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Vortices in the Magnetic Ginzburg-Landau Model
This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in a bounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.
Variational Methods : Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems
Gives a concise introduction to variational methods and presents an overview of areas of current research in the calculus of variations. The fourth edition gives a survey on new developments in the field. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Also the recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. Aside from these more significant additions, a number of smaller changes throughout the text have been made and the references have been updated.
Trends in Partial Differential Equations of Mathematical Physics
Vsevolod Alekseevich Solonnikov is known as one of the outstanding mathematicians from the St Petersburg Mathematical School. This book contains twenty original contributions on many topics related to V A Solonnikov's work, selected from the invited talks of an international conference held on the occasion of his 70th birthday.
The Role of Mathematics in Physical Sciences ; Interdisciplinary and Philosophical Aspects
Even though mathematics and physics have been related for centuries and this relation appears to be unproblematic, there are many questions still open: Is mathematics really necessary for physics, or could physics exist without mathematics? Should we think physically and then add the mathematics apt to formalise our physical intuition, or should we think mathematically and then interpret physically the obtained results? Do we get mathematical objects by abstraction from real objects, or vice versa? Why is mathematics effective into physics? These are all relevant questions, whose answers are necessary to fully understand the status of physics, particularly of contemporary physics. The aim of this book is to offer plausible answers to such questions through both historical analyses of relevant cases, and philosophical analyses of the relations between mathematics and physics.
The Mathematics of the Bose Gas and its Condensation
This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem.It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area .The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.
The Fourfold Way in Real Analysis : An Alternative to the Metaplectic Representation
The fourfold way starts with the consideration of entire functions of one variable satisfying specific estimates at infinity, both on the real line and the pure imaginary line. A major part of classical analysis, mainly that which deals with Fourier analysis and related concepts, can then be given a parameter-dependent analogue. The parameter is some real number modulo 2, the classical case being obtained when it is an integer. The space L2(R) has to give way to a pseudo-Hilbert space, on which a new translation-invariant integral still exists. All this extends to the n-dimensional case, and in the alternative to the metaplectic representation so obtained, it is the space of Lagrangian subspaces of R2n that plays the usual role of the complex Siegel domain. In fourfold analysis, the spectrum of the harmonic oscillator can be an arbitrary class modulo the integers.
The Challenge of Chance : A Multidisciplinary Approach from Science and the Humanities
Presents a multidisciplinary perspective on chance, with contributions from distinguished researchers in the areas of biology, cognitive neuroscience, economics, genetics, general history, law, linguistics, logic, mathematical physics, statistics, theology and philosophy. The individual chapters are bound together by a general introduction followed by an opening chapter that surveys 2500 years of linguistic, philosophical, and scientific reflections on chance, coincidence, fortune, randomness, luck and related concepts.
The Breadth of Symplectic and Poisson Geometry : Festschrift in Honor of Alan Weinstein
One of the world’s foremost geometers, Alan Weinstein has made deep contributions to symplectic and differential geometry, Lie theory, mechanics, and related fields. Written in his honor, the invited papers in this volume reflect the active and vibrant research in these areas and are a tribute to Weinstein’s ongoing influence.The text cover a broad range of topics: Induction and reduction for systems with symmetry, symplectic geometry and topology, geometric quantization, the Weinstein Conjecture, Poisson algebra and geometry, Dirac structures, deformations for Lie group actions, Kähler geometry of moduli spaces, theory and applications of Lagrangian and Hamiltonian mechanics and dynamics, symplectic and Poisson groupoids, and quantum representations.
Symmetry Breaking ; 1st ed.
Devoted to classical field theory, such a mechanism is explained in terms of the occurrence of disjoint sectors and their stability properties and of an improved version of the Noether theorem. For infinitely extended quantum systems, discussed in the second part, the mechanism is related to the occurrence of disjoint pure phases and characterized by a symmetry breaking order parameter, for which non perturbative criteria are discussed, following Wightman, and contrasted with the standard Goldstone perturbative strategy. The Goldstone theorem is discussed with a critical look at the hypotheses that emphasizes the crucial role of the dynamical delocalization induced by the interaction range. The Higgs mechanism in local gauges is explained in terms of the Gauss law constraint on the physical states.
Symmetry Breaking ; 2nd ed.
Presents a more detailed and thorough discussion of the mechanism of symmetry breaking in classical field theory in relation with the Noether theorem. Moreover, the link between symmetry breaking without massless Goldstone bosons in Coulomb systems and in gauge theories is made more explicit in terms of the delocalized Coulomb dynamics. Furthermore, the chapter on the Higgs mechanism has been significantly expanded with a non-perturbative treatment of the Higgs phenomenon, at the basis of the standard model of particle physics, in the local and in the Coulomb gauges.
Stable Approximate Evaluation of Unbounded Operators
Spectral theory of bounded linear operators teams up with von Neumann’s theory of unbounded operators in this monograph to provide a general framework for the study of stable methods for the evaluation of unbounded operators. An introductory chapter provides numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. Before the general theory of stabilization methods is developed, an extensive exposition of the necessary background material from the theory of operators on Hilbert space is provided. Several specific stabilization methods are studied in detail, with particular attention to the Tikhonov-Morozov method and its iterated version.
Self-Dual Gauge Field Vortices : An Analytical Approach
In modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity and supergravity. This monograph discusses specific examples of gauge field theories that exhibit a selfdual structure.
Selected Works of S.L. Sobolev : Vol. I : Equations of Mathematical Physics, Computational Mathematics, and Cubature Formulas
S.L. Sobolev (1908–1989) was a great mathematician of the twentieth century. His selected works included in this volume laid the foundations for intensive development of the modern theory of partial differential equations and equations of mathematical physics, and they were a gold mine for new directions of functional analysis and computational mathematics.
Quaternions, Clifford Algebras and Relativistic Physics
The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have priviledged a geometric approach, the author uses an algebraic approach which can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics.The book proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism and general relativity.
Quantum Riemannian Geometry
Provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points.Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators ....
Quantum Independent Increment Processes II : Structure of Quantum Lévy Processes, Classical Probability, and Physics
The present volume contains the following lectures: Random Walks on Finite Quantum Groups, Quantum Markov Processes and Applications in Physics and Classical and Free Infinite Divisibility and Lévy Processes.
Quantum Independent Increment Processes I : From Classical Probability to Quantum Stochastic Calculus
This volume is the first of two volumes containing the revised and completed notes lectures given at the school "Quantum Independent Increment Processes: Structure and Applications to Physics". The present first volume contains the following lectures: "Lévy Processes in Euclidean Spaces and Groups" by David Applebaum, "Locally Compact Quantum Groups" by Johan Kustermans, "Quantum Stochastic Analysis" by J. Martin Lindsay, and "Dilations, Cocycles and Product Systems" by B.V. Rajarama Bhat.
Quantum Field Theory I : Basics in Mathematics and Physics : A Bridge between Mathematicians and Physicists
This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics.
Quantization on Nilpotent Lie Groups
Presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups.
Pseudo-Differential Operators and Related Topics
The objective is to look at pseudo-differential operators and related topics and to report recent advances in a broad spectrum of topics such as partial differential equations, quantization, Wigner transforms, Weyl transforms on Lie groups, mathematical physics, time-frequency analysis, frames, and stochastic processes.
