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Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Author: Kurusch Ebrahimi-Fard

Publisher: Springer

Published: 2018-11-05

Total Pages: 361

ISBN-13: 3030013979

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This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

Book Synopsis Discrete Mechanics, Geometric Integration and Lie–Butcher Series by : Kurusch Ebrahimi-Fard

Download or read book Discrete Mechanics, Geometric Integration and Lie–Butcher Series written by Kurusch Ebrahimi-Fard and published by Springer. This book was released on 2018-11-05 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

Computation and Combinatorics in Dynamics, Stochastics and Control

Author: Elena Celledoni

Publisher: Springer

Published: 2019-01-13

Total Pages: 734

ISBN-13: 3030015939

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The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory. In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields. The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

Book Synopsis Computation and Combinatorics in Dynamics, Stochastics and Control by : Elena Celledoni

Download or read book Computation and Combinatorics in Dynamics, Stochastics and Control written by Elena Celledoni and published by Springer. This book was released on 2019-01-13 with total page 734 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory. In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields. The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

Mathematical Models and Integration Methods

Author: Oleg V. Kaptsov

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2024-09-23

Total Pages: 238

ISBN-13: 3111546802

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The book compiles works presented at a seminar aiming to attract global experts in differential equations, mathematical modeling, and integration methods. It covers classical and contemporary integration techniques for partial differential equations, including Monge and Darboux's approaches and their extensions. Additionally, it introduces a novel theoretical model for plane turbulent flows, presents gravitational equations derived from the principle of least action, and explores symmetry-preserving conservative finite-difference schemes for hydrodynamic-type equations. Analytical solutions for Maxwell's equations in incompressible viscoelastic mediums are examined, alongside theoretical-group analysis of wake mathematical models and reduction to ordinary differential equations. The book also delves into special classes of two-dimensional ideal fluid motion and advancements in discrete orthogonal polynomial theory, showcasing rapid decay properties near interval boundaries. In conclusion, this comprehensive collection is indispensable for researchers and practitioners in applied mathematics, fluid dynamics, and computational modeling, providing valuable insights into cutting-edge methods and solutions in the field.

Book Synopsis Mathematical Models and Integration Methods by : Oleg V. Kaptsov

Download or read book Mathematical Models and Integration Methods written by Oleg V. Kaptsov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2024-09-23 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book compiles works presented at a seminar aiming to attract global experts in differential equations, mathematical modeling, and integration methods. It covers classical and contemporary integration techniques for partial differential equations, including Monge and Darboux's approaches and their extensions. Additionally, it introduces a novel theoretical model for plane turbulent flows, presents gravitational equations derived from the principle of least action, and explores symmetry-preserving conservative finite-difference schemes for hydrodynamic-type equations. Analytical solutions for Maxwell's equations in incompressible viscoelastic mediums are examined, alongside theoretical-group analysis of wake mathematical models and reduction to ordinary differential equations. The book also delves into special classes of two-dimensional ideal fluid motion and advancements in discrete orthogonal polynomial theory, showcasing rapid decay properties near interval boundaries. In conclusion, this comprehensive collection is indispensable for researchers and practitioners in applied mathematics, fluid dynamics, and computational modeling, providing valuable insights into cutting-edge methods and solutions in the field.

From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory

Author: Fritz Gesztesy

Publisher: Springer Nature

Published: 2021-11-11

Total Pages: 388

ISBN-13: 3030754251

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The main topics of this volume, dedicated to Lance Littlejohn, are operator and spectral theory, orthogonal polynomials, combinatorics, number theory, and the various interplays of these subjects. Although the event, originally scheduled as the Baylor Analysis Fest, had to be postponed due to the pandemic, scholars from around the globe have contributed research in a broad range of mathematical fields. The collection will be of interest to both graduate students and professional mathematicians. Contributors are: G.E. Andrews, B.M. Brown, D. Damanik, M.L. Dawsey, W.D. Evans, J. Fillman, D. Frymark, A.G. García, L.G. Garza, F. Gesztesy, D. Gómez-Ullate, Y. Grandati, F.A. Grünbaum, S. Guo, M. Hunziker, A. Iserles, T.F. Jones, K. Kirsten, Y. Lee, C. Liaw, F. Marcellán, C. Markett, A. Martinez-Finkelshtein, D. McCarthy, R. Milson, D. Mitrea, I. Mitrea, M. Mitrea, G. Novello, D. Ong, K. Ono, J.L. Padgett, M.M.M. Pang, T. Poe, A. Sri Ranga, K. Schiefermayr, Q. Sheng, B. Simanek, J. Stanfill, L. Velázquez, M. Webb, J. Wilkening, I.G. Wood, M. Zinchenko.

