Algebra Books

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 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 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 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 monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer–Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong’s proof of Gabber’s theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer–Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer–Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.Trade Review“The book gives a comprehensive, clear, up-to date presentation of the theory, including most proofs. A particular strength is that it nicely collects many results, examples and counterexamples from various areas of algebraic and arithmetic geometry … . the book fills a wide gap and is a most welcome addition to the literature.” (Stefan Schröer, zbMATH 1490.14001, 2022)“This book has collected in one place much of the fundamental cohomological theory of the Brauer group, along with excellent references. It then gives some coverage of further results, especially on the two important topics of obstructions to rationality and obstructions to the Hasse principle. For whatever is not included in this book, it gives a thorough and coherent overview of the relevant literature. Approximately four hundred references are given.” (Thomas Benedict Williams, Mathematical Reviews, September, 2022)Table of Contents1 Galois Cohomology.- 2 Étale Cohomology.- 3 Brauer Groups of Schemes.- 4 Comparison of the Two Brauer Groups, II.- 5 Varieties Over a Field.- 6 Birational Invariance.- 7 Severi–Brauer Varieties and Hypersurfaces.- 8 Singular Schemes and Varieties.- 9 Varieties with a Group Action.- 10 Schemes Over Local Rings and Fields.- 11 Families of Varieties.- 12 Rationality in a Family.- 13 The Brauer–Manin Set and the Formal Lemma.- 14 Are Rational Points Dense in the Brauer–Manin Set?.- 15 The Brauer–Manin Obstruction for Zero-Cycles.- 16 Tate Conjecture, Abelian Varieties and K3 Surfaces.- Bibliography.- Index.

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Book SynopsisThis textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed. Beginning with a thorough treatment of the natural numbers via Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics. Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.Trade Review“Elements of mathematics is a curious book. The most challenging aspect of this volume to assess is its purpose.” (Jeff Johannes, Mathematical Reviews, October, 2022)“Transparency of explanation and gradually built material are outstanding features of the textbook. In addition, solutions to some problems are designed using more than one approach, making it adaptable to various students' backgrounds. … The book makes itself accessible to a vast population of students. The book can enhance the undergraduate curriculum or serve as a reflective resource for graduate mathematics students.” (Andrzej Sokolowski, MAA Reviews, March 20, 2022)“A historical concern is present throughout, with pieces of information on the history of concepts and theorems.” (Victor V. Pambuccian, zbMATH 1479.00002, 2022)Table of Contents0. Preliminaries: Sets, Relations, Maps.- 1. Natural, Integral and Rational Numbers.- 2. Real Numbers.- 3. Rational and Real Exponentiation.- 4. Limits of Real Functions.- 5. Real Analytic Plane Geometry.- 6. Polynomial Expressions.- 7. Polynomial Functions.- 8. Conics.- 9. Rational and Algebraic Expressions and Functions.- 10. Exponential and Logarithmic Functions.- 11. Trigonometry.- Further Reading.- Index.

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Book SynopsisThis self-contained monograph unifies theorems, applications and problem solving techniques of matrix inequalities. In addition to the frequent use of methods from Functional Analysis, Operator Theory, Global Analysis, Linear Algebra, Approximations Theory, Difference and Functional Equations and more, the reader will also appreciate techniques of classical analysis and algebraic arguments, as well as combinatorial methods. Subjects such as operator Young inequalities, operator inequalities for positive linear maps, operator inequalities involving operator monotone functions, norm inequalities, inequalities for sector matrices are investigated thoroughly throughout this book which provides an account of a broad collection of classic and recent developments. Detailed proofs for all the main theorems and relevant technical lemmas are presented, therefore interested graduate and advanced undergraduate students will find the book particularly accessible. In addition to several areas of theoretical mathematics, Matrix Analysis is applicable to a broad spectrum of disciplines including operations research, mathematical physics, statistics, economics, and engineering disciplines. It is hoped that graduate students as well as researchers in mathematics, engineering, physics, economics and other interdisciplinary areas will find the combination of current and classical results and operator inequalities presented within this monograph particularly useful.Trade Review“The book is written in a readable style and provides several interesting and nice techniques. It is very useful for graduate students and researchers interested in operator and norm inequalities.” (Mohammad Sal Moslehian, Mathematical Reviews, June, 2023)The book contains a bibliography of over 200 items and … the many inequalities presented, usually with full proofs provided. … if you are looking for an inequality in the areas covered, then this should be a useful source.” (John D. Dixon, zbMATH 1477.15001, 2022)Table of Contents1. Elementary linear algebra review.- 2. Interpolating the arithmetic-geometric mean inequality and its operator version.- 3. Operator inequalities for positive linear maps.- 4. Operator inequalities involving operator monotone functions.- 5. Inequalities for sector matrices.- 6. Positive partial transpose matrix inequalities.- References.- Index.

