Algebra Books

Book SynopsisMarge Lial (late) was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. Her interest in archeology led to trips to various digs and ruin sites, producing some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics education or journalism. His ultimate decision was to become a teacher, but afTable of ContentsR. Review of Basic Concepts R.1. Fractions, Decimals, and Percents R.2. Sets and Real Numbers R.3. Real Number Operations and Properties R.4. Integer and Rational Exponents R.5. Polynomials R.6. Factoring Polynomials R.7. Rational Expressions R.8. Radical Expressions 1. Equations and Inequalities 1.1. Linear Equations 1.2. Applications and Modeling with Linear Equations 1.3. Complex Numbers 1.4. Quadratic Equations 1.5. Applications and Modeling with Quadratic Equations 1.6. Other Types of Equations and Applications 1.7. Inequalities 1.8. Absolute Value Equations and Inequalities 2. Graphs and Functions 2.1. Rectangular Coordinates and Graphs 2.2. Circles 2.3. Functions 2.4. Linear Functions 2.5. Equations of Lines and Linear Models 2.6. Graphs of Basic Functions 2.7. Graphing Techniques 2.8. Function Operations and Composition 3. Polynomial and Rational Functions 3.1. Quadratic Functions and Models 3.2. Synthetic Division 3.3. Zeros of Polynomial Functions 3.4. Polynomial Functions: Graphs, Applications, and Models 3.5. Rational Functions: Graphs, Applications, and Models 3.6. Polynomial and Rational Inequalities 3.7. Variation 4. Inverse, Exponential, and Logarithmic Functions 4.1. Inverse Functions 4.2. Exponential Functions 4.3. Logarithmic Functions 4.4. Evaluating Logarithms and the Change-of-Base Theorem 4.5. Exponential and Logarithmic Equations 4.6. Applications of Models of Exponential Growth and Decay 5. Trigonometric Functions 5.1. Angles 5.2. Trigonometric Functions 5.3. Trigonometric Function Values and Angle Measures 5.4. Solutions and Applications of Right Triangles 6. The Circular Functions and Their Graphs 6.1. Radian Measure 6.2. The Unit Circle and Circular Functions 6.3. Graphs of the Sine and Cosine Functions 6.4. Translations of the Graphs of the Sine and Cosine Functions 6.5. Graphs of the Tangent and Cotangent Functions 6.6. Graphs of the Secant and Cosecant Functions 6.7. Harmonic Motion 7. Trigonometric Identities and Equations 7.1. Fundamental Identities 7.2. Verifying Trigonometric Identities 7.3. Sum and Difference Identities 7.4. Double-Angle and Half-Angle Identities 7.5. Inverse Circular Functions 7.6. Trigonometric Equations 7.7. Equations Involving Inverse Trigonometric Functions 8. Applications of Trigonometry 8.1. The Law of Sines 8.2. The Law of Cosines 8.3. Geometrically Defined Vectors and Applications 8.4. Algebraically Defined Vectors and the Dot Product 8.5. Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.6. De Moivre's Theorem; Powers and Roots of Complex Numbers 8.7. Polar Equations and Graphs 8.8. Parametric Equations, Graphs, and Applications 9. Systems and Matrices 9.1. Systems of Linear Equations 9.2. Matrix Solution of Linear Systems 9.3. Determinant Solution of Linear Systems 9.4. Partial Fractions 9.5. Nonlinear Systems of Equations 9.6. Systems of Inequalities and Linear Programming 9.7. Properties of Matrices 9.8. Matrix Inverses 10. Analytic Geometry 10.1. Parabolas 10.2. Ellipses 10.3. Hyperbolas 10.4. Summary of the Conic Sections 11. Further Topics in Algebra 11.1. Sequences and Series 11.2. Arithmetic Sequences and Series 11.3. Geometric Sequences and Series 11.4. The Binomial Theorem 11.5. Mathematical Induction 11.6. Basics of Probability Appendix A Polar Form of Conic Sections Appendix B Rotation of Axes Appendix C Geometry Formulas

