 Path Integrals 3.6.3 Point Interactions 3.6.4 No-Cloning Theorem 3.6.5 Boundary Conditions 3.6.6 Aharonov-Bergmann-Lebowitz Symmetry and Two-State Vector Formalism Summary Exercises References 4. Nonlocality 4.1 The Einstein-Podolsky-Rosen Argument 4.1.1 The EPR Argument 4.1.2 Further Conclusions 4.1.3 Bohm's Version of the EPR Argument Using Spin 4.1.4 Einstein's Boxes Argument 4.1.5 Too Good to Be True 4.2 Proof of Nonlocality 4.2.1 Bell's Experiment 4.2.2 Bell's 1964 Proof of Nonlocality 4.2.3 Bell's 1976 Proof of Nonlocality 4.2.4 Photons 4.3 Discussion of Nonlocality 4.3.1 Nonlocality in Bohmian Mechanics, GRW, Copenhagen, Many-Worlds 4.3.2 Popular Myths About Bell's Proof 4.3.3 Bohr's Reply to EPR Summary Exercises References 5. General Observables 5.1 POVMs: Generalized Observables 5.1.1 Definition 5.1.2 The Main Theorem About POVMs 5.1.3 Limitations to Knowledge 5.1.4 The Concept of Observable 5.2 Time of Detection 5.2.1 The Problem 5.2.2 The Quantum Zeno Effect 5.2.3 The Absorbing Boundary Rule 5.2.4 Historical Overview 5.3 Density Matrix 5.3.1 Trace 5.3.2 The Trace Formula in Quantum Mechanics 5.3.3 Limitations to Knowledge 5.3.4 Density Matrix and Dynamics 5.4 Reduced Density Matrix and Partial Trace 5.4.1 Partial Trace 5.4.2 The Trace Formula 5.4.3 Statistical Reduced Density Matrix 5.4.4 The Measurement Problem and Density Matrices 5.4.5 The No-Signaling Theorem 5.4.6 Completely Positive Superoperators 5.4.7 Canonical Typicality 5.4.8 The Possibility of a Fundamental Density Matrix 5.5 Quantum Logic 5.6 No-Hidden-Variables Theorems 5.6.1 Bell's NHVT 5.6.2 Von Neumann's NHVT 5.6.3 Gleason's NHVT 5.7 The Pusey-Barrett-Rudolph Theorem 5.8 The Decoherent Histories Interpretation Summary Exercises References 6. Particle Creation 6.1 Identical Particles 6.1.1 Symmetrization Postulate 6.1.2 Schrodinger Equation and Symmetry 6.1.3 The Space of Unordered Configurations 6.1.4 Identical Particles in Bohmian Mechanics 6.1.5 Identical Particles in GRW Theory 6.2 Particle Creation 6.2.1 Configuration Space of a Variable Number of Particles 6.2.2 Fock Space 6.2.3 Example: Emission-Absorption Model 6.2.4 Creation and Annihilation Operators 6.2.5 Ultraviolet Divergence 6.2.6 Bell's Jump Process 6.2.7 Determinism vs. Stochasticism 6.2.8 GRW Theory and Fock Space 6.2.9 Many Worlds and Fock Space 6.2.10 Interior-Boundary Conditions 6.3 A Brief Look at Quantum Field Theory 6.3.1 Historical Overview 6.3.2 Field Ontology vs. Particle Ontology 6.3.3 Scattering and the Dyson Series 6.3.4 Renormalization Summary Exercises References 7. Relativity 7.1 Brief Introduction to Relativity 7.1.1 Galilean Relativity 7.1.2 Minkowski Space 7.1.3 Arc Length 7.1.4 Classical Electrodynamics as a Paradigm of a Relativistic Theory 7.1.5 Cauchy Surfaces 7.1.6 Outlook on General Relativity 7.2 Relativistic Schrodinger Equations 7.2.1 The Klein-Gordon Equation 7.2.2 Two-Spinors and Four-Vectors 7.2.3 The Weyl Equation 7.2.4 The Dirac Equation 7.2.5 Bohmian Trajectories for the 1-Particle Weyl and Dirac Equations 7.2.6 Probability Conservation 7.2.7 Multi-Time Wave Functions 7.2.8 Hypersurface Wave Functions 7.2.9 The Maxwell Equation as the Schrodinger Equation for Photons 7.3 Bohmian Mechanics in Relativistic Space-Time 7.3.1 Law of Motion 7.3.2 Equivariance 7.3.3 Intersection Probability and Detection Probability 7.3.4 Possible Laws Governing the Time Foliation 7.3.5 Does This Count as Relativistic? 7.4 Predictions in Relativistic Space-Time 7.4.1 Is Collapse Incompatible with Relativity? 7.4.2 Joint Distribution of Outcomes of Local Experiments 7.4.3 The Aharonov-Albert Wave Function 7.4.4 Tunneling Times 7.5 GRW Theory in Relativistic Space-Time 7.5.1 1-Particle Case 7.5.2 The Case of N Non-Interacting Particles 7.5.3 Nonlocality in Relativistic GRW Theory 7.5.4 Interacting Particles 7.5.5 Primitive Ontology 7.5.6 Which Theories Count as Relativistic? 7.6 Copenhagen Interpretation in Relativistic Space-Time 7.7 Many-Worlds in Relativistic Space-Time 7.8 Special Topics 7.8.1 Multi-Time Equations of Particle Creation 7.8.2 The Tomonaga-Schwinger Equation 7.8.3 Born's Rule on Cauchy Surfaces 7.8.4 Negative Energy States and the Dirac Sea Summary Exercises References 8. Some Morals Drawn 8.1 Positivism vs. Realism 8.2 Limitations to Knowledge 8.3 What if Two Theories Are Empirically Equivalent? 8.4 Open Problems References Appendix · Topological View of the Symmetrization Postulate · Philosophical Topics · Free Will · Causation · Nelson's Stochastic Mechanics · Probability and Typicality in Bohmian Mechanics - The Law of Large Numbers in Bohmian Mechanics - The Explanation of Quantum Equilibrium - Quantum Non-Equilibrium · Vector Bundles - The Intuition Behind Vector Bundles - Electromagnetic Vector Potential - The Aharonov-Bohm Effect - Using Bundles for the Symmetrization Postulate Solutions Index

