Algebra Books

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Book SynopsisThis book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the first of three volumes specifically focusing on algebra and its applications. Algebra and Applications 1 centers on non-associative algebras and includes an introduction to derived categories. 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. Jordan superalgebras, Lie algebras, composition algebras, graded division algebras, non-associative C*- algebras, H*-algebras, Krichever-Novikov type algebras, preLie algebras and related structures, geometric structures on 3-Lie algebras and derived categories are all explored. Algebra and Applications 1 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.Table of ContentsForeword xiAbdenacer MAKHLOUF Chapter 1. Jordan Superalgebras 1Consuelo MARTINEZ and Efim ZELMANOV Chapter 2. Composition Algebras 27Alberto ELDUQUE Chapter 3. Graded-Division Algebras 59Yuri BAHTURIN, Mikhail KOCHETOV and Mikhail ZAICEV Chapter 4. Non-associative C∗-algebras 111Ángel RODRÍGUEZ PALACIOS and Miguel CABRERA GARCÍA Chapter 5. Structure of H∗-algebras 155José Antonio CUENCA MIRA Chapter 6. Krichever–Novikov Type Algebras: Definitions and Results 199Martin SCHLICHENMAIER Chapter 7. An Introduction to Pre-Lie Algebras 245Chengming BAI Chapter 8. Symplectic, Product and Complex Structures on 3-Lie Algebras 275Yunhe SHENG and Rong TANG Chapter 9. Derived Categories 321Bernhard KELLER List of Authors 347 Index 349

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Book SynopsisThis 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.Table of ContentsPrefacexiAbdenacer MAKHLOUF Chapter 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization 1Dominique MANCHON 1.1. Introduction 1 1.2. Hopf algebras: generalproperties 2 1.2.1. Algebras 2 1.2.2. Coalgebras 3 1.2.3. Convolution product 6 1.2.4. Bialgebras andHopf algebras 7 1.2.5. Some simple examples of Hopf algebras 8 1.2.6. Some basic properties of Hopf algebras 9 1.3. ConnectedHopf algebras 10 1.3.1. Connectedgradedbialgebras 10 1.3.2. An example: the Hopf algebra of decorated rooted trees 13 1.3.3. Connectedfiltered bialgebras 14 1.3.4. The convolution product 15 1.3.5. Characters 17 1.3.6. Group schemes and the Cartier–Milnor–Moore–Quillen theorem 19 1.3.7. Renormalization in connected filtered Hopf algebras 21 1.4. Pre-Lie algebras 24 1.4.1. Definition and general properties 24 1.4.2. The groupof formalflows 25 1.4.3. The pre-Lie Poincaré–Birkhoff–Witt theorem 26 1.5. Algebraicoperads 28 1.5.1. Manipulatingalgebraicoperations 28 1.5.2. The operad of multi-linear operations 29 1.5.3. A definition for linear operads 31 1.5.4. Afewexamplesof operads 32 1.6. Pre-Lie algebras (continued) 35 1.6.1. Pre-Lie algebras and augmented operads 35 1.6.2. A pedestrian approach to free pre-Lie algebra 36 1.6.3. Right-sided commutative Hopf algebras and theLoday–Roncotheorem 38 1.6.4. Pre-Lie algebras of vectorfields 40 1.6.5. B-series, composition and substitution 42 1.7. Other related algebraic structures 44 1.7.1. NAPalgebras 44 1.7.2. Novikovalgebras 48 1.7.3. Assosymmetric algebras 48 1.7.4. Dendriformalgebras 48 1.7.5. Post-Lie algebras 49 1.8. References 50 Chapter 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras 55Kurusch EBRAHIMI-FARD and Frédéric PATRAS 2.1. Introduction 55 2.2. Generalizediterated integrals 58 2.2.1. Permutations andsimplices 59 2.2.2. Descents,NCSFand theBCHformula 64 2.2.3. Rooted trees and nonlinear differential equations 67 2.2.4. Flows and Hopf algebraic structures 71 2.3. Advances in chronological calculus 74 2.3.1. Chronological calculus and half-shuffles 75 2.3.2. Chronological calculus and pre-Lie products 79 2.3.3. Time-ordered products and enveloping algebras 81 2.3.4. Formal flows and Hopf algebraic structures 83 2.4. Rota–Baxter algebras 87 2.4.1. Origin 87 2.4.2. Definition and examples 91 2.4.3. Related algebraic structures 95 2.4.4. Atkinson’s factorization and Bogoliubov’s recursion 101 2.4.5. Spitzer’s identity: commutative case 103 2.4.6. Free commutativeRota–Baxter algebras 107 2.4.7. Spitzer’s identity: noncommutative case 108 2.4.8. FreeRota–Baxter algebras 111 2.5. References 113 Chapter 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras 119Jean-Yves THIBON 3.1. Introduction 119 3.2. Classical symmetric functions 120 3.2.1. Symmetric polynomials 120 3.2.2. The Hopf algebra of symmetric functions 122 3.2.3. The λ-ringnotation 124 3.2.4. Symmetric functions and formal power series 125 3.2.5. Duality 126 3.3. Noncommutativesymmetric functions 129 3.3.1. Basic definitions 129 3.3.2. Generators andlinear bases 131 3.3.3. Duality 133 3.3.4. Solomon’sdescent algebras 136 3.4. Lie series andLie idempotents 139 3.4.1. Permutational operators on tensor spaces 139 3.4.2. TheHausdorff series 139 3.4.3. Lie idempotents in the descent algebra 143 3.5. Lie idempotents as noncommutative symmetric functions 144 3.5.1. Noncommutativepower-sums 144 3.5.2. The Magnus expansion 146 3.5.3. The continuous BCH expansion 148 3.5.4. Another proof of the Magnus expansion 150 3.5.5. The (1 − q)-transform 150 3.5.6. Hopf algebras enter the scene 151 3.5.7. A one-parameter family of Lie idempotents 152 3.5.8. The iterated q-bracketing and its diagonalization 153 3.6. Decompositionsof the descent algebras 155 3.6.1. Complete families of minimal orthogonal idempotents 155 3.6.2. Eulerianidempotents 156 3.6.3. GeneralizedEulerianidempotents 160 3.7. Decompositionsof the tensor algebra 160 3.8. General deformations 162 3.9. Lie quasi-idempotents as Lie polynomials 163 3.9.1. The left derivative 163 3.9.2. Multilinear Lie polynomials 164 3.9.3. Decompositions on other bases 167 3.10. Permutations and free quasi-symmetric functions 169 3.10.1. Free quasi-symmetricfunctions 169 3.11. Packed words and word quasi-symmetric functions 171 3.12. References 175 Chapter 4. From Runge–Kutta Methods to Hopf Algebras of Rooted Trees 179Ander MURUA 4.1. Numerical integration methods for ordinary differential equations 179 4.1.1. Introduction 179 4.1.2. Runge–Kutta methods 180 4.2. Algebraic theory of Runge–Kutta methods 182 4.2.1. The order conditions of RK methods 182 4.2.2. The independence of order conditions 186 4.2.3. Proof of necessary and sufficient order conditions 188 4.2.4. Composition of RK methods, rooted trees and forests 191 4.2.5. TheButchergroup 195 4.2.6. Equivalence classes of RK methods 197 4.2.7. Bibliographicalcomments 198 4.3. B-series and relatedformal expansions 198 4.3.1. B-series 198 4.3.2. Backward error analysis, the exponential and the logarithm 199 4.3.3. Series of linear differentialoperators 203 4.3.4. The Lie algebra of the Butcher group 205 4.3.5. The pre-Lie algebra structure on g 206 4.3.6. Bibliographicalcomments 209 4.4. Hopf algebrasof rootedtrees 209 4.4.1. The commutative Hopf algebra of rooted trees 210 4.4.2. The dual algebra H∗ and the dual Hopf algebra H◦ 212 4.4.3. B-series and series of differential operators revisited 213 4.4.4. A universal property of the commutative Hopf algebra of rootedtrees 215 4.4.5. The substitution law 216 4.4.6. Bibliographicalcomments 217 4.5. References 217 Chapter 5. Combinatorial Algebra in Controllability and Optimal Control 221Matthias KAWSKI 5.1. Introduction 221 5.1.1. Motivation: idealized examples 223 5.1.2. Controlled dynamical systems 225 5.1.3. Fundamental questions in control 226 5.2. Analytic foundations 228 5.2.1. State-space models and vector fields on manifolds 228 5.2.2. Chronological calculus 230 5.2.3. Piecewise constant controls and theBaker–Campbell–Hausdorff formula 233 5.2.4. Picard iterationand formal series solutions 235 5.2.5. The Chen–Fliess series and abstractions 237 5.3. Controllability and optimality 241 5.3.1. Reachable sets and accessibility 241 5.3.2. Small-time local controllability 243 5.3.3. Nilpotent approximatingsystems 247 5.3.4. Optimality and the maximum principle 251 5.3.5. Control variations and approximating cones 255 5.4. Product expansions and realizations 262 5.4.1. Variation of parameters and exponential products 263 5.4.2. Computations using Zinbiel products 267 5.4.3. Exponential products and normal forms for nilpotent systems 269 5.4.4. Logarithmof theChen series 273 5.5. References 279 Chapter 6. Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application 287Alexander SCHMEDING 6.1. The Butcher group and the Connes–Kreimer algebra 288 6.1.1. The Butcher group and B-series from numerical analysis 288 6.1.2. Beyond the Butcher group 291 6.2. Character groups of graded and connected Hopf algebras 292 6.2.1. The exponential and logarithm 294 6.3. Controlled groups of characters 297 6.3.1. Conventions for this section 297 6.3.2. Combinatorial Hopf algebras and the inverse factorial character 304 6.4. Appendix: Calculus in locally convex spaces 305 6.4.1. Cr-Manifolds and Cr-mappingsbetween them 306 6.5. References 306 List of Authors 311 Index 313