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Book SynopsisAlgebra is the gateway to college and careers, yet it functions as the eye of the needle because of low pass rates for the middle school/high school course and students’ struggles to understand. We have forty years of research that discusses the ways students think and their cognitive challenges as they engage with algebra. This book is a response to the National Council of Teachers of Mathematics’ (NCTM) call to better link research and practice by capturing what we have learned about students’ algebraic thinking in a way that is usable by teachers as they prepare lessons or reflect on their experiences in the classroom. Through a Fund for the Improvement of Post-Secondary Education (FIPSE) grant, 17 teachers and mathematics educators read through the past 40 years of research on students’ algebraic thinking to capture what might be useful information for teachers to know—over 1000 articles altogether. The resulting five domains addressed in the book (Variables & Expressions, Algebraic Relations, Analysis of Change, Patterns & Functions, and Modeling & Word Problems) are closely tied to CCSS topics.Over time, veteran math teachers develop extensive knowledge of how students engage with algebraic concepts—their misconceptions, ways of thinking, and when and how they are challenged to understand—and use that knowledge to anticipate students’ struggles with particular lessons and plan accordingly. Veteran teachers learn to evaluate whether an incorrect response is a simple error or the symptom of a faulty or naïve understanding of a concept. Novice teachers, on the other hand, lack the experience to anticipate important moments in the learning of their students. They often struggle to make sense of what students say in the classroom and determine whether the response is useful or can further discussion (Leatham, Stockero, Peterson, & Van Zoest 2011; Peterson & Leatham, 2009). The purpose of this book is to accelerate early career teachers’ “experience” with how students think when doing algebra in middle or high school as well as to supplement veteran teachers’ knowledge of content and students. The research that this book is based upon can provide teachers with insight into the nature of the student’s struggles with particular algebraic ideas—to help teachers identify patterns that imply underlying thinking.Our book, How Students Think When Doing Algebra, is not intended to be a “how to” book for teachers. Instead, it is intended to orient new teachers to the ways students think and be a book that teachers at all points in their career continually pull of the shelf when they wonder, “how might my students struggle with this algebraic concept I am about to teach?” The primary audience for this book is early career mathematics teachers who don’t have extensive experience working with students engaged in mathematics. However, the book can also be useful to veteran teachers to supplement their knowledge and is an ideal resource for mathematics educators who are preparing preservice teachers.Table of Contents Chapter 1: Introduction Chapter 2: Variables and Expressions Chapter 3: Algebraic Relations Chapter 4: Analysis of Change (Graphing) Chapter 5: Patterns & Functions Chapter 6: Modeling and Word Problems

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Book SynopsisLinear algebra has become the subject to know for people in quantitative disciplines of all kinds. No longer the exclusive domain of mathematicians and engineers, it is now used everywhere there is data and everybody who works with data needs to know more. This new book from Professor Gilbert Strang, author of the acclaimed Introduction to Linear Algebra, now in its fifth edition, makes linear algebra accessible to everybody, not just those with a strong background in mathematics. It takes a more active start, beginning by finding independent columns of small matrices, leading to the key concepts of linear combinations and rank and column space. From there it passes on to the classical topics of solving linear equations, orthogonality, linear transformations and subspaces, all clearly explained with many examples and exercises. The last major topics are eigenvalues and the important singular value decomposition, illustrated with applications to differential equations and image compression. A final optional chapter explores the ideas behind deep learning.

