Algebra Books

Book SynopsisThis book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more.As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level.The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.Table of Contents

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Book SynopsisThis English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas."This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander BeilinsonTrade Review“This slim volume is still a valuable introduction to schemes and nicely complements the textbooks on this topic which have appeared in the meantime.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“Throughout the text there are many instructive examples, remarks, and clarifying footnotes. The style of exposition is rather concise, very elegant, extremely lucid and enlightening, versatile, and – despite its venerable age of fifty years – absolutely modern and timely. … an excellent source for students, instructors, and mathematical physicists. No doubt, with this textbook, the mathematical community has another general standard reference in algebraic geometry at its disposal.” (Werner Kleinert, zbMATH 1390.14002, 2018)Table of ContentsEditor's Preface.- Author's Preface.- 1 Affine Schemes.- 2 Sheaves, Schemes, and Projective Spaces.- References.- Index.

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Book SynopsisGalois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.Trade Review“This book covers a lot of interesting material and is surely a valuable addition to the literature, but is certainly not for the timid. It brings together a broad array of sophisticated mathematics … and it does so in a very general and abstract way, with an exposition that gives whole new meaning to the word ‘concise’.” (Mark Hunacek, MAA Reviews, April 5, 2021)Table of ContentsIntroduction.- Chapter 1. Zorn’s Lemma.- Chapter 2. Categories and Functors.- Chapter 3. Linear Algebra.- Chapter 4. Coverings.- Chapter 5. Galois Theory.- Chapter 6. Riemann Surfaces.- Chapter 7. Dessins d’Enfants.- Bibliography.- Index of Notation

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Book SynopsisDieses Lehrbuch ist ein idealer Begleitband für eine vierstündige Vorlesung mit Übungen für angehende Naturwissenschaftlerinnen und Naturwissenschaftler, kann aber auch für eine Einführungsvorlesung in die höhere Mathematik in anderen Disziplinen eingesetzt werden. Aufbauend auf Vorkenntnissen aus dem Gymnasium werden zunächst die wichtigsten Begriffe nochmals repetiert. Im Hauptteil werden Vektoren, Differential- und Integralrechnung sowie Differentialgleichungen eingeführt und ausführlich behandelt. Abschließend wird auf Funktionen mehrerer Variablen eingegangen. Zahlreiche Übungsaufgaben mit Lösungen zu jedem Kapitel helfen, den Stoff zu festigen. Neben den Erklärungen für alle Leserinnen und Leser werden in speziell markierten Teilen weiterführende Fragen vertieft behandelt, welche nicht zwingend für das Verständnis notwendig sind, aber interessante Einblicke geben. Das Buch und Übungskonzept ist eine weitgehend überarbeitete Neuausgabe des Texts einer über ein Jahrzehnt erfolgreich gelehrten Vorlesung.Table of ContentsA. Vektorrechnung.- 1. Vektoren und ihre geometrische Bedeutung.- 2. Vektorrechnung mit Koordinaten.- B. Differentialrechnung.- 3. Beispiele zum Begriff der Ableitung.- 4. Die Ableitung.- 5. Technik des Differenzierens.- 6. Anwendungen der Ableitung.- 7. Linearisierung und das Differential.- 8. Die Ableitung einer Vektorfunktion.- C. Integralrechnung.- 9. Einleitende Beispiele zum Begriff des Integrals.- 10. Das bestimmte Integral.- 11. Der Hauptsatz der Differential- und Integralrechnung.- 12. Stammfunktionen und das unbestimmte Integral.- 13. Weitere Integrationsmethoden.- 14. Integration von Vektorfunktionen.- D. Differentialgleichungen.- 15. Der Begriff der Differentialgleichung.- 16. Einige Lösungsmethoden.- E. Ausbau der Infinitesimalrechnung.- 17. Umkehrfunktionen.- 18. Einige wichtige Funktionen und ihre Anwendungen.- 19. Potenzreihen.- 20. Uneigentliche Integrale.- 21. Numerische Methoden.- F. Funktionen von Mehreren Variablen.- 22. Allgemeines über Funktionen von mehreren Variablen.- 23. Differentialrechnung von Funktionen von mehreren Variablen.- 24. Das totale Differential.- 25. Mehrdimensionale Integrale.- G. Anhang.- 26. Zusammenstellung einiger Grundbegriffe.- 27. Einige Ergänzungen.- 28. Lösungen der Aufgaben.

