Algebra Books

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Book SynopsisThis book determines whether the general member of each family of smooth Fano threefolds admits a KählerEinstein metric, using K-stability. Complemented by appendices outlining results needed to understand this active area, it will be essential reading for researchers and graduate students working on algebraic and complex geometry.Trade Review'The notion of K-stability for Fano manifold has origins in differential geometry and geometric analysis but is now also of fundamental importance in algebraic geometry, with recent developments in moduli theory. This monograph gives an account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105 deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London'It is a difficult problem to check whether a given Fano variety is K-polystable. This book settles this problem for the general members of all the 105 deformation families of smooth Fano 3-folds. The book is recommended to anyone interested in K-stability and existence of Kähler-Einstein metrics on Fano varieties.' Caucher Birkar FRS, Tsinghua University and University of CambridgeTable of ContentsIntroduction; 1. K-stability; 2. Warm-up: smooth del Pezzo surfaces; 3. Proof of main theorem: known cases; 4. Proof of main theorem: special cases; 5. Proof of main theorem: remaining cases; 6. The big table; 7. Conclusion; Appendix. Technical results used in proof of main theorem; References; Index.

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Book SynopsisFrom social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.Trade Review'This is a well-planned book that is true to its title in that it is indeed accessible for anyone with just an undergraduate student's knowledge of enumerative combinatorics and probability.' Miklós Bóna, MAA ReviewsTable of ContentsPreface; Part I. Basic Models: 1. Random graphs; 2. Evolution; 3. Vertex degrees; 4. Connectivity; 5. Small subgraphs; 6. Spanning subgraphs; 7. Extreme characteristics; 8. Extremal properties; Part II. Basic Model Extensions: 9. Inhomogeneous graphs; 10. Fixed degree sequence; 11. Intersection graphs; 12. Digraphs; 13. Hypergraphs; Part III. Other Models: 14. Trees; 15. Mappings; 16. k-out; 17. Real-world networks; 18. Weighted graphs; 19. Brief notes on uncovered topics; Part IV. Tools and Methods: 20. Moments; 21. Inequalities; 22. Differential equations method; 23. Branching processes; 24. Entropy; References; Author index; Main index.

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Book SynopsisK3 surfaces are central objects in mathematics and connect to string theory in physics. By studying the many rich aspects of these surfaces, this book surveys powerful techniques in algebraic geometry. Working from the basics to recent breakthroughs, it is suitable as a graduate text and reference for researchers.Trade Review'K3 surfaces play something of a magical role in algebraic geometry and neighboring areas. They arise in astonishingly varied contexts, and the study of K3 surfaces has propelled the development of many of the most powerful tools in the field. The present lectures provide a comprehensive and wide-ranging survey of this fascinating subject. Suitable both for study and as a reference work, and written with Huybrechts's usual clarity of exposition, this book is destined to become the standard text on K3 surfaces.' Rob Lazarsfeld, State University of New York, Stony Brook'This book will be extremely valuable to all mathematicians who are interested in K3 surfaces and related topics. It not only serves as an excellent introduction, but also covers a wide variety of advanced subjects, ranging from complex geometry to derived geometry and arithmetic.' Klaus Hulek, Leibniz Universität Hannover'Since the nineteenth century, K3 surfaces have been a source of intriguing examples, problems and theorems. Huybrechts' book is a beautiful and reader-friendly presentation of the main results regarding this special class of varieties. The author fully succeeded in illustrating the richness of concepts and techniques which come into play in the theory of K3 surfaces.' Kieran G. O'Grady, Università degli Studi di Roma 'La Sapienza', Italy'K3 surfaces play a ubiquitous role in algebraic geometry. At first glance they seem to be well understood and easy to describe, still they provide non-trivial examples of the most fundamental concepts: Hodge structures, moduli spaces, Chow ring, vector bundles, Picard and Brauer groups … Huybrechts' book, written with the usual talent of the author, is the first to cover systematically all these aspects. It will be an invaluable reference for algebraic geometers.' Arnaud Beauville, Université de Nice, Sophia Antipolis'… the book covers many subjects and recent developments, and contains an encyclopedic total of 655 references, which will be very useful for researchers and graduate students. A reader who opens any page of the book will enjoy the subject there. This book will become one's favorite book.' Shigeyuki Kondo, MathSciNet'The book is a welcome addition to the literature, especially since its scope ranges from a very good introduction to K3 surfaces to the more recent advances on these surfaces and related topics.' Felipe Zaldivar, MAA ReviewsTable of ContentsPreface; 1. Basic definitions; 2. Linear systems; 3. Hodge structures; 4. Kuga-Satake construction; 5. Moduli spaces of polarised K3 surfaces; 6. Periods; 7. Surjectivity of the period map and Global Torelli; 8. Ample cone and Kähler cone; 9. Vector bundles on K3 surfaces; 10. Moduli spaces of sheaves on K3 surfaces; 11. Elliptic K3 surfaces; 12. Chow ring and Grothendieck group; 13. Rational curves on K3 surfaces; 14. Lattices; 15. Automorphisms; 16. Derived categories; 17. Picard group; 18. Brauer group.