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Book SynopsisThis textbook offers an algorithmic introduction to the field of computer algebra. A leading expert in the field, the author guides readers through numerous hands-on tutorials designed to build practical skills and algorithmic thinking. This implementation-oriented approach equips readers with versatile tools that can be used to enhance studies in mathematical theory, applications, or teaching. Presented using Mathematica code, the book is fully supported by downloadable sessions in Mathematica, Maple, and Maxima. Opening with an introduction to computer algebra systems and the basics of programming mathematical algorithms, the book goes on to explore integer arithmetic. A chapter on modular arithmetic completes the number-theoretic foundations, which are then applied to coding theory and cryptography. From here, the focus shifts to polynomial arithmetic and algebraic numbers, with modern algorithms allowing the efficient factorization of polynomials. The final chapters offer extensions into more advanced topics: simplification and normal forms, power series, summation formulas, and integration. Computer Algebra is an indispensable resource for mathematics and computer science students new to the field. Numerous examples illustrate algorithms and their implementation throughout, with online support materials to encourage hands-on exploration. Prerequisites are minimal, with only a knowledge of calculus and linear algebra assumed. In addition to classroom use, the elementary approach and detailed index make this book an ideal reference for algorithms in computer algebra.Trade Review“Strong interplay between the abstract exposition, which includes the relevant theorems as well as their proofs, and the practical utilization of those concepts in Mathematica is certainly a remarkable feature of this textbook. … Overall, the book is very well written and the approach to provide examples as actual Mathematica sessions is commendable.” (Andreas Maletti, zbMATH 1484.68004, 2022)Table of Contents

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Book SynopsisThis research monograph develops the theory of relative nonhomogeneous Koszul duality. Koszul duality is a fundamental phenomenon in homological algebra and related areas of mathematics, such as algebraic topology, algebraic geometry, and representation theory. Koszul duality is a popular subject of contemporary research. This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare–Birkhoff–Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first time in the history of Koszul duality the derived D-\Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.Trade Review“The book under review is pretty self-contained, and it is not necessary to be familiar with all the background material before reading it. It also contains many examples to illustrate the main concepts.” (Dag Oskar Madsen, Mathematical Reviews, October, 2023)Table of ContentsPreface.- Prologue.- Introduction.- Homogeneous Quadratic Duality over a Base Ring.- Flat and Finitely Projective Koszulity.- Relative Nonhomogeneous Quadratic Duality.- The Poincare-Birkhoff-Witt Theorem.- Comodules and Contramodules over Graded Rings.- Relative Nonhomogeneous Derived Koszul Duality: the Comodule Side.- Relative Nonhomogeneous Derived Koszul Duality: the Contramodule Side.- The Co-Contra Correspondence.- Koszul Duality and Conversion Functor.- Examples.- References.