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Book SynopsisThis textbook introduces linear algebra and optimization in the context of machine learning. Examples and exercises are provided throughout the book. A solution manual for the exercises at the end of each chapter is available to teaching instructors. This textbook targets graduate level students and professors in computer science, mathematics and data science. Advanced undergraduate students can also use this textbook. The chapters for this textbook are organized as follows:1. Linear algebra and its applications: The chapters focus on the basics of linear algebra together with their common applications to singular value decomposition, matrix factorization, similarity matrices (kernel methods), and graph analysis. Numerous machine learning applications have been used as examples, such as spectral clustering, kernel-based classification, and outlier detection. The tight integration of linear algebra methods with examples from machine learning differentiates this book from generic volumes on linear algebra. The focus is clearly on the most relevant aspects of linear algebra for machine learning and to teach readers how to apply these concepts.2. Optimization and its applications: Much of machine learning is posed as an optimization problem in which we try to maximize the accuracy of regression and classification models. The “parent problem” of optimization-centric machine learning is least-squares regression. Interestingly, this problem arises in both linear algebra and optimization, and is one of the key connecting problems of the two fields. Least-squares regression is also the starting point for support vector machines, logistic regression, and recommender systems. Furthermore, the methods for dimensionality reduction and matrix factorization also require the development of optimization methods. A general view of optimization in computational graphs is discussed together with its applications to back propagation in neural networks. A frequent challenge faced by beginners in machine learning is the extensive background required in linear algebra and optimization. One problem is that the existing linear algebra and optimization courses are not specific to machine learning; therefore, one would typically have to complete more course material than is necessary to pick up machine learning. Furthermore, certain types of ideas and tricks from optimization and linear algebra recur more frequently in machine learning than other application-centric settings. Therefore, there is significant value in developing a view of linear algebra and optimization that is better suited to the specific perspective of machine learning. Trade Review“Based on the topics covered and the excellent presentation, I would recommend Aggarwal's book over these other books for an advanced undergraduate or beginning graduate course on mathematics for data science.” (Brian Borchers, MAA Reviews, March 28, 2021)“This book should be of interest to graduate students in engineering, applied mathematics, and other fields requiring an understanding of the mathematical underpinnings of machine learning.” (IEEE Control Systems Magazine, Vol. 40 (6), December, 2020)Table of ContentsPreface.- 1 Linear Algebra and Optimization: An Introduction.- 2 Linear Transformations and Linear Systems.- 3 Eigenvectors and Diagonalizable Matrices.- 4 Optimization Basics: A Machine Learning View.- 5 Advanced Optimization Solutions.- 6 Constrained Optimization and Duality.- 7 Singular Value Decomposition.- 8 Matrix Factorization.- 9 The Linear Algebra of Similarity.- 10 The Linear Algebra of Graphs.- 11 Optimization in Computational Graphs.- Index.

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Book SynopsisThis book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group. The main components are: - a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism, - the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals, - algebraic representation theory in terms of category O, and - analytic representation theory of quantized complex semisimple groups. Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.Trade Review“The book is largely self-contained. … It is highly recommended for mathematicians of all levels wishing to learn about these topics, in the algebraic setting and/or in the analytic setting.” (Huafeng Zhang, zbMATH 1514.20006, 2023)Table of Contents- Introduction. - Multiplier Hopf Algebras. - Quantized Universal Enveloping Algebras. - Complex Semisimple Quantum Groups. - Category O. - Representation Theory of Complex Semisimple Quantum Groups.

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Book SynopsisThis undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics.Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise.This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts.From reviews of the first edition:Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATHTable of ContentsPreface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement.- 4 What's True in Mathematics?.- A Ten Famous Conjectures.-B Ten Famous Theorems.- II The Foundations of Mathematics.- 5 Let's Be Logical!.- 6 Setting Examples.- 7 Quantifier Mechanics.- 8 Let's Be Functional!.- C The Foundations of Set Theory.- III How to Prove It.- 9 Universal Proofs.- 10 The Domino Theory.- 11 More Domino Games.- 12 Existential Proofs.- D Ten Famous Problems.- IV Advanced Math for Beginners.- 13 Mathematical Structures.- 14 Working in the Fields (and Other Structures).- 15 Group Work.- 16 Good Relations.- 17 Order, Please!.- 18 Now That's the Limit!.- 19 Sizing It Up.- 20 Infinite Delights.- 21 Number Systems Systematically.- 22 Games Are Valuable!.- E Graphic Content.- F All Games Considered.- G A Top Forty List of Math Theorems.

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Book SynopsisThis undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics.Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise.This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts.From reviews of the first edition:Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATHTable of ContentsPreface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement.- 4 What's True in Mathematics?.- A Ten Famous Conjectures.-B Ten Famous Theorems.- II The Foundations of Mathematics.- 5 Let's Be Logical!.- 6 Setting Examples.- 7 Quantifier Mechanics.- 8 Let's Be Functional!.- C The Foundations of Set Theory.- III How to Prove It.- 9 Universal Proofs.- 10 The Domino Theory.- 11 More Domino Games.- 12 Existential Proofs.- D Ten Famous Problems.- IV Advanced Math for Beginners.- 13 Mathematical Structures.- 14 Working in the Fields (and Other Structures).- 15 Group Work.- 16 Good Relations.- 17 Order, Please!.- 18 Now That's the Limit!.- 19 Sizing It Up.- 20 Infinite Delights.- 21 Number Systems Systematically.- 22 Games Are Valuable!.- E Graphic Content.- F All Games Considered.- G A Top Forty List of Math Theorems.

