Algebra Books

Book SynopsisProvides an exploration of standard numerical analysis topics, as well as non-traditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. This textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering.Trade Review"Distinguishing features are the inclusion of many recent applications of numerical methods and the extensive discussion of methods based on Chebyshev interpolation. This book would be suitable for use in courses aimed at advanced undergraduate students in mathematics, the sciences, and engineering."--Choice "An instructor could assemble several different one-semester courses using this book--numerical linear algebra and interpolation, or numerical solutions of differential equations--or perhaps a two-semester sequence. This is a charming book, well worth consideration for the next numerical analysis course."--William J. Satzer, MAA FocusTable of ContentsPreface xiii Chapter 1: MATHEMATICAL MODELING 1 1.1 Modeling in Computer Animation 2 1.1.1 A Model Robe 2 1.2 Modeling in Physics: Radiation Transport 4 1.3 Modeling in Sports 6 1.4 Ecological Models 8 1.5 Modeling a Web Surfer and Google 11 1.5.1 The Vector Space Model 11 1.5.2 Google's PageRank 13 1.6 Chapter 1 Exercises 14 Chapter 2: BASIC OPERATIONS WITH MATLAB 19 2.1 Launching MATLAB 19 2.2 Vectors 20 2.3 Getting Help 22 2.4 Matrices 23 2.5 Creating and Running .m Files 24 2.6 Comments 25 2.7 Plotting 25 2.8 Creating Your Own Functions 27 2.9 Printing 28 2.10 More Loops and Conditionals 29 2.11 Clearing Variables 31 2.12 Logging Your Session 31 2.13 More Advanced Commands 31 2.14 Chapter 2 Exercises 32 Chapter 3: MONTE CARLO METHODS 41 3.1 A Mathematical Game of Cards 41 3.1.1 The Odds in Texas Holdem 42 3.2 Basic Statistics 46 3.2.1 Discrete Random Variables 48 3.2.2 Continuous Random Variables 51 3.2.3 The Central Limit Theorem 53 3.3 Monte Carlo Integration 56 3.3.1 Buffon's Needle 56 3.3.2 Estimating pi 58 3.3.3 Another Example of Monte Carlo Integration 60 3.4 Monte Carlo Simulation of Web Surfing 64 3.5 Chapter 3 Exercises 67 Chapter 4: SOLUTION OF A SINGLE NONLINEAR EQUATION IN ONE UNKNOWN 71 4.1 Bisection 75 4.2 Taylor's Theorem 80 4.3 Newton's Method 83 4.4 Quasi-Newton Methods 89 4.4.1 Avoiding Derivatives 89 4.4.2 Constant Slope Method 89 4.4.3 Secant Method 90 4.5 Analysis of Fixed Point Methods 93 4.6 Fractals, Julia Sets, and Mandelbrot Sets 98 4.7 Chapter 4 Exercises 102 Chapter 5: FLOATING-POINT ARITHMETIC 107 5.1 Costly Disasters Caused by Rounding Errors 108 5.2 Binary Representation and Base 2 Arithmetic 110 5.3 Floating-Point Representation 112 5.4 IEEE Floating-Point Arithmetic 114 5.5 Rounding 116 5.6 Correctly Rounded Floating-Point Operations 118 5.7 Exceptions 119 5.8 Chapter 5 Exercises 120 Chapter 6: CONDITIONING OF PROBLEMS; STABILITY OF ALGORITHMS 124 6.1 Conditioning of Problems 125 6.2 Stability of Algorithms 126 6.3 Chapter 6 Exercises 129 Chapter 7: DIRECT METHODS FOR SOLVING LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS 131 7.1 Review of Matrix Multiplication 132 7.2 Gaussian Elimination 133 7.2.1 Operation Counts 137 7.2.2 LU Factorization 139 7.2.3 Pivoting 141 7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required 144 7.2.5 Implementation Considerations for High Performance 148 7.3 Other Methods for Solving Ax = b 151 7.4 Conditioning of Linear Systems 154 7.4.1 Norms 154 7.4.2 Sensitivity of Solutions of Linear Systems 158 7.5 Stability of Gaussian Elimination with Partial Pivoting 164 7.6 Least Squares Problems 166 7.6.1 The Normal Equations 167 7.6.2 QR Decomposition 168 7.6.3 Fitting Polynomials to Data 171 7.7 Chapter 7 Exercises 175 Chapter 8: POLYNOMIAL AND PIECEWISE POLYNOMIAL INTERPOLATION 181 8.1 The Vandermonde System 181 8.2 The Lagrange Form of the Interpolation Polynomial 181 8.3 The Newton Form of the Interpolation Polynomial 185 8.3.1 Divided Differences 187 8.4 The Error in Polynomial Interpolation 190 8.5 Interpolation at Chebyshev Points and chebfun 192 8.6 Piecewise Polynomial