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Book SynopsisThis revised, updated edition provides a comprehensive and rigorous description of the application of Hamilton’s principle to continuous media. To introduce terminology and initial concepts, it begins with what is called the first problem of the calculus of variations. For both historical and pedagogical reasons, it first discusses the application of the principle to systems of particles, including conservative and non-conservative systems and systems with constraints. The foundations of mechanics of continua are introduced in the context of inner product spaces. With this basis, the application of Hamilton’s principle to the classical theories of fluid and solid mechanics are covered. Then recent developments are described, including materials with microstructure, mixtures, and continua with singular surfaces.Table of ContentsMechanics of Systems of Particles .- Mathematical Preliminaries.- Mechanics of Continuous Media.- Motions and Comparison Motions of a Mixture.- Singular Surfaces.- Index.

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Book SynopsisThis proceedings volume gathers selected, revised papers presented at the 51st Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 2020), held at Florida Atlantic University in Boca Raton, USA, on March 9-13, 2020. The SEICCGTC is broadly considered to be a trendsetter for other conferences around the world – many of the ideas and themes first discussed at it have subsequently been explored at other conferences and symposia.The conference has been held annually since 1970, in Baton Rouge, Louisiana and Boca Raton, Florida. Over the years, it has grown to become the major annual conference in its fields, and plays a major role in disseminating results and in fostering collaborative work.This volume is intended for the community of pure and applied mathematicians, in academia, industry and government, working in combinatorics and graph theory, as well as related areas of computer science and the interactions among these fields.Table of ContentsRatio Balancing Numbers(Bartz et al).- An Unexpected Digit Permutation from Multiplying in any Number Base(Qu et al).- A & Z Sequences for Double Riordan Arrays (Branch et al).- Constructing Clifford Algebras for Windmill and Dutch Windmill Graphs; A New Proof of The Friendship Theorem(Myers).- Finding Exact Values of a Character Sum (Peart et al).- On Minimum Index Stanton 4-cycle Designs (Bunge et al).- k-Plane Matroids and Whiteley’s Flattening Conjectures (Servatius et al).- Bounding the edge cover of a hypergraph (Shahrokhi).- A Generalization on Neighborhood Representatives (Holliday).- Harmonious Labelings of Disconnected Graphs involving Cycles and Multiple Components Consisting of Starlike Trees(Abueida et al).- On Rainbow Mean Colorings of Trees (Hallas et al).- Examples of Edge Critical Graphs in Peg Solitaire (Beeler et al).- Regular Tournaments with Minimum Split Domination Number and Cycle Extendability (Factor et al).- Independence and Domination of Chess Pieces on Triangular Boards and on the Surface of a Tetrahedron(Munger et al).- Efficient and Non-efficient Domination of Z-stacked Archimedean Lattices (Paskowitz et al).- On subdivision graphs which are 2-steps Hamiltonian graphs and hereditary non 2-steps Hamiltonian graphs (Lee et al).- On the Erd}os-S_os Conjecture for graphs with circumference at most k + 1 (Heissan et al).- Regular graph and some vertex-deleted subgraph (Egawa et al).- Connectivity and Extendability in Digraphs (Beasle).-On the extraconnectivity of arrangement graphs (Cheng et al).- k-Paths of k-Trees(Bickle).-Rearrangement of the Simple Random Walk(Skyers et al).- On the Energy of Transposition Graphs(DeDeo).- A Smaller Upper Bound for the (4; 82) Lattice Site Percolation Threshold(Wierman).

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Book SynopsisThis textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach. Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.Table of Contents1. Algebraic Preliminaries.- 2. Algebraic Numbers and Their Polynomials.- 3. Extending Fields.- 4. Irreducible Polynomials.- 5. Straightedge and Compass Constructions.- 6. Proofs of the Geometric Impossibilities.- 7. Zeros of Polynomials of Degrees 2, 3, and 4.- 8. Quintic Equations 1: Symmetric Polynomials.- 9. Quintic Equations II: The Abel–Ruffini Theorem.- 10. Transcendence of e and π.- 11. An Algebraic Postscript.- 12. Other Impossibilities: Regular Polygons and Integration in Finite Terms.- References.- Index.

