Attractors of Hamiltonian Nonlinear Partial Differential Equations

Attractors of Hamiltonian Nonlinear Partial Differential Equations

Author: Alexander Komech

Publisher: Cambridge University Press

Published: 2021-09-30

Total Pages:

ISBN-13: 100903605X

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This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.


Book Synopsis Attractors of Hamiltonian Nonlinear Partial Differential Equations by : Alexander Komech

Download or read book Attractors of Hamiltonian Nonlinear Partial Differential Equations written by Alexander Komech and published by Cambridge University Press. This book was released on 2021-09-30 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.

Nonlinear Semigroups, Partial Differential Equations and Attractors

Nonlinear Semigroups, Partial Differential Equations and Attractors

Author: T.L. Gill

Publisher: Springer

Published: 2006-11-15

Total Pages: 194

ISBN-13: 3540477918

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The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.


Book Synopsis Nonlinear Semigroups, Partial Differential Equations and Attractors by : T.L. Gill

Download or read book Nonlinear Semigroups, Partial Differential Equations and Attractors written by T.L. Gill and published by Springer. This book was released on 2006-11-15 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.

Lectures On Quantum Mechanics And Attractors

Lectures On Quantum Mechanics And Attractors

Author: Alexander Komech

Publisher: World Scientific

Published: 2022-02-18

Total Pages: 272

ISBN-13: 9811248915

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This book gives a concise introduction to Quantum Mechanics with a systematic, coherent, and in-depth explanation of related mathematical methods from the scattering theory and the theory of Partial Differential Equations.The book is aimed at graduate and advanced undergraduate students in mathematics, physics, and chemistry, as well as at the readers specializing in quantum mechanics, theoretical physics and quantum chemistry, and applications to solid state physics, optics, superconductivity, and quantum and high-frequency electronic devices.The book utilizes elementary mathematical derivations. The presentation assumes only basic knowledge of the origin of Hamiltonian mechanics, Maxwell equations, calculus, Ordinary Differential Equations and basic PDEs. Key topics include the Schrödinger, Pauli, and Dirac equations, the corresponding conservation laws, spin, the hydrogen spectrum, and the Zeeman effect, scattering of light and particles, photoelectric effect, electron diffraction, and relations of quantum postulates with attractors of nonlinear Hamiltonian PDEs. Featuring problem sets and accompanied by extensive contemporary and historical references, this book could be used for the course on Quantum Mechanics and is also suitable for individual study.


Book Synopsis Lectures On Quantum Mechanics And Attractors by : Alexander Komech

Download or read book Lectures On Quantum Mechanics And Attractors written by Alexander Komech and published by World Scientific. This book was released on 2022-02-18 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a concise introduction to Quantum Mechanics with a systematic, coherent, and in-depth explanation of related mathematical methods from the scattering theory and the theory of Partial Differential Equations.The book is aimed at graduate and advanced undergraduate students in mathematics, physics, and chemistry, as well as at the readers specializing in quantum mechanics, theoretical physics and quantum chemistry, and applications to solid state physics, optics, superconductivity, and quantum and high-frequency electronic devices.The book utilizes elementary mathematical derivations. The presentation assumes only basic knowledge of the origin of Hamiltonian mechanics, Maxwell equations, calculus, Ordinary Differential Equations and basic PDEs. Key topics include the Schrödinger, Pauli, and Dirac equations, the corresponding conservation laws, spin, the hydrogen spectrum, and the Zeeman effect, scattering of light and particles, photoelectric effect, electron diffraction, and relations of quantum postulates with attractors of nonlinear Hamiltonian PDEs. Featuring problem sets and accompanied by extensive contemporary and historical references, this book could be used for the course on Quantum Mechanics and is also suitable for individual study.