Interpolation 197 8.6.1 Piecewise Cubic Hermite Interpolation 200 8.6.2 Cubic Spline Interpolation 201 8.7 Some Applications 204 8.8 Chapter 8 Exercises 206 Chapter 9: NUMERICAL DIFFERENTIATION AND RICHARDSON EXTRAPOLATION 212 9.1 Numerical Differentiation 213 9.2 Richardson Extrapolation 221 9.3 Chapter 9 Exercises 225 Chapter 10: NUMERICAL INTEGRATION 227 10.1 Newton-Cotes Formulas 227 10.2 Formulas Based on Piecewise Polynomial Interpolation 232 10.3 Gauss Quadrature 234 10.3.1 Orthogonal Polynomials 236 10.4 Clenshaw-Curtis Quadrature 240 10.5 Romberg Integration 242 10.6 Periodic Functions and the Euler-Maclaurin Formula 243 10.7 Singularities 247 10.8 Chapter 10 Exercises 248 Chapter 11: NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 251 11.1 Existence and Uniqueness of Solutions 253 11.2 One-Step Methods 257 11.2.1 Euler's Method 257 11.2.2 Higher-Order Methods Based on Taylor Series 262 11.2.3 Midpoint Method 262 11.2.4 Methods Based on Quadrature Formulas 264 11.2.5 Classical Fourth-Order Runge-Kutta and Runge-Kutta-Fehlberg Methods 265 11.2.6 An Example Using MATLAB's ODE Solver 267 11.2.7 Analysis of One-Step Methods 270 11.2.8 Practical Implementation Considerations 272 11.2.9 Systems of Equations 274 11.3 Multistep Methods 275 11.3.1 Adams-Bashforth and Adams-Moulton Methods 275 11.3.2 General Linear m-Step Methods 277 11.3.3 Linear Difference Equations 280 11.3.4 The Dahlquist Equivalence Theorem 283 11.4 Stiff Equations 284 11.4.1 Absolute Stability 285 11.4.2 Backward Differentiation Formulas (BDF Methods) 289 11.4.3 Implicit Runge-Kutta (IRK) Methods 290 11.5 Solving Systems of Nonlinear Equations in Implicit Methods 291 11.5.1 Fixed Point Iteration 292 11.5.2 Newton's Method 293 11.6 Chapter 11 Exercises 295 Chapter 12: MORE NUMERICAL LINEAR ALGEBRA: EIGENVALUES AND ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 300 12.1 Eigenvalue Problems 300 12.1.1 The Power Method for Computing the Largest Eigenpair 310 12.1.2 Inverse Iteration 313 12.1.3 Rayleigh Quotient Iteration 315 12.1.4 The QR Algorithm 316 12.1.5 Google's PageRank 320 12.2 Iterative Methods for Solving Linear Systems 327 12.2.1 Basic Iterative Methods for Solving Linear Systems 327 12.2.2 Simple Iteration 328 12.2.3 Analysis of Convergence 332 12.2.4 The Conjugate Gradient Algorithm 336 12.2.5 Methods for Nonsymmetric Linear Systems 334 12.3 Chapter 12 Exercises 345 Chapter 13: NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEMS 350 13.1 An Application: Steady-State Temperature Distribution 350 13.2 Finite Difference Methods 352 13.2.1 Accuracy 354 13.2.2 More General Equations and Boundary Conditions 360 13.3 Finite Element Methods 365 13.3.1 Accuracy 372 13.4 Spectral Methods 374 13.5 Chapter 13 Exercises 376 Chapter 14: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 379 14.1 Elliptic Equations 381 14.1.1 Finite Difference Methods 381 14.1.2 Finite Element Methods 386 14.2 Parabolic Equations 388 14.2.1 Semidiscretization and the Method of Lines 389 14.2.2 Discretization in Time 389 14.3 Separation of Variables 396 14.3.1 Separation of Variables for Difference Equations 400 14.4 Hyperbolic Equations 402 14.4.1 Characteristics 402 14.4.2 Systems of Hyperbolic Equations 403 14.4.3 Boundary Conditions 404 14.4.4 Finite Difference Methods 404 14.5 Fast Methods for Poisson's Equation 409 14.5.1 The Fast Fourier Transform 411 14.6 Multigrid Methods 414 14.7 Chapter 14 Exercises 418 APPENDIX A REVIEW OF LINEAR ALGEBRA 421 A.1 Vectors and Vector Spaces 421 A.2 Linear Independence and Dependence 422 A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space 423 A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram-Schmidt Algorithm 423 A.5 Matrices and Linear Equations 425 A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility 427 A.7 Linear Transformations; the Matrix of a Linear Transformation 431 A.8 Similarity Transformations; Eigenvalues and Eigenvectors 432 APPENDIX B TAYLOR'S THEOREM IN MULTIDIMENSIONS 436 References 439 Index 445