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Book SynopsisThis textbook gradually introduces the reader to several topics related to black hole physics with a didactic approach. It starts with the most basic black hole solution, the Schwarzschild metric, and discusses the basic classical properties of black hole solutions as seen by different probes. Then it reviews various theorems about black hole properties as solutions to Einstein gravity coupled to matter fields, conserved charges associated with black holes, and laws of black hole thermodynamics. Next, it elucidates semiclassical and quantum aspects of black holes, which are relevant in ongoing and future research. The book is enriched with many exercises and solutions to assist in the learning.The textbook is designed for physics graduate students who want to start their research career in the field of black holes; postdocs who recently changed their research focus towards black holes and want to get up-to-date on recent and current research topics; advanced researchers intending to teach (or learn) basic and advanced aspects of black hole physics and the associated mathematical tools. Besides general relativity, the reader needs to be familiar with standard undergraduate physics, like thermodynamics, quantum mechanics, and statistical mechanics. Moreover, familiarity with basic quantum field theory in Minkowski space is assumed. The book covers the rest of the needed background material in the main text or the appendices.Table of ContentsChapter I: INTRODUCTION1. A brief review on essentials of General Relativity, from basic concepts, mathematical frameworkand Einstein equations Einstein-Hilbert action and classical tests of GR;2. Brief review of history and timeline of developments from Schwarzschild solution to black holemergers and to information paradox and rewall;3. Gravitational collapse and Chandrasekhar mass bound;4. Different schools of thought on black holes: high energy oriented, GR oriented and quantuminformation theory oriented; open issue how to merge these schoolsChapter II: BASIC CONCEPTS and TOOLS1. Schwarzschild metric and some basic facts and analysis;2. Analysis of geodesics, notion of Killing horizon and near horizon Rindler geometry;3. Kruskal coordinates, maximal extensions and Carter-Penrose diagram;4. Einstein-Maxwell theory and Reisner-Nordström solution and its basic analysis;5. Kerr solution and its basic analysis;6. Black holes in (A)dS backgrounds.Chapter III: CLASSICAL ASPECTS1. Lensing and black hole shadows;2. Super-radiance, Penrose process and black hole mining;3. Gravitational wave emission in black hole mergers;4. Accretion disk physics;5. Extremal black holes, their near horizon and basic analysis.Chapter IV: ADVANCED CONCEPTS1. Mathematical defnition of black holes, notion of various different horizons, Killing, event,cosmological, isolated; trapped surface.2. Conjectures and theorems (Cosmic censorship; Penrose mass inequality, singularity, uniquenessand topology theorems)3. Raychaudhuri equation and area theorem (2nd law); energy conditions;4. Linear and nonlinear stability of black hole solutions;5. More detailed analysis of collapse, Choptuik exponents and critical exponents;6. Canonical boundary charges (1st law), ADM, Brown-York, Regge-Teitelboim, Iyer-Wald-Zoupas,Barnich-Brandt and Hajian-Sh-J charges.7. Variation principle; Gibbons-Hawking-York boundary term; Brown-York stress tensor;8. Quasi-normal modes and black hole perturbations;9. Four laws of black hole thermodynamics and their new derivations a la Wald-Hajian-Sh-J;Chapter V: SEMICLASSICAL ASPECTS1. Quantization on black hole backgrounds;2. Unruh effect;3. Hawking effect;4. Bekenstein entropy and the area law, the Bekenstein bound;5. Parikh-Wilczek tunneling;6. Black hole evaporation;7. Membrane paradigm.Chapter VI: EXPERT TOPICS1. Gravity in lower dimensions (including various asymptotic symmetry algebras)2. Gravity in higher dimensions (including a brief discussion on supergravity);3. Higher dimensional black hole/ring/brane solutions.4. Aspects of holography - holographic renormalization, correlation functions and asymptoticsymmetries5. Extremal black holes and attractor mechanism6. Kerr/CFT and related topics7. Soft hair and black hole microstates.Chapter VII: QUANTUM ASPECTS1. Black holes in string theory;2. Microstate counting;3. Microstate identification/constructions, fuzzballs, fluffballs;4. Information paradox and black hole complementarity and firewalls;5. Black holes and quantum gravity;6. Information paradox and the AdS/CFT;7. Holography, Quantum information (entanglement entropy, Bousso bound, QNEC etc.) andgeneralized laws of black hole thermodynamics.Chapter VIII: OUTLOOK1. Summary;2. Outlook and open issues; - Experimental/observational prospects - Black holes as a window to Quantum Gravity - gravity may be emergent | what does it emerge from?Chapter IX: SOLUTIONS TO EXERCISESWe present numerous exercises throughout the book and in this chapter we give solutions to aselected subset of them.AppendicesWe intend to have some appendices in which we present some details of crucial mathematicalframeworks and formulations not fitting into the main text, in particular - Cartan formulation - Basics of QFT in curved spacetime - Covariant phase space method

