Algebra Books

Book SynopsisTrade Review"This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way."---Jonathan Shock, Mathemafrica"Reading Nahin is like reading through a select library of ancient Babylonian mathematical clay tablets. Surprises abound. . . . Nahin weaves much colorful history into his narrative."---Andrew Simoson, Mathematical Intelligencer"Engaging. . . . The book contains a wealth of original problems. . . . An enjoyable read."---Antonín Slavík, Zentralblatt MATH"This reviewer found himself being drawn to a variety of unfamiliar settings with much interest and even fascination." * Choice *"I certainly enjoyed [the book]!"---Alan Stevens, Mathematics Today"The potential audience for this book should be fairly large and go from highly talented high school students up through professionals in any STEM field."---Geoffrey Dietz, MAA Reviews

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Book SynopsisTrade Review"A Choice Outstanding Academic Title of the Year"

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Book SynopsisTrade Review"An excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians."---Raffaele Pisano, Metascience"Well written and engaging with a wealth of useful material and a substantial bibliography for further reading, this book is a valuable resource for anyone with a serious interest in the history of algebra. With Taming the Unknown, Victor Katz and Karen Parshall have created a comprehensive synthesis of recent research on the subject, accessible to mathematicians, historians of mathematics and anyone involved in the teaching of algebra."---Adrian Rice, BSHM Bulletin"The authors have . . . pitched their writing perfectly for their intended audience. The broad outline of the story is expressed in clear prose, combined with a judicious use of that other ‘native tongue' of the college mathematics graduate, symbolic algebra. . . . There is an extensive bibliography presenting the more detailed historical research that has been carried out. . . . You could base a really nice third-year course on this book."---John Hannah, Aestimatio

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Book SynopsisTrade Review"This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way."---Jonathan Shock, Mathemafrica"Reading Nahin is like reading through a select library of ancient Babylonian mathematical clay tablets. Surprises abound. . . . Nahin weaves much colorful history into his narrative."---Andrew Simoson, Mathematical Intelligencer"Engaging. . . . The book contains a wealth of original problems. . . . An enjoyable read."---Antonín Slavík, Zentralblatt MATH"This reviewer found himself being drawn to a variety of unfamiliar settings with much interest and even fascination." * Choice *"I certainly enjoyed [the book]!"---Alan Stevens, Mathematics Today"The potential audience for this book should be fairly large and go from highly talented high school students up through professionals in any STEM field."---Geoffrey Dietz, MAA Reviews

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Book SynopsisPresents the Russian tradition of expository mathematical writing. Suitable for students and teachers who want to study its various aspects, this book includes topics in number theory. It treats diverse aspects of analysis and algebra.Table of ContentsThe arithmetic of binomial coefficients by D. B. Fuchs and M. B. Fuchs Do you like messing around with integers? by M. I. Bashmakov On Bertrand's conjecture by M. I. Bashmakov On best approximations. I by D. B. Fuchs and M. B. Fuchs On best approximations. II by D. B. Fuchs and M. B. Fuchs On a certain property of binomial coefficients by A. I. Shirshov On $n!$ and the number $e$ (Several approaches to a certain problem) by L. G. Limanov Rational approximations and transcendence by D. B. Fuchs and M. B. Fuchs Close fractions by V. N. Vaguten On the equation $\binom{n}{m} = \binom{n+1}{m-1}$ by A. I. Shirshov On regular polygons, Euler's function, and Fermat numbers by A. Kirillov 2-adic numbers by B. Bekker, S. Vostokov, and Yu. Ionin On the number $e$ by E. Kuzmin and A. Shirshov Markov's Diophantine equation by M. G. Krein The arithmetic of Gaussian integers by A. B. Goncharov Three formulas of Ramanujan by V. S. Shevelev Amazing adventures in the land of repeating decimals by V. G. Stolyar, E. A. Kuraev, Z. K. Silogadze, G. A. Galperin, and A. V. Korlyukov.

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Book SynopsisAlgebraic Geometry has been at the centre of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology.

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Book SynopsisPresents twenty-nine cases, written by teachers describing real situations and actual student thinking in their classrooms, that provide the basis of each session's investigation of specific mathematical concepts and teaching strategies.