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Book Synopsis1 Cellular Automata.- 2 Residually Finite Groups.- 3 Surjunctive Groups.- 4 Amenable Groups.- 5 The Garden of Eden Theorem.- 6 Finitely Generated Amenable Groups.- 7 Local Embeddability and Sofic Groups.- 8 Linear Cellular Automata.

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Book SynopsisThis book is an English translation of an entirely revised version of the 1958 edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics.It is devoted to the study of certain classes of rings and of modules, in particular to the notions of Noetherian or Artinian modules and rings, as well as that of radical.This chapter studies Morita equivalence of module and algebras, it describes the structure of semisimple rings. Various Grothendieck groups are defined that play a universal role for module invariants.The chapter also presents two particular cases of algebras over a field. The theory of central simple algebras is discussed in detail; their classification involves the Brauer group, of which severaldescriptions are given. Finally, the chapter considers group algebras and applies the general theory to representations of finite groups.At the end of the volume, a historical note taken from the previous edition recounts the evolution of many of the developed notions.Table of ContentsArtinian Modules and Noetherian Modules.- The Structure of Modules of Finite Length.- Simple Modules.- Semisimple Modules.- Commutation.- Morita Equivalence of Modules and Algebras.- Simple Rings.- Semisimple Rings.- Radical.- Modules over an Artinian Ring.- Grothendieck Groups.- Tensor Products of Semisimple Modules.- Absolutely Semisimple Algebras.- Central Simple Algebras.- Brauer Groups.- Other Descriptions of the Brauer Group.- Reduced Norms and Traces.- Simple Algebras over a Finite Field.- Quaternion Algebras.- Linear Representations of Algebras.- Linear Representations of Finite Groups.- Algebras without Unit Element.- Determinants over a Noncommunitative Field.- Hilbert's Nullstellensatz.- Trace of an Endomorphism of Finite Rank.- Historical Note.- Bibliography.- Notation Index.- Terminology Index.

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Book SynopsisThis monograph provides an exhaustive treatment of several classes of Noetherian rings and morphisms of Noetherian local rings. Chapters carefully examine some of the most important topics in the area, including Nagata, F-finite and excellent rings, Bertini’s Theorem, and Cohen factorizations. Of particular interest is the presentation of Popescu’s Theorem on Neron Desingularization and the structure of regular morphisms, with a complete proof. Classes of Good Noetherian Rings will be an invaluable resource for researchers in commutative algebra, algebraic and arithmetic geometry, and number theory.Table of Contents1. Fibres of Noetherian Rings.- 2. Nagata Rings and Reduced Morphisms.- 3. Excellent Rings and Regular Morphisms.- 4. Localization and Lifting Theorems.- 5. Structure of Regular Morphisms.- 6. Further Results on Classes of Good Rings.

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Book SynopsisThis monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian.Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource for many researchers in geometry, analysis, and mathematical physics. Table of ContentsIntroduction.- Bott-Chern Cohomology and Characteristic Classes.- The Derived Category ${\mathrm{D^{b}_{\mathrm{coh}}}}$.- Preliminaries on Linear Algebra and Differential Geometry.- The Antiholomorphic Superconnections of Block.- An Equivalence of Categories.- Antiholomorphic Superconnections and Generalized Metrics.- Generalized Metrics and Chern Character Forms.- The Case of Embeddings.- Submersions and Elliptic Superconnections.- Elliptic Superconnection Forms and Direct Images.- A Proof of Theorem 10-1 when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$..- The Hypoelliptic Superconnections.- The Hypoelliptic Superconnection Forms.- The Hypoelliptic Superconnection Forms when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.- Exotic Superconnections and Riemann-Roch-Grothendieck.- Subject Index.- Index of Notation.- Bibliography.

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Book SynopsisPreface.- The Abel Prize Winners 2018-2022.- The Abel Laureate Presenters.- The Interviews with the Abel Laureates.- Addenda.