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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 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 SynopsisThis volume 6 of the Collected Works comprises 27 papers by V.I.Arnold, one of the most outstanding mathematicians of all times, written in 1991 to 1995. During this period Arnold's interests covered Vassiliev’s theory of invariants and knots, invariants and bifurcations of plane curves, combinatorics of Bernoulli, Euler and Springer numbers, geometry of wave fronts, the Berry phase and quantum Hall effect. The articles include a list of problems in dynamical systems, a discussion of the problem of (in)solvability of equations, papers on symplectic geometry of caustics and contact geometry of wave fronts, comments on problems of A.D.Sakharov, as well as a rather unusual paper on projective topology. The interested reader will certainly enjoy Arnold’s 1994 paper on mathematical problems in physics with the opening by-now famous phrase “Mathematics is the name for those domains of theoretical physics that are temporarily unfashionable.” The book will be of interest to the wide audience from college students to professionals in mathematics or physics and in the history of science. The volume also includes translations of two interviews given by Arnold to the French and Spanish media. One can see how worried he was about the fate of Russian and world mathematics and science in general.Table of Contents1 Bernoulli–Euler updown numbers associated with function singularities, their combinatorics and arithmetics.- 2 Congruences for Euler, Bernoulli and Springer numbers of Coxeter groups.- 3 The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups.- 4 Springer numbers and Morsification spaces.- 5 Polyintegrable flows.- 6 Bounds for Milnor numbers of intersections in holomorphic dynamical systems.- 7 Some remarks on symplectic monodromy of Milnor fibrations.- 8 Topological properties of Legendre projections in contact geometry of wave fronts [On topological properties of Legendre projections in contact geometry of wave fronts].- 9 Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques [On topological properties of Lagrangian projections in symplectic geometry of caustics].- 10 Plane curves, their invariants, perestroikas and classifications (with an appendix by F. Aicardi).- 11 Invariants and perestroikas of plane fronts.- 12 The Vassiliev theory of discriminants and knots.- 13 The geometry of spherical curves and the algebra of quaternions.- 14 Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect.- 15 Problems on singularities and dynamical systems.- 16 Sur quelques problèmes de la théorie des systèmes dynamiques [On some problems in the theory of dynamical systems].- 17 Mathematical problems in classical physics.- 18 Problèmes résolubles et problèmes irrésolubles analytiques et géométriques [Solvable and unsolvable analytic and geometric problems].- 19 Projective topology.- 20 Questions à V.I. Arnold (an interview with M. Audin and P. Iglésias) [Questions to V.I. Arnold].- 21 En todo matemático hay un ángel y un demonio (an interview with Marimar Jiménez) [In every mathematician, there is an angel and a devil].- 22 Will Russian mathematics survive?.- 23 Will mathematics survive? Report on the Zurich Congress.- 24 Why study mathematics? What mathematicians think about it.- 25 Preface to the Russian translation of the book by M.F. Atiyah “The Geometry and Physics of Knots”.- 26 A comment on one of A.D. Sakharov’s “Amateur Problems”.- 27 Comments on two of A.D. Sakharov’s “Amateur Problems”.- Acknowledgements.

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Book SynopsisNow available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.From the reviews of previous editions:Altogether, the text is a didactic masterpiece. — zbMATHThe determinant-free proofs are elegant and intuitive. — American Mathematical MonthlyThe most original linear algebra book to appear in years, it certainly belongs in every undergraduate library — CHOICETable of ContentsPreface for the Instructor-Preface for the Student-Acknowledgments-1. Vector Spaces.- 2. Finite-Dimensional Vector Spaces.- 3. Linear Maps.- 4. Polynomials.- 5. Eigenvalues, Eigenvectors, and Invariant Subspaces.- 6. Inner Product Spaces.- 7. Operators on Inner Product Spaces.- 8. Operators on Complex Vector Spaces.- 9. Operators on Real Vector Spaces.- 10. Trace and Determinant-Photo Credits-Symbol Index-Index.