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Book SynopsisIdeal for researchers and graduate students at the interface between mathematics and physics, this text develops quantum field theory from the ground up using a rich mix of modern mathematics. It provides a unified approach to deformation quantization, Hochschild homology, vertex algebras, conformal field theory, quantum groups, and gauge theory.Trade Review'The central achievement of the book is in its development of a formalism that leads to classical and quantum versions of Noether's theorem, itself a familiar topic in physics, using the language of factorization algebras … Institutions employing mathematicians and theoretical physicists actively working in this area should acquire the book … Recommended.' M. C. Ogilvie, Choice ConnectTable of Contents1. Introduction and overview; Part I. Classical Field Theory: 2. Introduction to classical field theory; 3. Elliptic moduli problems; 4. The classical Batalin–Vilkovisky formalism; 5. The observables of a classical field theory; Part II. Quantum Field Theory: 6. Introduction to quantum field theory; 7. Effective field theories and Batalin–Vilkovisky quantization; 8. The observables of a quantum field theory; 9. Further aspects of quantum observables; 10. Operator product expansions, with examples; Part III. A Factorization Enhancement of Noether's Theorem: 11. Introduction to Noether's theorems; 12. Noether's theorem in classical field theory; 13. Noether's theorem in quantum field theory; 14. Examples of the Noether theorems; Appendix A. Background; Appendix B. Functions on spaces of sections; Appendix C. A formal Darboux lemma; References; Index.

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Book SynopsisMohammed ben Musa (c.780âc.850) was a Persian mathematician and astronomer. The word 'algebra' derives from his Compendious Book on Calculation by Completion and Balancing, which introduced modern algebraic methods. First published in 1831, this translation from Arabic into English was prepared by the German orientalist Friedrich August Rosen (1805â37). The key algebraic methods introduced are reduction, completion and balancing. To reduce an equation is to change an expression to a simpler form; completion is to remove a negative quantity from one side of the equation and add it to the other; and balancing is to cancel like terms on opposite sides of the equation. An account is also given of solving polynomial equations up to the second degree. Rosen's introduction and notes accompany the translation, which remains relevant in the history of mathematics.Table of ContentsPreface; Author's preface; Mohammed ben Musa's compendium of calculating by completion and reduction; Notes; Arabic text.

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Book SynopsisScholar and East India Company administrator Henry Thomas Colebrooke (17651837) brought India's rich mathematical heritage to the attention of the wider world with this book, first published in 1817. It is a compilation of classic works by Indian mathematicians Brahmagupta (598668) and Bhascara (111485), translated into English.Table of ContentsDissertation; Part I. Bhascara: 1. Introductions; 2. Invocation; 3. Miscellaneous; 4. Mixture; 5. Progression; 6. Plane figure; 7. Excavations and contents of solids; 8. Stacks; 9. Saw; 10. Mounds of grain; 11. Shadow of a gnomon; 12. Pulverizer; 13. Combination; Part II. Algebra: 1. Invocation; 2. Pulverizer; 3. Affected square; 4. Simple equation; 5. Quadratic, etc., equations; 6. Multilateral equations; 7. Varieties of quadratics; 8. Equation involving a factum of unknown quantities; 9. Conclusion; Part III. Brahmegupta: 1. Arithmetic; 2. Algebra.