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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 SynopsisIn this undergraduate textbook, now in its 2nd edition, the author develops the quantum theory from first principles based on very simple experiments: a photon traveling through beam splitters to detectors, an electron moving through magnetic fields, and an atom emitting radiation. From the physical description of these experiments follows a natural mathematical description in terms of matrices and complex numbers.The first part of the book examines how experimental facts force us to let go of some deeply held preconceptions and develops this idea into a description of states, probabilities, observables, and time evolution. The quantum mechanical principles are illustrated using applications such as gravitational wave detection, magnetic resonance imaging, atomic clocks, scanning tunneling microscopy, and many more. The first part concludes with an overview of the complete quantum theory.The second part of the book covers more advanced topics, including the concept of entanglement, the process of decoherence or how quantum systems become classical, quantum computing and quantum communication, and quantum particles moving in space. Here, the book makes contact with more traditional approaches to quantum physics. The remaining chapters delve deeply into the idea of uncertainty relations and explore what the quantum theory says about the nature of reality.The book is an ideal accessible introduction to quantum physics, tested in the classroom, with modern examples and plenty of end-of-chapter exercises.Table of ContentsChapter 1: Three simple experiments.- The purpose of physical theories.- A laser and a detector.- A laser and a beam splitter.- A Mach-Zehnder interferometer.- The breakdown of classical concepts.- Chapter 2: Photons and Interference.- Photon paths and superpositions.- The beam splitter as a matrix.- The phase in an interferometer.- How to calculate probabilities.- Gravitational wave detection.- Chapter 3: Electrons with Spin.- The Stern-Gerlach experiment.- The spin observable.- The Bloch sphere.- The uncertainty principle.- Magnetic resonance imaging.- Chapter 4: Atoms and Energy.- The energy spectrum of atoms.- Changes over time.- The Hamiltonian.- Interactions.- Atomic clocks.- Chapter 5: Operators.- Eigenvalue problems.- Observables.- Evolution.- The commutator.- Projectors.- Chapter 6: Entanglement.- The state of two electrons.- Entanglement.- Quantum teleportation.- Quantum computers.- Chapter 7: Decoherence.- Classical and quantum uncertainty.- The density matrix.- Interactions with the environment.- Entropy and Landauer’s principle.- Chapter 8: The Motion of Particles.- A particle in a box.- The momentum of a particle.- The energy of a particle.- The scanning tunneling microscope.- Chemistry.- Chapter 9: Uncertainty Relations.- Quantum uncertainty revisited.- Position-momentum uncertainty.- The energy-time uncertainty relation.- The quantum mechanical pendulum.- Precision measurements.- Chapter 10: The Nature of Reality.- The emergent classical world.- The quantum state revisited.- Nonlocality.- Contextuality.- A compendium of interpretations.