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Book SynopsisThe seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins’ classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.Trade ReviewFrom the reviews:“The book is aimed at a readership with a solid background in algebra, in particular representation theory, commutative algebra and homological algebra. The volume comprises five chapters and an appendix, and each chapter is divided into four sections. Each chapter consists of the lecture material and the exercises handled during one day at the Oberwolfach Seminar (in 2010) with the same title. … The book ends with an appendix … and there is a comprehensive bibliography.” (Nadia P. Mazza, Mathematical Reviews, March, 2013)“The manuscript under review provides a quite nice introduction to the tools used in these classification theorems and offers an excellent starting point for someone new to the area. The manuscript is based on a week-long series of lectures given by the authors to introduce people to the ideas involved in the proof of the classification of localising subcategories of Mod(kG).” (Christopher P. Bendel, Zentralblatt MATH, Vol. 1246, 2012)Table of ContentsPreface.- 1 Monday.- 1.1 Overview.- 1.2 Modules over group algebras.- 1.3 Triangulated categories.- 1.4 Exercises.- 2 Tuesday.- 2.1 Perfect complexes over commutative rings.- 2.2 Brown representability and localization.- 2.3 The stable module category of a finite group.- 2.4 Exercises.- 3 Wednesday.- 3.1.- 3.2 Koszul objects and support.- 3.3 The homotopy category of injectives.- 3.4 Exercises.- 4 Thursday.- 4.1 Stratifying triangulated categories.- 4.2 Consequences of stratification.- 4.3 The Klein four group.- 4.4 Exercises.- 5 Friday.- 5.1 Localising subcategories of D(A).- 5.2 Elementary abelian 2-groups.- 5.3 Stratification for arbitrary finite groups.- 5.4 Exercises.- A Support for modules over commutative rings.- Bibliography.- Index.

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Book SynopsisThe book contains seven refereed research papers on locally compact quantum groups and groupoids by leading experts in the respective fields. These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg from February 21 to February 23, 2002. Topics covered are: various constructions of locally compact quantum groups and their multiplicative unitaries; duality theory for locally compact quantum groups; combinatorial quantization of flat connections associated with SL(2,c); quantum groupoids, especially coming from Depth 2 Extensions of von Neumann algebras, C*-algebras and Rings. Many mathematical results are motivated by problems in theoretical physics. Historical remarks set the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume will give an overview of a field of research in which great progress has been achieved in the last few years, with new ties to many other areas of mathematics and physics.Table of ContentsIntroduction · M. Enock: Quantum groupoids and pseudo-multiplicative unitaries · E. Koelink and J. Kustermans: Quantum SU(1,1) and its Pontryagin dual · P. Schauenburg: Morita base change in quantum groupoids · K. Szlachanyi: Galois actions by finite quantum groupoids · S. Vaes and L. Vainerman: On Low-Dimensional Locally Compact Quantum Groups · J.-M. Vallin: Multiplicative partial isometries and finite quantum groupoids · A. Van Daele: Mulitplier Hopf *-algebras with positive integrals: A laboratory for locally compact quantum groups

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Book SynopsisThis volume contains nine refereed research papers in various areas from combinatorics to dynamical systems, with computer algebra as an underlying and unifying theme. Topics covered include irregular connections, rank reduction and summability of solutions of differential systems, asymptotic behaviour of divergent series, integrability of Hamiltonian systems, multiple zeta values, quasi-polynomial formalism, Padé approximants related to analytic integrability, hybrid systems. The interactions between computer algebra, dynamical systems and combinatorics discussed in this volume should be useful for both mathematicians and theoretical physicists who are interested in effective computation.

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Book SynopsisThis volume is the proceedings of the conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, held at the Department of Mathematics of The Ohio State University, March 27–30, 2003, in honor of the 60th birthday of Steve Rallis. The theory of automorphic representations, automorphic L-functions and their applications to arithmetic continues to be an area of vigorous and fruitful research. The contributed papers in this volume represent many of the most recent developments and directions, including Rankin–Selberg L-functions (Bump, Ginzburg–Jiang–Rallis, Lapid–Rallis) the relative trace formula (Jacquet, Mao–Rallis) automorphic representations (Gan–Gurevich, Ginzburg–Rallis–Soudry) representation theory of p-adic groups (Baruch, Kudla–Rallis, Mœglin, Cogdell–Piatetski-Shapiro–Shahidi) p-adic methods (Harris–Li–Skinner, Vigneras), and arithmetic applications (Chinta–Friedberg–Hoffstein). The survey articles by Bump, on the Rankin–Selberg method, and by Jacquet, on the relative trace formula, should be particularly useful as an introduction to the key ideas about these important topics. This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field.Trade Review"This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field."L'Enseignement Mathematique 3-4/2005