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Book SynopsisThis is the first full-length book on the major theme of symmetry in graphs. Forming part of algebraic graph theory, this fast-growing field is concerned with the study of highly symmetric graphs, particularly vertex-transitive graphs, and other combinatorial structures, primarily by group-theoretic techniques. In practice the street goes both ways and these investigations shed new light on permutation groups and related algebraic structures. The book assumes a first course in graph theory and group theory but no specialized knowledge of the theory of permutation groups or vertex-transitive graphs. It begins with the basic material before introducing the field''s major problems and most active research themes in order to motivate the detailed discussion of individual topics that follows. Featuring many examples and over 450 exercises, it is an essential introduction to the field for graduate students and a valuable addition to any algebraic graph theorist''s bookshelf.Trade Review'The book is an excellent introduction to graph symmetry, assuming only first courses in each of group theory and graph theory. Illustrative and instructive examples of graphs with high symmetry are given along with motivating problems. The theory of group actions is interspersed throughout the book, as appropriate to the development of the graph story, and there are separate chapters treating different research directions, for example, vertex-transitive graphs and their automorphism groups, the Cayley Isomorphism Problem, and Hamiltonicity. The book provides a seamless entry for students and other interested people into this fascinating study of the interplay between symmetry and network theory, with extensive lists of exercises at the end of each chapter, and important research problems on graph symmetry discussed throughout the book, and especially in the final chapter.' Cheryl Praeger, University of Western Australia, Perth'Dobson, Malnič and Marušič have done us a real service. They offer a thorough treatment of graph symmetry, the first text book on the topic. What makes this even more useful is that their treatment is detailed, careful and gentle.' Chris Godsil, University of Waterloo, Ontario'A book like this is long overdue. It brings together a vast array of important and interesting material about graph symmetries, and is very well presented. Congratulations to the authors on a fine achievement.' Marston Conder, University of AucklandTable of Contents1. Introduction and constructions; 2. The Petersen graph, blocks, and actions of A5; 3. Some motivating problems; 4. Graphs with imprimitive automorphism group; 5. The end of the beginning; 6. Other classes of graphs; 7. The Cayley isomorphism problem; 8. Automorphism groups of vertex-transitive graphs; 9. Classifying vertex-transitive graphs; 10. Symmetric graphs; 11. Hamiltonicity; 12. Semiregularity; 13. Graphs with other types of symmetry: Half-arc-transitive graphs and semisymmetric graphs; 14. Fare you well; References; Author index; Index of graphs; Index of symbols;Index of terms.

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Book SynopsisThis graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. Written in an informal style, this is a contemporary introduction to the subject. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.Trade Review'… the exposition is very clear and logical. It has the advantage of giving the basic facts about Lie algebra theory with enough arguments but skipping the tedious technical details of proofs. Another excellent feature of the book is that many of the basic notions, properties and results are illustrated by a great number of exercises and examples. In my opinion this book is a nice addition to the landmarks in the field … I strongly recommend it to anyone wishing to enter into the beautiful and exciting field of Lie algebras and their applications.' Journal of Geometry and Symmetry in Physics'… very readable … I strongly recommend this book as a possible selection for graduate courses, as well as for independent study, or individual reading.' MAA Reviews'There are many exercises … The last appendix contains a useful detailed sample syllabus for a one-semester graduate course (two lectures a week).' EMS Newsletter'The book is a very concise and nice introduction to Lie groups and Lie algebras. It seems to be well suited for a course on the subject.' Mathematical ReviewsTable of ContentsPreface; 1. Introduction; 2. Lie groups: basic definitions; 3. Lie groups and Lie algebras; 4. Representations of Lie groups and Lie algebras; 5. Structure theory of Lie algebras; 6. Complex semisimple Lie algebras; 7. Root systems; 8. Representations of semisimple Lie Algebras; Overview of the literature; A. Root systems and simple Lie algebras; B. Sample syllabus; List of notation; Index; Bibliography.