Read more less

Book SynopsisTrade Review"Lane explores secondary, or hidden, mathematical gems that a player might discover upon mature reflection. . . . Just as most car drivers prefer not to inquire how the internal combustion engine works, most video-type users prefer not to ask how computer magic works. For the few who do ask questions, Lane assures us and as his book testifies, 'there's a lot of mathematics under the surface'."---Andrew James Simoson, MathSciNet"Lane explains some pretty technical concepts in an accessible way. . . . A fun survey of interesting maths related through the lens of video games."---Paul Taylor, Aperiodical"The examples [in Power-Up] were carefully chosen from very popular games, so even the most casual player will have heard of the vast majority of the games discussed. In general, Lane's writing is easy to digest, and the use of color and high-quality paper gives the book a nice look and feel." * Choice *"Power­Up is a very readable book based on examples taken from popular video games. . . . It is a pity that too many people are deprived of the pleasure of finding things out via the intellectual game of mathematics. Hopefully, the effort of the likes of Matthew Lane will someday solve the severe marketing problem of mathematics." * Computing Reviews *"Overall the book is excellent. Lane has written a high readable text with colorful illustrations. You won’t regret reading it and maybe Power-Up will add a new level of insight to your computer gaming." * MAA Reviews *"Matthew Lane explores the mathematical underpinning many popular video games in this well-written and very enjoyable book that is pitched at a very broad audience"---Dominic Thorrington, Mathematics TodayTable of ContentsAcknowledgments xi Introduction 1 1. Let's Get Physical 7 1.1 Platforming Perils 7 1.2 Platforming in Three Dimensions 10 1.3 LittleBigPlanet: Exploring Physics through Gameplay 12 1.4 From 2D to 3D: Bending Laws in Portal 14 1.5 Exploring Reality with A Slower Speed of Light 18 1.6 Exploring Alternative Realities 21 1.7 Beyond Physics: Minecraft or Mine Field? 26 1.8 Closing Remarks 27 1.9 Addendum: Describing Distortion 29 2. Repeat Offenders 34 2.1 Let's Play the Feud! 34 2.2 Game Shows and Birthdays 36 2.3 Beyond the First Duplicate 39 2.4 The Draw Something Debacle 41 2.5 Delayed Repetition: Increasing N 46 2.6 Delayed Repetition:Weight Lifting 48 2.7 The Completionist's Dilemma 53 2.8 Closing Remarks 55 2.9 Addendum: In Search of a Minimal k 55 3. Get Out the Voting System 58 3.1 Everybody Votes, but Not for Everything 58 3.2 Plurality Voting: An Example 60 3.3 Ranked-Choice Voting Systems and Arrow's Impossibility Theorem 61 3.4 An Escape from Impossibility? 66 3.5 Is There a "Best" System? 68 3.6 What Game Developers Know that Politicians Don't 71 3.7 The Best of the Rest 76 3.8 Closing Remarks 82 3.9 Addendum: TheWilson Score Confidence Interval 83 4. Knowing the Score 86 4.1 Ranking Players 86 4.2 Orisinal Original 87 4.3 What's in a Score? 91 4.4 Threes! Company 98 4.5 A Mathematical Model of Threes! 100 4.6 Invalid Scores 105 4.7 Lowest of the Low 109 4.8 Highest of the High 116 4.9 Closing Remarks 121 5. The Thrill of the Chase 122 5.1 I'ma GonnaWin! 122 5.2 Shell Games 123 5.3 Green-Shelled Monsters 125 5.4 Generalizations and Limitations 129 5.5 Seeing Red 131 5.6 Apollonius Circle Pursuit 134 5.7 Overview of aWinning Strategy 136 5.8 Pinpointing the Intersections 141 5.9 Blast Radius 145 5.10 The Pursuer and Pursued in Ms. Pac-Man 148 5.11 Concluding Remarks 153 5.12 Addendum: The Pursuit Curve for Red Shells and a Refined Inequality 153 6. Gaming Complexity 158 6.1 From Russia with Fun 158 6.2 P, NP, and Kevin Bacon 160 6.3 Desktop Diversions 165 6.4 Platforming Problems 169 6.5 Fetch Quests: An Overview 170 6.6 Fetch Quests and Traveling Salesmen 175 6.7 Closing Remarks 183 7. The Friendship Realm 184 7.1 Taking It to the Next Level 184 7.2 Friendship as Gameplay: The Sims and Beyond 186 7.3 A Game-Inspired Friendship Model 190 7.4 Approximations to the Model 193 7.5 The Cost of Maintaining a Friendship 195 7.6 From Virtual Friends to Realistic Romance 198 7.7 Modeling Different Personalities 200 7.8 Improving the Model (Again!) 203 7.9 Concluding Remarks 209 8. Order in Chaos 210 8.1 The Essence of Chaos 210 8.2 Love in the Time of Chaos 211 8.3 Shell Games Revisited 216 8.4 How's theWeather? 223 8.5 Concluding Remarks 225 9. The Value of Games 227 9.1 More Important Than Math 227 9.2 Why Games? 230 9.3 What Next? 242 Notes 244 Bibliography 269 Index 273

Read more less

Book SynopsisQuaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quateTrade ReviewOne of Choice's Outstanding Academic Titles for 2015 "Rodman fills a void in the monographic literature with this work."--Choice "The book is self-contained and well organized... Full and detailed proofs are supplied. Another exciting point is the presence of many open problems throughout the book."--Gisele C. Ducati, MatchSciNetTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. xi*Chapter One. Introduction, pg. 1*Chapter Two. The algebra of quaternions, pg. 9*Chapter Three. Vector spaces and matrices: Basic theory, pg. 28*Chapter Four. Symmetric matrices and congruence, pg. 64*Chapter Five. Invariant subspaces and Jordan form, pg. 83*Chapter Six. Invariant neutral and semidefinite subspaces, pg. 131*Chapter Seven. Smith form and Kronecker canonical form, pg. 153*Chapter Eight. Pencils of hermitian matrices, pg. 172*Chapter Nine. Skewhermitian and mixed pencils, pg. 194*Chapter Ten. Indefinite inner products: Conjugation, pg. 228*Chapter Eleven. Matrix pencils with symmetries: Nonstandard involution, pg. 261*Chapter Twelve. Mixed matrix pencils: Nonstandard involutions, pg. 279*Chapter Thirteen. Indefinite inner products: Nonstandard involution, pg. 300*Chapter Fourteen. Matrix equations, pg. 328*Chapter Fifteen. Appendix: Real and complex canonical forms, pg. 339*Bibliography, pg. 353*Index, pg. 361