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Book SynopsisThisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].Trade ReviewFrom the reviews: “The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. … This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions … . The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful.” (Felipe Zaldivar, The Mathematical Association of America, March, 2010)“One of the great achievements of mathematics was the classification of the finite simple groups … . the book brings much information to the classroom. It contains exactly those things one would like to know if one were to meet the individual simple groups for the first time. … perfectly suitable for an advanced course or seminar. … accessible also to those who are not great experts in group theory. Anyone interested in finite groups … should have this book on his or her bookshelf.” (Gernot Stroth, Mathematical Reviews, Issue 2011 e)“The author of this book has succeeded in giving an overview of all non-abelian finite simple groups which is accessible to non-experts. … For anyone who wants to get information on finite simple groups without having to tackle massive monographs this volume will be most welcome.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)“It is the first text at this level in which all the finite simple groups are treated together, pointing out their connections. … The text is very well organised. The introduction, which forms the first chapter, contains a brief history and the statement of the classification theorem, together with sections giving remarks on the applications and the proof of the theorem. … Consequently the book may also be useful to a reader who just wants an introduction to a particular group or family of groups.” (Peter Shiu, The Mathematical Gazette, Vol. 95 (532), March, 2011)“This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra. The section entitled ‘Further reading’ at the end of each chapter is a nice guide to further study of the subjects.” (Hiromichi Yamada, Zentralblatt MATH, Vol. 1203, 2011)Table of ContentsThe alternating groups.- The classical groups.- The exceptional groups.- The sporadic groups.