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Book SynopsisPraise for the First Edition . . .will certainly fascinate anyone interested in abstract algebra: a remarkable book! Monatshefte fur Mathematik Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel's theory of Abelian equations, casus irreducibili, and the Galois theory of origami. In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including: The contributions of Lagrange, Galois, and Kronecker How to compute Galois groups Galois''s results about irreducible polynomials of prime orprime-squared degreTrade Review“There is barely a better introduction to the subject, in all its theoretical and practical aspects, than the book under review.” (Zentralblatt MATH, 1 December 2012)"the great merit of this book (one of many expositions of the subject) is that everything is taken at a slow pace, with many examples to illustrate every idea. You get the (probably true) impression that the author loves this material, has taught it to undergraduates at Amherst College many times, has learned by experience the ideas which students find difficult, and has then taken great trouble to dissect these ideas and to pick out exactly the right examples and exercises to make them part of the reader’s mental equipment." (The Mathematical Gazette 2016) Table of ContentsPreface to the First Edition xvii Preface to the Second Edition xxi Notation xxiii 1 Basic Notation xxiii 2 Chapter-by-Chapter Notation xxv PART I POLYNOMIALS 1 Cubic Equations 3 1.1 Cardan's Formulas 4 1.2 Permutations of the Roots 10 1.3 Cubic Equations over the Real Numbers 15 2 Symmetric Polynomials 25 2.1 Polynomials of Several Variables 25 2.2 Symmetric Polynomials 30 2.3 Computing with Symmetric Polynomials (Optional) 42 2.4 The Discriminant 46 3 Roots of Polynomials 55 3.1 The Existence of Roots 55 3.2 The Fundamental Theorem of Algebra 62 PART II FIELDS 4 Extension Fields 73 4.1 Elements of Extension Fields 73 4.2 Irreducible Polynomials 81 4.3 The Degree of an Extension 89 4.4 Algebraic Extensions 95 5 Normal and Separable Extensions 101 5.1 Splitting Fields 101 5.2 Normal Extensions 107 5.3 Separable Extensions 109 5.4 Theorem of the Primitive Element 119 6 The Galois Group 125 6.1 Definition of the Galois Group 125 6.2 Galois Groups of Splitting Fields 130 6.3 Permutations of the Roots 132 6.4 Examples of Galois Groups 136 6.5 Abelian Equations (Optional) 143 7 The Galois Correspondence 147 7.1 Galois Extensions 147 7.2 Normal Subgroups and Normal Extensions 154 7.3 The Fundamental Theorem of Galois Theory 161 7.4 First Applications 167 7.5 Automorphisms and Geometry (Optional) 173 PART III APPLICATIONS 8 Solvability by Radicals 191 8.1 Solvable Groups 191 8.2 Radical and Solvable Extensions 196 8.3 Solvable Extensions and Solvable Groups 201 8.4 Simple Groups 210 8.5 Solving Polynomials by Radicals 215 8.6 The Casus Irreducbilis (Optional) 220 9 Cyclotomic Extensions 229 9.1 Cyclotomic Polynomials 229 9.2 Gauss and Roots of Unity (Optional) 238 10 Geometric Constructions 255 10.1 Constructible Numbers 255 10.2 Regular Polygons and Roots of Unity 270 10.3 Origami (Optional) 274 11 Finite Fields 291 11.1 The Structure of Finite Fields 291 11.2 Irreducible Polynomials over Finite Fields (Optional) 301 PART IV FURTHER TOPICS 12 Lagrange, Galois, and Kronecker 315 12.1 Lagrange 315 12.2 Galois 334 12.3 Kronecker 347 13 Computing Galois Groups 357 13.1 Quartic Polynomials 357 13.2 Quintic Polynomials 368 13.3 Resolvents 386 13.4 Other Methods 400 14 Solvable Permutation Groups 413 14.1 Polynomials of Prime Degree 413 14.2 Imprimitive Polynomials of Prime-Squared Degree 419 14.3 Primitive Permutation Groups 429 14.4 Primitive Polynomials of Prime-Squared Degree 444 15 The Lemniscate 463 15.1 Division Points and Arc Length 464 15.2 The Lemniscatic Function 470 15.3 The Complex Lemniscatic Function 482 15.4 Complex Multiplication 489 15.5 Abel's Theorem 504 A Abstract Algebra 515 A.1 Basic Algebra 515 A.2 Complex Numbers 524 A.3 Polynomials with Rational Coefficients 528 A.4 Group Actions 530 A.5 More Algebra 532 Index 557

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Book SynopsisPraise for the Third Edition . . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . .Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book''s unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text. The Fourth Edition features important concepts as well as specialized topics, including: The treatment of nilpotent groups, including the Frattini and Fitting subgroups Symmetric polynomials The pTrade Review “This could also be an excellent adjunct to more theoretically oriented textbooks used in more intensive courses.” (Computing Reviews, 5 November 2012) Table of ContentsPREFACE ix ACKNOWLEDGMENTS xvii NOTATION USED IN THE TEXT xix A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii 0 Preliminaries 1 0.1 Proofs / 1 0.2 Sets / 5 0.3 Mappings / 9 0.4 Equivalences / 17 1 Integers and Permutations 23 1.1 Induction / 24 1.2 Divisors and Prime Factorization / 32 1.3 Integers Modulo n / 42 1.4 Permutations / 53 1.5 An Application to Cryptography / 67 2 Groups 69 2.1 Binary Operations / 70 2.2 Groups / 76 2.3 Subgroups / 86 2.4 Cyclic Groups and the Order of an Element / 90 2.5 Homomorphisms and Isomorphisms / 99 2.6 Cosets and Lagrange’s Theorem / 108 2.7 Groups of Motions and Symmetries / 117 2.8 Normal Subgroups / 122 2.9 Factor Groups / 131 2.10 The Isomorphism Theorem / 137 2.11 An Application to Binary Linear Codes / 143 3 Rings 159 3.1 Examples and Basic Properties / 160 3.2 Integral Domains and Fields / 171 3.3 Ideals and Factor Rings / 180 3.4 Homomorphisms / 189 3.5 Ordered Integral Domains / 199 4 Polynomials 202 4.1 Polynomials / 203 4.2 Factorization of Polynomials Over a Field / 214 4.3 Factor Rings of Polynomials Over a Field / 227 4.4 Partial Fractions / 236 4.5 Symmetric Polynomials / 239 4.6 Formal Construction of Polynomials / 248 5 Factorization in Integral Domains 251 5.1 Irreducibles and Unique Factorization / 252 5.2 Principal Ideal Domains / 264 6 Fields 274 6.1 Vector Spaces / 275 6.2 Algebraic Extensions / 283 6.3 Splitting Fields / 291 6.4 Finite Fields / 298 6.5 Geometric Constructions / 304 6.6 The Fundamental Theorem of Algebra / 308 6.7 An Application to Cyclic and BCH Codes / 310 7 Modules over Principal Ideal Domains 324 7.1 Modules / 324 7.2 Modules Over a PID / 335 8 p-Groups and the Sylow Theorems 349 8.1 Products and Factors / 350 8.2 Cauchy’s Theorem / 357 8.3 Group Actions / 364 8.4 The Sylow Theorems / 371 8.5 Semidirect Products / 379 8.6 An Application to Combinatorics / 382 9 Series of Subgroups 388 9.1 The Jordan–H¨older Theorem / 389 9.2 Solvable Groups / 395 9.3 Nilpotent Groups / 401 10 Galois Theory 412 10.1 Galois Groups and Separability / 413 10.2 The Main Theorem of Galois Theory / 422 10.3 Insolvability of Polynomials / 434 10.4 Cyclotomic Polynomials and Wedderburn’s Theorem / 442 11 Finiteness Conditions for Rings and Modules 447 11.1 Wedderburn’s Theorem / 448 11.2 The Wedderburn–Artin Theorem / 457 Appendices 471 Appendix A Complex Numbers / 471 Appendix B Matrix Algebra / 478 Appendix C Zorn’s Lemma / 486 Appendix D Proof of the Recursion Theorem / 490 BIBLIOGRAPHY 492 SELECTED ANSWERS 495 INDEX 523