Book Synopsis From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory by : Fritz Gesztesy

Download or read book From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory written by Fritz Gesztesy and published by Springer Nature. This book was released on 2021-11-11 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main topics of this volume, dedicated to Lance Littlejohn, are operator and spectral theory, orthogonal polynomials, combinatorics, number theory, and the various interplays of these subjects. Although the event, originally scheduled as the Baylor Analysis Fest, had to be postponed due to the pandemic, scholars from around the globe have contributed research in a broad range of mathematical fields. The collection will be of interest to both graduate students and professional mathematicians. Contributors are: G.E. Andrews, B.M. Brown, D. Damanik, M.L. Dawsey, W.D. Evans, J. Fillman, D. Frymark, A.G. García, L.G. Garza, F. Gesztesy, D. Gómez-Ullate, Y. Grandati, F.A. Grünbaum, S. Guo, M. Hunziker, A. Iserles, T.F. Jones, K. Kirsten, Y. Lee, C. Liaw, F. Marcellán, C. Markett, A. Martinez-Finkelshtein, D. McCarthy, R. Milson, D. Mitrea, I. Mitrea, M. Mitrea, G. Novello, D. Ong, K. Ono, J.L. Padgett, M.M.M. Pang, T. Poe, A. Sri Ranga, K. Schiefermayr, Q. Sheng, B. Simanek, J. Stanfill, L. Velázquez, M. Webb, J. Wilkening, I.G. Wood, M. Zinchenko.

Geometric Science of Information

Author: Frank Nielsen

Publisher: Springer Nature

Published: 2023-07-31

Total Pages: 670

ISBN-13: 3031382994

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This book constitutes the proceedings of the 6th International Conference on Geometric Science of Information, GSI 2023, held in St. Malo, France, during August 30-September 1, 2023. The 125 full papers presented in this volume were carefully reviewed and selected from 161 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications. The papers are organized in the following topics: geometry and machine learning; divergences and computational information geometry; statistics, topology and shape spaces; geometry and mechanics; geometry, learning dynamics and thermodynamics; quantum information geometry; geometry and biological structures; geometry and applications.

Book Synopsis Geometric Science of Information by : Frank Nielsen

Download or read book Geometric Science of Information written by Frank Nielsen and published by Springer Nature. This book was released on 2023-07-31 with total page 670 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 6th International Conference on Geometric Science of Information, GSI 2023, held in St. Malo, France, during August 30-September 1, 2023. The 125 full papers presented in this volume were carefully reviewed and selected from 161 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications. The papers are organized in the following topics: geometry and machine learning; divergences and computational information geometry; statistics, topology and shape spaces; geometry and mechanics; geometry, learning dynamics and thermodynamics; quantum information geometry; geometry and biological structures; geometry and applications.

Algebra and Applications 2

Author: Abdenacer Makhlouf

Publisher: John Wiley & Sons

Published: 2021-12-07

Total Pages: 336

ISBN-13: 1119880904

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This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota–Baxter algebras are explored. Algebra and Applications 2 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.

Book Synopsis Algebra and Applications 2 by : Abdenacer Makhlouf

Download or read book Algebra and Applications 2 written by Abdenacer Makhlouf and published by John Wiley & Sons. This book was released on 2021-12-07 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota–Baxter algebras are explored. Algebra and Applications 2 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.

Geometric Numerical Integration

Author: Ernst Hairer

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 526

ISBN-13: 3662050188

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This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

Book Synopsis Geometric Numerical Integration by : Ernst Hairer

Download or read book Geometric Numerical Integration written by Ernst Hairer and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

The Structure of Compact Groups

Author: Karl H. Hofmann

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2020-06-08

Total Pages: 1034

ISBN-13: 3110695995

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This book is designed both as a textbook for high-level graduate courses and as a reference for researchers who need to apply the structure and representation theory of compact groups. A gentle introduction to compact groups and their representation theory is followed by self-contained courses on linear and compact Lie groups, and on locally compact abelian groups. This fourth edition was updated with the latest developments in the field.

Book Synopsis The Structure of Compact Groups by : Karl H. Hofmann

Download or read book The Structure of Compact Groups written by Karl H. Hofmann and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-06-08 with total page 1034 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is designed both as a textbook for high-level graduate courses and as a reference for researchers who need to apply the structure and representation theory of compact groups. A gentle introduction to compact groups and their representation theory is followed by self-contained courses on linear and compact Lie groups, and on locally compact abelian groups. This fourth edition was updated with the latest developments in the field.

A Concise Introduction to Geometric Numerical Integration

Author: Sergio Blanes

Publisher: CRC Press

Published: 2017-11-22

Total Pages: 233

ISBN-13: 1482263440

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Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.

Book Synopsis A Concise Introduction to Geometric Numerical Integration by : Sergio Blanes

Download or read book A Concise Introduction to Geometric Numerical Integration written by Sergio Blanes and published by CRC Press. This book was released on 2017-11-22 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.

The Problem of Integrable Discretization

Author: Yuri B. Suris

Publisher: Birkhäuser

Published: 2012-11-05

Total Pages: 0

ISBN-13: 9783034894043

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An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new. Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.

Book Synopsis The Problem of Integrable Discretization by : Yuri B. Suris

Download or read book The Problem of Integrable Discretization written by Yuri B. Suris and published by Birkhäuser. This book was released on 2012-11-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new. Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.