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Book SynopsisThis book concerns the use of dioid algebra as (max, +) algebra to treat the synchronization of tasks expressed by the maximum of the ends of the tasks conditioning the beginning of another task – a criterion of linear programming. A classical example is the departure time of a train which should wait for the arrival of other trains in order to allow for the changeover of passengers. The content focuses on the modeling of a class of dynamic systems usually called “discrete event systems” where the timing of the events is crucial. Events are viewed as sudden changes in a process which is, essentially, a man-made system, such as automated manufacturing lines or transportation systems. Its main advantage is its formalism which allows us to clearly describe complex notions and the possibilities to transpose theoretical results between dioids and practical applications.Table of ContentsChapter 1 Introduction 1 1.1 General introduction 1 1.2 History and three mainstays 2 1.3 Scientific context 2 1.3.1 Dioids 3 1.3.2 Petri nets 4 1.3.3 Time and algebraic models 5 1.4 Organization of the book 7 Chapter 2 Consistency 9 2.1 Introduction 9 2.1.1 Models 9 2.1.2 Physical point of view 11 2.1.3 Objectives 12 2.2 Preliminaries 14 2.3 Models and principle of the approach 17 2.3.1 P-time event graphs 17 2.3.2 Dater form 21 2.3.3 Principle of the approach (example 2) 23 2.4 Analysis in the “static” case 25 2.5 “Dynamic” model 28 2.6 Extremal acceptable trajectories by series of matrices 31 2.6.1 Lowest state trajectory 32 2.6.2 Greatest state trajectory 35 2.7 Consistency 36 2.7.1 Example 3 41 2.7.2 Maximal horizon of temporal consistency 44 2.7.3 Date of the first token deaths 47 2.7.4 Computational complexity 48 2.8 Conclusion 50 Chapter 3 Cycle Time 53 3.1 Objectives 53 3.2 Problem without optimization 55 3.2.1 Objective 55 3.2.2 Matrix expression of a P-time event graph 56 3.2.3 Matrix expression of P-time event graphs with interdependent residence durations 57 3.2.4 General form Ax ≤ b 59 3.2.5 Example 60 3.2.6 Existence of a 1-periodic behavior 61 3.2.7 Example continued 65 3.3 Optimization 67 3.3.1 Approach 1 67 3.3.2 Example continued 69 3.3.3 Approach 2 70 3.4 Conclusion 75 3.5 Appendix 76 Chapter 4 Control with Specifications 79 4.1 Introduction 79 4.2 Time interval systems 80 4.2.1 (min, max, +) algebraic models 81 4.2.2 Timed event graphs 82 4.2.3 P-time event graphs 83 4.2.4 Time stream event graphs 84 4.3 Control synthesis 88 4.3.1 Problem 88 4.3.2 Pedagogical example: education system 89 4.3.3 Algebraic models 91 4.4 Fixed-point approach 92 4.4.1 Fixed-point formulation 92 4.4.2 Existence 95 4.4.3 Structure 103 4.5 Algorithm 107 4.6 Example 111 4.6.1 Models 111 4.6.2 Fixed-point formulation 113 4.6.3 Existence 114 4.6.4 Optimal control with specifications 116 4.6.5 Initial conditions 117 4.7 Conclusion 118 Chapter 5 Online Aspect of Predictive Control 119 5.1 Introduction 119 5.1.1 Problem 119 5.1.2 Specific characteristics 120 5.2 Control without desired output (problem 1) 122 5.2.1 Objective 122 5.2.2 Example 1 123 5.2.3 Trajectory description 124 5.2.4 Relaxed system 125 5.3 Control with desired output (problem 2) 127 5.3.1 Objective 127 5.3.2 Fixed-point form 128 5.3.3 Relaxed system 129 5.4 Control on a sliding horizon (problem 3): online and offline aspects 130 5.4.1 CPU time of the online control 131 5.5 Kleene star of the block tri-diagonal matrix and formal expressions of the sub-matrices 132 5.6 Conclusion 138 Bibliography 141 List of Symbols 149 Index 153

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Book SynopsisRecent years have seen a significant rise of interest in max-linear theory and techniques. Specialised international conferences and seminars or special sessions devoted to max-algebra have been organised. This book aims to provide a first detailed and self-contained account of linear-algebraic aspects of max-algebra for general (that is both irreducible and reducible) matrices. Among the main features of the book is the presentation of the fundamental max-algebraic theory (Chapters 1-4), often scattered in research articles, reports and theses, in one place in a comprehensive and unified form. This presentation is made with all proofs and in full generality (that is for both irreducible and reducible matrices). Another feature is the presence of advanced material (Chapters 5-10), most of which has not appeared in a book before and in many cases has not been published at all. Intended for a wide-ranging readership, this book will be useful for anyone with basic mathematical knowledge (including undergraduate students) who wish to learn fundamental max-algebraic ideas and techniques. It will also be useful for researchers working in tropical geometry or idempotent analysis.Table of ContentsMax-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.