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Book SynopsisSemi-Infinite Geometry is a theory of "doubly infinite-dimensional" geometric or topological objects. In this book the author explains what should be meant by an algebraic variety of semi-infinite nature. Then he applies the framework of semiderived categories, suggested in his previous monograph titled Homological Algebra of Semimodules and Semicontramodules, (Birkhäuser, 2010), to the study of semi-infinite algebraic varieties. Quasi-coherent torsion sheaves and flat pro-quasi-coherent pro-sheaves on ind-schemes are discussed at length in this book, making it suitable for use as an introduction to the theory of quasi-coherent sheaves on ind-schemes. The main output of the homological theory developed in this monograph is the functor of semitensor product on the semiderived category of quasi-coherent torsion sheaves, endowing the semiderived category with the structure of a tensor triangulated category. The author offers two equivalent constructions of the semitensor product, as well as its particular case, the cotensor product, and shows that they enjoy good invariance properties. Several geometric examples are discussed in detail in the book, including the cotangent bundle to an infinite-dimensional projective space, the universal fibration of quadratic cones, and the important popular example of the loop group of an affine algebraic group.Table of Contents- 1. Ind-Schemes and Their Morphisms. - 2. Quasi-Coherent Torsion Sheaves. - 3. Flat Pro-Quasi-Coherent Pro-Sheaves. - 4. Dualizing Complexes on Ind-Noetherian Ind-Schemes. - 5. The Cotensor Product. - 6. Ind-Schemes of Ind-Finite Type and the factorial !-Tensor Product. - 7. X-Flat Pro-Quasi-Coherent Pro-Sheaves on Y. - 8. The Semitensor Product. - 9. Flat Affine Ind-Schemes over Ind-Schemes of Ind-Finite Type. - 10. Invariance Under Postcomposition with a Smooth Morphism. - 11. Some Infinite-Dimensional Geometric Examples.

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Book SynopsisThis book is about semisimple Banach algebras with a focus on those that are commutative. Some of the questions dealt with in the book are: Whether the introduced Banach algebras are BSE-algebras, whether they have BSE norms, whether they have the separating ball property or some variant of it, and whether they are Arens regular.

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Book SynopsisThe book intends to be a collection of research papers on algebra and related topics, most of which were presented at the international Workshop on Associative Rings and Algebras with additional structures (WARA22).

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Book Synopsis1 Singularity Theory.- 2 Local Deformation Theory.- 3 Singularities in Arbitrary Characteristics.- Appendix A: Sheaves.- Appendix B: Commutative Algebra.- Appendix C: Formal Deformation Theory.

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Book SynopsisPreface.- 1. Analytic Functions and Morse Theory.- 2. Normal Forms of Functions.- 3. Algebraic Topology of Manifolds.- 4. Topology and Monodromy of Functions.- 5. Integrals along Vanishing Cycles.- 6. Vector Fields and Abelian Integrals.- 7. Hodge Structures and Period Map.- 8. Linear Differential Systems.- 9. Holomorphic Foliations. Local Theory.- 10. Holomorphic Foliations. Global Aspects.- 11. The Galois Theory.- 12. Hypergeometric Functions.- Bibliography.- Index.

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Book SynopsisChapter 1. Finite Dimensional Division Algebras.- Chapter 2. Structure of Finite Dimensional Algebras.- Chapter 3. Modules and Semisimple Rings.- Chapter 4. Structure of Rings.- Chapter 5. Tensor Products in Noncommutative Algebra.- Chapter 6. Noncommutative Polynomials.- Chapter 7. Rings of Quotients and Structure of PI-Rings.

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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 SynopsisThis volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.Table of ContentsCohomological Theory of Crystals over Function Fields and Applications.- On Geometric Iwasawa Theory and Special Values of Zeta Functions.- The Ongoing Binomial Revolution.- Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields.- Curves and Jacobians over Function Fields.

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Book SynopsisThis is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p‒1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index. The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems.

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Book SynopsisThis 3. edition is an introduction to classical knot theory. It contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory. Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups and some basic results of combinatorial group theory are assumed to be known.