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Book SynopsisThis Student Solutions Manual to Accompany Linear Algebra: Ideas and Applications, Fourth Edition contains solutions to the odd numbered problems to further aid in reader comprehension, and an Instructor''s Solutions Manual (inclusive of suggested syllabi) is available via written request to the Publisher. Both the Student and Instructor Manuals have been enhanced with further discussions of the applications sections, which is ideal for readers who wish to obtain a deeper knowledge than that provided by pure algorithmic approaches. Linear Algebra: Ideas and Applications, Fourth Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, this book successfully helps readers to understand not only how to implement a technique, but why its use is important.Table of ContentsSTUDENT MANUAL 1 1 SYSTEMS OF LINEAR EQUATIONS 3 1.1 The Vector Space of m × n Matrices / 3 1.1.2 Applications to Graph Theory I / 7 1.2 Systems / 8 1.2.2 Applications to Circuit Theory / 11 1.3 Gaussian Elimination / 13 1.3.2 Applications to Traffic Flow / 18 1.4 Column Space and Nullspace / 19 2 LINEAR INDEPENDENCE AND DIMENSION 26 2.1 The Test for Linear Independence / 26 2.2 Dimension / 33 2.2.2 Applications to Differential Equations / 37 2.3 Row Space and the Rank-Nullity Theorem / 38 3 LINEAR TRANSFORMATIONS 43 3.1 The Linearity Properties / 43 3.2 Matrix Multiplication (Composition) / 49 3.2.2 Applications to Graph Theory II / 55 3.3 Inverses / 55 3.3.2 Applications to Economics / 60 3.4 The LU Factorization / 61 3.5 The Matrix of a Linear Transformation / 62 4 DETERMINANTS 67 4.1 Definition of the Determinant / 67 4.2 Reduction and Determinants / 69 4.2.1 Volume / 72 4.3 A Formula for Inverses / 74 5 EIGENVECTORS AND EIGENVALUES 76 5.1 Eigenvectors / 76 5.1.2 Application to Markov Processes / 79 5.2 Diagonalization / 80 5.2.1 Application to Systems of Differential Equations / 82 5.3 Complex Eigenvectors / 83 6 ORTHOGONALITY 85 6.1 The Scalar Product in ℝn / 85 6.2 Projections: The Gram-Schmidt Process / 87 6.3 Fourier Series: Scalar Product Spaces / 89 6.4 Orthogonal Matrices / 92 6.5 Least Squares / 93 6.6 Quadratic Forms: Orthogonal Diagonalization / 94 6.7 The Singular Value Decomposition (SVD) / 97 6.8 Hermitian Symmetric and Unitary Matrices / 98 7 GENERALIZED EIGENVECTORS 100 7.1 Generalized Eigenvectors / 100 7.2 Chain Bases / 104 8 NUMERICAL TECHNIQUES 107 8.1 Condition Number / 107 8.2 Computing Eigenvalues / 108