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Book SynopsisMost of the introductory courses on linear algebra develop the basic theory of finite­ dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num­ ber of illustrative and worked examples, as well as many exercises that are strategi­ cally placed throughout the text. Solutions to the exercises are also provided. Many applications of linear algebra require careful, and at times rather tedious, calculations by hand. Very often these are subject to error, so the assistance of a com­ puter is welcome. As far as computation in algebra is concerned, there are several packages available. Here we include, in the spirit of a tutorial, a chapter that gives 1 a brief introduction to the use of MAPLE in dealing with numerical and algebraic problems in linear algebra.Trade ReviewFrom the reviews of the first edition: MAA ONLINE "This book will be of interest to anyone who wishes to have a good grasp of linear llgebra and matrix theory. It can also be used as an advanced undergraduate textbook. Although this book does not treat infinite dimensional linear spaces, it provides the reader with a deep understanding of finite dimensional linear spaces. Many aspects of the theory of finite dimensional linear spaces can easily be generalized to the infinite dimensional case. Therefore, this book will also be helpful to those who intend to study infinite dimensional spaces later. " ZENTRALBLATT MATH "…it embodies a beautiful, concise and precise treatment of the subject, with succinct numerical and algebraic worked examples at the right points, and many exercises…This is an excellent textbook which, together with the earlier book, comprises a very nearly complete introduction to linear algebra which not only the undergraduate but also the advanced reader will enjoy studying." "The present book is a contribution of the volume Basic Linear Algebra of the same authors, which appeared in the same Springer series. … It is a very interesting, accessible book for undergraduates (or teacher professors), and the many examples and exercises should really help the undergraduate to study its contents." (Koen Thas, Bulletin of the Belgian Mathematical Society, Vol. 11 (4), 2004) "The book is a contribution of the authors’ Basic Linear Algebra published in the same series. … Besides numerous well-chosen examples scattered throughout the text, the reader can also enjoy short biographical profiles of twenty one eminent mathematicians associated with the subject." (European Mathematical Society Newsletter, December, 2003) "The present book is a natural sequel to the same authors’ successful SUMS volume ‘Basic Linear Algebra’. The most advanced topics here take the reader to the very heart of the subject … . To sum up, this textbook is well suited for self-study or for a one- or two-semester course. Therefore, we warmly recommend it to undergraduate students studying as well as professors teaching Linear Algebra at any level." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 69, 2003) "Further Linear Algebra is a natural sequel to the authors’ highly acclaimed SUMS volume Basis Linear Algebra. … An introductory chapter recaps the prerequisites (for those readers unfamiliar with the first volume), and a wide range of worked examples and exercises (with solutions) are strategically placed throughout the text to consolidate understanding." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)Table of ContentsThe story so far.- 1. Inner Product Spaces.- 2. Direct Sums of Subspaces.- 3. Primary Decomposition.- 4. Reduction to Triangular Form.- 5. Reduction to Jordan Form.- 6. Rational and Classical Forms.- 7. Dual Spaces.- 8. Orthogonal Direct Sums.- 9. Bilinear and Quadratic Forms.- 10. Real Normality.- 11. Computer Assistance.- 12. …. but who were they?.- 13. Solutions to the Exercises.

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Book SynopsisThis book offers a first taste of the theory of Lie groups, focusing mainly on matrix groups: closed subgroups of real and complex general linear groups. The first part studies examples and describes classical families of simply connected compact groups. The second section introduces the idea of a lie group and explores the associated notion of a homogeneous space using orbits of smooth actions. The emphasis throughout is on accessibility. Trade ReviewFrom the reviews of the first edition: MATHEMATICAL REVIEWS "This excellent book gives an easy introduction to the theory of Lie groups and Lie algebras by restricting the material to real and complex matrix groups. This provides the reader not only with a wealth of examples, but it also makes the key concepts much more concrete. This combination makes the material in this book more easily accessible for the readers with a limited background…The book is very easy to read and suitable for an elementary course in Lie theory aimed at advanced undergraduates or beginning graduate students…To summarize, this is a well-written book, which is highly suited as an introductory text for beginning graduate students without much background in differential geometry or for advanced undergraduates. It is a welcome addition to the literature in Lie theory." "This book is an introduction to Lie group theory with focus on the matrix case. … This book can be recommended to students, making Lie group theory more accessible to them." (A. Akutowicz, Zentralblatt MATH, Vol. 1009, 2003)Table of ContentsI. Basic Ideas and Examples.- 1. Real and Complex Matrix Groups.- 2. Exponentials, Differential Equations and One-parameter Subgroups.- 3. Tangent Spaces and Lie Algebras.- 4. Algebras, Quaternions and Quaternionic Symplectic Groups.- 5. Clifford Algebras and Spinor Groups.- 6. Lorentz Groups.- II. Matrix Groups as Lie Groups.- 7. Lie Groups.- 8. Homogeneous Spaces.- 9. Connectivity of Matrix Groups.- III. Compact Connected Lie Groups and their Classification.- 10. Maximal Tori in Compact Connected Lie Groups.- 11. Semi-simple Factorisation.- 12. Roots Systems, Weyl Groups and Dynkin Diagrams.- Hints and Solutions to Selected Exercises.

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Book SynopsisBasic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers: this will take the form of a tutorial on the use of the "LinearAlgebra" package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book.Trade ReviewFrom the reviews: "It embodies a beautiful, concise and precise treatment of the subject, with succint numerical and algebra worked examples at the right points... an excellent textbook which is also eminently suitable for self-study..." Zentralblatt MATH "This is the second edition of a text for first-year students which covers the main themes of linear algebra in a succinct and readable way. … The book is well-written, with a very high standard of proof-reading, and there are full answers to all the exercises … . It should be welcomed as an excellent introductory textbook which could be used either for self-study and to complement a course of lectures." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)Table of ContentsPreface Forward The Algebra of Matrices Some Applications of Matrices Systems of Linear Equations Invertible Matrices Vector Spaces Linear Mappings The Matrix Connection Determinants Eigenvalues and Eigenvectors The Minimum Polynomial Computer Assistance Solutions to the Exercises index