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Book SynopsisDieses Lehrbuch ist eine Einführung in die Techniken der Gruppentheorie und behandelt alle wichtigen Begriffe aus diesem Gebiet, wobei der Schwerpunkt im Bereich der endlichen Gruppen liegt. Es beginnt dort, wo die Gruppentheorie beginnt: bei den Permutationsgruppen. Danach werden wesentliche Strukturen und Methoden, wie das Arbeiten mit Kommutatoren und die Konstruktion von neuen aus gegebenen Gruppen behandelt. Nächstes Ziel sind die Fittinggruppe und ihre Verallgemeinerung, wozu nilpotente Gruppen studiert werden. Danach wendet sich der Text den einfachen Gruppen zu. Zu guter Letzt wird zunächst die Einfachheit der projektiven linearen Gruppen bewiesen und ein Überblick über orthogonale, symplektische und unitäre Gruppen gegeben. Weiter werden die sporadischen Mathieu-Gruppen und die Higman-Sims-Gruppe konstruiert. Das Buch ist geschrieben für Studierende im Bachelor- und Masterstudium. Es setzt den Besuch der üblichen Algebra-Vorlesungen und somit nur allgemeine Kenntnisse über Gruppen voraus.

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Book SynopsisIn a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.

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Book SynopsisThis text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.Trade Review“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)Table of ContentsPreface.- 1 Introduction and Overview.- 2 The Group U(1) and its Representations.- 3 Two-state Systems and SU(2).- 4 Linear Algebra Review, Unitary and Orthogonal Groups.- 5 Lie Algebras and Lie Algebra Representations.- 6 The Rotation and Spin Groups in 3 and 4 Dimensions.- 7 Rotations and the Spin 1/2 Particle in a Magnetic Field.- 8 Representations of SU(2) and SO(3).- 9 Tensor Products, Entanglement, and Addition of Spin.- 10 Momentum and the Free Particle.- 11 Fourier Analysis and the Free Particle.- 12 Position and the Free Particle.- 13 The Heisenberg group and the Schrödinger Representation.- 14 The Poisson Bracket and Symplectic Geometry.- 15 Hamiltonian Vector Fields and the Moment Map.- 16 Quadratic Polynomials and the Symplectic Group.- 17 Quantization.- 18 Semi-direct Products.- 19 The Quantum Free Particle as a Representation of the Euclidean Group.- 20 Representations of Semi-direct Products.- 21 Central Potentials and the Hydrogen Atom.- 22 The Harmonic Oscillator.- 23 Coherent States and the Propagator for the Harmonic Oscillator.- 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1.- 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d.- 26 Complex Structures and Quantization.- 27 The Fermionic Oscillator.- 28 Weyl and Clifford Algebras.- 29 Clifford Algebras and Geometry.- 30 Anticommuting Variables and Pseudo-classical Mechanics.- 31 Fermionic Quantization and Spinors.- 32 A Summary: Parallels Between Bosonic and Fermionic Quantization.- 33 Supersymmetry, Some Simple Examples.- 34 The Pauli Equation and the Dirac Operator.- 35 Lagrangian Methods and the Path Integral.- 36 Multi-particle Systems: Momentum Space Description.- 37 Multi-particle Systems and Field Quantization.- 38 Symmetries and Non-relativistic Quantum Fields.- 39 Quantization of Infinite dimensional Phase Spaces.- 40 Minkowski Space and the Lorentz Group.- 41 Representations of the Lorentz Group.- 42 The Poincaré Group and its Representations.- 43 The Klein-Gordon Equation and Scalar Quantum Fields.- 44 Symmetries and Relativistic Scalar Quantum Fields.- 45 U(1) Gauge Symmetry and Electromagnetic Field.- 46 Quantization of the Electromagnetic Field: the Photon.- 47 The Dirac Equation and Spin-1/2 Fields.- 48 An Introduction to the Standard Model.- 49 Further Topics.- A Conventions.- B Exercises.- Index.