Read more less

Book SynopsisTrade Review"Are you a video game enthusiast who is getting tired of being asked 'How can you waste time on such stuff?' This book is your answer! Matthew Lane skillfully weaves a tale of how video games can be important tools for teaching mathematics and physics. As a long-time video gamer, I highly recommend Power-Up."—Paul J. Nahin, author of In Praise of Simple Physics"What a delightful journey through the math of hidden worlds! This is much more than a book about video games. It's an exploration of interconnectedness and an invitation for the perpetually curious."—Karim Ani, founder of Mathalicious"A fun survey."—Paul Taylor, Aperiodical"A very readable book."—Computing Reviews

Read more less

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

Read more less

Book Synopsis

Read more less

Book SynopsisPresents a study of the theory of totally nonnegative matrices, defined by the nonnegativity of subdeterminants. This title explores methodological background, historical highlights of key ideas, and specialized topics.Trade Review"This book is a very useful new reference on the subject of TN matrices and it will be of interest to researchers on matrix theory as well as to researchers of any field where total positivity has applications."---Juan Manuel Pefia, Mathematical Reviews

Read more less

Book SynopsisA collection of lectures presented at the Fourth International Conference on Nonassociative Algebra and its Applications, held in Sao Paulo, Brazil. Topics in algebra theory include alternative, Bernstein, Jordan, lie, and Malcev algebras and superalgebras. The volume presents applications to population genetics theory, physics, and more.Table of ContentsPreface -- Contributors -- New approach to octonions and Cayley algebras /Helena Albuquerque and Shahn Majid -- Identities of graded alternative algebras /Y. Bahturin and M. Zaicev -- Invariant connections on symmetric spaces /Pilar Benito, Cristina Draper, and Alberto Elduque -- £-ideals in Bernstein algebras /A. Catalan -- Octonions and affine connections on spheres /Alberto Elduque and Hyo Chul Myung -- Immediate applications of a new classification of finite dimensional simple Lie algebras /Mohammed El Houari -- An approach to integrable theories in any dimension: the role of non-semi simple Lie algebras /Luiz A. Ferreira -- Versal deformations of Lie algebras /Alice Fialowski -- Derivations in second order Bernstein algebras /M. A. Garcia-Muniz and C. Martinez -- Algebraic structure of the non-abelian Toda models /J. F. Gomes, F. E. Mendonga da Silveira, G. M. Sotkov, and A. H. Zimerman -- Alternative rings of small order and the hunt for Moufang circle loops /Edgar G. Goodaire -- A note on linearization of some identities /H. Guzzo, Jr., and P. Vicente -- The commuting nucleus of right alternative rings /Irvin Roy Hentzel -- A construction of Jordan superalgebras from Jordan-Lie triple systems /Noriaki Kamiya and Susumu Okubo -- Rings with (x,y,z) = (x,z,y) /Erwin Kleinfeld and Margaret Kleinfeld -- On the semi simplicity of special Jordan superalgebras /Carlos Gomez-Ambrosi and Jesus Laliena -- Bernstein superalgebras of dimension four /S. Gonzalez, C. Lopez-Diaz, and C. Martinez -- Banach-Bemstein algebras and their applications /Yu. Lyubich -- Jordan superalgebras /C. Martinez -- Derivations of structurable / /*-algebras /J. Martinez Moreno -- Lie mappings of C* -algebras /Martin Mathieu -- Sur des anneaux commutatifs a puissances associatives de caracteristique premiere p> 0 /Joao Carlos da Motta Ferreira and Artibano Micali -- Principal ^-bundles /Alexander I. Nesterov -- Z x Z-graded Lie algebras containing a Heisenberg algebra /J. Marshall Osborn and Kaiming Zhao -- A„(l) Toda solitons: a relation between dressing transformations and vertex operators /H. Belich and R. Paunov -- Generalized quadratic rings /Luiz Antonio Peresi -- The derivation algebra of a division composition algebra /J. M. Perez-Izquierdo -- On the structure of the simple Moufang loop GLL (F2) /M. L. Merlini Giuliani and Cesar Polcino Milies -- Lie groups, differential geometry, and nonlinear integrable systems /A V. Razumov and M. V. Saveliev -- Smooth quasigroups and loops. Recent achievements and open problems /L. V. Sabinin -- On smooth left square distributive quasigroup /Liudmila Sabinina -- Algebraic structure of trans symmetric spaces /L. V. Sbitneva -- Bernstein algebras whose lattice of ideals is distributive /C. Martinez and J. Setd -- Specialty problem for Malcev algebras and Poisson Malcev algebras /Ivan P. Shestakov -- Applications of Lie ring methods to group theory /Pavel Shumyatsky -- Computable rings, groups and their isomorphisms /Bakhadyr Khoussainov and Arkadii Slinko -- Quantum F-algebras /J. F. Gomes, F. E. Mendonga da Silveira, G. M. Sotkov, and A. H. Zimerman -- Conditions d’existence d’une loi de Hardy-Weinberg periodique dans des populations selectivement neutres /C. Mallol and R. Varro -- Finite semi fields three-dimensional over the left nuclei /G. P. Wene -- Supplementing radicals of nonassociative rings /Richard Wiegandt.

