Download or read book KAM Theory and Semiclassical Approximations to Eigenfunctions written by Vladimir F. Lazutkin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is a surprising fact that so far almost no books have been published on KAM theory. The first part of this book seems to be the first monographic exposition of this subject, despite the fact that the discussion of KAM theory started as early as 1954 (Kolmogorov) and was developed later in 1962 by Arnold and Moser. Today, this mathematical field is very popular and well known among physicists and mathematicians. In the first part of this Ergebnisse-Bericht, Lazutkin succeeds in giving a complete and self-contained exposition of the subject, including a part on Hamiltonian dynamics. The main results concern the existence and persistence of KAM theory, their smooth dependence on the frequency, and the estimate of the measure of the set filled by KAM theory. The second part is devoted to the construction of the semiclassical asymptotics to the eigenfunctions of the generalized Schrödinger operator. The main result is the asymptotic formulae for eigenfunctions and eigenvalues, using Maslov`s operator, for the set of eigenvalues of positive density in the set of all eigenvalues. An addendum by Prof. A.I. Shnirelman treats eigenfunctions corresponding to the "chaotic component" of the phase space.
Download or read book KAM Theory and Semiclassical Approximations to Eigenfunctions written by Vladimir Fedorovich Lazutkin and published by Springer Verlag. This book was released on 1993 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Singularity Theory for Non Twist KAM Tori written by A. González-Enríquez and published by American Mathematical Soc.. This book was released on 2014-01-08 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph the authors introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameter-dependent Hamiltonian systems. It is based on Singularity Theory of critical points of a real-valued function which the authors call the potential. The potential is constructed in such a way that: nondegenerate critical points of the potential correspond to twist invariant tori (i.e. with nondegenerate torsion) and degenerate critical points of the potential correspond to non-twist invariant tori. Hence, bifurcating points correspond to non-twist tori.
Download or read book Eigenfunctions of the Laplacian on a Riemannian Manifold written by Steve Zelditch and published by American Mathematical Soc.. This book was released on 2017-12-12 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.
Download or read book Nonlinear Optical and Atomic Systems written by Christophe Besse and published by Springer. This book was released on 2015-08-26 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of nonlinear systems in optics and atomic physics. The text covers a wide spectrum of current research on the subject, which is an extremely active field in physics and mathematical physics, with a very broad range of implications, both for fundamental science and technological applications: light propagation in microstructured optical fibers, Bose-Einstein condensates, disordered systems, and the newly emerging field of nonlinear quantum mechanics. Accessible to PhD students, this book will also be of interest to post-doctoral researchers and seasoned academics.
Download or read book Perturbation Theory written by Giuseppe Gaeta and published by Springer Nature. This book was released on 2022-12-16 with total page 601 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomotion, Condensed Matter among others, while all other contributions have been revised and updated. Coverage includes the theory of (Poincare’-Birkhoff) Normal Forms, aspects of PT in specific mathematical settings (Hamiltonian, KAM theory, Nekhoroshev theory, and symmetric systems), technical problems arising in PT with solutions, convergence of series expansions, diagrammatic methods, parametric resonance, systems with nilpotent real part, PT for non-smooth systems, and on PT for PDEs [write out this acronym partial differential equations]. Another group of papers is focused specifically on applications to Celestial Mechanics, Quantum Mechanics and the related semiclassical PT, Quantum Bifurcations, Molecular Dynamics, the so-called choreographies in the N-body problem, as well as Evolutionary Theory. Overall, this unique volume serves to demonstrate the wide utility of PT, while creating a foundation for innovations from a new generation of graduate students and professionals in Physics, Mathematics, Mechanics, Engineering and the Biological Sciences.
Download or read book Partial Differential Equations V written by M.V. Fedoryuk and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution.
Download or read book KdV KAM written by Thomas Kappeler and published by Springer Science & Business Media. This book was released on 2003-05-19 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text treats the Korteweg-de Vries (KdV) equation with periodic boundary conditions. This equation models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. For the first time, these important results are comprehensively covered in book form, authored by internationally renowned experts in the field.
Download or read book Harmonic Analysis at Mount Holyoke written by William Beckner and published by American Mathematical Soc.. This book was released on 2003 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the conference on harmonic analysis and related areas. The conference provided an opportunity for researchers and students to exchange ideas and report on progress in this large and central field of modern mathematics. The volume is suitable for graduate students and research mathematicians interested in harmonic analysis and related areas.