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Book SynopsisA modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study Previous books by Howie in the SUMS series have attracted excellent reviews Trade ReviewFrom the reviews:“This is a short but very good introductory book on abstract algebra, with emphasis on Galois Theory. Very little background in mathematics is required, so that the potential audience for this book range from undergraduate and graduate students, researchers, computer professionals, and the math enthusiasts.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)"The author wrote this book to provide the reader with a treatment of classical Galois theory. … The book is well written. It contains many examples and over 100 exercises with solutions in the back of the book. Sprinkled throughout the book are interesting commentaries and historical comments. The book is suitable as a textbook for upper level undergraduate or beginning graduate students." (John N. Mordeson, Zentralblatt MATH, Vol. 1103 (5), 2007)"To write such a book on a widely known but genuinely non-trivial topic is a challenge. … J. M. Howie did exactly what it takes. And he did it with such vigour and skill that the outcome is indeed absorbing and astounding. … Every paragraph has been scheduled with utmost care and the proofs are crystal clear. … the reader will never feel forlorn amidst brilliant theorems, which makes the book such a good read." (J. Lang, Internationale Mathematische Nachrichten, Issue 206, 2007)"Howie’s book ... provides a rigorous and thorough introduction to Galois theory. ... this book would be an excellent choice for anyone with at least some backgound in abstract algebra who seeks an introduction to the study of Galois theory. Summing Up: Highly recommended. Upper-division undergraduates; graduate students." (D. S. Larson, CHOICE, Vol. 43 (10), June, 2006)"The latest addition to Springer’s Undergraduate Mathematics Series is John Howie’s Fields and Galois Theory. … Howie is a fine writer, and the book is very self-contained. … I know that many of my students would appreciate Howie’s approach much more as it is not as overwhelming. This book also has a large number of good exercises, all of which have solutions in the back of the book. All in all, Howie has done a fine job writing a book on field theory … ." (Darren Glass, MathDL, February, 2006)"The book can serve as a useful introduction to the theory of fields and their extensions. The relevant background material on groups and rings is covered. The text is interspersed with many worked examples, as well as more than 100 exercises, for which solutions are provided at the end." (Chandan Singh Dalawat, Mathematical Reviews, Issue 2006 g)Table of ContentsRings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.

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Book SynopsisMuch of modern algebra arose from attempts to prove Fermat's Last Theorem, which in turn has its roots in Diophantus' classification of Pythagorean triples. This book, designed for prospective and practising mathematics teachers, makes explicit connections between the ideas of abstract algebra and the mathematics taught at high-school level. Algebraic concepts are presented in historical order, and the book also demonstrates how other important themes in algebra arose from questions related to teaching. The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalisations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the work of Galois and Abel. Results are motivated with specific examples, and applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions.Trade ReviewThis book covers abstract algebra from a historical perspective by using mathematics from attempts to prove Fermat's last theorem, as the title indicates. The target audience is high school mathematics teachers. However, typical undergraduate students will also derive great benefit by studying this text. The book is permeated with fascinating mathematical nuggets that are clearly explained." - D. P. Turner, CHOICE"This book is destined for college students in the U.S. who intend to teach mathematics in high school. The reviewer finds it even more apt as a text for algebra courses. Special features in the book are side notes given and printed prominently at the margins of the pages, like: How to think about it, Historical notes, Etymology of notions and words. … The reviewer considers the book a refreshing read among the vast amount of books dealing with similar topics." - Robert W. van der Waall, Zentrallblatt MATH"Although this book is designed for college students who want to teach in high school," its mathematical richness fits it admirably as a text for a first abstract algebra course or a handbook for assiduous students working on their own. While definitions, examples, theorems and their proofs are organized formally, the book is enhanced by substantial historical notes, advice on "how to think about it," marginal comments, connections and etymology that are designed to "balance experience and formality." The book is tightly organized with the goal of elucidating developments leading to the solution of the Fermat conjecture and the theory of solvability by radicals." - E. J. Barbeau, Mathematical Reviews"The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. … In writing this book, the authors have obviously kept the needs of the student reader firmly in mind at all times. The writing style is not just clear; iit is often conversational and humorous. … There are lots of exercises covering a wide range of difficulty, some with hints (but none with complete solutions) and there is a pretty good 39-entry bibliography. … What might be a very interesting use for this book would be as a text for a senior seminar or “topics” course for students who already have some prior exposure to abstract algebra. And, of course, whatever may be the applicability of this book as a text for undergraduate course, it seems clear to me that it belongs in any good undergraduate library." - Mark Hunacek, MAA ReviewsTable of ContentsPreface; Some features of this book; A note to students; A note to instructors; Notation; 1. Early number theory; 2. Induction; 3. Renaissance; 4. Modular arithmetic; 5. Abstract algebra; 6. Arithmetic of polynomials; 7. Quotients, fields, and classical problems; 8. Cyclotomic integers; 9. Epilogue; References; Index.

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book SynopsisThis book enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. High school students have the potential to explore it, and undergraduate students can master it. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.Table of ContentsBasic Concepts.- Euclidean 3D Geometric Algebra.- Applications.- Geometric Algebra and Matrices.- Appendix.- Solutions for Some Problems.- Problems.- Why Geometric Algebra?.- Formulae.- Literature.- References.

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Book SynopsisThis second volume in a two-volume series provides an extensive collection of conjectures and open problems in graph theory. It is designed for both graduate students and established researchers in discrete mathematics who are searching for research ideas and references. Each chapter provides more than a simple collection of results on a particular topic; it captures the reader’s interest with techniques that worked and failed in attempting to solve particular conjectures. The history and origins of specific conjectures and the methods of researching them are also included throughout this volume. Students and researchers can discover how the conjectures have evolved and the various approaches that have been used in an attempt to solve them. An annotated glossary of nearly 300 graph theory parameters, 70 conjectures, and over 600 references is also included in this volume. This glossary provides an understanding of parameters beyond their definitions and enables readers to discover new ideas and new definitions in graph theory. The editors were inspired to create this series of volumes by the popular and well-attended special sessions entitled “My Favorite Graph Theory Conjectures,” which they organized at past AMS meetings. These sessions were held at the winter AMS/MAA Joint Meeting in Boston, January 2012, the SIAM Conference on Discrete Mathematics in Halifax in June 2012, as well as the winter AMS/MAA Joint Meeting in Baltimore in January 2014, at which many of the best-known graph theorists spoke. In an effort to aid in the creation and dissemination of conjectures and open problems, which is crucial to the growth and development of this field, the editors invited these speakers, as well as other experts in graph theory, to contribute to this series.Table of Contents1. Desert Island Conjectures (L.W. Beineke).- 2. Binding Number, Cycles and Cliques ( W. Goddard).- 3. On a Conjecture Involving Laplacian Eigenvalues of Trees (D. P. Jacobs and V. Trevison).- 4. Queens Around the World in Twenty-five Years ( D. Weakley).- 5. Reflections on a Theme of Ulam (R.Graham).- 6. Ulam Numbers of Graphs (S.T. Hedetniemi).- 7. Forbidden Trees (D. Sumner).- 8. Some of My Favorite Conjectures: Local Conditions Implying Global Cycle Properties (O. Oellermann).- 9. The Path Partition Conjecture (M. Frick and J. E. Dunbar).- 10. To the Moon and Beyond (E. Gethner).- 11. My Favorite Domination Game Conjectures (M. A. Henning).- 12. A De Bruijn–Erdos theorem in graphs? (V. Chvatal).- 13. An Annotated Glossary of Graph Theory Parameters, with Conjectures (R. Gera, T. W. Haynes, S. T. Hedetniemi, and M. A. Henning).