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Book SynopsisThis textbook is an introduction to the theory of infinity-categories, a tool used in many aspects of modern pure mathematics. It treats the basics of the theory and supplies all the necessary details while leading the reader along a streamlined path from the basic definitions to more advanced results such as the very important adjoint functor theorems. The book is based on lectures given by the author on the topic. While the material itself is well-known to experts, the presentation of the material is, in parts, novel and accessible to non-experts. Exercises complement this textbook that can be used both in a classroom setting at the graduate level and as an introductory text for the interested reader.Trade Review“This book is a concise and rather comprehensive introduction to the theory of ∞-categories, aimed at a wide audience. … The material developed in this book mostly originates from the work of Joyal and Lurie. It is presented here in a fresh, streamlined, and accessible way. The length and style of this book make it ideally suited for a lecture course or seminar on ∞-categories. … Finally, a nice collection of exercises is included as an appendix.” (Gijs Heuts, Mathematical Reviews, July, 2022)Table of ContentsPreface.- Categories, simplicial sets, and in nity-categories.- Joyal's theorem, applications, and Dwyer-Kan localizations.- (Co)Cartesian brations and the construction of functors.- Limits and Colimits.- Adjunctions and adjoint functor theorems.- Exercises.

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Book SynopsisThis book focuses on the Symmetric Informationally Complete quantum measurements (SICs) in dimensions 2 and 3, along with one set of SICs in dimension 8. These objects stand out in ways that have earned them the moniker of "sporadic SICs". By some standards, they are more approachable than the other known SICs, while by others they are simply atypical. The author forays into quantum information theory using them as examples, and the author explores their connections with other exceptional objects like the Leech lattice and integral octonions. The sporadic SICs take readers from the classification of finite simple groups to Bell's theorem and the discovery that "hidden variables" cannot explain away quantum uncertainty.While no one department teaches every subject to which the sporadic SICs pertain, the topic is approachable without too much background knowledge. The book includes exercises suitable for an elective at the graduate or advanced undergraduate level.Table of ContentsEquiangular Lines.- Sporadic SICs and the Exceptional Lie Algebras.- The Hoggar-type SICs.-SICs as Equicoherent Quantum States.- SICs and Bell Inequalities.

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Book SynopsisMahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book.One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.Trade Review“The reader at the graduate level having enough time and energy can learn a lot from this book about the Mahler measure, conjugate sets of algebraic integers, and related results. Some chapters of the book are quite accessible to undergraduate students as well, and may serve as an introduction to their research in this area.” (Arturas Dubickas, Mathematical Reviews, May, 2023)“It contains some material that is unavailable elsewhere. Each chapter is concluded by notes and a glossary of newly introduced definitions. … The reader at the graduate level having enough time and energy from this book can learn a lot about the Mahler measure, conjugate sets of algebraic integers and related results.” (Artūras Dubickas, zbMATH 1486.11003, 2022)Table of Contents1 Mahler Measures of Polynomials in One Variable.- 2 Mahler Measures of Polynomials in Several Variables.- 3 Dobrowolski's Theorem.- 4 The Schinzel–Zassenhaus Conjecture.- 5 Roots of Unity and Cyclotomic Polynomials.- 6 Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem.- 7 Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem.- 8 The Set of Cassels Heights.- 9 Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tesselations.- 10 The Transfinite Diameter and Conjugate Sets of Algebraic Integers.- 11 Restricted Mahler Measure Results.- 12 The Mahler Measure of Nonreciprocal Polynomials.- 13 Minimal Noncyclotomic Integer Symmetric Matrices.- 14 The Method of Explicit Auxiliary Functions.- 15 The Trace Problem For Integer Symmetric Matrices.- 16 Small-Span Integer Symmetric Matrices.- 17 Symmetrizable Matrices I: Introduction.- 18 Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices.- 19 Symmetrizable Matrices III: The Trace Problem.- 20 Salem Numbers from Graphs and Interlacing Quotients.- 21 Minimal Polynomials of Integer Symmetric Matrices.- 22 Breaking Symmetry.- A Algebraic Background.- B Combinatorial Background.- C Tools from the Theory of Functions.- D Tables.- References.- Index.

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Book SynopsisThis book provides a concise survey of the theory of zero product-determined algebras, which has been developed over the last 15 years. It is divided into three parts. The first part presents the purely algebraic branch of the theory, the second part presents the functional analytic branch, and the third part discusses various applications. The book is intended for researchers and graduate students in ring theory, Banach algebra theory, and nonassociative algebra.Trade Review“This book is about zero product determined algebras and is written in an attractive way. It deals with the introduction and study of this class of algebras. Most of this book is taken from research articles from the last 15 years and is suitable for researchers in this field and students with different backgrounds and can be used for self-study.” (Hoger Ghahramani, Mathematical Reviews, March, 2023)Table of Contents- Part I Algebraic Theory. - Zero Product Determined Nonassociative Algebras. - Zero Product Determined Rings and Algebras. - Zero Lie/Jordan Product Determined Algebras. - Part II Analytic Theory. - Zero Product Determined Nonassociative Banach Algebras. - Zero Product Determined Banach Algebras. - Zero Lie/Jordan Product Determined Banach Algebras. - Part III Applications. - Homomorphisms and Related Maps. - Derivations and Related Maps. - Miscellany.

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Book SynopsisThis book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today.The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.Table of Contents- Foreword.- Preface.- Part I Algebraic Theory: 1. Free Groups.- 2. Nilpotent Groups.- 3. Residual Finiteness and the Zassenhaus Filtration.- 4. Solvable Groups.- 5. Polycyclic Groups.- 6. The Burnside Problem.- Part II Geometric Theory: 7. Finitely Generated Groups and Their Growth Functions.- 8. Hyperbolic Plane Geometry and the Tits Alternative.- 9. Topological Groups, Lie Groups, and Hilbert Fifth Problem.- 10. Dimension Theory.- 11. Ultrafilters, Ultraproducts, Ultrapowers, and Asymptotic Cones.- 12. Gromov’s Theorem.- Part III Analytic and Probabilistic Theory: 13. The Theorems of Polya and Varopoulos.- 14. Amenability, Isoperimetric Profile, and Følner Functions.- 15. Solutions or Hints to Selected Exercises.- References.- Subject Index.- Index of Authors.