Download or read book Chaos written by Bertrand Duplantier and published by Springer Science & Business Media. This book was released on 2013-11-26 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This twelfth volume in the Poincaré Seminar Series presents a complete and interdisciplinary perspective on the concept of Chaos, both in classical mechanics in its deterministic version, and in quantum mechanics. This book expounds some of the most wide ranging questions in science, from uncovering the fingerprints of classical chaotic dynamics in quantum systems, to predicting the fate of our own planetary system. Its seven articles are also highly pedagogical, as befits their origin in lectures to a broad scientific audience. Highlights include a complete description by the mathematician É. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly effect as it is understood today, illuminating the fundamental mathematical issues at play with deterministic chaos; a detailed account by the experimentalist S. Fauve of the masterpiece experiment, the von Kármán Sodium or VKS experiment, which established in 2007 the spontaneous generation of a magnetic field in a strongly turbulent flow, including its reversal, a model of Earth’s magnetic field; a simple toy model by the theorist U. Smilansky – the discrete Laplacian on finite d-regular expander graphs – which allows one to grasp the essential ingredients of quantum chaos, including its fundamental link to random matrix theory; a review by the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the fascinating connection between the distribution of zeros of the Riemann ζ-function and the statistics of eigenvalues of random unitary matrices, which could ultimately provide a spectral interpretation for the zeros of the ζ-function, thus a proof of the celebrated Riemann Hypothesis itself; an article by a pioneer of experimental quantum chaos, H-J. Stöckmann, who shows in detail how experiments on the propagation of microwaves in 2D or 3D chaotic cavities beautifully verify theoretical predictions; a thorough presentation by the mathematical physicist S. Nonnenmacher of the “anatomy” of the eigenmodes of quantized chaotic systems, namely of their macroscopic localization properties, as ruled by the Quantum Ergodic theorem, and of the deep mathematical challenge posed by their fluctuations at the microscopic scale; a review, both historical and scientific, by the astronomer J. Laskar on the stability, hence the fate, of the chaotic Solar planetary system we live in, a subject where he made groundbreaking contributions, including the probabilistic estimate of possible planetary collisions. This book should be of broad general interest to both physicists and mathematicians.
Download or read book Quasi Periodic Motions in Families of Dynamical Systems written by Hendrik W. Broer and published by Springer Science & Business Media. This book was released on 1996 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is on Kolmogorov-Arnol'd-Moser theory for quasi-periodic tori in dynamical systems. It gives an up-to-date report on the role parameters play for persis- tence of such tori, typically occuring on Cantor sets of positive Hausdorff measure inside phase and parameter space. The cases with preservation of symplectic or volume forms or time-reversal symmetries are included. The concepts of Whitney-smoothness and Diophantine approximation of Cantor sets on submanifolds of Euclidean space are treated, as well as Bruno's theory on analytic continuation of tori. Partly this material is new to Western mathematicians. The reader should be familiar with dynamical systems theory, differen- tial equations and some analysis. The book is directed to researchers, but its entrance level is introductory.
Download or read book Visions in Mathematics written by Noga Alon and published by Springer Science & Business Media. This book was released on 2011-03-31 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.
Download or read book Nonlinear Dynamics Chaotic and Complex Systems written by Eryk Infeld and published by Cambridge University Press. This book was released on 1997-06-19 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: The physics and mathematics of nonlinear dynamics, chaotic and complex systems constitute some of the most fascinating developments of late twentieth century science. It turns out that chaotic bahaviour can be understood, and even utilized, to a far greater degree than had been suspected. Surprisingly, universal constants have been discovered. The implications have changed our understanding of important phenomena in physics, biology, chemistry, economics, medicine and numerous other fields of human endeavor. In this book, two dozen scientists and mathematicians who were deeply involved in the "nonlinear revolution" cover most of the basic aspects of the field.
Download or read book The Principle of Least Action in Geometry and Dynamics written by Karl Friedrich Siburg and published by Springer. This book was released on 2004-04-30 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather’s minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.
Download or read book Ergodic Theory Analysis and Efficient Simulation of Dynamical Systems written by Bernold Fiedler and published by Springer Science & Business Media. This book was released on 2001 with total page 840 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book summarizes and highlights progress in Dynamical Systems achieved during six years of the German Priority Research Program "Ergotic Theory, Analysis, and Efficient Simulation of Dynamical Systems", funded by the Deutsche Forschungsgemeinschaft (DFG). The three fundamental topics of large time behavior, dimension, and measure are tackled with by a rich circle of uncompromisingly rigorous mathematical concepts. The range of applied issues comprises such diverse areas as crystallization and dendrite growth, the dynamo effect, efficient simulation of biomolecules, fluid dynamics and reacting flows, mechanical problems involving friction, population biology, the spread of infectious diseases, and quantum chaos. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications fair into the neighboring disciplines of Science.
Download or read book Differential Equations Theory and Applications written by David Betounes and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 686 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way or in a more applied way. The accompanying CD contains Maple worksheets for the exercises, and special Maple code for performing various tasks. In addition to its use in a traditional one or two semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering.
Download or read book Emerging Applications of Number Theory written by Dennis A. Hejhal and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 693 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most people tend to view number theory as the very paradigm of pure mathematics. With the advent of computers, however, number theory has been finding an increasing number of applications in practical settings, such as in cryptography, random number generation, coding theory, and even concert hall acoustics. Yet other applications are still emerging - providing number theorists with some major new areas of opportunity. The 1996 IMA summer program on Emerging Applications of Number Theory was aimed at stimulating further work with some of these newest (and most attractive) applications. Concentration was on number theory's recent links with: (a) wave phenomena in quantum mechanics (more specifically, quantum chaos); and (b) graph theory (especially expander graphs and related spectral theory). This volume contains the contributed papers from that meeting and will be of interest to anyone intrigued by novel applications of modern number-theoretical techniques.