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Book SynopsisA world model: economies, trade, migration, security and development aid. This bookprovides the analytical capability to understand and explore the dynamics of globalisation. It is anchored in economic input-output models of over 200 countries and their relationships through trade, migration, security and development aid. The tools of complexity science are brought to bear and mathematical and computer models are developed both for the elements and for an integrated whole. Models are developed at a variety of scales ranging from the global and international trade through a European model of inter-sub-regional migration to piracy in the Gulf and the London riots of 2011. The models embrace the changing technology of international shipping, the impacts of migration on economic development along with changing patterns of military expenditure and development aid. A unique contribution is the level of spatial disaggregation which presents each of 200+ countries and their muTable of ContentsNotes on Contributors xiii Acknowledgements xvii Part I GLOBAL DYNAMICS AND THE TOOLS OF COMPLEXITY SCIENCE 1 Global Dynamics and the Tools of Complexity Science 3Alan Wilson Reference 7 Part II TRADE AND ECONOMIC DEVELOPMENT 2 The Global Trade System and Its Evolution 11Simone Caschili and Francesca Medda 2.1 The Evolution of the Shipping and Ports’ System 11 2.2 Analyses of the Cargo Ship Network 12 2.3 A Complex Adaptive Systems (CASs) Perspective 15 2.4 Conclusions: The Benefits of a Systems Perspective 20 References 21 Appendix 23 A.1 Complexity Science and Complex Adaptive Systems: Key Characteristics 23 A.1.1 Four Properties 24 A.1.2 Three Mechanisms 25 3 An Interdependent Multi-layer Model for Trade 26Simone Caschili, Francesca Medda, and Alan Wilson 3.1 Introduction 26 3.2 The Interdependent Multi-layer Model: Vertical Integration 27 3.3 Model Layers 30 3.3.1 Economic Layer 30 3.3.2 Social and Cultural Layer (Socio-cultural) 33 3.3.3 Physical Layer 34 3.4 The Workings of the Model 34 3.5 Model Calibration 35 3.6 Result 1: Steady State 39 3.7 Result 2: Estimation and Propagation of Shocks in the IMM 42 3.8 Discussion and Conclusions 48 References 48 4 A Global Inter-country Economic Model Based on Linked Input–Output Models 51Robert G. Levy, Thomas P. Oléron Evans, and Alan Wilson 4.1 Introduction 51 4.2 Existing Global Economic Models 52 4.3 Description of the Model 53 4.3.1 Outline 53 4.3.2 Introduction to Input–Output Tables 53 4.3.3 A Single Country Model 55 4.3.4 An International Trade Model 57 4.3.5 Setting Model Coefficients from Data 58 4.4 Solving the Model 58 4.4.1 The Leontief Equation 58 4.4.2 The Drawbacks of Mathematical Elegance 59 4.4.3 Algorithm for an Iterative Solution 59 4.5 Analysis 61 4.5.1 Introduction 61 4.5.2 Simple Modelling Approaches 61 4.5.3 A Unified Network Approach 64 4.5.4 Comparison with a Multi-region Input–Output Model 67 4.6 Conclusions 67 Acknowledgements 69 References 69 Appendix 71 A.1 Modelling the ‘Rest of the World’ 71 A.2 Services Trade Data 71 A.2.1 Importing Own Exports 72 A.2.2 The Rest of the World for Sectors 72 Part III MIGRATION 5 Global Migration Modelling: A Review of Key Policy Needs and Research Centres 75Adam Dennett and Pablo Mateos 5.1 Introduction 75 5.2 Policy and Migration Research 76 5.2.1 Key Policy Issues in Contemporary Migration Research 76 5.2.2 Linking Policy Issues to Modelling Challenges 81 5.2.3 Policy-related Research Questions for Modellers 82 5.2.4 Other International Migration Modelling Research 83 5.3 Conclusion 84 References 84 Appendix 87 A.1 United Kingdom 87 A.2 Rest of Europe 90 A.3 Rest of the World 94 6 Estimating Inter-regional Migration in Europe 97Adam Dennett and Alan Wilson 6.1 Introduction 97 6.2 The Spatial System and the Modelling Challenge 98 6.3 Biproportional Fitting Modelling Methodology 100 6.3.1 Model (i) 104 6.3.2 Model (ii) 105 6.3.3 Model (iii) 105 6.3.4 Model (iv) 108 6.3.5 Model (v) 109 6.3.6 Model (vi) 110 6.4 Model Parameter Calibration 110 6.5 Model Experiments 113 6.6 Results 118 6.7 Conclusions and Comments on the New Framework for Estimating Inter-regional, Inter-country Migration Flows in Europe 121 References 123 7 Estimating an Annual Time Series of Global Migration Flows – An Alternative Methodology for Using Migrant Stock Data 125Adam Dennett 7.1 Introduction 125 7.2 Methodology 129 7.2.1 Introduction 129 7.2.2 Calculating Migration Probabilities 129 7.2.3 Calculating Total Migrants in the Global System 130 7.2.4 Generating a Consistent Time Series of Migration Probabilities 133 7.2.5 Producing Annual Bilateral Estimates 135 7.3 Results and Validation 135 7.3.1 Introduction 135 7.3.2 IMEM comparison 135 7.3.3 UN Flow Data Comparison 136 7.4 Discussion 138 7.5 Conclusions 140 References 140 Part IV SECURITY 8 Conflict Modelling: Spatial Interaction as Threat 145Peter Baudains and Alan Wilson 8.1 Introduction 145 8.2 Conflict Intensity: Space–Time Patterning of Events 146 8.3 Understanding Conflict Onset: Simulation-based Models 148 8.4 Forecasting Global Conflict Hotspots 150 8.5 A Spatial Model of Threat 150 8.6 Discussion: The Use of a Spatial Threat Measure in Models of Conflict 153 8.6.1 Threat in Models for Operational Decision-Making 153 8.6.2 Threat in a Model of Conflict Escalation 154 8.6.3 Threat in Modelling Global Military Expenditure 156 8.6.4 Summary 156 References 157 9 Riots 159Peter Baudains 9.1 Introduction 159 9.2 The 2011 Riots in London 160 9.2.1 Space–Time Interaction 162 9.2.2 Journey to Crime 164 9.2.3 Characteristics of Rioters 165 9.3 Data-Driven Modelling of Riot Diffusion 166 9.4 Statistical Modelling of Target Choice 169 9.5 A Generative Model of the Riots 171 9.6 Discussion 172 References 173 10 Rebellions 175Peter Baudains, Jyoti Belur, Alex Braithwaite, Elio Marchione and Shane D. Johnson 10.1 Introduction 175 10.2 Data 176 10.3 Hawkes model 177 10.4 Results 181 10.5 Discussion 183 References 185 11 Spatial Interaction as Threat: Modelling Maritime Piracy 187Elio Marchione and Alan Wilson 11.1 The Model 187 11.2 The Test Case 188 11.3 Uses of the Model 189 Reference 191 Appendix 192 A.1 Volume Field of Type k Ship 192 A.2 Volume Field of Naval Units 193 A.3 Pirates Ports and Mother Ships 193 12 Space–Time Modelling of Insurgency and Counterinsurgency in Iraq 195Alex Braithwaite and Shane Johnson 12.1 Introduction 195 12.2 Counterinsurgency in Iraq 196 12.3 Counterinsurgency Data 200 12.4 Diagnoses of Space, Time and Space–Time Distributions 202 12.4.1 Introduction 202 12.4.2 Spatial Distribution 202 12.4.3 Temporal Distribution 203 12.4.4 Space–Time Distribution 203 12.4.5 Univariate Knox Analysis 206 12.4.6 Bivariate Knox Analysis 208 12.5 Concluding Comments 210 References 212 13 International Information Flows, Government Response and the Contagion of Ethnic Conflict 214Janina Beiser 13.1 Introduction 214 13.2 Global Information Flows 216 13.3 The Effect of Information Flows on Armed Civil Conflict 220 13.4 The Effect of Information Flows on Government Repression 225 13.5 Conclusion 226 References 226 Appendix 229 Part V AID AND DEVELOPMENT 14 International Development Aid: A Complex System 233Belinda Wu 14.1 Introduction: A Complex Systems’ Perspective 233 14.2 The International Development Aid System: Definitions 234 14.3 Features of International Development Aid as a Complex System 235 14.3.1 Introduction 235 14.3.2 Non-linearity 235 14.3.3 Connectedness 237 14.3.4 Self-Adapting and Self-Organising 238 14.3.5 Emergence 238 14.4 Complexity and Approaches to Research 238 14.4.1 Organisations 238 14.4.2 The Range of Issues 239 14.4.3 Research Approaches 240 14.4.4 The Complexity Science Approach 242 14.5 The Assessment of the Effectiveness of International Development Aid 242 14.5.1 Whether Aid Can Be Effective 242 14.5.2 Complexity in the Measurement of Aid Effectiveness 244 14.5.3 Complexity in Methods/Standards of Measurement of Aid Effectiveness 245 14.5.4 Standardising Aid Effectiveness 246 14.6 Relationships and Interactions 248 14.6.1 Relationships between Donor and Recipient Countries 248 14.6.2 Relationships between Aid and Other Systems 249 14.7 Conclusions 251 References 252 15 Model Building for the Complex System of International Development Aid 257Belinda Wu, Sean Hanna and Alan Wilson 15.1 Introduction 257 15.2 Data Collection 258 15.2.1 Introduction 258 15.2.2 Aid Data 258 15.2.3 Trade Data 260 15.2.4 Security Data 261 15.2.5 Migration Data 261 15.2.6 Geographical Data 261 15.2.7 Data Selected 262 15.3 Model Building 263 15.3.1 Modelling Approach 263 15.3.2 Alesina and Dollar Model 263 15.3.3 Our Models 264 15.3.4 Model B: Introducing Donor Interactions and Modification of the Model 267 15.3.5 Findings from Model B 267 15.3.6 Model C: Introducing Interactions with Trade System and Further Modification of the Model 267 15.3.7 Findings from Model C 268 15.4 Discussion and Future Work 268 References 269 16 Aid Allocation: A Complex Perspective 271Robert J. Downes and Steven R. Bishop 16.1 Aid Allocation Networks 271 16.1.1 Introduction 271 16.1.2 Why Networks? 272 16.1.3 Donor Motivation in Aid Allocation 273 16.2 Quantifying Aid via a Mathematical Model 273 16.2.1 Overview of Approach 273 16.2.2 Basic Set-Up 274 16.2.3 The Network of Nations 275 16.2.4 Preference Functions 275 16.2.5 Specifying the Preference Functions 275 16.2.6 Recipient Selection by Donors 276 16.3 Application of the Model 277 16.3.1 Introduction 277 16.3.2 Scenario 1. No Feedback 277 16.3.3 Scenario 2. Bandwagon Feedback 281 16.3.4 Scenario 3. Aid Effectiveness Feedback 283 16.3.5 Aid Usage Mechanism 284 16.3.6 Application 286 16.3.7 Conclusions 287 16.4 Remarks 287 Acknowledgements 288 References 288 Appendix 290 A.1 Common Functional Definitions 290 Part VI GLOBAL DYNAMICS: AN INTEGRATED MODEL AND POLICY CHALLENGES 17 An Integrated Model 293Robert G. Levy 17.1 Introduction 293 17.2 Adding Migration 294 17.2.1 Introduction 294 17.2.2 The Familiarity Effect 295 17.2.3 Consumption Similarity 301 17.2.4 Conclusions 304 17.3 Adding Aid 304 17.3.1 Introduction 304 17.3.2 Estimating ‘Exportness’ 305 17.3.3 Modelling Approach 306 17.3.4 Results 306 17.3.5 Conclusions 314 17.4 Adding Security 316 17.4.1 Introduction 316 17.4.2 Literature Review 316 17.4.3 Measures of Threat and the Global Dynamics Model 317 17.4.4 Trade during Changing Security Conditions 318 17.4.5 An Experiment of Increased Threat in the Global Dynamics Model 318 17.4.6 Conclusions 322 17.5 Concluding Comments 323 References 324 Index 327