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Book SynopsisHistorical introduction.- Part I Some Galois theorems for fields.- 1 The classical Galois theorem.- 2 The Galois theorem of Grothendieck.- 3 Profinite topological spaces.- 4 The Galois theorems in arbitrary dimension.- Part II The Galois theory of rings.- 5 Adjunctions and monads.- 6 Profinite groupoids and presheaves.- 7 The descent theory of rings.- 8 The Pierce spectrum of a ring.- 9 The Galois theorem for rings.- Further Reading.- Index.

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Book SynopsisA survey on applications of a new approach to Quillen rational homotopy theory.- The Yang problem for complete bounded complex submanifolds a survey.- Orthogonal polynomials and operator theory.- Open problems and applications.- A Sample of problems where ULRICH BUNDLES show up.

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Book SynopsisThe directed Cayley diameter and the Davenport constant.- Heaps and trusses.- Prüfer v-multiplication domains, a survey.- Some applications of a new approach to factorization.- Atomicity in integral domains.- Norms, normsets, and factorization.-On the Ratliff-Rush closure of an ideal of a one-dimensional ring.- Graded identities of infinite-dimensional Lie algebras.-On a class of quotients of Rees algebras: a survey.- P-adic approximation of algebraic integers and residue class rings of integer-valued polynomials.- On matrix superalgebras with pseudoautomorphism.- Krull Rings, Semistar Operations, and Bases of Regularity.- Descriptions of radicals in polynomial ring extensions.- Boolean inverse semigroups and their type monoids.-Generalized derivations characterized by their action on Jordan products.- Open problems on relations of numerical semigroups.- Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely.- The role of divisors in non-commutative ideal theory.- Essential-like properties for integer-valued polynomial rings: a survey with a note on valuation domains.- On the isomorphism problem for power semigroups.- Dimension-free matricial Nullstellensatze for noncommutative polynomials.

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Book SynopsisDistributed Fusion Estimation in the Presence of Measurement Quantization and Mixed Attacks.- p-Frobenius Numbers of Numerical Semigroups Generated by Three Consecutive Squares.- On Solutions of a Third Order Linear Difference Equation with Variable Coefficients Applications.- Binet-Fibonacci Calculus and N = 2 Supersymmetric Golden Quantum Oscillator.- Geometry and Entanglement of Super-Qubit Quantum States.- On the Bi-periodic Edouard and the Bi-periodic EdouardLucas Numbers.- The Investment Portfolio Selection with Social Network Decision-Making with Minimum Cost Consensus Model and Incomplete Fermatean Fuzzy Preference Relations.- Boundary Value Problems for the Bitsadze Equation on a Quarter Plane.- Galois Bundles and Automorphisms of the Principal Bundle Moduli Space.- Linear Algebra in Crystal Geometry, and Vice Versa.- On Dual Biquaternionic Sequence Involving Vietoris' Numbers.- A Quadratic Programming Model for Operating Rooms Scheduling based on Resources Availability.- Derivative method for solving cubic equations.- Improved Computational Techniques for Heat Sources Localization.- Open access fisheries model considering depensatory growth functions in the exploited resource.- An Empirical Comparison of Supervised Machine Learning Models in  Predicting Mathematics Performance in Somaliland.- Using SOLO Taxonomy to develop a structured design for mathematics exam questions.- Application of Supervised Machine Learning Algorithms to Identify the Prevalence and Determinants of Spontaneous Abortion among Ever-married Women in Somaliland: Insights from SLDHS Data 2020.- Modified Hungarian Method (MHM) in Optimizing Competency-Preference Scores in Lecturer-To-Course Assignment.- Effects of autonomous vehicles on particulate matter emissions.- Codimension-2 Bifurcation Analysis of a Modi?ed Non-degenerate Fisher Equation Introducing Two Unfolding Parameters.- Modelling Medfly Pest Management.- BETWEEN t*-CLOSED AND *IG-CLOSED SETS.- Fault Injection Attacks against RSA-CRT Digital Signature.- Analysis of Prognostic Factors in Prostate Cancer - A New Approach.- On k-order Jacobsthal Polynomials and Their Properties.- On Euclidean Norms Of Max On Euclidean Norms Of Max Matrices With Chebyshev Polynomials.- Bridging academic learning and community service.- Escape Rooms and Students Competencies.- Descriptive and Inferential Analysis of Renal Health and Patterns of Water Consumption: A case study of Ciudad Hidalgo.- Modeling Solar Energy Through Mathematics.- Problem-Based Learning, Teamwork and Entrepreneurship in the Numerical Methods Course.- Advanced Teaching Strategies for Enhancing STEM Education in Higher Institutions.- Numerical Methods: Artifacts in teaching in a mathematics degree.