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Book SynopsisThis volume contains survey articles and original research papers, presenting the state of the art on applying the symbolic approach of Gröbner bases and related methods to differential and difference equations. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods hosted by the Johann Radon Institute of Computational and Applied Mathematics, Linz, Austria, in May 2006.Trade Review"Many a researchers and many a mathematics graduate will find its pages filled with a contagious enthusiasm packing a treasure of wisdom and delight into eleven chapters."In: Current Engeneering Pratice 2009

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Book SynopsisThis volume contains survey and original articles presenting the state of the art on the application of Gröbner bases in control theory and signal processing. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods at the Johann Radon Institute of Computational and Applied Mathematics (RICAM), Linz, Austria, in May 2006.Trade Review"In particular, Grobner bases proved to be a powerful tool in designing multidimensional filter banks and have shown a good potential for solving partial differential equations."In: EMS Newsletter 9/2008

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Book SynopsisThis is the second of three volumes devoted to elementary finite p-group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of p-groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2n > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of p-groups all of whose subgroups of index p2 are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) p-groups with cyclic subgroups of index p2 are classified. This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra.Trade Review"Again, buy these books and learn from it!"R.W. van der Waall in: Zentralblatt MATH 1168

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Book SynopsisThis is the third volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume: impact of minimal nonabelian subgroups on the structure of p-groups, classification of groups all of whose nonnormal subgroups have the same order, degrees of irreducible characters of p-groups associated with finite algebras, groups covered by few proper subgroups, p-groups of element breadth 2 and subgroup breadth 1, exact number of subgroups of given order in a metacyclic p-group, soft subgroups, p-groups with a maximal elementary abelian subgroup of order p2, p-groups generated by certain minimal nonabelian subgroups, p-groups in which certain nonabelian subgroups are 2-generator. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

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Book SynopsisEigenvalues and eigenvectors of matrices and linear operators play an important role when solving problems from structural mechanics and electrodynamics, e.g., by describing the resonance frequencies of systems, when investigating the long-term behavior of stochastic processes, e.g., by describing invariant probability measures, and as a tool for solving more general mathematical problems, e.g., by diagonalizing ordinary differential equations or systems from control theory. This textbook presents a number of the most important numerical methods for finding eigenvalues and eigenvectors of matrices. The authors discuss the central ideas underlying the different algorithms and introduce the theoretical concepts required to analyze their behavior with the goal to present an easily accessible introduction to the field, including rigorous proofs of all important results, but not a complete overview of the vast body of research. Several programming examples allow the reader to experience the behavior of the different algorithms first-hand. The book addresses students and lecturers of mathematics, physics and engineering who are interested in the fundamental ideas of modern numerical methods and want to learn how to apply and extend these ideas to solve new problems.