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Book SynopsisA thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS, MATLAB, and R throughout This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also: Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices Covers the analysis of balanced linear models using direct products of matrices Analyzes multiresponse linTrade Review"Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra." Mathematical Reviews, Sept 2017 Table of ContentsPREFACE xvii PREFACE TO THE FIRST EDITION xix INTRODUCTION xxi ABOUT THE COMPANION WEBSITE xxxi PART I DEFINITIONS, BASIC CONCEPTS, AND MATRIX OPERATIONS 1 1 Vector Spaces, Subspaces, and Linear Transformations 3 1.1 Vector Spaces 3 1.2 Base of a Vector Space 5 1.3 Linear Transformations 7 2 Matrix Notation and Terminology 11 2.1 Plotting of a Matrix 14 2.2 Vectors and Scalars 16 2.3 General Notation 16 3 Determinants 21 3.1 Expansion by Minors 21 3.2 Formal Definition 25 3.3 Basic Properties 27 3.4 Elementary Row Operations 34 3.5 Examples 37 3.6 Diagonal Expansion 39 3.7 The Laplace Expansion 42 3.8 Sums and Differences of Determinants 44 3.9 A Graphical Representation of a 3 × 3 Determinant 45 4 Matrix Operations 51 4.1 The Transpose of a Matrix 51 4.2 Partitioned Matrices 52 4.3 The Trace of a Matrix 55 4.4 Addition 56 4.5 Scalar Multiplication 58 4.6 Equality and the Null Matrix 58 4.7 Multiplication 59 4.8 The Laws of Algebra 74 4.9 Contrasts With Scalar Algebra 76 4.10 Direct Sum of Matrices 77 4.11 Direct Product of Matrices 78 4.12 The Inverse of a Matrix 80 4.13 Rank of a Matrix—Some Preliminary Results 82 4.14 The Number of LIN Rows and Columns in a Matrix 84 4.15 Determination of the Rank of a Matrix 85 4.16 Rank and Inverse Matrices 87 4.17 Permutation Matrices 87 5 Special Matrices 97 5.1 Symmetric Matrices 97 5.2 Matrices Having All Elements Equal 102 5.3 Idempotent Matrices 104 5.4 Orthogonal Matrices 106 5.5 Parameterization of Orthogonal Matrices 109 5.6 Quadratic Forms 110 5.7 Positive Definite Matrices 113 6 Eigenvalues and Eigenvectors 119 6.1 Derivation of Eigenvalues 119 6.2 Elementary Properties of Eigenvalues 122 6.3 Calculating Eigenvectors 125 6.4 The Similar Canonical Form 128 6.5 Symmetric Matrices 131 6.6 Eigenvalues of Orthogonal and Idempotent Matrices 135 6.7 Eigenvalues of Direct Products and Direct Sums of Matrices 138 6.8 Nonzero Eigenvalues of AB and BA 140 7 Diagonalization of Matrices 145 7.1 Proving the Diagonability Theorem 145 7.2 Other Results for Symmetric Matrices 148 7.3 The Cayley–Hamilton Theorem 152 7.4 The Singular-Value Decomposition 153 8 Generalized Inverses 159 8.1 The Moore–Penrose Inverse 159 8.2 Generalized Inverses 160 8.3 Other Names and Symbols 164 8.4 Symmetric Matrices 165 9 Matrix Calculus 171 9.1 Matrix Functions 171 9.2 Iterative Solution of Nonlinear Equations 174 9.3 Vectors of Differential Operators 175 9.4 Vec and Vech Operators 179 9.5 Other Calculus Results 181 9.6 Matrices with Elements That Are Complex Numbers 188 9.7 Matrix Inequalities 189 PART II APPLICATIONS OF MATRICES IN STATISTICS 199 10 Multivariate Distributions and Quadratic Forms 201 10.1 Variance-Covariance Matrices 202 10.2 Correlation Matrices 203 10.3 Matrices of Sums of Squares and Cross-Products 204 10.4 The Multivariate Normal Distribution 207 10.5 Quadratic Forms and ;;2-Distributions 208 10.6 Computing the Cumulative Distribution Function of a Quadratic Form 213 11 Matrix Algebra of Full-Rank Linear Models 219 11.1 Estimation of ;; by the Method of Least Squares 220 11.2 Statistical Properties of the Least-Squares Estimator 226 11.3 Multiple Correlation Coefficient 229 11.4 Statistical Properties under the Normality Assumption 231 11.5 Analysis of Variance 233 11.6 The Gauss–Markov Theorem 234 11.7 Testing Linear Hypotheses 237 11.8 Fitting Subsets of the x-Variables 246 11.9 The Use of the R(.|.) Notation in Hypothesis Testing 247 12 Less-Than-Full-Rank Linear Models 253 12.1 General Description 253 12.2 The Normal Equations 256 12.3 Solving the Normal Equations 257 12.4 Expected Values and Variances 259 12.5 Predicted y-Values 260 12.6 Estimating the Error Variance 261 12.7 Partitioning the Total Sum of Squares 262 12.8 Analysis of Variance 263 12.9 The R(⋅|⋅) Notation 265 12.10 Estimable Linear Functions 266 12.11 Confidence Intervals 272 12.12 Some Particular Models 272 12.13 The R(⋅|⋅) Notation (Continued) 277 12.14 Reparameterization to a Full-Rank Model 281 13 Analysis of Balanced Linear Models Using Direct Products of Matrices 287 13.1 General Notation for Balanced Linear Models 289 13.2 Properties Associated with Balanced Linear Models 293 13.3 Analysis of Balanced Linear Models 298 14 Multiresponse Models 313 14.1 Multiresponse Estimation of Parameters 314 14.2 Linear Multiresponse Models 316 14.3 Lack of Fit of a Linear Multiresponse Model 318 PART III MATRIX COMPUTATIONS AND RELATED SOFTWARE 327 15 SAS/IML 329 15.1 Getting Started 329 15.2 Defining a Matrix 329 15.3 Creating a Matrix 330 15.4 Matrix Operations 331 15.5 Explanations of SAS Statements Used Earlier in the Text 354 16 Use of MATLAB in Matrix Computations 363 16.1 Arithmetic Operators 363 16.2 Mathematical Functions 364 16.3 Construction of Matrices 365 16.4 Two- and Three-Dimensional Plots 371 17 Use of R in Matrix Computations 383 17.1 Two- and Three-Dimensional Plots 396 Exercises 408 APPENDIX 413 INDEX 475