Lectures on Quantum Mechanics and Attractors

Lectures on Quantum Mechanics and Attractors

Author: A. I. Komech

Publisher: World Scientific Publishing Company

Published: 2022

Total Pages: 0

ISBN-13: 9789811248894

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This book gives a concise introduction to Quantum Mechanics with a systematic, coherent, and in-depth explanation of related mathematical methods from the scattering theory and the theory of Partial Differential Equations. The book is aimed at graduate and advanced undergraduate students in mathematics, physics, and chemistry, as well as at the readers specializing in quantum mechanics, theoretical physics and quantum chemistry, and applications to solid state physics, optics, superconductivity, and quantum and high-frequency electronic devices. The book utilizes elementary mathematical derivations. The presentation assumes only basic knowledge of the origin of Hamiltonian mechanics, Maxwell equations, calculus, Ordinary Differential Equations and basic PDEs. Key topics include the Schrö dinger, Pauli, and Dirac equations, the corresponding conservation laws, spin, the hydrogen spectrum, and the Zeeman effect, scattering of light and particles, photoelectric effect, electron diffraction, and relations of quantum postulates with attractors of nonlinear Hamiltonian PDEs. Featuring problem sets and accompanied by extensive contemporary and historical references, this book could be used for the course on Quantum Mechanics and is also suitable for individual study.


Book Synopsis Lectures on Quantum Mechanics and Attractors by : A. I. Komech

Download or read book Lectures on Quantum Mechanics and Attractors written by A. I. Komech and published by World Scientific Publishing Company. This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a concise introduction to Quantum Mechanics with a systematic, coherent, and in-depth explanation of related mathematical methods from the scattering theory and the theory of Partial Differential Equations. The book is aimed at graduate and advanced undergraduate students in mathematics, physics, and chemistry, as well as at the readers specializing in quantum mechanics, theoretical physics and quantum chemistry, and applications to solid state physics, optics, superconductivity, and quantum and high-frequency electronic devices. The book utilizes elementary mathematical derivations. The presentation assumes only basic knowledge of the origin of Hamiltonian mechanics, Maxwell equations, calculus, Ordinary Differential Equations and basic PDEs. Key topics include the Schrö dinger, Pauli, and Dirac equations, the corresponding conservation laws, spin, the hydrogen spectrum, and the Zeeman effect, scattering of light and particles, photoelectric effect, electron diffraction, and relations of quantum postulates with attractors of nonlinear Hamiltonian PDEs. Featuring problem sets and accompanied by extensive contemporary and historical references, this book could be used for the course on Quantum Mechanics and is also suitable for individual study.

Hamiltonian Dynamical Systems and Applications

Hamiltonian Dynamical Systems and Applications

Author: Walter Craig

Publisher: Springer Science & Business Media

Published: 2008-02-17

Total Pages: 450

ISBN-13: 1402069642

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This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.


Book Synopsis Hamiltonian Dynamical Systems and Applications by : Walter Craig

Download or read book Hamiltonian Dynamical Systems and Applications written by Walter Craig and published by Springer Science & Business Media. This book was released on 2008-02-17 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.

Partial Differential Equations and Functional Analysis

Partial Differential Equations and Functional Analysis

Author: Andrew Comech

Publisher: Springer Nature

Published: 2023-11-15

Total Pages: 334

ISBN-13: 303133681X

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Mark Vishik was one of the prominent figures in the theory of partial differential equations. His ground-breaking contributions were instrumental in integrating the methods of functional analysis into this theory. The book is based on the memoirs of his friends and students, as well as on the recollections of Mark Vishik himself, and contains a detailed description of his biography: childhood in Lwów, his connections with the famous Lwów school of Stefan Banach, a difficult several year long journey from Lwów to Tbilisi after the Nazi assault in June 1941, going to Moscow and forming his own school of differential equations, whose central role was played by the famous Vishik Seminar at the Department of Mechanics and Mathematics at Moscow State University. The reader is introduced to a number of remarkable scientists whose lives intersected with Vishik’s, including S. Banach, J. Schauder, I. N. Vekua, N. I. Muskhelishvili, L. A. Lyusternik, I. G. Petrovskii, S. L. Sobolev, I. M. Gelfand, M. G. Krein, A. N. Kolmogorov, N. I. Akhiezer, J. Leray, J.-L. Lions, L. Schwartz, L. Nirenberg, and many others. The book also provides a detailed description of the main research directions of Mark Vishik written by his students and colleagues, as well as several reviews of the recent development in these directions.