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Book SynopsisThis book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more.As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level.The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.Table of Contents

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Book SynopsisThis English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas."This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander BeilinsonTrade Review“This slim volume is still a valuable introduction to schemes and nicely complements the textbooks on this topic which have appeared in the meantime.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“Throughout the text there are many instructive examples, remarks, and clarifying footnotes. The style of exposition is rather concise, very elegant, extremely lucid and enlightening, versatile, and – despite its venerable age of fifty years – absolutely modern and timely. … an excellent source for students, instructors, and mathematical physicists. No doubt, with this textbook, the mathematical community has another general standard reference in algebraic geometry at its disposal.” (Werner Kleinert, zbMATH 1390.14002, 2018)Table of ContentsEditor's Preface.- Author's Preface.- 1 Affine Schemes.- 2 Sheaves, Schemes, and Projective Spaces.- References.- Index.

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Book SynopsisThis textbook offers an innovative approach to abstract algebra, based on a unified treatment of similar concepts across different algebraic structures. This makes it possible to express the main ideas of algebra more clearly and to avoid unnecessary repetition.The book consists of two parts: The Language of Algebra and Algebra in Action. The unified approach to different algebraic structures is a primary feature of the first part, which discusses the basic notions of algebra at an elementary level. The second part is mathematically more complex, covering topics such as the Sylow theorems, modules over principal ideal domains, and Galois theory.Intended for an undergraduate course or for self-study, the book is written in a readable, conversational style, is rich in examples, and contains over 700 carefully selected exercises.Trade Review “This book can be also used by incoming graduate students to refresh their knowledge of Algebra before taking graduate courses. I highly recommend this book for a standard undergraduate algebra course, as well as to students interested in independent study.” (Louisa Catalano, MAA Reviews, July 21, 2019)Table of Contents

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Book SynopsisThis book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics.The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis. Trade Review“The book is carefully prepared and well presented, and I recommend the book … for students who have just mastered vector calculus and Maxwellian electromagnetism.” (Hirokazu Nishimura, zbMATH 1433.58001, 2020)Table of ContentsPrelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

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Book SynopsisGalois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.Trade Review“This book covers a lot of interesting material and is surely a valuable addition to the literature, but is certainly not for the timid. It brings together a broad array of sophisticated mathematics … and it does so in a very general and abstract way, with an exposition that gives whole new meaning to the word ‘concise’.” (Mark Hunacek, MAA Reviews, April 5, 2021)Table of ContentsIntroduction.- Chapter 1. Zorn’s Lemma.- Chapter 2. Categories and Functors.- Chapter 3. Linear Algebra.- Chapter 4. Coverings.- Chapter 5. Galois Theory.- Chapter 6. Riemann Surfaces.- Chapter 7. Dessins d’Enfants.- Bibliography.- Index of Notation

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Book SynopsisDieses Lehrbuch ist ein idealer Begleitband für eine vierstündige Vorlesung mit Übungen für angehende Naturwissenschaftlerinnen und Naturwissenschaftler, kann aber auch für eine Einführungsvorlesung in die höhere Mathematik in anderen Disziplinen eingesetzt werden. Aufbauend auf Vorkenntnissen aus dem Gymnasium werden zunächst die wichtigsten Begriffe nochmals repetiert. Im Hauptteil werden Vektoren, Differential- und Integralrechnung sowie Differentialgleichungen eingeführt und ausführlich behandelt. Abschließend wird auf Funktionen mehrerer Variablen eingegangen. Zahlreiche Übungsaufgaben mit Lösungen zu jedem Kapitel helfen, den Stoff zu festigen. Neben den Erklärungen für alle Leserinnen und Leser werden in speziell markierten Teilen weiterführende Fragen vertieft behandelt, welche nicht zwingend für das Verständnis notwendig sind, aber interessante Einblicke geben. Das Buch und Übungskonzept ist eine weitgehend überarbeitete Neuausgabe des Texts einer über ein Jahrzehnt erfolgreich gelehrten Vorlesung.Table of ContentsA. Vektorrechnung.- 1. Vektoren und ihre geometrische Bedeutung.- 2. Vektorrechnung mit Koordinaten.- B. Differentialrechnung.- 3. Beispiele zum Begriff der Ableitung.- 4. Die Ableitung.- 5. Technik des Differenzierens.- 6. Anwendungen der Ableitung.- 7. Linearisierung und das Differential.- 8. Die Ableitung einer Vektorfunktion.- C. Integralrechnung.- 9. Einleitende Beispiele zum Begriff des Integrals.- 10. Das bestimmte Integral.- 11. Der Hauptsatz der Differential- und Integralrechnung.- 12. Stammfunktionen und das unbestimmte Integral.- 13. Weitere Integrationsmethoden.- 14. Integration von Vektorfunktionen.- D. Differentialgleichungen.- 15. Der Begriff der Differentialgleichung.- 16. Einige Lösungsmethoden.- E. Ausbau der Infinitesimalrechnung.- 17. Umkehrfunktionen.- 18. Einige wichtige Funktionen und ihre Anwendungen.- 19. Potenzreihen.- 20. Uneigentliche Integrale.- 21. Numerische Methoden.- F. Funktionen von Mehreren Variablen.- 22. Allgemeines über Funktionen von mehreren Variablen.- 23. Differentialrechnung von Funktionen von mehreren Variablen.- 24. Das totale Differential.- 25. Mehrdimensionale Integrale.- G. Anhang.- 26. Zusammenstellung einiger Grundbegriffe.- 27. Einige Ergänzungen.- 28. Lösungen der Aufgaben.