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Book SynopsisMathematical models have become a crucial way for the Earth scientist to understand and predict how our planet functions and evolves through time and space. The finite element method (FEM) is a remarkably flexible and powerful tool with enormous potential in the Earth Sciences.Table of ContentsPreface xiii Symbols xv About the Companion Website xvii Part I The Finite Element Method with Matlab 1 1 Preliminaries 3 1.1 Mathematical Models 3 1.2 Boundary and Initial Conditions 4 1.3 Analytical Solutions 5 1.4 Numerical Solutions 5 1.5 Numerical Solution Methods 7 1.6 Matlab Script 8 1.7 Exercises 10 Suggested Reading 12 2 Beginning with the Finite Element Method 13 2.1 The Governing PDE 13 2.2 Approximating the Continuous Variable 14 2.3 Minimizing the Residual 15 2.4 Evaluating the Element Matrices 17 2.5 Time Discretization 18 2.6 Assembly 19 2.7 Boundary and Initial Conditions 21 2.8 Solution of the Algebraic Equations 21 2.9 Exercises 22 Suggested Reading 23 3 Programming the Finite Element Method in Matlab 25 3.1 Program Structure and Philosophy 25 3.2 Summary of the Problem 25 3.3 Discretized Equations 26 3.4 The Program 27 3.4.1 Preprocessor Stage 27 3.4.2 Solution Stage 29 3.4.3 Postprocessor Stage 30 3.5 Matlab Script 30 3.6 Exercises 33 Suggested Reading 34 4 Numerical Integration and Local Coordinates 35 4.1 Gauss–Legendre Quadrature 36 4.2 Local Coordinates 37 4.3 Evaluating the Integrals 39 4.4 Variable Material Properties 40 4.5 Programming Considerations 41 4.6 Matlab Script 43 4.7 Exercises 45 Suggested Reading 47 5 The Finite Element Method in Two Dimensions 49 5.1 Discretization 50 5.2 Geometry and Nodal Connectivity 52 5.3 Integration of Element Matrices 54 5.4 Multielement Assembly 57 5.5 Boundary Conditions and Solution 60 5.6 Matlab Script 61 5.7 Exercises 65 Suggested Reading 66 6 The Finite Element Method in Three Dimensions 67 6.1 Discretization 67 6.2 Element Integration 69 6.3 Assembly for Multielement Mesh 72 6.4 Boundary Conditions and Solution 73 6.5 Matlab Program 74 6.6 Exercises 79 Suggested Reading 80 7 Generalization of Finite Element Concepts 81 7.1 The FEM for an Elliptic Problem 84 7.2 The FEM for a Hyperbolic Problem 96 7.3 The FEM for Systems of Equations 102 7.4 Exercises 116 Suggested Reading 116 Part II Applications of the Finite Element Method in Earth Science 119 8 Heat Transfer 121 8.1 Conductive Cooling in an Eroding Crust 122 8.2 Conductive Cooling of an Intrusion 126 Suggested Reading 135 9 Landscape Evolution 137 9.1 Evolution of a 1D River Profile 138 9.2 Evolution of a Fluvially Dissected Landscape 143 Suggested Reading 150 10 Fluid Flow in Porous Media 151 10.1 Fluid Flow Around a Fault 152 10.2 Viscous Fingering 157 Suggested Reading 166 11 Lithospheric Flexure 167 11.1 Governing Equations 167 11.2 FEM Discretization 168 11.3 Matlab Implementation 171 Suggested Reading 181 12 Deformation of Earth’s Crust 183 12.1 Governing Equations 183 12.2 Rate Formulation 185 12.3 FEM Discretization 186 12.4 Viscoelastoplasticity 188 12.5 Matlab Implementation 190 Suggested Reading 205 13 Going Further 207 13.1 Optimization 207 13.2 Using Other FEMs 213 13.3 Use of Existing Finite Element Software 215 Appendix A Derivation of the Diffusion Equation 217 Appendix B Basics of Linear Algebra with Matlab 221 Appendix C Comparison between Different Numerical Methods 227 Appendix D Integration by Parts 237 Appendix E Time Discretization 239 References 241 Index 245

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Book SynopsisTable of ContentsPreliminaries.- Incidence Matrix.- Adjacency Matrix.- Laplacian Matrix.- Cycles and Cuts.- Regular Graphs.- Line Graph of a Tree.- Algebraic Connectivity.- Distance Matrix of a Tree.- Resistance Distance.- Laplacian Eigenvalues of Threshold Graphs.- Positive Definite Completion Problem.- Matrix Games Based on Graphs.