Read more less

Book SynopsisThis volume publishes key proceedings from the recent International Conference on Hopf Algebras held at DePaul University, Chicago, Illinois. With contributions from leading researchers in the field, this collection deals with current topics ranging from categories of infinitesimal Hopf modules and bimodules to the construction of a Hopf algebraic Morita invariant. It uses the newly introduced theory of bi-Frobenius algebras to investigate a notion of group-like algebras and summarizes results on the classification of Hopf algebras of dimension pq. It also explores pre-Lie, dendriform, and Nichols algebras and discusses support cones for infinitesimal group schemes.Table of ContentsI. Infinitesimal Bialgebras, Pre-Lie and Dendriform Algebras. 2. Some Remarks on Nichols Algebras. Nicolas Andruskiewitsch. 3. The Coradical of the Dual of a Lifting of a Quantum Plane Nicolas Andruskiewitsch and M Beattie. 4. Representations of Two-Parameter Quantum Groups and Schur-Weyl Duality. 5. A New Proof of the Skolem-Noether Theorem. 6. Projectivity of a Relative Hopf Module over the Subring of Coinvariants. 7. A Brief Introduction to Coalgebra Representation Theory. 8. Some Examples of Integrals for Bialgebras. 9. Bi-Frobenius Algebras and Group-Like Algebras. 10. Bialgebras and Realizations. 11. Relatively Free Coalgebras. 12. Example of Almost Commutative Hopf Algebras Which Are Not Coquasitriangular. 13. Hopf Algebras of Dimension p2. 14. Support Cones for Infinitesimal Group Schemes. 15. Coalgebras from Formulas. 16. Fourier Theory for Coalgebras, Bicointegrals and Injectivity for Bicomodules. 17. Notes on the Classification of Hopf Algebras of Dimension pq. 18. A Hopf Algebraic Morita Invariant

Read more less

Book SynopsisAl-Khwarizmi was a mathematician, astronomer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad during the first half of the 9th century and is considered by many to be the father of algebra. This book deals with algebraic theory, and focuses on the calculation of inheritances and legacies.

Read more less

Book SynopsisThe first graduate-level introduction to the finite generation theorem of the canonical ring, a major achievement of modern algebraic geometry. Largely self-contained, this text explains the basics of minimal model theory, covering all the progress of the last three decades and assuming only the basics in algebraic geometry.

Read more less

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

Read more less

Book Synopsis

Read more less

Book SynopsisThe theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This is the second volume of Algebras, Rings and Modules: Non-commutative Algebras and Rings by M. Hazewinkel and N. Gubarenis, a continuation stressing the more important recent results on advanced topics of the structural theory of associative algebras, rings and modules. Table of ContentsRings related to finite posets. Distributive and semidistributive rings. The group of extensions. Modules over semiperfect rings. Representations of primitive posets. Representations of quivers, species and finite dimensional algebras. Artinian rings of finite representation type. O-species and SPSD-rings of bounded representation type.