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Book SynopsisThis book is intended as a teacher’s manual and as an independent-study handbook for students and mathematical competitors. Based on a traditional teaching philosophy and a non-traditional writing approach (the stair-step method), this book consists of new problems with solutions created by the authors. The main idea of this approach is to start from relatively easy problems and “step-by-step” increase the level of difficulty toward effectively maximizing students' learning potential. In addition to providing solutions, a separate table of answers is also given at the end of the book. A broad view of mathematics is covered, well beyond the typical elementary level, by providing more in depth treatment of Geometry and Trigonometry, Number Theory, Algebra, Calculus, and Combinatorics.Trade Review“This book is original, enticing, and highly stimulating, and it is a useful addition to the competition-oriented literature.” (Stephen Rout, The Mathematical Gazette, Vol. 104 (560), July, 2020)Table of ContentsGeometry and Trigonometry.- Number Theory.- Algebra.- Calculus.- Combinatorics.- Hints.- Solutions.- Answers.

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Book Synopsis This book presents two new decomposition methods to decompose a time series in intrinsic components of low and high frequencies. The methods are based on Singular Value Decomposition (SVD) of a Hankel matrix (HSVD). The proposed decomposition is used to improve the accuracy of linear and nonlinear auto-regressive models. Linear Auto-regressive models (AR, ARMA and ARIMA) and Auto-regressive Neural Networks (ANNs) have been found insufficient because of the highly complicated nature of some time series. Hybrid models are a recent solution to deal with non-stationary processes which combine pre-processing techniques with conventional forecasters, some pre-processing techniques broadly implemented are Singular Spectrum Analysis (SSA) and Stationary Wavelet Transform (SWT). Although the flexibility of SSA and SWT allows their usage in a wide range of forecast problems, there is a lack of standard methods to select their parameters. The proposed decomposition HSVD and Multilevel SVD are described in detail through time series coming from the transport and fishery sectors. Further, for comparison purposes, it is evaluated the forecast accuracy reached by SSA and SWT, both jointly with AR-based models and ANNs. Table of ContentsPreface 1. Time Series and Forecasting 1.1. Introduction 1.2. Time series 1.3. Linear Autoregressive Models 1.4. Artificial Neural Networks 1.5. Hybrid models 1.5.1. Singular Spectrum Analysis 1.5.2. Wavelet Transform 1.6. Forecasting Accuracy Measures 1.7. Empirical Applications 1.7.1. Traffic Accidents Forecasting based on AR, ANNs and Hybrid models. 1.7.2. Anchovy Stock Forecasting based on AR, ANNs and Hybrid models. 1.7.3. Sardine Stock Forecasting based on AR, ANNs and Hybrid models. 2. Decomposition methods based on Singular Value Decomposition of a Hankel matrix 2.1. Introduction 2.2. Eigenvalues and Eigenvectors 2.3. Theorem of Singular Values Decomposition 2.4. One-level Singular Value Decomposition of a Hankel matrix 2.4.1. Embedding 2.4.2. Decomposition 2.4.3. Unembedding 2.4.4. Window Length Selection 2.5. Multi-level Singular Value Decomposition of a Hankel matrix 2.5.1. Embedding 2.5.2. Decomposition 2.5.3. Unembedding 2.5.4. Singular Spectrum Rate 2.6. Empirical Applications 2.6.1. Extraction of Components from traffic accidents time series based on HSVD and MSVD 2.6.2. Extraction of Components from fishery time series based on HSVD and MSVD 3. Forecasting based on components 3.1. Introduction 3.2. One-step ahead forecasting 3.3. Multi-step ahead forecasting 3.3.1. Direct Strategy 3.3.2. MIMO Strategy 3.4. Empirical Applications 3.4.1. Forecasting of traffic accidents based on HSVD and MSVD 3.4.2. Forecasting of anchovy stock based on HSVD and MSVD 3.4.3. Forecasting of sardine stock based on HSVD and MSVD List of Figures List of Tables List of Acronyms List of Symbols References