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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 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 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 SynopsisThis monograph provides a comprehensive study of the Riemann problem for systems of conservation laws arising in continuum physics. It presents the state-of-the-art on the dynamics of compressible fluids and mixtures that undergo phase changes, while remaining accessible to applied mathematicians and engineers interested in shock waves, phase boundary propagation, and nozzle flows. A large selection of nonlinear hyperbolic systems is treated here, including the Saint-Venant, van der Waals, and Baer-Nunziato models. A central theme is the role of the kinetic relation for the selection of under-compressible interfaces in complex fluid flows. This book is recommended to graduate students and researchers who seek new mathematical perspectives on shock waves and phase dynamics. Table of Contents1 Overview of this monograph.- 2 Models arising in fluid and solid dynamics.- 3 Nonlinear hyperbolic systems of balance laws.- 4 Riemann problem for ideal fluids.- 5 Compressible fluids governed by a general equation of state.- 6 Nonclassical Riemann solver with prescribed kinetics. The hyperbolic regime.- 7 Nonclassical Riemann solver with prescribed kinetics. The hyperbolic-elliptic regime.- 8 Compressible fluids in a nozzle with discontinuous cross-section. Isentropic flows.- 9 Compressible fluids in a nozzle with discontinuous cross-section. General flows.- 10 Shallow water flows with discontinuous topography.- 11 Shallow water flows with temperature gradient.- 12 Baer-Nunziato model of two-phase flows.- References.- Index.

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Book SynopsisThis open access book bridges a gap between introductory Quantum Field Theory (QFT) courses and state-of-the-art research in scattering amplitudes. It covers the path from basic definitions of QFT to amplitudes, which are relevant for processes in the Standard Model of particle physics. The book begins with a concise yet self-contained introduction to QFT, including perturbative quantum gravity. It then presents modern methods for calculating scattering amplitudes, focusing on tree-level amplitudes, loop-level integrands and loop integration techniques. These methods help to reveal intriguing relations between gauge and gravity amplitudes and are of increasing importance for obtaining high-precision predictions for collider experiments, such as those at the Large Hadron Collider, as well as for foundational mathematical physics studies in QFT, including recent applications to gravitational wave physics.These course-tested lecture notes include numerous exercises with solutions. Requiring only minimal knowledge of QFT, they are well-suited for MSc and PhD students as a preparation for research projects in theoretical particle physics. They can be used as a one-semester graduate level course, or as a self-study guide for researchers interested in fundamental aspects of quantum field theory.Table of Contents1. Introduction & basics1.1 Poincaré group & representations 1.2. Weyl & Dirac spinors 1.3. Non-abelian gauge theories 1.4. Perturbative quantum gravity 1.5. Feynman-rules 1.6. Spinor helicity formalism for massless particles 1.7. Polarizations 1.8. Color decomposition 1.9. Color ordered amplitudes 1.10. Outlook 1: Massive spinor helicity 1.11. Outlook 2: Momentum twistors 2. Tree-level amplitudes 2.1. BCFW recursion 2.2. 3-point amplitudes 2.3. Factorizations2.4. Symmetries of scattering amplitudes 2.5. Dualities for gluons & gravitons 2.6. Massive BCFW2.7. Outlook 1: Scattering eqs. and the CHY Formalism 3. Loop-level integrands and amplitudes 3.1. Introduction 3.2. Unitarity and Cut-Construction 3.3. Generalised Unitarity3.4. Reduction methods 3.5. General method for one-loop amplitudes 3.5.1. The integral basis 3.5.2. Constructing integrand basis for box, triangle and bubble topologies 3.5.3. D-dimensional integrands and rational terms 3.5.4. Direct construction method (Forde) 3.6. Outlook: multi-loop integrand reduction 4. Loop integration techniques and special functions 4.1. Introduction 4.2. Conventions and Feynman parameter method 4.3. Ultraviolet and infrared divergences 4.4. Mellin-Barnes method4.5. Feynman integrals and transcedental weights 4.6. Differential equation method 4.7. Functional identities and symbol method 4.8. Other topics 4.9. Exercises 4.10. Outlook, suggested reading for student presentations 5. Exercises with 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 SynopsisThe primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers’ convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.Trade ReviewFrom the reviews:“The book is well written and not unnecessarily wordy. There is an up-to-date bibliography and a nice index. … a mathematician who wishes to know what the important issues concerning eq. (1) are and what has been achieved, would find this an excellent source. Equally, a mathematically-minded student, with a good grounding in analysis and who has decided to work in this area, or the teacher who wants to teach a course on this material would find this a valuable text.” (P. N. Shankar,Current Science, Vol. 85 (2), July, 2003)Table of ContentsPreliminary Results.- The Stationary Navier-Stokes Equations.- The Linearized Nonstationary Theory.- The Full Nonlinear Navier-Stokes Equations.

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

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

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