Read more less

Book SynopsisThe first and second editions of this successful textbook have been highly praised for their lucid and detailed coverage of abstract algebra. In this third edition, the author has carefully revised and extended his treatment, particularly the material on rings and fields, to provide an even more satisfying first course in abstract algebra.Table of ContentsChapter One SETS AND LOGIC -- 1. Some very general remarks -- 2. Introductory remarks on sets -- 3. Statements and conditions; quantifiers -- 4. The implies sign () -- 5. Proof by contradiction -- 6. Subsets -- 7. Unions and intersections -- 8. Cartesian product of sets -- EXERCISES -- Chapter Two SOME PROPERTIES OF -- 9. Introduction -- 10. The well-ordering principle -- I I. The division algorithm -- 12. Highest common factors and Euclid’s algorithm -- 13. The fundamental theorem of arithmetic -- 14. Congruence modulo m (mE f4J) -- EXERCISES -- Chapter Three EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES -- 15. Relations in general -- 16. Equivalence relations -- 17. Equivalence classes -- 18. Congruence classes -- 19. Properties of l,, as an algebraic system -- EXERCISES -- Chapter Four MAPPINGS -- 20. Introduction -- 21. The image of a subset of the domain; surjcctions -- 22. Injections; bijections; inverse of a bijection -- 23. Restriction of a mapping -- 24. Composition of mappings -- 25. Some further results and examples on mappings -- EXERCISES -- Chapter Five SEMIGROUPS -- 26. Introduction -- 27. Binary operations -- 28. Associativity and commutativity -- 29. Semigroups: definition and examples -- 30. Powers of an element in a semigroup -- 31. Identity elements and inverses -- 32. Subsemigroups -- EXERCISES -- Chapter Six AN INTRODUCTION TO GROUPS -- 33. The definition of a group -- 34. Examples of groups -- 35. Elementary consequences of the group axioms -- 36. Subgroups -- 37. Some important general examples of subgroups -- 38. Period of an element -- 39. Cyclic groups -- EXERCISES -- Chapter Seven COSETS AND LAGRANGE’S THEOREM ON FINITE GROUPS -- 40. Introduction -- 41. Multiplication of subsets of a group -- 42. Another approach to cosets -- 43. Lagrange’s theorem -- 44. Some consequences of Lagrange’s theorem -- EXERCISES -- Chapter Eight HOMOMORPHISMS, NORMAL SUBGROUPS, AND QUOTIENT GROUPS -- 45. Introduction -- 46. Isomorphic groups -- 47. Homomorphisms and their elementary properties -- 48. Conjugacy -- 49. Normal subgroups -- 50. Quotient groups -- 51. The quotient group G/Z -- 52. The first isomorphism theorem -- EXERCISES -- Chapter Nine THE SYMMETRIC GROUP S -- 53. Introduction -- 54. Cycles -- 55. Products of disjoint cycles -- 56. Periods of elements of Sft -- 57. Conjugacy in S1 -- 58. Arrangement of the objects 1,2,...,n -- 59. The alternating character, and alternating groups -- 60. The simplicity of A5 -- EXERCISES -- Chapter Ten THE FUNDAMENTALS OF RING THEORY -- 61. Introduction -- 62. The definition of a ring and its elementary consequences -- 63. Special types of ring and ring elements -- 64. Subrings and subtIelds -- 65. Ring homomorphisms -- 66. Ideals -- 67. Principal ideals in a commutative ring with a one -- 68. Factor rings -- 69. Characteristic of an integral domain or field -- 70. The field of fractions of an integral domain -- EXERCISES -- Chapter Eleven POLYNOMIALS AND FIELDS -- 71. Introduction -- 72. Polynomial rings -- 73. Some properties of F[X], where F is a field -- 74 Generalities on factorization -- 75. Further properties of F[XJ, where F is a field -- 76. Some matters of notation -- 77. Minimal polynomials and the structure of F(c) -- 78. Some elementary properties of finite fields -- 79. Construction of fields by root adjunction -- 80. Degrees of field extensions -- 81. Epilogue -- EXERCISES -- BIBLIOGRAPHY -- APPENDIX TO EXERCISES -- INDEX.

Read more less

Book Synopsisthis section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets.Trade Review“This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a ‘transition’ course.” (Margret Höft, zbMATH 1012.00013, 2021)“The contents of the book is organized in three parts … . this is a nice book, which also this reviewer has used with profit in his teaching of beginner students. It is written in a highly pedagogical style and based upon valuable didactical ideas.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)“Books in this category are meant to teach mathematical topics and techniques that will become valuable in more advanced courses. This book meets these criteria. … This book is well suited as a textbook for a transitional course between calculus and more theoretical courses. I also recommend it for academic libraries.” (Edgar R. Chavez, ACM Computing Reviews, February, 2012)“This is an improved edition of a good book that can serve in the undergraduate curriculum as a bridge between computationally oriented courses like calculus and more abstract courses like algebra.” (Teun Koetsier, Zentralblatt MATH, Vol. 1230, 2012)Table of ContentsPreface to the Second Edition Preface to the First Edition To the Student To the Instructor Part I. Proofs 1. Informal Logic 2. Strategies for Proofs Part II. Fundamentals 3. Sets 4. Functions 5. Relations 6. Finite and Infinite Sets Part III. Extras 7. Selected Topics 8. Explorations Appendix: Properties of Numbers Bibliography Index

Read more less

Book SynopsisA Classroom-Tested, Alternative Approach to Teaching Math for Liberal Arts Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking uses puzzles and paradoxes to introduce basic principles of mathematical thought. The text is designed for students in liberal arts mathematics courses. Decision-making situations that progress from recreational problems to important contemporary applications develop the critical-thinking skills of non-science and non-technical majors. The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. The text's five units focus on graphs, logic, probability, voting, and cryptography. The authors also cover related areas, such as operations research, game theory, number theory, combinatorics, statistics, and circuit design. The text uses a core set oTrade Review"… an interesting approach to the world of mathematics. … The layout is good, as is the coverage. … The reinforcing exercises are excellent. Using logic and other techniques, the text lays out methods to help students learn to think in a mathematical manner. Summing up: Recommended. General readers, lower- and upper-division undergraduates."—M. D. Sanford, Felician College, Lodi, New Jersey, USA for CHOICE, November 2015Table of ContentsGraphs: Puzzles and Optimization. Logic: Rational Inference and Computer Circuits. Probability: Predictions and Expectations. Counting: Voting Methods and Apportionment. Numbers: Cryptosystems and Security. Appendices. Index.