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Book SynopsisTable of ContentsInhalt: Normalformen: Überblick über die Klassifikation - Die Klassifikation nilpotenter Endomorphismen - Eigenwerte, Eigenräume, Jordan-Zerlegung - Die Jordan-Normalform - Elementarteiler - Die Klassifikation bis auf Konjugation - 1. Beispiel: GL (2,IR) - 2. Beispiel: GL (3,IR) - Anhang: Die schwingende Saite - Historische Bemerkungen zur Untersuchung der Struktur linearer Transformationen/ Vektorräume mit Hermiteschen Formen und ihre Endomorphismen: Sesquilinearformen - Selbstadjungierte und unitäre Endomorphismen- Orthogonalisierung - Isotropie - Klassifikation hermitescher und antihermitescher Formen - Euklidische und unitäre Vektorräume - Die Klassischen Gruppen - Bemerkungen zur Geschichte der Geometrie der klassischen Gruppen.

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Book SynopsisDie Simulation technischer Prozesse erfordert in der Regel die Lösung von linearen Gleichungssystemen großer Dimension. Hierfür werden moderne vorkonditionierte Iterationsverfahren (z.B. CG, GMRES, BiCGStab) hergeleitet und die zur Realisierung notwendigen Algorithmen beschrieben. Für Systeme mit strukturierten Matrizen werden effiziente direkte Lösungsverfahren angegeben. Numerische Beispiele für praktische Problemstellungen illustrieren die Effizienz der vorgestellten Verfahren.Table of Contents1 Grundlagen.- 1.1 Normen von Vektoren und Matrizen.- 1.2 Eigenwerte und Singulärwerte.- 1.3 Orthogonalisierung von Vektorsystemen.- 1.4 Tschebyscheff-Polynome.- 2 Lineare Gleichungssysteme.- 2.1 Interpolation.- 2.2 Projektionsmethoden.- 2.3 Finite Element Methoden.- 2.4 Randelementmethoden.- 3 Strukturierte Matrizen.- 3.1 Schnelle Fouriertransformation.- 3.2 Zirkulante Matrizen.- 3.3 Toeplitz Matrizen.- 3.4 Niedrig-Rang-Störung regulärer Matrizen.- 4 Klassische Iterationsverfahren.- 4.1 Stationäre Iterationsverfahren.- 4.2 Gradientenverfahren.- 5 Verfahren orthogonaler Richtungen.- 5.1 Verfahren konjugierter Gradienten.- 5.2 Verfahren des minimalen Residuums.- 5.3 Verfahren biorthogonaler Richtungen.- 6 Gleichungssysteme mit Blockstruktur.- 6.1 Symmetrische Gleichungssysteme.- 6.2 Blockschiefsymmetrische Systeme.- 6.3 Zweifache Sattelpunktprobleme.- 7 Hierarchische Matrizen.- 7.1 Partitionierte Matrizen.- 7.2 Approximation mit Niedrigrang-Matrizen.- 7.2.1 Approximation symmetrischer Matrizen.- 7.2.2 Approximation allgemeiner Matrizen.- 7.3 Arithmetik von Hierarchischen Matrizen.- 7.3.1 Matrix-Vektor-Multiplikation.- 7.3.2 Addition.- 7.3.3 Matrix-Matrix-Multiplikation.- 7.3.4 Invertierung.- 7.4 Geometrische Partitionierungen.- 7.4.1 Box-Clustering.- 7.4.2 Bisektionsverfahren.- 7.5 Niedrigrang-Approximation von Funktionen.- 7.5.1 Darstellung mit Taylor-Reihen.- 7.5.2 Explizite Reihendarstellung.- 7.5.3 Adaptive Cross-Approximation.- 7.6 Anwendungen in der FEM.- 7.6.1 L2-Projektion.- 7.6.2 Randwertprobleme zweiter Ordnung.- Literatur.

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