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Book Synopsis- 1. Basic Monoid Theory.- 2. The Formalism of Module and Ideal Systems.- 3. Prime and Primary Ideals and Noetherian Conditions.- 4. Invertibility, Cancellation and Integrality.- 5. Arithmetic of Cancellative Mori Monoids.- 6. Ideal Theory of Polynomial Rings.

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Book Synopsis1 Sistemi lineari.- 2 Numeri interi e razionali.- 3 Numeri reali e complessi.- 4 Spazi vettoriali.- 5 Applicazioni lineari.- 6 Operazioni con le matrici.- 7 Riduzioni a scala ed applicazioni.- 8 Determinanti.- 9 Endomorfismi e polinomi caratteristici.- 10 Polinomi minimi.- 11 Forme canoniche di Jordan e razionale.- 12 Spazi duali.- 13 Spazi quoziente.- 14 Spazi vettoriali euclidei ed hermitiani.- 15 Forme bilineari e quadratiche.- 16 Numeri trascendenti.- 17 Soluzioni e suggerimenti di alcuni esercizi.

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Book SynopsisQuotients of vector spaces.- Sequences and chain complexes.- Chain maps.- Abstract simplicial complexes.- Simplicial homology and homotopy.- Sequences and chain complexes of sequences.

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Book SynopsisAward-winning monograph of the Ferran Sunyer i Balaguer Prize 2001. Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index. In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged.As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure. A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and Strong Approximation, algebraic and analytic number theory, probability, and p-adic model theory. Relevant aspects of such topics are explained in self-contained 'windows'.Trade ReviewSubgroup Growth is an extremely well-written book and is a delight to read. It has a wealth of information making a rich and timely contribution to an emerging area in the theory of groups which has come to be known as Asymptotic Group Theory. This monograph and the challenging open problems with which it concludes are bound to play a fundamental role in the development of the subject for many years to come. —Journal Indian Inst of Science "[Subgroup growth] is one of the first books on Asymptotic Group Theory – a new, quickly developing direction in modern mathematics…The book of A. Lubotzky and D.Segal, leading specialists in group theory, answers…questions in a beautiful way .…It was natural to expect a text on the subject that would summarize the achievements in the field and we are very lucky to witness the appearance of this wonderful book. …Readers will be impressed with the encyclopedic scope of the text. It includes all, or almost all, topics related to subgroup growth ….The book also includes plenty of general information on topics that are well known to algebraic audiences and should be part of the background for every modern researcher in mathematics. …a wonderful methodological tool introduced by the authors. … The book ends [with] a section on open problems, which contains 35 problems related to subgroup growth. The list will be useful and interesting to both established mathematicians and young researchers. There is no doubt that the list includes the most important and illuminating problems in the area, and we eagerly anticipate solutions of at least some of them in the near future. The book will surely have [a] big impact on all readers interested in Group Theory, as well as in Algebra and Number Theory in general." —Bulletin of the AMS "The proofs in this book employ a remarkable variety of tools, from all branches of group theory, certainly, but also from number theory, logic, and analysis…. The authors supply surveys, and some proofs, of necessary results, in the "windows" at the end of the book. These comprise about one quarter of the full book, and they give the needy reader a handy reference, without interrupting the flow of argument in the main text…. Since the subject of this book is an active area of current research, there are many open problems in it…." —Mathematical ReviewsTable of Contents0 Introduction and Overview.- 0.1 Preliminary comments and definitions.- 0.2 Overview of the chapters.- 0.3 On CFSG.- 0.4 The windows.- 0.5 The ‘notes’.- 1 Basic Techniques of Subgroup Counting.- 1.1 Permutation representations.- 1.2 Quotients and subgroups.- 1.3 Group extensions.- 1.4 Nilpotent and soluble groups.- 1.5 Abelian groups I.- 1.6 Finite p-groups.- 1.7 Sylow’s theorem.- 1.8 Rest riet ing to soluble subgroups.- 1.9 Applications of the ‘minimal index’.- 1.10 Abelian groups II.- 1.11 Growth types.- Notes.- 2 Free Groups.- 2.1 The subgroup growth of free groups.- 2.2 Subnormal subgroups.- 2.3 Counting d-generator finite groups.- Notes.- 3 Groups with Exponential Subgroup Growth.- 3.1 Upper bounds.- 3.2 Lower bounds.- 3.3 Free pro-p groups.- 3.4 Normal subgroups in free pro-p groups.- 3.5 Relations in p-groups and Lie algebras.- Notes.- 4 Pro-p Groups.- 4.1 Pro-p groups with polynomial subgroup growth.- 4.2 Pro-p groups with slow subgroup growth.