“…After establishing the physical concept of alternating gradient focusing for those new to the field, the classical matrix theory and betatron functions are derived from a Hamiltonian. …”
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“…Statistical mechanics is one of the most exciting areas of physics today, and it also has applications to subjects as diverse as economics, social behavior, algorithmic theory, and evolutionary biology. <i>Statistical Mechanics in a Nutshell</i> offers the most concise, self-contained introduction to this rapidly developing field. …”
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“…From there arises a new category of classical and quantized field parts, apparently not treated in Quantum Electrodynamics before. …”
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“…Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this  text uniquely presents an exciting playing field  for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. …”
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“…A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. …”
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“…This book, now in its second edition, provides an introductory course on theoretical particle physics with the aim of filling the gap that exists between basic courses of classical and quantum mechanics and advanced courses of (relativistic) quantum mechanics and field theory. …”
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“…The text comprises three lectures covering: the theory of stochastic Hamilton–Jacobi equations, one of the most intriguing and rich new chapters of this subject; singular SPDEs, which are at the cutting edge of innovation in the field following the breakthroughs of regularity structures and related theories, with the KPZ equation as a central example; and the study of dispersive equations with random initial conditions, which gives new insights into classical problems and at the same time provides a surprising parallel to the theory of singular SPDEs, viewed from many different perspectives. …”
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“…The Lord–Shulman theory, Green–Lindsay theory, and the classical one can be outlined in this form. …”
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“…In such cases a simple analytical theory of particle production can be developed. Application of our results to the theory of quantum creation of moduli fields demonstrates that if the moduli mass is smaller than the Hubble constant then these fields are copiously produced during inflation. …”
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“…Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. …”
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“…The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. …”
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“…Although compared with research on the fabrication processes and applications of PAO, research on their formation mechanisms started late, several mainstream theories have been formed in the academic community, including the field-assisted dissolution (FAD) theory, the field-assisted ejection (FAE) theory, the self-organization theory, the ionic and electronic current theory and the oxygen bubble mould effect. …”
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“…An extension of the notion of classical equivalence of equivalence in the Batalin–Vilkovisky (BV) and Batalin–Fradkin–Vilkovisky (BFV) frameworks for local Lagrangian field theory on manifolds possibly with boundary is discussed. …”
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“…NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS Inadequacy of Classical Description for Small Systems Basis of Quantum Mechanics Representation of States Dual Vectors: Bra and Ket Vectors Linear Operators Adjoint of a Linear Operator Eigenvalues and Eigenvectors of a Linear Operator Physical Interpretation Observables and Completeness Criterion Commutativity and Compatibility of Observables Position and Momentum Commutation Relations Commutation Relation and the Uncertainty ProductAppendix: Basic Concepts in Classical MechanicsREPRESENTATION THEORY Meaning of Representation How to Set up a Representation Representatives of a Linear Operator Change of Representation Coordinate Representation Replacement of Momentum Observable p by -ih d/dqIntegral Representation of Dirac Bracket A2|F|A1> The Momentum Representation Dirac Delta FunctionRelation between the Coordinate and Momentum RepresentationsEQUATIONS OF MOTIONSchrödinger Equation of Motion Schrödinger Equation in the Coordinate Representation Equation of Continuity Stationary States Time-Independent Schrödinger Equation in the Coordinate Representation Time-Independent Schrödinger Equation in the Momentum Representation Time-Independent Schrödinger Equation in Matrix Form The Heisenberg Picture The Interaction Picture Appendix: MatricesPROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS Motion of a Particle across a Potential Step Passage of a Particle through a Potential Barrier of Finite Extent Tunneling of a Particle through a Potential Barrier Bound States in a One-Dimensional Square Potential Well Motion of a Particle in a Periodic PotentialBOUND STATES OF SIMPLE SYSTEMS Introduction Motion of a Particle in a Box Simple Harmonic Oscillator Operator Formulation of the Simple Harmonic Oscillator Problem Bound State of a Two-Particle System with Central Interaction Bound States of Hydrogen (or Hydrogen-Like) Atoms The Deuteron Problem Energy Levels in a Three-Dimensional Square Well: General Case Energy Levels in an Isotropic Harmonic Potential Well Appendix 1: Special FunctionsAppendix 2: Orthogonal Curvilinear Coordinate SystemsSYMMETRIES AND CONSERVATION LAWS Symmetries and Their Group Properties Symmetries in a Quantum