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Book SynopsisThis textbook introduces linear algebra and optimization in the context of machine learning. Examples and exercises are provided throughout the book. A solution manual for the exercises at the end of each chapter is available to teaching instructors. This textbook targets graduate level students and professors in computer science, mathematics and data science. Advanced undergraduate students can also use this textbook. The chapters for this textbook are organized as follows:1. Linear algebra and its applications: The chapters focus on the basics of linear algebra together with their common applications to singular value decomposition, matrix factorization, similarity matrices (kernel methods), and graph analysis. Numerous machine learning applications have been used as examples, such as spectral clustering, kernel-based classification, and outlier detection. The tight integration of linear algebra methods with examples from machine learning differentiates this book from generic volumes on linear algebra. The focus is clearly on the most relevant aspects of linear algebra for machine learning and to teach readers how to apply these concepts.2. Optimization and its applications: Much of machine learning is posed as an optimization problem in which we try to maximize the accuracy of regression and classification models. The “parent problem” of optimization-centric machine learning is least-squares regression. Interestingly, this problem arises in both linear algebra and optimization, and is one of the key connecting problems of the two fields. Least-squares regression is also the starting point for support vector machines, logistic regression, and recommender systems. Furthermore, the methods for dimensionality reduction and matrix factorization also require the development of optimization methods. A general view of optimization in computational graphs is discussed together with its applications to back propagation in neural networks. A frequent challenge faced by beginners in machine learning is the extensive background required in linear algebra and optimization. One problem is that the existing linear algebra and optimization courses are not specific to machine learning; therefore, one would typically have to complete more course material than is necessary to pick up machine learning. Furthermore, certain types of ideas and tricks from optimization and linear algebra recur more frequently in machine learning than other application-centric settings. Therefore, there is significant value in developing a view of linear algebra and optimization that is better suited to the specific perspective of machine learning. Trade Review“Based on the topics covered and the excellent presentation, I would recommend Aggarwal's book over these other books for an advanced undergraduate or beginning graduate course on mathematics for data science.” (Brian Borchers, MAA Reviews, March 28, 2021)“This book should be of interest to graduate students in engineering, applied mathematics, and other fields requiring an understanding of the mathematical underpinnings of machine learning.” (IEEE Control Systems Magazine, Vol. 40 (6), December, 2020)Table of ContentsPreface.- 1 Linear Algebra and Optimization: An Introduction.- 2 Linear Transformations and Linear Systems.- 3 Eigenvectors and Diagonalizable Matrices.- 4 Optimization Basics: A Machine Learning View.- 5 Advanced Optimization Solutions.- 6 Constrained Optimization and Duality.- 7 Singular Value Decomposition.- 8 Matrix Factorization.- 9 The Linear Algebra of Similarity.- 10 The Linear Algebra of Graphs.- 11 Optimization in Computational Graphs.- Index.

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Book SynopsisThis book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.Trade Review“This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc.” (Piotr Pokora, zbMATH 1445.13001, 2020)Table of Contents- Part I Associated Primes of Powers of Ideals - Associated Primes of Powers of Ideals. - Associated Primes of Powers of Squarefree Monomial Ideals. - Final Comments and Further Reading. - Part II Regularity of Powers of Ideals. - Regularity of Powers of Ideals and the Combinatorial Framework. - Problems, Questions, and Inductive Techniques. - Examples of the Inductive Techniques. - Final Comments and Further Reading. - Part III The Containment Problem. - The Containment Problem: Background. - The Containment Problem. - The Waldschmidt Constant of Squarefree Monomial Ideals. - Symbolic Defect. - Final Comments and Further Reading. - Part IV Unexpected Hypersurfaces. - Unexpected Hypersurfaces. - Final Comments and Further Reading.

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Book SynopsisThis book contains 296 exercises and solutions covering a wide variety of topics in linear model theory, including generalized inverses, estimability, best linear unbiased estimation and prediction, ANOVA, confidence intervals, simultaneous confidence intervals, hypothesis testing, and variance component estimation. The models covered include the Gauss-Markov and Aitken models, mixed and random effects models, and the general mixed linear model. Given its content, the book will be useful for students and instructors alike. Readers can also consult the companion textbook Linear Model Theory - With Examples and Exercises by the same author for the theory behind the exercises.Trade Review“This volume contains solutions to the book's exercises … Many of those exercises stand as useful applications of results stated in the theory volume. Some of them go one step beyond and extend the theoretical results. I found this to be a very interesting and unique feature of the book on linear models, making the whole set particularly useful for both graduate students and instructors.” (Vassilis G. S. Vasdekis, Mathematical Reviews, August 2022)Table of Contents1 A Brief Introduction.- 2 Selected Matrix Algebra Topics and Results.- 3 Generalized Inverses and Solutions to Sytems of Linear Equations.- 4 Moments of a Random Vector and of Linear and Quadratic Forms in a Random Vector.- 5 Types of Linear Models.- 6 Estimability.- 7 Least Squares Estimation for the Gauss-Markov Model.- 8 Least Squares Geometry and the Overall ANOVA.- 9 Least Squares Estimation and ANOVA for Partitioned Models.- 10 Constrained Least Squares Estimation and ANOVA.- 11 Best Linear Unbiased Estimation for the Aitken Model.- 12 Model Misspecification.- 13 Best Linear Unbiased Prediction.- 14 Distribution Theory.- 15 Inference for Estimable and Predictable Functions.- 16 Inference for Variance-Covariance Parameters.- 17 Empirical BLUE and BLUP.