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Book SynopsisOffers a rigorous and coherent introduction to the five basic number systems of mathematics - natural numbers, integers, rational numbers, real numbers, and complex numbers. The great merit of the book lies in its extensive list of exercises following each chapter. These exercises are designed to assist the instructor and to enhance the learning experience of the students.Table of Contents Natural numbers Integers Rational numbers Real numbers Complex numbers Sets, relations, functions Bibliography Index

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Book SynopsisTrade ReviewThe book under review contains six chapters that can be read independently, each one surveying one mathematical topic. ... It is carefully written, and it is better than a collection of lecture notes. Such books are needed for students, as a complement to the standard textbooks and to present more specialized applications of classical mathematics. The reviewer wishes there were many more such books." - Athanase Papadopoulos, Zentralblatt MATH"This book has its origins, we are told, in the authors' experiences teaching graduate students in computer science, who needed background in certain mathematical topics. Since these topics were not covered in the basic courses that these students had taken, the authors undertook to introduce them in courses spanning several semesters, the lecture notes of which, suitably expanded, became this text. ... I like expository books, because I think, particularly in these days of increasing specialization, that they serve a valuable purpose, not only for students but also professionals who want to see what's going on in other areas, or who need some background in one area for research in another. This book is a fine example of that genre." - Mark Hunacek, MAA ReviewsTable of Contents Measaure and integral High-dimensional geometry and measure concentration Fourier analysis Representations of finite groups Polynomials Topology Index

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Book SynopsisAddresses various aspects of the theory of automorphic forms and its relations with the theory of $L$-functions, the theory of elliptic curves, and representation theory. This volume is intended for researchers interested in expanding their own areas of focus, thus allowing them to build bridges to mathematical questions in other fields.Table of Contents S. Anni, A note on the minimal level of realization for a mod $\ell$ eigenvalue system A. Arnold-Roksandich, A discussion on the number eta-quotients of prime level C. Burrin, Dedekind sums, reciprocity, and non-arithmetic groups G. Chinta, I. Horozov, and C. O'Sullivan, Noncommutative modular symbols and Eisenstein series A. Espinosa, An annotated discussion of a panel presentation on improving diversity in mathematics J. S. Friedman, J. Jorgenson, and L. Smajlovic, Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps X. Guitart and M. Masdeu, Computing $p$-adic periods of abelian varieties from automorphic forms A. Haensch and B. Kane, An algebraic and analytic approach to spinor exceptional behavior in translated lattices A. K. Jha and B. Sahu, Differential operators on Jacobi forms and special values of certain Dirichlet series J. Jorgenson and L. Smajlovic, Some results in study of Kronecker limit formula and Dedekind sums D. Kelmer, Equidistribution of shears and their arithmetic applications K. Khuri-Makdisi, Fake proofs for identities involving products of Eisenstein series K. Khuri-Makdisi, Modular forms constructed from moduli of elliptic curves, with applications to explicit models of modular curves B. Kumar, J. Meher, and S. Pujahari, Some remarks on the coefficients of symmetric power $L$-functions J. Li, On primes in arithmetic progressions B. Linowitz and L. Thompson, The Fourier coefficients of Eisenstein series newforms K. Maurischat, Properties of Sturm's formula A. Odzak and L. Sceta, An application of a special form of a Tauberian theorem A. Odzak and L. Sceta, On the zeros of some $L$ functions from the extended Selberg class E. Ozman, Rational points on twisted modular curves B. Ramakrishnan, B. Sahu, and A. K. Singh, On the number of representations of certain quadratic forms in 8 variables M. Roy, Level of Siegel modular forms constructed via $\textrm{sym}^3$ lifting F. Stromberg, Dimension formulas and kernel functions for Hilbert modular forms H. Then, An explicit evaluation of the Hauptmoduli at elliptic points for certain arithmetic groups A. Trbovic, Torsion groups of elliptic curves over quadratic fields S. Wagh, Maass space for lifting from SL(2,$\mathbb{R}$) to GL(2,B) over a division quaternion algebra N. Walji, On the occurrence of large positive Hecke eigenvalues for GL(2) L. H. Walling, Representations by quadratic forms and the Eichler Commutation Relation S. Yamana, Degenerate principal series and Langlands classification.

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Book SynopsisContains the proceedings of the workshop ‘Crossing the Walls in Enumerative Geometry’, held in May 2018. The volume features a collection of both expository and research articles about mirror symmetry, quantized singularity theory (FJRW theory), and the gauged linear sigma model.Table of Contents R. Webb, Quasimaps and some examples of stacks for everybody E. Clader, Introduction to the gauged linear sigma model D. Ross, Localization and mirror symmetry W.-P. Li, A brief introduction to cosection localization and $P$-fields M. Shoemaker, Virtual classes for hypersurfaces via two-periodic complexes J. Oh, Localized Chern characters for 2-periodic complexes and virtual cycles T. Milanov, Singularity theory and mirror symmetry U. Whitcher, Counting points with Berglund-Hubsch-Kravitz mirror symmetry H. Fan and Y.-P. Lee, Variations on the theme of quantum Lefschetz R. Mi, Type II extremal transitions in Gromov-Witten theory C.-C. M. Liu, A lecture on holomorphic anomaly equations and extended holomorphic anomaly equation