Book Synopsis Partial Differential Equations and Functional Analysis by : Andrew Comech

Download or read book Partial Differential Equations and Functional Analysis written by Andrew Comech and published by Springer Nature. This book was released on 2023-11-15 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mark Vishik was one of the prominent figures in the theory of partial differential equations. His ground-breaking contributions were instrumental in integrating the methods of functional analysis into this theory. The book is based on the memoirs of his friends and students, as well as on the recollections of Mark Vishik himself, and contains a detailed description of his biography: childhood in Lwów, his connections with the famous Lwów school of Stefan Banach, a difficult several year long journey from Lwów to Tbilisi after the Nazi assault in June 1941, going to Moscow and forming his own school of differential equations, whose central role was played by the famous Vishik Seminar at the Department of Mechanics and Mathematics at Moscow State University. The reader is introduced to a number of remarkable scientists whose lives intersected with Vishik’s, including S. Banach, J. Schauder, I. N. Vekua, N. I. Muskhelishvili, L. A. Lyusternik, I. G. Petrovskii, S. L. Sobolev, I. M. Gelfand, M. G. Krein, A. N. Kolmogorov, N. I. Akhiezer, J. Leray, J.-L. Lions, L. Schwartz, L. Nirenberg, and many others. The book also provides a detailed description of the main research directions of Mark Vishik written by his students and colleagues, as well as several reviews of the recent development in these directions.

Nearly Integrable Infinite-Dimensional Hamiltonian Systems

Nearly Integrable Infinite-Dimensional Hamiltonian Systems

Author: Sergej B. Kuksin

Publisher: Springer

Published: 2006-11-15

Total Pages: 128

ISBN-13: 3540479201

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The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.


Book Synopsis Nearly Integrable Infinite-Dimensional Hamiltonian Systems by : Sergej B. Kuksin

Download or read book Nearly Integrable Infinite-Dimensional Hamiltonian Systems written by Sergej B. Kuksin and published by Springer. This book was released on 2006-11-15 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.

Transcendence and Linear Relations of 1-Periods

Transcendence and Linear Relations of 1-Periods

Author: Annette Huber

Publisher: Cambridge University Press

Published: 2022-05-26

Total Pages: 266

ISBN-13: 1009022717

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This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.


Book Synopsis Transcendence and Linear Relations of 1-Periods by : Annette Huber

Download or read book Transcendence and Linear Relations of 1-Periods written by Annette Huber and published by Cambridge University Press. This book was released on 2022-05-26 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.

The Mordell Conjecture

The Mordell Conjecture

Author: Hideaki Ikoma

Publisher: Cambridge University Press

Published: 2022-02-03

Total Pages: 180

ISBN-13: 1108998194

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The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell–Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.


Book Synopsis The Mordell Conjecture by : Hideaki Ikoma

Download or read book The Mordell Conjecture written by Hideaki Ikoma and published by Cambridge University Press. This book was released on 2022-02-03 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell–Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.

Point-Counting and the Zilber–Pink Conjecture

Point-Counting and the Zilber–Pink Conjecture

Author: Jonathan Pila

Publisher: Cambridge University Press

Published: 2022-06-09

Total Pages: 267

ISBN-13: 1009170325

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Explores the recent spectacular applications of point-counting in o-minimal structures to functional transcendence and diophantine geometry.


Book Synopsis Point-Counting and the Zilber–Pink Conjecture by : Jonathan Pila

Download or read book Point-Counting and the Zilber–Pink Conjecture written by Jonathan Pila and published by Cambridge University Press. This book was released on 2022-06-09 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explores the recent spectacular applications of point-counting in o-minimal structures to functional transcendence and diophantine geometry.