Algebra and Applications 2 E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Seitenzahl: 514
Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright
Preface
1 Algebraic Background for Numerical Methods, Control Theory and Renormalization
1.1. Introduction
1.2. Hopf algebras: general properties
1.3. Connected Hopf algebras
1.4. Pre-Lie algebras
1.5. Algebraic operads
1.6. Pre-Lie algebras (continued)
1.7. Other related algebraic structures
1.8. References
2 From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras
2.1. Introduction
2.2. Generalized iterated integrals
2.3. Advances in chronological calculus
2.4. Rota–Baxter algebras
2.5. References
3 Noncommutative Symmetric Functions, Lie Series and Descent Algebras
3.1. Introduction
3.2. Classical symmetric functions
3.3. Noncommutative symmetric functions
3.4. Lie series and Lie idempotents
3.5. Lie idempotents as noncommutative symmetric functions
3.6. Decompositions of the descent algebras
3.7. Decompositions of the tensor algebra
3.8. General deformations
3.9. Lie quasi-idempotents as Lie polynomials
3.10. Permutations and free quasi-symmetric functions
3.11. Packed words and word quasi-symmetric functions
3.12. References
4 From Runge–Kutta Methods to Hopf Algebras of Rooted Trees
4.1. Numerical integration methods for ordinary differential equations
4.2. Algebraic theory of Runge–Kutta methods
4.3. B-series and related formal expansions
4.4. Hopf algebras of rooted trees
4.5. References
5 Combinatorial Algebra in Controllability and Optimal Control
5.1. Introduction
5.2. Analytic foundations
5.3. Controllability and optimality
5.4. Product expansions and realizations
5.5. References
6 Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application
6.1. The Butcher group and the Connes–Kreimer algebra
6.2. Character groups of graded and connected Hopf algebras
6.3. Controlled groups of characters
6.4. Appendix: Calculus in locally convex spaces
6.5. References
List of Authors
Index
List of Illustrations
Chapter 5
Figure 5.1.
Parallel parking a car (bicycle). For a color version of this figure...
Figure 5.2.
The states of the bicycle
Figure 5.3.
An inverted pendulum
Figure 5.4.
Open-loop and closed-loop controls with a feedback controller K
Figure 5.5.
Parallel parking a car (bicycle). For a color version of this figure...
Figure 5.6.
Pontryagin maximum principle. For a color version of this figure, se...
Figure 5.7.
Needle variations also scaled by amplitude
Figure 5.8.
More complex family of control variations
Tables
Chapter 4
Table 4.1.
Functions
associated with rooted trees with up to four vertices
Table 4.2.
Elementary differentials F
u
and the values of u
! and σ(u) for rooted ...
Chapter 6
Table 6.1.
Standard examples for growth families (Dahmen and Schmeding 2018, Pro...
Guide
Cover
Table of Contents
Title page
Copyright
Preface
Begin Reading
List of Authors
Index
End User License Agreement
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SCIENCES
Mathematics, Field Director – Nikolaos Limnios
Algebra and Geometry, Subject Head – Abdenacer Makhlouf
Algebra and Applications 2
Combinatorial Algebra and Hopf Algebras
Coordinated by
Abdenacer Makhlouf
First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.wiley.com
© ISTE Ltd 2021
The rights of Abdenacer Makhlouf to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2021942616
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78945–018-7
ERC code:
PE1 Mathematics
PE1_2 Algebra
PE1_5 Lie groups, Lie algebras
PE1_12 Mathematical physics
Preface
Abdenacer MAKHLOUF
University of Haute Alsace, Mulhouse, France
The aim of this series of books is to report on the new trends of research in algebra and related topics. We provide an insight into the fast development of new concepts and theories related to algebra and present self-contained chapters on various topics, with each chapter combining some of the features of both graduate-level textbooks and research-level surveys. Each chapter includes an introduction with motivations and historical remarks, as well as the basic concepts, main results and perspectives. Moreover, the authors have commented on the relevance of the results in relation to other results and applications.
In this volume, the chapters encompass surveys of basic theories on non-associative algebras like Lie theories, using modern tools and more recent algebraic structures like Hopf algebras, which are related to Quantum groups and Mathematical Physics. The algebraic background of pre-Lie algebras, other non-associative algebras (non-associative permutative, assosymmetric, dendriform, etc.) and algebraic operads is presented. This volume also deals with noncommutative symmetric functions, Lie series, descent algebras, chronological and Rota–Baxter algebras. We focus on the increasing role played by Combinatorial algebra and Hopf algebras, as well as some non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods and control theory. It turns out that the Hopf algebra of rooted trees is an adequate tool, not only for vector fields, but also for studying the numerical approximation of their integral curves. Runge–Kutta methods form a group (called the Butcher group), which is the character group of the Connes–Kreimer Hopf algebra. The algebraic theory of Runge–Kutta methods B-series and related formal expansions are considered. Algebraic structures underlying calculus with iterated integrals lead naturally to the notions of descent (Hopf) algebra, as well as permutation Hopf algebra. In this volume we discuss the Lie-theoretic perspective and advances of chronological calculus. Chronological algebras and time-ordered products appear in an (almost) uncountable number of places, especially in theoretical physics and control theory. Noncommutative symmetric functions are applied to the study of formal power series with coefficients in a noncommutative algebra, in particular to Lie series. Moreover, Lie idempotents, Eulerian idempotents and Magnus expansion are considered. In addition, the interaction of algebra and (infinite-dimensional) geometry in the guise of Hopf algebras and certain associated character groups is examined. It turns out that fundamental concepts in control theory are inherently linked to combinatorial and algebraic structures. It is shown how modern combinatorial algebraic tools provide deeper insight and facilitate analysis, computations and design. The emphasis is on exhibiting the algebraic structures that map combinatorial structures to geometric and dynamic objects.
I thank Kurusch Ebrahimi-Fard for suggesting these topics, presented at Benasque Intensive School, and express my deep gratitude to all of the contributors of this volume and to ISTE for their support.
August 2021