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Book SynopsisThis book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group. The main components are: - a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism, - the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals, - algebraic representation theory in terms of category O, and - analytic representation theory of quantized complex semisimple groups. Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.Trade Review“The book is largely self-contained. … It is highly recommended for mathematicians of all levels wishing to learn about these topics, in the algebraic setting and/or in the analytic setting.” (Huafeng Zhang, zbMATH 1514.20006, 2023)Table of Contents- Introduction. - Multiplier Hopf Algebras. - Quantized Universal Enveloping Algebras. - Complex Semisimple Quantum Groups. - Category O. - Representation Theory of Complex Semisimple Quantum Groups.

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Book SynopsisThis textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques. Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section. Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.Trade Review“The book is well-organized. The main notions and results are well-presented, followed by a discussion and problems with detailed solutions. There are many helpful notes and examples. At the end of each section, the reader can frequently find several computational, true/false, or proof exercises. … There are several illustrative and colorful figures. For instance, those illustrating the examples and remarks about the Gershgorin disc theorem or about the geometric interpretation of the positive semidefiniteness are really helpful.” (Carlos M. da Fonseca, zbMATH 1471.15001, 2021)Table of ContentsChapter 1: Vector Spaces.- Chapter 2: Matrix Decompositions.- Chapter 3: Tensors and Multilinearity.- Appendix A: Mathematical Preliminaries.- Appendix B: Additional Proofs.- Appendix C: Selected Exercise Solutions.

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Book SynopsisThis textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques. Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section. Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.Trade Review“The book is well-organized. The main notions and results are well-presented, followed by a discussion and problems with detailed solutions. There are many helpful notes and examples. At the end of each section, the reader can frequently find several computational, true/false, or proof exercises. … There are several illustrative and colorful figures. For instance, those illustrating the examples and remarks about the Gershgorin disc theorem or about the geometric interpretation of the positive semidefiniteness are really helpful.” (Carlos M. da Fonseca, zbMATH 1471.15001, 2021)Table of ContentsChapter 1: Vector Spaces.- Chapter 2: Matrix Decompositions.- Chapter 3: Tensors and Multilinearity.- Appendix A: Mathematical Preliminaries.- Appendix B: Additional Proofs.- Appendix C: Selected Exercise Solutions.

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Book SynopsisThis open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.Trade Review“The book contains a huge amount of interesting and very well-chosen exercises. … This ‘encyclopedic’ character of the text may play an important role both as a guide to some special topics and as a source of information for both students and those whose research in related fields creates a need to familiarize themselves with the knowledge of the case when quaternion algebras are relevant.” (Juliusz Brzeziński, Mathematical Reviews, September, 2022)Table of Contents1. Introduction.- 2. Beginnings.- 3. Involutions.- 4. Quadratic Forms.- 5. Ternary Quadratic Forms.- 6. Characteristic 2.- 7. Simple Algebras.- 8. Simple Algebras and Involutions.- 9. Lattices and Integral Quadratic Forms.- 10. Orders.- 11. The Hurwitz Order.- 12. Ternary Quadratic Forms Over Local Fields.- 13. Quaternion Algebras Over Local Fields.- 14. Quaternion Algebras Over Global Fields.- 15. Discriminants.- 16. Quaternion Ideals and Invertability.- 17. Classes of Quaternion Ideals.- 18. Picard Group.- 19. Brandt Groupoids.- 20. Integral Representation Theory.- 21. Hereditary and Extremal Orders.- 22. Ternary Quadratic Forms.- 23. Quaternion Orders.- 24. Quaternion Orders: Second Meeting.- 25. The Eichler Mass Formula.- 26. Classical Zeta Functions.- 27. Adelic Framework.- 28. Strong Approximation.- 29. Idelic Zeta Functions.- 30. Optimal Embeddings.- 31. Selectivity.- 32. Unit Groups.- 33. Hyperbolic Plane.- 34. Discrete Group Actions.- 35. Classical Modular Group.- 36. Hyperbolic Space.- 37. Fundamental Domains.- 38. Quaternionic Arithmetic Groups.- 39. Volume Formula.- 40. Classical Modular Forms.- 41. Brandt Matrices.- 42. Supersingular Elliptic Curves.- 43. Abelian Surfaces with QM.

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Book SynopsisThis volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference “Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification”, held on the occasion of Chari’s 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari’s significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including: String diagrams and categorification Quantum affine algebras and cluster algebras Steinberg groups for Jordan pairs Dynamical quantum determinants and Pfaffians Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.Table of ContentsPublications of Vyjayanthi Chari.- Students of Vyjayanthi Chari.- Part I: Courses.- String Diagrams and Categorification.- Quantum Affine Algebras and Cluster Algebras.- Part II: Surveys.- Work of Vyjayanthi Chari.- Steinberg Groups for Jordan Pairs - An Introduction with Open Problems.- On the Hecke-Algebraic Approach for General Linear Groups over a p-adic Field.- Part III: Papers.- Categorical Representations and Classical p-adic Groups.- Formulae of l-Divided Powers in Uq(sl2),II.- Longest Weyl Group Elements in Action.- Dual Kashiwara Functions for the B(∞) Crystal in the Bipartite Case.- Lusztig's t-Analogue of weight multiplicity via Crystals.- Conormal Varieties on the Cominuscule Grassmannian.- Evaluation Modules for Quantum Toroidal gln Algebras.- Dynamical Quantum Determinants and Pfaffians.

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