- 4.3 The groups $$SL_r^1({\mathbb{F}_p}[[t]])$$.- 4.4 A-perfect groups.- 4.5 The Nottingham group.- 4.6 Finitely presented pro-p groups.- Notes.- 5 Finitely Generated Groups with Polynomial Subgroup Growth.- 5.1 Preliminary observations.- 5.2 Linear groups with PSG.- 5.3 Upper chief factors.- 5.4 Groups of prosoluble type.- 5.5 Groups of finite upper rank.- 5.6 The degree of polynomial subgroup growth.- Notes.- 6 Congruence Subgroups.- 6.1 The characteristic 0 case.- 6.2 The positive characteristic case.- 6.3 Perfect Lie algebras.- 6.4 Normal congruence subgroups.- Notes.- 7 The Generalized Congruence Subgroup Problem.- 7.1 The congruence subgroup problem.- 7.2 Subgroup growth of lattices.- 7.3 Counting hyperbolic manifolds.- Notes.- 8 Linear Groups.- 8.1 Subgroup growth, characteristic 0.- 8.2 Residually nilpotent groups.- 8.3 Subgroup growth, characteristic p.- 8.4 Normal subgroup growth.- Notes.- 9 Soluble Groups.- 9.1 Metabelian groups.- 9.2 Residually nilpotent groups.- 9.3 Some finitely presented metabelian groups.- 9.4 Normal subgroup growth in metabelian groups.- Notes.- 10 Profinite Groups with Polynomial Subgroup Growth.- 10.1 Upper rank.- 10.2 Profinite groups with wPSG: structure.- 10.3 Quasi-semisimple groups.- 10.4 Profinite groups with wPSG: characterization.- 10.5 Weak PSG = PSG.- Notes.- 11 Probabilistic Methods.- 11.1 The probability measure.- 11.2 Generation probabilities.- 11.3 Maximal subgroups.- 11.4 Further applications.- 11.5 Pro-p groups.- Notes.- 12 Other Growth Conditions.- 12.1 Rank and bounded generation.- 12.2 Adelic groups.- 12.3 The structure of finite linear groups.- 12.4 Composition factors.- 12.5 BG, PIG and subgroup growth.- 12.6 Residually nilpotent groups.- 12.7 Arithmetic groups and the CSP.- 12.8 Examples.- Notes.- 13 The Growth Spectrum.- 13.1 Products of alternating groups.- 13.2 Some finitely generated permutation groups.- 13.3 Some profinite groups with restricted composition factors.- 13.4 Automorphisms of rooted trees.- Notes.- 14 Explicit Formulas and Asymptotics.- 14.1 Free groups and the modular group.- 14.2 Free products of finite groups.- 14.3 Modular subgroup arithmetic.- 14.4 Surface groups.- Notes.- 15 Zeta Functions I: Nilpotent Groups.- 15.1 Local zeta functions as p-adic integrals.- 15.2 Alternative methods.- 15.3 The zeta function of a nilpotent group.- Notes.- 16 Zeta Functions II: p-adic Analytic Groups.- 16.1 Integration on pro-p groups.- 16.2 Counting subgroups in a p-adic analytic group.- 16.3 Counting orbits.- 16.4 Counting p-groups.- Notes.- Windows.- 1 Finite Group Theory.- 1 Hall subgroups and Sylow bases.- 2 Carter subgroups.- 3 The Fitting subgroup.- 4 The generalized Fitting subgroup.- 5 Tate’s theorem.- 6 Rank and p-rank.- 7 Schur multiplier.- 8 Powerful p-groups.- 9 GLn and Sym(n).- 2 Finite Simple Groups.- 1 The list.- 2 Generators.- 3 Subgroups.- 4 Representations.- 5 Automorphisms.- 6 Schur multipliers.- 7 An elementary proof.- 3 Permutation Groups.- 1 Primitive groups.- 2 Groups with restricted sections.- 3 Subgroups of alternating groups.- 4 Profinite Groups.- 1 Completions.- 2 Free profinite groups.- 3 Profinite presentations.- 5 Pro-p Groups.- 1 Generators and relations.- 2 Pro-p groups of finite rank.- 3 Linear pro-p groups over local fields.- 4 Automorphisms of finite p-groups.- 5 Hall’s enumeration principle.- 6 Soluble Groups.- 1 Nilpotent groups.- 2 Soluble groups of finite rank.- 3 Finitely generated metabelian groups.- 7 Linear Groups.- 1 Soluble groups.- 2 Jordan’s theorem.- 3 Monomial groups.- 4 Finitely generated groups.- 5 Lang’s theorem.- 8 Linearity Conditions for Infinite Groups.- 1 Variations on Mal’cev’s local theorem.- 2 Groups that are residually of bounded rank.- 3 Applications of Ado’s theorem.- 9 Strong Approximation for Linear Groups.- 1 A variant of the Strong Approximation Theorem.- 2 Subgroups of SLn(Fp).- 3 The ‘Lubotzky alternative’.- 4 Strong approximation in positive characteristic.- 10 Primes.- 1 The Prime Number Theorem.- 2 Arithmetic progressions and the Bombieri-Vinogradov theorem.- 3 Global fields and Chebotarev’s theorem.- 11 Probability.- 12 p-adic Integrals and Logic.- 1 Results.- 2 A peek inside the black box.- Open Problems.- 1 ‘Growth spectrum’.- 2 Normal subgroup growth in pro-p groups and metabelian groups.- 3 The degree of f.g. nilpotent groups.- 4 Finite extensions.- 5 Soluble groups.- 6 Isospectral groups.- 7 Congruence subgroups, lattices in Lie groups.- 8 Other growth conditions.- 9 Zeta functions.

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Book SynopsisA comprehensive and easy to understand introduction to a wide range of tools to help designers to optimize their projects. The authors are engineers and therefore many of the examples are on engineering applications, but the techniques presented are common to various areas of knowledge and pervade disciplinary divisions. The book describes the fundamental ideas, mathematical and graphic methods and shows how to use Matlab and EXCEL for optimization.