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Book SynopsisTodayâs Developmental Math students enter college needing more than just the math, and this has directly impacted the instructorâs role in the classroom. Instructors have to teach to different learning styles, within multiple teaching environments, and to a student population that is mostly unfamiliar with how to be a successful college student. Authors Andrea Hendricks and Pauline Chow have noticed this growing trend in their combined 30+ years of teaching at their respective community colleges, both in their face-to-face and online courses. As a result, they set out to create course materials that help todayâs students not only learn the mathematical concepts but also build life skills for future success. Understanding the time constraints for instructors, these authors have worked to integrate success strategies into both the print and digital materials, so that there is no sacrifice of time spent on the math. Furthermore, Andrea and Pauline have taken the time to write purpoTable of ContentsChapter S: Success StrategiesS.1: Time Management and Goal SettingS.2: Learning StylesS.3: Study SkillsS.4: Test TakingS.5: Blended and Online ClassesChapter 1: Real Numbers and Algebraic Expressions1.1: The Set of Real Numbers1.2: Fractions Review1.3: The Order of Operations, Algebraic Expressions and Equations1.4: Addition of Real Numbers1.5: Subtraction of Real Numbers1.6: Multiplication and Division of Real Numbers1.7: Properties of Real Numbers1.8: Algebraic ExpressionsChapter 2: Linear Equations & Inequalities in One Variable2.1: Equations and Their Solutions2.2: The Addition Property of Equality2.3: The Multiplication Property of Equality2.4: More on Solving Linear Equations2.5: Formulas and Applications from Geometry2.6: Percent, Rate, and Mixture Problems2.7: Linear Inequalities in One VariableChapter 3: Linear Equations in Two Variables3.1: Equations and the Rectangular Coordinate System3.2: Graphing Linear Equations3.3: The Slope of a Line3.4: More about Slope3.5: Writing Equations of Lines3.6: FunctionsChapter 4: Systems of Linear Equations in Two and Three Variables4.1: Solving Systems of Linear Equations Graphically4.2: Solving Systems of Linear Equations by Substitution4.3: Solving Systems of Linear Equations by Elimination4.4: Applications of Systems of Linear Equations in Two Variables 4.5: Solving Systems of Linear Equations in Three Variables and Their Applications Chapter 5: Exponents, Polynomials, and Polynomial Functions5.1: Rules of Exponents and Zero and Negative Exponents5.2: More Rules of Exponents and Scientific Notation5.3: Polynomial Functions, Addition and Subtraction of Polynomials5.4: Multiplication of Polynomials and Polynomial Functions5.5: Special Products5.6: Division of Polynomials5.7: Synthetic Division and the Remainder TheoremChapter 6: Factoring Polynomials and Polynomial Equations6.1: Greatest Common Factor and Grouping6.2: Factoring Trinomials6.3: More on Factoring Trinomials6.4: Factoring Binomials6.5: Solving Quadratic Equations and Other Polynomial Equations by Factoring6.6: Applications of Quadratic EquationsChapter 7: Rational Functions and Equations7.1: Rational Functions and Simplifying Rational Expressions7.2: Multiplication and Division of Rational Expressions7.3: Least Common Denominator and Equivalent Fractions7.4: Addition and Subtraction of Rational Expressions7.5: Complex Fractions7.6: Solving Rational Equations7.7: Proportions and Other Applications of Rational EquationsChapter 8: More on Functions and Graphs; Variation8.1: The Domain and Range of Functions8.2: Graphing and Writing Linear Functions8.3: Graphing Nonlinear Functions and Piecewise Defined Functions8.4: Variation and ApplicationsChapter 9: Inequalities and Absolute Value9.1: Compound Inequalities9.2: Absolute Value Equations9.3: Absolute Value Inequalities9.4: Linear Inequalities in Two Variables and Systems of Linear InequalitiesChapter 10: Rational Exponents, Radicals, and Complex Numbers10.1: Radicals and Radical Functions10.2: Rational Exponents10.3: Simplifying Radical Expressions10.4: Adding, Subtracting, and Multiplying Radical Expressions10.5: Dividing Radicals and Rationalizing10.6: Radical Equations and their Applications10.7: Complex NumbersChapter 11: Quadratic Equations and Functions11.1: Quadratic Functions and their Graphs11.2: Solving Quadratic Equations by the Square Root Property and Completing the Square11.3: Solving Quadratic Equations by the Quadratic Formula11.4: Solving Equations by Using Quadratic Methods11.5: More on Graphing Quadratic Functions11.6: Solving Quadratic and Rational Inequalities in One VariableChapter 12: Exponential and Logarithmic Functions12.1: Operations and Composition of Functions12.2: Inverse Functions12.3: Exponential Functions12.4: Logarithmic Functions12.5: Properties of Logarithms12.6: The Common Log, Natural Log, and Change of Base Formula12.7: Exponential and Logarithmic Equations and ApplicationsChapter 13: Conic Sections and Nonlinear Systems13.1: The Parabola and the Circle13.2: The Ellipse and the Hyperbola13.3: Solving Nonlinear Systems of Equations13.4: Solving Nonlinear Inequalities and Systems of InequalitiesChapter 14: Sequences, Series, and the Binomial Theorem14.1: Sequences14.2: Arithmetic Sequences and Series14.3: Geometric Sequences and Series14.4: The Binomial Theorem