Mechanical System Basic Symmetry Groups of the Hamiltonian and Conservation Laws Lie Groups and Their Generators Examples of Lie Group Appendix 1: Groups and RepresentationsANGULAR MOMENTUM IN QUANTUM MECHANICS Introduction Raising and Lowering Operators Matrix Representation of Angular Momentum Operators Matrix Representation of Eigenstates of Angular Momentum Coordinate Representation of Orbital Angular Momentum Operators and States General Rotation Group and Rotation Matrices Coupling of Two Angular Momenta Properties of Clebsch-Gordan Coefficients Coupling of Three Angular Momenta Coupling of Four Angular Momenta (L - S and j - j Coupling)APPROXIMATION METHODS Introduction Nondegenerate Time-Independent Perturbation Theory Time-Independent Degenerate Perturbation Theory The Zeeman Effect WKBJ Approximation Particle in a Potential Well Application of WKBJ Approximation to a-decay The Variational Method The Problem of the Hydrogen Molecule System of n Identical Particles: Symmetric and Antisymmetric States Excited States of the Helium Atom Statistical (Thomas-Fermi) Model of the Atom Hartree's Self-consistent Field Method for Multi-Electron Atoms Hartree-Fock Equations Occupation Number RepresentationQUANTUM THEORY OF SCATTERING Introduction Laboratory and Center-of-Mass (CM) Reference Frames Scattering Equation and the Scattering AmplitudePartial Waves and Phase Shifts Calculation of Phase Shift Phase Shifts for Some Simple Potential Forms Scattering due to Coulomb Potential The Integral Form of Scattering Equation Lippmann-Schwinger Equation and the Transition Operator Born Expansion Appendix: The Calculus of ResiduesTIME-DEPENDENT PERTURBATION METHODS Introduction Perturbation Constant over an Interval of Time Harmonic Perturbation: Semiclassical Theory of Radiation Einstein Coeffcients Multipole Transitions Electric Dipole Transitions in Atoms and Selection Rules Photo-Electric Effect Sudden and Adiabatic Approximations Second-Order EffectsTHE THREE-BODY PROBLEM Introduction Eyges Approach Mitra's Approach Faddeev's Approach Faddeev Equations in Momentum Representation Faddeev Equations for a Three-Body Bound System Alt, Grassberger, and Sandhas (AGS) EquationsRELATIVISTIC QUANTUM MECHANICS Introduction Dirac Equation Spin of the Electron Free Particle (Plane Wave) Solutions of Dirac Equation Dirac Equation for a Zero Mass Particle Zitterbewegung and Negative Energy Solutions Dirac Equation for an Electron in an Electromagnetic FieldInvariance of Dirac Equation Dirac Bilinear Covariants Dirac Electron in a Spherically Symmetric Potential Charge Conjugation, Parity, and Time-Reversal Invariance Appendix: Theory of Special RelativityQUANTIZATION OF RADIATION FIELD Introduction Radiation Field as a Swarm of Oscillators Quantization of Radiation Field Interaction of Matter with Quantized Radiation Field Applications Bethe's Treatment of Atomic Level Shift Due to the Self Energy of the Electron: (Lamb-Retherford Shift)Compton Scattering Appendix: Electromagnetic Field in Coulomb GaugeSECOND QUANTIZATION Introduction Classical Concept of Field Analogy of Field and Particle Mechanics Field Equations from Lagrangian DensityQuantization of a Real Scalar (KG) Field Quantization of Complex Scalar (KG) Field Dirac Field and Its Quantization Positron Operators and SpinorsInteracting Fields and the Covariant Perturbation Theory Second-Order Processes in Electrodynamics Amplitude for Compton Scattering Feynman Graphs Calculation of the Cross-Section of Compton Scattering Cross-Sections for Other Electromagnetic Processes Appendix 1: Calculus of Variation and Euler-Lagrange Equations Appendix 2: Functionals and Functional Derivatives Appendix 3: Interaction of the Electron and Radiation Fields Appendix 4: On the Convergence of Iterative Expansion of the S OperatorEPILOGUE Introduction Einstein-Podolsky-Rosen Gedanken Experiment Einstein-Podolsky-Rosen-Bohm Gedanken Experiment Theory of Hidden Variables and Bell's Inequality Clauser-Horne Form of Bell's Inequality and Its Violation in Two-Photon Correlation Experiments GENERAL REFERENCESINDEX.…”
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“…Moreover, we give the precise relationship of the Yukawa couplings of special geometry with three-point functions in topological field theory.…”
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“…Covering a range of subjects from operator theory and classical harmonic analysis to Banach space theory, this book contains survey and expository articles by leading experts in their corresponding fields, and features fully-refereed, high-quality papers exploring new results and trends in spectral theory, mathematical physics, geometric function theory, and partial differential equations. …”
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“…Using the theory of Lie groups as a unifying vehicle, concepts and results from several fields of physics can be expressed in an extremely economical way. …”
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“…Let $p$ be an odd prime. For any CM number field $K$ containing a primitive $p^{\rm th}$-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality $\lambda^+ \leq \lambda^-$ between the ``plus'' and``minus'' parts of the $\lambda$-invariant of $K$. …”
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“…During the last decade, scientists working in quantum theory have been engaging in promising new fields such as quantum computation and quantum information processing, and have also been reflecting on the possibilities of nonlinear behavior on the quantum level. …”
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“…Part II contains accounts of classical mean-field methods. The parallels between Landau expansions and catastrophe theory are discussed and Ginzburg—Landau theory is introduced. …”
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