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Book SynopsisThis book starts with the description of polarization in classical optics, including also a chapter on crystal optics, which is necessary to understand the use of nonlinear crystals. In addition, spatially non-uniform polarization states are introduced and described. Further, the role of polarization in nonlinear optics is discussed. The final chapters are devoted to the description and applications of polarization in quantum optics and quantum technologies.

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Book SynopsisThis work covers quantum mechanics by answering questions such as where did the Planck constant and Heisenberg algebra come from, what motivated Feynman to introduce his path integral and why does one distinguish two types of particles, the bosons and fermions. The author addresses all these topics with utter mathematical rigor. The high number of instructive Appendices and numerous Remark sections supply the necessary background knowledge.

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Book SynopsisIn the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory. A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications. Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.

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Book SynopsisOrdinary differential equations (ODEs) and differential-algebraic equations (DAEs) are widely used to model control systems in engineering, natural sciences, and economy. Optimal control plays a central role in optimizing such systems and to operate them effi ciently and safely. The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ODEs and DAEs. An emphasis is placed on the interplay between the optimal control problem, which typically is defi ned and analyzed in a Banach space setting, and discretizations thereof, which lead to finite dimensional optimization problems. The theoretical parts of the book require some knowledge of functional analysis, the numerically oriented parts require knowledge from linear algebra and numerical analysis. Practical examples are provided throughout the book for illustration purposes. The book addresses primarily master and PhD students as well as researchers in applied mathematics, but also engineers or scientists with a good background in mathematics. The book serves as a reference in research and teaching and hopefully helps to advance the state-of-the-art in optimal control.

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Book SynopsisThe primary audience for this book is students and the young researchers interested in the core of the discipline. Commutative algebra is by and large a self-contained discipline, which makes it quite dry for the beginner with a basic training in elementary algebra and calculus. A stable mathematical discipline such as this enshrines a vital number of topics to be learned at an early stage, more or less universally accepted and practiced. Naturally, authors tend to turn these topics into an increasingly short and elegant list of basic facts of the theory. So, the shorter the better. However, there is a subtle watershed between elegance and usefulness, especially if the target is the beginner. From my experience throughout years of teaching, elegance and terseness do not do it, except much later in the carrier. To become useful, the material ought to carry quite a bit of motivation through justification and usefulness pointers. On the other hand, it is difficult to contemplate these teaching devices in the writing of a short book. I have divided the material in three parts. starting with more elementary sections, then carrying an intermezzo on more difficult themes to make up for a smooth crescendo with additional tools and, finally, the more advanced part, versing on a reasonable chunk of present-day steering of commutative algebra. Historic notes at the end of each chapter provide insight into the original sources and background information on a particular subject or theorem. Exercises are provided and propose problems that apply the theory to solve concrete questions (yes, with concrete polynomials, and so forth).

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Book SynopsisThe present book is based on the extensive lecture notes of the author and contains a concise course on Linear Algebra. The sections begin with an intuitive presentation, aimed at the beginners, and then often include rather non-trivial topics and exercises. This makes the book suitable for introductory as well as advanced courses on Linear Algebra.The first part of the book deals with the general idea of systems of linear equations, matrices and eigenvectors. Linear systems of differential equations are developed carefully and in great detail. The last chapter gives an overview of applications to other areas of Mathematics, like calculus and differential geometry. A large number of exercises with selected solutions make this a valuable textbook for students of the topic as well as lecturers, preparing a course on Linear Algebra.

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Book SynopsisThere are numerous linear algebra textbooks available on the market. Yet, there are few that approach the notion of eigenvectors and eigenvalues across an operator's minimum polynomial. In this book, we take that approach. This book provides a thorough introduction to the fundamental concepts of linear algebra. The material is divided into two sections: Part I covers fundamental concepts in linear algebra, whereas Part II covers the theory of determinants, the theory of eigenvalues and eigenvectors, and fundamental results on Euclidean vector spaces. We highlight that: Consider hypothetical manufacturing models as a starting point for studying linear equations. There are two novel ideas in the book: the use of a production model to motivate the concept of matrix product and the use of an operator's minimal polynomial to describe the theory of eigenvalues and eigenvectors. Several examples incorporate the use of SageMath., allowing the reader to focus on conceptual comprehension rather than formulas.

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Book SynopsisThis undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform.The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.Trade Review“Undergraduate mathematics students need both to develop facility with numerical and symbolic calculation and comfort with abstraction. Algebraic number theory offers an ideal context for encountering the synthesis of these goals. One could compile a shelf of graduate-level expositions of algebraic number theory, and another shelf of undergraduate general number theory texts that culminate with a first exposure to it. … Summing Up: Highly recommended. Upper-division undergraduates.” (D. V. Feldman, Choice, Vol. 52 (8), April, 2015)“In this book, the author leads the readers from the theorem of unique factorization in elementary number theory to central results in algebraic number theory. … This book is designed for being used in undergraduate courses in algebraic number theory; the clarity of the exposition and the wealth of examples and exercises (with hints and solutions) also make it suitable for self-study and reading courses.” (Franz Lemmermeyer, zbMATH, Vol. 1303, 2015)Table of ContentsUnique factorisation in the natural numbers.- Number fields.- Fields, discriminants and integral bases.- Ideals.- Prime ideals and unique factorisation.- Imaginary quadratic fields.- Lattices and geometrical methods.- Other fields of small degree.- Cyclotomic fields and the Fermat equation.- Analytic methods.- The number field sieve.