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Book SynopsisProvides a gentle introduction into discrete Morse theory. Using a combinatorial approach, the author emphasizes acyclic matchings as the central object of study. The first two parts of the book can be used as a stand-alone introduction to homology, the last two parts delve into the core of discrete Morse theory.Table of Contents Preamble Preface The idea of homology The idea of discrete Morse theory A sample application How to use this book Prerequisites Guide to the literature Part 1 . Introduction to Homology Chapter 1. The First Steps Chapter 2. Simplicial Homology Chapter 3. Beyond the Simplicial Setting Part 2 . Further Aspects of Homology Theory Chapter 4. Category of Chain Complexes Chapter 5. Chain Homotopy Chapter 6. Connecting Homomorphism Chapter 7. Singular Homology Chapter 8. Cellular Homology Suggested further reading for Parts 1 and 2 Part 3 . Basic Discrete Morse Theory Chapter 9. Simplicial Collapses Chapter 10. Organizing Collapsing Sequences Chapter 11. Internal Collapses and Discrete Morse Theory Chapter 12. Explicit Homology Classes Associated to Critical Cells Chapter 13. The Critical Morse Complex Chapter 14. Implications and Variations Suggested further reading for Part 3 Part 4 . Extensions of Discrete Morse Theory Chapter 15. Algebraic Morse Theory Chapter 16. Discrete Morse Theory for Posets Chapter 17. Discrete Morse Theory for CW Complexes Chapter 18. Discrete Morse Theory and Persistence Suggested further reading for Part 4 Index List of Figures List of Tables Bibliography Index Preview Material Preface Table of Contents

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Book SynopsisFocuses on new and existing connections between three types of compactifications, thereby setting the stage for further research. The book draws on the discipline-specific expertise of all contributors, and at the same time gives a unified, self-contained reference for compactifications and related constructions in different contexts.Table of ContentsA. Balibanu, A quasi-Poisson structure on the multiplicative Grothendieck-Springer resolution; P. Brosnan, Volumes of definable sets in o-minimal expansions and affine GAGA theorems; P. Crooks and R. Roser, Hessenberg varieties and Poisson slices; G. Denham and A. Steiner, Geometry of logarithmic derivations of hyperplane arrangements; I. Halacheva, Shift of argument algebras and de Concini-Procesi spaces; B. Knudsen, Projection spaces and twisted Lie algebras; A. I. Suciu, Cohomology, Bocksteins, and resonance varieties in characteristic 2.

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Book SynopsisFocuses on group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems.Table of Contents Dynamical systems Group actions Groupoids Iterated monodromy groups Groups from groupoids Growth and amenability Bibliography Index

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Book SynopsisDonald Knuth's influence in computer science ranges from the invention of literate programming to the development of the TeX programming language. One of the foremost figures in the field of mathematical sciences, his papers are widely referenced and stand as milestones of development over a wide range of topics. This volume assembles more than three dozen of Professor Knuth's pioneering contributions to discrete mathematics. It includes a variety of topics in combinatorial mathematics (finite geometries, graph theory, enumeration, partitions, tableaux, matroids, codes); discrete algebra (finite fields, groupoids, closure operators, inequalities, convolutions, Pfaffians); and concrete mathematics (recurrence relations, special numbers and notations, identities, discrete probability). Of particular interest are two fundamental papers in which the evolution of random graphs is studied by means of generating functions.Table of Contents1. Discussion of Mr. Riordan's paper 'Abel identities and inverse relations'; 2. Duality in addition chains; 3. Combinatorial analysis and computers; 4. Tables of finite fields; 5. Finite semifields and projective planes; 6. A class of projective planes; 7. Construction of a random sequence; 8. Oriented subtrees of an arc digraph; 9. Another enumeration of trees; 10. Notes on central groupoids; 11. Permutations, matrices, and generalized Young tableaux; 12. A note on solid partitions; 13. Subspaces, subsets, and partitions; 14. Enumeration of plane partitions; 15. Complements and transitive closures; 16. Permutations with nonnegative partial sums; 17. Wheels within wheels; 18. The asymptotic number of geometries; 19. Random matroids; 20. Identities from partition involutions; 21. Huffman's algorithm via algebra; 22. A permanent inequality; 23. Efficient balanced codes; 24. The power of a prime that divides a generalized binomial coefficient; 25. The first cycles in an evolving graph; 26. The birth of the giant component; 27. Polynomials involving the floor function; 28. The sandwich theorem; 29. Aztec diamonds, checkerboard graphs, and spanning trees.

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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 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 SynopsisAn Introduction to Algebraic and Combinatorial Coding Theory is a comprehensive book that offers a thorough exploration of the principles and techniques of coding theory. It serves as a valuable resource for readers interested in gaining a deeper understanding of error detection and correction in communication systems. With its well-structured chapters covering coding theory fundamentals, algebraic codes, cyclic codes, block codes, and advanced coding techniques, this book caters to the needs of students, researchers, and professionals in the field. It provides a solid foundation in coding theory and showcases its practical applications in various domains, including telecommunications, data storage, and cryptography.Table of Contents Chapter 1 Introduction Chapter 2 Digital Image Information Hiding Algorithm Research Based on LDPC Code Chapter 3 Rate-Adaptive BCH Codes for Distributed Source Coding Chapter 4 Reducing the Overhead of BCH Codes: New Double Error Correction Codes Chapter 5 Ideals of Numerical Semigroups and Error-Correcting Codes Chapter 6 Turbo Codes for Multi-Hop Wireless Sensor Networks with Decode-and-Forward Mechanism Chapter 7 Entanglement-Assisted Quantum Codes from Cyclic Codes Chapter 8 Low-Complexity Chase Decoding of Reed–Solomon Codes Using Channel Evaluation Chapter 9 How Reed-Solomon Codes Can Improve Steganographic Schemes Chapter 10 A New Minimize Matrix Computation Coding Method for Distributed Storage Systems Chapter 11 Permutation-Based Block Code for Short Packet Communication Systems Chapter 12 A Simple Neural-Network-Based Decoder for Short Binary Linear Block Codes Chapter 13 An Analytical Approach to Error Detection and Correction for Onboard Nanosatellites Chapter 14 Application of Forward Error Correction (FEC) Codes in Wireless Acoustic Emission Structural Health Monitoring on Railway Infrastructures Chapter 15 Blind Recognition of Forward Error Correction Codes Based on Recurrent Neural Network

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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 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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