Read more less

Book SynopsisAlgebraic Operads: An Algorithmic Companion presents a systematic treatment of Gröbner bases in several contexts. The book builds up to the theory of Gröbner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. The authors present a variety of topics including: noncommutative Gröbner bases and their applications to the construction of universal enveloping algebras; Gröbner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Gröbner bases, important for applications to algebraic topology, and homological and homotopical algebra. The last chapters of the book combine classical commutative Gröbner bases with operadic ones to approach some classification problems for operads. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exerciseTrade Review"This book presents a systematic treatment of Gröbner bases, and more generally of the problem of normal forms, departing from linear algebra, going through commutative and noncommutative algebra, to operads. The algorithmic aspects are especially developed, with numerous examples and exercises."- Loϊc Foissy"By balancing computational methods and abstract reasoning, the authors of the book under review have written an excellent up-to-date introduction to Grobner basis methods applicable to associative structures, especially including operads. The book will be of interest to a wide range of readers, from undergraduates to experts in the field."~ Ralf Holtkamp, Mathematical Reviews, March 2018"This book presents a systematic treatment of Gröbner bases, and more generally of the problem of normal forms, departing from linear algebra, going through commutative and noncommutative algebra, to operads. The algorithmic aspects are especially developed, with numerous examples and exercises."- Loϊc Foissy"By balancing computational methods and abstract reasoning, the authors of the book under review have written an excellent up-to-date introduction to Grobner basis methods applicable to associative structures, especially including operads. The book will be of interest to a wide range of readers, from undergraduates to experts in the field."~ Ralf Holtkamp, Mathematical Reviews, March 2018Table of ContentsNormal Forms for Vectors and Univariate Polynomials. Noncommutative Associative Algebras. Nonsymmetric Operads. Twisted Associative Algebras and Shuffle Algebras. Symmetric Operads and Shuffle Operads. Operadic Homological Algebra and Gröbner Bases. Commutative Gröbner Bases. Linear Algebra over Polynomial Rings. Case Study of Nonsymmetric Binary Cubic Operads. Case Study of Nonsymmetric Ternary Quadratic Operads. Appendices: Maple Code for Buchberger’s Algorithm.

Read more less

Book SynopsisThe Essentials of a First Linear Algebra Course and MoreLinear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear transformations. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the spectral theorem.An Engaging Treatment of the Interplay among Algebra, Geometry, and MappingsThe text starts with basic questions about images and pre-images of mappings, injectivity, surjectivity, and distortion. In the process of answering these questions in the linear setting, the book covers all the standard topics for a first course on linear algebra, including linear systems, vector geometry, matrix algebra, subspaces, independence, dimension, orthogonality, eigenvectors, and diagonalization. A Smooth Transition to the Conceptual Realm of Higher MathematicsThis book guides students on a journey from Trade Review"All the standard topics of a first course are covered, but the treatment omits abstract vector spaces. … What is unusual is the author's aim to interpret every concept and result geometrically, thus motivating the student to learn to visualize what is going on, rather than just relying on calculations. This is a strong and useful feature. … The book has very many practice sections with over 500 exercises, most of them numerical. … As the author mentions in the preface, it was his aim to provide a sound mathematical introduction, and in the reviewer's opinion he has succeeded in doing this."—Zentralblatt MATH 1314Table of ContentsVectors, Mappings, and Linearity. Solving Linear Systems. Linear Geometry. The Algebra of Matrices. Subspaces. Orthogonality. Linear Transformation. Appendices. Index.

Read more less

Book SynopsisThe book reviews inequalities for weighted entry sums of matrix powers. Applications range from mathematics and CS to pure sciences. It unifies and generalizes several results for products and powers of sesquilinear forms derived from powers of Hermitian, positive-semidefinite, as well as nonnegative matrices. It shows that some inequalities are valid only in specific cases. How to translate the Hermitian matrix results into results for alternating powers of general rectangular matrices? Inequalities that compare the powers of the row and column sums to the row and column sums of the matrix powers are refined for nonnegative matrices. Lastly, eigenvalue bounds and derive results for iterated kernels are improved.Table of ContentsIntroduction. Notation and Basic Facts. Motivation. Diagonalization and Spectral Decomposition. Undirected Graphs / Hermitian Matrices. General Results. Restricted Graph Classes. Directed Graphs / Nonsymmetric. Walks and Alternating Walks in Directed Graphs. Powers of Row and Column Sums. Applications. Bounds for the Largest Eigenvalue. Iterated Kernels. Conclusion. Bibliography. Index.

Read more less

Book SynopsisInvitation to Linear Algebra is an informative, clearly written, flexible textbook for instructors and students. Based on over 30 years of experience as a mathematics professor, the author invites students to develop a more informed understanding of complex algebraic concepts using innovative, easy-to-follow methods. The book is organized into lessons rather than chapters. This limits the size of the mathematical morsels that students must digest, making it easier for instructors to budget class time. Each definition is carefully explained with detailed proofs of key theorems, including motivation for each step. This makes the book more flexible, allowing instructors to choose material that reflects their and their students' interests. A larger than normal amount of exercises illustrate how linear and nonlinear algebra apply in the students' areas of study. Features The book's unique lesson format enabTable of ContentsMatrices and Linear Systems. Introduction to Matrices. Matrix Multiplication. Additional Topics in Matrix Algebra. Introduction to Linear Systems. The Inverse of a Matrix. Determinants. Introduction to Determinants. Properties of Determinants. Applications of Determinants. A First Look at Vector Spaces. Introduction to Vector Spaces. Subspaces of Vector Spaces. Linear Dependence and Independence. Basis and Dimension. The Rank of a Matrix. Linear Systems Revisited. More About Vector Spaces. Sums and Direct Sums of Subspaces. Quotient Spaces. Change of Basis. Euclidean Spaces. Orthonormal Bases. Linear Transformations. Introduction to Linear Transformations. Isomorphisms of Vector Spaces. The Kernel and Range of a Linear Transformation. Matrices of Linear Transformations. Similar Matrices. Matrix Diagonalization. Eigenvalues and Eigenvectors. Diagonalization of Square Matrices. Diagonalization of Symmetric Matrices. Complex Vector Spaces. Complex Vector Spaces. Unitary and Hermitian Matrices. Advanced Topics. Powers of Matrices. Functions of a Square Matrix. Matrix Power Series. Minimal Polynomials. Direct Sum Decompositions. Jordan Canonical Form. Applications. Systems of First Order Differential Equations. Stability Analysis of First Order Systems. Coupled Oscillations. Appendix. Solutions and Hints to Selected Exercises.