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Book SynopsisProgress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.Trade ReviewThis is an excellent introduction to the Rokhlin and Casson invariants for homology 3-spheres [...], and in particular also to the necessary background material from the theory of 3- and 4-manifolds [...], so the book may serve also as a reasonable short and efficient introduction to some important parts of low-dimensional topology. It grew out of a course for second year graduate students and concentrates 19 lectures on less than 200 pages, including also a glossary on back-ground material from algebraic topology, a collection of exercises, open problems and comments on recent developments [...] To conclude, the author has succeeded in presenting a lot of material in a clear and efficient way, and the book is interesting and stimulating to read. Birge Zimmermann-Huisgen, Zentralblatt MATHTable of ContentsPreface Introduction Glossary 1 Heegaard splittings 1.1 Introduction 1.2 Existence of Heegaard splittings 1.3 Stable equivalence of Heegaard splittings 1.4 The mapping class group 1.5 Manifolds of Heegaard genus ≤ 1 1.6 Seifert manifolds 1.7 Heegaard diagrams 2 Dehn surgery 2.1 Knots and links in 3-manifolds 2.2 Surgery on links in S3 2.3 Surgery description of lens spaces and Seifert manifolds 2.4 Surgery and 4-manifolds 3 Kirby calculus 3.1 The linking number 3.2 Kirby moves 3.3 The linking matrix 3.4 Reversing orientation 4 Even surgeries 5 Review of 4-manifolds 5.1 Definition of the intersection form 5.2 The unimodular integral forms 5.3 Four-manifolds and intersection forms 6 Four-manifolds with boundary 6.1 The intersection form 6.2 Homology spheres via surgery on knots 6.3 Seifert homology spheres 6.4 The Rohlin invariant 7 Invariants of knots and links 7.1 Seifert surfaces 7.2 Seifert matrices 7.3 The Alexander polynomial 7.4 Other invariants from Seifert surfaces 7.5 Knots in homology spheres 7.6 Boundary links and the Alexander polynomial 8 Fibered knots 8.1 The definition of a fibered knot 8.2 The monodromy 8.3 More about torus knots 8.4 Joins 8.5 The monodromy of torus knots 8.6 Open book decompositions 9 The Arf-invariant 9.1 The Arf-invariant of a quadratic form 9.2 The Arf-invariant of a knot 10 Rohlin’s theorem 10.1 Characteristic surfaces 10.2 The definition of ˜q 10.3 Representing homology classes by surfaces 11 The Rohlin invariant 11.1 Definition of the Rohlin invariant 11.2 The Rohlin invariant of Seifert spheres 11.3 A surgery formula for the Rohlin invariant 11.4 The homology cobordism group 12 The Casson invariant 13 The group SU(2) 14 Representation spaces 14.1 The topology of representation spaces 14.2 Irreducible representations 14.3 Representations of free groups 14.4 Representations of surface groups 14.5 Representations for Seifert homology spheres 15 The local properties of representation spaces 16 Casson’s invariant for Heegaard splittings 16.1 The intersection product 16.2 The orientations 16.3 Independence of Heegaard splitting 17 Casson’s invariant for knots 17.1 Preferred Heegaard splittings 17.2 The Casson invariant for knots 17.3 The difference cycle 17.4 The Casson invariant for boundary links 17.5 The Casson invariant of a trefoil 18 An application of the Casson invariant 18.1 Triangulating 4-manifolds 18.2 Higher-dimensional manifolds 19 The Casson invariant of Seifert manifolds 19.1 The space R(p; q; r) 19.2 Calculation of the Casson invariant Conclusion Bibliography Index

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Book SynopsisGreen's functions represent one of the classical and widely used issues in the area of differential equations. This monograph is looking at applied elliptic and parabolic type partial differential equations in two variables. The elliptic type includes the Laplace, static Klein-Gordon and biharmonic equation. The parabolic type is represented by the classical heat equation and the Black-Scholes equation which has emerged as a mathematical model in financial mathematics. The book is attractive for practical needs: It contains many easily computable or computer friendly representations of Green's functions, includes all the standard Green's functions and many novel ones, and provides innovative and new approaches that might lead to Green's functions. The book is a useful source for everyone who is studying or working in the fields of science, finance, or engineering that involve practical solution of partial differential equations.Trade Review"[...] the book is a useful source for everyone who is studying or working in fields of science, finance, or engineering that involve practical solutions of partial differential equations."In: Enseignement Mathematique 2/2012

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Book SynopsisThe problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This is the last of three volumes. In this monograph the proof of the Classification Theorem presented in the first volume is concluded. It collects all the important results on the topic which can be found only in scattered scientific literature so far.