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Book SynopsisTable of ContentsCollege Algebra: Graphs & Models Chapter R: A Review of Basic Concepts and Skills R.1: Algebraic Expressions and the Properties of Real Numbers R.2: Exponents, Scientific Notation, and a Review of Polynomials R.3: Solving Linear Equations and Inequalities R.4: Factoring Polynomials and Solving Polynomial Equations by Factoring R.5: Rational Expressions and Equations R.6: Radicals, Rational Exponents, and Radical Equations Chapter 1: Functions and Graphs 1.1: Rectangular Coordinates, Graphing Circles and Other Relations1.2: Functions, Function Notation, and the Graph of a Function 1.3: Linear Equations and Rates of Change1.4: Linear Functions, Special Forms, and More on Rates of Change1.5: Solving Equations and Inequalities Graphically; Formulas and Problem Solving1.6: Linear Models and Real Data Chapter 2: Relations, More on Functions 2.1: Analyzing the Graph of a Function 2.2: The Toolbox Functions and Transformations 2.3: Absolute Value Functions, Equations, and Inequalities 2.4: Rational and Radical Functions; More on the Domain 2.5: Piecewise-Defined Functions 2.6: Variation: The Toolbox Functions in Action Chapter 3: Quadratic Functions and Operations on Functions 3.1: Complex Numbers 3.2: Solving Quadratic Equations and Inequalities 3.3: Quadratic Functions and Applications 3.4: Quadratic Models; More on Rates of Change 3.5: The Algebra of Functions 3.6: Composition of Functions and the Difference Quotient Chapter 4: Polynomial and Rational Functions 4.1: Synthetic Division; the Remainder and Factor Theorems 4.2: The Zeros of Polynomial Functions 4.3: Graphing Polynomial Functions 4.4: Graphing Rational Functions 4.5: Additional Insights into Rational Functions Chapter 5: Exponential and Logarithmic Functions 5.1: One-to-One and Inverse Functions 5.2: Exponential Functions 5.3: Logarithms and Logarithmic Functions 5.4: Properties of Logarithms 5.5: Solving Exponential/Logarithmic Equations 5.6: Applications from Business, Finance, and Science 5.7: Exponential, Logarithmic, and Logistic Equation Models Chapter 6: Systems of Equations and Inequalities 6.1: Linear Systems in Two Variables with Applications 6.2: Linear Systems in Three Variables with Applications 6.3: Nonlinear Systems of Equations and Inequalities 6.4: Systems of Inequalities and Linear Programming Chapter 7: Matrices and Matrix Applications 7.1: Solving Linear Systems Using Matrices and Row Operations 7.2: The Algebra of Matrices 7.3: Solving Linear Systems Using Matrix Equations 7.4: Applications of Matrices and Determinants: Cramer’s rule, Partial Fractions, and More 7.5: Matrix Applications and Technology Use Chapter 8: Analytic Geometry and the Conic Sections 8.1: A Brief Introduction to Analytic Geometry 8.2: The Circle and the Ellipse 8.3: The Hyperbola 8.4: The Analytic Parabola: More on Nonlinear Systems Chapter 9: Additional Topics in Algebra 9.1: Sequences and Series 9.2: Arithmetic Sequences 9.3: Geometric Sequences 9.4: Mathematical Induction 9.5: Counting Techniques 9.6: Introduction to Probability 9.7: The Binomial Theorem Appendices The Language, Notation, and Numbers of Mathematics Geometry Review with Unit Conversions More on Synthetic Division More on Matrices Deriving the Equation of a Conic Proof Positive - A Selection of Proofs from College Algebra

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