Read more less

Book SynopsisWhat comes to mind when you think about algebra? For many of us, it's memories of dull or frustrating classes in high school. Award-winning mathematics professor G. Arnell Williams is here to change that. Algebra the Beautiful is a journey into the heart of fundamental math that proves just how amazing this subject really is. Drawing on lessons from twenty-five years of teaching mathematics, Williams blends metaphor, history, and storytelling to uncover algebra's hidden grandeur. Whether you're a teacher looking to make math come alive for your students, a parent hoping to get your children engaged, a student trying to come to terms with a sometimes bewildering subject, or just a lover of mathematics, this book has something for you. With a passion that's contagious, G. Arnell Williams shows how each of us can grasp the beauty and harmony of algebra.

Read more less

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.

Read more less

Book SynopsisComprising a selection of expository and research papers, Harmonic Analysis and Integral Geometry grew from presentations offered at the July 1998 Summer University of Safi, Morocco-an annual, advanced research school and congress. This lively and very successful event drew the attendance of many top researchers, who offered both individual lectures and coordinated courses on specific research topics within this fast growing subject.Harmonic Analysis and Integral Geometry presents important recent advances in the fields of Radon transforms, integral geometry, and harmonic analysis on Lie groups and symmetric spaces. Several articles are devoted to the new theory of Radon transforms on trees.With its related presentations addressing recent developments in various aspects of these intriguing areas of study, Harmonic Analysis and Integral Geometry becomes an important addition not only to the Research Notes in Mathematics series, but to the general mathematics literature.Table of ContentsJohn's Equation and the Plane to Line Transform on R3. Radon Transforms on Compact Grassmann Manifolds and Invariant Differential Operators of Determinantal Type. Invariant Berezin Transforms. Integral Geometry on Hyperbolic Spaces. On Laguerre Polynomials of Two Variables. A Topological Obstruction for the Real Radon Transform. Integral Geometry in the Sphere Sd. The Distribution-Valued Horocyclic Radon Transform on Trees. The Geodesic Radon Transform on Trees. Integral Geometry on Affine Buildings. Poisson Transform on H3. Realization of a Holomorphic Discete-Series of the Lie Group SU(1,2) as Star-Representation. q-Analogue of Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials.

Read more less

Book SynopsisAbout the book…In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend.These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra. about the editors…H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group.OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visiting assistant professor in 1987. In 1988, he joined the Algebra research group at Auburn University.Table of ContentsSome Aspects of Noncommutative Geometry. Omega Groups. Cancellation for Quotient Divisible Mixed Abelian Groups. Abelian Groups and Topological Structures. Gorenstein Homological Algebra. Applications of Module Approximations. Picard Groups of Certain Module Categories. Some Trends in the Theory of Covers and Envelopes. On Degenerate B2 Groups. The Loewy Length of Modules over Almost Perfect Domains. Half-factorial Domains. Projective Presentations of Finitely Generated Modules over Integral Domains. E-locally Cyclic Abelian Groups and Maximal Near-rings of Mappings. Approximately Simultaneously Diagnoalizable Matrices. FI-extending Hulls for Abelian Groups. Torsion-free Modules over a Discrete Valuation Ring. Forcing a Finite Group to be Abelian. Torsion-free Modules over Non-commutative Rings. Axiom 3 and Coverings. Divisibility Properties in Ultrapowers of Commutative Rings. Localizations of Torsion-free Abelian Groups. Modular Representations of Algebraic Groups, Finite Groups, and Frobenius Kernels. Homological Properties of Universal Enveloping Algebras.

Read more less

Book SynopsisAlthough there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains. As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings. Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.Trade Review"All topics … are developed in a clear way and illustrated by many examples." – EMS Newsletter, September 2008Table of ContentsBirational Simple Extensions. Simple Extensions of High Degree. Subrings of Anti-Integral Extensions. Denominator Ideals and Excellent Elements. Unramified Extensions. The Unit Groups of Extensions. Exclusive Extensions of Noetherian Domains. Ultra-Primitive Extensions and Their Generators. Flatness and Contractions of Ideals. Anti-Integral Ideals and Super Primitive Polynomials. Semi Anti-Integral and Pseudo-Simple Extensions. References. Index.

Read more less

Book Synopsis

Read more less

Book Synopsis

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

Book SynopsisThis textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

Read more less