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Book SynopsisThis text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields and linearly recursive sequences.The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforward applications of the theory to problems that are devised to challenge the reader. Questions for further study are provided after selected exercises. Most proofs are given in detail, though a few proofs are omitted since they are beyond the scope of this book.Trade Review“The goal of the book under review is to introduce graduate students to some basic results on coalgebras, bialgebras, Hopf algebras, and their applications. The book may be used as the main text or as a supplementary text for a graduate course. … This book should be very useful as a first introduction for someone who wants to learn about Hopf algebras and their applications.” (Jörg Feldvoss, zbMATH 1341.16034, 2016)Table of ContentsPreface.- Notation.- 1. Algebras and Coalgebras.- 2. Bialgebras.- 3. Hopf Algebras.- 4. Applications of Hopf Algebras.- Bibliography.

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Book SynopsisThis textbook provides an introduction to the mathematics on which modern cryptology is based. It covers not only public key cryptography, the glamorous component of modern cryptology, but also pays considerable attention to secret key cryptography, its workhorse in practice. Modern cryptology has been described as the science of the integrity of information, covering all aspects like confidentiality, authenticity and non-repudiation and also including the protocols required for achieving these aims. In both theory and practice it requires notions and constructions from three major disciplines: computer science, electronic engineering and mathematics. Within mathematics, group theory, the theory of finite fields, and elementary number theory as well as some topics not normally covered in courses in algebra, such as the theory of Boolean functions and Shannon theory, are involved. Although essentially self-contained, a degree of mathematical maturity on the part of the reader is assumed, corresponding to his or her background in computer science or engineering. Algebra for Cryptologists is a textbook for an introductory course in cryptography or an upper undergraduate course in algebra, or for self-study in preparation for postgraduate study in cryptology.Trade Review“Cryptography is technically synonymous with information security. Providing security features for data and information is the most important concern in a business requirement. … this book provides rich mathematical treatment of cryptographic algorithms in a simplified way. … This book presents nice and valuable material for students, researchers, and practitioners working in the field of information security.” (S. Ramakrishnan, Computing Reviews, March, 2017)“The present book consolidates the mathematics these students need to know into a single volume and will help them greatly. Initially, this reviewer believed that the book would be mostly redundant for mathematics majors, but as he read further, he found that it could serve as a valuable supplement within a math course. … Summing Up: Highly recommended. Upper-division undergraduates and above; faculty and professionals.” (C. Bauer, Choice, Vol. 54 (7), March, 2017)“This book introduces the basic algebra used in modern cryptology. It is mainly addressed to computer scientists and engineers entering the field of mathematical cryptology, but it may also be used as an introduction to algebra and elementary number theory with emphasis on application in cryptology for undergraduate students in computer science, engineering and also mathematics. The book is easily accessible, enjoyable to read, and essentially self-contained.” (Wilfried Meidl, zbMATH 1364.94005, 2017)“First, the text also covers number theory and also introduces information theory and coding theory. This material is of course algebraic in nature, but does go beyond what one typically expects to see in a book … that is advertised as a text in algebra. Second, the book not only discusses these topics but actually shows how they are used in the study of cryptography. … this book might definitely have some appeal as a prospective text.” (Mark Hunacek, MAA Reviews, December, 2016)Table of ContentsPrerequisites and Notation.- Basic Properties of the Integers.- Groups, Rings and Ideals.- Applications to Public Key Cryptography.- Fields.- Properties of Finite Fields.- Applications to Stream Ciphers.- Boolean Functions.- Applications to Block Ciphers.- Number Theory in Public Key Cryptography.- Where do we go from here?.- Probability.

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Book SynopsisWritten by an experienced physicist who is active in applying computer algebra to relativistic astrophysics and education, this is the resource for mathematical methods in physics using MapleTM and MathematicaTM. Through in-depth problems from core courses in the physics curriculum, the author guides students to apply analytical and numerical techniques in mathematical physics, and present the results in interactive graphics. Around 180 simulating exercises are included to facilitate learning by examples. This book is a must-have for students of physics, electrical and mechanical engineering, materials scientists, lecturers in physics, and university libraries. * Free online MapleTM material at http://www.wiley-vch.de/templates/pdf/maplephysics.zip * Free online MathematicaTM material at http://www.wiley-vch.de/templates/pdf/physicswithmathematica.zip * Solutions manual for lecturers available at www.wiley-vch.de/supplements/Table of Contents1. Introduction 2. Oscillatory Motion 3. Calculus of Variations 4. Integration of Equations of Motion 5. Orthogonal Functions and Expansions 6. Electrostatics 7. Boundary-Value Problems 8. Magnetostatics 9. Electric Circuits 10. Waves 11. Physical Optics 12. Special Relativity 13. Quantum Phenomena 14. Schrodinger Equation in One Dimension I 15. Schrodinger Equation in One Dimension II 16. Schrodinger Equation in Three Dimensions 17. Quantum Statistics 18. General Relativity A1 Physical and Astrophysical Constants A2 Mathematical Notes

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