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Book SynopsisThis book focuses on the dynamic complexity of neural, genetic networks, and reaction diffusion systems. The author shows that all robust attractors can be realized in dynamics of such systems. In particular, a positive solution of the Ruelle-Takens hypothesis for on chaos existence for large class of reaction-diffusion systems is given. The book considers viability problems for such systems - viability under extreme random perturbations - and discusses an interesting hypothesis of M. Gromov and A. Carbone on biological evolution. There appears a connection with the Kolmogorov complexity theory. As applications, transcription-factors-microRNA networks are considered, patterning in biology, a new approach to estimate the computational power of neural and genetic networks, social and economical networks, and a connection with the hard combinatorial problems.Table of ContentsComplexity and evolution of spatially extended systems: analytical approach Chapter 1: Introduction Dynamical systems Attractors Strange attractors Neural and genetic networks Reaction diffusion systems Systems with random perturbations and Gromov-Carbone problem Chapter 2: Method to control dynamics: Invariant manifolds, realization of vector fields Invariant manifolds Method of realization of vector fields Control of attractor and inertial dynamics for neural networks Chapter 3: Complexity of patterns and attractors in genetic networks Centralized networks and attractor complexity in such network A connection with computational problems, Turing machines and finite automatons Graph theory, graph growth and computational power of neural and genetical networks Mathematical model that shows how positional information can be transformed into body plan of multicellular organism Applications to TF- microRNA networks. Bifurcation complexity in networks Chapter 4: Viability problem, Robustness under noise and evolution Here we consider neural and genetic networks under large random perturbations Viability problem We show that network should evolve to be viable, and network complexity should increase A connection with graph growth theory (Erdos-Renyi, Albert-Barabasi) Relation between robustness, attractor complexity and functioning speed Why Stalin and Putin's empires fall (as a simple illustration) The Kolmogorov complexity of multicellular organisms and genetic codes: nontrivial connections Robustness of multicellular organisms (Drosophila as an example) A connection with the Hopfield system Chapter 5: Complexity of attractors for reaction diffusion systems and systems with convection Existence of chemical waves with complex fronts Existence of complicated attractors for reaction diffusion systems Applications to Ginzburg Landau systems and natural computing Existence of complicated attractors for Navier Stokes equations

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Book SynopsisCombinatorial optimization is a fascinating topic. Combinatorial optimization problems arise in a wide variety of important fields such as transportation, telecommunications, computer networking, location, planning, distribution problems, etc. Important and significant results have been obtained on the theory, algorithms and applications over the last few decades. In combinatorial optimization, many network design problems can be generalized in a natural way by considering a related problem on a clustered graph, where the original problem's feasibility constraints are expressed in terms of the clusters, i.e., node sets instead of individual nodes. This class of problems is usually referred to as generalized network design problems (GNDPs) or generalized combinatorial optimization problems. The express purpose of this monograph is to describe a series of mathematical models, methods, propositions, algorithms developed in the last years on generalized network design problems in a unified manner. The book consists of seven chapters, where in addition to an introductory chapter, the following generalized network design problems are formulated and examined: the generalized minimum spanning tree problem, the generalized traveling salesman problem, the railway traveling salesman problem, the generalized vehicle routing problem, the generalized fixed-charge network design problem and the generalized minimum vertex-biconnected network problem. The book will be useful for researchers, practitioners, and graduate students in operations research, optimization, applied mathematics and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will find this book useful, too.Table of ContentsChapter 1: Introduction Chapter 2: The generalized minimum spanning tree problem Chapter 3: The generalized traveling salesman problem Chapter 4: The railway traveling salesman problem Chapter 5: The generalized vehicle routing problem Chapter 6: The generalized fixed-charge network design problem

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Book SynopsisIn recent years, there has been a tremendous amount of research activity in the general area of population dynamics, particularly the Lotka-Volterra system, which has been a rich source of mathematical ideas from both theoretical and application points of view. In spite of the technological advances, many authors seem to be unaware of the bulk of the work that has been done in this area recently. This often leads to duplication of work and frustration to the authors as well as to the editors of various journals. This book is built out of lecture notes and consists of three chapters written by four mathematicians with overlapping expertise that cover a broad sector of the research in this area. Each chapter consists of carefully written introductory exposition, main breakthroughs, open questions and bibliographies. The chapters present recent developments on topics involving the dynamic behavior of solutions and topics such as stability theory, permanence, persistence, extinction, existence of positive solutions for the Lotka-Volterra and related systems. This fills a void in the literature, by making available a source book of relevant information on the theory, methods and applications of an important area of research.

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Book SynopsisThis book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

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